# The AdS-Schwarzschild black hole solution

My question is from an AdS/CFT review: http://arxiv.org/abs/1112.5403

The AdS_5 metric in the article is written

$$ds^2=\frac{l^2}{z^2}(-dt^2+dz^2+dx^2),$$ where I'm denoting collectively three dimensions by $dx^2$.

Then the article says Schwarz black hole solution with a horizon radius $z_h$ in this spacetime is

$$ds^2=\frac{l^2}{z^2}\Big[-\big(1-\frac{z^4}{z_h^4}\big)dt^2+\frac{1}{1-\frac{z^4}{z_h^4}}dz^2+dx^2\Big].$$

I'm observing some weird things here:

when the black hole radius $z_h$ approaches infinity, the Schwarz metric becomes AdS,

so an infinitely large AdS-Schwarz black hole is AdS???

• How does this contradict the fact that the spacetime is asymptotically AdS? The Schwarzschild solution that you have written down is a singular solution in AdS_5 spacetime in the Fefferman-Graham form. What are the weird things that you are observing? – user106422 Sep 12 '16 at 21:37
• The weird thing is in the limit $z_h$ goes to infinity, the black hole solution reduces to AdS metric. An infinitely large black is same as an AdS spacetime having no black hole? – JamieBondi Sep 16 '16 at 5:59