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