# On the balance problem for two rotating and charged black holes

## TL;DR

This paper investigates whether two rotating, charged black holes can be in equilibrium by analyzing the most general axisymmetric solutions to Einstein-Maxwell equations, addressing a fundamental open problem in black hole physics.

## Contribution

It derives the most general form of axis potentials for equilibrium configurations of two rotating, charged black holes, expanding understanding of possible solutions.

## Key findings

- Derived the general form of axis potentials for equilibrium black holes.
- Clarified the scope of candidate solutions for black hole equilibrium.
- Addressed the open problem of black hole balance in Einstein-Maxwell theory.

## Abstract

It is an interesting open problem whether two non-extremal rotating and electrically charged black holes can be in physical equilibrium, which might be possible due to a balance between the gravitational attraction and the spin-spin and electrical repulsions. Exact candidate solutions were constructed, but it is unclear whether they are physically acceptable. These solutions were obtained by assuming a particular behaviour on the symmetry axis. However, it was not known whether the assumed form of the axis data covers the general case or whether data of some other type need to be considered as well. By studying a boundary value problem for the axisymmetric and stationary Einstein-Maxwell equations, we address this question and derive the most general form of permissible axis potentials for possible equilibrium configurations.