Stress-Energy Pseudotensors |
In General > s.a. energy
and perturbations in general relativity.
* Idea: The various two-index
objects tab that are
constructed for a metric gab,
or a linear perturbation hab
from Minkowski, and used as "stress-energy of the gravitational field"
to express the total 4-momentum or angular momentum as volume integrals.
* Issue: No true energy-momentum
tensor (that is, non-vanishing on shell) can be defined for the gravitational
field, as a consequence of the equivalence principle; Some people have, as a
result, doubted the validity of gravitational energy-momentum and angular
momentum transfer.
* Properties: They are not
geometrical objects – they can always be made to vanish at any given
point –, and they are not unique; These are reflections of the
non-localizability of gravitational energy-momentum; There is no actual
tensor which is appropriate.
* General expression:
tab:= \(1\over16\pi G\) [Hambn,mn(h) − 2G(2)ab(h)] ,
where Hambn is locally
constructed, quadratic in hab,
and satisfies Hambn
= H[am]bn
= Ham[bn]
= Hbnam.
* 3-forms: A similar role
is played by certain 3-forms which are conserved modulo the field
equations, such as the Sparling 3-form.
Canonical Stress-Energy Pseudotensor
* Expression:
The one obtained, in terms of the metric perturbation
hab:=
gab
− ηab,
h'ab:=
hab −
\(1\over2\)hηab, as
Hambn:= −(h' ab ηmn + h' mn ηab − h'mb ηan − h' an ηmb) .
Landau-Lifshitz Pseudotensor
* Idea: The symmetric
tLLab,
quadratic in the first derivatives of the metric, such that for an asymptotically
flat spacetime,
pa = ∫Σ TLL, effa0 d3x, where TLL, effab:= |g| (T ab + tLLab) .
@ References: Landau & Lifshitz v2; Trautman in(62); in Misner et al 73, §20.22; Persides & Papadopoulos GRG(79); Sardanashvily gq/94.
Freud Pseudotensor
$ Def: The expression
(which has often been incorrectly called "von Freud pseudotensor")
θab:= −\(1\over8\pi G\)tGab + tTab + \(1\over8\pi G\)[tgabgmn (Γrms Γsrn − Γrmn Γsrs) + grb (Γamn tgmn,r − Γmnm tgna,r)] .
(the "t" here denotes a densitized tensor, of weight 1).
@ References: Freud AM(39);
Frauendiener CQG(89);
Notte-Cuello & Rodrigues a0801;
Böhmer & Hehl PRD(18)-a1712 [in general relativity and Einstein-Cartan theory].
Gravitational Noether Operator
* Expression:
In terms of hmnab
= |g| (gma
gnb
− gna
gmb),
it is given by
T mn · Xn = −\(1\over8\pi G\)|g|1/2 Gmn Xn + \(1\over2\)∂a (|g|−1/2 hmnab,b Xn) .
@ References: Schutz & Sorkin AP(77); Bak et al PRD(94)ht/93.
Superenergy Tensors
> s.a. Bel-Robinson; Chevreton.
* Examples: The Bel tensor, Bel-Robinson
tensor Tmnab, the Chevreton tensor.
* Relationships: In Riemann normal
coordinates, Tmnab
= ∂2mn
tab, where
tab is a
combination of Einstein and Landau-Lifshitz pseudotensors.
@ General references: Roberts GRG(88);
Mashhoon et al PLA(97)gq/96;
Senovilla CQG(00)gq/99,
gq/99-proc,
mp/02-proc;
Balfagón & Jaén CQG(00) [computational];
Pozo & Parra CQG(02)gq/01;
Lazkoz et al CQG(03)gq [superenergy currents];
Garecki a1209-conf [reappraisal].
@ Related topics: Deser CQG(03)gq [higher-derivative generalizations];
Tintareanu-Mircea & Popa CEJP(05)gq/04 [from Killing-Yano tensors];
Bini & Geralico CQG(18)-a1809 [and energy content of electromagnetic and gravitational plane waves].
Other References
> s.a. conservation; gravitational
energy; schwarzschild spacetime.
@ General: in Wald 84, pp84ff;
Szabados CQG(92) [differential geometry formulation];
Magnano & Sokołowski CQG(02)gq/01 [and gauge];
Montesinos gq/03-in
[gravitational Tmn, covariance and equations of motion];
Deser FP(05)gq/04-in [and conservation];
So et al CQG(09)-a0901 [small regions];
Pitts a0910-conf.
@ Proposals: Anderson PRD(97)gq/96 [gravitational waves];
Babak & Grishchuk PRD(00)gq/99;
Nikishov PPN(01)gq/99;
Tung & Nester gq/00-proc [tetrad/spinor];
Lau gq/06,
Lo et al CQG(09) [and Bel-Robinson tensor].
@ 3-forms: Bonanos CQG(97);
> s.a. Sparling Forms.
@ Tolman energy-momentum complex: Radinschi MPLA(00)gq [static, spherical],
APS(00)gq [Bianchi I],
APS(99)gq/00,
MPLA(00)gq [dyonic black hole].
@ And quasilocal quantities: Chang et al PRL(99)gq/98.
@ In other gravity theories: de Andrade et al PRL(00)gq,
gq/00-GR9 [in teleparallel gravity];
Capozziello et al IJTP(10)-a1001 [f(R) gravity];
Capozziello et al AdP(17)-a1702 [higher-order theories];
Capozziello et al IJGMP(18)-a1804
[f(R) and f(T) gravity].
@ Related topics: Yilmaz NCB(92) [wrong];
Mashhoon et al CQG(99)gq/98 [gravitomagnetic];
Sokołowski APPB(04)gq/03 [fields carrying no energy];
Butcher et al PRD(12)-a1210 [linearized gravity, gravitational spin tensor].
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