Conserved quantities in general relativity: from the quasi-local level to spatial infinity

Po-Ning Chen Columbia University Mu-Tao Wang Columbia University Shing-Tung Yau Harvard University

Mathematical Physics mathscidoc:1608.10021

Gold Award Paper in 2017

Communications in Mathematical Physics, 338, (1), 31–80, 2015.8
We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in Wang and Yau (Commun Math Phys 288(3):919–942, 2009) 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 the gravitational 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 (Proc R Soc Lond Ser A 381(1780):53–63, 1982).
Conserved quantities, general relativity, optimal isometric embedding, Minkowski spacetime, Quasi-local angular momentum
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@inproceedings{po-ning2015conserved,
  title={Conserved quantities in general relativity: from the quasi-local level to spatial infinity},
  author={Po-Ning Chen, Mu-Tao Wang, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160820152944728260325},
  booktitle={Communications in Mathematical Physics},
  volume={338},
  number={1},
  pages={31–80},
  year={2015},
}
Po-Ning Chen, Mu-Tao Wang, and Shing-Tung Yau. Conserved quantities in general relativity: from the quasi-local level to spatial infinity. 2015. Vol. 338. In Communications in Mathematical Physics. pp.31–80. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160820152944728260325.
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