Ⅰ．INTRODUCTION 1
1．The Subject-Matter 1
2．The Method of Presentation 7
3．The Point of View 9
Ⅱ．THE SPECIAL THEORY OF RELATIVITY 12
Part Ⅰ．THE TWO POSTULATES AND THE LORENTZ TRANSFORMATION 12
4．Introduction 12
5．The First Postulate of Relativity 12
6．The Second Postulate of Relativity 15
7．Necessity for Modifying Older Ideas as to Space and Time 17
8．The Lorentz Transformation Equations 18
9．Transformation Equations for Spatial and Temporal Intervals Lorentz Contraction and Time Dilation 22
10．Transformation Equations for Velocity 25
11．Transformation Equation for the Lorentz Contraction Factor 27
12．Transformation Equations for Acceleration 27
Part Ⅱ．TREATMENT OF SPECIAL RELATIVITY WITH THE HELP OF A FOUR-DIMENSIONAL GEOMETRY 28
13．The Space-Time Continuum 28
14．The Three Plus One Dimensions of Space-Time 29
15．The Geometry Corresponding to Space-Time 30
16．The Signature of the Line Element and the Three Kinds of Interval 31
17．The Lorentz Rotation of Axes 32
18．The Transformation to Proper Coordinates 33
19．Use of Tensor Analysis in the Theory of Relativity 34
20．Simplification of Tensor Analysis in the Case of Special Relativity Galilean Coordinates 37
21．Correspondence of Four-Dimensional Treatment with the Postulates of Special Relativity 39
Ⅲ．SPECIAL RELATIVITY AND MECHANICS 42
Part Ⅰ．THE DYNAMICS OF A PARTICLE 42
22．The Principles of the Conservation of Mass and Momentum 42
23．The Mass of a Moving Particle 43
24．The Transformation Equations for Mass 45
25．The Definition and Transformation Equations for Force 45
26．Work and Kinetic Energy 47
27．The Relations between Mass，Energy，and Momentum 48
28．Four-Dimensional Expression of the Mechanics of a Particle 50
29．Applications of the Dynamics of a Particle 53
(a) The Mass of High-Velocity Electrons 53
(b) The Relation between Force and Acceleration 54
(c) Applications in Electromagnetic Theory 55
(d) Tests of the Interrelation of Mass，Energy，and Momentum 57
Part Ⅱ．THE DYNAMICS OF A CONTINUOUS MECHANICAL MEDIUM 59
30．The Principles Postulated 59
31．The Conservation of Momentum and the Components of Stress tij 60
32．The Equations of Motion in Terms of the Stresses tij 60
33．The Equation of Continuity 62
34．The Transformation Equations for the Stresses tij 62
35．The Transformation Equations for the Densities of Mass and Momentum 65
36．Restatement of Results in Terms of the (Absolute) Stresses pij 69
37．Four-Dimensional Expression of the Mechanics of a Continuous Medium 70
38．Applications of the Mechanics of a Continuous Medium 73
(a) The Mass and Momentum of a Finite System 74
(b) The Angular Momentum of a Finite System 77
(c) The Right-Angled Lever as an Example 79
(d) The Complete Static System 80
Ⅳ．SPECIAL RELATIVITY AND ELECTRODYNAMICS 84
Part Ⅰ．ELECTRON THEORY 84
39．The Maxwell-Lorentz Field Equations 84
40．The Transformation Equations for E，H，and ρ 86
41．The Force on a Moving Charge 88
42．The Energy and Momentum of the Electromagnetic Field 89
43．The Electromagnetic Stresses 91
44．Transformation Equations for Electromagnetic Densities and Stresses 92
45．Combined Result of Mechanical and Electromagnetic Actions 93
46．Four-Dimensional Expression of the Electron Theory 95
(a) The Field Equations 95
(b) Four-Dimensional Expression of Force on Moving Charge 98
(c) Four-Dimensional Expression of Electromagnetic Energy-Momentum Tensor 99
47．Applications of the Electron Theory 99
