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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:05:27 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:05:27 -0700 |
| commit | df6de2f74ebf90f9147218e91355b9517c895c11 (patch) | |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/36276-h/36276-h.htm b/36276-h/36276-h.htm new file mode 100644 index 0000000..b0b22a7 --- /dev/null +++ b/36276-h/36276-h.htm @@ -0,0 +1,5278 @@ +<!DOCTYPE html> +<html lang="en"> +<head> +<meta charset="UTF-8"> +<title>The meaning of Relativity | Project Gutenberg</title> + +<link href="images/cover.jpg" rel="icon" type="image/x-cover"> + +<style> + +body { + font-family: "Times New Roman", Times, serif; + margin-left: 10%; + margin-right: 10%; +} + + h1,h2,h3,h4,h5,h6 { + text-align: center; + clear: both; +} + +p { + margin-top: .51em; + text-align: justify; + margin-bottom: .49em; + text-indent:4%; +} + +.nind {text-indent:0%;} + +.hanging2 {padding-left: 2em; + text-indent: -1em; + } + +.align-center { +display: block; +text-align: center; +margin-top: 1em; +margin-bottom: 1em; +} + +code { + font-family: "Times New Roman", Times, serif; + padding: 1px; + font-size: 105%; +} + +.center {text-align: center;text-indent:0%;} + +hr { + width: 33%; + margin-top: 2em; + margin-bottom: 2em; + margin-left: auto; + margin-right: auto; + clear: both; +} +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} + +.footnote .label {position: absolute; right: 84%; text-align: right;} + +.fnanchor { + vertical-align: super; + font-size: .8em; + text-decoration: + none; +} + +.figcenter { + margin: 3% auto 3% auto; + clear: both; + text-align: center; + text-indent: 0% +} + +.caption {font-weight: normal; + font-size: 90%; + text-align: right; + padding-bottom: 1em;} + +.caption p +{ + text-align: center; + text-indent: 0; + margin: 0.25em 0; +} + +.dropcap { + float: left; + clear: left; + font-size: 250%; + margin-top:-.7%; + margin: 0 0.15em 0 0; + padding: 0; + line-height: 0.85em; + text-indent: 0 +} + +div.chapter { + page-break-before: always; + margin-top: 4em + } + +.pagenum { + position: absolute; + left: 92%; + font-size: small; + text-align: right; + font-style: normal; + font-weight: normal; + font-variant: normal; + text-indent: 0; +} + +.transnote {background-color: #E6E6FA; + color: black; + font-size:smaller; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; } + + </style> + + <style> + #pg-header div, #pg-footer div { + all: initial; + display: block; + margin-top: 1em; + margin-bottom: 1em; + margin-left: 2em; + } + #pg-footer div.agate { + font-size: 90%; + margin-top: 0; + margin-bottom: 0; + text-align: center; + } + #pg-footer li { + all: initial; + display: block; + margin-top: 1em; + margin-bottom: 1em; + text-indent: -0.6em; + } + #pg-footer div.secthead { + font-size: 110%; + font-weight: bold; + } + #pg-footer #project-gutenberg-license { + font-size: 110%; + margin-top: 0; + margin-bottom: 0; + text-align: center; + } + #pg-header h2 { + text-align: center; + font-size: 110%; + } + #pg-footer h2 { + text-align: center; + font-size: 120%; + font-weight: normal; + margin-top: 0; + margin-bottom: 0; + } + #pg-header #pg-machine-header p { + text-indent: -4em; + padding-left: 4em; + margin-top: 1em; + } + #pg-header #pg-header-authlist p { + margin-left: -2em; + margin-top: 0; + margin-bottom: 0; + } + #pg-header #pg-machine-header strong { + font-weight: normal; + } + #pg-header #pg-start-separator, #pg-footer #pg-end-separator { + margin-bottom: 3em; + margin-left: 0; + margin-right: auto; + margin-top: 2em; + text-align: center + } + </style> +</head> + +<body> +<div>*** START OF THE PROJECT GUTENBERG EBOOK 36276 ***</div> + +<div class="figcenter" style="width: 500px;"> +<img src="images/cover.jpg" width="500" alt="500"> +</div> + +<div class='chapter'> +<h1>THE MEANING OF<br> +RELATIVITY</h1> + +<p><br><br></p> + +<p class="center"><b>FOUR LECTURES DELIVERED AT +PRINCETON UNIVERSITY, MAY, 1921</b></p> + +<p><br><br></p> + +<p class="center"><b>BY</b></p> + +<div style='text-align:center; font-size:1.2em;'>ALBERT EINSTEIN</div> + +<p><br><br></p> + +<p class="center"><b>WITH FOUR DIAGRAMS</b></p> + +<p><br><br></p> + +<p class="center"><b>PRINCETON</b><br> +<b>PRINCETON UNIVERSITY PRESS</b><br> +<b>1923</b></p> +</div> + +<p><br><br></p> + +<div class='chapter'> +<p class="center"><i>Copyright 1922 +Princeton University Press +Published 1922</i></p> + +<p><br><br></p> + +<p class="hanging2"> +NOTE.—The translation of these lectures into English +was made by EDWIN PLIMPTON ADAMS, Professor +of Physics in Princeton University +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2>CONTENTS</h2> +<p class="nind"> +LECTURE I<br> +<a href="#chap01">SPACE AND TIME IN PRE-RELATIVITY PHYSICS</a><br> +LECTURE II<br> +<a href="#chap02">THE THEORY OF SPECIAL RELATIVITY</a><br> +LECTURE III<br> +<a href="#chap03">THE GENERAL THEORY OF RELATIVITY</a><br> +LECTURE IV<br> +<a href="#chap04">THE GENERAL THEORY OF RELATIVITY (<i>continued</i>)</a><br> +<a href="#INDEX">INDEX</a></p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2>THE MEANING OF RELATIVITY</h2> + +<p><br><br><br></p> + +<h2 title="LECTURE I: SPACE AND TIME IN PRE-RELATIVITY +PHYSICS"><a id="chap01"></a>LECTURE I +<br><br> +SPACE AND TIME IN PRE-RELATIVITY +PHYSICS</h2> + +<p class="nind"> +<span class="dropcap">T</span>HE theory of relativity is intimately connected with the theory +of space and time. I shall therefore begin with a brief investigation +of the origin of our ideas of space and time, although in +doing so I know that I introduce a controversial subject. The +object of all science, whether natural science or psychology, is +to co-ordinate our experiences and to bring them into a logical +system. How are our customary ideas of space and time related +to the character of our experiences? +</p> +<p> +The experiences of an individual appear to us arranged in a +series of events; in this series the single events which we remember +appear to be ordered according to the criterion of "earlier" +and "later," which cannot be analysed further. There exists, +therefore, for the individual, an I-time, or subjective time. This +in itself is not measurable. I can, indeed, associate numbers with +the events, in such a way that a greater number is associated +with the later event than with an earlier one; but the nature of +this association may be quite arbitrary. This association I can +define by means of a clock by comparing the order of events furnished +by the clock with the order of the given series of events. +We understand by a clock something which provides a series of +events which can be counted, and which has other properties of +which we shall speak later. +<span class="pagenum" id="Page_1">[Pg 1]</span> +</p> +<p> +By the aid of speech different individuals can, to a certain +extent, compare their experiences. In this way it is shown that +certain sense perceptions of different individuals correspond to +each other, while for other sense perceptions no such correspondence +can be established. We are accustomed to regard as real +those sense perceptions which are common to different individuals, +and which therefore are, in a measure, impersonal. The natural +sciences, and in particular, the most fundamental of them, +physics, deal with such sense perceptions. The conception of +physical bodies, in particular of rigid bodies, is a relatively constant +complex of such sense perceptions. A clock is also a body, +or a system, in the same sense, with the additional property that +the series of events which it counts is formed of elements all of +which can be regarded as equal. +</p> +<p> +The only justification for our concepts and system of concepts +is that they serve to represent the complex of our experiences; +beyond this they have no legitimacy. I am convinced that +the philosophers have had a harmful effect upon the progress +of scientific thinking in removing certain fundamental concepts +from the domain of empiricism, where they are under our control, +to the intangible heights of the <i>a priori</i>. For even if it should +appear that the universe of ideas cannot be deduced from experience +by logical means, but is, in a sense, a creation of the +human mind, without which no science is possible, nevertheless +this universe of ideas is just as little independent of the nature +of our experiences as clothes are of the form of the human body. +This is particularly true of our concepts of time and space, which +physicists have been obliged by the facts to bring down from the +Olympus of the <i>a priori</i> in order to adjust them and put them +in a serviceable condition. +<span class="pagenum" id="Page_2">[Pg 2]</span> +</p> +<p> +We now come to our concepts and judgments of space. It +is essential here also to pay strict attention to the relation of +experience to our concepts. It seems to me that Poincaré clearly +recognized the truth in the account he gave in his book, "La +Science et l'Hypothèse." Among all the changes which we can +perceive in a rigid body those are marked by their simplicity +which can be made reversibly by an arbitrary motion of the +body; Poincaré calls these, changes in position. By means of +simple changes in position we can bring two bodies into contact. +The theorems of congruence, fundamental in geometry, have to +do with the laws that govern such changes in position. For the +concept of space the following seems essential. We can form new +bodies by bringing bodies <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">, <img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">, ... up to body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">; we say that +we <i>continue</i> body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">. We can continue body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> in such a way that +it comes into contact with any other body, <img style="vertical-align: 0; width: 1.928ex; height: 1.545ex;" src="images/4.svg" alt=" " data-tex="X">. The ensemble of +all continuations of body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> we can designate as the "space of +the body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." Then it is true that all bodies are in the "space of +the (arbitrarily chosen) body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." In this sense we cannot speak +of space in the abstract, but only of the "space belonging to a +body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." The earth's crust plays such a dominant rôle in our +daily life in judging the relative positions of bodies that it has +led to an abstract conception of space which certainly cannot be +defended. In order to free ourselves from this fatal error we shall +speak only of "bodies of reference," or "space of reference." It +was only through the theory of general relativity that refinement +of these concepts became necessary, as we shall see later. +</p> +<p> +I shall not go into detail concerning those properties of the +space of reference which lead to our conceiving points as elements +of space, and space as a continuum. Nor shall I attempt +to analyse further the properties of space which justify the conception +<span class="pagenum" id="Page_3">[Pg 3]</span> +of continuous series of points, or lines. If these concepts +are assumed, together with their relation to the solid bodies of +experience, then it is easy to say what we mean by the three-dimensionality +of space; to each point three numbers, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> +(co-ordinates), may be associated, in such a way that this association +is uniquely reciprocal, and that <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}"> and <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> vary +continuously when the point describes a continuous series of points +(a line). +</p> +<p> +It is assumed in pre-relativity physics that the laws of the +orientation of ideal rigid bodies are consistent with Euclidean +geometry. What this means may be expressed as follows: Two +points marked on a rigid body form an <i>interval</i>. Such an interval +can be oriented at rest, relatively to our space of reference, in +a multiplicity of ways. If, now, the points of this space can +be referred to co-ordinates <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, in such a way that the +differences of the co-ordinates, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/8.svg" alt=" " data-tex="\Delta x_{1}">, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/9.svg" alt=" " data-tex="\Delta x_{2}">, +<img style="vertical-align: -0.375ex; width: 4.166ex; height: 1.994ex;" src="images/10.svg" alt=" " data-tex="\Delta x_{3}">, of the two ends +of the interval furnish the same sum of squares, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 34.482ex; height: 2.633ex;" src="images/11.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}, \qquad\text{(1)} +"></span> +for every orientation of the interval, then the space of reference +is called Euclidean, and the co-ordinates Cartesian.<a id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> It is +sufficient, indeed, to make this assumption in the limit for an +infinitely small interval. Involved in this assumption there are +some which are rather less special, to which we must call attention +on account of their fundamental significance. In the first +place, it is assumed that one can move an ideal rigid body in an +arbitrary manner. In the second place, it is assumed that the behaviour +of ideal rigid bodies towards orientation is independent +<span class="pagenum" id="Page_4">[Pg 4]</span> +of the material of the bodies and their changes of position, in the +sense that if two intervals can once be brought into coincidence, +they can always and everywhere be brought into coincidence. +Both of these assumptions, which are of fundamental importance +for geometry and especially for physical measurements, +naturally arise from experience; in the theory of general relativity +their validity needs to be assumed only for bodies and spaces +of reference which are infinitely small compared to astronomical +dimensions. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a>This relation must hold for an arbitrary choice of the origin and of the +direction +(ratios <img style="vertical-align: -0.375ex; width: 16.271ex; height: 1.994ex;" src="images/12.svg" alt=" " data-tex="\Delta x_{1}: \Delta x_{2}: \Delta x_{3}">) +of the interval.</p></div> + +<p> +The quantity <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> we call the length of the interval. In order +that this may be uniquely determined it is necessary to fix arbitrarily +the length of a definite interval; for example, we can put +it equal to 1 (unit of length). Then the lengths of all other intervals +may be determined. If we make the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> linearly dependent +upon a parameter <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda">, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 14.299ex; height: 2.009ex;" src="images/16.svg" alt=" " data-tex=" +x_{\nu} = a_{\nu} + \lambda b_{\nu}, +"></span> +we obtain a line which has all the properties of the straight +lines of the Euclidean geometry. In particular, it easily follows +that by laying off <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> times the interval <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> upon a straight line, an +interval of length <img style="vertical-align: -0.025ex; width: 4.053ex; height: 1.025ex;" src="images/18.svg" alt=" " data-tex="n·s"> is obtained. A length, therefore, means +the result of a measurement carried out along a straight line by +means of a unit measuring rod. It has a significance which is as +independent of the system of co-ordinates as that of a straight +line, as will appear in the sequel. +</p> +<p> +We come now to a train of thought which plays an analogous +role in the theories of special and general relativity. We ask +the question: besides the Cartesian co-ordinates which we have +used are there other equivalent co-ordinates? An interval has +<span class="pagenum" id="Page_5">[Pg 5]</span> +a physical meaning which is independent of the choice of co-ordinates; +and so has the spherical surface which we obtain as +the locus of the end points of all equal intervals that we lay off +from an arbitrary point of our space of reference. If <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> as well +as <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> (<img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> from 1 to 3) are Cartesian co-ordinates of our space +of reference, then the spherical surface will be expressed in our +two systems of co-ordinates by the equations +<span class="align-center"><img style="vertical-align: -3.006ex; width: 31.357ex; height: 7.142ex;" src="images/21.svg" alt=" " data-tex=" +\begin{align*} +\sum {\Delta x_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2)} \\ +\sum {{\Delta x'}_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2a)} +\end{align*} +"></span> +How must the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> be expressed in terms of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> in order that +equations (2) and (2a) may be equivalent to each other? Regarding +the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> expressed as functions of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, we can write, +by Taylor's theorem, for small values of the <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"><img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, +<span class="align-center"><img style="vertical-align: -3.06ex; width: 52.322ex; height: 6.477ex;" src="images/23.svg" alt=" " data-tex=" +{\Delta x'}_{\nu} + = \sum_{\alpha} \frac{\partial {x'}_{\nu}}{\partial x_{\alpha}}\, \Delta x_{\alpha} + + \frac{1}{2} \sum_{\alpha \text{,} \beta} + \frac{\partial^{2} {x'}_{\nu}}{\partial x_{\alpha}\, \partial x_{\beta}}\, + \Delta x_{\alpha}\, \Delta x_{\beta}.\ldots +"></span> +If we substitute (2a) in this equation and compare with (1), +we see that the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> must be linear functions of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If we +therefore put +<span class="align-center"><img style="vertical-align: -4.542ex; width: 42.282ex; height: 10.216ex;" src="images/24.svg" alt=" " data-tex=" +\begin{align*} +& & {x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}, +\qquad & &\text{(3)}\\ +&\text{or} & \Delta {x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}, +\qquad & &\text{(3a)} +\end{align*}"></span> +<span class="pagenum" id="Page_6">[Pg 6]</span> +then the equivalence of equations (2) and (2a) is expressed in +the form +<span class="align-center"><img style="vertical-align: -1.018ex; width: 59.118ex; height: 3.296ex;" src="images/25.svg" alt=" " data-tex=" +\sum{\Delta x'_{\nu}}^{2} = \lambda \sum{\Delta x_{\nu}}^{2} + \qquad\text{(\(\lambda\) independent of \(\Delta x_{\nu}\)).} +\qquad\text{(2b)} +"></span> +It therefore follows that <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> must be a constant. If we put <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> = 1, +(2b) and (3a) furnish the conditions +<span class="align-center"><img style="vertical-align: -2.602ex; width: 24.147ex; height: 4.751ex;" src="images/26.svg" alt=" " data-tex=" +\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}, +\qquad\text{(4)} +"></span> +in which <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> = 1, or <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> = 0, according <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> or +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> ≠<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">. The +conditions (4) are called the conditions of orthogonality, and the +transformations (3), (4), linear orthogonal transformations. If +we stipulate that <img style="vertical-align: -0.566ex; width: 13.034ex; height: 2.633ex;" src="images/30.svg" alt=" " data-tex="s^2 = \sum {\Delta x_{\nu}}^2"> shall be equal to the square of +the length in every system of co-ordinates, and if we always measure +with the same unit scale, then <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> must be equal to 1. Therefore +the linear orthogonal transformations are the only ones by +means of which we can pass from one Cartesian system of co-ordinates +in our space of reference to another. We see that in +applying such transformations the equations of a straight line +become equations of a straight line. Reversing equations (3a) +by multiplying both sides by <img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/31.svg" alt=" " data-tex="b_{\nu\beta}"> and summing for all +the <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">'s, we obtain +<span class="align-center"><img style="vertical-align: -2.912ex; width: 58.403ex; height: 5.061ex;" src="images/32.svg" alt=" " data-tex=" +\sum_{\nu} b_{\nu\beta}\, {\Delta x'}_{\nu} + = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} + = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} + = \Delta x_{\beta} +\qquad\text{(5)} +"></span> +The same coefficients, <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/33.svg" alt=" " data-tex="b">, also determine the inverse substitution +of <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"><img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. Geometrically, <img style="vertical-align: -0.357ex; width: 3.03ex; height: 1.927ex;" src="images/34.svg" alt=" " data-tex="b_{\nu\alpha}"> is the cosine of the angle between +the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> axis and the <img style="vertical-align: -0.357ex; width: 2.506ex; height: 1.357ex;" src="images/35.svg" alt=" " data-tex="x_{\alpha}"> axis. +<span class="pagenum" id="Page_7">[Pg 7]</span> +</p> +<p> +To sum up, we can say that in the Euclidean geometry +there are (in a given space of reference) preferred systems of +co-ordinates, the Cartesian systems, which transform into each +other by linear orthogonal transformations. The distance <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> between +two points of our space of reference, measured by a measuring +rod, is expressed in such co-ordinates in a particularly +simple manner. The whole of geometry may be founded upon +this conception of distance. In the present treatment, geometry +is related to actual things (rigid bodies), and its theorems are +statements concerning the behaviour of these things, which may +prove to be true or false. +</p> +<p> +One is ordinarily accustomed to study geometry divorced +from any relation between its concepts and experience. There +are advantages in isolating that which is purely logical and independent +of what is, in principle, incomplete empiricism. This +is satisfactory to the pure mathematician. He is satisfied if he +can deduce his theorems from axioms correctly, that is, without +errors of logic. The question as to whether Euclidean geometry +is true or not does not concern him. But for our purpose it +is necessary to associate the fundamental concepts of geometry +with natural objects; without such an association geometry is +worthless for the physicist. The physicist is concerned with the +question as to whether the theorems of geometry are true or +not. That Euclidean geometry, from this point of view, affirms +something more than the mere deductions derived logically from +definitions may be seen from the following simple consideration. +</p> +<p> +Between <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> points of space there are <img style="vertical-align: -1.552ex; width: 9.368ex; height: 4.855ex;" src="images/36.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2}"> distances, <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}">; +between these and the <img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/38.svg" alt=" " data-tex="3n"> co-ordinates we have the relations +<span class="align-center"><img style="vertical-align: -0.904ex; width: 43.644ex; height: 3.274ex;" src="images/39.svg" alt=" " data-tex=" +s_{\mu\nu}^{2} + = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + + .\dots +"></span> +<span class="pagenum" id="Page_8">[Pg 8]</span> +</p> +<p> +From these <img style="vertical-align: -1.552ex; width: 9.368ex; height: 4.855ex;" src="images/36.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2}"> equations the <img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/38.svg" alt=" " data-tex="3n"> co-ordinates may be +eliminated, and from this elimination at least <img style="vertical-align: -1.552ex; width: 14.622ex; height: 4.855ex;" src="images/40.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2} - 3n"> +equations in the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}">, will result.<a id="FNanchor_2_1"></a><a href="#Footnote_2_1" class="fnanchor">[2]</a> +Since the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}"> are measurable +quantities, and by definition are independent of each other, these +relations between the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}"> are not necessary <i>a priori</i>. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_2_1"></a><a href="#FNanchor_2_1"><span class="label">[2]</span></a>In reality there are <img style="vertical-align: -1.552ex; width: 18.519ex; height: 4.855ex;" src="images/41.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2} - 3n + 6"> equations.</p></div> + +<p> +From the foregoing it is evident that the equations of transformation +(3), (4) have a fundamental significance in Euclidean +geometry, in that they govern the transformation from one +Cartesian system of co-ordinates to another. The Cartesian +systems of co-ordinates are characterized by the property that +in them the measurable distance between two points, <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, is +expressed by the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 14.541ex; height: 3.167ex;" src="images/42.svg" alt=" " data-tex=" +s^{2} = \sum {\Delta x_{\nu}}^{2}. +"></span> +</p> +<p> +If <img style="vertical-align: -0.579ex; width: 3.756ex; height: 2.125ex;" src="images/43.svg" alt=" " data-tex="K_{x_{\nu}}"> and <img style="vertical-align: -0.799ex; width: 3.756ex; height: 2.516ex;" src="images/44.svg" alt=" " data-tex="K'_{x_{\nu}}"> are two Cartesian systems of co-ordinates, +then +<span class="align-center"><img style="vertical-align: -1.018ex; width: 21.338ex; height: 3.296ex;" src="images/45.svg" alt=" " data-tex=" +\sum {\Delta x_{\nu}}^{2} = \sum {\Delta x'_{\nu}}^{2}. +"></span> +</p> +<p> +The right-hand side is identically equal to the left-hand side +on account of the equations of the linear orthogonal transformation, +and the right-hand side differs from the left-hand side +only in that the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> are replaced by the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}">. This is expressed +by the statement that <img style="vertical-align: -0.566ex; width: 7.968ex; height: 2.633ex;" src="images/46.svg" alt=" " data-tex="\sum {\Delta x_{\nu}}^{2}"> is an invariant with respect to +linear orthogonal transformations. It is evident that in the Euclidean +geometry only such, and all such, quantities have an +objective significance, independent of the particular choice of +<span class="pagenum" id="Page_9">[Pg 9]</span> +the Cartesian co-ordinates, as can be expressed by an invariant +with respect to linear orthogonal transformations. This is +the reason that the theory of invariants, which has to do with +the laws that govern the form of invariants, is so important for +analytical geometry. +</p> +<p> +As a second example of a geometrical invariant, consider a +volume. This is expressed by +<span class="align-center"><img style="vertical-align: -1.948ex; width: 21.373ex; height: 5.027ex;" src="images/47.svg" alt=" " data-tex=" +V = \iiint dx_{1}\, dx_{2}\, dx_{3}. +"></span> +By means of Jacobi's theorem we may write +<span class="align-center"><img style="vertical-align: -2.172ex; width: 52.029ex; height: 5.495ex;" src="images/48.svg" alt=" " data-tex=" +\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} + = \iiint \frac{\partial({x'}_{1}, {x'}_{2}, {x'}_{3})}{\partial(x_{1}, x_{2}, x_{3})}\, + dx_{1}\, dx_{2}\, dx_{3} +"></span> +where the integrand in the last integral is the functional determinant +of the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> with respect to the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and this by (3) is equal +to the determinant <img style="vertical-align: -0.685ex; width: 4.229ex; height: 2.381ex;" src="images/49.svg" alt=" " data-tex="|b_{\mu\nu}|"> of the coefficients of substitution, <img style="vertical-align: -0.357ex; width: 3.03ex; height: 1.927ex;" src="images/34.svg" alt=" " data-tex="b_{\nu\alpha}">. If +we form the determinant of the <img style="vertical-align: -0.685ex; width: 3.181ex; height: 2.307ex;" src="images/50.svg" alt=" " data-tex="\delta_{\mu\alpha}"> from equation (4), we obtain, +by means of the theorem of multiplication of determinants, +<span class="align-center"><img style="vertical-align: -2.602ex; width: 52.322ex; height: 6.335ex;" src="images/51.svg" alt=" " data-tex=" +1 = |\delta_{\alpha\beta}| + = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| + = |b_{\mu\nu}|^{2};\quad +|b_{\mu\nu}| = ±1. +\qquad\text{(6)} +"></span> +If we limit ourselves to those transformations which have the determinant ++1,<a id="FNanchor_3_1"></a><a href="#Footnote_3_1" class="fnanchor">[3]</a> +and only these arise from continuous variations +of the systems of co-ordinates, then <img style="vertical-align: -0.05ex; width: 1.74ex; height: 1.595ex;" src="images/52.svg" alt=" " data-tex="V"> is an invariant. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_3_1"></a><a href="#FNanchor_3_1"><span class="label">[3]</span></a>There are thus two kinds of Cartesian systems which are designated as +"right-handed" and "left-handed" systems. The difference between these is +familiar to every physicist and engineer. It is interesting to note that these +two kinds of systems cannot be defined geometrically, but only the contrast +between them.</p></div> +<p><span class="pagenum" id="Page_10">[Pg 10]</span></p> +<p> +Invariants, however, are not the only forms by means of +which we can give expression to the independence of the particular +choice of the Cartesian co-ordinates. Vectors and tensors +are other forms of expression. Let us express the fact that the +point with the current co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> lies upon a straight line. +We have +<span class="align-center"><img style="vertical-align: -0.566ex; width: 31.899ex; height: 2.262ex;" src="images/53.svg" alt=" " data-tex=" +x_{\nu} - A_{\nu} = \lambda B_{\nu} \quad (\nu \text{ from 1 to 3}). +"></span> +Without limiting the generality we can put +<span class="align-center"><img style="vertical-align: -1.018ex; width: 12.162ex; height: 3.167ex;" src="images/54.svg" alt=" " data-tex=" +\sum {B_{\nu}}^{2} = 1. +"></span> +</p> +<p> +If we multiply the equations by <img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/55.svg" alt=" " data-tex="b_{\beta\nu}"> (compare (3a) and (5)) +and sum for all the <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">'s, we get +<span class="align-center"><img style="vertical-align: -0.869ex; width: 15.719ex; height: 2.699ex;" src="images/56.svg" alt=" " data-tex=" +x'_{\beta} - A'_{\beta} = \lambda B'_{\beta}, +"></span> +where we have written +<span class="align-center"><img style="vertical-align: -2.602ex; width: 34.129ex; height: 4.751ex;" src="images/57.svg" alt=" " data-tex=" +B'_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad +A'_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}. +"></span> +</p> +<p> +These are the equations of straight lines with respect to a +second Cartesian system of co-ordinates <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. They have the +same form as the equations with respect to the original system +of co-ordinates. It is therefore evident that straight lines +have a significance which is independent of the system of co-ordinates. +Formally, this depends upon the fact that the quantities (<img style="vertical-align: -0.339ex; width: 7.828ex; height: 1.959ex;" src="images/59.svg" alt=" " data-tex="x_{\nu} - A_{\nu}">) - <img style="vertical-align: -0.339ex; width: 4.072ex; height: 1.91ex;" src="images/60.svg" alt=" " data-tex="\lambda B_{\nu}"> are +transformed as the components of +an interval, <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">. The ensemble of three quantities, defined for +every system of Cartesian co-ordinates, and which transform as +the components of an interval, is called a vector. If the three +<span class="pagenum" id="Page_11">[Pg 11]</span> +components of a vector vanish for one system of Cartesian co-ordinates, +they vanish for all systems, because the equations of +transformation are homogeneous. We can thus get the meaning +of the concept of a vector without referring to a geometrical representation. +This behaviour of the equations of a straight line +can be expressed by saying that the equation of a straight line +is co-variant with respect to linear orthogonal transformations. +</p> +<p> +We shall now show briefly that there are geometrical entities +which lead to the concept of tensors. Let <img style="vertical-align: -0.375ex; width: 2.44ex; height: 1.92ex;" src="images/62.svg" alt=" " data-tex="P_{0}"> be the centre of a +surface of the second degree, <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> any point on the surface, and +<img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> the projections of the interval <img style="vertical-align: -0.375ex; width: 4.139ex; height: 1.92ex;" src="images/65.svg" alt=" " data-tex="P_{0}P"> upon the co-ordinate axes. +Then the equation of the surface is +<span class="align-center"><img style="vertical-align: -1.018ex; width: 15.789ex; height: 3.167ex;" src="images/66.svg" alt=" " data-tex=" +\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +"></span> +In this, and in analogous cases, we shall omit the sign of summation, +and understand that the summation is to be carried out +for those indices that appear twice. We thus write the equation +of the surface +<span class="align-center"><img style="vertical-align: -0.685ex; width: 12.144ex; height: 2.278ex;" src="images/67.svg" alt=" " data-tex=" +a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +"></span> +The quantities <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}"> determine the surface completely, for a given +position of the centre, with respect to the chosen system of +Cartesian co-ordinates. From the known law of transformation +for the <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> (3a) for linear orthogonal transformations, we easily +find the law of transformation for the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">:<a id="FNanchor_4_1"></a><a href="#Footnote_4_1" class="fnanchor">[4]</a> +<span class="align-center"><img style="vertical-align: -0.685ex; width: 16.466ex; height: 2.515ex;" src="images/69.svg" alt=" " data-tex=" +{a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}. +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_4_1"></a><a href="#FNanchor_4_1"><span class="label">[4]</span></a>The equation <img style="vertical-align: -0.565ex; width: 13.255ex; height: 2.282ex;" src="images/70.svg" alt=" " data-tex="{a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1"> +may, by (5), be replaced by +<img style="vertical-align: -0.58ex; width: 3.125ex; height: 2.297ex;" src="images/71.svg" alt=" " data-tex="a'_{\sigma\tau}"><img style="vertical-align: -0.685ex; width: 3.037ex; height: 2.255ex;" src="images/72.svg" alt=" " data-tex="b_{\mu\sigma}"><img style="vertical-align: -0.36ex; width: 2.833ex; height: 1.93ex;" src="images/73.svg" alt=" " data-tex="b_{\nu\tau}"><img style="vertical-align: -0.685ex; width: 2.143ex; height: 2.278ex;" src="images/74.svg" alt=" " data-tex="\xi_{\mu}"><img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> = 1, +from which the result stated immediately follows.</p></div> +<p><span class="pagenum" id="Page_12">[Pg 12]</span></p> + +<p class="nind"> +This transformation is homogeneous and of the first degree in +the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">. On account of this transformation, the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">, are called +components of a tensor of the second rank (the latter on account +of the double index). If all the components, <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">, of a tensor with +respect to any system of Cartesian co-ordinates vanish, they +vanish with respect to every other Cartesian system. The form +and the position of the surface of the second degree is described +by this tensor (<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/75.svg" alt=" " data-tex="a">). +</p> +<p> +Analytic tensors of higher rank (number of indices) may be +defined. It is possible and advantageous to regard vectors as +tensors of rank 1, and invariants (scalars) as tensors of rank 0. +In this respect, the problem of the theory of invariants may be so +formulated: according to what laws may new tensors be formed +from given tensors? We shall consider these laws now, in order +to be able to apply them later. We shall deal first only with the +properties of tensors with respect to the transformation from +one Cartesian system to another in the same space of reference, +by means of linear orthogonal transformations. As the laws are +wholly independent of the number of dimensions, we shall leave +this number, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n">, indefinite at first. +</p> + +<p><br></p> + +<p> +<i>Definition</i>. If a figure is defined with respect to every system +of Cartesian co-ordinates in a space of reference of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> dimensions +by the <img style="vertical-align: -0.025ex; width: 2.569ex; height: 1.553ex;" src="images/76.svg" alt=" " data-tex="n^{\alpha}"> numbers <img style="vertical-align: -0.685ex; width: 6.399ex; height: 2.305ex;" src="images/77.svg" alt=" " data-tex="A_{\mu\nu\rho\ldots}"> (<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = number of indices), then +these numbers are the components of a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> if the +transformation law is +<span class="align-center"><img style="vertical-align: -0.685ex; width: 42.162ex; height: 2.515ex;" src="images/78.svg" alt=" " data-tex=" +{A'}_{\mu'\nu'\rho'}\ldots + = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho}\ldots A_{\mu\nu\rho}.\ldots +\qquad\text{(7)} +"></span> +<span class="pagenum" id="Page_13">[Pg 13]</span> +</p> + +<p><br></p> + +<p> +<i>Remark</i>. From this definition it follows that +<span class="align-center"><img style="vertical-align: -0.685ex; width: 29.803ex; height: 2.382ex;" src="images/79.svg" alt=" " data-tex=" +A_{\mu\nu\rho}\ldots = B_{\mu} C_{\nu} D_{\rho}\dots +\qquad\text{(8)} +"></span> +is an invariant, provided that (<img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">), (<img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">), +(<img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/80.svg" alt=" " data-tex="D">) ... are vectors. +Conversely, the tensor character of (<img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">) may be inferred, if it +is known that the expression (8) leads to an invariant for an +arbitrary choice of the vectors (<img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">), (<img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">), etc. +</p> + +<p><br></p> + +<p> +<i>Addition and Subtraction</i>. By addition and subtraction of +the corresponding components of tensors of the same rank, a +tensor of equal rank results: +<span class="align-center"><img style="vertical-align: -0.685ex; width: 37.182ex; height: 2.382ex;" src="images/81.svg" alt=" " data-tex=" +A_{\mu\nu\rho}\ldots ± B_{\mu\nu\rho}\ldots = C_{\mu\nu\rho}\ldots. +\qquad\text{(9)} +"></span> +The proof follows from the definition of a tensor given above. +</p> + +<p><br></p> + +<p> +<i>Multiplication</i>. From a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and a tensor of +rank <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> we may obtain a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> + <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> by multiplying all +the components of the first tensor by all the components of the +second tensor: +<span class="align-center"><img style="vertical-align: -0.685ex; width: 40.555ex; height: 2.382ex;" src="images/82.svg" alt=" " data-tex=" +T_{\mu\nu\rho}\ldots_{\alpha\beta}\ldots += A_{\mu\nu\rho}\ldots B_{\alpha\beta\gamma}.\ldots +\qquad\text{(10)} +"></span> +</p> + +<p><br></p> + +<p> +<i>Contraction</i>. A tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> - 2 may be obtained from +one of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> by putting two definite indices equal to each other +and then summing for this single index: +<span class="align-center"><img style="vertical-align: -2.947ex; width: 41.445ex; height: 5.097ex;" src="images/83.svg" alt=" " data-tex=" +T_{\rho}\ldots + = A_{\mu\mu\rho}\ldots + ( = \sum_{\mu} A_{\mu\mu\rho}\ldots). +\qquad\text{(11)} +"></span> +</p> +<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> +<p class="nind"> +The proof is +<span class="align-center"><img style="vertical-align: -2.222ex; width: 45.796ex; height: 5.576ex;" src="images/84.svg" alt=" " data-tex=" +\begin{align*} +{A'}_{\mu\mu\rho\cdots} + = b_{\mu\alpha} b_{\mu\beta} b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} + = \delta_{\alpha\beta} & b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} \\ + {} ={} & b_{\rho\gamma\cdots} A_{\alpha\alpha\gamma\cdots}. +\end{align*} +"></span> +</p> +<p> +In addition to these elementary rules of operation there is +also the formation of tensors by differentiation ("erweiterung"): +<span class="align-center"><img style="vertical-align: -1.909ex; width: 28.293ex; height: 5.255ex;" src="images/85.svg" alt=" " data-tex=" +T_{\mu\nu\rho\cdots\alpha} + = \frac{\partial A_{\mu\nu\rho\cdots}}{\partial x_{\alpha}}. +\qquad\text{(12)} +"></span> +</p> +<p> +New tensors, in respect to linear orthogonal transformations, +may be formed from tensors according to these rules of operation. +</p> + +<p><br></p> + +<p> +<i>Symmetrical Properties of Tensors</i>. Tensors are called symmetrical +or skew-symmetrical in respect to two of their indices, +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, if both the components which result from interchanging +the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> are equal to each other or equal with +opposite signs. +<span class="align-center"><img style="vertical-align: -2.156ex; width: 46.294ex; height: 5.442ex;" src="images/87.svg" alt=" " data-tex=" +\begin{alignat*}{2} +&\text{Condition for symmetry:} & + A_{\mu\nu\rho} &= A_{{\nu\mu}\rho}. \\ +&\text{Condition for skew-symmetry:}\quad & + A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ +\end{alignat*} +"></span> +</p> + +<p><br></p> + +<p> +<i>Theorem</i>. The character of symmetry or skew-symmetry +exists independently of the choice of co-ordinates, and in this +lies its importance. The proof follows from the equation defining tensors. +</p> +<p> +<i>Special Tensors.</i> +</p> +<p> +I. The quantities <img style="vertical-align: -0.685ex; width: 2.933ex; height: 2.307ex;" src="images/88.svg" alt=" " data-tex="\delta_{\rho\sigma}"> (4) are tensor components +(fundamental tensor). +<span class="pagenum" id="Page_15">[Pg 15]</span> +</p> +<p> +<i>Proof</i>. If in the right-hand side of the equation of transformation +<img style="vertical-align: -0.904ex; width: 3.697ex; height: 2.622ex;" src="images/89.svg" alt=" " data-tex="A'_{\mu\nu}"> = <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"><img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/31.svg" alt=" " data-tex="b_{\nu\beta}"><img style="vertical-align: -0.65ex; width: 3.814ex; height: 2.27ex;" src="images/91.svg" alt=" " data-tex="A_{\alpha\beta}">, we substitute +for <img style="vertical-align: -0.65ex; width: 3.814ex; height: 2.27ex;" src="images/91.svg" alt=" " data-tex="A_{\alpha\beta}"> the quantities <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> (which are equal +to 1 or 0 according as <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> or <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> ≠<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">), +we get +<span class="align-center"><img style="vertical-align: -0.685ex; width: 20.17ex; height: 2.515ex;" src="images/92.svg" alt=" " data-tex=" +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}. +"></span> +The justification for the last sign of equality becomes evident if +one applies (4) to the inverse substitution (5). +</p> +<p> +II. There is a tensor (<img style="vertical-align: -0.685ex; width: 5.707ex; height: 2.307ex;" src="images/93.svg" alt=" " data-tex="\delta_{\mu\nu\rho\ldots}">) skew-symmetrical with respect +to all pairs of indices, whose rank is equal to the number of +dimensions, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n">, and whose components are equal to +1 or -1 +according as <img style="vertical-align: -0.489ex; width: 6.762ex; height: 1.489ex;" src="images/94.svg" alt=" " data-tex="\mu\nu\rho\ldots"> is an even or odd permutation of 1 2 3.... +</p> +<p> +The proof follows with the aid of the theorem proved above +<img style="vertical-align: -0.685ex; width: 8.305ex; height: 2.381ex;" src="images/95.svg" alt=" " data-tex="|b_{\rho\sigma}| = 1"> +</p> +<p> +These few simple theorems form the apparatus from the +theory of invariants for building the equations of pre-relativity +physics and the theory of special relativity. +</p> +<p> +We have seen that in pre-relativity physics, in order to specify +relations in space, a body of reference, or a space of reference, +is required, and, in addition, a Cartesian system of co-ordinates. +We can fuse both these concepts into a single one by thinking +of a Cartesian system of co-ordinates as a cubical frame-work +formed of rods each of unit length. The co-ordinates of the lattice +points of this frame are integral numbers. It follows from +the fundamental relation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 34.607ex; height: 2.633ex;" src="images/96.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} +\qquad\text{(13)} +"></span> +that the members of such a space-lattice are all of unit length. +To specify relations in time, we require in addition a standard +clock placed at the origin of our Cartesian system of co-ordinates +<span class="pagenum" id="Page_16">[Pg 16]</span> +or frame of reference. If an event takes place anywhere we can +assign to it three co-ordinates, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and a time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, as soon as +we have specified the time of the clock at the origin which is +simultaneous with the event. We therefore give an objective significance +to the statement of the simultaneity of distant events, +while previously we have been concerned only with the simultaneity +of two experiences of an individual. The time so specified +is at all events independent of the position of the system of co-ordinates +in our space of reference, and is therefore an invariant +with respect to the transformation (3). +</p> +<p> +It is postulated that the system of equations expressing the +laws of pre-relativity physics is co-variant with respect to the +transformation (3), as are the relations of Euclidean geometry. +The isotropy and homogeneity of space is expressed in this way.<a id="FNanchor_5_1"></a><a href="#Footnote_5_1" class="fnanchor">[5]</a> +We shall now consider some of the more important equations of +physics from this point of view. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_5_1"></a><a href="#FNanchor_5_1"><span class="label">[5]</span></a>The laws of physics could be expressed, even in case there were a +unique direction in space, in such a way as to be co-variant with respect +to the transformation (3); but such an expression would in this case be +unsuitable. If there were a unique direction in space it would simplify the +description of natural phenomena to orient the system of co-ordinates in +a definite way in this direction. But if, on the other hand, there is no +unique direction in space it is not logical to formulate the laws of nature +in such a way as to conceal the equivalence of systems of co-ordinates that +are oriented differently. We shall meet with this point of view again in the +theories of special and general relativity.</p></div> + +<p> +The equations of motion of a material particle are +<span class="align-center"><img style="vertical-align: -1.654ex; width: 22.955ex; height: 5.07ex;" src="images/98.svg" alt=" " data-tex=" +m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}; +\qquad\text{(14)} +"></span> +(<img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">) is a vector; <img style="vertical-align: -0.025ex; width: 1.993ex; height: 1.595ex;" src="images/100.svg" alt=" " data-tex="dt">, and therefore also <img style="vertical-align: -1.577ex; width: 2.989ex; height: 4.613ex;" src="images/101.svg" alt=" " data-tex="\dfrac{1}{dt}">, an invariant; thus +<span class="pagenum" id="Page_17">[Pg 17]</span> +(<img style="vertical-align: -1.577ex; width: 4.502ex; height: 4.676ex;" src="images/102.svg" alt=" " data-tex="\dfrac{dx_{\nu}}{dt}">) is a vector; in the same way it may be shown that +(<img style="vertical-align: -1.654ex; width: 5.489ex; height: 5.07ex;" src="images/103.svg" alt=" " data-tex="\dfrac{d^{2} x_{\nu}}{dt^{2}}">) is a vector. In general, the operation of differentiation +with respect to time does not alter the tensor character. Since +<img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/104.svg" alt=" " data-tex="m"> is an invariant (tensor of rank 0), <img style="vertical-align: -1.654ex; width: 8.356ex; height: 5.07ex;" src="images/105.svg" alt=" " data-tex="(m\dfrac{d^{2} x_{\nu}}{dt^{2}}">) +is a vector, or +tensor of rank 1 (by the theorem of the multiplication of tensors). +If the force (<img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">) has a vector character, the same holds for +the difference (<img style="vertical-align: -1.654ex; width: 14.031ex; height: 5.07ex;" src="images/107.svg" alt=" " data-tex="m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu})">. These equations of motion are +therefore valid in every other system of Cartesian co-ordinates +in the space of reference. In the case where the forces are conservative +we can easily recognize the vector character of (<img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">). +For a potential energy, <img style="vertical-align: 0; width: 1.633ex; height: 1.545ex;" src="images/108.svg" alt=" " data-tex="\Phi">, exists, which depends only upon the +mutual distances of the particles, and is therefore an invariant. +The vector character of the force, <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> = <img style="vertical-align: -1.891ex; width: 6.366ex; height: 5.038ex;" src="images/109.svg" alt=" " data-tex="-\dfrac{\partial \Phi}{\partial x_{\nu}}">, +is then a consequence of our general theorem about the derivative of a tensor +of rank 0. +</p> +<p> +Multiplying by the velocity, a tensor of rank 1, we obtain the +tensor equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 25.76ex; height: 5.564ex;" src="images/110.svg" alt=" " data-tex=" +\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0. +"></span> +By contraction and multiplication by the scalar <img style="vertical-align: -0.025ex; width: 1.993ex; height: 1.595ex;" src="images/100.svg" alt=" " data-tex="dt"> we obtain the +equation of kinetic energy +<span class="align-center"><img style="vertical-align: -2.148ex; width: 20.636ex; height: 5.564ex;" src="images/111.svg" alt=" " data-tex=" +d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\ dx_{\nu}. +"></span> +<span class="pagenum" id="Page_18">[Pg 18]</span> +</p> +<p> +If <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> denotes the difference of the co-ordinates of the material +particle and a point fixed in space, then the <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> have the +character of vectors. We evidently have +<img style="vertical-align: -1.654ex; width: 5.489ex; height: 5.07ex;" src="images/103.svg" alt=" " data-tex="\dfrac{d^{2} x_{\nu}}{dt^{2}}"> = <img style="vertical-align: -1.654ex; width: 5.186ex; height: 5.07ex;" src="images/112.svg" alt=" " data-tex="\dfrac{d^{2} \xi_{\nu}}{dt^{2}}">, so that +the equations of motion of the particle may be written +<span class="align-center"><img style="vertical-align: -1.654ex; width: 17.625ex; height: 5.07ex;" src="images/113.svg" alt=" " data-tex=" +m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0. +"></span> +</p> +<p> +Multiplying this equation by <img style="vertical-align: -0.685ex; width: 2.143ex; height: 2.278ex;" src="images/74.svg" alt=" " data-tex="\xi_{\mu}"> we obtain a tensor equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 23.098ex; height: 5.564ex;" src="images/114.svg" alt=" " data-tex=" +\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0. +"></span> +</p> +<p> +Contracting the tensor on the left and taking the time average +we obtain the virial theorem, which we shall not consider +further. By interchanging the indices and subsequent subtraction, +we obtain, after a simple transformation, the theorem of +moments, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 51.585ex; height: 5.467ex;" src="images/115.svg" alt=" " data-tex=" +\frac{d}{dt} \biggl[ + m \biggl(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\biggr)\biggr] += \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}. +\qquad\text{(15)} +"></span> +</p> +<p> +It is evident in this way that the moment of a vector is not a +vector but a tensor. On account of their skew-symmetrical character +there are not nine, but only three independent equations of +this system. The possibility of replacing skew-symmetrical tensors +of the second rank in space of three dimensions by vectors +depends upon the formation of the vector +<span class="align-center"><img style="vertical-align: -1.552ex; width: 16.145ex; height: 4.588ex;" src="images/116.svg" alt=" " data-tex=" +A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}. +"></span> +<span class="pagenum" id="Page_19">[Pg 19]</span> +</p> +<p> +If we multiply the skew-symmetrical tensor of rank 2 by the +special skew-symmetrical tensor <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/117.svg" alt=" " data-tex="\delta"> introduced above, and contract +twice, a vector results whose components are numerically +equal to those of the tensor. These are the so-called axial vectors +which transform differently, from a right-handed system to +a left-handed system, from the <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">. There is a gain in +picturesqueness in regarding a skew-symmetrical tensor of rank 2 +as a vector in space of three dimensions, but it does not represent +the exact nature of the corresponding quantity so well as +considering it a tensor. +</p> +<p> +We consider next the equations of motion of a continuous +medium. Let <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"> be the density, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}"> the velocity components +considered as functions of the co-ordinates and the time, <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> the +volume forces per unit of mass, and <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> the stresses upon a +surface perpendicular to the a-axis in the direction of increasing +<img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. Then the equations of motion are, by Newton's law, +<span class="align-center"><img style="vertical-align: -1.909ex; width: 23.285ex; height: 5.056ex;" src="images/121.svg" alt=" " data-tex=" +\rho \frac{du_{\nu}}{dt} + = -\frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + \rho X_{\nu}, +"></span> +in which <img style="vertical-align: -1.577ex; width: 4.502ex; height: 4.676ex;" src="images/122.svg" alt=" " data-tex="\dfrac{du_{\nu}}{dt}"> is the acceleration of the particle which +at time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> has the co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If we express this acceleration +by partial differential coefficients, we obtain, after dividing by <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, +<span class="align-center"><img style="vertical-align: -2.041ex; width: 42.284ex; height: 5.188ex;" src="images/123.svg" alt=" " data-tex=" +\frac{\partial u_{\nu}}{dt} + \frac{\partial u_{\nu}}{dx_{\sigma}} u_{\sigma} + = -\frac{1}{\rho}\, \frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + X_{\nu}. +\qquad\text{(16)} +"></span> +</p> +<p> +We must show that this equation holds independently of the +special choice of the Cartesian system of co-ordinates. (<img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}">) is a +vector, and therefore <img style="vertical-align: -1.602ex; width: 4.606ex; height: 4.749ex;" src="images/124.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial t}"> is also a vector. +<img style="vertical-align: -1.909ex; width: 4.671ex; height: 5.056ex;" src="images/125.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial x_{\sigma}}"> +<span class="pagenum" id="Page_20">[Pg 20]</span> +is a tensor of rank 2, <img style="vertical-align: -1.909ex; width: 6.98ex; height: 5.056ex;" src="images/126.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial x_{\sigma}} u_{\tau}"> +is a tensor of rank 3. The second term on the left +results from contraction in the indices <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau">. The vector character +of the second term on the right is obvious. In order that the first +term on the right may also be a vector it is necessary for <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> to be +a tensor. Then by differentiation and contraction <img style="vertical-align: -1.909ex; width: 5.363ex; height: 5.056ex;" src="images/129.svg" alt=" " data-tex="\dfrac{\partial p_{\nu\sigma}}{\partial x_{\sigma}}"> results, +and is therefore a vector, as it also is after multiplication by +the reciprocal scalar <img style="vertical-align: -2.041ex; width: 2.165ex; height: 5.077ex;" src="images/130.svg" alt=" " data-tex="\dfrac{1}{\rho}">. That <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> is a tensor, and therefore +transforms according to the equation +<span class="align-center"><img style="vertical-align: -0.864ex; width: 16.726ex; height: 2.695ex;" src="images/131.svg" alt=" " data-tex=" +{p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta}, +"></span> +is proved in mechanics by integrating this equation over an infinitely +small tetrahedron. It is also proved there, by application +of the theorem of moments to an infinitely small parallelopipedon, +that <img style="vertical-align: -0.439ex; width: 9.191ex; height: 1.758ex;" src="images/132.svg" alt=" " data-tex="p_{\nu\sigma} = p_{\sigma\nu}">, and hence that the tensor of the stress is +a symmetrical tensor. From what has been said it follows that, +with the aid of the rules given above, the equation is co-variant +with respect to orthogonal transformations in space (rotational +transformations); and the rules according to which the quantities +in the equation must be transformed in order that the +equation may be co-variant also become evident. +</p> +<p> +The co-variance of the equation of continuity, +<span class="align-center"><img style="vertical-align: -1.891ex; width: 27.449ex; height: 5.195ex;" src="images/133.svg" alt=" " data-tex=" +\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u_{\nu})}{\partial x_{\nu}} = 0, +\qquad\text{(17)} +"></span> +requires, from the foregoing, no particular discussion. +</p> +<p> +We shall also test for co-variance the equations which express +the dependence of the stress components upon the properties of +<span class="pagenum" id="Page_21">[Pg 21]</span> +the matter, and set up these equations for the case of a compressible +viscous fluid with the aid of the conditions of co-variance. +If we neglect the viscosity, the pressure, <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/134.svg" alt=" " data-tex="p">, will be a scalar, and +will depend only upon the density and the temperature of the +fluid. The contribution to the stress tensor is then evidently +<span class="align-center"><img style="vertical-align: -0.685ex; width: 4.143ex; height: 2.307ex;" src="images/135.svg" alt=" " data-tex=" +p \delta_{\mu\nu} +"></span> +in which <img style="vertical-align: -0.685ex; width: 3.005ex; height: 2.307ex;" src="images/136.svg" alt=" " data-tex="\delta_{\mu\nu}"> is the special symmetrical tensor. This term will +also be present in the case of a viscous fluid. But in this case +there will also be pressure terms, which depend upon the space +derivatives of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}">. We shall assume that this dependence is a +linear one. Since these terms must be symmetrical tensors, the +only ones which enter will be +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.199ex; height: 5.58ex;" src="images/137.svg" alt=" " data-tex=" +\alpha\left(\frac{\partial u_{\mu}}{\partial x_{\nu}} + + \frac{\partial u_{\nu}}{\partial x_{\mu}}\right) + + \beta\delta_{\mu\nu} \frac{\partial u_{\alpha}}{\partial x_{\alpha}} +"></span> +(for <img style="vertical-align: -1.909ex; width: 4.782ex; height: 5.056ex;" src="images/138.svg" alt=" " data-tex="\dfrac{\partial u_{\alpha}}{\partial x_{\alpha}}"> is a scalar). For +physical reasons (no slipping) it +is assumed that for symmetrical dilatations in all directions, +i.e. when +<span class="align-center"><img style="vertical-align: -1.927ex; width: 38.03ex; height: 5.074ex;" src="images/139.svg" alt=" " data-tex=" +\frac{\partial u_{1}}{\partial x_{1}} = +\frac{\partial u_{2}}{\partial x_{2}} = +\frac{\partial u_{3}}{\partial x_{3}};\quad +\frac{\partial u_{1}}{\partial x_{2}}, \text{ etc.,} = 0, +"></span> +there are no frictional forces present, from which it follows that +<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> = <img style="vertical-align: -1.602ex; width: 5.335ex; height: 4.638ex;" src="images/140.svg" alt=" " data-tex="-\dfrac{2}{3}\alpha">. If only <img style="vertical-align: -1.927ex; width: 4.558ex; height: 5.074ex;" src="images/141.svg" alt=" " data-tex="\dfrac{\partial u_{1}}{\partial x_{3}}"> +is different from zero, let <img style="vertical-align: -1.927ex; width: 13.709ex; height: 5.074ex;" src="images/142.svg" alt=" " data-tex="p_{31} = -\alpha \dfrac{\partial u_{1}}{\partial x_{3}}">, +by which <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> is determined. We then obtain for the complete +stress tensor, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 72.428ex; height: 5.58ex;" src="images/143.svg" alt=" " data-tex=" +p_{\mu\nu} = p \delta_{\mu\nu} - \alpha \biggl[ + \biggl(\frac{\partial u_{\mu}}{\partial x_{\nu}} + + \frac{\partial u_{\nu}}{\partial x_{\mu}}\biggr) + - \frac{2}{3} \biggl(\frac{\partial u_{1}}{\partial x_{1}} + + \frac{\partial u_{2}}{\partial x_{2}} + + \frac{\partial u_{3}}{\partial x_{3}}\biggr)\delta_{\mu\nu} +\biggr]. +\qquad\text{(18)} +"></span> +<span class="pagenum" id="Page_22">[Pg 22]</span> +</p> +<p> +The heuristic value of the theory of invariants, which arises +from the isotropy of space (equivalence of all directions), becomes +evident from this example. +</p> +<p> +We consider, finally, Maxwell's equations in the form which +are the foundation of the electron theory of Lorentz. +</p> +<span class="align-center"><img style="vertical-align: -22.069ex; width: 44.49ex; height: 45.27ex;" src="images/144.svg" alt=" " data-tex=" +\begin{align*} +\left. + \begin{aligned} + &\begin{alignedat}{3} + \frac{\partial h_{3}}{\partial x_{2}} &- + \frac{\partial h_{2}}{\partial x_{3}} + &&= \frac{1}{c}\, \frac{\partial e_{1}}{\partial t} + &&+ \frac{1}{c}\, i_{1}, +\qquad \\ + \frac{\partial h_{1}}{\partial x_{3}} &- + \frac{\partial h_{3}}{\partial x_{1}} + &&= \frac{1}{c}\, \frac{\partial e_{2}}{\partial t} + &&+ \frac{1}{c}\, i_{2}, \\ + \frac{\partial h_{2}}{\partial x_{1}} &- + \frac{\partial h_{1}}{\partial x_{2}} + &&= \frac{1}{c}\, \frac{\partial e_{3}}{\partial t} + &&+ \frac{1}{c}\, i_{3}, + \end{alignedat} \\ + & \frac{\partial e_{1}}{\partial x_{1}} + + \frac{\partial e_{2}}{\partial x_{2}} + + \frac{\partial e_{3}}{\partial x_{3}} = \rho; + \end{aligned} + \right\} +\qquad\text{(19)}\\ +\left. + \begin{aligned} + &\begin{alignedat}{2} + \frac{\partial e_{3}}{\partial x_{2}} &- \frac{\partial e_{2}}{\partial x_{3}} + &&= -\frac{1}{c}\, \frac{\partial h_{1}}{\partial t}, +\qquad \\ + \frac{\partial e_{1}}{\partial x_{3}} &- \frac{\partial e_{3}}{\partial x_{1}} + &&= -\frac{1}{c}\, \frac{\partial h_{2}}{\partial t}, \\ + \frac{\partial e_{2}}{\partial x_{1}} &- \frac{\partial e_{1}}{\partial x_{2}} + &&= -\frac{1}{c}\, \frac{\partial h_{3}}{\partial t}, + \end{alignedat} \\ + & \frac{\partial h_{1}}{\partial x_{1}} + + \frac{\partial h_{2}}{\partial x_{2}} + + \frac{\partial h_{3}}{\partial x_{3}} = 0. + \end{aligned} +\right\} +\qquad \text{(20)} +\end{align*} +"></span> +<p> +<img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/145.svg" alt=" " data-tex="\mathbf i"> is a vector, because the current density is defined as the +density of electricity multiplied by the vector velocity of the +electricity. According to the first three equations it is evident +that <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e"> is also to be regarded as a vector. Then <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h"> cannot be +regarded as a vector.<a id="FNanchor_6_1"></a><a href="#Footnote_6_1" class="fnanchor">[6]</a> The equations may, however, easily be +<span class="pagenum" id="Page_23">[Pg 23]</span> +interpreted if <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h"> is regarded as a skew-symmetrical tensor of the +second rank. In this sense, we write <img style="vertical-align: -0.375ex; width: 3.091ex; height: 1.945ex;" src="images/148.svg" alt=" " data-tex="h_{23}">, <img style="vertical-align: -0.375ex; width: 3.091ex; height: 1.945ex;" src="images/149.svg" alt=" " data-tex="h_{31}">, <img style="vertical-align: -0.339ex; width: 3.091ex; height: 1.91ex;" src="images/150.svg" alt=" " data-tex="h_{12}"> in place of +<img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.91ex;" src="images/151.svg" alt=" " data-tex="h_{1}">, <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.91ex;" src="images/152.svg" alt=" " data-tex="h_{2}">, <img style="vertical-align: -0.375ex; width: 2.291ex; height: 1.945ex;" src="images/153.svg" alt=" " data-tex="h_{3}"> respectively. Paying attention to the skew-symmetry +of <img style="vertical-align: -0.685ex; width: 3.304ex; height: 2.255ex;" src="images/154.svg" alt=" " data-tex="h_{\mu\nu}">, the first three equations of (19) and (20) may be written +in the form +<span class="align-center"><img style="vertical-align: -5.181ex; width: 39.788ex; height: 11.494ex;" src="images/155.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial h_{\mu\nu}}{\partial x_{\nu}} + = \frac{1}{c}\, \frac{\partial e_{\mu}}{\partial t} + \frac{1}{c} i_{\mu}, +\qquad & &\text{(19a)} \\ +\frac{\partial e_{\mu}}{\partial x_{\nu}} - \frac{\partial e_{\nu}}{\partial x_{\mu}} + = +\frac{1}{c}\, \frac{\partial h_{\mu\nu}}{\partial t}. +\qquad & &\text{(20a)} +\end{gather*} +"></span> +In contrast to <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e">, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h"> appears as a quantity which has the same type +of symmetry as an angular velocity. The divergence equations +then take the form +<span class="align-center"><img style="vertical-align: -5.083ex; width: 41.219ex; height: 11.297ex;" src="images/156.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial e_{\nu}}{\partial x_{\nu}} = \rho\, +\qquad & &\text{(19b)} \\ +\frac{\partial h_{\mu\nu}}{\partial x_{\rho}} + +\frac{\partial h_{\nu\rho}}{\partial x_{\mu}} + +\frac{\partial h_{\rho\mu}}{\partial x_{\nu}} = 0. +\qquad & &\text{(20b)} +\end{gather*} +"></span> +The last equation is a skew-symmetrical tensor equation of the +third rank (the skew-symmetry of the left-hand side with respect +to every pair of indices may easily be proved, if attention +is paid to the skew-symmetry of <img style="vertical-align: -0.685ex; width: 3.304ex; height: 2.255ex;" src="images/154.svg" alt=" " data-tex="h_{\mu\nu}">). This notation is more +natural than the usual one, because, in contrast to the latter, +it is applicable to Cartesian left-handed systems as well as to +right-handed systems without change of sign. +<span class="pagenum" id="Page_24">[Pg 24]</span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_6_1"></a><a href="#FNanchor_6_1"><span class="label">[6]</span></a>These considerations will make the reader familiar with tensor operations +without the special difficulties of the four-dimensional treatment; +corresponding considerations in the theory of special relativity (Minkowski's +interpretation of the field) will then offer fewer difficulties.</p></div> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title="LECTURE II: THE THEORY OF SPECIAL RELATIVITY"><a id="chap02"></a>LECTURE II +<br><br> +THE THEORY OF SPECIAL RELATIVITY</h2> + +<p class="nind"> +<span class="dropcap">T</span>HE previous considerations concerning the configuration of +rigid bodies have been founded, irrespective of the assumption +as to the validity of the Euclidean geometry, upon the hypothesis +that all directions in space, or all configurations of Cartesian systems +of co-ordinates, are physically equivalent. We may express +this as the "principle of relativity with respect to direction," and +it has been shown how equations (laws of nature) may be found, +in accord with this principle, by the aid of the calculus of tensors. +We now inquire whether there is a relativity with respect +to the state of motion of the space of reference; in other words, +whether there are spaces of reference in motion relatively to each +other which are physically equivalent. From the standpoint of +mechanics it appears that equivalent spaces of reference do exist. +For experiments upon the earth tell us nothing of the fact +that we are moving about the sun with a velocity of approximately +30 kilometres a second. On the other hand, this physical +equivalence does not seem to hold for spaces of reference in arbitrary +motion; for mechanical effects do not seem to be subject +to the same laws in a jolting railway train as in one moving with +uniform velocity; the rotation of the earth must be considered +in writing down the equations of motion relatively to the earth. +It appears, therefore, as if there were Cartesian systems of co-ordinates, +the so-called inertial systems, with reference to which +the laws of mechanics (more generally the laws of physics) are +expressed in the simplest form. We may infer the validity of +the following theorem: If <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is an inertial system, then every +<span class="pagenum" id="Page_25">[Pg 25]</span> +other system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which moves uniformly and without rotation +relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, is also an inertial system; the laws of nature are +in concordance for all inertial systems. This statement we shall +call the "principle of special relativity." We shall draw certain +conclusions from this principle of "relativity of translation" just +as we have already done for relativity of direction. +</p> +<p> +In order to be able to do this, we must first solve the following +problem. If we are given the Cartesian co-ordinates, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and +the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, of an event relatively to one inertial system, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +how can we calculate the co-ordinates, <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}">, and the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">', of +the same event relatively to an inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which moves +with uniform translation relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">? In the pre-relativity +physics this problem was solved by making unconsciously two +hypotheses:— +</p> +<p> +1. The time is absolute; the time of an event, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">', relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is the same as the time relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. If instantaneous +signals could be sent to a distance, and if one knew that the +state of motion of a clock had no influence on its rate, then this +assumption would be physically established. For then clocks, +similar to one another, and regulated alike, could be distributed +over the systems <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', at rest relatively to them, and their +indications would be independent of the state of motion of the +systems; the time of an event would then be given by the clock +in its immediate neighbourhood. +</p> +<p> +2. Length is absolute; if an interval, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +has a length <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, then it has the same length <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> relatively to a +system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which is in motion relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. +</p> +<p> +If the axes of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' are parallel to each other, a simple +calculation based on these two assumptions, gives the equations +<span class="pagenum" id="Page_26">[Pg 26]</span> +of transformation +<span class="align-center"><img style="vertical-align: -2.17ex; width: 29.765ex; height: 5.47ex;" src="images/157.svg" alt=" " data-tex=" +\left. +\begin{aligned} +{x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\text{,} \\ +t' &= t - b\text{.} +\end{aligned} +\right\} +\qquad \text{(21)} +"></span> +</p> +<p> +This transformation is known as the "Galilean Transformation." +Differentiating twice by the time, we get +<span class="align-center"><img style="vertical-align: -1.654ex; width: 15.253ex; height: 5.07ex;" src="images/158.svg" alt=" " data-tex=" +\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}. +"></span> +Further, it follows that for two simultaneous events, +<span class="align-center"><img style="vertical-align: -0.339ex; width: 28.681ex; height: 2.752ex;" src="images/159.svg" alt=" " data-tex=" +{{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}. +"></span> +The invariance of the distance between the two points results +from squaring and adding. From this easily follows the co-variance +of Newton's equations of motion with respect to the +Galilean transformation (21). Hence it follows that classical +mechanics is in accord with the principle of special relativity if +the two hypotheses respecting scales and clocks are made. +</p> +<p> +But this attempt to found relativity of translation upon the +Galilean transformation fails when applied to electromagnetic +phenomena. The Maxwell-Lorentz electromagnetic equations +are not co-variant with respect to the Galilean transformation. +In particular, we note, by (21), that a ray of light which referred +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> has a velocity <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/160.svg" alt=" " data-tex="c">, has a different velocity referred to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +depending upon its direction. The space of reference of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is +therefore distinguished, with respect to its physical properties, +from all spaces of reference which are in motion relatively to it +(quiescent æther). But all experiments have shown that electromagnetic +and optical phenomena, relatively to the earth as the +<span class="pagenum" id="Page_27">[Pg 27]</span> +body of reference, are not influenced by the translational velocity +of the earth. The most important of these experiments are +those of Michelson and Morley, which I shall assume are known. +The validity of the principle of special relativity can therefore +hardly be doubted. +</p> +<p> +On the other hand, the Maxwell-Lorentz equations have +proved their validity in the treatment of optical problems in +moving bodies. No other theory has satisfactorily explained the +facts of aberration, the propagation of light in moving bodies +(Fizeau), and phenomena observed in double stars (De Sitter). +The consequence of the Maxwell-Lorentz equations that in a +vacuum light is propagated with the velocity <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/160.svg" alt=" " data-tex="c">, at least with respect +to a definite inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, must therefore be regarded +as proved. According to the principle of special relativity, we +must also assume the truth of this principle for every other +inertial system. +</p> +<p> +Before we draw any conclusions from these two principles +we must first review the physical significance of the concepts +"time" and "velocity." It follows from what has gone before, that +co-ordinates with respect to an inertial system are physically +defined by means of measurements and constructions with the +aid of rigid bodies. In order to measure time, we have supposed +a clock, <img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/161.svg" alt=" " data-tex="U">, present somewhere, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. But +we cannot fix the time, by means of this clock, of an event +whose distance from the clock is not negligible; for there are no +"instantaneous signals" that we can use in order to compare the +time of the event with that of the clock. In order to complete the +definition of time we may employ the principle of the constancy +of the velocity of light in a vacuum. Let us suppose that we +place similar clocks at points of the system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, at rest relatively +<span class="pagenum" id="Page_28">[Pg 28]</span> +to it, and regulated according to the following scheme. A ray +of light is sent out from one of the clocks, <img style="vertical-align: -0.357ex; width: 3.138ex; height: 1.902ex;" src="images/162.svg" alt=" " data-tex="U_{m}">, at the instant +when it indicates the time <img style="vertical-align: -0.357ex; width: 2.409ex; height: 1.773ex;" src="images/163.svg" alt=" " data-tex="t_{m}">, and travels through a vacuum a +distance <img style="vertical-align: -0.357ex; width: 3.573ex; height: 1.357ex;" src="images/164.svg" alt=" " data-tex="r_{mn}">, to the clock <img style="vertical-align: -0.357ex; width: 2.693ex; height: 1.902ex;" src="images/165.svg" alt=" " data-tex="U_{n}">; at the instant when this ray meets +the clock <img style="vertical-align: -0.357ex; width: 2.693ex; height: 1.902ex;" src="images/165.svg" alt=" " data-tex="U_{n}"> the latter is set to indicate the time +<img style="vertical-align: -0.357ex; width: 1.964ex; height: 1.773ex;" src="images/166.svg" alt=" " data-tex="t_{n}"> = <img style="vertical-align: -0.357ex; width: 2.409ex; height: 1.773ex;" src="images/163.svg" alt=" " data-tex="t_{m}"> + <img style="vertical-align: -1.577ex; width: 4.568ex; height: 4.106ex;" src="images/167.svg" alt=" " data-tex="\dfrac{r_{mn}}{c}">.<a id="FNanchor_7_1"></a><a href="#Footnote_7_1" class="fnanchor">[7]</a> +The principle of the constancy of the velocity of light then states +that this adjustment of the clocks will not lead to contradictions. +With clocks so adjusted, we can assign the time to events which +take place near any one of them. It is essential to note that this +definition of time relates only to the inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, since +we have used a system of clocks at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. The +assumption which was made in the pre-relativity physics of the +absolute character of time (i.e. the independence of time of the +choice of the inertial system) does not follow at all from this +definition. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_7_1"></a><a href="#FNanchor_7_1"><span class="label">[7]</span></a>Strictly speaking, it would be more correct to define simultaneity first, +somewhat as follows: two events taking place at the points <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> and <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B"> of +the system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> are simultaneous if they appear at the same instant when +observed from the middle point, <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M">, of the interval <img style="vertical-align: 0; width: 3.414ex; height: 1.62ex;" src="images/169.svg" alt=" " data-tex="AB">. Time is then +defined as the ensemble of the indications of similar clocks, at rest relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, which register the same simultaneously.</p></div> + +<p> +The theory of relativity is often criticized for giving, without +justification, a central theoretical role to the propagation +of light, in that it founds the concept of time upon the law of +propagation of light. The situation, however, is somewhat as +follows. In order to give physical significance to the concept of +time, processes of some kind are required which enable relations +to be established between different places. It is immaterial what +kind of processes one chooses for such a definition of time. It +is advantageous, however, for the theory, to choose only those +<span class="pagenum" id="Page_29">[Pg 29]</span> +processes concerning which we know something certain. This +holds for the propagation of light <i>in vacuo</i> in a higher degree +than for any other process which could be considered, thanks to +the investigations of Maxwell and H. A. Lorentz. +</p> +<p> +From all of these considerations, space and time data have +a physically real, and not a mere fictitious, significance; in particular +this holds for all the relations in which co-ordinates and +time enter, e.g. the relations (21). There is, therefore, sense in +asking whether those equations are true or not, as well as in +asking what the true equations of transformation are by which +we pass from one inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> to another, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', moving relatively +to it. It may be shown that this is uniquely settled by +means of the principle of the constancy of the velocity of light +and the principle of special relativity. +</p> +<p> +To this end we think of space and time physically defined +with respect to two inertial systems, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in the way that +has been shown. Further, let a ray of light pass from one point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> +to another point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> through a vacuum. If <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r"> is the measured +distance between the two points, then the propagation of light +must satisfy the equation +<span class="align-center"><img style="vertical-align: -0.186ex; width: 9.982ex; height: 1.805ex;" src="images/173.svg" alt=" " data-tex=" +r = c·\Delta t. +"></span> +</p> +<p> +If we square this equation, and express <img style="vertical-align: -0.025ex; width: 2.008ex; height: 1.912ex;" src="images/174.svg" alt=" " data-tex="r^{2}"> by the differences +of the co-ordinates, <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">, in place of this equation we can write +<span class="align-center"><img style="vertical-align: -1.018ex; width: 31.976ex; height: 3.167ex;" src="images/175.svg" alt=" " data-tex=" +\sum (\Delta x_{\nu})^{2} - c^{2} \Delta t^{2} = 0. \qquad\text{(22)} +"></span> +This equation formulates the principle of the constancy of the +velocity of light relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. It must hold whatever may be +the motion of the source which emits the ray of light. +<span class="pagenum" id="Page_30">[Pg 30]</span> +</p> +<p> +The same propagation of light may also be considered relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in which case also the principle of the constancy of +the velocity of light must be satisfied. Therefore, with respect +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', we have the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 34.175ex; height: 3.167ex;" src="images/176.svg" alt=" " data-tex=" +\sum ({\Delta x'}_{\nu})^{2} - c^{2} \Delta t'^{2} = 0. +\qquad\text{(22a)} +"></span> +</p> +<p> +Equations (22a) and (22) must be mutually consistent with +each other with respect to the transformation which transforms +from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. A transformation which effects this we shall call +a "Lorentz transformation." +</p> +<p> +Before considering these transformations in detail we shall +make a few general remarks about space and time. In the pre-relativity +physics space and time were separate entities. Specifications +of time were independent of the choice of the space of +reference. The Newtonian mechanics was relative with respect +to the space of reference, so that, e.g. the statement that two +non-simultaneous events happened at the same place had no objective +meaning (that is, independent of the space of reference). +But this relativity had no role in building up the theory. One +spoke of points of space, as of instants of time, as if they were +absolute realities. It was not observed that the true element +of the space-time specification was the event, specified by the +four numbers <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">. The conception of something +happening was always that of a four-dimensional continuum; but +the recognition of this was obscured by the absolute character +of the pre-relativity time. Upon giving up the hypothesis of the +absolute character of time, particularly that of simultaneity, the +four-dimensionality of the time-space concept was immediately +recognized. It is neither the point in space, nor the instant in +<span class="pagenum" id="Page_31">[Pg 31]</span> +time, at which something happens that has physical reality, but +only the event itself. There is no absolute (independent of the +space of reference) relation in space, and no absolute relation +in time between two events, but there is an absolute (independent +of the space of reference) relation in space and time, as +will appear in the sequel. The circumstance that there is no +objective rational division of the four-dimensional continuum +into a three-dimensional space and a one-dimensional time continuum +indicates that the laws of nature will assume a form +which is logically most satisfactory when expressed as laws in +the four-dimensional space-time continuum. Upon this depends +the great advance in method which the theory of relativity owes +to Minkowski. Considered from this standpoint, we must regard +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> as the four co-ordinates of an event in the +four-dimensional continuum. We have far less success in picturing +to ourselves relations in this four-dimensional continuum than +in the three-dimensional Euclidean continuum; but it must be +emphasized that even in the Euclidean three-dimensional geometry +its concepts and relations are only of an abstract nature in +our minds, and are not at all identical with the images we form +visually and through our sense of touch. The non-divisibility of +the four-dimensional continuum of events does not at all, however, +involve the equivalence of the space co-ordinates with the +time co-ordinate. On the contrary, we must remember that the +time co-ordinate is defined physically wholly differently from the +space co-ordinates. The relations (22) and (22a) which when +equated define the Lorentz transformation show, further, a difference +in the role of the time co-ordinate from that of the space +co-ordinates; for the term <img style="vertical-align: -0.025ex; width: 3.689ex; height: 1.912ex;" src="images/177.svg" alt=" " data-tex="\Delta t^{2}"> has the opposite sign to the space +terms, <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/178.svg" alt=" " data-tex="\Delta x_{1}^{2}">, <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/179.svg" alt=" " data-tex="\Delta x_{2}^{2}">, <img style="vertical-align: -0.685ex; width: 4.166ex; height: 2.572ex;" src="images/180.svg" alt=" " data-tex="\Delta x_{3}^{2}">. +<span class="pagenum" id="Page_32">[Pg 32]</span> +</p> +<p> +Before we analyse further the conditions which define the +Lorentz transformation, we shall introduce the light-time, <img style="vertical-align: -0.186ex; width: 5.488ex; height: 1.756ex;" src="images/181.svg" alt=" " data-tex="l = ct">, +in place of the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, in order that the constant <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/160.svg" alt=" " data-tex="c"> shall not +enter explicitly into the formulas to be developed later. Then +the Lorentz transformation is defined in such a way that, first, +it makes the equation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 41.26ex; height: 2.633ex;" src="images/182.svg" alt=" " data-tex=" +{\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 +\qquad\text{(22b)} +"></span> +a co-variant equation, that is, an equation which is satisfied with +respect to every inertial system if it is satisfied in the inertial +system to which we refer the two given events (emission and +reception of the ray of light). Finally, with Minkowski, we introduce +in place of the real time co-ordinate <img style="vertical-align: -0.186ex; width: 5.488ex; height: 1.756ex;" src="images/181.svg" alt=" " data-tex="l = ct">, the imaginary +time co-ordinate +<span class="align-center"><img style="vertical-align: -0.566ex; width: 25.618ex; height: 2.754ex;" src="images/183.svg" alt=" " data-tex=" +x_{4} = il = ict\quad (\sqrt{-1} = i). +"></span> +Then the equation defining the propagation of light, which must +be co-variant with respect to the Lorentz transformation, becomes +<span class="align-center"><img style="vertical-align: -3.222ex; width: 55.106ex; height: 5.371ex;" src="images/184.svg" alt=" " data-tex=" +\sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} + = 0. +\qquad\text{(22c)} +"></span> +This condition is always satisfied<a id="FNanchor_8_1"></a><a href="#Footnote_8_1" class="fnanchor">[8]</a> if we satisfy the more general +condition that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 42.527ex; height: 2.633ex;" src="images/185.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} +\qquad\text{(23)} +"></span> +<span class="pagenum" id="Page_33">[Pg 33]</span> +shall be an invariant with respect to the transformation. This +condition is satisfied only by linear transformations, that is, +transformations of the type +<span class="align-center"><img style="vertical-align: -0.685ex; width: 25.407ex; height: 2.515ex;" src="images/186.svg" alt=" " data-tex=" +{x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} +\qquad\text{(24)} +"></span> +in which the summation over the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> is to be extended from <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = 1 +to <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = 4. A glance at equations (23) and (24) shows that the +Lorentz transformation so defined is identical with the translational +and rotational transformations of the Euclidean geometry, +if we disregard the number of dimensions and the relations of reality. +We can also conclude that the coefficients <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"> must satisfy +the conditions +<span class="align-center"><img style="vertical-align: -0.685ex; width: 30.947ex; height: 2.382ex;" src="images/187.svg" alt=" " data-tex=" +b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}. +\qquad\text{(25)} +"></span> +Since the ratios of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> are real, it follows that all the <img style="vertical-align: -0.685ex; width: 2.349ex; height: 1.683ex;" src="images/188.svg" alt=" " data-tex="a_{\mu}"> and +the <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"> are real, except <img style="vertical-align: -0.339ex; width: 2.185ex; height: 1.337ex;" src="images/189.svg" alt=" " data-tex="a_{4}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/190.svg" alt=" " data-tex="b_{41}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/191.svg" alt=" " data-tex="b_{42}">, <img style="vertical-align: -0.375ex; width: 2.758ex; height: 1.945ex;" src="images/192.svg" alt=" " data-tex="b_{43}">, +<img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/193.svg" alt=" " data-tex="b_{14}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/194.svg" alt=" " data-tex="b_{24}"> and <img style="vertical-align: -0.375ex; width: 2.758ex; height: 1.945ex;" src="images/195.svg" alt=" " data-tex="b_{34}">, which +are purely imaginary. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_8_1"></a><a href="#FNanchor_8_1"><span class="label">[8]</span></a>That this specialization lies in the nature of the case will be evident +later.</p></div> + +<p><br></p> + +<p> +<i>Special Lorentz Transformation</i>. We obtain the simplest +transformations of the type of (24) and (25) if only two of the +co-ordinates are to be transformed, and if all the <img style="vertical-align: -0.685ex; width: 2.349ex; height: 1.683ex;" src="images/188.svg" alt=" " data-tex="a_{\mu}">, which determine +the new origin, vanish. We obtain then for the indices +1 and 2, on account of the three independent conditions which +the relations (25) furnish, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 34.703ex; height: 11.62ex;" src="images/196.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi, \\ +{x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi, \\ +{x'}_{3} &= x_{3}, && \\ +{x'}_{4} &= x_{4}. && +\end{alignedat} +\right\} +\qquad\text{(26)} +"></span> +<span class="pagenum" id="Page_34">[Pg 34]</span> +</p> +<p> +This is a simple rotation in space of the (space) co-ordinate +system about <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">-axis. We see that the rotational transformation +in space (without the time transformation) which we studied +before is contained in the Lorentz transformation as a special +case. For the indices 1 and 4 we obtain, in an analogous manner, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 36.083ex; height: 11.62ex;" src="images/197.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi, \\ +{x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi, \\ +{x'}_{2} &= x_{2}, && \\ +{x'}_{3} &= x_{3}. && +\end{alignedat} +\right\} +\qquad\text{(26a)} +"></span> +</p> +<p> +On account of the relations of reality <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/198.svg" alt=" " data-tex="\psi"> must be taken as +imaginary. To interpret these equations physically, we introduce +the real light-time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> and the velocity <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +instead of the imaginary angle <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/198.svg" alt=" " data-tex="\psi">. We have, first, +<span class="align-center"><img style="vertical-align: -2.17ex; width: 26.107ex; height: 5.47ex;" src="images/201.svg" alt=" " data-tex=" +\begin{alignat*}{4} +{x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi, \\ +l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. +\end{alignat*} +"></span> +Since for the origin of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' i.e., for <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}"> = 0, we must have <img style="vertical-align: -0.339ex; width: 7.07ex; height: 1.91ex;" src="images/202.svg" alt=" " data-tex="x_{1} = vl">, +it follows from the first of these equations that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 19.945ex; height: 2.262ex;" src="images/203.svg" alt=" " data-tex=" +v = i\tan\psi, +\qquad\text{(27)} +"></span> +and also +<span class="align-center"><img style="vertical-align: -5.112ex; width: 27.989ex; height: 11.355ex;" src="images/204.svg" alt=" " data-tex=" +\begin{aligned} +\left. +\begin{aligned} +\sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}, \\ +\cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}, +\end{aligned} +\right\} +\qquad\text{(28)} +\end{aligned} +"></span> +<span class="pagenum" id="Page_35">[Pg 35]</span> +so that we obtain +<span class="align-center"><img style="vertical-align: -8.256ex; width: 26.022ex; height: 17.642ex;" src="images/205.svg" alt=" " data-tex=" +\left. +\begin{aligned} +{x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}, \\ +l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}, \\ +{x'}_{2} &= x_{2}, \\ +{x'}_{3} &= x_{3}. +\end{aligned} +\right\} +\qquad\text{(29)} +"></span> +</p> +<p> +These equations form the well-known special Lorentz transformation, +which in the general theory represents a rotation, +through an imaginary angle, of the four-dimensional system of +co-ordinates. If we introduce the ordinary time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, in place of the +light-time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">, then in (29) we must replace <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> by <img style="vertical-align: -0.025ex; width: 1.796ex; height: 1.441ex;" src="images/206.svg" alt=" " data-tex="ct"> and <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v"> by +<img style="vertical-align: -1.577ex; width: 2.093ex; height: 4.109ex;" src="images/207.svg" alt=" " data-tex="\dfrac{v}{c}">. +</p> +<p> +We must now fill in a gap. From the principle of the constancy +of the velocity of light it follows that the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 12.994ex; height: 3.167ex;" src="images/208.svg" alt=" " data-tex=" +\sum {\Delta x_{\nu}}^{2} = 0 +"></span> +has a significance which is independent of the choice of the inertial +system; but the invariance of the quantity <img style="vertical-align: -0.559ex; width: 4.214ex; height: 2.446ex;" src="images/209.svg" alt=" " data-tex="\Delta x_{\nu}^{2}"> does +not at all follow from this. This quantity might be transformed +with a factor. This depends upon the fact that the right-hand +side of (29) might be multiplied by a factor <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda">, independent of <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v">. +But the principle of relativity does not permit this factor to be +different from 1, as we shall now show. Let us assume that we +have a rigid circular cylinder moving in the direction of its axis. +If its radius, measured at rest with a unit measuring rod is equal +to <img style="vertical-align: -0.375ex; width: 2.705ex; height: 1.92ex;" src="images/210.svg" alt=" " data-tex="R_{0}">, its radius <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> in motion, might be different from <img style="vertical-align: -0.375ex; width: 2.705ex; height: 1.92ex;" src="images/210.svg" alt=" " data-tex="R_{0}">, since +the theory of relativity does not make the assumption that the +shape of bodies with respect to a space of reference is independent +of their motion relatively to this space of reference. But +<span class="pagenum" id="Page_36">[Pg 36]</span> +all directions in space must be equivalent to each other. <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> may +therefore depend upon the magnitude <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q"> of the velocity, but not +upon its direction; <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> must therefore be an even function of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q">. If +the cylinder is at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' the equation of its lateral +surface is +<span class="align-center"><img style="vertical-align: -0.464ex; width: 15.362ex; height: 2.464ex;" src="images/213.svg" alt=" " data-tex=" +x'^{2} + y'^{2} = {R_{0}}^{2}. +"></span> +If we write the last two equations of (29) more generally +<span class="align-center"><img style="vertical-align: -2.17ex; width: 10.156ex; height: 5.47ex;" src="images/214.svg" alt=" " data-tex=" +\begin{align*} +{x'}_{2} = \lambda x_{2}, \\ +{x'}_{3} = \lambda x_{3}, +\end{align*} +"></span> +then the lateral surface of the cylinder referred to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> satisfies the +equation +<span class="align-center"><img style="vertical-align: -1.656ex; width: 15.478ex; height: 5.179ex;" src="images/215.svg" alt=" " data-tex=" +x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}. +"></span> +The factor <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> therefore measures the lateral contraction of the +cylinder, and can thus, from the above, be only an even function +of <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v">. +</p> +<p> +If we introduce a third system of co-ordinates, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"><code>"</code>, which +moves relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' with velocity <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v"> in the direction of the +negative <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/216.svg" alt=" " data-tex="x">-axis of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, we obtain, by applying (29) twice, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 19.39ex; height: 11.62ex;" src="images/217.svg" alt=" " data-tex=" +\begin{align*} +{x''}_{1} &= \lambda(v) \lambda(-v) x_{1}, \\ +{x''}_{2} &= \lambda(v) \lambda(-v) x_{2}, \\ +{x''}_{3} &= \lambda(v) \lambda(-v) x_{3}, \\ +l'' &= \lambda(v) \lambda(-v) l. +\end{align*} +"></span> +Now, since <img style="vertical-align: -0.566ex; width: 4.176ex; height: 2.262ex;" src="images/218.svg" alt=" " data-tex="\lambda(v)"> must be equal to <img style="vertical-align: -0.566ex; width: 5.937ex; height: 2.262ex;" src="images/219.svg" alt=" " data-tex="\lambda(-v)"> and since we assume +that we use the same measuring rods in all the systems, it follows +that the transformation of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"><code>"</code> to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> must be the identical +<span class="pagenum" id="Page_37">[Pg 37]</span> +transformation (since the possibility <img style="vertical-align: -0.186ex; width: 7.228ex; height: 1.756ex;" src="images/220.svg" alt=" " data-tex="\lambda = -1"> does not need to +be considered). It is essential for these considerations to assume +that the behaviour of the measuring rods does not depend upon +the history of their previous motion. +</p> + +<p><br></p> + +<p> +<i>Moving Measuring Rods and Clocks</i>. At the definite <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">-time, +<img style="vertical-align: -0.186ex; width: 4.823ex; height: 1.756ex;" src="images/221.svg" alt=" " data-tex="l = 0">, the position of the points given by the integers +<img style="vertical-align: -0.583ex; width: 6.656ex; height: 2.3ex;" src="images/222.svg" alt=" " data-tex="x'_{1} = n">, is with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, given by +<img style="vertical-align: -0.339ex; width: 14.568ex; height: 2.526ex;" src="images/223.svg" alt=" " data-tex="x_{1} = n {\sqrt{1 - v^{2}}}">; this follows +from the first of equations (29) and expresses the Lorentz +contraction. A clock at rest at the origin <img style="vertical-align: -0.339ex; width: 6.43ex; height: 1.846ex;" src="images/224.svg" alt=" " data-tex="x_{1} = 0"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, whose +beats are characterized by <img style="vertical-align: -0.186ex; width: 5.049ex; height: 1.756ex;" src="images/225.svg" alt=" " data-tex="l = n">, will, when observed from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +have beats characterized by +<span class="align-center"><img style="vertical-align: -2.308ex; width: 13.855ex; height: 4.837ex;" src="images/226.svg" alt=" " data-tex=" +l' = \frac{n}{\sqrt{1 - v^{2}}}; +"></span> +this follows from the second of equations (29) and shows that +the clock goes slower than if it were at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. +These two consequences, which hold, <i>mutatis mutandis</i>, for every +system of reference, form the physical content, free from +convention, of the Lorentz transformation. +</p> + +<p><br></p> + +<p> +<i>Addition Theorem for Velocities</i>. If we combine two special +Lorentz transformations with the relative velocities <img style="vertical-align: -0.339ex; width: 2.085ex; height: 1.342ex;" src="images/227.svg" alt=" " data-tex="v_{1}"> and <img style="vertical-align: -0.339ex; width: 2.085ex; height: 1.342ex;" src="images/228.svg" alt=" " data-tex="v_{2}">, +then the velocity of the single Lorentz transformation which +takes the place of the two separate ones is, according to (27), +given by +<span class="align-center"><img style="vertical-align: -2.016ex; width: 62.69ex; height: 5.115ex;" src="images/229.svg" alt=" " data-tex=" +v_{12} = i \tan (\psi_{1} + \psi_{2}) + = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} + = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. +\qquad\text{(30)} +"></span> +<span class="pagenum" id="Page_38">[Pg 38]</span> +</p> + +<p><br></p> + +<p> +<i>General Statements about the Lorentz Transformation and +its Theory of Invariants</i>. The whole theory of invariants of the +special theory of relativity depends upon the invariant <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> (23). +Formally, it has the same rôle in the four-dimensional space-time +continuum as the invariant +<img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/178.svg" alt=" " data-tex="\Delta x_{1}^{2}"> + <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/179.svg" alt=" " data-tex="\Delta x_{2}^{2}"> + <img style="vertical-align: -0.685ex; width: 4.166ex; height: 2.572ex;" src="images/180.svg" alt=" " data-tex="\Delta x_{3}^{2}"> in the Euclidean +geometry and in the pre-relativity physics. The latter quantity +is not an invariant with respect to all the Lorentz transformations; +the quantity <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> of equation (23) assumes the rôle of this +invariant. With respect to an arbitrary inertial system, <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> may +be determined by measurements; with a given unit of measure +it is a completely determinate quantity, associated with an arbitrary +pair of events. +</p> +<p> +The invariant <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> differs, disregarding the number of dimensions, +from the corresponding invariant of the Euclidean geometry +in the following points. In the Euclidean geometry <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> is +necessarily positive; it vanishes only when the two points concerned +come together. On the other hand, from the vanishing +of +<span class="align-center"><img style="vertical-align: -3.222ex; width: 44.378ex; height: 5.371ex;" src="images/231.svg" alt=" " data-tex=" +s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2} +"></span> +it cannot be concluded that the two space-time points fall together; +the vanishing of this quantity <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}">, is the invariant condition +that the two space-time points can be connected by a light +signal <i>in vacuo</i>. If <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> a point (event) represented in the +four-dimensional space of the <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> then all the "points" which +can be connected to <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> by means of a light signal lie upon the +cone <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> = 0 (compare Fig. 1, in which the dimension <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> is suppressed). +The "upper" half of the cone may contain the "points" +to which light signals can be sent from <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">; then the "lower" half +<span class="pagenum" id="Page_39">[Pg 39]</span> +of the cone will contain the "points" from which light signals +can be sent to <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">. The points <img style="vertical-align: 0; width: 2.452ex; height: 1.717ex;" src="images/232.svg" alt=" " data-tex="P'"> enclosed by the conical surface +furnish, with <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">, a negative <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}">; <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="PP'"> as well as <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/234.svg" alt=" " data-tex="P'P"> is then, +according to Minkowski, of the nature of a time. Such intervals +represent elements of possible paths of motion, the velocity being +less than that of light.<a id="FNanchor_9_1"></a><a href="#Footnote_9_1" class="fnanchor">[9]</a> In +this case the <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">-axis may be drawn +<span class="pagenum" id="Page_40">[Pg 40]</span> +in the direction of <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="PP'"> by suitably choosing the state of motion +of the inertial system. If <img style="vertical-align: 0; width: 2.452ex; height: 1.717ex;" src="images/232.svg" alt=" " data-tex="P'"> lies outside of the "light-cone" then +<img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="PP'"> is of the nature of a space; in this case, by properly choosing +the inertial system, <img style="vertical-align: -0.025ex; width: 2.559ex; height: 1.645ex;" src="images/235.svg" alt=" " data-tex="\Delta l"> can be made to vanish. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure01.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG. 1.</p> +</div></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_9_1"></a><a href="#FNanchor_9_1"><span class="label">[9]</span></a>That material velocities exceeding that of light are not possible, follows from the appearance of the radical <img style="vertical-align: -0.212ex; width: 7.912ex; height: 2.398ex;" src="images/236.svg" alt=" " data-tex="\sqrt{1 - v^{2}}"> in the special Lorentz +transformation (29).</p></div> + +<p> +By the introduction of the imaginary time variable, <img style="vertical-align: -0.339ex; width: 6.754ex; height: 1.91ex;" src="images/237.svg" alt=" " data-tex="x_{4} = il">, +Minkowski has made the theory of invariants for the four-dimensional +continuum of physical phenomena fully analogous +to the theory of invariants for the three-dimensional continuum +of Euclidean space. The theory of four-dimensional tensors of +special relativity differs from the theory of tensors in three-dimensional +space, therefore, only in the number of dimensions +and the relations of reality. +</p> +<p> +A physical entity which is specified by four quantities, <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/238.svg" alt=" " data-tex="A_{\nu}">, +in an arbitrary inertial system of the +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">, is called +a 4-vector, with the components <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/238.svg" alt=" " data-tex="A_{\nu}">, if the <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/238.svg" alt=" " data-tex="A_{\nu}"> correspond in +their relations of reality and the properties of transformation to +the <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">; it may be of the nature of a space or of a time. The +sixteen quantities <img style="vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;" src="images/240.svg" alt=" " data-tex="A_{\mu\nu}">, then form the components of a tensor of +the second rank, if they transform according to the scheme +<span class="align-center"><img style="vertical-align: -0.685ex; width: 17.844ex; height: 2.515ex;" src="images/241.svg" alt=" " data-tex=" +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}. +"></span> +It follows from this that the <img style="vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;" src="images/240.svg" alt=" " data-tex="A_{\mu\nu}"> behave, with respect to +their properties of transformation and their properties of reality, +as the products of components, <img style="vertical-align: -0.685ex; width: 5.052ex; height: 2.23ex;" src="images/242.svg" alt=" " data-tex="U_{\mu}V_{\nu}"> of two 4-vectors, +(<img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/161.svg" alt=" " data-tex="U">) and (<img style="vertical-align: -0.05ex; width: 1.74ex; height: 1.595ex;" src="images/52.svg" alt=" " data-tex="V">). All the components are real except those which +contain the index 4 once, those being purely imaginary. Tensors +<span class="pagenum" id="Page_41">[Pg 41]</span> +of the third and higher ranks may be defined in an analogous +way. The operations of addition, subtraction, multiplication, +contraction and differentiation for these tensors are wholly +analogous to the corresponding operations for tensors in three-dimensional space. +</p> +<p> +Before we apply the tensor theory to the four-dimensional +space-time continuum, we shall examine more particularly the +skew-symmetrical tensors. The tensor of the second rank has, in +general, 16 = 4·4 components. In the case of skew-symmetry the +components with two equal indices vanish, and the components +with unequal indices are equal and opposite in pairs. There +exist, therefore, only six independent components, as is the case +in the electromagnetic field. In fact, it will be shown when we +consider Maxwell's equations that these may be looked upon as +tensor equations, provided we regard the electromagnetic field +as a skew-symmetrical tensor. Further, it is clear that the skew-symmetrical +tensor of the third rank (skew-symmetrical in all +pairs of indices) has only four independent components, since +there are only four combinations of three different indices. +</p> +<p> +We now turn to Maxwell's equations (19a), (19b), (20a), +(20b), and introduce the notation:<a id="FNanchor_10_1"></a><a href="#Footnote_10_1" class="fnanchor">[10]</a> +<span class="align-center"><img style="vertical-align: -6.265ex; width: 58.388ex; height: 13.661ex;" src="images/243.svg" alt=" " data-tex=" +\begin{align*} +&\left. +\begin{aligned} +&\phi_{23} && \phi_{31} && \phi_{12} && \phi_{14} && \phi_{24} && \phi_{34} \\ +&\mathbf h_{23} && \mathbf h_{31} && \mathbf h_{12} & -&i\mathbf e_{x} & -&i\mathbf e_{y} & -&i\mathbf e_{z} +\end{aligned} +\right\} +&&\text{(30a)}\\ +&\qquad\qquad\left. +\begin{aligned} +&J_{1} && J_{2} && J_{3} && J_{4} \\ +&\frac{1}{c}{\mathbf i_{x}} && +\frac{1}{c}{\mathbf i_{y}} && +\frac{1}{c}{\mathbf i_{z}} && i\rho +\end{aligned} +\right\} +&&\text{(31)} +\end{align*} +"></span> +with the convention that <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}"> shall be equal to <img style="vertical-align: -0.685ex; width: 5.109ex; height: 2.255ex;" src="images/245.svg" alt=" " data-tex="-\phi_{\mu\nu}">. Then +<span class="pagenum" id="Page_42">[Pg 42]</span> +Maxwell's equations may be combined into the forms +<span class="align-center"><img style="vertical-align: -5.181ex; width: 40.27ex; height: 11.494ex;" src="images/246.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial \phi_{\mu\nu}}{\partial x_{\nu}} = J_{\mu}, +\qquad & &\text{(32)}\\ +\frac{\partial \phi_{\mu\nu}}{\partial x_{\sigma}} + +\frac{\partial \phi_{\nu\sigma}}{\partial x_{\mu}} + +\frac{\partial \phi_{\sigma\mu}}{\partial x_{\nu}} = 0, +\qquad & &\text{(33)} +\end{gather*} +"></span> +as one can easily verify by substituting from (30a) and (31). +Equations (32) and (33) have a tensor character, and are +therefore co-variant with respect to Lorentz transformations, +if the <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}"> and the <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.885ex;" src="images/247.svg" alt=" " data-tex="J_{\nu}"> have a tensor character, which we assume. +Consequently, the laws for transforming these quantities from +one to another allowable (inertial) system of co-ordinates are +uniquely determined. The progress in method which electrodynamics +owes to the theory of special relativity lies principally +in this, that the number of independent hypotheses is diminished. +If we consider, for example, equations (19a) only from the +standpoint of relativity of direction, as we have done above, we +see that they have three logically independent terms. The way +in which the electric intensity enters these equations appears to +be wholly independent of the way in which the magnetic intensity +enters them; it would not be surprising if instead of +<img style="vertical-align: -1.602ex; width: 4.483ex; height: 4.945ex;" src="images/248.svg" alt=" " data-tex="\dfrac{\partial e_{\mu}}{\partial l}">, +we had, say, +<img style="vertical-align: -1.679ex; width: 5.588ex; height: 5.291ex;" src="images/249.svg" alt=" " data-tex="\dfrac{\partial^{2} e_{\mu}}{\partial l^{2}}"> +or if this term were absent. On the other +hand, only two independent terms appear in equation (32). The +electromagnetic field appears as a formal unit; the way in which +the electric field enters this equation is determined by the way in +which the magnetic field enters it. Besides the electromagnetic +field, only the electric current density appears as an independent +entity. This advance in method arises from the fact that the +<span class="pagenum" id="Page_43">[Pg 43]</span> +electric and magnetic fields draw their separate existences from +the relativity of motion. A field which appears to be purely an +electric field, judged from one system, has also magnetic field +components when judged from another inertial system. When +applied to an electromagnetic field, the general law of transformation +furnishes, for the special case of the special Lorentz +transformation, the equations +<span class="align-center"><img style="vertical-align: -6.897ex; width: 46.646ex; height: 14.925ex;" src="images/250.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{\mathbf e'}_{x} &= \mathbf e_{x}\qquad & {\mathbf h'}_{x} &= \mathbf h_{x}, \\ +{\mathbf e'}_{y} &= \frac{\mathbf e_{y} - v\mathbf h_{z}}{\sqrt{1 - v^{2}}}\qquad & +{\mathbf h'}_{y} &= \frac{\mathbf h_{y} + v\mathbf e_{z}}{\sqrt{1 - v^{2}}}, \\ +{\mathbf e'}_{z} &= \frac{\mathbf e_{z} + v\mathbf h_{y}}{\sqrt{1 - v^{2}}}\qquad & +{\mathbf h'}_{z} &= \frac{\mathbf h_{z} - v\mathbf e_{y}}{\sqrt{1 - v^{2}}}. +\end{alignedat} +\right\} +\qquad \text{(34)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_10_1"></a><a href="#FNanchor_10_1"><span class="label">[10]</span></a>In +order to avoid confusion from now on we shall use the three-dimensional +space indices, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/216.svg" alt=" " data-tex="x">, <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/251.svg" alt=" " data-tex="y">, <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/252.svg" alt=" " data-tex="z"> instead of 1, 2, 3, and we shall reserve the +numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum.</p></div> + +<p> +If there exists with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> only a magnetic field, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h">, but +no electric field, <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e">, then with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' there exists an electric +field <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e"> as well, which would act upon an electric particle at rest +relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. An observer at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> would +designate this force as the Biot-Savart force, or the Lorentz electromotive +force. It therefore appears as if this electromotive force +had become fused with the electric field intensity into a single +entity. +</p> +<p> +In order to view this relation formally, let us consider the +expression for the force acting upon unit volume of electricity, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 23.503ex; height: 2.262ex;" src="images/253.svg" alt=" " data-tex=" +\mathbf k = \rho\mathbf e + [\mathbf i, \mathbf h], +\qquad \text{(35)} +"></span> +in which <img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/145.svg" alt=" " data-tex="\mathbf i"> is the vector velocity of electricity, with the velocity +of light as the unit. If we introduce <img style="vertical-align: -0.685ex; width: 2.408ex; height: 2.23ex;" src="images/254.svg" alt=" " data-tex="J_{\mu}"> and <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}"> according to +(30a) and (31), we obtain for the first component the expression +<span class="align-center"><img style="vertical-align: -0.464ex; width: 22.298ex; height: 2.034ex;" src="images/255.svg" alt=" " data-tex=" +\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}. +"></span> +<span class="pagenum" id="Page_44">[Pg 44]</span> +Observing that <img style="vertical-align: -0.464ex; width: 3.136ex; height: 2.034ex;" src="images/256.svg" alt=" " data-tex="\phi_{11}"> vanishes on account of the skew-symmetry of +the tensor (<img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi">), the components of <img style="vertical-align: 0; width: 1.373ex; height: 1.57ex;" src="images/258.svg" alt=" " data-tex="\mathbf k"> are given by the first three +components of the four-dimensional vector +<span class="align-center"><img style="vertical-align: -0.685ex; width: 21.284ex; height: 2.382ex;" src="images/259.svg" alt=" " data-tex=" +K_{\mu} = \phi_{\mu\nu} J_{\nu}, +\qquad \text{(36)} +"></span> +and the fourth component is given by +<span class="align-center"><img style="vertical-align: -0.667ex; width: 65.109ex; height: 2.364ex;" src="images/260.svg" alt=" " data-tex=" +K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} + = i(\mathbf e_{x} \mathbf i_{x} + \mathbf e_{y} \mathbf i_{y} + \mathbf e_{z} \mathbf i_{z}) + = i \lambda. +\qquad \text{(37)} +"></span> +There is, therefore, a four-dimensional vector of force per unit +volume, whose first three components, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/261.svg" alt=" " data-tex="K_{1}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/262.svg" alt=" " data-tex="K_{2}">, <img style="vertical-align: -0.375ex; width: 2.908ex; height: 1.92ex;" src="images/263.svg" alt=" " data-tex="K_{3}">, are +the ponderomotive force components per unit volume, and whose fourth +component is the rate of working of the field per unit volume, +multiplied by <img style="vertical-align: -0.318ex; width: 4.821ex; height: 2.398ex;" src="images/264.svg" alt=" " data-tex="\sqrt{-1}">. +</p> +<p> +A comparison of (36) and (35) shows that the theory of relativity +formally unites the ponderomotive force of the electric +field, <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"><img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e">, and the Biot-Savart or Lorentz force [<img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/145.svg" alt=" " data-tex="\mathbf i">, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h">]. +</p> + +<p><br></p> + +<p> +<i>Mass and Energy</i>. An important conclusion can be drawn +from the existence and significance of the 4-vector <img style="vertical-align: -0.685ex; width: 3.073ex; height: 2.23ex;" src="images/265.svg" alt=" " data-tex="K_{\mu}">. Let us +imagine a body upon which the electromagnetic field acts for +a time. In the symbolic figure (Fig. 2) <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/266.svg" alt=" " data-tex="Ox_{1}"> designates the +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">-axis, and is at the same time a substitute for the three space axes +<img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/266.svg" alt=" " data-tex="Ox_{1}">, <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/267.svg" alt=" " data-tex="Ox_{2}">, <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/266.svg" alt=" " data-tex="Ox_{1}">; <img style="vertical-align: -0.05ex; width: 2.4ex; height: 1.643ex;" src="images/268.svg" alt=" " data-tex="Ol"> designates the real time axis. In this diagram +a body of finite extent is represented, at a definite time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">, by +the interval <img style="vertical-align: 0; width: 3.414ex; height: 1.62ex;" src="images/169.svg" alt=" " data-tex="AB"> the whole space-time existence of the body is +represented by a strip whose boundary is everywhere inclined +less than 45° to the <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">-axis. Between the time sections, +<img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/269.svg" alt=" " data-tex="l_{1}"> and <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{2}">, but not extending to them, +a portion of the strip is shaded. This represents the portion +of the space-time manifold +<span class="pagenum" id="Page_45">[Pg 45]</span> +in which the electromagnetic field acts upon the body, or upon +the electric charges contained in it, the action upon them being +transmitted to the body. We shall now consider the changes +which take place in the momentum and energy of the body as a +result of this action. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure02.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG. 2.</p> +</div></div> + +<p> +We shall assume that the principles of momentum and +energy are valid for the body. The change in momentum, +<img style="vertical-align: -0.357ex; width: 3.983ex; height: 1.977ex;" src="images/271.svg" alt=" " data-tex="\Delta I_{x}">, <img style="vertical-align: -0.667ex; width: 3.852ex; height: 2.287ex;" src="images/272.svg" alt=" " data-tex="\Delta I_{y}">, <img style="vertical-align: -0.357ex; width: 3.812ex; height: 1.977ex;" src="images/273.svg" alt=" " data-tex="\Delta I_{z}">, and the change +in energy, <img style="vertical-align: 0; width: 3.613ex; height: 1.62ex;" src="images/274.svg" alt=" " data-tex="\Delta E"> are then given +<span class="pagenum" id="Page_46">[Pg 46]</span> +by the expressions +<span class="align-center"><img style="vertical-align: -12.132ex; width: 60.141ex; height: 25.396ex;" src="images/275.svg" alt=" " data-tex=" +\begin{alignat*}{2} +\begin{aligned} +\Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{x}\, dx\, dy\, dz} + &&= \frac{1}{i}\int{K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{y}\, dx\, dy\, dz} + &&= \frac{1}{i}\int{K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{z}\, dx\, dy\, dz\,} + &&= \frac{1}{i}\int{K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta E &= \int_{l_{1}}^{l_{2}}dl \int{\lambda\, dx\, dy\, dz} + &&= \frac{1}{i}\int{ + \frac{1}{i}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}. +\end{aligned} +\end{alignat*} +"></span> +Since the four-dimensional element of volume is an invariant, +and (<img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/261.svg" alt=" " data-tex="K_{1}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/262.svg" alt=" " data-tex="K_{2}">, <img style="vertical-align: -0.375ex; width: 2.908ex; height: 1.92ex;" src="images/263.svg" alt=" " data-tex="K_{3}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/276.svg" alt=" " data-tex="K_{4}">) forms a 4-vector, the four-dimensional +integral extended over the shaded portion transforms as a 4-vector, +as does also the integral between the limits <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/269.svg" alt=" " data-tex="l_{1}"> and <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{2}">, because +the portion of the region which is not shaded contributes nothing +to the integral. It follows, therefore, that +<img style="vertical-align: -0.357ex; width: 3.983ex; height: 1.977ex;" src="images/271.svg" alt=" " data-tex="\Delta I_{x}">, <img style="vertical-align: -0.667ex; width: 3.852ex; height: 2.287ex;" src="images/272.svg" alt=" " data-tex="\Delta I_{y}">, <img style="vertical-align: -0.357ex; width: 3.812ex; height: 1.977ex;" src="images/273.svg" alt=" " data-tex="\Delta I_{z}">, <img style="vertical-align: -0.025ex; width: 4.394ex; height: 1.645ex;" src="images/277.svg" alt=" " data-tex="i \Delta E"> +form a 4-vector. Since the quantities themselves transform in +the same way as their increments, it follows that the aggregate +of the four quantities +<span class="align-center"><img style="vertical-align: -0.667ex; width: 13.217ex; height: 2.213ex;" src="images/278.svg" alt=" " data-tex=" +I_{x},\ I_{y},\ I_{z},\ iE +"></span> +has itself the properties of a vector; these quantities are referred +to an instantaneous condition of the body (e.g. at the time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/269.svg" alt=" " data-tex="l_{1}">). +</p> +<p> +This 4-vector may also be expressed in terms of the mass <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/104.svg" alt=" " data-tex="m">, +and the velocity of the body, considered as a material particle. +To form this expression, we note first, that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 66.28ex; height: 2.584ex;" src="images/279.svg" alt=" " data-tex=" +-ds^{2} = d\tau^{2} + = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} + = dl^{2}(1 - q^{2}) + \qquad \text{(38)} +"></span> +<span class="pagenum" id="Page_47">[Pg 47]</span> +is an invariant which refers to an infinitely short portion of the +four-dimensional line which represents the motion of the material +particle. The physical significance of the invariant <img style="vertical-align: -0.029ex; width: 2.346ex; height: 1.6ex;" src="images/280.svg" alt=" " data-tex="d \tau"> may +easily be given. If the time axis is chosen in such a way that it +has the direction of the line differential which we are considering, +or, in other words, if we reduce the material particle to rest, +we shall then have <img style="vertical-align: -0.186ex; width: 7.214ex; height: 1.756ex;" src="images/281.svg" alt=" " data-tex="d \tau = dl">; this will therefore be measured by +the light-seconds clock which is at the same place, and at rest +relatively to the material particle. We therefore call <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> the proper +time of the material particle. As opposed to <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl">, <img style="vertical-align: -0.029ex; width: 2.346ex; height: 1.6ex;" src="images/280.svg" alt=" " data-tex="d \tau"> is therefore an +invariant, and is practically equivalent to <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl"> for motions whose +velocity is small compared to that of light. Hence we see that +<span class="align-center"><img style="vertical-align: -1.581ex; width: 18.527ex; height: 4.681ex;" src="images/283.svg" alt=" " data-tex=" +u_{\sigma} = \frac{dx_{\sigma}}{d\tau} +\qquad \text{(39)} +"></span> +has, just as the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, the character of a vector; we shall designate +(<img style="vertical-align: -0.357ex; width: 2.395ex; height: 1.357ex;" src="images/284.svg" alt=" " data-tex="u_{\sigma}">) as the four-dimensional vector (in brief, 4-vector) of +velocity. Its components satisfy, by (38), the condition +<span class="align-center"><img style="vertical-align: -1.018ex; width: 22.112ex; height: 3.167ex;" src="images/285.svg" alt=" " data-tex=" +\sum {u_{\sigma}}^{2} = -1. +\qquad \text{(40)} +"></span> +We see that this 4-vector, whose components in the ordinary +notation are +<span class="align-center"><img style="vertical-align: -2.76ex; width: 55.726ex; height: 5.862ex;" src="images/286.svg" alt=" " data-tex=" +\frac{\mathbf q_{x}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf q_{y}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf q_{z}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf i}{\sqrt{1 - q^{2}}} +\qquad \text{(41)} +"></span> +is the only 4-vector which can be formed from the velocity components +of the material particle which are defined in three dimensions by +<span class="align-center"><img style="vertical-align: -1.577ex; width: 33.314ex; height: 4.676ex;" src="images/287.svg" alt=" " data-tex=" +\mathbf q_{x} = \frac{dx}{dl},\quad +\mathbf q_{y} = \frac{dy}{dl},\quad +\mathbf q_{z} = \frac{dz}{dl}. +"></span> +<span class="pagenum" id="Page_48">[Pg 48]</span> +We therefore see that +<span class="align-center"><img style="vertical-align: -2.148ex; width: 18.483ex; height: 5.444ex;" src="images/288.svg" alt=" " data-tex=" +\left(m \frac{dx_{\mu}}{d\tau}\right) +\qquad \text{(42)} +"></span> +must be that 4-vector which is to be equated to the 4-vector of +momentum and energy whose existence we have proved above. +By equating the components, we obtain, in three-dimensional +notation, +<span class="align-center"><img style="vertical-align: -11.148ex; width: 25.646ex; height: 23.428ex;" src="images/289.svg" alt=" " data-tex=" +\left. +\begin{aligned} +I_{x} = \frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}, \\ +I_{y} = \frac{m\mathbf q_{y}}{\sqrt{1 - q^{2}}}, \\ +I_{z} = \frac{m\mathbf q_{z}}{\sqrt{1 - q^{2}}}, \\ +E = \frac{m}{\sqrt{1 - q^{2}}}. +\end{aligned} +\right\} +\qquad \text{(43)} +"></span> +</p> +<p> +We recognize, in fact, that these components of momentum +agree with those of classical mechanics for velocities which are +small compared to that of light. For large velocities the momentum +increases more rapidly than linearly with the velocity, so as +to become infinite on approaching the velocity of light. +</p> +<p> +If we apply the last of equations (43) to a material particle +at rest (<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q"> = 0), we see that the energy, <img style="vertical-align: -0.375ex; width: 2.657ex; height: 1.913ex;" src="images/290.svg" alt=" " data-tex="E_{0}"> of a, body at rest is +equal to its mass. Had we chosen the second as our unit of time, +we would have obtained +<span class="align-center"><img style="vertical-align: -0.566ex; width: 19.182ex; height: 2.565ex;" src="images/291.svg" alt=" " data-tex=" +E_{0} = mc^{2}. + \qquad \text{(44)} +"></span> +Mass and energy are therefore essentially alike; they are only +different expressions for the same thing. The mass of a body +<span class="pagenum" id="Page_49">[Pg 49]</span> +is not a constant; it varies with changes in its energy.<a id="FNanchor_11_1"></a><a href="#Footnote_11_1" class="fnanchor">[11]</a> +We see from the last of equations (43) that <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/292.svg" alt=" " data-tex="E"> becomes infinite when +<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q"> approaches 1, the velocity of light. If we develop <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/292.svg" alt=" " data-tex="E"> in powers +of <img style="vertical-align: -0.439ex; width: 2.143ex; height: 2.326ex;" src="images/293.svg" alt=" " data-tex="q^{2}">, we obtain, +<span class="align-center"><img style="vertical-align: -1.602ex; width: 38.993ex; height: 4.636ex;" src="images/294.svg" alt=" " data-tex=" +E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots. + \qquad \text{(45)} +"></span> +The second term of this expansion corresponds to the kinetic +energy of the material particle in classical mechanics. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_11_1"></a><a href="#FNanchor_11_1"><span class="label">[11]</span></a>The emission of energy in radioactive processes is evidently connected +with the fact that the atomic weights are not integers. Attempts have been +made to draw conclusions from this concerning the structure and stability +of the atomic nuclei.</p></div> + +<p><br></p> + +<p> +<i>Equations of Motion of Material Particles</i>. From (43) we +obtain, by differentiating by the time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">, and using the principle +of momentum, in the notation of three-dimensional vectors, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 30.382ex; height: 6.785ex;" src="images/295.svg" alt=" " data-tex=" +\mathbf{K} = \frac{d}{dl}\left(\frac{m\mathbf q}{\sqrt{1 - q^{2}}}\right). +\qquad \text{(46)} +"></span> +</p> +<p> +This equation, which was previously employed by H. A. +Lorentz for the motion of electrons, has been proved to be true, +with great accuracy, by experiments with <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">-rays. +</p> + +<p><br></p> + +<p> +<i>Energy Tensor of the Electromagnetic Field</i>. Before the development +of the theory of relativity it was known that the principles +of energy and momentum could be expressed in a differential +form for the electromagnetic field. The four-dimensional +formulation of these principles leads to an important conception, +<span class="pagenum" id="Page_50">[Pg 50]</span> +that of the energy tensor, which is important for the further development +of the theory of relativity. +</p> +<p> +If in the expression for the 4-vector of force per unit volume, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 12.359ex; height: 2.255ex;" src="images/296.svg" alt=" " data-tex=" +K_{\mu} = \phi_{\mu\nu} J_{\nu}, +"></span> +using the field equations (32), we express <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.885ex;" src="images/247.svg" alt=" " data-tex="J_{\nu}"> in terms of the +field intensities, <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}">, we obtain, after some transformations and +repeated application of the field equations (32) and (33), the +expression +<span class="align-center"><img style="vertical-align: -1.891ex; width: 23.002ex; height: 5.235ex;" src="images/297.svg" alt=" " data-tex=" +K_{\mu} = -\frac{\partial T_{\mu\nu}}{\partial x_{\nu}}, +\qquad \text{(47)} +"></span> +where we have written<a id="FNanchor_12_1"></a><a href="#Footnote_12_1" class="fnanchor">[12]</a> +<span class="align-center"><img style="vertical-align: -0.781ex; width: 36.604ex; height: 2.799ex;" src="images/298.svg" alt=" " data-tex=" +T_{\mu\nu} + = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + + \phi_{\mu\alpha} \phi_{\nu\alpha}. +\qquad \text{(48)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_12_1"></a><a href="#FNanchor_12_1"><span class="label">[12]</span></a>To be summed for the indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">.</p></div> + +<p> +The physical meaning of equation (47) becomes evident if in +place of this equation we write, using a new notation, +<span class="align-center"><img style="vertical-align: -11.527ex; width: 55.328ex; height: 24.185ex;" src="images/299.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{4} +\mathbf k_{x} &= -\frac{\partial p_{xx}}{\partial x} + &&- \frac{\partial p_{xy}}{\partial y} + &&- \frac{\partial p_{xz}}{\partial z} + &&- \frac{\partial (i b_{x})}{\partial (il)}, \\ +\mathbf k_{y} &= -\frac{\partial p_{yx}}{\partial x} + &&- \frac{\partial p_{yy}}{\partial y} + &&- \frac{\partial p_{yz}}{\partial z} + &&- \frac{\partial (i b_{y})}{\partial (il)}, \\ +\mathbf k_{z} &= -\frac{\partial p_{zx}}{\partial x} + &&- \frac{\partial p_{zy}}{\partial y} + &&- \frac{\partial p_{zz}}{\partial z} + &&- \frac{\partial (i b_{z})}{\partial (il)}, \\ +i\lambda + &= -\frac{\partial (i\mathbf s_{x})}{\partial x} + &&- \frac{\partial (i\mathbf s_{y})}{\partial y} + &&- \frac{\partial (i\mathbf s_{z})}{\partial z} + &&- \frac{\partial (-\eta)}{\partial (i l)}; +\end{alignedat} +\right\} +\qquad \text{(47a)} +"></span> +<span class="pagenum" id="Page_51">[Pg 51]</span> +or, on eliminating the imaginary, +<span class="align-center"><img style="vertical-align: -11.136ex; width: 48.145ex; height: 23.402ex;" src="images/300.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{4} +k_{x} &= -\frac{\partial p_{xx}}{\partial x} + &&- \frac{\partial p_{xy}}{\partial y} + &&- \frac{\partial p_{xz}}{\partial z} + &&- \frac{\partial b_{x}}{\partial l}, \\ +k_{y} &= -\frac{\partial p_{yx}}{\partial x} + &&- \frac{\partial p_{yy}}{\partial y} + &&- \frac{\partial p_{yz}}{\partial z} + &&- \frac{\partial b_{y}}{\partial l}, \\ +k_{z} &= -\frac{\partial p_{zx}}{\partial x} + &&- \frac{\partial p_{zy}}{\partial y} + &&- \frac{\partial p_{zz}}{\partial z} + &&- \frac{\partial b_{z}}{\partial l}, \\ +\lambda + &= -\frac{\partial s_{x}}{\partial x} + &&- \frac{\partial s_{y}}{\partial y} + &&- \frac{\partial s_{z}}{\partial z} + &&- \frac{\partial \eta}{\partial l}; +\end{alignedat} +\right\} +\qquad \text{(47b)} +"></span> +</p> +<p> +When expressed in the latter form, we see that the first three +equations state the principle of momentum; <img style="vertical-align: -0.439ex; width: 3.156ex; height: 1.439ex;" src="images/301.svg" alt=" " data-tex="p_{xx}">,..., <img style="vertical-align: -0.439ex; width: 2.985ex; height: 1.439ex;" src="images/302.svg" alt=" " data-tex="p_{zx}"> are the +Maxwell stresses in the electromagnetic field, and (<img style="vertical-align: -0.357ex; width: 2.073ex; height: 1.927ex;" src="images/303.svg" alt=" " data-tex="b_{x}">, <img style="vertical-align: -0.667ex; width: 1.942ex; height: 2.237ex;" src="images/304.svg" alt=" " data-tex="b_{y}">, <img style="vertical-align: -0.357ex; width: 1.902ex; height: 1.927ex;" src="images/305.svg" alt=" " data-tex="b_{z}">) is +the vector momentum per unit volume of the field. The last of +equations (47b) expresses the energy principle; <img style="vertical-align: -0.014ex; width: 1.027ex; height: 1.038ex;" src="images/306.svg" alt=" " data-tex="\mathbf s"> is the vector +flow of energy, and <img style="vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;" src="images/307.svg" alt=" " data-tex="\eta"> the energy per unit volume of the field. In +fact, we get from (48) by introducing the well-known expressions +for the components of the field intensity from electrodynamics, +<span class="align-center"><img style="vertical-align: -14.002ex; width: 63.031ex; height: 29.135ex;" src="images/308.svg" alt=" " data-tex=" +\left. +\begin{aligned} +&\begin{alignedat}{4} +p_{xx} = &{} - \mathbf h_{x} \mathbf h_{x} + &&+ \tfrac{1}{2}({\mathbf h_{x}}^{2} &&+ {\mathbf h_{y}}^{2} &&+ {\mathbf h_{z}}^{2}) \\ + &{} - \mathbf e_{x} \mathbf e_{y} + &&+ \tfrac{1}{2}({\mathbf e_{x}}^{2} &&+ {\mathbf e_{y}}^{2} &&+ {\mathbf e_{z}}^{2}), +\end{alignedat} \\ +&\qquad\qquad\qquad\qquad\qquad +\begin{alignedat}{3} +p_{xy} = &{} - \mathbf h_{x} \mathbf h_{y}\quad + && p_{xz} = && {} - \mathbf h_{x} \mathbf h_{z} \\ + &{} - \mathbf e_{x} \mathbf e_{y},\quad + && && {} - \mathbf e_{x} \mathbf e_{z}, +\end{alignedat} \\ +&\qquad\qquad\qquad\qquad\qquad\quad\vdots \\ +&b_{x} = \mathbf s_{x} = \mathbf e_{y} \mathbf h_{z} - \mathbf e_{z}\mathbf h_{y}, \\ +&b_{y} = \mathbf s_{y} = \mathbf e_{z} \mathbf h_{x} - \mathbf e_{x}\mathbf h_{z}, \\ +&b_{z} = \mathbf s_{z} = \mathbf e_{x} \mathbf h_{y} - \mathbf e_{y}\mathbf h_{x}, \\ +&\eta = +\tfrac{1}{2}({\mathbf e_{x}}^{2} + {\mathbf e_{y}}^{2} + {\mathbf e_{z}}^{2} + + {\mathbf h_{x}}^{2} + {\mathbf h_{y}}^{2} + {\mathbf h_{z}}^{2}). +\end{aligned} +\right\} +\qquad \text{(48a)} +"></span> +<span class="pagenum" id="Page_52">[Pg 52]</span> +We conclude from (48) that the energy tensor of the electromagnetic +field is symmetrical; with this is connected the fact +that the momentum per unit volume and the how of energy are +equal to each other (relation between energy and inertia). +</p> +<p> +We therefore conclude from these considerations that the +energy per unit volume has the character of a tensor. This has +been proved directly only for an electromagnetic field, although +we may claim universal validity for it. Maxwell's equations determine +the electromagnetic field when the distribution of electric +charges and currents is known. But we do not know the +laws which govern the currents and charges. We do know, indeed, +that electricity consists of elementary particles (electrons, +positive nuclei), but from a theoretical point of view we cannot +comprehend this. We do not know the energy factors which +determine the distribution of electricity in particles of definite +size and charge, and all attempts to complete the theory in this +direction have failed. If then we can build upon Maxwell's equations +in general, the energy tensor of the electromagnetic field +is known only outside the charged particles.<a id="FNanchor_13_1"></a><a href="#Footnote_13_1" class="fnanchor">[13]</a> In these regions, +outside of charged particles, the only regions in which we can believe +that we have the complete expression for the energy tensor, +we have, by (47), +<span class="align-center"><img style="vertical-align: -1.891ex; width: 19.927ex; height: 5.235ex;" src="images/309.svg" alt=" " data-tex=" +\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = 0. +\qquad \text{(47c)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_13_1"></a><a href="#FNanchor_13_1"><span class="label">[13]</span></a>It has been attempted to remedy this lack of knowledge by considering +the charged particles as proper singularities. But in my opinion this means +giving up a real understanding of the structure of matter. It seems to me +much better to give in to our present inability rather than to be satisfied +by a solution that is only apparent.</p></div> +<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> +<p><br></p> + +<p> +<i>General Expressions for the Conservation Principles</i>. We +can hardly avoid making the assumption that in all other cases, +also, the space distribution of energy is given by a symmetrical +tensor, <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}">, and that this complete energy tensor everywhere +satisfies the relation (47c). At any rate we shall see that by +means of this assumption we obtain the correct expression for +the integral energy principle. +</p> +<p> +Let us consider a spatially bounded, closed system, which, +four-dimensionally, we may represent as a strip, outside of which +the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> vanish. Integrate equation (47c) over a space section. +Since the integrals of +<img style="vertical-align: -1.891ex; width: 5.55ex; height: 5.235ex;" src="images/311.svg" alt=" " data-tex="\dfrac{\partial T_{\mu1}}{\partial x_{1}}">, +<img style="vertical-align: -1.891ex; width: 5.55ex; height: 5.235ex;" src="images/312.svg" alt=" " data-tex="\dfrac{\partial T_{\mu2}}{\partial x_{2}}"> and +<img style="vertical-align: -1.927ex; width: 5.55ex; height: 5.27ex;" src="images/313.svg" alt=" " data-tex="\dfrac{\partial T_{\mu3}}{\partial x_{3}}"> +vanish because +the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> vanish at the limits of integration, we obtain +<span class="align-center"><img style="vertical-align: -2.148ex; width: 36.963ex; height: 5.428ex;" src="images/314.svg" alt=" " data-tex=" +\frac{\partial}{\partial l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} + = 0. +\qquad \text{(49)} +"></span> +Inside the parentheses are the expressions for the momentum of +the whole system, multiplied by <img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/315.svg" alt=" " data-tex="i">, together with the negative +energy of the system, so that (49) expresses the conservation +principles in their integral form. That this gives the right conception +of energy and the conservation principles will be seen +from the following considerations. +</p> + +<p><br></p> + +<p class="center"> +PHENOMENOLOGICAL REPRESENTATION OF THE +ENERGY TENSOR OF MATTER.</p> + +<p> +<i>Hydrodynamical Equations</i>. We know that matter is built +up of electrically charged particles, but we do not know the laws +which govern the constitution of these particles. In treating mechanical +problems, we are therefore obliged to make use of an +<span class="pagenum" id="Page_54">[Pg 54]</span> +inexact description of matter, which corresponds to that of classical +mechanics. The density <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, of a material substance and the +hydrodynamical pressures are the fundamental concepts upon +which such a description is based. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure03.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG. 3.</p> +</div></div> + +<p> +Let <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}"> be the density of matter at a place, estimated with +reference to a system of co-ordinates moving with the matter. +Then <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}">, the density at rest, is an invariant. If we think of the +matter in arbitrary motion and neglect the pressures (particles +of dust <i>in vacuo</i>, neglecting the size of the particles and the +temperature), then the energy tensor will depend only upon the +<span class="pagenum" id="Page_55">[Pg 55]</span> +velocity components, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}"> and <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}">. We secure the tensor character +of <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> by putting +<span class="align-center"><img style="vertical-align: -0.685ex; width: 22.948ex; height: 2.382ex;" src="images/317.svg" alt=" " data-tex=" +T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}, + \qquad \text{(50)} +"></span> +in which the <img style="vertical-align: -0.685ex; width: 2.447ex; height: 1.685ex;" src="images/318.svg" alt=" " data-tex="u_{\mu}">, in the three-dimensional representation, are +given by (41). In fact, it follows from (50) that for <img style="vertical-align: -0.439ex; width: 5.189ex; height: 1.946ex;" src="images/319.svg" alt=" " data-tex="q = 0">, +<img style="vertical-align: -0.375ex; width: 10.166ex; height: 1.906ex;" src="images/320.svg" alt=" " data-tex="T_{44} = -\sigma_{0}"> +(equal to the negative energy per unit volume), as it should, +according to the theorem of the equivalence of mass and energy, +and according to the physical interpretation of the energy tensor +given above. If an external force (four-dimensional vector, <img style="vertical-align: -0.685ex; width: 3.073ex; height: 2.23ex;" src="images/265.svg" alt=" " data-tex="K_{\mu}">) +acts upon the matter, by the principles of momentum and energy +the equation +<span class="align-center"><img style="vertical-align: -1.891ex; width: 11.688ex; height: 5.235ex;" src="images/321.svg" alt=" " data-tex=" +K_{\mu} = \frac{\partial T_{\mu\nu}}{\partial x_{\nu}} +"></span> +must hold. We shall now show that this equation leads to the +same law of motion of a material particle as that already obtained. +Let us imagine the matter to be of infinitely small extent +in space, that is, a four-dimensional thread; then by integration +over the whole thread with respect to the space co-ordinates +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, we obtain +<span class="align-center"><img style="vertical-align: -5.035ex; width: 54.942ex; height: 11.201ex;" src="images/322.svg" alt=" " data-tex=" +\begin{align*} +\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} + &= \int \frac{\partial T_{14}}{\partial x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ + &= -i \frac{d}{dl}\left\{ + \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} + \right\}. +\end{align*} +"></span> +</p> +<p> +Now <img style="vertical-align: -0.691ex; width: 16.724ex; height: 2.514ex;" src="images/323.svg" alt=" " data-tex="\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}"> is an invariant, as is, therefore, also +<img style="vertical-align: -0.691ex; width: 19.381ex; height: 2.514ex;" src="images/324.svg" alt=" " data-tex="\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}">. +We shall calculate this integral, first with +respect to the inertial system which we have chosen, and second, +with respect to a system relatively to which the matter has the +velocity zero. The integration is to be extended over a filament +<span class="pagenum" id="Page_56">[Pg 56]</span> +of the thread for which <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}"> may be regarded as constant over the +whole section. If the space volumes of the filament referred to +the two systems are <img style="vertical-align: -0.05ex; width: 2.916ex; height: 1.62ex;" src="images/325.svg" alt=" " data-tex="dV"> and <img style="vertical-align: -0.375ex; width: 3.483ex; height: 1.945ex;" src="images/326.svg" alt=" " data-tex="dV_{0}"> respectively, then we have +<span class="align-center"><img style="vertical-align: -1.948ex; width: 24.709ex; height: 5.027ex;" src="images/327.svg" alt=" " data-tex=" +\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau +"></span> +and therefore also +<span class="align-center"><img style="vertical-align: -1.948ex; width: 38.788ex; height: 5.048ex;" src="images/328.svg" alt=" " data-tex=" +\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} + = \int dm\, i\, \frac{d\tau}{dx_{4}}. +"></span> +</p> +<p> +If we substitute the right-hand side for the left-hand side in +the former integral, and put <img style="vertical-align: -1.581ex; width: 4.454ex; height: 4.681ex;" src="images/329.svg" alt=" " data-tex="\dfrac{dx_{1}}{d\tau}"> outside the sign of integration, +we obtain, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 38.194ex; height: 6.785ex;" src="images/330.svg" alt=" " data-tex=" +\mathbf K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) + = \frac{d}{dl}\left(\frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}\right). +"></span> +We see, therefore, that the generalized conception of the energy +tensor is in agreement with our former result. +</p> + +<p><br></p> + +<p> +<i>The Eulerian Equations for Perfect Fluids</i>. In order to get +nearer to the behaviour of real matter we must add to the energy +tensor a term which corresponds to the pressures. The simplest +case is that of a perfect fluid in which the pressure is determined +by a scalar <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/134.svg" alt=" " data-tex="p">. Since the tangential stresses <img style="vertical-align: -0.667ex; width: 3.025ex; height: 1.667ex;" src="images/331.svg" alt=" " data-tex="p_{xy}">, etc., vanish in +this case, the contribution to the energy tensor must be of the +form <img style="vertical-align: -0.685ex; width: 4.143ex; height: 2.307ex;" src="images/332.svg" alt=" " data-tex="p\delta_{\nu\mu}">. We must therefore put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 28.869ex; height: 2.382ex;" src="images/333.svg" alt=" " data-tex=" +T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}. +\qquad \text{(51)} +"></span> +<span class="pagenum" id="Page_57">[Pg 57]</span> +At rest, the density of the matter, or the energy per unit volume, +is in this case, not <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma"> but <img style="vertical-align: -0.439ex; width: 5.196ex; height: 1.758ex;" src="images/334.svg" alt=" " data-tex="\sigma - p">. For +<span class="align-center"><img style="vertical-align: -1.581ex; width: 35.761ex; height: 4.681ex;" src="images/335.svg" alt=" " data-tex=" +-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} + = \sigma - p. +"></span> +In the absence of any force, we have +<span class="align-center"><img style="vertical-align: -2.237ex; width: 42.095ex; height: 5.58ex;" src="images/336.svg" alt=" " data-tex=" +\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} + = \sigma u_{\nu} \frac{\partial u_{\mu}}{\partial x_{\nu}} + + u_{\mu} \frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + + \frac{\partial p}{\partial x_{\mu}} = 0. +"></span> +If we multiply this equation by <img style="vertical-align: -2.148ex; width: 13.161ex; height: 5.444ex;" src="images/337.svg" alt=" " data-tex="u_{\mu} \left(= \dfrac{dx_{\mu}}{d\tau}\right)"> +and sum for the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">'s we obtain, using (40), +<span class="align-center"><img style="vertical-align: -1.891ex; width: 29.227ex; height: 5.195ex;" src="images/338.svg" alt=" " data-tex=" +-\frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{dp}{d\tau} = 0, +\qquad \text{(52)} +"></span> +where we have put <img style="vertical-align: -2.237ex; width: 16.078ex; height: 5.533ex;" src="images/339.svg" alt=" " data-tex="\dfrac{\partial p}{\partial x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}">. +This is the equation of continuity, which differs from that of +classical mechanics by the term <img style="vertical-align: -1.581ex; width: 3.342ex; height: 4.681ex;" src="images/340.svg" alt=" " data-tex="\dfrac{dp}{d\tau}">, which, practically, +is vanishingly small. Observing (52), the conservation principles take the form +<span class="align-center"><img style="vertical-align: -2.237ex; width: 35.277ex; height: 5.533ex;" src="images/341.svg" alt=" " data-tex=" +\sigma \frac{du_{\mu}}{d\tau} + + u_{\mu} \frac{dp}{d\tau} + + \frac{\partial p}{\partial x_{\mu}} = 0. +\qquad \text{(53)} +"></span> +The equations for the first three indices evidently correspond to +the Eulerian equations. That the equations (52) and (53) correspond, +to a first approximation, to the hydrodynamical equations +of classical mechanics, is a further confirmation of the generalized +energy principle. The density of matter and of energy +has the character of a symmetrical tensor. +<span class="pagenum" id="Page_58">[Pg 58]</span> +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title="LECTURE III: THE GENERAL THEORY OF RELATIVITY"><a id="chap03"></a>LECTURE III +<br><br> +THE GENERAL THEORY OF RELATIVITY</h2> + +<p class="nind"> +<span class="dropcap">A</span>LL of the previous considerations have been based upon the +assumption that all inertial systems are equivalent for the description +of physical phenomena, but that they are preferred, for +the formulation of the laws of nature, to spaces of reference in a +different state of motion. We can think of no cause for this preference +for definite states of motion to all others, according to our +previous considerations, either in the perceptible bodies or in the +concept of motion; on the contrary, it must be regarded as an independent +property of the space-time continuum. The principle +of inertia, in particular, seems to compel us to ascribe physically +objective properties to the space-time continuum. Just as +it was necessary from the Newtonian standpoint to make both +the statements, <i>tempus est absolutum</i>, <i>spatium est absolutum</i>, so +from the standpoint of the special theory of relativity we must +say, <i>continuum spatii et temporis est absolutum</i>. In this latter +statement <i>absolutum</i> means not only "physically real," but also +"independent in its physical properties, having a physical effect, +but not itself influenced by physical conditions." +</p> +<p> +As long as the principle of inertia is regarded as the keystone +of physics, this standpoint is certainly the only one which +is justified. But there are two serious criticisms of the ordinary +conception. In the first place, it is contrary to the mode of thinking +in science to conceive of a thing (the space-time continuum) +which acts itself, but which cannot be acted upon. This is the +reason why E. Mach was led to make the attempt to eliminate +space as an active cause in the system of mechanics. According +<span class="pagenum" id="Page_59">[Pg 59]</span> +to him, a material particle does not move in unaccelerated +motion relatively to space, but relatively to the centre of all the +other masses in the universe; in this way the series of causes of +mechanical phenomena was closed, in contrast to the mechanics +of Newton and Galileo. In order to develop this idea within the +limits of the modern theory of action through a medium, the +properties of the space-time continuum which determine inertia +must be regarded as field properties of space, analogous to the +electromagnetic field. The concepts of classical mechanics afford +no way of expressing this. For this reason Mach's attempt +at a solution failed for the time being. We shall come back to +this point of view later. In the second place, classical mechanics +indicates a limitation which directly demands an extension of +the principle of relativity to spaces of reference which are not +in uniform motion relatively to each other. The ratio of the +masses of two bodies is defined in mechanics in two ways which +differ from each other fundamentally; in the first place, as the +reciprocal ratio of the accelerations which the same motional +force imparts to them (inert mass), and in the second place, as +the ratio of the forces which act upon them in the same gravitational +field (gravitational mass). The equality of these two +masses, so differently defined, is a fact which is confirmed by +experiments of very high accuracy (experiments of Eötvös), and +classical mechanics offers no explanation for this equality. It is, +however, clear that science is fully justified in assigning such a +numerical equality only after this numerical equality is reduced +to an equality of the real nature of the two concepts. +</p> +<p> +That this object may actually be attained by an extension +of the principle of relativity, follows from the following consideration. +A little reflection will show that the theorem of the +<span class="pagenum" id="Page_60">[Pg 60]</span> +equality of the inert and the gravitational mass is equivalent +to the theorem that the acceleration imparted to a body by a +gravitational field is independent of the nature of the body. For +Newton's equation of motion in a gravitational field, written out +in full, is +<span class="align-center"><img style="vertical-align: -2.036ex; width: 69.485ex; height: 5.204ex;" src="images/342.svg" alt=" " data-tex=" +\begin{aligned} +(\text{Inert mass})·(\text{Acceleration}) = \,(\text{Intensity of the} \\ +\qquad + \text{gravitational field}) &· (\text{Gravitational mass}). +\end{aligned} +"></span> +It is only when there is numerical equality between the inert +and gravitational mass that the acceleration is independent of +the nature of the body. Let now <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> be an inertial system. Masses +which are sufficiently far from each other and from other bodies +are then, with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, free from acceleration. We shall +also refer these masses to a system of co-ordinates <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' uniformly +accelerated with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. Relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' all the masses +have equal and parallel accelerations; with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' they +behave just as if a gravitational field were present and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' were +unaccelerated. Overlooking for the present the question as to the +"cause" of such a gravitational field, which will occupy us later, +there is nothing to prevent our conceiving this gravitational field +as real, that is, the conception that <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is "at rest" and a gravitational +field is present we may consider as equivalent to the conception +that only <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is an "allowable" system of co-ordinates and +no gravitational field is present. The assumption of the complete +physical equivalence of the systems of co-ordinates, +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +we call the "principle of equivalence;" this principle is evidently +intimately connected with the theorem of the equality between +the inert and the gravitational mass, and signifies an extension +of the principle of relativity to co-ordinate systems which are in +<span class="pagenum" id="Page_61">[Pg 61]</span> +non-uniform motion relatively to each other. In fact, through +this conception we arrive at the unity of the nature of inertia +and gravitation. For according to our way of looking at it, the +same masses may appear to be either under the action of inertia +alone (with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">) or under the combined action of +inertia and gravitation (with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'). The possibility of +explaining the numerical equality of inertia and gravitation by +the unity of their nature gives to the general theory of relativity, +according to my conviction, such a superiority over the conceptions +of classical mechanics, that all the difficulties encountered +in development must be considered as small in comparison. +</p> +<p> +What justifies us in dispensing with the preference for inertial +systems over all other co-ordinate systems, a preference +that seems so securely established by experiment based upon +the principle of inertia? The weakness of the principle of inertia +lies in this, that it involves an argument in a circle: a mass moves +without acceleration if it is sufficiently far from other bodies; we +know that it is sufficiently far from other bodies only by the fact +that it moves without acceleration. Are there, in general, any +inertial systems for very extended portions of the space-time +continuum, or, indeed, for the whole universe? We may look +upon the principle of inertia as established, to a high degree of +approximation, for the space of our planetary system, provided +that we neglect the perturbations due to the sun and planets. +Stated more exactly, there are finite regions, where, with respect +to a suitably chosen space of reference, material particles move +freely without acceleration, and in which the laws of the special +theory of relativity, which have been developed above, hold +with remarkable accuracy. Such regions we shall call "Galilean +regions." We shall proceed from the consideration of such regions +<span class="pagenum" id="Page_62">[Pg 62]</span> +as a special case of known properties. +</p> +<p> +The principle of equivalence demands that in dealing with +Galilean regions we may equally well make use of non-inertial +systems, that is, such co-ordinate systems as, relatively to inertial +systems, are not free from acceleration and rotation. If, +further, we are going to do away completely with the difficult +question as to the objective reason for the preference of certain +systems of co-ordinates, then we must allow the use of arbitrarily +moving systems of co-ordinates. As soon as we make this +attempt seriously we come into conflict with that physical interpretation +of space and time to which we were led by the special +theory of relativity. For let <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' be a system of co-ordinates whose +<img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/252.svg" alt=" " data-tex="z">'-axis coincides with the <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/252.svg" alt=" " data-tex="z">-axis of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, and which rotates about +the latter axis with constant angular velocity. Are the configurations +of rigid bodies, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in accordance with +the laws of Euclidean geometry? Since <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is not an inertial system, +we do not know directly the laws of configuration of rigid +bodies with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', nor the laws of nature, in general. But +we do know these laws with respect to the inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +and we can therefore estimate them with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. Imagine +a circle drawn about the origin in the <img style="vertical-align: -0.464ex; width: 6.424ex; height: 2.181ex;" src="images/343.svg" alt=" " data-tex="x'-y'"> plane of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' and a +diameter of this circle. Imagine, further, that we have given a +large number of rigid rods, all equal to each other. We suppose +these laid in series along the periphery and the diameter of the +circle, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. If <img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/161.svg" alt=" " data-tex="U"> is the number of these rods +along the periphery, <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/80.svg" alt=" " data-tex="D"> the number along the diameter, then, if +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' does not rotate relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, we shall have +<span class="align-center"><img style="vertical-align: -1.552ex; width: 7.804ex; height: 4.627ex;" src="images/344.svg" alt=" " data-tex=" +\frac{U}{D} = \pi. +"></span> +<span class="pagenum" id="Page_63">[Pg 63]</span> +But if <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' rotates we get a different result. Suppose that at +a definite time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> we determine the ends of all the rods. +With respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> all the rods upon the periphery experience +the Lorentz contraction, but the rods upon the diameter do not +experience this contraction (along their lengths!).