Phys. Rev. D 99, 101502(R) (2019)        arXiv link

Black holes that repel

H. Lü^1,*, Zhao-Long Wang^2,3,†, and Qing-Qing Zhao^1,‡

¹ Center for Joint Quantum Studies, School of Science, Tianjin University, Tianjin 300350, China

² Institute of Modern Physics, Northwest University, XiAn 710069, China

³ Shaanxi Key Laboratory for Theoretical Physics Frontiers, XiAn 710069, China

*mrhonglu@gmail.com
†zlwang@nwu.edu.cn
‡zhaoqq@tju.edu.cn

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 $\exp \left(a_i\varphi \right)$ in their kinetic terms. We calculate the gravielectrostatic 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 pointlike object. We find there is a potential barrier caused by the dilaton coupling at $r_0$ outside the black 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 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.