Bel–Robinson tensor explained

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} + (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} - (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. 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:\begin{align} T _{abcd} & = T _{(abcd)} \\ T ^{a} {} _{acd} & = 0 \end{align}

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