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 Σ, 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 ρ and current density j,
assumed to satisfy the energy condition ρ2 ≥
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 Σ, the distribution and flow of mass-energy, ρ,
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
\(\dot q\)ab = 2 N Kab , \(\dot K\)ab = −Da Db N − 2 N Kam Kbm − N KKab − 3Rab N + 8πG N qam qbm(T mn − Tgmn) .
* Hyperbolic, curvature-based:
A formulation for arbitrary lapse and shift based on a wave equation for curvature.
Other approaches: see initial-value
formalism and approaches [existence and uniqueness, characteristic problem].
References > s.a. constraints for general relativity;
numerical relativity; Peeling Properties.
@ General: Stellmacher MA(38)-GRG(10); 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;
York proc(06)-gq/04-MGX [qab
+ Kab];
Chruściel & Friedrich ed-04;
Brown PRD(05)gq [conformal-traceless].
@ Intros, reviews: Gourgoulhon gq/07-ln;
Gourgoulhon 12;
Isenberg a1304-ch;
Ringström 13
[CQG+(15)];
Minucci a1902-MS.
@ 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 PhD(04)gq/05 [constraint-preserving boundary conditions];
Hilditch & Richter PRD(12) [with Hamiltonian structure];
Fatibene & Garruto CQG(15)-a1507 [algebraic characterization].
@ Hyperbolic, curvature-based: Abrahams et al PRL(95)gq,
CQG(97)gq/96;
Abrahams & York gq/96;
Anderson et al gq/97,
gq/99-proc;
Choquet-Bruhat et al gq/98-MG8;
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].
@ With matter fields: Husain PRD(99)gq/98;
Pugliese & Valiente Kroon GRG(13)-a1301 [Einstein-Maxwell-Klein-Gordon system].
@ Related topics: Durrer & Straumann HPA(88) [applications];
Frittelli & Reula JMP(99)gq [conformally decoupled];
Rácz CQG(01)gq [and symmetries];
Khokhlov & Novikov CQG(02)gq/01 [gauge stability];
Alcubierre et al PRD(09)-a0907 [with Maxwell fields, and multi-black-holes].
Types of Spacetimes > s.a. minkowski
spacetime [classical stability]; spherical symmetry;
types of spacetimes [including stationary].
@ Black holes: Cole & Valiente Kroon AHP(17)-a1609 [Kerr-Newman spacetime, geometric invariant];
> s.a. black holes.
@ Black-hole binaries: Pfeiffer et al PRD(02)gq [possible initial data];
Giulini in(03)gq [pedagogical];
Rácz a1605 [data];
> 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 spacetime];
Lopes Costa JPA(10)-a0912 [upper bound for angular-momentum and charge];
Bäckdahl & Valiente Kroon PRL(10)-a1001,
AHP(10)-a1005 [deviation from Kerr spacetime].
@ Conformally flat: Wagh & Saraykar PRD(89);
Karkowski & Malec gq/04.
@ Higher-dimensional: Anderson & Tavakol gq/03 [including branes].
@ Other spacetimes and related topics: Beig LNP(00)gq [Bowen-York initial data];
Andersson et al AsJM-a1508 [initial data sets with horizons];
Beig et al a1901 [compact initial data of constant mean curvature];
> s.a. causality violations; constraints;
gravitating bodies; numerical relativity.
For Other Theories > s.a. higher-order theories;
scalar-tensor theories; supergravity.
@ Strong coupling limit: Salopek CQG(98)gq,
CQG(99)gq/98 [Hamilton-Jacobi].
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