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Einstein’s General Relativity predicts the existence of black holes. In the deep interior of a black hole a singularity arises, that implies the breakdown of the theory itself. Such ill-defined regions are expected to be eliminate by quantum gravitational effects. While a complete and satisfactory theory of Quantum Gravity is not yet available, it is possible to start from a semi-classical approach.
The basic idea is to mimic background-independent effects that allow the avoidance of the singularity, by building effective metrics solutions of the Einstein’s Equations such that the resulting SpaceTime is Schwarzschild-like in the outer region and, at the same time, singularity-free in the deep interior
The main goal of this work is to analyze, thanks to classic and new tools, both static and dynamic properties of such solutions, in order to verify their physical plausibility.

The analysis is divided in three Parts, each of which contains two Chapters.

Part I:
The First Part is devoted to a presentation of the principal features of classic black holes. Nevertheless, its aim is not to make a complete satisfactory review of the classical black hole physics, but to discuss those aspects that will be relevant in the following analysis.
Therefore, the First Chapter starts with the presentation of the Schwarzschild’s solution to the Einstein’s Equations from where the study of black holes initially arose. The following more mathematical Section is dedicated to the proof of the Singularity Theorems that lay at the basis of the motivations of this thesis.
Going beyond the purely classical properties, in Chapter 2 we introduce the machinery of Quantum Field Theory in Curved SpaceTime and Bogoliubov’s formalism for particle creation. The astonishing result of Hawking’s radiation naturally arises applying these formalisms to the Vaidya-Schwarzschild’s metric: black holes radiate energy away with a Planck spectrum. Their complete evaporation implies the so called information-loss paradox: the evolution of a pure quantum state propagating on Vaidya-Schwarzschild metric is not unitary. Moreover, a discussion on the importance of the Hawking’s result for the so called black hole thermodynamics introduces us to the concepts of black hole entropy and entanglement entropy. Since the latter plays as basis for the entire Chapter 6, we dedicate the last Section of this Part to its definition and basic features.

The Second and the Third Part represent the core of the thesis. As said before, the goal is to analyze non-singular black holes metrics both in their static and dynamic behavior. Therefore, we dedicate the Second Part to the study of the properties of such objects settled by the gravitational collapse of a spherical body and remaining in their static configuration. In the Third Part Hawking’s radiation is turned on and the dynamic features are analyzed.

Part II:
The Second Part starts, in its first Chapter (Chapter 3), with the introduction of what is known about non-singular black holes. The first Section presents the proof of a theorem by Irina Dymnikova asserting that, if such non-singular black holes exist, they must have a rather universal causal structure. This structure is deeply studied in the second Section, focusing on the particular example proposed by Sean Hayward and recently reconsidered by many authors.
Chapter 4, on the other hand, contains the first original results. We point out two physical requirements that are not satisfied by the current metrics, and show how to properly take them into account. Indeed, it seems physically unreasonable that a clock at the (regular) center of the star suffers no time delay with respect to a clock at infinity. Moreover, an effective metric that supposes to mimic quantum effects should capture the 1-loop quantum corrections to the Newton’s potential obtained by John Donoghue using effective field theory. In the last Section a relatively easy solution is proposed (Modified Hayward’s Metric), providing a more realistic description of a non-singular black hole.

Part III:
Static non-singular black holes form an event horizon. Therefore we expected Hawking’s radiation and consequent evaporation to take place and the after-formation system to become dynamic.
In the introductory brief Chapter 5 we introduce some first insights in the problem considering the so called quasi-statical approximation to hold during the entire evaporation process. As in the original Hawking’s evaporation case, the dynamics will be simply encoded allowing the mass of the black hole to decrease in time. Different scenarios are shortly discussed.
The main results, however, are presented in the last Chapter. Here the plausibility of evaporation processes is studied through the investigation of their entanglement entropy production, the so called Page’s curve. This analysis is made quantitative possible thanks to a new covariant definition of entanglement entropy developed by Eugenio Bianchi and Matteo Smerlak. From this definition follows the possibility to give a precise characterization of entanglement entropy production and to analytically compute the Page’s curve associated to any SpaceTime. In particular, applied to the Hayward’s metric, this analysis confirms the recover of unitarity, but at the same time shines a light on two non-easily solvable problems. Namely, (i) the total energy radiated by the hole turns out to be much bigger than the initial ADM mass, and (ii) the so called purification time does not satisfies a physical lower bound we can impose on it. These inconsistencies undermine the physical validity of the dynamic Hayward’s metric itself (and, because of the Dymnikova’s theorem, of almost all the metrics so far proposed) as a good semi-classical approach to the resolution of the singularity and of the information-loss paradox. Different ideas are needed. The new definition of entanglement entropy provides a powerful tool to analyze the physical plausibility of any semi-classical scenario of formation and consequent unitary evaporation that can be proposed, as for example the ‘black hole firework’ proposed by Hal Haggard and Carlo Rovelli studied in the last Section of this work. Up to now, however, no one of the proposal we encountered seems to satisfy all the requirements one can impose on it.

The study of Hawking’s radiation and evaporating black holes is a very active and fascinating field of research to which this thesis can contribute with original ideas and results.