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Eddington % +% % +% This eBook is for the use of anyone anywhere 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 www.gutenberg.org % +% % +% % +% Title: Space, Time and Gravitation % +% An Outline of the General Relativity Theory % +% % +% Author: A. S. Eddington % +% % +% Release Date: August 24, 2009 [EBook #29782] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK SPACE, TIME AND GRAVITATION *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{29782} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% textcomp: Better ditto marks. Optional. %% +%% fix-cm: For larger title page fonts. Optional. %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% array: Enhanced tabular features. Required. %% +%% %% +%% footmisc: Extended footnote capabilities. Required. %% +%% multicol: Multi-column environment for index. Required. %% +%% makeidx: Indexing capabilities. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% wrapfig: Illustrations surrounded by text. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% Minor spelling/punctuation changes, etc. are [** PP: noted] %% +%% in this file. On occasions where a word is hyphenated across a %% +%% line in the original and the word occurs only once, comparison %% +%% with similar words was made to decide whether to hyphenate. %% +%% Such instances are noted. %% +%% %% +%% The following spellings are retained: %% +%% debateable, idiosyncracies, and unbiassed. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behaviors may be controlled by boolean flags. %% +%% %% +%% ForPrinting (true by default): %% +%% Compile a print-optimized PDF file. Set to false for screen- %% +%% optimized file (pages cropped, one-sided, blue hyperlinks). %% +%% %% +%% IndexExtras (false by default): %% +%% Add a few entries to the index, e.g. Geometry, Euclidean. %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% Aviator's time table (PDF p. 22) is not separated from the %% +%% preceding sentence or broken across pages %% +%% %% +%% Wrapped images do not fall near a page bottom, lest indented %% +%% lines continue conspicuously past the break %% +%% Fig. 1, p. 15; Fig. 2, p. 42; Fig. 8, p. 50 %% +%% %% +%% %% +%% Spellcheck: .................................. OK %% +%% Smoothreading pool: ......................... yes %% +%% %% +%% lacheck: ..................................... OK %% +%% Numerous false positives from commented code %% +%% %% +%% PDF pages: 219 (if ForPrinting set to true) %% +%% PDF page size: US Letter %% +%% PDF bookmarks: created, point to ToC entries %% +%% PDF document info: filled in %% +%% Images: 18 pdf diagrams, 1 jpg (frontispiece) %% +%% %% +%% Summary of log file: %% +%% * One overfull hbox (0.7pt too wide). %% +%% * Three underfull vboxes (but good illo placement overall). %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% March, 2009: adhere (Andrew D. Hwang) %% +%% texlive2007, GNU/Linux %% +%% July, 2009: dcwilson %% +%% MiKTeX 2.7, WinXP Pro %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x3 (Run pdflatex three times) %% +%% makeindex %% +%% pdflatex %% +%% %% +%% %% +%% August 2009: pglatex. %% +%% Compile this project with: %% +%% pdflatex 29782-t.tex ..... THREE times %% +%% makeindex 29782-t.idx %% +%% pdflatex 29782-t.tex %% +%% %% +%% pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt]{book}[2005/09/16] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[utf8]{inputenc}[2006/05/05] %% DP standard encoding + +\IfFileExists{textcomp.sty}{% %% For ditto marks +\usepackage{textcomp}[2005/09/27]% +}{} + +\newlength{\MySkip} +\IfFileExists{fix-cm.sty}{% %% For larger title page fonts +\usepackage{fix-cm}[2006/03/24]% +\newcommand{\MyHuge}{\fontsize{38}{48}\selectfont}% +\setlength{\MySkip}{0.375in}}% else +{\newcommand{\MyHuge}{\Huge}% +\setlength{\MySkip}{0.25in}} + + +\usepackage{ifthen}[2001/05/26] %% Logical conditionals + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2002/01/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\usepackage{array}[2005/08/23] %% extended array/tabular features + + %% extended footnote capabilities +\usepackage[symbol,perpage]{footmisc}[2005/03/17] + +\usepackage{multicol}[2006/05/18] +\usepackage{makeidx}[2000/03/29] + +\usepackage{graphicx}[1999/02/16]%% For diagrams +\usepackage{wrapfig}[2003/01/31] %% and wrapping text around them + + +% for running heads; no package date available +\usepackage{fancyhdr} +\renewcommand{\headrulewidth}{0pt} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Conditional compilation switches %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Sets up a handful of additional index entries +\newboolean{IndexExtras} +% UNCOMMENT the next line for extra entries +%\setboolean{IndexExtras}{true} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ForPrinting=true (default) false +% Letterpaper Cropped pages +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +\newboolean{ForPrinting} + +%% COMMENT the next line for a SCREEN-OPTIMIZED VERSION of the text %% +\setboolean{ForPrinting}{true} + +%% Initialize values to ForPrinting=false +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon} +{ + Figures may have been moved with respect to the surrounding text. + Minor typographical corrections and presentational changes have + been made without comment. +} + +\newcommand{\TransNoteText} +{ + \TransNoteCommon + + This PDF file is formatted for screen viewing, but may be easily + formatted for printing. Please consult the preamble of the \LaTeX\ + source file for instructions. +} + +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is formatted for printing, but may be easily formatted + for screen viewing. Please see the preamble of the \LaTeX\ source + file for instructions. + } +}{} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Text block size carefully chosen to accommodate tall illos +\usepackage[body={5.2in,8.125in},\Margins]{geometry}[2002/07/08] + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: Space, Time and Gravitation}, + pdfauthor={Arthur Stanley Eddington}, + pdfkeywords={David Clarke, Andrew D. Hwang, + Project Gutenberg Online Distributed Proofreading Team, + The Internet Archive/American Libraries}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + +%%%% Re-crop screen-formatted version, omit blank verso pages %%%% +\ifthenelse{\boolean{ForPrinting}} + {} + {\hypersetup{pdfpagescrop = 100 90 512 780} + % If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COMMANDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% For various ad hoc alignment needs +\newlength{\TmpLen} + +%%%% Fixed-width environment to format PG boilerplate %%%% +% 9.2pt leaves no overfull hbox at 80 char line width +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{9.2}{10.5}\ttfamily\selectfont}% +{\end{alltt}} + +% Extra index entries, e.g., ``Geometry, Euclidean'' +\newcommand{\IndexExtra}[1]% +{% + \ifthenelse{\boolean{IndexExtras}}% + {\index{#1}}% + {}% +} + +% Copyright page formatting +\newcommand{\Publine}[2]{\makebox[\TmpLen][s]{\textsc{#1} : \textsc{#2}}} + + +% Cross-referencing: anchors +\newcommand{\Pagelabel}[1] + {\phantomsection\label{page:#1}} + +\newcommand{\Figlabel}[1] + {\phantomsection\label{fig:#1}} + +\newcommand{\Tag}[1]% + {\tag{#1}\phantomsection\label{eqn:#1}} + +% and links +\newcommand{\Pageref}[1] + {\hyperref[page:#1]{p.~\pageref{page:#1}}} + +\newcommand{\Figref}[1]% + {\hyperref[fig:#1]{\texorpdfstring{Fig.~#1}{Fig #1}}} + +\newcommand{\Noteref}[1]% + {\hyperref[appnote:#1]{\texorpdfstring{Note~#1}{Note #1}}} + + +% \Eqref{equation}{5}, \Eqref{formula}{2}, etc. +\newcommand{\Eqref}[2]{\hyperref[eqn:#2]{#1~(#2)}} + +\newcommand{\Chapref}[1]{\hyperref[chapter:#1]% + {Chapter~\textsc{\MakeLowercase{#1}}}} + + +%%%% Table of contents %%%% +% Dot leader for chapter-level entries +\makeatletter +\renewcommand{\l@chapter}{\@dottedtocline{0}{0pt}{0pt}} +\makeatother + +% No page numbers +\newcommand{\TableofContents}{{\let\thepage=\empty\tableofcontents}} + +% Centered chapter-level titles +\newlength{\ToCBox} +\settowidth{\ToCBox}{\scshape chapter~viii}% Widest title + +\newcommand{\ToCCenter}[2]% +{\normalfont\scshape\null\hfill% + \makebox[\ToCBox][l]{\scshape #1~\MakeLowercase{#2}}\hfill} + +% The table of contents may be set across two pages; we'll ensure the +% word ``page'' is printed at the top of the page number column, +% centered on the first page (to match the scan) and right-justified +% on subsequent pages (better visual appearance). +% +% Each chapter-like unit (Prologue, Chapter, Appendix) puts a \ToCLine +% into the toc file. The Prologue command defines a \ToCAnchor macro, +% which expands to the current page of the table of contents. +% +% Subsequent units put down a label and get the \pageref. If this has +% changed, the word ``page'' is written at the right margin, and the +% \ToCAnchor command is updated. +\newcommand{\ToCLine}[2]% +{\label{#1-toc:#2}% +\ifthenelse{\equal{#1}{prologue}}% + {\ToCCenter{#1}{#2}\makebox[0pt][c]{\footnotesize page}}% else... + {\ifthenelse{\not\equal{\pageref{#1-toc:#2}}{\ToCAnchor}}% + {\renewcommand{\ToCAnchor}{\pageref{#1-toc:#2}}% + \ToCCenter{#1}{#2}\makebox[0pt][r]{\footnotesize page}}% else... + {\ToCCenter{#1}{#2}}}} + + +% [** PP: ``Frontispiece'' heading in ToC would overlap dot leaders] +% To prevent this, write the code below into the toc file: +% \newlength{\FPlen} +% \settowidth{\FPlen}{\small\textit{Frontis}} +% \makeatletter +% \renewcommand{\@pnumwidth}{\FPlen} +% \makeatother +\newcommand{\FrontispieceToCEntry} +{ +\addtocontents{toc}{\protect\newlength{\protect\FPlen}} + +\ifthenelse{\boolean{ForPrinting}} +{\addtocontents{toc}{\protect\settowidth{\protect\FPlen}% + {\protect\small\protect\textit{Frontis}}}} % About half the width +{\addtocontents{toc}{\protect\settowidth{\protect\FPlen}% + {\protect\small\protect\textit{Frontispie}}}} % A bit wider + +\addtocontents{toc}{\protect\makeatletter} +\addtocontents{toc}{\protect\renewcommand{\protect\@pnumwidth}{\protect\FPlen}} +\addtocontents{toc}{\protect\makeatother} + +% The actual contents +\ifthenelse{\boolean{ForPrinting}} +{\addtocontents{toc}% +{\protect\contentsline{chapter}% + {\hyperref[frontispiece]{\protect\scshape Eclipse Instruments at Sobral}}% + {\protect\textit{\protect\makebox[12pt][c]{\protect\small Frontispiece}}}{}}} +% else move ``frontispiece'' heading slightly farther to the left +{\addtocontents{toc}% +{\protect\contentsline{chapter}% + {\hyperref[frontispiece]{\protect\scshape Eclipse Instruments at Sobral}}% + {\protect\textit{\protect\makebox[36pt][c]{\protect\small Frontispiece}}}{}}} + +% And restore \@pnumwidth +\addtocontents{toc}{\protect\settowidth{\protect\FPlen}{100}} +\addtocontents{toc}{\protect\makeatletter} +\addtocontents{toc}{\protect\renewcommand{\protect\@pnumwidth}{\protect\FPlen}} +\addtocontents{toc}{\protect\makeatother} +} + +% redefine hyperref's re-definition +% so that chapter anchor is above chapter title +\makeatletter +\AtBeginDocument{% in case hyperref clobbers this +\def\@schapter#1{% + \begingroup + \let\@mkboth\@gobbletwo + \Hy@GlobalStepCount\Hy@linkcounter + \xdef\@currentHref{\Hy@chapapp*.\the\Hy@linkcounter}% + \Hy@raisedlink{% + \hyper@anchorstart{\@currentHref}\hyper@anchorend + }% + \endgroup + \H@old@schapter{#1}% +}} +\makeatother + + + +% Sectioning: Paragraph, Chapter, Preface, Prologue, Appendix + +% Each chapter starts unindented, with smallcaps +\newcommand{\First}[1]{\noindent\textsc{#1}} + +% Appendix Notes are indented +\newcommand{\Indent}{\hspace*{\parindent}} + +\newcommand{\Paragraph}[1]{#1} % Null semantic markup + +\newcommand{\ChapterHead}[1]% +{\protect\centering\normalfont\upshape\textsc{\Large \MakeUppercase{#1}}} + +% \Chapter[title for ToC]{N}{Title} -- for numbered chapters +\newcommand{\Chapter}[3][]{% + % Clear stale heading on previous page + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RE]{}}{}% + + % page formatting + \chapter*{\ChapterHead{Chapter~#2}\\ % + \protect\centering\textsc{\large \MakeUppercase{#3}}} + + \label{chapter:#2} + + % ToC entry + \addtocontents{toc}{\protect\filbreak} + \addtocontents{toc}{\protect\ToCLine{chapter}{#2}} + + \ifthenelse{\equal{#1}{}} + {\addcontentsline{toc}{chapter} + {\texorpdfstring{\protect\scshape\protect{#3}}{#3}} + % Running heads + \fancyhead{} + \fancyhead[C]{\textsc{\MakeUppercase{#3}}}} + {\addcontentsline{toc}{chapter} + {\texorpdfstring{\protect\scshape\protect{#1}}{#1}} + \fancyhead{} + \fancyhead[C]{\textsc{\MakeUppercase{#1}}}} + + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage} + \fancyhead[RE]{\textsc{[ch.}} + \fancyhead[LO]{\textsc{\MakeLowercase{#2}]}}}% End of ForPrinting + {\fancyhead[R]{\thepage} + \fancyhead[L]{\textsc{[ch.~\MakeLowercase{#2}]}}} +} + +% Prologue +\newcommand{\Prologue}{% + % page formatting + \chapter*{\ChapterHead{Prologue}\\ % + \centering\textsc{\large WHAT IS GEOMETRY?}} + + \label{prologue}% Location in document + + \addtocontents{toc}% + {\protect\newcommand{\protect\ToCAnchor}{\protect\pageref{prolog-toc}}} + + % ToC entry + \addtocontents{toc}{\protect\ToCLine{prologue}{}} + \addtocontents{toc}{\protect\label{prolog-toc}}% Location in ToC + \addcontentsline{toc}{chapter}% + {\texorpdfstring{\protect\scshape What is Geometry?}{What is Geometry?}} + + % Running heads + \fancyhead{} + \fancyhead[CE]{\textsc{PROLOGUE}} + \fancyhead[CO]{\textsc{WHAT IS GEOMETRY?}} + + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage}}% End of ForPrinting + {\fancyhead[R]{\thepage}} +} + + +% Preface +\makeatletter +\newcommand{\Preface}{% + % page formatting + \chapter*{\ChapterHead{Preface}} + + % Bookmark; No ToC entry + % access the anchor created by the \chapter* command + \xdef\foo{chapter*.\the\Hy@linkcounter}% + \Hy@writebookmark{}{Preface}{\foo}{0}{toc}% + + \label{preface} + + % Running heads + \setlength{\headheight}{14.5pt} + + % add some stretch to paragraph breaks to reduce underfull pages + \setlength\parskip{0pt plus 3pt} + + \pagestyle{fancy} + \fancyhead{} + \fancyfoot{} + \fancyhead[C]{\textsc{PREFACE}} + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage}}% End of ForPrinting + {\fancyhead[R]{\thepage}} +} + + +% Appendix +\newcommand{\Appendix}{% + % Clear stale heading on previous page + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RE]{}}{}% + + % page formatting + % the preliminary \pdfbookmark ensures the anchor is above the heading + % can't re-use the \chapter* anchor because a parent and child bookmark + % must have distinct anchors + \chapter*{\pdfbookmark[-1]{Appendix}{Appendix}\ChapterHead{Appendix}\\ % + \centering\textsc{\large MATHEMATICAL NOTES}} + + \label{appendix} + + % ToC entry -- PDF bookmarks auto-generated + \addtocontents{toc}{\protect\filbreak} + \addtocontents{toc}{\protect\ToCLine{appendix}{}} + \addcontentsline{toc}{chapter} + {\texorpdfstring{\protect\scshape Mathematical Notes}{Mathematical Notes}} + + % Running heads + \fancyhead{} + \fancyhead[CE]{\textsc{APPENDIX}} + \fancyhead[CO]{\textsc{MATHEMATICAL NOTES}} + + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage}}% + {\fancyhead[R]{\thepage}} +} +\makeatother + + +% Appendix Notes \AppNote{2}{(p.~20)} +\newcommand{\AppNote}[2]{% + % page formatting + \section*{\centering\textbf{Note~#1} \textrm{\normalsize #2}} + \label{appnote:#1} +} + + +% Define custom index format +\makeatletter +\renewcommand{\@idxitem}{\par\hangindent 30\p@\global\let\idxbrk\nobreak} +\renewcommand\subitem{\idxbrk\@idxitem \hspace*{15\p@}\let\idxbrk\relax} +\renewcommand{\indexspace}{\par\penalty-3000 \vskip 10pt plus5pt minus3pt\relax} + +% raw TeX manipulations are to position the bookmark anchor above the heading +\renewenvironment{theindex} + {\setlength\columnseprule{0.5pt}\setlength\columnsep{18pt}% + \begin{multicols}{2}[% + \begin{center}\Large IN\setbox0=\hbox{\phantomsection + \vbox to40pt{\hsize=20pt\IndexBookmark\vss}\hss}% + \ht0=0pt\dp0=0pt\wd0=0pt\box0DEX\vspace*{12pt}\end{center}]% + \setlength\parindent{0pt}\setlength\parskip{0pt plus 0.3pt}% + \thispagestyle{empty}\let\item\@idxitem\raggedright } + {\end{multicols}\clearpage\fancyhead{}\cleardoublepage} +\makeatother +\newcommand\IndexBookmark{\pdfbookmark[-1]{Index}{Index}} + + +% Contents heading +\AtBeginDocument{\renewcommand{\contentsname}% + {\protect\centering\normalfont\large\scshape CONTENTS\protect\\[0pt]}} + +% Illustrations +\newcommand{\Graphic}[3][] + {\includegraphics[width=#2]{./images/#3.pdf}% + \ifthenelse{\not\equal{#1}{}}{\Figlabel{#1}}{}} + +% %%%%% GLOBAL STYLE PARAMETERS %%%%% +\setlength{\parindent}{1em} + +\newlength{\QIndent} +\setlength{\QIndent}{0.6\parindent} + +% Chapter quotations +% \Quote[break]{Author (Date)}{Text}, etc. +\newcommand{\Signature}[1]{\allowbreak\null\nobreak% + \hfill\nobreak\raisebox{-2ex}{#1}} + +\newcommand{\Quote}[3][]% +{{\par\noindent\hspace*{\QIndent}\small#3% +\ifthenelse{\equal{#1}{break}}{\hfill\break\null}{}% +\hfill\textsc{\footnotesize #2}% +\ifthenelse{\equal{#1}{break}}{}{\hspace*{\QIndent}}\medskip}} + + +% Macros used in only one or two locations +\newcommand{\Actor}[1]{\par\hangindent 4.5\parindent \hangafter 1\qquad #1} + +\newcommand{\Ditto}{\normalfont\ttfamily\textquotesingle$\!$\textquotesingle} + +\newcommand{\Magnitude}{{}^\text{m}} + +\newcommand{\Neg}{\phantom{-}} + +\DeclareInputMath{176}{\mbox{\textdegree}} +\DeclareInputMath{183}{\cdot} + + +\newcommand{\HalfTitleBlock}% +{\settowidth{\TmpLen}{\textbf{\MyHuge GRAVITATION}} +\begin{minipage}{\TmpLen} +\noindent{\textbf{\MyHuge SPACE \hfill TIME}}\\[\MySkip] +\textbf{\LARGE\null\hfill AND\hfill\null}\\[\MySkip] +\textbf{\MyHuge GRAVITATION} +\end{minipage}} + + +\makeindex + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DPALIGN %%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter +\providecommand\shortintertext\intertext +\newcount\DP@lign@no +\newtoks\DP@lignb@dy +\newif\ifDP@cr +\newif\ifbr@ce +\def\f@@zl@bar{\null} +\def\addto@DPbody#1{\global\DP@lignb@dy\@xp{\the\DP@lignb@dy#1}} +\def\parseb@dy#1{\ifx\f@@zl@bar#1\f@@zl@bar + \addto@DPbody{{}}\let\@next\parseb@dy + \else\ifx\end#1 + \let\@next\process@DPb@dy + \ifDP@cr\else\addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&}\fi + \addto@DPbody{\end} + \else\ifx\intertext#1 + \def\@next{\eat@command0}% + \else\ifx\shortintertext#1 + \def\@next{\eat@command1}% + \else\ifDP@cr\addto@DPbody{&\DP@lint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\DPh@@kl} + \DP@crfalse\fi + \ifx\begin#1\def\begin@stack{b} + \let\@next\eat@environment + \else\ifx\lintertext#1 + \let\@next\linter@text + \else\ifx\rintertext#1 + \let\@next\rinter@text + \else\ifx\\#1 + \addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\\}\DP@crtrue + \global\advance\DP@lign@no\@ne + \let\@next\parse@cr + \else\check@braces#1!Q!Q!Q!\ifbr@ce\addto@DPbody{{#1}}\else + \addto@DPbody{#1}\fi + \let\@next\parseb@dy + \fi\fi\fi\fi\fi\fi\fi\fi\@next} +\def\process@DPb@dy{\let\lintertext\@gobble\let\rintertext\@gobble + \@xp\start@align\@xp\tw@\@xp\st@rredtrue\@xp\m@ne\the\DP@lignb@dy} +\def\linter@text#1{\@xp\DPlint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\rinter@text#1{\@xp\DPrint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\DPlint#1#2{\@xp\def\csname DP@lint:#1\endcsname{\text{#2}}} +\def\DPrint#1#2{\@xp\def\csname DP@rint:#1\endcsname{\text{#2}}} +\def\DP@lint#1{\ifbalancedlrint\@xp\ifx\csname DP@lint:#1\endcsname\relax\phantom + {\csname DP@rint:#1\endcsname}\else\csname DP@lint:#1\endcsname\fi + \else\csname DP@lint:#1\endcsname\fi} +\def\DP@rint#1{\ifbalancedlrint\@xp\ifx\csname DP@rint:#1\endcsname\relax\phantom + {\csname DP@lint:#1\endcsname}\else\csname DP@rint:#1\endcsname\fi + \else\csname DP@rint:#1\endcsname\fi} +\def\eat@command#1#2{\ifcase#1\addto@DPbody{\intertext{#2}}\or + \addto@DPbody{\shortintertext{#2}}\fi\DP@crtrue + \global\advance\DP@lign@no\@ne\parseb@dy} +\def\parse@cr{\new@ifnextchar*{\parse@crst}{\parse@crst{}}} +\def\parse@crst#1{\addto@DPbody{#1}\new@ifnextchar[{\parse@crb}{\parseb@dy}} +\def\parse@crb[#1]{\addto@DPbody{[#1]}\parseb@dy} +{\catcode`\$=13\gdef\check@braces#1#2!Q!Q!Q!{\ifx#2$$\br@cefalse\else\br@cetrue\fi + }\gdef${\textbf{\huge ERROR}\GenericError{\space\space\space\@spaces\@spaces\@spaces}% + {!!! DPalign/gather brace-parsing problem}% + {Likely nested argument beginning with doubled character}% + {Try putting an empty group at the start of the argument}\let$\relax}} +\def\eat@environment#1{\addto@DPbody{\begin{#1}}\begingroup + \def\@currenvir{#1}\let\@next\digest@env\@next} +\def\digest@env#1\end#2{% + \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}% + \ifx\@empty\begin@stack + \@checkend{#2} + \endgroup\let\@next\parseb@dy\fi + \addto@DPbody{#1\end{#2}} + \@next} +\def\lintertext{lint}\def\rintertext{rint} +\newif\ifbalancedlrint +\let\DPh@@kl\empty\let\DPh@@kr\empty +\def\DPg@therl{&\omit\hfil$\displaystyle} +\def\DPg@therr{$\hfil} + +\newenvironment{DPalign*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \DP@lignb@dy{&\DP@lint0&}\parseb@dy +}{% + \endalign +} +\newenvironment{DPgather*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \let\DPh@@kl\DPg@therl + \let\DPh@@kr\DPg@therr + \DP@lignb@dy{&\DP@lint0&\DPh@@kl}\parseb@dy +}{% + \endalign +} +\makeatother + +%%%%%%%%%%%%%%%%%%%%%%%% END OF DPALIGN %%%%%%%%%%%%%%%%%%%%%%%%%% + +% to avoid over/underfull boxes without using explicit linebreaks +\def\stretchyspace{\spaceskip0.5em plus 0.5em minus 0.25em} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +Project Gutenberg's Space, Time and Gravitation, by A. S. Eddington + +This eBook is for the use of anyone anywhere 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 www.gutenberg.org + + +Title: Space, Time and Gravitation + An Outline of the General Relativity Theory + +Author: A. S. Eddington + +Release Date: August 24, 2009 [EBook #29782] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK SPACE, TIME AND GRAVITATION *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by David Clarke, Andrew D. Hwang and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/American Libraries.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +\normalsize + +%% -----File: 001.png---Folio -9------- + +%Title Page +\cleardoublepage + +\null\vfil +\begin{center} +\HalfTitleBlock +\end{center} +\vfil + +%% -----File: 002.png---Folio -8------- + +%Copyright Page + +\clearpage +\settowidth{\TmpLen}{\textsc{CAMBRIDGE UNIVERSITY PRESS}} +\addtolength{\TmpLen}{0.25in} % [** PP: Hard-coded padding] +\null +\vfill +\begin{center}\stretchyspace +\makebox[\TmpLen][s]{\textsc{CAMBRIDGE UNIVERSITY PRESS}} \\[0.15in] +\addtolength{\TmpLen}{-0.125in} +\begin{minipage}{\TmpLen} +\centering +\footnotesize +\textsc{C. F. CLAY, Manager} \\[0.125in] +\Publine{\small LONDON}{FETTER LANE, E.C.~4}\\[0.1in] + +\Graphic{1in}{device} %Publisher's device + +\Publine{NEW YORK}{THE MACMILLAN CO.} \\ +\textsc{BOMBAY}\hfill\break +\textsc{CALCUTTA}\smash{$\!\left.\rule[-12pt]{0pt}{12pt}\right\}$}\hfill +\textsc{MACMILLAN AND CO., Ltd.} \\ +\textsc{MADRAS}\hfill\break +\Publine{TORONTO}{THE MACMILLAN CO. OF} \\ +\textsc{CANADA, Ltd.} \\ +\Publine{TOKYO}{MARUZEN-KABUSHIKI-KAISHA} +\end{minipage} + +\vspace*{0.5in} +\textsc{\scriptsize ALL RIGHTS RESERVED} +\end{center} +\vfill + +%% -----File: 003.png---Folio -7------- +%[Blank Page] +%% -----File: 004.png---Folio -6------- + +% [** PP: Force frontispiece to verso if ForPrinting] +\ifthenelse{\boolean{ForPrinting}} +{\cleardoublepage +\null\vfill +\newpage}{} + +\begin{center} +\pdfbookmark[0]{Frontispiece}{Frontispiece} +\rotatebox[origin=c]{90}{% +\begin{minipage}{8in} % [** PP: hard-coded size] +\centering +\includegraphics[width=7.6in]{./images/frontis_bw.jpg}\\ +\hspace*{1em}{\scriptsize +\textit{C.~Davidson\hfill Frontispiece\hfill +See \hyperref[page:117]{page~\upshape{\pageref{page:117}}}}}\hspace*{1em}\null\\ +{\textsc{\small eclipse instruments at sobral}} +\end{minipage}} +\end{center} +\phantomsection +\label{frontispiece} + +\FrontispieceToCEntry + +%% -----File: 005.png---Folio -5------- + +%Title Page + +\cleardoublepage +\begin{center} +\HalfTitleBlock\\[\MySkip] + +\textbf{\Large AN OUTLINE OF THE GENERAL}\\[0.125in] +\textbf{\Large RELATIVITY THEORY}\\[0.5in] +{\large BY}\\[0.2in] +\textbf{\Large A.~S. EDDINGTON, M.A., M.Sc., F.R.S.}\\[0.15in] +{\footnotesize PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL\\ +PHILOSOPHY, CAMBRIDGE} + +\vfill + +\textbf{\Large CAMBRIDGE}\\[0.15in] +\textbf{\Large AT THE UNIVERSITY PRESS}\\[0.15in] +\textbf{\large 1920} +\end{center} + +%% -----File: 006.png---Folio -4------- + +\clearpage +\null\vfil +\begin{center} +\begin{minipage}{4.5in} +\begin{verse} +\hfil\qquad\qquad Perhaps to move \\ +His laughter at their quaint opinions wide \\ +Hereafter, when they come to model heaven \\ +And calculate the stars: how they will wield \\ +The mighty frame: how build, unbuild, contrive \\ +To save appearances.\Signature{\textit{Paradise Lost.}} +\end{verse} +\end{minipage} +\end{center} +\vfil + +%% -----File: 007.png---Folio -3------- + + +\Preface + +\First{By} his theory of relativity Albert Einstein has provoked a +revolution of thought in physical science. + +The achievement consists essentially in this:---Einstein has +succeeded in separating far more completely than hitherto the +share of the observer and the share of external nature in the +things we see happen. The perception of an object by an observer +depends on his own situation and circumstances; for example, +distance will make it appear smaller and dimmer. We make +allowance for this almost unconsciously in interpreting what we +see. But it now appears that the allowance made for the \textit{motion} +of the observer has hitherto been too crude---a fact overlooked +because in practice all observers share nearly the same motion, +that of the earth. Physical space and time are found to be +closely bound up with this motion of the observer; and only an +amorphous combination of the two is left inherent in the external +world. When space and time are relegated to their proper source---the +observer---the world of nature which remains appears +strangely unfamiliar; but it is in reality simplified, and the +underlying unity of the principal phenomena is now clearly +revealed. The deductions from this new outlook have, with one +doubtful exception, been confirmed when tested by experiment. + +It is my aim to give an account of this work without introducing +anything very technical in the way of mathematics, +physics, or philosophy. The new view of space and time, so +opposed to our habits of thought, must in any case demand +unusual mental exercise. The results appear strange; and the +incongruity is not without a humorous side. For the first nine +chapters the task is one of interpreting a clear-cut theory, +accepted in all its essentials by a large and growing school of +physicists---although perhaps not everyone would accept the +author's views of its meaning. +% [** PP: Reads ``Chapters X and XI'' in original; modify for screen version] +\ifthenelse{\boolean{ForPrinting}}% +{Chapters~\hyperref[chapter:X]{\textsc{x}} and~\hyperref[chapter:XI]{\textsc{xi}}}% +{\Chapref{X} and~\Chapref{XI}} +deal with +very recent advances, with regard to which opinion is more +fluid. As for the last chapter, containing the author's speculations +on the meaning of nature, since it touches on the rudiments +of a philosophical system, it is perhaps too sanguine to hope that +it can ever be other than controversial. +%% -----File: 008.png---Folio -2------- + +A non-mathematical presentation has necessary limitations; +and the reader who wishes to learn how certain exact results +follow from Einstein's, or even Newton's, law of gravitation is +bound to seek the reasons in a mathematical treatise. But this +limitation of range is perhaps less serious than the limitation of +intrinsic truth. There is a relativity of truth, as there is a +relativity of space.---% +\begin{center} +``For \textsc{is} and \textsc{is-not} though \textit{with} Rule and Line\\[-3ex] +\phantom{``}And \textsc{up-and-down} \textit{without}, I could define.'' +\end{center} +Alas! It is not so simple. We abstract from the phenomena that +which is peculiar to the position and motion of the observer; +but can we abstract that which is peculiar to the limited imagination +of the human brain? We think we can, but only in the +symbolism of mathematics. As the language of a poet rings with +a truth that eludes the clumsy explanations of his commentators, +so the geometry of relativity in its perfect harmony expresses a +truth of form and type in nature, which my bowdlerised version +misses. + +But the mind is not content to leave scientific Truth in a dry +husk of mathematical symbols, and demands that it shall be +alloyed with familiar images. The mathematician, who handles~$x$ +so lightly, may fairly be asked to state, not indeed the inscrutable +meaning of~$x$ in nature, but the meaning which~$x$ +conveys to \textit{him}. + +Although primarily designed for readers without technical +knowledge of the subject, it is hoped that the book may also +appeal to those who have gone into the subject more deeply. +A few notes have been added in the Appendix mainly to bridge +the gap between this and more mathematical treatises, and to +indicate the points of contact between the argument in the text +and the parallel analytical investigation. + +It is impossible adequately to express my debt to contemporary +literature and discussion. The writings of Einstein, +Minkowski, Hilbert, Lorentz, Weyl, Robb, and others, have +provided the groundwork; in the give and take of debate with +friends and correspondents, the extensive ramifications have +gradually appeared. \Signature{A.~S.~E.\qquad} +\medskip + +\qquad{\small 1~\textit{May}, 1920.} + +%% -----File: 009.png---Folio -1------- + +%[** Table of Contents] + +\cleardoublepage +\phantomsection +\pdfbookmark[0]{Contents}{Contents} +\TableofContents % Arrange for empty pagestyle + +\fancyhead{} + +\iffalse %%%%%%%%%% BEGIN DEAD CODE %%%%%%%%%% +CONTENTS + +ECLIPSE INSTRUMENTS AT SOBRAL . Frontispiece + +PROLOGUE PAGE + +WHAT IS GEOMETRY? . . . . . . 1 + +CHAPTER I + +THE FITZGERALD CONTRACTION . . . 17 + +CHAPTER II + +RELATIVITY . . . . . . . . 30 + +CHAPTER III + +THE WORLD OF FOUR DIMENSIONS . . 45 + +CHAPTER IV + +FIELDS OF FORCE . . . . . . . 63 + +CHAPTER V + +KINDS OF SPACE . . . . . . . 77 + +CHAPTER VI + +THE NEW LAW OF GRAVITATION AND THE +OLD LAW . . . . . . . . . 93 + +CHAPTER VII + +WEIGHING LIGHT . . . . . . . 110 + +CHAPTER VIII + +OTHER TESTS OF THE THEORY . . . 123 + +CHAPTER IX + +MOMENTUM AND ENERGY . . . . . 136 + +CHAPTER X + +TOWARDS INFINITY . . . . . . 152 + +CHAPTER XI + +ELECTRICITY AND GRAVITATION . . 167 + +CHAPTER XII + +ON THE NATURE OF THINGS . . . . 180 + +APPENDIX + +MATHEMATICAL NOTES . . . . . 202 + +HISTORICAL NOTE . . . . . . 210 +\fi %%%%%%%%%% END OF DEAD CODE %%%%%%%%%% + +%% -----File: 010.png---Folio 0------- +%[Blank Page] +%% -----File: 011.png---Folio 1------- + +\mainmatter +\pagenumbering{arabic} + +\phantomsection +\pdfbookmark[-1]{Main Matter}{Main Matter} + +\Prologue + +{\small +\qquad A conversation between---% +\Actor{An experimental \textsc{Physicist}.} +\Actor{A pure \textsc{Mathematician}.} +\Actor{A \textsc{Relativist}, who advocates the newer conceptions of time +and space in physics.} + +}\medskip +\index{Euclidean geometry}% +\index{Geometry!Euclidean}% + + +\textit{Rel}. There is a well-known proposition of Euclid which states +that ``Any two sides of a triangle are together greater than the +third side.'' Can either of you tell me whether nowadays there +is good reason to believe that this proposition is true? + +\textit{Math}. For my part, I am quite unable to say whether the +proposition is true or not. I can deduce it by trustworthy +reasoning from certain other propositions or axioms, which are +supposed to be still more elementary. If these axioms are true, +the proposition is true; if the axioms are not true, the proposition +is not true universally. Whether the axioms are true or not +I cannot say, and it is outside my province to consider. + +\textit{Phys}. But is it not claimed that the truth of these axioms is +self-evident? + +\textit{Math}. They are by no means self-evident to me; and I think +the claim has been generally abandoned. + +\textit{Phys}. Yet since on these axioms you have been able to found +a logical and self-consistent system of geometry, is not this +indirect evidence that they are true? + +\textit{Math}. No. Euclid's geometry is not the only self-consistent +system of geometry. By choosing a different set of axioms I can, +for example, arrive at Lobatchewsky's geometry, in which many +of the propositions of Euclid are not in general true. From my +point of view there is nothing to choose between these different +geometries.% +\index{Geometry!Lobatchewskian}% +\index{Lobatchewsky}% + +\textit{Rel}. How is it then that Euclid's geometry is so much the +most important system? + +\textit{Math}. I am scarcely prepared to admit that it is the most +important. But for reasons which I do not profess to understand, +my friend the Physicist is more interested in Euclidean geometry +%% -----File: 012.png---Folio 2------- +than in any other, and is continually setting us problems in it. +Consequently we have tended to give an undue share of attention +to the Euclidean system. There have, however, been great +geometers like Riemann who have done something to restore +a proper perspective.% +\index{Riemann}% + +\textit{Rel}. (to Physicist). Why are you specially interested in +Euclidean geometry? Do you believe it to be the true geometry? + +\textit{Phys}. Yes. Our experimental work proves it true. + +\textit{Rel}. How, for example, do you prove that any two sides of +a triangle are together greater than the third side? + +\textit{Phys}. I can, of course, only prove it by taking a very large +number of typical cases, and I am limited by the inevitable +inaccuracies of experiment. My proofs are not so general or so +perfect as those of the pure mathematician. But it is a recognised +principle in physical science that it is permissible to generalise +from a reasonably wide range of experiment; and this kind of +proof satisfies me. + +\textit{Rel}. It will satisfy me also. I need only trouble you with +a special case. Here is a triangle $ABC$; how will you prove that +$AB + BC$ is greater than~$AC$? + +\textit{Phys}. I shall take a scale and measure the three sides. + +\textit{Rel}. But we seem to be talking about different things. I was +speaking of a proposition of geometry---properties of space, not +of matter. Your experimental proof only shows how a material +scale behaves when you turn it into different positions. + +\textit{Phys}. I might arrange to make the measures with an optical +device. + +\textit{Rel}. That is worse and worse. Now you are speaking of +properties of light. + +\textit{Phys}. I really cannot tell you anything about it, if you will +not let me make measurements of any kind. Measurement is +my only means of finding out about nature. I am not a metaphysicist. + +\textit{Rel}. Let us then agree that by \textit{length} and \textit{distance} you always +mean a quantity arrived at by measurements with material or +optical appliances. You have studied experimentally the laws +obeyed by these \textit{measured lengths}, and have found the geometry +to which they conform. We will call this geometry ``Natural +Geometry''; +\index{Geometry!natural}% +\index{Natural geometry}% +\index{Length!definition of}% +and it evidently has much greater importance for +%% -----File: 013.png---Folio 3------- +you than any other of the systems which the brain of the +mathematician has invented. But we must remember that its +subject matter involves the behaviour of material scales---the +properties of matter. Its laws are just as much laws of physics +as, for example, the laws of electromagnetism. + +\textit{Phys}. Do you mean to compare space to a kind of magnetic +field? I scarcely understand.% +\index{Space!meaning of}% + +\textit{Rel}. You say that you cannot explore the world without +some kind of apparatus. If you explore with a scale, you find +out the natural geometry; if you explore with a magnetic needle, +you find out the magnetic field. What we may call the field of +extension, or space-field, is just as much a physical quality as +the magnetic field. You can think of them both existing together +in the aether, if you like. The laws of both must be determined +by experiment. Of course, certain approximate laws of the space-field +(Euclidean geometry) have been familiar to us from childhood; +but we must get rid of the idea that there is anything +inevitable about these laws, and that it would be impossible to +find in other parts of the universe space-fields where these laws +do not apply. As to how far space really resembles a magnetic +field, I do not wish to dogmatise; my point is that they present +themselves to experimental investigation in very much the same +way. + +Let us proceed to examine the laws of natural geometry. +I have a tape-measure, and here is the triangle. $AB = 39\frac{1}{2}$~in., +$BC = \frac{1}{8}$~in., $CA = 39\frac{7}{8}$~in. Why, your proposition does not hold! + +\textit{Phys}. You know very well what is wrong. You gave the +tape-measure a big stretch when you measured~$AB$. + +\textit{Rel}. Why shouldn't I? + +\textit{Phys}. Of course, a length must be measured with a rigid +scale. + +\textit{Rel}. That is an important addition to our definition of length. +But what is a rigid scale?% +\index{Rigid scale, definition of}% + +\textit{Phys}. A scale which always keeps the same length. + +\textit{Rel}. But we have just defined length as the quantity arrived +at by measures with a rigid scale; so you will want another rigid +scale to test whether the first one changes length; and a third +to test the second; and so \textit{ad infinitum}. You remind me of the +incident of the clock and time-gun in Egypt. The man in charge +%% -----File: 014.png---Folio 4------- +of the time-gun fired it by the clock; and the man in charge of +the clock set it right by the time-gun. No, you must not define +length by means of a rigid scale, and define a rigid scale by +means of length. + +\textit{Phys}. I admit I am hazy about strict definitions. There is +not time for everything; and there are so many interesting +things to find out in physics, which take up my attention. Are +you so sure that you are prepared with a logical definition of all +the terms you use? + +\textit{Rel}. Heaven forbid! I am not naturally inclined to be +rigorous about these things. Although I appreciate the value of +the work of those who are digging at the foundations of science, +my own interests are mainly in the upper structure. But sometimes, +if we wish to add another storey, it is necessary to deepen +the foundations. I have a definite object in trying to arrive at +the exact meaning of length. A strange theory is floating round, +to which you may feel initial objections; and you probably +would not wish to let your views go by default. And after all, +when you claim to determine lengths to eight significant figures, +you must have a pretty definite standard of right and wrong +measurements. + +\textit{Phys}. It is difficult to define what we mean by rigid; but in +practice we can tell if a scale is likely to change length appreciably +in different circumstances. + +\textit{Rel}. No. Do not bring in the idea of change of length in +describing the apparatus for defining length. Obviously the +adopted standard of length cannot change length, whatever it +is made of. If a metre is defined as the length of a certain bar, +that bar can never be anything but a metre long; and if we +assert that this bar changes length, it is clear that we must have +changed our minds as to the definition of length. You recognised +that my tape-measure was a defective standard---that it was +not rigid. That was not because it changed length, because, if +it was the standard of length, it could not change length. It +was lacking in some other quality. + +You know an approximately rigid scale when you see one. +What you are comparing it with is not some non-measurable +ideal of length, but some attainable, or at least approachable, +ideal of material constitution. Ordinary scales have defects---% +%% -----File: 015.png---Folio 5------- +flexure, expansion with temperature, etc.---which can be reduced +by suitable precautions; and the limit, to which you approach +as you reduce them, is your rigid scale. You can define these +defects without appealing to any extraneous definition of length; +for example, if you have two rods of the same material whose +extremities are just in contact with one another, and when one +of them is heated the extremities no longer can be adjusted to +coincide, then the material has a temperature-coefficient of +expansion. Thus you can compare experimentally the temperature-coefficients +of different metals and arrange them in +diminishing sequence. In this sort of way you can specify the +nature of your ideal rigid rod, before you introduce the term +length. + +\textit{Phys}. No doubt that is the way it should be defined. + +\textit{Rel}. We must recognise then that all our knowledge of space +rests on the behaviour of material measuring-scales free from +certain definable defects of constitution. + +\textit{Phys}. I am not sure that I agree. Surely there is a sense in +which the statement $AB = 2CD$ is true or false, even if we had +no conception of a material measuring-rod. For instance, there +is, so to speak, twice as much paper between $A$ and~$B$, as between +$C$ and~$D$. + +\textit{Rel}. Provided the paper is uniform. But then, what does +uniformity of the paper mean? That the amount in given length +is constant. We come back at once to the need of defining length. + +If you say instead that the amount of ``space'' between +$A$ and~$B$ is twice that between $C$ and~$D$, the same thing applies. +You imagine the intervals filled with uniform space; but the +uniformity simply means that the same amount of space corresponds +to each inch of your rigid measuring-rod. You have +arbitrarily used your rod to divide space into so-called equal +lumps. It all comes back to the rigid rod. + +I think you were right at first when you said that you could +not find out anything without measurement; and measurement +involves some specified material appliance. + +Now you admit that your measures cannot go beyond a +certain close approximation, and that you have not tried all +possible conditions. Supposing that one corner of your triangle +was in a very intense gravitational field---far stronger than any +%% -----File: 016.png---Folio 6------- +we have had experience of---I have good ground for believing +that under those conditions you might find the sum of two sides +of a triangle, as measured with a rigid rod, appreciably less than +the third side. In that case would you be prepared to give up +Euclidean geometry? + +\textit{Phys}. I think it would be risky to assume that the strong +force of gravitation made no difference to the experiment. + +\textit{Rel}. On my supposition it makes an important difference. + +\textit{Phys}. I mean that we might have to make corrections to the +measures, because the action of the strong force might possibly +distort the measuring-rod. + +\textit{Rel}. In a rigid rod we have eliminated any special response +to strain. + +\textit{Phys}. But this is rather different. The extension of the rod +is determined by the positions taken up by the molecules under +the forces to which they are subjected; and there might be a +response to the gravitational force which all kinds of matter +would share. This could scarcely be regarded as a defect; and +our so-called rigid rod would not be free from it any more than +any other kind of matter. + +\textit{Rel}. True; but what do you expect to obtain by correcting +the measures? You correct measures, when they are untrue to +standard. Thus you correct the readings of a hydrogen-thermometer +to obtain the readings of a perfect gas-thermometer, +because the hydrogen molecules have finite size, and exert special +attractions on one another, and you prefer to take as standard +an ideal gas with infinitely small molecules. But in the present +case, what is the standard you are aiming at when you propose +to correct measures made with the rigid rod? + +\textit{Phys}. I see the difficulty. I have no knowledge of space +apart from my measures, and I have no better standard than +the rigid rod. So it is difficult to see what the corrected measures +would mean. And yet it would seem to me more natural to +suppose that the failure of the proposition was due to the +measures going wrong rather than to an alteration in the character +of space. + +\textit{Rel}. Is not that because you are still a bit of a metaphysicist? +You keep some notion of a space which is superior to measurement, +and are ready to throw over the measures rather than let +%% -----File: 017.png---Folio 7------- +this space be distorted. Even if there were reason for believing +in such a space, what possible reason could there be for assuming +it to be Euclidean? Your sole reason for believing space to be +Euclidean is that hitherto your measures have made it appear so; +if now measures of certain parts of space prefer non-Euclidean +geometry, all reason for assuming Euclidean space disappears. +Mathematically and conceptually Euclidean and non-Euclidean +space are on the same footing; our preference for Euclidean +space was based on measures, and must stand or fall by +measures.% +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% + +\textit{Phys}. Let me put it this way. I believe that I am trying to +measure something called length, which has an absolute meaning +in nature, and is of importance in connection with the laws of +nature. This length obeys Euclidean geometry. I believe my +measures with a rigid rod determine it accurately when no +disturbance like gravitation is present; but in a gravitational +field it is not unreasonable to expect that the uncorrected +measures may not give it exactly. + +\textit{Rel}. You have three hypotheses there:---(1)~there is an +absolute thing in nature corresponding to length, (2)~the +geometry of these absolute lengths is Euclidean, and (3)~practical +measures determine this length accurately when there is no +gravitational force. I see no necessity for these hypotheses, and +propose to do without them. \textit{Hypotheses non fingo.} The second +hypothesis seems to me particularly objectionable. You assume +that this absolute thing in nature obeys the laws of Euclidean +geometry. Surely it is contrary to scientific principles to lay +down arbitrary laws for nature to obey; we must find out her +laws by experiment. In this case the only experimental evidence +is that measured lengths (which by your own admission are not +necessarily the same as this absolute thing) sometimes obey +Euclidean geometry and sometimes do not. Again it would +seem reasonable to doubt your third hypothesis beyond, say, +the sixth decimal place; and that would play havoc with your +more delicate measures. But where I fundamentally differ from +you is the first hypothesis. Is there some absolute quantity in +nature that we try to determine when we measure length? +When we try to determine the number of molecules in a given +piece of matter, we have to use indirect methods, and different +%% -----File: 018.png---Folio 8------- +methods may give systematically different results; but no one +doubts that there is a definite number of molecules, so that there +is some meaning in saying that certain methods are theoretically +good and others inaccurate. Counting appears to be an absolute +operation. But it seems to me that other physical measures are +on a different footing. Any physical quantity, such as length, +mass, force, etc., which is not a pure number, can only be defined +as the result arrived at by conducting a physical experiment +according to specified rules.% +\index{Space!meaning of}% + +So I cannot conceive of any ``length'' in nature independent +of a definition of the way of measuring length. And, if there is, +we may disregard it in physics, because it is beyond the range +of experiment. Of course, it is always possible that we may +come across some quantity, not given directly by experiment, +which plays a fundamental part in theory. If so, it will turn up +in due course in our theoretical formulae. But it is no good +assuming such a quantity, and laying down \textit{a~priori} laws for it +to obey, on the off-chance of its proving useful. + +\textit{Phys}. Then you will not let me blame the measuring-rod +when the proposition fails? + +\textit{Rel}. By all means put the responsibility on the measuring-rod. +Natural geometry is the theory of the behaviour of material +scales. Any proposition in natural geometry is an assertion as +to the behaviour of rigid scales, which must accordingly take +the blame or credit. But do not say that the rigid scale is +wrong, because that implies a standard of right which does not +exist. + +\textit{Phys}. The space which you are speaking of must be a sort of +abstraction of the extensional relations of matter.% +\index{Matter!extensional relations of}% + +\textit{Rel}. Exactly so. And when I ask you to believe that space +can be non-Euclidean, or, in popular phrase, warped, I am not +asking you for any violent effort of the imagination; +\index{Warping of space}% +I only +mean that the extensional relations of matter obey somewhat +modified laws. Whenever we investigate the properties of space +experimentally, it is these extensional relations that we are +finding. Therefore it seems logical to conclude that space as +known to us must be the abstraction of these material relations, +and not something more transcendental. The reformed methods +of teaching geometry in schools would be utterly condemned, +%% -----File: 019.png---Folio 9------- +and it would be misleading to set schoolboys to verify propositions +of geometry by measurement, if the space they are supposed to +be studying had not this meaning. + +I suspect that you are doubtful whether this abstraction of +extensional relations quite fulfils your general idea of space; and, +as a necessity of thought, you require something beyond. I do +not think I need disturb that impression, provided you realise +that it is not the properties of this more transcendental thing +we are speaking of when we describe geometry as Euclidean or +non-Euclidean. + +\textit{Math}. The view has been widely held that space is neither +physical nor metaphysical, but conventional. Here is a passage +from Poincaré's \textit{Science and Hypothesis}, which describes this +alternative idea of space:% +\index{Space!conventional}% + +``If Lobatchewsky's geometry is true, the parallax of a very +distant star will be finite. +\index{Geometry!Lobatchewskian}% +\index{Lobatchewsky}% +If Riemann's is true, it will be negative. +These are the results which seem within the reach of experiment, +and it is hoped that astronomical observations may enable us +to decide between the two geometries. But what we call a +straight line in astronomy is simply the path of a ray of light. +If, therefore, we were to discover negative parallaxes, or to +prove that all parallaxes are higher than a certain limit, we +should have a choice between two conclusions: we could give +up Euclidean geometry, or modify the laws of optics, and +suppose that light is not rigorously propagated in a straight +line. It is needless to add that everyone would look upon this +solution as the more advantageous. Euclidean geometry, +therefore, has nothing to fear from fresh experiments.'' + +\textit{Rel}. Poincaré's brilliant exposition is a great help in understanding +the problem now confronting us. +\index{Poincaré}% +He brings out the +interdependence between geometrical laws and physical laws, +which we have to bear in mind continually. We can add on to +one set of laws that which we subtract from the other set. +I admit that space is conventional---for that matter, the meaning +of every word in the language is conventional. Moreover, we +have actually arrived at the parting of the ways imagined by +Poincaré, though the crucial experiment is not precisely the +one he mentions. But I deliberately adopt the alternative, +which, he takes for granted, everyone would consider less +%% -----File: 020.png---Folio 10------- +advantageous. I call the space thus chosen \textit{physical space}, and +its geometry \textit{natural geometry}, thus admitting that other conventional +meanings of space and geometry are possible. If it +were only a question of the meaning of space---a rather vague +term---these other possibilities might have some advantages. +But the meaning assigned to length and distance has to go +along with the meaning assigned to space. Now these are +quantities which the physicist has been accustomed to measure +with great accuracy; and they enter fundamentally into the +whole of our experimental knowledge of the world. We have a +knowledge of the so-called extent of the stellar universe, which, +whatever it may amount to in terms of ultimate reality, is not +a mere description of location in a conventional and arbitrary +mathematical space. Are we to be robbed of the terms in which +we are accustomed to describe that knowledge? + +The law of Boyle states that the pressure of a gas is proportional +to its density. It is found by experiment that this law is +only approximately true. A certain mathematical simplicity +would be gained by conventionally redefining \textit{pressure} in such +a way that Boyle's law would be rigorously obeyed. But it +would be high-handed to appropriate the word pressure in this +way, unless it had been ascertained that the physicist had no +further use for it in its original meaning. + +\textit{Phys}. I have one other objection. Apart from measures, we +have a general perception of space, and the space we perceive +is at least approximately Euclidean. + +\textit{Rel}. Our perceptions are crude measures. +\index{Perceptions, as crude measures}% +It is true that our +perception of space is very largely a matter of optical measures +with the eyes. If in a strong gravitational field optical and +mechanical measures diverged, we should have to make up our +minds which was the preferable standard, and afterwards abide +by it. So far as we can ascertain, however, they agree in all +circumstances, and no such difficulty arises. So, if physical +measures give us a non-Euclidean space, the space of perception +will be non-Euclidean. If you were transplanted into an extremely +intense gravitational field, you would directly perceive +the non-Euclidean properties of space. + +\textit{Phys}. Non-Euclidean space seems contrary to reason. + +\textit{Math}. It is not contrary to reason, but contrary to common +%% -----File: 021.png---Folio 11------- +experience, which is a very different thing, since experience is +very limited. + +\textit{Phys}. I cannot imagine myself perceiving non-Euclidean space! + +\textit{Math}. Look at the reflection of the room in a polished doorknob, %[** PP: Hyphenated across a line in original] +and imagine yourself one of the actors in what you see +going on there. + +\textit{Rel}. I have another point to raise. The distance between +two points is to be the length measured with a rigid scale. Let +us mark the two points by particles of matter, because we must +somehow identify them by reference to material objects. For +simplicity we shall suppose that the two particles have no +relative motion, so that the distance---whatever it is---remains +constant. Now you will probably agree that there is no such +thing as absolute motion; consequently there is no standard +condition of the scale which we can call ``at rest.'' We may +measure with the scale moving in any way we choose, and if +results for different motions disagree, there is no criterion for +selecting the true one. Further, if the particles are sliding past +the scale, it makes all the difference what instants we choose +for making the two readings. + +\textit{Phys}. You can avoid that by defining distance as the measurement +made with a scale which has the same velocity as the two +points. Then they will always be in contact with two particular +divisions of the scale. + +\textit{Rel}. A very sound definition; but unfortunately it does not +agree with the meaning of distance in general use. When the +relativist wishes to refer to this length, he calls it the \textit{proper-length}; +\index{Proper-length}% +in non-relativity physics it does not seem to have been +used at all. You see it is not convenient to send your apparatus +hurling through the laboratory---after a pair of $\alpha$ particles, for +example. And you could scarcely measure the length of a wave +of light by this convention\footnote% +{The proper-length of a light-wave is actually infinite.}. So the physicist refers his lengths +to apparatus at rest on the earth; and the mathematician starts +with the words ``Choose unaccelerated rectangular axes~$Ox$, $Oy$, +$Oz$,~$\dotsc$'' and assumes that the measuring-scales are at rest +relatively to these axes. So when the term length is used some +arbitrary standard motion of the measuring apparatus must +always be implied. + +%% -----File: 022.png---Folio 12------- + +\textit{Phys}. Then if you have fixed your standard motion of the +measuring-rod, there will be no ambiguity if you take the +readings of both particles at the same moment. + +\textit{Rel}. What is the same moment at different places? The +conception of simultaneity in different places is a difficult one. +Is there a particular instant in the progress of time on another +world, Arcturus, which is the same as the present instant on the +Earth?% +\index{Simultaneity}% + +\textit{Phys}. I think so, if there is any connecting link. We can +observe an event, say a change of brightness, on Arcturus, and, +allowing for the time taken by light to travel the distance, +determine the corresponding instant on the earth. + +\textit{Rel}. But then you must know the speed of the earth through +the aether. It may have shortened the light-time by going some +way to meet the light coming from Arcturus. + +\textit{Phys}. Is not that a small matter? + +\textit{Rel}. At a very modest reckoning the motion of the earth in +the interval might alter the light-time by several days. Actually, +however, any speed of the earth through the aether up to the +velocity of light is admissible, without affecting anything observable. +At least, nothing has been discovered which contradicts +this. So the error may be months or years. + +\textit{Phys}. What you have shown is that we have not sufficient +knowledge to determine in practice which are simultaneous +events on the Earth and Arcturus. It does not follow that there +is no definite simultaneity.% +\index{Absolute simultaneity}% + +\textit{Rel}. That is true, but it is at least possible that the reason +why we are unable to determine simultaneity in practice (or, +what comes to pretty much the same thing, our motion through +the aether) in spite of many brilliant attempts, is that there is +no such thing as absolute simultaneity of distant events. It is +better therefore not to base our physics on this notion of absolute +simultaneity, which may turn out not to exist, and is in any +case out of reach at present. + +But what all this comes to is that time as well as space is +implied in all our measures. The fundamental measurement is +not the interval between two points of space, but between two +points of space associated with instants of time. + +Our natural geometry is incomplete at present. We must +%% -----File: 023.png---Folio 13------- +supplement it by bringing in time as well as space. We shall +need a perfect clock as well as a rigid scale for our measures. +\index{Clock!perfect}% +It may be difficult to choose an ideal standard clock; but whatever +definition we decide on must be a physical definition. We +must not dodge it by saying that a perfect clock is one which +keeps perfect time. Perhaps the best theoretical clock would be +a pulse of light travelling in vacuum to and fro between mirrors +at the ends of a rigid scale. The instants of arrival at one end +would define equal intervals of time.% +\index{Time!measurement of}% + +\textit{Phys}. I think your unit of time would change according to +the motion of your ``clock'' through the aether. + +\textit{Rel}. Then you are comparing it with some notion of absolute +time. I have no notion of time except as the result of measurement +with some kind of clock. (Our immediate perception of +the flight of time is presumably associated with molecular +processes in the brain which play the part of a material clock.) +If you know a better clock, let us adopt it; but, having once +fixed on our ideal clock there can be no appeal from its judgments. +You must remember too that if you wish to measure +a second \textit{at one place}, you must keep your clock fixed at what +you consider to be one place; so its motion is defined. The +necessity of defining the motion of the clock emphasises that +one cannot consider time apart from space; there is one geometry +comprising both. + +\textit{Phys}. Is it right to call this study \textit{geometry}? %[** PP: Changed . to ?] +Geometry deals +with space alone. + +\textit{Math}. I have no objection. It is only necessary to consider +time as a fourth dimension. Your complete natural geometry +will be a geometry of four dimensions.% +\index{Fourth dimension}% + +\textit{Phys}. Have we then found the long-sought fourth dimension? + +\textit{Math}. It depends what kind of a fourth dimension you were +seeking. Probably not in the sense you intend. For me it only +means adding a fourth variable,~$t$, to my three space-variables +$x$, $y$,~$z$. It is no concern of mine what these variables really +represent. You give me a few fundamental laws that they +satisfy, and I proceed to deduce other consequences that may +be of interest to you. The four variables may for all I know be +the pressure, density, temperature and entropy of a gas; that +is of no importance to me. But you would not say that a gas +%% -----File: 024.png---Folio 14------- +had four dimensions because four mathematical variables were +used to describe it. Your use of the term ``dimensions'' is +probably more restricted than mine. + +\textit{Phys}. I know that it is often a help to represent pressure +and volume as height and width on paper; and so geometry +may have applications to the theory of gases. But is it not going +rather far to say that geometry can deal directly with these +things and is not necessarily concerned with lengths in +space? + +\textit{Math}. No. Geometry is nowadays largely analytical, so that +in form as well as in effect, it deals with variables of an unknown +nature. It is true that I can often see results more easily by +taking my $x$ and~$y$ as lengths on a sheet of paper. Perhaps it +would be helpful in seeing other results if I took them as pressure +and density in a steam-engine; but a steam-engine is not so +handy as a pencil. It is literally true that I do not want to +know the significance of the variables $x$, $y$, $z$, $t$ that I am discussing. +That is lucky for the Relativist, because although he has defined +carefully how they are to be measured, he has certainly not +conveyed to me any notion of how I am to picture them, if my +picture of absolute space is an illusion. + +\textit{Phys}. Yours is a strange subject. You told us at the beginning +that you are not concerned as to whether your propositions are +true, and now you tell us you do not even care to know what +you are talking about. + +\textit{Math}. That is an excellent description of Pure Mathematics, +which has already been given by an eminent mathematician\footnotemark. +\index{Russell}% + \footnotetext{``Pure mathematics consists entirely of such asseverations as that, if such + and such a proposition is true of \textit{anything}, then such and such a proposition + is true of that thing. It is essential not to discuss whether the first proposition + is really true, and not to mention what the anything is of which it is supposed + to be true\ldots. Thus mathematics may be defined as the subject in which we + never know what we are talking about, nor whether what we are saying is true.'' + \Signature{\textsc{Bertrand Russell}.}} + +\textit{Rel}. I think there is a real sense in which time is a fourth +dimension---as distinct from a fourth variable. The term +dimension seems to be associated with relations of \textit{order}. +\index{Order and dimensions}% +I believe that the order of events in nature is one indissoluble +four-dimensional order. We may split it arbitrarily into space +and time, just as we can split the order of space into length, +%% -----File: 025.png---Folio 15------- +breadth and thickness. But space without time is as incomplete +as a surface without thickness. + +\textit{Math}. Do you argue that the real world behind the phenomena +is four-dimensional? + +\textit{Rel}. I think that in the real world there must be a set of +entities related to one another in a four-dimensional order, and +that these are the basis of the perceptual world so far as it is +yet explored by physics. But it is possible to pick out a four-dimensional +set of entities from a basal world of five dimensions, +or even of three dimensions. The straight lines in three-dimensional +space form a four-dimensional set of entities, i.e.\ +they have a four-fold % [** PP: Regularized fourfold] +order. So one cannot predict the ultimate +number of dimensions in the world---if indeed the expression +\textit{dimensions} is applicable. + +\textit{Phys}. What would a philosopher think of these conceptions? +Or is he solely concerned with a metaphysical space and time +which is not within reach of measurement. + +\textit{Rel}. In so far as he is a psychologist our results must concern +him. Perception is a kind of crude physical measurement; +\index{Perceptions, as crude measures}% +and +perceptual space and time is the same as the measured space +and time, which is the subject-matter of natural geometry. In +other respects he may not be so immediately concerned. +Physicists and philosophers have long agreed that motion +through absolute space can have no meaning; +\index{Space!meaning of}% +but in physics +the question is whether motion through aether has any meaning. +I consider that it has no meaning; but that answer, though it +brings philosophy and physics into closer relation, has no bearing +on the philosophic question of absolute motion. I think, +however, we are entitled to expect a benevolent interest from +philosophers, in that we are giving to their ideas a perhaps +unexpected practical application. +\bigskip + +%[** Thought break] + +Let me now try to sum up my conclusions from this conversation. +We have been trying to give a precise meaning to the +term \textit{space}, so that we may be able to determine exactly the +properties of the space we live in. There is no means of determining +the properties of our space by \textit{a~priori} reasoning, because +there are many possible kinds of space to choose from, no one +of which can be considered more likely than any other. For +%% -----File: 026.png---Folio 16------- +more than 2000 years we have believed in a Euclidean space, +because certain experiments favoured it; but there is now reason +to believe that these same experiments when pushed to greater +accuracy decide in favour of a slightly different space (in the +neighbourhood of massive bodies). The relativist sees no reason +to change the rules of the game because the result does not +agree with previous anticipations. Accordingly when he speaks +of space, he means the space revealed by measurement, whatever +its geometry. He points out that this is the space with which +physics is concerned; and, moreover, it is the space of everyday +perception. If his right to appropriate the term space in this +way is challenged, he would urge that this is the sense in which +the term has always been used in physics hitherto; it is only +recently that conservative physicists, frightened by the revolutionary +consequences of modern experiments, have begun to +play with the idea of a pre-existing space whose properties +cannot be ascertained by experiment---a metaphysical space, to +which they arbitrarily assign Euclidean properties, although it +is obvious that its geometry can never be ascertained by experiment. +But the relativist, in defining space as \textit{measured space}, +clearly recognises that all measurement involves the use of +material apparatus; the resulting geometry is specifically a study +of the extensional relations of matter. He declines to consider +anything more transcendental. + +My second point is that since natural geometry is the study +of extensional relations of natural objects, and since it is found +that their space-order cannot be discussed without reference to +their time-order as well, it has become necessary to extend our +geometry to four dimensions in order to include time. +%% -----File: 027.png---Folio 17------- + +\Chapter{I}{The FitzGerald Contraction} + +\Quote{Descartes.} +{In order to reach the Truth, it is necessary, once in one's life, to put every +thing in doubt---so far as possible.} + + +\First{Will} it take longer to swim to a point $100$~yards up-stream +and back, or to a point $100$~yards across-stream and back? + +In the first case there is a long toil up against the current, +and then a quick return helped by the current, which is all too +short to compensate. In the second case the current also hinders, +because part of the effort is devoted to overcoming the drift +down-stream. But no swimmer will hesitate to say that the +hindrance is the greater in the first case. + +Let us take a numerical example. Suppose the swimmer's +speed is $50$~yards a minute in still water, and the current is +$30$~yards a minute. Thus the speed against the current is~$20$, +and with the current $80$~yards a minute. The up journey then +takes $5$~minutes and the down journey $1\frac{1}{4}$ minutes. Total time, +$6\frac{1}{4}$ minutes. + +%[Illustration: Fig. 1.] +\begin{wrapfigure}{r}{1.5in} +\Graphic[1]{1.375in}{027a}% \Figlabel{1} +\end{wrapfigure} +Going across-stream the swimmer must aim at a point~$E$ above +the point~$B$ where he wishes to arrive, so +that $OE$ represents his distance travelled +in still water, and $EB$ the amount he has +drifted down. These must be in the ratio +$50$ to~$30$, and we then know from the right-angled +triangle $OBE$ that $OB$ will correspond +to~$40$. Since $OB$ is $100$~yards, $OE$ +is $125$~yards, and the time taken is $2\frac{1}{2}$ +minutes. Another $2\frac{1}{2}$ minutes will be +needed for the return journey. Total time, +$5$~minutes. + +In still water the time would have been $4$~minutes. + +The up-and-down swim is thus longer than the transverse +swim in the ratio $6\frac{1}{4}:5$ minutes. Or we may write the ratio +\[ +\dfrac{1}{\surd \bigl(1 - (\tfrac{30}{50})^2\bigr)} +\] +%% -----File: 028.png---Folio 18------- +which shows how the result depends on the ratio of the speed +of the current to the speed of the swimmer, viz.~$\frac{30}{50}$. + +A very famous experiment on these lines was tried in America +in the year~1887. The swimmer was a wave of light, which we +know swims through the aether with a speed of $186,330$ miles +a second. The aether was flowing through the laboratory like +a river past its banks. The light-wave was divided, by partial +reflection at a thinly silvered surface, into two parts, one of +which was set to perform the up-and-down stream journey and +the other the across-stream journey. When the two waves +reached their proper turning-points they were sent back to the +starting-point by mirrors. To judge the result of the race, there +was an optical device for studying interference fringes; because +the recomposition of the two waves after the journey would +reveal if one had been delayed more than the other, so that, for +example, the crest of one instead of fitting on to the crest of +the other coincided with its trough. + +To the surprise of Michelson and Morley, who conducted the +experiment, the result was a dead-heat. +\index{Michelson-Morley experiment}% +It is true that the +direction of the current of aether was not known---they hoped +to find it out by the experiment. That, however, was got over +by trying a number of different orientations. Also it was +possible that there might actually be no current at a particular +moment. But the earth has a velocity of $18\frac{1}{2}$ miles a second, +continually changing direction as it goes round the sun; so that +at some time during the year the motion of a terrestrial laboratory +through the aether must be at least $18\frac{1}{2}$ miles a second. +The experiment should have detected the delay by a much +smaller current; in a repetition of it by Morley and Miller +in~1905, a current of $2$~miles a second would have been +sufficient. + +If we have two competitors, one of whom is known to be +slower than the other, and yet they both arrive at the winning-post +at the same time, it is clear that they cannot have travelled +equal courses. To test this, the whole apparatus was rotated +through a right angle, so that what had been the up-and-down +course became the transverse course, and \textit{vice versa}. Our two +competitors interchanged courses, but still the result was a +dead-heat. + +%% -----File: 029.png---Folio 19------- + +The surprising character of this result can be appreciated by +contrasting it with a similar experiment on sound-waves. +Sound consists of waves in air or other material, as light consists +of waves in aether. It would be possible to make a precisely +similar experiment on sound, with a current of air past the +apparatus instead of a current of aether. In that case the greater +delay of the wave along the direction of the current would +certainly show itself experimentally. Why does light seem to +behave differently? + +The straightforward interpretation of this remarkable result +is that each course undergoes an automatic contraction when it +is swung from the transverse to the longitudinal position, so +that whichever arm of the apparatus is placed up-stream it +straightway becomes the shorter. The course is marked out in +the rigid material apparatus, and we have to suppose that the +length of any part of the apparatus changes as it is turned in +different directions with respect to the aether-current. It is +found that the kind of material---metal, stone or wood---makes +no difference to the experiment. The contraction must be the +same for all kinds of matter; the expected delay depends only +on the ratio of the speed of the aether current to the speed of +light, and the contraction which compensates it must be equally +definite.% +\index{Length!effect of motion on}% + +This explanation was proposed by FitzGerald, and at first +sight it seems a strange and arbitrary hypothesis. But it has +been rendered very plausible by subsequent theoretical researches +of Larmor and Lorentz. +\index{Larmor}% +\index{Lorentz}% +Under ordinary circumstances the form +and size of a solid body is maintained by the forces of cohesion +between its particles. What is the nature of cohesion? We guess +that it is made up of electric forces between the molecules. But +the aether is the medium in which electric force has its seat; +hence it will not be a matter of indifference to these forces how +the electric medium is flowing with respect to the molecules. +When the flow changes there will be a readjustment of cohesive +forces, and we must expect the body to take a new shape and +size. + +The theory of Larmor and Lorentz enables us to trace in +detail the readjustment. Taking the accepted formulae of +electromagnetic theory, they showed that the new form of +%% -----File: 030.png---Folio 20------- +equilibrium would be contracted in just such a way and by +just such an amount as FitzGerald's explanation requires\footnote% +{Appendix, \Noteref{1}.}.% +\Pagelabel{note1}% +\index{Contraction, FitzGerald}% +\index{FitzGerald Contraction}% + +The contraction in most cases is extremely minute. We have +seen that when the ratio of the speed of the current to that +of the swimmer is $\frac{3}{5}$, a contraction in the ratio +$\surd \bigl(1 - (\frac{3}{5})^2\bigr)$ +is needed to compensate for the delay. The earth's orbital +velocity is $\frac{1}{10000}$ of the velocity of light, so that it will give a +contraction of $\surd \bigl(1 - (\frac{1}{10000})^2\bigr)$, or $1$~part in~$200,000,000$. This +would mean that the earth's diameter in the direction of its +motion is shortened by $2\frac{1}{2}$ inches. + +The Michelson-Morley experiment has thus failed to detect +our motion through the aether, because the effect looked for---the +delay of one of the light waves---is exactly compensated by +an automatic contraction of the matter forming the apparatus. +Other ingenious experiments have been tried, electrical and +optical experiments of a more technical nature. They likewise +have failed, because there is always an automatic compensation +somewhere. We now believe there is something in the nature +of things which inevitably makes these compensations, so that +it will never be possible to determine our motion through the +aether. Whether we are at rest in it, or whether we are rushing +through it with a speed not much less than that of light, will +make no difference to anything that can possibly be observed. + +This may seem a rash generalization from the few experiments +actually performed; more particularly, since we can only experiment +with the small range of velocity caused by the earth's +orbital motion. With a larger range residual differences might +be disclosed. But there is another reason for believing that the +compensation is not merely approximate but exact. The compensation +has been traced theoretically to its source in the +well-known laws of electromagnetic force; and here it is mathematically +exact. Thus the generalization is justified, at least in +so far as the observed phenomena depend on electromagnetic +causes, and in so far as the universally accepted laws of electromagnetism +are accurate. + +The generalization here laid down is called the restricted +Principle of Relativity:---\textit{It is impossible by any experiment to +detect uniform motion relative to the aether.}\Pagelabel{20}% +\index{Principle of Relativity (restricted)}% +\index{Relativity!restricted Principle of}% + +%% -----File: 031.png---Folio 21------- + +There are other natural forces which have not as yet been +recognised as coming within the electromagnetic scheme---gravitation, +for example---and for these other tests are required. +Indeed we were scarcely justified in stating above that the +diameter of the earth would contract $2\frac{1}{2}$~inches, because the +figure of the earth is determined mainly by gravitation, whereas +the Michelson-Morley experiment relates to bodies held together +by cohesion. There is fair evidence of a rather technical kind +that the compensation exists also for phenomena in which +gravitation is concerned; and we shall assume that the principle +covers all the forces of nature. + +Suppose for a moment it were not so, and that it were possible +to determine a kind of absolute motion of the earth by experiments +or observations involving gravitation. Would this throw +light on our motion through the aether? I think not. It would +show that there is some standard of rest with respect to which +the law of gravitation takes a symmetrical and simple form; +presumably this standard corresponds to some gravitational +medium, and the motion determined would be motion with +respect to that medium. Similarly if the motion were revealed +by vital or psychical phenomena, it would be motion relative +to some vital or psychical medium. The aether, defined as the +seat of electric forces, must be revealed, if at all, by electric +phenomena. + +It is well to remember that there is reasonable justification +for adopting the principle of relativity even if the evidence is +insufficient to prove it. In Newtonian dynamics the phenomena +are independent of uniform motion of the system; no explanation +is asked for, because it is difficult to see any reason why there +should be an effect. +\index{Gravitation!relativity for uniform motion}% +If in other phenomena the principle fails, +then we must seek for an explanation of its failure---and no +doubt a plausible explanation can be devised; but so long as +experiment gives no indication of a failure, it is idle to anticipate +such a complication. Clearly physics cannot concern itself with +all the possible complexities which \textit{may} exist in nature, but have +not hitherto betrayed themselves in any experiment. + +The principle of relativity has implications of a most revolutionary +kind. Let us consider what is perhaps an exaggerated +case---or perhaps the actual case, for we cannot tell. Let the +%% -----File: 032.png---Folio 22------- +reader suppose that he is travelling through the aether at +$161,000$~miles a second vertically upwards; if he likes to make +the positive assertion that this is his velocity, no one will be +able to find any evidence to contradict him. For this speed the +FitzGerald contraction is just $\frac{1}{2}$, so that every object contracts +to half its original length when turned into the vertical position.% +\index{FitzGerald Contraction!consequences of}% + +As you lie in bed, you are, say, $6$~feet long. Now stand upright; +you are $3$~feet. You are incredulous? Well, let us prove it! +Take a yard-measure; when turned vertically it must undergo +the FitzGerald contraction, and become only half a yard. If you +measure yourself with it, you will find you are just two---\textit{half-yards}. +``But I can see that the yard-measure does not change +length when I turn it.'' What you perceive is an image of the +rod on the retina of your eye; you imagine that the image +occupies the same space in both positions; but your retina has +contracted in the vertical direction without your knowing it, so +that your visual estimates of vertical length are double what +they should be. And so on with every test you can devise. +Because everything is altered in the same way, nothing appears +to be altered at all. + +It is possible to devise electrical and optical tests; in that +case the argument is more complicated, because we must consider +the effect of the rapid current of aether on the electric +forces and on waves of light. But the final conclusion is always +the same; the tests will reveal nothing. Here is one illustration. +To avoid distortion of the retina, lie on your back on the floor, +and watch in a suitably inclined mirror someone turn the rod +from the horizontal to the vertical position. You will, of course, +see no change of length, and it is not possible to blame the +retina this time. But is the appearance in the mirror a faithful +reproduction of what is actually occurring? +\index{Mirror, distortion by moving}% +In a plane mirror +at rest the appearance is correct; the rays of light come off the +mirror at the same angle as they fall on to it, like billiard balls +rebounding from an elastic cushion. But if the cushion is in +rapid motion the angle of the billiard-ball will be altered; and +similarly the rapid motion of the mirror through the aether +alters the law of reflection. +\index{Reflection by moving mirror}% +Precise calculation shows that the +moving mirror will distort the image, so as to conceal exactly +the changes of length which occur. + +%% -----File: 033.png---Folio 23------- + +The mathematician does not need to go through all the +possible tests in detail; he knows that the complete compensation +is inherent in the fundamental laws of nature, and so must +occur in every case. So if any suggestion is made of a device +for detecting these effects, he starts at once to look for the +fallacy which must surely be there. Our motion through the +aether may be very much less than the value here adopted, and +the changes of length may be very small; but the essential point +is that they escape notice, not because they are small (if they +are small), but because from their very nature they are undetectable. + +There is a remarkable reciprocity about the effects of motion +on length, which can best be illustrated by another example. +Suppose that by development in the powers of aviation, a man +flies past us at the rate of $161,000$ miles a second. +\index{Aviator, space and time-reckoning of|(}% [** PP: Using range] +We shall +suppose that he is in a comfortable travelling conveyance in +which he can move about, and act normally and that his length +is in the direction of the flight. If we could catch an instantaneous +glimpse as he passed, we should see a figure about three feet +high, but with the breadth and girth of a normal human being. +And the strange thing is that he would be sublimely unconscious +of his own undignified appearance. If he looks in a mirror in +his conveyance, he sees his usual proportions; this is because of +the contraction of his retina, or the distortion by the moving +mirror, as already explained. But when he looks down on us, +he sees a strange race of men who have apparently gone through +some flattening-out process; one man looks barely $10$~inches +across the shoulders, another standing at right angles is almost +``length and breadth, without thickness.'' As they turn about +they change appearance like the figures seen in the old-fashioned +convex-mirrors. If the reader has watched a cricket-match +through a pair of prismatic binoculars, he will have seen this +effect exactly. + +It is the reciprocity of these appearances--that each party +should think the other has contracted---that is so difficult to +realise. Here is a paradox beyond even the imagination of +Dean Swift. Gulliver regarded the Lilliputians as a race of +dwarfs; and the Lilliputians regarded Gulliver as a giant. That +is natural. If the Lilliputians had appeared dwarfs to Gulliver, +%% -----File: 034.png---Folio 24------- +and Gulliver had appeared a dwarf to the Lilliputians---but no! +that is too absurd for fiction, and is an idea only to be found in +the sober pages of science. + +This reciprocity is easily seen to be a necessary consequence +of the Principle of Relativity. The aviator must detect a FitzGerald +contraction of objects moving rapidly relatively to him, +just as we detect the contraction of objects moving relatively to us, +and as an observer at rest in the aether detects the contraction +of objects moving relatively to the aether. Any other result +would indicate an observable effect due to his own motion +through the aether. + +Which is right? Are we or the aviator? Or are both the +victims of illusion? It is not illusion in the ordinary sense, +because the impressions of both would be confirmed by every +physical test or scientific calculation suggested. No one knows +which is right. No one will ever know, because we can never +find out which, if either, is truly at rest in the aether. + +It is not only in space but in time that these strange variations +occur. If we observed the aviator carefully we should infer that +he was unusually slow in his movements; and events in the +conveyance moving with him would be similarly retarded---as +though time had forgotten to go on. +\index{Retardation of time}% +\index{Time!for moving observer}% +His cigar lasts twice as +long as one of ours. I said ``infer'' deliberately; we should \textit{see} +a still more extravagant slowing down of time; but that is easily +explained, because the aviator is rapidly increasing his distance +from us and the light-impressions take longer and longer to +reach us. The more moderate retardation referred to remains +after we have allowed for the time of transmission of light. + +But here again reciprocity comes in, because in the aviator's +opinion it is we who are travelling at $161,000$ miles a second +past him; and when he has made all allowances, he finds that +it is we who are sluggish. Our cigar lasts twice as long as his. + +Let us examine more closely how the two views are to be +reconciled. Suppose we both light similar cigars at the instant +he passes us. At the end of $30$~minutes our cigar is finished. +This signal, borne on the waves of light, hurries out at the rate +of $186,000$ miles a second to overtake the aviator travelling at +$161,000$ miles a second, who has had $30$~minutes start. It will +take nearly $194$~minutes to overtake him, giving a total time of +%% -----File: 035.png---Folio 25------- +$224$~minutes after lighting the cigar. His watch like everything +else about him (including his cigar) is going at half-speed; so +it records only $112$~minutes elapsed when our signal arrives. +The aviator knows, of course, that this is not the true time when +our cigar was finished, and that he must correct for the time of +transmission of the light-signal. He sets himself this problem---that +man has travelled away from me at $161,000$~miles a second +for an unknown time $x$~minutes; he has then sent a signal which +travels the same distance back at $186,000$~miles a second; the +total time is $112$~minutes; problem, find~$x$. Answer, $x = 60$ +minutes. He therefore judges that our cigar lasted $60$~minutes, +or twice as long as his own. His cigar lasted $30$~minutes by his +watch (because the same retardation affects both watch and +cigar); and that was in our opinion twice as long as ours, because +his watch was going at half-speed. + +Here is the full time-table. +\begin{center} +{\footnotesize +\begin{tabular}{r@{}c% @{} signifies no inter-column separation + >{\centering\hspace{0pt}}m{1.5in}@{}% + >{\centering\hspace{0pt}}m{1.75in}r@{}c} % end of alignment preamble +% Two multicolumn headers and extra vertical space +\multicolumn{2}{c}{\parbox[c]{0.625in}% [** PP: 0.7pt overfull] + {\centering Stationary watch}} + & Stationary Observer & Aviator + & \multicolumn{2}{c}{\parbox[c]{0.625in}% + {\centering Aviator's watch}} \\[3ex] +% + $0$ & min. & Lights cigar & Lights cigar & $0$ & min. \\ + $30$ & \Ditto & Finishes cigar & \ldots & $15$ & \Ditto \\ + $60$ & \Ditto & Inferred time aviator's cigar finished + & Finishes cigar & $30$ & \Ditto \\ +% +$112$ & \Ditto & Receives signal aviator's cigar finished + & \ldots & $56$ & \Ditto \\ +% +$120$ & \Ditto & \ldots & Inferred time stationary cigar finished + & $60$ & \Ditto \\ +% +$224$ & \Ditto & \ldots & Receives signal stationary cigar finished + & $112$ & \Ditto \\ +\end{tabular} +}% End of \footnotesize +\end{center} + +This is analysed from our point of view, not the aviator's; +because it makes out that he was wrong in his inference and we +were right. But no one can tell which was really right. + +The argument will repay a careful examination, and it will +be recognised that the chief cause of the paradox is that we +assume that we are at rest in the aether, whereas the aviator +assumes that he is at rest. Consequently whereas in our opinion +the light-signal is overtaking him at merely the difference +between $186,000$ and $161,000$ miles a second, he considers that +it is coming to him through the relatively stationary aether at +the normal speed of light. It must be remembered that each +observer is furnished with complete experimental evidence in +support of his own assumption. If we suggest to the aviator +%% -----File: 036.png---Folio 26------- +that owing to his high velocity the relative speed of the wave +overtaking him can only be $25,000$ miles a second, he will reply +``I have determined the velocity of the wave relatively to me +by timing it as it passes two points in my conveyance; and it +turns out to be $186,000$ miles a second. So I know my correction +for light-time is right\footnotemark.'' + \footnotetext{We need not stop to prove this directly. If the aviator could detect anything + in his measurements inconsistent with the hypothesis that he was at rest + in the aether (e.g.\ a difference of velocity of overtaking waves of light and + waves meeting him) it would contradict the restricted principle of relativity.}% +His clocks and scales are all behaving +in an extraordinary way from our point of view, so it is not +surprising that he should arrive at a measure of the velocity of +the overtaking wave which differs from ours; but there is no +way of convincing him that our reckoning is preferable. + +Although not a very practical problem, it is of interest to +inquire what happens when the aviator's speed is still further +increased and approximates to the velocity of light. +\index{Light, velocity of!system moving with}% +\index{Time!``standing still''}% +\index{Velocity of light!system moving with}% +Lengths +in the direction of flight become smaller and smaller, until for +the speed of light they shrink to zero. The aviator and the +objects accompanying him shrink to two dimensions. We are +saved the difficulty of imagining how the processes of life can +go on in two dimensions, because nothing goes on. Time is +arrested altogether. This is the description according to the +terrestrial observer. The aviator himself detects nothing unusual; +he does not perceive that he has stopped moving. He is +merely waiting for the next instant to come before making the +next movement; and the mere fact that time is arrested means +that he does not perceive that the next instant is a long time +coming. + +It is a favourite device for bringing home the vast distances +of the stars to imagine a voyage through space with the velocity +of light. The youthful adventurer steps on to his magic carpet +loaded with provisions for a century. He reaches his journey's +end, say Arcturus, a decrepit centenarian. This is wrong. It is +quite true that the journey would last something like a hundred +years by terrestrial chronology; but the adventurer would arrive +at his destination no more aged than when he started, and he +would not have had time to think of eating. So long as he travels +with the speed of light he has immortality and eternal youth. +%% -----File: 037.png---Folio 27------- +If in some way his motion were reversed so that he returned to +the earth again, he would find that centuries had elapsed here, +whilst he himself did not feel a day older---for him the voyage +had lasted only an instant\footnotemark. + \footnotetext{Since the earth is moving relatively to our adventurer with the velocity + of light, we might be tempted to argue that from this point of view the terrestrial + observer would have perpetual youth whilst the voyager grew older. Evidently, + if they met again, they could disprove one or other of the two arguments. But + in order to meet again the velocity of one of them must be reversed by supernatural % [** PP: Hyphenated across a line in original] + means or by an intense gravitational force so that the conditions are + not symmetrical and reciprocity does not apply. The argument given in the + text appears to be the correct one.} + +Our reason for discussing at length the effects of these +improbably high velocities is simply in order that we may speak +of the results in terms of common experience; otherwise it +would be necessary to use the terms of refined technical measurement. +The relativist is sometimes suspected of an inordinate +fondness for paradox; but that is rather a misunderstanding of +his argument. The paradoxes exist when the new experimental +discoveries are woven into the scheme of physics hitherto +current, and the relativist is ready enough to point this out. +But the conclusion he draws is that a revised scheme of physics +is needed in which the new experimental results will find a natural +place without paradox. + +To sum up---on any planet moving with a great velocity +through the aether, extraordinary changes of length of objects +are continually occurring as they move about, and there is a +slowing down of all natural processes as though time were +retarded. These things cannot be perceived by anyone on the +planet; but similar effects would be detected by any observer +having a great velocity relative to the planet (who makes all +allowances for the effect of the motion on the observations, but +takes it %[** PP: Typo ``if''] +for granted that he himself is at rest in the aether\footnotemark). + \footnotetext{The last clause is perhaps unnecessary. The correction applied for light + transmission will naturally be based on the observer's own experimental determination + of the velocity of light. According to experiment the velocity of light + relatively to him is \textit{apparently} the same in all directions, and he will apply + the corrections accordingly. This is equivalent to assuming that he is at rest + in the aether; but he need not, and probably would not, make the assumption + explicitly.}% +There is complete reciprocity so that each of two observers in +relative motion will find the same strange phenomena occurring +%% -----File: 038.png---Folio 28------- +to the other; and there is nothing to help us to decide which is +right.% +\index{Aviator, space and time-reckoning of|)} % [** PP: Using range] + +I think that no one can contemplate these results without +feeling that the whole strangeness must arise from something +perverse and inappropriate in our ordinary point of view. +Changes go on on a planet, all nicely balanced by adjustments +of natural forces, in such a way that no one on the planet can +possibly detect what is taking place. Can we seriously imagine +that there is anything in the reality behind the phenomena, +which reflects these changes? Is it not more probable that we +ourselves introduce the complexity, because our method of +description is not well-adapted to give a simple and natural +statement of what is really occurring? + +The search for a more appropriate apparatus of description +leads us to the standpoint of relativity described in the next +chapter. +\index{Relativity, standpoint of}% +I draw a distinction between the principle and the +standpoint of relativity. The principle of relativity is a statement +of experimental fact, which may be right or wrong; the +first part of it---the restricted principle---has already been +enunciated. Its consequences can be deduced by mathematical +reasoning, as in the case of any other scientific generalization. +It postulates no particular mechanism of nature, \textit{and no particular +view as to the meaning of time and space}, though it may suggest +theories on the subject. The only question is whether it is +experimentally true or not. + +The standpoint of relativity is of a different character. It +asserts first that certain unproved hypotheses as to time and +space have insensibly crept into current physical theories, and +that these are the source of the difficulties described above. +Now the most dangerous hypotheses are those which are tacit +and unconscious. So the standpoint of relativity proposes +tentatively to do without these hypotheses (not making any +others in their place); and it discovers that they are quite +unnecessary and are not supported by any known fact. This in +itself appears to be sufficient justification for the standpoint. +Even if at some future time facts should be discovered which +confirm the rejected hypotheses, the relativist is not wrong in +reserving them until they are required. + +It is not our policy to take shelter in impregnable positions; +%% -----File: 039.png---Folio 29------- +and we shall not hesitate to draw reasonable conclusions as well +as absolutely proved conclusions from the knowledge available. +But to those who think that the relativity theory is a passing +phase of scientific thought, which may be reversed in the light +of future experimental discoveries, we would point out that, +though like other theories it may be developed and corrected, +there is a certain minimum statement possible which represents +irreversible progress. Certain hypotheses enter into all physical +descriptions and theories hitherto current, dating back in some +cases for 2000 years, in other cases for 200 years. It can now +be proved that these hypotheses have nothing to do with any +phenomena yet observed, and do not afford explanations of any +known fact. This is surely a discovery of the greatest importance---quite +apart from any question as to whether the hypotheses +are actually wrong. + +I am not satisfied with the view so often expressed that the +sole aim of scientific theory is ``economy of thought.'' I cannot +reject the hope that theory is by slow stages leading us nearer +to the truth of things. But unless science is to degenerate into +idle guessing, the test of value of any theory must be whether +it expresses with as little redundancy as possible the facts +which it is intended to cover. Accidental truth of a conclusion +is no compensation for erroneous deduction. + +The relativity standpoint is then a discarding of certain +hypotheses, which are uncalled for by any known facts, and +stand in the way of an understanding of the simplicity of nature. +%% -----File: 040.png---Folio 30------- + + +\Chapter{II}{Relativity} + +\Quote{H.~Minkowski (1908).} +{The views of time and space, which I have to set forth, have their foundation +in experimental physics. Therein is their strength. Their tendency is revolutionary. +From henceforth space in itself and time in itself sink to mere shadows, +and only a kind of union of the two preserves an independent existence.}% +\index{Minkowski}% + +\First{There} are two parties to every observation---the observed and +the observer.% +\index{Observer and observed}% + +What we see depends not only on the object looked at, but +on our own circumstances---position, motion, or more personal +idiosyncracies. Sometimes by instinctive habit, sometimes by +design, we attempt to eliminate our own share in the observation, +and so form a general picture of the world outside us, +which shall be common to all observers. A small speck on the +horizon of the sea is interpreted as a giant steamer. From the +window of our railway carriage we see a cow glide past at fifty +miles an hour, and remark that the creature is enjoying a rest. +We see the starry heavens revolve round the earth, but decide +that it is really the earth that is revolving, and so picture the +state of the universe in a way which would be acceptable to an +astronomer on any other planet. + +The first step in throwing our knowledge into a common +stock must be the elimination of the various individual standpoints +and the reduction to some specified standard observer. +The picture of the world so obtained is none the less relative. +We have not eliminated the observer's share; we have only +fixed it definitely. + +To obtain a conception of the world from the point of view +of no one in particular is a much more difficult task. The +position of the observer can be eliminated; we are able to grasp +the conception of a chair as an object in nature---looked at all +round, and not from any particular angle or distance. We can +think of it without mentally assigning ourselves some position +with respect to it. This is a remarkable faculty, which has +evidently been greatly assisted by the perception of solid relief +%% -----File: 041.png---Folio 31------- +with our two eyes. But the motion of the observer is not +eliminated so simply. We had thought that it was accomplished; +but the discovery in the last chapter that observers with +different motions use different space- and time-reckoning shows +that the matter is more complicated than was supposed. It may +well require a complete change in our apparatus of description, +because all the familiar terms of physics refer primarily to the +relations of the world to an observer in some specified circumstances. + +\Pagelabel{31}% +Whether we are able to go still further and obtain a knowledge +of the world, which not merely does not particularise the +observer, but does not postulate an observer at all; whether if +such knowledge could be obtained, it would convey any intelligible +meaning; and whether it could be of any conceivable +interest to anybody if it could be understood---these questions +need not detain us now. The answers are not necessarily +negative, but they lie outside the normal scope of physics. + +The circumstances of an observer which affect his observations +are his position, motion and gauge of magnitude. More personal +idiosyncracies disappear if, instead of relying on his crude +senses, he employs scientific measuring apparatus. But scientific +apparatus has position, motion and size, so that these are still +involved in the results of any observation. There is no essential +distinction between scientific measures and the measures of the +senses. In either case our acquaintance with the external world +comes to us through material channels; the observer's body can +be regarded as part of his laboratory equipment, and, so far as +we know, it obeys the same laws. We therefore group together +perceptions and scientific measures, and in speaking of ``a +particular observer'' we include all his measuring appliances.% +\index{Perceptions, as crude measures}% + +Position, motion, magnitude-scale---these factors have a profound +influence on the aspect of the world to us. Can we form +a picture of the world which shall be a synthesis of what is seen +by observers in all sorts of positions, having all sorts of velocities, +and all sorts of sizes? %[** PP: Changed . to ?] +\index{Synthesis of appearances}% +As already stated we have accomplished +the synthesis of positions. We have two eyes, which have +dinned into our minds from babyhood that the world has to be +looked at from more than one position. Our brains have so far +responded as to give us the idea of solid relief, which enables us +%% -----File: 042.png---Folio 32------- +to appreciate the three-dimensional world in a vivid way that +would be scarcely possible if we were only acquainted with +strictly two-dimensional pictures. We not merely deduce the +three-dimensional world; we see it. But we have no such aid +in synthesising different motions. Perhaps if we had been +endowed with two eyes moving with different velocities our +brains would have developed the necessary faculty; we should +have perceived a kind of relief in a fourth dimension so as to +combine into one picture the aspect of things seen with different +motions. Finally, if we had had two eyes of different sizes, we +might have evolved a faculty for combining the points of view +of the mammoth and the microbe. + +It will be seen that we are not fully equipped by our senses +for forming an impersonal picture of the world. And it is +because the deficiency is manifest that we do not hesitate to +advocate a conception of the world which transcends the images +familiar to the senses. Such a world can perhaps be grasped, +but not pictured by the brain. It would be unreasonable to +limit our thought of nature to what can be comprised in sense-pictures. +As Lodge has said, our senses were developed by the +struggle for existence, not for the purpose of philosophising on +the world.% +\index{Lodge}% + +Let us compare two well-known books, which might be +described as elementary treatises on relativity, \textit{Alice in Wonderland} +and \textit{Gulliver's Travels}. Alice was continually changing size, +sometimes growing, sometimes on the point of vanishing altogether. +Gulliver remained the same size, but on one occasion +he encountered a race of men of minute size with everything in +proportion, and on another voyage a land where everything was +gigantic. It does not require much reflection to see that both +authors are describing the same phenomenon---a relative change +of scale of observer and observed. Lewis Carroll took what is +probably the ordinary scientific view, that the observer had +changed, rather than that a simultaneous change had occurred +to all her surroundings. But it would never have appeared like +that to Alice; she could not have ``stepped outside and looked +at herself,'' picturing herself as a giant filling the room. She +would have said that the room had unaccountably shrunk. +Dean Swift took the truer view of the human mind when he +%% -----File: 043.png---Folio 33------- +made Gulliver attribute his own changes to the things around +him; it never occurred to Gulliver that his own size had altered; +and, if he had thought of the explanation, he could scarcely +have accustomed himself to that way of thinking. But both +points of view are legitimate. The size of a thing can only be +imagined as relative to something else; and there is no means of +assigning the change to one end of the relation rather than the +other.% +\index{Relativity of size}% + +We have seen in the theory of the Michelson-Morley experiment +that, according to current physical views, our standard of +size---the rigid meas\-uring-rod---must change according to the +circumstances of its motion; and the aviator's adventures +illustrated a similar change in the standard of duration of time. +Certain rather puzzling irregularities have been discovered in +the apparent motions of the Sun, Mercury, Venus and the Moon; +but there is a strong family resemblance between these, which +leads us to believe that the real phenomenon is a failure of the +time-keeping of our standard clock, the Earth. Instances could +be multiplied where a change of the observer or his standards +produces or conceals changes in the world around him. + +The object of the relativity theory, however, is not to attempt +the hopeless task of apportioning responsibility between the +observer and the external world, but to emphasise that in our +ordinary description and in our scientific description of natural +phenomena the two factors are indissolubly united. All the +familiar terms of physics---length, duration of time, motion, +force, mass, energy, and so on---refer primarily to this relative +knowledge of the world; and it remains to be seen whether any +of them can be retained in a description of the world which is +not relative to a particular observer. + +Our first task is a description of the world independent of +the motion of the observer. The question of the elimination of +his gauge of magnitude belongs to a later development of the +theory discussed in \Chapref{XI}. +\index{Gauge!effect on observations}% +Let us draw a square $ABCD$ on +a sheet of paper, making the sides equal, to the best of our +knowledge. We have seen that an aviator flying at $161,000$ +miles a second in the direction $AB$, would judge that the sides +$AB$, $DC$ had contracted to half their length, so that for him +the figure would be an oblong. If it were turned through a right +%% -----File: 044.png---Folio 34------- +angle $AB$ and $DC$ would expand and the other two sides contract---in +his judgment. For us, the lengths of $AB$ and $AC$ are +equal; for him, one length is twice the other. Clearly length +cannot be a property inherent in our drawing; it needs the +specification of some observer. + +We have seen further that duration of time also requires that +an observer should be specified. The stationary observer and +the aviator disagreed as to whose cigar lasted the longer time. + +Thus \textit{length} and \textit{duration} are not things inherent in the +external world; they are relations of things in the external +world to some specified observer. +\index{Duration, not inherent in external world}% +\index{Length!relativity of}% +\index{Relativity!of length and duration}% +\index{Space!relativity of}% +If we grasp this all the mystery +disappears from the phenomena described in \Chapref{I}. When +the rod in the Michelson-Morley experiment is turned through +a right angle it contracts; that naturally gives the impression +that something has happened to the rod itself. Nothing whatever +has happened to the rod---the object in the external world. +Its length has altered, but length is not an intrinsic property of +the rod, since it is quite indeterminate until some observer is +specified. Turning the rod through a right angle has altered the +relation to the observer (implied in the discussion of the experiment); +but the rod itself, or the relation of a molecule at one +end to a molecule at the other, is unchanged. Measurement of +length and duration is a comparison with partitions of space +and time drawn by the observer concerned, with the help of +apparatus which shares his motion. Nature is not concerned +with these partitions; it has, as we shall see later, a geometry +of its own which is of a different type. + +Current physics has hitherto assumed that all observers are +not to be regarded as on the same footing, and that there is +some absolute observer whose judgments of length and duration +are to be treated with respect, because nature pays attention to +\textit{his} space-time partitions. He is supposed to be at rest in the +aether, and the aether materialises his space-partitions so that +they have a real significance in the external world. This is +sheer hypothesis, and we shall find it is unsupported by any +facts. Evidently our proper course is to pursue our investigations, +and call in this hypothetical observer only if we find there +is something which he can help to explain. + +We have been leading up from the older physics to the new +%% -----File: 045.png---Folio 35------- +outlook of relativity, and the reader may feel some doubt as to +whether the strange phenomena of contraction and time-retardation, +that were described in the last chapter, are to be +taken seriously, or are part of a \textit{reductio ad absurdum} argument. +The answer is that we believe that the phenomena do occur as +described; only the description (like that of all observed phenomena) +concerns the relations of the external world to some +observer, and not the external world itself. The startling +character of the phenomena arises from the natural but fallacious +inference that they involve intrinsic changes in the objects +themselves. + +We have been considering chiefly the observer's end of the +observation; we must now turn to the other end---the thing +observed. Although length and duration have no exact counterparts +in the external world, it is clear that there is a certain +ordering of things and events outside us which we must now +find more appropriate terms to describe. The order of events is +a four-fold order; we can arrange them as right-and-left, backwards-and-forwards, +up-and-down, sooner-and-later. +\index{Ordering of events in external world}% +An individual +may at first consider these as four independent orders, +but he will soon attempt to combine some of them. It is +recognised at once that there is no essential distinction between +right-and-left and backwards-and-forwards. The observer has +merely to turn through a right angle and the two are interchanged. +If he turns through a smaller angle, he has first to +combine them, and then to redivide them in a different way. +Clearly it would be a nuisance to continually combine and redivide; +so we get accustomed to the thought of leaving them +combined in a two-fold or two-dimensional order. The amalgamation +of up-and-down is less simple. There are obvious reasons +for considering this dimension of the world as fundamentally +distinct from the other two. Yet it would have been a great +stumbling-block to science if the mind had refused to combine +space into a three-dimensional whole. The combination has not +concealed the real distinction of horizontal and vertical, but has +enabled us to understand more clearly its nature---for what +phenomena it is relevant, and for what irrelevant. We can +understand how an observer in another country redivides the +combination into a different vertical and horizontal. We must +%% -----File: 046.png---Folio 36------- +now go further and amalgamate the fourth order, sooner-and-later. +This is still harder for the mind. It does not imply that +there is no distinction between space and time; but it gives a +fresh unbiassed start by which to determine what the nature of +the distinction is. + +The idea of putting together space and time, so that time is +regarded as a fourth dimension, is not new. But until recently +it was regarded as merely a picturesque way of looking at things +without any deep significance. We can put together time and +temperature in a thermometer chart, or pressure and volume +on an indicator-diagram. It is quite non-committal. But our +theory is going to lead much further than that. We can lay +two dimensional surfaces---sheets of paper---on one another till +we build up a three-dimensional block; but there is a difference +between a block which is a pile of sheets and a solid block of +paper. The solid block is the true analogy for the four-dimensional +combination of space-time; it does not separate naturally +into a particular set of three-dimensional spaces piled in time-order. +It can be redivided into such a pile; \textit{but it can be redivided +in any direction we please}. + +Just as the observer by changing his orientation makes a new +division of the two-dimensional plane into right-and-left, backwards-and-forwards---just +as the observer by changing his +longitude makes a new division of three-dimensional space into +vertical and horizontal---so the observer by \textit{changing his motion} +makes a new division of the four-dimensional order into time +and space.% +\index{Four-dimensional order}% + +This will be justified in detail later; it indicates that observers +with different motions will have different time and space-reckoning---a +conclusion we have already reached from another +point of view. + +Although different observers separate the four orders differently, +they all agree that the order of events is four-fold; and +it appears that this undivided four-fold order is the same for +all observers. We therefore believe that it is inherent in the +external world; it is in fact the synthesis, which we have been +seeking, of the appearances seen by observers having all sorts of +positions and all sorts of (uniform) motions. It is therefore to +be regarded as a conception of the real world not relative to any +particularly circumstanced observer. +%% -----File: 047.png---Folio 37------- + +The term ``real world'' is used in the ordinary sense of physics, +without any intention of prejudging philosophical questions as +to reality. +\index{Real world of physics}% +It has the same degree of reality as was formerly +attributed to the three-dimensional world of scientific theory or +everyday conception, which by the advance of knowledge it +replaces. As I have already indicated, it is merely the accident +that we are not furnished with a pair of eyes in rapid relative +motion, which has allowed our brains to neglect to develop a +faculty for visualising this four-dimensional world as directly +as we visualise its three-dimensional section. + +It is now easy to see that length and duration must be the +components of a single entity in the four-dimensional world of +space-time. Just as we resolve a structure into plan and elevation, +so we resolve extension in the four-dimensional world into +length and duration. The structure has a size and shape +independent of our choice of vertical. Similarly with things in +space-time. Whereas length and duration are relative, the +single ``extension'' of which they are components has an absolute +significance in nature, independent of the particular decomposition +into space and time separately adopted by the observer.% +\index{Extension in four dimensions}% + +Consider two events; for example, the stroke of one o'clock +and the stroke of two o'clock by Big Ben. These occupy two +points in space-time, and there is a definite separation between +them. An observer at Westminster considers that they occur at +the same place, and that they are separated by an hour in time; +thus he resolves their four-dimensional separation into zero +distance in space and one hour distance in time. An observer +on the sun considers that they do not occur at the same place; +they are separated by about $70,000$ miles, that being the distance +travelled by the earth in its orbital motion with respect to the +sun. It is clear that he is not resolving in quite the same directions +as the terrestrial observer, since he finds the space-component +to be $70,000$ miles instead of zero. But if he alters one +component he must necessarily alter the other; so he will make +the time-component differ slightly from an hour. By analogy +with resolution into components in three-dimensions, we should +expect him to make it less than an hour---having, as it were, +borrowed from time to make space; but as a matter of fact he +makes it longer. This is because space-time has a different +%% -----File: 048.png---Folio 38------- +geometry, which will be described later. Our present point is +that there is but one separation of two events in four dimensions, +which can be resolved in any number of ways into the components +length and duration. + +We see further how motion must be purely relative. +\index{Relativity!of motion}% +Take +two events $A$ and~$B$ in the history of one particle. We can choose +any direction as the time-direction; let us choose it along~$AB$. +Then $A$ and~$B$ are separated only in time and not in space, so +the particle is at rest. If we choose a slightly inclined time-direction, +the separation $AB$ will have a component in space; +the two events then do not occur at the same place, that is to +say, the particle has moved. The negation of absolute motion +is thus associated with the possibility of choosing the time-direction +in any way we please. What determines the separation +of space and time for any particular observer can now be seen. +Let the observer place himself so that he is, to the best of his +knowledge, at rest. If he is a normal human being, he will +seat himself in an arm-chair; if he is an astronomer, he will +place himself on the sun or at the centre of the stellar universe. +Then all the events happening directly to him will in his opinion +occur at the same place. Their separation will have no space-component, +and they will accordingly be ranged solely in the +time-direction. This chain of events, marking his track through +the four-dimensional world, will be his time-direction. Each +observer bases his separation of space and time on his own track +through the world.% +\index{Time!depends on observer's track}% + +Since any separation of space and time is admissible, it is +possible for the astronomer to base his space and time on the +track of a solar observer instead of that of a terrestrial observer; +but it must be remembered that in practice the space and time +of the solar observer have to be inferred indirectly from those +of the terrestrial observer; and, if the corrections are made +according to the crude methods hitherto employed, they may +be inferred wrongly (if extreme accuracy is needed). + +The most formidable objection to this relativist view of the +world is the aether difficulty. We have seen that uniform motion +through the aether cannot be detected by experiment, and +therefore it is entirely in accordance with experiment that such +motion should have no counterpart in the four-dimensional +%% -----File: 049.png---Folio 39------- +world. Nevertheless, it would almost seem that such motion +must logically exist, if the aether exists; +\index{Aether!non-material nature of}% +and, even at the +expense of formal simplicity, it ought to be exhibited in any +theory which pretends to give a complete account of what is +going on in nature. If a substantial aether analogous to a +material ocean exists, it must rigidify, as it were, a definite +space; and whether the observer or whether nature pays any +attention to that space or not, a fundamental separation of +space and time must be there. Some would cut the knot by +denying the aether altogether. We do not consider that desirable, +or, so far as we can see, possible; but we do deny that the aether +need have such properties as to separate space and time in the +way supposed. It seems an abuse of language to speak of a +division existing, when nothing has ever been found to pay any +attention to the division. + +Mathematicians of the nineteenth century devoted much time +to theories of elastic solid and other material aethers. Waves of +light were supposed to be actual oscillations of this substance; +it was thought to have the familiar properties of rigidity and +density; it was sometimes even assigned a place in the table of +the elements. The real death-blow to this materialistic conception +of the aether was given when attempts were made to explain +matter as some state in the aether. For if matter is vortex-motion +or beknottedness in aether, the aether cannot be matter--some +state in itself. If any property of matter comes to be +regarded as a thing to be explained by a theory of its structure, +clearly that property need not be attributed to the aether. +If physics evolves a theory of matter which explains some +property, it stultifies itself when it postulates that the same +property exists unexplained in the primitive basis of matter. + +Moreover the aether has ceased to take any very active part +in physical theory and has, as it were, gone into reserve. A +modern writer on electromagnetic theory will generally start +with the postulate of an aether pervading all space; he will then +explain that at any point in it there is an electromagnetic vector +whose intensity can be measured; henceforth his sole dealings +are with this vector, and probably nothing more will be heard +of the aether itself. In a vague way it is supposed that this +vector represents some condition of the aether, and we need not +%% -----File: 050.png---Folio 40------- +dispute that without some such background the vector would +scarcely be intelligible---but the aether is now only a background +and not an active participant in the theory. + +There is accordingly no reason to transfer to this vague background +of aether the properties of a material ocean. Its properties +must be determined by experiment, not by analogy. In particular +there is no reason to suppose that it can partition out space in +a definite way, as a material ocean would do. We have seen in +the Prologue that natural geometry depends on laws of matter; +therefore it need not apply to the aether. Permanent identity +of particles is a property of matter, which Lord Kelvin sought +to explain in his vortex-ring hypothesis. +\index{Identity, permanent} % [** PP: Added comma] +\index{Permanent identity}% +This abandoned +hypothesis at least teaches us that permanence should not be +regarded as axiomatic, but may be the result of elaborate constitution. +There need not be anything corresponding to +permanent identity in the constituent portions of the aether; +we cannot lay our finger at one spot and say ``this piece of +aether was a few seconds ago over there.'' Without any continuity +of identity of the aether motion through the aether +becomes meaningless; and it seems likely that this is the true +reason why no experiment ever reveals it. + +This modern theory of the relativity of all uniform motion is +essentially a return to the original Newtonian view, temporarily +disturbed by the introduction of aether problems; for in Newton's +dynamics uniform motion of the whole system has not---and no +one would expect it to have---any effect. But there are considerable +difficulties in the limitation to uniform motion. Newton +himself seems to have appreciated the difficulty; but the experimental +evidence appeared to him to be against any extension +of the principle. Accordingly Newton's laws of mechanics are +not of the general type in which it is unnecessary to particularise +the observer; they hold only for observers with a special kind +of motion which is described as ``unaccelerated.'' +\index{Newton!relativity for uniform motion}% +\index{Relativity, Newtonian}% +The only +definition of this epithet that can be given is that an ``unaccelerated'' +observer is one for whom Newton's laws of motion +hold. On this theory, the phenomena are not indifferent to an +acceleration or non-uniform motion of the whole system. Yet +an absolute non-uniform motion through space is just as impossible +to imagine as an absolute uniform motion. The partial +%% -----File: 051.png---Folio 41------- +relativity of phenomena makes the difficulty all the greater. +If we deny a fundamental medium with continuous identity of +its parts, motion uniform or non-uniform should have no +significance; if we admit such a medium, motion uniform or +non-uniform should be detectable; but it is much more difficult +to devise a plan of the world according to which uniform motion +has no significance and non-uniform motion is significant.\Pagelabel{41} + +It is through experiment that we have been led back to the +principle of relativity for uniform motion. In seeking some kind +of extension of this principle to accelerated motion, we are led +by the feeling that, having got so far, it is difficult and arbitrary +to stop at this point. We now try to conceive a system of nature +for which all kinds of motion of the observer are indifferent. +It will be a completion of our synthesis of what is perceived by +observers having all kinds of motions with respect to one +another, removing the restriction to uniform motion. The +experimental tests must follow after the consequences of this +generalisation have been deduced. + +The task of formulating such a theory long appeared impossible. +It was pointed out by Newton that, whereas there is no criterion +for detecting whether a body is at rest or in uniform motion, it +is easy to detect whether it is in rotation. For example the +bulge of the earth's equator is a sign that the earth is rotating, +since a plastic body at rest would be spherical.% +\index{Newton!absolute rotation}% + +This problem of rotation affords a hint as to the cause of the +incomplete relativity of Newtonian mechanics. The laws of +motion are formulated with respect to an unaccelerated observer, +and do not apply to a frame of reference rotating with the earth. +Yet mathematicians frequently do use a rotating frame. Some +modification of the laws is then necessary; and the modification +is made by introducing a centrifugal force---not regarded as a +real force like gravitation, but as a mathematical fiction employed +to correct for the improper choice of a frame of reference. +\index{Centrifugal Force!compared with gravitation}% +The bulge of the earth's equator may be attributed indifferently +to the earth's rotation or to the outward pull of the centrifugal +force introduced when the earth is regarded as non-rotating. + +Now it is generally assumed that the centrifugal force is +something \textit{sui generis}, which could always be distinguished +experimentally from any other natural phenomenon. If then +%% -----File: 052.png---Folio 42------- +on choosing a frame of reference we find that a centrifugal force +is detected, we can at once infer that the frame of reference is +a ``wrong'' one; rotating and non-rotating frames can be distinguished +by experiment, and rotation is thus strictly absolute. +But this assumes that the observed effects of centrifugal force +cannot be produced in any other way than by rotation of the +observer's frame of reference. If once it is admitted that centrifugal +force may not be completely distinguishable by experiment +from another kind of force---gravitation---perceived even by +Newton's unaccelerated observer, the argument ceases to apply. +We can never determine exactly how much of the observed field +of force is centrifugal force and how much is gravitation; and +we cannot find experimentally any definite standard that is to +be considered absolutely non-rotating. + +The question then, whether there exists a distinction between +``right'' frames of reference and ``wrong'' frames, turns on +whether the use of a ``wrong'' frame produces effects experimentally +distinguishable from any natural effects which can be +perceived when a ``right'' frame is used. +\index{Frames@Frames of reference, ``right'' and ``wrong''}% +If there is no such +difference, all frames may be regarded as on the same footing +and equally right. In that case we can have a complete relativity +of natural phenomena. Since the effect of departing from +Newton's standard frame is the introduction of a field of force, +this generalised relativity theory must be largely occupied with +the nature of fields of force. + +The precise meaning of the statement that all frames of +reference are on the same footing is rather difficult to grasp. +We believe that there are absolute things in the world---not only +matter, but certain characteristics in empty space or aether. +In the atmosphere a frame of reference which moves with the +air is differentiated from other frames moving in a different +manner; this is because, besides discharging the normal functions +of a frame of reference, the air-frame embodies certain of the +absolute properties of the matter existing in the region. +Similarly, if in empty space we choose a frame of reference +which more or less follows the lines of the absolute structure in +the region, the frame will usurp some of the absolute qualities +of that structure. What we mean by the equivalence of all +frames is that they are not differentiated by any qualities +%% -----File: 053.png---Folio 43------- +formerly supposed to be intrinsic in the frames themselves---rest, +rectangularity, acceleration---independent of the absolute +structure of the world that is referred to them. Accordingly the +objection to attributing absolute properties to Newton's frame +of reference is not that it is impossible for a frame of reference +to acquire absolute properties, but that the Newtonian frame +has been laid down on the basis of relative knowledge without +any attempt to follow the lines of absolute structure. + +Force, as known to us observationally, is like the other +quantities of physics, a relation. The force, measured with a +spring-balance, for example, depends on the acceleration of the +observer holding the balance; and the term may, like length +and duration, have no exact counterpart in a description of +nature independent of the observer. Newton's view assumes +that there is such a counterpart, an active cause in nature +which is identical with the force perceived by his standard +unaccelerated observer. Although any other observer perceives +this force with additions of his own, it is implied that the +original force in nature and the observer's additions can in some +way be separated without ambiguity. There is no experimental +foundation for this separation, and the relativity view is that +a field of force can, like length and duration, be nothing but +a link between nature and the observer. There is, of course, +something at the far end of the link, just as we found an +extension in four dimensions at the far end of the relations +denoted by length and duration. We shall have to study the +nature of this unknown whose relation to us appears as force. +Meanwhile we shall realise that the alteration of perception of +force by non-uniform motion of the observer, as well as the +alteration of the perception of length by his uniform motion, is +what might be expected from the nature of these quantities as +relations solely.% +\index{Force!relativity of}% +\index{Relativity of Force}% + +We proceed now to a more detailed study of the four-dimensional +world, of the things which occur in it, and of the laws by +which they are regulated. It is necessary to dive into this +absolute world to seek the truth about nature; but the physicist's +object is always to obtain knowledge which can be applied to +the relative and familiar aspect of the world. The absolute +world is of so different a nature, that the relative world, with +%% -----File: 054.png---Folio 44------- +which we are acquainted, seems almost like a dream. But if +indeed we are dreaming, our concern is with the baseless fabric +of our vision. We do not suggest that physicists ought to +translate their results into terms of four-dimensional space for +the empty satisfaction of working in the realm of reality. It is +rather the opposite. They explore the new field and bring back +their spoils---a few simple generalisations---to apply them to the +practical world of three-dimensions. Some guiding light will be +given to the attempts to build a scheme of things entire. For +the rest, physics will continue undisturbedly to explore the +relative world, and to employ the terms applicable to relative +knowledge, but with a fuller appreciation of its relativity. +%% -----File: 055.png---Folio 45------- + + +\Chapter{III}{The World of Four Dimensions} + +\Quote{H.~G. Wells, \textit{The Time Machine.}} +{Here is a portrait of a man at eight years old, another at fifteen, another at +seventeen, another at twenty-three, and so on. All these are evidently sections, +as it were, Three-Dimensional representations of his Four-Dimensional being, +which is a fixed and unalterable thing.}% +\index{Four-dimensional space-time!geometry of}% + +\First{The} distinction between horizontal and vertical is not an +illusion; and the man who thinks it can be disregarded is likely +to come to an untimely end. Yet we cannot arrive at a comprehensive +view of nature unless we combine horizontal and +vertical dimensions into a three-dimensional space. By doing +this we obtain a better idea of what the distinction of horizontal +and vertical really is in those cases where it is relevant, e.g.\ the +phenomena of motion of a projectile. We recognise also that +vertical is not a universally differentiated direction in space, as +the flat-earth philosophers might have imagined. + +Similarly by combining the time-ordering and space-ordering +of the events of nature into a single order of four dimensions, +we shall not only obtain greater simplicity for the phenomena +in which the separation of time and space is irrelevant, but we +shall understand better the nature of the differentiation when it +is relevant. + +A point in this space-time, that is to say a given instant at +a given place, is called an ``event.'' +\index{Event, definition of}% +\index{Point-event}% +\index{Space-time}% +An event in its customary +meaning would be the physical happening which occurs at and +identifies a particular place and time. However, we shall use +the word in both senses, because it is scarcely possible to think +of a point in space-time without imagining some identifying +occurrence. + +In the ordinary geometry of two or three dimensions, the +distance between two points is something which can be measured, +usually with a rigid scale; it is supposed to be the same for all +observers, and there is no need to specify horizontal and vertical +directions or a particular system of coordinates. In four-dimensional +space-time there is likewise a certain extension or +%% -----File: 056.png---Folio 46------- +generalised distance between two events, of which the distance +in space and the separation in time are particular components. +This extension in space and time combined is called the +``interval'' between the two events; +\index{Extension in four dimensions}% +\index{Interval}% +it is the same for all +observers, however they resolve it into space and time separately. +We may think of the interval as something intrinsic in external +nature---an absolute relation of the two events, which postulates +no particular observer. Its practical measurement is suggested +by analogy with the distance of two points in space. + +%[Illustration: \textsc{Fig}. 2.] +\begin{wrapfigure}{l}{1.75in} +\Graphic[2]{1.75in}{056a} +\end{wrapfigure} +In two dimensions on a plane, two points $P_1$, $P_2$ (\Figref{2}) can +be specified by their rectangular +coordinates $(x_1, y_1)$ and $(x_2, y_2)$, +when arbitrary axes have been selected. +In the figure, $OX_1 = x_1$, +$OY_1 = y_1$, etc. We have +\begin{align*} +P_1 P_2{}^2 + &= P_1 M^2 + M P_2{}^2 \\ + &= X_1 X_2{}^2 + Y_1 Y_2{}^2 \\ + &= (x_2 - x_1)^2 + (y_2 - y_1)^2, +\end{align*} +so that if $s$ is the distance between +$P_1$ and~$P_2$ +\[ +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. +\] + +The extension to three dimensions is, as we should expect, +\[ +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2. +\] +Introducing the times of the events $t_1$, $t_2$, we should naturally +expect that the interval in the four-dimensional world would +be given by +\[ +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (t_2 - t_1)^2. +\] + +An important point arises here. It was, of course, assumed +that the same scale was used for measuring $x$ and $y$ and~$z$. But +how are we to use the same scale for measuring~$t$? We cannot +use a scale at all; some kind of clock is needed. The most +natural connection between the measure of time and length is +given by the fact that light travels $300,000$ kilometres in +$1$~second. For the four-dimensional world we shall accordingly +regard $1$~second as the equivalent of $300,000$ kilometres, and +measure lengths and times in seconds or kilometres indiscriminately; +in other words we make the velocity of light the unit of +%% -----File: 057.png---Folio 47------- +velocity. It is not essential to do this, but it greatly simplifies +the discussion. + +Secondly, the formulae here given for $s^2$ are the characteristic +formulae of Euclidean geometry. So far as three-dimensional +space is concerned the applicability of Euclidean geometry is +very closely confirmed by experiment. But space-time is not +Euclidean; it does, however, conform (at least approximately) +to a very simple modification of Euclidean geometry indicated +by the corrected formula +\index{Euclidean geometry}% +\IndexExtra{Geometry!Euclidean}% +\[ +s^2 = (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 - (t_2-t_1)^2. +\] +There is only a sign altered; but that minus sign is the secret +of the differences of the manifestations of time and space in +nature. + +This change of sign is often found puzzling at the start. We +could not define~$s$ by the expression originally proposed (with +the positive sign), because the expression does not define anything +objective. Using the space and time of one observer, one +value is obtained; for another observer, another value is +obtained. But if $s$ is defined by the expression now given, it is +found that the same result is obtained by all observers\footnote% + {Appendix, \Noteref{2}.