Beyond General Relativity: Modified Theories and Non-Singular Black Holes

Einstein's general relativity is an extremely elegant and successful theory. Recent observations coming from the LIGO/VIRGO collaboration as well as from the Event Horizon Telescope give us the possibility to perform precision test of general relativity in regimes never tested before. Even though all the current observations are in perfect agreement with the predictions of general relativity, there are several reasons to study extensions of the theory. From the experimental point of view, we are forced to include a dark sector for the matter and energy content of the universe to explain the cosmological data. Whereas from a conceptual point of view, the theory is not perturbatively renormalizable, and it predicts the formation of spacetime singularities. This thesis studies possible modifications of general relativity both considering specific theories of modified gravity and implementing a model-independent approach. In the first part of the thesis, we study a specific class of modified theory of gravity which has the peculiarity of propagating the same number of degrees of freedom of general relativity. The existence of these theories apparently challenges the distinctive role of general relativity as the unique non-linear theory of massless spin-2 particles. However, we provide strong evidence that these theories are actually equivalent to general relativity in vacuum. In the second part of the thesis, we focus on the problem of black hole singularities which are unavoidably present in general relativity. However, it is reasonable to assume that there will be a mechanism preventing their formation in a full theory of quantum gravity. Without specifying any theory of quantum gravity or the nature of such mechanism, simply assuming a minimal set of kinematical constraints, we classify and study the properties of non-singular spacetime with a trapping horizon. Contrary to what one might expect, the set of regular geometries that arises is remarkably limited. Furthermore, we show that it is very difficult to construct a self-consistent geometry without any long range effect. This gives us further motivation to study the phenomenology of non-singular black holes. To this end, we provide a set of parameters describing the deviations from classical black holes, and we review the possible observational channels that can measure or constrain them.