Bel and Bel-Robinson Tensors |

**Bel Tensor**

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

*T*_{abcd} := \(1\over2\)(
*R*_{amcn}* R*_{b}^{m}_{d}^{n}
+ **R*_{amcn} **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),

*T*_{abcd} :=
*R*_{amcn} *R*_{b}^{m}_{d}^{n} + **R*_{amcn} **R*_{b}^{m}_{d}^{n}

= *R*_{amcn} *R*_{b}^{m}_{d}^{n} + \(1\over4\)*ε*_{am}^{pq} *ε*_{b}^{mr}_{s} *R*_{pqcn}
*R*_{r}^{s}_{d}^{n}

= *R*_{amcn} *R*_{b}^{m}_{d}^{n} – \(3\over2\)*g*_{a[b} *R*_{pq]cn} *R*^{ pq}_{d}^{n}
.

* __Properties__: It is
symmetric, *T*_{abcd} = *T*_{(abcd)}, traceless,
*T*^{ a}_{acd} = 0, and conserved in vacuum,
∇^{a}* T*_{abcd}
= 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 *T*_{abcd} =
*ψ*_{ABCD} *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].

> __Online resources__: see Wikipedia page.

**References**

@ __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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