Kruskal coords for extreme RN black hole

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Homework Statement


I'm trying to find the kruskal coordinates for the extreme Reissner-Nordstrom black hole


Homework 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]


The Attempt at a Solution


Following a treatment of the Schwarzschild black hole, this is what I do.
Write
[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].
So
[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.
 
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