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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.08568 |
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| _version_ | 1866911499161698304 |
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| author | de Haro, Jaume |
| author_facet | de Haro, Jaume |
| contents | This work places the invariant $ds^2$ at the center of the gravitational interaction, interpreting it not as a purely geometric object but as the differential of proper time, endowed with direct physical meaning. Starting from the extension of Fermat's principle to massive particles--namely, the requirement that freely falling bodies follow trajectories that extremize proper time, which for timelike motion corresponds to a local maximum--and invoking the universality of Galilean free fall, we derive the form of $ds^2$ in a static gravitational field. Lorentz invariance then provides the natural framework to extend this result to systems involving moving matter. The invariant derived through this procedure matches the weak-field limit of General Relativity formulated in the harmonic gauge.
Within this linearized regime, we show that the structure of the theory already contains the seeds of its non-linear completion: any dynamically consistent extension to strong gravitational fields necessarily involves the Ricci tensor. From this viewpoint, Einstein's field equations appear not as a postulated geometric law, but as the unique covariant closure required to ensure energy momentum conservation and the self consistency of the gravitational interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08568 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A dynamical approach to General Relativity based on proper time de Haro, Jaume General Relativity and Quantum Cosmology This work places the invariant $ds^2$ at the center of the gravitational interaction, interpreting it not as a purely geometric object but as the differential of proper time, endowed with direct physical meaning. Starting from the extension of Fermat's principle to massive particles--namely, the requirement that freely falling bodies follow trajectories that extremize proper time, which for timelike motion corresponds to a local maximum--and invoking the universality of Galilean free fall, we derive the form of $ds^2$ in a static gravitational field. Lorentz invariance then provides the natural framework to extend this result to systems involving moving matter. The invariant derived through this procedure matches the weak-field limit of General Relativity formulated in the harmonic gauge. Within this linearized regime, we show that the structure of the theory already contains the seeds of its non-linear completion: any dynamically consistent extension to strong gravitational fields necessarily involves the Ricci tensor. From this viewpoint, Einstein's field equations appear not as a postulated geometric law, but as the unique covariant closure required to ensure energy momentum conservation and the self consistency of the gravitational interaction. |
| title | A dynamical approach to General Relativity based on proper time |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2603.08568 |