Renormalized thermal stress tensor for arbitrary static space-times

A method of constructing an analytical expression for the renormalized vacuum expectation value of the stress tensor $〈T_{\nu }^{\mu }〉$ and the mean-square field $〈{\phi }^2\mathrm{}\left(x\right)\mathrm{}〉$ for a conformally invariant scalar field propagating on static space-times is presented. Particular emphasis is given to the case where the background geometry corresponds to a general spherically symmetric black hole, and explicit results are given for an extremal Reissner-Nordström black hole. In the special case of a Schwarzschild black hole, Page's results are recovered. Possible extensions of the method to cover stationary black holes are briefly discussed.