O'Neill B. — The Geometry of Kerr Black Holes :: Электронная библиотека попечительского совета мехмата МГУ

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O'Neill B. — The Geometry of Kerr Black Holes

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Название: The Geometry of Kerr Black Holes

Автор: O'Neill B.

Язык:

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1995

Количество страниц: 381

Добавлена в каталог: 04.04.2009

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Предметный указатель

$\vartheta$ -equation      183
4-velocity      35
Acceleration      15
Achronal set      171—172
Action (of a Lie group      6—7
Allowed region, in an r-L plot      214
Analytic function      8
Analytic manifold      8
Angle function A for Kerr coordinates      80 122
Angular momentum of a geodesic around the Kerr axis      179
Angular momentum, per unit mass, of Kerr spacetime      58
Asymptotic flatness      59
Axial symmetry      59
Axis (of Kerr spacetime)      63 68
Axis (of Kerr spacetime), abstract      140 143
Axis (of Kerr spacetime), geometry of      87—88 102
Backward isometry      59 110 152—154 155—160 162—163
Barrier, in an r-L plot      230
Basis Hodge      303
Basis orthonormal      26
Basis positively oriented      303
Basis Theorem      3
Bianchi identity      19 341
Bianchi identity, in Newman — Penrose formalism      341—342
Biskew (0, 4) tensor      298
Bivector      360
Bivector decomposable      362
Bivector principal null      318
Black holes      see under “Schwarzschild and Kerr”
Bound orbit      see under “Orbit types and Global trajectories”
Boyer — Lindquist (co)frame field      90—91
Boyer — Lindquist (co)frame field, brackets      96
Boyer — Lindquist (co)frame field, connection forms      92
Boyer — Lindquist (co)frame field, covariant derivatives      95
Boyer — Lindquist (co)frame field, curvature forms      98
Boyer — Lindquist (co)frame field, signs for      90 92
Boyer — Lindquist blocks      65—67
Boyer — Lindquist blocks, canonical imbedding of      82
Boyer — Lindquist blocks, causal properties of      75—78
Boyer — Lindquist blocks, incompleteness of block III      101
Boyer — Lindquist blocks, stationary properties of      70—72
Boyer — Lindquist blocks, visualization      63
Boyer — Lindquist coordinates      57
Boyer — Lindquist coordinates, failure at horizons      74
Boyer — Lindquist coordinates, metric identities for      60
Boyer — Lindquist polynomial      282
Boyer, R.H.      121
Bracket operation (Lie bracket)      5
Bracket operation (Lie bracket), for the Boyer — Lindquist vector fields      96
Bracket operation (Lie bracket), in Newman — Penrose formalism      334
Brans, Carl      348
Bundle structure of Kerr spacetime      140—144
Caesar, Julius      56
Canonical frame      323
Canonical imbedding of blocks      82
Canonical Kerr vector fields      60 68
Cartan computations, for the Boyer — Lindquist frame field      90—99
Cartan computations, in general      49—55
Carter band, in an r-L plot      236
Carter constant $\mathcal{K}$       183
Carter constant $\mathcal{K}$ and $\mathcal{Q}$ and approach to the ring singularity      201
Carter constant $\mathcal{K}$ and $\mathcal{Q}$ , effect on forbidden regions      217
Carter constant $\mathcal{K}$ and $\mathcal{Q}$ , effect on latitudinal motion      203—207
Carter constant $\mathcal{K}$ and $\mathcal{Q}$ , general properties of      186—188
Carter constant $\mathcal{Q}$       188
Carter's (first-integral) theorem      183
Carter's (first-integral) theorem, proof of      357—359
Carter, Brandon      78 183 288
Causal (nonspacelike) vector      29
Causal character of a curve, vector field      33
Causal character of a submanifold      46
Causal character of a subspace      29
Causal character of a vector      27
Causal character of Boyer — Lindquist coordinate vector fields      66 70—72 78
Causal character of the canonical vector fields      66—67 see “Null” “Spacelike”
Causal spacetime      75
Causality      74—75 171
Central sphere      69 145
Chandrasekhar, S.      349
Christoffel symbols      15
Chronological future [past] set      171
Chronological spacetime      75 174
Chronology      171
Chronology of Kerr spacetimes      171—176
Circular coordinate      7—8
Closed submanifold      43
Coframe field      11
Colatitude $\vartheta$       43—44
Colatitude $\vartheta$ , as Boyer-Lindquist coordinate      57—58
Colatitude $\vartheta$ , as coordinate of a geodesic      201—207
Comparison of Schwarzschild and Kerr spacetimes      see “Schwar schild comparison”
Complete vector field      6
Components of tensors      10 11
Configuration of an r-L plot      see under “r-L plot”
Conformal flat spacetime      314
Congruence      45 see
Connection forms      50—52
Connection forms for the Boyer — Lindquist frame field      see under “Boyer — Lindquist”
Conservation lemma      20
Continents, in an r-L plot      223
Contraction of tensors      11—12 17 299 332—333
Coordinate function      2
Coordinate neighborhood      2
Coordinate one-form      6
Coordinate system      2
Coordinate vector      3 5
Covariant derivative      14—15
Covariant derivative in Newman — Penrose formalism      334
Covariant derivative of Boyer — Lindquist vector fields      94—95
Covector      6
Crossing spheres      119 121 130 135 138 140 146
Curvature form      52
Curvature of Kerr spacetime, Cartan computation      96—100
Curvature of Kerr spacetime, Newman-Penrose expression      3
Curvature singularity      101
Curvature transformation      300
Curvature, abstract      298
Curvature, Gaussian      19 54
Curvature, in Newman-Penrose terms      338—340
Curvature, Kretschmann      18
Curvature, relativistic significance of      35—36
Curvature, Ricci      18
Curvature, Riemannian      17—19
Curvature, scalar      18
Curvature, sectional      18
Curvature, symmetries of      18
Curvature, Weyl conformal      301
Curve      4
Cyclic symmetry      298
Deformation retract      164
Diffeomorphism      3
Differential forms      6 355—356
Differential map      4
Differential of a function      6
Dimension of a manifold      2
Discriminant polynomial      215 217—222
distribution      44—45
Distribution, integrability of      45
Dual basis (or coframe)      11
e-Q chart      219—220
Effective energy, of a geodesic      214
Effective potential, for axial geodesics      254
Effective potential, for polar geodesics      263
Effective potential, in the $\vartheta$ -equation      201—202
Einstein manifold      308 311
Einstein summation convention (weak)      10
Einstein, Albert      31 33 35
Energy extraction from a black hole      181—182
Energy, of a particle in Kerr spacetime      179
Energy-momentum 4-vector      35

