Consider a pair of (possibly rotating) charged black holes with masses m1, m2 and like charges q1, q2. It seems that under certain conditions gravitational attraction should exactly cancel electrostatic repulsion and a stationary spacetime will result.
What are these conditions?
The point charges analogy suggests the equation
k q1 q2 = G m1 m2
However, it is by no means obvious this equation is the correct condition except in the large distance limit. Also:
Is it possible to write down this solution of Einstein-Maxwell theory in closed form?