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 *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].

**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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send feedback and suggestions to bombelli at olemiss.edu – modified 18
sep 2016