Global existence of solutions to the Einstein–Maxwell–Massive scalar field system in 3 + 1 formulation on Bianchi spacetimes

Abstract

Global existence to the coupled Einstein-Maxwell-Massive Scalar Field system which rules the dynamics of a kind of charged pure matter in the presence of a massive scalar field is proved, in Bianchi I-VIII spacetimes; asymptotic behaviour, geodesic completeness, energy conditions are investigated in the case of a cosmological constant bounded from below by a strictly negative constant depending only on the massive scalar field.

Einstein-Scalar Field System with a cosmological constant on the type I Bianchi space-time

“…This is the focus of this paper. Several studies have already been carried out on the notion of scalar field like the works of [6], [7] [2] [4] and [5].…”

“…• The ordinary matter is modeling by (5), which represents the relativistic perfect fluid of pure radiation type, in which ρ ≥ 0 is an unknown function of single variable t, representing the matter density. For simplicity, we consider a co-moving fluid, which means that u i = u i = 0, where u = (u α ) is a future time-like unit vector (i.e g αβ u α u β = −1, u 0 > 0).…”

Global Properties of the Solution of the Einstein-Maxwell-Boltzmann-Scalar Field System with Pseudo-Tensor of Pressure on a Bianchi Type I Space-Time

“…As specified in [9] the components T 1 αβ , τ αβ , T 2 αβ , and H αβ defined by (6), (6'), (7) and (8) can clearly be written as follow:…”

“…(5) is the Boltzmann equation, where L X is the Lie derivative of f with respect to the vectors field X( F ) = p α , P α (F ) and Q(f, f ) the collision operator (For more details on Q(f, f ), see ( [9]). The massive particles have a rest mass m > 0, normalized to the unity.…”

“…• Using the change of variables as in [9], The Einstein-Maxwell-Boltzmann-Scalar field system is equivalent to the following system (20)-(28), which is a first order differential system:…”