Einstein-Gilbert-Straus solution of Einstein field equations: Timelike geodesic congruence with conventional and quantized fundamental metric tensor

In a cosmic background characterized by inhomogeneity, anisotropy, and spherical symmetry, a nonsingular solution to the Einstein field equations is established through the application of both conventional and quantized fundamental metric tensor. The Swiss cheese model describes the complex configuration of the Einstein–Gilbert–Straus (EGS) metric, particularly emphasizing the presently undefined correlation between the radial distance r and cosmic time t . This complexity necessitates two key approaches: first, to model the cosmic substance inside the inhomogeneously interspersed spherical holes, and second, to analytically derive and numerically estimate the timelike geodesic congruence via the chain rule. In this analysis, the derivative d r / d t is calculated at the proper horizon, incorporating the proper-time derivative as proposed by Misner and Sharp. The Raychaudhuri equation serves as a crucial lemma in the context of the Hawking–Penrose singularity theorems. It provides significant insights into the dynamics of various kinematic quantities, including the expansion (volume scalar), shearing (anisotropy), rotation (vorticity), and the Ricci identity (local gravitational field). The Raychaudhuri equation describes how these quantities correlate and contribute to the characteristics and onset of space and initial singularities. To conclude, we find that a) a singularity-free solution is achieved in both space and time using either the conventional or quantized fundamental metric tensor, and b) the quantized fundamental metric tensor, which provides a significant extension of general relativity into relativistic and quantum regimes, naturally leads to the quantization of the Raychaudhuri equation, thereby greatly enhancing the strength of such a solution. The EGS metric exhibits significant differences from both the Schwarzschild and FLRW metrics, suggesting that the conclusion regarding the nonsingularity of the EGS is likely independent of the specific model employed. Additionally, the Ricci and Kretschmann scalars seem to provide evidence that the proposed geometric quantization reveals nonsingularity which is fundamentally real, intrinsic, and essential rather than an artifact arising from a specific choice of coordinates.