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Purpose This paper aims to improve the computational efficiency and to achieve high-order accuracy for the computation of helicopter rotor unsteady flows in forward flight during the industrial preliminary design stage. Design/methodology/approach The integral arbitrary Lagrangian–Eulerian form of unsteady compressible Navier–Stokes equations with low Mach number preconditioned pseudo time terms based on non-inertial frame of reference undergoing rotating and translating was derived and discretized in the framework of multi-block structured finite volume grid using three types of spatial reconstruction schemes, i.e. the third-order accurate monotonic upwind scheme for conservation laws, the fifth-order accurate weighted essentially non-oscillatory and the fifth-order accurate weighted compact nonlinear schemes. Findings The results show that the present non-inertial computational method can obtain comparable results with other methods, such as the dynamic overset method, and make sure that the higher-order spatial schemes can significantly improve the tip vortex resolution. Originality/value The computational grid used by the present method remained static during the whole unsteady computation process, with only local deformations induced by blade cyclic pitch and other operating motions, which greatly reduced the complexity of grid motion and enhanced the efficiency and robustness.

Spherical reservoirs, as objects perfect with respect to their weight, are used in spacecrafts, where thin-walled elements are joined by frames into multifunction structures. The junctions are local, which results in origination of stress concentration regions and the corresponding rigidity problems. The thin-walled elements are reinforced by frame to decrease the stresses in them. To simplify the analysis of the mathematical model of common deformation of the shell (which is a mathematical idealization of the reservoir) and the frame, the assumption that the frame axial line is inextensible is used widely (in particular, in the manual literature). The unjustified use of this assumption significantly distorts the concept of the stress-strain state. In this paper, an example of a lens-shaped structure formed as two spherical shell segments connected by a frame of square profile is used to carry out a numerical comparative analysis of the solutions with and without the inextensible frame hypothesis taken into account. The scope of the hypothesis is shown depending on the structure geometric parameters and the load location degree. The obtained results can be used to determine the stress-strain state of the thin-walled structure with an a priori prescribed error, for example, in research and experimental design of aerospace systems.

A non-iterative direct forcing immersed boundary method is presented for the strongly-coupled simulations of fluid-solid interactions. While it retains many advantages of the immersed boundary framework by Yang and Stern (2012) 30], especially the simplified field extension strategy for moving boundary treatment and the pointwise integration of hydrodynamic force using the momentum forcing term, the present approach improves upon the previous method in several aspects including optimized computational cost for a strong coupling scheme, reduced algorithm complexity for a straightforward implementation, and enhanced numerical stability for low density ratio problems. Central to these improvements is a simple intermediate step in which the velocity fields around solid bodies are predicted on temporary non-inertial reference frames attached to moving solid bodies in a one-to-one manner. This step enables the explicit inverse of the implicit equations for rigid body dynamics, thus rendering unnecessary the previous predictor-corrector scheme for iteratively adjusting the displacements and velocities of the immersed bodies until reaching a convergence. In addition, a simple, generalized procedure is developed to obtain the interpolation coefficients in a local reconstruction stencil explicitly from the geometric relationship. For verification and validation, the vortex-induced vibration of a circular cylinder and the rotational galloping of a rectangular body are considered first; then several particulate flow problems, including settling and buoyant particles of low density ratios, a settling particle in a small container, and the kissing-drafting-tumbling problem of two settling particles, are studied. The agreement between the present results and the reference data in the literature is excellent. An overall second-order accuracy of the algorithm is verified in two systematic grid convergence tests. The present idea can be easily applied to similar methods for achieving a strong coupling scheme on top of a weak one with a nominal increase in the computational cost. Details of the algorithm are provided to facilitate its implementation in other solvers using non-boundary-conforming grids.

The paper studies the motion of the Foucault Pendulum in a rotating non-inertial reference frame and provides a closed form vector solution determined by vector and matrix calculus. The solution is determined through vector and matrix calculus in both cases, for both forms of the law of motion (for the Foucault Pendulum Problem and its “Reduced Form”). A complex vector which transforms the motion equation in a first order differential equation with constant coefficients is used. Also, a novel kinematic interpretation of the Foucault Pendulum motion is given.

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.

