On the Stress Tensor for Asymptotically Flat Gravity

TL;DR

The paper addresses the problem of defining conserved charges in asymptotically flat gravity via a boundary stress tensor $T_{ab}$ and reconciling it with ADM and Ashtekar–Hansen definitions. It uses a covariant counterterm action and the Beig–Schmidt expansion to analyze the extra terms $oldsymbol{ riangle}_{ab}$ and their contributions. The main results show that for $d>4$ the dangerous $oldsymbol{ riangle}_{ab}$ terms vanish at the relevant orders, and for $d=4$ the remaining pieces do not contribute to charges, preserving the equivalence with the electric part of the Weyl tensor. The work includes explicit Kerr and boosted-black-hole examples illustrating the charge construction and its agreement with the expected mass, momentum, and angular momentum.

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.