Abstract
The tunnelling mechanism is today considered a popular and widely used method in describing Hawking radiation. However, in relation to black hole (BH) emission, this mechanism is mostly used to obtain the Hawking temperature by comparing the probability of emission of an outgoing particle with the Boltzmann factor. On the other hand, Banerjee and Majhi reformulated the tunnelling framework deriving a black body spectrum through the density matrix for the outgoing modes for both the Bose-Einstein distribution and the Fermi-Dirac distribution. In contrast, Parikh and Wilczek introduced a correction term performing an exact calculation of the action for a tunnelling spherically symmetric particle and, as a result, the probability of emission of an outgoing particle corresponds to a non-strictly thermal radiation spectrum. Recently, one of us (C. Corda) introduced a BH effective state and was able to obtain a non-strictly black body spectrum from the tunnelling mechanism corresponding to the probability of emission of an outgoing particle found by Parikh and Wilczek. The present work introduces the quantum corrected effective temperature and the corresponding quantum corrected effective metric is written using Hawking’s periodicity arguments. Thus, we obtain further corrections to the non-strictly thermal BH radiation spectrum as the final distributions take into account both the BH dynamical geometry during the emission of the particle and the quantum corrections to the semiclassical Hawking temperature.
Highlights
Considering Hawking radiation [1] in the tunnelling approach, [2,3,4,5,6,7,8,9,10,11] particle creation mechanism caused by the vacuum fluctuations near the black hole (BH) horizon works as follows
A virtual particle pair is created just inside the horizon and the virtual particle with positive energy can tunnel out the BH horizon as a real particle
The virtual particle pair is created just outside the horizon and the negative energy particle can tunnel inwards. For both the possibilities, the particle with negative energy is absorbed by the BH and as a result the mass of the BH decreases
Summary
Considering Hawking radiation [1] in the tunnelling approach, [2,3,4,5,6,7,8,9,10,11] particle creation mechanism caused by the vacuum fluctuations near the BH horizon works as follows. The flow of positive energy particles towards infinity is considered as Hawking radiation Earlier, this approach was limited to obtain only the Hawking temperature through a comparison of the probability of emission of an outgoing particle with the Boltzmann factor rather than the actual radiation spectrum with the correspondent distributions. Considering contributions beyond semiclassical approximation in the tunnelling process, Parikh and Wilczek [2,3] found a probability of emission compatible with a non-thermal spectrum of the radiation from BH. Due to conservation of energy, in [2,3] the BH horizon contracts during the radiation process which deviates from the perfect black body spectrum This non-thermal spectrum has profound implications for realizing the underlying quantum gravity theory. The tunnelling points have zero separation, so there is no clear trajectory because there is no barrier [3,14,15]
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