Aspects of Gravitational Collapse and the formation of Spacetime Singularities
Possibilities emerging out of the dynamical evolutions of collapsing systems are addressed in this thesis through analytical investigations of the highly non linear Einstein Field Equations. Studies of exact solutions and their properties, play a non trivial role in general relativity, even in the current context. Finding non trivial solutions to the Einstein field equations requires some reduction of the problem, which usually is done by exploiting symmetries or other properties. Exact solutions of the Einstein's field equations describing an unhindered gravitational collapse are studied which generally predict an ultimate singular end state. In the vicinity of such a spacetime singularity, the energy densities, spacetime curvatures, and all other physical quantities blow up. Despite exhaustive attempts over decades, the famous conjecture that the formation of a singularity during stellar collapse necessarily accompanies the formation of an event horizon, thereby covering the central singularity, still remains without a proof. Moreover, there are examples of stellar collapse models with reasonable matter contribution in which an event horizon does not form at all, giving rise to a naked singularity from which both matter and radiation can fall in and come out. These examples suggest that the so called "cosmic censorship" conjecture may not be a general rule. Therefore one must embark upon analysis of realistic theoretical models of gravitational collapse and gradually generalizing previous efforts. Viable f (R) models are quite successful in providing a geometrical origin of the dark energy sector of the universe. However, they possess considerable problems in some other significant sectors, such as, difficulty to find exact solutions of the field equations which are fourth order differential equations in the metric components. Moreover, a recent proposition that homogeneous collapsing stellar models (e.g. Oppenheimer Snyder Datt model of a collapsing homogeneous dust ball with an exterior Schwarzschild spacetime) of General Relativity can not be viable models in f (R) theories, heavily constrict the set of useful astrophysical solutions. In this thesis, we address some collapsing models in f (R) gravity such that at the comoving boundary of the collapsing star, the interior spacetime matches smoothly with an exterior spacetime. The presence and importance of spatial inhomogeneity is duely noted and discussed. The ultimate spacetime singularity remains hidden or exposed to an exterior observer depending on initial conditions from which the collapse evolves. The study of collapsing solutions of the Einstein equations with a scalar field as the matter contribution owes special importance, because one would like to know if cosmic censorship is necessarily preserved or violated in gravitational collapse for fundamental matter fields, which are derived from a suitable Lagrangian. In this thesis we have studied some models of gravitational collapse under spherical symmetry, with a self interacting scalar field minimally coupled to gravity along with a fluid description. The field equations are solved under certain significant symmetry assumption at the outset (for instance, conformal flatness, self similarity) without assuming any particular equation of state for the matter contribution. The relevance of such investigations stems from the present importance of a scalar field as the dark energy vis a vis the fluid, whose distribution still remains unknown apart from the general belief that the dark energy does not cluster at any scale below the Hubble scale. The study of collapse of scalar fields, particularly in the presence of a fluid may in some way enlighten us regarding the possible clustering of dark energy. The collapsing models are studied in this thesis for certain popular and physically significant forms of the self interaction potential, for example, a power law or an exponential dependence over the scalar field. The end state of the collapse is investigated by analyzing the apparent horizon curve and existence of radial null geodesics emanating from the spacetime singularity.