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diff --git a/78343-src.zip b/78343-src.zip Binary files differdeleted file mode 100644 index 769dea3..0000000 --- a/78343-src.zip +++ /dev/null diff --git a/78343-src/78343-src.htm b/78343-src/78343-src.htm new file mode 100644 index 0000000..632743b --- /dev/null +++ b/78343-src/78343-src.htm @@ -0,0 +1,1294 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + <meta charset="UTF-8"> + <title> + A Simplified Presentation of Einstein's Unified Field Equations | Project Gutenberg + </title> + <link rel="icon" href="images/cover.jpg" type="image/x-cover"> + <style> + +body { + margin-left: 10%; + margin-right: 10%; +} + +/* General headers */ + +h1 { + text-align: center; + clear: both; +} +h2, h3 { + text-align: center; + font-weight: bold; + margin-top: 1em; + margin-bottom: 1em; + } + +p { + margin-top: .51em; + text-align: justify; + margin-bottom: .49em; + text-indent: 1.5em; +} + +.nind {text-indent:0;} + +.nindc {text-align:center; text-indent:0;} + +.space-above2 { margin-top: 2em; } +.space-below2 { margin-bottom: 2em; } + +hr { + width: 33%; + margin-top: 2em; + margin-bottom: 2em; + margin-left: 33.5%; + margin-right: 33.5%; + clear: both; +} + +hr.chap {width: 65%; margin-left: 17.5%; margin-right: 17.5%;} +@media print { hr.chap {display: none; visibility: hidden;} } + +hr.r5 {width: 5%; margin-top: 1em; margin-bottom: 1em; margin-left: 47.5%; margin-right: 47.5%;} + +div.chapter {page-break-before: always;} +h2.nobreak {page-break-before: avoid;} + +.allsmcap {font-variant: small-caps; text-transform: lowercase;} + +/* Images */ + +img {max-width: 100%; width: 100%; height: auto;} +.width500 {max-width: 500px;} +.x-ebookmaker .width500 {width: 100%;} + + +.figcenter { + margin: auto; + text-align: center; + page-break-inside: avoid; + max-width: 100%; +} + +/* Footnotes */ +.footnotes {border: 1px dashed;} + +.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; +} + +/* css needed in m2svg output: displayed equations and prevention of bad breaks*/ + .align-center { + display: block; + text-align: center; + text-indent: 0; + margin-top: 1em; + margin-bottom: 1em; + } + .nowrap { + white-space: nowrap; + } + + </style> +</head> +<body> +<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 78343 ***</div> + + + +<figure class="figcenter width500" id="cover" style="width: 1623px;"> +<img src="images/cover.jpg" width="1623" height="2560" alt="This book +is a technical monograph that provides a more accessible mathematical +treatment of Einstein's attempts to unify gravitational and +electromagnetic fields into a single geometric framework."> +</figure> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h1>A SIMPLIFIED PRESENTATION<br> +<span class="allsmcap">OF</span> +<br> +EINSTEIN'S<br> +UNIFIED FIELD EQUATIONS</h1> + + +<p class="nindc space-above2"> +<span class="allsmcap">By</span><br> +TULLIO LEVI-CIVITA</p> + +<p class="nindc space-below2">Professor of Rational Mechanics in the University of Rome<br> +Fellow of R. Accademia Nazionale del Lincei</p> + + +<p class="nindc space-above2 space-below2"> +<i>Authorized Translation by</i><br> +JOHN DOUGALL, <span class="allsmcap">M.A., D.Sc.</span></p> + + +<p class="nindc space-above2 space-below2"> +BLACKIE & SON LIMITED<br> +LONDON AND GLASGOW<br> +1929 +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc space-above2 space-below2"> +<span class="allsmcap">BLACKIE & SON LIMITED</span><br> +<i>50 Old Bailey, London</i><br> +<i>17 Stanhope Street, Glasgow</i><br> +<span class="allsmcap">BLACKIE & SON (INDIA) LIMITED</span><br> +<i>Warwick House, Fort Street, Bombay</i><br> +<span class="allsmcap">BLACKIE & SON (CANADA) LIMITED</span><br> +<i>1118 Bay Street, Toronto</i><br> +</p> + + +<p class="nindc space-above2 space-below2"> +<i>Printed in Great Britain by Blackie & Son, Ltd., Glasgow</i><br> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="A_SIMPLIFIED_PRESENTATION_OF">A SIMPLIFIED PRESENTATION OF +EINSTEIN'S +UNIFIED FIELD EQUATIONS</h2> +</div> + +<hr class="r5"> + +<p>In his recent paper, "Zur einheitlichen Feldtheorie",<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a> Einstein made +use of the fundamental idea that it is both possible and useful to give +a geometrical interpretation of the complete system of the sixteen +field equations (consisting of Einstein's celebrated gravitational +equations and Maxwell's equations) in such a way as to include the +definition (and the definition only) of an orthogonal quadruplet<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> +embedded in the space-time world.</p> + +<p>Conversely, the sixteen parameters determining a quadruplet are to give +a complete definition not only of the Riemannian metric of space (as is +well known, this takes place automatically), but of the phenomena of +electromagnetism as well.</p> + +<p>For this purpose the eminent author introduced covariant derivatives +with respect to the quadruplet, and suggested relationships between +them which to a first approximation lead to the required co-ordination +of gravitational and electromagnetic phenomena.</p> + +<p>It appears to me, however, that the root problem raised by Einstein can +be solved in a simpler and more general way by making use of perfectly +familiar methods of the absolute differential calculus on the one hand, +while, on the other hand, retaining unaltered all results previously +obtained.</p> + + +<p class="nindc space-above2"> +<b>1. Geometrical and formal preliminaries.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></b></p> + +<p>Let \(x^{\nu}(\nu=0,1, \ldots, n-1)\) be general co-ordinates of a +Riemannian space \(R_{n}\), and \(\lambda_{i}^{\nu} (i=0,1, \ldots, n-1)\) +the parameters of \(n\) congruences, which define a lattice of +lines in \(R_{n}\) and an \(n\)-uplet<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a> at every point.</p> + +<p>Following Einstein's example I shall use Greek letters for co-ordinate +indices (such as \(\nu\)), and Roman letters, on the other hand, for +indices referring to the \(n\)-uplet (such as \(i\)). I shall leave out +signs of summation with respect to Greek indices (provided they occur +once above and once below), but other \(\Sigma\)'s will be retained.