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arXiv · 2110.04386

A Lorentzian analog for Hausdorff dimension and measure

Robert J. McCann·Clemens Sämann
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Abstract

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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BibTeXRIS

Robert J. McCann, Clemens Sämann. 2021-10-08. A Lorentzian analog for Hausdorff dimension and measure. https://doi.org/10.2140/paa.2022.4.367

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