Higher-Dimensional Black Holes and Black Strings

With one compact spatial circle, vacuum general relativity admits both black holes localized on the circle and black strings whose horizons wrap it. The ratio of horizon thickness to circle length organizes the simplest branches, but does not by itself choose a unique endpoint: dimension, conserved tension, perturbation sector, ensemble, and nonlinear initial data all matter. This page separates exact solutions, linear instability, static numerical branches, and dynamical evidence.

Required background. Gravitational Actions and Boundary Terms supplies the action, asymptotic charges, and boundary variations; Gauge Theory and String-Theory Preparation supplies compact-circle and Kaluza–Klein terminology.

Helpful background. Einstein Equations supplies the vacuum equations, Formulations of General Relativity clarifies perturbative and numerical initial-boundary problems, and Exact Spacetimes supplies the global comparison of horizon topology and asymptotics.

A localized-hole approximation and an exact uniform string

Use $D\ge5$, signature $(-,+,\ldots,+)$, vacuum Einstein gravity, and asymptotics $\mathbb R^{D-2,1}\times S^1$ with circle coordinate $z\sim z+L$. A small localized object, $r_h/L\ll1$, is approximated near its horizon by the $D$-dimensional Schwarzschild–Tangherlini metric

$$ds_D^2=-f_D(r)dt^2+\frac{dr^2}{f_D(r)}+r^2d\Omega_{D-2}^2, \qquad f_D(r)=1-\left(\frac{r_h}{r}\right)^{D-3},$$

with

$$M=\frac{(D-2)\Omega_{D-2}}{16\pi G_D}r_h^{D-3}, \qquad T_H=\frac{D-3}{4\pi r_h}.$$

Here $\Omega_p=2\pi^{(p+1)/2}/\Gamma((p+1)/2)$ is the area of the unit $p$-sphere. The static exterior is $r\gt r_h$; the apparent pole at $r=r_h$ is removed by an ingoing horizon chart. Its horizon has topology $S^{D-2}$. The displayed mass and temperature are exact for the isolated Tangherlini solution and leading small-hole approximations for the compact-circle problem. Images around the circle correct the geometry and charges, so the exact localized Kaluza–Klein solution is not simply the isolated metric.

An exact uniform string is obtained instead by multiplying a $(D-1)$-dimensional Tangherlini solution by the circle:

$$ds_D^2=-f_s(r)dt^2+\frac{dr^2}{f_s(r)}+r^2d\Omega_{D-3}^2+dz^2, \qquad f_s(r)=1-\left(\frac{r_s}{r}\right)^{D-4}.$$

Its horizon is $S^{D-3}\times S^1$. Direct products preserve Ricci flatness, which independently checks the field equation. Emparan and Reall derive both the Tangherlini normalization and black-brane product in Emparan and Reall 2008, §3.2, Eqs. (25)–(30), PDF. The two topologies already show why four-dimensional uniqueness intuition is insufficient.

The thickness-to-length threshold

Perturb a uniform string by $h_{AB}\propto e^{\Omega t+ikz}$. Gregory and Laflamme found a threshold $k_{\rm GL}r_s$ depending on $D$: modes with sufficiently long wavelength, $0\lt |k|\lt k_{\rm GL}$, grow. The homogeneous $k=0$ sector must be treated separately; the unstable band has a nonzero circle wave number. Periodicity restricts $k_n=2\pi n/L$, so the first mode becomes unstable when

$$\frac{r_s}{L}\lt\frac{k_{\rm GL}r_s}{2\pi}.$$

Thus a thin, long string is unstable, while a sufficiently thick uniform string has no instability in this perturbation sector. The calculation assumes a neutral uniform background and specified regular-horizon and asymptotic boundary conditions; it neither constructs a localized hole nor determines the nonlinear endpoint. The mode content and threshold logic are reviewed in Emparan and Reall 2008, §3.4, PDF, based on the original analysis Gregory and Laflamme 1993, pp. 2–6, PDF.

At the marginal mode $\Omega=0$, a nonuniform static string branch bifurcates. Its horizon still wraps the circle but its radius varies with $z$. A localized-hole branch exists separately at small mass. The major frontier question is how these static branches meet and how, if at all, a dynamical unstable string approaches that phase structure.

Results and open questions through 2026-08-19

The evidence cutoff for this frontier assessment is 2026-08-19. “Result” below means a result within the stated formulation, not a universal claim about charged, rotating, AdS, higher-curvature, or quantum-corrected objects.

Formulation or evidence	Result	Scope and limitation
Exact neutral vacuum solutions with Kaluza–Klein asymptotics	Tangherlini-like localized and uniform wrapped geometries supply controlled small-$r_h/L$ and uniform-string limits.	The localized metric is only asymptotically approximated by isolated Tangherlini; exact endpoints do not select a phase at intermediate mass.
Linear perturbations of a uniform neutral string	A marginal wavelength $k_{\rm GL}^{-1}$ separates stable and growing Fourier sectors Emparan and Reall 2008, §3.4, PDF.	Linear theory identifies onset, not saturation, topology change, or the result for generic initial data.
Perturbative nonuniform strings across dimensions	At fixed mass, the entropy comparison changes between $D\le13$ and $D\ge14$ in the near-threshold calculation; Sorkin denotes the dividing integer by $D_*=13$ Sorkin 2004, pp. 1–6, Eqs. (1)–(8), PDF.	Near-threshold perturbation theory and ensemble choice do not establish global stability or a dynamical endpoint.
Five-dimensional nonlinear evolution with imposed symmetry	A self-similar sequence of thinner string segments was resolved, supporting finite-asymptotic-time curvature blow-up if the cascade persists Lehner and Pretorius 2010, pp. 1–4 and Table I, PDF.	Finite resolution cannot reach zero radius; the calculation assumes neutral vacuum data, periodicity, and $SO(3)$ symmetry, so the final singular limit remains an inference.
Static localized holes and nonuniform strings near merger	Independent spectral constructions in $D=5,6$ find matching spiral critical behavior and double-cone exponents Kalisch, Moeckel, and Ammon 2017, §§2–3.4 and Appendix B, PDF.	Both numerical branches terminate before the singular transit geometry; static, reflection-symmetric solutions do not decide dynamical genericity.