Equator (equatorial plane)      68 146—147
Equatorial geodesics      272—287
Equatorial geodesics, circular null      277—278
Equatorial geodesics, circular timelike      282—283
Equatorial geodesics, in r<0      273—274
Equatorial geodesics, longitudinal motion of      283—284
Equatorial geodesics, null      276—279
Equatorial geodesics, spacelike      285—286
Equatorial geodesics, timelike      279—283
Equatorial isometry      68
Ergosphere      72—73 181
Escape energy      211
Euclidean space      2
Euler equations      16
Event      32
Event horizon      86—87
expansion      see under “Optical scalars”
Extension of Kerr spacetimes      57
Extension of manifolds by coordinates      20—21
Extention of Kerr spacetimes across r=0      62
Extention of Kerr spacetimes over the axis      64
Extention of Kerr spacetimes, plan for      105 see
Extention of manifolds by gluing      21—25
Exterior derivative      356
Exterior of Kerr spacetime      66
Exterior product of forms and of vectors      see “wedge product”
Exterior product of vector spaces      299 361—364
Fiber bundle      7
Fiber spheres      144
Fiber spheres in horizons      145—146
First structural equation      50
First-integrals      178—179 183
First-integrals as coefficients of the polynomial R(r)      208
First-integrals effect on orbits      222—235
Flat manifold      36
Flow      6
Flyby orbit      see under “Orbit types and Global trajectories”
Foliation      45
Forbidden region, in an r-L plot      214
Frame field      10—11
Frame field, Boyer — Lindquist      see under “Boyer-Lindquist”
Frame field, orthonormal      17
Frobenius' Theorem      44 356
Fundamental groups of Kerr spacetimes      165 167 170
Future set      see “Chronological future set”
Future [past] complete geodesic      111
Future [past] side      172
Future-pointing      33
Futurecone      33
g-null      318
General relativity, introduction to      31—42
Geodesic completeness      15
Geodesic completeness modulo the ring singularity      189
Geodesics      15—16 48 334
Geodesics complete      111
Geodesics in Kerr spacetime, axial      250—255
Geodesics in Kerr spacetime, equations for: Colatitude equation ( $\vartheta$ -equation)      183
Geodesics in Kerr spacetime, equations for: Coordinate integral equations      190 191
Geodesics in Kerr spacetime, equations for: First-order equations for longitude and time      181 190
Geodesics in Kerr spacetime, equations for: Radial equation (r-equation)      183
Geodesics in Kerr spacetime, extensions of, across horizons      196—200
Geodesics in Kerr spacetime, extensions of, through a Boyer-Lindquist block      195
Geodesics in Kerr spacetime, hitting the ring, singularity      101
Geodesics in Kerr spacetime, hitting the ring, via the equator      288
Geodesics in Kerr spacetime, in horizons      255—262
Geodesics in Kerr spacetime, initial data for      185
Geodesics in Kerr spacetime, polar      262—272
Geodesics in Kerr spacetime, regular      189
Geodesics in Kerr spacetime, with Q=0      287—291 293 see “First-integrals” “Global “Principal
Geometric units      351—353
Global trajectories, of Kerr geodesics      243—250
Global trajectories, of Kerr geodesics, for standard particles      244
Global trajectories, of Kerr geodesics, timelike bound orbits: in block I      247
Global trajectories, of Kerr geodesics, timelike bound orbits: in block III      247
Global trajectories, of Kerr geodesics, timelike bound orbits: large      248—249
Global trajectories, of Kerr geodesics, timelike flyby orbits: exterior      245
