Clines and the Analytic Structure of Black Hole Perturbations

TL;DR

This paper introduces the concept of clines in the complex plane to organize and simplify black hole perturbation equations, enabling explicit solutions and insights into black hole properties.

Contribution

It presents a novel framework using clines and M"obius invariance to analyze black hole perturbations via Heun equations, leading to explicit solutions and new physical insights.

Findings

Explicit scalar perturbation solutions for 7D Myers-Perry black holes

Derivation of static scalar tidal Love numbers

Clines reveal M"obius-invariant structure in black hole perturbations

Abstract

We revisit black hole perturbations through Heun differential equations, focusing on Frobenius power-series solutions near regular singularities and their connection formulas. Central to our approach is the notion of a cline in the complex plane, which organizes singular points of the differential equations and remain invariant under M\"obius transformations. Building on the cline structure we identified in black hole horizons, we carry out a systematic reduction and relocation of poles in the differential equation to obtain explicit representations of the solutions. We illustrate our approach by extracting the scalar perturbation solutions for the 7-dimensional Myers-Perry black hole and deriving the static scalar tidal Love numbers. These results suggest that clines expose a M\"obius-invariant order within black hole perturbations, rendering black hole perturbation problems remarkably tractable.