The Cosmological Time Function

TL;DR

The paper studies the properties of the cosmological time function in Lorentzian manifolds, showing that its regularity implies strong global hyperbolicity and geometric features of spacetime.

Contribution

It establishes that regular cosmological time functions ensure global hyperbolicity and the existence of maximizing geodesics, with implications for spacetime structure.

Findings

Regularity of $ au$ implies global hyperbolicity.

Every point can be connected to the singularity by a maximizing geodesic.

$ au$ is a continuous, locally Lipschitz time function with almost everywhere second derivatives.

Abstract

Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$ and also $\tau \to 0$ along every past inextendible causal curve. If the cosmological time function $\tau$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $\tau$ is a time function in the usual sense, in particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second derivatives of $\tau$ exist almost everywhere.