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].
@ In cosmological backgrounds: Gowdy PRL(71) [closed universes]; Caldwell PRD(93) [Green functions in FLRW spacetime]; Bishop 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-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 to \(\mathbb R\)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 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(dxiU idu)(dxjU 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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