Conserved Quantities in General Relativity: From the Quasi-Local Level to Spatial Infinity

Abstract

We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [28] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group is assigned. Quasi-local angular momentum and quasi-local center of mass correspond to pairing this element with rotation Killing fields and boost Killing fields, respectively. They obey classical transformation laws under the action of the Poincaré group. We further justify these definitions by considering their limits as the total angular momentum and the total center of mass of an isolated system. These expressions were derived from the Hamilton-Jacobi analysis of gravitation action and thus satisfy conservation laws. As a result, we obtained an invariant total angular momentum theorem in the Kerr spacetime. For a vacuum asymptotically flat i! nitial data set of order 1, it is shown that the limits are always finite without any extra assumptions. We also study these total conserved quantities on a family of asymptotically flat initial data sets evolving by the vacuum Einstein evolution equation. It is shown that the total angular momentum is conserved under the evolution. For the total center of mass, the classical dynamical formula relating the center of mass, energy, and linear momentum is recovered, in the nonlinear context of initial data sets evolving by the vacuum Einstein evolution equation. The definition of quasi-local angular momentum provides an answer to the second problem in classical general relativity on Penrose's list [20].

Energy, momentum, and center of mass in general relativity

“…As fundamental as it is, this seems to be the first time when it is shown to be consistent with the nonlinear Einstein evolution. In addition, we prove an invariance of angular momentum theorem in the Kerr spacetime [14,Section 8]. It is likely that the finiteness theorem can be generalized to more general asymptotics (with the order expansion condition) using the gravitational conservation law in [14,Section 5].…”

“…In joint work with Po-Ning Chen and Shing-Tung Yau [14,15], we introduced new definitions of center of mass C i , i = 1, 2, 3 and angular momentum J i , i = 1, 2, 3 that satisfy rather remarkable properties. For example, we prove a finiteness theorem [14,Section 7] under an order expansion condition, and in particular, no parity assumption is needed.…”

“…For example, we prove a finiteness theorem [14,Section 7] under an order expansion condition, and in particular, no parity assumption is needed. More importantly, we show that…”

“…We assume H is spacelike and extract from it a function |H| and a connection one-form α H of the normal bundle in the mean curvature gauge (see [14] for the definition of α H ). (σ, |H|, α H ) determines the "best match" surfaceΣ ⊂ R 3,1 .…”

“…With these new definitions, the classical formula p = mv is shown to be consistent with Einstein's field equation for the first time. This paper is based on joint work [14,15] with Po-Ning Chen and Shing-Tung Yau. …”