Heun equation, Teukolsky equation, and type-D metrics

TL;DR

This paper demonstrates that the separated Teukolsky equation for various massless fields in type-D vacuum backgrounds with cosmological constant can be expressed as an Heun equation, revealing a hidden mathematical structure.

Contribution

It shows that the Teukolsky equation in type-D backgrounds with cosmological constant is equivalent to an Heun equation, uncovering a new mathematical connection.

Findings

Teukolsky equation is an Heun equation in type-D backgrounds

Applicable to fields with spins s=±2, ±1, ±1/2

Provides insight into the mathematical structure of gravitational and electromagnetic perturbations

Abstract

Starting with the whole class of type-D vacuum backgrounds with cosmological constant we show that the separated Teukolsky equation for zero rest-mass fields with spin $s=\pm 2$ (gravitational waves), $s=\pm 1$ (electromagnetic waves) and $s=\pm 1/2$ (neutrinos) is an Heun equation in disguise.