Penrose R., Rindler W. — Spinors and space-time. Spinor and twistor methods in space-time geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Penrose R., Rindler W. — Spinors and space-time. Spinor and twistor methods in space-time geometry

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Название: Spinors and space-time. Spinor and twistor methods in space-time geometry

Авторы: Penrose R., Rindler W.

Аннотация:

Spinor and Twistor Methods in Space-Time Geometry introduces the theory of twistors, and studies in detail how the theory of twistors and 2-spinors can be applied to the study of space-time. Twistors have, in recent years, attracted increasing attention as a mathematical tool and as a means of gaining new insights into the structure of physical laws. This volume also includes a comprehensive treatment of the conformal approach to space-time infinity with results on general-relativistic mass and angular momentum, a detailed spinorial classification of the full space-time curvature tensor, and an account of the geometry of null geodesics.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Издание: Repr

Год издания: 1988

Количество страниц: 512

Добавлена в каталог: 26.03.2011

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Предметный указатель

$(\omega^{A}(O)$ , $(\omega^{0}(O)$ (values at origin)      48
$A_{\alpha\beta} = 2E^{\gamma} {}_{(\alpha}I_{\beta)\gamma}$ (angular momentum twistor)      71
$b=\frac{1}{2}(p+q)$ (boost-weight)      29
$C_{abcd}$ (Weyl tensor)      20
$dx^{a} = g_{i_{1}} {}^{a}$       437
$d\phi\psi = (d\phi)\psi$ , $\partial\phi\psi = (\partial\phi)\psi$ etc.      16
$e_{abcd}$ (alternating tensor)      10 11
$E_{ab}$ (source for linear gravity)      38
$F_{ab}$ , $\varphi_{ab}$ , $\Phi_{a}$ , $J^{a}$ , $T_{ab}$ (Maxwell theory)      32
$F_{i...k}|_{l...n}$ (restriction)      159
$g_{0} {}^{AB^{'}}$ etc. (translation symbols)      8
$g_{ab} \mapsto \hat{g}_{ab} = \Omega^{2}g_{ab}$       35
$g_{ab}$ (metric tensor)      5
$I^{+}[\Sigma], I^{-}[\Sigma]$ (future-, past-set)      296
$I_{\alpha\beta}$ , $I^{\alpha\beta}$ (infinity twistors)      66
$K_{abcd}$ (linearized curvature)      38
$K_{ab\Omega} {}^{\Phi}$ , $F_{ab\Omega} {}^{\Phi}$ , $\varphi_{AB\Theta} {}^{\Psi}$ , $\chi_{AB\Theta} {}^{\Psi}$ (bundle curvature)      34 35
$l^{a}$ , $m^{a}$ , $n^{a}$ (null tetrad)      6
$N = 2^{\frac{1}{2}n}$ or $2^{\frac{1}{2}n-\frac{1}{2}}$ (spin-space dimension)      440
$o^{A}$ , $\iota^{A}$ (spinor basis)      6
$P_{ABA^{'}B} = \frac{1}{12}Rg_{ab}-\frac{1}{2}R_{ab}$       122
$P_{a}$ , $M^{ab}$ (momentum, angular momentum)      68
$Q^{ab}$ (spin-lowering tensor)      77
