2
$\begingroup$

Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler characteristics.

If the above question is too general, I am specifically interested in the two subcases:

  1. What if we require the Cauchy hypersurface to be compact?
  2. And/or what if we require the ambient manifold to be flat?

Motivation

(Copied from comments below)

I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d (2 space and 1 time) I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search.

I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including ones with compact Cauchy hypersurfaces (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows.

Improve this question
$\endgroup$
11
  • 4
    $\begingroup$ Without the restriction of (local) flatness, you could take any compact Riemannian manifold $(M,g)$ and consider $S^1\times M$ with the Lorentzian metric $-\mathrm{d}\theta^2 + g$. $\endgroup$
    – Robert Bryant
    Commented Feb 13, 2014 at 9:46
  • $\begingroup$ @WillieWong: Strictly speaking, you are right, but then a 'compact $n$-dimensional Lorentzian manifold' (as the OP originally specified) would never have any true Cauchy surfaces, so the OP's question (which, as it turns out, was not correctly formulated to reflect his intent) would not make any sense. Thus, I chose to interpret 'Cauchy surface' in this case as 'local Cauchy surface', i.e., a space-like slice that meets all of the time-like geodesics (i.e., light rays). $\endgroup$
    – Robert Bryant
    Commented Feb 14, 2014 at 11:30
  • $\begingroup$ @Robert, I rewrote the question based on OP's comments to Misha's answer. I hope I didn't miss anything in the copying! $\endgroup$
    – Willie Wong
    Commented Feb 14, 2014 at 11:32
  • $\begingroup$ @David: can you specify whether the dimension you mentioned are the space-time dimension or the spatial dimension? Your last comment on Misha's answer seems to suggest that when you wrote 3d you are talking about 2 spatial and 1 temporal directions, but I am not 100% sure I understood you right. $\endgroup$
    – Willie Wong
    Commented Feb 14, 2014 at 11:34
  • $\begingroup$ @WillieWong: thanks for improving my question! Yes, when I said 3d I meant 2 spatial, 1 temporal. $\endgroup$
    – David Hillman
    Commented Feb 14, 2014 at 13:13

1 Answer 1

Reset to default
4
$\begingroup$

I read your question as the one about compact Cauchy surfaces in locally flat space times. Then the answer is negative: Take quotient of upper hyperboloid $H$ in $R^{2,1}$ (i.e. the hyperbolic plane) by a torsion free discrete cocompact subgroup $\Gamma$ in $SO(2,1)$. The Euler characteristic will be negative. Now, take the future light cone $C$ in $R^{2,1}$ and take the quotient $C/\Gamma$. This is your locally flat space-time, containing $H/\Gamma$ as a Cauchy hypersurface. This manifold is, of course, incomplete, but, if I remember correctly, Geoff Mess proved that you cannot have a compact Cauchy hypersurface in a complete locally flat Lorentzian manifold.

Edit: On the other hand, there are topological obstructions for existence of locally flat Lorenztian metrics. For instance, suppose that $M$ is a compact n-dimensional simply-connected manifold which does not immerse in $R^{n+1}$. For instance, $CP^2$ is an example of such manifold. Then $M\times R$ does not admit a locally flat Lorentzian metric, even incomplete one. (This follows because the developing map to $R^{n,1}$ of such a structure would yield an immersion to $R^{n,1}$.

Talking about tiles: My guess is that you are actually trying to construct complete locally flat Lorentzian metrics. It is known (proved by Geoff Mess) that such metrics do not exist on manifolds of the form $M\times R$, where $M$ is compact and hyperbolic. I will check, by now there is probably a reasonably good description of $M$'s for which such metric exists. On the other hand, if you take $M$ which is a noncompact surface, then such a metric does exist; these are so called Margulis space-times. One even has some nice description of their fundamental domains (tiles) due to Todd Drumm. The trick is that the tiles are not convex. (Google "crooked planes" to learn more about them.) I am not sure if $M$ will be a Cauchy hypersurface in this case, I would have to check.

See e.g. here for a somewhat dated survey.

Improve this answer
$\endgroup$
17
  • $\begingroup$ However, your locally flat ambient Lorentzian manifold $C/\Gamma$ is not compact, which the OP wanted (though, of course, such an example would not be of physical interest). $\endgroup$
    – Robert Bryant
    Commented Feb 13, 2014 at 9:47
  • $\begingroup$ @RobertBryant: Maybe I misinterpreted the question, but my understanding of Cauchy hypersurfaces is that physicists want to have a manifold of the form $M\times R$, foliated by Cauchy hypersurfaces $M\times t$ ("time machines" are not allowed). Hopefully, OP will clarify if he wanted a compact space-time of a compact Cauchy hypersurface, that was unclear from the question. $\endgroup$
    – Misha
    Commented Feb 13, 2014 at 12:05
  • $\begingroup$ I think your understanding of what physicists would want (and what is physically permissible) is correct, but the OP wrote 'compact $n$-dimensional Lorentzian spacetime', implying that the spacetime itself is to be compact. As you say, perhaps the OP did not intend this and will so inform us. $\endgroup$
    – Robert Bryant
    Commented Feb 13, 2014 at 12:14
  • $\begingroup$ Thanks for your replies...my DG is not up to speed so it will take me some time to digest. Here is where the question came from. I'm trying to find a certain kind of tiling system of Minkowski space, where there are finitely many polyhedral tile shapes (up to Poincare equivalence). There are local rules for assembling them into a Cauchy surface, and rules for evolving the surface. In 3d I haven't found an interesting example and am wondering if there is something I should know about Euler characteristic that would restrict my search. (to be continued...) $\endgroup$
    – David Hillman
    Commented Feb 13, 2014 at 20:40
  • $\begingroup$ I think in general with tiles and local rules for assembling them you can get arbitrary flat Lorentz manifolds including compact ones (definitely seems true in 2d) so I'm looking for any results, compact or not, that might suggest something about my situation. I have a feeling that the finiteness of the tiling system might make compact results relevant, but who knows. $\endgroup$
    – David Hillman
    Commented Feb 13, 2014 at 20:43

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .