Emerging Geometry of Corpuscular Black Holes

Quantum physics lends a view of space-time geometry as an emergent structure that shows classical features only at some observational level. The space-time manifold can be viewed as a purely theoretical arena, where quantum states and observables are defined, with the additional freedom of changing coordinates. We focus on spherically symmetric quantum sources, and determine the probability they are black holes. The gravitational radius is promoted to quantum mechanical operator acting on the ``horizon wave-function''. This formalism is applied to several sources with mass around the fundamental scale, as natural candidates of quantum black holes. This horizon quantum mechanics supports some features of BEC models of black holes. The Klein-Gordon equation for a toy graviton field coupled to a static matter current classically reproduces the Newtonian potential, while the corresponding quantum state is given by a coherent superposition of scalar modes. When N such bosons are self-confined in a volume of the size of the Schwarzschild radius, the horizon shows that their radius corresponds to a proper horizon whose related uncertainty is connected to the typical energy of Hawking modes: it is suppressed as N increases, contrarily to a single very massive particle. The spectrum of these systems is formed by a discrete ground state and a continuous Planckian distribution at the Hawking temperature representing the radiation. Assuming the internal scatterings give rise to the Hawking radiation, the N-particle state can be collectively described by a single-particle wave-function. The partition function follows together with the usual entropy law, with a logarithmic correction related to the Hawking component. The backreaction of radiating modes is also shown to reduce the Hawking flux, and eventually stop it.