Vacuum〈Tμν〉in Schwarzschild spacetime

The vacuum expectation value of the stress-energy tensor for the Hartle-Hawking vacuum in Schwarzschild spacetime has been calculated by means of the method of covariant point separation. It is found that $〈{T_{\mu }}^{\nu }〉$ separates naturally into the sum of two terms. The first coincides with an approximate expression suggested by Page on the basis of a Gaussian approximation to the propertime Green's function. The second term is a "remainder" which comprises sums over mode functions that may be evaluated numerically. It is found that the total expression is in good qualitative agreement with Page's approximation. These results are at variance with earlier numerical results given by Fawcett which purported to show that the true value of $〈{T_{\mu }}^{\nu }〉$ differed in important respects from Page's approximation. The error in Fawcett's calculation is explained.