Killing approximation for vacuum and thermal stress-energy tensor in static space-times

The problem of the vacuum polarization of conformal massless fields in static space-times is considered. A tensor $T_{\mu \nu }$ constructed from the curvature, the Killing vector, and their covariant derivatives is proposed which can be used to approximate the average value of the stress-energy tensor 〈T${^}_{\mu \nu }$ ${〉}^{\mathrm{ren}}$ in such spaces. It is shown that if (i) its trace T ${}_{\epsilon }{}^{\epsilon }{}_{}{}^{}$ coincides with the trace anomaly 〈T^ ${}_{\epsilon }{}^{\epsilon }{}_{}{}^{}〉_{}^{\mathrm{ren}}{}_{}{}^{}$, (ii) it satisfies the conservation law $T^{\mu \epsilon }$ ${\mathrm{}}_{;\epsilon }$=0, and (iii) it has the correct behavior under the scale transformations, then it is uniquely defined up to a few arbitrary constants. These constants must be chosen to satisfy the boundary conditions. In the case of a static black hole in a vacuum these conditions single out the unique tensor $T_{\mu \nu }$ which provides a good approximation for 〈T${^}_{\mu \nu }$ ${〉}^{\mathrm{ren}}$ in the Hartle-Hawking vacuum. The relation between this approach and the Page-Brown-Ottewill approach is discussed.