Horizon of quantum black holes in various dimensions

Abstract

We adapt the horizon wave-function formalism to describe massive static spherically symmetric sources in a general $(1+D)$-dimensional space-time, for $D>3$ and including the $D=1$ case. We find that the probability $P_{\rm BH}$ that such objects are (quantum) black holes behaves similarly to the probability in the $(3+1)$ framework for $D> 3$. In fact, for $D\ge 3$, the probability increases towards unity as the mass grows above the relevant $D$-dimensional Planck scale $m_D$. At fixed mass, however, $P_{\rm BH}$ decreases with increasing $D$, so that a particle with mass $m\simeq m_D$ has just about $10\%$ probability to be a black hole in $D=5$, and smaller for larger $D$. This result has a potentially strong impact on estimates of black hole production in colliders. In contrast, for $D=1$, we find the probability is comparably larger for smaller masses, but $P_{\rm BH} < 0.5$, suggesting that such lower dimensional black holes are purely quantum and not classical objects. This result is consistent with recent observations that sub-Planckian black holes are governed by an effective two-dimensional gravitation theory. Lastly, we derive Generalised Uncertainty Principle relations for the black holes under consideration, and find a minimum length corresponding to a characteristic energy scale of the order of the fundamental gravitational mass $m_D$ in $D>3$. For $D=1$ we instead find the uncertainty due to the horizon fluctuations has the same form as the usual Heisenberg contribution, and therefore no fundamental scale exists.