Horizon Quantum Mechanics of Generalized Uncertainty Principle Black Holes

Abstract

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 $M=m\left(1+\frac{\beta }{2}\frac{M_{\mathrm{Pl}}^2}{m^2}\right)$. Considering the case of a wave-packet shaped by a Gaussian distribution, we compute the HWF and the probability ${\mathcal{P}}_{BH}$ that the source is a (quantum) black hole, i.e., that it lies within its horizon radius. The case $\beta <0$ is qualitatively similar to the standard Schwarzschild case, and the general shape of ${\mathcal{P}}_{BH}$ 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 $\beta$, and drops to 0 for a minimum mass value. The scenario differs in significantly for increasing $\beta >0$, where a minimum in ${\mathcal{P}}_{BH}$ is encountered, thus meaning that every particle has some probability of decaying to a black hole. Furthermore, for sufficiently large $\beta$ we find that every particle is a quantum black hole, in agreement with the intuitive effect of increasing $\beta$, which creates larger $M$ and $R_H$ 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 $R_H\sim M^{-1}$.

arXiv comment15 pages, 6 color figures

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