Catalog of AdS black holes

This page is a working catalog of black-hole geometries in asymptotically anti-de Sitter (AdS) spacetime. The organizing principle is holographic: for each family of solutions, the useful questions are not only what is the metric?, but also what boundary state does it describe?, what ensemble is natural?, which instabilities does it have?, and what phases does it compete with? Throughout, the bulk dimension is $D=d+1$, so the dual field theory lives in $d$ spacetime dimensions. The AdS radius is $L$, the Newton constant is $G_{d+1}$, and the boundary metric is defined only up to a Weyl factor.

Pure AdS gravity

The simplest theory is Einstein gravity with negative cosmological constant,

$$S_{\rm bulk}=\frac{1}{16\pi G_{d+1}} \int d^{d+1}x\sqrt{-g}\,(R-2\Lambda), \qquad \Lambda=-\frac{d(d-1)}{2L^2}.$$

For a well-defined variational principle and finite holographic one-point functions, one supplements this with the Gibbons—Hawking—York term and the standard local counterterms on the asymptotic boundary. The equations of motion are

$$R_{\mu\nu}-\frac12 g_{\mu\nu}R+\Lambda g_{\mu\nu}=0, \qquad\Longleftrightarrow\qquad R_{\mu\nu}+\frac{d}{L^2}g_{\mu\nu}=0.$$

A solution is called asymptotically locally AdS when, near the conformal boundary, it approaches AdS up to a choice of boundary conformal class. In Fefferman—Graham coordinates,

$$ds^2=\frac{L^2}{z^2}\left(dz^2+g_{ij}(z,x)dx^i dx^j\right), \qquad g_{ij}(z,x)=g^{(0)}_{ij}(x)+\cdots+z^d g^{(d)}_{ij}(x)+\cdots .$$

The leading coefficient $g^{(0)}_{ij}$ is the boundary metric, while $g^{(d)}_{ij}$ determines the CFT stress tensor after holographic renormalization.

Schwarzschild-AdS and topological AdS black holes

The basic static family is

$$ds^2=-f_k(r)dt^2+\frac{dr^2}{f_k(r)}+r^2 d\Sigma_{k,d-1}^2,$$

with

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

Here $d\Sigma_{k,d-1}^2$ is the metric on a unit sphere, plane, or hyperbolic space:

$$\Sigma_{+1}=S^{d-1}, \qquad \Sigma_0=\mathbb{R}^{d-1}\ \text{or a torus quotient}, \qquad \Sigma_{-1}=H^{d-1}\ \text{or a compact hyperbolic quotient}.$$

The horizon radius $r_h$ is the largest root of $f_k(r_h)=0$, so

$$\mu=r_h^{d-2}\left(k+\frac{r_h^2}{L^2}\right).$$

The temperature and entropy are

$$T=\frac{f'_k(r_h)}{4\pi} =\frac{1}{4\pi}\left(\frac{d r_h}{L^2}+\frac{(d-2)k}{r_h}\right), \qquad S=\frac{V_{k,d-1}r_h^{d-1}}{4G_{d+1}},$$

where $V_{k,d-1}$ is the regulated volume of $\Sigma_{k,d-1}$. The three values of $k$ have different physics.

Spherical black holes: $k=+1$

The boundary is the Einstein static universe $\mathbb{R}_t\times S^{d-1}$. There are two branches above the minimum temperature,

$$T_{\min}=\frac{\sqrt{d(d-2)}}{2\pi L}, \qquad r_h=L\sqrt{\frac{d-2}{d}},$$

for $d>2$. The small black hole has negative specific heat and is locally thermodynamically unstable. The large black hole has positive specific heat. In the canonical ensemble, the Hawking—Page transition occurs at

$$r_h=L, \qquad T_{\rm HP}=\frac{d-1}{2\pi L}.$$

Holographically, this is the large-$N$ confinement/deconfinement transition of the CFT on $S^{d-1}$.