}. +\Pagelabel{note2}% +The +quantity~$s$ is thus something which concerns solely the two +events chosen; we give it a name---the interval between the two +events. In ordinary space the distance between two points is +the corresponding property, which concerns only the two points +and not the extraneous coordinate system of location which is +used. Hence interval, as here defined, is the analogue of distance; +and the analogy is strengthened by the evident resemblance +of the formula for~$s$ in both cases. Moreover, when the difference +of time vanishes, the interval reduces to the distance. But the +discrepancy of sign introduces certain important differences. +These differences are summed up in the statement that the +geometry of space is Euclidean, but the geometry of space-time +is semi-Euclidean or ``hyperbolic.'' +\index{Geometry!hyperbolic}% +\index{Geometry!semi-Euclidean}% +\index{Hyperbolic geometry}% +The association of a geometry +with any continuum always implies the existence of some +uniquely measurable quantity like interval or distance; in +ordinary space, geometry without the idea of distance would be +meaningless. + +%% -----File: 058.png---Folio 48------- + +For the moment the difficulty of thinking in terms of an +unfamiliar geometry may be evaded by a dodge. Instead of +real time~$t$, consider imaginary time~$\tau$; that is to say, let +\index{Imaginary time}% +\index{Time!imaginary}% +\begin{DPgather*} +t = \tau \sqrt{-1}. \\ +\lintertext{Then} +(t_2 - t_1)^2 = -(\tau_2 - \tau_1)^2, \\ +\intertext{so that} +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (\tau_2 - \tau_1)^2. +\end{DPgather*} +Everything is now symmetrical and there is no distinction +between~$\tau$ and the other variables. The continuum formed of +space and imaginary time is completely isotropic for all measurements; +no direction can be picked out in it as fundamentally +distinct from any other. + +The observer's separation of this continuum into space and +time consists in slicing it in some direction, viz.\ that perpendicular +to the path along which he is himself travelling. The +section gives three-dimensional space at some moment, and the +perpendicular dimension is (imaginary) time. Clearly the slice +may be taken in any direction; there is no question of a true +separation and a fictitious separation. There is no conspiracy +of the forces of nature to conceal our absolute motion---because, +looked at from this broader point of view, there is nothing to +conceal. The observer is at liberty to orient his rectangular axes +of $x$, $y$, $z$ and~$\tau$ arbitrarily, just as in three-dimensions he can +orient his axes of $x$, $y$, $z$ arbitrarily. + +It can be shown that the different space and time used by +the aviator in \Chapref{I} correspond to an orientation of the +time-axis along his own course in the four-dimensional world, +whereas the ordinary time and space are given when the time-axis +is oriented along the course of a terrestrial observer. The +FitzGerald contraction and the change of time-measurement +are given exactly by the usual formulae for rotation of rectangular +axes\footnote{Appendix, \Noteref{3}.}. +\Pagelabel{note3} + +It is not very profitable to speculate on the implication of the +mysterious factor~$\sqrt{-1}$, which seems to have the property of +turning time into space. It can scarcely be regarded as more +than an analytical device. To follow out the theory of the four-% +%% -----File: 059.png---Folio 49------- +dimensional world in more detail, it is necessary to return to +real time, and face the difficulties of a strange geometry. + +Consider a particular observer, $S$, and represent time according +to his reckoning by distance up the page parallel to~$OT$. One +dimension of his space will be represented by horizontal distance +parallel to~$OX$; another will stand out at right angles from the +page; and the reader must imagine the third as best he can. +Fortunately it will be sufficient for us to consider only the one +dimension of space $OX$ and deal with the phenomena of ``line-land,'' +i.e.\ we limit ourselves to motion to and fro in one straight +line in space. +\begin{figure*}[hbt] +\begin{center} +\Graphic[3]{3.5in}{059a} +\end{center} +\end{figure*} +%[Illustration: \textsc{Fig}. 3.] + +The two lines $U'OU$, $V'OV$, at $45°$ to the axes, represent the +tracks of points which progress $1$~unit horizontally (in space) +for $1$~unit vertically (in time); thus they represent points moving +with unit velocity. We have chosen the velocity of light as unit +velocity; hence $U'OU$, $V'OV$ will be the tracks of pulses of light +in opposite directions along the straight line. + +Any event $P$ within the sector $UOV$ is indubitably after the +event~$O$, whatever system of time-reckoning is adopted. For it +would be possible for a material particle to travel from~$O$ to~$P$, +the necessary velocity being less than that of light; and no +%% -----File: 060.png---Folio 50------- +rational observer would venture to state that the particle had +completed its journey before it had begun it. It would, in fact, +be possible for an observer travelling along $NP$ to receive a +light-signal or wireless telegram announcing the event~$O$, just +as he reached~$N$, since $ON$ is the track of such a message; and +then after the time $NP$ he would have direct experience of the +event~$P$. To have actual evidence of the occurrence of one +event before experiencing the second is a clear proof of their +absolute order in nature, which should convince not merely +the observer concerned but any other observer with whom he +can communicate. + +% [** PP: 8.75pt overfull at 5in] +Similarly events in the sector $U'OV'$ are indubitably before +the event~$O$. + +With regard to an event~$P'$ in the sector $UOV'$ or~$VOU'$ we +cannot assert that it is absolutely before or after~$O$. According +to the time-reckoning of our chosen observer~$S$, $P'$ is after~$O$, +because it lies above the line~$OX$; but there is nothing absolute +about this. The track~$OP'$ corresponds to a velocity greater +than that of light, so that we know of no particle or physical +impulse which could follow the track. An observer experiencing +the event $P'$ could not get news of the event~$O$ by any known +means until after~$P'$ had happened. The order of the two events +can therefore only be inferred by estimating the delay of the +message and this estimate will depend on the observer's mode +of reckoning space and time. + +% [** PP: 4.3pt overfull at 5in] +Space-time is thus divided into three zones with respect to +the event~$O$. $U'OV'$ belongs to the indubitable past. $UOV$ is +the indubitable future. $UOV'$ and $VOU'$ are (absolutely) neither +past nor future, but simply ``elsewhere.'' +\index{Elsewhere@``Elsewhere''}% +It may be remarked +that, as we have no means of identifying points in space as ``the +same point,'' and as the events $O$ and~$P$ might quite well happen +to the same particle of matter, the events are not necessarily to +be regarded as in different places, though the observer~$S$ will +judge them so; but the events $O$ and~$P'$ cannot happen to the +same particle, and no observer could regard them as happening +at the same place. The main interest of this analysis is that it +shows that the arbitrariness of time-direction is not inconsistent +with the existence of regions of absolute past and future.% +\index{Absolute past and future}% +\index{Future, absolute}% +\index{Past, absolute}% + +Although there is an absolute past and future, there is between +%% -----File: 061.png---Folio 51------- +them an extended neutral zone; and simultaneity of events at +different places has no absolute meaning. +\index{Absolute simultaneity}% +\index{Simultaneity}% +For our selected +observer all events along~$OX$ are simultaneous with one another; +for another observer the line of events simultaneous with~$O$ +would lie in a different direction. The denial of absolute +simultaneity is a natural complement to the denial of absolute +motion. The latter asserts that we cannot find out what is the +same place at two different times; the former that we cannot +find out what is the same time at two different places. It is +curious that the philosophical denial of absolute motion is +readily accepted, whilst the denial of absolute simultaneity +appears to many people revolutionary. + +The division into past and future (a feature of time-order +which has no analogy in space-order) is closely associated with +our ideas of causation and free will. % [** PP: Removed hyphen] +\index{Causation and free will}% +\index{Free will}% [** PP: Added space] +\index{Time!past and future}% +In a perfectly determinate +scheme the past and future may be regarded as lying mapped +out---as much available to present exploration as the distant +parts of space. Events do not happen; they are just there, and +we come across them. ``The formality of taking place'' is merely +the indication that the observer has on his voyage of exploration +passed into the absolute future of the event in question; and it +has no important significance. We can be aware of an eclipse +in the year 1999, very much as we are aware of an unseen +companion to Algol. Our knowledge of things \textit{where} we are not, +and of things \textit{when} we are not, is essentially the same---an +inference (sometimes a mistaken inference) from brain impressions, +including memory, \textit{here} and \textit{now}. + +So, if events are determinate, there is nothing to prevent a +person from being \textit{aware} of an event before it happens; and an +event may cause other events previous to it. Thus the eclipse +of the Sun in May 1919 caused observers to embark in March. +It may be said that it was not the eclipse, but the calculations +of the eclipse, which caused the embarkation; but I do not +think any such distinction is possible, having regard to the +indirect character of our acquaintance with all events except +those at the precise point of space where we stand. A detached +observer contemplating our world would see some events +apparently causing events in their future, others apparently +causing events in their past---the truth being that all are linked +%% -----File: 062.png---Folio 52------- +by determinate laws, the so-called causal events being merely +conspicuous foci from which the links radiate. + +The recognition of an absolute past and future seems to +depend on the possibility of events which are not governed by +a determinate scheme. If, say, the event~$O$ is an ultimatum, +and the person describing the path~$NP$ is a ruler of the country +affected, then it may be manifest to all observers that it is his +knowledge of the actual occurrence of the event~$O$ which has +caused him to create the event~$P$. $P$~must then be in the absolute +%[Illustration: \textsc{Fig}. 4.] +\begin{figure*}[hbt]% +\begin{center}% +\Graphic[4]{3.5in}{062a} +\end{center}% +\end{figure*}% +future of~$O$, and, as we have seen, must lie in the sector~$UOV$. +But the inference is only permissible, if the event~$P$ could be +determined by the event~$O$, and was not predetermined by +causes anterior to both---if it was possible for it to happen or +not, consistently with the laws of nature. Since physics does +not attempt to cover indeterminate events of this kind, the +distinction of absolute past and future is not directly important +for physics; but it is of interest to show that the theory of four-dimensional +space-time provides an absolute past and future, in +accordance with common requirements, although this can +usually be ignored in applications to physics. + +%% -----File: 063.png---Folio 53------- + +Consider now all the events which are at an interval of one +unit from~$O$, according to the definition of the interval~$s$ +\Pagelabel{53} +\[ +s^2 = - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 + (t_2 - t_1)^2. +\Tag{1} +\] +We have changed the sign of $s^{2}$, because usually (though not +always) the original $s^{2}$ would have come out negative. In +Euclidean space points distant a unit interval lie on a circle; +but, owing to the change in geometry due to the altered sign +of $(t_{2} - t_{1})^{2}$, they now lie on a rectangular hyperbola with two +branches $KLM$, $K'L'M'$. Since the interval is an absolute +quantity, all observers will agree that these points are at unit +interval from~$O$. + +Now make the following construction:---draw a straight line +$OFT_{1}$ to meet the hyperbola in~$F$; draw the tangent~$FG$ at~$F$, +meeting the light-line $U'OU$ in~$G$; complete the parallelogram +$OFGH$; produce~$OH$ to~$X_{1}$. We now assert that an observer +$S_{1}$ who chooses $OT_{1}$ for his time-direction will regard $OX_{1}$ as +his space direction and will consider $OF$ and~$OH$ to be the units +of time and space. + +The two observers make their partitions of space and time +in different ways, as illustrated in Figs.~5 and~6, where in each +case the partitions are at unit distance (in space and time) +according to the observers' own reckoning. The same diagram +of events in the world will serve for both observers; $S_{1}$ merely +removes $S$'s partitions and overlays his own, locating the events +in his space and time accordingly. It will be seen at once that +the lines of unit velocity---progress of one unit of space for one +unit of time---agree, so that the velocity of a pulse of light is +unity for both observers. It can be shown from the properties +of the hyperbola that the locus of points at any interval~$s$ from~$O$, +given by \Eqref{equation}{1}, viz. +\[ +s^2 = (t - t_0)^2 - (x - x_0)^2, +\] +is the same locus (a hyperbola) for both systems of reckoning +$x$ and~$t$. The two observers will always agree on the measures +of intervals, though they will disagree about lengths, durations, +and the velocities of everything except light. This rather complex +transformation is mathematically equivalent to the simple +rotation of the axes required when imaginary time is used. + +It must not be supposed that there is any natural distinction +%% -----File: 064.png---Folio 54------- +corresponding to the difference between the square-partitions +of observer $S$ and the diamond-shaped partitions of observer~$S_1$. +We might say that $S_1$ transplants the space-time world unchanged +from \Figref{5} to \Figref{6}, and then distorts it until the +diamonds shown become squares; or we might equally well start +with this distorted space-time, partitioned by $S_1$ into squares, +and then $S$'s partitions would be represented by diamonds. +It cannot be said that either observer's space-time is distorted +absolutely, but one is distorted relatively to the other. It is the +relation of \textit{order} which is intrinsic in nature, and is the same +both for the squares and diamonds; \textit{shape} is put into nature by +the observer when he has chosen his partitions. +\index{Ordering of events in external world}% +\index{Space-time!partitions of}% +\begin{figure*}[htb] +\begin{center} +\Graphic[5]{4.5in}{064a} +\Figlabel{6}% +\end{center} +\end{figure*} +% [** PP: Moved up one paragraph] +%[Illustration: \textsc{Fig}. 5.] +%[Illustration: \textsc{Fig}. 6.] + +We can now deduce the FitzGerald contraction. +\index{Contraction, FitzGerald}% +Consider +a rod of unit length at rest relatively to the observer~$S$. The +two extremities are at rest in his space, and consequently remain +on the same space-partitions; hence their tracks in four dimensions +$PP'$, $QQ'$ (\Figref{7}) are entirely in the time-direction. The +real rod in nature is the four-dimensional object shown in section +as $P'PQQ'$. Overlay the same figure with $S_1$'s space and time +partitions, shown by the dotted lines. Taking a section at any +one ``time,'' the instantaneous rod is $P_1Q_1$, viz.\ the section of +$P'PQQ'$ by $S_1$'s time-line. Although on paper $P_1Q_1$ is actually +longer than $PQ$, it is seen that it is a little shorter than one of +$S_1$'s space-partitions; and accordingly $S_1$ judges that it is less +%% -----File: 065.png---Folio 55------- +than one unit long---it has contracted on account of its motion +relative to him. +%[Illustration: Fig. 7.] +\begin{figure*}[htb] +\begin{center} +\Graphic[7]{4in}{065a} +\end{center} +\end{figure*} + +Similarly $RR'-SS'$ is a rod of unit length at rest relatively +to $S_1$. Overlaying $S$'s partitions we see that it occupies $R_1S_1$ at +a particular instant for $S$; and this is less than one of $S$'s +partitions. Thus $S$ judges it to have contracted on account of +its motion relative to him.% +\index{Retardation of time}% + +%[Illustration: Fig. 8.] +\begin{wrapfigure}{r}{2.5in} +\Graphic[8]{2.5in}{065b} +\end{wrapfigure} +In the same way we can illustrate the problem of the duration +of the cigar; each observer +believed the other's cigar to +last the longer time. Taking +$LM$ (\Figref{8}) to represent the +duration of $S$'s cigar (two +units), we see that in $S_1$'s +reckoning it reaches over a +little more than two time-partitions. +Moreover it has not +kept to one space-partition, +i.e.\ it has moved. Similarly $L'N'$ is the duration of $S_1$'s cigar +(two time-units for him); and it lasts a little beyond two unit-partitions +in $S$'s time-reckoning. (Note, in comparing the two +diagrams, $L', M', N'$ are the same points as $L, M, N$.) + +If in \Figref{4} we had taken the line $OT_1$ very near to $OU$, our +%% -----File: 066.png---Folio 56------- +diamonds would have been very elongated, and the unit-divisions +$OF, OH$ very large. This kind of partition would be +made by an observer whose course through the world is $OT_1$, +and who is accordingly travelling with a velocity approaching +that of light relative to $S$. +\index{Light, velocity of!system moving with}% +\index{Velocity of light!system moving with}% +In the limit, when the velocity +reaches that of light, both space-unit and time-unit become +infinite, so that in the natural units for an observer travelling +with the speed of light, all the events in the finite experience of +$S$ take place ``in no time'' and the size of every object is zero. +This applies, however, only to the two dimensions $x$ and $t$; the +space-partitions parallel to the plane of the paper are not +affected by this motion along $x$. Consequently for an observer +travelling with the speed of light all ordinary objects become +two-dimensional, preserving their lateral dimensions, but infinitely +thin longitudinally. The fact that events take place ``in +no time'' is usually explained by saying that the inertia of any +particle moving with the velocity of light becomes infinite so +that all molecular processes in the observer must stop; many +things may happen in $S$'s world in a twinkling of an eye---of +$S_1$'s eye.% +\index{Inertia!infinite}% + +However successful the theory of a four-dimensional world +may be, it is difficult to ignore a voice inside us which whispers +``At the back of your mind, you know that a fourth dimension +is all nonsense.'' +\index{Four-dimensional order}% +I fancy that that voice must often have had a +busy time in the past history of physics. What nonsense to +say that this solid table on which I am writing is a collection +of electrons moving with prodigious speeds in empty spaces, +which relatively to electronic dimensions are as wide as the +spaces between the planets in the solar system! What nonsense +to say that the thin air is trying to crush my body with a load +of 14~lbs.\ to the square inch! What nonsense that the star-cluster, +which I see through the telescope obviously there now, +is a glimpse into a past age $50,000$ years ago! Let us not be +beguiled by this voice. It is discredited. + +But the statement that time is a fourth dimension may +suggest unnecessary difficulties which a more precise definition +avoids. It is in the external world that the four dimensions are +united---not in the relations of the external world to the +individual which constitute his direct acquaintance with space +%% -----File: 067.png---Folio 57------- +and time. Just in that process of relation to an individual, the +order falls apart into the distinct manifestations of space and time. +An individual is a four-dimensional object of greatly elongated +form; in ordinary language we say that he has considerable extension +in time and insignificant extension in space. Practically he +is represented by a line---his track through the world. When the +world is related to such an individual, his own asymmetry is +introduced into the relation; and that order of events which is +parallel with his track, that is to say with \textit{himself}, appears in +his experience to be differentiated from all other orders of events.% +\index{Observer, an unsymmetrical object}% +\index{Time!depends on observer's track}% + +Probably the best known exposition of the fourth dimension +is that given in E.~Abbott's popular book \textit{Flatland}. +\index{Flatland}% +It may be of +interest to see how far the four-dimensional world of space-time +conforms with his anticipations. He lays stress on three points. + +(1)~As a four-dimensional body moves, its section by the +three-dim\-en\-sional world may vary; thus a rigid body can alter +size and shape. + +(2)~It should be possible for a body to enter a completely +closed room, by travelling into it in the direction of the fourth +dimension, just as we can bring our pencil down on to any point +within a square without crossing its sides. + +(3)~It should be possible to see the inside of a solid, just as +we can see the inside of a square by viewing it from a point +outside its plane. + +The first phenomenon is manifested by the FitzGerald contraction.% +\index{FitzGerald contraction!relativity explanation of}% + +If quantity of matter is to be identified with its mass, the +second phenomenon does not happen. It could easily be conceived +of as happening, but it is provided against by a special +law of nature---the conservation of mass. It could happen, +but it does not happen. + +The third phenomenon does not happen for two reasons. +A natural body extends in time as well as in space, and is +therefore four-dimensional; but for the analogy to hold, the +object must have one dimension less than the world, like the +square seen from the third dimension. If the solid suddenly +went out of existence so as to present a plane section towards +time, we should still fail to see the interior of it; because light-tracks +in four-dimensions are restricted to certain lines like +%% -----File: 068.png---Folio 58------- +$UOV, U'OV'$ in \Figref{3}, whereas in three-dimensions light can +traverse any straight line. This could be remedied by interposing +some kind of dispersive medium, so that light of some wave-length +could be found travelling with every velocity and following +every track in space-time; then, looking at a solid which suddenly +went out of existence, we should receive at the same moment +light-impressions from every particle in its interior (supposing +them self-luminous). We actually should see the inside of it. + +How our poor eyes are to disentangle this overwhelming +experience is quite another question. + +The interval is a quantity so fundamental for us that we may +consider its measurement in some detail. Suppose we have a +scale $AB$ divided into kilometres, say, and at each division is +placed a clock also registering kilometres. (It will be remembered +that time can be measured in seconds or kilometres indifferently.) +%[Illustration: Fig. 9.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[9]{\textwidth}{068a} +\end{center} +\end{figure*}% +\index{Clock-scale}% [** PP: Presumed location of entry] +\index{Interval!practical measurement of}% +When the clocks are correctly set and viewed from $A$ the sum +of the readings of any clock and the division beside it is the +same for all, since the scale-reading gives the correction for the +time taken by light, travelling with unit velocity, to reach~$A$. +This is shown in \Figref{9} where the clock-readings are given as +though they were being viewed from~$A$. + +Now lay the scale in line with the two events; note the clock +and scale-readings $t_1, x_1$, of the first event, and the corresponding +readings $t_2, x_2$, of the second event. Then by the formula +already given +\index{Clock!affected by velocity}% +\[ +s^2 = (t_2 - t_1)^2 - (x_2 - x_1)^2. +\] +But suppose we took a different standard of rest, and set the +scale moving uniformly in the direction~$AB$. Then the divisions +would have advanced to meet the second event, and $(x_2 - x_1)$ +would be smaller. This is compensated, because $t_2-t_1$ also +becomes altered. $A$~is now advancing to meet the light coming +from any of the clocks along the rod; the light arrives too +%% -----File: 069.png---Folio 59------- +quickly, and in the initial adjustment described above the clock +must be set back a little. The clock-reading of the event is thus +smaller. There are other small corrections arising from the +FitzGerald contraction, etc.; and the net result is that, it does +not matter what uniform motion is given to the scale, the final +result for $s$ is always the same. + +In elementary mechanics we are taught that velocities can be +compounded by adding. +\index{Addition of velocities}% +\index{Velocity, addition-law}% +If $B$'s velocity relative to $A$ (as observed +by either of them) is $100$~km.\ per sec., and $C$'s velocity relative +to $B$ is $100$~km.\ per sec.\ in the same direction, then $C$'s velocity +relative to~$A$ should be $200$~km.\ per sec. This is not quite +accurate; the true answer is $199.999978$~km.\ per sec. The discrepancy +is not difficult to explain. The two velocities and their +resultant are not all reckoned with respect to the same partitions +of space and time. When $B$ measures $C$'s velocity relative to +him he uses his own space and time, and it must be corrected +to reduce to $A$'s space and time units, before it can be added +on to a velocity measured by $A$. + +If we continue the chain, introducing $D$ whose velocity +relative to $C$, and measured by $C$, is $100$~km.\ per sec., and so on +\textit{ad infinitum}, we never obtain an infinite velocity with respect +to~$A$, but gradually approach the limiting velocity of $300,000$ +km.\ per sec., the speed of light. This speed has the remarkable +property of being absolute, whereas every other speed is relative. +\index{Light, velocity of!an absolute velocity}% +If a speed of $100$~km.\ per sec.\ or of $100,000$~km.\ per sec.\ is +mentioned, we have to ask---speed relative to what? But if +a speed of $300,000$~km.\ per sec.\ is mentioned, there is no need +to ask the question; the answer is---relative to any and every +piece of matter. A $\beta$ particle shot off from radium can move at +more than $200,000$~km.\ per sec.; +\index{Beta particles}% +but the speed of light relative +to an observer travelling with it is still $300,000$~km.\ per sec. It +reminds us of the mathematicians' transfinite number Aleph; +you can subtract any number you like from it, and it still +remains the same. + +The velocity of light plays a conspicuous part in the relativity +theory, and it is of importance to understand what is the +property associated with it which makes it fundamental. +\index{Light, velocity of!importance of}% +\index{Velocity of light!importance of}% +The +fact that the velocity of light is the same for all observers is a +consequence rather than a cause of its pre-eminent character. +%% -----File: 070.png---Folio 60------- +Our first introduction of it, for the purpose of coordinating +units of length and time, was merely conventional with a view +to simplifying the algebraic expressions. Subsequently, considerable +use has been made of the fact that nothing is known +in physics which travels with greater speed, so that in practice +our determinations of simultaneity depend on signals transmitted +with this speed. If some new kind of ray with a higher +speed were discovered, it would perhaps tend to displace light-signals +and light-velocity in this part of the work, time-reckoning +being modified to correspond; on the other hand, this would +lead to greater complexity in the formulae, because the FitzGerald +contraction which affects space-measurement depends +on light-velocity. But the chief importance of the velocity of +light is that no material body can exceed this velocity. This +gives a general physical distinction between paths which are time-like +and space-like, respectively---those which can be traversed +by matter, and those which cannot. The material structure of +the four-dimensional world is fibrous, with the threads all running +along time-like tracks; it is a tangled warp without a woof. +Hence, even if the discovery of a new ray led us to modify the +reckoning of time and space, it would still be necessary in the +study of material systems to preserve the \textit{present} absolute +distinction of time-like and space-like intervals, under a new +name if necessary.% +\index{Space-like intervals}% +\index{Time-like intervals}% + +It may be asked whether it is possible for anything to have +a speed greater than the velocity of light. Certainly matter +cannot attain a greater speed; but there might be other things +in nature which could. ``Mr Speaker,'' said Sir Boyle Roche, +``not being a bird, I could not be in two places at the same time.'' +Any entity with a speed greater than light would have the +peculiarity of Sir Boyle Roche's bird. It can scarcely be said to +be a self-contradictory property to be in two places at the same +time any more than for an object to be at two times in the same +place. The perplexities of the quantum theory of energy sometimes +seem to suggest that the possibility ought not to be +overlooked; but, on the whole, the evidence seems to be against +the existence of anything moving with a speed beyond that of +light.% +\index{Quanta}% + +The standpoint of relativity and the principle of relativity +%% -----File: 071.png---Folio 61------- +are quite independent of any views as to the constitution of +matter or light. Hitherto our only reference to electrical theory +has been in connection with Larmor and Lorentz's explanation +of the FitzGerald contraction; but now from the discussion of +the four-dimensional world, we have found a more general +explanation of the change of length. The case for the electrical +theory of matter is actually weakened, because many experimental +effects formerly thought to depend on the peculiar +properties of electrical forces are now found to be perfectly +general consequences of the relativity of observational knowledge. + +Whilst the evidence for the electrical theory of matter is not +so conclusive, as at one time appeared, the theory may be +accepted without serious misgivings. To postulate two entities, +matter and electric charges, when one will suffice is an arbitrary +hypothesis, unjustifiable in our present state of knowledge. The +great contribution of the electrical theory to this subject is a +precise explanation of the property of inertia. It was shown +theoretically by J.~J. Thomson that if a charged conductor is +to be moved or stopped, additional effort will be necessary +simply on account of the charge. +\index{Thomson, J.~J.}% +The conductor has to carry +its electric field with it, and force is needed to set the field +moving. This property is called inertia, and it is measured by +\textit{mass}. If, keeping the charge constant, the size of the conductor +is diminished, this inertia increases. Since the smallest separable +particles of matter are found by experiment to be very minute +and to carry charges, the suggestion arises that these charges +may be responsible for the whole of the inertia detected in +matter. The explanation is sufficient; and there seems no reason +to doubt that all inertia is of this electrical kind.% +\index{Electrical theory of inertia}% +\index{Energy!inertia of}% +\index{Inertia!electrical theory of}% +\index{Mass!electrical theory of}% + +When the calculations are extended to charges moving with +high velocities, it is found that the electrical inertia is not +strictly constant but depends on the speed; in all cases the +variation is summed up in the statement that the inertia is +simply proportional to the total energy of the electromagnetic +field. We can say if we like that the mass of a charged particle +at rest belongs to its electrostatic energy; when the charge is +set in motion, kinetic energy is added, and this kinetic energy +also has mass. Hence it appears that mass (inertia) and energy +%% -----File: 072.png---Folio 62------- +are essentially the same thing, or, at the most, two aspects of +the same thing. It must be remembered that on this view the +greater part of the mass of matter is due to concealed energy, +which is not as yet releasable. + +The question whether electrical energy not bound to electric +charges has mass, is answered in the affirmative in the case of +light. Light has mass. +\index{Light!mass of}% +\index{Mass of light}% +Presumably also gravitational energy +has mass; or, if not, mass will be created when, as often happens, +gravitational energy is converted into kinetic energy. The mass +of the whole (negative) gravitational energy of the earth is of +the order \textit{minus} a billion tons. + +The theoretical increase of the mass of an electron with speed +has been confirmed experimentally, the agreement with calculation +being perfect if the electron undergoes the FitzGerald +contraction by its motion. +\index{Electron!Kaufmann's experiment on}% +\index{Kaufmann's experiment}% +This has been held to indicate that +the electron cannot have any inertia other than that due to the +electromagnetic field carried with it. +\index{Electron!inertia of}% +But the conclusion (though +probable enough) is not a fair inference; because these results, +obtained by special calculation for electrical inertia, are found +to be predicted by the theory of relativity for any kind of +inertia. This will be shown in \Chapref{IX}. The factor giving +the increase of mass with speed is the same as that which affects +length and time. Thus if a rod moves at such a speed that its +length is halved, its mass will be doubled. Its density will be +increased four-fold, since it is both heavier and less in volume.% +\index{Density, effect of motion on}% + +We have thought it necessary to include this brief summary +of the electrical theory of matter and mass, because, although +it is not required by the relativity theory, it is so universally +accepted in physics that we can scarcely ignore it. Later on we +shall reach in a more general way the identification of mass with +energy and the variation of mass with speed; but, since the +experimental measurement of inertia involves the study of a +body in non-uniform motion, it is not possible to enter on a +satisfactory discussion of mass until the more general theory of +relativity for non-uniform motion has been developed. +%% -----File: 073.png---Folio 63------- + + +\Chapter{IV}{Fields of Force} + +\Quote{Lucretius, \textit{De Natura Rerum.}} +{For whenever bodies fall through water and thin air, they must quicken their +descents in proportion to their weights, because the body of water and subtle +nature of air cannot retard everything in equal degree, but more readily give +way overpowered by the heavier; on the other hand empty void cannot offer +resistance to anything in any direction at any time, but must, as its nature +craves, continually give way; and for this reason all things must be moved and +borne along with equal velocities though of unequal weights through the +unresisting void.}% +\index{Force!elementary conception of}% + + +\First{The} primary conception of force is associated with the muscular +sensation felt when we make an effort to cause or prevent the +motion of matter. Similar effects on the motion of matter can +be caused by non-living agency, and these also are regarded as +due to forces. As is well known, the scientific measure of a force +is the momentum that it communicates to a body in given time. +There is nothing very abstract about a force transmitted by +material contact; modern physics shows that the momentum is +communicated by a process of molecular bombardment. We can +visualise the mechanism, and see the molecules carrying the +motion in small parcels across the boundary into the body that +is being acted on. Force is no mysterious agency; it is merely +a convenient summary of this flow of motion, which we can +trace continuously if we take the trouble. It is true that the +difficulties are only set back a stage, and the exact mode by +which the momentum is redistributed during a molecular +collision is not yet understood; but, so far as it goes, this analysis +gives a clear idea of the transmission of motion by ordinary +forces. + +But even in elementary mechanics an important natural force +appears, which does not seem to operate in this manner. Gravitation +is not resolvable into a succession of molecular blows. +A massive body, such as the earth, seems to be surrounded by +a field of latent force, ready, if another body enters the field, to +become active, and transmit motion. One usually thinks of this +influence as existing in the space round the earth even when +%% -----File: 074.png---Folio 64------- +there is no test-body to be affected, and in a rather vague way +it is suspected to be some state of strain or other condition of +an unperceived medium. + +Although gravitation has been recognised for thousands of +years, and its laws were formulated with sufficient accuracy for +almost all purposes more than 200 years ago, it cannot be said +that much progress has been made in explaining the nature or +mechanism of this influence. It is said that more than 200 +theories of gravitation have been put forward; but the most +plausible of these have all had the defect that they lead nowhere +and admit of no experimental test. Many of them would nowadays +be dismissed as too materialistic for our taste---filling space +with the hum of machinery---a procedure curiously popular in +the nineteenth century. Few would survive the recent discovery +that gravitation acts not only on the molecules of matter, but +on the undulations of light. + +The nature of gravitation has seemed very mysterious, yet it +is a remarkable fact that in a limited region it is possible to +create an artificial field of force which imitates a natural +gravitational field so exactly that, so far as experiments have +yet gone, no one can tell the difference. +\index{Artificial fields of force}% +\index{Fields of force!artificial}% +\index{Force!fields of}% +Those who seek for an +explanation of gravitation naturally aim to find a model which +will reproduce its effects; but no one before Einstein seems to +have thought of finding the clue in these artificial fields, familiar +as they are. + +When a lift starts to move upwards the occupants feel a +characteristic sensation, which is actually identical with a +sensation of increased weight. +\index{Lift, accelerated}% +The feeling disappears as soon +as the motion becomes uniform; it is associated only with the +change of motion of the lift, that is to say, the acceleration. +Increased weight is not only a matter of sensation; it is shown +by any physical experiments that can be performed. The usual +laboratory determination of the value of gravity by Atwood's +machine would, if carried out inside the accelerated lift, give +a higher value. A spring-balance would record higher weights. +Projectiles would follow the usual laws of motion but with a +higher value of gravity. In fact, the upward acceleration of +the lift is in its mechanical effects exactly similar to an additional +gravitational field superimposed on that normally present. + +%% -----File: 075.png---Folio 65------- + +Perhaps the equivalence is most easily seen when we produce +in this manner an artificial field which just neutralises the earth's +field of gravitation. Jules Verne's book \textit{Round the Moon} tells +the story of three men in a projectile shot from a cannon into +space. +\index{Projectile, Jules Verne's}% +The author enlarges on their amusing experiences when +their weight vanished altogether at the neutral point, where the +attraction of the earth and moon balance one another. As a +matter of fact they would not have had any feeling of weight +at any time during their journey after they left the earth's +atmosphere. +\index{Weight!vanishes inside free projectile}% +The projectile was responding freely to the pull of +gravity, and so were its occupants. When an occupant let go +of a plate, the plate could not ``fall'' any more than it was +doing already, and so it must remain poised. + +It will be seen that the sensation of weight is not felt when +we are free to respond to the force of gravitation; it is only +felt when something interferes to prevent our falling. It is +primarily the floor or the chair which causes the sensation of +weight by checking the fall. It seems literally true to say that +we never feel the force of the earth's gravitation; what we do +feel is the bombardment of the soles of our boots by the molecules +of the ground, and the consequent impulses spreading upwards +through the body. This point is of some importance, since the +idea of the force of gravitation as something which can be felt, +predisposes us to a materialistic view of its nature. + +Another example of an artificial field of force is the centrifugal +force of the earth's rotation. In most books of Physical Constants +will be found a table of the values of~``$g$,'' the acceleration +due to gravity, at different latitudes. +\index{Centrifugal Force!compared with gravitation}% +But the numbers given +do not relate to gravity alone; they are the resultant of gravity +and the centrifugal force of the earth's rotation. These are so +much alike in their effects that for practical purposes physicists +have not thought it worth while to distinguish them. + +Similar artificial fields are produced when an aeroplane +changes its course or speed; and one of the difficulties of navigation +is the impossibility of discriminating between these and the +true gravitation of the earth with which they combine. One +usually finds that the practical aviator requires little persuasion +of the relativity of force. + +To find a unifying idea as to the origin of these artificial +%% -----File: 076.png---Folio 66------- +fields of force, we must return to the four-dimensional world of +space-time. The observer is progressing along a certain track +in this world. Now his course need not necessarily be straight. +It must be remembered that straight in the four-dimensional +world means something more than straight in space; it implies +also uniform velocity, since the velocity determines the inclination +of the track to the time-axis. + +The observer in the accelerated lift travels upwards in a +straight line, say $1$~foot in the first second, $4$~feet in two seconds, +$9$~feet in three seconds, and so on. If we plot these points as +$x$ and~$t$ on a diagram we obtain a curved track. Presently the +speed of the lift becomes uniform and the track in the diagram +becomes straight. So long as the track is curved (accelerated +motion) a field of force is perceived; it disappears when the +track becomes straight (uniform motion). + +Again the observer on the earth is carried round in a circle +once a day by the earth's rotation; allowing for steady progress +through time, the track in four dimensions is a spiral. For an +observer at the north pole the track is straight, and there the +centrifugal force is zero. + +Clearly the artificial field of force is associated with curvature +of track, and we can lay down the following rule:--- + +Whenever the observer's track through the four-dimensional +world is curved he perceives an artificial field of force. + +The field of force is not only perceived by the observer in his +sensations, but reveals itself in his physical measures. It should +be understood, however, that the curvature of track must not +have been otherwise allowed for. Naturally if the observer in +the lift recognises that his measures are affected by his own +acceleration and applies the appropriate corrections, the artificial +force will be removed by the process. It only exists if he is +unaware of, or does not choose to consider, his acceleration. + +The centrifugal force is often called ``unreal.'' From the point +of view of an observer who does not rotate with the earth, there +is no centrifugal force; it only arises for the terrestrial observer +who is too lazy to make other allowance for the effects of the +earth's rotation. It is commonly thought that this ``unreality'' +quite differentiates it from a ``real'' force like gravity; but if +we try to find the grounds of this distinction they evade us. +%% -----File: 077.png---Folio 67------- +The centrifugal force is made to disappear if we choose a suitable +standard observer not rotating with the earth; the gravitational +force was made to disappear when we chose as standard observer +an occupant of Jules Verne's falling projectile. If the possibility +of annulling a field of force by choosing a suitable standard +observer is a test of unreality, then gravitation is equally unreal +with centrifugal force.% +\index{Fields of force!relativity of}% +\index{Force!relativity of}% +\IndexExtra{Relativity of Force}% + +It may be urged that we have not stated the case quite +fairly. When we choose the non-rotating observer the centrifugal +force disappears completely and everywhere. When we choose +the occupant of the falling projectile, gravitation disappears in +his immediate neighbourhood; but he would notice that, +although unsupported objects round him experienced no acceleration +relative to him, objects on the other side of the earth would +fall towards him. So far from getting rid of the field of force, +he has merely removed it from his own surroundings, and piled +it up elsewhere. Thus gravitation is removable locally, but +centrifugal force can be removed everywhere. The fallacy of +this argument is that it speaks as though gravitation and +centrifugal force were distinguishable experimentally. It presupposes +the distinction that we are challenging. Looking simply +at the resultant of gravitation and centrifugal force, which is all +that can be observed, neither observer can get rid of the resultant +force at all parts of space. Each has to be content with leaving +a residuum. The non-rotating observer claims that he has got +rid of all the unreal part, leaving a remainder (the usual gravitational +field) which he regards as really existing. We see no +justification for this claim, which might equally well be made +by Jules Verne's observer. + +It is not denied that the separation of centrifugal and gravitational +force generally adopted has many advantages for +mathematical calculation. If it were not so, it could not have +endured so long. But it is a mathematical separation only, +without physical basis; and it often happens that the separation +of a mathematical expression into two terms of distinct nature, +though useful for elementary work, becomes vitiated for more +accurate work by the occurrence of minute cross-terms which +have to be taken into account. + +Newtonian mechanics proceeds on the supposition that there +%% -----File: 078.png---Folio 68------- +is some super-observer. If \textit{he} feels a field of force, then that +force really exists. Lesser beings, such as the occupants of the +falling projectile, have other ideas, but they are the victims of +illusion. It is to this super-observer that the mathematician +appeals when he starts a dynamical investigation with the words +``Take unaccelerated rectangular axes, $Ox$, $Oy$, $Oz\dotsc$.'' +Unaccelerated rectangular axes are the measuring-appliances of the +super-observer.% +\index{Newton!super-observer}% +\index{Super-observer, Newton's}% + +It is quite possible that there might be a super-observer, +whose views have a natural right to be regarded as the truest, +or at least the simplest. A society of learned fishes would probably +agree that phenomena were best described from the point +of view of a fish at rest in the ocean. But relativity mechanics +finds that there is no evidence that the circumstances of any +observer can be such as to make his views pre-eminent. All are +on an equality. Consider an observer $A$ in a projectile falling +freely to the earth, and an observer $B$ in space out of range of +any gravitational attraction. Neither $A$ nor~$B$ feel any field of +force in their neighbourhood. Yet in Newtonian mechanics an +artificial distinction is drawn between their circumstances; $B$ is +in no field of force at all, but $A$ is really in a field of force, only +its effects are neutralised by his acceleration. But what is this +acceleration of~$A$? Primarily it is an acceleration relative to the +earth; but then that can equally well be described as an acceleration +of the earth relative to~$A$, and it is not fair to regard it as +something located with $A$. Its importance in Newtonian +philosophy is that it is an acceleration relative to what we have +called the super-observer. This potentate has drawn planes and +lines partitioning space, as space appears to him. I fear that +the time has come for his abdication. + +Suppose the whole system of the stars were falling freely +under the uniform gravitation of some vast external mass, like +a drop of rain falling to the ground. Would this make any +difference to phenomena? None at all. There would be a +gravitational field; but the consequent acceleration of the +observer and his landmarks would produce a field of force +annulling it. Who then shall say what is absolute acceleration?% +\index{Absolute acceleration}% + +We shall accordingly give up the attempt to separate artificial +fields of force and natural gravitational fields; and call the whole +%% -----File: 079.png---Folio 69------- +measured field of force the gravitational field, generalising the +expression. This field is not absolute, but always requires that +some observer should be specified. + +It may avoid some mystification if we state at once that there +are certain intricacies in the gravitational influence radiating +from heavy matter which are distinctive. A theory which did +not admit this would run counter to common sense. What our +argument has shown is that the characteristic symptom in a +region in the neighbourhood of matter is not the field of force; +it must be something more intricate. In due course we shall +have to explain the nature of this more complex effect of matter +on the condition of the world. + +Our previous rule, that the observer perceives an \textit{artificial} +field of force when he deviates from a straight track, must now +be superseded. We need rather a rule determining when he +perceives a field of force of any kind. +\index{Fields of force!due to disturbance of observer}% +Indeed the original rule +has become meaningless, because a straight track is no longer +an absolute conception. Uniform motion in a straight line is +not the same for an observer rotating with the earth as for a +non-rotating observer who takes into account the sinuosity of +the rotation. We have decided that these two observers are on +the same footing and their judgments merit the same respect. +A straight-line in space-time is accordingly not an absolute +conception, but is only defined relative to some observer. + +Now we have seen that so long as the observer and his +measuring-appliances are unconstrained (falling freely) the field +of force immediately round him vanishes. It is only when he is +deflected from his proper track that he finds himself in the +midst of a field of force. Leaving on one side the question of +the motion of electrically charged bodies, which must be reserved +for more profound treatment, the observer can only leave his +proper track if he is being disturbed by material impacts, e.g.\ +the molecules of the ground bombarding the soles of his boots. +We may say then that a body does not leave its natural track +without visible cause; and any field of force round an observer +is the result of his leaving his natural track by such cause. +There is nothing mysterious about this field of force; it is merely +the reflection in the phenomena of the observer's disturbance; +just as the flight of the houses and hedgerows past our railway-carriage +is the reflection of our motion with the train. + +%% -----File: 080.png---Folio 70------- + +Our attention is thus directed to the natural tracks of unconstrained +bodies, which appear to be marked out in some +absolute way in the four-dimensional world. There is no +question of an observer here; the body takes the same course +in the world whoever is watching it. Different observers will +describe the track as straight, parabolical, or sinuous, but it is +the same absolute locus.% +\index{Geodesic!absolute significance of}% +\index{Tracks, natural}% +\index{Natural tracks}% + +Now we cannot pretend to predict without reference to +experiment the laws determining the nature of these tracks; +but we can examine whether our knowledge of the four-dimensional +world is already sufficient to specify definite tracks of this +kind, or whether it will be necessary to introduce new hypothetical +factors. It will be found that it is already sufficient. So far we +have had to deal with only one quantity which is independent +of the observer and has therefore an absolute significance in the +world, namely the \textit{interval} between two events in space and time. +Let us choose two fairly distant events $P_1$ and~$P_2$. These can +be joined by a variety of tracks, and the interval-length from +$P_1$ to~$P_2$ along any track can be measured. In order to make +sure that the interval-length is actually being measured along +the selected track, the method is to take a large number of +intermediate points on the track, measure the interval corresponding +to each subdivision, and take the sum. It is virtually +the same process as measuring the length of a twisty road on +a map with a piece of cotton. The interval-length along a +particular track is thus something which can be measured +absolutely, since all observers agree as to the measurement of +the interval for each subdivision. It follows that all observers +will agree as to which track (if any) is the shortest track between +the two points, judged in terms of interval-length.% +\index{Interval-length!tracks of maximum}% +\index{Longest tracks}% + +This gives a means of defining certain tracks in space-time as +having an absolute significance, and we proceed tentatively to +identify them with the natural tracks taken by freely moving +particles. + +In one respect we have been caught napping. Dr A. A. Robb +has pointed out the curious fact that it is not the shortest track, +but the longest track, which is unique\footnotemark. + \footnotetext{It is here assumed that $P_2$ is in the future of $P_1$ so that it is possible for + a particle to travel from $P_1$ to~$P_2$. If $P_1$ and~$P_2$ are situated like $O$ and~$P'$ in + \Figref{3}, the interval-length is imaginary, and the \textit{shortest} track is unique.}% +There are any number +%% -----File: 081.png---Folio 71------- +of tracks from $P_1$ to~$P_2$ of zero interval-length; there is just one +which has maximum length. This is because of the peculiar +geometry which the minus sign of $(t_2-t_1)^2$ introduces. For +instance, it will be seen from \Eqref{equation}{1}, \Pageref{53}, that when +\index{Interval-length!zero for velocity of light}% +\[ +(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 = (t_2 - t_1)^2, +\] +that is to say when the resultant distance travelled in space is +equal to the distance travelled in time, then $s$ is zero. This +happens when the velocity is unity---the velocity of light. To +get from $P_1$ to $P_2$ by a path of no interval-length, we must +simply keep on travelling with the velocity of light, cruising +round if necessary, until the moment comes to turn up at~$P_2$. +On the other hand there is evidently an upper limit to the interval-length +of the track, because each portion of $s$ is always less than +the corresponding portion of $(t_2-t_1)$, and $s$ can never exceed +$t_2-t_1$. + +There is a physical interpretation of interval-length along the +path of a particle which helps to give a more tangible idea of +its meaning. It is the time as perceived by an observer, or +measured by a clock, carried on the particle. This is called the +proper-time; +\index{Clock!recording proper-time}% +\index{Interval-length!identified with proper time}% +\index{Proper-time}% +and, of course, it will not in general agree with the +time-reckoning of the independent onlooker who is supposed to +be watching the whole proceedings. To prove this, we notice +from \Eqref{equation}{1} that if $x_2 = x_1, y_2 = y_1$ and $z_2 = z_1$, then +$s = t_2-t_1$. The condition $x_2 = x_1$, etc.\ means that the particle +must remain stationary relative to the observer who is measuring +$x$, $y$, $z$,~$t$. To secure this we mount our observer on the particle +and then the interval-length $s$ will be $t_2-t_1$, which is the time +elapsed according to his clock. + +We can use proper-time as generally equivalent to interval-length; +but it must be admitted that the term is not very +logical unless the track in question is a natural track. For any +other track, the drawback to defining the interval-length as the +time measured by a clock which follows the track, is that no +clock could follow the track without violating the laws of nature. +We may force it into the track by continually hitting it; but +that treatment may not be good for its time-keeping qualities. +The original definition by \Eqref{equation}{1} is the more general +definition. + +%% -----File: 082.png---Folio 72------- + +We are now able to state formally our proposed law of motion---Every +particle moves so as to take the track of greatest interval-length +between two events, except in so far as it is disturbed by +impacts of other particles or electrical forces. + +This cannot be construed into a truism like Newton's first +law of motion. The reservation is not an undefined agency like +force, whose meaning can be extended to cover any breakdown +of the law. We reserve only direct material impacts and electromagnetic +causes, the latter being outside our present field of +discussion. + +Consider, for example, two events in space-time, viz.\ the +position of the earth at the present moment, and its position a +hundred years ago. Call these events $P_2$ and~$P_1$. In the interim +the earth (being undisturbed by impacts) has moved so as to +take the longest track from $P_1$ to $P_2$---or, if we prefer, so as to +take the longest possible proper-time over the journey. In the +weird geometry of the part of space-time through which it +passes (a geometry which is no doubt associated in some way +with our perception of the existence of a massive body, the sun) +this longest track is a spiral---a circle in space, drawn out into +a spiral by continuous displacement in time. Any other course +would have had shorter interval-length. + +In this way the study of fields of force is reduced to a study +of geometry. To a certain extent this is a retrograde step; we +adopt Kepler's description of the sun's gravitational field instead +of Newton's. The field of force is completely described if the +tracks through space and time of particles projected in every +possible way are prescribed. But we go back in order to go +forward in a new direction. To express this unmanageable mass +of detail in a unified way, a world-geometry is found in which +the tracks of greatest length are the actual tracks of the particles. +It only remains to express the laws of this geometry in a concise +form. The change from a mechanical to a geometrical theory of +fields of force is not so fundamental a change as might be +supposed. If we are now reducing mechanics to a branch of +natural geometry, we have to remember that natural geometry +is equally a branch of mechanics, since it is concerned with the +behaviour of material measuring-appliances. + +Reference has been made to weird geometry. There is no +%% -----File: 083.png---Folio 73------- +help for it, if the longest track can be a spiral like that known +to be described by the earth. Non-Euclidean geometry is +necessary. In Euclidean geometry the shortest track is always +a straight line; and the slight modification of Euclidean geometry +described in \Chapref{III} is found to give a straight line as the +longest track. The status of non-Euclidean geometry has already +been thrashed out in the Prologue; and there seems to be no +reason whatever for preferring Euclid's geometry unless observations +decide in its favour. +\index{Euclidean geometry}% +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% +\index{Semi-Euclidean geometry}% +\IndexExtra{Geometry!Euclidean}% +\Eqref{Equation}{1}, \Pageref{53}, is the expression +of the Euclidean (or semi-Euclidean) geometry we have hitherto +adopted; we shall have to modify it, if we adopt non-Euclidean +geometry. + +But the point arises that the geometry arrived at in \Chapref{III} +was not arbitrary. It was the synthesis of measures made with +clocks and scales, by observers with all kinds of uniform motion +relative to one another; we cannot modify it arbitrarily to fit +the behaviour of moving particles like the earth. Now, if the +worst came to the worst, and we could not reconcile a geometry +based on measures with clocks and scales and a geometry based +on the natural tracks of moving particles---if we had to select +one or the other and keep to it---I think we ought to prefer to +use the geometry based on the tracks of moving particles. The +free motion of a particle is an example of the simplest possible +kind of phenomenon; it is unanalysable; whereas, what the +readings of any kind of clock record, what the extension of a +material rod denotes, may evidently be complicated phenomena +involving the secrets of molecular constitution. Each geometry +would be right in its own sphere; but the geometry of moving +particles would be the more fundamental study. But it turns +out that there is probably no need to make the choice; clocks, +scales, moving particles, light-pulses, give the same geometry. +This might perhaps be expected since a clock must comprise +moving particles of some kind.% +\index{Clock-scale geometry, not fundamental}% + +A formula, such as \Eqref{equation}{1}, based on experiment can +only be verified to a certain degree of approximation. Within +certain limits it will be possible to introduce modifications. Now +it turns out that the free motion of a particle is a much more +sensitive way of exploring space-time, than any practicable +measures with scales and clocks. If then we employ our accurate +%% -----File: 084.png---Folio 74------- +knowledge of the motion of particles to correct the formula, we +shall find that the changes introduced are so small that they are +inappreciable in any practical measures with scales and clocks. +There is only one case where a possible detection of the modification +is indicated; this refers to the behaviour of a clock on the +surface of the sun, but the experiment is one of great difficulty +and no conclusive answer has been given. +\index{Clock!on sun}% +We conclude then +that the geometry of space and time based on the motions of +particles is accordant with the geometry based on the cruder +observations with clocks and scales; but if subsequent experiment +should reveal a discrepancy, we shall adhere to the moving +particle on account of its greater simplicity. + +The proposed modification can be regarded from another +point of view. \Eqref{Equation}{1} is the synthesis of the experiences +of all observers in uniform motion. But uniform motion means +that their four-dimensional tracks are straight lines. We must +suppose that the observers were moving in their natural tracks; +for, if not, they experienced fields of force, and presumably +allowed for these in their calculations, so that reduction was +made to the natural tracks. If then \Eqref{equation}{1} shows that +the natural tracks are straight lines, we are merely getting out +of the equation that which we originally put into it. + +The formula needs generalising in another way. Suppose there +is a region of space-time where, for some observer, the natural +tracks are all straight lines and \Eqref{equation}{1} holds rigorously. +For another (accelerated) observer the tracks will be curved, +and the equation will not hold. At the best it is of a form which +can only hold good for specially selected observers. + +Although it has become necessary to throw our formula into +the melting-pot, that does not create any difficulty in measuring +the interval. Without going into technical details, it may be +pointed out that the innovations arise solely from the introduction +of gravitational fields of force into our scheme. When +there is no force, the tracks of all particles are straight lines as +our previous geometry requires. In any small region we can +choose an observer (falling freely) for whom the force vanishes, +and accordingly the original formula holds good. Thus it is only +necessary to modify our rule for determining the interval by +two provisos (1)~that the interval measured must be small, +%% -----File: 085.png---Folio 75------- +(2)~that the scales and clocks used for measuring it must be +falling freely. The second proviso is natural, because, if we do +not leave our apparatus to fall freely, we must allow for the +strain that it undergoes. The first is not a serious disadvantage, +because a larger interval can be split up into a number of small +intervals and the parts measured separately. In mathematical +problems the same device is met with under the name of integration. +To emphasise that the formula is strictly true only for +infinitesimal intervals, it is written with a new notation +\Pagelabel{75} +\index{Interval!practical measurement of}% +\[ +ds^2 = - dx^2 - dy^2 - dz^2 + dt^2 +\Tag{2} +\] +where $dx$ stands for the small difference $x_2-x_1$, etc. + +The condition that the measuring appliances must not be +subjected to a field of force is illustrated by Ehrenfest's paradox. +\index{Ehrenfest's paradox}% +Consider a wheel revolving rapidly. Each portion of the +circumference is moving in the direction of its length, and +might be expected to undergo the FitzGerald contraction due +to its velocity; each portion of a radius is moving transversely +and would therefore have no longitudinal contraction. +\index{Acceleration!modifies FitzGerald contraction}% +\index{FitzGerald Contraction!modified by acceleration}% +It looks +as though the rim of the wheel should contract and the spokes +remain the same length, when the wheel is set revolving. The +conclusion is absurd, for a revolving wheel has no tendency to +buckle---which would be the only way of reconciling these +conditions. The point which the argument has overlooked is +that the results here appealed to apply to unconstrained bodies, +which have no acceleration relative to the natural tracks in +space. Each portion of the rim of the wheel has a radial acceleration, +and this affects its extensional properties. When accelerations +as well as velocities occur a more far-reaching theory is +needed to determine the changes of length. + +To sum up---the interval between two (near) events is something +quantitative which has an absolute significance in nature. +The track between two (distant) events which has the longest +interval-length must therefore have an absolute significance. +Such tracks are called \textit{geodesics}. +\index{Geodesic!definition of}% +Geodesics can be traced practically, +because they are the tracks of particles undisturbed by +material impacts. By the practical tracing of these geodesics +we have the best means of studying the character of the natural +geometry of the world. An auxiliary method is by scales and +%% -----File: 086.png---Folio 76------- +clocks, which, it is believed, when unconstrained, measure a +small interval according to \Eqref{formula}{2}. + +The identity of the two methods of exploring the geometry +of the world is connected with a principle which must now be +enunciated definitely. We have said that no experiments have +been able to detect a difference between a gravitational field +and an artificial field of force such as the centrifugal force. +\index{Force!relativity of}% +\index{Relativity of Force}% +This +is not quite the same thing as saying that it has been proved +that there is no difference. It is well to be explicit when a +positive generalisation is made from negative experimental +evidence. The generalisation which it is proposed to adopt is +known as the Principle of Equivalence.% +\index{Equivalence!Principle of}% +\index{Principle of Equivalence}% + +\textit{A gravitational field of force is precisely equivalent to an artificial +field of force, so that in any small region it is impossible by any +conceivable experiment to distinguish between them.} + +In other words, force is purely relative. +%% -----File: 087.png---Folio 77------- + + +\Chapter{V}{Kinds of Space} + +\Quote[break]{W.~K. Clifford ({\upshape and} K.~Pearson), \textit{Common Sense of the Exact Sciences.}} +{The danger of asserting dogmatically that an axiom based on the experience +of a limited region holds universally will now be to some extent apparent to +the reader. It may lead us to entirely overlook, or when suggested at once +reject, a possible explanation of phenomena. The hypothesis that space is not +homaloidal [flat], and again that its geometrical character may change with +the time, may or may not be destined to play a great part in the physics of the +future; yet we cannot refuse to consider them as possible explanations of +physical phenomena, because they may be opposed to the popular dogmatic +belief in the universality of certain geometrical axioms---a belief which has +risen from centuries of indiscriminating worship of the genius of Euclid.}% +\index{Clifford}% + +\First{On} any surface it requires two independent numbers or ``coordinates'' +to specify the position of a point. For this reason +a surface, whether flat or curved, is called a two-dimensional +space. Points in three-dimensional space require three, and in +four-dimensional space-time four numbers or coordinates.% +\index{Coordinates}% + +To locate a point on a surface by two numbers, we divide the +surface into meshes by any two systems of lines which cross one +another. Attaching consecutive numbers to the lines, or better +to the channels between them, one number from each system +will identify a particular mesh; and if the subdivision is sufficiently +fine any point can be specified in this way with all the accuracy +needed. This method is used, for example, in the Post Office +Directory of London for giving the location of streets on the +map. The point $(4, 2)$ will be a point in the mesh where channel +No.~$4$ of the first system crosses channel No.~$2$ of the second. +If this indication is not sufficiently accurate, we must divide +channel No.~$4$ into ten parts numbered $4.0$, $4.1$ etc. The subdivision +must be continued until the meshes are so small that +all points in one mesh can be considered identical within the +limits of experimental detection. + +The diagrams, Figs.~10, 11, 12, illustrate three of the many +kinds of mesh-systems commonly used on a flat surface.% +\index{Mesh-systems}% + +If we speak of the properties of the triangle formed by the +points $(1, 2)$, $(3, 0)$, $(4, 4)$, we shall be at once asked, What mesh-% +%% -----File: 088.png---Folio 78------- +system are you using? No one can form a picture of the triangle +until that information has been given. But if we speak of the +properties of a triangle whose sides are of lengths $2$, $3$, $4$~inches, +anyone with a graduated scale can draw the triangle, and follow +our discussion of its properties. The distance between two points +can be stated without referring to any mesh-system. For this +reason, if we use a mesh-system, it is important to find formulae +connecting the absolute distance with the particular system that +is being used. + +In the more complicated kinds of mesh-systems it makes a +great simplification if we content ourselves with the formulae for +very short distances. The mathematician then finds no difficulty +in extending the results to long distances by the process called +integration. We write $ds$ for the distance between two points +\begin{figure*}[hbt]% +\begin{center} +\Graphic[10]{\textwidth}{088a} +\Figlabel{11}% +\Figlabel{12}% +\end{center} +\end{figure*}% +close together, $x_1$ and $x_2$ for the two numbers specifying the +location of one of them, $dx_1$ and $dx_2$ for the small differences of +these numbers in passing from the first point to the second. +But in using one of the particular mesh-systems illustrated in +the diagrams, we usually replace $x_1$, $x_2$ by particular symbols +sanctioned by custom, viz.\ $(x_1, x_2)$ becomes $(x, y)$, $(r, \theta)$, $(\xi, \eta)$ +for Figs.~10, 11, 12, respectively. +%[Illustration: Fig. 10.] +%[Illustration: Fig. 11.] +%[Illustration: Fig. 12.] + +The formulae, found by geometry, are: + +For rectangular coordinates $(x, y)$, \Figref{10}, +\[ +ds^2 = dx^2 + dy^2. +\] + +For polar coordinates $(r, \theta)$, \Figref{11}, +\[ +ds^2 = dr^2 + r^2\, d\theta^2. +\] + +For oblique coordinates $(\xi, \eta)$, \Figref{12}, +\[ +ds^2 = d\xi^2 - 2\kappa\, d\xi d\eta + d\eta^2, +\] +where $\kappa$ is the cosine of the angle between the lines of partition. + +%% -----File: 089.png---Folio 79------- + +As an example of a mesh-system on a curved surface, we may +take the lines of latitude and longitude on a sphere. + +For latitude and longitude $(\beta, \lambda)$ +\Pagelabel{79} +\[ +ds^2 = d\beta^2 + \cos^2 \beta\, d\lambda^2. +\] + +These expressions form a test, and in fact the only possible +test, of the kind of coordinates we are using. It may perhaps +seem inconceivable that an observer should for an instant be in +doubt whether he was using the mesh-system of \Figref{10} or +\Figref{11}. He sees at a glance that \Figref{11} is not what he would +call a rectangular mesh-system. But in that glance, he makes +measures with his eye, that is to say he determines $ds$ for pairs +of points, and he notices how these values are related to the +number of intervening channels. In fact he is testing which +formula for $ds$ will fit. For centuries man was in doubt whether +the earth was flat or round---whether he was using plane rectangular +coordinates or some kind of spherical coordinates. In +some cases an observer adopts his mesh-system blindly and long +afterwards discovers by accurate measures that $ds$ does not fit +the formula he assumed---that his mesh-system is not exactly of +the nature he supposed it was. In other cases he deliberately +sets himself to plan out a mesh-system of a particular variety, +say rectangular coordinates; he constructs right angles and rules +parallel lines; but these constructions are all measurements of +the way the $x$-channels and $y$-channels ought to go, and the +rules of construction reduce to a formula connecting his measures +$ds$ with $x$ and~$y$. + +The use of special symbols for the coordinates, varying +according to the kind of mesh-system used, thus anticipates a +knowledge which is really derived from the form of the formulae. +In order not to give away the secret prematurely, it will be +better to use the symbols $x_1$, $x_2$ in all cases. The four kinds of +coordinates already considered then give respectively the relations, +\begin{align*} +ds^2 &= dx_1{}^2 + dx_2{}^2 && \text{(rectangular)}, \\ +ds^2 &= dx_1{}^2 + x_1{}^2\, dx_2{}^2 && \text{(polar)}, \\ +ds^2 &= dx_1{}^2 - 2 \kappa\, dx_1 dx_2 + dx_2{}^2 + && \text{(oblique)}, \\ +ds^2 &= dx_1{}^2 + \cos^2 x_1\, dx_2{}^2 && \text{(latitude and longitude)}. +\end{align*} +If we have any mesh-system and want to know its nature, we +%% -----File: 090.png---Folio 80------- +must make a number of measures of the length $ds$ between +adjacent points $(x_1, x_2)$ and $(x_1 + dx_1, x_2 + dx_2)$ and test which +formula fits. If, for example, we then find that $ds^2$ is always +equal to $dx_1{}^2 + x_1{}^2\, dx_2{}^2$, we know that our mesh-system is like +that in \Figref{11}, $x_1$ and $x_2$ being the numbers usually denoted by +the polar coordinates $r$, $\theta$. The statement that polar coordinates +are being used is unnecessary, because it adds nothing to our +knowledge which is not already contained in the formula. It is +merely a matter of giving a name; but, of course, the name calls +to our minds a number of familiar properties which otherwise +might not occur to us. + +For instance, it is characteristic of the polar coordinate system +that there is only one point for which $x_1$ (or~$r$) is equal to~$0$, +whereas in the other systems $x_1 = 0$ gives a line of points. This +is at once apparent from the formula; for if we have two points +for which $x_1 = 0$ and $x_1 + dx_1 = 0$, respectively, then +\[ +dx_1{}^2 + x_1{}^2\, dx_2{}^2 = 0. +\] +The distance $ds$ between the two points vanishes, and accordingly +they must be the same point. + +The examples given can all be summed up in one general +expression +\[ +ds^2 = g_{11}\, dx_1{}^2 + 2g_{12}\, dx_1 dx_2 + g_{22}\, dx_2{}^2, +\] +where $g_{11}$, $g_{12}$, $g_{22}$ may be constants or functions of $x_1$ and~$x_2$. +For instance, in the fourth example their values are $1$, $0$, $\cos^2 x_1$. +It is found that all possible mesh-systems lead to values of $ds^2$ +which can be included in an expression of this general form; so +that mesh-systems are distinguished by three functions of +position $g_{11}$, $g_{12}$, $g_{22}$ which can be determined by making physical +measurements. These three quantities are sometimes called +potentials.% +\index{Potentials}% + +We now come to a point of far-reaching importance. The +formula for $ds^2$ teaches us not only the character of the mesh-system, +but the nature of our two-dimensional space, which is +independent of any mesh-system. If $ds^2$ satisfies any one of the +first three formulae, then the space is like a flat surface; +\index{Flat space in two dimensions}% +if it +satisfies the last formula, then the space is a surface curved like +a sphere. Try how you will, you cannot draw a mesh-system on +a flat (Euclidean) surface which agrees with the fourth formula. + +%% -----File: 091.png---Folio 81------- + +If a being limited to a two-dimensional world finds that his +measures agree with the first formula, he can make them agree +with the second or third formulae by drawing the meshes +differently. But to obtain the fourth formula he must be translated +to a different world altogether. + +We thus see that there are different kinds of two-dimensional +space, betrayed by different metrical properties. +\index{Kinds of space}% +\index{Space!kinds of}% +They are +naturally visualised as different surfaces in Euclidean space of +three dimensions. This picture is helpful in some ways, but +perhaps misleading in others. The metrical relations on a plane +sheet of paper are not altered when the paper is rolled into a +cylinder---the measures being, of course, confined to the two-dimensional +world represented by the paper, and not allowed to +take a short cut through space. The formulae apply equally +well to a plane surface or a cylindrical surface; and in so far as +our picture draws a distinction between a plane and a cylinder, +it is misleading. +\index{Cylinder and plane, indistinguishable in two dimensions}% +But they do not apply to a sphere, because +a plane sheet of paper cannot be wrapped round a sphere. +A genuinely two-dimensional being could not be cognisant of +the difference between a cylinder\footnote% + {One should perhaps rather say a roll, to avoid any question of joining the + two edges.} and a plane; but a sphere +would appear as a different kind of space, and he would recognise +the difference by measurement. + +Of course there are many kinds of mesh-systems, and many +kinds of two-dimensional spaces, besides those illustrated in the +four examples. Clearly it is not going to be a simple matter to +discriminate the different kinds of spaces by the values of the~$g$'s. +There is no characteristic, visible to cursory inspection, +which suggests why the first three formulae should all belong to +the same kind of space, and the fourth to a different one. +Mathematical investigation has discovered what is the common +link between the first three formulae. The $g_{11}$, $g_{12}$, $g_{22}$ satisfy in +all three cases a certain differential equation\footnotemark; and whenever +this differential equation is satisfied, the same kind of space +occurs. +\footnotetext{Appendix, \Noteref{4}.} +\Pagelabel{note4} + +No doubt it seems a very clumsy way of approaching these +intrinsic differences of kinds of space---to introduce potentials +%% -----File: 092.png---Folio 82------- +which specifically refer to a particular mesh-system, although +the mesh-system can have nothing to do with the matter. It is +worrying not to be able to express the differences of space in a +purer form without mixing them up with irrelevant differences +of potential. But we have neither the vocabulary nor the +imagination for a description of absolute properties as such. +All physical knowledge is relative to space and time partitions; +and to gain an understanding of the absolute it is necessary to +approach it through the relative. +\index{Absolute@Absolute, approached through the relative}% +The absolute may be defined +as a relative which is always the same no matter what it is +relative to\footnote{Cf.\ \Pageref{31}, where a distinction was drawn between knowledge which does +not particularise the observer and knowledge which does not postulate an +observer at all.}. Although we think of it as self-existing, we cannot +give it a place in our knowledge without setting up some dummy +to relate it to. And similarly the absolute differences of space +always appear as related to some mesh-system, although the +mesh-system is only a dummy and has nothing to do with the +problem. + +The results for two dimensions can be generalised, and applied +to four-dimensional space-time. +\index{Four-dimensional space-time!geometry of}% +Distance must be replaced by +interval, which it will be remembered, is an absolute quantity, +and therefore independent of the mesh-system used. Partitioning +space-time by any system of meshes, a mesh being given by the +crossing of four channels, we must specify a point in space-time +by four coordinate numbers, $x_1$, $x_2$, $x_3$, $x_4$. By analogy the +general formula will be +\index{Interval!general expression for}% +\iffalse %%%%%%%%%% DEAD CODE (matches original) %%%%%%%%%% +\begin{multline*} +ds^2 = g_{11}\, dx_1{}^2 + g_{22}\, dx_2{}^2 + + g_{33}\, dx_3{}^2 + g_{44}\, dx_4{}^2 + + 2g_{12}\, dx_1 dx_2 \\ + + 2g_{13}\, dx_1 dx_3 + + 2g_{14}\, dx_1 dx_4 + + 2g_{23}\, dx_2 dx_3 \\ + + 2g_{24}\, dx_2 dx_4 + + 2g_{34}\, dx_3 dx_4. +\Tag{3} +\end{multline*} +\fi %%%%%%%%%% END OF DEAD CODE %%%%%%%%%% +\begin{align*} % [** PP: Re-breaking] +ds^2 &= g_{11}\, dx_1{}^2 + g_{22}\, dx_2{}^2 + + g_{33}\, dx_3{}^2 + g_{44}\, dx_4{}^2 \\ + &\quad+ 2g_{12}\, dx_1 dx_2 + 2g_{13}\, dx_1 dx_3 + 2g_{14}\, dx_1 dx_4 \\ + &\quad+ 2g_{23}\, dx_2 dx_3 + 2g_{24}\, dx_2 dx_4 + 2g_{34}\, dx_3 dx_4. +\Tag{3} +\end{align*} +The only difference is that there are now ten $g$'s, or potentials, +instead of three, to summarise the metrical properties of the +mesh-system. It is convenient in specifying special values of +the potentials to arrange them in the standard form +\[ +\begin{matrix} +g_{11} & g_{12} & g_{13} & g_{14} \\ + & g_{22} & g_{23} & g_{24} \\ + & & g_{33} & g_{34} \\ + & & & g_{44} +\end{matrix} +\] +%% -----File: 093.png---Folio 83------- +The space-time already discussed at length in \Chapref{III} +corresponded to the \Eqref{formula}{2}, \Pageref{75}, +\[ +ds^2 = - dx^2 - dy^2 - dz^2 + dt^2. +\] +Here $(x, y, z, t)$ are the conventional symbols for $(x_1, x_2, x_3, x_4)$ +when this special mesh-system is used, viz.\ rectangular coordinates +and time. Comparing with~(3) the potentials have the +special values +\index{Flat space-time}% +\index{Galilean potentials}% +\index{Potentials!Galilean values}% +\[ +\begin{matrix} +-1 & \Neg0 & \Neg0 & \Neg0 \\ + & -1 & \Neg0 & \Neg0 \\ + & & -1 & \Neg0 \\ + & & & +1 +\end{matrix} +\] +These are called the ``Galilean values.'' If the potentials have +these values everywhere, space-time may be called ``flat,'' +because the geometry is that of a plane surface drawn in +Euclidean space of five dimensions. +\index{Euclidean space of five dimensions}% +Recollecting what we found +for two dimensions, we shall realise that a quite different set +of values of the potentials may also belong to flat space-time, +because the meshes may be drawn in different ways. We must +clearly understand that + +(1)~The only way of discovering what kind of space-time is +being dealt with is from the values of the potentials, which are +determined practically by measurements of intervals, + +(2)~Different values of the potentials do not necessarily +indicate different kinds of space-time, + +(3)~There is some complicated mathematical property +common to all values of the potentials which belong to the +same space-time, which is not shared by those which belong to +a different kind of space-time. This property is expressed by +a set of differential equations. + +It can now be deduced that the space-time in which we live +is not quite flat. If it were, a mesh-system could be drawn for +which the $g$'s have the Galilean values, and the geometry with +respect to these partitions of space and time would be that +discussed in \Chapref{III}. For that geometry the geodesics, giving +the natural tracks of particles, are straight lines. + +Thus in flat space-time the law of motion is that (with +suitably chosen coordinates) every particle moves uniformly in +a straight line except when it is disturbed by the impacts of +%% -----File: 094.png---Folio 84------- +other particles. Clearly this is not true of our world; for example, +the planets do not move in straight lines although they do not +suffer any impacts. It is true that if we confine attention to a +small region like the interior of Jules Verne's projectile, all the +tracks become straight lines for an appropriate observer, or, +as we generally say, he detects no field of force. It needs a +large region to bring out the differences of geometry. That is +not surprising, because we cannot expect to tell whether a +surface is flat or curved unless we consider a reasonably large +portion of it.% +\index{Flat space-time!at infinity}% + +According to Newtonian ideas, at a great distance from all +matter beyond the reach of any gravitation, particles would all +move uniformly in straight lines. Thus at a great distance from +all matter space-time tends to become perfectly flat. This can +only be checked by experiment to a certain degree of accuracy, +and there is some doubt as to whether it is rigorously true. We +shall leave this afterthought to \Chapref{X}, meanwhile assuming +with Newton that space-time far enough away from everything +is flat, although near matter it is curved. It is this puckering +near matter which accounts for its gravitational effects. + +Just as we picture different kinds of two-dimensional space +as differently curved surfaces in our ordinary space of three-dimensions, +so we are now picturing different kinds of four-dimensional +space-time as differently curved surfaces in a +Euclidean space of \textit{five} dimensions. This is a picture only\footnotemark. + \footnotetext{A fifth dimension suffices for illustrating the property here considered; + \index{Curvature!merely illustrative}% + but for an exact representation of the geometry of the world, Euclidean space + of \textit{ten} dimensions is required. We may well ask whether there is merit in + Euclidean geometry sufficient to justify going to such extremes.}% +The fifth dimension is neither space nor time nor anything that +can be perceived; so far as we know, it is nonsense. I should not +describe it as a mathematical fiction, because it is of no great +advantage in a mathematical treatment. It is even liable to +mislead because it draws distinctions, like the distinction between +a plane and a roll, which have no meaning. It is, like +the notion of a field of force acting in space and time, merely +introduced to bolster up Euclidean geometry, when Euclidean +geometry has been found inappropriate. +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% +The real difference +between the various kinds of space-time is that they have +%% -----File: 095.png---Folio 85------- +different kinds of geometry, involving different properties of the~$g$'s. +It is no explanation to say that this is because the surfaces +are differently curved in a real Euclidean space of five dimensions. +We should naturally ask for an explanation why the space of +five dimensions is Euclidean; and presumably the answer would +be, because it is a plane in a real Euclidean space of six dimensions, +and so on \textit{ad infinitum}. + +The value of the picture to us is that it enables us to describe +important properties with common terms like ``pucker'' and +``curvature'' instead of technical terms like ``differential +invariant.'' +\index{Pucker in space-time}% +We have, however, to be on our guard, because +analogies based on three-dimensional space do not always apply +immediately to many-dimensional space. The writer has keen +recollections of a period of much perplexity, when he had not +realised that a four-dimensional space with ``no curvature'' is +not the same as a ``flat'' space! Three-dimensional geometry +does not prepare us for these surprises. + +Picturing the space-time in the gravitational field round the +earth as a pucker, we notice that we cannot locate the pucker +at a point; it is ``somewhere round'' the point. At any special +point the pucker can be pressed out flat, and the irregularity +runs off somewhere else. That is what the inhabitants of Jules +Verne's projectile did; they flattened out the pucker inside the +projectile so that they could not detect any field of force there; +but this only made things worse somewhere else, and they +would find an increased field of force (relative to them) on the +other side of the earth. + +What determines the existence of the pucker is not the values +of the $g$'s at any point, or, what comes to the same thing, the +field of force there. It is the way these values link on to those +at other points---the gradient of the~$g$'s, and more particularly +the gradient of the gradient. Or, as has already been said, the +kind of space-time is fixed by differential equations. + +Thus, although a gravitational field of force is not an absolute +thing, and can be imitated or annulled at any point by an +acceleration of the observer or a change of his mesh-system, +nevertheless the presence of a heavy particle does modify the +world around it in an absolute way which cannot be imitated +artificially. Gravitational force is relative; but there is this +%% -----File: 096.png---Folio 86------- +more complex character of gravitational influence which is +absolute. + +The question must now be put, Can every possible kind of +space-time occur in an empty region in nature? Suppose we +give the ten potentials perfectly arbitrary values at every point; +that will specify the geometry of some mathematically possible +space-time. But could that kind of space-time actually occur---by +any arrangement of the matter round the region? + +The answer is that only certain kinds of space-time can occur +in an empty region in nature. The law which determines what +kinds can occur is the law of gravitation. + +It is indeed clear that, since we have reduced the theory of +fields of force to a theory of the geometry of the world, if there +is any law governing fields of force (including the gravitational +field), that law must be of the nature of a restriction on the +possible geometries of the world. + +The choice of $g$'s in any special problem is thus arrived at by +a three-fold sorting out: (1)~many sets of values can be dismissed +because they can never occur in nature, (2)~others, while possible, +do not relate to the kind of space-time present in the problem +considered, (3)~of those which remain, one set of values relates +to the particular mesh-system that has been chosen. We have +now to find the law governing the first discrimination. What is +the criterion that decides what values of the $g$'s give a kind of +space-time possible in nature? + +In solving this problem Einstein had only two clues to guide +him. + +(1)~Since it is a question of whether the \textit{kind of space-time} is +possible, the criterion must refer to those properties of the $g$'s +which distinguish different kinds of space-time, not to those +which distinguish different kinds of mesh-system in the same +space-time. The formulae must therefore not be altered in any +way, if we change the mesh-system. + +(2)~We know that flat space-time \textit{can} occur in nature (at +great distances from all gravitating matter). Hence the criterion +must be satisfied by any values of the $g$'s belonging to flat +space-time. + +It is remarkable that these slender clues are sufficient to +indicate almost uniquely a particular law. Afterwards the +%% -----File: 097.png---Folio 87------- +further test must be applied---whether the law is confirmed by +observation. + +\index{Coincidences|(}% +The irrelevance of the mesh-system to the laws of nature is +sometimes expressed in a slightly different way. +\index{Mesh-systems!irrelevance to laws of nature}% +There is one +type of observation which, we can scarcely doubt, must be +independent of any possible circumstances of the observer, +namely a complete coincidence in space and time. The track of +a particle through four-dimensional space-time is called its +world-line. +\index{World-line}% +Now, the world-lines of two particles either intersect +or they do not intersect; the standpoint of the observer is not +involved. In so far as our knowledge of nature is a knowledge +of intersections of world-lines, it is absolute knowledge independent +of the observer. If we examine the nature of our +observations, distinguishing what is actually seen from what is +merely inferred, we find that, at least in all exact measurements, +our knowledge is primarily built up of intersections of world-lines +of two or more entities, that is to say their coincidences. +For example, an electrician states that he has observed a current +of $5$~milliamperes. This is his inference: his actual observation +was a \textit{coincidence} of the image of a wire in his galvanometer +with a division of a scale. A meteorologist finds that the temperature +of the air is~$75°$; his observation was the \textit{coincidence} of +the top of the mercury-thread with division $75$ on the scale of +his thermometer. It would be extremely clumsy to describe the +results of the simplest physical experiment entirely in terms of +coincidence. The absolute observation is, whether or not the +coincidence exists, not when or where or under what circumstances +the coincidence exists; unless we are to resort to relative +knowledge, the place, time and other circumstances must in +their turn be described by reference to other coincidences. But +it seems clear that if we could draw all the world-lines so as to +show all the intersections in their proper order, but otherwise +arbitrary, this would contain a complete history of the world, +and nothing within reach of observation would be omitted.% +\index{Coincidences|)}% + +Let us draw such a picture, and imagine it embedded in a +jelly. If we deform the jelly in any way, the intersections will +still occur in the same order along each world-line and no +additional intersections will be created. The deformed jelly will +represent a history of the world, just as accurate as the one +%% -----File: 098.png---Folio 88------- +originally drawn; there can be no criterion for distinguishing +which is the best representation. + +Suppose now we introduce space and time-partitions, which +we might do by drawing rectangular meshes in both jellies. +We have now two ways of locating the world-lines and events +in space and time, both on the same absolute footing. But +clearly it makes no difference in the result of the location whether +we first deform the jelly and then introduce regular meshes, or +whether we introduce irregular meshes in the undeformed jelly. +And so all mesh-systems are on the same footing. + +This account of our observational knowledge of nature shows +that there is no \textit{shape} inherent in the absolute world, so that +when we insert a mesh-system, it has no shape initially, and a +rectangular mesh-system is intrinsically no different from any +other mesh-system. + +Returning to our two clues, condition (1)~makes an extraordinarily +clean sweep of laws that might be suggested; among +them Newton's law is swept away. The mode of rejection can +be seen by an example; it will be sufficient to consider two +dimensions. If in one mesh-system $(x, y)$ +\begin{align*} +ds^2 &= g_{11} dx^2 + 2g_{12} dxdy + g_{22} dy^2, \\ +\intertext{and in another system $(x', y')$} +ds^2 &= g_{11}'\, {dx'}^2 + 2g_{12}'\, dx'dy' + g_{22}'\, dy'^2, +\end{align*} +the same law must be satisfied if the unaccented letters are +throughout replaced by accented letters. Suppose the law +$g_{11} = g_{22}$ is suggested. Change the mesh-system by spacing the +$y$-lines twice as far apart, that is to say take $y' = \frac{1}{2}y$, with +$x' = x$. Then +\begin{DPalign*} +ds^2 &= g_{11}\, dx^2 + 2g_{12}\, dx dy + g_{22}\, dy^2 \\ + &= g_{11}\, {dx'}^2 + 4g_{12}\, dx'dy' + 4g_{22}\, {dy'}^2, \\ +\lintertext{so that} + &\quad {g_{11}}' = g_{11},\qquad {g_{22}}' = 4g_{22}. +\end{DPalign*} +And if $g_{11}$ is equal to $g_{22}$, $g_{11}'$ cannot be equal to $g_{22}'$. + +After a few trials the reader will begin to be surprised that +any possible law could survive the test. It seems so easy to +defeat any formula that is set up by a simple change of mesh-system. +Certainly it is unlikely that anyone would hit on such +a law by trial. But there are such laws, composed of exceedingly +complicated mathematical expressions. The theory of these is +%% -----File: 099.png---Folio 89------- +called the ``theory of tensors,'' and had already been worked +out by the pure mathematicians Riemann, Christoffel, Ricci, +Levi-Civita who, it may be presumed, never dreamt of a physical +application for it.% +\index{Christoffel}% +\index{Levi-Civita}% +\index{Ricci}% +\index{Riemann}% + +One law of this kind is the condition for flat space-time, +which is generally written in the simple, but not very illuminating, +form +\index{Flat space-time!conditions for}% +\[ +B ^\rho _{\mu\nu\sigma} = 0. +\Tag{4} +\] +The quantity on the left is called the Riemann-Christoffel +tensor, and it is written out in a less abbreviated form in the +Appendix\footnote{Appendix, \Noteref{5}.}. % [** PP: Added ``Appendix''] +\Pagelabel{note5}% +\index{Riemann-Christoffel tensor}% +\index{Tensors}% +It must be explained that the letters $\mu$, $\nu$, $\sigma$, $\rho$ +indicate \textit{gaps}, which are to be filled up by any of the numbers +$1$, $2$, $3$, $4$, chosen at pleasure. (When the expression is written +out at length, the gaps are in the suffixes of the $x$'s and $g$'s.) +Filling the gaps in different ways, a large number of expressions, +$B^1_{111}$, $B^4_{123}$, $B^1_{432}$, etc., are obtained. The \Eqref{equation}{4} states that +all of these are zero. There are $4^4$, or~$256$, of these expressions +altogether, but many of them are repetitions. Only 20 of the +equations are really necessary; the others merely say the same +thing over again. + +It is clear that the \Eqref{law}{4} is not the law of gravitation for +which we are seeking, because it is much too drastic. If it were +a law of nature, then only flat space-time could exist in nature, +and there would be no such thing as gravitation. It is not the +general condition, but a special case---when all attracting +matter is infinitely remote. + +But in finding a general condition, it may be a great help to +know a special case. Would it do to select a certain number of +the 20~equations to be satisfied generally, leaving the rest to +be satisfied only in the special case? Unfortunately the equations +hang together; and, unless we take them all, it is found that +the condition is not independent of the mesh-system. But there +happens to be one way of building up out of the 20~conditions +a less stringent set of conditions independent of the mesh-system. +Let +\begin{align*} +G_{11} + &= B^1_{111} + B^2_{112} + B^3_{113} + B^4_{114}, \\ +\intertext{and, generally} +G_{\mu\nu} + &= B^1_{\mu\nu 1} + B^2_{\mu\nu 2} + B^3_{\mu\nu 3} + B^4_{\mu\nu 4}, +\end{align*} +%% -----File: 100.png---Folio 90------- +then the conditions +\[ +G_{\mu\nu} = 0 +\Tag{5} +\] +will satisfy our requirements for a general law of nature. + +This law is independent of the mesh-system, though this can +only be proved by elaborate mathematical analysis. Evidently, +when all the $B$'s vanish, \Eqref{equation}{5} is satisfied; so, when flat +space-time occurs, this law of nature is not violated. Further +it is not so stringent as the condition for flatness, and admits +of the occurrence of a limited variety of non-Euclidean geometries. +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% +Rejecting duplicates, it comprises 10~equations; but four +of these can be derived from the other six, so that it gives +six conditions, which happens to be the number required for a +law of gravitation\footnotemark. + \footnotetext{Isolate a region of empty space-time; and suppose that everywhere outside + the region the potentials are known. It should then be possible by the law of + gravitation to determine the nature of space-time in the region. Ten differential + equations together with the boundary-values would suffice to determine the + ten potentials throughout the region; but that would determine not only the + kind of space-time but the mesh-system, whereas the partitions of the mesh-system + can be continued across the region in any arbitrary way. The four + sets of partitions give a four-fold arbitrariness; and to admit of this, the number + of equations required is reduced to six.} + +The suggestion is thus reached that +\index{Gravitation, Einstein's law of!differential formula}% +\[ +G_{\mu\nu} = 0 +\] +may be the general law of gravitation. Whether it is so or not +can only be settled by experiment. In particular, it must in +ordinary cases reduce to something so near the Newtonian law, +that the remarkable confirmation of the latter by observation +is accounted for. Further it is necessary to examine whether +there are any exceptional cases in which the difference between +it and Newton's law can be tested. We shall see that these +tests are satisfied. + +What would have been the position if this suggested law had +failed? We might continue the search for other laws satisfying +the two conditions laid down; but these would certainly be far +more complicated mathematically. I believe too that they would +not help much, because practically they would be indistinguishable +from the simpler law here suggested---though this has not +been demonstrated rigorously. The other alternative is that +there is something causing force in nature not comprised in the +%% -----File: 101.png---Folio 91------- +geometrical scheme hitherto considered, so that force is not +purely relative, and Newton's super-observer exists. + +Perhaps the best survey of the meaning of our theory can be +obtained from the standpoint of a ten-dimensional Euclidean +continuum, in which space-time is conceived as a particular +four-dimensional surface. It has to be remarked that in ten +dimensions there are gradations intermediate between a flat +surface and a fully curved surface, which we shall speak of as +curved in the ``first degree'' or ``second degree\footnote{This is not a recognised nomenclature.}.'' The distinction +is something like that of curves in ordinary space, +which may be \textit{curved} like a circle, or \textit{twisted} like a helix; but the +analogy is not very close. The full ``curvature'' of a surface is a +single quantity called~$G$, built up out of the various terms $G_{\mu\nu}$ in +somewhat the same way as these are built up out of $B^\rho_{\mu\nu\sigma}$. +The following conclusions can be stated.% +\index{Curvature!degrees of}% + +\begin{DPgather*} +\lintertext{\indent If} + B^\rho_{\mu\nu\sigma} = 0 +\rintertext{(20 conditions)\quad} \\ +\intertext{space-time is flat. This is the state of the world at an infinite +distance from all matter and all forms of energy.} +% +\lintertext{\indent If} + G_{\mu\nu} = 0 +\rintertext{(6 conditions)\quad} \\ +\intertext{space-time is curved in the first degree. This is the state of the +world in an empty region---not containing matter, light or +electromagnetic fields, but in the neighbourhood of these forms +of energy.} +% +\lintertext{\indent If} + G = 0 +\rintertext{(1 condition)\quad} \\ +\intertext{space-time is curved in the second degree. This is the state of +the world in a region not containing matter or electrons (bound +energy), but containing light or electromagnetic fields (free +energy).} +% +\lintertext{\indent If} +G~\text{is not zero} +\end{DPgather*} +space-time is fully curved. This is the state of the world in a +region containing continuous matter.% +\index{Continuous matter}% +\index{Matter!continuous}% + +According to current physical theory continuous matter does +not exist, so that strictly speaking the last case never arises. +Matter is built of electrons or other nuclei. The regions lying +between the electrons are not fully curved, whilst the regions +inside the electrons must be cut out of space-time altogether. +\index{Electron!geometry inside}% +We cannot imagine ourselves exploring the inside of an electron +%% -----File: 102.png---Folio 92------- +with moving particles, light-waves, or material clocks and +measuring-rods; hence, without further definition, any geometry +of the interior, or any statement about space and time in the +interior, is meaningless. But in common life, and frequently in +physics, we are not concerned with this \textit{microscopic} structure of +matter. We need to know, not the actual values of the $g$'s at +a point, but their average values through a region, small from +the ordinary standpoint but large compared with the molecular +structure of matter. In this \textit{macroscopic} treatment molecular +matter is replaced by continuous matter, and uncurved space-time +studded with holes is replaced by an equivalent fully +curved space-time without holes.% +\index{Macroscopic!equations}% + +It is natural that our senses should have developed faculties +for perceiving some of these intrinsic distinctions of the possible +states of the world around us. I prefer to think of matter and +energy, not as agents causing the degrees of curvature of the +world, but as parts of our perceptions of the existence of the +curvature. + +It will be seen that the law of gravitation can be summed up +in the statement that in an empty region space-time can be +curved only in the first degree. +%% -----File: 103.png---Folio 93------- + +% [** PP: Re-breaking title to match running heads] +\Chapter[The New Law of Gravitation and the Old Law]{VI}% +{The New Law of Gravitation \break and the Old Law} + +% [** PP: Special headers] +\fancyhead[CE]{\textsc{THE NEW LAW OF GRAVITATION}} +\fancyhead[CO]{\textsc{AND THE OLD LAW}} + +\Quote{Sir Isaac Newton.} +{I don't know what I may seem to the world, but, as to myself, I seem to have +been only as a boy playing on the sea-shore, and diverting myself in now and +then finding a smoother pebble or a prettier shell than ordinary, whilst the +great ocean of truth lay all undiscovered before me.} + + +\First{Was} there any reason to feel dissatisfied with Newton's law of +gravitation? + +Observationally it had been subjected to the most stringent +tests, and had come to be regarded as the perfect model of an +exact law of nature. The cases, where a possible failure could +be alleged, were almost insignificant. There are certain unexplained +irregularities in the moon's motion; but astronomers +generally looked---and must still look---in other directions for +the cause of these discrepancies. One failure only had led to +a serious questioning of the law; this was the discordance of +motion of the perihelion of Mercury. How small was this discrepancy +may be judged from the fact that, to meet it, it was +proposed to amend \textit{square} of the distance to the $2.00000016$ +power of the distance. Further it seemed possible, though +unlikely, that the matter causing the zodiacal light might be of +sufficient mass to be responsible for this effect.% +\index{Gravitation, Newton's law of!ambiguity of}% +\index{Moon, motion of}% +\index{Newton!law of gravitation}% + +The most serious objection against the Newtonian law as an +exact law was that it had become ambiguous. The law refers +to the product of the masses of the two bodies; but the mass +depends on the velocity---a fact unknown in Newton's day. +Are we to take the variable mass, or the mass reduced to rest? +Perhaps a learned judge, interpreting Newton's statement like +a last will and testament, could give a decision; but that is +scarcely the way to settle an important point in scientific +theory. + +Further \textit{distance}, also referred to in the law, is something +relative to an observer. Are we to take the observer travelling +with the sun or with the other body concerned, or at rest in the +aether or in some gravitational medium? + +%% -----File: 104.png---Folio 94------- + +Finally is the force of gravitation propagated instantaneously, +or with the velocity of light, or some other velocity? +\index{Gravitation!propagation with velocity of light}% +\index{Propagation of Gravitation}% +\index{Velocity of gravitation}% +Until +comparatively recently it was thought that conclusive proof +had been given that the speed of gravitation must be far higher +than that of light. The argument was something like this. If +the Sun attracts Jupiter towards its present position~$S$, and +Jupiter attracts the Sun towards its present position~$J$, the two +forces are in the same line and balance. But if the Sun attracts +Jupiter towards its previous position~$S'$, and Jupiter attracts +the Sun towards its previous position~$J'$, when the force of +attraction started out to cross the gulf, then the two forces +%[Illustration: Fig. 13.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[13]{3.5in}{104a} +\end{center} +\end{figure*}% +give a couple. This couple will tend to increase the angular +momentum of the system, and, acting cumulatively, will soon +cause an appreciable change of period, disagreeing with observation +if the speed is at all comparable with that of light. The +argument is fallacious, because the effect of propagation will not +necessarily be that $S$ is attracted in the direction towards~$J'$. +Indeed it is found that if $S$ and~$J$ are two electric charges, $S$ will +be attracted very approximately towards~$J$ (not~$J'$) in spite of +the electric influence being propagated with the velocity of +light\footnotemark. In the theory given in this book, gravitation is propagated +with the speed of light, and there is no discordance with +observation. +\footnotetext{Appendix, \Noteref{6}.} +\Pagelabel{note6} + +It is often urged that Newton's law of gravitation is much +%% -----File: 105.png---Folio 95------- +simpler than Einstein's new law. That depends on the point of +view; and from the point of view of the four-dimensional world +Newton's law is far more complicated. Moreover, it will be seen +that if the ambiguities are to be cleared up, the statement of +Newton's law must be greatly expanded. + +Some attempts have been made to expand Newton's law on +the basis of the restricted principle of relativity (\Pageref{20}) alone. +This was insufficient to determine a definite amendment. Using +the principle of equivalence, or relativity of force, we have +arrived at a definite law proposed in the last chapter. Probably +the question has arisen in the reader's mind, why should it be +called the law of gravitation? It may be plausible as a law of +nature; but what has the degree of curvature of space-time to +do with attractive forces, whether real or apparent? + +A race of flat-fish once lived in an ocean in which there were +only two dimensions. +\index{Flatfish, analogy of} % [** PP: No hyphen in index] +It was noticed that in general fishes swam +in straight lines, unless there was something obviously interfering +with their free courses. This seemed a very natural behaviour. +But there was a certain region where all the fish seemed to be +bewitched; some passed through the region but changed the +direction of their swim, others swam round and round indefinitely. +One fish invented a theory of vortices, and said that +there were whirlpools in that region which carried everything +round in curves. By-and-by a far better theory was proposed; +it was said that the fishes were all attracted towards a particularly +large fish---a sun-fish---which was lying asleep in the middle +of the region; and that was what caused the deviation of their +paths. The theory might not have sounded particularly plausible +at first; but it was confirmed with marvellous exactitude by all +kinds of experimental tests. All fish were found to possess this +attractive power in proportion to their sizes; the law of attraction +was extremely simple, and yet it was found to explain all the +motions with an accuracy never approached before in any +scientific investigations. Some fish grumbled that they did not +see how there could be such an influence at a distance; but it +was generally agreed that the influence was communicated +through the ocean and might be better understood when more +was known about the nature of water. Accordingly, nearly +every fish who wanted to explain the attraction started by +%% -----File: 106.png---Folio 96------- +proposing some kind of mechanism for transmitting it through +the water. + +But there was one fish who thought of quite another plan. +He was impressed by the fact that whether the fish were big +or little they always took the same course, although it would +naturally take a bigger force to deflect the bigger fish. He therefore +concentrated attention on the courses rather than on the +forces. And then he arrived at a striking explanation of the +whole thing. There was a mound in the world round about +where the sun-fish lay. Flat-fish could not appreciate it directly +because they were two-dimensional; but whenever a fish went +swimming over the slopes of the mound, although he did his +best to swim straight on, he got turned round a bit. (If a traveller +goes over the left slope of a mountain, he must consciously +keep bearing away to the left if he wishes to keep to his original +direction relative to the points of the compass.) This was the +secret of the mysterious attraction, or bending of the paths, +which was experienced in the region. + +The parable is not perfect, because it refers to a hummock in +space alone, whereas we have to deal with hummocks in space-time. +But it illustrates how a curvature of the world we live +in may give an illusion of attractive force, and indeed can only +be discovered through some such effect. How this works out in +detail must now be considered. + +In the form $G_{\mu\nu} =0$, Einstein's law expresses conditions to be +satisfied in a gravitational field produced by any arbitrary +distribution of attracting matter. An analogous form of Newton's +law was given by Laplace in his celebrated expression $\nabla^{2} V = 0$. +\index{Laplace's equation}% +A more illuminating form of the law is obtained if, instead of +putting the question what kinds of space-time can exist under +the most general conditions in an empty region, we ask what +kind of space-time exists in the region round a single attracting +particle? We separate out the effect of a single particle, just as +Newton did. We can further simplify matters by introducing +some definite mesh-system, which, of course, must be of a type +which is not inconsistent with the kind of space-time found. + +We need only consider space of two dimensions---sufficient +for the so-called plane orbit of a planet---time being added as +the third dimension. The remaining dimension of space can +%% -----File: 107.png---Folio 97------- +always be added, if desired, by conditions of symmetry. The +result of long algebraic calculations\footnote{Appendix, \Noteref{7}.} +\Pagelabel{note7}% +is that, round a particle +\index{Gravitation, Einstein's law of!integrated formula for a particle}% +\[ +ds^{2} + = - \frac{1}{\gamma}\, dr^{2} + - r^{2}\, d\theta^{2} + + \gamma\, dt^{2} +\Tag{6} +\] +where $\gamma = 1 - \dfrac{2m} {r}$. + +The quantity $m$ is the gravitational mass of the particle---but +we are not supposed to know that at present. $r$~and $\theta$ are +polar coordinates, the mesh-system being as in \Figref{11}; or rather +they are the nearest thing to polar coordinates that can be +found in space which is not truly flat. + +The fact is that this expression for $ds^{2}$ is found in the first +place simply as a particular solution of Einstein's equations of +the gravitational field; it is a variety of hummock (apparently +the simplest variety) which is not curved beyond the first degree. +\index{Hummock in space-time}% +There \textit{could} be such a state of the world under suitable circumstances. +To find out what those circumstances are, we have to +trace some of the consequences, find out how any particle +moves when $ds^{2}$ is of this form, and then examine whether we +know of any case in which these consequences are found +observationally. It is only after having ascertained that this +form of $ds^{2}$ does correspond to the leading observed effects +attributable to a particle of mass $m$ at the origin that we have +the right to identify this particular solution with the one we +hoped to find. + +It will be a sufficient illustration of this procedure, if we +indicate how the position of the matter causing this particular +solution is located. Wherever the \Eqref{formula}{6} holds good there +can be no matter, because the law which applies to empty space +is satisfied. But if we try to approach the origin ($r = 0$), a +curious thing happens. Suppose we take a measuring-rod, and, +laying it radially, start marking off equal lengths with it along +a radius, gradually approaching the origin. Keeping the time +$t$ constant, and $d\theta$ being zero for radial measurements, the +\Eqref{formula}{6} reduces to +\begin{DPalign*} +ds^{2} &= - \frac{1}{\gamma}\, dr^{2} \\ +\lintertext{or} +dr^{2} &= - \gamma\, ds^{2}. +\end{DPalign*} +%% -----File: 108.png---Folio 98------- +We start with $r$ large. By-and-by we approach the point +where $r = 2m$. But here, from its definition, $\gamma$ is equal to~$0$. +So that, however large the measured interval $ds$ may be, $dr = 0$. +We can go on shifting the measuring-rod through its own length +time after time, but $dr$ is zero; that is to say, we do not reduce~$r$. +There is a magic circle which no measurement can bring us +inside. It is not unnatural that we should picture something +obstructing our closer approach, and say that a particle of +matter is filling up the interior.% +\index{Matter!definition of a particle}% + +The fact is that so long as we keep to space-time curved only +in the first degree, we can never round off the summit of the +hummock. It must end in an infinite chimney. In place of the +chimney, however, we round it off with a small region of greater +curvature. This region cannot be empty because the law applying +to empty space does not hold. We describe it therefore as containing +matter---a procedure which practically amounts to a +definition of matter. Those familiar with hydrodynamics may +be reminded of the problem of the irrotational rotation of a +fluid; the conditions cannot be satisfied at the origin, and it is +necessary to cut out a region which is filled by a vortex-filament. + +A word must also be said as to the coordinates $r$ and $t$ used +in~(6). They correspond to our ordinary notion of radial distance +and time---as well as any variables in a non-Euclidean world +can correspond to words which, as ordinarily used, presuppose +a Euclidean world. We shall thus call $r$ and~$t$, distance and time. % [** PP: Retaining comma] +But to give names to coordinates does not give more information---and +in this case gives considerably less information---than is +already contained in the formula for $ds^{2}$. If any question arises +as to the exact significance of $r$ and $t$ it must always be settled +by reference to \Eqref{equation}{6}. + +The want of flatness in the gravitational field is indicated by +the deviation of the coefficient $\gamma$ from unity. If the mass $m = 0$, +$\gamma = 1$, and space-time is perfectly flat. Even in the most intense +gravitational fields known, the deviation is extremely small. +For the sun, the quantity $m$, called the gravitational mass, is +only $1.47$~kilometres\footnote{Appendix, \Noteref{8}.}, +\Pagelabel{note8}% +for the earth it is $5$~millimetres. +\index{Gravitational field of Sun}% +\index{Mass!gravitational}% +In any +practical problem the ratio $2m/r$ must be exceedingly small. +%% -----File: 109.png---Folio 99------- +Yet it is on the small corresponding difference in $\gamma$ that the +whole of the phenomena of gravitation depend. + +The coefficient $\gamma$ appears twice in the formula, and so modifies +the flatness of space-time in two ways. But as a rule these two +ways are by no means equally important. Its appearance as a +coefficient of $dt^{2}$ produces much the most striking effects. +Suppose that it is wished to measure the interval between two +events in the history of a planet. If the events are, say $1$~second +apart in time, $dt = 1~\text{second} = 300,000~\text{kilometres}$. Thus +$dt^{2}= 90,000,000,000~\text{sq.\ km}$. Now no planet moves more than +$50$~kilometres in a second, so that the change $dr$ associated with +the lapse of $1$~second in the history of the planet will not be +more than $50$~km. Thus $dr^{2}$ is not more than $2500$ sq.~km. +Evidently the small term $2m/r$ has a much greater chance of +making an impression where it is multiplied by $dt^{2}$ than where +it is multiplied by~$dr^{2}$. + +Accordingly as a first approximation, we ignore the coefficient +of~$dr^{2}$, and consider only the meaning of +\[ +ds^{2} = - dr^{2} - r^{2}\, d\theta^{2} + (1 - 2m/r)\, dt^{2}. +\Tag{7} +\] +We shall now show that particles situated in this kind of space-time +will appear to be under the influence of an attractive force +directed towards the origin. + +Let us consider the problem of mapping a small portion of this +kind of world on a plane. + +It is first necessary to define carefully the distinction which is +here drawn between a ``picture'' and a ``map.'' If we are given +the latitudes and longitudes of a number of places on the earth, +we can make a picture by taking latitude and longitude as +vertical and horizontal distances, so that the lines of latitude +and longitude form a mesh-system of squares; but that does not +give a true map. In an ordinary map of Europe the lines of +longitude run obliquely and the lines of latitude are curved. +Why is this? Because the map aims at showing as accurately +as possible all distances in their true proportions\footnotemark. + \footnotetext{This is usually the object, though maps are sometimes made for a different + purpose, e.g.\ Mercator's Chart.}% +\index{Map of sun's gravitational field}% +Distance is +the important thing which it is desired to represent correctly. +In four dimensions interval is the analogue of distance, and a +map of the four-dimensional world will aim at showing all the +%% -----File: 110.png---Folio 100------- +intervals in their correct proportions. Our natural \textit{picture} of +space-time takes $r$ and $t$ as horizontal and vertical distances, +e.g.\ when we plot the graph of the motion of a particle; but in +a true \textit{map}, representing the intervals in their proper proportions, +the $r$ and $t$ lines run obliquely or in curves across the map. + +The instructions for drawing latitude and longitude lines $(\beta, \lambda)$ +on a map, are summed up in the formula for $ds$, \Pageref{79}, +\[ +ds^{2} = d\beta^{2} + \cos^{2}\beta\, d\lambda^{2}, +\] +%[Illustration: \textsc{Fig}. 14.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[14]{3in}{110a} +\end{center} +\end{figure*}% +and similarly the instructions for drawing the $r$ and $t$ lines are +given by the \Eqref{formula}{7}. + +The map is shown in \Figref{14}. It is not difficult to see why the +$t$-lines converge to the left of the diagram. The factor $1 - 2m/r$ +decreases towards the left where $r$ is small; and consequently +any change of $t$ corresponds to a shorter interval, and must be +represented in the map by a shorter distance on the left. It is +less easy to see why the $r$-lines take the courses shown; by +analogy with latitude and longitude we might expect them to +be curved the other way. But we discussed in \Chapref{III} how +%% -----File: 111.png---Folio 101------- +the slope of the time-direction is connected with the slope of +the space-direction; and it will be seen that the map gives +approximately diamond-shaped partitions of the kind represented +in \Figref{6}\footnotemark. + \footnotetext{The substitution $x = r + \frac{1}{2} t^{2} m/r^{2}$, + $y=t(1-m/r)$, gives $ds^{2}= -dx^{2} + dy^{2}$, if + squares of~$m$ are negligible. The map is drawn with $x$ and $y$ as rectangular + coordinates.} + +Like all maps of curved surfaces, the diagram is only accurate +in the limit when the area covered is very small. + +It is important to understand clearly the meaning of this map. +When we speak in the ordinary way of distance from the sun +and the time at a point in the solar system, we mean the two +variables $r$ and~$t$. These are not the result of any precise measures +with scales and clocks made at a point, but are mathematical +variables most appropriate for describing the whole solar system. +%[Illustration: \textsc{Fig}. 15.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[15]{3.5in}{111a} +\end{center} +\end{figure*}% +They represent a compromise, because it is necessary to deal +with a region too large for accurate representation on a plane +map. We should naturally picture them as rectangular coordinates +partitioning space-time into square meshes, as in +\Figref{15}; but such a picture is not a true map, because it does +not represent in their true proportions the intervals between the +various points in the picture. It is not possible to draw any +map of the whole curved region without distortion; but a small +enough portion can be represented without distortion if the +partitions of equal $r$ and $t$ are drawn as in \Figref{14}. To get back +%% -----File: 112.png---Folio 102------- +from the true map to the customary picture of $r$ and $t$ as perpendicular +space and time, we must strain \Figref{14} until all the +meshes become squares as in \Figref{15}. + +Now in the map the geometry is Euclidean and the tracks of +all material particles will be straight lines. Take such a straight +track~$PQ$, which will necessarily be nearly vertical, unless the +velocity is very large. Strain the figure so as to obtain the +customary representation of $r$ and~$t$ (in \Figref{15}), and the track +$PQ$ will become curved---curved towards the left, where the sun +lies. In each successive vertical interval (time), a successively +greater progress is made to the left horizontally (space). Thus +the velocity towards the sun increases. We say that the particle +is attracted to the sun. + +The mathematical reader should find no difficulty in proving +from the diagram that for a particle with small velocity the +acceleration towards the sun is approximately $m/r^{2}$, agreeing +with the Newtonian law.% +\index{Gravitational field of Sun!Newtonian attraction}% + +Tracks for very high speeds may be affected rather differently. +The track corresponding to a wave of light is represented by +a straight line at~$45°$ to the horizontal in \Figref{14}. It would +require very careful drawing to trace what happens to it when +the strain is made transforming to \Figref{15}; but actually, whilst +becoming more nearly vertical, it receives a curvature in the +opposite direction. The effect of the gravitation of the sun on +a light-wave, or very fast particle, proceeding radially is actually +a \textit{repulsion}!% +\index{Repulsion of light proceeding radially} + +The track of a transverse light-wave, coming out from the +plane of the paper, will be affected like that of a particle of +zero velocity in distorting from \Figref{14} to \Figref{15}. Hence the +sun's influence on a transverse light-wave is always an attraction. +The acceleration is simply $m/r^{2}$ as for a particle at rest. + +The result that the expression found for the geometry of the +gravitational field of a particle leads to Newton's law of attraction +is of great importance. It shows that the law, $G_{\mu\nu}= 0$, +proposed on theoretical grounds, agrees with observation at +least approximately. It is no drawback that the Newtonian +law applies only when the speed is small; all planetary speeds +are small compared with the velocity of light, and the considerations +mentioned at the beginning of this chapter suggest that +%% -----File: 113.png---Folio 103------- +some modification may be needed for speeds comparable with +that of light. + +Another important point to notice is that the attraction of +gravitation is simply a geometrical deformation of the straight +tracks. It makes no difference what body or influence is pursuing +the track, the deformation is a general discrepancy between the +``mental picture'' and the ``true map'' of the portion of space-time +considered. Hence light is subject to the same disturbance +of path as matter. This is involved in the Principle of Equivalence; +otherwise we could distinguish between the acceleration +of a lift and a true increase of gravitation by optical experiments; +in that case the observer for whom light-rays appear to +take straight tracks might be described as absolutely unaccelerated +and there could be no relativity theory. Physicists in +general have been prepared to admit the likelihood of an +influence of gravitation on light similar to that exerted on +matter; and the problem whether or not light has ``weight'' +has often been considered. + +The appearance of $\gamma$ as the coefficient of $dt^{2}$ is responsible for +the main features of Newtonian gravitation; the appearance of +$1/\gamma$ as the coefficient of $dr^{2}$ is responsible for the principal +deviations of the new law from the old. This classification seems +to be correct; but the Newtonian law is ambiguous and it is +difficult to say exactly what are to be regarded as discrepancies +from it. Leaving aside now the time-term as sufficiently discussed, +we consider the space-terms alone\footnote + {We change the sign of $ds^{2}$, so that $ds$, when real, means measured space + instead of measured time.} +\index{Gravitation, Newton's law of!approximation to Einstein's law}% +\[ +ds^{2} = \frac{1}{\gamma}\, dr^{2} + r^{2}\, d\theta^{2}. +\] + +The expression shows that space considered alone is non-Euclidean +in the neighbourhood of an attracting particle. This +is something entirely outside the scope of the old law of gravitation. +Time can only be explored by something moving, whether +a free particle or the parts of a clock, so that the non-Euclidean +character of space-time can be covered up by introducing a field +of force, suitably modifying the motion, as a convenient fiction. +But space can be explored by static methods; and theoretically +%% -----File: 114.png---Folio 104------- +its non-Euclidean character could be ascertained by sufficiently +precise measures with rigid scales. + +If we lay our measuring scale transversely and proceed to +measure the circumference of a circle of nominal radius~$r$, we +see from the formula that the measured length~$ds$ is equal to +$r\,d\theta$, so that, when we have gone right round the circle, $\theta$~has +increased by~$2\pi$ and the measured circumference is $2\pi r$. +\index{Circle in non-Euclidean space}% +But +when we lay the scale radially the measured length~$ds$ is equal +to $dr/\sqrt{\gamma}$, which is always greater than~$dr$. Thus, in measuring +a diameter, we obtain a result greater than~$2r$, each portion being +greater than the corresponding change of~$r$. + +Thus if we draw a circle, placing a massive particle near the +centre so as to produce a gravitational field, and measure with +a rigid scale the circumference and the diameter, the ratio of +the measured circumference to the measured diameter will not be +the famous number $\pi = 3.141592653589793238462643383279\ldots$ +but a little smaller. Or if we inscribe a regular hexagon in this +circle its sides will not be exactly equal to the radius of the +circle. Placing the particle near, instead of at, the centre, +avoids measuring the diameter \textit{through} the particle, and so +makes the experiment a practical one. But though practical, +it is not practicable to determine the non-Euclidean character +of space in this way. Sufficient refinement of measures is not +attainable. If the mass of a ton were placed inside a circle of +$5$~yards radius, the defect in the value of $\pi$ would only appear +in the twenty-fourth or twenty-fifth place of decimals. + +It is of value to put the result in this way, because it shows +that the relativist is not talking metaphysics when he says that +space in the gravitational field is non-Euclidean. His statement +has a plain physical meaning, which we may some day learn how +to test experimentally. Meanwhile we can test it by indirect +methods. + +Suppose that a plane field is uniformly studded with hurdles. +\index{Hurdles, analogy of counts of}% +The distance between any two points will be proportional to +the number of hurdles that must be passed over in getting from +one point to the other by the straight route---in fact the minimum +number of hurdles. We can use counts of hurdles as the equivalent +of distance, and map the field by these counts. The map +can be drawn on a plane sheet of paper without any inconsistency, +%% -----File: 115.png---Folio 105------- +since the field is plane. Let us now dismiss from our +minds all idea of distances in the field or straight lines in the +field, and assume that distances on the map merely represent +the minimum number of hurdles between two points; straight +lines on the map will represent the corresponding routes. This +has the advantage that if an earthquake occurs, deforming the +field, the map will still be correct. The path of fewest hurdles +will still cross the same hurdles as before the earthquake; it +will be twisted out of the straight line in the field; but we should +gain nothing by taking a straighter course, since that would +lead through a region where the hurdles are more crowded. +We do not alter the number of hurdles in any path by deforming +it. + +This can be illustrated by Figs.~14 and~15. \Figref{14} represents +the original undistorted field with the hurdles uniformly placed. +The straight line $PQ$ represents the path of fewest hurdles from +$P$ to~$Q$, and its length is proportional to the number of hurdles. +\Figref{15} represents the distorted field, with $PQ$ distorted into +a curve; but $PQ$ is still the path of fewest hurdles from $P$ to~$Q$, +and the number of hurdles in the path is the same as before. +If therefore we map according to hurdle-counts we arrive at +\Figref{14} again, just as though no deformation had taken place. + +To make any difference in the hurdle-counts, the hurdles +must be taken up and replanted. Starting from a given point +as centre, let us arrange them so that they gradually thin out +towards the boundaries of the field. Now choose a circle with +this point as centre;---but first, what is a circle? It has to be +defined in terms of hurdle-counts; and clearly it must be a +curve such that the minimum number of hurdles between any +point on it and the centre is a constant (the radius). With this +definition we can defy earthquakes. The number of hurdles in +the circumference of such a circle will not bear the same proportion +to the number in the radius as in the field of uniform +hurdles; owing to the crowding near the centre, the ratio will +be less. Thus we have a suitable analogy for a circle whose +circumference is less than $\pi$ times its diameter. + +This analogy enables us to picture the condition of space +round a heavy particle, where the ratio of the circumference of +a circle to the diameter is less than~$\pi$. Hurdle-counts will no +%% -----File: 116.png---Folio 106------- +longer be accurately mappable on a plane sheet of paper, +because they do not conform to Euclidean geometry. + +Now suppose a heavy particle wishes to cross this field, +passing near but not through the centre. In Euclidean space, +with the hurdles uniformly distributed, it travels in a straight +line, i.e.\ it goes between any two points by a path giving the +fewest hurdle jumps. We may assume that in the non-Euclidean +field with rearranged hurdles, the particle still goes by the path +of least effort. In fact, in any small portion we cannot distinguish +between the rearrangement and a distortion; so we may imagine +that the particle takes each portion as it comes according to the +rule, and is not troubled by the rearrangement which is only +visible to a general survey of the whole field\footnotemark. + \footnotetext{There must be some absolute track, and if absolute significance can only + be associated with hurdle-counts and not with distances in the field, the path + of fewest hurdles is the only track capable of absolute definition.} + +Now clearly it will pay not to go straight through the dense +portion, but to keep a little to the outside where the hurdles +are sparser---not too much, or the path will be unduly lengthened. +The particle's track will thus be a little concave to the centre, +and an onlooker will say that it has been attracted to the centre. +It is rather curious that we should call it attraction, when the +track has rather been avoiding the central region; but it is clear +that the direction of motion has been bent round in the way +attributable to an attractive force. + +This bending of the path is additional to that due to the +Newtonian force of gravitation which depends on the second +appearance of~$\gamma$ in the formula. As already explained it is in +general a far smaller effect and will appear only as a minute +correction to Newton's law. The only case where the two rise +to equal importance is when the track is that of a light-wave, +or of a particle moving with a speed approaching that of light; +for then $dr^{2}$ rises to the same order of magnitude as~$dt^{2}$. + +To sum up, a ray of light passing near a heavy particle will +be bent, firstly, owing to the non-Euclidean character of the +combination of time with space. This bending is equivalent to +that due to Newtonian gravitation, and may be calculated in +the ordinary way on the assumption that light has weight like +a material body. Secondly, it will be bent owing to the non-% +%% -----File: 117.png---Folio 107------- +Euclidean character of space alone, and this curvature is +additional to that predicted by Newton's law. If then we can +observe the amount of curvature of a ray of light, we can make +a crucial test of whether Einstein's or Newton's theory is +obeyed. + +This separation of the attraction into two parts is useful in +a comparison of the new theory with the old; but from the +point of view of relativity it is artificial. Our view is that light +is bent just in the same way as the track of a material particle +moving with the same velocity would be bent. +\index{Bending of light!theory of}% +\index{Light, bending of}% +\index{Weight!of light}% +Both causes of +bending may be ascribed either to weight or to non-Euclidean +space-time, according to the nomenclature preferred. The only +difference between the predictions of the old and new theories +is that in one case the weight is calculated according to Newton's +law of gravitation, in the other case according to Einstein's. + +There is an alternative way of viewing this effect on light +according to Einstein's theory, which, for many reasons is to +be preferred. This depends on the fact that the velocity of +light in the gravitational field is not a constant (unity) but +becomes smaller as we approach the sun. This does not mean +that an observer determining the velocity of light experimentally +at a spot near the sun would detect the decrease; if he performed +Fizeau's experiment, his result in kilometres per second would +be exactly the same as that of a terrestrial observer. It is the +coordinate velocity that is here referred to, described in terms +of the quantities $r$, $\theta$, $t$, introduced by the observer who is +contemplating the whole solar system at the same time.% +\index{Coordinate velocity}% +\index{Light!coordinate velocity of}% + +It will be remembered that in discussing the approximate +geometry of space-time in \Figref{3}, we found that certain events +like $P$ were in the absolute past or future of~$O$, and others like +$P'$ were neither before nor after~$O$, but elsewhere. Analytically +the distinction is that for the interval $OP$, $ds^{2}$ is positive; for +$OP'$, $ds^{2}$ is negative. In the first case the interval is real or +``time-like''; in the second it is imaginary or ``space-like.'' The +two regions are separated by lines (or strictly, cones) in crossing +which $ds^{2}$ changes from positive to negative; and along the lines +themselves $ds$ is zero. It is clear that these lines must have +important absolute significance in the geometry of the world. +Physically their most important property is that pulses of light +%% -----File: 118.png---Folio 108------- +travel along these tracks, and the motion of a light-pulse is +always given by the equation $ds = 0$. + +Using the expression for $ds^{2}$ in a gravitational field, we +accordingly have for light +\[ +0 = -\frac{1}{\gamma}\, dr^{2} + -r^{2}\, d\theta^{2} + \gamma\, dt^{2}. +\] +For radial motion, $d\theta = 0$, and therefore +\[ +\left(\frac{dr}{dt}\right)^{2} = \gamma^{2}. +\] +For transverse motion, $dr = 0$, and therefore +\[ +\left(\frac{r\, d\theta}{dt}\right)^{2} = \gamma. +\] +Thus the coordinate velocity of light travelling radially is $\gamma$, +and of light travelling transversely is $\surd\gamma$, in the coordinates +chosen.% +\index{Velocity of light!in gravitational field}% + +The coordinate velocity must depend on the coordinates +chosen; and it is more convenient to use a slightly different +system in which the velocity of light is the same in all directions\footnotemark, + \footnotetext{This is obtained by writing $r + m$ instead of~$r$, or diminishing the nominal + distance of the sun by $1\frac{1}{2}$~kilometres. This change of coordinates simplifies + the problem, but can, of course, make no difference to anything observable. + After we have traced the course of the light ray in the coordinates chosen, we + have to connect the results with experimental measures, using the corresponding + formula for~$ds^{2}$. This final connection of mathematical and experimental results + is, however, comparatively simple, because it relates to measuring operations + performed in a terrestrial observatory where the difference of $\gamma$ from unity is + negligible.