A note on circular geodesics and phase transitions of black holes

The circular motion of charged test particles in the gravitational field of a Reissner-Nordstrom black hole in Anti de Sitter space-time is investigated, using a set of independent parameters, such as charge Q, mass M and cosmological constant Lambda = -3/l(2) of the space-time, and charge to mass ratio epsilon = q/mu of the test particles. Classification of different spatial regions where circular motion is allowed, is presented, showing in particular, the presence of orbits at special limiting values, M= 4/root 6Q and l = 6Q. Thermodynamically, these values are known to occur when the black hole is on the verge of a second order phase transition, there by, giving an interesting connection between thermodynamics and geodesics of black holes. We also comment on the possibility of such a connection for black holes in flat spacetime in a box. (C) 2019 The Author(s). Published by Elsevier B.V.