THE MATHEMATICAL THEORY OF RELATIVITY BY A. S. EDDINGTON, M.A., M.SC., F.R.S. PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE With a Foreword by Tigran Aivazian THIRD EDITION ahb Bibles.org.uk, 2022 First Edition, Cambridge University Press 1923 Second Edition, Cambridge University Press 1924 Third Edition, Bibles.org.uk 2022 Please send all comments to: [email protected]  Typeset with XELATEX of TEX Live 2021 under Linux. PDF version/date: Colour 10”, Sunday 19th June, 2022 Source Code: https://github.com/tigran123/mtor CONTENTS EDITOR’S FOREWORD ............................................. 8 PREFACE ........................................................ 16 INTRODUCTION .................................................. 19 CHAPTER I ELEMENTARY PRINCIPLES 1. Indeterminateness of the space-time frame ......................... 30 2. The fundamental quadratic form................................... 34 3. Measurement of intervals ......................................... 35 4. Rectangular coordinates and time.................................. 38 5. The Lorentz transformation....................................... 44 6. The velocity of light .............................................. 46 7. Timelike and spacelike intervals ................................... 52 8. Immediate consciousness of time................................... 54 9. The “3+1 dimensional” world .................................... 59 10. The FitzGerald contraction........................................ 60 11. Simultaneity at different places .................................... 63 12. Momentum and mass............................................. 66 13. Energy.......................................................... 70 14. Density and temperature.......................................... 73 15. General transformations of coordinates............................. 75 16. Fields of force.................................................... 79 17. The Principle of Equivalence ...................................... 83 18. Retrospect....................................................... 85 CONTENTS 4 CHAPTER II THE TENSOR CALCULUS 19. Contravariant and covariant vectors................................ 88 20. The mathematical notion of a vector ............................... 90 21. The physical notion of a vector .................................... 94 22. The summation convention........................................ 99 23. Tensors.......................................................... 101 24. Inner multiplication and contraction. The quotient law .............. 103 25. The fundamental tensors.......................................... 108 26. Associated tensors................................................ 110 27. Christoffel’s 3-index symbols...................................... 114 28. Equations of a geodesic........................................... 115 29. Covariant derivative of a vector.................................... 117 30. Covariant derivative of a tensor.................................... 119 31. Alternative discussion of the covariant derivative.................... 123 32. Surface-elements and Stokes’s theorem............................. 125 33. Significance of covariant differentiation............................. 128 34. The Riemann-Christoffel tensor ................................... 133 35. Miscellaneous formulae........................................... 138 CHAPTER III THE LAW OF GRAVITATION 36. The condition for flat space-time. Natural coordinates............... 142 37. Einstein’s law of gravitation ....................................... 150 38. The gravitational field of an isolated particle........................ 153 39. Planetary orbits .................................................. 157 40. The advance of perihelion......................................... 160 41. The deflection of light ............................................ 164 42. Displacement of the Fraunhofer lines .............................. 166 43. Isotropic coordinates.............................................. 169 44. Problem of two bodies—Motion of the moon....................... 172 CONTENTS 5 45. Solution for a particle in a curved world............................ 180 46. Transition to continuous matter.................................... 181 47. Experiment and deductive theory.................................. 187 CHAPTER IV RELATIVITY MECHANICS 48. The antisymmetrical tensor of the fourth rank....................... 191 49. Element of volume. Tensor-density................................ 195 50. The problem of the rotating disc................................... 201 51. The divergence of a tensor......................................... 203 52. The four identities................................................ 205 53. The material energy-tensor........................................ 209 54. New derivation of Einstein’s law of gravitation...................... 212 55. The force ........................................................ 218 56. Dynamics of a particle............................................ 222 57. Equality of gravitational and inertial mass. Gravitational waves....... 227 58. Lagrangian form of the gravitational equations...................... 242 59. Pseudo-energy-tensor of the gravitational field...................... 245 60. Action........................................................... 250 61. A property of invariants........................................... 254 62. Alternative energy-tensors......................................... 256 63. Gravitational flux from a particle .................................. 260 64. Retrospect....................................................... 264 CHAPTER V CURVATURE OF SPACE AND TIME 65. Curvature of a four-dimensional manifold.......................... 268 66. Interpretation of Einstein’s law of gravitation ....................... 274 67. Cylindrical and spherical space-time ............................... 278 68. Elliptical space................................................... 282 69. Law of gravitation for curved space-time........................... 284 CONTENTS 6 70. Properties of de Sitter’s spherical world............................. 287 71. Properties of Einstein’s cylindrical world............................ 296 72. The problem of the homogeneous sphere........................... 300 CHAPTER VI ELECTRICITY 73. The electromagnetic equations..................................... 305 74. Electromagnetic waves............................................ 311 75. The Lorentz transformation of electromagnetic force ................ 320 76. Mechanical effects of the electromagnetic field...................... 321 77. The electromagnetic energy-tensor................................. 326 78. The gravitational field of an electron ............................... 330 79. Electromagnetic action............................................ 334 80. Explanation of the mechanical force................................ 337 81. Electromagnetic volume........................................... 344 82. Macroscopic equations............................................ 345 CHAPTER VII WORLD GEOMETRY PART I. WEYL’S THEORY 83. Natural geometry and world geometry.............................. 349 84. Non-integrability of length........................................ 353 85. Transformation of gauge-systems .................................. 355 86. Gauge-invariance................................................. 359 87. The generalised Riemann-Christoffel tensor......................... 362 88. The in-invariants of a region....................................... 363 89. The natural gauge ................................................ 366 90. Weyl’s action-principle............................................ 371 CONTENTS 7 PART II. GENERALISED THEORY 91. Parallel displacement.............................................. 376 92. Displacement round an infinitesimal circuit......................... 378 93. Introduction of a metric........................................... 381 94. Evaluation of the fundamental in-tensors........................... 386 95. The natural gauge of the world .................................... 388 96. The principle of identification ..................................... 392 97. The bifurcation of geometry and electrodynamics.................... 395 98. General relation-structure......................................... 396 99. The tensor B..................................................... 399 100. Dynamical consequences of the general properties of world-invariants. 402 101. The generalised volume ........................................... 415 102. Numerical values................................................. 420 103. Conclusion....................................................... 423 BIBLIOGRAPHY ................................................... 433 INDEX .......................................................... 438 EDITOR’S FOREWORD S it right and proper today, in the year 2022 A.D., standing I as we are on the brink of the New Epoch of brotherhood of all mankind, to publish a monograph on Special and General Relativity, written by a physicist, who flourished nearly a century ago? It is indeed. Sir Arthur Stanley Eddington (1882–1944) was not “merely” the father of modern stellar astrophysics, who started with the scraps of uncoordinated patchwork of guesses about the internal constitution of stars, which only marginally differed from the idle speculations of antiquity, and proceeded to build the entire body of observationally verified knowledge, which serves as the basis of all stellar astronomy today. Neither was he “merely” a lone genius physicist, who for the first time in history predicted theoretically the value of the mass ratio of protons and electrons, as well as that of the fine structure constant—a feat standing unrivalled to this day. No, he was, first and foremost, the greatest God-knowing physicist who ever lived on this planet. Theoretical physics of the XX century, alas, was not immune to the dangers besetting the economico-political structure of the world at the time. As is usually com- mon in the poisoned atmosphere permeated by the “spirit of democracy”, which, as we know, fosters the election of base and ignorant rulers and glorifies mediocrity by placing the universal suffrage in the hands of uneducated and in- dolent majorities, the science was forced to pursue the false materialistic goals of a few mediocre (relatively speaking) men like Niels Bohr and Max Born, forgetting for a season