Einstein's Field Equation for General Relativity |

**In General** \ s.a. canonical general relativity;
general relativity; energy-momentum tensor.

* __History__: Einstein
initially thought that it was not possible to find any exact non-trivial
solution, but then Schwarzschild came along.

$ __Def__: The equation

*R*_{ab} −
\(1\over2\)*R g*_{ab}
+ Λ *g*_{ab}
= (8π*G*/*c*^{4})
*T*_{ab} , or
*R*_{ab}
= (8π*G*/*c*^{4})
(*T*_{ab} −
\(1\over2\)*T g*_{ab}) .

* __Motivation__: The form
of the lhs in *G*_{ab}
= 8π*G* *T*_{ab}
is forced by the requirements that
(1) Flat spacetime have *G*_{ab}
= 0; (2) *G*_{ab} be constructed
from the metric and Riemann tensor only and is linear in the latter; (3)
*G*_{ab} be (symmetric and)
divergenceless.

* __Conditions__: In order for the equation
to be well-defined in a distributional sense, and guarantee existence and uniqueness
of geodesics, the metric must be at least of class C^{1,+},
although in general one takes it to be C^{2}.

* __Restrictions__: The stress-energy tensor
*T*_{ab} is usually thought to satisfy some energy
positivity condition, although we know all such conditions can be violated by quantum effects;
> see energy conditions; casimir
effect; quantum field theory effects.

@ __Derivations__: Chandrasekhar AJP(72)feb;
Barbour et al CQG(02)gq/00 [3D derivation];
Spaans a0705
[derivation without axiom of choice, and topological content];
Navarro & Sancho JGP(08)-a0709 [naturalness];
Li et al a1207 [from holographic thermodynamics and entropy];
Talshir a1412 [from symmetries and conservation laws];
Curiel a1601 [uniqueness, in all dimensions];
Oh et al PRD(18)-a1709 [first law of entanglement];
> s.a. curvature; Ether.

@ __Content, interpretation__: Sciama RMP(64);
Rosen PRD(87);
Mannheim FP(96)gq;
Anderson gq/99 [no need for metric];
Heighway gq/00 [??];
Frank gq/02 [pictorially];
Chryssomalakos & Sudarsky GRG(03)gq/02 [geometric nature];
Baez & Bunn AJP(05)jul-gq/01 [meaning];
Bouda & Belabbas IJTP(10)-a1012 [in Maxwell-equation form];
Mondino & Suhr a1810 [optimal transport formulation].

@ __Local freedom__: Maartens et al CQG(97)gq/96;
Pareja & MacCallum CQG(06)gq.

> __Solutions__: see solutions in general,
with matter [including effective equations] and generating methods.

**Alternative Formulations** > s.a. formulations
/ gravitational thermodynamics [as equation of state];
initial-value formulation [hyperbolic, existence and uniqueness].

* __Integral form__: The local
curved metric is obtained as an integral expression involving sources and
boundary conditions, which allows one to separate source-generated and
source-free contributions; As a consequence, an exact formulation of
Mach's Principle can be achieved.

* __"Relaxed Einstein equations"__:
Equations written in terms of the densitized inverse metric
[@ in Will LRR(06)].

@ __General references__: Schmidt G&C(96)gq;
Wild gq/98 [matrix];
Abbassi & Abbassi FPL(01)gq/00;
Brown PRD(11)-a1109 [generalized harmonic equations];
Fine et al NYJM-a1312 [gauge-theoretic approach];
Harte PRL(14)-a1409 [elimination of non-linearities beyond a particular order];
Rácz a1412
[using a two-parameter family of codimension-two surfaces];
Cimatti a1709 [boundary-value problems].

@ __Hyperbolic forms__: Fischer & Marsden CMP(72);
Frittelli & Reula PRL(96)gq [first-order];
Reula LRR(98);
Anderson & York PRL(99)gq [first-order];
Choquet-Bruhat gq/01 [non-strict],
& York LNP(02)gq;
Sarbach & Tiglio PRD(02)gq [gauge and constraints];
Alekseenko & Arnold PRD(03)gq/02;
Frittelli & Gómez CQG(03) [boundary conditions];
Alekseev gq/03-proc [characteristic initial-value problem];
Lindblom & Scheel PRD(03)gq [Einstein-gauge field].

@ __Second-order__: Nagy et al PRD(04)gq [strongly hyperbolic];
Frittelli PRD(04)gq [well-posedness];
Gundlach & Martín-García PRD(04) [symmetric hyperbolicity, boundary conditions];
Paschalidis PRD(08)-a0704 [mixed hyperbolic-second-order parabolic].

@ __Integral form__: Dantas GRG(05)gq/04 [in braneworld].

@ __Conformal form__: Friedrich JGP(95),
CQG(96);
> s.a. numerical general relativity.

> __ Related topics__: see propagation of
gravitational radiation [DIRE]; solution-generating methods [gravity/fluid correspondence].

**Approximation Schemes**
> s.a. black-hole solutions [skeleton]; general relativity;
gravitomagnetism; solutions.

* __No-radiation approximations__:
The motivation is to be able to analyze the evolution of a general multibody
system without the added difficulties associated with gravitational waves,
even numerically; This gives a very accurate approximation in many cases.

@ __General references__: Deruelle & Langlois PRD(95) [long-wavelength iteration];
Anderson gq/99 [Einstein-Infeld-Hoffmann].

@ __Perturbative__: Lessner IJTP(77);
Ichinose & Ikeda IJMPA(99)ht/98;
Detweiler & Brown PRD(97)gq/96 [in powers of *G*];
Comer CQG(97)ht/05 [long-wavelength iteration];
Bratchikov a0911 [diagrammatic].

@ __No-radiation approximations__:
Schäfer & Gopakumar PRD(04)gq/03 [analogous to magneto-hydrodynamics];
Isenberg IJMPD(08)gq/07.

@ __Other__: Maniopoulou & Andersson MNRAS(04)ap/03 [traditional approximation];
Kerner & Vitale a0801 [via deformation of embeddings];
Brizuela & Schaefer PRD(10)-a1002 [4PN-exact];
> s.a. perturbations.

@ __Modifications__: Brown & Lowe PRD(06)gq [modification off the constraint hypersurface].

**With Symmetries** > s.a. symmetric solutions of general relativity.

@ __Two Killing vector fields__: Whelan & Romano PRD(99)gq/98.

@ __Symmetries__: Ernst & Hauser JMP(93);
Gürses PRL(93);
Torre & Anderson PRL(93)gq;
Capovilla PRD(94)gq/93;
Torre gq/95-proc;
Anderson & Torre CMP(96)gq/94;
Zafiris AP(98) [decomposition].

@ __Classification of models obtained__:
Uggla et al CQG(95);
Zafiris JMP(97)gq [with spacetime symmetries].

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send feedback and suggestions to bombelli at olemiss.edu – modified 3 nov 2019