Eduardo Hafemann

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only C^0 ∩\Wloc^1,n and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs’ Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity is obtained for C^0 ∩\Wloc^1,p metrics with p>n via the comparison theory of \RCD-spaces and a rigidity theorem for compact manifolds with metrics of nonnegative distributional curvature by Jiang-Sheng-Zhang \citeJiang2023. The argument relies on either elementary techniques or generalisations of the standard argument. In essence, a version of the main theorem of Lee-LeFloch \citeLeeLeFloch2015 is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set.

The classical Penrose inequality, a relation between the ADM mass and the area of any cross section of the black hole event horizon, was introduced as a test of the weak cosmic censorship conjecture: if it fails, the trapped surface is not necessarily behind the event horizon and a naked singularity could form. Since that original derivation, a variety of proofs have developed, mainly focused on the initial data formulation on maximal spacelike slices of spacetime. Most of these proofs are applicable only for classical fields, as the energy conditions required are violated in the context of quantum field theory. In this work we provide two generalizations of the Penrose inequality for spherically symmetric spacetimes: a proof of a classical Penrose inequality using initial data and an average energy condition, and a proof of a modified Penrose inequality for evaporating black holes with a connection to the weak cosmic censorship conjecture. The latter case could also be applicable to quantum fields as it uses a condition inspired by quantum energy inequalities. Finally, we provide physically motivated examples for both.

We prove Hawking’s singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type lemma and (2) the replacement of the usual focusing techniques for timelike geodesics—which in the absence of a classical ODE-theory for the initial value problem are no longer available—by a worldvolume estimate based on a segment-type inequality that allows one to control the volume of the set of points in a spacelike surface that possess long maximisers.