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Spin connection

  1. Jun 28, 2015 #1

    I've worked through most of Carroll's appendix on the non-coordinate basis.
    I see and agree how the spin connection and tetrad one-forms are useful while calculating.

    However as an example he sets out to apply the formalism to a spatially flat, expanding universe.
    [tex]ds^2 = -dt^2 +a^2(t)\delta_{ij}dx^idx^j = -e^0\otimes e^0 + \sum_i e^i\otimes e^i [/tex]

    The choice of the vielbein one-forms is clear in this case as is the calculation.

    He states however that by using raising and lowering indices on the spin connection ##\omega_{ab}## we can show the following identities by using the antisymmetry. (I copied the expressions verbatim, i and j are different from 0)

    [tex]\omega^0_{\,\, j} = \omega^j_{\,\, 0}[/tex]

    Here I wrote ##\omega^0_{\,\, j} = \eta^{0a}\omega_{a j} = - \eta^{0a}\omega_{j a} = \omega_{j}^{\, \, 0}##
    This is similar but not the same.

    [tex]\omega^i_{\,\, j} = -\omega^j_{\,\, i}[/tex]
    For the other relation I can do exactly the same and get a similar result.
    With the same difference in index position.

    So I don't get the same relation. Is he abusing notation here?
    Or am I overlooking something? It bothers me without measure whenever I encounter such a problem.


  2. jcsd
  3. Jun 28, 2015 #2


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    ##-\eta^{0a}\omega_{ja} = -\eta^{00}\omega_{j0} = \omega_{j0} = \omega^j{}{}_0##
  4. Jun 29, 2015 #3
    Thanks, I don't know why I couldn't get it.
    But now I can continue to do the mixmaster universe in the tetrad formalism.
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