  Magnetic fields inside and around neutron stars are at the heart of pulsar
magnetospheric activity. Strong magnetic fields are responsible for quantum
effects, an essential ingredient to produce leptonic pairs and the subsequent
broadband radiation. The variety of electromagnetic field topologies could lead
to the observed diversity of neutron star classes. Thus it is important to
include multipolar components to a presumably dominant dipolar magnetic field.
Exact analytical solutions for these multipoles in Newtonian gravity have been
computed in recent literature. However, flat spacetime is not adequate to
describe physics in the immediate surrounding of neutron stars. We generalize
the multipole expressions to the strong gravity regime by using a slowly
rotating metric approximation such as the one expected around neutron stars.
Approximate formulas for the electromagnetic field including frame dragging are
computed from which we estimate the Poynting flux and the braking index.
Corrections to leading order in compactness and spin parameter are presented.
As far as spindown luminosity is concerned, it is shown that frame dragging
remains irrelevant. For high order multipoles starting from the quadrupole, the
electric part can radiate more efficiently than the magnetic part. Both
analytical and numerical tools are employed.
