Conformally invariant quantum field theory in static Einstein space-times

The conformal transformation law for the renormalized one-loop effective action is used to give approximate renormalized stress tensors for conformally invariant scalar, spinor, and vector field theories on static Einstein space-times, paying particular attention to the ambiguous coefficient of the term □R in the vector trace anomaly. In black-hole space-times these tensors for the Hartle-Hawking thermal state are shown to be regular on the horizon. Geometrical expressions are given for the approximate energy densities there; these can all be expressed in terms of the scalar invariant $C_{\mu \nu \rho \tau }$ $C^{\mu \nu \rho \tau }$. A number of explicit examples are presented, including Schwarzschild space-time. In flat and conformally flat space-times these approximate stress tensors agree with the exact results wherever the latter are known. In more complicated space-times, where we do not always obtain the exact results, a brief discussion is given of how to obtain a more accurate approximation.