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:= (1/16G) [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 – _ab, h'_ab:= h_ab – h_ab, as

H^ambn:= -(h' ^{ab mn} + h' ^{mn ab} – h'^{mb an} – h' ^{an 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= _Sigma T_{LL, eff}^a0 d³x,   where   T_{LL, eff}^ab:= |g| (T ^ab + t_LL^ab) .

@ References: Landau & Lifshitz on field theory; Trautman in(62); in Misner et al 73, 20.22; Persides & Papadopoulos GRG(79); Sardanashvili gq/94.

Freud Pseudotensor
$ Def: The expression (which has often been incorrectly called "von Freud pseudotensor")

^ab:= –(1/8G) tG^ab + tT^ab + (1/8G) [tg^abg^mn(^r_ms ^s_rn – ^r_mn ^s_rs) + g^rb(^a_mn tg^mn_,r – ^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.

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 = –(8G)^–1|g|^1/2 G^m_n Xⁿ + _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-Robinson tensor T_mnab.
* 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-in, mp/02-in; Balfagón & Jaén CQG(00) [computational]; Pozo & Parra CQG(02)gq/01; Lazkoz et al CQG(03)gq [superenergy currents].
@ Related topics: Deser CQG(03)gq [higher-derivative generalizations]; Tintareanu-Mircea & Popa CEJP(05)gq/04 [from Killing-Yano tensors].

Other References > s.a. conservation; gravitational energy; schwarzschild.
@ General: in Wald 84, pp 84 ff; Szabados CQG(92) [differential geometry formulation]; Magnano & Sokolowski CQG(02)gq/01 [and gauge]; Montesinos gq/03-in [gravitational T_mn, covariance and equations of motion]; Deser gq/04-in [and conservation]; So et al a0901 [small regions]; Pitts a0910-in.
@ Proposals: Anderson PRD(97)gq/96 [gravitational waves]; Babak & Grishchuk PRD(00)gq/99; Nikishov PPN(01)gq/99; Tung & Nester gq/00-in [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.
@ Related topics: Yilmaz NCB(92) [wrong]; Mashhoon et al CQG(99)gq/98 [gravitomagnetic]; de Andrade et al PRL(00)gq, gq/00-in [in teleparallel gravity]; Sokolowski APPB(04)gq/03 [fields carrying no energy].

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