On the center of mass of isolated systems

Abstract

We discuss the center of mass of asymptotically flat manifolds. Our main result is that for a class of metrics that includes those which near infinity are conformally flat with vanishing scalar curvature and positive mass, the Huisken–Yau geometric center of mass agrees with the center of mass defined by the ADM formulation of the initial-value problem for Einstein's equation.

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

“…From our construction, we not only prove that the geometric center converges, but we also show that it is equal to center of mass C. The following corollary generalizes the results in [7,9].…”

“…Using the unique foliation, they defined a geometric center of the foliation. Corvino and Wu [7] proved that the geometric center of the foliation is the center of mass if the metric is conformally flat near infinity. The condition that the metric is conformally flat near infinity is later removed by the author [9].…”

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…The geometric notion of center-of-mass introduced by Huisken and Yau [280] is another form of the Beig-Ó Murchadha center-of-mass [156]. …”

On the center of mass in general relativity

“…A more general uniqueness result was obtained by Qing and Tian [18]. For strongly asymptotically flat metric which is conformally flat near infinity, Corvino and Wu proved the geometric center of Huisken-Yau's foliation is equal to center of mass [9], and we later removed the condition of being conformally flat [11]. However, there are various interesting physical solutions to the constraint equations which are not strongly asymptotically flat.…”

“…Notice that when g is asymptotically flat, we can replace ν A detailed discussion on these conformal Killing vector fields can be found in, for example [9].…”