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Link btw manifolds and space-time

  1. Jan 30, 2007 #1

    quasar987

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    Fact: Spacetime is a curved pseudo-Riemannian manifold with a metric of signature (-+++).

    Fact: A manifold is a set together with a topology that is locally homeomorphic to R^n.

    Question: In the case of space-time, what is the set, what is the topology and what is n?
     
  2. jcsd
  3. Jan 30, 2007 #2
    Set: every event (even the boring ones)
    n: 4
    topology: isn't answering this what all the work is about?
    Also, the word "curved" seems redundant in the context of a these many-folded things..
     
    Last edited: Jan 30, 2007
  4. Jan 30, 2007 #3

    quasar987

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    What are events if not 4-tuples?
     
  5. Jan 30, 2007 #4
    well.. you additionally must specify which coordinate-chart the 4-tuple of coordinates is associated with.
     
  6. Jan 30, 2007 #5

    pervect

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    Well, the set in space-time is the set of all possible events. You can think of an event as a 4-tuple because for space-time, n=4.

    This leaves the question of "what is the topology".

    Let's talk about the topology of R, first.

    R is the set of all real numbers.

    A topological space (R,T) consists of the set R together with a collection of all subsets of R that satisfy the following three properties

    1) The union of an arbitrary collection of subsets (often called open balls) each of which is in T, is also in T

    2) The intersection of a finite number of subsets in T is also in T

    3) R is in T, and so is the empty set

    The usual topology of R is just (a,b), i.e. an "open ball" is defined as the set of points between a and b but not including the endpoints. Basically, the idea is that an open ball is the set of points "in the neighborhood" of some other point.

    If we consider R^2, I think we can consider open balls to be either circular (like an actual ball) or square, but you might want to get a mathemetican to advise you more fullly if you care about the details.

    You might also want to look up "homeomorphism", first one defines continuous maps as those maps f: X->Y such that every open set O in Y has an inverse image that is an open set in X. If f is continuous, one-one, onto, and its inverse is also continuous, f is a "homeomorphism".
     
  7. Jan 30, 2007 #6

    quasar987

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    Hey pervect,

    I am quite familiar with the maths of GR up to covariant derivatives of a general tensor field*. It is precisely how the link "math --> physics" is made that I'm wondering about.

    So now you two have made it clear that space-time is indeed R^4, but with an a priori unknown topology, but be make the hypothesis that is it locally homeomorphic to R^4 with the...standard open-ball topology. Is that it?

    *which I studied from "Relativity on Curved Manifold" by de Felice and Clarke. Do you know about it, and if so, what is your opinion of it?
     
    Last edited: Jan 31, 2007
  8. Jan 31, 2007 #7

    mjsd

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    So, I believe the connection is (which could be wrong of course)
    Spacetime is a curved pseudo-Riemannian manifold ....etc
    means spacetime is locally flat (from your definition that it is locally homomorphic to R^n, ie. Euclidean space)... but since the signature is -+++ not ++++ so it is "pseudo".
     
  9. Jan 31, 2007 #8

    quasar987

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    Hi mjsd,

    You bring up another concern of mine. I don't quite get why people translate "is locally homeomorphic" into "is locally flat". An homeomorphism is only a topological isomorphism, i.e. opens are preserved under application of the map and its inverse, and hence also some properties that rely on the notion of open sets. But it does not say anything about how the metric on the two manifolds is linked (supposing our R^4 with the standard topology is equipped with a metric). So even if we equip our R^4 with a flat metric (minkowski-flat or euclidian-flat), we cannot say anything about the "local flatness" of space-time.

    I think this connection is made by the principle of equivalence, but I'm not sure, I don't really understand what it says.
     
    Last edited: Jan 31, 2007
  10. Jan 31, 2007 #9

    vanesch

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    You are right: it would be silly to say that spacetime is locally flat ! Flatness of a manifold is essentially measured by the Riemann tensor (which is in 1-1 relationship with the metric on torsionless connections if I'm not mistaking), and that Riemann tensor can be non-vanishing. However, there's a way in which one can understand the statement, and that is that Riemanian curvature is "second-order" in displacements. So everything that is only concerned with first order in displacements will not see the difference between a curved and a flat piece of spacetime. Think of a "sphere which is locally flat", as a justification for the fact that earth looks flat to us. The curvature is not 0 nowhere on a sphere, but a small enough piece can manifest some kind of "flatness" for certain quantities, as long as they are first order in displacement.
     
