Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Volume comparison by timelike Lipschitz maps

Hikaru Kubota

TL;DR

This work develops a Lorentzian analog of Hausdorff measure by introducing a modified timelike measure $\mathcal{W}^{N}$ alongside McCann–S"a"mann's $\mathcal{V}^{N}$, and proves volume comparison inequalities for timelike Lipschitz maps with respect to both measures. It establishes that $\mathcal{W}^{N}$ and $\mathcal{V}^{N}$ coincide in several natural geometries, notably on globally hyperbolic smooth spacetimes via null distance, on compact continuous spacetimes with cylindrical neighborhoods, and on Lorentzian warped products. The paper also provides explicit constructions of timelike Lipschitz and causality-preserving maps, clarifying how causal structure interacts with volume control in non-smooth Lorentzian spaces. Collectively, these results lay groundwork for robust volume- and convergence-type analyses in low-regularity Lorentzian geometry and offer pathways to charting and dimension theory in Lorentzian settings.

Abstract

In this article, we introduce a modification of the timelike Hausdorff measure VN defined by McCann and Samann on Lorentzian pre-length spaces. We write the modification of VN as WN. We establish volume comparison inequalities by causality preserving and timelike Lipschitz maps for VN and WN, and discuss basic properties of both VN and WN. Moreover, we show the coincidence of WN and VN on smooth spacetimes and some Lorentzian pre-length spaces, and construct some examples of timelike Lipschitz maps and causality preserving maps.

Volume comparison by timelike Lipschitz maps

TL;DR

This work develops a Lorentzian analog of Hausdorff measure by introducing a modified timelike measure alongside McCann–S"a"mann's , and proves volume comparison inequalities for timelike Lipschitz maps with respect to both measures. It establishes that and coincide in several natural geometries, notably on globally hyperbolic smooth spacetimes via null distance, on compact continuous spacetimes with cylindrical neighborhoods, and on Lorentzian warped products. The paper also provides explicit constructions of timelike Lipschitz and causality-preserving maps, clarifying how causal structure interacts with volume control in non-smooth Lorentzian spaces. Collectively, these results lay groundwork for robust volume- and convergence-type analyses in low-regularity Lorentzian geometry and offer pathways to charting and dimension theory in Lorentzian settings.

Abstract

In this article, we introduce a modification of the timelike Hausdorff measure VN defined by McCann and Samann on Lorentzian pre-length spaces. We write the modification of VN as WN. We establish volume comparison inequalities by causality preserving and timelike Lipschitz maps for VN and WN, and discuss basic properties of both VN and WN. Moreover, we show the coincidence of WN and VN on smooth spacetimes and some Lorentzian pre-length spaces, and construct some examples of timelike Lipschitz maps and causality preserving maps.
Paper Structure (15 sections, 28 theorems, 56 equations)

This paper contains 15 sections, 28 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Lorentzian pre-length spaces and timelike Hausdorff measure
  3. Lorentzian pre-length spaces
  4. McCann and Sämann's timelike Hausdorff measure
  5. Volume comparison inequality
  6. Volume comparison inequality for $\mathcal{V}^{N}$
  7. A modification of $\mathcal{V}^{N}$
  8. Volume comparison inequality for $\mathcal{W}^{N}$
  9. Coincidence of $\mathcal{W}^{N}$ and $\mathcal{V}^{N}$
  10. Null distance and $\mathcal{W}^{N}=\mathcal{V}^{N}$ on Lorentzian pre-length spaces
  11. Null distance and $\mathrm{vol}^{g}=\mathcal{W}^{N}$ on globally hyperbolic smooth spacetimes
  12. $\mathcal{W}^{N}=\mathcal{V}^{N}$ on continuous spacetimes
  13. $\mathcal{W}^{N}=\mathcal{V}^{N}$ on Lorentzian warped products
  14. Examples of timelike Lipschitz maps and causality preserving maps
  15. Outlook

Key Result

Theorem 1.3

Let $(X,d_{X},\le_{X},\ll_{X},\tau_{X})$ and $(Y, d_{Y}, \le_{Y}, \ll_{Y}, \tau_{Y})$ be Lorentzian pre-length spaces. Take $A \subseteq X$. Let $f:X \to Y$ satisfy the following. Then, $\mathcal{V}^{N}(f(A)) \le \lambda^{N} \mathcal{V}^{N}(A)$ holds.

Theorems & Definitions (63)

  • Definition 1.1: Construction of $\mathcal{W}^{N}$
  • Definition 1.2: Timelike Lipschitz maps
  • Theorem 1.3: Volume comparison inequality for $\mathcal{V}^{N}$
  • Theorem 1.4: Volume comparison inequality for $\mathcal{W}^{N}$
  • Definition 2.1: Lorentzian pre-length space
  • Example 2.2: Smooth spacetimes
  • Example 2.3: Lorentzian warped products
  • Definition 2.4: Future and past
  • Example 2.5: $C^0$ as a Lorentzian pre-length space
  • Definition 2.6: Causal and timelike curves
  • ...and 53 more