[Paper Review] Einstein-Maxwell-Massive Scalar Field System in 3+1 formulation on Bianchi Spacetimes type I-VIII

Norbert Noutchegueme, Alexis Nangue|arXiv (Cornell University)|May 2, 2012

Cosmology and Gravitation Theories11 references3 citations

TL;DR

This paper establishes global existence and geodesic completeness for the coupled Einstein-Maxwell-massive scalar field system in 3+1 formulation on Bianchi spacetimes I–VIII, under the condition that the cosmological constant Λ > −α², where α depends on the scalar field potential. It demonstrates accelerated expansion via exponential growth of gravitational potentials and confirms the model satisfies weak, dominant, and, under additional constraints, strong energy conditions.

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.

Motivation & Objective

• To establish global in time existence of solutions for the coupled Einstein-Maxwell-massive scalar field system in Bianchi I–VIII spacetimes.
• To analyze the asymptotic behavior of the system, particularly the growth of gravitational potentials.
• To investigate geodesic completeness and energy conditions (weak, dominant, strong) under constraints on the cosmological constant and scalar field potential.
• To extend previous results by considering Λ > −α², including cases where Λ ≤ 0, not just Λ > 0.
• To provide a 3+1 formulation of the system on spatially homogeneous spacetimes, enabling time-evolution analysis of geometric and matter fields.

Proposed method

• Adopts the 3+1 formulation of the Einstein equations, evolving the first and second fundamental forms on spacelike hypersurfaces of constant time.
• Uses a 3-dimensional simply connected Lie group G to define the spatial geometry, with metric g_ij(t) depending only on time t.
• Imposes the condition that the initial mean curvature is strictly negative to ensure long-term evolution.
• Derives evolution equations for the extrinsic curvature k_ij and spatial Ricci curvature R_ij using the Gauss-Codazzi equations.
• Applies the Einstein-Maxwell-massive scalar field equations with a cosmological constant Λ > −α², where α depends on the scalar field potential.
• Employs constraint equations and conservation laws to verify consistency and derive evolution equations for matter and electromagnetic fields.

Experimental results

Research questions

• RQ1 Under what conditions on the cosmological constant does the Einstein-Maxwell-massive scalar field system admit global solutions in Bianchi I–VIII spacetimes?
• RQ2 How does the asymptotic behavior of the system reflect accelerated expansion, particularly through the growth of gravitational potentials?
• RQ3 What are the implications for geodesic completeness and the validity of energy conditions (weak, dominant, strong) in the presence of a massive scalar field and negative Λ?
• RQ4 Can the results be extended to cases where Λ ≤ 0, particularly when Λ > −α², and how does this relate to observed accelerated cosmic expansion?
• RQ5 How do the 3+1 evolution equations for the metric, extrinsic curvature, and matter fields interact under the constraints of spatial homogeneity?

Key findings

• Global existence of solutions is proven for the coupled Einstein-Maxwell-massive scalar field system in Bianchi I–VIII spacetimes when the cosmological constant satisfies Λ > −α², where α > 0 depends only on the scalar field potential.
• The asymptotic behavior reveals exponential growth of gravitational potentials, indicating accelerated expansion of the universe.
• The spacetime is geodesically complete, meaning no singularities form in finite time under the given conditions.
• The model satisfies the weak and dominant energy conditions for all Λ > −α².
• The strong energy condition holds if Λ > β², where β > 0 depends only on the initial mean curvature of the spacetime.
• The 3+1 formulation successfully reduces the system to a set of time-evolution equations for the metric, extrinsic curvature, and matter fields, consistent with the constraints and conservation laws.