A new Geometry for Einstein’s Theory of Relativity & Beyond

General Relativity, Einsteins famous theory of space, time and gravity, has one central message: gravity is the curvature of the universe or spacetime to be precise. The mathematical language in which we usually speak of spacetime curvature is Lorentzian Differential Geometry. It is the somewhat strangely behaved sister-theory of our everyday Euclidean (or Riemannian) Geometry: Lorentzian detours (measured in spacetime distance) are shorter rather than longer as is the case in the Riemannian setting using the usual notion of a distance. One essential drawback of Differential Geometry (Riemannian and Lorentzian alike) is that its central objects must be smooth: One may only speak of the curvature of very nice geometries without corners, edges or spikes. However, physics builds more often than not on rough, non-smooth models, thus providing a strong motivation for a non-smooth geometry. Luckily during the past decades, a powerful formalism that provides a very robust notion of curvature for the non-smooth Riemannian setting has been developed. Based on the mathematical theories of Metric Geometry and Optimal Transport, it has revolutionised Riemannian geometry. In this so-called synthetic setting the prime object is the distance function and curvature is encoded in the convexity properties of an entropy functional. In 2018 our research group formulated the foundations of a synthetic Lorentzian Geometry built on spacetime distance as its central concept. We have thereby built a bridge between the robust curvature framework of Metric Geometry/Optimal Transport and Lorentzian Geometry. Our vision is to cross this bridge and develop a new geometry to tackle some long-standing open problems in fundamental physics like the nature of spacetime singularities in General Relativity, and beyond it by providing a unifying language for approaches to Quantum Gravity that are fundamentally discrete.