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Ⅰ．INTRODUCTION1

1．The Subject-Matter1

2．The Method of Presentation7

3．The Point of View9

Ⅱ．THE SPECIAL THEORY OF RELATIVITY12

Part Ⅰ．THE TWO POSTULATES AND THE LORENTZ TRANSFORMATION12

4．Introduction12

5．The First Postulate of Relativity12

6．The Second Postulate of Relativity15

7．Necessity for Modifying Older Ideas as to Space and Time17

8．The Lorentz Transformation Equations18

9．Transformation Equations for Spatial and Temporal Intervals Lorentz Contraction and Time Dilation22

10．Transformation Equations for Velocity25

11．Transformation Equation for the Lorentz Contraction Factor27

12．Transformation Equations for Acceleration27

Part Ⅱ．TREATMENT OF SPECIAL RELATIVITY WITH THE HELP OF A FOUR-DIMENSIONAL GEOMETRY28

13．The Space-Time Continuum28

14．The Three Plus One Dimensions of Space-Time29

15．The Geometry Corresponding to Space-Time30

16．The Signature of the Line Element and the Three Kinds of Interval31

17．The Lorentz Rotation of Axes32

18．The Transformation to Proper Coordinates33

19．Use of Tensor Analysis in the Theory of Relativity34

20．Simplification of Tensor Analysis in the Case of Special Relativity Galilean Coordinates37

21．Correspondence of Four-Dimensional Treatment with the Postulates of Special Relativity39

Ⅲ．SPECIAL RELATIVITY AND MECHANICS42

Part Ⅰ．THE DYNAMICS OF A PARTICLE42

22．The Principles of the Conservation of Mass and Momentum42

23．The Mass of a Moving Particle43

24．The Transformation Equations for Mass45

25．The Definition and Transformation Equations for Force45

26．Work and Kinetic Energy47

27．The Relations between Mass，Energy，and Momentum48

28．Four-Dimensional Expression of the Mechanics of a Particle50

29．Applications of the Dynamics of a Particle53

(a) The Mass of High-Velocity Electrons53

(b) The Relation between Force and Acceleration54

(c) Applications in Electromagnetic Theory55

(d) Tests of the Interrelation of Mass，Energy，and Momentum57

Part Ⅱ．THE DYNAMICS OF A CONTINUOUS MECHANICAL MEDIUM59

30．The Principles Postulated59

31．The Conservation of Momentum and the Components of Stress tij60

32．The Equations of Motion in Terms of the Stresses tij60

33．The Equation of Continuity62

34．The Transformation Equations for the Stresses tij62

35．The Transformation Equations for the Densities of Mass and Momentum65

36．Restatement of Results in Terms of the (Absolute) Stresses pij69

37．Four-Dimensional Expression of the Mechanics of a Continuous Medium70

38．Applications of the Mechanics of a Continuous Medium73

(a) The Mass and Momentum of a Finite System74

(b) The Angular Momentum of a Finite System77

(c) The Right-Angled Lever as an Example79

(d) The Complete Static System80

Ⅳ．SPECIAL RELATIVITY AND ELECTRODYNAMICS84

Part Ⅰ．ELECTRON THEORY84

39．The Maxwell-Lorentz Field Equations84

40．The Transformation Equations for E，H，and ρ86

41．The Force on a Moving Charge88

42．The Energy and Momentum of the Electromagnetic Field89

43．The Electromagnetic Stresses91

44．Transformation Equations for Electromagnetic Densities and Stresses92

45．Combined Result of Mechanical and Electromagnetic Actions93

46．Four-Dimensional Expression of the Electron Theory95

(a) The Field Equations95

(b) Four-Dimensional Expression of Force on Moving Charge98

(c) Four-Dimensional Expression of Electromagnetic Energy-Momentum Tensor99

47．Applications of the Electron Theory99

Part Ⅱ．MACROSCOPIC THEORY101

48．The Field Equations for Stationary Matter101

49．The Constitutive Equations for Stationary Matter102

50．The Field Equations in Four-Dimensional Language102

51．The Constitutive Equations in Four-Dimensional Language104

52．The Field Equations for Moving Matter in Ordinary Vector Language105

53．The Constitutive Equations for Moving Matter in Ordinary Vector Language108

54．Applications of the Macroscopic Theory109

(a) The Conservation of Electric Charge109

(b) Boundary Conditions110

(c) The Joule Heating Effect112

(d) Electromagnetic Energy and Momentum113

(e) The Energy-Momentum Tensor115

(f) Applications to Experimental Observations116

Ⅴ．SPECIAL RELATIVITY AND THERMODYNAMICS118

Part Ⅰ．THE THERMODYNAMICS OF STATIONARY SYSTEMS118

55．Introduction118

56．The First Law of Thermodynamics and the Zero Point of Energy Content120

57．The Second Law of Thermodynamics and the Starting-Point for Entropy Content121

