A Lorentzian analog for Hausdorff dimension and measure

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.

Functors in Lorentzian geometry -- three variations on a theme

“…McCann and Sämann, in [14] show MCS •L = SEP, i.e. on the respective images of CST, the maps F and MCS are inverse to each other.…”

“…McCann and Sämann [14], along the central and very fruitful idea to replace in the definition of Hausdorff dimension and measure the balls with causal diamonds, construct a natural measure on Lorentzian pre-length spaces (and thus a functor MCS : ALL → POM), via the following:…”

“…The original definition in[14] is a bit weaker inasfar it only requires ≤ to be a quasiorder, i.e. only requires reflexiveness and transitiveness but not antisymmetry in order to include non-causal spacetimes.…”

A review of Lorentzian synthetic theory of timelike Ricci curvature bounds

“…Note that while the former is purely metric, the latter needs to fix a reference volume measure in order to be formulated. Natural candidates are the canonical volume measures on Lorentzian (pre-)length spaces constructed in [72].…”

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

“…A related topic was then to find canonical measures in the Lorentzian setting akin to Hausdorff measures in the metric one. This has been achieved recently in [MS22], where also a synthetic dimension for Lorentzian pre-length spaces is given, the compatibility of the Lorentzian measures with the volume measure of continuous spacetimes is shown and its relation to [CM20] is studied.…”