Temperatures of extremal black holes

The temperature of an extremal Reissner Nordstrom black hole is not restricted by the requirement of absence of a conical singularity. It is demonstrated how Kruskal-like coordinates may be constructed corresponding to any temperature whatsoever. A recently discovered stringy extremal black hole which apparently has an infinite temperature is also shown to have its temperature unrestricted by conical singularity arguments. A classical black hole has a horizon beyond which nothing can leak out. This suggests that it can be assigned a zero temperature. But the relation between the area of the horizon and the mass and other parameters like the charge indicates a close similarity [1] with the thermodynamical laws, thus allowing the definition of a temperature. This analogy was understood as being of quantum origin and made quantitative after the discovery of Hawking radiation [2]. The associated Hawking temperature vanishes only in the classical limit. The thermodynamics of black holes has been extensively studied since then. Most of the studies were first made for the simplest kind of black hole, viz., the Schwarzschild spacetime. Of more recent interest is the case of the so-called extremal black e-mail [email protected] e-mail [email protected] 1 holes which have peculiarities not always present in the corresponding non-extremal cases [3,4]. For extremal Reissner Nordstrom black holes, the näıvely defined temperature is zero, but the area, which is usually thought of as the entropy, is nonzero. For extremal dilatonic black holes, where the temperature is not zero, the area vanishes. In this note we shall reexamine the temperature of an extremal Reissner Nordstrom black hole, for which the definition in terms of the surface gravity leads to a zero temperature. We discuss the conical singularity approach in detail: there is no conical singularity in the extremal case, so that there is no constraint on the temperature from this point of view. Thereafter we consider Kruskal-like coordinates and show that they may be constructed for an arbitrary temperature. We also comment on some stringy black holes. One of them has an infinite temperature in the surface gravity approach. We show however that there is no conical singularity in this case. REISSNER NORDSTROM BLACK HOLES The metric of the Reissner Nordstrom spacetime is given by ds = −(1− 2M r + Q r2 )dt + (1− 2M r + Q r2 )dr + r(dθ + sin θdφ) (1) in general, with M and Q denoting the mass and the charge respectively. There are apparent singularities at