Note on counterterms in asymptotically flat spacetimes

TL;DR

The paper demonstrates how the Mann-Marolf covariant boundary counterterm can be implemented in cylindrical cut-offs to produce finite, intrinsic actions and conserved charges for asymptotically flat spacetimes. It derives explicit expressions in general $d$ and applies them to NUT-charged spacetimes and the Kerr black hole, showing that the renormalized action and boundary stress tensor are governed by the electric part of the Weyl tensor. The authors compare with alternative counterterms (e.g., Lau-Mann) and find consistent results for masses and actions, while highlighting differences in certain stress-tensor components. The work provides practical tools for gravitational thermodynamics in asymptotically flat spacetimes and sets the stage for higher-dimensional and more general stationary backgrounds.

Abstract

We consider in more detail the covariant counterterm proposed by Mann and Marolf in asymptotically flat spacetimes. With an eye to specific practical computations using this counterterm, we present explicit expressions in general $d$ dimensions that can be used in the so-called `cylindrical cut-off' to compute the action and the associated conserved quantities for an asymptotically flat spacetime. As applications, we show how to compute the action and the conserved quantities for the NUT-charged spacetime and for the Kerr black hole in four dimensions.