Galilei Group Research Articles

Linear surface gravity waves on current for a general inertial viewer

Marine measurement instrumentation, such as free-floating wave buoys, drones, and autonomous unmanned vehicles, often propagates in different directions and velocities relative to the fluid and waves. Convention assumes that these different instrumentations provide Galilean invariant descriptions of the wave field. Herein, it is shown that Galilean invariance exists for the water wave problem only in a restricted sense. The impact of this loss of invariance is investigated using a new formulation of the water wave problem, which is generalized for both current and an arbitrary inertial viewer. In the still water limit, the boundary value problem is shown to be non-invariant under Galilean transformations. This impacts the dispersion relation and interpretation of measurements. It also explains the appearance of wave modes on current, which have no analogy on still water. These modes do not appear in a still water formulation because it is a degenerate representation exhibiting a loss of Galilean symmetries. The approach provides a more complete solution of the wave–current boundary value problem by making a clear distinction between current and viewer velocity effects. Numerical examples that demonstrate the importance of the results on calculating wave characteristics are given.

Analytical Solution for the Gross-Pitaevskii Equation in Phase Space and Wigner Function

In this work, we study symplectic unitary representations for the Galilei group. As a consequence a nonlinear Schrödinger equation is derived in phase space. The formalism is based on the noncommutative structure of the star product, and using the group theory approach as a guide a physically consistent theory is constructed in phase space. The state is described by a quasi-probability amplitude that is in association with the Wigner function. With these results, we solve the Gross-Pitaevskii equation in phase space and obtained the Wigner function for the system considered.

Hall anomaly and moving vortex charge in layered superconductors

Magnetotransport theory of layered superconductors in the flux flow steady state is revisited. Longstanding controversies concerning observed Hall sign reversals are resolved. The conductivity separates into a Bardeen-Stephen vortex core contribution, and a Hall conductivity due to moving vortex charge. This charge, which is responsible for Hall anomaly, diverges logarithmically at weak magnetic field. Its values can be extracted from magetoresistivity data by extrapolation of vortex core Hall angle from the normal phase. Hall anomalies in YBa_22Cu_33O_{7}7, Bi_22Sr_22CaCu_22O_{8-x}8−x, and Nd_{1.85}1.85Ce_{0.15}0.15CuO_{4-y}4−y data are consistent with theoretical estimates based on doping dependence of London penetration depths. In the appendices, we derive the Streda formula for the hydrodynamical Hall conductivity, and refute previously assumed relevance of Galilean symmetry to Hall anomalies.

Seven formulations of the kinematics of special relativity

We present a comprehensive discussion of the formulation of the kinematics of special relativity, i.e., the Lorentz transformation. We begin with a concise new proof that the principle of relativity implies that the transformation of event coordinates between inertial reference frames is linear. We then give a clear derivation of the pre-Lorentz transformation, which follows from the principle of relativity. We then show that the pre-Lorentz transformation and the inertial invariance of the speed of light together result in the Lorentz transformation. This, of course, is essentially the traditional formulation. We next present two additional formulations, one using Lorentz–Fitzgerald contraction and one using time dilation, instead of inertial invariance. This is reasonable since Lorentz–Fitzgerald contraction and time dilation are about as well established as and are arguably less abstract than inertial invariance, and thus may profitably be used instead of inertial invariance to complete the formulation. We then present a complete proof that the pre-Lorentz transformation and the requirement of closure upon composition together imply that the transformation is either a Galilean transformation or a generalized Lorentz transformation. This is noteworthy in that it gets ever so close to the Lorentz transformation without invoking light. In the course of this, we obtain a generalized velocity addition rule, which reduces to the velocity addition rule of special relativity. We next show that the generalized Lorentz transformation, together with inertial invariance, Lorentz–Fitzgerald contraction, and time dilation, used one at a time, yields three more formulations. We then show that the unspecified, nonzero, constant speed in the generalized Lorentz transformation can be determined without any reference to light, thereby obtaining a seventh formulation. Light plays no explicit role in the four formulations employing Lorentz–Fitzgerald contraction and time dilation and plays no role whatsoever in the seventh formulation. Thus, and this is a fact which should be strongly emphasized, the formulation of special relativity in no way depends upon the nature of electromagnetic radiation. We conclude by briefly discussing these seven formulations of the kinematics of special relativity and some associated implications.

