On the equilibrium of a charged massive particle in the field of a   Reissner-Nordstr\"om black hole
  The multiyear problem of a two-body system consisting of a Reissner-Nordstr\"om black hole and a charged massive particle at rest is here solved by an exact perturbative solution of the full Einstein-Maxwell system of equations. The expressions of the metric and of the electromagnetic field, including the effects of the electromagnetically induced gravitational perturbation and of the gravitationally induced electromagnetic perturbation, are presented in closed analytic formulas. 
D. Bini, A. Geralico, R. Ruffini