Planar black branes: $k=0$

The planar black brane is the workhorse of finite-temperature holography. In a common normalization,

$$ds^2=\frac{r^2}{L^2}\left[-\left(1-\frac{r_h^d}{r^d}\right)dt^2+d\vec{x}^{\,2}\right] +\frac{L^2dr^2}{r^2\left(1-r_h^d/r^d\right)},$$

with

$$T=\frac{d\,r_h}{4\pi L^2}, \qquad s=\frac{1}{4G_{d+1}}\left(\frac{r_h}{L}\right)^{d-1}.$$

It describes a homogeneous deconfined plasma on Minkowski space. Perturbations of this geometry give the hydrodynamic modes of the dual CFT, including the universal two-derivative result $\eta/s=1/(4\pi)$ for Einstein gravity.

Hyperbolic black holes: $k=-1$

Hyperbolic AdS black holes have boundary geometry conformal to $\mathbb{R}_t\times H^{d-1}$. They are important in the holographic computation of Rényi entropies across spherical entangling surfaces, because the causal development of a ball in flat space is conformal to $\mathbb{R}\times H^{d-1}$. The massless hyperbolic black hole is locally pure AdS written in accelerated coordinates. Extremal hyperbolic black holes exist at

$$r_h^2=L^2\frac{d-2}{d},$$

where the near-horizon region develops an $\mathrm{AdS}_2$ factor.

BTZ black holes in $\mathrm{AdS}_3$

In three bulk dimensions, vacuum Einstein gravity has no local propagating gravitons, but it has black holes when the angular direction is periodically identified. The rotating BTZ metric may be written as

$$ds^2=-N^2dt^2+\frac{dr^2}{N^2}+r^2\left(d\phi+N^\phi dt\right)^2,$$

with

$$N^2=-M+\frac{r^2}{L^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2}.$$

Equivalently, in terms of outer and inner horizons $r_\pm$,

$$T=\frac{r_+^2-r_-^2}{2\pi L^2 r_+}, \qquad \Omega=\frac{r_-}{Lr_+}, \qquad S=\frac{2\pi r_+}{4G_3}.$$

The BTZ black hole is the cleanest laboratory for the microscopic origin of black-hole entropy: the Bekenstein—Hawking entropy follows from the Cardy formula using the Brown—Henneaux central charge

$$c=\frac{3L}{2G_3}.$$

It is also central to modern discussions of wormholes, multi-boundary geometries, and the relation between semiclassical gravity and ensembles.

Kerr-AdS black holes

Kerr-AdS black holes are rotating black holes in global AdS. In four bulk dimensions, one standard form uses

$$\Delta_r=(r^2+a^2)\left(1+\frac{r^2}{L^2}\right)-2mr, \qquad \Xi=1-\frac{a^2}{L^2},$$

with $|a|<L$ for a non-singular conformal boundary. The horizon is at $\Delta_r(r_h)=0$, and the physical angular velocity relative to a non-rotating frame at infinity is

$$\Omega_H=\frac{a\left(1+r_h^2/L^2\right)}{r_h^2+a^2}.$$

Holographically, Kerr-AdS describes a rotating thermal state on the boundary Einstein universe. It is a key example where global boundary structure matters: AdS acts like a confining box, so waves reflected from the boundary can repeatedly scatter off the rotating horizon. When the superradiant condition is met, rotating black holes can become unstable. Nonlinear endpoints include rotating hairy black holes, geons, and black resonators, depending on dimension and matter content.

In higher dimensions, Kerr-AdS black holes can carry several independent angular momenta, one for each rotation plane. They are essential in the phase diagram of rotating large-$N$ plasmas and in the study of ultraspinning instabilities.

AdS soliton: not a black hole, but often the competitor

The AdS soliton is obtained by a double analytic continuation of the planar black brane. It has no horizon, caps off smoothly in the interior, and is the lowest-energy solution for a boundary with one spatial circle and antiperiodic fermion boundary conditions. Although it is not a black hole, it competes thermodynamically with planar black branes and is frequently used as a holographic model of confinement.

For a boundary $\mathbb{R}^{d-2,1}\times S^1$, the black brane/AdS soliton transition is the analogue of the Hawking—Page transition. In bottom-up holographic QCD and holographic insulator/superconductor models, the soliton background is often the confining or insulating phase.

