Characterizing intrinsic Lorentzian length spaces via $\tau$-midpoints

TL;DR

This paper extends the concept of midpoints from metric geometry to Lorentzian pre-length spaces, providing new characterizations of intrinsic properties using $ au$-midpoints and null distance.

Contribution

It introduces a novel characterization of Lorentzian length spaces through $ au$-midpoints, connecting metric properties with Lorentzian geometry.

Findings

Spaces with $ au$-midpoints are strictly intrinsic.

Spaces with approximate $ au$-midpoints are merely intrinsic.

The approach utilizes the null distance framework.

Abstract

In metric geometry, the question of whether a distance metric is given by the length of curves can be decided via the existence of midpoints with respect to the metric $d$. We adapt a similar characterization to the setting of Lorentzian pre-length spaces. In particular, we show that a given space is strictly intrinsic provided it has $\tau$-midpoints and merely intrinsic provided it has approximate $\tau$-midpoints. Our approach is based on the null distance of C. Sormani and C. Vega.