Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates

Alexander Kritov

doi:10.3390/sym13030366

Alexander Kritov

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https://doi.org/10.3390/sym13030366

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Journal: Symmetry	Publication Date: Feb 24, 2021
License type: CC BY 4.0

Affiliation: Lomonosov Moscow State University

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This paper presents a novel approach to the cosmological constant problem by the use of the Clifford algebras of space Cl3,0 and anti-space Cl0,3 with a particular focus on the paravector representation, emphasizing the fact that both algebras have a center represented just by two coordinates. Since the paravector representation allows assigning the scalar element of grade 0 to the time coordinate, we consider the relativity in such two-dimensional spacetime for a uniformly accelerated frame with the constant acceleration 3H0c. Using the Rindler coordinate transformations in two-dimensional spacetime and then applying it to Minkowski coordinates, we obtain the FLRW metric, which in the case of the Clifford algebra of space Cl3,0 corresponds to the anti-de Sitter (AdS) flat (k=0) case, the negative cosmological term and an oscillating model of the universe. The approach with anti-Euclidean Clifford algebra Cl0,3 leads to the de Sitter model with the positive cosmological term and the exact form of the scale factor used in modern cosmology.

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The recently proposed model of the ”unified two-dimensional spacetime”[18] shows substantial conformity to Clifford algebras Cl3,0 and Cl0,3
The model considers the classical gravity as the uniform acceleration α = 3H0c in such coordinates, and the addition of the cosmological term appears as the relativistic effect attributed to the Rindler coordinate transformations
This paper demonstrates the fundamental significance of the approach based on the symmetry of underlying Clifford algebras

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The recently proposed model of the ”unified two-dimensional spacetime”[18] shows substantial conformity to Clifford algebras Cl3,0 and Cl0,3. Because the basis of Cl3,0 precisely corresponds to two copies of 4-dimensional vector spaces, all eight orthogonal coordinates can be split into two sets of Minkowski and anti-Minkowski spacetimes, spanned by two 4-vectors given by upper and lower raw as follows x0 x1 x2 x3 (2) The algebra Cl3,0 treated in such a way obtains the assigned scalar value to the time coordinate of Minkowski spacetime, where the volume element represents its ”anti-Minkowskian” counterpart2.

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