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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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