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Geodesic coordinates and tensor identities

  1. Dec 6, 2007 #1

    haushofer

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    Hi, I have a question about deriving tensor identities using geodesic coordinates ( coordinates in which one can put the connection to zero ). For the reference, I'm a master student physics and followed courses on general relativity and geometry. I'm busy with an article of Wald and Lee called "local symmetries and constraints", and that's where my question comes from.

    In short, Wald defines a collection of tensor fields [tex]\phi[/tex]:

    [tex]
    \phi : M \rightarrow M^{'}
    [/tex]

    where M can be seen as our space-time, and M' is finite-dimensional. He then states the following: "In a sufficiently small neighbourhood U' of any point [tex]\phi_{0} \in M^{'}[/tex] we may choose coordinates for M' such that the map [tex]\phi[/tex] can be represented locally as a collection of scalar functions [tex]\phi^{a}[/tex] of the space-time point x. Note that a change of coordinates in U' corresponds to an x-independent field redefinition [tex]\psi^{a} = f^{a}(\phi^{b})[/tex]. "

    He uses this to write down the variation of the Lagrangian [tex] L(\phi^{a}, \nabla_{\mu}\phi^{a},\ldots,\nabla_{\mu_{1} }\cdots \nabla_{\mu_{k}}\phi^{a} ; \gamma^{b} ) [/tex],
    where [tex]\gamma^{b}[/tex] is any nondynamical background field ( like the minkowski-metric in special relativistic field theories ). He uses only the symmetric part of the derivatives, because every antisymmetric part can be rewritten in terms of the Riemann-tensor. The variation of the Lagrangian should be

    [tex]
    \delta L = \frac{\partial L}{\partial \phi^{a}} \delta\phi^{a} + \frac{\partial L }{\partial (\nabla_{\mu}\phi^{a})}\delta(\nabla_{\mu}\phi^{a}) + \ldots
    [/tex]

    where the dots indicate higher derivatives. But it appears to me that he uses his specially chosen coordinates to write this variation as

    [tex]
    \delta L = \frac{\partial L}{\partial \phi^{a}} \delta\phi^{a} + \frac{\partial L }{\partial (\nabla_{\mu}\phi^{a})}\nabla_{\mu}\delta\phi^{a} + \ldots
    [/tex]

    and then states that the result doesn't depend on the coordinates, so it holds in general. It's quite obvious that for an arbitrary tensor field the variation and covariant derivative don't commute; you are left with variations in the connection. For example, for the first derivative one has that

    [tex]
    \delta\nabla\phi^{a} = \nabla\delta\phi^{a} + \sum\phi^{a}\delta\Gamma
    [/tex]

    in short-hand notation. As long as you treat the space-time metric as nondynamical background field everything is OK, but if we treat the metric as dynamical, this doesn't make any sense to me. Why is it possible to just say "let's choose special coordinates in which we can treat the tensor fields as scalar fields, so that we can commute the partial derivatives and variations, and then transform back to general coordinates and get the covariant derivative and variation commuting? Is this just simply chosing geodesic coordinates? In books like Inverno they use geodesic coordinates to get the identity for the variation of the Riemanntensor, which looks quite the same as this. What they do there, is:

    *choose geodesic coordinates where the connection is zero
    *write down the Riemann-tensor, where the connection terms are zero, but the derivatives are not
    *induce a variation in the connection
    *this induces a variation in the Riemann-tensor
    *commute the partial derivative and variation
    *observe that a tensor equation holds in every frame
    *observe that the result holds in every coordinate frame

    Could this be what Wald does in his article ?

    I hope I have given enough information, because the last time I had the idea that I was pretty unclear in my questioning :) I have the idea that there are some fundamental things about geometry which aren't completely clear to me.
     
    Last edited: Dec 6, 2007
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  3. Dec 6, 2007 #2

    Chris Hillman

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    Terminological confusion?

    You appear to be confusing several distinct concepts:
    • the Levi-Civita connection vanishes in some neighborhood only for a manifold which is locally flat in that neighborhood,
    • in Riemannian (Lorentzian) geometry, the term geodesic coordinates usually refers to a chart in which the integral curves of one of the coordinate vector fields, say [itex]\partial_x[/itex], are geodesic curves, with the coordinate playing the role of an arc-length parameter (examples include polar spherical chart on the ordinary sphere in which [itex]\partial_\theta[/itex] is a geodesic vector field with [itex]\theta[/itex] playing the role of arc length parameter),
    • in Riemannian (Lorentzian) geometry, the term Riemann normal coordinates usually refers to a chart in which the connection vanishes at one point, and coordinate "lines" issuing from that point do represent geodesics issuing from that point (note that due to geodesic convergence creating "caustics", such charts are rarely global charts); indeed a common trick does involve proving theorems by assuming a Riemann normal chart with some "base point", proving that some tensor equation holds at the point, and inferring the desired result (see the discussion in the textbook by Sean Carroll, Spacetime and Geometry),
    • in Lorentzian geometry, the term Fermi normal coordinates usually refers to a similar construction "centered" on a given geodesic rather than a given point.
    (Pedantic warning: in Riemannian geometry, Gaussian normal coordinates is usually synonymous with Riemann normal coordinates, but in the context of gtr, a Gaussian chart is sometimes taken to mean something else entirely.)

    Since you are asking for advice at the research level (apparently), it seems to me that it would be appropriate to obey such conventions as proper citations of books/papers. I guess (but IMO shouldn't have to guess) that you mean this paper:

    Lee, J., and R. M. Wald, "Local Symmetries and Constraints", Journal of Mathematical Physics 31 (1990): 725 – 743.

