Exploring the Einstein field equations in different space-time structures
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Date
2025
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UMT Lahore
Abstract
This thesis explores the behavior of Einstein’s field equations in different space-time structures, with a particular focus on the stability of the Einstein static universe in modified theories of gravity. Einstein’s field equations form the core of general relativity by describing how matter and energy determine the curvature of space-time, essentially linking the distribution of mass-energy to the geometry of the universe. In this work, the equations are extended through modifications involving curvature and matter couplings, and then applied within anisotropic cosmological models, specifically the Bianchi type IX framework. Small homogeneous and anisotropic perturbations are introduced in the scale factor and matter fields to examine how the universe reacts to deviations from a static state. Static and perturbed versions of the field equations are derived and analyzed using a fluid equation of state, which simplifies the relation between pressure and energy density. The study evaluates specific functional forms of the modified gravity models and investigates whether stable static solutions exist under these perturbations. Through analytical and graphical methods, it is shown that unlike in standard general relativity, stable Einstein static universes can emerge in both curvature-based and matter-coupled theories. This highlights the deeper flexibility of extended gravity models and shows how the structure of space-time, governed by the field equations, can support a wider range of cosmological behavior than previously thought.