A Weyl's law for black holes

Abstract

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 $N(\omega )$ of quasi-normal modes of $(d+1)$-dimensional black holes follows a power-law $N(\omega )\sim {\mathrm{Vol}}_d^{\mathrm{eff}}{\omega }^d$, with ${\mathrm{Vol}}_d^{\mathrm{eff}}$ an effective $d$-volume determined by the light-trapping properties of the black hole geometry. Concretely, the factorisation ${\mathrm{Vol}}_d^{\mathrm{eff}}\sim \left(8\pi /\kappa \right)\cdot {\mathrm{Vol}}_{d-1}^{\mathrm{trapped}}$ makes apparent the two underlying structural ingredients, namely the (local) redshift effect controlled by the surface gravity $\kappa$ and the volume ${\mathrm{Vol}}_{d-1}^{\mathrm{trapped}}$ 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.

21 pages, 4 figures, updated version matching the published manuscript