Zero Temperature Black Holes and the Failure of Semi-classical Analysis

The extreme Reissner-Nordström black holes have zero surface gravity. However, a semi-classical analysis seems to be ill-definite for these objects and apparently no notion of temperature exists for them. It is argued here that these properties are shared for all kind of black holes whose surface gravity is zero. Two examples are worked out explicitely: the scalar-tensor cold black holes and extreme black holes resulting from a gravity system coupled to a generalized Maxwell field in higher dimensions. The reasons for this anomolous behaviour are discussed as well as its thermodynamics implications. The temperature of a black hole may be obtained in different ways. It is generally stated that this temperature is proportional to the black hole surface gravity. When the surface gravity is not zero, this statement is consistent with the temperature obtained through other methods, like the euclideanization of the metric or the computation of the Bogolubov’s coefficients [1, 2, 3, 4]. However, these considerations seem to be more delicate for those black holes whose surface gravity is zero. In the case of the Reissner-Nordström solution, the extreme black holes have zero surface gravity suggesting that they have zero temperature. However, the computation of the temperature through the euclideanization of the metric indicates an arbitrary value [4, 5, 6]; moreover, the computation of the Bogolubov’s coefficients seems to be ill-definite in the sense that the normalization conditions are not satisfied indicating a break down of semi-classical analysis [7]. Hence, it seems that no notion of temperature can be assined to the RN extreme black holes. In this work, we will employ the term ”extreme black holes” to denote those black holes which are, in principle, the limit case of a more general class of black holes, like in the Reissner-Nordström solution. There are black holes which have zero surface gravity but do not match in this classification, like the so-called scalar-tensor black holes [8, 9]: in this case, all possible black hole solutions have zero surface gravity. Zero surface gravity are indeed very particular objects. In the case of the ReissnerNordström solutions, they are the limiting case asM → Q. However, it has been argued in [6] that the RN extreme black holes must be seen as a qualitatively different objects with