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__: An infinite-dimensional
transitive group of symmetries of cylindrically symmetric gravitational waves, which
acts by non-canonical transformations on the phase space of these waves; It is 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).

* __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];
Peraza et al a1906 [quantization].

@ __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 / Interpolation between Solutions**
> s.a. Kerr-Schild Solutions.

* __Result__: If
*g*_{ab} is a solution
of the Einstein equation and *l*_{a}
*l*_{b} (with
*l*_{a} null) solves the linearized
equation, then *g*_{ab}
+ *l*_{a}
*l*_{b} is also a solution.

@ __References__: Xanthopoulos JMP(78),
in(85),
CQG(86);
Gergely CQG(02)gq [Kerr-Schild with matter];
Halilsoy & Mazharimousavi IJMPD(19)-a1901 [interpolation].

**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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