}% +viz.\ $\gamma$ or~$1 - 2m/r$. This diminishes as we approach the sun---an +illustration of our previous remark that a pulse of light +proceeding radially is repelled by the sun.% +\index{Repulsion of light proceeding radially}% + +The wave-motion in a ray of light can be compared to a +succession of long straight waves rolling onward in the sea. If +the motion of the waves is slower at one end than the other, the +whole wave-front must gradually slew round, and the direction +in which it is rolling must change. +\index{Wave-front, slewing of}% +In the sea this happens when +one end of the wave reaches shallow water before the other, +because the speed in shallow water is slower. It is well known +that this causes waves proceeding diagonally across a bay to +slew round and come in parallel to the shore; the advanced end +%% -----File: 119.png---Folio 109------- +is delayed in the shallow water and waits for the other. In the +same way when the light waves pass near the sun, the end nearest +the sun has the smaller velocity and the wave-front slews round; +thus the course of the waves is bent. + +Light moves more slowly in a material medium than in +vacuum, the velocity being inversely proportional to the refractive +index of the medium. The phenomenon of refraction +is in fact caused by a slewing of the wave-front in passing into +a region of smaller velocity. We can thus imitate the gravitational +effect on light precisely, if we imagine the space round +the sun filled with a refracting medium which gives the +appropriate velocity of light. +\index{Refracting medium equivalent to gravitational field}% +To give the velocity $1-2m/r$, the +refractive index must be $1/(1-2m/r)$, or, very approximately, +$1 + 2m/r$. At the surface of the sun, $r= 697,000~\text{km.}$, $m = 1.47~\text{km.}$, +hence the necessary refractive index is~$1.00000424$. At a +height above the sun equal to the radius it is~$1.00000212$. + +Any problem on the paths of rays near the sun can now be +solved by the methods of geometrical optics applied to the +equivalent refracting medium. It is not difficult to show that +the total deflection of a ray of light passing at a distance $r$ from +the centre of the sun is (in circular measure) +\index{Deflection of light!theory of}% +\index{Gravitational field of Sun!deflection of light}% +\[ +\frac{4m}{r}, +\] +whereas the deflection of the same ray calculated on the +Newtonian theory would be +\index{Gravitation, Newton's law of!deflection of light}% +\[ +\frac{2m}{r}. +\] + +For a ray grazing the surface of the sun the numerical value +of this deflection is +\begin{align*} +&1''.75 \quad \text{(Einstein's theory)}, \\ +&0''.87 \quad \text{(Newtonian theory)}. +\end{align*} +%% -----File: 120.png---Folio 110------- + + +\Chapter{VII}{Weighing Light} + +\Quote{Newton, \textit{Opticks}.} +{Query~1. Do not Bodies act upon Light at a distance, and by their action +bend its Rays, and is not this action (\textit{caeteris paribus}) strongest at the least +distance?}% +\index{Newton!bending of light}% + +\First{We} come now to the experimental test of the influence of +gravitation on light discussed theoretically in the last chapter. +It is not the general purpose of this book to enter into details +of experiments; and if we followed this plan consistently, we +should, as hitherto, summarise the results of the observations +in a few lines. But it is this particular test which has turned +public attention towards the relativity theory, and there appears +to be widespread desire for information. We shall therefore tell +the story of the eclipse expeditions in some detail. It will make +a break in the long theoretical arguments, and will illustrate +the important applications of this theory to practical observations. + +It must be understood that there were two questions to +answer: firstly, whether light has weight (as suggested by +Newton), or is indifferent to gravitation; secondly, if it has +weight, is the amount of the deflection in accordance with +Einstein's or Newton's laws? + +It was already known that light possesses mass or inertia like +other forms of electromagnetic energy. +\index{Inertia!of light}% +\index{Light!mass of}% +\index{Mass of light}% +This is manifested in +the phenomena of radiation-pressure. +\index{Radiation-pressure}% +Some force is required to +stop a beam of light by holding an obstacle in its path; a searchlight % [** PP: Hyphenated across a line in original] +experiences a minute force of recoil just as if it were a +machine-gun firing material projectiles. The force, which is +predicted by orthodox electromagnetic theory, is exceedingly +minute; but delicate experiments have detected it. Probably +this inertia of radiation is of great cosmical importance, playing +a great part in the equilibrium of the more diffuse stars. Indeed +it is probably the agent which has carved the material of the +universe into stars of roughly uniform mass. Possibly the tails +of comets are a witness to the power of the momentum of sunlight, +%% -----File: 121.png---Folio 111------- +which drives outwards the smaller or the more absorptive +particles.% +\index{Comets!radiation-pressure in}% + +It is legitimate to speak of a pound of light as we speak of +a pound of any other substance. The mass of ordinary quantities +of light is however extremely small, and I have calculated that +at the low charge of~3\textit{d}.\ a unit, an Electric Light Company +would have to sell light at the rate of £140,000,000 a pound. +All the sunlight falling on the earth amounts to $160$~tons daily. + +It is perhaps not easy to realise how a wave-motion can have +inertia, and it is still more difficult to understand what is meant +by its having weight. Perhaps this will be better understood if +we put the problem in a concrete form. Imagine a hollow body, +with radiant heat or light-waves traversing the hollow; the +mass of the body will be the sum of the masses of the material +and of the radiant energy in the hollow; a greater force will be +required to shift it because of the light-waves contained in it. +Now let us weigh it with scales or a spring-balance. Will it also +weigh heavier on account of the radiation contained, or will the +weight be that of the solid material alone? If the former, then +clearly from this aspect light has weight; and it is not difficult +to deduce the effect of this weight on a freely moving light-beam +not enclosed within a hollow.% +\index{Light!weight of}% +\index{Momentum!of light}% +\index{Weight!of light}% +\index{Weight!of radio-active energy}% + +The effect of weight is that the radiation in the hollow body +acquires each second a downward momentum proportional to +its mass. This in the long run is transmitted to the material +enclosing it. For a free light-wave in space, the added momentum +combines with the original momentum, and the total +momentum determines the direction of the ray, which is +accordingly bent. Newton's theory suggests no means for +bringing about the bending, but contents itself with predicting +it on general principles. Einstein's theory provides a means, +viz.\ the variation of velocity of the waves. + +Hitherto mass and weight have always been found associated +in strict proportionality. One very important test had already +shown that this proportionality is not confined to material +energy. The substance uranium contains a great deal of radio-active +energy, presumably of an electromagnetic nature, which +it slowly liberates. The mass of this energy must be an appreciable +fraction of the whole mass of the substance. But it was shown +%% -----File: 122.png---Folio 112------- +by experiments with the Eötvös torsion-balance that the ratio +of weight to mass for uranium is the same as for all other +substances; so the energy of radio-activity has weight. +\index{Energy!weight of radio-active}% +\index{Eotvos@Eötvös torsion-balance}% +Still +even this experiment deals only with bound electromagnetic +energy, and we are not justified in deducing the properties of +the free energy of light. + +It is easy to see that a terrestrial experiment has at present +no chance of success. If the mass and weight of light are in the +same proportion as for matter, the ray of light will be bent +just like the trajectory of a material particle. On the earth a +rifle bullet, like everything else, drops $16$~feet in the first second, +$64$~feet in two seconds, and so on, below its original line of flight; +the rifle must thus be aimed above the target. Light would also +drop $16$~feet in the first second\footnotemark; + \footnotetext{Or $32$~feet according to Einstein's law. The fall increases with the speed of + the motion.}% +but, since it has travelled $186,000$ +miles along its course in that time, the bend is inappreciable. +% [Illustration: Fig. 16.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[16]{4.5in}{122a} +\end{center} +\end{figure*}% +In fact any terrestrial course is described so quickly that +gravitation has scarcely had time to accomplish anything. + +The experiment is therefore transferred to the neighbourhood +of the sun. There we get a pull of gravitation $27$~times more +intense than on the earth; and---what is more important---the +greater size of the sun permits a much longer trajectory throughout +which the gravitation is reasonably powerful. The deflection +in this case may amount to something of the order of a second +of arc, which for the astronomer is a fairly large quantity.% +\index{Deflection of light!effect on star's position}% +\index{Gravitation, Newton's law of!deflection of light}% [** PP: Index refers to p. 111] + +In \Figref{16} the line $EFQP$ shows the track of a ray of light +from a distant star~$P$ which reaches the earth~$E$. The main +part of the bending of the ray occurs as it passes the sun~$S$; +\index{Bending of light!effect on star's position}% +\index{Light, bending of}% +and the initial course~$PQ$ and the final course~$FE$ are practically +straight. Since the light rays enter the observer's eye or telescope +in the direction~$FE$, this will be the direction in which the star +appears. But its true direction from the earth is~$QP$, the initial +%% -----File: 123.png---Folio 113------- +course. So the star appears displaced outwards from its true +position by an angle equal to the total deflection of the light.% +\index{Displacement of star-images}% + +\Pagelabel{113}% +It must be noticed that this is only true because a star is so +remote that its true direction with respect to the earth~$E$ is +indistinguishable from its direction with respect to the point~$Q$. +For a source of light within the solar system, the apparent +displacement of the source is by no means equal to the deflection +of the light-ray. It is perhaps curious that the attraction of +light by the sun should produce an apparent displacement of +the star away from the sun; but the necessity for this is +clear. + +The bending affects stars seen near the sun, and accordingly +the only chance of making the observation is during a total +eclipse when the moon cuts off the dazzling light. +\index{Eclipse, observations during}% +Even then +there is a great deal of light from the sun's corona which stretches +far above the disc. It is thus necessary to have rather bright +stars near the sun, which will not be lost in the glare of the +corona. Further the displacements of these stars can only be +measured relatively to other stars, preferably more distant from +the sun and less displaced; we need therefore a reasonable +number of outer bright stars to serve as reference points. + +In a superstitious age a natural philosopher wishing to perform +an important experiment would consult an astrologer to ascertain +an auspicious moment for the trial. With better reason, an +astronomer to-day consulting the stars would announce that the +most favourable day of the year for weighing light is May 29. +The reason is that the sun in its annual journey round the +ecliptic goes through fields of stars of varying richness, but on +May~29 it is in the midst of a quite exceptional patch of bright +stars---part of the Hyades---by far the best star-field encountered. +Now if this problem had been put forward at some other period +of history, it might have been necessary to wait some thousands +of years for a total eclipse of the sun to happen on the lucky +date. But by strange good fortune an eclipse did happen on +May~29, 1919. Owing to the curious sequence of eclipses a +similar opportunity will recur in~1938; we are in the midst of +the most favourable cycle. It is not suggested that it is impossible +to make the test at other eclipses; but the work will +necessarily be more difficult. + +%% -----File: 124.png---Folio 114------- +Attention was called to this remarkable opportunity by the +Astron\-o\-mer Royal in March, 1917; and preparations were begun +by a Committee of the Royal Society and Royal Astronomical +Society for making the observations. Two expeditions were sent +to different places on the line of totality to minimise the risk +of failure by bad weather. Dr A.~C.~D. Crommelin +\index{Crommelin}% +and Mr~C. +Davidson went to Sobral in North Brazil; Mr E.~T. Cottingham +\index{Cottingham}% +\index{Davidson}% +and the writer went to the Isle of Principe in the Gulf of Guinea, +West Africa. +\index{Principe, eclipse expedition to}% +The instrumental equipment for both expeditions +was prepared at Greenwich Observatory under the care of the +Astronomer Royal; and here Mr Davidson made the arrangements +which were the main factor in the success of both +parties.% +\index{Greenwich, Royal Observatory}% + +The circumstances of the two expeditions were somewhat +different and it is scarcely possible to treat them together. We +shall at first follow the fortunes of the Principe observers. They +had a telescope of focal length $11$~feet $4$~inches. On their +photographs $1$~second of arc (which was about the largest displacement +to be measured) corresponds to about $\frac{1}{1500}$ inch---by +no means an inappreciable quantity. The aperture of the +object-glass was $13$~inches, but as used it was stopped down to +$8$~inches to give sharper images. It is necessary, even when the +exposure is only a few seconds, to allow for the diurnal motion +of the stars across the sky, making the telescope move so as to +follow them. But since it is difficult to mount a long and heavy +telescope in the necessary manner in a temporary installation +in a remote part of the globe, the usual practice at eclipses is +to keep the telescope rigid and reflect the stars into it by a +coelostat---a plane mirror kept revolving at the right rate by +clock-work. This arrangement was adopted by both expeditions. + +The observers had rather more than a month on the island +to make their preparations. On the day of the eclipse the +weather was unfavourable. When totality began the dark disc +of the moon surrounded by the corona was visible through cloud, +much as the moon often appears through cloud on a night when +no stars can be seen. There was nothing for it but to carry out +the arranged programme and hope for the best. One observer +was kept occupied changing the plates in rapid succession, whilst +the other gave the exposures of the required length with a screen +%% -----File: 125.png---Folio 115------- +held in front of the object-glass to avoid shaking the telescope in +any way. +\begin{verse} +For in and out, above, about, below \\ +'Tis nothing but a Magic \textit{Shadow}-show \\ +Played in a Box whose candle is the Sun \\ +Round which we Phantom Figures come and go. +\end{verse} +Our shadow-box takes up all our attention. There is a marvellous +spectacle above, and, as the photographs afterwards revealed, +a wonderful prominence-flame is poised a hundred thousand +miles above the surface of the sun. We have no time to snatch +a glance at it. We are conscious only of the weird half-light of +the landscape and the hush of nature, broken by the calls of the +observers, and beat of the metronome ticking out the $302$~seconds +of totality. + +Sixteen photographs were obtained, with exposures ranging +from $2$ to $20$~seconds. The earlier photographs showed no stars, +though they portrayed the remarkable prominence; but apparently +the cloud lightened somewhat towards the end of totality, +and a few images appeared on the later plates. In many cases +one or other of the most essential stars was missing through +cloud, and no use could be made of them; but one plate was +found showing fairly good images of five stars, which were +suitable for a determination. This was measured on the spot +a few days after the eclipse in a micrometric measuring-machine. +The problem was to determine how the apparent positions of +the stars, affected by the sun's gravitational field, compared +with the normal positions on a photograph taken when the sun +was out of the way. Normal photographs for comparison had +been taken with the same telescope in England in January. +The eclipse photograph and a comparison photograph were +placed film to film in the measuring-machine so that corresponding +images fell close together\footnotemark, + \footnotetext{This was possible because at Principe the field of stars was reflected in + the coelostat mirror, whereas in England it was photographed direct.}% +and the small distances +were measured in two rectangular directions. From these the +relative displacements of the stars could be ascertained. +\index{Displacement of star-images}% +In +comparing two plates, various allowances have to be made for +refraction, aberration, plate-orientation, etc.; but since these +occur equally in determinations of stellar parallax, for which +%% -----File: 126.png---Folio 116------- +much greater accuracy is required, the necessary procedure is +well-known to astronomers. + +The results from this plate gave a definite displacement, in +good accordance with Einstein's theory and disagreeing with +the Newtonian prediction. Although the material was very +meagre compared with what had been hoped for, the writer +(who it must be admitted was not altogether unbiassed) believed +it convincing. + +It was not until after the return to England that any further +confirmation was forthcoming. Four plates were brought home +undeveloped, as they were of a brand which would not stand +development in the hot climate. One of these was found to +show sufficient stars; and on measurement it also showed the +deflection predicted by Einstein, confirming the other plate. + +The bugbear of possible systematic error affects all investigations +of this kind. How do you know that there is not something +in your apparatus responsible for this apparent deflection? +Your object-glass has been shaken up by travelling; you have +introduced a mirror into your optical system; perhaps the $50°$ +rise of temperature between the climate at the equator and +England in winter has done some kind of mischief. To meet +this criticism, a different field of stars was photographed at +night in Principe and also in England at the same altitude as +the eclipse field. If the deflection were really instrumental, stars +on these plates should show relative displacements of a similar +kind to those on the eclipse plates. But on measuring these +check-plates no appreciable displacements were found. That +seems to be satisfactory evidence that the displacement observed +during the eclipse is really due to the sun being in the region, +and is not due to differences in instrumental conditions between +England and Principe. Indeed the only possible loophole is a +difference between the night conditions at Principe when the +check-plates were taken, and the day, or rather eclipse, conditions +when the eclipse photographs were taken. That seems +impossible since the temperature at Principe did not vary more +than $1°$ between day and night. + +The problem appeared to be settled almost beyond doubt; +and it was with some confidence that we awaited the return of +the other expedition from Brazil. +\index{Brazil, eclipse expedition to}% +The Brazil party had had +%% -----File: 127.png---Folio 117------- +fine weather and had gained far more extensive material on +their plates. They had remained two months after the eclipse +to photograph the same region before dawn, when clear of the +sun, in order that they might have comparison photographs +taken under exactly the same circumstances. One set of +photographs was secured with a telescope similar to that used +at Principe. In addition they used a longer telescope of 4 inches +aperture and $19$~feet focal length\footnotemark. + \footnotetext{See \hyperref[frontispiece]{Frontispiece}. The two telescopes are shown and the backs of the two + coelostat-mirrors which reflect the sky into them. The clock driving the larger + mirror is seen on the pedestal on the left.}% +The photographs obtained +with the former were disappointing. Although the full number +of stars expected (about~12) were shown, and numerous plates +had been obtained, the definition of the images had been spoiled +by some cause, probably distortion of the coelostat-mirror by +the heat of the sunshine falling on it. The observers were +pessimistic as to the value of these photographs; but they were +the first to be measured on return to England, and the results +came as a great surprise after the indications of the Principe +plates. The measures pointed with all too good agreement to +the ``half-deflection,'' that is to say, the Newtonian value which +is one-half the amount required by Einstein's theory. It seemed +difficult to pit the meagre material of Principe against the wealth +of data secured from the clear sky of Sobral. +\index{Sobral, eclipse expedition to}% +\Pagelabel{117}% +It is true the +Sobral images were condemned, but whether so far as to +invalidate their testimony on this point was not at first clear; +besides the Principe images were not particularly well-defined, +and were much enfeebled by cloud. Certain compensating +advantages of the latter were better appreciated later. Their +strong point was the satisfactory check against systematic error +afforded by the photographs of the check-field; there were +no check-plates taken at Sobral, and, since it was obvious +that the discordance of the two results depended on systematic +error and not on the wealth of material, this distinctly +favoured the Principe results. Further, at Principe there could +be no evil effects from the sun's rays on the mirror, for the +sun had withdrawn all too shyly behind the veil of cloud. +A further advantage was provided by the check-plates at +Principe, which gave an independent determination of the +%% -----File: 128.png---Folio 118------- +difference of scale of the telescope as used in England and at +the eclipse; for the Sobral plates this scale-difference was +eliminated by the method of reduction, with the consequence +that the results depended on the measurement of a much smaller +relative displacement. + +There remained a set of seven plates taken at Sobral with the +$4$-inch lens; +\index{Bending of light!observational results}% +\index{Light, bending of}% +their measurement had been delayed by the necessity +of modifying a micrometer to hold them, since they were of +unusual size. From the first no one entertained any doubt that +the final decision must rest with them, since the images were +almost ideal, and they were on a larger scale than the other +photographs. The use of this instrument must have presented +considerable difficulties---the unwieldy length of the telescope, +the slower speed of the lens necessitating longer exposures and +more accurate driving of the clock-work, the larger scale rendering +the focus more sensitive to disturbances---but the observers +achieved success, and the perfection of the negatives surpassed +anything that could have been hoped for. + +These plates were now measured and they gave a final verdict +definitely confirming Einstein's value of the deflection, in agreement +with the results obtained at Principe. + +It will be remembered that Einstein's theory predicts a +deflection of $1''.74$ at the edge of the sun\footnotemark, + \footnotetext{The predicted deflection of light from infinity to infinity is just over~$1''.745$, + from infinity to the earth it is just under.}% +the amount falling +off inversely as the distance from the sun's centre. The simple +Newtonian deflection is half this,~$0''.87$. The final results +(reduced to the edge of the sun) obtained at Sobral and Principe +with their ``probable accidental errors'' were +\index{Deflection of light!observational results}% +\index{Gravitational field of Sun!deflection of light}% +\begin{center} +\begin{tabular}{lc} +Sobral & $1''.98 ± 0''.12$, \\ +Principe & $1''.61 ± 0''.30$. +\end{tabular} +\end{center} +It is usual to allow a margin of safety of about twice the probable +error on either side of the mean. The evidence of the Principe +plates is thus just about sufficient to rule out the possibility of +the ``half-deflection,'' and the Sobral plates exclude it with +practical certainty. The value of the material found at Principe +cannot be put higher than about one-sixth of that at Sobral; +but it certainly makes it less easy to bring criticism against this +confirmation of Einstein's theory seeing that it was obtained +%% -----File: 129.png---Folio 119------- +independently with two different instruments at different places +and with different kinds of checks. + +The best check on the results obtained with the $4$-inch lens +at Sobral is the striking internal accordance of the measures for +different stars. The theoretical deflection should vary inversely +as the distance from the sun's centre; hence, if we plot the mean +radial displacement found for each star separately against the +inverse distance, the points should lie on a straight line. This +%[Illustration: \textsc{Fig}. 17.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[17]{4.5in}{129a} +\end{center} +\end{figure*}% +is shown in \Figref{17} where the broken line shows the theoretical +prediction of Einstein, the deviations being within the accidental +errors of the determinations. A line of half the slope representing +the half-deflection would clearly be inadmissible. + +Moreover, values of the deflection were deduced from the +measures in right ascension and declination independently. +These were in close agreement. + +%% -----File: 130.png---Folio 120------- + +A diagram showing the relative positions of the stars is given +in \Figref{18}. + +The square shows the limits of the plates used at Principe, +and the oblique rectangle the limits with the 4-inch lens at +Sobral. The centre of the sun moved from $S$ to~$P$ in the $2\frac{1}{4}$~hours +%[Illustration: \textsc{Fig}.~18.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[18]{5in}{130a} +\end{center} +\end{figure*}% +interval between totality at the two stations; the sun is +here represented for a time about midway between. The stars +measured on the Principe plates were Nos.~3, 4, 5, 6, 10, 11; those +at Sobral were 11, 10, 6, 5, 4, 2, 3 (in the order of the dots +from left to right in \Figref{17}). None of these were fainter than +$6\Magnitude.0$, the brightest $\kappa^1$~Tauri (No.~4) being $4\Magnitude.5$. + +It has been objected that although the observations establish +%% -----File: 131.png---Folio 121------- +a deflection of light in passing the sun equal to that predicted +by Einstein, it is not immediately obvious that this deflection +must necessarily be attributed to the sun's gravitational field. +It is suggested that it may not be an essential effect of the sun +as a massive body, but an accidental effect owing to the circumstance +that the sun is surrounded by a corona which acts as +a refracting atmosphere. +\index{Corona, refraction by}% +\index{Refraction of light in corona}% +It would be a strange coincidence if +this atmosphere imitated the theoretical law in the exact +quantitative way shown in \Figref{17}; and the suggestion appears +to us far-fetched. However the objection can be met in a more +direct way. We have already shown that the gravitational +effect on light is equivalent to that produced by a refracting +medium round the sun and have calculated the necessary +refractive index. At a height of $400,000$ miles above the surface +the refractive index required is~$1.0000021$. This corresponds to +air at $\frac{1}{140}$ atmosphere, hydrogen at $\frac{1}{70}$ atmosphere, helium at +$\frac{1}{20}$ atmospheric pressure. It seems obvious that there can be no +material of this order of density at such a distance from the sun. +The pressure on the sun's surface of the columns of material +involved would be of the order $10,000$ atmospheres; and we +know from spectroscopic evidence that there is no pressure of +this order. If it is urged that the mass could perhaps be supported +by electrical forces, the argument from absorption is +even more cogent. The light from the stars photographed during +the eclipse has passed through a depth of at least a million miles +of material of this order of density---or say the equivalent of +$10,000$ miles of air at atmospheric density. We know to our +cost what absorption the earth's $5$~miles of homogeneous +atmosphere can effect. And yet at the eclipse the stars appeared +on the photographs with their normal brightness. If the irrepressible +critic insists that the material round the sun may be +composed of some new element with properties unlike any +material known to us, we may reply that the mechanism of +refraction and of absorption is the same, and there is a limit to +the possibility of refraction without appreciable absorption. +Finally it would be necessary to arrange that the density of the +material falls off inversely as the distance from the sun's centre +in order to give the required variation of refractive index. + +Several comets have been known to approach the sun within +%% -----File: 132.png---Folio 122------- +the limits of distance here considered. If they had to pass +through an atmosphere of the density required to account for +the displacement, they would have suffered enormous resistance. +Dr~Crommelin has shown that a study of these comets sets an +upper limit to the density of the corona, which makes the +refractive effect quite negligible.% +\index{Comets!motion through coronal medium}% +\index{Crommelin}% + +Those who regard Einstein's law of gravitation as a natural +deduction from a theory based on the minimum of hypotheses +will be satisfied to find that his remarkable prediction is quantitatively +confirmed by observation, and that no unforeseen cause +has appeared to invalidate the test. +%% -----File: 133.png---Folio 123------- + + +\Chapter{VIII}{Other Tests of the Theory} + +\Quote[break]{\textit{Love's Labour's Lost.}} +{The words of Mercury are harsh after the songs of Apollo.} + +\First{We} have seen that the swift-moving light-waves possess great +advantages as a means of exploring the non-Euclidean property +of space. But there is an old fable about the hare and the +tortoise. The slow-moving planets have qualities which must +not be overlooked. The light-wave traverses the region in a few +minutes and makes its report; the planet plods on and on for +centuries going over the same ground again and again. Each +time it goes round it reveals a little about the space, and the +knowledge slowly accumulates. + +According to Newton's law a planet moves round the sun in +an ellipse, and if there are no other planets disturbing it, the +ellipse remains the same for ever. According to Einstein's law +the path is very nearly an ellipse, but it does not quite close up; +and in the next revolution the path has advanced slightly in the +same direction as that in which the planet was moving. The +orbit is thus an ellipse which very slowly revolves\footnote% +{Appendix, \Noteref{9}.}.% +\Pagelabel{note9}% +\index{Mercury, perihelion of}% +\index{Orbits under Einstein's law}% +\index{Perihelia of planets, motions of}% + +The exact prediction of Einstein's law is that in one revolution +of the planet the orbit will advance through a fraction of a +revolution equal to $3v^2/C^2$, where $v$ is the speed of the planet +and $C$ the speed of light. The earth has $1/10,000$ of the speed of +light; thus in one revolution (one year) the point where the +earth is at greatest distance from the sun will move on +$3/100,000,000$ of a revolution, or $0''.038$. We could not detect +this difference in a year, but we can let it add up for a century +at least. It would then be observable but for one thing---the +earth's orbit is very blunt, very nearly circular, and so we +cannot tell accurately enough which way it is pointing and how +its sharpest apses move. We can choose a planet with higher +speed so that the effect is increased, not only because $v^2$ is +increased, but because the revolutions take less time; but, what +%% -----File: 134.png---Folio 124------- +is perhaps more important, we need a planet with a sharp +elliptical orbit, so that it is easy to observe how its apses move +round. Both these conditions are fulfilled in the case of Mercury. +It is the fastest of the planets, and the predicted advance of the +orbit amounts to $43''$ per century; further the eccentricity of +its orbit is far greater than that of any of the other seven +planets. + +Now an unexplained advance of the orbit of Mercury had +long been known. It had occupied the attention of Le~Verrier, +\index{Le Verrier}% +who, having successfully predicted the planet Neptune from the +disturbances of Uranus, thought that the anomalous motion of +Mercury might be due to an interior planet, which was called +Vulcan in anticipation. But, though thoroughly sought for, +Vulcan has never turned up. Shortly before Einstein arrived +at his law of gravitation, the accepted figures were as follows. +The actual observed advance of the orbit was $574''$ per century; +the calculated perturbations produced by all the known planets +amounted to $532''$ per century. The excess of $42''$ per century +remained to be explained. Although the amount could scarcely +be relied on to a second of arc, it was at least thirty times as +great as the probable accidental error. + +The big discrepancy from the Newtonian gravitational theory +is thus in agreement with Einstein's prediction of an advance +of $43''$ per century. + +The derivation of this prediction from Einstein's law can only +be followed by mathematical analysis; but it may be remarked +that any slight deviation from the inverse square law is likely +to cause an advance or recession of the apse of the orbit. That +a particle, if it does not move in a circle, should oscillate between +two extreme distances is natural enough; it could scarcely do +anything else unless it had sufficient speed to break away +altogether. But the interval between the two extremes will not +in general be half a revolution. It is only under the exact +adjustment of the inverse square law that this happens, so that +the orbit closes up and the next revolution starts at the same +point. I do not think that any ``simple explanation'' of this +property of the inverse-square law has been given; and it seems +fair to remind those, who complain of the difficulty of understanding +Einstein's prediction of the advance of the perihelion, +\index{Gravitational field of Sun!motion of perihelion} % [** PP: Index entry reads p. 122] +%% -----File: 135.png---Folio 125------- +that the real trouble is that they have not yet succeeded in +making clear to the uninitiated this recondite result of the +Newtonian theory. The slight modifications introduced by +Einstein's law of gravitation upset this fine adjustment, so that +the oscillation between the extremes occupies slightly more than +a revolution. A simple example of this effect of a small deviation +from the inverse-square law was actually given by Newton. + +It had already been recognised that the change of mass with +velocity may cause an advance of perihelion; but owing to the +ambiguity of Newton's law of gravitation the discussion was +unsatisfactory. It was, however, clear that the effect was too +small to account for the motion of perihelion of Mercury, the +prediction being $\frac{1}{2} v^2/C^2$, or at most $v^2/C^2$. Einstein's theory is +the only one which gives the full amount $3v^2/C^2$.% +\index{Mercury, perihelion of}% + +It was suggested by Lodge that, % [** PP: Retaining awkward commas] +\index{Lodge}% +this variation of mass with +velocity might account for the whole motion of the orbit of +Mercury, if account were taken of the sun's unknown absolute +motion through the aether, combining sometimes additively and +sometimes negatively with the orbital motion. In a discussion +between him and the writer, it appeared that, if the absolute +motion were sufficient to produce this effect on Mercury, it +must give observable effects for Venus and the Earth; and these +do not exist. Indeed from the close accordance of Venus and +the Earth with observation, it is possible to conclude that, either +the sun's motion through the aether is improbably small, or +gravitation must conform to relativity, in the sense of the +restricted principle (\Pageref{20}), and conceal the effects of the +increase of mass with speed so far as an additive uniform motion +is concerned.% +\index{Gravitation!relativity for uniform motion}% + +Unfortunately it is not possible to obtain any further test of +Einstein's law of gravitation from the remaining planets. We +have to pass over Venus and the Earth, whose orbits are too +nearly circular to show the advance of the apses observationally. +Coming next to Mars with a moderately eccentric orbit, the +speed is very much smaller, and the predicted advance is only +$1''.3$ per century. Now the accepted figures show an observed +advance (additional to that produced by known causes) of $5''$ +per century, so that Einstein's correction improves the accordance +of observation with theory; but, since the result for Mars +%% -----File: 136.png---Folio 126------- +is in any case scarcely trustworthy to $5''$ owing to the inevitable +errors of observation, the improvement is not very important. +The main conclusion is that Einstein's theory brings Mercury +into line, without upsetting the existing good accordance of all +the other planets. + +We have tested Einstein's law of gravitation for fast movement +(light) and for moderately slow movement (Mercury). +For very slow movement it agrees with Newton's law, and the +general accordance of the latter with observation can be transferred +to Einstein's law. These tests appear to be sufficient to +establish the law firmly. We can express it in this way. + +Every particle or light-pulse moves so that the quantity $s$ +measured along its track between two points has the maximum +possible value, where +\[ +ds^2 = - (1-2m/r)^{-1}\, dr^2 -r^2\, d\theta^2 + (1-2m/r)\, dt^2. +\] +And the accuracy of the experimental test is sufficient to verify +the coefficients as far as terms of order $m/r$ in the coefficient of~$dr^2$, +and as far as terms of order $m^2/r^2$ in the coefficient of~$dt^2$\footnote{Appendix, \Noteref{10}.}. +\Pagelabel{note10} + +In this form the law appears to be firmly based on experiment, +and the revision or even the complete abandonment of the +general ideas of Einstein's theory would scarcely affect it. + +These experimental proofs, that space in the gravitational +field of the sun is non-Euclidean or curved, have appeared +puzzling to those unfamiliar with the theory. It is pointed out +that the experiments show that physical objects or loci are +``warped'' in the sun's field; but it is suggested that there is +nothing to show that the space in which they exist is warped. +\index{Gravitational field of Sun!result of observational verification}% +The answer is that it does not seem possible to draw any distinction +between the warping of physical space and the warping +of physical objects which define space. +\index{Warping of space}% +If our purpose were +merely to call attention to these phenomena of the gravitational +field as curiosities, it would, no doubt, be preferable to avoid +using words which are liable to be misconstrued. But if we wish +to arrive at an understanding of the conditions of the gravitational +field, we cannot throw over the vocabulary appropriate +for that purpose, merely because there may be some who insist +on investing the words with a metaphysical meaning which is +clearly inappropriate to the discussion. + +%% -----File: 137.png---Folio 127------- + +We come now to another kind of test. In the statement of +the law of gravitation just given, a quantity $s$ is mentioned; +and, so far as that statement goes, $s$ is merely an intermediary +quantity defined mathematically. But in our theory we have +been identifying $s$ with interval-length, measured with an +apparatus of scales and clocks; and it is very desirable to test +whether this identification can be confirmed---whether the +geometry of scales and clocks is the same as the geometry of +moving particles and light-pulses. + +The question has been mooted whether we may not divide +the present theory into two parts. Can we not accept the law +of gravitation in the form suggested above as a self-contained +result proved by observation, leaving the further possibility +that $s$ is to be identified with interval-length open to debate? +The motive is partly a desire to consolidate our gains, freeing +them from the least taint of speculation; but perhaps also it is +inspired by the wish to leave an opening by which clock-scale +geometry, i.e.\ the space and time of ordinary perception, may +remain Euclidean. Disregarding the connection of $s$ with +interval-length, there is no object in attributing any significance +of length to it; +\index{Interval-length!geometrical significance essential}% +it can be regarded as a dynamical quantity like +Action, and the new law of gravitation can be expressed after +the traditional manner without dragging in strange theories of +space and time. Thus interpreted, the law perhaps loses its +theoretical inevitability; but it remains strongly grounded on +observation. Unfortunately for this proposal, it is impossible +to make a clean division of the theory at the point suggested. +Without some geometrical interpretation of $s$ our conclusions as +to the courses of planets and light-waves cannot be connected +with the astronomical measurements which verify them. The +track of a light-wave in terms of the coordinates $r$, $\theta$, $t$ cannot +be tested directly; the coordinates afford only a temporary +resting-place; and the measurement of the displacement of the +star-image on the photographic plate involves a reconversion +from the coordinates to~$s$, which here appears in its significance +as the interval in clock-scale geometry. + +Thus even from the experimental standpoint, a rough correspondence +of the quantity $s$ occurring in the law of gravitation +with the clock-scale interval is an essential feature. We have +%% -----File: 138.png---Folio 128------- +now to examine whether experimental evidence can be found +as to the exactness of this correspondence. + +It seems reasonable to suppose that a vibrating atom is an +ideal type of clock. The beginning and end of a single vibration +constitute two events, and the interval $ds$ between two events +is an absolute quantity independent of any mesh-system. This +interval must be determined by the nature of the atom; and +hence atoms which are absolutely similar will measure by their +vibrations equal values of the absolute interval $ds$. Let us +adopt the usual mesh-system $(r, \theta, t)$ for the solar system, so +that +\[ +ds^2 = - \gamma^{-1}\, dr^2 -r^2\, d\theta^2 + \gamma\, dt^2. +\] +Consider an atom momentarily at rest at some point in the solar +system; we say \textit{momentarily}, because it must undergo the +acceleration of the gravitational field where it is. If $ds$ corresponds +to one vibration, then, since the atom has not moved, +the corresponding $dr$ and $d\theta$ will be zero, and we have +\[ +ds^2 = \gamma\, dt^2. +\] +The \textit{time} of vibration $dt$ is thus $1/\surd\gamma$ times the \textit{interval} of +vibration~$ds$. + +Accordingly, if we have two similar atoms at rest at different +points in the system, the interval of vibration will be the same +for both; but the time of vibration will be proportional to the +inverse square-root of~$\gamma$, which differs for the two atoms. Since +\begin{align*} + \gamma &= 1 - \frac{2m}{r} \\ +1/\surd \gamma &= 1 + \,\frac{m}{r}, \quad\text{very approximately.} +\end{align*} + +Take an atom on the surface of the sun, and a similar atom +in a terrestrial laboratory. +\index{Atom, vibrating on sun}% +\index{Clock!on sun}% +For the first, $1 + m/r = 1.00000212$, +and for the second $1 + m/r$ is practically~$1$. The time of vibration +of the solar atom is thus longer in the ratio $1.00000212$, and it +might be possible to test this by spectroscopic examination. + +There is one important point to consider. The spectroscopic +examination must take place in the terrestrial laboratory; and +we have to test the period of the solar atom by the period of +the waves emanating from it when they reach the earth. Will +they carry the period to us unchanged? Clearly they must. +%% -----File: 139.png---Folio 129------- +The first and second pulse have to travel the same distance ($r$), +and they travel with the same velocity ($dr/dt$); for the velocity +of light in the mesh-system used is $1- 2m/r$, and though this +velocity depends on~$r$, it does not depend on~$t$. Hence the difference +$dt$ at one end of the waves is the same as that at the other +end. + +Thus in the laboratory the light from a solar source should +be of greater period and greater wave-length (i.e.\ redder) than +that from a corresponding terrestrial source. Taking blue light +of wave-length $4000$~Ã…, the solar lines should be displaced +$4000 × .00000212$, or $0.008$~Ã… towards the red end of the +spectrum.% +\index{Displacement of spectral lines}% +\index{Gravitational field of Sun!displacement of spectral lines}% + +The properties of a gravitational field of force are similar to +those of a centrifugal field of force; and it may be of interest +to see how a corresponding shift of the spectral lines occurs for +an atom in a field of centrifugal force. +\index{Retardation of time!in centrifugal field}% +Suppose that, as we rotate +with the earth, we observe a very remote atom momentarily at +rest relative to our rotating axes. The case is just similar to +that of the solar atom; both are at rest relative to the respective +mesh-systems; the solar atom is in a field of gravitational force, +and the other is in a field of centrifugal force. The direction of +the force is in both cases the same---from the earth towards the +atom observed. Hence the atom in the centrifugal field ought +also to vibrate more slowly, and show a displacement to the red +in its spectral lines. +\index{Centrifugal Force!vibrating atom in field of}% +It does, if the theory hitherto given is +right. We can abolish the centrifugal force by choosing non-rotating +axes. But the distant atom was at rest relative to the +rotating axes, that is to say, it was whizzing round with them. +Thus from the ordinary standpoint the atom has a large velocity +relative to the observer, and, in accordance with \Chapref{I}, its +vibrations slow down just as the aviator's watch did. The shift +of spectral lines due to a field of centrifugal force is only another +aspect of a phenomenon already discussed. + +The expected shift of the spectral lines on the sun, compared +with the corresponding terrestrial lines, has been looked for; +but it has not been found. + +In estimating the importance of this observational result in +regard to the relativity theory, we must distinguish between +a failure of the test and a definite conclusion that the lines are +%% -----File: 140.png---Folio 130------- +undisplaced. The chief investigators St~John, Schwarzschild, +Evershed, and Grebe and Bachem, seem to be agreed that the +observed displacement is at any rate less than that predicted +by the theory. The theory can therefore in no case claim support +from the present evidence. But something more must be +established, if the observations are to be regarded as in the +slightest degree adverse to the theory. If for instance the mean +deflection is found to be $.004$ instead of $.008$ Angström units, +the only possible conclusion is that there are certain causes of +displacement of the lines, acting in the solar atmosphere and not +yet identified. No one could be much surprised if this were the +case; and it would, of course, render the test nugatory. The +case is not much altered if the observed displacement is $.002$ +units, provided the latter quantity is above the accidental error +of measurement; if we have to postulate some unexplained disturbance, +it may just as well produce a displacement $-.006$ as +$+.002$. For this reason Evershed's evidence is by no means +adverse to the theory, since he finds unexplained displacements +in any case. +\index{Evershed}% +\index{St John}% +One set of lines measured by St~John gave a mean +displacement of $.0036$~units; and this also shows that the test +has failed. The only evidence \textit{adverse} to the theory, and not +merely neutral, is a series of measures by St~John on $17$ cyanogen +lines, which he regarded as most dependable. These gave a mean +shift of exactly~$.000$. If this stood alone we should certainly be +disposed to infer that the test had gone against Einstein's +theory, and that nothing had intervened to cast doubt on the +validity of the test. The writer is unqualified to criticise these +mutually contradictory spectroscopic conclusions; but he has +formed the impression that the last-mentioned result obtained +by St~John has the greatest weight of any investigations up to +the present\footnotemark. + \footnotetext{A further paper by Grebe and Bachem (\textit{Zeitschrift für Physik}, 1920, p.~51), + \index{Grebe and Bachem}% + received whilst this is passing through the press, makes out a case strongly + favourable for the Einstein displacement, and reconciles the discordant results + found by most of the investigators. But it may still be the best counsel to + ``wait and see,'' and I have made no alteration in the discussion here given.} + +It seems that judgment must be reserved; but it may be well +to examine how the present theory would stand if the verdict +of this third crucial experiment finally went against it. + +It has become apparent that there is something illogical in +%% -----File: 141.png---Folio 131------- +the sequence we have followed in developing the theory, owing +to the necessity of proceeding from the common ideas of space +and time to the more fundamental properties of the absolute +world. +\index{Clock-scale geometry, not fundamental}% +We started with a definition of the interval by measurements +made with clocks and scales, and afterwards connected +it with the tracks of moving particles. Clearly this is an inversion +of the logical order. The simplest kind of clock is an elaborate +mechanism, and a material scale is a very complex piece of +apparatus. The best course then is to discover $ds$ by exploration +of space and time with a moving particle or light-pulse, rather +than by measures with scales and clocks. On this basis by +astronomical observation alone the formula for $ds$ in the gravitational +field of the sun has already been established. To proceed +from this to determine exactly what is measured by a scale and +a clock, it would at first seem necessary to have a detailed theory +of the mechanisms involved in a scale and clock. But there is +a short-cut which seems legitimate. This short-cut is in fact +the Principle of Equivalence. +\index{Equivalence!Principle of}% +\index{Principle of Equivalence}% +Whatever the mechanism of the +clock, whether it is a good clock or a bad clock, the intervals it +is beating must be something absolute; the clock cannot know +what mesh-system the observer is using, and therefore its +absolute rate cannot be altered by position or motion which is +relative merely to a mesh-system. Thus wherever it is placed, +and however it moves, provided it is not constrained by impacts +or electrical forces, it must always beat equal intervals as we +have previously assumed. Thus a clock may fairly be used to +measure intervals, even when the interval is defined in the new +manner; any other result seems to postulate that it pays heed +to some particular mesh-system\footnotemark. + \footnotetext{Of course, there is always the possibility that this might be the case, + though it seems unlikely. The essential point of the relativity theory is that + (contrary to the common opinion) no experiments yet made have revealed any + mesh-system of an absolute character, not that experiments never will reveal + such a system.} + +Three modes of escape from this conclusion seem to be left +open. A clock cannot pay any heed to the mesh-system used; +but it may be affected by the kind of space-time around it\footnote% + {Appendix, \Noteref{11}.}. +\Pagelabel{note11}% +The terrestrial atom is in a field of gravitation so weak that the +space-time may be considered practically flat; but the space-% +%% -----File: 142.png---Folio 132------- +time round the solar atom is not flat. It may happen that the +two atoms actually detect this absolute difference in the world +around them and do not vibrate with the same interval $ds$---contrary +to our assumption above. Then the prediction of the +shift of the lines in the solar spectrum is invalidated. Now it is +very doubtful if an atom can detect the curving of the region it +occupies, because curvature is only apparent when an extended +region is considered; still an atom has some extension, and it is +not impossible that its equations of motion involve the quantities +$B^{\rho}_{\mu\nu\sigma}$ which distinguish gravitational from flat space-time. An +apparently insuperable objection to this explanation is that the +effect of curvature on the period would almost certainly be +represented by terms of the form $m^{2}/r^{2}$, whereas to account for +a negative result for the shift of the spectral lines terms of much +greater order of magnitude $m/r$ are needed. + +The second possibility depends on the question whether it is +possible for an atom at rest on the sun to be precisely similar to +one on the earth. If an atom fell from the earth to the sun it +would acquire a velocity of $610$~km.\ per sec., and could only be +brought to rest by a systematic hammering by other atoms. +May not this have made a permanent alteration in its time-keeping +properties? It is true that every atom is continually +undergoing collisions, but it is just possible that the average +solar atom has a different period from the average terrestrial +atom owing to this systematic difference in its history. + +What are the two events which mark the beginning and end +of an atomic vibration? This question suggests a third possibility. +If they are two absolute events, like the explosions of +two detonators, then the interval between them will be a definite +quantity, and our argument applies. But if, for example, an +atomic vibration is determined by the revolution of an electron +around a nucleus, it is not marked by any definite events. A +revolution means a return to the same position as before; but +we cannot define what is the same position as before without +reference to some mesh-system. Hence it is not clear that there +is any absolute interval corresponding to the vibration of an +atom; an absolute interval only exists between two events +absolutely defined. + +It is unlikely that any of these three possibilities can negative %[** PP: OK] +%% -----File: 143.png---Folio 133------- +the expected shift of the spectral lines. The uncertainties introduced +by them are, so far as we can judge, of a much smaller +order of magnitude. But it will be realised that this third test +of Einstein's theory involves rather more complicated considerations +than the two simple tests with light-waves and the moving +planet. I think that a shift of the Fraunhofer lines is a highly +probable prediction from the theory and I anticipate that +experiment will ultimately confirm the prediction; but it is not +entirely free from guess-work. These theoretical uncertainties +are apart altogether from the great practical difficulties of the +test, including the exact allowance for the unfamiliar circumstances +of an absorbing atom in the sun's atmosphere. + +Outside the three leading tests, there appears to be little +chance of checking the theory unless our present methods of +measurement are greatly improved. It is not practicable to +measure the deflection of light by any body other than the sun. +The apparent displacement of a star just grazing the limb of +Jupiter should be~$0''.017$. +\index{Jupiter, deflection of light by}% +A hundredth of a second of arc is +just about within reach of the most refined measurements with +the largest telescopes. If the observation could be conducted +under the same conditions as the best parallax measurements, +the displacement could be detected but not measured with any +accuracy. The glare from the light of the planet ruins any chance +of success. + +% [** PP: Retaining commas] +Most astronomers, who look into the subject, are entrapped +sooner or later by a fallacy in connection with double stars. +\index{Double stars and Einstein effect}% +It is thought that when one component passes behind the other +it will appear displaced from its true position, like a star passing +behind the sun; if the size of the occulting star is comparable +with that of the sun, the displacement should be of the same +order, $1''.7$. This would cause a very conspicuous irregularity in +the apparent orbit of a double star. But reference to \Pageref{113} +shows that an essential point in the argument was the enormous +ratio of the distance $QP$ of the star from the sun to the distance +$EF$ of the sun from the earth. It is only in these conditions that +the apparent displacement of the object is equal to the deflection +undergone by its light. It is easy to see that where this ratio is +reversed, as in the case of the double star, the apparent displacement +is an extremely small fraction of the deflection of the light. +It would be quite imperceptible to observation. + +%% -----File: 144.png---Folio 134------- + +If two independent stars are seen in the same line of vision +within about~$1''$, one being a great distance behind the other, +the conditions seem at first more favourable. I do not know if +any such pairs exist. It would seem that we ought to see the +more distant star not only by the direct ray, which would be +practically undisturbed, but also by a ray passing round the +other side of the nearer star and bent by it to the necessary +extent. The second image would, of course, be indistinguishable +from that of the nearer star; but it would give it additional +brightness, which would disappear in time when the two stars +receded. But consider a pencil of light coming past the nearer +star; the inner edge will be bent more than the outer edge, so +that the divergence is increased. The increase is very small; +but then the whole divergence of a pencil from a source some +hundred billion miles away is very minute. It is easily calculated +that the increased divergence would so weaken the light as to +make it impossible to detect it when it reached us\footnote% + {Appendix, \Noteref{12}.}. +\Pagelabel{note12} + +If two unconnected stars approached the line of sight still +more closely, so that one almost occulted the other, observable +effects might be perceived. When the proximity was such that +the direct ray from the more distant star passed within about +$100$~million kilometres of the nearer star, it would begin to fade +appreciably. The course of the ray would not yet be appreciably +deflected, but the divergence of the pencil would be rapidly +increased, and less light from the star would enter our telescopes. +The test is scarcely likely to be an important one, since a +sufficiently close approach is not likely to occur; and in any +case it would be difficult to feel confident that the fading was +not due to a nebulous atmosphere around the nearer star. + +The theory gives small corrections to the motion of the moon +which have been investigated by de~Sitter. +\index{Moon, motion of}% +\index{de Sitter}% +Both the axis of +the orbit and its line of intersection with the ecliptic should +advance about $2''$ per century more than the Newtonian theory +indicates. Neither observation nor Newtonian theory are as yet +pushed to sufficient accuracy to test this; but a comparatively +small increase in accuracy would make a comparison possible. + +Since certain stars are perhaps ten times more massive than +the sun, without the radius being unduly increased, they should +show a greater shift of the spectral lines and might be more +%% -----File: 145.png---Folio 135------- +favourable for the third crucial test. Unfortunately the predicted +shift is indistinguishable from that caused by a velocity +of the star in the line-of-sight on Doppler's principle. Thus the +expected shift on the sun is equivalent to that caused by a receding +velocity of $0.634$ kilometres per second. In the case of the +sun we know by other evidence exactly what the line-of-sight +velocity should be; but we have not this knowledge for other +stars. The only indication that could be obtained would be the +detection of an \textit{average} motion of recession of the more massive +stars. It seems rather unlikely that there should be a real +preponderance of receding motions among stars taken indiscriminately +from all parts of the sky; and the apparent effect +might then be attributed to the Einstein shift. Actually the +most massive stars (those of spectral type~$B$) have been found +to show an average velocity of recession of about $4.5$~km.\ per +sec., which would be explained if the values of $m/r$ for these +stars are about seven times greater than the value for the sun---a +quite reasonable hypothesis. +\index{Displacement of spectral lines!in stars}% +\index{Receding velocities of B@Receding velocities!of $B$-type stars}% +This phenomenon was well-known +to astrophysicists some years before Einstein's theory +was published. But there are so many possible interpretations +that no stress should be placed on this evidence. Moreover the +very diffuse ``giant'' stars of type~$M$ have also a considerable +systematic velocity of recession, and for these $m/r$ must be much +less than for the sun. +%% -----File: 146.png---Folio 136------- + + +\Chapter{IX}{Momentum and Energy} + +\null\hspace{0.75in} +\begin{minipage}{4.25in} % [** PP: Hard-coded width; ~2.6pt overfull] +\Quote{Newman, \textit{Dream of Gerontius}.} +{\hspace*{-\QIndent}For spirits and men by different standards mete \\ +The less and greater in the flow of time. \\ +By sun and moon, primeval ordinances--- \\ +By stars which rise and set harmoniously--- \\ +By the recurring seasons, and the swing \\ +This way and that of the suspended rod \\ +Precise and punctual, men divide the hours, \\ +Equal, continuous, for their common use. \\ +Not so with us in the immaterial world; \\ +But intervals in their succession \\ +Are measured by the living thought alone \\ +And grow or wane with its intensity. \\ +And time is not a common property; \\ +But what is long is short, and swift is slow \\ +And near is distant, as received and grasped \\ +By this mind and by that.} +\end{minipage}% +\index{Motion!Newton's first law}% +\index{Newton!law of motion}% + +\First{One} of the most important consequences of the relativity theory +is the unification of inertia and gravitation. + +The beginner in mechanics does not accept Newton's first law +of motion without a feeling of hesitation. He readily agrees that +a body at rest will remain at rest unless something causes it to +move; but he is not satisfied that a body in motion will remain +in uniform motion so long as it is not interfered with. It is +quite natural to think that motion is an impulse which will +exhaust itself, and that the body will finally come to a stop. +The teacher easily disposes of the arguments urged in support +of this view, pointing out the friction which has to be overcome +when a train or a bicycle is kept moving uniformly. He shows +that if the friction is diminished, as when a stone is projected +across ice, the motion lasts for a longer time, so that if all interference +by friction were removed uniform motion might continue +indefinitely. But he glosses over the point that if there +were no interference with the motion---if the ice were abolished +altogether---the motion would be by no means uniform, but like +that of a falling body. The teacher probably insists that the +continuance of uniform motion does not require anything that +%% -----File: 147.png---Folio 137------- +can properly be called a \textit{cause}. The property is given a name +\textit{inertia}; but it is thought of as an innate tendency in contrast +to \textit{force} which is an active cause. +\index{Force!compared with inertia}% +\index{Inertia!compared with force}% +So long as forces are confined +to the thrusts and tensions of elementary mechanics, where there +is supposed to be direct contact of material, there is good ground +for this distinction; we can visualise the active hammering of +the molecules on the body, causing it to change its motion. But +when force is extended to include the gravitational field the +distinction is not so clear. + +For our part we deny the distinction in this last case. Gravitational +force is not an active agent working against the passive +tendency of inertia. Gravitation and inertia are one. The +uniform straight track is only relative to some mesh-system, +which is assigned by arbitrary convention. We cannot imagine +that a body looks round to see who is observing it and then feels +an innate tendency to move in that observer's straight line---probably +at the same time feeling an active force compelling +it to move some other way. If there is anything that can be +called an innate tendency it is the tendency to follow what we +have called the natural track---the longest track between two +points. We might restate the first law of motion in the form +``Every body tends to move in the track in which it actually +does move, except in so far as it is compelled by material impacts +to follow some other track than that in which it would otherwise +move.'' Probably no one will dispute this profound statement! + +Whether the natural track is straight or curved, whether the +motion is uniform or changing, a cause is in any case required. +This cause is in all cases the combined inertia-gravitation. +\index{Inertia-gravitation}% +To +have given it a name does not excuse us from attempting an +explanation of it in due time. Meanwhile this identification of +inertia and gravitation as arbitrary components of one property +explains why weight is always proportional to inertia. +\index{Weight!proportional to inertia}% +This +experimental fact verified to a very high degree of accuracy +would otherwise have to be regarded as a remarkable law of +nature. + +We have learnt that the natural track is the longest track +between two points; and since this is the only definable track +having an absolute significance in nature, we seem to have a +sufficient explanation of why an undisturbed particle must +%% -----File: 148.png---Folio 138------- +follow it. That is satisfactory, so far as it goes, but still we should +naturally wish for a clearer picture of the cause---inertia-gravitation---which +propels it in this track. + +It has been seen that the gravitational field round a body +involves a kind of curvature of space-time, and accordingly +round each particle there is a minute pucker. Now at each +successive instant a particle is displaced continuously in time if +not in space; and so in our four-dimensional representation +which gives a bird's-eye-view of all time, the pucker has the +form of a long groove along the track of the particle. Now such +a groove or pleat in a continuum cannot take an arbitrary +course---as every dress-maker knows. Einstein's law of gravitation +gives the rule according to which the curvatures at any +point of space-time link on to those at surrounding points; so +that when a groove is started in any direction the rest of its +course can be forecasted. We have hitherto thought of the law +of gravitation as showing how the pucker spreads out in space, +cf.\ Newton's statement that the corresponding force weakens as +the inverse square of the distance. But the law of Einstein +equally shows how the gravitational field spreads out in time, +since there is no absolute distinction of time and space. It can +be deduced mathematically from Einstein's law that a pucker +of the form corresponding to a particle necessarily runs along +the track of greatest interval-length between two points. + +The track of a particle of matter is thus determined by the +interaction of the minute gravitational field, which surrounds +and, so far as we know, constitutes it, with the general space-time +of the region. The various forms which it can take, find +their explanation in the new law of gravitation. The straight +tracks of the stars and the curved tracks of the planets are +placed on the same level, and receive the same kind of explanation. +The one universal law, that the space-time continuum +can be curved only in the first degree, is sufficient to prescribe +the forms of all possible grooves crossing it.% +\index{Geodesic!motion of particles in}% + +The application of Einstein's law to trace the gravitational +field not only through space but through time leads to a great +unification of mechanics. If we have given for a start a narrow +slice of space-time representing the state of the universe for a few +seconds, with all the little puckers belonging to particles of matter +%% -----File: 149.png---Folio 139------- +properly described, then step by step all space-time can be linked +on and the positions of the puckers shown at all subsequent +times (electrical forces being excluded). Nothing is needed for +this except the law of gravitation---that the curvature is only +of the first degree---and there can thus be nothing in the predictions +of mechanics which is not comprised in the law of +gravitation. The conservation of mass, of energy, and of +momentum must all be contained implicitly in Einstein's law.% +\index{Conservation!of energy and momentum}% +\index{Energy!conservation of}% +\IndexExtra{Conservation!of mass}% + +It may seem strange that Einstein's law of gravitation should +take over responsibility for the whole of mechanics; because in +many mechanical problems gravitation in the ordinary sense +can be neglected. But inertia and gravitation are unified; the +law is also the law of inertia, and inertia or mass appears in all +mechanical problems. When, as in many problems, we say that +gravitation is negligible, we mean only that the interaction of +the minute puckers with one another can be neglected; we do +not mean that the interaction of the pucker of a particle with +the general character of the space-time in which it lies can be +neglected, because this constitutes the inertia of the particle.% +\index{Inertia!relativity theory of}% + +The conservation of energy and the conservation of momentum +in three independent directions, constitute together four laws +or equations which are fundamental in all branches of mechanics. +Although they apply when gravitation in the ordinary sense is +not acting, they must be deducible like everything else in +mechanics from the law of gravitation. It is a great triumph for +Einstein's theory that his law gives correctly these experimental +principles, which have generally been regarded as unconnected +with gravitation. We cannot enter into the mathematical +deduction of these equations; but we shall examine generally +how they are arrived at. + +It has already been explained that although the values of +$G_{\mu\nu}$ are strictly zero everywhere in space-time, yet if we take +average values through a small region containing a large number +of particles of matter their average or ``macroscopic'' values +will not be zero\footnotemark. + \footnotetext{It is the $g$'s which are first averaged, then the $G_{\mu\nu}$ are calculated by the + formulae in \Noteref{5}.}% +\index{Macroscopic!equations}% +Expressions for these macroscopic values can +be found in terms of the number, masses and motions of the +particles. Since we have averaged the $G_{\mu\nu}$, we should also +%% -----File: 150.png---Folio 140------- +average the particles; that is to say, we replace them by a +distribution of continuous matter having equivalent properties. +We thus obtain macroscopic equations of the form +\index{Continuous matter}% +\[ +G_{\mu\nu} = K_{\mu\nu}, +\] +where on the one side we have the somewhat abstruse quantities +describing the kind of space-time, and on the other side we have +well-known physical quantities describing the density, momentum, +energy and internal stresses of the matter present. These +macroscopic equations are obtained solely from the law of +gravitation by the process of averaging. + +By an exactly similar process we pass from Laplace's equation +$\nabla^2\phi = 0$ to Poisson's equation for continuous matter $\nabla^2\phi = -4\pi\rho$, +in the Newtonian theory of gravitation.% +\index{Laplace's equation}% + +When continuous matter is admitted, \textit{any} kind of space-time +becomes possible. The law of gravitation instead of denying the +possibility of certain kinds, states what values of $K_{\mu\nu}$, i.e.\ what +distribution and motion of continuous matter in the region, are +a necessary accompaniment. This is no contradiction with the +original statement of the law, since that referred to the case in +which continuous matter is denied or excluded. Any set of +values of the potentials is now possible; we have only to calculate +by the formulae the corresponding values of $G_{\mu\nu}$, and we at +once obtain ten equations giving the $K_{\mu\nu}$ which define the +conditions of the matter necessary to produce these potentials. +But suppose the necessary distribution of matter through space +and time is an impossible one, violating the laws of mechanics! +No, there is only one law of mechanics, the law of gravitation; +we have specified the distribution of matter so as to satisfy +$G_{\mu\nu} = K_{\mu\nu}$, and there can be no other condition for it to fulfil. +The distribution must be mechanically possible; it might, however, +be unrealisable in practice, involving inordinately high or +even negative density of matter.% +\index{Gravitation, Einstein's law of!macroscopic equations}% + +In connection with the law for empty space, $G_{\mu\nu} = 0$, it was +noticed that whereas this apparently forms a set of ten equations, +only six of them can be independent. This was because ten +equations would suffice to determine the ten potentials precisely, +and so fix not only the kind of space-time but the mesh-system. +It is clear that we must preserve the right to draw the mesh-system +as we please; it is fixed by arbitrary choice not by a law +%% -----File: 151.png---Folio 141------- +of nature. To allow for the four-fold arbitrariness of choice, +there must be four relations always satisfied by the $G_{\mu\nu}$, so that +when six of the equations are given the remaining four become +tautological. + +These relations must be identities implied in the mathematical +definition of $G_{\mu\nu}$; that is to say, when the $G_{\mu\nu}$ have been written +out in full according to their definition, and the operations +indicated by the identities carried out, all the terms will cancel, +leaving only $0 = 0$. The essential point is that the four relations +follow from the mode of formation of the $G_{\mu\nu}$ from their simpler +constituents ($g_{\mu\nu}$ and their differential coefficients) and apply +universally. These four identical relations have actually been +discovered\footnote{Appendix, \Noteref{13}.}.% +\Pagelabel{note13}% +\index{Identities@Identities connecting $G_{\mu \nu}$}% + +When in continuous matter $G_{\mu\nu} = K_{\mu\nu}$ clearly the same four +relations must exist between the $K_{\mu\nu}$, not now as identities, +but as consequences of the law of gravitation, viz.\ the equality +of $G_{\mu\nu}$ and $K_{\mu\nu}$.% +\index{Matter!gravitational equations in}% + +Thus the four dimensions of the world bring about a four-fold +arbitrariness of choice of mesh-system; this in turn necessitates +four identical relations between the $G_{\mu\nu}$; and finally, in consequence +of the law of gravitation, these identities reveal four new +facts or laws relating to the density, energy, momentum or stress +of matter, summarised in the expressions $K_{\mu\nu}$. + +These four laws turn out to be the laws of conservation of +momentum and energy. + +The argument is so general that we can even assert that +corresponding to any \textit{absolute} property of a volume of a world +of four dimensions (in this case, \textit{curvature}), there must be four +\textit{relative} properties which are conserved. This might be made the +starting-point of a general inquiry into the necessary qualities +of a permanent perceptual world, i.e.\ a world whose substance +is conserved.% +\index{Conservation!of mass}% +\index{Mass!conservation of}% +\index{Momentum!conservation of}% +\index{Permanent perceptual world}% +\IndexExtra{Conservation!of energy and momentum}% + +There is another law of physics which was formerly regarded +as funda\-mental---the conservation of mass. Modern progress +has somewhat altered our position with regard to it; not that +its validity is denied, but it has been reinterpreted, and has +finally become merged in the conservation of energy. It will be +desirable to consider this in detail. + +%% -----File: 152.png---Folio 142------- + +%[Illustration: \textsc{Fig}. 19.] +% [** PP: Moved to top of paragraph] +\begin{figure*}[hbt] +\begin{center} +\Graphic[19]{3in}{152a} +\end{center} +\end{figure*}% +It was formerly supposed that the mass of a particle was a +number attached to the particle, expressing an intrinsic property, +which remained unaltered in all its vicissitudes. If $m$ is this +number, and $u$ the velocity of the particle, the momentum is~$mu$; +and it is through this relation, coupled with the law of conservation +of momentum that the mass~$m$ was defined. Let us take +for example two particles of masses $m_1 = 2$ and $m_2 = 3$, moving +in the same straight line. In the space-time diagram for an +observer $S$ the velocity of the first particle will be represented +by a direction~$OA$ (\Figref{19}). The first particle moves through +a space $MA$ in unit time, so that $MA$ is equal to its velocity +referred to the observer~$S$. Prolonging the line $OA$ to meet the +second time-partition, $NB$ is equal to the velocity multiplied +by the mass~$2$; thus the horizontal distance $NB$ represents the +momentum. Similarly, starting from $B$ and drawing $BC$ in the +direction of the velocity of~$m_2$, prolonged through three +time-partitions, the horizontal progress from $B$ represents the +momentum of the second particle. The length $PC$ then represents +the total momentum of the system of two particles. + +Suppose that some change of their velocities occurs, not +involving any transference of momentum from outside, e.g.\ a +collision. Since the total momentum $PC$ is unaltered, a similar +%% -----File: 153.png---Folio 143------- +construction made with the new velocities must again bring us +to~$C$; that is to say, the new velocities are represented by the +directions $OB'$, $B'C$, where $B'$ is some other point on the line~$NB$. + +%[Illustration: \textsc{Fig}.~20.] +% [** PP: Moved to top of paragraph] +\begin{figure*}[hbt] +\begin{center} +\Graphic[20]{3in}{153a} +\end{center} +\end{figure*}% +Now examine how this will appear to some other observer $S_1$ +in uniform motion relative to $S$. His transformation of space +and time has been described in \Chapref{III} and is represented in +\Figref{20}, which shows how his time-partitions run as compared +with those of~$S$. The same actual motion is, of course, represented +by parallel directions in the two diagrams; but the +interpretation as a velocity $MA$ is different in the two cases. +Carrying the velocity of $m_1$ through two time-partitions, and of +$m_2$ through three time-partitions, as before, we find that the total +momentum for the observer $S_1$ is represented by~$PC$ (\Figref{20}); +but making a similar construction with the velocities after +collision, we arrive at a different point~$C'$. Thus whilst momentum +is conserved for the observer~$S$, it has altered from $PC$ to +$PC'$ for the observer~$S_1$. + +The discrepancy arises because in the construction the lines +are prolonged to meet partitions which are different for the two +%% -----File: 154.png---Folio 144------- +observers. The rule for determining momentum ought to be +such that both observers make the same construction, independent +of their partitions, so that both arrive by the two routes +at the same point~$C$. Then it will not matter if, through their +different measures of time, one observer measures momentum +by horizontal progress and the other by oblique progress; both +will agree that the momentum has not been altered by the +collision. To describe such a construction, we must use the +interval which is alike for both observers; make the interval-length +of $OB$ equal to $2$~units, and that of $BC$ equal to $3$~units, +disregarding the mesh-system altogether. Then both observers +will make the same diagram and arrive at the same point~$C$ +(different from $C$ or~$C'$ in the previous diagrams). Then if +momentum is conserved for one observer, it will be conserved +for the other. + +This involves a modified definition of momentum. Momentum +must now be the mass multiplied by the change of position $\delta x$ +per lapse of interval $\delta s$, instead of per lapse of time $\delta t$. Thus +\index{Momentum!redefinition of}% +\begin{align*} + \text{momentum} &= m \frac{\delta x}{\delta s} \\ +\text{instead of momentum} &= m \frac{\delta x}{\delta t}, +\end{align*} +and the mass~$m$ still preserves its character as an invariant +number associated with the particle. + +Whether the momentum as now defined is actually conserved +or not, is a matter for experiment, or for theoretical deduction +from the law of gravitation. The point is that with the original +definition general conservation is impossible, because if it held +good for one observer it could not hold for another. The new +definition makes general conservation possible. Actually this +form of the momentum is the one deduced from the law of +gravitation through the identities already described. With +regard to experimental confirmation it is sufficient at present +to state that in all ordinary cases the interval and the time are +so nearly equal that such experimental foundation as existed +for the law of conservation of the old momentum is just as +applicable to the new momentum. + +Thus in the theory of relativity momentum appears as an +%% -----File: 155.png---Folio 145------- +invariant mass multiplied by a modified velocity $\delta x/\delta s$. The +physicist, however, prefers for practical purposes to keep to the +old definition of momentum as mass multiplied by the velocity +$\delta x/\delta t$. We have +\index{Invariant mass}% +\index{Mass!invariant}% +\[ +m \frac{\delta x}{\delta s} + = m \frac{\delta t}{\delta s} · \frac{\delta x}{\delta t}, +\] +accordingly the momentum is separated into two factors, the +velocity $\delta x/\delta t$, and a mass $M = m \delta t/\delta s$, which is no longer an +invariant for the particle but depends on its motion relative to +the observer's space and time. In accordance with the usual +practice of physicists the mass (unless otherwise qualified) is +taken to mean the quantity~$M$. + +Using unaccelerated rectangular axes, we have by definition +of~$s$ +\[ +\delta s^2 = \delta t^{2} - \delta x^{2} - \delta y^{2} - \delta z^{2}, +\] +so that +\begin{align*} +\left(\frac{\delta s}{\delta t}\right)^2 + &= 1 - \left(\frac{\delta x}{\delta t}\right)^2 + - \left(\frac{\delta y}{\delta t}\right)^2 + - \left(\frac{\delta z}{\delta t}\right)^2, \\ + &= 1-u^2, +\end{align*} +where $u$ is the resultant velocity of the particle (the velocity of +light being unity). Hence +\index{Mass!variation with velocity}% +\[ +M = \frac{m}{\surd(1-u^2)}. +\] +Thus the mass increases as the velocity increases, the factor +being the same as that which determines the FitzGerald contraction. + +The increase of mass with velocity is a property which challenges +experimental test. For success it is necessary to be able +to experiment with high velocities and to apply a known force +large enough to produce appreciable deflection in the fast-moving +particle. These conditions are conveniently fulfilled by +the small negatively charged particles emitted by radio-active % [** PP: Regularized radioactive] +substances, known as $\beta$~particles, or the similar particles which +constitute cathode rays. +\index{Beta particles}% +They attain speeds up to 0.8 of the +velocity of light, for which the increase of mass is in the ratio +1.66; and the negative charge enables a large electric or magnetic +force to be applied. Modern experiments fully confirm the +theoretical increase of mass, and show that the factor $1/\surd(1-u^2)$ +%% -----File: 156.png---Folio 146------- +is at least approximately correct. The experiment was originally +performed by Kaufmann; but much greater accuracy has been +obtained by recent modified methods.% +\index{Electron!Kaufmann's experiment on}% +\index{Kaufmann's experiment}% + +Unless the velocity is very great the mass~$M$ may be written +\[ +m/\surd(1-u^2) = m + \tfrac{1}{2} mu^2. +\] +Thus it consists of two parts, the mass when at rest, together +with the second term which is simply the energy of the motion. +If we can say that the term $m$ represents a kind of potential +energy concealed in the matter, mass can be identified with +energy. The increase of mass with velocity simply means that +the energy of motion has been added on. +\index{Energy!identified with mass}% +\index{Mass!identified with energy}% +We are emboldened +to do this because in the case of an electrical charge the electrical +mass is simply the energy of the static field. Similarly the mass +of light is simply the electromagnetic energy of the light. + +In our ordinary units the velocity of light is not unity, and +a rather artificial distinction between mass and energy is introduced. +They are measured by different units, and energy~$E$ has +a mass $E/C^2$ where $C$ is the velocity of light in the units used. +But it seems very probable that mass and energy are two ways +of measuring what is essentially the same thing, in the same +sense that the parallax and distance of a star are two ways of +expressing the same property of location. If it is objected that +they ought not to be confused inasmuch as they are distinct +properties, it must be pointed out that they are not sense-properties, +but mathematical terms expressing the dividend +and product of more immediately apprehensible properties, viz.\ +momentum and velocity. They are essentially mathematical +compositions, and are at the disposal of the mathematician. + +This proof of the variation of mass with velocity is much more +general than that based on the electrical theory of inertia. +\index{Energy!inertia of}% +It +applies immediately to matter in bulk. The masses $m_1$ and~$m_2$ +need not be particles; they can be bodies of any size or composition. +On the electrical theory alone, there is no means of +deducing the variation of mass of a planet from that of an +electron. + +It has to be remarked that, although the inertial mass of a +particle only comes under physical measurement in connection +with a change of its motion, it is just when the motion is changing +that the conception of its mass is least definite; because it is at +%% -----File: 157.png---Folio 147------- +that time that the kinetic energy, which forms part of the mass, +is being passed on to another particle or radiated into the +surrounding field; and it is scarcely possible to define the +moment at which this energy ceases to be associated with the +particle and must be reckoned as broken loose. The amount of +energy or mass in a given region is always a definite quantity; +but the amount attributable to a particle is only definite when +the motion is uniform. In rigorous work it is generally necessary +to consider the mass not of a particle but of a region. + +The motion of matter from one place to another causes an +alteration of the gravitational field in the surrounding space. +If the motion is uniform, the field is simply convected; but if +the motion is accelerated, something of the nature of a gravitational +wave is propagated outwards. The velocity of propagation +is the velocity of light. +\index{Gravitation!propagation with velocity of light}% +\index{Propagation of Gravitation}% +\index{Velocity of gravitation}% +The exact laws are not very simple +because we have seen that the gravitational field modifies the +velocity of light; and so the disturbance itself modifies the +velocity with which it is propagated. In the same way the +exact laws of propagation of sound are highly complicated, +because the disturbance of the air by sound modifies the speed +with which it is propagated. But the approximate laws of +propagation of gravitation are quite simple and are the same as +those of electromagnetic disturbances. + +After mass and energy there is one physical quantity which +plays a very fundamental part in modern physics, known as +\textit{Action}. +\index{Action}% +\textit{Action} here is a very technical term, and is not to be +confused with Newton's ``Action and Reaction.'' In the relativity +theory in particular this seems in many respects to be the most +fundamental thing of all. The reason is not difficult to see. If +we wish to speak of the continuous matter present \textit{at} any particular +point of space and time, we must use the term \textit{density}. +Density multiplied by volume in space gives us \textit{mass} or, what +appears to be the same thing, \textit{energy}. But from our space-time +point of view, a far more important thing is density multiplied +by a four-dimensional volume of space and time; this is \textit{action}. +The multiplication by three dimensions gives mass or energy; +and the fourth multiplication gives mass or energy multiplied +by time. Action is thus mass multiplied by time, or energy +multiplied by time, and is more fundamental than either. + +%% -----File: 158.png---Folio 148------- + +Action is the curvature of the world. +\index{Curvature!identified with action}% +It is scarcely possible +to visualise this statement, because our notion of curvature is +derived from surfaces of two dimensions in a three-dimensional +space, and this gives too limited an idea of the possibilities of a +four-dimensional surface in space of five or more dimensions. +In two dimensions there is just one total curvature, and if that +vanishes the surface is flat or at least can be unrolled into a +plane. In four dimensions there are many coefficients of +curvature; but there is one curvature \textit{par excellence}, which is, +of course, an invariant independent of our mesh-system. It is +the quantity we have denoted by~$G$. It does not follow that if +the curvature vanishes space-time is flat; we have seen in fact +that in a natural gravitational field space-time is not flat +although there may be no mass or energy and therefore no action +or curvature. + +Wherever there is matter\footnote% + {It is rather curious that there is no action in space containing only light. + Light has mass ($M$) of the ordinary kind; but the invariant mass ($m$) vanishes.} +\index{Invariant mass!of light}% +\index{Light!mass of}% +\index{Mass of light}% +there is action and therefore +curvature; and it is interesting to notice that in ordinary matter +the curvature of the space-time world is by no means insignificant. +For example, in water of ordinary density the curvature is the +same as that of space in the form of a sphere of radius $570,000,000$ +kilometres. The result is even more surprising if expressed in +time units; the radius is about half-an-hour. + +It is difficult to picture quite what this means; but at least +we can predict that a globe of water of $570,000,000$~km.\ radius +would have extraordinary properties. +\index{Curvature!of a globe of water}% +\index{Globe of water, limit to size of}% +Presumably there must +be an upper limit to the possible size of a globe of water. So +far as I can make out a homogeneous mass of water of about +this size (and no larger) could exist. It would have no centre, +and no boundary, every point of it being in the same position +with respect to the whole mass as every other point of it---like +points on the \textit{surface} of a sphere with respect to the surface. +Any ray of light after travelling for an hour or two would come +back to the starting point. Nothing could enter or leave the +mass, because there is no boundary to enter or leave by; in +fact, it is coextensive with space. There could not be any other +world anywhere else, because there isn't an ``anywhere else.'' + +The mass of this volume of water is not so great as the most +%% -----File: 159.png---Folio 149------- +moderate estimates of the mass of the stellar system. Some +physicists have predicted a distant future when all energy will +be degraded, and the stellar universe will gradually fall together +into one mass. Perhaps then these strange conditions will be +realised! + +The law of gravitation, the laws of mechanics, and the laws +of the electromagnetic field have all been summed up in a single +Principle of Least Action. +\index{Action, Principle of Least}% +\index{Principle of Least Action}% +For the most part this unification +was accomplished before the advent of the relativity theory, +and it is only the addition of gravitation to the scheme which is +novel. We can see now that if action is something absolute, +a configuration giving minimum action is capable of absolute +definition; and accordingly we should expect that the laws of +the world would be expressible in some such form. The argument +is similar to that by which we first identified the natural +tracks of particles with the tracks of greatest interval-length. +The fact that some such form of law is inevitable, rather discourages +us from seeking in it any clue to the structural details +of our world. + +Action is one of the two terms in pre-relativity physics which +survive unmodified in a description of the absolute world. The +only other survival is entropy. The coming theory of relativity +had cast its shadow before; and physics was already converging +to two great generalisations, the principle of least action and +the second law of thermodynamics or principle of maximum +entropy.% +\index{Entropy}% + +We are about to pass on to recent and more shadowy developments +of this subject; and this is an appropriate place to glance +back on the chief results that have emerged. The following +summary will recall some of the salient points. + +\Paragraph{1.} The order of events in the external world is a four-dimensional +order. + +\Paragraph{2.} The observer either intuitively or deliberately constructs +a system of meshes (space and time partitions) and locates the +events with respect to these. + +\Paragraph{3.} Although it seems to be theoretically possible to describe +phenomena without reference to any mesh-system (by a catalogue +of coincidences), such a description would be cumbersome. In +%% -----File: 160.png---Folio 150------- +practice, physics describes the relations of the events to our +mesh-system; and all the terms of elementary physics and of +daily life refer to this relative aspect of the world. + +\Paragraph{4.} Quantities like length, duration, mass, force, etc.\ have no +absolute significance; their values will depend on the mesh-system +to which they are referred. When this fact is realised, +the results of modern experiments relating to changes of length +of rigid bodies are no longer paradoxical. + +\Paragraph{5.} There is no fundamental mesh-system. In particular +problems, and more particularly in restricted regions, it may +be possible to choose a mesh-system which follows more or less +closely the lines of absolute structure in the world, and so +simplify the phenomena which are related to it. But the world-structure +is not of a kind which can be traced in an exact way +by mesh-systems, and in any large region the mesh-system +drawn must be considered arbitrary. In any case the systems +used in current physics are arbitrary. + +\Paragraph{6.} The study of the absolute structure of the world is based +on the ``interval'' between two events close together, which is +an absolute attribute of the events independent of any mesh-system. +A world-geometry is constructed by adopting the +interval as the analogue of distance in ordinary geometry.% +\index{Interval}% + +\Paragraph{7.} This world-geometry has a property unlike that of +Euclidean geometry in that the interval between two real +events may be real or imaginary. The necessity for a physical +distinction, corresponding to the mathematical distinction between +real and imaginary intervals, introduces us to the separation +of the four-dimensional order into time and space. But this +separation is not unique, and the separation commonly adopted +depends on the observer's track through the four-dimensional +world.% +\index{Imaginary intervals}% + +\Paragraph{8.} The geodesic, or track of maximum or minimum interval-length +between two distant events, has an absolute significance. +And since no other kind of track can be defined absolutely, it +is concluded that the tracks of freely moving particles are +geodesics.% +\index{Geodesic!absolute significance of}% +\index{Geodesic!motion of particles in}% +\index{Interval-length!tracks of maximum}% + +\Paragraph{9.} In Euclidean geometry the geodesics are straight lines. It +is evidently impossible to choose space and time-reckoning so +that all free particles in the solar system move in straight lines. +%% -----File: 161.png---Folio 151------- +Hence the geometry must be non-Euclidean in a field of gravitation. + +\Paragraph{10.} Since the tracks of particles in a gravitational field are +evidently governed by some law, the possible geometries must +be limited to certain types. + +\Paragraph{11.} The limitation concerns the absolute structure of the +world, and must be independent of the choice of mesh-system. +This narrows down the possible discriminating characters. +Practically the only reasonable suggestion is that the world +must (in empty space) be ``curved no higher than the first +degree''; and this is taken as the law of gravitation. + +\Paragraph{12.} The simplest type of hummock with this limited curvature +has been investigated. It has a kind of infinite chimney at the +summit, which we must suppose cut out and filled up with a +region where this law is not obeyed, i.e.\ with a particle of matter. + +\Paragraph{13.} The tracks of the geodesics on the hummock are such as +to give a very close accordance with the tracks computed by +Newton's law of gravitation. The slight differences from the +Newtonian law have been experimentally verified by the motion +of Mercury and the deflection of light. + +\Paragraph{14.} The hummock might more properly be described as a +ridge extending linearly. Since the interval-length along it is +real or time-like, the ridge can be taken as a time-direction. +Matter has thus a continued existence in time. Further, in +order to conform with the law, a small ridge must always follow +a geodesic in the general field of space-time, confirming the conclusion +arrived at under~(8). + +\Paragraph{15.} The laws of conservation of energy and momentum in +mechanics can be deduced from this law of world-curvature. + +\Paragraph{16.} Certain phenomena such as the FitzGerald contraction +and the variation of mass with velocity, which were formerly +thought to depend on the behaviour of electrical forces concerned, +are now seen to be general consequences of the relativity +of knowledge. That is to say, length and mass being the relations +of some absolute thing to the observer's mesh-system, we can +foretell how these relations will be altered when referred to +another mesh-system. +%% -----File: 162.png---Folio 152------- + + +\Chapter{X}{Towards Infinity} + +\Quote{W.~K. Clifford (1873).} +{The geometer of to-day knows nothing about the nature of actually existing +space at an infinite distance; he knows nothing about the properties of this +present space in a past or a future eternity. He knows, indeed, that the laws +assumed by Euclid are true with an accuracy that no direct experiment can +approach, not only in this place where we are, but in places at a distance from +us that no astronomer has conceived; but he knows this as of Here and Now; +beyond his range is a There and Then of which he knows nothing at present, +but may ultimately come to know more.}% +\index{Clifford}% + + +\First{The} great stumbling-block for a philosophy which denies +absolute space is the experimental detection of absolute rotation. +\index{Absolute rotation}% +\index{Relativity of rotation}% +\index{Rotation, absolute}% +The belief that the earth rotates on its axis was suggested by +the diurnal motions of the heavenly bodies; this observation is +essentially one of relative rotation, and, if the matter rested +there, no difficulty would be felt. But we can detect the same +rotation, or a rotation very closely equal to it, by methods +which do not seem to bring the heavenly bodies into consideration; +and such a rotation is apparently absolute. The planet +Jupiter is covered with cloud, so that an inhabitant would +probably be unaware of the existence of bodies outside; yet he +could quite well measure the rotation of Jupiter. By the gyrocompass +he would fix two points on the planet---the north and +south poles. Then by Foucault's experiment on the change of +the plane of motion of a freely suspended pendulum, he would +determine an angular velocity about the poles. +\index{Foucault's pendulum}% +\index{Gyrocompass} % [** PP: Removed hyphen for consistency] +Thus there is +certainly a definite physical constant, an angular velocity about +an axis, which has a fundamental importance for the inhabitants +of Jupiter; the only question is whether we are right in giving +it the name absolute rotation. + +Contrast this with absolute translation. Here it is not a +question of giving the right name to a physical constant; the +inhabitants of Jupiter would find no constant to name. We see +at once that a relativity theory of translation is on a different +footing from a relativity theory of rotation. The duty of the +%% -----File: 163.png---Folio 153------- +former is to explain facts; the duty of the latter is to explain +away facts. + +Our present theory seems to make a start at tackling this +problem, but gives it up. It permits the observer, if he wishes, +to consider the earth as non-rotating, but surrounded by a field +of centrifugal force; all the other bodies in the universe are then +revolving round the earth in orbits mainly controlled by this +field of centrifugal force. Astronomy on this basis is a little +cumbersome; but all the phenomena are explained perfectly. +The centrifugal force is part of the gravitational field, and obeys +Einstein's law of gravitation, so that the laws of nature are +completely satisfied by this representation. One awkward +question remains, What causes the centrifugal force? Certainly +not the earth which is here represented as non-rotating. As we +go further into space to look for a cause, the centrifugal force +becomes greater and greater, so that the more we defer the debt +the heavier the payment demanded in the end. Our present +theory is like the debtor who does not mind how big an obligation +accumulates satisfied that he can always put off the payment. +It chases the cause away to infinity, content that the laws of +nature---the relations between contiguous parts of the world---are +satisfied all the way. + +One suggested loophole must be explored. Our new law of +gravitation admits that a rapid motion of the attracting body +will affect the field of force. If the earth is non-rotating, the +stars must be going round it with terrific speed. May they not +in virtue of their high velocities produce gravitationally a +sensible field of force on the earth, which we recognise as the +centrifugal field? This would be a genuine elimination of +absolute rotation, attributing all effects indifferently to the +rotation of the earth the stars being at rest, or to the revolution +of the stars the earth being at rest; nothing matters except the +relative rotation. I doubt whether anyone will persuade himself +that the stars have anything to do with the phenomenon. We +do not believe that if the heavenly bodies were all annihilated +it would upset the gyrocompass. In any case, precise calculation +shows that the centrifugal force could not be produced by the +motion of the stars, so far as they are known.% +\index{Centrifugal Force!not caused by stars}% + +We are therefore forced to give up the idea that the signs of +%% -----File: 164.png---Folio 154------- +the earth's rotation---the protuberance of its equator, the +phenomena of the gyrocompass, etc.---are due to a rotation +relative to any matter we can recognise. The philosopher who +persists that a rotation which is not relative to matter is unthinkable, +will no doubt reply that the rotation must then be +relative to some matter which we have not yet recognised. We +have hitherto been greatly indebted to the suggestions of +philosophy in evolving this theory, because the suggestions +related to the things we know about; and, as it turned out, they +were confirmed by experiment. But as physicists we cannot +take the same interest in the new demand; we do not necessarily +challenge it, but it is outside our concern. Physics demands of +its scheme of nature something else besides truth, namely a +certain quality that we may call convergence. The law of +conservation of energy is only strictly true when the whole +universe is taken into account; but its value in physics lies in +the fact that it is \textit{approximately} true for a very limited system. +Physics is an exact science because the chief essentials of a +problem are limited to a few conditions; and it draws near to +the truth with ever-increasing approximation as it widens its +purview. The approximations of physics form a convergent +series. +\index{Convergence of physical approximations}% +History, on the other hand, is very often like a divergent +series; no approximation to its course is reached until the last +term of the infinite series has been included in the data of +prediction. Physics, if it wishes to retain its advantage, must +take its own course, formulating those laws which are approximately +true for the limited data of sense, and extending them +into the unknown. The relativity of rotation is not approximately +true for the data of sense, although it may possibly be +true when the unknown as well as the known are included. + +The same considerations that apply to rotation apply to +acceleration, although the difficulty is less striking. We can if +we like attribute to the sun some arbitrary acceleration, balancing +it by introducing a uniform gravitational field. Owing to this +field the rest of the stars will move with the same acceleration +and no phenomena will be altered. But then it seems necessary +to find a cause for this field. It is not produced by the gravitation +of the stars. Our only course is to pursue the cause further and +further towards infinity; the further we put it away, the greater +%% -----File: 165.png---Folio 155------- +the mass of attracting matter needed to produce it. On the +other hand, the earth's absolute acceleration does not intrude +on our attention in the way that its absolute rotation does\footnotemark. + \footnotetext{To determine even roughly the earth's absolute acceleration we should + need a fairly full knowledge of the disturbing effects of all the matter in the + universe. A similar knowledge would be required to determine the absolute + rotation \textit{accurately}; but all the matter likely to exist would have so small an + effect, that we can at once assume that the absolute rotation is very nearly + the same as the experimentally determined rotation.}% +\index{Absolute acceleration}% [** PP: Index reads p. 154] + + +We are vaguely conscious of a difficulty in these results; but +if we examine it closely, the difficulty does not seem to be a +very serious one. The theory of relativity, as we have understood +it, asserts that our partitions of space and time are introduced +by the observer and are irrelevant to the laws of nature; and +therefore the current quantities of physics, length, duration, +mass, force, etc., which are relative to these partitions, are not +things having an absolute significance in nature. But we have +never denied that there are features of the world having an +absolute significance; in fact, we have spent much time in finding +such features. The geodesics or natural tracks have been shown +to have an absolute significance; +\index{Geodesic structure!absolute character of}% +\index{Structure, geodesic!absolute character of}% +and it is possible in a limited +region of the world to choose space and time partitions such +that all geodesics become approximately straight lines. We may +call this a ``natural'' frame for that region, although it is not +as a rule the space and time adopted in practice; +\index{Natural frame}% +it is for example +the space and time of the observers in the falling projectile, not +of Newton's super-observer. It is capable of absolute definition, +except that it is ambiguous in regard to uniform motion. Now +the rotation of the earth determined by Foucault's pendulum +experiment is the rotation referred to this natural frame. But +we must have misunderstood our own theory of relativity +altogether, if we think there is anything inadmissible in an +absolute rotation of such a kind.% +\index{Relativity of rotation}% + +Material particles and geodesics are both features of the +absolute structure of the world; and a rotation relative to +geodesic structure does not seem to be on any different footing +from a velocity relative to matter. There is, however, the +striking feature that rotation seems to be relative not merely +to the local geodesic structure but to a generally accepted +universal frame; whereas it is necessary to specify precisely +%% -----File: 166.png---Folio 156------- +what matter a velocity is measured with respect to. This is +largely a question of how much accuracy is needed in specifying +velocities and rotations, respectively. If in stating the speed of +a $\beta$~particle we do not mind an error of $10,000$ kilometres a +second, we need not specify precisely what star or planet its +velocity is referred to. The moon's (local) angular velocity is +sometimes given to fourteen significant figures; I doubt if +any universal frame is well-defined enough for this accuracy. +There is no doubt much greater continuity in the geodesic +structure in different parts of the world than in the material +structure; but the difference is in degree rather than in +principle. + +It is probable that here we part company from many of the +continental relativists, who give prominent place to a principle +known as the \textit{law of causality}---that only those things are to be +regarded as being in causal connection which are capable of +being actually observed. +\index{Causality, law of}% +This seems to be interpreted as placing +matter on a plane above geodesic structure in regard to the +formulation of physical laws, though it is not easy to see in +what sense a distribution of matter can be regarded as more +observable than the field of influence in surrounding space +which makes us aware of its existence. The principle itself is +debateable; that which is observable to us is determined by +the accident of our own structure, and the law of causality +seems to impose our own limitations on the free interplay of +entities in the world outside us. In this book the tradition of +Faraday and Maxwell still rules our outlook; and for us matter +and electricity are but incidental points of complexity, the +activity of nature being primarily in the so-called empty spaces +between. + +The vague universal frame to which rotation is referred is +called the \textit{inertial frame}. +\index{Frame, inertial}% +\index{Inertial frame}% +It is definite in the flat space-time far +away from all matter. In the undulating country corresponding +to the stellar universe it is not a precise conception; it is rather +a rude outline, arbitrary within reasonable limits, but with the +general course indicated. The reason for the term inertial frame +is of interest. We can quite freely use a mesh-system deviating +widely from the inertial frame (e.g.\ rotating axes); but we have +seen that there is a postponed debt to pay in the shape of an +%% -----File: 167.png---Folio 157------- +apparently uncaused field of force. But is there no debt to pay, +even when the inertial frame is used? In that case there is no +gravitational or centrifugal force at infinity; +\index{Centrifugal Force!debt at infinity}% +\index{Geodesic!in regions at infinity}% +\index{Infinity, conditions at}% +\index{Structure, geodesic!behaviour at infinity}% +but there is still +inertia, which is of the same nature. The distinction between +force as requiring a cause and inertia as requiring no cause +cannot be sustained. We shall not become any more solvent by +commuting our debt into pure inertia. The debt is inevitable +whatever mesh-system is used; we are only allowed to choose +the form it shall take. + +The debt after all is a very harmless one. At infinity we have +the absolute geodesics in space-time, and we have our own +arbitrarily drawn mesh-system. The relation of the geodesics +to the mesh-system decides whether our axes shall be termed +rotating or non-rotating; and ideally it is this relation that is +determined when a so-called absolute rotation is measured. +No one could reasonably expect that there would be no determinable +relation. On the other hand uniform translation does +not affect the relation of the geodesics to the mesh-system---if +they were straight lines originally, they remain straight lines---thus +uniform translation cannot be measured except relative to +matter. + +We have been supposing that the conditions found in the +remotest parts of space accessible to observation can be extrapolated +to infinity; and that there are still definite natural +tracks in space-time far beyond the influence of matter. Feelings +of objection to this view arise in certain minds. It is urged that +as matter influences the course of geodesics it may well be +responsible for them altogether; so that a region outside the +field of action of matter could have no geodesics, and consequently +no intervals. All the potentials would then necessarily +be zero. Various modified forms of this objection arise; but the +main feeling seems to be that it is unsatisfactory to have certain +conditions prevailing in the world, which can be traced away to +infinity and so have, as it were, their source at infinity; and +there is a desire to find some explanation of the inertial frame +as built up through conditions at a finite distance.% +\index{Inertia!in regions at infinity}% + +Now if all intervals vanished space-time would shrink to a +point. Then there would be no space, no time, no inertia, no +anything. Thus a cause which creates intervals and geodesics +%% -----File: 168.png---Folio 158------- +must, so to speak, extend the world. We can imagine the world +stretched out like a plane sheet; but then the stretching cause---the +cause of the intervals---is relegated beyond the bounds of +space and time, i.e.\ to infinity. This is the view objected to, +though the writer does not consider that the objection has +much force. An alternative way is to inflate the world from +inside, as a balloon is blown out. In this case the stretching +force is not relegated to infinity, and ruled outside the scope of +experiment; it is acting at every point of space and time, curving +the world to a sphere. We thus get the idea that space-time +may have an essential curvature on a great scale independent +of the small hummocks due to recognised matter. + +It is not necessary to speculate whether the curvature is +produced (as in the balloon) by some pressure applied from a +fifth dimension. For us it will appear as an innate tendency of +four-dimensional space-time to curve. It may be asked, what +have we gained by substituting a natural curvature of space-time +for a natural stretched condition corresponding to the +inertial frame? +\index{Curvature!of space and time}% +As an explanation, nothing. But there is this +difference, that the theory of the inertial frame can now be +included in the differential law of gravitation instead of remaining +outside and additional to the law. + +It will be remembered that one clue by which we previously +reached the law of gravitation was that flat space-time must be +compatible with it. But if space-time is to have a small natural +curvature independent of matter this condition is now altered. +It is not difficult to find the necessary alteration of the law\footnote{Appendix, \Noteref{14}.}. +\Pagelabel{note14}% +It will contain an additional, and at present unknown, constant, +which determines the size of the world. + +Spherical space is not very easy to imagine. We have to +think of the properties of the surface of a sphere---the two-dimensional +case---and try to conceive something similar applied +to three-dimensional space. Stationing ourselves at a point let +us draw a series of spheres of successively greater radii. The +surface of a sphere of radius~$r$ should be proportional to~$r^{2}$; but +in spherical space the areas of the more distant spheres begin +to fall below the proper proportion. There is not so much room +out there as we expected to find. Ultimately we reach a sphere +%% -----File: 169.png---Folio 159------- +of biggest possible area, and beyond it the areas begin to decrease\footnotemark. + \footnotetext{The area is, of course, to be determined by measurement of some kind.}% +The last sphere of all shrinks to a point---our antipodes. +Is there nothing beyond this? Is there a kind of boundary +there? There is nothing beyond and yet there is no boundary. +On the earth's surface there is nothing beyond our own antipodes +but there is no boundary there. + +The difficulty is that we try to realise this spherical world by +imagining how it would appear to us and to our measurements. +There has been nothing in our experience to compare it with, +and it seems fantastic. But if we could get rid of the personal +point of view, and regard the sphericity of the world as a statement +of the type of order of events outside us, we should think +that it was a simple and natural order which is as likely as any +other to occur in the world. + +In such a world there is no difficulty about accumulated debt +at the boundary. There is no boundary. The centrifugal force +increases until we reach the sphere of greatest area, and then, +still obeying the law of gravitation, diminishes to zero at the +antipodes. The debt has paid itself automatically. + +We must not exaggerate what has been accomplished by this +modification of the theory. A new constant has been introduced +into the law of gravitation which gives the world a definite +extension. Previously there was nothing to fix the scale of the +world; it was simply given \textit{a~priori} that it was infinite. Granted +extension, so that the intervals are not invariably zero, we can +determine geodesics everywhere, and hence mark out the inertial +frame. + +Spherical space-time, that is to say a four-dimensional continuum +of space and imaginary time forming the surface of a +sphere in five dimensions, has been investigated by Prof.\ de~Sitter. +\index{de Sitter}% +\index{Spherical space-time}% +If real time is used the world is spherical in its space +dimensions, but open towards plus and minus infinity in its +time dimension, like an hyperboloid. This happily relieves us +of the necessity of supposing that as we progress in time we +shall ultimately come back to the instant we started from! +\index{Retardation of time!in spherical world} % [** PP: Index reads p. 160] +History never repeats itself. But in the space dimensions we +should, if we went on, ultimately come back to the starting +point. This would have interesting physical results, and we +%% -----File: 170.png---Folio 160------- +shall see presently that Einstein has a theory of the world in +which the return can actually happen; but in de~Sitter's theory +it is rather an abstraction, because, as he says, ``all the paradoxical +phenomena can only happen after the end or before the +beginning of eternity.'' + +The reason is this. Owing to curvature in the time dimension, +as we examine the condition of things further and further from +our starting point, our time begins to run faster and faster, or +to put it another way natural phenomena and natural clocks +slow down. The condition becomes like that described in +Mr~H.~G. Wells's story ``The new accelerator.'' + +When we reach half-way to the antipodal point, time stands +still. Like the Mad Hatter's tea party, it is always 6~o'clock; +and nothing whatever can happen however long we wait. There +is no possibility of getting any further, because everything +including light has come to rest here. All that lies beyond is +for ever cut off from us by this barrier of time; and light can +never complete its voyage round the world. + +That is what happens when the world is viewed from one +station; but if attracted by such a delightful prospect, we proceeded +to visit this scene of repose, we should be disappointed. +We should find nature there as active as ever. We thought time +was standing still, but it was really proceeding there at the +usual rate, as if in a fifth dimension of which we had no +cognisance. Casting an eye back on our old home we should see +that time apparently had stopped still there. Time in the two +places is proceeding in directions at right angles, so that the +progress of time at one point has no relation to the perception +of time at the other point. The reader will easily see that a being +confined to the surface of a sphere and not cognisant of a third +dimension, will, so to speak, lose one of his dimensions altogether +when he watches things occurring at a point $90°$ away. He +regains it if he visits the spot and so adapts himself to the two +dimensions which prevail there. + +It might seem that this kind of fantastic world-building can +have little to do with practical problems. But that is not quite +certain. May we not be able actually to observe the slowing +down of natural phenomena at great distances from us? The +most remote objects known are the spiral nebulae, whose +%% -----File: 171.png---Folio 161------- +distances may perhaps be of the order a million light years. +If natural phenomena are slowed down there, the vibrations of +an atom are slower, and its characteristic spectral lines will +appear displaced to the red. +\index{Displacement of spectral lines!in nebulae}% +\index{Nebulae, atomic vibrations in}% +We should generally interpret this +as a Doppler effect, implying that the nebula is receding. The +motions in the line-of-sight of a number of nebulae have been +determined, chiefly by Prof.\ Slipher. The data are not so ample +as we should like; but there is no doubt that large receding +motions greatly preponderate. +\index{Receding velocities!of spiral nebulae}% +This may be a genuine phenomenon +in the evolution of the material universe; but it is also +possible that the interpretation of spectral displacement as a +receding velocity is erroneous; and the effect is really the slowing +of atomic vibrations predicted by de~Sitter's theory. + +Prof.\ Einstein himself prefers a different theory of curved +space-time. His world is cylindrical---curved in the three space +dimensions and straight in the time dimension. +\index{Cylindrical world, Einstein's}% +Since time is no +longer curved, the slowing of phenomena at great distances +from the observer disappears, and with it the slight experimental +support given to the theory by the observations of spiral nebulae. +There is no longer a barrier of eternal rest, and a ray of light is +able to go round the world. + +In various ways crude estimates of the size of the world both +on de~Sitter's and Einstein's hypotheses have been made; and +in both cases the radius is thought to be of the order $10^{13}$ times +the distance of the earth from the sun. A ray of light from the +sun would thus take about $1000$~million years to go round the +world; and after the journey the rays would converge again at +the starting point, and then diverge for the next circuit. +\index{Light!voyage round the world}% +The +convergent would have all the characteristics of a real sun so +far as light and heat are concerned, only there would be no +substantial body present. Thus corresponding to the sun we +might see a series of ghosts occupying the positions where the +sun was $1000$, $2000$, $3000$, etc., million years ago, if (as seems +probable) the sun has been luminous for so long.% +\index{Ghosts of stars}% + +It is rather a pleasing speculation that records of the previous +states of the sidereal universe may be automatically reforming +themselves on the original sites. Perhaps one or more of the +many spiral nebulae are really phantoms of our own stellar +system. Or it may be that only a proportion of the stars are +%% -----File: 172.png---Folio 162------- +substantial bodies; the remainder are optical ghosts revisiting +their old haunts. It is, however, unlikely that the light rays +after their long journey would converge with the accuracy which +this theory would require. The minute deflections by the various +gravitational fields encountered on the way would turn them +aside, and the focus would be blurred. Moreover there is a +likelihood that the light would gradually be absorbed or +scattered by matter diffused in space, which is encountered on +the long journey. + +It is sometimes suggested that the return of the light-wave +to its starting point can most easily be regarded as due to the +force of gravitation, there being sufficient mass distributed +through the universe to control its path in a closed orbit. We +should have no objection in principle to this way of looking at +it; but we doubt whether it is correct in fact. It is quite possible +for light to return to its starting point in a world without +gravitation. We can roll flat space-time into a cylinder and join +the edges; its geometry will still be Euclidean and there will be +no gravitation; but a ray of light can go right round the cylinder +and return to the starting point in space. Similarly in Einstein's +more complex type of cylinder (three dimensions curved and +one dimension linear), it seems likely that the return of the +light is due as much to the connectivity of his space, as to +the non-Euclidean properties which express the gravitational +field. + +For Einstein's cylindrical world it is necessary to postulate +the existence of vast quantities of matter (not needed on de~Sitter's +theory) far in excess of what has been revealed by our +telescopes. This additional material may either be in the form +of distant stars and galaxies beyond our limits of vision, or it +may be uniformly spread through space and escape notice by +its low density. There is a definite relation between the average +density of matter and the radius of the world; the greater the +radius the smaller must be the average density. + +Two objections to this theory may be urged. In the first +place, absolute space and time are restored for phenomena on +a cosmical scale. The ghost of a star appears at the spot where +the star was a certain number of million years ago; and from +the ghost to the present position of the star is a definite distance% +%% -----File: 173.png---Folio 163------- +---the absolute motion of the star in the meantime\footnotemark. + \footnotetext{The ghost is not formed where the star is now. If two stars were near + together when the light left them their ghosts must be near together, although + the stars may now be widely separated.}% +The world +taken as a whole has one direction in which it is not curved; +that direction gives a kind of absolute time distinct from space. +\index{Absolute time, in cylindrical world}% +Relativity is reduced to a local phenomenon; and although this +is quite sufficient for the theory hitherto described, we are +inclined to look on the limitation rather grudgingly. But we +have already urged that the relativity theory is not concerned +to deny the possibility of an absolute time, but to deny that it +is concerned in any experimental knowledge yet found; and it +need not perturb us if the conception of absolute time turns up +in a new form in a theory of phenomena on a cosmical scale, +as to which no experimental knowledge is yet available. Just +as each limited observer has his own particular separation of +space and time, so a being coextensive with the world might +well have a special separation of space and time natural to him. +It is the time for this being that is here dignified by the title +``absolute.''% +\index{Time!absolute}% + +Secondly, the revised law of gravitation involves a new +constant which depends on the total amount of matter in the +world; or conversely the total amount of matter in the world +is determined by the law of gravitation. This seems very hard +to accept---at any rate without some plausible explanation of +how the adjustment is brought about. We can see that, the +constant in the law of gravitation being fixed, there may be +some upper limit to the amount of matter possible; as more +and more matter is added in the distant parts, space curves +round and ultimately closes; the process of adding more matter +must stop, because there is no more space, and we can only +return to the region already dealt with. But there seems nothing +to prevent a defect of matter, leaving space unclosed. Some +mechanism seems to be needed, whereby either gravitation +creates matter, or all the matter in the universe conspires to +define a law of gravitation. + +Although this appears to the writer rather bewildering, it is +welcomed by those philosophers who follow the lead of Mach. +For it leads to the result that the extension of space and time +%% -----File: 174.png---Folio 164------- +depends on the amount of matter in the world---partly by its +direct effect on the curvature and partly by its influence on the +constant of the law of gravitation. The more matter there is, +the more space is created to contain it, and if there were no +matter the world would shrink to a point. + +In the philosophy of Mach a world without \textit{matter} is unthinkable. +Matter in Mach's philosophy is not merely required as +a test body to display properties of something already there, +which have no physical meaning except in relation to matter; +it is an essential factor in causing those properties which it is +able to display. Inertia, for example, would not appear by the +insertion of one test body in the world; in some way the presence +of other matter is a necessary condition. It will be seen how +welcome to such a philosophy is the theory that space and the +inertial frame come into being with matter, and grow as it grows. +Since the laws of inertia are part of the law of gravitation, +Mach's philosophy was summed up---perhaps unconsciously---in +the profound saying ``If there were no matter in the universe, +the law of gravitation would fall to the ground.''% +\index{Inertia!Mach's views}% +\index{Mach's philosophy}% + +No doubt a world without matter, in which nothing could +ever happen, would be very uninteresting; and some might deny +its claim to be regarded as a world at all. But a world uniformly +filled with matter would be equally dull and unprofitable; so +there seems to be little object in denying the possibility of the +former and leaving the latter possible. + +The position can be summed up as follows:---in a space +without absolute features, an absolute rotation would be as +meaningless as an absolute translation; +\index{Absolute rotation}% +\index{Rotation, absolute}% +accordingly, the existence +of an experimentally determined quantity generally +identified with absolute rotation requires explanation. It was +remarked on \Pageref{41} that it would be difficult to devise a plan +of the world according to which uniform motion has no significance +but non-uniform motion is significant; but such a world has +been arrived at---a plenum, of which the absolute features are +intervals and geodesics. +\index{Aether!a plenum with geodesic structure}% +\index{Geodesic structure!absolute character of}% +\index{Structure, geodesic!absolute character of}% +In a limited region this plenum gives +a natural frame with respect to which an acceleration or rotation +(but not a velocity) capable of absolute definition can be +measured. In the case of rotation the local distortions of the +frame are of comparatively little account; and this explains +%% -----File: 175.png---Folio 165------- +why in practice rotation appears to have reference to some world-wide % [** PP: Hyphenated across a line in original] +inertial frame. + +Thus absolute rotation does not indicate any logical flaw in +the theory hitherto developed; and there is no need to accept +any modification of our views. Possibly there may be a still +wider relativity theory, in which our supposed plenum is to be +regarded as itself an abstraction of the relations of the matter +distributed throughout the world, and not existent apart from +such matter. This seems to exalt matter rather unnecessarily. +It may be true; but we feel no necessity for it, unless experiment +points that way. It is with some such underlying idea that +Einstein's cylindrical space-time was suggested, since this +cannot exist without matter to keep it stretched. Now we freely +admit that our assumption of perfect flatness in the remote +parts of space was arbitrary, and there is no justification for +insisting on it. A small curvature is possible both conceptually +and experimentally. The arguments on both sides have hitherto +been little more than prejudices, which would be dissipated by +any experimental or theoretical lead in one direction. Weyl's +theory of the electromagnetic field, discussed in the next +chapter, assigns a definite function to the curvature of space; +and this considerably alters the aspect of the question. We are +scarcely sufficiently advanced to offer a final opinion; but the +conception of cylindrical space-time seems to be favoured by +this new development of the theory. + +Some may be inclined to challenge the right of the Einstein +theory, at least as interpreted in this book, to be called a +relativity theory. Perhaps it has not all the characteristics +which have at one time or another been associated with that +name; but the reader, who has followed us so far, will see how +our search for an absolute world has been guided by a recognition +of the relativity of the measurements of physics. It may be +urged that our geodesics ought not to be regarded as fundamental; +a geodesic has no meaning in itself; what we are really concerned +with is the relation of a particle following a geodesic to all the +other matter of the world and the geodesic cannot be thought of +apart from such other matter. We would reply, ``Your particle +of matter is not fundamental; it has no meaning in itself; what +you are really concerned with is its `field'---the relation of the +%% -----File: 176.png---Folio 166------- +geodesics about it to the other geodesics in the world---and +matter cannot be thought of apart from its field.'' It is all +a tangle of relations; physical theory starts with the simplest +constituents, philosophical theory with the most familiar constituents. +They may reach the same goal; but their methods +are often incompatible. +%% -----File: 177.png---Folio 167------- + + +\Chapter{XI}{Electricity and Gravitation} + +% [** PP: Explicit formatting containg hard-coded dimensions] +\noindent\begin{minipage}{\textwidth} +\small +Thou shalt not have in thy bag divers weights, a great and a small. + +Thou shalt not have in thine house divers measures, a great and a small. + +\hangindent 2em +But thou shalt have a perfect and just weight, a perfect and just measure shalt thou have.\hfill\hbox{\textit{Book of Deuteronomy.}\hspace*{\QIndent}} +\end{minipage}% +\index{Electricity and gravitation}% + + +\First{The} relativity theory deduces from geometrical principles the +existence of gravitation and the laws of mechanics of matter. +Mechanics is derived from geometry, not by \textit{adding} arbitrary +hypotheses, but by \textit{removing} unnecessary assumptions, so that +a geometer like Riemann might almost have foreseen the more +important features of the actual world. But nature has in +reserve one great surprise---electricity.% +\index{Riemann}% + +Electrical phenomena are not in any way a misfit in the +relativity theory, and historically it is through them that it has +been developed. Yet we cannot rest satisfied until a deeper +unity between the gravitational and electrical properties of the +world is apparent. The electron, which seems to be the smallest +particle of matter, is a singularity in the gravitational field and +also a singularity in the electrical field. +\index{Electron!singularity in field}% +How can these two facts +be connected? The gravitational field is the expression of some +state of the world, which also manifests itself in the natural +geometry determined with measuring appliances; the electric +field must also express some state of the world, but we have not +as yet connected it with natural geometry. May there not still +be unnecessary assumptions to be removed, so that a yet more +comprehensive geometry can be found, in which gravitational +and electrical fields both have their place? + +There \textit{is} an arbitrary assumption in our geometry up to this +point, which it is desirable now to point out. We have based +everything on the ``interval,'' which, it has been said, is something +which all observers, whatever their motion or whatever +their mesh-system, can measure absolutely, agreeing on the +result. This assumes that they are provided with identical +standards of measurement---scales and clocks. But if $A$ is in +%% -----File: 178.png---Folio 168------- +motion relative to $B$ and wishes to hand his standards to $B$ to +check his measures, he must stop their motion; this means in +practice that he must bombard his standards with material +molecules until they come to rest. Is it fair to assume that no +alteration of the standard is caused by this process? Or if $A$ +measures time by the vibrations of a hydrogen atom, and space +by the wave-length of the vibration, still it is necessary to stop +the atom by a collision in which electrical forces are involved? % [** PP: Changed . to ?] + +The standard of length in physics is the length in the year +1799 of a bar deposited at Paris. +\index{Standard metre, comparison with}% +Obviously no interval is ever +compared directly with that length; there must be a continuous +chain of intermediate steps extending like a geodetic triangulation +through space and time, first along the past history of the +scale actually used, then through intermediate standards, and +finally along the history of the Paris metre itself. It may be +that these intermediate steps are of no importance---that the +same result is reached by whatever route we approach the +standard; but clearly we ought not to make that assumption +without due consideration. We ought to construct our geometry +in such a way as to show that there are intermediate steps, and +that the comparison of the interval with the ultimate standard +is not a kind of action at a distance. + +To compare intervals in different directions at a point in +space and time does not require this comparison with a distant +standard. The physicist's method of describing phenomena +near a point~$P$ is to lay down for comparison (1)~a mesh-system, +(2)~a unit of length (some kind of material standard), which can +also be used for measuring time, the velocity of light being unity. +With this system of reference he can measure in terms of his +unit small intervals $PP'$ running in any direction from~$P$, +summarising the results in the fundamental formula +\[ +ds^{2} = g_{11}\, dx_{1}^{2} + + g_{22}\, dx_{2}^{2} + \dotsb + + 2g_{12}\, dx_{1}dx_{2} + \dotsb . +\] +If now he wishes to measure intervals near a distant point~$Q$, he +must lay down a mesh-system and a unit of measure there. He +naturally tries to simplify matters by using what he would call +the \textit{same} unit of measure at $P$ and~$Q$, either by transporting a +material rod or some equivalent device. If it is immaterial by +what route the unit is carried from $P$ to~$Q$, and replicas of the +%% -----File: 179.png---Folio 169------- +unit carried by different routes all agree on arrival at~$Q$, this +method is at any rate explicit. The question whether the unit +at~$Q$ defined in this way is \textit{really} the same as that at $P$ is mere +metaphysics. But if the units carried by different routes disagree, +there is no unambiguous means of identifying a unit at +$Q$ with the unit at~$P$. Suppose $P$ is an event at Cambridge on +March~1, and $Q$ at London on May~1; we are contemplating the +possibility that there will be a difference in the results of measures +made with our standard in London on May~1, according as the +standard is taken up to London on March~1 and remains there, +or is left at Cambridge and taken up on May~1. This seems at +first very improbable; but our reasons for allowing for this +possibility will appear presently. If there is this ambiguity the +only possible course is to lay down (1)~a mesh-system filling all +the space and time considered, (2)~a definite unit of interval, or +gauge, \textit{at every point of space and time}. +\index{Gauge-system}% +The geometry of the +world referred to such a system will be more complicated than +that of Riemann hitherto used; and we shall see that it is +necessary to specify not only the 10~$g$'s, but four other functions +of position, which will be found to have an important physical +meaning.