<a id="FNanchor_14_1"></a><a href="#Footnote_14_1" class="fnanchor">[14]</a> +It therefore follows that +<span class="align-center"><img style="vertical-align: -1.552ex; width: 7.804ex; height: 4.627ex;" src="images/345.svg" alt=" " data-tex=" +\frac{U}{D} > \pi. +"></span> +</p> +<p> +It therefore follows that the laws of configuration of rigid +bodies with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' do not agree with the laws of configuration +of rigid bodies that are in accordance with Euclidean geometry. +If, further, we place two similar clocks (rotating with <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'), +one upon the periphery, and the other at the centre of the circle, +then, judged from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, the clock on the periphery will go +slower than the clock at the centre. The same thing must take +place, judged from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', if we define time with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' in a +not wholly unnatural way, that is, in such a way that the laws +with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' depend explicitly upon the time. Space and +time, therefore, cannot be defined with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' as they +were in the special theory of relativity with respect to inertial +systems. But, according to the principle of equivalence, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is +also to be considered as a system at rest, with respect to which +there is a gravitational field (field of centrifugal force, and force +of Coriolis). We therefore arrive at the result: the gravitational +field influences and even determines the metrical laws of the +space-time continuum. If the laws of configuration of ideal rigid +bodies are to be expressed geometrically, then in the presence +of a gravitational field the geometry is not Euclidean. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_14_1"></a><a href="#FNanchor_14_1"><span class="label">[14]</span></a>These considerations assume that the behaviour of rods and clocks +depends only upon velocities, and not upon accelerations, or, at least, that +the influence of acceleration does not counteract that of velocity.</p></div> +<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> +<p> +The case that we have been considering is analogous to that +which is presented in the two-dimensional treatment of surfaces. +It is impossible in the latter case also, to introduce co-ordinates +on a surface (e.g. the surface of an ellipsoid) which +have a simple metrical significance, while on a plane the Cartesian +co-ordinates, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, signify directly lengths measured by a +unit measuring rod. Gauss overcame this difficulty, in his theory +of surfaces, by introducing curvilinear co-ordinates which, +apart from satisfying conditions of continuity, were wholly arbitrary, +and afterwards these co-ordinates were related to the +metrical properties of the surface. In an analogous way we +shall introduce in the general theory of relativity arbitrary co-ordinates, +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">, which shall number uniquely the space-time +points, so that neighbouring events are associated with +neighbouring values of the co-ordinates; otherwise, the choice +of co-ordinates is arbitrary. We shall be true to the principle +of relativity in its broadest sense if we give such a form to the +laws that they are valid in every such four-dimensional system +of co-ordinates, that is, if the equations expressing the laws are +co-variant with respect to arbitrary transformations. +</p> +<p> +The most important point of contact between Gauss's theory +of surfaces and the general theory of relativity lies in the metrical +properties upon which the concepts of both theories, in the +main, are based. In the case of the theory of surfaces, Gauss's +argument is as follows. Plane geometry may be based upon the +concept of the distance <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds">, between two indefinitely near points. +The concept of this distance is physically significant because +the distance can be measured directly by means of a rigid measuring +rod. By a suitable choice of Cartesian co-ordinates this +<span class="pagenum" id="Page_65">[Pg 65]</span> +distance may be expressed by the formula <img style="vertical-align: -0.339ex; width: 17.9ex; height: 2.357ex;" src="images/347.svg" alt=" " data-tex="ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}">. +We may base upon this quantity the concepts of the straight +line as the geodesic (<img style="vertical-align: -0.691ex; width: 8.391ex; height: 2.514ex;" src="images/348.svg" alt=" " data-tex="\delta\! \int\!\! ds = 0">), the interval, the circle, and the +angle, upon which the Euclidean plane geometry is built. A +geometry may be developed upon another continuously curved +surface, if we observe that an infinitesimally small portion of the +surface may be regarded as plane, to within relatively infinitesimal +quantities. There are Cartesian co-ordinates, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/349.svg" alt=" " data-tex="X_{1}">, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/350.svg" alt=" " data-tex="X_{2}">, upon +such a small portion of the surface, and the distance between +two points, measured by a measuring rod, is given by +<span class="align-center"><img style="vertical-align: -0.339ex; width: 19.687ex; height: 2.357ex;" src="images/351.svg" alt=" " data-tex=" +ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}. +"></span> +If we introduce arbitrary curvilinear co-ordinates, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, on the +surface, then <img style="vertical-align: -0.339ex; width: 4.037ex; height: 1.91ex;" src="images/352.svg" alt=" " data-tex="dX_{1}">, <img style="vertical-align: -0.339ex; width: 4.037ex; height: 1.91ex;" src="images/353.svg" alt=" " data-tex="dX_{2}">, may be expressed linearly in terms of +<img style="vertical-align: -0.339ex; width: 3.458ex; height: 1.91ex;" src="images/354.svg" alt=" " data-tex="dx_{1}">, <img style="vertical-align: -0.339ex; width: 3.458ex; height: 1.91ex;" src="images/355.svg" alt=" " data-tex="dx_{2}">. Then everywhere upon the surface we have +<span class="align-center"><img style="vertical-align: -0.464ex; width: 39.454ex; height: 2.482ex;" src="images/356.svg" alt=" " data-tex=" +ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + + g_{22}\, {dx_{2}}^{2}, +"></span> +where <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/357.svg" alt=" " data-tex="g_{11}">, <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/358.svg" alt=" " data-tex="g_{12}">, <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/359.svg" alt=" " data-tex="g_{22}"> are determined by the nature of the surface +and the choice of co-ordinates; if these quantities are known, +then it is also known how networks of rigid rods may be laid +upon the surface. In other words, the geometry of surfaces may +be based upon this expression for <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> exactly as plane geometry +is based upon the corresponding expression. +</p> +<p> +There are analogous relations in the four-dimensional space-time +continuum of physics. In the immediate neighbourhood of +an observer, falling freely in a gravitational field, there exists no +gravitational field. We can therefore always regard an infinitesimally +small region of the space-time continuum as Galilean. +For such an infinitely small region there will be an inertial system +(with the space co-ordinates, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/349.svg" alt=" " data-tex="X_{1}">, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/350.svg" alt=" " data-tex="X_{2}">, <img style="vertical-align: -0.375ex; width: 2.861ex; height: 1.92ex;" src="images/361.svg" alt=" " data-tex="X_{3}">, and the time +<span class="pagenum" id="Page_66">[Pg 66]</span> +co-ordinate <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/362.svg" alt=" " data-tex="X_{4}">) relatively to which we are to regard the laws of +the special theory of relativity as valid. The quantity which is +directly measurable by our unit measuring rods and clocks, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 29.027ex; height: 2.457ex;" src="images/363.svg" alt=" " data-tex=" +{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2}, +"></span> +or its negative, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 45.954ex; height: 2.584ex;" src="images/364.svg" alt=" " data-tex=" +ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}, +\qquad \text{(54)} +"></span> +is therefore a uniquely determinate invariant for two neighbouring +events (points in the four-dimensional continuum), provided +that we use measuring rods that are equal to each other when +brought together and superimposed, and clocks whose rates are +the same when they are brought together. In this the physical +assumption is essential that the relative lengths of two measuring +rods and the relative rates of two clocks are independent, in +principle, of their previous history. But this assumption is certainly +warranted by experience; if it did not hold there could be +no sharp spectral lines; for the single atoms of the same element +certainly do not have the same history, and it would be absurd +to suppose any relative difference in the structure of the single +atoms due to their previous history if the mass and frequencies +of the single atoms of the same element were always the same. +</p> +<p> +Space-time regions of finite extent are, in general, not +Galilean, so that a gravitational field cannot be done away +with by any choice of co-ordinates in a finite region. There +is, therefore, no choice of co-ordinates for which the metrical +relations of the special theory of relativity hold in a finite region. +But the invariant <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> always exists for two neighbouring +points (events) of the continuum. This invariant <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> may be +<span class="pagenum" id="Page_67">[Pg 67]</span> +expressed in arbitrary co-ordinates. If one observes that the +local <img style="vertical-align: -0.339ex; width: 4.085ex; height: 1.91ex;" src="images/365.svg" alt=" " data-tex="dX_{\nu}"> may be expressed linearly in terms of the co-ordinate +differentials <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> may be expressed in the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.76ex; height: 2.685ex;" src="images/366.svg" alt=" " data-tex=" +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. +\qquad \text{(55)} +"></span> +</p> +<p> +The functions <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> describe, with respect to the arbitrarily +chosen system of co-ordinates, the metrical relations of the +space-time continuum and also the gravitational field. As in +the special theory of relativity, we have to discriminate between +time-like and space-like line elements in the four-dimensional +continuum; owing to the change of sign introduced, time-like line +elements have a real, space-like line elements an imaginary <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds">. +The time-like <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> can be measured directly by a suitably chosen +clock. +</p> +<p> +According to what has been said, it is evident that the formulation +of the general theory of relativity assumes a generalization +of the theory of invariants and the theory of tensors; the question +is raised as to the form of the equations which are co-variant +with respect to arbitrary point transformations. The generalized +calculus of tensors was developed by mathematicians long before +the theory of relativity. Riemann first extended Gauss's +train of thought to continua of any number of dimensions; with +prophetic vision he saw the physical meaning of this generalization +of Euclid's geometry. Then followed the development of +the theory in the form of the calculus of tensors, particularly by +Ricci and Levi-Civita. This is the place for a brief presentation +of the most important mathematical concepts and operations of +this calculus of tensors. +</p> +<p> +We designate four quantities, which are defined as functions +of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> with respect to every system of co-ordinates, as components, +<span class="pagenum" id="Page_68">[Pg 68]</span> +<img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}">, of a contra-variant vector, if they transform in a +change of co-ordinates as the co-ordinate differentials <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">. We +therefore have +<span class="align-center"><img style="vertical-align: -1.891ex; width: 24.13ex; height: 5.334ex;" src="images/369.svg" alt=" " data-tex=" +{{A'}^{\mu}} = \frac{{\partial x'}_{\mu}}{\partial x_{\nu}} A^{\nu}. +\qquad \text{(56)} +"></span> +Besides these contra-variant vectors, there are also co-variant +vectors. If <img style="vertical-align: -0.339ex; width: 2.753ex; height: 1.885ex;" src="images/370.svg" alt=" " data-tex="B_{\nu}"> are the components of a co-variant vector, these +vectors are transformed according to the rule +<span class="align-center"><img style="vertical-align: -2.237ex; width: 24.171ex; height: 5.384ex;" src="images/371.svg" alt=" " data-tex=" +{B'}_{\mu} = \frac{\partial x_{\nu}}{{\partial {x'}}_{\mu}} B_{\nu}. +\qquad \text{(57)} +"></span> +The definition of a co-variant vector is chosen in such a way that +a co-variant vector and a contra-variant vector together form a +scalar according to the scheme, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 32.939ex; height: 2.262ex;" src="images/372.svg" alt=" " data-tex=" +\phi = B_{\nu} A^{\nu}\quad \text{(summed over the } \nu). +"></span> +Accordingly, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 35.894ex; height: 5.68ex;" src="images/373.svg" alt=" " data-tex=" +{B'}_{\mu} {{A'}^{\mu}} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} + \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} B_{\alpha} A^{\beta} + = B_{\alpha} A^{\alpha}. +"></span> +In particular, the derivatives <img style="vertical-align: -1.909ex; width: 4.782ex; height: 5.056ex;" src="images/374.svg" alt=" " data-tex="\dfrac{\partial \phi}{\partial x_{\alpha}}"> of a scalar +<img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi">, are components +of a co-variant vector, which, with the co-ordinate differentials, +form the scalar <img style="vertical-align: -1.909ex; width: 8.842ex; height: 5.056ex;" src="images/375.svg" alt=" " data-tex="\dfrac{\partial \phi}{\partial x_{\alpha}}\, dx_{\alpha}">; +we see from this example how natural +is the definition of the co-variant vectors. +</p> +<p> +There are here, also, tensors of any rank, which may have +co-variant or contra-variant character with respect to each index; +as with vectors, the character is designated by the position +<span class="pagenum" id="Page_69">[Pg 69]</span> +of the index. For example, <img style="vertical-align: -0.904ex; width: 2.849ex; height: 2.524ex;" src="images/376.svg" alt=" " data-tex="{A_{\mu}^{\nu}}"> denotes a tensor of the second +rank, which is co-variant with respect to the index <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, and contra-variant +with respect to the index <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">. The tensor character indicates +that the equation of transformation is +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.54ex; height: 5.484ex;" src="images/377.svg" alt=" " data-tex=" +{{A'}_{\mu}^{\nu}} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} + \frac{{\partial x'}_{\nu}}{\partial x_{\beta}} A_{\alpha}^{\beta}. +\qquad \text{(58)} +"></span> +</p> +<p> +Tensors may be formed by the addition and subtraction of +tensors of equal rank and like character, as in the theory of +invariants of orthogonal linear substitutions, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 23.929ex; height: 2.601ex;" src="images/378.svg" alt=" " data-tex=" +A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}. +\qquad \text{(59)} +"></span> +The proof of the tensor character of <img style="vertical-align: -0.904ex; width: 2.873ex; height: 2.499ex;" src="images/379.svg" alt=" " data-tex="C_{\mu}^{\nu}"> depends upon (58). +</p> +<p> +Tensors may be formed by multiplication, keeping the character +of the indices, just as in the theory of invariants of linear +orthogonal transformations, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 23.576ex; height: 2.601ex;" src="images/380.svg" alt=" " data-tex=" +A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}. +\qquad \text{(60)} +"></span> +The proof follows directly from the rule of transformation. +</p> +<p> +Tensors may be formed by contraction with respect to two +indices of different character, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 20.806ex; height: 2.601ex;" src="images/381.svg" alt=" " data-tex=" +A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}. +\qquad \text{(61)} +"></span> +The tensor character of <img style="vertical-align: -0.66ex; width: 4.59ex; height: 2.513ex;" src="images/382.svg" alt=" " data-tex="A_{\mu\sigma\tau}^{\mu}"> determines the tensor character +of <img style="vertical-align: -0.36ex; width: 3.646ex; height: 1.905ex;" src="images/383.svg" alt=" " data-tex="B_{\sigma\tau}">. Proof— +<span class="align-center"><img style="vertical-align: -2.237ex; width: 52.078ex; height: 5.68ex;" src="images/384.svg" alt=" " data-tex=" +{{A'}_{\mu\sigma\tau}^{\mu}} += \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} + \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} + {A_{\alpha st}^{\beta}} += \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} + A_{\alpha st}^{\alpha}. +"></span> +<span class="pagenum" id="Page_70">[Pg 70]</span> +</p> +<p> +The properties of symmetry and skew-symmetry of a tensor +with respect to two indices of like character have the same +significance as in the theory of invariants. +</p> +<p> +With this, everything essential has been said with regard to +the algebraic properties of tensors. +</p> + +<p><br></p> + +<p> +<i>The Fundamental Tensor</i>. It follows from the invariance +of <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}">for an arbitrary choice of the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, in connexion with +the condition of symmetry consistent with (55), that the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> +are components of a symmetrical co-variant tensor (Fundamental +Tensor). Let us form the determinant, <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/385.svg" alt=" " data-tex="g">, of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">, and +also the minors, divided by <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/385.svg" alt=" " data-tex="g">, corresponding to the single <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. +These minors, divided by <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/385.svg" alt=" " data-tex="g">, will be denoted by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, and their +co-variant character is not yet known. Then we have +<span class="align-center"><img style="vertical-align: -2.148ex; width: 36.542ex; height: 5.428ex;" src="images/387.svg" alt=" " data-tex=" +g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} + = \begin{cases} + 1 & \text{if \(\alpha = \beta,\)} \\ + 0 & \text{if \(\alpha \neq \beta.\)} + \end{cases} +\qquad \text{(62)} +"></span> +</p> +<p> +If we form the infinitely small quantities (co-variant vectors) +<span class="align-center"><img style="vertical-align: -0.685ex; width: 23.206ex; height: 2.382ex;" src="images/388.svg" alt=" " data-tex=" +d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}, +\qquad \text{(63)} +"></span> +multiply by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> and sum over the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, we obtain, by the use +of (62), +<span class="align-center"><img style="vertical-align: -0.685ex; width: 22.969ex; height: 2.747ex;" src="images/389.svg" alt=" " data-tex=" +dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}. +\qquad \text{(64)} +"></span> +Since the ratios of the <img style="vertical-align: -0.685ex; width: 3.32ex; height: 2.278ex;" src="images/390.svg" alt=" " data-tex="d\xi_{\mu}">, are arbitrary, and the <img style="vertical-align: -0.65ex; width: 3.564ex; height: 2.22ex;" src="images/391.svg" alt=" " data-tex="dx_{\beta}"> as well as +the <img style="vertical-align: -0.685ex; width: 3.623ex; height: 2.255ex;" src="images/392.svg" alt=" " data-tex="dx_{\mu}"> are components of vectors, it follows that the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> are the +components of a contra-variant tensor<a id="FNanchor_15_1"></a><a href="#Footnote_15_1" class="fnanchor">[15]</a> +(contra-variant fundamental tensor). +The tensor character of <img style="vertical-align: -0.332ex; width: 2.216ex; height: 2.57ex;" src="images/393.svg" alt=" " data-tex="\delta_{\alpha}^{\beta}"> (mixed fundamental +<span class="pagenum" id="Page_71">[Pg 71]</span> +tensor) accordingly follows, by (62). By means of the fundamental +tensor, instead of tensors with co-variant index character, we +can introduce tensors with contra-variant index character, and +conversely. For example, +<span class="align-center"><img style="vertical-align: -3.736ex; width: 12.845ex; height: 8.603ex;" src="images/394.svg" alt=" " data-tex=" +\begin{align*} +A^{\mu} &= g^{\mu\alpha} A_{\alpha}, \\ +A_{\mu} &= g_{\mu\alpha} A^{\alpha}, \\ +T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. +\end{align*} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_15_1"></a><a href="#FNanchor_15_1"><span class="label">[15]</span></a>If we multiply (64) +by <img style="vertical-align: -2.202ex; width: 5.41ex; height: 5.448ex;" src="images/395.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\alpha}}{\partial x_{\beta}}">, +sum over the <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, and replace the <img style="vertical-align: -0.464ex; width: 3.32ex; height: 2.057ex;" src="images/396.svg" alt=" " data-tex="d\xi^{\mu}"> by +a transformation to the accented system, we obtain +<span class="align-center"><img style="vertical-align: -2.237ex; width: 26.832ex; height: 5.484ex;" src="images/397.svg" alt=" " data-tex=" +{dx'}_{\alpha} + = \frac{{\partial x'}_{\sigma}}{\partial x_{\mu}}\, + \frac{{\partial x'}_{\alpha}}{\partial x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}. +"></span> +The statement made above follows from this, since, by (64), we must also +have <img style="vertical-align: -0.561ex; width: 15.36ex; height: 2.597ex;" src="images/398.svg" alt=" " data-tex="{dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}"> and both equations +must hold for every choice of <img style="vertical-align: -0.561ex; width: 3.896ex; height: 2.279ex;" src="images/399.svg" alt=" " data-tex="{d\xi'}_{\sigma}">.</p></div> + +<p><br></p> + +<p> +<i>Volume Invariants</i>. The volume element +<span class="align-center"><img style="vertical-align: -1.948ex; width: 22.967ex; height: 5.027ex;" src="images/400.svg" alt=" " data-tex=" +\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx +"></span> +is not an invariant. For by Jacobi's theorem, +<span class="align-center"><img style="vertical-align: -2.312ex; width: 24.893ex; height: 5.754ex;" src="images/401.svg" alt=" " data-tex=" +dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx. +\qquad \text{(65)} +"></span> +But we can complement <img style="vertical-align: -0.025ex; width: 2.471ex; height: 1.595ex;" src="images/402.svg" alt=" " data-tex="dx"> so that it becomes an invariant. If +we form the determinant of the quantities +<span class="align-center"><img style="vertical-align: -2.237ex; width: 21.889ex; height: 5.545ex;" src="images/403.svg" alt=" " data-tex=" +{g'}_{\mu\nu} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}}\, + \frac{\partial x_{\beta}}{{\partial x'}_{\nu}}\, g_{\alpha\beta}, +"></span> +<span class="pagenum" id="Page_72">[Pg 72]</span> +we obtain, by a double application of the theorem of multiplication +of determinants, +<span class="align-center"><img style="vertical-align: -2.312ex; width: 49.262ex; height: 6.202ex;" src="images/404.svg" alt=" " data-tex=" +g' = |{g'}_{\mu\nu}| + = \left|\frac{\partial x_{\nu}}{{\partial x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| + = \left|\frac{{\partial x'}_{\mu}}{\partial x_{\nu}}\right|^{-2} g. +\qquad \text{(66)} +"></span> +We therefore get the invariant, +<span class="align-center"><img style="vertical-align: -0.554ex; width: 17.372ex; height: 2.851ex;" src="images/405.svg" alt=" " data-tex=" +\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx. +"></span> +</p> + +<p><br></p> + +<p> +<i>Formation of Tensors by Differentiation</i>. Although the algebraic +operations of tensor formation have proved to be as +simple as in the special case of invariance with respect to linear +orthogonal transformations, nevertheless in the general case, +the invariant differential operations are, unfortunately, considerably +more complicated. The reason for this is as follows. If +<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}"> is a contra-variant vector, the coefficients of its +transformation, <img style="vertical-align: -1.891ex; width: 5.35ex; height: 5.334ex;" src="images/407.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}">, are +independent of position only if the transformation +is a linear one. For then the vector components, +<img style="vertical-align: -1.909ex; width: 14.8ex; height: 5.058ex;" src="images/408.svg" alt=" " data-tex="A^{\mu} + \dfrac{\partial A^{\mu}}{\partial x_{\alpha}}\, dx_{\alpha}">, +at a neighbouring point transform in the same way as the <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">, +from which follows the vector character of the vector differentials, +and the tensor character of <img style="vertical-align: -1.909ex; width: 5.125ex; height: 5.058ex;" src="images/409.svg" alt=" " data-tex="\dfrac{\partial A^{\mu}}{\partial x_{\alpha}}">. +But if the <img style="vertical-align: -1.891ex; width: 5.35ex; height: 5.334ex;" src="images/407.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}"> +are variable this is no longer true. +</p> +<p> +That there are, nevertheless, in the general case, invariant +differential operations for tensors, is recognized most satisfactorily +in the following way, introduced by Levi-Civita and Weyl. +Let (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) be a contra-variant vector whose components are given +with respect to the co-ordinate system of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. +Let <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> and <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> +<span class="pagenum" id="Page_73">[Pg 73]</span> +be two infinitesimally near points of the continuum. For the infinitesimal +region surrounding the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}">, there is, according +to our way of considering the matter, a co-ordinate system of +the <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> (with imaginary <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/362.svg" alt=" " data-tex="X_{4}">-co-ordinate) for which the +continuum is Euclidean. Let <img style="vertical-align: -1.207ex; width: 3.929ex; height: 3.06ex;" src="images/410.svg" alt=" " data-tex="A_{(1)}^{\mu}"> be the co-ordinates of the vector at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}">. Imagine a vector drawn at the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, using the +local system of the <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">, with the same co-ordinates (parallel vector +through <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">), then this parallel vector is uniquely determined +by the vector at <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> and the displacement. We designate this operation, +whose uniqueness will appear in the sequel, the parallel +displacement of the vector <img style="vertical-align: -0.685ex; width: 2.849ex; height: 2.305ex;" src="images/411.svg" alt=" " data-tex="A_{\mu}"> from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> to the infinitesimally near +point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">. If we form the vector difference of the vector (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> and the vector obtained by parallel displacement +from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, we get a vector which may be regarded as the +differential of the vector (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) for the given displacement <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">. +</p> +<p> +This vector displacement can naturally also be considered +with respect to the co-ordinate system of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}"> are the +co-ordinates of the vector at <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}">, <img style="vertical-align: -0.186ex; width: 9.235ex; height: 1.808ex;" src="images/412.svg" alt=" " data-tex="A^{\nu} + \delta A^{\nu}"> the co-ordinates of +the vector displaced to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> along the interval (<img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">), then +the <img style="vertical-align: -0.023ex; width: 3.737ex; height: 1.645ex;" src="images/413.svg" alt=" " data-tex="\delta A^{\nu}"> +do not vanish in this case. We know of these quantities, which +do not have a vector character, that they must depend linearly +and homogeneously upon the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}"> and the <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}">. We therefore put +<span class="align-center"><img style="vertical-align: -0.911ex; width: 28.449ex; height: 2.608ex;" src="images/414.svg" alt=" " data-tex=" +\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}. +\qquad \text{(67)} +"></span> +</p> +<p> +In addition, we can state that the <img style="vertical-align: -1.024ex; width: 3.531ex; height: 2.563ex;" src="images/415.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\nu}"> must be symmetrical +with respect to the indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">. +For we can assume from +a representation by the aid of a Euclidean system of local co-ordinates +that the same parallelogram will be described by the +displacement of an element <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/416.svg" alt=" " data-tex="d^{(1)}x_{\nu}"> along a second element +<img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/417.svg" alt=" " data-tex="d^{(2)}x_{\nu}"> +<span class="pagenum" id="Page_74">[Pg 74]</span> +as by a displacement of <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/417.svg" alt=" " data-tex="d^{(2)}x_{\nu}"> along <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/416.svg" alt=" " data-tex="d^{(1)}x_{\nu}">. +We must therefore have +<span class="align-center"><img style="vertical-align: -2.819ex; width: 73.179ex; height: 6.769ex;" src="images/418.svg" alt=" " data-tex=" +\begin{aligned} +d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ + \quad + &= d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). +\end{aligned} +"></span> +The statement made above follows from this, after interchanging +the indices of summation, <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, on the right-hand side. +</p> +<p> +Since the quantities <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> determine all the metrical properties +of the continuum, they must also determine the <img style="vertical-align: -1.024ex; width: 3.531ex; height: 2.563ex;" src="images/415.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\nu}">. If we +consider the invariant of the vector <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}"> that is, the square of its +magnitude, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 9.29ex; height: 2.326ex;" src="images/419.svg" alt=" " data-tex=" +g_{\mu\nu} A^{\mu} A^{\nu}, +"></span> +which is an invariant, this cannot change in a parallel displacement. +We therefore have +<span class="align-center"><img style="vertical-align: -1.909ex; width: 58.452ex; height: 5.252ex;" src="images/420.svg" alt=" " data-tex=" +0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) + = \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu} +"></span> +or, by (67), +<span class="align-center"><img style="vertical-align: -2.148ex; width: 41.858ex; height: 5.491ex;" src="images/421.svg" alt=" " data-tex=" +\left(\frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} + - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} + - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} += 0. +"></span> +</p> +<p> +Owing to the symmetry of the expression in the brackets +with respect to the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, this equation can be valid +for an arbitrary choice of the vectors (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) and <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}"> only when +the expression in the brackets vanishes for all combinations of +the indices. By a cyclic interchange of the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, we +obtain thus altogether three equations, from which we obtain, +on taking into account the symmetrical property of the +<img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/422.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\alpha}">, +<span class="align-center"><img style="vertical-align: -1.577ex; width: 24.423ex; height: 4.178ex;" src="images/423.svg" alt=" " data-tex=" +\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] + = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}, +\qquad \text{(68)} +"></span> +<span class="pagenum" id="Page_75">[Pg 75]</span> +in which, following Christoffel, the abbreviation has been used, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 44.772ex; height: 5.58ex;" src="images/424.svg" alt=" " data-tex=" +\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] + = \tfrac{1}{2}\left( + \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} + - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} +\right). +\qquad \text{(69)} +"></span> +</p> +<p> +If we multiply (68) by <img style="vertical-align: -0.464ex; width: 3.204ex; height: 1.992ex;" src="images/425.svg" alt=" " data-tex="g^{\alpha\sigma}"> and sum over the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, we obtain +<span class="align-center"><img style="vertical-align: -2.237ex; width: 55.667ex; height: 5.58ex;" src="images/426.svg" alt=" " data-tex=" +\Gamma_{\mu\nu}^{\alpha} + = \tfrac{1}{2} g^{\sigma\alpha}\left( + \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} + - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} +\right) = \left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}, +\qquad \text{(70)} +"></span> +in which <img style="vertical-align: -0.798ex; width: 4.994ex; height: 2.72ex;" src="images/427.svg" alt=" " data-tex="\left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}"> is +the Christoffel symbol of the second kind. +Thus the quantities <img style="vertical-align: 0; width: 1.414ex; height: 1.538ex;" src="images/428.svg" alt=" " data-tex="\Gamma"> are deduced from the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. Equations +(67) and (70) are the foundation for the following discussion. +</p> + +<p><br></p> + +<p> +<i>Co-variant Differentiation of Tensors</i>. If (<img style="vertical-align: -0.186ex; width: 9.469ex; height: 1.808ex;" src="images/429.svg" alt=" " data-tex="A^{\mu} + \delta A^{\mu}">) is +the vector resulting from an infinitesimal parallel displacement +from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, and (<img style="vertical-align: -0.186ex; width: 9.641ex; height: 1.805ex;" src="images/430.svg" alt=" " data-tex="A^{\mu} + dA^{\mu}">) the vector <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}"> at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, +then the difference of these two, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 35.532ex; height: 5.428ex;" src="images/431.svg" alt=" " data-tex=" +dA^{\mu} - \delta A^{\mu} = \left( + \frac{\partial A^{\mu}}{\partial x_{\sigma}} + + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma}, +"></span> +is also a vector. Since this is the case for an arbitrary choice of +the <img style="vertical-align: -0.357ex; width: 3.572ex; height: 1.927ex;" src="images/432.svg" alt=" " data-tex="dx_{\sigma}">, it follows that +<span class="align-center"><img style="vertical-align: -1.909ex; width: 30.676ex; height: 5.058ex;" src="images/433.svg" alt=" " data-tex=" +{A^{\mu}}_{;\, \sigma} + = \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} +\qquad \text{(71)} +"></span> +is a tensor, which we designate as the co-variant derivative of +the tensor of the first rank (vector). Contracting this tensor, we +obtain the divergence of the contra-variant tensor <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">. In this +we must observe that according to (70), +<span class="align-center"><img style="vertical-align: -2.308ex; width: 39.215ex; height: 5.747ex;" src="images/434.svg" alt=" " data-tex=" +\Gamma_{\mu\sigma}^{\sigma} + = \tfrac{1}{2} g^{\sigma\alpha} \frac{\partial g_{\sigma\alpha}}{\partial x_{\mu}} + = \frac{1}{\sqrt{g}}\, \frac{\partial \sqrt{g}}{\partial x_{\mu}}. +\qquad \text{(72)} +"></span> +<span class="pagenum" id="Page_76">[Pg 76]</span> +If we put, further, +<span class="align-center"><img style="vertical-align: -0.602ex; width: 21.206ex; height: 2.398ex;" src="images/435.svg" alt=" " data-tex=" +A^{\mu} \sqrt{g} = \mathfrak A^{\mu}, +\qquad \text{(73)} +"></span> +a quantity designated by Weyl as the contra-variant tensor density<a id="FNanchor_16_1"></a><a href="#Footnote_16_1" class="fnanchor">[16]</a> +of the first rank, it follows that, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 18.242ex; height: 5.384ex;" src="images/436.svg" alt=" " data-tex=" +\mathfrak A = \frac{\partial \mathfrak A^{\mu}}{\partial x_{\mu}} +\qquad \text{(74)} +"></span> +is a scalar density. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_16_1"></a><a href="#FNanchor_16_1"><span class="label">[16]</span></a>This expression is justified, in that +<img style="vertical-align: -0.71ex; width: 17.691ex; height: 2.398ex;" src="images/437.svg" alt=" " data-tex="A^{\mu}\sqrt{g}\, dx = \mathbf A^{\mu}\, dx"> has a tensor +character. Every tensor, when multiplied by <img style="vertical-align: -0.71ex; width: 3.009ex; height: 2.398ex;" src="images/438.svg" alt=" " data-tex="\sqrt{g}">, changes into a tensor +density. We employ capital Gothic letters for tensor densities.</p></div> + +<p> +We get the law of parallel displacement for the co-variant +vector <img style="vertical-align: -0.685ex; width: 2.87ex; height: 2.23ex;" src="images/439.svg" alt=" " data-tex="B_{\mu}"> by stipulating that the parallel displacement shall be +effected in such a way that the scalar +<span class="align-center"><img style="vertical-align: -0.685ex; width: 10.084ex; height: 2.326ex;" src="images/440.svg" alt=" " data-tex=" +\phi = A^{\mu} B_{\mu} +"></span> +remains unchanged, and that therefore +<span class="align-center"><img style="vertical-align: -0.685ex; width: 16.968ex; height: 2.326ex;" src="images/441.svg" alt=" " data-tex=" +A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu} +"></span> +vanishes for every value assigned to (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">). We therefore get +<span class="align-center"><img style="vertical-align: -0.904ex; width: 26.783ex; height: 2.601ex;" src="images/442.svg" alt=" " data-tex=" +\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}. +\qquad \text{(75)} +"></span> +</p> +<p> +From this we arrive at the co-variant derivative of the co-variant +vector by the same process as that which led to (71), +<span class="align-center"><img style="vertical-align: -1.909ex; width: 31.497ex; height: 5.252ex;" src="images/443.svg" alt=" " data-tex=" +B_{\mu;\, \sigma} + = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}. +\qquad \text{(76)} +"></span> +<span class="pagenum" id="Page_77">[Pg 77]</span> +By interchanging the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, and subtracting, we get +the skew-symmetrical tensor, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 28.991ex; height: 5.58ex;" src="images/444.svg" alt=" " data-tex=" +\phi_{\mu\sigma} + = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \frac{\partial B_{\sigma}}{\partial x_{\mu}}. +\qquad \text{(77)} +"></span> +</p> +<p> +For the co-variant differentiation of tensors of the second +and higher ranks we may use the process by which (75) was +deduced. Let, for example, (<img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/445.svg" alt=" " data-tex="A_{\sigma\tau}">) be a co-variant tensor of the +second rank. Then <img style="vertical-align: -0.36ex; width: 9.406ex; height: 1.98ex;" src="images/446.svg" alt=" " data-tex="A_{\sigma\tau} E^{\sigma} F^{\tau}"> is a scalar, +if <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/292.svg" alt=" " data-tex="E"> and <img style="vertical-align: 0; width: 1.695ex; height: 1.538ex;" src="images/447.svg" alt=" " data-tex="F"> are vectors. +This expression must not be changed by the <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/117.svg" alt=" " data-tex="\delta">-displacement; +expressing this by a formula, we get, using (67), <img style="vertical-align: -0.36ex; width: 4.63ex; height: 1.982ex;" src="images/448.svg" alt=" " data-tex="\delta A_{\sigma\tau}">, whence +we get the desired co-variant derivative, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 43.434ex; height: 5.386ex;" src="images/449.svg" alt=" " data-tex=" +A_{\sigma\tau;\, \rho} + = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} + - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}. +\qquad \text{(78)} +"></span> +</p> +<p> +In order that the general law of co-variant differentiation of +tensors may be clearly seen, we shall write down two co-variant +derivatives deduced in an analogous way: +<span class="align-center"><img style="vertical-align: -5.204ex; width: 48.077ex; height: 11.539ex;" src="images/450.svg" alt=" " data-tex=" +\begin{align*} +A_{\sigma;\, \rho}^{\tau} + &= \frac{\partial A_{\sigma}^{\tau}}{\partial x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha},\, +\qquad & &\text{(79)} \\ +{A^{\sigma\tau}}_{;\, \rho} + &= \frac{\partial A^{\sigma\tau}}{\partial x_{\rho}} + + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}. +\qquad & &\text{(80)} +\end{align*} +"></span> +The general law of formation now becomes evident. From these +formulae we shall deduce some others which are of interest for +the physical applications of the theory. +</p> +<p> +In case <img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/445.svg" alt=" " data-tex="A_{\sigma\tau}"> is skew-symmetrical, we obtain the tensor +<span class="align-center"><img style="vertical-align: -2.237ex; width: 40.172ex; height: 5.583ex;" src="images/451.svg" alt=" " data-tex=" +A_{\sigma\tau\rho} + = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + + \frac{\partial A_{\tau\rho}}{\partial x_{\sigma}} + + \frac{\partial A_{\rho\sigma}}{\partial x_{\tau}}, +\qquad \text{(81)} +"></span> +<span class="pagenum" id="Page_78">[Pg 78]</span> +which is skew-symmetrical in all pairs of indices, by cyclic +interchange and addition. +</p> +<p> +If, in (78), we replace <img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/445.svg" alt=" " data-tex="A_{\sigma\tau}"> by the fundamental tensor, +<img style="vertical-align: -0.464ex; width: 3.008ex; height: 1.464ex;" src="images/452.svg" alt=" " data-tex="g_{\sigma\tau}">, +then the right-hand side vanishes identically; an analogous statement +holds for (80) with respect to <img style="vertical-align: -0.464ex; width: 3.008ex; height: 1.975ex;" src="images/453.svg" alt=" " data-tex="g^{\sigma\tau}">; that is, the co-variant +derivatives of the fundamental tensor vanish. That this must be +so we see directly in the local system of co-ordinates. +</p> +<p> +In case <img style="vertical-align: 0; width: 3.625ex; height: 1.62ex;" src="images/454.svg" alt=" " data-tex="A^{\sigma\tau}"> is skew-symmetrical, we obtain from (80), by +contraction with respect to <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, +<span class="align-center"><img style="vertical-align: -1.912ex; width: 21.125ex; height: 5.059ex;" src="images/455.svg" alt=" " data-tex=" +\mathfrak A^{\sigma} = \frac{\partial \mathfrak A^{\sigma\tau}}{\partial x_{\tau}}. +\qquad \text{(82)} +"></span> +</p> +<p> +In the general case, from (79) and (80), by contraction with +respect to <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, we obtain the equations, +<span class="align-center"><img style="vertical-align: -4.874ex; width: 35.508ex; height: 10.879ex;" src="images/456.svg" alt=" " data-tex=" +\begin{align*} +\mathfrak A_{\sigma} &= \frac{\partial \mathfrak A_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak A_{\alpha}^{\beta}, +\qquad & &\text{(83)} \\ +\mathfrak A^{\sigma} &= \frac{\partial \mathfrak A^{\sigma\alpha}}{\partial x_{\alpha}} + + \Gamma_{\alpha\beta}^{\sigma} \mathfrak A^{\alpha\beta}. +\qquad & &\text{(84)} +\end{align*} +"></span> +</p> + +<p><br></p> + +<p> +<i>The Riemann Tensor</i>. If we have given a curve extending +from the point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> to the point <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G"> of the continuum, then a +vector <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">, given at <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">, may, by a parallel displacement, be moved +along the curve to <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G">. If the continuum is Euclidean (more generally, +if by a suitable choice of co-ordinates the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">, are constants) +then the vector obtained at <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G"> as a result of this displacement +does not depend upon the choice of the curve joining +<img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> and <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G">. +But otherwise, the result depends upon the path of the displacement. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure04.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG.4.</p> +</div></div> + +<p class="nind"> +In this case, therefore, a vector suffers a change, +<img style="vertical-align: 0; width: 4.734ex; height: 1.62ex;" src="images/458.svg" alt=" " data-tex="\Delta A^{\mu}"> +(in its direction, not its magnitude), when it is carried from a +<span class="pagenum" id="Page_79">[Pg 79]</span> +point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> of a closed curve, along the curve, and back to P. We +shall now calculate this vector change: +<span class="align-center"><img style="vertical-align: -1.948ex; width: 14.747ex; height: 5.027ex;" src="images/459.svg" alt=" " data-tex=" +\Delta A^{\mu} = \oint \delta A^{\mu}. +"></span> +As in Stokes' theorem for the line integral of a vector around +a closed curve, this problem may be reduced to the integration +around a closed curve with infinitely small linear dimensions; we +shall limit ourselves to this case. +</p> +<p> +We have, first, by (67), +<span class="align-center"><img style="vertical-align: -1.948ex; width: 23.411ex; height: 5.027ex;" src="images/460.svg" alt=" " data-tex=" +\Delta A^{\mu} = -\oint \Gamma_{\alpha\beta}^{\mu} A^{\alpha}\, dx_{\beta}. +"></span> +</p> +<p> +In this, <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> is the value of +this quantity at the variable +point <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G"> of the path of integration. If we put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 19.174ex; height: 2.382ex;" src="images/462.svg" alt=" " data-tex=" +\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P} +"></span> +<span class="pagenum" id="Page_80">[Pg 80]</span> +and denote the value of +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> at <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> by +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 3.615ex;" src="images/463.svg" alt=" " data-tex="\overline{\Gamma_{\alpha\beta}^{\mu}}"> then we have, with +sufficient accuracy, +<span class="align-center"><img style="vertical-align: -1.891ex; width: 21.686ex; height: 6.547ex;" src="images/464.svg" alt=" " data-tex=" +\Gamma_{\alpha\beta}^{\mu} + = \overline{\Gamma_{\alpha\beta}^{\mu}} + + \frac{\overline{\partial \Gamma_{\alpha\beta}^{\mu}}}{\partial x_{\nu}}\, \xi^{\nu}. +"></span> +</p> +<p> +Let, further, <img style="vertical-align: 0; width: 2.908ex; height: 1.62ex;" src="images/465.svg" alt=" " data-tex="A^{\alpha}"> be the value obtained from +<img style="vertical-align: 0; width: 2.908ex; height: 2.337ex;" src="images/466.svg" alt=" " data-tex="\overline{A^{\alpha}}"> by a parallel +displacement along the curve from <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> to <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G">. It may now easily +be proved by means of (67) that +<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}"> - <img style="vertical-align: 0; width: 2.849ex; height: 2.337ex;" src="images/467.svg" alt=" " data-tex="\overline{A^{\mu}}"> is infinitely small of +the first order, while, for a curve of infinitely small dimensions +of the first order, <img style="vertical-align: 0; width: 4.734ex; height: 1.62ex;" src="images/458.svg" alt=" " data-tex="\Delta A^{\mu}"> is infinitely small of the second order. +Therefore there is an error of only the second order if we put +<span class="align-center"><img style="vertical-align: -0.58ex; width: 21.633ex; height: 2.917ex;" src="images/468.svg" alt=" " data-tex=" +A^{\alpha} = \overline{A^{\alpha}} + - \overline{\Gamma_{\sigma\tau}^{\alpha}}\; + \overline{A^{\sigma}}\; + \overline{\xi^{\tau}}. +"></span> +</p> +<p> +If we introduce these values of +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> and <img style="vertical-align: 0; width: 2.908ex; height: 1.62ex;" src="images/465.svg" alt=" " data-tex="A^{\alpha}"> into the integral, +we obtain, neglecting all quantities of a higher order of small +quantities than the second, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 49.891ex; height: 6.785ex;" src="images/469.svg" alt=" " data-tex=" +\Delta A^{\mu} + = - \left(\frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) + A^{\sigma} \oint \xi^{\alpha}\, d\xi^{\beta}. +\qquad \text{(85)} +"></span> +The quantity removed from under the sign of integration refers +to the point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">. Subtracting +<img style="vertical-align: -1.552ex; width: 9.35ex; height: 4.588ex;" src="images/470.svg" alt=" " data-tex="\dfrac{1}{2} d(\xi^{\alpha} \xi^{\beta})"> from the integrand, we +obtain +<span class="align-center"><img style="vertical-align: -1.948ex; width: 21.145ex; height: 5.027ex;" src="images/471.svg" alt=" " data-tex=" +\tfrac{1}{2} \oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}). +"></span> +This skew-symmetrical tensor of the second rank, +<img style="vertical-align: -0.464ex; width: 3.481ex; height: 2.413ex;" src="images/472.svg" alt=" " data-tex="f^{\alpha\beta}">, characterizes +the surface element bounded by the curve in magnitude +and position. If the expression in the brackets in (85) were +skew-symmetrical with respect to the indices +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, we could +<span class="pagenum" id="Page_81">[Pg 81]</span> +conclude its tensor character from (85). We can accomplish this +by interchanging the summation indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> in (85) and +adding the resulting equation to (85). We obtain +<span class="align-center"><img style="vertical-align: -1.045ex; width: 31.223ex; height: 3.108ex;" src="images/473.svg" alt=" " data-tex=" +2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}, +\qquad \text{(86)} +"></span> +in which +<span class="align-center"><img style="vertical-align: -2.202ex; width: 52.635ex; height: 6.141ex;" src="images/474.svg" alt=" " data-tex=" +R_{\sigma\alpha\beta}^{\mu} + = - \frac{\partial \Gamma_{\sigma\alpha}^{\mu}}{\partial x_{\beta}} + + \frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}. +\qquad \text{(87)} +"></span> +<p> +The tensor character of <img style="vertical-align: -1.045ex; width: 4.748ex; height: 2.898ex;" src="images/475.svg" alt=" " data-tex="R_{\sigma\alpha\beta}^{\mu}"> follows from (86); this is the +Riemann curvature tensor of the fourth rank, whose properties of +symmetry we do not need to go into. Its vanishing is a sufficient +condition (disregarding the reality of the chosen co-ordinates) +that the continuum is Euclidean. +</p> +<p> +By contraction of the Riemann tensor with respect to the +indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, we obtain the symmetrical tensor of the second +rank, +<span class="align-center"><img style="vertical-align: -1.909ex; width: 51.793ex; height: 5.472ex;" src="images/476.svg" alt=" " data-tex=" +R_{\mu\nu} + = - \frac{\partial \Gamma_{\mu\nu}^{\alpha}}{\partial x_{\alpha}} + + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + + \frac{\partial \Gamma_{\mu\alpha}^{\alpha}}{\partial x_{\nu}} + - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}. +\qquad \text{(88)} +"></span> +The last two terms vanish if the system of co-ordinates is so +chosen that <img style="vertical-align: -0.464ex; width: 12.531ex; height: 1.855ex;" src="images/477.svg" alt=" " data-tex="g = \text{constant}">. From <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}">, we can form the scalar, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 21.085ex; height: 2.382ex;" src="images/479.svg" alt=" " data-tex=" +R = g^{\mu\nu} R_{\mu\nu}. +\qquad \text{(89)} +"></span> +</p> + +<p><br></p> + +<p> +<i>Straightest Geodetic Lines</i>. A line may be constructed in +such a way that its successive elements arise from each other by +parallel displacements. This is the natural generalization of the +straight line of the Euclidean geometry. For such a line, we have +<span class="align-center"><img style="vertical-align: -2.148ex; width: 26.887ex; height: 5.444ex;" src="images/480.svg" alt=" " data-tex=" +\delta \left(\frac{dx_{\mu}}{ds}\right) + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}. +"></span> +<span class="pagenum" id="Page_82">[Pg 82]</span> +The left-hand side is to be replaced by +<img style="vertical-align: -1.651ex; width: 5.606ex; height: 5.264ex;" src="images/481.svg" alt=" " data-tex="\dfrac{d^{2} x_{\mu}}{ds^{2}}">,<a id="FNanchor_17_1"></a><a href="#Footnote_17_1" class="fnanchor">[17]</a> +so that we have +<span class="align-center"><img style="vertical-align: -1.651ex; width: 34.843ex; height: 5.264ex;" src="images/482.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0. +\qquad \text{(90)} +"></span> +We get the same line if we find the line which gives a stationary +value to the integral +<span class="align-center"><img style="vertical-align: -1.948ex; width: 27.455ex; height: 5.027ex;" src="images/483.svg" alt=" " data-tex=" +\int ds\quad\text{or}\quad +\int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}} +"></span> +between two points (geodetic line). +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_17_1"></a><a href="#FNanchor_17_1"><span class="label">[17]</span></a>The direction vector at a neighbouring point of the curve results, by +a parallel displacement along the line element (<img style="vertical-align: -0.65ex; width: 3.564ex; height: 2.22ex;" src="images/391.svg" alt=" " data-tex="dx_{\beta}">), from the direction +vector of each point considered.</p></div> +<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title="LECTURE IV: THE GENERAL THEORY OF RELATIVITY"><a id="chap04"></a>LECTURE IV +<br><br> +THE GENERAL THEORY OF RELATIVITY<br> +(<i>continued</i>)</h2> + +<p class="nind"> +<span class="dropcap">W</span>E are now in possession of the mathematical apparatus which +is necessary to formulate the laws of the general theory of relativity. +No attempt will be made in this presentation at systematic +completeness, but single results and possibilities will +be developed progressively from what is known and from the results +obtained. Such a presentation is most suited to the present +provisional state of our knowledge. +</p> +<p> +A material particle upon which no force acts moves, according +to the principle of inertia, uniformly in a straight line. In +the four-dimensional continuum of the special theory of relativity +(with real time co-ordinate) this is a real straight line. The +natural, that is, the simplest, generalization of the straight line +which is plausible in the system of concepts of Riemann's general +theory of invariants is that of the straightest, or geodetic, +line. We shall accordingly have to assume, in the sense of the +principle of equivalence, that the motion of a material particle, +under the action only of inertia and gravitation, is described by +the equation, +<span class="align-center"><img style="vertical-align: -1.651ex; width: 34.843ex; height: 5.264ex;" src="images/482.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0. +\qquad \text{(90)} +"></span> +In fact, this equation reduces to that of a straight line if all the +components, <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}">, of the gravitational field vanish. +</p> +<p> +How are these equations connected with Newton's equations +of motion? According to the special theory of relativity, +the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> as well as the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, have +the values, with respect to an inertial +<span class="pagenum" id="Page_84">[Pg 84]</span> +system (with real time co-ordinate and suitable choice of the +sign of <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}">), +<span class="align-center"><img style="vertical-align: -5.317ex; width: 28.158ex; height: 11.765ex;" src="images/484.svg" alt=" " data-tex=" +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & 1 +\end{array} +\right\}. +\qquad \text{(91)} +"></span> +The equations of motion then become +<span class="align-center"><img style="vertical-align: -1.651ex; width: 10.383ex; height: 5.264ex;" src="images/485.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} = 0. +"></span> +We shall call this the "first approximation" to the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field. In +considering approximations it is often useful, as in the special +theory of relativity, to use an imaginary <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">-co-ordinate, as then +the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. to the first approximation, assume the values +<span class="align-center"><img style="vertical-align: -5.317ex; width: 31.049ex; height: 11.765ex;" src="images/486.svg" alt=" " data-tex=" +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & -1 +\end{array} +\right\}. +\qquad \text{(91a)} +"></span> +These values may be collected in the relation +<span class="align-center"><img style="vertical-align: -0.685ex; width: 11.491ex; height: 2.307ex;" src="images/487.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu}. +"></span> +To the second approximation we must then put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.353ex; height: 2.382ex;" src="images/488.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, +\qquad \text{(92)} +"></span> +where the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> are to be regarded as small of the first order. +<span class="pagenum" id="Page_85">[Pg 85]</span> +</p> +<p> +Both terms of our equation of motion are then small of the +first order. If we neglect terms which, relatively to these, are +small of the first order, we have to put +<span class="align-center"><img style="vertical-align: -3.856ex; width: 67.308ex; height: 8.843ex;" src="images/490.svg" alt=" " data-tex=" +\begin{gather*} +ds^{2} = -{dx_{\nu}}^{2} = dl^{2} (1 - q^{2}), +\qquad & &\text{(93)}\\ +\Gamma_{\alpha\beta}^{\mu} + = -\delta_{\mu\sigma} [^{\alpha\beta}_{\,\delta}] + = -[^{\alpha\beta}_{\,\mu}] + = \frac{1}{2} \left( + \frac{\partial \gamma_{\alpha\beta}}{\partial x_{\mu}} + - \frac{\partial \gamma_{\alpha\mu}}{\partial x_{\beta}} + - \frac{\partial \gamma_{\beta\mu}}{\partial x_{\alpha}}\right). +\qquad & &\text{(94)} +\end{gather*} +"></span> +We shall now introduce an approximation of a second kind. Let +the velocity of the material particles be very small compared to +that of light. Then <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> will be the same as the time differential, <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl">. +Further, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/491.svg" alt=" " data-tex="\dfrac{dx_{1}}{ds}">, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/492.svg" alt=" " data-tex="\dfrac{dx_{2}}{ds}">, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/493.svg" alt=" " data-tex="\dfrac{dx_{3}}{ds}"> +will vanish compared to <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/494.svg" alt=" " data-tex="\dfrac{dx_{4}}{ds}">. +We shall assume, in addition, that the gravitational field varies +so little with the time that the derivatives of the +<img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> by <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}"> may +be neglected. Then the equation of motion (for <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> = 1,2,3) +reduces to +<span class="align-center"><img style="vertical-align: -2.237ex; width: 30.687ex; height: 5.849ex;" src="images/495.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{dl^{2}} + = \frac{\partial}{\partial x_{\mu}} \left(\frac{\gamma_{44}}{2}\right). +\qquad \text{(90a)} +"></span> +This equation is identical with Newton's equation of motion for +a material particle in a gravitational field, if we identify +<img style="vertical-align: -1.552ex; width: 6.656ex; height: 4.153ex;" src="images/496.svg" alt=" " data-tex="\left(\dfrac{\gamma_{44}}{2}\right)"> +with the potential of the gravitational field; whether or not this +is allowable, naturally depends upon the field equations of gravitation, +that is, it depends upon whether or not this quantity +satisfies, to a first approximation, the same laws of the field +as the gravitational potential in Newton's theory. A glance at +(90) and (90a) shows that the <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> actually do play the rôle of +the intensity of the gravitational field. These quantities do not +have a tensor character. +</p> +<p> +Equations (90) express the influence of inertia and gravitation +upon the material particle. The unity of inertia and gravitation +<span class="pagenum" id="Page_86">[Pg 86]</span> +is formally expressed by the fact that the whole left-hand +side of (90) has the character of a tensor (with respect to any +transformation of co-ordinates), but the two terms taken separately +do not have tensor character, so that, in analogy with +Newton's equations, the first term would be regarded as the expression +for inertia, and the second as the expression for the +gravitational force. +</p> +<p> +We must next attempt to find the laws of the gravitational +field. For this purpose, Poisson's equation, +<span class="align-center"><img style="vertical-align: -0.489ex; width: 11.852ex; height: 2.109ex;" src="images/497.svg" alt=" " data-tex=" +\Delta\phi = 4\pi K\rho +"></span> +of the Newtonian theory must serve as a model. This equation +has its foundation in the idea that the gravitational field +arises from the density <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"> of ponderable matter. It must also +be so in the general theory of relativity. But our investigations +of the special theory of relativity have shown that in place of +the scalar density of matter we have the tensor of energy per +unit volume. In the latter is included not only the tensor of +the energy of ponderable matter, but also that of the electromagnetic +energy. We have seen, indeed, that in a more complete +analysis the energy tensor can be regarded only as a provisional +means of representing matter. In reality, matter consists of electrically +charged particles, and is to be regarded itself as a part, +in fact, the principal part, of the electromagnetic field. It is +only the circumstance that we have not sufficient knowledge of +the electromagnetic field of concentrated charges that compels +us, provisionally, to leave undetermined in presenting the theory, +the true form of this tensor. From this point of view our +problem now is to introduce a tensor, <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}">. of the second rank, +<span class="pagenum" id="Page_87">[Pg 87]</span> +whose structure we do not know provisionally, and which includes +in itself the energy density of the electromagnetic field +and of ponderable matter; we shall denote this in the following +as the "energy tensor of matter." +</p> +<p> +According to our previous results, the principles of momentum +and energy are expressed by the statement that the divergence +of this tensor vanishes (47c). In the general theory of relativity, +we shall have to assume as valid the corresponding general +co-variant equation. If (<img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}">) denotes the co-variant energy tensor +of matter, <img style="vertical-align: -0.576ex; width: 2.615ex; height: 2.162ex;" src="images/498.svg" alt=" " data-tex="\mathfrak T_{\sigma}^{\nu}"> the corresponding mixed tensor density, then, +in accordance with (83), we must require that +<span class="align-center"><img style="vertical-align: -1.909ex; width: 26.609ex; height: 5.144ex;" src="images/499.svg" alt=" " data-tex=" +0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta} +\qquad \text{(95)} +"></span> +be satisfied. It must be remembered that besides the energy density +of the matter there must also be given an energy density of +the gravitational field, so that there can be no talk of principles +of conservation of energy and momentum for matter alone. This +is expressed mathematically by the presence of the second term +in (95), which makes it impossible to conclude the existence of +an integral equation of the form of (49). The gravitational field +transfers energy and momentum to the "matter," in that it exerts +forces upon it and gives it energy; this is expressed by the +second term in (95). +</p> +<p> +If there is an analogue of Poisson's equation in the general +theory of relativity, then this equation must be a tensor equation +for the tensor <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> of the gravitational potential; the energy +tensor of matter must appear on the right-hand side of this +equation. On the left-hand side of the equation there must be +a differential tensor in the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. We have to find this differential +<span class="pagenum" id="Page_88">[Pg 88]</span> +tensor. It is completely determined by the following three +conditions:— +</p> +<p> +1. It may contain no differential coefficients of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> higher +than the second. +</p> +<p> +2. It must be linear and homogeneous in these second differential +coefficients. +</p> +<p> +3. Its divergence must vanish identically. +</p> +<p> +The first two of these conditions are naturally taken from +Poisson's equation. Since it may be proved mathematically +that all such differential tensors can be formed algebraically +(i.e. without differentiation) from Riemann's tensor, our tensor +must be of the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 13.106ex; height: 2.23ex;" src="images/500.svg" alt=" " data-tex=" +R_{\mu\nu} + ag_{\mu\nu} R, +"></span> +in which <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}"> and <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> are defined by (88) and (89) respectively. +Further, it may be proved that the third condition requires a +to have the value <img style="vertical-align: -1.552ex; width: 3.887ex; height: 4.588ex;" src="images/501.svg" alt=" " data-tex="-\dfrac{1}{2}">. For the law of the gravitational field we +therefore get the equation +<span class="align-center"><img style="vertical-align: -0.781ex; width: 32.031ex; height: 2.737ex;" src="images/502.svg" alt=" " data-tex=" +R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}. +\qquad \text{(96)} +"></span> +Equation (95) is a consequence of this equation. <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/503.svg" alt=" " data-tex="\kappa"> denotes a +constant, which is connected with the Newtonian gravitation +constant. +</p> +<p> +In the following I shall indicate the features of the theory +which are interesting from the point of view of physics, using as +little as possible of the rather involved mathematical method. +It must first be shown that the divergence of the left-hand side +actually vanishes. The energy principle for matter may be expressed, by (83), +<span class="align-center"><img style="vertical-align: -1.909ex; width: 27.615ex; height: 5.144ex;" src="images/504.svg" alt=" " data-tex=" +0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta}, +\qquad \text{(97)} +"></span> +<span class="pagenum" id="Page_89">[Pg 89]</span> +in which +<span class="align-center"><img style="vertical-align: -0.669ex; width: 17.886ex; height: 2.851ex;" src="images/505.svg" alt=" " data-tex=" +\mathfrak T_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}. +"></span> +The analogous operation, applied to the left-hand side of (96), +will lead to an identity. +</p> +<p> +In the region surrounding each world-point there are systems +of co-ordinates for which, choosing the <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">-co-ordinate imaginary, +at the given point, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 34.922ex; height: 5.428ex;" src="images/506.svg" alt=" " data-tex=" +g_{\mu\nu} = g^{\mu\nu} + = -\delta_{\mu\nu} + = \begin{cases} + -1 & \text{if \(\mu = \nu\)}, \\ + \,\,\,\,0 & \text{if \(\mu \neq \nu\)}, + \end{cases} +"></span> +and for which the first derivatives of the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> and the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> vanish. +We shall verify the vanishing of the divergence of the left-hand +side at this point. At this point the components +<img style="vertical-align: -1.042ex; width: 3.421ex; height: 2.58ex;" src="images/507.svg" alt=" " data-tex="\Gamma_{\sigma\beta}^{\alpha}"> vanish, so +that we have to prove the vanishing only of +<span class="align-center"><img style="vertical-align: -1.909ex; width: 30.825ex; height: 5.056ex;" src="images/508.svg" alt=" " data-tex=" +\frac{\partial}{\partial x_{\sigma}} \left[ + \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) +\right]. +"></span> +Introducing (88) and (70) into this expression, we see that the +only terms that remain are those in which third derivatives of +the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> enter. Since the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> are to be replaced by +<img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/509.svg" alt=" " data-tex="-\delta_{\mu\nu}">, we obtain, +finally, only a few terms which may easily be seen to cancel +each other. Since the quantity that we have formed has a tensor +character, its vanishing is proved for every other system of co-ordinates +also, and naturally for every other four-dimensional +point. The energy principle of matter (97) is thus a mathematical +consequence of the field equations (96). +</p> +<p> +In order to learn whether the equations (96) are consistent +with experience, we must, above all else, find out whether they +<span class="pagenum" id="Page_90">[Pg 90]</span> +lead to the Newtonian theory as a first approximation. For this +purpose we must introduce various approximations into these +equations. We already know that Euclidean geometry and the +law of the constancy of the velocity of light are valid, to a certain +approximation, in regions of a great extent, as in the planetary +system. If, as in the special theory of relativity, we take the +fourth co-ordinate imaginary, this means that we must put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.353ex; height: 2.382ex;" src="images/510.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, +\qquad \text{(98)} +"></span> +in which the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> are so small compared to 1 that we can neglect +the higher powers of the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> and their derivatives. If we do this, +we learn nothing about the structure of the gravitational held, or +of metrical space of cosmical dimensions, but we do learn about +the influence of neighbouring masses upon physical phenomena. +</p> +<p> +Before carrying through this approximation we shall transform +(96). We multiply (96) by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, summed over the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> +observing the relation which follows from the definition of +the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 10.936ex; height: 2.326ex;" src="images/511.svg" alt=" " data-tex=" +g_{\mu\nu} g^{\mu\nu} = 4, +"></span> +we obtain the equation +<span class="align-center"><img style="vertical-align: -0.685ex; width: 18.981ex; height: 2.326ex;" src="images/512.svg" alt=" " data-tex=" +R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T. +"></span> +If we put this value of <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> in (96) we obtain +<span class="align-center"><img style="vertical-align: -0.904ex; width: 44.2ex; height: 2.861ex;" src="images/513.svg" alt=" " data-tex=" +R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) + = -\kappa T_{\mu\nu}^{*}. +\qquad \text{(96a)} +"></span> +When the approximation which has been mentioned is carried +out, we obtain for the left-hand side, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 48.357ex; height: 6.785ex;" src="images/514.svg" alt=" " data-tex=" +-\tfrac{1}{2}\left( + \frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + + \frac{\partial^{2} \gamma_{\alpha\alpha}}{\partial x_{\mu}\, \partial x_{\nu}} + - \frac{\partial^{2} \gamma_{\mu\alpha}}{\partial x_{\nu}\, \partial x_{\alpha}} + - \frac{\partial^{2} \gamma_{\nu\alpha}}{\partial x_{\mu}\, \partial x_{\alpha}} +\right) +"></span> +<span class="pagenum" id="Page_91">[Pg 91]</span> +or +<span class="align-center"><img style="vertical-align: -2.827ex; width: 48.834ex; height: 6.785ex;" src="images/515.svg" alt=" " data-tex=" +-\tfrac{1}{2}\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + + \tfrac{1}{2} \frac{\partial}{\partial x_{\nu}}\left( + \frac{{\partial \gamma'}_{\mu\alpha}}{\partial x_{\alpha}} + \right) + + \tfrac{1}{2} \frac{\partial}{\partial x_{\mu}}\left( + \frac{{\partial \gamma'}_{\nu\alpha}}{\partial x_{\alpha}} + \right), +"></span> +in which has been put +<span class="align-center"><img style="vertical-align: -0.914ex; width: 30.468ex; height: 2.871ex;" src="images/516.svg" alt=" " data-tex=" +{\gamma'}_{\mu\nu} + = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}. +\qquad \text{(99)} +"></span> +</p> +<p> +We must now note that equation (96) is valid for any system +of co-ordinates. We have already specialized the system of +co-ordinates in that we have chosen it so that within the region +considered the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> differ infinitely little from the constant values +<img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/509.svg" alt=" " data-tex="-\delta_{\mu\nu}">. But this condition remains satisfied in any infinitesimal +change of co-ordinates, so that there are still four conditions +to which the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> may be subjected, provided these conditions +do not conflict with the conditions for the order of magnitude of +the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}">. We shall now assume that the system of co-ordinates +is so chosen that the four relations— +<span class="align-center"><img style="vertical-align: -2.237ex; width: 38.565ex; height: 5.909ex;" src="images/517.svg" alt=" " data-tex=" +0 = \frac{{\partial\gamma'}_{\mu\nu}}{\partial x_{\nu}} + = \frac{\partial\gamma_{\mu\nu}}{\partial x_{\nu}} + - \tfrac{1}{2} \frac{\partial\gamma_{\sigma\sigma}}{\partial x_{\mu}} +\qquad \text{(100)} +"></span> +are satisfied. Then (96a) takes the form +<span class="align-center"><img style="vertical-align: -2.332ex; width: 26.138ex; height: 5.944ex;" src="images/518.svg" alt=" " data-tex=" +\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}. +\qquad \text{(96b)} +"></span> +These equations may be solved by the method, familiar in +electrodynamics, of retarded potentials; we get, in an easily +understood notation, +<span class="align-center"><img style="vertical-align: -1.948ex; width: 48.917ex; height: 5.59ex;" src="images/519.svg" alt=" " data-tex=" +\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int + \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}. +\qquad \text{(101)} +"></span> +<span class="pagenum" id="Page_92">[Pg 92]</span> +</p> +<p> +In order to see in what sense this theory contains the Newtonian +theory, we must consider in greater detail the energy +tensor of matter. Considered phenomenologically, this energy +tensor is composed of that of the electromagnetic field and of +matter in the narrower sense. If we consider the different parts +of this energy tensor with respect to their order of magnitude, +it follows from the results of the special theory of relativity that +the contribution of the electromagnetic field practically vanishes +in comparison to that of ponderable matter. In our system of +units, the energy of one gram of matter is equal to 1, compared +to which the energy of the electric fields may be ignored, and +also the energy of deformation of matter, and even the chemical +energy. We get an approximation that is fully sufficient for our +purpose if we put +<span class="align-center"><img style="vertical-align: -3.551ex; width: 30.02ex; height: 8.234ex;" src="images/520.svg" alt=" " data-tex=" +\left. +\begin{aligned} +T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}, \\ +ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. +\end{aligned} +\right\} +\qquad \text{(102)} +"></span> +In this, <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma"> is the density at rest, that is, the density of the ponderable +matter, in the ordinary sense, measured with the aid +of a unit measuring rod, and referred to a Galilean system of +co-ordinates moving with the matter. +</p> +<p> +We observe, further, that in the co-ordinates we have chosen, +we shall make only a relatively small error if we replace the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> +by <img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/509.svg" alt=" " data-tex="-\delta_{\mu\nu}">, so that we put +<span class="align-center"><img style="vertical-align: -1.018ex; width: 28.45ex; height: 3.167ex;" src="images/521.svg" alt=" " data-tex=" +ds^{2} = -\sum {dx_{\mu}}^{2}. +\qquad \text{(102a)} +"></span> +</p> +<p> +The previous developments are valid however rapidly the +masses which generate the field may move relatively to our chosen +system of quasi-Galilean co-ordinates. But in astronomy +<span class="pagenum" id="Page_93">[Pg 93]</span> +we have to do with masses whose velocities, relatively to the +co-ordinate system employed, are always small compared to the +velocity of light, that is, small compared to 1, with our choice +of the unit of time. We therefore get an approximation which is +sufficient for nearly all practical purposes if in (101) we replace +the retarded potential by the ordinary (non-retarded) potential, +and if, for the masses which generate the field, we put +<span class="align-center"><img style="vertical-align: -1.577ex; width: 60.851ex; height: 5.187ex;" src="images/522.svg" alt=" " data-tex=" +\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad +\frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}. +\qquad \text{(103)} +"></span> +Then we get for <img style="vertical-align: 0; width: 3.72ex; height: 1.532ex;" src="images/523.svg" alt=" " data-tex="T^{\mu\nu}"> and <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> the values +<span class="align-center"><img style="vertical-align: -5.317ex; width: 25.929ex; height: 11.765ex;" src="images/524.svg" alt=" " data-tex=" +\left. +\begin{array}{*{3}{>{\qquad}r}>{\quad}r} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & -\sigma +\end{array} +\right\}. +\qquad \text{(104)} +"></span> +For <img style="vertical-align: 0; width: 1.593ex; height: 1.532ex;" src="images/525.svg" alt=" " data-tex="T"> we get the value <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, and, finally, +for <img style="vertical-align: -0.904ex; width: 3.322ex; height: 2.47ex;" src="images/526.svg" alt=" " data-tex="T_{\mu\nu}^{*}"> the values, +<span class="align-center"><img style="vertical-align: -8.905ex; width: 31.524ex; height: 18.941ex;" src="images/527.svg" alt=" " data-tex=" +\left. +\begin{array}{*{3}{>{\qquad}c}>{\quad}r} +\dfrac{\sigma}{2} & 0 & 0 & 0 \\ +0 & \dfrac{\sigma}{2} & 0 & 0 \\ +0 & 0 & \dfrac{\sigma}{2} & 0 \\ +0 & 0 & 0 & -\dfrac{\sigma}{2} +\end{array} +\right\}. +\qquad \text{(104a)} +"></span> +</p> +<p> +We thus get, from (101), +<span class="align-center"><img style="vertical-align: -4.821ex; width: 45.595ex; height: 10.774ex;" src="images/528.svg" alt=" " data-tex=" +\left. +\begin{aligned} +\gamma_{11} = \gamma_{22} = \gamma_{33} + &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, +\end{aligned} +\right\} +\qquad \text{(101a)} +"></span> +<span class="pagenum" id="Page_94">[Pg 94]</span> +while all the other <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}">, vanish. The least of these equations, +in connexion with equation (90a), contains Newton's theory of +gravitation. If we replace <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> by <img style="vertical-align: -0.025ex; width: 1.796ex; height: 1.441ex;" src="images/206.svg" alt=" " data-tex="ct"> we get +<span class="align-center"><img style="vertical-align: -2.237ex; width: 40.856ex; height: 5.849ex;" src="images/529.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{dt^{2}} + = \frac{\kappa c^{2}}{8\pi}\, \frac{\partial}{\partial x_{\mu}} \left\{ + \int \frac{\sigma\, dV_{0}}{r} + \right\}. +\qquad \text{(90b)} +"></span> +We see that the Newtonian gravitation constant <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, is connected +with the constant <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/503.svg" alt=" " data-tex="\kappa"> that enters into our field equations by the +relation +<span class="align-center"><img style="vertical-align: -1.602ex; width: 19.979ex; height: 5.018ex;" src="images/530.svg" alt=" " data-tex=" +K = \frac{\kappa c^{2}}{8\pi}. +\qquad \text{(105)} +"></span> +From the known numerical value of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, it therefore follows that +<span class="align-center"><img style="vertical-align: -1.914ex; width: 53.752ex; height: 5.398ex;" src="images/531.svg" alt=" " data-tex=" +\kappa = \frac{8\pi K}{c^{2}} + = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} + = 1.86·10^{-27}. +\qquad \text{(105a)} +"></span> +From (101) we see that even in the first approximation the structure +of the gravitational field differs fundamentally from that +which is consistent with the Newtonian theory; this difference +lies in the fact that the gravitational potential has the character +of a tensor and not a scalar. This was not recognized in the past +because only the component <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/532.svg" alt=" " data-tex="g_{44}">, to a first approximation, enters +the equations of motion of material particles. +</p> +<p> +In order now to be able to judge the behaviour of measuring +rods and clocks from our results, we must observe the following. +According to the principle of equivalence, the metrical relations +of the Euclidean geometry are valid relatively to a Cartesian +system of reference of infinitely small dimensions, and in a suitable +state of motion (freely falling, and without rotation). We +can make the same statement for local systems of co-ordinates +<span class="pagenum" id="Page_95">[Pg 95]</span> +which, relatively to these, have small accelerations, and therefore +for such systems of co-ordinates as are at rest relatively to +the one we have selected. For such a local system, we have, for +two neighbouring point events, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 51.054ex; height: 2.457ex;" src="images/533.svg" alt=" " data-tex=" +ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} + = - dS^{2} + dT^{2}, +"></span> +where <img style="vertical-align: -0.05ex; width: 2.636ex; height: 1.645ex;" src="images/534.svg" alt=" " data-tex="dS"> is measured directly by a measuring rod and <img style="vertical-align: -0.023ex; width: 2.769ex; height: 1.593ex;" src="images/535.svg" alt=" " data-tex="dT"> by +a clock at rest relatively to the system; these are the naturally +measured lengths and times. Since <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}">, on the other hand, is +known in terms of the co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> employed in finite regions, +in the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 17.836ex; height: 2.685ex;" src="images/536.svg" alt=" " data-tex=" +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, +"></span> +we have the possibility of getting the relation between naturally +measured lengths and times, on the one hand, and the corresponding +differences of co-ordinates, on the other hand. As the +division into space and time is in agreement with respect to the +two systems of co-ordinates, so when we equate the two expressions +for <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> we get two relations. If, by (101a), we put +<span class="align-center"><img style="vertical-align: -5.201ex; width: 48.691ex; height: 11.534ex;" src="images/537.svg" alt=" " data-tex=" +\begin{aligned} +ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) +({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ +\qquad + + \mspace{26mu}\left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}, +\end{aligned} +"></span> +we obtain, to a sufficiently close approximation, +<span class="align-center"><img style="vertical-align: -7.645ex; width: 63.489ex; height: 16.421ex;" src="images/538.svg" alt=" " data-tex=" +\left. +\begin{aligned} + &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ + &\qquad + \begin{aligned} + &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}, \\ +dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. +\end{aligned} +\end{aligned} +\right\} +\qquad \text{(106)} +"></span> +<span class="pagenum" id="Page_96">[Pg 96]</span> +</p> +<p> +The unit measuring rod has therefore the length, +<span class="align-center"><img style="vertical-align: -1.948ex; width: 16.351ex; height: 5.048ex;" src="images/539.svg" alt=" " data-tex=" +1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +"></span> +in respect to the system of co-ordinates we have selected. The +particular system of co-ordinates we have selected insures that +this length shall depend only upon the place, and not upon the +direction. If we had chosen a different system of co-ordinates +this would not be so. But however we may choose a system of +co-ordinates, the laws of configuration of rigid rods do not agree +with those of Euclidean geometry; in other words, we cannot +choose any system of co-ordinates so that the co-ordinate differences, +<img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/8.svg" alt=" " data-tex="\Delta x_{1}">, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/9.svg" alt=" " data-tex="\Delta x_{2}">, <img style="vertical-align: -0.375ex; width: 4.166ex; height: 1.994ex;" src="images/10.svg" alt=" " data-tex="\Delta x_{3}">, corresponding +to the ends of a unit measuring rod, oriented in any way, +shall always satisfy the relation +<img style="vertical-align: -0.685ex; width: 22.179ex; height: 2.572ex;" src="images/540.svg" alt=" " data-tex="\Delta x_{1}^{2} + \Delta x_{2}^{2} + \Delta x_{3}^{2} = 1">. +In this sense space is not Euclidean, +but "curved." It follows from the second of the relations above +that the interval between two beats of the unit clock (<img style="vertical-align: -0.023ex; width: 2.769ex; height: 1.593ex;" src="images/535.svg" alt=" " data-tex="dT"> = 1) +corresponds to the "time" +<span class="align-center"><img style="vertical-align: -1.948ex; width: 16.351ex; height: 5.048ex;" src="images/541.svg" alt=" " data-tex=" +1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +"></span> +in the unit used in our system of co-ordinates. The rate of a +clock is accordingly slower the greater is the mass of the ponderable +matter in its neighbourhood. We therefore conclude that +spectral lines which are produced on the sun's surface will be +displaced towards the red, compared to the corresponding lines +produced on the earth, by about 2 • 10<sup>-6</sup> of their wave-lengths. +At first, this important consequence of the theory appeared to +conflict with experiment; but results obtained during the past +year seem to make the existence of this effect more probable, and +<span class="pagenum" id="Page_97">[Pg 97]</span> +it can hardly be doubted that this consequence of the theory will +be confirmed within the next year. +</p> +<p> +Another important consequence of the theory, which can be +tested experimentally, has to do with the path of rays of light. +In the general theory of relativity also the velocity of light is +everywhere the same, relatively to a local inertial system. This +velocity is unity in our natural measure of time. The law of +the propagation of light in general co-ordinates is therefore, +according to the general theory of relativity, characterized, by the +equation +<span class="align-center"><img style="vertical-align: -0.186ex; width: 7.374ex; height: 2.185ex;" src="images/542.svg" alt=" " data-tex=" +ds^{2} = 0 +"></span> +To within the approximation which we are using, and in the +system of co-ordinates which we have selected, the velocity of +light is characterized, according to (106), by the equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 66.477ex; height: 5.428ex;" src="images/543.svg" alt=" " data-tex=" +\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) + = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}. +"></span> +The velocity of light <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/544.svg" alt=" " data-tex="L">, is therefore expressed in our co-ordinates +by +<span class="align-center"><img style="vertical-align: -1.948ex; width: 52.226ex; height: 5.945ex;" src="images/545.svg" alt=" " data-tex=" +\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} + = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}. +\qquad \text{(107)} +"></span> +We can therefore draw the conclusion from this, that a ray of +light passing near a large mass is deflected. If we imagine the +sun, of mass <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M"> concentrated at the origin of our system of co-ordinates, +then a ray of fight, travelling parallel to the <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">-axis. +in the <img style="vertical-align: -0.375ex; width: 7.329ex; height: 1.694ex;" src="images/546.svg" alt=" " data-tex="x_{1}-x_{3}"> plane, at a distance <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"> from the origin, will be +deflected, in all, by an amount +<span class="align-center"><img style="vertical-align: -2.159ex; width: 21.475ex; height: 5.553ex;" src="images/547.svg" alt=" " data-tex=" +\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\partial L}{\partial x_{1}}\, dx_{3} +"></span> +<span class="pagenum" id="Page_98">[Pg 98]</span> +towards the sun. On performing the integration we get +<span class="align-center"><img style="vertical-align: -1.577ex; width: 20.451ex; height: 4.652ex;" src="images/548.svg" alt=" " data-tex=" +\alpha = \frac{\kappa M}{2\pi\Delta}. +\qquad \text{(108)} +"></span> +</p> +<p> +The existence of this deflection, which amounts to 1.7<code>''</code> for +<img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"> equal to the radius of the sun, was confirmed, with remarkable +accuracy, by the English Solar Eclipse Expedition in 1919, and +most careful preparations have been made to get more exact +observational data at the solar eclipse in 1922. It should be +noted that this result, also, of the theory is not influenced by +our arbitrary choice of a system of co-ordinates. +</p> +<p> +This is the place to speak of the third consequence of the +theory which can be tested by observation, namely, that which +concerns the motion of the perihelion of the planet Mercury. The +secular changes in the planetary orbits are known with such accuracy +that the approximation we have been using is no longer +sufficient for a comparison of theory and observation. It is necessary +to go back to the general field equations (96). To solve +this problem I made use of the method of successive approximations. +Since then, however, the problem of the central symmetrical +statical gravitational field has been completely solved by +Schwarzschild and others; the derivation given by H. Weyl in his +book, "Raum-Zeit-Materie," is particularly elegant. The calculation +can be simplified somewhat if we do not go back directly +to the equation (96), but base it upon a principle of variation +that is equivalent to this equation. I shall indicate the procedure +only in so far as is necessary for understanding the method. +<span class="pagenum" id="Page_99">[Pg 99]</span> +</p> +<p> +In the case of a statical field, <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> must have the form +<span class="align-center"><img style="vertical-align: -3.502ex; width: 33.95ex; height: 8.135ex;" src="images/549.svg" alt=" " data-tex=" +\left. +\begin{aligned} +ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}, \\ +d\sigma^{2} + &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}, +\end{aligned} +\right\} +\qquad \text{(109)} +"></span> +where the summation on the right-hand side of the last equation +is to be extended over the space variables only. The central +symmetry of the field requires the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}">, to be of the form, +<span class="align-center"><img style="vertical-align: -0.65ex; width: 30.455ex; height: 2.347ex;" src="images/550.svg" alt=" " data-tex=" +\gamma_{\alpha\beta} + = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}; +\qquad \text{(110)} +"></span> +<img style="vertical-align: -0.464ex; width: 2.352ex; height: 2.351ex;" src="images/551.svg" alt=" " data-tex="f^{2}">, <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> are functions of +<img style="vertical-align: -0.532ex; width: 21.685ex; height: 2.851ex;" src="images/552.svg" alt=" " data-tex="r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}"> only. One +of these three functions can be chosen arbitrarily, because our +system of co-ordinates is, <i>a priori</i>, completely arbitrary; for by +a substitution +<span class="align-center"><img style="vertical-align: -2.17ex; width: 13.76ex; height: 5.47ex;" src="images/553.svg" alt=" " data-tex=" +\begin{align*} +{x'}_{4} &= x_{4}, \\ +{x'}_{\alpha} &= F(r) x_{\alpha}, +\end{align*} +"></span> +we can always insure that one of these three functions shall be +an assigned function of <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r">'. In place of (110) we can therefore +put, without limiting the generality, +<span class="align-center"><img style="vertical-align: -0.65ex; width: 30.222ex; height: 2.347ex;" src="images/554.svg" alt=" " data-tex=" +\gamma_{\alpha\beta} + = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}. +\qquad \text{(110a)} +"></span> +</p> +<p> +In this way the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> are expressed in terms of the two quantities +<img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> and <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/555.svg" alt=" " data-tex="f">. These are to be determined as functions of <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r">, +by introducing them into equation (96), after first calculating +<span class="pagenum" id="Page_100">[Pg 100]</span> +the <img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/556.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\sigma}"> from (109) and (110a). We have +<span class="align-center"><img style="vertical-align: -7.006ex; width: 64.764ex; height: 15.143ex;" src="images/557.svg" alt=" " data-tex=" +\left. +\begin{aligned} +\Gamma_{\alpha\beta}^{\sigma} + &= \tfrac{1}{2} \frac{x_{\sigma}}{r} + · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} + {1 + \lambda r^{2}}\,\, + \text{(for \(\alpha, \beta, \sigma = 1, 2, 3)\)}, \\ +\Gamma_{44}^{4} + &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad + \text{(for \(\alpha, \beta = 1, 2, 3)\)}, \\ +\Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\partial f^{2}}{\partial x_{\alpha}},\quad +\Gamma_{44}^{\alpha} + = -\tfrac{1}{2}{g^{\alpha\beta}}\, + \frac{\partial f^{2}}{\partial {x_{\beta}}}. +\end{aligned} +\right\} +\qquad \text{(110b)} +"></span> +</p> +<p> +With the help of these results, the field equations furnish +Schwarzschild's solution: +<span class="align-center"><img style="vertical-align: -4.636ex; width: 65.804ex; height: 10.403ex;" src="images/558.svg" alt=" " data-tex=" +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ + \frac{dr^{2}} + {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) +\right], +\qquad \text{(109a)} +"></span> +in which we have put +<span class="align-center"><img style="vertical-align: -7.643ex; width: 29.619ex; height: 16.416ex;" src="images/559.svg" alt=" " data-tex=" +\left. +\begin{aligned} +x_{4} &= l, \\ +x_{1} &= r \sin\theta \sin\phi, \\ +x_{2} &= r \sin\theta \cos\phi, \\ +x_{3} &= r \cos\theta, \\ +A &= \frac{\kappa M}{4\pi}. +\end{aligned} +\right\} +\qquad \text{(109b)} +"></span> +</p> +<p> +<img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M"> denotes the sun's mass, centrally symmetrically placed +about the origin of co-ordinates; the solution (109) is valid only +outside of this mass, where all the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> vanish. If the motion +of the planet takes place in the <img style="vertical-align: -0.339ex; width: 7.329ex; height: 1.658ex;" src="images/560.svg" alt=" " data-tex="x_{1}-x_{2}"> plane then we +must replace (109a) by +<span class="align-center"><img style="vertical-align: -4.636ex; width: 49.704ex; height: 8.052ex;" src="images/561.svg" alt=" " data-tex=" +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} + - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}. +\qquad \text{(109c)} +"></span> +<span class="pagenum" id="Page_101">[Pg 101]</span> +</p> +<p> +The calculation of the planetary motion depends upon equation (90). +From the first of equations (110b) and (90) we get, +for the indices 1, 2, 3, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 28.237ex; height: 5.428ex;" src="images/562.svg" alt=" " data-tex=" +\frac{d}{ds} + \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0, +"></span> +or, if we integrate, and express the result in polar co-ordinates, +<span class="align-center"><img style="vertical-align: -1.575ex; width: 27.665ex; height: 4.674ex;" src="images/563.svg" alt=" " data-tex=" +r^{2} \frac{d\phi}{ds} = \text{constant}. +\qquad \text{(111)} +"></span> +</p> +<p> +From (90), for <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> = 4, we get +<span class="align-center"><img style="vertical-align: -2.093ex; width: 50.697ex; height: 5.509ex;" src="images/564.svg" alt=" " data-tex=" +0 = \frac{d^{2} l}{ds^{2}} + + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} +{\, \frac{dl}{ds}} + = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} +{\, \frac{dl}{ds}}. +"></span> +From this, after multiplication by <img style="vertical-align: -0.464ex; width: 2.352ex; height: 2.351ex;" src="images/551.svg" alt=" " data-tex="f^{2}"> and integration, we have +<span class="align-center"><img style="vertical-align: -1.575ex; width: 27.721ex; height: 4.674ex;" src="images/565.svg" alt=" " data-tex=" +f^{2} \frac{dl}{ds} = \text{constant}. +\qquad \text{(112)} +"></span> +</p> +<p> +In (109c), (111) and (112) we have three equations between +the four variables <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r">, <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> and <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi">, from which the motion of the +planet may be calculated in the same way as in classical mechanics. +The most important result we get from this is a secular +rotation of the elliptic orbit of the planet in the same sense as +the revolution of the planet, amounting in radians per revolution +to +<span class="align-center"><img style="vertical-align: -2.194ex; width: 24.054ex; height: 5.611ex;" src="images/566.svg" alt=" " data-tex=" +\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}, +\qquad \text{(113)} +"></span> +<span class="pagenum" id="Page_102">[Pg 102]</span> +where +<span class="align-center"><img style="vertical-align: -5.129ex; width: 59.981ex; height: 11.389ex;" src="images/567.svg" alt=" " data-tex=" +\begin{align*} +a &= \text{the semi-major axis of the planetary orbit in centimetres.} \\ +e &= \text{the numerical eccentricity.} \\ +c &= 3·10^{+10}, \text{the velocity of light}\, \mathit{in\, vacuo}. \\ +T &= \text{the period of revolution in seconds.} +\end{align*} +"></span> +This expression furnishes the explanation of the motion of the +perihelion of the planet Mercury, which has been known for a +hundred years (since Leverrier), and for which theoretical astronomy +has hitherto been unable satisfactorily to account. +</p> +<p> +There is no difficulty in expressing Maxwell's theory of the +electromagnetic field in terms of the general theory of relativity; +this is done by application of the tensor formation (81), (82) +and (77). Let <img style="vertical-align: -0.685ex; width: 2.501ex; height: 2.255ex;" src="images/568.svg" alt=" " data-tex="\phi_{\mu}"> be a tensor of the first rank, to be denoted +as an electromagnetic 4-potential; then an electromagnetic field +tensor may be defined by the relations, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.316ex; height: 5.58ex;" src="images/569.svg" alt=" " data-tex=" +\phi_{\mu\nu} + = \frac{\partial \phi_{\mu}}{\partial x_{\nu}} - \frac{\partial \phi_{\nu}}{\partial x_{\mu}}. +\qquad \text{(114)} +"></span> +The second of Maxwell's systems of equations is then defined by +the tensor equation, resulting from this, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 38.212ex; height: 5.58ex;" src="images/570.svg" alt=" " data-tex=" +\frac{\partial \phi_{\mu\nu}}{\partial x_{\rho}} + +\frac{\partial \phi_{\nu\rho}}{\partial x_{\mu}} + +\frac{\partial \phi_{\rho\mu}}{\partial x_{\nu}} = 0, +\qquad \text{(114a)} +"></span> +and the first of Maxwell's systems of equations is defined by the +tensor-density relation +<span class="align-center"><img style="vertical-align: -1.891ex; width: 21.762ex; height: 5.038ex;" src="images/571.svg" alt=" " data-tex=" +\frac{\partial \mathfrak F^{\mu\nu}}{\partial x_{\nu}} = \mathfrak J^{\mu}, +\qquad \text{(115)} +"></span> +<span class="pagenum" id="Page_103">[Pg 103]</span> +in which +<span class="align-center"><img style="vertical-align: -3.536ex; width: 21.852ex; height: 8.204ex;" src="images/572.svg" alt=" " data-tex=" +\begin{align*} +\mathfrak F^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}, \\ +\mathfrak J^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. +\end{align*} +"></span> +If we introduce the energy tensor of the electromagnetic field +into the right-hand side of (96), we obtain (115), for the special +case <img style="vertical-align: -0.314ex; width: 2.401ex; height: 1.867ex;" src="images/573.svg" alt=" " data-tex="\mathfrak J^{\mu}"> = 0, as a consequence of (96) by taking the divergence. +This inclusion of the theory of electricity in the scheme of the +general theory of relativity has been considered arbitrary and +unsatisfactory by many theoreticians. Nor can we in this way +conceive of the equilibrium of the electricity which constitutes +the elementary electrically charged particles. A theory in which +the gravitational field and the electromagnetic field enter as an +essential entity would be much preferable. H. Weyl, and recently +Th. Kaluza, have discovered some ingenious theorems along this +direction; but concerning them, I am convinced that they do not +bring us nearer to the true solution of the fundamental problem. +I shall not go into this further, but shall give a brief discussion +of the so-called cosmological problem, for without this, the considerations +regarding the general theory of relativity would, in +a certain sense, remain unsatisfactory. +</p> +<p> +Our previous considerations, based upon the field equations (96), +had for a foundation the conception that space on +the whole is Galilean-Euclidean, and that this character is disturbed +only by masses embedded in it. This conception was +certainly justified as long as we were dealing with spaces of the +order of magnitude of those that astronomy has to do with. +But whether portions of the universe, however large they may +be, are quasi-Euclidean, is a wholly different question. We can +<span class="pagenum" id="Page_104">[Pg 104]</span> +make this clear by using an example from the theory of surfaces +which we have employed many times. If a portion of a surface +is observed by the eye to be practically plane, it does not at all +follow that the whole surface has the form of a plane; the surface +might just as well be a sphere, for example, of sufficiently large +radius. The question as to whether the universe as a whole is +non-Euclidean was much discussed from the geometrical point of +view before the development of the theory of relativity. But with +the theory of relativity, this problem has entered upon a new +stage, for according to this theory the geometrical properties of +bodies are not independent, but depend upon the distribution +of masses. +</p> +<p> +If the universe were quasi-Euclidean, then Mach was wholly +wrong in his thought that inertia, as well as gravitation, depends +upon a kind of mutual action between bodies. For in this case, +with a suitably selected system of co-ordinates, the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> would +be constant at infinity, as they are in the special theory of relativity, +while within finite regions the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> would differ from these +constant values by small amounts only, with a suitable choice +of co-ordinates, as a result of the influence of the masses in finite +regions. The physical properties of space would not then be +wholly independent, that is, uninfluenced by matter, but in the +main they would be, and only in small measure, conditioned by +matter. Such a dualistic conception is even in itself not satisfactory; +there are, however, some important physical arguments +against it, which we shall consider. +</p> +<p> +The hypothesis that the universe is infinite and Euclidean +at infinity, is, from the relativistic point of view, a complicated +hypothesis. In the language of the general theory of relativity +it demands that the Riemann tensor of the fourth rank +<img style="vertical-align: -0.357ex; width: 5.172ex; height: 1.902ex;" src="images/574.svg" alt=" " data-tex="R_{iklm}"> +<span class="pagenum" id="Page_105">[Pg 105]</span> +shall vanish at infinity, which furnishes twenty independent conditions, +while only ten curvature components <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}">, enter into +the laws of the gravitational field. It is certainly unsatisfactory +to postulate such a far-reaching limitation without any physical +basis for it. +</p> +<p> +But in the second place, the theory of relativity makes it +appear probable that Mach was on the right road in his thought +that inertia depends upon a mutual action of matter. For we +shall show in the following that, according to our equations, inert +masses do act upon each other in the sense of the relativity of +inertia, even if only very feebly. What is to be expected along +the line of Mach's thought? +</p> +<p class="hanging2"> +1. The inertia of a body must increase when ponderable +masses are piled up in its neighbourhood. +</p> +<p class="hanging2"> +2. A body must experience an accelerating force when +neighbouring masses are accelerated, and, in fact, the +force must be in the same direction as the acceleration. +</p> +<p class="hanging2"> +3. A rotating hollow body must generate inside of itself +a "Coriolis field," which deflects moving bodies in the +sense of the rotation, and a radial centrifugal field as +well. +</p> +<p> +We shall now show that these three effects, which are to be +expected in accordance with Mach's ideas, are actually present +according to our theory, although their magnitude is so small +that confirmation of them by laboratory experiments is not to be +thought of. For this purpose we shall go back to the equations of +motion of a material particle (90), and carry the approximations +somewhat further than was done in equation (90a). +<span class="pagenum" id="Page_106">[Pg 106]</span> +</p> +<p> +First, we consider <img style="vertical-align: -0.489ex; width: 2.96ex; height: 1.486ex;" src="images/575.svg" alt=" " data-tex="\gamma_{44}"> as small of the first order. The square +of the velocity of masses moving under the influence of the gravitational +force is of the same order, according to the energy +equation. It is therefore logical to regard the velocities of the +material particles we are considering, as well as the velocities +of the masses which generate the field, as small, of the order +<img style="vertical-align: -1.552ex; width: 2.127ex; height: 4.588ex;" src="images/576.svg" alt=" " data-tex="\dfrac{1}{2}">. +We shall now carry out the approximation in the equations that +arise from the field equations (101) and the equations of motion (90) +so far as to consider terms, in the second member +of (90), that are linear in those velocities. Further, we shall not +put <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> and <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl"> equal to each other, but, corresponding to the +higher approximation, we shall put +<span class="align-center"><img style="vertical-align: -1.552ex; width: 28.33ex; height: 4.153ex;" src="images/577.svg" alt=" " data-tex=" +ds = \sqrt{g_{44}}\, dl + = \left(1 - \frac{\gamma_{44}}{2}\right) dl. +"></span> +From (90) we obtain, at first, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 59.568ex; height: 5.444ex;" src="images/578.svg" alt=" " data-tex=" +\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} + \left(1 + \frac{\gamma_{44}}{2}\right). +\qquad \text{(116)} +"></span> +</p> +<p> +From (101) we get, to the approximation sought for, +<span class="align-center"><img style="vertical-align: -7.642ex; width: 54.085ex; height: 16.414ex;" src="images/579.svg" alt=" " data-tex=" +\left. +\begin{aligned} +-\gamma_{11} = -\gamma_{22} = -\gamma_{33} + &= \gamma_{44} = \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\gamma_{4\alpha} + &= -\frac{i \kappa}{2} + \int \frac{\sigma \dfrac{dx_{\alpha}}{ds}\, dV_{0}}{r}, \\ +\gamma_{\alpha\beta} &= 0, +\end{aligned} +\right\} +\qquad \text{(117)} +"></span> +in which, in (117), <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> denote the space indices only. +<span class="pagenum" id="Page_107">[Pg 107]</span> +</p> +<p> +On the right-hand side of (116) we can replace +1 + <img style="vertical-align: -1.552ex; width: 6.656ex; height: 4.153ex;" src="images/496.svg" alt=" " data-tex="\left(\dfrac{\gamma_{44}}{2}\right)"> by 1 +and <img style="vertical-align: -1.045ex; width: 5.291ex; height: 2.898ex;" src="images/580.svg" alt=" " data-tex="-\Gamma_{\alpha\beta}^{\mu}"> by +<img style="vertical-align: -1.469ex; width: 4.608ex; height: 4.07ex;" src="images/581.svg" alt=" " data-tex="\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right]">. +It is easy to see, in addition, that to this +degree of approximation we must put +<span class="align-center"><img style="vertical-align: -8.407ex; width: 28.058ex; height: 17.946ex;" src="images/582.svg" alt=" " data-tex=" +\begin{align*} +\left[{{\genfrac{}{}{0pt}{}{44}{\mu}}}\right] + &= -\tfrac{1}{2} \frac{\partial \gamma_{44}}{\partial x_{\mu}} + + \frac{\partial \gamma_{4\mu}}{\partial x_{4}},\\ +\left[{{\genfrac{}{}{0pt}{}{\alpha 4}{\mu}}}\right] + &= \tfrac{1}{2} + \left(\frac{\partial \gamma_{4\mu}}{\partial x_{\alpha}} - \frac{\partial \gamma_{4\alpha}}{\partial x_{\mu}}\right), \\ +\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right] &= 0, +\end{align*} +"></span> +in which <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> and <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> denote space indices. +We therefore obtain +from (116), in the usual vector notation, +<span class="align-center"><img style="vertical-align: -8.844ex; width: 52.436ex; height: 18.819ex;" src="images/583.svg" alt=" " data-tex=" +\DeclareMathOperator{\grad}{grad} +\DeclareMathOperator{\rot}{rot} +\left. +\begin{aligned} +\frac{d}{dl}\bigl[(1 + \overline{\sigma}) \mathbf v \bigr] + &= \grad \overline{\sigma} + \frac{\partial \mathfrak A}{\partial l} + [\rot \mathfrak A, \mathbf v], \\ +\overline{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\mathfrak A &= \frac{\kappa}{2 \pi} + \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}. +\end{aligned} +\right\} +\qquad \text{(118)} +"></span> +</p> +<p> +The equations of motion, (118), show now, in fact, that +</p> +<p class="hanging"> +1. The inert mass is proportional to 1 + <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1.742ex;" src="images/584.svg" alt=" " data-tex="\overline{\sigma}"> and therefore +increases when ponderable masses approach the test +body. +</p> +<p class="hanging"> +2. There is an inductive action of accelerated masses, +of the same sign, upon the test body. This is the +term <img style="vertical-align: -1.602ex; width: 3.9ex; height: 4.749ex;" src="images/585.svg" alt=" " data-tex="\dfrac{\partial \mathfrak A}{\partial l}">. +<span class="pagenum" id="Page_108">[Pg 108]</span> +</p> +<p class="hanging"> +3. A material particle, moving perpendicularly to the axis +of rotation inside a rotating hollow body, is deflected in +the sense of the rotation (Coriolis field). The centrifugal +action, mentioned above, inside a rotating hollow +body, also follows from the theory, as has been shown +by Thirring.<a id="FNanchor_18_1"></a><a href="#Footnote_18_1" class="fnanchor">[18]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_18_1"></a><a href="#FNanchor_18_1"><span class="label">[18]</span></a>That the centrifugal action must be inseparably connected with the +existence of the Coriolis field may be recognized, even without calculation, +in the special case of a co-ordinate system rotating uniformly relatively to +an inertial system; our general co-variant equations naturally must apply +to such a case.</p></div> + +<p> +Although all of these effects are inaccessible to experiment, +because <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/503.svg" alt=" " data-tex="\kappa"> is so small, nevertheless they certainly exist according +to the general theory of relativity. We must see in them a +strong support for Mach's ideas as to the relativity of all inertial +actions. If we think these ideas consistently through to the end +we must expect the <i>whole</i> inertia, that is, the <i>whole</i> +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field, to +be determined by the matter of the universe, and not mainly by +the boundary conditions at infinity. +</p> +<p> +For a satisfactory conception of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field of cosmical dimensions, +the fact seems to be of significance that the relative +velocity of the stars is small compared to the velocity of light. +It follows from this that, with a suitable choice of co-ordinates, +<img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/532.svg" alt=" " data-tex="g_{44}"> is nearly constant in the universe, at least, in that part of +the universe in which there is matter. The assumption appears +natural, moreover, that there are stars in all parts of the universe, +so that we may well assume that the inconstancy of <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/532.svg" alt=" " data-tex="g_{44}"> depends +only upon the circumstance that matter is not distributed +continuously, but is concentrated in single celestial bodies and +systems of bodies. If we are willing to ignore these more local +<span class="pagenum" id="Page_109">[Pg 109]</span> +non-uniformities of the density of matter and of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field, in +order to learn something of the geometrical properties of the universe +as a whole, it appears natural to substitute for the actual +distribution of masses a continuous distribution, and furthermore +to assign to this distribution a uniform density <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">. In this +imagined universe all points with space directions will be geometrically +equivalent; with respect to its space extension it will +have a constant curvature, and will be cylindrical with respect +to its <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">-co-ordinate. The possibility seems to be particularly +satisfying that the universe is spatially bounded and thus, in +accordance with our assumption of the constancy of <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, is of +constant curvature, being either spherical or elliptical; for then +the boundary conditions at infinity which are so inconvenient +from the standpoint of the general theory of relativity, may be +replaced by the much more natural conditions for a closed surface. +</p> +<p> +According to what has been said, we are to put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 35.196ex; height: 2.703ex;" src="images/586.svg" alt=" " data-tex=" +ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, +\qquad \text{(119)} +"></span> +in which the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> run from 1 to 3 only. +The <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> +will be such functions of <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> as correspond to +a three-dimensional continuum of constant positive curvature. We must +now investigate whether such an assumption can satisfy the field +equations of gravitation. +</p> +<p> +In order to be able to investigate this, we must first find +what differential conditions the three-dimensional manifold of +constant curvature satisfies. A spherical manifold of three dimensions, +<span class="pagenum" id="Page_110">[Pg 110]</span> +embedded in a Euclidean continuum of four dimensions,<a id="FNanchor_19_1"></a><a href="#Footnote_19_1" class="fnanchor">[19]</a> +is given by the equations +<span class="align-center"><img style="vertical-align: -2.348ex; width: 32.952ex; height: 5.828ex;" src="images/587.svg" alt=" " data-tex=" +\begin{align*} +{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}, \\ +{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. +\end{align*} +"></span> +By eliminating <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">, we get +<span class="align-center"><img style="vertical-align: -2.003ex; width: 56.135ex; height: 5.496ex;" src="images/588.svg" alt=" " data-tex=" +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} + {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}. +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_19_1"></a><a href="#FNanchor_19_1"><span class="label">[19]</span></a>The aid of a fourth space dimension has naturally no significance except +that of a mathematical artifice.</p></div> + +<p> +As far as terms of the third and higher degrees in the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, we +can put, in the neighbourhood of the origin of co-ordinates, +<span class="align-center"><img style="vertical-align: -1.651ex; width: 28.622ex; height: 4.377ex;" src="images/589.svg" alt=" " data-tex=" +ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) + dx_{\mu}\, dx_{\nu}. +"></span> +</p> +<p> +Inside the brackets are the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> of the manifold in the neighbourhood +of the origin. Since the first derivatives of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">, +and therefore also the <img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/556.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\sigma}">, vanish at the origin, the calculation +of the <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}"> for this manifold, by (88), is very simple at the origin. +We have +<span class="align-center"><img style="vertical-align: -1.651ex; width: 24.585ex; height: 4.688ex;" src="images/590.svg" alt=" " data-tex=" +R_{\mu\nu} = -\frac{2}{a^{2}} \delta_{\mu\nu} + = \frac{2}{a^{2}} g_{\mu\nu}. +"></span> +</p> +<p> +Since the relation +<img style="vertical-align: -1.651ex; width: 12.994ex; height: 4.688ex;" src="images/591.svg" alt=" " data-tex="R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}"> is universally co-variant, +and since all points of the manifold are geometrically equivalent, +this relation holds for every system of co-ordinates, and +everywhere in the manifold. In order to avoid confusion with +<span class="pagenum" id="Page_111">[Pg 111]</span> +the four-dimensional continuum, we shall, in the following, designate +quantities that refer to the three-dimensional continuum +by Greek letters, and put +<span class="align-center"><img style="vertical-align: -1.651ex; width: 25.267ex; height: 4.688ex;" src="images/592.svg" alt=" " data-tex=" +P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}. +\qquad \text{(120)} +"></span> +</p> +<p> +We now proceed to apply the field equations (96) to our special +case. From (119) we get for the four-dimensional manifold, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 44.083ex; height: 5.428ex;" src="images/593.svg" alt=" " data-tex=" +\left. +\begin{aligned} +R_{\mu\nu} &= P_{\mu\nu} \quad \text{for the indices 1 to 3}, \\ +R_{14} &= R_{24} = R_{34} = R_{44} = 0. +\end{aligned} +\right\} +\qquad \text{(121)} +"></span> +</p> +<p> +For the right-hand side of (96) we have to consider the energy +tensor for matter distributed like a cloud of dust. According to +what has gone before we must therefore put +<span class="align-center"><img style="vertical-align: -1.575ex; width: 17.527ex; height: 4.87ex;" src="images/594.svg" alt=" " data-tex=" +T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds} +"></span> +specialized for the case of rest. But in addition, we shall add +a pressure term that may be physically established as follows. +Matter consists of electrically charged particles. On the basis +of Maxwell's theory these cannot be conceived of as electromagnetic +fields free from singularities. In order to be consistent +with the facts, it is necessary to introduce energy terms, not +contained in Maxwell's theory, so that the single electric particles +may hold together in spite of the mutual repulsions between +their elements, charged with electricity of one sign. For the sake +of consistency with this fact, Poincaré has assumed a pressure +<span class="pagenum" id="Page_112">[Pg 112]</span> +to exist inside these particles which balances the electrostatic +repulsion. It cannot, however, be asserted that this pressure +vanishes outside the particles. We shall be consistent with this +circumstance if, in our phenomenological presentation, we add +a pressure term. This must not, however, be confused with a +hydrodynamical pressure, as it serves only for the energetic presentation +of the dynamical relations inside matter. In this sense +we put +<span class="align-center"><img style="vertical-align: -1.575ex; width: 39.787ex; height: 4.835ex;" src="images/595.svg" alt=" " data-tex=" +T_{\mu\nu} + = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + - g_{\mu\nu} p. +\qquad \text{(122)} +"></span> +</p> +<p> +In our special case we have, therefore, to put +<span class="align-center"><img style="vertical-align: -3.626ex; width: 38.187ex; height: 8.383ex;" src="images/596.svg" alt=" " data-tex=" +\begin{align*} +T_{\mu\nu} + &= \gamma_{\mu\nu} p \quad \text{(for}\,\,\mu\,\, \text{and}\,\,\,\nu \,\,\text{from 1 to 3)}, \\ +T_{44} &= \sigma - p, \\ +T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p + = \sigma - 4p. +\end{align*} +"></span> +Observing that the field equation (96) may be written in the +form +<span class="align-center"><img style="vertical-align: -0.781ex; width: 24.742ex; height: 2.737ex;" src="images/597.svg" alt=" " data-tex=" +R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T), +"></span> +we get from (96) the equations, +<span class="align-center"><img style="vertical-align: -4.194ex; width: 25.503ex; height: 9.519ex;" src="images/598.svg" alt=" " data-tex=" +\begin{align*} ++\frac{2}{a^{2}} \gamma_{\mu\nu} + &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}, \\ +0 &= -\kappa \left(\frac{\sigma}{2} + p\right). +\end{align*} +"></span> +From this follows +<span class="align-center"><img style="vertical-align: -4.585ex; width: 22.431ex; height: 10.301ex;" src="images/599.svg" alt=" " data-tex=" +\left. +\begin{aligned} +p &= -\frac{\sigma}{2}, \\ +a &= \sqrt{\frac{2}{\kappa\sigma}}. +\end{aligned} +\right\} +\qquad \text{(123)} +"></span> +</p> +<p> +If the universe is quasi-Euclidean, and its radius of curvature +therefore infinite, then σ would vanish. But it is improbable that +<span class="pagenum" id="Page_113">[Pg 113]</span> +the mean density of matter in the universe is actually zero; this +is our third argument against the assumption that the universe +is quasi-Euclidean. Nor does it seem possible that our hypothetical +pressure can vanish; the physical nature of this pressure can +be appreciated only after we have a better theoretical knowledge +of the electromagnetic field. According to the second of equations +(123) the radius, <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/75.svg" alt=" " data-tex="a">, of the universe is determined in terms +of the total mass, <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M">, of matter, by the equation +<span class="align-center"><img style="vertical-align: -1.654ex; width: 19.575ex; height: 4.728ex;" src="images/600.svg" alt=" " data-tex=" +a = \frac{M\kappa}{4\pi^{2}}. +\qquad \text{(124)} +"></span> +The complete dependence of the geometrical upon the physical +properties becomes clearly apparent by means of this equation. +</p> +<p> +Thus we may present the following arguments against the +conception of a space-infinite, and for the conception of a space-bounded, +universe:— +</p> +<p> +1. From the standpoint of the theory of relativity, the condition +for a closed surface is very much simpler than the corresponding +boundary condition at infinity of the quasi-Euclidean +structure of the universe. +</p> +<p> +2. The idea that Mach expressed, that inertia depends upon +the mutual action of bodies, is contained, to a first approximation, +in the equations of the theory of relativity; it follows +from these equations that inertia depends, at least in part, upon +mutual actions between masses. As it is an unsatisfactory assumption +to make that inertia depends in part upon mutual +actions, and in part upon an independent property of space, +Mach's idea gains in probability. But this idea of Mach's corresponds +only to a finite universe, bounded in space, and not to a +quasi-Euclidean, infinite universe. From the standpoint of epistemology +<span class="pagenum" id="Page_114">[Pg 114]</span> +it is more satisfying to have the mechanical properties +of space completely determined by matter, and this is the case +only in a space-bounded universe. +</p> +<p> +3. An infinite universe is possible only if the mean density of +matter in the universe vanishes. Although such an assumption +is logically possible, it is less probable than the assumption that +there is a finite mean density of matter in the universe. +<span class="pagenum" id="Page_115">[Pg 115]</span> +</p> +</div> + +<p><br><br></p> + +<div class='chapter'> +<h2 title="INDEX"><a id="INDEX">INDEX</a></h2> + +<p>A</p> +<p class="nind"> +Accelerated masses, inductive<br> +<span style="margin-left: 1em;">action of, <a href="#Page_108">108</a></span><br> +Addition and subtraction of<br> +<span style="margin-left: 1em;">tensors, <a href="#Page_14">14</a></span><br> +—theorem of velocities, <a href="#Page_38">38</a> +</p> +<p><br></p> +<p>B</p> +<p class="nind"> +Biot-Savart force, <a href="#Page_44">44</a> +</p> +<p><br></p> +<p>C</p> +<p class="nind"> +Centrifugal force, <a href="#Page_64">64</a><br> +Clocks, moving, <a href="#Page_38">38</a><br> +Compressible viscous fluid, <a href="#Page_22">22</a><br> +Concept of space, <a href="#Page_3">3</a><br> +—time, <a href="#Page_28">28</a><br> +Conditions of orthogonality, <a href="#Page_7">7</a><br> +Congruence, theorems of, <a href="#Page_3">3</a><br> +Conservation principles, <a href="#Page_54">54</a><br> +Continuum, four-dimensional, <a href="#Page_31">31</a><br> +Contraction of tensors, <a href="#Page_14">14</a><br> +Contra-variant vectors, <a href="#Page_69">69</a><br> +—tensors, <a href="#Page_71">71</a><br> +Co-ordinates, preferred systems<br> +<span style="margin-left: 1em;">of, <a href="#Page_8">8</a></span><br> +Co-variance of equation of<br> +<span style="margin-left: 1em;">continuity, <a href="#Page_21">21</a></span><br> +Co-variant, <a href="#Page_12">12</a> <i>et seq.</i><br> +—vector, <a href="#Page_68">68</a><br> +Criticism of principle of inertia, <a href="#Page_62">62</a><br> +Criticisms of theory of<br> +<span style="margin-left: 1em;">relativity, <a href="#Page_29">29</a></span><br> +Curvilinear co-ordinates, <a href="#Page_65">65</a> +</p> +<p><br></p> +<p>D</p> +<p class="nind"> +Differentiation of tensors, <a href="#Page_73">73</a>, <a href="#Page_76">76</a><br> +Displacement of spectral lines, <a href="#Page_97">97</a><br> +</p> +<p><br></p> +<p>E</p> +<p class="nind"> +Energy and mass, <a href="#Page_45">45</a>, <a href="#Page_49">49</a><br> +—tensor of electromagnetic<br> +<span style="margin-left: 1em;">field, <a href="#Page_50">50</a></span><br> +—of matter, <a href="#Page_54">54</a><br> +Equation of continuity, co-variance<br> +<span style="margin-left: 1em;">of, <a href="#Page_21">21</a></span><br> +Equations of motion of material<br> +<span style="margin-left: 1em;">particle, <a href="#Page_50">50</a></span><br> +Equivalence of mass and<br> +<span style="margin-left: 1em;">energy, <a href="#Page_49">49</a></span><br> +Equivalent spaces of reference, <a href="#Page_25">25</a><br> +Euclidean geometry, <a href="#Page_4">4</a><br> +</p> +<p><br></p> +<p>F</p> +<p class="nind"> +Finiteness of universe, <a href="#Page_105">105</a><br> +Fizeau, <a href="#Page_28">28</a><br> +Four-dimensional continuum, <a href="#Page_31">31</a><br> +Four-vector, <a href="#Page_41">41</a><br> +Fundamental tensor, <a href="#Page_71">71</a> +</p> +<p><br></p> +<p>G</p> +<p class="nind"> +Galilean regions, <a href="#Page_62">62</a><br> +—transformation, <a href="#Page_27">27</a><br> +Gauss, <a href="#Page_65">65</a><br> +Geodetic lines, <a href="#Page_82">82</a><br> +Geometry, Euclidean, <a href="#Page_4">4</a><br> +Gravitation constant, <a href="#Page_95">95</a><br> +Gravitational mass, <a href="#Page_60">60</a> +</p> +<p><br></p> +<p>H</p> +<p class="nind"> +Homogeneity of space, <a href="#Page_17">17</a><br> +Hydrodynamical equations, <a href="#Page_54">54</a><br> +Hypotheses of pre-relativity<br> +<span style="margin-left: 1em;">physics, <a href="#Page_73">73</a></span> +</p> +<p><br></p> +<p>I</p> +<p class="nind"> +Inductive action of accelerated<br> +<span style="margin-left: 1em;">masses, <a href="#Page_108">108</a></span><br> +Inert and gravitational mass, equality<br> +<span style="margin-left: 1em;">of, <a href="#Page_60">60</a></span><br> +Invariant, <a href="#Page_9">9</a> <i>et seq.</i><br> +Isotropy of space, <a href="#Page_17">17</a><br> +</p> +<p><br></p> +<p>K</p> +<p class="nind"> +Kaluza, <a href="#Page_104">104</a> +</p> +<p><br></p> +<p>L</p> +<p class="nind"> +Levi-Civita, <a href="#Page_73">73</a><br> +Light-cone, <a href="#Page_41">41</a><br> +Light ray, path of, <a href="#Page_98">98</a><br> +Light-time, <a href="#Page_33">33</a><br> +Linear orthogonal<br> +<span style="margin-left: 1em;">transformation, <a href="#Page_7">7</a></span><br> +Lorentz electromotive force, <a href="#Page_44">44</a><br> +—transformation, <a href="#Page_31">31</a><br> +</p> +<p><br></p> +<p>M</p> +<p class="nind"> +Mach, <a href="#Page_59">59</a>, <a href="#Page_105">105</a>, <a href="#Page_106">106</a>, <a href="#Page_109">109</a>, <a href="#Page_114">114</a><br> +Mass and Energy, <a href="#Page_45">45</a>, <a href="#Page_49">49</a><br> +—equality of gravitational and<br> +<span style="margin-left: 1em;">inert, <a href="#Page_60">60</a></span><br> +—gravitational, <a href="#Page_60">60</a><br> +Maxwell's equations, <a href="#Page_23">23</a><br> +Mercury, perihelion of, <a href="#Page_99">99</a>, <a href="#Page_103">103</a><br> +Michelson and Morley, <a href="#Page_28">28</a><br> +Minkowski, <a href="#Page_32">32</a><br> +Motion of particle, equations of, <a href="#Page_50">50</a><br> +Moving measuring rods and<br> +<span style="margin-left: 1em;">clocks, <a href="#Page_38">38</a></span><br> +Multiplication of tensors, <a href="#Page_14">14</a><br> +</p> +<p><br></p> +<p>N</p> +<p class="nind"> +Newtonian gravitation<br> +<span style="margin-left: 1em;">constant, <a href="#Page_95">95</a></span><br> +</p> +<p><br></p> +<p>O</p> +<p class="nind"> +Operations on tensors, <a href="#Page_13">13</a> <i>et seq.</i><br> +Orthogonal transformations, linear, <a href="#Page_7">7</a><br> +Orthogonality, conditions of, <a href="#Page_7">7</a><br> +</p> +<p><br></p> +<p>P</p> +<p class="nind"> +Path of light ray, <a href="#Page_98">98</a><br> +Perihelion of Mercury, <a href="#Page_99">99</a>, <a href="#Page_103">103</a><br> +Poisson's equation, <a href="#Page_87">87</a><br> +Preferred systems of<br> +<span style="margin-left: 1em;">co-ordinates, <a href="#Page_8">8</a></span><br> +Pre-relativity physics, hypotheses<br> +<span style="margin-left: 1em;">of, <a href="#Page_26">26</a></span><br> +Principle of equivalence, <a href="#Page_61">61</a><br> +—inertia, criticism of, <a href="#Page_62">62</a><br> +Principles of conservation, <a href="#Page_54">54</a><br> +</p> +<p><br></p> +<p>R</p> +<p class="nind"> +Radius of Universe, <a href="#Page_113">113</a><br> +Rank of tensor, <a href="#Page_13">13</a><br> +Ray of light, path of, <a href="#Page_98">98</a><br> +Reference, space of, <a href="#Page_3">3</a><br> +Riemann, <a href="#Page_68">68</a><br> +—tensor, <a href="#Page_79">79</a>, <a href="#Page_82">82</a>, <a href="#Page_105">105</a><br> +Rods (measuring) and clocks in<br> +<span style="margin-left: 1em;">motion, <a href="#Page_38">38</a></span><br> +Rotation, <a href="#Page_63">63</a> +</p> +<p><br></p> +<p>S</p> +<p class="nind"> +Simultaneity, <a href="#Page_17">17</a>, <a href="#Page_29">29</a><br> +Sitter, <a href="#Page_28">28</a><br> +Skew-symmetrical tensor, <a href="#Page_15">15</a><br> +Solar Eclipse expedition (1919), <a href="#Page_99">99</a><br> +Space, concept of, <a href="#Page_2">2</a><br> +—Homogeneity of, <a href="#Page_17">17</a><br> +—Isotropy of, <a href="#Page_17">17</a><br> +Spaces of reference, <a href="#Page_3">3</a><br> +—equivalence of, <a href="#Page_25">25</a><br> +Special Lorentz transformation, <a href="#Page_34">34</a><br> +Spectral lines, displacement of, <a href="#Page_97">97</a><br> +Straightest lines, <a href="#Page_82">82</a><br> +Stress tensor, <a href="#Page_22">22</a><br> +Symmetrical tensor, <a href="#Page_15">15</a><br> +Systems of co-ordinates, <br> +preferred, <a href="#Page_8">8</a><br> +</p> +<p><br></p> +<p>T</p> +<p class="nind"> +Tensor, <a href="#Page_12">12</a> <i>et seq.</i>, <a href="#Page_68">68</a> <i>et seq.</i><br> +—Addition and subtraction of, <a href="#Page_14">14</a><br> +—Contraction of, <a href="#Page_14">14</a><br> +—Fundamental, <a href="#Page_71">71</a><br> +—Multiplication of, <a href="#Page_14">14</a><br> +—operations, <a href="#Page_13">13</a> <i>et seq.</i><br> +—Rank of, <a href="#Page_13">13</a><br> +—Symmetrical and<br> +<span style="margin-left: 1em;">Skew-symmetrical, <a href="#Page_15">15</a></span><br> +Tensors, formation by<br> +<span style="margin-left: 1em;">differentiation, <a href="#Page_73">73</a></span><br> +Theorem for addition of<br> +<span style="margin-left: 1em;">velocities, <a href="#Page_38">38</a></span><br> +Theorems of congruence, <a href="#Page_3">3</a><br> +Theory of relativity, criticisms<br> +<span style="margin-left: 1em;">of, <a href="#Page_29">29</a></span><br> +Thirring, <a href="#Page_109">109</a><br> +Time-concept, <a href="#Page_28">28</a><br> +Time-space concept, <a href="#Page_31">31</a><br> +Transformation, Galilean, <a href="#Page_27">27</a><br> +—Linear orthogonal, <a href="#Page_7">7</a><br> +</p> +<p><br></p> +<p>U</p> +<p class="nind"> +Universe, Finiteness of, <a href="#Page_105">105</a><br> +—Radius of, <a href="#Page_113">113</a><br> +</p> +<p><br></p> +<p>V</p> +<p class="nind"> +Vector, co-variant, <a href="#Page_69">69</a><br> +—contra-variant, <a href="#Page_69">69</a><br> +Velocities, addition theorem of, <a href="#Page_38">38</a><br> +Viscous compressible fluid, <a href="#Page_22">22</a><br> +</p> +<p><br></p> +<p>W</p> +<p class="nind"> +Weyl, <a href="#Page_69">73</a>, <a href="#Page_99">99</a>, <a href="#Page_104">104</a> +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<p class="center"> +PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN +</p> +</div> + +<p><br><br></p> + +<div class='chapter'> +<div class="transnote"> +<p class="center"><b>TRANSCRIBER'S NOTES</b></p> + +<p> +This ebook was produced using OCR text provided by Northeastern +University's Snell Library through the Internet Archive. +</p> +<p> +Minor typographical corrections and presentational changes have been +made without comment. +</p></div> +</div> + +<div>*** END OF THE PROJECT GUTENBERG EBOOK 36276 ***</div> +</body> + +</html> + + + + diff --git a/36276-h/images/1.svg b/36276-h/images/1.svg new file mode 100644 index 0000000..c5c203e --- /dev/null +++ b/36276-h/images/1.svg @@ -0,0 +1 @@ +<svg version="1.1" style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="759px" height="683px" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-304-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 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You may copy it, give it away or re-use it under the terms of the +Project Gutenberg License included with this ebook or online at +<a class="reference external" href="https://www.gutenberg.org">www.gutenberg.org</a>. +If you are not located in the United States, you’ll have to check the laws of the +country where you are located before using this eBook. +</div> +<div class="container" id="pg-machine-header"> +<p><strong>Title:</strong> The meaning of relativity<br> Four lectures delivered at Princeton University, May, 1921</p> +<div id='pg-header-authlist'> +<p><strong>Author:</strong> Albert Einstein</p> +<p><strong>Translator:</strong> Edwin Plimpton Adams</p> +</div> +<p><strong>Release Date:</strong> February 19, 2023 [eBook #36276]</p> +<p><strong>Language:</strong> English</p> +<p><strong>Original Publication:</strong> United States: Princeton University Press, Princeton, NJ, 1922</p> +<p><strong>Credit:</strong> Andrew D. Hwang, updated for HTML+SVG by Laura Natal Rodrigues</p> +</div> +<div id='pg-start-separator'> +<span>*** START OF THE PROJECT GUTENBERG EBOOK THE MEANING OF RELATIVITY ***</span> +</div> +</section> + +<div class="figcenter" style="width: 500px;"> +<img src="images/cover.jpg" width="500" alt="500"> +</div> + + +<h1>THE MEANING OF<br> +RELATIVITY</h1> + +<p><br><br></p> + +<p class="center"><b>FOUR LECTURES DELIVERED AT +PRINCETON UNIVERSITY, MAY, 1921</b></p> + +<p><br><br></p> + +<p class="center"><b>BY</b></p> + +<h2>ALBERT EINSTEIN</h2> + +<p><br><br></p> + +<p class="center"><b>WITH FOUR DIAGRAMS</b></p> + +<p><br><br></p> + +<p class="center"><b>PRINCETON</b><br> +<b>PRINCETON UNIVERSITY PRESS</b><br> +<b>1923</b></p> + +<p><br><br></p> + +<p class="center"><i>Copyright 1922 +Princeton University Press +Published 1922</i></p> + +<p><br><br></p> + +<p class="hanging2"> +NOTE.—The translation of these lectures into English +was made by EDWIN PLIMPTON ADAMS, Professor +of Physics in Princeton University +</p> + +<p><br><br><br></p> + +<h2>CONTENTS</h2> +<p class="nind"> +LECTURE I<br> +<a href="#chap01">SPACE AND TIME IN PRE-RELATIVITY PHYSICS</a><br> +LECTURE II<br> +<a href="#chap02">THE THEORY OF SPECIAL RELATIVITY</a><br> +LECTURE III<br> +<a href="#chap03">THE GENERAL THEORY OF RELATIVITY</a><br> +LECTURE IV<br> +<a href="#chap04">THE GENERAL THEORY OF RELATIVITY (<i>continued</i>)</a><br> +<a href="#">INDEX</a></p> + +<p><br><br><br></p> + +<h3>THE MEANING OF RELATIVITY</h3> + +<p><br><br><br></p> + +<h3><a id="chap01"></a>LECTURE I +<br><br> +SPACE AND TIME IN PRE-RELATIVITY +PHYSICS</h3> + +<p class="nind"> +<span class="dropcap">T</span>HE theory of relativity is intimately connected with the theory +of space and time. I shall therefore begin with a brief investigation +of the origin of our ideas of space and time, although in +doing so I know that I introduce a controversial subject. The +object of all science, whether natural science or psychology, is +to co-ordinate our experiences and to bring them into a logical +system. How are our customary ideas of space and time related +to the character of our experiences? +</p> +<p> +The experiences of an individual appear to us arranged in a +series of events; in this series the single events which we remember +appear to be ordered according to the criterion of "earlier" +and "later," which cannot be analysed further. There exists, +therefore, for the individual, an I-time, or subjective time. This +in itself is not measurable. I can, indeed, associate numbers with +the events, in such a way that a greater number is associated +with the later event than with an earlier one; but the nature of +this association may be quite arbitrary. This association I can +define by means of a clock by comparing the order of events furnished +by the clock with the order of the given series of events. +We understand by a clock something which provides a series of +events which can be counted, and which has other properties of +which we shall speak later. +<span class="pagenum" id="Page_1">[Pg 1]</span> +</p> +<p> +By the aid of speech different individuals can, to a certain +extent, compare their experiences. In this way it is shown that +certain sense perceptions of different individuals correspond to +each other, while for other sense perceptions no such correspondence +can be established. We are accustomed to regard as real +those sense perceptions which are common to different individuals, +and which therefore are, in a measure, impersonal. The natural +sciences, and in particular, the most fundamental of them, +physics, deal with such sense perceptions. The conception of +physical bodies, in particular of rigid bodies, is a relatively constant +complex of such sense perceptions. A clock is also a body, +or a system, in the same sense, with the additional property that +the series of events which it counts is formed of elements all of +which can be regarded as equal. +</p> +<p> +The only justification for our concepts and system of concepts +is that they serve to represent the complex of our experiences; +beyond this they have no legitimacy. I am convinced that +the philosophers have had a harmful effect upon the progress +of scientific thinking in removing certain fundamental concepts +from the domain of empiricism, where they are under our control, +to the intangible heights of the <i>a priori</i>. For even if it should +appear that the universe of ideas cannot be deduced from experience +by logical means, but is, in a sense, a creation of the +human mind, without which no science is possible, nevertheless +this universe of ideas is just as little independent of the nature +of our experiences as clothes are of the form of the human body. +This is particularly true of our concepts of time and space, which +physicists have been obliged by the facts to bring down from the +Olympus of the <i>a priori</i> in order to adjust them and put them +in a serviceable condition. +<span class="pagenum" id="Page_2">[Pg 2]</span> +</p> +<p> +We now come to our concepts and judgments of space. It +is essential here also to pay strict attention to the relation of +experience to our concepts. It seems to me that Poincaré clearly +recognized the truth in the account he gave in his book, "La +Science et l'Hypothèse." Among all the changes which we can +perceive in a rigid body those are marked by their simplicity +which can be made reversibly by an arbitrary motion of the +body; Poincaré calls these, changes in position. By means of +simple changes in position we can bring two bodies into contact. +The theorems of congruence, fundamental in geometry, have to +do with the laws that govern such changes in position. For the +concept of space the following seems essential. We can form new +bodies by bringing bodies <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">, <img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">, ... up to body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">; we say that +we <i>continue</i> body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">. We can continue body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> in such a way that +it comes into contact with any other body, <img style="vertical-align: 0; width: 1.928ex; height: 1.545ex;" src="images/4.svg" alt=" " data-tex="X">. The ensemble of +all continuations of body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> we can designate as the "space of +the body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." Then it is true that all bodies are in the "space of +the (arbitrarily chosen) body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." In this sense we cannot speak +of space in the abstract, but only of the "space belonging to a +body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." The earth's crust plays such a dominant rôle in our +daily life in judging the relative positions of bodies that it has +led to an abstract conception of space which certainly cannot be +defended. In order to free ourselves from this fatal error we shall +speak only of "bodies of reference," or "space of reference." It +was only through the theory of general relativity that refinement +of these concepts became necessary, as we shall see later. +</p> +<p> +I shall not go into detail concerning those properties of the +space of reference which lead to our conceiving points as elements +of space, and space as a continuum. Nor shall I attempt +to analyse further the properties of space which justify the conception +<span class="pagenum" id="Page_3">[Pg 3]</span> +of continuous series of points, or lines. If these concepts +are assumed, together with their relation to the solid bodies of +experience, then it is easy to say what we mean by the three-dimensionality +of space; to each point three numbers, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> +(co-ordinates), may be associated, in such a way that this association +is uniquely reciprocal, and that <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}"> and <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> vary +continuously when the point describes a continuous series of points +(a line). +</p> +<p> +It is assumed in pre-relativity physics that the laws of the +orientation of ideal rigid bodies are consistent with Euclidean +geometry. What this means may be expressed as follows: Two +points marked on a rigid body form an <i>interval</i>. Such an interval +can be oriented at rest, relatively to our space of reference, in +a multiplicity of ways. If, now, the points of this space can +be referred to co-ordinates <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, in such a way that the +differences of the co-ordinates, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/8.svg" alt=" " data-tex="\Delta x_{1}">, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/9.svg" alt=" " data-tex="\Delta x_{2}">, +<img style="vertical-align: -0.375ex; width: 4.166ex; height: 1.994ex;" src="images/10.svg" alt=" " data-tex="\Delta x_{3}">, of the two ends +of the interval furnish the same sum of squares, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 34.482ex; height: 2.633ex;" src="images/11.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}, \qquad\text{(1)} +"></span> +for every orientation of the interval, then the space of reference +is called Euclidean, and the co-ordinates Cartesian.<a id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> It is +sufficient, indeed, to make this assumption in the limit for an +infinitely small interval. Involved in this assumption there are +some which are rather less special, to which we must call attention +on account of their fundamental significance. In the first +place, it is assumed that one can move an ideal rigid body in an +arbitrary manner. In the second place, it is assumed that the behaviour +of ideal rigid bodies towards orientation is independent +<span class="pagenum" id="Page_4">[Pg 4]</span> +of the material of the bodies and their changes of position, in the +sense that if two intervals can once be brought into coincidence, +they can always and everywhere be brought into coincidence. +Both of these assumptions, which are of fundamental importance +for geometry and especially for physical measurements, +naturally arise from experience; in the theory of general relativity +their validity needs to be assumed only for bodies and spaces +of reference which are infinitely small compared to astronomical +dimensions. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a>This relation must hold for an arbitrary choice of the origin and of the +direction +(ratios <img style="vertical-align: -0.375ex; width: 16.271ex; height: 1.994ex;" src="images/12.svg" alt=" " data-tex="\Delta x_{1}: \Delta x_{2}: \Delta x_{3}">) +of the interval.</p></div> + +<p> +The quantity <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> we call the length of the interval. In order +that this may be uniquely determined it is necessary to fix arbitrarily +the length of a definite interval; for example, we can put +it equal to 1 (unit of length). Then the lengths of all other intervals +may be determined. If we make the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> linearly dependent +upon a parameter <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda">, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 14.299ex; height: 2.009ex;" src="images/16.svg" alt=" " data-tex=" +x_{\nu} = a_{\nu} + \lambda b_{\nu}, +"></span> +we obtain a fine which has all the properties of the straight +lines of the Euclidean geometry. In particular, it easily follows +that by laying off <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> times the interval <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> upon a straight fine, an +interval of length <img style="vertical-align: -0.025ex; width: 4.053ex; height: 1.025ex;" src="images/18.svg" alt=" " data-tex="n·s"> is obtained. A length, therefore, means +the result of a measurement carried out along a straight line by +means of a unit measuring rod. It has a significance which is as +independent of the system of co-ordinates as that of a straight +line, as will appear in the sequel. +</p> +<p> +We come now to a train of thought which plays an analogous +role in the theories of special and general relativity. We ask +the question: besides the Cartesian co-ordinates which we have +used are there other equivalent co-ordinates? An interval has +<span class="pagenum" id="Page_5">[Pg 5]</span> +a physical meaning which is independent of the choice of co-ordinates; +and so has the spherical surface which we obtain as +the locus of the end points of all equal intervals that we lay off +from an arbitrary point of our space of reference. If <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> as well +as <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> (<img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> from 1 to 3) are Cartesian co-ordinates of our space +of reference, then the spherical surface will be expressed in our +two systems of co-ordinates by the equations +<span class="align-center"><img style="vertical-align: -3.006ex; width: 31.357ex; height: 7.142ex;" src="images/21.svg" alt=" " data-tex=" +\begin{align*} +\sum {\Delta x_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2)} \\ +\sum {{\Delta x'}_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2a)} +\end{align*} +"></span> +How must the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> be expressed in terms of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> in order that +equations (2) and (2a) may be equivalent to each other? Regarding +the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> expressed as functions of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, we can write, +by Taylor's theorem, for small values of the <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"><img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, +<span class="align-center"><img style="vertical-align: -3.06ex; width: 52.322ex; height: 6.477ex;" src="images/23.svg" alt=" " data-tex=" +{\Delta x'}_{\nu} + = \sum_{\alpha} \frac{\partial {x'}_{\nu}}{\partial x_{\alpha}}\, \Delta x_{\alpha} + + \frac{1}{2} \sum_{\alpha \text{,} \beta} + \frac{\partial^{2} {x'}_{\nu}}{\partial x_{\alpha}\, \partial x_{\beta}}\, + \Delta x_{\alpha}\, \Delta x_{\beta}.\ldots +"></span> +If we substitute (2a) in this equation and compare with (1), +we see that the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> must be linear functions of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If we +therefore put +<span class="align-center"><img style="vertical-align: -4.542ex; width: 42.282ex; height: 10.216ex;" src="images/24.svg" alt=" " data-tex=" +\begin{align*} +& & {x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}, +\qquad & &\text{(3)}\\ +&\text{or} & \Delta {x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}, +\qquad & &\text{(3a)} +\end{align*}"></span> +<span class="pagenum" id="Page_6">[Pg 6]</span> +then the equivalence of equations (2) and (2a) is expressed in +the form +<span class="align-center"><img style="vertical-align: -1.018ex; width: 59.118ex; height: 3.296ex;" src="images/25.svg" alt=" " data-tex=" +\sum{\Delta x'_{\nu}}^{2} = \lambda \sum{\Delta x_{\nu}}^{2} + \qquad\text{(\(\lambda\) independent of \(\Delta x_{\nu}\)).} +\qquad\text{(2b)} +"></span> +It therefore follows that <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> must be a constant. If we put <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> = 1, +(2b) and (3a) furnish the conditions +<span class="align-center"><img style="vertical-align: -2.602ex; width: 24.147ex; height: 4.751ex;" src="images/26.svg" alt=" " data-tex=" +\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}, +\qquad\text{(4)} +"></span> +in which <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> = 1, or <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> = 0, according <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> or +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> ≠<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">. The +conditions (4) are called the conditions of orthogonality, and the +transformations (3), (4), linear orthogonal transformations. If +we stipulate that <img style="vertical-align: -0.566ex; width: 13.034ex; height: 2.633ex;" src="images/30.svg" alt=" " data-tex="s^2 = \sum {\Delta x_{\nu}}^2"> shall be equal to the square of +the length in every system of co-ordinates, and if we always measure +with the same unit scale, then <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> must be equal to 1. Therefore +the linear orthogonal transformations are the only ones by +means of which we can pass from one Cartesian system of co-ordinates +in our space of reference to another. We see that in +applying such transformations the equations of a straight line +become equations of a straight line. Reversing equations (3a) +by multiplying both sides by <img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/31.svg" alt=" " data-tex="b_{\nu\beta}"> and summing for all +the <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">'s, we obtain +<span class="align-center"><img style="vertical-align: -2.912ex; width: 58.403ex; height: 5.061ex;" src="images/32.svg" alt=" " data-tex=" +\sum_{\nu} b_{\nu\beta}\, {\Delta x'}_{\nu} + = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} + = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} + = \Delta x_{\beta} +\qquad\text{(5)} +"></span> +The same coefficients, <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/33.svg" alt=" " data-tex="b">, also determine the inverse substitution +of <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"><img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. Geometrically, <img style="vertical-align: -0.357ex; width: 3.03ex; height: 1.927ex;" src="images/34.svg" alt=" " data-tex="b_{\nu\alpha}"> is the cosine of the angle between +the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> axis and the <img style="vertical-align: -0.357ex; width: 2.506ex; height: 1.357ex;" src="images/35.svg" alt=" " data-tex="x_{\alpha}"> axis. +<span class="pagenum" id="Page_7">[Pg 7]</span> +</p> +<p> +To sum up, we can say that in the Euclidean geometry +there are (in a given space of reference) preferred systems of +co-ordinates, the Cartesian systems, which transform into each +other by linear orthogonal transformations. The distance <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> between +two points of our space of reference, measured by a measuring +rod, is expressed in such co-ordinates in a particularly +simple manner. The whole of geometry may be founded upon +this conception of distance. In the present treatment, geometry +is related to actual things (rigid bodies), and its theorems are +statements concerning the behaviour of these things, which may +prove to be true or false. +</p> +<p> +One is ordinarily accustomed to study geometry divorced +from any relation between its concepts and experience. There +are advantages in isolating that which is purely logical and independent +of what is, in principle, incomplete empiricism. This +is satisfactory to the pure mathematician. He is satisfied if he +can deduce his theorems from axioms correctly, that is, without +errors of logic. The question as to whether Euclidean geometry +is true or not does not concern him. But for our purpose it +is necessary to associate the fundamental concepts of geometry +with natural objects; without such an association geometry is +worthless for the physicist. The physicist is concerned with the +question as to whether the theorems of geometry are true or +not. That Euclidean geometry, from this point of view, affirms +something more than the mere deductions derived logically from +definitions may be seen from the following simple consideration. +</p> +<p> +Between <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> points of space there are <img style="vertical-align: -1.552ex; width: 9.368ex; height: 4.855ex;" src="images/36.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2}"> distances, <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}">; +between these and the <img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/38.svg" alt=" " data-tex="3n"> co-ordinates we have the relations +<span class="align-center"><img style="vertical-align: -0.904ex; width: 43.644ex; height: 3.274ex;" src="images/39.svg" alt=" " data-tex=" +s_{\mu\nu}^{2} + = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + + .\dots +"></span> +<span class="pagenum" id="Page_8">[Pg 8]</span> +</p> +<p> +From these <img style="vertical-align: -1.552ex; width: 9.368ex; height: 4.855ex;" src="images/36.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2}"> equations the <img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/38.svg" alt=" " data-tex="3n"> co-ordinates may be +eliminated, and from this elimination at least <img style="vertical-align: -1.552ex; width: 14.622ex; height: 4.855ex;" src="images/40.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2} - 3n"> +equations in the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}">, will result.<a id="FNanchor_2_1"></a><a href="#Footnote_2_1" class="fnanchor">[2]</a> +Since the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}"> are measurable +quantities, and by definition are independent of each other, these +relations between the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}"> are not necessary <i>a priori</i>. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_2_1"></a><a href="#FNanchor_2_1"><span class="label">[2]</span></a>In reality there are <img style="vertical-align: -1.552ex; width: 18.519ex; height: 4.855ex;" src="images/41.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2} - 3n + 6"> equations.</p></div> + +<p> +From the foregoing it is evident that the equations of transformation +(3), (4) have a fundamental significance in Euclidean +geometry, in that they govern the transformation from one +Cartesian system of co-ordinates to another. The Cartesian +systems of co-ordinates are characterized by the property that +in them the measurable distance between two points, <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, is +expressed by the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 14.541ex; height: 3.167ex;" src="images/42.svg" alt=" " data-tex=" +s^{2} = \sum {\Delta x_{\nu}}^{2}. +"></span> +</p> +<p> +If <img style="vertical-align: -0.579ex; width: 3.756ex; height: 2.125ex;" src="images/43.svg" alt=" " data-tex="K_{x_{\nu}}"> and <img style="vertical-align: -0.799ex; width: 3.756ex; height: 2.516ex;" src="images/44.svg" alt=" " data-tex="K'_{x_{\nu}}"> are two Cartesian systems of co-ordinates, +then +<span class="align-center"><img style="vertical-align: -1.018ex; width: 21.338ex; height: 3.296ex;" src="images/45.svg" alt=" " data-tex=" +\sum {\Delta x_{\nu}}^{2} = \sum {\Delta x'_{\nu}}^{2}. +"></span> +</p> +<p> +The right-hand side is identically equal to the left-hand side +on account of the equations of the linear orthogonal transformation, +and the right-hand side differs from the left-hand side +only in that the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> are replaced by the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}">. This is expressed +by the statement that <img style="vertical-align: -0.566ex; width: 7.968ex; height: 2.633ex;" src="images/46.svg" alt=" " data-tex="\sum {\Delta x_{\nu}}^{2}"> is an invariant with respect to +linear orthogonal transformations. It is evident that in the Euclidean +geometry only such, and all such, quantities have an +objective significance, independent of the particular choice of +<span class="pagenum" id="Page_9">[Pg 9]</span> +the Cartesian co-ordinates, as can be expressed by an invariant +with respect to linear orthogonal transformations. This is +the reason that the theory of invariants, which has to do with +the laws that govern the form of invariants, is so important for +analytical geometry. +</p> +<p> +As a second example of a geometrical invariant, consider a +volume. This is expressed by +<span class="align-center"><img style="vertical-align: -1.948ex; width: 21.373ex; height: 5.027ex;" src="images/47.svg" alt=" " data-tex=" +V = \iiint dx_{1}\, dx_{2}\, dx_{3}. +"></span> +By means of Jacobi's theorem we may write +<span class="align-center"><img style="vertical-align: -2.172ex; width: 52.029ex; height: 5.495ex;" src="images/48.svg" alt=" " data-tex=" +\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} + = \iiint \frac{\partial({x'}_{1}, {x'}_{2}, {x'}_{3})}{\partial(x_{1}, x_{2}, x_{3})}\, + dx_{1}\, dx_{2}\, dx_{3} +"></span> +where the integrand in the last integral is the functional determinant +of the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> with respect to the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and this by (3) is equal +to the determinant <img style="vertical-align: -0.685ex; width: 4.229ex; height: 2.381ex;" src="images/49.svg" alt=" " data-tex="|b_{\mu\nu}|"> of the coefficients of substitution, <img style="vertical-align: -0.357ex; width: 3.03ex; height: 1.927ex;" src="images/34.svg" alt=" " data-tex="b_{\nu\alpha}">. If +we form the determinant of the <img style="vertical-align: -0.685ex; width: 3.181ex; height: 2.307ex;" src="images/50.svg" alt=" " data-tex="\delta_{\mu\alpha}"> from equation (4), we obtain, +by means of the theorem of multiplication of determinants, +<span class="align-center"><img style="vertical-align: -2.602ex; width: 52.322ex; height: 6.335ex;" src="images/51.svg" alt=" " data-tex=" +1 = |\delta_{\alpha\beta}| + = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| + = |b_{\mu\nu}|^{2};\quad +|b_{\mu\nu}| = ±1. +\qquad\text{(6)} +"></span> +If we limit ourselves to those transformations which have the determinant ++1,<a id="FNanchor_3_1"></a><a href="#Footnote_3_1" class="fnanchor">[3]</a> +and only these arise from continuous variations +of the systems of co-ordinates, then <img style="vertical-align: -0.05ex; width: 1.74ex; height: 1.595ex;" src="images/52.svg" alt=" " data-tex="V"> is an invariant. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_3_1"></a><a href="#FNanchor_3_1"><span class="label">[3]</span></a>There are thus two kinds of Cartesian systems which are designated as +"right-handed" and "left-handed" systems. The difference between these is +familiar to every physicist and engineer. It is interesting to note that these +two kinds of systems cannot be defined geometrically, but only the contrast +between them.