Towards the Final Fate of an Unstable Black String

TL;DR

The paper investigates the non-linear evolution of the 5D Gregory-Laflamme unstable black string by solving the full vacuum Einstein equations using a 5D ADM framework with a periodically compactified direction and horizon excision. It demonstrates the reproduction of the GL instability threshold and provides evidence that, at intermediate times, the horizon evolves toward a neck–bulge geometry consistent with a sequence of spherical black holes connected by thin strings, though late-time coordinate pathologies prevent a definitive end-state conclusion. The study highlights the need for adaptive gauge choices and more general initial data to determine whether the end-state is a vanishing neck, a cascade of instabilities, or a stationary non-uniform solution, with implications for cosmic censorship in higher dimensions. Overall, it lays out a concrete non-linear numerical framework for exploring black string dynamics and outlines clear directions for improving coordinates and extending the analysis to resolve the final fate of the instability.

Abstract

Black strings, one class of higher dimensional analogues of black holes, were shown to be unstable to long wavelength perturbations by Gregory and Laflamme in 1992, via a linear analysis. We revisit the problem through numerical solution of the full equations of motion, and focus on trying to determine the end-state of a perturbed, unstable black string. Our preliminary results show that such a spacetime tends towards a solution resembling a sequence of spherical black holes connected by thin black strings, at least at intermediate times. However, our code fails then, primarily due to large gradients that develop in metric functions, as the coordinate system we use is not well adapted to the nature of the unfolding solution. We are thus unable to determine how close the solution we see is to the final end-state, though we do observe rich dynamical behavior of the system in the intermediate stages.

Figures (7)

• Figure 1: The maximum ($R_{{\rm max}}$) and minimum ($R_{{\rm min}}$) areal radii, and the corresponding function $\lambda$ of the apparent horizon as a function of time, from the evolution of perturbed black strings with $L=1.03L_c$ and $L=0.975L_c$. The initial fluctuation in the plots correspond to the effect of the initial gravitational wave perturbation, most of which either falls into the string, or escapes to infinity. This close to the threshold $L_c$, the growth/decay of the remnant perturbation is quite slow, and so we cannot feasibly (at the resolution of the these simulations---$800 \times 200$ points in $r \times z$) follow the evolution for much further than shown while maintaining reasonable accuracy (though we see no signs of numerical instabilities in the stable case, and such simulations have been followed to $10,000M$). However, the main purpose of this figure is to demonstrate the qualitative recovery of the expected threshold behavior for the onset of the instability at $L=L_c$.
• Figure 2: The maximum ($R_{{\rm max}}$) and minimum ($R_{{\rm min}}$) areal radii, and the corresponding function $\lambda = (R_{\rm max}/R_{\rm min} - 1)/2$, of the apparent horizon, as a function of time, from the evolution of a perturbed black string with $L=1.4L_c$. $h$ labels grid spacing; hence smaller $h$ corresponds to higher resolution. This plot, combined with the results shown in Fig. \ref{['hc_1000']} suggest that the code is in the convergent regime---in particular at later times---for the $h/2$ and higher resolution simulations.
• Figure 3: The logarithm of the $\ell_2$-norm of the Hamiltonian constraint as a function of time, evaluated on the portion of the computational domain lying exterior to the apparent horizon, and from simulations at several resolutions of a perturbed black string with $L=1.4L_c$. As with Fig. \ref{['lambda_1000']}, this plot provides evidence that convergence is quite good for the $h/2$ and higher resolution simulations (at least until very close to when the supposed coordinate singularity forms, near $t=165$).
• Figure 4: Embedding diagrams of the apparent horizon, with the two angular dimensions $\theta$ and $\phi$ suppressed, from the $h/4$ evolution of a perturbed black string with $L = 1.4 L_c$. These plots thus describe the intrinsic geometry of the apparent horizon, at the given instants of constant $t$, in a coordinate system with metric $ds^2=d\bar{r}^2 + d\bar{z}^2$. Here, $\bar{z}$ is a periodic coordinate, and $\bar{r}$ is the areal radius of $\bar{z}=constant$ sections of the horizon. To better illustrate the dynamics of the horizon, we have extended the solution using the $\bar{z}$-periodicity, showing roughly two periods of the solution. See Fig. \ref{['ah_embed_1000_length']} for a plot of the length of one period of the apparent horizon versus time.
• Figure 5: The proper length of the apparent horizon curve in the $(r,z)$ plane (between $z=0$ and $z=L$) as a function of time, from the $h/4$ evolution of a perturbed black string with $L = 1.4 L_c$.