Connecting 2-Forms, Conformal Transformations, Curvature Invariants and Topological Classes in Einstein Spacetimes
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909139743014912 |
|---|---|
| author | Hughes, Jack C. M. Kusmartsev, Fedor V. |
| author_facet | Hughes, Jack C. M. Kusmartsev, Fedor V. |
| contents | The unique Nature of the Lorentz group in four dimensions is the root cause of the many remarkable properties of the Einstein spacetimes, in particular their operational structure on the 2-forms. We show how this operational structure can be used for two ends. First, it allows for a simple generalization of the Birkhoff theorem to Schwarzschild (A)de-Sitter spacetime. Second, it provides the means to construct an Abelian endomorphism group on the space of 2-forms. It is observed that taking the trace over this group element-wise induces a further Abelian group which may be identified with a tensor representation of conformal transformations, giving Einstein spacetimes access to their own conformal equivalence class. A further trace over the group yields the curvature invariants of the spacetime. The Kretschmann scalar becomes the topological Euler density, which may be linked in a simple way to the Hawking temperature of horizons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connecting 2-Forms, Conformal Transformations, Curvature Invariants and Topological Classes in Einstein Spacetimes Hughes, Jack C. M. Kusmartsev, Fedor V. General Relativity and Quantum Cosmology High Energy Physics - Phenomenology High Energy Physics - Theory Mathematical Physics The unique Nature of the Lorentz group in four dimensions is the root cause of the many remarkable properties of the Einstein spacetimes, in particular their operational structure on the 2-forms. We show how this operational structure can be used for two ends. First, it allows for a simple generalization of the Birkhoff theorem to Schwarzschild (A)de-Sitter spacetime. Second, it provides the means to construct an Abelian endomorphism group on the space of 2-forms. It is observed that taking the trace over this group element-wise induces a further Abelian group which may be identified with a tensor representation of conformal transformations, giving Einstein spacetimes access to their own conformal equivalence class. A further trace over the group yields the curvature invariants of the spacetime. The Kretschmann scalar becomes the topological Euler density, which may be linked in a simple way to the Hawking temperature of horizons. |
| title | Connecting 2-Forms, Conformal Transformations, Curvature Invariants and Topological Classes in Einstein Spacetimes |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Phenomenology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2403.11527 |