Zero temperature black holes in semiclassical gravity

The semiclassical Einstein equations are solved to first order in \(\epsilon = \hbar/M^2\) for the case of an extreme or nearly extreme Reissner-Nordstr\"{o}m black hole perturbed by the vacuum stress-energy of quantized free fields. It is shown that, for realistic fields of spin 0, 1/2, or 1, any zero temperature black hole solution to the equations must have an event horizon at \(r_h < |Q|\), with \(Q\) the charge of the black hole. It is further shown that no black hole solutions with \(r_h < |Q|\) can be obtained by solving the semiclassical Einstein equations perturbatively.