Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry

Abstract

We find a new class of invariant inhomogeneous Bianchi type-I cosmological models in electromagnetic field with variable magnetic permeability. For this, Lie group analysis method is used to identify the generators that leave the given system of nonlinear partial differential equations (NLPDEs) (Einstein field equations) invariant. With the help of canonical variables associated with these generators, the assigned system of PDEs is reduced to ordinary differential equations (ODEs) whose simple solutions provide nontrivial solutions of the original system. A new class of exact (invariant-similarity) solutions have been obtained by considering the potentials of metric and displacement field as functions of coordinates $x$ and $t$. We have assumed that $F_{12}$ is only nonvanishing component of electromagnetic field tensor $F_{ij}$. The Maxwell equations show that $F_{12}$ is the function of $x$ alone whereas the magnetic permeability $\overline{\mu }$ is the function of $x$ and $t$ both. The physical behavior of the obtained model is discussed.