Bel–Robinson tensor

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

$T_{abcd}=C_{aecf}{C}_b{}^e{}_d{}^f+\frac{1}{4}{ϵ}_{ae}{}^{hi}{ϵ}_b{}^{ej}{}_k{C}_{hicf}{C}_j{}^k{}_d{}^f$

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. 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:

$\begin{array}{cc}{T}_{abcd}& =T_{(abcd)}\\ T^a{}_{acd}& =0\end{array}$

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:

${\mathrm{\nabla }}^a{T}_{abcd}=0$

References

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