The modification of the Bel-Robinson energy-momentum

Abstract

For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson `momentum' in a quasi-local small sphere limit, the Bel-Robinson tensor $B$ is desirable. However, we found this Bel-Robinson `momentum' can be modified such that it still satisfy the non-spacelike and future pointing requirement. These particular energy-momentum properties can be obtained from a linear combination between $B$ with other tensor $S$ in a small sphere limit. This implies that the Landau-Lifshitz pseudo-tensor is no longer disqualified for this non-spacelike and future pointing requirement. Moreover, we constructed a certain linear combination using tensors $B,S,T$ that gives the dominate energy condition in a small sphere region.