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Linearized general relativity problem

  1. May 31, 2006 #1
    I've got a problem where I'm not sure my solution is true!
    I have to proof that given the vacuum einstein's equation and the lorenz gauge condition imply that the stress energy tensor that generates the gravitational waves must have a vanishing divergence.

    The vacuum einstein's equation is given by [tex]-\partial_{\alpha}\partial^{\alpha}\bar{h}^{\mu\nu} = 16\pi T^{\mu\nu}[/tex], and the lorentz condition is given by [tex]\partial^{\nu}\bar{h}_{\mu\nu} = 0[/tex]

    Now if I just take the divergence to [itex]\nu[/tex] of this equation I obtain

    [tex]-\partial_{\nu}\partial_{\alpha}\partial^{\alpha}\bar{h}^{\mu\nu} = 16\pi \partial_{\nu}T^{\mu\nu}[/tex]

    Since we can swap the order of partial differentiation this becomes:

    [tex]-\partial_{\alpha}\partial^{\alpha}\left(\partial_{\nu}\bar{h}^{\mu\nu}\right)= 16\pi \partial_{\nu}T^{\mu\nu}[/tex]

    Now what I want to proof is that [tex]\partial_{\nu}\bar{h}^{\mu\nu} = \partial^{\nu}\bar{h}_{\mu\nu}[/tex]

    I think that is true because [tex]\partial_{\gamma}\bar{h}^{\mu\nu} = - \partial^{\gamma}\bar{h}_{\mu\nu}[/tex]. So if I just replace [itex]\gamma[/itex] by [itex]\nu[/itex] this should imply that the divergence of T vanishes.
    But I don't know for sure if I could just replace [itex]\gamma[/itex] by [itex]\nu[/itex]! :frown:

    Thanks in advance!
     
    Last edited: May 31, 2006
  2. jcsd
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