The Mathematical Theory of Relativity

The Project Gutenberg EBook of The Mathematical Theory of Relativity, by Arthur Stanley Eddington

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Title: The Mathematical Theory of Relativity

Author: Arthur Stanley Eddington

Release Date: April 11, 2019 [EBook #59248]

Language: English

*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL THEORY OF RELATIVITY ***

Produced by Andrew D. Hwang

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\raggedright \TransNoteText \end{minipage} %%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% \FrontMatter \PageSep{i} \begin{center} \null\vfill \LARGE THE MATHEMATICAL THEORY \medskip

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  \LARGE RELATIVITY

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  CAMBRIDGE UNIVERSITY PRESS \\

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  \LARGE THE MATHEMATICAL THEORY

  \medskip

  \large OF

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\LARGE RELATIVITY \bigskip

\normalsize BY \\ \Large A. S. EDDINGTON, M.A., \textsc{M.Sc.}, F.R.S. \bigskip

\scriptsize PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL \\[6pt] PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE \vfill

\large CAMBRIDGE \\[6pt] AT THE UNIVERSITY PRESS \\[6pt] 1923 \end{center} \newpage \PageSep{iv} \begin{center} \null\vfill \scriptsize PRINTED IN GREAT BRITAIN \end{center} \newpage \PageSep{v}

\Preface

\First{A FIRST} draft of this book was published in 1921 as a mathematical supplement to the French Edition of \Title{Space, Time and Gravitation}. During the ensuing eighteen months I have pursued my intention of developing it into a more systematic and comprehensive treatise on the mathematical theory of Relativity. The matter has been rewritten, the sequence of the argument rearranged in many places, and numerous additions made throughout; so that the work is now expanded to three times its former size. It is hoped that, as now enlarged, it may meet the needs of those who wish to enter fully into these problems of reconstruction of theoretical physics.

The reader is expected to have a general acquaintance with the less technical discussion of the theory given in \Title{Space, Time and Gravitation}, although there is not often occasion to make direct reference to it. But it is eminently desirable to have a general grasp of the revolution of thought associated with the theory of Relativity before approaching it along the narrow lines of strict mathematical deduction. In the former work we explained how the older conceptions of physics had become untenable, and traced the gradual ascent to the ideas which must supplant them. Here our task is to formulate mathematically this new conception of the world and to follow out the consequences to the fullest extent.

The present widespread interest in the theory arose from the verification of certain minute deviations from Newtonian laws. To those who are still hesitating and reluctant to leave the old faith, these deviations will remain the chief centre of interest; but for those who have caught the spirit of the new ideas the observational predictions form only a minor part of the subject. It is claimed for the theory that it leads to an understanding of the world of physics clearer and more penetrating than that previously attained, and it has been my aim to develop the theory in a form which throws most light on the origin and significance of the great laws of physics.

It is hoped that difficulties which are merely analytical have been minimised by giving rather fully the intermediate steps in all the proofs with abundant cross-references to the auxiliary formulae used.

For those who do not read the book consecutively attention may be called to the following points in the notation. The summation convention (\PageRef{50}) is used. German letters always denote the product of the corresponding English letter by~$\sqrt{-g}$ (\PageRef{111}). $\Ham$~is the symbol for ``Hamiltonian differentiation'' introduced on \PageRef{139}. An asterisk is prefixed to symbols generalised so as to be independent of or covariant with the gauge (\PageRef{203}). \PageSep{vi}

A selected list of original papers on the subject is given in the Bibliography at the end, and many of these are sources (either directly or at second-hand) of the developments here set forth. To fit these into a continuous chain of deduction has involved considerable modifications from their original form, so that it has not generally been found practicable to indicate the sources of the separate sections. A frequent cause of deviation in treatment is the fact that in the view of most contemporary writers the Principle of Stationary Action is the final governing law of the world; for reasons explained in the text I am unwilling to accord it so exalted a position. After the original papers of Einstein, and those of de~Sitter from which I first acquired an interest in the theory, I am most indebted to Weyl's \Title{Raum, Zeit, Materie}. Weyl's influence will be especially traced in \SecRefs{49}, \SecNum{58}, \SecNum{59}, \SecNum{61}, \SecNum{63}, as well as in the sections referring to his own theory.

I am under great obligations to the officers and staff of the University

Press for their help and care in the intricate printing.