AdS C-metric

The C-metric describes accelerating black holes. In AdS, acceleration can be supported by conical defects, cosmic strings, struts, or by the choice of boundary geometry. A common four-dimensional form is schematically

$$ds^2=\frac{1}{A^2(x-y)^2} \left[-F(y)dt^2+\frac{dy^2}{F(y)}+\frac{dx^2}{G(x)}+G(x)d\phi^2\right],$$

where $A$ is an acceleration scale and the zeros of the functions $F$ and $G$ determine horizons, axes, and possible conical singularities.

The AdS C-metric is unusually useful because it gives analytic access to physics that is otherwise numerical:

• black holes localized near the AdS boundary;
• boundary black holes coupled to strongly interacting CFT matter;
• braneworld black holes in Karch—Randall or Randall—Sundrum-type setups;
• transitions between droplet-like and funnel-like horizon topologies.

The main caution is that the global interpretation depends sensitively on parameter ranges and on how conical defects are treated. A C-metric that is perfectly regular in one patch may represent a boundary black hole, an accelerating pair of black holes, or a braneworld black hole in another description.

Black droplets and black funnels

Black droplets and black funnels are asymptotically AdS solutions whose conformal boundary contains a black hole. They are best understood as bulk duals of strongly coupled CFT states on a fixed black-hole background.

A typical boundary metric has the form

$$ds_{\partial}^2=-f_{\partial}(R)dt^2+\frac{dR^2}{f_{\partial}(R)}+R^2d\Omega_{d-2}^2,$$

with a boundary horizon at $f_{\partial}(R_h)=0$. In the bulk, there are two qualitatively different possibilities.

Black droplets

A black droplet is a horizon attached to the boundary black hole that caps off in the bulk. If a planar or hyperbolic plasma horizon exists deeper in the bulk, the droplet horizon is disconnected from it.

The dual interpretation is a CFT state in which the degrees of freedom near the boundary black hole are not efficiently exchanging heat with the distant plasma. At large $N$ and strong coupling, such states can resemble Hartle—Hawking, Unruh, or Boulware-like states depending on boundary conditions and horizon regularity.

Black funnels

A black funnel is a connected horizon joining the boundary black hole to an extended bulk horizon. The horizon has a neck-like region and reaches the deep infrared of the geometry.

The dual interpretation is a deconfined plasma that can transport energy efficiently between the boundary black hole and the ambient heat bath. Funnels are therefore natural gravitational duals of heat flow in strongly coupled quantum field theory on black-hole backgrounds.

Droplet/funnel transition

The droplet/funnel distinction is a topological and thermodynamic one. Roughly, large and hot boundary black holes tend to favor funnels, while small or cold boundary black holes tend to favor droplets. The precise phase boundary depends on the boundary geometry, dimension, ensemble, and possible matter fields.

Black tunnels and black hammocks

The terms black tunnel and black hammock are used for multi-horizon generalizations of droplets and funnels. The terminology is less standardized than for droplets and funnels, so the invariant data are the horizon topology, the boundary horizon components, and the connectedness of the bulk horizon.

A useful working distinction is:

• a black tunnel is a connected bulk horizon stretching between two or more boundary black holes, rather like a funnel with multiple mouths;
• a black hammock is a connected horizon suspended between localized boundary horizons and an extended infrared or plasma horizon, with several necks or attachment regions.

These geometries are mainly constructed numerically. They are interesting because they encode strongly coupled heat transport, nonlocal horizon connectivity, and possible topology-changing transitions in boundary black-hole backgrounds.

Black rings in AdS

In $D\ge5$, black holes need not have spherical horizon topology. A black ring has horizon topology

$$S^1\times S^{D-3}.$$

In asymptotically flat five-dimensional gravity, black rings are exact solutions. In AdS, the negative cosmological constant introduces a confining scale, and regular rings typically require rotation. Fully backreacted AdS black rings are much harder than their flat-space cousins and are often studied by perturbative, matched-asymptotic, blackfold, or numerical methods.

