The Cosmological Time Function

Details The Cosmological Time Function The Cosmological Time Function

1997-09-30

arxiv.org/abs/gr-qc/9709084v1

Class.Quant.Grav. 15 (1998) 309-322

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

19 pages, AEI preprint, latex2e with amsmath and amsthm

Scientific paper

10.1088/0264-9381/15/2/006

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.

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