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

A method is given to compute an approximation to the noise kernel, defined as the symmetrized connected 2-point function of the stress tensor, for the conformally invariant scalar field in any spacetime conformal to an ultra-static spacetime for the case in which the field is in a thermal state at an arbitrary temperature. The most useful applications of the method are flat space where the approximation is exact and Schwarzschild spacetime where the approximation is better than it is in most other spacetimes. The two points are assumed to be separated in a timelike or spacelike direction. The method involves the use of a Gaussian approximation which is of the same type as that used by Page to compute an approximate form of the stress tensor for this field in Schwarzschild spacetime. All components of the noise kernel have been computed exactly for hot flat space and one component is explicitly displayed. Several components have also been computed for Schwarzschild spacetime and again one component is explicitly displayed.