Change Of Reference Frame Research Articles

An analysis of students’ misconceptions on Special Relativity

Special Relativity is one of the key theories describing our reality but its accommodation among students at different level is still a critical issue. Even after instruction, students’ answers continue to be biased by Classical Mechanics. We present the analysis of high-school students’ answers to open questions concerning topics on Classical Mechanics and Special Relativity showing the persistence of pre-relativistic reasoning. This study is part of an experimentation on the teaching of Special Relativity with the use of a mechanical instrument that allows students to explore by hand the effects of a change of reference frame.

Chaotic coexistence of librational and rotational dynamics in the averaged planar three-body problem

Through an appropriate change of reference frame and rescalings of the variables and the parameters introduced, the Hamiltonian of the three-body problem is written as a perturbed Kepler problem. In this system, new Delaunay variables are defined and a suitable configuration of the phase space and the mass parameters is chosen. In such a system, wide regions of librational and rotational motions where orbits are regular and stable are found. Close to the separatrix of these regions, the existence of chaotic motions presenting a double rotational and librational dynamics is proved, numerically, through Poincaré sections and the use of FLI.

The Bulk Flow Motion and the Hubble-Lemaître Law in the Local Universe with the ALFALFA Survey

The knowledge of the main features of the bulk flow in the Local Universe is important for a better determination of the relative motions there, an information that would contribute to a precise calculation of the Hubble-Lemaître law at very low redshifts. We study how to obtain the Hubble-Lemaître law in two sky regions using the catalogue of HI sources of the ALFALFA survey, with data $$cz_{\odot } < 6000$$ km/s. Our methodology aims to compute $$H_0$$ in two regions — located in opposite galactic hemispheres — mapped by the ALFALFA survey, and look for dependence with distance, direction, and also test for reference frame changes. We calculate the Hubble constant, in the Cosmic Microwave Background reference frame, in opposite galactic hemispheres: $$H_0^N = 70.87 \pm 2.38$$ and $$H_0^S = 66.07 \pm 3.02$$ , which allows us to measure the bulk flow velocity $$V_{\mathrm {BF}} = 401.06 \pm 150.55$$ km/s at the effective distance $$31.3 \pm 6.26$$ Mpc, a novel result found analyzing the ALFALFA data at low redshift. We confirm the influence of the bulk flow on the structures of the Local Universe which manifests through a dipolar behavior of the Hubble constant in opposite hemispheres.

The Frenet Frame as a Generalization of the Park Transform

The paper proposes a generalization of the Park transform based on the Frenet frame, which is a special set of coordinates defined in differential geometry for space curves. The proposed geometric transform is first discussed for three dimensions, which correspond to the common three-phase circuits. Then, the expression of the time derivative of the proposed transform is discussed and the Frenet-Serret formulas and the Darboux vector are introduced. The change of reference frame and its differentiation based on Cartan's moving frames and attitude matrices are also described. Finally, the extension to circuits with more than three phases is presented. The features of the Frenet frame are illustrated through a variety of examples, including a case study based on the IEEE 39-bus system.

Teaching planetary retrograde motion using a local positioning system

For millennia, humans have observed planetary retrograde motion and attempted to explain the phenomena. Our modern understanding of the motions of bodies within the Solar System is built up from a rich history of past models, each of which tried to explain astronomical observations while also being influenced by the contemporary belief system. One of the pivotal observations that assisted astronomers in the development of our current planetary model is planetary retrograde motion. Complexities in heliocentric and geocentric planetary models coupled with the change of reference frame required to visualize retrograde can be challenging for students. For this reason, we developed a kinaesthetic learning activity for introductory physics and astronomy courses that requires students to physically walk planetary orbits and allows them to see apparent motions across a virtual sky in real-time. In this paper, we present multiple activities using a local positioning system that are based on changing historical models for the retrograde phenomenon. We aim for these activities to culminate in students’ holistic understanding of how astronomically observed retrograde patterns were explained by models stemming from the existing scientific knowledge and beliefs available at the time.

Free fall of a quantum many-body system

The quantum version of the free fall problem is a topic often skipped in undergraduate quantum mechanics courses, because its discussion usually requires wavepackets built on the Airy functions—a difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and for general many-body systems by making use of a gauge transformation that corresponds to a change of reference frame from the laboratory frame to the one comoving with the falling system. Using this approach, the quantum mechanics problem of a particle in an external gravitational potential reduces to a much simpler one where there is no longer any gravitational potential in the Schrödinger equation. It is instructive to see that the same procedure can be used for many-body systems subjected to an external gravitational potential and a two-body interparticle potential that is a function of the distance between the particles. This topic provides a helpful and pedagogical example of a quantum many-body system whose dynamics can be analytically described in simple terms.

