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 squared field and stress-energy tensor in the ultrastatic metric. Explicit formulas for the changes in and 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 is exact for de Sitter space and agrees closely with the numerical calculations of Fawcett and Whiting in the Schwarzschild metric. The result for 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