B..1 Effective potential for a charged particle in the Reißner-Nordström spacetime

The effective potential for the motion of a particle of charge $q$ and mass $\mu$ in the Reißner-Nordström spacetime can be obtained in many different ways. Probably the quickest one is to start from the more general result that can be found in the equation after equation (3) in Box 33.5 of [70], i.e. the effective potential for the orbits of test particles in the equatorial plane of a Kerr-Newman black hole. Specializing this result to a black hole with zero angular momentum we obtain the effective potential in the Reißner-Nordström case. Perhaps more instructive is to perform again the analysis until equation (6) of [8] adding the electrostatic contribution to the particle momentum or solving the Hamilton-Jacobi equation in the Reißner-Nordström metric. Anyway, the final result for the extremal case we are interested in is

$$\tilde{V} (x) = \frac{\tilde{\epsilon} \Theta}{x} + \le... ...eft( 1 + \frac{\tilde{\lambda} ^{2}}{x ^{2}} \right) ^{1/2} ,$$ (34)

where $\tilde{\epsilon}$ is the charge/mass ratio of the test particle and $\tilde{\lambda} = \tilde{L} / H$ is its angular momentum per unit mass ( $\tilde{L} = L / \mu)$ in the $H$ scale defined in (14) and (15).