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The problem of transformation of electromagnetic field from inertial frame of reference to non-inertial rotating frame of reference and vice versa is discussed. By the use of the tetrad method (moving reper method) it is shown that in cylindrical coordinates the usual Lorentz formulas can be applied. It is also shown that in cylindrical coordinates the form of Maxwell equations in rotating frame of reference is the same as in inertial one.

In this paper we study the occurrence of accelerating universe versus decelerating universe between the F(R) gravity frame (Jordan frame) and non-minimally coupled scalar field theory frame, and the minimally coupled scalar field theory frame (Einstein frame) for various models. As we show, if acceleration is imposed in one frame, it will not necessarily correspond to an accelerating metric when transformed in another frame. As we will demonstrate, this issue is model and frame-dependent but it seems there is no general scheme which permits to classify such cases., To appear in Physics Letters B. Some corrections added

It is commonly thought that the Michelson-Morley experiment from 1887 and Kennedy-Thorndike experiment from 1932 demonstrated that the universal frame of reference (ether) does not exist and that the velocity of light in vacuum is absolutely constant. The analysis of this experiment led to the creation of the Special Theory of Relativity (STR). The article explains why Michelson-Morley and Kennedy-Thorndike experiments could not detect the universal frame of reference. In this article, a different transformation of time and position than the Lorentz transformation is derived on the basis of the geometric analysis of the Michelson-Morley and Kennedy-Thorndike experiments. The transformation is derived based on the assumption that the universal frame of reference (UFR) exists. UFR is a frame of reference in which the velocity of light is constant in every direction. In inertial frames of reference moving in the UFR, the velocity of light may be different. The article has derived the formula for relative speed and patterns for the maximum and minimum speed of light that can be measured in the inertial system. Finally, the anisotropy of the microwave background radiation has been explained by using the presented theory. According to the body kinematics model presented in this article, anisotropy of cosmic microwave background is the Doppler effect for observer moving in the UFR.

The mechanical energy change of a system in an inertial frame of reference equals work done by the total nonconservative force in the same frame. This relation is covariant under the Galilean transformations from inertial frame S to S′, where S′ moves with constant velocity relative to S. In the presence of nonconservative forces, such as normal and tension forces, the mechanical energy of a system can be conserved in S and not be conserved in S′. In this paper we find useful relations between the mechanical energy changes in two inertial frames of reference.

The result of the Lorentz Transformation on a classical formula is that an observer in reference frame A observes the same phenomena in reference frame B like an observer in reference frame B observing the phenomena in reference frame A. Is that true? In this article I demonstrate at the Doppler effect that this is valid even for ‘absolute’ velocities belonging to a reference frame outside of A and B. I suggest to test whether the geometric means of frequencies belong to the frames A and B or to a reference frame outside of A and B by an experiment at CERN. In the latter case this would have significant implications in space-time modelling

In addition to the well-known Møller frame (or Rindler frame), we may construct another frame to describe the uniformly accelerating system. In the new frame, all "static" (i.e. spatial coordinates keeping constant) observers have the same proper acceleration but each has his own horizon. In contrast, the proper acceleration of a static observer in the Møller frame (or Rindler frame) depends on his position, but the horizon is (static-) observer-independent. We argue that the new uniformly accelerating reference frame is more suitable than the Møller frame to describe the system in an accelerating spacecraft. It is possible to distinguish the Møller frame and the new frame by high-precision experiments (such as arrival-time- and/or redshift-measurements) in an accelerating spacecraft. When the non-relativistic limit is taken, the second law of mechanics and Schrödinger equation in the new uniformly accelerating frame are all different from those in the Møller frame. The effects on the equivalence principle are discussed. We argue that even when the space–time curvature is ignored, it is still possible in some sense to distinguish gravity from acceleration at the next leading order.

Representations of centrifugal forces are derived in a variety of rotating frames. Although the rotating angle of a point mass relative to an inertial frame is often confused with the rotating angle of a rotating frame relative to the inertial frame, they should be differentiated.

