Kerr–Anti-de Sitter Black Holes

Kerr–anti-de Sitter is the rotating vacuum black-hole family with negative cosmological constant. Its defining subtlety is global: the Boyer–Lindquist azimuth rotates relative to a nonrotating frame at the timelike conformal boundary. Horizon angular velocity, mass, and angular momentum satisfy the first law only when they are referred to compatible asymptotic generators.

Kerr–AdS geometry

Property	Value
Object class	Exact solution family
Theory	Four-dimensional general relativity with $\Lambda=-3/L^2$
Parameters	metric mass $m$, rotation length $a$, AdS radius $L$
Regular global domain	$\lvert a\rvert\lt L$ and a regular largest positive root
Physical status	Exact $\Lambda$-vacuum solution; boundary-frame-dependent charges

Define

$$\rho^2=r^2+a^2\cos^2\theta, \quad \Delta_r=(r^2+a^2)\left(1+\frac{r^2}{L^2}\right)-2mr,$$ $$\Delta_\theta=1-\frac{a^2}{L^2}\cos^2\theta, \qquad \Xi=1-\frac{a^2}{L^2}.$$

The Boyer–Lindquist metric is

$$\begin{aligned} ds^2={}&-\frac{\Delta_r}{\rho^2} \left(dt-\frac{a\sin^2\theta}{\Xi}d\phi\right)^2 +\frac{\rho^2}{\Delta_r}dr^2+\frac{\rho^2}{\Delta_\theta}d\theta^2\\ &+\frac{\Delta_\theta\sin^2\theta}{\rho^2} \left(a\,dt-\frac{r^2+a^2}{\Xi}d\phi\right)^2 . \end{aligned}$$

The standard global quotient has $\phi\sim\phi+2\pi$ and requires $\Xi\gt0$. The exact charge normalization and its relation to boundary frames were fixed systematically by Gibbons, Perry, and Pope 2005, § 2.1, eqs. (2.5)–(2.15).

Horizon, frame, and extremality

The event horizon lies at the largest positive root $r_+$ of $\Delta_r$, with

$$m=\frac{(r_+^2+a^2)(1+r_+^2/L^2)}{2r_+}.$$

Ingoing rotating coordinates remove the Boyer–Lindquist singularity at a simple root. A root does not suffice: $\Delta_\theta$ must stay positive, the exterior must reach the declared boundary, and the azimuthal quotient must be regular.

The coordinate angular velocity and the velocity relative to nonrotating infinity are

$$\Omega'=\frac{a\Xi}{r_+^2+a^2}, \qquad \Omega=\frac{a(1+r_+^2/L^2)}{r_+^2+a^2}, \qquad \Omega-\Omega'=\frac{a}{L^2}.$$

The standard conserved charges, area, and surface gravity are

$$M=\frac{m}{G\Xi^2}, \qquad J=aM, \qquad A_H=\frac{4\pi(r_+^2+a^2)}{\Xi},$$ $$\kappa= \frac{r_+\left(1+a^2/L^2+3r_+^2/L^2-a^2/r_+^2\right)} {2(r_+^2+a^2)}.$$

Extremality sets $\kappa=0$ and makes the outer root double. Combining that relation with the standard-quotient condition $\Xi\gt0$ gives the sharp regular domain $r_+\lt L/\sqrt3$, with $a$ tuned by the double-root equation. Parameter choices with no appropriate outer root are horizonless or singular continuations, not Kerr–AdS black holes.

Mechanics and stability boundary

At fixed $L$, the nonrotating-boundary quantities obey

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

Pairing $M$ with $\Omega'$ misses the work associated with the changing asymptotic frame. This is not a cosmetic coordinate choice: the Hamiltonian generator changes.

Reflecting AdS boundary conditions can return amplified waves to the horizon. For a neutral bosonic mode, superradiance is possible when $0\lt\omega\lt m_\phi\Omega$, using the nonrotating boundary time. Hawking and Reall 2000, § 2.2, eqs. (2.13)–(2.14) identified a globally timelike generator that obstructs the corresponding energy growth in an appropriate parameter region. Outside that region, loss of the positive-energy estimate opens an instability channel but does not alone prove a growing mode. Explicit small-hole instabilities depend on the field and boundary condition, as demonstrated by Cardoso, Dias, Lemos, and Yoshida 2004, §§ 1 and 4.

Controlled limits

Nonrotating limit. Taking $a\to0$ gives $\Xi\to1$, $J,\Omega\to0$, and Schwarzschild–AdS with the same boundary-time normalization.

Flat limit. Taking $L\to\infty$ at fixed $(m,a)$ makes $\Xi\to1$, merges $\Omega$ and $\Omega'$, and recovers asymptotically flat Kerr.

Extremal limit. Approaching the double-root relation sends $\kappa\to0$ at finite area and produces a near-horizon throat after a coordinate scaling. It is not the same as the boundary ultraspinning limit.

Ultraspinning boundary. Formally taking $a\to L$ at fixed standard azimuth makes $A_H$, $M$, and $J$ diverge through powers of $\Xi^{-1}$. A finite ultraspinning solution requires a simultaneous coordinate rescaling and new quotient; substitution of $\Xi=0$ is not a regular member.

Boundary and stability diagnostics

The stationary-limit surface depends on latitude, as in Kerr, but rotation is measured relative to a boundary frame. The condition $\lvert a\rvert\lt L$ keeps the standard angular quotient and conformal boundary regular. It is independent of the double-root condition: angular regularity, horizon existence, and extremality constrain different features of parameter space.

Local thermodynamic response is distinct from superradiance. Heat capacities and rotational susceptibilities depend on whether angular momentum or angular velocity is fixed; dynamical growth depends on a field spectrum and reflecting falloff. A branch can pass one test and fail another.

The flat limit is nonuniform globally. Although $L\to\infty$ recovers Kerr on bounded regions, AdS normal-mode spectra, causal return times, and boundary charges do not follow by simply deleting $r^2/L^2$. A stability comparison must translate the complete initial-boundary-value problem.

Connections to the detailed theory

Kerr–AdS is exact within vacuum Einstein gravity with a fixed negative cosmological constant. It does not by itself specify a thermal ensemble, dual field theory, or perturbing field content. Its type-D family context is given in Plebański–Demiański and the Type-D Electrovacuum Family. Charges and frames are treated in Kerr–AdS Frames, Charges, and Horizons. The boundary-sensitive dynamical question belongs to Superradiant and Trapping Instabilities in AdS.

References

• Cardoso, Vitor, Óscar J. C. Dias, José P. S. Lemos, and Shijun Yoshida. “The Black Hole Bomb and Superradiant Instabilities.” Physical Review D 70 (2004): 044039. DOI.
• 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 Harvey S. Reall. “Charged and Rotating AdS Black Holes and Their CFT Duals.” Physical Review D 61 (2000): 024014. DOI.