Hawking radiation of massive fields in 2D

TL;DR

This paper investigates Hawking radiation for massive scalar fields in a two-dimensional black hole model, demonstrating the validity of the WKB approximation for large mass-radius products and confirming thermal radiation with Hawking temperature.

Contribution

It extends previous massless field analyses by including massive fields and shows the conditions under which semiclassical modes accurately describe the radiation.

Findings

WKB approximation valid for $mr_g \\gg 1$ outside the shell

Radiation near the shell and at infinity is thermal with Hawking temperature

Negative flux $T_{vv}$ observed near the shell, similar to massless case

Abstract

We extend the classic results of the paper P. C. W. Davies, S. A. Fulling, and W. G. Unruh "Energy-momentum tensor near an evaporating black hole" by considering a massive scalar field in a two dimensions in the presence of a thin shell collapse. We show that outside the shell the WKB approximation is valid for any value of $r$ if $mr_{g} \gg 1$, where $m$ is the mass of the field, and $r_{g}$ is the Schwarzschild radius. Thus, we use semiclassical modes to calculate the flux in the vicinity of the shell, and at spatial infinity, $r \rightarrow +\infty$ at the final stage of the collapse, $t \rightarrow +\infty$ with the use of the covariant point-splitting regularization. We get that near the shell and at the spatial infinity the radiation is thermal with Hawking temperature. We obtain the negative flux $T_{vv}$ in the vicinity of the shell, which is similar to the classic result in the massless case.