Noise kernel and the stress energy bitensor of quantum fields in hot flat space and the Schwarzschild black hole under the Gaussian approximation

We continue our investigation of the properties of the noise kernel in curved spacetimes [N. G. Phillips and B. L. Hu, Phys. Rev. D 63, 104001 (2001)] by working out explicit examples by means of the modified point-separation scheme and the Gaussian approximation for the Green functions in the manner of Bekenstein, Parker, and Page [J. D. Bekenstein and L. Parker, Phys. Rev. D 23, 2850 (1981); D. N. Page, ibid. 25, 1499 (1982)]. In the first part we consider the class of optical spacetimes. As a first example we derive the regularized noise kernel for a thermal field in flat space. It is useful for black hole nucleation considerations. In the second example of an optical-Schwarzschild spacetime we obtain after this procedure a finite expression for the noise kernel at the horizon. In the second part we consider the noise kernel for a scalar field in the Schwarzschild black hole. Knowledge of the noise kernel is essential for studying issues related to black hole horizon fluctuations and Hawking radiation back reaction. Much of the work in this part is to determine how the divergent piece conformally transforms under the point-separation scheme. 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 58, 125021 (1998); Int. J. Theor. Phys. 38, 1253 (1999)]. We also verify Page's result for the stress tensor, which, though used often, still lacks a rigorous proof, since in his original work the direct use of the conformal transformation was circumvented. We find that the noise kernel at the Schwarzschild horizon is finite. This dispels speculations in some recent papers that the black hole fluctuations diverge at the horizon. However, as already manifest in the optical case, the Gaussian approximated Green function which works surprisingly well for the stress tensor at the Schwarzschild horizon produces significant error in the noise kernel evaluated there. We check this using the trace anomaly expression and identify the failure as occurring at the fourth covariant derivative order