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Thermodynamic Features of Black Holes Dressed with Quantum Fields

The thermal properties of black holes in the presence of quantum fields can be revealed through solutions of the semi-classical Einstein equation. We present a brief but self-contained review of the main features of the semi-classical back reaction problem for a black hole in the microcanonical ensemble. The solutions, obtained for conformal scalars, massless spinors and U(1) gauge bosons, are used to calculate the $O(\hbar)$ corrections to the temperature and thermodynamical entropy of a Schwarzschild black hole. In each spin case considered, the entropy corrections $ΔS(r)$, are positive definite and monotone increasing with increasing distance $r$ from the hole, and are of the same order as the naive flat space radiation entropy.