Generating Solutions of Einstein's Equation  

In General > s.a. axisymmetry [Ernst]; black-hole solutions; duality; einstein's equation; hamilton-jacobi theory; Kaluza-Klein; riemann tensor.
* Fluid-gravity correspondence: The idea that Einstein's equation (with negative cosmological constant) in d+1 dimensions captures the (generalized) Navier-Stokes equation in d dimensions; It was developed in the context of the gauge/gravity duality.
* Newman-Janis algorithm: A method for finding new stationary solutions, used to obtain the Kerr metric from the Schwarzschild metric using a complex transformation within the framework of the Newman-Penrose formalism.
@ Newman-Janis algorithm: Newman & Janis JMP(65); Yazadjiev GRG(00)gq/99; Keane CQG(14)-a1407 [extended]; Erbin & Heurtier CQG(15)-a1411 [5D].
@ Other stationary from static: Clément PRD(98)gq/97, gq/98, G&C(99)gq-conf [Einstein-Maxwell].
@ Spatial symmetries: Deser & Tekin CQG(03); Chaisi & Maharaj Pra(06)gq [anisotropic from isotropic].
@ Matter from vacuum: Tangen a0705 [Einstein-scalar from vacuum + cosmological constant].
@ Fluid-gravity correspondence: Rangamani CQG(09)-a0905-ln; Hubeny CQG(11)-a1011-GR19 [rev]; Lysov a1310 [charged fluids]; > s.a. turbulence.
@ Thermodynamic approach: Zhang & Li PLB(14)-a1406; Zhang et al PRD(14) [Schwarzschild solution]; Tan et al a1609 [using the Komar mass].
@ Other: Singleton PLA(96)gq [from Yang-Mills solutions]; Torres del Castillo JMP(96) [from linearized solutions]; Quevedo & Ryan in(00)gq/03 [cosmological]; Iguchi & Mishima PRD(06) [5D vacuum]; Haesen et al gq/07 [with spacelike circle action]; Contopoulos et al a1501/JMP.
@ Other techniques: Denisova & Mehta GRG(97) [successive approximations]; Alekseev gq/99-conf, PhyD(01)gq/00-in [monodromy transform]; Scheel et al PRD(06)gq [using two coordinate systems]; Yunes CQG(07)gq/06 [gluing scheme, and binary black holes]; Vacaru IJGMP(08)-a0704, IJGMP(11) [anholonomic deformation method]; Starkl JMP(07) [GAP theory]; Vacaru IJTP(10)-a0909 [method of anholonomic frame deformations]; Beheshti & Tahvildar-Zadeh a1312 [dressing technique combined with a control-theory approach]; Korkina & Kopteva STEI-a1604 [mass-function method]; > s.a. numerical methods.

1 Killing Vector Field and Ehlers-Harrison Transformations > s.a. axisymmetry [electrovac].
* Idea: If one starts with a solution of Einstein's equation with a Killing vector field, one can give it equivalently as a set of fields on the manifold of orbits of this Killing vector, the induced metric, a covector field (the twist of the Killing vector) and a scalar field (its norm); Or, one can collect the last two into a single object and obtain a σ-model on this space of orbits; On these fields, one can perform some transformations (which form a group if one is not worried about superfluous transformations) and the new fields in turn define new full metrics on the original manifold; Can be generalized to more Killing vectors (see below).
@ General references: Buchdahl QJM(54); Ehlers 57 [discovery, stationary vacuum case]; Ehlers in(59) [1 hypersurface-orthogonal Killing vector]; Harrison JMP(68) [special electrovac]; Neugebauer & Kramer AdP(69) [electrovac, non-linear realization of SO(2,1)]; Geroch JMP(71) [4D vacuum, 1 Killing vector, SL(2, \(\mathbb R\))/N]; Kinnersley JMP(73) [4D, 1 Killing vector, SU(2,1)]; Mason & Woodhouse Nonlin(88); Fayos et al GRG(89); Mars CQG(01)gq.
@ Related topics: Garfinkle et al GRG(97)gq/96 [with perfect fluids]; Gustafsson & Haggi-Mani CQG(99)ht/98 [in supergravity].

2 Killing Vector Fields and Geroch Group > s.a. axisymmetry [Geroch conjecture]; einstein equation; solutions with symmetries.
* Idea: If one reduces the theory to a 2D one using two hypersurface-orthogonal, commuting Killing vectors (e.g., stationary axisymmetric solutions), one gets an infinite-dimensional group (the Geroch group) and an infinite number of conserved currents; The theory is integrable.
* Geroch group: A Kac-Moody group (isomorphic to the SL(2, \(\mathbb R\)) affine Kac-Moody group), parametrized by two real functions (one curve in a 3D vector space), one from the 2 Killing vectors, the other from conformal invariance (this is where 2D comes into play); It is conjectured to be transitive on this space of solutions.
* Alternative frameworks: Use the Belinskii and Zakharov soliton and monodromy transformations.
* Stationary axisymmetric: In this case, one gets the Ernst equation; > s.a. axisymmetry.
@ Geroch group: Geroch JMP(72) [vacuum]; Kinnersley JMP(77) [electrovac], & Chitre JMP(77), JMP(78), JMP(78); Maison PRL(78), JMP(79); Hauser & Ernst JMP(81); Breitenlohner & Maison AIHP(87); Mizoguchi PRD(95)gq/94 [Ashtekar variables]; Manko & Ruiz PTP(11)-a1101 [simple representation for Kinnersley-Chitre metrics]; Chakrabarty & Virmani JHEP(14)-a1408 [and black-sole solutions].
@ Stationary axisymmetric: Persides & Xanthopoulos JMP(88).
@ And mixmaster-like behavior: Grubisic & Moncrief PRD(94)gq/93.
@ Extension to Einstein spaces: Leigh et al CQG(14)-a1403 [with one Killing vector]; Petkou et al a1512 [and holographic integrability].
@ Related topics: Korotkin & Samtleben CQG(97)gq/96 [quantization]; Bernard & Julia NPB(99)ht/97 [alternative group].
> Specific solutions: see gowdy spacetime; Levi-Civita Solution; Weyl Solutions [precursors].

Superpositions of Solutions > s.a. Kerr-Schild Solutions.
* Result: If gab is a solution of the Einstein equation and la lb (with la null) solves the linearized equation, then gab + la lb is also a solution.
@ References: Xanthopoulos JMP(78), in(85), CQG(86); Gergely CQG(02)gq [Kerr-Schild with matter].

Euclidean and Complex Solutions > s.a. complex structure; gravitational instanton; Taub-Bolt Solution.
* Methods: They can be found from analytic continuation of known Lorentzian solutions, imposing symmetries, finding simple Kähler potentials, or looking at topologically simple manifolds.
@ References: Gibbons & Pope CMP(79); Robinson GRG(87) [complex]; Brill gq/95-ln; Valent ht/02 [self-dual, multi-center].

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