  11. Jan 31, 2007 #10

    mjsd

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    i think you mean "is locally homeomorphic to R^4" being translated to "being locally flat".... don't think the translation is that direct, as u suggested you need more... but perhaps the metric is assumed (ie.usual physicist's abuse of terminologies). btw, remember when Einstein did all these, he did so using only tensor algebras and very little modern differential geometry... In my opinion, it is more of the subject of differential geometry swallowing GR (making it more formal) rather than GR being "grow out of formal mathematics" in the first place. (ie. Top to bottom approach, rather than bottom to top)

    principle of equivalence (strong): laws of physics in a freely falling inertial frame are identical to their laws in Special Relativity

    Back to the topology/local flatness issue:
    start with understanding the logic behind the construction of a Reimanian manifold (semi-layman perspective)

    you begin with a set of things, then introduce the notion of an open set (ie. topology is introduced), then make sure that it is topological space (ie. satisfying all relevant axioms) . Next, force each open sets to look like a piece of R^n and that your coordinate charts are smoothly put together (ie. no weird singularities etc.) and now you have a Manifold.
    Give this manifold a metric (which automatically means that you have the affine connections), then you have now arrived at the Riemannian manifold ...and spacetime is pseudo-Riemannian because of the -+++.....phew!

    In a sense, local flatness does seems to remind us of the equivalent principle because Special Relativity operates in flat spacetime. Whether the link between the two is direct or accidental, I can't remember off top of my head.

    Anyway, going back to your original question of what is the topology.... as u said, it will depend on what is our notion of an open set..... mmm...i don't think I know enough to answer it :redface:
     
  12. Jan 31, 2007 #11

    robphy

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    Here's a good reference:

    "Global Structure of Spacetimes", by R. Geroch and G. Horowitz,
    in "General Relativity: An Einstein Centenary Survey", edited by
    Hawking and Israel, 1979.
    [Find in a library.]

    With regard to issue of "topology of spacetime", let me comment that there is usual "manifold topology", which can be argued isn't very physical from a spacetime viewpoint. Some alternative topologies (which are equivalent to the manifold topology under suitable causality conditions) have been studied by Alexandrov, Zeeman, and Hawking-King-McCarthy (http://link.aip.org/link/?JMAPAQ/17/174/1) [see also Malament (http://link.aip.org/link/?JMAPAQ/18/1399/1)].
     
  13. Feb 4, 2007 #12

    quasar987

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    Could it be that no topology is ever explicitely mentionned? We just assume out of physical consideration that space-time is 4-manifold with some topology that makes it paracompact, connected, Hausdorff and without boundaries. Furthermore, since quantum field theories should be described on them, we demand that it admits a spinor structure (Geroch, 1986). [all these conditions I copy/pasted from p.129 of De Felice & Clarke's 'Relativity on Curved Manifolds']

    After all, what does it matter to know these things explicitely. All we are concerned about are the measurable effects, i.e. the curvature of this manifold.

    It's just a guess. Comments?
     
  14. Feb 4, 2007 #13

    pervect

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    I meant to get back to this thread, but I got busy and forgot :-(.

    AFAIK we don't have any evidence on what the global topology of the universe is. If we see "circles in the sky" http://www.citebase.org/abstract?id=oai:arXiv.org:astro-ph/9801212
    for instance, we'll have some experimental evidence about the topology of the universe, but it appears that there is no clear evidence at this point for the existence of such circles. (There were some papers suggesting that there might be such circles, but IIRC after re-analysis the authors feel more data is needed).

    Observing a wormhole would also be an example of a measurement that would show us that the universe has a non-trivial global topology.

    I think that the "global toplogy", i.e. R^4 vs S x R^3, etc, is of what's interest to the original poster.

    As has been remarked, we can map a section of a curved sphere (S^2) to a section of a flat plane via a homeomorphism. Another way of saying this - curvature doesn't enter the picture with just a topology, one needs a metric to define the curvature.

    I'm not sure if it's been remarked yet that the formal defintion of a toplogy isn't a single mapping, but a collection of (local) homeomorphic mappings, i.e. a topology is defined so that it can contain many 'charts'. There are some conditions on the "seams" of how the charts are "glued together" in regions where they overlap as well - basically, the seams have to be continuous.
     
  15. Feb 4, 2007 #14

    quasar987

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    Hi pervect,

    What do you (and the authors of "circles in the sky") mean by "topology of the universe"? I feel there is a meaning that elludes me because to me, the question of the topology of space-time is purely abstract; it is just a specification of what subsets we choose to call "open sets". This is purely mathematical and I don't see how the topology could ever be "observed" experimentally !

    The question of wheter space-time is actually the whole of R^4 or just S^1 x R^3 sounds to me like the question of "what is the set" and not "what is the topology"! But what would be the physical implication of space-time being just this "cylinder" ?

