Spacetime positive mass theorems for initial data sets with noncompact boundary

Abstract

In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is lightlike. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

41 pages. Final version. Int. Math. Res. Notices 2020

References in corpus (23)

• Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy
• Quasilocal Energy and Conserved Charges Derived from the Gravitational Action
• The Gravitational Hamiltonian, Action, Entropy, and Surface Terms
• Holographic Dual of BCFT
• Covariant phase space with boundaries
• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Scalar Curvature Rigidity for asymptotically locally hyperbolic manifolds
• Gauge theories on manifolds with boundary
• Killing vectors in asymptotically flat space-times: I. Asymptotically translational Killing vectors and the rigid positive energy theorem
• On New Proposal for Holographic BCFT
• Geometric invariance of mass-like asymptotic invariants
• Rigid upper bounds for the angular momentum and centre of mass of non-singular asymptotically anti-de Sitter space-times
• Mass rigidity for hyperbolic manifolds
• Equality in the Spacetime Positive Mass Theorem
• Killing vectors in asymptotically flat space--times: II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions
• The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight
• Computing asymptotic invariants with the Ricci tensor on asymptotically flat and hyperbolic manifolds
• The Initial-Boundary Value Problem in General Relativity
• The hyperbolic positive energy theorem
• Boundary value problems for noncompact boundaries of Spin${}^c$ manifolds and spectral estimates
• A Feynman-Kac formula for differential forms on manifolds with boundary and applications
• The mass in terms of Einstein and Newton
• Geometric Boundary Data for the Gravitational Field

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• Doubling of asymptotically flat half-spaces and the Riemannian Penrose inequality
• Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields