Abstract
We examine knotted solutions, the most simple of which is the “Hopfion”, from the point of view of relations between electromagnetism and ideal fluid dynamics. A map between fluid dynamics and electromagnetism works for initial conditions or for linear perturbations, allowing us to find new knotted fluid solutions. Knotted solutions are also found to be solutions of nonlinear generalizations of electromagnetism, and of quantum-corrected actions for electromagnetism coupled to other modes. For null configurations, electromagnetism can be described as a null pressureless fluid, for which we can find solutions from the knotted solutions of electromagnetism. We also map them to solutions of Euler's equations, obtained from a type of nonrelativistic reduction of the relativistic fluid equations.
Highlights
Solutions with knotted topological structures play an important role in various areas of physics, but in this paper we will concern ourselves with two, electromagnetism and fluid dynamics
The theory of knots was developed in the 19th century based on knotted fluid lines, whose topological robustness was already discovered by Lord Kelvin, following the work of Helmholtz in 1858
To characterize the non-trivial topology of electromagnetic fields, a common set of observables are the helicities, that give a measure of the mean value of the linking number of the electromagnetic field lines
Summary
Solutions with knotted topological structures play an important role in various areas of physics, but in this paper we will concern ourselves with two, electromagnetism and fluid dynamics. There are null solutions, E2 − B2 = 0, as well as generically non-null solutions in Rañada’s construction, both of which are explicitly time dependent. Moffat [14] defined a “helicity” Hv for the fluid flow similar, as we will see, to a magnetic helicity for electromagnetism, and wrote some explicit solutions with Hv = 0. The properties of these were studied in [15,16]. We will use connections between electromagnetism and ideal fluid dynamics to find both new knotted solutions in electromagnetism, as well as new (time dependent) knotted solutions in fluid dynamics, that we believe have not been written explicitly before
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