Table Of ContentCore Principles of Special
and General Relativity
Core Principles of Special
and General Relativity
James H. Luscombe
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Contents
Preface xi
CHAPTER1(cid:4)Relativity: A theory of space, time, and gravity 1
1.1 THE PRINCIPLE OF RELATIVITY 1
1.2 THE LAW OF INERTIA: FOUNDATION OF SPECIAL RELATIVITY 2
1.3 SPACE, TIME, AND SPACETIME 7
1.4 SPACETIME DIAGRAMS 11
1.5 RELATIVITY OF CAUSALITY: SPACELIKE AND TIMELIKE 13
1.6 SEGUE TO GENERAL RELATIVITY: NONINERTIAL FRAMES 15
1.7 GENERAL RELATIVITY: A THEORY OF GRAVITATION 17
1.8 HASTA LA VISTA, GRAVITY 24
CHAPTER2(cid:4)Basic special relativity 27
2.1 COMPARISON OF TIME INTERVALS: THE BONDI K-FACTOR 27
2.2 TIME DILATION 29
2.3 VELOCITY ADDITION 30
2.4 LORENTZ TRANSFORMATION 32
2.5 LENGTH CONTRACTION 34
2.6 FOUNDATIONAL EXPERIMENTS 41
CHAPTER3(cid:4)Lorentz transformation, I 45
3.1 FRAMES IN STANDARD CONFIGURATION 45
3.2 FRAMES NOT IN STANDARD CONFIGURATION 51
3.3 TRANSFORMATION OF VELOCITY AND ACCELERATION 52
3.4 RELATIVISTIC ABERRATION AND DOPPLER EFFECT 54
v
vi (cid:4) Contents
CHAPTER4(cid:4)Geometry of Lorentz invariance 57
4.1 LORENTZ TRANSFORMATIONS AS SPACETIME ROTATIONS 57
4.2 KINEMATIC EFFECTS FROM THE INVARIANT HYPERBOLA 61
4.3 CLASSIFICATION OF LORENTZ TRANSFORMATIONS 62
4.4 SPACETIME GEOMETRY AND CAUSALITY 64
CHAPTER5(cid:4)Tensors on flat spaces 69
5.1 TRANSFORMATION PROPERTIES 69
5.2 TENSOR DENSITIES, INVARIANT VOLUME ELEMENT 85
5.3 DERIVATIVES OF TENSORS AND THE FOUR-WAVEVECTOR 87
5.4 INTERLUDE: DRAW A LINE HERE 89
5.5 TENSORS AS MULTILINEAR MAPPINGS 89
5.6 METRIC TENSOR REVISITED 91
5.7 SYMMETRY OPERATIONS ON TENSORS 92
5.8 LEVI-CIVITA TENSOR AND DETERMINANTS 93
5.9 PSEUDOTENSORS 95
5.10 TOTALLY ANTISYMMETRIC TENSORS 97
5.11 INTEGRATION ON MINKOWSKI SPACE 108
5.12 THE GHOST OF TENSORS YET TO COME 110
CHAPTER6(cid:4)Lorentz transformation, II 113
6.1 DECOMPOSITION INTO ROTATIONS AND BOOSTS 113
6.2 INFINITESIMAL LORENTZ TRANSFORMATION 119
6.3 SPINOR REPRESENTATION OF LORENTZ TRANSFORMATIONS 120
6.4 THOMAS-WIGNER ROTATION 123
CHAPTER7(cid:4)Particle dynamics 129
7.1 PROPER TIME, FOUR-VELOCITY, AND FOUR-ACCELERATION 129
7.2 THE ENERGY-MOMENTUM FOUR-VECTOR 132
7.3 ACTION PRINCIPLE FOR PARTICLES 134
7.4 KEPLER PROBLEM IN SPECIAL RELATIVITY 136
7.5 COVARIANT EULER-LAGRANGE EQUATION 138
7.6 PARTICLE CONSERVATION LAWS 140
7.7 ENERGY-MOMENTUM CONSERVATION 143
Contents (cid:4) vii
CHAPTER8(cid:4)Covariant electrodynamics 147
8.1 ELECTROMAGNETISM IN SPACE AND TIME 147
8.2 SOURCES IN SPACETIME: THE FOUR-CURRENT 150
8.3 CONSERVATION IN SPACETIME: SPACELIKE HYPERSURFACES 151
8.4 THE FOUR-POTENTIAL 153
8.5 MAXWELL EQUATIONS IN COVARIANT FORM: FIELD TENSOR 153
8.6 LORENTZ TRANSFORMATION OF E AND B FIELDS 155
8.7 LORENTZ FORCE AS A RELATIVISTIC EFFECT 156
8.8 INVARIANTS OF THE ELECTROMAGNETIC FIELD 157
8.9 ACTION PRINCIPLE FOR CHARGED PARTICLES 158
8.10 GAUGE INVARIANCE AND CHARGE CONSERVATION 161
CHAPTER9(cid:4)Energy-momentum of fields 165
9.1 SYMMETRIES AND CONSERVATION LAWS 165
9.2 SPACETIME HOMOGENEITY: ENERGY-MOMENTUM TENSOR 167
9.3 SPACETIME ISOTROPY: ANGULAR MOMENTUM TENSOR 169
9.4 SYMMETRIC ENERGY-MOMENTUM TENSOR 171
