Quasilocal energy-momentum for tensors B and V in small regions

Abstract

The Bel-Robinson tensor $B$ and the tensor $V$ have the same quasilocal energy-momentum in a small sphere. Using a pseudotensor approach to evaluate the energy-momentum in a half-cylinder, we find that $B$ and $V$ have different values, not proportional to the "Bel-Robinson energy-momentum". Furthermore, even if we arrange things so that we do get the same "Bel-Robinson energy-momentum" value, the angular momentum gives different values using $B$ and $V$ in a half cylinder. In addition, we find that $B$ and $V$ have a different number of independent components. The fully trace free property of $B$ and $V$ implies conservation of pure "Bel-Robinson energy-momentum" in small regions, and vice versa. In addition, we also demonstrate the tidal heating, rate of change of momentum and spin angular momentum flux by using these two tensors.