</p> + +<p>As usual, let the quantities \(\lambda_{i \mid \nu}\) be the elements +reciprocal to \(\lambda_{i}^{\nu}\) (normalized cofactors). For every +\(i\) they form a covariant system (moments of the \(n\)-uplet in +question). By composition with the quantities \(\lambda_{i}^{\nu}\), +\(\lambda_{i \mid \nu}\) we obtain, from every mixed tensor of rank +\(p + q\) with the components +\[ +A_{\mu_{1} \mu_{2} \ldots \mu_{p}}^{\nu_{1} \nu_{2} \ldots \nu_{q}} \quad\left(\mu_{1}, \ldots, \mu_{p}, \nu_{1}, + \ldots, \nu_{q}=0,1, \ldots, n-1\right), +\] +an "\(n\)-uplet tensor",<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a> the components of which are defined by the +formulæ +\[ +\eqalign +{A_{i_{1}} \cdots i_{p} k_{1} \cdots k_{q}=A_{\mu_{1}}^{\nu_{1}} \cdots \mu_{p}{ }_{p} + \lambda_{i_{1}}^{\mu_{1}} \ldots \lambda_{i_{p}}^{\mu_{p}} \lambda_{k_{1} \mid \nu_{1}} \ldots \lambda_{k_{q} + \mid \nu_{q}}, \qquad \text{(1)}} +\] +and conversely, since these formulæ can be solved for the co-ordinate +components in the form +\[ +A_{\mu_{1} \ldots \mu_{p}}^{\nu_{1} \ldots \nu{q}} = \sum_{i_{1}}^{n-1} \cdots_{i_{p} k_{1}} \cdots_{k_{q}} A_{i_{1}} + \cdots_{i_{p} k_{1}} \cdots_{k_{q}} \lambda_{i_{1} \mid \mu_{1}} \ldots \lambda_{i_{p} \mid \mu_{p}} + \lambda_{k_{1}}^{\nu_{1}} \ldots \lambda_{k_{q}}^{\nu_{q}}. +\]</p> + +<p>The components of the \(n\)-uplet tensor are pure invariants with +respect to transformations of co-ordinates; they essentially depend +on the \(n\)-uplet considered, but, as is easily verified, they also +behave like a tensor when the quantities \(\lambda_{i}^{\mu}\) and +\(\lambda_{k \mid \nu}\) are simultaneously subjected to orthogonal +transformations.</p> + +<p>If we put +\[ +\eqalign +{g_{\mu \nu}=\sum_{0}^{n-1} \lambda_{i \mid \mu} \lambda_{i \mid \nu}, \quad(\mu, \nu=0,1, \ldots, n-1) \qquad \text{(2)}} +\] +a definite metric +\[ +\eqalign +{d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu} \qquad \text{(3)}} +\] +(for real values of the quantities involved) is introduced into +\(R_{n}\) in such a way that our \(n\)-uplet turns out orthogonal. +Later (§3) I shall give the (unimportant) modifications required +to transfer the \(n\)-uplet theory, avoiding any appearance of +imaginaries, to an indefinite metric (with a given index of inertia).</p> + +<p>Meanwhile I suppose that the covariant derivatives of the moments +\(\lambda_{i \mid \nu}\) have been introduced, and, following Ricci, I +take the coefficients of rotation +\[ +\eqalign +{\gamma_{i k l}=\lambda_{i \mid \nu \rho} \lambda_k^{\nu} \lambda_l^{\rho} \qquad \text{(4)}} +\]</p> + +<p>In virtue of the identities +\[ +\eqalign +{\gamma_{i k l}+\gamma_{k i l}=0 \qquad \text{(5)}} +\] +(which result from the relationships between parameters and moments), +Ricci's quantities \(\gamma\) form \(n \frac{n(n-1)}{2}\) invariants +with respect to transformations of co-ordinates, which of course +essentially depend on the given \(n\)-uplet and necessarily include +all its geometrical differential properties of the first order. With +respect to orthogonal transformations <i>with constant coefficients</i> +the quantities \(\gamma\) behave like a tensor of the third rank. In +order to emphasize the limitation to transformations with constant +coefficients I shall call such systems <i>local</i> \(n\)-<i>uplet +tensors</i>. True \(n\)-uplet tensors behave as invariants with respect +to all orthogonal transformations whose coefficients can vary in any +way with the quantities \(x\).</p> + +<p>Perhaps it is not superfluous to remark that the explicit expressions +for the coefficients of rotation, \(\gamma\), can also be obtained +directly by ordinary differentiation without making use of the +covariant derivatives of the quantities \(\lambda_{i \mid \nu}\).</p> + +<p>In order to do this, we have to introduce either the Pfaffian +expressions +\[ +\psi_i=\lambda_{i \mid \nu} d x^{\nu}, +\] +or the operators +\[ +\frac{d f}{d s_i}=X_i f=\sum_0^{n-1} \lambda_i^{\nu} \frac{\partial f}{\partial x^{\nu}} +\] +(derivatives of a function \(f\left[x^{0}, \ldots, x^{n-1}\right]\) in +the direction of the lines of the congruences), and then to form the +corresponding bilinear covariants or Poisson brackets. We can, however, +attain the desired result even more rapidly by using (4) and noticing +that, according to the definition of covariant differentiation, we have +the identity +\[ +\lambda_{i \mid \nu \rho}-\lambda_{i \mid \rho \nu}=\frac{\partial \lambda_{i \mid \nu}}{\partial x^{\rho}}-\frac{\partial \lambda_{i \mid \rho}}{\partial + x^{\nu}} . +\] +We thus obtain +\[ +\gamma_{i k l}-\gamma_{i l k}=\sum_{0}^{n-1} \lambda_{k}^{\nu} \lambda_{l}^{\rho}\left\{ +\frac{\partial \lambda_{i \mid \nu}} +{\partial x^{\rho}} +-\frac{\partial \lambda_{i \mid \rho}}{\partial x^{\nu}}\right\}, +\] +and all the quantities \(\gamma\) are uniquely determined by these +equations together with (5).</p> + +<p>Equations (4) can be solved for the quantities \(\lambda_{i \mid \nu\rho}\), +giving +\[ +\eqalign +{\lambda_{i \mid v \rho}=\sum_{0}^{n-1} \gamma_{i j h} \lambda_{j \mid \nu} \lambda_{h \mid \rho} \qquad (4')} +\] +from which we obtain the conditions of integrability of \((4')\) by +repeated covariant differentiation and formation of differences. For +this we require the commutation-formula +\[ +\eqalign +{\lambda_{i \mid v \rho \sigma}-\lambda_{i \mid v \tau \rho}=R_{\mu v, \rho \sigma} \lambda_{i}^{\mu} \qquad \text{(6)}} +\] +where \(R_{\mu \nu, \rho \sigma}\) denotes the Riemannian tensor. In +this way we obtain +\[ +\eqalign +{\gamma_{i j, h k}=R_{\mu \nu, \rho \sigma} \lambda_{i}^{\mu} \lambda_{j}^{\nu} \lambda_{h}^{\rho} \lambda_{k}^{\sigma} + \qquad \text{(7)}} +\] +where for brevity we write +\[ +\eqalign +{\gamma_{i j, h k}= +\frac{d \gamma_{i j h}}{d s_{k}}-\frac{d \gamma_{i j k}}{d + s_{h}}+\Sigma_{0}^{n-1}\left[\gamma_{i j l}\left(\gamma_{l h k}-\gamma_{l k h}\right)+\gamma_{l i k} + \gamma_{l j h}-\gamma_{l i h} \gamma_{l j k}\right] . \qquad \text{(8)}} +\] +From (7) we conclude that the 4-index symbols, \(\gamma\), form a +(true) \(n\)-uplet tensor. In virtue of the well-known identities +satisfied by the Riemannian symbols the formulæ (7) lead to similar +identities for the 4-index symbols, \(\gamma\), namely +\[ +\left.\begin{array}{l} +\gamma_{i j, h k} = -\gamma_{j i, h k} = -\gamma_{i j, k h} = \gamma_{h k, i j} \\ +\gamma_{i j, h k}+\gamma_{i h, k j}+\gamma_{i k, j h} = 0 +\end{array}\right\}\qquad \text{(9)} +\]</p> + +<p>Now for the Einstein tensor +\[ +G_{\mu \sigma} = R_{\mu \nu, \rho \sigma} g^{\nu \rho} . +\]</p> + +<p>Its components \(G_{i k}\), with respect to the two members \(i\), +\(k\) of the \(n\)-uplet are expressed, by (1), by +\[ +G_{i k} = G_{\mu \sigma} \lambda_{i}^{\mu} \lambda_{k}^{\sigma} , +\] +whence, by (7), +\[ +\eqalign +{G_{i k} = \sum_{0}^{n-1}{_{h}} \gamma_{i h, h k} \qquad \text{(10)}} +\]</p> + +<p>The linear (co-ordinate and \(n\)-uplet) invariant +\[ +G = G_{\mu \sigma} g^{\mu \sigma}=\sum_{0}^{n-1}{_k} G_{k k} +\] +consequently takes the form +\[ +\eqalign +{G = \sum_{0}^{n-1}{_{k k}} \gamma_{k h, k k} \qquad \text{(11)}} +\]</p> + +<p>In conclusion, I shall emphasize one other fact, namely that +contraction of two indices in an \(n\)-uplet tensor leads to a reduced +tensor—of the \((m-2)\)th rank if the original tensor is of the +\(m\)th rank.