 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

Rab − $$1\over2$$R gab + Λ gab = (8πG/c4) Tab ,   or    Rab = (8πG/c4) (Tab − $$1\over2$$T gab) .

* Motivation: The form of the lhs in Gab = 8πG Tab is forced by the requirements that (1) Flat spacetime have Gab = 0; (2) Gab be constructed from the metric and Riemann tensor only and is linear in the latter; (3) Gab 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 C1,+, although in general one takes it to be C2.
* Restrictions: The stress-energy tensor Tab 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].