Gravitational Energy-Momentum Expressions  

Komar Integral > s.a. non-commutative gravity.
$ Def: The conserved quantity associated with a Killing vector field ξa is

H = − (8πG)−1 \(\oint_S\)εabcdc ξd = (4πG)−1 \(\oint_S\)D[a ξb] dvab = (4πG)−1 \(\oint_S\)Db D[b ξa] dSa .

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

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

Ka Pa = (8πG)−1\(\oint_S\)ωab dSab ,

where Ka = αA α*A' is a null vector determined by αA, and ωab = i εabcd ψCd ψC', with ψA = αA + O(r−1), an asymptotically constant spinor; Different components of P are obtained using different Ks.
@ References: Chen et al gq/02-proc.

Other Expressions > s.a. ADM formalism; quasilocal energy; asymptotic flatness at spatial infinity.
* Results: If ηa:= eabb r, then [@ Ashtekar in(84)]:

E = −(8πG)−1\(\oint_S\) r 3Rab ηa ηb d2v = (8πG)−1\(\oint_S\) Γabc ηa ηb ηc d2v
= (16πG)−1\(\oint_S\) r [3R − (2π)2 + (2πab) (2πab)] d2v .

@ General references: Sorkin in(88) [from first-order action]; Chruściel et al PRL(90)gq [uniqueness]; Lau PRD(00)gq [lightcone reference]; Rizzi gq/02 [and angular momentum]; Chruściel et al ATMP(04)gq/03 [Trautman-Bondi, initial data sets]; Nester CQG(04) [rev]; Chen & Zhu PRD(11)-a1006 [from metric decomposition].
@ Bondi mass / momentum: Ashtekar & Magnon-Ashtekar PRL(79) [and ADM]; Huang & Zhang ScCh(07)gq/06 [relationship with ADM]; Gallo et al PRD(08) [estimating from finite distance]; > s.a. asymptotic flatness at null infinity.
@ Møller complex: Xulu MPLA(00) [Kerr-Newman spacetime]; Radinschi gq/02, Gad ASS(06)gq/04, MPLA(04)gq [axisymmetric]; Nashed NCB(04)gq/05 [Kerr solutions]; Yang & Tsai gq/05 [evaluation]; Salti MPLA(05)gq [Kasner-type spacetime].
> Related topics: see gravitational energy-momentum [specific spacetimes]; angular momentum; Smarr Formula.

In Other Theories > s.a. kaluza-klein phenomenology.
@ General references: Ortín JHEP(17)-a1705 [coupling of gravity to the gravitational energy-momentum tensor, and the strong equivalence principle].
@ Teleparallel theory: Maluf & Ulhoa GRG(09)-a0810 [teleparallel equivalent of general relativity, rotating sources]; Ulhoa et al IJMPD(10)-a1010 [cosmological spacetimes]; Abedi & Salti GRG(15) [f(T) theory non-minimally coupled to a scalar field]; > s.a. teleparallel theories [including energy-momentum density].
@ Other theories: Eling PRD(06) [Einstein-aether]; Baryshev a0809-proc [Minkowski field theory]; > s.a. 3D general relativity; higher-order theories.

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