Holographically, AdS black rings describe rotating ring-shaped lumps of deconfined plasma. They are closely related to the broader question of how horizon topology is represented in the dual large-$N$ field theory. Depending on the boundary compactification and ensemble, they can compete with spherical black holes, black branes, plasma balls, and plasma rings.

Black binaries and multi-black-hole configurations

A black binary in AdS is a configuration with two localized black holes or two boundary black-hole sources. In pure vacuum Einstein gravity, static multi-black-hole equilibrium is highly constrained; without rotation, external fields, boundary anchoring, or conical defects, one typically expects a force-balance obstruction.

AdS boundary conditions make the problem richer. The boundary metric can hold two black holes at fixed separation, while the bulk may respond with:

• two disconnected droplets;
• a connected tunnel-like horizon;
• a funnel connecting one or both boundary black holes to a plasma horizon;
• time-dependent heat flow if the two boundary horizons have different temperatures.

These solutions are natural laboratories for nonequilibrium holography, strong-coupling Hawking radiation, and geometric measures of quantum connectivity.

Entanglement islands in higher dimensions

Entanglement islands are not a separate black-hole metric, but a way of computing fine-grained entropy in gravitational systems. The island formula states that the entropy of a non-gravitating radiation region $R$ is obtained by extremizing

$$S(R)=\min_I\,\mathrm{ext}\left[ \frac{\mathrm{Area}(\partial I)}{4G_N}+S_{\rm bulk}(R\cup I) \right],$$

where $I$ is a candidate island in the gravitating region.

In higher-dimensional AdS/CFT, islands often appear in double holography: a gravitating brane or lower-dimensional black hole is embedded in a higher-dimensional asymptotically AdS spacetime, and the island saddle becomes an ordinary classical extremal surface in the higher-dimensional geometry.

Important higher-dimensional questions include:

• how islands arise for localized braneworld black holes rather than eternal black branes;
• how droplet/funnel topology affects Page curves;
• whether the relevant extremal surfaces pass through a black funnel neck or wrap around a droplet;
• how higher-curvature terms and quantum bulk matter modify the generalized entropy.

The technical challenge is that realistic higher-dimensional island geometries are rarely analytic. They often require numerical Einstein equations, brane junction conditions, and extremal-surface searches in the resulting bulk spacetime.

With matter fields

Matter fields enlarge the AdS black-hole landscape and make contact with condensed matter, nuclear matter, and quantum information. A generic bottom-up action has the form

$$S=\frac{1}{16\pi G_{d+1}}\int d^{d+1}x\sqrt{-g}\left[ R+\frac{d(d-1)}{L^2}+\mathcal{L}_{\rm matter}(g,\Phi,A_\mu,\cdots) \right].$$

Different choices of $\mathcal{L}_{\rm matter}$ encode conserved currents, order parameters, momentum relaxation, flavor sectors, or relevant deformations of the CFT.

Reissner—Nordström-AdS black holes

Einstein—Maxwell theory is the canonical finite-density model:

$$S=\frac{1}{16\pi G_{d+1}}\int d^{d+1}x\sqrt{-g}\left[ R+\frac{d(d-1)}{L^2}-\frac{L^2}{4}F_{\mu\nu}F^{\mu\nu}\right].$$

A charged topological black hole has

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Sigma_{k,d-1}^2,$$

with

$$f(r)=k+\frac{r^2}{L^2}-\frac{m}{r^{d-2}}+\frac{q^2}{r^{2d-4}}, \qquad A=A_t(r)dt,$$

up to dimension-dependent normalizations of $q$ and $A_t$.

Holographically, the boundary value of $A_t$ is the chemical potential $\mu_{\rm chem}$, and the normalizable coefficient determines the charge density. The planar RN-AdS black brane is the standard large-$N$ finite-density saddle.

At zero temperature, the planar RN-AdS near-horizon region becomes $\mathrm{AdS}_2\times\mathbb{R}^{d-1}$. This emergent $\mathrm{AdS}_2$ controls many low-energy observables and is responsible for locally critical behavior. It is also a warning sign: the extremal RN-AdS black brane has a large ground-state entropy in the classical limit, and it is often unstable to additional charged or neutral matter fields.

