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 [26] 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 initial 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 [18].

55 pages. The dynamical formula now holds under much weaker asymptotic condition

References in corpus (6)

• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• Time evolution of ADM and CMC center of mass in general relativity
• Rigidity and minimizing properties of quasi-local mass
• Conserved Quantities of harmonic asymptotic initial data sets
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Cited by in corpus (25)

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• Quasi-local energy with respect to de Sitter/anti-de Sitter reference
• Gravity: a gauge theory perspective
• Four lectures on quasi-local mass
• Properties of Quasi-local mass in binary black hole mergers
• Aspects of Quasi-local energy for gravity coupled to gauge fields
• Conserved Quantities of harmonic asymptotic initial data sets
• Quasi-local mass on unit spheres at spatial infinity
• Quasi-local energy in presence of gravitational radiation
• Energy, momentum, and center of mass in general relativity
• Local space time constant mean curvature and constant expansion foliations
• Some Remarks on Wang-Yau Quasi-Local Mass
• On the validity of the definition of angular momentum in general relativity
• A note on Minkowski formula of conformal Killing-Yano $2$-form
• Evaluating small sphere limit of the Wang-Yau quasi-local energy
• Total mass and limits of quasi-local mass at future null infinity
• Quasi-local mass at null infinity in Bondi-Sachs coordinates
• Strong field behavior of Wang-Yau Quasi-local energy
• On the limiting behavior of the Brown-York quasi-local mass in asymptotically hyperbolic manifolds