# Kruskal coords for extreme RN black hole

## Homework Statement

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

## Homework Equations

The extreme RN metric is
$$ds^2=-\left(1-{M\over r} \right)^2 dt^2 + \left(1-{M\over r} \right)^{-2} dr^2 + r^2 d\Omega^2$$

## The Attempt at a Solution

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

Any help would be great.

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