The gravitational energy-momentum for the super-energy Bel-Robinson tensor

Abstract

Describing the gravitational energy-momentum, the super-energy Bel-Robinson tensor is the best candidate. In the past, people seems only explore the lowest order: the electric part $E_{ab}$ and magnetic part $B_{ab}$ for the Riemann tensor. These two components are related with the static case, however, for the energy transfer situation, one may need to consider the time varying $\dot{E}_{ab}$ and $\dot{B}_{ab}$. Here we use $(\dot{E}_{ab},\dot{B}_{ab}$) to study the energy-momentum for the Bel-Robinson tensor in a small sphere limit. Meanwhile, our result illustrates how the gravitational field carries the 4-momentum including this extra information.