Application of the Raychaudhuri Equation in Gravitational Systems

Gupta Choudhury, Shibendu (2022) Application of the Raychaudhuri Equation in Gravitational Systems. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Shibendu Gupta Choudhury (14IP003)) 14IP003.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

The works reported in this thesis primarily address the application of the Raychaudhuri equation in two intriguing problems in gravitational physics. These problems still lack universally accepted explanations. The first problem is related to the existence of spacetime singularities. We aim to find possible escape routes from these problematic singularities at the classical level. The second problem is associated with the late time accelerated expansion of the Universe. In this context, our goal is to find a possible explanation of this phenomenon without assuming the presence of any exotic contribution to the stress-energy tensor. To investigate the problem of singularities, we begin with the study of gravitational collapse within the scope of General Relativity. We consider two different gravitational collapse scenarios in our study. At first, we examine a self-similar gravitational collapse of a matter distribution containing a fluid and a scalar field. We work under the assumption of spherical symmetry and conformal flatness. We obtain the general focusing condition for this system. This condition is quite helpful in getting insights into the corresponding evolution. We show that the general condition leads to crucial constraints on the metric and matter variables for a few special cases. These constraints distinguish between circumstances where a singularity is inevitable and where it can be avoided. The results obtained from the focusing condition are verified against exact solutions whenever available. Using both approaches, i.e., employing the focusing condition and the exact solutions, we find that non-singular evolution is possible for the system under consideration. We also explore the connection between the Raychaudhuri equation and the critical phenomena in gravitational collapse. The second system addresses the role of magnetic fields in a gravitational collapse. There is an inherent tendency of magnetic fields to act against gravity due to the repulsion between magnetic field lines. Inhomogeneous models with no restriction on the magnetic field strength to begin with have not been studied significantly in the literature. We investigate such general models of gravitational collapse in our work. Our system consists of a charged fluid distribution collapsing in the presence of a magnetic field under cylindrical symmetry. For such systems, we find the necessary constraints that the magnetic field strength needs to obey to avert collapse. Next, we explore if it is possible to avoid singularities in a class of modified theories of gravity, namely the scalar-tensor theories of gravity. We study focusing of timelike geodesic congruences, particularly the fate of the timelike convergence condition within this framework. In General Relativity, the strong energy condition implies the timelike convergence condition. The latter is a necessary assumption in the proof of the singularity theorem. But for a scalar-tensor theory, such convergence conditions may not necessarily follow from the energy conditions as the field equations in this theory contain additional terms. We choose the Brans-Dicke theory and Bekenstein’s conformally coupled scalar-tensor theory as specific examples within the broad class of non-minimally coupled scalar-tensor theories. Additionally, we consider static, spherically symmetric and spatially homogeneous and isotropic backgrounds for our study. We observe that violation of the convergence condition is possible in both theories. This gives rise to the possibility of avoiding singularities. Finally, we examine the phenomenon of late time accelerated expansion of the Universe. We will look into this within the purview of the f (R)-gravity. In f(R)-gravity, repulsive effects can emerge from the geometry itself, and therefore we do not need to introduce any exotic matter to explain this phenomenon. We find a new strategy for reconstructing f(R)-gravity models for an accelerated universe employing the Raychaudhuri equation. We investigate two different models - one of them represents an ever-accelerating universe. In the other one, the evolution mimics the Lambda Cold Dark Matter (ΛCDM) expansion history. We start by characterizing these evolution histories in terms of the corresponding kinematical quantities, namely the deceleration parameter and the jerk parameter, respectively. For the first example, we find that a combination of power-law terms gives the expression for f(R). In the second example, the form of f(R) involves hypergeometric functions. Viability analysis of these models reveals that the corresponding f(R)-gravity models for both examples are unsuitable options.

Actions (login required)

View Item