Lorentzian coarea inequality

TL;DR

This paper introduces a new framework for Lorentzian geometry involving locally uniformly d-controlling maps and establishes a coarea inequality for Lorentzian Hausdorff measure, along with a covering lemma under a novel local assumption.

Contribution

It presents a new notion of controlling maps in Lorentzian spaces and proves a coarea inequality and a covering lemma under local causal enlargement conditions.

Findings

Established the coarea inequality for Lorentzian Hausdorff measure.

Introduced the concept of locally uniformly d-controlling maps.

Proved a covering lemma for Lorentzian pre-length spaces.

Abstract

In this article, we introduce the notion of locally uniformly d-controlling map between Lorentzian pre-length spaces which is preserving the diameters of causal diamonds, and through that we establish the coarea inequality for Lorentzian Hausdorff measure which is introduced by McCann and S\"{a}mann. Besides that we get a covering lemma for subsets in a Lorentzian pre-length space with a new local assumption named the local causal enlargement property, which enables us to enlarge causal diamonds.