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 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,r2U,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 (Kasner, Friedmann); + isotropic (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.


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