The authors aim to establish the theory necessary for developing the free gyroscopic compass and focus on mainly two points. One is to suggest north-finding principle by the angular velocity of the earth's rotation, and the other is to suggest orthogonal coordinate transformations of the motion rate of the spin axis, which transforms the components of motion rate in the free gyro frame into those in the platform frame and that this transformed rate is, in turn, transformed into the NED(north-east-down) navigation frame. Subsequently, ship's heading is obtained by using the fore-aft and athwartship components of the motion rate of the spin axis in the NED frame. In addition it was found how to solve the transformation matrix necessary for transforming each frame.

In a given inertial frame, a charged particle initially at rest moves in the central electric field caused by a fixed charge at the origin. The particle has no initial angular momentum and experiences no torque. However, in an inertial frame moving with a nonvanishing velocity with respect to the first, the charged particle’s motion is subject to the Lorentz force with both electric and magnetic fields produced by the “fixed” charge, which is now in uniform motion. The charged particle experiences a torque from the magnetic part of the force; its nonvanishing angular momentum changes in time. This puzzle is analyzed in detail from different reference frames. The chief characteristic of the motion, independent of the choice of the inertial frame, is the constancy in time of the orientation of the relative coordinate r as the particle moves with respect to the fixed (or uniformly moving) charge.

By analogy with the calculation of the path of a mass point in terms of the integral of the point velocity with respect to time, such that the point has a constant velocity V(ti) within a time interval dti, then changes this velocity stepwise by V(ti+1), moves with this velocity within a time interval dti+1, etc., an accelerated motion of an observer with a clock is represented by alternating states of rest in a sequence of inertial frames of reference and instantaneous jumps from one frame of reference into another. Lorentz transformations are used to calculate the readings of a “resting” clock observed from a noninertial frame of reference represented in this manner, during the rest of a noninertial observer in a next-in-turn inertial frame of reference belonging to the mentioned sequence, and upon a jump. For the observation from a noninertial frame of reference, the relation of the time interval counted by the “resting” clock to the time interval counted by the accelerated clock and to the acceleration has been obtained.

Pellegrini and Swift have recently suggested that the use of special relativity in the calculation of the electric dipole moment of a moving magnetic dipole cannot be applied to the classic experiment of Wilson and Wilson, which used rotational motion. This paper contests that view. The disagreement arises in the choice of coordinates used to represent physical quantities measured in the rotating frame. The arguments of this paper are based on Einstein’s discussion of the validity of arbitrary coordinates and the difficulty in their interpretation. Because of the lack of synchronization of clocks, caution must be used in assigning values to physical quantities in the usual coordinates that describe a rotating frame. This paper gives the detailed transformations to an inertial rest frame, where the interpretation of measurements is assured. Other aspects of the rotating frame are also discussed.

The physical significance of time symmetry and the difficulty of the definition of an inertia frame (reference frame) is considered for a quantum field of material points with special consideration of the electrodynamics of a superconducting body.

Newton’s Mathematical Principles of Natural Philosophy provided the foundation of classical physics. This paper reviews several of his critical definitions, his three axioms, key corollaries, and concept of inertial frames. Newton’s first axiom or law requires the vector addition of velocities by Corollary I to translate the equation of motion of a constantly moving body from one inertial frame to another inertial frame. His relativity principle in Corollary V is often expanded to mean that any equation retains the same form in all inertial frames. This is true if the equations involve only Newtonian forces, but equations that specify velocity need to include the mutual velocity between moving inertial frames to fully transform the results between all reference frames. The speed of light parameter must correctly incorporate the mutual velocity between moving inertial frames. It is assumed the speed of light is a constant in all directions only in absolutely stationary, nongravitated reference frames, which is less restrictive than the assumption of universal speed of light in all inertial frames. A test is outlined with suggested equipment to measure the one-way speed of light simultaneously in three dimensions. Equations are provided to convert the results into the instantaneous directional velocity of the laboratory frame. It may take a few years to collect data to separate the Earth’s rotation, precession, nutation, polar wobble, Earth’s orbital velocity around the Earth-Sun and Earth-Moon barycenters, and the solar system’s movement due to the Milky Way’s rotation and translational velocity within the universe.

