Stress-Energy Pseudotensors

In General > s.a. energy and perturbations in general relativity.
* Idea: The various two-index objects t^ab that are constructed for a metric g_ab, or a linear perturbation h_ab 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:

t^ab:= \(1\over16\pi G\) [H^ambn_,mn(h) − 2G^(2)ab(h)] ,

where H^ambn is locally constructed, quadratic in h_ab, and satisfies H_ambn = H_[am]bn = H_am[bn] = H_bnam.
* 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 h_ab:= g_ab − η_ab, h'_ab:= h_ab − \(1\over2\)hη_ab, as

H^ambn:= −(h' ^ab η^mn + h' ^mn η^ab − h'^mb η^an − h' ^an η^mb) .

Landau-Lifshitz Pseudotensor
* Idea: The symmetric t_LL^ab, quadratic in the first derivatives of the metric, such that for an asymptotically flat spacetime,

p^a = ∫_Σ T_{LL, eff}^a0 d³x,   where   T_{LL, eff}^ab:= |g| (T ^ab + t_LL^ab) .

@ 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")

θ^ab:= −\(1\over8\pi G\)tG^ab + tT^ab + \(1\over8\pi G\)[tg^abg^mn (Γ^r_ms Γ^s_rn − Γ^r_mn Γ^s_rs) + g^rb (Γ^a_mn tg^mn_,r − Γ^m_nm tg^na_,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 h^mnab = |g| (g^ma g^nb − g^na g^mb), it is given by

T ^m_n · Xⁿ = −\(1\over8\pi G\)|g|^1/2 G^m_n Xⁿ + \(1\over2\)∂_a (|g|^−1/2 h^mnab_,b X_n) .

@ 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 T_mnab, the Chevreton tensor.
* Relationships: In Riemann normal coordinates, T_mnab = ∂²_mn t_ab, where t_ab 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 T_mn, 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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