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arXiv · 2604.18680

Beyond Three Terms: Continued Fractions for Rotating Black Holes in Modified Gravity

Georgios Karikos·Jayana A. Saes·Pratik Wagle·Nicolás Yunes
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Abstract

Black-hole ringdown offers a clean probe of strong gravity, but one of its most accurate tools--Leaver's continued-fraction method--requires a three-term recurrence relation. Beyond general relativity, and more generally in non-Kerr spacetimes, Frobenius expansions of the perturbation equations generically produce higher-order recurrence relations and, often, couplings among the series coefficients, obstructing a direct application of Leaver's method. Here we develop a general reduction scheme that maps arbitrary scalar and matrix $N$-term recurrence relations to a three-term form, thereby extending continued fractions to a broad class of perturbation problems in modified gravity. As an application, we compute the quasinormal-mode spectrum of slowly-rotating black holes in dynamical Chern-Simons gravity, where the polar sector yields a 16-term, decoupled, scalar recurrence relation, and the axial sector yields a 12-term, coupled, matrix recurrence relation. After applying our reduction scheme, both systems can be solved with continued fractions. For the fundamental $(\ell,m)=(2,2)$ mode, our results agree well with independent calculations based on eigenvalue-perturbation and metric/spectral methods across the parameter range studied. This framework provides a robust and practical route to precision ringdown calculations beyond the standard three-term setting and supports tests of gravity with current and future gravitational-wave observations.

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Georgios Karikos, Jayana A. Saes, Pratik Wagle, Nicolás Yunes. 2026-04-20. Beyond Three Terms: Continued Fractions for Rotating Black Holes in Modified Gravity. https://arxiv.org/abs/2604.18680

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