Black Hole Scattering from Monodromy

TL;DR

This work develops a monodromy-based framework to extract black hole scattering data and quasinormal modes from the global analytic structure of complexified wave equations. By encoding boundary conditions in monodromy eigenvectors and exploiting the global constraint among horizon and infinity monodromies, the authors connect greybody factors and QNM spectra to local singularity data and Stokes phenomena, with Kerr as a concrete demonstration. The approach yields both perturbative and numerical methods, reveals transcendental structures in QNM conditions, and highlights the role of irregular singularities in non-analytic transport properties. The analysis also leverages symmetries of the confluent Heun equation to constrain the connection matrices, offering a path to broader applications across black hole spacetimes and higher-spin perturbations, including extremal and AdS backgrounds.

Abstract

We study scattering coefficients in black hole spacetimes using analytic properties of complexified wave equations. For a concrete example, we analyze the singularities of the Teukolsky equation and relate the corresponding monodromies to scattering data. These techniques, valid in full generality, provide insights into complex-analytic properties of greybody factors and quasinormal modes. This leads to new perturbative and numerical methods which are in good agreement with previous results.

Figures (2)

• Figure 1: The figure depicts the monodromy around the irregular singular point, $\alpha_{\textit{irr}}$, as a function of the frequency $\omega$ for fixed values of the mass $M=0.7$, angular momenta $a=0.2$, and $\ell=m=2$. The black line outlines the analytic perturbative results for $\alpha_{\textit{irr}}$ while the grey line represents a fit to the numerical data given by the grey dots.
• Figure 2: The figure shows the real part of the monodromy around the irregular singular point ${\rm Re}(\alpha_{\textit{irr}})$ as a function of the highly damped QNM frequency ${\rm Im}( \omega_\textsc{qnm})=\pi \, T_{BH} (2n-1)$ for fixed values of the mass $M=\frac{1}{2}$, angular momenta $a=0$, and $\ell=1$. The blue dots are the numerical values; for $n\gg 40$, the growth becomes linear and ${\rm Re}(\alpha_{irr}) \approx 0.43243\,{\rm Im}(\omega_\textsc{qnm})-5.51604$.