Graviton-Graviton Scattering, Bel-Robinson and Energy (Pseudo)-Tensors

TL;DR

This paper establishes an exact tensor-square root relation between the Bel-Robinson tensor and a combination of gravitational energy pseudo-tensors, linking scattering amplitudes with energy localization in gravity.

Contribution

It introduces a novel exact tensor-square root equality connecting the Bel-Robinson tensor with gravitational pseudo-tensors in Riemann normal coordinates.

Findings

Proves the tensor-square root relation $B_{\mu\nu\alpha\beta} = \partial^2_{\mu\nu} t_{\alpha\beta}$.

Relates superpotential pseudo-tensors to curvature polynomials on-shell.

Links graviton scattering amplitudes with energy pseudo-tensor descriptions.

Abstract

Motivated by recent work involving the graviton-graviton tree scattering amplitude, and its twin descriptions as the square of the Bel-Robinson tensor, $B_{\m\n\a\b}$, and as the "current-current interaction" square of gravitational energy pseudo-tensors $t_{\a\b}$,we find an exact tensor-square root equality $B_{\mn\a\b} = \pa^2_\mn t_{\a\b}$, for a combination of Einstein and Landau-Lifschitz $t_\ab$, in Riemann normal coordinates. In the process, we relate, on-shell, the usual superpotential basis for classifying pseudo-tensors with one spanned by polynomials in the curvature.