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Kruskal coords for extreme RN black hole

  1. Apr 13, 2008 #1
    1. The problem statement, all variables and given/known data
    I'm trying to find the kruskal coordinates for the extreme Reissner-Nordstrom black hole

    2. Relevant equations
    The extreme RN metric is
    [tex]ds^2=-\left(1-{M\over r} \right)^2 dt^2 + \left(1-{M\over r} \right)^{-2} dr^2 + r^2 d\Omega^2[/tex]

    3. The attempt at a solution
    Following a treatment of the Schwarzschild black hole, this is what I do.
    [tex]ds^2=-fdudv + r^2 d\Omega^2[/tex] where [tex]f=(1-M/r)^2[/tex] and
    [tex]u = t-r^*[/tex]
    [tex]v = t+r^*[/tex]
    [tex]r^* = \int {dr \over f}[/tex]
    The horizon is at r = M. So expand f about r=M and you get
    [tex]f \approx 2k(r-M)^2[/tex] where [tex]k = {1\over {2M^2}}[/tex]
    Near the horizon then,
    [tex]r^* = {v-u\over 2} \approx {-1\over{2k(r-M)}}[/tex].
    [tex]f \approx {1\over {k(v-u)^2}}[/tex].
    Now the metric is
    [tex]ds^2 = {-1\over k}{dudv\over {(v-u)^2}}+r^2 d\Omega^2[/tex].
    But this does not separate nicely as in the Schwarzschild case, where you would get (following the same method)
    [tex]ds^2 = -2(e^{kv}dv)(e^{-ku}du) + r^2 d\Omega^2[/tex],
    where we'd define the kruskal coordinates such
    [tex]dU=e^{-ku}du[/tex] and [tex]dV=e^{kv}dv[/tex]

    Any help would be great.
    Last edited: Apr 13, 2008
  2. jcsd
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