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Universe 2018, 4(9), 93; https://doi.org/10.3390/universe4090093

Greybody Factors for Schwarzschild Black Holes: Path-Ordered Exponentials and Product Integrals

School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
Current address: Perimeter Institute, Waterloo, ON N2L 2Y5, Canada; fgray@perimeterinstitute.ca and Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada.
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Received: 14 August 2018 / Revised: 28 August 2018 / Accepted: 28 August 2018 / Published: 3 September 2018
(This article belongs to the Collection Open Questions in Black Hole Physics)
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Abstract

In earlier work concerning the sparsity of the Hawking flux, 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. Focusing specifically on Schwarzschild black holes, we have re-calculated and re-assessed the greybody factors using a path-ordered-exponential approach, a technique which has the virtue of providing a pedagogically useful semi-explicit formula for the relevant Bogoliubov coefficients. These path-ordered-exponentials, being based on a variant of the “transfer matrix” formalism, are closely related to so-called “product integrals”, leading to quite straightforward and direct numerical evaluation, while side-stepping any need for numerically solving the relevant ordinary 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 actually concentrated. Working in a more general context, these path-ordered-exponential techniques are also likely to be of interest for generic barrier-penetration problems. View Full-Text
Keywords: Hawking flux; greybody factor; cross section; Regge–Wheeler potential; Bogoliubov coefficients; barrier penetration Hawking flux; greybody factor; cross section; Regge–Wheeler potential; Bogoliubov coefficients; barrier penetration
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Gray, F.; Visser, M. Greybody Factors for Schwarzschild Black Holes: Path-Ordered Exponentials and Product Integrals. Universe 2018, 4, 93.

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