% +\index{Geometry!non-Riemannian}% +\index{Non-Riemannian geometry}% + +The observer will naturally simplify things by making the +units of gauge at different points as nearly as possible equal, +judged by ordinary comparisons. But the fact remains that, +when the comparison depends on the route taken, exact equality +is not definable; and we have therefore to admit that the \textit{exact} +standards are laid down at every point independently. + +It is the same problem over again as occurs in regard to +mesh-systems. We lay down particular rectangular axes near +a point~$P$; presently we make some observations near a distant +point~$Q$. To what coordinates shall the latter be referred? The +natural answer is that we must use the same coordinates as we +were using at~$P$. But, except in the particular case of flat space, +there is no means of defining exactly what coordinates at $Q$ are +the \textit{same} as those at~$P$. In many cases the ambiguity may be +too trifling to trouble us; but in exact work the only course is +to lay down a definite mesh-system extending throughout space, +the precise route of the partitions being necessarily arbitrary. +We now find that we have to add to this by placing in each +%% -----File: 180.png---Folio 170------- +mesh a gauge whose precise length must be arbitrary. Having +done this the next step is to make measurements of intervals +(using our gauges). This connects the absolute properties of the +world with our arbitrarily drawn mesh-system and gauge-system. +And so by measurement we determine the $g$'s and the +new additional quantities, which determine the geometry of our +chosen system of reference, and at the same time contain within +themselves the absolute geometry of the world---the kind of +space-time which exists in the field of our experiments. + +Having laid down a unit-gauge at every point, we can speak +quite definitely of the change in interval-length of a measuring-rod +moved from point to point, meaning, of course, the change +compared with the unit-gauges. Let us take a rod of interval-length +$l$ at~$P$, and move it successively through the displacements +$dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$; and let the result be to increase its length +in terms of the gauges by the amount~$\lambda l$. The change depends +as much on the difference of the gauges at the two points as +on the behaviour of the rod; but there is no possibility of +separating the two factors. It is clear that $\lambda$ will not depend +on~$l$, because the change of length must be proportional to +the original length---unless indeed our whole idea of measurement +by comparison with a gauge is wrong\footnotemark. + \footnotetext{We refuse to contemplate the idea that when the metre rod changes its + length to two metres, each centimetre of it changes to three centimetres.}% +Further it will +not depend on the direction of the rod either in its initial or +final positions because the interval-length is independent of +direction. (Of course, the space-length would change, but that +is already taken care of by the~$g$'s.) $\lambda$~can thus only depend on +the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$, and we may write it +\[ +\lambda + = \kappa_{1}\, dx_{1} + \kappa_{2}\, dx_{2} + + \kappa_{3}\, dx_{3} + \kappa_{4}\, dx_{4}, +\] +so long as the displacements are small. The coefficients $\kappa_{1}$, $\kappa_{2}$, +$\kappa_{3}$, $\kappa_{4}$ apply to the neighbourhood of~$P$, and will in general be +different in different parts of space. + +This indeed assumes that the result is independent of the +order of the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$---that is to say +that the ambiguity of the comparison by different routes disappears +in the limit when the whole route is sufficiently small. +It is parallel with our previous implicit assumption that although +the length of the track from a point~$P$ to a distant point~$Q$ +%% -----File: 181.png---Folio 171------- +depends on the route, and no definite meaning can be attached +to the interval between them without specifying a route, yet in +the limit there is a definite small interval between $P$ and~$Q$ when +they are sufficiently close together. + +To understand the meaning of these new coefficients $\kappa$ let us +briefly recapitulate what we understand by the~$g$'s. Primarily +they are quantities derived from experimental measurements of +intervals, and describe the geometry of the space and time +partitions which the observer has chosen. As consequential +properties they describe the field of force, gravitational, centrifugal, +etc., with which he perceives himself surrounded. They +relate to the particular mesh-system of the observer; and by +altering his mesh-system, he can alter their values, though not +entirely at will. From their values can be deduced intrinsic +properties of the world---the \textit{kind} of space-time in which the +phenomena occur. Further they satisfy a definite condition---the +law of gravitation---so that not all mathematically possible +space-times and not all arbitrary values of the $g$'s are such as +can occur in nature.% +\index{Real world of physics}% + +All this applies equally to the~$\kappa$'s, if we substitute gauge-system +for mesh-system, and some at present unknown force +for gravitation. They can theoretically be determined by +interval-measurement; but they will be more conspicuously +manifested to the observer through their consequential property +of describing some kind of field of force surrounding him. The +$\kappa$'s refer to the arbitrary gauge-system of the observer; but he +cannot by altering his gauge-system alter their values entirely +at will. Intrinsic properties of the world are contained in their +values, unaffected by any change of gauge-system. Further we +may expect that they will have to satisfy some law corresponding +to the law of gravitation, so that not all arbitrary values of the +$\kappa$'s are such as can occur in nature. + +It is evident that the $\kappa$'s must refer to some type of phenomenon +which has not hitherto appeared in our discussion; and +the obvious suggestion is that they refer to the electromagnetic +field. +\index{Fields of force!electromagnetic}% +This hypothesis is strengthened when we recall that the +electromagnetic field is, in fact, specified at every point by the +values of four quantities, viz.\ the three components of electromagnetic +vector potential, and the scalar potential of electrostatics. +%% -----File: 182.png---Folio 172------- +Surely it is more than a coincidence that the physicist +needs just four more quantities to specify the state of the world +at a point in space, and four more quantities are provided by +removing a rather illogical restriction on our system of geometry +of natural measures. + +[The general reader will perhaps pardon a few words addressed +especially to the mathematical physicist. Taking the ordinary +unaccelerated rectangular coordinates $x$, $y$, $z$, $t$, let us write +$F$, $G$, $H$, $-\Phi$ for $\kappa_{1}$, $\kappa_{2}$, $\kappa_{3}$, $\kappa_{4}$, then +\[ +\frac{dl}{l} = \lambda = F\, dx + G\, dy + H\, dx - \Phi\, dt. +\] +From which, by integration, +\[ +\log l + \text{const.} = \int(F\, dx + G\, dy + H\, dz - \Phi\, dt). +\] + +The length $l$ will be independent of the route taken if +\[ +F\, dx + G\, dy + H\, dz - \Phi\, dt +\] +is a perfect differential. The condition for this is +\index{Electromagnetic potentials and forces}% +\index{Force!electromagnetic}% +\index{Potentials, electromagnetic}% +\begin{align*} +\frac{\partial H}{\partial y} - \frac{\partial G}{\partial z} &= 0, & +\frac{\partial F}{\partial z} - \frac{\partial H}{\partial x} &= 0, & +\frac{\partial G}{\partial x} - \frac{\partial F}{\partial y} &= 0, \\ +% +-\frac{\partial \Phi}{\partial x} - \frac{\partial F}{\partial t} &= 0, & +-\frac{\partial \Phi}{\partial y} - \frac{\partial G}{\partial t} &= 0, & +-\frac{\partial \Phi}{\partial z} - \frac{\partial H}{\partial t} &= 0. +\end{align*} +If $F$, $G$, $H$, $\Phi$ are the potentials of electromagnetic theory, these +are precisely the expressions for the three components of +magnetic force and the three components of electric force, given +in the text-books. Thus the condition that distant intervals can +be compared directly without specifying a particular route of +comparison is that the electric and magnetic forces are zero in +the intervening space and time. + +It may be noted that, even when the coordinate system has +been defined, the electromagnetic potentials are not unique in +value; but arbitrary additions can be made provided these +additions form a perfect differential. It is just this flexibility +which in our geometrical theory appears in the form of the +arbitrary choice of gauge-system. The electromagnetic \textit{forces} +on the other hand are independent of the gauge-system, which +is eliminated by ``curling.''] + +It thus appears that the four new quantities appearing in our +extended geometry may actually be the four potentials of +%% -----File: 183.png---Folio 173------- +electromagnetic theory; and further, when there is no electromagnetic +field our previous geometry is valid. But in the more +general case we have to adopt the more general geometry in +which there appear fourteen coefficients, ten describing the +gravitational and four the electrical conditions of the world. + +We ought now to seek the law of the electromagnetic field +on the same lines as we sought for the law of gravitation, laying +down the condition that it must be independent of mesh-system +and gauge-system since it seeks to limit the possible kinds of +world which can exist in nature. Happily this presents no +difficulty, because the law expressed by Maxwell's equations, +and universally adopted, fulfils the conditions. There is no +need to modify it fundamentally as we modified the law of +gravitation. We do, however, generalise it so that it applies +when a gravitational field is present at the same time---not +merely, as given by Maxwell, for flat space-time. The deflection +of electromagnetic waves (light) by a gravitational field is duly +contained in this generalised law. + +Strictly speaking the laws of gravitation and of the electromagnetic +field are not two laws but one law, as the geometry +of the $g$'s and the $\kappa$'s is one geometry. Although it is often +convenient to separate them, they are really parts of the general +condition limiting the possible kinds of metric that can occur in +empty space. + +It will be remembered that the four-fold arbitrariness of our +mesh-system involved four identities, which were found to +express the conservation of energy and momentum. In the new +geometry there is a fifth arbitrariness, namely that of the selected +gauge-system. This must also give rise to an identity; and it is +found that the new identity expresses the law of conservation of +electric charge.% +\index{Conservation!of electric charge}% + +A grasp of the new geometry may perhaps be assisted by a +further comparison. Suppose an observer has laid down a line +of a certain length and in a certain direction at a point~$P$, and +he wishes to lay down an exactly similar line at a distant point~$Q$. +\index{Integrability of length and direction} % [** PP: Index reads p. 174] +If he is in flat space there will be no difficulty; he will have +to proceed by steps, a kind of triangulation, but the route chosen +is of no importance. We know definitely that there is just one +direction at $Q$ parallel to the original direction at~$P$; and it is +%% -----File: 184.png---Folio 174------- +in ordinary geometry supposed that the length is equally +determinate. But if space is not flat the case is different. +Imagine a two-dimensional observer confined to the curved +surface of the earth trying to perform this task. As he does not +appreciate the third dimension he will not immediately perceive +the impossibility; but he will find that the direction which he has +transferred to $Q$ differs according to the route chosen. Or if he +went round a complete circuit he would find on arriving back +at $P$ that the direction he had so carefully tried to preserve on +the journey did not agree with that originally drawn\footnotemark. + \footnotetext{It might be thought that if the observer preserved mentally the original + direction in three-dimensional space, and obtained the direction at any point + in the two-dimensional space by projecting it, there would be no ambiguity. + But the three-dimensional space in which a curved two-dimensional space is + conceived to exist is quite arbitrary. A two-dimensional observer cannot + ascertain by any observation whether he is on a plane or a cylinder, a sphere + or any other convex surface of the same total curvature.}% +We +describe this by saying that in curved space, direction is not +integrable; and it is this non-integrability of direction which +characterises the gravitational field. In the case considered the +length would be preserved throughout the circuit; but it is +possible to conceive a more general kind of space in which the +length which it was attempted to preserve throughout the +circuit, as well as the direction, disagreed on return to the starting +point with that originally drawn. In that case length is not +integrable; and the non-integrability of length characterises the +electromagnetic field. Length associated with direction is called +a vector; and the combined gravitational and electric field +describe that influence of the world on our measurements by +which a vector carried by physical measurement round a closed +circuit changes insensibly into a different vector. + +The welding together of electricity and gravitation into one +geometry is the work of Prof.\ H.~Weyl, first published in 1918\footnote% +{Appendix, \Noteref{15}.}. +\Pagelabel{note15}% +\index{Vector, non-integrable on Weyl's theory}% +\index{Weyl}% +It appears to the writer to carry conviction, although up to the +present no experimental test has been proposed. It need scarcely +be said that the inconsistency of length for an ordinary circuit +would be extremely minute\footnote{I do not think that any numerical estimate has been made.}, and the ordinary manifestations +of the electromagnetic field are the consequential results of +%% -----File: 185.png---Folio 175------- +changes which would be imperceptible to direct measurement. +It will be remembered that the gravitational field is likewise +perceived by the consequential effects, and not by direct interval-measurement. + +But the theory does appear to require that, for example, the +time of vibration of an atom is not quite independent of its +previous history. It may be assumed that the previous histories +of terrestrial atoms are so much alike that there are no significant +differences in their periods. The possibility that the systematic +difference of history of solar and terrestrial atoms may have an +effect on the expected shift of the spectral lines on the sun has +already been alluded to. It seems doubtful, however, whether +the effect could attain the necessary magnitude. + +It may seem difficult to identify these abstract geometrical +qualities of the world with the physical forces of electricity and +magnetism. How, for instance, can the change in the length of +a rod taken round a circuit in space and time be responsible for +the sensations of an electric shock? The geometrical potentials +($\kappa$) obey the recognised laws of electromagnetic potentials, and +each entity in the physical theory---charge, electric force, +magnetic element, light, etc.---has its exact analogue in the +geometrical theory; but is this formal correspondence a sufficient +ground for identification? The doubt which arises in our minds +is due to a failure to recognise the formalism of all physical +knowledge. +\index{Formalism of knowledge}% +The suggestion ``This is not the thing I am speaking +of, though it behaves exactly like it in all respects'' carries no +physical meaning. Anything which behaves exactly like +electricity must manifest itself to us as electricity. Distinction +of form is the only distinction that physics can recognise; and +distinction of individuality, if it has any meaning at all, has no +bearing on physical manifestations. + +We can only explore the world with apparatus, which is itself +part of the world. Our idealised apparatus is reduced to a few +simple types---a neutral particle, a charged particle, a rigid +scale, etc. The absolute constituents of the world are related in +various ways, which we have studied, to the indications of these +test-bodies. The main features of the absolute world are so +simple that there is a redundancy of apparatus at our disposal; +and probably all that there is to be known could theoretically +%% -----File: 186.png---Folio 176------- +be found out by exploration with an uncharged particle. Actually +we prefer to look at the world as revealed by exploration with +scales and clocks---the former for measuring so-called imaginary +intervals, and the latter for real intervals; this gives us a unified +geometrical conception of the world. +\index{Geometrical conception of the world}% +Presumably, we could obtain +a unified mechanical conception by taking the moving uncharged +particle as standard indicator; or a unified electrical conception +by taking the charged particle. For particular purposes one +test-body is generally better adapted than others. The gravitational +field is more sensitively explored with a moving particle +than a scale. Although the electrical field can theoretically be +explored by the change of length of a scale taken round a circuit, +a far more sensitive way is to use a little bit of the scale---an +electron. And in general for practical efficiency, we do not use +any simple type of apparatus, but a complicated construction +built up with a view to a particular experiment. The reason for +emphasising the theoretical interchangeability of test-bodies is +that it brings out the unity and simplicity of the world; and for +that reason there is an importance in characterising the electromagnetic +condition of the world by reference to the indications +of a scale and clock, however inappropriate they may be as +practical test-bodies. + +Weyl's theory opens up interesting avenues for development. +The details of the further steps involve difficult mathematics; +but a general outline is possible. As on Einstein's more limited +theory there is at any point an important property of the world +called the curvature; but on the new theory it is not an absolute +quantity in the strictest sense of the word. +\index{Curvature!on Weyl's theory}% +It is independent of +the observer's mesh-system, but it depends on his gauge. It is +obvious that the number expressing the radius of curvature of +the world at a point must depend on the unit of length; so we +cannot say that the curvatures at two points are absolutely +equal, because they depend on the gauges assigned at the two +points. Conversely the radius of curvature of the world provides +a natural and absolute gauge at every point; +\index{Natural gauge}% +and it will presumably +introduce the greatest possible symmetry into our laws +if the observer chooses this, or some definite fraction of it, as +his gauge. He, so to speak, forces the world to be spherical by +adopting at every point a unit of length which will make it so. +%% -----File: 187.png---Folio 177------- +Actual rods as they are moved about change their lengths compared +with this absolute unit according to the route taken, and +the differences correspond to the electromagnetic field. Einstein's +curved space appears in a perfectly natural manner in this +theory; no part of space-time is flat, even in the absence of +ordinary matter, for that would mean infinite radius of curvature, +and there would be no natural gauge to determine, for +example, the dimensions of an electron---the electron could not +know how large it ought to be, unless it had something to +measure itself against.% +\index{Electron!dimensions of}% +\index{Gauge!provided by radius of space}% + +The connection between the form of the law of gravitation +and the total amount of matter in the world now appears less +mysterious. The curvature of space indirectly provides the +gauge which we use for measuring the amount of matter in the +world. + +Since the curvature is not independent of the gauge, Weyl +does not identify it with the most fundamental quantity in +nature. There is, however, a slightly more complicated invariant +which is a pure number, and this is taken to be Action\footnotemark. +\footnotetext{Appendix, \Noteref{16}.}% +\Pagelabel{note16}% +\index{Action!on Weyl's theory}% +\index{Atomicity!of Action}% +We +can thus mark out a definite volume of space and time, and +say that the action within it is~$5$, without troubling to define +coordinates or the unit of measure! It might be expected that +the action represented by the number~$1$ would have specially +interesting properties; it might, for instance, be an atom of +action and indivisible. Experiment has isolated what are believed +to be units of action, which at least in many phenomena +behave as indivisible atoms called quanta; +\index{Quanta}% +but the theory, as +at present developed, does not permit us to represent the +quantum of action by the number~$1$. The quantum is a very +minute fraction of the absolute unit. + +When we come across a pure number having some absolute +significance in the world it is natural to speculate on its possible +interpretation. It might represent a number of discrete entities; +but in that case it must necessarily be an integer, and it seems +clear that action can have fractional values. An angle is commonly +represented as a pure number, but it has not really this +character; an angle can only be measured in terms of a unit of +angle, just as a length is measured in terms of a unit of length. +%% -----File: 188.png---Folio 178------- +I can only think of one interpretation of a fractional number +which can have an absolute significance, though doubtless there +are others. The number may represent the \textit{probability} of something, +or some function of a probability. +\index{Probability, a pure number}% +The precise function +is easily found. We combine probabilities by multiplying, but +we combine the actions in two regions by adding; hence the +logarithm of a probability is indicated. Further, since the +logarithm of a probability is necessarily negative, we may +identify action provisionally with minus the logarithm of +the statistical probability of the state of the world which +exists. + +The suggestion is particularly attractive because the Principle +of Least Action now becomes the Principle of Greatest Probability. +\index{Action, Principle of Least}% +\index{Principle of Least Action}% +The law of nature is that the actual state of the world is +that which is statistically most probable. + +Weyl's theory also shows that the mass of a portion of matter +is necessarily positive; on the original theory no adequate reason +is given why negative matter should not exist. It is further +claimed that the theory shows to some extent why the world +is four-dimensional. To the mathematician it seems so easy to +generalise geometry to $n$~dimensions, that we naturally expect +a world of four dimensions to have an analogue in five dimensions. +Apparently this is not the case, and there are some essential +properties, without which it could scarcely be a world, which +exist only for four dimensions. Perhaps this may be compared +with the well-known difficulty of generalising the idea of a knot; +a knot can exist in space of any odd number of dimensions, but +not in space of an even number. + +Finally the theory suggests a mode of attacking the problem +of how the electric charge of an electron is held together; at +least it gives an explanation of why the gravitational force is so +extremely weak compared with the electric force. It will be +remembered that associated with the mass of the sun is a certain +length, called the gravitational mass, which is equal to $1.5$~kilometres. +In the same way the gravitational mass or radius of an +electron is $7·10^{-56}$~cms. +\index{Electron, gravitational mass of}% +Its electrical properties are similarly +associated with a length $2·10^{-13}$~cms., which is called the electrical +radius. The latter is generally supposed to correspond to the +electron's actual dimensions. The theory suggests that the ratio +%% -----File: 189.png---Folio 179------- +of the gravitational to the electrical radius, $3·10^{42}$, ought to be +of the same order as the ratio of the latter to the radius of +curvature of the world. This would require the radius of space +to be of the order $6·10^{29}$~cms., or $2·10^{11}$ parsecs., which though +somewhat larger than the provisional estimates made by de~Sitter, +is within the realm of possibility.% +\index{de Sitter}% + +%% -----File: 190.png---Folio 180------- + + +\Chapter{XII}{On the Nature of Things} + +% [** PP: Explicit formatting containing hard-coded dimensions]] +{\small% +\settowidth{\TmpLen}{\textit{Hippolyta}:\ } +\hspace*{\QIndent}\makebox[\TmpLen][l]{\textit{Hippolyta}.} This is the silliest stuff that ever I heard. \\ +\hspace*{\QIndent}\makebox[\TmpLen][l]{\textit{Theseus}.} +\begin{minipage}[t]{3.75in} +\hangindent 2em The best in this kind are but shadows; and the worst are +no worse, if imagination amend them. +\end{minipage} \\[1ex] +\null\hfill\textit{A Midsummer-Night's Dream.}}% End of \small +\medskip + +\First{The} constructive results of the theory of relativity are based +on two principles which have been enunciated---the restricted +principle of relativity, and the principle of equivalence. These +may be summed up in the statement that uniform motion and +fields of force are purely relative. In their more formal enunciations +they are experimental generalisations, which can be +admitted or denied; if admitted, all the observational results +obtained by us can be deduced mathematically without any +reference to the views of space, time, or force, described in this +book. In many respects this is the most attractive aspect of +Einstein's work; it deduces a great number of remarkable +phenomena solely from two general principles, aided by a +mathematical calculus of great power; and it leaves aside as +irrelevant all questions of mechanism. But this mode of development +of the theory cannot be described in a non-technical book. + +To avoid mathematical analysis we have had to resort to +geometrical illustrations, which run parallel with the mathematical +development and enable its processes to be understood +to some extent. The question arises, are these merely illustrations +of the mathematical argument, or illustrations of the actual +processes of nature. No doubt the safest course is to avoid the +thorny questions raised by the latter suggestion, and to say +that it is quite sufficient that the illustrations should correctly +replace the mathematical argument. But I think that this +would give a misleading view of what the theory of relativity +has accomplished in science. + +The physicist, so long as he thinks as a physicist, has a definite +belief in a real world outside him. For instance, he believes that +atoms and molecules really exist; they are not mere inventions +%% -----File: 191.png---Folio 181------- +that enable him to grasp certain laws of chemical combination. +That suggestion might have sufficed in the early days of the +atomic theory; but now the existence of atoms as entities in the +real world of physics is fully demonstrated. This confident +assertion is not inconsistent with philosophic doubts as to the +meaning of ultimate reality. + +When therefore we are asked whether the four-dimensional +world may not be regarded merely as an illustration of mathematical +processes, we must bear in mind that our questioner has +probably an ulterior motive. +\index{Four-dimensional space-time!reality of}% +He has already a belief in a real +world of three Euclidean dimensions, and he hopes to be allowed +to continue in this belief undisturbed. In that case our answer +must be definite; the real three-dimensional world is obsolete, +and must be replaced by the four-dimensional space-time with +non-Euclidean properties. In this book we have sometimes +employed illustrations which certainly do not correspond to any +physical reality---imaginary time, and an unperceived fifth +dimension. +\index{Imaginary time}% +But the four-dimensional world is no mere illustration; +it is the real world of physics, arrived at in the recognised +way by which physics has always (rightly or wrongly) sought for +reality. + +I hold a certain object before me, and see an outline of the +figure of Britannia; another observer on the other side sees a +picture of a monarch; a third observer sees only a thin rectangle. +Am I to say that the figure of Britannia is the real object; and +that the crude impressions of the other observers must be +corrected to make allowance for their positions? All the appearances +can be accounted for if we are all looking at a three-dimensional +object---a penny---and no reasonable person can +doubt that the penny is the corresponding physical reality. +Similarly, an observer on the earth sees and measures an oblong +block; an observer on another star contemplating the same +object finds it to be a cube. Shall we say that the oblong block +is the real thing, and that the other observer must correct his +measures to make allowance for his motion? All the appearances +are accounted for if the real object is four-dimensional, and the +observers are merely measuring different three-dimensional +appearances or sections; and it seems impossible to doubt that +this is the true explanation. He who doubts the reality of the +%% -----File: 192.png---Folio 182------- +four-dimensional world (for logical, as distinct from experimental, +reasons) can only be compared to a man who doubts the +reality of the penny, and prefers to regard one of its innumerable +appearances as the real object. + +Physical reality is the synthesis of all possible physical aspects +of nature. +\index{Synthesis of appearances}% +An illustration may be taken from the phenomena of +radiant-energy, or light. In a very large number of phenomena +the light coming from an atom appears to be a series of spreading +waves, extending so as to be capable of filling the largest +telescope yet made. In many other phenomena the light coming +from an atom appears to remain a minute bundle of energy, all +of which can enter and blow up a single atom. +\index{Quanta}% +There may be +some illusion in these experimental deductions; but if not, it +must be admitted that the physical reality corresponding to +light must be some synthesis comprehending both these appearances. +How to make this synthesis has hitherto baffled conception. +But the lesson is that a vast number of appearances +may be combined into one consistent whole---perhaps all +appearances that are directly perceived by terrestrial observers---and +yet the result may still be only an appearance. Reality +is only obtained when all conceivable points of view have been +combined. + +That is why it has been necessary to give up the reality of +the everyday world of three dimensions. Until recently it comprised +all the possible appearances that had been considered. +But now it has been discovered that there are new points of +view with new appearances; and the reality must contain them +all. It is by bringing in all these new points of view that we +have been able to learn the nature of the real world of +physics. + +Let us briefly recapitulate the steps of our synthesis. We +found one step already accomplished. The immediate perception +of the world with one eye is a two-dimensional appearance. But +we have two eyes, and these combine the appearances of the +world as seen from two positions; in some mysterious way the +brain makes the synthesis by suggesting solid relief, and we +obtain the familiar appearance of a three-dimensional world. +This suffices for all possible positions of the observer within the +parts of space hitherto explored. The next step was to combine +%% -----File: 193.png---Folio 183------- +the appearances for all possible states of uniform motion of the +observer. The result was to add another dimension to the world, +making it four-dimensional. Next the synthesis was extended +to include all possible variable motions of the observer. The +process of adding dimensions stopped, but the world became +non-Euclidean; a new geometry called Riemannian geometry +was adopted. Finally the points of view of observers varying +in size in any way were added; and the result was to replace +the Riemannian geometry by a still more general geometry +described in the last chapter. + +The search for physical reality is not necessarily utilitarian, +but it has been by no means profitless. As the geometry became +more complex, the physics became simpler; until finally it +almost appears that the physics has been absorbed into the +geometry. We did not consciously set out to construct a +geometrical theory of the world; we were seeking physical +reality by approved methods, and this is what has happened.% +\index{Geometrical conception of the world}% + +Is the point now reached the ultimate goal? Have the points +of view of all conceivable observers now been absorbed? We do +not assert that they have. But it seems as though a definite +task has been rounded off, and a natural halting-place reached. +So far as we know, the different possible impersonal points of +view have been exhausted---those for which the observer can be +regarded as a mechanical automaton, and can be replaced by +scientific measuring-appliances. A variety of more personal +points of view may indeed be needed for an ultimate reality; +but they can scarcely be incorporated in a real world of physics. +There is thus justification for stopping at this point but not for +stopping earlier. + +It may be asked whether it is necessary to take into account +all conceivable observers, many of whom, we suspect, have no +existence. Is not the \textit{real} world that which comprehends the +appearances to all \textit{real} observers? Whether or not it is a tenable +hypothesis that that which no one observes does not exist, +science uncompromisingly rejects it. If we deny the rights of +extra-terrestrial observers, we must take the side of the Inquisition +against Galileo. And if extra-terrestrial observers are +admitted, the other observers, whose results are here combined, +cannot be excluded. + +%% -----File: 194.png---Folio 184------- + +Our inquiry into the nature of things is subject to certain +limitations which it is important to realise. The best comparison +I can offer is with a future antiquarian investigation, which may +be dated about the year 5000~\textsc{a.d.} An interesting find has been +made relating to a vanished civilisation which flourished about +the twentieth century, namely a volume containing a large +number of games of chess, written out in the obscure symbolism +usually adopted for that purpose. +\index{Chess, analogy of}% +The antiquarians, to whom +the game was hitherto unknown, manage to discover certain +uniformities; and by long research they at last succeed in +establishing beyond doubt the nature of the moves and rules of +the game. But it is obvious that no amount of study of the +volume will reveal the true nature either of the participants in +the game---the chessmen---or the field of the game---the chess-board. +With regard to the former, all that is possible is to give +arbitrary names distinguishing the chessmen according to their +properties; but with regard to the chess-board something more +can be stated. The material of the board is unknown, so too +are the shapes of the meshes---whether squares or diamonds; +but it is ascertainable that the different points of the field are +connected with one another by relations of two-dimensional +order, and a large number of hypothetical types of chess-board +satisfying these relations of order can be constructed. In +spite of these gaps in their knowledge, our antiquarians may +fairly claim that they thoroughly understand the game of +chess.% +\index{Ordering of events in external world}% + +The application of this analogy is as follows. The recorded +games are our physical experiments. The rules of the game, +ascertained by study of them, are the laws of physics. The +hypothetical chess-board of 64~squares is the space and time of +some particular observer or player; whilst the more general +relations of two-fold order, are the absolute relations of order +in space-time which we have been studying. The chessmen are +the entities of physics---electrons, particles, or point-events; and +the range of movement may perhaps be compared to the fields +of relation radiating from them---electric and gravitational +fields, or intervals. By no amount of study of the experiments +can the absolute nature or appearance of these participants be +deduced; nor is this knowledge relevant, for without it we may +%% -----File: 195.png---Folio 185------- +yet learn ``the game'' in all its intricacy. Our knowledge of the +nature of things must be like the antiquarians' knowledge of +the nature of chessmen, viz.\ their nature as pawns and pieces +in the game, not as carved shapes of wood. In the latter aspect +they may have relations and significance transcending anything +dreamt of in physics. + +It is believed that the familiar things of experience are very +complex; and the scientific method is to analyse them into +simpler elements. Theories and laws of behaviour of these +simpler constituents are studied; and from these it becomes +possible to predict and explain phenomena. It seems a natural +procedure to explain the complex in terms of the simple, but +it carries with it the necessity of explaining the familiar in terms +of the unfamiliar. + +There are thus two reasons why the ultimate constituents of +the real world must be of an unfamiliar nature. Firstly, all +familiar objects are of a much too complex character. Secondly, +familiar objects belong not to the real world of physics, but to +a much earlier stage in the synthesis of appearances. The +ultimate elements in a theory of the world must be of a nature +impossible to define in terms recognisable to the mind. + +The fact that he has to deal with entities of unknown nature +presents no difficulty to the mathematician. As the mathematician +in the Prologue explained, he is never so happy as +when he does not know what he is talking about. But we ourselves +cannot take any interest in the chain of reasoning he is +producing, unless we can give it some meaning---a meaning, +which we find by experiment, it will bear. We have to be in +a position to make a sort of running comment on his work. +At first his symbols bring no picture of anything before our +eyes, and we watch in silence. Presently we can say ``Now he +is talking about a particle of matter''\ldots ``Now he is talking +about another particle''\ldots ``Now he is saying where they will +be at a certain time of day''\ldots ``Now he says that they will be +in the same spot at a certain time.'' We watch to see.---``Yes. +The two particles have collided. For once he is speaking about +something familiar, and speaking the truth, although, of course, +he does not know it.'' Evidently his chain of symbols can be +interpreted as describing what occurs in the world; we need not, +%% -----File: 196.png---Folio 186------- +and do not, form any idea of the meaning of each individual +symbol; it is only certain elaborate combinations of them that +we recognise. + +Thus, although the elementary concepts of the theory are of +undefined nature, at some later stage we must link the derivative +concepts to the familiar objects of experience. + +We shall now collect the results arrived at in the previous +chapters by successive steps, and set the theory out in more +logical order. The extension in \Chapref{XI} will not be considered +here, partly because it would increase the difficulty of grasping +the main ideas, partly because it is less certainly established. + +In the relativity theory of nature the most elementary concept +is the \textit{point-event}. +\index{Event, definition of}% +\index{Point-event}% +In ordinary language a point-event is an +instant of time at a point of space; but this is only one aspect +of the point-event, and it must not be taken as a definition. +Time and space---the familiar terms---are derived concepts to +be introduced much later in our theory. The first simple concepts +are necessarily undefinable, and their nature is beyond +human understanding. The aggregate of all the point-events is +called the \textit{world}. +\index{World}% +It is postulated that the world is four-dimensional, +which means that a particular point-event has to be +specified by the values of four variables or coordinates, though +there is entire freedom as to the way in which these four identifying +numbers are to be assigned. + +The meaning of the statement that the world is four-dimensional +is not so clear as it appears at first. An aggregate of a +large number of things has in itself no particular number of +dimensions. Consider, for example, the words on this page. To +a casual glance they form a two-dimensional distribution; but +they were written in the hope that the reader would regard +them as a one-dimensional distribution. In order to define the +number of dimensions we have to postulate some ordering +relation; and the result depends entirely on what this ordering +relation is---whether the words are ordered according to sense +or to position on the page. Thus the statement that the world +is four-dimensional contains an implicit reference to some ordering +relation. +\index{Four-dimensional order}% +\index{Order and dimensions}% +This relation appears to be the \textit{interval}, though I am +not sure whether that alone suffices without some relation +corresponding to \textit{proximity}. It must be remembered that if the +%% -----File: 197.png---Folio 187------- +interval $s$ between two events is small, the events are not +necessarily near together in the ordinary sense.% +\index{Time!``standing still''}% + +Between any two neighbouring point-events there exists a +certain relation known as the \textit{interval} between them. +\index{Interval}% +The +relation is a quantitative one which can be measured on a +definite scale of numerical values\footnotemark. +\index{Space-like intervals}% +\index{Time-like intervals}% + \footnotetext{There is also a qualitative distinction into two kinds, ultimately identified + as time-like and space-like, which for mathematical treatment are distinguished + by real and imaginary numbers.}% +But the term ``interval'' +is not to be taken as a guide to the real nature of the relation, +which is altogether beyond our conception. +\index{Imaginary intervals}% +Its geometrical +properties, which we have dwelt on so often in the previous +chapters, can only represent one aspect of the relation. It may +have other aspects associated with features of the world outside +the scope of physics. But in physics we are concerned not with +the nature of the relation but with the number assigned to +express its intensity; and this suggests a graphical representation, +leading to a geometrical theory of the world of physics.% +\index{World}% + +What we have here called the \textit{world} might perhaps have been +legitimately called the \textit{aether}; at least it is the universal substratum +of things which the relativity theory gives us in place +of the aether.% +\index{Aether!identified with the ``world''}% + +We have seen that the number expressing the intensity of +the interval-relation can be measured practically with scales and +clocks. Now, I think it is improbable that our coarse measures +can really get hold of the individual intervals of point-events; +our measures are not sufficiently microscopic for that. The +interval which has appeared in our analysis must be a \textit{macroscopic} +value; and the potentials and kinds of space deduced from +it are averaged properties of regions, perhaps small in comparison +even with the electron, but containing vast numbers of the +primitive intervals. We shall therefore pass at once to the +consideration of the macroscopic interval; but we shall not +forestall later results by assuming that it is measurable with +a scale and clock. That property must be introduced in its +logical order.% +\index{Macroscopic!interval}% + +Consider a small portion of the world. It consists of a large +(possibly infinite) number of point-events between every two of +which an interval exists. If we are given the intervals between +%% -----File: 198.png---Folio 188------- +a point $A$ and a sufficient number of other points, and also +between $B$ and the same points, can we calculate what will be +the interval between $A$ and~$B$? In ordinary geometry this +would be possible; but, since in the present case we know nothing +of the relation signified by the word interval, it is impossible to +predict any law \textit{a~priori}. But we have found in our previous +work that there is such a rule, expressed by the formula +\[ +ds^{2} + = g_{11}\, dx_{1}^{2} + g_{22}\, dx_{2}^{2} + \dotsb + + 2g_{12}\, dx_{1} dx_{2} + \dotsb. +\] +This means that, having assigned our identification numbers +% [** PP: overfull at 5in; set as individual entries] +$(x_{1}$, $x_{2}$, $x_{3}$, $x_{4})$ to the point-events, we have only to measure +ten different intervals to enable us to determine the ten coefficients, +$g_{11}$, etc., which in a small region may be considered to be +constants; then all other intervals in this region can be predicted +from the formula. For any other region we must make fresh +measures, and determine the coefficients for a new formula. + +I think it is unlikely that the \textit{individual} interval-relations of +point-events follow any such definite rule. A microscopic +examination would probably show them as quite arbitrary, the +relations of so-called intermediate points being not necessarily +intermediate. Perhaps even the primitive interval is not +quantitative, but simply $1$ for certain pairs of point-events and +$0$ for others. The formula given is just an average summary +which suffices for our coarse methods of investigation, and holds +true only statistically. Just as statistical averages of one community +may differ from those of another, so may this statistical +formula for one region of the world differ from that of another. +This is the starting point of the infinite variety of nature. + +Perhaps an example may make this clearer. Compare the +point-events to persons, and the intervals to the degree of +acquaintance between them. There is no means of forecasting +the degree of acquaintance between $A$ and $B$ from a knowledge +of the familiarity of both with $C$, $D$, $E$,~etc. But a statistician +may compute in any community a kind of average rule. In +most cases if $A$ and $B$ both know~$C$, it slightly increases the +probability of their knowing one another. A community in +which this correlation was very high would be described as +\textit{cliquish}. +\index{Cliquishness}% +There may be differences among communities in this +respect, corresponding to their degree of cliquishness; and so +%% -----File: 199.png---Folio 189------- +the statistical laws may be the means of expressing intrinsic +differences in communities. + +Now comes the difficulty which is by this time familiar to us. +The ten $g$'s are concerned, not only with intrinsic properties of +the world, but with our arbitrary system of identification-numbers +for the point-events; or, as we have previously expressed +it, they describe not only the kind of space-time, but +the nature of the arbitrary mesh-system that is used. Mathematics +shows the way of steering through this difficulty by fixing +attention on expressions called tensors, of which $B^{\rho}_{\mu\nu\sigma}$ and $G_{\mu\nu}$ +are examples.% +\index{Tensors}% + +A tensor does not express explicitly the measure of an intrinsic +quality of the world, for some kind of mesh-system is essential +to the idea of measurement of a property, except in certain very +special cases where the property is expressed by a single number +termed an invariant, e.g.\ the interval, or the total curvature. +But to state that a tensor vanishes, or that it is equal to another +tensor in the same region, is a statement of intrinsic property, +quite independent of the mesh-system chosen. Thus by keeping +entirely to tensors, we contrive that there shall be behind our +formulae an undercurrent of information having reference to the +intrinsic state of the world. + +In this way we have found two absolute formulae, which +appear to be fully confirmed by observation, namely +\begin{DPalign*} +\lintertext{in empty space,} +G_{\mu\nu} &= 0, \\ +\lintertext{in space containing matter,} +G_{\mu\nu} &= K_{\mu\nu}, +\end{DPalign*} +where $K_{\mu\nu}$ contains only physical quantities which are perfectly +familiar to us, viz.\ the density and state of motion of the matter +in the region. + +I think the usual view of these equations would be that the +first expresses some law existing in the world, so that the point-events +by natural necessity tend to arrange their relations in +conformity with this equation. But when matter intrudes it +causes a disturbance or strain of the natural linkages; and a +rearrangement takes place to the extent indicated by the second +equation. + +But let us examine more closely what the equation $G_{\mu\nu} = 0$ +tells us. We have been giving the mathematician a free hand +%% -----File: 200.png---Folio 190------- +with his indefinable intervals and point-events. He has arrived +at the quantity~$G_{\mu\nu}$; but as yet this means to us---absolutely +nothing. The pure mathematician left to himself never ``deviates +into sense.'' His work can never relate to the familiar things +around us, unless we boldly lay hold of some of his symbols and +\textit{give} them an intelligible meaning---tentatively at first, and then +definitely as we find that they satisfy all experimental knowledge. +We have decided that in empty space $G_{\mu\nu}$~vanishes. Here +is our opportunity. In default of any other suggestion as to +what the vanishing of $G_{\mu\nu}$ might mean, let us say that the +vanishing of $G_{\mu\nu}$ \textit{means} emptiness; so that $G_{\mu\nu}$, if it does not +vanish, is a condition of the world which distinguishes space +said to be occupied from space said to be empty. Hitherto $G_{\mu\nu}$ +was merely a formal outline to be filled with some undefined +contents; we are as far as ever from being able to explain what +those contents are; but we have now given a recognisable +meaning to the completed picture, so that we shall know it +when we come across it in the familiar world of experience. + +The two equations are accordingly merely definitions---definitions +of the way in which certain states of the world +(described in terms of the indefinables) impress themselves on +our perceptions. When we perceive that a certain region of the +world is empty, that is merely the mode in which our senses +recognise that it is curved no higher than the first degree. +\index{Curvature!perception of}% +When we perceive that a region contains matter we are recognising +the intrinsic curvature of the world; and when we believe we +are measuring the mass and momentum of the matter (relative +to some axes of reference) we are measuring certain components +of world-curvature (referred to those axes). The statistical +averages of something unknown, which have been used to +describe the state of the world, vary from point to point; and +it is out of these that the mind has constructed the familiar +notions of matter and emptiness.% +\index{Emptiness, perception of}% +\index{Matter!perception of}% + +The law of gravitation is not a law in the sense that it restricts +the possible behaviour of the substratum of the world; it is +merely the definition of a vacuum. +\index{Vacuum, defined by law of gravitation}% +We need not regard matter +as a foreign entity causing a disturbance in the gravitational +field; the disturbance is matter. In the same way we do not +regard light as an intruder in the electromagnetic field, causing +%% -----File: 201.png---Folio 191------- +the electromagnetic force to oscillate along its path; the oscillations +constitute the light. Nor is heat a fluid causing agitation +of the molecules of a body; the agitation is heat. + +This view, that matter is a symptom and not a cause, seems +so natural that it is surprising that it should be obscured in the +usual presentation of the theory. The reason is that the connection +of mathematical analysis with the things of experience +is usually made, not by determining what matter is, but by +what certain combinations of matter do. Hence the interval is +at once identified with something familiar to experience, namely +the thing that a scale and a clock measure. However advantageous +that may be for the sake of bringing the theory into +touch with experiment at the outset, we can scarcely hope to +build up a theory of the nature of things if we take a scale and +clock as the simplest unanalysable concepts. The result of this +logical inversion is that by the time the equation $G_{\mu\nu} = K_{\mu\nu}$ is +encountered, both sides of the equation are well-defined +quantities. Their \textit{necessary} identity is overlooked, and the +equation is regarded as a new law of nature. This is the fault +of introducing the scale and clock prematurely. For our part +we prefer first to define what matter is in terms of the elementary +concepts of the theory; then we can introduce any kind of +scientific apparatus; and finally determine what property of the +world that apparatus will measure.% +\index{Clock-scale geometry, not fundamental}% + +Matter defined in this way obeys all the laws of mechanics, +including conservation of energy and momentum. Proceeding +with a similar development of Weyl's more general theory of +the combined gravitational and electrical fields, we should find +that it has the familiar electrical and optical properties. It is +purely gratuitous to suppose that there is anything else present, +controlling but not to be identified with the relations of the +fourteen potentials ($g$'s and $\kappa$'s). % [** PP: Changed k to \kappa] + +There is only one further requirement that can be demanded +from matter. Our brains are constituted of matter, and they +feel and think---or at least feeling and thinking are closely +associated with motions or changes of the matter of the brain. +\index{Brain, constitution of}% +It would be difficult to say that any hypothesis as to the nature +of matter makes this process less or more easily understood; +and a brain constituted out of differential coefficients of $g$'s can +%% -----File: 202.png---Folio 192------- +scarcely be said to be less adapted to the purposes of thought +than one made, say, out of tiny billiard balls! But I think we +may even go a little beyond this negative justification. The +primary interval relation is of an undefined nature, and the +$g$'s contain this undefinable element. The expression $G_{\mu\nu}$ is +therefore of defined \textit{form}, but of undefined \textit{content}. +\index{Content contrasted with structural form}% +\index{Form contrasted with content}% +By its form +alone it is fitted to account for all the physical properties of +matter; and physical investigation can never penetrate beneath +the form. The matter of the brain in its physical aspects is +merely the form; but the reality of the brain includes the +content. We cannot expect the form to explain the activities of +the content, any more than we can expect the number~4 to +explain the activities of the Big Four at Versailles. + +Some of these views of matter were anticipated with marvellous +foresight by W.~K. Clifford forty years ago. +\index{Clifford}% +Whilst other English +physicists were distracted by vortex-atoms and other will-o'-the-wisps, +Clifford was convinced that matter and the motion +of matter were aspects of space-curvature \textit{and nothing more}. +And he was no less convinced that these geometrical notions +were only partial aspects of the relations of what he calls +``elements of feeling.''---% +\index{Feeling, elements of}% +\index{Matter!physical and psychological aspects}% +``The reality corresponding to our perception +of the motion of matter is an element of the complex +thing we call feeling. What we might perceive as a plexus of +nerve-disturbances is really in itself a feeling; and the succession +of feelings which constitutes a man's consciousness is the reality +which produces in our minds the perception of the motions of +his brain. These elements of feeling have relations of \textit{nextness} +or contiguity in space, which are exemplified by the sight-perceptions +of contiguous points; and relations of succession in +time which are exemplified by all perceptions. Out of these two +relations the future theorist has to build up the world as best +he may. Two things may perhaps help him. There are many +lines of mathematical thought which indicate that distance or +quantity may come to be expressed in terms of \textit{position} in the +wide sense of the \textit{analysis situs}. And the theory of space-curvature +hints at a possibility of describing matter and motion +in terms of extension only.'' (\textit{Fortnightly Review}, 1875.) + +The equation $G_{\mu\nu} = K_{\mu\nu}$ is a kind of dictionary explaining +what the different components of world-curvature mean in +%% -----File: 203.png---Folio 193------- +terms ordinarily used in mechanics. If we write it in the slightly +modified, but equivalent, form +\[ +G_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} G = - 8\pi T_{\mu\nu}, +\] +we have the following scheme of interpretation +\index{Gravitation, Einstein's law of!macroscopic equations}% +\[ +\begin{matrix} +T_{11}, & T_{12}, & T_{13}, & T_{14} \\ + & T_{22}, & T_{23}, & T_{24} \\ + & & T_{33}, & T_{34} \\ + & & & T_{44} +\end{matrix} += +\begin{matrix} +p_{11} + \rho u^{2}, & p_{12} + \rho uv, & p_{13} + \rho uw, & - \rho u, \\ + & p_{22} + \rho v^2, & p_{23} + \rho vw, & - \rho v, \\ + & & p_{33} + \rho w^2, & - \rho w, \\ + & & & \rho. +\end{matrix} +\] +Here we are using the partitions of space and time adopted in +ordinary mechanics; $\rho$ is the density of the matter, $u$, $v$, $w$ its +component velocities, and $p_{11}$, $p_{12}$, $\dotsc p_{33}$, the components of +the internal stresses which are believed to be analysable into +molecular movements.% +\index{Stresses in continuous matter}% + +Now the question arises, is it legitimate to make identifications +on such a wholesale scale? Having identified $T_{44}$ as density, +can we go on to identify another quantity $T_{34}$ as density +multiplied by velocity? It is as though we identified one ``thing'' +as \textit{air}, and a quite different ``thing'' as \textit{wind}. Yes, it is legitimate, +because we have not hitherto explained what is to be the +counterpart of velocity in our scheme of the world; and this is +the way we choose to introduce it. All identifications are at +this stage provisional, being subject to subsequent test by +observation. + +A definition of the velocity of matter in some such terms as +``\textit{wind} divided by \textit{air},'' does not correspond to the way in +which motion primarily manifests itself in our experience. +\index{Velocity, definition of}% +Motion is generally recognised by the disappearance of a particle +at one point of space and the appearance of an apparently +identical particle at a neighbouring point. This manifestation +of motion can be deduced mathematically from the identifying +definition here adopted. Remembering that in physical theory +it is necessary to proceed from the simple to the complex, +which is often opposed to the instinctive desire to proceed from +the familiar to the unfamiliar, this inversion of the order in +which the manifestations of motion appear need occasion no +surprise. Permanent identity of particles of matter (without +which the ordinary notion of velocity fails) is a very familiar +idea, but it appears to be a very complex feature of the world.% +\index{Identity, permanent} % [** PP: Added comma] +\index{Permanent identity}% + +%% -----File: 204.png---Folio 194------- + +A simple instance may be given where the familiar kinematical +conception of motion is insufficient. +\index{Motion!insufficiency of kinematical conception}% +Suppose a perfectly homogeneous +continuous ring is rotating like a wheel, what meaning +can we attach to its motion? +\index{Rotation of a continuous ring}% +The kinematical conception of +motion implies change---disappearance at one point and reappearance +at another point---but no change is detectable. The +state at any one moment is the same as at a previous moment, +and the matter occupying one position now is indistinguishable +from the matter in the same position a moment ago. At the +most it can only differ in a mysterious non-physical quality---that +of identity; but if, as most physicists are willing to believe, +matter is some state in the aether, what can we mean by saying +that two states are exactly alike, but are not identical? Is the +hotness of the room equal to, but not identical with, its hotness +yesterday? Considered kinematically, the rotation of the ring +appears to have no meaning; yet the revolving ring differs +mechanically from a stationary ring. For example, it has +gyrostatic properties. The fact that in nature a ring has atomic +and not continuous structure is scarcely relevant. A conception +of motion which affords a distinction between a rotating and +non-rotating continuous ring must be possible; otherwise this +would amount to an \textit{a~priori} proof that matter is atomic. +According to the conception now proposed, velocity of matter +is as much a static quality as density. +\index{Velocity!static character}% +Generally velocity is +accompanied by changes in the physical state of the world, +which afford the usual means of recognising its existence; but +the foregoing illustration shows that these symptoms do not +always occur. + +This definition of velocity enables us to understand why +velocity except in reference to matter is meaningless, whereas +acceleration and rotation have a meaning. +\index{Absolute acceleration}% +\index{Absolute rotation}% +\index{Rotation, absolute}% +The philosophical +argument, that velocity through space is meaningless, ceases to +apply as soon as we admit any kind of structure or aether in +empty regions; consequently the problem is by no means so +simple as is often supposed. But our definition of velocity is +dynamical, not kinematical. Velocity is the ratio of certain +components of~$T_{\mu\nu}$, and only exists when $T_{44}$ is not zero. Thus +matter (or electromagnetic energy) is the only thing that can +have a velocity relative to the frame of reference. The velocity +%% -----File: 205.png---Folio 195------- +of the world-structure or aether, where the $T_{\mu\nu}$ vanish, is always +of the indeterminate form $0÷0$. On the other hand acceleration +and rotation are defined by means of the $g_{\mu\nu}$ and exist wherever +these exist\footnotemark; + \footnotetext{Even in Newtonian mechanics we speak of the ``field of acceleration,'' and + think of it as existing even when there is no test body to display the acceleration. + In the present theory this field of acceleration is described by the $g_{\mu\nu}$. + There is no such thing as a ``field of velocity'' in empty space; + \index{Field of velocity}% + but there is in + a material ocean.}% +so that the acceleration and rotation of the world-structure +or aether relative to the frame of reference are determinate. +Notice that acceleration is not defined as change of +velocity; it is an independent entity, much simpler and more +universal than velocity. It is from a comparison of these two +entities that we ultimately obtain the definition of time. + +This finally resolves the difficulty encountered in \Chapref{X}---the +apparent difference in the Principle of Relativity as +applied to uniform and non-uniform motion. Fundamentally +velocity and acceleration are both static qualities of a region +of the world (referred to some mesh-system). Acceleration is a +comparatively simple quality present wherever there is geodesic +structure, that is to say everywhere. +\index{Acceleration!a simpler quality than velocity}% +\index{Geodesic structure!acceleration of}% +\index{Structure, geodesic!acceleration of}% +Velocity is a highly complex +quality existing only where the structure is itself more +than ordinarily complicated, viz.\ in matter. Both these qualities +commonly give physical manifestations, to which the terms +acceleration and velocity are more particularly applied; but it +is by examining their more fundamental meaning that we can +understand the universality of the one and the localisation of +the other. + +It has been shown that there are four identical relations +between the ten qualities of a piece of matter here identified, +which depend solely on the way the $G_{\mu\nu}$ were by definition +constructed out of simpler elements. These four relations state +that, \textit{provided the mesh-system is drawn in one of a certain number +of ways}, mass (or energy) and momentum will be conserved. +The conservation of mass is of great importance; +\index{Mass!conservation of}% +\IndexExtra{Conservation of mass} % +matter will +be permanent, and for every particle disappearing at any point +a corresponding mass will appear at a neighbouring point; the +change consists in the displacement of matter, not its creation +or destruction. This gives matter the right to be regarded, not +as a mere assemblage of symbols, but as the substance of a +%% -----File: 206.png---Folio 196------- +permanent world. But the permanent world so found demands +the partitioning of space-time in one of a certain number of +ways, viz.\ those discussed in \Chapref{III}\footnotemark; + \footnotetext{When the kind of space-time is such that a strict partition of this kind is + impossible, strict conservation does not exist; but we retain the principle as + formally satisfied by attributing energy and momentum to the gravitational + field.}% +from these a particular +space and time are selected, because the observer wishes to +consider himself, or some arbitrary body, at rest. This gives +the space and time used for ordinary descriptions of experience. +In this way we are able to introduce perceptual space and time +into the four-dimensional world, as derived concepts depending +on our desire that the new-found matter should be permanent. + +I think it is now possible to discern something of the reason +why the world must of necessity be as we have described it. +When the eye surveys the tossing waters of the ocean, the +eddying particles of water leave little impression; it is the waves +that strike the attention, because they have a certain degree +of permanence. The motion particularly noticed is the motion of +the wave-form, which is not a motion of the water at all. So +the mind surveying the world of point-events looks for the +permanent things. The simpler relations, the intervals and +potentials, are transient, and are not the stuff out of which +mind can build a habitation for itself. But the thing that has +been identified with matter is permanent, and because of its +permanence it must be for mind the substance of the world. +Practically no other choice was possible. + +It must be recognised that the conservation of mass is not +exactly equivalent to the permanence of matter. +\index{Conservation!of mass}% +\index{Permanence of matter}% +If a loaf of +bread suddenly transforms into a cabbage, our surprise is not +diminished by the fact that there may have been no change of +weight. It is not very easy to define this extra element of +permanence required, because we accept as quite natural +apparently similar transformations---an egg into an omelette, +or radium into lead. But at least it seems clear that some degree +of permanence of one quality, mass, would be the primary +property looked for in matter, and this gives sufficient reason +for the particular choice. + +We see now that the choice of a permanent substance for the +%% -----File: 207.png---Folio 197------- +world of perception necessarily carries with it the law of gravitation, +all the laws of mechanics, and the introduction of the +ordinary space and time of experience. Our whole theory has +really been a discussion of the most general way in which +permanent substance can be built up out of relations; and it is +the mind which, by insisting on regarding only the things that +are permanent, has actually imposed these laws on an indifferent +world. Nature has had very little to do with the matter; she +had to provide a basis---point-events; but practically anything +would do for that purpose if the relations were of a reasonable +degree of complexity. The relativity theory of physics reduces +everything to relations; that is to say, it is structure, not +material, which counts. The structure cannot be built up without +material; but the nature of the material is of no importance. +We may quote a passage from Bertrand Russell's \textit{Introduction +to Mathematical Philosophy}.% +\index{Mathematics, Russell's description of}% +\index{Russell}% +\index{Structure opposed to content}% + +``There has been a great deal of speculation in traditional +philosophy which might have been avoided if the importance +of structure, and the difficulty of getting behind it, had been +realised. For example it is often said that space and time are +subjective, but they have objective counterparts; or that +phenomena are subjective, but are caused by things in themselves, +which must have differences \textit{inter~se} corresponding with +the differences in the phenomena to which they give rise. Where +such hypotheses are made, it is generally supposed that we can +know very little about the objective counterparts. In actual +fact, however, if the hypotheses as stated were correct, the +objective counterparts would form a world having the same +structure as the phenomenal world\ldots. In short, every proposition +having a communicable significance must be true of both worlds +or of neither: the only difference must lie in just that essence +of individuality which always eludes words and baffles description, +but which for that very reason is irrelevant to science.'' + +This is how our theory now stands.---We have a world of +point-events with their primary interval-relations. Out of these +an unlimited number of more complicated relations and qualities +can be built up mathematically, describing various features of +the state of the world. These exist in nature in the same sense +as an unlimited number of walks exist on an open moor. But +%% -----File: 208.png---Folio 198------- +the existence is, as it were, latent unless someone gives a significance +to the walk by following it; and in the same way the +existence of any one of these qualities of the world only acquires +significance above its fellows, if a mind singles it out for +recognition. Mind filters out matter from the meaningless +jumble of qualities, as the prism filters out the colours of the +rainbow from the chaotic pulsations of white light. Mind exalts +the permanent and ignores the transitory; and it appears from +the mathematical study of relations that the only way in which +mind can achieve her object is by picking out one particular +quality as the permanent substance of the perceptual world, +partitioning a perceptual time and space for it to be permanent +in, and, as a necessary consequence of this Hobson's choice, the +laws of gravitation and mechanics and geometry have to be +obeyed. +\index{Permanent perceptual world}% +Is it too much to say that mind's search for permanence +has created the world of physics? So that the world we +perceive around us could scarcely have been other than it is\footnotemark? + \footnotetext{This summary is intended to indicate the direction in which the views + suggested by the relativity theory appear to me to be tending, rather than to + be a precise statement of what has been established. I am aware that there + are at present many gaps in the argument. Indeed the whole of this part of + the discussion should be regarded as suggestive rather than dogmatic.} + +The last sentence possibly goes too far, but it illustrates the +direction in which these views are tending. With Weyl's more +general theory of interval-relations, the laws of electrodynamics +appear in like manner to depend merely on the identification +of another permanent thing---electric charge. In this case the +identification is due, not to the rudimentary instinct of the +savage or the animal, but the more developed reasoning-power +of the scientist. But the conclusion is that the whole of those +laws of nature which have been woven into a unified scheme---mechanics, +gravitation, electrodynamics and optics---have their +origin, not in any special mechanism of nature, but in the +workings of the mind. + +``Give me matter and motion,'' said Descartes, ``and I will +construct the universe.'' The mind reverses this. ``Give me a +world---a world in which there are relations---and I will construct +matter and motion.'' + +Are there then no genuine laws in the external world? Laws +inherent in the substratum of events, which break through into +%% -----File: 209.png---Folio 199------- +the phenomena otherwise regulated by the despotism of the +mind? We cannot foretell what the final answer will be; but, +at present, we have to admit that there are laws which appear +to have their seat in external nature. The most important of +these, if not the only law, is a law of atomicity. +\index{Atomicity!law of}% +Why does that +quality of the world which distinguishes matter from emptiness +exist only in certain lumps called atoms or electrons, all of +comparable mass? Whence arises this discontinuity? At +present, there seems no ground for believing that discontinuity +is a law due to the mind; indeed the mind seems rather to take +pains to smooth the discontinuities of nature into continuous +perception. We can only suppose that there is something in +the nature of things that causes this aggregation into atoms. +Probably our analysis into point-events is not final; and if it +could be pushed further to reach something still more fundamental, +then atomicity and the remaining laws of physics would +be seen as identities. This indeed is the only kind of explanation +that a physicist could accept as ultimate. But this more ultimate +analysis stands on a different plane from that by which the +point-events were reached. The world \textit{may} be so constituted +that the laws of atomicity must necessarily hold; but, so far as +the mind is concerned, there seems no reason why it should +have been constituted in that way. We can conceive a world +constituted otherwise. But our argument hitherto has been +that, however the world is constituted, the necessary combinations +of things can be found which obey the laws of mechanics, +gravitation and electrodynamics, and these combinations are +ready to play the part of the world of perception for any mind +that is tuned to appreciate them; and further, any world of +perception of a different character would be rejected by the +mind as unsubstantial. + +If atomicity depends on laws inherent in nature, it seems at +first difficult to understand why it should relate to matter +especially; since matter is not of any great account in the +analytical scheme, and owes its importance to irrelevant considerations +introduced by the mind. It has appeared, however, +that atomicity is by no means confined to matter and electricity; +the quantum, which plays so great a part in recent physics, is +apparently an atom of action. +\index{Action!atomicity of}% +So nature cannot be accused of +%% -----File: 210.png---Folio 200------- +connivance with mind in singling out matter for special distinction. +Action is generally regarded as the most fundamental +thing in the real world of physics, although the mind passes it +over because of its lack of permanence; and it is vaguely believed +that the atomicity of action is the general law, and the appearance +of electrons is in some way dependent on this. But the +precise formulation of the theory of quanta of action has hitherto +baffled physicists.% +\index{Quanta}% + +There is a striking contrast between the triumph of the +scientific mind in formulating the great general scheme of +natural laws, nowadays summed up in the principle of least +action, and its present defeat by the newly discovered but equally +general phenomena depending on the laws of atomicity of +quanta. It is too early to cry failure in the latter case; but +possibly the contrast is significant. It is one thing for the human +mind to extract from the phenomena of nature the laws which +it has itself put into them; it may be a far harder thing to +extract laws over which it has had no control. It is even possible +that laws which have not their origin in the mind may be +irrational, and we can never succeed in formulating them. This +is, however, only a remote possibility; probably if they were +really irrational it would not have been possible to make the +limited progress that has been achieved. But if the laws of +quanta do indeed differentiate the actual world from other +worlds possible to the mind, we may expect the task of formulating +them to be far harder than anything yet accomplished +by physics. + +The theory of relativity has passed in review the whole subject-matter +of physics. It has unified the great laws, which by the +precision of their formulation and the exactness of their application +have won the proud place in human knowledge which +physical science holds to-day. And yet, in regard to the nature +of things, this knowledge is only an empty shell---a form of +symbols. It is knowledge of structural form, and not knowledge +of content. +\index{Content contrasted with structural form}% +\index{Form contrasted with content}% +\index{Structure opposed to content}% +All through the physical world runs that unknown +content, which must surely be the stuff of our consciousness. +Here is a hint of aspects deep within the world of physics, and +yet unattainable by the methods of physics. And, moreover, +we have found that where science has progressed the farthest, +%% -----File: 211.png---Folio 201------- +the mind has but regained from nature that which the mind has +put into nature. + +We have found a strange foot-print on the shores of the +unknown. We have devised profound theories, one after +another, to account for its origin. At last, we have succeeded +in reconstructing the creature that made the foot-print. And +Lo! it is our own. +%% -----File: 212.png---Folio 202------- + + +\Appendix + +\First{The} references marked ``Report'' are to the writer's ``Report +on the Relativity Theory of Gravitation'' for the Physical +Society of London (Fleetway Press), where fuller mathematical +details are given. + +Probably the most complete treatise on the mathematical +theory of the subject is H.~Weyl's \textit{Raum, Zeit, Materie} (Julius +Springer, Berlin). + +\AppNote{1}{(\Pageref{note1}).} % p.~20 + +\Indent It is not possible to predict the contraction rigorously from +the universally accepted electromagnetic equations, because +these do not cover the whole ground. There must be other forces +or conditions which govern the form and size of an electron; +under electromagnetic forces alone it would expand indefinitely. +The old electrodynamics is entirely vague as to these forces. + +The theory of Larmor and Lorentz shows that if any system +at rest in the aether is in equilibrium, a similar system in +uniform motion through the aether, but with all lengths in the +direction of motion diminished in FitzGerald's ratio, will also +be in equilibrium so far as the differential equations of the +electromagnetic field are concerned. There is thus a general +theoretical agreement with the observed contraction, provided +the boundary conditions at the surface of an electron behave in +the same way. The latter suggestion is confirmed by experiments +on isolated electrons in rapid motion (Kaufmann's experiment). +\IndexExtra{Electron!Kaufmann's experiment on} % +It turns out that this requires an electron to suffer the same +kind of contraction as a material rod; and thus, although the +theory throws light on the adjustments involved in material +contraction, it can scarcely be said to give an explanation of the +occurrence of contraction generally. +%% -----File: 213.png---Folio 203------- + + +\AppNote{2}{(\Pageref{note2}).} % p.~47 + +\Indent Suppose a particle moves from $(x_{1}, y_{1}, z_{1}, t_{1})$ to $(x_{2}, y_{2}, z_{2}, t_{2})$, +its velocity~$u$ is given by +\[ +u^{2} = \frac{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} + (z_{2}-z_{1})^{2}} + {(t_{2}-t_{1})^{2}}. +\] +Hence from the formula for $s^{2}$ +\[ +s = (t_{2}-t_{1}) \surd(1-u^{2}). +\] +(We omit a $\sqrt{-1}$, as the sign of $s^{2}$ is changed later in the +chapter.) + +If we take $t_{1}$ and $t_{2}$ to be the start and finish of the aviator's +cigar (\Chapref{I}), then as judged by a terrestrial observer, +$t_{2}-t_{1} = 60~\text{minutes}$, $\surd(1- u^{2}) = \text{FitzGerald contraction} = \frac{1}{2}$. + +As judged by the aviator, +\[ +t_{2} - t_{1} = 30~\text{minutes}, \quad \surd(1-u^{2}) = 1. +\] + +Thus for both observers $s = 30$ minutes, verifying that it is +an absolute quantity independent of the observer. + +\AppNote{3}{(\Pageref{note3}).} % p.~48 + +\Indent The formulae of transformation to axes with a different +orientation are +\[ + x = x' \cos\theta - \tau' \sin\theta, \quad + y = y', \quad + z = z', \quad +\tau = x' \sin\theta + \tau' \cos\theta, +\] +where $\theta$ is the angle turned through in the plane~$x\tau$. + +Let $u = i \tan \theta$, so that $\cos\theta = (1-u^{2})^{-\frac{1}{2}} = \beta$, say. The +formulae become +\[ + x = \beta (x'- iu\tau'), \quad + y = y', \quad + z = z', \quad +\tau = \beta (\tau' + iux'), +\] +or, reverting to real time by setting $i\tau = t$, +\[ +x = \beta (x' - ut'), \quad +y = y', \quad +z = z', \quad +t = \beta (t' - ux'), +\] +which gives the relation between the estimates of space and +time by two different observers. + +The factor $\beta$ gives in the first equation the FitzGerald contraction, +and in the fourth equation the retardation of time. +The terms $ut'$ and~$ux'$ correspond to the changed conventions +as to \textit{rest} and \textit{simultaneity}. + +A point at rest, $x = \text{const.}$, for the first observer corresponds +to a point moving with velocity~$u$, $x'-ut' = \text{const.}$, for the second +observer. Hence their relative velocity is~$u$. +%% -----File: 214.png---Folio 204------- + + +\AppNote{4}{(\Pageref{note4}).} % p.~81 + +\Indent The condition for flat space in two dimensions is +\begin{multline*} +\frac{\partial}{\partial x_1} + \left( \frac{g_{12}}{g_{11} + \surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{11}}{\partial x_2} + - \frac{1}{\surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{22}}{\partial x_1}\right) \\ % [** PP: Changed \delta to \partial] +% ++ \frac{\partial}{\partial x_2} + \left( \frac{2}{\surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{12}}{\partial x_1} + - \frac{1}{\surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{11}}{\partial x_2} \right. \\ +% +- \left.\frac{g_{12}}{g_{11} \surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{11}}{\partial x_1}\right) = 0. +\end{multline*} + + +\AppNote{5}{(\Pageref{note5}).} % p.~89 + +\Indent Let~$g$ be the determinant of four rows and columns formed +with the elements~$g_{\mu\nu}$. + +Let $g^{\mu\nu}$ be the minor of~$g_{\mu\nu}$, divided by~$g$. + +Let the ``$3$-index symbol'' $\{\mu\nu, \lambda\}$ denote +\[ +\tfrac{1}{2} g^{\lambda\alpha} + \left( \frac{\partial g_{\mu\alpha}}{\partial x_\nu} + + \frac{\partial g_{\nu\alpha}}{\partial x_\mu} + - \frac{\partial g_{\mu\nu}}{\partial x_\alpha} \right) +\] +summed for values of~$\alpha$ from~$1$ to~$4$. There will be $40$ different +$3$-index symbols. + +Then the Riemann-Christoffel tensor is +\[ +B^\rho_{\mu\nu\sigma} + = \{\mu\sigma, \epsilon\} \{\epsilon\nu, \rho\} + - \{\mu\nu, \epsilon\} \{\epsilon\sigma, \rho\} + + \frac{\partial}{\partial x_\nu} \{\mu\sigma, \rho\} + - \frac{\partial}{\partial x_\sigma} \{\mu\nu, \rho\}, +\] +the terms containing~$\epsilon$ being summed for values of~$\epsilon$ from~$1$ to~$4$. + +The ``contracted'' Riemann-Christoffel tensor~$G_{\mu\nu}$ can be +reduced to +\begin{multline*} +G_{\mu\nu} + = - \frac{\partial}{\partial x_\alpha} \{\mu\nu, \alpha\} + + \{\mu\alpha, \beta\} \{\nu\beta, \alpha\} \\ + + \frac{\partial^2}{\partial x_\mu \partial x_\nu} \log \sqrt{-g} + - \{\mu\nu, \alpha\} \frac{\partial}{\partial x_\alpha} \log \sqrt{-g}, +\end{multline*} +where in accordance with a general convention in this subject, +each term containing a suffix twice over ($\alpha$ and~$\beta$) must be +summed for the values $1$, $2$, $3$, $4$ of that suffix. + +The curvature $G = g^{\mu\nu} G_{\mu\nu}$, summed in accordance with the +foregoing convention. + + +\AppNote{6}{(\Pageref{note6}).} % p.~94 + +\Indent The electric potential due to a charge~$e$ is +\[ +\phi = \frac{e}{\bigl[r(1-v_r/C)\bigr]}, +\] +%% -----File: 215.png---Folio 205------- +where $v_r$ is the velocity of the charge in the direction of~$r$, $C$ the +velocity of light, and the square bracket signifies antedated +values. To the first order of $v_r/C$, the denominator is equal to +the \textit{present} distance~$r$, so the expression reduces to~$e/r$ in spite +of the time of propagation. The foregoing formula for the +potential was found by Liénard and Wiechert. + + +\AppNote{7}{(\Pageref{note7}).} % p.~97 + +\Indent It is found that the following scheme of potentials rigorously +satisfies the equations $G_{\mu\nu} = 0$, according to the values of $G_{\mu\nu}$ +in \Noteref{5}, +\[ +\begin{array}{cccc} +-1/\gamma & 0 & 0 & 0 \\ + & -x_{1}{}^{2} & 0 & 0 \\ + & & -x_{1}{}^{2}\sin^{2} x_{2}{}^{2} & 0 \\ + & & & \gamma +\end{array} +\] +where $\gamma = 1-\kappa/x_{1}$ and $\kappa$ is any constant (see Report, §~28). +Hence these potentials describe a kind of space-time which can +occur in nature referred to a possible mesh-system. If $\kappa = 0$, +the potentials reduce to those for flat space-time referred to +polar coordinates; and, since in the applications required $\kappa$ will +always be extremely small, our coordinates can scarcely be +distinguished from polar coordinates. We can therefore use the +familiar symbols $r$, $\theta$, $\phi$, $t$, instead of $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. It must, +however, be remembered that the identification with polar +coordinates is only approximate; and, for example, an equally +good approximation is obtained if we write $x_{1} = r + \frac{1}{2} \kappa$, a substitution +often used instead of $x_{1} = r$ since it has the advantage +of making the coordinate-velocity of light more symmetrical. + +We next work out analytically all the mechanical and optical +properties of this kind of space-time, and find that they agree +observationally with those existing round a particle at rest at +the origin with gravitational mass~$\frac{1}{2} \kappa$. The conclusion is that +the gravitational field here described is produced by a particle +of mass $\frac{1}{2} \kappa$---or, if preferred, a particle of matter at rest is +produced by the kind of space-time here described. + + +\AppNote{8}{(\Pageref{note8}).} % p.~98 + +\Indent Setting the gravitational constant equal to unity, we have for +a circular orbit +\begin{DPalign*} +m/r^{2} &= v^{2}/r, \\ +\lintertext{so that} + m &= v^{2}r. +\end{DPalign*} + +%% -----File: 216.png---Folio 206------- + +The earth's speed,~$v$, is approximately $30$~km.\ per sec., or +$\frac{1}{10000}$ in terms of the velocity of light. The radius of its orbit,~$r$, +is about $1.5 · 10^8$~km. Hence,~$m$, the gravitational mass of the +sun is approximately $1.5$~km. % [** PP: Changing scientific notation] + +The radius of the sun is $697,000$~kms., so that the quantity +$2m/r$ occurring in the formulae is, for the sun's surface, $.00000424$ +or~$0''.87$. + + +\AppNote{9}{(\Pageref{note9}).} % p.~123 + +\index{Gravitational field of Sun!motion of perihelion}% +\Indent See Report, §§~29, 30. The general equations of a geodesic are +\[ +\frac{d^2x_\mu}{ds^2} + + \{\alpha\beta, \mu\}\, \frac{dx_\alpha}{ds} \frac{dx_\beta}{ds} = 0\quad + (\mu = 1,\ 2,\ 3,\ 4). +\] + +From the formula for the line-element +\[ +ds^2 = -\gamma^{-1}\, dr^2 - r^2\, d\theta^2 + \gamma\, dt^2, +\tag{1} +\] +we calculate the three-index symbols and it is found that two +of the equations of the geodesic take the rather simple form +\begin{align*} +&\frac{d^2\theta}{ds^2} + + \frac{2}{r} · \frac{dr}{ds} \frac{d\theta}{ds} = 0, \\ +% +&\frac{d^2t}{ds^2} + + \frac{d(\log\gamma)}{dr} · \frac{dr}{ds} \frac{dt}{ds} = 0, +\end{align*} +which can be integrated giving +\begin{align*} +r^2\, \frac{d\theta}{ds} &= h, +\tag{2} \\ +% +\frac{dt}{ds} &= \frac{c}{\gamma}, +\tag{3} +\end{align*} +where $h$ and~$c$ are constants of integration. + +Eliminating $dt$ and~$ds$ from (1), (2) and~(3), we have +\[ +\left(\frac{h}{r^2} \frac{dr}{d\theta}\right)^2 + \frac{h^2}{r^2} + = c^2 - 1 + \frac{2m}{r} + \frac{2mh^2}{r^3}, +\] +or writing $u = 1/r$, % [** PP: Condensed intertext in original] +\[ +\left(\frac{du}{d\theta}\right)^2 + u^2 + = \frac{c^2 - 1}{h^2} + \frac{2mu}{h^2} + 2mu^3. +\] + +Differentiating with respect to~$\theta$ +\[ +\frac{d^2u}{d\theta^2} + u = \frac{m}{h^2} + 3mu^2, +\] +%% -----File: 217.png---Folio 207------- +which gives the equation of the orbit in the usual form in particle +dynamics. It differs from the equation of the Newtonian orbit +by the small term $3mu^2$, which is easily shown to give the motion +of perihelion. + +The track of a ray of light is also obtained from this formula, +since by the principle of equivalence it agrees with that of a +material particle moving with the speed of light. +\index{Bending of light!theory of}% +\index{Light, bending of}% +This case is +given by $ds = 0$, and therefore $h = \infty$. The differential equation +for the path of a light-ray is thus +\[ +\frac{d^2u}{d\theta^2} + u = 3mu^2. +\] + +An approximate solution is +\[ +u = \frac{\cos\theta}{R} + \frac{m}{R^2}(\cos^2\theta + 2\sin^2\theta), +\] +neglecting the very small quantity~$m^2/R^2$. Converting to +Cartesian coordinates, this becomes +\[ +x = R - \frac{m}{R} \frac{x^2 + 2y^2}{\surd(x^2 + y^2)}. +\] + +The asymptotes of the light-track are found by taking~$y$ +very large compared with~$x$, giving +\[ +x = R ± \frac{2m}{R} y +\] +so that the angle between them is~$4m/R$. + + +\AppNote{10}{(\Pageref{note10}).} % p.~126 + +\Indent Writing the line element in the form +\[ +ds^2 + = -\left(1 + a \frac{m}{r} + \dotsb\right) dr^2 + - r^2\, d\theta^2 + + \left(1 + b \frac{m}{r} + c \frac{m^2}{r^2} + \dotsb\right) dt^2, +\] +the approximate Newtonian attraction fixes~$b$ equal to~$-2$; +then the observed deflection of light fixes~$a$ equal to~$+2$; and +with these values the observed motion of Mercury fixes~$c$ equal +to~$0$.% +\index{Deflection of light!theory of}% +\index{Gravitational field of Sun!deflection of light}% + +To insert an arbitrary coefficient of $r^2\, d\theta^2$ would merely vary +the coordinate system. We cannot arrive at any intrinsically +different kind of space-time in that way. Hence, within the +limits of accuracy mentioned, the expression found by Einstein +is completely determinable by observation. + +%% -----File: 218.png---Folio 208------- + +It may be mentioned that the line-element +\[ +ds^2 = - dr^2 - r^2\, d\theta^2 + (1-2m/r)\, dt^2, +\] +gives one-half the observed deflection of light, and one-third +the motion of perihelion of Mercury. As both these can be +obtained on older theories, taking account of the variation of +mass with velocity, the coefficient~$\gamma^{-1}$ of~$dr^2$ is the essentially +novel point in Einstein's theory. + + +\AppNote{11}{(\Pageref{note11}).} % p.~131 + +\Indent It is often supposed that by the Principle of Equivalence any +invariant property which holds outside a gravitational field also +holds in a gravitational field; but there is necessarily some +limitation on this equivalence. Consider for instance the two +invariant equations +\begin{gather*} +ds^2 = 1, \\ +ds^2 (1 + k^4 B^\rho_{\mu\nu\sigma} B^{\mu\nu\sigma}_\rho) = 1, +\end{gather*} +where $k$ is some constant having the dimensions of a length. +Since $B^\rho_{\mu\nu\sigma}$ vanishes outside a gravitational field, if one of these +equations is true the other will be. But they cannot both hold +in a gravitational field, since there $B^\rho_{\mu\nu\sigma} B^{\mu\nu\sigma}_\rho$ does not vanish, +and is in fact equal to $24m^2/r^6$. (I believe that the numerical +factor~$24$ is correct; but there are 65,536~terms in the expression, +and the terms which do not vanish have to be picked out.) % [** F1: Added closing parenthesis] + +This ambiguity of the Principle of Equivalence is referred to +in Report, §§~14, 27; and an enunciation is given which makes +it definite. The enunciation however is merely an explicit statement, +and not a defence, of the assumptions commonly made in +applying the principle. + +So far as general reasoning goes there seems no ground for +choosing $ds^2$ rather than $ds^2\, (1 + 24k^4m^2/r^6)$, or any similar expression, +as the constant character in the vibration of an atom. + + +\AppNote{12}{(\Pageref{note12}).} % p.~134 + +\Indent Let two rays diverging from a point at a distance~$R$ pass at +distances $r$ and~$r + dr$ from a star of mass~$m$. The deflection +being~$4m/r$, their divergence will be increased by~$4m\, dr/r^2$. This +increase will be equal to the original divergence~$dr/R$ if +$r = \sqrt{4mR}$. Take for instance $4m = 10$~km., $R = 10^{15}$~km., then +$r = 10^8$~km. So that the divergence of the light will be doubled, +%% -----File: 219.png---Folio 209------- +when the actual deflection of the ray is only $10^{-7}$, or $0''.02$. +In the case of a star seen behind the sun the added divergence +has no time to take effect; but when the light has to travel a +stellar distance after the divergence is produced, it becomes +weakened by it. Generally in stellar phenomena the weakening +of the light should be more prominent than the actual deflection. + + +\AppNote{13}{(\Pageref{note13}).} % p.~141 + +\Indent The relations are (Report, §~39) +\[ +G^\nu_{\mu\nu} = \tfrac{1}{2} \frac{\partial G}{\partial x_\mu}\quad +(\mu = 1,\ 2,\ 3,\ 4), +\] +where $G^\nu_{\mu\nu}$ is the (contracted) covariant derivative of~$G^\nu_\mu$, +or~$g^{\nu\alpha} G_{\mu\alpha}$. + +I doubt whether anyone has performed the laborious task of +verifying these identities by straightforward algebra. + + +\AppNote{14}{(\Pageref{note14}).} % p.~158 + +\Indent The modified law for spherical space-time is in empty space +\[ +G_{\mu\nu} = \lambda g_{\mu\nu}. +\] + +In cylindrical space-time, matter is essential. The law in space +occupied by matter is +\index{Cylindrical world, Einstein's}% +\[ +G_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} (G - 2\lambda) = -8\pi T_{\mu\nu}, +\] +the term $2\lambda$ being the only modification. Spherical space-time +of radius~$R$ is given by $\lambda = 3/R^2$; cylindrical space-time by +$\lambda = 1/R^2$ provided matter of average density $\rho = 1/4 \pi R^2$ is +present. (See Report, §§~50, 51.) The total mass of matter in +the cylindrical world is $\frac{1}{2} \pi R$. This must be enormous, seeing +that the sun's mass is only $1\frac{1}{2}$~kilometres. + + +\AppNote{15}{(\Pageref{note15}).} % p.~174 + +\Indent Weyl's theory is given in \textit{Berlin.\ Sitzungsberichte}, 30~May, 1918; +\textit{Annalen der Physik}, Bd.~59 (1919), p.~101. + + +\AppNote{16}{(\Pageref{note16}).} % p.~177 + +\Indent The argument is rather more complicated than appears in +the text, where the distinction between action-density and +action in a region, curvature and total curvature in a region, +has not been elaborated. Taking a definitely marked out region +in space and time, its measured volume will be increased $16$-fold +%% -----File: 220.png---Folio 210------- +by halving the gauge. Therefore for action-density we must +take an expression which will be diminished $16$-fold by halving +the gauge. Now~$G$ is proportional to $1/R^{2}$, where~$R$ is the radius +of curvature, and so is diminished $4$-fold. The invariant $B^{\rho}_{\mu\nu\sigma} B^{\mu\nu\sigma}_{\rho}$ +has the same gauge-dimensions as~$G^{2}$; and hence when integrated +through a volume gives a pure number independent of the gauge. +In Weyl's theory this is only the gravitational part of the complete +invariant +\[ +(B^{\rho}_{\mu \nu \sigma} - \tfrac{1}{2} g^{\rho}_{\mu} F_{\nu \sigma}) +(B^{\mu \nu \sigma}_{\rho} - \tfrac{1}{2} g^{\mu}_{\rho} F^{\nu \sigma}), +\] +which reduces to +\[ +B^{\rho}_{\mu \nu \sigma} +B^{\mu \nu \sigma}_{\rho} + F_{\nu \sigma}F^{\nu \sigma}. +\] + +The second term gives actually the well-known expression for +the action-density of the electromagnetic field, and this evidently +strengthens the identification of this invariant with action-density. + +Einstein's theory, on the other hand, creates a difficulty here, +because although there may be action in an electromagnetic +field without electrons, the curvature is zero. + +\fancyhead[CO]{\textsc{HISTORICAL NOTE}} +\phantomsection\addcontentsline{toc}{chapter} +{\texorpdfstring{\scshape Historical Note}{Historical Note}} +\section*{\centering\normalfont\textsc{\large HISTORICAL NOTE}} + +\Indent Before the Michelson-Morley experiment the question had +been widely discussed whether the aether in and near the earth +was carried along by the earth in its motion, or whether it +slipped through the interstices between the atoms. Astronomical +aberration pointed decidedly to a stagnant aether; +\index{Aether!stagnant}% +but the +experiments of Arago and Fizeau on the effect of motion of +transparent media on the velocity of light in those media, +suggested a partial convection of the aether in such cases. These +experiments were first-order experiments, i.e.\ %[** PP: Add missing .] +they depended on +the ratio of the velocity of the transparent body to the velocity +of light. The Michelson-Morley experiment is the first example +of an experiment delicate enough to detect second-order effects, +depending on the square of the above ratio; the result, that no +current of aether past terrestrial objects could be detected, +appeared favourable to the view that the aether must be convected +by the earth. The difficulty of reconciling this with +astronomical aberration was recognised. + +%% -----File: 221.png---Folio 211------- + +An attempt was made by Stokes to reconcile mathematically +a convection of aether by the earth with the accurately verified +facts of astronomical aberration; but his theory cannot be +regarded as tenable. Lodge investigated experimentally the +question whether smaller bodies carried the aether with them +in their motion, and showed that the aether between two +whirling steel discs was undisturbed.% +\index{Lodge}% + +{\stretchyspace +The controversy, stagnant \textit{versus} convected aether, had now +reached an intensely interesting stage. In 1895, Lorentz discussed +the problem from the point of view of the electrical +theory of light and matter. By his famous transformation of +the electromagnetic equations, he cleared up the difficulties +associated with the first-order effects, showing that they could +all be reconciled with a stagnant aether. In 1900, Larmor carried +the theory as far as second-order effects, and obtained an exact +theoretical foundation for FitzGerald's hypothesis of contraction, +which had been suggested in 1892 as an explanation of the +Michelson-Morley experiment. The theory of a stagnant aether +was thus reconciled with all observational results; and henceforward % [** PP: Hyphenated across a line in original] +it held the field.% +\index{Larmor}% +\index{Lorentz}% + +Further second-order experiments were performed by Rayleigh +and Brace on double refraction (1902, 1904), Trouton and Noble +on a torsional effect on a charged condenser (1903), and Trouton +and Rankine on electric conductivity (1908). All showed that +the earth's motion has no effect on the phenomena. On the +theoretical side, Lorentz (1902) showed that the indifference of +the equations of the electromagnetic field to any velocity of the +axes of reference, which he had previously established to the +first order, and Larmor to the second order, was exact to all +orders. He was not, however, able to establish with the same +exactness a corresponding transformation for bodies containing +electrons. + +Both Larmor and Lorentz had introduced a ``local time'' for +the moving system. It was clear that for many phenomena this +local time would replace the ``real'' time; but it was not +suggested that the observer in the moving system would be +deceived into thinking that it was the real time. Einstein, in +1905 founded the modern principle of relativity by postulating +that this local time was \textit{the time} for the moving observer; no +%% -----File: 222.png---Folio 212------- +real or absolute time existed, but only the local times, different +for different observers. He showed that absolute simultaneity +and absolute location in space are inextricably bound together, +and the denial of the latter carries with it the denial of the +former. By realising that an observer in the moving system +would measure all velocities in terms of the local space and time +of that system, Einstein removed the last discrepancies from +Lorentz's transformation. + +The relation between the space and time coordinates in two +systems in relative motion was now obtained immediately from +the principles of space and time-measurement. It must hold +for all phenomena provided they do not postulate a medium +which can serve as a standard for absolute location and simultaneity. +The previous deduction of these formulae by lengthy +transformation of the electromagnetic equations now appears +as a particular case; it shows that electromagnetic phenomena +have no reference to a medium with such properties. + +The combination of the local spaces and times of Einstein +into an absolute space-time of four dimensions is the work of +Minkowski~(1908). +\index{Minkowski}% +\index{Space-time!due to Minkowski}% +\Chapref{III} is largely based on his researches. +Much progress was made in the four-dimensional vector-analysis +of the world; but the whole problem was greatly +simplified when Einstein and Grossmann introduced for this +purpose the more powerful mathematical calculus of Riemann, +Ricci, and Levi-Civita. + +}In 1911, Einstein put forward the Principle of Equivalence, +thus turning the subject towards gravitation for the first time. +\index{Equivalence!Principle of}% +\index{Principle of Equivalence}% +By postulating that not only mechanical but optical and +electrical phenomena in a field of gravitation and in a field +produced by acceleration of the observer were equivalent, he +deduced the displacement of the spectral lines on the sun and +the displacement of a star during a total eclipse. In the latter +case, however, he predicted only the half-deflection, since he +was still working with Newton's law of gravitation. Freundlich +\index{Freundlich}% +at once examined plates obtained at previous eclipses, but failed +to find sufficient data; he also prepared to observe the eclipse +of 1914 in Russia with this object, but was stopped by the outbreak +of war. Another attempt was made by the Lick Observatory +at the not very favourable eclipse of 1918. Only preliminary +%% -----File: 223.png---Folio 213------- +results have been published; according to the information given, +the probable accidental error of the mean result (reduced to the +sun's limb) was about $1''.6$, so that no conclusion was permissible. + +The principle of equivalence opened up the possibility of a +general theory of relativity not confined to uniform motion, for +it pointed a way out of the objections which had been urged +against such an extension from the time of Newton. At first +the opening seemed a very narrow one, merely indicating that +the objections could not be considered final until the possibilities +of complications by gravitation had been more fully exhausted. +By 1913, Einstein had surmounted the main difficulties. His +theory in a complete form was published in 1915; but it was not +generally accessible in England until a year or two later. As +this theory forms the main subject-matter of the book, we may +leave our historical survey at this point. + +%% -----File: 224.png---Folio 214------- +% [Blank Page] +%% -----File: 225.png---Folio 215------- +\iffalse +INDEX + +Absolute, approached through the +relative, 82 + +Absolute acceleration, 68, 154, 194 + +Absolute past and future, 50 + +Absolute rotation, 152, 164, 194 + +Absolute simultaneity, 12, 51 + +Absolute time, in cylindrical world, +163 + +Acceleration, a simpler quality than +velocity, 195; modifies FitzGerald +contraction, 75 + +Action, 147; atomicity of, 199; on +Weyl's theory, 177 + +Action, Principle of Least, 149, 178 + +Addition of velocities, 59 + +Aether, a plenum with geodesic structure, +164; identified with the +``world,'' 187; non-material nature +of, 39; stagnant, 210 + +Artificial fields of force, 64 + +Atom, vibrating on sun, 128 + +Atomicity, law of, 199; of Action, 177 + +Aviator, space and time-reckoning of, +23 + +Bending of light, effect on star's +position, 112; observational results, +118; theory of, 107, 207 + +Beta particles, 59, 145 + +Brain, constitution of, 191 + +Brazil, eclipse expedition to, 117 + +Causality, law of, 156 + +Causation and free will, 51 + +Centrifugal Force, compared with +gravitation, 41, 65; debt at infinity, +157; not caused by stars, 153; +vibrating atom in field of, 129 + +Chess, analogy of, 184 + +Christoffel, 89 + +Circle in non-Euclidean space, 104 + +Clifford, 77, 152, 192 + +Cliquishness, 188 + +Clock, affected by velocity, 58; on sun, +74, 128; perfect, 13; recording +proper-time, 71 + +Clock-scale, 58 + +Clock-scale geometry, not fundamental, +73, 131, 191 + +Coincidences, 87 + +Comets, motion through coronal medium, +121; radiation-pressure in, 110 + +Conservation of electric charge, 173; +of energy and momentum, 139; of +mass, 141, 196 + +Content contrasted with structural +form, 192, 200 + +Continuous matter, 91, 140 + +Contraction, FitzGerald, 19, 54 + +Convergence of physical approximations, +154 + +Coordinates, 77 + +Coordinate velocity, 107 + +Corona, refraction by, 121 + +Cottingham, 114 + +Crommelin, 114, 122 + +Curvature, degrees of, 91; identified +with action, 148; merely illustrative, +84; of a globe of water, 148; +of space and time, 158; on Weyl's +theory, 176; perception of, 190 + +Cylinder and plane, indistinguishable +in two dimensions, 81 + +Cylindrical world, Einstein's, 161, 177 + +Davidson, 114 + +Deflection of light, effect on star's +position, 112; observational results, +118; theory of, 107, 207 + +Density, effect of motion on, 62 + +Displacement of spectral lines, 129; in +nebulae, 161; in stars, 135 + +Displacement of star-images, 112, 115 + +Double stars and Einstein effect, 133 + +Duration, not inherent in external +world, 34 +\fi +%% -----File: 226.png---Folio 216------- +\iffalse +Eclipse, observations during, 113 + +Ehrenfest's paradox, 75 + +Electrical theory of inertia, 61 + +Electricity and gravitation, 167 + +Electromagnetic potentials and forces, +172 + +Electron, dimensions of, 177; geometry +inside, 91; gravitational mass of, +178; inertia of, 61; Kaufmann's +experiment on, 62, 146; singularity +in field, 167 + +``Elsewhere,'' 50 + +Emptiness, perception of, 190 + +Energy, conservation of, 139; identified +with mass, 146; inertia of, 61, +146; weight of radio-active, 112 + +Entropy, 149 + +Eötvös torsion-balance, 112 + +Equivalence, Principle of, 76, 131, 212 + +Euclidean geometry, 1, 47, 73 + +Euclidean space of five dimensions, 84 + +Event, definition of, 45, 186 + +Evershed, 130 + +Extension in four dimensions, 37, 46 + +Feeling, elements of, 192 + +Fields of force, artificial, 64; due to +disturbance of observer, 69; electromagnetic, +171; relativity of, 67 + +Field of velocity, 195 + +FitzGerald contraction, 19; consequences +of, 22; modified by +acceleration, 75; relativity explanation +of, 54 + +Flat space in two dimensions, 80 + +Flat space-time, 83; at infinity, 84; +conditions for, 89 + +Flatfish, analogy of, 95 + +Flatland, 57 + +Force, compared with inertia, 137; +electromagnetic, 172; elementary +conception of, 63; fields of, 64; +relativity of, 43, 67, 76 + +Form contrasted with content, 192, 200 + +Formalism of knowledge, 175 + +Foucault's pendulum, 152 + +Four-dimensional order, 35, 56, 186 + +Four-dimensional space-time, geometry +of, 45, 82; reality of, 181 + +Fourth dimension, 13 + +Frame, inertial, 156 + +Frames of reference, ``right'' and +``wrong,'' 42 + +Freewill, 51 + +Freundlich, 212 + +Future, absolute, 50 + +Galilean potentials, 83 + +Gauge, effect on observations, 31; +provided by radius of space, 177 + +Gauge-system, 169 + +Geodesic, absolute significance of, 70, +150; definition of, 75; motion of +particles in, 138, 151; in regions at +infinity, 157 + +Geodesic structure, absolute character +of, 155, 164; acceleration of, 195 + +Geometrical conception of the world, +176, 183 + +Geometry, Euclidean, 1; hyperbolic, +47; Lobatchewskian, 1, 9; natural, +2; non-Euclidean, or Riemannian, +6, 73, 84, 90; non-Riemannian, 169; +semi-Euclidean, 47 + +Ghosts of stars, 161 + +Globe of water, limit to size of, 148 + +Gravitation, Einstein's law of, differential +formula, 90; integrated formula +for a particle, 97; macroscopic +equations, 140, 193 + +Gravitation, Newton's law of, ambiguity +of, 93; approximation to +Einstein's law, 103; deflection of +light, 109, 111 + +Gravitation, propagation with velocity +of light, 94, 147; relativity for +uniform motion, 21, 125 + +Gravitational field of Sun, 97; deflection +of light, 107, 118, 207; displacement +of spectral lines, 129; +motion of perihelion, 122, 206; +Newtonian attraction, 102; result +of observational verification, 126 + +Grebe and Bachem, 130 + +Greenwich, Royal Observatory, 114 + +Gyro-compass, 152 + +Hummock in space-time, 97 + +Hurdles, analogy of counts of, 104 + +Hyperbolic geometry, 47 +\fi +%% -----File: 227.png---Folio 217------- +\iffalse +Identities connecting G_{\mu \nu}, 141 + +Identity permanent, 40, 193 + +Imaginary intervals, 150, 187 + +Imaginary time, 48, 181 + +Inertia, compared with force, 137; +electrical theory of, 61; in regions +at infinity, 157; infinite, 56; +Mach's views, 164; of light, 110; +relativity theory of, 139 + +Inertia-gravitation, 137 + +Inertial frame, 156 + +Infinity, conditions at, 157 + +Integrability of length and direction, +174 + +Interval, 46, 150, 187; general expression +for, 82; practical measurement +of, 58, 75 + +Interval-length, geometrical significance +essential, 127; identified with +proper time, 71; tracks of maximum, +70, 150; zero for velocity of light, 71 + +Invariant mass, 145; of light, 148 + +Jupiter, deflection of light by, 133 + +Kaufmann's experiment, 62, 146 + +Kinds of space, 81 + +Laplace's equation, 96, 140 + +Larmor, 19, 211 + +Length, definition of, 2; effect of +motion on, 19; relativity of, 34 + +Le Verrier, 124 + +Levi-Civita, 89 + +Lift, accelerated, 64 + +Light, bending of, 107, 112, 118, 207; +coordinate velocity of, 107; mass +of, 62, 110, 148; voyage round the +world, 161; weight of, 111 + +Light, velocity of, an absolute velocity, +59; importance of, 60; system +moving with, 26, 56 + +Lobatchewsky, 1, 9 + +Lodge, 32, 125, 211 + +Longest tracks, 70 + +Lorentz, 19, 211 + +Mach's philosophy, 163 + +Macroscopic equations, 92, 139; interval, +187 + +Map of sun's gravitational field, 99 + +Mass, conservation of, 141, 195; +electrical theory of, 61; gravitational, +98; identified with energy, +146; invariant, 145; of light, 62, +110, 148; variation with velocity, +145 + +Mathematics, Russell's description of, +14 + +Matter, continuous, 91; definition of a +particle, 98; extensional relations +of, 8; gravitational equations in, +141; perception of, 190; physical +and psychological aspects, 192 + +Mercury, perihelion of, 123, 125 + +Mesh-systems, 77; irrelevance to laws +of nature, 87 + +Michelson-Morley experiment, 18 + +Minkowski, 30, 212 + +Mirror, distortion by moving, 22 + +Momentum, conservation of, 141; redefinition +of, 144; of light, 111 + +Moon, motion of, 93, 134 + +Motion, insufficiency of kinematical +conception, 194; Newton's first +law, 136 + +Natural frame, 155 + +Natural gauge, 176 + +Natural geometry, 2 + +Natural tracks, 70 + +Nebulae, atomic vibrations in, 161 + +Newton, absolute rotation, 41; bending +of light, 110; law of gravitation, +93; law of motion, 136; relativity +for uniform motion, 40; super-observer, +68 + +Non-Euclidean geometry, 6, 73, 84, 90 + +Non-Riemannian geometry, 169 + +Observer, an unsymmetrical object, 57 + +Observer and observed, 30 + +Orbits under Einstein's law, 123 + +Order and dimensions, 14, 186 + +Ordering of events in external world, +35, 54, 184 + +Past, absolute, 50 + +Perceptions, as crude measures, 10, 15, +31 +\fi +%% -----File: 228.png---Folio 218------- +\iffalse +Perihelia of planets, motions of, 123 +Permanence of matter, 196 +Permanent identity, 40, 193 +Permanent perceptual world, 141, 198 +Poincaré, 9 +Point-event, 45, 186 +Potentials, 80; Galilean values, 83 +Potentials, electromagnetic, 172 +Principe, eclipse expedition to, 114 +Principle of Equivalence, 76, 131, 212 +Principle of Least Action, 149, 178 +Principle of Relativity (restricted), 20 +Probability, a pure number, 178 +Projectile, Jules Verne's, 65 +Propagation of Gravitation, 94, 147 +Proper-length, 11 +Proper-time, 71 +Pucker in space-time, 85 + +Quanta, 60, 177, 182, 200 + +Radiation-pressure, 110 +Real world of physics, 37, 181 +Receding velocities of B-type stars, + 135; of spiral nebulae, 161 +Reflection by moving mirror, 22 +Refracting medium equivalent to + gravitational field, 109 +Refraction of light in corona, 121 +Relativity of force, 43, 76; of length + and duration, 34; of motion, 38; of + rotation, 152, 155; of size, 33 +Relativity, Newtonian, 40; restricted + Principle of, 20; standpoint of, 28 +Repulsion of light proceeding radially, + 102, 108 +Retardation of time, 24, 55; in centrifugal + field, 129; in spherical world, + 160 +Ricci, 89 +Riemann, 2, 89, 167 +Riemann-Christoffel tensor, 89 +Riemannian, or non-Euclidean, geometry, + 6, 73, 84, 90 +Rigid scale, definition of, 3 +Rotation, absolute, 152, 164, 194 +Rotation of a continuous ring, 194 +Russell, 14, 197 + +St John, 130 +Semi-Euclidean geometry, 47 +Simultaneity, 12, 51 +de Sitter, 134, 159, 179 +Sobral, eclipse expedition to, 117 +Space, conventional, 9; kinds of, 81; + meaning of, 3, 8, 15; relativity of, + 34 +Space-like intervals, 60, 187 +Space-time, 45; due to Minkowski, 212; + partitions of, 54; +Spherical space-time, 159 +Standard metre, comparison with, 168 +Stresses in continuous matter, 193 +Structure opposed to content, 197, 200 +Structure, geodesic, absolute character + of, 155, 164; acceleration of, 195; + behaviour at infinity, 157 +Super-observer, Newton's, 68 +Synthesis of appearances, 31, 182 + +Tensors, 89, 189 +Thomson, J. J., 61 +Time, absolute, 163; depends on + observer's track, 38, 57; for moving + observer, 24; imaginary, 48; measurement + of, 13; past and future, +51; ``standing still,'' 26, 160 +Time-like intervals, 60, 187 +Tracks, natural, 70 + +Vacuum, defined by law of gravitation, + 190 +Vector, non-integrable on Weyl's + theory, 174 +Velocity, addition-law, 59; definition + of, 193; static character, 194 +Velocity of gravitation, 94, 147 +Velocity of light, importance of, 60; + in gravitational field, 108; system + moving with, 26, 56 + +Warping of space, 8, 126 +Wave-front, slewing of, 108 +Weight, of light, 107, 111; of radio-active + energy, 111; proportional to + inertia, 137; vanishes inside free + projectile, 65 +Weyl, 174 +World, 186, 187 +World-line, 87 +\fi + +\cleardoublepage + +\fancyhead[C]{\textsc{INDEX}} +{\small +\printindex + +} + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage + +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\fancyhead[C]{\textsc{LICENSING}} + +\begin{PGtext} +End of Project Gutenberg's Space, Time and Gravitation, by A. 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+\useasboundingbox (0in,0in) rectangle (1.6in,2in); +\pgftext[at={\pgfpoint{0.8in}{-0.110696in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~1.}}}} +\pgfsetlinewidth{0.8pt} +\draw (0in,0.2in)--(1.6in,0.2in)--(1.6in,1.4in)--(0in,0.2in)--cycle; +\pgftext[at={\pgfpoint{0in}{0.144652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $O$}}} +\pgftext[at={\pgfpoint{1.6in}{0.144652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $B$}}} +\pgftext[at={\pgfpoint{1.6in}{1.45535in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $E$}}} +\draw (0.32in,2in)--(0.32in,1.48in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](0.32in,1.52324in)--(0.285407in,1.56648in)-- + (0.32in,1.48in)--(0.354593in,1.56648in)--(0.32in,1.52324in)--cycle; +\draw (0.96in,2in)--(0.96in,1.48in); +\draw [fill](0.96in,1.52324in)--(0.925407in,1.56648in)-- + (0.96in,1.48in)--(0.994593in,1.56648in)--(0.96in,1.52324in)--cycle; +\end{tikzpicture} diff --git a/29782-t/images/sources/027a.xp b/29782-t/images/sources/027a.xp new file mode 100644 index 0000000..03885c6 --- /dev/null +++ b/29782-t/images/sources/027a.xp @@ -0,0 +1,31 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +int main() +{ + picture(P(0,-0.125), P(1,1.125), "1.6 x 2in"); + + begin(); + + label(P(0.5, ymin()), P(0,-8), "\\textsc{Fig.~1.}", b); + + arrow_inset(0.5); + arrow_width(5); + arrow_ratio(2.5); + + bold(); + P O(0,0), B(1,0), E(1,0.75); + triangle(O, B, E); + + font_size("footnotesize"); + label(O, P(0,-4), "$O$", b); + label(B, P(0,-4), "$B$", b); + label(E, P(0, 4), "$E$", t); + + arrow(P(0.2, 1.125), P(0.2, 0.8)); + arrow(P(0.6, 1.125), P(0.6, 0.8)); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/056a.eepic b/29782-t/images/sources/056a.eepic new file mode 100644 index 0000000..7adafcb --- /dev/null +++ b/29782-t/images/sources/056a.eepic @@ -0,0 +1,91 @@ +%% Generated from 056a.xp on Wed Mar 4 20:37:14 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 1.75 x 1.75in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (1.75in,1.75in); +\pgftext[at={\pgfpoint{0.875in}{-0.274098in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~2.}}}} +\pgfsetlinewidth{0.8pt} +\draw (0.6125in,0.7875in)--(1.4in,1.3125in); +\draw (0in,0in)--(1.75in,0in); +\draw (0in,0in)--(0in,1.75in); +\draw (0in,0.7875in)--(0.0388889in,0.7875in); +\draw (0.116667in,0.7875in)--(0.155556in,0.7875in); +\draw (0.155556in,0.7875in)--(0.194444in,0.7875in); +\draw (0.272222in,0.7875in)--(0.311111in,0.7875in); +\draw (0.311111in,0.7875in)--(0.35in,0.7875in); +\draw (0.427778in,0.7875in)--(0.466667in,0.7875in); +\draw (0.466667in,0.7875in)--(0.505556in,0.7875in); +\draw (0.583333in,0.7875in)--(0.622222in,0.7875in); +\draw (0.622222in,0.7875in)--(0.661111in,0.7875in); +\draw (0.738889in,0.7875in)--(0.777778in,0.7875in); +\draw (0.777778in,0.7875in)--(0.816667in,0.7875in); +\draw (0.894444in,0.7875in)--(0.933333in,0.7875in); +\draw (0.933333in,0.7875in)--(0.972222in,0.7875in); +\draw (1.05in,0.7875in)--(1.08889in,0.7875in); +\draw (1.08889in,0.7875in)--(1.12778in,0.7875in); +\draw (1.20556in,0.7875in)--(1.24444in,0.7875in); +\draw (1.24444in,0.7875in)--(1.28333in,0.7875in); +\draw (1.36111in,0.7875in)--(1.4in,0.7875in); +\draw (0in,1.3125in)--(0.0388889in,1.3125in); +\draw (0.116667in,1.3125in)--(0.155556in,1.3125in); +\draw (0.155556in,1.3125in)--(0.194444in,1.3125in); +\draw (0.272222in,1.3125in)--(0.311111in,1.3125in); +\draw (0.311111in,1.3125in)--(0.35in,1.3125in); +\draw (0.427778in,1.3125in)--(0.466667in,1.3125in); +\draw (0.466667in,1.3125in)--(0.505556in,1.3125in); +\draw (0.583333in,1.3125in)--(0.622222in,1.3125in); +\draw (0.622222in,1.3125in)--(0.661111in,1.3125in); +\draw (0.738889in,1.3125in)--(0.777778in,1.3125in); +\draw 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(1.4in,0.984375in)--(1.4in,1.02539in); +\draw (1.4in,1.10742in)--(1.4in,1.14844in); +\draw (1.4in,1.14844in)--(1.4in,1.18945in); +\draw (1.4in,1.27148in)--(1.4in,1.3125in); +\draw (0.6125in,0in)--(0.6125in,0.039375in); +\draw (0.6125in,0.118125in)--(0.6125in,0.1575in); +\draw (0.6125in,0.1575in)--(0.6125in,0.196875in); +\draw (0.6125in,0.275625in)--(0.6125in,0.315in); +\draw (0.6125in,0.315in)--(0.6125in,0.354375in); +\draw (0.6125in,0.433125in)--(0.6125in,0.4725in); +\draw (0.6125in,0.4725in)--(0.6125in,0.511875in); +\draw (0.6125in,0.590625in)--(0.6125in,0.63in); +\draw (0.6125in,0.63in)--(0.6125in,0.669375in); +\draw (0.6125in,0.748125in)--(0.6125in,0.7875in); +\pgftext[at={\pgfpoint{-0.055348in}{-0.055348in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}\footnotesize $O$}}} +\pgftext[at={\pgfpoint{1.75in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $x$}}} +\pgftext[at={\pgfpoint{0.6125in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $X_1$}}} +\pgftext[at={\pgfpoint{1.4in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $X_2$}}} +\pgftext[at={\pgfpoint{-0.055348in}{1.75in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $y$}}} +\pgftext[at={\pgfpoint{-0.055348in}{0.7875in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $Y_1$}}} +\pgftext[at={\pgfpoint{-0.055348in}{1.3125in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $Y_2$}}} +\pgftext[at={\pgfpoint{1.45535in}{0.7875in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $M$}}} +\pgftext[at={\pgfpoint{0.584826in}{0.842848in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $P_1$}}} +\pgftext[at={\pgfpoint{1.45535in}{1.3125in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $P_2$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/056a.xp b/29782-t/images/sources/056a.xp new file mode 100644 index 0000000..435d3cf --- /dev/null +++ b/29782-t/images/sources/056a.xp @@ -0,0 +1,49 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +int main() +{ + picture(P(0,0), P(1,1), "1.75 x 1.75in"); + + begin(); + + label(P(0.5, -0.125), P(0,-4), "\\textsc{Fig.~2.}", b); + + /* + arrow_inset(0.5); + arrow_width(5); + arrow_ratio(2.5); + */ + + bold(); + P O(0,0), pt1(0.35,0.45), pt2(0.8,0.75), M(pt2.x1(), pt1.x2()); + + line(pt1, pt2); + line(O, P(xmax())); + line(O, P(0,ymax())); + + dashed(); + line(P(0,pt1.x2()), M); + line(P(0,pt2.x2()), pt2); + + line(P(pt2.x1()), pt2); + line(P(pt1.x1()), pt1); + + font_size("footnotesize"); + label(O, P(-4,-4), "$O$", bl); + label(P(xmax()), P(0,-4), "$x$", b); + label(P(pt1.x1()), P(0,-4), "$X_1$", b); + label(P(pt2.x1()), P(0,-4), "$X_2$", b); + + label(P(0,ymax()), P(-4), "$y$", l); + label(P(0,pt1.x2()), P(-4), "$Y_1$", l); + label(P(0,pt2.x2()), P(-4), "$Y_2$", l); + + label(M, P(4), "$M$", r); + label(pt1, P(-2,4), "$P_1$", tl); + label(pt2, P(4), "$P_2$", r); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/059a.eepic b/29782-t/images/sources/059a.eepic new file mode 100644 index 0000000..8150ad4 --- /dev/null +++ b/29782-t/images/sources/059a.eepic @@ -0,0 +1,50 @@ +%% Generated from 059a.xp on Wed Mar 4 09:33:16 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-1,1] x [-1,1] +%% Actual size: 3.5 x 3.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.5in,3.5in); +\pgftext[at={\pgfpoint{1.75in}{-0.110696in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~3.}}}} +\pgfsetlinewidth{0.8pt} +\draw (3.325in,3.325in)--(0.175in,0.175in); +\draw (0.175in,3.325in)--(3.325in,0.175in); +\draw (1.75in,0in)--(1.75in,3.5in); +\draw (0in,1.75in)--(3.5in,1.75in); +\draw (1.05in,3.4125in)--(1.05in,3.37118in); +\draw (1.05in,3.28854in)--(1.05in,3.24722in); +\draw (1.05in,3.24722in)--(1.05in,3.2059in); +\draw (1.05in,3.12326in)--(1.05in,3.08194in); +\draw (1.05in,3.08194in)--(1.05in,3.04062in); +\draw (1.05in,2.95799in)--(1.05in,2.91667in); +\draw (1.05in,2.91667in)--(1.05in,2.87535in); +\draw (1.05in,2.79271in)--(1.05in,2.75139in); +\draw (1.05in,2.75139in)--(1.05in,2.71007in); +\draw (1.05in,2.62743in)--(1.05in,2.58611in); +\draw (1.05in,2.58611in)--(1.05in,2.54479in); +\draw (1.05in,2.46215in)--(1.05in,2.42083in); +\draw (1.05in,2.42083in)--(1.05in,2.37951in); +\draw (1.05in,2.29688in)--(1.05in,2.25556in); +\draw (1.05in,2.25556in)--(1.05in,2.21424in); +\draw (1.05in,2.1316in)--(1.05in,2.09028in); +\draw (1.05in,2.09028in)--(1.05in,2.04896in); +\draw (1.05in,1.96632in)--(1.05in,1.925in); +\pgftext[at={\pgfpoint{1.58396in}{1.69465in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}\footnotesize $O$}}} +\pgftext[at={\pgfpoint{3.55535in}{1.75in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $X$}}} +\pgftext[at={\pgfpoint{1.80535in}{3.5in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $T$}}} +\pgftext[at={\pgfpoint{3.35267in}{3.35267in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $U$}}} +\pgftext[at={\pgfpoint{0.147326in}{0.202674in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $U'$}}} +\pgftext[at={\pgfpoint{0.202674in}{3.35267in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $V$}}} +\pgftext[at={\pgfpoint{3.35267in}{0.202674in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $V'$}}} +\pgftext[at={\pgfpoint{1.10535in}{3.4125in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $P$}}} +\filldraw[color=rgb_000000] (1.05in,3.4125in) circle(0.0207555in); +\pgftext[at={\pgfpoint{1.10535in}{1.925in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $P'$}}} +\filldraw[color=rgb_000000] (1.05in,1.925in) circle(0.0207555in); +\pgftext[at={\pgfpoint{1.07767in}{2.47767in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $N$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/059a.xp b/29782-t/images/sources/059a.xp new file mode 100644 index 0000000..8c32f16 --- /dev/null +++ b/29782-t/images/sources/059a.xp @@ -0,0 +1,53 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +int main() +{ + picture(P(-1,-1), P(1,1), "3.5 x 3.5in"); + + begin(); + + label(P(0, ymin()), P(0,-8), "\\textsc{Fig.~3.}", b); + + /* + arrow_inset(0.5); + arrow_width(5); + arrow_ratio(2.5); + */ + + const double MAX(0.9); + const double X1(-0.4); + + bold(); + P O(0,0), U1(MAX,MAX), U2(-U1), V1(-MAX,MAX), V2(-V1); + P T(0,ymax()), X(xmax()), P1(X1, 0.5*(1+MAX)), P2(X1,0.1), N(X1,-X1); + + line(U1, U2); + line(V1, V2); + + line(-T, T); + line(-X, X); + + dashed(); + line(P1, P2); + + font_size("footnotesize"); + label(O, P(-12,-4), "$O$", bl); + label(X, P(4), "$X$", r); + label(T, P(4), "$T$", r); + + label(U1, P( 2,2), "$U$", tr); + label(U2, P(-2,2), "$U'$", tl); + + label(V1, P( 2,2), "$V$", tr); + label(V2, P( 2,2), "$V'$", tr); + + dot(P1, P(4), "$P$", r); + dot(P2, P(4), "$P'$", r); + + label(N, P(2,2), "$N$", tr); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/062a.eepic b/29782-t/images/sources/062a.eepic new file mode 100644 index 0000000..b40ec63 --- /dev/null +++ b/29782-t/images/sources/062a.eepic @@ -0,0 +1,174 @@ +%% Generated from 062a.xp on Sun Mar 15 09:32:31 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-3.5,3.5] x [-3.5,3.5] +%% Actual size: 3.5 x 3.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.5in,3.5in); +\pgftext[at={\pgfpoint{1.75in}{-0.180348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~4.}}}} +\pgfsetlinewidth{0.8pt} +\draw (0in,1.75in)--(3.5in,1.75in); +\draw (1.75in,0in)--(1.75in,3.5in); 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{\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $T$}}} +\pgftext[at={\pgfpoint{2.36613in}{3.39928in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $T_1$}}} +\pgftext[at={\pgfpoint{2.98884in}{2.96116in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $U$}}} +\pgftext[at={\pgfpoint{0.511163in}{0.538837in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $U'$}}} +\pgftext[at={\pgfpoint{0.497326in}{2.94733in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}\footnotesize $V$}}} +\pgftext[at={\pgfpoint{3.00267in}{0.552674in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $V'$}}} +\pgftext[at={\pgfpoint{1.73616in}{2.29151in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $L$}}} +\pgftext[at={\pgfpoint{1.72233in}{1.20849in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}\footnotesize $L'$}}} +\pgftext[at={\pgfpoint{1.95538in}{2.31821in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $F$}}} +\pgftext[at={\pgfpoint{2.52359in}{2.49591in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $G$}}} +\pgftext[at={\pgfpoint{2.29054in}{1.9277in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $H$}}} +\pgftext[at={\pgfpoint{2.96116in}{3.08695in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $K$}}} +\pgftext[at={\pgfpoint{0.552674in}{3.10079in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $M$}}} +\pgftext[at={\pgfpoint{0.538837in}{0.413051in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $K'$}}} +\pgftext[at={\pgfpoint{2.94733in}{0.399214in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}\footnotesize $M'$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/062a.xp b/29782-t/images/sources/062a.xp new file mode 100644 index 0000000..97fb3b2 --- /dev/null +++ b/29782-t/images/sources/062a.xp @@ -0,0 +1,89 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double tau(0.4); +double CSH(cosh(tau)), SNH(sinh(tau)); + +const double MAX(3.5); +const double MAX2(0.7*MAX); +const double TMAX(asinh(MAX2)); + +P boost(const P& arg) +{ + double x(arg.x1()), y(arg.x2()); + return P(CSH*x + SNH*y, CSH*y + SNH*x); +} + +P fp(double t) +{ + return P(sinh(t), cosh(t)); +} + +P fm(double t) +{ + return -fp(t); +} + +int main() +{ + picture(P(-MAX,-MAX), P(MAX,MAX), "3.5 x 3.5in"); + + begin(); + + label(P(0, ymin()-0.25), P(0,-4), "\\textsc{Fig.~4.}", b); + + P O(0,0), U1(MAX2,MAX2), U2(-U1), V1(-MAX2,MAX2), V2(-V1); + P T(0,3), T1(boost(T)), X(3), X1(boost(X)); + P L(0,1), F(boost(L)), G(boost(P(1,1))), H(boost(P(1,0))); + + bold(); + line(P(-MAX), P(MAX)); + line(P(0,-MAX), P(0,MAX)); + + h_axis(O, P(3), 3); + v_axis(O, P(0,3), 3); + + line(O, T1); + line(O, X1); + + line(G, H); + line(G, boost(P(-1,1))); + + plot(fp, -TMAX, TMAX, 40); + plot(fm, -TMAX, TMAX, 40); + + dashed(); + line(U1, U2); + line(V1, V2); + + font_size("footnotesize"); + label(O, P(-12,-4), "$O$", bl); + label(P(xmax()), P(0,-4), "$X$", b); + label(X1, P(2,0), "$X_1$", tr); + + label(P(0,ymax()), P(-1,0), "$T$", l); + label(T1, P(0,2), "$T_1$", tr); + + label(U1, P( 1,-1), "$U$", br); + label(U2, P(-1, 1), "$U'$", tl); + + label(V1, P(-2,-2), "$V$", bl); + label(V2, P( 2, 2), "$V'$", tr); + + label(L, P(-1, 3), "$L$", tl); + label(P(0,-1), P(-2, -3), "$L'$", bl); + label(F, P( 0, 2), "$F$", tl); + + label(G, P( 2, 0), "$G$", br); + label(H, P( 0,-2), "$H$", br); + + label(fp( TMAX), P(-1,1), "$K$", tl); + label(fp(-TMAX), P( 2,2), "$M$", tr); + + label(fm( TMAX), P( 1,-1), "$K'$", br); + label(fm(-TMAX), P(-2,-2), "$M'$", bl); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/064a.eepic b/29782-t/images/sources/064a.eepic new file mode 100644 index 0000000..7db0ba4 --- /dev/null +++ b/29782-t/images/sources/064a.eepic @@ -0,0 +1,43 @@ +%% Generated from 064a.xp on Wed Mar 4 09:33:26 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,2.5] x [0,1.5] +%% Actual size: 5 x 3in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (5in,3in); +\pgftext[at={\pgfpoint{1in}{-0.305348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~5.}}}} +\pgftext[at={\pgfpoint{3.25in}{-0.305348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~6.}}}} +\pgfsetlinewidth{0.8pt} +\draw (0in,0in)--(0in,1.8in); +\draw (0in,0in)--(1.8in,0in); +\draw (0.4in,0in)--(0.4in,1.8in); +\draw (0in,0.4in)--(1.8in,0.4in); +\draw (0.8in,0in)--(0.8in,1.8in); +\draw (0in,0.8in)--(1.8in,0.8in); +\draw (1.2in,0in)--(1.2in,1.8in); +\draw (0in,1.2in)--(1.8in,1.2in); +\draw (1.6in,0in)--(1.6in,1.8in); +\draw (0in,1.6in)--(1.8in,1.6in); +\pgftext[at={\pgfpoint{0in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $O$}}} +\pgftext[at={\pgfpoint{1.77233in}{-0.055348in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $X$}}} +\pgftext[at={\pgfpoint{-0.055348in}{1.8in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $T$}}} +\draw (2.2in,0in)--(2.93935in,1.94593in); +\draw (2.2in,0in)--(4.14593in,0.739354in); +\draw (2.63243in,0.164301in)--(3.37178in,2.11023in); +\draw (2.3643in,0.432429in)--(4.31023in,1.17178in); +\draw (3.06486in,0.328602in)--(3.80421in,2.27453in); +\draw (2.5286in,0.864858in)--(4.47453in,1.60421in); +\draw (3.49729in,0.492903in)--(4.23664in,2.43883in); +\draw (2.6929in,1.29729in)--(4.63883in,2.03664in); +\draw (3.92972in,0.657204in)--(4.66907in,2.60313in); +\draw (2.8572in,1.72972in)--(4.80313in,2.46907in); +\pgftext[at={\pgfpoint{2.2in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $O$}}} +\pgftext[at={\pgfpoint{4.11826in}{0.684006in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $X_1$}}} +\pgftext[at={\pgfpoint{2.88401in}{1.94593in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $T_1$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/064a.xp b/29782-t/images/sources/064a.xp new file mode 100644 index 0000000..0202c81 --- /dev/null +++ b/29782-t/images/sources/064a.xp @@ -0,0 +1,54 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double mult(4.5); + +const double tau(0.4); +double CSH(cosh(tau)), SNH(sinh(tau)); + +P boost(const P& arg) +{ + double x(arg.x1()), y(arg.x2()); + return P(CSH*x + SNH*y, CSH*y + SNH*x); +} + +void my_grid(const P& arg, const std::string& msg1, const std::string& msg2) +{ + P arg2(arg.x2(), arg.x1()); + for (int i=0; i < 5; ++i) + { + line(i*arg, i*arg + mult*arg2); + line(i*arg2, i*arg2 + mult*arg); + } + + label(P(0,0), P(0,-4), "$O$", b); + label(mult* arg, P(-2,-4), msg1, br); + label(mult*arg2, P(-4), msg2, tl); +} + +int main() +{ + picture(P(0,0), P(2.5,1.5), "5 x 3in"); + + begin(); + label(P(0.5, -0.125), P(0,-4), "\\textsc{Fig.~5.}", b); + label(P(1.625, -0.125), P(0,-4), "\\textsc{Fig.~6.}", b); + + screen scr1(P(0,0), P(5, 5)), scr2(P(0,0), P(7,7)); + + bold(); + font_size("footnotesize"); + activate(scr1); + my_grid(P(1,0), "$X$", "$T$"); + inset(P(0,0), P(1,1)); + deactivate(scr1); + + activate(scr2); + my_grid(boost(P(1,0)), "$X_1$", "$T_1$"); + inset(P(1.1,0), P(2.5,1.4)); + deactivate(scr2); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/065a.eepic b/29782-t/images/sources/065a.eepic new file mode 100644 index 0000000..8b48c0f --- /dev/null +++ b/29782-t/images/sources/065a.eepic @@ -0,0 +1,230 @@ +%% Generated from 065a.xp on Sun Mar 15 10:18:23 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-2.25,2.25] x [-0.2,3.2] +%% Actual size: 4.5 x 3.4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4.5in,3.4in); 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/dev/null +++ b/29782-t/images/sources/065a.xp @@ -0,0 +1,106 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double tau(0.4); + +P boost(const P& arg, double t) +{ + double CSH(cosh(t)), SNH(sinh(t)); + double x(arg.x1()), y(arg.x2()); + + return P(CSH*x + SNH*y, CSH*y + SNH*x); +} + +void time_grid(double t) +{ + P sw(boost(P(0,0), t)), se(boost(P( 1,0), t)); + P nw(boost(P(0, 1), t)), ne(boost(P( 1, 1), t)); + + line(sw, se, 120); + line(sw, nw, 120); + line(ne, se, 120); + line(ne, nw, 120); +} + +const double my_x1(0.15); +const double my_x2(1+my_x1); + +std::string math(const std::string& arg) +{ + return "$" + arg + "$"; +} + +void rod(double t, const std::string& lbl1, const std::string& lbl2) +{ + P sw(boost(P(my_x1,-1.5), t)), se(boost(P(my_x2,-1.5), t)); + P nw(boost(P(my_x1, 2.5), t)), ne(boost(P(my_x2, 2.5), t)); + + line(sw, se); + line(sw, nw); + line(ne, se); + + label(sw, P(-4,0), math(lbl1), bl); + label(se, P( 2,0), math(lbl2), br); + + label(nw, P(-2,0), math(lbl1 + "'"), tl); + label(ne, P( 2,0), math(lbl2 + "'"), tr); +} + +P left_end(double t) +{ + Segment tg(boost(P(0, 0), t), boost(P(1, 0), t)); + Segment rd(boost(P(my_x1, 0), tau-t), boost(P(my_x1, 1), tau-t)); + + return tg*rd; +} + +P right_end(double t) +{ + Segment tg(boost(P(0, 0), t), boost(P(1, 0), t)); + Segment rd(boost(P(my_x2, 0), tau-t), boost(P(my_x2, 1), tau-t)); + + return tg*rd; +} + +int main() +{ + picture(P(-2.25,-0.2), P(2.25,3.2), "4.5 x 3.4in"); + + begin(); + + label(P(0.5*(xmin()+xmax()), ymin()), P(0,-8), "\\textsc{Fig.~7.}", b); + + screen scr1(P(-1,-1.5), P(2,2.5)), scr2(scr1); + + bold(); + font_size("footnotesize"); + + activate(scr1); + rod(0, "P", "Q"); + + dashed(); + time_grid(tau); + solid(); + + dot(left_end(tau), P(2,-1), "$P_1$", br); + dot(right_end(tau), P(2,-1), "$Q_1$", br); + + inset(P(-2.5,0), P(-0.25,3)); + deactivate(scr1); + + activate(scr2); + rod(tau, "R", "S"); + + dashed(); + time_grid(0); + + dot(left_end(0), P(0,-3), "$R_1$", br); + dot(right_end(0), P(0,-3), "$S_1$", br); + + inset(P(0.25,0), P(2.5,3)); + deactivate(scr2); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/065b.eepic b/29782-t/images/sources/065b.eepic new file mode 100644 index 0000000..715800a --- /dev/null +++ b/29782-t/images/sources/065b.eepic @@ -0,0 +1,161 @@ +%% Generated from 065b.xp on Fri Mar 13 15:42:21 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,1.5] x [0,1] +%% Actual size: 3 x 2in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3in,2in); +\pgftext[at={\pgfpoint{1.5in}{-0.221392in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~8.}}}} +\pgfsetlinewidth{1.5pt} +\draw (0.666667in,0in)--(0.666667in,1.33333in); +\pgftext[at={\pgfpoint{0.666667in}{1.38868in}}] 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+1,88 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double tau(0.4); +double MAX(3.5); + +P boost(const P& arg, double t) +{ + double CSH(cosh(t)), SNH(sinh(t)); + double x(arg.x1()), y(arg.x2()); + + return P(CSH*x + SNH*y, CSH*y + SNH*x); +} + +std::string math(const std::string& arg) +{ + return "$" + arg + "$"; +} + +const double mx(0.75); +void time_grid(double t, + const std::string& lbl1, + const std::string& lbl2) +{ + P sw(boost(P(-mx,0), t)), se(boost(P(mx,0), t)); + P we(boost(P(-mx,1), t)), ea(boost(P(mx,1), t)); + P nw(boost(P(-mx,2), t)), ne(boost(P(mx,2), t)); + + line(sw, se); + line(we, ea); + line(ne, nw); + + line(P(0,0), boost(P(0,2.75), t)); + + label(P(0,0), P(0,-2), math(lbl1), b); + // label(nw, P(-4,2), math(lbl2), l); + label(boost(P(0,2), t), P(2,-2), math(lbl2), br); +} + + +void cigar(double T, const std::string& lbl) +{ + P end(boost(P(0, 2), T)); + line(P(0,0), end); + label(end, P(0,4), math(lbl), t); +} + + +int main() +{ + picture(P(0,0), P(1.5, 1), "3 x 2in"); + + begin(); + + label(P(0.5*xmax(), ymin()), P(0,-16), "\\textsc{Fig.~8.}", b); + + screen scr1(P(-1,0), P(2,3)), scr2(P(-0.75,0), P(0.75,3)); + + bold(); + font_size("footnotesize"); + + activate(scr1); + pen(1.5); + cigar(0, "M"); + + bold(); + dashed(); + time_grid(tau, "L", "N"); + solid(); + + inset(P(0,0), P(1,1)); + deactivate(scr1); + + activate(scr2); + pen(1.5); + cigar(tau, "N'"); + bold(); + + dashed(); + time_grid(0, "L'", "M'"); + + inset(P(1,0), P(1.5,1)); + deactivate(scr2); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/068a.eepic b/29782-t/images/sources/068a.eepic new file mode 100644 index 0000000..31344b1 --- /dev/null +++ b/29782-t/images/sources/068a.eepic @@ -0,0 +1,860 @@ +%% Generated from 068a.xp on Wed Mar 4 11:33:59 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-5.5,4.5] x [-3,-0.5] +%% Actual size: 5 x 1in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% 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*/ +#include "epix.h" +using namespace ePiX; + +const double rad(0.8); +const double th(M_PI/6); + +const double ydim(0.5); +const double zmax(5); +const double dx(0.6); +const double dz(1.66); + +void clock(const P& ctr, double t) +{ + fill(White()); + circle(ctr, 1); + for (int i=0; i<12; ++i) + line(ctr+polar(rad, i*th), ctr+cis(i*th)); + + fill(Black()); + circle(ctr, 0.05); + fill(Neutral()); + + arrow(ctr, ctr + polar(0.9, t*th)); +} + +std::string to_str(int i) +{ + std::stringstream buf; + buf << i; + return buf.str(); +} + +int main() +{ + // picture(P(-5.5,-3.5), P(4.5,-0.5), "5 x 1.5in"); + picture(P(-5.5,-3), P(4.5,-0.5), "5 x 1in"); + + begin(); + camera.at(P(2, -4, 1.75)).look_at(P(1.75,0,0)).range(100).roll(-M_PI_2); + arrow_inset(-0.5); + + label(P(-1-dx, 0), P(0,-16), "\\textsc{Fig.~9.}", b); + + font_size("footnotesize"); + + // fill(Black()); + // rect(P(-1, 0, zmax - 6*dz), P(-1, ydim, zmax)); + // fill(Neutral()); + // mild hackery to avoid tikz's mitre joints on sharp angles + pen(2); + line(P(-1, 0, zmax - 6*dz), P(-1, ydim, zmax - 6*dz)); + line(P(-1, ydim, zmax - 6*dz), P(-1, ydim, zmax)); + line(P(-1, 0, zmax - 6*dz), P(-1, 0, zmax)); + bold(); + + rect(P(-1, 0, zmax - 6*dz), P(-1-dx, 0, zmax)); + rect(P(-1, ydim, zmax), P(-1-dx, 0, zmax)); + + for (int i=0; i<6; ++i) + { + double ht(zmax - i*dz); + clock(P(0,0, ht), i); + line(P(-1, 0, ht), P(-1 - 0.5*dx, 0, ht)); + if (i != 0) + label(P(-1 - 0.5*dx, 0, ht), P(-2,0), to_str(i), l); + } + + label(P(-1-0.5*dx,ydim,zmax), P(-4,0), "$A$", l); + label(P(-1-0.5*dx,0,zmax-6*dz), P(4,0), "$B$", r); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/088a.eepic b/29782-t/images/sources/088a.eepic new file mode 100644 index 0000000..64f93c1 --- /dev/null +++ b/29782-t/images/sources/088a.eepic @@ -0,0 +1,347 @@ +%% Generated from 088a.xp on Wed Mar 4 12:39:10 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,3.5] x [0,1] +%% Actual size: 4.5 x 1.286in +%% Figure offset: 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(2.61934in,0.333016in)--(2.6809in,0.281352in)--(2.74246in,0.229688in); +\draw (2.25in,0.643in)--(2.32551in,0.61551in)-- + (2.40102in,0.58802in)--(2.47653in,0.56053in)-- + (2.55204in,0.533041in)--(2.62756in,0.505551in)-- + (2.70307in,0.478061in)--(2.77858in,0.450571in)--(2.85409in,0.423081in); +\draw (2.89286in,0.0861457in)--(3.53571in,1.19985in); +\draw (2.89286in,0.0861457in)--(4.17857in,0.0861457in); +\draw (3.15in,0.0861457in)--(3.79286in,1.19985in); +\draw (3.02143in,0.308887in)--(4.30714in,0.308887in); +\draw (3.40714in,0.0861457in)--(4.05in,1.19985in); +\draw (3.15in,0.531629in)--(4.43571in,0.531629in); +\draw (3.66429in,0.0861457in)--(4.30714in,1.19985in); +\draw (3.27857in,0.754371in)--(4.56429in,0.754371in); +\draw (3.92143in,0.0861457in)--(4.56429in,1.19985in); +\draw (3.40714in,0.977113in)--(4.69286in,0.977113in); +\draw (4.17857in,0.0861457in)--(4.82143in,1.19985in); +\draw (3.53571in,1.19985in)--(4.82143in,1.19985in); +\pgftext[at={\pgfpoint{0.642857in}{-0.110696in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~10}}}} +\pgftext[at={\pgfpoint{2.25in}{-0.110696in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~11}}}} +\pgftext[at={\pgfpoint{3.69643in}{-0.110696in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~12}}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/088a.xp b/29782-t/images/sources/088a.xp new file mode 100644 index 0000000..f40a923 --- /dev/null +++ b/29782-t/images/sources/088a.xp @@ -0,0 +1,46 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P arg1(0.4,0), arg2(polar(0.4, M_PI/3)), sw(-2.5*(arg1+arg2)); + +void my_grid() +{ + for (int i=0; i <= 5; ++i) + { + line(sw + i*arg1, sw + i*arg1 + 5*arg2); + line(sw + i*arg2, sw + i*arg2 + 5*arg1); + } +} + +int main() +{ + picture(P(0,0), P(3.5,1), "4.5 x 1.286in"); + + begin(); + + screen scr1(P(-1,-1), P(1,1)), scr2(scr1), scr3(scr1); + + bold(); + activate(scr1); + grid(5,5); + inset(P(0,0), P(1,1)); + deactivate(scr1); + + activate(scr2); + polar_grid(1, 4,18); + inset(P(1.25,0), P(2.25,1)); + deactivate(scr2); + + activate(scr3); + my_grid(); + inset(P(2.5,0), P(3.5,1)); + deactivate(scr3); + + label(P(0.5,0), P(0,-8), "\\textsc{Fig.