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

Abstract

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 energy-momentum density should be proportional to the Bel-Robinson tensor $B_{\alpha \beta \mu \nu }$. However, we propose a new tensor $V_{\alpha \beta \mu \nu }$ which 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" content as $B_{\alpha \beta \mu \nu }$ 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.

arXiv comment7

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