Corpuscular description of black hole interiors

The motivation from this work stems from the idea of Dvali and Gomez that the end-state of the gravitational collapse is a Bose-Einstein condensate at the critical point, constituted by a large number of soft, off-shell and maximally packed gravitons. This approach goes beyond the semiclassical picture and considers black holes as purely quantum objects. The result is a model without a central singularity that expresses crucial quantities, such as the self-coupling constant of gravitons and the mass of the black hole in terms of only one parameter, the number of gravitons N, and is able to account for Bekenstein entropy and Hawking radiation. Recently, an effective quantum description of the static gravitational potential for a spherically symmetric system up to post-Newtonian order has been constructed, relying on a toy model of scalar gravitons. This model allows to reproduce the classical Newtonian potential by employing a coherent state, establishing a connection between the corpuscular model and post-Newtonian corrections. These works constitute the starting point of this thesis. After recovering physical units in the Lagrangian for the field up to post-Newtonian order and finding its equations of motion, we move on to its linearisation, that models the field as a Newtonian background plus a small perturbation. The perturbation is parametrized by a spherical wave ansatz in the WKB approximation, where the wavelength of the wave is thought to be much smaller than the scale on which the background varies. After writing the equations of motion for the perturbation, we then find the correspondent dispersion relation, up to the first order in the graviton self-coupling, and make considerations about the behaviour of the perturbations. The end result is that the perturbations decay in time and end up reabsorbed in the background. If on the other hand the opposite, long-wavelength limit is considered, the perturbations get amplified and might signal an instability.