Classical Measurements in Curved Space-Times

Half-title......Page 3
Series-title......Page 4
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 13
Notation......Page 14
Physical dimensions......Page 17
Conversion factors......Page 18
1.1 Observers and physical measurements......Page 19
1.3 Clock synchronization and relativity of time......Page 20
2.1 The space-time......Page 29
Right and left contractions......Page 30
Change of frame......Page 31
Unit volume 4-form......Page 32
Symmetrization and antisymmetrization......Page 33
Hodge duality operation......Page 34
Covariant derivative and connection......Page 35
Covariant derivative and curvature......Page 36
Ricci tensor and curvature scalar......Page 37
Cotton tensor......Page 38
Parallel transport and geodesics......Page 39
Fermi-Walker derivative and transport......Page 40
Exterior derivative......Page 41
The divergence operator......Page 43
De Rham Laplacian......Page 44
2.3 Killing symmetries......Page 45
2.5 Einstein's equations......Page 46
Constant space-time curvature solutions......Page 47
Kasner solution......Page 48
Reissner-Nordstrom solution......Page 49
Kerr-Newman solution......Page 50
Single gravitational plane-wave solution......Page 51
3.1 Orthogonal decompositions......Page 52
Splitting of a vector......Page 53
Splitting of p-forms......Page 54
Splitting of differential operators......Page 56
3.2 Three-dimensional notation......Page 57
3.4 Adapted frames......Page 59
Spatial-Fermi-Walker and spatial-Lie temporal derivatives......Page 63
Frame components of the Riemann tensor......Page 64
3.5 Comparing families of observers......Page 66
Maps between LRSs......Page 67
Boost maps......Page 69
3.6 Splitting of derivatives along a time-like curve......Page 71
Projected absolute derivative......Page 73
Projected Fermi-Walker derivative......Page 75
Projected Lie derivative......Page 76
4.1 Orthonormal frames......Page 77
Absolute Frenet-Serret frames......Page 81
Relative Frenet-Serret frames......Page 82
Comoving relative Frenet-Serret frame......Page 83
Complex (Newman-Penrose) null frames......Page 87
Real null frames......Page 88
Fermi-Walker transport along the null world line......Page 91
5 The world function......Page 92
5.1 The connector......Page 93
5.2 Mathematical properties of the world function......Page 95
5.3 The world function in Fermi coordinates......Page 98
5.4 The world function in de Sitter space-time......Page 100
5.5 The world function in Gödel space-time......Page 102
5.6 The world function of a weak gravitational wave......Page 105
5.7 Applications of the world function: GPS or emission coordinates......Page 106
6.1 Measurements of time intervals and space distances......Page 109
6.2 Measurements of angles......Page 113
6.3 Measurements of spatial velocities......Page 114
6.4 Composition of velocities......Page 115
6.6 Measurements of frequencies......Page 117
6.7 Measurements of acceleration......Page 120
Longitudinal-tranverse splitting of the force equation......Page 122
6.9 Kinematical tensor change under observer transformations......Page 126
6.10 Measurements of electric and magnetic fields......Page 129
6.11 Local properties of an electromagnetic field......Page 131
Observers U(em) who measure a vanishing Poynting vector......Page 132
6.13 Gravitoelectromagnetism......Page 133
6.14 Physical properties of fluids......Page 134
Ordinary fluids: absolute dynamics......Page 136
Ordinary fluids: relative dynamics......Page 138
Transformation law for proper mechanical stresses......Page 139
Example: the perfect fluid......Page 140
Hydrodynamics with thermal flux: absolute formulation......Page 141
Hydrodynamics with thermal flux: relative formulation......Page 142
7.1 Measurement of the space-time curvature......Page 144
7.2 Vacuum Einstein's equations in 1+3 form......Page 147
7.3 Divergence of the Weyl tensor in 1+3 form......Page 148
7.4 Electric and magnetic parts of the Weyl tensor......Page 149
7.5 The Bel-Robinson tensor......Page 150
Old ideas and modern approaches......Page 153
The relative acceleration equation......Page 154
Geometrical meaning of the twist tensor......Page 158
7.7 Measurement of the magnetic part of the Riemann tensor......Page 159
7.8 Curvature contributions to spatial velocity......Page 161
7.9 Curvature contributions to the measurements of angles......Page 167
8.1 Schwarzschild space-time......Page 170
Various coordinate patches......Page 171
Principal null directions, Petrov type and the principal complex null frame......Page 173
Static observers......Page 174
Observers on spatially circular orbits......Page 175
Observers on equatorial spatially circular orbits......Page 181
8.3 Kerr space-time......Page 183
Various coordinate patches......Page 184
Curvature invariants......Page 185
Principal null directions, Petrov type and principal complex null frame......Page 186
Static observers......Page 187
Zero-angular-momentum observers......Page 188
Observers on general spatially circular orbits......Page 190
Carter’s observers......Page 194
Null spatially circular orbits......Page 195
Observers on equatorial circular orbits......Page 196
Observer-dependent embedding diagrams in Kerr space-time......Page 200
8.5 Gravitational plane-wave space-time......Page 202
Strain-induced rigidity......Page 204
Partially constrained circular motion......Page 209
9.2 The problem of space navigation......Page 212
Determining where the black hole is......Page 213
Probing the strength of the gravitational field......Page 216
9.3 Measurements in Kerr space-time......Page 217
Response of the internal structure......Page 219
Measurements and ambiguities......Page 220
9.4 Relativistic thrust anomaly......Page 224
9.5 Measurements of black-hole parameters......Page 230
9.6 Gravitationally induced time delay......Page 233
Null geodesics......Page 236
Images......Page 239
Direct image......Page 241
9.8 High-precision astrometry......Page 243
Fermi frame......Page 244
Attitude frame......Page 245
10.1 Behavior of spin in general space-times......Page 249
Test gyroscope: the projected spin vector......Page 250
Boosted spin vector......Page 252
10.2 Motion of a test gyroscope in a weak gravitational field......Page 255
10.3 Motion of a test gyroscope in Schwarzschild space-time......Page 257
10.4 Motion of a spinning body in Schwarzschild space-time......Page 259
Clock effect for spinning bodies......Page 264
10.5 Motion of a test gyroscope in Kerr space-time......Page 265
10.6 Motion of a spinning body in Kerr space-time......Page 266
Clock effect for spinning bodies......Page 272
Test gyroscopes in motion along a geodesic......Page 273
Test gyroscopes at rest in the TT-grid of a gravitational wave......Page 274
Test gyroscopes in general geodesic motion......Page 277
10.8 Motion of an extended body in a gravitational wave space-time......Page 279
Epilogue......Page 283
Exercises......Page 285
References......Page 317
Index......Page 323