Black hole in thermal equilibrium with a scalar field: The back-reaction

The accurate approximation found by Page for the expectation value of the renormalized thermal equilibrium stress-energy tensor of a free conformal scalar field in a Schwarzschild black-hole background is used as the source in the semiclassical Einstein equation. The back-reaction and new equilibrium metric are found perturbatively in order ħ. The new metric is not asymptotically flat unless the system is enclosed by a reflecting wall. Solutions are obtained for systems of finite radius using microcanonical (fixed energy) and canonical (fixed temperature) boundary conditions. Explicit effects of the back-reaction on the equilibrium temperature distribution inside the cavity are given. With microcanonical boundary conditions there is an asymptotically flat region where the temperature at infinity is defined. It is shown that this temperature does not have the Schwarzschild value ħ(8πM${)}^{\mathrm{-}1}$ for a black hole of mass M. Curvature invariants are computed and the order-${ħ}^2$ correction to the conformal scalar-field trace anomaly originating from the back-reaction that this field produces is found. The principal qualitative features of the results should be valid for any quantum field at one loop in the Schwarzschild geometry.