Quasi-normal modes of generic type D black holes via Heun functions

The increasing diversity and precision of gravitational-wave (GW) observations provide unprecedented opportunities to perform stringent tests of general relativity (GR). Although several conceptual and observational challenges motivate the development of modified theories of gravity, a crucial prerequisite is a complete understanding of all phenomena allowed within GR itself. A key result in this context is the no-hair theorem, whose classical formulation relies on the assumption of asymptotic flatness. When this assumption is relaxed, GR already admits a much richer space of solutions with additional “charges.”

In this work, we investigate asymptotically non-flat black-hole spacetimes within GR, focusing on the full Plebanski–Demianski (PD) class of algebraic type D solutions. This general family of metrics is characterized by up to seven independent parameters: mass, spin, electric charge, magnetic charge, acceleration, cosmological constant, and NUT charge. Our objective is to compute the quasi-normal modes (QNMs) of these black holes in their most general form.

Two notable subclasses—Kerr–de Sitter black holes and accelerating (C-metric) black holes—have previously been shown to yield Teukolsky equations that reduce to the general Heun equation with four regular singular points. Such equations can now be solved efficiently using Mathematica’s Heun function framework, allowing high-precision computation of QNM frequencies within seconds. Heun functions are generalizations of hypergeometric functions; they are the solutions to the general Heun equation.

We extend these results by demonstrating that the Teukolsky equation for the full PD metric also maps to the general Heun equation. Consequently, we obtain a unified method to compute QNM spectra across the entire seven-parameter family and investigate how each parameter influences the oscillation and damping frequencies.

This generalization significantly broadens the landscape of GR-consistent QNM predictions and should be taken into account when interpreting future GW observations and formulating robust tests of GR.