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Apr
19
comment GR - curve (in)completeness & (in)extendibility
The negative term "inextendible" is preferred, I think; and, yes - a curve is inextendible (in a particular direction) if it has no endpoint that way, i.e. it is extendible if it does have an endpoint. I think the idea is that a curve with an endpoint could be joined to another curve having the same endpoint, but Wald's point that the endpoint need not lie on the curve also confuses me - I'm not sure it's a problem for such a joining (if that is what is meant) but I'm also not sure it isn't. This is why I wrote out all four possible combinations...
Apr
18
comment GR - curve (in)completeness & (in)extendibility
IF I have understood correctly, per the detail in case 4, a geodesic is complete if and only if its affine parameter spans the interval (−∞,∞). The definition of an affine parameter means that given an affine paramater $\gamma$, $\gamma' = \alpha\gamma + \beta$ (where $\alpha$ and $\beta$ are $\in$ reals) is also an affine parameter, so the specific initial choice of $\gamma$ is immaterial.
Apr
14
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Re excised open ball (corrections expected :) Excising an open ball gives intersecting inextendible curves endpoints, so they cease to be inextendible - doesn't affect the Cauchy surface because it's still met exactly once by every timelike curve that remains inextendible (I had thought it broke things, but it seems it just removes those curves from consideration). But, all Cauchy surfaces in GH are homeomorphic and I don't see that the surface with the excision can be homeomorphic to one without the excisions, so result can't be GH. If true, no doubt you can express it better.
Apr
12
awarded  Tumbleweed
Apr
11
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Clarification? If in such proofs it is said that e.g. compact $J^-(p+\hat t,M')\cap J^+(q-\hat t,M')$ implies each part is closed, it does mean that each part is absolutely closed - and not both closed and open - doesn't it? Thus my miswording "it would have sufficed to show either was 'open'" should have been "it would have sufficed to show either was not closed and not (closed and open)"...?
Apr
6
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Finally got it! Thx. For others: see Wikipedia
Apr
6
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
I've got it now. Despite your clear wording I'd lost track of set openness AND had overlooked the "causal shadows" of the excision. But I was also confused by the overall construction: compact $J^-(p+\hat t,M')\cap J^+(q-\hat t,M')$ implies closedness of both parts and it would have sufficed to show that either was open. I wish there was some notational device to mark sets as open/closed, it would be harder to lose track if there were. Thank you - working through that was very instructive.
Apr
5
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
I've worked through it now and I have some questions - SE says avoid extended discussions in comments... how should I proceed? (I've illustrated the regions and would like to share the images.)
Apr
5
asked GR - curve (in)completeness & (in)extendibility
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Even I'd worked that one out :-) Any other way? (And, ps, what are the hats on the t's?)
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Thank you greatly; have to go AFK now but will be reading the edit carefully. If you are feeling generous... can this mutiliated spacetime be made GH (again the original question about joins of open sets)? [Apparently I can only +1 your answer once]
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
the CLOSED ball erratum was me correcting my 1st comment (out of time to re-edit); not disputing it didn't need patching to be valid.
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Erratum: Minkowski - CLOSED ball $\rightarrow$ valid etc.
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
[...] That curves that just stop relates to the incomplete/extensible part; could you say which category do they fall into and why?
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
Thank you, and, yes, that makes sense. Minkowski - (open) ball $\rightarrow$ valid (not what I thought) but not GH (and it can't be patched up...?) which was my intuition. "Proving it is not GH is hard..." that is what I would really now like to do/see done :) Is it not GH because: (i) the curves that just end are naked singularities; (ii) because the surgery makes $J^{+}(p)\cap J^{-}(q)$ non-compact? (In which case is it because $J^{\pm}$ are no longer closed?) or (iii) because a CTC has been created (by Geroch; hasn't the topology changed?) Are (i) and (ii) the same thing here?
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
@slereah Closed ball; if I removed an open ball I could identify points on the boundary, couldn't I? (Hence the emphasis on the closed ball.)
Apr
3
comment Spacetime manifold surgery: is this result still a valid etc. spacetime?
@slereah By valid I meant that which came after "i.e.". Since I'm no expert in this I thought it best not to attempt to be definitive, though given that GH typically includes the absence of naked singularities and compactness of the causal diamond IIRC, I might have said GH in place of "valid" but I didn't want to incorrectly exclude things I was ignorant of. I said suspect $M'$ is singular without being definitive about geodesics and that I am unclear about incompleteness/inextensibility...
Apr
3
asked Spacetime manifold surgery: is this result still a valid etc. spacetime?
May
23
awarded  Yearling
May
22
awarded  Nice Answer