Schwarzschild–Anti-de Sitter Black Holes

Schwarzschild–anti-de Sitter is the static, spherical black-hole family of general relativity with negative cosmological constant. The AdS curvature radius supplies a second scale, so black holes smaller and larger than that radius have qualitatively different mechanical and thermal response. The metric is an exact $\Lambda$-vacuum solution; holographic or ensemble interpretations require additional boundary and quantum input.

Schwarzschild–AdS geometry

Property	Value
Object class	Exact solution family
Theory	Four-dimensional $R_{\mu\nu}=-3g_{\mu\nu}/L^2$
Parameters	mass $M$ and AdS radius $L$
Asymptotics	Global asymptotically AdS with spherical boundary sections
Horizon topology	$S^2$
Physical status	Exact $\Lambda$-vacuum solution; boundary-condition-dependent dynamics

The metric is

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2GM}{r}+\frac{r^2}{L^2}.$$

For $M\gt0$, $f$ is strictly increasing on $r\gt0$ and has exactly one positive root $r_+$. Eliminating the mass gives

$$2GM=r_+\left(1+\frac{r_+^2}{L^2}\right).$$

“Small” and “large” therefore label different members at fixed $L$, not two horizons of one solution. The general AdS charge and first-law normalization are derived by Gibbons, Perry, and Pope 2005, § 2.1, eqs. (2.5)–(2.15).

Horizon, boundary, and regularity

The static chart is regularized at $r_+$ by $v=t+\int dr/f$, giving an ingoing cross term rather than the divergent $dr^2/f$. Curvature invariants are finite at the horizon and diverge at $r=0$. Outside, the spacetime reaches a timelike conformal boundary rather than null infinity. Evolution therefore requires a boundary condition; reflecting, transparent, or mixed conditions define different problems even on the same exact metric.

With the standard global AdS time at the boundary,

$$A_H=4\pi r_+^2, \qquad \kappa=\frac{1}{2r_+}\left(1+\frac{3r_+^2}{L^2}\right).$$

The surface gravity never vanishes for $r_+\gt0$: the uncharged spherical family has no extremal black-hole member. At fixed $L$,

$$dM=\frac{\kappa}{8\pi G}\,dA_H.$$

The function $\kappa(r_+)$ reaches a minimum at $r_+=L/\sqrt3$. Its two sides are the small and large branches at fixed surface gravity. This turning point is not a degenerate horizon, because $\kappa$ remains $\sqrt3/L$.

Thermal context and dynamics

If semiclassical identifications $T=\kappa/(2\pi)$ and $S=A_H/(4G)$ are made, the small branch has negative heat capacity and the sufficiently large branch positive heat capacity. Global dominance is a separate saddle comparison. Relative to thermal AdS with the same boundary data, the spherical black-hole free energy changes sign at $r_+=L$ in four dimensions, the Hawking–Page transition discovered in Hawking and Page 1983, § 2, eqs. (2.6)–(2.9), printed pp. 581–582. The minimum-temperature point and the free-energy crossing are not the same.

In holographic theories the transition can map to a finite-volume large-N confinement/deconfinement transition, as explained by Witten 1998, § 2.3. That dictionary is theory dependent; it is not a property of every classical AdS metric. Perturbative decay also depends on boundary conditions, and AdS trapping can make relaxation much slower than in asymptotically flat space.

Boundary and ensemble diagnostics

The conformal-boundary metric fixes the normalization of $\partial_t$ used for both $M$ and $\kappa$. Rescaling boundary time rescales energy and surface gravity together. Comparisons across different $L$ or different conformal representatives must therefore translate the generator, rather than comparing bare numbers.

The geometric first law alone does not decide thermal stability. Only after adopting the semiclassical identifications $T=\kappa/(2\pi)$ and $S=A_H/(4G)$ does the sign of $dT/dr_+$ diagnose local canonical response. Global dominance still requires a renormalized action comparison with thermal AdS at identical boundary data. Thus the minimum-temperature point, local heat-capacity change, and Hawking–Page crossing answer three distinct questions.

Spherical Schwarzschild–AdS must also be separated from planar and hyperbolic solutions. Changing the constant-curvature term changes horizon topology, charge subtraction, and possible negative-mass ranges. A planar black brane can emerge from a controlled scaling, but a finite spherical horizon does not change topology merely because $r_+/L$ is large.

Finally, statements about deconfinement, finite-N rounding, or quantum entropy require a specified holographic theory and approximation. They are not consequences of the classical metric alone.

Controlled limits

Flat limit. Taking $L\to\infty$ at fixed $M$ gives $r_+\to2GM$ and $\kappa\to1/(4GM)$, recovering Schwarzschild on bounded regions. The timelike boundary recedes and the global initial-boundary-value problem changes.

Small-hole limit. When $r_+/L\ll1$,

$$M=\frac{r_+}{2G}\left[1+O(r_+^2/L^2)\right], \qquad \kappa=\frac{1}{2r_+}\left[1+O(r_+^2/L^2)\right].$$

Near-horizon geometry is Schwarzschild-like, while the far region still knows about AdS.

Large-hole limit. For $r_+/L\gg1$, $M\sim r_+^3/(2GL^2)$ and $\kappa\sim3r_+/(2L^2)$. A separate planar scaling can yield a black brane, but a large spherical horizon does not change topology merely by being large.

The $M=0$ member is pure AdS and has no positive horizon. Negative $M$ in the spherical family exposes a singularity rather than producing a regular “superextremal” branch.

Connections to the detailed theory

The local uniqueness statement with $\Lambda$ is developed in Birkhoff’s Theorem with a Cosmological Constant. Root uniqueness, charge normalization, and branch mechanics are treated in Schwarzschild–AdS Horizon Families. The Euclidean saddle comparison and its holographic interpretation belong to Hawking–Page Transition and Deconfinement.

References

• Gibbons, Gary W., Malcolm J. Perry, and Christopher N. Pope. “The First Law of Thermodynamics for Kerr–Anti-de Sitter Black Holes.” Classical and Quantum Gravity 22 (2005): 1503–1526. DOI.
• Hawking, Stephen W., and Don N. Page. “Thermodynamics of Black Holes in Anti-de Sitter Space.” Communications in Mathematical Physics 87 (1983): 577–588. DOI.
• Witten, Edward. “Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories.” Advances in Theoretical and Mathematical Physics 2 (1998): 505–532. DOI.