Nonvanishing entropy of extremal charged black holes

Abstract

We calculate the entropy of a scalar field in terms of the brick-wall method by faithfully following the definition of the extremal black hole with zero Hawking temperature. Without recourse to the Euclidean formalism, which gives the arbitrary nonvanishing temperature, we obtain the nonvanishing and divergent entropy in the two-dimensional dilaton-Maxwell theory and the four-dimensional Reissner-Nordström solution even for the vanishing Hawking temperature in view of the microcanonical ensemble.