Price's law from quasinormal modes

TL;DR

This work links Price's late-time tail for perturbations of Schwarzschild spacetime to a sum over Schwarzschild-de Sitter quasinormal modes in the limit of vanishing cosmological constant, $\Lambda\to 0^+$. By recasting perturbations as a Heun equation and examining the near-horizon regime, the authors show that de Sitter QNM poles with frequencies $\omega_n^{(\text{dS})} = -i\sqrt{\Lambda/3}(\ell+n+1) + O(\Lambda)$ contribute a residue series that, when summed, yields a closed-form expression for the Green's function tail, reducing to Price's law $G(t)\propto (t)^{-2\ell-3}$ with the correct coefficient as $\Lambda\to 0^+$. They also demonstrate that the same tail arises from the branch-cut structure that emerges in the Schwarzschild limit, providing a cross-check between the pole-sum and cut- integral pictures. For generic observer/source locations, numerical residue summation at small $\Lambda$ reproduces the expected tail and matches direct time-domain PDE solutions, establishing a robust connection between QNMs, branch cuts, and late-time decay in black-hole perturbations. The work also connects to gauge-theory methods (AGT/ Liouville) via the Heun equation, enabling analytic control of connection data through Nekrasov partition functions in the zero-instanton sector.

Abstract

We show that Price's power-law tail for perturbations of Schwarzschild, $t^{-2\ell-3}$ as $t\to \infty$, can be obtained from a sum of Schwarzschild-de Sitter quasinormal modes in the limit $Λ\to 0^+$.

Figures (2)

• Figure 1: The analytic structure of the retarded Green's function in frequency space, $\widetilde{G}(\omega, r, r')$, at small $|\omega R_h|$, illustrating the change caused by heating up null infinity with a small $\Lambda > 0$. Left panel: Schwarzschild-de Sitter at small but finite $\Lambda >0$, which contains QNM poles at de Sitter frequencies \ref{['dSmodes']} (black points), as well as black hole QNMs at order-1 values of $\omega R_h$ (not shown). Right panel: Schwarzschild, showing a logarithmic branch cut obtained in the limit $\Lambda \to 0^+$, from the coalescence of poles in the left panel. We demonstrate that the residue sum (blue contour, left panel) reproduces the power-law tail, $t^{-2\ell-3}$, equal to the branch cut discontinuity integral in this limit (blue contour, right panel).
• Figure 2: Price's power-law tail derived from de Sitter QNM residue sums (red, dashed) compared to a time-domain numerical solution of the Schwarzschild Green's function PDE (\ref{['GreensPDE']} at $\Lambda = 0$) using Gaussian initial data of width $\sigma$ to approximate the delta function (black, solid). A power law of $(t-t')^{-2\ell - 3}$ with arbitrary normalisation is indicated in blue as reference. We use parameters $s= \ell = 2, \sigma = 1, R_h = 1$. Upper panel: Behaviour near the black hole horizon $r_* = -12.2006$, $r_*' = -10.1005$. Here, the dS mode infinite residue sum is computed analytically in the limit $\Lambda \to 0$, given by \ref{['priceslawintro']}, and is shown analytically to be equal to the corresponding cut discontinuity integral. Lower panel: Behaviour far from the black hole, $r_*' = 12.0506$, $r_* = 16.0508$. Here, a partial dS mode residue sum is computed numerically from solutions to the Heun differential equation, at small $R_h^2\Lambda = 10^{-6}$.