Modification of the Bel-Robinson type energy-momentum

TL;DR

This paper proposes modifications to the Bel-Robinson type energy-momentum tensors to ensure they satisfy Lorentz covariance, future pointing, and non-spacelike properties in a quasilocal setting.

Contribution

It introduces a way to modify Bel-Robinson type tensors to meet key physical properties in gravitational energy-momentum descriptions.

Findings

Modified energy-momentum tensors satisfy Lorentz covariance.

Ensured tensors are future pointing and non-spacelike.

Derived properties from small region limits of tensors.

Abstract

For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson type energy-momentum in a quasilocal 2-surface, both the Bel-Robinson tensor $B$ and tensor $V$ are suitable. We found that this Bel-Robinson type energy-momentum can be modified such that it satisfies the Lorentz covariant, future pointing and non-spacelike properties. We find that these particular energy-momentum properties can be obtained from (i): $B$ or $V$ plus a tensor $S$ in a quasilocal small cube limit, or (ii): directly evaluating the energy-momentum of $B$ or $V$ in a quasilocal small box region.