The temperature in Hawking radiation as tunneling

TL;DR

This paper investigates the derivation of Hawking radiation using a quasi-classical tunneling approach, emphasizing the importance of canonical invariance and the horizon as a two-way barrier, and highlights the benefits of Painlevé coordinates.

Contribution

It introduces a canonical invariant method for deriving Hawking radiation, treating the horizon as a two-way barrier and clarifying the role of ingoing negative energy particles.

Findings

Standard Hawking temperature is recovered.

Horizon should be viewed as a two-way barrier.

Painlevé coordinates offer advantages in tunneling analysis.

Abstract

The quasi-classical method of deriving Hawking radiation under the consideration of canonical invariance is investigated. We find that the horizon should be regarded as a two-way barrier and the ingoing amplitude should be calculated according to the negative energy particles tunneling into the black hole because of the whole space-time interchange and thus the standard Hawking temperature is recovered. We also discuss the advantage of the Painlev\'{e} coordinates in Hawking radiation as tunneling.