In ergonomics studies, linked segment models are commonly used for estimating dynamic L5/S1 joint moments during lifting tasks. The kinematics data input to these models are with respect to an arbitrary stationary reference frame. However, a body-centered reference frame, which is defined using the position and the orientation of human body segments, is sometimes used to conveniently identify the location of the load relative to the body. When a body-centered reference frame is moving with the body, it is a non-inertial reference frame and fictitious force exists. Directly applying a linked segment model to the kinematics data with respect to a body-centered non-inertial reference frame will ignore the effect of this fictitious force and introduce errors during L5/S1 moment estimation. In the current study, various lifting tasks were performed in the laboratory environment. The L5/S1 joint moments during the lifting tasks were calculated by a linked segment model with respect to a stationary reference frame and to a body-centered non-inertial reference frame. The results indicate that applying a linked segment model with respect to a body-centered non-inertial reference frame will result in overestimating the peak L5/S1 joint moments of the coronal plane, sagittal plane, and transverse plane during lifting tasks by 78%, 2%, and 59% on average, respectively. The instant when the peak moment occurred was delayed by 0.13, 0.03, and 0.09. s on average, correspondingly for the three planes. The root-mean-square errors of the L5/S1 joint moment for the three planes are 21. Nm, 19. Nm, and 9. Nm, correspondingly. © 2013 Elsevier Ltd.

Owing to lack of geometric method for calculating frames with the shape of T, the conversion method is used to resolve the frames in which no joint has lateral displacement. It puts forward the analysis steps of the solution of the frame. The concrete process of calculation belongs to primary computation of algebraic equation. It gets analytic solution. The calculation sequence of an example of a frame is advanced.

Owing to lack of geometric method for calculating frames without forces in one direction, the conversion method is used to resolve the slope of the frames in which no joint has lateral displacement as no force in this direction. It puts forward the twins equations of the displacement and the analysis steps of the solution of the frame. The concrete process of calculation belongs to primary computation of algebraic equation. It gets analytic solution. The calculation sequence of an example of a frame is advanced.

This paper deals with formulation of dynamics of a moving flexible body in a local frame of reference. In a conventional approach the local frame is normally fixed to the corresponding body and always represents the positions and angles of the body: the positions and angles are represented by Cartesian coordinates and Euler angles or Euler parameters, respectively. The elastic degrees of freedom are expressed by, e.g. nodal coordinates in a finite element analysis, modal coordinates, etc. However, the choice of these variables as the generalized coordinates makes the resulting equations of motion extremely complicated. This is because the representation of the rotation of a body is highly non-linear and this non-linearity makes the coefficient matrices dependent on the coordinates themselves. In this paper, we propose an alternative way of treating the issue by explicitly predicting the body motions and regularly updating the local frame. First, the motion of the local frame is assumed to explicitly follow the associated moving body. Then, the equations of motion are derived in a set of generalized coordinates that express both rigid-body and elastic degrees of freedom in the local frame. These equations are solved by a time integration with a given time interval. The motion of the local frame in the interval is estimated from a prediction of the rigid-body motions. Then, the gap between the predicted and the actual motions is evaluated. Finally, the predictions are iteratively corrected by the obtained responses in the rigid-body motions so that the gap should remain within an imposed tolerance. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper the method of transmission matrix is applied for investigating planar mechanisms [5-7]. As is known, in many engineering problems, for example, in the problem of synthesis and analysis of motion, the following problem is more and more interested. There is an object (a particle or body) connected with a frame of reference, which moves with respect to another frame. It is necessary to define the configuration (the position, velocity, acceleration) of the object under consideration in the latter frame. There are two types of the frame of reference. A frame that is fixed in time named inertia frame or global frame of reference and the frame attached to each body of the system- the body frame of reference. The problem is stated now as follows: Determine the configuration in global frame of a particle fixed in any body of the system.In this paper a method is constructed for solving such a problem by means of transmission matrices.The obtained results are illustrated by examples.

For the simulation of flow around an arbitrarily moving body, an immersed boundary method is developed in a non-inertial reference frame that is fixed to the body. The Navier-Stokes equation is formulated in a conservative form such that the force terms due to the rotation and the translational and rotational accelerations are included in the nonlinear term. In order to satisfy the no-slip condition on the body surface, momentum forcing and mass source/sink are applied on the body surface or inside the body. The numerical method is based on a finite volume approach on a staggered mesh together with a fractional-step method. The present numerical method is applied to both the forced motion and fluid-structure interaction problems. In the latter, we solve fully coupled Navier-Stokes and dynamic equations for the moving body without introducing any iteration. Four different flow problems are tested and the results are in excellent agreements with previous numerical and experimental ones.

