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Poisson E. A relativists toolkit
Poisson E.  A relativists toolkit







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: A relativists toolkit

: Poisson E.

:

This textbook fills a gap in the existing literature on general relativity by providing the advanced student with practical tools for the computation of many physically interesting quantities. The context is provided by the mathematical theory of black holes, one of the most successful and relevant applications of general relativity. Topics covered include congruences of timelike and null geodesics, the embedding of spacelike, timelike and null hypersurfaces in spacetime, and the Lagrangian and Hamiltonian formulations of general relativity.


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Acceleration      17 18 92 104 105 185
Accretion onto a black hole      ix 109 201 223
Action functional for gravitation      x 118 121 122 124 126 127 136139 141 145 157 158
Action functional in general      118122 125 128 129 132 138
ADM mass      see Mass Arnowitt
Advanced time      152 161 165 173 199 204 207 213 214 218
Affine parameter      1 7 10 16 17 20 21 45 48 5052 56 58 61 63 66 98 104106 108 110113 171 173 186 193 194 199 204 207
Angular momentum of a black hole      ix 109 163 187 190 192 200 201 206 209211 221223
Angular momentum of a particle      10 27 155 183 188 192
Angular momentum of a rotating shell      94 109
Angular momentum of an axisymmetric body      148 150 151
Angular momentum of an isolated body      ix 118 119 146 147 205 210
Angular momentum, definition in terms of gravitational Hamiltonian      ix x 147 156 177
Angular momentum, definition in terms of Komar integral      149 151 177 209
Angular momentum, flux vector      155
Angular momentum, transfer across a hypersurface      155 156 160162
Angular velocity of a rotating black hole      190 191 200 204 209 221
Angular velocity of a rotating shell      96 97 112
Angular velocity of a rotating star      82
Angular velocity of an inertial frame      94 97
Angular velocity of an observer      27 189 190 192
Angular velocity of null generators      110 112 191
Angular velocity of ZAMOs      188
Apparent horizon      171176 178 180 190 191 203 214217
Area theorem      see Black-hole mechanics second
Asymptotically-flat spacetime      x 54 118 119 126 127 138 146148 151 157 162 180 183 187 198 217
Auxiliary null vector      4651 56 57 63 66 99 107 207
Basis vectors      19 22 24 29 57 66 73 75 99 101 103 106108 110 113 130 150 207 222
Bianchi identities      15 16 127 128
Bifurcation two-sphere      176 199 200 204 208
Birkhoffs theorem      163 164
Black hole, extreme      178 184 190 199 200 203 213 214 216
Black hole, general      viii ix 84 151 163 201211
Black hole, Kerr      97 109 187201 205 206 208 209 211 213 215 218 219 221
Black hole, Kerr Newman      187 221223
Black hole, Reissner Nordstroem      176187 191 194 196198 206 213 216 217
Black hole, Schwarzschild      55 90 109 110 163176 180 184 205
Black hole, stationary      204206
Black-hole entropy      214216
Black-hole mechanics, first law      viii 192 209 211212 215 216 223
Black-hole mechanics, second law      viii 192 201 212213 215 216
Black-hole mechanics, third law      viii 213214 216
Black-hole mechanics, zeroth law      viii 208209 214216 220 221
Black-hole tunnel      180 181 183 184 198 217 218
Bondi Sachs mass      see Mass Bondi
Boundary      viii 69 90 92 117119 126 132 134 136 141 142 148 150 151 156 163 167 169 172 174 196 202 203 209 218
Boundary terms      x 118 119 121 124 137142 144 146 156 212
Boyer Lindquist coordinates      187 192 196
Brane worlds      114
Canonical momentum      118 128 129 131 133 140
Carter constant      192 219
Casimir effect      32
Cauchy horizon      217 218
Cauchy problem      218
Causal structure      163 168 170 174 194 198 202 203
Caustic      41 51 203 212
Centrifugal force      183
Chandrasekhar on the perfection of black holes      206
Charge      25 176 177 184 187 206 213 216 217 221 223
Christoffel symbols      6 7 22 23 86 87 101 122 219
Cofactor      65 69 131
Collapse, gravitational      see Gravitational collapse