</p> + +<p>As we have already seen, the quantities \(\gamma_{i k l}\) form a +local \(n\)-uplet tensor of the third rank, which in virtue of (5) is +skew-symmetrical with respect to the two first indices \(i\), \(k\). +The same is true for the differences \(\gamma_{l i k}-\gamma_{l k i}\), +which for \(i, k \neq l\) are called <i>anormalities</i> (i.e. +quantities which vanish when the \(l\)th congruence of the \(n\)-uplet +is normal).</p> + +<p>If we apply the differential operator \(\dfrac{d}{d s_{j}}\) to the +elements \(A_{(h)}\) (where \((h)\) stands for \(h_{1} h_{2} \ldots h_{m}\)) +of a local or true \(n\)-uplet tensor, we obtain a new local +\(n\)-uplet tensor \(\dfrac{d A_{(h)}}{d s_{j}}\), the rank of which +exceeds that of the original tensor by unity. In particular, we obtain +in this way the local \(n\)-uplet tensor of the fourth rank +\[ +\frac{d \gamma_{i k l}}{d s_{j}} +\] +which is skew-symmetrical with respect to \(i\) and \(k\). By +contraction we obtain +\[ +\eqalign +{\xi_{i k}=\sum_{0}^{n-1}{_{l}} \frac{d \gamma_{i k l}}{d s_{l}} \qquad \text{(12)}} +\] +so that we have obviously formed a skew-symmetrical local \(n\)-uplet +tensor \(\boldsymbol{\xi}\) of the second rank. Its covariant and contravariant +components are respectively +\[ +\eqalign +{\xi_{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} \lambda_{i \mid \mu} \lambda_{k \mid \nu}, + \quad \xi^{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} \lambda_{i}^{\mu} \lambda_{k}^{\nu} \qquad \text{(13)}} +\]</p> + +<p>We may mention in addition that the \(n\) quantities +\[ +\eqalign +{c_{l}=\sum_{0}^{n-1}{_{j}} \gamma_{j l j} \qquad \text{(14)}} +\] +may be interpreted as mean curvatures of the \(n-1\)-fold sections, +drawn orthogonally to the lines of the \(n\)-uplet. By what we have +said above, they are line-components of a local \(n\)-uplet vector. +From the tensor of the third rank, \(\gamma_{l i k}-\gamma_{l k i}\), +and this vector we obtain by contraction a new local \(n\)-uplet tensor +of the second rank, namely +\[ +\eqalign +{\eta_{i k}=\sum_{0}^{n-1}{_{l}} c_{l}\left(\gamma_{l i k}-\gamma_{l k i}\right) \qquad \text{(15)}} +\] +which is also skew-symmetrical.</p> + + +<p class="nindc space-above2"> +<b>2. Formation of divergences. The special case</b> \(n=4\).</p> + +<p>If \(v^{\nu}\) are the contravariant components of a vector +\(\mathbf{v}\), its divergence is defined by the invariant +\[ +\eqalign +{\operatorname{div} \mathbf{v}=v_{\mid \nu}^{\nu}=\frac{1}{\sqrt{ }|g|} \sum_{0}^{n-1} \frac{\partial\left(\sqrt{|g|} v^{\nu}\right)}{\partial x^{\nu}}, + \qquad \text{(16)}} +\] +where, as usual, \(g\) denotes the determinant \(\left\|g_{\mu\nu}\right\|\) +and \(|g|\)is written (instead of simply \(g\)) because +the formula is then valid as it stands even for an indefinite +\(ds^{2}\).</p> + +<p>For the divergence of a tensor \(\boldsymbol{\xi}\) of the second rank +with the contravariant components \(\xi^{\mu \nu}\) we obtain a vector +\(\chi\) with the contravariant components +\[ +\eqalign +{\chi^{\mu}=\xi_{\mid \nu}^{\mu \nu} \qquad \text{(17)}} +\]</p> + +<p>Following von Laue,<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a> we shall write simply +\[ +\eqalign +{\chi=\operatorname{Div} \xi \qquad (17')} +\]</p> + +<p>If we here replace the covariant derivatives \(\xi^{\mu \nu}_{\mid_{\rho}}\) +by their explicit values, we obtain +\[ +\eqalign +{\chi^{\mu}=\frac{1}{\sqrt{|g|}} \sum_{0}^{n-1}{_{\nu}} \frac{\partial}{\partial x^{\prime}}\left(\sqrt{|g|} \xi^{\mu \nu}\right) \qquad (17'')} +\] +in the case of a skew-symmetrical tensor \(\left(\xi^{\mu \nu}+\xi^{\nu \mu}=0\right)\); hence, by (16), +\[ +\eqalign +{\operatorname{div} \chi=\frac{1}{\sqrt{|g|}} \sum_{0}^{n-1}{_{\mu \nu}} \frac{\partial^{2}}{\partial x^{\mu} \partial x^{\nu}}\left(\sqrt{|g|} + \xi^{\mu \nu}\right) \qquad \text{(18)}} +\]</p> + +<p>Owing to the skew-symmetry of the quantities \(\xi^{\mu \nu}\), the +right-hand side vanishes identically.</p> + +<p>Thus if we again make use of covariant derivatives, we obtain the +identity +\[ +\chi^{\mu}{ }_{\mid \mu}=\xi^{\mu \nu}{ }_{\mid \nu \mu}=0 +\] +or finally, in tensor notation, +\[ +\eqalign +{\operatorname{div}(\operatorname{Div} \xi)=0 \qquad (18')} +\]</p> + +<p><i>That is, in an arbitrary Riemannian space the divergence of +the divergence of a skew-symmetrical tensor of the second rank is +identically zero.</i></p> + +<p>In order to express the right-hand sides of (16) and (17) in +\(n\)-uplet tensor components, it is sufficient to apply the operator +\[ +\frac{d}{d s_{l}}=\lambda_{l}^{\rho} \frac{\partial}{\partial x^{\rho}} +\] +to the formulæ of definition +\[ +\begin{aligned} +v_{k} & =v^{\nu} \lambda_{k \mid \nu}, \\ +\xi_{i k} & =\xi^{\mu \nu} \lambda_{i \mid \mu} \lambda_{k \mid \nu}. +\end{aligned} +\]</p> + +<p>By replacing ordinary differentiation by covariant differentiation +on the right-hand side (which is permissible, as we are dealing with +invariants), we obtain +\[ +\begin{align*} +\frac{d v_{k}}{d s_{l}}=v_{\mid \rho}^{\nu} \lambda_{k \mid \nu} \lambda_{l}^{\rho}+v^{\nu} \lambda_{k \mid \nu \rho} + \lambda_{l}^{\rho}, \\ +\frac{d \xi_{i k}}{d s_{l}}=\xi^{\mu \nu}{ }_{\mid \rho} \lambda_{i \mid \mu} \lambda_{k \mid \nu} \lambda_{l}^{\rho}+\xi^{\mu \nu} + \lambda_{l}^{\rho}\left(\lambda_{i \mid \mu \rho} \lambda_{k \mid \nu}+\lambda_{i \mid \mu} \lambda_{k \mid \nu \rho}\right), +\end{align*} +\] +whence, by \((4')\), (16), and (17), +\[ +\begin{align*} +\sum_{0}^{n-1}{_{k}} \frac{d v_{k}}{d s_{k}}&=\operatorname{div} \mathbf{v}+\sum_{0}^{n-1}{_{hk}} \gamma_{k h k} v_{h}, + &\qquad \text{(19)}\\ +\sum_{0}^{n-1}{_{k}} \frac{d \xi_{i k}}{d s_{k}}&=\chi_{i}+\sum_{0}^{n-1}{_{hk}}\left(\gamma_{i h k} + \xi_{h k}+\gamma_{k h k} \xi_{i h}\right), &\qquad \text{(20)} +\end{align*} +\] +which give the divergences \(\operatorname{div} \mathbf{v}\) and +\(\operatorname{Div} \boldsymbol{\xi}\) of \(n\)-uplet tensors (of the +first or second rank) directly by means of \(n\)-uplet components and +\(n\)-uplet operations.