A Weyl law for black holes

We discuss a Weyl's law for the quasinormal 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 $N(\ensuremath{\omega})$ of quasinormal modes of ($d+1$)-dimensional black holes follows a power-law $N(\ensuremath{\omega})\ensuremath{\sim}{\mathrm{Vol}}_{d}^{\mathrm{eff}}{\ensuremath{\omega}}^{d}$, with ${\mathrm{Vol}}_{d}^{\mathrm{eff}}$ an effective $d$-volume determined by the light-trapping properties of the black hole geometry. Concretely, the factorization ${\mathrm{Vol}}_{d}^{\mathrm{eff}}\ensuremath{\sim}(8\ensuremath{\pi}/\ensuremath{\kappa})\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{Vol}}_{d\ensuremath{-}1}^{\text{trapped}}$ makes apparent the two underlying structural ingredients, namely the (local) redshift effect controlled by the surface gravity $\ensuremath{\kappa}$ and the volume ${\mathrm{Vol}}_{d\ensuremath{-}1}^{\text{trapped}}$ of the (phase space) trapped set. In particular, this proposal extends the Weyl's law proved by Dyatlov and Zworski for the counting of slowest decaying quasinormal 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 quasinormal modes in the ringdown signal of binary black hole mergers.