The Confluent Heun functions in Black Hole Perturbation Theory: a spacetime interpretation

Abstract

This work provides a spacetime interpretation of the confluent Heun functions within black hole perturbation theory (BHPT) and explores their relationship to the hyperboloidal framework. In BHPT, the confluent Heun functions are solutions to the radial Teukolsky equation, but they are traditionally studied without an explicit reference to the underlying spacetime geometry. Here, we show that the distinct behaviour of these functions near their singular points reflects the structure of key surfaces in black hole spacetimes. By interpreting homotopic transformations of the confluent Heun functions as changes in the spacetime foliation, we connect these solutions to different regions of the black hole's global structure, such as the past and future event horizons, past and future null infinity, spatial infinity, and even past and future timelike infinity. We also discuss the relationship between the confluent Heun functions and the hyperboloidal formulation of the Teukolsky equation. Although neither confluent Heun form of the radial Teukolsky equation can be interpreted as hyperboloidal slices, this approach offers new insights into wave propagation and scattering from a global black hole spacetime perspective.

Figures (3)

• Figure 1: Carter-Penrose diagram representing hypersurfaces of constant Boyer-Lindquist time coordinate, upon which the radial Teukolsky function $\, {}_{\frak s}{\cal R}_{\ell m}( \omega; r)$ is defined. The corresponding radial Teukolsky equation assumes the generic form of the confluent Heun equation \ref{['eq:GenConfHeun']}. The derived characteristic exponent \ref{['eq:BL_coeff_C']}-\ref{['eq:BL_coeff_infty']} relate to the function's asymptotic behaviour as $r\rightarrow r_{\rm h}$ (or $r_{\rm c}$) towards the bifurcation sphere ${\cal B}$, and $r\rightarrow \infty$ at spatial infinity $i^0$.
• Figure 2: Carter-Penrose diagrams representing the $\tilde{t} = 0$ hypersurfaces corresponding to the $8$ possible combinations of $(\mu_c^\pm, \mu_h^\pm, \nu^\pm)$ yielding a Heun slice. All configurations are horizon penetrating, with the particular sign in $\mu_c^\pm$ and $\mu_h^\pm$ indicating if the horizon is generated by the outgoing $\tilde{\ell}^a_+$ or ingoing $\tilde{\ell}^a_-$ null vector. The geometrical meaning of $r\rightarrow \infty$ along $\tilde{t} = \text{constant}$ differs according to each configuration: it may represent future or past null infinity (top left), spatial infinity (top right and bottom left), or even timelike infinity (bottom right).
• Figure 3: Carter-Penrose diagrams representing hyperboloidal slices in the minimal gauge class. Left Panel: In the radial fixing gauge, the slices not only intersect future null infinity $\mathscr{I}^+$ and the black hole horizon ${\cal H}_{\rm f}^+$, but the Cauchy horizon ${\cal C}_{\rm f}^+$ is also a regular surface, with its coordinate value $\sigma_{\rm c} = \kappa^{-2}$ depending on the black hole rotation parameter. Across ${\cal C}_{\rm f}^+$, the hypersurfaces change character as and they all accumulate into singular point as $\sigma \rightarrow \infty$$(r\rightarrow 0)$. Right Panel: In the Cauchy fixing gauge, the Cauchy horizon is fixed at $\sigma \rightarrow \infty$ and all the slices accumulate into the singular point without crossing ${\cal C}_{\rm f}^+$. This accumulation into a singular point as $\sigma \rightarrow \infty$ in both cases leads to a violation of conditions \ref{['eq:cond_Allternative_Heun']} in the confluent Heun equation.