Evaluation of the ADM mass and center of mass via the Ricci tensor

Abstract

We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using the Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using the Ricci tensor are the same.

Quasi-Local Mass Integrals and the Total Mass

“…We first estimate Σρ G(a(ρ), ν)dσ. To do so, we note the following asymptotic formulae of Ric(g) and S g (see (2.2) and (2.6) in [17]):…”

Equality in the Spacetime Positive Mass Theorem

“…It is well-known that we can express E as a flux integral involving the Ricci curvature (see, for example, [14,18]) and thus S∞ −aR ij x i ν j dH n−1 = (n − 1)(n − 2)ω n−1 aE.…”

Static potentials and area minimizing hypersurfaces

“…(The strictly outer-minimizing assumption on ∂Ω guarantees that the competitors produced in [13, Corollary 1.2] still lie in PM.) By asymptotics of static potentials, either V goes to a constant or V is unbounded (see [5] and [23], or Proposition B.4 below). If V goes to a constant, the claim follows.…”

“…Most of the statement in Proposition B.4 below is known and can be found in [5, Appendix C] and [23]. We include the statement and the arguments here because it seems that the estimate (B.2) below used in the proof of Theorem 7 is not explicitly stated in the literature.…”