</p> + +<p>For \(n=4\) we have an elementary tensor of the fourth rank at our +disposal, namely the well-known Riccian \(\epsilon\)-system, the +covariant and contravariant components of which, \(\epsilon_{\mu \nu\rho \sigma}\), +\(\epsilon^{\mu \nu \rho \sigma}\) respectively, are +equal to zero if the four indices are not all different. The other +components have the respective values \(\pm \sqrt{|g|}\), +\(\pm d\frac{1}{\sqrt{|g|}}\), the upper or lower sign being taken +according as the permutation \((\mu \nu \rho \sigma)\) is even or odd +with respect to (0123).</p> + +<p>Let \(\boldsymbol{\xi}\) again be a skew-symmetrical tensor of the +second rank with the contravariant components \(\xi^{\nu \rho}\). If we +put +\[ +\eqalign +{p_{\mu}=\epsilon_{\mu \nu \rho \sigma} \xi^{\nu \rho \mid \sigma}, \qquad \text{(21)}} +\] +which means the same as +\[ +\begin{aligned} +p^{\mu}&=\epsilon^{\mu \nu \rho \sigma} \xi_{\nu \rho \mid \sigma}, &\qquad (21')\\ +\text{or}\quad \mathbf{p}&=\operatorname{Div}^{*} \boldsymbol{\xi} &\qquad (21'') +\end{aligned} +\] +in von Laue's notation, we are justified in calling the vector +\(\mathbf{p}\) with the above covariant and contravariant components +the Pfaffian divergence of \(\xi\), because the \(p^{\mu}\)'s vanish +identically if, and only if, the \(\xi_{\nu \rho}\)'s coincide with +the coefficients of the bilinear covariants of a Pfaffian expression +\(\phi_{\nu} d x^{\nu}\). This is most easily proved by replacing the +covariant derivatives \(\xi_{\nu \rho \mid \sigma}\) in \((21')\) by their +explicit values and noting that, owing to the skew-symmetry of the +quantities \(\xi_{\nu \rho}\), all that we have left is +\[ +\eqalign +{p^{\mu}=\sum_{0}^{3} \sum_{\nu \rho \sigma} \epsilon^{\mu \nu \rho \sigma} \frac{\partial \xi_{\nu \rho}}{\partial x^{\sigma}} + \qquad (21''')} +\] +The right-hand sides obviously vanish if the quantities +\(\dfrac{\partial \xi_{\nu \rho}}{\partial x^{\sigma}}\) are replaced +by the differences \(\dfrac{\partial^{2} \phi_{\nu}}{\partial x^{\rho} +\partial x^{\sigma}}-\dfrac{\partial^{2} \phi_{\rho}}{\partial x^{\nu} +\partial x^{\sigma}}\).</p> + +<p>By substituting the expression \((21''')\) for the \(p^{\mu}\)'s in the +second form (16) of the divergence of a vector we immediately obtain +\(\operatorname{div} \mathbf{p}=0\), which, bearing \((21'')\) in mind, may be written +\[ +\eqalign +{\operatorname{div}\left(\operatorname{Div}^{*} \xi\right)=0 ; \qquad \text{(22)}} +\] +<i>that is, the divergence of the Pfaffian divergence of a +skew-symmetrical tensor of the second rank in</i> \(\mathrm{R}_{4}\) +<i>vanishes identically</i>.</p> + +<p>Further, we shall proceed to represent the vector \(\mathbf{p}\) (the +Pfaffian divergence) directly in terms of the \(n\)-uplet components +\[ +\xi_{i k}=\xi_{\mu \nu} \lambda_{i}^{\mu} \lambda_{k}^{\nu} +\] +of the given tensor. Here it suggests itself to start from the solved +form of the equations which we have just written down, namely +\[ +\xi_{\nu \rho}=\sum_{0}^{3}{_{h k}} \xi_{h k} \lambda_{h \mid \nu} \lambda_{k \mid \rho} +\] +and to calculate the quantities \(\xi_{\nu \rho \mid \sigma}\) by +covariant differentiation of the right-hand side.</p> + +<p>From +\[ +\xi_{h k \mid \sigma}=\sum_{0}^{3}{_{l}} \frac{d \xi_{h k}}{d s_{l}} \lambda_{l \mid \sigma} +\] +and \((4')\) we obtain +\[ +\xi_{\nu \rho \mid \sigma}=\sum_{0}^{3}{_{hkl}} \frac{d \xi_{h k}}{d s_{l}} \lambda_{h \mid \nu} \lambda_{k \mid \rho} + \lambda_{l \mid \sigma}+\sum_{0}^{3}{_{hkjl}} \xi_{h k} \lambda_{l \mid \sigma}\left\{\gamma_{h j l} \lambda_{j \mid \nu} + \lambda_{k \mid \rho}+\gamma_{k j l} \lambda_{j \mid \rho} \lambda_{h \mid \nu}\right\}; +\] +hence, by \((21')\), +\[ +\begin{aligned} +p_{i}=p^{\mu} \lambda_{i \mid \mu} & =\epsilon^{\mu \nu \rho \sigma} \xi_{\nu \rho \mid \sigma} \lambda_{i \mid \mu} \\ +& =\sum_{0}^{3}{_{hkl}} \epsilon_{i h k l} \frac{d \xi_{h k}}{d s_{l}}+\sum_{0}^{3}{_{jhkl}} + \xi_{h k}\left(\epsilon_{i j k l} \gamma_{h j l}+\epsilon_{i h j l} \gamma_{k j l}\right) \\ +& =\sum_{0}^{3}{_{hkl}} \epsilon_{i h k l}\left\{\frac{d \xi_{h k}}{d s_{l}}+\sum_{0}^{3}{_{j}}\left(\gamma_{j h l} + \xi_{j k}+\gamma_{j k l} \xi_{h j}\right)\right\} +\end{aligned} +\] +where for brevity we have put +\[ +\begin{align*} +\epsilon_{i k k l}=\epsilon^{\mu \nu \rho \sigma} \lambda_{i \mid \mu} \lambda_{k \mid \nu} \lambda_{k \mid \rho} \lambda_{l \mid \sigma}\\ +& =\frac{1}{\sqrt{|g|}}\left|\begin{array}{llll} +\lambda_{0 \mid 0} & \lambda_{0 \mid 1} & \lambda_{0 \mid 2} & \lambda_{0 \mid 3} \\ +\lambda_{1 \mid 0} & \lambda_{1 \mid 1} & \lambda_{1 \mid 2} & \lambda_{1 \mid 3} \\ +\lambda_{2 \mid 0} & \lambda_{2 \mid 1} & \lambda_{2 \mid 2} & \lambda_{2 \mid 3} \\ +\lambda_{3 \mid 0} & \lambda_{3 \mid 1} & \lambda_{3 \mid 2} & \lambda_{3 \mid 3} +\end{array}\right|.\qquad \text{(23)} +\end{align*} +\] +Thus these quantities \(\epsilon_{i k k l}\) are equal to zero if +two of the four indices are equal. If, on the other hand, \(ihkl\) +is a permutation of the numbers 0123, \(\epsilon_{\text {ihkl }}\) +has the value \(\pm 1\), according as the class of the substitution +\(\binom{i h k l}{0123}\) is even or odd. We accordingly see that in +the expression which we have just obtained for the \(p_{i}\)'s the two +last terms are equal to each other, so that we finally obtain +\[ +p_{i}=\sum_{0}^{3}{_{h k l}} \epsilon_{i h k l}\left\{ +\frac{d \xi_{h k}}{d s_{l}}+2\sum_{0}^{3}{_{j}} \gamma_{j h l} \xi_{j k}\right\} \qquad \text{(24)} +\]</p> + + +<p class="nindc space-above2"> +<b>3. Transformations for an indefinite metric.</b></p> + +<p>According to Eisenhart<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a> all the formulæ of the \(n\)-uplet theory can +be transferred in a readily intelligible way to indefinite metrics, +without leaving the real region even temporarily.</p> + +<p>If we consider an indefinite +\[ +ds^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu} +\] +we (as is well known) call a (real) direction \(d x^{\nu}\) +<i>time-like</i> or <i>space-like</i>, according as the corresponding +\(ds^{2}\) turns out greater or less than zero; null directions are +those directions, \(\infty^{n-2}\) in number, for which \(ds^{2}=0\).</p> + +<p>In any case we call the ratios +\[ +\lambda^{\nu}=\frac{d x^{\nu}}{|ds|} \quad(\nu=0,1, \ldots, n-1) +\] +<i>parameters</i> of a proper (i.e. non-null) direction.</p> + +<p>Hence we have +\[ +\eqalign +{g_{\mu \nu} \lambda^{\mu} \lambda^{\nu}=\frac{d s^{2}}{\left|ds^{2}\right|}= \pm 1=e, \qquad \text{(25)}} +\] +if we henceforth denote positive or negative unity by \(e\).