Revisit emission spectrum and entropy quantum of the Reissner-Nordstr\"{o}m black hole

TL;DR

This paper refines the tunneling approach to black hole radiation by introducing a proper Kruskal extension for Reissner-Nordström black holes, successfully reproducing the Hawking spectrum and entropy quantization.

Contribution

It provides a correct Kruskal extension for charged black holes, enabling a consistent tunneling picture of Hawking radiation and entropy quantization.

Findings

Reproduces Hawking emission spectrum via tunneling

Derives a better approximation for entropy quantum

Shows tunneling method's robustness for charged black holes

Abstract

Banerjee and Majhi's recent work shows that black hole's emission spectrum could be fully reproduced in the tunneling picture, where, as an intriguing technique, the Kruskal extension was introduced to connect the left and right modes inside and outside the horizon. Some attempt, as an extension, was focused on producing the Hawking emission spectrum of the (charged) Reissner-Nordstr\"{o}m black hole in the Banerjee-Majhi's treatment. Unfortunately, the Kruskal extension in their observation was so badly defined that the ingoing mode was classically forbidden traveling towards the center of black hole, but could quantum tunnel across the horizon with the probability $\Gamma=e^{-\pi \omega_0/\kappa_+}$. This tunneling picture is unphysical. With this point as a central motivation, in this paper we first introduce such a suitable Kruskal extension for the (charged) Reissner-Nordstr\"{o}m black hole that a perfect tunneling picture can be provided during the charged particle's emission. Then, under the new Kruskal extension, we revisit the Hawking emission spectrum and entropy spectroscopy as tunneling from the charged black hole. The result shows that the tunneling method is so universally robust that the Hawking blackbody emission spectrum from a charged black hole can be well reproduced in the tunneling mechanism, and its induced entropy quantum is a much better approximation for the forthcoming quantum gravity theory.