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

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.