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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 λ. We find that although λ approaches universally the horizon surface gravity κ when the equilibria are close to the horizon, the proposed chaotic motion bound λ<κ 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 λ/κ can be arbitrarily large for sufficiently small κ. Our investigation does suggest a universal bound for sufficiently large κ, namely λ/κ

The recent observation that black holes in certain Einstein-Maxwell-dilaton (EMD) theories can violate the entropy super-additivity led to the suggestion that these black holes might repel each other. In this paper, we consider EMD theories with two Maxwell fields Ai, with general exponential couplings exp(aiϕ) in their kinetic terms. We calculate the gravielectrostatic force between charged black holes (m1,e1) and (M2,Q2); the former is sufficiently small and can be treated as a pointlike object. We find there is a potential barrier caused by the dilaton coupling at r0 outside the black hole horizon r+, provided that -a1a2>2(D-3)/(D-2). As the black hole approaches extremality, both r+ and r0 vanish, the barrier becomes infinitesimally thin but infinitely high, and the two black holes repel each other in the whole space. There is no electrostatic force between them; the dilaton is the antigravity agent. Furthermore we find that the exact constraint on a1a2 can be derived from the requirements that two-charged extremal black holes have a fission bomb like mass formula and the violation of entropy super-additivity can occur. The two very different approaches give a consistent picture of the black hole repulsion. [ABSTRACT FROM AUTHOR]