Bel–Robinson tensor

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

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

Alternatively,

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

where $C_{abcd}$ 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:

$T_{abcd}=T_{(abcd)}$

$T^{a}{}_{acd} = 0$

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:

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

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: 2043, doi:10.1023/A:1001906821162