STABILITY ANALYSIS: A CRUCIAL STUDY IN GENERAL RELATIVITY

Abstract

The objective of this thesis is to discuss stability of the collapsing self- gravitating objects as well as thin-shell wormholes. For this purpose, we consider two type of fluids, anisotropic fluid with expansion-free condition and collapsing dissipative fluids. The dynamical instability of expansion- free fluid is explored with and without charge in spherical and cylindrical backgrounds. We formulate the Einstein field equations, junction conditions and expansion-free dynamical equations by using the perturbation scheme in Newtonian and post-Newtonian regimes. We conclude that instability range of expansion-free fluid can be discussed in terms of energy density inhomogeneity, anisotropic fluid as well as electromagnetic field. We also investigate the dynamical instability of dissipative fluid in terms of adiabatic index in the framework of non-dissipative and dissipative per- turbations. The effects of electromagnetic field as well as physical variables in the onset of instability range are explored. For this purpose, we con- struct the collapse equation and explore the instability range in Newtonian and post-Newtonian regimes. We note that physical variables of the fluid can drastically change the instability range of the system which makes the system stable or more unstable. In particular, we find the critical value for which the collapsing cylinder will be stable or unstable. Finally, we construct thin-shell wormholes using the cut and paste pro- cedure and investigate their stability under the standard potential method. We formulate the general formalism to construct and investigate the sta- bility of spherical thin-shell wormholes supported by modified Chaplygin ixx gas. This is applied to construct general class of wormholes like Reissner- Nordstr ̈ o m, Schwarzschild with cosmological constant thin-shell wormholes and their stability under the radial perturbations are investigated. We find some extra stable as well unstable static wormhole solutions. Also, we construct charged black string thin-shell wormholes supported by modified Chaplygin gas and explore their stability. Interestingly, we find both stable and unstable solutions for cylindrical wormholes.