Greybody factors for Schwarzschild black holes: Path-ordered exponentials and product integrals

TL;DR

This paper introduces a path-ordered-exponential method to calculate greybody factors for Schwarzschild black holes, providing a semi-explicit, numerically efficient approach that improves understanding of Hawking radiation across frequency regimes.

Contribution

The authors develop a novel path-ordered-exponential technique for computing black hole greybody factors, offering a semi-explicit formula and straightforward numerical evaluation, extending previous methods from the 1970s.

Findings

Provides a semi-explicit formula for Bogoliubov coefficients.

Enables direct numerical evaluation without solving differential equations.

Facilitates modeling of greybody factors in the intermediate frequency regime.

Abstract

In recent work concerning the sparsity of the Hawking flux [arXiv:1506.03975v2], we found it necessary to re-examine what is known regarding the greybody factors of black holes, with a view to extending and expanding on some old results from the 1970s. Focussing specifically on Schwarzschild black holes, we re-calculated and re-assessed the greybody factors using a path-ordered-exponential approach, a technique which has the virtue of providing a semi-explicit formula for the relevant Bogoliubov coefficients. These path-ordered-exponentials, (being based on a "transfer matrix" formalism), are closely related to so-called "product integrals", leading to quite straightforward and direct numerical evaluation, while avoiding any need for numerically solving differential equations. Furthermore, while considerable analytic information is already available regarding both the high-frequency and low-frequency asymptotics of these greybody factors, numerical approaches seem better adapted to finding suitable "global models" for these greybody factors in the intermediate frequency regime, where most of the Hawking flux is concentrated. Working in a more general context, these path-ordered-exponential techniques are also likely to be of interest for generic barrier-penetration problems.