Initial-Value Formulation of GR – Formalism and Approaches| Initial-Value Formulation of GR – Formalism and Approaches |
Existence and Uniqueness of Solutions
> s.a. history of general relativity.
* 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 spacetime];
Choquet-Bruhat & Friedrich CQG(06)gq [Einstein-Maxwell-dust, compact support];
Parlongue a1004 [breakdown criterion];
Chruściel a1112 [vacuum, existence and uniqueness];
Czimek a1609
[compact support, asymptotically flat solutions].
@ Closed 3-manifolds:
Andersson in(04)gq/99;
Scannell CQG(01)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 dimensions];
Choquet-Bruhat et al CQG(06)gq [Einstein-Maxwell, higher dimensions].
@ Various types of matter:
Choquet-Bruhat et al gq/06 [Einstein-scalar];
Jia & Guo a1909
[Einstein-Yang-Mills-Higgs system, global existence result].
Characteristic and Related Problems
* Characteristic problem: A null
surface initial-value formulation, with initial data assigned on a null surface;
> s.a. null infinity.
@ 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;
Luk a1107 [local existence];
Chruściel & Jezierski JGP(12);
Tadmon a1203 [for the Einstein-Yang-Mills-Higgs system];
Chruściel & Paetz CQG(12)-a1203 [review];
Winicour PRD(13)-a1303 [new evolution algorithm, affine-null metric formulation];
Chruściel & Paetz AHP(15)-a1403,
Paetz JMP(14)-a1403 [and smoothness of scri];
Reisenberger CQG(18)-a1804 [free null data, Poisson brackets];
Hilditch et al a1911 [Newman-Penrose formalism];
> s.a. linearized gravity.
@ For other theories: Mongwane PRD(17)-a1707 [for f(R) gravity].
@ Cauchy-characteristic problem: Gómez et al PRD(96) [for Einstein-Klein-Gordon theory];
Kánnár PRS(96) [asymptotically characteristic].
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, ...
@ Initial-boundary value problem:
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(07)-a0707;
Friedrich GRG(09)-a0903 [geometric uniqueness];
Winicour GRG(09),
PRD(09)-a0909 [geometric formulation];
Reula & Sarbach IJMPD(11)-a1009-fs [rev];
Sarbach & Tiglio LRR(12)-a1203 [rev, continuum and discrete];
Kreiss & Winicour CQG(14)-a1302 [geometric boundary data];
Hilditch & Ruiz CQG(18)-a1609 [for free-evolution formulations];
An & Anderson a2103 [and quasi-local Hamiltonians].
@ 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-GR16;
Lindblom & Scheel PRD(02) [energy norms and stability];
Dafermos gq/02-CM [uniqueness and Reissner-Nordström stability];
Karp a0906 [harmonic gauge, well-posedness of problem];
Klainerman IJMPD(13)-MG13 [current state].
@ 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];
Alcubierre & Mendez GRG(11)-a1010 [in curvilinear coordinates];
Maxwell a1407 [expansion, conformal deformation and drift];
> s.a. Threading.
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