~10}", b); + label(P(1.75,0), P(0,-8), "\\textsc{Fig.~11}", b); + label(P(2.875,0), P(0,-8), "\\textsc{Fig.~12}", b); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/104a.eepic b/29782-t/images/sources/104a.eepic new file mode 100644 index 0000000..2c3de6c --- /dev/null +++ b/29782-t/images/sources/104a.eepic @@ -0,0 +1,196 @@ +%% Generated from 104a.xp on Wed Mar 4 12:39:39 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-1,1] x [-1,1] +%% Actual size: 3.5 x 3.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.5in,3.5in); +\pgftext[at={\pgfpoint{1.75in}{-0.110696in}}] 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(2.09952in,1.73168in)--(2.1in,1.75in)--cycle; +\draw (2.28868in,3.23002in)--(2.21424in,3.25503in)-- + (2.13864in,3.2763in)--(2.06208in,3.29377in)-- + (1.98474in,3.30741in)--(1.90682in,3.31717in)-- + (1.82851in,3.32304in)--(1.75in,3.325in)--(1.67149in,3.32304in)-- + (1.59318in,3.31717in)--(1.51526in,3.30741in)-- + (1.43792in,3.29377in)--(1.36136in,3.2763in)-- + (1.28576in,3.25503in)--(1.21132in,3.23002in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](1.25231in,3.24379in)--(1.28228in,3.29035in)-- + (1.21132in,3.23002in)--(1.30431in,3.22477in)--(1.25231in,3.24379in)--cycle; +\draw (2.79772in,2.15074in)--(2.77783in,2.14415in); +\draw [fill](2.81887in,2.15775in)--(2.84903in,2.20419in)-- + (2.77783in,2.14415in)--(2.8708in,2.13852in)--(2.81887in,2.15775in)--cycle; +\draw (2.91639in,1.8184in)--(2.93699in,1.8214in); +\draw [fill](2.8942in,1.81517in)--(2.85641in,1.7747in)-- + (2.93699in,1.8214in)--(2.84643in,1.84316in)--(2.8942in,1.81517in)--cycle; +\pgftext[at={\pgfpoint{1.40512in}{1.57441in}}] 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+\draw (2.78242in,2.14567in)--(2.82067in,2.15835in); +\draw (2.89718in,2.18371in)--(2.93543in,2.19639in); +\draw (2.93543in,2.19639in)--(2.97368in,2.20907in); +\draw (3.05019in,2.23442in)--(3.08844in,2.2471in); +\draw (3.08844in,2.2471in)--(3.12669in,2.25978in); +\draw (3.2032in,2.28514in)--(3.24145in,2.29782in); +\draw (3.24145in,2.29782in)--(3.2797in,2.3105in); +\draw (3.35621in,2.33586in)--(3.39446in,2.34854in); +\end{tikzpicture} diff --git a/29782-t/images/sources/104a.xp b/29782-t/images/sources/104a.xp new file mode 100644 index 0000000..a2a9e35 --- /dev/null +++ b/29782-t/images/sources/104a.xp @@ -0,0 +1,51 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double rad(0.2); // inner circle + +P midpt(const P& tail, const P& head, double t) +{ + return tail + t*(head - tail); +} + +int main() +{ + picture(P(-1,-1), P(1,1), "3.5 x 3.5in"); + + begin(); + degrees(); + + label(P(0, ymin()), P(0,-8), "\\textsc{Fig.~13.}", b); + + arrow_inset(0.5); + arrow_width(5); + arrow_ratio(2.5); + + P Sun(polar(rad, 205)), Sun2(polar(rad, 190)); + P Jp(cis(20)), Jp2(cis(5)); + + font_size("footnotesize"); + + bold(); + + circle(); + circle(P(0,0), rad); + + arc_arrow(P(0,0), 0.9, 70, 110); + arrow(midpt(Jp, Sun2, 0.3), midpt(Jp, Sun2, 0.31)); + arrow(midpt(Sun, Jp2, 0.72), midpt(Sun, Jp2, 0.73)); + + dot(Sun, P(-2,-2), "$S$", bl); + dot(Sun2, P(-2, 0), "$S'$", l); + + dot(Jp, P(2,2), "$J$", tr); + dot(Jp2, P(2,0), "$J'$", r); + + dashed(); + line(Sun, Jp2); + line(Sun2, Jp); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/110a.eepic b/29782-t/images/sources/110a.eepic new file mode 100644 index 0000000..925bf99 --- /dev/null +++ b/29782-t/images/sources/110a.eepic @@ -0,0 +1,223 @@ +%% Generated from 110a.xp on Wed Mar 18 21:23:51 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-0.625,1.375] x [-1,1] +%% Actual size: 3.5 x 3.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% 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(1.07448in,2.24277in)--(1.06719in,2.26156in); +\draw (1.05261in,2.29912in)--(1.04531in,2.3179in); +\draw (1.04531in,2.3179in)--(1.03802in,2.33668in); +\draw (1.02344in,2.37425in)--(1.01615in,2.39303in); +\draw (1.01615in,2.39303in)--(1.00886in,2.41181in); +\draw (0.994283in,2.44937in)--(0.986992in,2.46816in); +\draw (0.986992in,2.46816in)--(0.979702in,2.48694in); +\draw (0.965121in,2.5245in)--(0.957831in,2.54328in); +\pgftext[at={\pgfpoint{1.68538in}{0.74022in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $P$}}} +\pgftext[at={\pgfpoint{0.985505in}{2.54328in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $Q$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/110a.xp b/29782-t/images/sources/110a.xp new file mode 100644 index 0000000..9f2918d --- /dev/null +++ b/29782-t/images/sources/110a.xp @@ -0,0 +1,51 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; +const double C1(cosh(1)); + +P f(double x, double y) +{ + y *= 0.75; + return polar(pow(Sec(y), 3)*cosh(x), y) - P(C1,0); +} + +P f(const P& arg) { return f(arg.x1(), arg.x2()); } + +/* +// inverse map +P g(double u, double v) +{ + double y(Atan2(v, u+C1)); + return P(acosh(sqrt((u+C1)*(u+C1) + v*v)*pow(Cos(y), 3)), y/0.75); +} + +P g(const P& arg) { return g(arg.x1(), arg.x2()); } +*/ + +domain R(P(0.5, -0.5), P(1.5,0.5), mesh(4, 5), mesh(4, 60)); + +int main() +{ + picture(P(-0.625,-1), P(1.375,1), "3.5 x 3.5in"); + + begin(); + + label(P(0.5*(xmin()+xmax()), ymin()), P(0,-8), "\\textsc{Fig.~14.}", b); + + P ptP(1.125, -0.4), ptQ(0.8, 0.4); + + bold(); + + plot(f, R); + + dashed(); + dash_size(6); + line(f(ptP), f(ptQ)); + + font_size("footnotesize"); + label(f(ptP), P(2,0), "$P$", r); + label(f(ptQ), P(2,0), "$Q$", r); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/110a_new.xp b/29782-t/images/sources/110a_new.xp new file mode 100644 index 0000000..f1da6d9 --- /dev/null +++ b/29782-t/images/sources/110a_new.xp @@ -0,0 +1,52 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; +const double C1(cosh(1)); + +const double mass(0.1); + +P f(double x, double y) +{ + return P(x + 0.5*mass*pow(y/x, 2), y*(1 - mass/x)); +} + +P f(const P& arg) { return f(arg.x1(), arg.x2()); } + +/* +// inverse map +P g(double u, double v) +{ + double y(Atan2(v, u+C1)); + return P(acosh(sqrt((u+C1)*(u+C1) + v*v)*pow(Cos(y), 3)), y/0.75); +} + +P g(const P& arg) { return g(arg.x1(), arg.x2()); } +*/ + +domain R(P(0.5, -2), P(2,2), mesh(4, 5), mesh(4, 60)); + +int main() +{ + picture(P(0,-3), P(6,3), "3.5 x 3.5in"); + + begin(); + + label(P(0.5*(xmin()+xmax()), ymin()), P(0,-8), "\\textsc{Fig.~14.}", b); + + P ptP(1.125, -0.4), ptQ(0.8, 0.4); + + bold(); + + plot(f, R); + + dashed(); + dash_size(6); + line(f(ptP), f(ptQ)); + + font_size("footnotesize"); + label(f(ptP), P(2,0), "$P$", r); + label(f(ptQ), P(2,0), "$Q$", r); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/111a.eepic b/29782-t/images/sources/111a.eepic new file mode 100644 index 0000000..e46ce58 --- /dev/null +++ b/29782-t/images/sources/111a.eepic @@ -0,0 +1,74 @@ +%% Generated from 111a.xp on Wed Mar 4 13:04:44 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0.5,1.5] x [-0.5,0.5] +%% Actual size: 3 x 3in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3in,3in); +\pgftext[at={\pgfpoint{1.5in}{-0.110696in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~15.}}}} +\pgfsetlinewidth{0.8pt} +\draw (0in,0in)--(3in,0in)--(3in,3in)--(0in,3in)--(0in,0in)--cycle; +\draw (0.75in,0in)--(0.75in,3in); +\draw (1.5in,0in)--(1.5in,3in); +\draw (2.25in,0in)--(2.25in,3in); +\draw (0in,0.6in)--(3in,0.6in); +\draw (0in,1.2in)--(3in,1.2in); +\draw (0in,1.8in)--(3in,1.8in); +\draw (0in,2.4in)--(3in,2.4in); +\draw 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(1.25865in,2.32643in)--(1.24399in,2.34403in); +\draw (1.24399in,2.34403in)--(1.22933in,2.36163in); +\draw (1.20001in,2.39684in)--(1.18535in,2.41444in); +\draw (1.18535in,2.41444in)--(1.16867in,2.43222in); +\draw (1.13531in,2.46778in)--(1.11863in,2.48556in); +\draw (1.11863in,2.48556in)--(1.10195in,2.50334in); +\draw (1.06859in,2.5389in)--(1.0519in,2.55668in); +\draw (1.0519in,2.55668in)--(1.03292in,2.57459in); +\draw (0.994941in,2.61042in)--(0.975952in,2.62834in); +\draw (0.975952in,2.62834in)--(0.956964in,2.64625in); +\draw (0.918988in,2.68208in)--(0.9in,2.7in); +\pgftext[at={\pgfpoint{1.90267in}{0.3in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $P$}}} +\pgftext[at={\pgfpoint{0.927674in}{2.7in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $Q$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/111a.xp b/29782-t/images/sources/111a.xp new file mode 100644 index 0000000..6ec89ae --- /dev/null +++ b/29782-t/images/sources/111a.xp @@ -0,0 +1,54 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; +const double C1(cosh(1)); + +P f(double x, double y) +{ + y *= 0.75; + return polar(pow(Sec(y), 3)*cosh(x), y) - P(C1,0); +} + +P f(const P& arg) { return f(arg.x1(), arg.x2()); } + +// inverse map +P g(double u, double v) +{ + double y(Atan2(v, u+C1)); + return P(acosh(sqrt((u+C1)*(u+C1) + v*v)*pow(Cos(y), 3)), y/0.75); +} + +P g(const P& arg) { return g(arg.x1(), arg.x2()); } + +domain R(P(0.5, -0.5), P(1.5,0.5), mesh(4, 5), mesh(4, 60)); + +P ptP(1.125, -0.4), ptQ(0.8, 0.4); + +P cv(double t) +{ + P tl(f(ptP)), hd(f(ptQ)); + return g(tl + t*(hd - tl)); +} + +int main() +{ + picture(P(0.5,-0.5), P(1.5,0.5), "3 x 3in"); + + begin(); + + label(P(0.5*(xmin()+xmax()), ymin()), P(0,-8), "\\textsc{Fig.~15.}", b); + + bold(); + + grid(4, 5); + + dashed(); + plot(cv, 0, 1, 20); + + font_size("footnotesize"); + label(ptP, P(2,0), "$P$", r); + label(ptQ, P(2,0), "$Q$", tr); + + tikz_format(); + end(); +} diff --git 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+\pgftext[at={\pgfpoint{1.125in}{0.627674in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $F$}}} +\pgftext[at={\pgfpoint{4.5in}{0.627674in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $P'$}}} +\pgftext[at={\pgfpoint{2.25in}{0.459129in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $Q$}}} +\pgftext[at={\pgfpoint{4.5in}{0.177674in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $P$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/122a.xp b/29782-t/images/sources/122a.xp new file mode 100644 index 0000000..4aadc01 --- /dev/null +++ b/29782-t/images/sources/122a.xp @@ -0,0 +1,56 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double ht(0.6); +const double k(5); // tightness of bending + +P E(0, ht), ptP2(4.5, ht), Sun(1.5, 0.3); + +double f(double t) +{ + t -= Sun.x1(); + return log(1 + exp(k*t))/k; +} + +P straight(double t) +{ + return E + t*(ptP2 - E); +} + +P bent(double t) +{ + P tmp(straight(t)); + double tau(tmp.x1()); + return tmp - P(0, 0.15*f(tau)); +} + +int main() +{ + picture(P(0,0), P(4.5,1), "4.5 x 1in"); + + begin(); + + label(P(0.5*(xmin()+xmax()), ymin()), P(0,-8), "\\textsc{Fig.~16.}", b); + + bold(); + circle(Sun, 0.125); + plot(bent, 0, 1, 60); + + dashed(); + line(E, ptP2); + + font_size("footnotesize"); + label(Sun, "$S$"); + + dot(E, P(2,2), "$E$", tr); + label(straight(0.25), P(0,2), "$F$", t); + + label(straight(1), P(0,2), "$P'$", t); + + label(bent(0.5), P(0,-2), "$Q$", b); + label(bent(1), P(0,2), "$P$", t); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/129a.eepic b/29782-t/images/sources/129a.eepic new file mode 100644 index 0000000..e982ee9 --- /dev/null +++ b/29782-t/images/sources/129a.eepic @@ -0,0 +1,128 @@ +%% Generated from 119a.xp on Fri Mar 13 18:22:01 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 4 x 3.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,3.5in); +\pgftext[at={\pgfpoint{2in}{-0.442784in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~17.}}}} +\pgfsetlinewidth{0.8pt} +\draw (-0.147136in,0in)--(4.24714in,0in); +\draw (0in,0in)--(0in,3.85in); +\draw (0in,0in)--(0.0293528in,0.0281406in); +\draw (0.0880583in,0.0844218in)--(0.117411in,0.112562in); +\draw (0.117411in,0.112562in)--(0.146764in,0.140703in); +\draw (0.205469in,0.196984in)--(0.234822in,0.225125in); +\draw (0.234822in,0.225125in)--(0.264175in,0.253265in); +\draw (0.32288in,0.309547in)--(0.352233in,0.337687in); +\draw (0.352233in,0.337687in)--(0.381586in,0.365828in); +\draw (0.440291in,0.422109in)--(0.469644in,0.45025in); +\draw (0.469644in,0.45025in)--(0.498997in,0.47839in); +\draw (0.557702in,0.534671in)--(0.587055in,0.562812in); +\draw (0.587055in,0.562812in)--(0.616408in,0.590953in); +\draw 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{\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{0.35in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{0.7in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{1.05in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{1.4in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{1.75in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{2.1in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{2.45in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{2.8in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$\rule{10pt}{0.5pt}$}}} +\pgftext[at={\pgfpoint{0in}{3.15in}}] 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$\rule{0.5pt}{10pt}$}}} +\pgftext[at={\pgfpoint{3.34495in}{0in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $\rule{0.5pt}{10pt}$}}} +\pgftext[at={\pgfpoint{4in}{0in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $\rule{0.5pt}{10pt}$}}} +\pgftext[at={\pgfpoint{1.12892in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $90'$}}} +\pgftext[at={\pgfpoint{1.67247in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $60'$}}} +\pgftext[at={\pgfpoint{1.99303in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $50'$}}} +\pgftext[at={\pgfpoint{2.50871in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $40'$}}} +\pgftext[at={\pgfpoint{3.34495in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $30'$}}} +\pgftext[at={\pgfpoint{4in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $25'$}}} +\filldraw[color=rgb_000000] (1.17073in,0.710728in) circle(0.0207555in); +\filldraw[color=rgb_000000] (1.22648in,1.12644in) circle(0.0207555in); +\filldraw[color=rgb_000000] (1.46341in,1.95785in) circle(0.0207555in); +\filldraw[color=rgb_000000] (1.93728in,1.90421in) circle(0.0207555in); +\filldraw[color=rgb_000000] (2.74564in,2.93678in) circle(0.0207555in); +\filldraw[color=rgb_000000] (3.10801in,3.40613in) circle(0.0207555in); +\filldraw[color=rgb_000000] (3.23345in,3.56705in) circle(0.0207555in); +\pgftext[at={\pgfpoint{0in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize Distance}}} +\pgftext[at={\pgfpoint{0in}{3.90535in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize Displacement}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/129a.xp b/29782-t/images/sources/129a.xp new file mode 100644 index 0000000..753f918 --- /dev/null +++ b/29782-t/images/sources/129a.xp @@ -0,0 +1,107 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P to_pic(double x, double y) +{ + return P((x-62)/287, (330-y)/261); +} + +void star(double x, double y) +{ + dot(to_pic(x, y)); +} + +// ticks et al. at Cartesian location +void htick(const P& loc) +{ + h_axis_tick(loc, b); +} + +void vtick(const P& loc) +{ + v_axis_tick(loc, l); +} + +void hlab(const P& loc, const std::string& lbl) +{ + label(loc, P(0,-12), "$"+lbl+"'$", b); +} + +void vlab(const P& loc, const std::string& lbl) +{ + label(loc, P(-12,0), "$"+lbl+"$", l); +} + +int main() +{ + picture(P(0,0), P(1, 1), "4 x 3.5in"); + + begin(); + label(P(0.5*(xmin()+xmax()), 0), P(0,-32), "\\textsc{Fig.~17.}", b); + + tick_size(5); + + bold(); + // h_axis ticks + P pt90(to_pic(143, 330)); + P pt80(to_pic(153, 330)); + P pt70(to_pic(166, 330)); + P pt60(to_pic(182, 330)); + P pt50(to_pic(205, 330)); + P pt40(to_pic(242, 330)); + P pt30(to_pic(302, 330)); + P pt25(to_pic(349, 330)); + + line(P(0,0), P(1.025*xmax(),0), 10); + line(P(0,0), P(0,1.1*ymax())); + + dashed(); + line(to_pic(62, 330), to_pic(340, 53)); + + for (int i=0; i<=10; ++i) + vtick(P(0,0.1*i)); + + font_size("footnotesize"); + vlab(P(0,0.1), ".01"); + vlab(P(0,0.2), ".02"); + vlab(P(0,0.3), ".03"); + vlab(P(0,0.4), ".04"); + vlab(P(0,0.5), ".05"); + vlab(P(0,0.6), ".06"); + vlab(P(0,0.7), ".07"); + vlab(P(0,0.8), ".08"); + vlab(P(0,0.9), ".09"); + vlab(P(0,1), "1''.00"); + + htick(pt90); + htick(pt80); + htick(pt70); + htick(pt60); + htick(pt50); + htick(pt40); + htick(pt30); + htick(pt25); + + hlab(pt90, "90"); + hlab(pt60, "60"); + hlab(pt50, "50"); + hlab(pt40, "40"); + hlab(pt30, "30"); + hlab(pt25, "25"); + + // stars + star(146, 277); + star(150, 246); + star(167, 184); + star(201, 188); + star(259, 111); + star(285, 76); + star(294, 64); + + label(P(0,0), P(0,-12), "Distance", b); + label(P(0,1.1*ymax()), P(0,4), "Displacement", t); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/130a.eepic b/29782-t/images/sources/130a.eepic new file mode 100644 index 0000000..4f82d9f --- /dev/null +++ b/29782-t/images/sources/130a.eepic @@ -0,0 +1,113 @@ +%% Generated from 120a.xp on Fri Mar 13 19:35:18 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 5.5 x 5.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (5.5in,5.5in); +\pgftext[at={\pgfpoint{2.75in}{-0.332088in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~18.}}}} +\pgfsetlinewidth{0.8pt} +\draw (3.01847in,0.0144737in)--(5.47098in,2.40263in)-- + (2.48153in,5.48553in)--(0in,3.11184in)--(3.01847in,0.0144737in)--cycle; +\draw (5.36939in,0.390789in)--(5.34037in,4.92105in)-- + (0.841689in,4.92105in)--(0.841689in,0.390789in)--(5.36939in,0.390789in)--cycle; +\draw (2.80102in,2.27237in)--(2.80041in,2.29592in)-- + (2.79856in,2.31942in)--(2.79548in,2.34278in)-- + (2.79119in,2.36595in)--(2.78569in,2.38886in)-- + 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double y, + const P& offset, const std::string& lbl, epix_label_posn A) +{ + dot_size(5); + dot(to_pic(x, y), offset, lbl, A); +} + +void star3(double x, double y, + const P& offset, const std::string& lbl, epix_label_posn A) +{ + dot_size(4); + ddot(to_pic(x, y), offset, lbl, A); +} + +int main() +{ + picture(P(0,0), P(1,1), "5.5 x 5.5in"); + + begin(); + + bold(); + tick_size(5); + + label(P(0.5*(xmin() + xmax()), 0), P(0,-24), "\\textsc{Fig.~18.}", b); + + // sun locations + P pp(to_pic(187, 242)), ps(to_pic(198, 247)); + + // cardinal points + P ptE(to_pic( 44,220)), ptS(to_pic(246,413)); + P ptW(to_pic(414,214)), ptN(to_pic(226, 26)); + + quad(to_pic(238, 402), to_pic(407, 237), to_pic(201,24), to_pic(30,188)); + quad(to_pic(400, 376), to_pic(398,63), to_pic(88,63), to_pic(88,376)); + + // circle(to_pic(192, 246), 0.075); + P ctr(to_pic(192, 246)); + circle(ctr, norm(ctr-to_pic(223, 245))); + line(pp, ps); + + font_size("footnotesize"); + label(pp, P(-2,0), "P", l); + label(ps, P( 2,0), "S", r); + + star3(238, 241, P(3,0), "$1$", r); + star3(135, 254, P(3,0), "$2$", r); + star2(206, 186, P(4,0), "$3$", r); + star1(207, 175, P(6,0), "$4$", r); + star2(256, 329, P(4,0), "$5$", r); + star1(182, 114, P(6,0), "$6$", r); + + star3(310, 305, P(3,0), "$7$", r); + star3(312, 323, P(-3,0), "$8$", l); + star3(288, 348, P(3,-3), "$9$", br); + star2(154, 92, P(4,0), "$10$", r); + star2(364, 236, P(4,0), "$11$", r); + star2(369, 310, P(4,0), "$12$", r); + star3(213, 122, P(3,0), "$13$", r); + + h_axis_tick(ptN, b); + label(ptN, P(0,2), "N", t); + + h_axis_tick(ptS, t); + label(ptS, P(0,-2), "S", b); + + v_axis_tick(ptE, r); + label(ptE, P(-2,0), "E", l); + + v_axis_tick(ptW, l); + label(ptW, P(2,0), "W", r); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/152a.eepic b/29782-t/images/sources/152a.eepic new file mode 100644 index 0000000..3894b67 --- /dev/null +++ b/29782-t/images/sources/152a.eepic @@ -0,0 +1,224 @@ +%% Generated from 152a.xp on Fri Mar 13 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+\pgftext[at={\pgfpoint{-0.027674in}{3in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $T$}}} +\pgftext[at={\pgfpoint{0.126446in}{0.574305in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $A$}}} +\pgftext[at={\pgfpoint{0.2in}{1.11858in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}\footnotesize $B$}}} +\pgftext[at={\pgfpoint{0.904325in}{1.12054in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}\footnotesize $B'$}}} +\pgftext[at={\pgfpoint{1.5in}{2.75495in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $C$}}} +\pgftext[at={\pgfpoint{2.5in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $X$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/152a.xp b/29782-t/images/sources/152a.xp new file mode 100644 index 0000000..14f991a --- /dev/null +++ b/29782-t/images/sources/152a.xp @@ -0,0 +1,70 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double tau(0.4); +const double MAX(5.5); +const double M(3); +double m11(0.2), m12((M-2*m11)/3); // original momenta +double m21(0.85), m22((M-2*m21)/3); // new momenta + +P boost(const P& arg, double t) +{ + double CSH(cosh(t)), SNH(sinh(t)); + double x(arg.x1()), y(arg.x2()); + + return P(CSH*x + SNH*y, CSH*y + SNH*x); +} + +void time_grid(double t) +{ + P sw(boost(P(0,0), t)), se(boost(P(5,0), t)); + P nw(boost(P(0,5), t)), ne(boost(P(5,5), t)); + P dy(0.2*(nw-sw)); + + for (int i=0; i<= 5; ++i) + line(sw + i*dy, se + i*dy); +} + +int main() +{ + picture(P(0,0), P(5,MAX), "2.5 x 3in"); + + begin(); + + P A1(m11, 1), M11(2*A1), M12(3*m12, 3), C1(M11 + M12), O(0,0); + P A2(m21, 1), M21(2*A2), M22(3*m22, 3), C2(M21 + M22); + + label(P(0.5*xmax(), ymin()), P(0,-8), "\\textsc{Fig.~19.}", b); + + bold(); + line(O, P(xmax(),0)); + line(O, P(0,ymax())); + + line(O, M11); + line(O, M21); + line(M11, C1); + line(M21, C2); + + dashed(); + time_grid(0); + + font_size("footnotesize"); + label(O, P(-2,-2), "$O$", bl); + + label(P(0,1), P(-2,0), "$M$", l); + label(P(0,2), P(-2,0), "$N$", l); + label(P(0,5), P(-2,0), "$P$", l); + label(P(0,ymax()), P(-2,0), "$T$", l); + + label(A1, P(2,2), "$A$", tr); + label(M11, P(0,2), "$B$", tl); + label(M21, P(4,2), "$B'$", tr); + + label(C1, P(0,2), "$C$", t); + + label(P(xmax()), P(0,-4), "$X$", b); + + tikz_format(); + end(); +} diff --git a/29782-t/images/sources/153a.eepic b/29782-t/images/sources/153a.eepic new file mode 100644 index 0000000..062dc3d --- /dev/null +++ b/29782-t/images/sources/153a.eepic @@ -0,0 +1,249 @@ +%% Generated from 153a.xp on Fri Mar 13 16:35:09 EDT 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,5] x [0,5.5] +%% Actual size: 2.5 x 3in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture}[cap=round] +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (2.5in,3in); +\pgftext[at={\pgfpoint{1.25in}{-0.110696in}}] 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{\makebox(0,0)[tl]{\hbox{\color{rgb_000000}\footnotesize $B'$}}} +\pgftext[at={\pgfpoint{2.01955in}{3.37772in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $C$}}} +\pgftext[at={\pgfpoint{1.82572in}{3.30243in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $C'$}}} +\pgftext[at={\pgfpoint{2.70268in}{1.06489in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $X_1$}}} +\end{tikzpicture} diff --git a/29782-t/images/sources/153a.xp b/29782-t/images/sources/153a.xp new file mode 100644 index 0000000..05b887d --- /dev/null +++ b/29782-t/images/sources/153a.xp @@ -0,0 +1,82 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double tau(0.4); +const double MAX(5.5); +const double M(3); +double m11(0.2), m12((M-2*m11)/3); // original momenta +double m21(0.85), m22((M-2*m21)/3); // new momenta + +P boost(const P& arg, double t) +{ + double CSH(cosh(t)), SNH(sinh(t)); + double x(arg.x1()), y(arg.x2()); + + return P(CSH*x + SNH*y, CSH*y + SNH*x); +} + +void time_grid(double t) +{ + P sw(boost(P(0,0), t)), se(boost(P(5,0), t)); + P nw(boost(P(0,5), t)), ne(boost(P(5,5), t)); + P dy(0.2*(nw-sw) - 0.075*(se - sw)); + + for (int i=0; i<= 5; ++i) + line(sw + i*dy, se + i*dy); +} + +int main() +{ + picture(P(0,0), P(5,MAX), "2.5 x 3in"); + + begin(); + + Segment time2(boost(P(0,2), tau), boost(P(5,2), tau)); + Segment time3(boost(P(0,3), tau), boost(P(5,3), tau)); + Segment time5(boost(P(0,5), tau), boost(P(5,5), tau)); + + // particle directions + P A1(m11, 1), B1(m12, 1), A2(m21, 1), B2(m22, 1), O(0,0); + + P M11(Segment(O, A1)*time2), M12(Segment(O, B1)*time3); + + // P M21(Segment(O, A2)*time2), M22(Segment(O, B2)*time3); + P M21(Segment(O, A2)*time2), M22(Segment(O, B2)*time3); + + P C1(M11 + M12), C2(Segment(M21, M21+B2)*time5); + + label(P(0.5*xmax(), ymin()), P(0,-8), "\\textsc{Fig.~20.}", b); + + bold(); + line(O, boost(P(xmax(),0), tau)); + line(O, boost(P(0,ymax()), tau)); + + line(O, M11); + line(O, M21); + line(M11, C1); + line(M21, C2); + + dashed(); + time_grid(tau); + + font_size("footnotesize"); + label(O, P(-2,-2), "$O$", bl); + + label(boost(P(0,1), tau), P(0,-2), "$M$", br); + label(boost(P(0,2), tau), P(0,-2), "$N$", br); + label(boost(P(0,5), tau), P(-2,0), "$P$", tl); + label(boost(P(0,ymax()), tau), P(-2,0), "$T_1$", tl); + + label( A1, P(-2,2), "$A$", tl); + label(M11, P(0, 2), "$B$", tl); + label(M21, P(0,-2), "$B'$", br); + + label(C1, P( 2,2), "$C$", t); + label(C2, P(-4,0), "$C'$", t); + + label(boost(P(xmax()), tau), P(0,-4), "$X_1$", b); + + tikz_format(); + end(); +} diff --git a/29782-t/old/29782-t.tex b/29782-t/old/29782-t.tex new file mode 100644 index 0000000..c6abb44 --- /dev/null +++ b/29782-t/old/29782-t.tex @@ -0,0 +1,12619 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% Project Gutenberg's Space, Time and Gravitation, by A. S. Eddington % +% % +% This eBook is for the use of anyone anywhere 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 www.gutenberg.org % +% % +% % +% Title: Space, Time and Gravitation % +% An Outline of the General Relativity Theory % +% % +% Author: A. S. Eddington % +% % +% Release Date: August 24, 2009 [EBook #29782] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK SPACE, TIME AND GRAVITATION *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{29782} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% textcomp: Better ditto marks. Optional. %% +%% fix-cm: For larger title page fonts. Optional. %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% array: Enhanced tabular features. Required. %% +%% %% +%% footmisc: Extended footnote capabilities. Required. %% +%% multicol: Multi-column environment for index. Required. %% +%% makeidx: Indexing capabilities. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% wrapfig: Illustrations surrounded by text. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% Minor spelling/punctuation changes, etc. are [** PP: noted] %% +%% in this file. On occasions where a word is hyphenated across a %% +%% line in the original and the word occurs only once, comparison %% +%% with similar words was made to decide whether to hyphenate. %% +%% Such instances are noted. %% +%% %% +%% The following spellings are retained: %% +%% debateable, idiosyncracies, and unbiassed. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behaviors may be controlled by boolean flags. %% +%% %% +%% ForPrinting (true by default): %% +%% Compile a print-optimized PDF file. Set to false for screen- %% +%% optimized file (pages cropped, one-sided, blue hyperlinks). %% +%% %% +%% IndexExtras (false by default): %% +%% Add a few entries to the index, e.g. Geometry, Euclidean. %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% Aviator's time table (PDF p. 22) is not separated from the %% +%% preceding sentence or broken across pages %% +%% %% +%% Wrapped images do not fall near a page bottom, lest indented %% +%% lines continue conspicuously past the break %% +%% Fig. 1, p. 15; Fig. 2, p. 42; Fig. 8, p. 50 %% +%% %% +%% %% +%% Spellcheck: .................................. OK %% +%% Smoothreading pool: ......................... yes %% +%% %% +%% lacheck: ..................................... 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THREE times %% +%% makeindex 29782-t.idx %% +%% pdflatex 29782-t.tex %% +%% %% +%% pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt]{book}[2005/09/16] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[latin1]{inputenc}[2006/05/05] %% DP standard encoding + +\IfFileExists{textcomp.sty}{% %% For ditto marks +\usepackage{textcomp}[2005/09/27]% +}{} + +\newlength{\MySkip} +\IfFileExists{fix-cm.sty}{% %% For larger title page fonts +\usepackage{fix-cm}[2006/03/24]% +\newcommand{\MyHuge}{\fontsize{38}{48}\selectfont}% +\setlength{\MySkip}{0.375in}}% else +{\newcommand{\MyHuge}{\Huge}% +\setlength{\MySkip}{0.25in}} + + +\usepackage{ifthen}[2001/05/26] %% Logical conditionals + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2002/01/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\usepackage{array}[2005/08/23] %% extended array/tabular features + + %% extended footnote capabilities +\usepackage[symbol,perpage]{footmisc}[2005/03/17] + +\usepackage{multicol}[2006/05/18] +\usepackage{makeidx}[2000/03/29] + +\usepackage{graphicx}[1999/02/16]%% For diagrams +\usepackage{wrapfig}[2003/01/31] %% and wrapping text around them + + +% for running heads; no package date available +\usepackage{fancyhdr} +\renewcommand{\headrulewidth}{0pt} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Conditional compilation switches %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Sets up a handful of additional index entries +\newboolean{IndexExtras} +% UNCOMMENT the next line for extra entries +%\setboolean{IndexExtras}{true} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ForPrinting=true (default) false +% Letterpaper Cropped pages +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +\newboolean{ForPrinting} + +%% COMMENT the next line for a SCREEN-OPTIMIZED VERSION of the text %% +\setboolean{ForPrinting}{true} + +%% Initialize values to ForPrinting=false +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon} +{ + Figures may have been moved with respect to the surrounding text. + Minor typographical corrections and presentational changes have + been made without comment. +} + +\newcommand{\TransNoteText} +{ + \TransNoteCommon + + This PDF file is formatted for screen viewing, but may be easily + formatted for printing. Please consult the preamble of the \LaTeX\ + source file for instructions. +} + +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is formatted for printing, but may be easily formatted + for screen viewing. Please see the preamble of the \LaTeX\ source + file for instructions. + } +}{} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Text block size carefully chosen to accommodate tall illos +\usepackage[body={5.2in,8.125in},\Margins]{geometry}[2002/07/08] + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: Space, Time and Gravitation}, + pdfauthor={Arthur Stanley Eddington}, + pdfkeywords={David Clarke, Andrew D. Hwang, + Project Gutenberg Online Distributed Proofreading Team, + The Internet Archive/American Libraries}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + +%%%% Re-crop screen-formatted version, omit blank verso pages %%%% +\ifthenelse{\boolean{ForPrinting}} + {} + {\hypersetup{pdfpagescrop = 100 90 512 780} + % If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COMMANDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% For various ad hoc alignment needs +\newlength{\TmpLen} + +%%%% Fixed-width environment to format PG boilerplate %%%% +% 9.2pt leaves no overfull hbox at 80 char line width +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{9.2}{10.5}\ttfamily\selectfont}% +{\end{alltt}} + +% Extra index entries, e.g., ``Geometry, Euclidean'' +\newcommand{\IndexExtra}[1]% +{% + \ifthenelse{\boolean{IndexExtras}}% + {\index{#1}}% + {}% +} + +% Copyright page formatting +\newcommand{\Publine}[2]{\makebox[\TmpLen][s]{\textsc{#1} : \textsc{#2}}} + + +% Cross-referencing: anchors +\newcommand{\Pagelabel}[1] + {\phantomsection\label{page:#1}} + +\newcommand{\Figlabel}[1] + {\phantomsection\label{fig:#1}} + +\newcommand{\Tag}[1]% + {\tag{#1}\phantomsection\label{eqn:#1}} + +% and links +\newcommand{\Pageref}[1] + {\hyperref[page:#1]{p.~\pageref{page:#1}}} + +\newcommand{\Figref}[1]% + {\hyperref[fig:#1]{\texorpdfstring{Fig.~#1}{Fig #1}}} + +\newcommand{\Noteref}[1]% + {\hyperref[appnote:#1]{\texorpdfstring{Note~#1}{Note #1}}} + + +% \Eqref{equation}{5}, \Eqref{formula}{2}, etc. +\newcommand{\Eqref}[2]{\hyperref[eqn:#2]{#1~(#2)}} + +\newcommand{\Chapref}[1]{\hyperref[chapter:#1]% + {Chapter~\textsc{\MakeLowercase{#1}}}} + + +%%%% Table of contents %%%% +% Dot leader for chapter-level entries +\makeatletter +\renewcommand{\l@chapter}{\@dottedtocline{0}{0pt}{0pt}} +\makeatother + +% No page numbers +\newcommand{\TableofContents}{{\let\thepage=\empty\tableofcontents}} + +% Centered chapter-level titles +\newlength{\ToCBox} +\settowidth{\ToCBox}{\scshape chapter~viii}% Widest title + +\newcommand{\ToCCenter}[2]% +{\normalfont\scshape\null\hfill% + \makebox[\ToCBox][l]{\scshape #1~\MakeLowercase{#2}}\hfill} + +% The table of contents may be set across two pages; we'll ensure the +% word ``page'' is printed at the top of the page number column, +% centered on the first page (to match the scan) and right-justified +% on subsequent pages (better visual appearance). +% +% Each chapter-like unit (Prologue, Chapter, Appendix) puts a \ToCLine +% into the toc file. The Prologue command defines a \ToCAnchor macro, +% which expands to the current page of the table of contents. +% +% Subsequent units put down a label and get the \pageref. If this has +% changed, the word ``page'' is written at the right margin, and the +% \ToCAnchor command is updated. +\newcommand{\ToCLine}[2]% +{\label{#1-toc:#2}% +\ifthenelse{\equal{#1}{prologue}}% + {\ToCCenter{#1}{#2}\makebox[0pt][c]{\footnotesize page}}% else... + {\ifthenelse{\not\equal{\pageref{#1-toc:#2}}{\ToCAnchor}}% + {\renewcommand{\ToCAnchor}{\pageref{#1-toc:#2}}% + \ToCCenter{#1}{#2}\makebox[0pt][r]{\footnotesize page}}% else... + {\ToCCenter{#1}{#2}}}} + + +% [** PP: ``Frontispiece'' heading in ToC would overlap dot leaders] +% To prevent this, write the code below into the toc file: +% \newlength{\FPlen} +% \settowidth{\FPlen}{\small\textit{Frontis}} +% \makeatletter +% \renewcommand{\@pnumwidth}{\FPlen} +% \makeatother +\newcommand{\FrontispieceToCEntry} +{ +\addtocontents{toc}{\protect\newlength{\protect\FPlen}} + +\ifthenelse{\boolean{ForPrinting}} +{\addtocontents{toc}{\protect\settowidth{\protect\FPlen}% + {\protect\small\protect\textit{Frontis}}}} % About half the width +{\addtocontents{toc}{\protect\settowidth{\protect\FPlen}% + {\protect\small\protect\textit{Frontispie}}}} % A bit wider + +\addtocontents{toc}{\protect\makeatletter} +\addtocontents{toc}{\protect\renewcommand{\protect\@pnumwidth}{\protect\FPlen}} +\addtocontents{toc}{\protect\makeatother} + +% The actual contents +\ifthenelse{\boolean{ForPrinting}} +{\addtocontents{toc}% +{\protect\contentsline{chapter}% + {\hyperref[frontispiece]{\protect\scshape Eclipse Instruments at Sobral}}% + {\protect\textit{\protect\makebox[12pt][c]{\protect\small Frontispiece}}}{}}} +% else move ``frontispiece'' heading slightly farther to the left +{\addtocontents{toc}% +{\protect\contentsline{chapter}% + {\hyperref[frontispiece]{\protect\scshape Eclipse Instruments at Sobral}}% + {\protect\textit{\protect\makebox[36pt][c]{\protect\small Frontispiece}}}{}}} + +% And restore \@pnumwidth +\addtocontents{toc}{\protect\settowidth{\protect\FPlen}{100}} +\addtocontents{toc}{\protect\makeatletter} +\addtocontents{toc}{\protect\renewcommand{\protect\@pnumwidth}{\protect\FPlen}} +\addtocontents{toc}{\protect\makeatother} +} + +% redefine hyperref's re-definition +% so that chapter anchor is above chapter title +\makeatletter +\AtBeginDocument{% in case hyperref clobbers this +\def\@schapter#1{% + \begingroup + \let\@mkboth\@gobbletwo + \Hy@GlobalStepCount\Hy@linkcounter + \xdef\@currentHref{\Hy@chapapp*.\the\Hy@linkcounter}% + \Hy@raisedlink{% + \hyper@anchorstart{\@currentHref}\hyper@anchorend + }% + \endgroup + \H@old@schapter{#1}% +}} +\makeatother + + + +% Sectioning: Paragraph, Chapter, Preface, Prologue, Appendix + +% Each chapter starts unindented, with smallcaps +\newcommand{\First}[1]{\noindent\textsc{#1}} + +% Appendix Notes are indented +\newcommand{\Indent}{\hspace*{\parindent}} + +\newcommand{\Paragraph}[1]{#1} % Null semantic markup + +\newcommand{\ChapterHead}[1]% +{\protect\centering\normalfont\upshape\textsc{\Large \MakeUppercase{#1}}} + +% \Chapter[title for ToC]{N}{Title} -- for numbered chapters +\newcommand{\Chapter}[3][]{% + % Clear stale heading on previous page + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RE]{}}{}% + + % page formatting + \chapter*{\ChapterHead{Chapter~#2}\\ % + \protect\centering\textsc{\large \MakeUppercase{#3}}} + + \label{chapter:#2} + + % ToC entry + \addtocontents{toc}{\protect\filbreak} + \addtocontents{toc}{\protect\ToCLine{chapter}{#2}} + + \ifthenelse{\equal{#1}{}} + {\addcontentsline{toc}{chapter} + {\texorpdfstring{\protect\scshape\protect{#3}}{#3}} + % Running heads + \fancyhead{} + \fancyhead[C]{\textsc{\MakeUppercase{#3}}}} + {\addcontentsline{toc}{chapter} + {\texorpdfstring{\protect\scshape\protect{#1}}{#1}} + \fancyhead{} + \fancyhead[C]{\textsc{\MakeUppercase{#1}}}} + + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage} + \fancyhead[RE]{\textsc{[ch.}} + \fancyhead[LO]{\textsc{\MakeLowercase{#2}]}}}% End of ForPrinting + {\fancyhead[R]{\thepage} + \fancyhead[L]{\textsc{[ch.~\MakeLowercase{#2}]}}} +} + +% Prologue +\newcommand{\Prologue}{% + % page formatting + \chapter*{\ChapterHead{Prologue}\\ % + \centering\textsc{\large WHAT IS GEOMETRY?}} + + \label{prologue}% Location in document + + \addtocontents{toc}% + {\protect\newcommand{\protect\ToCAnchor}{\protect\pageref{prolog-toc}}} + + % ToC entry + \addtocontents{toc}{\protect\ToCLine{prologue}{}} + \addtocontents{toc}{\protect\label{prolog-toc}}% Location in ToC + \addcontentsline{toc}{chapter}% + {\texorpdfstring{\protect\scshape What is Geometry?}{What is Geometry?}} + + % Running heads + \fancyhead{} + \fancyhead[CE]{\textsc{PROLOGUE}} + \fancyhead[CO]{\textsc{WHAT IS GEOMETRY?}} + + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage}}% End of ForPrinting + {\fancyhead[R]{\thepage}} +} + + +% Preface +\makeatletter +\newcommand{\Preface}{% + % page formatting + \chapter*{\ChapterHead{Preface}} + + % Bookmark; No ToC entry + % access the anchor created by the \chapter* command + \xdef\foo{chapter*.\the\Hy@linkcounter}% + \Hy@writebookmark{}{Preface}{\foo}{0}{toc}% + + \label{preface} + + % Running heads + \setlength{\headheight}{14.5pt} + + % add some stretch to paragraph breaks to reduce underfull pages + \setlength\parskip{0pt plus 3pt} + + \pagestyle{fancy} + \fancyhead{} + \fancyfoot{} + \fancyhead[C]{\textsc{PREFACE}} + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage}}% End of ForPrinting + {\fancyhead[R]{\thepage}} +} + + +% Appendix +\newcommand{\Appendix}{% + % Clear stale heading on previous page + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RE]{}}{}% + + % page formatting + % the preliminary \pdfbookmark ensures the anchor is above the heading + % can't re-use the \chapter* anchor because a parent and child bookmark + % must have distinct anchors + \chapter*{\pdfbookmark[-1]{Appendix}{Appendix}\ChapterHead{Appendix}\\ % + \centering\textsc{\large MATHEMATICAL NOTES}} + + \label{appendix} + + % ToC entry -- PDF bookmarks auto-generated + \addtocontents{toc}{\protect\filbreak} + \addtocontents{toc}{\protect\ToCLine{appendix}{}} + \addcontentsline{toc}{chapter} + {\texorpdfstring{\protect\scshape Mathematical Notes}{Mathematical Notes}} + + % Running heads + \fancyhead{} + \fancyhead[CE]{\textsc{APPENDIX}} + \fancyhead[CO]{\textsc{MATHEMATICAL NOTES}} + + \thispagestyle{empty} + + \ifthenelse{\boolean{ForPrinting}}% + {\fancyhead[RO,LE]{\thepage}}% + {\fancyhead[R]{\thepage}} +} +\makeatother + + +% Appendix Notes \AppNote{2}{(p.~20)} +\newcommand{\AppNote}[2]{% + % page formatting + \section*{\centering\textbf{Note~#1} \textrm{\normalsize #2}} + \label{appnote:#1} +} + + +% Define custom index format +\makeatletter +\renewcommand{\@idxitem}{\par\hangindent 30\p@\global\let\idxbrk\nobreak} +\renewcommand\subitem{\idxbrk\@idxitem \hspace*{15\p@}\let\idxbrk\relax} +\renewcommand{\indexspace}{\par\penalty-3000 \vskip 10pt plus5pt minus3pt\relax} + +% raw TeX manipulations are to position the bookmark anchor above the heading +\renewenvironment{theindex} + {\setlength\columnseprule{0.5pt}\setlength\columnsep{18pt}% + \begin{multicols}{2}[% + \begin{center}\Large IN\setbox0=\hbox{\phantomsection + \vbox to40pt{\hsize=20pt\IndexBookmark\vss}\hss}% + \ht0=0pt\dp0=0pt\wd0=0pt\box0DEX\vspace*{12pt}\end{center}]% + \setlength\parindent{0pt}\setlength\parskip{0pt plus 0.3pt}% + \thispagestyle{empty}\let\item\@idxitem\raggedright } + {\end{multicols}\clearpage\fancyhead{}\cleardoublepage} +\makeatother +\newcommand\IndexBookmark{\pdfbookmark[-1]{Index}{Index}} + + +% Contents heading +\AtBeginDocument{\renewcommand{\contentsname}% + {\protect\centering\normalfont\large\scshape CONTENTS\protect\\[0pt]}} + +% Illustrations +\newcommand{\Graphic}[3][] + {\includegraphics[width=#2]{./images/#3.pdf}% + \ifthenelse{\not\equal{#1}{}}{\Figlabel{#1}}{}} + +% %%%%% GLOBAL STYLE PARAMETERS %%%%% +\setlength{\parindent}{1em} + +\newlength{\QIndent} +\setlength{\QIndent}{0.6\parindent} + +% Chapter quotations +% \Quote[break]{Author (Date)}{Text}, etc. +\newcommand{\Signature}[1]{\allowbreak\null\nobreak% + \hfill\nobreak\raisebox{-2ex}{#1}} + +\newcommand{\Quote}[3][]% +{{\par\noindent\hspace*{\QIndent}\small#3% +\ifthenelse{\equal{#1}{break}}{\hfill\break\null}{}% +\hfill\textsc{\footnotesize #2}% +\ifthenelse{\equal{#1}{break}}{}{\hspace*{\QIndent}}\medskip}} + + +% Macros used in only one or two locations +\newcommand{\Actor}[1]{\par\hangindent 4.5\parindent \hangafter 1\qquad #1} + +\newcommand{\Ditto}{\normalfont\ttfamily\textquotesingle$\!$\textquotesingle} + +\newcommand{\Magnitude}{{}^\text{m}} + +\newcommand{\Neg}{\phantom{-}} + +\DeclareInputMath{176}{\mbox{\textdegree}} +\DeclareInputMath{183}{\cdot} + + +\newcommand{\HalfTitleBlock}% +{\settowidth{\TmpLen}{\textbf{\MyHuge GRAVITATION}} +\begin{minipage}{\TmpLen} +\noindent{\textbf{\MyHuge SPACE \hfill TIME}}\\[\MySkip] +\textbf{\LARGE\null\hfill AND\hfill\null}\\[\MySkip] +\textbf{\MyHuge GRAVITATION} +\end{minipage}} + + +\makeindex + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DPALIGN %%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter +\providecommand\shortintertext\intertext +\newcount\DP@lign@no +\newtoks\DP@lignb@dy +\newif\ifDP@cr +\newif\ifbr@ce +\def\f@@zl@bar{\null} +\def\addto@DPbody#1{\global\DP@lignb@dy\@xp{\the\DP@lignb@dy#1}} +\def\parseb@dy#1{\ifx\f@@zl@bar#1\f@@zl@bar + \addto@DPbody{{}}\let\@next\parseb@dy + \else\ifx\end#1 + \let\@next\process@DPb@dy + \ifDP@cr\else\addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&}\fi + \addto@DPbody{\end} + \else\ifx\intertext#1 + \def\@next{\eat@command0}% + \else\ifx\shortintertext#1 + \def\@next{\eat@command1}% + \else\ifDP@cr\addto@DPbody{&\DP@lint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\DPh@@kl} + \DP@crfalse\fi + \ifx\begin#1\def\begin@stack{b} + \let\@next\eat@environment + \else\ifx\lintertext#1 + \let\@next\linter@text + \else\ifx\rintertext#1 + \let\@next\rinter@text + \else\ifx\\#1 + \addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\\}\DP@crtrue + \global\advance\DP@lign@no\@ne + \let\@next\parse@cr + \else\check@braces#1!Q!Q!Q!\ifbr@ce\addto@DPbody{{#1}}\else + \addto@DPbody{#1}\fi + \let\@next\parseb@dy + \fi\fi\fi\fi\fi\fi\fi\fi\@next} +\def\process@DPb@dy{\let\lintertext\@gobble\let\rintertext\@gobble + \@xp\start@align\@xp\tw@\@xp\st@rredtrue\@xp\m@ne\the\DP@lignb@dy} +\def\linter@text#1{\@xp\DPlint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\rinter@text#1{\@xp\DPrint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\DPlint#1#2{\@xp\def\csname DP@lint:#1\endcsname{\text{#2}}} +\def\DPrint#1#2{\@xp\def\csname DP@rint:#1\endcsname{\text{#2}}} +\def\DP@lint#1{\ifbalancedlrint\@xp\ifx\csname DP@lint:#1\endcsname\relax\phantom + {\csname DP@rint:#1\endcsname}\else\csname DP@lint:#1\endcsname\fi + \else\csname DP@lint:#1\endcsname\fi} +\def\DP@rint#1{\ifbalancedlrint\@xp\ifx\csname DP@rint:#1\endcsname\relax\phantom + {\csname DP@lint:#1\endcsname}\else\csname DP@rint:#1\endcsname\fi + \else\csname DP@rint:#1\endcsname\fi} +\def\eat@command#1#2{\ifcase#1\addto@DPbody{\intertext{#2}}\or + \addto@DPbody{\shortintertext{#2}}\fi\DP@crtrue + \global\advance\DP@lign@no\@ne\parseb@dy} +\def\parse@cr{\new@ifnextchar*{\parse@crst}{\parse@crst{}}} +\def\parse@crst#1{\addto@DPbody{#1}\new@ifnextchar[{\parse@crb}{\parseb@dy}} +\def\parse@crb[#1]{\addto@DPbody{[#1]}\parseb@dy} +{\catcode`\$=13\gdef\check@braces#1#2!Q!Q!Q!{\ifx#2$$\br@cefalse\else\br@cetrue\fi + }\gdef${\textbf{\huge ERROR}\GenericError{\space\space\space\@spaces\@spaces\@spaces}% + {!!! DPalign/gather brace-parsing problem}% + {Likely nested argument beginning with doubled character}% + {Try putting an empty group at the start of the argument}\let$\relax}} +\def\eat@environment#1{\addto@DPbody{\begin{#1}}\begingroup + \def\@currenvir{#1}\let\@next\digest@env\@next} +\def\digest@env#1\end#2{% + \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}% + \ifx\@empty\begin@stack + \@checkend{#2} + \endgroup\let\@next\parseb@dy\fi + \addto@DPbody{#1\end{#2}} + \@next} +\def\lintertext{lint}\def\rintertext{rint} +\newif\ifbalancedlrint +\let\DPh@@kl\empty\let\DPh@@kr\empty +\def\DPg@therl{&\omit\hfil$\displaystyle} +\def\DPg@therr{$\hfil} + +\newenvironment{DPalign*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \DP@lignb@dy{&\DP@lint0&}\parseb@dy +}{% + \endalign +} +\newenvironment{DPgather*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \let\DPh@@kl\DPg@therl + \let\DPh@@kr\DPg@therr + \DP@lignb@dy{&\DP@lint0&\DPh@@kl}\parseb@dy +}{% + \endalign +} +\makeatother + +%%%%%%%%%%%%%%%%%%%%%%%% END OF DPALIGN %%%%%%%%%%%%%%%%%%%%%%%%%% + +% to avoid over/underfull boxes without using explicit linebreaks +\def\stretchyspace{\spaceskip0.5em plus 0.5em minus 0.25em} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +Project Gutenberg's Space, Time and Gravitation, by A. S. Eddington + +This eBook is for the use of anyone anywhere 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 www.gutenberg.org + + +Title: Space, Time and Gravitation + An Outline of the General Relativity Theory + +Author: A. S. Eddington + +Release Date: August 24, 2009 [EBook #29782] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK SPACE, TIME AND GRAVITATION *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by David Clarke, Andrew D. Hwang and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/American Libraries.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +\normalsize + +%% -----File: 001.png---Folio -9------- + +%Title Page +\cleardoublepage + +\null\vfil +\begin{center} +\HalfTitleBlock +\end{center} +\vfil + +%% -----File: 002.png---Folio -8------- + +%Copyright Page + +\clearpage +\settowidth{\TmpLen}{\textsc{CAMBRIDGE UNIVERSITY PRESS}} +\addtolength{\TmpLen}{0.25in} % [** PP: Hard-coded padding] +\null +\vfill +\begin{center}\stretchyspace +\makebox[\TmpLen][s]{\textsc{CAMBRIDGE UNIVERSITY PRESS}} \\[0.15in] +\addtolength{\TmpLen}{-0.125in} +\begin{minipage}{\TmpLen} +\centering +\footnotesize +\textsc{C. F. CLAY, Manager} \\[0.125in] +\Publine{\small LONDON}{FETTER LANE, E.C.~4}\\[0.1in] + +\Graphic{1in}{device} %Publisher's device + +\Publine{NEW YORK}{THE MACMILLAN CO.} \\ +\textsc{BOMBAY}\hfill\break +\textsc{CALCUTTA}\smash{$\!\left.\rule[-12pt]{0pt}{12pt}\right\}$}\hfill +\textsc{MACMILLAN AND CO., Ltd.} \\ +\textsc{MADRAS}\hfill\break +\Publine{TORONTO}{THE MACMILLAN CO. OF} \\ +\textsc{CANADA, Ltd.} \\ +\Publine{TOKYO}{MARUZEN-KABUSHIKI-KAISHA} +\end{minipage} + +\vspace*{0.5in} +\textsc{\scriptsize ALL RIGHTS RESERVED} +\end{center} +\vfill + +%% -----File: 003.png---Folio -7------- +%[Blank Page] +%% -----File: 004.png---Folio -6------- + +% [** PP: Force frontispiece to verso if ForPrinting] +\ifthenelse{\boolean{ForPrinting}} +{\cleardoublepage +\null\vfill +\newpage}{} + +\begin{center} +\pdfbookmark[0]{Frontispiece}{Frontispiece} +\rotatebox[origin=c]{90}{% +\begin{minipage}{8in} % [** PP: hard-coded size] +\centering +\includegraphics[width=7.6in]{./images/frontis_bw.jpg}\\ +\hspace*{1em}{\scriptsize +\textit{C.~Davidson\hfill Frontispiece\hfill +See \hyperref[page:117]{page~\upshape{\pageref{page:117}}}}}\hspace*{1em}\null\\ +{\textsc{\small eclipse instruments at sobral}} +\end{minipage}} +\end{center} +\phantomsection +\label{frontispiece} + +\FrontispieceToCEntry + +%% -----File: 005.png---Folio -5------- + +%Title Page + +\cleardoublepage +\begin{center} +\HalfTitleBlock\\[\MySkip] + +\textbf{\Large AN OUTLINE OF THE GENERAL}\\[0.125in] +\textbf{\Large RELATIVITY THEORY}\\[0.5in] +{\large BY}\\[0.2in] +\textbf{\Large A.~S. EDDINGTON, M.A., M.Sc., F.R.S.}\\[0.15in] +{\footnotesize PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL\\ +PHILOSOPHY, CAMBRIDGE} + +\vfill + +\textbf{\Large CAMBRIDGE}\\[0.15in] +\textbf{\Large AT THE UNIVERSITY PRESS}\\[0.15in] +\textbf{\large 1920} +\end{center} + +%% -----File: 006.png---Folio -4------- + +\clearpage +\null\vfil +\begin{center} +\begin{minipage}{4.5in} +\begin{verse} +\hfil\qquad\qquad Perhaps to move \\ +His laughter at their quaint opinions wide \\ +Hereafter, when they come to model heaven \\ +And calculate the stars: how they will wield \\ +The mighty frame: how build, unbuild, contrive \\ +To save appearances.\Signature{\textit{Paradise Lost.}} +\end{verse} +\end{minipage} +\end{center} +\vfil + +%% -----File: 007.png---Folio -3------- + + +\Preface + +\First{By} his theory of relativity Albert Einstein has provoked a +revolution of thought in physical science. + +The achievement consists essentially in this:---Einstein has +succeeded in separating far more completely than hitherto the +share of the observer and the share of external nature in the +things we see happen. The perception of an object by an observer +depends on his own situation and circumstances; for example, +distance will make it appear smaller and dimmer. We make +allowance for this almost unconsciously in interpreting what we +see. But it now appears that the allowance made for the \textit{motion} +of the observer has hitherto been too crude---a fact overlooked +because in practice all observers share nearly the same motion, +that of the earth. Physical space and time are found to be +closely bound up with this motion of the observer; and only an +amorphous combination of the two is left inherent in the external +world. When space and time are relegated to their proper source---the +observer---the world of nature which remains appears +strangely unfamiliar; but it is in reality simplified, and the +underlying unity of the principal phenomena is now clearly +revealed. The deductions from this new outlook have, with one +doubtful exception, been confirmed when tested by experiment. + +It is my aim to give an account of this work without introducing +anything very technical in the way of mathematics, +physics, or philosophy. The new view of space and time, so +opposed to our habits of thought, must in any case demand +unusual mental exercise. The results appear strange; and the +incongruity is not without a humorous side. For the first nine +chapters the task is one of interpreting a clear-cut theory, +accepted in all its essentials by a large and growing school of +physicists---although perhaps not everyone would accept the +author's views of its meaning. +% [** PP: Reads ``Chapters X and XI'' in original; modify for screen version] +\ifthenelse{\boolean{ForPrinting}}% +{Chapters~\hyperref[chapter:X]{\textsc{x}} and~\hyperref[chapter:XI]{\textsc{xi}}}% +{\Chapref{X} and~\Chapref{XI}} +deal with +very recent advances, with regard to which opinion is more +fluid. As for the last chapter, containing the author's speculations +on the meaning of nature, since it touches on the rudiments +of a philosophical system, it is perhaps too sanguine to hope that +it can ever be other than controversial. +%% -----File: 008.png---Folio -2------- + +A non-mathematical presentation has necessary limitations; +and the reader who wishes to learn how certain exact results +follow from Einstein's, or even Newton's, law of gravitation is +bound to seek the reasons in a mathematical treatise. But this +limitation of range is perhaps less serious than the limitation of +intrinsic truth. There is a relativity of truth, as there is a +relativity of space.---% +\begin{center} +``For \textsc{is} and \textsc{is-not} though \textit{with} Rule and Line\\[-3ex] +\phantom{``}And \textsc{up-and-down} \textit{without}, I could define.'' +\end{center} +Alas! It is not so simple. We abstract from the phenomena that +which is peculiar to the position and motion of the observer; +but can we abstract that which is peculiar to the limited imagination +of the human brain? We think we can, but only in the +symbolism of mathematics. As the language of a poet rings with +a truth that eludes the clumsy explanations of his commentators, +so the geometry of relativity in its perfect harmony expresses a +truth of form and type in nature, which my bowdlerised version +misses. + +But the mind is not content to leave scientific Truth in a dry +husk of mathematical symbols, and demands that it shall be +alloyed with familiar images. The mathematician, who handles~$x$ +so lightly, may fairly be asked to state, not indeed the inscrutable +meaning of~$x$ in nature, but the meaning which~$x$ +conveys to \textit{him}. + +Although primarily designed for readers without technical +knowledge of the subject, it is hoped that the book may also +appeal to those who have gone into the subject more deeply. +A few notes have been added in the Appendix mainly to bridge +the gap between this and more mathematical treatises, and to +indicate the points of contact between the argument in the text +and the parallel analytical investigation. + +It is impossible adequately to express my debt to contemporary +literature and discussion. The writings of Einstein, +Minkowski, Hilbert, Lorentz, Weyl, Robb, and others, have +provided the groundwork; in the give and take of debate with +friends and correspondents, the extensive ramifications have +gradually appeared. \Signature{A.~S.~E.\qquad} +\medskip + +\qquad{\small 1~\textit{May}, 1920.} + +%% -----File: 009.png---Folio -1------- + +%[** Table of Contents] + +\cleardoublepage +\phantomsection +\pdfbookmark[0]{Contents}{Contents} +\TableofContents % Arrange for empty pagestyle + +\fancyhead{} + +\iffalse %%%%%%%%%% BEGIN DEAD CODE %%%%%%%%%% +CONTENTS + +ECLIPSE INSTRUMENTS AT SOBRAL . Frontispiece + +PROLOGUE PAGE + +WHAT IS GEOMETRY? . . . . . . 1 + +CHAPTER I + +THE FITZGERALD CONTRACTION . . . 17 + +CHAPTER II + +RELATIVITY . . . . . . . . 30 + +CHAPTER III + +THE WORLD OF FOUR DIMENSIONS . . 45 + +CHAPTER IV + +FIELDS OF FORCE . . . . . . . 63 + +CHAPTER V + +KINDS OF SPACE . . . . . . . 77 + +CHAPTER VI + +THE NEW LAW OF GRAVITATION AND THE +OLD LAW . . . . . . . . . 93 + +CHAPTER VII + +WEIGHING LIGHT . . . . . . . 110 + +CHAPTER VIII + +OTHER TESTS OF THE THEORY . . . 123 + +CHAPTER IX + +MOMENTUM AND ENERGY . . . . . 136 + +CHAPTER X + +TOWARDS INFINITY . . . . . . 152 + +CHAPTER XI + +ELECTRICITY AND GRAVITATION . . 167 + +CHAPTER XII + +ON THE NATURE OF THINGS . . . . 180 + +APPENDIX + +MATHEMATICAL NOTES . . . . . 202 + +HISTORICAL NOTE . . . . . . 210 +\fi %%%%%%%%%% END OF DEAD CODE %%%%%%%%%% + +%% -----File: 010.png---Folio 0------- +%[Blank Page] +%% -----File: 011.png---Folio 1------- + +\mainmatter +\pagenumbering{arabic} + +\phantomsection +\pdfbookmark[-1]{Main Matter}{Main Matter} + +\Prologue + +{\small +\qquad A conversation between---% +\Actor{An experimental \textsc{Physicist}.} +\Actor{A pure \textsc{Mathematician}.} +\Actor{A \textsc{Relativist}, who advocates the newer conceptions of time +and space in physics.} + +}\medskip +\index{Euclidean geometry}% +\index{Geometry!Euclidean}% + + +\textit{Rel}. There is a well-known proposition of Euclid which states +that ``Any two sides of a triangle are together greater than the +third side.'' Can either of you tell me whether nowadays there +is good reason to believe that this proposition is true? + +\textit{Math}. For my part, I am quite unable to say whether the +proposition is true or not. I can deduce it by trustworthy +reasoning from certain other propositions or axioms, which are +supposed to be still more elementary. If these axioms are true, +the proposition is true; if the axioms are not true, the proposition +is not true universally. Whether the axioms are true or not +I cannot say, and it is outside my province to consider. + +\textit{Phys}. But is it not claimed that the truth of these axioms is +self-evident? + +\textit{Math}. They are by no means self-evident to me; and I think +the claim has been generally abandoned. + +\textit{Phys}. Yet since on these axioms you have been able to found +a logical and self-consistent system of geometry, is not this +indirect evidence that they are true? + +\textit{Math}. No. Euclid's geometry is not the only self-consistent +system of geometry. By choosing a different set of axioms I can, +for example, arrive at Lobatchewsky's geometry, in which many +of the propositions of Euclid are not in general true. From my +point of view there is nothing to choose between these different +geometries.% +\index{Geometry!Lobatchewskian}% +\index{Lobatchewsky}% + +\textit{Rel}. How is it then that Euclid's geometry is so much the +most important system? + +\textit{Math}. I am scarcely prepared to admit that it is the most +important. But for reasons which I do not profess to understand, +my friend the Physicist is more interested in Euclidean geometry +%% -----File: 012.png---Folio 2------- +than in any other, and is continually setting us problems in it. +Consequently we have tended to give an undue share of attention +to the Euclidean system. There have, however, been great +geometers like Riemann who have done something to restore +a proper perspective.% +\index{Riemann}% + +\textit{Rel}. (to Physicist). Why are you specially interested in +Euclidean geometry? Do you believe it to be the true geometry? + +\textit{Phys}. Yes. Our experimental work proves it true. + +\textit{Rel}. How, for example, do you prove that any two sides of +a triangle are together greater than the third side? + +\textit{Phys}. I can, of course, only prove it by taking a very large +number of typical cases, and I am limited by the inevitable +inaccuracies of experiment. My proofs are not so general or so +perfect as those of the pure mathematician. But it is a recognised +principle in physical science that it is permissible to generalise +from a reasonably wide range of experiment; and this kind of +proof satisfies me. + +\textit{Rel}. It will satisfy me also. I need only trouble you with +a special case. Here is a triangle $ABC$; how will you prove that +$AB + BC$ is greater than~$AC$? + +\textit{Phys}. I shall take a scale and measure the three sides. + +\textit{Rel}. But we seem to be talking about different things. I was +speaking of a proposition of geometry---properties of space, not +of matter. Your experimental proof only shows how a material +scale behaves when you turn it into different positions. + +\textit{Phys}. I might arrange to make the measures with an optical +device. + +\textit{Rel}. That is worse and worse. Now you are speaking of +properties of light. + +\textit{Phys}. I really cannot tell you anything about it, if you will +not let me make measurements of any kind. Measurement is +my only means of finding out about nature. I am not a metaphysicist. + +\textit{Rel}. Let us then agree that by \textit{length} and \textit{distance} you always +mean a quantity arrived at by measurements with material or +optical appliances. You have studied experimentally the laws +obeyed by these \textit{measured lengths}, and have found the geometry +to which they conform. We will call this geometry ``Natural +Geometry''; +\index{Geometry!natural}% +\index{Natural geometry}% +\index{Length!definition of}% +and it evidently has much greater importance for +%% -----File: 013.png---Folio 3------- +you than any other of the systems which the brain of the +mathematician has invented. But we must remember that its +subject matter involves the behaviour of material scales---the +properties of matter. Its laws are just as much laws of physics +as, for example, the laws of electromagnetism. + +\textit{Phys}. Do you mean to compare space to a kind of magnetic +field? I scarcely understand.% +\index{Space!meaning of}% + +\textit{Rel}. You say that you cannot explore the world without +some kind of apparatus. If you explore with a scale, you find +out the natural geometry; if you explore with a magnetic needle, +you find out the magnetic field. What we may call the field of +extension, or space-field, is just as much a physical quality as +the magnetic field. You can think of them both existing together +in the aether, if you like. The laws of both must be determined +by experiment. Of course, certain approximate laws of the space-field +(Euclidean geometry) have been familiar to us from childhood; +but we must get rid of the idea that there is anything +inevitable about these laws, and that it would be impossible to +find in other parts of the universe space-fields where these laws +do not apply. As to how far space really resembles a magnetic +field, I do not wish to dogmatise; my point is that they present +themselves to experimental investigation in very much the same +way. + +Let us proceed to examine the laws of natural geometry. +I have a tape-measure, and here is the triangle. $AB = 39\frac{1}{2}$~in., +$BC = \frac{1}{8}$~in., $CA = 39\frac{7}{8}$~in. Why, your proposition does not hold! + +\textit{Phys}. You know very well what is wrong. You gave the +tape-measure a big stretch when you measured~$AB$. + +\textit{Rel}. Why shouldn't I? + +\textit{Phys}. Of course, a length must be measured with a rigid +scale. + +\textit{Rel}. That is an important addition to our definition of length. +But what is a rigid scale?% +\index{Rigid scale, definition of}% + +\textit{Phys}. A scale which always keeps the same length. + +\textit{Rel}. But we have just defined length as the quantity arrived +at by measures with a rigid scale; so you will want another rigid +scale to test whether the first one changes length; and a third +to test the second; and so \textit{ad infinitum}. You remind me of the +incident of the clock and time-gun in Egypt. The man in charge +%% -----File: 014.png---Folio 4------- +of the time-gun fired it by the clock; and the man in charge of +the clock set it right by the time-gun. No, you must not define +length by means of a rigid scale, and define a rigid scale by +means of length. + +\textit{Phys}. I admit I am hazy about strict definitions. There is +not time for everything; and there are so many interesting +things to find out in physics, which take up my attention. Are +you so sure that you are prepared with a logical definition of all +the terms you use? + +\textit{Rel}. Heaven forbid! I am not naturally inclined to be +rigorous about these things. Although I appreciate the value of +the work of those who are digging at the foundations of science, +my own interests are mainly in the upper structure. But sometimes, +if we wish to add another storey, it is necessary to deepen +the foundations. I have a definite object in trying to arrive at +the exact meaning of length. A strange theory is floating round, +to which you may feel initial objections; and you probably +would not wish to let your views go by default. And after all, +when you claim to determine lengths to eight significant figures, +you must have a pretty definite standard of right and wrong +measurements. + +\textit{Phys}. It is difficult to define what we mean by rigid; but in +practice we can tell if a scale is likely to change length appreciably +in different circumstances. + +\textit{Rel}. No. Do not bring in the idea of change of length in +describing the apparatus for defining length. Obviously the +adopted standard of length cannot change length, whatever it +is made of. If a metre is defined as the length of a certain bar, +that bar can never be anything but a metre long; and if we +assert that this bar changes length, it is clear that we must have +changed our minds as to the definition of length. You recognised +that my tape-measure was a defective standard---that it was +not rigid. That was not because it changed length, because, if +it was the standard of length, it could not change length. It +was lacking in some other quality. + +You know an approximately rigid scale when you see one. +What you are comparing it with is not some non-measurable +ideal of length, but some attainable, or at least approachable, +ideal of material constitution. Ordinary scales have defects---% +%% -----File: 015.png---Folio 5------- +flexure, expansion with temperature, etc.---which can be reduced +by suitable precautions; and the limit, to which you approach +as you reduce them, is your rigid scale. You can define these +defects without appealing to any extraneous definition of length; +for example, if you have two rods of the same material whose +extremities are just in contact with one another, and when one +of them is heated the extremities no longer can be adjusted to +coincide, then the material has a temperature-coefficient of +expansion. Thus you can compare experimentally the temperature-coefficients +of different metals and arrange them in +diminishing sequence. In this sort of way you can specify the +nature of your ideal rigid rod, before you introduce the term +length. + +\textit{Phys}. No doubt that is the way it should be defined. + +\textit{Rel}. We must recognise then that all our knowledge of space +rests on the behaviour of material measuring-scales free from +certain definable defects of constitution. + +\textit{Phys}. I am not sure that I agree. Surely there is a sense in +which the statement $AB = 2CD$ is true or false, even if we had +no conception of a material measuring-rod. For instance, there +is, so to speak, twice as much paper between $A$ and~$B$, as between +$C$ and~$D$. + +\textit{Rel}. Provided the paper is uniform. But then, what does +uniformity of the paper mean? That the amount in given length +is constant. We come back at once to the need of defining length. + +If you say instead that the amount of ``space'' between +$A$ and~$B$ is twice that between $C$ and~$D$, the same thing applies. +You imagine the intervals filled with uniform space; but the +uniformity simply means that the same amount of space corresponds +to each inch of your rigid measuring-rod. You have +arbitrarily used your rod to divide space into so-called equal +lumps. It all comes back to the rigid rod. + +I think you were right at first when you said that you could +not find out anything without measurement; and measurement +involves some specified material appliance. + +Now you admit that your measures cannot go beyond a +certain close approximation, and that you have not tried all +possible conditions. Supposing that one corner of your triangle +was in a very intense gravitational field---far stronger than any +%% -----File: 016.png---Folio 6------- +we have had experience of---I have good ground for believing +that under those conditions you might find the sum of two sides +of a triangle, as measured with a rigid rod, appreciably less than +the third side. In that case would you be prepared to give up +Euclidean geometry? + +\textit{Phys}. I think it would be risky to assume that the strong +force of gravitation made no difference to the experiment. + +\textit{Rel}. On my supposition it makes an important difference. + +\textit{Phys}. I mean that we might have to make corrections to the +measures, because the action of the strong force might possibly +distort the measuring-rod. + +\textit{Rel}. In a rigid rod we have eliminated any special response +to strain. + +\textit{Phys}. But this is rather different. The extension of the rod +is determined by the positions taken up by the molecules under +the forces to which they are subjected; and there might be a +response to the gravitational force which all kinds of matter +would share. This could scarcely be regarded as a defect; and +our so-called rigid rod would not be free from it any more than +any other kind of matter. + +\textit{Rel}. True; but what do you expect to obtain by correcting +the measures? You correct measures, when they are untrue to +standard. Thus you correct the readings of a hydrogen-thermometer +to obtain the readings of a perfect gas-thermometer, +because the hydrogen molecules have finite size, and exert special +attractions on one another, and you prefer to take as standard +an ideal gas with infinitely small molecules. But in the present +case, what is the standard you are aiming at when you propose +to correct measures made with the rigid rod? + +\textit{Phys}. I see the difficulty. I have no knowledge of space +apart from my measures, and I have no better standard than +the rigid rod. So it is difficult to see what the corrected measures +would mean. And yet it would seem to me more natural to +suppose that the failure of the proposition was due to the +measures going wrong rather than to an alteration in the character +of space. + +\textit{Rel}. Is not that because you are still a bit of a metaphysicist? +You keep some notion of a space which is superior to measurement, +and are ready to throw over the measures rather than let +%% -----File: 017.png---Folio 7------- +this space be distorted. Even if there were reason for believing +in such a space, what possible reason could there be for assuming +it to be Euclidean? Your sole reason for believing space to be +Euclidean is that hitherto your measures have made it appear so; +if now measures of certain parts of space prefer non-Euclidean +geometry, all reason for assuming Euclidean space disappears. +Mathematically and conceptually Euclidean and non-Euclidean +space are on the same footing; our preference for Euclidean +space was based on measures, and must stand or fall by +measures.% +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% + +\textit{Phys}. Let me put it this way. I believe that I am trying to +measure something called length, which has an absolute meaning +in nature, and is of importance in connection with the laws of +nature. This length obeys Euclidean geometry. I believe my +measures with a rigid rod determine it accurately when no +disturbance like gravitation is present; but in a gravitational +field it is not unreasonable to expect that the uncorrected +measures may not give it exactly. + +\textit{Rel}. You have three hypotheses there:---(1)~there is an +absolute thing in nature corresponding to length, (2)~the +geometry of these absolute lengths is Euclidean, and (3)~practical +measures determine this length accurately when there is no +gravitational force. I see no necessity for these hypotheses, and +propose to do without them. \textit{Hypotheses non fingo.} The second +hypothesis seems to me particularly objectionable. You assume +that this absolute thing in nature obeys the laws of Euclidean +geometry. Surely it is contrary to scientific principles to lay +down arbitrary laws for nature to obey; we must find out her +laws by experiment. In this case the only experimental evidence +is that measured lengths (which by your own admission are not +necessarily the same as this absolute thing) sometimes obey +Euclidean geometry and sometimes do not. Again it would +seem reasonable to doubt your third hypothesis beyond, say, +the sixth decimal place; and that would play havoc with your +more delicate measures. But where I fundamentally differ from +you is the first hypothesis. Is there some absolute quantity in +nature that we try to determine when we measure length? +When we try to determine the number of molecules in a given +piece of matter, we have to use indirect methods, and different +%% -----File: 018.png---Folio 8------- +methods may give systematically different results; but no one +doubts that there is a definite number of molecules, so that there +is some meaning in saying that certain methods are theoretically +good and others inaccurate. Counting appears to be an absolute +operation. But it seems to me that other physical measures are +on a different footing. Any physical quantity, such as length, +mass, force, etc., which is not a pure number, can only be defined +as the result arrived at by conducting a physical experiment +according to specified rules.% +\index{Space!meaning of}% + +So I cannot conceive of any ``length'' in nature independent +of a definition of the way of measuring length. And, if there is, +we may disregard it in physics, because it is beyond the range +of experiment. Of course, it is always possible that we may +come across some quantity, not given directly by experiment, +which plays a fundamental part in theory. If so, it will turn up +in due course in our theoretical formulae. But it is no good +assuming such a quantity, and laying down \textit{a~priori} laws for it +to obey, on the off-chance of its proving useful. + +\textit{Phys}. Then you will not let me blame the measuring-rod +when the proposition fails? + +\textit{Rel}. By all means put the responsibility on the measuring-rod. +Natural geometry is the theory of the behaviour of material +scales. Any proposition in natural geometry is an assertion as +to the behaviour of rigid scales, which must accordingly take +the blame or credit. But do not say that the rigid scale is +wrong, because that implies a standard of right which does not +exist. + +\textit{Phys}. The space which you are speaking of must be a sort of +abstraction of the extensional relations of matter.% +\index{Matter!extensional relations of}% + +\textit{Rel}. Exactly so. And when I ask you to believe that space +can be non-Euclidean, or, in popular phrase, warped, I am not +asking you for any violent effort of the imagination; +\index{Warping of space}% +I only +mean that the extensional relations of matter obey somewhat +modified laws. Whenever we investigate the properties of space +experimentally, it is these extensional relations that we are +finding. Therefore it seems logical to conclude that space as +known to us must be the abstraction of these material relations, +and not something more transcendental. The reformed methods +of teaching geometry in schools would be utterly condemned, +%% -----File: 019.png---Folio 9------- +and it would be misleading to set schoolboys to verify propositions +of geometry by measurement, if the space they are supposed to +be studying had not this meaning. + +I suspect that you are doubtful whether this abstraction of +extensional relations quite fulfils your general idea of space; and, +as a necessity of thought, you require something beyond. I do +not think I need disturb that impression, provided you realise +that it is not the properties of this more transcendental thing +we are speaking of when we describe geometry as Euclidean or +non-Euclidean. + +\textit{Math}. The view has been widely held that space is neither +physical nor metaphysical, but conventional. Here is a passage +from Poincaré's \textit{Science and Hypothesis}, which describes this +alternative idea of space:% +\index{Space!conventional}% + +``If Lobatchewsky's geometry is true, the parallax of a very +distant star will be finite. +\index{Geometry!Lobatchewskian}% +\index{Lobatchewsky}% +If Riemann's is true, it will be negative. +These are the results which seem within the reach of experiment, +and it is hoped that astronomical observations may enable us +to decide between the two geometries. But what we call a +straight line in astronomy is simply the path of a ray of light. +If, therefore, we were to discover negative parallaxes, or to +prove that all parallaxes are higher than a certain limit, we +should have a choice between two conclusions: we could give +up Euclidean geometry, or modify the laws of optics, and +suppose that light is not rigorously propagated in a straight +line. It is needless to add that everyone would look upon this +solution as the more advantageous. Euclidean geometry, +therefore, has nothing to fear from fresh experiments.'' + +\textit{Rel}. Poincaré's brilliant exposition is a great help in understanding +the problem now confronting us. +\index{Poincaré}% +He brings out the +interdependence between geometrical laws and physical laws, +which we have to bear in mind continually. We can add on to +one set of laws that which we subtract from the other set. +I admit that space is conventional---for that matter, the meaning +of every word in the language is conventional. Moreover, we +have actually arrived at the parting of the ways imagined by +Poincaré, though the crucial experiment is not precisely the +one he mentions. But I deliberately adopt the alternative, +which, he takes for granted, everyone would consider less +%% -----File: 020.png---Folio 10------- +advantageous. I call the space thus chosen \textit{physical space}, and +its geometry \textit{natural geometry}, thus admitting that other conventional +meanings of space and geometry are possible. If it +were only a question of the meaning of space---a rather vague +term---these other possibilities might have some advantages. +But the meaning assigned to length and distance has to go +along with the meaning assigned to space. Now these are +quantities which the physicist has been accustomed to measure +with great accuracy; and they enter fundamentally into the +whole of our experimental knowledge of the world. We have a +knowledge of the so-called extent of the stellar universe, which, +whatever it may amount to in terms of ultimate reality, is not +a mere description of location in a conventional and arbitrary +mathematical space. Are we to be robbed of the terms in which +we are accustomed to describe that knowledge? + +The law of Boyle states that the pressure of a gas is proportional +to its density. It is found by experiment that this law is +only approximately true. A certain mathematical simplicity +would be gained by conventionally redefining \textit{pressure} in such +a way that Boyle's law would be rigorously obeyed. But it +would be high-handed to appropriate the word pressure in this +way, unless it had been ascertained that the physicist had no +further use for it in its original meaning. + +\textit{Phys}. I have one other objection. Apart from measures, we +have a general perception of space, and the space we perceive +is at least approximately Euclidean. + +\textit{Rel}. Our perceptions are crude measures. +\index{Perceptions, as crude measures}% +It is true that our +perception of space is very largely a matter of optical measures +with the eyes. If in a strong gravitational field optical and +mechanical measures diverged, we should have to make up our +minds which was the preferable standard, and afterwards abide +by it. So far as we can ascertain, however, they agree in all +circumstances, and no such difficulty arises. So, if physical +measures give us a non-Euclidean space, the space of perception +will be non-Euclidean. If you were transplanted into an extremely +intense gravitational field, you would directly perceive +the non-Euclidean properties of space. + +\textit{Phys}. Non-Euclidean space seems contrary to reason. + +\textit{Math}. It is not contrary to reason, but contrary to common +%% -----File: 021.png---Folio 11------- +experience, which is a very different thing, since experience is +very limited. + +\textit{Phys}. I cannot imagine myself perceiving non-Euclidean space! + +\textit{Math}. Look at the reflection of the room in a polished doorknob, %[** PP: Hyphenated across a line in original] +and imagine yourself one of the actors in what you see +going on there. + +\textit{Rel}. I have another point to raise. The distance between +two points is to be the length measured with a rigid scale. Let +us mark the two points by particles of matter, because we must +somehow identify them by reference to material objects. For +simplicity we shall suppose that the two particles have no +relative motion, so that the distance---whatever it is---remains +constant. Now you will probably agree that there is no such +thing as absolute motion; consequently there is no standard +condition of the scale which we can call ``at rest.'' We may +measure with the scale moving in any way we choose, and if +results for different motions disagree, there is no criterion for +selecting the true one. Further, if the particles are sliding past +the scale, it makes all the difference what instants we choose +for making the two readings. + +\textit{Phys}. You can avoid that by defining distance as the measurement +made with a scale which has the same velocity as the two +points. Then they will always be in contact with two particular +divisions of the scale. + +\textit{Rel}. A very sound definition; but unfortunately it does not +agree with the meaning of distance in general use. When the +relativist wishes to refer to this length, he calls it the \textit{proper-length}; +\index{Proper-length}% +in non-relativity physics it does not seem to have been +used at all. You see it is not convenient to send your apparatus +hurling through the laboratory---after a pair of $\alpha$ particles, for +example. And you could scarcely measure the length of a wave +of light by this convention\footnote% +{The proper-length of a light-wave is actually infinite.}. So the physicist refers his lengths +to apparatus at rest on the earth; and the mathematician starts +with the words ``Choose unaccelerated rectangular axes~$Ox$, $Oy$, +$Oz$,~$\dotsc$'' and assumes that the measuring-scales are at rest +relatively to these axes. So when the term length is used some +arbitrary standard motion of the measuring apparatus must +always be implied. + +%% -----File: 022.png---Folio 12------- + +\textit{Phys}. Then if you have fixed your standard motion of the +measuring-rod, there will be no ambiguity if you take the +readings of both particles at the same moment. + +\textit{Rel}. What is the same moment at different places? The +conception of simultaneity in different places is a difficult one. +Is there a particular instant in the progress of time on another +world, Arcturus, which is the same as the present instant on the +Earth?% +\index{Simultaneity}% + +\textit{Phys}. I think so, if there is any connecting link. We can +observe an event, say a change of brightness, on Arcturus, and, +allowing for the time taken by light to travel the distance, +determine the corresponding instant on the earth. + +\textit{Rel}. But then you must know the speed of the earth through +the aether. It may have shortened the light-time by going some +way to meet the light coming from Arcturus. + +\textit{Phys}. Is not that a small matter? + +\textit{Rel}. At a very modest reckoning the motion of the earth in +the interval might alter the light-time by several days. Actually, +however, any speed of the earth through the aether up to the +velocity of light is admissible, without affecting anything observable. +At least, nothing has been discovered which contradicts +this. So the error may be months or years. + +\textit{Phys}. What you have shown is that we have not sufficient +knowledge to determine in practice which are simultaneous +events on the Earth and Arcturus. It does not follow that there +is no definite simultaneity.% +\index{Absolute simultaneity}% + +\textit{Rel}. That is true, but it is at least possible that the reason +why we are unable to determine simultaneity in practice (or, +what comes to pretty much the same thing, our motion through +the aether) in spite of many brilliant attempts, is that there is +no such thing as absolute simultaneity of distant events. It is +better therefore not to base our physics on this notion of absolute +simultaneity, which may turn out not to exist, and is in any +case out of reach at present. + +But what all this comes to is that time as well as space is +implied in all our measures. The fundamental measurement is +not the interval between two points of space, but between two +points of space associated with instants of time. + +Our natural geometry is incomplete at present. We must +%% -----File: 023.png---Folio 13------- +supplement it by bringing in time as well as space. We shall +need a perfect clock as well as a rigid scale for our measures. +\index{Clock!perfect}% +It may be difficult to choose an ideal standard clock; but whatever +definition we decide on must be a physical definition. We +must not dodge it by saying that a perfect clock is one which +keeps perfect time. Perhaps the best theoretical clock would be +a pulse of light travelling in vacuum to and fro between mirrors +at the ends of a rigid scale. The instants of arrival at one end +would define equal intervals of time.% +\index{Time!measurement of}% + +\textit{Phys}. I think your unit of time would change according to +the motion of your ``clock'' through the aether. + +\textit{Rel}. Then you are comparing it with some notion of absolute +time. I have no notion of time except as the result of measurement +with some kind of clock. (Our immediate perception of +the flight of time is presumably associated with molecular +processes in the brain which play the part of a material clock.) +If you know a better clock, let us adopt it; but, having once +fixed on our ideal clock there can be no appeal from its judgments. +You must remember too that if you wish to measure +a second \textit{at one place}, you must keep your clock fixed at what +you consider to be one place; so its motion is defined. The +necessity of defining the motion of the clock emphasises that +one cannot consider time apart from space; there is one geometry +comprising both. + +\textit{Phys}. Is it right to call this study \textit{geometry}? %[** PP: Changed . to ?] +Geometry deals +with space alone. + +\textit{Math}. I have no objection. It is only necessary to consider +time as a fourth dimension. Your complete natural geometry +will be a geometry of four dimensions.% +\index{Fourth dimension}% + +\textit{Phys}. Have we then found the long-sought fourth dimension? + +\textit{Math}. It depends what kind of a fourth dimension you were +seeking. Probably not in the sense you intend. For me it only +means adding a fourth variable,~$t$, to my three space-variables +$x$, $y$,~$z$. It is no concern of mine what these variables really +represent. You give me a few fundamental laws that they +satisfy, and I proceed to deduce other consequences that may +be of interest to you. The four variables may for all I know be +the pressure, density, temperature and entropy of a gas; that +is of no importance to me. But you would not say that a gas +%% -----File: 024.png---Folio 14------- +had four dimensions because four mathematical variables were +used to describe it. Your use of the term ``dimensions'' is +probably more restricted than mine. + +\textit{Phys}. I know that it is often a help to represent pressure +and volume as height and width on paper; and so geometry +may have applications to the theory of gases. But is it not going +rather far to say that geometry can deal directly with these +things and is not necessarily concerned with lengths in +space? + +\textit{Math}. No. Geometry is nowadays largely analytical, so that +in form as well as in effect, it deals with variables of an unknown +nature. It is true that I can often see results more easily by +taking my $x$ and~$y$ as lengths on a sheet of paper. Perhaps it +would be helpful in seeing other results if I took them as pressure +and density in a steam-engine; but a steam-engine is not so +handy as a pencil. It is literally true that I do not want to +know the significance of the variables $x$, $y$, $z$, $t$ that I am discussing. +That is lucky for the Relativist, because although he has defined +carefully how they are to be measured, he has certainly not +conveyed to me any notion of how I am to picture them, if my +picture of absolute space is an illusion. + +\textit{Phys}. Yours is a strange subject. You told us at the beginning +that you are not concerned as to whether your propositions are +true, and now you tell us you do not even care to know what +you are talking about. + +\textit{Math}. That is an excellent description of Pure Mathematics, +which has already been given by an eminent mathematician\footnotemark. +\index{Russell}% + \footnotetext{``Pure mathematics consists entirely of such asseverations as that, if such + and such a proposition is true of \textit{anything}, then such and such a proposition + is true of that thing. It is essential not to discuss whether the first proposition + is really true, and not to mention what the anything is of which it is supposed + to be true\ldots. Thus mathematics may be defined as the subject in which we + never know what we are talking about, nor whether what we are saying is true.'' + \Signature{\textsc{Bertrand Russell}.}} + +\textit{Rel}. I think there is a real sense in which time is a fourth +dimension---as distinct from a fourth variable. The term +dimension seems to be associated with relations of \textit{order}. +\index{Order and dimensions}% +I believe that the order of events in nature is one indissoluble +four-dimensional order. We may split it arbitrarily into space +and time, just as we can split the order of space into length, +%% -----File: 025.png---Folio 15------- +breadth and thickness. But space without time is as incomplete +as a surface without thickness. + +\textit{Math}. Do you argue that the real world behind the phenomena +is four-dimensional? + +\textit{Rel}. I think that in the real world there must be a set of +entities related to one another in a four-dimensional order, and +that these are the basis of the perceptual world so far as it is +yet explored by physics. But it is possible to pick out a four-dimensional +set of entities from a basal world of five dimensions, +or even of three dimensions. The straight lines in three-dimensional +space form a four-dimensional set of entities, i.e.\ +they have a four-fold % [** PP: Regularized fourfold] +order. So one cannot predict the ultimate +number of dimensions in the world---if indeed the expression +\textit{dimensions} is applicable. + +\textit{Phys}. What would a philosopher think of these conceptions? +Or is he solely concerned with a metaphysical space and time +which is not within reach of measurement. + +\textit{Rel}. In so far as he is a psychologist our results must concern +him. Perception is a kind of crude physical measurement; +\index{Perceptions, as crude measures}% +and +perceptual space and time is the same as the measured space +and time, which is the subject-matter of natural geometry. In +other respects he may not be so immediately concerned. +Physicists and philosophers have long agreed that motion +through absolute space can have no meaning; +\index{Space!meaning of}% +but in physics +the question is whether motion through aether has any meaning. +I consider that it has no meaning; but that answer, though it +brings philosophy and physics into closer relation, has no bearing +on the philosophic question of absolute motion. I think, +however, we are entitled to expect a benevolent interest from +philosophers, in that we are giving to their ideas a perhaps +unexpected practical application. +\bigskip + +%[** Thought break] + +Let me now try to sum up my conclusions from this conversation. +We have been trying to give a precise meaning to the +term \textit{space}, so that we may be able to determine exactly the +properties of the space we live in. There is no means of determining +the properties of our space by \textit{a~priori} reasoning, because +there are many possible kinds of space to choose from, no one +of which can be considered more likely than any other. For +%% -----File: 026.png---Folio 16------- +more than 2000 years we have believed in a Euclidean space, +because certain experiments favoured it; but there is now reason +to believe that these same experiments when pushed to greater +accuracy decide in favour of a slightly different space (in the +neighbourhood of massive bodies). The relativist sees no reason +to change the rules of the game because the result does not +agree with previous anticipations. Accordingly when he speaks +of space, he means the space revealed by measurement, whatever +its geometry. He points out that this is the space with which +physics is concerned; and, moreover, it is the space of everyday +perception. If his right to appropriate the term space in this +way is challenged, he would urge that this is the sense in which +the term has always been used in physics hitherto; it is only +recently that conservative physicists, frightened by the revolutionary +consequences of modern experiments, have begun to +play with the idea of a pre-existing space whose properties +cannot be ascertained by experiment---a metaphysical space, to +which they arbitrarily assign Euclidean properties, although it +is obvious that its geometry can never be ascertained by experiment. +But the relativist, in defining space as \textit{measured space}, +clearly recognises that all measurement involves the use of +material apparatus; the resulting geometry is specifically a study +of the extensional relations of matter. He declines to consider +anything more transcendental. + +My second point is that since natural geometry is the study +of extensional relations of natural objects, and since it is found +that their space-order cannot be discussed without reference to +their time-order as well, it has become necessary to extend our +geometry to four dimensions in order to include time. +%% -----File: 027.png---Folio 17------- + +\Chapter{I}{The FitzGerald Contraction} + +\Quote{Descartes.} +{In order to reach the Truth, it is necessary, once in one's life, to put every +thing in doubt---so far as possible.} + + +\First{Will} it take longer to swim to a point $100$~yards up-stream +and back, or to a point $100$~yards across-stream and back? + +In the first case there is a long toil up against the current, +and then a quick return helped by the current, which is all too +short to compensate. In the second case the current also hinders, +because part of the effort is devoted to overcoming the drift +down-stream. But no swimmer will hesitate to say that the +hindrance is the greater in the first case. + +Let us take a numerical example. Suppose the swimmer's +speed is $50$~yards a minute in still water, and the current is +$30$~yards a minute. Thus the speed against the current is~$20$, +and with the current $80$~yards a minute. The up journey then +takes $5$~minutes and the down journey $1\frac{1}{4}$ minutes. Total time, +$6\frac{1}{4}$ minutes. + +%[Illustration: Fig. 1.] +\begin{wrapfigure}{r}{1.5in} +\Graphic[1]{1.375in}{027a}% \Figlabel{1} +\end{wrapfigure} +Going across-stream the swimmer must aim at a point~$E$ above +the point~$B$ where he wishes to arrive, so +that $OE$ represents his distance travelled +in still water, and $EB$ the amount he has +drifted down. These must be in the ratio +$50$ to~$30$, and we then know from the right-angled +triangle $OBE$ that $OB$ will correspond +to~$40$. Since $OB$ is $100$~yards, $OE$ +is $125$~yards, and the time taken is $2\frac{1}{2}$ +minutes. Another $2\frac{1}{2}$ minutes will be +needed for the return journey. Total time, +$5$~minutes. + +In still water the time would have been $4$~minutes. + +The up-and-down swim is thus longer than the transverse +swim in the ratio $6\frac{1}{4}:5$ minutes. Or we may write the ratio +\[ +\dfrac{1}{\surd \bigl(1 - (\tfrac{30}{50})^2\bigr)} +\] +%% -----File: 028.png---Folio 18------- +which shows how the result depends on the ratio of the speed +of the current to the speed of the swimmer, viz.~$\frac{30}{50}$. + +A very famous experiment on these lines was tried in America +in the year~1887. The swimmer was a wave of light, which we +know swims through the aether with a speed of $186,330$ miles +a second. The aether was flowing through the laboratory like +a river past its banks. The light-wave was divided, by partial +reflection at a thinly silvered surface, into two parts, one of +which was set to perform the up-and-down stream journey and +the other the across-stream journey. When the two waves +reached their proper turning-points they were sent back to the +starting-point by mirrors. To judge the result of the race, there +was an optical device for studying interference fringes; because +the recomposition of the two waves after the journey would +reveal if one had been delayed more than the other, so that, for +example, the crest of one instead of fitting on to the crest of +the other coincided with its trough. + +To the surprise of Michelson and Morley, who conducted the +experiment, the result was a dead-heat. +\index{Michelson-Morley experiment}% +It is true that the +direction of the current of aether was not known---they hoped +to find it out by the experiment. That, however, was got over +by trying a number of different orientations. Also it was +possible that there might actually be no current at a particular +moment. But the earth has a velocity of $18\frac{1}{2}$ miles a second, +continually changing direction as it goes round the sun; so that +at some time during the year the motion of a terrestrial laboratory +through the aether must be at least $18\frac{1}{2}$ miles a second. +The experiment should have detected the delay by a much +smaller current; in a repetition of it by Morley and Miller +in~1905, a current of $2$~miles a second would have been +sufficient. + +If we have two competitors, one of whom is known to be +slower than the other, and yet they both arrive at the winning-post +at the same time, it is clear that they cannot have travelled +equal courses. To test this, the whole apparatus was rotated +through a right angle, so that what had been the up-and-down +course became the transverse course, and \textit{vice versa}. Our two +competitors interchanged courses, but still the result was a +dead-heat. + +%% -----File: 029.png---Folio 19------- + +The surprising character of this result can be appreciated by +contrasting it with a similar experiment on sound-waves. +Sound consists of waves in air or other material, as light consists +of waves in aether. It would be possible to make a precisely +similar experiment on sound, with a current of air past the +apparatus instead of a current of aether. In that case the greater +delay of the wave along the direction of the current would +certainly show itself experimentally. Why does light seem to +behave differently? + +The straightforward interpretation of this remarkable result +is that each course undergoes an automatic contraction when it +is swung from the transverse to the longitudinal position, so +that whichever arm of the apparatus is placed up-stream it +straightway becomes the shorter. The course is marked out in +the rigid material apparatus, and we have to suppose that the +length of any part of the apparatus changes as it is turned in +different directions with respect to the aether-current. It is +found that the kind of material---metal, stone or wood---makes +no difference to the experiment. The contraction must be the +same for all kinds of matter; the expected delay depends only +on the ratio of the speed of the aether current to the speed of +light, and the contraction which compensates it must be equally +definite.% +\index{Length!effect of motion on}% + +This explanation was proposed by FitzGerald, and at first +sight it seems a strange and arbitrary hypothesis. But it has +been rendered very plausible by subsequent theoretical researches +of Larmor and Lorentz. +\index{Larmor}% +\index{Lorentz}% +Under ordinary circumstances the form +and size of a solid body is maintained by the forces of cohesion +between its particles. What is the nature of cohesion? We guess +that it is made up of electric forces between the molecules. But +the aether is the medium in which electric force has its seat; +hence it will not be a matter of indifference to these forces how +the electric medium is flowing with respect to the molecules. +When the flow changes there will be a readjustment of cohesive +forces, and we must expect the body to take a new shape and +size. + +The theory of Larmor and Lorentz enables us to trace in +detail the readjustment. Taking the accepted formulae of +electromagnetic theory, they showed that the new form of +%% -----File: 030.png---Folio 20------- +equilibrium would be contracted in just such a way and by +just such an amount as FitzGerald's explanation requires\footnote% +{Appendix, \Noteref{1}.}.% +\Pagelabel{note1}% +\index{Contraction, FitzGerald}% +\index{FitzGerald Contraction}% + +The contraction in most cases is extremely minute. We have +seen that when the ratio of the speed of the current to that +of the swimmer is $\frac{3}{5}$, a contraction in the ratio +$\surd \bigl(1 - (\frac{3}{5})^2\bigr)$ +is needed to compensate for the delay. The earth's orbital +velocity is $\frac{1}{10000}$ of the velocity of light, so that it will give a +contraction of $\surd \bigl(1 - (\frac{1}{10000})^2\bigr)$, or $1$~part in~$200,000,000$. This +would mean that the earth's diameter in the direction of its +motion is shortened by $2\frac{1}{2}$ inches. + +The Michelson-Morley experiment has thus failed to detect +our motion through the aether, because the effect looked for---the +delay of one of the light waves---is exactly compensated by +an automatic contraction of the matter forming the apparatus. +Other ingenious experiments have been tried, electrical and +optical experiments of a more technical nature. They likewise +have failed, because there is always an automatic compensation +somewhere. We now believe there is something in the nature +of things which inevitably makes these compensations, so that +it will never be possible to determine our motion through the +aether. Whether we are at rest in it, or whether we are rushing +through it with a speed not much less than that of light, will +make no difference to anything that can possibly be observed. + +This may seem a rash generalization from the few experiments +actually performed; more particularly, since we can only experiment +with the small range of velocity caused by the earth's +orbital motion. With a larger range residual differences might +be disclosed. But there is another reason for believing that the +compensation is not merely approximate but exact. The compensation +has been traced theoretically to its source in the +well-known laws of electromagnetic force; and here it is mathematically +exact. Thus the generalization is justified, at least in +so far as the observed phenomena depend on electromagnetic +causes, and in so far as the universally accepted laws of electromagnetism +are accurate. + +The generalization here laid down is called the restricted +Principle of Relativity:---\textit{It is impossible by any experiment to +detect uniform motion relative to the aether.}\Pagelabel{20}% +\index{Principle of Relativity (restricted)}% +\index{Relativity!restricted Principle of}% + +%% -----File: 031.png---Folio 21------- + +There are other natural forces which have not as yet been +recognised as coming within the electromagnetic scheme---gravitation, +for example---and for these other tests are required. +Indeed we were scarcely justified in stating above that the +diameter of the earth would contract $2\frac{1}{2}$~inches, because the +figure of the earth is determined mainly by gravitation, whereas +the Michelson-Morley experiment relates to bodies held together +by cohesion. There is fair evidence of a rather technical kind +that the compensation exists also for phenomena in which +gravitation is concerned; and we shall assume that the principle +covers all the forces of nature. + +Suppose for a moment it were not so, and that it were possible +to determine a kind of absolute motion of the earth by experiments +or observations involving gravitation. Would this throw +light on our motion through the aether? I think not. It would +show that there is some standard of rest with respect to which +the law of gravitation takes a symmetrical and simple form; +presumably this standard corresponds to some gravitational +medium, and the motion determined would be motion with +respect to that medium. Similarly if the motion were revealed +by vital or psychical phenomena, it would be motion relative +to some vital or psychical medium. The aether, defined as the +seat of electric forces, must be revealed, if at all, by electric +phenomena. + +It is well to remember that there is reasonable justification +for adopting the principle of relativity even if the evidence is +insufficient to prove it. In Newtonian dynamics the phenomena +are independent of uniform motion of the system; no explanation +is asked for, because it is difficult to see any reason why there +should be an effect. +\index{Gravitation!relativity for uniform motion}% +If in other phenomena the principle fails, +then we must seek for an explanation of its failure---and no +doubt a plausible explanation can be devised; but so long as +experiment gives no indication of a failure, it is idle to anticipate +such a complication. Clearly physics cannot concern itself with +all the possible complexities which \textit{may} exist in nature, but have +not hitherto betrayed themselves in any experiment. + +The principle of relativity has implications of a most revolutionary +kind. Let us consider what is perhaps an exaggerated +case---or perhaps the actual case, for we cannot tell. Let the +%% -----File: 032.png---Folio 22------- +reader suppose that he is travelling through the aether at +$161,000$~miles a second vertically upwards; if he likes to make +the positive assertion that this is his velocity, no one will be +able to find any evidence to contradict him. For this speed the +FitzGerald contraction is just $\frac{1}{2}$, so that every object contracts +to half its original length when turned into the vertical position.% +\index{FitzGerald Contraction!consequences of}% + +As you lie in bed, you are, say, $6$~feet long. Now stand upright; +you are $3$~feet. You are incredulous? Well, let us prove it! +Take a yard-measure; when turned vertically it must undergo +the FitzGerald contraction, and become only half a yard. If you +measure yourself with it, you will find you are just two---\textit{half-yards}. +``But I can see that the yard-measure does not change +length when I turn it.'' What you perceive is an image of the +rod on the retina of your eye; you imagine that the image +occupies the same space in both positions; but your retina has +contracted in the vertical direction without your knowing it, so +that your visual estimates of vertical length are double what +they should be. And so on with every test you can devise. +Because everything is altered in the same way, nothing appears +to be altered at all. + +It is possible to devise electrical and optical tests; in that +case the argument is more complicated, because we must consider +the effect of the rapid current of aether on the electric +forces and on waves of light. But the final conclusion is always +the same; the tests will reveal nothing. Here is one illustration. +To avoid distortion of the retina, lie on your back on the floor, +and watch in a suitably inclined mirror someone turn the rod +from the horizontal to the vertical position. You will, of course, +see no change of length, and it is not possible to blame the +retina this time. But is the appearance in the mirror a faithful +reproduction of what is actually occurring? +\index{Mirror, distortion by moving}% +In a plane mirror +at rest the appearance is correct; the rays of light come off the +mirror at the same angle as they fall on to it, like billiard balls +rebounding from an elastic cushion. But if the cushion is in +rapid motion the angle of the billiard-ball will be altered; and +similarly the rapid motion of the mirror through the aether +alters the law of reflection. +\index{Reflection by moving mirror}% +Precise calculation shows that the +moving mirror will distort the image, so as to conceal exactly +the changes of length which occur. + +%% -----File: 033.png---Folio 23------- + +The mathematician does not need to go through all the +possible tests in detail; he knows that the complete compensation +is inherent in the fundamental laws of nature, and so must +occur in every case. So if any suggestion is made of a device +for detecting these effects, he starts at once to look for the +fallacy which must surely be there. Our motion through the +aether may be very much less than the value here adopted, and +the changes of length may be very small; but the essential point +is that they escape notice, not because they are small (if they +are small), but because from their very nature they are undetectable. + +There is a remarkable reciprocity about the effects of motion +on length, which can best be illustrated by another example. +Suppose that by development in the powers of aviation, a man +flies past us at the rate of $161,000$ miles a second. +\index{Aviator, space and time-reckoning of|(}% [** PP: Using range] +We shall +suppose that he is in a comfortable travelling conveyance in +which he can move about, and act normally and that his length +is in the direction of the flight. If we could catch an instantaneous +glimpse as he passed, we should see a figure about three feet +high, but with the breadth and girth of a normal human being. +And the strange thing is that he would be sublimely unconscious +of his own undignified appearance. If he looks in a mirror in +his conveyance, he sees his usual proportions; this is because of +the contraction of his retina, or the distortion by the moving +mirror, as already explained. But when he looks down on us, +he sees a strange race of men who have apparently gone through +some flattening-out process; one man looks barely $10$~inches +across the shoulders, another standing at right angles is almost +``length and breadth, without thickness.'' As they turn about +they change appearance like the figures seen in the old-fashioned +convex-mirrors. If the reader has watched a cricket-match +through a pair of prismatic binoculars, he will have seen this +effect exactly. + +It is the reciprocity of these appearances--that each party +should think the other has contracted---that is so difficult to +realise. Here is a paradox beyond even the imagination of +Dean Swift. Gulliver regarded the Lilliputians as a race of +dwarfs; and the Lilliputians regarded Gulliver as a giant. That +is natural. If the Lilliputians had appeared dwarfs to Gulliver, +%% -----File: 034.png---Folio 24------- +and Gulliver had appeared a dwarf to the Lilliputians---but no! +that is too absurd for fiction, and is an idea only to be found in +the sober pages of science. + +This reciprocity is easily seen to be a necessary consequence +of the Principle of Relativity. The aviator must detect a FitzGerald +contraction of objects moving rapidly relatively to him, +just as we detect the contraction of objects moving relatively to us, +and as an observer at rest in the aether detects the contraction +of objects moving relatively to the aether. Any other result +would indicate an observable effect due to his own motion +through the aether. + +Which is right? Are we or the aviator? Or are both the +victims of illusion? It is not illusion in the ordinary sense, +because the impressions of both would be confirmed by every +physical test or scientific calculation suggested. No one knows +which is right. No one will ever know, because we can never +find out which, if either, is truly at rest in the aether. + +It is not only in space but in time that these strange variations +occur. If we observed the aviator carefully we should infer that +he was unusually slow in his movements; and events in the +conveyance moving with him would be similarly retarded---as +though time had forgotten to go on. +\index{Retardation of time}% +\index{Time!for moving observer}% +His cigar lasts twice as +long as one of ours. I said ``infer'' deliberately; we should \textit{see} +a still more extravagant slowing down of time; but that is easily +explained, because the aviator is rapidly increasing his distance +from us and the light-impressions take longer and longer to +reach us. The more moderate retardation referred to remains +after we have allowed for the time of transmission of light. + +But here again reciprocity comes in, because in the aviator's +opinion it is we who are travelling at $161,000$ miles a second +past him; and when he has made all allowances, he finds that +it is we who are sluggish. Our cigar lasts twice as long as his. + +Let us examine more closely how the two views are to be +reconciled. Suppose we both light similar cigars at the instant +he passes us. At the end of $30$~minutes our cigar is finished. +This signal, borne on the waves of light, hurries out at the rate +of $186,000$ miles a second to overtake the aviator travelling at +$161,000$ miles a second, who has had $30$~minutes start. It will +take nearly $194$~minutes to overtake him, giving a total time of +%% -----File: 035.png---Folio 25------- +$224$~minutes after lighting the cigar. His watch like everything +else about him (including his cigar) is going at half-speed; so +it records only $112$~minutes elapsed when our signal arrives. +The aviator knows, of course, that this is not the true time when +our cigar was finished, and that he must correct for the time of +transmission of the light-signal. He sets himself this problem---that +man has travelled away from me at $161,000$~miles a second +for an unknown time $x$~minutes; he has then sent a signal which +travels the same distance back at $186,000$~miles a second; the +total time is $112$~minutes; problem, find~$x$. Answer, $x = 60$ +minutes. He therefore judges that our cigar lasted $60$~minutes, +or twice as long as his own. His cigar lasted $30$~minutes by his +watch (because the same retardation affects both watch and +cigar); and that was in our opinion twice as long as ours, because +his watch was going at half-speed. + +Here is the full time-table. +\begin{center} +{\footnotesize +\begin{tabular}{r@{}c% @{} signifies no inter-column separation + >{\centering\hspace{0pt}}m{1.5in}@{}% + >{\centering\hspace{0pt}}m{1.75in}r@{}c} % end of alignment preamble +% Two multicolumn headers and extra vertical space +\multicolumn{2}{c}{\parbox[c]{0.625in}% [** PP: 0.7pt overfull] + {\centering Stationary watch}} + & Stationary Observer & Aviator + & \multicolumn{2}{c}{\parbox[c]{0.625in}% + {\centering Aviator's watch}} \\[3ex] +% + $0$ & min. & Lights cigar & Lights cigar & $0$ & min. \\ + $30$ & \Ditto & Finishes cigar & \ldots & $15$ & \Ditto \\ + $60$ & \Ditto & Inferred time aviator's cigar finished + & Finishes cigar & $30$ & \Ditto \\ +% +$112$ & \Ditto & Receives signal aviator's cigar finished + & \ldots & $56$ & \Ditto \\ +% +$120$ & \Ditto & \ldots & Inferred time stationary cigar finished + & $60$ & \Ditto \\ +% +$224$ & \Ditto & \ldots & Receives signal stationary cigar finished + & $112$ & \Ditto \\ +\end{tabular} +}% End of \footnotesize +\end{center} + +This is analysed from our point of view, not the aviator's; +because it makes out that he was wrong in his inference and we +were right. But no one can tell which was really right. + +The argument will repay a careful examination, and it will +be recognised that the chief cause of the paradox is that we +assume that we are at rest in the aether, whereas the aviator +assumes that he is at rest. Consequently whereas in our opinion +the light-signal is overtaking him at merely the difference +between $186,000$ and $161,000$ miles a second, he considers that +it is coming to him through the relatively stationary aether at +the normal speed of light. It must be remembered that each +observer is furnished with complete experimental evidence in +support of his own assumption. If we suggest to the aviator +%% -----File: 036.png---Folio 26------- +that owing to his high velocity the relative speed of the wave +overtaking him can only be $25,000$ miles a second, he will reply +``I have determined the velocity of the wave relatively to me +by timing it as it passes two points in my conveyance; and it +turns out to be $186,000$ miles a second. So I know my correction +for light-time is right\footnotemark.'' + \footnotetext{We need not stop to prove this directly. If the aviator could detect anything + in his measurements inconsistent with the hypothesis that he was at rest + in the aether (e.g.\ a difference of velocity of overtaking waves of light and + waves meeting him) it would contradict the restricted principle of relativity.}% +His clocks and scales are all behaving +in an extraordinary way from our point of view, so it is not +surprising that he should arrive at a measure of the velocity of +the overtaking wave which differs from ours; but there is no +way of convincing him that our reckoning is preferable. + +Although not a very practical problem, it is of interest to +inquire what happens when the aviator's speed is still further +increased and approximates to the velocity of light. +\index{Light, velocity of!system moving with}% +\index{Time!``standing still''}% +\index{Velocity of light!system moving with}% +Lengths +in the direction of flight become smaller and smaller, until for +the speed of light they shrink to zero. The aviator and the +objects accompanying him shrink to two dimensions. We are +saved the difficulty of imagining how the processes of life can +go on in two dimensions, because nothing goes on. Time is +arrested altogether. This is the description according to the +terrestrial observer. The aviator himself detects nothing unusual; +he does not perceive that he has stopped moving. He is +merely waiting for the next instant to come before making the +next movement; and the mere fact that time is arrested means +that he does not perceive that the next instant is a long time +coming. + +It is a favourite device for bringing home the vast distances +of the stars to imagine a voyage through space with the velocity +of light. The youthful adventurer steps on to his magic carpet +loaded with provisions for a century. He reaches his journey's +end, say Arcturus, a decrepit centenarian. This is wrong. It is +quite true that the journey would last something like a hundred +years by terrestrial chronology; but the adventurer would arrive +at his destination no more aged than when he started, and he +would not have had time to think of eating. So long as he travels +with the speed of light he has immortality and eternal youth. +%% -----File: 037.png---Folio 27------- +If in some way his motion were reversed so that he returned to +the earth again, he would find that centuries had elapsed here, +whilst he himself did not feel a day older---for him the voyage +had lasted only an instant\footnotemark. + \footnotetext{Since the earth is moving relatively to our adventurer with the velocity + of light, we might be tempted to argue that from this point of view the terrestrial + observer would have perpetual youth whilst the voyager grew older. Evidently, + if they met again, they could disprove one or other of the two arguments. But + in order to meet again the velocity of one of them must be reversed by supernatural % [** PP: Hyphenated across a line in original] + means or by an intense gravitational force so that the conditions are + not symmetrical and reciprocity does not apply. The argument given in the + text appears to be the correct one.} + +Our reason for discussing at length the effects of these +improbably high velocities is simply in order that we may speak +of the results in terms of common experience; otherwise it +would be necessary to use the terms of refined technical measurement. +The relativist is sometimes suspected of an inordinate +fondness for paradox; but that is rather a misunderstanding of +his argument. The paradoxes exist when the new experimental +discoveries are woven into the scheme of physics hitherto +current, and the relativist is ready enough to point this out. +But the conclusion he draws is that a revised scheme of physics +is needed in which the new experimental results will find a natural +place without paradox. + +To sum up---on any planet moving with a great velocity +through the aether, extraordinary changes of length of objects +are continually occurring as they move about, and there is a +slowing down of all natural processes as though time were +retarded. These things cannot be perceived by anyone on the +planet; but similar effects would be detected by any observer +having a great velocity relative to the planet (who makes all +allowances for the effect of the motion on the observations, but +takes it %[** PP: Typo ``if''] +for granted that he himself is at rest in the aether\footnotemark). + \footnotetext{The last clause is perhaps unnecessary. The correction applied for light + transmission will naturally be based on the observer's own experimental determination + of the velocity of light. According to experiment the velocity of light + relatively to him is \textit{apparently} the same in all directions, and he will apply + the corrections accordingly. This is equivalent to assuming that he is at rest + in the aether; but he need not, and probably would not, make the assumption + explicitly.}% +There is complete reciprocity so that each of two observers in +relative motion will find the same strange phenomena occurring +%% -----File: 038.png---Folio 28------- +to the other; and there is nothing to help us to decide which is +right.% +\index{Aviator, space and time-reckoning of|)} % [** PP: Using range] + +I think that no one can contemplate these results without +feeling that the whole strangeness must arise from something +perverse and inappropriate in our ordinary point of view. +Changes go on on a planet, all nicely balanced by adjustments +of natural forces, in such a way that no one on the planet can +possibly detect what is taking place. Can we seriously imagine +that there is anything in the reality behind the phenomena, +which reflects these changes? Is it not more probable that we +ourselves introduce the complexity, because our method of +description is not well-adapted to give a simple and natural +statement of what is really occurring? + +The search for a more appropriate apparatus of description +leads us to the standpoint of relativity described in the next +chapter. +\index{Relativity, standpoint of}% +I draw a distinction between the principle and the +standpoint of relativity. The principle of relativity is a statement +of experimental fact, which may be right or wrong; the +first part of it---the restricted principle---has already been +enunciated. Its consequences can be deduced by mathematical +reasoning, as in the case of any other scientific generalization. +It postulates no particular mechanism of nature, \textit{and no particular +view as to the meaning of time and space}, though it may suggest +theories on the subject. The only question is whether it is +experimentally true or not. + +The standpoint of relativity is of a different character. It +asserts first that certain unproved hypotheses as to time and +space have insensibly crept into current physical theories, and +that these are the source of the difficulties described above. +Now the most dangerous hypotheses are those which are tacit +and unconscious. So the standpoint of relativity proposes +tentatively to do without these hypotheses (not making any +others in their place); and it discovers that they are quite +unnecessary and are not supported by any known fact. This in +itself appears to be sufficient justification for the standpoint. +Even if at some future time facts should be discovered which +confirm the rejected hypotheses, the relativist is not wrong in +reserving them until they are required. + +It is not our policy to take shelter in impregnable positions; +%% -----File: 039.png---Folio 29------- +and we shall not hesitate to draw reasonable conclusions as well +as absolutely proved conclusions from the knowledge available. +But to those who think that the relativity theory is a passing +phase of scientific thought, which may be reversed in the light +of future experimental discoveries, we would point out that, +though like other theories it may be developed and corrected, +there is a certain minimum statement possible which represents +irreversible progress. Certain hypotheses enter into all physical +descriptions and theories hitherto current, dating back in some +cases for 2000 years, in other cases for 200 years. It can now +be proved that these hypotheses have nothing to do with any +phenomena yet observed, and do not afford explanations of any +known fact. This is surely a discovery of the greatest importance---quite +apart from any question as to whether the hypotheses +are actually wrong. + +I am not satisfied with the view so often expressed that the +sole aim of scientific theory is ``economy of thought.'' I cannot +reject the hope that theory is by slow stages leading us nearer +to the truth of things. But unless science is to degenerate into +idle guessing, the test of value of any theory must be whether +it expresses with as little redundancy as possible the facts +which it is intended to cover. Accidental truth of a conclusion +is no compensation for erroneous deduction. + +The relativity standpoint is then a discarding of certain +hypotheses, which are uncalled for by any known facts, and +stand in the way of an understanding of the simplicity of nature. +%% -----File: 040.png---Folio 30------- + + +\Chapter{II}{Relativity} + +\Quote{H.~Minkowski (1908).} +{The views of time and space, which I have to set forth, have their foundation +in experimental physics. Therein is their strength. Their tendency is revolutionary. +From henceforth space in itself and time in itself sink to mere shadows, +and only a kind of union of the two preserves an independent existence.}% +\index{Minkowski}% + +\First{There} are two parties to every observation---the observed and +the observer.% +\index{Observer and observed}% + +What we see depends not only on the object looked at, but +on our own circumstances---position, motion, or more personal +idiosyncracies. Sometimes by instinctive habit, sometimes by +design, we attempt to eliminate our own share in the observation, +and so form a general picture of the world outside us, +which shall be common to all observers. A small speck on the +horizon of the sea is interpreted as a giant steamer. From the +window of our railway carriage we see a cow glide past at fifty +miles an hour, and remark that the creature is enjoying a rest. +We see the starry heavens revolve round the earth, but decide +that it is really the earth that is revolving, and so picture the +state of the universe in a way which would be acceptable to an +astronomer on any other planet. + +The first step in throwing our knowledge into a common +stock must be the elimination of the various individual standpoints +and the reduction to some specified standard observer. +The picture of the world so obtained is none the less relative. +We have not eliminated the observer's share; we have only +fixed it definitely. + +To obtain a conception of the world from the point of view +of no one in particular is a much more difficult task. The +position of the observer can be eliminated; we are able to grasp +the conception of a chair as an object in nature---looked at all +round, and not from any particular angle or distance. We can +think of it without mentally assigning ourselves some position +with respect to it. This is a remarkable faculty, which has +evidently been greatly assisted by the perception of solid relief +%% -----File: 041.png---Folio 31------- +with our two eyes. But the motion of the observer is not +eliminated so simply. We had thought that it was accomplished; +but the discovery in the last chapter that observers with +different motions use different space- and time-reckoning shows +that the matter is more complicated than was supposed. It may +well require a complete change in our apparatus of description, +because all the familiar terms of physics refer primarily to the +relations of the world to an observer in some specified circumstances. + +\Pagelabel{31}% +Whether we are able to go still further and obtain a knowledge +of the world, which not merely does not particularise the +observer, but does not postulate an observer at all; whether if +such knowledge could be obtained, it would convey any intelligible +meaning; and whether it could be of any conceivable +interest to anybody if it could be understood---these questions +need not detain us now. The answers are not necessarily +negative, but they lie outside the normal scope of physics. + +The circumstances of an observer which affect his observations +are his position, motion and gauge of magnitude. More personal +idiosyncracies disappear if, instead of relying on his crude +senses, he employs scientific measuring apparatus. But scientific +apparatus has position, motion and size, so that these are still +involved in the results of any observation. There is no essential +distinction between scientific measures and the measures of the +senses. In either case our acquaintance with the external world +comes to us through material channels; the observer's body can +be regarded as part of his laboratory equipment, and, so far as +we know, it obeys the same laws. We therefore group together +perceptions and scientific measures, and in speaking of ``a +particular observer'' we include all his measuring appliances.% +\index{Perceptions, as crude measures}% + +Position, motion, magnitude-scale---these factors have a profound +influence on the aspect of the world to us. Can we form +a picture of the world which shall be a synthesis of what is seen +by observers in all sorts of positions, having all sorts of velocities, +and all sorts of sizes? %[** PP: Changed . to ?] +\index{Synthesis of appearances}% +As already stated we have accomplished +the synthesis of positions. We have two eyes, which have +dinned into our minds from babyhood that the world has to be +looked at from more than one position. Our brains have so far +responded as to give us the idea of solid relief, which enables us +%% -----File: 042.png---Folio 32------- +to appreciate the three-dimensional world in a vivid way that +would be scarcely possible if we were only acquainted with +strictly two-dimensional pictures. We not merely deduce the +three-dimensional world; we see it. But we have no such aid +in synthesising different motions. Perhaps if we had been +endowed with two eyes moving with different velocities our +brains would have developed the necessary faculty; we should +have perceived a kind of relief in a fourth dimension so as to +combine into one picture the aspect of things seen with different +motions. Finally, if we had had two eyes of different sizes, we +might have evolved a faculty for combining the points of view +of the mammoth and the microbe. + +It will be seen that we are not fully equipped by our senses +for forming an impersonal picture of the world. And it is +because the deficiency is manifest that we do not hesitate to +advocate a conception of the world which transcends the images +familiar to the senses. Such a world can perhaps be grasped, +but not pictured by the brain. It would be unreasonable to +limit our thought of nature to what can be comprised in sense-pictures. +As Lodge has said, our senses were developed by the +struggle for existence, not for the purpose of philosophising on +the world.% +\index{Lodge}% + +Let us compare two well-known books, which might be +described as elementary treatises on relativity, \textit{Alice in Wonderland} +and \textit{Gulliver's Travels}. Alice was continually changing size, +sometimes growing, sometimes on the point of vanishing altogether. +Gulliver remained the same size, but on one occasion +he encountered a race of men of minute size with everything in +proportion, and on another voyage a land where everything was +gigantic. It does not require much reflection to see that both +authors are describing the same phenomenon---a relative change +of scale of observer and observed. Lewis Carroll took what is +probably the ordinary scientific view, that the observer had +changed, rather than that a simultaneous change had occurred +to all her surroundings. But it would never have appeared like +that to Alice; she could not have ``stepped outside and looked +at herself,'' picturing herself as a giant filling the room. She +would have said that the room had unaccountably shrunk. +Dean Swift took the truer view of the human mind when he +%% -----File: 043.png---Folio 33------- +made Gulliver attribute his own changes to the things around +him; it never occurred to Gulliver that his own size had altered; +and, if he had thought of the explanation, he could scarcely +have accustomed himself to that way of thinking. But both +points of view are legitimate. The size of a thing can only be +imagined as relative to something else; and there is no means of +assigning the change to one end of the relation rather than the +other.% +\index{Relativity of size}% + +We have seen in the theory of the Michelson-Morley experiment +that, according to current physical views, our standard of +size---the rigid meas\-uring-rod---must change according to the +circumstances of its motion; and the aviator's adventures +illustrated a similar change in the standard of duration of time. +Certain rather puzzling irregularities have been discovered in +the apparent motions of the Sun, Mercury, Venus and the Moon; +but there is a strong family resemblance between these, which +leads us to believe that the real phenomenon is a failure of the +time-keeping of our standard clock, the Earth. Instances could +be multiplied where a change of the observer or his standards +produces or conceals changes in the world around him. + +The object of the relativity theory, however, is not to attempt +the hopeless task of apportioning responsibility between the +observer and the external world, but to emphasise that in our +ordinary description and in our scientific description of natural +phenomena the two factors are indissolubly united. All the +familiar terms of physics---length, duration of time, motion, +force, mass, energy, and so on---refer primarily to this relative +knowledge of the world; and it remains to be seen whether any +of them can be retained in a description of the world which is +not relative to a particular observer. + +Our first task is a description of the world independent of +the motion of the observer. The question of the elimination of +his gauge of magnitude belongs to a later development of the +theory discussed in \Chapref{XI}. +\index{Gauge!effect on observations}% +Let us draw a square $ABCD$ on +a sheet of paper, making the sides equal, to the best of our +knowledge. We have seen that an aviator flying at $161,000$ +miles a second in the direction $AB$, would judge that the sides +$AB$, $DC$ had contracted to half their length, so that for him +the figure would be an oblong. If it were turned through a right +%% -----File: 044.png---Folio 34------- +angle $AB$ and $DC$ would expand and the other two sides contract---in +his judgment. For us, the lengths of $AB$ and $AC$ are +equal; for him, one length is twice the other. Clearly length +cannot be a property inherent in our drawing; it needs the +specification of some observer. + +We have seen further that duration of time also requires that +an observer should be specified. The stationary observer and +the aviator disagreed as to whose cigar lasted the longer time. + +Thus \textit{length} and \textit{duration} are not things inherent in the +external world; they are relations of things in the external +world to some specified observer. +\index{Duration, not inherent in external world}% +\index{Length!relativity of}% +\index{Relativity!of length and duration}% +\index{Space!relativity of}% +If we grasp this all the mystery +disappears from the phenomena described in \Chapref{I}. When +the rod in the Michelson-Morley experiment is turned through +a right angle it contracts; that naturally gives the impression +that something has happened to the rod itself. Nothing whatever +has happened to the rod---the object in the external world. +Its length has altered, but length is not an intrinsic property of +the rod, since it is quite indeterminate until some observer is +specified. Turning the rod through a right angle has altered the +relation to the observer (implied in the discussion of the experiment); +but the rod itself, or the relation of a molecule at one +end to a molecule at the other, is unchanged. Measurement of +length and duration is a comparison with partitions of space +and time drawn by the observer concerned, with the help of +apparatus which shares his motion. Nature is not concerned +with these partitions; it has, as we shall see later, a geometry +of its own which is of a different type. + +Current physics has hitherto assumed that all observers are +not to be regarded as on the same footing, and that there is +some absolute observer whose judgments of length and duration +are to be treated with respect, because nature pays attention to +\textit{his} space-time partitions. He is supposed to be at rest in the +aether, and the aether materialises his space-partitions so that +they have a real significance in the external world. This is +sheer hypothesis, and we shall find it is unsupported by any +facts. Evidently our proper course is to pursue our investigations, +and call in this hypothetical observer only if we find there +is something which he can help to explain. + +We have been leading up from the older physics to the new +%% -----File: 045.png---Folio 35------- +outlook of relativity, and the reader may feel some doubt as to +whether the strange phenomena of contraction and time-retardation, +that were described in the last chapter, are to be +taken seriously, or are part of a \textit{reductio ad absurdum} argument. +The answer is that we believe that the phenomena do occur as +described; only the description (like that of all observed phenomena) +concerns the relations of the external world to some +observer, and not the external world itself. The startling +character of the phenomena arises from the natural but fallacious +inference that they involve intrinsic changes in the objects +themselves. + +We have been considering chiefly the observer's end of the +observation; we must now turn to the other end---the thing +observed. Although length and duration have no exact counterparts +in the external world, it is clear that there is a certain +ordering of things and events outside us which we must now +find more appropriate terms to describe. The order of events is +a four-fold order; we can arrange them as right-and-left, backwards-and-forwards, +up-and-down, sooner-and-later. +\index{Ordering of events in external world}% +An individual +may at first consider these as four independent orders, +but he will soon attempt to combine some of them. It is +recognised at once that there is no essential distinction between +right-and-left and backwards-and-forwards. The observer has +merely to turn through a right angle and the two are interchanged. +If he turns through a smaller angle, he has first to +combine them, and then to redivide them in a different way. +Clearly it would be a nuisance to continually combine and redivide; +so we get accustomed to the thought of leaving them +combined in a two-fold or two-dimensional order. The amalgamation +of up-and-down is less simple. There are obvious reasons +for considering this dimension of the world as fundamentally +distinct from the other two. Yet it would have been a great +stumbling-block to science if the mind had refused to combine +space into a three-dimensional whole. The combination has not +concealed the real distinction of horizontal and vertical, but has +enabled us to understand more clearly its nature---for what +phenomena it is relevant, and for what irrelevant. We can +understand how an observer in another country redivides the +combination into a different vertical and horizontal. We must +%% -----File: 046.png---Folio 36------- +now go further and amalgamate the fourth order, sooner-and-later. +This is still harder for the mind. It does not imply that +there is no distinction between space and time; but it gives a +fresh unbiassed start by which to determine what the nature of +the distinction is. + +The idea of putting together space and time, so that time is +regarded as a fourth dimension, is not new. But until recently +it was regarded as merely a picturesque way of looking at things +without any deep significance. We can put together time and +temperature in a thermometer chart, or pressure and volume +on an indicator-diagram. It is quite non-committal. But our +theory is going to lead much further than that. We can lay +two dimensional surfaces---sheets of paper---on one another till +we build up a three-dimensional block; but there is a difference +between a block which is a pile of sheets and a solid block of +paper. The solid block is the true analogy for the four-dimensional +combination of space-time; it does not separate naturally +into a particular set of three-dimensional spaces piled in time-order. +It can be redivided into such a pile; \textit{but it can be redivided +in any direction we please}. + +Just as the observer by changing his orientation makes a new +division of the two-dimensional plane into right-and-left, backwards-and-forwards---just +as the observer by changing his +longitude makes a new division of three-dimensional space into +vertical and horizontal---so the observer by \textit{changing his motion} +makes a new division of the four-dimensional order into time +and space.