Classical dynamics from a unitary representation of the Galilei group

We give a formulation of classical mechanics in the language of operators acting on a Hilbert space. The formulation given comes from a unitary irreducible representation of the Galilei group that is compatible with the basic postulates of classical mechanics, particularly the absence of an uncertainty principle between the position and momentum of a particle. It is shown that the theory exposed in the article contains the Koopman–von Neumann formalism of classical mechanics as a particular case.

Emergent Spacetime and Gravitational Nieh-Yan Anomaly in Chiral p+ip Weyl Superfluids and Superconductors.

Momentum transport is anomalous in chiral p+ip superfluids and superconductors in the presence of textures and superflow. Using the gradient expansion of the semiclassical approximation, we show how gauge and Galilean symmetries induce an emergent curved spacetime with torsion and curvature for the quasirelativistic low-energy Majorana-Weyl quasiparticles. We explicitly show the emergence of the spin connection and curvature, in addition to torsion, using the superfluid hydrodynamics. The background constitutes an emergent quasirelativistic Riemann-Cartan spacetime for the Weyl quasiparticles which satisfy the conservation laws associated with local Lorentz symmetry restricted to the plane of uniaxial anisotropy of the superfluid (or superconductor). Moreover, we show that the anomalous Galilean momentum conservation is a consequence of the gravitational Nieh-Yan (NY) chiral anomaly the Weyl fermions experience on the background geometry. Notably, the NY anomaly coefficient features a nonuniversal ultraviolet cutoff scale Λ, with canonical dimensions of momentum. Comparison of the anomaly equation and the hydrodynamic equations suggests that the value of the cutoff parameter Λ is determined by the normal state Fermi liquid and nonrelativistic uniaxial symmetry of the p-wave superfluid or superconductor.

Galileon scalar electrodynamics

We construct a consistent model of Galileon scalar electrodynamics. The model satisfies three essential requirements: (1) The action contains higher-order derivative terms, and obey the Galilean symmetry, (2) Equations of motion also satisfy Galilean symmetry and contain only up to second-order derivative terms in the matter fields and, hence do not suffer from instability, and (3) local U(1) gauge invariance is preserved. We show that the non-minimal coupling terms in our model are different from that of the real scalar Galileon models; however, they match with the Galileon real scalar field action. We show that the model can lead to an accelerated expansion in the early Universe. We discuss the implications of the model for cosmological inflation.

Solitonic, super nonlinear, periodic, quasiperiodic, chaotic waves and conservation laws of modified Zakharov-Kuznetsov equation in transmission line

In this paper, modified Zakharov-Kuznetsov (mZK) equation is taken into consideration. Diverse variety of solitonic structures are computed by using extended algebraic method. Sufficient conditions for the existence of the computed solutions are presented. Then, Galilean transformation is utilized to transform the considered model in the planer dynamical system. All possible forms of phase portraits with respect to the parameters are plotted. Moreover, we utilized the numerical techniques to find out the nonlinear periodic structures of the discussed model and results are shown graphically. Moreover, after deploying an extrinsic periodic force the effect of frequency is observed then sensitive analysis is applied for different initial value problems to analyze the quasiperiodic and chaotic behaviour. In addition to this, Lie point symmetries and their corresponding conservation laws are reported at the end.

Galilei proposed the principle of relativity, but not the “Galilean transformation”

Galileo Galilei was the first scientist to hypothesize a universal principle of relativity, but he neither proposed, nor limited his principle to, the so-called “Galilean transformation.” The danger in calling the classical transformation “Galilean” is that students begin to believe that Galilei's principle of relativity is somehow limited to the classical transformation and classical mechanics, thus neglecting its universality. Because Galilei's 1632 statement [Dialogue Concerning the Two Chief World Systems—Ptolemaic and Copernican, 2nd ed., translated by S. Drake (University of California Press, Berkeley, 1967), pp. 186–188] of the principle is remarkably similar to Poincaré's well accepted statement [Proceedings of the Congress of Arts and Science, Universal Exposition, St. Louis, 1904, edited by H. J. Rogers (Houghton Mifflin and Company, Boston, 1904), Vol. 1, pp. 604–622], Galilei should be given greater recognition for it. However, the so-called Galilean transformation might be better named the “Euclidean space-time transformation” since it is shown here that universal Euclidean space actually requires time to be universal, that is, t′ = t. It is not usually realized that the Euclidean transformation assumes lightspeed to be infinite. Pedagogical thought should be given to combining the Galilean principle of relativity, the Euclidean transformation and Newton's mechanics as “Ordinary Relativity” to distinguish it from “Special Relativity.” This would have pedagogical and conceptual value as a model and precursor to special relativity. Finally, it is suggested that the current use of “relativistic” is incorrect when applied exclusively to special relativity, for it implies that classical mechanics is non-relativistic.