</p></div> +<p><span class="pagenum" id="Page_10">[Pg 10]</span></p> +<p> +Invariants, however, are not the only forms by means of +which we can give expression to the independence of the particular +choice of the Cartesian co-ordinates. Vectors and tensors +are other forms of expression. Let us express the fact that the +point with the current co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> lies upon a straight line. +We have +<span class="align-center"><img style="vertical-align: -0.566ex; width: 31.899ex; height: 2.262ex;" src="images/53.svg" alt=" " data-tex=" +x_{\nu} - A_{\nu} = \lambda B_{\nu} \quad (\nu \text{ from 1 to 3}). +"></span> +Without limiting the generality we can put +<span class="align-center"><img style="vertical-align: -1.018ex; width: 12.162ex; height: 3.167ex;" src="images/54.svg" alt=" " data-tex=" +\sum {B_{\nu}}^{2} = 1. +"></span> +</p> +<p> +If we multiply the equations by <img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/55.svg" alt=" " data-tex="b_{\beta\nu}"> (compare (3a) and (5)) +and sum for all the <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">'s, we get +<span class="align-center"><img style="vertical-align: -0.869ex; width: 15.719ex; height: 2.699ex;" src="images/56.svg" alt=" " data-tex=" +x'_{\beta} - A'_{\beta} = \lambda B'_{\beta}, +"></span> +where we have written +<span class="align-center"><img style="vertical-align: -2.602ex; width: 34.129ex; height: 4.751ex;" src="images/57.svg" alt=" " data-tex=" +B'_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad +A'_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}. +"></span> +</p> +<p> +These are the equations of straight lines with respect to a +second Cartesian system of co-ordinates <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. They have the +same form as the equations with respect to the original system +of co-ordinates. It is therefore evident that straight lines +have a significance which is independent of the system of co-ordinates. +Formally, this depends upon the fact that the quantities (<img style="vertical-align: -0.339ex; width: 7.828ex; height: 1.959ex;" src="images/59.svg" alt=" " data-tex="x_{\nu} - A_{\nu}">) - <img style="vertical-align: -0.339ex; width: 4.072ex; height: 1.91ex;" src="images/60.svg" alt=" " data-tex="\lambda B_{\nu}"> are +transformed as the components of +an interval, <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">. The ensemble of three quantities, defined for +every system of Cartesian co-ordinates, and which transform as +the components of an interval, is called a vector. If the three +<span class="pagenum" id="Page_11">[Pg 11]</span> +components of a vector vanish for one system of Cartesian co-ordinates, +they vanish for all systems, because the equations of +transformation are homogeneous. We can thus get the meaning +of the concept of a vector without referring to a geometrical representation. +This behaviour of the equations of a straight line +can be expressed by saying that the equation of a straight line +is co-variant with respect to linear orthogonal transformations. +</p> +<p> +We shall now show briefly that there are geometrical entities +which lead to the concept of tensors. Let <img style="vertical-align: -0.375ex; width: 2.44ex; height: 1.92ex;" src="images/62.svg" alt=" " data-tex="P_{0}"> be the centre of a +surface of the second degree, <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> any point on the surface, and +<img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> the projections of the interval <img style="vertical-align: -0.375ex; width: 4.139ex; height: 1.92ex;" src="images/65.svg" alt=" " data-tex="P_{0}P"> upon the co-ordinate axes. +Then the equation of the surface is +<span class="align-center"><img style="vertical-align: -1.018ex; width: 15.789ex; height: 3.167ex;" src="images/66.svg" alt=" " data-tex=" +\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +"></span> +In this, and in analogous cases, we shall omit the sign of summation, +and understand that the summation is to be carried out +for those indices that appear twice. We thus write the equation +of the surface +<span class="align-center"><img style="vertical-align: -0.685ex; width: 12.144ex; height: 2.278ex;" src="images/67.svg" alt=" " data-tex=" +a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +"></span> +The quantities <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}"> determine the surface completely, for a given +position of the centre, with respect to the chosen system of +Cartesian co-ordinates. From the known law of transformation +for the <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> (3a) for linear orthogonal transformations, we easily +find the law of transformation for the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">:<a id="FNanchor_4_1"></a><a href="#Footnote_4_1" class="fnanchor">[4]</a> +<span class="align-center"><img style="vertical-align: -0.685ex; width: 16.466ex; height: 2.515ex;" src="images/69.svg" alt=" " data-tex=" +{a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}. +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_4_1"></a><a href="#FNanchor_4_1"><span class="label">[4]</span></a>The equation <img style="vertical-align: -0.565ex; width: 13.255ex; height: 2.282ex;" src="images/70.svg" alt=" " data-tex="{a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1"> +may, by (5), be replaced by +<img style="vertical-align: -0.58ex; width: 3.125ex; height: 2.297ex;" src="images/71.svg" alt=" " data-tex="a'_{\sigma\tau}"><img style="vertical-align: -0.685ex; width: 3.037ex; height: 2.255ex;" src="images/72.svg" alt=" " data-tex="b_{\mu\sigma}"><img style="vertical-align: -0.36ex; width: 2.833ex; height: 1.93ex;" src="images/73.svg" alt=" " data-tex="b_{\nu\tau}"><img style="vertical-align: -0.685ex; width: 2.143ex; height: 2.278ex;" src="images/74.svg" alt=" " data-tex="\xi_{\mu}"><img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> = 1, +from which the result stated immediately follows.</p></div> +<p><span class="pagenum" id="Page_12">[Pg 12]</span></p> + +<p class="nind"> +This transformation is homogeneous and of the first degree in +the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">. On account of this transformation, the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">, are called +components of a tensor of the second rank (the latter on account +of the double index). If all the components, <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">, of a tensor with +respect to any system of Cartesian co-ordinates vanish, they +vanish with respect to every other Cartesian system. The form +and the position of the surface of the second degree is described +by this tensor (<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/75.svg" alt=" " data-tex="a">). +</p> +<p> +Analytic tensors of higher rank (number of indices) may be +defined. It is possible and advantageous to regard vectors as +tensors of rank 1, and invariants (scalars) as tensors of rank 0. +In this respect, the problem of the theory of invariants may be so +formulated: according to what laws may new tensors be formed +from given tensors? We shall consider these laws now, in order +to be able to apply them later. We shall deal first only with the +properties of tensors with respect to the transformation from +one Cartesian system to another in the same space of reference, +by means of linear orthogonal transformations. As the laws are +wholly independent of the number of dimensions, we shall leave +this number, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n">, indefinite at first. +</p> + +<p><br></p> + +<p> +<i>Definition</i>. If a figure is defined with respect to every system +of Cartesian co-ordinates in a space of reference of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> dimensions +by the <img style="vertical-align: -0.025ex; width: 2.569ex; height: 1.553ex;" src="images/76.svg" alt=" " data-tex="n^{\alpha}"> numbers <img style="vertical-align: -0.685ex; width: 6.399ex; height: 2.305ex;" src="images/77.svg" alt=" " data-tex="A_{\mu\nu\rho\ldots}"> (<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = number of indices), then +these numbers are the components of a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> if the +transformation law is +<span class="align-center"><img style="vertical-align: -0.685ex; width: 42.162ex; height: 2.515ex;" src="images/78.svg" alt=" " data-tex=" +{A'}_{\mu'\nu'\rho'}\ldots + = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho}\ldots A_{\mu\nu\rho}.\ldots +\qquad\text{(7)} +"></span> +<span class="pagenum" id="Page_13">[Pg 13]</span> +</p> + +<p><br></p> + +<p> +<i>Remark</i>. From this definition it follows that +<span class="align-center"><img style="vertical-align: -0.685ex; width: 29.803ex; height: 2.382ex;" src="images/79.svg" alt=" " data-tex=" +A_{\mu\nu\rho}\ldots = B_{\mu} C_{\nu} D_{\rho}\dots +\qquad\text{(8)} +"></span> +is an invariant, provided that (<img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">), (<img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">), +(<img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/80.svg" alt=" " data-tex="D">) ... are vectors. +Conversely, the tensor character of (<img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">) may be inferred, if it +is known that the expression (8) leads to an invariant for an +arbitrary choice of the vectors (<img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">), (<img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">), etc. +</p> + +<p><br></p> + +<p> +<i>Addition and Subtraction</i>. By addition and subtraction of +the corresponding components of tensors of the same rank, a +tensor of equal rank results: +<span class="align-center"><img style="vertical-align: -0.685ex; width: 37.182ex; height: 2.382ex;" src="images/81.svg" alt=" " data-tex=" +A_{\mu\nu\rho}\ldots ± B_{\mu\nu\rho}\ldots = C_{\mu\nu\rho}\ldots. +\qquad\text{(9)} +"></span> +The proof follows from the definition of a tensor given above. +</p> + +<p><br></p> + +<p> +<i>Multiplication</i>. From a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and a tensor of +rank <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> we may obtain a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> + <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> by multiplying all +the components of the first tensor by all the components of the +second tensor: +<span class="align-center"><img style="vertical-align: -0.685ex; width: 40.555ex; height: 2.382ex;" src="images/82.svg" alt=" " data-tex=" +T_{\mu\nu\rho}\ldots_{\alpha\beta}\ldots += A_{\mu\nu\rho}\ldots B_{\alpha\beta\gamma}.\ldots +\qquad\text{(10)} +"></span> +</p> + +<p><br></p> + +<p> +<i>Contraction</i>. A tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> - 2 may be obtained from +one of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> by putting two definite indices equal to each other +and then summing for this single index: +<span class="align-center"><img style="vertical-align: -2.947ex; width: 41.445ex; height: 5.097ex;" src="images/83.svg" alt=" " data-tex=" +T_{\rho}\ldots + = A_{\mu\mu\rho}\ldots + ( = \sum_{\mu} A_{\mu\mu\rho}\ldots). +\qquad\text{(11)} +"></span> +</p> +<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> +<p class="nind"> +The proof is +<span class="align-center"><img style="vertical-align: -2.222ex; width: 45.796ex; height: 5.576ex;" src="images/84.svg" alt=" " data-tex=" +\begin{align*} +{A'}_{\mu\mu\rho\cdots} + = b_{\mu\alpha} b_{\mu\beta} b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} + = \delta_{\alpha\beta} & b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} \\ + {} ={} & b_{\rho\gamma\cdots} A_{\alpha\alpha\gamma\cdots}. +\end{align*} +"></span> +</p> +<p> +In addition to these elementary rules of operation there is +also the formation of tensors by differentiation ("erweiterung"): +<span class="align-center"><img style="vertical-align: -1.909ex; width: 28.293ex; height: 5.255ex;" src="images/85.svg" alt=" " data-tex=" +T_{\mu\nu\rho\cdots\alpha} + = \frac{\partial A_{\mu\nu\rho\cdots}}{\partial x_{\alpha}}. +\qquad\text{(12)} +"></span> +</p> +<p> +New tensors, in respect to linear orthogonal transformations, +may be formed from tensors according to these rules of operation. +</p> + +<p><br></p> + +<p> +<i>Symmetrical Properties of Tensors</i>. Tensors are called symmetrical +or skew-symmetrical in respect to two of their indices, +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, if both the components which result from interchanging +the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> are equal to each other or equal with +opposite signs. +<span class="align-center"><img style="vertical-align: -2.156ex; width: 46.294ex; height: 5.442ex;" src="images/87.svg" alt=" " data-tex=" +\begin{alignat*}{2} +&\text{Condition for symmetry:} & + A_{\mu\nu\rho} &= A_{{\nu\mu}\rho}. \\ +&\text{Condition for skew-symmetry:}\quad & + A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ +\end{alignat*} +"></span> +</p> + +<p><br></p> + +<p> +<i>Theorem</i>. The character of symmetry or skew-symmetry +exists independently of the choice of co-ordinates, and in this +lies its importance. The proof follows from the equation defining tensors. +</p> +<p> +<i>Special Tensors.</i> +</p> +<p> +I. The quantities <img style="vertical-align: -0.685ex; width: 2.933ex; height: 2.307ex;" src="images/88.svg" alt=" " data-tex="\delta_{\rho\sigma}"> (4) are tensor components +(fundamental tensor). +<span class="pagenum" id="Page_15">[Pg 15]</span> +</p> +<p> +<i>Proof</i>. If in the right-hand side of the equation of transformation +<img style="vertical-align: -0.904ex; width: 3.697ex; height: 2.622ex;" src="images/89.svg" alt=" " data-tex="A'_{\mu\nu}"> = <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"><img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/31.svg" alt=" " data-tex="b_{\nu\beta}"><img style="vertical-align: -0.65ex; width: 3.814ex; height: 2.27ex;" src="images/91.svg" alt=" " data-tex="A_{\alpha\beta}">, we substitute +for <img style="vertical-align: -0.65ex; width: 3.814ex; height: 2.27ex;" src="images/91.svg" alt=" " data-tex="A_{\alpha\beta}"> the quantities <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> (which are equal +to 1 or 0 according as <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> or <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> ≠<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">), +we get +<span class="align-center"><img style="vertical-align: -0.685ex; width: 20.17ex; height: 2.515ex;" src="images/92.svg" alt=" " data-tex=" +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}. +"></span> +The justification for the last sign of equality becomes evident if +one applies (4) to the inverse substitution (5). +</p> +<p> +II. There is a tensor (<img style="vertical-align: -0.685ex; width: 5.707ex; height: 2.307ex;" src="images/93.svg" alt=" " data-tex="\delta_{\mu\nu\rho\ldots}">) skew-symmetrical with respect +to all pairs of indices, whose rank is equal to the number of +dimensions, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n">, and whose components are equal to +1 or -1 +according as <img style="vertical-align: -0.489ex; width: 6.762ex; height: 1.489ex;" src="images/94.svg" alt=" " data-tex="\mu\nu\rho\ldots"> is an even or odd permutation of 1 2 3.... +</p> +<p> +The proof follows with the aid of the theorem proved above +<img style="vertical-align: -0.685ex; width: 8.305ex; height: 2.381ex;" src="images/95.svg" alt=" " data-tex="|b_{\rho\sigma}| = 1"> +</p> +<p> +These few simple theorems form the apparatus from the +theory of invariants for building the equations of pre-relativity +physics and the theory of special relativity. +</p> +<p> +We have seen that in pre-relativity physics, in order to specify +relations in space, a body of reference, or a space of reference, +is required, and, in addition, a Cartesian system of co-ordinates. +We can fuse both these concepts into a single one by thinking +of a Cartesian system of co-ordinates as a cubical frame-work +formed of rods each of unit length. The co-ordinates of the lattice +points of this frame are integral numbers. It follows from +the fundamental relation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 34.607ex; height: 2.633ex;" src="images/96.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} +\qquad\text{(13)} +"></span> +that the members of such a space-lattice are all of unit length. +To specify relations in time, we require in addition a standard +clock placed at the origin of our Cartesian system of co-ordinates +<span class="pagenum" id="Page_16">[Pg 16]</span> +or frame of reference. If an event takes place anywhere we can +assign to it three co-ordinates, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and a time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, as soon as +we have specified the time of the clock at the origin which is +simultaneous with the event. We therefore give an objective significance +to the statement of the simultaneity of distant events, +while previously we have been concerned only with the simultaneity +of two experiences of an individual. The time so specified +is at all events independent of the position of the system of co-ordinates +in our space of reference, and is therefore an invariant +with respect to the transformation (3). +</p> +<p> +It is postulated that the system of equations expressing the +laws of pre-relativity physics is co-variant with respect to the +transformation (3), as are the relations of Euclidean geometry. +The isotropy and homogeneity of space is expressed in this way.<a id="FNanchor_5_1"></a><a href="#Footnote_5_1" class="fnanchor">[5]</a> +We shall now consider some of the more important equations of +physics from this point of view. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_5_1"></a><a href="#FNanchor_5_1"><span class="label">[5]</span></a>The laws of physics could be expressed, even in case there were a +unique direction in space, in such a way as to be co-variant with respect +to the transformation (3); but such an expression would in this case be +unsuitable. If there were a unique direction in space it would simplify the +description of natural phenomena to orient the system of co-ordinates in +a definite way in this direction. But if, on the other hand, there is no +unique direction in space it is not logical to formulate the laws of nature +in such a way as to conceal the equivalence of systems of co-ordinates that +are oriented differently. We shall meet with this point of view again in the +theories of special and general relativity.</p></div> + +<p> +The equations of motion of a material particle are +<span class="align-center"><img style="vertical-align: -1.654ex; width: 22.955ex; height: 5.07ex;" src="images/98.svg" alt=" " data-tex=" +m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}; +\qquad\text{(14)} +"></span> +(<img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">) is a vector; <img style="vertical-align: -0.025ex; width: 1.993ex; height: 1.595ex;" src="images/100.svg" alt=" " data-tex="dt">, and therefore also <img style="vertical-align: -1.577ex; width: 2.989ex; height: 4.613ex;" src="images/101.svg" alt=" " data-tex="\dfrac{1}{dt}">, an invariant; thus +<span class="pagenum" id="Page_17">[Pg 17]</span> +(<img style="vertical-align: -1.577ex; width: 4.502ex; height: 4.676ex;" src="images/102.svg" alt=" " data-tex="\dfrac{dx_{\nu}}{dt}">) is a vector; in the same way it may be shown that +(<img style="vertical-align: -1.654ex; width: 5.489ex; height: 5.07ex;" src="images/103.svg" alt=" " data-tex="\dfrac{d^{2} x_{\nu}}{dt^{2}}">) is a vector. In general, the operation of differentiation +with respect to time does not alter the tensor character. Since +<img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/104.svg" alt=" " data-tex="m"> is an invariant (tensor of rank 0), <img style="vertical-align: -1.654ex; width: 8.356ex; height: 5.07ex;" src="images/105.svg" alt=" " data-tex="(m\dfrac{d^{2} x_{\nu}}{dt^{2}}">) +is a vector, or +tensor of rank 1 (by the theorem of the multiplication of tensors). +If the force (<img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">) has a vector character, the same holds for +the difference (<img style="vertical-align: -1.654ex; width: 14.031ex; height: 5.07ex;" src="images/107.svg" alt=" " data-tex="m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu})">. These equations of motion are +therefore valid in every other system of Cartesian co-ordinates +in the space of reference. In the case where the forces are conservative +we can easily recognize the vector character of (<img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">). +For a potential energy, <img style="vertical-align: 0; width: 1.633ex; height: 1.545ex;" src="images/108.svg" alt=" " data-tex="\Phi">, exists, which depends only upon the +mutual distances of the particles, and is therefore an invariant. +The vector character of the force, <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> = <img style="vertical-align: -1.891ex; width: 6.366ex; height: 5.038ex;" src="images/109.svg" alt=" " data-tex="-\dfrac{\partial \Phi}{\partial x_{\nu}}">, +is then a consequence of our general theorem about the derivative of a tensor +of rank 0. +</p> +<p> +Multiplying by the velocity, a tensor of rank 1, we obtain the +tensor equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 25.76ex; height: 5.564ex;" src="images/110.svg" alt=" " data-tex=" +\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0. +"></span> +By contraction and multiplication by the scalar <img style="vertical-align: -0.025ex; width: 1.993ex; height: 1.595ex;" src="images/100.svg" alt=" " data-tex="dt"> we obtain the +equation of kinetic energy +<span class="align-center"><img style="vertical-align: -2.148ex; width: 20.636ex; height: 5.564ex;" src="images/111.svg" alt=" " data-tex=" +d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\ dx_{\nu}. +"></span> +<span class="pagenum" id="Page_18">[Pg 18]</span> +</p> +<p> +If <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> denotes the difference of the co-ordinates of the material +particle and a point fixed in space, then the <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> have the +character of vectors. We evidently have +<img style="vertical-align: -1.654ex; width: 5.489ex; height: 5.07ex;" src="images/103.svg" alt=" " data-tex="\dfrac{d^{2} x_{\nu}}{dt^{2}}"> = <img style="vertical-align: -1.654ex; width: 5.186ex; height: 5.07ex;" src="images/112.svg" alt=" " data-tex="\dfrac{d^{2} \xi_{\nu}}{dt^{2}}">, so that +the equations of motion of the particle may be written +<span class="align-center"><img style="vertical-align: -1.654ex; width: 17.625ex; height: 5.07ex;" src="images/113.svg" alt=" " data-tex=" +m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0. +"></span> +</p> +<p> +Multiplying this equation by <img style="vertical-align: -0.685ex; width: 2.143ex; height: 2.278ex;" src="images/74.svg" alt=" " data-tex="\xi_{\mu}"> we obtain a tensor equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 23.098ex; height: 5.564ex;" src="images/114.svg" alt=" " data-tex=" +\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0. +"></span> +</p> +<p> +Contracting the tensor on the left and taking the time average +we obtain the virial theorem, which we shall not consider +further. By interchanging the indices and subsequent subtraction, +we obtain, after a simple transformation, the theorem of +moments, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 51.585ex; height: 5.467ex;" src="images/115.svg" alt=" " data-tex=" +\frac{d}{dt} \biggl[ + m \biggl(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\biggr)\biggr] += \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}. +\qquad\text{(15)} +"></span> +</p> +<p> +It is evident in this way that the moment of a vector is not a +vector but a tensor. On account of their skew-symmetrical character +there are not nine, but only three independent equations of +this system. The possibility of replacing skew-symmetrical tensors +of the second rank in space of three dimensions by vectors +depends upon the formation of the vector +<span class="align-center"><img style="vertical-align: -1.552ex; width: 16.145ex; height: 4.588ex;" src="images/116.svg" alt=" " data-tex=" +A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}. +"></span> +<span class="pagenum" id="Page_19">[Pg 19]</span> +</p> +<p> +If we multiply the skew-symmetrical tensor of rank 2 by the +special skew-symmetrical tensor <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/117.svg" alt=" " data-tex="\delta"> introduced above, and contract +twice, a vector results whose components are numerically +equal to those of the tensor. These are the so-called axial vectors +which transform differently, from a right-handed system to +a left-handed system, from the <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">. There is a gain in +picturesqueness in regarding a skew-symmetrical tensor of rank 2 +as a vector in space of three dimensions, but it does not represent +the exact nature of the corresponding quantity so well as +considering it a tensor. +</p> +<p> +We consider next the equations of motion of a continuous +medium. Let <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"> be the density, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}"> the velocity components +considered as functions of the co-ordinates and the time, <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> the +volume forces per unit of mass, and <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> the stresses upon a +surface perpendicular to the a-axis in the direction of increasing +<img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. Then the equations of motion are, by Newton's law, +<span class="align-center"><img style="vertical-align: -1.909ex; width: 23.285ex; height: 5.056ex;" src="images/121.svg" alt=" " data-tex=" +\rho \frac{du_{\nu}}{dt} + = -\frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + \rho X_{\nu}, +"></span> +in which <img style="vertical-align: -1.577ex; width: 4.502ex; height: 4.676ex;" src="images/122.svg" alt=" " data-tex="\dfrac{du_{\nu}}{dt}"> is the acceleration of the particle which +at time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> has the co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If we express this acceleration +by partial differential coefficients, we obtain, after dividing by <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, +<span class="align-center"><img style="vertical-align: -2.041ex; width: 42.284ex; height: 5.188ex;" src="images/123.svg" alt=" " data-tex=" +\frac{\partial u_{\nu}}{dt} + \frac{\partial u_{\nu}}{dx_{\sigma}} u_{\sigma} + = -\frac{1}{\rho}\, \frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + X_{\nu}. +\qquad\text{(16)} +"></span> +</p> +<p> +We must show that this equation holds independently of the +special choice of the Cartesian system of co-ordinates. (<img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}">) is a +vector, and therefore <img style="vertical-align: -1.602ex; width: 4.606ex; height: 4.749ex;" src="images/124.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial t}"> is also a vector. +<img style="vertical-align: -1.909ex; width: 4.671ex; height: 5.056ex;" src="images/125.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial x_{\sigma}}"> +<span class="pagenum" id="Page_20">[Pg 20]</span> +is a tensor of rank 2, <img style="vertical-align: -1.909ex; width: 6.98ex; height: 5.056ex;" src="images/126.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial x_{\sigma}} u_{\tau}"> +is a tensor of rank 3. The second term on the left +results from contraction in the indices <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau">. The vector character +of the second term on the right is obvious. In order that the first +term on the right may also be a vector it is necessary for <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> to be +a tensor. Then by differentiation and contraction <img style="vertical-align: -1.909ex; width: 5.363ex; height: 5.056ex;" src="images/129.svg" alt=" " data-tex="\dfrac{\partial p_{\nu\sigma}}{\partial x_{\sigma}}"> results, +and is therefore a vector, as it also is after multiplication by +the reciprocal scalar <img style="vertical-align: -2.041ex; width: 2.165ex; height: 5.077ex;" src="images/130.svg" alt=" " data-tex="\dfrac{1}{\rho}">. That <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> is a tensor, and therefore +transforms according to the equation +<span class="align-center"><img style="vertical-align: -0.864ex; width: 16.726ex; height: 2.695ex;" src="images/131.svg" alt=" " data-tex=" +{p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta}, +"></span> +is proved in mechanics by integrating this equation over an infinitely +small tetrahedron. It is also proved there, by application +of the theorem of moments to an infinitely small parallelopipedon, +that <img style="vertical-align: -0.439ex; width: 9.191ex; height: 1.758ex;" src="images/132.svg" alt=" " data-tex="p_{\nu\sigma} = p_{\sigma\nu}">, and hence that the tensor of the stress is +a symmetrical tensor. From what has been said it follows that, +with the aid of the rules given above, the equation is co-variant +with respect to orthogonal transformations in space (rotational +transformations); and the rules according to which the quantities +in the equation must be transformed in order that the +equation may be co-variant also become evident. +</p> +<p> +The co-variance of the equation of continuity, +<span class="align-center"><img style="vertical-align: -1.891ex; width: 27.449ex; height: 5.195ex;" src="images/133.svg" alt=" " data-tex=" +\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u_{\nu})}{\partial x_{\nu}} = 0, +\qquad\text{(17)} +"></span> +requires, from the foregoing, no particular discussion. +</p> +<p> +We shall also test for co-variance the equations which express +the dependence of the stress components upon the properties of +<span class="pagenum" id="Page_21">[Pg 21]</span> +the matter, and set up these equations for the case of a compressible +viscous fluid with the aid of the conditions of co-variance. +If we neglect the viscosity, the pressure, <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/134.svg" alt=" " data-tex="p">, will be a scalar, and +will depend only upon the density and the temperature of the +fluid. The contribution to the stress tensor is then evidently +<span class="align-center"><img style="vertical-align: -0.685ex; width: 4.143ex; height: 2.307ex;" src="images/135.svg" alt=" " data-tex=" +p \delta_{\mu\nu} +"></span> +in which <img style="vertical-align: -0.685ex; width: 3.005ex; height: 2.307ex;" src="images/136.svg" alt=" " data-tex="\delta_{\mu\nu}"> is the special symmetrical tensor. This term will +also be present in the case of a viscous fluid. But in this case +there will also be pressure terms, which depend upon the space +derivatives of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}">. We shall assume that this dependence is a +linear one. Since these terms must be symmetrical tensors, the +only ones which enter will be +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.199ex; height: 5.58ex;" src="images/137.svg" alt=" " data-tex=" +\alpha\left(\frac{\partial u_{\mu}}{\partial x_{\nu}} + + \frac{\partial u_{\nu}}{\partial x_{\mu}}\right) + + \beta\delta_{\mu\nu} \frac{\partial u_{\alpha}}{\partial x_{\alpha}} +"></span> +(for <img style="vertical-align: -1.909ex; width: 4.782ex; height: 5.056ex;" src="images/138.svg" alt=" " data-tex="\dfrac{\partial u_{\alpha}}{\partial x_{\alpha}}"> is a scalar). For +physical reasons (no slipping) it +is assumed that for symmetrical dilatations in all directions, +i.e. when +<span class="align-center"><img style="vertical-align: -1.927ex; width: 38.03ex; height: 5.074ex;" src="images/139.svg" alt=" " data-tex=" +\frac{\partial u_{1}}{\partial x_{1}} = +\frac{\partial u_{2}}{\partial x_{2}} = +\frac{\partial u_{3}}{\partial x_{3}};\quad +\frac{\partial u_{1}}{\partial x_{2}}, \text{ etc.,} = 0, +"></span> +there are no frictional forces present, from which it follows that +<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> = <img style="vertical-align: -1.602ex; width: 5.335ex; height: 4.638ex;" src="images/140.svg" alt=" " data-tex="-\dfrac{2}{3}\alpha">. If only <img style="vertical-align: -1.927ex; width: 4.558ex; height: 5.074ex;" src="images/141.svg" alt=" " data-tex="\dfrac{\partial u_{1}}{\partial x_{3}}"> +is different from zero, let <img style="vertical-align: -0.439ex; width: 2.926ex; height: 1.439ex;" src="images/142.svg" alt=" " data-tex="p_{31}"> = -<img style="vertical-align: -1.927ex; width: 6.006ex; height: 5.074ex;" src="images/143.svg" alt=" " data-tex="\alpha \dfrac{\partial u_{1}}{\partial x_{3}}">, +by which <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> is determined. We then obtain for the complete +stress tensor, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 72.428ex; height: 5.58ex;" src="images/144.svg" alt=" " data-tex=" +p_{\mu\nu} = p \delta_{\mu\nu} - \alpha \biggl[ + \biggl(\frac{\partial u_{\mu}}{\partial x_{\nu}} + + \frac{\partial u_{\nu}}{\partial x_{\mu}}\biggr) + - \frac{2}{3} \biggl(\frac{\partial u_{1}}{\partial x_{1}} + + \frac{\partial u_{2}}{\partial x_{2}} + + \frac{\partial u_{3}}{\partial x_{3}}\biggr)\delta_{\mu\nu} +\biggr]. +\qquad\text{(18)} +"></span> +<span class="pagenum" id="Page_22">[Pg 22]</span> +</p> +<p> +The heuristic value of the theory of invariants, which arises +from the isotropy of space (equivalence of all directions), becomes +evident from this example. +</p> +<p> +We consider, finally, Maxwell's equations in the form which +are the foundation of the electron theory of Lorentz. +</p> +<span class="align-center"><img style="vertical-align: -22.069ex; width: 44.49ex; height: 45.27ex;" src="images/145.svg" alt=" " data-tex=" +\begin{align*} +\left. + \begin{aligned} + &\begin{alignedat}{3} + \frac{\partial h_{3}}{\partial x_{2}} &- + \frac{\partial h_{2}}{\partial x_{3}} + &&= \frac{1}{c}\, \frac{\partial e_{1}}{\partial t} + &&+ \frac{1}{c}\, i_{1}, +\qquad \\ + \frac{\partial h_{1}}{\partial x_{3}} &- + \frac{\partial h_{3}}{\partial x_{1}} + &&= \frac{1}{c}\, \frac{\partial e_{2}}{\partial t} + &&+ \frac{1}{c}\, i_{2}, \\ + \frac{\partial h_{2}}{\partial x_{1}} &- + \frac{\partial h_{1}}{\partial x_{2}} + &&= \frac{1}{c}\, \frac{\partial e_{3}}{\partial t} + &&+ \frac{1}{c}\, i_{3}, + \end{alignedat} \\ + & \frac{\partial e_{1}}{\partial x_{1}} + + \frac{\partial e_{2}}{\partial x_{2}} + + \frac{\partial e_{3}}{\partial x_{3}} = \rho; + \end{aligned} + \right\} +\qquad\text{(19)}\\ +\left. + \begin{aligned} + &\begin{alignedat}{2} + \frac{\partial e_{3}}{\partial x_{2}} &- \frac{\partial e_{2}}{\partial x_{3}} + &&= -\frac{1}{c}\, \frac{\partial h_{1}}{\partial t}, +\qquad \\ + \frac{\partial e_{1}}{\partial x_{3}} &- \frac{\partial e_{3}}{\partial x_{1}} + &&= -\frac{1}{c}\, \frac{\partial h_{2}}{\partial t}, \\ + \frac{\partial e_{2}}{\partial x_{1}} &- \frac{\partial e_{1}}{\partial x_{2}} + &&= -\frac{1}{c}\, \frac{\partial h_{3}}{\partial t}, + \end{alignedat} \\ + & \frac{\partial h_{1}}{\partial x_{1}} + + \frac{\partial h_{2}}{\partial x_{2}} + + \frac{\partial h_{3}}{\partial x_{3}} = 0. + \end{aligned} +\right\} +\qquad \text{(20)} +\end{align*} +"></span> +<p> +<img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/146.svg" alt=" " data-tex="\mathbf i"> is a vector, because the current density is defined as the +density of electricity multiplied by the vector velocity of the +electricity. According to the first three equations it is evident +that <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/147.svg" alt=" " data-tex="\mathbf e"> is also to be regarded as a vector. Then <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/148.svg" alt=" " data-tex="\mathbf h"> cannot be +regarded as a vector.<a id="FNanchor_6_1"></a><a href="#Footnote_6_1" class="fnanchor">[6]</a> The equations may, however, easily be +<span class="pagenum" id="Page_23">[Pg 23]</span> +interpreted if <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/148.svg" alt=" " data-tex="\mathbf h"> is regarded as a skew-symmetrical tensor of the +second rank. In this sense, we write <img style="vertical-align: -0.375ex; width: 3.091ex; height: 1.945ex;" src="images/149.svg" alt=" " data-tex="h_{23}">, <img style="vertical-align: -0.375ex; width: 3.091ex; height: 1.945ex;" src="images/150.svg" alt=" " data-tex="h_{31}">, <img style="vertical-align: -0.339ex; width: 3.091ex; height: 1.91ex;" src="images/151.svg" alt=" " data-tex="h_{12}"> in place of +<img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.91ex;" src="images/152.svg" alt=" " data-tex="h_{1}">, <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.91ex;" src="images/153.svg" alt=" " data-tex="h_{2}">, <img style="vertical-align: -0.375ex; width: 2.291ex; height: 1.945ex;" src="images/154.svg" alt=" " data-tex="h_{3}"> respectively. Paying attention to the skew-symmetry +of <img style="vertical-align: -0.685ex; width: 3.304ex; height: 2.255ex;" src="images/155.svg" alt=" " data-tex="h_{\mu\nu}">, the first three equations of (19) and (20) may be written +in the form +<span class="align-center"><img style="vertical-align: -5.181ex; width: 39.788ex; height: 11.494ex;" src="images/156.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial h_{\mu\nu}}{\partial x_{\nu}} + = \frac{1}{c}\, \frac{\partial e_{\mu}}{\partial t} + \frac{1}{c} i_{\mu}, +\qquad & &\text{(19a)} \\ +\frac{\partial e_{\mu}}{\partial x_{\nu}} - \frac{\partial e_{\nu}}{\partial x_{\mu}} + = +\frac{1}{c}\, \frac{\partial h_{\mu\nu}}{\partial t}. +\qquad & &\text{(20a)} +\end{gather*} +"></span> +In contrast to <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/147.svg" alt=" " data-tex="\mathbf e">, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/148.svg" alt=" " data-tex="\mathbf h"> appears as a quantity which has the same type +of symmetry as an angular velocity. The divergence equations +then take the form +<span class="align-center"><img style="vertical-align: -5.083ex; width: 41.219ex; height: 11.297ex;" src="images/157.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial e_{\nu}}{\partial x_{\nu}} = \rho\, +\qquad & &\text{(19b)} \\ +\frac{\partial h_{\mu\nu}}{\partial x_{\rho}} + +\frac{\partial h_{\nu\rho}}{\partial x_{\mu}} + +\frac{\partial h_{\rho\mu}}{\partial x_{\nu}} = 0. +\qquad & &\text{(20b)} +\end{gather*} +"></span> +The last equation is a skew-symmetrical tensor equation of the +third rank (the skew-symmetry of the left-hand side with respect +to every pair of indices may easily be proved, if attention +is paid to the skew-symmetry of <img style="vertical-align: -0.685ex; width: 3.304ex; height: 2.255ex;" src="images/155.svg" alt=" " data-tex="h_{\mu\nu}">). This notation is more +natural than the usual one, because, in contrast to the latter, +it is applicable to Cartesian left-handed systems as well as to +right-handed systems without change of sign. +<span class="pagenum" id="Page_24">[Pg 24]</span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_6_1"></a><a href="#FNanchor_6_1"><span class="label">[6]</span></a>These considerations will make the reader familiar with tensor operations +without the special difficulties of the four-dimensional treatment; +corresponding considerations in the theory of special relativity (Minkowski's +interpretation of the field) will then offer fewer difficulties.</p></div> + +<p><br><br><br></p> + +<h3><a id="chap02"></a>LECTURE II +<br><br> +THE THEORY OF SPECIAL RELATIVITY</h3> + +<p class="nind"> +<span class="dropcap">T</span>HE previous considerations concerning the configuration of +rigid bodies have been founded, irrespective of the assumption +as to the validity of the Euclidean geometry, upon the hypothesis +that all directions in space, or all configurations of Cartesian systems +of co-ordinates, are physically equivalent. We may express +this as the "principle of relativity with respect to direction," and +it has been shown how equations (laws of nature) may be found, +in accord with this principle, by the aid of the calculus of tensors. +We now inquire whether there is a relativity with respect +to the state of motion of the space of reference; in other words, +whether there are spaces of reference in motion relatively to each +other which are physically equivalent. From the standpoint of +mechanics it appears that equivalent spaces of reference do exist. +For experiments upon the earth tell us nothing of the fact +that we are moving about the sun with a velocity of approximately +30 kilometres a second. On the other hand, this physical +equivalence does not seem to hold for spaces of reference in arbitrary +motion; for mechanical effects do not seem to be subject +to the same laws in a jolting railway train as in one moving with +uniform velocity; the rotation of the earth must be considered +in writing down the equations of motion relatively to the earth. +It appears, therefore, as if there were Cartesian systems of co-ordinates, +the so-called inertial systems, with reference to which +the laws of mechanics (more generally the laws of physics) are +expressed in the simplest form. We may infer the validity of +the following theorem: If <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is an inertial system, then every +<span class="pagenum" id="Page_25">[Pg 25]</span> +other system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which moves uniformly and without rotation +relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, is also an inertial system; the laws of nature are +in concordance for all inertial systems. This statement we shall +call the "principle of special relativity." We shall draw certain +conclusions from this principle of "relativity of translation" just +as we have already done for relativity of direction. +</p> +<p> +In order to be able to do this, we must first solve the following +problem. If we are given the Cartesian co-ordinates, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and +the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, of an event relatively to one inertial system, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +how can we calculate the co-ordinates, <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}">, and the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">', of +the same event relatively to an inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which moves +with uniform translation relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">? In the pre-relativity +physics this problem was solved by making unconsciously two +hypotheses:— +</p> +<p> +1. The time is absolute; the time of an event, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">', relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is the same as the time relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. If instantaneous +signals could be sent to a distance, and if one knew that the +state of motion of a clock had no influence on its rate, then this +assumption would be physically established. For then clocks, +similar to one another, and regulated alike, could be distributed +over the systems <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', at rest relatively to them, and their +indications would be independent of the state of motion of the +systems; the time of an event would then be given by the clock +in its immediate neighbourhood. +</p> +<p> +2. Length is absolute; if an interval, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +has a length <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, then it has the same length <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> relatively to a +system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which is in motion relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. +</p> +<p> +If the axes of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' are parallel to each other, a simple +calculation based on these two assumptions, gives the equations +<span class="pagenum" id="Page_26">[Pg 26]</span> +of transformation +<span class="align-center"><img style="vertical-align: -2.17ex; width: 29.765ex; height: 5.47ex;" src="images/158.svg" alt=" " data-tex=" +\left. +\begin{aligned} +{x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\text{,} \\ +t' &= t - b\text{.} +\end{aligned} +\right\} +\qquad \text{(21)} +"></span> +</p> +<p> +This transformation is known as the "Galilean Transformation." +Differentiating twice by the time, we get +<span class="align-center"><img style="vertical-align: -1.654ex; width: 15.253ex; height: 5.07ex;" src="images/159.svg" alt=" " data-tex=" +\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}. +"></span> +Further, it follows that for two simultaneous events, +<span class="align-center"><img style="vertical-align: -0.339ex; width: 28.681ex; height: 2.752ex;" src="images/160.svg" alt=" " data-tex=" +{{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}. +"></span> +The invariance of the distance between the two points results +from squaring and adding. From this easily follows the co-variance +of Newton's equations of motion with respect to the +Galilean transformation (21). Hence it follows that classical +mechanics is in accord with the principle of special relativity if +the two hypotheses respecting scales and clocks are made. +</p> +<p> +But this attempt to found relativity of translation upon the +Galilean transformation fails when applied to electromagnetic +phenomena. The Maxwell-Lorentz electromagnetic equations +are not co-variant with respect to the Galilean transformation. +In particular, we note, by (21), that a ray of light which referred +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> has a velocity <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/161.svg" alt=" " data-tex="c">, has a different velocity referred to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +depending upon its direction. The space of reference of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is +therefore distinguished, with respect to its physical properties, +from all spaces of reference which are in motion relatively to it +(quiescent æther). But all experiments have shown that electromagnetic +and optical phenomena, relatively to the earth as the +<span class="pagenum" id="Page_27">[Pg 27]</span> +body of reference, are not influenced by the translational velocity +of the earth. The most important of these experiments are +those of Michelson and Morley, which I shall assume are known. +The validity of the principle of special relativity can therefore +hardly be doubted. +</p> +<p> +On the other hand, the Maxwell-Lorentz equations have +proved their validity in the treatment of optical problems in +moving bodies. No other theory has satisfactorily explained the +facts of aberration, the propagation of light in moving bodies +(Fizeau), and phenomena observed in double stars (De Sitter). +The consequence of the Maxwell-Lorentz equations that in a +vacuum light is propagated with the velocity <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/161.svg" alt=" " data-tex="c">, at least with respect +to a definite inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, must therefore be regarded +as proved. According to the principle of special relativity, we +must also assume the truth of this principle for every other +inertial system. +</p> +<p> +Before we draw any conclusions from these two principles +we must first review the physical significance of the concepts +"time" and "velocity." It follows from what has gone before, that +co-ordinates with respect to an inertial system are physically +defined by means of measurements and constructions with the +aid of rigid bodies. In order to measure time, we have supposed +a clock, <img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/162.svg" alt=" " data-tex="U">, present somewhere, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. But +we cannot fix the time, by means of this clock, of an event +whose distance from the clock is not negligible; for there are no +"instantaneous signals" that we can use in order to compare the +time of the event with that of the clock. In order to complete the +definition of time we may employ the principle of the constancy +of the velocity of fight in a vacuum. Let us suppose that we +place similar clocks at points of the system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, at rest relatively +<span class="pagenum" id="Page_28">[Pg 28]</span> +to it, and regulated according to the following scheme. A ray +of light is sent out from one of the clocks, <img style="vertical-align: -0.357ex; width: 3.138ex; height: 1.902ex;" src="images/163.svg" alt=" " data-tex="U_{m}">, at the instant +when it indicates the time <img style="vertical-align: -0.357ex; width: 2.409ex; height: 1.773ex;" src="images/164.svg" alt=" " data-tex="t_{m}">, and travels through a vacuum a +distance <img style="vertical-align: -0.357ex; width: 3.573ex; height: 1.357ex;" src="images/165.svg" alt=" " data-tex="r_{mn}">, to the clock <img style="vertical-align: -0.357ex; width: 2.693ex; height: 1.902ex;" src="images/166.svg" alt=" " data-tex="U_{n}">; at the instant when this ray meets +the clock <img style="vertical-align: -0.357ex; width: 2.693ex; height: 1.902ex;" src="images/166.svg" alt=" " data-tex="U_{n}"> the latter is set to indicate the time +<img style="vertical-align: -0.357ex; width: 1.964ex; height: 1.773ex;" src="images/167.svg" alt=" " data-tex="t_{n}"> = <img style="vertical-align: -0.357ex; width: 2.409ex; height: 1.773ex;" src="images/164.svg" alt=" " data-tex="t_{m}"> + <img style="vertical-align: -1.577ex; width: 4.568ex; height: 4.106ex;" src="images/168.svg" alt=" " data-tex="\dfrac{r_{mn}}{c}">.<a id="FNanchor_7_1"></a><a href="#Footnote_7_1" class="fnanchor">[7]</a> +The principle of the constancy of the velocity of light then states +that this adjustment of the clocks will not lead to contradictions. +With clocks so adjusted, we can assign the time to events which +take place near any one of them. It is essential to note that this +definition of time relates only to the inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, since +we have used a system of clocks at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. The +assumption which was made in the pre-relativity physics of the +absolute character of time (i.e. the independence of time of the +choice of the inertial system) does not follow at all from this +definition. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_7_1"></a><a href="#FNanchor_7_1"><span class="label">[7]</span></a>Strictly speaking, it would be more correct to define simultaneity first, +somewhat as follows: two events taking place at the points <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> and <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B"> of +the system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> are simultaneous if they appear at the same instant when +observed from the middle point, <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/169.svg" alt=" " data-tex="M">, of the interval <img style="vertical-align: 0; width: 3.414ex; height: 1.62ex;" src="images/170.svg" alt=" " data-tex="AB">. Time is then +defined as the ensemble of the indications of similar clocks, at rest relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, which register the same simultaneously.</p></div> + +<p> +The theory of relativity is often criticized for giving, without +justification, a central theoretical role to the propagation +of light, in that it founds the concept of time upon the law of +propagation of light. The situation, however, is somewhat as +follows. In order to give physical significance to the concept of +time, processes of some kind are required which enable relations +to be established between different places. It is immaterial what +kind of processes one chooses for such a definition of time. It +is advantageous, however, for the theory, to choose only those +<span class="pagenum" id="Page_29">[Pg 29]</span> +processes concerning which we know something certain. This +holds for the propagation of light <i>in vacuo</i> in a higher degree +than for any other process which could be considered, thanks to +the investigations of Maxwell and H. A. Lorentz. +</p> +<p> +From all of these considerations, space and time data have +a physically real, and not a mere fictitious, significance; in particular +this holds for all the relations in which co-ordinates and +time enter, e.g. the relations (21). There is, therefore, sense in +asking whether those equations are true or not, as well as in +asking what the true equations of transformation are by which +we pass from one inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> to another, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', moving relatively +to it. It may be shown that this is uniquely settled by +means of the principle of the constancy of the velocity of light +and the principle of special relativity. +</p> +<p> +To this end we think of space and time physically defined +with respect to two inertial systems, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in the way that +has been shown. Further, let a ray of light pass from one point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}"> +to another point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> through a vacuum. If <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/173.svg" alt=" " data-tex="r"> is the measured +distance between the two points, then the propagation of light +must satisfy the equation +<span class="align-center"><img style="vertical-align: -0.186ex; width: 9.982ex; height: 1.805ex;" src="images/174.svg" alt=" " data-tex=" +r = c·\Delta t. +"></span> +</p> +<p> +If we square this equation, and express <img style="vertical-align: -0.025ex; width: 2.008ex; height: 1.912ex;" src="images/175.svg" alt=" " data-tex="r^{2}"> by the differences +of the co-ordinates, <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">, in place of this equation we can write +<span class="align-center"><img style="vertical-align: -1.018ex; width: 31.976ex; height: 3.167ex;" src="images/176.svg" alt=" " data-tex=" +\sum (\Delta x_{\nu})^{2} - c^{2} \Delta t^{2} = 0. \qquad\text{(22)} +"></span> +This equation formulates the principle of the constancy of the +velocity of light relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. It must hold whatever may be +the motion of the source which emits the ray of light. +<span class="pagenum" id="Page_30">[Pg 30]</span> +</p> +<p> +The same propagation of light may also be considered relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in which case also the principle of the constancy of +the velocity of light must be satisfied. Therefore, with respect +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', we have the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 34.175ex; height: 3.167ex;" src="images/177.svg" alt=" " data-tex=" +\sum ({\Delta x'}_{\nu})^{2} - c^{2} \Delta t'^{2} = 0. +\qquad\text{(22a)} +"></span> +</p> +<p> +Equations (22a) and (22) must be mutually consistent with +each other with respect to the transformation which transforms +from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. A transformation which effects this we shall call +a "Lorentz transformation." +</p> +<p> +Before considering these transformations in detail we shall +make a few general remarks about space and time. In the pre-relativity +physics space and time were separate entities. Specifications +of time were independent of the choice of the space of +reference. The Newtonian mechanics was relative with respect +to the space of reference, so that, e.g. the statement that two +non-simultaneous events happened at the same place had no objective +meaning (that is, independent of the space of reference). +But this relativity had no role in building up the theory. One +spoke of points of space, as of instants of time, as if they were +absolute realities. It was not observed that the true element +of the space-time specification was the event, specified by the +four numbers <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">. The conception of something +happening was always that of a four-dimensional continuum; but +the recognition of this was obscured by the absolute character +of the pre-relativity time. Upon giving up the hypothesis of the +absolute character of time, particularly that of simultaneity, the +four-dimensionality of the time-space concept was immediately +recognized. It is neither the point in space, nor the instant in +<span class="pagenum" id="Page_31">[Pg 31]</span> +time, at which something happens that has physical reality, but +only the event itself. There is no absolute (independent of the +space of reference) relation in space, and no absolute relation +in time between two events, but there is an absolute (independent +of the space of reference) relation in space and time, as +will appear in the sequel. The circumstance that there is no +objective rational division of the four-dimensional continuum +into a three-dimensional space and a one-dimensional time continuum +indicates that the laws of nature will assume a form +which is logically most satisfactory when expressed as laws in +the four-dimensional space-time continuum. Upon this depends +the great advance in method which the theory of relativity owes +to Minkowski. Considered from this standpoint, we must regard +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> as the four co-ordinates of an event in the +four-dimensional continuum. We have far less success in picturing +to ourselves relations in this four-dimensional continuum than +in the three-dimensional Euclidean continuum; but it must be +emphasized that even in the Euclidean three-dimensional geometry +its concepts and relations are only of an abstract nature in +our minds, and are not at all identical with the images we form +visually and through our sense of touch. The non-divisibility of +the four-dimensional continuum of events does not at all, however, +involve the equivalence of the space co-ordinates with the +time co-ordinate. On the contrary, we must remember that the +time co-ordinate is defined physically wholly differently from the +space co-ordinates. The relations (22) and (22a) which when +equated define the Lorentz transformation show, further, a difference +in the role of the time co-ordinate from that of the space +co-ordinates; for the term <img style="vertical-align: -0.025ex; width: 3.689ex; height: 1.912ex;" src="images/178.svg" alt=" " data-tex="\Delta t^{2}"> has the opposite sign to the space +terms, <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/179.svg" alt=" " data-tex="\Delta x_{1}^{2}">, <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/180.svg" alt=" " data-tex="\Delta x_{2}^{2}">, <img style="vertical-align: -0.685ex; width: 4.166ex; height: 2.572ex;" src="images/181.svg" alt=" " data-tex="\Delta x_{3}^{2}">. +<span class="pagenum" id="Page_32">[Pg 32]</span> +</p> +<p> +Before we analyse further the conditions which define the +Lorentz transformation, we shall introduce the light-time, <img style="vertical-align: -0.186ex; width: 5.488ex; height: 1.756ex;" src="images/182.svg" alt=" " data-tex="l = ct">, +in place of the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, in order that the constant <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/161.svg" alt=" " data-tex="c"> shall not +enter explicitly into the formulas to be developed later. Then +the Lorentz transformation is defined in such a way that, first, +it makes the equation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 41.26ex; height: 2.633ex;" src="images/183.svg" alt=" " data-tex=" +{\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 +\qquad\text{(22b)} +"></span> +a co-variant equation, that is, an equation which is satisfied with +respect to every inertial system if it is satisfied in the inertial +system to which we refer the two given events (emission and +reception of the ray of light). Finally, with Minkowski, we introduce +in place of the real time co-ordinate <img style="vertical-align: -0.186ex; width: 5.488ex; height: 1.756ex;" src="images/182.svg" alt=" " data-tex="l = ct">, the imaginary +time co-ordinate +<span class="align-center"><img style="vertical-align: -0.566ex; width: 25.618ex; height: 2.754ex;" src="images/184.svg" alt=" " data-tex=" +x_{4} = il = ict\quad (\sqrt{-1} = i). +"></span> +Then the equation defining the propagation of light, which must +be co-variant with respect to the Lorentz transformation, becomes +<span class="align-center"><img style="vertical-align: -3.222ex; width: 55.106ex; height: 5.371ex;" src="images/185.svg" alt=" " data-tex=" +\sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} + = 0. +\qquad\text{(22c)} +"></span> +This condition is always satisfied<a id="FNanchor_8_1"></a><a href="#Footnote_8_1" class="fnanchor">[8]</a> if we satisfy the more general +condition that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 42.527ex; height: 2.633ex;" src="images/186.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} +\qquad\text{(23)} +"></span> +<span class="pagenum" id="Page_33">[Pg 33]</span> +shall be an invariant with respect to the transformation. This +condition is satisfied only by linear transformations, that is, +transformations of the type +<span class="align-center"><img style="vertical-align: -0.685ex; width: 25.407ex; height: 2.515ex;" src="images/187.svg" alt=" " data-tex=" +{x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} +\qquad\text{(24)} +"></span> +in which the summation over the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> is to be extended from <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = 1 +to <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = 4. A glance at equations (23) and (24) shows that the +Lorentz transformation so defined is identical with the translational +and rotational transformations of the Euclidean geometry, +if we disregard the number of dimensions and the relations of reality. +We can also conclude that the coefficients <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"> must satisfy +the conditions +<span class="align-center"><img style="vertical-align: -0.685ex; width: 30.947ex; height: 2.382ex;" src="images/188.svg" alt=" " data-tex=" +b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}. +\qquad\text{(25)} +"></span> +Since the ratios of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> are real, it follows that all the <img style="vertical-align: -0.685ex; width: 2.349ex; height: 1.683ex;" src="images/189.svg" alt=" " data-tex="a_{\mu}"> and +the <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"> are real, except <img style="vertical-align: -0.339ex; width: 2.185ex; height: 1.337ex;" src="images/190.svg" alt=" " data-tex="a_{4}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/191.svg" alt=" " data-tex="b_{41}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/192.svg" alt=" " data-tex="b_{42}">, <img style="vertical-align: -0.375ex; width: 2.758ex; height: 1.945ex;" src="images/193.svg" alt=" " data-tex="b_{43}">, +<img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/194.svg" alt=" " data-tex="b_{14}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/195.svg" alt=" " data-tex="b_{24}"> and <img style="vertical-align: -0.375ex; width: 2.758ex; height: 1.945ex;" src="images/196.svg" alt=" " data-tex="b_{34}">, which +are purely imaginary. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_8_1"></a><a href="#FNanchor_8_1"><span class="label">[8]</span></a>That this specialization lies in the nature of the case will be evident +later.</p></div> + +<p><br></p> + +<p> +<i>Special Lorentz Transformation</i>. We obtain the simplest +transformations of the type of (24) and (25) if only two of the +co-ordinates are to be transformed, and if all the <img style="vertical-align: -0.685ex; width: 2.349ex; height: 1.683ex;" src="images/189.svg" alt=" " data-tex="a_{\mu}">, which determine +the new origin, vanish. We obtain then for the indices +1 and 2, on account of the three independent conditions which +the relations (25) furnish, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 34.703ex; height: 11.62ex;" src="images/197.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi, \\ +{x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi, \\ +{x'}_{3} &= x_{3}, && \\ +{x'}_{4} &= x_{4}. && +\end{alignedat} +\right\} +\qquad\text{(26)} +"></span> +<span class="pagenum" id="Page_34">[Pg 34]</span> +</p> +<p> +This is a simple rotation in space of the (space) co-ordinate +system about <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">-axis. We see that the rotational transformation +in space (without the time transformation) which we studied +before is contained in the Lorentz transformation as a special +case. For the indices 1 and 4 we obtain, in an analogous manner, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 36.083ex; height: 11.62ex;" src="images/198.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi, \\ +{x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi, \\ +{x'}_{2} &= x_{2}, && \\ +{x'}_{3} &= x_{3}. && +\end{alignedat} +\right\} +\qquad\text{(26a)} +"></span> +</p> +<p> +On account of the relations of reality <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/199.svg" alt=" " data-tex="\psi"> must be taken as +imaginary. To interpret these equations physically, we introduce +the real light-time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> and the velocity <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/201.svg" alt=" " data-tex="v"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +instead of the imaginary angle <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/199.svg" alt=" " data-tex="\psi">. We have, first, +<span class="align-center"><img style="vertical-align: -2.17ex; width: 26.107ex; height: 5.47ex;" src="images/202.svg" alt=" " data-tex=" +\begin{alignat*}{4} +{x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi, \\ +l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. +\end{alignat*} +"></span> +Since for the origin of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' i.e., for <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}"> = 0, we must have <img style="vertical-align: -0.339ex; width: 7.07ex; height: 1.91ex;" src="images/203.svg" alt=" " data-tex="x_{1} = vl">, +it follows from the first of these equations that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 19.945ex; height: 2.262ex;" src="images/204.svg" alt=" " data-tex=" +v = i\tan\psi, +\qquad\text{(27)} +"></span> +and also +<span class="align-center"><img style="vertical-align: -5.112ex; width: 27.989ex; height: 11.355ex;" src="images/205.svg" alt=" " data-tex=" +\begin{aligned} +\left. +\begin{aligned} +\sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}, \\ +\cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}, +\end{aligned} +\right\} +\qquad\text{(28)} +\end{aligned} +"></span> +<span class="pagenum" id="Page_35">[Pg 35]</span> +so that we obtain +<span class="align-center"><img style="vertical-align: -8.256ex; width: 26.022ex; height: 17.642ex;" src="images/206.svg" alt=" " data-tex=" +\left. +\begin{aligned} +{x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}, \\ +l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}, \\ +{x'}_{2} &= x_{2}, \\ +{x'}_{3} &= x_{3}. +\end{aligned} +\right\} +\qquad\text{(29)} +"></span> +</p> +<p> +These equations form the well-known special Lorentz transformation, +which in the general theory represents a rotation, +through an imaginary angle, of the four-dimensional system of +co-ordinates. If we introduce the ordinary time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, in place of the +light-time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l">, then in (29) we must replace <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> by <img style="vertical-align: -0.025ex; width: 1.796ex; height: 1.441ex;" src="images/207.svg" alt=" " data-tex="ct"> and <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/201.svg" alt=" " data-tex="v"> by +<img style="vertical-align: -1.577ex; width: 2.093ex; height: 4.109ex;" src="images/208.svg" alt=" " data-tex="\dfrac{v}{c}">. +</p> +<p> +We must now fill in a gap. From the principle of the constancy +of the velocity of light it follows that the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 12.994ex; height: 3.167ex;" src="images/209.svg" alt=" " data-tex=" +\sum {\Delta x_{\nu}}^{2} = 0 +"></span> +has a significance which is independent of the choice of the inertial +system; but the invariance of the quantity <img style="vertical-align: -0.559ex; width: 4.214ex; height: 2.446ex;" src="images/210.svg" alt=" " data-tex="\Delta x_{\nu}^{2}"> does +not at all follow from this. This quantity might be transformed +with a factor. This depends upon the fact that the right-hand +side of (29) might be multiplied by a factor <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda">, independent of <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/201.svg" alt=" " data-tex="v">. +But the principle of relativity does not permit this factor to be +different from 1, as we shall now show. Let us assume that we +have a rigid circular cylinder moving in the direction of its axis. +If its radius, measured at rest with a unit measuring rod is equal +to <img style="vertical-align: -0.375ex; width: 2.705ex; height: 1.92ex;" src="images/211.svg" alt=" " data-tex="R_{0}">, its radius <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/212.svg" alt=" " data-tex="R"> in motion, might be different from <img style="vertical-align: -0.375ex; width: 2.705ex; height: 1.92ex;" src="images/211.svg" alt=" " data-tex="R_{0}">, since +the theory of relativity does not make the assumption that the +shape of bodies with respect to a space of reference is independent +of their motion relatively to this space of reference. But +<span class="pagenum" id="Page_36">[Pg 36]</span> +all directions in space must be equivalent to each other. <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/212.svg" alt=" " data-tex="R"> may +therefore depend upon the magnitude <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/213.svg" alt=" " data-tex="q"> of the velocity, but not +upon its direction; <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/212.svg" alt=" " data-tex="R"> must therefore be an even function of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/213.svg" alt=" " data-tex="q">. If +the cylinder is at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' the equation of its lateral +surface is +<span class="align-center"><img style="vertical-align: -0.464ex; width: 15.362ex; height: 2.464ex;" src="images/214.svg" alt=" " data-tex=" +x'^{2} + y'^{2} = {R_{0}}^{2}. +"></span> +If we write the last two equations of (29) more generally +<span class="align-center"><img style="vertical-align: -2.17ex; width: 10.156ex; height: 5.47ex;" src="images/215.svg" alt=" " data-tex=" +\begin{align*} +{x'}_{2} = \lambda x_{2}, \\ +{x'}_{3} = \lambda x_{3}, +\end{align*} +"></span> +then the lateral surface of the cylinder referred to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> satisfies the +equation +<span class="align-center"><img style="vertical-align: -1.656ex; width: 15.478ex; height: 5.179ex;" src="images/216.svg" alt=" " data-tex=" +x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}. +"></span> +The factor <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> therefore measures the lateral contraction of the +cylinder, and can thus, from the above, be only an even function +of <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/201.svg" alt=" " data-tex="v">. +</p> +<p> +If we introduce a third system of co-ordinates, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"><code>"</code>, which +moves relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' with velocity <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/201.svg" alt=" " data-tex="v"> in the direction of the +negative <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/217.svg" alt=" " data-tex="x">-axis of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, we obtain, by applying (29) twice, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 19.39ex; height: 11.62ex;" src="images/218.svg" alt=" " data-tex=" +\begin{align*} +{x''}_{1} &= \lambda(v) \lambda(-v) x_{1}, \\ +{x''}_{2} &= \lambda(v) \lambda(-v) x_{2}, \\ +{x''}_{3} &= \lambda(v) \lambda(-v) x_{3}, \\ +l'' &= \lambda(v) \lambda(-v) l. +\end{align*} +"></span> +Now, since <img style="vertical-align: -0.566ex; width: 4.176ex; height: 2.262ex;" src="images/219.svg" alt=" " data-tex="\lambda(v)"> must be equal to <img style="vertical-align: -0.566ex; width: 5.937ex; height: 2.262ex;" src="images/220.svg" alt=" " data-tex="\lambda(-v)"> and since we assume +that we use the same measuring rods in all the systems, it follows +that the transformation of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"><code>"</code> to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> must be the identical +<span class="pagenum" id="Page_37">[Pg 37]</span> +transformation (since the possibility <img style="vertical-align: -0.186ex; width: 7.228ex; height: 1.756ex;" src="images/221.svg" alt=" " data-tex="\lambda = -1"> does not need to +be considered). It is essential for these considerations to assume +that the behaviour of the measuring rods does not depend upon +the history of their previous motion. +</p> + +<p><br></p> + +<p> +<i>Moving Measuring Rods and Clocks</i>. At the definite <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">-time, +<img style="vertical-align: -0.186ex; width: 4.823ex; height: 1.756ex;" src="images/222.svg" alt=" " data-tex="l = 0">, the position of the points given by the integers +<img style="vertical-align: -0.583ex; width: 6.656ex; height: 2.3ex;" src="images/223.svg" alt=" " data-tex="x'_{1} = n">, is with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, given by +<img style="vertical-align: -0.339ex; width: 14.568ex; height: 2.526ex;" src="images/224.svg" alt=" " data-tex="x_{1} = n {\sqrt{1 - v^{2}}}">; this follows +from the first of equations (29) and expresses the Lorentz +contraction. A clock at rest at the origin <img style="vertical-align: -0.339ex; width: 6.43ex; height: 1.846ex;" src="images/225.svg" alt=" " data-tex="x_{1} = 0"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, whose +beats are characterized by <img style="vertical-align: -0.186ex; width: 5.049ex; height: 1.756ex;" src="images/226.svg" alt=" " data-tex="l = n">, will, when observed from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +have beats characterized by +<span class="align-center"><img style="vertical-align: -2.308ex; width: 13.855ex; height: 4.837ex;" src="images/227.svg" alt=" " data-tex=" +l' = \frac{n}{\sqrt{1 - v^{2}}}; +"></span> +this follows from the second of equations (29) and shows that +the clock goes slower than if it were at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. +These two consequences, which hold, <i>mutatis mutandis</i>, for every +system of reference, form the physical content, free from +convention, of the Lorentz transformation. +</p> + +<p><br></p> + +<p> +<i>Addition Theorem for Velocities</i>. If we combine two special +Lorentz transformations with the relative velocities <img style="vertical-align: -0.339ex; width: 2.085ex; height: 1.342ex;" src="images/228.svg" alt=" " data-tex="v_{1}"> and <img style="vertical-align: -0.339ex; width: 2.085ex; height: 1.342ex;" src="images/229.svg" alt=" " data-tex="v_{2}">, +then the velocity of the single Lorentz transformation which +takes the place of the two separate ones is, according to (27), +given by +<span class="align-center"><img style="vertical-align: -2.016ex; width: 62.69ex; height: 5.115ex;" src="images/230.svg" alt=" " data-tex=" +v_{12} = i \tan (\psi_{1} + \psi_{2}) + = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} + = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. +\qquad\text{(30)} +"></span> +<span class="pagenum" id="Page_38">[Pg 38]</span> +</p> + +<p><br></p> + +<p> +<i>General Statements about the Lorentz Transformation and +its Theory of Invariants</i>. The whole theory of invariants of the +special theory of relativity depends upon the invariant <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}"> (23). +Formally, it has the same rôle in the four-dimensional space-time +continuum as the invariant +<img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/179.svg" alt=" " data-tex="\Delta x_{1}^{2}"> + <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/180.svg" alt=" " data-tex="\Delta x_{2}^{2}"> + <img style="vertical-align: -0.685ex; width: 4.166ex; height: 2.572ex;" src="images/181.svg" alt=" " data-tex="\Delta x_{3}^{2}"> in the Euclidean +geometry and in the pre-relativity physics. The latter quantity +is not an invariant with respect to all the Lorentz transformations; +the quantity <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}"> of equation (23) assumes the rôle of this +invariant. With respect to an arbitrary inertial system, <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}"> may +be determined by measurements; with a given unit of measure +it is a completely determinate quantity, associated with an arbitrary +pair of events. +</p> +<p> +The invariant <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}"> differs, disregarding the number of dimensions, +from the corresponding invariant of the Euclidean geometry +in the following points. In the Euclidean geometry <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}"> is +necessarily positive; it vanishes only when the two points concerned +come together. On the other hand, from the vanishing +of +<span class="align-center"><img style="vertical-align: -3.222ex; width: 44.378ex; height: 5.371ex;" src="images/232.svg" alt=" " data-tex=" +s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2} +"></span> +it cannot be concluded that the two space-time points fall together; +the vanishing of this quantity <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}">, is the invariant condition +that the two space-time points can be connected by a light +signal <i>in vacuo</i>. If <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> a point (event) represented in the +four-dimensional space of the <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> then all the "points" which +can be connected to <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> by means of a light signal lie upon the +cone <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}"> = 0 (compare Fig. 1, in which the dimension <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> is suppressed). +The "upper" half of the cone may contain the "points" +to which light signals can be sent from <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">; then the "lower" half +<span class="pagenum" id="Page_39">[Pg 39]</span> +of the cone will contain the "points" from which light signals +can be sent to <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">. The points <img style="vertical-align: 0; width: 2.452ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="P'"> enclosed by the conical surface +furnish, with <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">, a negative <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/231.svg" alt=" " data-tex="s^{2}">; <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/234.svg" alt=" " data-tex="PP'"> as well as <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/235.svg" alt=" " data-tex="P'P"> is then, +according to Minkowski, of the nature of a time. Such intervals +represent elements of possible paths of motion, the velocity being +less than that of light.<a id="FNanchor_9_1"></a><a href="#Footnote_9_1" class="fnanchor">[9]</a> In +this case the <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l">-axis may be drawn +<span class="pagenum" id="Page_40">[Pg 40]</span> +in the direction of <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/234.svg" alt=" " data-tex="PP'"> by suitably choosing the state of motion +of the inertial system. If <img style="vertical-align: 0; width: 2.452ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="P'"> lies outside of the "light-cone" then +<img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/234.svg" alt=" " data-tex="PP'"> is of the nature of a space; in this case, by properly choosing +the inertial system, <img style="vertical-align: -0.025ex; width: 2.559ex; height: 1.645ex;" src="images/236.svg" alt=" " data-tex="\Delta l"> can be made to vanish. +</p> + +<div class="figcenter" style="width: 400px;"> +<img src="images/figure01.jpg" width="400" alt="400"> +<div class="caption"> +<p>FIG. 1.</p> +</div></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_9_1"></a><a href="#FNanchor_9_1"><span class="label">[9]</span></a>That material velocities exceeding that of light are not possible, fol- +lows from the appearance of the radical <img style="vertical-align: -0.212ex; width: 7.912ex; height: 2.398ex;" src="images/237.svg" alt=" " data-tex="\sqrt{1 - v^{2}}"> in the special Lorentz +transformation (29).</p></div> + +<p> +By the introduction of the imaginary time variable, <img style="vertical-align: -0.339ex; width: 6.754ex; height: 1.91ex;" src="images/238.svg" alt=" " data-tex="x_{4} = il">, +Minkowski has made the theory of invariants for the four-dimensional +continuum of physical phenomena fully analogous +to the theory of invariants for the three-dimensional continuum +of Euclidean space. The theory of four-dimensional tensors of +special relativity differs from the theory of tensors in three-dimensional +space, therefore, only in the number of dimensions +and the relations of reality. +</p> +<p> +A physical entity which is specified by four quantities, <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/239.svg" alt=" " data-tex="A_{\nu}">, +in an arbitrary inertial system of the +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}">, is called +a 4-vector, with the components <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/239.svg" alt=" " data-tex="A_{\nu}">, if the <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/239.svg" alt=" " data-tex="A_{\nu}"> correspond in +their relations of reality and the properties of transformation to +the <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">; it may be of the nature of a space or of a time. The +sixteen quantities <img style="vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;" src="images/241.svg" alt=" " data-tex="A_{\mu\nu}">, then form the components of a tensor of +the second rank, if they transform according to the scheme +<span class="align-center"><img style="vertical-align: -0.685ex; width: 17.844ex; height: 2.515ex;" src="images/242.svg" alt=" " data-tex=" +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}. +"></span> +It follows from this that the <img style="vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;" src="images/241.svg" alt=" " data-tex="A_{\mu\nu}"> behave, with respect to +their properties of transformation and their properties of reality, +as the products of components, <img style="vertical-align: -0.685ex; width: 5.052ex; height: 2.23ex;" src="images/243.svg" alt=" " data-tex="U_{\mu}V_{\nu}"> of two 4-vectors, +(<img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/162.svg" alt=" " data-tex="U">) and (<img style="vertical-align: -0.05ex; width: 1.74ex; height: 1.595ex;" src="images/52.svg" alt=" " data-tex="V">). All the components are real except those which +contain the index 4 once, those being purely imaginary. Tensors +<span class="pagenum" id="Page_41">[Pg 41]</span> +of the third and higher ranks may be defined in an analogous +way. The operations of addition, subtraction, multiplication, +contraction and differentiation for these tensors are wholly +analogous to the corresponding operations for tensors in three-dimensional space. +</p> +<p> +Before we apply the tensor theory to the four-dimensional +space-time continuum, we shall examine more particularly the +skew-symmetrical tensors. The tensor of the second rank has, in +general, 16 = 4·4 components. In the case of skew-symmetry the +components with two equal indices vanish, and the components +with unequal indices are equal and opposite in pairs. There +exist, therefore, only six independent components, as is the case +in the electromagnetic field. In fact, it will be shown when we +consider Maxwell's equations that these may be looked upon as +tensor equations, provided we regard the electromagnetic field +as a skew-symmetrical tensor. Further, it is clear that the skew-symmetrical +tensor of the third rank (skew-symmetrical in all +pairs of indices) has only four independent components, since +there are only four combinations of three different indices. +</p> +<p> +We now turn to Maxwell's equations (19a), (19b), (20a), +(20b), and introduce the notation:<a id="FNanchor_10_1"></a><a href="#Footnote_10_1" class="fnanchor">[10]</a> +<span class="align-center"><img style="vertical-align: -6.265ex; width: 58.388ex; height: 13.661ex;" src="images/244.svg" alt=" " data-tex=" +\begin{align*} +&\left. +\begin{aligned} +&\phi_{23} && \phi_{31} && \phi_{12} && \phi_{14} && \phi_{24} && \phi_{34} \\ +&\mathbf h_{23} && \mathbf h_{31} && \mathbf h_{12} & -&i\mathbf e_{x} & -&i\mathbf e_{y} & -&i\mathbf e_{z} +\end{aligned} +\right\} +&&\text{(30a)}\\ +&\qquad\qquad\left. +\begin{aligned} +&J_{1} && J_{2} && J_{3} && J_{4} \\ +&\frac{1}{c}{\mathbf i_{x}} && +\frac{1}{c}{\mathbf i_{y}} && +\frac{1}{c}{\mathbf i_{z}} && i\rho +\end{aligned} +\right\} +&&\text{(31)} +\end{align*} +"></span> +with the convention that <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/245.svg" alt=" " data-tex="\phi_{\mu\nu}"> shall be equal to <img style="vertical-align: -0.685ex; width: 5.109ex; height: 2.255ex;" src="images/246.svg" alt=" " data-tex="-\phi_{\mu\nu}">. Then +<span class="pagenum" id="Page_42">[Pg 42]</span> +Maxwell's equations may be combined into the forms +<span class="align-center"><img style="vertical-align: -5.181ex; width: 40.27ex; height: 11.494ex;" src="images/247.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial \phi_{\mu\nu}}{\partial x_{\nu}} = J_{\mu}, +\qquad & &\text{(32)}\\ +\frac{\partial \phi_{\mu\nu}}{\partial x_{\sigma}} + +\frac{\partial \phi_{\nu\sigma}}{\partial x_{\mu}} + +\frac{\partial \phi_{\sigma\mu}}{\partial x_{\nu}} = 0, +\qquad & &\text{(33)} +\end{gather*} +"></span> +as one can easily verify by substituting from (30a) and (31). +Equations (32) and (33) have a tensor character, and are +therefore co-variant with respect to Lorentz transformations, +if the <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/245.svg" alt=" " data-tex="\phi_{\mu\nu}"> and the <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.885ex;" src="images/248.svg" alt=" " data-tex="J_{\nu}"> have a tensor character, which we assume. +Consequently, the laws for transforming these quantities from +one to another allowable (inertial) system of co-ordinates are +uniquely determined. The progress in method which electrodynamics +owes to the theory of special relativity lies principally +in this, that the number of independent hypotheses is diminished. +If we consider, for example, equations (19a) only from the +standpoint of relativity of direction, as we have done above, we +see that they have three logically independent terms. The way +in which the electric intensity enters these equations appears to +be wholly independent of the way in which the magnetic intensity +enters them; it would not be surprising if instead of +<img style="vertical-align: -1.602ex; width: 4.483ex; height: 4.945ex;" src="images/249.svg" alt=" " data-tex="\dfrac{\partial e_{\mu}}{\partial l}">, +we had, say, +<img style="vertical-align: -1.679ex; width: 5.588ex; height: 5.291ex;" src="images/250.svg" alt=" " data-tex="\dfrac{\partial^{2} e_{\mu}}{\partial l^{2}}"> +or if this term were absent. On the other +hand, only two independent terms appear in equation (32). The +electromagnetic field appears as a formal unit; the way in which +the electric field enters this equation is determined by the way in +which the magnetic field enters it. Besides the electromagnetic +field, only the electric current density appears as an independent +entity. This advance in method arises from the fact that the +<span class="pagenum" id="Page_43">[Pg 43]</span> +electric and magnetic fields draw their separate existences from +the relativity of motion. A field which appears to be purely an +electric field, judged from one system, has also magnetic field +components when judged from another inertial system. When +applied to an electromagnetic field, the general law of transformation +furnishes, for the special case of the special Lorentz +transformation, the equations +<span class="align-center"><img style="vertical-align: -6.897ex; width: 46.646ex; height: 14.925ex;" src="images/251.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{\mathbf e'}_{x} &= \mathbf e_{x}\qquad & {\mathbf h'}_{x} &= \mathbf h_{x}, \\ +{\mathbf e'}_{y} &= \frac{\mathbf e_{y} - v\mathbf h_{z}}{\sqrt{1 - v^{2}}}\qquad & +{\mathbf h'}_{y} &= \frac{\mathbf h_{y} + v\mathbf e_{z}}{\sqrt{1 - v^{2}}}, \\ +{\mathbf e'}_{z} &= \frac{\mathbf e_{z} + v\mathbf h_{y}}{\sqrt{1 - v^{2}}}\qquad & +{\mathbf h'}_{z} &= \frac{\mathbf h_{z} - v\mathbf e_{y}}{\sqrt{1 - v^{2}}}. +\end{alignedat} +\right\} +\qquad \text{(34)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_10_1"></a><a href="#FNanchor_10_1"><span class="label">[10]</span></a>In +order to avoid confusion from now on we shall use the three-dimensional +space indices, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/217.svg" alt=" " data-tex="x">, <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/252.svg" alt=" " data-tex="y">, <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/253.svg" alt=" " data-tex="z"> instead of 1, 2, 3, and we shall reserve the +numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum.</p></div> + +<p> +If there exists with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> only a magnetic field, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/148.svg" alt=" " data-tex="\mathbf h">, but +no electric field, <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/147.svg" alt=" " data-tex="\mathbf e">, then with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' there exists an electric +field <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/147.svg" alt=" " data-tex="\mathbf e"> as well, which would act upon an electric particle at rest +relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. An observer at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> would +designate this force as the Biot-Savart force, or the Lorentz electromotive +force. It therefore appears as if this electromotive force +had become fused with the electric field intensity into a single +entity. +</p> +<p> +In order to view this relation formally, let us consider the +expression for the force acting upon unit volume of electricity, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 23.503ex; height: 2.262ex;" src="images/254.svg" alt=" " data-tex=" +\mathbf k = \rho\mathbf e + [\mathbf i, \mathbf h], +\qquad \text{(35)} +"></span> +in which <img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/146.svg" alt=" " data-tex="\mathbf i"> is the vector velocity of electricity, with the velocity +of light as the unit. If we introduce <img style="vertical-align: -0.685ex; width: 2.408ex; height: 2.23ex;" src="images/255.svg" alt=" " data-tex="J_{\mu}"> and <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/245.svg" alt=" " data-tex="\phi_{\mu\nu}"> according to +(30a) and (31), we obtain for the first component the expression +<span class="align-center"><img style="vertical-align: -0.464ex; width: 22.298ex; height: 2.034ex;" src="images/256.svg" alt=" " data-tex=" +\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}. +"></span> +<span class="pagenum" id="Page_44">[Pg 44]</span> +Observing that <img style="vertical-align: -0.464ex; width: 3.136ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi_{11}"> vanishes on account of the skew-symmetry of +the tensor (<img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/258.svg" alt=" " data-tex="\phi">), the components of <img style="vertical-align: 0; width: 1.373ex; height: 1.57ex;" src="images/259.svg" alt=" " data-tex="\mathbf k"> are given by the first three +components of the four-dimensional vector +<span class="align-center"><img style="vertical-align: -0.685ex; width: 21.284ex; height: 2.382ex;" src="images/260.svg" alt=" " data-tex=" +K_{\mu} = \phi_{\mu\nu} J_{\nu}, +\qquad \text{(36)} +"></span> +and the fourth component is given by +<span class="align-center"><img style="vertical-align: -0.667ex; width: 65.109ex; height: 2.364ex;" src="images/261.svg" alt=" " data-tex=" +K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} + = i(\mathbf e_{x} \mathbf i_{x} + \mathbf e_{y} \mathbf i_{y} + \mathbf e_{z} \mathbf i_{z}) + = i \lambda. +\qquad \text{(37)} +"></span> +There is, therefore, a four-dimensional vector of force per unit +volume, whose first three components, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/262.svg" alt=" " data-tex="K_{1}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/263.svg" alt=" " data-tex="K_{2}">, <img style="vertical-align: -0.375ex; width: 2.908ex; height: 1.92ex;" src="images/264.svg" alt=" " data-tex="K_{3}">, are +the ponderomotive force components per unit volume, and whose fourth +component is the rate of working of the field per unit volume, +multiplied by <img style="vertical-align: -0.318ex; width: 4.821ex; height: 2.398ex;" src="images/265.svg" alt=" " data-tex="\sqrt{-1}">. +</p> +<p> +A comparison of (36) and (35) shows that the theory of relativity +formally unites the ponderomotive force of the electric +field, <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"><img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/147.svg" alt=" " data-tex="\mathbf e">, and the Biot-Savart or Lorentz force [<img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/146.svg" alt=" " data-tex="\mathbf i">, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/148.svg" alt=" " data-tex="\mathbf h">]. +</p> + +<p><br></p> + +<p> +<i>Mass and Energy</i>. An important conclusion can be drawn +from the existence and significance of the 4-vector <img style="vertical-align: -0.685ex; width: 3.073ex; height: 2.23ex;" src="images/266.svg" alt=" " data-tex="K_{\mu}">. Let us +imagine a body upon which the electromagnetic field acts for +a time. In the symbolic figure (Fig. 2) <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/267.svg" alt=" " data-tex="Ox_{1}"> designates the +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">-axis, and is at the same time a substitute for the three space axes +<img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/267.svg" alt=" " data-tex="Ox_{1}">, <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/268.svg" alt=" " data-tex="Ox_{2}">, <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/267.svg" alt=" " data-tex="Ox_{1}">; <img style="vertical-align: -0.05ex; width: 2.4ex; height: 1.643ex;" src="images/269.svg" alt=" " data-tex="Ol"> designates the real time axis. In this diagram +a body of finite extent is represented, at a definite time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l">, by +the interval <img style="vertical-align: 0; width: 3.414ex; height: 1.62ex;" src="images/170.svg" alt=" " data-tex="AB"> the whole space-time existence of the body is +represented by a strip whose boundary is everywhere inclined +less than 45° to the <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l">-axis. Between the time sections, +<img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{1}"> and <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/271.svg" alt=" " data-tex="l_{2}">, but not extending to them, +a portion of the strip is shaded. This represents the portion +of the space-time manifold +<span class="pagenum" id="Page_45">[Pg 45]</span> +in which the electromagnetic field acts upon the body, or upon +the electric charges contained in it, the action upon them being +transmitted to the body. We shall now consider the changes +which take place in the momentum and energy of the body as a +result of this action. +</p> + +<div class="figcenter" style="width: 400px;"> +<img src="images/figure02.jpg" width="400" alt="400"> +<div class="caption"> +<p>FIG. 2.</p> +</div></div> + +<p> +We shall assume that the principles of momentum and +energy are valid for the body. The change in momentum, +<img style="vertical-align: -0.357ex; width: 3.983ex; height: 1.977ex;" src="images/272.svg" alt=" " data-tex="\Delta I_{x}">, <img style="vertical-align: -0.667ex; width: 3.852ex; height: 2.287ex;" src="images/273.svg" alt=" " data-tex="\Delta I_{y}">, <img style="vertical-align: -0.357ex; width: 3.812ex; height: 1.977ex;" src="images/274.svg" alt=" " data-tex="\Delta I_{z}">, and the change +in energy, <img style="vertical-align: 0; width: 3.613ex; height: 1.62ex;" src="images/275.svg" alt=" " data-tex="\Delta E"> are then given +<span class="pagenum" id="Page_46">[Pg 46]</span> +by the expressions +<span class="align-center"><img style="vertical-align: -12.132ex; width: 60.141ex; height: 25.396ex;" src="images/276.svg" alt=" " data-tex=" +\begin{alignat*}{2} +\begin{aligned} +\Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{x}\, dx\, dy\, dz} + &&= \frac{1}{i}\int{K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{y}\, dx\, dy\, dz} + &&= \frac{1}{i}\int{K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{z}\, dx\, dy\, dz\,} + &&= \frac{1}{i}\int{K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta E &= \int_{l_{1}}^{l_{2}}dl \int{\lambda\, dx\, dy\, dz} + &&= \frac{1}{i}\int{ + \frac{1}{i}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}. +\end{aligned} +\end{alignat*} +"></span> +Since the four-dimensional element of volume is an invariant, +and (<img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/262.svg" alt=" " data-tex="K_{1}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/263.svg" alt=" " data-tex="K_{2}">, <img style="vertical-align: -0.375ex; width: 2.908ex; height: 1.92ex;" src="images/264.svg" alt=" " data-tex="K_{3}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/277.svg" alt=" " data-tex="K_{4}">) forms a 4-vector, the four-dimensional +integral extended over the shaded portion transforms as a 4-vector, +as does also the integral between the limits <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{1}"> and <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/271.svg" alt=" " data-tex="l_{2}">, because +the portion of the region which is not shaded contributes nothing +to the integral. It follows, therefore, that +<img style="vertical-align: -0.357ex; width: 3.983ex; height: 1.977ex;" src="images/272.svg" alt=" " data-tex="\Delta I_{x}">, <img style="vertical-align: -0.667ex; width: 3.852ex; height: 2.287ex;" src="images/273.svg" alt=" " data-tex="\Delta I_{y}">, <img style="vertical-align: -0.357ex; width: 3.812ex; height: 1.977ex;" src="images/274.svg" alt=" " data-tex="\Delta I_{z}">, <img style="vertical-align: -0.025ex; width: 4.394ex; height: 1.645ex;" src="images/278.svg" alt=" " data-tex="i \Delta E"> +form a 4-vector. Since the quantities themselves transform in +the same way as their increments, it follows that the aggregate +of the four quantities +<span class="align-center"><img style="vertical-align: -0.667ex; width: 13.217ex; height: 2.213ex;" src="images/279.svg" alt=" " data-tex=" +I_{x},\ I_{y},\ I_{z},\ iE +"></span> +has itself the properties of a vector; these quantities are referred +to an instantaneous condition of the body (e.g. at the time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{1}">). +</p> +<p> +This 4-vector may also be expressed in terms of the mass <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/104.svg" alt=" " data-tex="m">, +and the velocity of the body, considered as a material particle. +To form this expression, we note first, that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 66.28ex; height: 2.584ex;" src="images/280.svg" alt=" " data-tex=" +-ds^{2} = d\tau^{2} + = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} + = dl^{2}(1 - q^{2}) + \qquad \text{(38)} +"></span> +<span class="pagenum" id="Page_47">[Pg 47]</span> +is an invariant which refers to an infinitely short portion of the +four-dimensional line which represents the motion of the material +particle. The physical significance of the invariant <img style="vertical-align: -0.029ex; width: 2.346ex; height: 1.6ex;" src="images/281.svg" alt=" " data-tex="d \tau"> may +easily be given. If the time axis is chosen in such a way that it +has the direction of the fine differential which we are considering, +or, in other words, if we reduce the material particle to rest, +we shall then have <img style="vertical-align: -0.186ex; width: 7.214ex; height: 1.756ex;" src="images/282.svg" alt=" " data-tex="d \tau = dl">; this will therefore be measured by +the light-seconds clock which is at the same place, and at rest +relatively to the material particle. We therefore call <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> the proper +time of the material particle. As opposed to <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/283.svg" alt=" " data-tex="dl">, <img style="vertical-align: -0.029ex; width: 2.346ex; height: 1.6ex;" src="images/281.svg" alt=" " data-tex="d \tau"> is therefore an +invariant, and is practically equivalent to <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/283.svg" alt=" " data-tex="dl"> for motions whose +velocity is small compared to that of fight. Hence we see that +<span class="align-center"><img style="vertical-align: -1.581ex; width: 18.527ex; height: 4.681ex;" src="images/284.svg" alt=" " data-tex=" +u_{\sigma} = \frac{dx_{\sigma}}{d\tau} +\qquad \text{(39)} +"></span> +has, just as the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, the character of a vector; we shall designate +(<img style="vertical-align: -0.357ex; width: 2.395ex; height: 1.357ex;" src="images/285.svg" alt=" " data-tex="u_{\sigma}">) as the four-dimensional vector (in brief, 4-vector) of +velocity. Its components satisfy, by (38), the condition +<span class="align-center"><img style="vertical-align: -1.018ex; width: 22.112ex; height: 3.167ex;" src="images/286.svg" alt=" " data-tex=" +\sum {u_{\sigma}}^{2} = -1. +\qquad \text{(40)} +"></span> +We see that this 4-vector, whose components in the ordinary +notation are +<span class="align-center"><img style="vertical-align: -2.76ex; width: 55.726ex; height: 5.862ex;" src="images/287.svg" alt=" " data-tex=" +\frac{\mathbf q_{x}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf q_{y}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf q_{z}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf i}{\sqrt{1 - q^{2}}} +\qquad \text{(41)} +"></span> +is the only 4-vector which can be formed from the velocity components +of the material particle which are defined in three dimensions by +<span class="align-center"><img style="vertical-align: -1.577ex; width: 33.314ex; height: 4.676ex;" src="images/288.svg" alt=" " data-tex=" +\mathbf q_{x} = \frac{dx}{dl},\quad +\mathbf q_{y} = \frac{dy}{dl},\quad +\mathbf q_{z} = \frac{dz}{dl}. +"></span> +<span class="pagenum" id="Page_48">[Pg 48]</span> +We therefore see that +<span class="align-center"><img style="vertical-align: -2.148ex; width: 18.483ex; height: 5.444ex;" src="images/289.svg" alt=" " data-tex=" +\left(m \frac{dx_{\mu}}{d\tau}\right) +\qquad \text{(42)} +"></span> +must be that 4-vector which is to be equated to the 4-vector of +momentum and energy whose existence we have proved above. +By equating the components, we obtain, in three-dimensional +notation, +<span class="align-center"><img style="vertical-align: -11.148ex; width: 25.646ex; height: 23.428ex;" src="images/290.svg" alt=" " data-tex=" +\left. +\begin{aligned} +I_{x} = \frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}, \\ +I_{y} = \frac{m\mathbf q_{y}}{\sqrt{1 - q^{2}}}, \\ +I_{z} = \frac{m\mathbf q_{z}}{\sqrt{1 - q^{2}}}, \\ +E = \frac{m}{\sqrt{1 - q^{2}}}. +\end{aligned} +\right\} +\qquad \text{(43)} +"></span> +</p> +<p> +We recognize, in fact, that these components of momentum +agree with those of classical mechanics for velocities which are +small compared to that of light. For large velocities the momentum +increases more rapidly than linearly with the velocity, so as +to become infinite on approaching the velocity of light. +</p> +<p> +If we apply the last of equations (43) to a material particle +at rest (<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/213.svg" alt=" " data-tex="q"> = 0), we see that the energy, <img style="vertical-align: -0.375ex; width: 2.657ex; height: 1.913ex;" src="images/291.svg" alt=" " data-tex="E_{0}"> of a, body at rest is +equal to its mass. Had we chosen the second as our unit of time, +we would have obtained +<span class="align-center"><img style="vertical-align: -0.566ex; width: 19.182ex; height: 2.565ex;" src="images/292.svg" alt=" " data-tex=" +E_{0} = mc^{2}. + \qquad \text{(44)} +"></span> +Mass and energy are therefore essentially alike; they are only +different expressions for the same thing. The mass of a body +<span class="pagenum" id="Page_49">[Pg 49]</span> +is not a constant; it varies with changes in its energy.<a id="FNanchor_11_1"></a><a href="#Footnote_11_1" class="fnanchor">[11]</a> +We see from the last of equations (43) that <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/293.svg" alt=" " data-tex="E"> becomes infinite when +<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/213.svg" alt=" " data-tex="q"> approaches 1, the velocity of light. If we develop <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/293.svg" alt=" " data-tex="E"> in powers +of <img style="vertical-align: -0.439ex; width: 2.143ex; height: 2.326ex;" src="images/294.svg" alt=" " data-tex="q^{2}">, we obtain, +<span class="align-center"><img style="vertical-align: -1.602ex; width: 38.993ex; height: 4.636ex;" src="images/295.svg" alt=" " data-tex=" +E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots. + \qquad \text{(45)} +"></span> +The second term of this expansion corresponds to the kinetic +energy of the material particle in classical mechanics. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_11_1"></a><a href="#FNanchor_11_1"><span class="label">[11]</span></a>The emission of energy in radioactive processes is evidently connected +with the fact that the atomic weights are not integers. Attempts have been +made to draw conclusions from this concerning the structure and stability +of the atomic nuclei.</p></div> + +<p><br></p> + +<p> +<i>Equations of Motion of Material Particles</i>. From (43) we +obtain, by differentiating by the time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l">, and using the principle +of momentum, in the notation of three-dimensional vectors, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 30.382ex; height: 6.785ex;" src="images/296.svg" alt=" " data-tex=" +\mathbf{K} = \frac{d}{dl}\left(\frac{m\mathbf q}{\sqrt{1 - q^{2}}}\right). +\qquad \text{(46)} +"></span> +</p> +<p> +This equation, which was previously employed by H. A. +Lorentz for the motion of electrons, has been proved to be true, +with great accuracy, by experiments with <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">-rays. +</p> + +<p><br></p> + +<p> +<i>Energy Tensor of the Electromagnetic Field</i>. Before the development +of the theory of relativity it was known that the principles +of energy and momentum could be expressed in a differential +form for the electromagnetic field. The four-dimensional +formulation of these principles leads to an important conception, +<span class="pagenum" id="Page_50">[Pg 50]</span> +that of the energy tensor, which is important for the further development +of the theory of relativity. +</p> +<p> +If in the expression for the 4-vector of force per unit volume, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 12.359ex; height: 2.255ex;" src="images/297.svg" alt=" " data-tex=" +K_{\mu} = \phi_{\mu\nu} J_{\nu}, +"></span> +using the field equations (32), we express <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.885ex;" src="images/248.svg" alt=" " data-tex="J_{\nu}"> in terms of the +field intensities, <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/245.svg" alt=" " data-tex="\phi_{\mu\nu}">, we obtain, after some transformations and +repeated application of the field equations (32) and (33), the +expression +<span class="align-center"><img style="vertical-align: -1.891ex; width: 23.002ex; height: 5.235ex;" src="images/298.svg" alt=" " data-tex=" +K_{\mu} = -\frac{\partial T_{\mu\nu}}{\partial x_{\nu}}, +\qquad \text{(47)} +"></span> +where we have written<a id="FNanchor_12_1"></a><a href="#Footnote_12_1" class="fnanchor">[12]</a> +<span class="align-center"><img style="vertical-align: -0.781ex; width: 36.604ex; height: 2.799ex;" src="images/299.svg" alt=" " data-tex=" +T_{\mu\nu} + = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + + \phi_{\mu\alpha} \phi_{\nu\alpha}. +\qquad \text{(48)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_12_1"></a><a href="#FNanchor_12_1"><span class="label">[12]</span></a>To be summed for the indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">.</p></div> + +<p> +The physical meaning of equation (47) becomes evident if in +place of this equation we write, using a new notation, +<span class="align-center"><img style="vertical-align: -11.527ex; width: 55.328ex; height: 24.185ex;" src="images/300.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{4} +\mathbf k_{x} &= -\frac{\partial p_{xx}}{\partial x} + &&- \frac{\partial p_{xy}}{\partial y} + &&- \frac{\partial p_{xz}}{\partial z} + &&- \frac{\partial (i b_{x})}{\partial (il)}, \\ +\mathbf k_{y} &= -\frac{\partial p_{yx}}{\partial x} + &&- \frac{\partial p_{yy}}{\partial y} + &&- \frac{\partial p_{yz}}{\partial z} + &&- \frac{\partial (i b_{y})}{\partial (il)}, \\ +\mathbf k_{z} &= -\frac{\partial p_{zx}}{\partial x} + &&- \frac{\partial p_{zy}}{\partial y} + &&- \frac{\partial p_{zz}}{\partial z} + &&- \frac{\partial (i b_{z})}{\partial (il)}, \\ +i\lambda + &= -\frac{\partial (i\mathbf s_{x})}{\partial x} + &&- \frac{\partial (i\mathbf s_{y})}{\partial y} + &&- \frac{\partial (i\mathbf s_{z})}{\partial z} + &&- \frac{\partial (-\eta)}{\partial (i l)}; +\end{alignedat} +\right\} +\qquad \text{(47a)} +"></span> +<span class="pagenum" id="Page_51">[Pg 51]</span> +or, on eliminating the imaginary, +<span class="align-center"><img style="vertical-align: -11.136ex; width: 48.145ex; height: 23.402ex;" src="images/301.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{4} +k_{x} &= -\frac{\partial p_{xx}}{\partial x} + &&- \frac{\partial p_{xy}}{\partial y} + &&- \frac{\partial p_{xz}}{\partial z} + &&- \frac{\partial b_{x}}{\partial l}, \\ +k_{y} &= -\frac{\partial p_{yx}}{\partial x} + &&- \frac{\partial p_{yy}}{\partial y} + &&- \frac{\partial p_{yz}}{\partial z} + &&- \frac{\partial b_{y}}{\partial l}, \\ +k_{z} &= -\frac{\partial p_{zx}}{\partial x} + &&- \frac{\partial p_{zy}}{\partial y} + &&- \frac{\partial p_{zz}}{\partial z} + &&- \frac{\partial b_{z}}{\partial l}, \\ +\lambda + &= -\frac{\partial s_{x}}{\partial x} + &&- \frac{\partial s_{y}}{\partial y} + &&- \frac{\partial s_{z}}{\partial z} + &&- \frac{\partial \eta}{\partial l}; +\end{alignedat} +\right\} +\qquad \text{(47b)} +"></span> +</p> +<p> +When expressed in the latter form, we see that the first three +equations state the principle of momentum; <img style="vertical-align: -0.439ex; width: 3.156ex; height: 1.439ex;" src="images/302.svg" alt=" " data-tex="p_{xx}">,..., <img style="vertical-align: -0.439ex; width: 2.985ex; height: 1.439ex;" src="images/303.svg" alt=" " data-tex="p_{zx}"> are the +Maxwell stresses in the electromagnetic field, and (<img style="vertical-align: -0.357ex; width: 2.073ex; height: 1.927ex;" src="images/304.svg" alt=" " data-tex="b_{x}">, <img style="vertical-align: -0.667ex; width: 1.942ex; height: 2.237ex;" src="images/305.svg" alt=" " data-tex="b_{y}">, <img style="vertical-align: -0.357ex; width: 1.902ex; height: 1.927ex;" src="images/306.svg" alt=" " data-tex="b_{z}">) is +the vector momentum per unit volume of the field. The last of +equations (47b) expresses the energy principle; <img style="vertical-align: -0.014ex; width: 1.027ex; height: 1.038ex;" src="images/307.svg" alt=" " data-tex="\mathbf s"> is the vector +flow of energy, and <img style="vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;" src="images/308.svg" alt=" " data-tex="\eta"> the energy per unit volume of the field. In +fact, we get from (48) by introducing the well-known expressions +for the components of the field intensity from electrodynamics, +<span class="align-center"><img style="vertical-align: -14.002ex; width: 63.031ex; height: 29.135ex;" src="images/309.svg" alt=" " data-tex=" +\left. +\begin{aligned} +&\begin{alignedat}{4} +p_{xx} = &{} - \mathbf h_{x} \mathbf h_{x} + &&+ \tfrac{1}{2}({\mathbf h_{x}}^{2} &&+ {\mathbf h_{y}}^{2} &&+ {\mathbf h_{z}}^{2}) \\ + &{} - \mathbf e_{x} \mathbf e_{y} + &&+ \tfrac{1}{2}({\mathbf e_{x}}^{2} &&+ {\mathbf e_{y}}^{2} &&+ {\mathbf e_{z}}^{2}), +\end{alignedat} \\ +&\qquad\qquad\qquad\qquad\qquad +\begin{alignedat}{3} +p_{xy} = &{} - \mathbf h_{x} \mathbf h_{y}\quad + && p_{xz} = && {} - \mathbf h_{x} \mathbf h_{z} \\ + &{} - \mathbf e_{x} \mathbf e_{y},\quad + && && {} - \mathbf e_{x} \mathbf e_{z}, +\end{alignedat} \\ +&\qquad\qquad\qquad\qquad\qquad\quad\vdots \\ +&b_{x} = \mathbf s_{x} = \mathbf e_{y} \mathbf h_{z} - \mathbf e_{z}\mathbf h_{y}, \\ +&b_{y} = \mathbf s_{y} = \mathbf e_{z} \mathbf h_{x} - \mathbf e_{x}\mathbf h_{z}, \\ +&b_{z} = \mathbf s_{z} = \mathbf e_{x} \mathbf h_{y} - \mathbf e_{y}\mathbf h_{x}, \\ +&\eta = +\tfrac{1}{2}({\mathbf e_{x}}^{2} + {\mathbf e_{y}}^{2} + {\mathbf e_{z}}^{2} + + {\mathbf h_{x}}^{2} + {\mathbf h_{y}}^{2} + {\mathbf h_{z}}^{2}). +\end{aligned} +\right\} +\qquad \text{(48a)} +"></span> +<span class="pagenum" id="Page_52">[Pg 52]</span> +We conclude from (48) that the energy tensor of the electromagnetic +field is symmetrical; with this is connected the fact +that the momentum per unit volume and the how of energy are +equal to each other (relation between energy and inertia). +</p> +<p> +We therefore conclude from these considerations that the +energy per unit volume has the character of a tensor. This has +been proved directly only for an electromagnetic field, although +we may claim universal validity for it. Maxwell's equations determine +the electromagnetic field when the distribution of electric +charges and currents is known. But we do not know the +laws which govern the currents and charges. We do know, indeed, +that electricity consists of elementary particles (electrons, +positive nuclei), but from a theoretical point of view we cannot +comprehend this. We do not know the energy factors which +determine the distribution of electricity in particles of definite +size and charge, and all attempts to complete the theory in this +direction have failed. If then we can build upon Maxwell's equations +in general, the energy tensor of the electromagnetic field +is known only outside the charged particles.<a id="FNanchor_13_1"></a><a href="#Footnote_13_1" class="fnanchor">[13]</a> In these regions, +outside of charged particles, the only regions in which we can believe +that we have the complete expression for the energy tensor, +we have, by (47), +<span class="align-center"><img style="vertical-align: -1.891ex; width: 19.927ex; height: 5.235ex;" src="images/310.svg" alt=" " data-tex=" +\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = 0. +\qquad \text{(47c)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_13_1"></a><a href="#FNanchor_13_1"><span class="label">[13]</span></a>It has been attempted to remedy this lack of knowledge by considering +the charged particles as proper singularities. But in my opinion this means +giving up a real understanding of the structure of matter. It seems to me +much better to give in to our present inability rather than to be satisfied +by a solution that is only apparent.</p></div> +<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> +<p><br></p> + +<p> +<i>General Expressions for the Conservation Principles</i>. We +can hardly avoid making the assumption that in all other cases, +also, the space distribution of energy is given by a symmetrical +tensor, <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}">, and that this complete energy tensor everywhere +satisfies the relation (47c). At any rate we shall see that by +means of this assumption we obtain the correct expression for +the integral energy principle. +</p> +<p> +Let us consider a spatially bounded, closed system, which, +four-dimensionally, we may represent as a strip, outside of which +the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}"> vanish. Integrate equation (47c) over a space section. +Since the integrals of +<img style="vertical-align: -1.891ex; width: 5.55ex; height: 5.235ex;" src="images/312.svg" alt=" " data-tex="\dfrac{\partial T_{\mu1}}{\partial x_{1}}">, +<img style="vertical-align: -1.891ex; width: 5.55ex; height: 5.235ex;" src="images/313.svg" alt=" " data-tex="\dfrac{\partial T_{\mu2}}{\partial x_{2}}"> and +<img style="vertical-align: -1.927ex; width: 5.55ex; height: 5.27ex;" src="images/314.svg" alt=" " data-tex="\dfrac{\partial T_{\mu3}}{\partial x_{3}}"> +vanish because +the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}"> vanish at the limits of integration, we obtain +<span class="align-center"><img style="vertical-align: -2.148ex; width: 36.963ex; height: 5.428ex;" src="images/315.svg" alt=" " data-tex=" +\frac{\partial}{\partial l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} + = 0. +\qquad \text{(49)} +"></span> +Inside the parentheses are the expressions for the momentum of +the whole system, multiplied by <img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/316.svg" alt=" " data-tex="i">, together with the negative +energy of the system, so that (49) expresses the conservation +principles in their integral form. That this gives the right conception +of energy and the conservation principles will be seen +from the following considerations. +</p> + +<p><br></p> + +<p class="center"> +PHENOMENOLOGICAL REPRESENTATION OF THE +ENERGY TENSOR OF MATTER.</p> + +<p> +<i>Hydrodynamical Equations</i>. We know that matter is built +up of electrically charged particles, but we do not know the laws +which govern the constitution of these particles. In treating mechanical +problems, we are therefore obliged to make use of an +<span class="pagenum" id="Page_54">[Pg 54]</span> +inexact description of matter, which corresponds to that of classical +mechanics. The density <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, of a material substance and the +hydrodynamical pressures are the fundamental concepts upon +which such a description is based. +</p> + +<div class="figcenter" style="width: 400px;"> +<img src="images/figure03.jpg" width="400" alt="400"> +<div class="caption"> +<p>FIG. 3.</p> +</div></div> + +<p> +Let <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/317.svg" alt=" " data-tex="\sigma_{0}"> be the density of matter at a place, estimated with +reference to a system of co-ordinates moving with the matter. +Then <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/317.svg" alt=" " data-tex="\sigma_{0}">, the density at rest, is an invariant. If we think of the +matter in arbitrary motion and neglect the pressures (particles +of dust <i>in vacuo</i>, neglecting the size of the particles and the +temperature), then the energy tensor will depend only upon the +<span class="pagenum" id="Page_55">[Pg 55]</span> +velocity components, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}"> and <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/317.svg" alt=" " data-tex="\sigma_{0}">. We secure the tensor character +of <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}"> by putting +<span class="align-center"><img style="vertical-align: -0.685ex; width: 22.948ex; height: 2.382ex;" src="images/318.svg" alt=" " data-tex=" +T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}, + \qquad \text{(50)} +"></span> +in which the <img style="vertical-align: -0.685ex; width: 2.447ex; height: 1.685ex;" src="images/319.svg" alt=" " data-tex="u_{\mu}">, in the three-dimensional representation, are +given by (41). In fact, it follows from (50) that for <img style="vertical-align: -0.439ex; width: 5.189ex; height: 1.946ex;" src="images/320.svg" alt=" " data-tex="q = 0">, +<img style="vertical-align: -0.375ex; width: 10.166ex; height: 1.906ex;" src="images/321.svg" alt=" " data-tex="T_{44} = -\sigma_{0}"> +(equal to the negative energy per unit volume), as it should, +according to the theorem of the equivalence of mass and energy, +and according to the physical interpretation of the energy tensor +given above. If an external force (four-dimensional vector, <img style="vertical-align: -0.685ex; width: 3.073ex; height: 2.23ex;" src="images/266.svg" alt=" " data-tex="K_{\mu}">) +acts upon the matter, by the principles of momentum and energy +the equation +<span class="align-center"><img style="vertical-align: -1.891ex; width: 11.688ex; height: 5.235ex;" src="images/322.svg" alt=" " data-tex=" +K_{\mu} = \frac{\partial T_{\mu\nu}}{\partial x_{\nu}} +"></span> +must hold. We shall now show that this equation leads to the +same law of motion of a material particle as that already obtained. +Let us imagine the matter to be of infinitely small extent +in space, that is, a four-dimensional thread; then by integration +over the whole thread with respect to the space co-ordinates +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, we obtain +<span class="align-center"><img style="vertical-align: -5.035ex; width: 54.942ex; height: 11.201ex;" src="images/323.svg" alt=" " data-tex=" +\begin{align*} +\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} + &= \int \frac{\partial T_{14}}{\partial x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ + &= -i \frac{d}{dl}\left\{ + \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} + \right\}. +\end{align*} +"></span> +</p> +<p> +Now <img style="vertical-align: -0.691ex; width: 16.724ex; height: 2.514ex;" src="images/324.svg" alt=" " data-tex="\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}"> is an invariant, as is, therefore, also +<img style="vertical-align: -0.691ex; width: 19.381ex; height: 2.514ex;" src="images/325.svg" alt=" " data-tex="\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}">. +We shall calculate this integral, first with +respect to the inertial system which we have chosen, and second, +with respect to a system relatively to which the matter has the +velocity zero. The integration is to be extended over a filament +<span class="pagenum" id="Page_56">[Pg 56]</span> +of the thread for which <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/317.svg" alt=" " data-tex="\sigma_{0}"> may be regarded as constant over the +whole section. If the space volumes of the filament referred to +the two systems are <img style="vertical-align: -0.05ex; width: 2.916ex; height: 1.62ex;" src="images/326.svg" alt=" " data-tex="dV"> and <img style="vertical-align: -0.375ex; width: 3.483ex; height: 1.945ex;" src="images/327.svg" alt=" " data-tex="dV_{0}"> respectively, then we have +<span class="align-center"><img style="vertical-align: -1.948ex; width: 24.709ex; height: 5.027ex;" src="images/328.svg" alt=" " data-tex=" +\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau +"></span> +and therefore also +<span class="align-center"><img style="vertical-align: -1.948ex; width: 38.788ex; height: 5.048ex;" src="images/329.svg" alt=" " data-tex=" +\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} + = \int dm\, i\, \frac{d\tau}{dx_{4}}. +"></span> +</p> +<p> +If we substitute the right-hand side for the left-hand side in +the former integral, and put <img style="vertical-align: -1.581ex; width: 4.454ex; height: 4.681ex;" src="images/330.svg" alt=" " data-tex="\dfrac{dx_{1}}{d\tau}"> outside the sign of integration, +we obtain, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 38.194ex; height: 6.785ex;" src="images/331.svg" alt=" " data-tex=" +\mathbf K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) + = \frac{d}{dl}\left(\frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}\right). +"></span> +We see, therefore, that the generalized conception of the energy +tensor is in agreement with our former result. +</p> + +<p><br></p> + +<p> +<i>The Eulerian Equations for Perfect Fluids</i>. In order to get +nearer to the behaviour of real matter we must add to the energy +tensor a term which corresponds to the pressures. The simplest +case is that of a perfect fluid in which the pressure is determined +by a scalar <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/134.svg" alt=" " data-tex="p">. Since the tangential stresses <img style="vertical-align: -0.667ex; width: 3.025ex; height: 1.667ex;" src="images/332.svg" alt=" " data-tex="p_{xy}">, etc., vanish in +this case, the contribution to the energy tensor must be of the +form <img style="vertical-align: -0.685ex; width: 4.143ex; height: 2.307ex;" src="images/333.svg" alt=" " data-tex="p\delta_{\nu\mu}">. We must therefore put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 28.869ex; height: 2.382ex;" src="images/334.svg" alt=" " data-tex=" +T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}. +\qquad \text{(51)} +"></span> +<span class="pagenum" id="Page_57">[Pg 57]</span> +At rest, the density of the matter, or the energy per unit volume, +is in this case, not <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma"> but <img style="vertical-align: -0.439ex; width: 5.196ex; height: 1.758ex;" src="images/335.svg" alt=" " data-tex="\sigma - p">. For +<span class="align-center"><img style="vertical-align: -1.581ex; width: 35.761ex; height: 4.681ex;" src="images/336.svg" alt=" " data-tex=" +-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} + = \sigma - p. +"></span> +In the absence of any force, we have +<span class="align-center"><img style="vertical-align: -2.237ex; width: 42.095ex; height: 5.58ex;" src="images/337.svg" alt=" " data-tex=" +\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} + = \sigma u_{\nu} \frac{\partial u_{\mu}}{\partial x_{\nu}} + + u_{\mu} \frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + + \frac{\partial p}{\partial x_{\mu}} = 0. +"></span> +If we multiply this equation by <img style="vertical-align: -2.148ex; width: 13.161ex; height: 5.444ex;" src="images/338.svg" alt=" " data-tex="u_{\mu} \left(= \dfrac{dx_{\mu}}{d\tau}\right)"> +and sum for the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">'s we obtain, using (40), +<span class="align-center"><img style="vertical-align: -1.891ex; width: 29.227ex; height: 5.195ex;" src="images/339.svg" alt=" " data-tex=" +-\frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{dp}{d\tau} = 0, +\qquad \text{(52)} +"></span> +where we have put <img style="vertical-align: -2.237ex; width: 16.078ex; height: 5.533ex;" src="images/340.svg" alt=" " data-tex="\dfrac{\partial p}{\partial x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}">. +This is the equation of continuity, which differs from that of +classical mechanics by the term <img style="vertical-align: -1.581ex; width: 3.342ex; height: 4.681ex;" src="images/341.svg" alt=" " data-tex="\dfrac{dp}{d\tau}">, which, practically, +is vanishingly small. Observing (52), the conservation principles take the form +<span class="align-center"><img style="vertical-align: -2.237ex; width: 35.277ex; height: 5.533ex;" src="images/342.svg" alt=" " data-tex=" +\sigma \frac{du_{\mu}}{d\tau} + + u_{\mu} \frac{dp}{d\tau} + + \frac{\partial p}{\partial x_{\mu}} = 0. +\qquad \text{(53)} +"></span> +The equations for the first three indices evidently correspond to +the Eulerian equations. That the equations (52) and (53) correspond, +to a first approximation, to the hydrodynamical equations +of classical mechanics, is a further confirmation of the generalized +energy principle. The density of matter and of energy +has the character of a symmetrical tensor. +<span class="pagenum" id="Page_58">[Pg 58]</span> +</p> + +<p><br><br><br></p> + +<h3><a id="chap03"></a>LECTURE III +<br><br> +THE GENERAL THEORY OF RELATIVITY</h3> + +<p class="nind"> +<span class="dropcap">A</span>LL of the previous considerations have been based upon the +assumption that all inertial systems are equivalent for the description +of physical phenomena, but that they are preferred, for +the formulation of the laws of nature, to spaces of reference in a +different state of motion. We can think of no cause for this preference +for definite states of motion to all others, according to our +previous considerations, either in the perceptible bodies or in the +concept of motion; on the contrary, it must be regarded as an independent +property of the space-time continuum. The principle +of inertia, in particular, seems to compel us to ascribe physically +objective properties to the space-time continuum. Just as +it was necessary from the Newtonian standpoint to make both +the statements, <i>tempus est absolutum</i>, <i>spatium est absolutum</i>, so +from the standpoint of the special theory of relativity we must +say, <i>continuum spatii et temporis est absolutum</i>. In this latter +statement <i>absolutum</i> means not only "physically real," but also +"independent in its physical properties, having a physical effect, +but not itself influenced by physical conditions." +</p> +<p> +As long as the principle of inertia is regarded as the keystone +of physics, this standpoint is certainly the only one which +is justified. But there are two serious criticisms of the ordinary +conception. In the first place, it is contrary to the mode of thinking +in science to conceive of a thing (the space-time continuum) +which acts itself, but which cannot be acted upon. This is the +reason why E. Mach was led to make the attempt to eliminate +space as an active cause in the system of mechanics. According +<span class="pagenum" id="Page_59">[Pg 59]</span> +to him, a material particle does not move in unaccelerated +motion relatively to space, but relatively to the centre of all the +other masses in the universe; in this way the series of causes of +mechanical phenomena was closed, in contrast to the mechanics +of Newton and Galileo. In order to develop this idea within the +limits of the modern theory of action through a medium, the +properties of the space-time continuum which determine inertia +must be regarded as field properties of space, analogous to the +electromagnetic field. The concepts of classical mechanics afford +no way of expressing this. For this reason Mach's attempt +at a solution failed for the time being. We shall come back to +this point of view later. In the second place, classical mechanics +indicates a limitation which directly demands an extension of +the principle of relativity to spaces of reference which are not +in uniform motion relatively to each other. The ratio of the +masses of two bodies is defined in mechanics in two ways which +differ from each other fundamentally; in the first place, as the +reciprocal ratio of the accelerations which the same motional +force imparts to them (inert mass), and in the second place, as +the ratio of the forces which act upon them in the same gravitational +field (gravitational mass). The equality of these two +masses, so differently defined, is a fact which is confirmed by +experiments of very high accuracy (experiments of Eötvös), and +classical mechanics offers no explanation for this equality. It is, +however, clear that science is fully justified in assigning such a +numerical equality only after this numerical equality is reduced +to an equality of the real nature of the two concepts. +</p> +<p> +That this object may actually be attained by an extension +of the principle of relativity, follows from the following consideration. +A little reflection will show that the theorem of the +<span class="pagenum" id="Page_60">[Pg 60]</span> +equality of the inert and the gravitational mass is equivalent +to the theorem that the acceleration imparted to a body by a +gravitational field is independent of the nature of the body. For +Newton's equation of motion in a gravitational field, written out +in full, is +<span class="align-center"><img style="vertical-align: -2.036ex; width: 69.485ex; height: 5.204ex;" src="images/343.svg" alt=" " data-tex=" +\begin{aligned} +(\text{Inert mass})·(\text{Acceleration}) = \,(\text{Intensity of the} \\ +\qquad + \text{gravitational field}) &· (\text{Gravitational mass}). +\end{aligned} +"></span> +It is only when there is numerical equality between the inert +and gravitational mass that the acceleration is independent of +the nature of the body. Let now <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> be an inertial system. Masses +which are sufficiently far from each other and from other bodies +are then, with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, free from acceleration. We shall +also refer these masses to a system of co-ordinates <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' uniformly +accelerated with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. Relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' all the masses +have equal and parallel accelerations; with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' they +behave just as if a gravitational field were present and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' were +unaccelerated. Overlooking for the present the question as to the +"cause" of such a gravitational field, which will occupy us later, +there is nothing to prevent our conceiving this gravitational field +as real, that is, the conception that <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is "at rest" and a gravitational +field is present we may consider as equivalent to the conception +that only <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is an "allowable" system of co-ordinates and +no gravitational field is present. The assumption of the complete +physical equivalence of the systems of co-ordinates, +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +we call the "principle of equivalence;" this principle is evidently +intimately connected with the theorem of the equality between +the inert and the gravitational mass, and signifies an extension +of the principle of relativity to co-ordinate systems which are in +<span class="pagenum" id="Page_61">[Pg 61]</span> +non-uniform motion relatively to each other. In fact, through +this conception we arrive at the unity of the nature of inertia +and gravitation. For according to our way of looking at it, the +same masses may appear to be either under the action of inertia +alone (with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">) or under the combined action of +inertia and gravitation (with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'). The possibility of +explaining the numerical equality of inertia and gravitation by +the unity of their nature gives to the general theory of relativity, +according to my conviction, such a superiority over the conceptions +of classical mechanics, that all the difficulties encountered +in development must be considered as small in comparison. +</p> +<p> +What justifies us in dispensing with the preference for inertial +systems over all other co-ordinate systems, a preference +that seems so securely established by experiment based upon +the principle of inertia? The weakness of the principle of inertia +lies in this, that it involves an argument in a circle: a mass moves +without acceleration if it is sufficiently far from other bodies; we +know that it is sufficiently far from other bodies only by the fact +that it moves without acceleration. Are there, in general, any +inertial systems for very extended portions of the space-time +continuum, or, indeed, for the whole universe? We may look +upon the principle of inertia as established, to a high degree of +approximation, for the space of our planetary system, provided +that we neglect the perturbations due to the sun and planets. +Stated more exactly, there are finite regions, where, with respect +to a suitably chosen space of reference, material particles move +freely without acceleration, and in which the laws of the special +theory of relativity, which have been developed above, hold +with remarkable accuracy. Such regions we shall call "Galilean +regions." We shall proceed from the consideration of such regions +<span class="pagenum" id="Page_62">[Pg 62]</span> +as a special case of known properties. +</p> +<p> +The principle of equivalence demands that in dealing with +Galilean regions we may equally well make use of non-inertial +systems, that is, such co-ordinate systems as, relatively to inertial +systems, are not free from acceleration and rotation. If, +further, we are going to do away completely with the difficult +question as to the objective reason for the preference of certain +systems of co-ordinates, then we must allow the use of arbitrarily +moving systems of co-ordinates. As soon as we make this +attempt seriously we come into conflict with that physical interpretation +of space and time to which we were led by the special +theory of relativity. For let <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' be a system of co-ordinates whose +<img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/253.svg" alt=" " data-tex="z">'-axis coincides with the <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/253.svg" alt=" " data-tex="z">-axis of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, and which rotates about +the latter axis with constant angular velocity. Are the configurations +of rigid bodies, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in accordance with +the laws of Euclidean geometry? Since <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is not an inertial system, +we do not know directly the laws of configuration of rigid +bodies with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', nor the laws of nature, in general. But +we do know these laws with respect to the inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +and we can therefore estimate them with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. Imagine +a circle drawn about the origin in the <img style="vertical-align: -0.464ex; width: 6.424ex; height: 2.181ex;" src="images/344.svg" alt=" " data-tex="x'-y'"> plane of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' and a +diameter of this circle. Imagine, further, that we have given a +large number of rigid rods, all equal to each other. We suppose +these laid in series along the periphery and the diameter of the +circle, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. If <img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/162.svg" alt=" " data-tex="U"> is the number of these rods +along the periphery, <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/80.svg" alt=" " data-tex="D"> the number along the diameter, then, if +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' does not rotate relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, we shall have +<span class="align-center"><img style="vertical-align: -1.552ex; width: 7.804ex; height: 4.627ex;" src="images/345.svg" alt=" " data-tex=" +\frac{U}{D} = \pi. +"></span> +<span class="pagenum" id="Page_63">[Pg 63]</span> +But if <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' rotates we get a different result. Suppose that at +a definite time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> we determine the ends of all the rods. +With respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> all the rods upon the periphery experience +the Lorentz contraction, but the rods upon the diameter do not +experience this contraction (along their lengths!).<a id="FNanchor_14_1"></a><a href="#Footnote_14_1" class="fnanchor">[14]</a> +It therefore follows that +<span class="align-center"><img style="vertical-align: -1.552ex; width: 7.804ex; height: 4.627ex;" src="images/346.svg" alt=" " data-tex=" +\frac{U}{D} > \pi. +"></span> +</p> +<p> +It therefore follows that the laws of configuration of rigid +bodies with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' do not agree with the laws of configuration +of rigid bodies that are in accordance with Euclidean geometry. +If, further, we place two similar clocks (rotating with <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'), +one upon the periphery, and the other at the centre of the circle, +then, judged from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, the clock on the periphery will go +slower than the clock at the centre. The same thing must take +place, judged from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', if we define time with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' in a +not wholly unnatural way, that is, in such a way that the laws +with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' depend explicitly upon the time. Space and +time, therefore, cannot be defined with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' as they +were in the special theory of relativity with respect to inertial +systems. But, according to the principle of equivalence, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is +also to be considered as a system at rest, with respect to which +there is a gravitational field (field of centrifugal force, and force +of Coriolis). We therefore arrive at the result: the gravitational +field influences and even determines the metrical laws of the +space-time continuum. If the laws of configuration of ideal rigid +bodies are to be expressed geometrically, then in the presence +of a gravitational field the geometry is not Euclidean. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_14_1"></a><a href="#FNanchor_14_1"><span class="label">[14]</span></a>These considerations assume that the behaviour of rods and clocks +depends only upon velocities, and not upon accelerations, or, at least, that +the influence of acceleration does not counteract that of velocity.</p></div> +<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> +<p> +The case that we have been considering is analogous to that +which is presented in the two-dimensional treatment of surfaces. +It is impossible in the latter case also, to introduce co-ordinates +on a surface (e.g. the surface of an ellipsoid) which +have a simple metrical significance, while on a plane the Cartesian +co-ordinates, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, signify directly lengths measured by a +unit measuring rod. Gauss overcame this difficulty, in his theory +of surfaces, by introducing curvilinear co-ordinates which, +apart from satisfying conditions of continuity, were wholly arbitrary, +and afterwards these co-ordinates were related to the +metrical properties of the surface. In an analogous way we +shall introduce in the general theory of relativity arbitrary co-ordinates, +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}">, which shall number uniquely the space-time +points, so that neighbouring events are associated with +neighbouring values of the co-ordinates; otherwise, the choice +of co-ordinates is arbitrary. We shall be true to the principle +of relativity in its broadest sense if we give such a form to the +laws that they are valid in every such four-dimensional system +of co-ordinates, that is, if the equations expressing the laws are +co-variant with respect to arbitrary transformations. +</p> +<p> +The most important point of contact between Gauss's theory +of surfaces and the general theory of relativity lies in the metrical +properties upon which the concepts of both theories, in the +main, are based. In the case of the theory of surfaces, Gauss's +argument is as follows. Plane geometry may be based upon the +concept of the distance <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds">, between two indefinitely near points. +The concept of this distance is physically significant because +the distance can be measured directly by means of a rigid measuring +rod. By a suitable choice of Cartesian co-ordinates this +<span class="pagenum" id="Page_65">[Pg 65]</span> +distance may be expressed by the formula <img style="vertical-align: -0.339ex; width: 17.9ex; height: 2.357ex;" src="images/348.svg" alt=" " data-tex="ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}">. +We may base upon this quantity the concepts of the straight +line as the geodesic (<img style="vertical-align: -0.691ex; width: 8.391ex; height: 2.514ex;" src="images/349.svg" alt=" " data-tex="\delta\! \int\!\! ds = 0">), the interval, the circle, and the +angle, upon which the Euclidean plane geometry is built. A +geometry may be developed upon another continuously curved +surface, if we observe that an infinitesimally small portion of the +surface may be regarded as plane, to within relatively infinitesimal +quantities. There are Cartesian co-ordinates, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/350.svg" alt=" " data-tex="X_{1}">, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/351.svg" alt=" " data-tex="X_{2}">, upon +such a small portion of the surface, and the distance between +two points, measured by a measuring rod, is given by +<span class="align-center"><img style="vertical-align: -0.339ex; width: 19.687ex; height: 2.357ex;" src="images/352.svg" alt=" " data-tex=" +ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}. +"></span> +If we introduce arbitrary curvilinear co-ordinates, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, on the +surface, then <img style="vertical-align: -0.339ex; width: 4.037ex; height: 1.91ex;" src="images/353.svg" alt=" " data-tex="dX_{1}">, <img style="vertical-align: -0.339ex; width: 4.037ex; height: 1.91ex;" src="images/354.svg" alt=" " data-tex="dX_{2}">, may be expressed linearly in terms of +<img style="vertical-align: -0.339ex; width: 3.458ex; height: 1.91ex;" src="images/355.svg" alt=" " data-tex="dx_{1}">, <img style="vertical-align: -0.339ex; width: 3.458ex; height: 1.91ex;" src="images/356.svg" alt=" " data-tex="dx_{2}">. Then everywhere upon the surface we have +<span class="align-center"><img style="vertical-align: -0.464ex; width: 39.454ex; height: 2.482ex;" src="images/357.svg" alt=" " data-tex=" +ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + + g_{22}\, {dx_{2}}^{2}, +"></span> +where <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/358.svg" alt=" " data-tex="g_{11}">, <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/359.svg" alt=" " data-tex="g_{12}">, <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/360.svg" alt=" " data-tex="g_{22}"> are determined by the nature of the surface +and the choice of co-ordinates; if these quantities are known, +then it is also known how networks of rigid rods may be laid +upon the surface. In other words, the geometry of surfaces may +be based upon this expression for <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}"> exactly as plane geometry +is based upon the corresponding expression. +</p> +<p> +There are analogous relations in the four-dimensional space-time +continuum of physics. In the immediate neighbourhood of +an observer, falling freely in a gravitational field, there exists no +gravitational field. We can therefore always regard an infinitesimally +small region of the space-time continuum as Galilean. +For such an infinitely small region there will be an inertial system +(with the space co-ordinates, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/350.svg" alt=" " data-tex="X_{1}">, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/351.svg" alt=" " data-tex="X_{2}">, <img style="vertical-align: -0.375ex; width: 2.861ex; height: 1.92ex;" src="images/362.svg" alt=" " data-tex="X_{3}">, and the time +<span class="pagenum" id="Page_66">[Pg 66]</span> +co-ordinate <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/363.svg" alt=" " data-tex="X_{4}">) relatively to which we are to regard the laws of +the special theory of relativity as valid. The quantity which is +directly measurable by our unit measuring rods and clocks, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 29.027ex; height: 2.457ex;" src="images/364.svg" alt=" " data-tex=" +{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2}, +"></span> +or its negative, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 45.954ex; height: 2.584ex;" src="images/365.svg" alt=" " data-tex=" +ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}, +\qquad \text{(54)} +"></span> +is therefore a uniquely determinate invariant for two neighbouring +events (points in the four-dimensional continuum), provided +that we use measuring rods that are equal to each other when +brought together and superimposed, and clocks whose rates are +the same when they are brought together. In this the physical +assumption is essential that the relative lengths of two measuring +rods and the relative rates of two clocks are independent, in +principle, of their previous history. But this assumption is certainly +warranted by experience; if it did not hold there could be +no sharp spectral lines; for the single atoms of the same element +certainly do not have the same history, and it would be absurd +to suppose any relative difference in the structure of the single +atoms due to their previous history if the mass and frequencies +of the single atoms of the same element were always the same. +</p> +<p> +Space-time regions of finite extent are, in general, not +Galilean, so that a gravitational field cannot be done away +with by any choice of co-ordinates in a finite region. There +is, therefore, no choice of co-ordinates for which the metrical +relations of the special theory of relativity hold in a finite region. +But the invariant <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds"> always exists for two neighbouring +points (events) of the continuum. This invariant <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds"> may be +<span class="pagenum" id="Page_67">[Pg 67]</span> +expressed in arbitrary co-ordinates. If one observes that the +local <img style="vertical-align: -0.339ex; width: 4.085ex; height: 1.91ex;" src="images/366.svg" alt=" " data-tex="dX_{\nu}"> may be expressed linearly in terms of the co-ordinate +differentials <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}"> may be expressed in the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.76ex; height: 2.685ex;" src="images/367.svg" alt=" " data-tex=" +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. +\qquad \text{(55)} +"></span> +</p> +<p> +The functions <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> describe, with respect to the arbitrarily +chosen system of co-ordinates, the metrical relations of the +space-time continuum and also the gravitational field. As in +the special theory of relativity, we have to discriminate between +time-like and space-like line elements in the four-dimensional +continuum; owing to the change of sign introduced, time-like line +elements have a real, space-like line elements an imaginary <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds">. +The time-like <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds"> can be measured directly by a suitably chosen +clock. +</p> +<p> +According to what has been said, it is evident that the formulation +of the general theory of relativity assumes a generalization +of the theory of invariants and the theory of tensors; the question +is raised as to the form of the equations which are co-variant +with respect to arbitrary point transformations. The generalized +calculus of tensors was developed by mathematicians long before +the theory of relativity. Riemann first extended Gauss's +train of thought to continua of any number of dimensions; with +prophetic vision he saw the physical meaning of this generalization +of Euclid's geometry. Then followed the development of +the theory in the form of the calculus of tensors, particularly by +Ricci and Levi-Civita. This is the place for a brief presentation +of the most important mathematical concepts and operations of +this calculus of tensors. +</p> +<p> +We designate four quantities, which are defined as functions +of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> with respect to every system of co-ordinates, as components, +<span class="pagenum" id="Page_68">[Pg 68]</span> +<img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/369.svg" alt=" " data-tex="A^{\nu}">, of a contra-variant vector, if they transform in a +change of co-ordinates as the co-ordinate differentials <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">. We +therefore have +<span class="align-center"><img style="vertical-align: -1.891ex; width: 24.13ex; height: 5.334ex;" src="images/370.svg" alt=" " data-tex=" +{{A'}^{\mu}} = \frac{{\partial x'}_{\mu}}{\partial x_{\nu}} A^{\nu}. +\qquad \text{(56)} +"></span> +Besides these contra-variant vectors, there are also co-variant +vectors. If <img style="vertical-align: -0.339ex; width: 2.753ex; height: 1.885ex;" src="images/371.svg" alt=" " data-tex="B_{\nu}"> are the components of a co-variant vector, these +vectors are transformed according to the rule +<span class="align-center"><img style="vertical-align: -2.237ex; width: 24.171ex; height: 5.384ex;" src="images/372.svg" alt=" " data-tex=" +{B'}_{\mu} = \frac{\partial x_{\nu}}{{\partial {x'}}_{\mu}} B_{\nu}. +\qquad \text{(57)} +"></span> +The definition of a co-variant vector is chosen in such a way that +a co-variant vector and a contra-variant vector together form a +scalar according to the scheme, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 32.939ex; height: 2.262ex;" src="images/373.svg" alt=" " data-tex=" +\phi = B_{\nu} A^{\nu}\quad \text{(summed over the } \nu). +"></span> +Accordingly, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 35.894ex; height: 5.68ex;" src="images/374.svg" alt=" " data-tex=" +{B'}_{\mu} {{A'}^{\mu}} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} + \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} B_{\alpha} A^{\beta} + = B_{\alpha} A^{\alpha}. +"></span> +In particular, the derivatives <img style="vertical-align: -1.909ex; width: 4.782ex; height: 5.056ex;" src="images/375.svg" alt=" " data-tex="\dfrac{\partial \phi}{\partial x_{\alpha}}"> of a scalar +<img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/258.svg" alt=" " data-tex="\phi">, are components +of a co-variant vector, which, with the co-ordinate differentials, +form the scalar <img style="vertical-align: -1.909ex; width: 8.842ex; height: 5.056ex;" src="images/376.svg" alt=" " data-tex="\dfrac{\partial \phi}{\partial x_{\alpha}}\, dx_{\alpha}">; +we see from this example how natural +is the definition of the co-variant vectors. +</p> +<p> +There are here, also, tensors of any rank, which may have +co-variant or contra-variant character with respect to each index; +as with vectors, the character is designated by the position +<span class="pagenum" id="Page_69">[Pg 69]</span> +of the index. For example, <img style="vertical-align: -0.904ex; width: 2.849ex; height: 2.524ex;" src="images/377.svg" alt=" " data-tex="{A_{\mu}^{\nu}}"> denotes a tensor of the second +rank, which is co-variant with respect to the index <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, and contra-variant +with respect to the index <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">. The tensor character indicates +that the equation of transformation is +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.54ex; height: 5.484ex;" src="images/378.svg" alt=" " data-tex=" +{{A'}_{\mu}^{\nu}} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} + \frac{{\partial x'}_{\nu}}{\partial x_{\beta}} A_{\alpha}^{\beta}. +\qquad \text{(58)} +"></span> +</p> +<p> +Tensors may be formed by the addition and subtraction of +tensors of equal rank and like character, as in the theory of +invariants of orthogonal linear substitutions, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 23.929ex; height: 2.601ex;" src="images/379.svg" alt=" " data-tex=" +A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}. +\qquad \text{(59)} +"></span> +The proof of the tensor character of <img style="vertical-align: -0.904ex; width: 2.873ex; height: 2.499ex;" src="images/380.svg" alt=" " data-tex="C_{\mu}^{\nu}"> depends upon (58). +</p> +<p> +Tensors may be formed by multiplication, keeping the character +of the indices, just as in the theory of invariants of linear +orthogonal transformations, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 23.576ex; height: 2.601ex;" src="images/381.svg" alt=" " data-tex=" +A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}. +\qquad \text{(60)} +"></span> +The proof follows directly from the rule of transformation. +</p> +<p> +Tensors may be formed by contraction with respect to two +indices of different character, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 20.806ex; height: 2.601ex;" src="images/382.svg" alt=" " data-tex=" +A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}. +\qquad \text{(61)} +"></span> +The tensor character of <img style="vertical-align: -0.66ex; width: 4.59ex; height: 2.513ex;" src="images/383.svg" alt=" " data-tex="A_{\mu\sigma\tau}^{\mu}"> determines the tensor character +of <img style="vertical-align: -0.36ex; width: 3.646ex; height: 1.905ex;" src="images/384.svg" alt=" " data-tex="B_{\sigma\tau}">. Proof— +<span class="align-center"><img style="vertical-align: -2.237ex; width: 52.078ex; height: 5.68ex;" src="images/385.svg" alt=" " data-tex=" +{{A'}_{\mu\sigma\tau}^{\mu}} += \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} + \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} + {A_{\alpha st}^{\beta}} += \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} + A_{\alpha st}^{\alpha}. +"></span> +<span class="pagenum" id="Page_70">[Pg 70]</span> +</p> +<p> +The properties of symmetry and skew-symmetry of a tensor +with respect to two indices of like character have the same +significance as in the theory of invariants. +</p> +<p> +With this, everything essential has been said with regard to +the algebraic properties of tensors. +</p> + +<p><br></p> + +<p> +<i>The Fundamental Tensor</i>. It follows from the invariance +of <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}">for an arbitrary choice of the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, in connexion with +the condition of symmetry consistent with (55), that the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> +are components of a symmetrical co-variant tensor (Fundamental +Tensor). Let us form the determinant, <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/386.svg" alt=" " data-tex="g">, of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">, and +also the minors, divided by <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/386.svg" alt=" " data-tex="g">, corresponding to the single <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">. +These minors, divided by <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/386.svg" alt=" " data-tex="g">, will be denoted by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}">, and their +co-variant character is not yet known. Then we have +<span class="align-center"><img style="vertical-align: -2.148ex; width: 36.542ex; height: 5.428ex;" src="images/388.svg" alt=" " data-tex=" +g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} + = \begin{cases} + 1 & \text{if \(\alpha = \beta,\)} \\ + 0 & \text{if \(\alpha \neq \beta.\)} + \end{cases} +\qquad \text{(62)} +"></span> +</p> +<p> +If we form the infinitely small quantities (co-variant vectors) +<span class="align-center"><img style="vertical-align: -0.685ex; width: 23.206ex; height: 2.382ex;" src="images/389.svg" alt=" " data-tex=" +d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}, +\qquad \text{(63)} +"></span> +multiply by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}"> and sum over the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, we obtain, by the use +of (62), +<span class="align-center"><img style="vertical-align: -0.685ex; width: 22.969ex; height: 2.747ex;" src="images/390.svg" alt=" " data-tex=" +dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}. +\qquad \text{(64)} +"></span> +Since the ratios of the <img style="vertical-align: -0.685ex; width: 3.32ex; height: 2.278ex;" src="images/391.svg" alt=" " data-tex="d\xi_{\mu}">, are arbitrary, and the <img style="vertical-align: -0.65ex; width: 3.564ex; height: 2.22ex;" src="images/392.svg" alt=" " data-tex="dx_{\beta}"> as well as +the <img style="vertical-align: -0.685ex; width: 3.623ex; height: 2.255ex;" src="images/393.svg" alt=" " data-tex="dx_{\mu}"> are components of vectors, it follows that the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}"> are the +components of a contra-variant tensor<a id="FNanchor_15_1"></a><a href="#Footnote_15_1" class="fnanchor">[15]</a> +(contra-variant fundamental tensor). +The tensor character of <img style="vertical-align: -0.332ex; width: 2.216ex; height: 2.57ex;" src="images/394.svg" alt=" " data-tex="\delta_{\alpha}^{\beta}"> (mixed fundamental +<span class="pagenum" id="Page_71">[Pg 71]</span> +tensor) accordingly follows, by (62). By means of the fundamental +tensor, instead of tensors with co-variant index character, we +can introduce tensors with contra-variant index character, and +conversely. For example, +<span class="align-center"><img style="vertical-align: -3.736ex; width: 12.845ex; height: 8.603ex;" src="images/395.svg" alt=" " data-tex=" +\begin{align*} +A^{\mu} &= g^{\mu\alpha} A_{\alpha}, \\ +A_{\mu} &= g_{\mu\alpha} A^{\alpha}, \\ +T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. +\end{align*} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_15_1"></a><a href="#FNanchor_15_1"><span class="label">[15]</span></a>If we multiply (64) +by <img style="vertical-align: -2.202ex; width: 5.41ex; height: 5.448ex;" src="images/396.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\alpha}}{\partial x_{\beta}}">, +sum over the <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, and replace the <img style="vertical-align: -0.464ex; width: 3.32ex; height: 2.057ex;" src="images/397.svg" alt=" " data-tex="d\xi^{\mu}"> by +a transformation to the accented system, we obtain +<span class="align-center"><img style="vertical-align: -2.237ex; width: 26.832ex; height: 5.484ex;" src="images/398.svg" alt=" " data-tex=" +{dx'}_{\alpha} + = \frac{{\partial x'}_{\sigma}}{\partial x_{\mu}}\, + \frac{{\partial x'}_{\alpha}}{\partial x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}. +"></span> +The statement made above follows from this, since, by (64), we must also +have <img style="vertical-align: -0.561ex; width: 15.36ex; height: 2.597ex;" src="images/399.svg" alt=" " data-tex="{dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}"> and both equations +must hold for every choice of <img style="vertical-align: -0.561ex; width: 3.896ex; height: 2.279ex;" src="images/400.svg" alt=" " data-tex="{d\xi'}_{\sigma}">.</p></div> + +<p><br></p> + +<p> +<i>Volume Invariants</i>. The volume element +<span class="align-center"><img style="vertical-align: -1.948ex; width: 22.967ex; height: 5.027ex;" src="images/401.svg" alt=" " data-tex=" +\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx +"></span> +is not an invariant. For by Jacobi's theorem, +<span class="align-center"><img style="vertical-align: -2.312ex; width: 24.893ex; height: 5.754ex;" src="images/402.svg" alt=" " data-tex=" +dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx. +\qquad \text{(65)} +"></span> +But we can complement <img style="vertical-align: -0.025ex; width: 2.471ex; height: 1.595ex;" src="images/403.svg" alt=" " data-tex="dx"> so that it becomes an invariant. If +we form the determinant of the quantities +<span class="align-center"><img style="vertical-align: -2.237ex; width: 21.889ex; height: 5.545ex;" src="images/404.svg" alt=" " data-tex=" +{g'}_{\mu\nu} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}}\, + \frac{\partial x_{\beta}}{{\partial x'}_{\nu}}\, g_{\alpha\beta}, +"></span> +<span class="pagenum" id="Page_72">[Pg 72]</span> +we obtain, by a double application of the theorem of multiplication +of determinants, +<span class="align-center"><img style="vertical-align: -2.312ex; width: 49.262ex; height: 6.202ex;" src="images/405.svg" alt=" " data-tex=" +g' = |{g'}_{\mu\nu}| + = \left|\frac{\partial x_{\nu}}{{\partial x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| + = \left|\frac{{\partial x'}_{\mu}}{\partial x_{\nu}}\right|^{-2} g. +\qquad \text{(66)} +"></span> +We therefore get the invariant, +<span class="align-center"><img style="vertical-align: -0.554ex; width: 17.372ex; height: 2.851ex;" src="images/406.svg" alt=" " data-tex=" +\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx. +"></span> +</p> + +<p><br></p> + +<p> +<i>Formation of Tensors by Differentiation</i>. Although the algebraic +operations of tensor formation have proved to be as +simple as in the special case of invariance with respect to linear +orthogonal transformations, nevertheless in the general case, +the invariant differential operations are, unfortunately, considerably +more complicated. The reason for this is as follows. If +<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}"> is a contra-variant vector, the coefficients of its +transformation, <img style="vertical-align: -1.891ex; width: 5.35ex; height: 5.334ex;" src="images/408.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}">, are +independent of position only if the transformation +is a linear one. For then the vector components, +<img style="vertical-align: -1.909ex; width: 14.8ex; height: 5.058ex;" src="images/409.svg" alt=" " data-tex="A^{\mu} + \dfrac{\partial A^{\mu}}{\partial x_{\alpha}}\, dx_{\alpha}">, +at a neighbouring point transform in the same way as the <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">, +from which follows the vector character of the vector differentials, +and the tensor character of <img style="vertical-align: -1.909ex; width: 5.125ex; height: 5.058ex;" src="images/410.svg" alt=" " data-tex="\dfrac{\partial A^{\mu}}{\partial x_{\alpha}}">. +But if the <img style="vertical-align: -1.891ex; width: 5.35ex; height: 5.334ex;" src="images/408.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}"> +are variable this is no longer true. +</p> +<p> +That there are, nevertheless, in the general case, invariant +differential operations for tensors, is recognized most satisfactorily +in the following way, introduced by Levi-Civita and Weyl. +Let (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">) be a contra-variant vector whose components are given +with respect to the co-ordinate system of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. +Let <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}"> and <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}"> +<span class="pagenum" id="Page_73">[Pg 73]</span> +be two infinitesimally near points of the continuum. For the infinitesimal +region surrounding the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}">, there is, according +to our way of considering the matter, a co-ordinate system of +the <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> (with imaginary <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/363.svg" alt=" " data-tex="X_{4}">-co-ordinate) for which the +continuum is Euclidean. Let <img style="vertical-align: -1.207ex; width: 3.929ex; height: 3.06ex;" src="images/411.svg" alt=" " data-tex="A_{(1)}^{\mu}"> be the co-ordinates of the vector at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}">. Imagine a vector drawn at the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}">, using the +local system of the <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">, with the same co-ordinates (parallel vector +through <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}">), then this parallel vector is uniquely determined +by the vector at <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}"> and the displacement. We designate this operation, +whose uniqueness will appear in the sequel, the parallel +displacement of the vector <img style="vertical-align: -0.685ex; width: 2.849ex; height: 2.305ex;" src="images/412.svg" alt=" " data-tex="A_{\mu}"> from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}"> to the infinitesimally near +point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}">. If we form the vector difference of the vector (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">) at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}"> and the vector obtained by parallel displacement +from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}"> to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}">, we get a vector which may be regarded as the +differential of the vector (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">) for the given displacement <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">. +</p> +<p> +This vector displacement can naturally also be considered +with respect to the co-ordinate system of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/369.svg" alt=" " data-tex="A^{\nu}"> are the +co-ordinates of the vector at <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}">, <img style="vertical-align: -0.186ex; width: 9.235ex; height: 1.808ex;" src="images/413.svg" alt=" " data-tex="A^{\nu} + \delta A^{\nu}"> the co-ordinates of +the vector displaced to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}"> along the interval (<img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">), then +the <img style="vertical-align: -0.023ex; width: 3.737ex; height: 1.645ex;" src="images/414.svg" alt=" " data-tex="\delta A^{\nu}"> +do not vanish in this case. We know of these quantities, which +do not have a vector character, that they must depend linearly +and homogeneously upon the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}"> and the <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/369.svg" alt=" " data-tex="A^{\nu}">. We therefore put +<span class="align-center"><img style="vertical-align: -0.911ex; width: 28.449ex; height: 2.608ex;" src="images/415.svg" alt=" " data-tex=" +\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}. +\qquad \text{(67)} +"></span> +</p> +<p> +In addition, we can state that the <img style="vertical-align: -1.024ex; width: 3.531ex; height: 2.563ex;" src="images/416.