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 is 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]; Bicak 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].
@ Books, reviews:
Bourguignon
80 [mathematical]; Islam 85; Stephani et al 02; Chrusciel gq/04-in
[GR17 results]; Friedrich AdP(06)gq/05-in;
MacCallum AIP(06)gq
[methods, applications]; Bicak gq/06-in;
Senovilla CQG-a0710-in
[GR18 report]; Mueller & Grave a0904 [Catalogue
of Spacetimes]; Griffiths & Podolsky 09.
@ Interpretation: Bonnor GRG(92),
et al GRG(94);
Bicak gq/02-in.
Types of Solutions > s.a. einstein's
equation; types
of metrics [degenerate] and spacetimes;
with matter [including cosmological constant].
@ 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]; > s.a.
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: Chrusciel 91 [uniqueness]; 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 a0807 [time-periodic].
> Related concepts:
see embeddings; horizons; petrov [algebraically
special, classification]; torsion.
> Vacuum solutions:
see Goldberg-Sachs
Theorem; gravitational-wave solutions; Minkowski
space.
> Other types:
see black holes; Lemaître-Tolman-Bondi;
relativistic cosmology; self-dual solutions;
Vaidya; Wahlquist;
with symmetries.
> In other theories:
see 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); Chrusciel
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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