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Pondering basis vectors and one forms

  1. Nov 11, 2011 #1

    Matterwave

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    So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey the property that:

    [tex]\tilde{\omega}^j(\vec{e}_i)=\delta_i^j[/tex]

    Then, we cannot require the basis vectors and one forms to be "dual" to each other in the sense that we raise and lower their indices with the metric tensor. I.e.:

    [tex]\tilde{\omega}^i=\bf{g}(\vec{e}_i,\quad)[/tex]

    Since, unless my metric is the Cartesian metric, I cannot get the first property from the second quantity.

    Is this true, or have I messed up somewhere?
     
    Last edited: Nov 11, 2011
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  3. Nov 11, 2011 #2

    atyy

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    The "duality" between vector and one forms that you cite does not require a metric. When the vector basis is changed, the dual forms are also changed.

    The "duality" between vectors and one forms that can be defined with a metric is different, as the duals do not change if you change basis.

    So you are right. The former has nothing to do with raising and lowering indices with a metric (it's defined even without a metric), the latter involves raising and lowering indices with a metric.
     
    Last edited: Nov 11, 2011
  4. Nov 11, 2011 #3

    Matterwave

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    The natural set of one form basis vectors then, is defined by the first property and not the second one then right?
     
  5. Nov 11, 2011 #4

    atyy

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    Yes.
     
  6. Nov 11, 2011 #5

    Matterwave

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    I have another question. If we choose to use the tetrad method to describe our manifold, then the metric is automatically diag(1,1,1,...)? Since we are choosing ortho-normal basis vectors, then it would make sense that that would have to be the case.

    I could never figure out Wald's take on the tetrad method...His abstract index notation there makes me really confused what his equations mean...>.>
     
  7. Nov 11, 2011 #6

    atyy

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    Just reading wikipedia, the metric is the Minkowski metric times the "square" of the vierbein field. http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

    I remember Andrew Hamilton had a good set of notes about this (including using non-orthonormal tetrads). http://casa.colorado.edu/~ajsh/phys5770_08/grtetrad.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
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