Speaker
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Introduction. The discovery of the accelerating expansion of the Universe by the two teams of Riess et al. and Perlmutter et al. independently in 1998 marked the beginning of a paradigm shift that sparked a significant new field of research in Cosmology. Dark Energy, in addition to the Dark Matter suggested by Zwicky et al. (1933) and later supported by Rubin (1970), was then postulated as responsible for this acceleration. The theoretical model underlying Dark Energy, is essentially based on the cosmological equations originally derived by Friedman (1922), which were deduced from Einstein field equations from Relative Gravitation (1915). Since then, Cosmological Constant was introduced to account for this Dark Energy. After more than two decades, the most recent and more and more accurate experimental observations, such as those from the DESI (Dark Energy Spectroscopic Instrument) and DES (Dark Energy Survey), have shown that the influence of this hypothetical Dark Energy is beginning to weaken and additionally it is not uniformly distributed throughout the Universe. Therefore it can no longer be considered a unique constant. Different approaches are possible to understand these new data, either phenomenological or by using quantum physics passing through Quantum Vacuum, and account for this temporal and spatial variation.
The objective of this work is to consider a modified mathematical model of Relative Gravitation in order to go back to the fundamental phenomenological fields equations of spacetime.
Methods. To do this, we derived a mathematical model of spacetime whose underlying geometry is based on Riemann-Cartan manifold: it takes into account not only its curvature but also its torsion It is important to note that torsion is not added ad hoc but constitutes an unknown field of the Universe. Foundations of this vacuum spacetime derive their properties from generalized continuum with singularity distributions, developed by the author over thirty years [1], [2], which has also been called Planck-Kleinert crystals (1999). The physical content of the modeled vacuum spacetime is defined by an action including the Einstein-Palatini Lagrangian (extension of the classical Einstein-Hilbert) and the symmetrized Yang-Mills Lagrangian (also extension of classical Yang-Mills), thus consequently taking into account Relative Gravitation and Electromagnetism. It is stressed that adding Electromagnetism was necessarily to obtain the model overall consistency, owing that both gravity and electromagnetic forces are the two possible actions-at-distance in classical physics.
Results. A complete system of four equations was derived including the two pairs of Maxwell covariant equations in a curved and twisted vacuum where unknowns are the electromagnetic potential and its dual, the metric and the torsion. It is shown that the extended Einstein-Cartan-Maxwell (third) equation includes new terms that play the role of an isotropic and an anisotropic energies driving the Universe expansion, they essentially depend on torsion. And importantly, we deduced a new fourth additional equation that explicitly links torsion to Chern-Simmons currents. In other words, this highlights the role of Optical Helicity and Spin Orbital Momentum of the electromagnetic wave pervading the Universe, on the torsion field [3]. This is not an assumption but a result.
Preliminary analysis of the spacetime equations has shown that localized dark energy (instead of a global Dark Energy), depending on torsion field, then on the linking, twisting, writhing of electromagnetic tubes in the vacuum occurs in the model. Therefore Dark Energy is not only variable in time but also is non homogeneous within space. It has two major contributions: an isotropic part that appears consistent with recent experimental observations on accelerated expansion, and an anisotropic part that surely requires further investigation.
By the way, this work allows us to derive the auto-parallel curves, based solely of vacuum geometry, generalizing the Jacobi equation of geodesic deviation in Relative Gravitation, to include the influence of both the twisting and the curving of the vacuum spacetime [3].
General Comment. This work, although limited to a mathematical model, does not introduce further additional physical constants other than classical ones. It might nevertheless contribute, as modified gravitation modeling, to investigate some aspects of Dark Energy, and hopefully Dark Matter in Cosmology.
References. [1] Rakotomanana L. A Geometric Approach to Thermomechanics of Dissipative Continua. In Progress in Mathemaical Physics Series, Birkhaüser, Boston, 2003.
[2] Rakotomanana L. Covariance and Gauge Invariance in Continuum Physics : Application to Mechanics, Gravitation, and Electromagnetism. In Progress in Mathematical Physics Series, Birkhaüser, Cham, 2018.
[3] Rakotomanana L. Continuum Physics with Application to Cosmology. World Scientific, To appear, 2026.