Black Holes That Repel

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.