  In the canonical model of a pulsar, rotational energy is transmitted through
the surrounding plasma via two electrical circuits, each connecting to the star
over a small region known as a "polar cap." For a dipole-magnetized star, the
polar caps coincide with the magnetic poles (hence the name), but in general,
they can occur at any place and take any shape. In light of their crucial
importance to most models of pulsar emission (from radio to X-ray to wind), we
develop a general technique for determining polar cap properties. We consider a
perfectly conducting star surrounded by a force-free magnetosphere and include
the effects of general relativity. Using a combined numerical-analytical
technique that leverages the rotation rate as a small parameter, we derive a
general analytic formula for the polar cap shape and charge-current
distribution as a function of the stellar mass, radius, rotation rate, moment
of inertia, and magnetic field. We present results for dipole and quadrudipole
fields (superposed dipole and quadrupole) inclined relative to the axis of
rotation. The inclined dipole polar cap results are the first to include
general relativity, and they confirm its essential role in the pulsar problem.
The quadrudipole pulsar illustrates the phenomenon of thin annular polar caps.
More generally, our method lays a foundation for detailed modeling of pulsar
emission with realistic magnetic fields.
