Solutions of Einstein's Equation with Symmetries| Solutions of Einstein's Equation with Symmetries |
In General
> s.a. einstein equation; generating methods;
killing fields; lorentzian geometry;
Maximally Symmetric Geometry; 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].
* Symmetry reduction and integrability:
The integrability of some symmetry-reduced vacuum and electrovacuum Einstein equations
has given rise to the construction of powerful solution generating methods, including
the inverse-scattering approach, soliton-generating techniques, Bäcklund and
symmetry transformations.
@ References: Beig & Chruściel CQG(97)gq/96 [asymptotically flat, Killing data],
CMP(97)gq/96 [asymptotically flat and empty, symmetry groups];
Antoci & Liebscher a1007 [interpreting solutions with Killing symmetries];
Alekseev a1011-MG12
[integrable symmetry reductions].
Static Solutions
> s.a. black-hole solutions; spherically-symmetric
solutions; schwarzschild spacetime; types
of spacetimes [cylindrical].
$ Def: Solutions of the 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
Γttt
= Γtij
= Γitj
= 0, for all i, j, and
R ttij
= 0.
* Static axisymmetric: They
can be parametrized as Weyl solutions by the line element
ds2 = −e2Uc2dt2 + e−2U [e2γ(dz2 + dr2) + r2 dφ2] ,
where U and γ are functions of r and z;
The vacuum field equations are U,rr
+ U,r
r−2
+ U,zz
= 0 (the Laplace equation ∇2U(r, z)
= 0), and dγ = 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)-a1408 [spinning test particles].
@ Other vacuum:
Beig CQG(91) [conformal properties];
Chruściel CQG(99)gq/98 [classification];
Dadhich & Date gq/00 [axisymmetric];
Gutsunaev et al G&C(02) [rev];
Chruściel APPB(05)gq/04 [analyticity at horizons];
Anderson & Khuri a1103-wd [with prescribed geometric or Bartnik boundary data];
Qing & Yuan JGP(13).
@ Electrovac: Chruściel CQG(99)gq/98 [re classification];
> s.a. solutions with matter.
@ Einstein-Yang-Mills: Kleihaus & Kunz PRD(98)gq/97 [+ dilaton, axisymmetric];
Radu PRD(02)gq/01 [Λ < 0].
@ With cosmological constant: Chruściel & Simon JMP(01)gq/00 [Λ < 0 generalized Kottler];
Anderson et al gq/04-proc.
@ 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 CQG(07)gq/06 [Einstein-Vlasov cylinders].
Stationary Solutions > s.a. axisymmetry [including
Ernst equation]; black-hole solutions; types
of spacetimes [pseudostationary, cylindrical].
* Idea: Solutions of Einstein's
equation with a timelike Killing vector field; Important examples are the axisymmetric
kerr and kerr-newman
black-hole solutions.
* Result: The only geodesically complete
stationary vacuum solution of the Einstein equation is Minkowski, or a quotient of it.
* Result: A stationary, analytic
black-hole spacetime satisfying Einstein's equation must be axisymmetric.
* Vacuum: The most general
stationary, axisymmetric, sourcefree see involves two arbitrary functions (mass
and angular momentum distribution on the axis); The only non-singular one is
Minkowski/discrete G [@ Anderson AHP(00)gq].
* Electrovac case: The solutions
depend on two complex Ernst potentials, \(\cal E\) and ψ.
@ General references: Anderson AHP(00)gq [uniqueness];
Beig & Schmidt LNP(00)gq [review];
Clément a1109-ln.
@ 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 PLB(99)gq/98 [singular rings];
Beig et al CQG(09)-a0907 [with reflection symmetry, non-existence results];
Beig CM-a1005-conf [stationary n-body problem];
Bičák et al CQG(10)-a1008 [time-periodic implies stationary];
> s.a. Inverse Scattering; numerical models.
Solutions with Spacelike Symmetries
> s.a. axisymmetry [including cylindrical symmetry];
gowdy spacetime; spherical symmetry.
* Spatially homogeneous: Bianchi models (Kasner,
Friedmann solutions); + isotropic solutions (FLRW 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:
Chruściel 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 AIP(08)-a0709 [orthogonally transitive, commuting].
@ Plane symmetry: Pradhan et al IJTP(07)gq/06 [pfluid];
Sharif & Aziz CQG(07)gq/06 [pfluid];
Jones et al AJP(08)jan-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-conf,
gq/00-MG9,
CQG(01)gq/00 [integrability];
Avakyan et al gq/01
["homogeneous gravitational 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 in(00)gq/99 [approximate symmetries].
> Other solutions:
see c-metric; gödel solution;
Lewis-Papapetrou; Melvin;
Oppenheimer-Snyder; Self-Similarity;
Taub-NUT Solution; Tolman Solutions.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 mar 2016