In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by:
T a b c d = C a e c f C b e d f + 1 4 ϵ a e h i ϵ b e j k C h i c f C j k d f Alternatively,
T a b c d = C a e c f C b e d f − 3 2 g a [ b C j k ] c f C j k d f where C a b c d is the Weyl tensor. It was introduced by Lluís Bel in 1959. 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 a b c d = T ( a b c d ) T a a c d = 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:
∇ a T a b c d = 0 