Motion of solid object in fluid is studied in moving frame of reference. Navier-Stokes (N-S) equations in a non-inertial frame of reference is coupled with the equations of linear and angular momentums for the solid object. It is shown that the numerical implementation for solving the non-inertial N-S equations is equivalent to that of the arbitrary Lagrangian-Eulerian (ALE) method. The simulation results by the two methods show reasonable agreements for a 2-D flow field including a cylinder. In the present work, a generalised convective boundary condition is constructed for non-inertial frame of reference. Also, an appropriate boundary condition is proposed for fluid phase based on a least square adjustment (LSA) method. It is shown that the LSA is essential for the non-inertial study fluid-solid interaction. Also, the non-inertial N-S equation is applied to a problem of a concave object falling in a fluid. It is shown that traditional time-advancement schemes undermine the ortho-normality of the rotation matrix, resulting in non-physical deformation of the object. However, our time-advancement procedure with quaternion is found to be robust and effective even under a violent manoeuvre of the object.

Frame indifference is intimately related to the idealization, in conventional continuum mechanics, of the surface on which the surface forces act as a two-dimensional continuum devoid of mass. In the kinetic theory of gases, the surface is effectively a layer with thickness of the order of magnitude of the mean free path. The calculated pressure matrix contains a frame dependent term that represents the Coriolis forces that emerge in this layer when the applied forces are transformed from a fixed to a rotating reference frame.

A clear comprehension of the Newtonian notion of force remains elusive to many students due to lack of experience with observations in a noninertial frame of reference. In this article is described a software that can be easily run on a desktop computer and which acquaints the student user with the Newtonian notion of force. Fundamental principles of causality and determinism in Newtonian mechanics are elucidated.

Light, when viewed as a particle, reacts in a determinable manner with reference to the gravitational potential existing within the reference frame viewed. The elementary quanta of light, expressed under the terms of Planck, and as derived via the expressions of Einstein as a particle, may not reach a speed exactly equating to the speed (electromagnetic) of light of c. Here c is viewed as an electromagnetic constancy in any gravitational frame of reference. The theory is that a relative particle of mass may not achieve the speed of light, for the energy of that particle would then equate to infinity or in that the force required allowing the relative particle to reach c would then be infinite. The theory is then totally reliant upon the tenants of what has become to be known as the Special Theory of Relativity. As per the General Theory, light would be “bent”, more or less, from one gravitational reference frame as compared to another gravitational reference frame. The theory then evolves that light, when viewed as a particle, forms a curvilinear light path through the gravitational reference frame viewed. However, until now, the light path has been solely described on a linear basis. It is the result of the theory that the light path may be described on a curvilinear basis, under the method of Lagrange. This method, or model, allows a particle of light (viewed as a projectile of mass under a constant velocity, therefore under a constant acceleration) to achieve Newton's description of the path of a projectile. Note that the following paper is applicable to a previous paper, which proposes a displacement of light within the gravitational field.

This work presents the formulation of governing equations for a curved frame. An analytic method is also presented to examine the vibration of the curved frame. The orthogonality of any two distinct sets of mode shape functions of the frame is also derived. The forced vibration of the frame owing to a moving concentrated force is examined. The strain energy characteristics of the frame induced by the moving force are investigated as well.

The theoretical foundation for the modelling of the turbulent pressure-strain correlations in rotating reference frame is presented. Based on the recently developed materially-frame indifference principle it is observed that most of the existing second-moment closures violate the Taylor-Proudman theorem in the limit of rapid rotation. It is shown that the application of the materially-frame indifference principle gives rise to the formation of a new pressure-strain correlation model which satisfies the Taylor-Proudman theorem.

