Fermi-Walker Transport Research Articles

Dynamics of a binary system around a supermassive black hole

We discuss the motion of a binary system around a supermassive black hole. Using Fermi-Walker transport, we construct a local inertial reference frame and set up a Newtonian binary system. Assuming a circular geodesic observer around a Schwarzschild black hole, we write down the equations of motion of a binary. Introducing a small acceleration of the observer, we remove the interaction terms between the center of mass (CM) of a binary and its relative coordinates. The CM follows the observer's orbit, but its motion deviates from an exact circular geodesic. We first solve the relative motion of a binary system, and then find the motion of the CM by the perturbation equations with the small acceleration. We show that von Zeipel-Lidov-Kozai (vZLK) oscillations appear when a binary is compact and the initial inclination is larger than a critical angle. In a hard binary system, vZLK oscillations are regular, whereas in a soft binary system, oscillations are irregular both in period and in amplitude, although stable. We find an orbital flip when the initial inclination is large. As for the motion of the CM, the radial deviations from a circular orbit become stable oscillations with very small amplitude.

Motion of a gyroscope on a closed timelike curve

We consider the motion of a gyroscope on a closed timelike curve (CTC). A gyroscope is identified with a unit-length spacelike vector - a spin-vector - orthogonal to the tangent to the CTC, and satisfying the equations of Fermi-Walker transport along the curve. We investigate the consequences of the periodicity of the coefficients of the transport equations, which arise from the periodicty of the CTC, which is assumed to be piecewise $C^2$. We show that every CTC with period $T$ admits at least one $T-$periodic spin-vector. Further, either every other spin-vector is $T-$periodic, or no others are. It follows that gyroscopes carried by CTCs are either all $T-$periodic, or are generically not $T-$periodic. We consider examples of spacetimes admitting CTCs, and address the question of whether $T-$periodicity of gyroscopic motion occurs generically or only on a negligible set for these CTCs. We discuss these results from the perspective of principles of consistency in spacetimes admitting CTCs.

Incisive Approach to Fermi-Walker Transport

A rational approach to the Fermi-Walker transport equation is proposed by deriving it from a condition of “non-rotation”. First, the condition is applied to a tetrad basis and then generalized to an arbitrary space-time four-vector. The method is conceptually simple and apart from the use of tetrad bases in four-dimensional space-time, does not require the effort of visualizing abstract geometrical constructs in spaces of more than three dimensions. The argument develops in the context of the flat space-time of special relativity theory but the final result can be immediately extended to curved space-time.

Defect production by pure phase twist injection as Aharonov-Bohm effect.

In this paper we demonstrate that new phase defects of the Gross-Pitaevskii equation (GPE) can be produced as a Aharonov-Bohm effect resulting from pure phase twist injection on existing defects. This is a phenomenon that has physical justification in the hydrodynamic interpretation of GPE. Here we give an analytical proof of its effects by using Fermi-Walker transport and Biot-Savart induction law. An analytical derivation of the dispersion relation is derived from the superposition of phase twist on the fundamental state. Since the extra twist corresponds to a topological change of the total linking number of the system, we show that the production of new defects is just another manifestation of the Aharonov-Bohm effect. We propose a laboratory experiment for Bose-Einstein condensates to test this phenomenon and to show that it can have useful applications in science and technology.

A straightforward method for deriving the Fermi–Walker transport law

A mathematically and conceptually simple approach that allows one to recover the Fermi–Walker transport law is proposed. The method is based on defining the Fermi–Walker transport law as the one for which the moving vector undergoes in its instantaneous rest frame only an infinitesimal Lorentz boost. The general Lorentz boosts are represented by 3 × 3 matrices, which significantly simplifies calculations. Also, a comment elucidating that the celebrated formula for the Thomas precession rate is not directly encapsulated in the expression representing this law is provided.

Fermi-Walker Parallel Transport According to Quasi Frame in Three Dimensional Minkowski Space

In this paper, we present the Fermi-Walker parallel transport and the generalized Fermi-Walker parallel transport according to quasi frame in three dimensional Minkowski space.

Quantification of GR effects in muon g-2, EDM and other spin precession experiments

Recently, Morishima, Futamase and Shimizu published a series of manuscripts, putting forward arguments, based on a post-Newtonian approximative calculation, that there can be a sizable general relativistic (GR) correction in the experimental determination of the muon magnetic moment based on spin precession, i.e. in muon g-2 experiments. In response, other authors argued that the effect must be much smaller than claimed. Further authors argued that the effect exactly cancels. Also, the known formulae for the de Sitter and Lense–Thirring effect do not apply due to the non-geodesic motion. All this indicates that it is difficult to estimate from first principles the influence of GR corrections in the problem of spin propagation. Therefore, in this paper we present a full general relativistic calculation in order to quantify this effect. The main methodology is the purely differential geometrical tool of Fermi–Walker transport over a Schwarzschild background. The Larmor precession due to the propagation in the electromagnetic field of the experimental apparatus is also included. For the muon g-2 experiments the GR correction turns out to be very small, well below the present sensitivity. However, in other similar storage ring experimental settings, such as electric dipole moment search experiments, where the so-called frozen spin method is used, GR gives a well detectable effect, and should be corrected for. All frozen spin scenarios are affected which intend to reach a sensitivity of 0.1 μrad s−1 for the spin precession in the vertical plane.

