Noise Kernel and Stress Energy Bi-Tensor of Quantum Fields in Conformally-Optical Metrics: Schwarzschild Black Holes

TL;DR

This paper analyzes the noise kernel and stress-energy bi-tensor of quantum scalar fields in Schwarzschild black hole spacetimes using a conformally-optical metric approach, verifying known results and highlighting limitations of Gaussian approximations near the horizon.

Contribution

It extends the regularization of the noise kernel to Schwarzschild black holes and assesses the accuracy of Gaussian approximations in this context.

Findings

Fluctuations match previous analytic results in the far field.

Verifies Page's stress tensor result with a rigorous approach.

Gaussian approximation shows significant errors at the horizon.

Abstract

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.