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The topology of spacetimes

by kevinferreira
Tags: spacetimes, topology
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micromass
#37
Jan3-13, 12:04 AM
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Another good reference is "Fundamental of differential geometry" by Serge Lang. He covers pseudo-Riemannian metrics on page 175. It's a fun book to read, so I recommend it.
zonde
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Jan3-13, 12:11 AM
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Quote Quote by micromass View Post
Let (X,d) be a pseudometric space. A set V is a neighborhood of [itex]x\in X[/itex] if there exists an [itex]\varepsilon>0[/itex] such that [itex]B(x,\varepsilon)\subseteq V[/itex].
What is [itex]\varepsilon[/itex] - a point or a set or an open set? And B()?

According to wikipedia http://en.wikipedia.org/wiki/Neighbo...mathematics%29 neighborhood should contain an open set containing the point. Given spacetime properties neighborhood of any event in spcetime should include it's lightcones. But for any two distinct points there will be some place where their lightcones (future or past or future with past) will intersect. So they can't have disjoint neighbourhoods which is required to say they belong to Hausdorff space.
WannabeNewton
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Jan3-13, 12:20 AM
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Quote Quote by zonde View Post
What is [itex]\varepsilon[/itex] - a point or a set or an open set? And B()?

According to wikipedia http://en.wikipedia.org/wiki/Neighbo...mathematics%29 neighborhood should contain an open set containing the point. Given spacetime properties neighborhood of any event in spcetime should include it's lightcones. But for any two distinct points there will be some place where their lightcones (future or past or future with past) will intersect. So they can't have disjoint neighbourhoods which is required to say they belong to Hausdorff space.
A good book on topology would probably clear up much of the confusion here. Hausdorff property states there exists a pair of neighborhoods for two distinct points that are themselves disjoint; a point on a manifold will have multiple neighborhoods. As for your comment on the causal structure of space - time, please take a look at chapter 8 of Wald which should clear up confusion or something else if anyone else has another reference. In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.
atyy
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Jan3-13, 12:51 AM
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All the usual spacetimes are of course Hausdorff. But just for interest, Hawking and Ellis mention one example of a non-Hausdorff spacetime, and mention a paper by Hajicek.
TrickyDicky
#41
Jan3-13, 03:13 AM
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Quote Quote by micromass View Post
A pseudo-Riemannian metric is a function

[tex]g:T_pM\times T_pM\rightarrow \mathbb{R}[/tex]

for each p.

A pseudometric is a function

[tex]d:M\times M\rightarrow \mathbb{R}[/tex]

So how can they be the same thing??
This is a key distinction IMO. The tangent space at a point and the manifold itself are two very different objects, and this is manifested even more clearly when the manifold is curved.
One shouldn't be able to draw conclusions about the global spacetime features from the purely local effect of the pseudoriemannian metric at a point, more so when the distance metric function that acts on the manifold doesn't coincide with the one that would be derived from the pseudoriemannian metric tensor, due to the smooth structure of the manifold.
When I mention the global structure of the manifold I refer to things like its maximal extended form, its singularities or its Killing vector fields nature(timelike, spacelike,lightlike).
zonde
#42
Jan3-13, 04:49 AM
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If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event. But then we should relay on some concept of nearness that is positive-definite and rather unrelated to spacetime distances.

It seems like a kind of double standard.
micromass
#43
Jan3-13, 05:09 AM
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Quote Quote by zonde View Post
If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event. But then we should relay on some concept of nearness that is positive-definite and rather unrelated to spacetime distances.

