A Spacetime Interpretation of the Confluent Heun Functions in Black Hole Perturbation Theory
Abstract
In Black Hole Perturbation Theory (BHPT), confluent Heun functions emerge as solutions to the radial Teukolsky equation, which governs the dynamics of perturbations in black hole spacetimes. While these functions have traditionally been studied for their analytic properties, their connection to the underlying spacetime geometry has been less explored. In this talk, I will present a spacetime interpretation of the confluent Heun functions, showing how their distinctive behaviour near their singular points reflects the structure of key surfaces in Kerr spacetimes. By interpreting homotopic transformations of the confluent Heun functions as changes in the spacetime foliation, I will establish a connection between these solutions and various regions of the black hole’s global structure. Additionally, I will examine the relationship between these solutions and the hyperboloidal formulation of the Teukolsky equation. Although confluent Heun functions are not directly tied to hyperboloidal slices, I will show that the spacetime framework retains a structure that can be analysed using methods akin to those employed for Heun functions.
