2003/01/31 by D. Baskaran, S. R. Lau, S.R. Lau +2
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Bounded function #Curvature #General relativity #Geometric Analysis and Curvature Flows #Gravitation #Hamiltonian (control theory) #Line integral #Minkowski space #Noncommutative and Quantum Gravity Theories #Riemann integral #Surface integral #Volume integral #gr-qc
paper · pdf · doi:10.1016/s0003-4916(03)00062-9
published as Annals Phys. 307 (2003) 90-131 · 52 pages, revtex4, uses amsmath and amssymb
openalex publication_date 2003/05/27 · arxiv created 2003/09/13 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 N ∼ 1 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 N ∼ xk 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 Σbounded by B. In the large--radius limit, we find agreement between HB and an integral introduced by Beig and O'Murchadha. Although both HB and the Beig--O'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 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.