الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
stract In the realm of general relativity, this thesis delves into the pursuit of nonsingular solutions within a homogeneous, isotropic, and spherically symmetric spacetime, employing quantum-improved metric tensors. The principal focus of this study is centered on the comprehensive analysis of timelike geodesic congruences within three distinct spacetime backgrounds: Schwarzschild, de Sitter–Schwarzschild, and Friedmann–Lemaitre–Robertson–Walker (FLRW) solutions. The investigation scrutinizes the intricate behavior of these geodesic congruences, treating them as flow lines generated by velocity fields, and rigorously characterizing alterations in the shape of the associated volumes. An incisive comparative analysis among these solutions yields profound insights into the dynamics of the generalized quantum-improved metric tensor. The findings underscore that the evolution of expansion, as dictated by the focusing theorem, is poised to manifest either convergence or divergence along the trajectory congruence, thereby suggesting the potential for singular or non-singular expansion. Analytically challenging, this phenomenon is robustly substantiated by meticulous numerical analysis, which unequivocally supports the assertion that the imposition of space and initial singularities can be averted through the proposed quantization. Furthermore, this research diligently establishes upper bounds for the averaged values of the proposed quantization, thus ensuring the unswerving preservation of the classical theory of relativity, particularly at larger, nonrelativistic scales. The quantum-improved metric tensor extends the scope of applicability of relativistic predictions to the quantum realm. Consequently, the eradication of space and initial singularities is attributed to a constellation of factors, including the introduction of a minimal measurable length, xi the incorporation of noncommutative algebra, the imposition of additional curvatures, and the embrace of quantum geometry. The thesis posits that this transformative shift signifies a quantum-induced reexamination of the fundamental metric. In a contrasting scenario, set against an inhomogeneous, anisotropic, and spherically symmetric cosmic backdrop, the study explores the prospect of non-singular solutions within Einstein’s field equations, leveraging both conventional and generalized metric tensors. Employing the Swiss-cheese model, this research addresses the intricate structure of the Einstein–Gilbert–Straus (EGS) metric, specifically the hitherto undefined interdependence of radial distance r and cosmic time t. It is this intricate interplay that serves as the backDROP for i) modeling the cosmic medium within the heterogeneously interspersed spherical voids and ii) deriving, both analytically and numerically, the timelike geodesic congruence. Utilizing the chain rule, wherein dr dt is determined at the proper horizon, and adopting the proper-time derivative as proposed by Misner and Sharp, the thesis arrives at a comprehensive understanding of the dynamics at play. The application of the Landau–Raychaudhuri equation, the fundamental cornerstone of Hawking–Penrose singularity theorems, provides crucial insights into the evolution of expansion (volume scalar), shearing (anisotropy), rotation (vorticity), and the Ricci identity (local gravitational field). It is through a meticulous analysis of these kinematic quantities that the thesis delineates the individual contributions that collectively pave the way for a resolution of the space and initial singularities. • The thesis emphasizes three fundamental conclusions: 1. The attainment of a singularity-free solution is achievable, whether using conventional or generalized fundamental tensors. 2. The generalized fundamental tensor has far-reaching implications, expanding the domain of general relativity into both relativistic and quantum physics. 3. The Ricci and Kretschmann scalars, central to this study, strongly confirm the achieved nonsingularity. This confirmation is both real and inherently essential, affirming the authenticity of these results xii as intrinsic properties, not artifacts of specific coordinate systems.