The cosmological time function

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