Noise kernel for a quantum field in Schwarzschild spacetime under the Gaussian approximation

A method is given to compute the noise kernel, defined as the symmetrized connected 2-point function of the stress tensor, for the conformally invariant scalar field in the Hartle-Hawking state in Schwarzschild spacetime. We consider the two points to be separated in a timelike or a spacelike direction and use a Gaussian approximation of the same type as that used by Page to compute an approximate form of the stress tensor for this field in Schwarzschild spacetime. Several components of the noise kernel have been explicitly computed and the expression for one of them is displayed