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\nu}"> must be symmetrical +with respect to the indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">. +For we can assume from +a representation by the aid of a Euclidean system of local co-ordinates +that the same parallelogram will be described by the +displacement of an element <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/417.svg" alt=" " data-tex="d^{(1)}x_{\nu}"> along a second element +<img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/418.svg" alt=" " data-tex="d^{(2)}x_{\nu}"> +<span class="pagenum" id="Page_74">[Pg 74]</span> +as by a displacement of <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/418.svg" alt=" " data-tex="d^{(2)}x_{\nu}"> along <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/417.svg" alt=" " data-tex="d^{(1)}x_{\nu}">. +We must therefore have +<span class="align-center"><img style="vertical-align: -2.819ex; width: 73.179ex; height: 6.769ex;" src="images/419.svg" alt=" " data-tex=" +\begin{aligned} +d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ + \quad + &= d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). +\end{aligned} +"></span> +The statement made above follows from this, after interchanging +the indices of summation, <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, on the right-hand side. +</p> +<p> +Since the quantities <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> determine all the metrical properties +of the continuum, they must also determine the <img style="vertical-align: -1.024ex; width: 3.531ex; height: 2.563ex;" src="images/416.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\nu}">. If we +consider the invariant of the vector <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/369.svg" alt=" " data-tex="A^{\nu}"> that is, the square of its +magnitude, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 9.29ex; height: 2.326ex;" src="images/420.svg" alt=" " data-tex=" +g_{\mu\nu} A^{\mu} A^{\nu}, +"></span> +which is an invariant, this cannot change in a parallel displacement. +We therefore have +<span class="align-center"><img style="vertical-align: -1.909ex; width: 58.452ex; height: 5.252ex;" src="images/421.svg" alt=" " data-tex=" +0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) + = \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu} +"></span> +or, by (67), +<span class="align-center"><img style="vertical-align: -2.148ex; width: 41.858ex; height: 5.491ex;" src="images/422.svg" alt=" " data-tex=" +\left(\frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} + - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} + - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} += 0. +"></span> +</p> +<p> +Owing to the symmetry of the expression in the brackets +with respect to the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, this equation can be valid +for an arbitrary choice of the vectors (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">) and <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}"> only when +the expression in the brackets vanishes for all combinations of +the indices. By a cyclic interchange of the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, we +obtain thus altogether three equations, from which we obtain, +on taking into account the symmetrical property of the +<img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/423.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\alpha}">, +<span class="align-center"><img style="vertical-align: -1.577ex; width: 24.423ex; height: 4.178ex;" src="images/424.svg" alt=" " data-tex=" +\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] + = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}, +\qquad \text{(68)} +"></span> +<span class="pagenum" id="Page_75">[Pg 75]</span> +in which, following Christoffel, the abbreviation has been used, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 44.772ex; height: 5.58ex;" src="images/425.svg" alt=" " data-tex=" +\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] + = \tfrac{1}{2}\left( + \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} + - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} +\right). +\qquad \text{(69)} +"></span> +</p> +<p> +If we multiply (68) by <img style="vertical-align: -0.464ex; width: 3.204ex; height: 1.992ex;" src="images/426.svg" alt=" " data-tex="g^{\alpha\sigma}"> and sum over the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, we obtain +<span class="align-center"><img style="vertical-align: -2.237ex; width: 55.667ex; height: 5.58ex;" src="images/427.svg" alt=" " data-tex=" +\Gamma_{\mu\nu}^{\alpha} + = \tfrac{1}{2} g^{\sigma\alpha}\left( + \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} + - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} +\right) = \left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}, +\qquad \text{(70)} +"></span> +in which <img style="vertical-align: -0.798ex; width: 4.994ex; height: 2.72ex;" src="images/428.svg" alt=" " data-tex="\left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}"> is +the Christoffel symbol of the second kind. +Thus the quantities <img style="vertical-align: 0; width: 1.414ex; height: 1.538ex;" src="images/429.svg" alt=" " data-tex="\Gamma"> are deduced from the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">. Equations +(67) and (70) are the foundation for the following discussion. +</p> + +<p><br></p> + +<p> +<i>Co-variant Differentiation of Tensors</i>. If (<img style="vertical-align: -0.186ex; width: 9.469ex; height: 1.808ex;" src="images/430.svg" alt=" " data-tex="A^{\mu} + \delta A^{\mu}">) is +the vector resulting from an infinitesimal parallel displacement +from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{1}"> to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}">, and (<img style="vertical-align: -0.186ex; width: 9.641ex; height: 1.805ex;" src="images/431.svg" alt=" " data-tex="A^{\mu} + dA^{\mu}">) the vector <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}"> at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/172.svg" alt=" " data-tex="P_{2}">, +then the difference of these two, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 35.532ex; height: 5.428ex;" src="images/432.svg" alt=" " data-tex=" +dA^{\mu} - \delta A^{\mu} = \left( + \frac{\partial A^{\mu}}{\partial x_{\sigma}} + + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma}, +"></span> +is also a vector. Since this is the case for an arbitrary choice of +the <img style="vertical-align: -0.357ex; width: 3.572ex; height: 1.927ex;" src="images/433.svg" alt=" " data-tex="dx_{\sigma}">, it follows that +<span class="align-center"><img style="vertical-align: -1.909ex; width: 30.676ex; height: 5.058ex;" src="images/434.svg" alt=" " data-tex=" +{A^{\mu}}_{;\, \sigma} + = \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} +\qquad \text{(71)} +"></span> +is a tensor, which we designate as the co-variant derivative of +the tensor of the first rank (vector). Contracting this tensor, we +obtain the divergence of the contra-variant tensor <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">. In this +we must observe that according to (70), +<span class="align-center"><img style="vertical-align: -2.308ex; width: 39.215ex; height: 5.747ex;" src="images/435.svg" alt=" " data-tex=" +\Gamma_{\mu\sigma}^{\sigma} + = \tfrac{1}{2} g^{\sigma\alpha} \frac{\partial g_{\sigma\alpha}}{\partial x_{\mu}} + = \frac{1}{\sqrt{g}}\, \frac{\partial \sqrt{g}}{\partial x_{\mu}}. +\qquad \text{(72)} +"></span> +<span class="pagenum" id="Page_76">[Pg 76]</span> +If we put, further, +<span class="align-center"><img style="vertical-align: -0.602ex; width: 21.206ex; height: 2.398ex;" src="images/436.svg" alt=" " data-tex=" +A^{\mu} \sqrt{g} = \mathfrak A^{\mu}, +\qquad \text{(73)} +"></span> +a quantity designated by Weyl as the contra-variant tensor density<a id="FNanchor_16_1"></a><a href="#Footnote_16_1" class="fnanchor">[16]</a> +of the first rank, it follows that, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 18.242ex; height: 5.384ex;" src="images/437.svg" alt=" " data-tex=" +\mathfrak A = \frac{\partial \mathfrak A^{\mu}}{\partial x_{\mu}} +\qquad \text{(74)} +"></span> +is a scalar density. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_16_1"></a><a href="#FNanchor_16_1"><span class="label">[16]</span></a>This expression is justified, in that +<img style="vertical-align: -0.71ex; width: 17.691ex; height: 2.398ex;" src="images/438.svg" alt=" " data-tex="A^{\mu}\sqrt{g}\, dx = \mathbf A^{\mu}\, dx"> has a tensor +character. Every tensor, when multiplied by <img style="vertical-align: -0.71ex; width: 3.009ex; height: 2.398ex;" src="images/439.svg" alt=" " data-tex="\sqrt{g}">, changes into a tensor +density. We employ capital Gothic letters for tensor densities.</p></div> + +<p> +We get the law of parallel displacement for the co-variant +vector <img style="vertical-align: -0.685ex; width: 2.87ex; height: 2.23ex;" src="images/440.svg" alt=" " data-tex="B_{\mu}"> by stipulating that the parallel displacement shall be +effected in such a way that the scalar +<span class="align-center"><img style="vertical-align: -0.685ex; width: 10.084ex; height: 2.326ex;" src="images/441.svg" alt=" " data-tex=" +\phi = A^{\mu} B_{\mu} +"></span> +remains unchanged, and that therefore +<span class="align-center"><img style="vertical-align: -0.685ex; width: 16.968ex; height: 2.326ex;" src="images/442.svg" alt=" " data-tex=" +A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu} +"></span> +vanishes for every value assigned to (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">). We therefore get +<span class="align-center"><img style="vertical-align: -0.904ex; width: 26.783ex; height: 2.601ex;" src="images/443.svg" alt=" " data-tex=" +\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}. +\qquad \text{(75)} +"></span> +</p> +<p> +From this we arrive at the co-variant derivative of the co-variant +vector by the same process as that which led to (71), +<span class="align-center"><img style="vertical-align: -1.909ex; width: 31.497ex; height: 5.252ex;" src="images/444.svg" alt=" " data-tex=" +B_{\mu;\, \sigma} + = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}. +\qquad \text{(76)} +"></span> +<span class="pagenum" id="Page_77">[Pg 77]</span> +By interchanging the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, and subtracting, we get +the skew-symmetrical tensor, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 28.991ex; height: 5.58ex;" src="images/445.svg" alt=" " data-tex=" +\phi_{\mu\sigma} + = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \frac{\partial B_{\sigma}}{\partial x_{\mu}}. +\qquad \text{(77)} +"></span> +</p> +<p> +For the co-variant differentiation of tensors of the second +and higher ranks we may use the process by which (75) was +deduced. Let, for example, (<img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/446.svg" alt=" " data-tex="A_{\sigma\tau}">) be a co-variant tensor of the +second rank. Then <img style="vertical-align: -0.36ex; width: 9.406ex; height: 1.98ex;" src="images/447.svg" alt=" " data-tex="A_{\sigma\tau} E^{\sigma} F^{\tau}"> is a scalar, +if <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/293.svg" alt=" " data-tex="E"> and <img style="vertical-align: 0; width: 1.695ex; height: 1.538ex;" src="images/448.svg" alt=" " data-tex="F"> are vectors. +This expression must not be changed by the <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/117.svg" alt=" " data-tex="\delta">-displacement; +expressing this by a formula, we get, using (67), <img style="vertical-align: -0.36ex; width: 4.63ex; height: 1.982ex;" src="images/449.svg" alt=" " data-tex="\delta A_{\sigma\tau}">, whence +we get the desired co-variant derivative, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 43.434ex; height: 5.386ex;" src="images/450.svg" alt=" " data-tex=" +A_{\sigma\tau;\, \rho} + = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} + - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}. +\qquad \text{(78)} +"></span> +</p> +<p> +In order that the general law of co-variant differentiation of +tensors may be clearly seen, we shall write down two co-variant +derivatives deduced in an analogous way: +<span class="align-center"><img style="vertical-align: -5.204ex; width: 48.077ex; height: 11.539ex;" src="images/451.svg" alt=" " data-tex=" +\begin{align*} +A_{\sigma;\, \rho}^{\tau} + &= \frac{\partial A_{\sigma}^{\tau}}{\partial x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha},\, +\qquad & &\text{(79)} \\ +{A^{\sigma\tau}}_{;\, \rho} + &= \frac{\partial A^{\sigma\tau}}{\partial x_{\rho}} + + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}. +\qquad & &\text{(80)} +\end{align*} +"></span> +The general law of formation now becomes evident. From these +formulae we shall deduce some others which are of interest for +the physical applications of the theory. +</p> +<p> +In case <img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/446.svg" alt=" " data-tex="A_{\sigma\tau}"> is skew-symmetrical, we obtain the tensor +<span class="align-center"><img style="vertical-align: -2.237ex; width: 40.172ex; height: 5.583ex;" src="images/452.svg" alt=" " data-tex=" +A_{\sigma\tau\rho} + = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + + \frac{\partial A_{\tau\rho}}{\partial x_{\sigma}} + + \frac{\partial A_{\rho\sigma}}{\partial x_{\tau}}, +\qquad \text{(81)} +"></span> +<span class="pagenum" id="Page_78">[Pg 78]</span> +which is skew-symmetrical in all pairs of indices, by cyclic +interchange and addition. +</p> +<p> +If, in (78), we replace <img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/446.svg" alt=" " data-tex="A_{\sigma\tau}"> by the fundamental tensor, +<img style="vertical-align: -0.464ex; width: 3.008ex; height: 1.464ex;" src="images/453.svg" alt=" " data-tex="g_{\sigma\tau}">, +then the right-hand side vanishes identically; an analogous statement +holds for (80) with respect to <img style="vertical-align: -0.464ex; width: 3.008ex; height: 1.975ex;" src="images/454.svg" alt=" " data-tex="g^{\sigma\tau}">; that is, the co-variant +derivatives of the fundamental tensor vanish. That this must be +so we see directly in the local system of co-ordinates. +</p> +<p> +In case <img style="vertical-align: 0; width: 3.625ex; height: 1.62ex;" src="images/455.svg" alt=" " data-tex="A^{\sigma\tau}"> is skew-symmetrical, we obtain from (80), by +contraction with respect to <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, +<span class="align-center"><img style="vertical-align: -1.912ex; width: 21.125ex; height: 5.059ex;" src="images/456.svg" alt=" " data-tex=" +\mathfrak A^{\sigma} = \frac{\partial \mathfrak A^{\sigma\tau}}{\partial x_{\tau}}. +\qquad \text{(82)} +"></span> +</p> +<p> +In the general case, from (79) and (80), by contraction with +respect to <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, we obtain the equations, +<span class="align-center"><img style="vertical-align: -4.874ex; width: 35.508ex; height: 10.879ex;" src="images/457.svg" alt=" " data-tex=" +\begin{align*} +\mathfrak A_{\sigma} &= \frac{\partial \mathfrak A_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak A_{\alpha}^{\beta}, +\qquad & &\text{(83)} \\ +\mathfrak A^{\sigma} &= \frac{\partial \mathfrak A^{\sigma\alpha}}{\partial x_{\alpha}} + + \Gamma_{\alpha\beta}^{\sigma} \mathfrak A^{\alpha\beta}. +\qquad & &\text{(84)} +\end{align*} +"></span> +</p> + +<p><br></p> + +<p> +<i>The Riemann Tensor</i>. If we have given a curve extending +from the point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> to the point <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/458.svg" alt=" " data-tex="G"> of the continuum, then a +vector <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}">, given at <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">, may, by a parallel displacement, be moved +along the curve to <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/458.svg" alt=" " data-tex="G">. If the continuum is Euclidean (more generally, +if by a suitable choice of co-ordinates the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">, are constants) +then the vector obtained at <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/458.svg" alt=" " data-tex="G"> as a result of this displacement +does not depend upon the choice of the curve joining +<img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> and <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/458.svg" alt=" " data-tex="G">. +But otherwise, the result depends upon the path of the displacement. +</p> + +<div class="figcenter" style="width: 400px;"> +<img src="images/figure04.jpg" width="400" alt="400"> +<div class="caption"> +<p>FIG.4.</p> +</div></div> + +<p class="nind"> +In this case, therefore, a vector suffers a change, +<img style="vertical-align: 0; width: 4.734ex; height: 1.62ex;" src="images/459.svg" alt=" " data-tex="\Delta A^{\mu}"> +(in its direction, not its magnitude), when it is carried from a +<span class="pagenum" id="Page_79">[Pg 79]</span> +point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> of a closed curve, along the curve, and back to P. We +shall now calculate this vector change: +<span class="align-center"><img style="vertical-align: -1.948ex; width: 14.747ex; height: 5.027ex;" src="images/460.svg" alt=" " data-tex=" +\Delta A^{\mu} = \oint \delta A^{\mu}. +"></span> +As in Stokes' theorem for the line integral of a vector around +a closed curve, this problem may be reduced to the integration +around a closed curve with infinitely small linear dimensions; we +shall limit ourselves to this case. +</p> +<p> +We have, first, by (67), +<span class="align-center"><img style="vertical-align: -1.948ex; width: 23.411ex; height: 5.027ex;" src="images/461.svg" alt=" " data-tex=" +\Delta A^{\mu} = -\oint \Gamma_{\alpha\beta}^{\mu} A^{\alpha}\, dx_{\beta}. +"></span> +</p> +<p> +In this, <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/462.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> is the value of +this quantity at the variable +point <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/458.svg" alt=" " data-tex="G"> of the path of integration. If we put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 19.174ex; height: 2.382ex;" src="images/463.svg" alt=" " data-tex=" +\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P} +"></span> +<span class="pagenum" id="Page_80">[Pg 80]</span> +and denote the value of +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/462.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> at <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> by +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 3.615ex;" src="images/464.svg" alt=" " data-tex="\overline{\Gamma_{\alpha\beta}^{\mu}}"> then we have, with +sufficient accuracy, +<span class="align-center"><img style="vertical-align: -1.891ex; width: 21.686ex; height: 6.547ex;" src="images/465.svg" alt=" " data-tex=" +\Gamma_{\alpha\beta}^{\mu} + = \overline{\Gamma_{\alpha\beta}^{\mu}} + + \frac{\overline{\partial \Gamma_{\alpha\beta}^{\mu}}}{\partial x_{\nu}}\, \xi^{\nu}. +"></span> +</p> +<p> +Let, further, <img style="vertical-align: 0; width: 2.908ex; height: 1.62ex;" src="images/466.svg" alt=" " data-tex="A^{\alpha}"> be the value obtained from +<img style="vertical-align: 0; width: 2.908ex; height: 2.337ex;" src="images/467.svg" alt=" " data-tex="\overline{A^{\alpha}}"> by a parallel +displacement along the curve from <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> to <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/458.svg" alt=" " data-tex="G">. It may now easily +be proved by means of (67) that +<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/407.svg" alt=" " data-tex="A^{\mu}"> - <img style="vertical-align: 0; width: 2.849ex; height: 2.337ex;" src="images/468.svg" alt=" " data-tex="\overline{A^{\mu}}"> is infinitely small of +the first order, while, for a curve of infinitely small dimensions +of the first order, <img style="vertical-align: 0; width: 4.734ex; height: 1.62ex;" src="images/459.svg" alt=" " data-tex="\Delta A^{\mu}"> is infinitely small of the second order. +Therefore there is an error of only the second order if we put +<span class="align-center"><img style="vertical-align: -0.58ex; width: 21.633ex; height: 2.917ex;" src="images/469.svg" alt=" " data-tex=" +A^{\alpha} = \overline{A^{\alpha}} + - \overline{\Gamma_{\sigma\tau}^{\alpha}}\; + \overline{A^{\sigma}}\; + \overline{\xi^{\tau}}. +"></span> +</p> +<p> +If we introduce these values of +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/462.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> and <img style="vertical-align: 0; width: 2.908ex; height: 1.62ex;" src="images/466.svg" alt=" " data-tex="A^{\alpha}"> into the integral, +we obtain, neglecting all quantities of a higher order of small +quantities than the second, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 49.891ex; height: 6.785ex;" src="images/470.svg" alt=" " data-tex=" +\Delta A^{\mu} + = - \left(\frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) + A^{\sigma} \oint \xi^{\alpha}\, d\xi^{\beta}. +\qquad \text{(85)} +"></span> +The quantity removed from under the sign of integration refers +to the point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">. Subtracting +<img style="vertical-align: -1.552ex; width: 9.35ex; height: 4.588ex;" src="images/471.svg" alt=" " data-tex="\dfrac{1}{2} d(\xi^{\alpha} \xi^{\beta})"> from the integrand, we +obtain +<span class="align-center"><img style="vertical-align: -1.948ex; width: 21.145ex; height: 5.027ex;" src="images/472.svg" alt=" " data-tex=" +\tfrac{1}{2} \oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}). +"></span> +This skew-symmetrical tensor of the second rank, +<img style="vertical-align: -0.464ex; width: 3.481ex; height: 2.413ex;" src="images/473.svg" alt=" " data-tex="f^{\alpha\beta}">, characterizes +the surface element bounded by the curve in magnitude +and position. If the expression in the brackets in (85) were +skew-symmetrical with respect to the indices +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, we could +<span class="pagenum" id="Page_81">[Pg 81]</span> +conclude its tensor character from (85). We can accomplish this +by interchanging the summation indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> in (85) and +adding the resulting equation to (85). We obtain +<span class="align-center"><img style="vertical-align: -1.045ex; width: 31.223ex; height: 3.108ex;" src="images/474.svg" alt=" " data-tex=" +2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}, +\qquad \text{(86)} +"></span> +in which +<span class="align-center"><img style="vertical-align: -2.202ex; width: 52.635ex; height: 6.141ex;" src="images/475.svg" alt=" " data-tex=" +R_{\sigma\alpha\beta}^{\mu} + = - \frac{\partial \Gamma_{\sigma\alpha}^{\mu}}{\partial x_{\beta}} + + \frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}. +\qquad \text{(87)} +"></span> +<p> +The tensor character of <img style="vertical-align: -1.045ex; width: 4.748ex; height: 2.898ex;" src="images/476.svg" alt=" " data-tex="R_{\sigma\alpha\beta}^{\mu}"> follows from (86); this is the +Riemann curvature tensor of the fourth rank, whose properties of +symmetry we do not need to go into. Its vanishing is a sufficient +condition (disregarding the reality of the chosen co-ordinates) +that the continuum is Euclidean. +</p> +<p> +By contraction of the Riemann tensor with respect to the +indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, we obtain the symmetrical tensor of the second +rank, +<span class="align-center"><img style="vertical-align: -1.909ex; width: 51.793ex; height: 5.472ex;" src="images/477.svg" alt=" " data-tex=" +R_{\mu\nu} + = - \frac{\partial \Gamma_{\mu\nu}^{\alpha}}{\partial x_{\alpha}} + + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + + \frac{\partial \Gamma_{\mu\alpha}^{\alpha}}{\partial x_{\nu}} + - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}. +\qquad \text{(88)} +"></span> +The last two terms vanish if the system of co-ordinates is so +chosen that <img style="vertical-align: -0.464ex; width: 12.531ex; height: 1.855ex;" src="images/478.svg" alt=" " data-tex="g = \text{constant}">. From <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/479.svg" alt=" " data-tex="R_{\mu\nu}">, we can form the scalar, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 21.085ex; height: 2.382ex;" src="images/480.svg" alt=" " data-tex=" +R = g^{\mu\nu} R_{\mu\nu}. +\qquad \text{(89)} +"></span> +</p> + +<p><br></p> + +<p> +<i>Straightest Geodetic Lines</i>. A line may be constructed in +such a way that its successive elements arise from each other by +parallel displacements. This is the natural generalization of the +straight line of the Euclidean geometry. For such a line, we have +<span class="align-center"><img style="vertical-align: -2.148ex; width: 26.887ex; height: 5.444ex;" src="images/481.svg" alt=" " data-tex=" +\delta \left(\frac{dx_{\mu}}{ds}\right) + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}. +"></span> +<span class="pagenum" id="Page_82">[Pg 82]</span> +The left-hand side is to be replaced by +<img style="vertical-align: -1.651ex; width: 5.606ex; height: 5.264ex;" src="images/482.svg" alt=" " data-tex="\dfrac{d^{2} x_{\mu}}{ds^{2}}">,<a id="FNanchor_17_1"></a><a href="#Footnote_17_1" class="fnanchor">[17]</a> +so that we have +<span class="align-center"><img style="vertical-align: -1.651ex; width: 34.843ex; height: 5.264ex;" src="images/483.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0. +\qquad \text{(90)} +"></span> +We get the same line if we find the line which gives a stationary +value to the integral +<span class="align-center"><img style="vertical-align: -1.948ex; width: 27.455ex; height: 5.027ex;" src="images/484.svg" alt=" " data-tex=" +\int ds\quad\text{or}\quad +\int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}} +"></span> +between two points (geodetic line). +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_17_1"></a><a href="#FNanchor_17_1"><span class="label">[17]</span></a>The direction vector at a neighbouring point of the curve results, by +a parallel displacement along the line element (<img style="vertical-align: -0.65ex; width: 3.564ex; height: 2.22ex;" src="images/392.svg" alt=" " data-tex="dx_{\beta}">), from the direction +vector of each point considered.</p></div> +<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> +<p><br><br><br></p> + +<h3><a id="chap04"></a>LECTURE IV +<br><br> +THE GENERAL THEORY OF RELATIVITY<br> +(<i>continued</i>)</h3> + +<p class="nind"> +<span class="dropcap">W</span>E are now in possession of the mathematical apparatus which +is necessary to formulate the laws of the general theory of relativity. +No attempt will be made in this presentation at systematic +completeness, but single results and possibilities will +be developed progressively from what is known and from the results +obtained. Such a presentation is most suited to the present +provisional state of our knowledge. +</p> +<p> +A material particle upon which no force acts moves, according +to the principle of inertia, uniformly in a straight line. In +the four-dimensional continuum of the special theory of relativity +(with real time co-ordinate) this is a real straight line. The +natural, that is, the simplest, generalization of the straight line +which is plausible in the system of concepts of Riemann's general +theory of invariants is that of the straightest, or geodetic, +line. We shall accordingly have to assume, in the sense of the +principle of equivalence, that the motion of a material particle, +under the action only of inertia and gravitation, is described by +the equation, +<span class="align-center"><img style="vertical-align: -1.651ex; width: 34.843ex; height: 5.264ex;" src="images/483.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0. +\qquad \text{(90)} +"></span> +In fact, this equation reduces to that of a straight line if all the +components, <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/462.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}">, of the gravitational field vanish. +</p> +<p> +How are these equations connected with Newton's equations +of motion? According to the special theory of relativity, +the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> as well as the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}">, have +the values, with respect to an inertial +<span class="pagenum" id="Page_84">[Pg 84]</span> +system (with real time co-ordinate and suitable choice of the +sign of <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}">), +<span class="align-center"><img style="vertical-align: -5.317ex; width: 28.158ex; height: 11.765ex;" src="images/485.svg" alt=" " data-tex=" +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & 1 +\end{array} +\right\}. +\qquad \text{(91)} +"></span> +The equations of motion then become +<span class="align-center"><img style="vertical-align: -1.651ex; width: 10.383ex; height: 5.264ex;" src="images/486.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} = 0. +"></span> +We shall call this the "first approximation" to the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">-field. In +considering approximations it is often useful, as in the special +theory of relativity, to use an imaginary <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}">-co-ordinate, as then +the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">. to the first approximation, assume the values +<span class="align-center"><img style="vertical-align: -5.317ex; width: 31.049ex; height: 11.765ex;" src="images/487.svg" alt=" " data-tex=" +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & -1 +\end{array} +\right\}. +\qquad \text{(91a)} +"></span> +These values may be collected in the relation +<span class="align-center"><img style="vertical-align: -0.685ex; width: 11.491ex; height: 2.307ex;" src="images/488.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu}. +"></span> +To the second approximation we must then put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.353ex; height: 2.382ex;" src="images/489.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, +\qquad \text{(92)} +"></span> +where the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}"> are to be regarded as small of the first order. +<span class="pagenum" id="Page_85">[Pg 85]</span> +</p> +<p> +Both terms of our equation of motion are then small of the +first order. If we neglect terms which, relatively to these, are +small of the first order, we have to put +<span class="align-center"><img style="vertical-align: -3.856ex; width: 67.308ex; height: 8.843ex;" src="images/491.svg" alt=" " data-tex=" +\begin{gather*} +ds^{2} = -{dx_{\nu}}^{2} = dl^{2} (1 - q^{2}), +\qquad & &\text{(93)}\\ +\Gamma_{\alpha\beta}^{\mu} + = -\delta_{\mu\sigma} [^{\alpha\beta}_{\,\delta}] + = -[^{\alpha\beta}_{\,\mu}] + = \frac{1}{2} \left( + \frac{\partial \gamma_{\alpha\beta}}{\partial x_{\mu}} + - \frac{\partial \gamma_{\alpha\mu}}{\partial x_{\beta}} + - \frac{\partial \gamma_{\beta\mu}}{\partial x_{\alpha}}\right). +\qquad & &\text{(94)} +\end{gather*} +"></span> +We shall now introduce an approximation of a second kind. Let +the velocity of the material particles be very small compared to +that of light. Then <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds"> will be the same as the time differential, <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/283.svg" alt=" " data-tex="dl">. +Further, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/492.svg" alt=" " data-tex="\dfrac{dx_{1}}{ds}">, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/493.svg" alt=" " data-tex="\dfrac{dx_{2}}{ds}">, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/494.svg" alt=" " data-tex="\dfrac{dx_{3}}{ds}"> +will vanish compared to <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/495.svg" alt=" " data-tex="\dfrac{dx_{4}}{ds}">. +We shall assume, in addition, that the gravitational field varies +so little with the time that the derivatives of the +<img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}"> by <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}"> may +be neglected. Then the equation of motion (for <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> = 1,2,3) +reduces to +<span class="align-center"><img style="vertical-align: -2.237ex; width: 30.687ex; height: 5.849ex;" src="images/496.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{dl^{2}} + = \frac{\partial}{\partial x_{\mu}} \left(\frac{\gamma_{44}}{2}\right). +\qquad \text{(90a)} +"></span> +This equation is identical with Newton's equation of motion for +a material particle in a gravitational field, if we identify +<img style="vertical-align: -1.552ex; width: 6.656ex; height: 4.153ex;" src="images/497.svg" alt=" " data-tex="\left(\dfrac{\gamma_{44}}{2}\right)"> +with the potential of the gravitational field; whether or not this +is allowable, naturally depends upon the field equations of gravitation, +that is, it depends upon whether or not this quantity +satisfies, to a first approximation, the same laws of the field +as the gravitational potential in Newton's theory. A glance at +(90) and (90a) shows that the <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/462.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> actually do play the rôle of +the intensity of the gravitational field. These quantities do not +have a tensor character. +</p> +<p> +Equations (90) express the influence of inertia and gravitation +upon the material particle. The unity of inertia and gravitation +<span class="pagenum" id="Page_86">[Pg 86]</span> +is formally expressed by the fact that the whole left-hand +side of (90) has the character of a tensor (with respect to any +transformation of co-ordinates), but the two terms taken separately +do not have tensor character, so that, in analogy with +Newton's equations, the first term would be regarded as the expression +for inertia, and the second as the expression for the +gravitational force. +</p> +<p> +We must next attempt to find the laws of the gravitational +field. For this purpose, Poisson's equation, +<span class="align-center"><img style="vertical-align: -0.489ex; width: 11.852ex; height: 2.109ex;" src="images/498.svg" alt=" " data-tex=" +\Delta\phi = 4\pi K\rho +"></span> +of the Newtonian theory must serve as a model. This equation +has its foundation in the idea that the gravitational field +arises from the density <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"> of ponderable matter. It must also +be so in the general theory of relativity. But our investigations +of the special theory of relativity have shown that in place of +the scalar density of matter we have the tensor of energy per +unit volume. In the latter is included not only the tensor of +the energy of ponderable matter, but also that of the electromagnetic +energy. We have seen, indeed, that in a more complete +analysis the energy tensor can be regarded only as a provisional +means of representing matter. In reality, matter consists of electrically +charged particles, and is to be regarded itself as a part, +in fact, the principal part, of the electromagnetic field. It is +only the circumstance that we have not sufficient knowledge of +the electromagnetic field of concentrated charges that compels +us, provisionally, to leave undetermined in presenting the theory, +the true form of this tensor. From this point of view our +problem now is to introduce a tensor, <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}">. of the second rank, +<span class="pagenum" id="Page_87">[Pg 87]</span> +whose structure we do not know provisionally, and which includes +in itself the energy density of the electromagnetic field +and of ponderable matter; we shall denote this in the following +as the "energy tensor of matter." +</p> +<p> +According to our previous results, the principles of momentum +and energy are expressed by the statement that the divergence +of this tensor vanishes (47c). In the general theory of relativity, +we shall have to assume as valid the corresponding general +co-variant equation. If (<img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}">) denotes the co-variant energy tensor +of matter, <img style="vertical-align: -0.576ex; width: 2.615ex; height: 2.162ex;" src="images/499.svg" alt=" " data-tex="\mathfrak T_{\sigma}^{\nu}"> the corresponding mixed tensor density, then, +in accordance with (83), we must require that +<span class="align-center"><img style="vertical-align: -1.909ex; width: 26.609ex; height: 5.144ex;" src="images/500.svg" alt=" " data-tex=" +0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta} +\qquad \text{(95)} +"></span> +be satisfied. It must be remembered that besides the energy density +of the matter there must also be given an energy density of +the gravitational field, so that there can be no talk of principles +of conservation of energy and momentum for matter alone. This +is expressed mathematically by the presence of the second term +in (95), which makes it impossible to conclude the existence of +an integral equation of the form of (49). The gravitational field +transfers energy and momentum to the "matter," in that it exerts +forces upon it and gives it energy; this is expressed by the +second term in (95). +</p> +<p> +If there is an analogue of Poisson's equation in the general +theory of relativity, then this equation must be a tensor equation +for the tensor <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> of the gravitational potential; the energy +tensor of matter must appear on the right-hand side of this +equation. On the left-hand side of the equation there must be +a differential tensor in the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">. We have to find this differential +<span class="pagenum" id="Page_88">[Pg 88]</span> +tensor. It is completely determined by the following three +conditions:— +</p> +<p> +1. It may contain no differential coefficients of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> higher +than the second. +</p> +<p> +2. It must be linear and homogeneous in these second differential +coefficients. +</p> +<p> +3. Its divergence must vanish identically. +</p> +<p> +The first two of these conditions are naturally taken from +Poisson's equation. Since it may be proved mathematically +that all such differential tensors can be formed algebraically +(i.e. without differentiation) from Riemann's tensor, our tensor +must be of the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 13.106ex; height: 2.23ex;" src="images/501.svg" alt=" " data-tex=" +R_{\mu\nu} + ag_{\mu\nu} R, +"></span> +in which <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/479.svg" alt=" " data-tex="R_{\mu\nu}"> and <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/212.svg" alt=" " data-tex="R"> are defined by (88) and (89) respectively. +Further, it may be proved that the third condition requires a +to have the value <img style="vertical-align: -1.552ex; width: 3.887ex; height: 4.588ex;" src="images/502.svg" alt=" " data-tex="-\dfrac{1}{2}">. For the law of the gravitational field we +therefore get the equation +<span class="align-center"><img style="vertical-align: -0.781ex; width: 32.031ex; height: 2.737ex;" src="images/503.svg" alt=" " data-tex=" +R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}. +\qquad \text{(96)} +"></span> +Equation (95) is a consequence of this equation. <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/504.svg" alt=" " data-tex="\kappa"> denotes a +constant, which is connected with the Newtonian gravitation +constant. +</p> +<p> +In the following I shall indicate the features of the theory +which are interesting from the point of view of physics, using as +little as possible of the rather involved mathematical method. +It must first be shown that the divergence of the left-hand side +actually vanishes. The energy principle for matter may be expressed, by (83), +<span class="align-center"><img style="vertical-align: -1.909ex; width: 27.615ex; height: 5.144ex;" src="images/505.svg" alt=" " data-tex=" +0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta}, +\qquad \text{(97)} +"></span> +<span class="pagenum" id="Page_89">[Pg 89]</span> +in which +<span class="align-center"><img style="vertical-align: -0.669ex; width: 17.886ex; height: 2.851ex;" src="images/506.svg" alt=" " data-tex=" +\mathfrak T_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}. +"></span> +The analogous operation, applied to the left-hand side of (96), +will lead to an identity. +</p> +<p> +In the region surrounding each world-point there are systems +of co-ordinates for which, choosing the <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}">-co-ordinate imaginary, +at the given point, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 34.922ex; height: 5.428ex;" src="images/507.svg" alt=" " data-tex=" +g_{\mu\nu} = g^{\mu\nu} + = -\delta_{\mu\nu} + = \begin{cases} + -1 & \text{if \(\mu = \nu\)}, \\ + \,\,\,\,0 & \text{if \(\mu \neq \nu\)}, + \end{cases} +"></span> +and for which the first derivatives of the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> and the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}"> vanish. +We shall verify the vanishing of the divergence of the left-hand +side at this point. At this point the components +<img style="vertical-align: -1.042ex; width: 3.421ex; height: 2.58ex;" src="images/508.svg" alt=" " data-tex="\Gamma_{\sigma\beta}^{\alpha}"> vanish, so +that we have to prove the vanishing only of +<span class="align-center"><img style="vertical-align: -1.909ex; width: 30.825ex; height: 5.056ex;" src="images/509.svg" alt=" " data-tex=" +\frac{\partial}{\partial x_{\sigma}} \left[ + \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) +\right]. +"></span> +Introducing (88) and (70) into this expression, we see that the +only terms that remain are those in which third derivatives of +the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}"> enter. Since the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> are to be replaced by +<img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/510.svg" alt=" " data-tex="-\delta_{\mu\nu}">, we obtain, +finally, only a few terms which may easily be seen to cancel +each other. Since the quantity that we have formed has a tensor +character, its vanishing is proved for every other system of co-ordinates +also, and naturally for every other four-dimensional +point. The energy principle of matter (97) is thus a mathematical +consequence of the field equations (96). +</p> +<p> +In order to learn whether the equations (96) are consistent +with experience, we must, above all else, find out whether they +<span class="pagenum" id="Page_90">[Pg 90]</span> +lead to the Newtonian theory as a first approximation. For this +purpose we must introduce various approximations into these +equations. We already know that Euclidean geometry and the +law of the constancy of the velocity of light are valid, to a certain +approximation, in regions of a great extent, as in the planetary +system. If, as in the special theory of relativity, we take the +fourth co-ordinate imaginary, this means that we must put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.353ex; height: 2.382ex;" src="images/511.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, +\qquad \text{(98)} +"></span> +in which the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}"> are so small compared to 1 that we can neglect +the higher powers of the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}"> and their derivatives. If we do this, +we learn nothing about the structure of the gravitational held, or +of metrical space of cosmical dimensions, but we do learn about +the influence of neighbouring masses upon physical phenomena. +</p> +<p> +Before carrying through this approximation we shall transform +(96). We multiply (96) by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}">, summed over the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> +observing the relation which follows from the definition of +the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/387.svg" alt=" " data-tex="g^{\mu\nu}">, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 10.936ex; height: 2.326ex;" src="images/512.svg" alt=" " data-tex=" +g_{\mu\nu} g^{\mu\nu} = 4, +"></span> +we obtain the equation +<span class="align-center"><img style="vertical-align: -0.685ex; width: 18.981ex; height: 2.326ex;" src="images/513.svg" alt=" " data-tex=" +R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T. +"></span> +If we put this value of <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/212.svg" alt=" " data-tex="R"> in (96) we obtain +<span class="align-center"><img style="vertical-align: -0.904ex; width: 44.2ex; height: 2.861ex;" src="images/514.svg" alt=" " data-tex=" +R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) + = -\kappa T_{\mu\nu}^{*}. +\qquad \text{(96a)} +"></span> +When the approximation which has been mentioned is carried +out, we obtain for the left-hand side, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 48.357ex; height: 6.785ex;" src="images/515.svg" alt=" " data-tex=" +-\tfrac{1}{2}\left( + \frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + + \frac{\partial^{2} \gamma_{\alpha\alpha}}{\partial x_{\mu}\, \partial x_{\nu}} + - \frac{\partial^{2} \gamma_{\mu\alpha}}{\partial x_{\nu}\, \partial x_{\alpha}} + - \frac{\partial^{2} \gamma_{\nu\alpha}}{\partial x_{\mu}\, \partial x_{\alpha}} +\right) +"></span> +<span class="pagenum" id="Page_91">[Pg 91]</span> +or +<span class="align-center"><img style="vertical-align: -2.827ex; width: 48.834ex; height: 6.785ex;" src="images/516.svg" alt=" " data-tex=" +-\tfrac{1}{2}\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + + \tfrac{1}{2} \frac{\partial}{\partial x_{\nu}}\left( + \frac{{\partial \gamma'}_{\mu\alpha}}{\partial x_{\alpha}} + \right) + + \tfrac{1}{2} \frac{\partial}{\partial x_{\mu}}\left( + \frac{{\partial \gamma'}_{\nu\alpha}}{\partial x_{\alpha}} + \right), +"></span> +in which has been put +<span class="align-center"><img style="vertical-align: -0.914ex; width: 30.468ex; height: 2.871ex;" src="images/517.svg" alt=" " data-tex=" +{\gamma'}_{\mu\nu} + = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}. +\qquad \text{(99)} +"></span> +</p> +<p> +We must now note that equation (96) is valid for any system +of co-ordinates. We have already specialized the system of +co-ordinates in that we have chosen it so that within the region +considered the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> differ infinitely little from the constant values +<img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/510.svg" alt=" " data-tex="-\delta_{\mu\nu}">. But this condition remains satisfied in any infinitesimal +change of co-ordinates, so that there are still four conditions +to which the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}"> may be subjected, provided these conditions +do not conflict with the conditions for the order of magnitude of +the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}">. We shall now assume that the system of co-ordinates +is so chosen that the four relations— +<span class="align-center"><img style="vertical-align: -2.237ex; width: 38.565ex; height: 5.909ex;" src="images/518.svg" alt=" " data-tex=" +0 = \frac{{\partial\gamma'}_{\mu\nu}}{\partial x_{\nu}} + = \frac{\partial\gamma_{\mu\nu}}{\partial x_{\nu}} + - \tfrac{1}{2} \frac{\partial\gamma_{\sigma\sigma}}{\partial x_{\mu}} +\qquad \text{(100)} +"></span> +are satisfied. Then (96a) takes the form +<span class="align-center"><img style="vertical-align: -2.332ex; width: 26.138ex; height: 5.944ex;" src="images/519.svg" alt=" " data-tex=" +\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}. +\qquad \text{(96b)} +"></span> +These equations may be solved by the method, familiar in +electrodynamics, of retarded potentials; we get, in an easily +understood notation, +<span class="align-center"><img style="vertical-align: -1.948ex; width: 48.917ex; height: 5.59ex;" src="images/520.svg" alt=" " data-tex=" +\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int + \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}. +\qquad \text{(101)} +"></span> +<span class="pagenum" id="Page_92">[Pg 92]</span> +</p> +<p> +In order to see in what sense this theory contains the Newtonian +theory, we must consider in greater detail the energy +tensor of matter. Considered phenomenologically, this energy +tensor is composed of that of the electromagnetic field and of +matter in the narrower sense. If we consider the different parts +of this energy tensor with respect to their order of magnitude, +it follows from the results of the special theory of relativity that +the contribution of the electromagnetic field practically vanishes +in comparison to that of ponderable matter. In our system of +units, the energy of one gram of matter is equal to 1, compared +to which the energy of the electric fields may be ignored, and +also the energy of deformation of matter, and even the chemical +energy. We get an approximation that is fully sufficient for our +purpose if we put +<span class="align-center"><img style="vertical-align: -3.551ex; width: 30.02ex; height: 8.234ex;" src="images/521.svg" alt=" " data-tex=" +\left. +\begin{aligned} +T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}, \\ +ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. +\end{aligned} +\right\} +\qquad \text{(102)} +"></span> +In this, <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma"> is the density at rest, that is, the density of the ponderable +matter, in the ordinary sense, measured with the aid +of a unit measuring rod, and referred to a Galilean system of +co-ordinates moving with the matter. +</p> +<p> +We observe, further, that in the co-ordinates we have chosen, +we shall make only a relatively small error if we replace the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> +by <img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/510.svg" alt=" " data-tex="-\delta_{\mu\nu}">, so that we put +<span class="align-center"><img style="vertical-align: -1.018ex; width: 28.45ex; height: 3.167ex;" src="images/522.svg" alt=" " data-tex=" +ds^{2} = -\sum {dx_{\mu}}^{2}. +\qquad \text{(102a)} +"></span> +</p> +<p> +The previous developments are valid however rapidly the +masses which generate the field may move relatively to our chosen +system of quasi-Galilean co-ordinates. But in astronomy +<span class="pagenum" id="Page_93">[Pg 93]</span> +we have to do with masses whose velocities, relatively to the +co-ordinate system employed, are always small compared to the +velocity of light, that is, small compared to 1, with our choice +of the unit of time. We therefore get an approximation which is +sufficient for nearly all practical purposes if in (101) we replace +the retarded potential by the ordinary (non-retarded) potential, +and if, for the masses which generate the field, we put +<span class="align-center"><img style="vertical-align: -1.577ex; width: 60.851ex; height: 5.187ex;" src="images/523.svg" alt=" " data-tex=" +\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad +\frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}. +\qquad \text{(103)} +"></span> +Then we get for <img style="vertical-align: 0; width: 3.72ex; height: 1.532ex;" src="images/524.svg" alt=" " data-tex="T^{\mu\nu}"> and <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}"> the values +<span class="align-center"><img style="vertical-align: -5.317ex; width: 25.929ex; height: 11.765ex;" src="images/525.svg" alt=" " data-tex=" +\left. +\begin{array}{*{3}{>{\qquad}r}>{\quad}r} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & -\sigma +\end{array} +\right\}. +\qquad \text{(104)} +"></span> +For <img style="vertical-align: 0; width: 1.593ex; height: 1.532ex;" src="images/526.svg" alt=" " data-tex="T"> we get the value <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, and, finally, +for <img style="vertical-align: -0.904ex; width: 3.322ex; height: 2.47ex;" src="images/527.svg" alt=" " data-tex="T_{\mu\nu}^{*}"> the values, +<span class="align-center"><img style="vertical-align: -8.905ex; width: 31.524ex; height: 18.941ex;" src="images/528.svg" alt=" " data-tex=" +\left. +\begin{array}{*{3}{>{\qquad}c}>{\quad}r} +\dfrac{\sigma}{2} & 0 & 0 & 0 \\ +0 & \dfrac{\sigma}{2} & 0 & 0 \\ +0 & 0 & \dfrac{\sigma}{2} & 0 \\ +0 & 0 & 0 & -\dfrac{\sigma}{2} +\end{array} +\right\}. +\qquad \text{(104a)} +"></span> +</p> +<p> +We thus get, from (101), +<span class="align-center"><img style="vertical-align: -4.821ex; width: 45.595ex; height: 10.774ex;" src="images/529.svg" alt=" " data-tex=" +\left. +\begin{aligned} +\gamma_{11} = \gamma_{22} = \gamma_{33} + &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, +\end{aligned} +\right\} +\qquad \text{(101a)} +"></span> +<span class="pagenum" id="Page_94">[Pg 94]</span> +while all the other <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}">, vanish. The least of these equations, +in connexion with equation (90a), contains Newton's theory of +gravitation. If we replace <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> by <img style="vertical-align: -0.025ex; width: 1.796ex; height: 1.441ex;" src="images/207.svg" alt=" " data-tex="ct"> we get +<span class="align-center"><img style="vertical-align: -2.237ex; width: 40.856ex; height: 5.849ex;" src="images/530.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{dt^{2}} + = \frac{\kappa c^{2}}{8\pi}\, \frac{\partial}{\partial x_{\mu}} \left\{ + \int \frac{\sigma\, dV_{0}}{r} + \right\}. +\qquad \text{(90b)} +"></span> +We see that the Newtonian gravitation constant <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, is connected +with the constant <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/504.svg" alt=" " data-tex="\kappa"> that enters into our field equations by the +relation +<span class="align-center"><img style="vertical-align: -1.602ex; width: 19.979ex; height: 5.018ex;" src="images/531.svg" alt=" " data-tex=" +K = \frac{\kappa c^{2}}{8\pi}. +\qquad \text{(105)} +"></span> +From the known numerical value of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, it therefore follows that +<span class="align-center"><img style="vertical-align: -1.914ex; width: 53.752ex; height: 5.398ex;" src="images/532.svg" alt=" " data-tex=" +\kappa = \frac{8\pi K}{c^{2}} + = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} + = 1.86·10^{-27}. +\qquad \text{(105a)} +"></span> +From (101) we see that even in the first approximation the structure +of the gravitational field differs fundamentally from that +which is consistent with the Newtonian theory; this difference +lies in the fact that the gravitational potential has the character +of a tensor and not a scalar. This was not recognized in the past +because only the component <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/533.svg" alt=" " data-tex="g_{44}">, to a first approximation, enters +the equations of motion of material particles. +</p> +<p> +In order now to be able to judge the behaviour of measuring +rods and clocks from our results, we must observe the following. +According to the principle of equivalence, the metrical relations +of the Euclidean geometry are valid relatively to a Cartesian +system of reference of infinitely small dimensions, and in a suitable +state of motion (freely falling, and without rotation). We +can make the same statement for local systems of co-ordinates +<span class="pagenum" id="Page_95">[Pg 95]</span> +which, relatively to these, have small accelerations, and therefore +for such systems of co-ordinates as are at rest relatively to +the one we have selected. For such a local system, we have, for +two neighbouring point events, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 51.054ex; height: 2.457ex;" src="images/534.svg" alt=" " data-tex=" +ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} + = - dS^{2} + dT^{2}, +"></span> +where <img style="vertical-align: -0.05ex; width: 2.636ex; height: 1.645ex;" src="images/535.svg" alt=" " data-tex="dS"> is measured directly by a measuring rod and <img style="vertical-align: -0.023ex; width: 2.769ex; height: 1.593ex;" src="images/536.svg" alt=" " data-tex="dT"> by +a clock at rest relatively to the system; these are the naturally +measured lengths and times. Since <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}">, on the other hand, is +known in terms of the co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> employed in finite regions, +in the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 17.836ex; height: 2.685ex;" src="images/537.svg" alt=" " data-tex=" +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, +"></span> +we have the possibility of getting the relation between naturally +measured lengths and times, on the one hand, and the corresponding +differences of co-ordinates, on the other hand. As the +division into space and time is in agreement with respect to the +two systems of co-ordinates, so when we equate the two expressions +for <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}"> we get two relations. If, by (101a), we put +<span class="align-center"><img style="vertical-align: -5.201ex; width: 48.691ex; height: 11.534ex;" src="images/538.svg" alt=" " data-tex=" +\begin{aligned} +ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) +({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ +\qquad + + \mspace{26mu}\left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}, +\end{aligned} +"></span> +we obtain, to a sufficiently close approximation, +<span class="align-center"><img style="vertical-align: -7.645ex; width: 63.489ex; height: 16.421ex;" src="images/539.svg" alt=" " data-tex=" +\left. +\begin{aligned} + &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ + &\qquad + \begin{aligned} + &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}, \\ +dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. +\end{aligned} +\end{aligned} +\right\} +\qquad \text{(106)} +"></span> +<span class="pagenum" id="Page_96">[Pg 96]</span> +</p> +<p> +The unit measuring rod has therefore the length, +<span class="align-center"><img style="vertical-align: -1.948ex; width: 16.351ex; height: 5.048ex;" src="images/540.svg" alt=" " data-tex=" +1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +"></span> +in respect to the system of co-ordinates we have selected. The +particular system of co-ordinates we have selected insures that +this length shall depend only upon the place, and not upon the +direction. If we had chosen a different system of co-ordinates +this would not be so. But however we may choose a system of +co-ordinates, the laws of configuration of rigid rods do not agree +with those of Euclidean geometry; in other words, we cannot +choose any system of co-ordinates so that the co-ordinate differences, +<img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/8.svg" alt=" " data-tex="\Delta x_{1}">, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/9.svg" alt=" " data-tex="\Delta x_{2}">, <img style="vertical-align: -0.375ex; width: 4.166ex; height: 1.994ex;" src="images/10.svg" alt=" " data-tex="\Delta x_{3}">, corresponding +to the ends of a unit measuring rod, oriented in any way, +shall always satisfy the relation +<img style="vertical-align: -0.685ex; width: 22.179ex; height: 2.572ex;" src="images/541.svg" alt=" " data-tex="\Delta x_{1}^{2} + \Delta x_{2}^{2} + \Delta x_{3}^{2} = 1">. +In this sense space is not Euclidean, +but "curved." It follows from the second of the relations above +that the interval between two beats of the unit clock (<img style="vertical-align: -0.023ex; width: 2.769ex; height: 1.593ex;" src="images/536.svg" alt=" " data-tex="dT"> = 1) +corresponds to the "time" +<span class="align-center"><img style="vertical-align: -1.948ex; width: 16.351ex; height: 5.048ex;" src="images/542.svg" alt=" " data-tex=" +1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +"></span> +in the unit used in our system of co-ordinates. The rate of a +clock is accordingly slower the greater is the mass of the ponderable +matter in its neighbourhood. We therefore conclude that +spectral lines which are produced on the sun's surface will be +displaced towards the red, compared to the corresponding lines +produced on the earth, by about 2 • 10<sup>-6</sup> of their wave-lengths. +At first, this important consequence of the theory appeared to +conflict with experiment; but results obtained during the past +year seem to make the existence of this effect more probable, and +<span class="pagenum" id="Page_97">[Pg 97]</span> +it can hardly be doubted that this consequence of the theory will +be confirmed within the next year. +</p> +<p> +Another important consequence of the theory, which can be +tested experimentally, has to do with the path of rays of light. +In the general theory of relativity also the velocity of fight is +everywhere the same, relatively to a local inertial system. This +velocity is unity in our natural measure of time. The law of +the propagation of light in general co-ordinates is therefore, +according to the general theory of relativity, characterized, by the +equation +<span class="align-center"><img style="vertical-align: -0.186ex; width: 7.374ex; height: 2.185ex;" src="images/543.svg" alt=" " data-tex=" +ds^{2} = 0 +"></span> +To within the approximation which we are using, and in the +system of co-ordinates which we have selected, the velocity of +light is characterized, according to (106), by the equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 66.477ex; height: 5.428ex;" src="images/544.svg" alt=" " data-tex=" +\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) + = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}. +"></span> +The velocity of light <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/545.svg" alt=" " data-tex="L">, is therefore expressed in our co-ordinates +by +<span class="align-center"><img style="vertical-align: -1.948ex; width: 52.226ex; height: 5.945ex;" src="images/546.svg" alt=" " data-tex=" +\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} + = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}. +\qquad \text{(107)} +"></span> +We can therefore draw the conclusion from this, that a ray of +light passing near a large mass is deflected. If we imagine the +sun, of mass <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/169.svg" alt=" " data-tex="M"> concentrated at the origin of our system of co-ordinates, +then a ray of fight, travelling parallel to the <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">-axis. +in the <img style="vertical-align: -0.375ex; width: 7.329ex; height: 1.694ex;" src="images/547.svg" alt=" " data-tex="x_{1}-x_{3}"> plane, at a distance <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"> from the origin, will be +deflected, in all, by an amount +<span class="align-center"><img style="vertical-align: -2.159ex; width: 21.475ex; height: 5.553ex;" src="images/548.svg" alt=" " data-tex=" +\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\partial L}{\partial x_{1}}\, dx_{3} +"></span> +<span class="pagenum" id="Page_98">[Pg 98]</span> +towards the sun. On performing the integration we get +<span class="align-center"><img style="vertical-align: -1.577ex; width: 20.451ex; height: 4.652ex;" src="images/549.svg" alt=" " data-tex=" +\alpha = \frac{\kappa M}{2\pi\Delta}. +\qquad \text{(108)} +"></span> +</p> +<p> +The existence of this deflection, which amounts to 1.7<code>''</code> for +<img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"> equal to the radius of the sun, was confirmed, with remarkable +accuracy, by the English Solar Eclipse Expedition in 1919, and +most careful preparations have been made to get more exact +observational data at the solar eclipse in 1922. It should be +noted that this result, also, of the theory is not influenced by +our arbitrary choice of a system of co-ordinates. +</p> +<p> +This is the place to speak of the third consequence of the +theory which can be tested by observation, namely, that which +concerns the motion of the perihelion of the planet Mercury. The +secular changes in the planetary orbits are known with such accuracy +that the approximation we have been using is no longer +sufficient for a comparison of theory and observation. It is necessary +to go back to the general field equations (96). To solve +this problem I made use of the method of successive approximations. +Since then, however, the problem of the central symmetrical +statical gravitational field has been completely solved by +Schwarzschild and others; the derivation given by H. Weyl in his +book, "Raum-Zeit-Materie," is particularly elegant. The calculation +can be simplified somewhat if we do not go back directly +to the equation (96), but base it upon a principle of variation +that is equivalent to this equation. I shall indicate the procedure +only in so far as is necessary for understanding the method. +<span class="pagenum" id="Page_99">[Pg 99]</span> +</p> +<p> +In the case of a statical field, <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/361.svg" alt=" " data-tex="ds^{2}"> must have the form +<span class="align-center"><img style="vertical-align: -3.502ex; width: 33.95ex; height: 8.135ex;" src="images/550.svg" alt=" " data-tex=" +\left. +\begin{aligned} +ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}, \\ +d\sigma^{2} + &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}, +\end{aligned} +\right\} +\qquad \text{(109)} +"></span> +where the summation on the right-hand side of the last equation +is to be extended over the space variables only. The central +symmetry of the field requires the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}">, to be of the form, +<span class="align-center"><img style="vertical-align: -0.65ex; width: 30.455ex; height: 2.347ex;" src="images/551.svg" alt=" " data-tex=" +\gamma_{\alpha\beta} + = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}; +\qquad \text{(110)} +"></span> +<img style="vertical-align: -0.464ex; width: 2.352ex; height: 2.351ex;" src="images/552.svg" alt=" " data-tex="f^{2}">, <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> are functions of +<img style="vertical-align: -0.532ex; width: 21.685ex; height: 2.851ex;" src="images/553.svg" alt=" " data-tex="r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}"> only. One +of these three functions can be chosen arbitrarily, because our +system of co-ordinates is, <i>a priori</i>, completely arbitrary; for by +a substitution +<span class="align-center"><img style="vertical-align: -2.17ex; width: 13.76ex; height: 5.47ex;" src="images/554.svg" alt=" " data-tex=" +\begin{align*} +{x'}_{4} &= x_{4}, \\ +{x'}_{\alpha} &= F(r) x_{\alpha}, +\end{align*} +"></span> +we can always insure that one of these three functions shall be +an assigned function of <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/173.svg" alt=" " data-tex="r">'. In place of (110) we can therefore +put, without limiting the generality, +<span class="align-center"><img style="vertical-align: -0.65ex; width: 30.222ex; height: 2.347ex;" src="images/555.svg" alt=" " data-tex=" +\gamma_{\alpha\beta} + = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}. +\qquad \text{(110a)} +"></span> +</p> +<p> +In this way the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> are expressed in terms of the two quantities +<img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> and <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/556.svg" alt=" " data-tex="f">. These are to be determined as functions of <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/173.svg" alt=" " data-tex="r">, +by introducing them into equation (96), after first calculating +<span class="pagenum" id="Page_100">[Pg 100]</span> +the <img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/557.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\sigma}"> from (109) and (110a). We have +<span class="align-center"><img style="vertical-align: -7.006ex; width: 64.764ex; height: 15.143ex;" src="images/558.svg" alt=" " data-tex=" +\left. +\begin{aligned} +\Gamma_{\alpha\beta}^{\sigma} + &= \tfrac{1}{2} \frac{x_{\sigma}}{r} + · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} + {1 + \lambda r^{2}}\,\, + \text{(for \(\alpha, \beta, \sigma = 1, 2, 3)\)}, \\ +\Gamma_{44}^{4} + &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad + \text{(for \(\alpha, \beta = 1, 2, 3)\)}, \\ +\Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\partial f^{2}}{\partial x_{\alpha}},\quad +\Gamma_{44}^{\alpha} + = -\tfrac{1}{2}{g^{\alpha\beta}}\, + \frac{\partial f^{2}}{\partial {x_{\beta}}}. +\end{aligned} +\right\} +\qquad \text{(110b)} +"></span> +</p> +<p> +With the help of these results, the field equations furnish +Schwarzschild's solution: +<span class="align-center"><img style="vertical-align: -4.636ex; width: 65.804ex; height: 10.403ex;" src="images/559.svg" alt=" " data-tex=" +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ + \frac{dr^{2}} + {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) +\right], +\qquad \text{(109a)} +"></span> +in which we have put +<span class="align-center"><img style="vertical-align: -7.643ex; width: 29.619ex; height: 16.416ex;" src="images/560.svg" alt=" " data-tex=" +\left. +\begin{aligned} +x_{4} &= l, \\ +x_{1} &= r \sin\theta \sin\phi, \\ +x_{2} &= r \sin\theta \cos\phi, \\ +x_{3} &= r \cos\theta, \\ +A &= \frac{\kappa M}{4\pi}. +\end{aligned} +\right\} +\qquad \text{(109b)} +"></span> +</p> +<p> +<img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/169.svg" alt=" " data-tex="M"> denotes the sun's mass, centrally symmetrically placed +about the origin of co-ordinates; the solution (109) is valid only +outside of this mass, where all the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/311.svg" alt=" " data-tex="T_{\mu\nu}"> vanish. If the motion +of the planet takes place in the <img style="vertical-align: -0.339ex; width: 7.329ex; height: 1.658ex;" src="images/561.svg" alt=" " data-tex="x_{1}-x_{2}"> plane then we +must replace (109a) by +<span class="align-center"><img style="vertical-align: -4.636ex; width: 49.704ex; height: 8.052ex;" src="images/562.svg" alt=" " data-tex=" +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} + - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}. +\qquad \text{(109c)} +"></span> +<span class="pagenum" id="Page_101">[Pg 101]</span> +</p> +<p> +The calculation of the planetary motion depends upon equation (90). +From the first of equations (110b) and (90) we get, +for the indices 1, 2, 3, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 28.237ex; height: 5.428ex;" src="images/563.svg" alt=" " data-tex=" +\frac{d}{ds} + \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0, +"></span> +or, if we integrate, and express the result in polar co-ordinates, +<span class="align-center"><img style="vertical-align: -1.575ex; width: 27.665ex; height: 4.674ex;" src="images/564.svg" alt=" " data-tex=" +r^{2} \frac{d\phi}{ds} = \text{constant}. +\qquad \text{(111)} +"></span> +</p> +<p> +From (90), for <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> = 4, we get +<span class="align-center"><img style="vertical-align: -2.093ex; width: 50.697ex; height: 5.509ex;" src="images/565.svg" alt=" " data-tex=" +0 = \frac{d^{2} l}{ds^{2}} + + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} +{\, \frac{dl}{ds}} + = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} +{\, \frac{dl}{ds}}. +"></span> +From this, after multiplication by <img style="vertical-align: -0.464ex; width: 2.352ex; height: 2.351ex;" src="images/552.svg" alt=" " data-tex="f^{2}"> and integration, we have +<span class="align-center"><img style="vertical-align: -1.575ex; width: 27.721ex; height: 4.674ex;" src="images/566.svg" alt=" " data-tex=" +f^{2} \frac{dl}{ds} = \text{constant}. +\qquad \text{(112)} +"></span> +</p> +<p> +In (109c), (111) and (112) we have three equations between +the four variables <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/173.svg" alt=" " data-tex="r">, <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/200.svg" alt=" " data-tex="l"> and <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/258.svg" alt=" " data-tex="\phi">, from which the motion of the +planet may be calculated in the same way as in classical mechanics. +The most important result we get from this is a secular +rotation of the elliptic orbit of the planet in the same sense as +the revolution of the planet, amounting in radians per revolution +to +<span class="align-center"><img style="vertical-align: -2.194ex; width: 24.054ex; height: 5.611ex;" src="images/567.svg" alt=" " data-tex=" +\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}, +\qquad \text{(113)} +"></span> +<span class="pagenum" id="Page_102">[Pg 102]</span> +where +<span class="align-center"><img style="vertical-align: -5.129ex; width: 59.981ex; height: 11.389ex;" src="images/568.svg" alt=" " data-tex=" +\begin{align*} +a &= \text{the semi-major axis of the planetary orbit in centimetres.} \\ +e &= \text{the numerical eccentricity.} \\ +c &= 3·10^{+10}, \text{the velocity of light}\, \mathit{in\, vacuo}. \\ +T &= \text{the period of revolution in seconds.} +\end{align*} +"></span> +This expression furnishes the explanation of the motion of the +perihelion of the planet Mercury, which has been known for a +hundred years (since Leverrier), and for which theoretical astronomy +has hitherto been unable satisfactorily to account. +</p> +<p> +There is no difficulty in expressing Maxwell's theory of the +electromagnetic field in terms of the general theory of relativity; +this is done by application of the tensor formation (81), (82) +and (77). Let <img style="vertical-align: -0.685ex; width: 2.501ex; height: 2.255ex;" src="images/569.svg" alt=" " data-tex="\phi_{\mu}"> be a tensor of the first rank, to be denoted +as an electromagnetic 4-potential; then an electromagnetic field +tensor may be defined by the relations, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.316ex; height: 5.58ex;" src="images/570.svg" alt=" " data-tex=" +\phi_{\mu\nu} + = \frac{\partial \phi_{\mu}}{\partial x_{\nu}} - \frac{\partial \phi_{\nu}}{\partial x_{\mu}}. +\qquad \text{(114)} +"></span> +The second of Maxwell's systems of equations is then defined by +the tensor equation, resulting from this, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 38.212ex; height: 5.58ex;" src="images/571.svg" alt=" " data-tex=" +\frac{\partial \phi_{\mu\nu}}{\partial x_{\rho}} + +\frac{\partial \phi_{\nu\rho}}{\partial x_{\mu}} + +\frac{\partial \phi_{\rho\mu}}{\partial x_{\nu}} = 0, +\qquad \text{(114a)} +"></span> +and the hrst of Maxwell's systems of equations is defined by the +tensor-density relation +<span class="align-center"><img style="vertical-align: -1.891ex; width: 21.762ex; height: 5.038ex;" src="images/572.svg" alt=" " data-tex=" +\frac{\partial \mathfrak F^{\mu\nu}}{\partial x_{\nu}} = \mathfrak J^{\mu}, +\qquad \text{(115)} +"></span> +<span class="pagenum" id="Page_103">[Pg 103]</span> +in which +<span class="align-center"><img style="vertical-align: -3.536ex; width: 21.852ex; height: 8.204ex;" src="images/573.svg" alt=" " data-tex=" +\begin{align*} +\mathfrak F^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}, \\ +\mathfrak J^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. +\end{align*} +"></span> +If we introduce the energy tensor of the electromagnetic field +into the right-hand side of (96), we obtain (115), for the special +case <img style="vertical-align: -0.314ex; width: 2.401ex; height: 1.867ex;" src="images/574.svg" alt=" " data-tex="\mathfrak J^{\mu}"> = 0, as a consequence of (96) by taking the divergence. +This inclusion of the theory of electricity in the scheme of the +general theory of relativity has been considered arbitrary and +unsatisfactory by many theoreticians. Nor can we in this way +conceive of the equilibrium of the electricity which constitutes +the elementary electrically charged particles. A theory in which +the gravitational field and the electromagnetic field enter as an +essential entity would be much preferable. H. Weyl, and recently +Th. Kaluza, have discovered some ingenious theorems along this +direction; but concerning them, I am convinced that they do not +bring us nearer to the true solution of the fundamental problem. +I shall not go into this further, but shall give a brief discussion +of the so-called cosmological problem, for without this, the considerations +regarding the general theory of relativity would, in +a certain sense, remain unsatisfactory. +</p> +<p> +Our previous considerations, based upon the field equations (96), +had for a foundation the conception that space on +the whole is Galilean-Euclidean, and that this character is disturbed +only by masses embedded in it. This conception was +certainly justified as long as we were dealing with spaces of the +order of magnitude of those that astronomy has to do with. +But whether portions of the universe, however large they may +be, are quasi-Euclidean, is a wholly different question. We can +<span class="pagenum" id="Page_104">[Pg 104]</span> +make this clear by using an example from the theory of surfaces +which we have employed many times. If a portion of a surface +is observed by the eye to be practically plane, it does not at all +follow that the whole surface has the form of a plane; the surface +might just as well be a sphere, for example, of sufficiently large +radius. The question as to whether the universe as a whole is +non-Euclidean was much discussed from the geometrical point of +view before the development of the theory of relativity. But with +the theory of relativity, this problem has entered upon a new +stage, for according to this theory the geometrical properties of +bodies are not independent, but depend upon the distribution +of masses. +</p> +<p> +If the universe were quasi-Euclidean, then Mach was wholly +wrong in his thought that inertia, as well as gravitation, depends +upon a kind of mutual action between bodies. For in this case, +with a suitably selected system of co-ordinates, the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> would +be constant at infinity, as they are in the special theory of relativity, +while within finite regions the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> would differ from these +constant values by small amounts only, with a suitable choice +of co-ordinates, as a result of the influence of the masses in finite +regions. The physical properties of space would not then be +wholly independent, that is, uninfluenced by matter, but in the +main they would be, and only in small measure, conditioned by +matter. Such a dualistic conception is even in itself not satisfactory; +there are, however, some important physical arguments +against it, which we shall consider. +</p> +<p> +The hypothesis that the universe is infinite and Euclidean +at infinity, is, from the relativistic point of view, a complicated +hypothesis. In the language of the general theory of relativity +it demands that the Riemann tensor of the fourth rank +<img style="vertical-align: -0.357ex; width: 5.172ex; height: 1.902ex;" src="images/575.svg" alt=" " data-tex="R_{iklm}"> +<span class="pagenum" id="Page_105">[Pg 105]</span> +shall vanish at infinity, which furnishes twenty independent conditions, +while only ten curvature components <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/479.svg" alt=" " data-tex="R_{\mu\nu}">, enter into +the laws of the gravitational field. It is certainly unsatisfactory +to postulate such a far-reaching limitation without any physical +basis for it. +</p> +<p> +But in the second place, the theory of relativity makes it +appear probable that Mach was on the right road in his thought +that inertia depends upon a mutual action of matter. For we +shall show in the following that, according to our equations, inert +masses do act upon each other in the sense of the relativity of +inertia, even if only very feebly. What is to be expected along +the line of Mach's thought? +</p> +<p class="hanging2"> +1. The inertia of a body must increase when ponderable +masses are piled up in its neighbourhood. +</p> +<p class="hanging2"> +2. A body must experience an accelerating force when +neighbouring masses are accelerated, and, in fact, the +force must be in the same direction as the acceleration. +</p> +<p class="hanging2"> +3. A rotating hollow body must generate inside of itself +a "Coriolis field," which deflects moving bodies in the +sense of the rotation, and a radial centrifugal field as +well. +</p> +<p> +We shall now show that these three effects, which are to be +expected in accordance with Mach's ideas, are actually present +according to our theory, although their magnitude is so small +that confirmation of them by laboratory experiments is not to be +thought of. For this purpose we shall go back to the equations of +motion of a material particle (90), and carry the approximations +somewhat further than was done in equation (90a). +<span class="pagenum" id="Page_106">[Pg 106]</span> +</p> +<p> +First, we consider <img style="vertical-align: -0.489ex; width: 2.96ex; height: 1.486ex;" src="images/576.svg" alt=" " data-tex="\gamma_{44}"> as small of the first order. The square +of the velocity of masses moving under the influence of the gravitational +force is of the same order, according to the energy +equation. It is therefore logical to regard the velocities of the +material particles we are considering, as well as the velocities +of the masses which generate the field, as small, of the order +<img style="vertical-align: -1.552ex; width: 2.127ex; height: 4.588ex;" src="images/577.svg" alt=" " data-tex="\dfrac{1}{2}">. +We shall now carry out the approximation in the equations that +arise from the field equations (101) and the equations of motion (90) +so far as to consider terms, in the second member +of (90), that are linear in those velocities. Further, we shall not +put <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/347.svg" alt=" " data-tex="ds"> and <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/283.svg" alt=" " data-tex="dl"> equal to each other, but, corresponding to the +higher approximation, we shall put +<span class="align-center"><img style="vertical-align: -1.552ex; width: 28.33ex; height: 4.153ex;" src="images/578.svg" alt=" " data-tex=" +ds = \sqrt{g_{44}}\, dl + = \left(1 - \frac{\gamma_{44}}{2}\right) dl. +"></span> +From (90) we obtain, at first, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 59.568ex; height: 5.444ex;" src="images/579.svg" alt=" " data-tex=" +\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} + \left(1 + \frac{\gamma_{44}}{2}\right). +\qquad \text{(116)} +"></span> +</p> +<p> +From (101) we get, to the approximation sought for, +<span class="align-center"><img style="vertical-align: -7.642ex; width: 54.085ex; height: 16.414ex;" src="images/580.svg" alt=" " data-tex=" +\left. +\begin{aligned} +-\gamma_{11} = -\gamma_{22} = -\gamma_{33} + &= \gamma_{44} = \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\gamma_{4\alpha} + &= -\frac{i \kappa}{2} + \int \frac{\sigma \dfrac{dx_{\alpha}}{ds}\, dV_{0}}{r}, \\ +\gamma_{\alpha\beta} &= 0, +\end{aligned} +\right\} +\qquad \text{(117)} +"></span> +in which, in (117), <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> denote the space indices only. +<span class="pagenum" id="Page_107">[Pg 107]</span> +</p> +<p> +On the right-hand side of (116) we can replace +1 + <img style="vertical-align: -1.552ex; width: 6.656ex; height: 4.153ex;" src="images/497.svg" alt=" " data-tex="\left(\dfrac{\gamma_{44}}{2}\right)"> by 1 +and <img style="vertical-align: -1.045ex; width: 5.291ex; height: 2.898ex;" src="images/581.svg" alt=" " data-tex="-\Gamma_{\alpha\beta}^{\mu}"> by +<img style="vertical-align: -1.469ex; width: 4.608ex; height: 4.07ex;" src="images/582.svg" alt=" " data-tex="\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right]">. +It is easy to see, in addition, that to this +degree of approximation we must put +<span class="align-center"><img style="vertical-align: -8.407ex; width: 28.058ex; height: 17.946ex;" src="images/583.svg" alt=" " data-tex=" +\begin{align*} +\left[{{\genfrac{}{}{0pt}{}{44}{\mu}}}\right] + &= -\tfrac{1}{2} \frac{\partial \gamma_{44}}{\partial x_{\mu}} + + \frac{\partial \gamma_{4\mu}}{\partial x_{4}},\\ +\left[{{\genfrac{}{}{0pt}{}{\alpha 4}{\mu}}}\right] + &= \tfrac{1}{2} + \left(\frac{\partial \gamma_{4\mu}}{\partial x_{\alpha}} - \frac{\partial \gamma_{4\alpha}}{\partial x_{\mu}}\right), \\ +\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right] &= 0, +\end{align*} +"></span> +in which <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> and <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> denote space indices. +We therefore obtain +from (116), in the usual vector notation, +<span class="align-center"><img style="vertical-align: -8.844ex; width: 52.436ex; height: 18.819ex;" src="images/584.svg" alt=" " data-tex=" +\DeclareMathOperator{\grad}{grad} +\DeclareMathOperator{\rot}{rot} +\left. +\begin{aligned} +\frac{d}{dl}\bigl[(1 + \overline{\sigma}) \mathbf v \bigr] + &= \grad \overline{\sigma} + \frac{\partial \mathfrak A}{\partial l} + [\rot \mathfrak A, \mathbf v], \\ +\overline{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\mathfrak A &= \frac{\kappa}{2 \pi} + \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}. +\end{aligned} +\right\} +\qquad \text{(118)} +"></span> +</p> +<p> +The equations of motion, (118), show now, in fact, that +</p> +<p class="hanging"> +1. The inert mass is proportional to 1 + <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1.742ex;" src="images/585.svg" alt=" " data-tex="\overline{\sigma}"> and therefore +increases when ponderable masses approach the test +body. +</p> +<p class="hanging"> +2. There is an inductive action of accelerated masses, +of the same sign, upon the test body. This is the +term <img style="vertical-align: -1.602ex; width: 3.9ex; height: 4.749ex;" src="images/586.svg" alt=" " data-tex="\dfrac{\partial \mathfrak A}{\partial l}">. +<span class="pagenum" id="Page_108">[Pg 108]</span> +</p> +<p class="hanging"> +3. A material particle, moving perpendicularly to the axis +of rotation inside a rotating hollow body, is deflected in +the sense of the rotation (Coriolis field). The centrifugal +action, mentioned above, inside a rotating hollow +body, also follows from the theory, as has been shown +by Thirring.<a id="FNanchor_18_1"></a><a href="#Footnote_18_1" class="fnanchor">[18]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_18_1"></a><a href="#FNanchor_18_1"><span class="label">[18]</span></a>That the centrifugal action must be inseparably connected with the +existence of the Coriolis field may be recognized, even without calculation, +in the special case of a co-ordinate system rotating uniformly relatively to +an inertial system; our general co-variant equations naturally must apply +to such a case.</p></div> + +<p> +Although all of these effects are inaccessible to experiment, +because <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/504.svg" alt=" " data-tex="\kappa"> is so small, nevertheless they certainly exist according +to the general theory of relativity. We must see in them a +strong support for Mach's ideas as to the relativity of all inertial +actions. If we think these ideas consistently through to the end +we must expect the <i>whole</i> inertia, that is, the <i>whole</i> +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">-field, to +be determined by the matter of the universe, and not mainly by +the boundary conditions at infinity. +</p> +<p> +For a satisfactory conception of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">-field of cosmical dimensions, +the fact seems to be of significance that the relative +velocity of the stars is small compared to the velocity of light. +It follows from this that, with a suitable choice of co-ordinates, +<img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/533.svg" alt=" " data-tex="g_{44}"> is nearly constant in the universe, at least, in that part of +the universe in which there is matter. The assumption appears +natural, moreover, that there are stars in all parts of the universe, +so that we may well assume that the inconstancy of <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/533.svg" alt=" " data-tex="g_{44}"> depends +only upon the circumstance that matter is not distributed +continuously, but is concentrated in single celestial bodies and +systems of bodies. If we are willing to ignore these more local +<span class="pagenum" id="Page_109">[Pg 109]</span> +non-uniformities of the density of matter and of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">-field, in +order to learn something of the geometrical properties of the universe +as a whole, it appears natural to substitute for the actual +distribution of masses a continuous distribution, and furthermore +to assign to this distribution a uniform density <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">. In this +imagined universe all points with space directions will be geometrically +equivalent; with respect to its space extension it will +have a constant curvature, and will be cylindrical with respect +to its <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}">-co-ordinate. The possibility seems to be particularly +satisfying that the universe is spatially bounded and thus, in +accordance with our assumption of the constancy of <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, is of +constant curvature, being either spherical or elliptical; for then +the boundary conditions at infinity which are so inconvenient +from the standpoint of the general theory of relativity, may be +replaced by the much more natural conditions for a closed surface. +</p> +<p> +According to what has been said, we are to put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 35.196ex; height: 2.703ex;" src="images/587.svg" alt=" " data-tex=" +ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, +\qquad \text{(119)} +"></span> +in which the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> run from 1 to 3 only. +The <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/490.svg" alt=" " data-tex="\gamma_{\mu\nu}"> +will be such functions of <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> as correspond to +a three-dimensional continuum of constant positive curvature. We must +now investigate whether such an assumption can satisfy the field +equations of gravitation. +</p> +<p> +In order to be able to investigate this, we must first find +what differential conditions the three-dimensional manifold of +constant curvature satisfies. A spherical manifold of three dimensions, +<span class="pagenum" id="Page_110">[Pg 110]</span> +embedded in a Euclidean continuum of four dimensions,<a id="FNanchor_19_1"></a><a href="#Footnote_19_1" class="fnanchor">[19]</a> +is given by the equations +<span class="align-center"><img style="vertical-align: -2.348ex; width: 32.952ex; height: 5.828ex;" src="images/588.svg" alt=" " data-tex=" +\begin{align*} +{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}, \\ +{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. +\end{align*} +"></span> +By eliminating <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/240.svg" alt=" " data-tex="x_{4}">, we get +<span class="align-center"><img style="vertical-align: -2.003ex; width: 56.135ex; height: 5.496ex;" src="images/589.svg" alt=" " data-tex=" +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} + {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}. +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_19_1"></a><a href="#FNanchor_19_1"><span class="label">[19]</span></a>The aid of a fourth space dimension has naturally no significance except +that of a mathematical artifice.</p></div> + +<p> +As far as terms of the third and higher degrees in the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, we +can put, in the neighbourhood of the origin of co-ordinates, +<span class="align-center"><img style="vertical-align: -1.651ex; width: 28.622ex; height: 4.377ex;" src="images/590.svg" alt=" " data-tex=" +ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) + dx_{\mu}\, dx_{\nu}. +"></span> +</p> +<p> +Inside the brackets are the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}"> of the manifold in the neighbourhood +of the origin. Since the first derivatives of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/368.svg" alt=" " data-tex="g_{\mu\nu}">, +and therefore also the <img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/557.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\sigma}">, vanish at the origin, the calculation +of the <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/479.svg" alt=" " data-tex="R_{\mu\nu}"> for this manifold, by (88), is very simple at the origin. +We have +<span class="align-center"><img style="vertical-align: -1.651ex; width: 24.585ex; height: 4.688ex;" src="images/591.svg" alt=" " data-tex=" +R_{\mu\nu} = -\frac{2}{a^{2}} \delta_{\mu\nu} + = \frac{2}{a^{2}} g_{\mu\nu}. +"></span> +</p> +<p> +Since the relation +<img style="vertical-align: -1.651ex; width: 12.994ex; height: 4.688ex;" src="images/592.svg" alt=" " data-tex="R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}"> is universally co-variant, +and since all points of the manifold are geometrically equivalent, +this relation holds for every system of co-ordinates, and +everywhere in the manifold. In order to avoid confusion with +<span class="pagenum" id="Page_111">[Pg 111]</span> +the four-dimensional continuum, we shall, in the following, designate +quantities that refer to the three-dimensional continuum +by Greek letters, and put +<span class="align-center"><img style="vertical-align: -1.651ex; width: 25.267ex; height: 4.688ex;" src="images/593.svg" alt=" " data-tex=" +P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}. +\qquad \text{(120)} +"></span> +</p> +<p> +We now proceed to apply the field equations (96) to our special +case. From (119) we get for the four-dimensional manifold, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 44.083ex; height: 5.428ex;" src="images/594.svg" alt=" " data-tex=" +\left. +\begin{aligned} +R_{\mu\nu} &= P_{\mu\nu} \quad \text{for the indices 1 to 3}, \\ +R_{14} &= R_{24} = R_{34} = R_{44} = 0. +\end{aligned} +\right\} +\qquad \text{(121)} +"></span> +</p> +<p> +For the right-hand side of (96) we have to consider the energy +tensor for matter distributed like a cloud of dust. According to +what has gone before we must therefore put +<span class="align-center"><img style="vertical-align: -1.575ex; width: 17.527ex; height: 4.87ex;" src="images/595.svg" alt=" " data-tex=" +T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds} +"></span> +specialized for the case of rest. But in addition, we shall add +a pressure term that may be physically established as follows. +Matter consists of electrically charged particles. On the basis +of Maxwell's theory these cannot be conceived of as electromagnetic +fields free from singularities. In order to be consistent +with the facts, it is necessary to introduce energy terms, not +contained in Maxwell's theory, so that the single electric particles +may hold together in spite of the mutual repulsions between +their elements, charged with electricity of one sign. For the sake +of consistency with this fact, Poincaré has assumed a pressure +<span class="pagenum" id="Page_112">[Pg 112]</span> +to exist inside these particles which balances the electrostatic +repulsion. It cannot, however, be asserted that this pressure +vanishes outside the particles. We shall be consistent with this +circumstance if, in our phenomenological presentation, we add +a pressure term. This must not, however, be confused with a +hydrodynamical pressure, as it serves only for the energetic presentation +of the dynamical relations inside matter. In this sense +we put +<span class="align-center"><img style="vertical-align: -1.575ex; width: 39.787ex; height: 4.835ex;" src="images/596.svg" alt=" " data-tex=" +T_{\mu\nu} + = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + - g_{\mu\nu} p. +\qquad \text{(122)} +"></span> +</p> +<p> +In our special case we have, therefore, to put +<span class="align-center"><img style="vertical-align: -3.626ex; width: 38.187ex; height: 8.383ex;" src="images/597.svg" alt=" " data-tex=" +\begin{align*} +T_{\mu\nu} + &= \gamma_{\mu\nu} p \quad \text{(for}\,\,\mu\,\, \text{and}\,\,\,\nu \,\,\text{from 1 to 3)}, \\ +T_{44} &= \sigma - p, \\ +T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p + = \sigma - 4p. +\end{align*} +"></span> +Observing that the field equation (96) may be written in the +form +<span class="align-center"><img style="vertical-align: -0.781ex; width: 24.742ex; height: 2.737ex;" src="images/598.svg" alt=" " data-tex=" +R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T), +"></span> +we get from (96) the equations, +<span class="align-center"><img style="vertical-align: -4.194ex; width: 25.503ex; height: 9.519ex;" src="images/599.svg" alt=" " data-tex=" +\begin{align*} ++\frac{2}{a^{2}} \gamma_{\mu\nu} + &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}, \\ +0 &= -\kappa \left(\frac{\sigma}{2} + p\right). +\end{align*} +"></span> +From this follows +<span class="align-center"><img style="vertical-align: -4.585ex; width: 22.431ex; height: 10.301ex;" src="images/600.svg" alt=" " data-tex=" +\left. +\begin{aligned} +p &= -\frac{\sigma}{2}, \\ +a &= \sqrt{\frac{2}{\kappa\sigma}}. +\end{aligned} +\right\} +\qquad \text{(123)} +"></span> +</p> +<p> +If the universe is quasi-Euclidean, and its radius of curvature +therefore infinite, then a would vanish. But it is improbable that +<span class="pagenum" id="Page_113">[Pg 113]</span> +the mean density of matter in the universe is actually zero; this +is our third argument against the assumption that the universe +is quasi-Euclidean. Nor does it seem possible that our hypothetical +pressure can vanish; the physical nature of this pressure can +be appreciated only after we have a better theoretical knowledge +of the electromagnetic field. According to the second of equations +(123) the radius, <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/75.svg" alt=" " data-tex="a">, of the universe is determined in terms +of the total mass, <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/169.svg" alt=" " data-tex="M">, of matter, by the equation +<span class="align-center"><img style="vertical-align: -1.654ex; width: 19.575ex; height: 4.728ex;" src="images/601.svg" alt=" " data-tex=" +a = \frac{M\kappa}{4\pi^{2}}. +\qquad \text{(124)} +"></span> +The complete dependence of the geometrical upon the physical +properties becomes clearly apparent by means of this equation. +</p> +<p> +Thus we may present the following arguments against the +conception of a space-infinite, and for the conception of a space-bounded, +universe:— +</p> +<p> +1. From the standpoint of the theory of relativity, the condition +for a closed surface is very much simpler than the corresponding +boundary condition at infinity of the quasi-Euclidean +structure of the universe. +</p> +<p> +2. The idea that Mach expressed, that inertia depends upon +the mutual action of bodies, is contained, to a first approximation, +in the equations of the theory of relativity; it follows +from these equations that inertia depends, at least in part, upon +mutual actions between masses. As it is an unsatisfactory assumption +to make that inertia depends in part upon mutual +actions, and in part upon an independent property of space, +Mach's idea gains in probability. But this idea of Mach's corresponds +only to a finite universe, bounded in space, and not to a +quasi-Euclidean, infinite universe. From the standpoint of epistemology +<span class="pagenum" id="Page_114">[Pg 114]</span> +it is more satisfying to have the mechanical properties +of space completely determined by matter, and this is the case +only in a space-bounded universe. +</p> +<p> +3. An infinite universe is possible only if the mean density of +matter in the universe vanishes. Although such an assumption +is logically possible, it is less probable than the assumption that +there is a finite mean density of matter in the universe. +<span class="pagenum" id="Page_115">[Pg 115]</span> +</p> + +<p><br><br></p> + +<h2><a id="INDEX">INDEX</a></h2> + +<p>A</p> +<p class="nind"> +Accelerated masses, inductive<br> +<span style="margin-left: 1em;">action of, <a href="#Page_108">108</a></span><br> +Addition and subtraction of<br> +<span style="margin-left: 1em;">tensors, <a href="#Page_14">14</a></span><br> +—theorem of velocities, <a href="#Page_38">38</a> +</p> +<p><br></p> +<p>B</p> +<p class="nind"> +Biot-Savart force, <a href="#Page_44">44</a> +</p> +<p><br></p> +<p>C</p> +<p class="nind"> +Centrifugal force, <a href="#Page_64">64</a><br> +Clocks, moving, <a href="#Page_38">38</a><br> +Compressible viscous fluid, <a href="#Page_22">22</a><br> +Concept of space, <a href="#Page_3">3</a><br> +—time, <a href="#Page_28">28</a><br> +Conditions of orthogonality, <a href="#Page_7">7</a><br> +Congruence, theorems of, <a href="#Page_3">3</a><br> +Conservation principles, <a href="#Page_54">54</a><br> +Continuum, four-dimensional, <a href="#Page_31">31</a><br> +Contraction of tensors, <a href="#Page_14">14</a><br> +Contra-variant vectors, <a href="#Page_69">69</a><br> +—tensors, <a href="#Page_71">71</a><br> +Co-ordinates, preferred systems<br> +<span style="margin-left: 1em;">of, <a href="#Page_8">8</a></span><br> +Co-variance of equation of<br> +<span style="margin-left: 1em;">continuity, <a href="#Page_21">21</a></span><br> +Co-variant, <a href="#Page_12">12</a> <i>et seq.</i><br> +—vector, <a href="#Page_68">68</a><br> +Criticism of principle of inertia, <a href="#Page_62">62</a><br> +Criticisms of theory of<br> +<span style="margin-left: 1em;">relativity, <a href="#Page_29">29</a></span><br> +Curvilinear co-ordinates, <a href="#Page_65">65</a> +</p> +<p><br></p> +<p>D</p> +<p class="nind"> +Differentiation of tensors, <a href="#Page_73">73</a>, <a href="#Page_76">76</a><br> +Displacement of spectral lines, <a href="#Page_97">97</a><br> +</p> +<p><br></p> +<p>E</p> +<p class="nind"> +Energy and mass, <a href="#Page_45">45</a>, <a href="#Page_49">49</a><br> +—tensor of electromagnetic<br> +<span style="margin-left: 1em;">field, <a href="#Page_50">50</a></span><br> +—of matter, <a href="#Page_54">54</a><br> +Equation of continuity, co-variance<br> +<span style="margin-left: 1em;">of, <a href="#Page_21">21</a></span><br> +Equations of motion of material<br> +<span style="margin-left: 1em;">particle, <a href="#Page_50">50</a></span><br> +Equivalence of mass and<br> +<span style="margin-left: 1em;">energy, <a href="#Page_49">49</a></span><br> +Equivalent spaces of reference, <a href="#Page_25">25</a><br> +Euclidean geometry, <a href="#Page_4">4</a><br> +</p> +<p><br></p> +<p>F</p> +<p class="nind"> +Finiteness of universe, <a href="#Page_105">105</a><br> +Fizeau, <a href="#Page_28">28</a><br> +Four-dimensional continuum, <a href="#Page_31">31</a><br> +Four-vector, <a href="#Page_41">41</a><br> +Fundamental tensor, <a href="#Page_71">71</a> +</p> +<p><br></p> +<p>G</p> +<p class="nind"> +Galilean regions, <a href="#Page_62">62</a><br> +—transformation, <a href="#Page_27">27</a><br> +Gauss, <a href="#Page_65">65</a><br> +Geodetic lines, <a href="#Page_82">82</a><br> +Geometry, Euclidean, <a href="#Page_4">4</a><br> +Gravitation constant, <a href="#Page_95">95</a><br> +Gravitational mass, <a href="#Page_60">60</a> +</p> +<p><br></p> +<p>H</p> +<p class="nind"> +Homogeneity of space, <a href="#Page_17">17</a><br> +Hydrodynamical equations, <a href="#Page_54">54</a><br> +Hypotheses of pre-relativity<br> +<span style="margin-left: 1em;">physics, <a href="#Page_73">73</a></span> +</p> +<p><br></p> +<p>I</p> +<p class="nind"> +Inductive action of accelerated<br> +<span style="margin-left: 1em;">masses, <a href="#Page_108">108</a></span><br> +Inert and gravitational mass, equality<br> +<span style="margin-left: 1em;">of, <a href="#Page_60">60</a></span><br> +Invariant, <a href="#Page_9">9</a> <i>et seq.</i><br> +Isotropy of space, <a href="#Page_17">17</a><br> +</p> +<p><br></p> +<p>K</p> +<p class="nind"> +Kaluza, <a href="#Page_104">104</a> +</p> +<p><br></p> +<p>L</p> +<p class="nind"> +Levi-Civita, <a href="#Page_73">73</a><br> +Light-cone, <a href="#Page_41">41</a><br> +Light ray, path of, <a href="#Page_98">98</a><br> +Light-time, <a href="#Page_33">33</a><br> +Linear orthogonal<br> +<span style="margin-left: 1em;">transformation, <a href="#Page_7">7</a></span><br> +Lorentz electromotive force, <a href="#Page_44">44</a><br> +—transformation, <a href="#Page_31">31</a><br> +</p> +<p><br></p> +<p>M</p> +<p class="nind"> +Mach, <a href="#Page_59">59</a>, <a href="#Page_105">105</a>, <a href="#Page_106">106</a>, <a href="#Page_109">109</a>, <a href="#Page_114">114</a><br> +Mass and Energy, <a href="#Page_45">45</a>, <a href="#Page_49">49</a><br> +—equality of gravitational and<br> +<span style="margin-left: 1em;">inert, <a href="#Page_60">60</a></span><br> +—gravitational, <a href="#Page_60">60</a><br> +Maxwell's equations, <a href="#Page_23">23</a><br> +Mercury, perihelion of, <a href="#Page_99">99</a>, <a href="#Page_103">103</a><br> +Michelson and Morley, <a href="#Page_28">28</a><br> +Minkowski, <a href="#Page_32">32</a><br> +Motion of particle, equations of, <a href="#Page_50">50</a><br> +Moving measuring rods and<br> +<span style="margin-left: 1em;">clocks, <a href="#Page_38">38</a></span><br> +Multiplication of tensors, <a href="#Page_14">14</a><br> +</p> +<p><br></p> +<p>N</p> +<p class="nind"> +Newtonian gravitation<br> +<span style="margin-left: 1em;">constant, <a href="#Page_95">95</a></span><br> +</p> +<p><br></p> +<p>O</p> +<p class="nind"> +Operations on tensors, <a href="#Page_13">13</a> <i>et seq.</i><br> +Orthogonal transformations, linear, <a href="#Page_7">7</a><br> +Orthogonality, conditions of, <a href="#Page_7">7</a><br> +</p> +<p><br></p> +<p>P</p> +<p class="nind"> +Path of light ray, <a href="#Page_98">98</a><br> +Perihelion of Mercury, <a href="#Page_99">99</a>, <a href="#Page_103">103</a><br> +Poisson's equation, <a href="#Page_87">87</a><br> +Preferred systems of<br> +<span style="margin-left: 1em;">co-ordinates, <a href="#Page_8">8</a></span><br> +Pre-relativity physics, hypotheses<br> +<span style="margin-left: 1em;">of, <a href="#Page_26">26</a></span><br> +Principle of equivalence, <a href="#Page_61">61</a><br> +—inertia, criticism of, <a href="#Page_62">62</a><br> +Principles of conservation, <a href="#Page_54">54</a><br> +</p> +<p><br></p> +<p>R</p> +<p class="nind"> +Radius of Universe, <a href="#Page_113">113</a><br> +Rank of tensor, <a href="#Page_13">13</a><br> +Ray of light, path of, <a href="#Page_98">98</a><br> +Reference, space of, <a href="#Page_3">3</a><br> +Riemann, <a href="#Page_68">68</a><br> +—tensor, <a href="#Page_79">79</a>, <a href="#Page_82">82</a>, <a href="#Page_105">105</a><br> +Rods (measuring) and clocks in<br> +<span style="margin-left: 1em;">motion, <a href="#Page_38">38</a></span><br> +Rotation, <a href="#Page_63">63</a> +</p> +<p><br></p> +<p>S</p> +<p class="nind"> +Simultaneity, <a href="#Page_17">17</a>, <a href="#Page_29">29</a><br> +Sitter, <a href="#Page_28">28</a><br> +Skew-symmetrical tensor, <a href="#Page_15">15</a><br> +Solar Eclipse expedition (1919), <a href="#Page_99">99</a><br> +Space, concept of, <a href="#Page_2">2</a><br> +—Homogeneity of, <a href="#Page_17">17</a><br> +—Isotropy of, <a href="#Page_17">17</a><br> +Spaces of reference, <a href="#Page_3">3</a><br> +—equivalence of, <a href="#Page_25">25</a><br> +Special Lorentz transformation, <a href="#Page_34">34</a><br> +Spectral lines, displacement of, <a href="#Page_97">97</a><br> +Straightest lines, <a href="#Page_82">82</a><br> +Stress tensor, <a href="#Page_22">22</a><br> +Symmetrical tensor, <a href="#Page_15">15</a><br> +Systems of co-ordinates, <br> +preferred, <a href="#Page_8">8</a><br> +</p> +<p><br></p> +<p>T</p> +<p class="nind"> +Tensor, <a href="#Page_12">12</a> <i>et seq.</i>, <a href="#Page_68">68</a> <i>et seq.</i><br> +—Addition and subtraction of, <a href="#Page_14">14</a><br> +—Contraction of, <a href="#Page_14">14</a><br> +—Fundamental, <a href="#Page_71">71</a><br> +—Multiplication of, <a href="#Page_14">14</a><br> +—operations, <a href="#Page_13">13</a> <i>et seq.</i><br> +—Rank of, <a href="#Page_13">13</a><br> +—Symmetrical and<br> +<span style="margin-left: 1em;">Skew-symmetrical, <a href="#Page_15">15</a></span><br> +Tensors, formation by<br> +<span style="margin-left: 1em;">differentiation, <a href="#Page_73">73</a></span><br> +Theorem for addition of<br> +<span style="margin-left: 1em;">velocities, <a href="#Page_38">38</a></span><br> +Theorems of congruence, <a href="#Page_3">3</a><br> +Theory of relativity, criticisms<br> +<span style="margin-left: 1em;">of, <a href="#Page_29">29</a></span><br> +Thirring, <a href="#Page_109">109</a><br> +Time-concept, <a href="#Page_28">28</a><br> +Time-space concept, <a href="#Page_31">31</a><br> +Transformation, Galilean, <a href="#Page_27">27</a><br> +—Linear orthogonal, <a href="#Page_7">7</a><br> +</p> +<p><br></p> +<p>U</p> +<p class="nind"> +Universe, Finiteness of, <a href="#Page_105">105</a><br> +—Radius of, <a href="#Page_113">113</a><br> +</p> +<p><br></p> +<p>V</p> +<p class="nind"> +Vector, co-variant, <a href="#Page_69">69</a><br> +—contra-variant, <a href="#Page_69">69</a><br> +Velocities, addition theorem of, <a href="#Page_38">38</a><br> +Viscous compressible fluid, <a href="#Page_22">22</a><br> +</p> +<p><br></p> +<p>W</p> +<p class="nind"> +Weyl, <a href="#Page_69">73</a>, <a href="#Page_99">99</a>, <a href="#Page_104">104</a> +</p> + +<p><br><br><br></p> + +<p class="center"> +PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN +</p> + +<p><br><br></p> + +<div class="transnote"> +<p class="center"><b>TRANSCRIBER'S NOTES</b></p> + 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.net + + +Title: The Meaning of Relativity + Four lectures delivered at Princeton University, May, 1921 + +Author: Albert Einstein + +Translator: Edwin Plimpton Adams + +Release Date: May 29, 2011 [EBook #36276] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE MEANING OF RELATIVITY *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang. (This ebook was produced using +OCR text generously provided by Northeastern University's +Snell Library through the Internet Archive.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\frontmatter +\pagestyle{empty} +\PageSep{i} +\null\vfill +\noindent\textbf{\large THE MEANING OF RELATIVITY} +\vfill +\cleardoublepage +\PageSep{ii} +%[Blank page] +\PageSep{iii} +% Title page +\begin{center} +\textbf{\huge THE MEANING OF \\[8pt] +RELATIVITY} +\bigskip + +\normalsize +FOUR LECTURES DELIVERED AT \\ +PRINCETON UNIVERSITY, MAY, 1921 +\bigskip + +{\footnotesize BY} \\[8pt] +\textbf{\huge ALBERT EINSTEIN} +\vfill\vfill + +{\footnotesize WITH FOUR DIAGRAMS} +\vfill\vfill\vfill + +\normalsize +PRINCETON \\ +PRINCETON UNIVERSITY PRESS \\ +1923 +\end{center} +\newpage +\PageSep{iv} +\begin{center} +\null +\vfill\vfill +\itshape\footnotesize +Copyright 1922 \\ +Princeton University Press \\ +Published 1922 +\vfill +\upshape +PRINTED IN GREAT BRITAIN \\ +AT THE ABERDEEN UNIVERSITY PRESS \\ +ABERDEEN +\vfill + +\TranslatorsNote{% + \textsc{Note}.---The translation of these lectures into English + was made by \textsc{Edwin Plimpton Adams}, Professor + of Physics in Princeton University} +\cleardoublepage +\end{center} +\PageSep{v} +\TableofContents +\iffalse %%%% Table of contents text %%%% +CONTENTS + +LECTURE I PAGE +Space and Time in Pre-Relativity Physics 1 + +LECTURE II +The Theory of Special Relativity 26 + +LECTURE III +The General Theory of Relativity 61 + +LECTURE IV +The General Theory of Relativity (continued) 87 + +Index 121 +\fi %%%% Table of contents text %%%% +\newpage +\PageSep{1} + + +\mainmatter +\BookMark{-1}{Main Matter} +%[** TN: Text printed by \Lecture macro: THE MEANING OF RELATIVITY] +\Lecture{I}{Space and Time in Pre-Relativity +Physics}{Pre-Relativity Physics} + +\First{The} theory of relativity is intimately connected with +the theory of space and time. I shall therefore begin +with a brief investigation of the origin of our ideas of space +and time, although in doing so I know that I introduce a +controversial subject. The object of all science, whether +natural science or psychology, is to co-ordinate our experiences +and to bring them into a logical system. How are +our customary ideas of space and time related to the +character of our experiences? + +The experiences of an individual appear to us arranged +in a series of events; in this series the single events which +we remember appear to be ordered according to the criterion +of ``earlier'' and ``later,'' which cannot be analysed further. +There exists, therefore, for the individual, an I-time, or +subjective time. This in itself is not measurable. I can, +indeed, associate numbers with the events, in such a way +that a greater number is associated with the later event +than with an earlier one; but the nature of this association +may be quite arbitrary. This association I can define by +means of a clock by comparing the order of events furnished +\PageSep{2} +by the clock with the order of the given series of events. +We understand by a clock something which provides a +series of events which can be counted, and which has other +properties of which we shall speak later. + +By the aid of speech different individuals can, to a certain +extent, compare their experiences. In this way it is shown +that certain sense perceptions of different individuals +correspond to each other, while for other sense perceptions +no such correspondence can be established. We are accustomed +to regard as real those sense perceptions which +are common to different individuals, and which therefore +are, in a measure, impersonal. The natural sciences, and +in particular, the most fundamental of them, physics, deal +with such sense perceptions. The conception of physical +bodies, in particular of rigid bodies, is a relatively constant +complex of such sense perceptions. A clock is also a body, +or a system, in the same sense, with the additional property +that the series of events which it counts is formed of +elements all of which can be regarded as equal. + +The only justification for our concepts and system of +concepts is that they serve to represent the complex of +our experiences; beyond this they have no legitimacy. I +am convinced that the philosophers have had a harmful +effect upon the progress of scientific thinking in removing +certain fundamental concepts from the domain of empiricism, +where they are under our control, to the intangible +heights of the \textit{a~priori}. For even if it should appear that +the universe of ideas cannot be deduced from experience +by logical means, but is, in a sense, a creation of the human +mind, without which no science is possible, nevertheless +\PageSep{3} +this universe of ideas is just as little independent of the +nature of our experiences as clothes are of the form of +the human body. This is particularly true of our concepts +of time and space, which physicists have been +\index{Space@{Space, Concept of}}% +obliged by the facts to bring down from the Olympus of +the \textit{a~priori} in order to adjust them and put them in a +serviceable condition. + +We now come to our concepts and judgments of space. +It is essential here also to pay strict attention to the +relation of experience to our concepts. It seems to me +that Poincaré clearly recognized the truth in the account +he gave in his book, ``La Science et l'Hypothese.'' +Among all the changes which we can perceive in a rigid +body those are marked by their simplicity which can be +made reversibly by an arbitrary motion of the body; +Poincaré calls these, changes in position. By means of +simple changes in position we can bring two bodies into +contact. The theorems of congruence, fundamental in +\index{Congruence, theorems of}% +\index{Theorems of congruence}% +geometry, have to do with the laws that govern such +changes in position. For the concept of space the following +\index{Concept@{Concept of space}}% +seems essential. We can form new bodies by bringing +bodies $B$,~$C$,~\dots\ up to body~$A$; we say that we \emph{continue} +body~$A$. We can continue body~$A$ in such a way that +it comes into contact with any other body,~$X$. The +ensemble of all continuations of body~$A$ we can designate +as the ``space of the body~$A$.'' Then it is true that all +bodies are in the ``space of the (arbitrarily chosen) body~$A$.'' +In this sense we cannot speak of space in the +abstract, but only of the ``space belonging to a body~$A$.'' +The earth's crust plays such a dominant rôle in our daily +\PageSep{4} +life in judging the relative positions of bodies that it has +led to an abstract conception of space which certainly +cannot be defended. In order to free ourselves from this +fatal error we shall speak only of ``bodies of reference,'' +or ``space of reference.'' It was only through the theory +\index{Reference, space of}% +\index{Spaces of reference}% +of general relativity that refinement of these concepts +became necessary, as we shall see later. + +I shall not go into detail concerning those properties +of the space of reference which lead to our conceiving +points as elements of space, and space as a continuum. +Nor shall I attempt to analyse further the properties of +space which justify the conception of continuous series +of points, or lines. If these concepts are assumed, together +with their relation to the solid bodies of experience, then +it is easy to say what we mean by the three-dimensionality +of space; to each point three numbers, $x_{1}$,~$x_{2}$,~$x_{3}$ (co-ordinates), +may be associated, in such a way that this +association is uniquely reciprocal, and that $x_{1}$,~$x_{2}$\Change{,}{} and~$x_{3}$ +vary continuously when the point describes a continuous +series of points (a line). + +It is assumed in pre-relativity physics that the laws of +the orientation of ideal rigid bodies are consistent with +Euclidean geometry. What this means may be expressed +\index{Euclidean geometry}% +\index{Geometry, Euclidean}% +as follows: Two points marked on a rigid body form +an \emph{interval}. Such an interval can be oriented at rest, +relatively to our space of reference, in a multiplicity of +ways. If, now, the points of this space can be referred +to co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, in such a way that the differences +of the co-ordinates, $\Delta x_{1}$,~$\Delta x_{2}$,~$\Delta x_{3}$, of the two ends of the +interval furnish the same sum of squares, +\[ +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}\Add{,} +\Tag{(1)} +\] +\PageSep{5} +for every orientation of the interval, then the space of +reference is called Euclidean, and the co-ordinates +Cartesian.\footnote + {This relation must hold for an arbitrary choice of the origin and of the + direction (ratios $\Delta x_{1} : \Delta x_{2} : \Delta x_{3}$) of the interval.} +It is sufficient, indeed, to make this assumption +in the limit for an infinitely small interval. Involved +in this assumption there are some which are rather less +special, to which we must call attention on account of +their fundamental significance. In the first place, it is +assumed that one can move an ideal rigid body in an +arbitrary manner. In the second place, it is assumed +that the behaviour of ideal rigid bodies towards orientation +is independent of the material of the bodies and their +changes of position, in the sense that if two intervals can +once be brought into coincidence, they can always and +everywhere be brought into coincidence. Both of these +assumptions, which are of fundamental importance for +geometry and especially for physical measurements, +naturally arise from experience; in the theory of general +relativity their validity needs to be assumed only for +bodies and spaces of reference which are infinitely small +compared to astronomical dimensions. + +The quantity~$s$ we call the length of the interval. In +order that this may be uniquely determined it is necessary +to fix arbitrarily the length of a definite interval; for +example, we can put it equal to~$1$ (unit of length). Then +the lengths of all other intervals may be determined. If +we make the~$x_{\nu}$ linearly dependent upon a parameter~$\lambda$, +\[ +x_{\nu} = a_{\nu} + \lambda b_{\nu}, +\] +\PageSep{6} +we obtain a line which has all the properties of the straight +lines of the Euclidean geometry. In particular, it easily +follows that by laying off $n$~times the interval~$s$ upon a +straight line, an interval of length~$n·s$ is obtained. A +length, therefore, means the result of a measurement +carried out along a straight line by means of a unit +measuring rod. It has a significance which is as independent +of the system of co-ordinates as that of a straight +line, as will appear in the sequel. + +We come now to a train of thought which plays an +analogous rôle in the theories of special and general +relativity. We ask the question: besides the Cartesian +co-ordinates which we have used are there other equivalent +co-ordinates? An interval has a physical meaning which +is independent of the choice of co-ordinates; and so has +the spherical surface which we obtain as the locus of the +end points of all equal intervals that we lay off from an +arbitrary point of our space of reference. If~$x_{\nu}$ as well as~${x'}_{\nu}$ +($\nu$~from $1$~to~$3$) are Cartesian co-ordinates of our space +of reference, then the spherical surface will be expressed +in our two systems of co-ordinates by the equations +\begin{align*} +\sum {\Delta x_{\nu}}^{2} &= \text{const.} +\Tag{(2)} \\ +\sum {\Delta {x'}_{\nu}}^{2} &= \text{const.} +\Tag{(2a)} +\end{align*} +How must the~${x'}_{\nu}$ be expressed in terms of the~$x_{\nu}$ in order +that equations \Eqref{(2)}~and~\Eqref{(2a)} may be equivalent to each +other? Regarding the~${x'}_{\nu}$ expressed as functions of the~$x_{\nu}$, +we can write, by Taylor's theorem, for small values of +the~${\Delta x}_{\nu}$, +\PageSep{7} +\[ +{\Delta x'}_{\nu} + = \sum_{\alpha} \frac{\dd {x'}_{\nu}}{\dd x_{\alpha}}\, \Delta x_{\alpha} + + \frac{1}{2} \sum_{\alpha\Add{,}\beta} + \frac{\dd^{2} {x'}_{\nu}}{\dd x_{\alpha}\, \dd x_{\beta}}\, + \Delta x_{\alpha}\, \Delta x_{\beta}\ldots\Add{.} +\] +If we substitute~\Eqref{(2a)} in this equation and compare with~\Eqref{(1)}, +we see that the~${x'}_{\nu}$ must be linear functions of the~$x_{\nu}$. +If we therefore put +\begin{align*} +{x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}\Add{,} +\Tag{(3)} \\ +\intertext{or} +{\Delta x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}\Add{,} +\Tag{(3a)} +\end{align*} +then the equivalence of equations \Eqref{(2)}~and~\Eqref{(2a)} is expressed +in the form +\[ +\sum {{\Delta x'}_{\nu}}^{2} = \lambda \sum {\Delta x_{\nu}}^{2}\quad +\text{($\lambda$~independent of~$\Delta x_{\nu}$).} +\Tag{(2b)} +\] +It therefore follows that $\lambda$~must be a constant. If we put +$\lambda = 1$, \Eqref{(2b)}~and~\Eqref{(3a)} furnish the conditions +\[ +\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}\Add{,} +\Tag{(4)} +\] +in which $\delta_{\alpha\beta} = 1$, or $\delta_{\alpha\beta} = 0$, according as $\alpha = \beta$ or +$\alpha \neq \beta$. The conditions~\Eqref{(4)} are called the conditions of orthogonality, +\index{Conditions of orthogonality}% +\index{Linear orthogonal transformation}% +\index{Orthogonality, conditions of}% +\index{Orthogonal transformations, linear}% +\index{Transformation@{Transformation, Galilean}!Linear orthogonal}% +and the transformations \Eqref{(3)},~\Eqref{(4)}, linear orthogonal +transformations. If we stipulate that $s^{2} = \sum {\Delta x_{\nu}}^{2}$ shall be +equal to the square of the length in every system of +co-ordinates, and if we always measure with the same unit +scale, then $\lambda$~must be equal to~$1$. Therefore the linear +orthogonal transformations are the only ones by means of +which we can pass from one Cartesian system of co-ordinates +in our space of reference to another. We see +\PageSep{8} +that in applying such transformations the equations of +a straight line become equations of a straight line. +Reversing equations~\Eqref{(3a)} by multiplying both sides by~$b_{\nu\beta}$ +and summing for all the~$\nu$'s, we obtain +\[ +\Change{\sum}{\sum_{\nu}} b_{\nu\beta}\, {\Delta x'}_{\nu} + = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} + = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} + = \Delta x_{\beta}\Add{.} +\Tag{(5)} +\] +The same coefficients,~$b$, also determine the inverse +substitution of~$\Delta x_{\nu}$. Geometrically, $b_{\nu\alpha}$~is the cosine of the +angle between the ${x'}_{\nu}$~axis and the $x_{\alpha}$~axis. + +To sum up, we can say that in the Euclidean geometry +\index{Coordinates@{Co-ordinates, preferred systems of}}% +\index{Preferred systems of co-ordinates}% +\index{Systems of co-ordinates, preferred}% +there are (in a given space of reference) preferred systems +of co-ordinates, the Cartesian systems, which transform +into each other by linear orthogonal transformations. +The distance~$s$ between two points of our space of +reference, measured by a measuring rod, is expressed in +such co-ordinates in a particularly simple manner. The +whole of geometry may be founded upon this conception +of distance. In the present treatment, geometry is +related to actual things (rigid bodies), and its theorems +are statements concerning the behaviour of these things, +which may prove to be true or false. + +One is ordinarily accustomed to study geometry +divorced from any relation between its concepts and +experience. There are advantages in isolating that +which is purely logical and independent of what is, in +principle, incomplete empiricism. This is satisfactory to +the pure mathematician. He is satisfied if he can deduce +his theorems from axioms correctly, that is, without +errors of logic. The question as to whether Euclidean +\PageSep{9} +geometry is true or not does not concern him. But for +our purpose it is necessary to associate the fundamental +concepts of geometry with natural objects; without such +an association geometry is worthless for the physicist. +The physicist is concerned with the question as to +whether the theorems of geometry are true or not. That +Euclidean geometry, from this point of view, affirms +something more than the mere deductions derived +logically from definitions may be seen from the following +simple consideration. + +Between $n$~points of space there are $\dfrac{n(n - 1)}{2}$ distances,~$s_{\mu\nu}$; +between these and the $3n$~co-ordinates we have the +relations +\[ +{s_{\mu\nu}}^{2} + = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + + \dots\Add{.} +\] + +From these $\dfrac{n(n - 1)}{2}$~equations the $3n$~co-ordinates +may be eliminated, and from this elimination at least +$\dfrac{n(n - 1)}{2} - 3n$ equations in the~$s_{\mu\nu}$ will result.\footnote + {In reality there are $\dfrac{n(n - 1)}{2} - 3n + 6$ equations.} +Since +the~$s_{\mu\nu}$ are measurable quantities, and by definition are +independent of each other, these relations between the~$s_{\mu\nu}$ +are not necessary \textit{a~priori}. + +From the foregoing it is evident that the equations of +transformation \Eqref{(3)},~\Eqref{(4)} have a fundamental significance in +Euclidean geometry, in that they govern the transformation +from one Cartesian system of co-ordinates to another. +The Cartesian systems of co-ordinates are characterized +\PageSep{10} +by the property that in them the measurable distance +between two points,~$s$, is expressed by the equation +\[ +s^{2} = \sum {\Delta x_{\nu}}^{2}. +\] + +If $K_{(x_{\nu})}$~and~$K'_{(x_{\nu})}$ are two Cartesian systems of co-ordinates, +then +\[ +\sum {\Delta x_{\nu}}^{2} = \sum {{\Delta x'}_{\nu}}^{2}. +\] + +The right-hand side is identically equal to the left-hand +side on account of the equations of the linear orthogonal +transformation, and the right-hand side differs from the +left-hand side only in that the~$x_{\nu}$ are replaced by the~${x'}_{\nu}$. +This is expressed by the statement that~$\sum {\Delta x_{\nu}}^{2}$ is an +invariant with respect to linear orthogonal transformations. +\index{Invariant|etseq}% +It is evident that in the Euclidean geometry only +such, and all such, quantities have an objective significance, +independent of the particular choice of the Cartesian +co-ordinates, as can be expressed by an invariant with +respect to linear orthogonal transformations. This is the +reason that the theory of invariants, which has to do with +the laws that govern the form of invariants, is so important +for analytical geometry. + +As a second example of a geometrical invariant, consider +a volume. This is expressed by +\[ +V = \iiint dx_{1}\, dx_{2}\, dx_{3}. +\] +By means of Jacobi's theorem we may write +\[ +\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} + = \iiint \frac{\dd({x'}_{1}, {x'}_{2}, {x'}_{3})}{\dd(x_{1}, x_{2}, x_{3})}\, + dx_{1}\, dx_{2}\, dx_{3} +\] +\PageSep{11} +where the integrand in the last integral is the functional +determinant of the~${x'}_{\nu}$ with respect to the~$x_{\nu}$, and this by~\Eqref{(3)} +is equal to the determinant~$|b_{\mu\nu}|$ of the coefficients +of substitution,~$b_{\nu\alpha}$. If we form the determinant of the~$\delta_{\mu\alpha}$ +from equation~\Eqref{(4)}, we obtain, by means of the theorem +of multiplication of determinants, +\[ +1 = |\delta_{\alpha\beta}| + = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| + = |b_{\mu\nu}|^{2};\quad +|b_{\mu\nu}| = ±1\Add{.} +\Tag{(6)} +\] +If we limit ourselves to those transformations which have +the determinant~$+1$,\footnote + {There are thus two kinds of Cartesian systems which are designated + as ``right-handed'' and ``left-handed'' systems. The difference between + these is familiar to every physicist and engineer. It is interesting to note + that these two kinds of systems cannot be defined geometrically, but only + the contrast between them.} +and only these arise from continuous +variations of the systems of co-ordinates, then $V$~is +an invariant. + +Invariants, however, are not the only forms by means +of which we can give expression to the independence of +the particular choice of the Cartesian co-ordinates. Vectors +and tensors are other forms of expression. Let us express +the fact that the point with the current co-ordinates~$x_{\nu}$ lies +upon a straight line. We have +\[ +x_{\nu} - A_{\nu} = \lambda B_{\nu}\quad + \text{($\nu$~from $1$~to~$3$).} +\] +Without limiting the generality we can put +\[ +\sum {B_{\nu}}^{2} = 1. +\] + +If we multiply the equations by~$b_{\beta\nu}$ (compare \Eqref{(3a)} and~\Eqref{(5)}) +and sum for all the~$\nu$'s, we get +\[ +{x'}_{\beta} - {A'}_{\beta} = \lambda {B'}_{\beta}\Add{,} +\] +\PageSep{12} +where we have written +\[ +{B'}_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad +{A'}_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}. +\] + +These are the equations of straight lines with respect +to a second Cartesian system of co-ordinates~$K'$. They +have the same form as the equations with respect to the +original system of co-ordinates. It is therefore evident +that straight lines have a significance which is independent +of the system of co-ordinates. Formally, this depends +upon the fact that the quantities $(x_{\nu} - A_{\nu}) - \lambda B_{\nu}$ are +transformed as the components of an interval,~$\Delta x_{\nu}$. The +ensemble of three quantities, defined for every system of +Cartesian co-ordinates, and which transform as the components +of an interval, is called a vector. If the three +components of a vector vanish for one system of Cartesian +co-ordinates, they vanish for all systems, because the equations +of transformation are homogeneous. We can thus +get the meaning of the concept of a vector without referring +to a geometrical representation. This behaviour of the +equations of a straight line can be expressed by saying +that the equation of a straight line is co-variant with respect +\index{Covariant@{Co-variant}|etseq}% +to linear orthogonal transformations. + +We shall now show briefly that there are geometrical +\index{Tensor|etseq}% +entities which lead to the concept of tensors. Let $P_{0}$ be +the centre of a surface of the second degree, $P$~any point +on the surface, and $\xi_{\nu}$~the projections of the interval~$P_{0}P$ +upon the co-ordinate axes. Then the equation of the +surface is +\[ +\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +\] +\PageSep{13} +In this, and in analogous cases, we shall omit the sign of +summation, and understand that the summation is to be +carried out for those indices that appear twice. We thus +write the equation of the surface +\[ +a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +\] +The quantities~$a_{\mu\nu}$ determine the surface completely, for +a given position of the centre, with respect to the chosen +system of Cartesian co-ordinates. From the known law +of transformation for the~$\xi_{\nu}$ \Eqref{(3a)} for linear orthogonal +transformations, we easily find the law of transformation +%[** TN: Footnote mark comes before the colon in the original] +for the~$a_{\mu\nu}$:\footnote + {The equation ${a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1$ may, by~\Eqref{(5)}, be replaced by + ${a'}_{\sigma\tau} b_{\mu\sigma} b_{\nu\tau} \xi_{\Change{\sigma}{\mu}} \xi_{\Change{\tau}{\nu}} = 1$, + from which the result stated immediately follows.} +\[ +{a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}. +\] +This transformation is homogeneous and of the first degree +in the~$a_{\mu\nu}$. On account of this transformation, the~$a_{\mu\nu}$ +are called components of a tensor of the second rank (the +latter on account of the double index). If all the components,~$a_{\mu\nu}$, +of a tensor with respect to any system of +Cartesian co-ordinates vanish, they vanish with respect to +every other Cartesian system. The form and the position +of the surface of the second degree is described by this +tensor~$(a)$. + +Analytic tensors of higher rank (number of indices) +may be defined. It is possible and advantageous to +regard vectors as tensors of rank~$1$, and invariants (scalars) +as tensors of rank~$0$. In this respect, the problem of the +theory of invariants may be so formulated: according to +what laws may new tensors be formed from given tensors? +\PageSep{14} +\index{Operations on tensors|etseq}% +\index{Tensor!operations|etseq}% +We shall consider these laws now, in order to be able to +apply them later. We shall deal first only with the +properties of tensors with respect to the transformation +from one Cartesian system to another in the same space +of reference, by means of linear orthogonal transformations. +As the laws are wholly independent of the number +of dimensions, we shall leave this number,~$n$, indefinite at +first. + +\Paragraph{Definition.} If a figure is defined with respect to every +system of Cartesian co-ordinates in a space of reference of +$n$~dimensions by the $n^{\alpha}$~numbers $A_{\mu\nu\rho\Subdots}$ ($\alpha = \text{number +of indices}$), then these numbers are the components of a +tensor of rank~$\alpha$ if the transformation law is +\index{Rank of tensor}% +\index{Tensor!Rank of}% +\[ +{A'}_{\mu'\nu'\rho'\Subdots} + = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho} \dots A_{\mu\nu\rho\Subdots}\Add{.} +\Tag{(7)} +\] + +\Paragraph{Remark.} From this definition it follows that +\[ +A_{\mu\nu\rho\Subdots} = B_{\mu} C_{\nu} D_{\rho}\dots +\Tag{(8)} +\] +is an invariant, provided that $(B)$,~$(C)$,~$(D)$~\dots\ are +vectors. Conversely, the tensor character of~$(A)$ may be +inferred, if it is known that the expression~\Eqref{(8)} leads to an +invariant for an arbitrary choice of the vectors $(B)$,~$(C)$,~etc. + +\Paragraph{Addition and Subtraction.} By addition and subtraction +\index{Addition@{Addition and subtraction of tensors}}% +\index{Tensor!Addition and subtraction of}% +of the corresponding components of tensors of the same +rank, a tensor of equal rank results: +\[ +A_{\mu\nu\rho\Subdots} ± B_{\mu\nu\rho\Subdots} = C_{\mu\nu\rho\Subdots}. +\Tag{(9)} +\] +The proof follows from the definition of a tensor given +above. + +\Paragraph{Multiplication.} From a tensor of rank~$\alpha$ and a tensor +\index{Multiplication of tensors}% +\index{Tensor!Multiplication of}% +\PageSep{15} +of rank~$\beta$ we may obtain a tensor of rank~$\alpha + \beta$ by +multiplying all the components of the first tensor by all +the components of the second tensor: +\[ +T_{\mu\nu\rho\Subdots\alpha\beta\Subdots} += A_{\mu\nu\rho\Subdots} B_{\alpha\beta\gamma\Subdots}\Add{.} +\Tag{(10)} +\] + +\Paragraph{Contraction.} A tensor of rank~$\alpha - 2$ may be obtained +\index{Contraction of tensors}% +\index{Tensor!Contraction of}% +from one of rank~$\alpha$ by putting two definite indices equal +to each other and then summing for this single index: +\[ +T_{\rho\Subdots} + = A_{\mu\mu\rho\Subdots} + ( = \sum_{\mu} A_{\mu\mu\rho\Subdots})\Add{.} +\Tag{(11)} +\] +The proof is +\begin{align*} +{A'}_{\mu\mu\rho\Subdots} + = b_{\mu\alpha} b_{\mu\beta} b_{\rho\gamma}\dots A_{\alpha\beta\gamma\Subdots} + = \delta_{\alpha\beta} & b_{\rho\gamma}\dots A_{\alpha\beta\gamma\Subdots} \\ + {} ={} & b_{\rho\gamma}\dots A_{\alpha\alpha\gamma\Subdots}\Add{.} +\end{align*} + +In addition to these elementary rules of operation +there is also the formation of tensors by differentiation +(``erweiterung''): +\[ +T_{\mu\nu\rho\Subdots\alpha} + = \frac{\dd A_{\mu\nu\rho\Subdots}}{\dd x_{\alpha}}\Add{.} +\Tag{(12)} +\] + +New tensors, in respect to linear orthogonal transformations, +may be formed from tensors according to these rules +of operation. + +\Paragraph{Symmetrical Properties of Tensors.} Tensors are called +\index{Skew-symmetrical tensor}% +\index{Symmetrical tensor}% +\index{Tensor!Symmetrical and Skew-symmetrical}% +symmetrical or skew-symmetrical in respect to two of +their indices, $\mu$~and~$\nu$, if both the components which result +from interchanging the indices $\mu$~and~$\nu$ are equal to each +other or equal with opposite signs. +\begin{alignat*}{2} +&\text{Condition for symmetry:} & + A_{\mu\nu\rho} &= A_{\Change{\mu\nu}{\nu\mu}\rho}. \\ +&\text{Condition for skew-symmetry:}\quad & + A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ +\end{alignat*} + +\Paragraph{Theorem.} The character of symmetry or skew-symmetry +exists independently of the choice of co-ordinates, and in +\PageSep{16} +this lies its importance. The proof follows from the +equation defining tensors. + +\Subsection{Special Tensors.} + +I\@. The quantities~$\delta_{\rho\sigma}$ \Eqref{(4)} are tensor components (fundamental +tensor). + +\Paragraph{Proof.} If in the right-hand side of the equation of +transformation ${A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}$, we substitute for~$A_{\alpha\beta}$ the +quantities~$\delta_{\alpha\beta}$ (which are equal to $1$~or~$0$ according as +$\alpha = \beta$ or $\alpha \Change{ }{\neq} \beta$), we get +\[ +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}. +\] +The justification for the last sign of equality becomes +evident if one applies~\Eqref{(4)} to the inverse substitution~\Eqref{(5)}. + +II\@. There is a tensor $(\delta_{\mu\nu\rho\Subdots})$ skew-symmetrical with +respect to all pairs of indices, whose rank is equal to the +number of dimensions,~$n$, and whose components are +equal to $+1$~or~$-1$ according as $\mu\,\nu\,\rho\dots$~is an even +or odd permutation of~$1\,2\,3\dots$\Add{.} + +The proof follows with the aid of the theorem proved +above $|b_{\rho\sigma}| = 1$. + +These few simple theorems form the apparatus from +the theory of invariants for building the equations of pre-relativity +physics and the theory of special relativity. + +We have seen that in pre-relativity physics, in order to +specify relations in space, a body of reference, or a space +of reference, is required, and, in addition, a Cartesian +system of co-ordinates. We can fuse both these concepts +into a single one by thinking of a Cartesian system of +co-ordinates as a cubical frame-work formed of rods each +of unit length. The co-ordinates of the lattice points of +\PageSep{17} +this frame are integral numbers. It follows from the +fundamental relation +\[ +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} +\Change{}{\Tag{(13)}} +\] +that the members of such a space-lattice are all of unit +length. To specify relations in time, we require in +addition a standard clock placed at the origin of our +Cartesian system of co-ordinates or frame of reference. +If an event takes place anywhere we can assign to it three +co-ordinates,~$x_{\nu}$, and a time~$t$, as soon as we have specified +the time of the clock at the origin which is simultaneous +with the event. We therefore give an objective significance +to the statement of the simultaneity of distant +\index{Simultaneity}% +events, while previously we have been concerned only +with the simultaneity of two experiences of an individual. +The time so specified is at all events independent of the +position of the system of co-ordinates in our space of +reference, and is therefore an invariant with respect to +the transformation~\Eqref{(3)}. + +It is postulated that the system of equations expressing +the laws of pre-relativity physics is co-variant with respect +to the transformation~\Eqref{(3)}, as are the relations of Euclidean +geometry. The isotropy and homogeneity of space is +\index{Homogeneity of space}% +\index{Isotropy of space}% +\index{Space@{Space, Concept of}!Homogeneity of}% +\index{Space@{Space, Concept of}!Isotropy of}% +expressed in this way.\footnote + {The laws of physics could be expressed, even in case there were a + unique direction in space, in such a way as to be co-variant with respect to + the transformation~\Eqref{(3)}; but such an expression would in this case be unsuitable. + If there were a unique direction in space it would simplify the + description of natural phenomena to orient the system of co-ordinates in a + definite way in this direction. But if, on the other hand, there is no unique + direction in space it is not logical to formulate the laws of nature in such + a way as to conceal the equivalence of systems of co-ordinates that are + oriented differently. We shall meet with this point of view again in the + theories of special and general relativity.} +We shall now consider some of +\PageSep{18} +the more important equations of physics from this point +of view. + +The equations of motion of a material particle are +\[ +m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}\Add{;} +\Tag{(14)} +\] +$(dx_{\nu})$~is a vector; $dt$,~and therefore also~$\dfrac{1}{dt}$, an invariant; +thus $\left(\dfrac{dx_{\nu}}{dt}\right)$~is a vector; in the same way it may be shown +that $\left(\dfrac{d^{2} x_{\nu}}{dt^{2}}\right)$~is a vector. In general, the operation of differentiation +with respect to time does not alter the tensor +character. Since $m$~is an invariant (tensor of rank~$0$), +$\left(m\dfrac{d^{2} x_{\nu}}{dt^{2}}\right)$~is a vector, or tensor of rank~$1$ (by the theorem +of the multiplication of tensors). If the force~$(X_{\nu})$ has +a vector character, the same holds for the difference +$\left(m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right)$. These equations of motion are therefore +valid in every other system of Cartesian co-ordinates in +the space of reference. In the case where the forces are +conservative we can easily recognize the vector character +of~$(X_{\nu})$. For a potential energy,~$\Phi$, exists, which depends +only upon the mutual distances of the particles, and is +therefore an invariant. The vector character of the force, +$X_{\nu} = -\dfrac{\dd \Phi}{\dd x_{\nu}}$, is then a consequence of our general theorem +about the derivative of a tensor of rank~$0$. +\PageSep{19} + +Multiplying by the velocity, a tensor of rank~$1$, we +obtain the tensor equation +\[ +\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0. +\] +By contraction and multiplication by the scalar~$dt$ we +obtain the equation of kinetic energy +\[ +d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\, dx_{\nu}. +\] + +If $\xi_{\nu}$~denotes the difference of the co-ordinates of +the material particle and a point fixed in space, then +the~$\xi_{\nu}$ have the character of vectors. We evidently +have $\dfrac{d^{2} x_{\nu}}{dt^{2}} = \dfrac{d^{2} \xi_{\nu}}{dt^{2}}$, so that the equations of motion of the +particle may be written +\[ +m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0. +\] + +Multiplying this equation by~$\xi_{\mu}$ we obtain a tensor +equation +\[ +\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0. +\] + +Contracting the tensor on the left and taking the time +average we obtain the virial theorem, which we shall +not consider further. By interchanging the indices and +subsequent subtraction, we obtain, after a simple transformation, +the theorem of moments, +\[ +\frac{d}{dt} \left[ + m \left(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\right) +\right] = \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}\Add{.} +\Tag{(15)} +\] + +It is evident in this way that the moment of a vector +\PageSep{20} +is not a vector but a tensor. On account of their skew-symmetrical +character there are not nine, but only three +independent equations of this system. The possibility of +replacing skew-symmetrical tensors of the second rank in +space of three dimensions by vectors depends upon the +formation of the vector +\[ +A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}. +\] + +If we multiply the skew-symmetrical tensor of rank~$2$ +by the special skew-symmetrical tensor~$\delta$ introduced +above, and contract twice, a vector results whose components +are numerically equal to those of the tensor. These +are the so-called axial vectors which transform differently, +from a right-handed system to a left-handed system, +from the~$\Delta x_{\nu}$. There is a gain in picturesqueness in +regarding a skew-symmetrical tensor of rank~$2$ as a vector +in space of three dimensions, but it does not represent +the exact nature of the corresponding quantity so well as +considering it a tensor. + +We consider next the equations of motion of a continuous +medium. Let $\rho$~be the density, $u_{\nu}$~the velocity +components considered as functions of the co-ordinates and +the time, $X_{\nu}$~the volume forces per unit of mass, and $p_{\nu\sigma}$~the +stresses upon a surface perpendicular to the $\sigma$-axis +in the direction of increasing~$x_{\nu}$. Then the equations of +motion are, by Newton's law, +\[ +\rho \frac{du_{\nu}}{dt} + = -\frac{\dd p_{\nu\sigma}}{\dd x_{\sigma}} + \rho X_{\nu}\Add{,} +\] +in which $\dfrac{du_{\nu}}{dt}$~is the acceleration of the particle which at +\PageSep{21} +time~$t$ has the co-ordinates~$x_{\nu}$. If we express this +acceleration by partial differential coefficients, we obtain, +after dividing by~$\rho$, +\[ +\frac{\dd u_{\nu}}{dt} + \frac{\dd u_{\nu}}{dx_{\sigma}} u_{\sigma} + = -\frac{1}{\rho}\, \frac{\dd p_{\nu\sigma}}{\dd x_{\sigma}} + X_{\nu}\Add{.} +\Tag{(16)} +\] + +We must show that this equation holds independently +of the special choice of the Cartesian system of co-ordinates. +$(u_{\nu})$ is a vector, and therefore $\dfrac{\dd u_{\nu}}{\dd t}$~is also a vector. $\dfrac{\dd u_{\nu}}{\dd x_{\sigma}}$~is +a tensor of rank~$2$, $\dfrac{\dd u_{\nu}}{\dd x_{\sigma}} u_{\tau}$~is a tensor of rank~$3$. The second +term on the left results from contraction in the indices +$\sigma$,~$\tau$. The vector character of the second term on the right +is obvious. In order that the first term on the right may +also be a vector it is necessary for~$p_{\nu\sigma}$ to be a tensor. +Then by differentiation and contraction $\dfrac{\dd p_{\nu\sigma}}{\dd x_{\sigma}}$~results, and +is therefore a vector, as it also is after multiplication by +the reciprocal scalar~$\dfrac{1}{\rho}$. That $p_{\nu\sigma}$~is a tensor, and therefore +transforms according to the equation +\[ +{p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta}, +\] +is proved in mechanics by integrating this equation over +an infinitely small tetrahedron. It is also proved there, +by application of the theorem of moments to an infinitely +small parallelopipedon, that $p_{\nu\sigma} = p_{\sigma\nu}$, and hence that the +tensor of the stress is a symmetrical tensor. From what +has been said it follows that, with the aid of the rules +\PageSep{22} +given above, the equation is co-variant with respect to +orthogonal transformations in space (rotational transformations); +and the rules according to which the +quantities in the equation must be transformed in order +that the equation may be co-variant also become evident. + +The co-variance of the equation of continuity, +\index{Covariance@{Co-variance of equation of continuity}}% +\index{Equation of continuity, co-variance of}% +\[ +\frac{\dd\rho}{\dd t} + \frac{\dd(\rho u_{\nu})}{\dd x_{\nu}} = 0\Add{,} +\Tag{(17)} +\] +requires, from the foregoing, no particular discussion. + +We shall also test for co-variance the equations which +express the dependence of the stress components upon +the properties of the matter, and set up these equations +for the case of a compressible viscous fluid with the aid +\index{Compressible viscous fluid}% +\index{Viscous compressible fluid}% +of the conditions of co-variance. If we neglect the viscosity, +the pressure,~$p$, will be a scalar, and will depend +only upon the density and the temperature of the fluid. +The contribution to the stress tensor is then evidently +\index{Stress tensor}% +\[ +p \delta_{\mu\nu} +\] +in which $\delta_{\mu\nu}$~is the special symmetrical tensor. This term +will also be present in the case of a viscous fluid. But in +this case there will also be pressure terms, which depend +upon the space derivatives of the~$u_{\nu}$. We shall assume +that this dependence is a linear one. Since these terms +must be symmetrical tensors, the only ones which enter +will be +\[ +\alpha\left(\frac{\dd u_{\mu}}{\dd x_{\nu}} + + \frac{\dd u_{\nu}}{\dd x_{\mu}}\right) + + \beta\delta_{\mu\nu} \frac{\dd u_{\alpha}}{\dd x_{\alpha}} +\] +(for $\dfrac{\dd u_{\alpha}}{\dd x_{\alpha}}$~is a scalar). For physical reasons (no slipping) +\PageSep{23} +it is assumed that for symmetrical dilatations in all +directions, i.e.~when +\[ +\frac{\dd u_{1}}{\dd x_{1}} = +\frac{\dd u_{2}}{\dd x_{2}} = +\frac{\dd u_{3}}{\dd x_{3}};\quad +\frac{\dd u_{1}}{\dd x_{2}}, \text{ etc.,} = 0, +\] +there are no frictional forces present, from which it +follows that $\beta = -\dfrac{2}{3}\alpha$. If only $\dfrac{\dd u_{1}}{\dd x_{\Change{2}{3}}}$~is different from +zero, let $p_{31} = -\Change{\eta}{\alpha} \dfrac{\dd u_{1}}{\dd x_{3}}$, by which $\alpha$~is determined. We +then obtain for the complete stress tensor, +\[ +p_{\mu\nu} = p \delta_{\mu\nu} - \Change{\eta}{\alpha} \biggl[ + \biggl(\frac{\dd u_{\mu}}{\dd x_{\nu}} + + \frac{\dd u_{\nu}}{\dd x_{\mu}}\biggr) + - \frac{2}{3} \biggl(\frac{\dd u_{1}}{\dd x_{1}} + + \frac{\dd u_{2}}{\dd x_{2}} + + \frac{\dd u_{3}}{\dd x_{3}}\biggr)\delta_{\mu\nu} +\biggr]\Add{.} +\Tag{(18)} +\] + +The heuristic value of the theory of invariants, which +arises from the isotropy of space (equivalence of all +directions), becomes evident from this example. + +We consider, finally, Maxwell's equations in the form +\index{Maxwell's equations}% +which are the foundation of the electron theory of Lorentz. +\begin{align*} +\left. + \begin{aligned} + &\begin{alignedat}{3} + \frac{\dd h_{3}}{\dd x_{2}} &- \frac{\dd h_{2}}{\dd x_{3}} + &&= \frac{1}{c}\, \frac{\dd e_{1}}{\dd t} + &&+ \frac{1}{c}\, i_{1}\Add{,} \\ +% + \frac{\dd h_{1}}{\dd x_{3}} &- \frac{\dd h_{3}}{\dd x_{1}} + &&= \frac{1}{c}\, \frac{\dd e_{2}}{\dd t} + &&+ \frac{1}{c}\, i_{2}\Add{,} \\ +% +% [** TN: Adding code for third equation (elided in original)] + \frac{\dd h_{2}}{\dd x_{1}} &- \frac{\dd h_{1}}{\dd x_{2}} + &&= \frac{1}{c}\, \frac{\dd e_{3}}{\dd t} + &&+ \frac{1}{c}\, i_{3}\Add{,} + \end{alignedat} \\ + & \frac{\dd e_{1}}{\dd x_{1}} + + \frac{\dd e_{2}}{\dd x_{2}} + + \frac{\dd e_{3}}{\dd x_{3}} = \rho\Add{;} + \end{aligned} +\right\} +\Tag{(19)} \\[8pt] +\left. + \begin{aligned} + &\begin{alignedat}{2} + \frac{\dd e_{3}}{\dd x_{2}} &- \frac{\dd e_{2}}{\dd x_{3}} + &&= -\frac{1}{c}\, \frac{\dd h_{1}}{\dd t}\Add{,}\qquad \\ +% + \frac{\dd e_{1}}{\dd x_{3}} &- \frac{\dd e_{3}}{\dd x_{1}} + &&= -\frac{1}{c}\, \frac{\dd h_{2}}{\dd t}\Add{,} \\ +% +% [** TN: Adding code for third equation (elided in original)] + \frac{\dd e_{2}}{\dd x_{1}} &- \frac{\dd e_{1}}{\dd x_{2}} + &&= -\frac{1}{c}\, \frac{\dd h_{3}}{\dd t}\Add{,} + \end{alignedat} \\ + & \frac{\dd h_{1}}{\dd x_{1}} + + \frac{\dd h_{2}}{\dd x_{2}} + + \frac{\dd h_{3}}{\dd x_{3}} = 0\Add{.} + \end{aligned} +\right\} +\Tag{(20)} +\end{align*} +\PageSep{24} + +$\veci$~is a vector, because the current density is defined as +the density of electricity multiplied by the vector velocity +of the electricity. According to the first three equations +it is evident that $\e$~is also to be regarded as a vector. +Then $\h$~cannot be regarded as a vector.\footnote + {These considerations will make the reader familiar with tensor operations + without the special difficulties of the four-dimensional treatment; + corresponding considerations in the theory of special relativity (Minkowski's + interpretation of the field) will then offer fewer difficulties.} +The equations +may, however, easily be interpreted if $\h$~is regarded as a +\Change{}{skew-}symmetrical tensor of the second rank. In this sense, we +write $h_{23}$,~$h_{31}$,~$h_{12}$, in place of $h_{1}$,~$h_{2}$,~$h_{3}$ respectively. Paying +attention to the skew-symmetry of~$h_{\mu\nu}$, the first three +equations of \Eqref{(19)}~and~\Eqref{(20)} may be written in the form +\begin{gather*} +\frac{\dd h_{\mu\nu}}{\dd x_{\nu}} + = \frac{1}{c}\, \frac{\dd e_{\mu}}{\dd t} + \frac{1}{c} i_{\mu}\Add{,} +\Tag{(19a)} \\ +\frac{\dd e_{\mu}}{\dd x_{\nu}} - \frac{\dd e_{\nu}}{\dd x_{\mu}} + = +\frac{1}{c}\, \frac{\dd h_{\mu\nu}}{\dd t}\Add{.} +\Tag{(20a)} +\end{gather*} +In contrast to~$\e$, $\h$~appears as a quantity which has the +same type of symmetry as an angular velocity. The +divergence equations then take the form +\begin{gather*} +\frac{\dd e_{\nu}}{\dd x_{\nu}} = \rho\Add{,} +\Tag{(19b)} \\ +\frac{\dd h_{\mu\nu}}{\dd x_{\rho}} + +\frac{\dd h_{\nu\rho}}{\dd x_{\mu}} + +\frac{\dd h_{\rho\mu}}{\dd x_{\nu}} = 0\Add{.} +\Tag{(20b)} +\end{gather*} +The last equation is a skew-symmetrical tensor equation +of the third rank (the skew-symmetry of the left-hand +side with respect to every pair of indices may easily be +\PageSep{25} +proved, if attention is paid to the skew-symmetry of~$h_{\mu\nu}$). +This notation is more natural than the usual one, because, +in contrast to the latter, it is applicable to Cartesian left-handed +systems as well as to right-handed systems without +change of sign. +\PageSep{26} + + +\Lecture{II}{The Theory of Special Relativity}{Special Relativity} + +\First{The} previous considerations concerning the configuration +of rigid bodies have been founded, irrespective +of the assumption as to the validity of the Euclidean +geometry, upon the hypothesis that all directions in space, +or all configurations of Cartesian systems of co-ordinates, +are physically equivalent. We may express this as the +\index{Equivalent spaces of reference}% +``principle of relativity with respect to direction,'' and it +has been shown how equations (laws of nature) may be +found, in accord with this principle, by the aid of the +calculus of tensors. We now inquire whether there is a +relativity with respect to the state of motion of the space +of reference; in other words, whether there are spaces of +reference in motion relatively to each other which are +physically equivalent. From the standpoint of mechanics +it appears that equivalent spaces of reference do exist. +\index{Spaces of reference!equivalence of}% +For experiments upon the earth tell us nothing of the +fact that we are moving about the sun with a velocity of +approximately $30$~kilometres a second. On the other +hand, this physical equivalence does not seem to hold for +spaces of reference in arbitrary motion; for mechanical +effects do not seem to be subject to the same laws in a +jolting railway train as in one moving with uniform +\PageSep{27} +velocity; the rotation of the earth must be considered in +writing down the equations of motion relatively to the +earth. It appears, therefore, as if there were Cartesian +systems of co-ordinates, the so-called inertial systems, with +reference to which the laws of mechanics (more generally +the laws of physics) are expressed in the simplest form. +We may infer the validity of the following theorem: If +$K$~is an inertial system, then every other system~$K'$ which +moves uniformly and without rotation relatively to~$K$, is +also an inertial system; the laws of nature are in concordance +for all inertial systems. This statement we shall +call the ``principle of special relativity.'' We shall draw +certain conclusions from this principle of ``relativity of +translation'' just as we have already done for relativity of +direction. + +In order to be able to do this, we must first solve the +following problem. If we are given the Cartesian co-ordinates,~$x_{\nu}$, +and the time,~$t$, of an event relatively to one +inertial system,~$K$, how can we calculate the co-ordinates,~${x'}_{\nu}$, +and the time,~$t'$, of the same event relatively to an +inertial system~$K'$ which moves with uniform translation +relatively to~$K$? In the pre-relativity physics +\index{Prerelativity@{Pre-relativity physics, hypotheses of}}% +this problem was solved by making unconsciously two +hypotheses:--- + +1.~The time is absolute; the time of an event,~$t'$, +relatively to~$K'$ is the same as the time relatively to~$K$. +If instantaneous signals could be sent to a distance, and +if one knew that the state of motion of a clock had no +influence on its rate, then this assumption would be +physically established. For then clocks, similar to one +\PageSep{28} +another, and regulated alike, could be distributed over +the systems $K$~and~$K'$, at rest relatively to them, and +their indications would be independent of the state of +motion of the systems; the time of an event would then +be given by the clock in its immediate neighbourhood. + +2.~Length is absolute; if an interval, at rest relatively +to~$K$, has a length~$s$, then it has the same length~$s$ +relatively to a system~$K'$ which is in motion relatively +to~$K$. + +If the axes of $K$~and~$K'$ are parallel to each other, a +simple calculation based on these two assumptions, gives +the equations of transformation +\[ +\left. +\begin{aligned} +{x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\Add{,} \\ +t' &= t - b\Add{.} +\end{aligned} +\right\} +\Tag{(21)} +\] + +This transformation is known as the ``Galilean Transformation.'' +\index{Galilean@{Galilean regions}!transformation}% +\index{Transformation@{Transformation, Galilean}}% +Differentiating twice by the time, we get +\[ +\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}. +\] +Further, it follows that for two simultaneous events, +\[ +{{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}. +\] +The invariance of the distance between the two points +results from squaring and adding. From this easily +follows the co-variance of Newton's equations of motion +with respect to the Galilean transformation~\Eqref{(21)}. Hence +it follows that classical mechanics is in accord with the +principle of special relativity if the two hypotheses +respecting scales and clocks are made. + +But this attempt to found relativity of translation upon +the Galilean transformation fails when applied to electromagnetic +\PageSep{29} +phenomena. The Maxwell-Lorentz electromagnetic +equations are not co-variant with respect to the +Galilean transformation. In particular, we note, by~\Eqref{(21)}, +that a ray of light which referred to~$K$ has a velocity~$c$, +has a different velocity referred to~$K'$, depending upon +its direction. The space of reference of~$K$ is therefore +distinguished, with respect to its physical properties, from +all spaces of reference which are in motion relatively to it +(quiescent æther). But all experiments have shown that +electro\Change{-}{}magnetic\TNote{** TN: Only hyphenated instance} and optical phenomena, relatively to the +earth as the body of reference, are not influenced by the +translational velocity of the earth. The most important +of these experiments are those of Michelson and Morley, +\index{Michelson and Morley}% +which I shall assume are known. The validity of the +principle of special relativity can therefore hardly be +doubted. + +On the other hand, the Maxwell-Lorentz equations +have proved their validity in the treatment of optical +problems in moving bodies. No other theory has +satisfactorily explained the facts of aberration, the +propagation of light in moving bodies (Fizeau), and +\index{Fizeau}% +phenomena observed in double stars (De~Sitter). The +\index{Sitter}% +consequence of the Maxwell-Lorentz equations that in a +vacuum light is propagated with the velocity~$c$, at least +with respect to a definite inertial system~$K$, must therefore +be regarded as proved. According to the principle +of special relativity, we must also assume the truth of +this principle for every other inertial system. + +Before we draw any conclusions from these two +principles we must first review the physical significance +\PageSep{30} +\index{Concept@{Concept of space}!time}% +\index{Time-concept}% +of the concepts ``time'' and ``velocity.'' It follows from +what has gone before, that co-ordinates with respect to +an inertial system are physically defined by means of +measurements and constructions with the aid of rigid +bodies. In order to measure time, we have supposed a +clock,~$U$, present somewhere, at rest relatively to~$K$. But +we cannot fix the time, by means of this clock, of an event +whose distance from the clock is not negligible; for there +are no ``instantaneous signals'' that we can use in order +to compare the time of the event with that of the clock. +In order to complete the definition of time we may +employ the principle of the constancy of the velocity of +light in a vacuum. Let us suppose that we place similar +clocks at points of the system~$K$, at rest relatively to it, +and regulated according to the following scheme. A ray +of light is sent out from one of the clocks,~$U_{m}$, at the +instant when it indicates the time~$t_{m}$, and travels through +a vacuum a distance~$r_{mn}$, to the clock~$U_{n}$; at the instant +when this ray meets the clock~$U_{n}$ the latter is set to +indicate the time $t_{n} = t_{m} + \dfrac{r_{mn}}{c}$.\footnote + {Strictly speaking, it would be more correct to define simultaneity first, +\index{Simultaneity}% + somewhat as follows: two events taking place at the points $A$~and~$B$ of + the system~$K$ are simultaneous if they appear at the same instant when + observed from the middle point,~$M$, of the interval~$AB$. Time is then + defined as the ensemble of the indications of similar clocks, at rest + relatively to~$K$, which register the same simultaneously.} +The principle of the +constancy of the velocity of light then states that this +adjustment of the clocks will not lead to contradictions. +With clocks so adjusted, we can assign the time to events +which take place near any one of them. It is essential to +\PageSep{31} +note that this definition of time relates only to the inertial +system~$K$, since we have used a system of clocks at rest +relatively to~$K$. The assumption which was made in the +pre-relativity physics of the absolute character of time +(i.e.~the independence of time of the choice of the inertial +system) does not follow at all from this definition. + +The theory of relativity is often criticized for giving, +\index{Criticisms of theory of relativity}% +\index{Theory of relativity, criticisms of}% +without justification, a central theoretical rôle to the +propagation of light, in that it founds the concept of time +upon the law of propagation of light. The situation, +however, is somewhat as follows. In order to give +physical significance to the concept of time, processes of +some kind are required which enable relations to be +established between different places. It is immaterial +what kind of processes one chooses for such a definition +of time. It is advantageous, however, for the theory, +to choose only those processes concerning which we know +something certain. This holds for the propagation of +light \textit{in vacuo} in a higher degree than for any other process +which could be considered, thanks to the investigations +of Maxwell and H.~A.~Lorentz. + +From all of these considerations, space and time data +have a physically real, and not a mere fictitious, significance; +in particular this holds for all the relations in +which co-ordinates and time enter, e.g.~the relations~\Eqref{(21)}. +There is, therefore, sense in asking whether those +equations are true or not, as well as in asking what the +true equations of transformation are by which we pass +from one inertial system~$K$ to another,~$K'$, moving +relatively to it. It may be shown that this is uniquely +\PageSep{32} +settled by means of the principle of the constancy of the +velocity of light and the principle of special relativity. + +To this end we think of space and time physically +defined with respect to two inertial systems, $K$~and~$K'$, in +the way that has been shown. Further, let a ray of light +pass from one point~$P_{1}$ to another point~$P_{2}$ of~$K$ through +a vacuum. If $r$~is the measured distance between the two +points, then the propagation of light must satisfy the +equation +\[ +r = c·\Delta t. +\] + +If we square this equation, and express~$r^{2}$ by the +differences of the co-ordinates,~$\Delta x_{\nu}$, in place of this equation +we can write +\[ +\sum (\Delta x_{\nu})^{2} - c^{2}\, \Delta t^{2} = 0\Add{.} +\Tag{(22)} +\] +This equation formulates the principle of the constancy +of the velocity of light relatively to~$K$. It must hold +whatever may be the motion of the source which emits +the ray of light. + +The same propagation of light may also be considered +relatively to~$K'$, in which case also the principle of the +constancy of the velocity of light must be satisfied. +Therefore, with respect to~$K'$, we have the equation +\[ +\sum ({\Delta x'}_{\nu})^{2} - c^{2}\, \Delta t'^{2} = 0\Add{.} +\Tag{(22a)} +\] + +Equations \Eqref{(22a)}~and~\Eqref{(22)} must be mutually consistent +with each other with respect to the transformation which +transforms from~$K$ to~$K'$. A transformation which effects +this we shall call a ``Lorentz transformation.'' +\index{Lorentz@{Lorentz electromotive force}!transformation}% + +Before considering these transformations in detail we +\PageSep{33} +shall make a few general remarks about space and time. +In the pre-relativity physics space and time were separate +entities. Specifications of time were independent of +the choice of the space of reference. The Newtonian +mechanics was relative with respect to the space of +reference, so that, e.g.