The modification of the Bel-Robinson type energy-momentum

TL;DR

This paper proposes a modified Bel-Robinson type energy-momentum that satisfies key physical properties, using tensor modifications and small region evaluations, with applications to gravitational energy in specific spacetime scenarios.

Contribution

It introduces a modification to the Bel-Robinson type energy-momentum to ensure Lorentz covariance, future pointing, and non-spacelike properties, expanding its applicability in quasi-local gravitational energy.

Findings

Modified energy-momentum satisfies Lorentz covariance and causality.

Achieved quasi-local energy properties via tensor combinations and small region limits.

Applied to gravitational energy calculations in Schwarzschild spacetime.

Abstract

For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson type energy-momentum in a quasi-local 2-surface, both the Bel-Robinson tensor $B$ and tensor $V$ are suitable. We have 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 quasi-local energy-momentum properties can be obtained from (i): $B$ or $V$ plus a tensor $S$ in a small sphere limit, or (ii): directly evaluating the energy-momentum of $B$ or $V$ in a small ellipsoid region. (iii): calculate the total energy using the Landau-Lifshitz pseudotensor in a small ellipsoid, from Jupiter's tidal force to Io in Schwarzchild spacetime, in an elliptic orbit.