Universality in chaos of particle motion near black hole horizon

The motion of a particle near a horizon of a spherically symmetric static black hole is shown to possess a universal Lyapunov exponent of chaos bounded by its surface gravity. To probe the horizon, we introduce an electromagnetic or scalar force to the particle so that it does not fall into the horizon. There appears an unstable maximum of the total potential where the evaluated maximal Lyapunov exponent is found to be to the surface gravity of the black hole. This value is independent of the external forces, the particle mass and background geometry, and in this sense this Lyapunov exponent is universal. Unless there are other sources of chaos, the Lyapunov exponent is subject to an inequality $\lambda \le 2\pi T_{\mathrm{BH}}/\hslash$, which is identical to the bound recently discovered by Maldacena, Shenker, and Stanford.

Figure 1

A schematic plot of the effective potential $V_{\text{eff}}$ given in (13). The origin of the plot is the horizon $x=0$. The potential has a maximum near the horizon, due to the competition between the gravitational potential and the external force.

Figure 2

The Poincaré sections of the particle motion near the black hole horizon. As the energy increases, the Kolmogorov-Arnold-Moser (KAM) tori tend to break. The energies are $E=15$, $E=40$, and $E=49.5$, respectively, from left to right.

Figure 3

Numerical analysis of the Lyapunov exponents. The horizontal axis is discrete time steps for numerical calculations, whose value is equal to $t$. The vertical axis is the value of the Lyapunov (characteristic) exponents. After time steps, the Lyapunov exponents converge to a set of values.

Figure 4

The Poincaré sections of the model (b1). The sections are the $(x,p_x)$ plane with $y=0$ and $\dot{y}>0$. The energies are $E=40$, $E=49$, and $E=49.99$ from left to right. For larger energies, the particle motion comes closer to the line $x=0$, where the KAM tori start to break to form a scattered plot, and chaos appears. For the numerical simulation, we have chosen $x_c=1$ and $\omega =10$.