The floating frame of reference formulation which is widely used in flexible multibody simulations leads to a highly non-linear inertia matrix. This matrix which exhibits a strong inertia coupling between the reference motion and the elastic deformation can be expressed in terms of a unique set of inertia shape integrals that depend on the assumed displacement field. In this paper, we demonstrate that the floating frame of reference formulation leads to the same dynamic relationship obtained using large deformation finite element formulations. This result clearly demonstrates that the floating frame of reference formulation can also be used in the large deformation analysis of constrained flexible bodies provided that no infinitesimal or finite rotations are used as nodal coordinates; instead, displacements and slopes are used to describe the element configuration. The relationship between the local and the global slopes is defined and used to establish a coordinate transformation which is used to demonstrate the equivalence of the dynamic relationships and the inertia forces obtained using two different finite element formulations. Using this equivalence relationship, a simple and systematic procedure for evaluating all the inertia shape integrals that appear in the floating frame of reference formulation from the constant mass matrix that appears in linear structural dynamics is developed. In order to develop this procedure, the concepts of the local and global shape functions are introduced. The local shape function, which is used in the floating frame of reference formulation, does not have any rigid body modes and is defined using an appropriate set of reference conditions. The global shape function, on the other hand, has a complete set of rigid body modes that can describe an arbitrary rigid body displacement. The analysis presented in this paper demonstrates that the floating frame of reference formulation does not lead to a separation between the rigid body motion and the elastic deformation, and in such a formulation, the reference motion cannot be considered as the rigid body motion of the deformable body.

Formulation of incompressible boundary layers over generally accelerating two-dimensional bodies is addressed from the viewpoint of a non-inertial reference frame. Starting from the Navier-Stokes equations written in terms of a set of arbitrarily moving coordinates, the boundary-layer equations with reference to a body-fixed frame are derived that shows the equations retain essentially the same forms as in inertial cases. Expression of the outer inviscid flow needed to complete the description of the boundary layer around an arbitrarily accelerating body can be obtained by appealing to the concept of the ‘boundary stream function’ and techniques from the complex theory. As an example, the formulation of boundary layers over thin elliptic cylinders undergoing pitching/plunging motions is presented. A comparison of the inviscid-flow responses to the two motions is also made, and the validity of the classical linearized aerodynamic theory is discussed.

This article investigates the performance of second-moment closures in a rotating reference frame and presents a new closure for the rapid pressure-strain rate correlation based on the recently developed materially frame indifference principle. It is observed here that the existing second-moment closures with appropriate near-wall treatment can adequately predict flows in a rotating channel and in an axially rotating pipe for moderate rotation rate. Analysis of the newly proposed model indicates that it is capable of reflecting flow features under strong rotation.

Is is argued that the expression for the friction tensor of a sphere immerssed in a fluid depends on the choice of the frame of reference and in particular its rotation. The relation between the friction tensor in the laboratory frame and the corotating frame which have been derived by different authors respectively is explicitly given. Memory effects are found to play a crucial role in the transformation from one frame of reference to another.

An outline is given as to how gauge transformations in a frame fiber can be interpreted as defining various types of transport of a moving frame along a path. The cases of general linear, parallel, Lorentz, and other transport groups are examined in Minkowski space-time. A specific set of frame coordinates is introduced. A number of results are obtained including a generalization of Frenet-Serret transport, an extension of Fermi-Walker transport, a relation between frame spaces and certain types of Finsler space, and a derivation of a Kaluza-Klein type metric. Frame transport in general Riemannian space-time is also discussed.

The new variational principle of Gauss's form of nonlinear nonholonomic nonpotential system relative to non-inertial reference frame is established by constructing generalized inertial potentials. Noether's theorem and Noether's inverse Theorem of the system above is presented and proved. Finally, one example is given to illustrate the application.

Given a two-parameter of three-dimensional orbits, we construct the unit tangent vector, the normal and the binormal which define the Frenet reference frame. In this frame, by writing that the force is conservative, we explicitly obtain the potential as a function of the energy along the trajectories and of its derivatives.

The derivation and linearization of equations of motion for spinning vehicles is presented in this paper. Two reference frames are used for the basis of the derivations. The non-rotating body frame and the rotating body frame are used to establish state-space representations. An eigenvalue and eigenvector analysis presents the modes of flight for each reference frame using an asymmetrical body. The non-rotating body frame equations of motion results in five roots; three stable real poles and two identical complex conjugate pairs. This frame is found to be useful for control and linearised simulation of axisymmetrical vehicles, where the control mechanism can be decoupled from the spinning body. This frame is not useful for modelling an asymmetrical spinning vehicle. The analysis of the rotating body frame equations results in six non-zero roots; three real, stable poles, two different complex conjugate pairs and a marginally unstable real pole. Two of the real poles are dependant on the reference roll angle chosen for the analysis. It was found that the rotating body frame is appropriate for controller design, where the controller input is a requested angle of attack, sideslip angle, pitch rate or yaw rate. It is not applicable to a controller with a requested pitch or yaw angle.

