Conformal Klein-Gordon Equations and Quasinormal Modes

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 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. IntroductionQuasi-normal modes (QNM) arise in the context of general relativity as electromagnetic or gravitational perturbations occurring in the neighborhood of, e.g., Schwarzschild, Kerr, Reisnerr-Nordstrøm [5], and KerrNewman spacetimes, and their investigation in the Schwarzschild background was started in [3,4]. It is well known that no normal mode oscillations is produced in the process of emission of gravitational waves, but only quasi-normal oscillation modes, representing a oscillatory damped wave [1,2,4,6,9,14,15] while, as yet nothing is known for nonlinear stellar oscillations in general relativity [10] and in the collapse of a star to form a black hole [4]. As defined in, e.g. [2,16], QNMs are the eigenmodes of the homogeneous wave equations, describing these perturbations -with the boundary conditions corresponding to outgoing waves at the spatial infinity and incoming waves at the horizon. The paramount interests to QNMs have been mainly introduced by [8,9]. QNMs can bring imprints of black holes and can be detected in the gravitational wave framework [2].In one hand, QNMs equations can be derived if we introduce a projective approach, namely the theory of hyperspherical universes, developed by Arcidiacono[17] several years ago and, more specifically, in the socalled conformal case. When we write Maxwell equations in six dimensions, with six projective coordinates (we have, in these coordinates, a Pythagorean metric) a natural problem arises, namely, to provide a physical version of the formalism, i.e. to ascribe a physical meaning to the coordinates. For this theory, there are two possible different physical interpretations: a bitemporal interpretation and a biprojective interpretation. In the first case (bitemporal) we introduce a new universal constant c ′ and the coordinate x 5 = ic ′ t ′ where t ′ is interpreted as a second time; we thus obtain in cosmic scale the so-called multitemporal relativity, proposed by Kalitzen[18]. The set of Maxwell equations obtained in this theory generalizes the equations of the unitary theory of electromagnetism and gravitation, as proposed by Corben[19]...

Physical Effects of Extra Dimension and Concomitant Map between Photons and Gravitons in RS Brane-World Scenario

“…These corrections can also be achieved and generalized, using most general -e.g., Kerr-Newman -black holes and branes possessing different codimension [24]. Also, the relationship between mutual transformations of EM and gravitational fields can be described and investigated in terms of quasinormal modes [12,25].…”