A unified-description of curvature, torsion, and non-metricity of the metric-affine geometry with the Möbius representation
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913531182448640 |
|---|---|
| author | Tomonari, Kyosuke |
| author_facet | Tomonari, Kyosuke |
| contents | We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan's structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü-Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11558 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A unified-description of curvature, torsion, and non-metricity of the metric-affine geometry with the Möbius representation Tomonari, Kyosuke General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan's structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü-Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed. |
| title | A unified-description of curvature, torsion, and non-metricity of the metric-affine geometry with the Möbius representation |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2312.11558 |