Geodesic motion in the Reissner-Nordström space-time - Capobianco, Rogério Augusto (2019)
The motion of neutral test particles, both massive and massless, in the space time of a charged source described by the Reissner-Nordström solution is studied. This solution is characterized by two parameters, mass and charge, which defines the horizons of the source. When the mass is larger than the charge, the solution describes a black hole, with two distinct horizons. When the mass and charge are equal there is an extremal black hole, and both horizons merge to one. Finally, when the charge is larger than the mass there is a naked singularity, with no horizon. The structure and properties of these different type of solution are presented and discussed. A general solution of the equations of motion is presented in function of the Weierstrass elliptic function ℘. In addition, the possible orbits for test particles are discussed, and the conditions for existence of closed, circular or escape orbits are presented. The classifications is made based on the particles energy, and the mass and charge of the source. We find that all mentioned orbits are allowed for the three different type of solutions. In particular, for extremal black holes and naked singularities, we find stable circular orbits located outside the event horizon and hence being visible for an external observer.