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

TL;DR

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, this constitutes a generic violation of cosmic censorship.

Abstract

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.

Figures (4)

• 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 $t,\theta,\phi$) 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 $\delta Z=20M$ ($\delta Z=27.2M$).
• Figure 3: Curvature invariants evaluated on the apparent horizon at the last time of the simulation depicted in Fig. \ref{['fig:AH_embed']}. The invariant $K$ evaluates to $1$ for an exact black string, and $6$ for an exact spherical black hole; similarly for $S$ (\ref{['inv_def']}).
• Figure 4: Logarithm of the areal radius vs. logarithm of time for select points on the apparent horizon from the simulation depicted in Fig. \ref{['fig:AH_embed']}. 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 string-like throughout the evolution.