Construction of energy-momentum tensor of gravitation

TL;DR

This paper constructs a gravitational energy-momentum tensor in general relativity using Noether's theorem, demonstrating its tensorial nature and conservation properties, with implications for energy cancellation in specific space-times.

Contribution

It provides an explicit construction of the gravitational energy-momentum tensor that transforms as a tensor under coordinate changes and verifies its conservation in general relativity.

Findings

The constructed tensor can vary as a tensor under coordinate transformations.

Energy-momentum conservation holds when one index is in the local Lorentz frame.

Gravitational and matter energies cancel in certain space-times.

Abstract

We construct the gravitational energy-momentum tensor in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy-momentum conservation is satisfied because one of the two indices of the energy-momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.