The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915722356064256 |
|---|---|
| author | Dong, Kai |
| author_facet | Dong, Kai |
| contents | In this paper, we give a new proof to a past stability result established in Fournodavlos-Rodnianski-Speck (arXiv:2012.05888), for Kasner solutions of the $(3+1)$-dimensional Einstein vacuum equations under polarized $U(1)$-symmetry. Our method, inspired by Beyer-Oliynyk-Olvera-Santamar{\'ı}a-Zheng (arXiv:1907.04071, arXiv:2502.09210), relies on a newly developed $(2+1)$ orthonormal-frame decomposition and a careful symmetrization argument, after which the Fuchsian techniques can be applied.
We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete and crushing at the Big Bang singularity. They are achieved by reducing the $(3+1)$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of constraints propagation, we obtain global existence and precise asymptotics of the solutions up to the singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06957 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry Dong, Kai General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry In this paper, we give a new proof to a past stability result established in Fournodavlos-Rodnianski-Speck (arXiv:2012.05888), for Kasner solutions of the $(3+1)$-dimensional Einstein vacuum equations under polarized $U(1)$-symmetry. Our method, inspired by Beyer-Oliynyk-Olvera-Santamar{\'ı}a-Zheng (arXiv:1907.04071, arXiv:2502.09210), relies on a newly developed $(2+1)$ orthonormal-frame decomposition and a careful symmetrization argument, after which the Fuchsian techniques can be applied. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete and crushing at the Big Bang singularity. They are achieved by reducing the $(3+1)$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of constraints propagation, we obtain global existence and precise asymptotics of the solutions up to the singularities. |
| title | The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2601.06957 |