58．Heat Content，Free Energy，and Thermodynamic Potential123

59．General Conditions for Thermodynamic Change and Equilibrium125

60．Conditions for Change and Equilibrium in Homogeneous Systems127

61．Uniformity of Temperature at Thermal Equilibrium130

62．Irreversibility and Rate of Change132

63．Final State of an Isolated System134

64．Energy and Entropy of a Perfect Monatomic Gas136

65．Energy and Entropy of Black-Body Radiation139

66．The Equilibrium between Hydrogen and Helium140

67．The Equilibrium between Matter and Radiation146

Part Ⅱ．THE THERMODYNAMICS OF MOVING SYSTEMS152

68．The Two Laws of Thermodynamics for a Moving System152

69．The Lorentz Transformation for Thermodynamic Quantities153

(a) Volume and Pressure153

(b) Energy154

(c) Work156

(d) Heat156

(e) Entropy157

(f) Temperature158

70．Thermodynamic Applications159

(a) Carnot Cycle Involving Change in Velocity159

(b) The Dynamics of Thermal Radiation161

71．Use of Four-Dimensional Language in Thermodynamics162

Ⅵ．THE GENERAL THEORY OF RELATIVITY165

Part Ⅰ．THE FUNDAMENTAL PRINCIPLES OF GENERAL RELATIVITY165

72．Introduction165

73．The Principle of Covariance166

(a) Justification for the Principle of Covariance166

(b) Consequences of the Principle of Covariance167

(c) Method of Obtaining Covariant Expressions168

(d) Covariant Expression for Interval169

(e) Covariant Expression for the Trajectories of Free Particles and Light Rays171

74．The Principle of Equivalence174

(a) Formulation of the Principle of Equivalence．Metric and Gravitation174

(b) Principle of Equivalence and Relativity of Motion176

(c) Justification for the Principle of Equivalence179

(d) Use of the Principle of Equivalence in Generalizing the Principles of Special Relativity．Natural and Proper Coordinates180

(e) Interval and Trajectory in the Presence of Gravitational Fields181

75．The Dependence of Gravitational Field and Metric on the Distribution of Matter and Energy．Principle of Mach184

76．The Field Corresponding to the Special Theory of Relativity．The Riemann-Christoffel Tensor185

77．The Gravitational Field in Empty Space．The Contracted Riemann-Christoffel Tensor187

78．The Gravitational Field in the Presence of Matter and Energy188

Part Ⅱ．ELEMENTARY APPLICATIONS OF GENERAL RELATIVITY192

79．Simple Consequences of the Principle of Equivalence192

(a) The Proportionality of Weight and Mass192

(b) Effect of Gravitational Potential on the Rate of a Clock192

(c) The Clock Paradox194

80．Newton＇s Theory as a First Approximation198

(a) Motion of Free Particle in a Weak Gravitational Field198

(b) Poisson＇s Equation as an Approximation for Einstein＇s Field Equations199

81．Units to be Used in Relativistic Calculations201

82．The Schwarzschild Line Element202

83．The Three Crucial Tests of Relativity205

(a) The Advance of Perihelion208

(b) The Gravitational Deflexion of Light209

(c) Gravitational Shift in Spectral Lines211

Ⅶ．RELATIVISTIC MECHANICS214

Part Ⅰ．SOME GENERAL MECHANICAL PRINCIPLES214

84．The Fundamental Equations of Relativistic Mechanics214

85．The Nature of the Energy-Momentum Tensor．General Expression in the Case of a Perfect Fluid215

86．The Mechanical Behaviour of a Perfect Fluid218

87．Re-expression of the Equations of Mechanics in the Form of an Ordinary Divergence222

88．The Energy-Momentum Principle for Finite Systems225

89．The Densities of Energy and Momentum Expressed as Divergences229

90．Limiting Values for Certain Quantities at a Large Distance from an Isolated System230

91．The Mass，Energy and Momentum of an Isolated System232

92．The Energy of a Quasi-Static Isolated System Expressed by an Integral Extending Only Over the Occupied Space234

Part Ⅱ．SOLUTIONS OF THE FIELD EQUATIONS236

93．Einstein＇s General Solution of the Field Equations in the Case of Weak Fields236

