Quantum electrodynamical corrections to a magnetic dipole in general relativity

Magnetized neutron stars are privileged places where strong electromagnetic fields as high as $\text{BQ}=4.4\times {10}^9$~T exist, giving rise to non-linear corrections to Maxwell equations described by quantum electrodynamics (QED). These corrections need to be included to the general relativistic (GR) description of a magnetic dipole supposed to be anchored in the neutron star. In this paper, these QED and GR perturbations to the standard flat space-time dipole are calculated to the lowest order in the fine structure constant~${\alpha }_{\mathrm{s}\mathrm{f}}$ and to any order in the ratio $\text{Rs}\mathrm{/}R$ where $R$ is the neutron star radius and $\text{Rs}$ its Schwarzschild radius. Following our new 3+1~formalism developed in a previous work, we compute the multipolar non-linear corrections to this dipole and demonstrate the presence of a small dipolar~$\mathrm{\ell }=1$ and hexapolar~$\mathrm{\ell }=3$ component.