Circular Geodesics and Phase Transitions of Charged Black Holes in AdS

The motion of neutral and charged test particles in the gravitational field of a Reissner-Nordstr\{o}m black hole in Anti de Sitter space-time is investigated, concentrating on the case of circular orbits. A classification of circular orbits based on black hole parameters depending on charge Q, mass M and cosmological constant parameter $l$, together with charge to mass ratio $\epsilon=q/m$ of the test particles is presented. The analysis shows the presence of circular orbits at special limiting values, $M_{\rm cr}=4/\sqrt{6} Q$ and $l_{\rm cr}=6 Q$. Thermodynamically, these values are known to occur when the black hole is critical and on the verge of a second order phase transition. We also extend the correspondence of thermodynamics and phase transitions with the radius and impact parameter of circular geodesics, to the case of charged particles.