Gravitational Energy-Momentum Expressions| 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 \(\oint_S\)εabcd ∇c 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
ψC
∇d
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:=
eab
∂b 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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