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

TL;DR

This paper introduces a new tensor and modifies existing pseudotensors to ensure positive gravitational energy in small regions, aligning with the Bel-Robinson tensor's properties.

Contribution

It proposes a new tensor $V_{ ext{alpha beta mu nu}}$ and modifies Einstein and Landau-Lifshitz pseudotensors to guarantee positive gravitational energy in small regions.

Findings

The new tensor $V_{ ext{alpha beta mu nu}}$ mimics the Bel-Robinson tensor's energy properties.

Modified Einstein pseudotensor aligns with Chen-Nester quasilocal expressions.

Modified Landau-Lifshitz pseudotensor becomes the Papapetrou pseudotensor.

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.