Zero and near-zero temperature black holes in semiclassical gravity

The semiclassical Einstein equations are solved to first order in ǫ = h̄/M for the case of an extreme or nearly extreme Reissner-Nordström black hole perturbed by the vacuum stress-energy of quantized free fields. It is shown that the perturbed solution at first order is not unique; there exists a sequence of solutions with varying values of the position of the event horizon, rh. The smaller the value of rh the smaller the surface gravity of the black hole. The validity of the solutions is discussed and it is shown that solutions which have rh < |Q|, with Q the charge of the black hole, cannot be obtained by solving the semiclassical equations perturbatively. Static spherically symmetric zero temperature black holes have proven to be very interesting and important at the classical, semiclassical, and quantum levels. Classically the only static spherically symmetric 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 sta-