Non-inertial Frame Research Articles

Mapping of Translating, Rotating Icebergs With an Autonomous Underwater Vehicle

This paper presents a method for mapping translating, rotating icebergs with an autonomous underwater vehicle (AUV). To map an iceberg, the AUV first circumnavigates it, collecting multibeam sonar ranges and iceberg-relative Doppler sonar velocities from the submerged iceberg surface. The primary challenge is then to estimate the trajectory of the mapping vehicle in a noninertial reference frame attached to the moving iceberg. The collected multibeam ranges may then be projected from this trajectory to form a map of the iceberg's submerged surface. The approach of the method involves identifying the iceberg-frame locations of all the Doppler sonar measurements made during circumnavigation, allowing the AUV's iceberg-relative trajectory to be computed from those locations. The measurement locations are estimated simultaneously with the trajectory of the iceberg to be most consistent with the inertial-space positions, inertial-space velocities, distances between points on the iceberg surface as measured by the Doppler sonar, and alignment of multibeam ranges measured at the beginning and end of the circumnavigation. The measurements depend nonlinearly on the modeled positions and iceberg trajectory, and the paper presents a solution formulation that deals efficiently with the nonlinearity. By incorporating iceberg-relative vehicle velocity into the estimation, the method achieves two significant advances beyond prior work by the authors. First, and most significantly, the method adds ice-relative vehicle velocity measurements (e.g., using a Doppler velocity logger). This makes the method robust to common vehicle inertial navigation errors. Second, inclusion of iceberg-relative vehicle velocity data allows for the identification of a more general model of iceberg trajectory, making the method robust to changes in iceberg translation and rotation rates. Currently, no iceberg circumnavigation data sets are available that include iceberg-relative velocity from Doppler sonar. However, this paper includes results from simulated free-drifting icebergs, and experimental results from an AUV seafloor mapping dive. The simulation data provide a moving iceberg testbed with known ground truth for the mapping results. The seafloor data provide a qualitative verification that the method works with real vehicle data.

Environmental effects on star formation in dwarf galaxies and star clusters

We develop a simple analytical criterion to investigate the role of the environment on the onset of star formation. We will consider the main external agents that influence the star formation (i.e. ram pressure, tidal interaction, Rayleigh-Taylor and Kelvin-Helmholtz instabilities) in a spherical galaxy moving through an external environment. The theoretical framework developed here has direct applications to the cases of dwarf galaxies in galaxy clusters and dwarf galaxies orbiting our Milky Way system, as well as any primordial gas-rich cluster of stars orbiting within its host galaxy. We develop an analytic formalism to solve the fluid dynamics equations in a non-inertial reference frame mapped with spherical coordinates. The two-fluids instability at the interface between a stellar system and its surrounding hotter and less dense environment is related to the star formation processes through a set of differential equations. The solution presented here is quite general, allowing us to investigate most kinds of orbits allowed in a gravitationally bound system of stars in interaction with a major massive companion. We present an analytical criterion to elucidate the dependence of star formation in a spherical stellar system (as a dwarf galaxy or a globular cluster) on its surrounding environment useful in theoretical interpretations of numerical results as well as observational applications. We show how spherical coordinates naturally enlighten the interpretation of the two-fluids instability in a geometry that directly applies to astrophysical case. This criterion predicts the threshold value for the onset of star formation in a mass vs. size space for any orbit of interest. Moreover, we show for the first time the theoretical dependencies of the different instability phenomena acting on a system in a fully analytical way.

Two often disregarded aspects of Foucaultʼs pendulum

This paper revisits the famous Foucaultʼs pendulum by highlighting two often-disregarded aspects in mechanics courses. The first one concerns the existence of a local accelerated reference frame to express the law of dynamics without the Coriolis force. The second aspect deals with the geometrical phase that appears in pendulum dynamics. This last point, which could appear banal, should be related to analogous consideration in quantum physics. It is also linked to vectorial parallel transport of the pendulum angular momentum eigenvector. Numerical simulations with MATLAB are proposed.

Non-standard Lagrangians in rotational dynamics and the modified Navier–Stokes equation

We report some of the implications of non-standard Lagrangians in rotational dynamics. After deriving a new form of the Euler–Lagrange equation from the variational principle for the case of a particle moving in a non-inertial frame and subject to a velocity constraint, we deduce the modified Coriolis force, the modified centrifugal force and the modified transverse force. We then discuss the modified equation of motion relative to Earth, the free-fall problem and the Foucault pendulum issue. We show that the modified dynamics results on a modified Navier–Stokes equation where some features were raised and discussed accordingly.

Non-inertial frames in Minkowski space-time, accelerated either mathematical or dynamical observers and comments on non-inertial relativistic quantum mechanics

After a review of the existing theory of non-inertial frames and mathematical observers in Minkowski space-time we give the explicit expression of a family of such frames obtained from the inertial ones by means of point-dependent Lorentz transformations as suggested by the locality principle. These non-inertial frames have non-Euclidean 3-spaces and contain the differentially rotating ones in Euclidean 3-spaces as a subcase. Then we discuss how to replace mathematical accelerated observers with dynamical ones (their world-lines belong to interacting particles in an isolated system) and how to define Unruh–DeWitt detectors without using mathematical Rindler uniformly accelerated observers. Also some comments are done on the transition from relativistic classical mechanics to relativistic quantum mechanics in non-inertial frames.

