Gravitational Energy-Momentum  

In General > s.a. energy positivity; mass; stress-energy pseudotensors.
* Motivation: Just as in Newtonian theory one gets mass from the gravitational field, in general relativity it is reasonable to get also momentum and angular momentum; If there is matter, the expressions we will write down give the total (i.e., including matter) energy-momentum.
* Open universes: One could try, bearing in mind the Einstein equation,

Pm = Sigma T m0 dv = (8G)–1 Sigma Gm0 dv ,

but, unless ...
* Problems with local definition: If spacetime is stationary, one can get an energy-momentum from the stress-energy tensor Tab, but otherwise one really faces the gravitational field energy problem and finds at most pseudotensors.
* In closed universes: There is no (non-trivial) definition, no platform; This does not mean there is no evolution.

References > s.a. anti-de sitter; charges; conservation; energy [self-energy]; teleparallel [including energy-momentum density].
@ General: Palmer PRD(78) [conservation laws]; Peters AJP(81)jun; in Wald 84; Nissani & Leibowitz PLA(88), IJTP(89); Bondi PRS(90); Ferraris & Francaviglia in(91); Nissani & Leibowitz IJTP(91) [covariant, localized]; Gibbs gq/97 [covariant]; Katz CQG(05)gq; Aldrovandi a0812.
@ Doubting the reality: Infeld AP(59); Zel'dovich & Grishchuk SPU(88) + refs [debate]; Cooperstock FP(92), MPLA(99)gq, AP(00)gq/99, FP(01); Hoefer SHPMP(00) [conceptual].
@ With boundaries: Binz & Sniatycki CQG(86); Francaviglia & Raiteri CQG(02)gq/01.
@ Negative energy density? Bonnor & Cooperstock PLA(89).
@ The background question: Bombelli et al NPB(87); Hawking et al PRD(95) [Melvin]; Hawking & Hunter CQG(96)gq; Katz & Lerer CQG(97)gq/96 [null infinity].
@ Related topics: Geroch & Perng JMP(94)gq [arbitrary data]; Bozhkov & Rodrigues GRG(95) [definition of inertial mass].

Komar Integral
$ Def: The conserved quantity associated with a Killing vector field a is

H = – (8G)–1 S abcd c d = (4G)–1 S D[a b] dvab = (4G)–1 S Db D[b a] dSa .

* Limitations: One sometimes gets a factor 2 wrong.
@ References: Komar PR(59); Katz CQG(85); Chrusciel AIHP(85); Bazanski & Zyla GRG(90) [with a cosmological constant]; Glass & Naber JMP(94) [comparison]; Ansorg & Petroff CQG(06)gq, a0708-in [negative]; Kastor CQG(08) [in higher-order gravity].

Nester's and Other Spinor Expressions
$ Def: The energy-momentum component associated with a constant spinor A is

Ka Pa = (8G)–1  ab dSab ,

where Ka = A *A' is a null vector determined by A, and ab = i abcd C d C', with A = A + O(r–1), an asymptotically constant spinor; Different components of P are obtained using different K's.
@ References: Chen et al gq/02-in.

Other Expressions and Theories > s.a. ADM formalism; quasilocal energy; [angular momentum].
* Results: If a:= eab b r, then [@ Ashtekar in(84)]:

E = –(8G)–1 S_infty r 3Rab a b d2v = (8G)–1 S_infty abc a b c d2v
= (16G)–1 S_infty r [3R – (2)2 + (2ab)(2ab)] d2v .

@ General references: Sorkin in(88) [from first-order action]; Chrusciel et al PRL(90)gq [uniqueness]; Lau PRD(00)gq [lightcone reference]; Rizzi gq/02 [and angular momentum]; Chrusciel et al ATMP(04)gq/03 [Trautman-Bondi, initial data sets]; Nester CQG(04) [rev].
@ Bondi momentum: Ashtekar & Magnon-Ashtekar PRL(79) [and ADM]; Huang & Zhang gq/06 [relationship with ADM]; Gallo et al PRD(08) [estimating from finite distance]; > s.a. asymptotic flatness at null infinity.
@ Møller: Xulu MPLA(00) [Kerr-Newman spacetime]; Radinschi gq/02, Gad ASS(06)gq/04, MPLA(04)gq [axisymmetric]; Nashed NCB(04)gq/05 [Kerr]; Yang & Tsai gq/05 [evaluation]; Salti MPLA(05)gq [Kasner-type spacetime].
@ Other theories: Eling PRD(06) [Einstein-aether]; Baryshev a0809-in [Minkowski field theory]; Maluf & Ulhoa GRG-a0810 [teleparallel equivalent]; > s.a. higher-order theories, kaluza-klein phenomenology.
> Related topics: see asymptotic flatness at null infinity and spatial infinity; kerr [Smarr].

Specific Types of Spacetimes > s.a. asymptotic flatness; gravitational radiation.
@ Isolated: Chrusciel CMP(88); Dadhich & Narayan GRG(98)gq/97 [vanishing gravitational mass]; Sharif NCB(04)ht/03 [black hole]; Patashnick IJMPD(05)gq/04; Sharif & Fatima NCB(05)gq [Weyl metrics]; > s.a. minkowski, solitons, teleparallel.
@ Spherical symmetry: Sosnovskiy gq/05 [and cylindrical].
@ Cosmological: Faraoni & Cooperstock ApJ(03)ap/02 [open FRW models]; Gerhardt ATMP(06)m.DG/04 [asymptotically FRW spacetime]; Sharif & Fatima IJMPA(05) [Einstein-Maxwell]; Garecki gq/06-in, a0708 [FRW models]; Nester et al PRD(08)-a0803 [homogeneous]; > s.a. AdS, de sitter, quasilocal energy.
@ With conical defects: Maluf & Kneip JMP(97)gq/95; Nucamendi & Sudarsky CQG(97)gq/96 [ADM]; > s.a. singularities.

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