Pole-skipping of gravitational waves in the backgrounds of four-dimensional massive black holes

TL;DR

This work classifies pole-skipping points for gravitational perturbations in four-dimensional massive black-hole backgrounds across flat, AdS, and dS spacetimes. By exploiting the Darboux-integrable structure, it separates pole-skipping into algebraically special points, solvable analytically, and common points shared by even/odd parity channels, with a detailed determinant-based method to locate the latter. The analysis yields explicit conditions μ(μ−2K)=−3nτ or +3nτ for algebraically special points, plus general determinant results for common points, and realises hydrodynamic pole-skipping in AdS black branes as a special case. The results also clarify pole-skipping at the cosmological horizon in de Sitter space and illuminate the non-total-transmissivity of algebraically special pole-skipping modes. Overall, the paper provides a coherent 4d gravity-based framework tying horizon boundary conditions, Darboux transformations, and pole-skipping spectra across diverse horizon topologies and Λ values, with implications for both gravitational wave propagation and holographic transport.

Abstract

Pole-skipping is a property of gravitational waves dictated by their behaviour at horizons of black holes. It stems from the inability to unambiguously impose ingoing boundary conditions at the horizon at an infinite discrete set of Fourier modes. The phenomenon has been best understood, when such a description exists, in terms of dual holographic (AdS/CFT) correlation function that take the value of '0/0' at these special points. In this work, we investigate details of pole-skipping purely from the point of view of classical gravity in 4d massive black hole geometries with flat, spherical and hyperbolic horizons, and with an arbitrary cosmological constant. We show that pole-skipping points naturally fall into two categories: the algebraically special points and a set of pole-skipping points that is common to the even and odd channels of perturbations. Our analysis utilises and generalises (to arbitrary maximally symmetric horizon topology and cosmological constant) the 'integrable' structure of the Darboux transformations, which relate the master field equations that describe the evolution of gravitational perturbations in the two channels. Finally, we provide new insights into a number of special cases: spherical black holes, asymptotically Anti-de Sitter black branes and pole-skipping at the cosmological horizon in de Sitter space.

Figures (4)

• Figure 1: A representation of the solutions to Eq. \ref{['taulambda']} for spherical black holes with $K=1$ (blue), black branes with $K=0$ (red) and hyperbolic black holes with $K=-1$ (green). The event horizon branches are depicted with thick solid lines and the cosmological horizon with a dashed blue line. The event horizon branch is bijective for all physical $\tau$ while the cosmological horizon branch is not bijective. Black branes and hyperbolic black holes only exist in asymptotically AdS spaces with $\Lambda< 0$. Spherical black holes exist for any $\Lambda$. The unphysical solutions with $r_0<0$ (thin solid lines) will not be studied in this work.
• Figure 2: Locations of the common pole-skipping points (black dots) along the real $\mu$ axis for spherical black holes ($K=1$) at different Matsubara levels $n$. Three panels correspond to three choices of the parameter $\tau$, giving geometries that are asymptotically dS, flat and AdS (respectively, from left to right).
• Figure 3: All pole-skipping points plotted in the complex $\mu$ plane for six choices of the Matsubara level $n$ of the asymptotically flat spherical Schwarzschild black hole with $K=1$ and $\tau = 1$. Red and blue dots depict the algebraically special solutions in the even and odd channels, respectively. Black dots are the common points shared between the two channels.
• Figure 4: The cosmological horizon pole-skipping points in the complex $\mu$ plane plotted for $n=-6$ and for three different positions of the cosmological horizon. Red and blue dots depict the algebraically special solutions in the even and odd channels, respectively. Black dots are the common points shared between the two channels. In the even channel, the algebraically special points can overlap with common points, yielding anomalous pole-skipping points, which are plotted with green dots. For a given $\tau$, the solutions in the two channels can be related by taking $\mu\rightarrow 1-\mu$. Moreover, note that for the 'extreme' value of $\tau=-1/3$, the spectrum is symmetric across the line of $\text{Re}\mu=1$.