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].