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 the spatial manifold Σ and \(p^{ab}\) a tensor density of weight 1, related to the extrinsic curvature \(K_{ab}^~\) (in a solution) by

pab = (q1/2/2κ) (KabKqab) ,    or   Kab = 2κ q−1/2(pab − \(1\over2\)pqab) .

* Boundary conditions: The spacetime (M, gab) is asymptotically flat at spatial infinity if the data (qab, Kab) induced on a spacelike hypersurface Σ are such that

qabeab = O(r−1) ;   ∂c qab & Kab = O(r−2) ;   ∂cd qab & ∂cKab = O(r−3);   etc ;

One can then find a chart such that qab = (1 + M/2r)4 eab + O(r−2); In covariant form, gab = ηab + γab, γab = \(O(\rho^{-1}\)) in the Cartesian chart for κ.
* Symplectic structure:

Ω|(q, p)[(h, π), (h', π')] = Σ (hab π'ab − \(h'_{ab}\, \pi^{ab}\)) dv .

* Constraints and Hamiltonian: For a given choice of the lapse function N and shift vector Na,

\(\cal H\):= q1/2 3Rq−1/2 (pab pab − \(1\over2\)p2) ,   \(\cal H\)b:= Da pab ,   H = Σ d3x (N\(\cal H\) + Na\(\cal H\)a) .

* Constraint algebra: If the smeared scalar and vector constraints are, respectively \(\cal C\)[f] = Σ d3x f\(\cal H\) and \(\cal C\)[f a] = Σ d3x f a \(\cal H\)a, then

{\(\cal C\)[f a], \(\cal C\)[ga]} = \(\cal C\)[\(\cal L\)f ga] ;     {\(\cal C\)[f a], \(\cal C\)[g]} = −\(\cal C\)[\(\cal L\)f g] ;     {\(\cal C\)[f], \(\cal C\)[g]} = −\(\cal C\)[qab (fb ggb f)] .

@ General articles: Dirac PRS(58), PR(59); Arnowitt et al JMP(60), PR(60), in(62) [+ GRG(08)]; Anderson RMP(64); Kuchař 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]; Kiriushcheva et al a1108 [change of field variables and covariance]; Perlov PLB(15)-a1412 [scalar massless field as time]; > s.a. history of gravity.
@ Constraint algebra: Teitelboim AP(73); Kouletsis CQG(96)gq; Markopoulou CQG(96)gq; Kiriushcheva et al IJTP(12)-a1107 [group properties of the Lagrangian symmetries of the action]; > s.a. constraints.
@ Embedding variables: Hojman et al AP(76); Kuchař JMP(76); Isham & Kuchař AP(85); Braham JMP(93); Ambrus & Hájíček PRD(01)gq/00 [relationship with ADM]; > s.a. models in canonical general relativity [shells].

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

E = (16πG)−1 limr → ∞ \(\oint\)(∂a qbc − ∂b qac) eac dS b
pm = (8πG)−1 limr → ∞ \(\oint\)(KabKcc qab) Na dS b = (8πG)−1 limr → ∞ \(\oint\) pab Na dS b,

where r2 = ∑i (xi)2, with x1, x2, x3 asymptotically Euclidean coordinates for eab, the integrals are taken over constant r spheres, and N is an asymptotic translation; The results are independent of the choice of eab, and the vector Pa = −Ena+pa is independent of S (i.e., it is conserved), where na is the future-directed unit timelike normal to Σ at spacelike infinity.
@ Asymptotically flat: Arnowitt et al PR(59), PR(60), PR(61), in(62); Ashtekar & Horowitz PLA(82) [cannot be null]; Chruściel in(86)-a1312 [as geometric invariants]; Baskaran et al AP(03)gq [relationships]; Shi & Tam m.DG/04 [mass estimates]; Brewin GRG(07)gq/06 [simple mass expression]; Michel JMP(11)-a1012 [mass, invariance]; Lopes de Lima & Girão TAMS-a1108 [manifolds with warped product structure]; Cheng & Zhu a1109 [behavior under the Yamabe flow].
@ 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; quasilocal general relativity [2+2].
@ Various theories: York gq/98; Menotti & Seminara AP(00)ht/99, NPPS(00)ht/99 [2+1 with particles]; Barbashov et al IJMPA(08)ap/05-in; Lacquaniti & Montani IJMPD(06)gq, gq/06-MGXI [5D Kaluza-Klein]; Chakrabarti et al GRG(11)-a0908 [f(R) gravity]; Kiriushcheva et al a1111 [comment on Chaichian et al's "covariant renormalizable gravity"]; Kastikainen a1908 [Lovelock gravity]; > s.a. modified general relativity; teleparallel gravity.
@ Extended objects: Capovilla et al NPPS(00)ht; Steinhoff et al PRD(08), a1002-MGXII, Steinhoff & Schäfer EPL(09)-a0907 [spinning objects]; > s.a. black-hole entropy.
@ 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 Λ]; Wang PRD(05)gq [conformal]; Brown in(09)-a0802 [strongly hyperbolic extension]; Ghalati a0901; Gao PLB(10)-a0905 [f(R) and K-essence gravity]; Dengiz MS-a1103 [and conformal decomposition]; Golovnev et al JCAP(15)-a1412 [bimetric theory]; > s.a. massive gravity; numerical relativity [BSSN form]; Weyl Geometry.

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