Center of mass integral in canonical general relativity
The authors of this interesting paper consider the Brown-York boundary integral \(H_B\) for a two-surface \(B\) tending to an infinite-radius round sphere at spatial infinity. The integral belongs to the energy sector of the gravitational Hamiltonian. It is assumed that the lapse function behaves as \(N\sim 1\) in this limit. An agreement between \(H_B\) and the total Arnowitt-Deser-Misner energy is found, which was first noted by Braden, Brown, Whiting and York. Here the authors argue that the Arnowitt-Deser-Misner mass-aspect differs from a gauge invariant mass-aspect by a pure divergence on the unit sphere. The boundary integral \(H_B\) corresponding to the Hamiltonian generator of an asymptotic boost is examined as well. In this case the lapse \(N\sim x^k\) grows like one of the asymptotically Cartesian coordinate functions. Such a two-surface integral defines the \(k\) th component of the center of mass for a Cauchy surface \(\Sigma \) bounded by \(B\). In the large-radius limit it is found agreement between \(H_B\) and an integral introduced by Beig and O'Murchadha as an improvement upon the center-of-mass integral first shown 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. The relationship between \(H_B\) and a certain two-surface integral which is linear in the space-time Riemann curvature tensor is studied. Within the canonical \(3+1\) formalism, it is defined gravitational energy and center of mass as certain moments of Riemann curvature.
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