Gravitational-Wave Solutions of Einstein's Equation| Gravitational-Wave Solutions of Einstein's Equation |
Gravitational-Wave Solutions in General > s.a. gravitational
radiation; solutions of einstein's equation.
@ General references: Jordan et al AWL(60);
Repchenkov JETP(79) [without convergence];
Bini et al NCB(89) [wave packets];
Chandrasekhar & Ferrari PRS(93) [spherical];
Alì & Hunter JMP(99) [large-amplitude];
Bičák in(02)gq [existence, examples];
Hubeny & Rangamani JHEP(03)ht/02 [asymptotically plane wave];
Bondi PRS(04) [z-independent standing waves];
Keane & Tupper CQG(04)-a1308 [conformal symmetry classes];
Hervik CQG(04) [solvegeometric];
Edgar & Ramos GRG(07)gq/06 [type O, with cosmological constant].
@ Axisymmetric:
Kuchař PRD(71) [cylindrical];
Herrera & Jiménez JMP(86);
Kramer CQG(99).
@ Other types: Bičák et al PRD(12)-a1207 [rotating, to second order in the amplitude];
Swearngin a1302,
Thompson et al JPA(14)-a1402 [linked and knotted solutions];
Tucker & Walton CQG(17)
+ CQG+ [spatially compact pulsed gravitational waves, and astrophysical jets];
Bini et al PRD(18)-a1801,
Firouzjahi & Mashhoon a1904 [twisted].
@ In cosmological backgrounds: Gowdy PRL(71) [closed universes];
Caldwell PRD(93) [Green functions in FLRW spacetime];
Bishop PRD(16)-a1512 [in de Sitter spacetime];
> s.a. green functions.
@ With matter and in other theories: Simon CQG(92) [Einstein-Maxwell theory].
Solutions with Vanishing Curvature Invariants > s.a. Brinkman's
; chaotic motion; killing tensors.
$ pp-wave solutions:
Metrics representing plane-fronted waves with parallel rays, in which
all scalar curvature invariants vanish; They are of the form
ds2 = 2 du dv + 2 dζ dζ* + (f + f*) du2, with f = f(u, ζ), where ζ ∈ \(\mathbb C\) spans u = const ;
Special cases are those in which f is linear in ζ,
which gives Minkowski spacetime, and f = g(u)
ζ2, which gives plane waves, see below.
* Gyraton: A beam-pulse of
spinning gravitational radiation, for which all scalar invariants constructed
from the curvature and its covariant derivatives vanish.
@ pp-wave solutions: Peres PRL(59)ht/02;
in Kramer et al 80;
Szabados CQG(96) [spacelike 2-surface geometry];
Balakin et al G&CS(02)gq [precession];
Hubeny & Rangamani JHEP(02)ht [causal structure];
Coley et al PRD(03)gq/02 [higher-dimensional];
Nutku CQG(05)gq [electrovac];
Balasin & Aichelburg GRG(07)-a0705 [canonical formulation];
Milson et al JMP(13)-a1209 [invariant classification];
Hervik et al CQG(14)-a1311
[universal metrics, which solve the vacuum equations of all gravitational theories with Lagrangian
constructed from the metric, the Riemann tensor and its derivatives of arbitrary order].
@ pp-waves, in other theories: Baykal TJP(16)-a1510 [in modified gravity, rev];
> s.a. finsler geometry.
@ Impulsive: Steinbauer gq/98-proc;
Kunzinger & Steinbauer JMP(99)gq/98,
CQG(99)gq/98 [distributional diffeomorphisms];
Luk & Rodnianski a1209 [propagation].
@ Gyraton: Frolov & Fursaev PRD(05) [arbitrary D];
Frolov et al PRD(05)ht [arbitrary dimension];
Frolov & Zelnikov CQG(06) [charged].
Plane Wave Solutions > s.a. null
infinity; quasilocal general relativity.
