$AdS_2$ Black Hole

Question

I know that $AdS_2$ black hole has the following metric:

$$ ds^2=(r^2-a^2)dt^2+\frac{dr^2}{r^2-a^2}.\tag{1} $$

Here $a$ is constant.

On the other hand I am regularly facing with the following expression for black holes in $AdS_{d}$:

$$ ds^2=\frac{1}{z^2}\left[(-1-\mu z^{d-1})dt^2+\frac{dz^2}{1-\mu z^{d-1}}+d\vec{x}^2\right]. $$ So for $d=2$ one would have

$$ ds^2=\frac{1}{z^2}\left[(-1-\mu z)dt^2+\frac{dz^2}{1-\mu z}\right].\tag{2} $$

As I understand, $(2)$ is the expression for metric in so-called Poincare coordinates, am i right? If so, can you help me to show the equivalence of these two expressions?

I know that using the follwoing transformation we can get global $AdS_{2}$ metric from metric in Poincare coordinates:

If $$ a=\frac{\sqrt{1+r^2}\cos t}{r+\sqrt{1+r^2}\sin t} $$ $$ b=\frac{1}{r+\sqrt{1+r^2}\sin t} $$ Then $$ \frac{1}{b^2}\left[-da^2+db^2\right] \longrightarrow -(1+r^2)dt^2+\frac{dr^2}{1+r^2} $$

I tried to use these transformations, but I haven't succeed.

This post imported from StackExchange Physics at 2015-05-18 21:01 (UTC), posted by SE-user xxxxx