Static equilibria of charged particles around charged black holes: Chaos bound and its violations

We study the static equilibrium of a charged massive particle around a charged black hole, balanced by the Lorentz force. For a given black hole, the equilibrium surface is determined by the charge/mass ratio of the particle. By investigating a large class of charged black holes, we find that the equilibria can be stable, marginal or unstable. We focus on the unstable equilibria which signal chaotic motions and we obtain the corresponding Lyapunov exponents $\lambda$. We find that although $\lambda$ approaches universally the horizon surface gravity $\kappa$ when the equilibria are close to the horizon, the proposed chaotic motion bound $\lambda <\kappa$ is satisfied only by some specific black holes including the Reissner-Nordström (RN) and RN-anti–de Sitter (AdS) black holes. The bound can be violated by a large number of black holes including the RN-dS black holes or black holes in Einstein-Maxwell-dilaton, Einstein-Born-Infeld and Einstein-Gauss-Bonnet-Maxwell gravities. We find that unstable equilibria can even exist in extremal black holes, implying that the ratio $\lambda /\kappa$ can be arbitrarily large for sufficiently small $\kappa$. Our investigation does suggest a universal bound for sufficiently large $\kappa$, namely $\lambda /\kappa <\mathcal{C}$ for some order-one constant $\mathcal{C}$.

Figure 1

An explicit example of the violation of the bound (1.2) by a nonextremal black hole in EBI theory. The black hole parameters are given by $r_{+}=1$, $b=5$ and $Q=2$. It can be seen that the bound is satisfied in the vicinity of the black hole horizon $r_0\in (1,1.2)$, violated in the region (0.2, 0.4), and satisfied again in the region (0.4, 2). The $r_0>2$ equilibria become stable.