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

• Main Author: Alexander Kritov
• Format: Article
• Language: English
• Published: MDPI AG 2021-02-01
• Series: Symmetry
• Subjects: cosmological constant problem; Clifford algebras; Cl(3,0); Cl(0,3); two-dimensional spacetime; time-volume coordinates
• Online Access: https://www.mdpi.com/2073-8994/13/3/366

This paper presents a novel approach to the cosmological constant problem by the use of the Clifford algebras of space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>l</mi><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow></msub></mrow></semantics></math></inline-formula> and anti-space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>l</mi><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></msub></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><msub><mi>H</mi><mn>0</mn></msub><mi>c</mi></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>l</mi><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow></msub></mrow></semantics></math></inline-formula> corresponds to the anti-de Sitter (AdS) flat (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) case, the negative cosmological term and an oscillating model of the universe. The approach with anti-Euclidean Clifford algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>l</mi><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></msub></mrow></semantics></math></inline-formula> leads to the de Sitter model with the positive cosmological term and the exact form of the scale factor used in modern cosmology.