Evolution and End Point of the Black String Instability: Large $D$ Solution

We derive a simple set of nonlinear, $(1+1)$-dimensional partial differential equations that describe the dynamical evolution of black strings and branes to leading order in the expansion in the inverse of the number of dimensions $D$. These equations are easily solved numerically. Their solution shows that thin enough black strings are unstable to developing inhomogeneities along their length, and at late times they asymptote to stable nonuniform black strings. This proves an earlier conjecture about the end point of the instability of black strings in a large enough number of dimensions. If the initial black string is very thin, the final configuration is highly nonuniform and resembles a periodic array of localized black holes joined by short necks. We also present the equations that describe the nonlinear dynamics of anti–de Sitter black branes at large $D$.

Figure 1

Dynamical evolution of a perturbed string with $k_L=0.98$. The horizontal axis is the $S^{n+1}$-area function, $m(t,z)$. For this simulation, the final state was reached before $t=500$. The time it takes depends on the size and shape of the initial perturbation, but the final configuration is independent of them.

Figure 2

Dynamical evolution of a perturbed string with $k_L=0.55$. For the last two plots we display $\ln m(t,z)$ in insets, which show that the blobs extend to almost fill the compact direction. For this simulation, the final state was reached around $t=40$.