Solutions of Einstein's Equation| 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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