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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2508.05683 |
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| _version_ | 1866913979747532800 |
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| author | Reid, James A. |
| author_facet | Reid, James A. |
| contents | In the MacDowell-Mansouri formulation of general relativity, the spin connection and coframe variables are incorporated into a single Lie algebra-valued connection called the MacDowell-Mansouri connection, $ω$. From the curvature form $F$ of $ω$ and an auxiliary field, $B$, one may formulate general relativity as a deformed topological field theory by constructing an action functional whose variation yields a set of field equations that are equivalent to the Einstein equations on shell. In this article, we show that when the fundamental length scale of the MacDowell-Mansouri connection is regarded as a dynamical variable -- a cosmological scalar field -- the field equations obtained from the variation of the resulting action are equivalent to the conformal Einstein equations on shell. Through the lens of Cartan geometry, we then discuss a notable geometrical difference between general relativity and its conformally transformed counterpart. Specifically, for the latter, we show that points in spacetime are infinitesimally approximated by homogeneous spaces (restricted to a point) whose radii are parameterised by the value of the cosmological scalar field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformal Gravity as a Deformed Topological Field Theory Reid, James A. General Relativity and Quantum Cosmology In the MacDowell-Mansouri formulation of general relativity, the spin connection and coframe variables are incorporated into a single Lie algebra-valued connection called the MacDowell-Mansouri connection, $ω$. From the curvature form $F$ of $ω$ and an auxiliary field, $B$, one may formulate general relativity as a deformed topological field theory by constructing an action functional whose variation yields a set of field equations that are equivalent to the Einstein equations on shell. In this article, we show that when the fundamental length scale of the MacDowell-Mansouri connection is regarded as a dynamical variable -- a cosmological scalar field -- the field equations obtained from the variation of the resulting action are equivalent to the conformal Einstein equations on shell. Through the lens of Cartan geometry, we then discuss a notable geometrical difference between general relativity and its conformally transformed counterpart. Specifically, for the latter, we show that points in spacetime are infinitesimally approximated by homogeneous spaces (restricted to a point) whose radii are parameterised by the value of the cosmological scalar field. |
| title | Conformal Gravity as a Deformed Topological Field Theory |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2508.05683 |