Covariant Counterterms and Conserved Charges in Asymptotically Flat Spacetimes

TL;DR

The paper analyzes covariant counterterms for asymptotically flat spacetimes by introducing a boundary term with $\hat{K}$ to the Einstein–Hilbert action. It proves two key results: (i) in spacetime dimensions $d\ge 4$, the covariant action reduces to the ADM action under a space–time split, aligning the boundary stress-tensor charges with ADM charges, and (ii) in $d=4$, the counterterm charges coincide with the Ashtekar–Hansen conserved charges at spatial infinity, including Lorentz generators, via Beig–Schmidt expansions and electric/magnetic Weyl relations. Altogether, the work unifies covariant, background-independent boundary methods with the traditional ADM and Ashtekar–Hansen formalisms for conserved quantities in asymptotically flat spacetimes. This reinforces the robustness of covariant counterterm approaches and clarifies the precise matching of charges across these foundational frameworks.

Abstract

Recent work has shown that the addition of an appropriate covariant boundary term to the gravitational action yields a well-defined variational principle for asymptotically flat spacetimes and thus leads to a natural definition of conserved quantities at spatial infinity. Here we connect such results to other formalisms by showing explicitly i) that for spacetime dimension $d \ge 4$ the canonical form of the above-mentioned covariant action is precisely the ADM action, with the familiar ADM boundary terms and ii) that for $d=4$ the conserved quantities defined by counter-term methods agree precisely with the Ashtekar-Hansen conserved charges at spatial infinity.