Clines and the Analytic Structure of Black Hole Perturbations

TL;DR

The paper identifies clines—Möbius-invariant arrangements of singular points in the complex plane—as a unifying structure for black hole perturbations described by generalized Heun equations. By exploiting Möbius covariance and a sequence of coordinate and field redefinitions, the authors reduce generalized Heun problems to standard Heun form, enabling exact analytic constructions of solutions via Wronskian connection formulas. They concretely apply the framework to massless scalar perturbations of seven-dimensional Myers-Perry black holes, deriving a full static solution and extracting the tidal Love numbers, including nonzero results for certain non-integer $\hat{\ell}$ and the vanishing of Love numbers for integer $\hat{\ell}$, with consistency checks against Schwarzschild limits. The work demonstrates a Möbius-invariant, cline-guided route to tractable black hole perturbation analyses and suggests broader applicability to higher-dimensional spacetimes and confluent limits, potentially unifying diverse perturbation problems under Heun-function techniques.

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 Möbius 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 Möbius-invariant order within black hole perturbations, rendering black hole perturbation problems remarkably tractable.

Figures (6)

• Figure 1: Complex-plane representation of the black hole horizons for Myers--Perry solutions in $D=9,11,13$ dimensions with all angular momentum parameters set equal. We used as mass and angular momentum parameters $\{r_s, a\} = \{2, 0.1\}$, respectively. The seven-dimensional case will be discussed separately in Section \ref{['sec:3']}. Double poles are indicated by the same color, and the point at infinity is not shown in the plots.
• Figure 2: Complex-plane representation of the black hole horizons for the five-dimensional Myers–Perry solution belonging to a cline. After applying specific Möbius and diffeomorphism transformations, the singular points can be rearranged along the real axis in the complex plane. This procedure effectively reduces the corresponding differential equation to a hypergeometric form, rendering the problem analytically solvable. This approach was first illustrated in the work of Cvetič et. al Cvetic:1997uw and further employed for the computation of Love numbers in Rodriguez:2023xjd.
• Figure 3: Complex-plane representation of the black hole horizons for the five-, six-, and seven-dimensional Myers--Perry solutions Myers:2011yc. For the five-dimensional case, the mass and angular momentum parameters are $\{\mu,a,b\} = \{1, 1.5, 7\}$ respectively. For the solution in 6 dimensions, we have $\{\mu, a, b\}=\{7, 1,1.5\}$. In seven dimensions we used $\{\mu, a, b, c\} = \{40, 1, 1.5, 2\}$. The resulting horizons are found to lie along clines.
• Figure 4: Regular singular points for the scalar wave equation in Boyer-Lindquist coordinates.
• Figure 5: Regular singular points of the scalar wave equation on z coordinates.