Timeline for Lens-shaped vs globally hyperbolic
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 1, 2011 at 17:04 | comment | added | Igor Khavkine | FYI, if it hasn't reached you yet, I've sent an email to your Cambridge address. | |
| Sep 30, 2011 at 9:04 | comment | added | Willie Wong | Ah yes, I actually started to write something along this direction last year, but for some reason I stopped before the proof that causal relations form a preorder. I don't remember whether there is a technical annoyance there. But I think once you get past that then Geroch's proof can be taken almost word-for-word. | |
| Sep 30, 2011 at 8:46 | comment | added | Willie Wong | @Igor: Would you like to take this to e-mail? It has been a while since I looked at Geroch's splitting theorem, and I should look at it again before I say anything more. The one worry I had in mind is that a generic causal structure is not going to be metric-compatible (if one is lucky, it would be a higher order metric like something that is Finsler; if one is not lucky, the "light cone" may have corners). So it really depends on how much of the Lorentzian structure is needed in the GR case. | |
| Sep 29, 2011 at 22:58 | comment | added | Igor Khavkine | The converse direction, or my part (b), does seem more subtle. It requires the construction of a Cauchy foliation for a globally hyperbolic spacetime. It appears to me that this is precisely the kind of result achieved by Geroch's splitting theorem. Though it's formulated for Lorentzian spacetimes, from my limited understanding of it, I'd say that it might be extended to this more general kind of causal structure. What do you think of this. For every $p\in D^+(S)$, $I^-(p)\cap D(S)$ is globally hyperbolic and thus Cauchy-foliable, with each Cauchy leaf a compact deformation of $I^-(p)\cap S$. | |
| Sep 29, 2011 at 22:50 | comment | added | Igor Khavkine | Willie, thanks a lot for writing this out in detail. Your argument is very similar to the rough one I had in mind. So the argument in this direction, my part (a), is fairly direct. BTW, I like your notion of causal structure, as it's essentially the same as the one I settled on. Unfortunately, it is presented this way almost nowhere in the literature. For the record, when I was referring to timelike directions, I specifically meant $C=(C')^*$. | |
| Sep 29, 2011 at 9:50 | history | edited | Willie Wong | CC BY-SA 3.0 |
added 680 characters in body
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| Sep 29, 2011 at 9:34 | history | answered | Willie Wong | CC BY-SA 3.0 |