The ultimate fate of black holes subject to the Gregory–Laflamme instability has been an open question for almost two decades. In this chapter we discuss the behavior of an unstable five-dimensional black string and elucidate its final state. Our studies reveal that the instability unfolds in a self-similar fashion, in which the horizon at any given time can be seen as thin strings connected by hyperspherical black holes of different radii. As the evolution proceeds pieces of the string shrink while others give rise to further spherical black holes, and consequently the horizon develops a fractal structure. At this stage its overall topology is still ℝ × S2; the fractal geometry arises along ℝ and has an estimated Hausdorff dimension d ≈ 1.05. However, the ever-thinning string regions eventually shrink to zero size, revealing a (massless) naked singularity. Consequently, this spacetime provides a generic counterexample to the cosmic censorship conjecture, albeit in five dimensions. While we restrict to the five-dimensional case for reasons of computational cost, our observations are intuitively applicable to higher dimensions.
To capture the late-time nonlinear dynamics of the system correctly requires numerical solution of the full Einstein equations. In this chapter, following a brief historical account (section 3.2)we describe details of our numerical implementation (section 3.3) as well as the behavior of the obtained solution (section 3.4). We discuss some additional properties of the solution, including speculation on when quantum corrections are expected to become important, and future directions in section 3.5.