Homothetic Vectors of Bianchi Type I Spacetimes in Lyra Geometry and General Relativity

In this paper Bianchi type I spacetimes are completely classified by their homothetic vectors in the context of Lyra geometry. The non-linear coupled Lyra homothetic equations are obtained and solved completely for different cases. In some cases, Bianchi type I spacetimes admit proper Lyra homothetic vectors (LHVs) for special choices of the metric functions, while there exist other cases where the spacetime under consideration admits only Lyra Killing vectors (LKVs). In all the possible cases where Bianchi type I spacetimes admit proper LHVs or LKVs, we obtained homothetic and Killing vectors for Bianchi type I spacetimes in general relativity by taking the displacement vector of Lyra geometry as zero. Matter collineation symmetry is explained by taking the matter field as a perfect fluid. The obtained proper LHVs and LKVs are used in matter collineation equations and a barotropic equation of state having $\rho (t)=\gamma p(t)$, $0\le \gamma \le 1$ form is always obtained when the displacement vector is considered as a function of $t$ or treated as a constant.