Dynamics of non-axisymmetric Beltrami flows

In previous works [R. González, G. Sarasua, and A. Costa, “Kelvin waves with helical Beltrami flow structure,” Phys. Fluids 20, 024106 (2008) and R. González, A. Costa, and E. S. Santini, “On a variational principle for Beltrami flows,” Phys. Fluids 22, 074102 (2010)], we analyzed the Beltrami flow structure of Kelvin waves in an ideal fluid. As a result, we were able to show an important feature of Beltrami flows: their stability for Beltrami perturbations with the same eigenvalue as the basic flow. Here, instead, we study the dynamics of Beltrami perturbations by performing a modal analysis. In the first place, we study the modes that are generated by perturbing a uniformly translating and solidly rotating basic flow. In order to simplify the analysis, we consider the non-inertial frame in which this basic flow is at rest. In the second place, we analyze a basic Beltrami flow that is stationary in the non-inertial frame considered and is perturbed with Beltrami modes. We find that the last case is only possible when the perturbation eigenvalue is the same as that of the basic Beltrami flow. This is what we have called dynamical property. In both cases, the dynamics are represented by progressive waves in the moving frame. We apply this analysis to a rotating flow in an infinite cylinder and to an axisymmetric rotating Beltrami flow in a semi-infinite cylinder. In both cases, the development of secondary Beltrami modes is possible due to the dynamical property.

Publisher’s Note: Nonrelativistic limit of quantum field theory in inertial and noninertial frames and the principle of equivalence [Phys. Rev. D84, 085018 (2011)

Publisher’s Note: Nonrelativistic limit of quantum field theory in inertial and noninertial frames and the principle of equivalence [Phys. Rev. D<b>84</b>, 085018 (2011)

Theory of stellar convection: removing the mixing-length parameter

Stellar convection is customarily described by Mixing-Length Theory, which makes use of the mixing-length scale to express the convective flux, velocity, and temperature gradients of the convective elements and stellar medium. The mixing-length scale is taken to be proportional to the local pressure scale height, and the proportionality factor (the mixing-length parameter) must be determined by comparing the stellar models to some calibrator, usually the Sun. No strong arguments exist to suggest that the mixing-length parameter is the same in all stars and at all evolutionary phases. The aim of this study is to present a new theory of stellar convection that does not require the mixing length parameter. We present a self-consistent analytical formulation of stellar convection that determines the properties of stellar convection as a function of the physical behaviour of the convective elements themselves and of the surrounding medium. This new theory is formulated starting from a conventional solution of the Navier-Stokes/Euler equations, i.e. the Bernoulli equation for a perfect fluid, but expressed in a non-inertial reference frame co-moving with the convective elements. In our formalism the motion of stellar convective cells inside convectively-unstable layers is fully determined by a new system of equations for convection in a non-local and time-dependent formalism. We obtain an analytical, non-local, time-dependent sub-sonic solution for the convective energy transport that does not depend on any free parameter. The theory is suitable for the outer convective zones of solar type stars and stars of all mass on the main sequence band. The predictions of the new theory are compared with those from the standard mixing-length paradigm for the most accurate calibrator, the Sun, with very satisfactory results.

Protecting entanglement from amplitude damping in non-inertial frames by weak measurement and reversal

Suppression of quantum decoherence is a critical issue in quantum information processing. Weak measurement can protect entanglement under decoherence in inertial frames. Here, we study how to protect entanglement in non-inertial frames, because there is no strict inertial frames, decoherence and loss of the entanglement generated by the Unruh effect will influence each other remarkably, so it is important to discover some methods to protect entanglement in non-inertial frames.

Neutrino interaction with matter in a noninertial frame

We study the system of massive and mixed neutrinos interacting with background matter moving with an acceleration. We start with the derivation of the Dirac equation for a single neutrino in the noninertial frame where matter is at rest. A particular case of matter rotating with a constant angular velocity is considered. The Dirac equation is solved and the neutrino energy levels are found for ultrarelativistic particles propagating in rotating matter. Then we generalize our results to include several neutrino generations and consider mixing between them. Using the relativistic quantum mechanics approach we derive the effective Schr\"{o}dinger equation for the description of neutrino flavor oscillations in rotating matter. We obtain the resonance condition for neutrino oscillations and examine how it can be affected by the matter rotation. We also compare our results with the findings of other authors who studied analogous problem previously.

The concept of negative mass and its application to Helium-balloon-type dynamics

We propose that by using the concept of negative mass, the dynamics of objects embedded in fluids can be considerably simplified. For example, helium balloons, whose motion can be puzzling in a non-inertial reference frame, can be easily explained using the concept of negative mass. Two experiments have been carried out to demonstrate this idea. The same conceptual idea can serve as a model for other circumstances, when it is simpler to focus our attention on the lack of matter rather than on its presence.

