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

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.