Singularity Theorems and Comparison Geometry

The singularity theorems of General Relativity, initiated by R. Penrose and S. W. Hawking, are cornerstones of the theory and have given rise to many important developments and applications. They predict that under certain physically realistic assumption a spacetime (i.e., a mathematical model of the universe) will develop singularities in the sense that there necessarily exist incomplete causal geodesics in it, i.e., either light rays or material particles do not exist for all times. On a local scale, a typical such singularity is a so-called Black Hole, a spacetime-region from which even light cannot escape. On a cosmological scale, the Big Bang or Big Crunch are notable examples. It has been considered a weakness of these results that they do in fact not make any general statement on the nature of these singularities (e.g., whether they lead to regions of unbounded curvature in spacetime). In principle they might simply predict a drop in the regularity of the mathematical model used to describe the physical situation. There has therefore for a long time been a strong interest in determining the lowest possible regularity in which the theorems hold. In the past few years, decisive advances have been made in the understanding of low regularity causality theory, which have allowed members of our research group to prove that both the Hawking and the Penrose singularity theorem remain valid in low regularity. A natural next step in this program is to also extend the most general singularity theorem, namely the so-called Hawking and Penrose theorem to this regularity. This is the first aim of the proposed project. The second main thread of this proposal is to develop new geometrical techniques to address these problems. In particular, we intend to use comparison geometry for low regularity semi-Riemannian manifolds, a mathematical tool that allows to compare general geometries with rather simple (highly symmetric) model geometries, e.g. with respect to curvature, areas or volumes of spacetime regions. In this way we hope to gain new qualitative and quantitative insight into the singularity theorems and to develop a new mathematical language for addressing this important problem on the intersection of general relativity and differential geometry. The core team of the proposed project will consist of James D.E. Grant, M. Kunzinger, R. Steinbauer, and J.A. Vickers, all of whom are experienced researchers who over last years have made substantial contributions to the field.

The singularity theorems of Roger Penrose (Nobel lauraeate, 2020) and Stephen Hawking are among the milestones of theoretical physics in the 20th century. In their most general form, they were formulated jointly by Hawking and Penrose in 1970. The resulting theorem predicts the existence of gravitational singularities in our universe under very general and physically reasonable conditions. The main goal of this project was to generalize this theorem to spacetimes of low regularity, a question originally raised by Hawking and Ellis in the 1970ies. This part of the project was successfully concluded by proving the result while only assuming reduced differentiability of the spacetime-metric, which basically shows that even when allowing the spacetime to be less regular than classically admitted, the formation of singularities is still unavoidable. Our proof required the development of new mathematical techniques extending Lorentzian geometry to this more general situation. In particular, we established new techniques in the field of comparison geometry, to find new ways of describing focusing effects (focal and conjugate points of geodesic congruences) of light rays in very strong gravitational fields, emanating from trapped surfaces. Motivated by the success of this first part of the project, we went on to develop a new approach to the notion of curvature in low regularity spacetimes in maximal generality. This was achieved by introducing the new notion of Lorentzian length spaces, which are mathematical spaces that are significantly more general than the differentiable manifolds that are used in classical general relativity and differential geometry to model our universe. In our new theory, just as in the metric precursor theory (the so-called Alexandrov spaces), curvature is measured by comparing geodesic triangles in the low regularity spacetime with triangles in modes spaces of constant curvature. The role of the metric is now taken over by the time-separation function, which in general relativity measures the path of maximal length between two events (as opposed to the metric context, where one searches for smallest distances). The idea behind this emphasis on the time separation function is that it measures the length of the path ideally taken by a light ray or an observer in spacetime. Another main branch of the project was the study of physically relevant concrete spacetimes of very low regularity, namely impulsive gravitational waves. Such spacetimes are theoretical models of short but violent bursts of gravitational radiation and are described by a metric of low regularity, hence are an ideal testing ground for the newly developed methods in this project. In addition, we also explored mathematical methods for studying observables in quantum field theories of low regularity.