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.

In the thought experiment in this paper, we considered inertial frames M and A moving at a constant velocity relative to each other. A light signal emitted from inertial frame A, when time of a clock in inertial frame A was 1(s), arrived at inertial frame M when time of a clock in inertial frame M was 2(s). In this paper, the time in inertial frame A when the time in inertial frame M was 2(s) was predicted by observers in inertial frames M and A by applying the special theory of relativity (STR). Predictions of the two observers did not match. Einstein regarded all inertial frames as equivalent, but there are cases where a velocity vector is attached to some inertial frame. Einstein overlooked this fact, and thus a discrepancy appeared in the values predicted by the two observers. It is not the case that all inertial frames are equivalent. This paper concludes that the STR is a theory incorporating a contradiction which must be corrected.

The electrodynamic equations in a rotating reference frame (with velocities of rotation much less than the speed of light) are derived for the cases of a medium stationary in the rotating frame, and a medium with a general velocity relative to the rotating frame. The second case is considered as a necessary formalism for use in the modelling of large-scale electromagnetic fields induced by a prescribed ocean circulation. The terms that include rotational effects depend not only on the relative velocities but also on the electric properties of the material media. Hence, the assumption that rotational terms can be neglected simply because the velocities involved are much less than that of light in a vacuum is not supported. For typical values of the parameters describing the electric properties of the ocean, three of the four Maxwell equations (in Minkowskian form) are similar in form to those in an inertial frame. A governing induction equation in one vector variable (the magnetic field, b) is derived and written out explicitly in various coordinate systems.

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.

We discuss a geometrical method to define a preferred reference frame for precessing binary systems and the gravitational waves they emit. This minimal-rotation frame is aligned with the angular-momentum axis and fixes the rotation about that axis up to a constant angle, resulting in an essentially invariant frame. Gravitational waveforms decomposed in this frame are similarly invariant under rotations of the inertial frame and exhibit relatively smoothly varying phase. By contrast, earlier prescriptions for radiation-aligned frames induce extraneous features in the gravitational-wave phase which depend on the orientation of the inertial frame, leading to fluctuations in the frequency that may compound to many gravitational-wave cycles. We explore a simplified description of post-Newtonian approximations for precessing systems using the minimal-rotation frame, and describe the construction of analytical/numerical hybrid waveforms for such systems., Minor clarifications and journal reference

It is suggested that the mathematically abstract coordinate frames of reference commonly visualized to be centered at the celestial bodies have real counterparts in the shape of well-defined rigid spatial resonant singularities of infinite extension, which accommodate the matter waves from the superimposition of which the body residing at the coordinate origin results. A universally valid inertial reference frame system is proposed. Qualitative explanations are offered for the inertial and gravitational forces, their observed proportionality, and for the occurrence of second-order gravitational effects in the vicinity of massive bodies. The universal redshift is assumed to result from a closure condition of the eigenspaces introduced.

The origin of inertia has intrigued scientists and philosophers for centuries. Inertial frames of reference permeate our daily life. The inertial and centrifugal forces, such as the pull and push that we feel when our vehicle accelerates, brakes and turns, arise because of changes in velocity relative to uniformly moving inertial frames. A classical interpretation ascribed these forces to acceleration relative to some absolute frame independent of the cosmological matter, whereas an opposite view related them to acceleration relative to all the masses and 'fixed stars' in the Universe. An echo and partial realization of the latter idea can be found in Einstein's general theory of relativity, which predicts that a spinning mass will 'drag' inertial frames along with it. Here I review the recent measurements of frame dragging using satellites orbiting Earth.

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.

Analysis of particle interaction in the laboratory frame of storage rings is often complicated by the fact that particle, motion is relativistic, and that reference particle trajectory is curved. Rest frame of the reference particle is a convenient coordinate system to work with, within which particle motion is non-relativistic. We have derived the equations of motion in the beam rest frame from the general relativity formalism, and have successfully applied them to the analysis of crystalline beams.