$R^{\ast}_{abcd}$ , $^{\ast}R_{abcd}$ , $^{\ast}R^{\mathbf{\ast}}_{abcd}$       19
$R_{ab}$ , R      20
$R_{\alpha\beta\gamma} {}^{\delta}$ (curvature tensor)      15 19
$R_{\alpha\beta}$ (dual of $R^{\alpha\beta}$ )      65
$s = \frac{1}{2}\mathbf{Z}^{\alpha}\mathbf{\overline{Z}}_{\alpha}$ (helicity)      56
$s=\frac{1}{2}(p-q)$ (spin-weight)      29
$S^{+}$ (anti-celestial sphere)      1 380
$S^{+}$ (anti-celestial sphere) and general symmetric spinor      265
$S^{+}$ (anti-celestial sphere) and Weyl spinor      226 249
$S^{-}$ (celestial sphere)      2
$S_{a}$ (Pauli — Lubanski)      69
$S_{b} = o^{A}\nabla_{b}o_{A}$       174
$T_{\alpha\beta} {}^{\gamma}$ (torsion tensor)      15
$x^{a}$ (Minkowski coordinates)      1 46
$x^{a}$ (position vector)      45
$X_{ABCD}$ , $\Phi_{ABC'D'}$ , $\Lambda$ (curvature spinors)      19
$[^{1}_{0}]$ , $[^{0}_{1}],..., [^{p}_{q}]$ (twistor valence)      46 48 51
$[^{r}_{p} {}^{s}_{q}]$ (spinor valence symbol)      10
$\alpha$ -plane description of twistors      64
$\approx$ (weak equality)      354
$\beta$ (Sparling 3-form)      434
$\beta$ -plane description of dual twistors      64
$\chi = o^{A} \iota^{A}$       6
$\chi...(...)...,\chi...[...]...,\chi...|...|...$       9
$\chi=\{\zeta_{1}, \zeta_{2}, \zeta_{3}, \zeta_{4}\}$ (cross-ratio)      2
$\cup_{\alpha}$ , $\cup_{\alpha} + \delta\cup_{\alpha}$ (neighbouring null twistors)      181
$\delta$ (coboundary operator)      161
$\delta^{\alpha}_{\mathbf{\alpha}}$ , $\delta^{\mathbf{\alpha}}_{\alpha}$ (basis vectors)      4
$\Delta_{\alpha\beta}$ (derivative commutator)      15
$\eta$ (correction factor)      402
$\gamma$ (ray)      358
$\gamma_{ab...d} = \gamma_{[a}\gamma_{b}...\gamma_{d]}$ , $\gamma_{ab...d\rho} {}^{\sigma}$       444 445
$\gamma_{\mathbf{AA^{'}C}} {}^{B}$ (spin-coefficients)      18
$\hat{N}_{a}$ (normal to $\mathscr{I}$ )      352
$\hat{T}_{ab}$ (trace-reversed $T_{ab}$ )      11
$\hat{\varepsilon}_{AB} = \Omega\varepsilon_{AB}$       35
$\hbar$ (Planck's constant/ $2\pi$ )      142
$\int_{\mathscr{P}}\mathbf{A}$ (integral of p-form)      17
$\kappa$ (spin-vector)      2 3
$\lambda$ (cosmological constant)      20
$\lambda_{A}$ (Witten spinor)      430
$\mathbb{C}$ (= complexified)      64
$\mathbb{H}^{\alpha}_{\beta} = \hat{\mathbb{H}}^{\alpha}_{\beta} \oplus \mathbb{R}\delta^{\alpha}_{\beta}$ (Hermitian twistors)      95
$\mathbb{M} (\mathscr{S})$ , $\mathbb{CM} (\mathscr{S})$ (associated Minkowski spaces)      418
$\mathbb{M}$ (Minkowski space)      1
$\mathbb{M}^{#}$ (compactified $\mathbb{M}$ )      298
$\mathbb{N}$ , $\mathbb{N}^{\alpha}$ , $\mathbb{N}^{\bullet}$ , $\mathbb{N}_{\bullet}$ (space of rays in $\mathbb{M}$ )      207
