The Cosmological Time Function

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.

19 pages, AEI preprint, latex2e with amsmath and amsthm

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