The effect of vacuum polarization on the magnetic fields around a Schwarzschild black hole

It is a well known result that the effect of vacuum polarization in gravitational fields will lead to a non-minimal coupling between gravity and electromagnetism. We investigate this phenomenon further by considering the description of static magnetic field around a Schwarzschild black hole. It is found that close to the Schwarzschild horizon the magnetic fields can be strongly modified with respect to both cases of magnetic fields on flat spacetime and magnetic fields minimally coupled on curved spacetime. Under the proper sign of the non-minimal coupling parameter, $q$, the effective fields can undergo large amplifications. Furthermore, we discuss the physical meaning of the singularities that arise in the considered problem. We conclude by discussing the potential observational effects of vacuum polarization on the magnetic fields. In the case of astrophysical black holes, depending on the value of the coupling parameter, significant modifications of the magnetic near the black hole horizons are possible -- which could be used to detect the vacuum polarization effect or at least to put constraints on the values of the coupling parameter. Moreover, we show how the considered effect directly constraints the viability of primordial black holes of sizes smaller than that of the Compton wavelength for the electron, and also impacts the distribution of magnetic fields in the early Universe.