Black Holes in Anti-de Sitter Spacetimes

Anti-de Sitter asymptotics change more than the far end of a black-hole metric. The curvature scale competes with the horizon radius, constant- curvature horizons need not be spherical, conformal infinity is timelike, rotation requires a declared frame there, and varying the cosmological constant enlarges the mechanical phase space. This chapter develops those changes as classical geometry and mechanics. It does not use a boundary field theory, identify horizon area with quantum entropy, or infer stability from a thermodynamic response function alone.

Helpful background. Exact Solutions of Einstein’s Equations supplies the named metric families and their coordinate domains; Causality and Global Structure supplies conformal completion and causal pasts; The Einstein Equation supplies the vacuum equations with cosmological constant; and The GHY Boundary Term supplies the boundary variation needed when the asymptotic phase space is changed.

Geometry with a timelike conformal boundary

In $D$ spacetime dimensions, write

$$\Lambda=-\frac{(D-1)(D-2)}{2L^2}.$$

The scale $L$ enters both the radial horizon equation and the asymptotic normalization of stationary generators. For a four-dimensional static constant-curvature family,

$$f_k(r)=k-\frac{\mu}{r}+\frac{r^2}{L^2}, \qquad k\in\{1,0,-1\}.$$

The value of $k$ fixes the local curvature of a horizon section, but it does not by itself fix global topology. Planar and hyperbolic sections acquire finite total area and total charge only after a compact quotient is specified; otherwise the natural quantities are densities. The allowed mass range also depends on $k$: the hyperbolic family admits regular negative values relative to the conventional zero of energy, including a double-root endpoint. These facts follow directly from the root equation and quotient geometry (Birmingham 1999, §§2–3, Open PDF, pp. 1–4).

The causal definition is a separate layer. Once a conformal completion and a chosen component $\mathscr I_0$ of timelike infinity have been fixed, the black-hole region is

$$\mathcal B =\mathcal M\setminus J^-_{\widetilde{\mathcal M}}(\mathscr I_0).$$

Reflecting, Robin, or other admissible boundary conditions determine how a field evolves at that timelike boundary; they do not move $J^-(\mathscr I_0)$ in a fixed metric. Ishibashi and Wald formulate well-posed static field dynamics through positive self-adjoint extensions of the spatial operator, under assumptions that must be checked field by field (Ishibashi and Wald 2004, §1, Open PDF, pp. 2–4; §3.3, pp. 32–35).

Six pages in working order

1. Schwarzschild–AdS Horizon Families derives the unique spherical horizon, surface gravity, and the small- and large-radius branches at fixed $L$.
2. Planar and Hyperbolic AdS Horizons separates local curvature, quotient topology, charge density, and the negative-mass hyperbolic range.
3. Kerr–AdS Frames, Charges, and Horizons matches energy and angular velocity in the nonrotating frame at infinity and checks the differential first law.
4. AdS Boundary Conditions and the Black-Hole Region distinguishes the metric causal past from the boundary data required for field evolution.
5. Extended AdS Black-Hole Mechanics allows $\Lambda$ to vary, derives the $V\,\delta P$ term and Smarr scaling, and recovers fixed-$\Lambda$ mechanics as a restriction.
6. Ultraspinning Limits and Classical Phase Structure compares singular parameter limits, quotient limits, stationary branches, perturbative thresholds, and nonlinear evolutions through 2026-08-12.

The sequence deliberately fixes roots and normalizations before using them in causal or variational statements. The last page is a frontier comparison, not a universal phase diagram: its claims retain their dimension, boundary conditions, symmetry class, and evidence type.

Three routes and three independent choices

The horizon-geometry route reads the first two pages and then the frontier comparison. It is the shortest route to understanding why spherical, planar, hyperbolic, and ultraspinning horizons cannot be compared using a bare metric parameter or an unqualified word such as “topology.” The stationary-mechanics route reads Schwarzschild–AdS, Kerr–AdS, and extended mechanics. It follows the progression from a root relation to charges, conjugate variables, and Smarr scaling. The causal-evolution route starts with the boundary page, uses Kerr–AdS to fix the asymptotic time generator, and then reads the frontier page to separate boundary conditions, mode evidence, stationary solutions, and finite-amplitude evolution.

