Chaotic dynamics of strings in charged black hole backgrounds

We study the motion of a string in the background of a Reissner-Nordstrom black hole, in both anti–de Sitter as well as asymptotically flat spacetimes. We describe the phase space of this dynamical system through the largest Lyapunov exponent, Poincaré sections and basins of attraction. We observe that string motion in these settings is particularly chaotic and comment on its characteristics.

Figure 1

Ring string in the charge black hole background

Figure 2

Plots showing the ring string trajectory in an asymptotically flat RN black hole background for $Q=0.001$ with the initial conditions $E_n=12$, $\theta =0$, $\frac{d}{d\tau }(r\cos \theta )=0$ and $r=(10,8,6)$ respectively from top to bottom. The top plot shows the backscattering of the string, the middle plot shows the escape to infinity and the bottom plot shows the capture of the string by the black hole.

Figure 3

Plots showing the ring string trajectory in an asymptotically flat RN black hole background for $Q=(0.001,0.2,0.45)$ with the initial conditions $E_n=12$, $\theta =0$, $\frac{d}{d\tau }(r\cos \theta )=0$, $r=10$ in the left panel and with the initial conditions $E_n=12$, $\theta =0$, $\frac{d}{d\tau }(r\cos \theta )=0$, $r=15$ in the right panel of the figure.

Figure 4

Basin boundaries, plotted for different values of the charge of the black hole $Q=0.001$ (top) and $Q=0.2$ (bottom) with the initial conditions $E_n=12$, $\frac{d}{d\tau }(r\cos \theta )=0$ with $r\in [2,17]$ on the horizontal axis and $\theta \in [0,\pi /2]$ on the vertical axis. The figure on the right side shows a detailed section of the basin boundary diagram shown on the left.

Figure 5

Basin boundaries, plotted for the initial conditions $Q=0.45$ (top) and $Q=0.4987$ (extremal value, bottom) with the initial conditions $E_n=12$, $\frac{d}{d\tau }(r\cos \theta )=0$ with $r\in [2,17]$ on the horizontal axis and $\theta \in [0,\pi /2]$ on the vertical axis.The figure on the right side shows a detailed section of the basin boundary diagram shown on the left.

Figure 6

Plots from top to bottom depict the solution $r(\tau )$ and phase space plots for the ring string in the RN-AdS black hole background for increasing values of the charge $Q=0.001$ and $Q=0.45$ of the black hole respectively, with the initial conditions $E_n=12$, $P_r=3.6$, $r=10$ and $\theta =0$. The Hamiltonian $(\mathcal{H})$ as a function of proper time $\tau$ (with the error tolerance $\delta ={10}^{-10}$) for the corresponding solutions $r(\tau )$ is plotted in the right panel of the figure.

Figure 7

Plots depict the Poincaré sections at $\theta (\tau )=0$ for the solutions $r(\tau )$ with increasing values of the charge $Q=(0.001,0.2,0.45)$ and $Q=0.4987$ (extremal case) of the RN-AdS black hole and initial conditions $E_n=12$, $P_r=3.6$, $r=10$ and $\theta =0$.

Figure 8

The red, green and blue curves depict the Poincaré sections at $\theta (\tau )=0$ for the solutions $r(\tau )$ with increasing values of the energy of the ring string $E_n=11$, 11.5, 12 respectively with initial conditions $Q=0.2$, $P_r=3.6$, $r=10$ and $\theta =0$.

Figure 9

Plots from top to bottom depict the LLE as a function of proper time $\tau$ for the solutions $r(\tau )$ with increasing values of the charge $Q=(0.001,0.2,0.45)$ and $Q=0.4987$ (extremal case) of the RN-AdS black hole and the energy $E_n=12$ of the ring string as given in Fig. 6.

Figure 10

Basin boundaries for the dynamics of the ring string in the RN-AdS black hole background, plotted for the initial conditions $E_n=12$, $l=15$, $P_r(\tau )=0$ with $r\in [5,7.5]$ on the horizontal axis and $\theta \in [0,1]$ on the vertical axis. The graphs from left to right correspond to the values $Q=0.001$, 0.02, 0.045 of the black hole charge respectively.

Figure 11

A zoom in of the Basin boundaries for the dynamics of the ring string in the RN-AdS black hole background, plotted for the initial conditions $E_n=12$, $l=15$, $P_r(\tau )=0$ with $r\in [6.5,7]$ on the horizontal axis and $\theta \in [0.835,0.955]$ on the vertical axis. The graphs from left to right correspond to the values $Q=0.001$, 0.02, 0.045 of the black hole charge respectively.