Английская Википедия:Bel–Robinson tensor
In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by:
- <math>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>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>C_{abcd}</math> 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:
- <math>\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>\nabla^{a} T_{abcd} = 0</math>
References
Шаблон:Relativity-stub
Шаблон:Differential-geometry-stub