\Signature{A. S. E.}{10 \emph{August} 1922.}

\PageSep{vii}

\TableofContents

\iffalse%****

CONTENTS

PAGE

INTRODUCTION 1

CHAPTER I

ELEMENTARY PRINCIPLES

SECTION

1. Indeterminateness of the space-time frame

2. The fundamental quadratic form

3. Measurement of intervals

4. Rectangular coordinates and time

5. The Lorentz transformation

6. The velocity of light

7. Timelike and spacelike intervals

8. Immediate consciousness of time

9. The ``3 + 1 dimensional'' world

10. The FitzC4erald contraction

11. Simultaneity at different places

12. Momentum and Mass

13. Energy

14. Density and temperature

15. General transformations of coordinates

16. Fields of force

17. The Principle of Equivalence

18. Retrospect

CHAPTER II

THE TENSOR CALCULUS

19. Contravariant and covariant vectors

20. The mathematical notion of a vector

21. The physical notion of a vector

22. The summation convention

23. Tensors

24. Inner multiplication and contraction

25. The fundamental tensors

26. Associated tensors

27. Christoffel's 3-index symbols

28. Equations of a geodesic

29. Covariant derivative of a vector

30. Covariant derivative of a tensor

31. Alternative discussion of the covariant derivative

32. Surface-elements and Stokes's theorem

33. Significance of covariant differentiation

34. The Riemann-Christoffel tensor

35. Miscellaneous formulae \PageSep{viii}

CONTENTS

CHAPTER III

THE LAW OF GRAVITATION

SECTION PAGE

36. The condition for flat space-time. Natural coordinates

37. Einstein's law of gravitation

38. The gravitational field of an isolated particle

39. Planetary orbits

40. The advance of perihelion

41. The deflection of light

42. Displacement of the Fraunhofer lines

43. Isotropic coordinates

44. Problem of two bodies—-Motion of the moon

45. Solution for a particle in a curved world

46. Transition to continuous matter

47. Experiment and deductive theory

CHAPTER IV

RELATIVITY MECHANICS

48. The antisymmetrical tensor of the fourth rank 107

49. Element of volume. Tensor-density 109

50. The problem of the rotating disc 112

51. The divergence of a tensor 113

52. The four identities 115

53. The material energy-tensor 116

54. New derivation of Einstein's law of gravitation 119

55. The force 122

56. Dynamics of a particle 125

57. Equality of gravitational and inertial mass. Gravitational waves 128

58. Lagrangian form of the gravitational equations 131

59. Pseudo-energy-tensor of the gravitational field 134

60. Action 137

61. A property of invariants 140

62. Alternative energy-tensors 141

63. Gravitational flux from a particle 144

64. Retrospect 146

CHAPTER V

CURVATURE OF SPACE AND TIME

65. Curvature of a four-dimensional manifold

66. Interpretation of Einstein's law of gravitation

67. Cylindrical and spherical space-time

68. Elliptical space

69. Law of gravitation for curved space-time

70. Properties of de Sitter's spherical world

71. Properties of Einstein's cylindrical world

72. The problem of the homogeneous sphere \PageSep{ix}

CONTENTS

CHAPTER VI

ELECTRICITY

SECTION PAGE

73. The electromagnetic equations

74. Electromagnetic waves

75. The Lorentz transformation of electromagnetic force

76. Mechanical effects of the electromagnetic field

77. The electromagnetic energy-tensor

78. The gravitational field of an electron

79. Electromagnetic action

80. Explanation of the mechanical force

81. Electromagnetic volume

82. Macroscopic equations

CHAPTER VII

WORLD GEOMETRY

Part I. Weyl's Theory

83. Natural geometry and world geometry

84. Non-integrability of length

85. Transformation of gauge-systems

86. Gauge-invariance

87. The generalised Riemann-Christoffel tensor

88. The in-invariants of a region

89. The natural gauge

90. Weyl's action-principle

Part II. Generalised Theory

91. Parallel displacement

92. Displacement round an infinitesimal circuit

93. Introduction of a metric

94. Evaluation of the fundamental in-tensors

95. The natural gauge of the world

96. The principle of identification

97. The bifurcation of geometry and electrodynamics

98. General relation-structure

99. The tensor *B_{\mu\nu\sigma}^{\eps}.

100. Dynamical consequences of the general properties of world-invariants

101. The generalised volume

102. Numerical values

103. Conclusion

Bibliography

Index

\fi%****

\PageSep{x}

\PageSep{1}

\MainMatter

\Matter{Introduction}

\index{Differentiation|seealso{Derivative}} \index{Distance|see{Length}} \index{Gravitation|seealso{Einstein's law}} \index{Ponderomotive force|see{Mechanical force}} \index{Proper-|see{Invariant mass \emph{and} Density}}

