Initial-Value
Formulation of General Relativity| Initial-Value
Formulation of General Relativity |
Based on a Spacelike Hypersurface > s.a. canonical
formulation and connection variables; constraints; Sandwich
Conjecture.
* Idea: We assign the
value of the 3-metric qab and
the extrinsic curvature Kab on
an initial 3D spacelike hypersurface 
,
subject to some constraints, and then evolve them and find the full 4-metric
in spacetime;
If matter is present, we assign also the local mass density 
and
current density j, assumed to satisfy the energy condition 2 
j aja.
* Variables: The usual
dynamical variables are qab and Kab,
and the choice of gauge is represented by the lapse N and shift Na
(and an
initial choice of gauge/coordinates at t = 0); > s.a. gauge
transformations.
* Alternatively: One
can give, on ,
the distribution and flow of mass-energy, , j a and Sab,
up to a conformal factor, and the
conformal 3-geometry and its rate of change.
* Evolution equations:
For vanishing shift Na =
0, the metric
and extrinsic curvature (satisfying the constraints) evolve according to
qab· =
2 N Kab , Kab· =
–DaDb N – 2 N Kam Kbm – N KKab – 3Rab N +
8
G N qam qbm(T mn – Tgmn)
.
* Hyperbolic, curvature-based: A formulation for arbitrary lapse and shift based on a wave equation for curvature.
References > s.a. constraints
for general relativity; numerical relativity; Peeling
Properties.
@ General: Bonnor JMM(60) [re uniqueness];
Choquet-Bruhat & Geroch CMP(69);
York PRL(71);
Fischer & Marsden
JMP(72);
York JMP(72), PRL(72), JMP(73),
in(79); O'Murchadha & York JMP(73),
PRD(74), PRD(74);
Smarr & York PRD(78);
Fischer & Marsden in(79); Choquet-Bruhat & York
in(80);
York
in(82);
Isenberg
FP(86); Choquet-Bruhat
& York gq/95,
gq/96;
York gq/98;
Friedrich & Rendall LNP(00)gq-in;
York gq/04-in
[qab
+ Kab]; Chrusciel & Friedrich
ed-04; Brown PRD(05)
[conformal-traceless].
@ Intros, reviews: Gourgoulhon gq/07-ln.
@ Hyperbolic: Estabrook et al CQG(97)gq [first-order];
Yoneda & Shinkai
PRL(99)gq/98,
IJMPD(00)gq/99 [Ashtekar
variables]; Alvi CQG(02)
[dynamical N and Na];
Tarfulea gq/05-PhD
[constraint-preserving boundary conditions].
@ Hyperbolic, curvature-based: Abrahams et al PRL(95)gq,
CQG(97)gq/96;
Abrahams & York
gq/96; Anderson
et al gq/97, gq/99-in;
Choquet-Bruhat et al gq/98-in;
Lau IJMPD(98)gq/96 [based on forms]; Anderson & York
PRL(99)gq.
@ Other gauges: Andersson & Moncrief AHP(03)gq/01 [elliptic-hyperbolic];
Paschalidis et al PRD(07)gq/05
[well-posedness].
@ Related topics: Durrer & Straumann HPA(88)
[applications]; Husain PRD(99)gq/98 [matter
fields]; Frittelli & Reula JMP(99)gq [conformally
decoupled]; Rácz CQG(01)gq [and
symmetries];
Khokhlov & Novikov CQG(02)gq/01 [gauge
stability]; Alcubierre et al a0907 [with Maxwell fields, and multi-black-holes].
Existence and Uniqueness of Solutions
* Situation: The initial-value
problem is known to be well-defined for globally hyperbolic spacetimes, time-non-orientable
spacetimes whose orientable
double
cover is globally hyperbolic, and some cases of spacetimes with closed timelike
curves.
* Different 3-topologies:
They
all
admit good (vacuum) data, but generically
evolve
into spacetimes that are locally de Sitter and develop singularities.
@ Reviews: Rendall LRR(98), LRR(00)gq,
LRR(02)gq +
LRR(05)gq.
@ Long-time evolution: Friedrich CMP(86); Christodoulou CQG(99)A
[and singularities]; Anderson CMP(01)
[and 3-geometry]; Klainerman & Nicolò 02; Lindblad & Rodnianski
CMP(05)m.AP/03 [global
existence], m.AP/04 [global
stability of Minkowski]; Choquet-Bruhat & Friedrich CQG(06)gq [Einstein-dust
and
Einstein-Maxwell-dust,
compact support].
