Einstein's
Field Equation for General Relativity| 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
Rab – 
R gab + 
gab =
(8
G/c4)
Tab , or Rab =
(8G/c4)
(Tab – T gab)
.
* Motivation: The form
of the lhs in Gab =
8G 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 and
quantum field theory effects).
@ Derivations, content: Sciama RMP(64);
Chandrasekhar AJP(72)feb;
Rosen
PRD(87);
Mannheim FP(96)gq;
Anderson gq/99 [no
need for metric];
Heighway
gq/00 [??];
Frank gq/02 [pictorially];
Baez & Bunn AJP(05)jul-gq/01 [meaning];
Barbour et al CQG(02)gq/00 [3D
derivation]; Chryssomalakos & Sudarsky
GRG(03)gq/02 [geometric
nature]; Spaans a0705 [derivation
without axiom of choice, and topological content]; Navarro & Sancho JGP(08)-a0709 [naturalness];
Dadhich
a0802 [from Bianchi-like
identities]; > s.a. curvature.
@ Local freedom: Maartens et al CQG(97)gq/96;
Pareja
& MacCallum CQG(06)gq.
> Solutions: see initial-value
formulation [hyperbolic,
existence and uniqueness]; solutions
of
general relativity.
Alternative Formulations > s.a. gravitational
radiation [DIRE]; 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.
@ Alternative forms: Schmidt G&C(96)gq;
Wild gq/98 [matrix];
Abbassi & Abbassi
FPL(01)gq/00.
@ 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
gq/02-in; Sarbach & Tiglio
PRD(02)gq [gauge
and constraints]; Alekseenko & Arnold
PRD(03)gq/02;
Frittelli & Gómez CQG(03)
[boundary conditions]; Alekseev gq/03-in
[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.
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: 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]; > 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-in;
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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nov 2009