ADM Formulation of General Relativity

In General > s.a. 3D gravity; fluids; initial-value formulation; lattice gravity; metric decomposition; numerical relativity.
$ Variables: Phase space is the set of pairs (q_ab, p^ab), where q_ab is a positive-definite metric on and p^ab a tensor density of weight 1, related to the extrinsic curvature K_ab (in a solution) by

p^ab = (q^1/2/2) (K^ab – Kq^ab) ,   or   K^ab = 2 q^–1/2(p^ab – pq^ab) .

* Boundary conditions: The spacetime (M, g_ab) is asymptotically flat at spatial infinity if the data (q_ab, K_ab) induced on a spacelike hypersurface are such that

q_ab – e_ab = O(r^–1) ;   _c q_ab & K_ab = O(r^–2) ;   _cd q_ab & _cK_ab = O(r^–3);   etc ;

One can then find a chart such that q_ab = (1 + M/2r)⁴ e_ab + O(r^–2); In covariant form, g_ab = _ab + _ab, _ab = O(^–1) in the Cartesian chart for .
* Symplectic structure:

|_{(q, p)}[(h, ), (h', ')] = _Sigma (h_ab '^ab – h'_ab ^ab) dv .

* Constraints and Hamiltonian:

:= q^{1/2 3}R – q^–1/2 (p^abp_ab – p²) ,   ^b:= D_a p^ab ,   H = _Sigma d³x (N + N^a_a) .

* Constraint algebra: If the smeared scalar and vector constraints are, respectively [f] = _Sigma d³x f and [f ^a] = _Sigma d³x f ^a_a, then

{[f ^a], [g^a]} = [_f g^a] ;     {[f ^a], [g]} = –[_f g] ;     {[f], [g]} = –[q^ab (f_b g – g_b f] .

@ General articles: Dirac PRS(58), PR(59); Arnowitt et al JMP(60), PR(60), in(62) [+ GRG(08)]; Anderson RMP(64); Kuchar JMP(72) [bubble-time formalism]; Regge & Teitelboim AP(74) [boundary terms]; Isenberg & Nester in(80); Ashtekar PhyA(84) [good summary]; Beig & O'Murchadha AP(87) [boundary conditions]; Grishchuk & Petrov JETP(87); Vulcanov & Ciobanu AUVT(01)gq/00 [Maple routines]; Brewin PRD(09)-a0903 [equations from second variation of arclength].
@ Constraint algebra: Teitelboim AP(73); Kouletsis CQG(96)gq; Markopoulou CQG(96)gq; > s.a. constraints.
@ Embedding variables: Hojman et al AP(76); Kuchar JMP(76); Isham & Kuchar AP(85); Braham JMP(93); Ambrus & Hájícek PRD(01)gq/00 [relationship with ADM]; > s.a. models in canonical general relativity [shells].

Energy-Momentum > s.a. canonical general relativity; angular momentum.
$ Def: Given a spacelike surface in spacetime which is asymptotically flat at spatial infinity, with induced metric q_ab and a reference flat metric e_ab, the ADM four-momentum associated with it is defined by

E = (16G)^–1 lim_{r to infty} (_a q_bc – _b q_ac) e^ac dS ^b
p_m = (8G)^–1 lim_{r to infty} (K_ab– K^c_c q_ab) N^a dS ^b = (8G)^–1 lim_{r to infty} p_ab N^a dS ^b,

where r² = _i (xⁱ)², with x¹, x², x³ asymptotically Euclidean coordinates for e_ab, the integrals are taken over constant r spheres, and N is an asymptotic translation; The results are independent of the choice of e_ab, and the vector P_a = –En_a+p_a is independent of S (i.e., it is conserved), where n^a is the future-directed unit timelike
normal to at spacelike infinity.
@ Asymptotically flat: Arnowitt et al PR(59), PR(60), PR(61), in(62); Ashtekar & Horowitz PLA(82) [cannot be null]; Baskaran et al AP(03)gq [relationships]; Shi & Tam m.DG/04 [mass estimates]; Brewin GRG(07)gq/06 [simple mass expression].
@ More general spacetimes: Nucamendi & Sudarsky CQG(97)gq/96 [quasi-asymptotically flat].

Variations > s.a. 3D general relativity; non-standard approaches to canonical general relativity.
@ Various theories: York gq/98; Menotti & Seminara AP(00)ht/99, ht/99-in [2+1 with particles]; Barbashov et al IJMPA(08)ap/05-in; Lacquaniti & Montani IJMPD(06)gq, gq/06-in [5D Kaluza-Klein]; Chakrabarti et al a0908 [f(R) gravity].
@ Extended objects: Capovilla et al NPPS(00)ht; Steinhoff et al PRD(08), Steinhoff & Schäfer EPL(09)-a0907 [spinning objects].
@ Null surfaces: Goldberg in(86), pr(86); Torre CQG(86); Goldberg et al CQG(92).
@ Other theories and variations: Gunnarsen CQG(89) [weak field]; Brown & Marolf PRD(96)gq/95 [material reference systems]; Watson & Klauder CQG(02)gq/01 [metric on phase space]; Bonanno et al CQG(04)gq [variable G and ]; Wang PRD(05)gq [conformal]; Brown a0802 [strongly hyperbolic extension]; Ghalati a0901; > s.a. modified general relativity; numerical relativity [BSSN form].

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