Causal Isomorphism as Homeomorphism, or Diffeomorphism or a Conformal Isometry? Drunken Risibility.

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes.

We say that a bijection φ : M → M′ between their underlying point sets is a ≪-causal isomorphism if, ∀ p and q in M,

p ≪ q ⇐⇒ φ(p) ≪ φ(q).

Then we can ask the following: Does a ≪-causal isomorphism have to be a homeomorphism? A diffeomorphism? A conformal isometry? (We know in advance that a causal isomorphism need not be a (full) isometry because conformally equivalent metrics gab and Ω2gab on a manifold M determine the same relation ≪. The best one can ask for is that it be a conformal isometry – i.e. that it be a diffeomorphism that preserves the metric up to a conformal factor.) Without further restrictions on (M, gab) and (M′, g′ab), the answer is certainly “no” to all three questions. Unless the “causal structure” of a spacetime (i.e., the structure determined by ≪) is reasonably well behaved, it provides no useful information at all. For example, let us say that a spacetime is causally degenerate if p ≪ q for all points p and q. Any bijection between two causally degenerate spacetimes qualifies, trivially, as a ≪-causal isomorphism. But we can certainly find causally degenerate spacetimes whose underlying manifolds have different topologies. But a suitably “rolled-up” version of Minkowski spacetime is also causally degenerate, and the latter has the manifold structure S1 × R3.

There is a hierarchy of “causality conditions” that is relevant here. Hawking and Ellis impose, with varying degrees of stringency, the requirement that there exist no closed, or “almost closed,” timelike curves. Here are three.

Chronology: There do not exist smooth closed timelike curves. (Equivalently, for all p, it is not the case that p ≪ p.)

Future (respectively, past) distinguishablity: ∀ points p, and all sufficiently small open sets O containing p, no smooth future-directed (respectively, past-directed) timelike curve that starts at p, and leaves O, ever returns to O.

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Strong causality: For all points p, and all sufficiently small open sets O containing p, no smooth future-directed timelike curve that starts in O, and leaves O, ever returns to O.

It is clear that strong causality implies both future distinguishability and past distinguishability, and that each of the distinguishability conditions (alone) implies chronology.

The names “future distinguishability” and “past distinguishability” are easily explained. For any p, let I+(p) be the set {q: p ≪ q} and let I(p) be the set {q : q ≪ p}. It turns out that future distinguishability is equivalent to the requirement that, ∀ p and q,

I+(p) = I+(q) =⇒ p = q.

And the counterpart requirement with I+ replaced by I is equivalent to past distinguishability.

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