general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

and differential geometry, the Bel–Robinson tensor is a tensor defined in the

abstract index notation Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeh ...

by: :

T_=C_C_ ^ _ ^ + \frac\epsilon_^ \epsilon_^_ C_ C_^_^

Alternatively, :

T_ = C_C_ ^ _ ^ - \frac g_ C_ C^_^

where

C_

is the

Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal forc ...

. 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 In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...

. Like the electromagnetic stress–energy tensor, the Bel–Robinson tensor is totally symmetric and traceless: :

\begin
       T_ &= T_ \\
  T^_ &= 0
\end

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 In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

vanishes (i.e. in vacuum), the Bel–Robinson tensor is divergence-free: :

\nabla^ T_ = 0

References

Tensors in general relativity Differential geometry {{differential-geometry-stub