Vacuum〈φ2〉in Schwarzschild spacetime

For a conformally coupled scalar field we calculate the renormalized value of $〈{\phi }^2〉$ for the Hartle-Hawking vacuum in Schwarzschild spacetime for the region exterior to the horizon. We find that the mode-sum expression for ${〈{\phi }^2〉}_{\mathrm{ren}}$ separates naturally into two parts, a part that has a simple analytic form coinciding with the approximate expression of Whiting and Page, and a remainder that is small. We evaluate the remainder numerically and also provide an analytic approximation to it. Our results agree with, but are substantially more accurate than, those previously given by Fawcett.