Waves in deep water based on the nonlinear Schrödinger equation with variable coefficients

It has been shown that progression of waves in deep water is described by the nonlinear Schrödinger equation with time-dependent diffraction and nonlinearity coefficients. Investigation of the solutions is done here in the two cases when the coefficients are proportional or otherwise. In the first case, it is shown that the water waves are traveling at time-dependent speed and are periodic waves, which are coupled to solitons or elliptic waves seen in the noninertial frames. In the inertial frames wave modulation instability is visualized. In the second case, and when the diffraction coefficient dominates the nonlinearity, water waves collapse with unbounded amplitude at finite time. Exact solutions are found here by using the extended unified method together, while presenting a new algorithm for treating nonlinear coupled partial differential equations.

Generalization of the Lagrange Equations Formalism, for Motions with Respect to Non-Inertial Reference Frames

The paper presents an original generalization of the Lagrange equations of the second kind, for motions with respect to reference frames in arbitrary motions. The traditional form of the equations is modified by adding certain supplementary terms. Calculation methods are presented for these terms and some conclusions useful in practical applications are drawn. For validation, the new formalism is illustrated on the application of the Foucault pendulum. The results are in full agreement with those obtained by traditional methods. The formalism can be used to determine the motions of various mechanical systems, with respect to large size movable bodies (such as planets or orbital stations), that are not influenced by the relative motion of the studied parts.

Fluctuations of Entanglement Entropy in Noninertial Frames

The fluctuation and relative fluctuation of entanglement entropy of a bipartite system for Dirac fields in noninertial frames are investigated. It is shown that the fluctuation and relative fluctuation of entanglement entropy are observer-dependent, which depend on their observed frames. It is found that both the fluctuation and relative fluctuation of entanglement entropy become more noticeable with the increasing of the subsystem's acceleration. We also find that the entanglement entropy always has fluctuation regardless of the initial state parameter and for any magnitude of the acceleration. We argue that the statistical mean for the measurement of entanglement entropy relates to the accelerated motion of the observer.

Discrete Lorentz covariance for quantum walks and quantum cellular automata

We formalize a notion of discrete Lorentz transforms for quantum walks (QW) and quantum cellular automata (QCA), in -dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the clock QW and the clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.

Probing the quantum entanglement under finite temperature environment in nonineritial frames

In this paper, we investigate the dynamics of quantum entanglement of a two-qubit quantum system coupled with generalized amplitude damping (GAD) channel of nonzero temperature in noninertial frames. The results show that the concurrence decreases with the increase of acceleration and channel parameter r has a decisive impact on the entanglement. Accidentally, we manifest the inequivalence of the quantization for a Dirac field under the GAD channel in the noninertial frames.

Quantum correlations of helicity entangled states in non-inertial frames beyond single mode approximation

A helicity entangled tripartite state is considered in which the degree of entanglement is preserved in non-inertial frames. It is shown that Quantum Entanglement remains observer independent. As another measure of quantum correlation, Quantum Discord has been investigated. It is explicitly shown that acceleration has no effect on the degree of quantum correlation for the bipartite and tripartite helicity entangled states. Geometric Quantum Discord as a Hilbert–Schmidt distance is computed for helicity entangled states. It is shown that living in non-inertial frames does not make any influence on this distance, either. In addition, the analysis has been extended beyond single mode approximation to show that acceleration does not have any impact on the quantum features in the limit beyond the single mode. As an interesting result, while the density matrix depends on the right and left Unruh modes, the Negativity as a measure of Quantum Entanglement remains constant. Also, Quantum Discord does not change beyond single mode approximation.

The Machian origin of linear inertial forces from our gravitationally radiating black hole Universe

Gravitational radiation of the Universe (U), accelerated with respect to linear non-inertial frames, accounts for linear inertial forces provided accessible masses, i.e. those within the Hubble radius, obey $$\tfrac{{2GM_U }} {{c^2 R_U }} = 1$$ . This implies that we must be living in a black hole Universe, whose Schwarzschild radius is determined by the Hubble constant. Thus non-inertial frames represent the probe of the existence and substantial role of gravitational waves in the Universe, accounting for part of its of dark energy content.

Three-dimensional Rayleigh–Bénard convection of molten gallium in a rotating cuboid under the influence of a vertical magnetic field

The present work deals with magnetoconvection of molten gallium in a cuboid rotating about a vertical axis passing through its center. The governing equations are derived in a non-inertial frame of reference considering both centrifugal and Coriolis forces. A vertical magnetic field is applied through the center opposite to the direction of gravity. The cuboid is heated from below and cooled at top, while the remaining walls of the cuboid are thermally insulated. The modified Marker and Cell method is adopted for the numerical solution of the governing equations. The gradient dependent consistent hybrid upwinding scheme of second order is adopted for the discretization of the convective terms in the momentum equations. The operator splitting algorithm is used for the numerical treatment of the energy equation. The effects of cavity rotation and applied magnetic field on heat and momentum transport processes have been investigated. The uniform thorough mixing of fluids by rotation and regularization of flow by magnetic field are observed. The governing flow field and temperature distribution are shown graphically to elucidate the intricate physics of the phenomenon.

Physical theories in hypercomplex geometric description

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.