Across every route, keep three choices independent:

• Horizon geometry: specify $k$, the quotient if any, dimension, and the component of the root set being studied.
• Asymptotic frame and charges: specify the timelike generator and use the energy and angular velocity conjugate to that same generator. For Kerr–AdS, the nonrotating angular velocity differs from the Boyer–Lindquist coordinate value by $a/L^2$; mixing the two spoils the first law (Gibbons, Perry, and Pope 2005, §2.1, Open PDF, pp. 7–9).
• Boundary and phase-space data: distinguish a field boundary condition on a fixed spacetime from a variation that changes $\Lambda$, boundary geometry, angular identifications, or conserved charges.

When $\Lambda$ is varied, define

$$P=-\frac{\Lambda}{8\pi G}$$

and include its conjugate $V$ in the first law. In four dimensions the uncharged stationary relation becomes

$$\delta M=\frac{\kappa}{8\pi G}\,\delta A +\Omega_H\,\delta J+V\,\delta P, \qquad M=\frac{\kappa A}{4\pi G}+2\Omega_HJ-2PV.$$

Calling $M$ enthalpy is useful only in this declared extended phase space; it is not a claim about a microscopic statistical system (Kubizňák, Mann, and Teo 2017, §§2.2–3, Open PDF, pp. 5–16).

Review the chapter

A calculation of a four-dimensional Kerr–AdS family makes three claims. It uses the energy $M=m/(G\Xi^2)$ associated with the nonrotating time at infinity but pairs it with the coordinate angular velocity $\Omega'_H$; it then varies $L$ while writing only the fixed-$\Lambda$ first law; finally, it changes a scalar Robin boundary condition and says that the event horizon has moved even though the metric and conformal completion are unchanged. Diagnose each claim and give the corresponding correction.

Solution

The first claim mixes asymptotic frames. With $\Xi=1-a^2/L^2$, the angular velocity conjugate to $M$ is the value measured relative to a nonrotating frame,

$$\Omega_H=\Omega'_H+\frac{a}{L^2} =\frac{a(1+r_+^2/L^2)}{r_+^2+a^2}.$$

One may instead use a rotating time generator, but its energy and angular potential must be transformed together.

The second claim varies the theory’s length scale while omitting its work term. Since $P=3/(8\pi GL^2)$ in four dimensions, $\delta L\ne0$ implies $\delta P\ne0$ and the law needs $V\,\delta P$. The fixed-$\Lambda$ formula is recovered only on the subspace $\delta P=0$.

The third claim confuses the causal geometry with a field-evolution rule. The Robin condition can change the scalar spectrum, flux, and stability of the chosen dynamics. It cannot change $M\setminus J^-(\mathscr I_0)$ when the metric, the conformal completion, and the selected boundary component are held fixed. A moved event horizon would require a different spacetime geometry or a different causal endpoint, not merely a different boundary condition for a test field.

After this chapter, you should be able to solve and compare AdS horizon roots, state quotient-dependent charge conventions, match Kerr–AdS charges to an asymptotic frame, separate causal and field-theoretic boundary choices, and decide whether a proposed first law is fixed- or variable-$\Lambda$. Continue to Higher-Dimensional Black Holes and Black Objects to examine which topology, uniqueness, and stability statements survive when dimension and horizon geometry are varied more broadly.

References

• Birmingham, D. “Topological Black Holes in Anti-de Sitter Space.” Classical and Quantum Gravity 16 (1999): 1197–1205. doi:10.1088/0264-9381/16/4/009. Open PDF.
• 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.
• Ishibashi, A., and Wald, R. M. “Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime.” Classical and Quantum Gravity 21 (2004): 2981–3013. doi:10.1088/0264-9381/21/12/012. 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.