Constraint Equations in General Relativity

In General > s.a. canonical [constraint algebra]; initial value formulation; ADM and connection formulation.
* Idea: They are the Einstein equation with one or more indices projected arthogonally to a spacelike hypersurface or, in differential geometry terms, just the Gauss-Codazzi equations with the Ricci tensor substituted for in terms of the matter stress-energy.
* In terms of first and second fundamental form: For real, Lorentzian gravity

G_ab q_m^a n^b = D^a K_am – D_m K = 8G T_ab q_m^a n^b = 8G j_m

2 G_ab n^a n^b = ³R – K^ab K_ab + K² = 16G T_ab n^a n^b = 16G ,

where N:= [–(^a t) (_a t)]^–1/2 is the lapse, n_a:= –N_a t is the unit normal to , and K:= K^a_a = q^ab K_ab.
* For complex/euclidean general relativity: Flip the signs of the two KK terms in the second constraint equation.
@ References: Moncrief PRD(72) [redundancy], & Teitelboim PRD(72) [Hamiltonian and diffeomorphism]; Dittrich CQG(06)gq/05 [diff-invariant Hamiltonian constraints].

Variables and Solution Methods
* Lichnerowicz-York solution method: A conformal technique initiated by Lichnerowicz and perfected by York, the only efficient and robust method of generating consistent initial data; In the spatially compact case, the complete scheme consists of the ADM Hamiltonian and momentum constraints, the ADM Euler-Lagrange equations, York's constant-mean-curvature (CMC) condition, and a lapse-fixing equation (LFE) that ensures propagation of the CMC condition by the Euler-Lagrange equations; The variables are a conformal factor , a spatial metric g_ij, a symmetric tensor A_ij and a scalar , in terms of which the physical metric and extrinsic curvature are

h_ij = ⁴ g_ij,     K_ij = ^–2 A_ij +  ⁴ g_ij ;

The Hamiltonian constraint is rewritten as the Lichnerowicz-York equation for the conformal factor of the physical metric ⁴ g_ij, given an initial unphysical 3-metric g_ij; The CMC condition and LFE introduce a distinguished foliation (definition of simultaneity) on spacetime, and separate scaling laws for the canonical momenta and their trace are used.
@ General references: Isenberg & Marsden JGP(84) [York map]; Maxwell CMP(05)gq/03 [constant mean curvature conformal method]; Anderson et al CQG(05)gq/04 [physical degrees of freedom]; Pfeiffer et al PRD(05) [stationary + gravitational wave]; O'Murchadha APPB(05)gq [uses of Lichnerowicz-York equation]; Tiemblo & Tresguerres GRG(06)gq/05 [and Poincaré gauge theory, single condition]; Martin et al a0709-GRG [framework for solutions]; Maxwell a0804 [freely specified mean curvature].
@ And conformal structure: Beig & Ó Murchadha CMP(96)gq/94 [and spatial conformal Killing vectors]; Szabados CQG(02)gq/01 [Hamiltonian and Chern-Simons functional]; Butscher CMP(07)gq/02 [conformal constraint equations]; Pfeiffer gq/04-in [conformal method]; Pfeiffer & York PRL(05), Walsh CQG(07) [non-uniqueness of solutions]; Gourgoulhon a0704-in [and 3+1 numerical initial data]; Holst et al a0708.
@ Black holes: Brandt & Brügmann PRL(97) [multi-black-hole]; Loustó & Price PRD(97), PRD(98) [data for binary collisions]; Baker & Puzio PRD(99)gq/98 [axisymmetric]; Dain et al PRD(05)gq/04 [multi-black-hole].
@ Binary black holes: Baumgarte PRD(00)gq; Marronetti et al PRD(00)gq, & Matzner PRL(00)gq [arbitrary P, L]; Dain PRD(01)gq/00 [2 Kerr, head-on].
@ Closed 3-manifolds: Isenberg CQG(95) [constant mean curvature], & Moncrief CQG(96) [non-constant mean curvature]; Maxwell gq/05-in [low-regularity]; Holst et al PRL(08) [far-from-constant mean curvature].
@ Other solutions: Isenberg & Park CQG(97)gq/96 [asymptotically hyperbolic]; Maxwell CMP(05)gq/03 [with apparent horizon boundaries]; Choquet-Bruhat et al gq/05, CQG(07)gq/06, gq/06 [Einstein-scalar]; Korzynski PRD(06)gq [on dynamical horizons].

References > s.a. formulations of general relativity; linearized general relativity; numerical relativity; canonical quantum gravity.
@ Reviews: Bartnik & Isenberg gq/04-in.
@ Gluing solutions: Isenberg et al CMP(02)gq/01 [and wormholes], AHP(03)gq/02; Isenberg gq/02-in; Chrusciel et al CMP(05)gq/04, PRL(04)gq [more general]; Isenberg et al ATMP(05)gq [with matter]; > s.a. solution methods.
@ Space of solutions: Ó Murchadha CQG(87) [ADM energy as Morse function]; Chrusciel & Delay JGP(04)gq/03 [diff structure].
@ Related topics: Kuchar & Romano PRD(95)gq [sets that generate true Lie algebras]; Frittelli PRD(97) [propagation]; York gq/98 [propagation, and canonical formalism]; Choquet-Bruhat CQG(04)gq/03-in [compact ⁿ]; Dain gq/04-in [black holes as boundaries]; Frauendiener & Vogel CQG(05)gq/04 [instability of constraint surface]; Gambini & Pullin GRG(05)gq-GRF [getting rid of constraints in discretization]; Corvino & Schoen JDG(06) [vacuum, asymptotics]; Bojowald et al PRD(06)gq [effective constraints from lqg]; Szabados CQG(08)-a0711 [Hamiltonian constraint for Einstein-Yang-Mills as Poisson bracket].

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