General relativistic electric potential drops above pulsar polar caps

We study the general relativistic electrodynamics of an isolated, rotating, magnetic neutron star. We consider the region of a neutron star magnetosphere with steady, space charge limited flow along open magnetic field lines. The explicit solutions to the Maxwell equations are obtained. Being the simplest, this model enables one to carry out analytically a general relativistic treatment, and to demonstrate the influence of the effects of General Relativity on the creation of an electric field in the aforementioned region. Of particular importance is the effect of dragging of inertial frames of reference. The incorporation of this effect leads to a new result: we demonstrated the possibility of generation of an electric field component, which is purely general relativistic in origin and is proportional to cos χ (where χ is the inclination angle of a pulsar). For an oblique rotator the characteristic magnitude of this electric field is, approximately, $$4\times {10}^{2} ({r}_\text{g}/a){P}^{1/2}$$ times stronger than that (which is proportional to sin χ) in a flat space-time limit. An interesting consequence of this analysis is that for typical pulsar parameters $$(P\simeq 1\,\text{s},\,B\simeq {10}^{12}\,\text{G})$$ the development of an electron-positron avalanche is possible at a characteristic height of approximately a stellar radius above the surface. In addition, we have pointed out that the heating of a polar cap by the return flow of positrons can be less efficient, because the power put into their acceleration is reduced by a factor of $$6\times {10}^{-3}({r}_\text{g}/a){P}^{-1/2}$$ compared to the total energy loss rate on the acceleration of electrons.