    Please correct me if I my guess is wrong, and no, I haven't seen this, I simply found a citation which may or may not be the paper you have in mind.

    I didn't read the rest of your post because it seems to be based entirely upon a serious terminological confusion. Obvious question: shouldn't your professor be helping you read the literature? E.g. terminological questions certainly seem appropriate because as we see above one could easily guess wrong if an experienced researcher doesn't clue you in, and it probably best to obtain such crucial advice from faculty whenever possible.

    But regardless, hope this helps!
     
    Last edited: Dec 6, 2007
  4. Dec 6, 2007 #3

    haushofer

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    Ok, maybe that is part of the confusion. Thank you for clearing that up, obviously I was talking about Riemann normal coordinates ( Inverno calls this geodesic coordinates, or I am mixing things up ) :)

    Yes, that is the paper, I should have given the correct reference.

    Well, I have the idea that it is not only terminological confusion. Ofcourse I asked my professor for help, but he had some problems with this commutation too. What the author does in the paper, is simply commuting the variation and the covariant derivative. He states that in the field variations, it is understood that any nondynamical background field is held fixed.

    A simple example:

    [tex]
    L = L(g_{ab},F_{ab},\nabla_{c}F_{ab})
    [/tex]

    of which the variation yields

    [tex]
    \delta L = \frac{\partial L}{\partial g_{ab}}\delta g_{ab} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}\partial_{c}\delta F_{ab} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}(\Gamma^{d}_{ca}F_{db} + \Gamma^{d}_{cb}F_{ad})
    [/tex]

    The problem lies in the construction; I can't see why one would leave out those connection-terms in the variation ( the last term on the RHS ).

    But thank you for your comments !
     
    Last edited: Dec 6, 2007
  5. Dec 6, 2007 #4

    Chris Hillman

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    So he does, and how annoying! :grumpy: Well, the terminology appears to be more completely standardized in mathematical literature than the gtr literature, so I guess you need to check carefully when reading gtr papers that you correctly understand all the technical terms for special coordinate systems.

    OK, haven't read the paper myself, but Jack Lee is accessible so if you and your prof are both puzzled I think it would be OK to email him if after mulling the terminological confusion for a bit things still are not clear.
     
  6. Dec 15, 2007 #5

    samalkhaiat

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    Metric variations & Field redefinitions

     
    Last edited: Dec 15, 2007
  7. Dec 17, 2007 #6

    haushofer

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    Last edited: Dec 17, 2007
  8. Dec 18, 2007 #7

    samalkhaiat

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    Last edited: Dec 18, 2007
  9. Dec 19, 2007 #8

    haushofer

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    My mistake; indeed I wrote something silly in that last variation. It should read

    [tex]
    \delta L = \frac{\partial L}{\partial g_{ab}}\delta g_{ab} + \frac{\partial L}{\partial R_{abcd}}\delta R_{abcd} + \frac{\partial L}{\partial (\nabla_{c}F_{ab})}\delta(\nabla_{c}F_{ab}) \ \ \ \ \ \ \ (1)
    [/tex]

    in which we could rewrite the last variation as

    [tex]
    \delta(\nabla_{c}F_{ab}) = \nabla_{c}(\delta F_{ab}) - \delta\Gamma^{h}_{ca}F_{hb}
    -\delta\Gamma^{h}_{cb}F_{ah}
    [/tex]

    I call this the total variation in the sense one does if one wants to derive the equations of motion.

    So, if we define the variation in the "lambda-way",
    [tex]
    \delta\phi^{a} = \frac{\partial\phi^{a}(\lambda ; x)}{\partial \lambda} |_{\lambda = 0}
    [/tex]
    then taking variations of tensor fields and taking covariant derivatives commute. But this is not true for the kind of variations I used in my equation (1), for there we have

    [tex]
    \delta\nabla\phi^{a} = \nabla\delta\phi^{a} - \sum\phi^{a}\delta\Gamma
    [/tex]

    where the tensor field has covariant indices. So now I'm a little confused :) It's clear that in taking these kind of variations, we can't commute variations and covariant derivatives; you always end up with variations in the connection. So what is the precise difference between the variation you take in deriving the equations of motion and identities like

    [tex]
    \delta L = E \delta\phi + \nabla_{a}\Theta^{a}
    [/tex]
    ( where E stands for the equations of motion and [tex]\Theta[/tex] is the so-called symplectic potential ) and the variation with the lambda?
     
  10. Dec 20, 2007 #9

    haushofer

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    I believe I start to understand where the confusion comes from; the lambda notion of a variation can be regarded as a tangent vector in the configuration space, while the coordinate notion of a variation can be regarded as a vector in coordinate-space ( space-time ). Those are two completely different spaces, and in that light I understand why taking covariant derivatives and taking "lambda-variations" commute; you are considering two different spaces.

    But in the coordinate space we have that varying our fields and taking covariant derivatives don't commute, due to the connection terms... Am I going into the right direction?
     
    Last edited: Dec 20, 2007
  11. Dec 21, 2007 #10

    samalkhaiat

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    Last edited: Dec 21, 2007
  12. Dec 22, 2007 #11

    haushofer

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    Thank you for all your effort, it is very much appreciated. I now have the feeling that I see the point here. I will write some cases with higher derivatives explicitly out, and try to get the hang of it.

    This is not something which is absolutely crucial for my thesis, but I just wanted to have these things clear for myself. I find this kind of math quite a hard subject as you have noticed by my questions ( and my last topic, in which you also made some things clear for me ). All I can do now is to take some explicit examples, and ofcourse to wish you nice holidays ! Christmas greetings,

    Haushofer.
     
  13. Dec 22, 2007 #12

    samalkhaiat

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