Entropy of extremal black holes in two dimensions.

Entropy for two dimensional extremal black holes is computed explicitly in a finite-space formulation of the black hole thermodynamics and is shown to be zero locally. Our results are in conformity with the recent one by Hawking et al in four dimensions. It is known that black holes have temperature and entropy and are therefore amenable to a thermodynamic description. For an asymptotic observer, the temperature of the black hole is given by the surface gravity on the horizon and the entropy is one quarter the horizon area. It was shown by Gibbons and Hawking[1] that the entropy of the black hole coincides with the value of the Euclideanized action at infinity. The laws of black hole thermodynamics were also formulated [2, 3]. It was also thought that for charged black hole, e.g. Reissner–Nordström solution, one had a non-vanishing entropy even in the extremal limit (viz. charge → mass) although the temperature tends to zero. This is essentially due to the fact that extreme black holes have non-zero area. As a result, although the extremal limit was known to be unattainable due to the cosmic censorship[4], there was no purely thermodynamic way to establish this. In thermodynamics, unattainability of absolute zero by adiabatic processes follows from Nernst’s postulate of the vanishing of entropy in this limit[5]. It has been realized only recently[6, 7] that extreme black holes in four space-time dimensions do have zero entropy as measured at infinity. Recently we have studied the thermodynamics of several types of black hole solutions in two dimensions in a finite-space formulation[8] devised by Gibbons and Perry[9]. It was also commented that our results applied only to non-extremal black holes. The extremal black holes needed separate treatment. In this paper we apply the method of [8] to the extremal black holes in 2D and show that the entropy in these cases is also zero which corroborates the results of [6]. However in the present formulation the local entropy itself turns out to be zero. In this regard the extreme black holes are thermodynamically similar to the linear dilaton vacuum solutions. We also compute the energy of the extreme black holes from thermodynamic considerations and reproduce the ADM mass. We deal with two extremal black hole solutions. One of these is an asymptotically flat solution for