Knotted solutions for linear and nonlinear theories: Electromagnetism and fluid dynamics

Alves, Daniel W. F.; Hoyos, Carlos; Năstase, Horaţiu; Sonnenschein, Jacob

doi:10.1016/j.physletb.2017.08.063

Cited by 21 publications

(20 citation statements)

References 48 publications

(60 reference statements)

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“…In particular, embedded the same null Hopfion solution in fluid dynamics, with

P = 0

, for which the energy‐momentum tensor is

T_{μ ν} = ρ u_{μ} u_{ν},

and the velocity is null,

u^{μ} u_{μ} = 0

. As we see, in this case also we have the same condition

{T^{μ}}_{μ} = 0, {(T^{n})}^{μ}_{ν} = 0 . \forall n \geq 2 .

This suggests that we can generalize the condition to other systems.…”

Section: Symmetries and Conserved Chargesmentioning

confidence: 57%

See 1 more Smart Citation

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Năstase

Sonnenschein

2018

Fortschritte der Physik

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Knotted solutions to electromagnetism are investigated as an independent subsector of the theory. We write down a Lagrangian and a Hamiltonian formulation of Bateman's construction for the knotted electromagnetic solutions. We introduce a general definition of the null condition and generalize the construction of Maxwell's theory to massless free complex scalar, its dual two form field, and to a massless DBI scalar. We set up the framework for quantizing the theory both in a path integral approach, as well as the canonical Dirac method for a constrained system. We make several observations about the semi-classical quantization of systems of null configurations. IntroductionElectromagnetism is a free (non-selfinteracting) theory so, according to standard lore, we wouldn't expect topologically nontrivial solutions. Indeed, solitons are usually found in interacting theories, as in the case of water solitons, which started the field, with John Scott Russel's observation of a solitonic wave in a canal in Scotland. Sometimes there is a topological reason for the existence and stability of a soliton, which is the case for "kinks" in 1+1 dimensional scalar theories, vortices in 2+1 dimensional gauge theories, or monopoles in 3+1 dimesional gauge theories, for instance.But the existence of a topological constraint, of a fixed topological number, turns out to be possible even in a free theory like Maxwell electromagnetism without sources. Thus it was realized rather late that there exist solutions with a nonzero Hopf index, or "Hopfions," and the explicit solutions were written only in [1,2] by Rañada, after the early work by Trautman in [3]. The standard Hopfion solution is null in the sense of the Riemann-Silberstein (RS) vector F = E + i B, i.e. F 2 = 0, corresponding to E 2 = B 2 and E · B = 0, but there are also partially null solutions, as we will explain in the following.

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“…In particular, embedded the same null Hopfion solution in fluid dynamics, with

P = 0

, for which the energy‐momentum tensor is

T_{μ ν} = ρ u_{μ} u_{ν},

and the velocity is null,

u^{μ} u_{μ} = 0

. As we see, in this case also we have the same condition

{T^{μ}}_{μ} = 0, {(T^{n})}^{μ}_{ν} = 0 . \forall n \geq 2 .

This suggests that we can generalize the condition to other systems.…”

Section: Symmetries and Conserved Chargesmentioning

confidence: 57%

“…Note that a plane electromagnetic wave is also null, and one can apply such a procedure on it as well. In [], a connection of null electromagnetism with fluid dynamics was used in order to explore other ways of finding solutions in both theories.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Năstase

Sonnenschein

2018

Fortschritte der Physik

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“…In this paper we extend and elaborate on results already presented in the letter [27], using relations between electromagnetism and fluid dynamics to find more (time dependent) knotted solutions for both. First, we find that fluid dynamics can be rewritten in electromagnetism language, with some restrictions.…”

Section: Introductionmentioning

confidence: 68%

Knotted solutions, from electromagnetism to fluid dynamics

Alves

Hoyos

Năstase

et al. 2017

Int. J. Mod. Phys. A

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Knotted solutions to electromagnetism and fluid dynamics are investigated, based on relations we find between the two subjects. We can write fluid dynamics in electromagnetism language, but only on an initial surface, or for linear perturbations, and we use this map to find knotted fluid solutions, as well as new electromagnetic solutions. We find that knotted solutions of Maxwell electromagnetism are also solutions of more general nonlinear theories, like Born-Infeld, and including ones which contain quantum corrections from couplings with other modes, like Euler-Heisenberg and string theory DBI. Null configurations in electromagnetism can be described as a null pressureless fluid, and from this map we can find null fluid knotted solutions. A type of nonrelativistic reduction of the relativistic fluid equations is described, which allows us to find also solutions of the (nonrelativistic) Euler's equations.Comment: 36 pages, 3 figure

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“…Another problem is to study the Finsler geometries corresponding to the polynomial action which implies including the constraints in the fundamental Finsler function. New generalizations of the Rañada fields have been presented recently in the literature, see e. g. [5][6][7][8] for topological solutions in the presence of the gravitational field and [9][10][11][12][13][14] for generalization to the non-linear electrodynamics. It should be interesting to generalize the construction presented in this paper to determine the Finsler geometries associated to these systems.…”

Section: Discussionmentioning

confidence: 99%

“…The study of the topological solutions to Maxwell's equations in vacuum, firstly proposed by Trautman and Rañada in [1][2][3], has revealed so far a rich interplay between physical systems and mathematical structures which was previously unexpected in the realm of classical electrodynamics and classical field theory [4]. Since then, the subject of the topological electromagnetic fields has gain momentum with very interesting problems investigated recently, such as the existence of topological solutions of the Einstein-Maxwell theory [5][6][7][8] and of the non-linear electrodynamics [9][10][11][12][13][14]. Also, it has been shown that there are interesting mathematical structures that can be associated to the physical systems a e-mail: adina.crisan@mep.utcluj.ro b e-mail: ionvancea@ufrrj.br (corresponding author) with topological electromagnetic fields and play an important role in their dynamics, such as twistors [15], fibrations [16] and rational functions [17,18] (see for recent reviews [19][20][21]).…”

Section: Introductionmentioning

confidence: 99%

Finsler geometries from topological electromagnetism

Crișan

Vancea

2020

Eur. Phys. J. C

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We analyse the Finsler geometries of the kinematic space of spinless and spinning electrically charged particles in an external Rañada field. We consider the most general actions that are invariant under the Lorentz, electromagnetic gauge and reparametrization transformations. The Finsler geometries form a set parametrized by the gauge fields in each case. We give a simple method to calculate the fundamental objects of the Finsler geometry of the kinematic space of a particle in a generic electromagnetic field. Then we apply this method to calculate the geodesic equations of the spinless and spinning particles. Also, we show that the electromagnetic duality in the Rañada background induces a simple dual map in the set of Finsler geometries. The duality map has a simple interpretation in terms of an electrically charged particle that interacts with the electromagnetic potential and a magnetically charged particle that interacts with the dual magnetoelectric potential. We exemplify the action of the duality map by calculating the dual geodesic equation.

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Knotted solutions for linear and nonlinear theories: Electromagnetism and fluid dynamics

Cited by 21 publications

References 48 publications

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Knotted solutions, from electromagnetism to fluid dynamics

Finsler geometries from topological electromagnetism

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