Abstract
We study the space of Killing fields on the four dimensional AdS spacetime \(AdS^{3,1}\). Two subsets \({\mathcal {S}}\) and \({\mathcal {O}}\) are identified: \({\mathcal {S}}\) (the spinor Killing fields) is constructed from imaginary Killing spinors, and \({\mathcal {O}}\) (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of \( {\mathcal {O}}\) is properly contained in that of \({\mathcal {S}}\). This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. Chruściel et al. (J High Energy Phys 2006(11):084, 2006) proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the “rest mass” of an asymptotically AdS spacetime is then defined by minimizing the observer energy and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime.
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Communicated by James A. Isenberg.
P.-N. Chen is supported by NSF Grant DMS-1308164, M.-T. Wang is supported by NSF Grants DMS-1105483 and DMS-1405152, and S.-T. Yau is supported by NSF Grants PHY-0714648 and DMS-1308244. This work was partially supported by a grant from the Simons Foundation (#305519 to Mu-Tao Wang). Part of this work was carried out when P.-N. Chen and M.-T. Wang were visiting the Department of Mathematics and the Center of Mathematical Sciences and Applications at Harvard University.
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Chen, PN., Hung, PK., Wang, MT. et al. The Rest Mass of an Asymptotically Anti-de Sitter Spacetime. Ann. Henri Poincaré 18, 1493–1518 (2017). https://doi.org/10.1007/s00023-017-0555-4
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DOI: https://doi.org/10.1007/s00023-017-0555-4