Gravitational energy in small regions for the quasilocal expressions in orthonormal frames
Abstract
Description
The M$ø$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 $B_{αβμν}$. Treating a modified quasilocal expressions in a similar way, we found one unique combination that gives a multiple of $B_{αβμν}$ which provides a non-negative gravitational energy-momentum in the small sphere approximation. Moreover, in addition to $B_{αβμν}$, we found a certain tensor $S_{αβμν}+K_{αβμν}$ 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.
6
6