Homothetic maps of the space-time distance function and differentiability

Abstract

Let (M 1,g 1) and (M 2,g 1) be time-oriented space-times. Letd i(p,q) be the supremum of lengths of future directed causal curves inM i fromp toq. Ifq is not in the future ofp, thend i (p, q)=0. A distance homothetic mapf is a function fromM 1 ontoM 2 which is not assumed to be continuous, but which satisfiesd 2(f(p),f(q))=cd 1(p,q) for allp,q eM 1. IfM 1 is strongly causal, then the distance homothetic mapf is a diffeomorphism which mapsg 1 to a scalar multiple ofg 2. Thus for strongly causal space-times, distance homothetic maps are homothetic in the usual sense. WhenM 1 is not strongly causal, distance homothetic maps are not necessarily differentiable nor even continuous. An example is given of a space-time which has discontinuous maps which are one to one, onto, and distance preserving.