Time flat surfaces and the monotonicity of the spacetime Hawking mass

Abstract

We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from below by the quasi-local mass of a spacelike 2-surface contained therein.

23 pages; sign error fixed from previous version, statement of Theorem 1.1 changed accordingly

References in corpus (6)

• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• Generalized inverse mean curvature flows in spacetime
• Time flat surfaces and the monotonicity of the spacetime Hawking mass II
• Rigidity of time-flat surfaces in the Minkowski spacetime
• On curves with nonnegative torsion

Cited by in corpus (9)

• Geometrical inequalities bounding angular momentum and charges in General Relativity
• The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces
• Time flat surfaces and the monotonicity of the spacetime Hawking mass II
• Four lectures on quasi-local mass
• Rigidity of time-flat surfaces in the Minkowski spacetime
• RG-2 flow, mass and entropy
• Uniformly Area Expanding Flows in Spacetimes
• Null Geometry and the Penrose Conjecture
• Recent Developments in the Penrose Conjecture