Solutions
of Einstein's Equation with Symmetries| Solutions
of Einstein's Equation with Symmetries |
In General > s.a. einstein
equation; generating methods; killing fields; types
of spacetimes.
* Result: Any 2D Lie transformation group which contains a 1D subgroup
whose orbits are circles must be Abelian, and any 3D such group containing
a subgroup with closed orbits must have associated Lie algebra of Bianchi type
I (Abelian), II, III, VII0, VIII or IX [@ Barnes CQG(00)gq].
@ References: Beig & Chrusciel CQG(97)gq/96 [asymptotically
flat, Killing
data], CMP(97)gq/96 [asymptotically
flat
and empty, symmetry groups].
Static Solutions > s.a. black
hole solutions; spherical;
schwarzschild; types
of spacetimes [cylindrical].
$ Def: Solutions of Einstein
equation with a hypersurface-orthogonal timelike Killing vector field; The
line
element can be written in the form
ds2 = –V 2(x) dt2 + hij(x) dxi dxj .
* Properties: They always have 
ttt
= tij
= itj
= 0, for all i, j, and R ttij
= 0.
* Static axisymmetric: They can be parametrized as Weyl
solutions by the line element
ds2 = –exp{2U}c2dt2 +
exp{–2U}
[exp{2
}(dz2 +
dr2) + r2 d
2]
,
where U and are
functions of r and z;
The vacuum field equations are U,rr + U,r /r + U,zz
= 0 (the Laplace equation 
2U(r, z)
= 0), and d = r (U,r2 – U,z2)
dr + 2r U,r U,z
dz.
* Other special cases: Ultrastatic; In the electrovac case, Papapetrou-Majumdar.
@ Weyl spacetimes:
in Vieira & Letelier PRL(96); Emparan & Reall PRD(02) [D 
4]; Bini
et al JPA(05)
[spinning test particles].
@ Other vacuum: Beig CQG(91) [conformal properties]; Chrusciel CQG(99)gq/98 [classification];
Dadhich & Date gq/00 [axisymmetric];
Gutsunaev
et al G&C(02)
[rev]; Chrusciel APPB(05)gq/04 [analyticity
at horizons].
@ Electrovac: Chrusciel CQG(99)gq/98 [re classification].
@ Einstein-Yang-Mills: Kleihaus & Kunz PRD(98)gq/97
[+ dilaton, axisymmetric]; Radu PRD(02)gq/01 [
< 0].
@ With cosmological constant: Chrusciel & Simon JMP(01)gq/00 [ < 0
generalized Kottler];
Anderson et al gq/04-in.
@ Other matter and/or properties: Beig & Simon CMP(92)
[pfluid, uniqueness];
Allison & Ünal JGP(03)
[geodesics]; Gaudin
et al IJMPD(06)gq/05 [massless
scalar]; Fjällborg gq/06 [Einstein-Vlasov
cylinders].
Stationary Solutions > s.a. axisymmetry [including
Ernst equation]; scattering [inverse]; types
of spacetimes [pseudostationary].
* Idea: Solutions of
Einstein's equation with a timelike Killing vector field; Important examples
are the axisymmetric kerr, kerr-newman,
...
* Result: The only geodesically
complete stationary vacuum solution of
the Einstein equation is Minkowski, or a quotient of it.
* Vacuum: The most general
stationary, axisymmetric, sourcefree see involves two arbitrary functions (mass
and angular momentum distribution on the axis);
The only
nonsingular one is Minkowski/discrete G [@ Anderson AHP(00)gq].
* Electrovac case: The solutions depend on two complex Ernst potentials,
and 
.
@ Initial data:
Dain CQG(01)gq [asymptotically
flat]; Pfeiffer et al PRD(05)gq/04 [+
gravitational waves]; Dain PRL(04)
[departure from stationarity measure].
@ With fluid: Mars & Senovilla CQG(96)gq/02 [axisymmetric,
1 conformal Killing vector field];
Rácz & Zsigrai CQG(96), CQG(97).
@ Related topics: Clément gq/98 [singular
rings]; Anderson AHP(00)gq [uniqueness];
Beig & Schmidt
LNP(00)gq [review].
Solutions with Spacelike Symmetries > s.a. axisymmetry;
gowdy spacetime; spherical
symmetry.
* Spatially homogeneous:
Bianchi (Kasner, Friedmann); + isotropic (FRW models,
de Sitter and anti-de Sitter).
* Flat: Minkowski space
and
all the ones obtained by identifications
in it [@ Fried JDG(87)].
@ Spatially homogeneous: Christodoulakis et al JPA(04) [4+1]; Apostolopoulos
CQG(05)gq [expansion-normalized
variables]; > s.a. bianchi
models for 3+1.
Other Special Solutions > s.a. horizons [apparent];
einstein equation; radiating
solutions; solutions with matter; types
of metrics [degenerate].
* "Integrable" types:
Self-dual, stationary axisymmetric vacuum and electrovac
equations have transitive symmetry groups.
@ One Killing vector field: Moncrief AP(86)
[U(1)]; McIntosh & Arianrhod GRG(90),
Rácz
JMP(97)gq/93 [non-null];
Isenberg & Moncrief CQG(92).
@ Two Killing vector fields: Chrusciel
AP(90)
[U(1) × U(1)]; Husain LMP(96)
[commuting]; Mars & Wolf CQG(97)gq/02 [with
conformal Killing vector field]; Zagermann CQG(98)gq/97 [metric
vs connection reduction]; Alekseev & Griffiths
PRL(00)gq [spacelike];
Sparano et al PLB(01)gq [non-abelian],
DG&A(02)gq/03,
DG&A(02)gq/03;
Isenberg & Weaver
CQG(03)gq [T2 vacuum,
global existence]; Szereszewski & Tafel
CQG(04)gq/03 [pfluid];
Alekseev TMP(05)gq [spaces
of local solutions of integrable reductions]; Marvan & Stolin a0709 [orthogonally
transitive, commuting].
@ Plane symmetry: Pradhan et al gq/06/IJTP
[pfluid]; Sharif & Aziz CQG(07)gq/06
[pfluid]; Jones et al a0708 [infinite-plane-like].
@ Other symmetries: Taub AM(51) [3 commuting
Killing vector fields]; Misner AP(63)
[time-symmetric]; Zafiris JMP(97)gq [general];
Senovilla & Vera CQG(99)gq [classification];
Fayos & Sopuerta gq/00-in,
gq/00-MG9,
CQG(01)gq/00 [integrability];
Avakyan et al gq/01 ["homogeneous
gravitationa field"].
@ Features of special solutions: Dietz FP(88); Hoenselaers & Dietz
PLA(88);
Hoenselaers & Skea GRG(89)
[Petrov II electromagnetic null field]; Araujo et al GRG(92); Zalaletdinov gq/99-in
[approximate symmetries].
> Other: see c-metric; gödel; Lewis-Papapetrou;
Melvin; Oppenheimer-Snyder; Self-Similarity; Taub-NUT;
Tolman.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
12 jun 2008