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In the canonical model of a pulsar, rotational energy is transmitted throughthe surrounding plasma via two electrical circuits, each connecting to the starover a small region known as a "polar cap." For a dipole-magnetized star, thepolar 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 crucialimportance to most models of pulsar emission (from radio to X-ray to wind), wedevelop a general technique for determining polar cap properties. We consider aperfectly conducting star surrounded by a force-free magnetosphere and includethe effects of general relativity. Using a combined numerical-analyticaltechnique that leverages the rotation rate as a small parameter, we derive ageneral analytic formula for the polar cap shape and charge-currentdistribution as a function of the stellar mass, radius, rotation rate, momentof inertia, and magnetic field. We present results for dipole and quadrudipolefields (superposed dipole and quadrupole) inclined relative to the axis ofrotation. The inclined dipole polar cap results are the first to includegeneral 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 pulsaremission with realistic magnetic fields.

Inclined Pulsar Magnetospheres in General Relativity: Polar Caps for the Dipole, Quadrudipole and Beyond