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Writing the continuity equations in a non-coordinate basis

  1. Jun 29, 2015 #1

    pervect

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    The continuity equations are ##\nabla_a T^{ab} = 0##. In a coordinate basis, we can write out the resulting differential equations as:

    ##\nabla_a T^{ab} = \partial_a T^{ab} + \Gamma^a{}_{ac}T^{cb} + \Gamma^b{}_{ac}T^{ac}##

    (modulo possible typos, though I tried to be careful-ish).

    What do we do in a non-coordinate basis, ##T^{\hat{a}\hat{b}}## where we might have the ricci rotation coefficients ##\omega^a{}_{bc}## (or perhaps ##\omega_{abc}## if that's simpler), rather than the Christoffel symbols?

    Right now, I'd just convert back to a coordinate basis, take the covariant deriatiave in that coordinate basis, then reconvert back to the non-coordinate basis, but I'd like to see if there is a more direct way.
     
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  3. Jun 29, 2015 #2

    Mentz114

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    In section 11.9 of the text cited below, Hamilton defines the tetrad covariant derivative which I think is what you are looking for.

    Good luck calculating the RRCs !

    General Relativity, Black Holes, and Cosmology
    Andrew J. S. Hamilton

    available here http://jila.colorado.edu/~ajsh/astr5770_14/grbook.pdf
     
  4. Jun 29, 2015 #3

    WannabeNewton

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    You just compute the structure constants ##C^{\hat{\alpha}}{}{}_{\hat{\beta}\hat{\gamma}}## of the tetrad and use the formula for the connection coefficients in terms of the structure constants: ##\Gamma^{\hat{\alpha}}_{\hat{\beta}\hat{\gamma}} = \frac{1}{2}g^{\hat{\alpha}\hat{\delta}}(C_{\hat{\beta}\hat{\delta}\hat{\gamma}} + C_{\hat{\gamma}\hat{\delta}\hat{\beta}} - C_{\hat{\beta}\hat{\gamma}\hat{\delta}})+ \frac{1}{2}g^{\hat{\alpha}\hat{\delta}}(g_{\hat{\beta}\hat{\delta}, \hat{\gamma}} + g_{\hat{\gamma}\hat{\delta}, \hat{\beta}} - g_{\hat{\beta}\hat{\gamma}, \hat{\delta}}) ##.

    For a local Lorentz frame the second term vanishes.
     
  5. Jun 30, 2015 #4

    martinbn

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    Presumably the connection forms had been calculated at this point and the gammas are just the coefficients of the connection forms expressed in terms of the 1-form basis
    ##\omega^\hat\alpha_\hat\beta = \Gamma^\hat\alpha_{\hat\beta\hat\gamma}\omega^\hat\gamma##
     
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