Part Ⅱ．MACROSCOPIC THEORY 101
48．The Field Equations for Stationary Matter 101
49．The Constitutive Equations for Stationary Matter 102
50．The Field Equations in Four-Dimensional Language 102
51．The Constitutive Equations in Four-Dimensional Language 104
52．The Field Equations for Moving Matter in Ordinary Vector Language 105
53．The Constitutive Equations for Moving Matter in Ordinary Vector Language 108
54．Applications of the Macroscopic Theory 109
(a) The Conservation of Electric Charge 109
(b) Boundary Conditions 110
(c) The Joule Heating Effect 112
(d) Electromagnetic Energy and Momentum 113
(e) The Energy-Momentum Tensor 115
(f) Applications to Experimental Observations 116
Ⅴ．SPECIAL RELATIVITY AND THERMODYNAMICS 118
Part Ⅰ．THE THERMODYNAMICS OF STATIONARY SYSTEMS 118
55．Introduction 118
56．The First Law of Thermodynamics and the Zero Point of Energy Content 120
57．The Second Law of Thermodynamics and the Starting-Point for Entropy Content 121
58．Heat Content，Free Energy，and Thermodynamic Potential 123
59．General Conditions for Thermodynamic Change and Equilibrium 125
60．Conditions for Change and Equilibrium in Homogeneous Systems 127
61．Uniformity of Temperature at Thermal Equilibrium 130
62．Irreversibility and Rate of Change 132
63．Final State of an Isolated System 134
64．Energy and Entropy of a Perfect Monatomic Gas 136
65．Energy and Entropy of Black-Body Radiation 139
66．The Equilibrium between Hydrogen and Helium 140
67．The Equilibrium between Matter and Radiation 146
Part Ⅱ．THE THERMODYNAMICS OF MOVING SYSTEMS 152
68．The Two Laws of Thermodynamics for a Moving System 152
69．The Lorentz Transformation for Thermodynamic Quantities 153
(a) Volume and Pressure 153
(b) Energy 154
(c) Work 156
(d) Heat 156
(e) Entropy 157
(f) Temperature 158
70．Thermodynamic Applications 159
(a) Carnot Cycle Involving Change in Velocity 159
(b) The Dynamics of Thermal Radiation 161
71．Use of Four-Dimensional Language in Thermodynamics 162
Ⅵ．THE GENERAL THEORY OF RELATIVITY 165
Part Ⅰ．THE FUNDAMENTAL PRINCIPLES OF GENERAL RELATIVITY 165
72．Introduction 165
73．The Principle of Covariance 166
(a) Justification for the Principle of Covariance 166
(b) Consequences of the Principle of Covariance 167
(c) Method of Obtaining Covariant Expressions 168
(d) Covariant Expression for Interval 169
(e) Covariant Expression for the Trajectories of Free Particles and Light Rays 171
74．The Principle of Equivalence 174
(a) Formulation of the Principle of Equivalence．Metric and Gravitation 174
(b) Principle of Equivalence and Relativity of Motion 176
(c) Justification for the Principle of Equivalence 179
(d) Use of the Principle of Equivalence in Generalizing the Principles of Special Relativity．Natural and Proper Coordinates 180
(e) Interval and Trajectory in the Presence of Gravitational Fields 181
75．The Dependence of Gravitational Field and Metric on the Distribution of Matter and Energy．Principle of Mach 184
76．The Field Corresponding to the Special Theory of Relativity．The Riemann-Christoffel Tensor 185
77．The Gravitational Field in Empty Space．The Contracted Riemann-Christoffel Tensor 187
78．The Gravitational Field in the Presence of Matter and Energy 188
Part Ⅱ．ELEMENTARY APPLICATIONS OF GENERAL RELATIVITY 192
79．Simple Consequences of the Principle of Equivalence 192
(a) The Proportionality of Weight and Mass 192
