Null distance on a spacetime

الأصل · EN

Given a time function τ on a spacetime M, we define a `null distance function', dτ, built from and closely related to the causal structure of M. In basic models with timelike ∇ τ, we show that 1) dτ is a definite distance function, which induces the manifold topology, 2) the causal structure of M is completely encoded in dτ and τ. In general, dτ is a conformally invariant pseudometric, which may be indefinite. We give an `anti-Lipschitz' condition on τ, which ensures that dτ is definite, and show this condition to be satisfied whenever τ has gradient vectors ∇ τ almost everywhere, with ∇ τ locally `bounded away from the light cones'. As a consequence, we show that the cosmological time function of [1] 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'.