Volume comparison by timelike Lipschitz maps

TL;DR

This paper introduces a modified Lorentzian measure and establishes volume comparison inequalities using causality-preserving and timelike Lipschitz maps, with applications to smooth spacetimes and Lorentzian pre-length spaces.

Contribution

It defines a new measure WN, compares it with VN, and proves volume inequalities and properties for timelike Lipschitz maps in Lorentzian geometry.

Findings

WN coincides with VN on smooth spacetimes

Established volume comparison inequalities for causality-preserving maps

Constructed examples of timelike Lipschitz maps

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.