Force-Free Fields are Conformally Geodesic

Abstract

In this paper, we establish an equivalence between force-free fields and conformally geodesic fields, and between harmonic fields and conformally eikonal fields in the context of conformal geometry. In contrast to previous work, our approach and equivalence results generalize to arbitrary dimensions. In accordance with three-dimensional theory, our defining equations emerge as the Euler-Lagrange equations of hierarchies of variational principles - distinguished by the topological constraints they impose - and retain the known inclusions of the special cases from each other. Specifically, we relate stationary points of hierarchies of $L^2$ resp. $L^1$-optimization problems by a conformal change of metric, provide an explicit construction of the conformal factors relating the relevant metrics and identify the field lines of physical vector fields fields as conformal geodesics. Despite the allowed topological complexity of the fields under consideration, these observations reveal geometric order which is obtained by merely pointwise rescaling of the metric.

20 pages, 8 figures

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