94．Line Elements for Systems with Spherical Symmetry239

95．Static Line Element with Spherical Symmetry241

96．Schwarzschild＇s Exterior and Interior Solutions245

97．The Energy of a Sphere of Perfect Fluid247

98．Non-Static Line Elements with Spherical Symmetry250

99．Birkhoff＇s Theorem252

100．A More General Line Element253

Ⅷ．RELATIVISTIC ELECTRODYNAMICS258

Part Ⅰ．THE COVARIANT GENERALIZATION OF ELECTRICAL THEORY258

101．Introduction258

102．The Generalized Lorentz Electron Theory．The Field Equations258

103．The Motion of a Charged Particle259

104．The Energy-Momentum Tensor261

105．The Generalized Macroscopic Theory261

Part Ⅱ．SOME APPLICATIONS OF RELATIVISTIC ELECTRODYNAMICS264

106．The Conservation of Electric Charge264

107．The Gravitational Field of a Charged Particle265

108．The Propagation of Electromagnetic Waves267

109．The Energy-Momentum Tensor for Disordered Radiation269

110．The Gravitational Mass of Disordered Radiation271

111．The Energy-Momentum Tensor Corresponding to a Directed Flow of Radiation272

112．The Gravitational Field Corresponding to a Directed Flow of Radiation273

113．The Gravitational Action of a Pencil of Light274

(a) The Line Element in the Neighbourhood of a Limited Pencil of Light274

(b) Velocity of a Test Ray of Light in the Neighbourhood of the Pencil275

(c) Acceleration of a Test Particle in the Neighbourhood of the Pencil277

114．The Gravitational Action of a Pulse of Light279

(a) The Line Element in the Neighbourhood of the Limited Track of a Pulse of Light279

(b) Velocity of a Test Ray of Light in the Neighbourhood of the Pulse281

(c) Acceleration of a Test Particle in the Neighbourhood of the Pulse282

115．Discussion of the Gravitational Interaction of Light Rays and Particles285

116．The Ceneralized Doppler Effect288

Ⅸ．RELATIVISTIC THERMODYNAMICS291

Part Ⅰ．THE EXTENSION OF THERMODYNAMICS TO GENERAL RELATIVITY291

117．Introduction291

118．The Relativistic Analogue of the First Law of Thermodynamics292

119．The Relativistic Analogue of the Second Law of Thermodynamics293

120．On the Interpretation of the Relativistic Second Law of Thermodynamics296

121．On the Interpretation of Heat in Relativistic Thermodynamics297

122．On the Use of Co-Moving Coordinates in Thermodynamic Considerations301

Part Ⅱ．APPLICATIONS OF RELATIVISTIC THERMODYNAMICS304

123．Application of the First Law to Changes in the Static State of a System304

124．Application of the Second Law to Changes in the Static State of a System306

125．The Conditions for Static Thermodynamic Equilibrium307

126．Static Equilibrium in the Case of a Spherical Distribution of Fluid308

127．Chemical Equilibrium in a Gravitating Sphere of Fluid311

128．Thermal Equilibrium in a Gravitating Sphere of Fluid312

129．Thermal Equilibrium in a General Static Field315

130．On the Increased Possibility in Relativistic Thermodynamics for Reversible Processes at a Finite Rate319

131．On the Possibility for Irreversible Processes without Reaching a Final State of Maximum Entropy326

