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.