Bel–Robinson tensor

In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by:

${\displaystyle T_{abcd}=C_{aecf}C_{b}{}^{e}{}_{d}{}^{f}+{\frac {1}{4}}\epsilon _{ae}{}^{hi}\epsilon _{b}{}^{ej}{}_{k}C_{hicf}C_{j}{}^{k}{}_{d}{}^{f}}$

Alternatively,

${\displaystyle T_{abcd}=C_{aecf}C_{b}{}^{e}{}_{d}{}^{f}-{\frac {3}{2}}g_{a[b}C_{jk]cf}C^{jk}{}_{d}{}^{f}}$

where ${\displaystyle C_{abcd}}$ is the Weyl tensor. It was introduced by Lluís Bel in 1959.^[1][2] The Bel–Robinson tensor is constructed from the Weyl tensor in a manner analogous to the way the electromagnetic stress–energy tensor is built from the electromagnetic tensor. Like the electromagnetic stress–energy tensor, the Bel–Robinson tensor is totally symmetric and traceless:

${\displaystyle {\begin{aligned}T_{abcd}&=T_{(abcd)}\\T^{a}{}_{acd}&=0\end{aligned}}}$

In general relativity, there is no unique definition of the local energy of the gravitational field. The Bel–Robinson tensor is a possible definition for local energy, since it can be shown that whenever the Ricci tensor vanishes (i.e. in vacuum), the Bel–Robinson tensor is divergence-free:

${\displaystyle \nabla ^{a}T_{abcd}=0}$

References

1. Bel, L. (1959), "Introduction d'un tenseur du quatrième ordre", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 248: 1297
2. Senovilla, J. M. M. (2000), "Editor's Note: Radiation States and the Problem of Energy in General Relativity by Louis Bel", General Relativity and Gravitation, 32 (10): 2043, Bibcode:2000GReGr..32.2043S, doi:10.1023/A:1001906821162, S2CID 116937193