</p> + +<p>As in the definite case we introduce as moments of a given direction +the covariant quantities +\[ +\eqalign +{\lambda_{\nu}=g_{\mu \nu} \lambda^{\mu}, \qquad \text{(26)}} +\] +so that the quadratic identity (25) takes the form +\[ +\eqalign +{\lambda_{v} \lambda^{\nu}=e. \qquad \text{(27)}} +\]</p> + +<p>If the quantities \(\lambda_{i}^{\nu}(i=0,1, \ldots, n-1)\) are the +parameters of an orthogonal \(n\)-uplet consisting of proper directions +only, we have +\[ +\lambda_{i \mid \nu} \lambda_{k}^{\nu}=0 \quad(i \neq k) +\] +on account of the orthogonality of the \(n\)-uplet, and also +\[ +\lambda_{i \mid \nu} \lambda_{i}^{\nu}= \pm 1=e_{i}, +\] +by (27).</p> + +<p>The total number of negative (and consequently also of the remaining +positive) quantities \(e_{i}\) for a given \(ds^{2}\) is always equal +to its index of inertia, and hence is always the same no matter what +(proper) \(n\)-uplet is considered.</p> + +<p>The two groups of relationships between parameters and moments of an +\(n\)-uplet which we have just written down may be summarized in the +single formula +\[ +\eqalign +{\lambda_{i}^{\nu} \lambda_{k \mid \nu}=e_{k} \delta_{i k}=e_{i} \delta_{i k}, \qquad \text{(28)}} +\] +where the symbols \(\delta_{i k}\) have their usual meaning; or, since +\(e_{k}^{2}=1\), +\[ +\lambda_{i}^{\nu} e_{k} \lambda_{k \mid \nu}=\delta_{i k} . +\]</p> + +<p>From this we conclude that the elements reciprocal to the parameters +\(\lambda_{i}^{\nu}\) are not exactly equal to the moments \(\lambda_{i\mid \nu}\), +but to \(e_{i} \lambda_{i \mid \nu}\). Thus the quantities +\(e_{i} \lambda_{i}^{\nu}\) are the elements reciprocal to the moments +\(\lambda_{i \mid \nu}\). If we imagine the equations (26) written down +for every \(n\)-uplet, we have +\[ +\lambda_{i \mid \nu}=g_{\rho \nu} \lambda_{i}^{\rho} +\] +(denoting the index of summation by \(\rho\)). By multiplying by +\(e_{i} \lambda_{i \mid \mu}\) and summing with respect to \(i\) we +obtain +\[ +g_{\mu \nu}=\sum_{0}^{n-1}{_{i}} e_{i} \lambda_{i \mid \mu} \lambda_{i \mid \nu}, +\] +which replaces formula (2) for the definite case, and so on.</p> + +<p>From this point it will suffice if I confine myself to quite brief +hints, and I shall of course write down only those formulæ which do +not remain unaltered throughout. These will be marked with an asterisk +and given the same number as the corresponding formula referring to a +definite metric.</p> + +<p>In the first place, \(n\)-uplet components of any given tensor +and coefficients of rotation \(\gamma_{i k l}\) must in any case be +introduced by the equations of definition (1) and (4); the solved +expressions for the quantities \(\lambda_{i \mid \nu \rho}\), on the +other hand, are in general +\[ +\eqalign +{\lambda_{i \mid \nu \rho}=\sum_{0}^{n-1} {_{j h}} \gamma_{i j h} e_{j} e_{h} \lambda_{j \mid \nu} \lambda_{h \mid \rho} + \qquad (4'*)} +\]</p> + +<p>The covariant equations (6), and also the equations of definition of +the 4-index symbols \(\gamma\) (7) are true without restriction; but +the \(n\)-uplet tensor expressions for the quantities \(\gamma_{i j, h k}\) +suffer a small modification. In fact we must in general put +\[ +\begin{align*} +\gamma_{i j, h k}= & \frac{d \gamma_{i j h}}{d s_{k}}-\frac{d \gamma_{i j k}}{d s_{h}} \\ +& +\sum_{0}^{n-1}{_{l}} e_{l}\left[\gamma_{i j l}\left(\gamma_{l h k}-\gamma_{l k h}\right)+\gamma_{l i k} + \gamma_{l j h}-\gamma_{l i h} \gamma_{l j k}\right] \qquad (8*) +\end{align*} +\]</p> + +<p>Of course these quantities are still connected by the relationships +(9), in virtue of equations (7).</p> + +<p>It is essential to note, however, that the local transference from +one \(n\)-uplet to another does not correspond to any orthogonal +transformation, but to a pseudo-orthogonal transformation, i.e. to a +transformation which leaves the quadratic form +\[ +Q(z)=\sum_{0}^{n-1}{_{i}} e_{i} z_{i}^{2} +\] +invariant. Thus the coefficients \(\alpha_{i k}\) of a +pseudo-orthogonal transformation of this kind must satisfy the +conditions +\[ +\sum_{0}^{n-1}{_{l}} e_{l} \alpha_{i l} \alpha_{k l}=\sum_{0}^{n-1}{_{l}} e_{l} \alpha_{l i} + \alpha_{l k}=\delta_{i k}. +\]</p> + +<p>The most general expression which can be attributed to the coefficients +\(\alpha_{i k}\) in the case of infinitesimal pseudo-orthogonal +transformations follows immediately from the condition that the form +\(Q(z)\) is to be invariant. We have merely to put +\[ +\alpha_{i k}=\delta_{i k}+e_{i} \beta_{i k} +\] +and to regard the quantities \(\beta_{i k}\) as indefinitely small. If +in \(Q\) we carry out the substitution +\[ +\eqalign +{z_{i}=\sum_{0}^{n-1}{_{k}} \alpha_{i k} z_{k}^{\prime}=z_{i}^{\prime}+e_{i} \sum_{0}^{n-1}{_{k}} + \beta_{i k} z_{k}^{\prime} \qquad \text{(29)}} +\] +and require that \(Q\left(z^{\prime}\right)\) should retain the form +\[ +\sum_{0}^{n-1}{_{i}} e_{i} z_{i}^{2}, +\] +what we obtain (as in the case of pure orthogonal substitutions) is the +condition of skew-symmetry, namely +\[ +\eqalign +{\beta_{i k}+\beta_{k i}=0 \qquad \text{(30)}} +\]</p> + +<p>The components of an \(n\)-uplet tensor are systems of numbers which +behave like tensors with respect to pseudo-orthogonal transformations; +for <i>local</i> \(n\)-uplet tensors this behaviour is maintained only +with respect to pseudo-orthogonal transformations with <i>constant</i> +coefficients. The operators +\[ +\frac{d f}{d s_{i}}=X_{i} f=\sum_{0}^{n-1}{_{\nu}} \lambda_{i}^{\nu} \frac{\partial f}{\partial x^{\nu}} +\] +behave like \(n\)-uplet vectors.</p> + +<p>If (\(i\)) and (\(k\)) denote any group of \(n\)-uplet indices and +\(A_{(i) j}\), \(B_{(k) l}\) two local \(n\)-uplet tensors, then +contraction with respect to \(j\), \(l\) is defined by the formula +\[ +\sum_{0}^{n-1}{_{l}} e_{l} A_{(i) l} B_{(k) l}. +\]</p> + +<p>We accordingly obtain +\[ +\begin{align*} +G_{i k} & =\sum_{0}^{n-1}{_{h}} e_{h} \gamma_{i h, h k}, &\qquad \text{(10*)}\\ +G & =\sum_{0}^{n-1}{_{hk}} e_{h} e_{k} \gamma_{k h, h k} &\qquad \text{(11*)} +\end{align*} +\] +instead of (10) and (11).</p> + +<p>Further, the formulæ (12), (14), and (15) must be replaced by and +\[ +\begin{align*} +\xi_{i k} & =\sum_{0}^{n-1}{_{l}} e_{l} \frac{d \gamma_{i k l}}{d s_{l}} &\qquad \text{(12*)}\\ +c_{l} & =\sum_{0}^{n-1}{_{j}} e_{j} \gamma_{j l j} &\qquad \text{(14*)}\\ +\text{and}\quad \eta_{i k} & =\sum_{0}^{n-1}{_{l}} e_{l} c_{l}\left(\gamma_{l i k}-\gamma_{l k i}\right) &\qquad \text{(15*)} +\end{align*} +\] +while the expressions (13) for covariant and contravariant components +in terms of the \(n\)-uplet components \(\xi_{i k}\) are to be +deduced from (1), the universally valid definition of the \(n\)-uplet +components of a tensor. Hence they become +\[ +\left.