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| Format: | Preprint |
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2022
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| Online-Zugang: | https://arxiv.org/abs/2212.05570 |
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| _version_ | 1866911290532823040 |
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| author | Jaramillo, José Luis Macedo, Rodrigo P. Meneses-Rojas, Oscar Raffaelli, Bernard Sheikh, Lamis Al |
| author_facet | Jaramillo, José Luis Macedo, Rodrigo P. Meneses-Rojas, Oscar Raffaelli, Bernard Sheikh, Lamis Al |
| contents | We discuss a Weyl's law for the quasi-normal modes of black holes that recovers the structural features of the standard Weyl's law for the eigenvalues of Laplacian-like operators in compact regions. Specifically, we propose that the asymptotics of the counting function $N(ω)$ of quasi-normal modes of $(d+1)$-dimensional black holes follows a power-law $N(ω)\sim \mathrm{Vol}_d^{\mathrm{eff}}ω^d$, with $\mathrm{Vol}_d^{\mathrm{eff}}$ an effective $d$-volume determined by the light-trapping properties of the black hole geometry. Concretely, the factorisation $\mathrm{Vol}_d^{\mathrm{eff}} \sim \left(8π/κ\right) \cdot \mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ makes apparent the two underlying structural ingredients, namely the (local) redshift effect controlled by the surface gravity $κ$ and the volume $\mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ of the (phase space) trapped set. In particular, this proposal extends the Weyl's law proved by Dyatlov & Zworski for the counting of slowest decaying quasi-normal modes, to include overtones. As an application, these Weyl's laws could provide a probe into the effective spacetime dimensionality, upon the counting of sufficiently many quasi-normal modes in the ringdown signal of binary black hole mergers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05570 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Weyl's law for black holes Jaramillo, José Luis Macedo, Rodrigo P. Meneses-Rojas, Oscar Raffaelli, Bernard Sheikh, Lamis Al General Relativity and Quantum Cosmology High Energy Physics - Theory We discuss a Weyl's law for the quasi-normal modes of black holes that recovers the structural features of the standard Weyl's law for the eigenvalues of Laplacian-like operators in compact regions. Specifically, we propose that the asymptotics of the counting function $N(ω)$ of quasi-normal modes of $(d+1)$-dimensional black holes follows a power-law $N(ω)\sim \mathrm{Vol}_d^{\mathrm{eff}}ω^d$, with $\mathrm{Vol}_d^{\mathrm{eff}}$ an effective $d$-volume determined by the light-trapping properties of the black hole geometry. Concretely, the factorisation $\mathrm{Vol}_d^{\mathrm{eff}} \sim \left(8π/κ\right) \cdot \mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ makes apparent the two underlying structural ingredients, namely the (local) redshift effect controlled by the surface gravity $κ$ and the volume $\mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ of the (phase space) trapped set. In particular, this proposal extends the Weyl's law proved by Dyatlov & Zworski for the counting of slowest decaying quasi-normal modes, to include overtones. As an application, these Weyl's laws could provide a probe into the effective spacetime dimensionality, upon the counting of sufficiently many quasi-normal modes in the ringdown signal of binary black hole mergers. |
| title | A Weyl's law for black holes |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2212.05570 |