$\mathbb{P}$ (= projective)      56 57
$\mathbb{P}^{5}$       300
$\mathbb{R}$ (real locus)      266
$\mathbb{R}^{[\alpha,\beta]}$ (twistor-real)      95
$\mathbb{S}^{A}(\mathscr{S})_{1},..., \mathbb{S}_{A^{'}}(\mathscr{S})$ (asymptotic twistor spin-spaces)      417
$\mathbb{S}^{R}$ , $\mathbb{S}^{R^{'}}$ (reduced spin-spaces)      443
$\mathbb{S}^{\rho}$ (spin-space for n dimensions)      441
$\mathbb{S}_{A...H^{'}...}$ (constant spinors)      30
$\mathbb{T}^{\alpha}$ ( $\mathscr{S}$ ) (2-surface twistor space)      399
$\mathbb{T}^{\alpha}, \mathbb{T}^{\beta},..., \mathbb{T}_{\alpha}, \mathbb{T}^{\gamma\delta}_{\beta},...$ (twistor spaces)      46 47
$\mathbb{T}^{\bullet}$ , $\mathbb{T}^{0}$ = $\mathbb{N}$ , $\mathbb{T}^{+}$ , $\mathbb{T}^{-}$ , $\mathbb{T}_{\bullet}$ etc.      56 (57)
$\mathbb{U}^{a}$ , $\mathbb{PU}$ (hyperplane)      456
$\mathbb{V}$ (Minkowski vector space)      1
$\mathbb{V}_{a}$ (n-dimensional vector space)      441
$\mathbf{A} \wedge \mathbf{C}$ (exterior product)      16
$\mathbf{a}$ , $\mathbf{A}$ , $\mathbf{\alpha}$ , $\mathbf{\Gamma}$ (bold upright indices: numerical)      4
$\mathbf{B}_{\alpha\beta}$ , $\mathbf{C}_{\alpha\beta}$ , $\mathbf{J}_{\alpha\beta}$ (bang, crunch etc.)      343 344
$\mathbf{F}^{\alpha} {}_{\beta} = \mathbf{S}^{\alpha\gamma}\mathbf{I}_{\gamma\beta} + \mathbf{\overline{S}}_{\beta\gamma}\mathbf{I}^{\gamma\alpha}$ (Killing vector twistor)      87
$\mathbf{h} = i\bar{\cup}^{\alpha}d\cup_{\alpha}$ , $\Sigma = id\bar{\cup}^{\alpha} \wedge d\cup_{\alpha}$       209
$\mathbf{Z}^{A}$ , $\mathbf{Z}_{A^{'}}$ (spinor parts)      47
$\mathbf{Z}^{\alpha}$ , $[\omega^{A}]$ , $(\omega^{A}, \pi_{A^{'}})$ (twistor)      46 47
$\mathbf{\eta} = \gamma_{1}\gamma_{2}...\gamma_{n}$       442
$\mathbf{\gamma}_{a}$ , $\mathbf{\gamma}_{a}$ , $\gamma_{a\rho} {}^{\sigma}$ , $\gamma_{aR} {}^{S^{'}}$ ( $\gamma$ -matrices)      441 443
$\mathbf{\mathscr{S}} = i \mathbf{\bar{m}} \wedge \mathbf{m}$ (surface-area 2-form)      27
$\mathbf{\Pi}, \mathbf{\tilde{\Pi}} \Pi_{\rho} {}^{\sigma}, \bar{\Pi}_{\rho} {}^{\sigma}$ (projections to reduced spaces)      443
$\mathbf{\varepsilon}_{\alpha\beta\gamma\delta}$ (alternating twistor)      54
$\mathfrak{C}$ (Clifford algebra)      442
$\mathfrak{F}_{A...E}$ , $\mathfrak{D}_{A...D}$ (spinor sheaves)      117
$\mathfrak{p}$ (th), $\partial$ (eth), $\mathfrak{p}^{'}$ , $\partial^{'}$       23
$\mathfrak{p}_{c}$ , $\partial_{c}$ , $\mathfrak{p}^{'}_{c}$ , $\partial^{'}_{c}$ (conformally invariant: notation change from volume 1!)      36
$\mathfrak{S}, \mathfrak{S^{\mathscr{A}}}, ... ,\mathfrak{I}$       4