\begin{array}{l} +\xi_{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} e_{i} e_{k} \lambda_{i \mid \mu} \lambda_{k \mid \nu}, \\ +\xi^{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} e_{i} e_{k} \lambda_{i}^{\mu} \lambda_{k}^{\nu} +\end{array}\right\}\qquad \text{(13*)} +\]</p> + +<p>As contraction of pseudo-orthogonal \(n\)-uplet tensors is brought +about by inserting the factor \(e\) with the appropriate index, it is +at once clear that (19), (20), and (24) take the forms +\[ +\begin{align*} +\operatorname{div} \mathbf{v} & =\sum_{0}^{n-1}{_{k}} e_{k} \frac{d v_{k}}{d s_{k}}-\sum_{0}^{n-1}{_{hk}} e_{h k} + e_{k} \gamma_{k h k} v_{h}, &\qquad \text{(19*)}\\ +\chi_{i} & =\sum_{0}^{n-1}{_{h}} e_{k} \frac{d \xi_{i k}}{d s_{k}}-\sum_{0}^{n-1}{_{hk}} e_{h k} e_{h} + e_{l}\left(\gamma_{i h k} \xi_{h k}+\gamma_{k h k} \xi_{i h}\right) &\qquad \text{(20*)}\\ +p_{i} & =\sum_{0}^{3} \sum_{h k l}{_{h k l}} e_{h} e_{k} e_{l} \epsilon_{i h k l}\left\{\frac{d \xi_{h k}}{d + s_{l}}+2 \sum_{0}^{3}{_{j}} e_{j} \xi_{j k}\right\} &\qquad \text{(24*)} +\end{align*} +\]</p> + +<p>Of course the equations \((18')\) and (22), i.e. +\[ +\eqalign +{\operatorname{div}(\operatorname{Div} \xi)=0, \quad \operatorname{div}\left(\operatorname{Div}^{*} \xi\right)=0, \qquad \text{(31)}} +\] +which express invariant properties, always remain valid.</p> + + +<p class="nindc space-above2"> +<b>4. Gravitational equations.</b></p> + +<p>As usual, let the covariant components of the energy tensor be denoted +by \(T_{\mu \nu}\). If influences of any origin are admitted, these +quantities \(T_{\mu \nu}\) are to be imagined broken up into two parts, +one of which, \(\tau_{\mu \nu}\), is purely electromagnetic, and the +other, \(\mathbf{T}_{\mu \nu}\), represents the remainder, if any. We +therefore put +\[ +\eqalign +{T_{\mu \nu}=\tau_{\mu \nu}+\mathbf{T}_{\mu \nu}, \qquad \text{(32)}} +\] +where \(\boldsymbol{\tau}\) is the well-known Maxwell tensor; further, +for empty space \(\mathbf{T}_{\mu \nu}\) is of course equal to zero.</p> + +<p>As is well known, the Einstein equations (without the cosmological +term) are +\[ +G_{\mu \nu}-\frac{1}{2} G g_{\mu \nu}=-\kappa T_{\mu \nu} +\] +where the constant of proportionality \(\kappa\) may be expressed in +terms of \(f\), the gravitational constant, and \(c\), the velocity of +light \(\left(\kappa=\frac{8 \pi f}{c^{4}}\right)\).</p> + +<p>If we introduce the corresponding \(n\)-uplet tensors in accordance +with the formulæ +\[ +\begin{aligned} +G_{i k} & =G_{\mu \nu} \lambda_{i}^{\mu} \lambda_{k}^{i} \\ +T_{i k} & =T_{\mu \nu} \lambda_{i}^{\mu} \lambda_{k}^{\nu}, \quad\& \mathrm{c}, +\end{aligned} +\] +we have, on the one hand, +\[ +\eqalign +{T_{i k}=\tau_{i k}+\mathbf{T}_{i k}, \qquad (32')} +\] +from (32), and (what is most important) the gravitational equations in +the \(n\)-uplet tensor form <a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a> +\[ +\eqalign +{G_{i k}-\frac{1}{2} \delta_{i k} G=-\kappa T_{i k}, \quad(i, k=0,1,2,3) \qquad \text{(I)}} +\] +where, in accordance with (10*) and (11*), +\[ +G_{i k}=\sum_{0}^{n-1}{_{h}} e_{h} \gamma_{i h, h k}, \quad G=\sum_{0}^{n}{_{k}} e_{k} + G_{k k}=\sum_{0}^{3}{_{hk}} e_{h} e_{k} \gamma_{k h, h k} . +\]</p> + +<p>As the space-time manifold on which the general theory of relativity is +to be based possesses an indefinite metric with an index of inertia 3, +we have to put +\[ +\eqalign +{e_{0}=1,\quad e_{1}=e_{2}=e_{3}=-1. \qquad \text{(33)}} +\]</p> + +<p>The quantities \(\gamma_{i j, h k}\) are introduced by the equations +(8*) as lattice differential elements of the second order. Their +combinations \(G_{i k}\) behave like tensors with respect to all +pseudo-orthogonal (i.e. in the present case Lorentz) transformations +(even if the coefficients are permitted to vary in any way with +position).</p> + +<p>Accordingly, as indeed is clear from the outset, the ten equations (I) +do not, as far as their original form is concerned, favour any special +quadruplet. They are valid in one and the same form for all orthogonal +quadruplets of the relativistic \(R_{4}\), and, as is well-known, serve +to define their metric.</p> + +<p>As in every case they give ten relationships between the sixteen +parameters \(\lambda_{i}^{\nu}\), we need only find six other +apparently reasonable conditions connecting the latter, in order +to mark out a special lattice (the world lattice) from among +all the possible quadruplets and lattices corresponding to the +space-time-manifold \(R_{4}\).</p> + +<p>We shall shortly (§6) carry out this final step, which is in fact the +only essential one. Meanwhile we may appropriately lead up to it by +putting Maxwell's equations into a suitable form.</p> + + +<p class="nindc space-above2"> +<b>5. Electromagnetic equations.</b></p> + +<p>Let \(F_{\mu \nu}, F^{\mu \nu}, F_{i k}\) be the (covariant, +contravariant, and \(n\)-uplet) components of the skew-symmetrical +tensor \(\mathbf{F}\) which defines the electromagnetic field +in the space-time world; let \(\mathbf{S}\) (a vector) be the +current-vector<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a> and \(S_{\mu}\), \(\&c.\), its four components, where +all the quantities are understood to be measured in so-called rational +units.</p> + +<p>Maxwell's equations (as adopted in the general theory of relativity +after Einstein) then take the forms +\[ +\eqalign +{\operatorname{Div} \mathbf{F}=\mathbf{S}, \quad \operatorname{Div}^{*} \mathbf{F}=0 . \qquad \text{(34)}} +\]</p> + +<p>Each group contains four equations, so that at first glance one would +take the total number of equations to be eight. But we necessarily +have \(\operatorname{div} \mathbf{S}=0\), so that by (31) there must +exist two identical relationships, namely those which express the +fact that the divergences in question vanish. Thus two equations +of the system (34) may (with appropriate subsidiary conditions) be +regarded as resulting from the other six; and in fact we know that if +\(\mathbf{S}\) is regarded as given or as associated in some other way +with the tensor \(\mathbf{F}\), then the equations (34) merely serve to +determine the six components of \(\mathbf{F}\) for \(x^{0}+d x^{0}\) +uniquely from their values for a given \(x^{0}\) (and any \(x^{1}, x^{2}, x^{3}\) ).