Integral Equations of Fields on the Rotating Black Hole

TL;DR

This paper explains why the coefficients in the expansions of radial equations for fields around Kerr black holes are universal, by analyzing the integral equations and kernels involving hypergeometric functions.

Contribution

It demonstrates that the kernel of the integral equation can be expressed as a product of hypergeometric functions, unifying different expansion methods.

Findings

The kernel of the radial integral equation is written as a product of confluent hypergeometric functions.

The integral equation links expansions in confluent and hypergeometric functions, explaining coefficient universality.

The analysis clarifies the mathematical structure underlying solutions to the radial equations in Kerr black hole physics.

Abstract

It is known that the radial equation of the massless fields with spin around Kerr black holes cannot be solved by special functions. Recently, the analytic solution was obtained by use of the expansion in terms of the special functions and various astrophysical application have been discussed. It was pointed out that the coefficients of the expansion by the confluent hypergeometric functions are identical to those of the expansion by the hypergeometric functions. We explain the reason of this fact by using the integral equations of the radial equation. It is shown that the kernel of the equation can be written by the product of confluent hypergeometric functions. The integral equaton transforms the expansion in terms of the confluent hypergeometric functions to that of the hypergeometric functions and vice versa,which explains the reason why the expansion coefficients are universal.