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, 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 rh < |Q|, with Q the charge of the black hole. It is further shown that no black hole solutions with rh < |Q| can be obtained by solving the semiclassical Einstein 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 statistical mechanical entropy of zero temperature (extreme) black holes has been calculated in string theory [1] and shown to be identical to the usual Bekenstein-Hawking formula for the thermodynamic entropy. The usual semiclassical temperature and entropy calculations 1 for ERN black holes have all been made in the test field approximation where the effects of quantized fields on the spacetime geometry are not considered. 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 [2–4]. In a previous paper [5] we examined the semiclassical backreaction due to the vacuum stress-energy of massless and massive free quantized fields with spin 0, 1/2, and 1 on a static Reissner-Nordström (RN) black hole. We calculated the first-order (in h̄) corrections to the RN background and found, in all cases examined, that the surface gravity was nonzero because the energy density of the quantized field on the horizon is negative, 〈T t t 〉 > 0. [6] This led us to conclude that macroscopic zero temperature black holes might not exist in semiclassical gravity. After this work was published a paper appeared by Lowe [7] which finds a zero temperature solution to the linearized semiclassical backreaction equations. In this paper we investigate the perturbation series more generally than we did previously [5]. We find that for nearly extreme black holes, the surface gravity (and temperature) of the black hole decreases as the radius of the event horizon rh decreases. Thus these solutions form a sequence and the sequence appears to terminate at the zero temperature solution found by Lowe [7]. In our previous paper the lowest temperature solution examined had rh = |Q| and κ = 4π|Q|〈T t t 〉 > 0; Lowe’s proposed solution has rh = |Q|(1−4πQ2〈T t t 〉) < |Q| and κ = 0. We discuss the validity of the solutions in this sequence and show that it is not possible to use perturbation theory to obtain those with rh < |Q|, including the one proposed by Lowe. However it is possible that they nevertheless approximate exact solutions to the full nonlinear semiclassical equations. We also discuss a potential problem that may occur for all zero temperature black hole solutions to the semiclassical backreaction equations. The general static spherically symmetric metric can be written in the form [6]: ds = −f(r)dt + h(r)dr + rdΩ , (1) where dΩ is the metric of the two-sphere. The metric can describe a black hole with an event horizon at r = rh if f(rh) = 0. To avoid having a scalar curvature singularity at the 2 event horizon it is necessary that h(rh) = 0 as well. The surface gravity of such a black hole is κ = ( 1 2 ) f ′ √ fh ∣