Electromagnetic perturbations of black holes in general relativity coupled to nonlinear electrodynamics: Polar perturbations

The axial electromagnetic (EM) perturbations of the black hole (BH) solutions in general relativity coupled to nonlinear electrodynamics (NED) were studied for both electrically and magnetically charged BHs, assuming that the EM perturbations do not alter the spacetime geometry in our preceding paper [Phys. Rev. D 97, 084058 (2018)]. Here, as a continuation of that work, the formalism for the polar EM perturbations of the BHs in general relativity coupled to the NED is presented. We show that the quasinormal modes (QNMs) spectra of polar EM perturbations of the electrically and magnetically charged BHs in the NED are not isospectral, contrary to the case of the standard Reissner-Nordstr\"om BHs in the classical linear electrodynamics. It is shown by the detailed study of QNMs properties in the eikonal approximation that the EM perturbations can be a powerful tool to confirm that in the NED light ray does not follow the null geodesics of the spacetime. By specifying the NED model and comparing axial and polar EM perturbations of the electrically and magnetically charged BHs, it is shown that QNM spectra of the axial EM perturbations of magnetically (electrically) charged BH and polar EM perturbations of the electrically (magnetically) charged BH are isospectral, i.e., ${\ensuremath{\omega}}_{\mathrm{mag}}^{\mathrm{ax}}\ensuremath{\approx}{\ensuremath{\omega}}_{\mathrm{el}}^{\mathrm{pol}}$ (${\ensuremath{\omega}}_{\mathrm{mag}}^{\mathrm{pol}}\ensuremath{\approx}{\ensuremath{\omega}}_{\mathrm{el}}^{\mathrm{ax}}$).

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