Higher Dimensional Black Holes

• Higher-dimensional black holes are solutions of Einstein gravity in d>4, exhibiting diverse horizon topologies, multi-plane rotations, and extended black objects.
• Advanced methods like the Belinsky–Zakharov inverse scattering and rod structure analysis enable the generation and classification of these complex spacetimes.
• Instabilities such as the Gregory–Laflamme and ultraspinning effects drive rich dynamical phases, linking gravitational theory with string and gauge/gravity frameworks.

Higher dimensional black holes are exact and numerically constructed solutions of Einstein gravity and its extensions in spacetime dimensions $d > 4$, exhibiting a vast spectrum of horizon topologies, dynamical phases, and stability features far exceeding the possibilities in four dimensions. They are studied both as a natural mathematical extension of General Relativity, and because they play central roles in supergravity, string theory, and the gauge/gravity correspondence.

1. Key Classes and Properties of Higher-Dimensional Black Holes

In $d > 4$ spacetime dimensions, the Einstein field equations admit a much richer set of black objects whose properties starkly contrast their four-dimensional analogues.

• Myers–Perry black holes generalize the Kerr solution and, in $d > 4$, support up to $N = \lfloor (d-1)/2 \rfloor$ independent angular momenta. Rotation can occur simultaneously in multiple orthogonal planes, a property absent in $d=4$ (0801.3471).
• Non-spherical horizon topologies become allowed. While in $d=4$ Hawking's topology theorem enforces $S^2$ event horizons, in $d=5$, explicit solutions exist for black objects with horizon topology $S^1\times S^2$ (black rings), and more exotic multi-black hole composites such as black Saturns or concentric black rings (0801.3471, Tomizawa et al., 2011).
• Extended black objects include black strings ($S^{d-3} \times S^1$), black branes ($S^{d-p-2} \times \mathbb{R}^p$), and Kaluza–Klein black holes, which may be locally asymptotically flat but compact in one or more directions (Tomizawa et al., 2011).
• Ultra-spinning regimes for $d \geq 6$ allow for solutions with arbitrarily large rotation parameters, resulting in horizon "pancaking" along the rotation plane and local horizon geometries resembling black membranes (0801.3471, Pereñiguez, 2018). In some cases, the near-horizon geometry matches that of lower-dimensional black branes.

These phenomena are enabled by the higher dimensional Einstein equations’ increased number of gravitational degrees of freedom and the balance between gravitational and centrifugal forces, which differs fundamentally from the four-dimensional case (Pereñiguez, 2018).

2. Solution-Generating Techniques and Classification

The complexity of the field equations in higher dimensions motivates the systematic use of symmetry-adapted methodologies and solution-generating algorithms.

• Belinsky-Zakharov (BZ) inverse scattering method: For spacetimes with $(d-3)$ or more commuting Killing fields, the metric can be cast in a "generalized Weyl" form where all but two coordinates are ignorable. In this sector, the Einstein equations reduce to soliton equations for a symmetric matrix $g_{ab}(r,z)$ on a $(r,z)$ base, with $\det g_{ab} = -r^2$. New solutions are built by "dressing" a seed solution using a matrix $\chi(\lambda, r, z)$ depending on a spectral parameter $\lambda$, producing solutions such as Myers–Perry black holes and black rings from static seeds (0801.3471, Tomizawa et al., 2011).
• Rod (interval) structure analysis: The classification and balance conditions of higher-dimensional solutions, especially in five dimensions, are encoded in the so-called rod structure, which tracks the degeneracy loci of Killing vectors on the symmetry axis. This provides a framework for classifying black hole topologies, including multi-black holes, black rings, and lens spaces (Tomizawa et al., 2011, Ida et al., 2011).
• Twistor techniques and the generalized Ernst potential: In higher dimensions, the algebraically rich structure of symmetry-reduced vacuum solutions can be "linearized" via the construction of a higher-dimensional Ernst potential tied to twist data and Killing vector inner products, facilitating twistor approaches to solution construction and uniqueness studies (Metzner, 2012).
• Squashing transformations and Kaluza–Klein compactification: Deforming five- (or higher-) dimensional solutions to match compactified asymptotics, often using "squashing" of the fiber directions, generates Kaluza–Klein black holes and sequences of "caged" or localized black objects (Tomizawa et al., 2011, Kunz, 2013).

