Lorentzian Metric Geometry and Optimal Transport

In recent years, a new approach to the mathematical language underlying Einsteins theory of general relativity has emerged, strongly influenced by foundational works of our research group. The basic idea here is to give a central role to the function that assigns to any two causally related events in spacetime the maximal proper time an observer can experience when going from one event to the other. This function is called the time separation function, and as can be seen from the famous twin paradox, it takes into account that detours in spacetime result in shorter paths (in sharp contrast to the situation in standard Euclidean geometry). Considering very general spaces equipped with such a time separation function, the so-called Lorentzian length spaces, allows one to mimic many constructions from General Relativity in a much more general context. In particular, curvature can be measured by comparing geodesic triangles with triangles in model spaces of constant (classical) curvature. The project has three main goals: First, to further develop the foundations of the mathematical theory of Lorentzian length spaces by establishing new results on measuring curvature in this setting. Second, we want to deepen the connection to the theory of Optimal Transport, a mathematical field that is concerned with finding optimal solutions to transport problems in a very general sense. Combined with the theory of Lorentzian length spaces it becomes possible to study changes of volumes in spacetime to infer information about the curvature (and thereby, according to Einsteins theory, also about the energy-content) of the spacetime. Finally, we want to prove so- called rigidity results, stating that under certain assumptions on the curvature of a spacetime, in combination with knowledge about the long-term existence of geodesic paths, one can conclude that the spacetime is of a particularly simple structure, namely that of a product.