Gregory–Laflamme and Correlated Stability

A uniform black string or brane can be unstable to a long-wavelength modulation along its extended direction even though each transverse slice resembles a stationary black hole. The Gregory–Laflamme instability is a concrete dynamical result for specified backgrounds. Correlated stability is the broader proposal that, for suitable translationally invariant horizons, the dynamical threshold coincides with loss of local thermodynamic stability. The proposal is useful but ensemble-, matter-, and boundary-condition-dependent.

Required background. Gauge Theory and String-Theory Preparation supplies black branes and their charge sectors; Differential Forms and Integration supplies the conserved fluxes that determine which perturbations may redistribute charge.

Helpful background. Quantum Gravity frames the phase structure; Quantum Information supports entropy comparisons; and Exact Spacetimes provides the perturbation and horizon geometry.

The Gregory–Laflamme mode

For a uniform neutral black string with coordinate $z$, perturb a gauge-invariant metric sector as

$$\delta g_{\mu\nu}=e^{\Omega t+ikz}h_{\mu\nu}(r).$$

Regularity at the future horizon and normalizability at infinity turn the linearized Einstein equations into an eigenvalue problem. The threshold mode has $\Omega=0$ at $k=k_c$. Modes with sufficiently small $k$ have $\operatorname{Re}\Omega\gt0$ and grow. The original calculation establishes this instability and its dimension-dependent threshold for neutral strings; see Gregory and Laflamme 1993, Eqs. 1–17 and pp. 2837–2840.

If $z$ is compact with length $L$, allowed wavenumbers are $k_n=2\pi n/L$. The string is linearly unstable only if at least one nonzero allowed $k_n$ lies below $k_c$. Compactification can therefore remove a long-wavelength mode without changing the local horizon geometry.

The thermodynamic Hessian

Local thermodynamic stability is a statement about the second variation in the variables that can fluctuate. For energy and redistributable charges $x^i=(\mathcal E,\mathcal Q_A)$, entropy concavity requires

$$\delta^2S =\frac12\frac{\partial^2S}{\partial x^i\partial x^j} \delta x^i\delta x^j\lt0$$

on the allowed subspace, apart from possible marginal directions. Equivalently, the energy is convex in the corresponding entropy and charge densities at positive temperature. Free energies have different signs in their intensive variables: for example, $(\partial^2\mathcal F/\partial T^2)_{\mathcal Q}=-C_{\mathcal Q}/T$, so positive heat capacity requires concavity in $T$, not a universally positive free-energy Hessian. Choosing the wrong ensemble can reverse the diagnosis. A charge topologically fixed on each slice cannot be varied like a charge that diffuses along the brane.

The correlated-stability statement is strongest for noncompact, translationally invariant black branes with no additional hair and with all relevant conserved densities included. Under related assumptions, an Euclidean negative mode can be connected to a static threshold perturbation, as shown in Reall 2001, §§2–5, pp. 2–16.

What the proposed correlation says

The conservative formulation is:

For an appropriate family of translationally invariant black branes, with declared matter content, boundary conditions, and redistributable charges, a negative direction of the local thermodynamic Hessian is expected to accompany a normalizable long-wavelength classical mode.

This is not a theorem for every extended horizon. Compact directions discretize $k$; scalar fields can add dynamical directions not captured by a reduced Hessian; gauge constraints restrict charge variations; and boundary conditions in AdS or a cavity change both spectra. A threshold comparison must use the same family and ensemble on both sides.

The mechanism is physically plausible: a locally entropy-increasing redistribution can seed clumping along the brane. Yet thermodynamics determines neither the growth rate nor the nonlinear endpoint. Pinch-off, nonuniform strings, localized holes, or another phase require separate dynamical and global analysis.

Results and limitations through 2026-08-20

Evidence or source	Supported result or position	Assumptions or discriminator	Limitation or adverse case
Gregory and Laflamme 1993, Eqs. 1–17 and pp. 2837–2840	Uniform neutral black strings possess a long-wavelength growing gravitational mode and a static threshold	Linear perturbations, asymptotic flatness, specified dimension and horizon boundary conditions	It does not prove a thermodynamic equivalence or determine the nonlinear endpoint
Reall 2001, §§2–5, pp. 2–16	Euclidean negative modes and threshold modes are related for broad classes of translationally invariant branes	Appropriate action, noncompact brane, declared charge variations and regular boundary data	Extra matter, compactness, or an incomplete thermodynamic state space can evade the implication
Friess, Gubser, and Mitra 2005, §§2–4, pp. 3–13	Scalar-hair examples expose failures of an unrestricted correlated-stability conjecture	Specific gauged-supergravity matter sectors and boundary conditions	The adverse cases do not erase the correlation in the narrower Einstein–brane domain; they restrict its hypotheses
Matched numerical threshold and Hessian tests	Many charged-brane families show coincident sign changes	Same solution family, ensemble, charge transport, and perturbation sector	Agreement of critical points alone does not establish a general proof or equal growth data

The established result is the specified dynamical instability. Evidence supports a thermodynamic correlation in a controlled class, whereas a universal equivalence is not established. Competing positions differ over whether exceptions merely require a complete thermodynamic state space or show that additional dynamical fields fundamentally escape the criterion.

Checks and failure modes

The primary comparison computes $k_c$ from the static perturbation and an eigenvalue of the thermodynamic Hessian along the same parameter family. Where the long-wavelength instability disappears continuously, $k_c$ approaches zero as the relevant Hessian eigenvalue crosses zero. A nonnegative wavenumber does not itself change sign. A deliberately mismatched fixed-charge versus fixed-potential Hessian is an adverse check: it can predict the wrong stability boundary.

Other failure modes include a compact circle too short to support $k\lt k_c$, nonuniform background data, scalar boundary conditions that add a negative mode, or a perturbation sector prohibited from redistributing the nominally unstable charge. The unresolved question is how far a precise implication can be extended while retaining useful, checkable hypotheses.

What could change these conclusions

The conclusions could change if a theorem proves or refutes the implication for a larger action and boundary class, if an independent threshold calculation disagrees with the matched Hessian, or if a new counterexample survives inclusion of all thermodynamic variables. A numerical simulation that changes the nonlinear endpoint, or compact and charge boundary conditions that create a new mode family, would change the conclusions for that brane without establishing a result for every brane.

Exercises

A neutral black string has threshold $k_cr_0=0.88$ and is compactified on a circle of length $L$. Determine the first unstable Fourier mode and the minimum $L/r_0$ for instability.

Solution

The allowed nonzero wavenumbers are $k_n=2\pi n/L$. The first candidate is $n=1$, and it is unstable when $2\pi/L\lt k_c=0.88/r_0$. Therefore $L/r_0\gt2\pi/0.88\approx7.14$. Higher $n$ have larger wavenumber and cannot become unstable before $n=1$. This calculation checks an important limitation of a local thermodynamic argument: even if the infinite brane has a negative Hessian direction, a short compact circle can remove every wavelength capable of realizing the Gregory–Laflamme mode.

References

• Friess, Joshua J., Steven S. Gubser, and Indrajit Mitra. “Counter-Examples to the Correlated Stability Conjecture.” Physical Review D 72 (2005): 104019. Open manuscript.
• Gregory, Ruth, and Raymond Laflamme. “Black Strings and p-Branes Are Unstable.” Physical Review Letters 70 (1993): 2837–2840. Open manuscript.
• Reall, Harvey S. “Classical and Thermodynamic Stability of Black Branes.” Physical Review D 64 (2001): 044005. Open manuscript.