^{1, 2, 3}

^{1}

^{2}

^{3}

^{3}.

Some recent important results on black hole (BH) quantum physics concerning the

An intriguing and largely used framework to obtain Hawking radiation [

The nonprecise black body spectrum has important implications for the BH information puzzle. In fact, arguments that information is lost during BH evaporation partially rely on the assumption of strictly black body spectrum [

It is important to recall that Bohr model is an approximated model of the hydrogen atom with respect to the valence shell atom model of full quantum mechanics. In the same way, one expects the Bohr-like BH model to be an approximated model with respect to the definitive, but at the present time unknown, BH model arising from a definitive theory of quantum gravity.

The time evolution of the Bohr-like BH model obeys a

A problem on the tunnelling approach for Hawking radiation was that, in [

For the sake of simplicity, in all this paper we work with Planck units; that is,

Considering a Schwarzschild BH, the Schwarzschild line element is [

In fact, the tunnelling is a

If one wants to take into due account the dynamical geometry of the BH during the emission of the particle, a BH

After the introduction of

Now, let us rewrite (

Resuming, in [

In this section we review the results in [

BH QNMs are modes of radial perturbations obeying the time independent Schrödinger-like equation [

The relation between the Regge-Wheeler “tortoise” coordinate

Considering a strictly thermal approximation, BH QNMs are usually labelled as

Let us discuss another argument which emphasizes the correctness of the analysis in the Appendix of [

Equation (

As BHs cannot emit more energy than their total mass, the physical solution for the absolute values of the frequencies (

Considering an emission from the ground state (i.e., a BH which is not excited) to a state with large

Now, following [

Bekenstein [

On the other hand, the total BH entropy contains at least three parts which are necessary to realize the underlying theory of quantum gravity [

Let us resume the way in which the BH model analysed in this section works. If

One finds three different physical situations for excited BHs (

At the Planck scale,

Another key point is as follows. In Hawking’s original computation [

In his famous paper [

In Section

In addition, it will be shown that the present approach permits also solving the entanglement problem connected with the information paradox [

The BH information paradox is considered one of the most famous and intriguing scientific controversies in the whole history of Science [

As we previously recalled, a key point, concerning not only the BH information paradox, but the whole BH quantum physics, is that the BH radiation spectrum is not strictly thermal [

Let us start by recalling that

In a quantum mechanical framework, we physically interpret emissions of Hawking quanta like quantum jumps among the unperturbed levels (

We observe that the final state of (

Our analysis is strictly correct only for excited BHs, that is, for

Now, let us discuss another key point, which concerns quantum entanglement. We could think that although previous analysis discusses a very natural model of Hawking radiation and BH evaporation, there is no reference to the BH spacetime, where information is assumed to be conserved. There are indeed authors who claim that the real challenge in solving the information paradox is to reconcile models of Hawking radiation with the spacetime structure within the BH horizon, where the quantum information falling into the singularity is causally separated from the outgoing Hawking quanta; see the work by Mathur [

Now, we consider a second emission. This new emission corresponds to the transition from the state with

Let us consider a third emission, corresponding to the transition from the state with

The process will continue again, and again, and again… till the

In any case, we emphasize again that the energy

Therefore, all the quantum physical information which has fallen into the singularity is not causally separated from the outgoing Hawking radiation, but is instead recovered and codified in (

In this review paper some recent important results in BH quantum physics, which concern the BH effective state and the Bohr-like model for BHs in [

Again, we stress that Bohr model is an approximated model of the hydrogen atom with respect to the valence shell atom model of full quantum mechanics. Then, one expects the Bohr-like BH model to be an approximated model with respect to the definitive, but at the present time unknown, BH model arising from a complete theory of quantum gravity.

Important consequences on the BH information puzzle have been also discussed, reviewing the independent solution to the paradox found in [

Finally, for the sake of completeness, we recall that in some cases in extended gravity [

The author declares that there is no conflict of interests regarding the publication of this paper.

It is a pleasure to thank Professor Maxim Khlopov for his invitation to write this review paper. The author thanks the unknown reviewers for useful comments and Professor J. D. Bekenstein for pointing out some important issues concerning the history of BH thermodynamics.