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 := 
(
Ramcn Rbmdn
+ *Ramcn *Rbmdn + R*amcn R*bmdn + *R*amcn *R*bmdn)
.
Bel-Robinson Tensor > s.a. Chevreton
Tensor;
gowdy spacetime; stress-energy
pseudotensors; types of singularities.
$ Def: The tensor, constructed from the Riemann tensor (or Weyl tensor,
in the vacuum case),
Tabcd :=
Ramcn Rbmdn + *Ramcn *Rbmdn
= Ramcn Rbmdn + 
ampq bmrs Rpqcn
Rrsdn
= Ramcn Rbmdn – 
ga[b Rpq]cn R pqdn
.
* Properties: Symmetric, Tabcd = T(abcd);
Traceless,
T aacd =
0; 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 =
ABCD '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.
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 a0707-CQG.
@ Generalization: Deser in(87),
gq/99-in.
@ Related topics: Brown et al PRD(99)gq/98 [and
quasilocal energy]; Deser et al CQG(99)gq [graviton-graviton
scattering];
Douglas GRG(99)
[eigentensors]; Lazkoz et al gq/01-in
[currents]; Saha et al MPLA(06)gq/05 [and
Bianchi I evolution].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
18 jun 2008