Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship

We describe the behavior of 5-dimensional black strings, subject to the Gregory-Laflamme instability. Beyond the linear level, the evolving strings exhibit a rich dynamics, where at intermediate stages the horizon can be described as a sequence of 3-dimensional spherical black holes joined by black string segments. These segments are themselves subject to a Gregory-Laflamme instability, resulting in a self-similar cascade, where ever-smaller satellite black holes form connected by ever-thinner string segments. This behavior is akin to satellite formation in low-viscosity fluid streams subject to the Rayleigh-Plateau instability. The simulation results imply that the string segments will reach zero radius in finite asymptotic time, whence the classical space-time terminates in a naked singularity. Since no fine-tuning is required to excite the instability, this constitutes a generic violation of cosmic censorship.

Figure 1

(Normalized) apparent horizon area vs time.

Figure 2

Embedding diagram of the apparent horizon at several instances in the evolution of the perturbed black string, from the medium resolution run. $R$ is areal radius, and the embedding coordinate $Z$ is defined so that the proper length of the horizon in the space-time $z$ direction (for a fixed $t$, $\theta$, $\varphi$) is exactly equal to the Euclidean length of $R(Z)$ in the above figure. For visual aid, copies of the diagrams reflected about $R=0$ have also been drawn in. The light (dark) lines denote the first (last) time from the time segment depicted in the corresponding panel. The computational domain is periodic in $z$ with period $\delta z=20M$; at the initial (final) time of the simulation $\delta Z=20M$ ($\delta Z=27.2M$). See 28.

Figure 3

Curvature invariants evaluated on the apparent horizon at the last time of the simulation depicted in Fig. 2. The invariant $K$ evaluates to 1 for an exact black string and 6 for an exact spherical black hole, similarly for $S$ (1).

Figure 4

Logarithm of the areal radius vs logarithm of time for select points on the apparent horizon from the simulation depicted in Fig. 2. We have shifted the time axis assuming self-similar behavior; the putative naked singularity forms at asymptotic time $t/M\approx 231$. The coordinates at $z=15$, 5, and 4.06 correspond to the maxima of the areal radii of the first and second generation satellites and one of the third generation satellites at the time the simulation stopped. The value $z=6.5$ is a representative slice in the middle of a piece of the horizon that remains stringlike throughout the evolution.