Noise Kernel and Stress Energy Bi-Tensor of Quantum Fields in Hot Flat Space and Gaussian Approximation in the Optical Schwarzschild Metric

TL;DR

This paper develops a regularization method for the noise kernel of quantum fields in curved spacetimes, applies it to optical Schwarzschild metrics, and assesses the Gaussian approximation's accuracy near black hole horizons.

Contribution

It introduces a modified point separation scheme for optical spacetimes and evaluates the Gaussian approximation's effectiveness for the noise kernel near horizons.

Findings

Regularized noise kernel derived for thermal fields in flat space.

Finite noise kernel expression obtained at the Schwarzschild horizon.

Gaussian approximation shows significant errors at the horizon, especially at the fourth derivative order.

Abstract

Continuing our investigation of the regularization of the noise kernel in curved spacetimes [N. G. Phillips and B. L. Hu, Phys. Rev. D {\bf 63}, 104001 (2001)] we adopt the modified point separation scheme for the class of optical spacetimes using the Gaussian approximation for the Green functions a la Bekenstein-Parker-Page. In the 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 a finite expression for the noise kernel at the horizon and recover the hot flat space result at infinity. Knowledge of the noise kernel is essential for studying issues related to black hole horizon fluctuations and Hawking radiation backreaction. We show that the Gaussian approximated Green function which works surprisingly well for the stress tensor at the Schwarzschild horizon produces significant error in the noise kernel there. We identify the failure as occurring at the fourth covariant derivative order.