TITLE: In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure?
QUESTION [7 upvotes]: Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of Hawking, King, and McCarthy (see Thm 5) says that $M$ is determined up to conformal isomorphism by its class of null geodesic curves (where the parameterization of the null geodesic is forgotten) and thence by the causal relation $J^+$ on $M$.
Question: Does this theorem also hold in dimensions other than 4?
My hunch would be that it continues to hold in higher dimensions, but in lower dimensions I'm not so sure. At any rate, I don't see how to adapt Hawking, King, and McCarthy's argument (which involves choosing coordinates using a certain configuration of null geodesics) to dimension 2.
EDIT: As came up in the comments, I should clarify that a priori I don't want to assume anything about the smooth structure on $M$. In fact, I would prefer not to assume anything about the topology either -- just take $M$ as a set of points, equipped with the relation $J^+$. From this data, when can one recover the topological, smooth, and conformal structures on $M$?

REPLY [3 votes]: In 2 dimensions the question posted in this comment has a negative answer.
Consider the standard Minkowski space with double null coordinates $(u,v)$ in which the metric is $ds^2 = - du~dv$.
Any strictly increasing bijection $\phi: \mathbb{R}\to \mathbb{R}$ induces a mapping $(u,v) \mapsto (\phi(u),v)$ that is an causal isomorphism.
Choosing $\phi$ to be continuous and not differentiable you get an example of an $\mathscr{M}$-homeomorphism that preserves null geodesics and is not a diffeomorphism (in the terminology of Hawking-King-McCarthy; compare to Theorem 5).