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.

arXiv comment45 pages, 4 figures; v2: minor improvements

References in corpus (31)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• The causal set approach to quantum gravity
• The causal hierarchy of spacetimes
• Causality theory for closed cone structures with applications
• On Lorentzian causality with continuous metrics
• Convex neighborhoods for Lipschitz connections and sprays
• The Geometry of Small Causal Diamonds
• An optimal transport formulation of the Einstein equations of general relativity
• Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions
• Global hyperbolicity for spacetimes with continuous metrics
• Timelike completeness as an obstruction to $C^0$-extensions
• Null distance on a spacetime
• Hawking's singularity theorem for $C^{1,1}$-metrics
• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• Singularity theorems for $C^1$-Lorentzian metrics
• The Penrose singularity theorem in regularity $C^{1,1}$
• The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
• Some Remarks on the $C^0$-(in)extendibility of Spacetimes
• Inextendibility of spacetimes and Lorentzian length spaces
• Null distance and convergence of Lorentzian length spaces
• A regularisation approach to causality theory for $C^{1,1}$-Lorentzian metrics
• Maximizers in Lipschitz spacetimes are either timelike or null
• The future is not always open
• On the causal hierarchy of Lorentzian length spaces
• Properties of the Null Distance and Spacetime Convergence
• Causal Fermion Systems: A Primer for Lorentzian Geometers
• On geodesics in low regularity
• Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
• Some regularity results for Lorentz-Finsler spaces
• Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
• On holonomy singularities in general relativity and the $C_{\mathrm{loc}}^{0,1}$-inextendibility of spacetimes

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• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• The Null distance encodes causality
• Gluing constructions for Lorentzian length spaces
• Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes
• On the asymptotic assumptions for Milne-like spacetimes
• Gluing of Lorentzian length spaces and the causal ladder
• A Toponogov globalisation result for Lorentzian length spaces