Black Strings

A black string is a higher-dimensional horizon extended along one spatial direction. The uniform vacuum string is an exact product of a Schwarzschild–Tangherlini geometry with a line or circle; nonuniform strings are distinct, usually numerical, branches. Their central lesson is that a perfectly regular stationary horizon can be dynamically unstable when its length is large compared with its transverse radius.

Uniform black-string geometry

Property	Statement
Object class	Exact solution family
Theory	$D$-dimensional vacuum general relativity, usually $D\geq5$
Compactification	$z\sim z+L$ for a Kaluza–Klein circle
Horizon topology	$S^{D-3}\times S^1$ for the uniform compact string
Defining scales	Transverse radius $r_0$ and circumference $L$
Physical status	Exact and numerical solutions; no observational evidence

With $z$ periodically identified, the uniform solution is

$$ds_D^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_{D-3}^2+dz^2, \qquad f(r)=1-\left(\frac{r_0}{r}\right)^{D-4}.$$

The product solves $R_{\mu\nu}=0$ because each factor does. Its horizon is at $r=r_0$; it is not an array of separated spherical holes. For the asymptotic normalization shown,

$$\begin{aligned} M&=\frac{(D-3)\Omega_{D-3}L}{16\pi G_D}\,r_0^{D-4}, &\mathcal T&=\frac{\Omega_{D-3}}{16\pi G_D}\,r_0^{D-4},\\ A_H&=L\Omega_{D-3}r_0^{D-3}, &T_H&=\frac{D-4}{4\pi r_0}. \end{aligned}$$

Here $\mathcal T$ is the ADM tension conjugate to $L$, not a material-string tension. The relative tension $n=\mathcal TL/M=1/(D-3)$ is dimensionless and provides a useful phase-diagram coordinate. The derivation and charge normalizations belong with Uniform Black Strings and Branes.

Gregory–Laflamme threshold

Perturb a uniform string by a mode $e^{\Omega t+ikz}$. For a band of sufficiently small $k r_0$, the coupled gravitational equations admit $\Re\Omega>0$; the string is unstable when the compact circle can accommodate such a wavelength. At $k=k_{\mathrm{GL}}$ a static zero mode appears and generates a branch of nonuniform strings. This is a classical instability, not Hawking evaporation. The original calculation is Gregory and Laflamme 1993, complete article, printed pp. 2837–2840.

The threshold depends on $D$ and on charges, boosts, rotation, boundary conditions, and compactification. Consequently the phrase “thin strings are unstable” means thin relative to the available longitudinal wavelength. On an infinite line arbitrarily long modes exist; on a circle, the discrete values $k=2\pi q/L$ determine whether the unstable band is sampled.

Thermodynamics gives a valuable check but not a universal proof. For extended branes, local thermodynamic instability and long-wavelength dynamical instability often coincide under stated ensemble and translational assumptions. Those assumptions can fail, so the Lichnerowicz eigenvalue problem remains the decisive calculation.

Nonuniform and localized phases

A nonuniform string retains topology $S^{D-3}\times S^1$ but its transverse radius varies around the circle. A useful invariant deformation measure compares the largest and smallest horizon-sphere radii. The uniform limit sends that measure to zero and approaches the Gregory–Laflamme zero mode. Finite-amplitude solutions require solving an elliptic boundary-value problem with regular-axis, horizon, and Kaluza–Klein asymptotic conditions; Wiseman 2003, §§ 3–5 constructed a representative branch numerically.

Localized Kaluza–Klein black holes have horizon topology $S^{D-2}$ instead. Phase diagrams indicate that the localized and nonuniform branches approach a topology-changing merger described locally by a double-cone geometry. The critical behavior and which branch is entropically favored depend on dimension; one should not extrapolate a five-dimensional diagram to all $D$. Harmark and Obers 2005, § 3 review the Kaluza–Klein phase coordinates, while Kol 2006, § 3.4 explains the merger and critical-dimension picture.

Linear instability also does not determine the nonlinear endpoint. In five dimensions, simulations find a cascade toward ever thinner string segments, raising a classical-cosmic-censorship issue (Lehner and Pretorius 2010, complete article, printed pp. 101102-1–101102-4). Different dimensions, charges, or boundary conditions can change the evolution. A stationary phase diagram and a time-dependent endpoint are separate results.

Controlled limits and reductions

Decompactification. Taking $L\to\infty$ at fixed $r_0$ gives a string on a line. Total $M$ and $A_H$ diverge, while densities $M/L$ and $A_H/L$ remain finite. This is the correct comparison, rather than treating divergent total charges as a singular geometry.

Short-circle regime. When $L/r_0$ is small, the first allowed Fourier mode lies above the unstable band and the uniform string is linearly stable against the Gregory–Laflamme sector. This statement is boundary-condition dependent and does not establish stability against every perturbation in an extended theory.

Dimensional reduction. Reduction along $z$ turns the unboosted product into a lower-dimensional black hole plus Kaluza–Klein fields. Boosting before reduction produces charge. The dilaton normalization, asymptotic circle, and which higher-dimensional perturbations survive the truncation must be stated; reduction is not evidence that astrophysical four-dimensional holes are black strings.

The horizon definitions, topology, mass, and tension conventions are developed in Black Strings, Branes, and Kaluza–Klein Horizons.

Limitations

Uniform strings are exact analytic solutions. Generic nonuniform and localized branches, merger points, and nonlinear endpoints rely on numerical evidence in specified dimensions. Their existence in compactified models does not establish that nature has large or warped extra dimensions, nor that a produced string would survive competing instabilities. Any phenomenological use must first fix $D$, $L$, asymptotics, matter content, and the allowed perturbation sector.

References

• Gregory, Ruth, and Raymond Laflamme. “Black Strings and p-Branes Are Unstable.” Physical Review Letters 70 (1993): 2837–2840. DOI.
• Harmark, Troels, and Niels A. Obers. “Phases of Kaluza–Klein Black Holes: A Brief Review.” 2005. arXiv:hep-th/0503020.
• Kol, Barak. “The Phase Transition between Caged Black Holes and Black Strings: A Review.” Physics Reports 422 (2006): 119–165. DOI.
• Lehner, Luis, and Frans Pretorius. “Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship.” Physical Review Letters 105 (2010): 101102. DOI.
• Wiseman, Toby. “Static Axisymmetric Vacuum Solutions and Non-Uniform Black Strings.” Classical and Quantum Gravity 20 (2003): 1137–1176. DOI.