Ghost and singularity free theories of gravity

Albert Einstein's General Relativity (GR) from 1916 has become the widely accepted theory of gravity and been tested observationally to a very high precision at different scales of energy and distance. At the same time, there still remain important questions to resolve. At the classical level cosmological and black hole singularities are examples of problems which let us notice that GR is incomplete at short distances (high energy). Furthermore, at the quantum level GR is not ultraviolet (UV) complete, namely it is not perturbatively renormalizable. Most of the work try to solve these problems modifying GR by considering finite higher order derivative terms. Fourth Derivative Gravity, for example, turns out to be renormalizable, but at the same time it introduces ghost. To avoid both UV divergence and presence of ghost one could consider sets of infinite higher derivative terms that can be expressed in the form of entire functions satisfying the special property do not introduce new poles other than GR graviton one. By making a special choice for these entire functions, one could show that such a theory describes a gravity that, at least in the linear regime, can avoid both the presence of ghost and classical singularities (both black hole and cosmological singularities). In this master's thesis we review some of these aspects regarding gravitational interaction, focusing more on the classical level. Most of the calculations are done in detail and an extended treatment of the formalism of the spin projector operators is presented.