The logical difficulty surrounding the definition of an inertial frame and Newton’s first law of motion can be circumvented by defining the inertial frame in terms of its spatial and temporal properties, as embodied in the special theory of relativity. Whether or not these properties apply can be determined, in principle, by experiment. The mass of a body is measured through a completely inelastic collision with a body whose mass is taken as the standard. Rather than attempt to define force by a means not involving Newton’s second law, we use the second law as the definition.

A rest frame in a stationary, axially symmetric space−time is defined as a synchronizable frame which is as nearly Killing as possible. This is a possible generalization of the Newtonian rest frame. A kinematical theorem giving the condition for the existence of a rest frame whose time vector is a linear combination of the Killing vectors is proved. The frame is also unique. The condition is shown to be weaker than the assumption of orthogonal transitivity. The surfaces of simultaneity of the rest frame are the surfaces of constancy of a particular Killing time coordinate, and its time vector is the component of the time Killing vector orthogonal to the angular Killing vector. Some properties of the rest frame are then discussed; it is shown that the frame is well−behaved down to the event horizon, where its time vector becomes null. Under a suitable condition on the event horizon, the time vector coincides with a Killing vector there. The gravitational redshift relation in the rest frame is derived. There is a dependence on the angular momentum of the geodesic. Furthermore, the event horizon is shown to be an infinite redshift surface for the rest frame observers. Finally, the three−vector potential of Landau and Lifshitz is interpreted and shown to be closely related to the rest frame, and a corresponding four−vector potential is invariantly defined.

The dynamics of interacting molecules involve the translation and simultaneous rotation of each molecule in the ensemble. This means that the frame of reference defined by the principal molecular moments of inertia, frame (1,2,3), is a noninertial frame with respect to the laboratory frame (x, y,z). Furthermore, a rotating frame of reference (1,2,3)’ can be generated from (1,2,3) a by translation for each molecule from the center of mass of the origin of (x, y,z). The equations relating velocities, accelerations, and their derivatives in one of these frames to those of another are described and then used to calculate the cross correlation functions.

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...

The field equations of general relativity are solved to post-Newtonian order for a rotating frame of reference. A new method of approximation is used based on a 3+1 decomposition of the equations. The results are expressed explicitly in terms of the gravitational potentials. The space-time is asymptotically flat but not locally flat. The space-time metric contains gravitational terms, inertial terms, and coupled gravitational-inertial terms. The inertial terms in the equation of motion are in agreement with terms obtained by other authors using kinematic methods. The metric and equation of motion reduce to those for an inertial frame of reference under a simple coordinate transformation. The total energy of a particle is given. For the restricted three-body problem this represents the relativistic extension of Jacobi's integral to post-Newtonian order.

Special relativity relates measurements in a ‘‘moving’’ frame to measurements in a ‘‘rest’’ frame. Navigation, as it is performed in practice, uses neither pure a ‘‘rest’’ nor pure a ‘‘moving’’ frame, but instead measures distance in the fixed, Earth, frame and time in the moving, ship, frame. Brehme diagrams arise naturally when relative motion is considered using navigational coordinates. In navigational coordinates the Lorentz transformation is expressible as an ordinary orthogonal transformation.

A uniformly accelerated reference frame S is defined as a set of observers who remain at rest with respect to a given observer A who is accelerating at a constant rate with respect to the instantaneously comoving inertial frames. The one‐dimensional uniformly accelerated reference frame S is considered. The world lines of A and the other observers making up S are determined. Coordinates useful for describing events in S are carefully defined and the transformation equations between different sets of them are derived. The variation with position in S of the speed and frequency of light waves is determined. The motion of a free‐particle in S is determined. Various phenomena in S, ordinarily associated with general relativity, are considered, in particular the asymmetric aging of twins at rest at different positions and the existence of horizons.

