Kruskal coords for extreme RN black hole

[Pacopag]

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.

 

[Jhero]

Hi there,

The Kruskal coordinates for the extreme Reissner-Nordstrom black hole can be found by following a similar method as you did for the Schwarzschild black hole. However, there are a few differences that need to be taken into account.

Firstly, the extreme Reissner-Nordstrom metric is given by:

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

Here, M is the mass of the black hole and r is the radial coordinate. The horizon of the black hole is located at r=M.

To find the Kruskal coordinates, we need to introduce new coordinates u and v, defined as:

u = t - r^*
v = t + r^*

where r^* is the tortoise coordinate, given by:

r^* = \int {dr \over \left(1-{M\over r} \right)^2}

Using these coordinates, the metric can be written as:

ds^2=-fdudv + r^2 d\Omega^2

where f(r) = \left(1-{M\over r} \right)^2. Note that in the Schwarzschild case, f(r) = 1- {2M\over r} and the tortoise coordinate is given by r^* = \int {dr \over \left(1-{2M\over r} \right)}.

Now, let's expand f(r) around the horizon (r=M):

f(r) \approx 2k(r-M)^2

where k = {1\over {2M^2}}.

Near the horizon, the tortoise coordinate can be approximated as:

r^* \approx {-1\over{2k(r-M)}}.

Using this, we can rewrite the metric as:

ds^2 = {-1\over k}{dudv\over {(v-u)^2}}+r^2 d\Omega^2

This is similar to the metric for the Schwarzschild black hole, except for the factor of -1/k. To make it look like the Schwarzschild metric, we define new coordinates U and V as:

dU = -{1\over k}du and dV = {1