Common instabilities include:

• charged scalar condensation, leading to holographic superconductivity;
• spatially modulated phases when Chern—Simons, axion, or pseudoscalar couplings are present;
• electron-star or fermion-fluid phases when charged fermions backreact;
• dilatonic flows that replace the $\mathrm{AdS}_2$ throat by a hyperscaling-violating infrared geometry.

Dyonic and magnetic AdS black branes

In four bulk dimensions, one can turn on both electric and magnetic charge,

$$F=E(r)\,dr\wedge dt+B\,dx\wedge dy.$$

The dual $2+1$-dimensional theory is at finite charge density and background magnetic field. Dyonic black branes are simple holographic models for Hall transport, magnetohydrodynamics, cyclotron modes, and quantum Hall-like response.

The magnetic field explicitly breaks parity and time-reversal symmetry. In the presence of Chern—Simons couplings or axions, it can trigger chiral magnetic, anomalous Hall, or spatially modulated phases.

Einstein—Maxwell—dilaton black holes

Einstein—Maxwell—dilaton models introduce a neutral scalar $\phi$ that controls the effective gauge coupling and scalar potential:

$$S=\frac{1}{16\pi G_{d+1}}\int d^{d+1}x\sqrt{-g}\left[ R-\frac12(\partial\phi)^2-V(\phi)-\frac14 Z(\phi)F^2\right].$$

These models are flexible enough to realize renormalization-group flows from an AdS ultraviolet fixed point to nontrivial infrared scaling geometries. The infrared may have Lifshitz or hyperscaling-violating form,

$$ds^2\sim r^{2\theta/(d-1)}\left( -\frac{dt^2}{r^{2z}}+\frac{dr^2+d\vec{x}^{\,2}}{r^2} \right),$$

where $z$ is a dynamical critical exponent and $\theta$ is a hyperscaling-violation exponent.

These geometries are used as effective holographic models for strange metals, compressible phases, and QCD-like plasmas. The price of flexibility is that physical conclusions depend strongly on the choice of $V(\phi)$ and $Z(\phi)$.

Axion and massive-gravity black branes

Translational invariance makes the DC conductivity of a finite-density holographic plasma infinite: momentum cannot dissipate. A minimal way to relax momentum is to introduce linear axions,

$$\chi_i=k\,x_i, \qquad i=1,\dots,d-1,$$

with action

$$S=\frac{1}{16\pi G_{d+1}}\int d^{d+1}x\sqrt{-g}\left[ R+\frac{d(d-1)}{L^2}-\frac14F^2-\frac12\sum_i(\partial\chi_i)^2\right].$$

The stress tensor remains homogeneous even though translations are broken. This makes the model analytically tractable and gives closed-form horizon formulae for DC electric, thermoelectric, and thermal conductivities.

Related massive-gravity and Q-lattice models provide controlled ways to study metal-insulator crossovers, incoherent transport, charge diffusion, and bounds on diffusion constants.

Holographic superconductors

A minimal holographic superconductor contains gravity, a $U(1)$ gauge field, and a charged scalar:

$$S = \int d^{d+1}x\,\sqrt{-g}\left[ \frac{1}{16\pi G_{d+1}}\left(R+\frac{d(d-1)}{L^2}\right) -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} -|D\Psi|^2-m^2|\Psi|^2 \right],$$

where $D_\mu=\nabla_\mu-iqA_\mu$. Near the AdS boundary,

$$\Psi(z,x)=z^{d-\Delta}\Psi_{(s)}(x)+z^\Delta\Psi_{(v)}(x)+\cdots, \qquad \Delta(\Delta-d)=m^2L^2.$$

In standard quantization, $\Psi_{(s)}$ is the source and $\Psi_{(v)}$ is the expectation value of the charged operator $\mathcal{O}$. A spontaneous superconducting phase has

$$\Psi_{(s)}=0, \qquad \langle\mathcal{O}\rangle\ne0.$$

The mechanism is often visible in the near-horizon region of an extremal RN-AdS black brane: the effective mass of the charged scalar is lowered by the electric field. If it violates the appropriate near-horizon Breitenlohner—Freedman bound, the normal phase becomes unstable and scalar hair forms.

