# Black Holes That Repel

## TL;DR

This paper investigates how certain charged black holes in Einstein-Maxwell-Dilaton theories can repel each other due to dilaton coupling, especially near extremality, challenging traditional notions of black hole interactions.

## Contribution

It demonstrates that dilaton coupling can create a repulsive force between black holes, supported by force calculations and entropy considerations, revealing new black hole interaction dynamics.

## Key findings

- Black holes can repel each other due to dilaton coupling.
- A potential barrier causes repulsion near extremality.
- The force behavior aligns with entropy super-additivity violations.

## Abstract

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 $A_i$, with general exponential couplings $\exp(a_i \phi)$ in their kinetic terms. We calculate the gravi-electrostatic force between charged black holes $(m_1,e_1)$ and $(M_2,Q_2)$; the former is sufficiently small and can be treated as a point-like object. We find there is a potential barrier caused by the dilaton coupling at $r_0$ outside the back hole horizon $r_+$, provided that $-a_1 a_2> 2(D-3)/(D-2)$. As the black hole approaches extremality, both $r_+$ and $r_0$ 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 $a_1 a_2$ can be derived from the requirements that two-charged extremal black holes have a fusion 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.