Hawking radiation and the Stefan-Boltzmann law: The effective radius of the black-hole quantum atmosphere

TL;DR

This paper investigates the effective radius of the quantum atmosphere responsible for Hawking radiation in higher-dimensional Schwarzschild black holes, finding it approaches the horizon as the number of dimensions increases, supporting near-horizon origin of emissions.

Contribution

It extends Giddings's analysis to higher dimensions, showing the quantum atmosphere's size relative to the horizon decreases with more spacetime dimensions.

Findings

The effective radius ratio decreases with increasing dimensions.

In large dimensions, the quantum atmosphere is very close to the horizon.

Supports the near-horizon origin of Hawking radiation in high dimensions.

Abstract

It has recently been suggested [S. B. Giddings, Phys. Lett. B {\bf 754}, 39 (2016)] that the Hawking black-hole radiation spectrum originates from an effective quantum "atmosphere" which extends well outside the black-hole horizon. In particular, comparing the Hawking radiation power of a $(3+1)$-dimensional Schwarzschild black hole of horizon radius $r_{\text{H}}$ with the familiar Stefan-Boltzmann radiation power of a $(3+1)$-dimensional flat space perfect blackbody emitter, Giddings concluded that the source of the Hawking semi-classical black-hole radiation is a quantum region outside the Schwarzschild black-hole horizon whose effective radius $r_{\text{A}}$ is characterized by the relation $\Delta r\equiv r_{\text{A}}-r_{\text{H}}\sim r_{\text{H}}$. It is of considerable physical interest to test the general validity of Giddings's intriguing conclusion. To this end, we study the Hawking radiation of $(D+1)$-dimensional Schwarzschild black holes. We find that the dimensionless radii $r_{\text{A}}/r_{\text{H}}$ which characterize the black-hole quantum atmospheres, as determined from the Hawking black-hole radiation power and the $(D+1)$-dimensional Stefan-Boltzmann radiation law, are a decreasing function of the number $D+1$ of spacetime dimensions. In particular, it is shown that radiating $(D+1)$-dimensional Schwarzschild black holes are characterized by the relation $(r_{\text{A}}-r_{\text{H}})/r_{\text{H}}\ll1$ in the large $D\gg1$ regime. Our results therefore suggest that, at least in some physical cases, the Hawking emission spectrum originates from quantum excitations very near the black-hole horizon.