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Center of mass integral in canonical general relativity
General Relativity and Quantum Cosmology · Submitted 20 Jan 2003
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Within the canonical 3+1 formalism, we define gravitational energy and center--of--mass as certain moments of Riemann curvature.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.51 | 0.45 | 0.04 | asserted 0.74 |
| s1 | 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. | result 0.87 | 0.63 | 0.09 | asserted 1.00 |
| s2 | 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. | result 0.92 | 0.63 | 0.29 | asserted 0.91 |
| s3 | 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. | contribution 0.91 | 0.67 | 0.04 | asserted 0.94 |
| s4 | Such an integral defines the kth component of the center of mass for a Cauchy surface \Sigma bounded by B. | interpretation 0.32 | 0.61 | 0.11 | asserted 1.00 |
| s5 | In the large--radius limit, we find agreement between H_B and an integral introduced by Beig and O'Murchadha. | result 1.00 | 0.91 | 0.12 | asserted 1.00 |
| s6 | Although both H_B and the Beig--O'Murchadha integral are naively divergent, they are in fact finite modulo the Hamiltonian constraint. | result 0.96 | 0.83 | 0.14 | asserted 1.00 |
| s7 | Furthermore, we examine the relationship between H_B and a certain two--surface integral linear in the spacetime Riemann curvature tensor. | contribution 0.98 | 0.75 | 0.06 | asserted 0.55 |
| s8 | Similar integrals featuring the curvature appear in works by Ashtekar and Hansen, Penrose, Goldberg, and Hayward. | background 1.00 | 0.05 | 0.03 | asserted 0.99 |
| s9 | Within the canonical 3+1 formalism, we define gravitational energy and center--of--mass as certain moments of Riemann curvature. | contribution 0.80 | 0.85 | 0.40 | asserted 1.00 |
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Original 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 \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.
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