132．Conclusion330

Ⅹ．APPLICATIONS TO COSMOLOGY331

Part Ⅰ．STATIC COSMOLOGICAL MODELS331

133．Introduction331

134．The Three Possibilities for a Homogeneous Static Universe333

135．The Einstein Line Element335

136．The de Sitter Line Element335

137．The Special Relativity Line Element336

138．The Geometry of the Einstein Universe337

139．Density and Pressure of Material in the Einstein Universe339

140．Behaviour of Particles and Light Rays in the Einstein Universe341

141．Comparison of Einstein Model with Actual Universe344

142．The Geometry of the de Sitter Universe346

143．Absence of Matter and Radiation from the de Sitter Universe348

144．Behaviour of Test Particles and Light Rays in the de Sitter Universe349

(a) The Geodesic Equations349

(b) Orbits of Particles351

(c) Behaviour of Light Rays in the de Sitter Universe353

(d) Doppler Effect in the de Sitter Universe354

145．Comparison of de Sitter Model with Actual Universe359

Part Ⅱ．THE APPLICATION OF RELATIVISTIC MECHANICS TO NON-STATIC HOMOGENEOUS COSMOLOGICAL MODELS361

146．Reasons for Changing to Non-Static Models361

147．Assumption Employed in Deriving Non-Static Line Element362

148．Derivation of Line Element from Assumption of Spatial Isotropy364

149．General Properties of the Line Element370

(a) Different Forms of Expression for the Line Element370

(b) Geometry Corresponding to Line Element371

(c) Result of Transfer of Origin of Coordinates372

(d) Physical Interpretation of Line Element375

150．Density and Pressure in Non-Static Universe376

151．Change in Energy with Time379

152．Change in Matter with Time381

153．Behaviour of Particles in the MOdel383

154．Behaviour of Light Rays in the Model387

155．The Doppler Effect in the Model389

156．Change in Doppler Effect with Distance392

157．General Discussion of Dependence on Time for Closed Models394

(a) General Features of Time Dependence，R real，ρ00 ？ 0，p0 ？ 0395

(b) Curve for the Critical Function of R396

(c) Monotonic Universes of Type M1，for ？ > ？E399

(d) Asymptotic Universes of Types A1 and A2，for ？ = ？E400

(e) Monotonic Universes of Type M2 and Oscillating Universes of Types O1 and O2，for 0 < ？ < ？E401

(f) Oscillating Universes of Type O1，for ？ 0402

158．General Discussion of Dependence on Time for Open Models403

159．On the Instability of the Einstein Static Universe405

160．Models in Which the Amount of Matter is Constant407

161．Models Which Expand from an Original Static State409

162．Ever Expanding Models Which do not Start from a Static State412

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 Models416

Part Ⅲ．THE APPLICATION OF RELATIVISTIC THERMODYNAMICS TO NON-STATIC HOMOGENEOUS COSMOLOGICAL MODELS420

166．Application of the Relativistic First Law420

167．Application of the Relativistic Second Law421

168．The Conditions for Thermodynamic Equilibrium in a Static Einstein Universe423

169．The Conditions for Reversible and Irreversible Changes in Non-Static Models424

170．Model Filled with Incoherent Matter Exerting No Pressure as an Example of Reversible Behaviour426

171．Model Filled with Black-Body Radiation as an Example of Reversible Behaviour427

172．Discussion of Failure to Obtain Periodic Motions without Singular States429

173．Interpretation of Reversible Expansions by an Ordinary Observer432

174．Analytical Treatment of a Succession of Expansions and Contractions for a Closed Model with ？ = 0435

(a) The Upper Boundary of Expansion436

(b) Time Necessary to Reach Maximum436

(c) Time Necessary to Complete Contraction437

(d) Behaviour at Lower Limit of Contraction438

175．Application of Thermodynamics to a Succession of Irreversible Expansions and Contractions439

Part Ⅳ．CORRELATION OF PHENOMENA IN THE ACTUAL UNIVERSE WITH THE HELP OF NON-STATIC HOMOGENEOUS MODELS445

176．Introduction445

177．The Observational Data446

(a) The Absolute Magnitudes of the Nearer Nebulae446

(b) The Corrected Apparent Magnitudes for more Distant Nebulae448

(c) Nebular Distances Calculated from Apparent Magnitudes453

(d) Relation of Observed Red-Shift to Magnitude and Distance454

(e) Relation of Apparent Diameter to Magnitude and Distance457

(f) Actual Diameters and Masses of Nebulae458

(g) Distribution of Nebulae in Space459

(h) Density of Matter in Space461

178．The Relation between Coordinate Position and Luminosity462

179．The Relation between Coordinate Position and Astronomically Determined Distance465

180．The Relation between Coordinate Position and Apparent Diameter467

181．The Relation between Coordinate Position and Counts of Nebular Distribution468

182．The Relation between Coordinate Position and Red-shift469

183．The Relation of Density to Spatial Curvature and Cosmological Constant473

184．The Relation between Red-shift and Rate of Disappearance of Matter475

185．Summary of Correspondences between Model and Actual Universe478

186．Some General Remarks Concerning Cosmological Models482

(a) Homogeneity482

(b) Spatial Curvature483

(c) Temporal Behaviour484

187．Our Neighbourhood as a Sample of the Universe as a Whole486

Appendix Ⅰ．SYMBOLS FOR QUANTITIES489

Scalar Quantities489

Vector Quantities490

Tensors490

Tensor Densities491

Appendix Ⅱ．SOME FORMULAE OF VECTOR ANALYSIS491

Appendix Ⅲ．SOME FORMULAE OF TENSOR ANALYSIS493

(a) General Notation493

(b) The Fundamental Metrical Tensor and its Properties494

(c) Tensor Manipulations495

(d) Miscellaneous Formulae496

(e) Formulae Involving Tensor Densities496

(f) Four-Dimensional Volume．Proper Spatial Volume496

Appendix Ⅳ．USEFUL CONSTANTS497

Subject Index499

Name Index502