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 HB belonging to the energy sector of the gravitational Hamiltonian. Assuming that the lapse function behaves as N1 in the limit, we find agreement between HB 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 HB corresponding to the Hamiltonian generator of an asymptotic boost, in which case the lapse Nxk 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 ? bounded by B. In the large-radius limit, we find agreement between HB and an integral introduced by Beig and Murchadha as an improvement upon the center-of-mass integral first written down by Regge and Teitelboim. Although both HB and the Beig– Murchadha integral are naively divergent, they are in fact finite modulo the Hamiltonian constraint. Furthermore, we examine the relationship between HB 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