The Fermi–Walker Derivative and Non-rotating Frame in Dual Space

In this study, we defined Fermi–Walker derivative in dual space $$\mathbb {D}^3$$ . Fermi–Walker transport, non-rotating frame and Fermi–Walker termed Darboux vector by using Fermi–Walker derivative are given in dual space $$\mathbb {D}^3$$ . Being conditions of Fermi–Walker transport and non-rotating frame are investigated along any dual curve for dual Frenet frame, dual Darboux frame and dual Bishop frame.

Fermi–Walker transport and Thomas precession

An exact derivation of the Thomas precession formula is presented based on the Fermi–Walker transport equation. Given that the Thomas precession effect is not a particularly intuitive phenomenon, such that when discovered in 1925 it took by surprise even experts in relativity theory, Einstein included, an alternative perspective can be useful at an intermediate level for physics students. The existing literature linking the Thomas precession to Fermi–Walker transport use geometric algebra as mathematical tool. Here the mathematics is kept within the limits of the usual vector and tensor algebra commonly used in special relativity theory at a level appropriate for advanced undergraduate and beginning graduate students.

Uniform acceleration in general relativity

We extend de la Fuente and Romero's defining equation for uniform acceleration in a general curved spacetime from linear acceleration to the full Lorentz covariant uniform acceleration. In a flat spacetime background, we have explicit solutions. We use generalized Fermi-Walker transport to parallel transport the Frenet basis along the trajectory. In flat spacetime, we obtain velocity and acceleration transformations from a uniformly accelerated system to an inertial system. We obtain the time dilation between accelerated clocks. We apply our acceleration transformations to the motion of a charged particle in a constant electromagnetic field and recover the Lorentz-Abraham-Dirac equation.

Lorentz symmetry breaking effects on relativistic EPR correlations

Lorentz symmetry breaking effects on relativistic EPR (Einstein-Podolsky-Rosen) correlations are discussed. From the modified Maxwell theory coupled to gravity, we establish a possible scenario of the Lorentz symmetry violation and write an effective metric for the Minkowski spacetime. Then, we obtain the Wigner rotation angle via the Fermi-Walker transport of spinors and consider the WKB ((Wentzel-Kramers-Brillouin) approximation in order to study the influence of Lorentz symmetry breaking effects on the relativistic EPR correlations.

Wigner rotation via Fermi–Walker transport and relativistic EPR correlations in the Schwarzschild spacetime

The Wigner rotation angle for a particle in a circular motion in the Schwarzschild spacetime is obtained via the Fermi–Walker transport of spinors. Then, by applying the Wentzel, Kramers, Brillouin (WKB) approximation, a possible application of the Fermi–Walker transport of spinors in relativistic Einstein–Podolsky–Rosen (EPR) correlations is discussed, where it is shown that the spins of the correlated particle undergo a precession in an analogous way to that obtained by Terashima and Ueda [H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004)] via the application of successive infinitesimal Lorentz transformations. Moreover, from the WKB approach, it is also shown that the degree of violation of the Bell inequality depends on the Wigner rotation angle obtained via the Fermi–Walker transport. Finally, the relativistic effects from the geometry of the spacetime and the accelerated motion of the correlated particles is discussed in the nonrelativistic limit.

Inertial forces in General Relativity

An exact equation describing the inertial forces for arbitrary orthonormal frames on arbitrary spacetimes is presented. It manifests that the so-called gravitomagnetic field consists of a combination of two effects of independent origin: the vorticity of the observer congruence, plus the angular velocity of rotation, relative to Fermi-Walker transport, of the triad of spatial axes that each observer “carries” with it. Such formulation encompasses different gravitomagnetic fields in the literature. The formalism is applied to notable reference frames in Kerr spacetime, and their inertial forces shown to have familiar Newtonian analogues.

