The cosmological time function

Let (M, g) be a time-oriented Lorentzian manifold and d the Lorentzian distance on M. The function tau(q) := SUPp<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) < infinity for all q and also tau --> 0 along every past inextendible causal curve. If the cosmological time function tau of a spacetime (M, g) is regular it has several pleasant consequences: (i) it forces (M, g) to be globally hyperbolic; (ii) 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); (iii) the function tau is a time function in the usual sense; in particular, (iv) tau is continuous, in fact, locally Lipschitz and the second derivatives of tau exist almost everywhere.