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A Lorentzian analog for Hausdorff dimension and measure

Robert J. McCann and Clemens Sämann

Vol. 4 (2022), No. 2, 367–400
DOI: 10.2140/paa.2022.4.367
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.

Keywords
metric geometry, Lorentz geometry, Lorentzian length spaces, Hausdorff dimension, synthetic curvature bounds , continuous spacetimes, doubling measures
Mathematical Subject Classification
Primary: 28A75, 51K10, 53C23, 53C50, 53B30
Secondary: 53C80, 83C99
Milestones
Received: 2 November 2021
Revised: 12 January 2022
Accepted: 21 February 2022
Published: 16 October 2022
Authors
Robert J. McCann
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Clemens Sämann
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Faculty of Mathematics
University of Vienna
Vienna
Austria