Local orbital formulation of the Floquet theory of projectile electronic stopping

A recently proposed theoretical framework for the description of electronic quantum friction for constant-velocity nuclear projectiles traversing periodic crystals is here implemented using a local basis representation. The theory requires a change of reference frame to that of the projectile, and a basis set transformation for the target basis functions to a gliding basis is presented, which is time-periodic but does not displace in space with respect to the projectile, allowing a local-basis Floquet impurity-scattering formalism to be used. It is illustrated for a one-dimensional single-band tight-binding model, as the simplest paradigmatic example, displaying the qualitative behaviour of the formalism. The time-dependent non-orthogonality of the gliding basis requires care in the proper (simplest) definition of a local projectile perturbation. The Fermi level is tilted with a slope given by the projectile velocity, which complicates integration over occupied states. It is solved by a recurrent application of the Lippmann-Schwinger equation, in analogy with previous non-equilibrium treatment of electron ballistic transport. Aiming towards a first-principles mean-field-like implementation, the final result is the time-periodic particle density in the region around the projectile, describing the stroboscopically stationary perturbation cloud around the projectile, out of which other quantities can be obtained, such as the electronic stopping power.

Viscous Shock Solutions to the Stochastic Burgers Equation

We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock solutions from solutions initially connecting two constants, as time goes to infinity.

Galilean relativity for Brownian dynamics and energetics

We study experimentally the impact of inertial reference frame changes on overdamped Brownian motion. The reference frame changes are implemented by inducing, with a laser, laminar convection flows in a column of fluid where Brownian microspheres are dispersed. The convection flow plays the role of the relative velocity between the laboratory and the fluid comoving frames, and enables us to analyse the consequences of Galilean transformations on Brownian diffusion. We verify in particular how the Brownian dynamics remains ‘weakly’ Galilean invariant, in agreement with recent discussions Cairoli et al (2018 Proc. Natl Acad. Sci. USA 115 5714). We also carefully look at the consequences of Galilean relativity on the Brownian energetics. This leads us to derive a Galilean invariant expression of the stochastic thermodynamic first law, consistent with existing theoretical results Speck et al (2008 Phys. Rev. Lett. 100 178302). We finally discuss a potential ambiguity of the Galilean relativistic features of diffusive systems that has obvious practical implications in the context of force measurements in external flows.

Quantum mechanics for non-inertial reference frames

In quantum mechanics courses, the concept of a reference frame is frequently hidden. In this article, we present a basic perspective to introduce the transformations of a reference frame based on unitary transformations. These unitary transformations are inspired in canonical transformation for reference frame from the perspective of classical mechanics. The manner we deal with this change in reference frame allows us to discuss some fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. These unitary transformations are inspired in a reference-frame canonical transformation from the perspective of classical mechanics. The approach we take for this reference-frame change allows for a discussion on fundamental concepts in Schrödinger and Heisenberg pictures, as well as the ideas of passive and active description in quantum mechanics. We apply this unitary transformation in five examples.

Topological modes in relativistic hydrodynamics

We show that gapless modes in relativistic hydrodynamics could become topologically nontrivial by weakly breaking the conservation of energy momentum tensor in a specific way. This system has topological semimetal-like crossing nodes in the spectrum of hydrodynamic modes that require the protection of a special combination of translational and boost symmetries in two spatial directions. We confirm the nontrivial topology from the existence of an undetermined Berry phase. These energy momentum non-conservation terms could naturally be produced by an external gravitational field that comes from a reference frame change from the original inertial frame, i.e. by fictitious forces in a non-inertial reference frame. This non-inertial frame is the rest frame of an accelerating observer moving along a trajectory of a helix. This suggests that topologically trivial modes could become nontrivial by being observed in a special non-inertial reference frame, and this fact could be verified in laboratories, in principle. Finally, we propose a holographic realization of this system.

Quantum reference frames for general symmetry groups

A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry groupG, we define a general operator for reversibly changing between quantum reference frames associated to a groupG. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are `in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations ofG. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry groupGis a semi-direct productG=N⋊Por a direct productG=N×P, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.

Symmetries of the Dirac quantum walk and emergence of the de Sitter group

A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non-linear realization of SO+(2,1)⋉R3; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2 + 1 dimensional de Sitter group. This group also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the limit of small wave-vectors.

Thermal activation barriers for creation and annihilation of magnetic droplet solitons in the presence of spin transfer torque

We study noise-induced creation and annihilation of magnetic droplet solitons in experimental parameter regions in which they are linearly stable against drift. Exploiting the rotational symmetry of the problem, we transform to the reference frame rotating with the droplet soliton and introduce an effective potential energy that accounts for the work done against spin-transfer torque to rotate the magnetization between two different orientations. We use this function to compute the activation barrier in both directions between the uniform magnetization state and the droplet soliton state for a variety of nanocontact radii and currents. We investigate droplet soliton structures with both zero and nonzero spin-torque asymmetry parameter. Our approach can be applied to estimate activation barriers for dynamical systems where non-gradient terms can be absorbed by changes of reference frames, and suggests a technique applicable to extended systems that may not be uniformly magnetized.

