Black Hole in Thermal Equilibrium with a Spin-2 Quantum Field

Abstract

An approximate form for the vacuum averaged stress-energy tensor of a conformal spin-2 quantum field on a black hole background is employed as a source term in the semiclassical Einstein equations. Analytic corrections to the Schwarzschild metric are obtained to first order in $\epsilon = {\hbar}/M^2$, where $M$ denotes the mass of the black hole. The approximate tensor possesses the exact trace anomaly and the proper asymptotic behavior at spatial infinity, is conserved with respect to the background metric and is uniquely defined up to a free parameter $\hat c_2$, which relates to the average quantum fluctuation of the field at the horizon. We are able to determine and calculate an explicit upper bound on $\hat c_2$ by requiring that the entropy due to the back-reaction be a positive increasing function in $r$. A lower bound for $\hat c_2$ can be established by requiring that the metric perturbations be uniformly small throughout the region $2M \leq r < r_o$, where $r_o$ is the radius of perturbative validity of the modified metric. Additional insight into the nature of the perturbed spacetime outside the black hole is provided by studying the effective potential for test particles in the vicinity of the horizon.