Completeness relations      29 63 67 68 78 79 99 102 107 123 134 135 207 222
Conformal flatness      84
Congruence of null geodesies      viii 28 4554 5658 105 116 163 171175 186 190 192194 203 205 207 212 220 221
Congruence of timelike curves      54 129 135
Congruence of timelike geodesics      viii 28 3645 54 55 76 85 99 100 104 114
Connection      46 8 9 11 12 22 73 74 86
Conservation laws      16 71 91 116 125 153 155 185
Constraint equations      7984 116 145 146
Coordinates, Boyer Lindquist      see Boyer Lindquist coordinates
Coordinates, Eddington Finkelstein      see Eddington Finkelstein coordinates
Coordinates, Fermi normal      see Fermi normal coordinates
Coordinates, Kerr Schild      see Kerr Schild coordinates
Coordinates, Kruskal      see Kruskal coordinates
Coordinates, Painleve Gullstrand      see Painleve Gullstrand coordinates
Coordinates, Riemann normal      see Riemann normal coordinates
Cosmological constant      117
Cosmology      ix 41 54 80 112113
Covariant differentiation      x 1 46 8 9 47 73 74 76 123 139 143 159
Cross section      4245 48 5153 66 105 174 200 202 207209 212
Curvature tensors, Einstein      x 2 16 78 88 89 101 142 144 152
Curvature tensors, extrinsic      see Extrinsic curvature
Curvature tensors, Ricci      x 2 16 26 31 55 78 79 82 84 88 98 115 121 137 146 207 220
Curvature tensors, Riemann      x 2 1517 19 23 24 26 59 7678 8689 98 101 115 116 188 204
Curvature tensors, Weyl      x 26 55 58 98
Curve      16 8 9 17 18 21 22 36 39 46 52 53 60 62 83 111 129 130 135 160 164 166 170 171 202 217
de Sitter spacetime      117
Delta function      see Dirac distribution
density      see Mass density or Energy density
Derivative, covariant      see Covariant differentiation
Derivative, Lie      see Lie differentiation
Determinant      2 1214 45 53 6468 71 121 122 131 135 207
Deviation vector      viii 1618 21 23 36 37 4547
Deviation, geodesic      see Geodesic deviation equation
Dirac distribution      8589 98 101 104 113 158
Divergence formula      13 70 72 133
Dragging of inertial frames      97 110 188 189
Dual vector      13 5 9
Dust      90 91 153 155 173 175 213 216218
Eddington Finkelstein coordinates      167 168 173
Einstein field equations      viii ix 2 16 26 31 40 50 59 78 80 84 86 88 90 91 101 115 117 118 122 124126 145 146 151154 157 158 163 164 173 177 187 201 205 207 209 220
Einstein tensor      see Curvature tensors Einstein
Einstein Hilbert priority dispute      157
Einstein Maxwell field equations      176 187 217 221
electromagnetic field      32 152 157 176 177 214 216 222
Embedding      25 59 67 68 76 115 127 134 135 138 147 148 153 160
Energy conditions in general      ix 2833 54 173 174 214
Energy conditions, averaged      32
Energy conditions, dominant      32 209 220
Energy conditions, null      31 50 174 203205 212 215
Energy conditions, strong      31 40 41 204
Energy conditions, violations      32 54
Energy conditions, weak      3031 213 214
Energy density      29 30 32 55 113 153 156 173 214 218
Energy of a particle      10 27 42 92 109 155 181 191 192
Energy, binding      93
Energy, electrostatic      184
Energy, flux vector      155
Energy, gravitational potential      159
Energy, internal      214
Energy, kinetic      93 159
Energy, mass-energy      see Mass
Energy, rotational      192
entropy      see Black-hole entropy
Equivalence principle      see Principle of equivalence
Ergosphere      189 191 192
Euler Lagrange equation      6 7 120 121 128
Event horizon      viii ix 28 163 170176 178 180 185 186 189192 196 197 199 200 202205 207209 211213 215217 220223
Expansion scalar      28 34 35 37 4043 46 48 51 52 55 56 58 76 105 113 114 116 171175 190 203 212 220
Extrinsic curvature      ix 59 7577 7981 83 89 92 95 102 103 114 115 118 121 124 127 134136 138140 145 147150 153 158 160
Fermi normal coordinates      2 1824 26
Field equations, Einstein      see Einstein field equations
Field equations, Einstein Maxwell      see Einstein Maxwell field equations
Field equations, Klein Gordon      see Klein Gordon equation
Field equations, Maxwell      see Maxwell field equations
Flat space      25 60 70 80 8284 115 138 147 149 154 159 160 168 217
Flat spacetime      17 26 32 51 56 63 65 66 83 93 95 107 115 116 126 127 138 146 153 160 165 194 195 217 219