(b) Effect of Gravitational Potential on the Rate of a Clock 192
(c) The Clock Paradox 194
80．Newton＇s Theory as a First Approximation 198
(a) Motion of Free Particle in a Weak Gravitational Field 198
(b) Poisson＇s Equation as an Approximation for Einstein＇s Field Equations 199
81．Units to be Used in Relativistic Calculations 201
82．The Schwarzschild Line Element 202
83．The Three Crucial Tests of Relativity 205
(a) The Advance of Perihelion 208
(b) The Gravitational Deflexion of Light 209
(c) Gravitational Shift in Spectral Lines 211
Ⅶ．RELATIVISTIC MECHANICS 214
Part Ⅰ．SOME GENERAL MECHANICAL PRINCIPLES 214
84．The Fundamental Equations of Relativistic Mechanics 214
85．The Nature of the Energy-Momentum Tensor．General Expression in the Case of a Perfect Fluid 215
86．The Mechanical Behaviour of a Perfect Fluid 218
87．Re-expression of the Equations of Mechanics in the Form of an Ordinary Divergence 222
88．The Energy-Momentum Principle for Finite Systems 225
89．The Densities of Energy and Momentum Expressed as Divergences 229
90．Limiting Values for Certain Quantities at a Large Distance from an Isolated System 230
91．The Mass，Energy and Momentum of an Isolated System 232
92．The Energy of a Quasi-Static Isolated System Expressed by an Integral Extending Only Over the Occupied Space 234
Part Ⅱ．SOLUTIONS OF THE FIELD EQUATIONS 236
93．Einstein＇s General Solution of the Field Equations in the Case of Weak Fields 236
94．Line Elements for Systems with Spherical Symmetry 239
95．Static Line Element with Spherical Symmetry 241
96．Schwarzschild＇s Exterior and Interior Solutions 245
97．The Energy of a Sphere of Perfect Fluid 247
98．Non-Static Line Elements with Spherical Symmetry 250
99．Birkhoff＇s Theorem 252
100．A More General Line Element 253
Ⅷ．RELATIVISTIC ELECTRODYNAMICS 258
Part Ⅰ．THE COVARIANT GENERALIZATION OF ELECTRICAL THEORY 258
101．Introduction 258
102．The Generalized Lorentz Electron Theory．The Field Equations 258
103．The Motion of a Charged Particle 259
104．The Energy-Momentum Tensor 261
105．The Generalized Macroscopic Theory 261
Part Ⅱ．SOME APPLICATIONS OF RELATIVISTIC ELECTRODYNAMICS 264
106．The Conservation of Electric Charge 264
107．The Gravitational Field of a Charged Particle 265
108．The Propagation of Electromagnetic Waves 267
109．The Energy-Momentum Tensor for Disordered Radiation 269
110．The Gravitational Mass of Disordered Radiation 271
111．The Energy-Momentum Tensor Corresponding to a Directed Flow of Radiation 272
112．The Gravitational Field Corresponding to a Directed Flow of Radiation 273
113．The Gravitational Action of a Pencil of Light 274
(a) The Line Element in the Neighbourhood of a Limited Pencil of Light 274
(b) Velocity of a Test Ray of Light in the Neighbourhood of the Pencil 275
(c) Acceleration of a Test Particle in the Neighbourhood of the Pencil 277
114．The Gravitational Action of a Pulse of Light 279
(a) The Line Element in the Neighbourhood of the Limited Track of a Pulse of Light 279
(b) Velocity of a Test Ray of Light in the Neighbourhood of the Pulse 281
(c) Acceleration of a Test Particle in the Neighbourhood of the Pulse 282
115．Discussion of the Gravitational Interaction of Light Rays and Particles 285
116．The Ceneralized Doppler Effect 288
Ⅸ．RELATIVISTIC THERMODYNAMICS 291
Part Ⅰ．THE EXTENSION OF THERMODYNAMICS TO GENERAL RELATIVITY 291
117．Introduction 291
118．The Relativistic Analogue of the First Law of Thermodynamics 292
