Pulsar magnetospheres with parallel electric fields

It is demonstrated that the conventional force-free approach to the physics of pulsar magnetospheres contains a fundamental inconsistency, when applied to the region beyond the light cylinder. The component of electric current parallel to the magnetic field that follows from Ampere's law is incompatible with the corresponding expression derived from the appropriate particle equations of motion, for systems radiating finite electromagnetic energy. The suggested remedy, while still force-free in an extended sense, nevertheless shows that a small parallel determinate component of the electric field must exist in the plasma. In this paper, the first of a series, the extended set of equations is derived and discussed. Properties of the solutions will be considered in the later papers. (author)