# A gravitational energy-momentum and the thermodynamic description of gravity

## TL;DR

This paper investigates a specific gravitational energy-momentum tensor derived from the Weyl tensor, exploring its thermodynamic interpretation and the implications of its non-conservation and potential dissipation in vacuum conditions.

## Contribution

It provides a detailed analysis of the square root of Bel-Robinson tensor as a gravitational energy-momentum tensor and explores its thermodynamic significance in gravitational fields.

## Key findings

- The tensor is constructed from the Weyl tensor and relates to optical scalars of null congruences.
- The tensor is not necessarily conserved, allowing for dissipative effects in vacuum.
- Thermodynamic interpretation of geometric quantities in gravitational fields is proposed.

## Abstract

A proposal for the gravitational energy-momentum tensor, known in the literature as the square root of Bel-Robinson tensor, is analyzed in detail. Being constructed exclusively from the Weyl part of the Riemann tensor, such tensor encapsulates the geometric properties of free gravitational fields in terms of optical scalars of null congruences: making use of the general decomposition of any energymomentum tensor, we explore the thermodynamic interpretation of such geometric quantities. While the matter energy-momentum is identically conserved due to Einstein's field equations, the SQBR is not necessarily conserved and dissipative terms could arise in its vacuum continuity equation. We discuss the possible physical interpretations of such mathematical properties.