Important variants include:

• s-wave superconductors, where the order parameter is a scalar;
• p-wave superconductors, where a vector or non-Abelian gauge field condenses;
• d-wave models, built from charged spin-two fields or effective tensor order parameters;
• Josephson junctions, vortices, and superfluid flows;
• holographic insulator/superconductor transitions, often using the AdS soliton as the insulating phase.

Charge density waves, pair density waves, and striped phases

To model charge density waves (CDW), pair density waves (PDW), or related spatially ordered phases, one must allow black holes that break translation invariance. There are two broad mechanisms.

First, translation symmetry can be broken explicitly by a lattice source:

$$\mu(x)=\mu_0+\lambda\cos(kx),$$

or by Q-lattice/axion fields. This models a material with an imposed ionic lattice or disorder-like momentum relaxation.

Second, translation symmetry can break spontaneously. In that case the boundary sources remain homogeneous, but the normal phase has an unstable quasinormal mode at nonzero wave number:

$$\omega=0, \qquad k=k_\star\ne0.$$

The new black hole is spatially modulated, and the modulation wavelength is chosen dynamically.

A CDW has a modulated charge density,

$$\rho(x)=\rho_0+\rho_1\cos(kx)+\cdots,$$

while a PDW has a modulated superconducting order parameter,

$$\langle\mathcal{O}(x)\rangle=\Delta_1\cos(kx)+\cdots.$$

In many holographic models, CDW, PDW, and uniform superconducting orders coexist or compete. The fully backreacted phases are nonlinear inhomogeneous black holes and are usually constructed with the Einstein—DeTurck method, pseudospectral collocation, or finite-element methods.

Holographic QCD matter

Holographic QCD models use asymptotically AdS black holes to describe strongly coupled gauge theories that resemble QCD. The most common bottom-up five-dimensional action is an Einstein—Maxwell—dilaton model,

$$S=\frac{1}{16\pi G_5}\int d^5x\sqrt{-g}\left[ R-\frac12(\partial\phi)^2-V(\phi)-\frac14 Z(\phi)F^2\right]+S_{\rm flavor}+S_{\rm CS}+\cdots .$$

Here:

• $\phi$ is dual to a relevant scalar operator or an effective running coupling;
• $A_t$ is dual to baryon-number chemical potential;
• magnetic fields are introduced through spatial components of $F$;
• flavor branes or tachyon fields may encode chiral symmetry breaking.

The black-hole phase describes the deconfined quark-gluon plasma. The thermal gas or AdS-soliton-like phase describes confinement. By tuning $V(\phi)$ and $Z(\phi)$ to lattice QCD thermodynamics at $\mu=0$, one can model the equation of state, speed of sound, bulk viscosity, baryon susceptibility, and transport at finite temperature and density.

Top-down constructions, such as D3/D7 systems and the Sakai—Sugimoto model, give more constrained embeddings but are usually less flexible phenomenologically. Bottom-up models are more adaptable but must be interpreted as effective descriptions rather than controlled duals of real-world QCD.

Supersymmetric and string-theoretic AdS black holes

Many AdS black holes arise as solutions of gauged supergravity, which is the low-energy limit of string or M-theory compactified on spaces such as $S^5$, $S^7$, or Sasaki—Einstein manifolds. These black holes can carry electric charges, magnetic charges, angular momenta, and scalar hair.

Important classes include:

• charged rotating black holes in $\mathrm{AdS}_5$ dual to states of $\mathcal{N}=4$ super Yang—Mills theory;
• BPS black holes whose entropy can be matched to supersymmetric indices;
• magnetically charged AdS$_4$ black holes related to topologically twisted field theories;
• black strings and black branes whose near-horizon geometries include $\mathrm{AdS}_2$ or $\mathrm{AdS}_3$ factors.