    I hear what you're saying: this is just a restatement of the fact that a manifold can be homeomorphic to flat space in a neighborhood of a point and yet have a non-vanishing curvature there. I.e. the statement "a manifold is a topological space that is locally flat" is not right.

    This sounds like the definition of a (C^0) differentiable manifold, rather than that of a topology does it not?!? You yourself gave the definition of a topology in post #5.
     
  16. Feb 4, 2007 #15

    robphy

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    Apparently, you didn't follow up on the references I posted earlier.

    At the small-scale, topology is about what points are "close enough" to a given point. In principle, one can ask how one might go about determining experimentally what spacetime events are close to a given event. It would seem that your tools for measuring separations between spacetime events are of a different nature than those used to measure separation of points in (say) a Euclidean plane.



    Consider an (interval I) x R3 vs. S1 x R3.
    They describe the same underlying point set. However, they have different topologies... they are connected differently.

    If your spacetime has the topology of S1 x R3, you might be describing a universe with a periodic time, which allows closed timelike curves.
     
  17. Feb 4, 2007 #16

    quasar987

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    I checked them out and saved them on my computer for later because they are over my head right now. Thank you for those.

    Which events are close to which other events... isn't that the job of the metric?!?! (which we agreed had nothing to do with the topology)
     
  18. Feb 4, 2007 #17

    robphy

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    The metric tells you "how close".
     
  19. Feb 4, 2007 #18

    pervect

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    Hi - actually, I was trying to guess what aspect of topology you were interested in - and it looks like I might have guessed wrong.

    The results I was talking about intially were "point set topoology" results. See for instance http://mathworld.wolfram.com/Point-SetTopology.html

    This is the sort to define the fundamental structure of manifolds, where are defined by a set of charts (or coordinate systems). It is, as robphy points out, fundamentally driven by the concept of the "neighborhood" of a point, which is formalized by the notion of "open balls" or "open sets".

    These very basic defintions are needed as the starting point.

    But point set topology isn't the end of the story. While locally one can (by defintion) pick a particular chart of a topology and have the geometry of that part of the topology be equal to R^n, it's not in general possible to represent an arbitrary topology by a single chart. There's a fairly simple proof, for example, that it takes at least 2 charts to cover a sphere with a 1:1 mapping. Note that using lattitude and longitude as coordinates fails to be a 1:1 mapping at the poles, the pole corresponds to 0 degrees lattitude and many longitudes, not just one longitude.

    So I thought you might have been interested in the more global aspects. Some famous results in this area (I'm not quite sure what the correct name for this subtype of topology) would be Euler characteristic numbers and problems like the "Seven bridgnes of Konigsberg".
     
  20. Feb 5, 2007 #19

    pervect

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    Since I'm still not sure what you're interested in (perhaps you are just interested in learning more about a lot of unrelated things), I'll take the opportunity to present some more motivational material about the original topic of "point set" topology and why it's set up the way it is.

    First some history. It has been proven by Cantor that there are the same number of points on a line as points on a plane - i.e. there is a 1:1 reverisble mapping between points on a line and points on a plane.

    See for instance http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node21.html

    Avoding these sorts of mappings of points on a line to points on a plane is the motivation behind the "open balls" formalism and the defintion of homeomorphisms. Explaining exactly how it avoids it would take too much time to do well and probably be confusing, but if you want something to think about, try thinking about why Cantor's mapping from R to R^2 (or if you are really ambitious, look up "space filling curves", another different mapping from R to R^2) do not satisfy the defintion of a homeomorphism.

    All of point set topology as I have outlined it above is ultimately based on ZFC, the branch of mathematics that can handle infinite sets. http://en.wikipedia.org/wiki/ZFC
     
  21. Feb 5, 2007 #20

    George Jones

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    I woundn't say this. R^4 and S^1 x R^3 are isomorphic in the category of sets, but they (with their usual topologies) are not isomorphic in the category of topological spaces.

    In other words there is a bijection between R^4 and S^1 x R^3, but there is no homeomorphism between them. As robphy says, R^4 is simply connected, while S^1 x R^3 is not, and simply-connectedness is a topological property, i.e., simply-connectedness is preserved by homeomorphisms.

    Not just in the neighbourhood of a point - this can be true for the entire manifold. Examples: Minkowski space with a point removed is the topological space S^3 x R, the underlying space for the manifold of closed Friedmann-Robertson-Walker universes, and Minkowski space with a straight line removed is S^2 x R^2, the underlying space for the manifold of extended Schwarzschild.
     
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