% +\index{Four-dimensional order}% + +This will be justified in detail later; it indicates that observers +with different motions will have different time and space-reckoning---a +conclusion we have already reached from another +point of view. + +Although different observers separate the four orders differently, +they all agree that the order of events is four-fold; and +it appears that this undivided four-fold order is the same for +all observers. We therefore believe that it is inherent in the +external world; it is in fact the synthesis, which we have been +seeking, of the appearances seen by observers having all sorts of +positions and all sorts of (uniform) motions. It is therefore to +be regarded as a conception of the real world not relative to any +particularly circumstanced observer. +%% -----File: 047.png---Folio 37------- + +The term ``real world'' is used in the ordinary sense of physics, +without any intention of prejudging philosophical questions as +to reality. +\index{Real world of physics}% +It has the same degree of reality as was formerly +attributed to the three-dimensional world of scientific theory or +everyday conception, which by the advance of knowledge it +replaces. As I have already indicated, it is merely the accident +that we are not furnished with a pair of eyes in rapid relative +motion, which has allowed our brains to neglect to develop a +faculty for visualising this four-dimensional world as directly +as we visualise its three-dimensional section. + +It is now easy to see that length and duration must be the +components of a single entity in the four-dimensional world of +space-time. Just as we resolve a structure into plan and elevation, +so we resolve extension in the four-dimensional world into +length and duration. The structure has a size and shape +independent of our choice of vertical. Similarly with things in +space-time. Whereas length and duration are relative, the +single ``extension'' of which they are components has an absolute +significance in nature, independent of the particular decomposition +into space and time separately adopted by the observer.% +\index{Extension in four dimensions}% + +Consider two events; for example, the stroke of one o'clock +and the stroke of two o'clock by Big Ben. These occupy two +points in space-time, and there is a definite separation between +them. An observer at Westminster considers that they occur at +the same place, and that they are separated by an hour in time; +thus he resolves their four-dimensional separation into zero +distance in space and one hour distance in time. An observer +on the sun considers that they do not occur at the same place; +they are separated by about $70,000$ miles, that being the distance +travelled by the earth in its orbital motion with respect to the +sun. It is clear that he is not resolving in quite the same directions +as the terrestrial observer, since he finds the space-component +to be $70,000$ miles instead of zero. But if he alters one +component he must necessarily alter the other; so he will make +the time-component differ slightly from an hour. By analogy +with resolution into components in three-dimensions, we should +expect him to make it less than an hour---having, as it were, +borrowed from time to make space; but as a matter of fact he +makes it longer. This is because space-time has a different +%% -----File: 048.png---Folio 38------- +geometry, which will be described later. Our present point is +that there is but one separation of two events in four dimensions, +which can be resolved in any number of ways into the components +length and duration. + +We see further how motion must be purely relative. +\index{Relativity!of motion}% +Take +two events $A$ and~$B$ in the history of one particle. We can choose +any direction as the time-direction; let us choose it along~$AB$. +Then $A$ and~$B$ are separated only in time and not in space, so +the particle is at rest. If we choose a slightly inclined time-direction, +the separation $AB$ will have a component in space; +the two events then do not occur at the same place, that is to +say, the particle has moved. The negation of absolute motion +is thus associated with the possibility of choosing the time-direction +in any way we please. What determines the separation +of space and time for any particular observer can now be seen. +Let the observer place himself so that he is, to the best of his +knowledge, at rest. If he is a normal human being, he will +seat himself in an arm-chair; if he is an astronomer, he will +place himself on the sun or at the centre of the stellar universe. +Then all the events happening directly to him will in his opinion +occur at the same place. Their separation will have no space-component, +and they will accordingly be ranged solely in the +time-direction. This chain of events, marking his track through +the four-dimensional world, will be his time-direction. Each +observer bases his separation of space and time on his own track +through the world.% +\index{Time!depends on observer's track}% + +Since any separation of space and time is admissible, it is +possible for the astronomer to base his space and time on the +track of a solar observer instead of that of a terrestrial observer; +but it must be remembered that in practice the space and time +of the solar observer have to be inferred indirectly from those +of the terrestrial observer; and, if the corrections are made +according to the crude methods hitherto employed, they may +be inferred wrongly (if extreme accuracy is needed). + +The most formidable objection to this relativist view of the +world is the aether difficulty. We have seen that uniform motion +through the aether cannot be detected by experiment, and +therefore it is entirely in accordance with experiment that such +motion should have no counterpart in the four-dimensional +%% -----File: 049.png---Folio 39------- +world. Nevertheless, it would almost seem that such motion +must logically exist, if the aether exists; +\index{Aether!non-material nature of}% +and, even at the +expense of formal simplicity, it ought to be exhibited in any +theory which pretends to give a complete account of what is +going on in nature. If a substantial aether analogous to a +material ocean exists, it must rigidify, as it were, a definite +space; and whether the observer or whether nature pays any +attention to that space or not, a fundamental separation of +space and time must be there. Some would cut the knot by +denying the aether altogether. We do not consider that desirable, +or, so far as we can see, possible; but we do deny that the aether +need have such properties as to separate space and time in the +way supposed. It seems an abuse of language to speak of a +division existing, when nothing has ever been found to pay any +attention to the division. + +Mathematicians of the nineteenth century devoted much time +to theories of elastic solid and other material aethers. Waves of +light were supposed to be actual oscillations of this substance; +it was thought to have the familiar properties of rigidity and +density; it was sometimes even assigned a place in the table of +the elements. The real death-blow to this materialistic conception +of the aether was given when attempts were made to explain +matter as some state in the aether. For if matter is vortex-motion +or beknottedness in aether, the aether cannot be matter--some +state in itself. If any property of matter comes to be +regarded as a thing to be explained by a theory of its structure, +clearly that property need not be attributed to the aether. +If physics evolves a theory of matter which explains some +property, it stultifies itself when it postulates that the same +property exists unexplained in the primitive basis of matter. + +Moreover the aether has ceased to take any very active part +in physical theory and has, as it were, gone into reserve. A +modern writer on electromagnetic theory will generally start +with the postulate of an aether pervading all space; he will then +explain that at any point in it there is an electromagnetic vector +whose intensity can be measured; henceforth his sole dealings +are with this vector, and probably nothing more will be heard +of the aether itself. In a vague way it is supposed that this +vector represents some condition of the aether, and we need not +%% -----File: 050.png---Folio 40------- +dispute that without some such background the vector would +scarcely be intelligible---but the aether is now only a background +and not an active participant in the theory. + +There is accordingly no reason to transfer to this vague background +of aether the properties of a material ocean. Its properties +must be determined by experiment, not by analogy. In particular +there is no reason to suppose that it can partition out space in +a definite way, as a material ocean would do. We have seen in +the Prologue that natural geometry depends on laws of matter; +therefore it need not apply to the aether. Permanent identity +of particles is a property of matter, which Lord Kelvin sought +to explain in his vortex-ring hypothesis. +\index{Identity, permanent} % [** PP: Added comma] +\index{Permanent identity}% +This abandoned +hypothesis at least teaches us that permanence should not be +regarded as axiomatic, but may be the result of elaborate constitution. +There need not be anything corresponding to +permanent identity in the constituent portions of the aether; +we cannot lay our finger at one spot and say ``this piece of +aether was a few seconds ago over there.'' Without any continuity +of identity of the aether motion through the aether +becomes meaningless; and it seems likely that this is the true +reason why no experiment ever reveals it. + +This modern theory of the relativity of all uniform motion is +essentially a return to the original Newtonian view, temporarily +disturbed by the introduction of aether problems; for in Newton's +dynamics uniform motion of the whole system has not---and no +one would expect it to have---any effect. But there are considerable +difficulties in the limitation to uniform motion. Newton +himself seems to have appreciated the difficulty; but the experimental +evidence appeared to him to be against any extension +of the principle. Accordingly Newton's laws of mechanics are +not of the general type in which it is unnecessary to particularise +the observer; they hold only for observers with a special kind +of motion which is described as ``unaccelerated.'' +\index{Newton!relativity for uniform motion}% +\index{Relativity, Newtonian}% +The only +definition of this epithet that can be given is that an ``unaccelerated'' +observer is one for whom Newton's laws of motion +hold. On this theory, the phenomena are not indifferent to an +acceleration or non-uniform motion of the whole system. Yet +an absolute non-uniform motion through space is just as impossible +to imagine as an absolute uniform motion. The partial +%% -----File: 051.png---Folio 41------- +relativity of phenomena makes the difficulty all the greater. +If we deny a fundamental medium with continuous identity of +its parts, motion uniform or non-uniform should have no +significance; if we admit such a medium, motion uniform or +non-uniform should be detectable; but it is much more difficult +to devise a plan of the world according to which uniform motion +has no significance and non-uniform motion is significant.\Pagelabel{41} + +It is through experiment that we have been led back to the +principle of relativity for uniform motion. In seeking some kind +of extension of this principle to accelerated motion, we are led +by the feeling that, having got so far, it is difficult and arbitrary +to stop at this point. We now try to conceive a system of nature +for which all kinds of motion of the observer are indifferent. +It will be a completion of our synthesis of what is perceived by +observers having all kinds of motions with respect to one +another, removing the restriction to uniform motion. The +experimental tests must follow after the consequences of this +generalisation have been deduced. + +The task of formulating such a theory long appeared impossible. +It was pointed out by Newton that, whereas there is no criterion +for detecting whether a body is at rest or in uniform motion, it +is easy to detect whether it is in rotation. For example the +bulge of the earth's equator is a sign that the earth is rotating, +since a plastic body at rest would be spherical.% +\index{Newton!absolute rotation}% + +This problem of rotation affords a hint as to the cause of the +incomplete relativity of Newtonian mechanics. The laws of +motion are formulated with respect to an unaccelerated observer, +and do not apply to a frame of reference rotating with the earth. +Yet mathematicians frequently do use a rotating frame. Some +modification of the laws is then necessary; and the modification +is made by introducing a centrifugal force---not regarded as a +real force like gravitation, but as a mathematical fiction employed +to correct for the improper choice of a frame of reference. +\index{Centrifugal Force!compared with gravitation}% +The bulge of the earth's equator may be attributed indifferently +to the earth's rotation or to the outward pull of the centrifugal +force introduced when the earth is regarded as non-rotating. + +Now it is generally assumed that the centrifugal force is +something \textit{sui generis}, which could always be distinguished +experimentally from any other natural phenomenon. If then +%% -----File: 052.png---Folio 42------- +on choosing a frame of reference we find that a centrifugal force +is detected, we can at once infer that the frame of reference is +a ``wrong'' one; rotating and non-rotating frames can be distinguished +by experiment, and rotation is thus strictly absolute. +But this assumes that the observed effects of centrifugal force +cannot be produced in any other way than by rotation of the +observer's frame of reference. If once it is admitted that centrifugal +force may not be completely distinguishable by experiment +from another kind of force---gravitation---perceived even by +Newton's unaccelerated observer, the argument ceases to apply. +We can never determine exactly how much of the observed field +of force is centrifugal force and how much is gravitation; and +we cannot find experimentally any definite standard that is to +be considered absolutely non-rotating. + +The question then, whether there exists a distinction between +``right'' frames of reference and ``wrong'' frames, turns on +whether the use of a ``wrong'' frame produces effects experimentally +distinguishable from any natural effects which can be +perceived when a ``right'' frame is used. +\index{Frames@Frames of reference, ``right'' and ``wrong''}% +If there is no such +difference, all frames may be regarded as on the same footing +and equally right. In that case we can have a complete relativity +of natural phenomena. Since the effect of departing from +Newton's standard frame is the introduction of a field of force, +this generalised relativity theory must be largely occupied with +the nature of fields of force. + +The precise meaning of the statement that all frames of +reference are on the same footing is rather difficult to grasp. +We believe that there are absolute things in the world---not only +matter, but certain characteristics in empty space or aether. +In the atmosphere a frame of reference which moves with the +air is differentiated from other frames moving in a different +manner; this is because, besides discharging the normal functions +of a frame of reference, the air-frame embodies certain of the +absolute properties of the matter existing in the region. +Similarly, if in empty space we choose a frame of reference +which more or less follows the lines of the absolute structure in +the region, the frame will usurp some of the absolute qualities +of that structure. What we mean by the equivalence of all +frames is that they are not differentiated by any qualities +%% -----File: 053.png---Folio 43------- +formerly supposed to be intrinsic in the frames themselves---rest, +rectangularity, acceleration---independent of the absolute +structure of the world that is referred to them. Accordingly the +objection to attributing absolute properties to Newton's frame +of reference is not that it is impossible for a frame of reference +to acquire absolute properties, but that the Newtonian frame +has been laid down on the basis of relative knowledge without +any attempt to follow the lines of absolute structure. + +Force, as known to us observationally, is like the other +quantities of physics, a relation. The force, measured with a +spring-balance, for example, depends on the acceleration of the +observer holding the balance; and the term may, like length +and duration, have no exact counterpart in a description of +nature independent of the observer. Newton's view assumes +that there is such a counterpart, an active cause in nature +which is identical with the force perceived by his standard +unaccelerated observer. Although any other observer perceives +this force with additions of his own, it is implied that the +original force in nature and the observer's additions can in some +way be separated without ambiguity. There is no experimental +foundation for this separation, and the relativity view is that +a field of force can, like length and duration, be nothing but +a link between nature and the observer. There is, of course, +something at the far end of the link, just as we found an +extension in four dimensions at the far end of the relations +denoted by length and duration. We shall have to study the +nature of this unknown whose relation to us appears as force. +Meanwhile we shall realise that the alteration of perception of +force by non-uniform motion of the observer, as well as the +alteration of the perception of length by his uniform motion, is +what might be expected from the nature of these quantities as +relations solely.% +\index{Force!relativity of}% +\index{Relativity of Force}% + +We proceed now to a more detailed study of the four-dimensional +world, of the things which occur in it, and of the laws by +which they are regulated. It is necessary to dive into this +absolute world to seek the truth about nature; but the physicist's +object is always to obtain knowledge which can be applied to +the relative and familiar aspect of the world. The absolute +world is of so different a nature, that the relative world, with +%% -----File: 054.png---Folio 44------- +which we are acquainted, seems almost like a dream. But if +indeed we are dreaming, our concern is with the baseless fabric +of our vision. We do not suggest that physicists ought to +translate their results into terms of four-dimensional space for +the empty satisfaction of working in the realm of reality. It is +rather the opposite. They explore the new field and bring back +their spoils---a few simple generalisations---to apply them to the +practical world of three-dimensions. Some guiding light will be +given to the attempts to build a scheme of things entire. For +the rest, physics will continue undisturbedly to explore the +relative world, and to employ the terms applicable to relative +knowledge, but with a fuller appreciation of its relativity. +%% -----File: 055.png---Folio 45------- + + +\Chapter{III}{The World of Four Dimensions} + +\Quote{H.~G. Wells, \textit{The Time Machine.}} +{Here is a portrait of a man at eight years old, another at fifteen, another at +seventeen, another at twenty-three, and so on. All these are evidently sections, +as it were, Three-Dimensional representations of his Four-Dimensional being, +which is a fixed and unalterable thing.}% +\index{Four-dimensional space-time!geometry of}% + +\First{The} distinction between horizontal and vertical is not an +illusion; and the man who thinks it can be disregarded is likely +to come to an untimely end. Yet we cannot arrive at a comprehensive +view of nature unless we combine horizontal and +vertical dimensions into a three-dimensional space. By doing +this we obtain a better idea of what the distinction of horizontal +and vertical really is in those cases where it is relevant, e.g.\ the +phenomena of motion of a projectile. We recognise also that +vertical is not a universally differentiated direction in space, as +the flat-earth philosophers might have imagined. + +Similarly by combining the time-ordering and space-ordering +of the events of nature into a single order of four dimensions, +we shall not only obtain greater simplicity for the phenomena +in which the separation of time and space is irrelevant, but we +shall understand better the nature of the differentiation when it +is relevant. + +A point in this space-time, that is to say a given instant at +a given place, is called an ``event.'' +\index{Event, definition of}% +\index{Point-event}% +\index{Space-time}% +An event in its customary +meaning would be the physical happening which occurs at and +identifies a particular place and time. However, we shall use +the word in both senses, because it is scarcely possible to think +of a point in space-time without imagining some identifying +occurrence. + +In the ordinary geometry of two or three dimensions, the +distance between two points is something which can be measured, +usually with a rigid scale; it is supposed to be the same for all +observers, and there is no need to specify horizontal and vertical +directions or a particular system of coordinates. In four-dimensional +space-time there is likewise a certain extension or +%% -----File: 056.png---Folio 46------- +generalised distance between two events, of which the distance +in space and the separation in time are particular components. +This extension in space and time combined is called the +``interval'' between the two events; +\index{Extension in four dimensions}% +\index{Interval}% +it is the same for all +observers, however they resolve it into space and time separately. +We may think of the interval as something intrinsic in external +nature---an absolute relation of the two events, which postulates +no particular observer. Its practical measurement is suggested +by analogy with the distance of two points in space. + +%[Illustration: \textsc{Fig}. 2.] +\begin{wrapfigure}{l}{1.75in} +\Graphic[2]{1.75in}{056a} +\end{wrapfigure} +In two dimensions on a plane, two points $P_1$, $P_2$ (\Figref{2}) can +be specified by their rectangular +coordinates $(x_1, y_1)$ and $(x_2, y_2)$, +when arbitrary axes have been selected. +In the figure, $OX_1 = x_1$, +$OY_1 = y_1$, etc. We have +\begin{align*} +P_1 P_2{}^2 + &= P_1 M^2 + M P_2{}^2 \\ + &= X_1 X_2{}^2 + Y_1 Y_2{}^2 \\ + &= (x_2 - x_1)^2 + (y_2 - y_1)^2, +\end{align*} +so that if $s$ is the distance between +$P_1$ and~$P_2$ +\[ +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. +\] + +The extension to three dimensions is, as we should expect, +\[ +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2. +\] +Introducing the times of the events $t_1$, $t_2$, we should naturally +expect that the interval in the four-dimensional world would +be given by +\[ +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (t_2 - t_1)^2. +\] + +An important point arises here. It was, of course, assumed +that the same scale was used for measuring $x$ and $y$ and~$z$. But +how are we to use the same scale for measuring~$t$? We cannot +use a scale at all; some kind of clock is needed. The most +natural connection between the measure of time and length is +given by the fact that light travels $300,000$ kilometres in +$1$~second. For the four-dimensional world we shall accordingly +regard $1$~second as the equivalent of $300,000$ kilometres, and +measure lengths and times in seconds or kilometres indiscriminately; +in other words we make the velocity of light the unit of +%% -----File: 057.png---Folio 47------- +velocity. It is not essential to do this, but it greatly simplifies +the discussion. + +Secondly, the formulae here given for $s^2$ are the characteristic +formulae of Euclidean geometry. So far as three-dimensional +space is concerned the applicability of Euclidean geometry is +very closely confirmed by experiment. But space-time is not +Euclidean; it does, however, conform (at least approximately) +to a very simple modification of Euclidean geometry indicated +by the corrected formula +\index{Euclidean geometry}% +\IndexExtra{Geometry!Euclidean}% +\[ +s^2 = (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 - (t_2-t_1)^2. +\] +There is only a sign altered; but that minus sign is the secret +of the differences of the manifestations of time and space in +nature. + +This change of sign is often found puzzling at the start. We +could not define~$s$ by the expression originally proposed (with +the positive sign), because the expression does not define anything +objective. Using the space and time of one observer, one +value is obtained; for another observer, another value is +obtained. But if $s$ is defined by the expression now given, it is +found that the same result is obtained by all observers\footnote% + {Appendix, \Noteref{2}.}. +\Pagelabel{note2}% +The +quantity~$s$ is thus something which concerns solely the two +events chosen; we give it a name---the interval between the two +events. In ordinary space the distance between two points is +the corresponding property, which concerns only the two points +and not the extraneous coordinate system of location which is +used. Hence interval, as here defined, is the analogue of distance; +and the analogy is strengthened by the evident resemblance +of the formula for~$s$ in both cases. Moreover, when the difference +of time vanishes, the interval reduces to the distance. But the +discrepancy of sign introduces certain important differences. +These differences are summed up in the statement that the +geometry of space is Euclidean, but the geometry of space-time +is semi-Euclidean or ``hyperbolic.'' +\index{Geometry!hyperbolic}% +\index{Geometry!semi-Euclidean}% +\index{Hyperbolic geometry}% +The association of a geometry +with any continuum always implies the existence of some +uniquely measurable quantity like interval or distance; in +ordinary space, geometry without the idea of distance would be +meaningless. + +%% -----File: 058.png---Folio 48------- + +For the moment the difficulty of thinking in terms of an +unfamiliar geometry may be evaded by a dodge. Instead of +real time~$t$, consider imaginary time~$\tau$; that is to say, let +\index{Imaginary time}% +\index{Time!imaginary}% +\begin{DPgather*} +t = \tau \sqrt{-1}. \\ +\lintertext{Then} +(t_2 - t_1)^2 = -(\tau_2 - \tau_1)^2, \\ +\intertext{so that} +s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (\tau_2 - \tau_1)^2. +\end{DPgather*} +Everything is now symmetrical and there is no distinction +between~$\tau$ and the other variables. The continuum formed of +space and imaginary time is completely isotropic for all measurements; +no direction can be picked out in it as fundamentally +distinct from any other. + +The observer's separation of this continuum into space and +time consists in slicing it in some direction, viz.\ that perpendicular +to the path along which he is himself travelling. The +section gives three-dimensional space at some moment, and the +perpendicular dimension is (imaginary) time. Clearly the slice +may be taken in any direction; there is no question of a true +separation and a fictitious separation. There is no conspiracy +of the forces of nature to conceal our absolute motion---because, +looked at from this broader point of view, there is nothing to +conceal. The observer is at liberty to orient his rectangular axes +of $x$, $y$, $z$ and~$\tau$ arbitrarily, just as in three-dimensions he can +orient his axes of $x$, $y$, $z$ arbitrarily. + +It can be shown that the different space and time used by +the aviator in \Chapref{I} correspond to an orientation of the +time-axis along his own course in the four-dimensional world, +whereas the ordinary time and space are given when the time-axis +is oriented along the course of a terrestrial observer. The +FitzGerald contraction and the change of time-measurement +are given exactly by the usual formulae for rotation of rectangular +axes\footnote{Appendix, \Noteref{3}.}. +\Pagelabel{note3} + +It is not very profitable to speculate on the implication of the +mysterious factor~$\sqrt{-1}$, which seems to have the property of +turning time into space. It can scarcely be regarded as more +than an analytical device. To follow out the theory of the four-% +%% -----File: 059.png---Folio 49------- +dimensional world in more detail, it is necessary to return to +real time, and face the difficulties of a strange geometry. + +Consider a particular observer, $S$, and represent time according +to his reckoning by distance up the page parallel to~$OT$. One +dimension of his space will be represented by horizontal distance +parallel to~$OX$; another will stand out at right angles from the +page; and the reader must imagine the third as best he can. +Fortunately it will be sufficient for us to consider only the one +dimension of space $OX$ and deal with the phenomena of ``line-land,'' +i.e.\ we limit ourselves to motion to and fro in one straight +line in space. +\begin{figure*}[hbt] +\begin{center} +\Graphic[3]{3.5in}{059a} +\end{center} +\end{figure*} +%[Illustration: \textsc{Fig}. 3.] + +The two lines $U'OU$, $V'OV$, at $45°$ to the axes, represent the +tracks of points which progress $1$~unit horizontally (in space) +for $1$~unit vertically (in time); thus they represent points moving +with unit velocity. We have chosen the velocity of light as unit +velocity; hence $U'OU$, $V'OV$ will be the tracks of pulses of light +in opposite directions along the straight line. + +Any event $P$ within the sector $UOV$ is indubitably after the +event~$O$, whatever system of time-reckoning is adopted. For it +would be possible for a material particle to travel from~$O$ to~$P$, +the necessary velocity being less than that of light; and no +%% -----File: 060.png---Folio 50------- +rational observer would venture to state that the particle had +completed its journey before it had begun it. It would, in fact, +be possible for an observer travelling along $NP$ to receive a +light-signal or wireless telegram announcing the event~$O$, just +as he reached~$N$, since $ON$ is the track of such a message; and +then after the time $NP$ he would have direct experience of the +event~$P$. To have actual evidence of the occurrence of one +event before experiencing the second is a clear proof of their +absolute order in nature, which should convince not merely +the observer concerned but any other observer with whom he +can communicate. + +% [** PP: 8.75pt overfull at 5in] +Similarly events in the sector $U'OV'$ are indubitably before +the event~$O$. + +With regard to an event~$P'$ in the sector $UOV'$ or~$VOU'$ we +cannot assert that it is absolutely before or after~$O$. According +to the time-reckoning of our chosen observer~$S$, $P'$ is after~$O$, +because it lies above the line~$OX$; but there is nothing absolute +about this. The track~$OP'$ corresponds to a velocity greater +than that of light, so that we know of no particle or physical +impulse which could follow the track. An observer experiencing +the event $P'$ could not get news of the event~$O$ by any known +means until after~$P'$ had happened. The order of the two events +can therefore only be inferred by estimating the delay of the +message and this estimate will depend on the observer's mode +of reckoning space and time. + +% [** PP: 4.3pt overfull at 5in] +Space-time is thus divided into three zones with respect to +the event~$O$. $U'OV'$ belongs to the indubitable past. $UOV$ is +the indubitable future. $UOV'$ and $VOU'$ are (absolutely) neither +past nor future, but simply ``elsewhere.'' +\index{Elsewhere@``Elsewhere''}% +It may be remarked +that, as we have no means of identifying points in space as ``the +same point,'' and as the events $O$ and~$P$ might quite well happen +to the same particle of matter, the events are not necessarily to +be regarded as in different places, though the observer~$S$ will +judge them so; but the events $O$ and~$P'$ cannot happen to the +same particle, and no observer could regard them as happening +at the same place. The main interest of this analysis is that it +shows that the arbitrariness of time-direction is not inconsistent +with the existence of regions of absolute past and future.% +\index{Absolute past and future}% +\index{Future, absolute}% +\index{Past, absolute}% + +Although there is an absolute past and future, there is between +%% -----File: 061.png---Folio 51------- +them an extended neutral zone; and simultaneity of events at +different places has no absolute meaning. +\index{Absolute simultaneity}% +\index{Simultaneity}% +For our selected +observer all events along~$OX$ are simultaneous with one another; +for another observer the line of events simultaneous with~$O$ +would lie in a different direction. The denial of absolute +simultaneity is a natural complement to the denial of absolute +motion. The latter asserts that we cannot find out what is the +same place at two different times; the former that we cannot +find out what is the same time at two different places. It is +curious that the philosophical denial of absolute motion is +readily accepted, whilst the denial of absolute simultaneity +appears to many people revolutionary. + +The division into past and future (a feature of time-order +which has no analogy in space-order) is closely associated with +our ideas of causation and free will. % [** PP: Removed hyphen] +\index{Causation and free will}% +\index{Free will}% [** PP: Added space] +\index{Time!past and future}% +In a perfectly determinate +scheme the past and future may be regarded as lying mapped +out---as much available to present exploration as the distant +parts of space. Events do not happen; they are just there, and +we come across them. ``The formality of taking place'' is merely +the indication that the observer has on his voyage of exploration +passed into the absolute future of the event in question; and it +has no important significance. We can be aware of an eclipse +in the year 1999, very much as we are aware of an unseen +companion to Algol. Our knowledge of things \textit{where} we are not, +and of things \textit{when} we are not, is essentially the same---an +inference (sometimes a mistaken inference) from brain impressions, +including memory, \textit{here} and \textit{now}. + +So, if events are determinate, there is nothing to prevent a +person from being \textit{aware} of an event before it happens; and an +event may cause other events previous to it. Thus the eclipse +of the Sun in May 1919 caused observers to embark in March. +It may be said that it was not the eclipse, but the calculations +of the eclipse, which caused the embarkation; but I do not +think any such distinction is possible, having regard to the +indirect character of our acquaintance with all events except +those at the precise point of space where we stand. A detached +observer contemplating our world would see some events +apparently causing events in their future, others apparently +causing events in their past---the truth being that all are linked +%% -----File: 062.png---Folio 52------- +by determinate laws, the so-called causal events being merely +conspicuous foci from which the links radiate. + +The recognition of an absolute past and future seems to +depend on the possibility of events which are not governed by +a determinate scheme. If, say, the event~$O$ is an ultimatum, +and the person describing the path~$NP$ is a ruler of the country +affected, then it may be manifest to all observers that it is his +knowledge of the actual occurrence of the event~$O$ which has +caused him to create the event~$P$. $P$~must then be in the absolute +%[Illustration: \textsc{Fig}. 4.] +\begin{figure*}[hbt]% +\begin{center}% +\Graphic[4]{3.5in}{062a} +\end{center}% +\end{figure*}% +future of~$O$, and, as we have seen, must lie in the sector~$UOV$. +But the inference is only permissible, if the event~$P$ could be +determined by the event~$O$, and was not predetermined by +causes anterior to both---if it was possible for it to happen or +not, consistently with the laws of nature. Since physics does +not attempt to cover indeterminate events of this kind, the +distinction of absolute past and future is not directly important +for physics; but it is of interest to show that the theory of four-dimensional +space-time provides an absolute past and future, in +accordance with common requirements, although this can +usually be ignored in applications to physics. + +%% -----File: 063.png---Folio 53------- + +Consider now all the events which are at an interval of one +unit from~$O$, according to the definition of the interval~$s$ +\Pagelabel{53} +\[ +s^2 = - (x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2 + (t_2 - t_1)^2. +\Tag{1} +\] +We have changed the sign of $s^{2}$, because usually (though not +always) the original $s^{2}$ would have come out negative. In +Euclidean space points distant a unit interval lie on a circle; +but, owing to the change in geometry due to the altered sign +of $(t_{2} - t_{1})^{2}$, they now lie on a rectangular hyperbola with two +branches $KLM$, $K'L'M'$. Since the interval is an absolute +quantity, all observers will agree that these points are at unit +interval from~$O$. + +Now make the following construction:---draw a straight line +$OFT_{1}$ to meet the hyperbola in~$F$; draw the tangent~$FG$ at~$F$, +meeting the light-line $U'OU$ in~$G$; complete the parallelogram +$OFGH$; produce~$OH$ to~$X_{1}$. We now assert that an observer +$S_{1}$ who chooses $OT_{1}$ for his time-direction will regard $OX_{1}$ as +his space direction and will consider $OF$ and~$OH$ to be the units +of time and space. + +The two observers make their partitions of space and time +in different ways, as illustrated in Figs.~5 and~6, where in each +case the partitions are at unit distance (in space and time) +according to the observers' own reckoning. The same diagram +of events in the world will serve for both observers; $S_{1}$ merely +removes $S$'s partitions and overlays his own, locating the events +in his space and time accordingly. It will be seen at once that +the lines of unit velocity---progress of one unit of space for one +unit of time---agree, so that the velocity of a pulse of light is +unity for both observers. It can be shown from the properties +of the hyperbola that the locus of points at any interval~$s$ from~$O$, +given by \Eqref{equation}{1}, viz. +\[ +s^2 = (t - t_0)^2 - (x - x_0)^2, +\] +is the same locus (a hyperbola) for both systems of reckoning +$x$ and~$t$. The two observers will always agree on the measures +of intervals, though they will disagree about lengths, durations, +and the velocities of everything except light. This rather complex +transformation is mathematically equivalent to the simple +rotation of the axes required when imaginary time is used. + +It must not be supposed that there is any natural distinction +%% -----File: 064.png---Folio 54------- +corresponding to the difference between the square-partitions +of observer $S$ and the diamond-shaped partitions of observer~$S_1$. +We might say that $S_1$ transplants the space-time world unchanged +from \Figref{5} to \Figref{6}, and then distorts it until the +diamonds shown become squares; or we might equally well start +with this distorted space-time, partitioned by $S_1$ into squares, +and then $S$'s partitions would be represented by diamonds. +It cannot be said that either observer's space-time is distorted +absolutely, but one is distorted relatively to the other. It is the +relation of \textit{order} which is intrinsic in nature, and is the same +both for the squares and diamonds; \textit{shape} is put into nature by +the observer when he has chosen his partitions. +\index{Ordering of events in external world}% +\index{Space-time!partitions of}% +\begin{figure*}[htb] +\begin{center} +\Graphic[5]{4.5in}{064a} +\Figlabel{6}% +\end{center} +\end{figure*} +% [** PP: Moved up one paragraph] +%[Illustration: \textsc{Fig}. 5.] +%[Illustration: \textsc{Fig}. 6.] + +We can now deduce the FitzGerald contraction. +\index{Contraction, FitzGerald}% +Consider +a rod of unit length at rest relatively to the observer~$S$. The +two extremities are at rest in his space, and consequently remain +on the same space-partitions; hence their tracks in four dimensions +$PP'$, $QQ'$ (\Figref{7}) are entirely in the time-direction. The +real rod in nature is the four-dimensional object shown in section +as $P'PQQ'$. Overlay the same figure with $S_1$'s space and time +partitions, shown by the dotted lines. Taking a section at any +one ``time,'' the instantaneous rod is $P_1Q_1$, viz.\ the section of +$P'PQQ'$ by $S_1$'s time-line. Although on paper $P_1Q_1$ is actually +longer than $PQ$, it is seen that it is a little shorter than one of +$S_1$'s space-partitions; and accordingly $S_1$ judges that it is less +%% -----File: 065.png---Folio 55------- +than one unit long---it has contracted on account of its motion +relative to him. +%[Illustration: Fig. 7.] +\begin{figure*}[htb] +\begin{center} +\Graphic[7]{4in}{065a} +\end{center} +\end{figure*} + +Similarly $RR'-SS'$ is a rod of unit length at rest relatively +to $S_1$. Overlaying $S$'s partitions we see that it occupies $R_1S_1$ at +a particular instant for $S$; and this is less than one of $S$'s +partitions. Thus $S$ judges it to have contracted on account of +its motion relative to him.% +\index{Retardation of time}% + +%[Illustration: Fig. 8.] +\begin{wrapfigure}{r}{2.5in} +\Graphic[8]{2.5in}{065b} +\end{wrapfigure} +In the same way we can illustrate the problem of the duration +of the cigar; each observer +believed the other's cigar to +last the longer time. Taking +$LM$ (\Figref{8}) to represent the +duration of $S$'s cigar (two +units), we see that in $S_1$'s +reckoning it reaches over a +little more than two time-partitions. +Moreover it has not +kept to one space-partition, +i.e.\ it has moved. Similarly $L'N'$ is the duration of $S_1$'s cigar +(two time-units for him); and it lasts a little beyond two unit-partitions +in $S$'s time-reckoning. (Note, in comparing the two +diagrams, $L', M', N'$ are the same points as $L, M, N$.) + +If in \Figref{4} we had taken the line $OT_1$ very near to $OU$, our +%% -----File: 066.png---Folio 56------- +diamonds would have been very elongated, and the unit-divisions +$OF, OH$ very large. This kind of partition would be +made by an observer whose course through the world is $OT_1$, +and who is accordingly travelling with a velocity approaching +that of light relative to $S$. +\index{Light, velocity of!system moving with}% +\index{Velocity of light!system moving with}% +In the limit, when the velocity +reaches that of light, both space-unit and time-unit become +infinite, so that in the natural units for an observer travelling +with the speed of light, all the events in the finite experience of +$S$ take place ``in no time'' and the size of every object is zero. +This applies, however, only to the two dimensions $x$ and $t$; the +space-partitions parallel to the plane of the paper are not +affected by this motion along $x$. Consequently for an observer +travelling with the speed of light all ordinary objects become +two-dimensional, preserving their lateral dimensions, but infinitely +thin longitudinally. The fact that events take place ``in +no time'' is usually explained by saying that the inertia of any +particle moving with the velocity of light becomes infinite so +that all molecular processes in the observer must stop; many +things may happen in $S$'s world in a twinkling of an eye---of +$S_1$'s eye.% +\index{Inertia!infinite}% + +However successful the theory of a four-dimensional world +may be, it is difficult to ignore a voice inside us which whispers +``At the back of your mind, you know that a fourth dimension +is all nonsense.'' +\index{Four-dimensional order}% +I fancy that that voice must often have had a +busy time in the past history of physics. What nonsense to +say that this solid table on which I am writing is a collection +of electrons moving with prodigious speeds in empty spaces, +which relatively to electronic dimensions are as wide as the +spaces between the planets in the solar system! What nonsense +to say that the thin air is trying to crush my body with a load +of 14~lbs.\ to the square inch! What nonsense that the star-cluster, +which I see through the telescope obviously there now, +is a glimpse into a past age $50,000$ years ago! Let us not be +beguiled by this voice. It is discredited. + +But the statement that time is a fourth dimension may +suggest unnecessary difficulties which a more precise definition +avoids. It is in the external world that the four dimensions are +united---not in the relations of the external world to the +individual which constitute his direct acquaintance with space +%% -----File: 067.png---Folio 57------- +and time. Just in that process of relation to an individual, the +order falls apart into the distinct manifestations of space and time. +An individual is a four-dimensional object of greatly elongated +form; in ordinary language we say that he has considerable extension +in time and insignificant extension in space. Practically he +is represented by a line---his track through the world. When the +world is related to such an individual, his own asymmetry is +introduced into the relation; and that order of events which is +parallel with his track, that is to say with \textit{himself}, appears in +his experience to be differentiated from all other orders of events.% +\index{Observer, an unsymmetrical object}% +\index{Time!depends on observer's track}% + +Probably the best known exposition of the fourth dimension +is that given in E.~Abbott's popular book \textit{Flatland}. +\index{Flatland}% +It may be of +interest to see how far the four-dimensional world of space-time +conforms with his anticipations. He lays stress on three points. + +(1)~As a four-dimensional body moves, its section by the +three-dim\-en\-sional world may vary; thus a rigid body can alter +size and shape. + +(2)~It should be possible for a body to enter a completely +closed room, by travelling into it in the direction of the fourth +dimension, just as we can bring our pencil down on to any point +within a square without crossing its sides. + +(3)~It should be possible to see the inside of a solid, just as +we can see the inside of a square by viewing it from a point +outside its plane. + +The first phenomenon is manifested by the FitzGerald contraction.% +\index{FitzGerald contraction!relativity explanation of}% + +If quantity of matter is to be identified with its mass, the +second phenomenon does not happen. It could easily be conceived +of as happening, but it is provided against by a special +law of nature---the conservation of mass. It could happen, +but it does not happen. + +The third phenomenon does not happen for two reasons. +A natural body extends in time as well as in space, and is +therefore four-dimensional; but for the analogy to hold, the +object must have one dimension less than the world, like the +square seen from the third dimension. If the solid suddenly +went out of existence so as to present a plane section towards +time, we should still fail to see the interior of it; because light-tracks +in four-dimensions are restricted to certain lines like +%% -----File: 068.png---Folio 58------- +$UOV, U'OV'$ in \Figref{3}, whereas in three-dimensions light can +traverse any straight line. This could be remedied by interposing +some kind of dispersive medium, so that light of some wave-length +could be found travelling with every velocity and following +every track in space-time; then, looking at a solid which suddenly +went out of existence, we should receive at the same moment +light-impressions from every particle in its interior (supposing +them self-luminous). We actually should see the inside of it. + +How our poor eyes are to disentangle this overwhelming +experience is quite another question. + +The interval is a quantity so fundamental for us that we may +consider its measurement in some detail. Suppose we have a +scale $AB$ divided into kilometres, say, and at each division is +placed a clock also registering kilometres. (It will be remembered +that time can be measured in seconds or kilometres indifferently.) +%[Illustration: Fig. 9.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[9]{\textwidth}{068a} +\end{center} +\end{figure*}% +\index{Clock-scale}% [** PP: Presumed location of entry] +\index{Interval!practical measurement of}% +When the clocks are correctly set and viewed from $A$ the sum +of the readings of any clock and the division beside it is the +same for all, since the scale-reading gives the correction for the +time taken by light, travelling with unit velocity, to reach~$A$. +This is shown in \Figref{9} where the clock-readings are given as +though they were being viewed from~$A$. + +Now lay the scale in line with the two events; note the clock +and scale-readings $t_1, x_1$, of the first event, and the corresponding +readings $t_2, x_2$, of the second event. Then by the formula +already given +\index{Clock!affected by velocity}% +\[ +s^2 = (t_2 - t_1)^2 - (x_2 - x_1)^2. +\] +But suppose we took a different standard of rest, and set the +scale moving uniformly in the direction~$AB$. Then the divisions +would have advanced to meet the second event, and $(x_2 - x_1)$ +would be smaller. This is compensated, because $t_2-t_1$ also +becomes altered. $A$~is now advancing to meet the light coming +from any of the clocks along the rod; the light arrives too +%% -----File: 069.png---Folio 59------- +quickly, and in the initial adjustment described above the clock +must be set back a little. The clock-reading of the event is thus +smaller. There are other small corrections arising from the +FitzGerald contraction, etc.; and the net result is that, it does +not matter what uniform motion is given to the scale, the final +result for $s$ is always the same. + +In elementary mechanics we are taught that velocities can be +compounded by adding. +\index{Addition of velocities}% +\index{Velocity, addition-law}% +If $B$'s velocity relative to $A$ (as observed +by either of them) is $100$~km.\ per sec., and $C$'s velocity relative +to $B$ is $100$~km.\ per sec.\ in the same direction, then $C$'s velocity +relative to~$A$ should be $200$~km.\ per sec. This is not quite +accurate; the true answer is $199.999978$~km.\ per sec. The discrepancy +is not difficult to explain. The two velocities and their +resultant are not all reckoned with respect to the same partitions +of space and time. When $B$ measures $C$'s velocity relative to +him he uses his own space and time, and it must be corrected +to reduce to $A$'s space and time units, before it can be added +on to a velocity measured by $A$. + +If we continue the chain, introducing $D$ whose velocity +relative to $C$, and measured by $C$, is $100$~km.\ per sec., and so on +\textit{ad infinitum}, we never obtain an infinite velocity with respect +to~$A$, but gradually approach the limiting velocity of $300,000$ +km.\ per sec., the speed of light. This speed has the remarkable +property of being absolute, whereas every other speed is relative. +\index{Light, velocity of!an absolute velocity}% +If a speed of $100$~km.\ per sec.\ or of $100,000$~km.\ per sec.\ is +mentioned, we have to ask---speed relative to what? But if +a speed of $300,000$~km.\ per sec.\ is mentioned, there is no need +to ask the question; the answer is---relative to any and every +piece of matter. A $\beta$ particle shot off from radium can move at +more than $200,000$~km.\ per sec.; +\index{Beta particles}% +but the speed of light relative +to an observer travelling with it is still $300,000$~km.\ per sec. It +reminds us of the mathematicians' transfinite number Aleph; +you can subtract any number you like from it, and it still +remains the same. + +The velocity of light plays a conspicuous part in the relativity +theory, and it is of importance to understand what is the +property associated with it which makes it fundamental. +\index{Light, velocity of!importance of}% +\index{Velocity of light!importance of}% +The +fact that the velocity of light is the same for all observers is a +consequence rather than a cause of its pre-eminent character. +%% -----File: 070.png---Folio 60------- +Our first introduction of it, for the purpose of coordinating +units of length and time, was merely conventional with a view +to simplifying the algebraic expressions. Subsequently, considerable +use has been made of the fact that nothing is known +in physics which travels with greater speed, so that in practice +our determinations of simultaneity depend on signals transmitted +with this speed. If some new kind of ray with a higher +speed were discovered, it would perhaps tend to displace light-signals +and light-velocity in this part of the work, time-reckoning +being modified to correspond; on the other hand, this would +lead to greater complexity in the formulae, because the FitzGerald +contraction which affects space-measurement depends +on light-velocity. But the chief importance of the velocity of +light is that no material body can exceed this velocity. This +gives a general physical distinction between paths which are time-like +and space-like, respectively---those which can be traversed +by matter, and those which cannot. The material structure of +the four-dimensional world is fibrous, with the threads all running +along time-like tracks; it is a tangled warp without a woof. +Hence, even if the discovery of a new ray led us to modify the +reckoning of time and space, it would still be necessary in the +study of material systems to preserve the \textit{present} absolute +distinction of time-like and space-like intervals, under a new +name if necessary.% +\index{Space-like intervals}% +\index{Time-like intervals}% + +It may be asked whether it is possible for anything to have +a speed greater than the velocity of light. Certainly matter +cannot attain a greater speed; but there might be other things +in nature which could. ``Mr Speaker,'' said Sir Boyle Roche, +``not being a bird, I could not be in two places at the same time.'' +Any entity with a speed greater than light would have the +peculiarity of Sir Boyle Roche's bird. It can scarcely be said to +be a self-contradictory property to be in two places at the same +time any more than for an object to be at two times in the same +place. The perplexities of the quantum theory of energy sometimes +seem to suggest that the possibility ought not to be +overlooked; but, on the whole, the evidence seems to be against +the existence of anything moving with a speed beyond that of +light.% +\index{Quanta}% + +The standpoint of relativity and the principle of relativity +%% -----File: 071.png---Folio 61------- +are quite independent of any views as to the constitution of +matter or light. Hitherto our only reference to electrical theory +has been in connection with Larmor and Lorentz's explanation +of the FitzGerald contraction; but now from the discussion of +the four-dimensional world, we have found a more general +explanation of the change of length. The case for the electrical +theory of matter is actually weakened, because many experimental +effects formerly thought to depend on the peculiar +properties of electrical forces are now found to be perfectly +general consequences of the relativity of observational knowledge. + +Whilst the evidence for the electrical theory of matter is not +so conclusive, as at one time appeared, the theory may be +accepted without serious misgivings. To postulate two entities, +matter and electric charges, when one will suffice is an arbitrary +hypothesis, unjustifiable in our present state of knowledge. The +great contribution of the electrical theory to this subject is a +precise explanation of the property of inertia. It was shown +theoretically by J.~J. Thomson that if a charged conductor is +to be moved or stopped, additional effort will be necessary +simply on account of the charge. +\index{Thomson, J.~J.}% +The conductor has to carry +its electric field with it, and force is needed to set the field +moving. This property is called inertia, and it is measured by +\textit{mass}. If, keeping the charge constant, the size of the conductor +is diminished, this inertia increases. Since the smallest separable +particles of matter are found by experiment to be very minute +and to carry charges, the suggestion arises that these charges +may be responsible for the whole of the inertia detected in +matter. The explanation is sufficient; and there seems no reason +to doubt that all inertia is of this electrical kind.% +\index{Electrical theory of inertia}% +\index{Energy!inertia of}% +\index{Inertia!electrical theory of}% +\index{Mass!electrical theory of}% + +When the calculations are extended to charges moving with +high velocities, it is found that the electrical inertia is not +strictly constant but depends on the speed; in all cases the +variation is summed up in the statement that the inertia is +simply proportional to the total energy of the electromagnetic +field. We can say if we like that the mass of a charged particle +at rest belongs to its electrostatic energy; when the charge is +set in motion, kinetic energy is added, and this kinetic energy +also has mass. Hence it appears that mass (inertia) and energy +%% -----File: 072.png---Folio 62------- +are essentially the same thing, or, at the most, two aspects of +the same thing. It must be remembered that on this view the +greater part of the mass of matter is due to concealed energy, +which is not as yet releasable. + +The question whether electrical energy not bound to electric +charges has mass, is answered in the affirmative in the case of +light. Light has mass. +\index{Light!mass of}% +\index{Mass of light}% +Presumably also gravitational energy +has mass; or, if not, mass will be created when, as often happens, +gravitational energy is converted into kinetic energy. The mass +of the whole (negative) gravitational energy of the earth is of +the order \textit{minus} a billion tons. + +The theoretical increase of the mass of an electron with speed +has been confirmed experimentally, the agreement with calculation +being perfect if the electron undergoes the FitzGerald +contraction by its motion. +\index{Electron!Kaufmann's experiment on}% +\index{Kaufmann's experiment}% +This has been held to indicate that +the electron cannot have any inertia other than that due to the +electromagnetic field carried with it. +\index{Electron!inertia of}% +But the conclusion (though +probable enough) is not a fair inference; because these results, +obtained by special calculation for electrical inertia, are found +to be predicted by the theory of relativity for any kind of +inertia. This will be shown in \Chapref{IX}. The factor giving +the increase of mass with speed is the same as that which affects +length and time. Thus if a rod moves at such a speed that its +length is halved, its mass will be doubled. Its density will be +increased four-fold, since it is both heavier and less in volume.% +\index{Density, effect of motion on}% + +We have thought it necessary to include this brief summary +of the electrical theory of matter and mass, because, although +it is not required by the relativity theory, it is so universally +accepted in physics that we can scarcely ignore it. Later on we +shall reach in a more general way the identification of mass with +energy and the variation of mass with speed; but, since the +experimental measurement of inertia involves the study of a +body in non-uniform motion, it is not possible to enter on a +satisfactory discussion of mass until the more general theory of +relativity for non-uniform motion has been developed. +%% -----File: 073.png---Folio 63------- + + +\Chapter{IV}{Fields of Force} + +\Quote{Lucretius, \textit{De Natura Rerum.}} +{For whenever bodies fall through water and thin air, they must quicken their +descents in proportion to their weights, because the body of water and subtle +nature of air cannot retard everything in equal degree, but more readily give +way overpowered by the heavier; on the other hand empty void cannot offer +resistance to anything in any direction at any time, but must, as its nature +craves, continually give way; and for this reason all things must be moved and +borne along with equal velocities though of unequal weights through the +unresisting void.}% +\index{Force!elementary conception of}% + + +\First{The} primary conception of force is associated with the muscular +sensation felt when we make an effort to cause or prevent the +motion of matter. Similar effects on the motion of matter can +be caused by non-living agency, and these also are regarded as +due to forces. As is well known, the scientific measure of a force +is the momentum that it communicates to a body in given time. +There is nothing very abstract about a force transmitted by +material contact; modern physics shows that the momentum is +communicated by a process of molecular bombardment. We can +visualise the mechanism, and see the molecules carrying the +motion in small parcels across the boundary into the body that +is being acted on. Force is no mysterious agency; it is merely +a convenient summary of this flow of motion, which we can +trace continuously if we take the trouble. It is true that the +difficulties are only set back a stage, and the exact mode by +which the momentum is redistributed during a molecular +collision is not yet understood; but, so far as it goes, this analysis +gives a clear idea of the transmission of motion by ordinary +forces. + +But even in elementary mechanics an important natural force +appears, which does not seem to operate in this manner. Gravitation +is not resolvable into a succession of molecular blows. +A massive body, such as the earth, seems to be surrounded by +a field of latent force, ready, if another body enters the field, to +become active, and transmit motion. One usually thinks of this +influence as existing in the space round the earth even when +%% -----File: 074.png---Folio 64------- +there is no test-body to be affected, and in a rather vague way +it is suspected to be some state of strain or other condition of +an unperceived medium. + +Although gravitation has been recognised for thousands of +years, and its laws were formulated with sufficient accuracy for +almost all purposes more than 200 years ago, it cannot be said +that much progress has been made in explaining the nature or +mechanism of this influence. It is said that more than 200 +theories of gravitation have been put forward; but the most +plausible of these have all had the defect that they lead nowhere +and admit of no experimental test. Many of them would nowadays +be dismissed as too materialistic for our taste---filling space +with the hum of machinery---a procedure curiously popular in +the nineteenth century. Few would survive the recent discovery +that gravitation acts not only on the molecules of matter, but +on the undulations of light. + +The nature of gravitation has seemed very mysterious, yet it +is a remarkable fact that in a limited region it is possible to +create an artificial field of force which imitates a natural +gravitational field so exactly that, so far as experiments have +yet gone, no one can tell the difference. +\index{Artificial fields of force}% +\index{Fields of force!artificial}% +\index{Force!fields of}% +Those who seek for an +explanation of gravitation naturally aim to find a model which +will reproduce its effects; but no one before Einstein seems to +have thought of finding the clue in these artificial fields, familiar +as they are. + +When a lift starts to move upwards the occupants feel a +characteristic sensation, which is actually identical with a +sensation of increased weight. +\index{Lift, accelerated}% +The feeling disappears as soon +as the motion becomes uniform; it is associated only with the +change of motion of the lift, that is to say, the acceleration. +Increased weight is not only a matter of sensation; it is shown +by any physical experiments that can be performed. The usual +laboratory determination of the value of gravity by Atwood's +machine would, if carried out inside the accelerated lift, give +a higher value. A spring-balance would record higher weights. +Projectiles would follow the usual laws of motion but with a +higher value of gravity. In fact, the upward acceleration of +the lift is in its mechanical effects exactly similar to an additional +gravitational field superimposed on that normally present. + +%% -----File: 075.png---Folio 65------- + +Perhaps the equivalence is most easily seen when we produce +in this manner an artificial field which just neutralises the earth's +field of gravitation. Jules Verne's book \textit{Round the Moon} tells +the story of three men in a projectile shot from a cannon into +space. +\index{Projectile, Jules Verne's}% +The author enlarges on their amusing experiences when +their weight vanished altogether at the neutral point, where the +attraction of the earth and moon balance one another. As a +matter of fact they would not have had any feeling of weight +at any time during their journey after they left the earth's +atmosphere. +\index{Weight!vanishes inside free projectile}% +The projectile was responding freely to the pull of +gravity, and so were its occupants. When an occupant let go +of a plate, the plate could not ``fall'' any more than it was +doing already, and so it must remain poised. + +It will be seen that the sensation of weight is not felt when +we are free to respond to the force of gravitation; it is only +felt when something interferes to prevent our falling. It is +primarily the floor or the chair which causes the sensation of +weight by checking the fall. It seems literally true to say that +we never feel the force of the earth's gravitation; what we do +feel is the bombardment of the soles of our boots by the molecules +of the ground, and the consequent impulses spreading upwards +through the body. This point is of some importance, since the +idea of the force of gravitation as something which can be felt, +predisposes us to a materialistic view of its nature. + +Another example of an artificial field of force is the centrifugal +force of the earth's rotation. In most books of Physical Constants +will be found a table of the values of~``$g$,'' the acceleration +due to gravity, at different latitudes. +\index{Centrifugal Force!compared with gravitation}% +But the numbers given +do not relate to gravity alone; they are the resultant of gravity +and the centrifugal force of the earth's rotation. These are so +much alike in their effects that for practical purposes physicists +have not thought it worth while to distinguish them. + +Similar artificial fields are produced when an aeroplane +changes its course or speed; and one of the difficulties of navigation +is the impossibility of discriminating between these and the +true gravitation of the earth with which they combine. One +usually finds that the practical aviator requires little persuasion +of the relativity of force. + +To find a unifying idea as to the origin of these artificial +%% -----File: 076.png---Folio 66------- +fields of force, we must return to the four-dimensional world of +space-time. The observer is progressing along a certain track +in this world. Now his course need not necessarily be straight. +It must be remembered that straight in the four-dimensional +world means something more than straight in space; it implies +also uniform velocity, since the velocity determines the inclination +of the track to the time-axis. + +The observer in the accelerated lift travels upwards in a +straight line, say $1$~foot in the first second, $4$~feet in two seconds, +$9$~feet in three seconds, and so on. If we plot these points as +$x$ and~$t$ on a diagram we obtain a curved track. Presently the +speed of the lift becomes uniform and the track in the diagram +becomes straight. So long as the track is curved (accelerated +motion) a field of force is perceived; it disappears when the +track becomes straight (uniform motion). + +Again the observer on the earth is carried round in a circle +once a day by the earth's rotation; allowing for steady progress +through time, the track in four dimensions is a spiral. For an +observer at the north pole the track is straight, and there the +centrifugal force is zero. + +Clearly the artificial field of force is associated with curvature +of track, and we can lay down the following rule:--- + +Whenever the observer's track through the four-dimensional +world is curved he perceives an artificial field of force. + +The field of force is not only perceived by the observer in his +sensations, but reveals itself in his physical measures. It should +be understood, however, that the curvature of track must not +have been otherwise allowed for. Naturally if the observer in +the lift recognises that his measures are affected by his own +acceleration and applies the appropriate corrections, the artificial +force will be removed by the process. It only exists if he is +unaware of, or does not choose to consider, his acceleration. + +The centrifugal force is often called ``unreal.'' From the point +of view of an observer who does not rotate with the earth, there +is no centrifugal force; it only arises for the terrestrial observer +who is too lazy to make other allowance for the effects of the +earth's rotation. It is commonly thought that this ``unreality'' +quite differentiates it from a ``real'' force like gravity; but if +we try to find the grounds of this distinction they evade us. +%% -----File: 077.png---Folio 67------- +The centrifugal force is made to disappear if we choose a suitable +standard observer not rotating with the earth; the gravitational +force was made to disappear when we chose as standard observer +an occupant of Jules Verne's falling projectile. If the possibility +of annulling a field of force by choosing a suitable standard +observer is a test of unreality, then gravitation is equally unreal +with centrifugal force.% +\index{Fields of force!relativity of}% +\index{Force!relativity of}% +\IndexExtra{Relativity of Force}% + +It may be urged that we have not stated the case quite +fairly. When we choose the non-rotating observer the centrifugal +force disappears completely and everywhere. When we choose +the occupant of the falling projectile, gravitation disappears in +his immediate neighbourhood; but he would notice that, +although unsupported objects round him experienced no acceleration +relative to him, objects on the other side of the earth would +fall towards him. So far from getting rid of the field of force, +he has merely removed it from his own surroundings, and piled +it up elsewhere. Thus gravitation is removable locally, but +centrifugal force can be removed everywhere. The fallacy of +this argument is that it speaks as though gravitation and +centrifugal force were distinguishable experimentally. It presupposes +the distinction that we are challenging. Looking simply +at the resultant of gravitation and centrifugal force, which is all +that can be observed, neither observer can get rid of the resultant +force at all parts of space. Each has to be content with leaving +a residuum. The non-rotating observer claims that he has got +rid of all the unreal part, leaving a remainder (the usual gravitational +field) which he regards as really existing. We see no +justification for this claim, which might equally well be made +by Jules Verne's observer. + +It is not denied that the separation of centrifugal and gravitational +force generally adopted has many advantages for +mathematical calculation. If it were not so, it could not have +endured so long. But it is a mathematical separation only, +without physical basis; and it often happens that the separation +of a mathematical expression into two terms of distinct nature, +though useful for elementary work, becomes vitiated for more +accurate work by the occurrence of minute cross-terms which +have to be taken into account. + +Newtonian mechanics proceeds on the supposition that there +%% -----File: 078.png---Folio 68------- +is some super-observer. If \textit{he} feels a field of force, then that +force really exists. Lesser beings, such as the occupants of the +falling projectile, have other ideas, but they are the victims of +illusion. It is to this super-observer that the mathematician +appeals when he starts a dynamical investigation with the words +``Take unaccelerated rectangular axes, $Ox$, $Oy$, $Oz\dotsc$.'' +Unaccelerated rectangular axes are the measuring-appliances of the +super-observer.% +\index{Newton!super-observer}% +\index{Super-observer, Newton's}% + +It is quite possible that there might be a super-observer, +whose views have a natural right to be regarded as the truest, +or at least the simplest. A society of learned fishes would probably +agree that phenomena were best described from the point +of view of a fish at rest in the ocean. But relativity mechanics +finds that there is no evidence that the circumstances of any +observer can be such as to make his views pre-eminent. All are +on an equality. Consider an observer $A$ in a projectile falling +freely to the earth, and an observer $B$ in space out of range of +any gravitational attraction. Neither $A$ nor~$B$ feel any field of +force in their neighbourhood. Yet in Newtonian mechanics an +artificial distinction is drawn between their circumstances; $B$ is +in no field of force at all, but $A$ is really in a field of force, only +its effects are neutralised by his acceleration. But what is this +acceleration of~$A$? Primarily it is an acceleration relative to the +earth; but then that can equally well be described as an acceleration +of the earth relative to~$A$, and it is not fair to regard it as +something located with $A$. Its importance in Newtonian +philosophy is that it is an acceleration relative to what we have +called the super-observer. This potentate has drawn planes and +lines partitioning space, as space appears to him. I fear that +the time has come for his abdication. + +Suppose the whole system of the stars were falling freely +under the uniform gravitation of some vast external mass, like +a drop of rain falling to the ground. Would this make any +difference to phenomena? None at all. There would be a +gravitational field; but the consequent acceleration of the +observer and his landmarks would produce a field of force +annulling it. Who then shall say what is absolute acceleration?% +\index{Absolute acceleration}% + +We shall accordingly give up the attempt to separate artificial +fields of force and natural gravitational fields; and call the whole +%% -----File: 079.png---Folio 69------- +measured field of force the gravitational field, generalising the +expression. This field is not absolute, but always requires that +some observer should be specified. + +It may avoid some mystification if we state at once that there +are certain intricacies in the gravitational influence radiating +from heavy matter which are distinctive. A theory which did +not admit this would run counter to common sense. What our +argument has shown is that the characteristic symptom in a +region in the neighbourhood of matter is not the field of force; +it must be something more intricate. In due course we shall +have to explain the nature of this more complex effect of matter +on the condition of the world. + +Our previous rule, that the observer perceives an \textit{artificial} +field of force when he deviates from a straight track, must now +be superseded. We need rather a rule determining when he +perceives a field of force of any kind. +\index{Fields of force!due to disturbance of observer}% +Indeed the original rule +has become meaningless, because a straight track is no longer +an absolute conception. Uniform motion in a straight line is +not the same for an observer rotating with the earth as for a +non-rotating observer who takes into account the sinuosity of +the rotation. We have decided that these two observers are on +the same footing and their judgments merit the same respect. +A straight-line in space-time is accordingly not an absolute +conception, but is only defined relative to some observer. + +Now we have seen that so long as the observer and his +measuring-appliances are unconstrained (falling freely) the field +of force immediately round him vanishes. It is only when he is +deflected from his proper track that he finds himself in the +midst of a field of force. Leaving on one side the question of +the motion of electrically charged bodies, which must be reserved +for more profound treatment, the observer can only leave his +proper track if he is being disturbed by material impacts, e.g.\ +the molecules of the ground bombarding the soles of his boots. +We may say then that a body does not leave its natural track +without visible cause; and any field of force round an observer +is the result of his leaving his natural track by such cause. +There is nothing mysterious about this field of force; it is merely +the reflection in the phenomena of the observer's disturbance; +just as the flight of the houses and hedgerows past our railway-carriage +is the reflection of our motion with the train. + +%% -----File: 080.png---Folio 70------- + +Our attention is thus directed to the natural tracks of unconstrained +bodies, which appear to be marked out in some +absolute way in the four-dimensional world. There is no +question of an observer here; the body takes the same course +in the world whoever is watching it. Different observers will +describe the track as straight, parabolical, or sinuous, but it is +the same absolute locus.% +\index{Geodesic!absolute significance of}% +\index{Tracks, natural}% +\index{Natural tracks}% + +Now we cannot pretend to predict without reference to +experiment the laws determining the nature of these tracks; +but we can examine whether our knowledge of the four-dimensional +world is already sufficient to specify definite tracks of this +kind, or whether it will be necessary to introduce new hypothetical +factors. It will be found that it is already sufficient. So far we +have had to deal with only one quantity which is independent +of the observer and has therefore an absolute significance in the +world, namely the \textit{interval} between two events in space and time. +Let us choose two fairly distant events $P_1$ and~$P_2$. These can +be joined by a variety of tracks, and the interval-length from +$P_1$ to~$P_2$ along any track can be measured. In order to make +sure that the interval-length is actually being measured along +the selected track, the method is to take a large number of +intermediate points on the track, measure the interval corresponding +to each subdivision, and take the sum. It is virtually +the same process as measuring the length of a twisty road on +a map with a piece of cotton. The interval-length along a +particular track is thus something which can be measured +absolutely, since all observers agree as to the measurement of +the interval for each subdivision. It follows that all observers +will agree as to which track (if any) is the shortest track between +the two points, judged in terms of interval-length.% +\index{Interval-length!tracks of maximum}% +\index{Longest tracks}% + +This gives a means of defining certain tracks in space-time as +having an absolute significance, and we proceed tentatively to +identify them with the natural tracks taken by freely moving +particles. + +In one respect we have been caught napping. Dr A. A. Robb +has pointed out the curious fact that it is not the shortest track, +but the longest track, which is unique\footnotemark. + \footnotetext{It is here assumed that $P_2$ is in the future of $P_1$ so that it is possible for + a particle to travel from $P_1$ to~$P_2$. If $P_1$ and~$P_2$ are situated like $O$ and~$P'$ in + \Figref{3}, the interval-length is imaginary, and the \textit{shortest} track is unique.}% +There are any number +%% -----File: 081.png---Folio 71------- +of tracks from $P_1$ to~$P_2$ of zero interval-length; there is just one +which has maximum length. This is because of the peculiar +geometry which the minus sign of $(t_2-t_1)^2$ introduces. For +instance, it will be seen from \Eqref{equation}{1}, \Pageref{53}, that when +\index{Interval-length!zero for velocity of light}% +\[ +(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 = (t_2 - t_1)^2, +\] +that is to say when the resultant distance travelled in space is +equal to the distance travelled in time, then $s$ is zero. This +happens when the velocity is unity---the velocity of light. To +get from $P_1$ to $P_2$ by a path of no interval-length, we must +simply keep on travelling with the velocity of light, cruising +round if necessary, until the moment comes to turn up at~$P_2$. +On the other hand there is evidently an upper limit to the interval-length +of the track, because each portion of $s$ is always less than +the corresponding portion of $(t_2-t_1)$, and $s$ can never exceed +$t_2-t_1$. + +There is a physical interpretation of interval-length along the +path of a particle which helps to give a more tangible idea of +its meaning. It is the time as perceived by an observer, or +measured by a clock, carried on the particle. This is called the +proper-time; +\index{Clock!recording proper-time}% +\index{Interval-length!identified with proper time}% +\index{Proper-time}% +and, of course, it will not in general agree with the +time-reckoning of the independent onlooker who is supposed to +be watching the whole proceedings. To prove this, we notice +from \Eqref{equation}{1} that if $x_2 = x_1, y_2 = y_1$ and $z_2 = z_1$, then +$s = t_2-t_1$. The condition $x_2 = x_1$, etc.\ means that the particle +must remain stationary relative to the observer who is measuring +$x$, $y$, $z$,~$t$. To secure this we mount our observer on the particle +and then the interval-length $s$ will be $t_2-t_1$, which is the time +elapsed according to his clock. + +We can use proper-time as generally equivalent to interval-length; +but it must be admitted that the term is not very +logical unless the track in question is a natural track. For any +other track, the drawback to defining the interval-length as the +time measured by a clock which follows the track, is that no +clock could follow the track without violating the laws of nature. +We may force it into the track by continually hitting it; but +that treatment may not be good for its time-keeping qualities. +The original definition by \Eqref{equation}{1} is the more general +definition. + +%% -----File: 082.png---Folio 72------- + +We are now able to state formally our proposed law of motion---Every +particle moves so as to take the track of greatest interval-length +between two events, except in so far as it is disturbed by +impacts of other particles or electrical forces. + +This cannot be construed into a truism like Newton's first +law of motion. The reservation is not an undefined agency like +force, whose meaning can be extended to cover any breakdown +of the law. We reserve only direct material impacts and electromagnetic +causes, the latter being outside our present field of +discussion. + +Consider, for example, two events in space-time, viz.\ the +position of the earth at the present moment, and its position a +hundred years ago. Call these events $P_2$ and~$P_1$. In the interim +the earth (being undisturbed by impacts) has moved so as to +take the longest track from $P_1$ to $P_2$---or, if we prefer, so as to +take the longest possible proper-time over the journey. In the +weird geometry of the part of space-time through which it +passes (a geometry which is no doubt associated in some way +with our perception of the existence of a massive body, the sun) +this longest track is a spiral---a circle in space, drawn out into +a spiral by continuous displacement in time. Any other course +would have had shorter interval-length. + +In this way the study of fields of force is reduced to a study +of geometry. To a certain extent this is a retrograde step; we +adopt Kepler's description of the sun's gravitational field instead +of Newton's. The field of force is completely described if the +tracks through space and time of particles projected in every +possible way are prescribed. But we go back in order to go +forward in a new direction. To express this unmanageable mass +of detail in a unified way, a world-geometry is found in which +the tracks of greatest length are the actual tracks of the particles. +It only remains to express the laws of this geometry in a concise +form. The change from a mechanical to a geometrical theory of +fields of force is not so fundamental a change as might be +supposed. If we are now reducing mechanics to a branch of +natural geometry, we have to remember that natural geometry +is equally a branch of mechanics, since it is concerned with the +behaviour of material measuring-appliances. + +Reference has been made to weird geometry. There is no +%% -----File: 083.png---Folio 73------- +help for it, if the longest track can be a spiral like that known +to be described by the earth. Non-Euclidean geometry is +necessary. In Euclidean geometry the shortest track is always +a straight line; and the slight modification of Euclidean geometry +described in \Chapref{III} is found to give a straight line as the +longest track. The status of non-Euclidean geometry has already +been thrashed out in the Prologue; and there seems to be no +reason whatever for preferring Euclid's geometry unless observations +decide in its favour. +\index{Euclidean geometry}% +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% +\index{Semi-Euclidean geometry}% +\IndexExtra{Geometry!Euclidean}% +\Eqref{Equation}{1}, \Pageref{53}, is the expression +of the Euclidean (or semi-Euclidean) geometry we have hitherto +adopted; we shall have to modify it, if we adopt non-Euclidean +geometry. + +But the point arises that the geometry arrived at in \Chapref{III} +was not arbitrary. It was the synthesis of measures made with +clocks and scales, by observers with all kinds of uniform motion +relative to one another; we cannot modify it arbitrarily to fit +the behaviour of moving particles like the earth. Now, if the +worst came to the worst, and we could not reconcile a geometry +based on measures with clocks and scales and a geometry based +on the natural tracks of moving particles---if we had to select +one or the other and keep to it---I think we ought to prefer to +use the geometry based on the tracks of moving particles. The +free motion of a particle is an example of the simplest possible +kind of phenomenon; it is unanalysable; whereas, what the +readings of any kind of clock record, what the extension of a +material rod denotes, may evidently be complicated phenomena +involving the secrets of molecular constitution. Each geometry +would be right in its own sphere; but the geometry of moving +particles would be the more fundamental study. But it turns +out that there is probably no need to make the choice; clocks, +scales, moving particles, light-pulses, give the same geometry. +This might perhaps be expected since a clock must comprise +moving particles of some kind.% +\index{Clock-scale geometry, not fundamental}% + +A formula, such as \Eqref{equation}{1}, based on experiment can +only be verified to a certain degree of approximation. Within +certain limits it will be possible to introduce modifications. Now +it turns out that the free motion of a particle is a much more +sensitive way of exploring space-time, than any practicable +measures with scales and clocks. If then we employ our accurate +%% -----File: 084.png---Folio 74------- +knowledge of the motion of particles to correct the formula, we +shall find that the changes introduced are so small that they are +inappreciable in any practical measures with scales and clocks. +There is only one case where a possible detection of the modification +is indicated; this refers to the behaviour of a clock on the +surface of the sun, but the experiment is one of great difficulty +and no conclusive answer has been given. +\index{Clock!on sun}% +We conclude then +that the geometry of space and time based on the motions of +particles is accordant with the geometry based on the cruder +observations with clocks and scales; but if subsequent experiment +should reveal a discrepancy, we shall adhere to the moving +particle on account of its greater simplicity. + +The proposed modification can be regarded from another +point of view. \Eqref{Equation}{1} is the synthesis of the experiences +of all observers in uniform motion. But uniform motion means +that their four-dimensional tracks are straight lines. We must +suppose that the observers were moving in their natural tracks; +for, if not, they experienced fields of force, and presumably +allowed for these in their calculations, so that reduction was +made to the natural tracks. If then \Eqref{equation}{1} shows that +the natural tracks are straight lines, we are merely getting out +of the equation that which we originally put into it. + +The formula needs generalising in another way. Suppose there +is a region of space-time where, for some observer, the natural +tracks are all straight lines and \Eqref{equation}{1} holds rigorously. +For another (accelerated) observer the tracks will be curved, +and the equation will not hold. At the best it is of a form which +can only hold good for specially selected observers. + +Although it has become necessary to throw our formula into +the melting-pot, that does not create any difficulty in measuring +the interval. Without going into technical details, it may be +pointed out that the innovations arise solely from the introduction +of gravitational fields of force into our scheme. When +there is no force, the tracks of all particles are straight lines as +our previous geometry requires. In any small region we can +choose an observer (falling freely) for whom the force vanishes, +and accordingly the original formula holds good. Thus it is only +necessary to modify our rule for determining the interval by +two provisos (1)~that the interval measured must be small, +%% -----File: 085.png---Folio 75------- +(2)~that the scales and clocks used for measuring it must be +falling freely. The second proviso is natural, because, if we do +not leave our apparatus to fall freely, we must allow for the +strain that it undergoes. The first is not a serious disadvantage, +because a larger interval can be split up into a number of small +intervals and the parts measured separately. In mathematical +problems the same device is met with under the name of integration. +To emphasise that the formula is strictly true only for +infinitesimal intervals, it is written with a new notation +\Pagelabel{75} +\index{Interval!practical measurement of}% +\[ +ds^2 = - dx^2 - dy^2 - dz^2 + dt^2 +\Tag{2} +\] +where $dx$ stands for the small difference $x_2-x_1$, etc. + +The condition that the measuring appliances must not be +subjected to a field of force is illustrated by Ehrenfest's paradox. +\index{Ehrenfest's paradox}% +Consider a wheel revolving rapidly. Each portion of the +circumference is moving in the direction of its length, and +might be expected to undergo the FitzGerald contraction due +to its velocity; each portion of a radius is moving transversely +and would therefore have no longitudinal contraction. +\index{Acceleration!modifies FitzGerald contraction}% +\index{FitzGerald Contraction!modified by acceleration}% +It looks +as though the rim of the wheel should contract and the spokes +remain the same length, when the wheel is set revolving. The +conclusion is absurd, for a revolving wheel has no tendency to +buckle---which would be the only way of reconciling these +conditions. The point which the argument has overlooked is +that the results here appealed to apply to unconstrained bodies, +which have no acceleration relative to the natural tracks in +space. Each portion of the rim of the wheel has a radial acceleration, +and this affects its extensional properties. When accelerations +as well as velocities occur a more far-reaching theory is +needed to determine the changes of length. + +To sum up---the interval between two (near) events is something +quantitative which has an absolute significance in nature. +The track between two (distant) events which has the longest +interval-length must therefore have an absolute significance. +Such tracks are called \textit{geodesics}. +\index{Geodesic!definition of}% +Geodesics can be traced practically, +because they are the tracks of particles undisturbed by +material impacts. By the practical tracing of these geodesics +we have the best means of studying the character of the natural +geometry of the world. An auxiliary method is by scales and +%% -----File: 086.png---Folio 76------- +clocks, which, it is believed, when unconstrained, measure a +small interval according to \Eqref{formula}{2}. + +The identity of the two methods of exploring the geometry +of the world is connected with a principle which must now be +enunciated definitely. We have said that no experiments have +been able to detect a difference between a gravitational field +and an artificial field of force such as the centrifugal force. +\index{Force!relativity of}% +\index{Relativity of Force}% +This +is not quite the same thing as saying that it has been proved +that there is no difference. It is well to be explicit when a +positive generalisation is made from negative experimental +evidence. The generalisation which it is proposed to adopt is +known as the Principle of Equivalence.% +\index{Equivalence!Principle of}% +\index{Principle of Equivalence}% + +\textit{A gravitational field of force is precisely equivalent to an artificial +field of force, so that in any small region it is impossible by any +conceivable experiment to distinguish between them.} + +In other words, force is purely relative. +%% -----File: 087.png---Folio 77------- + + +\Chapter{V}{Kinds of Space} + +\Quote[break]{W.~K. Clifford ({\upshape and} K.~Pearson), \textit{Common Sense of the Exact Sciences.}} +{The danger of asserting dogmatically that an axiom based on the experience +of a limited region holds universally will now be to some extent apparent to +the reader. It may lead us to entirely overlook, or when suggested at once +reject, a possible explanation of phenomena. The hypothesis that space is not +homaloidal [flat], and again that its geometrical character may change with +the time, may or may not be destined to play a great part in the physics of the +future; yet we cannot refuse to consider them as possible explanations of +physical phenomena, because they may be opposed to the popular dogmatic +belief in the universality of certain geometrical axioms---a belief which has +risen from centuries of indiscriminating worship of the genius of Euclid.}% +\index{Clifford}% + +\First{On} any surface it requires two independent numbers or ``coordinates'' +to specify the position of a point. For this reason +a surface, whether flat or curved, is called a two-dimensional +space. Points in three-dimensional space require three, and in +four-dimensional space-time four numbers or coordinates.% +\index{Coordinates}% + +To locate a point on a surface by two numbers, we divide the +surface into meshes by any two systems of lines which cross one +another. Attaching consecutive numbers to the lines, or better +to the channels between them, one number from each system +will identify a particular mesh; and if the subdivision is sufficiently +fine any point can be specified in this way with all the accuracy +needed. This method is used, for example, in the Post Office +Directory of London for giving the location of streets on the +map. The point $(4, 2)$ will be a point in the mesh where channel +No.~$4$ of the first system crosses channel No.~$2$ of the second. +If this indication is not sufficiently accurate, we must divide +channel No.~$4$ into ten parts numbered $4.0$, $4.1$ etc. The subdivision +must be continued until the meshes are so small that +all points in one mesh can be considered identical within the +limits of experimental detection. + +The diagrams, Figs.~10, 11, 12, illustrate three of the many +kinds of mesh-systems commonly used on a flat surface.% +\index{Mesh-systems}% + +If we speak of the properties of the triangle formed by the +points $(1, 2)$, $(3, 0)$, $(4, 4)$, we shall be at once asked, What mesh-% +%% -----File: 088.png---Folio 78------- +system are you using? No one can form a picture of the triangle +until that information has been given. But if we speak of the +properties of a triangle whose sides are of lengths $2$, $3$, $4$~inches, +anyone with a graduated scale can draw the triangle, and follow +our discussion of its properties. The distance between two points +can be stated without referring to any mesh-system. For this +reason, if we use a mesh-system, it is important to find formulae +connecting the absolute distance with the particular system that +is being used. + +In the more complicated kinds of mesh-systems it makes a +great simplification if we content ourselves with the formulae for +very short distances. The mathematician then finds no difficulty +in extending the results to long distances by the process called +integration. We write $ds$ for the distance between two points +\begin{figure*}[hbt]% +\begin{center} +\Graphic[10]{\textwidth}{088a} +\Figlabel{11}% +\Figlabel{12}% +\end{center} +\end{figure*}% +close together, $x_1$ and $x_2$ for the two numbers specifying the +location of one of them, $dx_1$ and $dx_2$ for the small differences of +these numbers in passing from the first point to the second. +But in using one of the particular mesh-systems illustrated in +the diagrams, we usually replace $x_1$, $x_2$ by particular symbols +sanctioned by custom, viz.\ $(x_1, x_2)$ becomes $(x, y)$, $(r, \theta)$, $(\xi, \eta)$ +for Figs.~10, 11, 12, respectively. +%[Illustration: Fig. 10.] +%[Illustration: Fig. 11.] +%[Illustration: Fig. 12.] + +The formulae, found by geometry, are: + +For rectangular coordinates $(x, y)$, \Figref{10}, +\[ +ds^2 = dx^2 + dy^2. +\] + +For polar coordinates $(r, \theta)$, \Figref{11}, +\[ +ds^2 = dr^2 + r^2\, d\theta^2. +\] + +For oblique coordinates $(\xi, \eta)$, \Figref{12}, +\[ +ds^2 = d\xi^2 - 2\kappa\, d\xi d\eta + d\eta^2, +\] +where $\kappa$ is the cosine of the angle between the lines of partition. + +%% -----File: 089.png---Folio 79------- + +As an example of a mesh-system on a curved surface, we may +take the lines of latitude and longitude on a sphere. + +For latitude and longitude $(\beta, \lambda)$ +\Pagelabel{79} +\[ +ds^2 = d\beta^2 + \cos^2 \beta\, d\lambda^2. +\] + +These expressions form a test, and in fact the only possible +test, of the kind of coordinates we are using. It may perhaps +seem inconceivable that an observer should for an instant be in +doubt whether he was using the mesh-system of \Figref{10} or +\Figref{11}. He sees at a glance that \Figref{11} is not what he would +call a rectangular mesh-system. But in that glance, he makes +measures with his eye, that is to say he determines $ds$ for pairs +of points, and he notices how these values are related to the +number of intervening channels. In fact he is testing which +formula for $ds$ will fit. For centuries man was in doubt whether +the earth was flat or round---whether he was using plane rectangular +coordinates or some kind of spherical coordinates. In +some cases an observer adopts his mesh-system blindly and long +afterwards discovers by accurate measures that $ds$ does not fit +the formula he assumed---that his mesh-system is not exactly of +the nature he supposed it was. In other cases he deliberately +sets himself to plan out a mesh-system of a particular variety, +say rectangular coordinates; he constructs right angles and rules +parallel lines; but these constructions are all measurements of +the way the $x$-channels and $y$-channels ought to go, and the +rules of construction reduce to a formula connecting his measures +$ds$ with $x$ and~$y$. + +The use of special symbols for the coordinates, varying +according to the kind of mesh-system used, thus anticipates a +knowledge which is really derived from the form of the formulae. +In order not to give away the secret prematurely, it will be +better to use the symbols $x_1$, $x_2$ in all cases. The four kinds of +coordinates already considered then give respectively the relations, +\begin{align*} +ds^2 &= dx_1{}^2 + dx_2{}^2 && \text{(rectangular)}, \\ +ds^2 &= dx_1{}^2 + x_1{}^2\, dx_2{}^2 && \text{(polar)}, \\ +ds^2 &= dx_1{}^2 - 2 \kappa\, dx_1 dx_2 + dx_2{}^2 + && \text{(oblique)}, \\ +ds^2 &= dx_1{}^2 + \cos^2 x_1\, dx_2{}^2 && \text{(latitude and longitude)}. +\end{align*} +If we have any mesh-system and want to know its nature, we +%% -----File: 090.png---Folio 80------- +must make a number of measures of the length $ds$ between +adjacent points $(x_1, x_2)$ and $(x_1 + dx_1, x_2 + dx_2)$ and test which +formula fits. If, for example, we then find that $ds^2$ is always +equal to $dx_1{}^2 + x_1{}^2\, dx_2{}^2$, we know that our mesh-system is like +that in \Figref{11}, $x_1$ and $x_2$ being the numbers usually denoted by +the polar coordinates $r$, $\theta$. The statement that polar coordinates +are being used is unnecessary, because it adds nothing to our +knowledge which is not already contained in the formula. It is +merely a matter of giving a name; but, of course, the name calls +to our minds a number of familiar properties which otherwise +might not occur to us. + +For instance, it is characteristic of the polar coordinate system +that there is only one point for which $x_1$ (or~$r$) is equal to~$0$, +whereas in the other systems $x_1 = 0$ gives a line of points. This +is at once apparent from the formula; for if we have two points +for which $x_1 = 0$ and $x_1 + dx_1 = 0$, respectively, then +\[ +dx_1{}^2 + x_1{}^2\, dx_2{}^2 = 0. +\] +The distance $ds$ between the two points vanishes, and accordingly +they must be the same point. + +The examples given can all be summed up in one general +expression +\[ +ds^2 = g_{11}\, dx_1{}^2 + 2g_{12}\, dx_1 dx_2 + g_{22}\, dx_2{}^2, +\] +where $g_{11}$, $g_{12}$, $g_{22}$ may be constants or functions of $x_1$ and~$x_2$. +For instance, in the fourth example their values are $1$, $0$, $\cos^2 x_1$. +It is found that all possible mesh-systems lead to values of $ds^2$ +which can be included in an expression of this general form; so +that mesh-systems are distinguished by three functions of +position $g_{11}$, $g_{12}$, $g_{22}$ which can be determined by making physical +measurements. These three quantities are sometimes called +potentials.% +\index{Potentials}% + +We now come to a point of far-reaching importance. The +formula for $ds^2$ teaches us not only the character of the mesh-system, +but the nature of our two-dimensional space, which is +independent of any mesh-system. If $ds^2$ satisfies any one of the +first three formulae, then the space is like a flat surface; +\index{Flat space in two dimensions}% +if it +satisfies the last formula, then the space is a surface curved like +a sphere. Try how you will, you cannot draw a mesh-system on +a flat (Euclidean) surface which agrees with the fourth formula. + +%% -----File: 091.png---Folio 81------- + +If a being limited to a two-dimensional world finds that his +measures agree with the first formula, he can make them agree +with the second or third formulae by drawing the meshes +differently. But to obtain the fourth formula he must be translated +to a different world altogether. + +We thus see that there are different kinds of two-dimensional +space, betrayed by different metrical properties. +\index{Kinds of space}% +\index{Space!kinds of}% +They are +naturally visualised as different surfaces in Euclidean space of +three dimensions. This picture is helpful in some ways, but +perhaps misleading in others. The metrical relations on a plane +sheet of paper are not altered when the paper is rolled into a +cylinder---the measures being, of course, confined to the two-dimensional +world represented by the paper, and not allowed to +take a short cut through space. The formulae apply equally +well to a plane surface or a cylindrical surface; and in so far as +our picture draws a distinction between a plane and a cylinder, +it is misleading. +\index{Cylinder and plane, indistinguishable in two dimensions}% +But they do not apply to a sphere, because +a plane sheet of paper cannot be wrapped round a sphere. +A genuinely two-dimensional being could not be cognisant of +the difference between a cylinder\footnote% + {One should perhaps rather say a roll, to avoid any question of joining the + two edges.} and a plane; but a sphere +would appear as a different kind of space, and he would recognise +the difference by measurement. + +Of course there are many kinds of mesh-systems, and many +kinds of two-dimensional spaces, besides those illustrated in the +four examples. Clearly it is not going to be a simple matter to +discriminate the different kinds of spaces by the values of the~$g$'s. +There is no characteristic, visible to cursory inspection, +which suggests why the first three formulae should all belong to +the same kind of space, and the fourth to a different one. +Mathematical investigation has discovered what is the common +link between the first three formulae. The $g_{11}$, $g_{12}$, $g_{22}$ satisfy in +all three cases a certain differential equation\footnotemark; and whenever +this differential equation is satisfied, the same kind of space +occurs. +\footnotetext{Appendix, \Noteref{4}.} +\Pagelabel{note4} + +No doubt it seems a very clumsy way of approaching these +intrinsic differences of kinds of space---to introduce potentials +%% -----File: 092.png---Folio 82------- +which specifically refer to a particular mesh-system, although +the mesh-system can have nothing to do with the matter. It is +worrying not to be able to express the differences of space in a +purer form without mixing them up with irrelevant differences +of potential. But we have neither the vocabulary nor the +imagination for a description of absolute properties as such. +All physical knowledge is relative to space and time partitions; +and to gain an understanding of the absolute it is necessary to +approach it through the relative. +\index{Absolute@Absolute, approached through the relative}% +The absolute may be defined +as a relative which is always the same no matter what it is +relative to\footnote{Cf.\ \Pageref{31}, where a distinction was drawn between knowledge which does +not particularise the observer and knowledge which does not postulate an +observer at all.}. Although we think of it as self-existing, we cannot +give it a place in our knowledge without setting up some dummy +to relate it to. And similarly the absolute differences of space +always appear as related to some mesh-system, although the +mesh-system is only a dummy and has nothing to do with the +problem. + +The results for two dimensions can be generalised, and applied +to four-dimensional space-time. +\index{Four-dimensional space-time!geometry of}% +Distance must be replaced by +interval, which it will be remembered, is an absolute quantity, +and therefore independent of the mesh-system used. Partitioning +space-time by any system of meshes, a mesh being given by the +crossing of four channels, we must specify a point in space-time +by four coordinate numbers, $x_1$, $x_2$, $x_3$, $x_4$. By analogy the +general formula will be +\index{Interval!general expression for}% +\iffalse %%%%%%%%%% DEAD CODE (matches original) %%%%%%%%%% +\begin{multline*} +ds^2 = g_{11}\, dx_1{}^2 + g_{22}\, dx_2{}^2 + + g_{33}\, dx_3{}^2 + g_{44}\, dx_4{}^2 + + 2g_{12}\, dx_1 dx_2 \\ + + 2g_{13}\, dx_1 dx_3 + + 2g_{14}\, dx_1 dx_4 + + 2g_{23}\, dx_2 dx_3 \\ + + 2g_{24}\, dx_2 dx_4 + + 2g_{34}\, dx_3 dx_4. +\Tag{3} +\end{multline*} +\fi %%%%%%%%%% END OF DEAD CODE %%%%%%%%%% +\begin{align*} % [** PP: Re-breaking] +ds^2 &= g_{11}\, dx_1{}^2 + g_{22}\, dx_2{}^2 + + g_{33}\, dx_3{}^2 + g_{44}\, dx_4{}^2 \\ + &\quad+ 2g_{12}\, dx_1 dx_2 + 2g_{13}\, dx_1 dx_3 + 2g_{14}\, dx_1 dx_4 \\ + &\quad+ 2g_{23}\, dx_2 dx_3 + 2g_{24}\, dx_2 dx_4 + 2g_{34}\, dx_3 dx_4. +\Tag{3} +\end{align*} +The only difference is that there are now ten $g$'s, or potentials, +instead of three, to summarise the metrical properties of the +mesh-system. It is convenient in specifying special values of +the potentials to arrange them in the standard form +\[ +\begin{matrix} +g_{11} & g_{12} & g_{13} & g_{14} \\ + & g_{22} & g_{23} & g_{24} \\ + & & g_{33} & g_{34} \\ + & & & g_{44} +\end{matrix} +\] +%% -----File: 093.png---Folio 83------- +The space-time already discussed at length in \Chapref{III} +corresponded to the \Eqref{formula}{2}, \Pageref{75}, +\[ +ds^2 = - dx^2 - dy^2 - dz^2 + dt^2. +\] +Here $(x, y, z, t)$ are the conventional symbols for $(x_1, x_2, x_3, x_4)$ +when this special mesh-system is used, viz.\ rectangular coordinates +and time. Comparing with~(3) the potentials have the +special values +\index{Flat space-time}% +\index{Galilean potentials}% +\index{Potentials!Galilean values}% +\[ +\begin{matrix} +-1 & \Neg0 & \Neg0 & \Neg0 \\ + & -1 & \Neg0 & \Neg0 \\ + & & -1 & \Neg0 \\ + & & & +1 +\end{matrix} +\] +These are called the ``Galilean values.'' If the potentials have +these values everywhere, space-time may be called ``flat,'' +because the geometry is that of a plane surface drawn in +Euclidean space of five dimensions. +\index{Euclidean space of five dimensions}% +Recollecting what we found +for two dimensions, we shall realise that a quite different set +of values of the potentials may also belong to flat space-time, +because the meshes may be drawn in different ways. We must +clearly understand that + +(1)~The only way of discovering what kind of space-time is +being dealt with is from the values of the potentials, which are +determined practically by measurements of intervals, + +(2)~Different values of the potentials do not necessarily +indicate different kinds of space-time, + +(3)~There is some complicated mathematical property +common to all values of the potentials which belong to the +same space-time, which is not shared by those which belong to +a different kind of space-time. This property is expressed by +a set of differential equations. + +It can now be deduced that the space-time in which we live +is not quite flat. If it were, a mesh-system could be drawn for +which the $g$'s have the Galilean values, and the geometry with +respect to these partitions of space and time would be that +discussed in \Chapref{III}. For that geometry the geodesics, giving +the natural tracks of particles, are straight lines. + +Thus in flat space-time the law of motion is that (with +suitably chosen coordinates) every particle moves uniformly in +a straight line except when it is disturbed by the impacts of +%% -----File: 094.png---Folio 84------- +other particles. Clearly this is not true of our world; for example, +the planets do not move in straight lines although they do not +suffer any impacts. It is true that if we confine attention to a +small region like the interior of Jules Verne's projectile, all the +tracks become straight lines for an appropriate observer, or, +as we generally say, he detects no field of force. It needs a +large region to bring out the differences of geometry. That is +not surprising, because we cannot expect to tell whether a +surface is flat or curved unless we consider a reasonably large +portion of it.% +\index{Flat space-time!at infinity}% + +According to Newtonian ideas, at a great distance from all +matter beyond the reach of any gravitation, particles would all +move uniformly in straight lines. Thus at a great distance from +all matter space-time tends to become perfectly flat. This can +only be checked by experiment to a certain degree of accuracy, +and there is some doubt as to whether it is rigorously true. We +shall leave this afterthought to \Chapref{X}, meanwhile assuming +with Newton that space-time far enough away from everything +is flat, although near matter it is curved. It is this puckering +near matter which accounts for its gravitational effects. + +Just as we picture different kinds of two-dimensional space +as differently curved surfaces in our ordinary space of three-dimensions, +so we are now picturing different kinds of four-dimensional +space-time as differently curved surfaces in a +Euclidean space of \textit{five} dimensions. This is a picture only\footnotemark. + \footnotetext{A fifth dimension suffices for illustrating the property here considered; + \index{Curvature!merely illustrative}% + but for an exact representation of the geometry of the world, Euclidean space + of \textit{ten} dimensions is required. We may well ask whether there is merit in + Euclidean geometry sufficient to justify going to such extremes.}% +The fifth dimension is neither space nor time nor anything that +can be perceived; so far as we know, it is nonsense. I should not +describe it as a mathematical fiction, because it is of no great +advantage in a mathematical treatment. It is even liable to +mislead because it draws distinctions, like the distinction between +a plane and a roll, which have no meaning. It is, like +the notion of a field of force acting in space and time, merely +introduced to bolster up Euclidean geometry, when Euclidean +geometry has been found inappropriate. +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% +The real difference +between the various kinds of space-time is that they have +%% -----File: 095.png---Folio 85------- +different kinds of geometry, involving different properties of the~$g$'s. +It is no explanation to say that this is because the surfaces +are differently curved in a real Euclidean space of five dimensions. +We should naturally ask for an explanation why the space of +five dimensions is Euclidean; and presumably the answer would +be, because it is a plane in a real Euclidean space of six dimensions, +and so on \textit{ad infinitum}. + +The value of the picture to us is that it enables us to describe +important properties with common terms like ``pucker'' and +``curvature'' instead of technical terms like ``differential +invariant.'' +\index{Pucker in space-time}% +We have, however, to be on our guard, because +analogies based on three-dimensional space do not always apply +immediately to many-dimensional space. The writer has keen +recollections of a period of much perplexity, when he had not +realised that a four-dimensional space with ``no curvature'' is +not the same as a ``flat'' space! Three-dimensional geometry +does not prepare us for these surprises. + +Picturing the space-time in the gravitational field round the +earth as a pucker, we notice that we cannot locate the pucker +at a point; it is ``somewhere round'' the point. At any special +point the pucker can be pressed out flat, and the irregularity +runs off somewhere else. That is what the inhabitants of Jules +Verne's projectile did; they flattened out the pucker inside the +projectile so that they could not detect any field of force there; +but this only made things worse somewhere else, and they +would find an increased field of force (relative to them) on the +other side of the earth. + +What determines the existence of the pucker is not the values +of the $g$'s at any point, or, what comes to the same thing, the +field of force there. It is the way these values link on to those +at other points---the gradient of the~$g$'s, and more particularly +the gradient of the gradient. Or, as has already been said, the +kind of space-time is fixed by differential equations. + +Thus, although a gravitational field of force is not an absolute +thing, and can be imitated or annulled at any point by an +acceleration of the observer or a change of his mesh-system, +nevertheless the presence of a heavy particle does modify the +world around it in an absolute way which cannot be imitated +artificially. Gravitational force is relative; but there is this +%% -----File: 096.png---Folio 86------- +more complex character of gravitational influence which is +absolute. + +The question must now be put, Can every possible kind of +space-time occur in an empty region in nature? Suppose we +give the ten potentials perfectly arbitrary values at every point; +that will specify the geometry of some mathematically possible +space-time. But could that kind of space-time actually occur---by +any arrangement of the matter round the region? + +The answer is that only certain kinds of space-time can occur +in an empty region in nature. The law which determines what +kinds can occur is the law of gravitation. + +It is indeed clear that, since we have reduced the theory of +fields of force to a theory of the geometry of the world, if there +is any law governing fields of force (including the gravitational +field), that law must be of the nature of a restriction on the +possible geometries of the world. + +The choice of $g$'s in any special problem is thus arrived at by +a three-fold sorting out: (1)~many sets of values can be dismissed +because they can never occur in nature, (2)~others, while possible, +do not relate to the kind of space-time present in the problem +considered, (3)~of those which remain, one set of values relates +to the particular mesh-system that has been chosen. We have +now to find the law governing the first discrimination. What is +the criterion that decides what values of the $g$'s give a kind of +space-time possible in nature? + +In solving this problem Einstein had only two clues to guide +him. + +(1)~Since it is a question of whether the \textit{kind of space-time} is +possible, the criterion must refer to those properties of the $g$'s +which distinguish different kinds of space-time, not to those +which distinguish different kinds of mesh-system in the same +space-time. The formulae must therefore not be altered in any +way, if we change the mesh-system. + +(2)~We know that flat space-time \textit{can} occur in nature (at +great distances from all gravitating matter). Hence the criterion +must be satisfied by any values of the $g$'s belonging to flat +space-time. + +It is remarkable that these slender clues are sufficient to +indicate almost uniquely a particular law. Afterwards the +%% -----File: 097.png---Folio 87------- +further test must be applied---whether the law is confirmed by +observation. + +\index{Coincidences|(}% +The irrelevance of the mesh-system to the laws of nature is +sometimes expressed in a slightly different way. +\index{Mesh-systems!irrelevance to laws of nature}% +There is one +type of observation which, we can scarcely doubt, must be +independent of any possible circumstances of the observer, +namely a complete coincidence in space and time. The track of +a particle through four-dimensional space-time is called its +world-line. +\index{World-line}% +Now, the world-lines of two particles either intersect +or they do not intersect; the standpoint of the observer is not +involved. In so far as our knowledge of nature is a knowledge +of intersections of world-lines, it is absolute knowledge independent +of the observer. If we examine the nature of our +observations, distinguishing what is actually seen from what is +merely inferred, we find that, at least in all exact measurements, +our knowledge is primarily built up of intersections of world-lines +of two or more entities, that is to say their coincidences. +For example, an electrician states that he has observed a current +of $5$~milliamperes. This is his inference: his actual observation +was a \textit{coincidence} of the image of a wire in his galvanometer +with a division of a scale. A meteorologist finds that the temperature +of the air is~$75°$; his observation was the \textit{coincidence} of +the top of the mercury-thread with division $75$ on the scale of +his thermometer. It would be extremely clumsy to describe the +results of the simplest physical experiment entirely in terms of +coincidence. The absolute observation is, whether or not the +coincidence exists, not when or where or under what circumstances +the coincidence exists; unless we are to resort to relative +knowledge, the place, time and other circumstances must in +their turn be described by reference to other coincidences. But +it seems clear that if we could draw all the world-lines so as to +show all the intersections in their proper order, but otherwise +arbitrary, this would contain a complete history of the world, +and nothing within reach of observation would be omitted.% +\index{Coincidences|)}% + +Let us draw such a picture, and imagine it embedded in a +jelly. If we deform the jelly in any way, the intersections will +still occur in the same order along each world-line and no +additional intersections will be created. The deformed jelly will +represent a history of the world, just as accurate as the one +%% -----File: 098.png---Folio 88------- +originally drawn; there can be no criterion for distinguishing +which is the best representation. + +Suppose now we introduce space and time-partitions, which +we might do by drawing rectangular meshes in both jellies. +We have now two ways of locating the world-lines and events +in space and time, both on the same absolute footing. But +clearly it makes no difference in the result of the location whether +we first deform the jelly and then introduce regular meshes, or +whether we introduce irregular meshes in the undeformed jelly. +And so all mesh-systems are on the same footing. + +This account of our observational knowledge of nature shows +that there is no \textit{shape} inherent in the absolute world, so that +when we insert a mesh-system, it has no shape initially, and a +rectangular mesh-system is intrinsically no different from any +other mesh-system. + +Returning to our two clues, condition (1)~makes an extraordinarily +clean sweep of laws that might be suggested; among +them Newton's law is swept away. The mode of rejection can +be seen by an example; it will be sufficient to consider two +dimensions. If in one mesh-system $(x, y)$ +\begin{align*} +ds^2 &= g_{11} dx^2 + 2g_{12} dxdy + g_{22} dy^2, \\ +\intertext{and in another system $(x', y')$} +ds^2 &= g_{11}'\, {dx'}^2 + 2g_{12}'\, dx'dy' + g_{22}'\, dy'^2, +\end{align*} +the same law must be satisfied if the unaccented letters are +throughout replaced by accented letters. Suppose the law +$g_{11} = g_{22}$ is suggested. Change the mesh-system by spacing the +$y$-lines twice as far apart, that is to say take $y' = \frac{1}{2}y$, with +$x' = x$. Then +\begin{DPalign*} +ds^2 &= g_{11}\, dx^2 + 2g_{12}\, dx dy + g_{22}\, dy^2 \\ + &= g_{11}\, {dx'}^2 + 4g_{12}\, dx'dy' + 4g_{22}\, {dy'}^2, \\ +\lintertext{so that} + &\quad {g_{11}}' = g_{11},\qquad {g_{22}}' = 4g_{22}. +\end{DPalign*} +And if $g_{11}$ is equal to $g_{22}$, $g_{11}'$ cannot be equal to $g_{22}'$. + +After a few trials the reader will begin to be surprised that +any possible law could survive the test. It seems so easy to +defeat any formula that is set up by a simple change of mesh-system. +Certainly it is unlikely that anyone would hit on such +a law by trial. But there are such laws, composed of exceedingly +complicated mathematical expressions. The theory of these is +%% -----File: 099.png---Folio 89------- +called the ``theory of tensors,'' and had already been worked +out by the pure mathematicians Riemann, Christoffel, Ricci, +Levi-Civita who, it may be presumed, never dreamt of a physical +application for it.% +\index{Christoffel}% +\index{Levi-Civita}% +\index{Ricci}% +\index{Riemann}% + +One law of this kind is the condition for flat space-time, +which is generally written in the simple, but not very illuminating, +form +\index{Flat space-time!conditions for}% +\[ +B ^\rho _{\mu\nu\sigma} = 0. +\Tag{4} +\] +The quantity on the left is called the Riemann-Christoffel +tensor, and it is written out in a less abbreviated form in the +Appendix\footnote{Appendix, \Noteref{5}.}. % [** PP: Added ``Appendix''] +\Pagelabel{note5}% +\index{Riemann-Christoffel tensor}% +\index{Tensors}% +It must be explained that the letters $\mu$, $\nu$, $\sigma$, $\rho$ +indicate \textit{gaps}, which are to be filled up by any of the numbers +$1$, $2$, $3$, $4$, chosen at pleasure. (When the expression is written +out at length, the gaps are in the suffixes of the $x$'s and $g$'s.) +Filling the gaps in different ways, a large number of expressions, +$B^1_{111}$, $B^4_{123}$, $B^1_{432}$, etc., are obtained. The \Eqref{equation}{4} states that +all of these are zero. There are $4^4$, or~$256$, of these expressions +altogether, but many of them are repetitions. Only 20 of the +equations are really necessary; the others merely say the same +thing over again. + +It is clear that the \Eqref{law}{4} is not the law of gravitation for +which we are seeking, because it is much too drastic. If it were +a law of nature, then only flat space-time could exist in nature, +and there would be no such thing as gravitation. It is not the +general condition, but a special case---when all attracting +matter is infinitely remote. + +But in finding a general condition, it may be a great help to +know a special case. Would it do to select a certain number of +the 20~equations to be satisfied generally, leaving the rest to +be satisfied only in the special case? Unfortunately the equations +hang together; and, unless we take them all, it is found that +the condition is not independent of the mesh-system. But there +happens to be one way of building up out of the 20~conditions +a less stringent set of conditions independent of the mesh-system. +Let +\begin{align*} +G_{11} + &= B^1_{111} + B^2_{112} + B^3_{113} + B^4_{114}, \\ +\intertext{and, generally} +G_{\mu\nu} + &= B^1_{\mu\nu 1} + B^2_{\mu\nu 2} + B^3_{\mu\nu 3} + B^4_{\mu\nu 4}, +\end{align*} +%% -----File: 100.png---Folio 90------- +then the conditions +\[ +G_{\mu\nu} = 0 +\Tag{5} +\] +will satisfy our requirements for a general law of nature. + +This law is independent of the mesh-system, though this can +only be proved by elaborate mathematical analysis. Evidently, +when all the $B$'s vanish, \Eqref{equation}{5} is satisfied; so, when flat +space-time occurs, this law of nature is not violated. Further +it is not so stringent as the condition for flatness, and admits +of the occurrence of a limited variety of non-Euclidean geometries. +\index{Geometry!non-Euclidean, or Riemannian}% +\index{Non-Euclidean geometry}% +\index{Riemannian, or non-Euclidean, geometry}% +Rejecting duplicates, it comprises 10~equations; but four +of these can be derived from the other six, so that it gives +six conditions, which happens to be the number required for a +law of gravitation\footnotemark. + \footnotetext{Isolate a region of empty space-time; and suppose that everywhere outside + the region the potentials are known. It should then be possible by the law of + gravitation to determine the nature of space-time in the region. Ten differential + equations together with the boundary-values would suffice to determine the + ten potentials throughout the region; but that would determine not only the + kind of space-time but the mesh-system, whereas the partitions of the mesh-system + can be continued across the region in any arbitrary way. The four + sets of partitions give a four-fold arbitrariness; and to admit of this, the number + of equations required is reduced to six.} + +The suggestion is thus reached that +\index{Gravitation, Einstein's law of!differential formula}% +\[ +G_{\mu\nu} = 0 +\] +may be the general law of gravitation. Whether it is so or not +can only be settled by experiment. In particular, it must in +ordinary cases reduce to something so near the Newtonian law, +that the remarkable confirmation of the latter by observation +is accounted for. Further it is necessary to examine whether +there are any exceptional cases in which the difference between +it and Newton's law can be tested. We shall see that these +tests are satisfied. + +What would have been the position if this suggested law had +failed? We might continue the search for other laws satisfying +the two conditions laid down; but these would certainly be far +more complicated mathematically. I believe too that they would +not help much, because practically they would be indistinguishable +from the simpler law here suggested---though this has not +been demonstrated rigorously. The other alternative is that +there is something causing force in nature not comprised in the +%% -----File: 101.png---Folio 91------- +geometrical scheme hitherto considered, so that force is not +purely relative, and Newton's super-observer exists. + +Perhaps the best survey of the meaning of our theory can be +obtained from the standpoint of a ten-dimensional Euclidean +continuum, in which space-time is conceived as a particular +four-dimensional surface. It has to be remarked that in ten +dimensions there are gradations intermediate between a flat +surface and a fully curved surface, which we shall speak of as +curved in the ``first degree'' or ``second degree\footnote{This is not a recognised nomenclature.}.'' The distinction +is something like that of curves in ordinary space, +which may be \textit{curved} like a circle, or \textit{twisted} like a helix; but the +analogy is not very close. The full ``curvature'' of a surface is a +single quantity called~$G$, built up out of the various terms $G_{\mu\nu}$ in +somewhat the same way as these are built up out of $B^\rho_{\mu\nu\sigma}$. +The following conclusions can be stated.% +\index{Curvature!degrees of}% + +\begin{DPgather*} +\lintertext{\indent If} + B^\rho_{\mu\nu\sigma} = 0 +\rintertext{(20 conditions)\quad} \\ +\intertext{space-time is flat. This is the state of the world at an infinite +distance from all matter and all forms of energy.} +% +\lintertext{\indent If} + G_{\mu\nu} = 0 +\rintertext{(6 conditions)\quad} \\ +\intertext{space-time is curved in the first degree. This is the state of the +world in an empty region---not containing matter, light or +electromagnetic fields, but in the neighbourhood of these forms +of energy.} +% +\lintertext{\indent If} + G = 0 +\rintertext{(1 condition)\quad} \\ +\intertext{space-time is curved in the second degree. This is the state of +the world in a region not containing matter or electrons (bound +energy), but containing light or electromagnetic fields (free +energy).} +% +\lintertext{\indent If} +G~\text{is not zero} +\end{DPgather*} +space-time is fully curved. This is the state of the world in a +region containing continuous matter.% +\index{Continuous matter}% +\index{Matter!continuous}% + +According to current physical theory continuous matter does +not exist, so that strictly speaking the last case never arises. +Matter is built of electrons or other nuclei. The regions lying +between the electrons are not fully curved, whilst the regions +inside the electrons must be cut out of space-time altogether. +\index{Electron!geometry inside}% +We cannot imagine ourselves exploring the inside of an electron +%% -----File: 102.png---Folio 92------- +with moving particles, light-waves, or material clocks and +measuring-rods; hence, without further definition, any geometry +of the interior, or any statement about space and time in the +interior, is meaningless. But in common life, and frequently in +physics, we are not concerned with this \textit{microscopic} structure of +matter. We need to know, not the actual values of the $g$'s at +a point, but their average values through a region, small from +the ordinary standpoint but large compared with the molecular +structure of matter. In this \textit{macroscopic} treatment molecular +matter is replaced by continuous matter, and uncurved space-time +studded with holes is replaced by an equivalent fully +curved space-time without holes.% +\index{Macroscopic!equations}% + +It is natural that our senses should have developed faculties +for perceiving some of these intrinsic distinctions of the possible +states of the world around us. I prefer to think of matter and +energy, not as agents causing the degrees of curvature of the +world, but as parts of our perceptions of the existence of the +curvature. + +It will be seen that the law of gravitation can be summed up +in the statement that in an empty region space-time can be +curved only in the first degree. +%% -----File: 103.png---Folio 93------- + +% [** PP: Re-breaking title to match running heads] +\Chapter[The New Law of Gravitation and the Old Law]{VI}% +{The New Law of Gravitation \break and the Old Law} + +% [** PP: Special headers] +\fancyhead[CE]{\textsc{THE NEW LAW OF GRAVITATION}} +\fancyhead[CO]{\textsc{AND THE OLD LAW}} + +\Quote{Sir Isaac Newton.} +{I don't know what I may seem to the world, but, as to myself, I seem to have +been only as a boy playing on the sea-shore, and diverting myself in now and +then finding a smoother pebble or a prettier shell than ordinary, whilst the +great ocean of truth lay all undiscovered before me.} + + +\First{Was} there any reason to feel dissatisfied with Newton's law of +gravitation? + +Observationally it had been subjected to the most stringent +tests, and had come to be regarded as the perfect model of an +exact law of nature. The cases, where a possible failure could +be alleged, were almost insignificant. There are certain unexplained +irregularities in the moon's motion; but astronomers +generally looked---and must still look---in other directions for +the cause of these discrepancies. One failure only had led to +a serious questioning of the law; this was the discordance of +motion of the perihelion of Mercury. How small was this discrepancy +may be judged from the fact that, to meet it, it was +proposed to amend \textit{square} of the distance to the $2.00000016$ +power of the distance. Further it seemed possible, though +unlikely, that the matter causing the zodiacal light might be of +sufficient mass to be responsible for this effect.% +\index{Gravitation, Newton's law of!ambiguity of}% +\index{Moon, motion of}% +\index{Newton!law of gravitation}% + +The most serious objection against the Newtonian law as an +exact law was that it had become ambiguous. The law refers +to the product of the masses of the two bodies; but the mass +depends on the velocity---a fact unknown in Newton's day. +Are we to take the variable mass, or the mass reduced to rest? +Perhaps a learned judge, interpreting Newton's statement like +a last will and testament, could give a decision; but that is +scarcely the way to settle an important point in scientific +theory. + +Further \textit{distance}, also referred to in the law, is something +relative to an observer. Are we to take the observer travelling +with the sun or with the other body concerned, or at rest in the +aether or in some gravitational medium? + +%% -----File: 104.png---Folio 94------- + +Finally is the force of gravitation propagated instantaneously, +or with the velocity of light, or some other velocity? +\index{Gravitation!propagation with velocity of light}% +\index{Propagation of Gravitation}% +\index{Velocity of gravitation}% +Until +comparatively recently it was thought that conclusive proof +had been given that the speed of gravitation must be far higher +than that of light. The argument was something like this. If +the Sun attracts Jupiter towards its present position~$S$, and +Jupiter attracts the Sun towards its present position~$J$, the two +forces are in the same line and balance. But if the Sun attracts +Jupiter towards its previous position~$S'$, and Jupiter attracts +the Sun towards its previous position~$J'$, when the force of +attraction started out to cross the gulf, then the two forces +%[Illustration: Fig. 13.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[13]{3.5in}{104a} +\end{center} +\end{figure*}% +give a couple. This couple will tend to increase the angular +momentum of the system, and, acting cumulatively, will soon +cause an appreciable change of period, disagreeing with observation +if the speed is at all comparable with that of light. The +argument is fallacious, because the effect of propagation will not +necessarily be that $S$ is attracted in the direction towards~$J'$. +Indeed it is found that if $S$ and~$J$ are two electric charges, $S$ will +be attracted very approximately towards~$J$ (not~$J'$) in spite of +the electric influence being propagated with the velocity of +light\footnotemark. In the theory given in this book, gravitation is propagated +with the speed of light, and there is no discordance with +observation. +\footnotetext{Appendix, \Noteref{6}.} +\Pagelabel{note6} + +It is often urged that Newton's law of gravitation is much +%% -----File: 105.png---Folio 95------- +simpler than Einstein's new law. That depends on the point of +view; and from the point of view of the four-dimensional world +Newton's law is far more complicated. Moreover, it will be seen +that if the ambiguities are to be cleared up, the statement of +Newton's law must be greatly expanded. + +Some attempts have been made to expand Newton's law on +the basis of the restricted principle of relativity (\Pageref{20}) alone. +This was insufficient to determine a definite amendment. Using +the principle of equivalence, or relativity of force, we have +arrived at a definite law proposed in the last chapter. Probably +the question has arisen in the reader's mind, why should it be +called the law of gravitation? It may be plausible as a law of +nature; but what has the degree of curvature of space-time to +do with attractive forces, whether real or apparent? + +A race of flat-fish once lived in an ocean in which there were +only two dimensions. +\index{Flatfish, analogy of} % [** PP: No hyphen in index] +It was noticed that in general fishes swam +in straight lines, unless there was something obviously interfering +with their free courses. This seemed a very natural behaviour. +But there was a certain region where all the fish seemed to be +bewitched; some passed through the region but changed the +direction of their swim, others swam round and round indefinitely. +One fish invented a theory of vortices, and said that +there were whirlpools in that region which carried everything +round in curves. By-and-by a far better theory was proposed; +it was said that the fishes were all attracted towards a particularly +large fish---a sun-fish---which was lying asleep in the middle +of the region; and that was what caused the deviation of their +paths. The theory might not have sounded particularly plausible +at first; but it was confirmed with marvellous exactitude by all +kinds of experimental tests. All fish were found to possess this +attractive power in proportion to their sizes; the law of attraction +was extremely simple, and yet it was found to explain all the +motions with an accuracy never approached before in any +scientific investigations. Some fish grumbled that they did not +see how there could be such an influence at a distance; but it +was generally agreed that the influence was communicated +through the ocean and might be better understood when more +was known about the nature of water. Accordingly, nearly +every fish who wanted to explain the attraction started by +%% -----File: 106.png---Folio 96------- +proposing some kind of mechanism for transmitting it through +the water. + +But there was one fish who thought of quite another plan. +He was impressed by the fact that whether the fish were big +or little they always took the same course, although it would +naturally take a bigger force to deflect the bigger fish. He therefore +concentrated attention on the courses rather than on the +forces. And then he arrived at a striking explanation of the +whole thing. There was a mound in the world round about +where the sun-fish lay. Flat-fish could not appreciate it directly +because they were two-dimensional; but whenever a fish went +swimming over the slopes of the mound, although he did his +best to swim straight on, he got turned round a bit. (If a traveller +goes over the left slope of a mountain, he must consciously +keep bearing away to the left if he wishes to keep to his original +direction relative to the points of the compass.) This was the +secret of the mysterious attraction, or bending of the paths, +which was experienced in the region. + +The parable is not perfect, because it refers to a hummock in +space alone, whereas we have to deal with hummocks in space-time. +But it illustrates how a curvature of the world we live +in may give an illusion of attractive force, and indeed can only +be discovered through some such effect. How this works out in +detail must now be considered. + +In the form $G_{\mu\nu} =0$, Einstein's law expresses conditions to be +satisfied in a gravitational field produced by any arbitrary +distribution of attracting matter. An analogous form of Newton's +law was given by Laplace in his celebrated expression $\nabla^{2} V = 0$. +\index{Laplace's equation}% +A more illuminating form of the law is obtained if, instead of +putting the question what kinds of space-time can exist under +the most general conditions in an empty region, we ask what +kind of space-time exists in the region round a single attracting +particle? We separate out the effect of a single particle, just as +Newton did. We can further simplify matters by introducing +some definite mesh-system, which, of course, must be of a type +which is not inconsistent with the kind of space-time found. + +We need only consider space of two dimensions---sufficient +for the so-called plane orbit of a planet---time being added as +the third dimension. The remaining dimension of space can +%% -----File: 107.png---Folio 97------- +always be added, if desired, by conditions of symmetry. The +result of long algebraic calculations\footnote{Appendix, \Noteref{7}.