Universality in Chaos of Particle Motion near Black Hole Horizon

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Abstract

Motion of a particle near a horizon of a spherically symmetric black hole is shown to possess a universal Lyapunov exponent of a chaos provided by its surface gravity. To probe the horizon, we introduce 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 independent of the external forces and the particle mass. The Lyapunov exponent is universally given by the surface gravity of the black hole. Unless there are other sources of a chaos, the Lyapunov exponent is subject to an inequality $\lambda \leq 2\pi T_{\rm BH}/\hbar$, which is identical to the bound recently discovered by Maldacena, Shenker and Stanford.