Bel and Bel-Robinson Tensors| 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.
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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