Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of [Hawking, King, and McCarthy](https://aip.scitation.org/doi/abs/10.1063/1.522874) (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 $M$?



In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure?
Trying to recover as much of the topology/geometry from the causal order as possible has been studied quit a bit since the early paper of Hawking et al that you cite. A quick summary of my understanding of the situation is that there are some purely order-theoretic topologies (like Alexander or Scott topologies) that reproduce the topology of the original Lorentzian manifold when the causal relation is sufficiently non-pathological. However, it is very difficult to find a set of conditions on partially ordered sets (posets) that single out precisely those posets that come from the causal order of smooth (or even topological) Lorentzian manifolds. AFAIK, that is an open problem . (Please correct me if I'm wrong!)


You can find a relatively recent collection of results in that direction here:



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> *Martin, Keye; Panangaden, Prakash*, [**A domain of spacetime intervals in general relativity**](http://dx.doi.org/10.1007/s00220-006-0066-5), Commun. Math. Phys. 267, No. 3, 563-586 (2006). [ZBL1188.83071](https://zbmath.org/?q=an:1188.83071) [arXiv:gr-qc/0407094](https://arxiv.org/abs/gr-qc/0407094)
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One can start walking up and down the citation tree from here to find more recent progress.


**Added:** If one restricts attention to generalizing just Thm 5 of Hawking-King-McCarthy to other dimensions, as requested by Tim, then this was done in Thm 1.2 here



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> *Peleska, Jan*, [**A characterization for isometries and conformal mappings of pseudo- Riemannian manifolds**](http://dx.doi.org/10.1007/BF02192656), Aequationes Math. 27, 20-31 (1984). [ZBL0539.53017](https://zbmath.org/?q=an:0539.53017).
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All dimension higher than 2 are covered, which agrees with Willie's counter example.


