A Lorentzian analog for Hausdorff dimension and measure

TL;DR

This paper introduces a Lorentzian analog of Hausdorff dimension and measure, providing a new way to analyze the geometric structure of Lorentzian spaces, with applications to synthetic spacetimes and curvature bounds.

Contribution

It defines a canonical volume measure and a geometric dimension for Lorentzian spaces, extending metric measure concepts to the Lorentzian setting with new tools like causal doubling measures.

Findings

Defined a Lorentzian volume measure and geometric dimension.

Introduced a doubling condition for causal diamonds.

Connected the framework to synthetic timelike curvature bounds.

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.