Weinberg's energy-momentum pseudotensor for Schwarzschild field

TL;DR

This paper derives Weinberg's energy-momentum pseudotensor for the Schwarzschild metric in harmonic coordinates, highlighting singularities at the horizon and discussing implications for gravitational energy definitions.

Contribution

It provides a specific derivation of Weinberg's pseudotensor in harmonic coordinates for Schwarzschild spacetime and analyzes its singular behavior at the horizon.

Findings

Singularities at the horizon prevent total energy integration

Implication for defining gravitational energy in collapse scenarios

Discussion on selecting a unique energy-momentum tensor

Abstract

Weinberg's energy-momentum pseudotensor is obtained for Schwarzschild metric in harmonic coordinates. On the horizon it possesses unintegrable singularities. For this reason the total energy of a collapsar can't be obtained by integrating energy density over the system's volume. The implication for gravity theories is noted. A thought on how to choose unique energy-momentum tensor is given.