Force-free Electrodynamics and Foliations in an arbitrary Spacetime

Abstract

In this paper we formulate the relationship between force-free electrodynamics and foliations. The background metric, is considered predetermined and electrically neutral, but otherwise arbitrary. As it turns out, solutions to force-free electrodynamics is intimately connected to the existence of foliations of a spacetime with prescribed properties. We also prove a local existence and uniqueness theorem and provide a recipe for constructing the unique solution/class of solutions when certain conditions are met. We clarify the theorem with examples. We are also able to also prove a singularity theorem for when non-null solutions approach the null limit. Here too, we construct an explicit example to illustrate the singularity theorem.

The Null and Force-Free Electromagnetic Field

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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.

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“…We shall refer to such a chart as a field sheet adapted chart for F . As mentioned in my previous work ( [1]), there is no preference here for a timelike coordinate, and so we label the adapted coordinates with indices ranging from 1 − 4, rather than the usual 0 − 3. In the adapted chart, define quantities M r = g r3 g 34 − g 33 g r4 , and N r = g r3 g 44 − g 34 g r4 , for r = 1 − 4.…”

“…In the adapted chart, define quantities M r = g r3 g 34 − g 33 g r4 , and N r = g r3 g 44 − g 34 g r4 , for r = 1 − 4. It was shown in [1] that the equations of force-free electrodynamics are given by…”

“…A theoretical framework, however, will guide the search for new solutions. As shown in [1], force-free electrodynamics is completely determined by gravity (geometry) alone. In the non-null case, (i.e., when the field is electrically or magnetically dominated), picking initial conditions is restricted to a choice of an integration constant.…”

“…This work begins with a geometric recasting of the recent paper, [1], that describes the structural aspects of force-free electrodynamics in albeit an adapted coordinate chart. The adapted chart had the advantage of simplifying the relevant partial differential equations of force-free electrodynamics in a way one can study initial data surfaces, and the evolution of the field.…”