</p> + +<p>We have still to write down the symmetrical stress-energy tensor +explicitly. As is well known, its covariant components are defined as +follows: +\[ +\tau_{\mu \nu}=-g^{\rho \sigma} F_{\mu \rho} F_{\nu \sigma}+\frac{1}{4} g_{\mu \nu} F_{\rho \sigma} F^{\rho \sigma} . +\]</p> + +<p>By composition with \(\lambda_{i}^{\mu} \lambda_{l c}^{\nu}\) +(by replacing \(g^{\rho \sigma}\) on the right-hand side by +\(\sum_{0}^{3}{_{l}} e_{l} \lambda_{l}^{\rho} \lambda_{l}^{\sigma}\) +and \(F^{\rho \sigma}\) by \(\sum_{0}^{3}{_{jh}} e_{j} e_{h} F_{j h}\lambda_{j}^{\rho} \lambda_{h}^{\sigma}\)) +we obtain the required \(n\)-uplet tensor formula: +\[ +\eqalign +{\tau_{i k}=-\sum_{0}^{3}{_{l}} e_{l} F_{i l} F_{k l}+\frac{1}{4} \delta_{i k} \sum_{0}^{3}{_{jh}} e_{j} + e_{h} F_{j h}^{2} . \qquad \text{(35)}} +\]</p> + + +<p class="nindc space-above2"> +<b>6. Interpretation of the electromagnetic tensor in the world +lattice. Purely geometrical formulation of the field equations.</b></p> + +<p>A priori we may quite arbitrarily connect the six \(n\)-uplet +components \(\boldsymbol{F}_{i k}\) of the electromagnetic field +with any geometrical properties of a quadruplet (thereby defined) +of the \(R_{4}\). A very simple way of doing this is to make the +quantities \(F_{i k}\) proportional to the corresponding elements of a +(differential) skew-symmetrical local \(n\)-uplet tensor, e.g. to the +differential expressions, of the second or first order respectively, +which are defined by the equations +\[ +\begin{aligned} +\xi_{i k}&=\sum_{0}^{3}{_{l}} e_{l} \frac{d \gamma_{i k l}}{d s_{l}} &\qquad \text{(12*)}\\ +\text{or}\quad \eta_{i k}&=\sum_{0}^{3}{_{l}} e_{l} c_{l}\left(\gamma_{l i k}-\gamma_{l k i}\right) . &\qquad \text{(15*)} +\end{aligned} +\] +of §3.</p> + +<p>As we shall see, the best way is to select the first expression, and we +accordingly put +\[ +\eqalign +{F_{i k}=v \xi_{i k} \qquad \text{(P)}} +\] +where \(v\) denotes a constant.</p> + +<p>As the Ricci coefficients of rotation \(\gamma_{i k l}\) are merely +ratios of an angle and a length, the quantities \(\xi_{i k}\) are of +dimensions \(l^{-2}\). The quantities \(\boldsymbol{F}_{i k}\), on +the other hand, behave like the square root of an energy-density. +Consequently we have +\[ +\left[F_{i k}\right]=l^{-\tfrac{1}{2}} t^{-1} m^{\tfrac{1}{2}} . +\]</p> + +<p>Hence the factor of proportionality \(v\) has dimensions +\[ +l^{\tfrac{2}{2}} t^{-1} m^{\tfrac{1}{2}}, +\] +which are those of an electric charge \(e\), e.g. the electronic +charge, so that we may write +\[ +\eqalign +{v=z e, \qquad \text{(36)}} +\] +where the factor of proportionality \(z\) is now a pure number. +Moreover, we may also replace \(e\) in (36) by any other quantity of +the same dimensions; e.g. we may put +\[ +\eqalign +{v=i_{1} \sqrt{hc} \qquad (36')} +\] +where \(h\) is Planck's constant, \(c\) the velocity of light in empty +space, and \(i_{1}\) a pure number.</p> + +<p>Hence the final forms of the geometrical equations which arise from the +Maxwellian system (34) and our proposed addition (P), are +\[ +\eqalign +{\operatorname{Div} \xi=\frac{1}{v} \mathbf{S}, \quad \operatorname{Div}^{*} \boldsymbol{\xi}=0, \qquad \text{(II)}} +\] +where \(\boldsymbol{\xi}\) means the local \(n\)-uplet tensor (12*). +<i>In conclusion, then, the geometrical definition of the quadruplet +(world lattice) associated with the field is to be taken from the +two systems</i> (I) <i>and</i> (II), <i>which together give sixteen +(apparently eighteen, but in reality only sixteen) differential +equations (of the second and third order respectively) involving the +sixteen</i> \(n\)-<i>uplet parameters</i> \(\lambda_{i}^{\nu}\).</p> + + +<p class="nindc space-above2"> +<b>7. The case of empty space: absence of an electromagnetic field.</b></p> + +<p>In empty space \((T_{i k}=0, \mathbf{S}=0)\), (I) reduces in virtue of +(32) to the form +\[ +\eqalign +{G_{i k}-\frac{1}{2} \delta_{i k} G=-\kappa \tau_{i k}, \qquad \text{(I)}'} +\] +where the term \(\tau_{i k}\) on the right-hand side is given by +\[ +\eqalign +{\tau_{i k}=-v^{2} \sum_{0}^{3}{_{l}} e_{l} \xi_{i l} \xi_{k l}+\frac{1}{4} v^{2} \delta_{i k} + \sum_{0}^{3}{_{jh}} e_{j} e_{l h} \xi_{j h}{ }^{2} \qquad (35')} +\] +by (35) and (P); while the system (II) becomes +\[ +\eqalign +{\operatorname{Div} \boldsymbol{\xi}=0, \quad \operatorname{Div}^{*} \boldsymbol{\xi}=0 . \qquad \text{(II)}'} +\]</p> + +<p>If the electromagnetic field vanishes in addition to the external +energy tensor \(\mathbf{T}_{i k}\), the quantities \(\xi_{i k}\), and +hence, by \((35')\), the quantities \(\boldsymbol{\tau}_{i k}\) also, are +equal to zero. If this happens everywhere in the space-time world, we +know <a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a> that the equations \((I')\), which simply become \(G_{i k}=0\), +necessarily imply that the metric of the space is Euclidean or, more +correctly, pseudo-Euclidean.</p> + +<p>What, then, is the geometrical meaning of the absence of +electromagnetic phenomena in this limiting case, i.e. what is the +geometrical meaning of the equations +\[ +\eqalign +{\boldsymbol{\xi}_{i k}=0 . \qquad \text{(37)}} +\]</p> + +<p><i>They simply state the fact that the world lattice is Cartesian or, +more correctly, pseudo-Cartesian.</i></p> + +<p>In order to give as concise a proof of this as possible, I shall only +consider quadruplets in which the deviations from a pseudo-Cartesian +lattice are infinitely small.</p> + +<p>If, in particular, we take the co-ordinates \(x^{\nu}\) to be Cartesian +co-ordinates with respect to that lattice, we have +\[ +\lambda_{i}^{\prime \nu}=\delta_{i \nu} +\] +for the parameters of the corresponding quadruplet.</p> + +<p>Let \(\lambda_{i}^{\nu}\) be the parameters of any +neighbouring quadruplet. Since the passage from the quantities +\(\lambda_{i}^{\gamma^{{}\nu}}\) to the quantities \(\lambda_{i}^{\nu}\) +corresponds to an infinitesimal pseudo-orthogonal transformation, the +quantities \(\lambda_{i}^{\nu}\) must, by (29), be expressible as +follows: +\[ +\eqalign +{\lambda_{i}^{\nu}=\delta_{i \nu}+e_{i} \sum_{0}^{3}{_{k}} \beta_{i k} \delta_{k \nu}=\delta_{i \nu}+e_{i} + \beta_{i \nu}, \qquad \text{(38)}} +\] +where the quantities \(\beta_{i k}\) <i>form a skew-symmetrical</i> +\(n\)-<i>uplet tensor</i>. From this we can immediately calculate the +reciprocal elements. To a first approximation we obtain +\[ +e_{i} \lambda_{i \mid \nu}=\delta_{i \nu}+e_{\nu} \beta_{i \nu}, +\] +whence, multiplying by \(e_{i}\), +\[ +\eqalign +{\lambda_{i \mid \nu}=\delta_{i \nu} e_{i}+e_{i} e_{\nu} \beta_{i \nu} . \qquad (38')} +\]</p> + +<p>On the other hand, if we altogether neglect infinitely small +quantities, the operators +\[ +\frac{d}{d s_{l}}=\sum_{0}^{3}{_{\nu}} \lambda_{l}^{\nu} \frac{\partial}{\partial x^{\nu}} +\] +reduce to the simple form +\[ +\frac{\partial}{\partial x^{l}}, +\] +and the covariant derivatives reduce to their usual forms.