Hawking-like radiation and the density matrix for an infalling observer during gravitational collapse

TL;DR

This paper investigates the time-dependent Hawking-like radiation experienced by an infalling observer during gravitational collapse, revealing non-thermal features and the importance of off-diagonal correlations in the density matrix.

Contribution

It provides a detailed calculation of particle occupation numbers and the density matrix for an infalling observer, highlighting the significance of off-diagonal terms in the radiation's quantum description.

Findings

Radiation is approximately thermal near the Schwarzschild radius.

Temperature diverges as the shell approaches the singularity.

Off-diagonal density matrix terms are crucial for a complete quantum description.

Abstract

We study time-dependant Hawking-like radiation as seen by an infalling observer during gravitational collapse of a thin shell. We calculate the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in Eddington-Finkelstein coordinates. We solve the equations for the whole process from the beginning of the collapse till the moment when the collapsing shell reaches zero radius. The radiation distribution is not thermal in the whole frequency regime, but it is approximately thermal for the wavelengths of the order of the Schwarzschild radius of the collapsing shell. After the Schwarzschild radius is crossed, the temperature increases without limits as the singularity is approached. We also calculate the density matrix associated with this radiation. It turns out that the off-diagonal correlation terms to the diagonal Hawking's leading order terms are very important. While the trace of the diagonal (Hawking's) density matrix squared decreases during the evolution, the trace of the total density matrix squared remains unity at all times and all frequencies.