\First{The} subject of this mathematical treatise is not pure mathematics but \index{Mathematics contrasted with physics}% physics. The vocabulary of the physicist comprises a number of words such as length, angle, velocity, force, work, potential, current, etc., which we shall \index{Length!definition of}% call briefly ``physical quantities.'' Some of these terms occur in pure mathematics \index{Physical quantities}% also; in that subject they may have a generalised meaning which does not concern us here. The pure mathematician deals with ideal quantities defined as having the properties which he deliberately assigns to them. But in an experimental science we have to discover properties not to assign them; and physical quantities are defined primarily according to the way in which we recognise them when confronted by them in our observation of the world around us.

Consider, for example, a length or distance between two points. It is a numerical quantity associated with the two points; and we all know the procedure followed in practice in assigning this numerical quantity to two points in nature. A definition of distance will be obtained by stating the exact procedure; that clearly must be the primary definition if we are to make sure of using the word in the sense familiar to everybody. The pure mathematician proceeds differently; he defines distance as an attribute of the two points which obeys certain laws—-the axioms of the geometry which he happens to have chosen—-and he is not concerned with the question how this ``distance'' would exhibit itself in practical observation. So far as his own investigations are concerned, he takes care to use the word self-consistently; but it does not necessarily denote the thing which the rest of mankind are accustomed to recognise as the distance of the two points.

To find out any physical quantity we perform certain practical operations followed by calculations; the operations are called experiments or observations according as the conditions are more or less closely under our control. The physical quantity so discovered is primarily the result of the operations and calculations; it is, so to speak, \emph{a manufactured article}—-manufactured by \index{Manufacture of physical quantities}% our operations. But the physicist is not generally content to believe that the quantity he arrives at is something whose nature is inseparable from the kind of operations which led to it; he has an idea that if he could become a god contemplating the external world, he would see his manufactured physical quantity forming a distinct feature of the picture. By finding that he can lay $x$~unit measuring-rods in a line between two points, he has manufactured the quantity~$x$ which he calls the distance between the points; but he believes that that distance~$x$ is something already existing in the picture of the world—-a gulf which would be apprehended by a superior intelligence as existing in itself without reference to the notion of operations with measuring-rods. \PageSep{2} Yet he makes curious and apparently illogical discriminations. The parallax of a star is found by a well-known series of operations and calculations; the distance across the room is found by operations with a tape-measure. Both parallax and distance are quantities manufactured by our operations; but for some reason we do not expect parallax to appear as a distinct element in the true picture of nature in the same way that distance does. Or again, instead of cutting short the astronomical calculations when we reach the parallax, we might go on to take the cube of the result, and so obtain another manufactured quantity, a ``cubic parallax.'' For some obscure reason we expect to see distance appearing plainly as a gulf in the true world-picture; parallax does not appear directly, though it can be exhibited as an angle by a comparatively simple construction; and cubic parallax is not in the picture at all. The physicist would say that he \emph{finds} a length, and \emph{manufactures} a cubic parallax; but it is only because he has inherited a preconceived theory of the world that he makes the distinction. We shall venture to challenge this distinction.

Distance, parallax and cubic parallax have the same kind of potential existence even when the operations of measurement are not actually made—-\emph{if} you will move sideways you will be able to determine the angular shift, \emph{if} you will lay measuring-rods in a line to the object you will be able to count their number. Any one of the three is an indication to us of some existent condition or relation in the world outside us—-a condition not created by our operations. But there seems no reason to conclude that this world-condition \emph{resembles} distance any more closely than it resembles parallax or cubic parallax. Indeed any notion of ``resemblance'' between physical quantities and the world-conditions underlying them seems to be inappropriate. If the length~$AB$ is double the length~$CD$, the parallax of~$B$ from~$A$ is half the parallax of~$D$ from~$C$; there is undoubtedly some world-relation which is different for $AB$ and~$CD$, but there is no reason to regard the world-relation of~$AB$ as being better represented by double than by half the world-relation of~$CD$.