$ Of the full Einstein
equation: Solutions of the vacuum Einstein equation homeomorphic
4, of the form
ds2 = 2 du dv + dy2 + dz2 + H(y, z, u) du2 , H = (y2 − z2) f(u) − 2 yz g(u) ,
with f(u) and g(u) arbitrary C2
functions (representing amplitude and polarization of the waves).
* Form of the metric: Strong-field
gravitational plane waves are often represented in either the Rosen or Brinkmann forms.
* Properties: They admit a
5-parameter (or 6-parameter in some cases) group of isometries which acts
transitively on u = constant surfaces; They satisfy the causality
condition, but they do not admit global Cauchy surfaces; They have the same scalar
invariants as flat space; To distinguish them, have to use the frame bundle.
* Of the linearized equation:
Perturbations of the form γab
= Hab
exp(i kmxm),
with added gauge conditions.
@ General references:
Einstein & Rosen JFI(37);
Bondi et al PRS(59);
Penrose RMP(65),
in(68);
in Hawking & Ellis 73, 178-179;
Cropp & Visser CQG(11)-a1004 [Rosen form, general polarization modes];
Hinterleitner & Major PRD(11)-a1006 [in real connection variables].
@ Geometry:
Matzner & Tipler PRD(84) [curvature singularities];
Neville PRD(97) [intrinsic spin];
Torre GRG(06)gq/99 [symmetries];
David JHEP(03) [with weak singularities];
Shore JHEP(17)-a1705 [twisted null geodesic congruences].
@ Electrovac: Montanari & Calura AP(00);
Hervik CQG(03)gq/02 [from 5D vacuum].
Bondi-Sachs Metric
* Idea: Describes gravitational radiation from an
isolated source, and is valid in the vicinity of \(\cal I\)+.
* Line element: It is of the form
ds2 = W du2 − 2 exp{2β} du dr + r2 hij(dxi − U idu)(dxj − U jdu) ,
where hij is a specific spacelike metric.
@ References: in d'Inverno 92;
in Shore CP(03)gq;
Korbicz & Tafel CQG(04) [action and Hamiltonian];
Mädler & Winicour a1609-en [formalism, rev].
Impulsive and Colliding-Wave Solutions
> s.a. geodesics; Robinson-Trautman Spacetimes.
* Bell-Szekeres:
Electrovacuum solutions representing the collision of pure electromagnetic
plane waves with collinear polarization.
@ Impulsive: Aliev & Nutku CQG(01)gq/00 [spherical];
LeFloch in(11)-a1009 [rev];
van de Meent CQG(11)-a1106 [piecewise flat, impulsive plane wavefront];
Sämann & Steinbauer CQG(12)-a1207 [completeness].
@ Bell-Szekeres: Halil IJTP(81) [particle motion];
Clarke & Hayward CQG(89);
Dorca PRD(98)gq [\(\langle\)Tab\(\rangle\)];
Gürses et al PRD(03) [higher-dimensions];
Barrabès & Hogan CQG(06)gq [generalizations].
@ Other colliding: Khan & Penrose Nat(71)jan;
Chandrasekhar & Xanthopoulos PRS(86),
PRS(87),
PRS(88);
Yurtsever PRD(88),
PRD(89);
Griffiths 91;
Chakrabarti IJMPD(93);
Barrabès et al PTP(99)gq/00;
Gürses et al PRD(02)gq,
Gutperle & Pioline JHEP(03)ht [higher dimensions];
Barrabès & Hogan GRG(14) [collision of two homogeneous, plane, gravitational shock waves].
@ Colliding, with matter:
Hogan et al LMP(98)gq/97 [Einstein-Maxwell];
Gurtug et al GRG(03)gq [Einstein-Maxwell + scalar];
Gurtug & Halilsoy IJMPA(09)-a0802 [Einstein-Yang-Mills];
> s.a. solutions with matter.
> Related topics: see canonical quantum gravity;
harmonic maps; higher-order gravity.
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