Dynamic mode decomposition based analysis of flow over a sinusoidally pitching airfoil

Dynamic mode decomposition (DMD) has proven to be a valuable tool for the analysis of complex flow-fields but the application of this technique to flows with moving boundaries is not straightforward. This is due to the difficulty in accounting in the DMD formulation, for a body of non-zero thickness moving through the field of interest. This work presents a method for decomposing the flow on or near a moving boundary by a change of reference frame, followed by a correction to the computed modes that is determined by the frequency spectrum of the motion. The correction serves to recover the modes of the underlying flow dynamics, while removing the effect of change in reference frame. This method is applied to flow over sinusoidally pitching airfoils, and the DMD analysis is used to derive useful insights regarding flow-induced pitch oscillations of these airfoils.

Vortices with massive cores in a binary mixture of Bose-Einstein condensates

We analyze a notable class of states relevant to an immiscible bosonic binary mixture loaded in a rotating box-like circular trap, i.e. states where vortices in one species host the atoms of the other species, which thus play the role of massive cores. Within a fully-analytical framework, we calculate the equilibrium distance distinguishing the motion of precession of two corotating massive vortices, the angular momentum of each component, the vortices healing length and the characteristic size of the cores. We then compare these previsions with the measures extracted from the numerical solutions of the associated coupled Gross-Pitaevskii equations. Interestingly, making use of a suitable change of reference frame, we show that vortices drag the massive cores which they host thus conveying them their same motion of precession, but that there is no evidence of tangential entrainment between the two fluids, since the cores keep their orientation constant while orbiting.

Regularities in the transformation of the oscillating decay rate in moving unstable quantum systems

Decay laws of moving unstable quantum systems with oscillating decay rates are analyzed over intermediate times. The transformations of the decay laws at rest and of the intermediate times at rest, which are induced by the change of reference frame, are obtained by decomposing the modulus of the survival amplitude at rest into purely exponential and exponentially damped oscillating modes. The mass distribution density is considered to be approximately symmetric with respect to the mass of resonance. Under determined conditions, the modal decay widths at rest, $\Gamma_j$, and the modal frequencies of oscillations at rest, $\Omega_j$, reduce regularly, $\Gamma_j/\gamma$ and $\Omega_j/\gamma$, in the laboratory reference frame. Consequently, the survival probability at rest, the intermediate times at rest and, if the oscillations are periodic, the period of the oscillations at rest transform regularly in the laboratory reference frame according to the same time scaling, over a determined time window. The time scaling reproduces the relativistic dilation of times if the mass of resonance is considered to be the effective mass at rest of the moving unstable quantum system with relativistic Lorentz factor $\gamma$.

Transformation of intermediate times in the decays of moving unstable quantum systems via the exponential modes

The transformation of canonical decay laws of moving unstable quantum systems is derived from the basic principles of quantum theory and special relativity by approximating, over intermediate times, the decay laws at rest with superpositions of exponential modes via the Prony analysis. The survival probability , which is detected in the laboratory reference frame where the unstable system moves with constant linear momentum p , is represented by the transformed form of the survival probability at rest . The transformation of the intermediate times, which is induced by the change of reference frame, is obtained by evaluating the function . Under determined conditions, this function grows linearly and the survival probability transforms, approximately, according to a scaling law over an estimated time window. The relativistic dilation of times holds, approximately, over the time window if the mass of resonance of the mass distribution density is considered to be the effective mass at rest of the moving unstable quantum system.

Azimuthal-mode solutions of two-dimensional Euler flows and the Chaplygin–Lamb dipole

Exact solutions for multipolar azimuthal-mode vortices in two-dimensional Euler flows are presented. Flow solutions with non-vanishing far-field velocity are provided for any set of azimuthal wavenumbers $m$ and arbitrary number $n$ of vorticity shells. For azimuthal wavenumbers $m=0$ and $m=1$, the far-field velocity is a rigid motion and unsteady flow solutions with vanishing far-field velocity are obtained by means of a time-dependent change of reference frame. Addition of these first two modes, in the case of $n=1$, results in a particular Chaplygin–Lamb (C–L) dipole, with continuous and vanishing vorticity at the vortex boundary. Numerical simulations suggest that this particular C–L dipole is stable.

Relativistic quantum optics: The relativistic invariance of the light-matter interaction models

In this article we discuss the invariance under general changes of reference frame of all the physical predictions of particle detector models in quantum field theory in general and, in particular, of those used in quantum optics to model atoms interacting with light. We find explicitly how the light-matter interaction Hamiltonians change under general coordinate transformations, and analyze the subtleties of the Hamiltonians commonly used to describe the light-matter interaction when relativistic motion is taken into account.