How is the mass of the black hole defined in a asymptotcally AdS solution of the black hole? How can I find it? Beacause in asymptotcally flat solution I can read it from the $g_{tt}$ component of the metric.
Ps: Someone knows a good lecture about AdS black hole?
There are two complementary correspondences to black holes. One is a metric where far from the black hole horizon there is $AdS_3$ spacetime. The other occurs quite oppositely where the $AdS_2\times S^2$ spacetime occurs in the near horizon condition of an extremal black hole.
For the asymptotic condition, usually seen as the AdS black hole one simply has the metric $$ ds^2 = -\left(1 - \frac{2m}{r} + a^2r^2\right)dt^2 + \left(1 - \frac{2m}{r} + a^2r^2\right)^{-1}dr^2 + r^2d\Omega^2 $$ In the limit the radius $r$ is very large this reduces to $$ ds^2 = -r^2(a^2dt^2 - d\Omega^2), $$ that is the metric for $AdS_3$.
It is worth noting what I see as the complementary relationship between black holes and $AdS$ spacetime. The condition for an accelerated observer near the horizon is given by a constant radial distance. We then considerable $$ \rho = \int dr \sqrt{g_{rr}} = \int \frac{dr}{\sqrt{1 - 2m/r + Q^2/r^2}} $$ with low and upper limits on integration $r_+$ and $r$. The result is $$ \rho = m log[\sqrt{r^2 - 2mr + Q^2} + r - m] + \sqrt{r^2 - 2mr + Q^2} $$ $$ = m log[\sqrt{r^2 - 2mr + Q^2} + r - m] + r \sqrt{g_{tt}} - Λ. $$ Here $\Lambda$ is a large number evaluated within an infinitesimal distance from the horizon We write the metric at this position $$ ds^2 = \left(1 - \frac{2m}{r(\rho)} + \frac{Q^2}{r(\rho)^2}\right)dt^2 - d\rho^2 - r(\rho)^2dΩ^2. $$ With the near horizon condition we may set $r^2 - 2mr + Q^2 \simeq 0$ in the log so that $$ \rho \simeq m log(r - m) + r \sqrt{g_{tt}} – \Lambda. $$ The divergence of the log cancels the arbitrarily large $\Lambda$ $$ \rho/r_+ =~ \sqrt{g_{tt}}. $$ We now write the metric as $$ ds^2 = \frac{\rho^2}{r_+^2}dt^2 - d\rho^2 - m^2dΩ^2 $$ We may now observe that $d\rho^2 = dr^2/g_{tt}^2$ and substitute in $\rho/m$ for $g_{tt}$ for $r_+ = m$ and obtain $$ ds^2 = \left(\frac{\rho}{m}\right)^2dt^2 - \left(\frac{m}{\rho}\right)^2dr^2 - m^2dΩ^2. $$ Or $$ ds^2 = \left(\frac{\rho}{m}\right)^2dt^2 - \left(\frac{m}{\rho}\right)^2 d\rho^2 - m^2dΩ^2~ for~ \rho \simeq r. $$ This is the metric for $AdS_2$ in the $(t, r)$ variables and a sphere $S^2$ of constant radius $= m$ in the angular variables.
This is considerably easier to see than the approach taken by Carroll and Randall, which admittedly is a more exact derivation. I mention this because is should not escape one's attention that in $10$ dimensions this is $AdS_5\times S^5$ in the $AdS/CFT$ correspondence. This connects with the AdS black hole which reduces by a dimension and the $CFT$ information in the anti-de Sitter spacetime is also on the horizon of the black hole.
