The linearised conformal Einstein field equations around a Petrov-type~D spacetime: the conformal Teukolsky equation

TL;DR

This paper develops a conformal formulation of black-hole perturbation theory based on the conformal Einstein field equations, deriving a regularized conformal Teukolsky equation that connects traditional and conformal approaches.

Contribution

It introduces a conformal Teukolsky equation within the CEFE framework, linking curvature-based perturbation theory with conformal methods around Petrov-type D spacetimes.

Findings

Derives a regular conformal Teukolsky equation decoupled from the conformal factor.

Establishes a geometric interpretation of hyperboloidal master variables.

Provides a conformal representation of Kerr spacetime with spatial infinity as a blown-up cylinder.

Abstract

While the Teukolsky equation plays a central role in traditional treatments of perturbations of algebraically special spacetimes, its relation to Friedrich's conformal Einstein field equations (CEFEs) remains largely unexplored. Here we develop a conformal formulation of black-hole perturbation theory based on the CEFEs and derive the conformal Teukolsky equation. Starting from a transparent review of Friedrich's regularisation strategy, this work establishes a direct connection between mainstream curvature-based linear perturbation theory and conformal formulations of general relativity. This perspective is timely given the growing relevance of hyperboloidal frameworks in black-hole perturbation theory, where conformal compactification is introduced at the level of an already linearised effective wave equation. Here instead, the conformal factor is a dynamical variable within the field equations. In the non-linear equations there is a coupling between conformal and curvature perturbations; however, when linearised around a Petrov-type D background, the conformal factor decouples from the equations governing the Newman-Penrose components $\phi_0$ and $\phi_4$ of the rescaled Weyl tensor. The resulting equation preserves the structural form of the classical Teukolsky equation while remaining regular at the conformal boundary. This provides a geometric interpretation of the hyperboloidal master variable and an entry point into the CEFE framework. We further derive the conformal Teukolsky equation for a conformal representation of Kerr spacetime where spatial infinity is realised as a blown-up cylinder. By bridging conformal and traditional approaches to black-hole perturbation theory, the framework highlights a geometrically regular representation of perturbative dynamics that may inform extensions beyond the linear regime.