$\mathfrak{\sigma^{\mathscr{A}}}[P]$ , $\mathfrak{\sigma^{\mathscr{A}}}[\mathscr{U}]$       5
$\mathscr{A}$ (anti-Einstein)      333
$\mathscr{A}, \mathscr{B}, ...$ (clumped indices)      4
$\mathscr{A}_{1}, \mathscr{A}_{2}, ...$ (similarly clumped indices)      5
$\mathscr{B}$ (BMS group)      381
$\mathscr{B}$ (spin-vector bundle)      437
$\mathscr{C}$ (complex conjugation)      313 459
$\mathscr{E}$ (Einstein cylinder)      294
$\mathscr{H}$ (hypersurface)      215
$\mathscr{H}$ -space (Newman's 'Heaven')      129 389
$\mathscr{I}$ ('scri'), $\mathscr{I}^{\pm}$ , $i^{\pm}$ , $i^{0}$       291
$\mathscr{K}$ (null cone in $\mathbb{E}^{6}$ )      301
$\mathscr{L} \subset\mathbb{PV}$ (quadric in projective space)      452
$\mathscr{L}$ (labelling set)      4
$\mathscr{L}$ (Lorentz group)      383
$\mathscr{M}$ (manifold, space-time)      3
$\mathscr{N}$ (null volume 3-form)      28
$\mathscr{N}, \mathscr{N}^{\bullet}, \mathscr{N}_{\bullet}$ (space of rays in $\mathscr{M}$ )      208
$\mathscr{P}$ (Poincare group)      384
$\mathscr{P}$ , $\mathscr{P}^{#}$ (Minkowski 3-space)      187 188
$\mathscr{R}$ (Lorentz rotations)      382
$\mathscr{S}$ (spacelike 2-surface)      26 398
$\mathscr{T}$ (translations)      381
$\mathscr{U}$ (super translations)      381
$\mathscr{\bar{M}}$ (manifold-with-boundary)      291
$\mu\in\mathscr{C}$ (ray of null congruence)      169
$\mu^{A^{'}B^{'}}$ (angular momentum spinor)      71
$\nabla$ (covariant derivative)      15
$\omega$ (function on $\mathbb{C}S^{+}$ ), $\Omega$       266
$\overline{P^{\alpha} {}_{\beta}} = \overline{P}_{\alpha} {}^{\beta}$       56
$\overset{o}{\leftrightarrow}$ (origin-dependent correspondence)      47
$\overset{\circ}{\omega}^{A}$ , $\overset{\circ}{\pi}_{A^{'}}$ (constant spinors)      46
$\partial$ (boundary)      17
$\Phi(P, Q)$ (squared interval)      1
$\Phi_{00},..., \Phi_{22}$ (Ricci spinor components)      22
$\phi_{r}$ (components of massless field)      26
$\Pi = \chi\bar{\chi}\Lambda$       22
$\Pi_{ABCD}$ (Plebanski spinor)      265
$\pounds$ (Lie derivative)      16
$\Psi^{0}_{0},..., \Psi^{0}_{4}$ (Weyl components at $\mathscr{I}$ )      364
$\Psi_{0}, ..., \Psi_{4}$ (Weyl spinor components)      22
$\Psi_{ABCD} = X_{(ABCD)}$ (Weyl spinor)      20
$\rho = R\oplus R^{'}$ (spinor abstract indices reduced)      443
$\rho, \sigma, \kappa, \tau,..., \varepsilon^{'}$ (spin coefficients)      18
$\Sigma$ (symplectic invariant)      184
$\Sigma-, \Sigma-^{'}$ generator (on $\mathbb{C}S^{+}$ )      267
$\sigma...(...)...,\sigma...[...]...$       9
$\sigma^{AB},..., \xi^{AB^{'}}$ ,... (primary parts)      55
$\sqcap = \nabla^{a}\nabla_{a}$ (D'Alembertian)      39

