Charged scalar fields on Reissner--Nordström spacetimes I: integrated energy estimates
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918418836357120 |
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| author | Gajic, Dejan |
| author_facet | Gajic, Dejan |
| contents | This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordström spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner--Nordström(--de Sitter) spacetimes. These estimates extend to Reissner--Nordström spacetimes away from extremality under the assumption of mode stability on the real axis.
Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner--Nordström spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28874 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Charged scalar fields on Reissner--Nordström spacetimes I: integrated energy estimates Gajic, Dejan General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordström spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner--Nordström(--de Sitter) spacetimes. These estimates extend to Reissner--Nordström spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner--Nordström spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes. |
| title | Charged scalar fields on Reissner--Nordström spacetimes I: integrated energy estimates |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2603.28874 |