GLOBAL SOLUTIONS TO THE EINSTEIN - SCALAR FIELD EQUATION IN A MAGNETIZED BIANCHI MODEL

We prove the global existence of solutions to the coupled Einstein-Scalar field Equation, with the cosmological constant ^ in a Magnetized Bianchi type I space-time. We discuss this global existence using the sign of the derivatives of the initial data of potentials of gravitation a; b and the cosmological constant. To obtain the global existence, we make assumption that the unknown massive scalar field Φ is positive. This is possible because, physically; Φ∼ G -1; G standing for the variable gravitational “constant". We also consider in this work, the case in which the electromagnetic field F derives from a potential vector A = (Aλ) ; imposing to simplify, on A the Lorentz gauge ∇ αAα = 0:Thereafter, we transform the resulting Magnetized Einstein-Scalar field system, which is a second order differential system, into a first order differential system and we apply the standard theory . We then solve the problem of constraints and investigate a time global solution (a; b; Φ; F;A) of the resulting system.