D ec 1 99 9 Thermodynamics of charged anti - de Sitter black holes in canonical ensemble

As in the grand canonical treatment of Reissner-Nordström black holes in anti-de Sitter spacetime, the canonical ensemble formulation also shows that non-extremal black holes tend to have lower action than extremal ones. However, some small non-extremal black holes have higher action, leading to the possibility of transitions between non-extremal and extremal black holes. Black hole thermodynamics continues to be an interesting field of study. A special area of recent interest is that of extremal black holes. While the entropy of ordinary (non-extremal) black holes has been taken to be a quarter of the horizon area for a long time, recently there has been some confusion in the case of extremal black holes. The semiclassical derivations of the en-tropy formula for non-extremal black holes do not directly apply to extremal black holes, and because of the difference in topology of euclidean extremal and non-extremal black holes, one cannot rely on extrapolation. In fact, eu-clidean studies indicate that extremal black holes should have zero entropy [1] even though the horizon area is nonzero. On the other hand, practical-minded people have tended to expect that extremal black holes should satisfy