@ Closed 3-manifolds: Andersson gq/99-in;
Scannell m.DG/00 [flat
spacetime].
@ Different 3-topologies: Witt PRL(86);
Morrow-Jones; Bengtsson & Holst
CQG(99)gq [locally
de Sitter]; Isenberg et al AHP(03)gq/02 [all D's];
Choquet-Bruhat et al CQG(06)gq [Einstein-Maxwell,
higher dimensions].
@ Various types of matter: Choquet-Bruhat et al gq/06 [Einstein-scalar].
Types of Spacetimes > s.a. black
holes; minkowski [classical
stability];
spherical; types
of spacetimes [including
stationary].
@ Black hole binaries: Pfeiffer et al PRD(02)gq [possible
initial data]; Giulini in(03)gq
[pedagogical]; > s.a. embedding [diagrams].
@ Asymptotically flat: Penrose PRS(65)
[at scri]; Kánnár
CQG(00)gq [with
Killing vectors]; Valiente Kroon CQG(05)gq/04 [near
spi and scri], PRD(05)gq [Schwarzschildean];
García-Parrado & Valiente Kroon PRD(07)gq/06 [Schwarzschild].
@ Conformally flat: Wagh & Saraykar PRD(89); Karkowski & Malec
gq/04.
@ Higher-dimensional: Anderson & Tavakol gq/03 [including
branes].
@ Other spacetimes and related topics: Beig gq/00-in
[Bowen-York initial data]; > s.a. causality
violations; constraints; numerical
relativity.
Different Approaches and Issues > s.a. [general
relativity]; asymptotic
flatness; einstein's equation; holography; numerical
relativity.
* Possibile variables: ADM, Conformal ADM, Einstein-Bianchi, connection
(Ashtekar) variables, ...
* Characteristic problem: A null surface initial-value formulation,
with
initial data assigned on a null surface.
@ Characteristic problem: Bondi et al PRS(62);
Sachs JMP(62), PR(62);
Penrose
in(64); Müller zum Hagen & Seifert in(79);
Penrose GRG(80);
Bartnik CQG(97)gq/96 [null
quasi-spherical gauge]; Klainerman & Nicolò CQG(99)
[double null, vacuum asymptotically flat]; Winicour LRR(98)–LRR(01)–LRR(05);
Gómez & Frittelli PRD(03)gq [first-order
quasilinear]; Nicolò NCB(04);
Frittelli
PRD(06)
[ADM version of Bondi Sachs]; Caciotta & Nicolò gq/06 [vacuum,
small data]; Reisenberger PRL(08)-a0712; > s.a. linearized.
@ Cauchy-characteristic: Gómez et al PRD(96)
[for Einstein-Klein-Gordon]; Kánnár PRS(96)
[asymptotically characteristic].
@ Initial-boundary: Friedrich & Nagy CMP(99);
Szilagyi & Winicour PRD(03)gq/02;
Frittelli & Gómez
CQG(03),
PRD(03)gq,
PRD(04)gq/03,
PRD(04)gq [boundary
conditions]; Kreiss et al CQG-a0707;
Friedrich GRG(09)-a0903 [geometric
uniqueness]; Winicour GRG(09)
[geometric formulation].
@ Mathematical: Beig & Szabados CQG(97)gq [global
conformal invariant
Y]; Esposito & Stornaiolo gq/00 [and
elliptic operators]; Etesi JMP(02)gq/01 [rigidity
theorems]; Rendall gq/01-in;
Lindblom & Scheel
PRD(02)
[energy norms and stability]; Dafermos gq/02-in
[uniqueness and RN
stability]; Karp a0906 [harmonic gauge, well-posedness of problem].
@ Modified ADM formulation: York PRL(99)gq/98,
Esposito & Stornaiolo FPL(00)gq, gq/00;
Pfeiffer & York PRD(03);
Jantzen NCB(04)gq/05 [Taub
function = densitized lapse]; > s.a. canonical
general relativity.
@ Other formulations: Seriu
PRD(00)gq [Laplace
eigenvalues]; Bona et al PRD(03)
[generally covariant, with vector field Z]; Garfinkle & Gundlach CQG(05)gq [tetrad];
Anderson et al CQG(05)
[evolving conformal geometry]; > s.a. Threading.
For Other Theories > s.a. higher-order
theories;
scalar-tensor theories.
@ Strong coupling limit: Salopek CQG(98)gq, CQG(99)gq/98 [Hamilton-Jacobi].
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aug 2009