Precise model of Hawking radiation from the tunnelling mechanism

TL;DR

This paper refines the Hawking radiation tunnelling model by accurately determining the pre-factor of the emission probability, linking it to black hole quantum levels and unitarity, thus enhancing the understanding of black hole evaporation.

Contribution

It provides the exact pre-factor for the Parikh and Wilczek emission probability, incorporating black hole quantum levels and unitarity into the tunnelling framework.

Findings

Pre-factor depends on black hole quantum level n.

Emission probability expressed in terms of quantum levels.

Pre-factor approximates to 1 but varies with n.

Abstract

We recently improved the famous result of Parikh and Wilczek, who found a probability of emission of Hawking radiation which is compatible with a non-strictly thermal spectrum, showing that such a probability of emission is really associated to two non-strictly thermal distributions for boson and fermions. Here we finalize the model by finding the correct value of the pre-factor of the Parikh and Wilczek probability of emission. In fact, that expression has the "of order" sign instead of the equality. In general, in this kind of leading order tunnelling calculations, the exponent arises indeed from the classical action and the pre-factor is an order Planck constant correction. But in the case of emissions of Hawking quanta, the variation of the Bekenstein-Hawking entropy is order 1 for an emitted particle having energy of order the Hawking temperature. As a consequence, the exponent in the Parikh and Wilczek probability of emission is order unity and one asks what is the real significance of that scaling if the pre-factor is unknown. Here we solve the problem assuming the unitarity of the black hole (BH) quantum evaporation and considering the natural correspondence between Hawking radiation and quasi-normal modes (QNMs) of excited BHs, in a "Bohr-like model" that we recently discussed in a series of papers. In that papers, QNMs are interpreted as natural BH quantum levels (the "electron states" in the "Bohr-like model"). Here we find the intriguing result that, although in general it is well approximated by 1, the pre-factor of the Parikh and Wilczek probability of emission depends on the BH quantum level n. We also write down an elegant expression of the probability of emission in terms of the BH quantum levels.