Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates
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><...
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MDPI AG
2021-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/3/366 |
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| author | Alexander Kritov |
| author_facet | Alexander Kritov |
| author_sort | Alexander Kritov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T00:34:03Z |
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| spelling | doaj.art-e738a5851451450babe24e9f13a91c112023-12-11T18:18:42ZengMDPI AGSymmetry2073-89942021-02-0113336610.3390/sym13030366Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford CoordinatesAlexander Kritov0Faculty of Physics Moscow, Lomonosov Moscow State University, 119991 Moscow, RussiaThis 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.https://www.mdpi.com/2073-8994/13/3/366cosmological constant problemClifford algebras<i>Cl(3,0)</i><i>Cl(0,3)</i>two-dimensional spacetimetime-volume coordinates |
| spellingShingle | Alexander Kritov Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates Symmetry cosmological constant problem Clifford algebras <i>Cl(3,0)</i> <i>Cl(0,3)</i> two-dimensional spacetime time-volume coordinates |
| title | Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates |
| title_full | Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates |
| title_fullStr | Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates |
| title_full_unstemmed | Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates |
| title_short | Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates |
| title_sort | gravitation with cosmological term expansion of the universe as uniform acceleration in clifford coordinates |
| topic | cosmological constant problem Clifford algebras <i>Cl(3,0)</i> <i>Cl(0,3)</i> two-dimensional spacetime time-volume coordinates |
| url | https://www.mdpi.com/2073-8994/13/3/366 |
| work_keys_str_mv | AT alexanderkritov gravitationwithcosmologicaltermexpansionoftheuniverseasuniformaccelerationincliffordcoordinates |