Integrability of the Dirac equation on backgrounds that are the direct product of bidimensional spaces

Abstract

The field equation for a spin 1/2 massive charged particle propagating in spacetimes that are the direct product of 2-dimensional spaces is separated. Moreover, we use this result to attain the separability of the Dirac equation in some specific static black hole solutions whose horizons have topology R × S 2 × · · · × S 2 .

Quasinormal modes in generalized Nariai spacetimes

“…Nonetheless, we should note that the spacetime considered here is the direct product of two-dimensional spaces, which is exactly the class of spaces studied in Ref. [47]. Indeed, the main goal of this reference is to show that the Dirac equation is separable in such backgrounds.…”

“…Following the procedure of Ref. [47], we must introduce an orthonormal frame of vector fields, which in the case of our background is given by…”

“…it follows from Ref. [47] that the component Ψ s1 1 (t, r ) obeys the following differential equation…”

“…The separation of the Dirac equation for spacetimes that are the direct product of 2-dimensional spaces has been done at Ref. [47]. Applying Eq.…”

Quasinormal Modes of Dirac Field in Generalized Nariai Spacetimes

“…In this chapter, we consider a higher-dimensional generalization of the charged Nariai spacetime [31], namely, dS 2 Â S 2 Â … Â S 2 , and investigate the dynamics of perturbations of the electrically charged Dirac field (spin 1/2). In such a geometry, the spinorial formalism [32][33][34] is used to show that the Dirac equation is separable [35] and can be reduced to a Schrödinger-like equation [36] whose potential is contained in the Rosen-Morse class of integrable potentials, which has the so-called Pöschl-Teller potential as a particular case [37,38]. Finally, the boundary conditions leading to QNMs are analyzed, and the quasinormal frequencies (QNFs) are analytically obtained [5,39].…”