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Conservation law using Killing vector

  1. Jul 8, 2008 #1
    In Hartle's GR book (p. 177), there is a derivation of [tex]\xi \cdot u = constant[/tex], where [tex]\xi[/tex] is a Killing vector, [tex]u[/tex] is four-velocity along a geodesic in an arbitrary metric, and

    [tex]L = (-g_{\alpha\beta}\frac{dx^\alpha}{d\sigma}\frac{dx^\beta}{d\sigma})^\frac{1}{2}[/tex]

    The derivation goes:

    [tex] \frac{\partial}{\partial \sigma}\frac{\partial L}{\partial \frac{dx^1}{d\sigma}}} = 0 \\ \Rightarrow \frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = -g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma} = ... = -\xi \cdot u [/tex]

    is conserved along the geodesic. (Here the symmetry associated with [tex]\xi[/tex] is in [tex]x^1[/tex].) It seems to be saying that

    [tex]\frac{\partial L}{\partial \frac{dx^1}{d\sigma}} = \frac{1}{2L}({-g_{\alpha 1}\frac{1}{L}\frac{dx^\alpha}{d\sigma}-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}) = {-g_{1\beta}\frac{1}{L}\frac{dx^\beta}{d\sigma}[/tex]

    but it appears to me that [tex]\frac{\partial L}{\partial \frac{dx^1}{d\sigma}}[/tex] has only seven terms, not eight, since [tex]-g_{11}\frac{dx^1}{d\sigma}}\frac{dx^1}{d\sigma}}[/tex] appears only once. I'd appreciate it if someone could point out where I went wrong.

    Last edited: Jul 8, 2008
  2. jcsd
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