Center of mass integral in canonical general relativity

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.