Center of mass integral in canonical general relativity

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∼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∼ x k grows like one of the asymptotically Cartesian coordinate functions. Such a two-surface integral defines the kth component of the center of mass for (the initial data belonging to) a Cauchy surface Σ bounded by B. In the large-radius limit, we find agreement between H B and an integral introduced by Beig and o ́ Murchadha as an improvement upon the center-of-mass integral first written down by Regge and Teitelboim. 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 which is 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.