Gravitational Energy-Momentum  

In General > s.a. energy-momentum tensors; expressions for gravitational energy-momentum [including other theories]; 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 = Σ T m0 dv = \(1\over8\pi G\)Σ Gm0 dv ,

but, unless ...
* The situation: 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 general, gavitational energy-momentum and related quantities cannot be given by integrals of a local density; Rather, they are quasi-local (associated with a closed 2-surface), have no unique formula, and have no reference-frame-independent description.
* In closed universes: There is no generally accepted definition, but this does not mean that there is no evolution; 2012, Proposal by L B Szabados.
> Related topics: see energy positivity; mass.

Specific Types of Spacetimes > s.a. asymptotic flatness; gravitational radiation.
@ Asymptotically-flat spacetimes / isolated objects: Chruściel 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]; Ge et al a1211; > s.a. minkowski space; solitons; teleparallel theories.
@ Spherical symmetry: Sosnovskiy gq/05 [and cylindrical]; Mirshekari & Abbassi MPLA(09)-a0808 [comparing energy-momentum prescriptions].
@ FLRW spacetimes: Faraoni & Cooperstock ApJ(03)ap/02 [open]; Garecki gq/06-MGXI, APPB(08)-a0708; Mitra GRG(10).
@ Other cosmological spacetimes: Gerhardt ATMP(06)m.DG/04 [asymptotically FLRW]; Sharif & Fatima IJMPA(05) [Einstein-Maxwell]; Nester et al PRD(08)-a0803 [homogeneous]; Davis SA(10)jul [energy conservation and expansion-related redshift]; Penrose GRG(11) [retarded mass/energy with a positive Λ]; Nourinezhad & Mehdipour IJP(12)-a1202 [Bianchi IX models]; Amsel & Gorbonos PRD(13)-a1209 [with a constant-curvature background, Wald-like formula]; Szabados & Tod IJMPD(18)-a1808 [with a positive cosmological constant]; > s.a. anti-de sitter spacetime [asymptotically AdS]; de sitter spacetime; quasilocal energy.
@ In closed universes: Szabados GRG(13)-a1212-proc [closed models, Bianchi I and IX]; Szabados CQG(13)-a1306 [with a positive cosmological constant].
@ With conical defects: Maluf & Kneip JMP(97)gq/95; Nucamendi & Sudarsky CQG(97)gq/96 [ADM]; > s.a. singularities.

References > s.a. charges; energy [self-energy].
@ General: Einstein PZ(14), PZ(18); Peters AJP(81)jun; in Wald 84; Ferraris & Francaviglia in(91); Nissani & Leibowitz IJTP(91) [covariant, localized]; Gibbs gq/97 [covariant]; Katz CQG(05)gq; Aldrovandi et al a0812 [gravitational and inertial, teleparallel gravity]; Jaramillo & Gourgoulhon ln(10)-a1001 [and angular momentum, rev]; Rodrigues RPMP(12)-a1109-conf; Chen et al IJMPD(15)-a1507-in [covariant Hamiltonian formalism, and Poincaré gauge theories]; Padmanabhan GRG(15)-a1506 [momentum density of spacetime]; Shimizu MPLA(16)-a1601 [proposal]; Bičák & Schmidt PRD(16)-a1602 [uniqueness]; Wang a1605-conf [and center of mass]; Acquaviva et al CQG(18)-a1802 [and gravitational thermodynamics]; Goswami & Ellis CQG(18)-a1805 [energy-momentum transfer by the gravitational field]; Chen et al IJMPD(18)-a1805-GRF [it is well defined]; Curiel SHPMP-a1808; Lopes de Lima et al a1811; Wu & Zhang a1811-in; Chen et al a1912, a1912 [translation of 1918 papers by Felix Klein].
@ Conservation laws: Palmer PRD(78); Nissani & Leibowitz PLA(88), IJTP(89); Bondi PRS(90); Wiesendanger a1102, a1103, a1103 [gravitational vs inertial energy-momentum]; Epp et al CQG(13) [local vs quasilocal]; Palese & Winterroth MG14(17)-a1601 [nature of the law of conservation of energy, a problem posed by Hilbert, and Noether's theorem]; Rowe a1912 [Emmy Noether's work]; > s.a. conservation laws; gravity [theories with non-conserved energy-momentum].
@ And Noether's theorem: Bamba & Shimizu IJGMP(16)-a1506; Deser a1902; De Haro a2103 [and energy-momentum pseudo-tensor].
@ 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].
@ Other dimensionalities: Haslhofer JGP(11) [mass-decreasing flow in dimension three]; Barzegar et al PRD(17)-a1708 [higher-dimensional].
@ With boundaries: Binz & Śniatycki 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]; Lam PhSc(11) [need for a background structure].
@ Related topics: Geroch & Perng JMP(94)gq [arbitrary data]; Bozhkov & Rodrigues GRG(95) [definition of inertial mass].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 31 mar 2021