Motion4D-library extended

The new version of the Motion4D library is mainly an update to work with the four-dimensional ray tracing code GeoViS which makes full use of all the implemented metrics and geodesic integrators. New version program summaryProgram title: Motion4D-libraryCatalogue identifier: AEEX_v3_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_1.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 189793No. of bytes in distributed program, including test data, etc.: 6808546Distribution format: tar.gzProgramming language: C++.Computer: All platforms with a C++ compiler.Operating system: Linux, Windows.RAM: 61 MBytesClassification: 1.5.Catalogue identifier of previous version: AEEX_v3_0Journal reference of previous version: Comput. Phys. Comm. 182(2011)1386External routines: GNU Scientific Library (GSL) (http://www.gnu.org/software/gsl/)Does the new version supersede the previous version?: YesNature of problem:Solve geodesic equation, parallel and Fermi–Walker transport in four-dimensional Lorentzian spacetimes. Determine gravitational lensing by integration of Jacobi equation and parallel transport of Sachs basis.Solution method:Integration of ordinary differential equations.Reasons for new version:The main reason for the new version is the update of some methods to work with the four-dimensional ray tracing code GeoViS. Furthermore, some new metrics and integrators were implemented.Summary of revisions:The four-dimensional ray tracing code GeoViS [1] is based on the Motion4D library. All of the metrics and geodesic integrators of the library can be accessed by means of GeoViS’ scheme-based scripting language. For that, some methods had to be updated.In the following, a list of newly implemented metrics is given: •AlcubierreSimple: This metric uses a simplified warp bubble function compared to the original one by Alcubierre [2], see McMonigal et al. [3].•ChazyCurzonRot: The metric of the rotational Chazy–Curzon solution is taken from Stephani et al. [4].•Curzon: The Curzon metric in cylindrical coordinates is taken from Scott and Szekeres [5].•EinsteinRosenWaveWWB: A detailed discussion of the Einstein–Rosen wave with a Weber–Wheeler–Bonnor pulse can be found in Griffiths and Micciche [6].•ErezRosenVar: The original Erez–Rosen metric is quite intricate, see e.g. Krori and Sarmah [7]. Hence, we use here a reduced version with a simpler quadrupole term.•KastorTraschen: The Kastor–Traschen metric with two black holes is taken from Griffiths and Podolský [8] Furthermore, the Dormand–Prince 5(4) and 6(5) integrators taken from Guthmann [8] were implemented.Running time:The test runs provided with the distribution require only a few seconds to run.

Covariant Uniform Acceleration

We derive a 4D covariant Relativistic Dynamics Equation. This equation canonically extends the 3D relativistic dynamics equation , where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck.We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational, and general. For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion.We use Generalized Fermi-Walker transport to construct a uniformly accelerated family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K' to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame.We obtain velocity and acceleration transformations from a uniformly accelerated system K' to an inertial frame K. We introduce the 4D velocity, an adaptation of Horwitz and Piron s notion of "off-shell." We derive the general formula for the time dilation between accelerated clocks. We obtain a formula for the angular velocity of a uniformly accelerated object. Every rest point of K' is uniformly accelerated, and its acceleration is a function of the observer's acceleration and its position. We obtain an interpretation of the Lorentz-Abraham-Dirac equation as an acceleration transformation from K' to K.

Spacetime transformations from a uniformly accelerated frame

We use the generalized Fermi–Walker transport to construct a one-parameter family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the weak hypothesis of locality, we obtain local spacetime transformations from a uniformly accelerated frame K′ to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame.

Locally inertial null normal coordinates

Locally inertial coordinates are constructed by carrying Riemann normal coordinates on a codimension-2 spacelike surface along the geodesics normal to it. Since the normal tangents are labelled by components with respect to a null basis, these coordinates are referred to as null normal coordinates. They are convenient in the study of local causal horizons. As an application, the coordinate system is used to specify a vector field that satisfies the Killing equation approximately in a small region and the Killing identity exactly on a single null geodesic. We also construct a vector field on a surface, starting from a vector at a given point on the surface. This construction may be regarded as a generalization of the notion of Fermi–Walker transport.

Motion4D-library extended

Motion4D-library extended

RELATIVISTIC EINSTEIN–PODOLSKY–ROSEN CORRELATIONS IN COSMIC STRING SPACE–TIME VIA FERMI–WALKER TRANSPORT

We present a geometric approach to study the relativistic EPR correlations in curved space–time background given by the application of the Fermi–Walker transport in the relativistic EPR states and show that its result has the same effect as the applications of successive infinitesimal Lorentz boosts in the relativistic EPR states. We also show that the expression for the Bell inequality due to the Fermi–Walker transport is equivalent to the expression demonstrated by Terashima and Ueda,20 where the degree of violation of the Bell inequality depends on the angle of the Wigner rotation. This geometrical approach for study of the relativistic EPR correlations is a promising formulation to investigate the EPR correlations in the general relativity background.

Anandan quantum phase for a neutral particle with Fermi–Walker reference frame in the cosmic string background

We study geometric quantum phases in the relativistic and non-relativistic quantum dynamics of a neutral particle with a permanent magnetic dipole moment interacting with two distinct field configurations in a cosmic string spacetime. We consider the local reference frames of the observers are transported via Fermi–Walker transport and study the influence of the non-inertial effects on the phase shift of the wave function of the neutral particle due to the choice of this local frame. We show that the wave function of the neutral particle acquires non-dispersive relativistic and non-relativistic quantum geometric phases due to the topology of the spacetime, the interaction between the magnetic dipole moment with external fields and the spin–rotation coupling. However, due to the Fermi–Walker reference frame, no phase shift associated to the Sagnac effect appears in the quantum dynamics of a neutral particle. We show that in the absence of topological defect, the contribution to the quantum phase due to the spin–rotation coupling is equivalent to the Mashhoon effect in non-relativistic dynamics.