Conformal Klein-Gordon Equations and Quasinormal Modes

Using conformal coordinates associated with conformal relativity—associated with de Sitter spacetime homeomorphic projection into Minkowski spacetime—we obtain a conformal Klein-Gordon partial differential equation, which is intimately related to the production of quasi-normal modes (QNMs) oscillations, in the context of electromagnetic and/or gravitational perturbations around, e.g., black holes. While QNMs arise as the solution of a wave-like equation with a Pöschl-Teller potential, here we deduce and analytically solve a conformal ‘radial’ d’Alembert-like equation, from which we derive QNMs formal solutions, in a proposed alternative to more completely describe QNMs. As a by-product we show that this ‘radial’ equation can be identified with a Schrödinger-like equation in which the potential is exactly the second Pöschl-Teller potential, and it can shed some new light on the investigations concerning QNMs.