The motion of a rigid body (RB) is defined when the position vector, velocity, and acceleration of all points of the rigid body are defined as functions of time with respect to a fixed reference frame with the origin at O 0 . The velocity of an arbitrary point M of the rigid body, with respect to the fixed reference frame Ox 0 y 0 z 0 , is the derivative with respect to time of the position vector r 1 . The acceleration of an arbitrary point M Є (RB), with respect to a fixed reference frame O 0 x 0 y 0 z 0 , represents the double derivative with respect to time of the position vector r 1 . A reference frame that moves with the rigid body is a body fixed reference frame. The primary reference frame is a fixed reference frame or an earth fixed reference frame. The unit vectors of the primary reference frame are constant. The body fixed reference frame, Oxyz, has its origin at a point O of the rigid body and is a moving reference frame relative to the primary reference.

Publisher Summary A frame is made of straight or linear members. An indeterminate frame is obtained by adding extra restraints at the supports of a determinate frame, or by increasing the number of frame members. This chapter introduces flame analysis by IFM and the stiffness method. The treatment includes mechanical load, thermal load, and support settling. A frame member carries three internal forces: an axial force (F), a shear force (V), and a bending moment (M); and a node of a flame has three displacements: a translation, a transverse displacement, and a rotation. A flame member can be formulated by combining the action of the bar and the beam members. The analysis of frame structures is considerably more difficult than that of beam or truss. The complexity is greater because more algebraic equations are required even though the same four types of equations are used—the equilibrium equation, compatibility condition, deformation displacement relation, and force deformation relations. Further, this chapter illustrates the analysis for support settling for IFM and the stiffness method by considering the portal frame as an example. In IFM, support settling is handled by calculating an initial deformation vector and then the deformation vector.

This note presents a theoretical study on Kolmogorov's −5/3 scaling law under a change of frame of reference and, in particular, its corresponding form in a non-inertial frame of reference. First, we show that the Kolmogorov scales in the universal equilibrium range and the turbulent kinetic energy spectrum function E(κ, t) are not frame-indifferent, i.e., frame-dependent, in the sense of Truesdell and Noll (1965). Moreover, we investigate the aspects of the scaling coefficient C K under a change of frame of reference and arrive at a generalized Kolmogorov's −5/3 scaling law in a non-inertial frame of reference, which is frame-dependent and in an inertial frame of reference reduces to the form stemming from Kolmogorov's 1941 paper.

An exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time‐dependent, nongravitational acceleration and an arbitrary time‐dependent angular velocity is given. This transformation is a generalization of the Lorentz transformation and is obtained in two steps. First, the Minkowski metric is transformed under an intermediate coordinate transformation to obtain a new set of noninertial metric coefficients in which one can easily identify the Thomas precession, as well as the expression for the acceleration of the moving frame with respect to the instantaneous rest frame in terms of the acceleration as seen from a stationary inertial frame. Second, a rotation of axes is performed to absorb the Thomas precession and to add an ordinary spatial rotation. The coordinate transformation obtained by combining these effects is nonlinear, since certain terms involve time integrals, and leads to the appropriate space‐time metric for an accelerate...

This paper reexamines: (i) the analytical and physical structure of (linear, first-order) nonholonomic constraints and their associated reaction forces, and (ii) the effect of the motion of the frame of reference on the forms of these constraints and forces, in parallel with the constitutive equation theory of continuum mechanics; all the results are expressed in both particle (vectorial) and system forms. The theory clearly shows the precise way in which the acatastatic terms (in the system form of the nonholonomic constraints and in inertial system coordinates) are generated by the motion of the frame of reference relative to inertial space. The general theory is finally applied to the well-known knife-edge/sled problem in a uniformly rotating frame of reference.

Transformation equations between inertial and linearly accelerated frames of reference are derived and these transformation equations are shown to be compatible, where applicable, with those of special relativity. The physical nature of an accelerated frame of reference is unambiguously defined by means of an equation which relates the velocity of all points within the accelerated frame of reference to measurements made in an inertial frame of reference.

