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Confusing expression in GR

  1. Jul 20, 2012 #1
    Can anyone help me simplify the expression [itex]\frac{\delta g_{\mu\nu}}{\delta g^{\kappa\lambda}}[/itex]? I haven't seen a term like this before and I don't know how to proceed. It seems like it might be the product of some kronecker delta's but I'm not sure.
     
  2. jcsd
  3. Jul 21, 2012 #2

    tom.stoer

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    You get a pair of Kronecker deltas
     
  4. Jul 21, 2012 #3

    Bill_K

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    Write the Kronecker delta as δμν = gμα gνα. take the variation: 0 = δgμα gνα + gμα δgνα. Now multiply by gμβ to solve for δgνα.
     
  5. Jul 21, 2012 #4
    This is a functional derivative, defined as follows:
    $$
    \frac{\delta F[g(x)]}{\delta g(y)} \equiv \lim_{\epsilon\rightarrow 0}\frac{F[g(x)+\epsilon \delta(x-y)]-F[g(x)]}{\epsilon}
    $$
    In your case this leads to,
    $$
    \frac{\delta g_{\mu\nu}(x)}{\delta g_{\rho\sigma}(y)} =\frac{1}{2}\Big(\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}+\delta_{\nu}^{\rho}\delta_{\mu}^{\sigma} \Big) \delta^D(x-y),
    $$
    with D the dimensionality of spacetime. Notice that both sides of the equation have the same symmetries on the indices.
     
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