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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