A Weyl's law for black holes

TL;DR

This work proposes a Weyl-like law for black hole quasi-normal modes, stating that the counting function N(ω) grows as N(ω) ∼ Vol_d^{eff} ω^d in a stationary (d+1)-dimensional BH spacetime, with Vol_d^{eff} capturing redshift and light trapping via κ and the trapped set. The authors connect the BH QNM spectrum to the phase-space geometry of trapped null geodesics, extending Dyatlov & Zworski’s results for slowest decaying modes to the full overtone spectrum, and they illustrate the Schwarzschild case where N(ω) ∼ const × ω^3 with a geometric constant dependent on horizon area and light-ring area. A key insight is interpreting Vol_d^{eff} as an intrinsic spacetime volume, factorizing into a radial redshift term ∝ 1/κ and an angular trapped-set term ∝ Vol^{trapped}_{d-1}, yielding a robust, geometry-driven leading term. If observationally feasible to extract many QNM overtones from ringdown data, this Weyl law could serve as a probe of the effective spacetime dimensionality and the dynamical scales governing BH resonances, with potential extensions to non-spherical, charged, rotating, and AdS/dS BHs.

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 $N(ω)$ 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.

Figures (4)

• Figure 1: Illustration of the "convex hull" $\mathrm{chsupp(V)}$ of the support of a potential $V(x)$. This notion, namely the "diameter" of the region where the potential does not vanish, provides the relevant multiplicative scale in QNM Weyl's law for compact support potentials.
• Figure 2: Sketch of the figure used in the "counting argument" of QNMs in the Schwarzschild BHs. The key features are the asymptotic distribution with a constant gap of QNMs along a line parallel to the imaginary axis, as well as the bound on the real part of QNM frequencies $\omega_n$'s in terms of the real part of the fundamental QNM.
• Figure 3: Numerical study of the QNM Weyl's law for Schwarzschild. Top panel: Numerically calculated BH QNMs for a gravitational perturbation ($s$=$|$spin$|$ = 2) on the $(3+1)$-dimensional Schwarzschild spacetime. Within the circle $|r_{\rm h} \omega| = 8.1$, with $r_{\rm h}=2M$, one counts the number of QNMs for the angular modes $\ell=2,\cdots, 20$. Bottom panel: Weyl's law for $N(\omega)$ in $(3+1)$-dimensional Schwarzschild BHs for scalar ($s$=$|$spin$|$ = 0), electromagnetic ($s$=$|$spin$|$ = 1) and gravitational ($s$=$|$spin$|$ = 2) perturbations. The asymptotic $N(\omega) \sim |\omega|^3$ is recovered, with a multiplicative constant $\approx 24.0 \; r_{\rm h}^{3.0}$.
• Figure 4: Numerical study of the QNM Weyl's law for the $(3+1)$-Reissner-Nordström BH. The Weyl's law asymptotics for $N(\omega)$ for QNMs of gravitational ($s$=$|$spin$|$ = 2) perturbations are presented for different values of $Q/M$ (charge/mass ratio). We recover the asymptotic $N(\omega) \sim |\omega|^3$. However, contamination from the branch cut eigenvalues in our numerical scheme prevents us from recovering the multiplicative constant, that we have normalised to $1$ for all $Q/M$'s.