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-Nordström 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 Λ = − 3 / l 2 of the space-time, and charge to mass ratio ϵ = q / μ 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 / 6 Q and l = 6 Q . 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.