The Triple-Continuum Mechanics of Conductors as the Basis for the Theory of Electromagnetic Waves and Conduction

A new principle of the theory of electromagnetic waves and conduction is formulated. It models a metallic conductor as a set of interacting neutral atoms, each consisting of a positively charged nucleus, some electrons bound to it, and free electrons. The macroscopic model of the conductor is three interpenetrating interacting continua: a positively charged set of nuclei, a negatively charged set of electrons bound to the nuclei, and a negatively charged set of free electrons (electron gas). Densities of charge carriers, partial displacements, and partial stresses are introduced. The balance equations for the densities of charge carriers, the equations of conservation of momentum, and the equations of state that relate the dynamic and kinematic parameters are derived. Based on the charge conservation equations and the Gauss–Ostrogradsky theorem, the equations of triple-continuum mechanics of the conductor are transformed into a system of coupled dynamic equations for the macroscopic displacements of the skeleton of bound charges, electric field strengths due to bound and free charges, and conduction current density. These equations are invariant under Galilean transformations. Ohm’s law and Maxwell’s equations follow from them as special cases.

The Rotational Kinematic Formula Viewed from a Tetrad-based Anholonomic Frame

We give a fresh look into the age-old rotational kinematic formula which was originally devised by L. Euler in the eighteenth century. Instead of some verbose explanations for its logical validity, an argument of a covariant differentiation with respect to coordinate time with a tetrad-based relativistic account in an anholonomic frame will be given as a viewpoint that is mathematically sound and self-explanatory. The familiar “ω(t) × q(t)” term is replaced by a linear combination of space-frame fields of a tetrad with ‘Ricci’s connection coefficients’ of infinitesimal generators, so as to be expressed by sumnolimits_{i,j}^3 { = 1} e(i)c{gamma ^i}j0mathop qlimits^{oj} (t) with one subscript index set to zero, ‘o’, for the time coordinate. This recognition gives a new interpretation of the “time derivative for the space set of rotated axes” as the formal covariant time derivative in this tetrad-based coordinate transformation of the four-dimensional space-time that is curved due to an implicit Galilean transformation.

Symmetries of Post-Galilean Expansions.

In this Letter we study an infinite extension of the Galilei symmetry group in any dimension that can be thought of as a nonrelativistic or post-Galilean expansion of the Poincaré symmetry. We find an infinite-dimensional vector space on which this generalized Galilei group acts and usual Minkowski space can be modeled by our construction. We also construct particle and string actions that are invariant under these transformations.

Einstein-Nordström theory as a frame for a theory of quantum gravity

The Einstein-Nordström theory is proposed as a framework for a theory of quantum gravity, dividing the space of physical reality into a part longer than the radius of a proton and described by Einstein’s classical field theory of gravitation and one smaller than the radius of a proton, for the Yang–Mills theories of elementary particles, with both of them separated by a common event horizon. This division of space is reflected into the division of classical Newtonian point particle dynamics and Helmholtz line vortex dynamics, leading to a non-dimensional number for stable vortex line configurations, as for the stability of the Karman vortex street in classical fluid dynamics, and here for the non-dimensional mass ratio, of the Higgs particle mass to the Planck mass. Setting the vortex core radius of the line vortices equal to the Planck length, Lorentz invariance is broken for energies larger than the Planck energy, whereby the underlying more fundamental symmetry is the Galilei group, establishing a preferred reference system. With the Hamiltonian operator there commuting with the particle number operator, this permits the existence of negative, as well as positive masses.

A shear-rate-dependent flow generated via magnetically controlled metachronal motion of artificial cilia.

