Horizon Quantum Mechanics of Generalized Uncertainty Principle Black Holes

We study the Horizon Wavefunction (HWF) description of a generalized uncertainty principle inspired metric that admits sub-Planckian black holes, where the black hole mass m is replaced by M = m ( 1 + β2 M2 Pl m2 ) . Considering the case of a wave-packet shaped by a Gaussian distribution, we compute the HWF and the probability PBH that the source is a (quantum) black hole, i.e. that it lies within its horizon radius. The case β < 0 is qualitatively similar to the standard Schwarzschild case, and the general shape of PBH is maintained when decreasing the free parameter, but shifted to reduce the probability for the particle to be a black hole accordingly. The probability grows with increasing mass slowly for more negative β, and drops to 0 for a minimum mass value. The scenario differs in significantly for increasing β > 0, where a minimum in PBH is encountered, thus meaning that every particle has some probability of decaying to a black hole. Furthermore, for sufficiently large β we find that every particle is a quantum black hole, in agreement with the intuitive effect of increasing β, which creates larger M and RH terms. This is likely due to a “dimensional reduction” feature of the model, where the black hole characteristics for sub-Planckian black holes mimic those in (1 + 1)-dimensions and the horizon size grows as RH ∼M−1.