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 $\mathrm{Vol}_d^{\mathrm{eff}} \sim \left(8\pi/\kappa\right) \cdot \mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ makes apparent the two underlying structural ingredients, namely the (local) redshift effect controlled by the surface gravity $\kappa$ 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.