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What is the ontology of general relativity ?

  1. Dec 13, 2005 #1

    vanesch

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    What is the "ontology" of general relativity ?

    Hi,

    First of all, though I know some GR, I'm far from an expert on it. I used to think of the "ontology" of GR as being a 4-dimensional manifold with a metric connection on it, but apparently that doesn't fly, because of the "hole argument":

    http://plato.stanford.edu/entries/spacetime-holearg/

    I used to think that general covariance simply meant that we could choose arbitrary coordinates on this manifold, but it seems that this is not sufficient.
    So my question is: if the "ontology" of GR is NOT a 4-dim. manifold + metric connection, then what is this ontology ? What, according to (classical) GR, is "out there" ; how do relativists see their subject ?

    Or can we still see the ontology of GR as being a 4-dim manifold + metric connection, and the different physical descriptions as "gauge-equivalent" ?
     
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  3. Dec 13, 2005 #2

    selfAdjoint

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    Equivalence classes of diffeomorphisms on the manifold. At least I think that's what Einstein finallly came to after years of musing on the hole argument.
     
  4. Dec 13, 2005 #3

    robphy

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  5. Dec 13, 2005 #4

    vanesch

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    I'm trying to imagine what that is. Ok, this question belongs probably more in the differential geometry section, but I'm having a hard time seeing how two diffeomorphic manifolds are actually different manifolds. Of course, if two manifolds have other structure (for instance, are the result of a certain construction), they can be "different" in a certain respect, and nevertheless have a diffeomorphism between them. But a smooth manifold, to me, IS already an equivalence class of atlasses. As such, a diffeomorphism just shuffles the atlasses around within the equivalence class, so two smooth manifolds which are diffeomorphic ARE the same manifold to me, no ?
     
  6. Dec 13, 2005 #5

    JesseM

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    Not being well-versed in GR or differential geometry, I was confused by this explanation, in particular the diagram here:

    [​IMG]

    What if you decided that the event "E" in the diagram should represent a particular time-reading on the internal clock of a test particle--surely then there could be no ambiguity over whether this event coincides with a point along the worldline of a galaxy? If so, then in general, why can't you unambiguously decide whether two points in different coordinate systems represent the "same event" or not by imagining a test particle whose path travels through that point in one coordinate system, and seeing if the particle's worldline passes through the corresponding point (at the same time, according to its internal clock) in the second coordinate system? Or does the problem that the "hole argument" illustrates not have anything to do with ambiguity over whether two points in different coordinate systems represent the "same event" or not?
     
    Last edited: Dec 13, 2005
  7. Dec 13, 2005 #6

    Hurkyl

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    If I've interpreted correctly what I've read, this was essentially a measurement problem for General Relativity, and was thought to be a fatal flaw until it was figured out how one might possibly be able to "internally" perform a measurement. (Which, I suppose, is an explanation for the popularity of measuring things as they traverse a loop)
     
  8. Dec 13, 2005 #7

    JesseM

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    This page has a shorter summary of the hole argument...it's still beyond me, but it may help others understand the details better:
    The summary seems to say that the problem arises when you try to imagine "events" in the spacetime manifold having distinct identities before you impose a gravitational field on the manifold...so maybe there is no problem with thinking of each event having a distinct identify after you put in the gravitational field, which is what was confusing me earlier.
     
    Last edited: Dec 13, 2005
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