Null distance on a spacetime

Given a time function tau on a spacetime M, we define a null distance function, <(d)(tau)over cap>, built from and closely related to the causal structure of M. In basic models with timelike. tau we show that (1) (d(tau)) over cap is a definite distance function, which induces the manifold topology, (2) the causal structure of M is completely encoded in (d(tau)) over cap and tau. In general, (d(tau)) over cap is a conformally invariant pseudometric, which may be indefinite. We give an 'anti-Lipschitz' condition on tau, which ensures that (d(tau)) over cap is definite, and show this condition to be satisfied whenever tau has gradient vectors del tau almost everywhere, with del tau locally 'bounded away from the light cones'. As a consequence, we show that the cosmological time function of Andersson et al (1998 Class. Quantum Grav. 15 309-22) is anti-Lipschitz when 'regular', and hence induces a definite null distance function. This provides what may be interpreted as a canonical metric space structure on spacetimes which emanate from a common initial singularity, e.g. a 'big bang'.