~the statement that two non-simultaneous +events happened at the same place had no objective +meaning (that is, independent of the space of reference). +But this relativity had no rôle in building up the theory. +One spoke of points of space, as of instants of time, as if +they were absolute realities. It was not observed that +the true element of the space-time specification was the +event, specified by the four numbers $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$. The +conception of something happening was always that of a +four-dimensional continuum; but the recognition of this +\index{Continuum, four-dimensional}% +\index{Four-dimensional continuum}% +was obscured by the absolute character of the pre-relativity +time. Upon giving up the hypothesis of the absolute +character of time, particularly that of simultaneity, the +four-dimensionality of the time-space concept was immediately +\index{Time-space concept}% +recognized. It is neither the point in space, +nor the instant in time, at which something happens that +has physical reality, but only the event itself. There is +no absolute (independent of the space of reference) relation +in space, and no absolute relation in time between two +events, but there is an absolute (independent of the space +of reference) relation in space and time, as will appear in +the sequel. The circumstance that there is no objective +rational division of the four-dimensional continuum into +a three-dimensional space and a one-dimensional time +continuum indicates that the laws of nature will assume +\PageSep{34} +a form which is logically most satisfactory when expressed +as laws in the four-dimensional space-time continuum. +Upon this depends the great advance in method which +the theory of relativity owes to Minkowski. Considered +\index{Minkowski}% +from this standpoint, we must regard $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$ as the +four co-ordinates of an event in the four-dimensional continuum. +We have far less success in picturing to ourselves +relations in this four-dimensional continuum than in the +three-dimensional Euclidean continuum; but it must be +emphasized that even in the Euclidean three-dimensional +geometry its concepts and relations are only of an abstract +nature in our minds, and are not at all identical with the +images we form visually and through our sense of touch. +The non-divisibility of the four-dimensional continuum +of events does not at all, however, involve the equivalence +of the space co-ordinates with the time co-ordinate. On +the contrary, we must remember that the time co-ordinate +is defined physically wholly differently from the space +co-ordinates. The relations \Eqref{(22)}~and~\Eqref{(22a)} which when +equated define the Lorentz transformation show, further, +a difference in the rôle of the time co-ordinate from that +of the space co-ordinates; for the term~$\Delta t^{2}$ has the opposite +sign to the space terms, ${\Delta x_{1}}^{2}$,~${\Delta x_{2}}^{2}$,~${\Delta x_{3}}^{2}$. + +Before we analyse further the conditions which define +the Lorentz transformation, we shall introduce the light-time, +\index{Light time@{Light-time}}% +$l = ct$, in place of the time,~$t$, in order that the +constant~$c$ shall not enter explicitly into the formulas to +be developed later. Then the Lorentz transformation is +defined in such a way that, first, it makes the equation +\[ +{\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 +\Tag{(22b)} +\] +\PageSep{35} +a co-variant equation, that is, an equation which is satisfied +with respect to every inertial system if it is satisfied in +the inertial system to which we refer the two given events +(emission and reception of the ray of light). Finally, +with Minkowski, we introduce in place of the real time +co-ordinate $l = ct$, the imaginary time co-ordinate +\[ +x_{4} = il = ict\quad (\sqrt{-1} = i). +\] +Then the equation defining the propagation of light, +which must be co-variant with respect to the Lorentz +transformation, becomes +\[ +\sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} + = 0\Add{.} +\Tag{(22c)} +\] +This condition is always satisfied\footnote + {That this specialization lies in the nature of the case will be evident + later.} +if we satisfy the more +general condition that +\[ +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} +\Tag{(23)} +\] +shall be an invariant with respect to the transformation. +This condition is satisfied only by linear transformations, +that is, transformations of the type +\[ +{x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} +\Tag{(24)} +\] +in which the summation over the~$\alpha$ is to be extended +from $\alpha = 1$ to $\alpha = 4$. A glance at equations \Eqref{(23)}~and~\Eqref{(24)} +shows that the Lorentz transformation so defined is +identical with the translational and rotational transformations +of the Euclidean geometry, if we disregard the +number of dimensions and the relations of reality. We +\PageSep{36} +can also conclude that the coefficients~$b_{\mu\alpha}$ must satisfy the +conditions +\[ +b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}\Add{.} +\Tag{(25)} +\] +Since the ratios of the~$x_{\nu}$ are real, it follows that all the~$a_{\mu}$ +and the~$b_{\mu\alpha}$ are real, except $a_{4}$, $b_{41}$,~$b_{42}$,~$b_{43}$, $b_{14}$,~$b_{24}$\Change{,}{} and~$b_{34}$, +which are purely imaginary. + +\Paragraph{Special Lorentz Transformation.} We obtain the +\index{Special Lorentz transformation}% +simplest transformations of the type of \Eqref{(24)}~and~\Eqref{(25)} if +only two of the co-ordinates are to be transformed, and if +all the~$a_{\mu}$, which determine the new origin, vanish. We +obtain then for the indices $1$~and~$2$, on account of the +three independent conditions which the relations~\Eqref{(25)} +furnish, +\[ +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi\Add{,} \\ +{x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi\Add{,} \\ +{x'}_{3} &= x_{3}\Add{,} && \\ +{x'}_{4} &= x_{4}\Add{.} && +\end{alignedat} +\right\} +\Tag{(26)} +\] + +This is a simple rotation in space of the (space) +co-ordinate system about $x_{3}$-axis. We see that the +rotational transformation in space (without the time +transformation) which we studied before is contained in +the Lorentz transformation as a special case. For the +indices $1$~and~$4$ we obtain, in an analogous manner, +\[ +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi\Add{,} \\ +{x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi\Add{,} \\ +{x'}_{2} &= x_{2}\Add{,} && \\ +{x'}_{3} &= x_{3}\Add{.} && +\end{alignedat} +\right\} +\Tag{(26a)} +\] + +On account of the relations of reality $\psi$~must be taken +as imaginary. To interpret these equations physically, +we introduce the real light-time~$l$ and the velocity~$v$ of~$K'$ +\PageSep{37} +relatively to~$K$, instead of the imaginary angle~$\psi$. We +have, first, +\begin{alignat*}{4} +{x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi\Add{,} \\ +l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. +\end{alignat*} +Since for the origin of~$K'$ i.e., for $x_{1} = 0$, we must have +$x_{1} = vl$, it follows from the first of these equations that +\[ +v = i\tan\psi\Add{,} +\Tag{(27)} +\] +and also +\[ +\left. +\begin{aligned} +\sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}\Add{,} \\ +\cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}\Add{,} +\end{aligned} +\right\} +\Tag{(28)} +\] +so that we obtain +\[ +\left. +\begin{aligned} +{x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}\Add{,} \\ +l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}\Add{,} \\ +{x'}_{2} &= x_{2}\Add{,} \\ +{x'}_{3} &= x_{3}\Add{.} +\end{aligned} +\right\} +\Tag{(29)} +\] + +These equations form the well-known special Lorentz +transformation, which in the general theory represents a +rotation, through an imaginary angle, of the four-dimensional +system of co-ordinates. If we introduce the ordinary +time~$t$, in place of the light-time~$l$, then in~\Eqref{(29)} we must +replace $l$~by~$ct$ and $v$~by~$\dfrac{v}{c}$. + +We must now fill in a gap. From the principle of the +constancy of the velocity of light it follows that the +equation +\[ +\sum {\Delta x_{\nu}}^{2} = 0 +\] +\PageSep{38} +has a significance which is independent of the choice of +the inertial system; but the invariance of the quantity +$\sum {\Delta x_{\nu}}^{2}$ does not at all follow from this. This quantity +might be transformed with a factor. This depends upon +the fact that the right-hand side of~\Eqref{(29)} might be multiplied +by a factor~$\lambda$, independent of~$v$. But the principle +of relativity does not permit this factor to be different from~$1$, +as we shall now show. Let us assume that we have +a rigid circular cylinder moving in the direction of its +axis. If its radius, measured at rest with a unit measuring +rod is equal to~$R_{0}$, its radius~$R$ in motion, might be +different from~$R_{0}$, since the theory of relativity does not +make the assumption that the shape of bodies with respect +to a space of reference is independent of their motion +relatively to this space of reference. But all directions +in space must be equivalent to each other. $R$~may therefore +depend upon the magnitude~$q$ of the velocity, but +not upon its direction; $R$~must therefore be an even +function of~$q$. If the cylinder is at rest relatively to~$K'$ +the equation of its lateral surface is +\[ +%[** TN: Inconsistently upright R here, below in the original] +x'^{2} + y'^{2} = {R_{0}}^{2}. +\] +If we write the last two equations of~\Eqref{(29)} more generally +\begin{align*} +{x'}_{2} = \lambda x_{2}\Add{,} \\ +{x'}_{3} = \lambda x_{3}\Add{,} +\end{align*} +then the lateral surface of the cylinder referred to~$K$ +satisfies the equation +\[ +x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}. +\] +\PageSep{39} +The factor~$\lambda$ therefore measures the lateral contraction of +the cylinder, and can thus, from the above, be only an +even function of~$v$. + +If we introduce a third system of co-ordinates,~$K''$, +which moves relatively to~$K'$ with velocity~$v$ in the direction +of the negative $x$-axis of~$K$, we obtain, by applying +\Eqref{(29)} twice, +\begin{align*} +{x''}_{1} &= \lambda(v) \lambda(-v) x_{1}\Add{,} \\ +%[** TN: Added code for next two equations, elided in the original] +{x''}_{2} &= \lambda(v) \lambda(-v) x_{2}\Add{,} \\ +{x''}_{3} &= \lambda(v) \lambda(-v) x_{3}\Add{,} \\ +l'' &= \lambda(v) \lambda(-v) l. +\end{align*} +Now, since $\lambda(v)$~must be equal to~$\lambda(-v)$, and since we +assume that we use the same measuring rods in all the +systems, it follows that the transformation of~$K''$ to~$K$ +must be the identical transformation (since the possibility +$\lambda = -1$ does not need to be considered). It is essential +for these considerations to assume that the behaviour of +the measuring rods does not depend upon the history of +their previous motion. + +\Paragraph{Moving Measuring Rods and Clocks.} At the definite $K$-time, +\index{Clocks, moving}% +\index{Moving measuring rods and clocks}% +\index{Rods (measuring) and clocks in motion}% +$l = 0$, the position of the points given by the integers +${x'}_{1} = n$, is with respect to~$K$, given by $x_{1} = n\sqrt{1 - v^{2}}$; +this follows from the first of equations~\Eqref{(29)} and expresses +the Lorentz contraction. A clock at rest at the origin +$x_{1} = 0$ of~$K$, whose beats are characterized by $l = n$, will, +when observed from~$K'$, have beats characterized by +\[ +l' = \frac{n}{\sqrt{1 - v^{2}}}; +\] +this follows from the second of equations~\Eqref{(29)} and shows +\PageSep{40} +that the clock goes slower than if it were at rest relatively +to~$K'$. These two consequences, which hold, \textit{mutatis +mutandis}, for every system of reference, form the physical +content, free from convention, of the Lorentz transformation. + +\Paragraph{Addition Theorem for Velocities.} If we combine two +\index{Addition@{Addition and subtraction of tensors}!theorem of velocities}% +\index{Theorem for addition of velocities}% +\index{Velocities, addition theorem of}% +special Lorentz transformations with the relative velocities +$v_{1}$~and~$v_{2}$, then the velocity of the single Lorentz transformation +which takes the place of the two separate ones +is, according to~\Eqref{(27)}, given by +\[ +v_{12} = i \tan (\psi_{1} + \psi_{2}) + = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} + = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. +\Tag{(30)} +\] + +\Paragraph{General Statements about the Lorentz Transformation +and its Theory of Invariants.} The whole theory of +invariants of the special theory of relativity depends upon +the invariant~$s^{2}$~\Eqref{(23)}. Formally, it has the same rôle in +the four-dimensional space-time continuum as the invariant +${\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}$ in the Euclidean geometry +and in the pre-relativity physics. The latter quantity is +not an invariant with respect to all the Lorentz transformations; +the quantity~$s^{2}$ of equation~\Eqref{(23)} assumes the +rôle of this invariant. With respect to an arbitrary +inertial system, $s^{2}$~may be determined by measurements; +with a given unit of measure it is a completely determinate +quantity, associated with an arbitrary pair of events. + +The invariant~$s^{2}$ differs, disregarding the number of +dimensions, from the corresponding invariant of the +Euclidean geometry in the following points. In the +Euclidean geometry $s^{2}$~is necessarily positive; it vanishes +\PageSep{41} +only when the two points concerned come together. On +the other hand, from the vanishing of +\[ +s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2} +\] +it cannot be concluded that the two space-time points +%[Illustration: Fig. 1.] +\Figure{041} +fall together; the vanishing of this quantity~$s^{2}$, is the +invariant condition that the two space-time points can be +connected by a light signal \textit{in vacuo}. If $P$~is a point +\PageSep{42} +(event) represented in the four-dimensional space of the +$x_{1}$,~$x_{2}$,~$x_{3}$,~$l$, then all the ``points'' which can be connected +to~$P$ by means of a light signal lie upon the cone $s^{2} = 0$ +(compare \Figref{1}, in which the dimension~$x_{3}$ is suppressed). +The ``upper'' half of the cone may contain the ``points'' +to which light signals can be sent from~$P$; then the +``lower'' half of the cone will contain the ``points'' from +which light signals can be sent to~$P$. The points~$P'$ +enclosed by the conical surface furnish, with~$P$, a negative~$s^{2}$; +$PP'$,~as well as~$P'P$ is then, according to Minkowski, +of the nature of a time. Such intervals represent elements +of possible paths of motion, the velocity being less than +that of light.\footnote + {That material velocities exceeding that of light are not possible, + follows from the appearance of the radical $\sqrt{1 - v^{2}}$ in the special Lorentz + transformation~\Eqref{(29)}.} +In this case the $l$-axis may be drawn in +the direction of~$PP'$ by suitably choosing the state of +motion of the inertial system. If $P'$~lies outside of the +``light-cone'' then $PP'$~is of the nature of a space; in +\index{Light cone@{Light-cone}}% +this case, by properly choosing the inertial system, $\Delta l$~can +be made to vanish. + +By the introduction of the imaginary time variable, +$x_{4} = il$, Minkowski has made the theory of invariants for +the four-dimensional continuum of physical phenomena +fully analogous to the theory of invariants for the three-dimensional +continuum of Euclidean space. The theory +of four-dimensional tensors of special relativity differs from +the theory of tensors in three-dimensional space, therefore, +only in the number of dimensions and the relations of +reality. +\PageSep{43} + +A physical entity which is specified by four quantities,~$A_{\nu}$, +\index{Four-vector}% +in an arbitrary inertial system of the $x_{1}$,~$x_{2}$,~$x_{3}$,~$x_{4}$, is +called a $4$-vector, with the components~$A_{\nu}$, if the $A_{\nu}$ +correspond in their relations of reality and the properties +of transformation to the~$\Delta x_{\nu}$; it may be of the nature of +a space or of a time. The sixteen quantities~$A_{\mu\nu}$ then +form the components of a tensor of the second rank, if +they transform according to the scheme +\[ +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}. +\] +It follows from this that the~$A_{\mu\nu}$ behave, with respect to +their properties of transformation and their properties +of reality, as the products of components,~$U_{\mu}V_{\nu}$, of two +$4$-vectors, $(U)$~and~$(V)$. All the components are real +except those which contain the index~$4$ once, those being +purely imaginary. Tensors of the third and higher ranks +may be defined in an analogous way. The operations +of addition, subtraction, multiplication, contraction and +differentiation for these tensors are wholly analogous to +the corresponding operations for tensors in three-dimensional +space. + +Before we apply the tensor theory to the four-dimensional +space-time continuum, we shall examine more +particularly the skew-symmetrical tensors. The tensor +of the second rank has, in general, $16 = 4·4$ components. +In the case of skew-symmetry the components with two +equal indices vanish, and the components with unequal +indices are equal and opposite in pairs. There exist, +therefore, only six independent components, as is the +case in the electromagnetic field. In fact, it will be shown +\PageSep{44} +when we consider Maxwell's equations that these may +be looked upon as tensor equations, provided we regard +the electromagnetic field as a skew-symmetrical tensor. +Further, it is clear that the skew-symmetrical tensor of +the third rank (skew-symmetrical in all pairs of indices) +has only four independent components, since there are +only four combinations of three different indices. + +We now turn to Maxwell's equations \Eqref{(19a)}, \Eqref{(19b)}, \Eqref{(20a)}, +\Eqref{(20b)}, and introduce the notation:\footnote + {In order to avoid confusion from now on we shall use the three-dimensional + space indices, $x$,~$y$,~$z$ instead of $1$,~$2$,~$3$, and we shall reserve + the numeral indices $1$,~$2$, $3$,~$4$ for the four-dimensional space-time continuum.} +\begin{gather*} +\left. +\begin{alignedat}{6} +\phi_{23}\qquad & \phi_{31}\qquad && \phi_{12}\quad && +\Neg\phi_{14}\quad && \Neg\phi_{24}\quad && \Neg\phi_{34} \\ +\h_{23} \qquad & \h_{31} \qquad && \h_{12} \quad && +-\Change{\veci}{i}\e_{x}\quad && -\Change{\veci}{i}\e_{y}\quad && -\Change{\veci}{i}\e_{z} +\end{alignedat} +\right\} +\Tag{(30a)}\displaybreak[1] \\ +% +\left. +\begin{alignedat}{4} +& J_{1}\qquad && J_{2}\qquad && J_{3}\qquad && J_{4} \\ +&\frac{1}{c}\veci_{x}\qquad && +\frac{1}{c}\veci_{y}\qquad && +\frac{1}{c}\veci_{z}\qquad && \Change{\veci}{i}\rho +\end{alignedat} +\right\} +\Tag{(31)} +\end{gather*} +with the convention that $\phi_{\mu\nu}$~shall be equal to~$-\phi_{\nu\mu}$. +Then Maxwell's equations may be combined into the +forms +\begin{gather*} +\frac{\dd \phi_{\mu\nu}}{\dd x_{\nu}} = J_{\mu}\Add{,} +\Tag{(32)} \\ +\frac{\dd \phi_{\mu\nu}}{\dd x_{\sigma}} + +\frac{\dd \phi_{\nu\sigma}}{\dd x_{\mu}} + +\frac{\dd \phi_{\sigma\mu}}{\dd x_{\nu}} = 0\Add{,} +\Tag{(33)} +\end{gather*} +as one can easily verify by substituting from \Eqref{(30a)}~and~\Eqref{(31)}. +Equations \Eqref{(32)}~and~\Eqref{(33)} have a tensor character, +and are therefore co-variant with respect to Lorentz +transformations, if the~$\phi_{\mu\nu}$ and the~$J_{\nu}$ have a tensor +character, which we assume. Consequently, the laws for +\PageSep{45} +transforming these quantities from one to another allowable +(inertial) system of co-ordinates are uniquely +determined. The progress in method which electrodynamics +owes to the theory of special relativity lies +principally in this, that the number of independent +hypotheses is diminished. If we consider, for example, +equations~\Eqref{(19a)} only from the standpoint of relativity of +direction, as we have done above, we see that they have +three logically independent terms. The way in which +the electric intensity enters these equations appears to +be wholly independent of the way in which the magnetic +intensity enters them; it would not be surprising if instead +of~$\dfrac{\dd \e_{\mu}}{\dd l}$, we had, say,~$\dfrac{\dd^{2} \e_{\mu}}{\dd l^{2}}$, or if this term were absent. On +the other hand, only two independent terms appear in +equation~\Eqref{(32)}. The electromagnetic field appears as a +formal unit; the way in which the electric field enters +this equation is determined by the way in which the +magnetic field enters it. Besides the electromagnetic +field, only the electric current density appears as an +independent entity. This advance in method arises from +the fact that the electric and magnetic fields draw their +separate existences from the relativity of motion. A +field which appears to be purely an electric field, judged +from one system, has also magnetic field components +when judged from another inertial system. When applied +to an electromagnetic field, the general law of transformation +furnishes, for the special case of the special Lorentz +transformation, the equations +\PageSep{46} +\[ +\left. +\begin{alignedat}{2} +{\e'}_{x} &= \e_{x}\qquad & {\h'}_{x} &= \h_{x}\Add{,} \\ +{\e'}_{y} &= \frac{\e_{y} - v\h_{z}}{\sqrt{1 - v^{2}}}\qquad & +{\h'}_{y} &= \frac{\h_{y} + v\e_{z}}{\sqrt{1 - v^{2}}}\Add{,} \\ +{\e'}_{z} &= \frac{\e_{z} + v\h_{y}}{\sqrt{1 - v^{2}}}\qquad & +{\h'}_{z} &= \frac{\h_{z} - v\e_{y}}{\sqrt{1 - v^{2}}}\Add{.} +\end{alignedat} +\right\} +\Tag{(34)} +\] + +If there exists with respect to~$K$ only a magnetic field,~$\h$, +but no electric field,~$\e$, then with respect to~$K'$ there +exists an electric field~$\e'$ as well, which would act upon +an electric particle at rest relatively to~$K'$. An observer +at rest relatively to~$K$ would designate this force as the +Biot-Savart force, or the Lorentz electromotive force. It +\index{Biot-Savart force}% +\index{Lorentz@{Lorentz electromotive force}}% +therefore appears as if this electromotive force had become +fused with the electric field intensity into a single entity. + +In order to view this relation formally, let us consider +the expression for the force acting upon unit volume of +electricity, +\[ +\k = \rho\e + [\veci, \h]\Add{,} +\Tag{(35)} +\] +in which $\veci$~is the vector velocity of electricity, with the +velocity of light as the unit. If we introduce $J_{\mu}$~and~$\phi_{\mu\Change{}{\nu}}$ +according to \Eqref{(30a)}~and~\Eqref{(31)}, we obtain for the first +component the expression +\[ +\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}. +\] +Observing that $\phi_{11}$~vanishes on account of the skew-symmetry +of the tensor~$(\phi)$, the components of~$\Change{k}{\k}$ are given +by the first three components of the four-dimensional +vector +\[ +K_{\mu} = \phi_{\mu\nu} J_{\nu}\Add{,} +\Tag{(36)} +\] +and the fourth component is given by +\[ +K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} + = \Change{\veci}{i}(\e_{x} \veci_{x} + \e_{y} \veci_{y} + \e_{z} \veci_{z}) + = \Change{\veci}{i}\lambda\Add{.} +\Tag{(37)} +\] +\PageSep{47} +There is, therefore, a four-dimensional vector of force per +unit volume, whose first three components, $\Change{k}{K}_{1}$,~$\Change{k}{K}_{2}$,~$\Change{k}{K}_{3}$, are +the ponderomotive force components per unit volume, and +whose fourth component is the rate of working of the field +per unit volume, multiplied by~$\sqrt{-1}$. +%[Illustration: Fig. 2.] +\Figure{047} + +A comparison of \Eqref{(36)}~and~\Eqref{(35)} shows that the theory +of relativity formally unites the ponderomotive force of +the electric field,~$\rho\e$, and the Biot-Savart or Lorentz +force~$[\veci, \h]$. +\PageSep{48} + +\Paragraph{Mass and Energy.} An important conclusion can be +\index{Energy@{Energy and mass}}% +\index{Mass and Energy}% +drawn from the existence and significance of the $4$-vector~$K_{\mu}$. +Let us imagine a body upon which the electromagnetic +field acts for a time. In the symbolic figure +(\Figref{2}) $Ox_{1}$~designates the $x_{1}$-axis, and is at the same +time a substitute for the three space axes $Ox_{1}$,~$Ox_{2}$,~$Ox_{3}$; +$Ol$~designates the real time axis. In this diagram a body +of finite extent is represented, at a definite time~$l$, by the +interval~$AB$; the whole space-time existence of the body +is represented by a strip whose boundary is everywhere +inclined less than~$45°$ to the $l$-axis. Between the time +sections, $l = l_{1}$ and $l = l_{2}$, but not extending to them, +a portion of the strip is shaded. This represents the +portion of the space-time manifold in which the electromagnetic +field acts upon the body, or upon the electric +charges contained in it, the action upon them being +transmitted to the body. We shall now consider the +changes which take place in the momentum and energy +of the body as a result of this action. + +We shall assume that the principles of momentum +and energy are valid for the body. The change in +momentum, $\Delta I_{x}$,~$\Delta I_{y}$,~$\Delta I_{z}$, and the change in energy,~$\Delta E$, +are then given by the expressions +\begin{alignat*}{2} +%[** TN: \int_{l_{0}}^{l_{1}} in the original] +\Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int \k_{x}\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{,} \\ +% +%[** TN: Added code for next two equations, elided in the original] +\Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int \k_{y}\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{,} \\ +% +\Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int \k_{z}\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{,} \\ +% +\Delta E &= \int_{l_{1}}^{l_{2}}dl \int \lambda\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int + \frac{1}{\Change{\veci}{i}}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{.} +\end{alignat*} +\PageSep{49} +Since the four-dimensional element of volume is an +invariant, and $(K_{1}, K_{2}, K_{3}, K_{4})$ forms a $4$-vector, the four-dimensional +integral extended over the shaded portion +transforms as a $4$-vector, as does also the integral between +the limits $l_{1}$~and~$l_{2}$, because the portion of the region which +is not shaded contributes nothing to the integral. It +follows, therefore, that $\Delta I_{x}$,~$\Delta I_{y}$,~$\Delta I_{z}$,~$i\Delta E$ form a $4$-vector. +Since the quantities themselves transform in the same +way as their increments, it follows that the aggregate of +the four quantities +\[ +I_{x},\ I_{y},\ I_{z},\ \Change{\veci}{i} E +\] +has itself the properties of a vector; these quantities are +referred to an instantaneous condition of the body (e.g.~at +the time $l = l_{1}$). + +This $4$-vector may also be expressed in terms of the +mass~$m$, and the velocity of the body, considered as a +material particle. To form this expression, we note first, +that +\[ +-ds^{2} = d\tau^{2} + = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} + = dl^{2}(1 - q^{2}) +\Tag{(38)} +\] +is an invariant which refers to an infinitely short portion +of the four-dimensional line which represents the motion +of the material particle. The physical significance of the +invariant~$d\tau$ may easily be given. If the time axis is +chosen in such a way that it has the direction of the line +differential which we are considering, or, in other words, +if we reduce the material particle to rest, we shall then +have $d\tau = dl$; this will therefore be measured by the +light-seconds clock which is at the same place, and at +rest relatively to the material particle. We therefore call +\PageSep{50} +$\tau$~the proper time of the material particle. As opposed +to~$dl$, $d\tau$~is therefore an invariant, and is practically +equivalent to~$dl$ for motions whose velocity is small +compared to that of light. Hence we see that +\[ +u_{\sigma} = \frac{dx_{\sigma}}{d\tau} +\Tag{(39)} +\] +has, just as the~$dx_{\nu}$, the character of a vector; we shall +designate~$(u_{\sigma})$ as the four-dimensional vector (in brief, +$4$-vector) of velocity. Its components satisfy, by~\Eqref{(38)}, +the condition +\[ +\sum {u_{\sigma}}^{2} = -1\Add{.} +\Tag{(40)} +\] +We see that this $4$-vector, whose components in the +ordinary notation are +\[ +\frac{\q_{x}}{\sqrt{1 - q^{2}}},\quad +\frac{\q_{y}}{\sqrt{1 - q^{2}}},\quad +\frac{\q_{z}}{\sqrt{1 - q^{2}}},\quad +\frac{\Change{\veci}{i}}{\sqrt{1 - q^{2}}} +\Tag{(41)} +\] +is the only $4$-vector which can be formed from the velocity +components of the material particle which are defined in +three dimensions by +\[ +\q_{x} = \frac{dx}{dl},\quad +\q_{y} = \frac{dy}{dl},\quad +\q_{z} = \frac{dz}{dl}. +\] +We therefore see that +\[ +\left(m \frac{dx_{\mu}}{d\tau}\right) +\Tag{(42)} +\] +must be that $4$-vector which is to be equated to the +$4$-vector of momentum and energy whose existence we +have proved above. By equating the components, we +obtain, in three-dimensional notation, +\PageSep{51} +\[ +\left. +\begin{aligned} +I_{x} &= \frac{m\q_{x}}{\sqrt{1 - q^{2}}}\Add{,} \\ +%[** TN: Added code for next two equations, elided in the original] +I_{y} &= \frac{m\q_{y}}{\sqrt{1 - q^{2}}}\Add{,} \\ +I_{z} &= \frac{m\q_{z}}{\sqrt{1 - q^{2}}}\Add{,} \\ +E &= \frac{m}{\sqrt{1 - q^{2}}}\Add{.} +\end{aligned} +\right\} +\Tag{(43)} +\] + +We recognize, in fact, that these components of +momentum agree with those of classical mechanics for +velocities which are small compared to that of light. For +large velocities the momentum increases more rapidly +than linearly with the velocity, so as to become infinite +on approaching the velocity of light. + +If we apply the last of equations~\Eqref{(43)} to a material +particle at rest ($q = 0$), we see that the energy,~$E_{0}$, of a +\index{Energy@{Energy and mass}}% +body at rest is equal to its mass. Had we chosen the +second as our unit of time, we would have obtained +\[ +E_{0} = mc^{2}\Add{.} +\Tag{(44)} +\] +Mass and energy are therefore essentially alike; they are +\index{Equivalence of mass and energy}% +\index{Mass and Energy}% +only different expressions for the same thing. The mass +of a body is not a constant; it varies with changes in its +energy.\footnote + {The emission of energy in radioactive processes is evidently connected + with the fact that the atomic weights are not integers. Attempts have + been made to draw conclusions from this concerning the structure and + stability of the atomic nuclei.} +We see from the last of equations~\Eqref{(43)} that $E$ +becomes infinite when $q$~approaches~$1$, the velocity of +light. If we develop~$E$ in powers of~$q^{2}$, we obtain, +\[ +E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots\Add{.} +\Tag{(45)} +\] +\PageSep{52} +The second term of this expansion corresponds to the +kinetic energy of the material particle in classical +mechanics. + +\Paragraph{Equations of Motion of Material Particles.} From~\Eqref{(43)} +\index{Equations of motion of \Change{materia}{material} particle}% +\index{Motion of particle, equations of}% +we obtain, by differentiating by the time~$l$, and using +the principle of momentum, in the notation of three-dimensional +vectors, +\[ +\K = \frac{d}{dl}\left(\frac{m\q}{\sqrt{1 - q^{2}}}\right)\Add{.} +\Tag{(46)} +\] + +This equation, which was previously employed by +H.~A. Lorentz for the motion of electrons, has been +proved to be true, with great accuracy, by experiments +with $\beta$-rays. + +\Paragraph{Energy Tensor of the Electromagnetic Field.} Before the +\index{Energy@{Energy and mass}!tensor@{tensor of electromagnetic field}}% +development of the theory of relativity it was known +that the principles of energy and momentum could +be expressed in a differential form for the electromagnetic +field. The four-dimensional formulation of +these principles leads to an important conception, that of +the energy tensor, which is important for the further +development of the theory of relativity. + +If in the expression for the $4$-vector of force per unit +volume, +\[ +K_{\mu} = \phi_{\mu\nu} J_{\nu}\Add{,} +\] +using the field equations~\Eqref{(32)}, we express~$J_{\nu}$ in terms of +the field intensities,~$\phi_{\mu\nu}$, we obtain, after some transformations +and repeated application of the field equations +\Eqref{(32)}~and~\Eqref{(33)}, the expression +\[ +K_{\mu} = -\frac{\dd T_{\mu\nu}}{\dd x_{\nu}}\Add{,} +\Tag{(47)} +\] +\PageSep{53} +where we have written\footnote + {To be summed for the indices $\alpha$~and~$\beta$.} +\[ +T_{\mu\nu} + = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + + \phi_{\mu\alpha} \phi_{\nu\alpha}\Add{.} +\Tag{(48)} +\] + +The physical meaning of equation~\Eqref{(47)} becomes evident +if in place of this equation we write, using a new +notation, +\[ +\left. +\begin{alignedat}{4} +\k_{x} &= -\frac{\dd p_{xx}}{\dd x} + &&- \frac{\dd p_{xy}}{\dd y} + &&- \frac{\dd p_{xz}}{\dd z} + &&- \frac{\dd (\Change{\veci}{i} b_{x})}{\dd (\Change{\veci}{i} l)}\Add{,} \\ +% +%[** TN: Added code for next two equations] +\k_{y} &= -\frac{\dd p_{yx}}{\dd x} + &&- \frac{\dd p_{yy}}{\dd y} + &&- \frac{\dd p_{yz}}{\dd z} + &&- \frac{\dd (\Change{\veci}{i} b_{y})}{\dd (\Change{\veci}{i} l)}\Add{,} \\ +% +\k_{z} &= -\frac{\dd p_{zx}}{\dd x} + &&- \frac{\dd p_{zy}}{\dd y} + &&- \frac{\dd p_{zz}}{\dd z} + &&- \frac{\dd (\Change{\veci}{i} b_{z})}{\dd (\Change{\veci}{i} l)}\Add{,} \\ +% +\Change{\veci}{i}\lambda + &= -\frac{\dd (\Change{\veci}{i} \s_{x})}{\dd x} + &&- \frac{\dd (\Change{\veci}{i} \s_{y})}{\dd y} + &&- \frac{\dd (\Change{\veci}{i} \s_{z})}{\dd z} + &&- \frac{\dd (-\eta)}{\dd (\Change{\veci}{i} l)}\Add{;} +\end{alignedat} +\right\} +\Tag{(47a)} +\] +or, on eliminating the imaginary, +\[ +\left. +\begin{alignedat}{4} +\k_{x} &= -\frac{\dd p_{xx}}{\Change{\dd_{x}}{\dd x}} + &&- \frac{\dd p_{xy}}{\dd y} + &&- \frac{\dd p_{xz}}{\dd z} + &&- \frac{\dd b_{x}}{\dd l}\Add{,} \\ +% +%[** TN: Added code for next two equations] +\k_{y} &= -\frac{\dd p_{yx}}{\dd x} + &&- \frac{\dd p_{yy}}{\dd y} + &&- \frac{\dd p_{yz}}{\dd z} + &&- \frac{\dd b_{y}}{\dd l}\Add{,} \\ +% +\k_{z} &= -\frac{\dd p_{zx}}{\dd x} + &&- \frac{\dd p_{zy}}{\dd y} + &&- \frac{\dd p_{zz}}{\dd z} + &&- \frac{\dd b_{z}}{\dd l}\Add{,} \\ +% +\lambda + &= -\frac{\dd \s_{x}}{\dd x} + &&- \frac{\dd \s_{y}}{\dd y} + &&- \frac{\dd \s_{z}}{\dd z} + &&- \frac{\dd \eta}{\dd l}\Add{.} +\end{alignedat} +\right\} +\Tag{(47b)} +\] + +When expressed in the latter form, we see that the +first three equations state the principle of momentum; +$p_{xx}$\Add{,}\ldots\Add{,}~$p_{zx}$ are the Maxwell stresses in the electromagnetic +field, and $(b_{x}, b_{y}, b_{z})$ is the vector momentum +per unit volume of the field. The last of equations~\Eqref{(47b)} +expresses the energy principle; $\s$~is the vector flow of +energy, and $\eta$~the energy per unit volume of the field. +In fact, we get from~\Eqref{(48)} by introducing the well-known +expressions for the components of the field intensity from +electrodynamics, +\PageSep{54} +\[ +\left. +\begin{aligned} +&\begin{alignedat}{4} +p_{xx} = &{} - \h_{x} \h_{x} + &&+ \tfrac{1}{2}({\h_{x}}^{2} &&+ {\h_{y}}^{2} &&+ {\h_{z}}^{2}) \\ + &{} - \e_{x} \e_{\Change{y}{x}} + &&+ \tfrac{1}{2}({\e_{x}}^{2} &&+ {\e_{y}}^{2} &&+ {\e_{z}}^{2})\Add{,} +\end{alignedat} \\ +% +&\qquad\qquad\qquad\qquad\qquad +\begin{alignedat}{3} +p_{xy} = &{} - \h_{x} \h_{y}\quad + && p_{xz} = && {} - \h_{x} \h_{z} \\ + &{} - \e_{x} \e_{y}\Add{,}\quad + && && {} - \e_{x} \e_{z}\Add{,} +\end{alignedat} \\ +% +%**** Hard-coded width +& \makebox[3.5in][c]{$\vdots$} \\ +&b_{x} = \s_{x} = \e_{y}\h_{z} - \e_{z}\h_{y}\Add{,} \\ +%[** TN: Added code for next two equations, elided in the original] +&b_{y} = \s_{y} = \e_{z}\h_{x} - \e_{x}\h_{z}\Add{,} \\ +&b_{z} = \s_{z} = \e_{x}\h_{y} - \e_{y}\h_{x}\Add{,} \\ +% +&\eta = +\tfrac{1}{2}({\e_{x}}^{2} + {\e_{y}}^{2} + {\e_{z}}^{2} + + {\h_{x}}^{2} + {\h_{y}}^{2} + {\h_{z}}^{2})\Add{.} +\end{aligned} +\right\} +\Tag{(48a)} +\] +We conclude from~\Eqref{(48)} that the energy tensor of the +electromagnetic field is symmetrical; with this is connected +the fact that the momentum per unit volume and +the flow of energy are equal to each other (relation +between energy and inertia). + +We therefore conclude from these considerations that +the energy per unit volume has the character of a tensor. +This has been proved directly only for an electromagnetic +field, although we may claim universal validity for it. +Maxwell's equations determine the electromagnetic field +when the distribution of electric charges and currents is +known. But we do not know the laws which govern +the currents and charges. We do know, indeed, that +electricity consists of elementary particles (electrons, +positive nuclei), but from a theoretical point of view we +cannot comprehend this. We do not know the energy +factors which determine the distribution of electricity in +particles of definite size and charge, and all attempts to +complete the theory in this direction have failed. If then +we can build upon Maxwell's equations in general, the +\PageSep{55} +energy tensor of the electromagnetic field is known only +outside the charged particles.\footnote + {It has been attempted to remedy this lack of knowledge by considering + the charged particles as proper singularities. But in my opinion this means + giving up a real understanding of the structure of matter. It seems to me + much better to give in to our present inability rather than to be satisfied + by a solution that is only apparent.} +In these regions, outside +of charged particles, the only regions in which we can +believe that we have the complete expression for the +energy tensor, we have, by~\Eqref{(47)}, +\[ +\frac{\dd T_{\mu\nu}}{\dd x_{\nu}} = 0\Add{.} +\Tag{(47c)} +\] + +\Paragraph{General Expressions for the Conservation Principles.} We +\index{Conservation principles}% +\index{Principles of conservation}% +can hardly avoid making the assumption that in all other +cases, also, the space distribution of energy is given by a +symmetrical tensor,~$T_{\mu\Change{}{\nu}}$, and that this complete energy +tensor everywhere satisfies the relation~\Eqref{(47c)}. At any +rate we shall see that by means of this assumption we +obtain the correct expression for the integral energy +principle. + +Let us consider a spatially bounded, closed system, +which, four-dimensionally, we may represent as a strip, +outside of which the~$T_{\mu\nu}$ vanish. Integrate equation~\Eqref{(47c)} +over a space section. Since the integrals of +$\dfrac{\dd T_{\mu1}}{\dd x_{1}}$, $\dfrac{\dd T_{\mu2}}{\dd x_{2}}$ and $\dfrac{\dd T_{\mu3}}{\dd x_{3}}$ vanish because the~$T_{\mu\nu}$ vanish at the +limits of integration, we obtain +\[ +\frac{\dd}{\dd l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} + = 0\Add{.} +\Tag{(49)} +\] +Inside the parentheses are the expressions for the +\PageSep{56} +momentum of the whole system, multiplied by~$i$, together +with the negative energy of the system, so that \Eqref{(49)} +expresses the conservation principles in their integral +form. That this gives the right conception of energy and +%[Illustration: Fig. 3.] +\Figure{056} +the conservation principles will be seen from the following +considerations. + +\Section{Phenomenological Representation of the +Energy Tensor of Matter.} +\index{Energy@{Energy and mass}!tensor@{tensor of electromagnetic field}!of matter}% + +\Paragraph{Hydrodynamical Equations.} We know that matter is +\index{Hydrodynamical equations}% +built up of electrically charged particles, but we do not +\PageSep{57} +know the laws which govern the constitution of these +particles. In treating mechanical problems, we are therefore +obliged to make use of an inexact description of +matter, which corresponds to that of classical mechanics. +The density~$\sigma$, of a material substance and the hydrodynamical +pressures are the fundamental concepts upon +which such a description is based. + +Let~$\sigma_{0}$ be the density of matter at a place, estimated +with reference to a system of co-ordinates moving with +the matter. Then~$\sigma_{0}$, the density at rest, is an invariant. +If we think of the matter in arbitrary motion and neglect +the pressures (particles of dust \textit{in vacuo}, neglecting the +size of the particles and the temperature), then the energy +tensor will depend only upon the velocity components, +$u_{\nu}$~and~$\sigma_{0}$. We secure the tensor character of~$T_{\mu\nu}$ by +putting +\[ +T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}\Add{,} +\Tag{(50)} +\] +in which the~$u_{\mu}$, in the three-dimensional representation, +are given by~\Eqref{(41)}. In fact, it follows from~\Eqref{(50)} that for +$q = 0$, $T_{44} = -\sigma_{0}$ (equal to the negative energy per unit +volume), as it should, according to the theorem of the +equivalence of mass and energy, and according to the +physical interpretation of the energy tensor given above. +If an external force (four-dimensional vector,~$K_{\mu}$) acts +upon the matter, by the principles of momentum and +energy the equation +\[ +K_{\mu} = \frac{\dd T_{\mu\nu}}{\dd x_{\nu}} +\] +\PageSep{58} +must hold. We shall now show that this equation leads +to the same law of motion of a material particle as that +already obtained. Let us imagine the matter to be of +infinitely small extent in space, that is, a four-dimensional +thread; then by integration over the whole thread with +respect to the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, we obtain +\begin{align*} +%[** TN: Moved equals sign to the second line] +\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} + &= \int \frac{\dd T_{14}}{\dd x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ + &= -\Change{\veci}{i} \frac{d}{dl}\left\{ + \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} + \right\}. +\end{align*} + +Now $\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$ is an invariant, as is, therefore, also +$\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$. We shall calculate this integral, first +with respect to the inertial system which we have chosen, +and second, with respect to a system relatively to which +the matter has the velocity zero. The integration is to +be extended over a filament of the thread for which $\sigma_{0}$ +may be regarded as constant over the whole section. If +the space volumes of the filament referred to the two +systems are $dV$~and~$dV_{0}$ respectively, then we have +\[ +\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau +\] +and therefore also +\[ +\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} + = \int dm\, \Change{\veci}{i}\, \frac{d\tau}{dx_{4}}. +\] + +If we substitute the right-hand side for the left-hand +side in the former integral, and put $\dfrac{dx_{1}}{d\tau}$ outside the sign +\PageSep{59} +of integration, we obtain, +\[ +\K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) \\ + = \frac{d}{dl}\left(\frac{m\q_{x}}{\sqrt{1 - q^{2}}}\right). +\] +We see, therefore, that the generalized conception of the +energy tensor is in agreement with our former result. + +\Paragraph{The Eulerian Equations for Perfect Fluids.} In order +to get nearer to the behaviour of real matter we must add +to the energy tensor a term which corresponds to the +pressures. The simplest case is that of a perfect fluid in +which the pressure is determined by a scalar~$p$. Since +the tangential stresses $p_{xy}$,~etc., vanish in this case, the +contribution to the energy tensor must be of the form~$p\delta_{\nu\mu}$. +We must therefore put +\[ +T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}\Add{.} +\Tag{(51)} +\] +At rest, the density of the matter, or the energy per unit +volume, is in this case, not~$\sigma$ but~$\sigma - p$. For +\[ +-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} + = \sigma - p. +\] +In the absence of any force, we have +\[ +\frac{\dd T_{\mu\nu}}{\dd x_{\nu}} + = \sigma u_{\nu} \frac{\dd u_{\mu}}{\dd x_{\nu}} + + u_{\mu} \frac{\dd (\sigma u_{\nu})}{\dd x_{\nu}} + + \frac{\dd p}{\dd x_{\mu}} = 0. +\] +If we multiply this equation by~$u_{\mu}$ $\left(= \dfrac{dx_{\mu}}{d\tau}\right)$ and sum for +the $\mu$'s~we obtain, using~\Eqref{(40)}, +\[ +-\frac{\dd (\sigma u_{\nu})}{\dd x_{\nu}} + \frac{dp}{d\tau} = 0\Add{,} +\Tag{(52)} +\] +\PageSep{60} +where we have put $\dfrac{\dd p}{\dd x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}$. This is the equation of +continuity, which differs from that of classical mechanics +by the term~$\dfrac{dp}{d\tau}$, which, practically, is vanishingly small. +Observing~\Eqref{(52)}, the conservation principles take the form +\[ +\sigma \frac{du_{\mu}}{d\tau} + + u_{\mu} \frac{dp}{d\tau} + + \frac{\dd p}{\dd x_{\mu}} = 0\Add{.} +\Tag{(53)} +\] +The equations for the first three indices evidently correspond +to the Eulerian equations. That the equations +\Eqref{(52)}~and~\Eqref{(53)} correspond, to a first approximation, to the +hydrodynamical equations of classical mechanics, is a +further confirmation of the generalized energy principle. +The density of matter and of energy has the character of +a symmetrical tensor. +\PageSep{61} + + +\Lecture{III}{The General Theory of Relativity}{The General Theory} + +\First{All} of the previous considerations have been based +upon the assumption that all inertial systems are +equivalent for the description of physical phenomena, but +that they are preferred, for the formulation of the laws +of nature, to spaces of reference in a different state of +motion. We can think of no cause for this preference +for definite states of motion to all others, according to +our previous considerations, either in the perceptible +bodies or in the concept of motion; on the contrary, it +must be regarded as an independent property of the +space-time continuum. The principle of inertia, in +particular, seems to compel us to ascribe physically +objective properties to the space-time continuum. Just +as it was necessary from the Newtonian standpoint to +make both the statements, \textit{tempus est absolutum}, \textit{spatium +est absolutum}, so from the standpoint of the special theory +of relativity we must say, \textit{continuum spatii et temporis est +absolutum}. In this latter statement \textit{absolutum} means not +only ``physically real,'' but also ``independent in its +physical properties, having a physical effect, but not itself +influenced by physical conditions.'' + +As long as the principle of inertia is regarded as the +\PageSep{62} +keystone of physics, this standpoint is certainly the only +one which is justified. But there are two serious criticisms +of the ordinary conception. In the first place, it is contrary +to the mode of thinking in science to conceive of a thing +(the space-time continuum) which acts itself, but which +cannot be acted upon. This is the reason why E.~Mach +\index{Mach}% +was led to make the attempt to eliminate space as an +active cause in the system of mechanics. According to +him, a material particle does not move in unaccelerated +motion relatively to space, but relatively to the centre of +all the other masses in the universe; in this way the +series of causes of mechanical phenomena was closed, in +contrast to the mechanics of Newton and Galileo. In +order to develop this idea within the limits of the modern +theory of action through a medium, the properties of +the space-time continuum which determine inertia must +be regarded as field properties of space, analogous to +the electromagnetic field. The concepts of classical +mechanics afford no way of expressing this. For this +reason Mach's attempt at a solution failed for the time +being. We shall come back to this point of view later. +In the second place, classical mechanics indicates a +limitation which directly demands an extension of the +principle of relativity to spaces of reference which are not +in uniform motion relatively to each other. The ratio of +the masses of two bodies is defined in mechanics in two +ways which differ from each other fundamentally; in the +first place, as the reciprocal ratio of the accelerations +which the same motional force imparts to them (inert +mass), and in the second place, as the ratio of the forces +\PageSep{63} +which act upon them in the same gravitational field +(gravitational mass). The equality of these two masses, +\index{Gravitational mass}% +\index{Inert and gravitational mass, equality of}% +\index{Mass and Energy!equality of gravitational and inert}% +\index{Mass and Energy!gravitational}% +so differently defined, is a fact which is confirmed by +experiments of very high accuracy (experiments of Eötvös), +and classical mechanics offers no explanation for this +equality. It is, however, clear that science is fully justified +in assigning such a numerical equality only after this +numerical equality is reduced to an equality of the real +nature of the two concepts. + +That this object may actually be attained by an extension +of the principle of relativity, follows from the following +consideration. A little reflection will show that the +theorem of the equality of the inert and the gravitational +mass is equivalent to the theorem that the acceleration +imparted to a body by a gravitational field is independent +of the nature of the body. For Newton's equation of +motion in a gravitational field, written out in full, is +\begin{multline*} +(\text{Inert mass})·(\text{Acceleration}) = (\text{Intensity of the} \\ + \text{gravitational field})·(\text{Gravitational mass}). +\end{multline*} +It is only when there is numerical equality between the +inert and gravitational mass that the acceleration is independent +of the nature of the body. Let now $K$~be an +inertial system. Masses which are sufficiently far from +each other and from other bodies are then, with respect +to~$K$, free from acceleration. We shall also refer these +masses to a system of co-ordinates~$K'$, uniformly accelerated +with respect to~$K$. Relatively to~$K'$ all the masses +have equal and parallel accelerations; with respect to~$K'$ +they behave just as if a gravitational field were present and +\PageSep{64} +$K'$~were unaccelerated. Overlooking for the present the +question as to the ``cause'' of such a gravitational field, +which will occupy us later, there is nothing to prevent +our conceiving this gravitational field as real, that is, the +conception that $K'$~is ``at rest'' and a gravitational field +is present we may consider as equivalent to the conception +that only $K$~is an ``allowable'' system of co-ordinates +and no gravitational field is present. The assumption of +the complete physical equivalence of the systems of co-ordinates, +$K$~and~$K'$, we call the ``principle of equivalence;'' +\index{Principle@{Principle of equivalence}}% +this principle is evidently intimately connected +with the theorem of the equality between the inert and +the gravitational mass, and signifies an extension of the +principle of relativity to co-ordinate systems which are +in non-uniform motion relatively to each other. In fact, +through this conception we arrive at the unity of the +nature of inertia and gravitation. For according to our +way of looking at it, the same masses may appear to be +either under the action of inertia alone (with respect to~$K$) +or under the combined action of inertia and gravitation +(with respect to~$K'$). The possibility of explaining +the numerical equality of inertia and gravitation by the +unity of their nature gives to the general theory of +relativity, according to my conviction, such a superiority +over the conceptions of classical mechanics, that all the +difficulties encountered in development must be considered +as small in comparison. + +What justifies us in dispensing with the preference +for inertial systems over all other co-ordinate systems, a +preference that seems so securely established by experiment +\PageSep{65} +based upon the principle of inertia? The weakness +\index{Criticism of principle of inertia}% +\index{Principle@{Principle of equivalence}!inertia, criticism of}% +of the principle of inertia lies in this, that it involves an +argument in a circle: a mass moves without acceleration +if it is sufficiently far from other bodies; we know that +it is sufficiently far from other bodies only by the fact +that it moves without acceleration. Are there, in general, +any inertial systems for very extended portions of the +space-time continuum, or, indeed, for the whole universe? +We may look upon the principle of inertia as established, +to a high degree of approximation, for the space of our +planetary system, provided that we neglect the perturbations +due to the sun and planets. Stated more exactly, +there are finite regions, where, with respect to a suitably +chosen space of reference, material particles move freely +without acceleration, and in which the laws of the special +theory of relativity, which have been developed above, +hold with remarkable accuracy. Such regions we shall +call ``Galilean regions.'' We shall proceed from the +\index{Galilean@{Galilean regions}}% +consideration of such regions as a special case of known +properties. + +The principle of equivalence demands that in dealing +with Galilean regions we may equally well make use of +non-inertial systems, that is, such co-ordinate systems as, +relatively to inertial systems, are not free from acceleration +and rotation. If, further, we are going to do away +completely with the difficult question as to the objective +reason for the preference of certain systems of co-ordinates, +then we must allow the use of arbitrarily moving systems +of co-ordinates. As soon as we make this attempt seriously +\PageSep{66} +\index{Rotation}% +we come into conflict with that physical interpretation of +space and time to which we were led by the special theory +of relativity. For let $K'$ be a system of co-ordinates whose +$z'$-axis coincides with the $z$-axis of~$K$, and which rotates +about the latter axis with constant angular velocity. Are +the configurations of rigid bodies, at rest relatively to~$K'$, +in accordance with the laws of Euclidean geometry? +Since $K'$~is not an inertial system, we do not know +directly the laws of configuration of rigid bodies with +respect to~$K'$, nor the laws of nature, in general. But +we do know these laws with respect to the inertial system~$K$, +and we can therefore estimate them with respect to~$K'$. +Imagine a circle drawn about the origin in the \Change{$x'y'$}{$x'$-$y'$}~plane +of~$K'$, and a diameter of this circle. Imagine, further, that +we have given a large number of rigid rods, all equal to +each other. We suppose these laid in series along the +periphery and the diameter of the circle, at rest relatively +to~$K'$. If $U$~is the number of these rods along the periphery, +$D$~the number along the diameter, then, if $K'$~does +not rotate relatively to~$K$, we shall have +\[ +\frac{U}{D} = \pi. +\] +But if $K'$~rotates we get a different result. Suppose +that at a definite time~$t$\Change{,}{} of~$K$ we determine the ends of +all the rods. With respect to~$K$ all the rods upon the +periphery experience the Lorentz contraction, but the +rods upon the diameter do not experience this contraction +\PageSep{67} +(along their lengths!).\footnote + {These considerations assume that the behaviour of rods and clocks + depends only upon velocities, and not upon accelerations, or, at least, that + the influence of acceleration does not counteract that of velocity.} +It therefore follows that +\[ +\frac{U}{D} > \pi. +\] + +It therefore follows that the laws of configuration of +rigid bodies with respect to~$K'$ do not agree with the +laws of configuration of rigid bodies that are in accordance +with Euclidean geometry. If, further, we place two +similar clocks (rotating with~$K'$), one upon the periphery, +and the other at the centre of the circle, then, judged +from~$K$, the clock on the periphery will go slower than +the clock at the centre. The same thing must take place, +judged from~$K'$, if we define time with respect to~$K'$ in +a not wholly unnatural way, that is, in such a way that +the laws with respect to~$K'$ depend explicitly upon the +time. Space and time, therefore, cannot be defined +with respect to~$K'$ as they were in the special theory of +relativity with respect to inertial systems. But, according +to the principle of equivalence, $K'$~is also to be considered +as a system at rest, with respect to which there +is a gravitational field (field of centrifugal force, and +\index{Centrifugal force}% +force of Coriolis). We therefore arrive at the result: +the gravitational field influences and even determines the +metrical laws of the space-time continuum. If the laws +of configuration of ideal rigid bodies are to be expressed +geometrically, then in the presence of a gravitational +field the geometry is not Euclidean. +\PageSep{68} + +The case that we have been considering is analogous +to that which is presented in the two-dimensional treatment +of surfaces. It is impossible in the latter case +also, to introduce co-ordinates on a surface (e.g.~the +surface of an ellipsoid) which have a simple metrical +significance, while on a plane the Cartesian co-ordinates, +$x_{1}$,~$x_{2}$, signify directly lengths measured by a unit +measuring rod. Gauss overcame this difficulty, in his +\index{Gauss}% +theory of surfaces, by introducing curvilinear co-ordinates +\index{Curvilinear co-ordinates}% +which, apart from satisfying conditions of continuity, +were wholly arbitrary, and afterwards these co-ordinates +were related to the metrical properties of the surface. +In an analogous way we shall introduce in the general +theory of relativity arbitrary co-ordinates, $x_{1}$,~$x_{2}$,~$x_{3}$,~$x_{4}$, +which shall number uniquely the space-time points, so +that neighbouring events are associated with neighbouring +values of the co-ordinates; otherwise, the choice of +co-ordinates is arbitrary. We shall be true to the +principle of relativity in its broadest sense if we give +such a form to the laws that they are valid in every +such four-dimensional system of co-ordinates, that is, if +the equations expressing the laws are co-variant with +respect to arbitrary transformations. + +The most important point of contact between Gauss's +theory of surfaces and the general theory of relativity +lies in the metrical properties upon which the concepts +of both theories, in the main, are based. In the case +of the theory of surfaces, Gauss's argument is as follows. +Plane geometry may be based upon the concept of the +distance~$ds$, between two indefinitely near points. The +\PageSep{69} +concept of this distance is physically significant because +the distance can be measured directly by means of a +rigid measuring rod. By a suitable choice of Cartesian +co-ordinates this distance may be expressed by the +formula $ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}$. We may base upon this +quantity the concepts of the straight line as the geodesic +($\delta\! \int\!\! ds = 0$), the interval, the circle, and the angle, upon +which the Euclidean plane geometry is built. A +geometry may be developed upon another continuously +curved surface, if we observe that an infinitesimally +small portion of the surface may be regarded as plane, +to within relatively infinitesimal quantities. There are +Cartesian co-ordinates, $X_{1}$,~$X_{2}$, upon such a small +portion of the surface, and the distance between two +points, measured by a measuring rod, is given by +\[ +ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}. +\] +If we introduce arbitrary curvilinear co-ordinates, $x_{1}$,~$x_{2}$, +on the surface, then $dX_{1}$,~$dX_{2}$, may be expressed linearly +in terms of $dx_{1}$,~$dx_{2}$. Then everywhere upon the surface +we have +\[ +ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + + g_{22}\, {dx_{2}}^{2}\Add{,} +\] +where $g_{11}$,~$g_{12}$,~$g_{22}$ are determined by the nature of the +surface and the choice of co-ordinates; if these quantities +are known, then it is also known how networks of rigid +rods may be laid upon the surface. In other words, the +geometry of surfaces may be based upon this expression +for~$ds^{2}$ exactly as plane geometry is based upon the +corresponding expression. + +There are analogous relations in the four-dimensional +\PageSep{70} +space-time continuum of physics. In the immediate +neighbourhood of an observer, falling freely in a gravitational +field, there exists no gravitational field. We +can therefore always regard an infinitesimally small +region of the space-time continuum as Galilean. For +such an infinitely small region there will be an inertial +system (with the space co-ordinates, $X_{1}$,~$X_{2}$,~$X_{3}$, and the +time co-ordinate~$X_{4}$) relatively to which we are to regard +the laws of the special theory of relativity as valid. The +quantity which is directly measurable by our unit +measuring rods and clocks, +\[ +{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2}\Add{,} +\] +or its negative, +\[ +ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}\Add{,} +\Tag{(54)} +\] +is therefore a uniquely determinate invariant for two +neighbouring events (points in the four-dimensional +continuum), provided that we use measuring rods that +are equal to each other when brought together and +superimposed, and clocks whose rates are the same +when they are brought together. In this the physical +assumption is essential that the relative lengths of two +measuring rods and the relative rates of two clocks are +independent, in principle, of their previous history. But +this assumption is certainly warranted by experience; +if it did not hold there could be no sharp spectral lines; +for the single atoms of the same element certainly do +not have the same history, and it would be absurd to +suppose any relative difference in the structure of the +\PageSep{71} +single atoms due to their previous history if the mass +and frequencies of the single atoms of the same element +were always the same. + +Space-time regions of finite extent are, in general, +not Galilean, so that a gravitational field cannot be done +away with by any choice of co-ordinates in a finite +region. There is, therefore, no choice of co-ordinates +for which the metrical relations of the special theory of +relativity hold in a finite region. But the invariant~$ds$ +always exists for two neighbouring points (events) of +the continuum. This invariant~$ds$ may be expressed in +arbitrary co-ordinates. If one observes that the local~$dX_{\nu}$ +may be expressed linearly in terms of the co-ordinate +differentials~$dx_{\nu}$, $ds^{2}$~may be expressed in the +form +\[ +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{.} +\Tag{(55)} +\] + +The functions~$g_{\mu\nu}$ describe, with respect to the arbitrarily +chosen system of co-ordinates, the metrical relations +of the space-time continuum and also the +gravitational field. As in the special theory of relativity, +we have to discriminate between time-like and space-like +line elements in the four-dimensional continuum; +owing to the change of sign introduced, time-like +line elements have a real, space-like line elements an +imaginary~$ds$. The time-like~$ds$ can be measured directly +by a suitably chosen clock. + +According to what has been said, it is evident that +the formulation of the general theory of relativity +assumes a generalization of the theory of invariants and +the theory of tensors; the question is raised as to the +\PageSep{72} +\index{Tensor|etseq}% +form of the equations which are co-variant with respect +\index{Covariant@{Co-variant}!vector}% +to arbitrary point transformations. The generalized +calculus of tensors was developed by mathematicians +long before the theory of relativity. Riemann first +\index{Riemann}% +extended Gauss's train of thought to continua of any +number of dimensions; with prophetic vision he saw +the physical meaning of this generalization of Euclid's +geometry. Then followed the development of the theory +in the form of the calculus of tensors, particularly by +Ricci and Levi-Civita. This is the place for a brief +presentation of the most important mathematical concepts +and operations of this calculus of tensors. + +We designate four quantities, which are defined as +functions of the~$x_{\nu}$ with respect to every system of co-ordinates, +as components,~$A^{\nu}$, of a contra-variant vector, +\index{Contravariant@{Contra-variant vectors}}% +\index{Vector@{Vector, co-variant}!contra-variant}% +if they transform in a change of co-ordinates as the co-ordinate +differentials~$dx_{\nu}$. We therefore have +\[ +%[** TN: Place prime directly on the tensor name throughout, for consistency] +\Change{A^{\mu'}}{{A'}^{\mu}} = \frac{{\dd x'}_{\mu}}{\dd x_{\nu}} A^{\nu}\Add{.} +\Tag{(56)} +\] +Besides these contra-variant vectors, there are also co-variant +vectors. If $B_{\nu}$~are the components of a co-variant +vector, these vectors are transformed according to the +rule +\[ +{B'}_{\mu} = \frac{\dd x_{\nu}}{{\dd \Change{x}{x'}}_{\mu}} B_{\nu}\Add{.} +\Tag{(57)} +\] +The definition of a co-variant vector is chosen in such a +\index{Vector@{Vector, co-variant}}% +way that a co-variant vector and a contra-variant vector +together form a scalar according to the scheme, +\[ +\phi = B_{\nu} A^{\nu}\quad \text{(summed over the~$\nu$).} +\] +\PageSep{73} +Accordingly, +\[ +{B'}_{\mu} \Change{A^{\mu'}}{{A'}^{\mu}} + = \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}} + \frac{{\dd x'}_{\mu}}{\dd x_{\beta}} B_{\alpha} A^{\beta} + = B_{\alpha} A^{\alpha}\Add{.} +\] +In particular, the derivatives~$\dfrac{\dd \phi}{\dd x_{\alpha}}$ of a scalar~$\phi$, are components +of a co-variant vector, which, with the co-ordinate +differentials, form the scalar~$\dfrac{\dd \phi}{\dd x_{\alpha}}\, dx_{\alpha}$; we see from this +example how natural is the definition of the co-variant +vectors. + +There are here, also, tensors of any rank, which may +have co-variant or contra-variant character with respect +to each index; as with vectors, the character is designated +by the position of the index. For example, $\Change{{A_{\mu}}^{\nu}}{A_{\mu}^{\nu}}$~denotes +a tensor of the second rank, which is co-variant +with respect to the index~$\mu$, and contra-variant with respect +to the index~$\nu$. The tensor character indicates +that the equation of transformation is +\[ +\Change{A_{\mu}^{\nu'}}{{A'}_{\mu}^{\nu}} + = \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}} + \frac{{\dd x'}_{\nu}}{\dd x_{\beta}} A_{\alpha}^{\beta}\Add{.} +\Tag{(58)} +\] + +Tensors may be formed by the addition and subtraction +of tensors of equal rank and like character, as in the +theory of invariants of orthogonal linear substitutions, for +example, +\[ +A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}\Add{.} +\Tag{(59)} +\] +The proof of the tensor character of~$C_{\mu}^{\nu}$ depends upon~\Eqref{(58)}. + +Tensors may be formed by multiplication, keeping the +character of the indices, just as in the theory of invariants +of linear orthogonal transformations, for example, +\[ +A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}\Add{.} +\Tag{(60)} +\] +\PageSep{74} +The proof follows directly from the rule of transformation. + +Tensors may be formed by contraction with respect to +two indices of different character, for example, +\[ +A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}\Add{.} +\Tag{(61)} +\] +The tensor character of~$A_{\mu\sigma\tau}^{\mu}$ determines the tensor +character of~$B_{\sigma\tau}$. Proof--- +\[ +\Change{A_{\mu\sigma\tau}^{\mu'}}{{A'}_{\mu\sigma\tau}^{\mu}} += \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}} \frac{{\dd x'}_{\mu}}{\dd x_{\beta}} + \frac{\dd x_{s}}{{\dd x'}_{\sigma}} \frac{\dd x_{t}}{{\dd x'}_{\tau}} + \Change{}{A_{\alpha st}^{\beta}} += \frac{\dd x_{s}}{{\dd x'}_{\sigma}} \frac{\dd x_{t}}{{\dd x'}_{\tau}} + A_{\alpha st}^{\alpha}. +\] + +The properties of symmetry and skew-symmetry of a +tensor with respect to two indices of like character have +the same significance as in the theory of invariants. + +With this, everything essential has been said with +regard to the algebraic properties of tensors. + +\Paragraph{The Fundamental Tensor.} It follows from the invariance +\index{Fundamental tensor}% +\index{Tensor!Fundamental}% +of~$ds^{2}$ for an arbitrary choice of the~$dx_{\nu}$, in connexion +with the condition of symmetry consistent with~\Eqref{(55)}, that +the~$g_{\mu\nu}$ are components of a symmetrical co-variant tensor +(Fundamental Tensor). Let us form the determinant,~$g$, +of the~$g_{\mu\nu}$, and also the minors, divided by~$g$, corresponding +to the single~$g_{\mu\nu}$. These minors, divided by~$g$, +will be denoted by~$g^{\mu\nu}$, and their co-variant character +is not yet known. Then we have +\[ +g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} +%[** TN: No brace in original] + = \begin{cases} + 1 & \text{if $\alpha = \beta$\Add{,}} \\ + 0 & \text{if $\alpha \neq \beta$\Add{.}} + \end{cases} +\Tag{(62)} +\] + +If we form the infinitely small quantities (co-variant +vectors) +\[ +d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}\Add{,} +\Tag{(63)} +\] +\PageSep{75} +multiply by~$g^{\mu\beta}$ and sum over the~$\mu$, we obtain, by the +use of~\Eqref{(62)}, +\[ +dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}\Add{.} +\Tag{(64)} +\] +Since the ratios of the~$d\xi_{\mu}$ are arbitrary, and the~$dx_{\beta}$ as +well as the~$dx_{\mu}$ are components of vectors, it follows that +\index{Contravariant@{Contra-variant vectors}!tensors}% +the~$g^{\mu\nu}$ are the components of a contra-variant tensor\footnote + {If we multiply~\Eqref{(64)} by~$\dfrac{{\dd x'}_{\alpha}}{\dd x_{\beta}}$, sum over the~$\beta$, and replace the~$d\xi^{\mu}$ by a + transformation to the accented system, we obtain + \[ + {dx'}_{\alpha} + = \frac{{\dd x'}_{\sigma}}{\dd x_{\mu}}\, + \frac{{\dd x'}_{\alpha}}{\dd x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}. + \] + The statement made above follows from this, since, by~\Eqref{(64)}, we must also + have ${dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}$, and both equations must hold for every choice of the~${d\xi'}_{\sigma}$.} +(contra-variant fundamental tensor). The tensor character +of~$\delta_{\alpha}^{\beta}$ (mixed fundamental tensor) accordingly follows, +by~\Eqref{(62)}. By means of the fundamental tensor, instead +of tensors with co-variant index character, we can +introduce tensors with contra-variant index character, +and conversely. For example, +\begin{align*} +A^{\mu} &= g^{\mu\alpha} A_{\alpha}\Add{,} \\ +A_{\mu} &= g_{\mu\alpha} A^{\alpha}\Add{,} \\ +T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. +\end{align*} + +\Paragraph{Volume Invariants.} The volume element +\[ +\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx +\] +is not an invariant. For by Jacobi's theorem, +\[ +dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx\Add{.} +\Tag{(65)} +\] +\PageSep{76} +But we can complement~$dx$ so that it becomes an invariant. +If we form the determinant of the quantities +\[ +{g'}_{\mu\nu} + = \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}}\, + \frac{\dd x_{\beta}}{{\dd x'}_{\nu}}\, g_{\alpha\beta}\Add{,} +\] +we obtain, by a double application of the theorem of +multiplication of determinants, +\[ +g' = |{g'}_{\mu\nu}| + = \left|\frac{\dd x_{\nu}}{{\dd x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| + = \left|\frac{{\dd x'}_{\mu}}{\dd x_{\nu}}\right|^{-2} g. +\Change{}{\Tag{(66)}} +\] +We therefore get the invariant, +\[ +\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx. +\] + +\Paragraph{Formation of Tensors by Differentiation.} Although +\index{Differentiation of tensors}% +\index{Tensors, formation by differentiation}% +the algebraic operations of tensor formation have proved +to be as simple as in the special case of invariance with +respect to linear orthogonal transformations, nevertheless +in the general case, the invariant differential operations +are, unfortunately, considerably more complicated. The +reason for this is as follows. If $A^{\mu}$~is a contra-variant +vector, the coefficients of its transformation,~$\dfrac{{\dd x'}_{\mu}}{\dd x_{\nu}}$, are independent +of position only if the transformation is a linear +one. For then the vector components, $A^{\mu} + \dfrac{\dd A^{\mu}}{\dd x_{\alpha}}\, dx_{\alpha}$, at +a neighbouring point transform in the same way as the~$A^{\mu}$, +from which follows the vector character of the vector +differentials, and the tensor character of~$\dfrac{\dd A^{\mu}}{\dd x_{\alpha}}$. But if the +$\dfrac{{\dd x'}_{\mu}}{\dd x_{\nu}}$ are variable this is no longer true. +\PageSep{77} +\index{Hypotheses of pre-relativity physics}% + +That there are, nevertheless, in the general case, invariant +differential operations for tensors, is recognized +most satisfactorily in the following way, introduced by +Levi-Civita and Weyl. Let $(A^{\mu})$ be a contra-variant vector +\index{Levi-Civita}% +\index{Weyl}% +whose components are given with respect to the co-ordinate +system of the~$x_{\nu}$. Let $P_{1}$~and~$P_{2}$ be two infinitesimally +near points of the continuum. For the +infinitesimal region surrounding the point~$P_{1}$, there is, +according to our way of considering the matter, a co-ordinate +system of the~$X_{\nu}$ (with imaginary $X_{\Change{\nu}{4}}$-\Change{co-ordinates}{co-ordinate}) +for which the continuum is Euclidean. Let +$A_{(1)}^{\mu}$~be the co-ordinates of the vector at the point~$P_{1}$. +Imagine a vector drawn at the point~$P_{2}$, using the local +system of the~$X_{\nu}$, with the same co-ordinates (parallel +vector through~$P_{2}$), then this parallel vector is uniquely +determined by the vector at~$P_{1}$ and the displacement. +We designate this operation, whose uniqueness will appear +in the sequel, the parallel displacement of the vector~$A_{\mu}$ +from~$P_{1}$ to the infinitesimally near point~$P_{2}$\Change{}{.} If we form +the vector difference of the vector~$(A^{\mu})$ at the point~$P_{2}$ +and the vector obtained by parallel displacement from~$P_{1}$ +to~$P_{2}$, we get a vector which may be regarded as the +differential of the vector~$(A^{\mu})$ for the given displacement~$(dx_{\nu})$. + +This vector displacement can naturally also be considered +with respect to the co-ordinate system of the~$x_{\nu}$. +If $A^{\nu}$~are the co-ordinates of the vector at~$P_{1}$, $A^{\nu} + \delta A^{\nu}$ +the co-ordinates of the vector displaced to~$P_{2}$ along the +interval~$(dx_{\nu})$, then the~$\delta A^{\nu}$ do not vanish in this case. +We know of these quantities, which do not have a vector +\PageSep{78} +character, that they must depend linearly and homogeneously +upon the~$dx_{\nu}$ and the~$A^{\nu}$. We therefore put +\[ +\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}\Add{.} +\Tag{(67)} +\] + +In addition, we can state that the~$\Gamma_{\alpha\beta}^{\nu}$ must be symmetrical +with respect to the indices $\alpha$~and~$\beta$. For we +can assume from a representation by the aid of a Euclidean +system of local co-ordinates that the same parallelogram +will be described by the displacement of an element~$d^{(1)}x_{\nu}$ +along a second element~$d^{(2)}x_{\nu}$ as by a displacement +of~$d^{(2)}x_{\nu}$ along~$d^{(1)}x_{\nu}$. We must therefore have +\begin{multline*} +d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ + = d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). +\end{multline*} +The statement made above follows from this, after interchanging +the indices of summation, $\alpha$~and~$\beta$, on the +right-hand side. + +Since the quantities~$g_{\mu\nu}$ determine all the metrical +properties of the continuum, they must also determine +the~$\Gamma_{\alpha\beta}^{\nu}$. If we consider the invariant of the vector~$A^{\nu}$, +that is, the square of its magnitude, +\[ +g_{\mu\nu} A^{\mu} A^{\nu}\Add{,} +\] +which is an invariant, this cannot change in a parallel +displacement. We therefore have +\[ +0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) + = \frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu} +\] +or, by~\Eqref{(67)}, +\[ +\left(\frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} + - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} + - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} += 0. +\] +\PageSep{79} + +Owing to the symmetry of the expression in the +brackets with respect to the indices $\mu$~and~$\nu$, this equation +can be valid for an arbitrary choice of the vectors $(A^{\mu})$ +and~$dx_{\nu}$ only when the expression in the brackets vanishes +for all combinations of the indices. By a cyclic interchange +of the indices $\mu$,~$\nu$,~$\alpha$, we obtain thus altogether +three equations, from which we obtain, on taking into +account the symmetrical property of the~$\Gamma_{\mu\nu}^{\alpha}$, +\[ +\left[\Chr{\mu}{\nu}{\alpha}\right] + = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}\Add{,} +\Tag{(68)} +\] +in which, following Christoffel, the abbreviation has been +used, +\[ +\left[\Chr{\mu}{\nu}{\alpha}\right] + = \tfrac{1}{2}\left( + \frac{\dd g_{\mu\alpha}}{\dd x_{\nu}} + + \frac{\dd g_{\nu\alpha}}{\dd x_{\mu}} + - \frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} +\right)\Add{.} +\Tag{(69)} +\] + +If we multiply~\Eqref{(68)} by~$g^{\alpha\sigma}$ and sum over the~$\alpha$, we +obtain +\[ +\Gamma_{\mu\nu}^{\alpha} + = \tfrac{1}{2} g^{\sigma\alpha}\left( + \frac{\dd g_{\mu\alpha}}{\dd x_{\nu}} + + \frac{\dd g_{\nu\alpha}}{\dd x_{\mu}} + - \frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} +\right) = \left\{\Chr{\mu}{\nu}{\sigma}\right\}\Add{,} +\Tag{(70)} +\] +in which $\left\{\Chr{\mu}{\nu}{\sigma}\right\}$ is the Christoffel symbol of the second +kind. Thus the quantities~$\Gamma$ are deduced from the~$g_{\mu\nu}$. +Equations \Eqref{(67)}~and~\Eqref{(70)} are the foundation for the +following discussion. + +\Paragraph{Co-variant Differentiation of Tensors.} If $(A^{\mu} + \delta A^{\mu})$ is +\index{Differentiation of tensors}% +the vector resulting from an infinitesimal parallel displacement +from~$P_{1}$ to~$P_{2}$, and $(A^{\mu} + dA^{\mu})$~the vector~$A^{\mu}$ at the +point~$P_{2}$, then the difference of these two, +\[ +dA^{\mu} - \delta A^{\mu} = \left( + \frac{\Change{\delta}{\dd} A^{\mu}}{\Change{\delta}{\dd} x_{\sigma}} + + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma}\Add{,} +\] +\PageSep{80} +is also a vector. Since this is the case for an arbitrary +choice of the~$dx_{\sigma}$, it follows that +\[ +%[** TN: Covariant derivative index not subscripted in original] +{A^{\mu}}_{;\, \sigma} + = \frac{\dd A^{\mu}}{\dd x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} +\Tag{(71)} +\] +is a tensor, which we designate as the co-variant derivative +of the tensor of the first rank (vector). Contracting this +tensor, we obtain the divergence of the contra-variant +tensor~$A^{\mu}$. In this we must observe that according to~\Eqref{(70)}, +\[ +\Gamma_{\mu\sigma}^{\sigma} + = \tfrac{1}{2} g^{\sigma\alpha} \frac{\dd g_{\sigma\alpha}}{\dd x_{\mu}} + = \frac{1}{\sqrt{g}}\, \frac{\dd \sqrt{g}}{\dd x_{\mu}}\Add{.} +\Tag{(72)} +\] +If we put, further, +\[ +A^{\mu} \sqrt{g} = \tA^{\mu}\Add{,} +\Tag{(73)} +\] +a quantity designated by Weyl as the contra-variant tensor +density\footnote + {This expression is justified, in that $A^{\mu}\sqrt{g}\, dx = \tA^{\mu}\, dx$ has a tensor + character. Every tensor, when multiplied by~$\sqrt{g}$, changes into a tensor + density. We employ capital Gothic letters for tensor densities.} +of the first rank, it follows that, +\[ +\tA = \frac{\dd \tA^{\mu}}{\dd x_{\mu}} +\Tag{(74)} +\] +is a scalar density. + +We get the law of parallel displacement for the +co-variant vector~$B_{\mu}$ by stipulating that the parallel +displacement shall be effected in such a way that the +scalar +\[ +\phi = A^{\mu} B_{\mu} +\] +remains unchanged, and that therefore +\[ +A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu} +\] +\PageSep{81} +vanishes for every value assigned to~$(A^{\mu})$. We therefore +get +\[ +\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}\Add{.} +\Tag{(75)} +\] + +From this we arrive at the co-variant derivative of the +co-variant vector by the same process as that which led +to~\Eqref{(71)}, +\[ +%[** TN: Covariant derivative index not subscripted in original] +B_{\mu;\, \sigma} + = \frac{\dd B_{\mu}}{\dd x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}\Add{.} +\Tag{(76)} +\] +By interchanging the indices $\mu$~and~$\beta$, and subtracting, +we get the skew-symmetrical tensor, +\[ +\phi_{\mu\sigma} + = \frac{\dd B_{\mu}}{\dd x_{\sigma}} - \frac{\dd B_{\sigma}}{\dd x_{\mu}}\Add{.} +\Tag{(77)} +\] + +For the co-variant differentiation of tensors of the +second and higher ranks we may use the process by +which \Eqref{(75)}~was deduced. Let, for example, $(A_{\sigma\tau})$~be a +co-variant tensor of the second rank. Then $A_{\sigma\tau} E^{\sigma} F^{\tau}$~is +a scalar, if $E$~and~$F$ are vectors. This expression must +not be changed by the $\delta$-displacement; expressing this +by a formula, we get, using~\Eqref{(67)},~$\delta A_{\sigma\tau}$, whence we get the +desired co-variant derivative, +\[ +A_{\sigma\tau;\, \rho} + = \frac{\dd A_{\sigma\tau}}{\dd x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} + - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}\Add{.} +\Tag{(78)} +\] + +In order that the general law of co-variant differentiation +of tensors may be clearly seen, we shall write +down two co-variant derivatives deduced in an analogous +way: +\begin{align*} +A_{\sigma;\, \rho}^{\tau} + &= \frac{\dd A_{\sigma}^{\tau}}{\dd x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha}\Add{,} +\Tag{(79)} \\ +% +{A^{\sigma\tau}}_{;\, \rho} + &= \frac{\dd A^{\sigma\tau}}{\dd x_{\rho}} + + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}\Add{.} +\Tag{(80)} +\end{align*} +\PageSep{82} +The general law of formation now becomes evident. +From these formulæ we shall deduce some others which +are of interest for the physical applications of the theory. + +In case $A_{\sigma\tau}$~is skew-symmetrical, we obtain the tensor +\[ +A_{\sigma\tau\rho} + = \frac{\dd A_{\sigma\tau}}{\dd x_{\rho}} + + \frac{\dd A_{\tau\rho}}{\dd x_{\sigma}} + + \frac{\dd A_{\rho\sigma}}{\dd x_{\tau}}\Add{,} +\Tag{(81)} +\] +which is skew-symmetrical in all pairs of indices, by cyclic +interchange and addition. + +If, in~\Eqref{(78)}, we replace~$A_{\sigma\tau}$ by the fundamental tensor,~$g_{\sigma\tau}$, +then the right-hand side vanishes identically; an +analogous statement holds for~\Eqref{(80)} with respect to~$g^{\sigma\tau}$; +that is, the co-variant derivatives of the fundamental +tensor vanish. That this must be so we see directly in +the local system of co-ordinates. + +In case $A^{\sigma\tau}$~is skew-symmetrical, we obtain from~\Eqref{(80)}, +by contraction with respect to $\tau$~and~$\rho$, +\[ +\tA^{\sigma} = \frac{\dd \tA^{\sigma\tau}}{\dd x_{\tau}}\Add{.} +\Tag{(82)} +\] + +In the general case, from \Eqref{(79)}~and~\Eqref{(80)}, by contraction +with respect to $\tau$~and~$\rho$, we obtain the equations, +\begin{align*} +\tA_{\sigma} &= \frac{\dd \tA_{\sigma}^{\alpha}}{\dd x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \tA_{\alpha}^{\beta}\Add{,} +\Tag{(83)} \\ +% +\tA^{\sigma} &= \frac{\dd \tA^{\sigma\alpha}}{\dd x_{\alpha}} + + \Gamma_{\alpha\beta}^{\sigma} \tA^{\alpha\beta}\Add{.} +\Tag{(84)} +\end{align*} + +\Paragraph{The Riemann Tensor.} If we have given a curve extending +\index{Riemann!tensor}% +from the point~$P$ to the point~$G$ of the continuum, +then a vector~$A^{\mu}$, given at~$P$, may, by a parallel displacement, +be moved along the curve to~$G$. If the continuum +\PageSep{83} +is Euclidean (more generally, if by a suitable choice of +co-ordinates the~$g_{\mu\nu}$ are constants) then the vector obtained +at~$G$ as a result of this displacement does not depend +upon the choice of the curve joining $P$~and~$G$. But +otherwise, the result depends upon the path of the displacement. +In this case, therefore, a vector suffers a +change,~$\Delta A^{\mu}$ (in its direction, not its magnitude), when it +is carried from a point~$P$ of a closed curve, along the +%[Illustration: Fig. 4.] +\Figure[0.66\textwidth]{083} +curve, and back to~$P$. We shall now calculate this vector +change: +\[ +\Delta A^{\mu} = \Oint \delta A^{\mu}. +\] +As in Stokes' theorem for the line integral of a vector +around a closed curve, this problem may be reduced to +the integration around a closed curve with infinitely small +linear dimensions; we shall limit ourselves to this case. +\PageSep{84} + +We have, first, by~\Eqref{(67)}, +\[ +\Delta A^{\mu} = -\Oint \Gamma_{\alpha\beta}^{\mu} A^{\alpha}\, dx_{\beta}. +\] + +In this, $\Gamma_{\alpha\beta}^{\mu}$~is the value of this quantity at the variable +point~$G$ of the path of integration. If we put +\[ +\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P} +\] +and denote the value of~$\Gamma_{\alpha\beta}^{\mu}$ at~$P$ by~$\Bar{\Gamma_{\alpha\beta}^{\mu}}$, then we have, +with sufficient accuracy, +\[ +\Gamma_{\alpha\beta}^{\mu} + = \Bar{\Gamma_{\alpha\beta}^{\mu}} + + \frac{\Bar{\dd \Gamma_{\alpha\beta}^{\mu}}}{\dd x_{\nu}}\, \xi^{\nu}. +\] + +Let, further, $A^{\alpha}$~be the value obtained from~$\Bar{A^{\alpha}}$ by a +parallel displacement along the curve from~$P$ to~$G$. It +may now easily be proved by means of~\Eqref{(67)} that $A^{\mu} - \Bar{A^{\mu}}$ +is infinitely small of the first order, while, for a curve of +infinitely small dimensions of the first order, $\Delta A^{\mu}$~is +infinitely small of the second order. Therefore there is +an error of only the second order if we put +\[ +A^{\alpha} = \Bar{A^{\alpha}} + - \Bar{\Gamma_{\sigma\tau}^{\alpha}}\; + \Bar{A^{\Change{\alpha}{\sigma}}}\; + \Bar{\xi^{\Change{\beta}{\tau}}}. +\] + +If we introduce these values of $\Gamma_{\alpha\beta}^{\mu}$~and~$A^{\alpha}$ into the +integral, we obtain, neglecting all quantities of a higher +order of small quantities than the second, +\[ +\Delta A^{\mu} + = - \left(\frac{\dd \Gamma_{\sigma\beta}^{\mu}}{\dd x_{\alpha}} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) + A^{\sigma} \Oint \xi^{\alpha}\, d\xi^{\beta}\Add{.} +\Tag{(85)} +\] +The quantity removed from under the sign of integration +\PageSep{85} +refers to the point~$P$. Subtracting~$\frac{1}{2} d(\xi^{\alpha} \xi^{\beta})$ from the +integrand, we obtain +\[ +\tfrac{1}{2} \Oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}). +\] +This skew-symmetrical tensor of the second rank,~$\Change{f_{\alpha\beta}}{f^{\alpha\beta}}$, +characterizes the surface element bounded by the curve +in magnitude and position. If the expression in the +brackets in~\Eqref{(85)} were skew-symmetrical with respect to +the indices $\alpha$~and~$\beta$, we could conclude its tensor character +from~\Eqref{(85)}. We can accomplish this by interchanging +the summation indices $\alpha$~and~$\beta$ in~\Eqref{(85)} and adding the +resulting equation to~\Eqref{(85)}. We obtain +\[ +2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}\Add{,} +\Tag{(86)} +\] +in which +\[ +R_{\sigma\alpha\beta}^{\mu} + = - \frac{\dd \Gamma_{\sigma\alpha}^{\mu}}{\dd x_{\beta}} + + \frac{\dd \Gamma_{\sigma\beta}^{\mu}}{\dd x_{\alpha}} + + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\Add{.} +\Tag{(87)} +\] + +The tensor character of~$R_{\sigma\alpha\beta}^{\mu}$ follows from~\Eqref{(86)}; this is +the Riemann curvature tensor of the fourth rank, whose +\index{Riemann!tensor}% +properties of symmetry we do not need to go into. Its +vanishing is a sufficient condition (disregarding the reality +of the chosen co-ordinates) that the continuum is +Euclidean. + +By contraction of the Riemann tensor with respect to +the indices $\mu$,~$\beta$, we obtain the symmetrical tensor of the +second rank, +\[ +R_{\mu\nu} + = - \frac{\dd \Gamma_{\mu\nu}^{\alpha}}{\dd x_{\alpha}} + + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + + \frac{\dd \Gamma_{\mu\alpha}^{\alpha}}{\dd x_{\nu}} + - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}\Add{.} +\Tag{(88)} +\] +The last two terms vanish if the system of co-ordinates +\PageSep{86} +is so chosen that $g = \text{constant}$. From~$R_{\mu\nu}$ we can form +the scalar, +\[ +R = g^{\mu\nu} R_{\mu\nu}\Add{.} +\Tag{(89)} +\] + +\Paragraph{Straightest \(Geodetic\) Lines.} A line may be constructed +\index{Geodetic lines}% +\index{Straightest lines}% +in such a way that its successive elements arise from each +other by parallel displacements. This is the natural +generalization of the straight line of the Euclidean +geometry. For such a line, we have +\[ +\delta \left(\frac{dx_{\mu}}{ds}\right) + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}. +\] +The left-hand side is to be replaced by~$\dfrac{d^{2} x_{\mu}}{ds^{2}}$,\footnote + {The direction vector at a neighbouring point of the curve results, by a + parallel displacement along the line element~$(dx_{\beta})$, from the direction vector + of each point considered.} +so that we +have +\[ +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0\Add{.} +\Tag{(90)} +\] +We get the same line if we find the line which gives a +stationary value to the integral +\[ +\int ds\quad\text{or}\quad +\int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}} +\] +between two points (geodetic line). +\PageSep{87} + + +\Lecture[ (\textit{continued})]{IV}{The General Theory of Relativity} +{The General Theory} + +\First{We} are now in possession of the mathematical +apparatus which is necessary to formulate the +laws of the general theory of relativity. No attempt +will be made in this presentation at systematic completeness, +but single results and possibilities will be developed +progressively from what is known and from the +results obtained. Such a presentation is most suited +to the present provisional state of our knowledge. + +A material particle upon which no force acts moves, +according to the principle of inertia, uniformly in a +straight line. In the four-dimensional continuum of the +special theory of relativity (with real time co-ordinate) +this is a real straight line. The natural, that is, the +simplest, generalization of the straight line which is +plausible in the system of concepts of Riemann's general +theory of invariants is that of the straightest, or geodetic, +line. We shall accordingly have to assume, in the sense +of the principle of equivalence, that the motion of a +material particle, under the action only of inertia and +gravitation, is described by the equation, +\[ +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0\Add{.} +\tag*{(90)} %[** TN: Repeated from previous lecture] +\] +\PageSep{88} +In fact, this equation reduces to that of a straight line +if all the components,~$\Gamma_{\alpha\beta}^{\mu}$, of the gravitational field +vanish. + +How are these equations connected with Newton's +equations of motion? According to the special theory +of relativity, the~$g_{\mu\nu}$ as well as the~$g^{\mu\nu}$, have the values, +with respect to an inertial system (with real time co-ordinate +and suitable choice of the sign of~$ds^{2}$), +\[ +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & \Neg[]1 +\end{array} +\right\}\Add{.} +\Tag{(91)} +\] +The equations of motion then become +\[ +\frac{d^{2} x_{\mu}}{ds^{2}} = 0. +\] +We shall call this the ``first approximation'' to the $g_{\mu\nu}$-field. +In considering approximations it is often useful, +as in the special theory of relativity, to use an imaginary +$x_{4}$-co-ordinate, as then the~$g_{\mu\nu}$, to the first approximation, +assume the values +\[ +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & -1 +\end{array} +\right\}\Add{.} +\Tag{(91a)} +\] +These values may be collected in the relation +\[ +g_{\mu\nu} = -\delta_{\mu\nu}. +\] +To the second approximation we must then put +\[ +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}\Add{,} +\Tag{(92)} +\] +\PageSep{89} +where the~$\gamma_{\mu\nu}$ are to be regarded as small of the first +order. + +Both terms of our equation of motion are then small +of the first order. If we neglect terms which, relatively +to these, are small of the first order, we have to put +\begin{gather*} +ds^{2} = \Change{}{-}{dx_{\nu}}^{2} = dl^{2} (1 - q^{2})\Add{,} +\Tag{(93)} \\ +% +\Gamma_{\alpha\beta}^{\mu} + = -\delta_{\mu\sigma} \left[\Chr{\alpha}{\beta}{\sigma}\right] + = -\left[\Chr{\alpha}{\beta}{\mu}\right] + = \frac{1}{2} \left( + \frac{\dd \gamma_{\alpha\beta}}{\dd x_{\mu}} + - \frac{\dd \gamma_{\alpha\mu}}{\dd x_{\beta}} + - \frac{\dd \gamma_{\beta\mu}}{\dd x_{\alpha}}\right)\Add{.} +\Tag{(94)} +\end{gather*} +We shall now introduce an approximation of a second +kind. Let the velocity of the material particles be very +small compared to that of light. Then $ds$~will be the +same as the time differential,~$dl$. Further, $\dfrac{dx_{1}}{ds}$, $\dfrac{dx_{2}}{ds}$, $\dfrac{dx_{3}}{ds}$ +will vanish compared to~$\dfrac{dx_{4}}{ds}$. We shall assume, in addition, +that the gravitational field varies so little with the +time that the derivatives of the~$\gamma_{\mu\nu}$ by~$x_{4}$ may be +neglected. Then the equation of motion (for $\mu = 1, 2, 3$) +reduces to +\[ +\frac{d^{2} x_{\mu}}{dl^{2}} + = \frac{\dd}{\dd x_{\mu}} \left(\frac{\gamma_{44}}{2}\right)\Add{.} +\Tag{(90a)} +\] +This equation is identical with Newton's equation of +motion for a material particle in a gravitational field, if +we identify~$\left(\dfrac{\gamma_{44}}{2}\right)$ with the potential of the gravitational +field; whether or not this is allowable, naturally depends +upon the field equations of gravitation, that is, it depends +upon whether or not this quantity satisfies, to a +first approximation, the same laws of the field as the +\PageSep{90} +gravitational potential in Newton's theory. A glance +at \Eqref{(90)}~and~\Eqref{(90a)} shows that the~$\Gamma_{\alpha\beta}^{\mu}$ actually do play +the rôle of the intensity of the gravitational field. +These quantities do not have a tensor character. + +Equations~\Eqref{(90)} express the influence of inertia and +gravitation upon the material particle. The unity of +inertia and gravitation is formally expressed by the fact +that the whole left-hand side of~\Eqref{(90)} has the character +of a tensor (with respect to any transformation of co-ordinates), +but the two terms taken separately do not +have tensor character, so that, in analogy with Newton's +equations, the first term would be regarded as the expression +for inertia, and the second as the expression +for the gravitational force. + +We must next attempt to find the laws of the gravitational +field. For this purpose, Poisson's equation, +\index{Poisson's equation}% +\[ +\Delta\phi = 4\pi K\rho +\] +of the Newtonian theory must serve as a model. This +equation has its foundation in the idea that the gravitational +field arises from the density~$\rho$ of ponderable +matter. It must also be so in the general theory of +relativity. But our investigations of the special theory +of relativity have shown that in place of the scalar +density of matter we have the tensor of energy per unit +volume. In the latter is included not only the tensor +of the energy of ponderable matter, but also that of the +electromagnetic energy. We have seen, indeed, that +in a more complete analysis the energy tensor can be +regarded only as a provisional means of representing +\PageSep{91} +matter. In reality, matter consists of electrically charged +particles, and is to be regarded itself as a part, in fact, +the principal part, of the electromagnetic field. It is +only the circumstance that we have not sufficient knowledge +of the electromagnetic field of concentrated charges +that compels us, provisionally, to leave undetermined +in presenting the theory, the true form of this tensor. +From this point of view our problem now is to introduce +a tensor,~$T_{\mu\nu}$, of the second rank, whose structure we do +not know provisionally, and which includes in itself the +energy density of the electromagnetic field and of ponderable +matter; we shall denote this in the following as +the ``energy tensor of matter.'' + +According to our previous results, the principles of +momentum and energy are expressed by the statement +that the divergence of this tensor vanishes~\Eqref{(47c)}. In +the general theory of relativity, we shall have to assume +as valid the corresponding general co-variant equation. +If $(T_{\mu\nu})$~denotes the co-variant energy tensor of matter, +$\tT_{\sigma}^{\nu}$~the corresponding mixed tensor density, then, in +accordance with~\Eqref{(83)}, we must require that +\[ +0 = \frac{\dd \tT_{\sigma}^{\alpha}}{\dd x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \tT_{\alpha}^{\beta} +\Tag{(95)} +\] +be satisfied. It must be remembered that besides the +energy density of the matter there must also be given +an energy density of the gravitational field, so that there +can be no talk of principles of conservation of energy +and momentum for matter alone. This is expressed +mathematically by the presence of the second term in~\Eqref{(95)}, +\PageSep{92} +which makes it impossible to conclude the existence +of an integral equation of the form of~\Eqref{(49)}. The gravitational +field transfers energy and momentum to the +``matter,'' in that it exerts forces upon it and gives it +energy; this is expressed by the second term in~\Eqref{(95)}. + +If there is an analogue of Poisson's equation in the +general theory of relativity, then this equation must be +a tensor equation for the tensor~$g_{\mu\nu}$ of the gravitational +potential; the energy tensor of matter must appear on +the right-hand side of this equation. On the left-hand +side of the equation there must be a differential tensor +in the~$g_{\mu\nu}$. We have to find this differential tensor. +It is completely determined by the following three +conditions:--- + +1.~It may contain no differential coefficients of the~$g_{\mu\nu}$ +higher than the second. + +2.~It must be linear and homogeneous in these second +differential coefficients. + +3.~Its divergence must vanish identically. + +The first two of these conditions are naturally taken +from Poisson's equation. Since it may be proved +mathematically that all such differential tensors can be +formed algebraically (i.e.~without differentiation) from +Riemann's tensor, our tensor must be of the form +\[ +R_{\mu\nu} + ag_{\mu\nu} R\Add{,} +\] +in which $R_{\mu\nu}$~and~$R$ are defined by \Eqref{(88)}~and~\Eqref{(89)} respectively. +Further, it may be proved that the third condition +requires~$a$ to have the value~$-\frac{1}{2}$. For the law +\PageSep{93} +of the gravitational field we therefore get the equation +\[ +R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}\Add{.} +\Tag{(96)} +\] +Equation~\Eqref{(95)} is a consequence of this equation. $\kappa$~denotes +a constant, which is connected with the Newtonian +gravitation constant. + +In the following I shall indicate the features of the +theory which are interesting from the point of view of +physics, using as little as possible of the rather involved +mathematical method. It must first be shown that the +divergence of the left-hand side actually vanishes. The +energy principle for matter may be expressed, by~\Eqref{(83)}, +\[ +0 = \frac{\dd \tT_{\sigma}^{\alpha}}{\dd x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \tT_{\alpha}^{\beta}\Add{,} +\Tag{(97)} +\] +in which +\[ +\tT_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}. +\] +The analogous operation, applied to the left-hand side +of~\Eqref{(96)}, will lead to an identity. + +In the region surrounding each world-point there are +systems of co-ordinates for which, choosing the $x_{\Change{\mu}{4}}$-co-ordinate +imaginary, at the given point, +\[ +g_{\mu\nu} = g^{\mu\nu} + = -\delta_{\mu\nu} +%[** TN: Moved "="s outside brace] + = \begin{cases} + -1 & \text{if $\mu = \nu$\Add{,}} \\ + \Neg[]0 & \text{if $\mu \neq \nu$,} + \end{cases} +\] +and for which the first derivatives of the~$g_{\mu\nu}$ and the~$g^{\mu\nu}$ +vanish. We shall verify the vanishing of the divergence +of the left-hand side at this point. At this point +the components~$\Gamma_{\sigma\beta}^{\alpha}$ vanish, so that we have to prove +the vanishing only of +\[ +\frac{\dd}{\dd x_{\sigma}} \left[ + \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) +\right]. +\] +\PageSep{94} +Introducing \Eqref{(88)}~and~\Eqref{(70)} into this expression, we see +that the only terms that remain are those in which third +derivatives of the~$g^{\mu\nu}$ enter. Since the~$g_{\mu\nu}$ are to be +replaced by~$-\delta_{\mu\nu}$, we obtain, finally, only a few terms +which may easily be seen to cancel each other. Since +the quantity that we have formed has a tensor character, +its vanishing is proved for every other system of co-ordinates +also, and naturally for every other four-dimensional +point. The energy principle of matter~\Eqref{(97)} is thus a +mathematical consequence of the field equations~\Eqref{(96)}. + +In order to learn whether the equations~\Eqref{(96)} are +consistent with experience, we must, above all else, find +out whether they lead to the Newtonian theory as a +first approximation. For this purpose we must introduce +various approximations into these equations. We +already know that Euclidean geometry and the law of the +constancy of the velocity of light are valid, to a certain +approximation, in regions of a great extent, as in the +planetary system. If, as in the special theory of relativity, +we take the fourth co-ordinate imaginary, this +means that we must put +\[ +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}\Add{,} +\Tag{(98)} +\] +in which the~$\gamma_{\mu\nu}$ are so small compared to~$1$ that we +can neglect the higher powers of the~$\gamma_{\mu\nu}$ and their +derivatives. If we do this, we learn nothing about the +structure of the gravitational field, or of metrical space of +cosmical dimensions, but we do learn about the influence +of neighbouring masses upon physical phenomena. + +Before carrying through this approximation we shall +\PageSep{95} +transform~\Eqref{(96)}. We multiply~\Eqref{(96)} by~$g^{\mu\nu}$, summed over +the $\mu$~and~$\nu$; observing the relation which follows from +the definition of the~$g^{\mu\nu}$, +\[ +g_{\mu\nu} g^{\mu\nu} = 4\Add{,} +\] +we obtain the equation +\[ +R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T. +\] +If we put this value of~$R$ in~\Eqref{(96)} we obtain +\[ +R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) + = -\kappa T_{\mu\nu}^{*}\Add{.} +\Tag{(96a)} +\] +When the approximation which has been mentioned is +carried out, we obtain for the left-hand side, +\[ +-\tfrac{1}{2}\left( + \frac{\dd^{2} \gamma_{\mu\nu}}{{\dd x_{\alpha}}^{2}} + + \frac{\dd^{2} \gamma_{\alpha\alpha}}{\dd x_{\mu}\, \dd x_{\nu}} + - \frac{\dd^{2} \gamma_{\mu\alpha}}{\dd x_{\nu}\, \dd x_{\alpha}} + - \frac{\dd^{2} \gamma_{\nu\alpha}}{\dd x_{\mu}\, \dd x_{\alpha}} +\right) +\] +or +\[ +-\tfrac{1}{2}\frac{\dd^{2} \gamma_{\mu\nu}}{{\dd x_{\alpha}}^{2}} + + \tfrac{1}{2} \frac{\dd}{\dd x_{\nu}}\left( + \frac{{\dd \gamma'}_{\mu\alpha}}{\dd x_{\alpha}} + \right) + + \tfrac{1}{2} \frac{\dd}{\dd x_{\mu}}\left( + \frac{{\dd \gamma'}_{\nu\alpha}}{\dd x_{\alpha}} + \right)\Add{,} +\] +in which has been put +\[ +{\gamma'}_{\mu\nu} + = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}\Add{.} +\Tag{(99)} +\] + +We must now note that equation~\Eqref{(96)} is valid for any +system of co-ordinates. We have already specialized the +system of co-ordinates in that we have chosen it so that +within the region considered the $g_{\mu\nu}$~differ infinitely little +from the constant values~$-\delta_{\mu\nu}$. But this condition +remains satisfied in any infinitesimal change of co-ordinates, +so that there are still four conditions to which +the~$\gamma_{\mu\nu}$ may be subjected, provided these conditions do +not conflict with the conditions for the order of magnitude +\PageSep{96} +of the~$\gamma_{\mu\nu}$. We shall now assume that the system of co-ordinates +is so chosen that the four relations--- +\[ +0 = \frac{{\dd\gamma'}_{\mu\nu}}{\dd x_{\nu}} + = \frac{\dd\gamma_{\mu\nu}}{\dd x_{\nu}} + - \tfrac{1}{2} \frac{\dd\gamma_{\sigma\sigma}}{\dd x_{\mu}} +\Tag{(100)} +\] +are satisfied. Then \Eqref{(96a)}~takes the form +\[ +\frac{\dd^{2} \gamma_{\mu\nu}}{{\dd x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}\Add{.} +\Tag{(96b)} +\] +These equations may be solved by the method, familiar +in electrodynamics, of retarded potentials; we get, in an +easily understood notation, +\[ +\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int + \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}\Add{.} +\Tag{(101)} +\] + +In order to see in what sense this theory contains the +Newtonian theory, we must consider in greater detail +the energy tensor of matter. Considered phenomenologically, +this energy tensor is composed of that of the +electromagnetic field and of matter in the narrower sense. +If we consider the different parts of this energy tensor +with respect to their order of magnitude, it follows +from the results of the special theory of relativity that +the contribution of the electromagnetic field practically +vanishes in comparison to that of ponderable matter. In +our system of units, the energy of one gram of matter is +equal to~$1$, compared to which the energy of the electric +fields may be ignored, and also the energy of deformation +of matter, and even the chemical energy. We get an +approximation that is fully sufficient for our purpose if +\PageSep{97} +we put +\[ +\left. +\begin{aligned} +T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}\Add{,} \\ +ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{.} +\end{aligned} +\right\} +\Tag{(102)} +\] +In this, $\sigma$~is the density at rest, that is, the density of the +ponderable matter, in the ordinary sense, measured with +the aid of a unit measuring rod, and referred to a Galilean +system of co-ordinates moving with the matter. + +We observe, further, that in the co-ordinates we have +chosen, we shall make only a relatively small error if we +replace the~$g_{\mu\nu}$ by~$-\delta_{\mu\nu}$, so that we put +\[ +ds^{2} = -\sum {dx_{\mu}}^{2}\Add{.} +\Tag{(102a)} +\] + +The previous developments are valid however rapidly +the masses which generate the field may move relatively +to our chosen system of quasi-Galilean co-ordinates. But +in astronomy we have to do with masses whose velocities, +relatively to the co-ordinate system employed, are always +small compared to the velocity of light, that is, small +compared to~$1$, with our choice of the unit of time. +We therefore get an approximation which is sufficient +for nearly all practical purposes if in~\Eqref{(101)} we replace +the retarded potential by the ordinary (non-retarded) +potential, and if, for the masses which generate the field, +we put +\[ +\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad +\frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}\Add{.} +\Tag{(\Change{103a}{103})} +\] +\PageSep{98} +Then we get for $T^{\mu\nu}$~and~$T_{\mu\nu}$ the values +\[ +\left. +\begin{array}{*{3}{>{\qquad}r}>{\quad}r} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & -\sigma +\end{array} +\right\}\Add{.} +\Tag{(104)} +\] +For $T$ we get the value~$\sigma$, and, finally, for~$T_{\mu\nu}^{*}$ the +values, +\[ +\left. +\begin{array}{*{3}{>{\qquad}c}>{\quad}r} +\dfrac{\sigma}{2} & 0 & 0 & 0 \\ +0 & \dfrac{\sigma}{2} & 0 & 0 \\ +0 & 0 & \dfrac{\sigma}{2} & 0 \\ +0 & 0 & 0 & -\dfrac{\sigma}{2} +\end{array} +\right\}\Add{.} +\Tag{(104a)} +\] + +We thus get, from~\Eqref{(101)}, +\[ +\left. +\begin{aligned} +\gamma_{11} = \gamma_{22} = \gamma_{33} + &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} \\ +\gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} +\end{aligned} +\right\} +\Tag{(101a)} +\] +while all the other~$\gamma_{\mu\nu}$ vanish. The least of these equations, +in connexion with equation~\Eqref{(90a)}, contains Newton's +theory of gravitation. If we replace~$l$ by~$ct$ we +get +\[ +\frac{d^{2} x_{\mu}}{dt^{2}} + = \frac{\kappa c^{2}}{8\pi}\, \frac{\dd}{\dd x_{\mu}} \left\{ + \int \frac{\sigma\, dV_{0}}{r} + \right\}\Add{.} +\Tag{(90b)} +\] +We see that the Newtonian gravitation constant~$K$, is +\index{Gravitation constant}% +\index{Newtonian gravitation constant}% +connected with the constant~$\kappa$ that enters into our field +equations by the relation +\[ +K = \frac{\kappa c^{2}}{8\pi}\Add{.} +\Tag{(105)} +\] +\PageSep{99} +From the known numerical value of~$K$, it therefore +follows that +\[ +\kappa = \frac{8\pi K}{c^{2}} + = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} + = 1.86·10^{-27}\Add{.} +\Tag{(105a)} +\] +From \Eqref{(101)} we see that even in the first approximation +the structure of the gravitational field differs fundamentally +from that which is consistent with the Newtonian theory; +this difference lies in the fact that the gravitational +potential has the character of a tensor and not a scalar. +This was not recognized in the past because only the +component~$g_{44}$, to a first approximation, enters the equations +of motion of material particles. + +In order now to be able to judge the behaviour of +measuring rods and clocks from our results, we must +observe the following. According to the principle of +equivalence, the metrical relations of the Euclidean +geometry are valid relatively to a Cartesian system of +reference of infinitely small dimensions, and in a suitable +state of motion (freely falling, and without rotation). +We can make the same statement for local systems of +co-ordinates which, relatively to these, have small accelerations, +and therefore for such systems of co-ordinates +as are at rest relatively to the one we have selected. For +such a local system, we have, for two neighbouring point +events, +\[ +ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} + = - dS^{2} + dT^{2}\Add{,} +\] +where $dS$~is measured directly by a measuring rod and +$dT$~by a clock at rest relatively to the system; these are +\PageSep{100} +the naturally measured lengths and times. Since~$ds^{2}$, on +the other hand, is known in terms of the co-ordinates~$x_{\nu}$ +employed in finite regions, in the form +\[ +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{,} +\] +we have the possibility of getting the relation between +naturally measured lengths and times, on the one hand, +and the corresponding differences of co-ordinates, on the +other hand. As the division into space and time is in +agreement with respect to the two systems of co-ordinates, +so when we equate the two expressions for~$ds^{2}$ we get +two relations. If, by~\Eqref{(101a)}, we put +\begin{multline*} +ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ + + \left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}\Add{,} +\end{multline*} +we obtain, to a sufficiently close approximation, +\[ +\left. +\begin{aligned} + &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ + &\qquad + \begin{aligned} + &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}\Add{,} \\ +dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. +\end{aligned} +\end{aligned} +\right\} +\Tag{(106)} +\] + +The unit measuring rod has therefore the length, +\[ +1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +\] +in respect to the system of co-ordinates we have selected. +The particular system of co-ordinates we have selected +\PageSep{101} +insures that this length shall depend only upon +the place, and not upon the direction. If we had +chosen a different system of co-ordinates this would not +be so. But however we may choose a system of co-ordinates, +the laws of configuration of rigid rods do not +agree with those of Euclidean geometry; in other words, +we cannot choose any system of co-ordinates so that the +co-ordinate differences, $\Delta x_{1}$,~$\Delta x_{2}$,~$\Delta x_{3}$, corresponding to the +ends of a unit measuring rod, oriented in any way, shall +always satisfy the relation ${\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} = 1$. In +this sense space is not Euclidean, but ``curved.'' It +follows from the second of the relations above that the +interval between two beats of the unit clock ($dT = 1$) +corresponds to the ``time'' +\[ +1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +\] +in the unit used in our system of co-ordinates. The rate +of a clock is accordingly slower the greater is the mass of +the ponderable matter in its neighbourhood. We therefore +\index{Displacement of spectral lines}% +\index{Spectral lines, displacement of}% +conclude that spectral lines which are produced on +the sun's surface will be displaced towards the red, +compared to the corresponding lines produced on the +earth, by about $2·10^{-6}$~of their wave-lengths. At first, +this important consequence of the theory appeared to +conflict with experiment; but results obtained during the +past year seem to make the existence of this effect more +probable, and it can hardly be doubted that this consequence +of the theory will be confirmed within the next +year. +\PageSep{102} + +Another important consequence of the theory, which +\index{Light ray, path of}% +\index{Path of light ray}% +\index{Ray of light, path of}% +can be tested experimentally, has to do with the path of +rays of light. In the general theory of relativity also +the velocity of light is everywhere the same, relatively to +a local inertial system. This velocity is unity in our +natural measure of time. The law of the propagation of +light in general co-ordinates is therefore, according to the +general theory of relativity, characterized, by the equation +\[ +ds^{2} = 0. +\] +To within the approximation which we are using, and in +the system of co-ordinates which we have selected, the +velocity of light is characterized, according to~\Eqref{(106)}, by +the equation +\[ +\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) + = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}. +\] +The velocity of light~$L$, is therefore expressed in our +co-ordinates by +\[ +\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} + = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{.} +\Tag{(107)} +\] +We can therefore draw the conclusion from this, that a +ray of light passing near a large mass is deflected. If +we imagine the sun, of mass~$M$ concentrated at the +origin of our system of co-ordinates, then a ray of light, +travelling parallel to the $x_{3}$-axis, in the $x_{1}$-$x_{3}$~plane, +at a distance~$\Delta$ from the origin, will be deflected, in all, +by an amount +\PageSep{103} +\[ +\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\dd L}{\dd x_{1}}\, dx_{3} +\] +towards the sun. On performing the integration we get +\[ +\alpha = \frac{\kappa M}{2\pi\Delta}\Add{.} +\Tag{(108)} +\] + +The existence of this deflection, which amounts to~$1.7''$ +for $\Delta$~equal to the radius of the sun, was confirmed, +with remarkable accuracy, by the English Solar Eclipse +\index{Solar Eclipse expedition (1919)}% +Expedition in~1919, and most careful preparations have +been made to get more exact observational data at the +solar eclipse in~1922. It should be noted that this +result, also, of the theory is not influenced by our +arbitrary choice of a system of co-ordinates. + +This is the place to speak of the third consequence of +the theory which can be tested by observation, namely, +that which concerns the motion of the perihelion +of the planet Mercury. The secular changes in the +\index{Mercury, perihelion of}% +\index{Perihelion of Mercury}% +planetary orbits are known with such accuracy that the +approximation we have been using is no longer sufficient +for a comparison of theory and observation. It is necessary +to go back to the general field equations~\Eqref{(96)}. To +solve this problem I made use of the method of successive +approximations. Since then, however, the problem +of the central symmetrical statical gravitational field has +been completely solved by Schwarzschild and others; +the derivation given by H.~Weyl in his book, ``Raum-Zeit-Materie,'' +\index{Weyl}% +is particularly elegant. The calculation +can be simplified somewhat if we do not go back directly +\PageSep{104} +to the equation~\Eqref{(96)}, but base it upon a principle of +variation that is equivalent to this equation. I shall +indicate the procedure only in so far as is necessary for +understanding the method. + +In the case of a statical field, $ds^{2}$~must have the form +\[ +\left. +\begin{aligned} +ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}\Add{,} \\ +d\sigma^{2} + &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}\Add{,} +\end{aligned} +\right\} +\Tag{(109)} +\] +where the summation on the right-hand side of the last +equation is to be extended over the space variables only\Change{,}{.} +The central symmetry of the field requires the~$\gamma_{\mu\nu}$ to be +of the form, +\[ +\gamma_{\alpha\beta} + = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}\Add{;} +\Tag{(110)} +\] +$f^{2}$,~$\mu$ and~$\lambda$ are functions of $r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}$ only. +One of these three functions can be chosen arbitrarily, +because our system of co-ordinates is, \textit{a~priori}, completely +arbitrary; for by a substitution +\begin{align*} +{x'}_{4} &= x_{4}\Add{,} \\ +{x'}_{\alpha} &= F(r) x_{\alpha}\Add{,} +\end{align*} +we can always insure that one of these three functions +shall be an assigned function of~$r'$. In place of~\Eqref{(110)} we +can therefore put, without limiting the generality, +\[ +\gamma_{\alpha\beta} + = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}\Add{.} +\Tag{(110a)} +\] + +In this way the~$g_{\mu\nu}$ are expressed in terms of the two +quantities $\lambda$~and~$f$. These are to be determined as functions +of~$r$, by introducing them into equation~\Eqref{(96)}, after +\PageSep{105} +first calculating the~$\Change{{\Gamma_{\sigma}}^{\mu\nu}}{\Gamma_{\mu\nu}^{\sigma}}$ from \Eqref{(\Change{107}{109})}~and~\Eqref{(\Change{108a}{110a})}. We +have +\[ +\left. +\begin{aligned} +\Gamma_{\alpha\beta}^{\sigma} + &= \tfrac{1}{2} \frac{x_{\sigma}}{r} + · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} + {1 + \lambda r^{2}} + \text{ (for $\alpha, \beta, \sigma = 1, 2, 3$)}\Add{,} \\ +% +\Gamma_{44}^{4} + &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad + \text{(for $\alpha, \beta = 1, 2, 3$)}\Add{,} \\ +% +\Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\dd f^{2}}{\dd x_{\alpha}},\quad +\Gamma_{44}^{\alpha} + = -\tfrac{1}{2} \Change{f^{-2}}{g^{\alpha\beta}}\, + \frac{\dd f^{2}}{\dd \Change{x_{\alpha}}{x_{\beta}}}\Add{.} +\end{aligned} +\right\} +\Tag{(\Change{108b}{110b})} +\] + +With the help of these results, the field equations +furnish Schwarzschild's solution: +\[ +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ + \frac{dr^{2}} + {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) +\right]\Add{,} +\Tag{(\Change{109}{109a})} +\] +in which we have put +\[ +\left. +\begin{aligned} +x_{4} &= l\Add{,} \\ +x_{1} &= r \sin\theta \sin\phi\Add{,} \\ +x_{2} &= r \sin\theta \cos\phi\Add{,} \\ +x_{3} &= r \cos\theta \Add{,} \\ +A &= \frac{\kappa M}{4\pi}\Add{.} +\end{aligned} +\right\} +\Tag{(\Change{109a}{109b})} +\] + +$M$~denotes the sun's mass, centrally symmetrically +placed about the origin of co-ordinates; the solution~\Eqref{(109)} +is valid only outside of this mass, where all the~$T_{\mu\nu}$ vanish. +If the motion of the planet takes place in the $x_{1}$-$x_{2}$~plane +then we must replace~\Eqref{(\Change{109}{109a})} by +\[ +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} + - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}\Add{.} +\Tag{(\Change{109b}{109c})} +\] +\PageSep{106} + +The calculation of the planetary motion depends upon +equation~\Eqref{(90)}. From the first of equations \Eqref{(\Change{108b}{110b})}~and~\Eqref{(90)} +we get, for the indices $1, 2, 3$, +\[ +\frac{d}{ds} + \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0\Add{,} +\] +or, if we integrate, and express the result in polar co-ordinates, +\[ +r^{2} \frac{d\phi}{ds} = \text{constant}. +\Tag{(111)} +\] + +From~\Eqref{(90)}, for $\mu = 4$, we get +\[ +0 = \frac{d^{2} l}{ds^{2}} + + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} + \Change{}{\, \frac{dl}{ds}} + = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} + \Change{}{\, \frac{dl}{ds}}. +\] +From this, after multiplication by~$f^{2}$ and integration, we +have +\[ +f^{2} \frac{dl}{ds} = \text{constant}. +\Tag{(112)} +\] + +In \Eqref{(\Change{109b}{109c})},~\Eqref{(111)} and~\Eqref{(112)} we have three equations +between the four variables $s$,~$r$,~$l$ and~$\phi$, from which the +motion of the planet may be calculated in the same way +as in classical mechanics. The most important result we +get from this is a secular rotation of the elliptic orbit of +the planet in the same sense as the revolution of the +planet, amounting in radians per revolution to +\[ +\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}\Add{,} +\Tag{(113)} +\] +\PageSep{107} +where +\begin{align*} +a &= \text{the semi-major axis of the planetary orbit in +centimetres.} \\ +e &= \text{the numerical eccentricity.} \\ +c &= 3·10^{+10}, \text{ the velocity of light \textit{in vacuo}.} \\ +T &= \text{the period of revolution in seconds.} +\end{align*} +This expression furnishes the explanation of the motion +of the perihelion of the planet Mercury, which has been +\index{Mercury, perihelion of}% +\index{Perihelion of Mercury}% +known for a hundred years (since Leverrier), and for +which theoretical astronomy has hitherto been unable +satisfactorily to account. + +There is no difficulty in expressing Maxwell's theory +of the electromagnetic field in terms of the general theory +of relativity; this is done by application of the tensor +formation \Eqref{(81)},~\Eqref{(82)} and~\Eqref{(77)}. Let $\phi_{\mu}$~be a tensor of the +first rank, to be denoted as an electromagnetic $4$-potential; +then an electromagnetic field tensor may be defined by +the relations, +\[ +\phi_{\mu\nu} + = \frac{\dd \phi_{\mu}}{\dd x_{\nu}} - \frac{\dd \phi_{\nu}}{\dd x_{\mu}}\Add{.} +\Tag{(114)} +\] +The second of Maxwell's systems of equations is then +defined by the tensor equation, resulting from this, +\[ +\frac{\dd \phi_{\mu\nu}}{\dd x_{\rho}} + +\frac{\dd \phi_{\nu\rho}}{\dd x_{\mu}} + +\frac{\dd \phi_{\rho\mu}}{\dd x_{\nu}} = 0\Add{,} +\Tag{(114a)} +\] +and the first of Maxwell's systems of equations is defined +by the tensor-density relation +\[ +\frac{\dd \tF^{\mu\nu}}{\dd x_{\nu}} = \tJ^{\mu}\Add{,} +\Tag{(115)} +\] +\PageSep{108} +in which +\begin{align*} +\tF^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}\Add{,} \\ +\tJ^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. +\end{align*} +If we introduce the energy tensor of the electromagnetic +field into the right-hand side of~\Eqref{(96)}, we obtain~\Eqref{(115)}, +for the special case $\tJ^{\mu} = 0$, as a consequence of~\Eqref{(96)} by +taking the divergence. This inclusion of the theory of +electricity in the scheme of the general theory of relativity +has been considered arbitrary and unsatisfactory by +many theoreticians. Nor can we in this way conceive of +the equilibrium of the electricity which constitutes the +elementary electrically charged particles. A theory in +which the gravitational field and the electromagnetic field +enter as an essential entity would be much preferable. +H.~Weyl, and recently Th.~Kaluza, have discovered some +\index{Kaluza}% +\index{Weyl}% +ingenious theorems along this direction; but concerning +them, I am convinced that they do not bring us nearer to +the true solution of the fundamental problem. I shall +not go into this further, but shall give a brief discussion +of the so-called cosmological problem, for without this, +the considerations regarding the general theory of relativity +would, in a certain sense, remain unsatisfactory. + +Our previous considerations, based upon the field +equations~\Eqref{(96)}, had for a foundation the conception that +space on the whole is Galilean-Euclidean, and that this +character is disturbed only by masses embedded in it. +This conception was certainly justified as long as we were +dealing with spaces of the order of magnitude of those +\PageSep{109} +that astronomy has to do with. But whether portions of +the universe, however large they may be, are quasi-Euclidean, +is a wholly different question. We can make +this clear by using an example from the theory of surfaces +which we have employed many times. If a portion of a +surface is observed by the eye to be practically plane, it +does not at all follow that the whole surface has the form +of a plane; the surface might just as well be a sphere, for +example, of sufficiently large radius. The question as to +whether the universe as a whole is non-Euclidean was +much discussed from the geometrical point of view before +the development of the theory of relativity. But with the +theory of relativity, this problem has entered upon a +new stage, for according to this theory the geometrical +properties of bodies are not independent, but depend +upon the distribution of masses. + +If the universe were quasi-Euclidean, then Mach was +\index{Mach}% +wholly wrong in his thought that inertia, as well as +gravitation, depends upon a kind of mutual action between +bodies. For in this case, with a suitably selected system +of co-ordinates, the~$g_{\mu\nu}$ would be constant at infinity, as +they are in the special theory of relativity, while within +finite regions the~$g_{\mu\nu}$ would differ from these constant +values by small amounts only, with a suitable choice of +co-ordinates, as a result of the influence of the masses in +finite regions. The physical properties of space would +not then be wholly independent, that is, uninfluenced by +matter, but in the main they would be, and only in +small measure, conditioned by matter. Such a dualistic +conception is even in itself not satisfactory; there are, +\PageSep{110} +however, some important physical arguments against it, +which we shall consider. + +The hypothesis that the universe is infinite and +\index{Finiteness of universe}% +\index{Universe@{Universe, Finiteness of}}% +Euclidean at infinity, is, from the relativistic point of +view, a complicated hypothesis. In the language of the +general theory of relativity it demands that the Riemann +\index{Riemann!tensor}% +tensor of the fourth rank~$R_{iklm}$ shall vanish at infinity, +which furnishes twenty independent conditions, while only +ten curvature components~$R_{\mu\nu}$, enter into the laws of the +gravitational field. It is certainly unsatisfactory to +postulate such a far-reaching limitation without any +physical basis for it. + +But in the second place, the theory of relativity makes +it appear probable that Mach was on the right road in +\index{Mach}% +his thought that inertia depends upon a mutual action of +matter. For we shall show in the following that, according +to our equations, inert masses do act upon each other +in the sense of the relativity of inertia, even if only very +feebly. What is to be expected along the line of Mach's +thought? +\begin{itemize} +\item[1.] The inertia of a body must increase when ponderable + masses are piled up in its neighbourhood. + +\item[2.] A body must experience an accelerating force when + neighbouring masses are accelerated, and, in fact, + the force must be in the same direction as the + acceleration. + +\item[3.] A rotating hollow body must generate inside of + itself a ``Coriolis field,'' which deflects moving + bodies in the sense of the rotation, and a radial + centrifugal field as well. +\end{itemize} +\PageSep{111} + +We shall now show that these three effects, which are +to be expected in accordance with Mach's ideas, are +\index{Mach}% +actually present according to our theory, although their +magnitude is so small that confirmation of them by +laboratory experiments is not to be thought of. For this +purpose we shall go back to the equations of motion of +a material particle~\Eqref{(90)}, and carry the approximations +somewhat further than was done in equation~\Eqref{(90a)}. + +First, we consider~$\gamma_{44}$ as small of the first order. The +square of the velocity of masses moving under the influence +of the gravitational force is of the same order, according +to the energy equation. It is therefore logical to regard +the velocities of the material particles we are considering, +as well as the velocities of the masses which generate the +field, as small, of the order~$\frac{1}{2}$. We shall now carry out the +approximation in the equations that arise from the field +equations~\Eqref{(101)} and the equations of motion~\Eqref{(90)} so far +as to consider terms, in the second member of~\Eqref{(90)}, that +are linear in those velocities. Further, we shall not put +$ds$~and~$dl$ equal to each other, but, corresponding to the +higher approximation, we shall put +\[ +ds = \sqrt{g_{44}}\, dl + = \left(1 - \frac{\gamma_{44}}{2}\right) dl. +\] +From~\Eqref{(90)} we obtain, at first, +\[ +\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} + \left(1 + \frac{\gamma_{44}}{2}\right). +\Tag{(116)} +\] + +From~\Eqref{(101)} we get, to the approximation sought for, +\PageSep{112} +\[ +\left. +\begin{aligned} +-\gamma_{11} = -\gamma_{22} = -\gamma_{33} + &= \gamma_{44} = \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} \\ +% +\gamma_{4\alpha} + &= -\frac{\Change{\veci}{i} \kappa}{2} + \int \frac{\sigma \dfrac{dx_{\alpha}}{ds}\, dV_{0}}{r}\Add{,} \\ +% +\gamma_{\alpha\beta} &= 0\Add{,} +\end{aligned} +\right\} +\Tag{(117)} +\] +in which, in~\Eqref{(117)}, $\alpha$~and~$\beta$ denote the space indices only. + +On the right-hand side of~\Eqref{(116)} we can replace +$1 + \dfrac{\gamma_{44}}{2}$ by~$1$ and $-\Change{\Gamma_{\mu}^{\alpha\beta}}{\Gamma_{\alpha\beta}^{\mu}}$ by~$\left[\Chr{\alpha}{\beta}{\mu}\right]$. It is easy to see, in +addition, that to this degree of approximation we must +put +\begin{align*} +\left[\Chr{4}{4}{\mu}\right] + &= -\tfrac{1}{2} \frac{\dd \gamma_{44}}{\dd x_{\mu}} + + \frac{\dd \gamma_{4\mu}}{\dd x_{4}}\Add{,} \\ +% +\left[\Chr{\alpha}{4}{\mu}\right] + &= \tfrac{1}{2} \left( + \frac{\dd \gamma_{4\mu}}{\dd x_{\alpha}} - \frac{\dd \gamma_{4\alpha}}{\dd x_{\mu}} + \right)\Add{,} \\ +\left[\Chr{\alpha}{\beta}{\mu}\right] &= 0\Add{,} +\end{align*} +in which $\alpha$,~$\beta$ and~$\mu$ denote space indices. We therefore +obtain from~\Eqref{(116)}, in the usual vector notation, +\[ +\left. +\begin{aligned} +\frac{d}{dl}\bigl[(1 + \Bar[7pt]{\sigma}) \v\bigr] + &= \grad \Bar[7pt]{\sigma} + \frac{\dd \tA}{\dd l} + [\rot \tA, \v]\Add{,} \\ +\Bar[7pt]{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} \\ +\tA &= \frac{\kappa}{2\Change{}{\pi}} + \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}\Add{.} +\end{aligned} +\right\} +\Tag{(118)} +\] + +The equations of motion,~\Eqref{(118)}, show now, in fact, that +\PageSep{113} +\begin{itemize} +\item[1.] The inert mass is proportional to $1 + \Bar[7pt]{\sigma}$, and + therefore increases when ponderable masses + approach the test body. + +\item[2.] There is an inductive action of accelerated masses, +\index{Accelerated masses, inductive action of}% +\index{Inductive action of accelerated masses}% + of the same sign, upon the test body. This is + the term~$\dfrac{\dd \tA}{\dd l}$. + +\item[3.] A material particle, moving perpendicularly to the + axis of rotation inside a rotating hollow body, + is deflected in the sense of the rotation (Coriolis + field). The centrifugal action, mentioned above, + inside a rotating hollow body, also follows from + the theory, as has been shown by Thirring.\footnotemark +\index{Thirring}% +\end{itemize} +\footnotetext{That the centrifugal action must be inseparably connected with the + existence of the Coriolis field may be recognized, even without calculation, + in the special case of a co-ordinate system rotating uniformly relatively to + an inertial system; our general co-variant equations naturally must apply + to such a case.} + +Although all of these effects are inaccessible to experiment, +because $\kappa$~is so small, nevertheless they certainly +exist according to the general theory of relativity. We +must see in them a strong support for Mach's ideas as to +\index{Mach}% +the relativity of all inertial actions. If we think these +ideas consistently through to the end we must expect the +\emph{whole} inertia, that is, the \emph{whole} $g_{\mu\nu}$-field, to be determined +by the matter of the universe, and not mainly by the +boundary conditions at infinity. + +For a satisfactory conception of the $g_{\mu\nu}$-field of cosmical +dimensions, the fact seems to be of significance that the +relative velocity of the stars is small compared to the +velocity of light. It follows from this that, with a suitable +\PageSep{114} +choice of co-ordinates, $g_{44}$~is nearly constant in the +universe, at least, in that part of the universe in which +there is matter. The assumption appears natural, moreover, +that there are stars in all parts of the universe, so +that we may well assume that the inconstancy of~$g_{44}$ +depends only upon the circumstance that matter is not +distributed continuously, but is concentrated in single +celestial bodies and systems of bodies. If we are willing +to ignore these more local non-uniformities of the density +of matter and of the $g_{\mu\nu}$-field, in order to learn something +of the geometrical properties of the universe as a whole, +it appears natural to substitute for the actual distribution +of masses a continuous distribution, and furthermore to +assign to this distribution a uniform density~$\sigma$. In this +imagined universe all points with space directions will +be geometrically equivalent; with respect to its space +extension it will have a constant curvature, and will be +cylindrical with respect to its $x_{4}$-co-ordinate. The possibility +seems to be particularly satisfying that the universe +is spatially bounded and thus, in accordance with our +assumption of the constancy of~$\sigma$, is of constant curvature, +being either spherical or elliptical; for then the boundary +conditions at infinity which are so inconvenient from the +standpoint of the general theory of relativity, may be +replaced by the much more natural conditions for a closed +surface. + +According to what has been said, we are to put +\[ +ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{,} +\Tag{(119)} +\] +in which the indices $\mu$~and~$\nu$ run from~$1$ to~$3$ only. The~$\gamma_{\mu\nu}$ +\PageSep{115} +will be such functions of $x_{1}$,~$x_{2}$,~$x_{3}$ as correspond +to a three-dimensional continuum of constant positive +curvature. We must now investigate whether such an +assumption can satisfy the field equations of gravitation. + +In order to be able to investigate this, we must first +find what differential conditions the three-dimensional +manifold of constant curvature satisfies. A spherical +manifold of three dimensions, embedded in a Euclidean +continuum of four dimensions,\footnote + {The aid of a fourth space dimension has naturally no significance + except that of a mathematical artifice.} +is given by the equations +\begin{align*} +{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}\Add{,} \\ +{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. +\end{align*} +By eliminating~$x_{4}$, we get +\[ +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} + {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}. +\] + +As far as terms of the third and higher degrees in the~$x_{\nu}$, +we can put, in the neighbourhood of the origin of +co-ordinates, +\[ +ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) + dx_{\mu}\, dx_{\nu}. +\] + +Inside the brackets are the~$g_{\mu\nu}$ of the manifold in the +neighbourhood of the origin. Since the first derivatives +of the~$g_{\mu\nu}$, and therefore also the~$\Gamma_{\mu\nu}^{\sigma}$, vanish at the +origin, the calculation of the~$R_{\mu\nu}$ for this manifold, by~\Eqref{(88)}, +is very simple at the origin. We have +\[ +R_{\mu\nu} = \Change{}{-}\frac{2}{a^{2}} \delta_{\mu\nu} + = \frac{2}{a^{2}} g_{\mu\nu}. +\] +\PageSep{116} + +Since the relation $R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}$ is universally co-variant, +and since all points of the manifold are geometrically +equivalent, this relation holds for every system of co-ordinates, +and everywhere in the manifold. In order to +avoid confusion with the four-dimensional continuum, +we shall, in the following, designate quantities that refer +to the three-dimensional continuum by Greek letters, +and put +\[ +\P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}\Add{.} +\Tag{(120)} +\] + +We now proceed to apply the field equations~\Eqref{(96)} to +our special case. From~\Eqref{(119)} we get for the four-dimensional +manifold, +\[ +\left. +\begin{aligned} +R_{\mu\nu} &= \text{$\P_{\mu\nu}$ for the indices~$1$ to~$3$}\Add{,} \\ +R_{14} &= R_{24} = R_{34} = R_{44} = 0\Add{.} +\end{aligned} +\right\} +\Tag{(121)} +\] + +For the right-hand side of~\Eqref{(96)} we have to consider +the energy tensor for matter distributed like a cloud of +dust. According to what has gone before we must +therefore put +\[ +T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds} +\] +specialized for the case of rest. But in addition, we +shall add a pressure term that may be physically established +as follows. Matter consists of electrically charged +particles. On the basis of Maxwell's theory these +cannot be conceived of as electromagnetic fields free +from singularities. In order to be consistent with the +\PageSep{117} +facts, it is necessary to introduce energy terms, not contained +in Maxwell's theory, so that the single electric +particles may hold together in spite of the mutual repulsions +between their elements, charged with electricity +of one sign. For the sake of consistency with this fact, +Poincaré has assumed a pressure to exist inside these +particles which balances the electrostatic repulsion. It +cannot, however, be asserted that this pressure vanishes +outside the particles. We shall be consistent with this +circumstance if, in our phenomenological presentation, +we add a pressure term. This must not, however, be +confused with a hydrodynamical pressure, as it serves +only for the energetic presentation of the dynamical +relations inside matter. In this sense we put +\[ +T_{\mu\nu} + = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + - g_{\mu\nu} p\Add{.} +\Tag{(122)} +\] + +In our special case we have, therefore, to put +\begin{align*} +T_{\mu\nu} + &= \gamma_{\mu\nu} p \text{ (for $\mu$~and~$\nu$ from~$1$ to~$3$)}\Add{,} \\ +T_{44} &= \sigma - p\Add{,} \\ +T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p + = \sigma - 4p. +\end{align*} +Observing that the field equation~\Eqref{(96)} may be written +in the form +\[ +R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T)\Add{,} +\] +we get from~\Eqref{(96)} the equations, +\begin{align*} ++\frac{2}{a^{2}} \gamma_{\mu\nu} + &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}\Add{,} \\ +0 &= -\kappa \left(\frac{\sigma}{2} + p\right). +\end{align*} +\PageSep{118} +From this follows +\[ +\left. +\begin{aligned} +p &= -\frac{\sigma}{2}\Add{,} \\ +a &= \sqrt{\frac{2}{\kappa\sigma}}\Add{.} +\end{aligned} +\right\} +\Tag{(123)} +\] + +If the universe is quasi-Euclidean, and its radius of +\index{Radius of Universe}% +\index{Universe@{Universe, Finiteness of}!Radius of}% +curvature therefore infinite, then $\sigma$~would vanish. But +it is improbable that the mean density of matter in the +universe is actually zero; this is our third argument +against the assumption that the universe is quasi-Euclidean. +Nor does it seem possible that our hypothetical +pressure can vanish; the physical nature of this +pressure can be appreciated only after we have a better +theoretical knowledge of the electromagnetic field. +According to the second of equations~\Eqref{(123)} the radius,~$a$, +of the universe is determined in terms of the total +mass,~$M$, of matter, by the equation +\[ +a = \frac{M\kappa}{4\pi^{2}}\Add{.} +\Tag{(124)} +\] +The complete dependence of the geometrical upon the +physical properties becomes clearly apparent by means +of this equation. + +Thus we may present the following arguments against +the conception of a space-infinite, and for the conception +of a space-bounded, universe:--- + +1.~From the standpoint of the theory of relativity, +the condition for a closed surface is very much simpler +than the corresponding boundary condition at infinity +of the quasi-Euclidean structure of the universe. +\PageSep{119} + +2.~The idea that Mach expressed, that inertia depends +\index{Mach}% +upon the mutual action of bodies, is contained, to a +first approximation, in the equations of the theory of +relativity; it follows from these equations that inertia +depends, at least in part, upon mutual actions between +masses. As it is an unsatisfactory assumption to make +that inertia depends in part upon mutual actions, and +in part upon an independent property of space, Mach's +idea gains in probability. But this idea of Mach's +corresponds only to a finite universe, bounded in space, +and not to a quasi-Euclidean, infinite universe. From +the standpoint of epistemology it is more satisfying to +have the mechanical properties of space completely determined +by matter, and this is the case only in a space-bounded +universe. + +3.~An infinite universe is possible only if the mean +density of matter in the universe vanishes. Although +such an assumption is logically possible, it is less probable +than the assumption that there is a finite mean +density of matter in the universe. +\PageSep{120} +%[Blank page] +\PageSep{121} + + +\PrintIndex +\iffalse %%%% Index text %%%% +%INDEX +% A + +Accelerated masses, inductive action of 113 + +Addition@{Addition and subtraction of tensors}#addition 14 + theorem of velocities 40 +% B + +Biot-Savart force 46 +% C + +Centrifugal force 67 + +Clocks, moving#clocks 39 + +Compressible viscous fluid 22 + +Concept@{Concept of space}#concept 3 + time 30 + +Conditions of orthogonality 7 + +Congruence, theorems of#congruence 3 + +Conservation principles 55 + +Continuum, four-dimensional#continuum 33 + +Contraction of tensors 15 + +Contravariant@{Contra-variant vectors}#contra-variant 72 + tensors 75 + +Coordinates@{Co-ordinates, preferred systems of}#co-ordinates 8 + +Covariance@{Co-variance of equation of continuity}#co-variance 22 + +Covariant@{Co-variant}#co-variant 12 %** et seq. + vector 72 + +Criticism of principle of inertia#criticism 65 + +Criticisms of theory of relativity#criticisms 31 + +Curvilinear co-ordinates 68 +% D + +Differentiation of tensors 76, 79 + +Displacement of spectral lines 101 +% E + +Energy@{Energy and mass}#energy 48, 51 + tensor@{tensor of electromagnetic field}#tensor 52 + of matter 56 + +Equation of continuity, co-variance of#co-variance 22 + +Equations of motion of \Change{materia}{material} particle#equations of motion 52 + +Equivalence of mass and energy#equivalence 51 + +Equivalent spaces of reference#equivalent 26 + +Euclidean geometry 4 +% F + +Finiteness of universe 110 + +Fizeau 29 + +Four-dimensional continuum 33 + +Four-vector 43 + +Fundamental tensor 74 +% G + +Galilean@{Galilean regions}#Galilean 65 + transformation 28 + +Gauss 68 + +Geodetic lines 86 + +Geometry, Euclidean 4 + +Gravitational mass 63 + +Gravitation constant 98 +% H + +Homogeneity of space 17 + +Hydrodynamical equations 56 + +Hypotheses of pre-relativity physics 77 +%\PageSep{122} +% I + +Inductive action of accelerated masses 113 + +Inert and gravitational mass, equality of#gravitational mass 63 + +Invariant 10 %** et seq. + +Isotropy of space 17 +% K + +Kaluza 108 +% L + +Levi-Civita 77 + +Light cone@{Light-cone}#light-cone 42 + +Light ray, path of 102 + +Light time@{Light-time}#light-time 34 + +Linear orthogonal transformation 7 + +Lorentz@{Lorentz electromotive force}#Lorentz 46 + transformation 32 +% M + +Mach 62, 109, 110, 111, 113, 119 + +Mass and Energy 48, 51 + equality of gravitational and inert#gravitational 63 + gravitational 63 + +Maxwell's equations 23 + +Mercury, perihelion of#Mercury 103, 107 + +Michelson and Morley 29 + +Minkowski 34 + +Motion of particle, equations of#equations of motion 52 + +Moving measuring rods and clocks#clocks 39 + +Multiplication of tensors#multiplication 14 +% N + +Newtonian gravitation constant#Newtonian 98 +% O + +Operations on tensors 14 %** et seq. + +Orthogonality, conditions of#orthogonality 7 + +Orthogonal transformations, linear 7 +% P + +Path of light ray 102 + +Perihelion of Mercury 103, 107 + +Poisson's equation 90 + +Preferred systems of co-ordinates 8 + +Prerelativity@{Pre-relativity physics, hypotheses of}#pre-relativity 27 + +Principle@{Principle of equivalence}#principle of 64 + inertia, criticism of 65 + +Principles of conservation 55 +% R + +Radius of Universe#radius 118 + +Rank of tensor#rank 14 + +Ray of light, path of 102 + +Reference, space of 4 + +Riemann 72 + tensor 82, 85, 110 + +Rods (measuring) and clocks in motion#clocks 39 + +Rotation 66 +% S + +Simultaneity 17, 30 + +Sitter 29 + +Skew-symmetrical tensor 15 + +Solar Eclipse expedition (1919)#eclipse 103 + +Space@{Space, Concept of}#space 3 %[** TN: "concept of" in the original] + Homogeneity of 17 %[** TN: "homogeneity of" in the original] + Isotropy of 17 + +Spaces of reference 4 + equivalence of 26 + +Special Lorentz transformation 36 + +Spectral lines, displacement of 101 + +Straightest lines 86 + +Stress tensor 22 + +Symmetrical tensor 15 + +Systems of co-ordinates, preferred#preferred 8 + +% T + +%Tensor, 12 et seq., 72 et seq. +Tensor 12, 72 + Addition and subtraction of 14 + Contraction of 15 + Fundamental 74 + Multiplication of 14 +%\PageSep{123} +%Tensor, + operations 14 %** et seq. + Rank of 14 + Symmetrical and Skew-symmetrical 15 + +Tensors, formation by differentiation 76 + +Theorem for addition of velocities 40 + +Theorems of congruence 3 + +Theory of relativity, criticisms of#criticisms 31 + +Thirring 113 + +Time-concept 30 + +Time-space concept 33 + +Transformation@{Transformation, Galilean}#transformation 28 + Linear orthogonal 7 +% U + +Universe@{Universe, Finiteness of}#universe 110 %[** TN: "finiteness of" in the original] + Radius of 118 %[** TN: "radius of" in the original] +% V + +Vector@{Vector, co-variant}#vector 72 + contra-variant 72 + +Velocities, addition theorem of 40 + +Viscous compressible fluid 22 +% W + +Weyl 77, 103, 108 +\fi %%%% End of index text %%%% +\PageSep{124} +\cleardoublepage +\newpage +\begin{CenterPage} +\scriptsize +PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN +\end{CenterPage} + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\PGLicenseInit +\begin{PGtext} +End of Project Gutenberg's The Meaning of Relativity, by Albert Einstein + +*** END OF THIS PROJECT GUTENBERG EBOOK THE MEANING OF RELATIVITY *** + +***** This file should be named 36276-pdf.pdf or 36276-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/2/7/36276/ + +Produced by Andrew D. 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(1.875in,1.8125in)--(1.875in,1.85in); +\draw (1.875in,1.85in)--(1.875in,1.8875in); +\draw (1.875in,1.9625in)--(1.875in,2in); +\draw (1.875in,2in)--(1.875in,2.0375in); +\draw (1.875in,2.1125in)--(1.875in,2.15in); +\draw (1.875in,2.15in)--(1.875in,2.1875in); +\draw (1.875in,2.2625in)--(1.875in,2.3in); +\draw (1.875in,2.3in)--(1.875in,2.3375in); +\draw (1.875in,2.4125in)--(1.875in,2.45in); +\draw (1.875in,2.45in)--(1.875in,2.4875in); +\draw (1.875in,2.5625in)--(1.875in,2.6in); +\draw (1.875in,2.6in)--(1.875in,2.6375in); +\draw (1.875in,2.7125in)--(1.875in,2.75in); +\draw (1.875in,2.75in)--(1.875in,2.7875in); +\draw (1.875in,2.8625in)--(1.875in,2.9in); +\draw (1.875in,2.9in)--(1.875in,2.9375in); +\draw (1.875in,3.0125in)--(1.875in,3.05in); +\draw (1.875in,3.05in)--(1.875in,3.0875in); +\draw (1.875in,3.1625in)--(1.875in,3.2in); +\draw (1.875in,3.2in)--(1.875in,3.2375in); +\draw (1.875in,3.3125in)--(1.875in,3.35in); +\draw (1.875in,3.35in)--(1.875in,3.3875in); +\draw (1.875in,3.4625in)--(1.875in,3.5in); +\draw (1.875in,0.125in)--(1.875in,0.136719in); +\draw (1.875in,0.160156in)--(1.875in,0.171875in); +\draw (1.875in,0.171875in)--(1.875in,0.183594in); +\draw (1.875in,0.207031in)--(1.875in,0.21875in); +\draw (1.875in,0.21875in)--(1.875in,0.230469in); +\draw (1.875in,0.253906in)--(1.875in,0.265625in); +\draw (1.875in,0.265625in)--(1.875in,0.277344in); +\draw (1.875in,0.300781in)--(1.875in,0.3125in); +\draw (1.875in,0.3125in)--(1.875in,0.324219in); +\draw (1.875in,0.347656in)--(1.875in,0.359375in); +\draw (1.875in,0.359375in)--(1.875in,0.371094in); +\draw (1.875in,0.394531in)--(1.875in,0.40625in); +\draw (1.875in,0.40625in)--(1.875in,0.417969in); +\draw (1.875in,0.441406in)--(1.875in,0.453125in); +\draw (1.875in,0.453125in)--(1.875in,0.464844in); +\draw (1.875in,0.488281in)--(1.875in,0.5in); +\pgftext[at={\pgfpoint{3.80535in}{2in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$x_{1}$}}} +\pgftext[at={\pgfpoint{2.83927in}{2.32855in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$x_{2}$}}} +\pgftext[at={\pgfpoint{1.875in}{4.05535in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$l$}}} +\end{tikzpicture} diff --git a/old/36276-t-2021-06-11/images/sources/041.xp b/old/36276-t-2021-06-11/images/sources/041.xp new file mode 100644 index 0000000..83b2f14 --- /dev/null +++ b/old/36276-t-2021-06-11/images/sources/041.xp @@ -0,0 +1,49 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +int main() +{ + // aspect ratio hack to get perpendicular cone generators + picture(P(-2,-2), P(2,2), "3.75 x 4in"); + + begin(); + double ratio(0.25), ht(1.5), m(0.3), th(Atan(ratio)); + + P Px2(1, m), + Pht1( ht*Cos(th), ht*(1 - ratio*Sin(th))), + Pht2(-ht*Cos(th), ht*(1 - ratio*Sin(th))); + + bold(); + + line(P(xmin(), 0), P(xmax(), 0)); + line(P(0, ymin()), P(0, -ht*(1 + ratio))); + line(P(0, ht), P(0, ymax())); + line(-Px2, Px2); + + line(-Pht1, Pht1); + line(-Pht2, Pht2); + + ellipse(P(0, -ht), P(ht,0), P(0, ratio*ht), -M_PI-th, th, 60); + ellipse(P(0, ht), ht*P(1, ratio)); + + ddot(P(0,-ht)); + ddot(P(0, ht)); + + dashed(); + ellipse(P(0, -ht), P(ht,0), P(0, ratio*ht), th, M_PI-th, 35); + white(); + dot(P(0, ht*(-1 + ratio))); + black(); + line(P(0, -ht), P(0, ht)); + + dash_size(2); + line(P(0, -ht*(1 + ratio)), P(0, -ht)); + + label(P(xmax(), 0), P(4,0), "$x_{1}$", r); + label(Px2, P(2,2), "$x_{2}$", tr); + label(P(0, ymax()), P(0,4), "$l$", t); + + tikz_format(); + end(); +} diff --git a/old/36276-t-2021-06-11/images/sources/047.eepic b/old/36276-t-2021-06-11/images/sources/047.eepic new file mode 100644 index 0000000..541d042 --- /dev/null +++ b/old/36276-t-2021-06-11/images/sources/047.eepic @@ -0,0 +1,138 @@ +%% Generated from 047.xp on Wed May 18 07:53:50 EDT 2011 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,2] x [0,2] +%% Actual size: 4 x 4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\xdefinecolor{rgb_cccccc}{rgb}{0.8,0.8,0.8}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,4in); +\pgfsetlinewidth{0.8pt} +\draw (0in,0in)--(2in,0in)--(4in,0in); +\draw (0in,0in)--(0in,2in)--(0in,4in); +\draw (0in,1in)--(1.75in,1in)--(3.5in,1in); +\draw (0.906in,0.4in)--(0.995008in,0.57in)--(1.0753in,0.74in)-- + (1.14791in,0.91in)--(1.21387in,1.08in)--(1.27423in,1.25in)-- + (1.33002in,1.42in)--(1.38226in,1.59in)--(1.43201in,1.76in)-- + (1.4803in,1.93in)--(1.52816in,2.1in)--(1.57663in,2.27in)-- + (1.62674in,2.44in)--(1.67954in,2.61in)--(1.73606in,2.78in)-- + (1.79733in,2.95in)--(1.86439in,3.12in)--(1.93828in,3.29in)-- + (2.02004in,3.46in)--(2.11069in,3.63in)--(2.21128in,3.8in); +\draw (1.28872in,0.2in)--(1.38931in,0.37in)--(1.47996in,0.54in)-- + (1.56172in,0.71in)--(1.63561in,0.88in)--(1.70267in,1.05in)-- + (1.76394in,1.22in)--(1.82046in,1.39in)--(1.87326in,1.56in)-- + (1.92337in,1.73in)--(1.97184in,1.9in)--(2.0197in,2.07in)-- + (2.06799in,2.24in)--(2.11774in,2.41in)--(2.16998in,2.58in)-- + (2.22577in,2.75in)--(2.28613in,2.92in)--(2.35209in,3.09in)-- + (2.4247in,3.26in)--(2.50499in,3.43in)--(2.594in,3.6in); +\pgfsetfillcolor{rgb_cccccc} +\draw [fill](1.41468in,1.7in)--(1.44347in,1.8in)-- + (1.47184in,1.9in)--(1.5in,2in)--(1.52816in,2.1in)-- + (1.55653in,2.2in)--(1.58532in,2.3in)--(1.61475in,2.4in)-- + (1.64502in,2.5in)--(1.67634in,2.6in)--(1.70893in,2.7in)-- + (2.14195in,2.49in)--(2.1088in,2.38in)--(2.07663in,2.27in)-- + (2.04514in,2.16in)--(2.01407in,2.05in)--(1.98312in,1.94in)-- + (1.95201in,1.83in)--(1.92048in,1.72in)--(1.88823in,1.61in)-- + (1.85498in,1.5in)--(1.41468in,1.7in)--cycle; +\draw (0in,0.6in)--(0.0397727in,0.6in); +\draw (0.119318in,0.6in)--(0.159091in,0.6in); +\draw (0.159091in,0.6in)--(0.198864in,0.6in); +\draw (0.278409in,0.6in)--(0.318182in,0.6in); +\draw (0.318182in,0.6in)--(0.357955in,0.6in); +\draw (0.4375in,0.6in)--(0.477273in,0.6in); +\draw (0.477273in,0.6in)--(0.517045in,0.6in); +\draw 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(2.34659in,0.6in)--(2.38636in,0.6in); +\draw (2.38636in,0.6in)--(2.42614in,0.6in); +\draw (2.50568in,0.6in)--(2.54545in,0.6in); +\draw (2.54545in,0.6in)--(2.58523in,0.6in); +\draw (2.66477in,0.6in)--(2.70455in,0.6in); +\draw (2.70455in,0.6in)--(2.74432in,0.6in); +\draw (2.82386in,0.6in)--(2.86364in,0.6in); +\draw (2.86364in,0.6in)--(2.90341in,0.6in); +\draw (2.98295in,0.6in)--(3.02273in,0.6in); +\draw (3.02273in,0.6in)--(3.0625in,0.6in); +\draw (3.14205in,0.6in)--(3.18182in,0.6in); +\draw (3.18182in,0.6in)--(3.22159in,0.6in); +\draw (3.30114in,0.6in)--(3.34091in,0.6in); +\draw (3.34091in,0.6in)--(3.38068in,0.6in); +\draw (3.46023in,0.6in)--(3.5in,0.6in); +\draw (0in,3in)--(0.0397727in,3in); +\draw (0.119318in,3in)--(0.159091in,3in); +\draw (0.159091in,3in)--(0.198864in,3in); +\draw (0.278409in,3in)--(0.318182in,3in); +\draw (0.318182in,3in)--(0.357955in,3in); +\draw (0.4375in,3in)--(0.477273in,3in); +\draw (0.477273in,3in)--(0.517045in,3in); +\draw (0.596591in,3in)--(0.636364in,3in); +\draw (0.636364in,3in)--(0.676136in,3in); +\draw (0.755682in,3in)--(0.795455in,3in); +\draw (0.795455in,3in)--(0.835227in,3in); +\draw (0.914773in,3in)--(0.954545in,3in); +\draw (0.954545in,3in)--(0.994318in,3in); +\draw (1.07386in,3in)--(1.11364in,3in); +\draw (1.11364in,3in)--(1.15341in,3in); +\draw (1.23295in,3in)--(1.27273in,3in); +\draw (1.27273in,3in)--(1.3125in,3in); +\draw (1.39205in,3in)--(1.43182in,3in); +\draw (1.43182in,3in)--(1.47159in,3in); +\draw (1.55114in,3in)--(1.59091in,3in); +\draw (1.59091in,3in)--(1.63068in,3in); +\draw (1.71023in,3in)--(1.75in,3in); +\draw (1.75in,3in)--(1.78977in,3in); +\draw (1.86932in,3in)--(1.90909in,3in); +\draw (1.90909in,3in)--(1.94886in,3in); +\draw (2.02841in,3in)--(2.06818in,3in); +\draw (2.06818in,3in)--(2.10795in,3in); +\draw (2.1875in,3in)--(2.22727in,3in); +\draw (2.22727in,3in)--(2.26705in,3in); +\draw (2.34659in,3in)--(2.38636in,3in); +\draw (2.38636in,3in)--(2.42614in,3in); +\draw (2.50568in,3in)--(2.54545in,3in); +\draw (2.54545in,3in)--(2.58523in,3in); +\draw (2.66477in,3in)--(2.70455in,3in); +\draw (2.70455in,3in)--(2.74432in,3in); +\draw (2.82386in,3in)--(2.86364in,3in); +\draw (2.86364in,3in)--(2.90341in,3in); +\draw (2.98295in,3in)--(3.02273in,3in); +\draw (3.02273in,3in)--(3.0625in,3in); +\draw (3.14205in,3in)--(3.18182in,3in); +\draw (3.18182in,3in)--(3.22159in,3in); +\draw (3.30114in,3in)--(3.34091in,3in); +\draw (3.34091in,3in)--(3.38068in,3in); +\draw (3.46023in,3in)--(3.5in,3in); +\pgftext[at={\pgfpoint{4.05535in}{0in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$x_{1}$}}} +\pgftext[at={\pgfpoint{-0.110696in}{4in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$l$}}} +\pgftext[at={\pgfpoint{-0.055348in}{0.6in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$l_{1}$}}} +\pgftext[at={\pgfpoint{-0.110696in}{1in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$l}$}}} +\pgftext[at={\pgfpoint{-0.055348in}{3in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$l_{2}$}}} +\pgftext[at={\pgfpoint{-0.027674in}{-0.027674in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$O$}}} +\pgftext[at={\pgfpoint{1.18359in}{1.02767in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{1.76662in}{1.02767in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$B$}}} +\end{tikzpicture} diff --git a/old/36276-t-2021-06-11/images/sources/047.xp b/old/36276-t-2021-06-11/images/sources/047.xp new file mode 100644 index 0000000..3ed9ffc --- /dev/null +++ b/old/36276-t-2021-06-11/images/sources/047.xp @@ -0,0 +1,61 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +double dx(0.25), ml1(0.3), ml(0.5), ml2(1.5), mX(1.75), + tmin(-0.25), dt(0.1), tmax(-tmin + dt); + +P fA(double t) +{ + return P(0.75*(1 + 0.375*t*(1 + 0.5*t*t)), 1 + t); +} + +P fB(double t) +{ + return fA(t) + P(dx,0); +} + +int main() +{ + picture(P(0, 0), P(2,2), "4 x 4in"); + + begin(); + + P pA(fA(ml - 1)), pB(fB(ml - 1)); + + bold(); + + line(P(0, 0), P(xmax(), 0)); + line(P(0, 0), P(0, ymax())); + + line(P(0, ml), P(mX, ml)); + + plot(fA, -0.9 + dt, 0.9, 20); + plot(fB, -0.9, 0.9 - dt, 20); + + fill(Black(0.2)); + path bd(fA, tmin + dt, tmax, 10); + bd -= path(fB, tmin, tmax - 0.5*dt, 10); + bd.close().fill().draw(); + nofill(); + + dashed(); + + line(P(0, ml1), P(mX, ml1)); + line(P(0, ml2), P(mX, ml2)); + + label(P(xmax(), 0), P(4,0), "$x_{1}$", r); + label(P(0, ymax()), P(-8,0), "$l$", l); + + label(P(0, ml1), P(-4,0), "$l_{1}$", l); + label(P(0, ml), P(-8,0), "$l}$", l); + label(P(0, ml2), P(-4,0), "$l_{2}$", l); + + label(P(0,0), P(-2,-2), "$O$", bl); + + label(pA, P(0,2), "$A$", tl); + label(pB, P(6,2), "$B$", tr); + + tikz_format(); + end(); +} diff --git a/old/36276-t-2021-06-11/images/sources/056.eepic b/old/36276-t-2021-06-11/images/sources/056.eepic new file mode 100644 index 0000000..1ec6d17 --- /dev/null +++ b/old/36276-t-2021-06-11/images/sources/056.eepic @@ -0,0 +1,34 @@ +%% Generated from 056.xp on Tue May 17 15:56:19 EDT 2011 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,2] x [0,2] +%% Actual size: 4 x 4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,4in); +\pgfsetlinewidth{0.8pt} +\draw (0in,0in)--(2in,0in)--(4in,0in); +\draw (0in,0in)--(0in,2in)--(0in,4in); +\draw (0in,1in)--(1.75in,1in)--(3.5in,1in); +\draw (1.008in,0.4in)--(1.0948in,0.57in)--(1.16998in,0.74in)-- + (1.23492in,0.91in)--(1.291in,1.08in)--(1.3396in,1.25in)-- + (1.3821in,1.42in)--(1.41989in,1.59in)--(1.45435in,1.76in)-- + (1.48686in,1.93in)--(1.5188in,2.1in)--(1.55155in,2.27in)-- + (1.58649in,2.44in)--(1.62501in,2.61in)--(1.66849in,2.78in)-- + (1.71831in,2.95in)--(1.77586in,3.12in)--(1.8425in,3.29in)-- + (1.91963in,3.46in)--(2.00863in,3.63in)--(2.11088in,3.8in); +\draw (1.38912in,0.2in)--(1.49137in,0.37in)--(1.58037in,0.54in)-- + (1.6575in,0.71in)--(1.72414in,0.88in)--(1.78169in,1.05in)-- + (1.83151in,1.22in)--(1.87499in,1.39in)--(1.91351in,1.56in)-- + (1.94845in,1.73in)--(1.9812in,1.9in)--(2.01314in,2.07in)-- + (2.04565in,2.24in)--(2.08011in,2.41in)--(2.1179in,2.58in)-- + (2.1604in,2.75in)--(2.209in,2.92in)--(2.26508in,3.09in)-- + (2.33002in,3.26in)--(2.4052in,3.43in)--(2.492in,3.6in); +\pgftext[at={\pgfpoint{4.05535in}{0in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$x_{1}$}}} +\pgftext[at={\pgfpoint{-0.110696in}{4in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$l$}}} +\end{tikzpicture} diff --git a/old/36276-t-2021-06-11/images/sources/056.xp b/old/36276-t-2021-06-11/images/sources/056.xp new file mode 100644 index 0000000..50367a0 --- /dev/null +++ b/old/36276-t-2021-06-11/images/sources/056.xp @@ -0,0 +1,39 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +double dx(0.25), ml(0.5), mX(1.75), + tmin(-0.25), dt(0.1), tmax(-tmin + dt); + +P fA(double t) +{ + return P(0.75*(1 + 0.25*t*(1 + t*t)), 1 + t); +} + +P fB(double t) +{ + return fA(t) + P(dx,0); +} + +int main() +{ + picture(P(0, 0), P(2,2), "4 x 4in"); + + begin(); + + bold(); + + line(P(0, 0), P(xmax(), 0)); + line(P(0, 0), P(0, ymax())); + + line(P(0, ml), P(mX, ml)); + + plot(fA, -0.9 + dt, 0.9, 20); + plot(fB, -0.9, 0.9 - dt, 20); + + label(P(xmax(), 0), P(4,0), "$x_{1}$", r); + label(P(0, ymax()), P(-8,0), "$l$", l); + + tikz_format(); + end(); +} diff --git a/old/36276-t-2021-06-11/images/sources/083-300.eps b/old/36276-t-2021-06-11/images/sources/083-300.eps new file mode 100644 index 0000000..214f31d --- /dev/null +++ b/old/36276-t-2021-06-11/images/sources/083-300.eps @@ -0,0 +1,88 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: potrace 1.8, written by Peter Selinger 2001-2007 +%%LanguageLevel: 2 +%%BoundingBox: 0 0 468 364 +%%Pages: 1 +%%EndComments +%%Page: 1 1 +currentfile /ASCII85Decode filter /LZWDecode filter cvx exec +J.'GRZOs"gA.TUeVISs@bR,/NO(T\_Yh4@J6j[M4)8+mQN'n!XZ\3Wk0/t#m@KDR@U`_@Z+ 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.net + + +Title: The Meaning of Relativity + Four lectures delivered at Princeton University, May, 1921 + +Author: Albert Einstein + +Translator: Edwin Plimpton Adams + +Release Date: May 29, 2011 [EBook #36276] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE MEANING OF RELATIVITY *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang. (This ebook was produced using +OCR text generously provided by Northeastern University's +Snell Library through the Internet Archive.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\frontmatter +\pagestyle{empty} +\PageSep{i} +\null\vfill +\noindent\textbf{\large THE MEANING OF RELATIVITY} +\vfill +\cleardoublepage +\PageSep{ii} +%[Blank page] +\PageSep{iii} +% Title page +\begin{center} +\textbf{\huge THE MEANING OF \\[8pt] +RELATIVITY} +\bigskip + +\normalsize +FOUR LECTURES DELIVERED AT \\ +PRINCETON UNIVERSITY, MAY, 1921 +\bigskip + +{\footnotesize BY} \\[8pt] +\textbf{\huge ALBERT EINSTEIN} +\vfill\vfill + +{\footnotesize WITH FOUR DIAGRAMS} +\vfill\vfill\vfill + +\normalsize +PRINCETON \\ +PRINCETON UNIVERSITY PRESS \\ +1923 +\end{center} +\newpage +\PageSep{iv} +\begin{center} +\null +\vfill\vfill +\itshape\footnotesize +Copyright 1922 \\ +Princeton University Press \\ +Published 1922 +\vfill +\upshape +PRINTED IN GREAT BRITAIN \\ +AT THE ABERDEEN UNIVERSITY PRESS \\ +ABERDEEN +\vfill + +\TranslatorsNote{% + \textsc{Note}.---The translation of these lectures into English + was made by \textsc{Edwin Plimpton Adams}, Professor + of Physics in Princeton University} +\cleardoublepage +\end{center} +\PageSep{v} +\TableofContents +\iffalse %%%% Table of contents text %%%% +CONTENTS + +LECTURE I PAGE +Space and Time in Pre-Relativity Physics 1 + +LECTURE II +The Theory of Special Relativity 26 + +LECTURE III +The General Theory of Relativity 61 + +LECTURE IV +The General Theory of Relativity (continued) 87 + +Index 121 +\fi %%%% Table of contents text %%%% +\newpage +\PageSep{1} + + +\mainmatter +\BookMark{-1}{Main Matter} +%[** TN: Text printed by \Lecture macro: THE MEANING OF RELATIVITY] +\Lecture{I}{Space and Time in Pre-Relativity +Physics}{Pre-Relativity Physics} + +\First{The} theory of relativity is intimately connected with +the theory of space and time. I shall therefore begin +with a brief investigation of the origin of our ideas of space +and time, although in doing so I know that I introduce a +controversial subject. The object of all science, whether +natural science or psychology, is to co-ordinate our experiences +and to bring them into a logical system. How are +our customary ideas of space and time related to the +character of our experiences? + +The experiences of an individual appear to us arranged +in a series of events; in this series the single events which +we remember appear to be ordered according to the criterion +of ``earlier'' and ``later,'' which cannot be analysed further. +There exists, therefore, for the individual, an I-time, or +subjective time. This in itself is not measurable. I can, +indeed, associate numbers with the events, in such a way +that a greater number is associated with the later event +than with an earlier one; but the nature of this association +may be quite arbitrary. This association I can define by +means of a clock by comparing the order of events furnished +\PageSep{2} +by the clock with the order of the given series of events. +We understand by a clock something which provides a +series of events which can be counted, and which has other +properties of which we shall speak later. + +By the aid of speech different individuals can, to a certain +extent, compare their experiences. In this way it is shown +that certain sense perceptions of different individuals +correspond to each other, while for other sense perceptions +no such correspondence can be established. We are accustomed +to regard as real those sense perceptions which +are common to different individuals, and which therefore +are, in a measure, impersonal. The natural sciences, and +in particular, the most fundamental of them, physics, deal +with such sense perceptions. The conception of physical +bodies, in particular of rigid bodies, is a relatively constant +complex of such sense perceptions. A clock is also a body, +or a system, in the same sense, with the additional property +that the series of events which it counts is formed of +elements all of which can be regarded as equal. + +The only justification for our concepts and system of +concepts is that they serve to represent the complex of +our experiences; beyond this they have no legitimacy. I +am convinced that the philosophers have had a harmful +effect upon the progress of scientific thinking in removing +certain fundamental concepts from the domain of empiricism, +where they are under our control, to the intangible +heights of the \textit{a~priori}. For even if it should appear that +the universe of ideas cannot be deduced from experience +by logical means, but is, in a sense, a creation of the human +mind, without which no science is possible, nevertheless +\PageSep{3} +this universe of ideas is just as little independent of the +nature of our experiences as clothes are of the form of +the human body. This is particularly true of our concepts +of time and space, which physicists have been +\index{Space@{Space, Concept of}}% +obliged by the facts to bring down from the Olympus of +the \textit{a~priori} in order to adjust them and put them in a +serviceable condition. + +We now come to our concepts and judgments of space. +It is essential here also to pay strict attention to the +relation of experience to our concepts. It seems to me +that Poincaré clearly recognized the truth in the account +he gave in his book, ``La Science et l'Hypothese.'' +Among all the changes which we can perceive in a rigid +body those are marked by their simplicity which can be +made reversibly by an arbitrary motion of the body; +Poincaré calls these, changes in position. By means of +simple changes in position we can bring two bodies into +contact. The theorems of congruence, fundamental in +\index{Congruence, theorems of}% +\index{Theorems of congruence}% +geometry, have to do with the laws that govern such +changes in position. For the concept of space the following +\index{Concept@{Concept of space}}% +seems essential. We can form new bodies by bringing +bodies $B$,~$C$,~\dots\ up to body~$A$; we say that we \emph{continue} +body~$A$. We can continue body~$A$ in such a way that +it comes into contact with any other body,~$X$. The +ensemble of all continuations of body~$A$ we can designate +as the ``space of the body~$A$.'' Then it is true that all +bodies are in the ``space of the (arbitrarily chosen) body~$A$.'' +In this sense we cannot speak of space in the +abstract, but only of the ``space belonging to a body~$A$.'' +The earth's crust plays such a dominant rôle in our daily +\PageSep{4} +life in judging the relative positions of bodies that it has +led to an abstract conception of space which certainly +cannot be defended. In order to free ourselves from this +fatal error we shall speak only of ``bodies of reference,'' +or ``space of reference.'' It was only through the theory +\index{Reference, space of}% +\index{Spaces of reference}% +of general relativity that refinement of these concepts +became necessary, as we shall see later. + +I shall not go into detail concerning those properties +of the space of reference which lead to our conceiving +points as elements of space, and space as a continuum. +Nor shall I attempt to analyse further the properties of +space which justify the conception of continuous series +of points, or lines. If these concepts are assumed, together +with their relation to the solid bodies of experience, then +it is easy to say what we mean by the three-dimensionality +of space; to each point three numbers, $x_{1}$,~$x_{2}$,~$x_{3}$ (co-ordinates), +may be associated, in such a way that this +association is uniquely reciprocal, and that $x_{1}$,~$x_{2}$\Change{,}{} and~$x_{3}$ +vary continuously when the point describes a continuous +series of points (a line). + +It is assumed in pre-relativity physics that the laws of +the orientation of ideal rigid bodies are consistent with +Euclidean geometry. What this means may be expressed +\index{Euclidean geometry}% +\index{Geometry, Euclidean}% +as follows: Two points marked on a rigid body form +an \emph{interval}. Such an interval can be oriented at rest, +relatively to our space of reference, in a multiplicity of +ways. If, now, the points of this space can be referred +to co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, in such a way that the differences +of the co-ordinates, $\Delta x_{1}$,~$\Delta x_{2}$,~$\Delta x_{3}$, of the two ends of the +interval furnish the same sum of squares, +\[ +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}\Add{,} +\Tag{(1)} +\] +\PageSep{5} +for every orientation of the interval, then the space of +reference is called Euclidean, and the co-ordinates +Cartesian.\footnote + {This relation must hold for an arbitrary choice of the origin and of the + direction (ratios $\Delta x_{1} : \Delta x_{2} : \Delta x_{3}$) of the interval.