The imposition of periodic boundary conditions on the space‐time of special relativity is shown to lead to the existence of a preferred frame of reference. One characteristic of the preferred frame is that it is the frame in which the resulting finite universe has the smallest size. In a frame moving with respect to the preferred frame, global synchronization of clocks cannot be carried out and the transit time to traverse the universe depends on the direction of travel. As measured in a moving frame, a person at rest in the preferred frame can be found at two or more different places at the same time, although the age of the person will be different at each of those places. A ‘‘pole in the universe’’ paradox is presented, the resolution of which requires use of some of these bizarre properties of the universe of periodic boundary conditions.

We start from a reference frame Σ′ for which the pseudo-Euclidean geometry holds, i.e., all laws in Σ′ have the form of special relativity. The geometry of a reference frame Σ, moving with constant velocity with regard to Σ′, is received by using the well-known Galilei-transformation for the covariant space-time vector. In Σ the geometry is different from the pseudo-Euclidean one, i.e., the principle of special relativity does not hold. The velocity of light in Σ is direction dependent but the null result of the Michelson-Morley experiment is received. Several useful transformation formulae are given. In particular, we obtain the Marinov-transformation as transformation from the frame Σ′ to Σ, and conversely. Maxwell's equations and the equations of motion in the reference frame Σ are given. The theory is applied to some examples, e.g., the Doppler effect, the Fizeau experiment, the field of a charged particle, etc. The method can be generalized to an accelerated reference frame Σ.

The results of an investigation into the relation between the time measurements performed in an inertial frame and in an accelerated frame are presented. A wholly kinematic and not a gravitational scheme is used in this study of accelerated frames in empty space-time. (R.E.U.)

A coordinate transformation from a one-dimensional inertial frame of reference to an accelerated one in empty space-time has recently been proposed by Lass. His derivation displays the remarkable feature that the motion of the accelerated frame relative to the inertial frame is fully determined by a set of assumptions, none of which implies any requirement regarding the individual motions of the different points of one frame relative to the other frame. This peculiarity is ascribed to the conjunction of several independent constraints imposed on the mutual relative motions of such points among themselves. The properties involved in Lass' transformation are analyzed, and their mutual implications are studied. It is found that one property (namely, that the speed of light with respect to the noninertial frame, at each point of this frame and in both directions, is time-independent) has a special status. A comment is made regarding the extension of the transformation to a three-dimensional system.

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.

The non-linear conform-invariant spinor wave equation proposed in an earlier paper is given a simple geometrical interpretation within the frame of general relativity. At each point of a conformal space-time manifold, an orthogonal frame of reference is defined, this frame being associated with a 4-spinorψ. It is shown that if the contracted torsion tensor ια relative to the frame vanishes (as in the case of a co-ordinate transformation) and the pseudo-vectorLα derived from the same torsion tensor remains constant and space-like, then the associated 4-spinorψ satisfies just the above mentioned wave equation which, in turn, determines the orthogonal frame of reference at every space-time point to within a conformal transformation of the co-ordinate system. In particular, if the pseudo-vectorLα also vanishes one obtains Dirac’s equation for a particle of rest mass zero. In a discussion of the physical meaning of the orthogonal frame of reference, the time-like basis vector and one of the space-like basis vectors are respectively related to the current density and the spin density 4-vectors associated with the wave equation.

In the gravitational field produced by a uniformly rotating charged ring, a frame of reference that is the analog of an inertial frame in flat space is constructed. Expressions are obtained for the energy in a sphere of radius r and the total energy in such a frame of reference. The results are compared with the results of other papers.

A frame of reference is defined as inertial if the coordinates of bodies, predicted with respect to it on the basis of some mechanical principles, agree with those actually observed. It is pointed out that within the framework of Newtonian mechanics the concept of an inertial frame of reference is global, but within the framework of general relativity only local. The relativistic local inertial frame of the Solar System moves with respect to the Newtonian global frame by the amount of “geodetic precession”. In all rigor, the concept of a globally privileged inertial system has been rendered meaningless by general relativity, even though the concept of an inertial reference frame remains useful in contemporary celestial mechanics. It is further pointed out that none of the practically determined approximations to an inertial frame of reference (e.g., the dynamical reference frame of celestial mechanics, the FK5) must be regarded as a definition of an inertial system itself.

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.