It seems like a kind of double standard.
Please read this again:

Quote Quote by WannabeNewton View Post
In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.
TrickyDicky
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Jan3-13, 05:14 AM
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Quote Quote by micromass View Post
Please read this again:
In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.
Exactly, but a complete light cone structure is usually attributed in GR not only to the point p and its neighbourhood, but to the whole manifold. This is the double standard IMO.
WannabeNewton
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Jan3-13, 05:15 AM
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Quote Quote by TrickyDicky View Post
Exactly, but a complete light cone structure is usually attributed in GR not only to the point p and its neighbourhood (TpM), but to the manifold. This is the double standard IMO.
Tp(M) is not a neighborhood it is the tangent space to M at p.
zonde
#46
Jan3-13, 05:16 AM
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Quote Quote by micromass View Post
Please read this again:

Quote Quote by WannabeNewton View Post
In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.
Done.

So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positive-definite? Or what?
WannabeNewton
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Jan3-13, 05:18 AM
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Quote Quote by zonde View Post
Done.

So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positive-definite? Or what?
That is the subset of M generated by null geodesics emanating from p but you are talking about light cones as they relate to causal structure. Also, I'm not sure how you are concluding that the metric tensor must suddenly be positive - definite.
micromass
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Jan3-13, 05:21 AM
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Quote Quote by zonde View Post
Done.

So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positive-definite? Or what?
I'm saying that the topology of [itex]T_pM[/itex] determined by the pseudo-Riemannian metric is of course non-Hausdorff. But this is a topology on [itex]T_pM[/itex] and not on M. The topology on M is Hausdorff and has nothing to do with the metric tensor.
TrickyDicky
#49
Jan3-13, 05:27 AM
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Quote Quote by WannabeNewton View Post
Tp(M) is not a neighborhood it is the tangent space to M at p.
Yes, strictly you are right, but note that the whole justification of the concept of manifold depends upon the possibility of making the neighbourhood of a point and its tangent space "equivalent" in the sense of homeomorphic to R^n.
zonde
#50
Jan3-13, 05:29 AM
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Quote Quote by micromass View Post
The topology on M is Hausdorff and has nothing to do with the metric tensor.
So we do relay on some positive-definite concept of nearness when we speak about topology of M, right?
micromass
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Jan3-13, 05:33 AM
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Quote Quote by zonde View Post
So we do relay on some positive-definite concept of nearness when we speak about topology of M, right?
A topology has nothing to do with "positive-definiteness". Positive-definite is a property about inner products.
TrickyDicky
#52
Jan3-13, 05:35 AM
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Quote Quote by WannabeNewton View Post
That is the subset of M generated by null geodesics emanating from p but you are talking about light cones as they relate to causal structure.
That is the point, it is hard to find (at least for me) mathematical justification for deriving a causal structure for the whole manifold only from the local action of the pseudoriemannian metric at the tangent space, when the distance function that prevails in smooth manifolds is not even the same as the one that integrates from the pseudoriemannian metric tensor.
George Jones
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Jan3-13, 05:49 AM
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Quote Quote by zonde View Post
If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event.
As has been indicated, some care is needed with respect to the meaning of "lightcone". Depending on the context and reference, "lighcone at [itex]p[/itex]" can either mean a subset of [itex]T_p \left(M\right)[/itex], or it can mean a subset of [itex]M[/itex]. I think that you mean the latter. In this case, [itex]M[/itex] is a neighbourhood of [itex]p[/itex] that contains its light cone.
George Jones
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Jan3-13, 06:41 AM
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Quote Quote by zonde View Post
Fine there is someone else who thinks like you.
Now can you provide arguments? In that link there is only definition (belief) and no arguments.

Why do you believe that all spacetimes are Hausdorff?
Quote Quote by atyy View Post
All the usual spacetimes are of course Hausdorff. But just for interest, Hawking and Ellis mention one example of a non-Hausdorff spacetime, and mention a paper by Hajicek.
We want to model physics. For most situations, spacetime Hausdorffness seems to be a reasonable, physical separation axiom. Two distinct physical events always admit distinct neighbouhoods.

Having said this, we have strayed far off-topic with respect to the original post. Physics Forums rules advises that, instead of posts that are off-topic, new threads should be started.


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