} +It is sufficient, indeed, to make this assumption +in the limit for an infinitely small interval. Involved +in this assumption there are some which are rather less +special, to which we must call attention on account of +their fundamental significance. In the first place, it is +assumed that one can move an ideal rigid body in an +arbitrary manner. In the second place, it is assumed +that the behaviour of ideal rigid bodies towards orientation +is independent of the material of the bodies and their +changes of position, in the sense that if two intervals can +once be brought into coincidence, they can always and +everywhere be brought into coincidence. Both of these +assumptions, which are of fundamental importance for +geometry and especially for physical measurements, +naturally arise from experience; in the theory of general +relativity their validity needs to be assumed only for +bodies and spaces of reference which are infinitely small +compared to astronomical dimensions. + +The quantity~$s$ we call the length of the interval. In +order that this may be uniquely determined it is necessary +to fix arbitrarily the length of a definite interval; for +example, we can put it equal to~$1$ (unit of length). Then +the lengths of all other intervals may be determined. If +we make the~$x_{\nu}$ linearly dependent upon a parameter~$\lambda$, +\[ +x_{\nu} = a_{\nu} + \lambda b_{\nu}, +\] +\PageSep{6} +we obtain a line which has all the properties of the straight +lines of the Euclidean geometry. In particular, it easily +follows that by laying off $n$~times the interval~$s$ upon a +straight line, an interval of length~$n·s$ is obtained. A +length, therefore, means the result of a measurement +carried out along a straight line by means of a unit +measuring rod. It has a significance which is as independent +of the system of co-ordinates as that of a straight +line, as will appear in the sequel. + +We come now to a train of thought which plays an +analogous rôle in the theories of special and general +relativity. We ask the question: besides the Cartesian +co-ordinates which we have used are there other equivalent +co-ordinates? An interval has a physical meaning which +is independent of the choice of co-ordinates; and so has +the spherical surface which we obtain as the locus of the +end points of all equal intervals that we lay off from an +arbitrary point of our space of reference. If~$x_{\nu}$ as well as~${x'}_{\nu}$ +($\nu$~from $1$~to~$3$) are Cartesian co-ordinates of our space +of reference, then the spherical surface will be expressed +in our two systems of co-ordinates by the equations +\begin{align*} +\sum {\Delta x_{\nu}}^{2} &= \text{const.} +\Tag{(2)} \\ +\sum {\Delta {x'}_{\nu}}^{2} &= \text{const.} +\Tag{(2a)} +\end{align*} +How must the~${x'}_{\nu}$ be expressed in terms of the~$x_{\nu}$ in order +that equations \Eqref{(2)}~and~\Eqref{(2a)} may be equivalent to each +other? Regarding the~${x'}_{\nu}$ expressed as functions of the~$x_{\nu}$, +we can write, by Taylor's theorem, for small values of +the~${\Delta x}_{\nu}$, +\PageSep{7} +\[ +{\Delta x'}_{\nu} + = \sum_{\alpha} \frac{\dd {x'}_{\nu}}{\dd x_{\alpha}}\, \Delta x_{\alpha} + + \frac{1}{2} \sum_{\alpha\Add{,}\beta} + \frac{\dd^{2} {x'}_{\nu}}{\dd x_{\alpha}\, \dd x_{\beta}}\, + \Delta x_{\alpha}\, \Delta x_{\beta}\ldots\Add{.} +\] +If we substitute~\Eqref{(2a)} in this equation and compare with~\Eqref{(1)}, +we see that the~${x'}_{\nu}$ must be linear functions of the~$x_{\nu}$. +If we therefore put +\begin{align*} +{x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}\Add{,} +\Tag{(3)} \\ +\intertext{or} +{\Delta x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}\Add{,} +\Tag{(3a)} +\end{align*} +then the equivalence of equations \Eqref{(2)}~and~\Eqref{(2a)} is expressed +in the form +\[ +\sum {{\Delta x'}_{\nu}}^{2} = \lambda \sum {\Delta x_{\nu}}^{2}\quad +\text{($\lambda$~independent of~$\Delta x_{\nu}$).} +\Tag{(2b)} +\] +It therefore follows that $\lambda$~must be a constant. If we put +$\lambda = 1$, \Eqref{(2b)}~and~\Eqref{(3a)} furnish the conditions +\[ +\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}\Add{,} +\Tag{(4)} +\] +in which $\delta_{\alpha\beta} = 1$, or $\delta_{\alpha\beta} = 0$, according as $\alpha = \beta$ or +$\alpha \neq \beta$. The conditions~\Eqref{(4)} are called the conditions of orthogonality, +\index{Conditions of orthogonality}% +\index{Linear orthogonal transformation}% +\index{Orthogonality, conditions of}% +\index{Orthogonal transformations, linear}% +\index{Transformation@{Transformation, Galilean}!Linear orthogonal}% +and the transformations \Eqref{(3)},~\Eqref{(4)}, linear orthogonal +transformations. If we stipulate that $s^{2} = \sum {\Delta x_{\nu}}^{2}$ shall be +equal to the square of the length in every system of +co-ordinates, and if we always measure with the same unit +scale, then $\lambda$~must be equal to~$1$. Therefore the linear +orthogonal transformations are the only ones by means of +which we can pass from one Cartesian system of co-ordinates +in our space of reference to another. We see +\PageSep{8} +that in applying such transformations the equations of +a straight line become equations of a straight line. +Reversing equations~\Eqref{(3a)} by multiplying both sides by~$b_{\nu\beta}$ +and summing for all the~$\nu$'s, we obtain +\[ +\Change{\sum}{\sum_{\nu}} b_{\nu\beta}\, {\Delta x'}_{\nu} + = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} + = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} + = \Delta x_{\beta}\Add{.} +\Tag{(5)} +\] +The same coefficients,~$b$, also determine the inverse +substitution of~$\Delta x_{\nu}$. Geometrically, $b_{\nu\alpha}$~is the cosine of the +angle between the ${x'}_{\nu}$~axis and the $x_{\alpha}$~axis. + +To sum up, we can say that in the Euclidean geometry +\index{Coordinates@{Co-ordinates, preferred systems of}}% +\index{Preferred systems of co-ordinates}% +\index{Systems of co-ordinates, preferred}% +there are (in a given space of reference) preferred systems +of co-ordinates, the Cartesian systems, which transform +into each other by linear orthogonal transformations. +The distance~$s$ between two points of our space of +reference, measured by a measuring rod, is expressed in +such co-ordinates in a particularly simple manner. The +whole of geometry may be founded upon this conception +of distance. In the present treatment, geometry is +related to actual things (rigid bodies), and its theorems +are statements concerning the behaviour of these things, +which may prove to be true or false. + +One is ordinarily accustomed to study geometry +divorced from any relation between its concepts and +experience. There are advantages in isolating that +which is purely logical and independent of what is, in +principle, incomplete empiricism. This is satisfactory to +the pure mathematician. He is satisfied if he can deduce +his theorems from axioms correctly, that is, without +errors of logic. The question as to whether Euclidean +\PageSep{9} +geometry is true or not does not concern him. But for +our purpose it is necessary to associate the fundamental +concepts of geometry with natural objects; without such +an association geometry is worthless for the physicist. +The physicist is concerned with the question as to +whether the theorems of geometry are true or not. That +Euclidean geometry, from this point of view, affirms +something more than the mere deductions derived +logically from definitions may be seen from the following +simple consideration. + +Between $n$~points of space there are $\dfrac{n(n - 1)}{2}$ distances,~$s_{\mu\nu}$; +between these and the $3n$~co-ordinates we have the +relations +\[ +{s_{\mu\nu}}^{2} + = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + + \dots\Add{.} +\] + +From these $\dfrac{n(n - 1)}{2}$~equations the $3n$~co-ordinates +may be eliminated, and from this elimination at least +$\dfrac{n(n - 1)}{2} - 3n$ equations in the~$s_{\mu\nu}$ will result.\footnote + {In reality there are $\dfrac{n(n - 1)}{2} - 3n + 6$ equations.} +Since +the~$s_{\mu\nu}$ are measurable quantities, and by definition are +independent of each other, these relations between the~$s_{\mu\nu}$ +are not necessary \textit{a~priori}. + +From the foregoing it is evident that the equations of +transformation \Eqref{(3)},~\Eqref{(4)} have a fundamental significance in +Euclidean geometry, in that they govern the transformation +from one Cartesian system of co-ordinates to another. +The Cartesian systems of co-ordinates are characterized +\PageSep{10} +by the property that in them the measurable distance +between two points,~$s$, is expressed by the equation +\[ +s^{2} = \sum {\Delta x_{\nu}}^{2}. +\] + +If $K_{(x_{\nu})}$~and~$K'_{(x_{\nu})}$ are two Cartesian systems of co-ordinates, +then +\[ +\sum {\Delta x_{\nu}}^{2} = \sum {{\Delta x'}_{\nu}}^{2}. +\] + +The right-hand side is identically equal to the left-hand +side on account of the equations of the linear orthogonal +transformation, and the right-hand side differs from the +left-hand side only in that the~$x_{\nu}$ are replaced by the~${x'}_{\nu}$. +This is expressed by the statement that~$\sum {\Delta x_{\nu}}^{2}$ is an +invariant with respect to linear orthogonal transformations. +\index{Invariant|etseq}% +It is evident that in the Euclidean geometry only +such, and all such, quantities have an objective significance, +independent of the particular choice of the Cartesian +co-ordinates, as can be expressed by an invariant with +respect to linear orthogonal transformations. This is the +reason that the theory of invariants, which has to do with +the laws that govern the form of invariants, is so important +for analytical geometry. + +As a second example of a geometrical invariant, consider +a volume. This is expressed by +\[ +V = \iiint dx_{1}\, dx_{2}\, dx_{3}. +\] +By means of Jacobi's theorem we may write +\[ +\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} + = \iiint \frac{\dd({x'}_{1}, {x'}_{2}, {x'}_{3})}{\dd(x_{1}, x_{2}, x_{3})}\, + dx_{1}\, dx_{2}\, dx_{3} +\] +\PageSep{11} +where the integrand in the last integral is the functional +determinant of the~${x'}_{\nu}$ with respect to the~$x_{\nu}$, and this by~\Eqref{(3)} +is equal to the determinant~$|b_{\mu\nu}|$ of the coefficients +of substitution,~$b_{\nu\alpha}$. If we form the determinant of the~$\delta_{\mu\alpha}$ +from equation~\Eqref{(4)}, we obtain, by means of the theorem +of multiplication of determinants, +\[ +1 = |\delta_{\alpha\beta}| + = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| + = |b_{\mu\nu}|^{2};\quad +|b_{\mu\nu}| = ±1\Add{.} +\Tag{(6)} +\] +If we limit ourselves to those transformations which have +the determinant~$+1$,\footnote + {There are thus two kinds of Cartesian systems which are designated + as ``right-handed'' and ``left-handed'' systems. The difference between + these is familiar to every physicist and engineer. It is interesting to note + that these two kinds of systems cannot be defined geometrically, but only + the contrast between them.} +and only these arise from continuous +variations of the systems of co-ordinates, then $V$~is +an invariant. + +Invariants, however, are not the only forms by means +of which we can give expression to the independence of +the particular choice of the Cartesian co-ordinates. Vectors +and tensors are other forms of expression. Let us express +the fact that the point with the current co-ordinates~$x_{\nu}$ lies +upon a straight line. We have +\[ +x_{\nu} - A_{\nu} = \lambda B_{\nu}\quad + \text{($\nu$~from $1$~to~$3$).} +\] +Without limiting the generality we can put +\[ +\sum {B_{\nu}}^{2} = 1. +\] + +If we multiply the equations by~$b_{\beta\nu}$ (compare \Eqref{(3a)} and~\Eqref{(5)}) +and sum for all the~$\nu$'s, we get +\[ +{x'}_{\beta} - {A'}_{\beta} = \lambda {B'}_{\beta}\Add{,} +\] +\PageSep{12} +where we have written +\[ +{B'}_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad +{A'}_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}. +\] + +These are the equations of straight lines with respect +to a second Cartesian system of co-ordinates~$K'$. They +have the same form as the equations with respect to the +original system of co-ordinates. It is therefore evident +that straight lines have a significance which is independent +of the system of co-ordinates. Formally, this depends +upon the fact that the quantities $(x_{\nu} - A_{\nu}) - \lambda B_{\nu}$ are +transformed as the components of an interval,~$\Delta x_{\nu}$. The +ensemble of three quantities, defined for every system of +Cartesian co-ordinates, and which transform as the components +of an interval, is called a vector. If the three +components of a vector vanish for one system of Cartesian +co-ordinates, they vanish for all systems, because the equations +of transformation are homogeneous. We can thus +get the meaning of the concept of a vector without referring +to a geometrical representation. This behaviour of the +equations of a straight line can be expressed by saying +that the equation of a straight line is co-variant with respect +\index{Covariant@{Co-variant}|etseq}% +to linear orthogonal transformations. + +We shall now show briefly that there are geometrical +\index{Tensor|etseq}% +entities which lead to the concept of tensors. Let $P_{0}$ be +the centre of a surface of the second degree, $P$~any point +on the surface, and $\xi_{\nu}$~the projections of the interval~$P_{0}P$ +upon the co-ordinate axes. Then the equation of the +surface is +\[ +\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +\] +\PageSep{13} +In this, and in analogous cases, we shall omit the sign of +summation, and understand that the summation is to be +carried out for those indices that appear twice. We thus +write the equation of the surface +\[ +a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +\] +The quantities~$a_{\mu\nu}$ determine the surface completely, for +a given position of the centre, with respect to the chosen +system of Cartesian co-ordinates. From the known law +of transformation for the~$\xi_{\nu}$ \Eqref{(3a)} for linear orthogonal +transformations, we easily find the law of transformation +%[** TN: Footnote mark comes before the colon in the original] +for the~$a_{\mu\nu}$:\footnote + {The equation ${a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1$ may, by~\Eqref{(5)}, be replaced by + ${a'}_{\sigma\tau} b_{\mu\sigma} b_{\nu\tau} \xi_{\Change{\sigma}{\mu}} \xi_{\Change{\tau}{\nu}} = 1$, + from which the result stated immediately follows.} +\[ +{a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}. +\] +This transformation is homogeneous and of the first degree +in the~$a_{\mu\nu}$. On account of this transformation, the~$a_{\mu\nu}$ +are called components of a tensor of the second rank (the +latter on account of the double index). If all the components,~$a_{\mu\nu}$, +of a tensor with respect to any system of +Cartesian co-ordinates vanish, they vanish with respect to +every other Cartesian system. The form and the position +of the surface of the second degree is described by this +tensor~$(a)$. + +Analytic tensors of higher rank (number of indices) +may be defined. It is possible and advantageous to +regard vectors as tensors of rank~$1$, and invariants (scalars) +as tensors of rank~$0$. In this respect, the problem of the +theory of invariants may be so formulated: according to +what laws may new tensors be formed from given tensors? +\PageSep{14} +\index{Operations on tensors|etseq}% +\index{Tensor!operations|etseq}% +We shall consider these laws now, in order to be able to +apply them later. We shall deal first only with the +properties of tensors with respect to the transformation +from one Cartesian system to another in the same space +of reference, by means of linear orthogonal transformations. +As the laws are wholly independent of the number +of dimensions, we shall leave this number,~$n$, indefinite at +first. + +\Paragraph{Definition.} If a figure is defined with respect to every +system of Cartesian co-ordinates in a space of reference of +$n$~dimensions by the $n^{\alpha}$~numbers $A_{\mu\nu\rho\Subdots}$ ($\alpha = \text{number +of indices}$), then these numbers are the components of a +tensor of rank~$\alpha$ if the transformation law is +\index{Rank of tensor}% +\index{Tensor!Rank of}% +\[ +{A'}_{\mu'\nu'\rho'\Subdots} + = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho} \dots A_{\mu\nu\rho\Subdots}\Add{.} +\Tag{(7)} +\] + +\Paragraph{Remark.} From this definition it follows that +\[ +A_{\mu\nu\rho\Subdots} = B_{\mu} C_{\nu} D_{\rho}\dots +\Tag{(8)} +\] +is an invariant, provided that $(B)$,~$(C)$,~$(D)$~\dots\ are +vectors. Conversely, the tensor character of~$(A)$ may be +inferred, if it is known that the expression~\Eqref{(8)} leads to an +invariant for an arbitrary choice of the vectors $(B)$,~$(C)$,~etc. + +\Paragraph{Addition and Subtraction.} By addition and subtraction +\index{Addition@{Addition and subtraction of tensors}}% +\index{Tensor!Addition and subtraction of}% +of the corresponding components of tensors of the same +rank, a tensor of equal rank results: +\[ +A_{\mu\nu\rho\Subdots} ± B_{\mu\nu\rho\Subdots} = C_{\mu\nu\rho\Subdots}. +\Tag{(9)} +\] +The proof follows from the definition of a tensor given +above. + +\Paragraph{Multiplication.} From a tensor of rank~$\alpha$ and a tensor +\index{Multiplication of tensors}% +\index{Tensor!Multiplication of}% +\PageSep{15} +of rank~$\beta$ we may obtain a tensor of rank~$\alpha + \beta$ by +multiplying all the components of the first tensor by all +the components of the second tensor: +\[ +T_{\mu\nu\rho\Subdots\alpha\beta\Subdots} += A_{\mu\nu\rho\Subdots} B_{\alpha\beta\gamma\Subdots}\Add{.} +\Tag{(10)} +\] + +\Paragraph{Contraction.} A tensor of rank~$\alpha - 2$ may be obtained +\index{Contraction of tensors}% +\index{Tensor!Contraction of}% +from one of rank~$\alpha$ by putting two definite indices equal +to each other and then summing for this single index: +\[ +T_{\rho\Subdots} + = A_{\mu\mu\rho\Subdots} + ( = \sum_{\mu} A_{\mu\mu\rho\Subdots})\Add{.} +\Tag{(11)} +\] +The proof is +\begin{align*} +{A'}_{\mu\mu\rho\Subdots} + = b_{\mu\alpha} b_{\mu\beta} b_{\rho\gamma}\dots A_{\alpha\beta\gamma\Subdots} + = \delta_{\alpha\beta} & b_{\rho\gamma}\dots A_{\alpha\beta\gamma\Subdots} \\ + {} ={} & b_{\rho\gamma}\dots A_{\alpha\alpha\gamma\Subdots}\Add{.} +\end{align*} + +In addition to these elementary rules of operation +there is also the formation of tensors by differentiation +(``erweiterung''): +\[ +T_{\mu\nu\rho\Subdots\alpha} + = \frac{\dd A_{\mu\nu\rho\Subdots}}{\dd x_{\alpha}}\Add{.} +\Tag{(12)} +\] + +New tensors, in respect to linear orthogonal transformations, +may be formed from tensors according to these rules +of operation. + +\Paragraph{Symmetrical Properties of Tensors.} Tensors are called +\index{Skew-symmetrical tensor}% +\index{Symmetrical tensor}% +\index{Tensor!Symmetrical and Skew-symmetrical}% +symmetrical or skew-symmetrical in respect to two of +their indices, $\mu$~and~$\nu$, if both the components which result +from interchanging the indices $\mu$~and~$\nu$ are equal to each +other or equal with opposite signs. +\begin{alignat*}{2} +&\text{Condition for symmetry:} & + A_{\mu\nu\rho} &= A_{\Change{\mu\nu}{\nu\mu}\rho}. \\ +&\text{Condition for skew-symmetry:}\quad & + A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ +\end{alignat*} + +\Paragraph{Theorem.} The character of symmetry or skew-symmetry +exists independently of the choice of co-ordinates, and in +\PageSep{16} +this lies its importance. The proof follows from the +equation defining tensors. + +\Subsection{Special Tensors.} + +I\@. The quantities~$\delta_{\rho\sigma}$ \Eqref{(4)} are tensor components (fundamental +tensor). + +\Paragraph{Proof.} If in the right-hand side of the equation of +transformation ${A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}$, we substitute for~$A_{\alpha\beta}$ the +quantities~$\delta_{\alpha\beta}$ (which are equal to $1$~or~$0$ according as +$\alpha = \beta$ or $\alpha \Change{ }{\neq} \beta$), we get +\[ +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}. +\] +The justification for the last sign of equality becomes +evident if one applies~\Eqref{(4)} to the inverse substitution~\Eqref{(5)}. + +II\@. There is a tensor $(\delta_{\mu\nu\rho\Subdots})$ skew-symmetrical with +respect to all pairs of indices, whose rank is equal to the +number of dimensions,~$n$, and whose components are +equal to $+1$~or~$-1$ according as $\mu\,\nu\,\rho\dots$~is an even +or odd permutation of~$1\,2\,3\dots$\Add{.} + +The proof follows with the aid of the theorem proved +above $|b_{\rho\sigma}| = 1$. + +These few simple theorems form the apparatus from +the theory of invariants for building the equations of pre-relativity +physics and the theory of special relativity. + +We have seen that in pre-relativity physics, in order to +specify relations in space, a body of reference, or a space +of reference, is required, and, in addition, a Cartesian +system of co-ordinates. We can fuse both these concepts +into a single one by thinking of a Cartesian system of +co-ordinates as a cubical frame-work formed of rods each +of unit length. The co-ordinates of the lattice points of +\PageSep{17} +this frame are integral numbers. It follows from the +fundamental relation +\[ +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} +\Change{}{\Tag{(13)}} +\] +that the members of such a space-lattice are all of unit +length. To specify relations in time, we require in +addition a standard clock placed at the origin of our +Cartesian system of co-ordinates or frame of reference. +If an event takes place anywhere we can assign to it three +co-ordinates,~$x_{\nu}$, and a time~$t$, as soon as we have specified +the time of the clock at the origin which is simultaneous +with the event. We therefore give an objective significance +to the statement of the simultaneity of distant +\index{Simultaneity}% +events, while previously we have been concerned only +with the simultaneity of two experiences of an individual. +The time so specified is at all events independent of the +position of the system of co-ordinates in our space of +reference, and is therefore an invariant with respect to +the transformation~\Eqref{(3)}. + +It is postulated that the system of equations expressing +the laws of pre-relativity physics is co-variant with respect +to the transformation~\Eqref{(3)}, as are the relations of Euclidean +geometry. The isotropy and homogeneity of space is +\index{Homogeneity of space}% +\index{Isotropy of space}% +\index{Space@{Space, Concept of}!Homogeneity of}% +\index{Space@{Space, Concept of}!Isotropy of}% +expressed in this way.\footnote + {The laws of physics could be expressed, even in case there were a + unique direction in space, in such a way as to be co-variant with respect to + the transformation~\Eqref{(3)}; but such an expression would in this case be unsuitable. + If there were a unique direction in space it would simplify the + description of natural phenomena to orient the system of co-ordinates in a + definite way in this direction. But if, on the other hand, there is no unique + direction in space it is not logical to formulate the laws of nature in such + a way as to conceal the equivalence of systems of co-ordinates that are + oriented differently. We shall meet with this point of view again in the + theories of special and general relativity.} +We shall now consider some of +\PageSep{18} +the more important equations of physics from this point +of view. + +The equations of motion of a material particle are +\[ +m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}\Add{;} +\Tag{(14)} +\] +$(dx_{\nu})$~is a vector; $dt$,~and therefore also~$\dfrac{1}{dt}$, an invariant; +thus $\left(\dfrac{dx_{\nu}}{dt}\right)$~is a vector; in the same way it may be shown +that $\left(\dfrac{d^{2} x_{\nu}}{dt^{2}}\right)$~is a vector. In general, the operation of differentiation +with respect to time does not alter the tensor +character. Since $m$~is an invariant (tensor of rank~$0$), +$\left(m\dfrac{d^{2} x_{\nu}}{dt^{2}}\right)$~is a vector, or tensor of rank~$1$ (by the theorem +of the multiplication of tensors). If the force~$(X_{\nu})$ has +a vector character, the same holds for the difference +$\left(m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right)$. These equations of motion are therefore +valid in every other system of Cartesian co-ordinates in +the space of reference. In the case where the forces are +conservative we can easily recognize the vector character +of~$(X_{\nu})$. For a potential energy,~$\Phi$, exists, which depends +only upon the mutual distances of the particles, and is +therefore an invariant. The vector character of the force, +$X_{\nu} = -\dfrac{\dd \Phi}{\dd x_{\nu}}$, is then a consequence of our general theorem +about the derivative of a tensor of rank~$0$. +\PageSep{19} + +Multiplying by the velocity, a tensor of rank~$1$, we +obtain the tensor equation +\[ +\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0. +\] +By contraction and multiplication by the scalar~$dt$ we +obtain the equation of kinetic energy +\[ +d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\, dx_{\nu}. +\] + +If $\xi_{\nu}$~denotes the difference of the co-ordinates of +the material particle and a point fixed in space, then +the~$\xi_{\nu}$ have the character of vectors. We evidently +have $\dfrac{d^{2} x_{\nu}}{dt^{2}} = \dfrac{d^{2} \xi_{\nu}}{dt^{2}}$, so that the equations of motion of the +particle may be written +\[ +m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0. +\] + +Multiplying this equation by~$\xi_{\mu}$ we obtain a tensor +equation +\[ +\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0. +\] + +Contracting the tensor on the left and taking the time +average we obtain the virial theorem, which we shall +not consider further. By interchanging the indices and +subsequent subtraction, we obtain, after a simple transformation, +the theorem of moments, +\[ +\frac{d}{dt} \left[ + m \left(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\right) +\right] = \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}\Add{.} +\Tag{(15)} +\] + +It is evident in this way that the moment of a vector +\PageSep{20} +is not a vector but a tensor. On account of their skew-symmetrical +character there are not nine, but only three +independent equations of this system. The possibility of +replacing skew-symmetrical tensors of the second rank in +space of three dimensions by vectors depends upon the +formation of the vector +\[ +A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}. +\] + +If we multiply the skew-symmetrical tensor of rank~$2$ +by the special skew-symmetrical tensor~$\delta$ introduced +above, and contract twice, a vector results whose components +are numerically equal to those of the tensor. These +are the so-called axial vectors which transform differently, +from a right-handed system to a left-handed system, +from the~$\Delta x_{\nu}$. There is a gain in picturesqueness in +regarding a skew-symmetrical tensor of rank~$2$ as a vector +in space of three dimensions, but it does not represent +the exact nature of the corresponding quantity so well as +considering it a tensor. + +We consider next the equations of motion of a continuous +medium. Let $\rho$~be the density, $u_{\nu}$~the velocity +components considered as functions of the co-ordinates and +the time, $X_{\nu}$~the volume forces per unit of mass, and $p_{\nu\sigma}$~the +stresses upon a surface perpendicular to the $\sigma$-axis +in the direction of increasing~$x_{\nu}$. Then the equations of +motion are, by Newton's law, +\[ +\rho \frac{du_{\nu}}{dt} + = -\frac{\dd p_{\nu\sigma}}{\dd x_{\sigma}} + \rho X_{\nu}\Add{,} +\] +in which $\dfrac{du_{\nu}}{dt}$~is the acceleration of the particle which at +\PageSep{21} +time~$t$ has the co-ordinates~$x_{\nu}$. If we express this +acceleration by partial differential coefficients, we obtain, +after dividing by~$\rho$, +\[ +\frac{\dd u_{\nu}}{dt} + \frac{\dd u_{\nu}}{dx_{\sigma}} u_{\sigma} + = -\frac{1}{\rho}\, \frac{\dd p_{\nu\sigma}}{\dd x_{\sigma}} + X_{\nu}\Add{.} +\Tag{(16)} +\] + +We must show that this equation holds independently +of the special choice of the Cartesian system of co-ordinates. +$(u_{\nu})$ is a vector, and therefore $\dfrac{\dd u_{\nu}}{\dd t}$~is also a vector. $\dfrac{\dd u_{\nu}}{\dd x_{\sigma}}$~is +a tensor of rank~$2$, $\dfrac{\dd u_{\nu}}{\dd x_{\sigma}} u_{\tau}$~is a tensor of rank~$3$. The second +term on the left results from contraction in the indices +$\sigma$,~$\tau$. The vector character of the second term on the right +is obvious. In order that the first term on the right may +also be a vector it is necessary for~$p_{\nu\sigma}$ to be a tensor. +Then by differentiation and contraction $\dfrac{\dd p_{\nu\sigma}}{\dd x_{\sigma}}$~results, and +is therefore a vector, as it also is after multiplication by +the reciprocal scalar~$\dfrac{1}{\rho}$. That $p_{\nu\sigma}$~is a tensor, and therefore +transforms according to the equation +\[ +{p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta}, +\] +is proved in mechanics by integrating this equation over +an infinitely small tetrahedron. It is also proved there, +by application of the theorem of moments to an infinitely +small parallelopipedon, that $p_{\nu\sigma} = p_{\sigma\nu}$, and hence that the +tensor of the stress is a symmetrical tensor. From what +has been said it follows that, with the aid of the rules +\PageSep{22} +given above, the equation is co-variant with respect to +orthogonal transformations in space (rotational transformations); +and the rules according to which the +quantities in the equation must be transformed in order +that the equation may be co-variant also become evident. + +The co-variance of the equation of continuity, +\index{Covariance@{Co-variance of equation of continuity}}% +\index{Equation of continuity, co-variance of}% +\[ +\frac{\dd\rho}{\dd t} + \frac{\dd(\rho u_{\nu})}{\dd x_{\nu}} = 0\Add{,} +\Tag{(17)} +\] +requires, from the foregoing, no particular discussion. + +We shall also test for co-variance the equations which +express the dependence of the stress components upon +the properties of the matter, and set up these equations +for the case of a compressible viscous fluid with the aid +\index{Compressible viscous fluid}% +\index{Viscous compressible fluid}% +of the conditions of co-variance. If we neglect the viscosity, +the pressure,~$p$, will be a scalar, and will depend +only upon the density and the temperature of the fluid. +The contribution to the stress tensor is then evidently +\index{Stress tensor}% +\[ +p \delta_{\mu\nu} +\] +in which $\delta_{\mu\nu}$~is the special symmetrical tensor. This term +will also be present in the case of a viscous fluid. But in +this case there will also be pressure terms, which depend +upon the space derivatives of the~$u_{\nu}$. We shall assume +that this dependence is a linear one. Since these terms +must be symmetrical tensors, the only ones which enter +will be +\[ +\alpha\left(\frac{\dd u_{\mu}}{\dd x_{\nu}} + + \frac{\dd u_{\nu}}{\dd x_{\mu}}\right) + + \beta\delta_{\mu\nu} \frac{\dd u_{\alpha}}{\dd x_{\alpha}} +\] +(for $\dfrac{\dd u_{\alpha}}{\dd x_{\alpha}}$~is a scalar). For physical reasons (no slipping) +\PageSep{23} +it is assumed that for symmetrical dilatations in all +directions, i.e.~when +\[ +\frac{\dd u_{1}}{\dd x_{1}} = +\frac{\dd u_{2}}{\dd x_{2}} = +\frac{\dd u_{3}}{\dd x_{3}};\quad +\frac{\dd u_{1}}{\dd x_{2}}, \text{ etc.,} = 0, +\] +there are no frictional forces present, from which it +follows that $\beta = -\dfrac{2}{3}\alpha$. If only $\dfrac{\dd u_{1}}{\dd x_{\Change{2}{3}}}$~is different from +zero, let $p_{31} = -\Change{\eta}{\alpha} \dfrac{\dd u_{1}}{\dd x_{3}}$, by which $\alpha$~is determined. We +then obtain for the complete stress tensor, +\[ +p_{\mu\nu} = p \delta_{\mu\nu} - \Change{\eta}{\alpha} \biggl[ + \biggl(\frac{\dd u_{\mu}}{\dd x_{\nu}} + + \frac{\dd u_{\nu}}{\dd x_{\mu}}\biggr) + - \frac{2}{3} \biggl(\frac{\dd u_{1}}{\dd x_{1}} + + \frac{\dd u_{2}}{\dd x_{2}} + + \frac{\dd u_{3}}{\dd x_{3}}\biggr)\delta_{\mu\nu} +\biggr]\Add{.} +\Tag{(18)} +\] + +The heuristic value of the theory of invariants, which +arises from the isotropy of space (equivalence of all +directions), becomes evident from this example. + +We consider, finally, Maxwell's equations in the form +\index{Maxwell's equations}% +which are the foundation of the electron theory of Lorentz. +\begin{align*} +\left. + \begin{aligned} + &\begin{alignedat}{3} + \frac{\dd h_{3}}{\dd x_{2}} &- \frac{\dd h_{2}}{\dd x_{3}} + &&= \frac{1}{c}\, \frac{\dd e_{1}}{\dd t} + &&+ \frac{1}{c}\, i_{1}\Add{,} \\ +% + \frac{\dd h_{1}}{\dd x_{3}} &- \frac{\dd h_{3}}{\dd x_{1}} + &&= \frac{1}{c}\, \frac{\dd e_{2}}{\dd t} + &&+ \frac{1}{c}\, i_{2}\Add{,} \\ +% +% [** TN: Adding code for third equation (elided in original)] + \frac{\dd h_{2}}{\dd x_{1}} &- \frac{\dd h_{1}}{\dd x_{2}} + &&= \frac{1}{c}\, \frac{\dd e_{3}}{\dd t} + &&+ \frac{1}{c}\, i_{3}\Add{,} + \end{alignedat} \\ + & \frac{\dd e_{1}}{\dd x_{1}} + + \frac{\dd e_{2}}{\dd x_{2}} + + \frac{\dd e_{3}}{\dd x_{3}} = \rho\Add{;} + \end{aligned} +\right\} +\Tag{(19)} \\[8pt] +\left. + \begin{aligned} + &\begin{alignedat}{2} + \frac{\dd e_{3}}{\dd x_{2}} &- \frac{\dd e_{2}}{\dd x_{3}} + &&= -\frac{1}{c}\, \frac{\dd h_{1}}{\dd t}\Add{,}\qquad \\ +% + \frac{\dd e_{1}}{\dd x_{3}} &- \frac{\dd e_{3}}{\dd x_{1}} + &&= -\frac{1}{c}\, \frac{\dd h_{2}}{\dd t}\Add{,} \\ +% +% [** TN: Adding code for third equation (elided in original)] + \frac{\dd e_{2}}{\dd x_{1}} &- \frac{\dd e_{1}}{\dd x_{2}} + &&= -\frac{1}{c}\, \frac{\dd h_{3}}{\dd t}\Add{,} + \end{alignedat} \\ + & \frac{\dd h_{1}}{\dd x_{1}} + + \frac{\dd h_{2}}{\dd x_{2}} + + \frac{\dd h_{3}}{\dd x_{3}} = 0\Add{.} + \end{aligned} +\right\} +\Tag{(20)} +\end{align*} +\PageSep{24} + +$\veci$~is a vector, because the current density is defined as +the density of electricity multiplied by the vector velocity +of the electricity. According to the first three equations +it is evident that $\e$~is also to be regarded as a vector. +Then $\h$~cannot be regarded as a vector.\footnote + {These considerations will make the reader familiar with tensor operations + without the special difficulties of the four-dimensional treatment; + corresponding considerations in the theory of special relativity (Minkowski's + interpretation of the field) will then offer fewer difficulties.} +The equations +may, however, easily be interpreted if $\h$~is regarded as a +\Change{}{skew-}symmetrical tensor of the second rank. In this sense, we +write $h_{23}$,~$h_{31}$,~$h_{12}$, in place of $h_{1}$,~$h_{2}$,~$h_{3}$ respectively. Paying +attention to the skew-symmetry of~$h_{\mu\nu}$, the first three +equations of \Eqref{(19)}~and~\Eqref{(20)} may be written in the form +\begin{gather*} +\frac{\dd h_{\mu\nu}}{\dd x_{\nu}} + = \frac{1}{c}\, \frac{\dd e_{\mu}}{\dd t} + \frac{1}{c} i_{\mu}\Add{,} +\Tag{(19a)} \\ +\frac{\dd e_{\mu}}{\dd x_{\nu}} - \frac{\dd e_{\nu}}{\dd x_{\mu}} + = +\frac{1}{c}\, \frac{\dd h_{\mu\nu}}{\dd t}\Add{.} +\Tag{(20a)} +\end{gather*} +In contrast to~$\e$, $\h$~appears as a quantity which has the +same type of symmetry as an angular velocity. The +divergence equations then take the form +\begin{gather*} +\frac{\dd e_{\nu}}{\dd x_{\nu}} = \rho\Add{,} +\Tag{(19b)} \\ +\frac{\dd h_{\mu\nu}}{\dd x_{\rho}} + +\frac{\dd h_{\nu\rho}}{\dd x_{\mu}} + +\frac{\dd h_{\rho\mu}}{\dd x_{\nu}} = 0\Add{.} +\Tag{(20b)} +\end{gather*} +The last equation is a skew-symmetrical tensor equation +of the third rank (the skew-symmetry of the left-hand +side with respect to every pair of indices may easily be +\PageSep{25} +proved, if attention is paid to the skew-symmetry of~$h_{\mu\nu}$). +This notation is more natural than the usual one, because, +in contrast to the latter, it is applicable to Cartesian left-handed +systems as well as to right-handed systems without +change of sign. +\PageSep{26} + + +\Lecture{II}{The Theory of Special Relativity}{Special Relativity} + +\First{The} previous considerations concerning the configuration +of rigid bodies have been founded, irrespective +of the assumption as to the validity of the Euclidean +geometry, upon the hypothesis that all directions in space, +or all configurations of Cartesian systems of co-ordinates, +are physically equivalent. We may express this as the +\index{Equivalent spaces of reference}% +``principle of relativity with respect to direction,'' and it +has been shown how equations (laws of nature) may be +found, in accord with this principle, by the aid of the +calculus of tensors. We now inquire whether there is a +relativity with respect to the state of motion of the space +of reference; in other words, whether there are spaces of +reference in motion relatively to each other which are +physically equivalent. From the standpoint of mechanics +it appears that equivalent spaces of reference do exist. +\index{Spaces of reference!equivalence of}% +For experiments upon the earth tell us nothing of the +fact that we are moving about the sun with a velocity of +approximately $30$~kilometres a second. On the other +hand, this physical equivalence does not seem to hold for +spaces of reference in arbitrary motion; for mechanical +effects do not seem to be subject to the same laws in a +jolting railway train as in one moving with uniform +\PageSep{27} +velocity; the rotation of the earth must be considered in +writing down the equations of motion relatively to the +earth. It appears, therefore, as if there were Cartesian +systems of co-ordinates, the so-called inertial systems, with +reference to which the laws of mechanics (more generally +the laws of physics) are expressed in the simplest form. +We may infer the validity of the following theorem: If +$K$~is an inertial system, then every other system~$K'$ which +moves uniformly and without rotation relatively to~$K$, is +also an inertial system; the laws of nature are in concordance +for all inertial systems. This statement we shall +call the ``principle of special relativity.'' We shall draw +certain conclusions from this principle of ``relativity of +translation'' just as we have already done for relativity of +direction. + +In order to be able to do this, we must first solve the +following problem. If we are given the Cartesian co-ordinates,~$x_{\nu}$, +and the time,~$t$, of an event relatively to one +inertial system,~$K$, how can we calculate the co-ordinates,~${x'}_{\nu}$, +and the time,~$t'$, of the same event relatively to an +inertial system~$K'$ which moves with uniform translation +relatively to~$K$? In the pre-relativity physics +\index{Prerelativity@{Pre-relativity physics, hypotheses of}}% +this problem was solved by making unconsciously two +hypotheses:--- + +1.~The time is absolute; the time of an event,~$t'$, +relatively to~$K'$ is the same as the time relatively to~$K$. +If instantaneous signals could be sent to a distance, and +if one knew that the state of motion of a clock had no +influence on its rate, then this assumption would be +physically established. For then clocks, similar to one +\PageSep{28} +another, and regulated alike, could be distributed over +the systems $K$~and~$K'$, at rest relatively to them, and +their indications would be independent of the state of +motion of the systems; the time of an event would then +be given by the clock in its immediate neighbourhood. + +2.~Length is absolute; if an interval, at rest relatively +to~$K$, has a length~$s$, then it has the same length~$s$ +relatively to a system~$K'$ which is in motion relatively +to~$K$. + +If the axes of $K$~and~$K'$ are parallel to each other, a +simple calculation based on these two assumptions, gives +the equations of transformation +\[ +\left. +\begin{aligned} +{x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\Add{,} \\ +t' &= t - b\Add{.} +\end{aligned} +\right\} +\Tag{(21)} +\] + +This transformation is known as the ``Galilean Transformation.'' +\index{Galilean@{Galilean regions}!transformation}% +\index{Transformation@{Transformation, Galilean}}% +Differentiating twice by the time, we get +\[ +\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}. +\] +Further, it follows that for two simultaneous events, +\[ +{{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}. +\] +The invariance of the distance between the two points +results from squaring and adding. From this easily +follows the co-variance of Newton's equations of motion +with respect to the Galilean transformation~\Eqref{(21)}. Hence +it follows that classical mechanics is in accord with the +principle of special relativity if the two hypotheses +respecting scales and clocks are made. + +But this attempt to found relativity of translation upon +the Galilean transformation fails when applied to electromagnetic +\PageSep{29} +phenomena. The Maxwell-Lorentz electromagnetic +equations are not co-variant with respect to the +Galilean transformation. In particular, we note, by~\Eqref{(21)}, +that a ray of light which referred to~$K$ has a velocity~$c$, +has a different velocity referred to~$K'$, depending upon +its direction. The space of reference of~$K$ is therefore +distinguished, with respect to its physical properties, from +all spaces of reference which are in motion relatively to it +(quiescent æther). But all experiments have shown that +electro\Change{-}{}magnetic\TNote{** TN: Only hyphenated instance} and optical phenomena, relatively to the +earth as the body of reference, are not influenced by the +translational velocity of the earth. The most important +of these experiments are those of Michelson and Morley, +\index{Michelson and Morley}% +which I shall assume are known. The validity of the +principle of special relativity can therefore hardly be +doubted. + +On the other hand, the Maxwell-Lorentz equations +have proved their validity in the treatment of optical +problems in moving bodies. No other theory has +satisfactorily explained the facts of aberration, the +propagation of light in moving bodies (Fizeau), and +\index{Fizeau}% +phenomena observed in double stars (De~Sitter). The +\index{Sitter}% +consequence of the Maxwell-Lorentz equations that in a +vacuum light is propagated with the velocity~$c$, at least +with respect to a definite inertial system~$K$, must therefore +be regarded as proved. According to the principle +of special relativity, we must also assume the truth of +this principle for every other inertial system. + +Before we draw any conclusions from these two +principles we must first review the physical significance +\PageSep{30} +\index{Concept@{Concept of space}!time}% +\index{Time-concept}% +of the concepts ``time'' and ``velocity.'' It follows from +what has gone before, that co-ordinates with respect to +an inertial system are physically defined by means of +measurements and constructions with the aid of rigid +bodies. In order to measure time, we have supposed a +clock,~$U$, present somewhere, at rest relatively to~$K$. But +we cannot fix the time, by means of this clock, of an event +whose distance from the clock is not negligible; for there +are no ``instantaneous signals'' that we can use in order +to compare the time of the event with that of the clock. +In order to complete the definition of time we may +employ the principle of the constancy of the velocity of +light in a vacuum. Let us suppose that we place similar +clocks at points of the system~$K$, at rest relatively to it, +and regulated according to the following scheme. A ray +of light is sent out from one of the clocks,~$U_{m}$, at the +instant when it indicates the time~$t_{m}$, and travels through +a vacuum a distance~$r_{mn}$, to the clock~$U_{n}$; at the instant +when this ray meets the clock~$U_{n}$ the latter is set to +indicate the time $t_{n} = t_{m} + \dfrac{r_{mn}}{c}$.\footnote + {Strictly speaking, it would be more correct to define simultaneity first, +\index{Simultaneity}% + somewhat as follows: two events taking place at the points $A$~and~$B$ of + the system~$K$ are simultaneous if they appear at the same instant when + observed from the middle point,~$M$, of the interval~$AB$. Time is then + defined as the ensemble of the indications of similar clocks, at rest + relatively to~$K$, which register the same simultaneously.} +The principle of the +constancy of the velocity of light then states that this +adjustment of the clocks will not lead to contradictions. +With clocks so adjusted, we can assign the time to events +which take place near any one of them. It is essential to +\PageSep{31} +note that this definition of time relates only to the inertial +system~$K$, since we have used a system of clocks at rest +relatively to~$K$. The assumption which was made in the +pre-relativity physics of the absolute character of time +(i.e.~the independence of time of the choice of the inertial +system) does not follow at all from this definition. + +The theory of relativity is often criticized for giving, +\index{Criticisms of theory of relativity}% +\index{Theory of relativity, criticisms of}% +without justification, a central theoretical rôle to the +propagation of light, in that it founds the concept of time +upon the law of propagation of light. The situation, +however, is somewhat as follows. In order to give +physical significance to the concept of time, processes of +some kind are required which enable relations to be +established between different places. It is immaterial +what kind of processes one chooses for such a definition +of time. It is advantageous, however, for the theory, +to choose only those processes concerning which we know +something certain. This holds for the propagation of +light \textit{in vacuo} in a higher degree than for any other process +which could be considered, thanks to the investigations +of Maxwell and H.~A.~Lorentz. + +From all of these considerations, space and time data +have a physically real, and not a mere fictitious, significance; +in particular this holds for all the relations in +which co-ordinates and time enter, e.g.~the relations~\Eqref{(21)}. +There is, therefore, sense in asking whether those +equations are true or not, as well as in asking what the +true equations of transformation are by which we pass +from one inertial system~$K$ to another,~$K'$, moving +relatively to it. It may be shown that this is uniquely +\PageSep{32} +settled by means of the principle of the constancy of the +velocity of light and the principle of special relativity. + +To this end we think of space and time physically +defined with respect to two inertial systems, $K$~and~$K'$, in +the way that has been shown. Further, let a ray of light +pass from one point~$P_{1}$ to another point~$P_{2}$ of~$K$ through +a vacuum. If $r$~is the measured distance between the two +points, then the propagation of light must satisfy the +equation +\[ +r = c·\Delta t. +\] + +If we square this equation, and express~$r^{2}$ by the +differences of the co-ordinates,~$\Delta x_{\nu}$, in place of this equation +we can write +\[ +\sum (\Delta x_{\nu})^{2} - c^{2}\, \Delta t^{2} = 0\Add{.} +\Tag{(22)} +\] +This equation formulates the principle of the constancy +of the velocity of light relatively to~$K$. It must hold +whatever may be the motion of the source which emits +the ray of light. + +The same propagation of light may also be considered +relatively to~$K'$, in which case also the principle of the +constancy of the velocity of light must be satisfied. +Therefore, with respect to~$K'$, we have the equation +\[ +\sum ({\Delta x'}_{\nu})^{2} - c^{2}\, \Delta t'^{2} = 0\Add{.} +\Tag{(22a)} +\] + +Equations \Eqref{(22a)}~and~\Eqref{(22)} must be mutually consistent +with each other with respect to the transformation which +transforms from~$K$ to~$K'$. A transformation which effects +this we shall call a ``Lorentz transformation.'' +\index{Lorentz@{Lorentz electromotive force}!transformation}% + +Before considering these transformations in detail we +\PageSep{33} +shall make a few general remarks about space and time. +In the pre-relativity physics space and time were separate +entities. Specifications of time were independent of +the choice of the space of reference. The Newtonian +mechanics was relative with respect to the space of +reference, so that, e.g.~the statement that two non-simultaneous +events happened at the same place had no objective +meaning (that is, independent of the space of reference). +But this relativity had no rôle in building up the theory. +One spoke of points of space, as of instants of time, as if +they were absolute realities. It was not observed that +the true element of the space-time specification was the +event, specified by the four numbers $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$. The +conception of something happening was always that of a +four-dimensional continuum; but the recognition of this +\index{Continuum, four-dimensional}% +\index{Four-dimensional continuum}% +was obscured by the absolute character of the pre-relativity +time. Upon giving up the hypothesis of the absolute +character of time, particularly that of simultaneity, the +four-dimensionality of the time-space concept was immediately +\index{Time-space concept}% +recognized. It is neither the point in space, +nor the instant in time, at which something happens that +has physical reality, but only the event itself. There is +no absolute (independent of the space of reference) relation +in space, and no absolute relation in time between two +events, but there is an absolute (independent of the space +of reference) relation in space and time, as will appear in +the sequel. The circumstance that there is no objective +rational division of the four-dimensional continuum into +a three-dimensional space and a one-dimensional time +continuum indicates that the laws of nature will assume +\PageSep{34} +a form which is logically most satisfactory when expressed +as laws in the four-dimensional space-time continuum. +Upon this depends the great advance in method which +the theory of relativity owes to Minkowski. Considered +\index{Minkowski}% +from this standpoint, we must regard $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$ as the +four co-ordinates of an event in the four-dimensional continuum. +We have far less success in picturing to ourselves +relations in this four-dimensional continuum than in the +three-dimensional Euclidean continuum; but it must be +emphasized that even in the Euclidean three-dimensional +geometry its concepts and relations are only of an abstract +nature in our minds, and are not at all identical with the +images we form visually and through our sense of touch. +The non-divisibility of the four-dimensional continuum +of events does not at all, however, involve the equivalence +of the space co-ordinates with the time co-ordinate. On +the contrary, we must remember that the time co-ordinate +is defined physically wholly differently from the space +co-ordinates. The relations \Eqref{(22)}~and~\Eqref{(22a)} which when +equated define the Lorentz transformation show, further, +a difference in the rôle of the time co-ordinate from that +of the space co-ordinates; for the term~$\Delta t^{2}$ has the opposite +sign to the space terms, ${\Delta x_{1}}^{2}$,~${\Delta x_{2}}^{2}$,~${\Delta x_{3}}^{2}$. + +Before we analyse further the conditions which define +the Lorentz transformation, we shall introduce the light-time, +\index{Light time@{Light-time}}% +$l = ct$, in place of the time,~$t$, in order that the +constant~$c$ shall not enter explicitly into the formulas to +be developed later. Then the Lorentz transformation is +defined in such a way that, first, it makes the equation +\[ +{\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 +\Tag{(22b)} +\] +\PageSep{35} +a co-variant equation, that is, an equation which is satisfied +with respect to every inertial system if it is satisfied in +the inertial system to which we refer the two given events +(emission and reception of the ray of light). Finally, +with Minkowski, we introduce in place of the real time +co-ordinate $l = ct$, the imaginary time co-ordinate +\[ +x_{4} = il = ict\quad (\sqrt{-1} = i). +\] +Then the equation defining the propagation of light, +which must be co-variant with respect to the Lorentz +transformation, becomes +\[ +\sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} + = 0\Add{.} +\Tag{(22c)} +\] +This condition is always satisfied\footnote + {That this specialization lies in the nature of the case will be evident + later.} +if we satisfy the more +general condition that +\[ +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} +\Tag{(23)} +\] +shall be an invariant with respect to the transformation. +This condition is satisfied only by linear transformations, +that is, transformations of the type +\[ +{x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} +\Tag{(24)} +\] +in which the summation over the~$\alpha$ is to be extended +from $\alpha = 1$ to $\alpha = 4$. A glance at equations \Eqref{(23)}~and~\Eqref{(24)} +shows that the Lorentz transformation so defined is +identical with the translational and rotational transformations +of the Euclidean geometry, if we disregard the +number of dimensions and the relations of reality. We +\PageSep{36} +can also conclude that the coefficients~$b_{\mu\alpha}$ must satisfy the +conditions +\[ +b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}\Add{.} +\Tag{(25)} +\] +Since the ratios of the~$x_{\nu}$ are real, it follows that all the~$a_{\mu}$ +and the~$b_{\mu\alpha}$ are real, except $a_{4}$, $b_{41}$,~$b_{42}$,~$b_{43}$, $b_{14}$,~$b_{24}$\Change{,}{} and~$b_{34}$, +which are purely imaginary. + +\Paragraph{Special Lorentz Transformation.} We obtain the +\index{Special Lorentz transformation}% +simplest transformations of the type of \Eqref{(24)}~and~\Eqref{(25)} if +only two of the co-ordinates are to be transformed, and if +all the~$a_{\mu}$, which determine the new origin, vanish. We +obtain then for the indices $1$~and~$2$, on account of the +three independent conditions which the relations~\Eqref{(25)} +furnish, +\[ +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi\Add{,} \\ +{x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi\Add{,} \\ +{x'}_{3} &= x_{3}\Add{,} && \\ +{x'}_{4} &= x_{4}\Add{.} && +\end{alignedat} +\right\} +\Tag{(26)} +\] + +This is a simple rotation in space of the (space) +co-ordinate system about $x_{3}$-axis. We see that the +rotational transformation in space (without the time +transformation) which we studied before is contained in +the Lorentz transformation as a special case. For the +indices $1$~and~$4$ we obtain, in an analogous manner, +\[ +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi\Add{,} \\ +{x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi\Add{,} \\ +{x'}_{2} &= x_{2}\Add{,} && \\ +{x'}_{3} &= x_{3}\Add{.} && +\end{alignedat} +\right\} +\Tag{(26a)} +\] + +On account of the relations of reality $\psi$~must be taken +as imaginary. To interpret these equations physically, +we introduce the real light-time~$l$ and the velocity~$v$ of~$K'$ +\PageSep{37} +relatively to~$K$, instead of the imaginary angle~$\psi$. We +have, first, +\begin{alignat*}{4} +{x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi\Add{,} \\ +l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. +\end{alignat*} +Since for the origin of~$K'$ i.e., for $x_{1} = 0$, we must have +$x_{1} = vl$, it follows from the first of these equations that +\[ +v = i\tan\psi\Add{,} +\Tag{(27)} +\] +and also +\[ +\left. +\begin{aligned} +\sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}\Add{,} \\ +\cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}\Add{,} +\end{aligned} +\right\} +\Tag{(28)} +\] +so that we obtain +\[ +\left. +\begin{aligned} +{x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}\Add{,} \\ +l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}\Add{,} \\ +{x'}_{2} &= x_{2}\Add{,} \\ +{x'}_{3} &= x_{3}\Add{.} +\end{aligned} +\right\} +\Tag{(29)} +\] + +These equations form the well-known special Lorentz +transformation, which in the general theory represents a +rotation, through an imaginary angle, of the four-dimensional +system of co-ordinates. If we introduce the ordinary +time~$t$, in place of the light-time~$l$, then in~\Eqref{(29)} we must +replace $l$~by~$ct$ and $v$~by~$\dfrac{v}{c}$. + +We must now fill in a gap. From the principle of the +constancy of the velocity of light it follows that the +equation +\[ +\sum {\Delta x_{\nu}}^{2} = 0 +\] +\PageSep{38} +has a significance which is independent of the choice of +the inertial system; but the invariance of the quantity +$\sum {\Delta x_{\nu}}^{2}$ does not at all follow from this. This quantity +might be transformed with a factor. This depends upon +the fact that the right-hand side of~\Eqref{(29)} might be multiplied +by a factor~$\lambda$, independent of~$v$. But the principle +of relativity does not permit this factor to be different from~$1$, +as we shall now show. Let us assume that we have +a rigid circular cylinder moving in the direction of its +axis. If its radius, measured at rest with a unit measuring +rod is equal to~$R_{0}$, its radius~$R$ in motion, might be +different from~$R_{0}$, since the theory of relativity does not +make the assumption that the shape of bodies with respect +to a space of reference is independent of their motion +relatively to this space of reference. But all directions +in space must be equivalent to each other. $R$~may therefore +depend upon the magnitude~$q$ of the velocity, but +not upon its direction; $R$~must therefore be an even +function of~$q$. If the cylinder is at rest relatively to~$K'$ +the equation of its lateral surface is +\[ +%[** TN: Inconsistently upright R here, below in the original] +x'^{2} + y'^{2} = {R_{0}}^{2}. +\] +If we write the last two equations of~\Eqref{(29)} more generally +\begin{align*} +{x'}_{2} = \lambda x_{2}\Add{,} \\ +{x'}_{3} = \lambda x_{3}\Add{,} +\end{align*} +then the lateral surface of the cylinder referred to~$K$ +satisfies the equation +\[ +x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}. +\] +\PageSep{39} +The factor~$\lambda$ therefore measures the lateral contraction of +the cylinder, and can thus, from the above, be only an +even function of~$v$. + +If we introduce a third system of co-ordinates,~$K''$, +which moves relatively to~$K'$ with velocity~$v$ in the direction +of the negative $x$-axis of~$K$, we obtain, by applying +\Eqref{(29)} twice, +\begin{align*} +{x''}_{1} &= \lambda(v) \lambda(-v) x_{1}\Add{,} \\ +%[** TN: Added code for next two equations, elided in the original] +{x''}_{2} &= \lambda(v) \lambda(-v) x_{2}\Add{,} \\ +{x''}_{3} &= \lambda(v) \lambda(-v) x_{3}\Add{,} \\ +l'' &= \lambda(v) \lambda(-v) l. +\end{align*} +Now, since $\lambda(v)$~must be equal to~$\lambda(-v)$, and since we +assume that we use the same measuring rods in all the +systems, it follows that the transformation of~$K''$ to~$K$ +must be the identical transformation (since the possibility +$\lambda = -1$ does not need to be considered). It is essential +for these considerations to assume that the behaviour of +the measuring rods does not depend upon the history of +their previous motion. + +\Paragraph{Moving Measuring Rods and Clocks.} At the definite $K$-time, +\index{Clocks, moving}% +\index{Moving measuring rods and clocks}% +\index{Rods (measuring) and clocks in motion}% +$l = 0$, the position of the points given by the integers +${x'}_{1} = n$, is with respect to~$K$, given by $x_{1} = n\sqrt{1 - v^{2}}$; +this follows from the first of equations~\Eqref{(29)} and expresses +the Lorentz contraction. A clock at rest at the origin +$x_{1} = 0$ of~$K$, whose beats are characterized by $l = n$, will, +when observed from~$K'$, have beats characterized by +\[ +l' = \frac{n}{\sqrt{1 - v^{2}}}; +\] +this follows from the second of equations~\Eqref{(29)} and shows +\PageSep{40} +that the clock goes slower than if it were at rest relatively +to~$K'$. These two consequences, which hold, \textit{mutatis +mutandis}, for every system of reference, form the physical +content, free from convention, of the Lorentz transformation. + +\Paragraph{Addition Theorem for Velocities.} If we combine two +\index{Addition@{Addition and subtraction of tensors}!theorem of velocities}% +\index{Theorem for addition of velocities}% +\index{Velocities, addition theorem of}% +special Lorentz transformations with the relative velocities +$v_{1}$~and~$v_{2}$, then the velocity of the single Lorentz transformation +which takes the place of the two separate ones +is, according to~\Eqref{(27)}, given by +\[ +v_{12} = i \tan (\psi_{1} + \psi_{2}) + = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} + = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. +\Tag{(30)} +\] + +\Paragraph{General Statements about the Lorentz Transformation +and its Theory of Invariants.} The whole theory of +invariants of the special theory of relativity depends upon +the invariant~$s^{2}$~\Eqref{(23)}. Formally, it has the same rôle in +the four-dimensional space-time continuum as the invariant +${\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}$ in the Euclidean geometry +and in the pre-relativity physics. The latter quantity is +not an invariant with respect to all the Lorentz transformations; +the quantity~$s^{2}$ of equation~\Eqref{(23)} assumes the +rôle of this invariant. With respect to an arbitrary +inertial system, $s^{2}$~may be determined by measurements; +with a given unit of measure it is a completely determinate +quantity, associated with an arbitrary pair of events. + +The invariant~$s^{2}$ differs, disregarding the number of +dimensions, from the corresponding invariant of the +Euclidean geometry in the following points. In the +Euclidean geometry $s^{2}$~is necessarily positive; it vanishes +\PageSep{41} +only when the two points concerned come together. On +the other hand, from the vanishing of +\[ +s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2} +\] +it cannot be concluded that the two space-time points +%[Illustration: Fig. 1.] +\Figure{041} +fall together; the vanishing of this quantity~$s^{2}$, is the +invariant condition that the two space-time points can be +connected by a light signal \textit{in vacuo}. If $P$~is a point +\PageSep{42} +(event) represented in the four-dimensional space of the +$x_{1}$,~$x_{2}$,~$x_{3}$,~$l$, then all the ``points'' which can be connected +to~$P$ by means of a light signal lie upon the cone $s^{2} = 0$ +(compare \Figref{1}, in which the dimension~$x_{3}$ is suppressed). +The ``upper'' half of the cone may contain the ``points'' +to which light signals can be sent from~$P$; then the +``lower'' half of the cone will contain the ``points'' from +which light signals can be sent to~$P$. The points~$P'$ +enclosed by the conical surface furnish, with~$P$, a negative~$s^{2}$; +$PP'$,~as well as~$P'P$ is then, according to Minkowski, +of the nature of a time. Such intervals represent elements +of possible paths of motion, the velocity being less than +that of light.\footnote + {That material velocities exceeding that of light are not possible, + follows from the appearance of the radical $\sqrt{1 - v^{2}}$ in the special Lorentz + transformation~\Eqref{(29)}.} +In this case the $l$-axis may be drawn in +the direction of~$PP'$ by suitably choosing the state of +motion of the inertial system. If $P'$~lies outside of the +``light-cone'' then $PP'$~is of the nature of a space; in +\index{Light cone@{Light-cone}}% +this case, by properly choosing the inertial system, $\Delta l$~can +be made to vanish. + +By the introduction of the imaginary time variable, +$x_{4} = il$, Minkowski has made the theory of invariants for +the four-dimensional continuum of physical phenomena +fully analogous to the theory of invariants for the three-dimensional +continuum of Euclidean space. The theory +of four-dimensional tensors of special relativity differs from +the theory of tensors in three-dimensional space, therefore, +only in the number of dimensions and the relations of +reality. +\PageSep{43} + +A physical entity which is specified by four quantities,~$A_{\nu}$, +\index{Four-vector}% +in an arbitrary inertial system of the $x_{1}$,~$x_{2}$,~$x_{3}$,~$x_{4}$, is +called a $4$-vector, with the components~$A_{\nu}$, if the $A_{\nu}$ +correspond in their relations of reality and the properties +of transformation to the~$\Delta x_{\nu}$; it may be of the nature of +a space or of a time. The sixteen quantities~$A_{\mu\nu}$ then +form the components of a tensor of the second rank, if +they transform according to the scheme +\[ +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}. +\] +It follows from this that the~$A_{\mu\nu}$ behave, with respect to +their properties of transformation and their properties +of reality, as the products of components,~$U_{\mu}V_{\nu}$, of two +$4$-vectors, $(U)$~and~$(V)$. All the components are real +except those which contain the index~$4$ once, those being +purely imaginary. Tensors of the third and higher ranks +may be defined in an analogous way. The operations +of addition, subtraction, multiplication, contraction and +differentiation for these tensors are wholly analogous to +the corresponding operations for tensors in three-dimensional +space. + +Before we apply the tensor theory to the four-dimensional +space-time continuum, we shall examine more +particularly the skew-symmetrical tensors. The tensor +of the second rank has, in general, $16 = 4·4$ components. +In the case of skew-symmetry the components with two +equal indices vanish, and the components with unequal +indices are equal and opposite in pairs. There exist, +therefore, only six independent components, as is the +case in the electromagnetic field. In fact, it will be shown +\PageSep{44} +when we consider Maxwell's equations that these may +be looked upon as tensor equations, provided we regard +the electromagnetic field as a skew-symmetrical tensor. +Further, it is clear that the skew-symmetrical tensor of +the third rank (skew-symmetrical in all pairs of indices) +has only four independent components, since there are +only four combinations of three different indices. + +We now turn to Maxwell's equations \Eqref{(19a)}, \Eqref{(19b)}, \Eqref{(20a)}, +\Eqref{(20b)}, and introduce the notation:\footnote + {In order to avoid confusion from now on we shall use the three-dimensional + space indices, $x$,~$y$,~$z$ instead of $1$,~$2$,~$3$, and we shall reserve + the numeral indices $1$,~$2$, $3$,~$4$ for the four-dimensional space-time continuum.} +\begin{gather*} +\left. +\begin{alignedat}{6} +\phi_{23}\qquad & \phi_{31}\qquad && \phi_{12}\quad && +\Neg\phi_{14}\quad && \Neg\phi_{24}\quad && \Neg\phi_{34} \\ +\h_{23} \qquad & \h_{31} \qquad && \h_{12} \quad && +-\Change{\veci}{i}\e_{x}\quad && -\Change{\veci}{i}\e_{y}\quad && -\Change{\veci}{i}\e_{z} +\end{alignedat} +\right\} +\Tag{(30a)}\displaybreak[1] \\ +% +\left. +\begin{alignedat}{4} +& J_{1}\qquad && J_{2}\qquad && J_{3}\qquad && J_{4} \\ +&\frac{1}{c}\veci_{x}\qquad && +\frac{1}{c}\veci_{y}\qquad && +\frac{1}{c}\veci_{z}\qquad && \Change{\veci}{i}\rho +\end{alignedat} +\right\} +\Tag{(31)} +\end{gather*} +with the convention that $\phi_{\mu\nu}$~shall be equal to~$-\phi_{\nu\mu}$. +Then Maxwell's equations may be combined into the +forms +\begin{gather*} +\frac{\dd \phi_{\mu\nu}}{\dd x_{\nu}} = J_{\mu}\Add{,} +\Tag{(32)} \\ +\frac{\dd \phi_{\mu\nu}}{\dd x_{\sigma}} + +\frac{\dd \phi_{\nu\sigma}}{\dd x_{\mu}} + +\frac{\dd \phi_{\sigma\mu}}{\dd x_{\nu}} = 0\Add{,} +\Tag{(33)} +\end{gather*} +as one can easily verify by substituting from \Eqref{(30a)}~and~\Eqref{(31)}. +Equations \Eqref{(32)}~and~\Eqref{(33)} have a tensor character, +and are therefore co-variant with respect to Lorentz +transformations, if the~$\phi_{\mu\nu}$ and the~$J_{\nu}$ have a tensor +character, which we assume. Consequently, the laws for +\PageSep{45} +transforming these quantities from one to another allowable +(inertial) system of co-ordinates are uniquely +determined. The progress in method which electrodynamics +owes to the theory of special relativity lies +principally in this, that the number of independent +hypotheses is diminished. If we consider, for example, +equations~\Eqref{(19a)} only from the standpoint of relativity of +direction, as we have done above, we see that they have +three logically independent terms. The way in which +the electric intensity enters these equations appears to +be wholly independent of the way in which the magnetic +intensity enters them; it would not be surprising if instead +of~$\dfrac{\dd \e_{\mu}}{\dd l}$, we had, say,~$\dfrac{\dd^{2} \e_{\mu}}{\dd l^{2}}$, or if this term were absent. On +the other hand, only two independent terms appear in +equation~\Eqref{(32)}. The electromagnetic field appears as a +formal unit; the way in which the electric field enters +this equation is determined by the way in which the +magnetic field enters it. Besides the electromagnetic +field, only the electric current density appears as an +independent entity. This advance in method arises from +the fact that the electric and magnetic fields draw their +separate existences from the relativity of motion. A +field which appears to be purely an electric field, judged +from one system, has also magnetic field components +when judged from another inertial system. When applied +to an electromagnetic field, the general law of transformation +furnishes, for the special case of the special Lorentz +transformation, the equations +\PageSep{46} +\[ +\left. +\begin{alignedat}{2} +{\e'}_{x} &= \e_{x}\qquad & {\h'}_{x} &= \h_{x}\Add{,} \\ +{\e'}_{y} &= \frac{\e_{y} - v\h_{z}}{\sqrt{1 - v^{2}}}\qquad & +{\h'}_{y} &= \frac{\h_{y} + v\e_{z}}{\sqrt{1 - v^{2}}}\Add{,} \\ +{\e'}_{z} &= \frac{\e_{z} + v\h_{y}}{\sqrt{1 - v^{2}}}\qquad & +{\h'}_{z} &= \frac{\h_{z} - v\e_{y}}{\sqrt{1 - v^{2}}}\Add{.} +\end{alignedat} +\right\} +\Tag{(34)} +\] + +If there exists with respect to~$K$ only a magnetic field,~$\h$, +but no electric field,~$\e$, then with respect to~$K'$ there +exists an electric field~$\e'$ as well, which would act upon +an electric particle at rest relatively to~$K'$. An observer +at rest relatively to~$K$ would designate this force as the +Biot-Savart force, or the Lorentz electromotive force. It +\index{Biot-Savart force}% +\index{Lorentz@{Lorentz electromotive force}}% +therefore appears as if this electromotive force had become +fused with the electric field intensity into a single entity. + +In order to view this relation formally, let us consider +the expression for the force acting upon unit volume of +electricity, +\[ +\k = \rho\e + [\veci, \h]\Add{,} +\Tag{(35)} +\] +in which $\veci$~is the vector velocity of electricity, with the +velocity of light as the unit. If we introduce $J_{\mu}$~and~$\phi_{\mu\Change{}{\nu}}$ +according to \Eqref{(30a)}~and~\Eqref{(31)}, we obtain for the first +component the expression +\[ +\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}. +\] +Observing that $\phi_{11}$~vanishes on account of the skew-symmetry +of the tensor~$(\phi)$, the components of~$\Change{k}{\k}$ are given +by the first three components of the four-dimensional +vector +\[ +K_{\mu} = \phi_{\mu\nu} J_{\nu}\Add{,} +\Tag{(36)} +\] +and the fourth component is given by +\[ +K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} + = \Change{\veci}{i}(\e_{x} \veci_{x} + \e_{y} \veci_{y} + \e_{z} \veci_{z}) + = \Change{\veci}{i}\lambda\Add{.} +\Tag{(37)} +\] +\PageSep{47} +There is, therefore, a four-dimensional vector of force per +unit volume, whose first three components, $\Change{k}{K}_{1}$,~$\Change{k}{K}_{2}$,~$\Change{k}{K}_{3}$, are +the ponderomotive force components per unit volume, and +whose fourth component is the rate of working of the field +per unit volume, multiplied by~$\sqrt{-1}$. +%[Illustration: Fig. 2.] +\Figure{047} + +A comparison of \Eqref{(36)}~and~\Eqref{(35)} shows that the theory +of relativity formally unites the ponderomotive force of +the electric field,~$\rho\e$, and the Biot-Savart or Lorentz +force~$[\veci, \h]$. +\PageSep{48} + +\Paragraph{Mass and Energy.} An important conclusion can be +\index{Energy@{Energy and mass}}% +\index{Mass and Energy}% +drawn from the existence and significance of the $4$-vector~$K_{\mu}$. +Let us imagine a body upon which the electromagnetic +field acts for a time. In the symbolic figure +(\Figref{2}) $Ox_{1}$~designates the $x_{1}$-axis, and is at the same +time a substitute for the three space axes $Ox_{1}$,~$Ox_{2}$,~$Ox_{3}$; +$Ol$~designates the real time axis. In this diagram a body +of finite extent is represented, at a definite time~$l$, by the +interval~$AB$; the whole space-time existence of the body +is represented by a strip whose boundary is everywhere +inclined less than~$45°$ to the $l$-axis. Between the time +sections, $l = l_{1}$ and $l = l_{2}$, but not extending to them, +a portion of the strip is shaded. This represents the +portion of the space-time manifold in which the electromagnetic +field acts upon the body, or upon the electric +charges contained in it, the action upon them being +transmitted to the body. We shall now consider the +changes which take place in the momentum and energy +of the body as a result of this action. + +We shall assume that the principles of momentum +and energy are valid for the body. The change in +momentum, $\Delta I_{x}$,~$\Delta I_{y}$,~$\Delta I_{z}$, and the change in energy,~$\Delta E$, +are then given by the expressions +\begin{alignat*}{2} +%[** TN: \int_{l_{0}}^{l_{1}} in the original] +\Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int \k_{x}\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{,} \\ +% +%[** TN: Added code for next two equations, elided in the original] +\Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int \k_{y}\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{,} \\ +% +\Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int \k_{z}\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{,} \\ +% +\Delta E &= \int_{l_{1}}^{l_{2}}dl \int \lambda\, dx\, dy\, dz + &&= \frac{1}{\Change{\veci}{i}}\int + \frac{1}{\Change{\veci}{i}}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\Add{.} +\end{alignat*} +\PageSep{49} +Since the four-dimensional element of volume is an +invariant, and $(K_{1}, K_{2}, K_{3}, K_{4})$ forms a $4$-vector, the four-dimensional +integral extended over the shaded portion +transforms as a $4$-vector, as does also the integral between +the limits $l_{1}$~and~$l_{2}$, because the portion of the region which +is not shaded contributes nothing to the integral. It +follows, therefore, that $\Delta I_{x}$,~$\Delta I_{y}$,~$\Delta I_{z}$,~$i\Delta E$ form a $4$-vector. +Since the quantities themselves transform in the same +way as their increments, it follows that the aggregate of +the four quantities +\[ +I_{x},\ I_{y},\ I_{z},\ \Change{\veci}{i} E +\] +has itself the properties of a vector; these quantities are +referred to an instantaneous condition of the body (e.g.~at +the time $l = l_{1}$). + +This $4$-vector may also be expressed in terms of the +mass~$m$, and the velocity of the body, considered as a +material particle. To form this expression, we note first, +that +\[ +-ds^{2} = d\tau^{2} + = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} + = dl^{2}(1 - q^{2}) +\Tag{(38)} +\] +is an invariant which refers to an infinitely short portion +of the four-dimensional line which represents the motion +of the material particle. The physical significance of the +invariant~$d\tau$ may easily be given. If the time axis is +chosen in such a way that it has the direction of the line +differential which we are considering, or, in other words, +if we reduce the material particle to rest, we shall then +have $d\tau = dl$; this will therefore be measured by the +light-seconds clock which is at the same place, and at +rest relatively to the material particle. We therefore call +\PageSep{50} +$\tau$~the proper time of the material particle. As opposed +to~$dl$, $d\tau$~is therefore an invariant, and is practically +equivalent to~$dl$ for motions whose velocity is small +compared to that of light. Hence we see that +\[ +u_{\sigma} = \frac{dx_{\sigma}}{d\tau} +\Tag{(39)} +\] +has, just as the~$dx_{\nu}$, the character of a vector; we shall +designate~$(u_{\sigma})$ as the four-dimensional vector (in brief, +$4$-vector) of velocity. Its components satisfy, by~\Eqref{(38)}, +the condition +\[ +\sum {u_{\sigma}}^{2} = -1\Add{.} +\Tag{(40)} +\] +We see that this $4$-vector, whose components in the +ordinary notation are +\[ +\frac{\q_{x}}{\sqrt{1 - q^{2}}},\quad +\frac{\q_{y}}{\sqrt{1 - q^{2}}},\quad +\frac{\q_{z}}{\sqrt{1 - q^{2}}},\quad +\frac{\Change{\veci}{i}}{\sqrt{1 - q^{2}}} +\Tag{(41)} +\] +is the only $4$-vector which can be formed from the velocity +components of the material particle which are defined in +three dimensions by +\[ +\q_{x} = \frac{dx}{dl},\quad +\q_{y} = \frac{dy}{dl},\quad +\q_{z} = \frac{dz}{dl}. +\] +We therefore see that +\[ +\left(m \frac{dx_{\mu}}{d\tau}\right) +\Tag{(42)} +\] +must be that $4$-vector which is to be equated to the +$4$-vector of momentum and energy whose existence we +have proved above. By equating the components, we +obtain, in three-dimensional notation, +\PageSep{51} +\[ +\left. +\begin{aligned} +I_{x} &= \frac{m\q_{x}}{\sqrt{1 - q^{2}}}\Add{,} \\ +%[** TN: Added code for next two equations, elided in the original] +I_{y} &= \frac{m\q_{y}}{\sqrt{1 - q^{2}}}\Add{,} \\ +I_{z} &= \frac{m\q_{z}}{\sqrt{1 - q^{2}}}\Add{,} \\ +E &= \frac{m}{\sqrt{1 - q^{2}}}\Add{.} +\end{aligned} +\right\} +\Tag{(43)} +\] + +We recognize, in fact, that these components of +momentum agree with those of classical mechanics for +velocities which are small compared to that of light. For +large velocities the momentum increases more rapidly +than linearly with the velocity, so as to become infinite +on approaching the velocity of light. + +If we apply the last of equations~\Eqref{(43)} to a material +particle at rest ($q = 0$), we see that the energy,~$E_{0}$, of a +\index{Energy@{Energy and mass}}% +body at rest is equal to its mass. Had we chosen the +second as our unit of time, we would have obtained +\[ +E_{0} = mc^{2}\Add{.} +\Tag{(44)} +\] +Mass and energy are therefore essentially alike; they are +\index{Equivalence of mass and energy}% +\index{Mass and Energy}% +only different expressions for the same thing. The mass +of a body is not a constant; it varies with changes in its +energy.\footnote + {The emission of energy in radioactive processes is evidently connected + with the fact that the atomic weights are not integers. Attempts have + been made to draw conclusions from this concerning the structure and + stability of the atomic nuclei.} +We see from the last of equations~\Eqref{(43)} that $E$ +becomes infinite when $q$~approaches~$1$, the velocity of +light. If we develop~$E$ in powers of~$q^{2}$, we obtain, +\[ +E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots\Add{.} +\Tag{(45)} +\] +\PageSep{52} +The second term of this expansion corresponds to the +kinetic energy of the material particle in classical +mechanics. + +\Paragraph{Equations of Motion of Material Particles.} From~\Eqref{(43)} +\index{Equations of motion of \Change{materia}{material} particle}% +\index{Motion of particle, equations of}% +we obtain, by differentiating by the time~$l$, and using +the principle of momentum, in the notation of three-dimensional +vectors, +\[ +\K = \frac{d}{dl}\left(\frac{m\q}{\sqrt{1 - q^{2}}}\right)\Add{.} +\Tag{(46)} +\] + +This equation, which was previously employed by +H.~A. Lorentz for the motion of electrons, has been +proved to be true, with great accuracy, by experiments +with $\beta$-rays. + +\Paragraph{Energy Tensor of the Electromagnetic Field.} Before the +\index{Energy@{Energy and mass}!tensor@{tensor of electromagnetic field}}% +development of the theory of relativity it was known +that the principles of energy and momentum could +be expressed in a differential form for the electromagnetic +field. The four-dimensional formulation of +these principles leads to an important conception, that of +the energy tensor, which is important for the further +development of the theory of relativity. + +If in the expression for the $4$-vector of force per unit +volume, +\[ +K_{\mu} = \phi_{\mu\nu} J_{\nu}\Add{,} +\] +using the field equations~\Eqref{(32)}, we express~$J_{\nu}$ in terms of +the field intensities,~$\phi_{\mu\nu}$, we obtain, after some transformations +and repeated application of the field equations +\Eqref{(32)}~and~\Eqref{(33)}, the expression +\[ +K_{\mu} = -\frac{\dd T_{\mu\nu}}{\dd x_{\nu}}\Add{,} +\Tag{(47)} +\] +\PageSep{53} +where we have written\footnote + {To be summed for the indices $\alpha$~and~$\beta$.} +\[ +T_{\mu\nu} + = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + + \phi_{\mu\alpha} \phi_{\nu\alpha}\Add{.} +\Tag{(48)} +\] + +The physical meaning of equation~\Eqref{(47)} becomes evident +if in place of this equation we write, using a new +notation, +\[ +\left. +\begin{alignedat}{4} +\k_{x} &= -\frac{\dd p_{xx}}{\dd x} + &&- \frac{\dd p_{xy}}{\dd y} + &&- \frac{\dd p_{xz}}{\dd z} + &&- \frac{\dd (\Change{\veci}{i} b_{x})}{\dd (\Change{\veci}{i} l)}\Add{,} \\ +% +%[** TN: Added code for next two equations] +\k_{y} &= -\frac{\dd p_{yx}}{\dd x} + &&- \frac{\dd p_{yy}}{\dd y} + &&- \frac{\dd p_{yz}}{\dd z} + &&- \frac{\dd (\Change{\veci}{i} b_{y})}{\dd (\Change{\veci}{i} l)}\Add{,} \\ +% +\k_{z} &= -\frac{\dd p_{zx}}{\dd x} + &&- \frac{\dd p_{zy}}{\dd y} + &&- \frac{\dd p_{zz}}{\dd z} + &&- \frac{\dd (\Change{\veci}{i} b_{z})}{\dd (\Change{\veci}{i} l)}\Add{,} \\ +% +\Change{\veci}{i}\lambda + &= -\frac{\dd (\Change{\veci}{i} \s_{x})}{\dd x} + &&- \frac{\dd (\Change{\veci}{i} \s_{y})}{\dd y} + &&- \frac{\dd (\Change{\veci}{i} \s_{z})}{\dd z} + &&- \frac{\dd (-\eta)}{\dd (\Change{\veci}{i} l)}\Add{;} +\end{alignedat} +\right\} +\Tag{(47a)} +\] +or, on eliminating the imaginary, +\[ +\left. +\begin{alignedat}{4} +\k_{x} &= -\frac{\dd p_{xx}}{\Change{\dd_{x}}{\dd x}} + &&- \frac{\dd p_{xy}}{\dd y} + &&- \frac{\dd p_{xz}}{\dd z} + &&- \frac{\dd b_{x}}{\dd l}\Add{,} \\ +% +%[** TN: Added code for next two equations] +\k_{y} &= -\frac{\dd p_{yx}}{\dd x} + &&- \frac{\dd p_{yy}}{\dd y} + &&- \frac{\dd p_{yz}}{\dd z} + &&- \frac{\dd b_{y}}{\dd l}\Add{,} \\ +% +\k_{z} &= -\frac{\dd p_{zx}}{\dd x} + &&- \frac{\dd p_{zy}}{\dd y} + &&- \frac{\dd p_{zz}}{\dd z} + &&- \frac{\dd b_{z}}{\dd l}\Add{,} \\ +% +\lambda + &= -\frac{\dd \s_{x}}{\dd x} + &&- \frac{\dd \s_{y}}{\dd y} + &&- \frac{\dd \s_{z}}{\dd z} + &&- \frac{\dd \eta}{\dd l}\Add{.} +\end{alignedat} +\right\} +\Tag{(47b)} +\] + +When expressed in the latter form, we see that the +first three equations state the principle of momentum; +$p_{xx}$\Add{,}\ldots\Add{,}~$p_{zx}$ are the Maxwell stresses in the electromagnetic +field, and $(b_{x}, b_{y}, b_{z})$ is the vector momentum +per unit volume of the field. The last of equations~\Eqref{(47b)} +expresses the energy principle; $\s$~is the vector flow of +energy, and $\eta$~the energy per unit volume of the field. +In fact, we get from~\Eqref{(48)} by introducing the well-known +expressions for the components of the field intensity from +electrodynamics, +\PageSep{54} +\[ +\left. +\begin{aligned} +&\begin{alignedat}{4} +p_{xx} = &{} - \h_{x} \h_{x} + &&+ \tfrac{1}{2}({\h_{x}}^{2} &&+ {\h_{y}}^{2} &&+ {\h_{z}}^{2}) \\ + &{} - \e_{x} \e_{\Change{y}{x}} + &&+ \tfrac{1}{2}({\e_{x}}^{2} &&+ {\e_{y}}^{2} &&+ {\e_{z}}^{2})\Add{,} +\end{alignedat} \\ +% +&\qquad\qquad\qquad\qquad\qquad +\begin{alignedat}{3} +p_{xy} = &{} - \h_{x} \h_{y}\quad + && p_{xz} = && {} - \h_{x} \h_{z} \\ + &{} - \e_{x} \e_{y}\Add{,}\quad + && && {} - \e_{x} \e_{z}\Add{,} +\end{alignedat} \\ +% +%**** Hard-coded width +& \makebox[3.5in][c]{$\vdots$} \\ +&b_{x} = \s_{x} = \e_{y}\h_{z} - \e_{z}\h_{y}\Add{,} \\ +%[** TN: Added code for next two equations, elided in the original] +&b_{y} = \s_{y} = \e_{z}\h_{x} - \e_{x}\h_{z}\Add{,} \\ +&b_{z} = \s_{z} = \e_{x}\h_{y} - \e_{y}\h_{x}\Add{,} \\ +% +&\eta = +\tfrac{1}{2}({\e_{x}}^{2} + {\e_{y}}^{2} + {\e_{z}}^{2} + + {\h_{x}}^{2} + {\h_{y}}^{2} + {\h_{z}}^{2})\Add{.} +\end{aligned} +\right\} +\Tag{(48a)} +\] +We conclude from~\Eqref{(48)} that the energy tensor of the +electromagnetic field is symmetrical; with this is connected +the fact that the momentum per unit volume and +the flow of energy are equal to each other (relation +between energy and inertia). + +We therefore conclude from these considerations that +the energy per unit volume has the character of a tensor. +This has been proved directly only for an electromagnetic +field, although we may claim universal validity for it. +Maxwell's equations determine the electromagnetic field +when the distribution of electric charges and currents is +known. But we do not know the laws which govern +the currents and charges. We do know, indeed, that +electricity consists of elementary particles (electrons, +positive nuclei), but from a theoretical point of view we +cannot comprehend this. We do not know the energy +factors which determine the distribution of electricity in +particles of definite size and charge, and all attempts to +complete the theory in this direction have failed. If then +we can build upon Maxwell's equations in general, the +\PageSep{55} +energy tensor of the electromagnetic field is known only +outside the charged particles.\footnote + {It has been attempted to remedy this lack of knowledge by considering + the charged particles as proper singularities. But in my opinion this means + giving up a real understanding of the structure of matter. It seems to me + much better to give in to our present inability rather than to be satisfied + by a solution that is only apparent.} +In these regions, outside +of charged particles, the only regions in which we can +believe that we have the complete expression for the +energy tensor, we have, by~\Eqref{(47)}, +\[ +\frac{\dd T_{\mu\nu}}{\dd x_{\nu}} = 0\Add{.} +\Tag{(47c)} +\] + +\Paragraph{General Expressions for the Conservation Principles.} We +\index{Conservation principles}% +\index{Principles of conservation}% +can hardly avoid making the assumption that in all other +cases, also, the space distribution of energy is given by a +symmetrical tensor,~$T_{\mu\Change{}{\nu}}$, and that this complete energy +tensor everywhere satisfies the relation~\Eqref{(47c)}. At any +rate we shall see that by means of this assumption we +obtain the correct expression for the integral energy +principle. + +Let us consider a spatially bounded, closed system, +which, four-dimensionally, we may represent as a strip, +outside of which the~$T_{\mu\nu}$ vanish. Integrate equation~\Eqref{(47c)} +over a space section. Since the integrals of +$\dfrac{\dd T_{\mu1}}{\dd x_{1}}$, $\dfrac{\dd T_{\mu2}}{\dd x_{2}}$ and $\dfrac{\dd T_{\mu3}}{\dd x_{3}}$ vanish because the~$T_{\mu\nu}$ vanish at the +limits of integration, we obtain +\[ +\frac{\dd}{\dd l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} + = 0\Add{.} +\Tag{(49)} +\] +Inside the parentheses are the expressions for the +\PageSep{56} +momentum of the whole system, multiplied by~$i$, together +with the negative energy of the system, so that \Eqref{(49)} +expresses the conservation principles in their integral +form. That this gives the right conception of energy and +%[Illustration: Fig. 3.] +\Figure{056} +the conservation principles will be seen from the following +considerations. + +\Section{Phenomenological Representation of the +Energy Tensor of Matter.} +\index{Energy@{Energy and mass}!tensor@{tensor of electromagnetic field}!of matter}% + +\Paragraph{Hydrodynamical Equations.} We know that matter is +\index{Hydrodynamical equations}% +built up of electrically charged particles, but we do not +\PageSep{57} +know the laws which govern the constitution of these +particles. In treating mechanical problems, we are therefore +obliged to make use of an inexact description of +matter, which corresponds to that of classical mechanics. +The density~$\sigma$, of a material substance and the hydrodynamical +pressures are the fundamental concepts upon +which such a description is based. + +Let~$\sigma_{0}$ be the density of matter at a place, estimated +with reference to a system of co-ordinates moving with +the matter. Then~$\sigma_{0}$, the density at rest, is an invariant. +If we think of the matter in arbitrary motion and neglect +the pressures (particles of dust \textit{in vacuo}, neglecting the +size of the particles and the temperature), then the energy +tensor will depend only upon the velocity components, +$u_{\nu}$~and~$\sigma_{0}$. We secure the tensor character of~$T_{\mu\nu}$ by +putting +\[ +T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}\Add{,} +\Tag{(50)} +\] +in which the~$u_{\mu}$, in the three-dimensional representation, +are given by~\Eqref{(41)}. In fact, it follows from~\Eqref{(50)} that for +$q = 0$, $T_{44} = -\sigma_{0}$ (equal to the negative energy per unit +volume), as it should, according to the theorem of the +equivalence of mass and energy, and according to the +physical interpretation of the energy tensor given above. +If an external force (four-dimensional vector,~$K_{\mu}$) acts +upon the matter, by the principles of momentum and +energy the equation +\[ +K_{\mu} = \frac{\dd T_{\mu\nu}}{\dd x_{\nu}} +\] +\PageSep{58} +must hold. We shall now show that this equation leads +to the same law of motion of a material particle as that +already obtained. Let us imagine the matter to be of +infinitely small extent in space, that is, a four-dimensional +thread; then by integration over the whole thread with +respect to the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, we obtain +\begin{align*} +%[** TN: Moved equals sign to the second line] +\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} + &= \int \frac{\dd T_{14}}{\dd x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ + &= -\Change{\veci}{i} \frac{d}{dl}\left\{ + \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} + \right\}. +\end{align*} + +Now $\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$ is an invariant, as is, therefore, also +$\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$. We shall calculate this integral, first +with respect to the inertial system which we have chosen, +and second, with respect to a system relatively to which +the matter has the velocity zero. The integration is to +be extended over a filament of the thread for which $\sigma_{0}$ +may be regarded as constant over the whole section. If +the space volumes of the filament referred to the two +systems are $dV$~and~$dV_{0}$ respectively, then we have +\[ +\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau +\] +and therefore also +\[ +\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} + = \int dm\, \Change{\veci}{i}\, \frac{d\tau}{dx_{4}}. +\] + +If we substitute the right-hand side for the left-hand +side in the former integral, and put $\dfrac{dx_{1}}{d\tau}$ outside the sign +\PageSep{59} +of integration, we obtain, +\[ +\K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) \\ + = \frac{d}{dl}\left(\frac{m\q_{x}}{\sqrt{1 - q^{2}}}\right). +\] +We see, therefore, that the generalized conception of the +energy tensor is in agreement with our former result. + +\Paragraph{The Eulerian Equations for Perfect Fluids.} In order +to get nearer to the behaviour of real matter we must add +to the energy tensor a term which corresponds to the +pressures. The simplest case is that of a perfect fluid in +which the pressure is determined by a scalar~$p$. Since +the tangential stresses $p_{xy}$,~etc., vanish in this case, the +contribution to the energy tensor must be of the form~$p\delta_{\nu\mu}$. +We must therefore put +\[ +T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}\Add{.} +\Tag{(51)} +\] +At rest, the density of the matter, or the energy per unit +volume, is in this case, not~$\sigma$ but~$\sigma - p$. For +\[ +-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} + = \sigma - p. +\] +In the absence of any force, we have +\[ +\frac{\dd T_{\mu\nu}}{\dd x_{\nu}} + = \sigma u_{\nu} \frac{\dd u_{\mu}}{\dd x_{\nu}} + + u_{\mu} \frac{\dd (\sigma u_{\nu})}{\dd x_{\nu}} + + \frac{\dd p}{\dd x_{\mu}} = 0. +\] +If we multiply this equation by~$u_{\mu}$ $\left(= \dfrac{dx_{\mu}}{d\tau}\right)$ and sum for +the $\mu$'s~we obtain, using~\Eqref{(40)}, +\[ +-\frac{\dd (\sigma u_{\nu})}{\dd x_{\nu}} + \frac{dp}{d\tau} = 0\Add{,} +\Tag{(52)} +\] +\PageSep{60} +where we have put $\dfrac{\dd p}{\dd x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}$. This is the equation of +continuity, which differs from that of classical mechanics +by the term~$\dfrac{dp}{d\tau}$, which, practically, is vanishingly small. +Observing~\Eqref{(52)}, the conservation principles take the form +\[ +\sigma \frac{du_{\mu}}{d\tau} + + u_{\mu} \frac{dp}{d\tau} + + \frac{\dd p}{\dd x_{\mu}} = 0\Add{.} +\Tag{(53)} +\] +The equations for the first three indices evidently correspond +to the Eulerian equations. That the equations +\Eqref{(52)}~and~\Eqref{(53)} correspond, to a first approximation, to the +hydrodynamical equations of classical mechanics, is a +further confirmation of the generalized energy principle. +The density of matter and of energy has the character of +a symmetrical tensor. +\PageSep{61} + + +\Lecture{III}{The General Theory of Relativity}{The General Theory} + +\First{All} of the previous considerations have been based +upon the assumption that all inertial systems are +equivalent for the description of physical phenomena, but +that they are preferred, for the formulation of the laws +of nature, to spaces of reference in a different state of +motion. We can think of no cause for this preference +for definite states of motion to all others, according to +our previous considerations, either in the perceptible +bodies or in the concept of motion; on the contrary, it +must be regarded as an independent property of the +space-time continuum. The principle of inertia, in +particular, seems to compel us to ascribe physically +objective properties to the space-time continuum. Just +as it was necessary from the Newtonian standpoint to +make both the statements, \textit{tempus est absolutum}, \textit{spatium +est absolutum}, so from the standpoint of the special theory +of relativity we must say, \textit{continuum spatii et temporis est +absolutum}. In this latter statement \textit{absolutum} means not +only ``physically real,'' but also ``independent in its +physical properties, having a physical effect, but not itself +influenced by physical conditions.'' + +As long as the principle of inertia is regarded as the +\PageSep{62} +keystone of physics, this standpoint is certainly the only +one which is justified. But there are two serious criticisms +of the ordinary conception. In the first place, it is contrary +to the mode of thinking in science to conceive of a thing +(the space-time continuum) which acts itself, but which +cannot be acted upon. This is the reason why E.~Mach +\index{Mach}% +was led to make the attempt to eliminate space as an +active cause in the system of mechanics. According to +him, a material particle does not move in unaccelerated +motion relatively to space, but relatively to the centre of +all the other masses in the universe; in this way the +series of causes of mechanical phenomena was closed, in +contrast to the mechanics of Newton and Galileo. In +order to develop this idea within the limits of the modern +theory of action through a medium, the properties of +the space-time continuum which determine inertia must +be regarded as field properties of space, analogous to +the electromagnetic field. The concepts of classical +mechanics afford no way of expressing this. For this +reason Mach's attempt at a solution failed for the time +being. We shall come back to this point of view later. +In the second place, classical mechanics indicates a +limitation which directly demands an extension of the +principle of relativity to spaces of reference which are not +in uniform motion relatively to each other. The ratio of +the masses of two bodies is defined in mechanics in two +ways which differ from each other fundamentally; in the +first place, as the reciprocal ratio of the accelerations +which the same motional force imparts to them (inert +mass), and in the second place, as the ratio of the forces +\PageSep{63} +which act upon them in the same gravitational field +(gravitational mass). The equality of these two masses, +\index{Gravitational mass}% +\index{Inert and gravitational mass, equality of}% +\index{Mass and Energy!equality of gravitational and inert}% +\index{Mass and Energy!gravitational}% +so differently defined, is a fact which is confirmed by +experiments of very high accuracy (experiments of Eötvös), +and classical mechanics offers no explanation for this +equality. It is, however, clear that science is fully justified +in assigning such a numerical equality only after this +numerical equality is reduced to an equality of the real +nature of the two concepts. + +That this object may actually be attained by an extension +of the principle of relativity, follows from the following +consideration. A little reflection will show that the +theorem of the equality of the inert and the gravitational +mass is equivalent to the theorem that the acceleration +imparted to a body by a gravitational field is independent +of the nature of the body. For Newton's equation of +motion in a gravitational field, written out in full, is +\begin{multline*} +(\text{Inert mass})·(\text{Acceleration}) = (\text{Intensity of the} \\ + \text{gravitational field})·(\text{Gravitational mass}). +\end{multline*} +It is only when there is numerical equality between the +inert and gravitational mass that the acceleration is independent +of the nature of the body. Let now $K$~be an +inertial system. Masses which are sufficiently far from +each other and from other bodies are then, with respect +to~$K$, free from acceleration. We shall also refer these +masses to a system of co-ordinates~$K'$, uniformly accelerated +with respect to~$K$. Relatively to~$K'$ all the masses +have equal and parallel accelerations; with respect to~$K'$ +they behave just as if a gravitational field were present and +\PageSep{64} +$K'$~were unaccelerated. Overlooking for the present the +question as to the ``cause'' of such a gravitational field, +which will occupy us later, there is nothing to prevent +our conceiving this gravitational field as real, that is, the +conception that $K'$~is ``at rest'' and a gravitational field +is present we may consider as equivalent to the conception +that only $K$~is an ``allowable'' system of co-ordinates +and no gravitational field is present. The assumption of +the complete physical equivalence of the systems of co-ordinates, +$K$~and~$K'$, we call the ``principle of equivalence;'' +\index{Principle@{Principle of equivalence}}% +this principle is evidently intimately connected +with the theorem of the equality between the inert and +the gravitational mass, and signifies an extension of the +principle of relativity to co-ordinate systems which are +in non-uniform motion relatively to each other. In fact, +through this conception we arrive at the unity of the +nature of inertia and gravitation. For according to our +way of looking at it, the same masses may appear to be +either under the action of inertia alone (with respect to~$K$) +or under the combined action of inertia and gravitation +(with respect to~$K'$). The possibility of explaining +the numerical equality of inertia and gravitation by the +unity of their nature gives to the general theory of +relativity, according to my conviction, such a superiority +over the conceptions of classical mechanics, that all the +difficulties encountered in development must be considered +as small in comparison. + +What justifies us in dispensing with the preference +for inertial systems over all other co-ordinate systems, a +preference that seems so securely established by experiment +\PageSep{65} +based upon the principle of inertia? The weakness +\index{Criticism of principle of inertia}% +\index{Principle@{Principle of equivalence}!inertia, criticism of}% +of the principle of inertia lies in this, that it involves an +argument in a circle: a mass moves without acceleration +if it is sufficiently far from other bodies; we know that +it is sufficiently far from other bodies only by the fact +that it moves without acceleration. Are there, in general, +any inertial systems for very extended portions of the +space-time continuum, or, indeed, for the whole universe? +We may look upon the principle of inertia as established, +to a high degree of approximation, for the space of our +planetary system, provided that we neglect the perturbations +due to the sun and planets. Stated more exactly, +there are finite regions, where, with respect to a suitably +chosen space of reference, material particles move freely +without acceleration, and in which the laws of the special +theory of relativity, which have been developed above, +hold with remarkable accuracy. Such regions we shall +call ``Galilean regions.'' We shall proceed from the +\index{Galilean@{Galilean regions}}% +consideration of such regions as a special case of known +properties. + +The principle of equivalence demands that in dealing +with Galilean regions we may equally well make use of +non-inertial systems, that is, such co-ordinate systems as, +relatively to inertial systems, are not free from acceleration +and rotation. If, further, we are going to do away +completely with the difficult question as to the objective +reason for the preference of certain systems of co-ordinates, +then we must allow the use of arbitrarily moving systems +of co-ordinates. As soon as we make this attempt seriously +\PageSep{66} +\index{Rotation}% +we come into conflict with that physical interpretation of +space and time to which we were led by the special theory +of relativity. For let $K'$ be a system of co-ordinates whose +$z'$-axis coincides with the $z$-axis of~$K$, and which rotates +about the latter axis with constant angular velocity. Are +the configurations of rigid bodies, at rest relatively to~$K'$, +in accordance with the laws of Euclidean geometry? +Since $K'$~is not an inertial system, we do not know +directly the laws of configuration of rigid bodies with +respect to~$K'$, nor the laws of nature, in general. But +we do know these laws with respect to the inertial system~$K$, +and we can therefore estimate them with respect to~$K'$. +Imagine a circle drawn about the origin in the \Change{$x'y'$}{$x'$-$y'$}~plane +of~$K'$, and a diameter of this circle. Imagine, further, that +we have given a large number of rigid rods, all equal to +each other. We suppose these laid in series along the +periphery and the diameter of the circle, at rest relatively +to~$K'$. If $U$~is the number of these rods along the periphery, +$D$~the number along the diameter, then, if $K'$~does +not rotate relatively to~$K$, we shall have +\[ +\frac{U}{D} = \pi. +\] +But if $K'$~rotates we get a different result. Suppose +that at a definite time~$t$\Change{,}{} of~$K$ we determine the ends of +all the rods. With respect to~$K$ all the rods upon the +periphery experience the Lorentz contraction, but the +rods upon the diameter do not experience this contraction +\PageSep{67} +(along their lengths!).\footnote + {These considerations assume that the behaviour of rods and clocks + depends only upon velocities, and not upon accelerations, or, at least, that + the influence of acceleration does not counteract that of velocity.} +It therefore follows that +\[ +\frac{U}{D} > \pi. +\] + +It therefore follows that the laws of configuration of +rigid bodies with respect to~$K'$ do not agree with the +laws of configuration of rigid bodies that are in accordance +with Euclidean geometry. If, further, we place two +similar clocks (rotating with~$K'$), one upon the periphery, +and the other at the centre of the circle, then, judged +from~$K$, the clock on the periphery will go slower than +the clock at the centre. The same thing must take place, +judged from~$K'$, if we define time with respect to~$K'$ in +a not wholly unnatural way, that is, in such a way that +the laws with respect to~$K'$ depend explicitly upon the +time. Space and time, therefore, cannot be defined +with respect to~$K'$ as they were in the special theory of +relativity with respect to inertial systems. But, according +to the principle of equivalence, $K'$~is also to be considered +as a system at rest, with respect to which there +is a gravitational field (field of centrifugal force, and +\index{Centrifugal force}% +force of Coriolis). We therefore arrive at the result: +the gravitational field influences and even determines the +metrical laws of the space-time continuum. If the laws +of configuration of ideal rigid bodies are to be expressed +geometrically, then in the presence of a gravitational +field the geometry is not Euclidean. +\PageSep{68} + +The case that we have been considering is analogous +to that which is presented in the two-dimensional treatment +of surfaces. It is impossible in the latter case +also, to introduce co-ordinates on a surface (e.g.~the +surface of an ellipsoid) which have a simple metrical +significance, while on a plane the Cartesian co-ordinates, +$x_{1}$,~$x_{2}$, signify directly lengths measured by a unit +measuring rod. Gauss overcame this difficulty, in his +\index{Gauss}% +theory of surfaces, by introducing curvilinear co-ordinates +\index{Curvilinear co-ordinates}% +which, apart from satisfying conditions of continuity, +were wholly arbitrary, and afterwards these co-ordinates +were related to the metrical properties of the surface. +In an analogous way we shall introduce in the general +theory of relativity arbitrary co-ordinates, $x_{1}$,~$x_{2}$,~$x_{3}$,~$x_{4}$, +which shall number uniquely the space-time points, so +that neighbouring events are associated with neighbouring +values of the co-ordinates; otherwise, the choice of +co-ordinates is arbitrary. We shall be true to the +principle of relativity in its broadest sense if we give +such a form to the laws that they are valid in every +such four-dimensional system of co-ordinates, that is, if +the equations expressing the laws are co-variant with +respect to arbitrary transformations. + +The most important point of contact between Gauss's +theory of surfaces and the general theory of relativity +lies in the metrical properties upon which the concepts +of both theories, in the main, are based. In the case +of the theory of surfaces, Gauss's argument is as follows. +Plane geometry may be based upon the concept of the +distance~$ds$, between two indefinitely near points. The +\PageSep{69} +concept of this distance is physically significant because +the distance can be measured directly by means of a +rigid measuring rod. By a suitable choice of Cartesian +co-ordinates this distance may be expressed by the +formula $ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}$. We may base upon this +quantity the concepts of the straight line as the geodesic +($\delta\! \int\!\! ds = 0$), the interval, the circle, and the angle, upon +which the Euclidean plane geometry is built. A +geometry may be developed upon another continuously +curved surface, if we observe that an infinitesimally +small portion of the surface may be regarded as plane, +to within relatively infinitesimal quantities. There are +Cartesian co-ordinates, $X_{1}$,~$X_{2}$, upon such a small +portion of the surface, and the distance between two +points, measured by a measuring rod, is given by +\[ +ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}. +\] +If we introduce arbitrary curvilinear co-ordinates, $x_{1}$,~$x_{2}$, +on the surface, then $dX_{1}$,~$dX_{2}$, may be expressed linearly +in terms of $dx_{1}$,~$dx_{2}$. Then everywhere upon the surface +we have +\[ +ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + + g_{22}\, {dx_{2}}^{2}\Add{,} +\] +where $g_{11}$,~$g_{12}$,~$g_{22}$ are determined by the nature of the +surface and the choice of co-ordinates; if these quantities +are known, then it is also known how networks of rigid +rods may be laid upon the surface. In other words, the +geometry of surfaces may be based upon this expression +for~$ds^{2}$ exactly as plane geometry is based upon the +corresponding expression. + +There are analogous relations in the four-dimensional +\PageSep{70} +space-time continuum of physics. In the immediate +neighbourhood of an observer, falling freely in a gravitational +field, there exists no gravitational field. We +can therefore always regard an infinitesimally small +region of the space-time continuum as Galilean. For +such an infinitely small region there will be an inertial +system (with the space co-ordinates, $X_{1}$,~$X_{2}$,~$X_{3}$, and the +time co-ordinate~$X_{4}$) relatively to which we are to regard +the laws of the special theory of relativity as valid. The +quantity which is directly measurable by our unit +measuring rods and clocks, +\[ +{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2}\Add{,} +\] +or its negative, +\[ +ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}\Add{,} +\Tag{(54)} +\] +is therefore a uniquely determinate invariant for two +neighbouring events (points in the four-dimensional +continuum), provided that we use measuring rods that +are equal to each other when brought together and +superimposed, and clocks whose rates are the same +when they are brought together. In this the physical +assumption is essential that the relative lengths of two +measuring rods and the relative rates of two clocks are +independent, in principle, of their previous history. But +this assumption is certainly warranted by experience; +if it did not hold there could be no sharp spectral lines; +for the single atoms of the same element certainly do +not have the same history, and it would be absurd to +suppose any relative difference in the structure of the +\PageSep{71} +single atoms due to their previous history if the mass +and frequencies of the single atoms of the same element +were always the same. + +Space-time regions of finite extent are, in general, +not Galilean, so that a gravitational field cannot be done +away with by any choice of co-ordinates in a finite +region. There is, therefore, no choice of co-ordinates +for which the metrical relations of the special theory of +relativity hold in a finite region. But the invariant~$ds$ +always exists for two neighbouring points (events) of +the continuum. This invariant~$ds$ may be expressed in +arbitrary co-ordinates. If one observes that the local~$dX_{\nu}$ +may be expressed linearly in terms of the co-ordinate +differentials~$dx_{\nu}$, $ds^{2}$~may be expressed in the +form +\[ +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{.} +\Tag{(55)} +\] + +The functions~$g_{\mu\nu}$ describe, with respect to the arbitrarily +chosen system of co-ordinates, the metrical relations +of the space-time continuum and also the +gravitational field. As in the special theory of relativity, +we have to discriminate between time-like and space-like +line elements in the four-dimensional continuum; +owing to the change of sign introduced, time-like +line elements have a real, space-like line elements an +imaginary~$ds$. The time-like~$ds$ can be measured directly +by a suitably chosen clock. + +According to what has been said, it is evident that +the formulation of the general theory of relativity +assumes a generalization of the theory of invariants and +the theory of tensors; the question is raised as to the +\PageSep{72} +\index{Tensor|etseq}% +form of the equations which are co-variant with respect +\index{Covariant@{Co-variant}!vector}% +to arbitrary point transformations. The generalized +calculus of tensors was developed by mathematicians +long before the theory of relativity. Riemann first +\index{Riemann}% +extended Gauss's train of thought to continua of any +number of dimensions; with prophetic vision he saw +the physical meaning of this generalization of Euclid's +geometry. Then followed the development of the theory +in the form of the calculus of tensors, particularly by +Ricci and Levi-Civita. This is the place for a brief +presentation of the most important mathematical concepts +and operations of this calculus of tensors. + +We designate four quantities, which are defined as +functions of the~$x_{\nu}$ with respect to every system of co-ordinates, +as components,~$A^{\nu}$, of a contra-variant vector, +\index{Contravariant@{Contra-variant vectors}}% +\index{Vector@{Vector, co-variant}!contra-variant}% +if they transform in a change of co-ordinates as the co-ordinate +differentials~$dx_{\nu}$. We therefore have +\[ +%[** TN: Place prime directly on the tensor name throughout, for consistency] +\Change{A^{\mu'}}{{A'}^{\mu}} = \frac{{\dd x'}_{\mu}}{\dd x_{\nu}} A^{\nu}\Add{.} +\Tag{(56)} +\] +Besides these contra-variant vectors, there are also co-variant +vectors. If $B_{\nu}$~are the components of a co-variant +vector, these vectors are transformed according to the +rule +\[ +{B'}_{\mu} = \frac{\dd x_{\nu}}{{\dd \Change{x}{x'}}_{\mu}} B_{\nu}\Add{.} +\Tag{(57)} +\] +The definition of a co-variant vector is chosen in such a +\index{Vector@{Vector, co-variant}}% +way that a co-variant vector and a contra-variant vector +together form a scalar according to the scheme, +\[ +\phi = B_{\nu} A^{\nu}\quad \text{(summed over the~$\nu$).} +\] +\PageSep{73} +Accordingly, +\[ +{B'}_{\mu} \Change{A^{\mu'}}{{A'}^{\mu}} + = \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}} + \frac{{\dd x'}_{\mu}}{\dd x_{\beta}} B_{\alpha} A^{\beta} + = B_{\alpha} A^{\alpha}\Add{.} +\] +In particular, the derivatives~$\dfrac{\dd \phi}{\dd x_{\alpha}}$ of a scalar~$\phi$, are components +of a co-variant vector, which, with the co-ordinate +differentials, form the scalar~$\dfrac{\dd \phi}{\dd x_{\alpha}}\, dx_{\alpha}$; we see from this +example how natural is the definition of the co-variant +vectors. + +There are here, also, tensors of any rank, which may +have co-variant or contra-variant character with respect +to each index; as with vectors, the character is designated +by the position of the index. For example, $\Change{{A_{\mu}}^{\nu}}{A_{\mu}^{\nu}}$~denotes +a tensor of the second rank, which is co-variant +with respect to the index~$\mu$, and contra-variant with respect +to the index~$\nu$. The tensor character indicates +that the equation of transformation is +\[ +\Change{A_{\mu}^{\nu'}}{{A'}_{\mu}^{\nu}} + = \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}} + \frac{{\dd x'}_{\nu}}{\dd x_{\beta}} A_{\alpha}^{\beta}\Add{.} +\Tag{(58)} +\] + +Tensors may be formed by the addition and subtraction +of tensors of equal rank and like character, as in the +theory of invariants of orthogonal linear substitutions, for +example, +\[ +A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}\Add{.} +\Tag{(59)} +\] +The proof of the tensor character of~$C_{\mu}^{\nu}$ depends upon~\Eqref{(58)}. + +Tensors may be formed by multiplication, keeping the +character of the indices, just as in the theory of invariants +of linear orthogonal transformations, for example, +\[ +A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}\Add{.} +\Tag{(60)} +\] +\PageSep{74} +The proof follows directly from the rule of transformation. + +Tensors may be formed by contraction with respect to +two indices of different character, for example, +\[ +A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}\Add{.} +\Tag{(61)} +\] +The tensor character of~$A_{\mu\sigma\tau}^{\mu}$ determines the tensor +character of~$B_{\sigma\tau}$. Proof--- +\[ +\Change{A_{\mu\sigma\tau}^{\mu'}}{{A'}_{\mu\sigma\tau}^{\mu}} += \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}} \frac{{\dd x'}_{\mu}}{\dd x_{\beta}} + \frac{\dd x_{s}}{{\dd x'}_{\sigma}} \frac{\dd x_{t}}{{\dd x'}_{\tau}} + \Change{}{A_{\alpha st}^{\beta}} += \frac{\dd x_{s}}{{\dd x'}_{\sigma}} \frac{\dd x_{t}}{{\dd x'}_{\tau}} + A_{\alpha st}^{\alpha}. +\] + +The properties of symmetry and skew-symmetry of a +tensor with respect to two indices of like character have +the same significance as in the theory of invariants. + +With this, everything essential has been said with +regard to the algebraic properties of tensors. + +\Paragraph{The Fundamental Tensor.} It follows from the invariance +\index{Fundamental tensor}% +\index{Tensor!Fundamental}% +of~$ds^{2}$ for an arbitrary choice of the~$dx_{\nu}$, in connexion +with the condition of symmetry consistent with~\Eqref{(55)}, that +the~$g_{\mu\nu}$ are components of a symmetrical co-variant tensor +(Fundamental Tensor). Let us form the determinant,~$g$, +of the~$g_{\mu\nu}$, and also the minors, divided by~$g$, corresponding +to the single~$g_{\mu\nu}$. These minors, divided by~$g$, +will be denoted by~$g^{\mu\nu}$, and their co-variant character +is not yet known. Then we have +\[ +g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} +%[** TN: No brace in original] + = \begin{cases} + 1 & \text{if $\alpha = \beta$\Add{,}} \\ + 0 & \text{if $\alpha \neq \beta$\Add{.}} + \end{cases} +\Tag{(62)} +\] + +If we form the infinitely small quantities (co-variant +vectors) +\[ +d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}\Add{,} +\Tag{(63)} +\] +\PageSep{75} +multiply by~$g^{\mu\beta}$ and sum over the~$\mu$, we obtain, by the +use of~\Eqref{(62)}, +\[ +dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}\Add{.} +\Tag{(64)} +\] +Since the ratios of the~$d\xi_{\mu}$ are arbitrary, and the~$dx_{\beta}$ as +well as the~$dx_{\mu}$ are components of vectors, it follows that +\index{Contravariant@{Contra-variant vectors}!tensors}% +the~$g^{\mu\nu}$ are the components of a contra-variant tensor\footnote + {If we multiply~\Eqref{(64)} by~$\dfrac{{\dd x'}_{\alpha}}{\dd x_{\beta}}$, sum over the~$\beta$, and replace the~$d\xi^{\mu}$ by a + transformation to the accented system, we obtain + \[ + {dx'}_{\alpha} + = \frac{{\dd x'}_{\sigma}}{\dd x_{\mu}}\, + \frac{{\dd x'}_{\alpha}}{\dd x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}. + \] + The statement made above follows from this, since, by~\Eqref{(64)}, we must also + have ${dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}$, and both equations must hold for every choice of the~${d\xi'}_{\sigma}$.} +(contra-variant fundamental tensor). The tensor character +of~$\delta_{\alpha}^{\beta}$ (mixed fundamental tensor) accordingly follows, +by~\Eqref{(62)}. By means of the fundamental tensor, instead +of tensors with co-variant index character, we can +introduce tensors with contra-variant index character, +and conversely. For example, +\begin{align*} +A^{\mu} &= g^{\mu\alpha} A_{\alpha}\Add{,} \\ +A_{\mu} &= g_{\mu\alpha} A^{\alpha}\Add{,} \\ +T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. +\end{align*} + +\Paragraph{Volume Invariants.} The volume element +\[ +\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx +\] +is not an invariant. For by Jacobi's theorem, +\[ +dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx\Add{.} +\Tag{(65)} +\] +\PageSep{76} +But we can complement~$dx$ so that it becomes an invariant. +If we form the determinant of the quantities +\[ +{g'}_{\mu\nu} + = \frac{\dd x_{\alpha}}{{\dd x'}_{\mu}}\, + \frac{\dd x_{\beta}}{{\dd x'}_{\nu}}\, g_{\alpha\beta}\Add{,} +\] +we obtain, by a double application of the theorem of +multiplication of determinants, +\[ +g' = |{g'}_{\mu\nu}| + = \left|\frac{\dd x_{\nu}}{{\dd x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| + = \left|\frac{{\dd x'}_{\mu}}{\dd x_{\nu}}\right|^{-2} g. +\Change{}{\Tag{(66)}} +\] +We therefore get the invariant, +\[ +\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx. +\] + +\Paragraph{Formation of Tensors by Differentiation.} Although +\index{Differentiation of tensors}% +\index{Tensors, formation by differentiation}% +the algebraic operations of tensor formation have proved +to be as simple as in the special case of invariance with +respect to linear orthogonal transformations, nevertheless +in the general case, the invariant differential operations +are, unfortunately, considerably more complicated. The +reason for this is as follows. If $A^{\mu}$~is a contra-variant +vector, the coefficients of its transformation,~$\dfrac{{\dd x'}_{\mu}}{\dd x_{\nu}}$, are independent +of position only if the transformation is a linear +one. For then the vector components, $A^{\mu} + \dfrac{\dd A^{\mu}}{\dd x_{\alpha}}\, dx_{\alpha}$, at +a neighbouring point transform in the same way as the~$A^{\mu}$, +from which follows the vector character of the vector +differentials, and the tensor character of~$\dfrac{\dd A^{\mu}}{\dd x_{\alpha}}$. But if the +$\dfrac{{\dd x'}_{\mu}}{\dd x_{\nu}}$ are variable this is no longer true. +\PageSep{77} +\index{Hypotheses of pre-relativity physics}% + +That there are, nevertheless, in the general case, invariant +differential operations for tensors, is recognized +most satisfactorily in the following way, introduced by +Levi-Civita and Weyl. Let $(A^{\mu})$ be a contra-variant vector +\index{Levi-Civita}% +\index{Weyl}% +whose components are given with respect to the co-ordinate +system of the~$x_{\nu}$. Let $P_{1}$~and~$P_{2}$ be two infinitesimally +near points of the continuum. For the +infinitesimal region surrounding the point~$P_{1}$, there is, +according to our way of considering the matter, a co-ordinate +system of the~$X_{\nu}$ (with imaginary $X_{\Change{\nu}{4}}$-\Change{co-ordinates}{co-ordinate}) +for which the continuum is Euclidean. Let +$A_{(1)}^{\mu}$~be the co-ordinates of the vector at the point~$P_{1}$. +Imagine a vector drawn at the point~$P_{2}$, using the local +system of the~$X_{\nu}$, with the same co-ordinates (parallel +vector through~$P_{2}$), then this parallel vector is uniquely +determined by the vector at~$P_{1}$ and the displacement. +We designate this operation, whose uniqueness will appear +in the sequel, the parallel displacement of the vector~$A_{\mu}$ +from~$P_{1}$ to the infinitesimally near point~$P_{2}$\Change{}{.} If we form +the vector difference of the vector~$(A^{\mu})$ at the point~$P_{2}$ +and the vector obtained by parallel displacement from~$P_{1}$ +to~$P_{2}$, we get a vector which may be regarded as the +differential of the vector~$(A^{\mu})$ for the given displacement~$(dx_{\nu})$. + +This vector displacement can naturally also be considered +with respect to the co-ordinate system of the~$x_{\nu}$. +If $A^{\nu}$~are the co-ordinates of the vector at~$P_{1}$, $A^{\nu} + \delta A^{\nu}$ +the co-ordinates of the vector displaced to~$P_{2}$ along the +interval~$(dx_{\nu})$, then the~$\delta A^{\nu}$ do not vanish in this case. +We know of these quantities, which do not have a vector +\PageSep{78} +character, that they must depend linearly and homogeneously +upon the~$dx_{\nu}$ and the~$A^{\nu}$. We therefore put +\[ +\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}\Add{.} +\Tag{(67)} +\] + +In addition, we can state that the~$\Gamma_{\alpha\beta}^{\nu}$ must be symmetrical +with respect to the indices $\alpha$~and~$\beta$. For we +can assume from a representation by the aid of a Euclidean +system of local co-ordinates that the same parallelogram +will be described by the displacement of an element~$d^{(1)}x_{\nu}$ +along a second element~$d^{(2)}x_{\nu}$ as by a displacement +of~$d^{(2)}x_{\nu}$ along~$d^{(1)}x_{\nu}$. We must therefore have +\begin{multline*} +d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ + = d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). +\end{multline*} +The statement made above follows from this, after interchanging +the indices of summation, $\alpha$~and~$\beta$, on the +right-hand side. + +Since the quantities~$g_{\mu\nu}$ determine all the metrical +properties of the continuum, they must also determine +the~$\Gamma_{\alpha\beta}^{\nu}$. If we consider the invariant of the vector~$A^{\nu}$, +that is, the square of its magnitude, +\[ +g_{\mu\nu} A^{\mu} A^{\nu}\Add{,} +\] +which is an invariant, this cannot change in a parallel +displacement. We therefore have +\[ +0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) + = \frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu} +\] +or, by~\Eqref{(67)}, +\[ +\left(\frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} + - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} + - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} += 0. +\] +\PageSep{79} + +Owing to the symmetry of the expression in the +brackets with respect to the indices $\mu$~and~$\nu$, this equation +can be valid for an arbitrary choice of the vectors $(A^{\mu})$ +and~$dx_{\nu}$ only when the expression in the brackets vanishes +for all combinations of the indices. By a cyclic interchange +of the indices $\mu$,~$\nu$,~$\alpha$, we obtain thus altogether +three equations, from which we obtain, on taking into +account the symmetrical property of the~$\Gamma_{\mu\nu}^{\alpha}$, +\[ +\left[\Chr{\mu}{\nu}{\alpha}\right] + = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}\Add{,} +\Tag{(68)} +\] +in which, following Christoffel, the abbreviation has been +used, +\[ +\left[\Chr{\mu}{\nu}{\alpha}\right] + = \tfrac{1}{2}\left( + \frac{\dd g_{\mu\alpha}}{\dd x_{\nu}} + + \frac{\dd g_{\nu\alpha}}{\dd x_{\mu}} + - \frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} +\right)\Add{.} +\Tag{(69)} +\] + +If we multiply~\Eqref{(68)} by~$g^{\alpha\sigma}$ and sum over the~$\alpha$, we +obtain +\[ +\Gamma_{\mu\nu}^{\alpha} + = \tfrac{1}{2} g^{\sigma\alpha}\left( + \frac{\dd g_{\mu\alpha}}{\dd x_{\nu}} + + \frac{\dd g_{\nu\alpha}}{\dd x_{\mu}} + - \frac{\dd g_{\mu\nu}}{\dd x_{\alpha}} +\right) = \left\{\Chr{\mu}{\nu}{\sigma}\right\}\Add{,} +\Tag{(70)} +\] +in which $\left\{\Chr{\mu}{\nu}{\sigma}\right\}$ is the Christoffel symbol of the second +kind. Thus the quantities~$\Gamma$ are deduced from the~$g_{\mu\nu}$. +Equations \Eqref{(67)}~and~\Eqref{(70)} are the foundation for the +following discussion. + +\Paragraph{Co-variant Differentiation of Tensors.} If $(A^{\mu} + \delta A^{\mu})$ is +\index{Differentiation of tensors}% +the vector resulting from an infinitesimal parallel displacement +from~$P_{1}$ to~$P_{2}$, and $(A^{\mu} + dA^{\mu})$~the vector~$A^{\mu}$ at the +point~$P_{2}$, then the difference of these two, +\[ +dA^{\mu} - \delta A^{\mu} = \left( + \frac{\Change{\delta}{\dd} A^{\mu}}{\Change{\delta}{\dd} x_{\sigma}} + + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma}\Add{,} +\] +\PageSep{80} +is also a vector. Since this is the case for an arbitrary +choice of the~$dx_{\sigma}$, it follows that +\[ +%[** TN: Covariant derivative index not subscripted in original] +{A^{\mu}}_{;\, \sigma} + = \frac{\dd A^{\mu}}{\dd x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} +\Tag{(71)} +\] +is a tensor, which we designate as the co-variant derivative +of the tensor of the first rank (vector). Contracting this +tensor, we obtain the divergence of the contra-variant +tensor~$A^{\mu}$. In this we must observe that according to~\Eqref{(70)}, +\[ +\Gamma_{\mu\sigma}^{\sigma} + = \tfrac{1}{2} g^{\sigma\alpha} \frac{\dd g_{\sigma\alpha}}{\dd x_{\mu}} + = \frac{1}{\sqrt{g}}\, \frac{\dd \sqrt{g}}{\dd x_{\mu}}\Add{.} +\Tag{(72)} +\] +If we put, further, +\[ +A^{\mu} \sqrt{g} = \tA^{\mu}\Add{,} +\Tag{(73)} +\] +a quantity designated by Weyl as the contra-variant tensor +density\footnote + {This expression is justified, in that $A^{\mu}\sqrt{g}\, dx = \tA^{\mu}\, dx$ has a tensor + character. Every tensor, when multiplied by~$\sqrt{g}$, changes into a tensor + density. We employ capital Gothic letters for tensor densities.} +of the first rank, it follows that, +\[ +\tA = \frac{\dd \tA^{\mu}}{\dd x_{\mu}} +\Tag{(74)} +\] +is a scalar density. + +We get the law of parallel displacement for the +co-variant vector~$B_{\mu}$ by stipulating that the parallel +displacement shall be effected in such a way that the +scalar +\[ +\phi = A^{\mu} B_{\mu} +\] +remains unchanged, and that therefore +\[ +A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu} +\] +\PageSep{81} +vanishes for every value assigned to~$(A^{\mu})$. We therefore +get +\[ +\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}\Add{.} +\Tag{(75)} +\] + +From this we arrive at the co-variant derivative of the +co-variant vector by the same process as that which led +to~\Eqref{(71)}, +\[ +%[** TN: Covariant derivative index not subscripted in original] +B_{\mu;\, \sigma} + = \frac{\dd B_{\mu}}{\dd x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}\Add{.} +\Tag{(76)} +\] +By interchanging the indices $\mu$~and~$\beta$, and subtracting, +we get the skew-symmetrical tensor, +\[ +\phi_{\mu\sigma} + = \frac{\dd B_{\mu}}{\dd x_{\sigma}} - \frac{\dd B_{\sigma}}{\dd x_{\mu}}\Add{.} +\Tag{(77)} +\] + +For the co-variant differentiation of tensors of the +second and higher ranks we may use the process by +which \Eqref{(75)}~was deduced. Let, for example, $(A_{\sigma\tau})$~be a +co-variant tensor of the second rank. Then $A_{\sigma\tau} E^{\sigma} F^{\tau}$~is +a scalar, if $E$~and~$F$ are vectors. This expression must +not be changed by the $\delta$-displacement; expressing this +by a formula, we get, using~\Eqref{(67)},~$\delta A_{\sigma\tau}$, whence we get the +desired co-variant derivative, +\[ +A_{\sigma\tau;\, \rho} + = \frac{\dd A_{\sigma\tau}}{\dd x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} + - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}\Add{.} +\Tag{(78)} +\] + +In order that the general law of co-variant differentiation +of tensors may be clearly seen, we shall write +down two co-variant derivatives deduced in an analogous +way: +\begin{align*} +A_{\sigma;\, \rho}^{\tau} + &= \frac{\dd A_{\sigma}^{\tau}}{\dd x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha}\Add{,} +\Tag{(79)} \\ +% +{A^{\sigma\tau}}_{;\, \rho} + &= \frac{\dd A^{\sigma\tau}}{\dd x_{\rho}} + + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}\Add{.} +\Tag{(80)} +\end{align*} +\PageSep{82} +The general law of formation now becomes evident. +From these formulæ we shall deduce some others which +are of interest for the physical applications of the theory. + +In case $A_{\sigma\tau}$~is skew-symmetrical, we obtain the tensor +\[ +A_{\sigma\tau\rho} + = \frac{\dd A_{\sigma\tau}}{\dd x_{\rho}} + + \frac{\dd A_{\tau\rho}}{\dd x_{\sigma}} + + \frac{\dd A_{\rho\sigma}}{\dd x_{\tau}}\Add{,} +\Tag{(81)} +\] +which is skew-symmetrical in all pairs of indices, by cyclic +interchange and addition. + +If, in~\Eqref{(78)}, we replace~$A_{\sigma\tau}$ by the fundamental tensor,~$g_{\sigma\tau}$, +then the right-hand side vanishes identically; an +analogous statement holds for~\Eqref{(80)} with respect to~$g^{\sigma\tau}$; +that is, the co-variant derivatives of the fundamental +tensor vanish. That this must be so we see directly in +the local system of co-ordinates. + +In case $A^{\sigma\tau}$~is skew-symmetrical, we obtain from~\Eqref{(80)}, +by contraction with respect to $\tau$~and~$\rho$, +\[ +\tA^{\sigma} = \frac{\dd \tA^{\sigma\tau}}{\dd x_{\tau}}\Add{.} +\Tag{(82)} +\] + +In the general case, from \Eqref{(79)}~and~\Eqref{(80)}, by contraction +with respect to $\tau$~and~$\rho$, we obtain the equations, +\begin{align*} +\tA_{\sigma} &= \frac{\dd \tA_{\sigma}^{\alpha}}{\dd x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \tA_{\alpha}^{\beta}\Add{,} +\Tag{(83)} \\ +% +\tA^{\sigma} &= \frac{\dd \tA^{\sigma\alpha}}{\dd x_{\alpha}} + + \Gamma_{\alpha\beta}^{\sigma} \tA^{\alpha\beta}\Add{.} +\Tag{(84)} +\end{align*} + +\Paragraph{The Riemann Tensor.} If we have given a curve extending +\index{Riemann!tensor}% +from the point~$P$ to the point~$G$ of the continuum, +then a vector~$A^{\mu}$, given at~$P$, may, by a parallel displacement, +be moved along the curve to~$G$. If the continuum +\PageSep{83} +is Euclidean (more generally, if by a suitable choice of +co-ordinates the~$g_{\mu\nu}$ are constants) then the vector obtained +at~$G$ as a result of this displacement does not depend +upon the choice of the curve joining $P$~and~$G$. But +otherwise, the result depends upon the path of the displacement. +In this case, therefore, a vector suffers a +change,~$\Delta A^{\mu}$ (in its direction, not its magnitude), when it +is carried from a point~$P$ of a closed curve, along the +%[Illustration: Fig. 4.] +\Figure[0.66\textwidth]{083} +curve, and back to~$P$. We shall now calculate this vector +change: +\[ +\Delta A^{\mu} = \Oint \delta A^{\mu}. +\] +As in Stokes' theorem for the line integral of a vector +around a closed curve, this problem may be reduced to +the integration around a closed curve with infinitely small +linear dimensions; we shall limit ourselves to this case. +\PageSep{84} + +We have, first, by~\Eqref{(67)}, +\[ +\Delta A^{\mu} = -\Oint \Gamma_{\alpha\beta}^{\mu} A^{\alpha}\, dx_{\beta}. +\] + +In this, $\Gamma_{\alpha\beta}^{\mu}$~is the value of this quantity at the variable +point~$G$ of the path of integration. If we put +\[ +\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P} +\] +and denote the value of~$\Gamma_{\alpha\beta}^{\mu}$ at~$P$ by~$\Bar{\Gamma_{\alpha\beta}^{\mu}}$, then we have, +with sufficient accuracy, +\[ +\Gamma_{\alpha\beta}^{\mu} + = \Bar{\Gamma_{\alpha\beta}^{\mu}} + + \frac{\Bar{\dd \Gamma_{\alpha\beta}^{\mu}}}{\dd x_{\nu}}\, \xi^{\nu}. +\] + +Let, further, $A^{\alpha}$~be the value obtained from~$\Bar{A^{\alpha}}$ by a +parallel displacement along the curve from~$P$ to~$G$. It +may now easily be proved by means of~\Eqref{(67)} that $A^{\mu} - \Bar{A^{\mu}}$ +is infinitely small of the first order, while, for a curve of +infinitely small dimensions of the first order, $\Delta A^{\mu}$~is +infinitely small of the second order. Therefore there is +an error of only the second order if we put +\[ +A^{\alpha} = \Bar{A^{\alpha}} + - \Bar{\Gamma_{\sigma\tau}^{\alpha}}\; + \Bar{A^{\Change{\alpha}{\sigma}}}\; + \Bar{\xi^{\Change{\beta}{\tau}}}. +\] + +If we introduce these values of $\Gamma_{\alpha\beta}^{\mu}$~and~$A^{\alpha}$ into the +integral, we obtain, neglecting all quantities of a higher +order of small quantities than the second, +\[ +\Delta A^{\mu} + = - \left(\frac{\dd \Gamma_{\sigma\beta}^{\mu}}{\dd x_{\alpha}} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) + A^{\sigma} \Oint \xi^{\alpha}\, d\xi^{\beta}\Add{.} +\Tag{(85)} +\] +The quantity removed from under the sign of integration +\PageSep{85} +refers to the point~$P$. Subtracting~$\frac{1}{2} d(\xi^{\alpha} \xi^{\beta})$ from the +integrand, we obtain +\[ +\tfrac{1}{2} \Oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}). +\] +This skew-symmetrical tensor of the second rank,~$\Change{f_{\alpha\beta}}{f^{\alpha\beta}}$, +characterizes the surface element bounded by the curve +in magnitude and position. If the expression in the +brackets in~\Eqref{(85)} were skew-symmetrical with respect to +the indices $\alpha$~and~$\beta$, we could conclude its tensor character +from~\Eqref{(85)}. We can accomplish this by interchanging +the summation indices $\alpha$~and~$\beta$ in~\Eqref{(85)} and adding the +resulting equation to~\Eqref{(85)}. We obtain +\[ +2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}\Add{,} +\Tag{(86)} +\] +in which +\[ +R_{\sigma\alpha\beta}^{\mu} + = - \frac{\dd \Gamma_{\sigma\alpha}^{\mu}}{\dd x_{\beta}} + + \frac{\dd \Gamma_{\sigma\beta}^{\mu}}{\dd x_{\alpha}} + + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\Add{.} +\Tag{(87)} +\] + +The tensor character of~$R_{\sigma\alpha\beta}^{\mu}$ follows from~\Eqref{(86)}; this is +the Riemann curvature tensor of the fourth rank, whose +\index{Riemann!tensor}% +properties of symmetry we do not need to go into. Its +vanishing is a sufficient condition (disregarding the reality +of the chosen co-ordinates) that the continuum is +Euclidean. + +By contraction of the Riemann tensor with respect to +the indices $\mu$,~$\beta$, we obtain the symmetrical tensor of the +second rank, +\[ +R_{\mu\nu} + = - \frac{\dd \Gamma_{\mu\nu}^{\alpha}}{\dd x_{\alpha}} + + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + + \frac{\dd \Gamma_{\mu\alpha}^{\alpha}}{\dd x_{\nu}} + - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}\Add{.} +\Tag{(88)} +\] +The last two terms vanish if the system of co-ordinates +\PageSep{86} +is so chosen that $g = \text{constant}$. From~$R_{\mu\nu}$ we can form +the scalar, +\[ +R = g^{\mu\nu} R_{\mu\nu}\Add{.} +\Tag{(89)} +\] + +\Paragraph{Straightest \(Geodetic\) Lines.} A line may be constructed +\index{Geodetic lines}% +\index{Straightest lines}% +in such a way that its successive elements arise from each +other by parallel displacements. This is the natural +generalization of the straight line of the Euclidean +geometry. For such a line, we have +\[ +\delta \left(\frac{dx_{\mu}}{ds}\right) + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}. +\] +The left-hand side is to be replaced by~$\dfrac{d^{2} x_{\mu}}{ds^{2}}$,\footnote + {The direction vector at a neighbouring point of the curve results, by a + parallel displacement along the line element~$(dx_{\beta})$, from the direction vector + of each point considered.} +so that we +have +\[ +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0\Add{.} +\Tag{(90)} +\] +We get the same line if we find the line which gives a +stationary value to the integral +\[ +\int ds\quad\text{or}\quad +\int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}} +\] +between two points (geodetic line). +\PageSep{87} + + +\Lecture[ (\textit{continued})]{IV}{The General Theory of Relativity} +{The General Theory} + +\First{We} are now in possession of the mathematical +apparatus which is necessary to formulate the +laws of the general theory of relativity. No attempt +will be made in this presentation at systematic completeness, +but single results and possibilities will be developed +progressively from what is known and from the +results obtained. Such a presentation is most suited +to the present provisional state of our knowledge. + +A material particle upon which no force acts moves, +according to the principle of inertia, uniformly in a +straight line. In the four-dimensional continuum of the +special theory of relativity (with real time co-ordinate) +this is a real straight line. The natural, that is, the +simplest, generalization of the straight line which is +plausible in the system of concepts of Riemann's general +theory of invariants is that of the straightest, or geodetic, +line. We shall accordingly have to assume, in the sense +of the principle of equivalence, that the motion of a +material particle, under the action only of inertia and +gravitation, is described by the equation, +\[ +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0\Add{.} +\tag*{(90)} %[** TN: Repeated from previous lecture] +\] +\PageSep{88} +In fact, this equation reduces to that of a straight line +if all the components,~$\Gamma_{\alpha\beta}^{\mu}$, of the gravitational field +vanish. + +How are these equations connected with Newton's +equations of motion? According to the special theory +of relativity, the~$g_{\mu\nu}$ as well as the~$g^{\mu\nu}$, have the values, +with respect to an inertial system (with real time co-ordinate +and suitable choice of the sign of~$ds^{2}$), +\[ +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & \Neg[]1 +\end{array} +\right\}\Add{.} +\Tag{(91)} +\] +The equations of motion then become +\[ +\frac{d^{2} x_{\mu}}{ds^{2}} = 0. +\] +We shall call this the ``first approximation'' to the $g_{\mu\nu}$-field. +In considering approximations it is often useful, +as in the special theory of relativity, to use an imaginary +$x_{4}$-co-ordinate, as then the~$g_{\mu\nu}$, to the first approximation, +assume the values +\[ +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & -1 +\end{array} +\right\}\Add{.} +\Tag{(91a)} +\] +These values may be collected in the relation +\[ +g_{\mu\nu} = -\delta_{\mu\nu}. +\] +To the second approximation we must then put +\[ +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}\Add{,} +\Tag{(92)} +\] +\PageSep{89} +where the~$\gamma_{\mu\nu}$ are to be regarded as small of the first +order. + +Both terms of our equation of motion are then small +of the first order. If we neglect terms which, relatively +to these, are small of the first order, we have to put +\begin{gather*} +ds^{2} = \Change{}{-}{dx_{\nu}}^{2} = dl^{2} (1 - q^{2})\Add{,} +\Tag{(93)} \\ +% +\Gamma_{\alpha\beta}^{\mu} + = -\delta_{\mu\sigma} \left[\Chr{\alpha}{\beta}{\sigma}\right] + = -\left[\Chr{\alpha}{\beta}{\mu}\right] + = \frac{1}{2} \left( + \frac{\dd \gamma_{\alpha\beta}}{\dd x_{\mu}} + - \frac{\dd \gamma_{\alpha\mu}}{\dd x_{\beta}} + - \frac{\dd \gamma_{\beta\mu}}{\dd x_{\alpha}}\right)\Add{.} +\Tag{(94)} +\end{gather*} +We shall now introduce an approximation of a second +kind. Let the velocity of the material particles be very +small compared to that of light. Then $ds$~will be the +same as the time differential,~$dl$. Further, $\dfrac{dx_{1}}{ds}$, $\dfrac{dx_{2}}{ds}$, $\dfrac{dx_{3}}{ds}$ +will vanish compared to~$\dfrac{dx_{4}}{ds}$. We shall assume, in addition, +that the gravitational field varies so little with the +time that the derivatives of the~$\gamma_{\mu\nu}$ by~$x_{4}$ may be +neglected. Then the equation of motion (for $\mu = 1, 2, 3$) +reduces to +\[ +\frac{d^{2} x_{\mu}}{dl^{2}} + = \frac{\dd}{\dd x_{\mu}} \left(\frac{\gamma_{44}}{2}\right)\Add{.} +\Tag{(90a)} +\] +This equation is identical with Newton's equation of +motion for a material particle in a gravitational field, if +we identify~$\left(\dfrac{\gamma_{44}}{2}\right)$ with the potential of the gravitational +field; whether or not this is allowable, naturally depends +upon the field equations of gravitation, that is, it depends +upon whether or not this quantity satisfies, to a +first approximation, the same laws of the field as the +\PageSep{90} +gravitational potential in Newton's theory. A glance +at \Eqref{(90)}~and~\Eqref{(90a)} shows that the~$\Gamma_{\alpha\beta}^{\mu}$ actually do play +the rôle of the intensity of the gravitational field. +These quantities do not have a tensor character. + +Equations~\Eqref{(90)} express the influence of inertia and +gravitation upon the material particle. The unity of +inertia and gravitation is formally expressed by the fact +that the whole left-hand side of~\Eqref{(90)} has the character +of a tensor (with respect to any transformation of co-ordinates), +but the two terms taken separately do not +have tensor character, so that, in analogy with Newton's +equations, the first term would be regarded as the expression +for inertia, and the second as the expression +for the gravitational force. + +We must next attempt to find the laws of the gravitational +field. For this purpose, Poisson's equation, +\index{Poisson's equation}% +\[ +\Delta\phi = 4\pi K\rho +\] +of the Newtonian theory must serve as a model. This +equation has its foundation in the idea that the gravitational +field arises from the density~$\rho$ of ponderable +matter. It must also be so in the general theory of +relativity. But our investigations of the special theory +of relativity have shown that in place of the scalar +density of matter we have the tensor of energy per unit +volume. In the latter is included not only the tensor +of the energy of ponderable matter, but also that of the +electromagnetic energy. We have seen, indeed, that +in a more complete analysis the energy tensor can be +regarded only as a provisional means of representing +\PageSep{91} +matter. In reality, matter consists of electrically charged +particles, and is to be regarded itself as a part, in fact, +the principal part, of the electromagnetic field. It is +only the circumstance that we have not sufficient knowledge +of the electromagnetic field of concentrated charges +that compels us, provisionally, to leave undetermined +in presenting the theory, the true form of this tensor. +From this point of view our problem now is to introduce +a tensor,~$T_{\mu\nu}$, of the second rank, whose structure we do +not know provisionally, and which includes in itself the +energy density of the electromagnetic field and of ponderable +matter; we shall denote this in the following as +the ``energy tensor of matter.'' + +According to our previous results, the principles of +momentum and energy are expressed by the statement +that the divergence of this tensor vanishes~\Eqref{(47c)}. In +the general theory of relativity, we shall have to assume +as valid the corresponding general co-variant equation. +If $(T_{\mu\nu})$~denotes the co-variant energy tensor of matter, +$\tT_{\sigma}^{\nu}$~the corresponding mixed tensor density, then, in +accordance with~\Eqref{(83)}, we must require that +\[ +0 = \frac{\dd \tT_{\sigma}^{\alpha}}{\dd x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \tT_{\alpha}^{\beta} +\Tag{(95)} +\] +be satisfied. It must be remembered that besides the +energy density of the matter there must also be given +an energy density of the gravitational field, so that there +can be no talk of principles of conservation of energy +and momentum for matter alone. This is expressed +mathematically by the presence of the second term in~\Eqref{(95)}, +\PageSep{92} +which makes it impossible to conclude the existence +of an integral equation of the form of~\Eqref{(49)}. The gravitational +field transfers energy and momentum to the +``matter,'' in that it exerts forces upon it and gives it +energy; this is expressed by the second term in~\Eqref{(95)}. + +If there is an analogue of Poisson's equation in the +general theory of relativity, then this equation must be +a tensor equation for the tensor~$g_{\mu\nu}$ of the gravitational +potential; the energy tensor of matter must appear on +the right-hand side of this equation. On the left-hand +side of the equation there must be a differential tensor +in the~$g_{\mu\nu}$. We have to find this differential tensor. +It is completely determined by the following three +conditions:--- + +1.~It may contain no differential coefficients of the~$g_{\mu\nu}$ +higher than the second. + +2.~It must be linear and homogeneous in these second +differential coefficients. + +3.~Its divergence must vanish identically. + +The first two of these conditions are naturally taken +from Poisson's equation. Since it may be proved +mathematically that all such differential tensors can be +formed algebraically (i.e.~without differentiation) from +Riemann's tensor, our tensor must be of the form +\[ +R_{\mu\nu} + ag_{\mu\nu} R\Add{,} +\] +in which $R_{\mu\nu}$~and~$R$ are defined by \Eqref{(88)}~and~\Eqref{(89)} respectively. +Further, it may be proved that the third condition +requires~$a$ to have the value~$-\frac{1}{2}$. For the law +\PageSep{93} +of the gravitational field we therefore get the equation +\[ +R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}\Add{.} +\Tag{(96)} +\] +Equation~\Eqref{(95)} is a consequence of this equation. $\kappa$~denotes +a constant, which is connected with the Newtonian +gravitation constant. + +In the following I shall indicate the features of the +theory which are interesting from the point of view of +physics, using as little as possible of the rather involved +mathematical method. It must first be shown that the +divergence of the left-hand side actually vanishes. The +energy principle for matter may be expressed, by~\Eqref{(83)}, +\[ +0 = \frac{\dd \tT_{\sigma}^{\alpha}}{\dd x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \tT_{\alpha}^{\beta}\Add{,} +\Tag{(97)} +\] +in which +\[ +\tT_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}. +\] +The analogous operation, applied to the left-hand side +of~\Eqref{(96)}, will lead to an identity. + +In the region surrounding each world-point there are +systems of co-ordinates for which, choosing the $x_{\Change{\mu}{4}}$-co-ordinate +imaginary, at the given point, +\[ +g_{\mu\nu} = g^{\mu\nu} + = -\delta_{\mu\nu} +%[** TN: Moved "="s outside brace] + = \begin{cases} + -1 & \text{if $\mu = \nu$\Add{,}} \\ + \Neg[]0 & \text{if $\mu \neq \nu$,} + \end{cases} +\] +and for which the first derivatives of the~$g_{\mu\nu}$ and the~$g^{\mu\nu}$ +vanish. We shall verify the vanishing of the divergence +of the left-hand side at this point. At this point +the components~$\Gamma_{\sigma\beta}^{\alpha}$ vanish, so that we have to prove +the vanishing only of +\[ +\frac{\dd}{\dd x_{\sigma}} \left[ + \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) +\right]. +\] +\PageSep{94} +Introducing \Eqref{(88)}~and~\Eqref{(70)} into this expression, we see +that the only terms that remain are those in which third +derivatives of the~$g^{\mu\nu}$ enter. Since the~$g_{\mu\nu}$ are to be +replaced by~$-\delta_{\mu\nu}$, we obtain, finally, only a few terms +which may easily be seen to cancel each other. Since +the quantity that we have formed has a tensor character, +its vanishing is proved for every other system of co-ordinates +also, and naturally for every other four-dimensional +point. The energy principle of matter~\Eqref{(97)} is thus a +mathematical consequence of the field equations~\Eqref{(96)}. + +In order to learn whether the equations~\Eqref{(96)} are +consistent with experience, we must, above all else, find +out whether they lead to the Newtonian theory as a +first approximation. For this purpose we must introduce +various approximations into these equations. We +already know that Euclidean geometry and the law of the +constancy of the velocity of light are valid, to a certain +approximation, in regions of a great extent, as in the +planetary system. If, as in the special theory of relativity, +we take the fourth co-ordinate imaginary, this +means that we must put +\[ +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}\Add{,} +\Tag{(98)} +\] +in which the~$\gamma_{\mu\nu}$ are so small compared to~$1$ that we +can neglect the higher powers of the~$\gamma_{\mu\nu}$ and their +derivatives. If we do this, we learn nothing about the +structure of the gravitational field, or of metrical space of +cosmical dimensions, but we do learn about the influence +of neighbouring masses upon physical phenomena. + +Before carrying through this approximation we shall +\PageSep{95} +transform~\Eqref{(96)}. We multiply~\Eqref{(96)} by~$g^{\mu\nu}$, summed over +the $\mu$~and~$\nu$; observing the relation which follows from +the definition of the~$g^{\mu\nu}$, +\[ +g_{\mu\nu} g^{\mu\nu} = 4\Add{,} +\] +we obtain the equation +\[ +R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T. +\] +If we put this value of~$R$ in~\Eqref{(96)} we obtain +\[ +R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) + = -\kappa T_{\mu\nu}^{*}\Add{.} +\Tag{(96a)} +\] +When the approximation which has been mentioned is +carried out, we obtain for the left-hand side, +\[ +-\tfrac{1}{2}\left( + \frac{\dd^{2} \gamma_{\mu\nu}}{{\dd x_{\alpha}}^{2}} + + \frac{\dd^{2} \gamma_{\alpha\alpha}}{\dd x_{\mu}\, \dd x_{\nu}} + - \frac{\dd^{2} \gamma_{\mu\alpha}}{\dd x_{\nu}\, \dd x_{\alpha}} + - \frac{\dd^{2} \gamma_{\nu\alpha}}{\dd x_{\mu}\, \dd x_{\alpha}} +\right) +\] +or +\[ +-\tfrac{1}{2}\frac{\dd^{2} \gamma_{\mu\nu}}{{\dd x_{\alpha}}^{2}} + + \tfrac{1}{2} \frac{\dd}{\dd x_{\nu}}\left( + \frac{{\dd \gamma'}_{\mu\alpha}}{\dd x_{\alpha}} + \right) + + \tfrac{1}{2} \frac{\dd}{\dd x_{\mu}}\left( + \frac{{\dd \gamma'}_{\nu\alpha}}{\dd x_{\alpha}} + \right)\Add{,} +\] +in which has been put +\[ +{\gamma'}_{\mu\nu} + = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}\Add{.} +\Tag{(99)} +\] + +We must now note that equation~\Eqref{(96)} is valid for any +system of co-ordinates. We have already specialized the +system of co-ordinates in that we have chosen it so that +within the region considered the $g_{\mu\nu}$~differ infinitely little +from the constant values~$-\delta_{\mu\nu}$. But this condition +remains satisfied in any infinitesimal change of co-ordinates, +so that there are still four conditions to which +the~$\gamma_{\mu\nu}$ may be subjected, provided these conditions do +not conflict with the conditions for the order of magnitude +\PageSep{96} +of the~$\gamma_{\mu\nu}$. We shall now assume that the system of co-ordinates +is so chosen that the four relations--- +\[ +0 = \frac{{\dd\gamma'}_{\mu\nu}}{\dd x_{\nu}} + = \frac{\dd\gamma_{\mu\nu}}{\dd x_{\nu}} + - \tfrac{1}{2} \frac{\dd\gamma_{\sigma\sigma}}{\dd x_{\mu}} +\Tag{(100)} +\] +are satisfied. Then \Eqref{(96a)}~takes the form +\[ +\frac{\dd^{2} \gamma_{\mu\nu}}{{\dd x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}\Add{.} +\Tag{(96b)} +\] +These equations may be solved by the method, familiar +in electrodynamics, of retarded potentials; we get, in an +easily understood notation, +\[ +\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int + \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}\Add{.} +\Tag{(101)} +\] + +In order to see in what sense this theory contains the +Newtonian theory, we must consider in greater detail +the energy tensor of matter. Considered phenomenologically, +this energy tensor is composed of that of the +electromagnetic field and of matter in the narrower sense. +If we consider the different parts of this energy tensor +with respect to their order of magnitude, it follows +from the results of the special theory of relativity that +the contribution of the electromagnetic field practically +vanishes in comparison to that of ponderable matter. In +our system of units, the energy of one gram of matter is +equal to~$1$, compared to which the energy of the electric +fields may be ignored, and also the energy of deformation +of matter, and even the chemical energy. We get an +approximation that is fully sufficient for our purpose if +\PageSep{97} +we put +\[ +\left. +\begin{aligned} +T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}\Add{,} \\ +ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{.} +\end{aligned} +\right\} +\Tag{(102)} +\] +In this, $\sigma$~is the density at rest, that is, the density of the +ponderable matter, in the ordinary sense, measured with +the aid of a unit measuring rod, and referred to a Galilean +system of co-ordinates moving with the matter. + +We observe, further, that in the co-ordinates we have +chosen, we shall make only a relatively small error if we +replace the~$g_{\mu\nu}$ by~$-\delta_{\mu\nu}$, so that we put +\[ +ds^{2} = -\sum {dx_{\mu}}^{2}\Add{.} +\Tag{(102a)} +\] + +The previous developments are valid however rapidly +the masses which generate the field may move relatively +to our chosen system of quasi-Galilean co-ordinates. But +in astronomy we have to do with masses whose velocities, +relatively to the co-ordinate system employed, are always +small compared to the velocity of light, that is, small +compared to~$1$, with our choice of the unit of time. +We therefore get an approximation which is sufficient +for nearly all practical purposes if in~\Eqref{(101)} we replace +the retarded potential by the ordinary (non-retarded) +potential, and if, for the masses which generate the field, +we put +\[ +\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad +\frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}\Add{.} +\Tag{(\Change{103a}{103})} +\] +\PageSep{98} +Then we get for $T^{\mu\nu}$~and~$T_{\mu\nu}$ the values +\[ +\left. +\begin{array}{*{3}{>{\qquad}r}>{\quad}r} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & -\sigma +\end{array} +\right\}\Add{.} +\Tag{(104)} +\] +For $T$ we get the value~$\sigma$, and, finally, for~$T_{\mu\nu}^{*}$ the +values, +\[ +\left. +\begin{array}{*{3}{>{\qquad}c}>{\quad}r} +\dfrac{\sigma}{2} & 0 & 0 & 0 \\ +0 & \dfrac{\sigma}{2} & 0 & 0 \\ +0 & 0 & \dfrac{\sigma}{2} & 0 \\ +0 & 0 & 0 & -\dfrac{\sigma}{2} +\end{array} +\right\}\Add{.} +\Tag{(104a)} +\] + +We thus get, from~\Eqref{(101)}, +\[ +\left. +\begin{aligned} +\gamma_{11} = \gamma_{22} = \gamma_{33} + &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} \\ +\gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} +\end{aligned} +\right\} +\Tag{(101a)} +\] +while all the other~$\gamma_{\mu\nu}$ vanish. The least of these equations, +in connexion with equation~\Eqref{(90a)}, contains Newton's +theory of gravitation. If we replace~$l$ by~$ct$ we +get +\[ +\frac{d^{2} x_{\mu}}{dt^{2}} + = \frac{\kappa c^{2}}{8\pi}\, \frac{\dd}{\dd x_{\mu}} \left\{ + \int \frac{\sigma\, dV_{0}}{r} + \right\}\Add{.} +\Tag{(90b)} +\] +We see that the Newtonian gravitation constant~$K$, is +\index{Gravitation constant}% +\index{Newtonian gravitation constant}% +connected with the constant~$\kappa$ that enters into our field +equations by the relation +\[ +K = \frac{\kappa c^{2}}{8\pi}\Add{.} +\Tag{(105)} +\] +\PageSep{99} +From the known numerical value of~$K$, it therefore +follows that +\[ +\kappa = \frac{8\pi K}{c^{2}} + = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} + = 1.86·10^{-27}\Add{.} +\Tag{(105a)} +\] +From \Eqref{(101)} we see that even in the first approximation +the structure of the gravitational field differs fundamentally +from that which is consistent with the Newtonian theory; +this difference lies in the fact that the gravitational +potential has the character of a tensor and not a scalar. +This was not recognized in the past because only the +component~$g_{44}$, to a first approximation, enters the equations +of motion of material particles. + +In order now to be able to judge the behaviour of +measuring rods and clocks from our results, we must +observe the following. According to the principle of +equivalence, the metrical relations of the Euclidean +geometry are valid relatively to a Cartesian system of +reference of infinitely small dimensions, and in a suitable +state of motion (freely falling, and without rotation). +We can make the same statement for local systems of +co-ordinates which, relatively to these, have small accelerations, +and therefore for such systems of co-ordinates +as are at rest relatively to the one we have selected. For +such a local system, we have, for two neighbouring point +events, +\[ +ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} + = - dS^{2} + dT^{2}\Add{,} +\] +where $dS$~is measured directly by a measuring rod and +$dT$~by a clock at rest relatively to the system; these are +\PageSep{100} +the naturally measured lengths and times. Since~$ds^{2}$, on +the other hand, is known in terms of the co-ordinates~$x_{\nu}$ +employed in finite regions, in the form +\[ +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{,} +\] +we have the possibility of getting the relation between +naturally measured lengths and times, on the one hand, +and the corresponding differences of co-ordinates, on the +other hand. As the division into space and time is in +agreement with respect to the two systems of co-ordinates, +so when we equate the two expressions for~$ds^{2}$ we get +two relations. If, by~\Eqref{(101a)}, we put +\begin{multline*} +ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ + + \left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}\Add{,} +\end{multline*} +we obtain, to a sufficiently close approximation, +\[ +\left. +\begin{aligned} + &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ + &\qquad + \begin{aligned} + &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}\Add{,} \\ +dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. +\end{aligned} +\end{aligned} +\right\} +\Tag{(106)} +\] + +The unit measuring rod has therefore the length, +\[ +1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +\] +in respect to the system of co-ordinates we have selected. +The particular system of co-ordinates we have selected +\PageSep{101} +insures that this length shall depend only upon +the place, and not upon the direction. If we had +chosen a different system of co-ordinates this would not +be so. But however we may choose a system of co-ordinates, +the laws of configuration of rigid rods do not +agree with those of Euclidean geometry; in other words, +we cannot choose any system of co-ordinates so that the +co-ordinate differences, $\Delta x_{1}$,~$\Delta x_{2}$,~$\Delta x_{3}$, corresponding to the +ends of a unit measuring rod, oriented in any way, shall +always satisfy the relation ${\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} = 1$. In +this sense space is not Euclidean, but ``curved.'' It +follows from the second of the relations above that the +interval between two beats of the unit clock ($dT = 1$) +corresponds to the ``time'' +\[ +1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +\] +in the unit used in our system of co-ordinates. The rate +of a clock is accordingly slower the greater is the mass of +the ponderable matter in its neighbourhood. We therefore +\index{Displacement of spectral lines}% +\index{Spectral lines, displacement of}% +conclude that spectral lines which are produced on +the sun's surface will be displaced towards the red, +compared to the corresponding lines produced on the +earth, by about $2·10^{-6}$~of their wave-lengths. At first, +this important consequence of the theory appeared to +conflict with experiment; but results obtained during the +past year seem to make the existence of this effect more +probable, and it can hardly be doubted that this consequence +of the theory will be confirmed within the next +year. +\PageSep{102} + +Another important consequence of the theory, which +\index{Light ray, path of}% +\index{Path of light ray}% +\index{Ray of light, path of}% +can be tested experimentally, has to do with the path of +rays of light. In the general theory of relativity also +the velocity of light is everywhere the same, relatively to +a local inertial system. This velocity is unity in our +natural measure of time. The law of the propagation of +light in general co-ordinates is therefore, according to the +general theory of relativity, characterized, by the equation +\[ +ds^{2} = 0. +\] +To within the approximation which we are using, and in +the system of co-ordinates which we have selected, the +velocity of light is characterized, according to~\Eqref{(106)}, by +the equation +\[ +\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) + = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}. +\] +The velocity of light~$L$, is therefore expressed in our +co-ordinates by +\[ +\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} + = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{.} +\Tag{(107)} +\] +We can therefore draw the conclusion from this, that a +ray of light passing near a large mass is deflected. If +we imagine the sun, of mass~$M$ concentrated at the +origin of our system of co-ordinates, then a ray of light, +travelling parallel to the $x_{3}$-axis, in the $x_{1}$-$x_{3}$~plane, +at a distance~$\Delta$ from the origin, will be deflected, in all, +by an amount +\PageSep{103} +\[ +\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\dd L}{\dd x_{1}}\, dx_{3} +\] +towards the sun. On performing the integration we get +\[ +\alpha = \frac{\kappa M}{2\pi\Delta}\Add{.} +\Tag{(108)} +\] + +The existence of this deflection, which amounts to~$1.7''$ +for $\Delta$~equal to the radius of the sun, was confirmed, +with remarkable accuracy, by the English Solar Eclipse +\index{Solar Eclipse expedition (1919)}% +Expedition in~1919, and most careful preparations have +been made to get more exact observational data at the +solar eclipse in~1922. It should be noted that this +result, also, of the theory is not influenced by our +arbitrary choice of a system of co-ordinates. + +This is the place to speak of the third consequence of +the theory which can be tested by observation, namely, +that which concerns the motion of the perihelion +of the planet Mercury. The secular changes in the +\index{Mercury, perihelion of}% +\index{Perihelion of Mercury}% +planetary orbits are known with such accuracy that the +approximation we have been using is no longer sufficient +for a comparison of theory and observation. It is necessary +to go back to the general field equations~\Eqref{(96)}. To +solve this problem I made use of the method of successive +approximations. Since then, however, the problem +of the central symmetrical statical gravitational field has +been completely solved by Schwarzschild and others; +the derivation given by H.~Weyl in his book, ``Raum-Zeit-Materie,'' +\index{Weyl}% +is particularly elegant. The calculation +can be simplified somewhat if we do not go back directly +\PageSep{104} +to the equation~\Eqref{(96)}, but base it upon a principle of +variation that is equivalent to this equation. I shall +indicate the procedure only in so far as is necessary for +understanding the method. + +In the case of a statical field, $ds^{2}$~must have the form +\[ +\left. +\begin{aligned} +ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}\Add{,} \\ +d\sigma^{2} + &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}\Add{,} +\end{aligned} +\right\} +\Tag{(109)} +\] +where the summation on the right-hand side of the last +equation is to be extended over the space variables only\Change{,}{.} +The central symmetry of the field requires the~$\gamma_{\mu\nu}$ to be +of the form, +\[ +\gamma_{\alpha\beta} + = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}\Add{;} +\Tag{(110)} +\] +$f^{2}$,~$\mu$ and~$\lambda$ are functions of $r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}$ only. +One of these three functions can be chosen arbitrarily, +because our system of co-ordinates is, \textit{a~priori}, completely +arbitrary; for by a substitution +\begin{align*} +{x'}_{4} &= x_{4}\Add{,} \\ +{x'}_{\alpha} &= F(r) x_{\alpha}\Add{,} +\end{align*} +we can always insure that one of these three functions +shall be an assigned function of~$r'$. In place of~\Eqref{(110)} we +can therefore put, without limiting the generality, +\[ +\gamma_{\alpha\beta} + = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}\Add{.} +\Tag{(110a)} +\] + +In this way the~$g_{\mu\nu}$ are expressed in terms of the two +quantities $\lambda$~and~$f$. These are to be determined as functions +of~$r$, by introducing them into equation~\Eqref{(96)}, after +\PageSep{105} +first calculating the~$\Change{{\Gamma_{\sigma}}^{\mu\nu}}{\Gamma_{\mu\nu}^{\sigma}}$ from \Eqref{(\Change{107}{109})}~and~\Eqref{(\Change{108a}{110a})}. We +have +\[ +\left. +\begin{aligned} +\Gamma_{\alpha\beta}^{\sigma} + &= \tfrac{1}{2} \frac{x_{\sigma}}{r} + · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} + {1 + \lambda r^{2}} + \text{ (for $\alpha, \beta, \sigma = 1, 2, 3$)}\Add{,} \\ +% +\Gamma_{44}^{4} + &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad + \text{(for $\alpha, \beta = 1, 2, 3$)}\Add{,} \\ +% +\Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\dd f^{2}}{\dd x_{\alpha}},\quad +\Gamma_{44}^{\alpha} + = -\tfrac{1}{2} \Change{f^{-2}}{g^{\alpha\beta}}\, + \frac{\dd f^{2}}{\dd \Change{x_{\alpha}}{x_{\beta}}}\Add{.} +\end{aligned} +\right\} +\Tag{(\Change{108b}{110b})} +\] + +With the help of these results, the field equations +furnish Schwarzschild's solution: +\[ +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ + \frac{dr^{2}} + {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) +\right]\Add{,} +\Tag{(\Change{109}{109a})} +\] +in which we have put +\[ +\left. +\begin{aligned} +x_{4} &= l\Add{,} \\ +x_{1} &= r \sin\theta \sin\phi\Add{,} \\ +x_{2} &= r \sin\theta \cos\phi\Add{,} \\ +x_{3} &= r \cos\theta \Add{,} \\ +A &= \frac{\kappa M}{4\pi}\Add{.} +\end{aligned} +\right\} +\Tag{(\Change{109a}{109b})} +\] + +$M$~denotes the sun's mass, centrally symmetrically +placed about the origin of co-ordinates; the solution~\Eqref{(109)} +is valid only outside of this mass, where all the~$T_{\mu\nu}$ vanish. +If the motion of the planet takes place in the $x_{1}$-$x_{2}$~plane +then we must replace~\Eqref{(\Change{109}{109a})} by +\[ +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} + - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}\Add{.} +\Tag{(\Change{109b}{109c})} +\] +\PageSep{106} + +The calculation of the planetary motion depends upon +equation~\Eqref{(90)}. From the first of equations \Eqref{(\Change{108b}{110b})}~and~\Eqref{(90)} +we get, for the indices $1, 2, 3$, +\[ +\frac{d}{ds} + \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0\Add{,} +\] +or, if we integrate, and express the result in polar co-ordinates, +\[ +r^{2} \frac{d\phi}{ds} = \text{constant}. +\Tag{(111)} +\] + +From~\Eqref{(90)}, for $\mu = 4$, we get +\[ +0 = \frac{d^{2} l}{ds^{2}} + + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} + \Change{}{\, \frac{dl}{ds}} + = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} + \Change{}{\, \frac{dl}{ds}}. +\] +From this, after multiplication by~$f^{2}$ and integration, we +have +\[ +f^{2} \frac{dl}{ds} = \text{constant}. +\Tag{(112)} +\] + +In \Eqref{(\Change{109b}{109c})},~\Eqref{(111)} and~\Eqref{(112)} we have three equations +between the four variables $s$,~$r$,~$l$ and~$\phi$, from which the +motion of the planet may be calculated in the same way +as in classical mechanics. The most important result we +get from this is a secular rotation of the elliptic orbit of +the planet in the same sense as the revolution of the +planet, amounting in radians per revolution to +\[ +\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}\Add{,} +\Tag{(113)} +\] +\PageSep{107} +where +\begin{align*} +a &= \text{the semi-major axis of the planetary orbit in +centimetres.} \\ +e &= \text{the numerical eccentricity.} \\ +c &= 3·10^{+10}, \text{ the velocity of light \textit{in vacuo}.} \\ +T &= \text{the period of revolution in seconds.} +\end{align*} +This expression furnishes the explanation of the motion +of the perihelion of the planet Mercury, which has been +\index{Mercury, perihelion of}% +\index{Perihelion of Mercury}% +known for a hundred years (since Leverrier), and for +which theoretical astronomy has hitherto been unable +satisfactorily to account. + +There is no difficulty in expressing Maxwell's theory +of the electromagnetic field in terms of the general theory +of relativity; this is done by application of the tensor +formation \Eqref{(81)},~\Eqref{(82)} and~\Eqref{(77)}. Let $\phi_{\mu}$~be a tensor of the +first rank, to be denoted as an electromagnetic $4$-potential; +then an electromagnetic field tensor may be defined by +the relations, +\[ +\phi_{\mu\nu} + = \frac{\dd \phi_{\mu}}{\dd x_{\nu}} - \frac{\dd \phi_{\nu}}{\dd x_{\mu}}\Add{.