Center of mass integral in canonical general relativity

TL;DR

This paper investigates boundary integrals in canonical general relativity, establishing their relation to ADM energy and center of mass, and expressing gravitational energy and center of mass as moments of spacetime curvature.

Contribution

It demonstrates the equivalence of boundary integrals to ADM energy and center of mass, and introduces curvature-based definitions within the 3+1 formalism.

Findings

Boundary integral H_B agrees with ADM energy at infinity.

H_B related to Beig and O'Murchadha's center of mass integral.

Gravitational energy and center of mass expressed as moments of Riemann curvature.

Abstract

For a two-surface B tending to an infinite--radius round sphere at spatial infinity, we consider the Brown--York boundary integral H_B belonging to the energy sector of the gravitational Hamiltonian. Assuming that the lapse function behaves as N \sim 1 in the limit, we find agreement between H_B and the total Arnowitt--Deser--Misner energy, an agreement first noted by Braden, Brown, Whiting, and York. However, we argue that the Arnowitt--Deser--Misner mass--aspect differs from a gauge invariant mass--aspect by a pure divergence on the unit sphere. We also examine the boundary integral H_B corresponding to the Hamiltonian generator of an asymptotic boost, in which case the lapse N \sim x^k grows like one of the asymptotically Cartesian coordinate functions. Such an integral defines the kth component of the center of mass for a Cauchy surface \Sigma bounded by B. In the large--radius limit, we find agreement between H_B and an integral introduced by Beig and O'Murchadha. Although both H_B and the Beig--O'Murchadha integral are naively divergent, they are in fact finite modulo the Hamiltonian constraint. Furthermore, we examine the relationship between H_B and a certain two--surface integral linear in the spacetime Riemann curvature tensor. Similar integrals featuring the curvature appear in works by Ashtekar and Hansen, Penrose, Goldberg, and Hayward. Within the canonical 3+1 formalism, we define gravitational energy and center--of--mass as certain moments of Riemann curvature.