# Kerr-Newman-AdS black hole The metric of the Kerr-Newman-AdS black hole reads$\mathrm{d} s^2=-\frac{\Delta_r}{\rho^2}\left[\mathrm{~d} t-\frac{a \sin ^2 \theta}{\Xi} \mathrm{d} \phi\right]^2+\frac{\rho^2}{\Delta_r} \mathrm{~d} r^2+\frac{\rho^2}{\Delta_\theta} \mathrm{d} \theta^2+\frac{\Delta_\theta \sin ^2 \theta}{\rho^2}\left[a \mathrm{~d} t-\frac{r^2+a^2}{\Xi} \mathrm{d} \phi\right]^2,$where$\begin{aligned} \rho^2&=r^2+a^2 \cos ^2 \theta,\\ \Xi&=1-a^2 / \ell^2,\\ \Delta_r&=\left(r^2+a^2\right)\left(1+\frac{r^2}{\ell^2}\right)-2 m r+q^2,\\ \Delta_\theta&=1-\frac{a^2}{\ell^2} \cos ^2 \theta. \end{aligned}$And the gauge field is given by$A=-\frac{q r}{\rho^2}\left(\mathrm{~d} t-\frac{a \sin ^2 \theta}{\Xi} \mathrm{d} \phi\right).$Sometimes a transformation $A\to A+\alpha \mathrm{d} t$ for some constant $\alpha$ is performed to make the gauge connection smooth at the horizon. The energy, angular momentum, and charge are given by$E=\frac{m}{\Xi^2}, \quad J=a E, \quad Q=\frac{q}{\Xi}.$Taking the $l\to\infty$ limit leads to the asymptotically flat Kerr-Newman; taking $a\to0$ and $q\to0$ leads to Reissner-Nordström and Kerr respectively. (We have set $G_N=1$.) ## Charged black hole Taking the $a\to0$ limit, we obtain the metric of the RN-AdS black hole given by$\mathrm{d} s^2=-f(r)\mathrm{~d} t^2+\frac{1}{f(r)} \mathrm{~d} r^2+r^2(\mathrm{d} \theta^2+{ \sin ^2 \theta \,\mathrm{d} \phi^2}),$where$f(r)=\frac{r^2}{\ell^2}+1-\frac{2 m}{ r}+\frac{q^2}{r^2}.$And the gauge field is given by$A=\left(-\frac{q }{r}+\alpha\right)\mathrm{~d} t.$The energy and charge in this case are simply given by$E={m}, \quad Q={q}.$ In higher dimensions (AdS${}_{d+1}$), it is convenient to use a slightly different convention for the parameter $m$; the relation between $E$ and $m$ obtains a dimension-dependent constant (similarly for $Q$ and $q$). Explicitly, we use$d s^2=-f(r) \mathrm{d} t^2+\frac{\mathrm{d} r^2}{f(r)}+r^2 \mathrm{d} \Omega_{d-1}^2,$where$f(r)=\frac{r^2}{l^2}+1-\frac{m}{r^{d-2}}+\frac{q^2}{r^{2 d-4}}.$The gauge field is$A=\left(-\sqrt{\frac{d-1}{2(d-2)}} \frac{q}{r^{d-2}}+\Phi\right) \mathrm{d} t,$where $\Phi$ can be picked so that $A_{t}=0$ at the (outer) horizon $r=r_+$:$\Phi=\sqrt{\frac{d-1}{2(d-2)}} \frac{q}{r_{+}^{d-2}}.$In this convention and reinstating Newton's constant $G_N$,$M=\frac{(d-1) \omega_{d-1}}{16 \pi G} m,\quad Q=\sqrt{2(d-1)(d-2)}\left(\frac{\omega_{d-1}}{8 \pi G}\right) q,$where $\omega_{d-1}$ is the volume of the unit $(d-1)$-sphere.