Bel and Bel-Robinson Tensors  

Bel Tensor
$ Def: The tensor, constructed from the Riemann tensor, its left and right duals, and its double dual,

Tabcd := \(1\over2\)(Ramcn \(R_b{}^m{}_d{}^n\) + *Ramcn *\(R_b{}^m{}_d{}^n\) + R*amcn \(R^*{}_b{}^m{}_d{}^n\) + *R*amcn *\(R^*{}_b{}^m{}_d{}^n\)) .

Bel-Robinson Tensor > s.a. Chevreton Tensor; gowdy spacetime; gravitational energy-momentum; stress-energy pseudotensors; types of singularities.
$ Def: The tensor, constructed from the Riemann tensor (or Weyl tensor, in the vacuum case),

Tabcd := Ramcn \(R_b{}^m{}_d{}^n\) + *Ramcn *\(R_b{}^m{}_d{}^n\)

      = Ramcn \(R_b{}^m{}_d{}^n\) + \(1\over4\)εampq \(\epsilon_b{}^{mr}{}_s\) Rpqcn \(R_r{}^s{}_d{}^n\)

= Ramcn \(R_b{}^m{}_d{}^n\) − \(3\over2\)ga[b Rpq]cn \(R^{\,pq}{}_d{}^n\) .

* Properties: It is totally symmetric, Tabcd = T(abcd), traceless, T aacd = 0, and conserved in vacuum, ∇a Tabcd = 0; When it is contracted with four future-pointing vectors one always obtains a non-negative value.
* Spinorial expression: In vacuum it can be expressed as Tabcd = ψABCD ψ'A'B'C'D' .
* Applications: Used formally to construct an "energy" to control the evolution of certain Sobolev norms in general relativity; More physically, it appears in differences between the gravitational energy-momentum calculated in different coordinates and is interpreted as the effective super-energy-momentum tensor of free gravitational fields.
@ References: Ferrando & Sáez GRG(09)-a0807 [algebraic types]; So a0901 [alternative tensor], a1006 [tensorial proof of symmetry].
@ And quasilocal energy: Brown et al PRD(99)gq/98; So a1006 [non-negative, alternative to Bel-Robinson tensor]; García-Parrado CQG(14)-a1308 [geometric identity and conservation].
@ Applications: Hacyan a1612 [and gravitational radiation].
@ Generalizations: Deser in(87), gq/99-conf; Deser & Franklin CQG(11)-a1011 [for topologically massive gravity], CQG(11)-a1108 [none for quadratic curvature theories]; So a1812 [modification].
> Online resources: see Wikipedia page.

@ Reviews: Douglas GRG(03).
@ General: Bel CRAS(59); Penrose AP(60); in Penrose & Rindler 86; Robinson CQG(97); Bergqvist JMP(98) [positivity]; Senovilla CQG(00)gq/99; Bergqvist & Lankinen CQG(04)gq [characterization].
@ Interpretation: Garecki CQG(85), AdP(01)gq/00; Bonilla & Senovilla GRG(97); Bergqvist GRG(98); García-Parrado CQG(08)-a0707.
@ Related topics: Deser et al CQG(99)gq [graviton-graviton scattering]; Douglas GRG(99) [eigentensors]; Lazkoz et al gq/01-proc [currents]; Saha et al MPLA(06)gq/05 [and Bianchi I evolution].

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