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_ν^μ〉 and the mean-square field 〈φ²(x)〉 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-Nordstr\"om 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.