Lorentzian length spaces

The project, led by Clemens Sämann, has the goal to develop a completely new approach to General Relativity, Einsteins theory of gravity. In fact, the aim is to establish a geometry that is so robust that it can appropriately describe extreme situations, like black holes. Here it is helpful to transfer the philosophy of metric geometry to General Relativity. This has not been considered until a few years ago and only our research group took this conceptual leap. Since then it came apparent that this is a very fruitful approach and many different research directions have been initiated. In metric geometry the central notion is distance. For example, one can find out if one is in a curved space by comparing triangles to triangles in model spaces. However, in General Relativity, there is no natural notion of distance. Instead, there is a the so-called time-distance-function, which incorporates both space and time. This time-distance-function does not behave like a distance in usual three-dimensional space. For example, detours are shorter, which is illustrated by the famous twin paradox of Special Relativity. Since the time-distance-function is not a distance in the sense of metric geometry, we developed a kind of metric geometry for General Relativity from scratch, which has the time-distance-function as central object. For instance, it enables one to detect curvature by triangle comparison (as in the case of distances), where now the model spaces only come equipped with a time-distance-function themselves. The goal of the project is to develop this new kind of geometry further to obtain new far-reaching results about General Relativity. In particular, the task is to introduce a notion of convergence and thus study limiting procedures in General Relativity. Here it is of enormous importance that the objects obtained in such limiting procedures can be non-classical. This and further advancements of the theory have applications in central problems of contemporary mathematical physics, like the cosmic censorship hypothesis of Roger Penrose.