} +\Tag{(114)} +\] +The second of Maxwell's systems of equations is then +defined by the tensor equation, resulting from this, +\[ +\frac{\dd \phi_{\mu\nu}}{\dd x_{\rho}} + +\frac{\dd \phi_{\nu\rho}}{\dd x_{\mu}} + +\frac{\dd \phi_{\rho\mu}}{\dd x_{\nu}} = 0\Add{,} +\Tag{(114a)} +\] +and the first of Maxwell's systems of equations is defined +by the tensor-density relation +\[ +\frac{\dd \tF^{\mu\nu}}{\dd x_{\nu}} = \tJ^{\mu}\Add{,} +\Tag{(115)} +\] +\PageSep{108} +in which +\begin{align*} +\tF^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}\Add{,} \\ +\tJ^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. +\end{align*} +If we introduce the energy tensor of the electromagnetic +field into the right-hand side of~\Eqref{(96)}, we obtain~\Eqref{(115)}, +for the special case $\tJ^{\mu} = 0$, as a consequence of~\Eqref{(96)} by +taking the divergence. This inclusion of the theory of +electricity in the scheme of the general theory of relativity +has been considered arbitrary and unsatisfactory by +many theoreticians. Nor can we in this way conceive of +the equilibrium of the electricity which constitutes the +elementary electrically charged particles. A theory in +which the gravitational field and the electromagnetic field +enter as an essential entity would be much preferable. +H.~Weyl, and recently Th.~Kaluza, have discovered some +\index{Kaluza}% +\index{Weyl}% +ingenious theorems along this direction; but concerning +them, I am convinced that they do not bring us nearer to +the true solution of the fundamental problem. I shall +not go into this further, but shall give a brief discussion +of the so-called cosmological problem, for without this, +the considerations regarding the general theory of relativity +would, in a certain sense, remain unsatisfactory. + +Our previous considerations, based upon the field +equations~\Eqref{(96)}, had for a foundation the conception that +space on the whole is Galilean-Euclidean, and that this +character is disturbed only by masses embedded in it. +This conception was certainly justified as long as we were +dealing with spaces of the order of magnitude of those +\PageSep{109} +that astronomy has to do with. But whether portions of +the universe, however large they may be, are quasi-Euclidean, +is a wholly different question. We can make +this clear by using an example from the theory of surfaces +which we have employed many times. If a portion of a +surface is observed by the eye to be practically plane, it +does not at all follow that the whole surface has the form +of a plane; the surface might just as well be a sphere, for +example, of sufficiently large radius. The question as to +whether the universe as a whole is non-Euclidean was +much discussed from the geometrical point of view before +the development of the theory of relativity. But with the +theory of relativity, this problem has entered upon a +new stage, for according to this theory the geometrical +properties of bodies are not independent, but depend +upon the distribution of masses. + +If the universe were quasi-Euclidean, then Mach was +\index{Mach}% +wholly wrong in his thought that inertia, as well as +gravitation, depends upon a kind of mutual action between +bodies. For in this case, with a suitably selected system +of co-ordinates, the~$g_{\mu\nu}$ would be constant at infinity, as +they are in the special theory of relativity, while within +finite regions the~$g_{\mu\nu}$ would differ from these constant +values by small amounts only, with a suitable choice of +co-ordinates, as a result of the influence of the masses in +finite regions. The physical properties of space would +not then be wholly independent, that is, uninfluenced by +matter, but in the main they would be, and only in +small measure, conditioned by matter. Such a dualistic +conception is even in itself not satisfactory; there are, +\PageSep{110} +however, some important physical arguments against it, +which we shall consider. + +The hypothesis that the universe is infinite and +\index{Finiteness of universe}% +\index{Universe@{Universe, Finiteness of}}% +Euclidean at infinity, is, from the relativistic point of +view, a complicated hypothesis. In the language of the +general theory of relativity it demands that the Riemann +\index{Riemann!tensor}% +tensor of the fourth rank~$R_{iklm}$ shall vanish at infinity, +which furnishes twenty independent conditions, while only +ten curvature components~$R_{\mu\nu}$, enter into the laws of the +gravitational field. It is certainly unsatisfactory to +postulate such a far-reaching limitation without any +physical basis for it. + +But in the second place, the theory of relativity makes +it appear probable that Mach was on the right road in +\index{Mach}% +his thought that inertia depends upon a mutual action of +matter. For we shall show in the following that, according +to our equations, inert masses do act upon each other +in the sense of the relativity of inertia, even if only very +feebly. What is to be expected along the line of Mach's +thought? +\begin{itemize} +\item[1.] The inertia of a body must increase when ponderable + masses are piled up in its neighbourhood. + +\item[2.] A body must experience an accelerating force when + neighbouring masses are accelerated, and, in fact, + the force must be in the same direction as the + acceleration. + +\item[3.] A rotating hollow body must generate inside of + itself a ``Coriolis field,'' which deflects moving + bodies in the sense of the rotation, and a radial + centrifugal field as well. +\end{itemize} +\PageSep{111} + +We shall now show that these three effects, which are +to be expected in accordance with Mach's ideas, are +\index{Mach}% +actually present according to our theory, although their +magnitude is so small that confirmation of them by +laboratory experiments is not to be thought of. For this +purpose we shall go back to the equations of motion of +a material particle~\Eqref{(90)}, and carry the approximations +somewhat further than was done in equation~\Eqref{(90a)}. + +First, we consider~$\gamma_{44}$ as small of the first order. The +square of the velocity of masses moving under the influence +of the gravitational force is of the same order, according +to the energy equation. It is therefore logical to regard +the velocities of the material particles we are considering, +as well as the velocities of the masses which generate the +field, as small, of the order~$\frac{1}{2}$. We shall now carry out the +approximation in the equations that arise from the field +equations~\Eqref{(101)} and the equations of motion~\Eqref{(90)} so far +as to consider terms, in the second member of~\Eqref{(90)}, that +are linear in those velocities. Further, we shall not put +$ds$~and~$dl$ equal to each other, but, corresponding to the +higher approximation, we shall put +\[ +ds = \sqrt{g_{44}}\, dl + = \left(1 - \frac{\gamma_{44}}{2}\right) dl. +\] +From~\Eqref{(90)} we obtain, at first, +\[ +\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} + \left(1 + \frac{\gamma_{44}}{2}\right). +\Tag{(116)} +\] + +From~\Eqref{(101)} we get, to the approximation sought for, +\PageSep{112} +\[ +\left. +\begin{aligned} +-\gamma_{11} = -\gamma_{22} = -\gamma_{33} + &= \gamma_{44} = \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} \\ +% +\gamma_{4\alpha} + &= -\frac{\Change{\veci}{i} \kappa}{2} + \int \frac{\sigma \dfrac{dx_{\alpha}}{ds}\, dV_{0}}{r}\Add{,} \\ +% +\gamma_{\alpha\beta} &= 0\Add{,} +\end{aligned} +\right\} +\Tag{(117)} +\] +in which, in~\Eqref{(117)}, $\alpha$~and~$\beta$ denote the space indices only. + +On the right-hand side of~\Eqref{(116)} we can replace +$1 + \dfrac{\gamma_{44}}{2}$ by~$1$ and $-\Change{\Gamma_{\mu}^{\alpha\beta}}{\Gamma_{\alpha\beta}^{\mu}}$ by~$\left[\Chr{\alpha}{\beta}{\mu}\right]$. It is easy to see, in +addition, that to this degree of approximation we must +put +\begin{align*} +\left[\Chr{4}{4}{\mu}\right] + &= -\tfrac{1}{2} \frac{\dd \gamma_{44}}{\dd x_{\mu}} + + \frac{\dd \gamma_{4\mu}}{\dd x_{4}}\Add{,} \\ +% +\left[\Chr{\alpha}{4}{\mu}\right] + &= \tfrac{1}{2} \left( + \frac{\dd \gamma_{4\mu}}{\dd x_{\alpha}} - \frac{\dd \gamma_{4\alpha}}{\dd x_{\mu}} + \right)\Add{,} \\ +\left[\Chr{\alpha}{\beta}{\mu}\right] &= 0\Add{,} +\end{align*} +in which $\alpha$,~$\beta$ and~$\mu$ denote space indices. We therefore +obtain from~\Eqref{(116)}, in the usual vector notation, +\[ +\left. +\begin{aligned} +\frac{d}{dl}\bigl[(1 + \Bar[7pt]{\sigma}) \v\bigr] + &= \grad \Bar[7pt]{\sigma} + \frac{\dd \tA}{\dd l} + [\rot \tA, \v]\Add{,} \\ +\Bar[7pt]{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\Add{,} \\ +\tA &= \frac{\kappa}{2\Change{}{\pi}} + \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}\Add{.} +\end{aligned} +\right\} +\Tag{(118)} +\] + +The equations of motion,~\Eqref{(118)}, show now, in fact, that +\PageSep{113} +\begin{itemize} +\item[1.] The inert mass is proportional to $1 + \Bar[7pt]{\sigma}$, and + therefore increases when ponderable masses + approach the test body. + +\item[2.] There is an inductive action of accelerated masses, +\index{Accelerated masses, inductive action of}% +\index{Inductive action of accelerated masses}% + of the same sign, upon the test body. This is + the term~$\dfrac{\dd \tA}{\dd l}$. + +\item[3.] A material particle, moving perpendicularly to the + axis of rotation inside a rotating hollow body, + is deflected in the sense of the rotation (Coriolis + field). The centrifugal action, mentioned above, + inside a rotating hollow body, also follows from + the theory, as has been shown by Thirring.\footnotemark +\index{Thirring}% +\end{itemize} +\footnotetext{That the centrifugal action must be inseparably connected with the + existence of the Coriolis field may be recognized, even without calculation, + in the special case of a co-ordinate system rotating uniformly relatively to + an inertial system; our general co-variant equations naturally must apply + to such a case.} + +Although all of these effects are inaccessible to experiment, +because $\kappa$~is so small, nevertheless they certainly +exist according to the general theory of relativity. We +must see in them a strong support for Mach's ideas as to +\index{Mach}% +the relativity of all inertial actions. If we think these +ideas consistently through to the end we must expect the +\emph{whole} inertia, that is, the \emph{whole} $g_{\mu\nu}$-field, to be determined +by the matter of the universe, and not mainly by the +boundary conditions at infinity. + +For a satisfactory conception of the $g_{\mu\nu}$-field of cosmical +dimensions, the fact seems to be of significance that the +relative velocity of the stars is small compared to the +velocity of light. It follows from this that, with a suitable +\PageSep{114} +choice of co-ordinates, $g_{44}$~is nearly constant in the +universe, at least, in that part of the universe in which +there is matter. The assumption appears natural, moreover, +that there are stars in all parts of the universe, so +that we may well assume that the inconstancy of~$g_{44}$ +depends only upon the circumstance that matter is not +distributed continuously, but is concentrated in single +celestial bodies and systems of bodies. If we are willing +to ignore these more local non-uniformities of the density +of matter and of the $g_{\mu\nu}$-field, in order to learn something +of the geometrical properties of the universe as a whole, +it appears natural to substitute for the actual distribution +of masses a continuous distribution, and furthermore to +assign to this distribution a uniform density~$\sigma$. In this +imagined universe all points with space directions will +be geometrically equivalent; with respect to its space +extension it will have a constant curvature, and will be +cylindrical with respect to its $x_{4}$-co-ordinate. The possibility +seems to be particularly satisfying that the universe +is spatially bounded and thus, in accordance with our +assumption of the constancy of~$\sigma$, is of constant curvature, +being either spherical or elliptical; for then the boundary +conditions at infinity which are so inconvenient from the +standpoint of the general theory of relativity, may be +replaced by the much more natural conditions for a closed +surface. + +According to what has been said, we are to put +\[ +ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\Add{,} +\Tag{(119)} +\] +in which the indices $\mu$~and~$\nu$ run from~$1$ to~$3$ only. The~$\gamma_{\mu\nu}$ +\PageSep{115} +will be such functions of $x_{1}$,~$x_{2}$,~$x_{3}$ as correspond +to a three-dimensional continuum of constant positive +curvature. We must now investigate whether such an +assumption can satisfy the field equations of gravitation. + +In order to be able to investigate this, we must first +find what differential conditions the three-dimensional +manifold of constant curvature satisfies. A spherical +manifold of three dimensions, embedded in a Euclidean +continuum of four dimensions,\footnote + {The aid of a fourth space dimension has naturally no significance + except that of a mathematical artifice.} +is given by the equations +\begin{align*} +{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}\Add{,} \\ +{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. +\end{align*} +By eliminating~$x_{4}$, we get +\[ +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} + {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}. +\] + +As far as terms of the third and higher degrees in the~$x_{\nu}$, +we can put, in the neighbourhood of the origin of +co-ordinates, +\[ +ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) + dx_{\mu}\, dx_{\nu}. +\] + +Inside the brackets are the~$g_{\mu\nu}$ of the manifold in the +neighbourhood of the origin. Since the first derivatives +of the~$g_{\mu\nu}$, and therefore also the~$\Gamma_{\mu\nu}^{\sigma}$, vanish at the +origin, the calculation of the~$R_{\mu\nu}$ for this manifold, by~\Eqref{(88)}, +is very simple at the origin. We have +\[ +R_{\mu\nu} = \Change{}{-}\frac{2}{a^{2}} \delta_{\mu\nu} + = \frac{2}{a^{2}} g_{\mu\nu}. +\] +\PageSep{116} + +Since the relation $R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}$ is universally co-variant, +and since all points of the manifold are geometrically +equivalent, this relation holds for every system of co-ordinates, +and everywhere in the manifold. In order to +avoid confusion with the four-dimensional continuum, +we shall, in the following, designate quantities that refer +to the three-dimensional continuum by Greek letters, +and put +\[ +\P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}\Add{.} +\Tag{(120)} +\] + +We now proceed to apply the field equations~\Eqref{(96)} to +our special case. From~\Eqref{(119)} we get for the four-dimensional +manifold, +\[ +\left. +\begin{aligned} +R_{\mu\nu} &= \text{$\P_{\mu\nu}$ for the indices~$1$ to~$3$}\Add{,} \\ +R_{14} &= R_{24} = R_{34} = R_{44} = 0\Add{.} +\end{aligned} +\right\} +\Tag{(121)} +\] + +For the right-hand side of~\Eqref{(96)} we have to consider +the energy tensor for matter distributed like a cloud of +dust. According to what has gone before we must +therefore put +\[ +T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds} +\] +specialized for the case of rest. But in addition, we +shall add a pressure term that may be physically established +as follows. Matter consists of electrically charged +particles. On the basis of Maxwell's theory these +cannot be conceived of as electromagnetic fields free +from singularities. In order to be consistent with the +\PageSep{117} +facts, it is necessary to introduce energy terms, not contained +in Maxwell's theory, so that the single electric +particles may hold together in spite of the mutual repulsions +between their elements, charged with electricity +of one sign. For the sake of consistency with this fact, +Poincaré has assumed a pressure to exist inside these +particles which balances the electrostatic repulsion. It +cannot, however, be asserted that this pressure vanishes +outside the particles. We shall be consistent with this +circumstance if, in our phenomenological presentation, +we add a pressure term. This must not, however, be +confused with a hydrodynamical pressure, as it serves +only for the energetic presentation of the dynamical +relations inside matter. In this sense we put +\[ +T_{\mu\nu} + = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + - g_{\mu\nu} p\Add{.} +\Tag{(122)} +\] + +In our special case we have, therefore, to put +\begin{align*} +T_{\mu\nu} + &= \gamma_{\mu\nu} p \text{ (for $\mu$~and~$\nu$ from~$1$ to~$3$)}\Add{,} \\ +T_{44} &= \sigma - p\Add{,} \\ +T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p + = \sigma - 4p. +\end{align*} +Observing that the field equation~\Eqref{(96)} may be written +in the form +\[ +R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T)\Add{,} +\] +we get from~\Eqref{(96)} the equations, +\begin{align*} ++\frac{2}{a^{2}} \gamma_{\mu\nu} + &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}\Add{,} \\ +0 &= -\kappa \left(\frac{\sigma}{2} + p\right). +\end{align*} +\PageSep{118} +From this follows +\[ +\left. +\begin{aligned} +p &= -\frac{\sigma}{2}\Add{,} \\ +a &= \sqrt{\frac{2}{\kappa\sigma}}\Add{.} +\end{aligned} +\right\} +\Tag{(123)} +\] + +If the universe is quasi-Euclidean, and its radius of +\index{Radius of Universe}% +\index{Universe@{Universe, Finiteness of}!Radius of}% +curvature therefore infinite, then $\sigma$~would vanish. But +it is improbable that the mean density of matter in the +universe is actually zero; this is our third argument +against the assumption that the universe is quasi-Euclidean. +Nor does it seem possible that our hypothetical +pressure can vanish; the physical nature of this +pressure can be appreciated only after we have a better +theoretical knowledge of the electromagnetic field. +According to the second of equations~\Eqref{(123)} the radius,~$a$, +of the universe is determined in terms of the total +mass,~$M$, of matter, by the equation +\[ +a = \frac{M\kappa}{4\pi^{2}}\Add{.} +\Tag{(124)} +\] +The complete dependence of the geometrical upon the +physical properties becomes clearly apparent by means +of this equation. + +Thus we may present the following arguments against +the conception of a space-infinite, and for the conception +of a space-bounded, universe:--- + +1.~From the standpoint of the theory of relativity, +the condition for a closed surface is very much simpler +than the corresponding boundary condition at infinity +of the quasi-Euclidean structure of the universe. +\PageSep{119} + +2.~The idea that Mach expressed, that inertia depends +\index{Mach}% +upon the mutual action of bodies, is contained, to a +first approximation, in the equations of the theory of +relativity; it follows from these equations that inertia +depends, at least in part, upon mutual actions between +masses. As it is an unsatisfactory assumption to make +that inertia depends in part upon mutual actions, and +in part upon an independent property of space, Mach's +idea gains in probability. But this idea of Mach's +corresponds only to a finite universe, bounded in space, +and not to a quasi-Euclidean, infinite universe. From +the standpoint of epistemology it is more satisfying to +have the mechanical properties of space completely determined +by matter, and this is the case only in a space-bounded +universe. + +3.~An infinite universe is possible only if the mean +density of matter in the universe vanishes. Although +such an assumption is logically possible, it is less probable +than the assumption that there is a finite mean +density of matter in the universe. +\PageSep{120} +%[Blank page] +\PageSep{121} + + +\PrintIndex +\iffalse %%%% Index text %%%% +%INDEX +% A + +Accelerated masses, inductive action of 113 + +Addition@{Addition and subtraction of tensors}#addition 14 + theorem of velocities 40 +% B + +Biot-Savart force 46 +% C + +Centrifugal force 67 + +Clocks, moving#clocks 39 + +Compressible viscous fluid 22 + +Concept@{Concept of space}#concept 3 + time 30 + +Conditions of orthogonality 7 + +Congruence, theorems of#congruence 3 + +Conservation principles 55 + +Continuum, four-dimensional#continuum 33 + +Contraction of tensors 15 + +Contravariant@{Contra-variant vectors}#contra-variant 72 + tensors 75 + +Coordinates@{Co-ordinates, preferred systems of}#co-ordinates 8 + +Covariance@{Co-variance of equation of continuity}#co-variance 22 + +Covariant@{Co-variant}#co-variant 12 %** et seq. + vector 72 + +Criticism of principle of inertia#criticism 65 + +Criticisms of theory of relativity#criticisms 31 + +Curvilinear co-ordinates 68 +% D + +Differentiation of tensors 76, 79 + +Displacement of spectral lines 101 +% E + +Energy@{Energy and mass}#energy 48, 51 + tensor@{tensor of electromagnetic field}#tensor 52 + of matter 56 + +Equation of continuity, co-variance of#co-variance 22 + +Equations of motion of \Change{materia}{material} particle#equations of motion 52 + +Equivalence of mass and energy#equivalence 51 + +Equivalent spaces of reference#equivalent 26 + +Euclidean geometry 4 +% F + +Finiteness of universe 110 + +Fizeau 29 + +Four-dimensional continuum 33 + +Four-vector 43 + +Fundamental tensor 74 +% G + +Galilean@{Galilean regions}#Galilean 65 + transformation 28 + +Gauss 68 + +Geodetic lines 86 + +Geometry, Euclidean 4 + +Gravitational mass 63 + +Gravitation constant 98 +% H + +Homogeneity of space 17 + +Hydrodynamical equations 56 + +Hypotheses of pre-relativity physics 77 +%\PageSep{122} +% I + +Inductive action of accelerated masses 113 + +Inert and gravitational mass, equality of#gravitational mass 63 + +Invariant 10 %** et seq. + +Isotropy of space 17 +% K + +Kaluza 108 +% L + +Levi-Civita 77 + +Light cone@{Light-cone}#light-cone 42 + +Light ray, path of 102 + +Light time@{Light-time}#light-time 34 + +Linear orthogonal transformation 7 + +Lorentz@{Lorentz electromotive force}#Lorentz 46 + transformation 32 +% M + +Mach 62, 109, 110, 111, 113, 119 + +Mass and Energy 48, 51 + equality of gravitational and inert#gravitational 63 + gravitational 63 + +Maxwell's equations 23 + +Mercury, perihelion of#Mercury 103, 107 + +Michelson and Morley 29 + +Minkowski 34 + +Motion of particle, equations of#equations of motion 52 + +Moving measuring rods and clocks#clocks 39 + +Multiplication of tensors#multiplication 14 +% N + +Newtonian gravitation constant#Newtonian 98 +% O + +Operations on tensors 14 %** et seq. + +Orthogonality, conditions of#orthogonality 7 + +Orthogonal transformations, linear 7 +% P + +Path of light ray 102 + +Perihelion of Mercury 103, 107 + +Poisson's equation 90 + +Preferred systems of co-ordinates 8 + +Prerelativity@{Pre-relativity physics, hypotheses of}#pre-relativity 27 + +Principle@{Principle of equivalence}#principle of 64 + inertia, criticism of 65 + +Principles of conservation 55 +% R + +Radius of Universe#radius 118 + +Rank of tensor#rank 14 + +Ray of light, path of 102 + +Reference, space of 4 + +Riemann 72 + tensor 82, 85, 110 + +Rods (measuring) and clocks in motion#clocks 39 + +Rotation 66 +% S + +Simultaneity 17, 30 + +Sitter 29 + +Skew-symmetrical tensor 15 + +Solar Eclipse expedition (1919)#eclipse 103 + +Space@{Space, Concept of}#space 3 %[** TN: "concept of" in the original] + Homogeneity of 17 %[** TN: "homogeneity of" in the original] + Isotropy of 17 + +Spaces of reference 4 + equivalence of 26 + +Special Lorentz transformation 36 + +Spectral lines, displacement of 101 + +Straightest lines 86 + +Stress tensor 22 + +Symmetrical tensor 15 + +Systems of co-ordinates, preferred#preferred 8 + +% T + +%Tensor, 12 et seq., 72 et seq. +Tensor 12, 72 + Addition and subtraction of 14 + Contraction of 15 + Fundamental 74 + Multiplication of 14 +%\PageSep{123} +%Tensor, + operations 14 %** et seq. + Rank of 14 + Symmetrical and Skew-symmetrical 15 + +Tensors, formation by differentiation 76 + +Theorem for addition of velocities 40 + +Theorems of congruence 3 + +Theory of relativity, criticisms of#criticisms 31 + +Thirring 113 + +Time-concept 30 + +Time-space concept 33 + +Transformation@{Transformation, Galilean}#transformation 28 + Linear orthogonal 7 +% U + +Universe@{Universe, Finiteness of}#universe 110 %[** TN: "finiteness of" in the original] + Radius of 118 %[** TN: "radius of" in the original] +% V + +Vector@{Vector, co-variant}#vector 72 + contra-variant 72 + +Velocities, addition theorem of 40 + +Viscous compressible fluid 22 +% W + +Weyl 77, 103, 108 +\fi %%%% End of index text %%%% +\PageSep{124} +\cleardoublepage +\newpage +\begin{CenterPage} +\scriptsize +PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN +\end{CenterPage} + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\PGLicenseInit +\begin{PGtext} +End of Project Gutenberg's The Meaning of Relativity, by Albert Einstein + +*** END OF THIS PROJECT GUTENBERG EBOOK THE MEANING OF RELATIVITY *** + +***** This file should be named 36276-pdf.pdf or 36276-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/2/7/36276/ + +Produced by Andrew D. 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+} + +.transnote {background-color: #E6E6FA; + color: black; + font-size:smaller; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; } + + </style> + + <style> + #pg-header div, #pg-footer div { + all: initial; + display: block; + margin-top: 1em; + margin-bottom: 1em; + margin-left: 2em; + } + #pg-footer div.agate { + font-size: 90%; + margin-top: 0; + margin-bottom: 0; + text-align: center; + } + #pg-footer li { + all: initial; + display: block; + margin-top: 1em; + margin-bottom: 1em; + text-indent: -0.6em; + } + #pg-footer div.secthead { + font-size: 110%; + font-weight: bold; + } + #pg-footer #project-gutenberg-license { + font-size: 110%; + margin-top: 0; + margin-bottom: 0; + text-align: center; + } + #pg-header h2 { + text-align: center; + font-size: 110%; + } + #pg-footer h2 { + text-align: center; + font-size: 120%; + font-weight: normal; + margin-top: 0; + margin-bottom: 0; + } + #pg-header #pg-machine-header p { + text-indent: -4em; + padding-left: 4em; + margin-top: 1em; + } + #pg-header #pg-header-authlist p { + margin-left: -2em; + margin-top: 0; + margin-bottom: 0; + } + #pg-header #pg-machine-header strong { + font-weight: normal; + } + #pg-header #pg-start-separator, #pg-footer #pg-end-separator { + margin-bottom: 3em; + margin-left: 0; + margin-right: auto; + margin-top: 2em; + text-align: center + } + </style> +</head> + +<body> + + <section class="pg-boilerplate pgheader" id="pg-header" lang="en"> +<h2>The Project Gutenberg eBook of The meaning of relativity</h2> +<div> +This ebook is for the use of anyone anywhere in the United States and +most other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms of the +Project Gutenberg License included with this ebook or online at +<a class="reference external" href="https://www.gutenberg.org">www.gutenberg.org</a>. +If you are not located in the United States, you’ll have to check the laws of the +country where you are located before using this eBook. +</div> +<div class="container" id="pg-machine-header"> +<p><strong>Title:</strong> The meaning of relativity<br> Four lectures delivered at Princeton University, May, 1921</p> +<div id='pg-header-authlist'> +<p><strong>Author:</strong> Albert Einstein</p> +<p><strong>Translator:</strong> Edwin Plimpton Adams</p> +</div> +<p><strong>Release Date:</strong> February 19, 2023 [eBook #36276]<br> +[Last updated: March 2, 2023]</p> +<p><strong>Language:</strong> English</p> +<p><strong>Original Publication:</strong> United States: Princeton University Press, Princeton, NJ, 1922</p> +<p><strong>Credit:</strong> Andrew D. Hwang, updated for HTML+SVG by Laura Natal Rodrigues</p> +</div> +<div id='pg-start-separator'> +<span>*** START OF THE PROJECT GUTENBERG EBOOK THE MEANING OF RELATIVITY ***</span> +</div> +</section> + +<div class="figcenter" style="width: 500px;"> +<img src="images/cover.jpg" width="500" alt="500"> +</div> + +<div class='chapter'> +<h1>THE MEANING OF<br> +RELATIVITY</h1> + +<p><br><br></p> + +<p class="center"><b>FOUR LECTURES DELIVERED AT +PRINCETON UNIVERSITY, MAY, 1921</b></p> + +<p><br><br></p> + +<p class="center"><b>BY</b></p> + +<div style='text-align:center; font-size:1.2em;'>ALBERT EINSTEIN</div> + +<p><br><br></p> + +<p class="center"><b>WITH FOUR DIAGRAMS</b></p> + +<p><br><br></p> + +<p class="center"><b>PRINCETON</b><br> +<b>PRINCETON UNIVERSITY PRESS</b><br> +<b>1923</b></p> +</div> + +<p><br><br></p> + +<div class='chapter'> +<p class="center"><i>Copyright 1922 +Princeton University Press +Published 1922</i></p> + +<p><br><br></p> + +<p class="hanging2"> +NOTE.—The translation of these lectures into English +was made by EDWIN PLIMPTON ADAMS, Professor +of Physics in Princeton University +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2>CONTENTS</h2> +<p class="nind"> +LECTURE I<br> +<a href="#chap01">SPACE AND TIME IN PRE-RELATIVITY PHYSICS</a><br> +LECTURE II<br> +<a href="#chap02">THE THEORY OF SPECIAL RELATIVITY</a><br> +LECTURE III<br> +<a href="#chap03">THE GENERAL THEORY OF RELATIVITY</a><br> +LECTURE IV<br> +<a href="#chap04">THE GENERAL THEORY OF RELATIVITY (<i>continued</i>)</a><br> +<a href="#INDEX">INDEX</a></p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2>THE MEANING OF RELATIVITY</h2> + +<p><br><br><br></p> + +<h2 title="LECTURE I: SPACE AND TIME IN PRE-RELATIVITY +PHYSICS"><a id="chap01"></a>LECTURE I +<br><br> +SPACE AND TIME IN PRE-RELATIVITY +PHYSICS</h2> + +<p class="nind"> +<span class="dropcap">T</span>HE theory of relativity is intimately connected with the theory +of space and time. I shall therefore begin with a brief investigation +of the origin of our ideas of space and time, although in +doing so I know that I introduce a controversial subject. The +object of all science, whether natural science or psychology, is +to co-ordinate our experiences and to bring them into a logical +system. How are our customary ideas of space and time related +to the character of our experiences? +</p> +<p> +The experiences of an individual appear to us arranged in a +series of events; in this series the single events which we remember +appear to be ordered according to the criterion of "earlier" +and "later," which cannot be analysed further. There exists, +therefore, for the individual, an I-time, or subjective time. This +in itself is not measurable. I can, indeed, associate numbers with +the events, in such a way that a greater number is associated +with the later event than with an earlier one; but the nature of +this association may be quite arbitrary. This association I can +define by means of a clock by comparing the order of events furnished +by the clock with the order of the given series of events. +We understand by a clock something which provides a series of +events which can be counted, and which has other properties of +which we shall speak later. +<span class="pagenum" id="Page_1">[Pg 1]</span> +</p> +<p> +By the aid of speech different individuals can, to a certain +extent, compare their experiences. In this way it is shown that +certain sense perceptions of different individuals correspond to +each other, while for other sense perceptions no such correspondence +can be established. We are accustomed to regard as real +those sense perceptions which are common to different individuals, +and which therefore are, in a measure, impersonal. The natural +sciences, and in particular, the most fundamental of them, +physics, deal with such sense perceptions. The conception of +physical bodies, in particular of rigid bodies, is a relatively constant +complex of such sense perceptions. A clock is also a body, +or a system, in the same sense, with the additional property that +the series of events which it counts is formed of elements all of +which can be regarded as equal. +</p> +<p> +The only justification for our concepts and system of concepts +is that they serve to represent the complex of our experiences; +beyond this they have no legitimacy. I am convinced that +the philosophers have had a harmful effect upon the progress +of scientific thinking in removing certain fundamental concepts +from the domain of empiricism, where they are under our control, +to the intangible heights of the <i>a priori</i>. For even if it should +appear that the universe of ideas cannot be deduced from experience +by logical means, but is, in a sense, a creation of the +human mind, without which no science is possible, nevertheless +this universe of ideas is just as little independent of the nature +of our experiences as clothes are of the form of the human body. +This is particularly true of our concepts of time and space, which +physicists have been obliged by the facts to bring down from the +Olympus of the <i>a priori</i> in order to adjust them and put them +in a serviceable condition. +<span class="pagenum" id="Page_2">[Pg 2]</span> +</p> +<p> +We now come to our concepts and judgments of space. It +is essential here also to pay strict attention to the relation of +experience to our concepts. It seems to me that Poincaré clearly +recognized the truth in the account he gave in his book, "La +Science et l'Hypothèse." Among all the changes which we can +perceive in a rigid body those are marked by their simplicity +which can be made reversibly by an arbitrary motion of the +body; Poincaré calls these, changes in position. By means of +simple changes in position we can bring two bodies into contact. +The theorems of congruence, fundamental in geometry, have to +do with the laws that govern such changes in position. For the +concept of space the following seems essential. We can form new +bodies by bringing bodies <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">, <img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">, ... up to body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">; we say that +we <i>continue</i> body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">. We can continue body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> in such a way that +it comes into contact with any other body, <img style="vertical-align: 0; width: 1.928ex; height: 1.545ex;" src="images/4.svg" alt=" " data-tex="X">. The ensemble of +all continuations of body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> we can designate as the "space of +the body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." Then it is true that all bodies are in the "space of +the (arbitrarily chosen) body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." In this sense we cannot speak +of space in the abstract, but only of the "space belonging to a +body <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">." The earth's crust plays such a dominant rôle in our +daily life in judging the relative positions of bodies that it has +led to an abstract conception of space which certainly cannot be +defended. In order to free ourselves from this fatal error we shall +speak only of "bodies of reference," or "space of reference." It +was only through the theory of general relativity that refinement +of these concepts became necessary, as we shall see later. +</p> +<p> +I shall not go into detail concerning those properties of the +space of reference which lead to our conceiving points as elements +of space, and space as a continuum. Nor shall I attempt +to analyse further the properties of space which justify the conception +<span class="pagenum" id="Page_3">[Pg 3]</span> +of continuous series of points, or lines. If these concepts +are assumed, together with their relation to the solid bodies of +experience, then it is easy to say what we mean by the three-dimensionality +of space; to each point three numbers, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> +(co-ordinates), may be associated, in such a way that this association +is uniquely reciprocal, and that <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}"> and <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> vary +continuously when the point describes a continuous series of points +(a line). +</p> +<p> +It is assumed in pre-relativity physics that the laws of the +orientation of ideal rigid bodies are consistent with Euclidean +geometry. What this means may be expressed as follows: Two +points marked on a rigid body form an <i>interval</i>. Such an interval +can be oriented at rest, relatively to our space of reference, in +a multiplicity of ways. If, now, the points of this space can +be referred to co-ordinates <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, in such a way that the +differences of the co-ordinates, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/8.svg" alt=" " data-tex="\Delta x_{1}">, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/9.svg" alt=" " data-tex="\Delta x_{2}">, +<img style="vertical-align: -0.375ex; width: 4.166ex; height: 1.994ex;" src="images/10.svg" alt=" " data-tex="\Delta x_{3}">, of the two ends +of the interval furnish the same sum of squares, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 34.482ex; height: 2.633ex;" src="images/11.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}, \qquad\text{(1)} +"></span> +for every orientation of the interval, then the space of reference +is called Euclidean, and the co-ordinates Cartesian.<a id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> It is +sufficient, indeed, to make this assumption in the limit for an +infinitely small interval. Involved in this assumption there are +some which are rather less special, to which we must call attention +on account of their fundamental significance. In the first +place, it is assumed that one can move an ideal rigid body in an +arbitrary manner. In the second place, it is assumed that the behaviour +of ideal rigid bodies towards orientation is independent +<span class="pagenum" id="Page_4">[Pg 4]</span> +of the material of the bodies and their changes of position, in the +sense that if two intervals can once be brought into coincidence, +they can always and everywhere be brought into coincidence. +Both of these assumptions, which are of fundamental importance +for geometry and especially for physical measurements, +naturally arise from experience; in the theory of general relativity +their validity needs to be assumed only for bodies and spaces +of reference which are infinitely small compared to astronomical +dimensions. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a>This relation must hold for an arbitrary choice of the origin and of the +direction +(ratios <img style="vertical-align: -0.375ex; width: 16.271ex; height: 1.994ex;" src="images/12.svg" alt=" " data-tex="\Delta x_{1}: \Delta x_{2}: \Delta x_{3}">) +of the interval.</p></div> + +<p> +The quantity <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> we call the length of the interval. In order +that this may be uniquely determined it is necessary to fix arbitrarily +the length of a definite interval; for example, we can put +it equal to 1 (unit of length). Then the lengths of all other intervals +may be determined. If we make the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> linearly dependent +upon a parameter <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda">, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 14.299ex; height: 2.009ex;" src="images/16.svg" alt=" " data-tex=" +x_{\nu} = a_{\nu} + \lambda b_{\nu}, +"></span> +we obtain a fine which has all the properties of the straight +lines of the Euclidean geometry. In particular, it easily follows +that by laying off <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> times the interval <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> upon a straight fine, an +interval of length <img style="vertical-align: -0.025ex; width: 4.053ex; height: 1.025ex;" src="images/18.svg" alt=" " data-tex="n·s"> is obtained. A length, therefore, means +the result of a measurement carried out along a straight line by +means of a unit measuring rod. It has a significance which is as +independent of the system of co-ordinates as that of a straight +line, as will appear in the sequel. +</p> +<p> +We come now to a train of thought which plays an analogous +role in the theories of special and general relativity. We ask +the question: besides the Cartesian co-ordinates which we have +used are there other equivalent co-ordinates? An interval has +<span class="pagenum" id="Page_5">[Pg 5]</span> +a physical meaning which is independent of the choice of co-ordinates; +and so has the spherical surface which we obtain as +the locus of the end points of all equal intervals that we lay off +from an arbitrary point of our space of reference. If <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> as well +as <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> (<img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> from 1 to 3) are Cartesian co-ordinates of our space +of reference, then the spherical surface will be expressed in our +two systems of co-ordinates by the equations +<span class="align-center"><img style="vertical-align: -3.006ex; width: 31.357ex; height: 7.142ex;" src="images/21.svg" alt=" " data-tex=" +\begin{align*} +\sum {\Delta x_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2)} \\ +\sum {{\Delta x'}_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2a)} +\end{align*} +"></span> +How must the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> be expressed in terms of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> in order that +equations (2) and (2a) may be equivalent to each other? Regarding +the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> expressed as functions of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, we can write, +by Taylor's theorem, for small values of the <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"><img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, +<span class="align-center"><img style="vertical-align: -3.06ex; width: 52.322ex; height: 6.477ex;" src="images/23.svg" alt=" " data-tex=" +{\Delta x'}_{\nu} + = \sum_{\alpha} \frac{\partial {x'}_{\nu}}{\partial x_{\alpha}}\, \Delta x_{\alpha} + + \frac{1}{2} \sum_{\alpha \text{,} \beta} + \frac{\partial^{2} {x'}_{\nu}}{\partial x_{\alpha}\, \partial x_{\beta}}\, + \Delta x_{\alpha}\, \Delta x_{\beta}.\ldots +"></span> +If we substitute (2a) in this equation and compare with (1), +we see that the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> must be linear functions of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If we +therefore put +<span class="align-center"><img style="vertical-align: -4.542ex; width: 42.282ex; height: 10.216ex;" src="images/24.svg" alt=" " data-tex=" +\begin{align*} +& & {x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}, +\qquad & &\text{(3)}\\ +&\text{or} & \Delta {x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}, +\qquad & &\text{(3a)} +\end{align*}"></span> +<span class="pagenum" id="Page_6">[Pg 6]</span> +then the equivalence of equations (2) and (2a) is expressed in +the form +<span class="align-center"><img style="vertical-align: -1.018ex; width: 59.118ex; height: 3.296ex;" src="images/25.svg" alt=" " data-tex=" +\sum{\Delta x'_{\nu}}^{2} = \lambda \sum{\Delta x_{\nu}}^{2} + \qquad\text{(\(\lambda\) independent of \(\Delta x_{\nu}\)).} +\qquad\text{(2b)} +"></span> +It therefore follows that <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> must be a constant. If we put <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> = 1, +(2b) and (3a) furnish the conditions +<span class="align-center"><img style="vertical-align: -2.602ex; width: 24.147ex; height: 4.751ex;" src="images/26.svg" alt=" " data-tex=" +\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}, +\qquad\text{(4)} +"></span> +in which <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> = 1, or <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> = 0, according <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> or +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> ≠<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">. The +conditions (4) are called the conditions of orthogonality, and the +transformations (3), (4), linear orthogonal transformations. If +we stipulate that <img style="vertical-align: -0.566ex; width: 13.034ex; height: 2.633ex;" src="images/30.svg" alt=" " data-tex="s^2 = \sum {\Delta x_{\nu}}^2"> shall be equal to the square of +the length in every system of co-ordinates, and if we always measure +with the same unit scale, then <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> must be equal to 1. Therefore +the linear orthogonal transformations are the only ones by +means of which we can pass from one Cartesian system of co-ordinates +in our space of reference to another. We see that in +applying such transformations the equations of a straight line +become equations of a straight line. Reversing equations (3a) +by multiplying both sides by <img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/31.svg" alt=" " data-tex="b_{\nu\beta}"> and summing for all +the <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">'s, we obtain +<span class="align-center"><img style="vertical-align: -2.912ex; width: 58.403ex; height: 5.061ex;" src="images/32.svg" alt=" " data-tex=" +\sum_{\nu} b_{\nu\beta}\, {\Delta x'}_{\nu} + = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} + = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} + = \Delta x_{\beta} +\qquad\text{(5)} +"></span> +The same coefficients, <img style="vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;" src="images/33.svg" alt=" " data-tex="b">, also determine the inverse substitution +of <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"><img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. Geometrically, <img style="vertical-align: -0.357ex; width: 3.03ex; height: 1.927ex;" src="images/34.svg" alt=" " data-tex="b_{\nu\alpha}"> is the cosine of the angle between +the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> axis and the <img style="vertical-align: -0.357ex; width: 2.506ex; height: 1.357ex;" src="images/35.svg" alt=" " data-tex="x_{\alpha}"> axis. +<span class="pagenum" id="Page_7">[Pg 7]</span> +</p> +<p> +To sum up, we can say that in the Euclidean geometry +there are (in a given space of reference) preferred systems of +co-ordinates, the Cartesian systems, which transform into each +other by linear orthogonal transformations. The distance <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> between +two points of our space of reference, measured by a measuring +rod, is expressed in such co-ordinates in a particularly +simple manner. The whole of geometry may be founded upon +this conception of distance. In the present treatment, geometry +is related to actual things (rigid bodies), and its theorems are +statements concerning the behaviour of these things, which may +prove to be true or false. +</p> +<p> +One is ordinarily accustomed to study geometry divorced +from any relation between its concepts and experience. There +are advantages in isolating that which is purely logical and independent +of what is, in principle, incomplete empiricism. This +is satisfactory to the pure mathematician. He is satisfied if he +can deduce his theorems from axioms correctly, that is, without +errors of logic. The question as to whether Euclidean geometry +is true or not does not concern him. But for our purpose it +is necessary to associate the fundamental concepts of geometry +with natural objects; without such an association geometry is +worthless for the physicist. The physicist is concerned with the +question as to whether the theorems of geometry are true or +not. That Euclidean geometry, from this point of view, affirms +something more than the mere deductions derived logically from +definitions may be seen from the following simple consideration. +</p> +<p> +Between <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> points of space there are <img style="vertical-align: -1.552ex; width: 9.368ex; height: 4.855ex;" src="images/36.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2}"> distances, <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}">; +between these and the <img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/38.svg" alt=" " data-tex="3n"> co-ordinates we have the relations +<span class="align-center"><img style="vertical-align: -0.904ex; width: 43.644ex; height: 3.274ex;" src="images/39.svg" alt=" " data-tex=" +s_{\mu\nu}^{2} + = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + + .\dots +"></span> +<span class="pagenum" id="Page_8">[Pg 8]</span> +</p> +<p> +From these <img style="vertical-align: -1.552ex; width: 9.368ex; height: 4.855ex;" src="images/36.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2}"> equations the <img style="vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;" src="images/38.svg" alt=" " data-tex="3n"> co-ordinates may be +eliminated, and from this elimination at least <img style="vertical-align: -1.552ex; width: 14.622ex; height: 4.855ex;" src="images/40.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2} - 3n"> +equations in the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}">, will result.<a id="FNanchor_2_1"></a><a href="#Footnote_2_1" class="fnanchor">[2]</a> +Since the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}"> are measurable +quantities, and by definition are independent of each other, these +relations between the <img style="vertical-align: -0.685ex; width: 3.061ex; height: 1.685ex;" src="images/37.svg" alt=" " data-tex="s_{\mu\nu}"> are not necessary <i>a priori</i>. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_2_1"></a><a href="#FNanchor_2_1"><span class="label">[2]</span></a>In reality there are <img style="vertical-align: -1.552ex; width: 18.519ex; height: 4.855ex;" src="images/41.svg" alt=" " data-tex="\dfrac{n(n - 1)}{2} - 3n + 6"> equations.</p></div> + +<p> +From the foregoing it is evident that the equations of transformation +(3), (4) have a fundamental significance in Euclidean +geometry, in that they govern the transformation from one +Cartesian system of co-ordinates to another. The Cartesian +systems of co-ordinates are characterized by the property that +in them the measurable distance between two points, <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, is +expressed by the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 14.541ex; height: 3.167ex;" src="images/42.svg" alt=" " data-tex=" +s^{2} = \sum {\Delta x_{\nu}}^{2}. +"></span> +</p> +<p> +If <img style="vertical-align: -0.579ex; width: 3.756ex; height: 2.125ex;" src="images/43.svg" alt=" " data-tex="K_{x_{\nu}}"> and <img style="vertical-align: -0.799ex; width: 3.756ex; height: 2.516ex;" src="images/44.svg" alt=" " data-tex="K'_{x_{\nu}}"> are two Cartesian systems of co-ordinates, +then +<span class="align-center"><img style="vertical-align: -1.018ex; width: 21.338ex; height: 3.296ex;" src="images/45.svg" alt=" " data-tex=" +\sum {\Delta x_{\nu}}^{2} = \sum {\Delta x'_{\nu}}^{2}. +"></span> +</p> +<p> +The right-hand side is identically equal to the left-hand side +on account of the equations of the linear orthogonal transformation, +and the right-hand side differs from the left-hand side +only in that the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> are replaced by the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}">. This is expressed +by the statement that <img style="vertical-align: -0.566ex; width: 7.968ex; height: 2.633ex;" src="images/46.svg" alt=" " data-tex="\sum {\Delta x_{\nu}}^{2}"> is an invariant with respect to +linear orthogonal transformations. It is evident that in the Euclidean +geometry only such, and all such, quantities have an +objective significance, independent of the particular choice of +<span class="pagenum" id="Page_9">[Pg 9]</span> +the Cartesian co-ordinates, as can be expressed by an invariant +with respect to linear orthogonal transformations. This is +the reason that the theory of invariants, which has to do with +the laws that govern the form of invariants, is so important for +analytical geometry. +</p> +<p> +As a second example of a geometrical invariant, consider a +volume. This is expressed by +<span class="align-center"><img style="vertical-align: -1.948ex; width: 21.373ex; height: 5.027ex;" src="images/47.svg" alt=" " data-tex=" +V = \iiint dx_{1}\, dx_{2}\, dx_{3}. +"></span> +By means of Jacobi's theorem we may write +<span class="align-center"><img style="vertical-align: -2.172ex; width: 52.029ex; height: 5.495ex;" src="images/48.svg" alt=" " data-tex=" +\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} + = \iiint \frac{\partial({x'}_{1}, {x'}_{2}, {x'}_{3})}{\partial(x_{1}, x_{2}, x_{3})}\, + dx_{1}\, dx_{2}\, dx_{3} +"></span> +where the integrand in the last integral is the functional determinant +of the <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}"> with respect to the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and this by (3) is equal +to the determinant <img style="vertical-align: -0.685ex; width: 4.229ex; height: 2.381ex;" src="images/49.svg" alt=" " data-tex="|b_{\mu\nu}|"> of the coefficients of substitution, <img style="vertical-align: -0.357ex; width: 3.03ex; height: 1.927ex;" src="images/34.svg" alt=" " data-tex="b_{\nu\alpha}">. If +we form the determinant of the <img style="vertical-align: -0.685ex; width: 3.181ex; height: 2.307ex;" src="images/50.svg" alt=" " data-tex="\delta_{\mu\alpha}"> from equation (4), we obtain, +by means of the theorem of multiplication of determinants, +<span class="align-center"><img style="vertical-align: -2.602ex; width: 52.322ex; height: 6.335ex;" src="images/51.svg" alt=" " data-tex=" +1 = |\delta_{\alpha\beta}| + = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| + = |b_{\mu\nu}|^{2};\quad +|b_{\mu\nu}| = ±1. +\qquad\text{(6)} +"></span> +If we limit ourselves to those transformations which have the determinant ++1,<a id="FNanchor_3_1"></a><a href="#Footnote_3_1" class="fnanchor">[3]</a> +and only these arise from continuous variations +of the systems of co-ordinates, then <img style="vertical-align: -0.05ex; width: 1.74ex; height: 1.595ex;" src="images/52.svg" alt=" " data-tex="V"> is an invariant. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_3_1"></a><a href="#FNanchor_3_1"><span class="label">[3]</span></a>There are thus two kinds of Cartesian systems which are designated as +"right-handed" and "left-handed" systems. The difference between these is +familiar to every physicist and engineer. It is interesting to note that these +two kinds of systems cannot be defined geometrically, but only the contrast +between them.</p></div> +<p><span class="pagenum" id="Page_10">[Pg 10]</span></p> +<p> +Invariants, however, are not the only forms by means of +which we can give expression to the independence of the particular +choice of the Cartesian co-ordinates. Vectors and tensors +are other forms of expression. Let us express the fact that the +point with the current co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> lies upon a straight line. +We have +<span class="align-center"><img style="vertical-align: -0.566ex; width: 31.899ex; height: 2.262ex;" src="images/53.svg" alt=" " data-tex=" +x_{\nu} - A_{\nu} = \lambda B_{\nu} \quad (\nu \text{ from 1 to 3}). +"></span> +Without limiting the generality we can put +<span class="align-center"><img style="vertical-align: -1.018ex; width: 12.162ex; height: 3.167ex;" src="images/54.svg" alt=" " data-tex=" +\sum {B_{\nu}}^{2} = 1. +"></span> +</p> +<p> +If we multiply the equations by <img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/55.svg" alt=" " data-tex="b_{\beta\nu}"> (compare (3a) and (5)) +and sum for all the <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">'s, we get +<span class="align-center"><img style="vertical-align: -0.869ex; width: 15.719ex; height: 2.699ex;" src="images/56.svg" alt=" " data-tex=" +x'_{\beta} - A'_{\beta} = \lambda B'_{\beta}, +"></span> +where we have written +<span class="align-center"><img style="vertical-align: -2.602ex; width: 34.129ex; height: 4.751ex;" src="images/57.svg" alt=" " data-tex=" +B'_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad +A'_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}. +"></span> +</p> +<p> +These are the equations of straight lines with respect to a +second Cartesian system of co-ordinates <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. They have the +same form as the equations with respect to the original system +of co-ordinates. It is therefore evident that straight lines +have a significance which is independent of the system of co-ordinates. +Formally, this depends upon the fact that the quantities (<img style="vertical-align: -0.339ex; width: 7.828ex; height: 1.959ex;" src="images/59.svg" alt=" " data-tex="x_{\nu} - A_{\nu}">) - <img style="vertical-align: -0.339ex; width: 4.072ex; height: 1.91ex;" src="images/60.svg" alt=" " data-tex="\lambda B_{\nu}"> are +transformed as the components of +an interval, <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">. The ensemble of three quantities, defined for +every system of Cartesian co-ordinates, and which transform as +the components of an interval, is called a vector. If the three +<span class="pagenum" id="Page_11">[Pg 11]</span> +components of a vector vanish for one system of Cartesian co-ordinates, +they vanish for all systems, because the equations of +transformation are homogeneous. We can thus get the meaning +of the concept of a vector without referring to a geometrical representation. +This behaviour of the equations of a straight line +can be expressed by saying that the equation of a straight line +is co-variant with respect to linear orthogonal transformations. +</p> +<p> +We shall now show briefly that there are geometrical entities +which lead to the concept of tensors. Let <img style="vertical-align: -0.375ex; width: 2.44ex; height: 1.92ex;" src="images/62.svg" alt=" " data-tex="P_{0}"> be the centre of a +surface of the second degree, <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> any point on the surface, and +<img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> the projections of the interval <img style="vertical-align: -0.375ex; width: 4.139ex; height: 1.92ex;" src="images/65.svg" alt=" " data-tex="P_{0}P"> upon the co-ordinate axes. +Then the equation of the surface is +<span class="align-center"><img style="vertical-align: -1.018ex; width: 15.789ex; height: 3.167ex;" src="images/66.svg" alt=" " data-tex=" +\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +"></span> +In this, and in analogous cases, we shall omit the sign of summation, +and understand that the summation is to be carried out +for those indices that appear twice. We thus write the equation +of the surface +<span class="align-center"><img style="vertical-align: -0.685ex; width: 12.144ex; height: 2.278ex;" src="images/67.svg" alt=" " data-tex=" +a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1. +"></span> +The quantities <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}"> determine the surface completely, for a given +position of the centre, with respect to the chosen system of +Cartesian co-ordinates. From the known law of transformation +for the <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> (3a) for linear orthogonal transformations, we easily +find the law of transformation for the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">:<a id="FNanchor_4_1"></a><a href="#Footnote_4_1" class="fnanchor">[4]</a> +<span class="align-center"><img style="vertical-align: -0.685ex; width: 16.466ex; height: 2.515ex;" src="images/69.svg" alt=" " data-tex=" +{a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}. +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_4_1"></a><a href="#FNanchor_4_1"><span class="label">[4]</span></a>The equation <img style="vertical-align: -0.565ex; width: 13.255ex; height: 2.282ex;" src="images/70.svg" alt=" " data-tex="{a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1"> +may, by (5), be replaced by +<img style="vertical-align: -0.58ex; width: 3.125ex; height: 2.297ex;" src="images/71.svg" alt=" " data-tex="a'_{\sigma\tau}"><img style="vertical-align: -0.685ex; width: 3.037ex; height: 2.255ex;" src="images/72.svg" alt=" " data-tex="b_{\mu\sigma}"><img style="vertical-align: -0.36ex; width: 2.833ex; height: 1.93ex;" src="images/73.svg" alt=" " data-tex="b_{\nu\tau}"><img style="vertical-align: -0.685ex; width: 2.143ex; height: 2.278ex;" src="images/74.svg" alt=" " data-tex="\xi_{\mu}"><img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> = 1, +from which the result stated immediately follows.</p></div> +<p><span class="pagenum" id="Page_12">[Pg 12]</span></p> + +<p class="nind"> +This transformation is homogeneous and of the first degree in +the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">. On account of this transformation, the <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">, are called +components of a tensor of the second rank (the latter on account +of the double index). If all the components, <img style="vertical-align: -0.685ex; width: 3.197ex; height: 1.683ex;" src="images/68.svg" alt=" " data-tex="a_{\mu\nu}">, of a tensor with +respect to any system of Cartesian co-ordinates vanish, they +vanish with respect to every other Cartesian system. The form +and the position of the surface of the second degree is described +by this tensor (<img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/75.svg" alt=" " data-tex="a">). +</p> +<p> +Analytic tensors of higher rank (number of indices) may be +defined. It is possible and advantageous to regard vectors as +tensors of rank 1, and invariants (scalars) as tensors of rank 0. +In this respect, the problem of the theory of invariants may be so +formulated: according to what laws may new tensors be formed +from given tensors? We shall consider these laws now, in order +to be able to apply them later. We shall deal first only with the +properties of tensors with respect to the transformation from +one Cartesian system to another in the same space of reference, +by means of linear orthogonal transformations. As the laws are +wholly independent of the number of dimensions, we shall leave +this number, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n">, indefinite at first. +</p> + +<p><br></p> + +<p> +<i>Definition</i>. If a figure is defined with respect to every system +of Cartesian co-ordinates in a space of reference of <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n"> dimensions +by the <img style="vertical-align: -0.025ex; width: 2.569ex; height: 1.553ex;" src="images/76.svg" alt=" " data-tex="n^{\alpha}"> numbers <img style="vertical-align: -0.685ex; width: 6.399ex; height: 2.305ex;" src="images/77.svg" alt=" " data-tex="A_{\mu\nu\rho\ldots}"> (<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = number of indices), then +these numbers are the components of a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> if the +transformation law is +<span class="align-center"><img style="vertical-align: -0.685ex; width: 42.162ex; height: 2.515ex;" src="images/78.svg" alt=" " data-tex=" +{A'}_{\mu'\nu'\rho'}\ldots + = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho}\ldots A_{\mu\nu\rho}.\ldots +\qquad\text{(7)} +"></span> +<span class="pagenum" id="Page_13">[Pg 13]</span> +</p> + +<p><br></p> + +<p> +<i>Remark</i>. From this definition it follows that +<span class="align-center"><img style="vertical-align: -0.685ex; width: 29.803ex; height: 2.382ex;" src="images/79.svg" alt=" " data-tex=" +A_{\mu\nu\rho}\ldots = B_{\mu} C_{\nu} D_{\rho}\dots +\qquad\text{(8)} +"></span> +is an invariant, provided that (<img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">), (<img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">), +(<img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/80.svg" alt=" " data-tex="D">) ... are vectors. +Conversely, the tensor character of (<img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A">) may be inferred, if it +is known that the expression (8) leads to an invariant for an +arbitrary choice of the vectors (<img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B">), (<img style="vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;" src="images/2.svg" alt=" " data-tex="C">), etc. +</p> + +<p><br></p> + +<p> +<i>Addition and Subtraction</i>. By addition and subtraction of +the corresponding components of tensors of the same rank, a +tensor of equal rank results: +<span class="align-center"><img style="vertical-align: -0.685ex; width: 37.182ex; height: 2.382ex;" src="images/81.svg" alt=" " data-tex=" +A_{\mu\nu\rho}\ldots ± B_{\mu\nu\rho}\ldots = C_{\mu\nu\rho}\ldots. +\qquad\text{(9)} +"></span> +The proof follows from the definition of a tensor given above. +</p> + +<p><br></p> + +<p> +<i>Multiplication</i>. From a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and a tensor of +rank <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> we may obtain a tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> + <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> by multiplying all +the components of the first tensor by all the components of the +second tensor: +<span class="align-center"><img style="vertical-align: -0.685ex; width: 40.555ex; height: 2.382ex;" src="images/82.svg" alt=" " data-tex=" +T_{\mu\nu\rho}\ldots_{\alpha\beta}\ldots += A_{\mu\nu\rho}\ldots B_{\alpha\beta\gamma}.\ldots +\qquad\text{(10)} +"></span> +</p> + +<p><br></p> + +<p> +<i>Contraction</i>. A tensor of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> - 2 may be obtained from +one of rank <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> by putting two definite indices equal to each other +and then summing for this single index: +<span class="align-center"><img style="vertical-align: -2.947ex; width: 41.445ex; height: 5.097ex;" src="images/83.svg" alt=" " data-tex=" +T_{\rho}\ldots + = A_{\mu\mu\rho}\ldots + ( = \sum_{\mu} A_{\mu\mu\rho}\ldots). +\qquad\text{(11)} +"></span> +</p> +<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> +<p class="nind"> +The proof is +<span class="align-center"><img style="vertical-align: -2.222ex; width: 45.796ex; height: 5.576ex;" src="images/84.svg" alt=" " data-tex=" +\begin{align*} +{A'}_{\mu\mu\rho\cdots} + = b_{\mu\alpha} b_{\mu\beta} b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} + = \delta_{\alpha\beta} & b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} \\ + {} ={} & b_{\rho\gamma\cdots} A_{\alpha\alpha\gamma\cdots}. +\end{align*} +"></span> +</p> +<p> +In addition to these elementary rules of operation there is +also the formation of tensors by differentiation ("erweiterung"): +<span class="align-center"><img style="vertical-align: -1.909ex; width: 28.293ex; height: 5.255ex;" src="images/85.svg" alt=" " data-tex=" +T_{\mu\nu\rho\cdots\alpha} + = \frac{\partial A_{\mu\nu\rho\cdots}}{\partial x_{\alpha}}. +\qquad\text{(12)} +"></span> +</p> +<p> +New tensors, in respect to linear orthogonal transformations, +may be formed from tensors according to these rules of operation. +</p> + +<p><br></p> + +<p> +<i>Symmetrical Properties of Tensors</i>. Tensors are called symmetrical +or skew-symmetrical in respect to two of their indices, +<img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, if both the components which result from interchanging +the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> are equal to each other or equal with +opposite signs. +<span class="align-center"><img style="vertical-align: -2.156ex; width: 46.294ex; height: 5.442ex;" src="images/87.svg" alt=" " data-tex=" +\begin{alignat*}{2} +&\text{Condition for symmetry:} & + A_{\mu\nu\rho} &= A_{{\nu\mu}\rho}. \\ +&\text{Condition for skew-symmetry:}\quad & + A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ +\end{alignat*} +"></span> +</p> + +<p><br></p> + +<p> +<i>Theorem</i>. The character of symmetry or skew-symmetry +exists independently of the choice of co-ordinates, and in this +lies its importance. The proof follows from the equation defining tensors. +</p> +<p> +<i>Special Tensors.</i> +</p> +<p> +I. The quantities <img style="vertical-align: -0.685ex; width: 2.933ex; height: 2.307ex;" src="images/88.svg" alt=" " data-tex="\delta_{\rho\sigma}"> (4) are tensor components +(fundamental tensor). +<span class="pagenum" id="Page_15">[Pg 15]</span> +</p> +<p> +<i>Proof</i>. If in the right-hand side of the equation of transformation +<img style="vertical-align: -0.904ex; width: 3.697ex; height: 2.622ex;" src="images/89.svg" alt=" " data-tex="A'_{\mu\nu}"> = <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"><img style="vertical-align: -0.65ex; width: 2.912ex; height: 2.22ex;" src="images/31.svg" alt=" " data-tex="b_{\nu\beta}"><img style="vertical-align: -0.65ex; width: 3.814ex; height: 2.27ex;" src="images/91.svg" alt=" " data-tex="A_{\alpha\beta}">, we substitute +for <img style="vertical-align: -0.65ex; width: 3.814ex; height: 2.27ex;" src="images/91.svg" alt=" " data-tex="A_{\alpha\beta}"> the quantities <img style="vertical-align: -0.65ex; width: 3.122ex; height: 2.272ex;" src="images/27.svg" alt=" " data-tex="\delta_{\alpha\beta}"> (which are equal +to 1 or 0 according as <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> or <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> ≠<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">), +we get +<span class="align-center"><img style="vertical-align: -0.685ex; width: 20.17ex; height: 2.515ex;" src="images/92.svg" alt=" " data-tex=" +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}. +"></span> +The justification for the last sign of equality becomes evident if +one applies (4) to the inverse substitution (5). +</p> +<p> +II. There is a tensor (<img style="vertical-align: -0.685ex; width: 5.707ex; height: 2.307ex;" src="images/93.svg" alt=" " data-tex="\delta_{\mu\nu\rho\ldots}">) skew-symmetrical with respect +to all pairs of indices, whose rank is equal to the number of +dimensions, <img style="vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;" src="images/17.svg" alt=" " data-tex="n">, and whose components are equal to +1 or -1 +according as <img style="vertical-align: -0.489ex; width: 6.762ex; height: 1.489ex;" src="images/94.svg" alt=" " data-tex="\mu\nu\rho\ldots"> is an even or odd permutation of 1 2 3.... +</p> +<p> +The proof follows with the aid of the theorem proved above +<img style="vertical-align: -0.685ex; width: 8.305ex; height: 2.381ex;" src="images/95.svg" alt=" " data-tex="|b_{\rho\sigma}| = 1"> +</p> +<p> +These few simple theorems form the apparatus from the +theory of invariants for building the equations of pre-relativity +physics and the theory of special relativity. +</p> +<p> +We have seen that in pre-relativity physics, in order to specify +relations in space, a body of reference, or a space of reference, +is required, and, in addition, a Cartesian system of co-ordinates. +We can fuse both these concepts into a single one by thinking +of a Cartesian system of co-ordinates as a cubical frame-work +formed of rods each of unit length. The co-ordinates of the lattice +points of this frame are integral numbers. It follows from +the fundamental relation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 34.607ex; height: 2.633ex;" src="images/96.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} +\qquad\text{(13)} +"></span> +that the members of such a space-lattice are all of unit length. +To specify relations in time, we require in addition a standard +clock placed at the origin of our Cartesian system of co-ordinates +<span class="pagenum" id="Page_16">[Pg 16]</span> +or frame of reference. If an event takes place anywhere we can +assign to it three co-ordinates, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and a time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, as soon as +we have specified the time of the clock at the origin which is +simultaneous with the event. We therefore give an objective significance +to the statement of the simultaneity of distant events, +while previously we have been concerned only with the simultaneity +of two experiences of an individual. The time so specified +is at all events independent of the position of the system of co-ordinates +in our space of reference, and is therefore an invariant +with respect to the transformation (3). +</p> +<p> +It is postulated that the system of equations expressing the +laws of pre-relativity physics is co-variant with respect to the +transformation (3), as are the relations of Euclidean geometry. +The isotropy and homogeneity of space is expressed in this way.<a id="FNanchor_5_1"></a><a href="#Footnote_5_1" class="fnanchor">[5]</a> +We shall now consider some of the more important equations of +physics from this point of view. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_5_1"></a><a href="#FNanchor_5_1"><span class="label">[5]</span></a>The laws of physics could be expressed, even in case there were a +unique direction in space, in such a way as to be co-variant with respect +to the transformation (3); but such an expression would in this case be +unsuitable. If there were a unique direction in space it would simplify the +description of natural phenomena to orient the system of co-ordinates in +a definite way in this direction. But if, on the other hand, there is no +unique direction in space it is not logical to formulate the laws of nature +in such a way as to conceal the equivalence of systems of co-ordinates that +are oriented differently. We shall meet with this point of view again in the +theories of special and general relativity.</p></div> + +<p> +The equations of motion of a material particle are +<span class="align-center"><img style="vertical-align: -1.654ex; width: 22.955ex; height: 5.07ex;" src="images/98.svg" alt=" " data-tex=" +m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}; +\qquad\text{(14)} +"></span> +(<img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">) is a vector; <img style="vertical-align: -0.025ex; width: 1.993ex; height: 1.595ex;" src="images/100.svg" alt=" " data-tex="dt">, and therefore also <img style="vertical-align: -1.577ex; width: 2.989ex; height: 4.613ex;" src="images/101.svg" alt=" " data-tex="\dfrac{1}{dt}">, an invariant; thus +<span class="pagenum" id="Page_17">[Pg 17]</span> +(<img style="vertical-align: -1.577ex; width: 4.502ex; height: 4.676ex;" src="images/102.svg" alt=" " data-tex="\dfrac{dx_{\nu}}{dt}">) is a vector; in the same way it may be shown that +(<img style="vertical-align: -1.654ex; width: 5.489ex; height: 5.07ex;" src="images/103.svg" alt=" " data-tex="\dfrac{d^{2} x_{\nu}}{dt^{2}}">) is a vector. In general, the operation of differentiation +with respect to time does not alter the tensor character. Since +<img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/104.svg" alt=" " data-tex="m"> is an invariant (tensor of rank 0), <img style="vertical-align: -1.654ex; width: 8.356ex; height: 5.07ex;" src="images/105.svg" alt=" " data-tex="(m\dfrac{d^{2} x_{\nu}}{dt^{2}}">) +is a vector, or +tensor of rank 1 (by the theorem of the multiplication of tensors). +If the force (<img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">) has a vector character, the same holds for +the difference (<img style="vertical-align: -1.654ex; width: 14.031ex; height: 5.07ex;" src="images/107.svg" alt=" " data-tex="m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu})">. These equations of motion are +therefore valid in every other system of Cartesian co-ordinates +in the space of reference. In the case where the forces are conservative +we can easily recognize the vector character of (<img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">). +For a potential energy, <img style="vertical-align: 0; width: 1.633ex; height: 1.545ex;" src="images/108.svg" alt=" " data-tex="\Phi">, exists, which depends only upon the +mutual distances of the particles, and is therefore an invariant. +The vector character of the force, <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> = <img style="vertical-align: -1.891ex; width: 6.366ex; height: 5.038ex;" src="images/109.svg" alt=" " data-tex="-\dfrac{\partial \Phi}{\partial x_{\nu}}">, +is then a consequence of our general theorem about the derivative of a tensor +of rank 0. +</p> +<p> +Multiplying by the velocity, a tensor of rank 1, we obtain the +tensor equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 25.76ex; height: 5.564ex;" src="images/110.svg" alt=" " data-tex=" +\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0. +"></span> +By contraction and multiplication by the scalar <img style="vertical-align: -0.025ex; width: 1.993ex; height: 1.595ex;" src="images/100.svg" alt=" " data-tex="dt"> we obtain the +equation of kinetic energy +<span class="align-center"><img style="vertical-align: -2.148ex; width: 20.636ex; height: 5.564ex;" src="images/111.svg" alt=" " data-tex=" +d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\ dx_{\nu}. +"></span> +<span class="pagenum" id="Page_18">[Pg 18]</span> +</p> +<p> +If <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> denotes the difference of the co-ordinates of the material +particle and a point fixed in space, then the <img style="vertical-align: -0.464ex; width: 2.027ex; height: 2.057ex;" src="images/64.svg" alt=" " data-tex="\xi_{\nu}"> have the +character of vectors. We evidently have +<img style="vertical-align: -1.654ex; width: 5.489ex; height: 5.07ex;" src="images/103.svg" alt=" " data-tex="\dfrac{d^{2} x_{\nu}}{dt^{2}}"> = <img style="vertical-align: -1.654ex; width: 5.186ex; height: 5.07ex;" src="images/112.svg" alt=" " data-tex="\dfrac{d^{2} \xi_{\nu}}{dt^{2}}">, so that +the equations of motion of the particle may be written +<span class="align-center"><img style="vertical-align: -1.654ex; width: 17.625ex; height: 5.07ex;" src="images/113.svg" alt=" " data-tex=" +m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0. +"></span> +</p> +<p> +Multiplying this equation by <img style="vertical-align: -0.685ex; width: 2.143ex; height: 2.278ex;" src="images/74.svg" alt=" " data-tex="\xi_{\mu}"> we obtain a tensor equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 23.098ex; height: 5.564ex;" src="images/114.svg" alt=" " data-tex=" +\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0. +"></span> +</p> +<p> +Contracting the tensor on the left and taking the time average +we obtain the virial theorem, which we shall not consider +further. By interchanging the indices and subsequent subtraction, +we obtain, after a simple transformation, the theorem of +moments, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 51.585ex; height: 5.467ex;" src="images/115.svg" alt=" " data-tex=" +\frac{d}{dt} \biggl[ + m \biggl(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\biggr)\biggr] += \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}. +\qquad\text{(15)} +"></span> +</p> +<p> +It is evident in this way that the moment of a vector is not a +vector but a tensor. On account of their skew-symmetrical character +there are not nine, but only three independent equations of +this system. The possibility of replacing skew-symmetrical tensors +of the second rank in space of three dimensions by vectors +depends upon the formation of the vector +<span class="align-center"><img style="vertical-align: -1.552ex; width: 16.145ex; height: 4.588ex;" src="images/116.svg" alt=" " data-tex=" +A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}. +"></span> +<span class="pagenum" id="Page_19">[Pg 19]</span> +</p> +<p> +If we multiply the skew-symmetrical tensor of rank 2 by the +special skew-symmetrical tensor <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/117.svg" alt=" " data-tex="\delta"> introduced above, and contract +twice, a vector results whose components are numerically +equal to those of the tensor. These are the so-called axial vectors +which transform differently, from a right-handed system to +a left-handed system, from the <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">. There is a gain in +picturesqueness in regarding a skew-symmetrical tensor of rank 2 +as a vector in space of three dimensions, but it does not represent +the exact nature of the corresponding quantity so well as +considering it a tensor. +</p> +<p> +We consider next the equations of motion of a continuous +medium. Let <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"> be the density, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}"> the velocity components +considered as functions of the co-ordinates and the time, <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> the +volume forces per unit of mass, and <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> the stresses upon a +surface perpendicular to the a-axis in the direction of increasing +<img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. Then the equations of motion are, by Newton's law, +<span class="align-center"><img style="vertical-align: -1.909ex; width: 23.285ex; height: 5.056ex;" src="images/121.svg" alt=" " data-tex=" +\rho \frac{du_{\nu}}{dt} + = -\frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + \rho X_{\nu}, +"></span> +in which <img style="vertical-align: -1.577ex; width: 4.502ex; height: 4.676ex;" src="images/122.svg" alt=" " data-tex="\dfrac{du_{\nu}}{dt}"> is the acceleration of the particle which +at time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> has the co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If we express this acceleration +by partial differential coefficients, we obtain, after dividing by <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, +<span class="align-center"><img style="vertical-align: -2.041ex; width: 42.284ex; height: 5.188ex;" src="images/123.svg" alt=" " data-tex=" +\frac{\partial u_{\nu}}{dt} + \frac{\partial u_{\nu}}{dx_{\sigma}} u_{\sigma} + = -\frac{1}{\rho}\, \frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + X_{\nu}. +\qquad\text{(16)} +"></span> +</p> +<p> +We must show that this equation holds independently of the +special choice of the Cartesian system of co-ordinates. (<img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}">) is a +vector, and therefore <img style="vertical-align: -1.602ex; width: 4.606ex; height: 4.749ex;" src="images/124.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial t}"> is also a vector. +<img style="vertical-align: -1.909ex; width: 4.671ex; height: 5.056ex;" src="images/125.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial x_{\sigma}}"> +<span class="pagenum" id="Page_20">[Pg 20]</span> +is a tensor of rank 2, <img style="vertical-align: -1.909ex; width: 6.98ex; height: 5.056ex;" src="images/126.svg" alt=" " data-tex="\dfrac{\partial u_{\nu}}{\partial x_{\sigma}} u_{\tau}"> +is a tensor of rank 3. The second term on the left +results from contraction in the indices <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau">. The vector character +of the second term on the right is obvious. In order that the first +term on the right may also be a vector it is necessary for <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> to be +a tensor. Then by differentiation and contraction <img style="vertical-align: -1.909ex; width: 5.363ex; height: 5.056ex;" src="images/129.svg" alt=" " data-tex="\dfrac{\partial p_{\nu\sigma}}{\partial x_{\sigma}}"> results, +and is therefore a vector, as it also is after multiplication by +the reciprocal scalar <img style="vertical-align: -2.041ex; width: 2.165ex; height: 5.077ex;" src="images/130.svg" alt=" " data-tex="\dfrac{1}{\rho}">. That <img style="vertical-align: -0.439ex; width: 3.087ex; height: 1.439ex;" src="images/120.svg" alt=" " data-tex="p_{\nu\sigma}"> is a tensor, and therefore +transforms according to the equation +<span class="align-center"><img style="vertical-align: -0.864ex; width: 16.726ex; height: 2.695ex;" src="images/131.svg" alt=" " data-tex=" +{p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta}, +"></span> +is proved in mechanics by integrating this equation over an infinitely +small tetrahedron. It is also proved there, by application +of the theorem of moments to an infinitely small parallelopipedon, +that <img style="vertical-align: -0.439ex; width: 9.191ex; height: 1.758ex;" src="images/132.svg" alt=" " data-tex="p_{\nu\sigma} = p_{\sigma\nu}">, and hence that the tensor of the stress is +a symmetrical tensor. From what has been said it follows that, +with the aid of the rules given above, the equation is co-variant +with respect to orthogonal transformations in space (rotational +transformations); and the rules according to which the quantities +in the equation must be transformed in order that the +equation may be co-variant also become evident. +</p> +<p> +The co-variance of the equation of continuity, +<span class="align-center"><img style="vertical-align: -1.891ex; width: 27.449ex; height: 5.195ex;" src="images/133.svg" alt=" " data-tex=" +\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u_{\nu})}{\partial x_{\nu}} = 0, +\qquad\text{(17)} +"></span> +requires, from the foregoing, no particular discussion. +</p> +<p> +We shall also test for co-variance the equations which express +the dependence of the stress components upon the properties of +<span class="pagenum" id="Page_21">[Pg 21]</span> +the matter, and set up these equations for the case of a compressible +viscous fluid with the aid of the conditions of co-variance. +If we neglect the viscosity, the pressure, <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/134.svg" alt=" " data-tex="p">, will be a scalar, and +will depend only upon the density and the temperature of the +fluid. The contribution to the stress tensor is then evidently +<span class="align-center"><img style="vertical-align: -0.685ex; width: 4.143ex; height: 2.307ex;" src="images/135.svg" alt=" " data-tex=" +p \delta_{\mu\nu} +"></span> +in which <img style="vertical-align: -0.685ex; width: 3.005ex; height: 2.307ex;" src="images/136.svg" alt=" " data-tex="\delta_{\mu\nu}"> is the special symmetrical tensor. This term will +also be present in the case of a viscous fluid. But in this case +there will also be pressure terms, which depend upon the space +derivatives of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}">. We shall assume that this dependence is a +linear one. Since these terms must be symmetrical tensors, the +only ones which enter will be +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.199ex; height: 5.58ex;" src="images/137.svg" alt=" " data-tex=" +\alpha\left(\frac{\partial u_{\mu}}{\partial x_{\nu}} + + \frac{\partial u_{\nu}}{\partial x_{\mu}}\right) + + \beta\delta_{\mu\nu} \frac{\partial u_{\alpha}}{\partial x_{\alpha}} +"></span> +(for <img style="vertical-align: -1.909ex; width: 4.782ex; height: 5.056ex;" src="images/138.svg" alt=" " data-tex="\dfrac{\partial u_{\alpha}}{\partial x_{\alpha}}"> is a scalar). For +physical reasons (no slipping) it +is assumed that for symmetrical dilatations in all directions, +i.e. when +<span class="align-center"><img style="vertical-align: -1.927ex; width: 38.03ex; height: 5.074ex;" src="images/139.svg" alt=" " data-tex=" +\frac{\partial u_{1}}{\partial x_{1}} = +\frac{\partial u_{2}}{\partial x_{2}} = +\frac{\partial u_{3}}{\partial x_{3}};\quad +\frac{\partial u_{1}}{\partial x_{2}}, \text{ etc.,} = 0, +"></span> +there are no frictional forces present, from which it follows that +<img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> = <img style="vertical-align: -1.602ex; width: 5.335ex; height: 4.638ex;" src="images/140.svg" alt=" " data-tex="-\dfrac{2}{3}\alpha">. If only <img style="vertical-align: -1.927ex; width: 4.558ex; height: 5.074ex;" src="images/141.svg" alt=" " data-tex="\dfrac{\partial u_{1}}{\partial x_{3}}"> +is different from zero, let <img style="vertical-align: -1.927ex; width: 13.709ex; height: 5.074ex;" src="images/142.svg" alt=" " data-tex="p_{31} = -\alpha \dfrac{\partial u_{1}}{\partial x_{3}}">, +by which <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> is determined. We then obtain for the complete +stress tensor, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 72.428ex; height: 5.58ex;" src="images/143.svg" alt=" " data-tex=" +p_{\mu\nu} = p \delta_{\mu\nu} - \alpha \biggl[ + \biggl(\frac{\partial u_{\mu}}{\partial x_{\nu}} + + \frac{\partial u_{\nu}}{\partial x_{\mu}}\biggr) + - \frac{2}{3} \biggl(\frac{\partial u_{1}}{\partial x_{1}} + + \frac{\partial u_{2}}{\partial x_{2}} + + \frac{\partial u_{3}}{\partial x_{3}}\biggr)\delta_{\mu\nu} +\biggr]. +\qquad\text{(18)} +"></span> +<span class="pagenum" id="Page_22">[Pg 22]</span> +</p> +<p> +The heuristic value of the theory of invariants, which arises +from the isotropy of space (equivalence of all directions), becomes +evident from this example. +</p> +<p> +We consider, finally, Maxwell's equations in the form which +are the foundation of the electron theory of Lorentz. +</p> +<span class="align-center"><img style="vertical-align: -22.069ex; width: 44.49ex; height: 45.27ex;" src="images/144.svg" alt=" " data-tex=" +\begin{align*} +\left. + \begin{aligned} + &\begin{alignedat}{3} + \frac{\partial h_{3}}{\partial x_{2}} &- + \frac{\partial h_{2}}{\partial x_{3}} + &&= \frac{1}{c}\, \frac{\partial e_{1}}{\partial t} + &&+ \frac{1}{c}\, i_{1}, +\qquad \\ + \frac{\partial h_{1}}{\partial x_{3}} &- + \frac{\partial h_{3}}{\partial x_{1}} + &&= \frac{1}{c}\, \frac{\partial e_{2}}{\partial t} + &&+ \frac{1}{c}\, i_{2}, \\ + \frac{\partial h_{2}}{\partial x_{1}} &- + \frac{\partial h_{1}}{\partial x_{2}} + &&= \frac{1}{c}\, \frac{\partial e_{3}}{\partial t} + &&+ \frac{1}{c}\, i_{3}, + \end{alignedat} \\ + & \frac{\partial e_{1}}{\partial x_{1}} + + \frac{\partial e_{2}}{\partial x_{2}} + + \frac{\partial e_{3}}{\partial x_{3}} = \rho; + \end{aligned} + \right\} +\qquad\text{(19)}\\ +\left. + \begin{aligned} + &\begin{alignedat}{2} + \frac{\partial e_{3}}{\partial x_{2}} &- \frac{\partial e_{2}}{\partial x_{3}} + &&= -\frac{1}{c}\, \frac{\partial h_{1}}{\partial t}, +\qquad \\ + \frac{\partial e_{1}}{\partial x_{3}} &- \frac{\partial e_{3}}{\partial x_{1}} + &&= -\frac{1}{c}\, \frac{\partial h_{2}}{\partial t}, \\ + \frac{\partial e_{2}}{\partial x_{1}} &- \frac{\partial e_{1}}{\partial x_{2}} + &&= -\frac{1}{c}\, \frac{\partial h_{3}}{\partial t}, + \end{alignedat} \\ + & \frac{\partial h_{1}}{\partial x_{1}} + + \frac{\partial h_{2}}{\partial x_{2}} + + \frac{\partial h_{3}}{\partial x_{3}} = 0. + \end{aligned} +\right\} +\qquad \text{(20)} +\end{align*} +"></span> +<p> +<img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/145.svg" alt=" " data-tex="\mathbf i"> is a vector, because the current density is defined as the +density of electricity multiplied by the vector velocity of the +electricity. According to the first three equations it is evident +that <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e"> is also to be regarded as a vector. Then <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h"> cannot be +regarded as a vector.<a id="FNanchor_6_1"></a><a href="#Footnote_6_1" class="fnanchor">[6]</a> The equations may, however, easily be +<span class="pagenum" id="Page_23">[Pg 23]</span> +interpreted if <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h"> is regarded as a skew-symmetrical tensor of the +second rank. In this sense, we write <img style="vertical-align: -0.375ex; width: 3.091ex; height: 1.945ex;" src="images/148.svg" alt=" " data-tex="h_{23}">, <img style="vertical-align: -0.375ex; width: 3.091ex; height: 1.945ex;" src="images/149.svg" alt=" " data-tex="h_{31}">, <img style="vertical-align: -0.339ex; width: 3.091ex; height: 1.91ex;" src="images/150.svg" alt=" " data-tex="h_{12}"> in place of +<img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.91ex;" src="images/151.svg" alt=" " data-tex="h_{1}">, <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.91ex;" src="images/152.svg" alt=" " data-tex="h_{2}">, <img style="vertical-align: -0.375ex; width: 2.291ex; height: 1.945ex;" src="images/153.svg" alt=" " data-tex="h_{3}"> respectively. Paying attention to the skew-symmetry +of <img style="vertical-align: -0.685ex; width: 3.304ex; height: 2.255ex;" src="images/154.svg" alt=" " data-tex="h_{\mu\nu}">, the first three equations of (19) and (20) may be written +in the form +<span class="align-center"><img style="vertical-align: -5.181ex; width: 39.788ex; height: 11.494ex;" src="images/155.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial h_{\mu\nu}}{\partial x_{\nu}} + = \frac{1}{c}\, \frac{\partial e_{\mu}}{\partial t} + \frac{1}{c} i_{\mu}, +\qquad & &\text{(19a)} \\ +\frac{\partial e_{\mu}}{\partial x_{\nu}} - \frac{\partial e_{\nu}}{\partial x_{\mu}} + = +\frac{1}{c}\, \frac{\partial h_{\mu\nu}}{\partial t}. +\qquad & &\text{(20a)} +\end{gather*} +"></span> +In contrast to <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e">, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h"> appears as a quantity which has the same type +of symmetry as an angular velocity. The divergence equations +then take the form +<span class="align-center"><img style="vertical-align: -5.083ex; width: 41.219ex; height: 11.297ex;" src="images/156.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial e_{\nu}}{\partial x_{\nu}} = \rho\, +\qquad & &\text{(19b)} \\ +\frac{\partial h_{\mu\nu}}{\partial x_{\rho}} + +\frac{\partial h_{\nu\rho}}{\partial x_{\mu}} + +\frac{\partial h_{\rho\mu}}{\partial x_{\nu}} = 0. +\qquad & &\text{(20b)} +\end{gather*} +"></span> +The last equation is a skew-symmetrical tensor equation of the +third rank (the skew-symmetry of the left-hand side with respect +to every pair of indices may easily be proved, if attention +is paid to the skew-symmetry of <img style="vertical-align: -0.685ex; width: 3.304ex; height: 2.255ex;" src="images/154.svg" alt=" " data-tex="h_{\mu\nu}">). This notation is more +natural than the usual one, because, in contrast to the latter, +it is applicable to Cartesian left-handed systems as well as to +right-handed systems without change of sign. +<span class="pagenum" id="Page_24">[Pg 24]</span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_6_1"></a><a href="#FNanchor_6_1"><span class="label">[6]</span></a>These considerations will make the reader familiar with tensor operations +without the special difficulties of the four-dimensional treatment; +corresponding considerations in the theory of special relativity (Minkowski's +interpretation of the field) will then offer fewer difficulties.</p></div> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title="LECTURE II: THE THEORY OF SPECIAL RELATIVITY"><a id="chap02"></a>LECTURE II +<br><br> +THE THEORY OF SPECIAL RELATIVITY</h2> + +<p class="nind"> +<span class="dropcap">T</span>HE previous considerations concerning the configuration of +rigid bodies have been founded, irrespective of the assumption +as to the validity of the Euclidean geometry, upon the hypothesis +that all directions in space, or all configurations of Cartesian systems +of co-ordinates, are physically equivalent. We may express +this as the "principle of relativity with respect to direction," and +it has been shown how equations (laws of nature) may be found, +in accord with this principle, by the aid of the calculus of tensors. +We now inquire whether there is a relativity with respect +to the state of motion of the space of reference; in other words, +whether there are spaces of reference in motion relatively to each +other which are physically equivalent. From the standpoint of +mechanics it appears that equivalent spaces of reference do exist. +For experiments upon the earth tell us nothing of the fact +that we are moving about the sun with a velocity of approximately +30 kilometres a second. On the other hand, this physical +equivalence does not seem to hold for spaces of reference in arbitrary +motion; for mechanical effects do not seem to be subject +to the same laws in a jolting railway train as in one moving with +uniform velocity; the rotation of the earth must be considered +in writing down the equations of motion relatively to the earth. +It appears, therefore, as if there were Cartesian systems of co-ordinates, +the so-called inertial systems, with reference to which +the laws of mechanics (more generally the laws of physics) are +expressed in the simplest form. We may infer the validity of +the following theorem: If <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is an inertial system, then every +<span class="pagenum" id="Page_25">[Pg 25]</span> +other system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which moves uniformly and without rotation +relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, is also an inertial system; the laws of nature are +in concordance for all inertial systems. This statement we shall +call the "principle of special relativity." We shall draw certain +conclusions from this principle of "relativity of translation" just +as we have already done for relativity of direction. +</p> +<p> +In order to be able to do this, we must first solve the following +problem. If we are given the Cartesian co-ordinates, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, and +the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, of an event relatively to one inertial system, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +how can we calculate the co-ordinates, <img style="vertical-align: -0.559ex; width: 2.33ex; height: 2.276ex;" src="images/19.svg" alt=" " data-tex="x'_{\nu}">, and the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">', of +the same event relatively to an inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which moves +with uniform translation relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">? In the pre-relativity +physics this problem was solved by making unconsciously two +hypotheses:— +</p> +<p> +1. The time is absolute; the time of an event, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">', relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is the same as the time relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. If instantaneous +signals could be sent to a distance, and if one knew that the +state of motion of a clock had no influence on its rate, then this +assumption would be physically established. For then clocks, +similar to one another, and regulated alike, could be distributed +over the systems <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', at rest relatively to them, and their +indications would be independent of the state of motion of the +systems; the time of an event would then be given by the clock +in its immediate neighbourhood. +</p> +<p> +2. Length is absolute; if an interval, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +has a length <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, then it has the same length <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s"> relatively to a +system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' which is in motion relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. +</p> +<p> +If the axes of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' are parallel to each other, a simple +calculation based on these two assumptions, gives the equations +<span class="pagenum" id="Page_26">[Pg 26]</span> +of transformation +<span class="align-center"><img style="vertical-align: -2.17ex; width: 29.765ex; height: 5.47ex;" src="images/157.svg" alt=" " data-tex=" +\left. +\begin{aligned} +{x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\text{,} \\ +t' &= t - b\text{.} +\end{aligned} +\right\} +\qquad \text{(21)} +"></span> +</p> +<p> +This transformation is known as the "Galilean Transformation." +Differentiating twice by the time, we get +<span class="align-center"><img style="vertical-align: -1.654ex; width: 15.253ex; height: 5.07ex;" src="images/158.svg" alt=" " data-tex=" +\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}. +"></span> +Further, it follows that for two simultaneous events, +<span class="align-center"><img style="vertical-align: -0.339ex; width: 28.681ex; height: 2.752ex;" src="images/159.svg" alt=" " data-tex=" +{{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}. +"></span> +The invariance of the distance between the two points results +from squaring and adding. From this easily follows the co-variance +of Newton's equations of motion with respect to the +Galilean transformation (21). Hence it follows that classical +mechanics is in accord with the principle of special relativity if +the two hypotheses respecting scales and clocks are made. +</p> +<p> +But this attempt to found relativity of translation upon the +Galilean transformation fails when applied to electromagnetic +phenomena. The Maxwell-Lorentz electromagnetic equations +are not co-variant with respect to the Galilean transformation. +In particular, we note, by (21), that a ray of light which referred +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> has a velocity <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/160.svg" alt=" " data-tex="c">, has a different velocity referred to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +depending upon its direction. The space of reference of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is +therefore distinguished, with respect to its physical properties, +from all spaces of reference which are in motion relatively to it +(quiescent æther). But all experiments have shown that electromagnetic +and optical phenomena, relatively to the earth as the +<span class="pagenum" id="Page_27">[Pg 27]</span> +body of reference, are not influenced by the translational velocity +of the earth. The most important of these experiments are +those of Michelson and Morley, which I shall assume are known. +The validity of the principle of special relativity can therefore +hardly be doubted. +</p> +<p> +On the other hand, the Maxwell-Lorentz equations have +proved their validity in the treatment of optical problems in +moving bodies. No other theory has satisfactorily explained the +facts of aberration, the propagation of light in moving bodies +(Fizeau), and phenomena observed in double stars (De Sitter). +The consequence of the Maxwell-Lorentz equations that in a +vacuum light is propagated with the velocity <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/160.svg" alt=" " data-tex="c">, at least with respect +to a definite inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, must therefore be regarded +as proved. According to the principle of special relativity, we +must also assume the truth of this principle for every other +inertial system. +</p> +<p> +Before we draw any conclusions from these two principles +we must first review the physical significance of the concepts +"time" and "velocity." It follows from what has gone before, that +co-ordinates with respect to an inertial system are physically +defined by means of measurements and constructions with the +aid of rigid bodies. In order to measure time, we have supposed +a clock, <img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/161.svg" alt=" " data-tex="U">, present somewhere, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. But +we cannot fix the time, by means of this clock, of an event +whose distance from the clock is not negligible; for there are no +"instantaneous signals" that we can use in order to compare the +time of the event with that of the clock. In order to complete the +definition of time we may employ the principle of the constancy +of the velocity of fight in a vacuum. Let us suppose that we +place similar clocks at points of the system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, at rest relatively +<span class="pagenum" id="Page_28">[Pg 28]</span> +to it, and regulated according to the following scheme. A ray +of light is sent out from one of the clocks, <img style="vertical-align: -0.357ex; width: 3.138ex; height: 1.902ex;" src="images/162.svg" alt=" " data-tex="U_{m}">, at the instant +when it indicates the time <img style="vertical-align: -0.357ex; width: 2.409ex; height: 1.773ex;" src="images/163.svg" alt=" " data-tex="t_{m}">, and travels through a vacuum a +distance <img style="vertical-align: -0.357ex; width: 3.573ex; height: 1.357ex;" src="images/164.svg" alt=" " data-tex="r_{mn}">, to the clock <img style="vertical-align: -0.357ex; width: 2.693ex; height: 1.902ex;" src="images/165.svg" alt=" " data-tex="U_{n}">; at the instant when this ray meets +the clock <img style="vertical-align: -0.357ex; width: 2.693ex; height: 1.902ex;" src="images/165.svg" alt=" " data-tex="U_{n}"> the latter is set to indicate the time +<img style="vertical-align: -0.357ex; width: 1.964ex; height: 1.773ex;" src="images/166.svg" alt=" " data-tex="t_{n}"> = <img style="vertical-align: -0.357ex; width: 2.409ex; height: 1.773ex;" src="images/163.svg" alt=" " data-tex="t_{m}"> + <img style="vertical-align: -1.577ex; width: 4.568ex; height: 4.106ex;" src="images/167.svg" alt=" " data-tex="\dfrac{r_{mn}}{c}">.<a id="FNanchor_7_1"></a><a href="#Footnote_7_1" class="fnanchor">[7]</a> +The principle of the constancy of the velocity of light then states +that this adjustment of the clocks will not lead to contradictions. +With clocks so adjusted, we can assign the time to events which +take place near any one of them. It is essential to note that this +definition of time relates only to the inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, since +we have used a system of clocks at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. The +assumption which was made in the pre-relativity physics of the +absolute character of time (i.e. the independence of time of the +choice of the inertial system) does not follow at all from this +definition. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_7_1"></a><a href="#FNanchor_7_1"><span class="label">[7]</span></a>Strictly speaking, it would be more correct to define simultaneity first, +somewhat as follows: two events taking place at the points <img style="vertical-align: 0; width: 1.697ex; height: 1.62ex;" src="images/3.svg" alt=" " data-tex="A"> and <img style="vertical-align: 0; width: 1.717ex; height: 1.545ex;" src="images/1.svg" alt=" " data-tex="B"> of +the system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> are simultaneous if they appear at the same instant when +observed from the middle point, <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M">, of the interval <img style="vertical-align: 0; width: 3.414ex; height: 1.62ex;" src="images/169.svg" alt=" " data-tex="AB">. Time is then +defined as the ensemble of the indications of similar clocks, at rest relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, which register the same simultaneously.</p></div> + +<p> +The theory of relativity is often criticized for giving, without +justification, a central theoretical role to the propagation +of light, in that it founds the concept of time upon the law of +propagation of light. The situation, however, is somewhat as +follows. In order to give physical significance to the concept of +time, processes of some kind are required which enable relations +to be established between different places. It is immaterial what +kind of processes one chooses for such a definition of time. It +is advantageous, however, for the theory, to choose only those +<span class="pagenum" id="Page_29">[Pg 29]</span> +processes concerning which we know something certain. This +holds for the propagation of light <i>in vacuo</i> in a higher degree +than for any other process which could be considered, thanks to +the investigations of Maxwell and H. A. Lorentz. +</p> +<p> +From all of these considerations, space and time data have +a physically real, and not a mere fictitious, significance; in particular +this holds for all the relations in which co-ordinates and +time enter, e.g. the relations (21). There is, therefore, sense in +asking whether those equations are true or not, as well as in +asking what the true equations of transformation are by which +we pass from one inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> to another, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', moving relatively +to it. It may be shown that this is uniquely settled by +means of the principle of the constancy of the velocity of light +and the principle of special relativity. +</p> +<p> +To this end we think of space and time physically defined +with respect to two inertial systems, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in the way that +has been shown. Further, let a ray of light pass from one point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> +to another point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> through a vacuum. If <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r"> is the measured +distance between the two points, then the propagation of light +must satisfy the equation +<span class="align-center"><img style="vertical-align: -0.186ex; width: 9.982ex; height: 1.805ex;" src="images/173.svg" alt=" " data-tex=" +r = c·\Delta t. +"></span> +</p> +<p> +If we square this equation, and express <img style="vertical-align: -0.025ex; width: 2.008ex; height: 1.912ex;" src="images/174.svg" alt=" " data-tex="r^{2}"> by the differences +of the co-ordinates, <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">, in place of this equation we can write +<span class="align-center"><img style="vertical-align: -1.018ex; width: 31.976ex; height: 3.167ex;" src="images/175.svg" alt=" " data-tex=" +\sum (\Delta x_{\nu})^{2} - c^{2} \Delta t^{2} = 0. \qquad\text{(22)} +"></span> +This equation formulates the principle of the constancy of the +velocity of light relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. It must hold whatever may be +the motion of the source which emits the ray of light. +<span class="pagenum" id="Page_30">[Pg 30]</span> +</p> +<p> +The same propagation of light may also be considered relatively +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in which case also the principle of the constancy of +the velocity of light must be satisfied. Therefore, with respect +to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', we have the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 34.175ex; height: 3.167ex;" src="images/176.svg" alt=" " data-tex=" +\sum ({\Delta x'}_{\nu})^{2} - c^{2} \Delta t'^{2} = 0. +\qquad\text{(22a)} +"></span> +</p> +<p> +Equations (22a) and (22) must be mutually consistent with +each other with respect to the transformation which transforms +from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. A transformation which effects this we shall call +a "Lorentz transformation." +</p> +<p> +Before considering these transformations in detail we shall +make a few general remarks about space and time. In the pre-relativity +physics space and time were separate entities. Specifications +of time were independent of the choice of the space of +reference. The Newtonian mechanics was relative with respect +to the space of reference, so that, e.g. the statement that two +non-simultaneous events happened at the same place had no objective +meaning (that is, independent of the space of reference). +But this relativity had no role in building up the theory. One +spoke of points of space, as of instants of time, as if they were +absolute realities. It was not observed that the true element +of the space-time specification was the event, specified by the +four numbers <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">. The conception of something +happening was always that of a four-dimensional continuum; but +the recognition of this was obscured by the absolute character +of the pre-relativity time. Upon giving up the hypothesis of the +absolute character of time, particularly that of simultaneity, the +four-dimensionality of the time-space concept was immediately +recognized. It is neither the point in space, nor the instant in +<span class="pagenum" id="Page_31">[Pg 31]</span> +time, at which something happens that has physical reality, but +only the event itself. There is no absolute (independent of the +space of reference) relation in space, and no absolute relation +in time between two events, but there is an absolute (independent +of the space of reference) relation in space and time, as +will appear in the sequel. The circumstance that there is no +objective rational division of the four-dimensional continuum +into a three-dimensional space and a one-dimensional time continuum +indicates that the laws of nature will assume a form +which is logically most satisfactory when expressed as laws in +the four-dimensional space-time continuum. Upon this depends +the great advance in method which the theory of relativity owes +to Minkowski. Considered from this standpoint, we must regard +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> as the four co-ordinates of an event in the +four-dimensional continuum. We have far less success in picturing +to ourselves relations in this four-dimensional continuum than +in the three-dimensional Euclidean continuum; but it must be +emphasized that even in the Euclidean three-dimensional geometry +its concepts and relations are only of an abstract nature in +our minds, and are not at all identical with the images we form +visually and through our sense of touch. The non-divisibility of +the four-dimensional continuum of events does not at all, however, +involve the equivalence of the space co-ordinates with the +time co-ordinate. On the contrary, we must remember that the +time co-ordinate is defined physically wholly differently from the +space co-ordinates. The relations (22) and (22a) which when +equated define the Lorentz transformation show, further, a difference +in the role of the time co-ordinate from that of the space +co-ordinates; for the term <img style="vertical-align: -0.025ex; width: 3.689ex; height: 1.912ex;" src="images/177.svg" alt=" " data-tex="\Delta t^{2}"> has the opposite sign to the space +terms, <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/178.svg" alt=" " data-tex="\Delta x_{1}^{2}">, <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/179.svg" alt=" " data-tex="\Delta x_{2}^{2}">, <img style="vertical-align: -0.685ex; width: 4.166ex; height: 2.572ex;" src="images/180.svg" alt=" " data-tex="\Delta x_{3}^{2}">. +<span class="pagenum" id="Page_32">[Pg 32]</span> +</p> +<p> +Before we analyse further the conditions which define the +Lorentz transformation, we shall introduce the light-time, <img style="vertical-align: -0.186ex; width: 5.488ex; height: 1.756ex;" src="images/181.svg" alt=" " data-tex="l = ct">, +in place of the time, <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, in order that the constant <img style="vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;" src="images/160.svg" alt=" " data-tex="c"> shall not +enter explicitly into the formulas to be developed later. Then +the Lorentz transformation is defined in such a way that, first, +it makes the equation +<span class="align-center"><img style="vertical-align: -0.566ex; width: 41.26ex; height: 2.633ex;" src="images/182.svg" alt=" " data-tex=" +{\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 +\qquad\text{(22b)} +"></span> +a co-variant equation, that is, an equation which is satisfied with +respect to every inertial system if it is satisfied in the inertial +system to which we refer the two given events (emission and +reception of the ray of light). Finally, with Minkowski, we introduce +in place of the real time co-ordinate <img style="vertical-align: -0.186ex; width: 5.488ex; height: 1.756ex;" src="images/181.svg" alt=" " data-tex="l = ct">, the imaginary +time co-ordinate +<span class="align-center"><img style="vertical-align: -0.566ex; width: 25.618ex; height: 2.754ex;" src="images/183.svg" alt=" " data-tex=" +x_{4} = il = ict\quad (\sqrt{-1} = i). +"></span> +Then the equation defining the propagation of light, which must +be co-variant with respect to the Lorentz transformation, becomes +<span class="align-center"><img style="vertical-align: -3.222ex; width: 55.106ex; height: 5.371ex;" src="images/184.svg" alt=" " data-tex=" +\sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} + = 0. +\qquad\text{(22c)} +"></span> +This condition is always satisfied<a id="FNanchor_8_1"></a><a href="#Footnote_8_1" class="fnanchor">[8]</a> if we satisfy the more general +condition that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 42.527ex; height: 2.633ex;" src="images/185.svg" alt=" " data-tex=" +s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} +\qquad\text{(23)} +"></span> +<span class="pagenum" id="Page_33">[Pg 33]</span> +shall be an invariant with respect to the transformation. This +condition is satisfied only by linear transformations, that is, +transformations of the type +<span class="align-center"><img style="vertical-align: -0.685ex; width: 25.407ex; height: 2.515ex;" src="images/186.svg" alt=" " data-tex=" +{x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} +\qquad\text{(24)} +"></span> +in which the summation over the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> is to be extended from <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = 1 +to <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> = 4. A glance at equations (23) and (24) shows that the +Lorentz transformation so defined is identical with the translational +and rotational transformations of the Euclidean geometry, +if we disregard the number of dimensions and the relations of reality. +We can also conclude that the coefficients <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"> must satisfy +the conditions +<span class="align-center"><img style="vertical-align: -0.685ex; width: 30.947ex; height: 2.382ex;" src="images/187.svg" alt=" " data-tex=" +b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}. +\qquad\text{(25)} +"></span> +Since the ratios of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> are real, it follows that all the <img style="vertical-align: -0.685ex; width: 2.349ex; height: 1.683ex;" src="images/188.svg" alt=" " data-tex="a_{\mu}"> and +the <img style="vertical-align: -0.685ex; width: 3.147ex; height: 2.255ex;" src="images/90.svg" alt=" " data-tex="b_{\mu\alpha}"> are real, except <img style="vertical-align: -0.339ex; width: 2.185ex; height: 1.337ex;" src="images/189.svg" alt=" " data-tex="a_{4}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/190.svg" alt=" " data-tex="b_{41}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/191.svg" alt=" " data-tex="b_{42}">, <img style="vertical-align: -0.375ex; width: 2.758ex; height: 1.945ex;" src="images/192.svg" alt=" " data-tex="b_{43}">, +<img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/193.svg" alt=" " data-tex="b_{14}">, <img style="vertical-align: -0.339ex; width: 2.758ex; height: 1.91ex;" src="images/194.svg" alt=" " data-tex="b_{24}"> and <img style="vertical-align: -0.375ex; width: 2.758ex; height: 1.945ex;" src="images/195.svg" alt=" " data-tex="b_{34}">, which +are purely imaginary. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_8_1"></a><a href="#FNanchor_8_1"><span class="label">[8]</span></a>That this specialization lies in the nature of the case will be evident +later.</p></div> + +<p><br></p> + +<p> +<i>Special Lorentz Transformation</i>. We obtain the simplest +transformations of the type of (24) and (25) if only two of the +co-ordinates are to be transformed, and if all the <img style="vertical-align: -0.685ex; width: 2.349ex; height: 1.683ex;" src="images/188.svg" alt=" " data-tex="a_{\mu}">, which determine +the new origin, vanish. We obtain then for the indices +1 and 2, on account of the three independent conditions which +the relations (25) furnish, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 34.703ex; height: 11.62ex;" src="images/196.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi, \\ +{x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi, \\ +{x'}_{3} &= x_{3}, && \\ +{x'}_{4} &= x_{4}. && +\end{alignedat} +\right\} +\qquad\text{(26)} +"></span> +<span class="pagenum" id="Page_34">[Pg 34]</span> +</p> +<p> +This is a simple rotation in space of the (space) co-ordinate +system about <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">-axis. We see that the rotational transformation +in space (without the time transformation) which we studied +before is contained in the Lorentz transformation as a special +case. For the indices 1 and 4 we obtain, in an analogous manner, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 36.083ex; height: 11.62ex;" src="images/197.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi, \\ +{x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi, \\ +{x'}_{2} &= x_{2}, && \\ +{x'}_{3} &= x_{3}. && +\end{alignedat} +\right\} +\qquad\text{(26a)} +"></span> +</p> +<p> +On account of the relations of reality <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/198.svg" alt=" " data-tex="\psi"> must be taken as +imaginary. To interpret these equations physically, we introduce +the real light-time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> and the velocity <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +instead of the imaginary angle <img style="vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;" src="images/198.svg" alt=" " data-tex="\psi">. We have, first, +<span class="align-center"><img style="vertical-align: -2.17ex; width: 26.107ex; height: 5.47ex;" src="images/201.svg" alt=" " data-tex=" +\begin{alignat*}{4} +{x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi, \\ +l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. +\end{alignat*} +"></span> +Since for the origin of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' i.e., for <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}"> = 0, we must have <img style="vertical-align: -0.339ex; width: 7.07ex; height: 1.91ex;" src="images/202.svg" alt=" " data-tex="x_{1} = vl">, +it follows from the first of these equations that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 19.945ex; height: 2.262ex;" src="images/203.svg" alt=" " data-tex=" +v = i\tan\psi, +\qquad\text{(27)} +"></span> +and also +<span class="align-center"><img style="vertical-align: -5.112ex; width: 27.989ex; height: 11.355ex;" src="images/204.svg" alt=" " data-tex=" +\begin{aligned} +\left. +\begin{aligned} +\sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}, \\ +\cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}, +\end{aligned} +\right\} +\qquad\text{(28)} +\end{aligned} +"></span> +<span class="pagenum" id="Page_35">[Pg 35]</span> +so that we obtain +<span class="align-center"><img style="vertical-align: -8.256ex; width: 26.022ex; height: 17.642ex;" src="images/205.svg" alt=" " data-tex=" +\left. +\begin{aligned} +{x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}, \\ +l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}, \\ +{x'}_{2} &= x_{2}, \\ +{x'}_{3} &= x_{3}. +\end{aligned} +\right\} +\qquad\text{(29)} +"></span> +</p> +<p> +These equations form the well-known special Lorentz transformation, +which in the general theory represents a rotation, +through an imaginary angle, of the four-dimensional system of +co-ordinates. If we introduce the ordinary time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t">, in place of the +light-time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">, then in (29) we must replace <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> by <img style="vertical-align: -0.025ex; width: 1.796ex; height: 1.441ex;" src="images/206.svg" alt=" " data-tex="ct"> and <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v"> by +<img style="vertical-align: -1.577ex; width: 2.093ex; height: 4.109ex;" src="images/207.svg" alt=" " data-tex="\dfrac{v}{c}">. +</p> +<p> +We must now fill in a gap. From the principle of the constancy +of the velocity of light it follows that the equation +<span class="align-center"><img style="vertical-align: -1.018ex; width: 12.994ex; height: 3.167ex;" src="images/208.svg" alt=" " data-tex=" +\sum {\Delta x_{\nu}}^{2} = 0 +"></span> +has a significance which is independent of the choice of the inertial +system; but the invariance of the quantity <img style="vertical-align: -0.559ex; width: 4.214ex; height: 2.446ex;" src="images/209.svg" alt=" " data-tex="\Delta x_{\nu}^{2}"> does +not at all follow from this. This quantity might be transformed +with a factor. This depends upon the fact that the right-hand +side of (29) might be multiplied by a factor <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda">, independent of <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v">. +But the principle of relativity does not permit this factor to be +different from 1, as we shall now show. Let us assume that we +have a rigid circular cylinder moving in the direction of its axis. +If its radius, measured at rest with a unit measuring rod is equal +to <img style="vertical-align: -0.375ex; width: 2.705ex; height: 1.92ex;" src="images/210.svg" alt=" " data-tex="R_{0}">, its radius <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> in motion, might be different from <img style="vertical-align: -0.375ex; width: 2.705ex; height: 1.92ex;" src="images/210.svg" alt=" " data-tex="R_{0}">, since +the theory of relativity does not make the assumption that the +shape of bodies with respect to a space of reference is independent +of their motion relatively to this space of reference. But +<span class="pagenum" id="Page_36">[Pg 36]</span> +all directions in space must be equivalent to each other. <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> may +therefore depend upon the magnitude <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q"> of the velocity, but not +upon its direction; <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> must therefore be an even function of <img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q">. If +the cylinder is at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' the equation of its lateral +surface is +<span class="align-center"><img style="vertical-align: -0.464ex; width: 15.362ex; height: 2.464ex;" src="images/213.svg" alt=" " data-tex=" +x'^{2} + y'^{2} = {R_{0}}^{2}. +"></span> +If we write the last two equations of (29) more generally +<span class="align-center"><img style="vertical-align: -2.17ex; width: 10.156ex; height: 5.47ex;" src="images/214.svg" alt=" " data-tex=" +\begin{align*} +{x'}_{2} = \lambda x_{2}, \\ +{x'}_{3} = \lambda x_{3}, +\end{align*} +"></span> +then the lateral surface of the cylinder referred to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> satisfies the +equation +<span class="align-center"><img style="vertical-align: -1.656ex; width: 15.478ex; height: 5.179ex;" src="images/215.svg" alt=" " data-tex=" +x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}. +"></span> +The factor <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> therefore measures the lateral contraction of the +cylinder, and can thus, from the above, be only an even function +of <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v">. +</p> +<p> +If we introduce a third system of co-ordinates, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"><code>"</code>, which +moves relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' with velocity <img style="vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;" src="images/200.svg" alt=" " data-tex="v"> in the direction of the +negative <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/216.svg" alt=" " data-tex="x">-axis of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, we obtain, by applying (29) twice, +<span class="align-center"><img style="vertical-align: -5.244ex; width: 19.39ex; height: 11.62ex;" src="images/217.svg" alt=" " data-tex=" +\begin{align*} +{x''}_{1} &= \lambda(v) \lambda(-v) x_{1}, \\ +{x''}_{2} &= \lambda(v) \lambda(-v) x_{2}, \\ +{x''}_{3} &= \lambda(v) \lambda(-v) x_{3}, \\ +l'' &= \lambda(v) \lambda(-v) l. +\end{align*} +"></span> +Now, since <img style="vertical-align: -0.566ex; width: 4.176ex; height: 2.262ex;" src="images/218.svg" alt=" " data-tex="\lambda(v)"> must be equal to <img style="vertical-align: -0.566ex; width: 5.937ex; height: 2.262ex;" src="images/219.svg" alt=" " data-tex="\lambda(-v)"> and since we assume +that we use the same measuring rods in all the systems, it follows +that the transformation of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"><code>"</code> to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> must be the identical +<span class="pagenum" id="Page_37">[Pg 37]</span> +transformation (since the possibility <img style="vertical-align: -0.186ex; width: 7.228ex; height: 1.756ex;" src="images/220.svg" alt=" " data-tex="\lambda = -1"> does not need to +be considered). It is essential for these considerations to assume +that the behaviour of the measuring rods does not depend upon +the history of their previous motion. +</p> + +<p><br></p> + +<p> +<i>Moving Measuring Rods and Clocks</i>. At the definite <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">-time, +<img style="vertical-align: -0.186ex; width: 4.823ex; height: 1.756ex;" src="images/221.svg" alt=" " data-tex="l = 0">, the position of the points given by the integers +<img style="vertical-align: -0.583ex; width: 6.656ex; height: 2.3ex;" src="images/222.svg" alt=" " data-tex="x'_{1} = n">, is with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, given by +<img style="vertical-align: -0.339ex; width: 14.568ex; height: 2.526ex;" src="images/223.svg" alt=" " data-tex="x_{1} = n {\sqrt{1 - v^{2}}}">; this follows +from the first of equations (29) and expresses the Lorentz +contraction. A clock at rest at the origin <img style="vertical-align: -0.339ex; width: 6.43ex; height: 1.846ex;" src="images/224.svg" alt=" " data-tex="x_{1} = 0"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, whose +beats are characterized by <img style="vertical-align: -0.186ex; width: 5.049ex; height: 1.756ex;" src="images/225.svg" alt=" " data-tex="l = n">, will, when observed from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +have beats characterized by +<span class="align-center"><img style="vertical-align: -2.308ex; width: 13.855ex; height: 4.837ex;" src="images/226.svg" alt=" " data-tex=" +l' = \frac{n}{\sqrt{1 - v^{2}}}; +"></span> +this follows from the second of equations (29) and shows that +the clock goes slower than if it were at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. +These two consequences, which hold, <i>mutatis mutandis</i>, for every +system of reference, form the physical content, free from +convention, of the Lorentz transformation. +</p> + +<p><br></p> + +<p> +<i>Addition Theorem for Velocities</i>. If we combine two special +Lorentz transformations with the relative velocities <img style="vertical-align: -0.339ex; width: 2.085ex; height: 1.342ex;" src="images/227.svg" alt=" " data-tex="v_{1}"> and <img style="vertical-align: -0.339ex; width: 2.085ex; height: 1.342ex;" src="images/228.svg" alt=" " data-tex="v_{2}">, +then the velocity of the single Lorentz transformation which +takes the place of the two separate ones is, according to (27), +given by +<span class="align-center"><img style="vertical-align: -2.016ex; width: 62.69ex; height: 5.115ex;" src="images/229.svg" alt=" " data-tex=" +v_{12} = i \tan (\psi_{1} + \psi_{2}) + = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} + = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. +\qquad\text{(30)} +"></span> +<span class="pagenum" id="Page_38">[Pg 38]</span> +</p> + +<p><br></p> + +<p> +<i>General Statements about the Lorentz Transformation and +its Theory of Invariants</i>. The whole theory of invariants of the +special theory of relativity depends upon the invariant <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> (23). +Formally, it has the same rôle in the four-dimensional space-time +continuum as the invariant +<img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/178.svg" alt=" " data-tex="\Delta x_{1}^{2}"> + <img style="vertical-align: -0.651ex; width: 4.166ex; height: 2.538ex;" src="images/179.svg" alt=" " data-tex="\Delta x_{2}^{2}"> + <img style="vertical-align: -0.685ex; width: 4.166ex; height: 2.572ex;" src="images/180.svg" alt=" " data-tex="\Delta x_{3}^{2}"> in the Euclidean +geometry and in the pre-relativity physics. The latter quantity +is not an invariant with respect to all the Lorentz transformations; +the quantity <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> of equation (23) assumes the rôle of this +invariant. With respect to an arbitrary inertial system, <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> may +be determined by measurements; with a given unit of measure +it is a completely determinate quantity, associated with an arbitrary +pair of events. +</p> +<p> +The invariant <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> differs, disregarding the number of dimensions, +from the corresponding invariant of the Euclidean geometry +in the following points. In the Euclidean geometry <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> is +necessarily positive; it vanishes only when the two points concerned +come together. On the other hand, from the vanishing +of +<span class="align-center"><img style="vertical-align: -3.222ex; width: 44.378ex; height: 5.371ex;" src="images/231.svg" alt=" " data-tex=" +s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} + = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2} +"></span> +it cannot be concluded that the two space-time points fall together; +the vanishing of this quantity <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}">, is the invariant condition +that the two space-time points can be connected by a light +signal <i>in vacuo</i>. If <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> a point (event) represented in the +four-dimensional space of the <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> then all the "points" which +can be connected to <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> by means of a light signal lie upon the +cone <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}"> = 0 (compare Fig. 1, in which the dimension <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> is suppressed). +The "upper" half of the cone may contain the "points" +to which light signals can be sent from <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">; then the "lower" half +<span class="pagenum" id="Page_39">[Pg 39]</span> +of the cone will contain the "points" from which light signals +can be sent to <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">. The points <img style="vertical-align: 0; width: 2.452ex; height: 1.717ex;" src="images/232.svg" alt=" " data-tex="P'"> enclosed by the conical surface +furnish, with <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">, a negative <img style="vertical-align: -0.023ex; width: 2.049ex; height: 1.909ex;" src="images/230.svg" alt=" " data-tex="s^{2}">; <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="PP'"> as well as <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/234.svg" alt=" " data-tex="P'P"> is then, +according to Minkowski, of the nature of a time. Such intervals +represent elements of possible paths of motion, the velocity being +less than that of light.<a id="FNanchor_9_1"></a><a href="#Footnote_9_1" class="fnanchor">[9]</a> In +this case the <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">-axis may be drawn +<span class="pagenum" id="Page_40">[Pg 40]</span> +in the direction of <img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="PP'"> by suitably choosing the state of motion +of the inertial system. If <img style="vertical-align: 0; width: 2.452ex; height: 1.717ex;" src="images/232.svg" alt=" " data-tex="P'"> lies outside of the "light-cone" then +<img style="vertical-align: 0; width: 4.151ex; height: 1.717ex;" src="images/233.svg" alt=" " data-tex="PP'"> is of the nature of a space; in this case, by properly choosing +the inertial system, <img style="vertical-align: -0.025ex; width: 2.559ex; height: 1.645ex;" src="images/235.svg" alt=" " data-tex="\Delta l"> can be made to vanish. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure01.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG. 1.</p> +</div></div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_9_1"></a><a href="#FNanchor_9_1"><span class="label">[9]</span></a>That material velocities exceeding that of light are not possible, fol- +lows from the appearance of the radical <img style="vertical-align: -0.212ex; width: 7.912ex; height: 2.398ex;" src="images/236.svg" alt=" " data-tex="\sqrt{1 - v^{2}}"> in the special Lorentz +transformation (29).</p></div> + +<p> +By the introduction of the imaginary time variable, <img style="vertical-align: -0.339ex; width: 6.754ex; height: 1.91ex;" src="images/237.svg" alt=" " data-tex="x_{4} = il">, +Minkowski has made the theory of invariants for the four-dimensional +continuum of physical phenomena fully analogous +to the theory of invariants for the three-dimensional continuum +of Euclidean space. The theory of four-dimensional tensors of +special relativity differs from the theory of tensors in three-dimensional +space, therefore, only in the number of dimensions +and the relations of reality. +</p> +<p> +A physical entity which is specified by four quantities, <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/238.svg" alt=" " data-tex="A_{\nu}">, +in an arbitrary inertial system of the +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">, is called +a 4-vector, with the components <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/238.svg" alt=" " data-tex="A_{\nu}">, if the <img style="vertical-align: -0.339ex; width: 2.733ex; height: 1.959ex;" src="images/238.svg" alt=" " data-tex="A_{\nu}"> correspond in +their relations of reality and the properties of transformation to +the <img style="vertical-align: -0.339ex; width: 4.214ex; height: 1.959ex;" src="images/61.svg" alt=" " data-tex="\Delta x_{\nu}">; it may be of the nature of a space or of a time. The +sixteen quantities <img style="vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;" src="images/240.svg" alt=" " data-tex="A_{\mu\nu}">, then form the components of a tensor of +the second rank, if they transform according to the scheme +<span class="align-center"><img style="vertical-align: -0.685ex; width: 17.844ex; height: 2.515ex;" src="images/241.svg" alt=" " data-tex=" +{A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}. +"></span> +It follows from this that the <img style="vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;" src="images/240.svg" alt=" " data-tex="A_{\mu\nu}"> behave, with respect to +their properties of transformation and their properties of reality, +as the products of components, <img style="vertical-align: -0.685ex; width: 5.052ex; height: 2.23ex;" src="images/242.svg" alt=" " data-tex="U_{\mu}V_{\nu}"> of two 4-vectors, +(<img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/161.svg" alt=" " data-tex="U">) and (<img style="vertical-align: -0.05ex; width: 1.74ex; height: 1.595ex;" src="images/52.svg" alt=" " data-tex="V">). All the components are real except those which +contain the index 4 once, those being purely imaginary. Tensors +<span class="pagenum" id="Page_41">[Pg 41]</span> +of the third and higher ranks may be defined in an analogous +way. The operations of addition, subtraction, multiplication, +contraction and differentiation for these tensors are wholly +analogous to the corresponding operations for tensors in three-dimensional space. +</p> +<p> +Before we apply the tensor theory to the four-dimensional +space-time continuum, we shall examine more particularly the +skew-symmetrical tensors. The tensor of the second rank has, in +general, 16 = 4·4 components. In the case of skew-symmetry the +components with two equal indices vanish, and the components +with unequal indices are equal and opposite in pairs. There +exist, therefore, only six independent components, as is the case +in the electromagnetic field. In fact, it will be shown when we +consider Maxwell's equations that these may be looked upon as +tensor equations, provided we regard the electromagnetic field +as a skew-symmetrical tensor. Further, it is clear that the skew-symmetrical +tensor of the third rank (skew-symmetrical in all +pairs of indices) has only four independent components, since +there are only four combinations of three different indices. +</p> +<p> +We now turn to Maxwell's equations (19a), (19b), (20a), +(20b), and introduce the notation:<a id="FNanchor_10_1"></a><a href="#Footnote_10_1" class="fnanchor">[10]</a> +<span class="align-center"><img style="vertical-align: -6.265ex; width: 58.388ex; height: 13.661ex;" src="images/243.svg" alt=" " data-tex=" +\begin{align*} +&\left. +\begin{aligned} +&\phi_{23} && \phi_{31} && \phi_{12} && \phi_{14} && \phi_{24} && \phi_{34} \\ +&\mathbf h_{23} && \mathbf h_{31} && \mathbf h_{12} & -&i\mathbf e_{x} & -&i\mathbf e_{y} & -&i\mathbf e_{z} +\end{aligned} +\right\} +&&\text{(30a)}\\ +&\qquad\qquad\left. +\begin{aligned} +&J_{1} && J_{2} && J_{3} && J_{4} \\ +&\frac{1}{c}{\mathbf i_{x}} && +\frac{1}{c}{\mathbf i_{y}} && +\frac{1}{c}{\mathbf i_{z}} && i\rho +\end{aligned} +\right\} +&&\text{(31)} +\end{align*} +"></span> +with the convention that <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}"> shall be equal to <img style="vertical-align: -0.685ex; width: 5.109ex; height: 2.255ex;" src="images/245.svg" alt=" " data-tex="-\phi_{\mu\nu}">. Then +<span class="pagenum" id="Page_42">[Pg 42]</span> +Maxwell's equations may be combined into the forms +<span class="align-center"><img style="vertical-align: -5.181ex; width: 40.27ex; height: 11.494ex;" src="images/246.svg" alt=" " data-tex=" +\begin{gather*} +\frac{\partial \phi_{\mu\nu}}{\partial x_{\nu}} = J_{\mu}, +\qquad & &\text{(32)}\\ +\frac{\partial \phi_{\mu\nu}}{\partial x_{\sigma}} + +\frac{\partial \phi_{\nu\sigma}}{\partial x_{\mu}} + +\frac{\partial \phi_{\sigma\mu}}{\partial x_{\nu}} = 0, +\qquad & &\text{(33)} +\end{gather*} +"></span> +as one can easily verify by substituting from (30a) and (31). +Equations (32) and (33) have a tensor character, and are +therefore co-variant with respect to Lorentz transformations, +if the <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}"> and the <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.885ex;" src="images/247.svg" alt=" " data-tex="J_{\nu}"> have a tensor character, which we assume. +Consequently, the laws for transforming these quantities from +one to another allowable (inertial) system of co-ordinates are +uniquely determined. The progress in method which electrodynamics +owes to the theory of special relativity lies principally +in this, that the number of independent hypotheses is diminished. +If we consider, for example, equations (19a) only from the +standpoint of relativity of direction, as we have done above, we +see that they have three logically independent terms. The way +in which the electric intensity enters these equations appears to +be wholly independent of the way in which the magnetic intensity +enters them; it would not be surprising if instead of +<img style="vertical-align: -1.602ex; width: 4.483ex; height: 4.945ex;" src="images/248.svg" alt=" " data-tex="\dfrac{\partial e_{\mu}}{\partial l}">, +we had, say, +<img style="vertical-align: -1.679ex; width: 5.588ex; height: 5.291ex;" src="images/249.svg" alt=" " data-tex="\dfrac{\partial^{2} e_{\mu}}{\partial l^{2}}"> +or if this term were absent. On the other +hand, only two independent terms appear in equation (32). The +electromagnetic field appears as a formal unit; the way in which +the electric field enters this equation is determined by the way in +which the magnetic field enters it. Besides the electromagnetic +field, only the electric current density appears as an independent +entity. This advance in method arises from the fact that the +<span class="pagenum" id="Page_43">[Pg 43]</span> +electric and magnetic fields draw their separate existences from +the relativity of motion. A field which appears to be purely an +electric field, judged from one system, has also magnetic field +components when judged from another inertial system. When +applied to an electromagnetic field, the general law of transformation +furnishes, for the special case of the special Lorentz +transformation, the equations +<span class="align-center"><img style="vertical-align: -6.897ex; width: 46.646ex; height: 14.925ex;" src="images/250.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{2} +{\mathbf e'}_{x} &= \mathbf e_{x}\qquad & {\mathbf h'}_{x} &= \mathbf h_{x}, \\ +{\mathbf e'}_{y} &= \frac{\mathbf e_{y} - v\mathbf h_{z}}{\sqrt{1 - v^{2}}}\qquad & +{\mathbf h'}_{y} &= \frac{\mathbf h_{y} + v\mathbf e_{z}}{\sqrt{1 - v^{2}}}, \\ +{\mathbf e'}_{z} &= \frac{\mathbf e_{z} + v\mathbf h_{y}}{\sqrt{1 - v^{2}}}\qquad & +{\mathbf h'}_{z} &= \frac{\mathbf h_{z} - v\mathbf e_{y}}{\sqrt{1 - v^{2}}}. +\end{alignedat} +\right\} +\qquad \text{(34)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_10_1"></a><a href="#FNanchor_10_1"><span class="label">[10]</span></a>In +order to avoid confusion from now on we shall use the three-dimensional +space indices, <img style="vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;" src="images/216.svg" alt=" " data-tex="x">, <img style="vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;" src="images/251.svg" alt=" " data-tex="y">, <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/252.svg" alt=" " data-tex="z"> instead of 1, 2, 3, and we shall reserve the +numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum.</p></div> + +<p> +If there exists with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> only a magnetic field, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h">, but +no electric field, <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e">, then with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' there exists an electric +field <img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e"> as well, which would act upon an electric particle at rest +relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. An observer at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> would +designate this force as the Biot-Savart force, or the Lorentz electromotive +force. It therefore appears as if this electromotive force +had become fused with the electric field intensity into a single +entity. +</p> +<p> +In order to view this relation formally, let us consider the +expression for the force acting upon unit volume of electricity, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 23.503ex; height: 2.262ex;" src="images/253.svg" alt=" " data-tex=" +\mathbf k = \rho\mathbf e + [\mathbf i, \mathbf h], +\qquad \text{(35)} +"></span> +in which <img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/145.svg" alt=" " data-tex="\mathbf i"> is the vector velocity of electricity, with the velocity +of light as the unit. If we introduce <img style="vertical-align: -0.685ex; width: 2.408ex; height: 2.23ex;" src="images/254.svg" alt=" " data-tex="J_{\mu}"> and <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}"> according to +(30a) and (31), we obtain for the first component the expression +<span class="align-center"><img style="vertical-align: -0.464ex; width: 22.298ex; height: 2.034ex;" src="images/255.svg" alt=" " data-tex=" +\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}. +"></span> +<span class="pagenum" id="Page_44">[Pg 44]</span> +Observing that <img style="vertical-align: -0.464ex; width: 3.136ex; height: 2.034ex;" src="images/256.svg" alt=" " data-tex="\phi_{11}"> vanishes on account of the skew-symmetry of +the tensor (<img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi">), the components of <img style="vertical-align: 0; width: 1.373ex; height: 1.57ex;" src="images/258.svg" alt=" " data-tex="\mathbf k"> are given by the first three +components of the four-dimensional vector +<span class="align-center"><img style="vertical-align: -0.685ex; width: 21.284ex; height: 2.382ex;" src="images/259.svg" alt=" " data-tex=" +K_{\mu} = \phi_{\mu\nu} J_{\nu}, +\qquad \text{(36)} +"></span> +and the fourth component is given by +<span class="align-center"><img style="vertical-align: -0.667ex; width: 65.109ex; height: 2.364ex;" src="images/260.svg" alt=" " data-tex=" +K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} + = i(\mathbf e_{x} \mathbf i_{x} + \mathbf e_{y} \mathbf i_{y} + \mathbf e_{z} \mathbf i_{z}) + = i \lambda. +\qquad \text{(37)} +"></span> +There is, therefore, a four-dimensional vector of force per unit +volume, whose first three components, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/261.svg" alt=" " data-tex="K_{1}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/262.svg" alt=" " data-tex="K_{2}">, <img style="vertical-align: -0.375ex; width: 2.908ex; height: 1.92ex;" src="images/263.svg" alt=" " data-tex="K_{3}">, are +the ponderomotive force components per unit volume, and whose fourth +component is the rate of working of the field per unit volume, +multiplied by <img style="vertical-align: -0.318ex; width: 4.821ex; height: 2.398ex;" src="images/264.svg" alt=" " data-tex="\sqrt{-1}">. +</p> +<p> +A comparison of (36) and (35) shows that the theory of relativity +formally unites the ponderomotive force of the electric +field, <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"><img style="vertical-align: -0.014ex; width: 1.192ex; height: 1.036ex;" src="images/146.svg" alt=" " data-tex="\mathbf e">, and the Biot-Savart or Lorentz force [<img style="vertical-align: 0; width: 0.722ex; height: 1.572ex;" src="images/145.svg" alt=" " data-tex="\mathbf i">, <img style="vertical-align: 0; width: 1.446ex; height: 1.57ex;" src="images/147.svg" alt=" " data-tex="\mathbf h">]. +</p> + +<p><br></p> + +<p> +<i>Mass and Energy</i>. An important conclusion can be drawn +from the existence and significance of the 4-vector <img style="vertical-align: -0.685ex; width: 3.073ex; height: 2.23ex;" src="images/265.svg" alt=" " data-tex="K_{\mu}">. Let us +imagine a body upon which the electromagnetic field acts for +a time. In the symbolic figure (Fig. 2) <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/266.svg" alt=" " data-tex="Ox_{1}"> designates the +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">-axis, and is at the same time a substitute for the three space axes +<img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/266.svg" alt=" " data-tex="Ox_{1}">, <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/267.svg" alt=" " data-tex="Ox_{2}">, <img style="vertical-align: -0.339ex; width: 4.008ex; height: 1.932ex;" src="images/266.svg" alt=" " data-tex="Ox_{1}">; <img style="vertical-align: -0.05ex; width: 2.4ex; height: 1.643ex;" src="images/268.svg" alt=" " data-tex="Ol"> designates the real time axis. In this diagram +a body of finite extent is represented, at a definite time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">, by +the interval <img style="vertical-align: 0; width: 3.414ex; height: 1.62ex;" src="images/169.svg" alt=" " data-tex="AB"> the whole space-time existence of the body is +represented by a strip whose boundary is everywhere inclined +less than 45° to the <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">-axis. Between the time sections, +<img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/269.svg" alt=" " data-tex="l_{1}"> and <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{2}">, but not extending to them, +a portion of the strip is shaded. This represents the portion +of the space-time manifold +<span class="pagenum" id="Page_45">[Pg 45]</span> +in which the electromagnetic field acts upon the body, or upon +the electric charges contained in it, the action upon them being +transmitted to the body. We shall now consider the changes +which take place in the momentum and energy of the body as a +result of this action. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure02.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG. 2.</p> +</div></div> + +<p> +We shall assume that the principles of momentum and +energy are valid for the body. The change in momentum, +<img style="vertical-align: -0.357ex; width: 3.983ex; height: 1.977ex;" src="images/271.svg" alt=" " data-tex="\Delta I_{x}">, <img style="vertical-align: -0.667ex; width: 3.852ex; height: 2.287ex;" src="images/272.svg" alt=" " data-tex="\Delta I_{y}">, <img style="vertical-align: -0.357ex; width: 3.812ex; height: 1.977ex;" src="images/273.svg" alt=" " data-tex="\Delta I_{z}">, and the change +in energy, <img style="vertical-align: 0; width: 3.613ex; height: 1.62ex;" src="images/274.svg" alt=" " data-tex="\Delta E"> are then given +<span class="pagenum" id="Page_46">[Pg 46]</span> +by the expressions +<span class="align-center"><img style="vertical-align: -12.132ex; width: 60.141ex; height: 25.396ex;" src="images/275.svg" alt=" " data-tex=" +\begin{alignat*}{2} +\begin{aligned} +\Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{x}\, dx\, dy\, dz} + &&= \frac{1}{i}\int{K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{y}\, dx\, dy\, dz} + &&= \frac{1}{i}\int{K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{z}\, dx\, dy\, dz\,} + &&= \frac{1}{i}\int{K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ +\Delta E &= \int_{l_{1}}^{l_{2}}dl \int{\lambda\, dx\, dy\, dz} + &&= \frac{1}{i}\int{ + \frac{1}{i}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}. +\end{aligned} +\end{alignat*} +"></span> +Since the four-dimensional element of volume is an invariant, +and (<img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/261.svg" alt=" " data-tex="K_{1}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/262.svg" alt=" " data-tex="K_{2}">, <img style="vertical-align: -0.375ex; width: 2.908ex; height: 1.92ex;" src="images/263.svg" alt=" " data-tex="K_{3}">, <img style="vertical-align: -0.339ex; width: 2.908ex; height: 1.885ex;" src="images/276.svg" alt=" " data-tex="K_{4}">) forms a 4-vector, the four-dimensional +integral extended over the shaded portion transforms as a 4-vector, +as does also the integral between the limits <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/269.svg" alt=" " data-tex="l_{1}"> and <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/270.svg" alt=" " data-tex="l_{2}">, because +the portion of the region which is not shaded contributes nothing +to the integral. It follows, therefore, that +<img style="vertical-align: -0.357ex; width: 3.983ex; height: 1.977ex;" src="images/271.svg" alt=" " data-tex="\Delta I_{x}">, <img style="vertical-align: -0.667ex; width: 3.852ex; height: 2.287ex;" src="images/272.svg" alt=" " data-tex="\Delta I_{y}">, <img style="vertical-align: -0.357ex; width: 3.812ex; height: 1.977ex;" src="images/273.svg" alt=" " data-tex="\Delta I_{z}">, <img style="vertical-align: -0.025ex; width: 4.394ex; height: 1.645ex;" src="images/277.svg" alt=" " data-tex="i \Delta E"> +form a 4-vector. Since the quantities themselves transform in +the same way as their increments, it follows that the aggregate +of the four quantities +<span class="align-center"><img style="vertical-align: -0.667ex; width: 13.217ex; height: 2.213ex;" src="images/278.svg" alt=" " data-tex=" +I_{x},\ I_{y},\ I_{z},\ iE +"></span> +has itself the properties of a vector; these quantities are referred +to an instantaneous condition of the body (e.g. at the time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> = <img style="vertical-align: -0.339ex; width: 1.662ex; height: 1.91ex;" src="images/269.svg" alt=" " data-tex="l_{1}">). +</p> +<p> +This 4-vector may also be expressed in terms of the mass <img style="vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;" src="images/104.svg" alt=" " data-tex="m">, +and the velocity of the body, considered as a material particle. +To form this expression, we note first, that +<span class="align-center"><img style="vertical-align: -0.566ex; width: 66.28ex; height: 2.584ex;" src="images/279.svg" alt=" " data-tex=" +-ds^{2} = d\tau^{2} + = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} + = dl^{2}(1 - q^{2}) + \qquad \text{(38)} +"></span> +<span class="pagenum" id="Page_47">[Pg 47]</span> +is an invariant which refers to an infinitely short portion of the +four-dimensional line which represents the motion of the material +particle. The physical significance of the invariant <img style="vertical-align: -0.029ex; width: 2.346ex; height: 1.6ex;" src="images/280.svg" alt=" " data-tex="d \tau"> may +easily be given. If the time axis is chosen in such a way that it +has the direction of the fine differential which we are considering, +or, in other words, if we reduce the material particle to rest, +we shall then have <img style="vertical-align: -0.186ex; width: 7.214ex; height: 1.756ex;" src="images/281.svg" alt=" " data-tex="d \tau = dl">; this will therefore be measured by +the light-seconds clock which is at the same place, and at rest +relatively to the material particle. We therefore call <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> the proper +time of the material particle. As opposed to <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl">, <img style="vertical-align: -0.029ex; width: 2.346ex; height: 1.6ex;" src="images/280.svg" alt=" " data-tex="d \tau"> is therefore an +invariant, and is practically equivalent to <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl"> for motions whose +velocity is small compared to that of fight. Hence we see that +<span class="align-center"><img style="vertical-align: -1.581ex; width: 18.527ex; height: 4.681ex;" src="images/283.svg" alt=" " data-tex=" +u_{\sigma} = \frac{dx_{\sigma}}{d\tau} +\qquad \text{(39)} +"></span> +has, just as the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, the character of a vector; we shall designate +(<img style="vertical-align: -0.357ex; width: 2.395ex; height: 1.357ex;" src="images/284.svg" alt=" " data-tex="u_{\sigma}">) as the four-dimensional vector (in brief, 4-vector) of +velocity. Its components satisfy, by (38), the condition +<span class="align-center"><img style="vertical-align: -1.018ex; width: 22.112ex; height: 3.167ex;" src="images/285.svg" alt=" " data-tex=" +\sum {u_{\sigma}}^{2} = -1. +\qquad \text{(40)} +"></span> +We see that this 4-vector, whose components in the ordinary +notation are +<span class="align-center"><img style="vertical-align: -2.76ex; width: 55.726ex; height: 5.862ex;" src="images/286.svg" alt=" " data-tex=" +\frac{\mathbf q_{x}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf q_{y}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf q_{z}}{\sqrt{1 - q^{2}}},\quad +\frac{\mathbf i}{\sqrt{1 - q^{2}}} +\qquad \text{(41)} +"></span> +is the only 4-vector which can be formed from the velocity components +of the material particle which are defined in three dimensions by +<span class="align-center"><img style="vertical-align: -1.577ex; width: 33.314ex; height: 4.676ex;" src="images/287.svg" alt=" " data-tex=" +\mathbf q_{x} = \frac{dx}{dl},\quad +\mathbf q_{y} = \frac{dy}{dl},\quad +\mathbf q_{z} = \frac{dz}{dl}. +"></span> +<span class="pagenum" id="Page_48">[Pg 48]</span> +We therefore see that +<span class="align-center"><img style="vertical-align: -2.148ex; width: 18.483ex; height: 5.444ex;" src="images/288.svg" alt=" " data-tex=" +\left(m \frac{dx_{\mu}}{d\tau}\right) +\qquad \text{(42)} +"></span> +must be that 4-vector which is to be equated to the 4-vector of +momentum and energy whose existence we have proved above. +By equating the components, we obtain, in three-dimensional +notation, +<span class="align-center"><img style="vertical-align: -11.148ex; width: 25.646ex; height: 23.428ex;" src="images/289.svg" alt=" " data-tex=" +\left. +\begin{aligned} +I_{x} = \frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}, \\ +I_{y} = \frac{m\mathbf q_{y}}{\sqrt{1 - q^{2}}}, \\ +I_{z} = \frac{m\mathbf q_{z}}{\sqrt{1 - q^{2}}}, \\ +E = \frac{m}{\sqrt{1 - q^{2}}}. +\end{aligned} +\right\} +\qquad \text{(43)} +"></span> +</p> +<p> +We recognize, in fact, that these components of momentum +agree with those of classical mechanics for velocities which are +small compared to that of light. For large velocities the momentum +increases more rapidly than linearly with the velocity, so as +to become infinite on approaching the velocity of light. +</p> +<p> +If we apply the last of equations (43) to a material particle +at rest (<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q"> = 0), we see that the energy, <img style="vertical-align: -0.375ex; width: 2.657ex; height: 1.913ex;" src="images/290.svg" alt=" " data-tex="E_{0}"> of a, body at rest is +equal to its mass. Had we chosen the second as our unit of time, +we would have obtained +<span class="align-center"><img style="vertical-align: -0.566ex; width: 19.182ex; height: 2.565ex;" src="images/291.svg" alt=" " data-tex=" +E_{0} = mc^{2}. + \qquad \text{(44)} +"></span> +Mass and energy are therefore essentially alike; they are only +different expressions for the same thing. The mass of a body +<span class="pagenum" id="Page_49">[Pg 49]</span> +is not a constant; it varies with changes in its energy.<a id="FNanchor_11_1"></a><a href="#Footnote_11_1" class="fnanchor">[11]</a> +We see from the last of equations (43) that <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/292.svg" alt=" " data-tex="E"> becomes infinite when +<img style="vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;" src="images/212.svg" alt=" " data-tex="q"> approaches 1, the velocity of light. If we develop <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/292.svg" alt=" " data-tex="E"> in powers +of <img style="vertical-align: -0.439ex; width: 2.143ex; height: 2.326ex;" src="images/293.svg" alt=" " data-tex="q^{2}">, we obtain, +<span class="align-center"><img style="vertical-align: -1.602ex; width: 38.993ex; height: 4.636ex;" src="images/294.svg" alt=" " data-tex=" +E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots. + \qquad \text{(45)} +"></span> +The second term of this expansion corresponds to the kinetic +energy of the material particle in classical mechanics. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_11_1"></a><a href="#FNanchor_11_1"><span class="label">[11]</span></a>The emission of energy in radioactive processes is evidently connected +with the fact that the atomic weights are not integers. Attempts have been +made to draw conclusions from this concerning the structure and stability +of the atomic nuclei.</p></div> + +<p><br></p> + +<p> +<i>Equations of Motion of Material Particles</i>. From (43) we +obtain, by differentiating by the time <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l">, and using the principle +of momentum, in the notation of three-dimensional vectors, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 30.382ex; height: 6.785ex;" src="images/295.svg" alt=" " data-tex=" +\mathbf{K} = \frac{d}{dl}\left(\frac{m\mathbf q}{\sqrt{1 - q^{2}}}\right). +\qquad \text{(46)} +"></span> +</p> +<p> +This equation, which was previously employed by H. A. +Lorentz for the motion of electrons, has been proved to be true, +with great accuracy, by experiments with <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">-rays. +</p> + +<p><br></p> + +<p> +<i>Energy Tensor of the Electromagnetic Field</i>. Before the development +of the theory of relativity it was known that the principles +of energy and momentum could be expressed in a differential +form for the electromagnetic field. The four-dimensional +formulation of these principles leads to an important conception, +<span class="pagenum" id="Page_50">[Pg 50]</span> +that of the energy tensor, which is important for the further development +of the theory of relativity. +</p> +<p> +If in the expression for the 4-vector of force per unit volume, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 12.359ex; height: 2.255ex;" src="images/296.svg" alt=" " data-tex=" +K_{\mu} = \phi_{\mu\nu} J_{\nu}, +"></span> +using the field equations (32), we express <img style="vertical-align: -0.339ex; width: 2.291ex; height: 1.885ex;" src="images/247.svg" alt=" " data-tex="J_{\nu}"> in terms of the +field intensities, <img style="vertical-align: -0.685ex; width: 3.349ex; height: 2.255ex;" src="images/244.svg" alt=" " data-tex="\phi_{\mu\nu}">, we obtain, after some transformations and +repeated application of the field equations (32) and (33), the +expression +<span class="align-center"><img style="vertical-align: -1.891ex; width: 23.002ex; height: 5.235ex;" src="images/297.svg" alt=" " data-tex=" +K_{\mu} = -\frac{\partial T_{\mu\nu}}{\partial x_{\nu}}, +\qquad \text{(47)} +"></span> +where we have written<a id="FNanchor_12_1"></a><a href="#Footnote_12_1" class="fnanchor">[12]</a> +<span class="align-center"><img style="vertical-align: -0.781ex; width: 36.604ex; height: 2.799ex;" src="images/298.svg" alt=" " data-tex=" +T_{\mu\nu} + = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + + \phi_{\mu\alpha} \phi_{\nu\alpha}. +\qquad \text{(48)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_12_1"></a><a href="#FNanchor_12_1"><span class="label">[12]</span></a>To be summed for the indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">.</p></div> + +<p> +The physical meaning of equation (47) becomes evident if in +place of this equation we write, using a new notation, +<span class="align-center"><img style="vertical-align: -11.527ex; width: 55.328ex; height: 24.185ex;" src="images/299.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{4} +\mathbf k_{x} &= -\frac{\partial p_{xx}}{\partial x} + &&- \frac{\partial p_{xy}}{\partial y} + &&- \frac{\partial p_{xz}}{\partial z} + &&- \frac{\partial (i b_{x})}{\partial (il)}, \\ +\mathbf k_{y} &= -\frac{\partial p_{yx}}{\partial x} + &&- \frac{\partial p_{yy}}{\partial y} + &&- \frac{\partial p_{yz}}{\partial z} + &&- \frac{\partial (i b_{y})}{\partial (il)}, \\ +\mathbf k_{z} &= -\frac{\partial p_{zx}}{\partial x} + &&- \frac{\partial p_{zy}}{\partial y} + &&- \frac{\partial p_{zz}}{\partial z} + &&- \frac{\partial (i b_{z})}{\partial (il)}, \\ +i\lambda + &= -\frac{\partial (i\mathbf s_{x})}{\partial x} + &&- \frac{\partial (i\mathbf s_{y})}{\partial y} + &&- \frac{\partial (i\mathbf s_{z})}{\partial z} + &&- \frac{\partial (-\eta)}{\partial (i l)}; +\end{alignedat} +\right\} +\qquad \text{(47a)} +"></span> +<span class="pagenum" id="Page_51">[Pg 51]</span> +or, on eliminating the imaginary, +<span class="align-center"><img style="vertical-align: -11.136ex; width: 48.145ex; height: 23.402ex;" src="images/300.svg" alt=" " data-tex=" +\left. +\begin{alignedat}{4} +k_{x} &= -\frac{\partial p_{xx}}{\partial x} + &&- \frac{\partial p_{xy}}{\partial y} + &&- \frac{\partial p_{xz}}{\partial z} + &&- \frac{\partial b_{x}}{\partial l}, \\ +k_{y} &= -\frac{\partial p_{yx}}{\partial x} + &&- \frac{\partial p_{yy}}{\partial y} + &&- \frac{\partial p_{yz}}{\partial z} + &&- \frac{\partial b_{y}}{\partial l}, \\ +k_{z} &= -\frac{\partial p_{zx}}{\partial x} + &&- \frac{\partial p_{zy}}{\partial y} + &&- \frac{\partial p_{zz}}{\partial z} + &&- \frac{\partial b_{z}}{\partial l}, \\ +\lambda + &= -\frac{\partial s_{x}}{\partial x} + &&- \frac{\partial s_{y}}{\partial y} + &&- \frac{\partial s_{z}}{\partial z} + &&- \frac{\partial \eta}{\partial l}; +\end{alignedat} +\right\} +\qquad \text{(47b)} +"></span> +</p> +<p> +When expressed in the latter form, we see that the first three +equations state the principle of momentum; <img style="vertical-align: -0.439ex; width: 3.156ex; height: 1.439ex;" src="images/301.svg" alt=" " data-tex="p_{xx}">,..., <img style="vertical-align: -0.439ex; width: 2.985ex; height: 1.439ex;" src="images/302.svg" alt=" " data-tex="p_{zx}"> are the +Maxwell stresses in the electromagnetic field, and (<img style="vertical-align: -0.357ex; width: 2.073ex; height: 1.927ex;" src="images/303.svg" alt=" " data-tex="b_{x}">, <img style="vertical-align: -0.667ex; width: 1.942ex; height: 2.237ex;" src="images/304.svg" alt=" " data-tex="b_{y}">, <img style="vertical-align: -0.357ex; width: 1.902ex; height: 1.927ex;" src="images/305.svg" alt=" " data-tex="b_{z}">) is +the vector momentum per unit volume of the field. The last of +equations (47b) expresses the energy principle; <img style="vertical-align: -0.014ex; width: 1.027ex; height: 1.038ex;" src="images/306.svg" alt=" " data-tex="\mathbf s"> is the vector +flow of energy, and <img style="vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;" src="images/307.svg" alt=" " data-tex="\eta"> the energy per unit volume of the field. In +fact, we get from (48) by introducing the well-known expressions +for the components of the field intensity from electrodynamics, +<span class="align-center"><img style="vertical-align: -14.002ex; width: 63.031ex; height: 29.135ex;" src="images/308.svg" alt=" " data-tex=" +\left. +\begin{aligned} +&\begin{alignedat}{4} +p_{xx} = &{} - \mathbf h_{x} \mathbf h_{x} + &&+ \tfrac{1}{2}({\mathbf h_{x}}^{2} &&+ {\mathbf h_{y}}^{2} &&+ {\mathbf h_{z}}^{2}) \\ + &{} - \mathbf e_{x} \mathbf e_{y} + &&+ \tfrac{1}{2}({\mathbf e_{x}}^{2} &&+ {\mathbf e_{y}}^{2} &&+ {\mathbf e_{z}}^{2}), +\end{alignedat} \\ +&\qquad\qquad\qquad\qquad\qquad +\begin{alignedat}{3} +p_{xy} = &{} - \mathbf h_{x} \mathbf h_{y}\quad + && p_{xz} = && {} - \mathbf h_{x} \mathbf h_{z} \\ + &{} - \mathbf e_{x} \mathbf e_{y},\quad + && && {} - \mathbf e_{x} \mathbf e_{z}, +\end{alignedat} \\ +&\qquad\qquad\qquad\qquad\qquad\quad\vdots \\ +&b_{x} = \mathbf s_{x} = \mathbf e_{y} \mathbf h_{z} - \mathbf e_{z}\mathbf h_{y}, \\ +&b_{y} = \mathbf s_{y} = \mathbf e_{z} \mathbf h_{x} - \mathbf e_{x}\mathbf h_{z}, \\ +&b_{z} = \mathbf s_{z} = \mathbf e_{x} \mathbf h_{y} - \mathbf e_{y}\mathbf h_{x}, \\ +&\eta = +\tfrac{1}{2}({\mathbf e_{x}}^{2} + {\mathbf e_{y}}^{2} + {\mathbf e_{z}}^{2} + + {\mathbf h_{x}}^{2} + {\mathbf h_{y}}^{2} + {\mathbf h_{z}}^{2}). +\end{aligned} +\right\} +\qquad \text{(48a)} +"></span> +<span class="pagenum" id="Page_52">[Pg 52]</span> +We conclude from (48) that the energy tensor of the electromagnetic +field is symmetrical; with this is connected the fact +that the momentum per unit volume and the how of energy are +equal to each other (relation between energy and inertia). +</p> +<p> +We therefore conclude from these considerations that the +energy per unit volume has the character of a tensor. This has +been proved directly only for an electromagnetic field, although +we may claim universal validity for it. Maxwell's equations determine +the electromagnetic field when the distribution of electric +charges and currents is known. But we do not know the +laws which govern the currents and charges. We do know, indeed, +that electricity consists of elementary particles (electrons, +positive nuclei), but from a theoretical point of view we cannot +comprehend this. We do not know the energy factors which +determine the distribution of electricity in particles of definite +size and charge, and all attempts to complete the theory in this +direction have failed. If then we can build upon Maxwell's equations +in general, the energy tensor of the electromagnetic field +is known only outside the charged particles.