Ricci Solitons and Their Associated Vector Fields in Locally Rotationally Symmetric Bianchi Type I Spacetime

Abstract

This article presents a complete classification of Ricci solitons and their associated vector fields in the context of locally rotationally symmetric (LRS) Bianchi type I spacetime, a crucial model in cosmological studies. To systematically address the complexities inherent in the Ricci soliton equations, we adopt the Rif tree technique. The equations defining the Ricci
soliton and its vector field are transformed into a reduced involutive form using a computational algorithm, which assists in dividing the integration process into a collection of cases organized in a tree-like structure. Each of these cases is governed by specific constraints on the metric functions, which facilitates the solution process. Definite expressions for the metric functions and the corresponding vector field of the Ricci soliton are obtained by efficiently solving the system of equations characterizing the soliton vector field through the application of these constraints. This powerful approach enables us to derive novel and exact solutions that previous methods have overlooked. Our results demonstrate that this spacetime admits Ricci solitons of shrinking, steady, and expanding natures, characterized by vector fields with up to 11 free parameters. Crucially, we conduct a thorough physical analysis of the resulting models, determining their matter content through the equation of state and testing their physical viability via the standard energy conditions. We find specific families of solutions that correspond to physically significant scenarios, such as a spacetime filled with vacuum energy (a cosmological constant). This work not only provides a comprehensive mathematical classification but also establishes a direct link between these geometric structures and potentially realistic cosmological models.