These solutions are central to the microscopic counting of AdS black-hole entropy. The modern picture uses supersymmetric localization, refined indices, and attractor mechanisms to connect the gravitational entropy

$$S_{\rm BH}=\frac{A_h}{4G_N}$$

to a saddle-point evaluation of a dual field-theory partition function.

Stability and phase structure

AdS black holes are not isolated objects; the reflecting AdS boundary makes stability and phase competition especially important.

Thermodynamic stability

In a chosen ensemble, a saddle is locally thermodynamically stable when the Hessian of the appropriate thermodynamic potential is positive. Examples:

• small spherical Schwarzschild-AdS black holes have negative specific heat;
• large spherical Schwarzschild-AdS black holes are locally stable;
• RN-AdS black branes may be locally stable but dynamically unstable to charged scalar condensation;
• rotating Kerr-AdS black holes can be thermodynamically stable but superradiantly unstable.

Dynamical stability

Dynamical instabilities are detected by quasinormal modes with

$$\operatorname{Im}\omega>0.$$

Common examples include:

• Gregory—Laflamme-type instabilities of extended horizons;
• superradiant instabilities of rotating horizons;
• scalar condensation when an effective BF bound is violated;
• spatially modulated instabilities at nonzero wave number;
• turbulence-like cascades in global AdS.

Topology-changing transitions

Several AdS phase transitions involve a change in horizon topology or connectivity:

• thermal AdS $\leftrightarrow$ spherical black hole;
• AdS soliton $\leftrightarrow$ planar black brane;
• droplet $\leftrightarrow$ funnel;
• disconnected droplets $\leftrightarrow$ tunnel/hammock geometries;
• plasma balls/rings $\leftrightarrow$ black holes/rings in confining backgrounds.

At the classical level, topology change is usually mediated by a singular merger cone or by a limiting geometry. Numerically, resolving this regime requires careful gauge choice and high resolution near the neck.

How to read or extend this catalog

When adding a new AdS black-hole entry, it is useful to record the following data.

Geometric data

• bulk dimension $D=d+1$;
• action and matter content;
• asymptotic boundary metric;
• horizon topology and connectedness;
• regularity conditions at axes, horizons, and boundaries;
• analytic form, perturbative expansion, or numerical ansatz.

Thermodynamic data

• ensemble: microcanonical, canonical, grand canonical, mixed, or rotating;
• temperature, entropy, angular velocities, chemical potentials;
• conserved charges from holographic renormalization;
• free energy relative to competing saddles;
• local and global stability.

Holographic data

• dual field-theory state or phase;
• sources and expectation values;
• stress tensor and currents;
• transport coefficients and quasinormal modes;
• order parameters and symmetry breaking;
• entanglement entropy or quantum extremal surfaces, if relevant.

Suggested starting references

This list is intentionally selective. It is meant as a starting point for the families above, not as a complete bibliography.

• S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in Anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577.
• E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, arXiv:hep-th/9803131.
• M. Bañados, C. Teitelboim and J. Zanelli, The Black Hole in Three Dimensional Space-Time, arXiv:hep-th/9204099.
• S. W. Hawking, C. J. Hunter and M. Taylor-Robinson, Rotation and the AdS/CFT correspondence, arXiv:hep-th/9811056.
• G. W. Gibbons, H. Lü, D. N. Page and C. N. Pope, The general Kerr-de Sitter metrics in all dimensions, arXiv:hep-th/0404008.
• A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Charged AdS black holes and catastrophic holography, arXiv:hep-th/9902170.
• V. E. Hubeny, D. Marolf and M. Rangamani, Hawking radiation in large N strongly-coupled field theories.
• T. Wiseman, Numerical construction of static and stationary black holes, a useful entry point to Einstein—DeTurck methods.
• S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Building a Holographic Superconductor, arXiv:0803.3295.
• S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, arXiv:0903.3246.
• J. McGreevy, Holographic duality with a view toward many-body physics, arXiv:0909.0518.
• A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole.
• G. Penington, Entanglement Wedge Reconstruction and the Information Paradox.