Probing the regularization of spacetime singularities

In General relativity (GR), the formation of spacetime singularities is proved to be inevitable in the gravitational collapse of very compact objects and in the early universe. These singularities are considered the sign of the breakdown of the theory and the proof of its intrinsic incompleteness. The common belief is that their formation will be prevented in a full, potentially quantum, completion of GR. In this view, it is reasonable to assume that non-singular, or regular, metrics can provide an effective description of the outcome of gravitational collapse and of the early universe, providing a valid alternative to the Big Bang and to singular black holes. This thesis covers various aspects of this theme. Firstly we deal with a classification of the possible local regularization of spacetime singularities. Indeed, in recent years, a systematic, quantum gravity agnostic, study has been carried out to catalogue all the conceivable non-singular, continuous geometries arising from evading Penrose’s focusing theorem in gravitational collapse. In this study, we extend this inquiry by systematically examining all potential non-singular, continuous extensions into the past of an expanding Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric. Subsequently, we discuss the construction of global regular metrics that embody the aforementioned possible local regularizations of black holes singularities. In particular, we present a new rotating regular black hole whose inner horizon has zero surface gravity for any value of the spin parameter, and is therefore stable against mass inflation. Finally we study the phenomenological properties of these regular alternatives to sin- gular black holes, to which we refer as black hole mimickers. Namely we investigate the gravitational waves signal generated by their merger, in particular their ringdown signal. This is characterized by a different spectrum of quasinormal modes (QNMs) with respect to singular black holes and, in the case of horizonless mimickers, by the appearance of echoes after the prompt ringdown, that is a series of secondary pulses with progressively smaller amplitudes. We compute this spectrum of QNMs enlightening a common misunderstanding regarding the study of gravitational perturbations in such regular spacetimes. We also investigate the effect of non-linear interactions on the echoes part of the signal and its sensitivity to the internal structure of the emitting object. In particular, we find that if one considers the increase of the central object mass due to the partial absorption of the energy carried by perturbations, the echo signal can be quite different and non-periodic. Furthermore, when gravitational perturbations are assumed to travel thorough the object instead of being reflected at the surface, the echoes signal appears to be very sensitive to the structure of the innermost region of the mimicker. This research fits into the broader context of exploring alternatives to classical cosmological and black hole models, aiming to test new (quantum) physics beyond General Relativity in extreme density and curvature environments.