Cilia beating is a naturally occurring phenomenon that can be utilized in fluid transport in designing several biomechanical devices. Inspired by the ubiquity of bio-fluids (which are non-Newtonian), we report the characteristics of shear-rate-dependent viscosities on fluid flow generated by the wavy propulsion of magnetic cilia. We assume that the metachronal waves of these cilia form a two-dimensional wavy channel, which is filled with generalized Newtonian Carreau liquid. Galilean transformation is employed to relate fixed and moving frames. The constitutive equations are reduced under the classical lubrication assumption. The resulting fourth-order nonlinear differential equations are solved via a perturbation approach using the stream function. The effects of four dominant fluid parameters (shear thinning/thickening, power-law index, and zero- and infinite-shear-rate viscosity), magnetic parameter (Hartmann number), and metachronal wave parameters on fluid velocity, pressure rise per wavelength, and trapping phenomenon are shown in graphical results and explained thoroughly. This study could play an advisory role in designing a magnetic micro-bot useful in the biomedical industry.

Equivalence of Paths in Galilean Geometry

In this paper, we present an explicit description of finite transcendence bases in the differential field of differential rational functions that are invariant under the action of the Galilean transformation group in a real finite-dimensional space. Necessary and sufficient conditions of the equivalence of paths in the n-dimensional Galilean space are obtained.

Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity

In this work, we consider a nonlinear Schrodinger equation with stimulated Raman scattering and quintic nonlinearity, which works as a model for the propagation of ultrashort pulse in optical fiber and also corresponds to one-dimensional anisotropic Heisenberg ferromagnetic spin chain. Introducing a Galilean transformation, we transform the model into a fifth-order complex modified KdV equation, the integrability of which has been considered in the AKNS technique framework. The N-fold Darboux transformation for the model is derived in terms of the gauge transformation of the associated $$3 \times 3$$ matrix spectral problem. As applications, abundant intriguing types of nonlinear waves are obtained in zero and nonzero boundary conditions. The dynamical evolution of these nonlinear waves can be well controlled under stimulated Raman scattering and quintic nonlinearity management. Beside that, we also find some novel and interesting conversion phenomena, where a high-order term plays a pivotal role.

Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces

In this paper, we first introduce a new function space MHθ,p whose norm is given by the ℓp-sum of modulated Hθ-norms of a given function. In particular, when θ<−12, we show that the space MHθ,p agrees with the modulation space M2,p(R) on the real line and the Fourier-Lebesgue space FLp(T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on R is globally well-posed in M2,p(R) for any p<∞, while the renormalized cubic NLS on T is globally well-posed in FLp(T) for any p<∞.In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.

Collisions in Classical Mechanics in Terms of Mass-Momentum “Vectors” with Galilean Transformations

We present the usefulness of mass-momentum “vectors” to analyze the collision problems in classical mechanics for both one and two dimensions with Galilean transformations. The Galilean transformations connect the mass-momentum “vectors” in the center-of-mass and the laboratory systems. We show that just moving the two systems to and fro, we obtain the final states in the laboratory systems. This gives a simple way of obtaining them, in contrast with the usual way in which we have to solve the simultaneous equations. For one dimensional collision, the coefficient of restitution is introduced in the center-of-mass system. This clearly shows the meaning of the coefficient of restitution. For two dimensional collisions, we only discuss the elastic collision case. We also discuss the case of which the target particle is at rest before the collision. In addition to this, we discuss the case of which the two particles have the same masses.

Gauge, canonical and Galilean transformations and the non-uniqueness of Lagrange functions

The non-uniqueness of Lagrange functions is a standard topic in classical mechanics. However, it is still troublesome in undergraduate and graduate teaching. In this paper, we show that this non-uniqueness is equivalent to gauge freedom, and thus the gauge, canonical, and Galilean transformations are equivalent. In this sense, we also discuss how to solve the problem of selecting one out of many gauge-equivalent Hamiltonians to represent energy.

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach.

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with δ-correlated noise (denoted SR-KPZ). Thus, it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using nonperturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical timescale τ, and a power-law one with a varying exponent θ. We show that for the short-range noise with any finite τ, the symmetries (the Galilean symmetry, and the time-reversal one in 1+1 dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for θ below a critical value θ_{th}, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for θ>θ_{th}, and we evaluate the θ-dependent critical exponents at this long-range fixed point, in both 1+1 and 2+1 dimensions. While the results in 1+1 dimension can be compared with previous studies, no other prediction was available in 2+1 dimension. We finally report in 1+1 dimension the emergence of anomalous scaling in the long-range phase.