Global trajectories, of Kerr geodesics, timelike flyby orbits: in block III      245
Global trajectories, of Kerr geodesics, timelike flyby orbits: long      245—247
Global trajectories, of Kerr geodesics, timelike transit orbit      249—250
Gluing data      21
Gluing of manifolds      21—25
Goldberg — Sachs theorem      345—348
Hausdorff condition (for gluing)      23—24
Hitting the ring singularity      see under “Geodesics in Kerr spacetime”
Hodge basis      303
Hodge double indices      304 310
Hodge identity      304
Hodge star operator      303—308
Hodge star operator, as complex structure      305
Hodge star operator, in complex scalar product      306
Homotopy equivalence      164
Homotopy type      164
Homotopy type of extreme Kerr spacetime      168
Homotopy type of fast Kerr spacetime      164—165
Homotopy type of slow Kerr spacetime      166—168
Homotopy type of small Kerr spacetimes      168—170
Horizon function $\Delta$       58 62
Horizon sphere      145
Horizon, in Kerr spacetime      58 62—63
Horizon, in Kerr spacetime, as one-way membrane      84 172
Horizon, in Kerr spacetime, crossing, by geodesics      196—200
Horizon, in Kerr spacetime, geometry of      83—85
Horizon, in Kerr spacetime, long      see “Long horizon”
Hypersurface      42—43 45—46 48
Hypersurface, achronal      171
Hypersurface, one-and two-sided      46
Hypersurface, separating      46
Hypersurface-orthogonal vector field      48 71 356
Index of a scalar product      26
Inextendibility of Kerr spacetimes      293—295
Inner product      12
Instantaneous observer      38
Integral curve      5
Integral manifold      44—45
Isometries of Boyer — Lindquist blocks      149—154
Isometries of extreme Kerr spacetime      155—158
Isometries of Kruskal domains      158—160
Isometries of slow Kerr spacetime      161—163
Isometry      13—14 48
Jacobi identity      5
KBL coordinates      121—130
KBL coordinates, related to Boyer — Lindquist coordinates      131—132 137
Kerr black hole      see also “Kerr spacetime”
Kerr black hole, overview      55—56
Kerr black hole, physical properties summarized      102—103
Kerr exterior (block I)      66
Kerr metric on      131 134—135
Kerr metric tensor (or line-element), compared with Minkowski and Schwarzschild metrics      58
Kerr metric tensor (or line-element), in Boyer — Lindquist coordinates      58—59
Kerr metric tensor (or line-element), in Boyer — Lindquist coordinates, alternative formulation of      91
Kerr metric tensor (or line-element), in KBL coordinates      134—135
Kerr metric tensor (or line-element), in Kerr-star coordinates      81
Kerr metric tensor (or line-element), in star-Kerr coordinates      107
Kerr metric tensor (or line-element), on Kruskal domains      131—135
Kerr metric tensor (or line-element), on the axis      64—65
Kerr metric tensor (or line-element), Petrov type D character of      315—317
Kerr observers      71—74
Kerr patch      114 138
Kerr spacetime, maximal analytic extensions of      see “Maximal Kerr spacetimes”
Kerr spacetime, slow, extreme, fast      see “Rotation types”
Kerr spacetime, working definition of      67—68
Kerr, Roy      31
Kerr-Newman metric      61
Kerr-star spacetime      79—83
Kerr-star spacetime, definition      82
Killing isometry (group)      149—150
Killing orbit      69
Killing vector field      19—20 48
Killing vector field on Kerr spacetime      59 68 69 70—71 155
Kinnersley, William      344

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