Horizon Quantum Mechanics of Generalized Uncertainty Principle Black Holes

TL;DR

This paper explores how a generalized uncertainty principle affects the quantum properties of black holes, revealing that the probability of a particle being a black hole depends on a parameter that influences black hole size and formation likelihood.

Contribution

It introduces a modified mass parameter inspired by GUP into the Horizon Wavefunction framework, analyzing its impact on black hole probability and characteristics for sub-Planckian scales.

Findings

Probability of black hole formation varies with the GUP parameter β.

For negative β, black hole probability decreases with mass, similar to Schwarzschild.

For positive β, all particles can become black holes, indicating a dimensional reduction effect.

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 $M = m\left( 1 + \frac{\beta}{2} \frac{M_{\rm Pl}^2}{m^2} \right)$. Considering the case of a wave-packet shaped by a Gaussian distribution, we compute the HWF and the probability ${\cal {P}}_{BH}$ 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 ${\cal {P}}_{BH}$ 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 ${\cal {P}}_{BH}$ 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 $R_H \sim M^{-1}$.