Dominant property for the Bel-Robinson tensor and tensor S

TL;DR

This paper explores a linear combination of the Bel-Robinson tensor and tensor S that maintains the dominant property, simplifying the verification process across Petrov types for gravitational energy positivity.

Contribution

It introduces a new tensor combination that preserves the dominant property and simplifies the verification criteria for gravitational energy conditions.

Findings

The combined tensor maintains the dominant property.

Verification reduces to checking specific components in Petrov types.

Simplified conditions facilitate analysis of gravitational energy.

Abstract

The Bel-Robinson tensor contains many nice mathematical properties and its dominant energy condition is desirable for describing the positive gravitational energy. The dominant property is a basic requirement for the quasi-local mass, i.e., in small sphere limit. We claim that there exists another option, a linear combination between the Bel-Robinson tensor $B$ and tensor $S$, which contributes the same dominant property. Moreover, using the 5 Petrov types as the verification, we found that this dominant property justification for the Bel-Robinson tensor can be simplified as examining $B_{0000}\geq|B_{\alpha\beta{}00}|$ and $B_{0000}\geq|B_{0123}|$, instead of $B_{0000}\geq|B_{\alpha\beta\lambda\sigma}|$ for all $\alpha,\beta,\lambda,\sigma=0,1,2,3$.