Origin of black string instability

TL;DR

This paper tackles the Gregory-Laflamme instability of nonextremal black strings by developing a gauge-invariant perturbation framework that classifies perturbations into tensor, vector, and scalar types on the transverse sphere. It derives master equations for tensor modes and analyzes vector and scalar modes, using S-deformation techniques and numerical shooting to test stability across all multipoles, including exceptional cases. The main finding is that, aside from the exceptional s-wave ($\ell=0$) scalar GL mode, all perturbations are dynamically stable, clarifying the instability's origin and its relation to the breakdown of Birkhoff's theorem in higher dimensions. This work informs the stability landscape of black strings/braness, with implications for gauge/gravity dualities, phase transitions, and the validity of the correlated-stability conjecture.

Abstract

It is argued that many nonextremal black branes exhibit a classical Gregory-Laflamme (GL) instability. Why does the universal instability exist? To find an answer to this question and explore other possible instabilities, we study stability of black strings for all possible types of gravitational perturbation. The perturbations are classified into tensor-, vector-, and scalar-types, according to their behavior on the spherical section of the background metric. The vector and scalar perturbations have exceptional multipole moments, and we have paid particular attention to them. It is shown that for each type of perturbations there is no normalizable negative (unstable) modes, apart from the exceptional mode known as s-wave perturbation which is exactly the GL mode. We discuss the origin of instability and comment on the implication for the correlated-stability conjecture.

Figures (4)

• Figure 1: Plot of $\Omega$ as a function of $k_z$ for black strings with spacetime dimensions $D=n+3=5,~6, \cdots, 14$. The wave number $k_z$ is normalized by the horizon radius $r_h$.
• Figure 2: Static mode search. The possible asymptotic solutions of ${F^t_t}$ are ${F^t_t} \propto e^{\pm k_z r}$. In the figure, we plot ${F^t_t}$ at some $r/r_h \gg 1$ with respect to the single shooting parameter $k_z$. Since the normalizable solution decays exponentially, each narrow "throat" corresponds to a normalizable mode. The critical wave numbers agree precisely with the static limit $\Omega=0$ in Fig. \ref{['fig:zeromode1']}.
• Figure 3: Stable-unstable phase on $k_z^2$-$k_S^2$ plane. The zero mode $k_z^2=0$ corresponds to the perturbations of higher dimensional Schwarzschild BHs, which are stable. The Gregory-Laflamme mode $(\ell=0)$ is at $k_S^2=0$ with $k_z^2 < k_{\mathrm{crit}}^2$. On the plane, the shaded upper-right corner with $k_S \gg 1$ or $k_z^2 \gg r_h^{-2}$ is shown to be stable analytically. The stability of other generic modes is confirmed numerically.
• Figure 4: Search for critical static mode for $\ell = 2, 3$. (See Fig. \ref{['fig:zeromode2']} for $\ell=1$ mode.) This is a two-parameter shooting problem. The two parameters are the wave number $k_z$ and the derivative of $p$ at the horizon. The figure shows a plot of $(p^2 + q^2)$ at some asymptotic region with respect to the two parameters. Possible asymptotic solutions are $p,q \propto e^{\pm k_z r}$, and normalizable solutions will decay at $r \gg r_h$. No narrow "throat" appears so that there is no normalizable static mode. For other higher multipoles ($\ell \ge 2$), we obtained the same results.