0 O ct 1 99 4 ip / bbsr / 94 - 54 hep - th / 9410068 Thermodynamics of Two Dimensional Black Holes

Thermodynamic relations for a class of 2D black holes are obtained corresponding to observations made from finite spatial distances. We also study the thermodynamics of the charged version of the Jackiw-Teitelboim black holes found recently by Lowe and Strominger. Our results corroborate, in appropriate limits, to those obtained previously by other methods. We also analyze the stability of these black holes thermodynamically. It is customary in the study of black holes to write the laws of thermodynamics in the asymptotic space-time[1]. For a static uncharged black hole in four dimensions, the Hawking temperature is T c = 1 8πM , where M is the mass of the black hole. This temperature is measured at spatial infinity. Then Hawking's interpretation of the gravitational action as the entropy leads to the following asymptotically valid equation for this black hole: ST c = M 2. (1) Similar laws also exist for other black holes. However, if this equation is true for all M, then the specific heat of the black hole is negative; thus rendering the space-time unstable to radiation. It will lose mass with increasing temperature and is destined to meet with a catastrophic end. It has recently been pointed out [2], that the negativity of the specific heat of this black hole could be tracked down to ignoring the O(1 r) term in the action. Retention of this term is tantamount to measuring the thermodynamic quantities at a finite r. Therefore one needs a finite-space formulation of black hole thermodynamics. Such a formulation in the context of an effective 2D string theory has been proposed in [3]. One of the major motivations for analysing these 2D black hole solutions in string theory is to gain some insight into the nature of problems in thoroughly un-understood 4D theory of gravity with quantum effects. The 2D solutions can be obtained from higher dimensional ones, via some compactification scheme, say, and are simpler to deal with. In string theory, as is well-known, a scalar field, namely dilaton, plays an important role in obtaining the black hole solutions and their physics. For example, in 2D, pure Einstein gravity does not have any nontrivial solutions. But the introduction of dilaton gives rise to many interesting solutions, such as black holes[4], 1 cosmological universes[5] etc. Dilaton field also plays an important role in defining the thermodynamics of the black holes, as noted earlier[3]. The root of many …