The scale(s) of quantum gravity and integrable black holes

It is often argued that the Planck length (or mass) is the scale of quantum gravity, as shown by comparing the Compton length with the gravitational radius of a particle. However, the Compton length is relevant in scattering processes but does not play a significant role in bound states. We will derive a possible ground state for a dust ball composed of a large number of quantum particles entailing a core with the size of a fraction of the horizon radius. This suggests that quantum gravity becomes physically relevant for systems with compactness of order one for which the nonlinearity of General Relativity cannot be discarded. A quantum corrected geometry can then be obtained from the effective energy-momentum tensor of the core or from quantum coherent states for the effective gravitational degrees of freedom. These descriptions replace the classical singularity of black holes with integrable structures in which tidal forces remain finite and there is no inner Cauchy horizon. The extension to rotating systems is briefly mentioned.