} +\Pagelabel{note7}% +is that, round a particle +\index{Gravitation, Einstein's law of!integrated formula for a particle}% +\[ +ds^{2} + = - \frac{1}{\gamma}\, dr^{2} + - r^{2}\, d\theta^{2} + + \gamma\, dt^{2} +\Tag{6} +\] +where $\gamma = 1 - \dfrac{2m} {r}$. + +The quantity $m$ is the gravitational mass of the particle---but +we are not supposed to know that at present. $r$~and $\theta$ are +polar coordinates, the mesh-system being as in \Figref{11}; or rather +they are the nearest thing to polar coordinates that can be +found in space which is not truly flat. + +The fact is that this expression for $ds^{2}$ is found in the first +place simply as a particular solution of Einstein's equations of +the gravitational field; it is a variety of hummock (apparently +the simplest variety) which is not curved beyond the first degree. +\index{Hummock in space-time}% +There \textit{could} be such a state of the world under suitable circumstances. +To find out what those circumstances are, we have to +trace some of the consequences, find out how any particle +moves when $ds^{2}$ is of this form, and then examine whether we +know of any case in which these consequences are found +observationally. It is only after having ascertained that this +form of $ds^{2}$ does correspond to the leading observed effects +attributable to a particle of mass $m$ at the origin that we have +the right to identify this particular solution with the one we +hoped to find. + +It will be a sufficient illustration of this procedure, if we +indicate how the position of the matter causing this particular +solution is located. Wherever the \Eqref{formula}{6} holds good there +can be no matter, because the law which applies to empty space +is satisfied. But if we try to approach the origin ($r = 0$), a +curious thing happens. Suppose we take a measuring-rod, and, +laying it radially, start marking off equal lengths with it along +a radius, gradually approaching the origin. Keeping the time +$t$ constant, and $d\theta$ being zero for radial measurements, the +\Eqref{formula}{6} reduces to +\begin{DPalign*} +ds^{2} &= - \frac{1}{\gamma}\, dr^{2} \\ +\lintertext{or} +dr^{2} &= - \gamma\, ds^{2}. +\end{DPalign*} +%% -----File: 108.png---Folio 98------- +We start with $r$ large. By-and-by we approach the point +where $r = 2m$. But here, from its definition, $\gamma$ is equal to~$0$. +So that, however large the measured interval $ds$ may be, $dr = 0$. +We can go on shifting the measuring-rod through its own length +time after time, but $dr$ is zero; that is to say, we do not reduce~$r$. +There is a magic circle which no measurement can bring us +inside. It is not unnatural that we should picture something +obstructing our closer approach, and say that a particle of +matter is filling up the interior.% +\index{Matter!definition of a particle}% + +The fact is that so long as we keep to space-time curved only +in the first degree, we can never round off the summit of the +hummock. It must end in an infinite chimney. In place of the +chimney, however, we round it off with a small region of greater +curvature. This region cannot be empty because the law applying +to empty space does not hold. We describe it therefore as containing +matter---a procedure which practically amounts to a +definition of matter. Those familiar with hydrodynamics may +be reminded of the problem of the irrotational rotation of a +fluid; the conditions cannot be satisfied at the origin, and it is +necessary to cut out a region which is filled by a vortex-filament. + +A word must also be said as to the coordinates $r$ and $t$ used +in~(6). They correspond to our ordinary notion of radial distance +and time---as well as any variables in a non-Euclidean world +can correspond to words which, as ordinarily used, presuppose +a Euclidean world. We shall thus call $r$ and~$t$, distance and time. % [** PP: Retaining comma] +But to give names to coordinates does not give more information---and +in this case gives considerably less information---than is +already contained in the formula for $ds^{2}$. If any question arises +as to the exact significance of $r$ and $t$ it must always be settled +by reference to \Eqref{equation}{6}. + +The want of flatness in the gravitational field is indicated by +the deviation of the coefficient $\gamma$ from unity. If the mass $m = 0$, +$\gamma = 1$, and space-time is perfectly flat. Even in the most intense +gravitational fields known, the deviation is extremely small. +For the sun, the quantity $m$, called the gravitational mass, is +only $1.47$~kilometres\footnote{Appendix, \Noteref{8}.}, +\Pagelabel{note8}% +for the earth it is $5$~millimetres. +\index{Gravitational field of Sun}% +\index{Mass!gravitational}% +In any +practical problem the ratio $2m/r$ must be exceedingly small. +%% -----File: 109.png---Folio 99------- +Yet it is on the small corresponding difference in $\gamma$ that the +whole of the phenomena of gravitation depend. + +The coefficient $\gamma$ appears twice in the formula, and so modifies +the flatness of space-time in two ways. But as a rule these two +ways are by no means equally important. Its appearance as a +coefficient of $dt^{2}$ produces much the most striking effects. +Suppose that it is wished to measure the interval between two +events in the history of a planet. If the events are, say $1$~second +apart in time, $dt = 1~\text{second} = 300,000~\text{kilometres}$. Thus +$dt^{2}= 90,000,000,000~\text{sq.\ km}$. Now no planet moves more than +$50$~kilometres in a second, so that the change $dr$ associated with +the lapse of $1$~second in the history of the planet will not be +more than $50$~km. Thus $dr^{2}$ is not more than $2500$ sq.~km. +Evidently the small term $2m/r$ has a much greater chance of +making an impression where it is multiplied by $dt^{2}$ than where +it is multiplied by~$dr^{2}$. + +Accordingly as a first approximation, we ignore the coefficient +of~$dr^{2}$, and consider only the meaning of +\[ +ds^{2} = - dr^{2} - r^{2}\, d\theta^{2} + (1 - 2m/r)\, dt^{2}. +\Tag{7} +\] +We shall now show that particles situated in this kind of space-time +will appear to be under the influence of an attractive force +directed towards the origin. + +Let us consider the problem of mapping a small portion of this +kind of world on a plane. + +It is first necessary to define carefully the distinction which is +here drawn between a ``picture'' and a ``map.'' If we are given +the latitudes and longitudes of a number of places on the earth, +we can make a picture by taking latitude and longitude as +vertical and horizontal distances, so that the lines of latitude +and longitude form a mesh-system of squares; but that does not +give a true map. In an ordinary map of Europe the lines of +longitude run obliquely and the lines of latitude are curved. +Why is this? Because the map aims at showing as accurately +as possible all distances in their true proportions\footnotemark. + \footnotetext{This is usually the object, though maps are sometimes made for a different + purpose, e.g.\ Mercator's Chart.}% +\index{Map of sun's gravitational field}% +Distance is +the important thing which it is desired to represent correctly. +In four dimensions interval is the analogue of distance, and a +map of the four-dimensional world will aim at showing all the +%% -----File: 110.png---Folio 100------- +intervals in their correct proportions. Our natural \textit{picture} of +space-time takes $r$ and $t$ as horizontal and vertical distances, +e.g.\ when we plot the graph of the motion of a particle; but in +a true \textit{map}, representing the intervals in their proper proportions, +the $r$ and $t$ lines run obliquely or in curves across the map. + +The instructions for drawing latitude and longitude lines $(\beta, \lambda)$ +on a map, are summed up in the formula for $ds$, \Pageref{79}, +\[ +ds^{2} = d\beta^{2} + \cos^{2}\beta\, d\lambda^{2}, +\] +%[Illustration: \textsc{Fig}. 14.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[14]{3in}{110a} +\end{center} +\end{figure*}% +and similarly the instructions for drawing the $r$ and $t$ lines are +given by the \Eqref{formula}{7}. + +The map is shown in \Figref{14}. It is not difficult to see why the +$t$-lines converge to the left of the diagram. The factor $1 - 2m/r$ +decreases towards the left where $r$ is small; and consequently +any change of $t$ corresponds to a shorter interval, and must be +represented in the map by a shorter distance on the left. It is +less easy to see why the $r$-lines take the courses shown; by +analogy with latitude and longitude we might expect them to +be curved the other way. But we discussed in \Chapref{III} how +%% -----File: 111.png---Folio 101------- +the slope of the time-direction is connected with the slope of +the space-direction; and it will be seen that the map gives +approximately diamond-shaped partitions of the kind represented +in \Figref{6}\footnotemark. + \footnotetext{The substitution $x = r + \frac{1}{2} t^{2} m/r^{2}$, + $y=t(1-m/r)$, gives $ds^{2}= -dx^{2} + dy^{2}$, if + squares of~$m$ are negligible. The map is drawn with $x$ and $y$ as rectangular + coordinates.} + +Like all maps of curved surfaces, the diagram is only accurate +in the limit when the area covered is very small. + +It is important to understand clearly the meaning of this map. +When we speak in the ordinary way of distance from the sun +and the time at a point in the solar system, we mean the two +variables $r$ and~$t$. These are not the result of any precise measures +with scales and clocks made at a point, but are mathematical +variables most appropriate for describing the whole solar system. +%[Illustration: \textsc{Fig}. 15.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[15]{3.5in}{111a} +\end{center} +\end{figure*}% +They represent a compromise, because it is necessary to deal +with a region too large for accurate representation on a plane +map. We should naturally picture them as rectangular coordinates +partitioning space-time into square meshes, as in +\Figref{15}; but such a picture is not a true map, because it does +not represent in their true proportions the intervals between the +various points in the picture. It is not possible to draw any +map of the whole curved region without distortion; but a small +enough portion can be represented without distortion if the +partitions of equal $r$ and $t$ are drawn as in \Figref{14}. To get back +%% -----File: 112.png---Folio 102------- +from the true map to the customary picture of $r$ and $t$ as perpendicular +space and time, we must strain \Figref{14} until all the +meshes become squares as in \Figref{15}. + +Now in the map the geometry is Euclidean and the tracks of +all material particles will be straight lines. Take such a straight +track~$PQ$, which will necessarily be nearly vertical, unless the +velocity is very large. Strain the figure so as to obtain the +customary representation of $r$ and~$t$ (in \Figref{15}), and the track +$PQ$ will become curved---curved towards the left, where the sun +lies. In each successive vertical interval (time), a successively +greater progress is made to the left horizontally (space). Thus +the velocity towards the sun increases. We say that the particle +is attracted to the sun. + +The mathematical reader should find no difficulty in proving +from the diagram that for a particle with small velocity the +acceleration towards the sun is approximately $m/r^{2}$, agreeing +with the Newtonian law.% +\index{Gravitational field of Sun!Newtonian attraction}% + +Tracks for very high speeds may be affected rather differently. +The track corresponding to a wave of light is represented by +a straight line at~$45°$ to the horizontal in \Figref{14}. It would +require very careful drawing to trace what happens to it when +the strain is made transforming to \Figref{15}; but actually, whilst +becoming more nearly vertical, it receives a curvature in the +opposite direction. The effect of the gravitation of the sun on +a light-wave, or very fast particle, proceeding radially is actually +a \textit{repulsion}!% +\index{Repulsion of light proceeding radially} + +The track of a transverse light-wave, coming out from the +plane of the paper, will be affected like that of a particle of +zero velocity in distorting from \Figref{14} to \Figref{15}. Hence the +sun's influence on a transverse light-wave is always an attraction. +The acceleration is simply $m/r^{2}$ as for a particle at rest. + +The result that the expression found for the geometry of the +gravitational field of a particle leads to Newton's law of attraction +is of great importance. It shows that the law, $G_{\mu\nu}= 0$, +proposed on theoretical grounds, agrees with observation at +least approximately. It is no drawback that the Newtonian +law applies only when the speed is small; all planetary speeds +are small compared with the velocity of light, and the considerations +mentioned at the beginning of this chapter suggest that +%% -----File: 113.png---Folio 103------- +some modification may be needed for speeds comparable with +that of light. + +Another important point to notice is that the attraction of +gravitation is simply a geometrical deformation of the straight +tracks. It makes no difference what body or influence is pursuing +the track, the deformation is a general discrepancy between the +``mental picture'' and the ``true map'' of the portion of space-time +considered. Hence light is subject to the same disturbance +of path as matter. This is involved in the Principle of Equivalence; +otherwise we could distinguish between the acceleration +of a lift and a true increase of gravitation by optical experiments; +in that case the observer for whom light-rays appear to +take straight tracks might be described as absolutely unaccelerated +and there could be no relativity theory. Physicists in +general have been prepared to admit the likelihood of an +influence of gravitation on light similar to that exerted on +matter; and the problem whether or not light has ``weight'' +has often been considered. + +The appearance of $\gamma$ as the coefficient of $dt^{2}$ is responsible for +the main features of Newtonian gravitation; the appearance of +$1/\gamma$ as the coefficient of $dr^{2}$ is responsible for the principal +deviations of the new law from the old. This classification seems +to be correct; but the Newtonian law is ambiguous and it is +difficult to say exactly what are to be regarded as discrepancies +from it. Leaving aside now the time-term as sufficiently discussed, +we consider the space-terms alone\footnote + {We change the sign of $ds^{2}$, so that $ds$, when real, means measured space + instead of measured time.} +\index{Gravitation, Newton's law of!approximation to Einstein's law}% +\[ +ds^{2} = \frac{1}{\gamma}\, dr^{2} + r^{2}\, d\theta^{2}. +\] + +The expression shows that space considered alone is non-Euclidean +in the neighbourhood of an attracting particle. This +is something entirely outside the scope of the old law of gravitation. +Time can only be explored by something moving, whether +a free particle or the parts of a clock, so that the non-Euclidean +character of space-time can be covered up by introducing a field +of force, suitably modifying the motion, as a convenient fiction. +But space can be explored by static methods; and theoretically +%% -----File: 114.png---Folio 104------- +its non-Euclidean character could be ascertained by sufficiently +precise measures with rigid scales. + +If we lay our measuring scale transversely and proceed to +measure the circumference of a circle of nominal radius~$r$, we +see from the formula that the measured length~$ds$ is equal to +$r\,d\theta$, so that, when we have gone right round the circle, $\theta$~has +increased by~$2\pi$ and the measured circumference is $2\pi r$. +\index{Circle in non-Euclidean space}% +But +when we lay the scale radially the measured length~$ds$ is equal +to $dr/\sqrt{\gamma}$, which is always greater than~$dr$. Thus, in measuring +a diameter, we obtain a result greater than~$2r$, each portion being +greater than the corresponding change of~$r$. + +Thus if we draw a circle, placing a massive particle near the +centre so as to produce a gravitational field, and measure with +a rigid scale the circumference and the diameter, the ratio of +the measured circumference to the measured diameter will not be +the famous number $\pi = 3.141592653589793238462643383279\ldots$ +but a little smaller. Or if we inscribe a regular hexagon in this +circle its sides will not be exactly equal to the radius of the +circle. Placing the particle near, instead of at, the centre, +avoids measuring the diameter \textit{through} the particle, and so +makes the experiment a practical one. But though practical, +it is not practicable to determine the non-Euclidean character +of space in this way. Sufficient refinement of measures is not +attainable. If the mass of a ton were placed inside a circle of +$5$~yards radius, the defect in the value of $\pi$ would only appear +in the twenty-fourth or twenty-fifth place of decimals. + +It is of value to put the result in this way, because it shows +that the relativist is not talking metaphysics when he says that +space in the gravitational field is non-Euclidean. His statement +has a plain physical meaning, which we may some day learn how +to test experimentally. Meanwhile we can test it by indirect +methods. + +Suppose that a plane field is uniformly studded with hurdles. +\index{Hurdles, analogy of counts of}% +The distance between any two points will be proportional to +the number of hurdles that must be passed over in getting from +one point to the other by the straight route---in fact the minimum +number of hurdles. We can use counts of hurdles as the equivalent +of distance, and map the field by these counts. The map +can be drawn on a plane sheet of paper without any inconsistency, +%% -----File: 115.png---Folio 105------- +since the field is plane. Let us now dismiss from our +minds all idea of distances in the field or straight lines in the +field, and assume that distances on the map merely represent +the minimum number of hurdles between two points; straight +lines on the map will represent the corresponding routes. This +has the advantage that if an earthquake occurs, deforming the +field, the map will still be correct. The path of fewest hurdles +will still cross the same hurdles as before the earthquake; it +will be twisted out of the straight line in the field; but we should +gain nothing by taking a straighter course, since that would +lead through a region where the hurdles are more crowded. +We do not alter the number of hurdles in any path by deforming +it. + +This can be illustrated by Figs.~14 and~15. \Figref{14} represents +the original undistorted field with the hurdles uniformly placed. +The straight line $PQ$ represents the path of fewest hurdles from +$P$ to~$Q$, and its length is proportional to the number of hurdles. +\Figref{15} represents the distorted field, with $PQ$ distorted into +a curve; but $PQ$ is still the path of fewest hurdles from $P$ to~$Q$, +and the number of hurdles in the path is the same as before. +If therefore we map according to hurdle-counts we arrive at +\Figref{14} again, just as though no deformation had taken place. + +To make any difference in the hurdle-counts, the hurdles +must be taken up and replanted. Starting from a given point +as centre, let us arrange them so that they gradually thin out +towards the boundaries of the field. Now choose a circle with +this point as centre;---but first, what is a circle? It has to be +defined in terms of hurdle-counts; and clearly it must be a +curve such that the minimum number of hurdles between any +point on it and the centre is a constant (the radius). With this +definition we can defy earthquakes. The number of hurdles in +the circumference of such a circle will not bear the same proportion +to the number in the radius as in the field of uniform +hurdles; owing to the crowding near the centre, the ratio will +be less. Thus we have a suitable analogy for a circle whose +circumference is less than $\pi$ times its diameter. + +This analogy enables us to picture the condition of space +round a heavy particle, where the ratio of the circumference of +a circle to the diameter is less than~$\pi$. Hurdle-counts will no +%% -----File: 116.png---Folio 106------- +longer be accurately mappable on a plane sheet of paper, +because they do not conform to Euclidean geometry. + +Now suppose a heavy particle wishes to cross this field, +passing near but not through the centre. In Euclidean space, +with the hurdles uniformly distributed, it travels in a straight +line, i.e.\ it goes between any two points by a path giving the +fewest hurdle jumps. We may assume that in the non-Euclidean +field with rearranged hurdles, the particle still goes by the path +of least effort. In fact, in any small portion we cannot distinguish +between the rearrangement and a distortion; so we may imagine +that the particle takes each portion as it comes according to the +rule, and is not troubled by the rearrangement which is only +visible to a general survey of the whole field\footnotemark. + \footnotetext{There must be some absolute track, and if absolute significance can only + be associated with hurdle-counts and not with distances in the field, the path + of fewest hurdles is the only track capable of absolute definition.} + +Now clearly it will pay not to go straight through the dense +portion, but to keep a little to the outside where the hurdles +are sparser---not too much, or the path will be unduly lengthened. +The particle's track will thus be a little concave to the centre, +and an onlooker will say that it has been attracted to the centre. +It is rather curious that we should call it attraction, when the +track has rather been avoiding the central region; but it is clear +that the direction of motion has been bent round in the way +attributable to an attractive force. + +This bending of the path is additional to that due to the +Newtonian force of gravitation which depends on the second +appearance of~$\gamma$ in the formula. As already explained it is in +general a far smaller effect and will appear only as a minute +correction to Newton's law. The only case where the two rise +to equal importance is when the track is that of a light-wave, +or of a particle moving with a speed approaching that of light; +for then $dr^{2}$ rises to the same order of magnitude as~$dt^{2}$. + +To sum up, a ray of light passing near a heavy particle will +be bent, firstly, owing to the non-Euclidean character of the +combination of time with space. This bending is equivalent to +that due to Newtonian gravitation, and may be calculated in +the ordinary way on the assumption that light has weight like +a material body. Secondly, it will be bent owing to the non-% +%% -----File: 117.png---Folio 107------- +Euclidean character of space alone, and this curvature is +additional to that predicted by Newton's law. If then we can +observe the amount of curvature of a ray of light, we can make +a crucial test of whether Einstein's or Newton's theory is +obeyed. + +This separation of the attraction into two parts is useful in +a comparison of the new theory with the old; but from the +point of view of relativity it is artificial. Our view is that light +is bent just in the same way as the track of a material particle +moving with the same velocity would be bent. +\index{Bending of light!theory of}% +\index{Light, bending of}% +\index{Weight!of light}% +Both causes of +bending may be ascribed either to weight or to non-Euclidean +space-time, according to the nomenclature preferred. The only +difference between the predictions of the old and new theories +is that in one case the weight is calculated according to Newton's +law of gravitation, in the other case according to Einstein's. + +There is an alternative way of viewing this effect on light +according to Einstein's theory, which, for many reasons is to +be preferred. This depends on the fact that the velocity of +light in the gravitational field is not a constant (unity) but +becomes smaller as we approach the sun. This does not mean +that an observer determining the velocity of light experimentally +at a spot near the sun would detect the decrease; if he performed +Fizeau's experiment, his result in kilometres per second would +be exactly the same as that of a terrestrial observer. It is the +coordinate velocity that is here referred to, described in terms +of the quantities $r$, $\theta$, $t$, introduced by the observer who is +contemplating the whole solar system at the same time.% +\index{Coordinate velocity}% +\index{Light!coordinate velocity of}% + +It will be remembered that in discussing the approximate +geometry of space-time in \Figref{3}, we found that certain events +like $P$ were in the absolute past or future of~$O$, and others like +$P'$ were neither before nor after~$O$, but elsewhere. Analytically +the distinction is that for the interval $OP$, $ds^{2}$ is positive; for +$OP'$, $ds^{2}$ is negative. In the first case the interval is real or +``time-like''; in the second it is imaginary or ``space-like.'' The +two regions are separated by lines (or strictly, cones) in crossing +which $ds^{2}$ changes from positive to negative; and along the lines +themselves $ds$ is zero. It is clear that these lines must have +important absolute significance in the geometry of the world. +Physically their most important property is that pulses of light +%% -----File: 118.png---Folio 108------- +travel along these tracks, and the motion of a light-pulse is +always given by the equation $ds = 0$. + +Using the expression for $ds^{2}$ in a gravitational field, we +accordingly have for light +\[ +0 = -\frac{1}{\gamma}\, dr^{2} + -r^{2}\, d\theta^{2} + \gamma\, dt^{2}. +\] +For radial motion, $d\theta = 0$, and therefore +\[ +\left(\frac{dr}{dt}\right)^{2} = \gamma^{2}. +\] +For transverse motion, $dr = 0$, and therefore +\[ +\left(\frac{r\, d\theta}{dt}\right)^{2} = \gamma. +\] +Thus the coordinate velocity of light travelling radially is $\gamma$, +and of light travelling transversely is $\surd\gamma$, in the coordinates +chosen.% +\index{Velocity of light!in gravitational field}% + +The coordinate velocity must depend on the coordinates +chosen; and it is more convenient to use a slightly different +system in which the velocity of light is the same in all directions\footnotemark, + \footnotetext{This is obtained by writing $r + m$ instead of~$r$, or diminishing the nominal + distance of the sun by $1\frac{1}{2}$~kilometres. This change of coordinates simplifies + the problem, but can, of course, make no difference to anything observable. + After we have traced the course of the light ray in the coordinates chosen, we + have to connect the results with experimental measures, using the corresponding + formula for~$ds^{2}$. This final connection of mathematical and experimental results + is, however, comparatively simple, because it relates to measuring operations + performed in a terrestrial observatory where the difference of $\gamma$ from unity is + negligible.}% +viz.\ $\gamma$ or~$1 - 2m/r$. This diminishes as we approach the sun---an +illustration of our previous remark that a pulse of light +proceeding radially is repelled by the sun.% +\index{Repulsion of light proceeding radially}% + +The wave-motion in a ray of light can be compared to a +succession of long straight waves rolling onward in the sea. If +the motion of the waves is slower at one end than the other, the +whole wave-front must gradually slew round, and the direction +in which it is rolling must change. +\index{Wave-front, slewing of}% +In the sea this happens when +one end of the wave reaches shallow water before the other, +because the speed in shallow water is slower. It is well known +that this causes waves proceeding diagonally across a bay to +slew round and come in parallel to the shore; the advanced end +%% -----File: 119.png---Folio 109------- +is delayed in the shallow water and waits for the other. In the +same way when the light waves pass near the sun, the end nearest +the sun has the smaller velocity and the wave-front slews round; +thus the course of the waves is bent. + +Light moves more slowly in a material medium than in +vacuum, the velocity being inversely proportional to the refractive +index of the medium. The phenomenon of refraction +is in fact caused by a slewing of the wave-front in passing into +a region of smaller velocity. We can thus imitate the gravitational +effect on light precisely, if we imagine the space round +the sun filled with a refracting medium which gives the +appropriate velocity of light. +\index{Refracting medium equivalent to gravitational field}% +To give the velocity $1-2m/r$, the +refractive index must be $1/(1-2m/r)$, or, very approximately, +$1 + 2m/r$. At the surface of the sun, $r= 697,000~\text{km.}$, $m = 1.47~\text{km.}$, +hence the necessary refractive index is~$1.00000424$. At a +height above the sun equal to the radius it is~$1.00000212$. + +Any problem on the paths of rays near the sun can now be +solved by the methods of geometrical optics applied to the +equivalent refracting medium. It is not difficult to show that +the total deflection of a ray of light passing at a distance $r$ from +the centre of the sun is (in circular measure) +\index{Deflection of light!theory of}% +\index{Gravitational field of Sun!deflection of light}% +\[ +\frac{4m}{r}, +\] +whereas the deflection of the same ray calculated on the +Newtonian theory would be +\index{Gravitation, Newton's law of!deflection of light}% +\[ +\frac{2m}{r}. +\] + +For a ray grazing the surface of the sun the numerical value +of this deflection is +\begin{align*} +&1''.75 \quad \text{(Einstein's theory)}, \\ +&0''.87 \quad \text{(Newtonian theory)}. +\end{align*} +%% -----File: 120.png---Folio 110------- + + +\Chapter{VII}{Weighing Light} + +\Quote{Newton, \textit{Opticks}.} +{Query~1. Do not Bodies act upon Light at a distance, and by their action +bend its Rays, and is not this action (\textit{caeteris paribus}) strongest at the least +distance?}% +\index{Newton!bending of light}% + +\First{We} come now to the experimental test of the influence of +gravitation on light discussed theoretically in the last chapter. +It is not the general purpose of this book to enter into details +of experiments; and if we followed this plan consistently, we +should, as hitherto, summarise the results of the observations +in a few lines. But it is this particular test which has turned +public attention towards the relativity theory, and there appears +to be widespread desire for information. We shall therefore tell +the story of the eclipse expeditions in some detail. It will make +a break in the long theoretical arguments, and will illustrate +the important applications of this theory to practical observations. + +It must be understood that there were two questions to +answer: firstly, whether light has weight (as suggested by +Newton), or is indifferent to gravitation; secondly, if it has +weight, is the amount of the deflection in accordance with +Einstein's or Newton's laws? + +It was already known that light possesses mass or inertia like +other forms of electromagnetic energy. +\index{Inertia!of light}% +\index{Light!mass of}% +\index{Mass of light}% +This is manifested in +the phenomena of radiation-pressure. +\index{Radiation-pressure}% +Some force is required to +stop a beam of light by holding an obstacle in its path; a searchlight % [** PP: Hyphenated across a line in original] +experiences a minute force of recoil just as if it were a +machine-gun firing material projectiles. The force, which is +predicted by orthodox electromagnetic theory, is exceedingly +minute; but delicate experiments have detected it. Probably +this inertia of radiation is of great cosmical importance, playing +a great part in the equilibrium of the more diffuse stars. Indeed +it is probably the agent which has carved the material of the +universe into stars of roughly uniform mass. Possibly the tails +of comets are a witness to the power of the momentum of sunlight, +%% -----File: 121.png---Folio 111------- +which drives outwards the smaller or the more absorptive +particles.% +\index{Comets!radiation-pressure in}% + +It is legitimate to speak of a pound of light as we speak of +a pound of any other substance. The mass of ordinary quantities +of light is however extremely small, and I have calculated that +at the low charge of~3\textit{d}.\ a unit, an Electric Light Company +would have to sell light at the rate of £140,000,000 a pound. +All the sunlight falling on the earth amounts to $160$~tons daily. + +It is perhaps not easy to realise how a wave-motion can have +inertia, and it is still more difficult to understand what is meant +by its having weight. Perhaps this will be better understood if +we put the problem in a concrete form. Imagine a hollow body, +with radiant heat or light-waves traversing the hollow; the +mass of the body will be the sum of the masses of the material +and of the radiant energy in the hollow; a greater force will be +required to shift it because of the light-waves contained in it. +Now let us weigh it with scales or a spring-balance. Will it also +weigh heavier on account of the radiation contained, or will the +weight be that of the solid material alone? If the former, then +clearly from this aspect light has weight; and it is not difficult +to deduce the effect of this weight on a freely moving light-beam +not enclosed within a hollow.% +\index{Light!weight of}% +\index{Momentum!of light}% +\index{Weight!of light}% +\index{Weight!of radio-active energy}% + +The effect of weight is that the radiation in the hollow body +acquires each second a downward momentum proportional to +its mass. This in the long run is transmitted to the material +enclosing it. For a free light-wave in space, the added momentum +combines with the original momentum, and the total +momentum determines the direction of the ray, which is +accordingly bent. Newton's theory suggests no means for +bringing about the bending, but contents itself with predicting +it on general principles. Einstein's theory provides a means, +viz.\ the variation of velocity of the waves. + +Hitherto mass and weight have always been found associated +in strict proportionality. One very important test had already +shown that this proportionality is not confined to material +energy. The substance uranium contains a great deal of radio-active +energy, presumably of an electromagnetic nature, which +it slowly liberates. The mass of this energy must be an appreciable +fraction of the whole mass of the substance. But it was shown +%% -----File: 122.png---Folio 112------- +by experiments with the Eötvös torsion-balance that the ratio +of weight to mass for uranium is the same as for all other +substances; so the energy of radio-activity has weight. +\index{Energy!weight of radio-active}% +\index{Eotvos@Eötvös torsion-balance}% +Still +even this experiment deals only with bound electromagnetic +energy, and we are not justified in deducing the properties of +the free energy of light. + +It is easy to see that a terrestrial experiment has at present +no chance of success. If the mass and weight of light are in the +same proportion as for matter, the ray of light will be bent +just like the trajectory of a material particle. On the earth a +rifle bullet, like everything else, drops $16$~feet in the first second, +$64$~feet in two seconds, and so on, below its original line of flight; +the rifle must thus be aimed above the target. Light would also +drop $16$~feet in the first second\footnotemark; + \footnotetext{Or $32$~feet according to Einstein's law. The fall increases with the speed of + the motion.}% +but, since it has travelled $186,000$ +miles along its course in that time, the bend is inappreciable. +% [Illustration: Fig. 16.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[16]{4.5in}{122a} +\end{center} +\end{figure*}% +In fact any terrestrial course is described so quickly that +gravitation has scarcely had time to accomplish anything. + +The experiment is therefore transferred to the neighbourhood +of the sun. There we get a pull of gravitation $27$~times more +intense than on the earth; and---what is more important---the +greater size of the sun permits a much longer trajectory throughout +which the gravitation is reasonably powerful. The deflection +in this case may amount to something of the order of a second +of arc, which for the astronomer is a fairly large quantity.% +\index{Deflection of light!effect on star's position}% +\index{Gravitation, Newton's law of!deflection of light}% [** PP: Index refers to p. 111] + +In \Figref{16} the line $EFQP$ shows the track of a ray of light +from a distant star~$P$ which reaches the earth~$E$. The main +part of the bending of the ray occurs as it passes the sun~$S$; +\index{Bending of light!effect on star's position}% +\index{Light, bending of}% +and the initial course~$PQ$ and the final course~$FE$ are practically +straight. Since the light rays enter the observer's eye or telescope +in the direction~$FE$, this will be the direction in which the star +appears. But its true direction from the earth is~$QP$, the initial +%% -----File: 123.png---Folio 113------- +course. So the star appears displaced outwards from its true +position by an angle equal to the total deflection of the light.% +\index{Displacement of star-images}% + +\Pagelabel{113}% +It must be noticed that this is only true because a star is so +remote that its true direction with respect to the earth~$E$ is +indistinguishable from its direction with respect to the point~$Q$. +For a source of light within the solar system, the apparent +displacement of the source is by no means equal to the deflection +of the light-ray. It is perhaps curious that the attraction of +light by the sun should produce an apparent displacement of +the star away from the sun; but the necessity for this is +clear. + +The bending affects stars seen near the sun, and accordingly +the only chance of making the observation is during a total +eclipse when the moon cuts off the dazzling light. +\index{Eclipse, observations during}% +Even then +there is a great deal of light from the sun's corona which stretches +far above the disc. It is thus necessary to have rather bright +stars near the sun, which will not be lost in the glare of the +corona. Further the displacements of these stars can only be +measured relatively to other stars, preferably more distant from +the sun and less displaced; we need therefore a reasonable +number of outer bright stars to serve as reference points. + +In a superstitious age a natural philosopher wishing to perform +an important experiment would consult an astrologer to ascertain +an auspicious moment for the trial. With better reason, an +astronomer to-day consulting the stars would announce that the +most favourable day of the year for weighing light is May 29. +The reason is that the sun in its annual journey round the +ecliptic goes through fields of stars of varying richness, but on +May~29 it is in the midst of a quite exceptional patch of bright +stars---part of the Hyades---by far the best star-field encountered. +Now if this problem had been put forward at some other period +of history, it might have been necessary to wait some thousands +of years for a total eclipse of the sun to happen on the lucky +date. But by strange good fortune an eclipse did happen on +May~29, 1919. Owing to the curious sequence of eclipses a +similar opportunity will recur in~1938; we are in the midst of +the most favourable cycle. It is not suggested that it is impossible +to make the test at other eclipses; but the work will +necessarily be more difficult. + +%% -----File: 124.png---Folio 114------- +Attention was called to this remarkable opportunity by the +Astron\-o\-mer Royal in March, 1917; and preparations were begun +by a Committee of the Royal Society and Royal Astronomical +Society for making the observations. Two expeditions were sent +to different places on the line of totality to minimise the risk +of failure by bad weather. Dr A.~C.~D. Crommelin +\index{Crommelin}% +and Mr~C. +Davidson went to Sobral in North Brazil; Mr E.~T. Cottingham +\index{Cottingham}% +\index{Davidson}% +and the writer went to the Isle of Principe in the Gulf of Guinea, +West Africa. +\index{Principe, eclipse expedition to}% +The instrumental equipment for both expeditions +was prepared at Greenwich Observatory under the care of the +Astronomer Royal; and here Mr Davidson made the arrangements +which were the main factor in the success of both +parties.% +\index{Greenwich, Royal Observatory}% + +The circumstances of the two expeditions were somewhat +different and it is scarcely possible to treat them together. We +shall at first follow the fortunes of the Principe observers. They +had a telescope of focal length $11$~feet $4$~inches. On their +photographs $1$~second of arc (which was about the largest displacement +to be measured) corresponds to about $\frac{1}{1500}$ inch---by +no means an inappreciable quantity. The aperture of the +object-glass was $13$~inches, but as used it was stopped down to +$8$~inches to give sharper images. It is necessary, even when the +exposure is only a few seconds, to allow for the diurnal motion +of the stars across the sky, making the telescope move so as to +follow them. But since it is difficult to mount a long and heavy +telescope in the necessary manner in a temporary installation +in a remote part of the globe, the usual practice at eclipses is +to keep the telescope rigid and reflect the stars into it by a +coelostat---a plane mirror kept revolving at the right rate by +clock-work. This arrangement was adopted by both expeditions. + +The observers had rather more than a month on the island +to make their preparations. On the day of the eclipse the +weather was unfavourable. When totality began the dark disc +of the moon surrounded by the corona was visible through cloud, +much as the moon often appears through cloud on a night when +no stars can be seen. There was nothing for it but to carry out +the arranged programme and hope for the best. One observer +was kept occupied changing the plates in rapid succession, whilst +the other gave the exposures of the required length with a screen +%% -----File: 125.png---Folio 115------- +held in front of the object-glass to avoid shaking the telescope in +any way. +\begin{verse} +For in and out, above, about, below \\ +'Tis nothing but a Magic \textit{Shadow}-show \\ +Played in a Box whose candle is the Sun \\ +Round which we Phantom Figures come and go. +\end{verse} +Our shadow-box takes up all our attention. There is a marvellous +spectacle above, and, as the photographs afterwards revealed, +a wonderful prominence-flame is poised a hundred thousand +miles above the surface of the sun. We have no time to snatch +a glance at it. We are conscious only of the weird half-light of +the landscape and the hush of nature, broken by the calls of the +observers, and beat of the metronome ticking out the $302$~seconds +of totality. + +Sixteen photographs were obtained, with exposures ranging +from $2$ to $20$~seconds. The earlier photographs showed no stars, +though they portrayed the remarkable prominence; but apparently +the cloud lightened somewhat towards the end of totality, +and a few images appeared on the later plates. In many cases +one or other of the most essential stars was missing through +cloud, and no use could be made of them; but one plate was +found showing fairly good images of five stars, which were +suitable for a determination. This was measured on the spot +a few days after the eclipse in a micrometric measuring-machine. +The problem was to determine how the apparent positions of +the stars, affected by the sun's gravitational field, compared +with the normal positions on a photograph taken when the sun +was out of the way. Normal photographs for comparison had +been taken with the same telescope in England in January. +The eclipse photograph and a comparison photograph were +placed film to film in the measuring-machine so that corresponding +images fell close together\footnotemark, + \footnotetext{This was possible because at Principe the field of stars was reflected in + the coelostat mirror, whereas in England it was photographed direct.}% +and the small distances +were measured in two rectangular directions. From these the +relative displacements of the stars could be ascertained. +\index{Displacement of star-images}% +In +comparing two plates, various allowances have to be made for +refraction, aberration, plate-orientation, etc.; but since these +occur equally in determinations of stellar parallax, for which +%% -----File: 126.png---Folio 116------- +much greater accuracy is required, the necessary procedure is +well-known to astronomers. + +The results from this plate gave a definite displacement, in +good accordance with Einstein's theory and disagreeing with +the Newtonian prediction. Although the material was very +meagre compared with what had been hoped for, the writer +(who it must be admitted was not altogether unbiassed) believed +it convincing. + +It was not until after the return to England that any further +confirmation was forthcoming. Four plates were brought home +undeveloped, as they were of a brand which would not stand +development in the hot climate. One of these was found to +show sufficient stars; and on measurement it also showed the +deflection predicted by Einstein, confirming the other plate. + +The bugbear of possible systematic error affects all investigations +of this kind. How do you know that there is not something +in your apparatus responsible for this apparent deflection? +Your object-glass has been shaken up by travelling; you have +introduced a mirror into your optical system; perhaps the $50°$ +rise of temperature between the climate at the equator and +England in winter has done some kind of mischief. To meet +this criticism, a different field of stars was photographed at +night in Principe and also in England at the same altitude as +the eclipse field. If the deflection were really instrumental, stars +on these plates should show relative displacements of a similar +kind to those on the eclipse plates. But on measuring these +check-plates no appreciable displacements were found. That +seems to be satisfactory evidence that the displacement observed +during the eclipse is really due to the sun being in the region, +and is not due to differences in instrumental conditions between +England and Principe. Indeed the only possible loophole is a +difference between the night conditions at Principe when the +check-plates were taken, and the day, or rather eclipse, conditions +when the eclipse photographs were taken. That seems +impossible since the temperature at Principe did not vary more +than $1°$ between day and night. + +The problem appeared to be settled almost beyond doubt; +and it was with some confidence that we awaited the return of +the other expedition from Brazil. +\index{Brazil, eclipse expedition to}% +The Brazil party had had +%% -----File: 127.png---Folio 117------- +fine weather and had gained far more extensive material on +their plates. They had remained two months after the eclipse +to photograph the same region before dawn, when clear of the +sun, in order that they might have comparison photographs +taken under exactly the same circumstances. One set of +photographs was secured with a telescope similar to that used +at Principe. In addition they used a longer telescope of 4 inches +aperture and $19$~feet focal length\footnotemark. + \footnotetext{See \hyperref[frontispiece]{Frontispiece}. The two telescopes are shown and the backs of the two + coelostat-mirrors which reflect the sky into them. The clock driving the larger + mirror is seen on the pedestal on the left.}% +The photographs obtained +with the former were disappointing. Although the full number +of stars expected (about~12) were shown, and numerous plates +had been obtained, the definition of the images had been spoiled +by some cause, probably distortion of the coelostat-mirror by +the heat of the sunshine falling on it. The observers were +pessimistic as to the value of these photographs; but they were +the first to be measured on return to England, and the results +came as a great surprise after the indications of the Principe +plates. The measures pointed with all too good agreement to +the ``half-deflection,'' that is to say, the Newtonian value which +is one-half the amount required by Einstein's theory. It seemed +difficult to pit the meagre material of Principe against the wealth +of data secured from the clear sky of Sobral. +\index{Sobral, eclipse expedition to}% +\Pagelabel{117}% +It is true the +Sobral images were condemned, but whether so far as to +invalidate their testimony on this point was not at first clear; +besides the Principe images were not particularly well-defined, +and were much enfeebled by cloud. Certain compensating +advantages of the latter were better appreciated later. Their +strong point was the satisfactory check against systematic error +afforded by the photographs of the check-field; there were +no check-plates taken at Sobral, and, since it was obvious +that the discordance of the two results depended on systematic +error and not on the wealth of material, this distinctly +favoured the Principe results. Further, at Principe there could +be no evil effects from the sun's rays on the mirror, for the +sun had withdrawn all too shyly behind the veil of cloud. +A further advantage was provided by the check-plates at +Principe, which gave an independent determination of the +%% -----File: 128.png---Folio 118------- +difference of scale of the telescope as used in England and at +the eclipse; for the Sobral plates this scale-difference was +eliminated by the method of reduction, with the consequence +that the results depended on the measurement of a much smaller +relative displacement. + +There remained a set of seven plates taken at Sobral with the +$4$-inch lens; +\index{Bending of light!observational results}% +\index{Light, bending of}% +their measurement had been delayed by the necessity +of modifying a micrometer to hold them, since they were of +unusual size. From the first no one entertained any doubt that +the final decision must rest with them, since the images were +almost ideal, and they were on a larger scale than the other +photographs. The use of this instrument must have presented +considerable difficulties---the unwieldy length of the telescope, +the slower speed of the lens necessitating longer exposures and +more accurate driving of the clock-work, the larger scale rendering +the focus more sensitive to disturbances---but the observers +achieved success, and the perfection of the negatives surpassed +anything that could have been hoped for. + +These plates were now measured and they gave a final verdict +definitely confirming Einstein's value of the deflection, in agreement +with the results obtained at Principe. + +It will be remembered that Einstein's theory predicts a +deflection of $1''.74$ at the edge of the sun\footnotemark, + \footnotetext{The predicted deflection of light from infinity to infinity is just over~$1''.745$, + from infinity to the earth it is just under.}% +the amount falling +off inversely as the distance from the sun's centre. The simple +Newtonian deflection is half this,~$0''.87$. The final results +(reduced to the edge of the sun) obtained at Sobral and Principe +with their ``probable accidental errors'' were +\index{Deflection of light!observational results}% +\index{Gravitational field of Sun!deflection of light}% +\begin{center} +\begin{tabular}{lc} +Sobral & $1''.98 ± 0''.12$, \\ +Principe & $1''.61 ± 0''.30$. +\end{tabular} +\end{center} +It is usual to allow a margin of safety of about twice the probable +error on either side of the mean. The evidence of the Principe +plates is thus just about sufficient to rule out the possibility of +the ``half-deflection,'' and the Sobral plates exclude it with +practical certainty. The value of the material found at Principe +cannot be put higher than about one-sixth of that at Sobral; +but it certainly makes it less easy to bring criticism against this +confirmation of Einstein's theory seeing that it was obtained +%% -----File: 129.png---Folio 119------- +independently with two different instruments at different places +and with different kinds of checks. + +The best check on the results obtained with the $4$-inch lens +at Sobral is the striking internal accordance of the measures for +different stars. The theoretical deflection should vary inversely +as the distance from the sun's centre; hence, if we plot the mean +radial displacement found for each star separately against the +inverse distance, the points should lie on a straight line. This +%[Illustration: \textsc{Fig}. 17.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[17]{4.5in}{129a} +\end{center} +\end{figure*}% +is shown in \Figref{17} where the broken line shows the theoretical +prediction of Einstein, the deviations being within the accidental +errors of the determinations. A line of half the slope representing +the half-deflection would clearly be inadmissible. + +Moreover, values of the deflection were deduced from the +measures in right ascension and declination independently. +These were in close agreement. + +%% -----File: 130.png---Folio 120------- + +A diagram showing the relative positions of the stars is given +in \Figref{18}. + +The square shows the limits of the plates used at Principe, +and the oblique rectangle the limits with the 4-inch lens at +Sobral. The centre of the sun moved from $S$ to~$P$ in the $2\frac{1}{4}$~hours +%[Illustration: \textsc{Fig}.~18.] +\begin{figure*}[hbt] +\begin{center} +\Graphic[18]{5in}{130a} +\end{center} +\end{figure*}% +interval between totality at the two stations; the sun is +here represented for a time about midway between. The stars +measured on the Principe plates were Nos.~3, 4, 5, 6, 10, 11; those +at Sobral were 11, 10, 6, 5, 4, 2, 3 (in the order of the dots +from left to right in \Figref{17}). None of these were fainter than +$6\Magnitude.0$, the brightest $\kappa^1$~Tauri (No.~4) being $4\Magnitude.5$. + +It has been objected that although the observations establish +%% -----File: 131.png---Folio 121------- +a deflection of light in passing the sun equal to that predicted +by Einstein, it is not immediately obvious that this deflection +must necessarily be attributed to the sun's gravitational field. +It is suggested that it may not be an essential effect of the sun +as a massive body, but an accidental effect owing to the circumstance +that the sun is surrounded by a corona which acts as +a refracting atmosphere. +\index{Corona, refraction by}% +\index{Refraction of light in corona}% +It would be a strange coincidence if +this atmosphere imitated the theoretical law in the exact +quantitative way shown in \Figref{17}; and the suggestion appears +to us far-fetched. However the objection can be met in a more +direct way. We have already shown that the gravitational +effect on light is equivalent to that produced by a refracting +medium round the sun and have calculated the necessary +refractive index. At a height of $400,000$ miles above the surface +the refractive index required is~$1.0000021$. This corresponds to +air at $\frac{1}{140}$ atmosphere, hydrogen at $\frac{1}{70}$ atmosphere, helium at +$\frac{1}{20}$ atmospheric pressure. It seems obvious that there can be no +material of this order of density at such a distance from the sun. +The pressure on the sun's surface of the columns of material +involved would be of the order $10,000$ atmospheres; and we +know from spectroscopic evidence that there is no pressure of +this order. If it is urged that the mass could perhaps be supported +by electrical forces, the argument from absorption is +even more cogent. The light from the stars photographed during +the eclipse has passed through a depth of at least a million miles +of material of this order of density---or say the equivalent of +$10,000$ miles of air at atmospheric density. We know to our +cost what absorption the earth's $5$~miles of homogeneous +atmosphere can effect. And yet at the eclipse the stars appeared +on the photographs with their normal brightness. If the irrepressible +critic insists that the material round the sun may be +composed of some new element with properties unlike any +material known to us, we may reply that the mechanism of +refraction and of absorption is the same, and there is a limit to +the possibility of refraction without appreciable absorption. +Finally it would be necessary to arrange that the density of the +material falls off inversely as the distance from the sun's centre +in order to give the required variation of refractive index. + +Several comets have been known to approach the sun within +%% -----File: 132.png---Folio 122------- +the limits of distance here considered. If they had to pass +through an atmosphere of the density required to account for +the displacement, they would have suffered enormous resistance. +Dr~Crommelin has shown that a study of these comets sets an +upper limit to the density of the corona, which makes the +refractive effect quite negligible.% +\index{Comets!motion through coronal medium}% +\index{Crommelin}% + +Those who regard Einstein's law of gravitation as a natural +deduction from a theory based on the minimum of hypotheses +will be satisfied to find that his remarkable prediction is quantitatively +confirmed by observation, and that no unforeseen cause +has appeared to invalidate the test. +%% -----File: 133.png---Folio 123------- + + +\Chapter{VIII}{Other Tests of the Theory} + +\Quote[break]{\textit{Love's Labour's Lost.}} +{The words of Mercury are harsh after the songs of Apollo.} + +\First{We} have seen that the swift-moving light-waves possess great +advantages as a means of exploring the non-Euclidean property +of space. But there is an old fable about the hare and the +tortoise. The slow-moving planets have qualities which must +not be overlooked. The light-wave traverses the region in a few +minutes and makes its report; the planet plods on and on for +centuries going over the same ground again and again. Each +time it goes round it reveals a little about the space, and the +knowledge slowly accumulates. + +According to Newton's law a planet moves round the sun in +an ellipse, and if there are no other planets disturbing it, the +ellipse remains the same for ever. According to Einstein's law +the path is very nearly an ellipse, but it does not quite close up; +and in the next revolution the path has advanced slightly in the +same direction as that in which the planet was moving. The +orbit is thus an ellipse which very slowly revolves\footnote% +{Appendix, \Noteref{9}.}.% +\Pagelabel{note9}% +\index{Mercury, perihelion of}% +\index{Orbits under Einstein's law}% +\index{Perihelia of planets, motions of}% + +The exact prediction of Einstein's law is that in one revolution +of the planet the orbit will advance through a fraction of a +revolution equal to $3v^2/C^2$, where $v$ is the speed of the planet +and $C$ the speed of light. The earth has $1/10,000$ of the speed of +light; thus in one revolution (one year) the point where the +earth is at greatest distance from the sun will move on +$3/100,000,000$ of a revolution, or $0''.038$. We could not detect +this difference in a year, but we can let it add up for a century +at least. It would then be observable but for one thing---the +earth's orbit is very blunt, very nearly circular, and so we +cannot tell accurately enough which way it is pointing and how +its sharpest apses move. We can choose a planet with higher +speed so that the effect is increased, not only because $v^2$ is +increased, but because the revolutions take less time; but, what +%% -----File: 134.png---Folio 124------- +is perhaps more important, we need a planet with a sharp +elliptical orbit, so that it is easy to observe how its apses move +round. Both these conditions are fulfilled in the case of Mercury. +It is the fastest of the planets, and the predicted advance of the +orbit amounts to $43''$ per century; further the eccentricity of +its orbit is far greater than that of any of the other seven +planets. + +Now an unexplained advance of the orbit of Mercury had +long been known. It had occupied the attention of Le~Verrier, +\index{Le Verrier}% +who, having successfully predicted the planet Neptune from the +disturbances of Uranus, thought that the anomalous motion of +Mercury might be due to an interior planet, which was called +Vulcan in anticipation. But, though thoroughly sought for, +Vulcan has never turned up. Shortly before Einstein arrived +at his law of gravitation, the accepted figures were as follows. +The actual observed advance of the orbit was $574''$ per century; +the calculated perturbations produced by all the known planets +amounted to $532''$ per century. The excess of $42''$ per century +remained to be explained. Although the amount could scarcely +be relied on to a second of arc, it was at least thirty times as +great as the probable accidental error. + +The big discrepancy from the Newtonian gravitational theory +is thus in agreement with Einstein's prediction of an advance +of $43''$ per century. + +The derivation of this prediction from Einstein's law can only +be followed by mathematical analysis; but it may be remarked +that any slight deviation from the inverse square law is likely +to cause an advance or recession of the apse of the orbit. That +a particle, if it does not move in a circle, should oscillate between +two extreme distances is natural enough; it could scarcely do +anything else unless it had sufficient speed to break away +altogether. But the interval between the two extremes will not +in general be half a revolution. It is only under the exact +adjustment of the inverse square law that this happens, so that +the orbit closes up and the next revolution starts at the same +point. I do not think that any ``simple explanation'' of this +property of the inverse-square law has been given; and it seems +fair to remind those, who complain of the difficulty of understanding +Einstein's prediction of the advance of the perihelion, +\index{Gravitational field of Sun!motion of perihelion} % [** PP: Index entry reads p. 122] +%% -----File: 135.png---Folio 125------- +that the real trouble is that they have not yet succeeded in +making clear to the uninitiated this recondite result of the +Newtonian theory. The slight modifications introduced by +Einstein's law of gravitation upset this fine adjustment, so that +the oscillation between the extremes occupies slightly more than +a revolution. A simple example of this effect of a small deviation +from the inverse-square law was actually given by Newton. + +It had already been recognised that the change of mass with +velocity may cause an advance of perihelion; but owing to the +ambiguity of Newton's law of gravitation the discussion was +unsatisfactory. It was, however, clear that the effect was too +small to account for the motion of perihelion of Mercury, the +prediction being $\frac{1}{2} v^2/C^2$, or at most $v^2/C^2$. Einstein's theory is +the only one which gives the full amount $3v^2/C^2$.% +\index{Mercury, perihelion of}% + +It was suggested by Lodge that, % [** PP: Retaining awkward commas] +\index{Lodge}% +this variation of mass with +velocity might account for the whole motion of the orbit of +Mercury, if account were taken of the sun's unknown absolute +motion through the aether, combining sometimes additively and +sometimes negatively with the orbital motion. In a discussion +between him and the writer, it appeared that, if the absolute +motion were sufficient to produce this effect on Mercury, it +must give observable effects for Venus and the Earth; and these +do not exist. Indeed from the close accordance of Venus and +the Earth with observation, it is possible to conclude that, either +the sun's motion through the aether is improbably small, or +gravitation must conform to relativity, in the sense of the +restricted principle (\Pageref{20}), and conceal the effects of the +increase of mass with speed so far as an additive uniform motion +is concerned.% +\index{Gravitation!relativity for uniform motion}% + +Unfortunately it is not possible to obtain any further test of +Einstein's law of gravitation from the remaining planets. We +have to pass over Venus and the Earth, whose orbits are too +nearly circular to show the advance of the apses observationally. +Coming next to Mars with a moderately eccentric orbit, the +speed is very much smaller, and the predicted advance is only +$1''.3$ per century. Now the accepted figures show an observed +advance (additional to that produced by known causes) of $5''$ +per century, so that Einstein's correction improves the accordance +of observation with theory; but, since the result for Mars +%% -----File: 136.png---Folio 126------- +is in any case scarcely trustworthy to $5''$ owing to the inevitable +errors of observation, the improvement is not very important. +The main conclusion is that Einstein's theory brings Mercury +into line, without upsetting the existing good accordance of all +the other planets. + +We have tested Einstein's law of gravitation for fast movement +(light) and for moderately slow movement (Mercury). +For very slow movement it agrees with Newton's law, and the +general accordance of the latter with observation can be transferred +to Einstein's law. These tests appear to be sufficient to +establish the law firmly. We can express it in this way. + +Every particle or light-pulse moves so that the quantity $s$ +measured along its track between two points has the maximum +possible value, where +\[ +ds^2 = - (1-2m/r)^{-1}\, dr^2 -r^2\, d\theta^2 + (1-2m/r)\, dt^2. +\] +And the accuracy of the experimental test is sufficient to verify +the coefficients as far as terms of order $m/r$ in the coefficient of~$dr^2$, +and as far as terms of order $m^2/r^2$ in the coefficient of~$dt^2$\footnote{Appendix, \Noteref{10}.}. +\Pagelabel{note10} + +In this form the law appears to be firmly based on experiment, +and the revision or even the complete abandonment of the +general ideas of Einstein's theory would scarcely affect it. + +These experimental proofs, that space in the gravitational +field of the sun is non-Euclidean or curved, have appeared +puzzling to those unfamiliar with the theory. It is pointed out +that the experiments show that physical objects or loci are +``warped'' in the sun's field; but it is suggested that there is +nothing to show that the space in which they exist is warped. +\index{Gravitational field of Sun!result of observational verification}% +The answer is that it does not seem possible to draw any distinction +between the warping of physical space and the warping +of physical objects which define space. +\index{Warping of space}% +If our purpose were +merely to call attention to these phenomena of the gravitational +field as curiosities, it would, no doubt, be preferable to avoid +using words which are liable to be misconstrued. But if we wish +to arrive at an understanding of the conditions of the gravitational +field, we cannot throw over the vocabulary appropriate +for that purpose, merely because there may be some who insist +on investing the words with a metaphysical meaning which is +clearly inappropriate to the discussion. + +%% -----File: 137.png---Folio 127------- + +We come now to another kind of test. In the statement of +the law of gravitation just given, a quantity $s$ is mentioned; +and, so far as that statement goes, $s$ is merely an intermediary +quantity defined mathematically. But in our theory we have +been identifying $s$ with interval-length, measured with an +apparatus of scales and clocks; and it is very desirable to test +whether this identification can be confirmed---whether the +geometry of scales and clocks is the same as the geometry of +moving particles and light-pulses. + +The question has been mooted whether we may not divide +the present theory into two parts. Can we not accept the law +of gravitation in the form suggested above as a self-contained +result proved by observation, leaving the further possibility +that $s$ is to be identified with interval-length open to debate? +The motive is partly a desire to consolidate our gains, freeing +them from the least taint of speculation; but perhaps also it is +inspired by the wish to leave an opening by which clock-scale +geometry, i.e.\ the space and time of ordinary perception, may +remain Euclidean. Disregarding the connection of $s$ with +interval-length, there is no object in attributing any significance +of length to it; +\index{Interval-length!geometrical significance essential}% +it can be regarded as a dynamical quantity like +Action, and the new law of gravitation can be expressed after +the traditional manner without dragging in strange theories of +space and time. Thus interpreted, the law perhaps loses its +theoretical inevitability; but it remains strongly grounded on +observation. Unfortunately for this proposal, it is impossible +to make a clean division of the theory at the point suggested. +Without some geometrical interpretation of $s$ our conclusions as +to the courses of planets and light-waves cannot be connected +with the astronomical measurements which verify them. The +track of a light-wave in terms of the coordinates $r$, $\theta$, $t$ cannot +be tested directly; the coordinates afford only a temporary +resting-place; and the measurement of the displacement of the +star-image on the photographic plate involves a reconversion +from the coordinates to~$s$, which here appears in its significance +as the interval in clock-scale geometry. + +Thus even from the experimental standpoint, a rough correspondence +of the quantity $s$ occurring in the law of gravitation +with the clock-scale interval is an essential feature. We have +%% -----File: 138.png---Folio 128------- +now to examine whether experimental evidence can be found +as to the exactness of this correspondence. + +It seems reasonable to suppose that a vibrating atom is an +ideal type of clock. The beginning and end of a single vibration +constitute two events, and the interval $ds$ between two events +is an absolute quantity independent of any mesh-system. This +interval must be determined by the nature of the atom; and +hence atoms which are absolutely similar will measure by their +vibrations equal values of the absolute interval $ds$. Let us +adopt the usual mesh-system $(r, \theta, t)$ for the solar system, so +that +\[ +ds^2 = - \gamma^{-1}\, dr^2 -r^2\, d\theta^2 + \gamma\, dt^2. +\] +Consider an atom momentarily at rest at some point in the solar +system; we say \textit{momentarily}, because it must undergo the +acceleration of the gravitational field where it is. If $ds$ corresponds +to one vibration, then, since the atom has not moved, +the corresponding $dr$ and $d\theta$ will be zero, and we have +\[ +ds^2 = \gamma\, dt^2. +\] +The \textit{time} of vibration $dt$ is thus $1/\surd\gamma$ times the \textit{interval} of +vibration~$ds$. + +Accordingly, if we have two similar atoms at rest at different +points in the system, the interval of vibration will be the same +for both; but the time of vibration will be proportional to the +inverse square-root of~$\gamma$, which differs for the two atoms. Since +\begin{align*} + \gamma &= 1 - \frac{2m}{r} \\ +1/\surd \gamma &= 1 + \,\frac{m}{r}, \quad\text{very approximately.} +\end{align*} + +Take an atom on the surface of the sun, and a similar atom +in a terrestrial laboratory. +\index{Atom, vibrating on sun}% +\index{Clock!on sun}% +For the first, $1 + m/r = 1.00000212$, +and for the second $1 + m/r$ is practically~$1$. The time of vibration +of the solar atom is thus longer in the ratio $1.00000212$, and it +might be possible to test this by spectroscopic examination. + +There is one important point to consider. The spectroscopic +examination must take place in the terrestrial laboratory; and +we have to test the period of the solar atom by the period of +the waves emanating from it when they reach the earth. Will +they carry the period to us unchanged? Clearly they must. +%% -----File: 139.png---Folio 129------- +The first and second pulse have to travel the same distance ($r$), +and they travel with the same velocity ($dr/dt$); for the velocity +of light in the mesh-system used is $1- 2m/r$, and though this +velocity depends on~$r$, it does not depend on~$t$. Hence the difference +$dt$ at one end of the waves is the same as that at the other +end. + +Thus in the laboratory the light from a solar source should +be of greater period and greater wave-length (i.e.\ redder) than +that from a corresponding terrestrial source. Taking blue light +of wave-length $4000$~Å, the solar lines should be displaced +$4000 × .00000212$, or $0.008$~Å towards the red end of the +spectrum.% +\index{Displacement of spectral lines}% +\index{Gravitational field of Sun!displacement of spectral lines}% + +The properties of a gravitational field of force are similar to +those of a centrifugal field of force; and it may be of interest +to see how a corresponding shift of the spectral lines occurs for +an atom in a field of centrifugal force. +\index{Retardation of time!in centrifugal field}% +Suppose that, as we rotate +with the earth, we observe a very remote atom momentarily at +rest relative to our rotating axes. The case is just similar to +that of the solar atom; both are at rest relative to the respective +mesh-systems; the solar atom is in a field of gravitational force, +and the other is in a field of centrifugal force. The direction of +the force is in both cases the same---from the earth towards the +atom observed. Hence the atom in the centrifugal field ought +also to vibrate more slowly, and show a displacement to the red +in its spectral lines. +\index{Centrifugal Force!vibrating atom in field of}% +It does, if the theory hitherto given is +right. We can abolish the centrifugal force by choosing non-rotating +axes. But the distant atom was at rest relative to the +rotating axes, that is to say, it was whizzing round with them. +Thus from the ordinary standpoint the atom has a large velocity +relative to the observer, and, in accordance with \Chapref{I}, its +vibrations slow down just as the aviator's watch did. The shift +of spectral lines due to a field of centrifugal force is only another +aspect of a phenomenon already discussed. + +The expected shift of the spectral lines on the sun, compared +with the corresponding terrestrial lines, has been looked for; +but it has not been found. + +In estimating the importance of this observational result in +regard to the relativity theory, we must distinguish between +a failure of the test and a definite conclusion that the lines are +%% -----File: 140.png---Folio 130------- +undisplaced. The chief investigators St~John, Schwarzschild, +Evershed, and Grebe and Bachem, seem to be agreed that the +observed displacement is at any rate less than that predicted +by the theory. The theory can therefore in no case claim support +from the present evidence. But something more must be +established, if the observations are to be regarded as in the +slightest degree adverse to the theory. If for instance the mean +deflection is found to be $.004$ instead of $.008$ Angström units, +the only possible conclusion is that there are certain causes of +displacement of the lines, acting in the solar atmosphere and not +yet identified. No one could be much surprised if this were the +case; and it would, of course, render the test nugatory. The +case is not much altered if the observed displacement is $.002$ +units, provided the latter quantity is above the accidental error +of measurement; if we have to postulate some unexplained disturbance, +it may just as well produce a displacement $-.006$ as +$+.002$. For this reason Evershed's evidence is by no means +adverse to the theory, since he finds unexplained displacements +in any case. +\index{Evershed}% +\index{St John}% +One set of lines measured by St~John gave a mean +displacement of $.0036$~units; and this also shows that the test +has failed. The only evidence \textit{adverse} to the theory, and not +merely neutral, is a series of measures by St~John on $17$ cyanogen +lines, which he regarded as most dependable. These gave a mean +shift of exactly~$.000$. If this stood alone we should certainly be +disposed to infer that the test had gone against Einstein's +theory, and that nothing had intervened to cast doubt on the +validity of the test. The writer is unqualified to criticise these +mutually contradictory spectroscopic conclusions; but he has +formed the impression that the last-mentioned result obtained +by St~John has the greatest weight of any investigations up to +the present\footnotemark. + \footnotetext{A further paper by Grebe and Bachem (\textit{Zeitschrift für Physik}, 1920, p.~51), + \index{Grebe and Bachem}% + received whilst this is passing through the press, makes out a case strongly + favourable for the Einstein displacement, and reconciles the discordant results + found by most of the investigators. But it may still be the best counsel to + ``wait and see,'' and I have made no alteration in the discussion here given.} + +It seems that judgment must be reserved; but it may be well +to examine how the present theory would stand if the verdict +of this third crucial experiment finally went against it. + +It has become apparent that there is something illogical in +%% -----File: 141.png---Folio 131------- +the sequence we have followed in developing the theory, owing +to the necessity of proceeding from the common ideas of space +and time to the more fundamental properties of the absolute +world. +\index{Clock-scale geometry, not fundamental}% +We started with a definition of the interval by measurements +made with clocks and scales, and afterwards connected +it with the tracks of moving particles. Clearly this is an inversion +of the logical order. The simplest kind of clock is an elaborate +mechanism, and a material scale is a very complex piece of +apparatus. The best course then is to discover $ds$ by exploration +of space and time with a moving particle or light-pulse, rather +than by measures with scales and clocks. On this basis by +astronomical observation alone the formula for $ds$ in the gravitational +field of the sun has already been established. To proceed +from this to determine exactly what is measured by a scale and +a clock, it would at first seem necessary to have a detailed theory +of the mechanisms involved in a scale and clock. But there is +a short-cut which seems legitimate. This short-cut is in fact +the Principle of Equivalence. +\index{Equivalence!Principle of}% +\index{Principle of Equivalence}% +Whatever the mechanism of the +clock, whether it is a good clock or a bad clock, the intervals it +is beating must be something absolute; the clock cannot know +what mesh-system the observer is using, and therefore its +absolute rate cannot be altered by position or motion which is +relative merely to a mesh-system. Thus wherever it is placed, +and however it moves, provided it is not constrained by impacts +or electrical forces, it must always beat equal intervals as we +have previously assumed. Thus a clock may fairly be used to +measure intervals, even when the interval is defined in the new +manner; any other result seems to postulate that it pays heed +to some particular mesh-system\footnotemark. + \footnotetext{Of course, there is always the possibility that this might be the case, + though it seems unlikely. The essential point of the relativity theory is that + (contrary to the common opinion) no experiments yet made have revealed any + mesh-system of an absolute character, not that experiments never will reveal + such a system.} + +Three modes of escape from this conclusion seem to be left +open. A clock cannot pay any heed to the mesh-system used; +but it may be affected by the kind of space-time around it\footnote% + {Appendix, \Noteref{11}.}. +\Pagelabel{note11}% +The terrestrial atom is in a field of gravitation so weak that the +space-time may be considered practically flat; but the space-% +%% -----File: 142.png---Folio 132------- +time round the solar atom is not flat. It may happen that the +two atoms actually detect this absolute difference in the world +around them and do not vibrate with the same interval $ds$---contrary +to our assumption above. Then the prediction of the +shift of the lines in the solar spectrum is invalidated. Now it is +very doubtful if an atom can detect the curving of the region it +occupies, because curvature is only apparent when an extended +region is considered; still an atom has some extension, and it is +not impossible that its equations of motion involve the quantities +$B^{\rho}_{\mu\nu\sigma}$ which distinguish gravitational from flat space-time. An +apparently insuperable objection to this explanation is that the +effect of curvature on the period would almost certainly be +represented by terms of the form $m^{2}/r^{2}$, whereas to account for +a negative result for the shift of the spectral lines terms of much +greater order of magnitude $m/r$ are needed. + +The second possibility depends on the question whether it is +possible for an atom at rest on the sun to be precisely similar to +one on the earth. If an atom fell from the earth to the sun it +would acquire a velocity of $610$~km.\ per sec., and could only be +brought to rest by a systematic hammering by other atoms. +May not this have made a permanent alteration in its time-keeping +properties? It is true that every atom is continually +undergoing collisions, but it is just possible that the average +solar atom has a different period from the average terrestrial +atom owing to this systematic difference in its history. + +What are the two events which mark the beginning and end +of an atomic vibration? This question suggests a third possibility. +If they are two absolute events, like the explosions of +two detonators, then the interval between them will be a definite +quantity, and our argument applies. But if, for example, an +atomic vibration is determined by the revolution of an electron +around a nucleus, it is not marked by any definite events. A +revolution means a return to the same position as before; but +we cannot define what is the same position as before without +reference to some mesh-system. Hence it is not clear that there +is any absolute interval corresponding to the vibration of an +atom; an absolute interval only exists between two events +absolutely defined. + +It is unlikely that any of these three possibilities can negative %[** PP: OK] +%% -----File: 143.png---Folio 133------- +the expected shift of the spectral lines. The uncertainties introduced +by them are, so far as we can judge, of a much smaller +order of magnitude. But it will be realised that this third test +of Einstein's theory involves rather more complicated considerations +than the two simple tests with light-waves and the moving +planet. I think that a shift of the Fraunhofer lines is a highly +probable prediction from the theory and I anticipate that +experiment will ultimately confirm the prediction; but it is not +entirely free from guess-work. These theoretical uncertainties +are apart altogether from the great practical difficulties of the +test, including the exact allowance for the unfamiliar circumstances +of an absorbing atom in the sun's atmosphere. + +Outside the three leading tests, there appears to be little +chance of checking the theory unless our present methods of +measurement are greatly improved. It is not practicable to +measure the deflection of light by any body other than the sun. +The apparent displacement of a star just grazing the limb of +Jupiter should be~$0''.017$. +\index{Jupiter, deflection of light by}% +A hundredth of a second of arc is +just about within reach of the most refined measurements with +the largest telescopes. If the observation could be conducted +under the same conditions as the best parallax measurements, +the displacement could be detected but not measured with any +accuracy. The glare from the light of the planet ruins any chance +of success. + +% [** PP: Retaining commas] +Most astronomers, who look into the subject, are entrapped +sooner or later by a fallacy in connection with double stars. +\index{Double stars and Einstein effect}% +It is thought that when one component passes behind the other +it will appear displaced from its true position, like a star passing +behind the sun; if the size of the occulting star is comparable +with that of the sun, the displacement should be of the same +order, $1''.7$. This would cause a very conspicuous irregularity in +the apparent orbit of a double star. But reference to \Pageref{113} +shows that an essential point in the argument was the enormous +ratio of the distance $QP$ of the star from the sun to the distance +$EF$ of the sun from the earth. It is only in these conditions that +the apparent displacement of the object is equal to the deflection +undergone by its light. It is easy to see that where this ratio is +reversed, as in the case of the double star, the apparent displacement +is an extremely small fraction of the deflection of the light. +It would be quite imperceptible to observation. + +%% -----File: 144.png---Folio 134------- + +If two independent stars are seen in the same line of vision +within about~$1''$, one being a great distance behind the other, +the conditions seem at first more favourable. I do not know if +any such pairs exist. It would seem that we ought to see the +more distant star not only by the direct ray, which would be +practically undisturbed, but also by a ray passing round the +other side of the nearer star and bent by it to the necessary +extent. The second image would, of course, be indistinguishable +from that of the nearer star; but it would give it additional +brightness, which would disappear in time when the two stars +receded. But consider a pencil of light coming past the nearer +star; the inner edge will be bent more than the outer edge, so +that the divergence is increased. The increase is very small; +but then the whole divergence of a pencil from a source some +hundred billion miles away is very minute. It is easily calculated +that the increased divergence would so weaken the light as to +make it impossible to detect it when it reached us\footnote% + {Appendix, \Noteref{12}.}. +\Pagelabel{note12} + +If two unconnected stars approached the line of sight still +more closely, so that one almost occulted the other, observable +effects might be perceived. When the proximity was such that +the direct ray from the more distant star passed within about +$100$~million kilometres of the nearer star, it would begin to fade +appreciably. The course of the ray would not yet be appreciably +deflected, but the divergence of the pencil would be rapidly +increased, and less light from the star would enter our telescopes. +The test is scarcely likely to be an important one, since a +sufficiently close approach is not likely to occur; and in any +case it would be difficult to feel confident that the fading was +not due to a nebulous atmosphere around the nearer star. + +The theory gives small corrections to the motion of the moon +which have been investigated by de~Sitter. +\index{Moon, motion of}% +\index{de Sitter}% +Both the axis of +the orbit and its line of intersection with the ecliptic should +advance about $2''$ per century more than the Newtonian theory +indicates. Neither observation nor Newtonian theory are as yet +pushed to sufficient accuracy to test this; but a comparatively +small increase in accuracy would make a comparison possible. + +Since certain stars are perhaps ten times more massive than +the sun, without the radius being unduly increased, they should +show a greater shift of the spectral lines and might be more +%% -----File: 145.png---Folio 135------- +favourable for the third crucial test. Unfortunately the predicted +shift is indistinguishable from that caused by a velocity +of the star in the line-of-sight on Doppler's principle. Thus the +expected shift on the sun is equivalent to that caused by a receding +velocity of $0.634$ kilometres per second. In the case of the +sun we know by other evidence exactly what the line-of-sight +velocity should be; but we have not this knowledge for other +stars. The only indication that could be obtained would be the +detection of an \textit{average} motion of recession of the more massive +stars. It seems rather unlikely that there should be a real +preponderance of receding motions among stars taken indiscriminately +from all parts of the sky; and the apparent effect +might then be attributed to the Einstein shift. Actually the +most massive stars (those of spectral type~$B$) have been found +to show an average velocity of recession of about $4.5$~km.\ per +sec., which would be explained if the values of $m/r$ for these +stars are about seven times greater than the value for the sun---a +quite reasonable hypothesis. +\index{Displacement of spectral lines!in stars}% +\index{Receding velocities of B@Receding velocities!of $B$-type stars}% +This phenomenon was well-known +to astrophysicists some years before Einstein's theory +was published. But there are so many possible interpretations +that no stress should be placed on this evidence. Moreover the +very diffuse ``giant'' stars of type~$M$ have also a considerable +systematic velocity of recession, and for these $m/r$ must be much +less than for the sun. +%% -----File: 146.png---Folio 136------- + + +\Chapter{IX}{Momentum and Energy} + +\null\hspace{0.75in} +\begin{minipage}{4.25in} % [** PP: Hard-coded width; ~2.6pt overfull] +\Quote{Newman, \textit{Dream of Gerontius}.} +{\hspace*{-\QIndent}For spirits and men by different standards mete \\ +The less and greater in the flow of time. \\ +By sun and moon, primeval ordinances--- \\ +By stars which rise and set harmoniously--- \\ +By the recurring seasons, and the swing \\ +This way and that of the suspended rod \\ +Precise and punctual, men divide the hours, \\ +Equal, continuous, for their common use. \\ +Not so with us in the immaterial world; \\ +But intervals in their succession \\ +Are measured by the living thought alone \\ +And grow or wane with its intensity. \\ +And time is not a common property; \\ +But what is long is short, and swift is slow \\ +And near is distant, as received and grasped \\ +By this mind and by that.} +\end{minipage}% +\index{Motion!Newton's first law}% +\index{Newton!law of motion}% + +\First{One} of the most important consequences of the relativity theory +is the unification of inertia and gravitation. + +The beginner in mechanics does not accept Newton's first law +of motion without a feeling of hesitation. He readily agrees that +a body at rest will remain at rest unless something causes it to +move; but he is not satisfied that a body in motion will remain +in uniform motion so long as it is not interfered with. It is +quite natural to think that motion is an impulse which will +exhaust itself, and that the body will finally come to a stop. +The teacher easily disposes of the arguments urged in support +of this view, pointing out the friction which has to be overcome +when a train or a bicycle is kept moving uniformly. He shows +that if the friction is diminished, as when a stone is projected +across ice, the motion lasts for a longer time, so that if all interference +by friction were removed uniform motion might continue +indefinitely. But he glosses over the point that if there +were no interference with the motion---if the ice were abolished +altogether---the motion would be by no means uniform, but like +that of a falling body. The teacher probably insists that the +continuance of uniform motion does not require anything that +%% -----File: 147.png---Folio 137------- +can properly be called a \textit{cause}. The property is given a name +\textit{inertia}; but it is thought of as an innate tendency in contrast +to \textit{force} which is an active cause. +\index{Force!compared with inertia}% +\index{Inertia!compared with force}% +So long as forces are confined +to the thrusts and tensions of elementary mechanics, where there +is supposed to be direct contact of material, there is good ground +for this distinction; we can visualise the active hammering of +the molecules on the body, causing it to change its motion. But +when force is extended to include the gravitational field the +distinction is not so clear. + +For our part we deny the distinction in this last case. Gravitational +force is not an active agent working against the passive +tendency of inertia. Gravitation and inertia are one. The +uniform straight track is only relative to some mesh-system, +which is assigned by arbitrary convention. We cannot imagine +that a body looks round to see who is observing it and then feels +an innate tendency to move in that observer's straight line---probably +at the same time feeling an active force compelling +it to move some other way. If there is anything that can be +called an innate tendency it is the tendency to follow what we +have called the natural track---the longest track between two +points. We might restate the first law of motion in the form +``Every body tends to move in the track in which it actually +does move, except in so far as it is compelled by material impacts +to follow some other track than that in which it would otherwise +move.'' Probably no one will dispute this profound statement! + +Whether the natural track is straight or curved, whether the +motion is uniform or changing, a cause is in any case required. +This cause is in all cases the combined inertia-gravitation. +\index{Inertia-gravitation}% +To +have given it a name does not excuse us from attempting an +explanation of it in due time. Meanwhile this identification of +inertia and gravitation as arbitrary components of one property +explains why weight is always proportional to inertia. +\index{Weight!proportional to inertia}% +This +experimental fact verified to a very high degree of accuracy +would otherwise have to be regarded as a remarkable law of +nature. + +We have learnt that the natural track is the longest track +between two points; and since this is the only definable track +having an absolute significance in nature, we seem to have a +sufficient explanation of why an undisturbed particle must +%% -----File: 148.png---Folio 138------- +follow it. That is satisfactory, so far as it goes, but still we should +naturally wish for a clearer picture of the cause---inertia-gravitation---which +propels it in this track. + +It has been seen that the gravitational field round a body +involves a kind of curvature of space-time, and accordingly +round each particle there is a minute pucker. Now at each +successive instant a particle is displaced continuously in time if +not in space; and so in our four-dimensional representation +which gives a bird's-eye-view of all time, the pucker has the +form of a long groove along the track of the particle. Now such +a groove or pleat in a continuum cannot take an arbitrary +course---as every dress-maker knows. Einstein's law of gravitation +gives the rule according to which the curvatures at any +point of space-time link on to those at surrounding points; so +that when a groove is started in any direction the rest of its +course can be forecasted. We have hitherto thought of the law +of gravitation as showing how the pucker spreads out in space, +cf.\ Newton's statement that the corresponding force weakens as +the inverse square of the distance. But the law of Einstein +equally shows how the gravitational field spreads out in time, +since there is no absolute distinction of time and space. It can +be deduced mathematically from Einstein's law that a pucker +of the form corresponding to a particle necessarily runs along +the track of greatest interval-length between two points. + +The track of a particle of matter is thus determined by the +interaction of the minute gravitational field, which surrounds +and, so far as we know, constitutes it, with the general space-time +of the region. The various forms which it can take, find +their explanation in the new law of gravitation. The straight +tracks of the stars and the curved tracks of the planets are +placed on the same level, and receive the same kind of explanation. +The one universal law, that the space-time continuum +can be curved only in the first degree, is sufficient to prescribe +the forms of all possible grooves crossing it.% +\index{Geodesic!motion of particles in}% + +The application of Einstein's law to trace the gravitational +field not only through space but through time leads to a great +unification of mechanics. If we have given for a start a narrow +slice of space-time representing the state of the universe for a few +seconds, with all the little puckers belonging to particles of matter +%% -----File: 149.png---Folio 139------- +properly described, then step by step all space-time can be linked +on and the positions of the puckers shown at all subsequent +times (electrical forces being excluded). Nothing is needed for +this except the law of gravitation---that the curvature is only +of the first degree---and there can thus be nothing in the predictions +of mechanics which is not comprised in the law of +gravitation. The conservation of mass, of energy, and of +momentum must all be contained implicitly in Einstein's law.% +\index{Conservation!of energy and momentum}% +\index{Energy!conservation of}% +\IndexExtra{Conservation!of mass}% + +It may seem strange that Einstein's law of gravitation should +take over responsibility for the whole of mechanics; because in +many mechanical problems gravitation in the ordinary sense +can be neglected. But inertia and gravitation are unified; the +law is also the law of inertia, and inertia or mass appears in all +mechanical problems. When, as in many problems, we say that +gravitation is negligible, we mean only that the interaction of +the minute puckers with one another can be neglected; we do +not mean that the interaction of the pucker of a particle with +the general character of the space-time in which it lies can be +neglected, because this constitutes the inertia of the particle.% +\index{Inertia!relativity theory of}% + +The conservation of energy and the conservation of momentum +in three independent directions, constitute together four laws +or equations which are fundamental in all branches of mechanics. +Although they apply when gravitation in the ordinary sense is +not acting, they must be deducible like everything else in +mechanics from the law of gravitation. It is a great triumph for +Einstein's theory that his law gives correctly these experimental +principles, which have generally been regarded as unconnected +with gravitation. We cannot enter into the mathematical +deduction of these equations; but we shall examine generally +how they are arrived at. + +It has already been explained that although the values of +$G_{\mu\nu}$ are strictly zero everywhere in space-time, yet if we take +average values through a small region containing a large number +of particles of matter their average or ``macroscopic'' values +will not be zero\footnotemark. + \footnotetext{It is the $g$'s which are first averaged, then the $G_{\mu\nu}$ are calculated by the + formulae in \Noteref{5}.}% +\index{Macroscopic!equations}% +Expressions for these macroscopic values can +be found in terms of the number, masses and motions of the +particles. Since we have averaged the $G_{\mu\nu}$, we should also +%% -----File: 150.png---Folio 140------- +average the particles; that is to say, we replace them by a +distribution of continuous matter having equivalent properties. +We thus obtain macroscopic equations of the form +\index{Continuous matter}% +\[ +G_{\mu\nu} = K_{\mu\nu}, +\] +where on the one side we have the somewhat abstruse quantities +describing the kind of space-time, and on the other side we have +well-known physical quantities describing the density, momentum, +energy and internal stresses of the matter present. These +macroscopic equations are obtained solely from the law of +gravitation by the process of averaging. + +By an exactly similar process we pass from Laplace's equation +$\nabla^2\phi = 0$ to Poisson's equation for continuous matter $\nabla^2\phi = -4\pi\rho$, +in the Newtonian theory of gravitation.% +\index{Laplace's equation}% + +When continuous matter is admitted, \textit{any} kind of space-time +becomes possible. The law of gravitation instead of denying the +possibility of certain kinds, states what values of $K_{\mu\nu}$, i.e.\ what +distribution and motion of continuous matter in the region, are +a necessary accompaniment. This is no contradiction with the +original statement of the law, since that referred to the case in +which continuous matter is denied or excluded. Any set of +values of the potentials is now possible; we have only to calculate +by the formulae the corresponding values of $G_{\mu\nu}$, and we at +once obtain ten equations giving the $K_{\mu\nu}$ which define the +conditions of the matter necessary to produce these potentials. +But suppose the necessary distribution of matter through space +and time is an impossible one, violating the laws of mechanics! +No, there is only one law of mechanics, the law of gravitation; +we have specified the distribution of matter so as to satisfy +$G_{\mu\nu} = K_{\mu\nu}$, and there can be no other condition for it to fulfil. +The distribution must be mechanically possible; it might, however, +be unrealisable in practice, involving inordinately high or +even negative density of matter.% +\index{Gravitation, Einstein's law of!macroscopic equations}% + +In connection with the law for empty space, $G_{\mu\nu} = 0$, it was +noticed that whereas this apparently forms a set of ten equations, +only six of them can be independent. This was because ten +equations would suffice to determine the ten potentials precisely, +and so fix not only the kind of space-time but the mesh-system. +It is clear that we must preserve the right to draw the mesh-system +as we please; it is fixed by arbitrary choice not by a law +%% -----File: 151.png---Folio 141------- +of nature. To allow for the four-fold arbitrariness of choice, +there must be four relations always satisfied by the $G_{\mu\nu}$, so that +when six of the equations are given the remaining four become +tautological. + +These relations must be identities implied in the mathematical +definition of $G_{\mu\nu}$; that is to say, when the $G_{\mu\nu}$ have been written +out in full according to their definition, and the operations +indicated by the identities carried out, all the terms will cancel, +leaving only $0 = 0$. The essential point is that the four relations +follow from the mode of formation of the $G_{\mu\nu}$ from their simpler +constituents ($g_{\mu\nu}$ and their differential coefficients) and apply +universally. These four identical relations have actually been +discovered\footnote{Appendix, \Noteref{13}.}.% +\Pagelabel{note13}% +\index{Identities@Identities connecting $G_{\mu \nu}$}% + +When in continuous matter $G_{\mu\nu} = K_{\mu\nu}$ clearly the same four +relations must exist between the $K_{\mu\nu}$, not now as identities, +but as consequences of the law of gravitation, viz.\ the equality +of $G_{\mu\nu}$ and $K_{\mu\nu}$.% +\index{Matter!gravitational equations in}% + +Thus the four dimensions of the world bring about a four-fold +arbitrariness of choice of mesh-system; this in turn necessitates +four identical relations between the $G_{\mu\nu}$; and finally, in consequence +of the law of gravitation, these identities reveal four new +facts or laws relating to the density, energy, momentum or stress +of matter, summarised in the expressions $K_{\mu\nu}$. + +These four laws turn out to be the laws of conservation of +momentum and energy. + +The argument is so general that we can even assert that +corresponding to any \textit{absolute} property of a volume of a world +of four dimensions (in this case, \textit{curvature}), there must be four +\textit{relative} properties which are conserved. This might be made the +starting-point of a general inquiry into the necessary qualities +of a permanent perceptual world, i.e.\ a world whose substance +is conserved.% +\index{Conservation!of mass}% +\index{Mass!conservation of}% +\index{Momentum!conservation of}% +\index{Permanent perceptual world}% +\IndexExtra{Conservation!of energy and momentum}% + +There is another law of physics which was formerly regarded +as funda\-mental---the conservation of mass. Modern progress +has somewhat altered our position with regard to it; not that +its validity is denied, but it has been reinterpreted, and has +finally become merged in the conservation of energy. It will be +desirable to consider this in detail. + +%% -----File: 152.png---Folio 142------- + +%[Illustration: \textsc{Fig}. 19.] +% [** PP: Moved to top of paragraph] +\begin{figure*}[hbt] +\begin{center} +\Graphic[19]{3in}{152a} +\end{center} +\end{figure*}% +It was formerly supposed that the mass of a particle was a +number attached to the particle, expressing an intrinsic property, +which remained unaltered in all its vicissitudes. If $m$ is this +number, and $u$ the velocity of the particle, the momentum is~$mu$; +and it is through this relation, coupled with the law of conservation +of momentum that the mass~$m$ was defined. Let us take +for example two particles of masses $m_1 = 2$ and $m_2 = 3$, moving +in the same straight line. In the space-time diagram for an +observer $S$ the velocity of the first particle will be represented +by a direction~$OA$ (\Figref{19}). The first particle moves through +a space $MA$ in unit time, so that $MA$ is equal to its velocity +referred to the observer~$S$. Prolonging the line $OA$ to meet the +second time-partition, $NB$ is equal to the velocity multiplied +by the mass~$2$; thus the horizontal distance $NB$ represents the +momentum. Similarly, starting from $B$ and drawing $BC$ in the +direction of the velocity of~$m_2$, prolonged through three +time-partitions, the horizontal progress from $B$ represents the +momentum of the second particle. The length $PC$ then represents +the total momentum of the system of two particles. + +Suppose that some change of their velocities occurs, not +involving any transference of momentum from outside, e.g.\ a +collision. Since the total momentum $PC$ is unaltered, a similar +%% -----File: 153.png---Folio 143------- +construction made with the new velocities must again bring us +to~$C$; that is to say, the new velocities are represented by the +directions $OB'$, $B'C$, where $B'$ is some other point on the line~$NB$. + +%[Illustration: \textsc{Fig}.~20.] +% [** PP: Moved to top of paragraph] +\begin{figure*}[hbt] +\begin{center} +\Graphic[20]{3in}{153a} +\end{center} +\end{figure*}% +Now examine how this will appear to some other observer $S_1$ +in uniform motion relative to $S$. His transformation of space +and time has been described in \Chapref{III} and is represented in +\Figref{20}, which shows how his time-partitions run as compared +with those of~$S$. The same actual motion is, of course, represented +by parallel directions in the two diagrams; but the +interpretation as a velocity $MA$ is different in the two cases. +Carrying the velocity of $m_1$ through two time-partitions, and of +$m_2$ through three time-partitions, as before, we find that the total +momentum for the observer $S_1$ is represented by~$PC$ (\Figref{20}); +but making a similar construction with the velocities after +collision, we arrive at a different point~$C'$. Thus whilst momentum +is conserved for the observer~$S$, it has altered from $PC$ to +$PC'$ for the observer~$S_1$. + +The discrepancy arises because in the construction the lines +are prolonged to meet partitions which are different for the two +%% -----File: 154.png---Folio 144------- +observers. The rule for determining momentum ought to be +such that both observers make the same construction, independent +of their partitions, so that both arrive by the two routes +at the same point~$C$. Then it will not matter if, through their +different measures of time, one observer measures momentum +by horizontal progress and the other by oblique progress; both +will agree that the momentum has not been altered by the +collision. To describe such a construction, we must use the +interval which is alike for both observers; make the interval-length +of $OB$ equal to $2$~units, and that of $BC$ equal to $3$~units, +disregarding the mesh-system altogether. Then both observers +will make the same diagram and arrive at the same point~$C$ +(different from $C$ or~$C'$ in the previous diagrams). Then if +momentum is conserved for one observer, it will be conserved +for the other. + +This involves a modified definition of momentum. Momentum +must now be the mass multiplied by the change of position $\delta x$ +per lapse of interval $\delta s$, instead of per lapse of time $\delta t$. Thus +\index{Momentum!redefinition of}% +\begin{align*} + \text{momentum} &= m \frac{\delta x}{\delta s} \\ +\text{instead of momentum} &= m \frac{\delta x}{\delta t}, +\end{align*} +and the mass~$m$ still preserves its character as an invariant +number associated with the particle. + +Whether the momentum as now defined is actually conserved +or not, is a matter for experiment, or for theoretical deduction +from the law of gravitation. The point is that with the original +definition general conservation is impossible, because if it held +good for one observer it could not hold for another. The new +definition makes general conservation possible. Actually this +form of the momentum is the one deduced from the law of +gravitation through the identities already described. With +regard to experimental confirmation it is sufficient at present +to state that in all ordinary cases the interval and the time are +so nearly equal that such experimental foundation as existed +for the law of conservation of the old momentum is just as +applicable to the new momentum. + +Thus in the theory of relativity momentum appears as an +%% -----File: 155.png---Folio 145------- +invariant mass multiplied by a modified velocity $\delta x/\delta s$. The +physicist, however, prefers for practical purposes to keep to the +old definition of momentum as mass multiplied by the velocity +$\delta x/\delta t$. We have +\index{Invariant mass}% +\index{Mass!invariant}% +\[ +m \frac{\delta x}{\delta s} + = m \frac{\delta t}{\delta s} · \frac{\delta x}{\delta t}, +\] +accordingly the momentum is separated into two factors, the +velocity $\delta x/\delta t$, and a mass $M = m \delta t/\delta s$, which is no longer an +invariant for the particle but depends on its motion relative to +the observer's space and time. In accordance with the usual +practice of physicists the mass (unless otherwise qualified) is +taken to mean the quantity~$M$. + +Using unaccelerated rectangular axes, we have by definition +of~$s$ +\[ +\delta s^2 = \delta t^{2} - \delta x^{2} - \delta y^{2} - \delta z^{2}, +\] +so that +\begin{align*} +\left(\frac{\delta s}{\delta t}\right)^2 + &= 1 - \left(\frac{\delta x}{\delta t}\right)^2 + - \left(\frac{\delta y}{\delta t}\right)^2 + - \left(\frac{\delta z}{\delta t}\right)^2, \\ + &= 1-u^2, +\end{align*} +where $u$ is the resultant velocity of the particle (the velocity of +light being unity). Hence +\index{Mass!variation with velocity}% +\[ +M = \frac{m}{\surd(1-u^2)}. +\] +Thus the mass increases as the velocity increases, the factor +being the same as that which determines the FitzGerald contraction. + +The increase of mass with velocity is a property which challenges +experimental test. For success it is necessary to be able +to experiment with high velocities and to apply a known force +large enough to produce appreciable deflection in the fast-moving +particle. These conditions are conveniently fulfilled by +the small negatively charged particles emitted by radio-active % [** PP: Regularized radioactive] +substances, known as $\beta$~particles, or the similar particles which +constitute cathode rays. +\index{Beta particles}% +They attain speeds up to 0.8 of the +velocity of light, for which the increase of mass is in the ratio +1.66; and the negative charge enables a large electric or magnetic +force to be applied. Modern experiments fully confirm the +theoretical increase of mass, and show that the factor $1/\surd(1-u^2)$ +%% -----File: 156.png---Folio 146------- +is at least approximately correct. The experiment was originally +performed by Kaufmann; but much greater accuracy has been +obtained by recent modified methods.% +\index{Electron!Kaufmann's experiment on}% +\index{Kaufmann's experiment}% + +Unless the velocity is very great the mass~$M$ may be written +\[ +m/\surd(1-u^2) = m + \tfrac{1}{2} mu^2. +\] +Thus it consists of two parts, the mass when at rest, together +with the second term which is simply the energy of the motion. +If we can say that the term $m$ represents a kind of potential +energy concealed in the matter, mass can be identified with +energy. The increase of mass with velocity simply means that +the energy of motion has been added on. +\index{Energy!identified with mass}% +\index{Mass!identified with energy}% +We are emboldened +to do this because in the case of an electrical charge the electrical +mass is simply the energy of the static field. Similarly the mass +of light is simply the electromagnetic energy of the light. + +In our ordinary units the velocity of light is not unity, and +a rather artificial distinction between mass and energy is introduced. +They are measured by different units, and energy~$E$ has +a mass $E/C^2$ where $C$ is the velocity of light in the units used. +But it seems very probable that mass and energy are two ways +of measuring what is essentially the same thing, in the same +sense that the parallax and distance of a star are two ways of +expressing the same property of location. If it is objected that +they ought not to be confused inasmuch as they are distinct +properties, it must be pointed out that they are not sense-properties, +but mathematical terms expressing the dividend +and product of more immediately apprehensible properties, viz.\ +momentum and velocity. They are essentially mathematical +compositions, and are at the disposal of the mathematician. + +This proof of the variation of mass with velocity is much more +general than that based on the electrical theory of inertia. +\index{Energy!inertia of}% +It +applies immediately to matter in bulk. The masses $m_1$ and~$m_2$ +need not be particles; they can be bodies of any size or composition. +On the electrical theory alone, there is no means of +deducing the variation of mass of a planet from that of an +electron. + +It has to be remarked that, although the inertial mass of a +particle only comes under physical measurement in connection +with a change of its motion, it is just when the motion is changing +that the conception of its mass is least definite; because it is at +%% -----File: 157.png---Folio 147------- +that time that the kinetic energy, which forms part of the mass, +is being passed on to another particle or radiated into the +surrounding field; and it is scarcely possible to define the +moment at which this energy ceases to be associated with the +particle and must be reckoned as broken loose. The amount of +energy or mass in a given region is always a definite quantity; +but the amount attributable to a particle is only definite when +the motion is uniform. In rigorous work it is generally necessary +to consider the mass not of a particle but of a region. + +The motion of matter from one place to another causes an +alteration of the gravitational field in the surrounding space. +If the motion is uniform, the field is simply convected; but if +the motion is accelerated, something of the nature of a gravitational +wave is propagated outwards. The velocity of propagation +is the velocity of light. +\index{Gravitation!propagation with velocity of light}% +\index{Propagation of Gravitation}% +\index{Velocity of gravitation}% +The exact laws are not very simple +because we have seen that the gravitational field modifies the +velocity of light; and so the disturbance itself modifies the +velocity with which it is propagated. In the same way the +exact laws of propagation of sound are highly complicated, +because the disturbance of the air by sound modifies the speed +with which it is propagated. But the approximate laws of +propagation of gravitation are quite simple and are the same as +those of electromagnetic disturbances. + +After mass and energy there is one physical quantity which +plays a very fundamental part in modern physics, known as +\textit{Action}. +\index{Action}% +\textit{Action} here is a very technical term, and is not to be +confused with Newton's ``Action and Reaction.'' In the relativity +theory in particular this seems in many respects to be the most +fundamental thing of all. The reason is not difficult to see. If +we wish to speak of the continuous matter present \textit{at} any particular +point of space and time, we must use the term \textit{density}. +Density multiplied by volume in space gives us \textit{mass} or, what +appears to be the same thing, \textit{energy}. But from our space-time +point of view, a far more important thing is density multiplied +by a four-dimensional volume of space and time; this is \textit{action}. +The multiplication by three dimensions gives mass or energy; +and the fourth multiplication gives mass or energy multiplied +by time. Action is thus mass multiplied by time, or energy +multiplied by time, and is more fundamental than either. + +%% -----File: 158.png---Folio 148------- + +Action is the curvature of the world. +\index{Curvature!identified with action}% +It is scarcely possible +to visualise this statement, because our notion of curvature is +derived from surfaces of two dimensions in a three-dimensional +space, and this gives too limited an idea of the possibilities of a +four-dimensional surface in space of five or more dimensions. +In two dimensions there is just one total curvature, and if that +vanishes the surface is flat or at least can be unrolled into a +plane. In four dimensions there are many coefficients of +curvature; but there is one curvature \textit{par excellence}, which is, +of course, an invariant independent of our mesh-system. It is +the quantity we have denoted by~$G$. It does not follow that if +the curvature vanishes space-time is flat; we have seen in fact +that in a natural gravitational field space-time is not flat +although there may be no mass or energy and therefore no action +or curvature. + +Wherever there is matter\footnote% + {It is rather curious that there is no action in space containing only light. + Light has mass ($M$) of the ordinary kind; but the invariant mass ($m$) vanishes.} +\index{Invariant mass!of light}% +\index{Light!mass of}% +\index{Mass of light}% +there is action and therefore +curvature; and it is interesting to notice that in ordinary matter +the curvature of the space-time world is by no means insignificant. +For example, in water of ordinary density the curvature is the +same as that of space in the form of a sphere of radius $570,000,000$ +kilometres. The result is even more surprising if expressed in +time units; the radius is about half-an-hour. + +It is difficult to picture quite what this means; but at least +we can predict that a globe of water of $570,000,000$~km.\ radius +would have extraordinary properties. +\index{Curvature!of a globe of water}% +\index{Globe of water, limit to size of}% +Presumably there must +be an upper limit to the possible size of a globe of water. So +far as I can make out a homogeneous mass of water of about +this size (and no larger) could exist. It would have no centre, +and no boundary, every point of it being in the same position +with respect to the whole mass as every other point of it---like +points on the \textit{surface} of a sphere with respect to the surface. +Any ray of light after travelling for an hour or two would come +back to the starting point. Nothing could enter or leave the +mass, because there is no boundary to enter or leave by; in +fact, it is coextensive with space. There could not be any other +world anywhere else, because there isn't an ``anywhere else.'' + +The mass of this volume of water is not so great as the most +%% -----File: 159.png---Folio 149------- +moderate estimates of the mass of the stellar system. Some +physicists have predicted a distant future when all energy will +be degraded, and the stellar universe will gradually fall together +into one mass. Perhaps then these strange conditions will be +realised! + +The law of gravitation, the laws of mechanics, and the laws +of the electromagnetic field have all been summed up in a single +Principle of Least Action. +\index{Action, Principle of Least}% +\index{Principle of Least Action}% +For the most part this unification +was accomplished before the advent of the relativity theory, +and it is only the addition of gravitation to the scheme which is +novel. We can see now that if action is something absolute, +a configuration giving minimum action is capable of absolute +definition; and accordingly we should expect that the laws of +the world would be expressible in some such form. The argument +is similar to that by which we first identified the natural +tracks of particles with the tracks of greatest interval-length. +The fact that some such form of law is inevitable, rather discourages +us from seeking in it any clue to the structural details +of our world. + +Action is one of the two terms in pre-relativity physics which +survive unmodified in a description of the absolute world. The +only other survival is entropy. The coming theory of relativity +had cast its shadow before; and physics was already converging +to two great generalisations, the principle of least action and +the second law of thermodynamics or principle of maximum +entropy.% +\index{Entropy}% + +We are about to pass on to recent and more shadowy developments +of this subject; and this is an appropriate place to glance +back on the chief results that have emerged. The following +summary will recall some of the salient points. + +\Paragraph{1.} The order of events in the external world is a four-dimensional +order. + +\Paragraph{2.} The observer either intuitively or deliberately constructs +a system of meshes (space and time partitions) and locates the +events with respect to these. + +\Paragraph{3.} Although it seems to be theoretically possible to describe +phenomena without reference to any mesh-system (by a catalogue +of coincidences), such a description would be cumbersome. In +%% -----File: 160.png---Folio 150------- +practice, physics describes the relations of the events to our +mesh-system; and all the terms of elementary physics and of +daily life refer to this relative aspect of the world. + +\Paragraph{4.} Quantities like length, duration, mass, force, etc.\ have no +absolute significance; their values will depend on the mesh-system +to which they are referred. When this fact is realised, +the results of modern experiments relating to changes of length +of rigid bodies are no longer paradoxical. + +\Paragraph{5.} There is no fundamental mesh-system. In particular +problems, and more particularly in restricted regions, it may +be possible to choose a mesh-system which follows more or less +closely the lines of absolute structure in the world, and so +simplify the phenomena which are related to it. But the world-structure +is not of a kind which can be traced in an exact way +by mesh-systems, and in any large region the mesh-system +drawn must be considered arbitrary. In any case the systems +used in current physics are arbitrary. + +\Paragraph{6.} The study of the absolute structure of the world is based +on the ``interval'' between two events close together, which is +an absolute attribute of the events independent of any mesh-system. +A world-geometry is constructed by adopting the +interval as the analogue of distance in ordinary geometry.% +\index{Interval}% + +\Paragraph{7.} This world-geometry has a property unlike that of +Euclidean geometry in that the interval between two real +events may be real or imaginary. The necessity for a physical +distinction, corresponding to the mathematical distinction between +real and imaginary intervals, introduces us to the separation +of the four-dimensional order into time and space. But this +separation is not unique, and the separation commonly adopted +depends on the observer's track through the four-dimensional +world.% +\index{Imaginary intervals}% + +\Paragraph{8.} The geodesic, or track of maximum or minimum interval-length +between two distant events, has an absolute significance. +And since no other kind of track can be defined absolutely, it +is concluded that the tracks of freely moving particles are +geodesics.% +\index{Geodesic!absolute significance of}% +\index{Geodesic!motion of particles in}% +\index{Interval-length!tracks of maximum}% + +\Paragraph{9.} In Euclidean geometry the geodesics are straight lines. It +is evidently impossible to choose space and time-reckoning so +that all free particles in the solar system move in straight lines. +%% -----File: 161.png---Folio 151------- +Hence the geometry must be non-Euclidean in a field of gravitation. + +\Paragraph{10.} Since the tracks of particles in a gravitational field are +evidently governed by some law, the possible geometries must +be limited to certain types. + +\Paragraph{11.} The limitation concerns the absolute structure of the +world, and must be independent of the choice of mesh-system. +This narrows down the possible discriminating characters. +Practically the only reasonable suggestion is that the world +must (in empty space) be ``curved no higher than the first +degree''; and this is taken as the law of gravitation. + +\Paragraph{12.} The simplest type of hummock with this limited curvature +has been investigated. It has a kind of infinite chimney at the +summit, which we must suppose cut out and filled up with a +region where this law is not obeyed, i.e.\ with a particle of matter. + +\Paragraph{13.} The tracks of the geodesics on the hummock are such as +to give a very close accordance with the tracks computed by +Newton's law of gravitation. The slight differences from the +Newtonian law have been experimentally verified by the motion +of Mercury and the deflection of light. + +\Paragraph{14.} The hummock might more properly be described as a +ridge extending linearly. Since the interval-length along it is +real or time-like, the ridge can be taken as a time-direction. +Matter has thus a continued existence in time. Further, in +order to conform with the law, a small ridge must always follow +a geodesic in the general field of space-time, confirming the conclusion +arrived at under~(8). + +\Paragraph{15.} The laws of conservation of energy and momentum in +mechanics can be deduced from this law of world-curvature. + +\Paragraph{16.} Certain phenomena such as the FitzGerald contraction +and the variation of mass with velocity, which were formerly +thought to depend on the behaviour of electrical forces concerned, +are now seen to be general consequences of the relativity +of knowledge. That is to say, length and mass being the relations +of some absolute thing to the observer's mesh-system, we can +foretell how these relations will be altered when referred to +another mesh-system. +%% -----File: 162.png---Folio 152------- + + +\Chapter{X}{Towards Infinity} + +\Quote{W.~K. Clifford (1873).} +{The geometer of to-day knows nothing about the nature of actually existing +space at an infinite distance; he knows nothing about the properties of this +present space in a past or a future eternity. He knows, indeed, that the laws +assumed by Euclid are true with an accuracy that no direct experiment can +approach, not only in this place where we are, but in places at a distance from +us that no astronomer has conceived; but he knows this as of Here and Now; +beyond his range is a There and Then of which he knows nothing at present, +but may ultimately come to know more.}% +\index{Clifford}% + + +\First{The} great stumbling-block for a philosophy which denies +absolute space is the experimental detection of absolute rotation. +\index{Absolute rotation}% +\index{Relativity of rotation}% +\index{Rotation, absolute}% +The belief that the earth rotates on its axis was suggested by +the diurnal motions of the heavenly bodies; this observation is +essentially one of relative rotation, and, if the matter rested +there, no difficulty would be felt. But we can detect the same +rotation, or a rotation very closely equal to it, by methods +which do not seem to bring the heavenly bodies into consideration; +and such a rotation is apparently absolute. The planet +Jupiter is covered with cloud, so that an inhabitant would +probably be unaware of the existence of bodies outside; yet he +could quite well measure the rotation of Jupiter. By the gyrocompass +he would fix two points on the planet---the north and +south poles. Then by Foucault's experiment on the change of +the plane of motion of a freely suspended pendulum, he would +determine an angular velocity about the poles. +\index{Foucault's pendulum}% +\index{Gyrocompass} % [** PP: Removed hyphen for consistency] +Thus there is +certainly a definite physical constant, an angular velocity about +an axis, which has a fundamental importance for the inhabitants +of Jupiter; the only question is whether we are right in giving +it the name absolute rotation. + +Contrast this with absolute translation. Here it is not a +question of giving the right name to a physical constant; the +inhabitants of Jupiter would find no constant to name. We see +at once that a relativity theory of translation is on a different +footing from a relativity theory of rotation. The duty of the +%% -----File: 163.png---Folio 153------- +former is to explain facts; the duty of the latter is to explain +away facts. + +Our present theory seems to make a start at tackling this +problem, but gives it up. It permits the observer, if he wishes, +to consider the earth as non-rotating, but surrounded by a field +of centrifugal force; all the other bodies in the universe are then +revolving round the earth in orbits mainly controlled by this +field of centrifugal force. Astronomy on this basis is a little +cumbersome; but all the phenomena are explained perfectly. +The centrifugal force is part of the gravitational field, and obeys +Einstein's law of gravitation, so that the laws of nature are +completely satisfied by this representation. One awkward +question remains, What causes the centrifugal force? Certainly +not the earth which is here represented as non-rotating. As we +go further into space to look for a cause, the centrifugal force +becomes greater and greater, so that the more we defer the debt +the heavier the payment demanded in the end. Our present +theory is like the debtor who does not mind how big an obligation +accumulates satisfied that he can always put off the payment. +It chases the cause away to infinity, content that the laws of +nature---the relations between contiguous parts of the world---are +satisfied all the way. + +One suggested loophole must be explored. Our new law of +gravitation admits that a rapid motion of the attracting body +will affect the field of force. If the earth is non-rotating, the +stars must be going round it with terrific speed. May they not +in virtue of their high velocities produce gravitationally a +sensible field of force on the earth, which we recognise as the +centrifugal field? This would be a genuine elimination of +absolute rotation, attributing all effects indifferently to the +rotation of the earth the stars being at rest, or to the revolution +of the stars the earth being at rest; nothing matters except the +relative rotation. I doubt whether anyone will persuade himself +that the stars have anything to do with the phenomenon. We +do not believe that if the heavenly bodies were all annihilated +it would upset the gyrocompass. In any case, precise calculation +shows that the centrifugal force could not be produced by the +motion of the stars, so far as they are known.% +\index{Centrifugal Force!not caused by stars}% + +We are therefore forced to give up the idea that the signs of +%% -----File: 164.png---Folio 154------- +the earth's rotation---the protuberance of its equator, the +phenomena of the gyrocompass, etc.---are due to a rotation +relative to any matter we can recognise. The philosopher who +persists that a rotation which is not relative to matter is unthinkable, +will no doubt reply that the rotation must then be +relative to some matter which we have not yet recognised. We +have hitherto been greatly indebted to the suggestions of +philosophy in evolving this theory, because the suggestions +related to the things we know about; and, as it turned out, they +were confirmed by experiment. But as physicists we cannot +take the same interest in the new demand; we do not necessarily +challenge it, but it is outside our concern. Physics demands of +its scheme of nature something else besides truth, namely a +certain quality that we may call convergence. The law of +conservation of energy is only strictly true when the whole +universe is taken into account; but its value in physics lies in +the fact that it is \textit{approximately} true for a very limited system. +Physics is an exact science because the chief essentials of a +problem are limited to a few conditions; and it draws near to +the truth with ever-increasing approximation as it widens its +purview. The approximations of physics form a convergent +series. +\index{Convergence of physical approximations}% +History, on the other hand, is very often like a divergent +series; no approximation to its course is reached until the last +term of the infinite series has been included in the data of +prediction. Physics, if it wishes to retain its advantage, must +take its own course, formulating those laws which are approximately +true for the limited data of sense, and extending them +into the unknown. The relativity of rotation is not approximately +true for the data of sense, although it may possibly be +true when the unknown as well as the known are included. + +The same considerations that apply to rotation apply to +acceleration, although the difficulty is less striking. We can if +we like attribute to the sun some arbitrary acceleration, balancing +it by introducing a uniform gravitational field. Owing to this +field the rest of the stars will move with the same acceleration +and no phenomena will be altered. But then it seems necessary +to find a cause for this field. It is not produced by the gravitation +of the stars. Our only course is to pursue the cause further and +further towards infinity; the further we put it away, the greater +%% -----File: 165.png---Folio 155------- +the mass of attracting matter needed to produce it. On the +other hand, the earth's absolute acceleration does not intrude +on our attention in the way that its absolute rotation does\footnotemark. + \footnotetext{To determine even roughly the earth's absolute acceleration we should + need a fairly full knowledge of the disturbing effects of all the matter in the + universe. A similar knowledge would be required to determine the absolute + rotation \textit{accurately}; but all the matter likely to exist would have so small an + effect, that we can at once assume that the absolute rotation is very nearly + the same as the experimentally determined rotation.}% +\index{Absolute acceleration}% [** PP: Index reads p. 154] + + +We are vaguely conscious of a difficulty in these results; but +if we examine it closely, the difficulty does not seem to be a +very serious one. The theory of relativity, as we have understood +it, asserts that our partitions of space and time are introduced +by the observer and are irrelevant to the laws of nature; and +therefore the current quantities of physics, length, duration, +mass, force, etc., which are relative to these partitions, are not +things having an absolute significance in nature. But we have +never denied that there are features of the world having an +absolute significance; in fact, we have spent much time in finding +such features. The geodesics or natural tracks have been shown +to have an absolute significance; +\index{Geodesic structure!absolute character of}% +\index{Structure, geodesic!absolute character of}% +and it is possible in a limited +region of the world to choose space and time partitions such +that all geodesics become approximately straight lines. We may +call this a ``natural'' frame for that region, although it is not +as a rule the space and time adopted in practice; +\index{Natural frame}% +it is for example +the space and time of the observers in the falling projectile, not +of Newton's super-observer. It is capable of absolute definition, +except that it is ambiguous in regard to uniform motion. Now +the rotation of the earth determined by Foucault's pendulum +experiment is the rotation referred to this natural frame. But +we must have misunderstood our own theory of relativity +altogether, if we think there is anything inadmissible in an +absolute rotation of such a kind.% +\index{Relativity of rotation}% + +Material particles and geodesics are both features of the +absolute structure of the world; and a rotation relative to +geodesic structure does not seem to be on any different footing +from a velocity relative to matter. There is, however, the +striking feature that rotation seems to be relative not merely +to the local geodesic structure but to a generally accepted +universal frame; whereas it is necessary to specify precisely +%% -----File: 166.png---Folio 156------- +what matter a velocity is measured with respect to. This is +largely a question of how much accuracy is needed in specifying +velocities and rotations, respectively. If in stating the speed of +a $\beta$~particle we do not mind an error of $10,000$ kilometres a +second, we need not specify precisely what star or planet its +velocity is referred to. The moon's (local) angular velocity is +sometimes given to fourteen significant figures; I doubt if +any universal frame is well-defined enough for this accuracy. +There is no doubt much greater continuity in the geodesic +structure in different parts of the world than in the material +structure; but the difference is in degree rather than in +principle. + +It is probable that here we part company from many of the +continental relativists, who give prominent place to a principle +known as the \textit{law of causality}---that only those things are to be +regarded as being in causal connection which are capable of +being actually observed. +\index{Causality, law of}% +This seems to be interpreted as placing +matter on a plane above geodesic structure in regard to the +formulation of physical laws, though it is not easy to see in +what sense a distribution of matter can be regarded as more +observable than the field of influence in surrounding space +which makes us aware of its existence. The principle itself is +debateable; that which is observable to us is determined by +the accident of our own structure, and the law of causality +seems to impose our own limitations on the free interplay of +entities in the world outside us. In this book the tradition of +Faraday and Maxwell still rules our outlook; and for us matter +and electricity are but incidental points of complexity, the +activity of nature being primarily in the so-called empty spaces +between. + +The vague universal frame to which rotation is referred is +called the \textit{inertial frame}. +\index{Frame, inertial}% +\index{Inertial frame}% +It is definite in the flat space-time far +away from all matter. In the undulating country corresponding +to the stellar universe it is not a precise conception; it is rather +a rude outline, arbitrary within reasonable limits, but with the +general course indicated. The reason for the term inertial frame +is of interest. We can quite freely use a mesh-system deviating +widely from the inertial frame (e.g.\ rotating axes); but we have +seen that there is a postponed debt to pay in the shape of an +%% -----File: 167.png---Folio 157------- +apparently uncaused field of force. But is there no debt to pay, +even when the inertial frame is used? In that case there is no +gravitational or centrifugal force at infinity; +\index{Centrifugal Force!debt at infinity}% +\index{Geodesic!in regions at infinity}% +\index{Infinity, conditions at}% +\index{Structure, geodesic!behaviour at infinity}% +but there is still +inertia, which is of the same nature. The distinction between +force as requiring a cause and inertia as requiring no cause +cannot be sustained. We shall not become any more solvent by +commuting our debt into pure inertia. The debt is inevitable +whatever mesh-system is used; we are only allowed to choose +the form it shall take. + +The debt after all is a very harmless one. At infinity we have +the absolute geodesics in space-time, and we have our own +arbitrarily drawn mesh-system. The relation of the geodesics +to the mesh-system decides whether our axes shall be termed +rotating or non-rotating; and ideally it is this relation that is +determined when a so-called absolute rotation is measured. +No one could reasonably expect that there would be no determinable +relation. On the other hand uniform translation does +not affect the relation of the geodesics to the mesh-system---if +they were straight lines originally, they remain straight lines---thus +uniform translation cannot be measured except relative to +matter. + +We have been supposing that the conditions found in the +remotest parts of space accessible to observation can be extrapolated +to infinity; and that there are still definite natural +tracks in space-time far beyond the influence of matter. Feelings +of objection to this view arise in certain minds. It is urged that +as matter influences the course of geodesics it may well be +responsible for them altogether; so that a region outside the +field of action of matter could have no geodesics, and consequently +no intervals. All the potentials would then necessarily +be zero. Various modified forms of this objection arise; but the +main feeling seems to be that it is unsatisfactory to have certain +conditions prevailing in the world, which can be traced away to +infinity and so have, as it were, their source at infinity; and +there is a desire to find some explanation of the inertial frame +as built up through conditions at a finite distance.% +\index{Inertia!in regions at infinity}% + +Now if all intervals vanished space-time would shrink to a +point. Then there would be no space, no time, no inertia, no +anything. Thus a cause which creates intervals and geodesics +%% -----File: 168.png---Folio 158------- +must, so to speak, extend the world. We can imagine the world +stretched out like a plane sheet; but then the stretching cause---the +cause of the intervals---is relegated beyond the bounds of +space and time, i.e.