Thermal stress tensors in static Einstein spaces

The Bekenstein-Parker Gaussian path-integral approximation is used to evaluate the thermal propagator for a conformally invariant scalar field in an ultrastatic metric. If the ultrastatic metric is conformal to a static Einstein metric, the trace anomaly vanishes and the Gaussian approximation is especially good. One then gets the ordinary flat-space expressions for the renormalized mean-square field and stress-energy tensor in the ultrastatic metric. Explicit formulas for the changes in $〈{\phi }^2〉$ and $〈T_{\mu \nu }〉$ resulting from a conformal transformation of an arbitrary metric are found and used to take the Gaussian approximations for these quantities in the ultrastatic metric over to the Einstein metric. The result for $〈{\phi }^2〉$ is exact for de Sitter space and agrees closely with the numerical calculations of Fawcett and Whiting in the Schwarzschild metric. The result for $〈T_{\mu \nu }〉$ is exact in de Sitter space and the Nariai metric and is close to Candelas's values on the bifurcation two-sphere in the Schwarzschild metric. Thus one gets a good closed-form approximation for the energy density and stresses of a conformal scalar field in the Hartle-Hawking state everywhere outside a static black hole.