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By causal/conformal structure I mean the context of Malament's 1977 theorem. If I understand correctly this means that any two spacetimes which agree about all of the future-directed continuous timelike curves differ only by a conformal factor. Great, so the causal structure of metric is fixed by its conformal equivalence class. And we know that the Weyl tensor $C$ is preserved by Weyl transformations. But these conformal equivalence classes aren't simply labeled by their Weyl tensors, are they?

I know that determining whether or not a metric is conformally flat just amounts to checking whether or not its Weyl tensor vanishes. But outside of this case, I don't think this is true. Let $g$ be a generic metric with $C\neq0$. Let $g'$ be a metric with the same Weyl tensor $C'=C\neq 0$ but with but with Ricci tensor, $R=0$. (Am I right in thinking that $g'$ exists because of the algebraic independence of $C'$ and $R'$?) Now if conformal equivalence classes are simply labeled by their Weyl tensors then we would have that $g$ and $g'$ are conformally related. But I am pretty sure this isn't true in general. See, for instance, slides 5 and 9 of this presentation related to https://arxiv.org/abs/1304.7772.

So what more is needed in addition to $C$ to determine a metric's causal structure? Part of why I am interested in this is that from the Malament paper it follows that the causal structure of spacetime is actually a function of its topology and not its geometry (edit: see note below). So I am trying to determine which parts of the metric are actually fixed in advance by the spacetime's topological/causal structure.

Note: The topology that I have in mind here comes from Hawking, King, McCarthy (1976) whose abstract says: "A new topology is proposed for strongly causal space–times. Unlike the standard manifold topology (which merely characterizes continuity properties), the new topology determines the causal, differential, and conformal structures of space–time."

Edit: The presentation slides I mentioned are titled "How to recognize a conformally Einstein metric?" by Maciej Dunajski at the University of Cambridge. His paper with Paul Tod is called "Self-Dual Conformal Gravity" with reference number arXiv:1304.7772.

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  • $\begingroup$ "Part of why I am interested in this is that from the Malament paper it follows that the causal structure of spacetime is actually a function of its topology and not its geometry." I disagree. Malament's theorem seems to mean that the causal structure determines the topology, not the other way around. Pick Kruskal coordinates in Schwarzschild spacetime. Flip the signs of $dT^2$ and $d\theta^2$. The new spacetime has closed timeline curves (and hence a different causal structure), but the same topology $\mathbb{R}^2 \times \mathbb{S}^2$ $\endgroup$
    – Níckolas Alves
    Commented May 11 at 17:40
  • $\begingroup$ I agree that in the Malament paper, the causal structure determines topology which, in turn, fixes the metric up to a conformal factor. But you can easily turn this on its head. Suppose that I give you a spacetime manifold with some topology and no metric (yet). Now there is only a limited class of metrics which you could put on that spacetime. Namely, they are restricted to be in some specific conformal class and relatedly to have some specific causal structure. $\endgroup$
    – Daniel Grimmer
    Commented May 14 at 20:57
  • $\begingroup$ They aren't. I gave you an example: change the timelike direction in Schwarzschild spacetime and you have a completely different causal structure. Another example: pick the cylinder manifold $\mathbb{R} \times \mathbb{S}^1$. Consider the metrics $ds^2 = - d\rho^2 + \rho^2 d\phi^2$ and $ds^2 = d\rho^2 - \rho^2 d\phi^2$ (negative direction is time in both cases). They lead to completely different causal structures: one is globally hyperbolic, the other has closed timelike curves $\endgroup$
    – Níckolas Alves
    Commented May 14 at 21:09
  • $\begingroup$ Maybe we are talking past each other. The topology which I have in mind is not the standard manifold topology which you would normally put on $\mathbb{R}\times\mathbb{S}^1$. The relevant topology instead comes from Hawking, King, McCarthy (1976) whose abstract says: "A new topology is proposed for strongly causal space–times. Unlike the standard manifold topology (which merely characterizes continuity properties), the new topology determines the causal, differential, and conformal structures of space–time." Here is a link to the paper: authors.library.caltech.edu/records/6apqj-ep466 $\endgroup$
    – Daniel Grimmer
    Commented May 15 at 10:28
  • $\begingroup$ Oh, fair enough. Now that makes much more sense $\endgroup$
    – Níckolas Alves
    Commented May 15 at 13:56

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