The representation of spacetime through steep time functions

TL;DR

This paper explores how smooth steep time functions can reconstruct spacetime's causal structure, topology, and Lorentz-Finsler distance, providing new proofs for the distance formula in globally hyperbolic spacetimes.

Contribution

It introduces a novel proof of the Lorentz-Finsler distance formula using the product trick, enhancing understanding of spacetime geometry.

Findings

Steep time functions recover spacetime order and topology

The product trick converts metric statements into causal ones

A second proof of the distance formula in globally hyperbolic spacetimes

Abstract

In a recent work I showed that the family of smooth steep time functions can be used to recover the order, the topology and the (Lorentz-Finsler) distance of spacetime. In this work I present the main ideas entering the proof of the (smooth) distance formula, particularly the product trick which converts metric statements into causal ones. The paper ends with a second proof of the distance formula valid in globally hyperbolic Lorentzian spacetimes.