On the existence of smooth Cauchy steep time functions

TL;DR

This paper provides a straightforward proof that all globally hyperbolic spacetimes possess a smooth Cauchy steep time function, facilitating their embedding into Minkowski spacetime and their product structure.

Contribution

It introduces a simple proof for the existence of smooth Cauchy steep time functions in globally hyperbolic spacetimes, leveraging recent differentiability results.

Findings

Every globally hyperbolic spacetime admits a smooth Cauchy steep time function

Such spacetimes can be isometrically embedded in Minkowski spacetime

They can be decomposed as a product space

Abstract

A simple proof is given that every globally hyperbolic spacetime admits a smooth Cauchy steep time function. This result is useful in order to show that globally hyperbolic spacetimes can be isometrically embedded in Minkowski spacetimes and that they spit as a product. The proof is based on a recent result on the differentiability of Geroch's volume functions.