\ to infinity. This is the view objected to, +though the writer does not consider that the objection has +much force. An alternative way is to inflate the world from +inside, as a balloon is blown out. In this case the stretching +force is not relegated to infinity, and ruled outside the scope of +experiment; it is acting at every point of space and time, curving +the world to a sphere. We thus get the idea that space-time +may have an essential curvature on a great scale independent +of the small hummocks due to recognised matter. + +It is not necessary to speculate whether the curvature is +produced (as in the balloon) by some pressure applied from a +fifth dimension. For us it will appear as an innate tendency of +four-dimensional space-time to curve. It may be asked, what +have we gained by substituting a natural curvature of space-time +for a natural stretched condition corresponding to the +inertial frame? +\index{Curvature!of space and time}% +As an explanation, nothing. But there is this +difference, that the theory of the inertial frame can now be +included in the differential law of gravitation instead of remaining +outside and additional to the law. + +It will be remembered that one clue by which we previously +reached the law of gravitation was that flat space-time must be +compatible with it. But if space-time is to have a small natural +curvature independent of matter this condition is now altered. +It is not difficult to find the necessary alteration of the law\footnote{Appendix, \Noteref{14}.}. +\Pagelabel{note14}% +It will contain an additional, and at present unknown, constant, +which determines the size of the world. + +Spherical space is not very easy to imagine. We have to +think of the properties of the surface of a sphere---the two-dimensional +case---and try to conceive something similar applied +to three-dimensional space. Stationing ourselves at a point let +us draw a series of spheres of successively greater radii. The +surface of a sphere of radius~$r$ should be proportional to~$r^{2}$; but +in spherical space the areas of the more distant spheres begin +to fall below the proper proportion. There is not so much room +out there as we expected to find. Ultimately we reach a sphere +%% -----File: 169.png---Folio 159------- +of biggest possible area, and beyond it the areas begin to decrease\footnotemark. + \footnotetext{The area is, of course, to be determined by measurement of some kind.}% +The last sphere of all shrinks to a point---our antipodes. +Is there nothing beyond this? Is there a kind of boundary +there? There is nothing beyond and yet there is no boundary. +On the earth's surface there is nothing beyond our own antipodes +but there is no boundary there. + +The difficulty is that we try to realise this spherical world by +imagining how it would appear to us and to our measurements. +There has been nothing in our experience to compare it with, +and it seems fantastic. But if we could get rid of the personal +point of view, and regard the sphericity of the world as a statement +of the type of order of events outside us, we should think +that it was a simple and natural order which is as likely as any +other to occur in the world. + +In such a world there is no difficulty about accumulated debt +at the boundary. There is no boundary. The centrifugal force +increases until we reach the sphere of greatest area, and then, +still obeying the law of gravitation, diminishes to zero at the +antipodes. The debt has paid itself automatically. + +We must not exaggerate what has been accomplished by this +modification of the theory. A new constant has been introduced +into the law of gravitation which gives the world a definite +extension. Previously there was nothing to fix the scale of the +world; it was simply given \textit{a~priori} that it was infinite. Granted +extension, so that the intervals are not invariably zero, we can +determine geodesics everywhere, and hence mark out the inertial +frame. + +Spherical space-time, that is to say a four-dimensional continuum +of space and imaginary time forming the surface of a +sphere in five dimensions, has been investigated by Prof.\ de~Sitter. +\index{de Sitter}% +\index{Spherical space-time}% +If real time is used the world is spherical in its space +dimensions, but open towards plus and minus infinity in its +time dimension, like an hyperboloid. This happily relieves us +of the necessity of supposing that as we progress in time we +shall ultimately come back to the instant we started from! +\index{Retardation of time!in spherical world} % [** PP: Index reads p. 160] +History never repeats itself. But in the space dimensions we +should, if we went on, ultimately come back to the starting +point. This would have interesting physical results, and we +%% -----File: 170.png---Folio 160------- +shall see presently that Einstein has a theory of the world in +which the return can actually happen; but in de~Sitter's theory +it is rather an abstraction, because, as he says, ``all the paradoxical +phenomena can only happen after the end or before the +beginning of eternity.'' + +The reason is this. Owing to curvature in the time dimension, +as we examine the condition of things further and further from +our starting point, our time begins to run faster and faster, or +to put it another way natural phenomena and natural clocks +slow down. The condition becomes like that described in +Mr~H.~G. Wells's story ``The new accelerator.'' + +When we reach half-way to the antipodal point, time stands +still. Like the Mad Hatter's tea party, it is always 6~o'clock; +and nothing whatever can happen however long we wait. There +is no possibility of getting any further, because everything +including light has come to rest here. All that lies beyond is +for ever cut off from us by this barrier of time; and light can +never complete its voyage round the world. + +That is what happens when the world is viewed from one +station; but if attracted by such a delightful prospect, we proceeded +to visit this scene of repose, we should be disappointed. +We should find nature there as active as ever. We thought time +was standing still, but it was really proceeding there at the +usual rate, as if in a fifth dimension of which we had no +cognisance. Casting an eye back on our old home we should see +that time apparently had stopped still there. Time in the two +places is proceeding in directions at right angles, so that the +progress of time at one point has no relation to the perception +of time at the other point. The reader will easily see that a being +confined to the surface of a sphere and not cognisant of a third +dimension, will, so to speak, lose one of his dimensions altogether +when he watches things occurring at a point $90°$ away. He +regains it if he visits the spot and so adapts himself to the two +dimensions which prevail there. + +It might seem that this kind of fantastic world-building can +have little to do with practical problems. But that is not quite +certain. May we not be able actually to observe the slowing +down of natural phenomena at great distances from us? The +most remote objects known are the spiral nebulae, whose +%% -----File: 171.png---Folio 161------- +distances may perhaps be of the order a million light years. +If natural phenomena are slowed down there, the vibrations of +an atom are slower, and its characteristic spectral lines will +appear displaced to the red. +\index{Displacement of spectral lines!in nebulae}% +\index{Nebulae, atomic vibrations in}% +We should generally interpret this +as a Doppler effect, implying that the nebula is receding. The +motions in the line-of-sight of a number of nebulae have been +determined, chiefly by Prof.\ Slipher. The data are not so ample +as we should like; but there is no doubt that large receding +motions greatly preponderate. +\index{Receding velocities!of spiral nebulae}% +This may be a genuine phenomenon +in the evolution of the material universe; but it is also +possible that the interpretation of spectral displacement as a +receding velocity is erroneous; and the effect is really the slowing +of atomic vibrations predicted by de~Sitter's theory. + +Prof.\ Einstein himself prefers a different theory of curved +space-time. His world is cylindrical---curved in the three space +dimensions and straight in the time dimension. +\index{Cylindrical world, Einstein's}% +Since time is no +longer curved, the slowing of phenomena at great distances +from the observer disappears, and with it the slight experimental +support given to the theory by the observations of spiral nebulae. +There is no longer a barrier of eternal rest, and a ray of light is +able to go round the world. + +In various ways crude estimates of the size of the world both +on de~Sitter's and Einstein's hypotheses have been made; and +in both cases the radius is thought to be of the order $10^{13}$ times +the distance of the earth from the sun. A ray of light from the +sun would thus take about $1000$~million years to go round the +world; and after the journey the rays would converge again at +the starting point, and then diverge for the next circuit. +\index{Light!voyage round the world}% +The +convergent would have all the characteristics of a real sun so +far as light and heat are concerned, only there would be no +substantial body present. Thus corresponding to the sun we +might see a series of ghosts occupying the positions where the +sun was $1000$, $2000$, $3000$, etc., million years ago, if (as seems +probable) the sun has been luminous for so long.% +\index{Ghosts of stars}% + +It is rather a pleasing speculation that records of the previous +states of the sidereal universe may be automatically reforming +themselves on the original sites. Perhaps one or more of the +many spiral nebulae are really phantoms of our own stellar +system. Or it may be that only a proportion of the stars are +%% -----File: 172.png---Folio 162------- +substantial bodies; the remainder are optical ghosts revisiting +their old haunts. It is, however, unlikely that the light rays +after their long journey would converge with the accuracy which +this theory would require. The minute deflections by the various +gravitational fields encountered on the way would turn them +aside, and the focus would be blurred. Moreover there is a +likelihood that the light would gradually be absorbed or +scattered by matter diffused in space, which is encountered on +the long journey. + +It is sometimes suggested that the return of the light-wave +to its starting point can most easily be regarded as due to the +force of gravitation, there being sufficient mass distributed +through the universe to control its path in a closed orbit. We +should have no objection in principle to this way of looking at +it; but we doubt whether it is correct in fact. It is quite possible +for light to return to its starting point in a world without +gravitation. We can roll flat space-time into a cylinder and join +the edges; its geometry will still be Euclidean and there will be +no gravitation; but a ray of light can go right round the cylinder +and return to the starting point in space. Similarly in Einstein's +more complex type of cylinder (three dimensions curved and +one dimension linear), it seems likely that the return of the +light is due as much to the connectivity of his space, as to +the non-Euclidean properties which express the gravitational +field. + +For Einstein's cylindrical world it is necessary to postulate +the existence of vast quantities of matter (not needed on de~Sitter's +theory) far in excess of what has been revealed by our +telescopes. This additional material may either be in the form +of distant stars and galaxies beyond our limits of vision, or it +may be uniformly spread through space and escape notice by +its low density. There is a definite relation between the average +density of matter and the radius of the world; the greater the +radius the smaller must be the average density. + +Two objections to this theory may be urged. In the first +place, absolute space and time are restored for phenomena on +a cosmical scale. The ghost of a star appears at the spot where +the star was a certain number of million years ago; and from +the ghost to the present position of the star is a definite distance% +%% -----File: 173.png---Folio 163------- +---the absolute motion of the star in the meantime\footnotemark. + \footnotetext{The ghost is not formed where the star is now. If two stars were near + together when the light left them their ghosts must be near together, although + the stars may now be widely separated.}% +The world +taken as a whole has one direction in which it is not curved; +that direction gives a kind of absolute time distinct from space. +\index{Absolute time, in cylindrical world}% +Relativity is reduced to a local phenomenon; and although this +is quite sufficient for the theory hitherto described, we are +inclined to look on the limitation rather grudgingly. But we +have already urged that the relativity theory is not concerned +to deny the possibility of an absolute time, but to deny that it +is concerned in any experimental knowledge yet found; and it +need not perturb us if the conception of absolute time turns up +in a new form in a theory of phenomena on a cosmical scale, +as to which no experimental knowledge is yet available. Just +as each limited observer has his own particular separation of +space and time, so a being coextensive with the world might +well have a special separation of space and time natural to him. +It is the time for this being that is here dignified by the title +``absolute.''% +\index{Time!absolute}% + +Secondly, the revised law of gravitation involves a new +constant which depends on the total amount of matter in the +world; or conversely the total amount of matter in the world +is determined by the law of gravitation. This seems very hard +to accept---at any rate without some plausible explanation of +how the adjustment is brought about. We can see that, the +constant in the law of gravitation being fixed, there may be +some upper limit to the amount of matter possible; as more +and more matter is added in the distant parts, space curves +round and ultimately closes; the process of adding more matter +must stop, because there is no more space, and we can only +return to the region already dealt with. But there seems nothing +to prevent a defect of matter, leaving space unclosed. Some +mechanism seems to be needed, whereby either gravitation +creates matter, or all the matter in the universe conspires to +define a law of gravitation. + +Although this appears to the writer rather bewildering, it is +welcomed by those philosophers who follow the lead of Mach. +For it leads to the result that the extension of space and time +%% -----File: 174.png---Folio 164------- +depends on the amount of matter in the world---partly by its +direct effect on the curvature and partly by its influence on the +constant of the law of gravitation. The more matter there is, +the more space is created to contain it, and if there were no +matter the world would shrink to a point. + +In the philosophy of Mach a world without \textit{matter} is unthinkable. +Matter in Mach's philosophy is not merely required as +a test body to display properties of something already there, +which have no physical meaning except in relation to matter; +it is an essential factor in causing those properties which it is +able to display. Inertia, for example, would not appear by the +insertion of one test body in the world; in some way the presence +of other matter is a necessary condition. It will be seen how +welcome to such a philosophy is the theory that space and the +inertial frame come into being with matter, and grow as it grows. +Since the laws of inertia are part of the law of gravitation, +Mach's philosophy was summed up---perhaps unconsciously---in +the profound saying ``If there were no matter in the universe, +the law of gravitation would fall to the ground.''% +\index{Inertia!Mach's views}% +\index{Mach's philosophy}% + +No doubt a world without matter, in which nothing could +ever happen, would be very uninteresting; and some might deny +its claim to be regarded as a world at all. But a world uniformly +filled with matter would be equally dull and unprofitable; so +there seems to be little object in denying the possibility of the +former and leaving the latter possible. + +The position can be summed up as follows:---in a space +without absolute features, an absolute rotation would be as +meaningless as an absolute translation; +\index{Absolute rotation}% +\index{Rotation, absolute}% +accordingly, the existence +of an experimentally determined quantity generally +identified with absolute rotation requires explanation. It was +remarked on \Pageref{41} that it would be difficult to devise a plan +of the world according to which uniform motion has no significance +but non-uniform motion is significant; but such a world has +been arrived at---a plenum, of which the absolute features are +intervals and geodesics. +\index{Aether!a plenum with geodesic structure}% +\index{Geodesic structure!absolute character of}% +\index{Structure, geodesic!absolute character of}% +In a limited region this plenum gives +a natural frame with respect to which an acceleration or rotation +(but not a velocity) capable of absolute definition can be +measured. In the case of rotation the local distortions of the +frame are of comparatively little account; and this explains +%% -----File: 175.png---Folio 165------- +why in practice rotation appears to have reference to some world-wide % [** PP: Hyphenated across a line in original] +inertial frame. + +Thus absolute rotation does not indicate any logical flaw in +the theory hitherto developed; and there is no need to accept +any modification of our views. Possibly there may be a still +wider relativity theory, in which our supposed plenum is to be +regarded as itself an abstraction of the relations of the matter +distributed throughout the world, and not existent apart from +such matter. This seems to exalt matter rather unnecessarily. +It may be true; but we feel no necessity for it, unless experiment +points that way. It is with some such underlying idea that +Einstein's cylindrical space-time was suggested, since this +cannot exist without matter to keep it stretched. Now we freely +admit that our assumption of perfect flatness in the remote +parts of space was arbitrary, and there is no justification for +insisting on it. A small curvature is possible both conceptually +and experimentally. The arguments on both sides have hitherto +been little more than prejudices, which would be dissipated by +any experimental or theoretical lead in one direction. Weyl's +theory of the electromagnetic field, discussed in the next +chapter, assigns a definite function to the curvature of space; +and this considerably alters the aspect of the question. We are +scarcely sufficiently advanced to offer a final opinion; but the +conception of cylindrical space-time seems to be favoured by +this new development of the theory. + +Some may be inclined to challenge the right of the Einstein +theory, at least as interpreted in this book, to be called a +relativity theory. Perhaps it has not all the characteristics +which have at one time or another been associated with that +name; but the reader, who has followed us so far, will see how +our search for an absolute world has been guided by a recognition +of the relativity of the measurements of physics. It may be +urged that our geodesics ought not to be regarded as fundamental; +a geodesic has no meaning in itself; what we are really concerned +with is the relation of a particle following a geodesic to all the +other matter of the world and the geodesic cannot be thought of +apart from such other matter. We would reply, ``Your particle +of matter is not fundamental; it has no meaning in itself; what +you are really concerned with is its `field'---the relation of the +%% -----File: 176.png---Folio 166------- +geodesics about it to the other geodesics in the world---and +matter cannot be thought of apart from its field.'' It is all +a tangle of relations; physical theory starts with the simplest +constituents, philosophical theory with the most familiar constituents. +They may reach the same goal; but their methods +are often incompatible. +%% -----File: 177.png---Folio 167------- + + +\Chapter{XI}{Electricity and Gravitation} + +% [** PP: Explicit formatting containg hard-coded dimensions] +\noindent\begin{minipage}{\textwidth} +\small +Thou shalt not have in thy bag divers weights, a great and a small. + +Thou shalt not have in thine house divers measures, a great and a small. + +\hangindent 2em +But thou shalt have a perfect and just weight, a perfect and just measure shalt thou have.\hfill\hbox{\textit{Book of Deuteronomy.}\hspace*{\QIndent}} +\end{minipage}% +\index{Electricity and gravitation}% + + +\First{The} relativity theory deduces from geometrical principles the +existence of gravitation and the laws of mechanics of matter. +Mechanics is derived from geometry, not by \textit{adding} arbitrary +hypotheses, but by \textit{removing} unnecessary assumptions, so that +a geometer like Riemann might almost have foreseen the more +important features of the actual world. But nature has in +reserve one great surprise---electricity.% +\index{Riemann}% + +Electrical phenomena are not in any way a misfit in the +relativity theory, and historically it is through them that it has +been developed. Yet we cannot rest satisfied until a deeper +unity between the gravitational and electrical properties of the +world is apparent. The electron, which seems to be the smallest +particle of matter, is a singularity in the gravitational field and +also a singularity in the electrical field. +\index{Electron!singularity in field}% +How can these two facts +be connected? The gravitational field is the expression of some +state of the world, which also manifests itself in the natural +geometry determined with measuring appliances; the electric +field must also express some state of the world, but we have not +as yet connected it with natural geometry. May there not still +be unnecessary assumptions to be removed, so that a yet more +comprehensive geometry can be found, in which gravitational +and electrical fields both have their place? + +There \textit{is} an arbitrary assumption in our geometry up to this +point, which it is desirable now to point out. We have based +everything on the ``interval,'' which, it has been said, is something +which all observers, whatever their motion or whatever +their mesh-system, can measure absolutely, agreeing on the +result. This assumes that they are provided with identical +standards of measurement---scales and clocks. But if $A$ is in +%% -----File: 178.png---Folio 168------- +motion relative to $B$ and wishes to hand his standards to $B$ to +check his measures, he must stop their motion; this means in +practice that he must bombard his standards with material +molecules until they come to rest. Is it fair to assume that no +alteration of the standard is caused by this process? Or if $A$ +measures time by the vibrations of a hydrogen atom, and space +by the wave-length of the vibration, still it is necessary to stop +the atom by a collision in which electrical forces are involved? % [** PP: Changed . to ?] + +The standard of length in physics is the length in the year +1799 of a bar deposited at Paris. +\index{Standard metre, comparison with}% +Obviously no interval is ever +compared directly with that length; there must be a continuous +chain of intermediate steps extending like a geodetic triangulation +through space and time, first along the past history of the +scale actually used, then through intermediate standards, and +finally along the history of the Paris metre itself. It may be +that these intermediate steps are of no importance---that the +same result is reached by whatever route we approach the +standard; but clearly we ought not to make that assumption +without due consideration. We ought to construct our geometry +in such a way as to show that there are intermediate steps, and +that the comparison of the interval with the ultimate standard +is not a kind of action at a distance. + +To compare intervals in different directions at a point in +space and time does not require this comparison with a distant +standard. The physicist's method of describing phenomena +near a point~$P$ is to lay down for comparison (1)~a mesh-system, +(2)~a unit of length (some kind of material standard), which can +also be used for measuring time, the velocity of light being unity. +With this system of reference he can measure in terms of his +unit small intervals $PP'$ running in any direction from~$P$, +summarising the results in the fundamental formula +\[ +ds^{2} = g_{11}\, dx_{1}^{2} + + g_{22}\, dx_{2}^{2} + \dotsb + + 2g_{12}\, dx_{1}dx_{2} + \dotsb . +\] +If now he wishes to measure intervals near a distant point~$Q$, he +must lay down a mesh-system and a unit of measure there. He +naturally tries to simplify matters by using what he would call +the \textit{same} unit of measure at $P$ and~$Q$, either by transporting a +material rod or some equivalent device. If it is immaterial by +what route the unit is carried from $P$ to~$Q$, and replicas of the +%% -----File: 179.png---Folio 169------- +unit carried by different routes all agree on arrival at~$Q$, this +method is at any rate explicit. The question whether the unit +at~$Q$ defined in this way is \textit{really} the same as that at $P$ is mere +metaphysics. But if the units carried by different routes disagree, +there is no unambiguous means of identifying a unit at +$Q$ with the unit at~$P$. Suppose $P$ is an event at Cambridge on +March~1, and $Q$ at London on May~1; we are contemplating the +possibility that there will be a difference in the results of measures +made with our standard in London on May~1, according as the +standard is taken up to London on March~1 and remains there, +or is left at Cambridge and taken up on May~1. This seems at +first very improbable; but our reasons for allowing for this +possibility will appear presently. If there is this ambiguity the +only possible course is to lay down (1)~a mesh-system filling all +the space and time considered, (2)~a definite unit of interval, or +gauge, \textit{at every point of space and time}. +\index{Gauge-system}% +The geometry of the +world referred to such a system will be more complicated than +that of Riemann hitherto used; and we shall see that it is +necessary to specify not only the 10~$g$'s, but four other functions +of position, which will be found to have an important physical +meaning.% +\index{Geometry!non-Riemannian}% +\index{Non-Riemannian geometry}% + +The observer will naturally simplify things by making the +units of gauge at different points as nearly as possible equal, +judged by ordinary comparisons. But the fact remains that, +when the comparison depends on the route taken, exact equality +is not definable; and we have therefore to admit that the \textit{exact} +standards are laid down at every point independently. + +It is the same problem over again as occurs in regard to +mesh-systems. We lay down particular rectangular axes near +a point~$P$; presently we make some observations near a distant +point~$Q$. To what coordinates shall the latter be referred? The +natural answer is that we must use the same coordinates as we +were using at~$P$. But, except in the particular case of flat space, +there is no means of defining exactly what coordinates at $Q$ are +the \textit{same} as those at~$P$. In many cases the ambiguity may be +too trifling to trouble us; but in exact work the only course is +to lay down a definite mesh-system extending throughout space, +the precise route of the partitions being necessarily arbitrary. +We now find that we have to add to this by placing in each +%% -----File: 180.png---Folio 170------- +mesh a gauge whose precise length must be arbitrary. Having +done this the next step is to make measurements of intervals +(using our gauges). This connects the absolute properties of the +world with our arbitrarily drawn mesh-system and gauge-system. +And so by measurement we determine the $g$'s and the +new additional quantities, which determine the geometry of our +chosen system of reference, and at the same time contain within +themselves the absolute geometry of the world---the kind of +space-time which exists in the field of our experiments. + +Having laid down a unit-gauge at every point, we can speak +quite definitely of the change in interval-length of a measuring-rod +moved from point to point, meaning, of course, the change +compared with the unit-gauges. Let us take a rod of interval-length +$l$ at~$P$, and move it successively through the displacements +$dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$; and let the result be to increase its length +in terms of the gauges by the amount~$\lambda l$. The change depends +as much on the difference of the gauges at the two points as +on the behaviour of the rod; but there is no possibility of +separating the two factors. It is clear that $\lambda$ will not depend +on~$l$, because the change of length must be proportional to +the original length---unless indeed our whole idea of measurement +by comparison with a gauge is wrong\footnotemark. + \footnotetext{We refuse to contemplate the idea that when the metre rod changes its + length to two metres, each centimetre of it changes to three centimetres.}% +Further it will +not depend on the direction of the rod either in its initial or +final positions because the interval-length is independent of +direction. (Of course, the space-length would change, but that +is already taken care of by the~$g$'s.) $\lambda$~can thus only depend on +the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$, and we may write it +\[ +\lambda + = \kappa_{1}\, dx_{1} + \kappa_{2}\, dx_{2} + + \kappa_{3}\, dx_{3} + \kappa_{4}\, dx_{4}, +\] +so long as the displacements are small. The coefficients $\kappa_{1}$, $\kappa_{2}$, +$\kappa_{3}$, $\kappa_{4}$ apply to the neighbourhood of~$P$, and will in general be +different in different parts of space. + +This indeed assumes that the result is independent of the +order of the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$---that is to say +that the ambiguity of the comparison by different routes disappears +in the limit when the whole route is sufficiently small. +It is parallel with our previous implicit assumption that although +the length of the track from a point~$P$ to a distant point~$Q$ +%% -----File: 181.png---Folio 171------- +depends on the route, and no definite meaning can be attached +to the interval between them without specifying a route, yet in +the limit there is a definite small interval between $P$ and~$Q$ when +they are sufficiently close together. + +To understand the meaning of these new coefficients $\kappa$ let us +briefly recapitulate what we understand by the~$g$'s. Primarily +they are quantities derived from experimental measurements of +intervals, and describe the geometry of the space and time +partitions which the observer has chosen. As consequential +properties they describe the field of force, gravitational, centrifugal, +etc., with which he perceives himself surrounded. They +relate to the particular mesh-system of the observer; and by +altering his mesh-system, he can alter their values, though not +entirely at will. From their values can be deduced intrinsic +properties of the world---the \textit{kind} of space-time in which the +phenomena occur. Further they satisfy a definite condition---the +law of gravitation---so that not all mathematically possible +space-times and not all arbitrary values of the $g$'s are such as +can occur in nature.% +\index{Real world of physics}% + +All this applies equally to the~$\kappa$'s, if we substitute gauge-system +for mesh-system, and some at present unknown force +for gravitation. They can theoretically be determined by +interval-measurement; but they will be more conspicuously +manifested to the observer through their consequential property +of describing some kind of field of force surrounding him. The +$\kappa$'s refer to the arbitrary gauge-system of the observer; but he +cannot by altering his gauge-system alter their values entirely +at will. Intrinsic properties of the world are contained in their +values, unaffected by any change of gauge-system. Further we +may expect that they will have to satisfy some law corresponding +to the law of gravitation, so that not all arbitrary values of the +$\kappa$'s are such as can occur in nature. + +It is evident that the $\kappa$'s must refer to some type of phenomenon +which has not hitherto appeared in our discussion; and +the obvious suggestion is that they refer to the electromagnetic +field. +\index{Fields of force!electromagnetic}% +This hypothesis is strengthened when we recall that the +electromagnetic field is, in fact, specified at every point by the +values of four quantities, viz.\ the three components of electromagnetic +vector potential, and the scalar potential of electrostatics. +%% -----File: 182.png---Folio 172------- +Surely it is more than a coincidence that the physicist +needs just four more quantities to specify the state of the world +at a point in space, and four more quantities are provided by +removing a rather illogical restriction on our system of geometry +of natural measures. + +[The general reader will perhaps pardon a few words addressed +especially to the mathematical physicist. Taking the ordinary +unaccelerated rectangular coordinates $x$, $y$, $z$, $t$, let us write +$F$, $G$, $H$, $-\Phi$ for $\kappa_{1}$, $\kappa_{2}$, $\kappa_{3}$, $\kappa_{4}$, then +\[ +\frac{dl}{l} = \lambda = F\, dx + G\, dy + H\, dx - \Phi\, dt. +\] +From which, by integration, +\[ +\log l + \text{const.} = \int(F\, dx + G\, dy + H\, dz - \Phi\, dt). +\] + +The length $l$ will be independent of the route taken if +\[ +F\, dx + G\, dy + H\, dz - \Phi\, dt +\] +is a perfect differential. The condition for this is +\index{Electromagnetic potentials and forces}% +\index{Force!electromagnetic}% +\index{Potentials, electromagnetic}% +\begin{align*} +\frac{\partial H}{\partial y} - \frac{\partial G}{\partial z} &= 0, & +\frac{\partial F}{\partial z} - \frac{\partial H}{\partial x} &= 0, & +\frac{\partial G}{\partial x} - \frac{\partial F}{\partial y} &= 0, \\ +% +-\frac{\partial \Phi}{\partial x} - \frac{\partial F}{\partial t} &= 0, & +-\frac{\partial \Phi}{\partial y} - \frac{\partial G}{\partial t} &= 0, & +-\frac{\partial \Phi}{\partial z} - \frac{\partial H}{\partial t} &= 0. +\end{align*} +If $F$, $G$, $H$, $\Phi$ are the potentials of electromagnetic theory, these +are precisely the expressions for the three components of +magnetic force and the three components of electric force, given +in the text-books. Thus the condition that distant intervals can +be compared directly without specifying a particular route of +comparison is that the electric and magnetic forces are zero in +the intervening space and time. + +It may be noted that, even when the coordinate system has +been defined, the electromagnetic potentials are not unique in +value; but arbitrary additions can be made provided these +additions form a perfect differential. It is just this flexibility +which in our geometrical theory appears in the form of the +arbitrary choice of gauge-system. The electromagnetic \textit{forces} +on the other hand are independent of the gauge-system, which +is eliminated by ``curling.''] + +It thus appears that the four new quantities appearing in our +extended geometry may actually be the four potentials of +%% -----File: 183.png---Folio 173------- +electromagnetic theory; and further, when there is no electromagnetic +field our previous geometry is valid. But in the more +general case we have to adopt the more general geometry in +which there appear fourteen coefficients, ten describing the +gravitational and four the electrical conditions of the world. + +We ought now to seek the law of the electromagnetic field +on the same lines as we sought for the law of gravitation, laying +down the condition that it must be independent of mesh-system +and gauge-system since it seeks to limit the possible kinds of +world which can exist in nature. Happily this presents no +difficulty, because the law expressed by Maxwell's equations, +and universally adopted, fulfils the conditions. There is no +need to modify it fundamentally as we modified the law of +gravitation. We do, however, generalise it so that it applies +when a gravitational field is present at the same time---not +merely, as given by Maxwell, for flat space-time. The deflection +of electromagnetic waves (light) by a gravitational field is duly +contained in this generalised law. + +Strictly speaking the laws of gravitation and of the electromagnetic +field are not two laws but one law, as the geometry +of the $g$'s and the $\kappa$'s is one geometry. Although it is often +convenient to separate them, they are really parts of the general +condition limiting the possible kinds of metric that can occur in +empty space. + +It will be remembered that the four-fold arbitrariness of our +mesh-system involved four identities, which were found to +express the conservation of energy and momentum. In the new +geometry there is a fifth arbitrariness, namely that of the selected +gauge-system. This must also give rise to an identity; and it is +found that the new identity expresses the law of conservation of +electric charge.% +\index{Conservation!of electric charge}% + +A grasp of the new geometry may perhaps be assisted by a +further comparison. Suppose an observer has laid down a line +of a certain length and in a certain direction at a point~$P$, and +he wishes to lay down an exactly similar line at a distant point~$Q$. +\index{Integrability of length and direction} % [** PP: Index reads p. 174] +If he is in flat space there will be no difficulty; he will have +to proceed by steps, a kind of triangulation, but the route chosen +is of no importance. We know definitely that there is just one +direction at $Q$ parallel to the original direction at~$P$; and it is +%% -----File: 184.png---Folio 174------- +in ordinary geometry supposed that the length is equally +determinate. But if space is not flat the case is different. +Imagine a two-dimensional observer confined to the curved +surface of the earth trying to perform this task. As he does not +appreciate the third dimension he will not immediately perceive +the impossibility; but he will find that the direction which he has +transferred to $Q$ differs according to the route chosen. Or if he +went round a complete circuit he would find on arriving back +at $P$ that the direction he had so carefully tried to preserve on +the journey did not agree with that originally drawn\footnotemark. + \footnotetext{It might be thought that if the observer preserved mentally the original + direction in three-dimensional space, and obtained the direction at any point + in the two-dimensional space by projecting it, there would be no ambiguity. + But the three-dimensional space in which a curved two-dimensional space is + conceived to exist is quite arbitrary. A two-dimensional observer cannot + ascertain by any observation whether he is on a plane or a cylinder, a sphere + or any other convex surface of the same total curvature.}% +We +describe this by saying that in curved space, direction is not +integrable; and it is this non-integrability of direction which +characterises the gravitational field. In the case considered the +length would be preserved throughout the circuit; but it is +possible to conceive a more general kind of space in which the +length which it was attempted to preserve throughout the +circuit, as well as the direction, disagreed on return to the starting +point with that originally drawn. In that case length is not +integrable; and the non-integrability of length characterises the +electromagnetic field. Length associated with direction is called +a vector; and the combined gravitational and electric field +describe that influence of the world on our measurements by +which a vector carried by physical measurement round a closed +circuit changes insensibly into a different vector. + +The welding together of electricity and gravitation into one +geometry is the work of Prof.\ H.~Weyl, first published in 1918\footnote% +{Appendix, \Noteref{15}.}. +\Pagelabel{note15}% +\index{Vector, non-integrable on Weyl's theory}% +\index{Weyl}% +It appears to the writer to carry conviction, although up to the +present no experimental test has been proposed. It need scarcely +be said that the inconsistency of length for an ordinary circuit +would be extremely minute\footnote{I do not think that any numerical estimate has been made.}, and the ordinary manifestations +of the electromagnetic field are the consequential results of +%% -----File: 185.png---Folio 175------- +changes which would be imperceptible to direct measurement. +It will be remembered that the gravitational field is likewise +perceived by the consequential effects, and not by direct interval-measurement. + +But the theory does appear to require that, for example, the +time of vibration of an atom is not quite independent of its +previous history. It may be assumed that the previous histories +of terrestrial atoms are so much alike that there are no significant +differences in their periods. The possibility that the systematic +difference of history of solar and terrestrial atoms may have an +effect on the expected shift of the spectral lines on the sun has +already been alluded to. It seems doubtful, however, whether +the effect could attain the necessary magnitude. + +It may seem difficult to identify these abstract geometrical +qualities of the world with the physical forces of electricity and +magnetism. How, for instance, can the change in the length of +a rod taken round a circuit in space and time be responsible for +the sensations of an electric shock? The geometrical potentials +($\kappa$) obey the recognised laws of electromagnetic potentials, and +each entity in the physical theory---charge, electric force, +magnetic element, light, etc.---has its exact analogue in the +geometrical theory; but is this formal correspondence a sufficient +ground for identification? The doubt which arises in our minds +is due to a failure to recognise the formalism of all physical +knowledge. +\index{Formalism of knowledge}% +The suggestion ``This is not the thing I am speaking +of, though it behaves exactly like it in all respects'' carries no +physical meaning. Anything which behaves exactly like +electricity must manifest itself to us as electricity. Distinction +of form is the only distinction that physics can recognise; and +distinction of individuality, if it has any meaning at all, has no +bearing on physical manifestations. + +We can only explore the world with apparatus, which is itself +part of the world. Our idealised apparatus is reduced to a few +simple types---a neutral particle, a charged particle, a rigid +scale, etc. The absolute constituents of the world are related in +various ways, which we have studied, to the indications of these +test-bodies. The main features of the absolute world are so +simple that there is a redundancy of apparatus at our disposal; +and probably all that there is to be known could theoretically +%% -----File: 186.png---Folio 176------- +be found out by exploration with an uncharged particle. Actually +we prefer to look at the world as revealed by exploration with +scales and clocks---the former for measuring so-called imaginary +intervals, and the latter for real intervals; this gives us a unified +geometrical conception of the world. +\index{Geometrical conception of the world}% +Presumably, we could obtain +a unified mechanical conception by taking the moving uncharged +particle as standard indicator; or a unified electrical conception +by taking the charged particle. For particular purposes one +test-body is generally better adapted than others. The gravitational +field is more sensitively explored with a moving particle +than a scale. Although the electrical field can theoretically be +explored by the change of length of a scale taken round a circuit, +a far more sensitive way is to use a little bit of the scale---an +electron. And in general for practical efficiency, we do not use +any simple type of apparatus, but a complicated construction +built up with a view to a particular experiment. The reason for +emphasising the theoretical interchangeability of test-bodies is +that it brings out the unity and simplicity of the world; and for +that reason there is an importance in characterising the electromagnetic +condition of the world by reference to the indications +of a scale and clock, however inappropriate they may be as +practical test-bodies. + +Weyl's theory opens up interesting avenues for development. +The details of the further steps involve difficult mathematics; +but a general outline is possible. As on Einstein's more limited +theory there is at any point an important property of the world +called the curvature; but on the new theory it is not an absolute +quantity in the strictest sense of the word. +\index{Curvature!on Weyl's theory}% +It is independent of +the observer's mesh-system, but it depends on his gauge. It is +obvious that the number expressing the radius of curvature of +the world at a point must depend on the unit of length; so we +cannot say that the curvatures at two points are absolutely +equal, because they depend on the gauges assigned at the two +points. Conversely the radius of curvature of the world provides +a natural and absolute gauge at every point; +\index{Natural gauge}% +and it will presumably +introduce the greatest possible symmetry into our laws +if the observer chooses this, or some definite fraction of it, as +his gauge. He, so to speak, forces the world to be spherical by +adopting at every point a unit of length which will make it so. +%% -----File: 187.png---Folio 177------- +Actual rods as they are moved about change their lengths compared +with this absolute unit according to the route taken, and +the differences correspond to the electromagnetic field. Einstein's +curved space appears in a perfectly natural manner in this +theory; no part of space-time is flat, even in the absence of +ordinary matter, for that would mean infinite radius of curvature, +and there would be no natural gauge to determine, for +example, the dimensions of an electron---the electron could not +know how large it ought to be, unless it had something to +measure itself against.% +\index{Electron!dimensions of}% +\index{Gauge!provided by radius of space}% + +The connection between the form of the law of gravitation +and the total amount of matter in the world now appears less +mysterious. The curvature of space indirectly provides the +gauge which we use for measuring the amount of matter in the +world. + +Since the curvature is not independent of the gauge, Weyl +does not identify it with the most fundamental quantity in +nature. There is, however, a slightly more complicated invariant +which is a pure number, and this is taken to be Action\footnotemark. +\footnotetext{Appendix, \Noteref{16}.}% +\Pagelabel{note16}% +\index{Action!on Weyl's theory}% +\index{Atomicity!of Action}% +We +can thus mark out a definite volume of space and time, and +say that the action within it is~$5$, without troubling to define +coordinates or the unit of measure! It might be expected that +the action represented by the number~$1$ would have specially +interesting properties; it might, for instance, be an atom of +action and indivisible. Experiment has isolated what are believed +to be units of action, which at least in many phenomena +behave as indivisible atoms called quanta; +\index{Quanta}% +but the theory, as +at present developed, does not permit us to represent the +quantum of action by the number~$1$. The quantum is a very +minute fraction of the absolute unit. + +When we come across a pure number having some absolute +significance in the world it is natural to speculate on its possible +interpretation. It might represent a number of discrete entities; +but in that case it must necessarily be an integer, and it seems +clear that action can have fractional values. An angle is commonly +represented as a pure number, but it has not really this +character; an angle can only be measured in terms of a unit of +angle, just as a length is measured in terms of a unit of length. +%% -----File: 188.png---Folio 178------- +I can only think of one interpretation of a fractional number +which can have an absolute significance, though doubtless there +are others. The number may represent the \textit{probability} of something, +or some function of a probability. +\index{Probability, a pure number}% +The precise function +is easily found. We combine probabilities by multiplying, but +we combine the actions in two regions by adding; hence the +logarithm of a probability is indicated. Further, since the +logarithm of a probability is necessarily negative, we may +identify action provisionally with minus the logarithm of +the statistical probability of the state of the world which +exists. + +The suggestion is particularly attractive because the Principle +of Least Action now becomes the Principle of Greatest Probability. +\index{Action, Principle of Least}% +\index{Principle of Least Action}% +The law of nature is that the actual state of the world is +that which is statistically most probable. + +Weyl's theory also shows that the mass of a portion of matter +is necessarily positive; on the original theory no adequate reason +is given why negative matter should not exist. It is further +claimed that the theory shows to some extent why the world +is four-dimensional. To the mathematician it seems so easy to +generalise geometry to $n$~dimensions, that we naturally expect +a world of four dimensions to have an analogue in five dimensions. +Apparently this is not the case, and there are some essential +properties, without which it could scarcely be a world, which +exist only for four dimensions. Perhaps this may be compared +with the well-known difficulty of generalising the idea of a knot; +a knot can exist in space of any odd number of dimensions, but +not in space of an even number. + +Finally the theory suggests a mode of attacking the problem +of how the electric charge of an electron is held together; at +least it gives an explanation of why the gravitational force is so +extremely weak compared with the electric force. It will be +remembered that associated with the mass of the sun is a certain +length, called the gravitational mass, which is equal to $1.5$~kilometres. +In the same way the gravitational mass or radius of an +electron is $7·10^{-56}$~cms. +\index{Electron, gravitational mass of}% +Its electrical properties are similarly +associated with a length $2·10^{-13}$~cms., which is called the electrical +radius. The latter is generally supposed to correspond to the +electron's actual dimensions. The theory suggests that the ratio +%% -----File: 189.png---Folio 179------- +of the gravitational to the electrical radius, $3·10^{42}$, ought to be +of the same order as the ratio of the latter to the radius of +curvature of the world. This would require the radius of space +to be of the order $6·10^{29}$~cms., or $2·10^{11}$ parsecs., which though +somewhat larger than the provisional estimates made by de~Sitter, +is within the realm of possibility.% +\index{de Sitter}% + +%% -----File: 190.png---Folio 180------- + + +\Chapter{XII}{On the Nature of Things} + +% [** PP: Explicit formatting containing hard-coded dimensions]] +{\small% +\settowidth{\TmpLen}{\textit{Hippolyta}:\ } +\hspace*{\QIndent}\makebox[\TmpLen][l]{\textit{Hippolyta}.} This is the silliest stuff that ever I heard. \\ +\hspace*{\QIndent}\makebox[\TmpLen][l]{\textit{Theseus}.} +\begin{minipage}[t]{3.75in} +\hangindent 2em The best in this kind are but shadows; and the worst are +no worse, if imagination amend them. +\end{minipage} \\[1ex] +\null\hfill\textit{A Midsummer-Night's Dream.}}% End of \small +\medskip + +\First{The} constructive results of the theory of relativity are based +on two principles which have been enunciated---the restricted +principle of relativity, and the principle of equivalence. These +may be summed up in the statement that uniform motion and +fields of force are purely relative. In their more formal enunciations +they are experimental generalisations, which can be +admitted or denied; if admitted, all the observational results +obtained by us can be deduced mathematically without any +reference to the views of space, time, or force, described in this +book. In many respects this is the most attractive aspect of +Einstein's work; it deduces a great number of remarkable +phenomena solely from two general principles, aided by a +mathematical calculus of great power; and it leaves aside as +irrelevant all questions of mechanism. But this mode of development +of the theory cannot be described in a non-technical book. + +To avoid mathematical analysis we have had to resort to +geometrical illustrations, which run parallel with the mathematical +development and enable its processes to be understood +to some extent. The question arises, are these merely illustrations +of the mathematical argument, or illustrations of the actual +processes of nature. No doubt the safest course is to avoid the +thorny questions raised by the latter suggestion, and to say +that it is quite sufficient that the illustrations should correctly +replace the mathematical argument. But I think that this +would give a misleading view of what the theory of relativity +has accomplished in science. + +The physicist, so long as he thinks as a physicist, has a definite +belief in a real world outside him. For instance, he believes that +atoms and molecules really exist; they are not mere inventions +%% -----File: 191.png---Folio 181------- +that enable him to grasp certain laws of chemical combination. +That suggestion might have sufficed in the early days of the +atomic theory; but now the existence of atoms as entities in the +real world of physics is fully demonstrated. This confident +assertion is not inconsistent with philosophic doubts as to the +meaning of ultimate reality. + +When therefore we are asked whether the four-dimensional +world may not be regarded merely as an illustration of mathematical +processes, we must bear in mind that our questioner has +probably an ulterior motive. +\index{Four-dimensional space-time!reality of}% +He has already a belief in a real +world of three Euclidean dimensions, and he hopes to be allowed +to continue in this belief undisturbed. In that case our answer +must be definite; the real three-dimensional world is obsolete, +and must be replaced by the four-dimensional space-time with +non-Euclidean properties. In this book we have sometimes +employed illustrations which certainly do not correspond to any +physical reality---imaginary time, and an unperceived fifth +dimension. +\index{Imaginary time}% +But the four-dimensional world is no mere illustration; +it is the real world of physics, arrived at in the recognised +way by which physics has always (rightly or wrongly) sought for +reality. + +I hold a certain object before me, and see an outline of the +figure of Britannia; another observer on the other side sees a +picture of a monarch; a third observer sees only a thin rectangle. +Am I to say that the figure of Britannia is the real object; and +that the crude impressions of the other observers must be +corrected to make allowance for their positions? All the appearances +can be accounted for if we are all looking at a three-dimensional +object---a penny---and no reasonable person can +doubt that the penny is the corresponding physical reality. +Similarly, an observer on the earth sees and measures an oblong +block; an observer on another star contemplating the same +object finds it to be a cube. Shall we say that the oblong block +is the real thing, and that the other observer must correct his +measures to make allowance for his motion? All the appearances +are accounted for if the real object is four-dimensional, and the +observers are merely measuring different three-dimensional +appearances or sections; and it seems impossible to doubt that +this is the true explanation. He who doubts the reality of the +%% -----File: 192.png---Folio 182------- +four-dimensional world (for logical, as distinct from experimental, +reasons) can only be compared to a man who doubts the +reality of the penny, and prefers to regard one of its innumerable +appearances as the real object. + +Physical reality is the synthesis of all possible physical aspects +of nature. +\index{Synthesis of appearances}% +An illustration may be taken from the phenomena of +radiant-energy, or light. In a very large number of phenomena +the light coming from an atom appears to be a series of spreading +waves, extending so as to be capable of filling the largest +telescope yet made. In many other phenomena the light coming +from an atom appears to remain a minute bundle of energy, all +of which can enter and blow up a single atom. +\index{Quanta}% +There may be +some illusion in these experimental deductions; but if not, it +must be admitted that the physical reality corresponding to +light must be some synthesis comprehending both these appearances. +How to make this synthesis has hitherto baffled conception. +But the lesson is that a vast number of appearances +may be combined into one consistent whole---perhaps all +appearances that are directly perceived by terrestrial observers---and +yet the result may still be only an appearance. Reality +is only obtained when all conceivable points of view have been +combined. + +That is why it has been necessary to give up the reality of +the everyday world of three dimensions. Until recently it comprised +all the possible appearances that had been considered. +But now it has been discovered that there are new points of +view with new appearances; and the reality must contain them +all. It is by bringing in all these new points of view that we +have been able to learn the nature of the real world of +physics. + +Let us briefly recapitulate the steps of our synthesis. We +found one step already accomplished. The immediate perception +of the world with one eye is a two-dimensional appearance. But +we have two eyes, and these combine the appearances of the +world as seen from two positions; in some mysterious way the +brain makes the synthesis by suggesting solid relief, and we +obtain the familiar appearance of a three-dimensional world. +This suffices for all possible positions of the observer within the +parts of space hitherto explored. The next step was to combine +%% -----File: 193.png---Folio 183------- +the appearances for all possible states of uniform motion of the +observer. The result was to add another dimension to the world, +making it four-dimensional. Next the synthesis was extended +to include all possible variable motions of the observer. The +process of adding dimensions stopped, but the world became +non-Euclidean; a new geometry called Riemannian geometry +was adopted. Finally the points of view of observers varying +in size in any way were added; and the result was to replace +the Riemannian geometry by a still more general geometry +described in the last chapter. + +The search for physical reality is not necessarily utilitarian, +but it has been by no means profitless. As the geometry became +more complex, the physics became simpler; until finally it +almost appears that the physics has been absorbed into the +geometry. We did not consciously set out to construct a +geometrical theory of the world; we were seeking physical +reality by approved methods, and this is what has happened.% +\index{Geometrical conception of the world}% + +Is the point now reached the ultimate goal? Have the points +of view of all conceivable observers now been absorbed? We do +not assert that they have. But it seems as though a definite +task has been rounded off, and a natural halting-place reached. +So far as we know, the different possible impersonal points of +view have been exhausted---those for which the observer can be +regarded as a mechanical automaton, and can be replaced by +scientific measuring-appliances. A variety of more personal +points of view may indeed be needed for an ultimate reality; +but they can scarcely be incorporated in a real world of physics. +There is thus justification for stopping at this point but not for +stopping earlier. + +It may be asked whether it is necessary to take into account +all conceivable observers, many of whom, we suspect, have no +existence. Is not the \textit{real} world that which comprehends the +appearances to all \textit{real} observers? Whether or not it is a tenable +hypothesis that that which no one observes does not exist, +science uncompromisingly rejects it. If we deny the rights of +extra-terrestrial observers, we must take the side of the Inquisition +against Galileo. And if extra-terrestrial observers are +admitted, the other observers, whose results are here combined, +cannot be excluded. + +%% -----File: 194.png---Folio 184------- + +Our inquiry into the nature of things is subject to certain +limitations which it is important to realise. The best comparison +I can offer is with a future antiquarian investigation, which may +be dated about the year 5000~\textsc{a.d.} An interesting find has been +made relating to a vanished civilisation which flourished about +the twentieth century, namely a volume containing a large +number of games of chess, written out in the obscure symbolism +usually adopted for that purpose. +\index{Chess, analogy of}% +The antiquarians, to whom +the game was hitherto unknown, manage to discover certain +uniformities; and by long research they at last succeed in +establishing beyond doubt the nature of the moves and rules of +the game. But it is obvious that no amount of study of the +volume will reveal the true nature either of the participants in +the game---the chessmen---or the field of the game---the chess-board. +With regard to the former, all that is possible is to give +arbitrary names distinguishing the chessmen according to their +properties; but with regard to the chess-board something more +can be stated. The material of the board is unknown, so too +are the shapes of the meshes---whether squares or diamonds; +but it is ascertainable that the different points of the field are +connected with one another by relations of two-dimensional +order, and a large number of hypothetical types of chess-board +satisfying these relations of order can be constructed. In +spite of these gaps in their knowledge, our antiquarians may +fairly claim that they thoroughly understand the game of +chess.% +\index{Ordering of events in external world}% + +The application of this analogy is as follows. The recorded +games are our physical experiments. The rules of the game, +ascertained by study of them, are the laws of physics. The +hypothetical chess-board of 64~squares is the space and time of +some particular observer or player; whilst the more general +relations of two-fold order, are the absolute relations of order +in space-time which we have been studying. The chessmen are +the entities of physics---electrons, particles, or point-events; and +the range of movement may perhaps be compared to the fields +of relation radiating from them---electric and gravitational +fields, or intervals. By no amount of study of the experiments +can the absolute nature or appearance of these participants be +deduced; nor is this knowledge relevant, for without it we may +%% -----File: 195.png---Folio 185------- +yet learn ``the game'' in all its intricacy. Our knowledge of the +nature of things must be like the antiquarians' knowledge of +the nature of chessmen, viz.\ their nature as pawns and pieces +in the game, not as carved shapes of wood. In the latter aspect +they may have relations and significance transcending anything +dreamt of in physics. + +It is believed that the familiar things of experience are very +complex; and the scientific method is to analyse them into +simpler elements. Theories and laws of behaviour of these +simpler constituents are studied; and from these it becomes +possible to predict and explain phenomena. It seems a natural +procedure to explain the complex in terms of the simple, but +it carries with it the necessity of explaining the familiar in terms +of the unfamiliar. + +There are thus two reasons why the ultimate constituents of +the real world must be of an unfamiliar nature. Firstly, all +familiar objects are of a much too complex character. Secondly, +familiar objects belong not to the real world of physics, but to +a much earlier stage in the synthesis of appearances. The +ultimate elements in a theory of the world must be of a nature +impossible to define in terms recognisable to the mind. + +The fact that he has to deal with entities of unknown nature +presents no difficulty to the mathematician. As the mathematician +in the Prologue explained, he is never so happy as +when he does not know what he is talking about. But we ourselves +cannot take any interest in the chain of reasoning he is +producing, unless we can give it some meaning---a meaning, +which we find by experiment, it will bear. We have to be in +a position to make a sort of running comment on his work. +At first his symbols bring no picture of anything before our +eyes, and we watch in silence. Presently we can say ``Now he +is talking about a particle of matter''\ldots ``Now he is talking +about another particle''\ldots ``Now he is saying where they will +be at a certain time of day''\ldots ``Now he says that they will be +in the same spot at a certain time.'' We watch to see.---``Yes. +The two particles have collided. For once he is speaking about +something familiar, and speaking the truth, although, of course, +he does not know it.'' Evidently his chain of symbols can be +interpreted as describing what occurs in the world; we need not, +%% -----File: 196.png---Folio 186------- +and do not, form any idea of the meaning of each individual +symbol; it is only certain elaborate combinations of them that +we recognise. + +Thus, although the elementary concepts of the theory are of +undefined nature, at some later stage we must link the derivative +concepts to the familiar objects of experience. + +We shall now collect the results arrived at in the previous +chapters by successive steps, and set the theory out in more +logical order. The extension in \Chapref{XI} will not be considered +here, partly because it would increase the difficulty of grasping +the main ideas, partly because it is less certainly established. + +In the relativity theory of nature the most elementary concept +is the \textit{point-event}. +\index{Event, definition of}% +\index{Point-event}% +In ordinary language a point-event is an +instant of time at a point of space; but this is only one aspect +of the point-event, and it must not be taken as a definition. +Time and space---the familiar terms---are derived concepts to +be introduced much later in our theory. The first simple concepts +are necessarily undefinable, and their nature is beyond +human understanding. The aggregate of all the point-events is +called the \textit{world}. +\index{World}% +It is postulated that the world is four-dimensional, +which means that a particular point-event has to be +specified by the values of four variables or coordinates, though +there is entire freedom as to the way in which these four identifying +numbers are to be assigned. + +The meaning of the statement that the world is four-dimensional +is not so clear as it appears at first. An aggregate of a +large number of things has in itself no particular number of +dimensions. Consider, for example, the words on this page. To +a casual glance they form a two-dimensional distribution; but +they were written in the hope that the reader would regard +them as a one-dimensional distribution. In order to define the +number of dimensions we have to postulate some ordering +relation; and the result depends entirely on what this ordering +relation is---whether the words are ordered according to sense +or to position on the page. Thus the statement that the world +is four-dimensional contains an implicit reference to some ordering +relation. +\index{Four-dimensional order}% +\index{Order and dimensions}% +This relation appears to be the \textit{interval}, though I am +not sure whether that alone suffices without some relation +corresponding to \textit{proximity}. It must be remembered that if the +%% -----File: 197.png---Folio 187------- +interval $s$ between two events is small, the events are not +necessarily near together in the ordinary sense.% +\index{Time!``standing still''}% + +Between any two neighbouring point-events there exists a +certain relation known as the \textit{interval} between them. +\index{Interval}% +The +relation is a quantitative one which can be measured on a +definite scale of numerical values\footnotemark. +\index{Space-like intervals}% +\index{Time-like intervals}% + \footnotetext{There is also a qualitative distinction into two kinds, ultimately identified + as time-like and space-like, which for mathematical treatment are distinguished + by real and imaginary numbers.}% +But the term ``interval'' +is not to be taken as a guide to the real nature of the relation, +which is altogether beyond our conception. +\index{Imaginary intervals}% +Its geometrical +properties, which we have dwelt on so often in the previous +chapters, can only represent one aspect of the relation. It may +have other aspects associated with features of the world outside +the scope of physics. But in physics we are concerned not with +the nature of the relation but with the number assigned to +express its intensity; and this suggests a graphical representation, +leading to a geometrical theory of the world of physics.% +\index{World}% + +What we have here called the \textit{world} might perhaps have been +legitimately called the \textit{aether}; at least it is the universal substratum +of things which the relativity theory gives us in place +of the aether.% +\index{Aether!identified with the ``world''}% + +We have seen that the number expressing the intensity of +the interval-relation can be measured practically with scales and +clocks. Now, I think it is improbable that our coarse measures +can really get hold of the individual intervals of point-events; +our measures are not sufficiently microscopic for that. The +interval which has appeared in our analysis must be a \textit{macroscopic} +value; and the potentials and kinds of space deduced from +it are averaged properties of regions, perhaps small in comparison +even with the electron, but containing vast numbers of the +primitive intervals. We shall therefore pass at once to the +consideration of the macroscopic interval; but we shall not +forestall later results by assuming that it is measurable with +a scale and clock. That property must be introduced in its +logical order.% +\index{Macroscopic!interval}% + +Consider a small portion of the world. It consists of a large +(possibly infinite) number of point-events between every two of +which an interval exists. If we are given the intervals between +%% -----File: 198.png---Folio 188------- +a point $A$ and a sufficient number of other points, and also +between $B$ and the same points, can we calculate what will be +the interval between $A$ and~$B$? In ordinary geometry this +would be possible; but, since in the present case we know nothing +of the relation signified by the word interval, it is impossible to +predict any law \textit{a~priori}. But we have found in our previous +work that there is such a rule, expressed by the formula +\[ +ds^{2} + = g_{11}\, dx_{1}^{2} + g_{22}\, dx_{2}^{2} + \dotsb + + 2g_{12}\, dx_{1} dx_{2} + \dotsb. +\] +This means that, having assigned our identification numbers +% [** PP: overfull at 5in; set as individual entries] +$(x_{1}$, $x_{2}$, $x_{3}$, $x_{4})$ to the point-events, we have only to measure +ten different intervals to enable us to determine the ten coefficients, +$g_{11}$, etc., which in a small region may be considered to be +constants; then all other intervals in this region can be predicted +from the formula. For any other region we must make fresh +measures, and determine the coefficients for a new formula. + +I think it is unlikely that the \textit{individual} interval-relations of +point-events follow any such definite rule. A microscopic +examination would probably show them as quite arbitrary, the +relations of so-called intermediate points being not necessarily +intermediate. Perhaps even the primitive interval is not +quantitative, but simply $1$ for certain pairs of point-events and +$0$ for others. The formula given is just an average summary +which suffices for our coarse methods of investigation, and holds +true only statistically. Just as statistical averages of one community +may differ from those of another, so may this statistical +formula for one region of the world differ from that of another. +This is the starting point of the infinite variety of nature. + +Perhaps an example may make this clearer. Compare the +point-events to persons, and the intervals to the degree of +acquaintance between them. There is no means of forecasting +the degree of acquaintance between $A$ and $B$ from a knowledge +of the familiarity of both with $C$, $D$, $E$,~etc. But a statistician +may compute in any community a kind of average rule. In +most cases if $A$ and $B$ both know~$C$, it slightly increases the +probability of their knowing one another. A community in +which this correlation was very high would be described as +\textit{cliquish}. +\index{Cliquishness}% +There may be differences among communities in this +respect, corresponding to their degree of cliquishness; and so +%% -----File: 199.png---Folio 189------- +the statistical laws may be the means of expressing intrinsic +differences in communities. + +Now comes the difficulty which is by this time familiar to us. +The ten $g$'s are concerned, not only with intrinsic properties of +the world, but with our arbitrary system of identification-numbers +for the point-events; or, as we have previously expressed +it, they describe not only the kind of space-time, but +the nature of the arbitrary mesh-system that is used. Mathematics +shows the way of steering through this difficulty by fixing +attention on expressions called tensors, of which $B^{\rho}_{\mu\nu\sigma}$ and $G_{\mu\nu}$ +are examples.% +\index{Tensors}% + +A tensor does not express explicitly the measure of an intrinsic +quality of the world, for some kind of mesh-system is essential +to the idea of measurement of a property, except in certain very +special cases where the property is expressed by a single number +termed an invariant, e.g.\ the interval, or the total curvature. +But to state that a tensor vanishes, or that it is equal to another +tensor in the same region, is a statement of intrinsic property, +quite independent of the mesh-system chosen. Thus by keeping +entirely to tensors, we contrive that there shall be behind our +formulae an undercurrent of information having reference to the +intrinsic state of the world. + +In this way we have found two absolute formulae, which +appear to be fully confirmed by observation, namely +\begin{DPalign*} +\lintertext{in empty space,} +G_{\mu\nu} &= 0, \\ +\lintertext{in space containing matter,} +G_{\mu\nu} &= K_{\mu\nu}, +\end{DPalign*} +where $K_{\mu\nu}$ contains only physical quantities which are perfectly +familiar to us, viz.\ the density and state of motion of the matter +in the region. + +I think the usual view of these equations would be that the +first expresses some law existing in the world, so that the point-events +by natural necessity tend to arrange their relations in +conformity with this equation. But when matter intrudes it +causes a disturbance or strain of the natural linkages; and a +rearrangement takes place to the extent indicated by the second +equation. + +But let us examine more closely what the equation $G_{\mu\nu} = 0$ +tells us. We have been giving the mathematician a free hand +%% -----File: 200.png---Folio 190------- +with his indefinable intervals and point-events. He has arrived +at the quantity~$G_{\mu\nu}$; but as yet this means to us---absolutely +nothing. The pure mathematician left to himself never ``deviates +into sense.'' His work can never relate to the familiar things +around us, unless we boldly lay hold of some of his symbols and +\textit{give} them an intelligible meaning---tentatively at first, and then +definitely as we find that they satisfy all experimental knowledge. +We have decided that in empty space $G_{\mu\nu}$~vanishes. Here +is our opportunity. In default of any other suggestion as to +what the vanishing of $G_{\mu\nu}$ might mean, let us say that the +vanishing of $G_{\mu\nu}$ \textit{means} emptiness; so that $G_{\mu\nu}$, if it does not +vanish, is a condition of the world which distinguishes space +said to be occupied from space said to be empty. Hitherto $G_{\mu\nu}$ +was merely a formal outline to be filled with some undefined +contents; we are as far as ever from being able to explain what +those contents are; but we have now given a recognisable +meaning to the completed picture, so that we shall know it +when we come across it in the familiar world of experience. + +The two equations are accordingly merely definitions---definitions +of the way in which certain states of the world +(described in terms of the indefinables) impress themselves on +our perceptions. When we perceive that a certain region of the +world is empty, that is merely the mode in which our senses +recognise that it is curved no higher than the first degree. +\index{Curvature!perception of}% +When we perceive that a region contains matter we are recognising +the intrinsic curvature of the world; and when we believe we +are measuring the mass and momentum of the matter (relative +to some axes of reference) we are measuring certain components +of world-curvature (referred to those axes). The statistical +averages of something unknown, which have been used to +describe the state of the world, vary from point to point; and +it is out of these that the mind has constructed the familiar +notions of matter and emptiness.% +\index{Emptiness, perception of}% +\index{Matter!perception of}% + +The law of gravitation is not a law in the sense that it restricts +the possible behaviour of the substratum of the world; it is +merely the definition of a vacuum. +\index{Vacuum, defined by law of gravitation}% +We need not regard matter +as a foreign entity causing a disturbance in the gravitational +field; the disturbance is matter. In the same way we do not +regard light as an intruder in the electromagnetic field, causing +%% -----File: 201.png---Folio 191------- +the electromagnetic force to oscillate along its path; the oscillations +constitute the light. Nor is heat a fluid causing agitation +of the molecules of a body; the agitation is heat. + +This view, that matter is a symptom and not a cause, seems +so natural that it is surprising that it should be obscured in the +usual presentation of the theory. The reason is that the connection +of mathematical analysis with the things of experience +is usually made, not by determining what matter is, but by +what certain combinations of matter do. Hence the interval is +at once identified with something familiar to experience, namely +the thing that a scale and a clock measure. However advantageous +that may be for the sake of bringing the theory into +touch with experiment at the outset, we can scarcely hope to +build up a theory of the nature of things if we take a scale and +clock as the simplest unanalysable concepts. The result of this +logical inversion is that by the time the equation $G_{\mu\nu} = K_{\mu\nu}$ is +encountered, both sides of the equation are well-defined +quantities. Their \textit{necessary} identity is overlooked, and the +equation is regarded as a new law of nature. This is the fault +of introducing the scale and clock prematurely. For our part +we prefer first to define what matter is in terms of the elementary +concepts of the theory; then we can introduce any kind of +scientific apparatus; and finally determine what property of the +world that apparatus will measure.% +\index{Clock-scale geometry, not fundamental}% + +Matter defined in this way obeys all the laws of mechanics, +including conservation of energy and momentum. Proceeding +with a similar development of Weyl's more general theory of +the combined gravitational and electrical fields, we should find +that it has the familiar electrical and optical properties. It is +purely gratuitous to suppose that there is anything else present, +controlling but not to be identified with the relations of the +fourteen potentials ($g$'s and $\kappa$'s). % [** PP: Changed k to \kappa] + +There is only one further requirement that can be demanded +from matter. Our brains are constituted of matter, and they +feel and think---or at least feeling and thinking are closely +associated with motions or changes of the matter of the brain. +\index{Brain, constitution of}% +It would be difficult to say that any hypothesis as to the nature +of matter makes this process less or more easily understood; +and a brain constituted out of differential coefficients of $g$'s can +%% -----File: 202.png---Folio 192------- +scarcely be said to be less adapted to the purposes of thought +than one made, say, out of tiny billiard balls! But I think we +may even go a little beyond this negative justification. The +primary interval relation is of an undefined nature, and the +$g$'s contain this undefinable element. The expression $G_{\mu\nu}$ is +therefore of defined \textit{form}, but of undefined \textit{content}. +\index{Content contrasted with structural form}% +\index{Form contrasted with content}% +By its form +alone it is fitted to account for all the physical properties of +matter; and physical investigation can never penetrate beneath +the form. The matter of the brain in its physical aspects is +merely the form; but the reality of the brain includes the +content. We cannot expect the form to explain the activities of +the content, any more than we can expect the number~4 to +explain the activities of the Big Four at Versailles. + +Some of these views of matter were anticipated with marvellous +foresight by W.~K. Clifford forty years ago. +\index{Clifford}% +Whilst other English +physicists were distracted by vortex-atoms and other will-o'-the-wisps, +Clifford was convinced that matter and the motion +of matter were aspects of space-curvature \textit{and nothing more}. +And he was no less convinced that these geometrical notions +were only partial aspects of the relations of what he calls +``elements of feeling.''---% +\index{Feeling, elements of}% +\index{Matter!physical and psychological aspects}% +``The reality corresponding to our perception +of the motion of matter is an element of the complex +thing we call feeling. What we might perceive as a plexus of +nerve-disturbances is really in itself a feeling; and the succession +of feelings which constitutes a man's consciousness is the reality +which produces in our minds the perception of the motions of +his brain. These elements of feeling have relations of \textit{nextness} +or contiguity in space, which are exemplified by the sight-perceptions +of contiguous points; and relations of succession in +time which are exemplified by all perceptions. Out of these two +relations the future theorist has to build up the world as best +he may. Two things may perhaps help him. There are many +lines of mathematical thought which indicate that distance or +quantity may come to be expressed in terms of \textit{position} in the +wide sense of the \textit{analysis situs}. And the theory of space-curvature +hints at a possibility of describing matter and motion +in terms of extension only.'' (\textit{Fortnightly Review}, 1875.) + +The equation $G_{\mu\nu} = K_{\mu\nu}$ is a kind of dictionary explaining +what the different components of world-curvature mean in +%% -----File: 203.png---Folio 193------- +terms ordinarily used in mechanics. If we write it in the slightly +modified, but equivalent, form +\[ +G_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} G = - 8\pi T_{\mu\nu}, +\] +we have the following scheme of interpretation +\index{Gravitation, Einstein's law of!macroscopic equations}% +\[ +\begin{matrix} +T_{11}, & T_{12}, & T_{13}, & T_{14} \\ + & T_{22}, & T_{23}, & T_{24} \\ + & & T_{33}, & T_{34} \\ + & & & T_{44} +\end{matrix} += +\begin{matrix} +p_{11} + \rho u^{2}, & p_{12} + \rho uv, & p_{13} + \rho uw, & - \rho u, \\ + & p_{22} + \rho v^2, & p_{23} + \rho vw, & - \rho v, \\ + & & p_{33} + \rho w^2, & - \rho w, \\ + & & & \rho. +\end{matrix} +\] +Here we are using the partitions of space and time adopted in +ordinary mechanics; $\rho$ is the density of the matter, $u$, $v$, $w$ its +component velocities, and $p_{11}$, $p_{12}$, $\dotsc p_{33}$, the components of +the internal stresses which are believed to be analysable into +molecular movements.% +\index{Stresses in continuous matter}% + +Now the question arises, is it legitimate to make identifications +on such a wholesale scale? Having identified $T_{44}$ as density, +can we go on to identify another quantity $T_{34}$ as density +multiplied by velocity? It is as though we identified one ``thing'' +as \textit{air}, and a quite different ``thing'' as \textit{wind}. Yes, it is legitimate, +because we have not hitherto explained what is to be the +counterpart of velocity in our scheme of the world; and this is +the way we choose to introduce it. All identifications are at +this stage provisional, being subject to subsequent test by +observation. + +A definition of the velocity of matter in some such terms as +``\textit{wind} divided by \textit{air},'' does not correspond to the way in +which motion primarily manifests itself in our experience. +\index{Velocity, definition of}% +Motion is generally recognised by the disappearance of a particle +at one point of space and the appearance of an apparently +identical particle at a neighbouring point. This manifestation +of motion can be deduced mathematically from the identifying +definition here adopted. Remembering that in physical theory +it is necessary to proceed from the simple to the complex, +which is often opposed to the instinctive desire to proceed from +the familiar to the unfamiliar, this inversion of the order in +which the manifestations of motion appear need occasion no +surprise. Permanent identity of particles of matter (without +which the ordinary notion of velocity fails) is a very familiar +idea, but it appears to be a very complex feature of the world.% +\index{Identity, permanent} % [** PP: Added comma] +\index{Permanent identity}% + +%% -----File: 204.png---Folio 194------- + +A simple instance may be given where the familiar kinematical +conception of motion is insufficient. +\index{Motion!insufficiency of kinematical conception}% +Suppose a perfectly homogeneous +continuous ring is rotating like a wheel, what meaning +can we attach to its motion? +\index{Rotation of a continuous ring}% +The kinematical conception of +motion implies change---disappearance at one point and reappearance +at another point---but no change is detectable. The +state at any one moment is the same as at a previous moment, +and the matter occupying one position now is indistinguishable +from the matter in the same position a moment ago. At the +most it can only differ in a mysterious non-physical quality---that +of identity; but if, as most physicists are willing to believe, +matter is some state in the aether, what can we mean by saying +that two states are exactly alike, but are not identical? Is the +hotness of the room equal to, but not identical with, its hotness +yesterday? Considered kinematically, the rotation of the ring +appears to have no meaning; yet the revolving ring differs +mechanically from a stationary ring. For example, it has +gyrostatic properties. The fact that in nature a ring has atomic +and not continuous structure is scarcely relevant. A conception +of motion which affords a distinction between a rotating and +non-rotating continuous ring must be possible; otherwise this +would amount to an \textit{a~priori} proof that matter is atomic. +According to the conception now proposed, velocity of matter +is as much a static quality as density. +\index{Velocity!static character}% +Generally velocity is +accompanied by changes in the physical state of the world, +which afford the usual means of recognising its existence; but +the foregoing illustration shows that these symptoms do not +always occur. + +This definition of velocity enables us to understand why +velocity except in reference to matter is meaningless, whereas +acceleration and rotation have a meaning. +\index{Absolute acceleration}% +\index{Absolute rotation}% +\index{Rotation, absolute}% +The philosophical +argument, that velocity through space is meaningless, ceases to +apply as soon as we admit any kind of structure or aether in +empty regions; consequently the problem is by no means so +simple as is often supposed. But our definition of velocity is +dynamical, not kinematical. Velocity is the ratio of certain +components of~$T_{\mu\nu}$, and only exists when $T_{44}$ is not zero. Thus +matter (or electromagnetic energy) is the only thing that can +have a velocity relative to the frame of reference. The velocity +%% -----File: 205.png---Folio 195------- +of the world-structure or aether, where the $T_{\mu\nu}$ vanish, is always +of the indeterminate form $0÷0$. On the other hand acceleration +and rotation are defined by means of the $g_{\mu\nu}$ and exist wherever +these exist\footnotemark; + \footnotetext{Even in Newtonian mechanics we speak of the ``field of acceleration,'' and + think of it as existing even when there is no test body to display the acceleration. + In the present theory this field of acceleration is described by the $g_{\mu\nu}$. + There is no such thing as a ``field of velocity'' in empty space; + \index{Field of velocity}% + but there is in + a material ocean.}% +so that the acceleration and rotation of the world-structure +or aether relative to the frame of reference are determinate. +Notice that acceleration is not defined as change of +velocity; it is an independent entity, much simpler and more +universal than velocity. It is from a comparison of these two +entities that we ultimately obtain the definition of time. + +This finally resolves the difficulty encountered in \Chapref{X}---the +apparent difference in the Principle of Relativity as +applied to uniform and non-uniform motion. Fundamentally +velocity and acceleration are both static qualities of a region +of the world (referred to some mesh-system). Acceleration is a +comparatively simple quality present wherever there is geodesic +structure, that is to say everywhere. +\index{Acceleration!a simpler quality than velocity}% +\index{Geodesic structure!acceleration of}% +\index{Structure, geodesic!acceleration of}% +Velocity is a highly complex +quality existing only where the structure is itself more +than ordinarily complicated, viz.\ in matter. Both these qualities +commonly give physical manifestations, to which the terms +acceleration and velocity are more particularly applied; but it +is by examining their more fundamental meaning that we can +understand the universality of the one and the localisation of +the other. + +It has been shown that there are four identical relations +between the ten qualities of a piece of matter here identified, +which depend solely on the way the $G_{\mu\nu}$ were by definition +constructed out of simpler elements. These four relations state +that, \textit{provided the mesh-system is drawn in one of a certain number +of ways}, mass (or energy) and momentum will be conserved. +The conservation of mass is of great importance; +\index{Mass!conservation of}% +\IndexExtra{Conservation of mass} % +matter will +be permanent, and for every particle disappearing at any point +a corresponding mass will appear at a neighbouring point; the +change consists in the displacement of matter, not its creation +or destruction. This gives matter the right to be regarded, not +as a mere assemblage of symbols, but as the substance of a +%% -----File: 206.png---Folio 196------- +permanent world. But the permanent world so found demands +the partitioning of space-time in one of a certain number of +ways, viz.\ those discussed in \Chapref{III}\footnotemark; + \footnotetext{When the kind of space-time is such that a strict partition of this kind is + impossible, strict conservation does not exist; but we retain the principle as + formally satisfied by attributing energy and momentum to the gravitational + field.}% +from these a particular +space and time are selected, because the observer wishes to +consider himself, or some arbitrary body, at rest. This gives +the space and time used for ordinary descriptions of experience. +In this way we are able to introduce perceptual space and time +into the four-dimensional world, as derived concepts depending +on our desire that the new-found matter should be permanent. + +I think it is now possible to discern something of the reason +why the world must of necessity be as we have described it. +When the eye surveys the tossing waters of the ocean, the +eddying particles of water leave little impression; it is the waves +that strike the attention, because they have a certain degree +of permanence. The motion particularly noticed is the motion of +the wave-form, which is not a motion of the water at all. So +the mind surveying the world of point-events looks for the +permanent things. The simpler relations, the intervals and +potentials, are transient, and are not the stuff out of which +mind can build a habitation for itself. But the thing that has +been identified with matter is permanent, and because of its +permanence it must be for mind the substance of the world. +Practically no other choice was possible. + +It must be recognised that the conservation of mass is not +exactly equivalent to the permanence of matter. +\index{Conservation!of mass}% +\index{Permanence of matter}% +If a loaf of +bread suddenly transforms into a cabbage, our surprise is not +diminished by the fact that there may have been no change of +weight. It is not very easy to define this extra element of +permanence required, because we accept as quite natural +apparently similar transformations---an egg into an omelette, +or radium into lead. But at least it seems clear that some degree +of permanence of one quality, mass, would be the primary +property looked for in matter, and this gives sufficient reason +for the particular choice. + +We see now that the choice of a permanent substance for the +%% -----File: 207.png---Folio 197------- +world of perception necessarily carries with it the law of gravitation, +all the laws of mechanics, and the introduction of the +ordinary space and time of experience. Our whole theory has +really been a discussion of the most general way in which +permanent substance can be built up out of relations; and it is +the mind which, by insisting on regarding only the things that +are permanent, has actually imposed these laws on an indifferent +world. Nature has had very little to do with the matter; she +had to provide a basis---point-events; but practically anything +would do for that purpose if the relations were of a reasonable +degree of complexity. The relativity theory of physics reduces +everything to relations; that is to say, it is structure, not +material, which counts. The structure cannot be built up without +material; but the nature of the material is of no importance. +We may quote a passage from Bertrand Russell's \textit{Introduction +to Mathematical Philosophy}.% +\index{Mathematics, Russell's description of}% +\index{Russell}% +\index{Structure opposed to content}% + +``There has been a great deal of speculation in traditional +philosophy which might have been avoided if the importance +of structure, and the difficulty of getting behind it, had been +realised. For example it is often said that space and time are +subjective, but they have objective counterparts; or that +phenomena are subjective, but are caused by things in themselves, +which must have differences \textit{inter~se} corresponding with +the differences in the phenomena to which they give rise. Where +such hypotheses are made, it is generally supposed that we can +know very little about the objective counterparts. In actual +fact, however, if the hypotheses as stated were correct, the +objective counterparts would form a world having the same +structure as the phenomenal world\ldots. In short, every proposition +having a communicable significance must be true of both worlds +or of neither: the only difference must lie in just that essence +of individuality which always eludes words and baffles description, +but which for that very reason is irrelevant to science.'' + +This is how our theory now stands.---We have a world of +point-events with their primary interval-relations. Out of these +an unlimited number of more complicated relations and qualities +can be built up mathematically, describing various features of +the state of the world. These exist in nature in the same sense +as an unlimited number of walks exist on an open moor. But +%% -----File: 208.png---Folio 198------- +the existence is, as it were, latent unless someone gives a significance +to the walk by following it; and in the same way the +existence of any one of these qualities of the world only acquires +significance above its fellows, if a mind singles it out for +recognition. Mind filters out matter from the meaningless +jumble of qualities, as the prism filters out the colours of the +rainbow from the chaotic pulsations of white light. Mind exalts +the permanent and ignores the transitory; and it appears from +the mathematical study of relations that the only way in which +mind can achieve her object is by picking out one particular +quality as the permanent substance of the perceptual world, +partitioning a perceptual time and space for it to be permanent +in, and, as a necessary consequence of this Hobson's choice, the +laws of gravitation and mechanics and geometry have to be +obeyed. +\index{Permanent perceptual world}% +Is it too much to say that mind's search for permanence +has created the world of physics? So that the world we +perceive around us could scarcely have been other than it is\footnotemark? + \footnotetext{This summary is intended to indicate the direction in which the views + suggested by the relativity theory appear to me to be tending, rather than to + be a precise statement of what has been established. I am aware that there + are at present many gaps in the argument. Indeed the whole of this part of + the discussion should be regarded as suggestive rather than dogmatic.} + +The last sentence possibly goes too far, but it illustrates the +direction in which these views are tending. With Weyl's more +general theory of interval-relations, the laws of electrodynamics +appear in like manner to depend merely on the identification +of another permanent thing---electric charge. In this case the +identification is due, not to the rudimentary instinct of the +savage or the animal, but the more developed reasoning-power +of the scientist. But the conclusion is that the whole of those +laws of nature which have been woven into a unified scheme---mechanics, +gravitation, electrodynamics and optics---have their +origin, not in any special mechanism of nature, but in the +workings of the mind. + +``Give me matter and motion,'' said Descartes, ``and I will +construct the universe.'' The mind reverses this. ``Give me a +world---a world in which there are relations---and I will construct +matter and motion.'' + +Are there then no genuine laws in the external world? Laws +inherent in the substratum of events, which break through into +%% -----File: 209.png---Folio 199------- +the phenomena otherwise regulated by the despotism of the +mind? We cannot foretell what the final answer will be; but, +at present, we have to admit that there are laws which appear +to have their seat in external nature. The most important of +these, if not the only law, is a law of atomicity. +\index{Atomicity!law of}% +Why does that +quality of the world which distinguishes matter from emptiness +exist only in certain lumps called atoms or electrons, all of +comparable mass? Whence arises this discontinuity? At +present, there seems no ground for believing that discontinuity +is a law due to the mind; indeed the mind seems rather to take +pains to smooth the discontinuities of nature into continuous +perception. We can only suppose that there is something in +the nature of things that causes this aggregation into atoms. +Probably our analysis into point-events is not final; and if it +could be pushed further to reach something still more fundamental, +then atomicity and the remaining laws of physics would +be seen as identities. This indeed is the only kind of explanation +that a physicist could accept as ultimate. But this more ultimate +analysis stands on a different plane from that by which the +point-events were reached. The world \textit{may} be so constituted +that the laws of atomicity must necessarily hold; but, so far as +the mind is concerned, there seems no reason why it should +have been constituted in that way. We can conceive a world +constituted otherwise. But our argument hitherto has been +that, however the world is constituted, the necessary combinations +of things can be found which obey the laws of mechanics, +gravitation and electrodynamics, and these combinations are +ready to play the part of the world of perception for any mind +that is tuned to appreciate them; and further, any world of +perception of a different character would be rejected by the +mind as unsubstantial. + +If atomicity depends on laws inherent in nature, it seems at +first difficult to understand why it should relate to matter +especially; since matter is not of any great account in the +analytical scheme, and owes its importance to irrelevant considerations +introduced by the mind. It has appeared, however, +that atomicity is by no means confined to matter and electricity; +the quantum, which plays so great a part in recent physics, is +apparently an atom of action. +\index{Action!atomicity of}% +So nature cannot be accused of +%% -----File: 210.png---Folio 200------- +connivance with mind in singling out matter for special distinction. +Action is generally regarded as the most fundamental +thing in the real world of physics, although the mind passes it +over because of its lack of permanence; and it is vaguely believed +that the atomicity of action is the general law, and the appearance +of electrons is in some way dependent on this. But the +precise formulation of the theory of quanta of action has hitherto +baffled physicists.% +\index{Quanta}% + +There is a striking contrast between the triumph of the +scientific mind in formulating the great general scheme of +natural laws, nowadays summed up in the principle of least +action, and its present defeat by the newly discovered but equally +general phenomena depending on the laws of atomicity of +quanta. It is too early to cry failure in the latter case; but +possibly the contrast is significant. It is one thing for the human +mind to extract from the phenomena of nature the laws which +it has itself put into them; it may be a far harder thing to +extract laws over which it has had no control. It is even possible +that laws which have not their origin in the mind may be +irrational, and we can never succeed in formulating them. This +is, however, only a remote possibility; probably if they were +really irrational it would not have been possible to make the +limited progress that has been achieved. But if the laws of +quanta do indeed differentiate the actual world from other +worlds possible to the mind, we may expect the task of formulating +them to be far harder than anything yet accomplished +by physics. + +The theory of relativity has passed in review the whole subject-matter +of physics. It has unified the great laws, which by the +precision of their formulation and the exactness of their application +have won the proud place in human knowledge which +physical science holds to-day. And yet, in regard to the nature +of things, this knowledge is only an empty shell---a form of +symbols. It is knowledge of structural form, and not knowledge +of content. +\index{Content contrasted with structural form}% +\index{Form contrasted with content}% +\index{Structure opposed to content}% +All through the physical world runs that unknown +content, which must surely be the stuff of our consciousness. +Here is a hint of aspects deep within the world of physics, and +yet unattainable by the methods of physics. And, moreover, +we have found that where science has progressed the farthest, +%% -----File: 211.png---Folio 201------- +the mind has but regained from nature that which the mind has +put into nature. + +We have found a strange foot-print on the shores of the +unknown. We have devised profound theories, one after +another, to account for its origin. At last, we have succeeded +in reconstructing the creature that made the foot-print. And +Lo! it is our own. +%% -----File: 212.png---Folio 202------- + + +\Appendix + +\First{The} references marked ``Report'' are to the writer's ``Report +on the Relativity Theory of Gravitation'' for the Physical +Society of London (Fleetway Press), where fuller mathematical +details are given. + +Probably the most complete treatise on the mathematical +theory of the subject is H.~Weyl's \textit{Raum, Zeit, Materie} (Julius +Springer, Berlin). + +\AppNote{1}{(\Pageref{note1}).} % p.~20 + +\Indent It is not possible to predict the contraction rigorously from +the universally accepted electromagnetic equations, because +these do not cover the whole ground. There must be other forces +or conditions which govern the form and size of an electron; +under electromagnetic forces alone it would expand indefinitely. +The old electrodynamics is entirely vague as to these forces. + +The theory of Larmor and Lorentz shows that if any system +at rest in the aether is in equilibrium, a similar system in +uniform motion through the aether, but with all lengths in the +direction of motion diminished in FitzGerald's ratio, will also +be in equilibrium so far as the differential equations of the +electromagnetic field are concerned. There is thus a general +theoretical agreement with the observed contraction, provided +the boundary conditions at the surface of an electron behave in +the same way. The latter suggestion is confirmed by experiments +on isolated electrons in rapid motion (Kaufmann's experiment). +\IndexExtra{Electron!Kaufmann's experiment on} % +It turns out that this requires an electron to suffer the same +kind of contraction as a material rod; and thus, although the +theory throws light on the adjustments involved in material +contraction, it can scarcely be said to give an explanation of the +occurrence of contraction generally. +%% -----File: 213.png---Folio 203------- + + +\AppNote{2}{(\Pageref{note2}).} % p.~47 + +\Indent Suppose a particle moves from $(x_{1}, y_{1}, z_{1}, t_{1})$ to $(x_{2}, y_{2}, z_{2}, t_{2})$, +its velocity~$u$ is given by +\[ +u^{2} = \frac{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} + (z_{2}-z_{1})^{2}} + {(t_{2}-t_{1})^{2}}. +\] +Hence from the formula for $s^{2}$ +\[ +s = (t_{2}-t_{1}) \surd(1-u^{2}). +\] +(We omit a $\sqrt{-1}$, as the sign of $s^{2}$ is changed later in the +chapter.) + +If we take $t_{1}$ and $t_{2}$ to be the start and finish of the aviator's +cigar (\Chapref{I}), then as judged by a terrestrial observer, +$t_{2}-t_{1} = 60~\text{minutes}$, $\surd(1- u^{2}) = \text{FitzGerald contraction} = \frac{1}{2}$. + +As judged by the aviator, +\[ +t_{2} - t_{1} = 30~\text{minutes}, \quad \surd(1-u^{2}) = 1. +\] + +Thus for both observers $s = 30$ minutes, verifying that it is +an absolute quantity independent of the observer. + +\AppNote{3}{(\Pageref{note3}).} % p.~48 + +\Indent The formulae of transformation to axes with a different +orientation are +\[ + x = x' \cos\theta - \tau' \sin\theta, \quad + y = y', \quad + z = z', \quad +\tau = x' \sin\theta + \tau' \cos\theta, +\] +where $\theta$ is the angle turned through in the plane~$x\tau$. + +Let $u = i \tan \theta$, so that $\cos\theta = (1-u^{2})^{-\frac{1}{2}} = \beta$, say. The +formulae become +\[ + x = \beta (x'- iu\tau'), \quad + y = y', \quad + z = z', \quad +\tau = \beta (\tau' + iux'), +\] +or, reverting to real time by setting $i\tau = t$, +\[ +x = \beta (x' - ut'), \quad +y = y', \quad +z = z', \quad +t = \beta (t' - ux'), +\] +which gives the relation between the estimates of space and +time by two different observers. + +The factor $\beta$ gives in the first equation the FitzGerald contraction, +and in the fourth equation the retardation of time. +The terms $ut'$ and~$ux'$ correspond to the changed conventions +as to \textit{rest} and \textit{simultaneity}. + +A point at rest, $x = \text{const.}$, for the first observer corresponds +to a point moving with velocity~$u$, $x'-ut' = \text{const.}$, for the second +observer. Hence their relative velocity is~$u$. +%% -----File: 214.png---Folio 204------- + + +\AppNote{4}{(\Pageref{note4}).} % p.~81 + +\Indent The condition for flat space in two dimensions is +\begin{multline*} +\frac{\partial}{\partial x_1} + \left( \frac{g_{12}}{g_{11} + \surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{11}}{\partial x_2} + - \frac{1}{\surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{22}}{\partial x_1}\right) \\ % [** PP: Changed \delta to \partial] +% ++ \frac{\partial}{\partial x_2} + \left( \frac{2}{\surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{12}}{\partial x_1} + - \frac{1}{\surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{11}}{\partial x_2} \right. \\ +% +- \left.\frac{g_{12}}{g_{11} \surd(g_{11} g_{22} - g_{12}^2)}\, + \frac{\partial g_{11}}{\partial x_1}\right) = 0. +\end{multline*} + + +\AppNote{5}{(\Pageref{note5}).} % p.~89 + +\Indent Let~$g$ be the determinant of four rows and columns formed +with the elements~$g_{\mu\nu}$. + +Let $g^{\mu\nu}$ be the minor of~$g_{\mu\nu}$, divided by~$g$. + +Let the ``$3$-index symbol'' $\{\mu\nu, \lambda\}$ denote +\[ +\tfrac{1}{2} g^{\lambda\alpha} + \left( \frac{\partial g_{\mu\alpha}}{\partial x_\nu} + + \frac{\partial g_{\nu\alpha}}{\partial x_\mu} + - \frac{\partial g_{\mu\nu}}{\partial x_\alpha} \right) +\] +summed for values of~$\alpha$ from~$1$ to~$4$. There will be $40$ different +$3$-index symbols. + +Then the Riemann-Christoffel tensor is +\[ +B^\rho_{\mu\nu\sigma} + = \{\mu\sigma, \epsilon\} \{\epsilon\nu, \rho\} + - \{\mu\nu, \epsilon\} \{\epsilon\sigma, \rho\} + + \frac{\partial}{\partial x_\nu} \{\mu\sigma, \rho\} + - \frac{\partial}{\partial x_\sigma} \{\mu\nu, \rho\}, +\] +the terms containing~$\epsilon$ being summed for values of~$\epsilon$ from~$1$ to~$4$. + +The ``contracted'' Riemann-Christoffel tensor~$G_{\mu\nu}$ can be +reduced to +\begin{multline*} +G_{\mu\nu} + = - \frac{\partial}{\partial x_\alpha} \{\mu\nu, \alpha\} + + \{\mu\alpha, \beta\} \{\nu\beta, \alpha\} \\ + + \frac{\partial^2}{\partial x_\mu \partial x_\nu} \log \sqrt{-g} + - \{\mu\nu, \alpha\} \frac{\partial}{\partial x_\alpha} \log \sqrt{-g}, +\end{multline*} +where in accordance with a general convention in this subject, +each term containing a suffix twice over ($\alpha$ and~$\beta$) must be +summed for the values $1$, $2$, $3$, $4$ of that suffix. + +The curvature $G = g^{\mu\nu} G_{\mu\nu}$, summed in accordance with the +foregoing convention. + + +\AppNote{6}{(\Pageref{note6}).} % p.~94 + +\Indent The electric potential due to a charge~$e$ is +\[ +\phi = \frac{e}{\bigl[r(1-v_r/C)\bigr]}, +\] +%% -----File: 215.png---Folio 205------- +where $v_r$ is the velocity of the charge in the direction of~$r$, $C$ the +velocity of light, and the square bracket signifies antedated +values. To the first order of $v_r/C$, the denominator is equal to +the \textit{present} distance~$r$, so the expression reduces to~$e/r$ in spite +of the time of propagation. The foregoing formula for the +potential was found by Liénard and Wiechert. + + +\AppNote{7}{(\Pageref{note7}).} % p.~97 + +\Indent It is found that the following scheme of potentials rigorously +satisfies the equations $G_{\mu\nu} = 0$, according to the values of $G_{\mu\nu}$ +in \Noteref{5}, +\[ +\begin{array}{cccc} +-1/\gamma & 0 & 0 & 0 \\ + & -x_{1}{}^{2} & 0 & 0 \\ + & & -x_{1}{}^{2}\sin^{2} x_{2}{}^{2} & 0 \\ + & & & \gamma +\end{array} +\] +where $\gamma = 1-\kappa/x_{1}$ and $\kappa$ is any constant (see Report, §~28). +Hence these potentials describe a kind of space-time which can +occur in nature referred to a possible mesh-system. If $\kappa = 0$, +the potentials reduce to those for flat space-time referred to +polar coordinates; and, since in the applications required $\kappa$ will +always be extremely small, our coordinates can scarcely be +distinguished from polar coordinates. We can therefore use the +familiar symbols $r$, $\theta$, $\phi$, $t$, instead of $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. It must, +however, be remembered that the identification with polar +coordinates is only approximate; and, for example, an equally +good approximation is obtained if we write $x_{1} = r + \frac{1}{2} \kappa$, a substitution +often used instead of $x_{1} = r$ since it has the advantage +of making the coordinate-velocity of light more symmetrical. + +We next work out analytically all the mechanical and optical +properties of this kind of space-time, and find that they agree +observationally with those existing round a particle at rest at +the origin with gravitational mass~$\frac{1}{2} \kappa$. The conclusion is that +the gravitational field here described is produced by a particle +of mass $\frac{1}{2} \kappa$---or, if preferred, a particle of matter at rest is +produced by the kind of space-time here described. + + +\AppNote{8}{(\Pageref{note8}).} % p.~98 + +\Indent Setting the gravitational constant equal to unity, we have for +a circular orbit +\begin{DPalign*} +m/r^{2} &= v^{2}/r, \\ +\lintertext{so that} + m &= v^{2}r. +\end{DPalign*} + +%% -----File: 216.png---Folio 206------- + +The earth's speed,~$v$, is approximately $30$~km.\ per sec., or +$\frac{1}{10000}$ in terms of the velocity of light. The radius of its orbit,~$r$, +is about $1.5 · 10^8$~km. Hence,~$m$, the gravitational mass of the +sun is approximately $1.5$~km. % [** PP: Changing scientific notation] + +The radius of the sun is $697,000$~kms., so that the quantity +$2m/r$ occurring in the formulae is, for the sun's surface, $.00000424$ +or~$0''.87$. + + +\AppNote{9}{(\Pageref{note9}).} % p.~123 + +\index{Gravitational field of Sun!motion of perihelion}% +\Indent See Report, §§~29, 30. The general equations of a geodesic are +\[ +\frac{d^2x_\mu}{ds^2} + + \{\alpha\beta, \mu\}\, \frac{dx_\alpha}{ds} \frac{dx_\beta}{ds} = 0\quad + (\mu = 1,\ 2,\ 3,\ 4). +\] + +From the formula for the line-element +\[ +ds^2 = -\gamma^{-1}\, dr^2 - r^2\, d\theta^2 + \gamma\, dt^2, +\tag{1} +\] +we calculate the three-index symbols and it is found that two +of the equations of the geodesic take the rather simple form +\begin{align*} +&\frac{d^2\theta}{ds^2} + + \frac{2}{r} · \frac{dr}{ds} \frac{d\theta}{ds} = 0, \\ +% +&\frac{d^2t}{ds^2} + + \frac{d(\log\gamma)}{dr} · \frac{dr}{ds} \frac{dt}{ds} = 0, +\end{align*} +which can be integrated giving +\begin{align*} +r^2\, \frac{d\theta}{ds} &= h, +\tag{2} \\ +% +\frac{dt}{ds} &= \frac{c}{\gamma}, +\tag{3} +\end{align*} +where $h$ and~$c$ are constants of integration. + +Eliminating $dt$ and~$ds$ from (1), (2) and~(3), we have +\[ +\left(\frac{h}{r^2} \frac{dr}{d\theta}\right)^2 + \frac{h^2}{r^2} + = c^2 - 1 + \frac{2m}{r} + \frac{2mh^2}{r^3}, +\] +or writing $u = 1/r$, % [** PP: Condensed intertext in original] +\[ +\left(\frac{du}{d\theta}\right)^2 + u^2 + = \frac{c^2 - 1}{h^2} + \frac{2mu}{h^2} + 2mu^3. +\] + +Differentiating with respect to~$\theta$ +\[ +\frac{d^2u}{d\theta^2} + u = \frac{m}{h^2} + 3mu^2, +\] +%% -----File: 217.png---Folio 207------- +which gives the equation of the orbit in the usual form in particle +dynamics. It differs from the equation of the Newtonian orbit +by the small term $3mu^2$, which is easily shown to give the motion +of perihelion. + +The track of a ray of light is also obtained from this formula, +since by the principle of equivalence it agrees with that of a +material particle moving with the speed of light. +\index{Bending of light!theory of}% +\index{Light, bending of}% +This case is +given by $ds = 0$, and therefore $h = \infty$. The differential equation +for the path of a light-ray is thus +\[ +\frac{d^2u}{d\theta^2} + u = 3mu^2. +\] + +An approximate solution is +\[ +u = \frac{\cos\theta}{R} + \frac{m}{R^2}(\cos^2\theta + 2\sin^2\theta), +\] +neglecting the very small quantity~$m^2/R^2$. Converting to +Cartesian coordinates, this becomes +\[ +x = R - \frac{m}{R} \frac{x^2 + 2y^2}{\surd(x^2 + y^2)}. +\] + +The asymptotes of the light-track are found by taking~$y$ +very large compared with~$x$, giving +\[ +x = R ± \frac{2m}{R} y +\] +so that the angle between them is~$4m/R$. + + +\AppNote{10}{(\Pageref{note10}).} % p.~126 + +\Indent Writing the line element in the form +\[ +ds^2 + = -\left(1 + a \frac{m}{r} + \dotsb\right) dr^2 + - r^2\, d\theta^2 + + \left(1 + b \frac{m}{r} + c \frac{m^2}{r^2} + \dotsb\right) dt^2, +\] +the approximate Newtonian attraction fixes~$b$ equal to~$-2$; +then the observed deflection of light fixes~$a$ equal to~$+2$; and +with these values the observed motion of Mercury fixes~$c$ equal +to~$0$.% +\index{Deflection of light!theory of}% +\index{Gravitational field of Sun!deflection of light}% + +To insert an arbitrary coefficient of $r^2\, d\theta^2$ would merely vary +the coordinate system. We cannot arrive at any intrinsically +different kind of space-time in that way. Hence, within the +limits of accuracy mentioned, the expression found by Einstein +is completely determinable by observation. + +%% -----File: 218.png---Folio 208------- + +It may be mentioned that the line-element +\[ +ds^2 = - dr^2 - r^2\, d\theta^2 + (1-2m/r)\, dt^2, +\] +gives one-half the observed deflection of light, and one-third +the motion of perihelion of Mercury. As both these can be +obtained on older theories, taking account of the variation of +mass with velocity, the coefficient~$\gamma^{-1}$ of~$dr^2$ is the essentially +novel point in Einstein's theory. + + +\AppNote{11}{(\Pageref{note11}).} % p.~131 + +\Indent It is often supposed that by the Principle of Equivalence any +invariant property which holds outside a gravitational field also +holds in a gravitational field; but there is necessarily some +limitation on this equivalence. Consider for instance the two +invariant equations +\begin{gather*} +ds^2 = 1, \\ +ds^2 (1 + k^4 B^\rho_{\mu\nu\sigma} B^{\mu\nu\sigma}_\rho) = 1, +\end{gather*} +where $k$ is some constant having the dimensions of a length. +Since $B^\rho_{\mu\nu\sigma}$ vanishes outside a gravitational field, if one of these +equations is true the other will be. But they cannot both hold +in a gravitational field, since there $B^\rho_{\mu\nu\sigma} B^{\mu\nu\sigma}_\rho$ does not vanish, +and is in fact equal to $24m^2/r^6$. (I believe that the numerical +factor~$24$ is correct; but there are 65,536~terms in the expression, +and the terms which do not vanish have to be picked out.) % [** F1: Added closing parenthesis] + +This ambiguity of the Principle of Equivalence is referred to +in Report, §§~14, 27; and an enunciation is given which makes +it definite. The enunciation however is merely an explicit statement, +and not a defence, of the assumptions commonly made in +applying the principle. + +So far as general reasoning goes there seems no ground for +choosing $ds^2$ rather than $ds^2\, (1 + 24k^4m^2/r^6)$, or any similar expression, +as the constant character in the vibration of an atom. + + +\AppNote{12}{(\Pageref{note12}).} % p.~134 + +\Indent Let two rays diverging from a point at a distance~$R$ pass at +distances $r$ and~$r + dr$ from a star of mass~$m$. The deflection +being~$4m/r$, their divergence will be increased by~$4m\, dr/r^2$. This +increase will be equal to the original divergence~$dr/R$ if +$r = \sqrt{4mR}$. Take for instance $4m = 10$~km., $R = 10^{15}$~km., then +$r = 10^8$~km. So that the divergence of the light will be doubled, +%% -----File: 219.png---Folio 209------- +when the actual deflection of the ray is only $10^{-7}$, or $0''.02$. +In the case of a star seen behind the sun the added divergence +has no time to take effect; but when the light has to travel a +stellar distance after the divergence is produced, it becomes +weakened by it. Generally in stellar phenomena the weakening +of the light should be more prominent than the actual deflection. + + +\AppNote{13}{(\Pageref{note13}).} % p.~141 + +\Indent The relations are (Report, §~39) +\[ +G^\nu_{\mu\nu} = \tfrac{1}{2} \frac{\partial G}{\partial x_\mu}\quad +(\mu = 1,\ 2,\ 3,\ 4), +\] +where $G^\nu_{\mu\nu}$ is the (contracted) covariant derivative of~$G^\nu_\mu$, +or~$g^{\nu\alpha} G_{\mu\alpha}$. + +I doubt whether anyone has performed the laborious task of +verifying these identities by straightforward algebra. + + +\AppNote{14}{(\Pageref{note14}).} % p.~158 + +\Indent The modified law for spherical space-time is in empty space +\[ +G_{\mu\nu} = \lambda g_{\mu\nu}. +\] + +In cylindrical space-time, matter is essential. The law in space +occupied by matter is +\index{Cylindrical world, Einstein's}% +\[ +G_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} (G - 2\lambda) = -8\pi T_{\mu\nu}, +\] +the term $2\lambda$ being the only modification. Spherical space-time +of radius~$R$ is given by $\lambda = 3/R^2$; cylindrical space-time by +$\lambda = 1/R^2$ provided matter of average density $\rho = 1/4 \pi R^2$ is +present. (See Report, §§~50, 51.) The total mass of matter in +the cylindrical world is $\frac{1}{2} \pi R$. This must be enormous, seeing +that the sun's mass is only $1\frac{1}{2}$~kilometres. + + +\AppNote{15}{(\Pageref{note15}).} % p.~174 + +\Indent Weyl's theory is given in \textit{Berlin.\ Sitzungsberichte}, 30~May, 1918; +\textit{Annalen der Physik}, Bd.~59 (1919), p.~101. + + +\AppNote{16}{(\Pageref{note16}).} % p.~177 + +\Indent The argument is rather more complicated than appears in +the text, where the distinction between action-density and +action in a region, curvature and total curvature in a region, +has not been elaborated. Taking a definitely marked out region +in space and time, its measured volume will be increased $16$-fold +%% -----File: 220.png---Folio 210------- +by halving the gauge. Therefore for action-density we must +take an expression which will be diminished $16$-fold by halving +the gauge. Now~$G$ is proportional to $1/R^{2}$, where~$R$ is the radius +of curvature, and so is diminished $4$-fold. The invariant $B^{\rho}_{\mu\nu\sigma} B^{\mu\nu\sigma}_{\rho}$ +has the same gauge-dimensions as~$G^{2}$; and hence when integrated +through a volume gives a pure number independent of the gauge. +In Weyl's theory this is only the gravitational part of the complete +invariant +\[ +(B^{\rho}_{\mu \nu \sigma} - \tfrac{1}{2} g^{\rho}_{\mu} F_{\nu \sigma}) +(B^{\mu \nu \sigma}_{\rho} - \tfrac{1}{2} g^{\mu}_{\rho} F^{\nu \sigma}), +\] +which reduces to +\[ +B^{\rho}_{\mu \nu \sigma} +B^{\mu \nu \sigma}_{\rho} + F_{\nu \sigma}F^{\nu \sigma}. +\] + +The second term gives actually the well-known expression for +the action-density of the electromagnetic field, and this evidently +strengthens the identification of this invariant with action-density. + +Einstein's theory, on the other hand, creates a difficulty here, +because although there may be action in an electromagnetic +field without electrons, the curvature is zero. + +\fancyhead[CO]{\textsc{HISTORICAL NOTE}} +\phantomsection\addcontentsline{toc}{chapter} +{\texorpdfstring{\scshape Historical Note}{Historical Note}} +\section*{\centering\normalfont\textsc{\large HISTORICAL NOTE}} + +\Indent Before the Michelson-Morley experiment the question had +been widely discussed whether the aether in and near the earth +was carried along by the earth in its motion, or whether it +slipped through the interstices between the atoms. Astronomical +aberration pointed decidedly to a stagnant aether; +\index{Aether!stagnant}% +but the +experiments of Arago and Fizeau on the effect of motion of +transparent media on the velocity of light in those media, +suggested a partial convection of the aether in such cases. These +experiments were first-order experiments, i.e.\ %[** PP: Add missing .] +they depended on +the ratio of the velocity of the transparent body to the velocity +of light. The Michelson-Morley experiment is the first example +of an experiment delicate enough to detect second-order effects, +depending on the square of the above ratio; the result, that no +current of aether past terrestrial objects could be detected, +appeared favourable to the view that the aether must be convected +by the earth. The difficulty of reconciling this with +astronomical aberration was recognised. + +%% -----File: 221.png---Folio 211------- + +An attempt was made by Stokes to reconcile mathematically +a convection of aether by the earth with the accurately verified +facts of astronomical aberration; but his theory cannot be +regarded as tenable. Lodge investigated experimentally the +question whether smaller bodies carried the aether with them +in their motion, and showed that the aether between two +whirling steel discs was undisturbed.% +\index{Lodge}% + +{\stretchyspace +The controversy, stagnant \textit{versus} convected aether, had now +reached an intensely interesting stage. In 1895, Lorentz discussed +the problem from the point of view of the electrical +theory of light and matter. By his famous transformation of +the electromagnetic equations, he cleared up the difficulties +associated with the first-order effects, showing that they could +all be reconciled with a stagnant aether. In 1900, Larmor carried +the theory as far as second-order effects, and obtained an exact +theoretical foundation for FitzGerald's hypothesis of contraction, +which had been suggested in 1892 as an explanation of the +Michelson-Morley experiment. The theory of a stagnant aether +was thus reconciled with all observational results; and henceforward % [** PP: Hyphenated across a line in original] +it held the field.% +\index{Larmor}% +\index{Lorentz}% + +Further second-order experiments were performed by Rayleigh +and Brace on double refraction (1902, 1904), Trouton and Noble +on a torsional effect on a charged condenser (1903), and Trouton +and Rankine on electric conductivity (1908). All showed that +the earth's motion has no effect on the phenomena. On the +theoretical side, Lorentz (1902) showed that the indifference of +the equations of the electromagnetic field to any velocity of the +axes of reference, which he had previously established to the +first order, and Larmor to the second order, was exact to all +orders. He was not, however, able to establish with the same +exactness a corresponding transformation for bodies containing +electrons. + +Both Larmor and Lorentz had introduced a ``local time'' for +the moving system. It was clear that for many phenomena this +local time would replace the ``real'' time; but it was not +suggested that the observer in the moving system would be +deceived into thinking that it was the real time. Einstein, in +1905 founded the modern principle of relativity by postulating +that this local time was \textit{the time} for the moving observer; no +%% -----File: 222.png---Folio 212------- +real or absolute time existed, but only the local times, different +for different observers. He showed that absolute simultaneity +and absolute location in space are inextricably bound together, +and the denial of the latter carries with it the denial of the +former. By realising that an observer in the moving system +would measure all velocities in terms of the local space and time +of that system, Einstein removed the last discrepancies from +Lorentz's transformation. + +The relation between the space and time coordinates in two +systems in relative motion was now obtained immediately from +the principles of space and time-measurement. It must hold +for all phenomena provided they do not postulate a medium +which can serve as a standard for absolute location and simultaneity. +The previous deduction of these formulae by lengthy +transformation of the electromagnetic equations now appears +as a particular case; it shows that electromagnetic phenomena +have no reference to a medium with such properties. + +The combination of the local spaces and times of Einstein +into an absolute space-time of four dimensions is the work of +Minkowski~(1908). +\index{Minkowski}% +\index{Space-time!due to Minkowski}% +\Chapref{III} is largely based on his researches. +Much progress was made in the four-dimensional vector-analysis +of the world; but the whole problem was greatly +simplified when Einstein and Grossmann introduced for this +purpose the more powerful mathematical calculus of Riemann, +Ricci, and Levi-Civita. + +}In 1911, Einstein put forward the Principle of Equivalence, +thus turning the subject towards gravitation for the first time. +\index{Equivalence!Principle of}% +\index{Principle of Equivalence}% +By postulating that not only mechanical but optical and +electrical phenomena in a field of gravitation and in a field +produced by acceleration of the observer were equivalent, he +deduced the displacement of the spectral lines on the sun and +the displacement of a star during a total eclipse. In the latter +case, however, he predicted only the half-deflection, since he +was still working with Newton's law of gravitation. Freundlich +\index{Freundlich}% +at once examined plates obtained at previous eclipses, but failed +to find sufficient data; he also prepared to observe the eclipse +of 1914 in Russia with this object, but was stopped by the outbreak +of war. Another attempt was made by the Lick Observatory +at the not very favourable eclipse of 1918. Only preliminary +%% -----File: 223.png---Folio 213------- +results have been published; according to the information given, +the probable accidental error of the mean result (reduced to the +sun's limb) was about $1''.6$, so that no conclusion was permissible. + +The principle of equivalence opened up the possibility of a +general theory of relativity not confined to uniform motion, for +it pointed a way out of the objections which had been urged +against such an extension from the time of Newton. At first +the opening seemed a very narrow one, merely indicating that +the objections could not be considered final until the possibilities +of complications by gravitation had been more fully exhausted. +By 1913, Einstein had surmounted the main difficulties. His +theory in a complete form was published in 1915; but it was not +generally accessible in England until a year or two later. As +this theory forms the main subject-matter of the book, we may +leave our historical survey at this point. + +%% -----File: 224.png---Folio 214------- +% [Blank Page] +%% -----File: 225.png---Folio 215------- +\iffalse +INDEX + +Absolute, approached through the +relative, 82 + +Absolute acceleration, 68, 154, 194 + +Absolute past and future, 50 + +Absolute rotation, 152, 164, 194 + +Absolute simultaneity, 12, 51 + +Absolute time, in cylindrical world, +163 + +Acceleration, a simpler quality than +velocity, 195; modifies FitzGerald +contraction, 75 + +Action, 147; atomicity of, 199; on +Weyl's theory, 177 + +Action, Principle of Least, 149, 178 + +Addition of velocities, 59 + +Aether, a plenum with geodesic structure, +164; identified with the +``world,'' 187; non-material nature +of, 39; stagnant, 210 + +Artificial fields of force, 64 + +Atom, vibrating on sun, 128 + +Atomicity, law of, 199; of Action, 177 + +Aviator, space and time-reckoning of, +23 + +Bending of light, effect on star's +position, 112; observational results, +118; theory of, 107, 207 + +Beta particles, 59, 145 + +Brain, constitution of, 191 + +Brazil, eclipse expedition to, 117 + +Causality, law of, 156 + +Causation and free will, 51 + +Centrifugal Force, compared with +gravitation, 41, 65; debt at infinity, +157; not caused by stars, 153; +vibrating atom in field of, 129 + +Chess, analogy of, 184 + +Christoffel, 89 + +Circle in non-Euclidean space, 104 + +Clifford, 77, 152, 192 + +Cliquishness, 188 + +Clock, affected by velocity, 58; on sun, +74, 128; perfect, 13; recording +proper-time, 71 + +Clock-scale, 58 + +Clock-scale geometry, not fundamental, +73, 131, 191 + +Coincidences, 87 + +Comets, motion through coronal medium, +121; radiation-pressure in, 110 + +Conservation of electric charge, 173; +of energy and momentum, 139; of +mass, 141, 196 + +Content contrasted with structural +form, 192, 200 + +Continuous matter, 91, 140 + +Contraction, FitzGerald, 19, 54 + +Convergence of physical approximations, +154 + +Coordinates, 77 + +Coordinate velocity, 107 + +Corona, refraction by, 121 + +Cottingham, 114 + +Crommelin, 114, 122 + +Curvature, degrees of, 91; identified +with action, 148; merely illustrative, +84; of a globe of water, 148; +of space and time, 158; on Weyl's +theory, 176; perception of, 190 + +Cylinder and plane, indistinguishable +in two dimensions, 81 + +Cylindrical world, Einstein's, 161, 177 + +Davidson, 114 + +Deflection of light, effect on star's +position, 112; observational results, +118; theory of, 107, 207 + +Density, effect of motion on, 62 + +Displacement of spectral lines, 129; in +nebulae, 161; in stars, 135 + +Displacement of star-images, 112, 115 + +Double stars and Einstein effect, 133 + +Duration, not inherent in external +world, 34 +\fi +%% -----File: 226.png---Folio 216------- +\iffalse +Eclipse, observations during, 113 + +Ehrenfest's paradox, 75 + +Electrical theory of inertia, 61 + +Electricity and gravitation, 167 + +Electromagnetic potentials and forces, +172 + +Electron, dimensions of, 177; geometry +inside, 91; gravitational mass of, +178; inertia of, 61; Kaufmann's +experiment on, 62, 146; singularity +in field, 167 + +``Elsewhere,'' 50 + +Emptiness, perception of, 190 + +Energy, conservation of, 139; identified +with mass, 146; inertia of, 61, +146; weight of radio-active, 112 + +Entropy, 149 + +Eötvös torsion-balance, 112 + +Equivalence, Principle of, 76, 131, 212 + +Euclidean geometry, 1, 47, 73 + +Euclidean space of five dimensions, 84 + +Event, definition of, 45, 186 + +Evershed, 130 + +Extension in four dimensions, 37, 46 + +Feeling, elements of, 192 + +Fields of force, artificial, 64; due to +disturbance of observer, 69; electromagnetic, +171; relativity of, 67 + +Field of velocity, 195 + +FitzGerald contraction, 19; consequences +of, 22; modified by +acceleration, 75; relativity explanation +of, 54 + +Flat space in two dimensions, 80 + +Flat space-time, 83; at infinity, 84; +conditions for, 89 + +Flatfish, analogy of, 95 + +Flatland, 57 + +Force, compared with inertia, 137; +electromagnetic, 172; elementary +conception of, 63; fields of, 64; +relativity of, 43, 67, 76 + +Form contrasted with content, 192, 200 + +Formalism of knowledge, 175 + +Foucault's pendulum, 152 + +Four-dimensional order, 35, 56, 186 + +Four-dimensional space-time, geometry +of, 45, 82; reality of, 181 + +Fourth dimension, 13 + +Frame, inertial, 156 + +Frames of reference, ``right'' and +``wrong,'' 42 + +Freewill, 51 + +Freundlich, 212 + +Future, absolute, 50 + +Galilean potentials, 83 + +Gauge, effect on observations, 31; +provided by radius of space, 177 + +Gauge-system, 169 + +Geodesic, absolute significance of, 70, +150; definition of, 75; motion of +particles in, 138, 151; in regions at +infinity, 157 + +Geodesic structure, absolute character +of, 155, 164; acceleration of, 195 + +Geometrical conception of the world, +176, 183 + +Geometry, Euclidean, 1; hyperbolic, +47; Lobatchewskian, 1, 9; natural, +2; non-Euclidean, or Riemannian, +6, 73, 84, 90; non-Riemannian, 169; +semi-Euclidean, 47 + +Ghosts of stars, 161 + +Globe of water, limit to size of, 148 + +Gravitation, Einstein's law of, differential +formula, 90; integrated formula +for a particle, 97; macroscopic +equations, 140, 193 + +Gravitation, Newton's law of, ambiguity +of, 93; approximation to +Einstein's law, 103; deflection of +light, 109, 111 + +Gravitation, propagation with velocity +of light, 94, 147; relativity for +uniform motion, 21, 125 + +Gravitational field of Sun, 97; deflection +of light, 107, 118, 207; displacement +of spectral lines, 129; +motion of perihelion, 122, 206; +Newtonian attraction, 102; result +of observational verification, 126 + +Grebe and Bachem, 130 + +Greenwich, Royal Observatory, 114 + +Gyro-compass, 152 + +Hummock in space-time, 97 + +Hurdles, analogy of counts of, 104 + +Hyperbolic geometry, 47 +\fi +%% -----File: 227.png---Folio 217------- +\iffalse +Identities connecting G_{\mu \nu}, 141 + +Identity permanent, 40, 193 + +Imaginary intervals, 150, 187 + +Imaginary time, 48, 181 + +Inertia, compared with force, 137; +electrical theory of, 61; in regions +at infinity, 157; infinite, 56; +Mach's views, 164; of light, 110; +relativity theory of, 139 + +Inertia-gravitation, 137 + +Inertial frame, 156 + +Infinity, conditions at, 157 + +Integrability of length and direction, +174 + +Interval, 46, 150, 187; general expression +for, 82; practical measurement +of, 58, 75 + +Interval-length, geometrical significance +essential, 127; identified with +proper time, 71; tracks of maximum, +70, 150; zero for velocity of light, 71 + +Invariant mass, 145; of light, 148 + +Jupiter, deflection of light by, 133 + +Kaufmann's experiment, 62, 146 + +Kinds of space, 81 + +Laplace's equation, 96, 140 + +Larmor, 19, 211 + +Length, definition of, 2; effect of +motion on, 19; relativity of, 34 + +Le Verrier, 124 + +Levi-Civita, 89 + +Lift, accelerated, 64 + +Light, bending of, 107, 112, 118, 207; +coordinate velocity of, 107; mass +of, 62, 110, 148; voyage round the +world, 161; weight of, 111 + +Light, velocity of, an absolute velocity, +59; importance of, 60; system +moving with, 26, 56 + +Lobatchewsky, 1, 9 + +Lodge, 32, 125, 211 + +Longest tracks, 70 + +Lorentz, 19, 211 + +Mach's philosophy, 163 + +Macroscopic equations, 92, 139; interval, +187 + +Map of sun's gravitational field, 99 + +Mass, conservation of, 141, 195; +electrical theory of, 61; gravitational, +98; identified with energy, +146; invariant, 145; of light, 62, +110, 148; variation with velocity, +145 + +Mathematics, Russell's description of, +14 + +Matter, continuous, 91; definition of a +particle, 98; extensional relations +of, 8; gravitational equations in, +141; perception of, 190; physical +and psychological aspects, 192 + +Mercury, perihelion of, 123, 125 + +Mesh-systems, 77; irrelevance to laws +of nature, 87 + +Michelson-Morley experiment, 18 + +Minkowski, 30, 212 + +Mirror, distortion by moving, 22 + +Momentum, conservation of, 141; redefinition +of, 144; of light, 111 + +Moon, motion of, 93, 134 + +Motion, insufficiency of kinematical +conception, 194; Newton's first +law, 136 + +Natural frame, 155 + +Natural gauge, 176 + +Natural geometry, 2 + +Natural tracks, 70 + +Nebulae, atomic vibrations in, 161 + +Newton, absolute rotation, 41; bending +of light, 110; law of gravitation, +93; law of motion, 136; relativity +for uniform motion, 40; super-observer, +68 + +Non-Euclidean geometry, 6, 73, 84, 90 + +Non-Riemannian geometry, 169 + +Observer, an unsymmetrical object, 57 + +Observer and observed, 30 + +Orbits under Einstein's law, 123 + +Order and dimensions, 14, 186 + +Ordering of events in external world, +35, 54, 184 + +Past, absolute, 50 + +Perceptions, as crude measures, 10, 15, +31 +\fi +%% -----File: 228.png---Folio 218------- +\iffalse +Perihelia of planets, motions of, 123 +Permanence of matter, 196 +Permanent identity, 40, 193 +Permanent perceptual world, 141, 198 +Poincaré, 9 +Point-event, 45, 186 +Potentials, 80; Galilean values, 83 +Potentials, electromagnetic, 172 +Principe, eclipse expedition to, 114 +Principle of Equivalence, 76, 131, 212 +Principle of Least Action, 149, 178 +Principle of Relativity (restricted), 20 +Probability, a pure number, 178 +Projectile, Jules Verne's, 65 +Propagation of Gravitation, 94, 147 +Proper-length, 11 +Proper-time, 71 +Pucker in space-time, 85 + +Quanta, 60, 177, 182, 200 + +Radiation-pressure, 110 +Real world of physics, 37, 181 +Receding velocities of B-type stars, + 135; of spiral nebulae, 161 +Reflection by moving mirror, 22 +Refracting medium equivalent to + gravitational field, 109 +Refraction of light in corona, 121 +Relativity of force, 43, 76; of length + and duration, 34; of motion, 38; of + rotation, 152, 155; of size, 33 +Relativity, Newtonian, 40; restricted + Principle of, 20; standpoint of, 28 +Repulsion of light proceeding radially, + 102, 108 +Retardation of time, 24, 55; in centrifugal + field, 129; in spherical world, + 160 +Ricci, 89 +Riemann, 2, 89, 167 +Riemann-Christoffel tensor, 89 +Riemannian, or non-Euclidean, geometry, + 6, 73, 84, 90 +Rigid scale, definition of, 3 +Rotation, absolute, 152, 164, 194 +Rotation of a continuous ring, 194 +Russell, 14, 197 + +St John, 130 +Semi-Euclidean geometry, 47 +Simultaneity, 12, 51 +de Sitter, 134, 159, 179 +Sobral, eclipse expedition to, 117 +Space, conventional, 9; kinds of, 81; + meaning of, 3, 8, 15; relativity of, + 34 +Space-like intervals, 60, 187 +Space-time, 45; due to Minkowski, 212; + partitions of, 54; +Spherical space-time, 159 +Standard metre, comparison with, 168 +Stresses in continuous matter, 193 +Structure opposed to content, 197, 200 +Structure, geodesic, absolute character + of, 155, 164; acceleration of, 195; + behaviour at infinity, 157 +Super-observer, Newton's, 68 +Synthesis of appearances, 31, 182 + +Tensors, 89, 189 +Thomson, J. J., 61 +Time, absolute, 163; depends on + observer's track, 38, 57; for moving + observer, 24; imaginary, 48; measurement + of, 13; past and future, +51; ``standing still,'' 26, 160 +Time-like intervals, 60, 187 +Tracks, natural, 70 + +Vacuum, defined by law of gravitation, + 190 +Vector, non-integrable on Weyl's + theory, 174 +Velocity, addition-law, 59; definition + of, 193; static character, 194 +Velocity of gravitation, 94, 147 +Velocity of light, importance of, 60; + in gravitational field, 108; system + moving with, 26, 56 + +Warping of space, 8, 126 +Wave-front, slewing of, 108 +Weight, of light, 107, 111; of radio-active + energy, 111; proportional to + inertia, 137; vanishes inside free + projectile, 65 +Weyl, 174 +World, 186, 187 +World-line, 87 +\fi + +\cleardoublepage + +\fancyhead[C]{\textsc{INDEX}} +{\small +\printindex + +} + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage + +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\fancyhead[C]{\textsc{LICENSING}} + +\begin{PGtext} +End of Project Gutenberg's Space, Time and Gravitation, by A. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..398005a --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #29782 (https://www.gutenberg.org/ebooks/29782) |
