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 PBH that such objects are (quantum) black holes behaves similarly to the probability in the (3+1) framework for D>3. In fact, for D≥3, the probability increases towards unity as the mass grows above the relevant D-dimensional Planck scale mD. At fixed mass, however, PBH decreases with increasing D, so that a particle with mass m≃mD 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 PBH<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 mD 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.