Classification of electromagnetic and gravitational hopfions by algebraic type

Abstract

We extend the definition of hopfions to include a class of spin-h fields and use this to introduce the electromagnetic and gravitational hopfions of different algebraic types. The fields are constructed through the Penrose contour integral transform, thus the singularities of the generating functions are directly related to the geometry of the resulting physical fields. We discuss this relationship and how the topological structure of the fields is related to the Robinson congruence. Since the topology appears in the lines of force for both electromagnetism and gravity, the gravito-electromagnetic formalism is used to analyze the gravitational hopfions and describe the time evolution of their tendex and vortex lines. The correspondence between fields of different spin results in analogous configurations based on the same topological structure. The null and type N fields propagate at the speed of light, while the non-null and type D fields radiate energy outward from the center. Finally we discuss the type III gravitational hopfion, which has no direct electromagnetic analog, but find that it still exhibits some of the characteristic features common to the other hopfion fields.

Gravitational waves carrying orbital angular momentum

“…In a recent publication in the New Journal of Physics Stephen Barnett [2] addressed the problem of such a correspondence. In the present paper we employ an alternative electromagnetic-gravitational correspondence that has been already used in previous studies of this problem in general relativity [3][4][5][6][7] and also in the construction of gravitational Hopfions-knotted gravitational waves [8][9][10]. This paradigm is built starting from the correspondence between the wave equations for photons and gravitons.…”

“…The construction of electromagnetic and gravitational waves with the HDBWP method yields also immediately the gravitational counterpart of the Hopfion studied extensively in [8][9][10]. The generating function r t , a ( ) c for the Hopfion is:…”

Ferromagnetic Switching of Knotted Vector Fields in Liquid Crystal Colloids

“…1c) assemblies into lattices with and without other defects [5]. The experimental platform we have developed may allow for probing topological transformation of solitonic field configurations driven by weak external stimuli, with potential uses ranging from modeling similar phenomena in other branches of physics, such as particle physics and cosmology [23][24][25][26][27][28][29][30][31], to applications in magneto-optic devices [32,33]. |H|.…”

Spin-Orbital Momentum Decomposition and Helicity Exchange in a Set of Non-Null Knotted Electromagnetic Fields

“…That duality, termed "electromagnetic democracy" [10], has been central in the work of knotted field configurations [5,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Related field configurations have also appeared in plasma physics [27][28][29][30], optics [31][32][33][34][35], classical field theory [36], quantum physics [37,38], various states of matter [39][40][41][42][43] and twistors [44,45].…”