</p> + +<p>Thus (4), the definition of the rotational invariants \(\gamma\), gives +(except for infinitely small quantities of the second order) +\[ +\gamma_{i k l}=e_{i} e_{k} \frac{\partial \beta_{i k}}{\partial x^{l}}, +\] +and from (12*) we further obtain +\[ +\xi_{i k}=e_{i} e_{k} \sum_{0}^{3}{_{l}} e_{l} \frac{\partial^{2} \beta_{i k}}{\left(\partial x^{l}\right)^{2}}. +\]</p> + +<p>The differential operator \(\sum_{0}^{3}{_{l}} e_{l} \frac{\partial^{2}\beta_{i k}}{\left(\partial x^{l}\right)^{2}}\) +is none other than the Dalembertian or Lorentz operator \(\square\). +Thus the equations (37) take the form +\[ +\eqalign +{\square \beta_{i k}=0, \qquad (37')} +\] +and together with suitable initial and boundary conditions they give +\[ +\beta_{i k}=0, +\] +i.e. <i>the Cartesian</i> (or, more correctly, pseudo-Cartesian) +<i>character of the world lattice</i>. I think that this conclusion +justifies our assumption (P). If we had put, say, +\[ +F_{i k}=v^{\prime} \eta_{i k}, \quad\left(v^{\prime}=\text { constant }\right) +\] +where the quantities \(\eta_{i k}\) are given by the expressions (15*), +we should not have obtained any satisfactory result.</p> + +<p>A more general assumption, such as +\[ +F_{i k}=v \xi_{i k}+v^{\prime} \eta_{i k}, +\] +would, on the other hand, be more complicated, though just as +admissible as (A) from the logical point of view. To a first +approximation, in fact, we should obtain the same result, as the +\(\eta\)'s are of higher order than the \(\xi\)'s.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> +<i>Berliner Berichte</i>, I, 1929, pp. 1-8.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> +Ger. <i>Vierbein.</i></p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> +See in particular my <i>Absolute Differential Calculus</i> +(English translation by Miss Long), Chap. III. Blackie & Son, Ltd., +1927.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> +Ger. <i>n-Bein</i>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> +Ger. <i>Beintensor</i>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> +<i>Die Relativitätstheorie</i>, Bd. II (2nd edition, +Vieweg. Brunswick, 1923), § 14.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> +<i>Riemannian Geometry</i>, Princeton University Press, +1926, Chap. III.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> +Given in 1918 by Cisotti (<i>Rend. Acc. Lincei</i>, Vol. +XXVII, pp. 366-371), but confined to the (imaginary) notation of (8), +(10), (11).</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> +Ger. <i>Viererstrom</i>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> +Cf. Serini, <i>Rend. Acc. Lincei</i>, Vol. XX VII, 1918, +pp. 235-238.</p> + +</div> +</div> + +<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 78343 ***</div> +</body> +</html> diff --git a/78343-src/README-math.txt b/78343-src/README-math.txt new file mode 100644 index 0000000..adabf5a --- /dev/null +++ b/78343-src/README-math.txt @@ -0,0 +1,96 @@ +MathJax HTML source file instructions +================================== +This project is a math heavy eBook. The source is a preliminary HTML file that +uses MathJax to define mathematical expressions, which is processed to generate +a final HTML file with SVG images. + +This source file is kept for the purpose of applying errata fixes. Although the +MathJax takes some learning, it is clearer than the generated final. This also +allows the SVG images to be regenerated with changes. + + +Tools +===== +See the ppmath GitHub repository: + https://github.com/DistributedProofreaders/ppmath. +Follow the instructions to install m2svg. + +Command line: + m2svg -i input.htm -o output.htm + +- The SVG files will be placed in a subdirectory of the working directory + called "images". + +- In the converted file, the maths expressions, delimited by the tags `\[` + and `\]` for *display* expressions or `\(` and `\)` for *inline* + expressions, are replaced by `<img>` links. + +- The "data-tex" attribute will contain the original maths expression. + + +Inline code example +=================== +For the expression \(\mathrm{AB}^{2} = \mathrm{AG} \times \mathrm{BD}\), the +input `\(\mathrm{AB}^{2} = \mathrm{AG} \times \mathrm{BD}\)` + +becomes + `<span class="nowrap"><img style="vertical-align: -0.186ex; width: 16.872ex; + height: 2.253ex;" src="images/4.svg" alt="" data-tex="\mathrm{AB}^{2} + = \mathrm{AG} \times \mathrm{BD}">,</span>` + +The file images/4.svg displays the desired expression. + + +Source files structure +====================== +(eBook 75107 is used as an example) + +- 75107/ + - README-math.txt (this file) + - 75107-h/ + - 75107-h.htm (final HTML file) + - images/ + - 75107-src/ + - 75107-src.htm (source HTML file with MathJax) + + +SVG fixup for ebookmaker +======================== +Now, the SVG files contain a "data-variant" attribute that causes errors. +It needs to be removed by downloading and running this utility: +https://github.com/user-attachments/files/25548572/remove_data_variant_attribute.py + +Command line: + python remove_data_variant_attribute.py images + +Hopefully, this step will be removed in the future. + + +Submission process +================== +- Generated final HTML and images should be submitted as normal. +- In addition, the source HTML will be included, and needs to be renamed + to #####-src.htm by the whitewasher or the Workflow app. +- This readme will need to be added by the whitewasher or the Workflow app. + - Having it with the eBook makes it obvious, and avoids issues with + procedures changing in the future. + + +Errata process +============== +(eBook 75107 is used as an example) + +1. Download the project files using Errata Workbench, and unzip. +2. Install m2svg if not already done. +3. Make desired changes to 75107-src.htm. +4. Execute command line `m2svg -i 75107-src.htm -o 75107-h.htm` + - The image files will be placed in a subdirectory of the working directory + called images. +5. Move 75107-h.htm to the 75107-h directory. +6. Move the contents of the images directory to the 75107-h/images directory. + - Rename the existing 75107-h/images directory to images-old. + - Move the new images directory to 75107-h. + - Check images-old, move any non-generated images (JPG, PNG, etc.). to + images. + - Remove images-old and any other temporary files. +7. Zip the project directory and upload to Errata Workbench. |