119．The Relativistic Analogue of the Second Law of Thermodynamics 293
120．On the Interpretation of the Relativistic Second Law of Thermodynamics 296
121．On the Interpretation of Heat in Relativistic Thermodynamics 297
122．On the Use of Co-Moving Coordinates in Thermodynamic Considerations 301
Part Ⅱ．APPLICATIONS OF RELATIVISTIC THERMODYNAMICS 304
123．Application of the First Law to Changes in the Static State of a System 304
124．Application of the Second Law to Changes in the Static State of a System 306
125．The Conditions for Static Thermodynamic Equilibrium 307
126．Static Equilibrium in the Case of a Spherical Distribution of Fluid 308
127．Chemical Equilibrium in a Gravitating Sphere of Fluid 311
128．Thermal Equilibrium in a Gravitating Sphere of Fluid 312
129．Thermal Equilibrium in a General Static Field 315
130．On the Increased Possibility in Relativistic Thermodynamics for Reversible Processes at a Finite Rate 319
131．On the Possibility for Irreversible Processes without Reaching a Final State of Maximum Entropy 326
132．Conclusion 330
Ⅹ．APPLICATIONS TO COSMOLOGY 331
Part Ⅰ．STATIC COSMOLOGICAL MODELS 331
133．Introduction 331
134．The Three Possibilities for a Homogeneous Static Universe 333
135．The Einstein Line Element 335
136．The de Sitter Line Element 335
137．The Special Relativity Line Element 336
138．The Geometry of the Einstein Universe 337
139．Density and Pressure of Material in the Einstein Universe 339
140．Behaviour of Particles and Light Rays in the Einstein Universe 341
141．Comparison of Einstein Model with Actual Universe 344
142．The Geometry of the de Sitter Universe 346
143．Absence of Matter and Radiation from the de Sitter Universe 348
144．Behaviour of Test Particles and Light Rays in the de Sitter Universe 349
(a) The Geodesic Equations 349
(b) Orbits of Particles 351
(c) Behaviour of Light Rays in the de Sitter Universe 353
(d) Doppler Effect in the de Sitter Universe 354
145．Comparison of de Sitter Model with Actual Universe 359
Part Ⅱ．THE APPLICATION OF RELATIVISTIC MECHANICS TO NON-STATIC HOMOGENEOUS COSMOLOGICAL MODELS 361
146．Reasons for Changing to Non-Static Models 361
147．Assumption Employed in Deriving Non-Static Line Element 362
148．Derivation of Line Element from Assumption of Spatial Isotropy 364
149．General Properties of the Line Element 370
(a) Different Forms of Expression for the Line Element 370
(b) Geometry Corresponding to Line Element 371
(c) Result of Transfer of Origin of Coordinates 372
(d) Physical Interpretation of Line Element 375
150．Density and Pressure in Non-Static Universe 376
151．Change in Energy with Time 379
152．Change in Matter with Time 381
153．Behaviour of Particles in the MOdel 383
154．Behaviour of Light Rays in the Model 387
155．The Doppler Effect in the Model 389
156．Change in Doppler Effect with Distance 392
157．General Discussion of Dependence on Time for Closed Models 394
(a) General Features of Time Dependence，R real，ρ00 ？ 0，p0 ？ 0 395
(b) Curve for the Critical Function of R 396
(c) Monotonic Universes of Type M1，for ？ > ？E 399
(d) Asymptotic Universes of Types A1 and A2，for ？ = ？E 400
(e) Monotonic Universes of Type M2 and Oscillating Universes of Types O1 and O2，for 0 < ？ < ？E 401
(f) Oscillating Universes of Type O1，for ？ 0 402
158．General Discussion of Dependence on Time for Open Models 403
159．On the Instability of the Einstein Static Universe 405