<a id="FNanchor_13_1"></a><a href="#Footnote_13_1" class="fnanchor">[13]</a> In these regions, +outside of charged particles, the only regions in which we can believe +that we have the complete expression for the energy tensor, +we have, by (47), +<span class="align-center"><img style="vertical-align: -1.891ex; width: 19.927ex; height: 5.235ex;" src="images/309.svg" alt=" " data-tex=" +\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = 0. +\qquad \text{(47c)} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_13_1"></a><a href="#FNanchor_13_1"><span class="label">[13]</span></a>It has been attempted to remedy this lack of knowledge by considering +the charged particles as proper singularities. But in my opinion this means +giving up a real understanding of the structure of matter. It seems to me +much better to give in to our present inability rather than to be satisfied +by a solution that is only apparent.</p></div> +<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> +<p><br></p> + +<p> +<i>General Expressions for the Conservation Principles</i>. We +can hardly avoid making the assumption that in all other cases, +also, the space distribution of energy is given by a symmetrical +tensor, <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}">, and that this complete energy tensor everywhere +satisfies the relation (47c). At any rate we shall see that by +means of this assumption we obtain the correct expression for +the integral energy principle. +</p> +<p> +Let us consider a spatially bounded, closed system, which, +four-dimensionally, we may represent as a strip, outside of which +the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> vanish. Integrate equation (47c) over a space section. +Since the integrals of +<img style="vertical-align: -1.891ex; width: 5.55ex; height: 5.235ex;" src="images/311.svg" alt=" " data-tex="\dfrac{\partial T_{\mu1}}{\partial x_{1}}">, +<img style="vertical-align: -1.891ex; width: 5.55ex; height: 5.235ex;" src="images/312.svg" alt=" " data-tex="\dfrac{\partial T_{\mu2}}{\partial x_{2}}"> and +<img style="vertical-align: -1.927ex; width: 5.55ex; height: 5.27ex;" src="images/313.svg" alt=" " data-tex="\dfrac{\partial T_{\mu3}}{\partial x_{3}}"> +vanish because +the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> vanish at the limits of integration, we obtain +<span class="align-center"><img style="vertical-align: -2.148ex; width: 36.963ex; height: 5.428ex;" src="images/314.svg" alt=" " data-tex=" +\frac{\partial}{\partial l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} + = 0. +\qquad \text{(49)} +"></span> +Inside the parentheses are the expressions for the momentum of +the whole system, multiplied by <img style="vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;" src="images/315.svg" alt=" " data-tex="i">, together with the negative +energy of the system, so that (49) expresses the conservation +principles in their integral form. That this gives the right conception +of energy and the conservation principles will be seen +from the following considerations. +</p> + +<p><br></p> + +<p class="center"> +PHENOMENOLOGICAL REPRESENTATION OF THE +ENERGY TENSOR OF MATTER.</p> + +<p> +<i>Hydrodynamical Equations</i>. We know that matter is built +up of electrically charged particles, but we do not know the laws +which govern the constitution of these particles. In treating mechanical +problems, we are therefore obliged to make use of an +<span class="pagenum" id="Page_54">[Pg 54]</span> +inexact description of matter, which corresponds to that of classical +mechanics. The density <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, of a material substance and the +hydrodynamical pressures are the fundamental concepts upon +which such a description is based. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure03.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG. 3.</p> +</div></div> + +<p> +Let <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}"> be the density of matter at a place, estimated with +reference to a system of co-ordinates moving with the matter. +Then <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}">, the density at rest, is an invariant. If we think of the +matter in arbitrary motion and neglect the pressures (particles +of dust <i>in vacuo</i>, neglecting the size of the particles and the +temperature), then the energy tensor will depend only upon the +<span class="pagenum" id="Page_55">[Pg 55]</span> +velocity components, <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/119.svg" alt=" " data-tex="u_{\nu}"> and <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}">. We secure the tensor character +of <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> by putting +<span class="align-center"><img style="vertical-align: -0.685ex; width: 22.948ex; height: 2.382ex;" src="images/317.svg" alt=" " data-tex=" +T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}, + \qquad \text{(50)} +"></span> +in which the <img style="vertical-align: -0.685ex; width: 2.447ex; height: 1.685ex;" src="images/318.svg" alt=" " data-tex="u_{\mu}">, in the three-dimensional representation, are +given by (41). In fact, it follows from (50) that for <img style="vertical-align: -0.439ex; width: 5.189ex; height: 1.946ex;" src="images/319.svg" alt=" " data-tex="q = 0">, +<img style="vertical-align: -0.375ex; width: 10.166ex; height: 1.906ex;" src="images/320.svg" alt=" " data-tex="T_{44} = -\sigma_{0}"> +(equal to the negative energy per unit volume), as it should, +according to the theorem of the equivalence of mass and energy, +and according to the physical interpretation of the energy tensor +given above. If an external force (four-dimensional vector, <img style="vertical-align: -0.685ex; width: 3.073ex; height: 2.23ex;" src="images/265.svg" alt=" " data-tex="K_{\mu}">) +acts upon the matter, by the principles of momentum and energy +the equation +<span class="align-center"><img style="vertical-align: -1.891ex; width: 11.688ex; height: 5.235ex;" src="images/321.svg" alt=" " data-tex=" +K_{\mu} = \frac{\partial T_{\mu\nu}}{\partial x_{\nu}} +"></span> +must hold. We shall now show that this equation leads to the +same law of motion of a material particle as that already obtained. +Let us imagine the matter to be of infinitely small extent +in space, that is, a four-dimensional thread; then by integration +over the whole thread with respect to the space co-ordinates +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, we obtain +<span class="align-center"><img style="vertical-align: -5.035ex; width: 54.942ex; height: 11.201ex;" src="images/322.svg" alt=" " data-tex=" +\begin{align*} +\int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} + &= \int \frac{\partial T_{14}}{\partial x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ + &= -i \frac{d}{dl}\left\{ + \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} + \right\}. +\end{align*} +"></span> +</p> +<p> +Now <img style="vertical-align: -0.691ex; width: 16.724ex; height: 2.514ex;" src="images/323.svg" alt=" " data-tex="\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}"> is an invariant, as is, therefore, also +<img style="vertical-align: -0.691ex; width: 19.381ex; height: 2.514ex;" src="images/324.svg" alt=" " data-tex="\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}">. +We shall calculate this integral, first with +respect to the inertial system which we have chosen, and second, +with respect to a system relatively to which the matter has the +velocity zero. The integration is to be extended over a filament +<span class="pagenum" id="Page_56">[Pg 56]</span> +of the thread for which <img style="vertical-align: -0.375ex; width: 2.28ex; height: 1.35ex;" src="images/316.svg" alt=" " data-tex="\sigma_{0}"> may be regarded as constant over the +whole section. If the space volumes of the filament referred to +the two systems are <img style="vertical-align: -0.05ex; width: 2.916ex; height: 1.62ex;" src="images/325.svg" alt=" " data-tex="dV"> and <img style="vertical-align: -0.375ex; width: 3.483ex; height: 1.945ex;" src="images/326.svg" alt=" " data-tex="dV_{0}"> respectively, then we have +<span class="align-center"><img style="vertical-align: -1.948ex; width: 24.709ex; height: 5.027ex;" src="images/327.svg" alt=" " data-tex=" +\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau +"></span> +and therefore also +<span class="align-center"><img style="vertical-align: -1.948ex; width: 38.788ex; height: 5.048ex;" src="images/328.svg" alt=" " data-tex=" +\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} + = \int dm\, i\, \frac{d\tau}{dx_{4}}. +"></span> +</p> +<p> +If we substitute the right-hand side for the left-hand side in +the former integral, and put <img style="vertical-align: -1.581ex; width: 4.454ex; height: 4.681ex;" src="images/329.svg" alt=" " data-tex="\dfrac{dx_{1}}{d\tau}"> outside the sign of integration, +we obtain, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 38.194ex; height: 6.785ex;" src="images/330.svg" alt=" " data-tex=" +\mathbf K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) + = \frac{d}{dl}\left(\frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}\right). +"></span> +We see, therefore, that the generalized conception of the energy +tensor is in agreement with our former result. +</p> + +<p><br></p> + +<p> +<i>The Eulerian Equations for Perfect Fluids</i>. In order to get +nearer to the behaviour of real matter we must add to the energy +tensor a term which corresponds to the pressures. The simplest +case is that of a perfect fluid in which the pressure is determined +by a scalar <img style="vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;" src="images/134.svg" alt=" " data-tex="p">. Since the tangential stresses <img style="vertical-align: -0.667ex; width: 3.025ex; height: 1.667ex;" src="images/331.svg" alt=" " data-tex="p_{xy}">, etc., vanish in +this case, the contribution to the energy tensor must be of the +form <img style="vertical-align: -0.685ex; width: 4.143ex; height: 2.307ex;" src="images/332.svg" alt=" " data-tex="p\delta_{\nu\mu}">. We must therefore put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 28.869ex; height: 2.382ex;" src="images/333.svg" alt=" " data-tex=" +T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}. +\qquad \text{(51)} +"></span> +<span class="pagenum" id="Page_57">[Pg 57]</span> +At rest, the density of the matter, or the energy per unit volume, +is in this case, not <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma"> but <img style="vertical-align: -0.439ex; width: 5.196ex; height: 1.758ex;" src="images/334.svg" alt=" " data-tex="\sigma - p">. For +<span class="align-center"><img style="vertical-align: -1.581ex; width: 35.761ex; height: 4.681ex;" src="images/335.svg" alt=" " data-tex=" +-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} + = \sigma - p. +"></span> +In the absence of any force, we have +<span class="align-center"><img style="vertical-align: -2.237ex; width: 42.095ex; height: 5.58ex;" src="images/336.svg" alt=" " data-tex=" +\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} + = \sigma u_{\nu} \frac{\partial u_{\mu}}{\partial x_{\nu}} + + u_{\mu} \frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + + \frac{\partial p}{\partial x_{\mu}} = 0. +"></span> +If we multiply this equation by <img style="vertical-align: -2.148ex; width: 13.161ex; height: 5.444ex;" src="images/337.svg" alt=" " data-tex="u_{\mu} \left(= \dfrac{dx_{\mu}}{d\tau}\right)"> +and sum for the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">'s we obtain, using (40), +<span class="align-center"><img style="vertical-align: -1.891ex; width: 29.227ex; height: 5.195ex;" src="images/338.svg" alt=" " data-tex=" +-\frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{dp}{d\tau} = 0, +\qquad \text{(52)} +"></span> +where we have put <img style="vertical-align: -2.237ex; width: 16.078ex; height: 5.533ex;" src="images/339.svg" alt=" " data-tex="\dfrac{\partial p}{\partial x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}">. +This is the equation of continuity, which differs from that of +classical mechanics by the term <img style="vertical-align: -1.581ex; width: 3.342ex; height: 4.681ex;" src="images/340.svg" alt=" " data-tex="\dfrac{dp}{d\tau}">, which, practically, +is vanishingly small. Observing (52), the conservation principles take the form +<span class="align-center"><img style="vertical-align: -2.237ex; width: 35.277ex; height: 5.533ex;" src="images/341.svg" alt=" " data-tex=" +\sigma \frac{du_{\mu}}{d\tau} + + u_{\mu} \frac{dp}{d\tau} + + \frac{\partial p}{\partial x_{\mu}} = 0. +\qquad \text{(53)} +"></span> +The equations for the first three indices evidently correspond to +the Eulerian equations. That the equations (52) and (53) correspond, +to a first approximation, to the hydrodynamical equations +of classical mechanics, is a further confirmation of the generalized +energy principle. The density of matter and of energy +has the character of a symmetrical tensor. +<span class="pagenum" id="Page_58">[Pg 58]</span> +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title="LECTURE III: THE GENERAL THEORY OF RELATIVITY"><a id="chap03"></a>LECTURE III +<br><br> +THE GENERAL THEORY OF RELATIVITY</h2> + +<p class="nind"> +<span class="dropcap">A</span>LL of the previous considerations have been based upon the +assumption that all inertial systems are equivalent for the description +of physical phenomena, but that they are preferred, for +the formulation of the laws of nature, to spaces of reference in a +different state of motion. We can think of no cause for this preference +for definite states of motion to all others, according to our +previous considerations, either in the perceptible bodies or in the +concept of motion; on the contrary, it must be regarded as an independent +property of the space-time continuum. The principle +of inertia, in particular, seems to compel us to ascribe physically +objective properties to the space-time continuum. Just as +it was necessary from the Newtonian standpoint to make both +the statements, <i>tempus est absolutum</i>, <i>spatium est absolutum</i>, so +from the standpoint of the special theory of relativity we must +say, <i>continuum spatii et temporis est absolutum</i>. In this latter +statement <i>absolutum</i> means not only "physically real," but also +"independent in its physical properties, having a physical effect, +but not itself influenced by physical conditions." +</p> +<p> +As long as the principle of inertia is regarded as the keystone +of physics, this standpoint is certainly the only one which +is justified. But there are two serious criticisms of the ordinary +conception. In the first place, it is contrary to the mode of thinking +in science to conceive of a thing (the space-time continuum) +which acts itself, but which cannot be acted upon. This is the +reason why E. Mach was led to make the attempt to eliminate +space as an active cause in the system of mechanics. According +<span class="pagenum" id="Page_59">[Pg 59]</span> +to him, a material particle does not move in unaccelerated +motion relatively to space, but relatively to the centre of all the +other masses in the universe; in this way the series of causes of +mechanical phenomena was closed, in contrast to the mechanics +of Newton and Galileo. In order to develop this idea within the +limits of the modern theory of action through a medium, the +properties of the space-time continuum which determine inertia +must be regarded as field properties of space, analogous to the +electromagnetic field. The concepts of classical mechanics afford +no way of expressing this. For this reason Mach's attempt +at a solution failed for the time being. We shall come back to +this point of view later. In the second place, classical mechanics +indicates a limitation which directly demands an extension of +the principle of relativity to spaces of reference which are not +in uniform motion relatively to each other. The ratio of the +masses of two bodies is defined in mechanics in two ways which +differ from each other fundamentally; in the first place, as the +reciprocal ratio of the accelerations which the same motional +force imparts to them (inert mass), and in the second place, as +the ratio of the forces which act upon them in the same gravitational +field (gravitational mass). The equality of these two +masses, so differently defined, is a fact which is confirmed by +experiments of very high accuracy (experiments of Eötvös), and +classical mechanics offers no explanation for this equality. It is, +however, clear that science is fully justified in assigning such a +numerical equality only after this numerical equality is reduced +to an equality of the real nature of the two concepts. +</p> +<p> +That this object may actually be attained by an extension +of the principle of relativity, follows from the following consideration. +A little reflection will show that the theorem of the +<span class="pagenum" id="Page_60">[Pg 60]</span> +equality of the inert and the gravitational mass is equivalent +to the theorem that the acceleration imparted to a body by a +gravitational field is independent of the nature of the body. For +Newton's equation of motion in a gravitational field, written out +in full, is +<span class="align-center"><img style="vertical-align: -2.036ex; width: 69.485ex; height: 5.204ex;" src="images/342.svg" alt=" " data-tex=" +\begin{aligned} +(\text{Inert mass})·(\text{Acceleration}) = \,(\text{Intensity of the} \\ +\qquad + \text{gravitational field}) &· (\text{Gravitational mass}). +\end{aligned} +"></span> +It is only when there is numerical equality between the inert +and gravitational mass that the acceleration is independent of +the nature of the body. Let now <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> be an inertial system. Masses +which are sufficiently far from each other and from other bodies +are then, with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, free from acceleration. We shall +also refer these masses to a system of co-ordinates <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' uniformly +accelerated with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">. Relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' all the masses +have equal and parallel accelerations; with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' they +behave just as if a gravitational field were present and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' were +unaccelerated. Overlooking for the present the question as to the +"cause" of such a gravitational field, which will occupy us later, +there is nothing to prevent our conceiving this gravitational field +as real, that is, the conception that <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is "at rest" and a gravitational +field is present we may consider as equivalent to the conception +that only <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> is an "allowable" system of co-ordinates and +no gravitational field is present. The assumption of the complete +physical equivalence of the systems of co-ordinates, +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> and <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', +we call the "principle of equivalence;" this principle is evidently +intimately connected with the theorem of the equality between +the inert and the gravitational mass, and signifies an extension +of the principle of relativity to co-ordinate systems which are in +<span class="pagenum" id="Page_61">[Pg 61]</span> +non-uniform motion relatively to each other. In fact, through +this conception we arrive at the unity of the nature of inertia +and gravitation. For according to our way of looking at it, the +same masses may appear to be either under the action of inertia +alone (with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">) or under the combined action of +inertia and gravitation (with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'). The possibility of +explaining the numerical equality of inertia and gravitation by +the unity of their nature gives to the general theory of relativity, +according to my conviction, such a superiority over the conceptions +of classical mechanics, that all the difficulties encountered +in development must be considered as small in comparison. +</p> +<p> +What justifies us in dispensing with the preference for inertial +systems over all other co-ordinate systems, a preference +that seems so securely established by experiment based upon +the principle of inertia? The weakness of the principle of inertia +lies in this, that it involves an argument in a circle: a mass moves +without acceleration if it is sufficiently far from other bodies; we +know that it is sufficiently far from other bodies only by the fact +that it moves without acceleration. Are there, in general, any +inertial systems for very extended portions of the space-time +continuum, or, indeed, for the whole universe? We may look +upon the principle of inertia as established, to a high degree of +approximation, for the space of our planetary system, provided +that we neglect the perturbations due to the sun and planets. +Stated more exactly, there are finite regions, where, with respect +to a suitably chosen space of reference, material particles move +freely without acceleration, and in which the laws of the special +theory of relativity, which have been developed above, hold +with remarkable accuracy. Such regions we shall call "Galilean +regions." We shall proceed from the consideration of such regions +<span class="pagenum" id="Page_62">[Pg 62]</span> +as a special case of known properties. +</p> +<p> +The principle of equivalence demands that in dealing with +Galilean regions we may equally well make use of non-inertial +systems, that is, such co-ordinate systems as, relatively to inertial +systems, are not free from acceleration and rotation. If, +further, we are going to do away completely with the difficult +question as to the objective reason for the preference of certain +systems of co-ordinates, then we must allow the use of arbitrarily +moving systems of co-ordinates. As soon as we make this +attempt seriously we come into conflict with that physical interpretation +of space and time to which we were led by the special +theory of relativity. For let <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' be a system of co-ordinates whose +<img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/252.svg" alt=" " data-tex="z">'-axis coincides with the <img style="vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;" src="images/252.svg" alt=" " data-tex="z">-axis of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, and which rotates about +the latter axis with constant angular velocity. Are the configurations +of rigid bodies, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', in accordance with +the laws of Euclidean geometry? Since <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is not an inertial system, +we do not know directly the laws of configuration of rigid +bodies with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', nor the laws of nature, in general. But +we do know these laws with respect to the inertial system <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, +and we can therefore estimate them with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. Imagine +a circle drawn about the origin in the <img style="vertical-align: -0.464ex; width: 6.424ex; height: 2.181ex;" src="images/343.svg" alt=" " data-tex="x'-y'"> plane of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' and a +diameter of this circle. Imagine, further, that we have given a +large number of rigid rods, all equal to each other. We suppose +these laid in series along the periphery and the diameter of the +circle, at rest relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'. If <img style="vertical-align: -0.05ex; width: 1.735ex; height: 1.595ex;" src="images/161.svg" alt=" " data-tex="U"> is the number of these rods +along the periphery, <img style="vertical-align: 0; width: 1.873ex; height: 1.545ex;" src="images/80.svg" alt=" " data-tex="D"> the number along the diameter, then, if +<img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' does not rotate relatively to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, we shall have +<span class="align-center"><img style="vertical-align: -1.552ex; width: 7.804ex; height: 4.627ex;" src="images/344.svg" alt=" " data-tex=" +\frac{U}{D} = \pi. +"></span> +<span class="pagenum" id="Page_63">[Pg 63]</span> +But if <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' rotates we get a different result. Suppose that at +a definite time <img style="vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;" src="images/97.svg" alt=" " data-tex="t"> of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> we determine the ends of all the rods. +With respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K"> all the rods upon the periphery experience +the Lorentz contraction, but the rods upon the diameter do not +experience this contraction (along their lengths!).<a id="FNanchor_14_1"></a><a href="#Footnote_14_1" class="fnanchor">[14]</a> +It therefore follows that +<span class="align-center"><img style="vertical-align: -1.552ex; width: 7.804ex; height: 4.627ex;" src="images/345.svg" alt=" " data-tex=" +\frac{U}{D} > \pi. +"></span> +</p> +<p> +It therefore follows that the laws of configuration of rigid +bodies with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' do not agree with the laws of configuration +of rigid bodies that are in accordance with Euclidean geometry. +If, further, we place two similar clocks (rotating with <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">'), +one upon the periphery, and the other at the centre of the circle, +then, judged from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, the clock on the periphery will go +slower than the clock at the centre. The same thing must take +place, judged from <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">', if we define time with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' in a +not wholly unnatural way, that is, in such a way that the laws +with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' depend explicitly upon the time. Space and +time, therefore, cannot be defined with respect to <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' as they +were in the special theory of relativity with respect to inertial +systems. But, according to the principle of equivalence, <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">' is +also to be considered as a system at rest, with respect to which +there is a gravitational field (field of centrifugal force, and force +of Coriolis). We therefore arrive at the result: the gravitational +field influences and even determines the metrical laws of the +space-time continuum. If the laws of configuration of ideal rigid +bodies are to be expressed geometrically, then in the presence +of a gravitational field the geometry is not Euclidean. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_14_1"></a><a href="#FNanchor_14_1"><span class="label">[14]</span></a>These considerations assume that the behaviour of rods and clocks +depends only upon velocities, and not upon accelerations, or, at least, that +the influence of acceleration does not counteract that of velocity.</p></div> +<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> +<p> +The case that we have been considering is analogous to that +which is presented in the two-dimensional treatment of surfaces. +It is impossible in the latter case also, to introduce co-ordinates +on a surface (e.g. the surface of an ellipsoid) which +have a simple metrical significance, while on a plane the Cartesian +co-ordinates, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, signify directly lengths measured by a +unit measuring rod. Gauss overcame this difficulty, in his theory +of surfaces, by introducing curvilinear co-ordinates which, +apart from satisfying conditions of continuity, were wholly arbitrary, +and afterwards these co-ordinates were related to the +metrical properties of the surface. In an analogous way we +shall introduce in the general theory of relativity arbitrary co-ordinates, +<img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">, which shall number uniquely the space-time +points, so that neighbouring events are associated with +neighbouring values of the co-ordinates; otherwise, the choice +of co-ordinates is arbitrary. We shall be true to the principle +of relativity in its broadest sense if we give such a form to the +laws that they are valid in every such four-dimensional system +of co-ordinates, that is, if the equations expressing the laws are +co-variant with respect to arbitrary transformations. +</p> +<p> +The most important point of contact between Gauss's theory +of surfaces and the general theory of relativity lies in the metrical +properties upon which the concepts of both theories, in the +main, are based. In the case of the theory of surfaces, Gauss's +argument is as follows. Plane geometry may be based upon the +concept of the distance <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds">, between two indefinitely near points. +The concept of this distance is physically significant because +the distance can be measured directly by means of a rigid measuring +rod. By a suitable choice of Cartesian co-ordinates this +<span class="pagenum" id="Page_65">[Pg 65]</span> +distance may be expressed by the formula <img style="vertical-align: -0.339ex; width: 17.9ex; height: 2.357ex;" src="images/347.svg" alt=" " data-tex="ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}">. +We may base upon this quantity the concepts of the straight +line as the geodesic (<img style="vertical-align: -0.691ex; width: 8.391ex; height: 2.514ex;" src="images/348.svg" alt=" " data-tex="\delta\! \int\!\! ds = 0">), the interval, the circle, and the +angle, upon which the Euclidean plane geometry is built. A +geometry may be developed upon another continuously curved +surface, if we observe that an infinitesimally small portion of the +surface may be regarded as plane, to within relatively infinitesimal +quantities. There are Cartesian co-ordinates, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/349.svg" alt=" " data-tex="X_{1}">, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/350.svg" alt=" " data-tex="X_{2}">, upon +such a small portion of the surface, and the distance between +two points, measured by a measuring rod, is given by +<span class="align-center"><img style="vertical-align: -0.339ex; width: 19.687ex; height: 2.357ex;" src="images/351.svg" alt=" " data-tex=" +ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}. +"></span> +If we introduce arbitrary curvilinear co-ordinates, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, on the +surface, then <img style="vertical-align: -0.339ex; width: 4.037ex; height: 1.91ex;" src="images/352.svg" alt=" " data-tex="dX_{1}">, <img style="vertical-align: -0.339ex; width: 4.037ex; height: 1.91ex;" src="images/353.svg" alt=" " data-tex="dX_{2}">, may be expressed linearly in terms of +<img style="vertical-align: -0.339ex; width: 3.458ex; height: 1.91ex;" src="images/354.svg" alt=" " data-tex="dx_{1}">, <img style="vertical-align: -0.339ex; width: 3.458ex; height: 1.91ex;" src="images/355.svg" alt=" " data-tex="dx_{2}">. Then everywhere upon the surface we have +<span class="align-center"><img style="vertical-align: -0.464ex; width: 39.454ex; height: 2.482ex;" src="images/356.svg" alt=" " data-tex=" +ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + + g_{22}\, {dx_{2}}^{2}, +"></span> +where <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/357.svg" alt=" " data-tex="g_{11}">, <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/358.svg" alt=" " data-tex="g_{12}">, <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/359.svg" alt=" " data-tex="g_{22}"> are determined by the nature of the surface +and the choice of co-ordinates; if these quantities are known, +then it is also known how networks of rigid rods may be laid +upon the surface. In other words, the geometry of surfaces may +be based upon this expression for <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> exactly as plane geometry +is based upon the corresponding expression. +</p> +<p> +There are analogous relations in the four-dimensional space-time +continuum of physics. In the immediate neighbourhood of +an observer, falling freely in a gravitational field, there exists no +gravitational field. We can therefore always regard an infinitesimally +small region of the space-time continuum as Galilean. +For such an infinitely small region there will be an inertial system +(with the space co-ordinates, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/349.svg" alt=" " data-tex="X_{1}">, <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/350.svg" alt=" " data-tex="X_{2}">, <img style="vertical-align: -0.375ex; width: 2.861ex; height: 1.92ex;" src="images/361.svg" alt=" " data-tex="X_{3}">, and the time +<span class="pagenum" id="Page_66">[Pg 66]</span> +co-ordinate <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/362.svg" alt=" " data-tex="X_{4}">) relatively to which we are to regard the laws of +the special theory of relativity as valid. The quantity which is +directly measurable by our unit measuring rods and clocks, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 29.027ex; height: 2.457ex;" src="images/363.svg" alt=" " data-tex=" +{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2}, +"></span> +or its negative, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 45.954ex; height: 2.584ex;" src="images/364.svg" alt=" " data-tex=" +ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}, +\qquad \text{(54)} +"></span> +is therefore a uniquely determinate invariant for two neighbouring +events (points in the four-dimensional continuum), provided +that we use measuring rods that are equal to each other when +brought together and superimposed, and clocks whose rates are +the same when they are brought together. In this the physical +assumption is essential that the relative lengths of two measuring +rods and the relative rates of two clocks are independent, in +principle, of their previous history. But this assumption is certainly +warranted by experience; if it did not hold there could be +no sharp spectral lines; for the single atoms of the same element +certainly do not have the same history, and it would be absurd +to suppose any relative difference in the structure of the single +atoms due to their previous history if the mass and frequencies +of the single atoms of the same element were always the same. +</p> +<p> +Space-time regions of finite extent are, in general, not +Galilean, so that a gravitational field cannot be done away +with by any choice of co-ordinates in a finite region. There +is, therefore, no choice of co-ordinates for which the metrical +relations of the special theory of relativity hold in a finite region. +But the invariant <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> always exists for two neighbouring +points (events) of the continuum. This invariant <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> may be +<span class="pagenum" id="Page_67">[Pg 67]</span> +expressed in arbitrary co-ordinates. If one observes that the +local <img style="vertical-align: -0.339ex; width: 4.085ex; height: 1.91ex;" src="images/365.svg" alt=" " data-tex="dX_{\nu}"> may be expressed linearly in terms of the co-ordinate +differentials <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> may be expressed in the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.76ex; height: 2.685ex;" src="images/366.svg" alt=" " data-tex=" +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. +\qquad \text{(55)} +"></span> +</p> +<p> +The functions <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> describe, with respect to the arbitrarily +chosen system of co-ordinates, the metrical relations of the +space-time continuum and also the gravitational field. As in +the special theory of relativity, we have to discriminate between +time-like and space-like line elements in the four-dimensional +continuum; owing to the change of sign introduced, time-like line +elements have a real, space-like line elements an imaginary <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds">. +The time-like <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> can be measured directly by a suitably chosen +clock. +</p> +<p> +According to what has been said, it is evident that the formulation +of the general theory of relativity assumes a generalization +of the theory of invariants and the theory of tensors; the question +is raised as to the form of the equations which are co-variant +with respect to arbitrary point transformations. The generalized +calculus of tensors was developed by mathematicians long before +the theory of relativity. Riemann first extended Gauss's +train of thought to continua of any number of dimensions; with +prophetic vision he saw the physical meaning of this generalization +of Euclid's geometry. Then followed the development of +the theory in the form of the calculus of tensors, particularly by +Ricci and Levi-Civita. This is the place for a brief presentation +of the most important mathematical concepts and operations of +this calculus of tensors. +</p> +<p> +We designate four quantities, which are defined as functions +of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> with respect to every system of co-ordinates, as components, +<span class="pagenum" id="Page_68">[Pg 68]</span> +<img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}">, of a contra-variant vector, if they transform in a +change of co-ordinates as the co-ordinate differentials <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">. We +therefore have +<span class="align-center"><img style="vertical-align: -1.891ex; width: 24.13ex; height: 5.334ex;" src="images/369.svg" alt=" " data-tex=" +{{A'}^{\mu}} = \frac{{\partial x'}_{\mu}}{\partial x_{\nu}} A^{\nu}. +\qquad \text{(56)} +"></span> +Besides these contra-variant vectors, there are also co-variant +vectors. If <img style="vertical-align: -0.339ex; width: 2.753ex; height: 1.885ex;" src="images/370.svg" alt=" " data-tex="B_{\nu}"> are the components of a co-variant vector, these +vectors are transformed according to the rule +<span class="align-center"><img style="vertical-align: -2.237ex; width: 24.171ex; height: 5.384ex;" src="images/371.svg" alt=" " data-tex=" +{B'}_{\mu} = \frac{\partial x_{\nu}}{{\partial {x'}}_{\mu}} B_{\nu}. +\qquad \text{(57)} +"></span> +The definition of a co-variant vector is chosen in such a way that +a co-variant vector and a contra-variant vector together form a +scalar according to the scheme, +<span class="align-center"><img style="vertical-align: -0.566ex; width: 32.939ex; height: 2.262ex;" src="images/372.svg" alt=" " data-tex=" +\phi = B_{\nu} A^{\nu}\quad \text{(summed over the } \nu). +"></span> +Accordingly, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 35.894ex; height: 5.68ex;" src="images/373.svg" alt=" " data-tex=" +{B'}_{\mu} {{A'}^{\mu}} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} + \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} B_{\alpha} A^{\beta} + = B_{\alpha} A^{\alpha}. +"></span> +In particular, the derivatives <img style="vertical-align: -1.909ex; width: 4.782ex; height: 5.056ex;" src="images/374.svg" alt=" " data-tex="\dfrac{\partial \phi}{\partial x_{\alpha}}"> of a scalar +<img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi">, are components +of a co-variant vector, which, with the co-ordinate differentials, +form the scalar <img style="vertical-align: -1.909ex; width: 8.842ex; height: 5.056ex;" src="images/375.svg" alt=" " data-tex="\dfrac{\partial \phi}{\partial x_{\alpha}}\, dx_{\alpha}">; +we see from this example how natural +is the definition of the co-variant vectors. +</p> +<p> +There are here, also, tensors of any rank, which may have +co-variant or contra-variant character with respect to each index; +as with vectors, the character is designated by the position +<span class="pagenum" id="Page_69">[Pg 69]</span> +of the index. For example, <img style="vertical-align: -0.904ex; width: 2.849ex; height: 2.524ex;" src="images/376.svg" alt=" " data-tex="{A_{\mu}^{\nu}}"> denotes a tensor of the second +rank, which is co-variant with respect to the index <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, and contra-variant +with respect to the index <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">. The tensor character indicates +that the equation of transformation is +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.54ex; height: 5.484ex;" src="images/377.svg" alt=" " data-tex=" +{{A'}_{\mu}^{\nu}} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} + \frac{{\partial x'}_{\nu}}{\partial x_{\beta}} A_{\alpha}^{\beta}. +\qquad \text{(58)} +"></span> +</p> +<p> +Tensors may be formed by the addition and subtraction of +tensors of equal rank and like character, as in the theory of +invariants of orthogonal linear substitutions, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 23.929ex; height: 2.601ex;" src="images/378.svg" alt=" " data-tex=" +A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}. +\qquad \text{(59)} +"></span> +The proof of the tensor character of <img style="vertical-align: -0.904ex; width: 2.873ex; height: 2.499ex;" src="images/379.svg" alt=" " data-tex="C_{\mu}^{\nu}"> depends upon (58). +</p> +<p> +Tensors may be formed by multiplication, keeping the character +of the indices, just as in the theory of invariants of linear +orthogonal transformations, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 23.576ex; height: 2.601ex;" src="images/380.svg" alt=" " data-tex=" +A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}. +\qquad \text{(60)} +"></span> +The proof follows directly from the rule of transformation. +</p> +<p> +Tensors may be formed by contraction with respect to two +indices of different character, for example, +<span class="align-center"><img style="vertical-align: -0.904ex; width: 20.806ex; height: 2.601ex;" src="images/381.svg" alt=" " data-tex=" +A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}. +\qquad \text{(61)} +"></span> +The tensor character of <img style="vertical-align: -0.66ex; width: 4.59ex; height: 2.513ex;" src="images/382.svg" alt=" " data-tex="A_{\mu\sigma\tau}^{\mu}"> determines the tensor character +of <img style="vertical-align: -0.36ex; width: 3.646ex; height: 1.905ex;" src="images/383.svg" alt=" " data-tex="B_{\sigma\tau}">. Proof— +<span class="align-center"><img style="vertical-align: -2.237ex; width: 52.078ex; height: 5.68ex;" src="images/384.svg" alt=" " data-tex=" +{{A'}_{\mu\sigma\tau}^{\mu}} += \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} + \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} + {A_{\alpha st}^{\beta}} += \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} + A_{\alpha st}^{\alpha}. +"></span> +<span class="pagenum" id="Page_70">[Pg 70]</span> +</p> +<p> +The properties of symmetry and skew-symmetry of a tensor +with respect to two indices of like character have the same +significance as in the theory of invariants. +</p> +<p> +With this, everything essential has been said with regard to +the algebraic properties of tensors. +</p> + +<p><br></p> + +<p> +<i>The Fundamental Tensor</i>. It follows from the invariance +of <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}">for an arbitrary choice of the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">, in connexion with +the condition of symmetry consistent with (55), that the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> +are components of a symmetrical co-variant tensor (Fundamental +Tensor). Let us form the determinant, <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/385.svg" alt=" " data-tex="g">, of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">, and +also the minors, divided by <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/385.svg" alt=" " data-tex="g">, corresponding to the single <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. +These minors, divided by <img style="vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;" src="images/385.svg" alt=" " data-tex="g">, will be denoted by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, and their +co-variant character is not yet known. Then we have +<span class="align-center"><img style="vertical-align: -2.148ex; width: 36.542ex; height: 5.428ex;" src="images/387.svg" alt=" " data-tex=" +g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} + = \begin{cases} + 1 & \text{if \(\alpha = \beta,\)} \\ + 0 & \text{if \(\alpha \neq \beta.\)} + \end{cases} +\qquad \text{(62)} +"></span> +</p> +<p> +If we form the infinitely small quantities (co-variant vectors) +<span class="align-center"><img style="vertical-align: -0.685ex; width: 23.206ex; height: 2.382ex;" src="images/388.svg" alt=" " data-tex=" +d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}, +\qquad \text{(63)} +"></span> +multiply by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> and sum over the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, we obtain, by the use +of (62), +<span class="align-center"><img style="vertical-align: -0.685ex; width: 22.969ex; height: 2.747ex;" src="images/389.svg" alt=" " data-tex=" +dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}. +\qquad \text{(64)} +"></span> +Since the ratios of the <img style="vertical-align: -0.685ex; width: 3.32ex; height: 2.278ex;" src="images/390.svg" alt=" " data-tex="d\xi_{\mu}">, are arbitrary, and the <img style="vertical-align: -0.65ex; width: 3.564ex; height: 2.22ex;" src="images/391.svg" alt=" " data-tex="dx_{\beta}"> as well as +the <img style="vertical-align: -0.685ex; width: 3.623ex; height: 2.255ex;" src="images/392.svg" alt=" " data-tex="dx_{\mu}"> are components of vectors, it follows that the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> are the +components of a contra-variant tensor<a id="FNanchor_15_1"></a><a href="#Footnote_15_1" class="fnanchor">[15]</a> +(contra-variant fundamental tensor). +The tensor character of <img style="vertical-align: -0.332ex; width: 2.216ex; height: 2.57ex;" src="images/393.svg" alt=" " data-tex="\delta_{\alpha}^{\beta}"> (mixed fundamental +<span class="pagenum" id="Page_71">[Pg 71]</span> +tensor) accordingly follows, by (62). By means of the fundamental +tensor, instead of tensors with co-variant index character, we +can introduce tensors with contra-variant index character, and +conversely. For example, +<span class="align-center"><img style="vertical-align: -3.736ex; width: 12.845ex; height: 8.603ex;" src="images/394.svg" alt=" " data-tex=" +\begin{align*} +A^{\mu} &= g^{\mu\alpha} A_{\alpha}, \\ +A_{\mu} &= g_{\mu\alpha} A^{\alpha}, \\ +T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. +\end{align*} +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_15_1"></a><a href="#FNanchor_15_1"><span class="label">[15]</span></a>If we multiply (64) +by <img style="vertical-align: -2.202ex; width: 5.41ex; height: 5.448ex;" src="images/395.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\alpha}}{\partial x_{\beta}}">, +sum over the <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, and replace the <img style="vertical-align: -0.464ex; width: 3.32ex; height: 2.057ex;" src="images/396.svg" alt=" " data-tex="d\xi^{\mu}"> by +a transformation to the accented system, we obtain +<span class="align-center"><img style="vertical-align: -2.237ex; width: 26.832ex; height: 5.484ex;" src="images/397.svg" alt=" " data-tex=" +{dx'}_{\alpha} + = \frac{{\partial x'}_{\sigma}}{\partial x_{\mu}}\, + \frac{{\partial x'}_{\alpha}}{\partial x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}. +"></span> +The statement made above follows from this, since, by (64), we must also +have <img style="vertical-align: -0.561ex; width: 15.36ex; height: 2.597ex;" src="images/398.svg" alt=" " data-tex="{dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}"> and both equations +must hold for every choice of <img style="vertical-align: -0.561ex; width: 3.896ex; height: 2.279ex;" src="images/399.svg" alt=" " data-tex="{d\xi'}_{\sigma}">.</p></div> + +<p><br></p> + +<p> +<i>Volume Invariants</i>. The volume element +<span class="align-center"><img style="vertical-align: -1.948ex; width: 22.967ex; height: 5.027ex;" src="images/400.svg" alt=" " data-tex=" +\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx +"></span> +is not an invariant. For by Jacobi's theorem, +<span class="align-center"><img style="vertical-align: -2.312ex; width: 24.893ex; height: 5.754ex;" src="images/401.svg" alt=" " data-tex=" +dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx. +\qquad \text{(65)} +"></span> +But we can complement <img style="vertical-align: -0.025ex; width: 2.471ex; height: 1.595ex;" src="images/402.svg" alt=" " data-tex="dx"> so that it becomes an invariant. If +we form the determinant of the quantities +<span class="align-center"><img style="vertical-align: -2.237ex; width: 21.889ex; height: 5.545ex;" src="images/403.svg" alt=" " data-tex=" +{g'}_{\mu\nu} + = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}}\, + \frac{\partial x_{\beta}}{{\partial x'}_{\nu}}\, g_{\alpha\beta}, +"></span> +<span class="pagenum" id="Page_72">[Pg 72]</span> +we obtain, by a double application of the theorem of multiplication +of determinants, +<span class="align-center"><img style="vertical-align: -2.312ex; width: 49.262ex; height: 6.202ex;" src="images/404.svg" alt=" " data-tex=" +g' = |{g'}_{\mu\nu}| + = \left|\frac{\partial x_{\nu}}{{\partial x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| + = \left|\frac{{\partial x'}_{\mu}}{\partial x_{\nu}}\right|^{-2} g. +\qquad \text{(66)} +"></span> +We therefore get the invariant, +<span class="align-center"><img style="vertical-align: -0.554ex; width: 17.372ex; height: 2.851ex;" src="images/405.svg" alt=" " data-tex=" +\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx. +"></span> +</p> + +<p><br></p> + +<p> +<i>Formation of Tensors by Differentiation</i>. Although the algebraic +operations of tensor formation have proved to be as +simple as in the special case of invariance with respect to linear +orthogonal transformations, nevertheless in the general case, +the invariant differential operations are, unfortunately, considerably +more complicated. The reason for this is as follows. If +<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}"> is a contra-variant vector, the coefficients of its +transformation, <img style="vertical-align: -1.891ex; width: 5.35ex; height: 5.334ex;" src="images/407.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}">, are +independent of position only if the transformation +is a linear one. For then the vector components, +<img style="vertical-align: -1.909ex; width: 14.8ex; height: 5.058ex;" src="images/408.svg" alt=" " data-tex="A^{\mu} + \dfrac{\partial A^{\mu}}{\partial x_{\alpha}}\, dx_{\alpha}">, +at a neighbouring point transform in the same way as the <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">, +from which follows the vector character of the vector differentials, +and the tensor character of <img style="vertical-align: -1.909ex; width: 5.125ex; height: 5.058ex;" src="images/409.svg" alt=" " data-tex="\dfrac{\partial A^{\mu}}{\partial x_{\alpha}}">. +But if the <img style="vertical-align: -1.891ex; width: 5.35ex; height: 5.334ex;" src="images/407.svg" alt=" " data-tex="\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}"> +are variable this is no longer true. +</p> +<p> +That there are, nevertheless, in the general case, invariant +differential operations for tensors, is recognized most satisfactorily +in the following way, introduced by Levi-Civita and Weyl. +Let (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) be a contra-variant vector whose components are given +with respect to the co-ordinate system of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. +Let <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> and <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> +<span class="pagenum" id="Page_73">[Pg 73]</span> +be two infinitesimally near points of the continuum. For the infinitesimal +region surrounding the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}">, there is, according +to our way of considering the matter, a co-ordinate system of +the <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}"> (with imaginary <img style="vertical-align: -0.339ex; width: 2.861ex; height: 1.885ex;" src="images/362.svg" alt=" " data-tex="X_{4}">-co-ordinate) for which the +continuum is Euclidean. Let <img style="vertical-align: -1.207ex; width: 3.929ex; height: 3.06ex;" src="images/410.svg" alt=" " data-tex="A_{(1)}^{\mu}"> be the co-ordinates of the vector at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}">. Imagine a vector drawn at the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, using the +local system of the <img style="vertical-align: -0.339ex; width: 2.909ex; height: 1.885ex;" src="images/106.svg" alt=" " data-tex="X_{\nu}">, with the same co-ordinates (parallel vector +through <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">), then this parallel vector is uniquely determined +by the vector at <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> and the displacement. We designate this operation, +whose uniqueness will appear in the sequel, the parallel +displacement of the vector <img style="vertical-align: -0.685ex; width: 2.849ex; height: 2.305ex;" src="images/411.svg" alt=" " data-tex="A_{\mu}"> from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> to the infinitesimally near +point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">. If we form the vector difference of the vector (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> and the vector obtained by parallel displacement +from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, we get a vector which may be regarded as the +differential of the vector (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) for the given displacement <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">. +</p> +<p> +This vector displacement can naturally also be considered +with respect to the co-ordinate system of the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">. If <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}"> are the +co-ordinates of the vector at <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}">, <img style="vertical-align: -0.186ex; width: 9.235ex; height: 1.808ex;" src="images/412.svg" alt=" " data-tex="A^{\nu} + \delta A^{\nu}"> the co-ordinates of +the vector displaced to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}"> along the interval (<img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}">), then +the <img style="vertical-align: -0.023ex; width: 3.737ex; height: 1.645ex;" src="images/413.svg" alt=" " data-tex="\delta A^{\nu}"> +do not vanish in this case. We know of these quantities, which +do not have a vector character, that they must depend linearly +and homogeneously upon the <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}"> and the <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}">. We therefore put +<span class="align-center"><img style="vertical-align: -0.911ex; width: 28.449ex; height: 2.608ex;" src="images/414.svg" alt=" " data-tex=" +\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}. +\qquad \text{(67)} +"></span> +</p> +<p> +In addition, we can state that the <img style="vertical-align: -1.024ex; width: 3.531ex; height: 2.563ex;" src="images/415.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\nu}"> must be symmetrical +with respect to the indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">. +For we can assume from +a representation by the aid of a Euclidean system of local co-ordinates +that the same parallelogram will be described by the +displacement of an element <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/416.svg" alt=" " data-tex="d^{(1)}x_{\nu}"> along a second element +<img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/417.svg" alt=" " data-tex="d^{(2)}x_{\nu}"> +<span class="pagenum" id="Page_74">[Pg 74]</span> +as by a displacement of <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/417.svg" alt=" " data-tex="d^{(2)}x_{\nu}"> along <img style="vertical-align: -0.339ex; width: 5.739ex; height: 2.36ex;" src="images/416.svg" alt=" " data-tex="d^{(1)}x_{\nu}">. +We must therefore have +<span class="align-center"><img style="vertical-align: -2.819ex; width: 73.179ex; height: 6.769ex;" src="images/418.svg" alt=" " data-tex=" +\begin{aligned} +d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ + \quad + &= d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} + - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). +\end{aligned} +"></span> +The statement made above follows from this, after interchanging +the indices of summation, <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, on the right-hand side. +</p> +<p> +Since the quantities <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> determine all the metrical properties +of the continuum, they must also determine the <img style="vertical-align: -1.024ex; width: 3.531ex; height: 2.563ex;" src="images/415.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\nu}">. If we +consider the invariant of the vector <img style="vertical-align: 0; width: 2.733ex; height: 1.62ex;" src="images/368.svg" alt=" " data-tex="A^{\nu}"> that is, the square of its +magnitude, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 9.29ex; height: 2.326ex;" src="images/419.svg" alt=" " data-tex=" +g_{\mu\nu} A^{\mu} A^{\nu}, +"></span> +which is an invariant, this cannot change in a parallel displacement. +We therefore have +<span class="align-center"><img style="vertical-align: -1.909ex; width: 58.452ex; height: 5.252ex;" src="images/420.svg" alt=" " data-tex=" +0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) + = \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu} +"></span> +or, by (67), +<span class="align-center"><img style="vertical-align: -2.148ex; width: 41.858ex; height: 5.491ex;" src="images/421.svg" alt=" " data-tex=" +\left(\frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} + - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} + - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} += 0. +"></span> +</p> +<p> +Owing to the symmetry of the expression in the brackets +with respect to the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, this equation can be valid +for an arbitrary choice of the vectors (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">) and <img style="vertical-align: -0.339ex; width: 3.506ex; height: 1.91ex;" src="images/99.svg" alt=" " data-tex="dx_{\nu}"> only when +the expression in the brackets vanishes for all combinations of +the indices. By a cyclic interchange of the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu">, <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, we +obtain thus altogether three equations, from which we obtain, +on taking into account the symmetrical property of the +<img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/422.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\alpha}">, +<span class="align-center"><img style="vertical-align: -1.577ex; width: 24.423ex; height: 4.178ex;" src="images/423.svg" alt=" " data-tex=" +\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] + = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}, +\qquad \text{(68)} +"></span> +<span class="pagenum" id="Page_75">[Pg 75]</span> +in which, following Christoffel, the abbreviation has been used, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 44.772ex; height: 5.58ex;" src="images/424.svg" alt=" " data-tex=" +\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] + = \tfrac{1}{2}\left( + \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} + - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} +\right). +\qquad \text{(69)} +"></span> +</p> +<p> +If we multiply (68) by <img style="vertical-align: -0.464ex; width: 3.204ex; height: 1.992ex;" src="images/425.svg" alt=" " data-tex="g^{\alpha\sigma}"> and sum over the <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, we obtain +<span class="align-center"><img style="vertical-align: -2.237ex; width: 55.667ex; height: 5.58ex;" src="images/426.svg" alt=" " data-tex=" +\Gamma_{\mu\nu}^{\alpha} + = \tfrac{1}{2} g^{\sigma\alpha}\left( + \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} + - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} +\right) = \left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}, +\qquad \text{(70)} +"></span> +in which <img style="vertical-align: -0.798ex; width: 4.994ex; height: 2.72ex;" src="images/427.svg" alt=" " data-tex="\left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}"> is +the Christoffel symbol of the second kind. +Thus the quantities <img style="vertical-align: 0; width: 1.414ex; height: 1.538ex;" src="images/428.svg" alt=" " data-tex="\Gamma"> are deduced from the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. Equations +(67) and (70) are the foundation for the following discussion. +</p> + +<p><br></p> + +<p> +<i>Co-variant Differentiation of Tensors</i>. If (<img style="vertical-align: -0.186ex; width: 9.469ex; height: 1.808ex;" src="images/429.svg" alt=" " data-tex="A^{\mu} + \delta A^{\mu}">) is +the vector resulting from an infinitesimal parallel displacement +from <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/170.svg" alt=" " data-tex="P_{1}"> to <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, and (<img style="vertical-align: -0.186ex; width: 9.641ex; height: 1.805ex;" src="images/430.svg" alt=" " data-tex="A^{\mu} + dA^{\mu}">) the vector <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}"> at +the point <img style="vertical-align: -0.339ex; width: 2.44ex; height: 1.885ex;" src="images/171.svg" alt=" " data-tex="P_{2}">, +then the difference of these two, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 35.532ex; height: 5.428ex;" src="images/431.svg" alt=" " data-tex=" +dA^{\mu} - \delta A^{\mu} = \left( + \frac{\partial A^{\mu}}{\partial x_{\sigma}} + + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma}, +"></span> +is also a vector. Since this is the case for an arbitrary choice of +the <img style="vertical-align: -0.357ex; width: 3.572ex; height: 1.927ex;" src="images/432.svg" alt=" " data-tex="dx_{\sigma}">, it follows that +<span class="align-center"><img style="vertical-align: -1.909ex; width: 30.676ex; height: 5.058ex;" src="images/433.svg" alt=" " data-tex=" +{A^{\mu}}_{;\, \sigma} + = \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} +\qquad \text{(71)} +"></span> +is a tensor, which we designate as the co-variant derivative of +the tensor of the first rank (vector). Contracting this tensor, we +obtain the divergence of the contra-variant tensor <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">. In this +we must observe that according to (70), +<span class="align-center"><img style="vertical-align: -2.308ex; width: 39.215ex; height: 5.747ex;" src="images/434.svg" alt=" " data-tex=" +\Gamma_{\mu\sigma}^{\sigma} + = \tfrac{1}{2} g^{\sigma\alpha} \frac{\partial g_{\sigma\alpha}}{\partial x_{\mu}} + = \frac{1}{\sqrt{g}}\, \frac{\partial \sqrt{g}}{\partial x_{\mu}}. +\qquad \text{(72)} +"></span> +<span class="pagenum" id="Page_76">[Pg 76]</span> +If we put, further, +<span class="align-center"><img style="vertical-align: -0.602ex; width: 21.206ex; height: 2.398ex;" src="images/435.svg" alt=" " data-tex=" +A^{\mu} \sqrt{g} = \mathfrak A^{\mu}, +\qquad \text{(73)} +"></span> +a quantity designated by Weyl as the contra-variant tensor density<a id="FNanchor_16_1"></a><a href="#Footnote_16_1" class="fnanchor">[16]</a> +of the first rank, it follows that, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 18.242ex; height: 5.384ex;" src="images/436.svg" alt=" " data-tex=" +\mathfrak A = \frac{\partial \mathfrak A^{\mu}}{\partial x_{\mu}} +\qquad \text{(74)} +"></span> +is a scalar density. +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_16_1"></a><a href="#FNanchor_16_1"><span class="label">[16]</span></a>This expression is justified, in that +<img style="vertical-align: -0.71ex; width: 17.691ex; height: 2.398ex;" src="images/437.svg" alt=" " data-tex="A^{\mu}\sqrt{g}\, dx = \mathbf A^{\mu}\, dx"> has a tensor +character. Every tensor, when multiplied by <img style="vertical-align: -0.71ex; width: 3.009ex; height: 2.398ex;" src="images/438.svg" alt=" " data-tex="\sqrt{g}">, changes into a tensor +density. We employ capital Gothic letters for tensor densities.</p></div> + +<p> +We get the law of parallel displacement for the co-variant +vector <img style="vertical-align: -0.685ex; width: 2.87ex; height: 2.23ex;" src="images/439.svg" alt=" " data-tex="B_{\mu}"> by stipulating that the parallel displacement shall be +effected in such a way that the scalar +<span class="align-center"><img style="vertical-align: -0.685ex; width: 10.084ex; height: 2.326ex;" src="images/440.svg" alt=" " data-tex=" +\phi = A^{\mu} B_{\mu} +"></span> +remains unchanged, and that therefore +<span class="align-center"><img style="vertical-align: -0.685ex; width: 16.968ex; height: 2.326ex;" src="images/441.svg" alt=" " data-tex=" +A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu} +"></span> +vanishes for every value assigned to (<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">). We therefore get +<span class="align-center"><img style="vertical-align: -0.904ex; width: 26.783ex; height: 2.601ex;" src="images/442.svg" alt=" " data-tex=" +\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}. +\qquad \text{(75)} +"></span> +</p> +<p> +From this we arrive at the co-variant derivative of the co-variant +vector by the same process as that which led to (71), +<span class="align-center"><img style="vertical-align: -1.909ex; width: 31.497ex; height: 5.252ex;" src="images/443.svg" alt=" " data-tex=" +B_{\mu;\, \sigma} + = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}. +\qquad \text{(76)} +"></span> +<span class="pagenum" id="Page_77">[Pg 77]</span> +By interchanging the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, and subtracting, we get +the skew-symmetrical tensor, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 28.991ex; height: 5.58ex;" src="images/444.svg" alt=" " data-tex=" +\phi_{\mu\sigma} + = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \frac{\partial B_{\sigma}}{\partial x_{\mu}}. +\qquad \text{(77)} +"></span> +</p> +<p> +For the co-variant differentiation of tensors of the second +and higher ranks we may use the process by which (75) was +deduced. Let, for example, (<img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/445.svg" alt=" " data-tex="A_{\sigma\tau}">) be a co-variant tensor of the +second rank. Then <img style="vertical-align: -0.36ex; width: 9.406ex; height: 1.98ex;" src="images/446.svg" alt=" " data-tex="A_{\sigma\tau} E^{\sigma} F^{\tau}"> is a scalar, +if <img style="vertical-align: 0; width: 1.729ex; height: 1.538ex;" src="images/292.svg" alt=" " data-tex="E"> and <img style="vertical-align: 0; width: 1.695ex; height: 1.538ex;" src="images/447.svg" alt=" " data-tex="F"> are vectors. +This expression must not be changed by the <img style="vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;" src="images/117.svg" alt=" " data-tex="\delta">-displacement; +expressing this by a formula, we get, using (67), <img style="vertical-align: -0.36ex; width: 4.63ex; height: 1.982ex;" src="images/448.svg" alt=" " data-tex="\delta A_{\sigma\tau}">, whence +we get the desired co-variant derivative, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 43.434ex; height: 5.386ex;" src="images/449.svg" alt=" " data-tex=" +A_{\sigma\tau;\, \rho} + = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} + - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}. +\qquad \text{(78)} +"></span> +</p> +<p> +In order that the general law of co-variant differentiation of +tensors may be clearly seen, we shall write down two co-variant +derivatives deduced in an analogous way: +<span class="align-center"><img style="vertical-align: -5.204ex; width: 48.077ex; height: 11.539ex;" src="images/450.svg" alt=" " data-tex=" +\begin{align*} +A_{\sigma;\, \rho}^{\tau} + &= \frac{\partial A_{\sigma}^{\tau}}{\partial x_{\rho}} + - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha},\, +\qquad & &\text{(79)} \\ +{A^{\sigma\tau}}_{;\, \rho} + &= \frac{\partial A^{\sigma\tau}}{\partial x_{\rho}} + + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}. +\qquad & &\text{(80)} +\end{align*} +"></span> +The general law of formation now becomes evident. From these +formulae we shall deduce some others which are of interest for +the physical applications of the theory. +</p> +<p> +In case <img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/445.svg" alt=" " data-tex="A_{\sigma\tau}"> is skew-symmetrical, we obtain the tensor +<span class="align-center"><img style="vertical-align: -2.237ex; width: 40.172ex; height: 5.583ex;" src="images/451.svg" alt=" " data-tex=" +A_{\sigma\tau\rho} + = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + + \frac{\partial A_{\tau\rho}}{\partial x_{\sigma}} + + \frac{\partial A_{\rho\sigma}}{\partial x_{\tau}}, +\qquad \text{(81)} +"></span> +<span class="pagenum" id="Page_78">[Pg 78]</span> +which is skew-symmetrical in all pairs of indices, by cyclic +interchange and addition. +</p> +<p> +If, in (78), we replace <img style="vertical-align: -0.36ex; width: 3.625ex; height: 1.98ex;" src="images/445.svg" alt=" " data-tex="A_{\sigma\tau}"> by the fundamental tensor, +<img style="vertical-align: -0.464ex; width: 3.008ex; height: 1.464ex;" src="images/452.svg" alt=" " data-tex="g_{\sigma\tau}">, +then the right-hand side vanishes identically; an analogous statement +holds for (80) with respect to <img style="vertical-align: -0.464ex; width: 3.008ex; height: 1.975ex;" src="images/453.svg" alt=" " data-tex="g^{\sigma\tau}">; that is, the co-variant +derivatives of the fundamental tensor vanish. That this must be +so we see directly in the local system of co-ordinates. +</p> +<p> +In case <img style="vertical-align: 0; width: 3.625ex; height: 1.62ex;" src="images/454.svg" alt=" " data-tex="A^{\sigma\tau}"> is skew-symmetrical, we obtain from (80), by +contraction with respect to <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, +<span class="align-center"><img style="vertical-align: -1.912ex; width: 21.125ex; height: 5.059ex;" src="images/455.svg" alt=" " data-tex=" +\mathfrak A^{\sigma} = \frac{\partial \mathfrak A^{\sigma\tau}}{\partial x_{\tau}}. +\qquad \text{(82)} +"></span> +</p> +<p> +In the general case, from (79) and (80), by contraction with +respect to <img style="vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;" src="images/128.svg" alt=" " data-tex="\tau"> and <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho">, we obtain the equations, +<span class="align-center"><img style="vertical-align: -4.874ex; width: 35.508ex; height: 10.879ex;" src="images/456.svg" alt=" " data-tex=" +\begin{align*} +\mathfrak A_{\sigma} &= \frac{\partial \mathfrak A_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak A_{\alpha}^{\beta}, +\qquad & &\text{(83)} \\ +\mathfrak A^{\sigma} &= \frac{\partial \mathfrak A^{\sigma\alpha}}{\partial x_{\alpha}} + + \Gamma_{\alpha\beta}^{\sigma} \mathfrak A^{\alpha\beta}. +\qquad & &\text{(84)} +\end{align*} +"></span> +</p> + +<p><br></p> + +<p> +<i>The Riemann Tensor</i>. If we have given a curve extending +from the point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> to the point <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G"> of the continuum, then a +vector <img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}">, given at <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">, may, by a parallel displacement, be moved +along the curve to <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G">. If the continuum is Euclidean (more generally, +if by a suitable choice of co-ordinates the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">, are constants) +then the vector obtained at <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G"> as a result of this displacement +does not depend upon the choice of the curve joining +<img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> and <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G">. +But otherwise, the result depends upon the path of the displacement. +</p> + +<div class="figcenter" style="width: 300px;"> +<img src="images/figure04.jpg" width="300" alt="300"> +<div class="caption"> +<p>FIG.4.</p> +</div></div> + +<p class="nind"> +In this case, therefore, a vector suffers a change, +<img style="vertical-align: 0; width: 4.734ex; height: 1.62ex;" src="images/458.svg" alt=" " data-tex="\Delta A^{\mu}"> +(in its direction, not its magnitude), when it is carried from a +<span class="pagenum" id="Page_79">[Pg 79]</span> +point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> of a closed curve, along the curve, and back to P. We +shall now calculate this vector change: +<span class="align-center"><img style="vertical-align: -1.948ex; width: 14.747ex; height: 5.027ex;" src="images/459.svg" alt=" " data-tex=" +\Delta A^{\mu} = \oint \delta A^{\mu}. +"></span> +As in Stokes' theorem for the line integral of a vector around +a closed curve, this problem may be reduced to the integration +around a closed curve with infinitely small linear dimensions; we +shall limit ourselves to this case. +</p> +<p> +We have, first, by (67), +<span class="align-center"><img style="vertical-align: -1.948ex; width: 23.411ex; height: 5.027ex;" src="images/460.svg" alt=" " data-tex=" +\Delta A^{\mu} = -\oint \Gamma_{\alpha\beta}^{\mu} A^{\alpha}\, dx_{\beta}. +"></span> +</p> +<p> +In this, <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> is the value of +this quantity at the variable +point <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G"> of the path of integration. If we put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 19.174ex; height: 2.382ex;" src="images/462.svg" alt=" " data-tex=" +\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P} +"></span> +<span class="pagenum" id="Page_80">[Pg 80]</span> +and denote the value of +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> at <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> by +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 3.615ex;" src="images/463.svg" alt=" " data-tex="\overline{\Gamma_{\alpha\beta}^{\mu}}"> then we have, with +sufficient accuracy, +<span class="align-center"><img style="vertical-align: -1.891ex; width: 21.686ex; height: 6.547ex;" src="images/464.svg" alt=" " data-tex=" +\Gamma_{\alpha\beta}^{\mu} + = \overline{\Gamma_{\alpha\beta}^{\mu}} + + \frac{\overline{\partial \Gamma_{\alpha\beta}^{\mu}}}{\partial x_{\nu}}\, \xi^{\nu}. +"></span> +</p> +<p> +Let, further, <img style="vertical-align: 0; width: 2.908ex; height: 1.62ex;" src="images/465.svg" alt=" " data-tex="A^{\alpha}"> be the value obtained from +<img style="vertical-align: 0; width: 2.908ex; height: 2.337ex;" src="images/466.svg" alt=" " data-tex="\overline{A^{\alpha}}"> by a parallel +displacement along the curve from <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P"> to <img style="vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;" src="images/457.svg" alt=" " data-tex="G">. It may now easily +be proved by means of (67) that +<img style="vertical-align: 0; width: 2.849ex; height: 1.62ex;" src="images/406.svg" alt=" " data-tex="A^{\mu}"> - <img style="vertical-align: 0; width: 2.849ex; height: 2.337ex;" src="images/467.svg" alt=" " data-tex="\overline{A^{\mu}}"> is infinitely small of +the first order, while, for a curve of infinitely small dimensions +of the first order, <img style="vertical-align: 0; width: 4.734ex; height: 1.62ex;" src="images/458.svg" alt=" " data-tex="\Delta A^{\mu}"> is infinitely small of the second order. +Therefore there is an error of only the second order if we put +<span class="align-center"><img style="vertical-align: -0.58ex; width: 21.633ex; height: 2.917ex;" src="images/468.svg" alt=" " data-tex=" +A^{\alpha} = \overline{A^{\alpha}} + - \overline{\Gamma_{\sigma\tau}^{\alpha}}\; + \overline{A^{\sigma}}\; + \overline{\xi^{\tau}}. +"></span> +</p> +<p> +If we introduce these values of +<img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> and <img style="vertical-align: 0; width: 2.908ex; height: 1.62ex;" src="images/465.svg" alt=" " data-tex="A^{\alpha}"> into the integral, +we obtain, neglecting all quantities of a higher order of small +quantities than the second, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 49.891ex; height: 6.785ex;" src="images/469.svg" alt=" " data-tex=" +\Delta A^{\mu} + = - \left(\frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) + A^{\sigma} \oint \xi^{\alpha}\, d\xi^{\beta}. +\qquad \text{(85)} +"></span> +The quantity removed from under the sign of integration refers +to the point <img style="vertical-align: 0; width: 1.699ex; height: 1.545ex;" src="images/63.svg" alt=" " data-tex="P">. Subtracting +<img style="vertical-align: -1.552ex; width: 9.35ex; height: 4.588ex;" src="images/470.svg" alt=" " data-tex="\dfrac{1}{2} d(\xi^{\alpha} \xi^{\beta})"> from the integrand, we +obtain +<span class="align-center"><img style="vertical-align: -1.948ex; width: 21.145ex; height: 5.027ex;" src="images/471.svg" alt=" " data-tex=" +\tfrac{1}{2} \oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}). +"></span> +This skew-symmetrical tensor of the second rank, +<img style="vertical-align: -0.464ex; width: 3.481ex; height: 2.413ex;" src="images/472.svg" alt=" " data-tex="f^{\alpha\beta}">, characterizes +the surface element bounded by the curve in magnitude +and position. If the expression in the brackets in (85) were +skew-symmetrical with respect to the indices +<img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, we could +<span class="pagenum" id="Page_81">[Pg 81]</span> +conclude its tensor character from (85). We can accomplish this +by interchanging the summation indices <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> in (85) and +adding the resulting equation to (85). We obtain +<span class="align-center"><img style="vertical-align: -1.045ex; width: 31.223ex; height: 3.108ex;" src="images/473.svg" alt=" " data-tex=" +2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}, +\qquad \text{(86)} +"></span> +in which +<span class="align-center"><img style="vertical-align: -2.202ex; width: 52.635ex; height: 6.141ex;" src="images/474.svg" alt=" " data-tex=" +R_{\sigma\alpha\beta}^{\mu} + = - \frac{\partial \Gamma_{\sigma\alpha}^{\mu}}{\partial x_{\beta}} + + \frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} + - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}. +\qquad \text{(87)} +"></span> +<p> +The tensor character of <img style="vertical-align: -1.045ex; width: 4.748ex; height: 2.898ex;" src="images/475.svg" alt=" " data-tex="R_{\sigma\alpha\beta}^{\mu}"> follows from (86); this is the +Riemann curvature tensor of the fourth rank, whose properties of +symmetry we do not need to go into. Its vanishing is a sufficient +condition (disregarding the reality of the chosen co-ordinates) +that the continuum is Euclidean. +</p> +<p> +By contraction of the Riemann tensor with respect to the +indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu">, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta">, we obtain the symmetrical tensor of the second +rank, +<span class="align-center"><img style="vertical-align: -1.909ex; width: 51.793ex; height: 5.472ex;" src="images/476.svg" alt=" " data-tex=" +R_{\mu\nu} + = - \frac{\partial \Gamma_{\mu\nu}^{\alpha}}{\partial x_{\alpha}} + + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + + \frac{\partial \Gamma_{\mu\alpha}^{\alpha}}{\partial x_{\nu}} + - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}. +\qquad \text{(88)} +"></span> +The last two terms vanish if the system of co-ordinates is so +chosen that <img style="vertical-align: -0.464ex; width: 12.531ex; height: 1.855ex;" src="images/477.svg" alt=" " data-tex="g = \text{constant}">. From <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}">, we can form the scalar, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 21.085ex; height: 2.382ex;" src="images/479.svg" alt=" " data-tex=" +R = g^{\mu\nu} R_{\mu\nu}. +\qquad \text{(89)} +"></span> +</p> + +<p><br></p> + +<p> +<i>Straightest Geodetic Lines</i>. A line may be constructed in +such a way that its successive elements arise from each other by +parallel displacements. This is the natural generalization of the +straight line of the Euclidean geometry. For such a line, we have +<span class="align-center"><img style="vertical-align: -2.148ex; width: 26.887ex; height: 5.444ex;" src="images/480.svg" alt=" " data-tex=" +\delta \left(\frac{dx_{\mu}}{ds}\right) + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}. +"></span> +<span class="pagenum" id="Page_82">[Pg 82]</span> +The left-hand side is to be replaced by +<img style="vertical-align: -1.651ex; width: 5.606ex; height: 5.264ex;" src="images/481.svg" alt=" " data-tex="\dfrac{d^{2} x_{\mu}}{ds^{2}}">,<a id="FNanchor_17_1"></a><a href="#Footnote_17_1" class="fnanchor">[17]</a> +so that we have +<span class="align-center"><img style="vertical-align: -1.651ex; width: 34.843ex; height: 5.264ex;" src="images/482.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0. +\qquad \text{(90)} +"></span> +We get the same line if we find the line which gives a stationary +value to the integral +<span class="align-center"><img style="vertical-align: -1.948ex; width: 27.455ex; height: 5.027ex;" src="images/483.svg" alt=" " data-tex=" +\int ds\quad\text{or}\quad +\int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}} +"></span> +between two points (geodetic line). +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_17_1"></a><a href="#FNanchor_17_1"><span class="label">[17]</span></a>The direction vector at a neighbouring point of the curve results, by +a parallel displacement along the line element (<img style="vertical-align: -0.65ex; width: 3.564ex; height: 2.22ex;" src="images/391.svg" alt=" " data-tex="dx_{\beta}">), from the direction +vector of each point considered.</p></div> +<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<h2 title="LECTURE IV: THE GENERAL THEORY OF RELATIVITY"><a id="chap04"></a>LECTURE IV +<br><br> +THE GENERAL THEORY OF RELATIVITY<br> +(<i>continued</i>)</h2> + +<p class="nind"> +<span class="dropcap">W</span>E are now in possession of the mathematical apparatus which +is necessary to formulate the laws of the general theory of relativity. +No attempt will be made in this presentation at systematic +completeness, but single results and possibilities will +be developed progressively from what is known and from the results +obtained. Such a presentation is most suited to the present +provisional state of our knowledge. +</p> +<p> +A material particle upon which no force acts moves, according +to the principle of inertia, uniformly in a straight line. In +the four-dimensional continuum of the special theory of relativity +(with real time co-ordinate) this is a real straight line. The +natural, that is, the simplest, generalization of the straight line +which is plausible in the system of concepts of Riemann's general +theory of invariants is that of the straightest, or geodetic, +line. We shall accordingly have to assume, in the sense of the +principle of equivalence, that the motion of a material particle, +under the action only of inertia and gravitation, is described by +the equation, +<span class="align-center"><img style="vertical-align: -1.651ex; width: 34.843ex; height: 5.264ex;" src="images/482.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} + + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + = 0. +\qquad \text{(90)} +"></span> +In fact, this equation reduces to that of a straight line if all the +components, <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}">, of the gravitational field vanish. +</p> +<p> +How are these equations connected with Newton's equations +of motion? According to the special theory of relativity, +the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> as well as the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, have +the values, with respect to an inertial +<span class="pagenum" id="Page_84">[Pg 84]</span> +system (with real time co-ordinate and suitable choice of the +sign of <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}">), +<span class="align-center"><img style="vertical-align: -5.317ex; width: 28.158ex; height: 11.765ex;" src="images/484.svg" alt=" " data-tex=" +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & 1 +\end{array} +\right\}. +\qquad \text{(91)} +"></span> +The equations of motion then become +<span class="align-center"><img style="vertical-align: -1.651ex; width: 10.383ex; height: 5.264ex;" src="images/485.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{ds^{2}} = 0. +"></span> +We shall call this the "first approximation" to the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field. In +considering approximations it is often useful, as in the special +theory of relativity, to use an imaginary <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">-co-ordinate, as then +the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. to the first approximation, assume the values +<span class="align-center"><img style="vertical-align: -5.317ex; width: 31.049ex; height: 11.765ex;" src="images/486.svg" alt=" " data-tex=" +\left. +\begin{array}{*{4}{>{\quad}r}} +-1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & -1 +\end{array} +\right\}. +\qquad \text{(91a)} +"></span> +These values may be collected in the relation +<span class="align-center"><img style="vertical-align: -0.685ex; width: 11.491ex; height: 2.307ex;" src="images/487.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu}. +"></span> +To the second approximation we must then put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.353ex; height: 2.382ex;" src="images/488.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, +\qquad \text{(92)} +"></span> +where the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> are to be regarded as small of the first order. +<span class="pagenum" id="Page_85">[Pg 85]</span> +</p> +<p> +Both terms of our equation of motion are then small of the +first order. If we neglect terms which, relatively to these, are +small of the first order, we have to put +<span class="align-center"><img style="vertical-align: -3.856ex; width: 67.308ex; height: 8.843ex;" src="images/490.svg" alt=" " data-tex=" +\begin{gather*} +ds^{2} = -{dx_{\nu}}^{2} = dl^{2} (1 - q^{2}), +\qquad & &\text{(93)}\\ +\Gamma_{\alpha\beta}^{\mu} + = -\delta_{\mu\sigma} [^{\alpha\beta}_{\,\delta}] + = -[^{\alpha\beta}_{\,\mu}] + = \frac{1}{2} \left( + \frac{\partial \gamma_{\alpha\beta}}{\partial x_{\mu}} + - \frac{\partial \gamma_{\alpha\mu}}{\partial x_{\beta}} + - \frac{\partial \gamma_{\beta\mu}}{\partial x_{\alpha}}\right). +\qquad & &\text{(94)} +\end{gather*} +"></span> +We shall now introduce an approximation of a second kind. Let +the velocity of the material particles be very small compared to +that of light. Then <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> will be the same as the time differential, <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl">. +Further, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/491.svg" alt=" " data-tex="\dfrac{dx_{1}}{ds}">, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/492.svg" alt=" " data-tex="\dfrac{dx_{2}}{ds}">, <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/493.svg" alt=" " data-tex="\dfrac{dx_{3}}{ds}"> +will vanish compared to <img style="vertical-align: -1.575ex; width: 4.454ex; height: 4.674ex;" src="images/494.svg" alt=" " data-tex="\dfrac{dx_{4}}{ds}">. +We shall assume, in addition, that the gravitational field varies +so little with the time that the derivatives of the +<img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> by <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}"> may +be neglected. Then the equation of motion (for <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> = 1,2,3) +reduces to +<span class="align-center"><img style="vertical-align: -2.237ex; width: 30.687ex; height: 5.849ex;" src="images/495.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{dl^{2}} + = \frac{\partial}{\partial x_{\mu}} \left(\frac{\gamma_{44}}{2}\right). +\qquad \text{(90a)} +"></span> +This equation is identical with Newton's equation of motion for +a material particle in a gravitational field, if we identify +<img style="vertical-align: -1.552ex; width: 6.656ex; height: 4.153ex;" src="images/496.svg" alt=" " data-tex="\left(\dfrac{\gamma_{44}}{2}\right)"> +with the potential of the gravitational field; whether or not this +is allowable, naturally depends upon the field equations of gravitation, +that is, it depends upon whether or not this quantity +satisfies, to a first approximation, the same laws of the field +as the gravitational potential in Newton's theory. A glance at +(90) and (90a) shows that the <img style="vertical-align: -1.045ex; width: 3.531ex; height: 2.898ex;" src="images/461.svg" alt=" " data-tex="\Gamma_{\alpha\beta}^{\mu}"> actually do play the rôle of +the intensity of the gravitational field. These quantities do not +have a tensor character. +</p> +<p> +Equations (90) express the influence of inertia and gravitation +upon the material particle. The unity of inertia and gravitation +<span class="pagenum" id="Page_86">[Pg 86]</span> +is formally expressed by the fact that the whole left-hand +side of (90) has the character of a tensor (with respect to any +transformation of co-ordinates), but the two terms taken separately +do not have tensor character, so that, in analogy with +Newton's equations, the first term would be regarded as the expression +for inertia, and the second as the expression for the +gravitational force. +</p> +<p> +We must next attempt to find the laws of the gravitational +field. For this purpose, Poisson's equation, +<span class="align-center"><img style="vertical-align: -0.489ex; width: 11.852ex; height: 2.109ex;" src="images/497.svg" alt=" " data-tex=" +\Delta\phi = 4\pi K\rho +"></span> +of the Newtonian theory must serve as a model. This equation +has its foundation in the idea that the gravitational field +arises from the density <img style="vertical-align: -0.489ex; width: 1.17ex; height: 1.489ex;" src="images/118.svg" alt=" " data-tex="\rho"> of ponderable matter. It must also +be so in the general theory of relativity. But our investigations +of the special theory of relativity have shown that in place of +the scalar density of matter we have the tensor of energy per +unit volume. In the latter is included not only the tensor of +the energy of ponderable matter, but also that of the electromagnetic +energy. We have seen, indeed, that in a more complete +analysis the energy tensor can be regarded only as a provisional +means of representing matter. In reality, matter consists of electrically +charged particles, and is to be regarded itself as a part, +in fact, the principal part, of the electromagnetic field. It is +only the circumstance that we have not sufficient knowledge of +the electromagnetic field of concentrated charges that compels +us, provisionally, to leave undetermined in presenting the theory, +the true form of this tensor. From this point of view our +problem now is to introduce a tensor, <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}">. of the second rank, +<span class="pagenum" id="Page_87">[Pg 87]</span> +whose structure we do not know provisionally, and which includes +in itself the energy density of the electromagnetic field +and of ponderable matter; we shall denote this in the following +as the "energy tensor of matter." +</p> +<p> +According to our previous results, the principles of momentum +and energy are expressed by the statement that the divergence +of this tensor vanishes (47c). In the general theory of relativity, +we shall have to assume as valid the corresponding general +co-variant equation. If (<img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}">) denotes the co-variant energy tensor +of matter, <img style="vertical-align: -0.576ex; width: 2.615ex; height: 2.162ex;" src="images/498.svg" alt=" " data-tex="\mathfrak T_{\sigma}^{\nu}"> the corresponding mixed tensor density, then, +in accordance with (83), we must require that +<span class="align-center"><img style="vertical-align: -1.909ex; width: 26.609ex; height: 5.144ex;" src="images/499.svg" alt=" " data-tex=" +0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta} +\qquad \text{(95)} +"></span> +be satisfied. It must be remembered that besides the energy density +of the matter there must also be given an energy density of +the gravitational field, so that there can be no talk of principles +of conservation of energy and momentum for matter alone. This +is expressed mathematically by the presence of the second term +in (95), which makes it impossible to conclude the existence of +an integral equation of the form of (49). The gravitational field +transfers energy and momentum to the "matter," in that it exerts +forces upon it and gives it energy; this is expressed by the +second term in (95). +</p> +<p> +If there is an analogue of Poisson's equation in the general +theory of relativity, then this equation must be a tensor equation +for the tensor <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> of the gravitational potential; the energy +tensor of matter must appear on the right-hand side of this +equation. On the left-hand side of the equation there must be +a differential tensor in the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">. We have to find this differential +<span class="pagenum" id="Page_88">[Pg 88]</span> +tensor. It is completely determined by the following three +conditions:— +</p> +<p> +1. It may contain no differential coefficients of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> higher +than the second. +</p> +<p> +2. It must be linear and homogeneous in these second differential +coefficients. +</p> +<p> +3. Its divergence must vanish identically. +</p> +<p> +The first two of these conditions are naturally taken from +Poisson's equation. Since it may be proved mathematically +that all such differential tensors can be formed algebraically +(i.e. without differentiation) from Riemann's tensor, our tensor +must be of the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 13.106ex; height: 2.23ex;" src="images/500.svg" alt=" " data-tex=" +R_{\mu\nu} + ag_{\mu\nu} R, +"></span> +in which <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}"> and <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> are defined by (88) and (89) respectively. +Further, it may be proved that the third condition requires a +to have the value <img style="vertical-align: -1.552ex; width: 3.887ex; height: 4.588ex;" src="images/501.svg" alt=" " data-tex="-\dfrac{1}{2}">. For the law of the gravitational field we +therefore get the equation +<span class="align-center"><img style="vertical-align: -0.781ex; width: 32.031ex; height: 2.737ex;" src="images/502.svg" alt=" " data-tex=" +R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}. +\qquad \text{(96)} +"></span> +Equation (95) is a consequence of this equation. <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/503.svg" alt=" " data-tex="\kappa"> denotes a +constant, which is connected with the Newtonian gravitation +constant. +</p> +<p> +In the following I shall indicate the features of the theory +which are interesting from the point of view of physics, using as +little as possible of the rather involved mathematical method. +It must first be shown that the divergence of the left-hand side +actually vanishes. The energy principle for matter may be expressed, by (83), +<span class="align-center"><img style="vertical-align: -1.909ex; width: 27.615ex; height: 5.144ex;" src="images/504.svg" alt=" " data-tex=" +0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} + - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta}, +\qquad \text{(97)} +"></span> +<span class="pagenum" id="Page_89">[Pg 89]</span> +in which +<span class="align-center"><img style="vertical-align: -0.669ex; width: 17.886ex; height: 2.851ex;" src="images/505.svg" alt=" " data-tex=" +\mathfrak T_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}. +"></span> +The analogous operation, applied to the left-hand side of (96), +will lead to an identity. +</p> +<p> +In the region surrounding each world-point there are systems +of co-ordinates for which, choosing the <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">-co-ordinate imaginary, +at the given point, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 34.922ex; height: 5.428ex;" src="images/506.svg" alt=" " data-tex=" +g_{\mu\nu} = g^{\mu\nu} + = -\delta_{\mu\nu} + = \begin{cases} + -1 & \text{if \(\mu = \nu\)}, \\ + \,\,\,\,0 & \text{if \(\mu \neq \nu\)}, + \end{cases} +"></span> +and for which the first derivatives of the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> and the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> vanish. +We shall verify the vanishing of the divergence of the left-hand +side at this point. At this point the components +<img style="vertical-align: -1.042ex; width: 3.421ex; height: 2.58ex;" src="images/507.svg" alt=" " data-tex="\Gamma_{\sigma\beta}^{\alpha}"> vanish, so +that we have to prove the vanishing only of +<span class="align-center"><img style="vertical-align: -1.909ex; width: 30.825ex; height: 5.056ex;" src="images/508.svg" alt=" " data-tex=" +\frac{\partial}{\partial x_{\sigma}} \left[ + \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) +\right]. +"></span> +Introducing (88) and (70) into this expression, we see that the +only terms that remain are those in which third derivatives of +the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}"> enter. Since the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> are to be replaced by +<img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/509.svg" alt=" " data-tex="-\delta_{\mu\nu}">, we obtain, +finally, only a few terms which may easily be seen to cancel +each other. Since the quantity that we have formed has a tensor +character, its vanishing is proved for every other system of co-ordinates +also, and naturally for every other four-dimensional +point. The energy principle of matter (97) is thus a mathematical +consequence of the field equations (96). +</p> +<p> +In order to learn whether the equations (96) are consistent +with experience, we must, above all else, find out whether they +<span class="pagenum" id="Page_90">[Pg 90]</span> +lead to the Newtonian theory as a first approximation. For this +purpose we must introduce various approximations into these +equations. We already know that Euclidean geometry and the +law of the constancy of the velocity of light are valid, to a certain +approximation, in regions of a great extent, as in the planetary +system. If, as in the special theory of relativity, we take the +fourth co-ordinate imaginary, this means that we must put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 26.353ex; height: 2.382ex;" src="images/510.svg" alt=" " data-tex=" +g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, +\qquad \text{(98)} +"></span> +in which the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> are so small compared to 1 that we can neglect +the higher powers of the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> and their derivatives. If we do this, +we learn nothing about the structure of the gravitational held, or +of metrical space of cosmical dimensions, but we do learn about +the influence of neighbouring masses upon physical phenomena. +</p> +<p> +Before carrying through this approximation we shall transform +(96). We multiply (96) by <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, summed over the <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> +observing the relation which follows from the definition of +the <img style="vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;" src="images/386.svg" alt=" " data-tex="g^{\mu\nu}">, +<span class="align-center"><img style="vertical-align: -0.685ex; width: 10.936ex; height: 2.326ex;" src="images/511.svg" alt=" " data-tex=" +g_{\mu\nu} g^{\mu\nu} = 4, +"></span> +we obtain the equation +<span class="align-center"><img style="vertical-align: -0.685ex; width: 18.981ex; height: 2.326ex;" src="images/512.svg" alt=" " data-tex=" +R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T. +"></span> +If we put this value of <img style="vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;" src="images/211.svg" alt=" " data-tex="R"> in (96) we obtain +<span class="align-center"><img style="vertical-align: -0.904ex; width: 44.2ex; height: 2.861ex;" src="images/513.svg" alt=" " data-tex=" +R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) + = -\kappa T_{\mu\nu}^{*}. +\qquad \text{(96a)} +"></span> +When the approximation which has been mentioned is carried +out, we obtain for the left-hand side, +<span class="align-center"><img style="vertical-align: -2.827ex; width: 48.357ex; height: 6.785ex;" src="images/514.svg" alt=" " data-tex=" +-\tfrac{1}{2}\left( + \frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + + \frac{\partial^{2} \gamma_{\alpha\alpha}}{\partial x_{\mu}\, \partial x_{\nu}} + - \frac{\partial^{2} \gamma_{\mu\alpha}}{\partial x_{\nu}\, \partial x_{\alpha}} + - \frac{\partial^{2} \gamma_{\nu\alpha}}{\partial x_{\mu}\, \partial x_{\alpha}} +\right) +"></span> +<span class="pagenum" id="Page_91">[Pg 91]</span> +or +<span class="align-center"><img style="vertical-align: -2.827ex; width: 48.834ex; height: 6.785ex;" src="images/515.svg" alt=" " data-tex=" +-\tfrac{1}{2}\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + + \tfrac{1}{2} \frac{\partial}{\partial x_{\nu}}\left( + \frac{{\partial \gamma'}_{\mu\alpha}}{\partial x_{\alpha}} + \right) + + \tfrac{1}{2} \frac{\partial}{\partial x_{\mu}}\left( + \frac{{\partial \gamma'}_{\nu\alpha}}{\partial x_{\alpha}} + \right), +"></span> +in which has been put +<span class="align-center"><img style="vertical-align: -0.914ex; width: 30.468ex; height: 2.871ex;" src="images/516.svg" alt=" " data-tex=" +{\gamma'}_{\mu\nu} + = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}. +\qquad \text{(99)} +"></span> +</p> +<p> +We must now note that equation (96) is valid for any system +of co-ordinates. We have already specialized the system of +co-ordinates in that we have chosen it so that within the region +considered the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> differ infinitely little from the constant values +<img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/509.svg" alt=" " data-tex="-\delta_{\mu\nu}">. But this condition remains satisfied in any infinitesimal +change of co-ordinates, so that there are still four conditions +to which the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> may be subjected, provided these conditions +do not conflict with the conditions for the order of magnitude of +the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}">. We shall now assume that the system of co-ordinates +is so chosen that the four relations— +<span class="align-center"><img style="vertical-align: -2.237ex; width: 38.565ex; height: 5.909ex;" src="images/517.svg" alt=" " data-tex=" +0 = \frac{{\partial\gamma'}_{\mu\nu}}{\partial x_{\nu}} + = \frac{\partial\gamma_{\mu\nu}}{\partial x_{\nu}} + - \tfrac{1}{2} \frac{\partial\gamma_{\sigma\sigma}}{\partial x_{\mu}} +\qquad \text{(100)} +"></span> +are satisfied. Then (96a) takes the form +<span class="align-center"><img style="vertical-align: -2.332ex; width: 26.138ex; height: 5.944ex;" src="images/518.svg" alt=" " data-tex=" +\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}. +\qquad \text{(96b)} +"></span> +These equations may be solved by the method, familiar in +electrodynamics, of retarded potentials; we get, in an easily +understood notation, +<span class="align-center"><img style="vertical-align: -1.948ex; width: 48.917ex; height: 5.59ex;" src="images/519.svg" alt=" " data-tex=" +\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int + \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}. +\qquad \text{(101)} +"></span> +<span class="pagenum" id="Page_92">[Pg 92]</span> +</p> +<p> +In order to see in what sense this theory contains the Newtonian +theory, we must consider in greater detail the energy +tensor of matter. Considered phenomenologically, this energy +tensor is composed of that of the electromagnetic field and of +matter in the narrower sense. If we consider the different parts +of this energy tensor with respect to their order of magnitude, +it follows from the results of the special theory of relativity that +the contribution of the electromagnetic field practically vanishes +in comparison to that of ponderable matter. In our system of +units, the energy of one gram of matter is equal to 1, compared +to which the energy of the electric fields may be ignored, and +also the energy of deformation of matter, and even the chemical +energy. We get an approximation that is fully sufficient for our +purpose if we put +<span class="align-center"><img style="vertical-align: -3.551ex; width: 30.02ex; height: 8.234ex;" src="images/520.svg" alt=" " data-tex=" +\left. +\begin{aligned} +T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}, \\ +ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. +\end{aligned} +\right\} +\qquad \text{(102)} +"></span> +In this, <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma"> is the density at rest, that is, the density of the ponderable +matter, in the ordinary sense, measured with the aid +of a unit measuring rod, and referred to a Galilean system of +co-ordinates moving with the matter. +</p> +<p> +We observe, further, that in the co-ordinates we have chosen, +we shall make only a relatively small error if we replace the +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> +by <img style="vertical-align: -0.685ex; width: 4.765ex; height: 2.307ex;" src="images/509.svg" alt=" " data-tex="-\delta_{\mu\nu}">, so that we put +<span class="align-center"><img style="vertical-align: -1.018ex; width: 28.45ex; height: 3.167ex;" src="images/521.svg" alt=" " data-tex=" +ds^{2} = -\sum {dx_{\mu}}^{2}. +\qquad \text{(102a)} +"></span> +</p> +<p> +The previous developments are valid however rapidly the +masses which generate the field may move relatively to our chosen +system of quasi-Galilean co-ordinates. But in astronomy +<span class="pagenum" id="Page_93">[Pg 93]</span> +we have to do with masses whose velocities, relatively to the +co-ordinate system employed, are always small compared to the +velocity of light, that is, small compared to 1, with our choice +of the unit of time. We therefore get an approximation which is +sufficient for nearly all practical purposes if in (101) we replace +the retarded potential by the ordinary (non-retarded) potential, +and if, for the masses which generate the field, we put +<span class="align-center"><img style="vertical-align: -1.577ex; width: 60.851ex; height: 5.187ex;" src="images/522.svg" alt=" " data-tex=" +\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad +\frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}. +\qquad \text{(103)} +"></span> +Then we get for <img style="vertical-align: 0; width: 3.72ex; height: 1.532ex;" src="images/523.svg" alt=" " data-tex="T^{\mu\nu}"> and <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> the values +<span class="align-center"><img style="vertical-align: -5.317ex; width: 25.929ex; height: 11.765ex;" src="images/524.svg" alt=" " data-tex=" +\left. +\begin{array}{*{3}{>{\qquad}r}>{\quad}r} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & -\sigma +\end{array} +\right\}. +\qquad \text{(104)} +"></span> +For <img style="vertical-align: 0; width: 1.593ex; height: 1.532ex;" src="images/525.svg" alt=" " data-tex="T"> we get the value <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, and, finally, +for <img style="vertical-align: -0.904ex; width: 3.322ex; height: 2.47ex;" src="images/526.svg" alt=" " data-tex="T_{\mu\nu}^{*}"> the values, +<span class="align-center"><img style="vertical-align: -8.905ex; width: 31.524ex; height: 18.941ex;" src="images/527.svg" alt=" " data-tex=" +\left. +\begin{array}{*{3}{>{\qquad}c}>{\quad}r} +\dfrac{\sigma}{2} & 0 & 0 & 0 \\ +0 & \dfrac{\sigma}{2} & 0 & 0 \\ +0 & 0 & \dfrac{\sigma}{2} & 0 \\ +0 & 0 & 0 & -\dfrac{\sigma}{2} +\end{array} +\right\}. +\qquad \text{(104a)} +"></span> +</p> +<p> +We thus get, from (101), +<span class="align-center"><img style="vertical-align: -4.821ex; width: 45.595ex; height: 10.774ex;" src="images/528.svg" alt=" " data-tex=" +\left. +\begin{aligned} +\gamma_{11} = \gamma_{22} = \gamma_{33} + &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, +\end{aligned} +\right\} +\qquad \text{(101a)} +"></span> +<span class="pagenum" id="Page_94">[Pg 94]</span> +while all the other <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}">, vanish. The least of these equations, +in connexion with equation (90a), contains Newton's theory of +gravitation. If we replace <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> by <img style="vertical-align: -0.025ex; width: 1.796ex; height: 1.441ex;" src="images/206.svg" alt=" " data-tex="ct"> we get +<span class="align-center"><img style="vertical-align: -2.237ex; width: 40.856ex; height: 5.849ex;" src="images/529.svg" alt=" " data-tex=" +\frac{d^{2} x_{\mu}}{dt^{2}} + = \frac{\kappa c^{2}}{8\pi}\, \frac{\partial}{\partial x_{\mu}} \left\{ + \int \frac{\sigma\, dV_{0}}{r} + \right\}. +\qquad \text{(90b)} +"></span> +We see that the Newtonian gravitation constant <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, is connected +with the constant <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/503.svg" alt=" " data-tex="\kappa"> that enters into our field equations by the +relation +<span class="align-center"><img style="vertical-align: -1.602ex; width: 19.979ex; height: 5.018ex;" src="images/530.svg" alt=" " data-tex=" +K = \frac{\kappa c^{2}}{8\pi}. +\qquad \text{(105)} +"></span> +From the known numerical value of <img style="vertical-align: 0; width: 2.011ex; height: 1.545ex;" src="images/58.svg" alt=" " data-tex="K">, it therefore follows that +<span class="align-center"><img style="vertical-align: -1.914ex; width: 53.752ex; height: 5.398ex;" src="images/531.svg" alt=" " data-tex=" +\kappa = \frac{8\pi K}{c^{2}} + = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} + = 1.86·10^{-27}. +\qquad \text{(105a)} +"></span> +From (101) we see that even in the first approximation the structure +of the gravitational field differs fundamentally from that +which is consistent with the Newtonian theory; this difference +lies in the fact that the gravitational potential has the character +of a tensor and not a scalar. This was not recognized in the past +because only the component <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/532.svg" alt=" " data-tex="g_{44}">, to a first approximation, enters +the equations of motion of material particles. +</p> +<p> +In order now to be able to judge the behaviour of measuring +rods and clocks from our results, we must observe the following. +According to the principle of equivalence, the metrical relations +of the Euclidean geometry are valid relatively to a Cartesian +system of reference of infinitely small dimensions, and in a suitable +state of motion (freely falling, and without rotation). We +can make the same statement for local systems of co-ordinates +<span class="pagenum" id="Page_95">[Pg 95]</span> +which, relatively to these, have small accelerations, and therefore +for such systems of co-ordinates as are at rest relatively to +the one we have selected. For such a local system, we have, for +two neighbouring point events, +<span class="align-center"><img style="vertical-align: -0.439ex; width: 51.054ex; height: 2.457ex;" src="images/533.svg" alt=" " data-tex=" +ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} + = - dS^{2} + dT^{2}, +"></span> +where <img style="vertical-align: -0.05ex; width: 2.636ex; height: 1.645ex;" src="images/534.svg" alt=" " data-tex="dS"> is measured directly by a measuring rod and <img style="vertical-align: -0.023ex; width: 2.769ex; height: 1.593ex;" src="images/535.svg" alt=" " data-tex="dT"> by +a clock at rest relatively to the system; these are the naturally +measured lengths and times. Since <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}">, on the other hand, is +known in terms of the co-ordinates <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}"> employed in finite regions, +in the form +<span class="align-center"><img style="vertical-align: -0.685ex; width: 17.836ex; height: 2.685ex;" src="images/536.svg" alt=" " data-tex=" +ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, +"></span> +we have the possibility of getting the relation between naturally +measured lengths and times, on the one hand, and the corresponding +differences of co-ordinates, on the other hand. As the +division into space and time is in agreement with respect to the +two systems of co-ordinates, so when we equate the two expressions +for <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> we get two relations. If, by (101a), we put +<span class="align-center"><img style="vertical-align: -5.201ex; width: 48.691ex; height: 11.534ex;" src="images/537.svg" alt=" " data-tex=" +\begin{aligned} +ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) +({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ +\qquad + + \mspace{26mu}\left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}, +\end{aligned} +"></span> +we obtain, to a sufficiently close approximation, +<span class="align-center"><img style="vertical-align: -7.645ex; width: 63.489ex; height: 16.421ex;" src="images/538.svg" alt=" " data-tex=" +\left. +\begin{aligned} + &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ + &\qquad + \begin{aligned} + &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) + \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}, \\ +dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. +\end{aligned} +\end{aligned} +\right\} +\qquad \text{(106)} +"></span> +<span class="pagenum" id="Page_96">[Pg 96]</span> +</p> +<p> +The unit measuring rod has therefore the length, +<span class="align-center"><img style="vertical-align: -1.948ex; width: 16.351ex; height: 5.048ex;" src="images/539.svg" alt=" " data-tex=" +1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +"></span> +in respect to the system of co-ordinates we have selected. The +particular system of co-ordinates we have selected insures that +this length shall depend only upon the place, and not upon the +direction. If we had chosen a different system of co-ordinates +this would not be so. But however we may choose a system of +co-ordinates, the laws of configuration of rigid rods do not agree +with those of Euclidean geometry; in other words, we cannot +choose any system of co-ordinates so that the co-ordinate differences, +<img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/8.svg" alt=" " data-tex="\Delta x_{1}">, <img style="vertical-align: -0.339ex; width: 4.166ex; height: 1.959ex;" src="images/9.svg" alt=" " data-tex="\Delta x_{2}">, <img style="vertical-align: -0.375ex; width: 4.166ex; height: 1.994ex;" src="images/10.svg" alt=" " data-tex="\Delta x_{3}">, corresponding +to the ends of a unit measuring rod, oriented in any way, +shall always satisfy the relation +<img style="vertical-align: -0.685ex; width: 22.179ex; height: 2.572ex;" src="images/540.svg" alt=" " data-tex="\Delta x_{1}^{2} + \Delta x_{2}^{2} + \Delta x_{3}^{2} = 1">. +In this sense space is not Euclidean, +but "curved." It follows from the second of the relations above +that the interval between two beats of the unit clock (<img style="vertical-align: -0.023ex; width: 2.769ex; height: 1.593ex;" src="images/535.svg" alt=" " data-tex="dT"> = 1) +corresponds to the "time" +<span class="align-center"><img style="vertical-align: -1.948ex; width: 16.351ex; height: 5.048ex;" src="images/541.svg" alt=" " data-tex=" +1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r} +"></span> +in the unit used in our system of co-ordinates. The rate of a +clock is accordingly slower the greater is the mass of the ponderable +matter in its neighbourhood. We therefore conclude that +spectral lines which are produced on the sun's surface will be +displaced towards the red, compared to the corresponding lines +produced on the earth, by about 2 • 10<sup>-6</sup> of their wave-lengths. +At first, this important consequence of the theory appeared to +conflict with experiment; but results obtained during the past +year seem to make the existence of this effect more probable, and +<span class="pagenum" id="Page_97">[Pg 97]</span> +it can hardly be doubted that this consequence of the theory will +be confirmed within the next year. +</p> +<p> +Another important consequence of the theory, which can be +tested experimentally, has to do with the path of rays of light. +In the general theory of relativity also the velocity of fight is +everywhere the same, relatively to a local inertial system. This +velocity is unity in our natural measure of time. The law of +the propagation of light in general co-ordinates is therefore, +according to the general theory of relativity, characterized, by the +equation +<span class="align-center"><img style="vertical-align: -0.186ex; width: 7.374ex; height: 2.185ex;" src="images/542.svg" alt=" " data-tex=" +ds^{2} = 0 +"></span> +To within the approximation which we are using, and in the +system of co-ordinates which we have selected, the velocity of +light is characterized, according to (106), by the equation +<span class="align-center"><img style="vertical-align: -2.148ex; width: 66.477ex; height: 5.428ex;" src="images/543.svg" alt=" " data-tex=" +\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) + ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) + = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}. +"></span> +The velocity of light <img style="vertical-align: 0; width: 1.541ex; height: 1.545ex;" src="images/544.svg" alt=" " data-tex="L">, is therefore expressed in our co-ordinates +by +<span class="align-center"><img style="vertical-align: -1.948ex; width: 52.226ex; height: 5.945ex;" src="images/545.svg" alt=" " data-tex=" +\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} + = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}. +\qquad \text{(107)} +"></span> +We can therefore draw the conclusion from this, that a ray of +light passing near a large mass is deflected. If we imagine the +sun, of mass <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M"> concentrated at the origin of our system of co-ordinates, +then a ray of fight, travelling parallel to the <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}">-axis. +in the <img style="vertical-align: -0.375ex; width: 7.329ex; height: 1.694ex;" src="images/546.svg" alt=" " data-tex="x_{1}-x_{3}"> plane, at a distance <img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"> from the origin, will be +deflected, in all, by an amount +<span class="align-center"><img style="vertical-align: -2.159ex; width: 21.475ex; height: 5.553ex;" src="images/547.svg" alt=" " data-tex=" +\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\partial L}{\partial x_{1}}\, dx_{3} +"></span> +<span class="pagenum" id="Page_98">[Pg 98]</span> +towards the sun. On performing the integration we get +<span class="align-center"><img style="vertical-align: -1.577ex; width: 20.451ex; height: 4.652ex;" src="images/548.svg" alt=" " data-tex=" +\alpha = \frac{\kappa M}{2\pi\Delta}. +\qquad \text{(108)} +"></span> +</p> +<p> +The existence of this deflection, which amounts to 1.7<code>''</code> for +<img style="vertical-align: 0; width: 1.885ex; height: 1.62ex;" src="images/22.svg" alt=" " data-tex="\Delta"> equal to the radius of the sun, was confirmed, with remarkable +accuracy, by the English Solar Eclipse Expedition in 1919, and +most careful preparations have been made to get more exact +observational data at the solar eclipse in 1922. It should be +noted that this result, also, of the theory is not influenced by +our arbitrary choice of a system of co-ordinates. +</p> +<p> +This is the place to speak of the third consequence of the +theory which can be tested by observation, namely, that which +concerns the motion of the perihelion of the planet Mercury. The +secular changes in the planetary orbits are known with such accuracy +that the approximation we have been using is no longer +sufficient for a comparison of theory and observation. It is necessary +to go back to the general field equations (96). To solve +this problem I made use of the method of successive approximations. +Since then, however, the problem of the central symmetrical +statical gravitational field has been completely solved by +Schwarzschild and others; the derivation given by H. Weyl in his +book, "Raum-Zeit-Materie," is particularly elegant. The calculation +can be simplified somewhat if we do not go back directly +to the equation (96), but base it upon a principle of variation +that is equivalent to this equation. I shall indicate the procedure +only in so far as is necessary for understanding the method. +<span class="pagenum" id="Page_99">[Pg 99]</span> +</p> +<p> +In the case of a statical field, <img style="vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;" src="images/360.svg" alt=" " data-tex="ds^{2}"> must have the form +<span class="align-center"><img style="vertical-align: -3.502ex; width: 33.95ex; height: 8.135ex;" src="images/549.svg" alt=" " data-tex=" +\left. +\begin{aligned} +ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}, \\ +d\sigma^{2} + &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}, +\end{aligned} +\right\} +\qquad \text{(109)} +"></span> +where the summation on the right-hand side of the last equation +is to be extended over the space variables only. The central +symmetry of the field requires the <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}">, to be of the form, +<span class="align-center"><img style="vertical-align: -0.65ex; width: 30.455ex; height: 2.347ex;" src="images/550.svg" alt=" " data-tex=" +\gamma_{\alpha\beta} + = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}; +\qquad \text{(110)} +"></span> +<img style="vertical-align: -0.464ex; width: 2.352ex; height: 2.351ex;" src="images/551.svg" alt=" " data-tex="f^{2}">, <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> are functions of +<img style="vertical-align: -0.532ex; width: 21.685ex; height: 2.851ex;" src="images/552.svg" alt=" " data-tex="r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}"> only. One +of these three functions can be chosen arbitrarily, because our +system of co-ordinates is, <i>a priori</i>, completely arbitrary; for by +a substitution +<span class="align-center"><img style="vertical-align: -2.17ex; width: 13.76ex; height: 5.47ex;" src="images/553.svg" alt=" " data-tex=" +\begin{align*} +{x'}_{4} &= x_{4}, \\ +{x'}_{\alpha} &= F(r) x_{\alpha}, +\end{align*} +"></span> +we can always insure that one of these three functions shall be +an assigned function of <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r">'. In place of (110) we can therefore +put, without limiting the generality, +<span class="align-center"><img style="vertical-align: -0.65ex; width: 30.222ex; height: 2.347ex;" src="images/554.svg" alt=" " data-tex=" +\gamma_{\alpha\beta} + = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}. +\qquad \text{(110a)} +"></span> +</p> +<p> +In this way the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> are expressed in terms of the two quantities +<img style="vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;" src="images/15.svg" alt=" " data-tex="\lambda"> and <img style="vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;" src="images/555.svg" alt=" " data-tex="f">. These are to be determined as functions of <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r">, +by introducing them into equation (96), after first calculating +<span class="pagenum" id="Page_100">[Pg 100]</span> +the <img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/556.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\sigma}"> from (109) and (110a). We have +<span class="align-center"><img style="vertical-align: -7.006ex; width: 64.764ex; height: 15.143ex;" src="images/557.svg" alt=" " data-tex=" +\left. +\begin{aligned} +\Gamma_{\alpha\beta}^{\sigma} + &= \tfrac{1}{2} \frac{x_{\sigma}}{r} + · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} + {1 + \lambda r^{2}}\,\, + \text{(for \(\alpha, \beta, \sigma = 1, 2, 3)\)}, \\ +\Gamma_{44}^{4} + &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad + \text{(for \(\alpha, \beta = 1, 2, 3)\)}, \\ +\Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\partial f^{2}}{\partial x_{\alpha}},\quad +\Gamma_{44}^{\alpha} + = -\tfrac{1}{2}{g^{\alpha\beta}}\, + \frac{\partial f^{2}}{\partial {x_{\beta}}}. +\end{aligned} +\right\} +\qquad \text{(110b)} +"></span> +</p> +<p> +With the help of these results, the field equations furnish +Schwarzschild's solution: +<span class="align-center"><img style="vertical-align: -4.636ex; width: 65.804ex; height: 10.403ex;" src="images/558.svg" alt=" " data-tex=" +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ + \frac{dr^{2}} + {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) +\right], +\qquad \text{(109a)} +"></span> +in which we have put +<span class="align-center"><img style="vertical-align: -7.643ex; width: 29.619ex; height: 16.416ex;" src="images/559.svg" alt=" " data-tex=" +\left. +\begin{aligned} +x_{4} &= l, \\ +x_{1} &= r \sin\theta \sin\phi, \\ +x_{2} &= r \sin\theta \cos\phi, \\ +x_{3} &= r \cos\theta, \\ +A &= \frac{\kappa M}{4\pi}. +\end{aligned} +\right\} +\qquad \text{(109b)} +"></span> +</p> +<p> +<img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M"> denotes the sun's mass, centrally symmetrically placed +about the origin of co-ordinates; the solution (109) is valid only +outside of this mass, where all the <img style="vertical-align: -0.685ex; width: 3.322ex; height: 2.217ex;" src="images/310.svg" alt=" " data-tex="T_{\mu\nu}"> vanish. If the motion +of the planet takes place in the <img style="vertical-align: -0.339ex; width: 7.329ex; height: 1.658ex;" src="images/560.svg" alt=" " data-tex="x_{1}-x_{2}"> plane then we +must replace (109a) by +<span class="align-center"><img style="vertical-align: -4.636ex; width: 49.704ex; height: 8.052ex;" src="images/561.svg" alt=" " data-tex=" +ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} + - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}. +\qquad \text{(109c)} +"></span> +<span class="pagenum" id="Page_101">[Pg 101]</span> +</p> +<p> +The calculation of the planetary motion depends upon equation (90). +From the first of equations (110b) and (90) we get, +for the indices 1, 2, 3, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 28.237ex; height: 5.428ex;" src="images/562.svg" alt=" " data-tex=" +\frac{d}{ds} + \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0, +"></span> +or, if we integrate, and express the result in polar co-ordinates, +<span class="align-center"><img style="vertical-align: -1.575ex; width: 27.665ex; height: 4.674ex;" src="images/563.svg" alt=" " data-tex=" +r^{2} \frac{d\phi}{ds} = \text{constant}. +\qquad \text{(111)} +"></span> +</p> +<p> +From (90), for <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> = 4, we get +<span class="align-center"><img style="vertical-align: -2.093ex; width: 50.697ex; height: 5.509ex;" src="images/564.svg" alt=" " data-tex=" +0 = \frac{d^{2} l}{ds^{2}} + + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} +{\, \frac{dl}{ds}} + = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} +{\, \frac{dl}{ds}}. +"></span> +From this, after multiplication by <img style="vertical-align: -0.464ex; width: 2.352ex; height: 2.351ex;" src="images/551.svg" alt=" " data-tex="f^{2}"> and integration, we have +<span class="align-center"><img style="vertical-align: -1.575ex; width: 27.721ex; height: 4.674ex;" src="images/565.svg" alt=" " data-tex=" +f^{2} \frac{dl}{ds} = \text{constant}. +\qquad \text{(112)} +"></span> +</p> +<p> +In (109c), (111) and (112) we have three equations between +the four variables <img style="vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;" src="images/13.svg" alt=" " data-tex="s">, <img style="vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;" src="images/172.svg" alt=" " data-tex="r">, <img style="vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;" src="images/199.svg" alt=" " data-tex="l"> and <img style="vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;" src="images/257.svg" alt=" " data-tex="\phi">, from which the motion of the +planet may be calculated in the same way as in classical mechanics. +The most important result we get from this is a secular +rotation of the elliptic orbit of the planet in the same sense as +the revolution of the planet, amounting in radians per revolution +to +<span class="align-center"><img style="vertical-align: -2.194ex; width: 24.054ex; height: 5.611ex;" src="images/566.svg" alt=" " data-tex=" +\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}, +\qquad \text{(113)} +"></span> +<span class="pagenum" id="Page_102">[Pg 102]</span> +where +<span class="align-center"><img style="vertical-align: -5.129ex; width: 59.981ex; height: 11.389ex;" src="images/567.svg" alt=" " data-tex=" +\begin{align*} +a &= \text{the semi-major axis of the planetary orbit in centimetres.} \\ +e &= \text{the numerical eccentricity.} \\ +c &= 3·10^{+10}, \text{the velocity of light}\, \mathit{in\, vacuo}. \\ +T &= \text{the period of revolution in seconds.} +\end{align*} +"></span> +This expression furnishes the explanation of the motion of the +perihelion of the planet Mercury, which has been known for a +hundred years (since Leverrier), and for which theoretical astronomy +has hitherto been unable satisfactorily to account. +</p> +<p> +There is no difficulty in expressing Maxwell's theory of the +electromagnetic field in terms of the general theory of relativity; +this is done by application of the tensor formation (81), (82) +and (77). Let <img style="vertical-align: -0.685ex; width: 2.501ex; height: 2.255ex;" src="images/568.svg" alt=" " data-tex="\phi_{\mu}"> be a tensor of the first rank, to be denoted +as an electromagnetic 4-potential; then an electromagnetic field +tensor may be defined by the relations, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 29.316ex; height: 5.58ex;" src="images/569.svg" alt=" " data-tex=" +\phi_{\mu\nu} + = \frac{\partial \phi_{\mu}}{\partial x_{\nu}} - \frac{\partial \phi_{\nu}}{\partial x_{\mu}}. +\qquad \text{(114)} +"></span> +The second of Maxwell's systems of equations is then defined by +the tensor equation, resulting from this, +<span class="align-center"><img style="vertical-align: -2.237ex; width: 38.212ex; height: 5.58ex;" src="images/570.svg" alt=" " data-tex=" +\frac{\partial \phi_{\mu\nu}}{\partial x_{\rho}} + +\frac{\partial \phi_{\nu\rho}}{\partial x_{\mu}} + +\frac{\partial \phi_{\rho\mu}}{\partial x_{\nu}} = 0, +\qquad \text{(114a)} +"></span> +and the hrst of Maxwell's systems of equations is defined by the +tensor-density relation +<span class="align-center"><img style="vertical-align: -1.891ex; width: 21.762ex; height: 5.038ex;" src="images/571.svg" alt=" " data-tex=" +\frac{\partial \mathfrak F^{\mu\nu}}{\partial x_{\nu}} = \mathfrak J^{\mu}, +\qquad \text{(115)} +"></span> +<span class="pagenum" id="Page_103">[Pg 103]</span> +in which +<span class="align-center"><img style="vertical-align: -3.536ex; width: 21.852ex; height: 8.204ex;" src="images/572.svg" alt=" " data-tex=" +\begin{align*} +\mathfrak F^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}, \\ +\mathfrak J^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. +\end{align*} +"></span> +If we introduce the energy tensor of the electromagnetic field +into the right-hand side of (96), we obtain (115), for the special +case <img style="vertical-align: -0.314ex; width: 2.401ex; height: 1.867ex;" src="images/573.svg" alt=" " data-tex="\mathfrak J^{\mu}"> = 0, as a consequence of (96) by taking the divergence. +This inclusion of the theory of electricity in the scheme of the +general theory of relativity has been considered arbitrary and +unsatisfactory by many theoreticians. Nor can we in this way +conceive of the equilibrium of the electricity which constitutes +the elementary electrically charged particles. A theory in which +the gravitational field and the electromagnetic field enter as an +essential entity would be much preferable. H. Weyl, and recently +Th. Kaluza, have discovered some ingenious theorems along this +direction; but concerning them, I am convinced that they do not +bring us nearer to the true solution of the fundamental problem. +I shall not go into this further, but shall give a brief discussion +of the so-called cosmological problem, for without this, the considerations +regarding the general theory of relativity would, in +a certain sense, remain unsatisfactory. +</p> +<p> +Our previous considerations, based upon the field equations (96), +had for a foundation the conception that space on +the whole is Galilean-Euclidean, and that this character is disturbed +only by masses embedded in it. This conception was +certainly justified as long as we were dealing with spaces of the +order of magnitude of those that astronomy has to do with. +But whether portions of the universe, however large they may +be, are quasi-Euclidean, is a wholly different question. We can +<span class="pagenum" id="Page_104">[Pg 104]</span> +make this clear by using an example from the theory of surfaces +which we have employed many times. If a portion of a surface +is observed by the eye to be practically plane, it does not at all +follow that the whole surface has the form of a plane; the surface +might just as well be a sphere, for example, of sufficiently large +radius. The question as to whether the universe as a whole is +non-Euclidean was much discussed from the geometrical point of +view before the development of the theory of relativity. But with +the theory of relativity, this problem has entered upon a new +stage, for according to this theory the geometrical properties of +bodies are not independent, but depend upon the distribution +of masses. +</p> +<p> +If the universe were quasi-Euclidean, then Mach was wholly +wrong in his thought that inertia, as well as gravitation, depends +upon a kind of mutual action between bodies. For in this case, +with a suitably selected system of co-ordinates, the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> would +be constant at infinity, as they are in the special theory of relativity, +while within finite regions the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> would differ from these +constant values by small amounts only, with a suitable choice +of co-ordinates, as a result of the influence of the masses in finite +regions. The physical properties of space would not then be +wholly independent, that is, uninfluenced by matter, but in the +main they would be, and only in small measure, conditioned by +matter. Such a dualistic conception is even in itself not satisfactory; +there are, however, some important physical arguments +against it, which we shall consider. +</p> +<p> +The hypothesis that the universe is infinite and Euclidean +at infinity, is, from the relativistic point of view, a complicated +hypothesis. In the language of the general theory of relativity +it demands that the Riemann tensor of the fourth rank +<img style="vertical-align: -0.357ex; width: 5.172ex; height: 1.902ex;" src="images/574.svg" alt=" " data-tex="R_{iklm}"> +<span class="pagenum" id="Page_105">[Pg 105]</span> +shall vanish at infinity, which furnishes twenty independent conditions, +while only ten curvature components <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}">, enter into +the laws of the gravitational field. It is certainly unsatisfactory +to postulate such a far-reaching limitation without any physical +basis for it. +</p> +<p> +But in the second place, the theory of relativity makes it +appear probable that Mach was on the right road in his thought +that inertia depends upon a mutual action of matter. For we +shall show in the following that, according to our equations, inert +masses do act upon each other in the sense of the relativity of +inertia, even if only very feebly. What is to be expected along +the line of Mach's thought? +</p> +<p class="hanging2"> +1. The inertia of a body must increase when ponderable +masses are piled up in its neighbourhood. +</p> +<p class="hanging2"> +2. A body must experience an accelerating force when +neighbouring masses are accelerated, and, in fact, the +force must be in the same direction as the acceleration. +</p> +<p class="hanging2"> +3. A rotating hollow body must generate inside of itself +a "Coriolis field," which deflects moving bodies in the +sense of the rotation, and a radial centrifugal field as +well. +</p> +<p> +We shall now show that these three effects, which are to be +expected in accordance with Mach's ideas, are actually present +according to our theory, although their magnitude is so small +that confirmation of them by laboratory experiments is not to be +thought of. For this purpose we shall go back to the equations of +motion of a material particle (90), and carry the approximations +somewhat further than was done in equation (90a). +<span class="pagenum" id="Page_106">[Pg 106]</span> +</p> +<p> +First, we consider <img style="vertical-align: -0.489ex; width: 2.96ex; height: 1.486ex;" src="images/575.svg" alt=" " data-tex="\gamma_{44}"> as small of the first order. The square +of the velocity of masses moving under the influence of the gravitational +force is of the same order, according to the energy +equation. It is therefore logical to regard the velocities of the +material particles we are considering, as well as the velocities +of the masses which generate the field, as small, of the order +<img style="vertical-align: -1.552ex; width: 2.127ex; height: 4.588ex;" src="images/576.svg" alt=" " data-tex="\dfrac{1}{2}">. +We shall now carry out the approximation in the equations that +arise from the field equations (101) and the equations of motion (90) +so far as to consider terms, in the second member +of (90), that are linear in those velocities. Further, we shall not +put <img style="vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;" src="images/346.svg" alt=" " data-tex="ds"> and <img style="vertical-align: -0.025ex; width: 1.851ex; height: 1.595ex;" src="images/282.svg" alt=" " data-tex="dl"> equal to each other, but, corresponding to the +higher approximation, we shall put +<span class="align-center"><img style="vertical-align: -1.552ex; width: 28.33ex; height: 4.153ex;" src="images/577.svg" alt=" " data-tex=" +ds = \sqrt{g_{44}}\, dl + = \left(1 - \frac{\gamma_{44}}{2}\right) dl. +"></span> +From (90) we obtain, at first, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 59.568ex; height: 5.444ex;" src="images/578.svg" alt=" " data-tex=" +\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] + = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} + \left(1 + \frac{\gamma_{44}}{2}\right). +\qquad \text{(116)} +"></span> +</p> +<p> +From (101) we get, to the approximation sought for, +<span class="align-center"><img style="vertical-align: -7.642ex; width: 54.085ex; height: 16.414ex;" src="images/579.svg" alt=" " data-tex=" +\left. +\begin{aligned} +-\gamma_{11} = -\gamma_{22} = -\gamma_{33} + &= \gamma_{44} = \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\gamma_{4\alpha} + &= -\frac{i \kappa}{2} + \int \frac{\sigma \dfrac{dx_{\alpha}}{ds}\, dV_{0}}{r}, \\ +\gamma_{\alpha\beta} &= 0, +\end{aligned} +\right\} +\qquad \text{(117)} +"></span> +in which, in (117), <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha"> and <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> denote the space indices only. +<span class="pagenum" id="Page_107">[Pg 107]</span> +</p> +<p> +On the right-hand side of (116) we can replace +1 + <img style="vertical-align: -1.552ex; width: 6.656ex; height: 4.153ex;" src="images/496.svg" alt=" " data-tex="\left(\dfrac{\gamma_{44}}{2}\right)"> by 1 +and <img style="vertical-align: -1.045ex; width: 5.291ex; height: 2.898ex;" src="images/580.svg" alt=" " data-tex="-\Gamma_{\alpha\beta}^{\mu}"> by +<img style="vertical-align: -1.469ex; width: 4.608ex; height: 4.07ex;" src="images/581.svg" alt=" " data-tex="\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right]">. +It is easy to see, in addition, that to this +degree of approximation we must put +<span class="align-center"><img style="vertical-align: -8.407ex; width: 28.058ex; height: 17.946ex;" src="images/582.svg" alt=" " data-tex=" +\begin{align*} +\left[{{\genfrac{}{}{0pt}{}{44}{\mu}}}\right] + &= -\tfrac{1}{2} \frac{\partial \gamma_{44}}{\partial x_{\mu}} + + \frac{\partial \gamma_{4\mu}}{\partial x_{4}},\\ +\left[{{\genfrac{}{}{0pt}{}{\alpha 4}{\mu}}}\right] + &= \tfrac{1}{2} + \left(\frac{\partial \gamma_{4\mu}}{\partial x_{\alpha}} - \frac{\partial \gamma_{4\alpha}}{\partial x_{\mu}}\right), \\ +\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right] &= 0, +\end{align*} +"></span> +in which <img style="vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;" src="images/28.svg" alt=" " data-tex="\alpha">, <img style="vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;" src="images/29.svg" alt=" " data-tex="\beta"> and <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> denote space indices. +We therefore obtain +from (116), in the usual vector notation, +<span class="align-center"><img style="vertical-align: -8.844ex; width: 52.436ex; height: 18.819ex;" src="images/583.svg" alt=" " data-tex=" +\DeclareMathOperator{\grad}{grad} +\DeclareMathOperator{\rot}{rot} +\left. +\begin{aligned} +\frac{d}{dl}\bigl[(1 + \overline{\sigma}) \mathbf v \bigr] + &= \grad \overline{\sigma} + \frac{\partial \mathfrak A}{\partial l} + [\rot \mathfrak A, \mathbf v], \\ +\overline{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ +\mathfrak A &= \frac{\kappa}{2 \pi} + \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}. +\end{aligned} +\right\} +\qquad \text{(118)} +"></span> +</p> +<p> +The equations of motion, (118), show now, in fact, that +</p> +<p class="hanging"> +1. The inert mass is proportional to 1 + <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1.742ex;" src="images/584.svg" alt=" " data-tex="\overline{\sigma}"> and therefore +increases when ponderable masses approach the test +body. +</p> +<p class="hanging"> +2. There is an inductive action of accelerated masses, +of the same sign, upon the test body. This is the +term <img style="vertical-align: -1.602ex; width: 3.9ex; height: 4.749ex;" src="images/585.svg" alt=" " data-tex="\dfrac{\partial \mathfrak A}{\partial l}">. +<span class="pagenum" id="Page_108">[Pg 108]</span> +</p> +<p class="hanging"> +3. A material particle, moving perpendicularly to the axis +of rotation inside a rotating hollow body, is deflected in +the sense of the rotation (Coriolis field). The centrifugal +action, mentioned above, inside a rotating hollow +body, also follows from the theory, as has been shown +by Thirring.<a id="FNanchor_18_1"></a><a href="#Footnote_18_1" class="fnanchor">[18]</a> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_18_1"></a><a href="#FNanchor_18_1"><span class="label">[18]</span></a>That the centrifugal action must be inseparably connected with the +existence of the Coriolis field may be recognized, even without calculation, +in the special case of a co-ordinate system rotating uniformly relatively to +an inertial system; our general co-variant equations naturally must apply +to such a case.</p></div> + +<p> +Although all of these effects are inaccessible to experiment, +because <img style="vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;" src="images/503.svg" alt=" " data-tex="\kappa"> is so small, nevertheless they certainly exist according +to the general theory of relativity. We must see in them a +strong support for Mach's ideas as to the relativity of all inertial +actions. If we think these ideas consistently through to the end +we must expect the <i>whole</i> inertia, that is, the <i>whole</i> +<img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field, to +be determined by the matter of the universe, and not mainly by +the boundary conditions at infinity. +</p> +<p> +For a satisfactory conception of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field of cosmical dimensions, +the fact seems to be of significance that the relative +velocity of the stars is small compared to the velocity of light. +It follows from this that, with a suitable choice of co-ordinates, +<img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/532.svg" alt=" " data-tex="g_{44}"> is nearly constant in the universe, at least, in that part of +the universe in which there is matter. The assumption appears +natural, moreover, that there are stars in all parts of the universe, +so that we may well assume that the inconstancy of <img style="vertical-align: -0.464ex; width: 2.867ex; height: 1.464ex;" src="images/532.svg" alt=" " data-tex="g_{44}"> depends +only upon the circumstance that matter is not distributed +continuously, but is concentrated in single celestial bodies and +systems of bodies. If we are willing to ignore these more local +<span class="pagenum" id="Page_109">[Pg 109]</span> +non-uniformities of the density of matter and of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">-field, in +order to learn something of the geometrical properties of the universe +as a whole, it appears natural to substitute for the actual +distribution of masses a continuous distribution, and furthermore +to assign to this distribution a uniform density <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">. In this +imagined universe all points with space directions will be geometrically +equivalent; with respect to its space extension it will +have a constant curvature, and will be cylindrical with respect +to its <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">-co-ordinate. The possibility seems to be particularly +satisfying that the universe is spatially bounded and thus, in +accordance with our assumption of the constancy of <img style="vertical-align: -0.025ex; width: 1.292ex; height: 1ex;" src="images/127.svg" alt=" " data-tex="\sigma">, is of +constant curvature, being either spherical or elliptical; for then +the boundary conditions at infinity which are so inconvenient +from the standpoint of the general theory of relativity, may be +replaced by the much more natural conditions for a closed surface. +</p> +<p> +According to what has been said, we are to put +<span class="align-center"><img style="vertical-align: -0.685ex; width: 35.196ex; height: 2.703ex;" src="images/586.svg" alt=" " data-tex=" +ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, +\qquad \text{(119)} +"></span> +in which the indices <img style="vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;" src="images/86.svg" alt=" " data-tex="\mu"> and <img style="vertical-align: 0; width: 1.199ex; height: 1ex;" src="images/20.svg" alt=" " data-tex="\nu"> run from 1 to 3 only. +The <img style="vertical-align: -0.685ex; width: 3.172ex; height: 1.683ex;" src="images/489.svg" alt=" " data-tex="\gamma_{\mu\nu}"> +will be such functions of <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/5.svg" alt=" " data-tex="x_{1}">, <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/6.svg" alt=" " data-tex="x_{2}">, <img style="vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;" src="images/7.svg" alt=" " data-tex="x_{3}"> as correspond to +a three-dimensional continuum of constant positive curvature. We must +now investigate whether such an assumption can satisfy the field +equations of gravitation. +</p> +<p> +In order to be able to investigate this, we must first find +what differential conditions the three-dimensional manifold of +constant curvature satisfies. A spherical manifold of three dimensions, +<span class="pagenum" id="Page_110">[Pg 110]</span> +embedded in a Euclidean continuum of four dimensions,<a id="FNanchor_19_1"></a><a href="#Footnote_19_1" class="fnanchor">[19]</a> +is given by the equations +<span class="align-center"><img style="vertical-align: -2.348ex; width: 32.952ex; height: 5.828ex;" src="images/587.svg" alt=" " data-tex=" +\begin{align*} +{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}, \\ +{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. +\end{align*} +"></span> +By eliminating <img style="vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;" src="images/239.svg" alt=" " data-tex="x_{4}">, we get +<span class="align-center"><img style="vertical-align: -2.003ex; width: 56.135ex; height: 5.496ex;" src="images/588.svg" alt=" " data-tex=" +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} + {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}. +"></span> +</p> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_19_1"></a><a href="#FNanchor_19_1"><span class="label">[19]</span></a>The aid of a fourth space dimension has naturally no significance except +that of a mathematical artifice.</p></div> + +<p> +As far as terms of the third and higher degrees in the <img style="vertical-align: -0.339ex; width: 2.33ex; height: 1.339ex;" src="images/14.svg" alt=" " data-tex="x_{\nu}">, we +can put, in the neighbourhood of the origin of co-ordinates, +<span class="align-center"><img style="vertical-align: -1.651ex; width: 28.622ex; height: 4.377ex;" src="images/589.svg" alt=" " data-tex=" +ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) + dx_{\mu}\, dx_{\nu}. +"></span> +</p> +<p> +Inside the brackets are the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}"> of the manifold in the neighbourhood +of the origin. Since the first derivatives of the <img style="vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;" src="images/367.svg" alt=" " data-tex="g_{\mu\nu}">, +and therefore also the <img style="vertical-align: -0.904ex; width: 3.414ex; height: 2.443ex;" src="images/556.svg" alt=" " data-tex="\Gamma_{\mu\nu}^{\sigma}">, vanish at the origin, the calculation +of the <img style="vertical-align: -0.685ex; width: 3.718ex; height: 2.23ex;" src="images/478.svg" alt=" " data-tex="R_{\mu\nu}"> for this manifold, by (88), is very simple at the origin. +We have +<span class="align-center"><img style="vertical-align: -1.651ex; width: 24.585ex; height: 4.688ex;" src="images/590.svg" alt=" " data-tex=" +R_{\mu\nu} = -\frac{2}{a^{2}} \delta_{\mu\nu} + = \frac{2}{a^{2}} g_{\mu\nu}. +"></span> +</p> +<p> +Since the relation +<img style="vertical-align: -1.651ex; width: 12.994ex; height: 4.688ex;" src="images/591.svg" alt=" " data-tex="R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}"> is universally co-variant, +and since all points of the manifold are geometrically equivalent, +this relation holds for every system of co-ordinates, and +everywhere in the manifold. In order to avoid confusion with +<span class="pagenum" id="Page_111">[Pg 111]</span> +the four-dimensional continuum, we shall, in the following, designate +quantities that refer to the three-dimensional continuum +by Greek letters, and put +<span class="align-center"><img style="vertical-align: -1.651ex; width: 25.267ex; height: 4.688ex;" src="images/592.svg" alt=" " data-tex=" +P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}. +\qquad \text{(120)} +"></span> +</p> +<p> +We now proceed to apply the field equations (96) to our special +case. From (119) we get for the four-dimensional manifold, +<span class="align-center"><img style="vertical-align: -2.148ex; width: 44.083ex; height: 5.428ex;" src="images/593.svg" alt=" " data-tex=" +\left. +\begin{aligned} +R_{\mu\nu} &= P_{\mu\nu} \quad \text{for the indices 1 to 3}, \\ +R_{14} &= R_{24} = R_{34} = R_{44} = 0. +\end{aligned} +\right\} +\qquad \text{(121)} +"></span> +</p> +<p> +For the right-hand side of (96) we have to consider the energy +tensor for matter distributed like a cloud of dust. According to +what has gone before we must therefore put +<span class="align-center"><img style="vertical-align: -1.575ex; width: 17.527ex; height: 4.87ex;" src="images/594.svg" alt=" " data-tex=" +T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds} +"></span> +specialized for the case of rest. But in addition, we shall add +a pressure term that may be physically established as follows. +Matter consists of electrically charged particles. On the basis +of Maxwell's theory these cannot be conceived of as electromagnetic +fields free from singularities. In order to be consistent +with the facts, it is necessary to introduce energy terms, not +contained in Maxwell's theory, so that the single electric particles +may hold together in spite of the mutual repulsions between +their elements, charged with electricity of one sign. For the sake +of consistency with this fact, Poincaré has assumed a pressure +<span class="pagenum" id="Page_112">[Pg 112]</span> +to exist inside these particles which balances the electrostatic +repulsion. It cannot, however, be asserted that this pressure +vanishes outside the particles. We shall be consistent with this +circumstance if, in our phenomenological presentation, we add +a pressure term. This must not, however, be confused with a +hydrodynamical pressure, as it serves only for the energetic presentation +of the dynamical relations inside matter. In this sense +we put +<span class="align-center"><img style="vertical-align: -1.575ex; width: 39.787ex; height: 4.835ex;" src="images/595.svg" alt=" " data-tex=" +T_{\mu\nu} + = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} + - g_{\mu\nu} p. +\qquad \text{(122)} +"></span> +</p> +<p> +In our special case we have, therefore, to put +<span class="align-center"><img style="vertical-align: -3.626ex; width: 38.187ex; height: 8.383ex;" src="images/596.svg" alt=" " data-tex=" +\begin{align*} +T_{\mu\nu} + &= \gamma_{\mu\nu} p \quad \text{(for}\,\,\mu\,\, \text{and}\,\,\,\nu \,\,\text{from 1 to 3)}, \\ +T_{44} &= \sigma - p, \\ +T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p + = \sigma - 4p. +\end{align*} +"></span> +Observing that the field equation (96) may be written in the +form +<span class="align-center"><img style="vertical-align: -0.781ex; width: 24.742ex; height: 2.737ex;" src="images/597.svg" alt=" " data-tex=" +R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T), +"></span> +we get from (96) the equations, +<span class="align-center"><img style="vertical-align: -4.194ex; width: 25.503ex; height: 9.519ex;" src="images/598.svg" alt=" " data-tex=" +\begin{align*} ++\frac{2}{a^{2}} \gamma_{\mu\nu} + &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}, \\ +0 &= -\kappa \left(\frac{\sigma}{2} + p\right). +\end{align*} +"></span> +From this follows +<span class="align-center"><img style="vertical-align: -4.585ex; width: 22.431ex; height: 10.301ex;" src="images/599.svg" alt=" " data-tex=" +\left. +\begin{aligned} +p &= -\frac{\sigma}{2}, \\ +a &= \sqrt{\frac{2}{\kappa\sigma}}. +\end{aligned} +\right\} +\qquad \text{(123)} +"></span> +</p> +<p> +If the universe is quasi-Euclidean, and its radius of curvature +therefore infinite, then a would vanish. But it is improbable that +<span class="pagenum" id="Page_113">[Pg 113]</span> +the mean density of matter in the universe is actually zero; this +is our third argument against the assumption that the universe +is quasi-Euclidean. Nor does it seem possible that our hypothetical +pressure can vanish; the physical nature of this pressure can +be appreciated only after we have a better theoretical knowledge +of the electromagnetic field. According to the second of equations +(123) the radius, <img style="vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;" src="images/75.svg" alt=" " data-tex="a">, of the universe is determined in terms +of the total mass, <img style="vertical-align: 0; width: 2.378ex; height: 1.545ex;" src="images/168.svg" alt=" " data-tex="M">, of matter, by the equation +<span class="align-center"><img style="vertical-align: -1.654ex; width: 19.575ex; height: 4.728ex;" src="images/600.svg" alt=" " data-tex=" +a = \frac{M\kappa}{4\pi^{2}}. +\qquad \text{(124)} +"></span> +The complete dependence of the geometrical upon the physical +properties becomes clearly apparent by means of this equation. +</p> +<p> +Thus we may present the following arguments against the +conception of a space-infinite, and for the conception of a space-bounded, +universe:— +</p> +<p> +1. From the standpoint of the theory of relativity, the condition +for a closed surface is very much simpler than the corresponding +boundary condition at infinity of the quasi-Euclidean +structure of the universe. +</p> +<p> +2. The idea that Mach expressed, that inertia depends upon +the mutual action of bodies, is contained, to a first approximation, +in the equations of the theory of relativity; it follows +from these equations that inertia depends, at least in part, upon +mutual actions between masses. As it is an unsatisfactory assumption +to make that inertia depends in part upon mutual +actions, and in part upon an independent property of space, +Mach's idea gains in probability. But this idea of Mach's corresponds +only to a finite universe, bounded in space, and not to a +quasi-Euclidean, infinite universe. From the standpoint of epistemology +<span class="pagenum" id="Page_114">[Pg 114]</span> +it is more satisfying to have the mechanical properties +of space completely determined by matter, and this is the case +only in a space-bounded universe. +</p> +<p> +3. An infinite universe is possible only if the mean density of +matter in the universe vanishes. Although such an assumption +is logically possible, it is less probable than the assumption that +there is a finite mean density of matter in the universe. +<span class="pagenum" id="Page_115">[Pg 115]</span> +</p> +</div> + +<p><br><br></p> + +<div class='chapter'> +<h2 title="INDEX"><a id="INDEX">INDEX</a></h2> + +<p>A</p> +<p class="nind"> +Accelerated masses, inductive<br> +<span style="margin-left: 1em;">action of, <a href="#Page_108">108</a></span><br> +Addition and subtraction of<br> +<span style="margin-left: 1em;">tensors, <a href="#Page_14">14</a></span><br> +—theorem of velocities, <a href="#Page_38">38</a> +</p> +<p><br></p> +<p>B</p> +<p class="nind"> +Biot-Savart force, <a href="#Page_44">44</a> +</p> +<p><br></p> +<p>C</p> +<p class="nind"> +Centrifugal force, <a href="#Page_64">64</a><br> +Clocks, moving, <a href="#Page_38">38</a><br> +Compressible viscous fluid, <a href="#Page_22">22</a><br> +Concept of space, <a href="#Page_3">3</a><br> +—time, <a href="#Page_28">28</a><br> +Conditions of orthogonality, <a href="#Page_7">7</a><br> +Congruence, theorems of, <a href="#Page_3">3</a><br> +Conservation principles, <a href="#Page_54">54</a><br> +Continuum, four-dimensional, <a href="#Page_31">31</a><br> +Contraction of tensors, <a href="#Page_14">14</a><br> +Contra-variant vectors, <a href="#Page_69">69</a><br> +—tensors, <a href="#Page_71">71</a><br> +Co-ordinates, preferred systems<br> +<span style="margin-left: 1em;">of, <a href="#Page_8">8</a></span><br> +Co-variance of equation of<br> +<span style="margin-left: 1em;">continuity, <a href="#Page_21">21</a></span><br> +Co-variant, <a href="#Page_12">12</a> <i>et seq.</i><br> +—vector, <a href="#Page_68">68</a><br> +Criticism of principle of inertia, <a href="#Page_62">62</a><br> +Criticisms of theory of<br> +<span style="margin-left: 1em;">relativity, <a href="#Page_29">29</a></span><br> +Curvilinear co-ordinates, <a href="#Page_65">65</a> +</p> +<p><br></p> +<p>D</p> +<p class="nind"> +Differentiation of tensors, <a href="#Page_73">73</a>, <a href="#Page_76">76</a><br> +Displacement of spectral lines, <a href="#Page_97">97</a><br> +</p> +<p><br></p> +<p>E</p> +<p class="nind"> +Energy and mass, <a href="#Page_45">45</a>, <a href="#Page_49">49</a><br> +—tensor of electromagnetic<br> +<span style="margin-left: 1em;">field, <a href="#Page_50">50</a></span><br> +—of matter, <a href="#Page_54">54</a><br> +Equation of continuity, co-variance<br> +<span style="margin-left: 1em;">of, <a href="#Page_21">21</a></span><br> +Equations of motion of material<br> +<span style="margin-left: 1em;">particle, <a href="#Page_50">50</a></span><br> +Equivalence of mass and<br> +<span style="margin-left: 1em;">energy, <a href="#Page_49">49</a></span><br> +Equivalent spaces of reference, <a href="#Page_25">25</a><br> +Euclidean geometry, <a href="#Page_4">4</a><br> +</p> +<p><br></p> +<p>F</p> +<p class="nind"> +Finiteness of universe, <a href="#Page_105">105</a><br> +Fizeau, <a href="#Page_28">28</a><br> +Four-dimensional continuum, <a href="#Page_31">31</a><br> +Four-vector, <a href="#Page_41">41</a><br> +Fundamental tensor, <a href="#Page_71">71</a> +</p> +<p><br></p> +<p>G</p> +<p class="nind"> +Galilean regions, <a href="#Page_62">62</a><br> +—transformation, <a href="#Page_27">27</a><br> +Gauss, <a href="#Page_65">65</a><br> +Geodetic lines, <a href="#Page_82">82</a><br> +Geometry, Euclidean, <a href="#Page_4">4</a><br> +Gravitation constant, <a href="#Page_95">95</a><br> +Gravitational mass, <a href="#Page_60">60</a> +</p> +<p><br></p> +<p>H</p> +<p class="nind"> +Homogeneity of space, <a href="#Page_17">17</a><br> +Hydrodynamical equations, <a href="#Page_54">54</a><br> +Hypotheses of pre-relativity<br> +<span style="margin-left: 1em;">physics, <a href="#Page_73">73</a></span> +</p> +<p><br></p> +<p>I</p> +<p class="nind"> +Inductive action of accelerated<br> +<span style="margin-left: 1em;">masses, <a href="#Page_108">108</a></span><br> +Inert and gravitational mass, equality<br> +<span style="margin-left: 1em;">of, <a href="#Page_60">60</a></span><br> +Invariant, <a href="#Page_9">9</a> <i>et seq.</i><br> +Isotropy of space, <a href="#Page_17">17</a><br> +</p> +<p><br></p> +<p>K</p> +<p class="nind"> +Kaluza, <a href="#Page_104">104</a> +</p> +<p><br></p> +<p>L</p> +<p class="nind"> +Levi-Civita, <a href="#Page_73">73</a><br> +Light-cone, <a href="#Page_41">41</a><br> +Light ray, path of, <a href="#Page_98">98</a><br> +Light-time, <a href="#Page_33">33</a><br> +Linear orthogonal<br> +<span style="margin-left: 1em;">transformation, <a href="#Page_7">7</a></span><br> +Lorentz electromotive force, <a href="#Page_44">44</a><br> +—transformation, <a href="#Page_31">31</a><br> +</p> +<p><br></p> +<p>M</p> +<p class="nind"> +Mach, <a href="#Page_59">59</a>, <a href="#Page_105">105</a>, <a href="#Page_106">106</a>, <a href="#Page_109">109</a>, <a href="#Page_114">114</a><br> +Mass and Energy, <a href="#Page_45">45</a>, <a href="#Page_49">49</a><br> +—equality of gravitational and<br> +<span style="margin-left: 1em;">inert, <a href="#Page_60">60</a></span><br> +—gravitational, <a href="#Page_60">60</a><br> +Maxwell's equations, <a href="#Page_23">23</a><br> +Mercury, perihelion of, <a href="#Page_99">99</a>, <a href="#Page_103">103</a><br> +Michelson and Morley, <a href="#Page_28">28</a><br> +Minkowski, <a href="#Page_32">32</a><br> +Motion of particle, equations of, <a href="#Page_50">50</a><br> +Moving measuring rods and<br> +<span style="margin-left: 1em;">clocks, <a href="#Page_38">38</a></span><br> +Multiplication of tensors, <a href="#Page_14">14</a><br> +</p> +<p><br></p> +<p>N</p> +<p class="nind"> +Newtonian gravitation<br> +<span style="margin-left: 1em;">constant, <a href="#Page_95">95</a></span><br> +</p> +<p><br></p> +<p>O</p> +<p class="nind"> +Operations on tensors, <a href="#Page_13">13</a> <i>et seq.</i><br> +Orthogonal transformations, linear, <a href="#Page_7">7</a><br> +Orthogonality, conditions of, <a href="#Page_7">7</a><br> +</p> +<p><br></p> +<p>P</p> +<p class="nind"> +Path of light ray, <a href="#Page_98">98</a><br> +Perihelion of Mercury, <a href="#Page_99">99</a>, <a href="#Page_103">103</a><br> +Poisson's equation, <a href="#Page_87">87</a><br> +Preferred systems of<br> +<span style="margin-left: 1em;">co-ordinates, <a href="#Page_8">8</a></span><br> +Pre-relativity physics, hypotheses<br> +<span style="margin-left: 1em;">of, <a href="#Page_26">26</a></span><br> +Principle of equivalence, <a href="#Page_61">61</a><br> +—inertia, criticism of, <a href="#Page_62">62</a><br> +Principles of conservation, <a href="#Page_54">54</a><br> +</p> +<p><br></p> +<p>R</p> +<p class="nind"> +Radius of Universe, <a href="#Page_113">113</a><br> +Rank of tensor, <a href="#Page_13">13</a><br> +Ray of light, path of, <a href="#Page_98">98</a><br> +Reference, space of, <a href="#Page_3">3</a><br> +Riemann, <a href="#Page_68">68</a><br> +—tensor, <a href="#Page_79">79</a>, <a href="#Page_82">82</a>, <a href="#Page_105">105</a><br> +Rods (measuring) and clocks in<br> +<span style="margin-left: 1em;">motion, <a href="#Page_38">38</a></span><br> +Rotation, <a href="#Page_63">63</a> +</p> +<p><br></p> +<p>S</p> +<p class="nind"> +Simultaneity, <a href="#Page_17">17</a>, <a href="#Page_29">29</a><br> +Sitter, <a href="#Page_28">28</a><br> +Skew-symmetrical tensor, <a href="#Page_15">15</a><br> +Solar Eclipse expedition (1919), <a href="#Page_99">99</a><br> +Space, concept of, <a href="#Page_2">2</a><br> +—Homogeneity of, <a href="#Page_17">17</a><br> +—Isotropy of, <a href="#Page_17">17</a><br> +Spaces of reference, <a href="#Page_3">3</a><br> +—equivalence of, <a href="#Page_25">25</a><br> +Special Lorentz transformation, <a href="#Page_34">34</a><br> +Spectral lines, displacement of, <a href="#Page_97">97</a><br> +Straightest lines, <a href="#Page_82">82</a><br> +Stress tensor, <a href="#Page_22">22</a><br> +Symmetrical tensor, <a href="#Page_15">15</a><br> +Systems of co-ordinates, <br> +preferred, <a href="#Page_8">8</a><br> +</p> +<p><br></p> +<p>T</p> +<p class="nind"> +Tensor, <a href="#Page_12">12</a> <i>et seq.</i>, <a href="#Page_68">68</a> <i>et seq.</i><br> +—Addition and subtraction of, <a href="#Page_14">14</a><br> +—Contraction of, <a href="#Page_14">14</a><br> +—Fundamental, <a href="#Page_71">71</a><br> +—Multiplication of, <a href="#Page_14">14</a><br> +—operations, <a href="#Page_13">13</a> <i>et seq.</i><br> +—Rank of, <a href="#Page_13">13</a><br> +—Symmetrical and<br> +<span style="margin-left: 1em;">Skew-symmetrical, <a href="#Page_15">15</a></span><br> +Tensors, formation by<br> +<span style="margin-left: 1em;">differentiation, <a href="#Page_73">73</a></span><br> +Theorem for addition of<br> +<span style="margin-left: 1em;">velocities, <a href="#Page_38">38</a></span><br> +Theorems of congruence, <a href="#Page_3">3</a><br> +Theory of relativity, criticisms<br> +<span style="margin-left: 1em;">of, <a href="#Page_29">29</a></span><br> +Thirring, <a href="#Page_109">109</a><br> +Time-concept, <a href="#Page_28">28</a><br> +Time-space concept, <a href="#Page_31">31</a><br> +Transformation, Galilean, <a href="#Page_27">27</a><br> +—Linear orthogonal, <a href="#Page_7">7</a><br> +</p> +<p><br></p> +<p>U</p> +<p class="nind"> +Universe, Finiteness of, <a href="#Page_105">105</a><br> +—Radius of, <a href="#Page_113">113</a><br> +</p> +<p><br></p> +<p>V</p> +<p class="nind"> +Vector, co-variant, <a href="#Page_69">69</a><br> +—contra-variant, <a href="#Page_69">69</a><br> +Velocities, addition theorem of, <a href="#Page_38">38</a><br> +Viscous compressible fluid, <a href="#Page_22">22</a><br> +</p> +<p><br></p> +<p>W</p> +<p class="nind"> +Weyl, <a href="#Page_69">73</a>, <a href="#Page_99">99</a>, <a href="#Page_104">104</a> +</p> +</div> + +<p><br><br><br></p> + +<div class='chapter'> +<p class="center"> +PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN +</p> +</div> + 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