Black hole quantum spectrum

Introducing a black hole (BH) effective temperature, which takes into account both the non-strictly thermal character of Hawking radiation and the countable behavior of emissions of subsequent Hawking quanta, we recently re-analysed BH quasi-normal modes (QNMs) and interpreted them naturally in terms of quantum levels. In this work we improve such an analysis removing some approximations that have been implicitly used in our previous works and obtaining the corrected expressions for the formulas of the horizon's area quantization and the number of quanta of area and hence also for Bekenstein-Hawking entropy, its sub-leading corrections and the number of micro-states, i.e. quantities which are fundamental to realize the underlying quantum gravity theory, like functions of the QNMs quantum "overtone" number n and, in turn, of the BH quantum excited level. An approximation concerning the maximum value of n is also corrected. On the other hand, our previous results were strictly corrected only for scalar and gravitational perturbations. Here we show that the discussion holds also for vector perturbations. The analysis is totally consistent with the general conviction that BHs result in highly excited states representing both the "hydrogen atom" and the "quasi-thermal emission" in quantum gravity. Our BH model is somewhat similar to the semi-classical Bohr's model of the structure of a hydrogen atom. The thermal approximation of previous results in the literature is consistent with the results in this paper. In principle, such results could also have important implications for the BH information paradox.