Dr. Lalaonirina Rakotomanana

Abstract:

There are numerous aspects in the mathematical modeling of vacuum spacetime in Cosmology. Gravitation and electromagnetism are the two actions-at-distance phenomenological fields occurring in a vacuum (without mediating matter with infinite radius).  Nowadays, since the works of Permutter et al. [1] and Riess et al. [2], one of the biggest challenges in cosmology is to understand the physics behind the acceleration of the universe expansion, assumed to be due to an unknown dark energy and also the Universe missing mass assumed to be a dark matter required for maintaining the whole Universe. The Cosmological Constant was classically introduced ad hoc to explain the dark energy. 

The main motivation of the present paper is to develop a mathematical model of Generalized Continuum for analyzing the link between spacetime continuum, gravitation and electromagnetism with the only necessary three phenomenological fields on vacuum spacetime by avoiding micro-particles physics and cosmological fluids, despite their preeminent role in cosmology. The main application is to attempt to explain the concept of dark energy and dark matter. The work rather focuses only on phenomenological fields occurring in a vacuum Universe as a continuum. 

The present paper is based some fundamental assumptions to define the geometrical background of a Generalized Continuum model and the physical events occurring within it [3] : (1) the spacetime has a structure of differentiable four-dimensional manifold endowed with a metric, and independent connection with torsion; (2) only gravitation and electromagnetism are considered as physical fields, since they are the only actions-at-distance among the four universal fundamental forces [4]. The action is composed of the Einstein-Hilbert-Palatini (for gravitation) and Yang-Mills (for electromagnetism) Lagrangians.

The general methodology consists of exploiting the geometric structure of spacetime continuum by reminding Riemann and developing Riemann-Cartan manifolds. Accounting for the torsion field in addition to macroscopic deformation (metric and strain) was inspired from the work of Rainich (1925) [5] and Misner & Wheeler (1957) [6] by adding the property of multiply-connectedness to usual Riemann manifold in the framework of continuum mechanics. These two works were themselves inspired by the works of V. Volterra on dislocations and disclinations (1901) [7]. 

The idea is to reduce the phenomena of gravitation and electromagnetism to the geometric variables  as curvature and torsion fields on the continuum. Torsion will be a matter of concern all along this work, and implicitly we show that spacetime is more and more assimilated to an infinitely small sets of microcosms, as due to brusque cooling of the Universe at the beginning. Mathematical models extending the usual framework for field equation in classical continuum mechanics are developed within the Einstein-Cartan geometric background [8]. 

For the application in Cosmology, the introduction of an ad hoc hypothetical Cosmological Constant is no more necessary as shown by our results. Models nevertheless show the presence of non homogeneous and anisotropic fields definitely replacing an hypothetical Cosmological Constant. 

In sum, only electromagnetic and gravitational fields coupled with Generalized Continuum model might be sufficient to describe dark energy and by the way dark matter by means of the torsion field of the vacuum spacetime [8].