Einstein's
Field Equation for General Relativity |

**In General** [> s.a. canonical general
relativity; general
relativity;
energy-momentum.]

* __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]; > 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].

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

> __Solutions__: see initial-value
formulation [hyperbolic,
existence and uniqueness]; solutions
in
general relativity and solution-generating methods.

**Alternative Formulations** > s.a. formulations of general relativity; gravitational thermodynamics [as equation of state].

* __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].

@ __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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jan 2016