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

摘要

The purpose of the classical Einstein and Landau-Lifshitz and other pseudotensors is to determine 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 energy-momentum density should be proportional to the Bel-Robinson tensor B-alpha beta mu nu. However, we propose a new tensor V-alpha beta mu nu. that is the sum of certain tensors S-alpha beta mu nu. and K-alpha beta mu nu; it has certain properties so that it gives the same gravitational 'energy-momentum' density content as B-alpha beta mu nu. does. In comparison, the main difference is that B-alpha beta mu nu. fulfils the dominant energy condition while V-alpha beta mu nu. does not. Moreover, we show that a modified Einstein pseudotensor turns out to be a generalization of 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.