Timeline for Euler characteristic of Cauchy surface in Lorentz manifold
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2014 at 12:48 | comment | added | Willie Wong | @RobertBryant: no problem! I expected as such. I expanded a little bit on my response mostly for the benefit of lookers-on. | |
| Feb 17, 2014 at 12:09 | comment | added | Robert Bryant | @WillieWong: Oh, you are right; I read 'time-like' and thought 'null'. (I do know the difference, of course; I just had an 'input error'.) | |
| Feb 17, 2014 at 10:10 | comment | added | Willie Wong | @Robert: time-like is not the same as null. The reason why I (and most sources) state relative to the (open) "time-like" condition rather than the (closed) "null" condition is precisely that the given a closed time-like curve that intersects the surface once, you can always perturb it slightly to make it intersect (in the set-theoretic sense) the surface twice, giving a curve that demonstrably violates the "Cauchy" condition. | |
| Feb 14, 2014 at 16:02 | comment | added | Robert Bryant | @WillieWong: Well, I think you have to be careful about what you mean by 'intersect': For example, if you take the Lorentzian metric $dx\circ dy$ on the (compact) torus $\mathbb{R}^2/\mathbb{Z}^2$, then the space-like curve $x=y$ intersects each null curve exactly once in the set-theoretic sense, but, apparently, you don't want that to be a Cauchy surface since that would violate the `theorem' you quoted in your (now deleted) comment. | |
| Feb 14, 2014 at 14:47 | history | edited | Willie Wong | CC BY-SA 3.0 |
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| Feb 14, 2014 at 14:47 | comment | added | Willie Wong | @David: a Cauchy surface requires that "inextensible time-like curves intersect the surface exactly once". I objected (briefly) to Robert Bryant's comment because most inextensible time-like curves in his example intersects the surface infinitely often. | |
| Feb 14, 2014 at 13:33 | comment | added | David Hillman | @RobertBryant: what is a "true Cauchy surface"? I guess I've been thinking of "local Cauchy surface." For instance if one identifies the points {x,t}<-->{x,t+1} and {x,t}<-->{x+1,t} in 2d Minkowski space (I'm sure there's a better way to say this), you get a compact Lorentz manifold, no? in which case I'm thinking of the points {x,0} as forming a compact Cauchy surface. | |
| Feb 14, 2014 at 13:27 | history | edited | David Hillman | CC BY-SA 3.0 |
I meant compact to refer to Cauchy hypersurfaces, so now it says that
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| Feb 14, 2014 at 13:13 | comment | added | David Hillman | @WillieWong: thanks for improving my question! Yes, when I said 3d I meant 2 spatial, 1 temporal. | |
| Feb 14, 2014 at 11:34 | comment | added | Willie Wong | @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. | |
| Feb 14, 2014 at 11:32 | comment | added | Willie Wong | @Robert, I rewrote the question based on OP's comments to Misha's answer. I hope I didn't miss anything in the copying! | |
| Feb 14, 2014 at 11:31 | history | edited | Willie Wong | CC BY-SA 3.0 |
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| Feb 14, 2014 at 11:30 | comment | added | Robert Bryant | @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). | |
| Feb 13, 2014 at 9:46 | comment | added | Robert Bryant | 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 $-\mathrm{d}\theta^2 + g$. | |
| Feb 13, 2014 at 4:49 | answer | added | Misha | timeline score: 4 | |
| Feb 12, 2014 at 22:27 | history | asked | David Hillman | CC BY-SA 3.0 |