Non-smooth spacetime geometry

General Relativity (GR), Einsteins theory of space, time and matter describes gravity via spacetime curvature: Every form of mass or energy curves the surrounding spacetime and this curvature in turn governs the motion of particles through spacetime. GR continues to be a spectacular success also in the 21st century. Indeed, around the time of its centenary in 2015, one of its mayor predictions, the existence of gravitational waves, has been directly confirmed. On the other hand, GR, over the decades, has also exerted a strong influence on various fields of mathematics with many cross-fertilizations leading to milestone results as e.g. the singularity theorems of Penrose and Hawking. These results predict, under physically reasonable conditions, the occurrence of spacetime singularities such as black holes. One of the problems on this interface between physics and mathematics concerns the very language in which GR is formulated: Lorentzian geometry (LG) is the mathematical theory that describes curved geometries, which are the stage for GR. Its central object is the metric, which describes how lengths and angles are measured at different points in spacetime. Traditionally, LG has been formulated for smooth metrics only, that is, for metrics that vary continuously when one travels from point to point in spacetime. On the other hand, physically realistic models of e.g. stars, demand that the mass density jumps at their surface. This however, via the fundamental principles of GR, forces the metric to vary abruptly as well and hence to be non-smooth. This fundamental issue has of course been well-known for a long time, but has only rarely been addressed in the literature. This has dramatically changed roughly a decade ago when researchers in Mathematical Relativity started to systematically investigate non-smooth metrics. Since then a growing number of woks has appeared and turned the study of non-smooth metrics into a thriving line of research, with the project team playing a key role. In a previous project we have, on the one hand, proved the classical singularity theorems of Penrose and Hawking for (certain) non-smooth metrics. On the other hand, we have put forward a new approach to LG which is based on methods of synthetic geometry, where e.g. curvature is measured by how much the sum of angles in triangles deviates from the flat case (180 degrees). In this new project we aim at enhancing and generalizing the methods developed previously proving advanced singularity theorems, especially where the energy might become negative locally. The latter provides an interface to the (yet unfinished) quantum theory of gravity where energies are expected to fluctuate and to become negative. Also we want to turn the new synthetic approach to LG, the so-called Lorentzian length spaces into a mature theory. In short, we will develop a synthesis of analytic and synthetic methods to significantly push the understanding of non-smooth spacetime geometries.

General Relativity (GR), Einstein's theory of space, time, and matter, describes gravity through curvature: Every form of mass or energy curves the surrounding spacetime, and this curvature in turn determines how particles move in spacetime. GR has a long success story that continues into the 21st century. Right on time for its hundredth anniversary, gravitational waves, one of its most spectacular predictions, were directly detected in 2015. Furthermore, GR has exerted strong influences on many mathematical subfields over the years, and through mutual developments, important results such as the singularity theorems of Penrose and Hawking have emerged. These state that under physically realistic conditions, such as those occurring during the stellar collapse or in an expanding universe, spacetime singularities necessarily arise. One problem at this interface of physics and mathematics is the language in which the GR is formulated: Lorentzian geometry (LG) is the mathematical theory for describing curved spaces. Its central object, the metric, encodes how lengths and angles are measured at a specific point in spacetime. Traditionally, LG is formulated for smooth metrics, i.e., for those metrics that only change continuously when moving from point to point. On the other hand, physically realistic models, e.g., of stars, require that the mass density changes abruptly at their surface. The fundamental equations of the GR then lead to the metric also changing abruptly, thus requiring it to be non-smooth. While this fundamental problem has long been known, it was rarely addressed in the literature. This changed about 15 years ago when researchers in Mathematical Relativity began to systematically investigate non-smooth metrics. Since then, a number of works has appeared, transforming the study of non-smooth metrics into a viable research direction. Within the framework of the current research project, we have (1) extended the singularity theorems of Penrose and Hawking to non-smooth spacetimes, significantly generalizing earlier results of our research group. Only this extension ensures that the theorems really do describe physically realistic scenarios. In this strand of the project, we have significantly advanced the existing mathematical techniques for the analytical description of non-smooth spacetimes. However, as significant hurdles have become recognizable on the horizon of this development, we have (2) further developed the synthetic approach to LG that was developed within our research group around 2018. In particular, we have expanded the initial approaches into a broad foundation and performed important groundwork. It has turned out that this approach has generated great interest in the community, as it opens the door to a variety of new applications and also provides an interface to quantum theories of gravity. Overall, we have significantly expanded the understanding of non-smooth spacetime geometries with a mixture of analytical and synthetic methods.