  Electromagnetic waves arise in many area of physics. Solutions are difficult
to find in the general case. In this paper, we numerically integrate Maxwell
equations in a 3D spherical polar coordinate system. Straightforward finite
difference methods would lead to a coordinate singularity along the polar axis.
Spectral methods are better suited to deal with such artificial singularities
related to the choice of a coordinate system. When the radiating object is
rotating like for instance a star, special classes of solutions to Maxwell
equations are worthwhile to study such as quasi-stationary regimes. Moreover,
in high-energy astrophysics, strong gravitational and magnetic fields are
present especially around rotating neutron stars. In order to study such
systems, we designed an algorithm to solve the time-dependent Maxwell equations
in spherical polar coordinates including general relativity as well as quantum
electrodynamical corrections to leading order. As a diagnostic, we compute the
spindown luminosity expected from these stars and compare it to the classical
i.e. non relativistic and non quantum mechanical results. It is shown that
quantum electrodynamics leads to an irrelevant change in the spindown
luminosity even for magnetic field around the critical value of
$\numprint{4.4e9}$~\si{\tesla}. Therefore the braking index remains close to
its value for a point dipole in vacuum namely $n=3$. The same conclusion holds
for a general-relativistic quantum electrodynamically corrected force-free
magnetosphere.
