In the Rindler spacetime I am used to the proper time of an extended accelerated object running at different rates along the object (i.e. slower at the rear where the acceleration is greatest). Are you just saying that proper time passes at different rates with radius? It sounds like more than that to be honest but I had to ask.Actually, forget the Euclidean part. There is no spacelike slice (at all) through the world tube that corresponds to intuitive notions of "surface of the disk". Specifically, no spacelike slice that is a local time slice for every particle of the disk surface. Mathematically, the stationary rotating congruence has no hypersuface orthogonal spatial slices at all.