The connection of manufactured physical quantities with the existent world-condition can be expressed by saying that the physical quantities are \emph{measure-numbers} of the world-condition. Measure-numbers may be assigned \index{Measure-code}% according to any code, the only requirement being that the same measure-number always indicates the same world-condition and that different world-conditions receive different measure-numbers. Two or more physical quantities may thus be measure-numbers of the same world-condition, \emph{but in different codes}, e.g.\ parallax and distance; mass and energy; stellar magnitude and luminosity. The constant formulae connecting these pairs of physical quantities give the relation between the respective codes. But in admitting that physical quantities can be used as measure-numbers of world-conditions existing independently of our operations, we do not alter their status as manufactured quantities. The same series of operations will naturally manufacture the \PageSep{3} same result when world-conditions are the same, and different results when they are different. (Differences of world-conditions which do not influence the results of experiment and observation are \Foreign{ipso facto} excluded from the domain of physical knowledge.) The size to which a crystal grows may be a measure-number of the temperature of the mother-liquor; but it is none the less a manufactured size, and we do not conclude that the true nature of size is caloric.

The study of physical quantities, although they are the results of our \index{Physical quantities!definition of}% own operations (actual or potential), gives us some kind of knowledge of the world-conditions, since the same operations will give different results in different world-conditions. It seems that this indirect knowledge is all that we can ever attain, and that it is only through its influences on such operations that we can represent to ourselves a ``condition of the world.'' Any \index{Condition of the world}% attempt to describe a condition of the world otherwise is either mathematical symbolism or meaningless jargon. To grasp a condition of the world as completely as it is in our power to grasp it, we must have in our minds a symbol which comprehends at the same time its influence on the results of all possible kinds of operations. Or, what comes to the same thing, we must contemplate its measures according to all possible measure-codes—-of course, without confusing the different codes. It might well seem impossible to realise so comprehensive an outlook; but we shall find that the mathematical calculus of tensors does represent and deal with world-conditions precisely in this way. A tensor expresses simultaneously the whole group of measure-numbers associated with any world-condition; and machinery is provided for keeping the various codes distinct. For this reason the somewhat difficult tensor calculus is not to be regarded as an evil necessity in this subject, which ought if possible to be replaced by simpler analytical devices; our knowledge of conditions in the external world, as it comes to us through observation and experiment, is precisely of the kind which can be expressed by a tensor and not otherwise. And, just as in arithmetic we can deal freely with a billion objects without trying to visualise the enormous collection; so the tensor calculus enables us to deal with the world-condition in the totality of its aspects without attempting to picture it.

Having regard to this distinction between physical quantities and world-conditions, we shall not define a physical quantity as though it were a feature in the world-picture which had to be sought out. \emph{A physical quantity is defined by the series of operations and calculations of which it is the result.} The tendency to this kind of definition had progressed far even in pre-relativity physics. Force had become ``$\text{mass} × \text{acceleration}$,'' and was no longer an invisible agent in the world-picture, at least so far as its definition was concerned. Mass is defined by experiments on inertial properties, no longer as ``quantity of matter.'' But for some terms the older kind of definition (or lack of definition) has been obstinately adhered to; and for these the relativity \PageSep{4} theory must find new definitions. In most cases there is no great difficulty in framing them. We do not need to ask the physicist what conception he attaches to ``length''; we watch him measuring length, and frame our definition according to the operations he performs. There may sometimes be cases in which theory outruns experiment and requires us to decide between two definitions, either of which would be consistent with present experimental practice; but usually we can foresee which of them corresponds to the ideal which the experimentalist has set before himself. For example, until recently the practical man was never confronted with problems of non-Euclidean space, and it might be suggested that he would be uncertain how to construct a straight line when so confronted; but as a matter of fact he showed no hesitation, and the eclipse observers measured without ambiguity the bending of light from the ``straight line.'' The appropriate practical definition was so obvious that there was never any danger of different people meaning different loci by this term. Our guiding rule will be that a physical quantity must be defined by prescribing operations and calculations which will lead to an unambiguous result, and that due heed must be paid to existing practice; the last clause should secure that everyone uses the term to denote the same \emph{quantity}, however much disagreement there may be as to the \emph{conception} attached to it.

