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:= gabηab, h'ab:= hab − \(1\over2\)ab, as

Hambn:= −(h' ab ηmn + h' mn ηabh'mb ηanh' 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 + \(1\over8\pi G\)[tgabgmnrms Γsrn − Γrmn Γsrs) + grbamn 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 gnbgna 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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