Bel–Robinson tensor

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

[math]\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} }[/math]

Alternatively,

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

where [math]\displaystyle{ C_{abcd} }[/math] 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:

[math]\displaystyle{ \begin{align} T_{abcd} &= T_{(abcd)} \\ T^{a}{}_{acd} &= 0 \end{align} }[/math]

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:

[math]\displaystyle{ \nabla^{a} T_{abcd} = 0 }[/math]