When defined in this way, there can be no question as to whether the operations give us the real physical quantity or whether some theoretical correction (not mentioned in the definition) is needed. The physical quantity is the measure-number of a world-condition in some code; we cannot assert that a code is right or wrong, or that a measure-number is real or unreal; what we require is that the code should be the accepted code, and the measure-number the number in current use. For example, what is the real difference of time between two events at distant places? The operation of determining time has been entrusted to astronomers, who (perhaps for mistaken reasons) have elaborated a regular procedure. If the times of the two events are found in accordance with this procedure, the difference must be the real difference of time; the phrase has no other meaning. But there is a certain generalisation to be noticed. In cataloguing the operations of the astronomers, so as to obtain a definition of time, we remark that one condition is adhered to in practice evidently from necessity and not from design—-the observer and his apparatus are placed on the earth and move with the earth. This condition is so accidental and parochial that we are reluctant to insist on it in our definition of time; yet it so happens that the motion of the apparatus makes an important difference in the measurement, and without this restriction the operations lead to no definite result and cannot define anything. We adopt what seems to be the commonsense solution of the difficulty. We decide that time is \emph{relative to an observer}; that is to say, we admit that an observer on another star, who carries out all the rest of the operations and calculations \PageSep{5} as specified in our definition, is also measuring time—-not our time, but a time relative to himself. The same relativity affects the great majority of elementary physical quantities\footnotemark;\footnotetext {The most important exceptions are number (of discrete entities), action, and entropy.\index{Absolute change!physical quantities@physical quantities|indexfn}} the description of the operations is insufficient to lead to a unique answer unless we arbitrarily prescribe a particular motion of the observer and his apparatus.

In this example we have had a typical illustration of ``relativity,'' the \index{Relativity of physical quantities}% recognition of which has had far-reaching results revolutionising the outlook of physics. Any operation of measurement involves a comparison between a measuring-appliance and the thing measured. Both play an equal part in the comparison and are theoretically, and indeed often practically, interchangeable; for example, the result of an observation with the meridian circle gives the right ascension of the star or the error of the clock indifferently, and we can regard either the clock or the star as the instrument or the object of measurement. Remembering that physical quantities are results of comparisons of this kind, it is clear that they cannot be considered to belong solely to one partner in the comparison. It is true that we standardise the measuring appliance as far as possible (the method of standardisation being explained or implied in the definition of the physical quantity) so that in general the variability of the measurement can only indicate a variability of the object measured. To that extent there is no great practical harm in regarding the measurement as belonging solely to the second partner in the relation. But even so we have often puzzled ourselves needlessly over paradoxes, which disappear when we realise that the physical quantities are not properties of certain external objects but are relations between these objects and something else. Moreover, we have seen that the standardisation of the measuring-appliance is usually left incomplete, as regards the specification of its motion; and rather than complete it in a way which would be arbitrary and pernicious, we prefer to recognise explicitly that our physical quantities belong not solely to the objects measured but have reference also to the particular frame of motion that we choose.

The principle of relativity goes still further. Even if the measuring-appliances were standardised completely, the physical quantities would still involve the properties of the constant standard. We have seen that the world-condition or object which is surveyed can only be apprehended in our knowledge as the sum total of all the measurements in which it can be concerned; any \emph{intrinsic} property of the object must appear as a uniformity or law in these measures. When one partner in the comparison is fixed and the other partner varied widely, whatever is common to all the measurements may be ascribed exclusively to the first partner and regarded as an intrinsic property of it. Let us apply this to the converse comparison; that is to say, keep the measuring-appliance constant or standardised, and vary as widely as possible the objects measured—-or, in simpler terms, make a particular \PageSep{6} kind of measurement in all parts of the field. Intrinsic properties of the measuring-appliance should appear as uniformities or laws in these measures. We are familiar with several such uniformities; but we have not generally recognised them as properties of the measuring-appliance. We have called them \emph{laws of nature}.

The development of physics is progressive, and as the theories of the external world become crystallised, we often tend to replace the elementary physical quantities defined through operations of measurement by theoretical quantities believed to have a more fundamental significance in the external world. Thus the \Foreign{vis viva} $mv^{2}$, which is immediately determinable by experiment, becomes replaced by a generalised energy, virtually defined by having the property of conservation; and our problem becomes inverted—-we have not to discover the properties of a thing which we have recognised in nature, but to discover how to recognise in nature a thing whose properties we have assigned. This development seems to be inevitable; but it has grave drawbacks especially when theories have to be reconstructed. Fuller knowledge may show that there is nothing in nature having precisely the properties assigned; or it may turn out that the thing having these properties has entirely lost its importance when the new theoretical standpoint is adopted\footnotemark.\footnotetext {We shall see in \SecRef{59} that this has happened in the case of energy. The dead-hand of a superseded theory continues to embarrass us, because in this case the recognised terminology still has implicit reference to it. This, however, is only a slight drawback to set off against the many advantages obtained from the classical generalisation of energy as a step towards the more complete theory.}%

When we decide to throw the older theories into the melting-pot and make a clean start, it is best to relegate to the background terminology associated with special hypotheses of physics. Physical quantities defined by operations of measurement are independent of theory, and form the proper starting-point for