A Lorentzian analog for Hausdorff dimension and measure

15.02. bis 15.02.2022, 12:15-13:45  –  Room Campus Golm, C9A03 Tübingen
Geometric Analysis, Differential Geometry and Relativity

Clemens Sämann

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that distinguishes between e.g. spacelike and null subspaces of Minkowski spacetime. The volume measure corresponding to its geometric dimension gives a natural reference measure on a synthetic or limiting spacetime, and allows us to define what it means for such a spacetime to be collapsed (in analogy with metric measure geometry and the theory of Riemannian Ricci limit spaces). As a crucial tool we introduce a doubling condition for causal diamonds and a notion of causal doubling measures. Moreover, applications to continuous spacetimes and connections to synthetic timelike curvature bounds are given.

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