160．Models in Which the Amount of Matter is Constant 407
161．Models Which Expand from an Original Static State 409
162．Ever Expanding Models Which do not Start from a Static State 412
163．Oscillating Models (？ = 0) 412
164．The Open Model of Einstein and de Sitter (？ = 0，R0 = ？) 415
165．Discussion of Factors which were Neglected in Studying Special Models 416
Part Ⅲ．THE APPLICATION OF RELATIVISTIC THERMODYNAMICS TO NON-STATIC HOMOGENEOUS COSMOLOGICAL MODELS 420
166．Application of the Relativistic First Law 420
167．Application of the Relativistic Second Law 421
168．The Conditions for Thermodynamic Equilibrium in a Static Einstein Universe 423
169．The Conditions for Reversible and Irreversible Changes in Non-Static Models 424
170．Model Filled with Incoherent Matter Exerting No Pressure as an Example of Reversible Behaviour 426
171．Model Filled with Black-Body Radiation as an Example of Reversible Behaviour 427
172．Discussion of Failure to Obtain Periodic Motions without Singular States 429
173．Interpretation of Reversible Expansions by an Ordinary Observer 432
174．Analytical Treatment of a Succession of Expansions and Contractions for a Closed Model with ？ = 0 435
(a) The Upper Boundary of Expansion 436
(b) Time Necessary to Reach Maximum 436
(c) Time Necessary to Complete Contraction 437
(d) Behaviour at Lower Limit of Contraction 438
175．Application of Thermodynamics to a Succession of Irreversible Expansions and Contractions 439
Part Ⅳ．CORRELATION OF PHENOMENA IN THE ACTUAL UNIVERSE WITH THE HELP OF NON-STATIC HOMOGENEOUS MODELS 445
176．Introduction 445
177．The Observational Data 446
(a) The Absolute Magnitudes of the Nearer Nebulae 446
(b) The Corrected Apparent Magnitudes for more Distant Nebulae 448
(c) Nebular Distances Calculated from Apparent Magnitudes 453
(d) Relation of Observed Red-Shift to Magnitude and Distance 454
(e) Relation of Apparent Diameter to Magnitude and Distance 457
(f) Actual Diameters and Masses of Nebulae 458
(g) Distribution of Nebulae in Space 459
(h) Density of Matter in Space 461
178．The Relation between Coordinate Position and Luminosity 462
179．The Relation between Coordinate Position and Astronomically Determined Distance 465
180．The Relation between Coordinate Position and Apparent Diameter 467
181．The Relation between Coordinate Position and Counts of Nebular Distribution 468
182．The Relation between Coordinate Position and Red-shift 469
183．The Relation of Density to Spatial Curvature and Cosmological Constant 473
184．The Relation between Red-shift and Rate of Disappearance of Matter 475
185．Summary of Correspondences between Model and Actual Universe 478
186．Some General Remarks Concerning Cosmological Models 482
(a) Homogeneity 482
(b) Spatial Curvature 483
(c) Temporal Behaviour 484
187．Our Neighbourhood as a Sample of the Universe as a Whole 486
Appendix Ⅰ．SYMBOLS FOR QUANTITIES 489
Scalar Quantities 489
Vector Quantities 490
Tensors 490
Tensor Densities 491
Appendix Ⅱ．SOME FORMULAE OF VECTOR ANALYSIS 491
Appendix Ⅲ．SOME FORMULAE OF TENSOR ANALYSIS 493
(a) General Notation 493
(b) The Fundamental Metrical Tensor and its Properties 494
(c) Tensor Manipulations 495
(d) Miscellaneous Formulae 496
(e) Formulae Involving Tensor Densities 496
(f) Four-Dimensional Volume．Proper Spatial Volume 496
Appendix Ⅳ．USEFUL CONSTANTS 497
Subject Index 499
Name Index 502