Birkhoff rigidity from a covariant optical seed
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915931805974528 |
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| author | Easson, D. A. |
| author_facet | Easson, D. A. |
| contents | We present a local seed-to--Kerr--Schild route to Birkhoff rigidity in four-dimensional spherical vacuum gravity. On the two-dimensional orbit space, the areal radius \(r\) determines a scalar \(F:=-(\nabla r)^2\), and the reduced vacuum equations imply \(F(r)=1-2M/r\). We show that the normalized one-forms \(dr/F\) and \((*dr)/F\) are closed, so that the null combinations \(F^{-1}(dr\pm *dr)\) are exact null seed forms. Integrating these yields local Eddington--Finkelstein coordinates in which the metric takes Kerr--Schild form over a flat background. We then prove the corresponding uniqueness statement in the stationary optical sector: spherical symmetry forces the inverse optical seed \(\mathcal R\) to equal \(\pm r\), equivalently the optical seed \(ρ\) to equal \(\mp 1/r\), and the resulting seed data reconstruct the Schwarzschild family. Thus, Birkhoff rigidity is paired with a spherical converse theorem in the stationary optical framework: Schwarzschild is the unique spherically symmetric stationary vacuum Kerr--Schild geometry generated by a nowhere-vanishing optical seed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09830 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Birkhoff rigidity from a covariant optical seed Easson, D. A. General Relativity and Quantum Cosmology High Energy Physics - Theory We present a local seed-to--Kerr--Schild route to Birkhoff rigidity in four-dimensional spherical vacuum gravity. On the two-dimensional orbit space, the areal radius \(r\) determines a scalar \(F:=-(\nabla r)^2\), and the reduced vacuum equations imply \(F(r)=1-2M/r\). We show that the normalized one-forms \(dr/F\) and \((*dr)/F\) are closed, so that the null combinations \(F^{-1}(dr\pm *dr)\) are exact null seed forms. Integrating these yields local Eddington--Finkelstein coordinates in which the metric takes Kerr--Schild form over a flat background. We then prove the corresponding uniqueness statement in the stationary optical sector: spherical symmetry forces the inverse optical seed \(\mathcal R\) to equal \(\pm r\), equivalently the optical seed \(ρ\) to equal \(\mp 1/r\), and the resulting seed data reconstruct the Schwarzschild family. Thus, Birkhoff rigidity is paired with a spherical converse theorem in the stationary optical framework: Schwarzschild is the unique spherically symmetric stationary vacuum Kerr--Schild geometry generated by a nowhere-vanishing optical seed. |
| title | Birkhoff rigidity from a covariant optical seed |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2604.09830 |