ADM Formulation of General Relativity| 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) (Kab − Kqab) , 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
qab − eab = O(r−1) ; ∂c qab & Kab = O(r−2) ; ∂c ∂d 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 3R − q−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 (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);
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 →
∞ \(\oint\)(Kab
− Kcc
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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