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The spacetime Penrose inequality under a quasi final state hypothesis
Ahmed Ellithy
Uppsala University, Sweden
Penrose's heuristic argument for the spacetime Penrose inequality is fundamentally dynamical: the area of an apparent horizon should be compared with the mass measured at infinity after the black-hole exterior has settled down. In this talk, I will explain how this heuristic can be made precise without assuming convergence of the spacetime to Kerr. More precisely, I will describe a new proof of the inequality under a precise ``quasi-final-state hypothesis'', a late-time hypothesis which asks for much less than convergence of the spacetime to Kerr. The approach is new and formulated directly in spacetime. The main new geometric ingredient is the definition of a ``tangentially maximal'' hypersurface, carrying a foliation by spacelike spheres whose timelike mean curvature vanishes. We show that these hypersurfaces are governed by a quasilinear inward-parabolic PDE, and we develop the corresponding a priori theory and prove global existence. On these hypersurfaces, the spacetime Hawking mass reduces to the Riemannian Hawking mass, and the dominant energy condition gives nonnegative scalar curvature. This reduces the spacetime Penrose inequality to its Riemannian counterpart. Together with the horizon area law, this yields the desired inequality and gives a rigorous realization of Penrose's original dynamical heuristic.
