The linearised conformal Einstein field equations around a Petrov-type~D spacetime: the conformal Teukolsky equation
Edgar Gasperin, Rodrigo Panosso Macedo, Justin Feng
TL;DR
This work builds a covariant, conformal formulation of black-hole perturbation theory by deriving a CEFE-based version of the Teukolsky equation. By promoting the conformal factor $\Xi$ to a dynamical field within Friedrich’s conformal Einstein field equations and employing Newman-Penrose formalism, the authors obtain a conformal Teukolsky equation that mirrors the classical Teukolsky equation while remaining regular at the conformal boundary. Linearising about a Petrov type D background (e.g., Kerr) reveals decoupling of the conformal perturbations from the rescaled Weyl perturbations, enabling a clean master equation for spin weights $\mathfrak{s}=\pm2$. The paper provides two concrete applications: a hyperboloidal Kerr representation where the master variable is identified with a NP component of the rescaled Weyl tensor, and a cylinder-at-spatial-infinity representation where the CEFE-Teukolsky equation reduces to transport equations along the cylinder. This framework bridges traditional BH perturbation theory with a fully covariant conformal approach, paving the way for non-linear and metric-reconstruction extensions within a regular, geometrically transparent setting.
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 $φ_0$ and $φ_4$ 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.