$\sqcap_{AB}$ , $\sqcap_{A^{'}B^{'}}$ (spinor commutator)      21
$\tilde{I}(R^{\alpha\beta}),..., I(\cup_{\alpha}, \vee_{\alpha})$ (infinity functions)      341
$\underset{X}{\nabla}$ (directional derivative)      17
$\Upsilon_{a} = \nabla_{a}log \Omega$       35
$\varepsilon^{RS},..., \tilde{\varepsilon}_{RS^{'}}$       449
$\varepsilon_{AB}, ..., \varepsilon^{A^{'}B^{'}}$       5
$\varepsilon_{\rho\sigma},..., {}^{(\pm)}\varepsilon^{\rho\sigma}$ , $\tilde{\varepsilon}^{\rho\sigma}$       447 448
$\xi$ (holomorphic coordinate for $\mathscr{L}$ )      27
$\Xi$ (Witten 2-form)      430
$\zeta$ (complex stereographic coordinate)      1 31
$^{(r)}G^{\kappa\tau}_{\rho\sigma} = \frac{1}{r}\gamma_{a...c\rho} {}^{\kappa}\gamma^{a...c} {}_{\sigma} {}^{\tau}$       454
$^{(\pm)}E^{\kappa\tau}_{\rho\sigma}$ , $^{(\pm)}\mathbf{E}$       445
$^{-} F_{ab}$ , $^{+} F_{ab}$       12
$^{-}C_{abcd}$ , $^{+}C_{abcd}$ , $^{(\eta)}C_{abcd}$       21
$^{V}A=A_{\iota_{1}\iota_{2}...\iota_{p}}=A_{\alpha_{1}\alpha_{2}...\alpha_{p}}dx^{\alpha_{1}} \wedge ... \wedge dx^{\alpha_{p}}$ (p-form)      16 17
$^{\ast} F_{ab}$ , $^{\dag} J_{abc\mathscr{A}}$ , $^{\ddag} K_{a\mathscr{B}}$ (duals)      11
$_{s}\mathbf{Y}_{j, m}$ (basis for harmonics)      31
'Copies' of spin-space in odd dimension      (440) 458 459
(...)'      19
(p, q)-curve      267
A (scaling at $\mathscr{I}$ )      355 371
a = AA', b = BB', etc.      5
a, A, $\alpha$ , $\Gamma$ (lightface sloping indices: abstract)      4
Abreastness in twistor terms      181 182
Abreastness of non-abreast rays      185
Abreastness of rays      173 176 183 207 215 216
Abstract-index formalism      3
Abstract-index formalism for TV-dimensional spinors      441
Achronal hypersurface      431
Adams, J.F.      441 461
Addition of twistors      47
ADM (Arnowitt — Deser — Misner) mass      404 433
Advanced time coordinate      292 348 356
Affine parameter      170 292
Affine parameter in compacted formalism      (182)
Affine parameter under conformal rescaling      360
Affine parametrization of ray (null geodesic)      170 172
Aitken, A.C.      (285)
Algebraically special      190 191 205 206
Almost complex manifold      213 214
Alpha-curve      218 390
Alpha-plane (surface)      64 164 309
Alpha-plane (surface), higher-dimensional      453 457 458 460 461
Alternating tensor      10 11
Alternating tensor, n-dimensional      442
Alternating twistor      54 65 (328)
Alternating twistor for 2-surface twistors      408 418 419
Alternative law      462
Ambitwistor      (164) 462
Analyticity (real-analytic)      128 (191) 201 215 (351) see
Angular momentum (moment, kinematic) twistor      71 73 85 401
Angular momentum at $\mathscr{I}$       420
Angular momentum for asymptotically flat space      44 392 420
Angular momentum for linear gravity      75 395
Angular momentum in general relativity      395
Angular momentum of Kerr solution      109 204 205
Angular momentum of Kerr space-time      109 205
Angular momentum, (anti)-self-dual      419
Angular momentum, Quantized      144
Angular momentum, tensor      68
Angular momentum, total      110
Angular momentum, twistor      71 85 92
Angular momentum, twistor and Kerr's theorem      203 204
Angular momentum, twistor, apparent paradox      77 88 93
Angular momentum, twistor, quantized      144 148
Angular momentum, twistor, quasi-local version      402
Angular momentum, twistor, twistor version      (149)
Anti-celestial sphere      1
