= Schwarzschild--anti-de Sitter black hole
{c}
{title2=Schwarzschild-AdS black hole}
The four-dimensional <Schwarzschild--anti-de Sitter black hole> is a static, spherically symmetric <vacuum spacetime> with negative <cosmological constant> $\Lambda=-3/L^2$. In units $G=c=\hbar=k_B=1$, its <metric tensor> is
$$
ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega_2^2,
\qquad f(r)=1+\frac{r^2}{L^2}-\frac{2M}{r}.
$$
Here $L$ is the <Anti-de Sitter> curvature length and $M>0$ is the mass parameter. Setting $f(r_+)=0$ gives $M=r_+(1+r_+^2/L^2)/2$. The <surface gravity> $\kappa=f'(r_+)/2$ then gives the <Hawking temperature>
$$
T_H=\frac{1+3r_+^2/L^2}{4\pi r_+}.
$$
Unlike its asymptotically flat limit $L\to\infty$, its <temperature> eventually increases with horizon radius. Differentiation gives a minimum at $r_+=L/\sqrt3$, separating negative- and positive-<heat capacity> branches.<Thermodynamics of black holes in anti-de Sitter space by Stephen Hawking and Don N. Page>[Sections 2–3, metric, Euclidean period and canonical ensemble]{ref}
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