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

TL;DR

This paper offers a spacetime interpretation of confluent Heun functions in black hole perturbation theory, linking their mathematical properties to the physical structure of black hole spacetimes and their global geometry.

Contribution

It introduces a novel spacetime perspective on confluent Heun functions, connecting their behavior to black hole surfaces and the hyperboloidal framework in BHPT.

Findings

Confluent Heun functions reflect key black hole surfaces near singular points.

Homotopic transformations correspond to changes in spacetime foliation.

The approach provides new insights into wave propagation and scattering in black hole spacetimes.

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.