The dynamic response of a simple two-member frame subjected to a constant force moving across its girder with a constant velocity is discussed. After determining the orthogonality condition for the frame as well as its eigenfrequencies and normal modes, one can easily establish the dynamic response of the frame. Dynamic and static influence lines of characteristic displacements and internal forces are assessed for two different velocities of the moving force. Comparisons between dynamic and static solutions indicate that the dynamic effect of the moving load may be considerable.

The problem of relating measurements made in an inertial (laboratory) frame to measurements made in a rotating frame is attacked through the derivation from first principles of the appropriate transformation equations. These are then used to derive a metric for a rotating system in which the energy tensor is everywhere zero, i.e. a rotating massless system. This metric is used to obtain descriptions of a variety of phenomena in rotating systems. Similarities between results produced by the metric approach and results produced by the technique of associating linearly-moving Lorentz frames instantaneously with points within the rotating system are indicated. Emphasis is laid throughout upon the way in which measurements made by different techniques are interpreted and related.

A general method for attitude dynamics investigation of spinning deformable systems is presented. The stability of given equilibrium configurations is determined by use of the Liapunov technique. Necessary and sufficient conditions are obtained in most cases. The choice of the body reference frame is considered, and the motion of various frames, such as principal axes frame, mean frame, and Tisserand's frame, is investigated.

When relativistic theory amalgamates space and time into “event”, other magnitudes of classical mechanics merge or superimpose. In special relativity it is still easy to find them again by referring to the inertial frame of reference, but general relativity presents them to us as already identified in the curvatures of the Riemannian universe, which admits of no inertial frames, except local ones. It is a search for particular magnitudes and relations (such as momentum and conservation laws), physically interpretablein the classical sense, which specifies a privileged frame. Starting from this frame of reference, the present Note sets out to define the energy-momentum density and super-potentials that give rise, in the same frame, to the known integral results due to the Einstein and Freud psendotensors. Thus the snags arising out of the transformation laws of these pseudotensors are eliminated at the same time.

The energy formula quoted by Green, Ludlam and McIlveen (1966) requires an additional term if it refers to a frame moving over the Earth's surface. The correction is derived for steady motion of the frame and tested observationally. The error incurred by ignoring the motion of a weather system is discussed.

Space and force diagrams are drawn pertaining to some simple two-dimensional, minimum weight frames for particular dispositions of load, which are analogous respectively to slip-line fields and their associated hodographs for certain metal working processes. Displacement diagrams for these frames are drawn, following the well-known method suggested by Williot. 1 It is assumed that the strain in each member of a frame is ± e . A one-to-one mapping between the nodal points of a frame and the corresponding displacement field is observed and certain relevant co-ordinates are given in tabular form. A precise analogy is also established between the displacements of a minimum weight frame and the resultant forces in the corresponding slip-line field. The aim of this paper is to emphasize that the analogy used to define the layout of certain minimum weight frames directly from slip-line fields is valid only for the loaded, i.e. deflected condition of the frame. It is pointed out that for certain classes of frame the unloaded, or constructed form of the frame must be considered.

Two thought experiments are discussed which suggest, first, a geometric interpretation of the concept of a (say, vector) potential (i.e., as a kinematic quantity associated with a transformation between moving frames of reference suitably related to the problem) and, second, that, in a quantum treatment one should extend the notion of the equivalence principle to include not only the equivalence of inertial forces with suitable “real” forces, but also the equivalence of potentials of such inertial forces and the potentials of suitable real forces. The two types of cancellation are physically independent of each other, because of the Aharonov-Bohm effect. Finally, we show that the latter effect itself can be understood “geometrically” as a kinematic effect arising upon the transformation between the two reference frames.

This chapter describes the principle of special relativity. Considerations of relativity require that the uniformity of expansion of the wavefront in one inertial frame entail its uniformity in every inertial frame. The alternative would be to abandon the concept of inertial frame. The preservation of uniformity provides the valuable clue that the transformation be linear. Combining the property of linearity with the natural requirement of dimensional homogeneity that distances be transformed into distances and times into times, it follows that there can be no relative acceleration between the two frames. It is of some methodological significance that the absence of mutual acceleration need not be employed in the definition of an inertial frame, although most writers do so, but may be deduced as a consequence of the two frames being inertial. The time interval that is determined by the spreading of time through space, by spatially separated clocks which have been synchronized by means of a procedure which is appropriate to the particular reference frame in which the clocks are at rest, is called coordinate time.