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2nd order correction to gauge transformation

  1. Nov 20, 2011 #1
    In the weak field approximation,

    [itex]g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}[/itex]

    If we make a coordinate transformation of the form

    [itex]x^{\mu'}=x^{\mu}+\xi^{\mu}(x)[\itex]

    it changes [itex]h_{\mu\nu}[\itex] to

    [itex]h'_{\mu\nu}=h_{\mu\nu}+\xi_{\mu,\nu}+\xi_{\nu,\mu}+O(\xi^{2})[\itex]

    I was wondering if anyone could shed some light on what form the higher order terms take. I have an inkling it's terms from a taylor series expansion but i'm not sure.

    Thanks
     
  2. jcsd
  3. Nov 20, 2011 #2

    atyy

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    Last edited by a moderator: May 5, 2017
  4. Nov 20, 2011 #3
    Thanks for your reply but i'm afraid it doesn't shed any more light on it. I'm not sure if [itex]\xi^{\mu}(x)[\itex] being a Killing vector has anything to do with it.
     
  5. Nov 20, 2011 #4

    Bill_K

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    The exact transformation is xμ = x'μ + ξμ. Applied to the metric this is

    gμν = ∂xα/∂x'μ ∂xβ/∂x'ν gαβ

    = (δαμ + ξα)(δβν + ξβ)(ηαβ + hαβ)

    = ημν + (hμν + ξα hαν + ξβ hμβ + ξαξβhαβ)

    = ημν + h'μν
     
  6. Nov 20, 2011 #5
    A clear reference is http://arxiv.org/abs/gr-qc/9609040, although this is not that accessible to the beginner. One subtlety is that if you take the transformation x = x' + xi to be exact, then xi is no longer the generator of the diffeomorphism at higher than linear order. So most people use equation (1.1) of the above reference for the coordinate transformation. With that form you can express everything in terms of Lie derivatives, equation (1.3).
     
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