Gravitational energy in a small region for the modified Einstein and Landau-Lifshitz pseudotensors

The purpose of the classical Einstein and Landau-Lifshitz pseudotensors is for determining the gravitational energy. Neither of them can guarantee a positive energy in holonomic frames. In the small sphere approximation, it has been required that the quasilocal expression for the gravitational energymomentum density should be proportional to the Bel-Robinson tensor Bαβμν . However, we propose a new tensor Vαβμν which is the sum of certain tensors Sαβμν and Kαβμν , it has certain properties so that it gives the same gravitational “energy-momentum” content as Bαβμν does. Moreover, we show that a modified Einstein pseudotensor turns out to be one of the Chen-Nester quasilocal expressions, while the modified Landau-Lifshitz pseudotensor becomes the Papapetrou pseudotensor; these two modified pseudotensors have positive gravitational energy in a small region.