Flatness, asymptotic      see Asymptotically-flat spacetime
Flatness, conformal      see Conformal flatness
Flatness, local      see Local flatness theorem
Fluid      30 55 90 96 97 111113 117 153 155 158 173 218
Focusing action of gravity      ix 28 109 116 175 205
Focusing theorem for null geodesics      28 5051 175 212
Focusing theorem for timelike geodesics      28 4041 43
Friedmann Robertson Walker spacetime      54 81 90
Frobenius theorem      28 3840 4850 186
Gauss Codazzi equations      7679 115
Gauss Weingarten equation      75
Gauss law      177
Gauss Theorem      69 71 120 133 209
Generators of a null hypersurface      49 51 6163 65 66 98 104113
Generators of an event horizon      viii 174176 185 186 191 199 200 203205 207 208 212 220 223
Geodesic deviation equation      2 1618 21 23
Geodesic equation in affine parameter form      7 10 17 18 20 22 25 39 52 56 92 100 155 158 173 186 192 200 204 219
Geodesic equation in general form      7 25 50 61 186 204
Geodesic, defined      68
Gravitational collapse in general      167 176 200 205 209
Gravitational collapse of a pressureless star      9092
Gravitational collapse of a thin shell      93 107109
Gravitational collapse of null dust      216 217
Gravitational repulsion      184
Gravitational waves      152 156 164 223
GRTensorII      x
Hamiltonian in general      118 128 129 131 132
Hamiltonian, density      131 132 140
Hamiltonian, formulation of general relativity      x 118 136146 156
Hamiltonian, gravitational      ix 118 119 136147 157
Hamiltons equations      128 132 133 145146
Hawking evaporation      206 214216
Hawking temperature      214
Hawking Hartle formula      223
Hilbert action      121 122
Hilbert Einstein priority dispute      see Einstein Hilbert priority dispute
Horizon, apparent      see Apparent horizon
Horizon, Cauchy      see Cauchy horizon
Horizon, event      see Event horizon
Horizon, Killing      see Killing horizon
Horizon, trapping      see Trapping horizon
Hypersurface orthogonal      28 3840 42 43 4850 52 56 57 76 82 105 129 186 190 205
Hypersurface, defined      60
Ideal gas      159
Induced metric      ix 59 6263 68 71 73 74 76 80 82 84 86 91 9496 102 103 106 108 110 113115 117 118 121 122 124 126 131 134136 139 145 148 153 158 160 168 200
Inertial frame      94 97 110 112
Infinity, null      152 153 161 169 170 202 206
Infinity, spatial      ix 71 97 110 119 126 147 151 153 168170 185 188 191 209
Infinity, timelike      169 170
Initial-value problem      ix 7984 114 145
Integration over a hypersurface      ix 59 6469 118 150 151 155 156 163 209211
Jacobian      12 14 36
Junction conditions      ix 60 8490 92
Kerr spacetime      94 109 163 187201 205 206 208 211 215 218 219 221
Kerr Newman spacetime      187 221223
Kerr Schild coordinates      194196
Kerr Schild metric      195 219
Killing horizon      175176 185 191 204
Killing tensor      192 219
Killing vector      x 1 1011 2527 82 96 149 150 176 185189 191 192 204 207 208 219 220
Klein Gordon equation      121 125 133
Komar integral for angular momentum      see Angular momentum definition
Komar integral for mass      see Mass definition
Kruskal coordinates      164168 171 176 178180 184 186 196 197 216
Kruskal diagram      166168 170 172 176 180
Lagrangian in general      6 118120 124
Lagrangian, density      118 120122 125 131 133 139 157
Lagrangian, formulation of general relativity      x 118 121128 156
Laplaces equation      84 217
Lapse function      118 119 130 131 139 145147
Lense Thirring effect      see Dragging of inertial frames
Levi-Civita tensor      2 1315 26 64 65
Lie differentiation      x 1 810 17 25 44 53 76 80 103 127 130 131 139 146 158 207
Lie transport      9 45 75 103 160
Light cone      51 6364 66 170
Local flatness theorem      1 1112 18 19 22
Local Lorentz frame      2 11 12 15 37 46 123 185
Lorentz transformation      11
Machs principle      97
Majumdar Papapetrou spacetime      216 217
Manifold      13 8 12 26 59 60 68 69 76 120 126 127 138 146 166 169 178 202
mass density      84 9093 96 97 102 104 106 109 111113 116 117 155 159 164
Mass in spherical symmetry      82 115 152 158 164 173 184 217
Mass of a black hole      ix 55 109 110 163 176 177 184 187 200 201 206 209211 213 221223
