Entropy of Extremal Dyonic Black Holes

For extremal charged black holes, the thermodynamic entropy is proportional not to the area but to the mass or charges. This is demonstrated here for dyonic extremal black hole solutions of string theory. It is pointed out that these solutions have zero classical action although the area is nonzero. By combining the general form of the entropy allowed by thermodynamics with recent observations in the literature it is possible to fix the entropy almost completely. 1 electronic address: [email protected] electronic address: [email protected] 1. Black hole thermodynamics has been an intriguing subject for many years. The laws of classical black hole physics suggested definitions of temperature and entropy purely by analogy with the laws of thermodynamics, but the scale of these quantities could not be determined that way [1]. It was only with the introduction of quantum theoretical, or more precisely semiclassical, ideas that the scale could be set in terms of Planck’s constant [2]. The temperature defined in this way, related to surface gravity, was later rederived in a euclidean approach where there is a requirement of periodicity on the euclidean time coordinate if conical singularities are to be avoided. Apart from the obvious question about the origin of a nonzero entropy in this context, the expression for the entropy has itself been a cause for wonder. For ordinary, or what are now called non-extremal black holes, the entropy is proportional to the area of the horizon. Explanations have been sought to be given for this dependence. For instance [3] the statistical entropy of matter outside the black hole is proportional to the area of the horizon. There has been a lot of interest lately in the special case of extremal black holes [4, 5, 6]. The temperature and the entropy behave differently from the case of nonextremal black holes. Thus, when the temperature defined through the surface gravity is zero or infinity, it is found that there is no conical singularity, so that the temperature may really be arbitrary. Again, the thermodynamical entropy fails to be proportional to the area of the horizon. Another direction which recent research has taken involves black hole solutions of string theory. It has been possible to identify the states corresponding to extremal black hole solutions of string theory [7, 8, 9]. This presents an opportunity of reaching a better understanding of the entropy of black holes in terms of the underlying string theory. The entropy has indeed been calculated [8, 9] from the number of states. The result is sometimes consistent with the area formula but sometimes nonzero even when the area of the horizon vanishes. In the latter case, a new interpretation of the word “horizon” can be developed to match the area with the nonzero entropy. However, as mentioned earlier, the thermodynamic entropy of extremal black holes is in general not proportional to the area. Instead of seeking an area interpretation, the string result can be shown [10] to be consistent with a modified thermodynamical formula that can be justified for extremal black holes. To be precise, the expression proportional to the mass that we advocated earlier [6] (see also [11]) for the thermodynamic entropy