Gravitational Wave Solutions of Einstein's Equation  

pp-Wave Solutions > s.a. [gravitational radiation; solutions]; Brinkman's Theorem; chaotic motion.
* Idea: Plane-fronted waves with parallel rays, in which all scalar curvature invariants vanish.
$ Def: Metrics of the form ds2 = 2 du dv + 2 d d* + (f + f*) du2, with f = f(u,), where C spans u = const.
* Special cases: If f is linear in , Minkowski; if f = g(u) 2, plane waves, see below.
@ General references: 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].
@ Impulsive: Steinbauer gq/98-in; Kunzinger & Steinbauer JMP(99)gq/98, CQG(99)gq/98 [distributional diffeomorphisms].

Plane Wave Solutions > s.a. null infinity; quasilocal general relativity.
$ Of the full Einstein equation: Solutions of the vacuum Einstein equation homeomorphic to R4, 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).
* 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 ab = Hab exp(i kmxm), with added gauge conditions.
@ References: Einstein & Rosen JFI(37); Bondi et al PRS(59); Penrose RMP(65), in(68); in Hawking & Ellis 73, 178-179; Matzner & Tipler PRD(84) [curvature singularities]; Neville PRD(97) [intrinsic spin]; Torre GRG(06)gq/99 [symmetries]; Montanari & Calura AP(00) [and electromagnetic fields]; David JHEP(03) [with weak singularities].

Bondi-Sachs Metric
* Idea: Describes gravitational radiation from an isolated source, and is valid in the vicinity of +.
* Line element: It is of the form

ds2 = W du2 – 2 exp{2} 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].

Other Gravitational Wave Solutions > s.a. canonical quantum gravity; harmonic maps; higher-order gravity; Robinson-Trautman.
* Bell-Szekeres: Electrovacuum solutions representing the collision of pure electromagnetic plane waves with collinear polarization.
* Gyraton: A beam-pulse of spinning gravitational radiation, for which all scalar invariants constructed from the curvature and its covariant derivatives vanish.
@ Axisymmetric: Kuchar PRD(71) [cylindrical]; Herrera & Jiménez JMP(86); Kramer CQG(99).
@ Types: Repchenkov JETP(79) [without convergence]; Bini et al NCB(89) [wave packets]; Chandrasekhar & Ferrari PRS(93) [spherical]; Alì & Hunter JMP(99) [large-amplitude]; Bicak in(02)gq [existence, examples]; Hubeny & Rangamani JHEP(03)ht/02 [asymptotically plane wave]; Bondi PRS(04) [z-independent standing waves]; Hervik CQG(04) [solvegeometric]; Edgar & Ramos GRG(07)gq/06 [type O, with cosmological constant].
@ Impulsive: Aliev & Nutku CQG(01)gq/00 [spherical].
@ Bell-Szekeres: Clarke & Hayward CQG(89); Dorca PRD(98)gq [Tab]; 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].
@ Gyraton: Frolov & Fursaev PRD(05) [arbitrary D]; Frolov et al PRD(05)ht [arbitrary dimension]; Frolov & Zelnikov CQG(06) [charged].
@ Cosmological backgrounds: Gowdy PRL(71) [closed universes]; Caldwell PRD(93) [Green functions in FRW spacetime]; > s.a. green functions.
@ Einstein-Maxwell: Simon CQG(92); Barrabès et al LMP(98)gq/97 [colliding]; Hervik CQG(03)gq/02 [from 5D vacuum]; Gurtug et al GRG(03)gq [+ scalar, colliding].
@ Einstein-Yang-Mills: Gurtug & Halilsoy a0802 [colliding].

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