The Singularity Problem in Gravitational Theory. The Spherically Symmetric Case

In this work we discuss some specific features related to the concept of singularity in the gravitational theory. We give a brief review of some various definitions for singularity, then we explore some "negative" results, in the sense they are not able to reproduce, in general, a regular solution. We present some of these approaches, namely the non-commutative geometry; the Non-Linear Electrodynamics; and the conformal approach. We later generalize these results into a no-go theorem, which is actually a fully original result. In the second part of this work, we discuss some working examples of regular solutions: we present three of them already present in literature (non-minimal Yang-Mills coupling, mimetic field approach and non-polynomial gravity), then we use such results to build up a model of a regular cosmological solution. Its generating mechanism and its main features are described, replacing the Big Bang with a bounce; the inflationary behavior at large time is also recovered. In the following two chapters, we present some different schemes to build regular solutions from the coupling between gravity and a scalar field. In particular, in chapter 7, we use a minimal coupling, while in chapter 8 we find some sufficient (though not necessary) conditions to build a regular solution, within the framework of the Horndeski theory. In both cases we are not able to find explicit results. In the ninth chapter we discuss a model of a regular black hole, coupling gravity with some fluid: in this case, an exact solution is found. We prove it is regular and we show some of its general features; we also discuss the time-dependent case, although we are only able to discuss its asymptotic behavior. We also discuss some of its problems, mainly due to instability. In the appendices we try to extend the no-go theorem to $F(R)$ theories and try to solve the instabilities of the fluid approach respectively.