Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity
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arXiv
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| Format: | Preprint |
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2024
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| author | Gajic, Dejan Van de Moortel, Maxime |
| author_facet | Gajic, Dejan Van de Moortel, Maxime |
| contents | We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential.
We present a novel, geometric, physical-space approach for determining late-time asymptotics, based around embedding Minkowski spacetime conformally into the spacetime $AdS_2 \times \mathbb{S}^2$ (with $AdS_2$ the two-dimensional anti de-Sitter spacetime) to turn a global late-time asymptotics problem into a local existence problem for the wave equation in $AdS_2 \times \mathbb{S}^2$. Our approach is inspired by the treatment of the near-horizon geometry of extremal black holes in the physics literature.
We moreover apply our method to another scale-invariant model: the (complex-valued) charged wave equation on Minkowski spacetime in the presence of a static electric field, which can be viewed as a simplification of the charged Maxwell-Klein-Gordon equations on a black hole spacetime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity Gajic, Dejan Van de Moortel, Maxime Analysis of PDEs General Relativity and Quantum Cosmology Mathematical Physics We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential. We present a novel, geometric, physical-space approach for determining late-time asymptotics, based around embedding Minkowski spacetime conformally into the spacetime $AdS_2 \times \mathbb{S}^2$ (with $AdS_2$ the two-dimensional anti de-Sitter spacetime) to turn a global late-time asymptotics problem into a local existence problem for the wave equation in $AdS_2 \times \mathbb{S}^2$. Our approach is inspired by the treatment of the near-horizon geometry of extremal black holes in the physics literature. We moreover apply our method to another scale-invariant model: the (complex-valued) charged wave equation on Minkowski spacetime in the presence of a static electric field, which can be viewed as a simplification of the charged Maxwell-Klein-Gordon equations on a black hole spacetime. |
| title | Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2401.13047 |