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Spin connection

  1. Sep 15, 2014 #1
    1. The problem statement, all variables and given/known data

    I want to know wether the component [itex]\omega_{c}^{ab}[itex] of the spin connection is zero or not?

    2. Relevant equations
    [itex]de^{a}=-\omega^{a}_{b}\wedge e^{b}[itex]


    3. The attempt at a solution
    [itex]de^{a}=-\omega^{a}_{b}\wedge e^{b}[itex]
    ##=-\omega^{a}_{b,\nu} e^{b}_{\mu} dx^{\nu}\wedge dx^{\mu}##
    ##=-\omega^{a}_{b,c} e^{b}_{[\mu}e^{c}_{\nu]} dx^{\nu}\wedge dx^{\mu}##
    as it is antisymmetric in ##\mu## and ##\nu##.So it is also antisymmetric in b and c.Thus one can conclude from here.
     
    Last edited: Sep 15, 2014
  2. jcsd
  3. Sep 15, 2014 #2

    nrqed

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    I have reformatted your question so that the TeX commands would be visible.
     
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