On the Stress Tensor for Asymptotically Flat Gravity

TL;DR

This paper clarifies the relationship between boundary stress tensor methods and other approaches for defining conserved charges in asymptotically flat spacetimes, confirming their equivalence and exploring connections to the Weyl tensor.

Contribution

It corrects previous comparisons by including overlooked terms, showing they do not affect conserved charges, and links the stress tensor to the electric part of the Weyl tensor.

Findings

Boundary stress tensor yields the same conserved charges as other methods.

Overlooked terms in earlier definitions vanish or are non-contributing in relevant dimensions.

Connections established between the stress tensor and the electric part of the Weyl tensor.

Abstract

The recent introduction of a boundary stress tensor for asymptotically flat spacetimes enabled a new construction of energy, momentum, and Lorentz charges. These charges are known to generate the asymptotic symmetries of the theory, but their explicit formulas are not identical to previous constructions in the literature. This paper corrects an earlier comparison with other approaches, including terms in the definition of the stress tensor charges that were previously overlooked. We show that these terms either vanish identically (for d > 4) or take a form that does not contribute to the conserved charges (for d=4). This verifies the earlier claim that boundary stress tensor methods for asymptotically flat spacetimes yield the same conserved charges as other approaches. We also derive some additional connections between the boundary stress tensor and the electric part of the Weyl tensor.