# The modification of the Bel-Robinson energy-momentum

## TL;DR

This paper proposes a modification to the Bel-Robinson energy-momentum tensor, enabling it to satisfy non-spacelike and future pointing conditions, and introduces a linear combination with other tensors to meet energy conditions in small regions.

## Contribution

It introduces a modified Bel-Robinson tensor framework that satisfies key energy-momentum properties and energy conditions in small sphere limits, challenging previous disqualifications.

## Key findings

- Modified Bel-Robinson tensor satisfies non-spacelike, future pointing conditions.
- Linear combination with other tensors meets dominant energy condition.
- Landau-Lifshitz pseudo-tensor can be adapted for these properties.

## Abstract

For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson `momentum' in a quasi-local small sphere limit, the Bel-Robinson tensor $B$ is desirable. However, we found this Bel-Robinson `momentum' can be modified such that it still satisfy the non-spacelike and future pointing requirement. These particular energy-momentum properties can be obtained from a linear combination between $B$ with other tensor $S$ in a small sphere limit. This implies that the Landau-Lifshitz pseudo-tensor is no longer disqualified for this non-spacelike and future pointing requirement. Moreover, we constructed a certain linear combination using tensors $B,S,T$ that gives the dominate energy condition in a small sphere region.