A numerical examination of an evolving black string horizon

TL;DR

This paper analyzes the fate of a perturbed five-dimensional black string by extracting intrinsic horizon properties from a prior numerical simulation. By constructing the horizon generators and computing the affine parameter $\lambda$, expansion $\theta$, and shear $\sigma_{ab}$, and by employing a stable Raychaudhuri-based integration for the rescaled variables, the authors show that $\lambda$ grows to about $10^{21}$ (with $s=\ln\lambda \approx 50$) while $\theta$ and $\sigma_{ab}$ become extremely small ($\sim 10^{−22}$). The results indicate a slow, dynamic approach to a possible asymptotic regime that could correspond to pinch-off at infinite affine parameter, though they do not decisively rule out other outcomes such as a static wiggly end-state or a finite-affine-time pinch-off; longer simulations and more robust numerics are required. Overall, the work highlights the utility of the affine-parameter perspective and suggests that the logarithm of the affine parameter may serve as a natural dynamical time for studying horizon evolution in higher-dimensional black-string spacetimes.

Abstract

We use the numerical solution describing the evolution of a perturbed black string presented in Choptuik et al. (2003) to elucidate the intrinsic behavior of the horizon. It is found that by the end of the simulation, the affine parameter on the horizon has become very large and the expansion and shear of the horizon in turn very small. This suggests the possibility that the horizon might pinch off in infinite affine parameter.

Figures (4)

• Figure 1: $s$ vs $t$ for the horizon generator of minimum radius, from three simulations of identical initial data though differing resolution (the curves are labelled by $N_r$x$N_z$, where $N_r$($N_z$) is the number of grid points in the $r$($z$) direction). We show results from three different resolutions to demonstrate that we are in the convergent regime. Recall that the affine parameter $\lambda=e^s$, and therefore $\lambda$ is growing very rapidly with simulation time $t$.
• Figure 2: The plot at the top depicts $\tilde{A}$ as a function of $s$, while the plot at the bottom shows $\tilde{A}$ and $\tilde{B}$ vs. $s$ from the highest resolution simulation, both for the horizon generator of minimum radius. What the latter plot demonstrates is that $\tilde{A}$ and $\tilde{B}$ are very similar in magnitude; in particular the difference $\tilde{A} - \tilde{B}$ (which is just $\tilde{\theta}$) at late times is completely dominated by numerical error, the magnitude of which can be estimated by using the data from the three simulations. Specifically, the error in $\tilde{A}$ from the highest resolution simulation is quite small initially, is near $5\%$ near the minimum of $\tilde{A}$, and has grown to around $40\%$ by the end of the simulation ($\tilde{B}$ has similar error).
• Figure 3: ${{\tilde{\sigma}}^{ab}}{{\tilde{\sigma}}_{ab}}$ vs. $s$ for the horizon generator of minimum radius.
• Figure 4: $\tilde{\theta}$ vs. $s$ for the horizon generator of minimum radius, calculated by integrating (\ref{['Rayscale_b']}). For the integrations shown on the top figure, the initial value of $\tilde{\theta}$ (which is at the largest$s$, as we integrate from large to small $s$) is $0$, while on the bottom figure the initial $\tilde{\theta}$ was calculated using (\ref{['thetatilde']}). We show curves calculated using two different sets of initial conditions to demonstrate that the integration is fairly insensitive to the initial value of $\tilde{\theta}$; hence, expect near the end of the simulation, the error in the integrated $\tilde{\theta}$ is comparable to that of ${{\tilde{\sigma}}^{ab}}{{\tilde{\sigma}}_{ab}}$ shown in Figure \ref{['fig3']}---specifically around $10\%$ near the maxima for the highest resolution simulation, assuming second order convergence.