Bel–Robinson tensor

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 {\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}}

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

where C a b c d {\displaystyle 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 a b c d = T ( a b c d ) T a a c d = 0 {\displaystyle {\begin{aligned}T_{abcd}&=T_{(abcd)}\\T^{a}{}_{acd}&=0\end{aligned}}}

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,^[3] 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 {\displaystyle \nabla ^{a}T_{abcd}=0}

See also

• Bel decomposition

References