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 ρ2j 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 KbmN KKab3Rab N + 8πG N qam qbm(T mnTgmn) .

* Hyperbolic, curvature-based: A formulation for arbitrary lapse and shift based on a wave equation for curvature.
blue bullet 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].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 13 may 2019