Theory of force-free electromagnetic fields. I. General theory

A general method to deal with the (relativistic) force-free electromagnetic field is developed. We formulate the theory without assuming symmetry of the electromagnetic field configuration. Thus we can apply it to any object where the force-free approximation is justified, e.g., the pulsar magnetosphere, the black-hole magnetosphere, and the magnetosphere around the accretion disk. We describe the force-free electromagnetic field by a classical field theory. The basic variables are the Euler potentials extended to the relativistic degenerate electromagnetic field. The basic equation is given by a two-component nonlinear equation for two Euler potentials. The theory has a close connection with geometry. It is based on the concept of the flux surface. The flux surface is a geometric entity corresponding to the world sheet of the magnetic field line. We give both the covariant and the 3+1 expression of the basic equation. By the latter form, the causal development of the force-free electromagnetic field is discussed. It is shown that the theory describes the causal development of the force-free electromagnetic field self-consistently as far as $F\cdot F>\mathrm{}0\mathrm{}$. The basic equation contains arbitrariness. It does not determine the solution uniquely. Although this arbitrariness originates from the gauge freedom of the electromagnetic field, it differs from the arbitrariness in the ordinary gauge field theories. Namely, the dynamics of the Euler potentials itself does not contain arbitrariness. It appears from nonuniqueness in correspondence between the Euler potentials and the electromagnetic field. Further, we discuss the breakdown of the force-free approximation.