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Metric tensor, infitesimal transformation

  1. Jan 31, 2008 #1

    I don't think this belongs in the homework section since this is a graduate course.
    My question is regarding making a field theory generally covariant by including a metric tensor [tex]g_{\mu\nu}(x)[/tex]in the Lagransian density, and it's transformation under infinitesimal coordinate change.

    If I transform the coordinates according to:
    [tex]x^{\mu}\rightarrow x^{\mu}'=x^{\mu}+\epsilon^{\mu}(x)[/tex]

    The metric must be transformed according to:
    [tex]g_{\mu\nu}(x)\rightarrow g'_{\mu'\nu'}(x')=\frac{\partial x^{\alpha}}{\partial x^{\mu}'}\frac{\partial x^{\beta}}{\partial x^{\nu}'}g_{\alpha\beta}(x)[/tex]

    which I understand well. But according to the textbook I'm reading the infitesimal result is:
    [tex]\delta g_{\mu\nu}(x)=\epsilon_{\mu;\nu}+\epsilon_{\nu;\mu}[/tex]
    (with covariant derivatives).

    I have tried using:
    [tex]\frac{\partial x^{\alpha}}{\partial x^{\mu}'}\simeq\delta_{\mu}^{\alpha}-\frac{\partial\epsilon^{\alpha}}{\partial x^{\mu}}+O(\epsilon^{2})[/tex]

    to obtain:
    [tex]\delta g_{\mu\nu}(x)=-g_{\mu\alpha}\partial_{\nu}\epsilon^{\alpha}-g_{\nu\alpha}\partial_{\mu}\epsilon^{\alpha}[/tex]

    but I don't know how to obtain the textbook result.
    Can someone clue me in on how to do it?

  2. jcsd
  3. Jan 31, 2008 #2
    Never mind, I solved it...
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