Conformal Klein-Gordon equations and quasinormal modes

TL;DR

This paper derives and solves a conformal Klein-Gordon equation related to quasinormal modes in black hole perturbations, offering an analytical approach that connects conformal relativity with QNM analysis.

Contribution

It introduces a conformal Klein-Gordon equation in de Sitter spacetime and provides an analytical solution for QNMs using a Schrödinger-like equation with Poschl-Teller potential.

Findings

Derived an analytical solution for QNMs in a conformal framework.

Connected the radial equation to a Schrödinger-like equation with Poschl-Teller potential.

Provided insights into the mathematical structure of QNMs in black hole physics.

Abstract

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 Poschl-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 Schrodinger-like equation in which the potential is exactly the second Poschl-Teller potential, and it can shed some new light on the investigations concerning QNMs.