Mass of a collapsing shell      93 94 108 109 111
Mass of a particle      10 27 42 157 181 185
Mass of an isolated body      ix 82 91 92 116119 146 148 150 164 198 210
Mass transfer across a hypersurface      155 156 160 161
Mass, Arnowitt Deser Misner      147 151156 158 159 177 187 217
Mass, Bondi Sachs      151156
Mass, definition in terms of gravitational Hamiltonian      ix x 147 156 177
Mass, definition in terms of Kotnar integral      149 177 209
Mass-energy      see Energy
Matrix      1114 34 35 5658 65 66
Matter distribution      16 26 30 59 82 88 90 93 109 121 175 205 210 211 213
Maximal extension of a spacetime      167 176 196198
Maxwell field equations      157 176 177
Membrane paradigm      216
Metric determinant      see Determinant
Minkowski metric      11 19 22 23 29 56 64 95 195 219
Minkowski spacetime      see Flat spacetime
Moment of time symmetry      8184 115
Momentum vector      32 191
Multipole moments      206
Normal coordinates, Fermi      see Fermi normal coordinates
Normal coordinates, Riemann      see Riemann normal coordinates
Normal vector      38 48 6061 63 6569 71 75 8082 85 87 92 95 99 100 103 109 114 115 123 126 129 130 134 136 147149 153 160 174 185 190
Numerical relativity      84 114 146 157
Oppenheimer Snyder collapse      see Gravitational collapse of a pressureless star
Painleve Gullstrand coordinates      168
Parallel transport      1 4 6 7 19 22 24 25 37 47 57 58
Penrose process      191192
Penrose Carter diagram      168170 180 198 216
Penroses description of the event horizon      203 212 215
Permutation symbol      13
Perturbation of a black hole      184 201 211 212
Poissons equation      84
Pressure      29 55 90 91 93 96 97 102 104107 109 111113 116 117 153 159 173
Prices theorem      206
Principal null congruences of the Kerr spacetime      192194
Principle of equivalence      6 11
Projection of a tensor      24 47 56 73 74 77 96 100 101 139
Proper distance      7 8 1921 85 115 160 185
Proper time      7 8 19 20 24 25 36 41 43 85 91 99 100 104 116 147 158 168 181 188 192
Radiation      112 152155 160162 173 185 206 213 215 218 223
Raychaudhuris equation      28 4041 5051 55 57 58 105 116 205 207 212
Redshift      185
Reissner Nordstroem spacetime      163 176187 194 196 197 217
Retarded time      152 153 155 160 161 165
Ricci tensor      see Curvature tensors Ricci
Riemann normal coordinates      26
Riemann tensor      see Curvature tensors Riemann
Rigid rotation      161
Rotation tensor      28 3541 48 50 51 5557 205 220
Rotation, gravitational effects      97
Schwarzschild spacetime      26 42 52 55 90 93 108 114117 152 158 163176 205 217
Shear tensor      28 3437 40 41 48 50 51 5558 105 205 220 223
Shift vector      118 119 130 139 145147
Singularity of a congruence      see Caustic
Singularity of null-dust collapse      217
Singularity of the Boyer Lindquist coordinates      187 192
Singularity of the Eddington Finkelstein coordinates      182
Singularity of the Kerr spacetime      187 190 195196 198
Singularity of the Kruskal coordinates      179 180 197
Singularity of the Reissner Nordstroem spacetime      178 180
Singularity of the Schwarzschild coordinates      164 165 178
Singularity of the Schwarzschild spacetime      169 170
Singularity theorems      28 204 215
Smarrs formula      200 209211 215 222
Stokes Theorem      69 71 150
Stress-energy tensor in general      ix 16 29 105 113 125 155 205 207 211 213 215
Stress-energy tensor of a perfect fluid      30 96 155 158
Stress-energy tensor of a point particle      158
Stress-energy tensor of a scalar field      125 161
Stress-energy tensor of a surface layer      8790 93 95 96 98 100104 107 111 113 116
Stress-energy tensor of an electromagnetic field      157 177
Stress-energy tensor of null dust      153 173 213 214 218
Stress-energy tensor, decomposed in terms of density and principal pressures      29
Surface area of a black hole      ix 163 200 201 209 211213 222 223
Surface element      ix 59 6469 72 106 133 149 207 210 211
Surface gravity of a black hole      184187 198200 204 208 209 213215 220 222 223
Surface integral      see Integration over hypersurfaces
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