A simple tensorial proof for the completely symmetric property of the Bel-Robinson tensor

TL;DR

This paper offers a straightforward tensorial proof of the complete symmetry of the Bel-Robinson tensor, an important object in gravitational energy, and also derives a key vacuum identity using tensor methods.

Contribution

It introduces an alternative tensorial approach to prove the symmetry of the Bel-Robinson tensor and derives a known vacuum identity, expanding the toolkit beyond spinor methods.

Findings

Proved the complete symmetry of the Bel-Robinson tensor tensorially.

Derived the vacuum identity involving the Riemann tensor.

Provided an alternative proof method for fundamental gravitational tensor properties.

Abstract

The Bel-Robinson tensor $T_{\alpha\beta\mu\nu}$ was proposed in 1958. The main application of this tensor is for describing gravitational energy. It is known that $T_{\alpha\beta\mu\nu}$ has many nice properties such as being completely symmetric. It is easy to prove this property using spinors as shown in Penrose's book. The present paper provides an alternative way, using the tensorial method to prove that $T_{\alpha\beta\mu\nu}$ is indeed totally symmetric. Moreover, we also found that the well known identity in vacuum, $R_{\alpha\lambda\sigma\tau}R_{\beta}{}^{\lambda\sigma\tau} \equiv 1/4g_{\alpha\beta}R_{\rho\lambda\sigma\tau} R^{\rho\lambda\sigma\tau}$, which can be proven by the same tensorial method.