  In Paper II [N. G. Phillips and B. L. Hu, previous abstract] we presented the
details for the regularization of the noise kernel of a quantum scalar field in
optical spacetimes by the modified point separation scheme, and a Gaussian
approximation for the Green function. We worked out the regularized noise
kernel for two examples: hot flat space and optical Schwarzschild metric. In
this paper we consider noise kernels for a scalar field in the Schwarzschild
black hole. Much of the work in the point separation approach is to determine
how the divergent piece conformally transforms. For the Schwarzschild metric we
find that the fluctuations of the stress tensor of the Hawking flux in the far
field region checks with the analytic results given by Campos and Hu earlier
[A. Campos and B. L. Hu, Phys. Rev. D {\bf 58} (1998) 125021; Int. J. Theor.
Phys. {\bf 38} (1999) 1253]. We also verify Page's result [D. N. Page, Phys.
Rev. {\bf D25}, 1499 (1982)] for the stress tensor, which, though used often,
still lacks a rigorous proof, as in his original work the direct use of the
conformal transformation was circumvented. However, as in the optical case, we
show that the Gaussian approximation applied to the Green function produces
significant error in the noise kernel on the Schwarzschild horizon. As before
we identify the failure as occurring at the fourth covariant derivative order.
