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.
Highlights
In this paper we extend the classic results of P
III B we find the behavior of in-harmonics at the final stage of the thin shell collapse
At the final stage of the collapse, as we show later, such a separation of t and r is not possible because the background depends on time, and the harmonics have a more complicated form
Summary
In this paper we extend the classic results of P. Our goal is to calculate the expectation value of the stressenergy tensor at the final stage of the thin shell collapse using the covariant point-splitting regularization [4,5,6,7,8,9,10] In this classic paper [1], the authors calculate the expectation value, Tμν, of the stress-energy tensor for the massless scalar field in the two-dimensional model of the gravitational collapse. To make the paper self-contained the details of calculations are present in the Appendix A–C
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