Schwarzschild–AdS Horizon Families

Schwarzschild–anti-de Sitter spacetime is the simplest setting in which the cosmological curvature scale competes directly with the horizon scale. In four-dimensional vacuum Einstein gravity with $\Lambda=-3/L^2$, a positive mass produces exactly one spherical event horizon. Its radius determines the mass, area, and surface gravity, while the ratio $r_+/L$ separates a Schwarzschild-like regime from a curvature-dominated regime. This page concerns that classical geometry; it does not identify surface gravity or area with quantum temperature or entropy.

Helpful background. Exact Solutions of Einstein’s Equations supplies the static metric ansatz and horizon-root method; Causality and Global Structure supplies the event-horizon interpretation; The Einstein Equation supplies the vacuum equation with cosmological constant; The Gibbons-Hawking-York Boundary Term supplies the boundary conventions behind gravitational charges; and Black-Hole Geometry and Mechanics supplies the definitions of area and surface gravity used below.

The spherical Schwarzschild–AdS family

In static coordinates the metric is

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

Here $M$ is the conserved mass in the standard asymptotically AdS normalization, $L$ is the AdS radius, and $d\Omega_2^2$ is the unit round-sphere metric. The same function follows from the $k=1$ member of the general topological AdS solution; it is also the zero-rotation limit of Kerr–AdS Kubizňák, Mann, and Teo 2017, §2.2, equations (2.9)–(2.13), Open PDF, pp. 5–7 Gibbons, Perry, and Pope 2005, §2.1, equations (2.1)–(2.4), Open PDF, pp. 7–8.

A Killing horizon occurs at a positive zero $r=r_+$ of $f$. Eliminating $M$ in favor of that zero gives the especially useful horizon equation

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

For $M\gt0$, the zero is unique. Indeed, $f(r)$ tends to $-\infty$ as $r\to0^+$ and to $+\infty$ as $r\to\infty$, while

$$f'(r)=\frac{2GM}{r^2}+\frac{2r}{L^2}\gt0$$

throughout $r\gt0$. Thus “small” and “large” Schwarzschild–AdS black holes do not mean an inner and an outer root of one geometry. They mean different members of a one-parameter family, compared at a fixed $L$.

Monotonicity also fixes the static regions. Outside $r_+$ one has $f\gt0$, so $\partial_t$ is timelike and the constant-$t$ slices have positive radial metric. Inside, $f\lt0$ and the areal radius becomes timelike, as in Schwarzschild. The outward region does not end at an asymptotically null infinity: after conformal rescaling it meets the timelike AdS boundary. That global difference will matter when boundary data and predictability are discussed, even though the local test for this Killing horizon is still simply $f(r_+)=0$.

Radius, area, and surface gravity

The horizon cross-section is a round two-sphere, so

$$A=4\pi r_+^2.$$

With the static Killing field normalized by the standard AdS time at conformal infinity, its surface gravity is

$$\kappa=\frac{1}{2}f'(r_+) =\frac{1}{2r_+}\left(1+\frac{3r_+^2}{L^2}\right).$$

These are geometric relations. In particular, the branch language below can be stated entirely in terms of $\kappa(r_+)$, without invoking a thermal ensemble. The corresponding formulas and their small/large-radius behavior are reviewed in Kubizňák, Mann, and Teo 2017, §4.1, equations (4.3)–(4.7), Open PDF, pp. 23–26.

Surface gravity still depends on the normalization of the Killing generator. The value above uses the same boundary time that defines $M$. Rescaling $t$ would rescale $\kappa$ and the energy together, leaving the fixed-normalization first law consistent but changing a bare numerical comparison. AdS supplies a preferred standard through the chosen conformal-boundary metric, so that convention must remain fixed when two members of the family are compared.

Differentiation shows

$$\frac{d\kappa}{dr_+} =-\frac{1}{2r_+^2}+\frac{3}{2L^2}.$$

Consequently $\kappa$ decreases until $r_+=L/\sqrt3$, reaches

$$\kappa_{\min}=\frac{\sqrt3}{L},$$

and then increases. A value $\kappa L\gt\sqrt3$ can therefore label two distinct solutions: a small-radius solution and a large-radius solution. At the minimum the two radius branches meet. This turning point is a statement about the map from horizon radius to surface gravity, not a bifurcation of the horizon within a fixed spacetime.

The mass itself has no turning point. At fixed $L$,

$$\frac{dM}{dr_+}=\frac{1}{2G}\left(1+\frac{3r_+^2}{L^2}\right)\gt0,$$

so increasing $r_+$ always increases both $M$ and $A$. This derivative also supplies a classical-mechanics check. Since $dA=8\pi r_+dr_+$,

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

along the fixed-$L$ static family. The equality tests the normalization of $M$ and of the asymptotic Killing field independently of the field equation used to obtain $f$. It also clarifies why the minimum of $\kappa$ is not an extremal horizon: there $dM/dr_+$ remains positive and $\kappa$ remains nonzero.

Independent limits and scale hierarchy

When $r_+\ll L$, the cosmological term is a perturbation near the horizon:

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

Taking $L\to\infty$ at fixed $M$ makes this exact: $r_+=2GM$ and $\kappa=1/(4GM)$, the Schwarzschild values. This flat limit is an independent check on both normalization and sign.

For $r_+\gg L$, AdS curvature controls the scaling:

$$M\sim\frac{r_+^3}{2GL^2}, \qquad \kappa\sim\frac{3r_+}{2L^2}.$$

The large-radius mass is therefore cubic rather than linear in $r_+$. The horizon remains spherical, however: “large” refers to its size relative to $L$, not to a planar change of topology. A planar geometry can arise from an additional scaling limit or as a separate constant-curvature family, which requires an explicit choice of quotient and charge density.

Two qualifications matter. First, the $M=0$ member is pure AdS in these coordinates and has no positive horizon root. Second, $M\lt0$ produces no regular spherical black-hole exterior of this family; the singular center is not hidden by a horizon. Negative mass parameters can occur for compact hyperbolic horizons, but that is a different topology and normalization problem.

Worked transfer diagnostic

At fixed $L$, suppose the measured geometric surface gravity satisfies $\kappa L=2$. Find every compatible spherical horizon radius and mass. Explain why the result does not contradict uniqueness of the horizon.

Solution

Set $x=r_+/L$. The surface-gravity equation becomes

$$2=\frac{1+3x^2}{2x}, \qquad 3x^2-4x+1=0.$$

Hence $x=1$ or $x=1/3$. Substitution into $GM/L=x(1+x^2)/2$ gives

$$\frac{GM_{\mathrm{large}}}{L}=1, \qquad \frac{GM_{\mathrm{small}}}{L}=\frac{5}{27}.$$

These are two different spacetimes with the same normalized surface gravity. In either spacetime $f'(r)\gt0$ for every positive $r$, so its own horizon root remains unique.

Next, Planar and Hyperbolic AdS Horizons separates local horizon curvature from quotient topology and charge normalization.

References

• Gibbons, G. W., Perry, M. J., and Pope, C. N. “The First Law of Thermodynamics for Kerr–anti-de Sitter Black Holes.” Classical and Quantum Gravity 22 (2005): 1503–1526. doi:10.1088/0264-9381/22/9/002. Open PDF.
• Kubizňák, D., Mann, R. B., and Teo, M. “Black Hole Chemistry: Thermodynamics with Lambda.” Classical and Quantum Gravity 34 (2017): 063001. doi:10.1088/1361-6382/aa5c69. Open PDF.