Zero Temperature Black Holes in Semiclassical Gravity

Static spherically symmetric zero temperature black holes are very interesting and important at the classical, semiclassical, and quantum levels. Classically the only static black hole solution to Einstein's equations with zero surface gravity (and hence zero temperature) is the extreme Reissner-Nordström (ERN) black hole, which possesses a charge equal in magnitude to its mass. At the quantum level, the statistical mechanical entropy of zero temperature (extreme) black holes has been calculated in string theory¹ and shown to be identical to the usual Bekenstein-Hawking thermodynamic entropy. The usual semiclassical temperature and entropy calculations for ERN black holes have all been made in the test field approximation where the effects of quantized fields on the spacetime geometry are ignored. However, it is well known that quantum effects alter the spacetime geometry near the event horizon of a black hole. In particular they can change its surface gravity and hence its temperature^2,3,4...