The Null and Force-Free Electromagnetic Field

Abstract

In this paper we describe the electrodynamics of a null and force-free field in completely geometric terms. As was previously established in [1], solutions to force-free electrodynamics are governed by the existence of certain special types of foliations of spacetime. Here we prescribe the nature of the foliations in a coordinate free formalism in the null case. Finally, we illustrate all of our general results by constructing a null, force-free electrodynamic field in an Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. II. THE FORCE-FREE ELECTROMAGNETIC FIELDAs per general relativity, spacetime is a 4-dimensional smooth manifold M endowed with a metric g of Lorentz signature which we choose as (−1, 1, 1, 1). In this work we will consider the metric as fixed and predetermined. The only restriction we place is that the background metric is free of any electromagnetic contribution. The electromagnetic field tensor can be written as a 2-form F which satisfies the following Maxwell's equations:andHere * is the Hodge-Star operator and d is the exterior derivatives on forms. Also, j denotes the current density

The null and force-free electromagnetic field

“…The original derivation of the class of solutions presented here can be found in [16]. In [20] we showed that null foliations of spacetime admitting an equipartition of null mean curvature allows for a class of null force-free solutions. However, in the non-null case this is not true.…”

“…It was also clear that null and non-null solutions to FFE belonged to two very different categories of foliations. In [20], the theory of null and force-free fields and its connection to foliations of spacetime was formulated in a geometric and covariant manner. This works, then, completes the task of recasting the remaining non-null FFE and its correspondence with foliations in a covariant formalism.…”

“…By the appearance of eqs. ( 12) and ( 26) it might appear that we may obtain a dual solution under a mild restriction as in case of the null and force-free electromagnetic field (see [20]). This is true here as well, however, as we shall see, this happens only in the vacuum case.…”

“…In a recent paper ( [20]), the results of [19] was completely rewritten in geometric terms for the case of the null and force-free field. Here, it was shown that the leaves of the foliation contain a unique null pregeodesic, and the properties of the null mean curvature of the null congruence determined whether the foliation would allow a null and force-free solution; but when it does, the solutions came as a class of solutions with exactly two free parameters.…”

“…This paper is written in the vein of [19] and [20]. Its sole purpose is to develop a deeper understanding of the theory of FFE.…”