Gravitational energy in small regions for the quasilocal expressions in orthonormal frames

Abstract

The M$\o$ller tetrad gravitational energy-momentum expression was recently evaluated for a small vacuum region using orthonormal frames adapted to Riemann normal coordinates. However the result was not proportional to the Bel-Robinson tensor $B_{\alpha\beta\mu\nu}$. Treating a modified quasilocal expressions in a similar way, we found one unique combination that gives a multiple of $B_{\alpha\beta\mu\nu}$ which provides a non-negative gravitational energy-momentum in the small sphere approximation. Moreover, in addition to $B_{\alpha\beta\mu\nu}$, we found a certain tensor $S_{\alpha\beta\mu\nu}+K_{\alpha\beta\mu\nu}$ which gives the same "energy-momentum" density in vacuum. Using this tensor combination, we obtained an infinite set of solutions that provides a positive gravitational energy within the same limit.