Anti-de Sitter space      336
Anti-Einstein universe      333
Anti-Grgin behaviour of massless free field      330
Anti-self-dual      12 242 452
Anti-self-dual, Weyl curvature      21 129 164 390
Anti-self-dual, Yang — Mills curvature      35 164
Anti-symmetric (skew) tensor      11 12 14 see form"
Anti-symmetric (skew) tensor, simple      14 247 451 452
Arnol'd, V.I.      211
Arnowitt — Deser — Misner (ADM) mass      404 429 433
Arnowitt, R.      404 429
Aronson, B.      389
Ashtekar — Hansen angular momentum      404
Ashtekar, A.      (353) 404 429
Astigmatism of curvature      230 (231)
Asymptotic Einstein condition      352 354 363
Asymptotic Einstein condition eliminates logarithms      362
Asymptotic Einstein condition, strong      357 367 409
Asymptotic flatness      350 see conformal"
Asymptotic simplicity      292 347 351 see conformal"
Asymptotic simplicity, future      353 367 408
Asymptotic simplicity, weak      351 353
Asymptotic spin space      411 417
Asymptotic symmetry group      366
Asymptotic symmetry group for FRW (Friedmann — Robertson — Walker) models      366
Asymptotic twistors      212 389 390
Asymptotically plane wave front      377 378
Atiyah — Singer index (theorem)      399 454
Atiyah, M.F.      44 (129) 390 434 464
Bach tensor      127 (133) 137
Bach, R.      127
Backward tube      316
Bad cut      384
Bailey, T.N.      (149) 152
Bang twistor      342
Bardeen, J.M.      350 428
Basis      4
Basis 1-forms in abstract-index formalism      17 437 438
Basis for twistors      48
Basis, 2-spinor basis      6
Basis, n-dimensional (spinor) basis      442
Basis, twistor basis      48
Basis, twistor basis, dual      50
Bateman, H.      (139)
Bel — Robinson tensor      (85)
Bel, L.      (85)
Bergmann, P.G.      77
Beta-curve      217 219 390 406
Beta-plane      64 217 309
Beta-plane, higher-dimensional      453 457 458 460 461
Bianchi identity      22 121 196 356
Bianchi identity in compacted spin-coefficient (GHP)      25 26
Bianchi identity in terms of $P_{ab}$       123 387
Bianchi identity, linearized      38
Bianchi identity, spin-coefficient form (compacted)      25 26 424
Bianchi identity, spinor form      22 424
Bianchi identity, spinor form of Yang — Mills theory      35
Bianchi identity, spinor form, Einstein — Maxwell case      34
big bang      335 343 345
Big crunch      335 336 343
Bivectors      11 242
Bivectors, simple      14
Black and white regions on $S^{+}$ (anti-celestial sphere)      275
Black hole      204 351
BMS (Bondi — Metzner — Sachs) group      366 369
BMS (Bondi — Metzner — Sachs) group and (super-) translations      381
BMS (Bondi — Metzner — Sachs) group and Lorentz rotations      382
BMS (Bondi — Metzner — Sachs) group for $\mathbb{M}$       380 381
Bocher, M.      298
Bondi (retarded) time coordinate      373 380 393
Bondi parameter      367 368
Bondi parameter for infinity (conformal)      367 373
Bondi system      386
Bondi time coordinate for infinity (conformal)      373 380 393
Bondi — Metzner — Sachs group      369 370 see
Bondi — Sachs mass, momentum      44 392 396 (406) 415 423 431 see
Bondi — Sachs mass, momentum from Hawking expression      416
Bondi — Sachs mass, momentum, mass loss theorem      424
Bondi — Sachs mass, momentum, positivity      429
Bondi — Sachs metrics      350
Bondi — Sachs news function      386 425
Bondi, H.      179 185 336 350 (396) 415 (424)

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