Schwarzschild--anti-de Sitter black hole (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 $G=c=\hslash =k_B=1$, its metric tensor isHere $L$ 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 $\kappa =f^{\prime }(r_{+})/2$ then gives the Hawking temperatureUnlike its asymptotically flat limit $L\to \infty$, its temperature eventually increases with horizon radius. Differentiation gives a minimum at $r_{+}=L/\sqrt{3}$, separating negative- and positive-heat capacity branches.^[ref]

The spherical five-dimensional Schwarzschild--anti-de Sitter metric has $f(r)=1+r^2/L^2-\mu /r^2$. Its horizon satisfies $\mu =r_H^2(1+r_H^2/L^2)$ and has area $2{\pi }^2{r}_H^3$.

The Schwarzschild--anti-de Sitter heat capacity has the sign of $d{T}_H/d{r}_{+}$. It is positive precisely for $r_{+}>L/\sqrt{3}$, so these large black holes are locally stable in a fixed-temperature reservoir.