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_(µναβ), and as the 'current-current interaction' square of gravitational energy pseudo-tensors t_(αβ), we find an exact tensor-square root equality B_(µναβ)= ((∂2)/(µν))t_(αβ), for a combination of Einstein and Landau-Lifschitz t_(αβ), 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.