3. Topology, Symmetry, and Uniqueness

Several key results and open problems hinge on the interplay of topology, symmetry, and uniqueness:

• Event horizon topology in $d > 4$ is much less restricted. Under the dominant energy condition, cross-sections of the event horizon must admit metrics of positive scalar curvature (positive Yamabe type), but this includes a wealth of possibilities: $S^3$, $S^1\times S^2$, and certain lens spaces in $d=5$ and beyond (Ida et al., 2011).
• Rigidity theorem in higher dimensions ensures the existence of at least one extra $U(1)$ symmetry if the stationary Killing field is not tangent to the horizon generators, but additional axisymmetries are typically assumed for uniqueness results (Ida et al., 2011).
• Uniqueness fails generically. While Schwarzschild–Tangherlini black holes are unique among static, spherically symmetric solutions, even the stationary vacuum case allows for continuous non-uniqueness once rotation is included—notably in $d=5$ where both the Myers–Perry black hole and the black ring can share the same mass and angular momenta (0801.3471, Tomizawa et al., 2011).
• Classification problems utilize not only global charges (mass, angular momenta) but also local data (rod structure, horizon topology, local invariants) (Tomizawa et al., 2011, Ida et al., 2011). Complete classification remains unresolved, especially in $d \geq 6$ and for partially broken symmetries.

4. Instabilities and Dynamical Phases

Higher-dimensional black holes exhibit an array of stability and phase structure phenomena:

• Gregory–Laflamme instability: Uniform black strings and extended black branes are unstable below a critical wavelength, leading to non-uniform, possibly fractal horizon geometries and complicating the endpoint of classical gravitational evolution (Reall, 2012, Kleihaus et al., 2016).
• Ultraspinning instability: For large angular momentum, Myers–Perry black holes become unstable to non-axisymmetric perturbations, initiating a cascade towards non-uniform or lumpy horizon geometries (Reall, 2012).
• Black rings and critical mergers: Solutions in the black ring and black string sectors display "thin" and "fat" phases with merging points, suggesting topology-changing transitions and the possible existence of "pinched" or "lumpy" black holes (Kleihaus et al., 2016).
• Super-entropic black holes: In AdS, certain ultraspinning limits produce black holes with noncompact horizons of finite area, violating the isoperimetric inequality (i.e., their entropy per thermodynamic volume exceeds that of the Schwarzschild–AdS hole) (Kleihaus et al., 2016).

5. Black Holes in Supergravity and Gauge/Gravity Correspondence

In supergravity and string/M-theory reductions, higher-dimensional black holes acquire additional structure:

• Charged Myers–Perry generalizations: Gauge charges, dipoles, and magnetic charges can be included, yielding BPS and non-BPS solutions such as the BMPV black hole, as well as multiply charged black rings (0801.3471).
• Black holes in AdS and the AdS/CFT framework: Gauged supergravity admits asymptotically AdS black holes with rich thermodynamics and horizon topology, providing gravitational duals for thermal states in CFTs (0801.3471). Notably, global Killing fields can remain timelike to infinity, allowing for Hartle–Hawking states and the definition of global corotating frames.

6. Thermodynamics, Hawking Radiation, and Black Hole Microphysics

The inclusion of extra dimensions and the multitude of black object solutions create an expanded landscape of thermodynamic and quantum properties:

• Generalized laws of black hole mechanics incorporate work terms due to dipole charges (e.g., in black rings) and may admit multiple distinct entropy–area relations dependent on the solution branch and topology (0801.3471, Bravetti et al., 2012).
• Phase structure richness: Competing phases (e.g., small vs. large AdS black holes, black rings vs. Myers–Perry black holes) and transitions such as the Hawking–Page phase transition appear with added complexity in higher dimensions (0801.3471, Bravetti et al., 2012).
• Microscopic entropy via string theory: For certain BPS black holes, microscopically counting the D-brane state degeneracies precisely matches the Bekenstein–Hawking area law (0801.3471).
• Hawking radiation and stability: The emission spectra, superradiant instability, and energy loss channels scale sensitively with the number of dimensions, modifying evaporation and end-state scenarios when compared to $d=4$.

7. Open Problems and Future Directions

Substantial challenges remain in higher-dimensional black hole theory:

• Classification of stationary black holes in $d \geq 6$, especially for solutions lacking maximal rotational symmetry or with exotic horizon topology, is unresolved (0801.3471).
• Physical parameterization: The correct set of physical charges, local invariants, and "hair" characterizing black hole solutions in all regimes is not fully known.
• Dynamical evolution: The endpoints of Gregory–Laflamme and ultraspinning instabilities, the formation and merger of horizons with distinct topology, and the possible violation of cosmic censorship are active areas of research.
• Exotic solutions and stability: The existence of AdS black rings, hairy or lumpy black holes in supergravity, and their stability and quantum properties remain under investigation.
• Microscopic origin of entropy: Understanding beyond supersymmetric and BPS backgrounds, particularly for the non-BPS or near-extremal regime in string/M-theory, is ongoing.

The paper of higher-dimensional black holes, at the intersection of mathematical relativity, string theory, and field theory, continues to expose novel geometrical, topological, and dynamical structures, illuminating the fundamental relativistic and quantum properties of gravity (0801.3471).