Entropy of Extremal Black Holes in Asymptotically Anti-de Sitter Spacetime

Unlike the extremal Reissner-Nordström black hole in ordinary spacetime, the one in anti-de Sitter spacetime is a minimum of action and has zero entropy if quantization is carried out after extremaliza-tion. However, if extremalization is carried out after quantization, then the entropy is a quarter of the area as in the usual case. While the entropy of ordinary (non-extremal) black holes has been known to be given by a quarter of the horizon area for a long time, there has been some uncertainty in the case of extremal black holes. The usual derivations do not go through in a straightforward manner, and because of the difference in topology of euclidean extremal black holes and euclidean non-extremal black holes, one cannot fall back on extrapolation. In fact, it has been suggested that extremal black holes should have zero entropy [1] even though the horizon area is nonzero. On the other hand, some microscopic models have indicated that ex-tremal black holes could satisfy the area law just like non-extremal black