Conformal Weyl Tensor Dynamics and Stability Analysis in Rotating Black Hole Spacetimes: A Novel Approach to Quasinormal Mode Spectra

TL;DR

This paper introduces a new conformal invariant-based framework for analyzing the stability and quasinormal modes of rotating black holes, unifying existing formalisms and revealing new spectral features relevant to gravitational wave observations.

Contribution

The paper develops a novel conformal invariant approach that unifies perturbation formalisms and establishes new theorems on black hole stability and isospectrality, with numerical validation for Kerr black holes.

Findings

Predicts new quasinormal mode branches with frequencies up to 3.7% different in near-extremal Kerr black holes.

Establishes a conformal stability criterion based on the sign of the invariant.

Proves an isospectrality theorem for conformally related black hole spacetimes.

Abstract

We present a novel theoretical framework for analysing the stability of rotating black hole spacetimes through the conformal properties of the Weyl tensor. By introducing a new conformal invariant constructed from the electric and magnetic parts of the Weyl tensor, we derive a master equation governing perturbations that unifies the Teukolsky and Regge-Wheeler- Zerilli formalisms. Our approach reveals previously unrecognised relationships between quasinormal mode frequencies and the conformal structure of the spacetime.We prove two fundamental theorems: (i) the conformal stability criterion, which relates mode stability to the sign-definiteness of our conformal invariant, and (ii) the isospectrality theorem for conformally related black hole spacetimes. Numerical calculations for Kerr black holes demonstrate that our formalism predicts new branches in the quasinormal mode spectrum, with frequencies differing from standard predictions by up to 3.7% in the near-extremal regime. These results have significant implications for gravitational wave astronomy and tests of general relativity in the strong-field regime.