Solutions of Einstein's Equation  

In General > s.a. asymptotic flatness; generating methods; initial-value formulation [existence, stability].
* History: Up to the 1970s, very few exact solutions were known; Now the field has become very rich.
@ General references: Jordan et al AWL(60), re GRG(09); Hoenselaers & Dietz ed-84; Bonnor GRG(82); Xanthopoulos in(85); MacCallum in(88); Rendall LRR(98)-LRR(00)-LRR(02)-LRR(05) [existence]; Bičák LNP(00)gq-in; Schmidt ed-00; Pollney et al CQG(00), CQG(00), CQG(00) [computer database]; Ishak & Lake CQG(02)gq/01 [online database]; Ehlers GRG(06) [remarks]; Isenberg a1610-in [modeling astrophysical events and cosmology, solutions in which determinism and causal relationships break down]; Podolský CQG+(18); Etesi a1903 [solvability result].
@ Books, reviews: Bourguignon 80 [mathematical]; Islam 85; Stephani et al 04; Chruściel gq/04-GR17; Friedrich AdP(06)gq/05-in; MacCallum AIP(06)gq [methods, applications]; Bičák in(06)gq; Senovilla CQG(08)-a0710-GR18; Müller & Grave a0904 [Catalogue of Spacetimes]; Griffiths & Podolský 09; Barrow 11.
@ Interpretation: Bonnor GRG(92), et al GRG(94); Bičák gq/02-GR16.
> Online resources: see Differential Geometry Library page; Maple blog(15)oct.

Types of Solutions > s.a. einstein's equation; types of metrics [degenerate] and spacetimes; with matter [including cosmological constant].
* Universal metrics: Solutions of the vacuum field equations for general relativity and all higher-order theories of gravity.
@ General references: Chruściel 91 [uniqueness].
@ Universal metrics: Hervik et al CQG(15)-a1503 [type II]; Gürses et al CQG(17)-a1603 [Kerr-Schild-Kundt metrics]; Hervik et al JHEP(17)-a1707 [4D].
@ Non-trivial topology: Isenberg AP(80); Harriott & Williams JMP(88), IJTP(89); > s.a. geons [including kinks].
@ Approximate: Bel GRG(87); Garriga & Verdaguer GRG(88); Ellis in(93); Detweiler PRD(94)gq/93 [binaries], & Brown PRD(97)gq/96 [post-Minkowski expansion]; Manko & Ruiz CQG(04)gq [stationary axisymmetric]; Adler & Overduin GRG(05) [approximately flat]; Reiterer & Trubowitz a2005 [formal power series]; > s.a. gravitating matter.
@ Inhomogeneous: MacCallum in(79); Feinstein et al CQG(95).
@ No symmetries: Robinson & Robinson IJTP(69); Koutras & McIntosh CQG(96); Sussman & Triginer CQG(99)gq/98.
@ With defects: Edelen IJTP(94); Letelier & Wang JMP(95); > s.a. solutions with matter [distributional].
@ Other: Hayward CQG(93) [on a null 3-surface]; Puszkarz gq/97 [dipole]; Katanaev et al AP(99) [warped products]; Lewandowski CQG(00)gq/99 [with isolated horizons]; Robinson GRG(02) [holomorphic]; Headrick & Wiseman CQG(05)ht [on Calabi-Yau manifolds]; Klainerman & Rodnianski AM(05) [rough]; Kong et al ScCh(10)-a0807 [time-periodic]; Cropp MS-a1108.
> Related concepts: see embeddings; horizons; petrov classification [algebraically special]; quantum gravity [corrections]; torsion in physical theories.
> Vacuum solutions: see Goldberg-Sachs Theorem; gravitational-wave solutions [including impulsive, weakly regular solutions]; Minkowski space.
> Black holes and collapsing solutions: see black holes; Vaidya Metric; Lemaître-Tolman-Bondi Solutions; Wahlquist Metric.
> Other types: see relativistic cosmology; self-dual solutions; solutions with symmetries.
> In other theories: see 3D general relativity; 3D gravity; higher-order gravity; kaluza-klein theory.

Space of Solutions > s.a. generating methods; perturbations [stability]; space of lorentzian metrics and riemannian metrics.
* Smoothness: In the globally hyperbolic spatially compact case it is a smooth manifold near g iff g has no Killing vector fields; If g has a k-dimensional space of Killing vector fields and a compact Cauchy surface of constant mean curvature, then it has a conical singularity.
* Isolated solutions: There is no isolated solution in the space of Lorentzian metrics on a manifold M [@ Lerner CMP(73)].
@ General references: Lerner CMP(73); Fischer et al in(80); in Marsden 81; Isenberg & Marsden PRP(82); Andersson JGP(87) [action of diffeomorphisms]; Isenberg PRL(87); Saraykar JMP(88).
@ With symmetries: Fischer et al AIHP(80); Arms et al AP(82); Chruściel AP(90) [U(1) × U(1)]; McIntosh & Arianrhod GRG(90).
@ Conformal symmetries: Eardley et al CMP(86); Garfinkle JMP(87); Garfinkle & Tian CQG(87).
@ Related topics: Moncrief JGP(84) [generalized Taub-NUT]; Kriele gq/96 [with signature change]; Anderson JMP(03)gq/02 [curvature bounds].

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