This matrix supports a strong but conditional phase picture. It does not establish a theorem that every source with a given scale ratio forms one topology, nor that every unstable string follows the same cascade. The topology-changing transit is singular in classical GR, and how ultraviolet physics resolves it is unresolved.

Competing interpretations and negative limits

The conservative position accepts the exact branches, the Gregory–Laflamme instability, and the reported numerical solutions within their symmetry and error controls, but treats merger and endpoint claims as formulation-dependent. Static branch proximity is not time evolution, and a finite numerical cascade is not a proof of its infinite continuation.

The more permissive position observes that three independent structures cohere: a marginal mode generates nonuniform strings, localized and nonuniform static branches approach the same double-cone scaling, and dynamical evolution generates progressively thinner string necks. Taken together, this is compelling evidence for the standard phase diagram in neutral vacuum $D=5,6$. It is still evidence for a specified theory class and background, not a universal verdict.

Several adverse cases prevent overgeneralization. A thick uniform string lies on the linearly stable side even though it has the same topology. Near and above the critical dimension, the nonuniform branch has different thermodynamic behavior. Canonical and microcanonical comparisons need not select the same preferred phase. Charge, rotation, a cosmological constant, altered compactification, or higher-curvature terms change both the spectrum and available branches.

The controlled lower-dimensional limit also separates objects. Take $L\to0$ while holding $G_{D-1}=G_D/L$ fixed and probing energies well below $L^{-1}$. The uniform-string sector with vanishing KK vector and constant radion reduces to a $(D-1)$-dimensional vacuum GR black hole. A general zero-mode configuration also retains the vector and radion, so compactification alone does not yield pure GR. A localized hole requiring $r_h\ll L$ cannot remain in that same limit; its assumed scale hierarchy disappears. Decompactification $L\to\infty$ is another singular reorganization of the spectrum, not the lower-dimensional GR limit.

For stationary numerical solutions, the Kaluza–Klein Smarr relation provides an independent constraint,

$$(D-3)M=(D-2)T_HS+\mathcal T L,$$

where $\mathcal T$ is the asymptotic tension. A branch curve that solves discretized equations but violates this relation or the first law has not passed a global normalization check. Kalisch, Moeckel, and Ammon use asymptotic charges, horizon data, Smarr residuals, and convergence together in Kalisch, Moeckel, and Ammon 2017, §§2.2–2.3 and §3.4, PDF.

The conclusions could change if a generic nonlinear evolution changes the proposed cascade, if an existence or stability theorem alters the connection between localized and nonuniform branches, if independent near-merger numerics fail the double-cone scaling, or if controlled higher-curvature physics resolves the transit before classical curvature diverges.

Common pitfalls

Treating the GL threshold as a formation theorem. It is a linear instability of one uniform background. It does not say that arbitrary collapsing data form a localized hole or string.

Comparing only masses. Circle length, asymptotic tension, temperature or entropy ensemble, dimension, and topology are necessary comparison axes.

Equating a static merger with a dynamical endpoint. The static phase diagram and time evolution answer different questions, even when their geometries suggest a common mechanism.

Exercises

In a five-dimensional example take $k_{\rm GL}r_s=0.88$. Determine whether the lowest Fourier mode is unstable for (a) $L=10r_s$ and (b) $L=5r_s$. State exactly what the calculation does and does not prove.

Solution

The allowed lowest wavenumber is $k_1r_s=2\pi r_s/L$. For $L=10r_s$, this is $0.628\lt0.88$, so the lowest mode lies in the growing Gregory–Laflamme band. For $L=5r_s$, it is $1.257\gt0.88$, and no allowed $n\ge1$ mode satisfies the threshold, so this sector is linearly stable. The diagnostic proves only the presence or absence of that linear mode on a uniform neutral string. It does not choose a nonlinear endpoint, establish stability against every perturbation, or predict which topology generic collapse forms.

References

• Emparan, R., and H. S. Reall, “Black Holes in Higher Dimensions,” Living Reviews in Relativity 11 (2008) 6, DOI, Open PDF.
• Gregory, R., and R. Laflamme, “Black Strings and p-Branes Are Unstable,” Physical Review Letters 70 (1993) 2837–2840, DOI, Open PDF.
• Kalisch, M., S. Moeckel, and M. Ammon, “Critical Behavior of the Black Hole/Black String Transition,” Journal of High Energy Physics 08 (2017) 049, DOI, Open PDF.
• Lehner, L., and F. Pretorius, “Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship,” Physical Review Letters 105 (2010) 101102, DOI, Open PDF.
• Sorkin, E., “A Critical Dimension in the Black-String Phase Transition,” Physical Review Letters 93 (2004) 031601, DOI, Open PDF.