9.5 THE ELECTROMAGNETIC FIELD 172
CHAPTER10(cid:4)Relativistic hydrodynamics 177
10.1 NONRELATIVISTIC HYDRODYNAMICS 177
10.2 ENERGY-MOMENTUM TENSOR FOR PERFECT FLUIDS 180
10.3 ENERGY-MOMENTUM CONSERVATION 181
10.4 PARTICLE NUMBER CONSERVATION 182
10.5 COVARIANT EQUATION OF MOTION 182
10.6 LAGRANGIAN DENSITY 183
CHAPTER11(cid:4)Equivalence of local gravity and acceleration 185
11.1 THE EO¨TVO¨S EXPERIMENT 185
11.2 THE EQUIVALENCE PRINCIPLE 187
11.3 TIDAL FORCES AND REFERENCE FRAMES 187
11.4 WEAK AND STRONG EQUIVALENCE PRINCIPLES 191
11.5 SPACETIME IS GLOBALLY CURVED, LOCALLY FLAT 192
11.6 ENERGY COUPLES TO GRAVITY 192
11.7 GRAVITY AFFECTS TIME 194
viii (cid:4) Contents
CHAPTER12(cid:4)Acceleration in special relativity 199
12.1 LINEAR ACCELERATION 199
12.2 TWIN PARADOX 205
12.3 ROTATING REFERENCE FRAME 207
12.4 THE SAGNAC EFFECT 211
12.5 RELATIVISTIC DESCRIPTION OF SPIN 212
12.6 COVARIANT SPIN DYNAMICS 214
CHAPTER13(cid:4)Tensors on manifolds 217
13.1 MANIFOLDS 217
13.2 VECTOR AND TENSOR FIELDS 221
13.3 INTEGRAL CURVES, CONGRUENCES, AND FLOWS 225
13.4 MAPPINGS OF TENSORS 227
13.5 THE LIE DERIVATIVE 228
13.6 SUBMANIFOLDS, EMBEDDINGS, AND HYPERSURFACES 230
13.7 DIFFERENTIAL FORMS AND EXTERIOR DIFFERENTIATION 233
13.8 INTEGRATION ON MANIFOLDS 235
CHAPTER14(cid:4)Differential geometry 241
14.1 COVARIANT DIFFERENTIATION 241
14.2 WHAT DO THE CONNECTION COEFFICIENTS TELL US? 250
14.3 PARALLEL TRANSPORT AND GEODESIC CURVES 252
14.4 THE RIEMANN TENSOR 257
14.5 THE RICCI TENSOR AND SCALAR FIELD 265
14.6 THE EINSTEIN TENSOR 267
14.7 ISOMETRIES, KILLING VECTORS, AND CONSERVATION LAWS 268
14.8 MAXIMALLY SYMMETRIC SPACES 270
CHAPTER15(cid:4)General relativity 275
15.1 INTRODUCTION 275
15.2 WEAK, STATIC GRAVITY 276
15.3 THE EINSTEIN FIELD EQUATION 277
15.4 LAGRANGIAN FORMULATION 281
15.5 DUST 286
Contents (cid:4) ix
CHAPTER16(cid:4)The Schwarzschild metric 289
16.1 STATIC, SPHERICALLY SYMMETRIC SPACETIME METRICS 289
16.2 RICCI TENSOR FOR THE SCHWARZSCHILD METRIC 290
16.3 THE VACUUM SOLUTION 291
16.4 BIRKHOFF’S THEOREM 292
16.5 SPATIAL GEOMETRY OF THE SCHWARZSCHILD METRIC 293
CHAPTER17(cid:4)Physical effects of Schwarzschild spacetime 297
17.1 GEODESICS IN SCHWARZSCHILD SPACETIME 297
17.2 PARTICLE TRAJECTORIES 300
17.3 RADIAL NULL GEODESICS: KRUSKAL COORDINATES 303
17.4 GRAVITATIONAL DEFLECTION OF LIGHT 304
17.5 APSIDAL PRECESSION 309
17.6 GRAVITATIONAL TIME DELAY 311
17.7 PARAMETERIZED POST-NEWTONIAN FRAMEWORK 312
17.8 THE GLOBAL POSITIONING SYSTEM 313
17.9 SPIN PRECESSION I: GEODETIC EFFECT 315
17.10 WEIGHT OF AN AT-REST OBSERVER 316
CHAPTER18(cid:4)Linearized gravity 319
18.1 LINEARIZED FIELD EQUATION 319
18.2 STATIC SOURCE 322
18.3 FAR FROM A SLOWLY VARYING SOURCE 323
18.4 GRAVITOMAGNETISM: STATIONARY SOURCES 325
18.5 FRAME DRAGGING 326
18.6 SLOWLY ROTATING SOURCE 327
18.7 SPIN PRECESSION II: THE LENSE-THIRRING EFFECT 328
18.8 GRAVITATIONAL WAVES 331
18.9 ENERGY-MOMENTUM OF GRAVITATION 333
CHAPTER19(cid:4)Relativistic cosmology 339
19.1 THE COSMOLOGICAL PRINCIPLE 339
19.2 A COORDINATE SYSTEM FOR COSMOLOGY 340
19.3 SPACES OF CONSTANT CURVATURE 340
19.4 FRIEDMANN-ROBERTSON-WALKER SPACETIME 342
19.5 SPATIAL GEOMETRIES 343
19.6 THE FRIEDMANN EQUATIONS 346
