Non-Inertial Frames in Special Relativity

International Journal of Theoretical and Mathematical Physics, 2021

Existing Special Relativity (SR) theories involving non-inertial frames are shown to produce erroneous Space-time Transformation Relations (STR). Formulation of SR shows that specifying the trajectory in the traveller’s proper time makes his frame the reference frame. Traveller velocity change implies a new reference frame requiring correction for STR. But commonly used Successive Incremental Lorentz Transformation (SILT) or equivalently the Fermi Coordinates (FC) principle with Lorentz Boost and earlier theories fail to include corrections and simultaneity. In stationary (inertial) frames the time for light propagation between two points is simply the distance divided by the speed of light, but with changing velocities in non-inertial frames that relation doesn’t hold. Establishing simultaneity, which relies upon the information propagation from an event to the observation points, is difficult when the trajectory is in the traveller’s time. For an appropriate Kinematic Special Relativity (KSR) theory, (non-inertial SR), inclusion of corrections in earlier theories without consideration of the simultaneity and the information propagation time is inadequate. Moreover, it would be paradoxical to designate who is the traveller in SILT and FC if both frames are non-inertial. Therefore, KSR theory is developed here using simultaneity which is central to SR with the trajectory in the stationary frame’s proper time.

This paper presents a reformulation of special relativity, whose kinematic and dynamic magnitudes are invariant under transformations between inertial and non-inertial reference frames, which can be applied in massive and non-massive particles, and where the relationship between net force and special acceleration is as in Newton's second law. Additionally, new universal forces are proposed.

Science & Philosophy, 2022

The concept of inertial frame of reference in classical physics and special theory of relativity is analysed. It has been shown that this fundamental concept of physics is not clear enough. A definition of inertial frame of reference is proposed which expresses its key inherent property. The definition is operational and powerful. Many other properties of inertial frames follow from the definition, or it makes them plausible. In particular, the definition shows why physical laws obey space and time symmetries and the principle of relativity, it resolves the problem of clock synchronization and the role of light in it, as well as the problem of the geometry of inertial frames.

In present paper, we have proposed an alternative theory on the spacetime of non-inertial reference frame (NRF) which bases on the requirement of general completeness (RGC) and the principle of equality of all reference frames (PERF). The RGC is that the physical equations used to describe the dynamics of matter and/or fields should include the descriptions that not only the matter and/or fields are at rest, but also they move relative to this reference frame, and the structure of the spacetime of reference frame has been considered. The PERF is that any reference frame can be used to describe the motion of matter and/or fields. The spacetime of NRF is inhomogeneous and deformed caused by the accelerating motion of the reference frame. The inertial force is the manifestation of deformed spacetime. The Riemann curvature tensor of the spacetime of NRF equals zero, but the Riemann-Christoffel symbol never vanishs no matter what coordinate system is selected in the NRF. The physical equations satisfied the RGC remain covariance under the coordinate transformation between the reference frames. Mach's principle is incorrect. The problem of spacetime of NRF can be solved without considering gravitation.

The success of Special Relativity (SR) comes from the requirement of Lorentz covariance to all physical equations. The explanation with regard to the Lorentz covariance is based on two hypotheses, namely the principle of special relativity and the constancy of the speed of light. However, the statements of the principle of special relativity are various and confusing. The covariance of physical equations and the equality of inertial frames of reference are mixed up. The equality of inertial frames of reference is obvious, but the covariance of the physical equations is a more advanced requirement. Additionally, the way that the propagation property of light is placed in a central position of SR has caused people misunderstandings towards space-time, and also there is a logical circularity between the measurement of speed of light and the synchronization of clocks. These have obstructed to correctly extend the theory of space-time from an inertial frame of reference to a non-inertial frame of reference. These are the main reasons why many people criticize SR. In present paper, the two hypotheses have been discussed in detail and a new requirement to the equations of Physics has been proposed. The requirement is the Requirement of Special Completeness, namely, the physical equations used to describe the dynamics of matter and/or fields should include the descriptions that not only the matter and/or fields are at rest relative to an inertial frame of reference, but also they move relative to this frame. Basing on this requirement and the equality of the inertial frames of reference, we can approach to SR. Thereby let the theory of Lorentz covariance has a clear and solid foundation. The constancy of the speed of light is just a deduction, not a premise. The Lorentz covariance is just a characteristic of the Special Complete equations. Maxwell equations automatically satisfy the Lorentz transformations without any modification, while Newton law of gravity does not, because Newton law of gravity is not Special Complete and Maxwell equations are. The new approach has paved a road leading towards the generalizing of the theory of space-time from the inertial frame of reference to non-inertial frame of reference without considering gravitation. Résumé: Le succès de la relativité restreinte (RR) provient de l'exigence de la covariance de Lorentz à toutes les équations physiques. L'explication en ce qui concerne la covariance de Lorentz est fondée sur deux hypothèses, à saa

2021

Physics of non-inertial reference frames is a generalizing of Newton’s laws to any reference frames. The first, Law of Kinematic in non-inertial reference frames reads: the kinematic state of a body free of forces conserves and determinates a constant n-th order derivative with respect to time being equal in absolute value to an invariant of the observer’s reference frame. The second, Law of Dynamic extended Newton’s second law to non-inertial reference frames and also contains additional variables there are higher derivatives of coordinates. Dynamics Law in non-inertial reference frames reads: a force induces a change in the kinematic state of the body and is proportional to the rate of its change. It is mean that if the kinematic invariant of the reference frame is n-th derivative with respect the time, then the dynamics of a body being affected by the force F is described by the (n+1)-th differential equation. The third, Law of Static in non-inertial reference frames reads: the s...

The results of Special Relativity without Special Relativity, 2024

In this paper, we rederive the key results of Special Relativity, such as time dilation, the Doppler effect, and the formula for kinetic energy (E=mc²+1/mv²+…), without employing the central assumption of Special Relativity (invariance of Galilean systems), but instead using the prevailing hypothesis prior to it, the existence of a universal frame. We explain the Lorentz transformation within this framework. The explanation for the 'failure' of the Michelson-Morley experiment is proposed in an other paper.

Old and New Concepts of Physics, 2007

Several new ideas related to Special and General Relativity are proposed. The black-box method is used for the synchronization of the clocks and the space axes between two inertial systems or two accelerated systems and for the derivation of the transformations between them. There are two consistent ways of defining the inputs and outputs to describe the transformations and relative motion between the systems. The standard approach uses a mixture of the two ways. By formulating the principle of special and general relativity as a symmetry principle we are able to specify these transformations to depend only on a constant.

Physical Review A, 2000

Although there is no relative motion among different points on a rotating disc, each point belongs to a different noninertial frame. This fact, not recognized in previous approaches to the Ehrenfest paradox and related problems, is exploited to give a correct treatment of a rotating ring and a rotating disc. Tensile stresses are recovered, but, contrary to the prediction of the standard approach, it is found that an observer on the rim of the disc will see equal lengths of other differently moving objects as an inertial observer whose instantaneous position and velocity are equal to that of the observer on the rim. The rate of clocks at various positions, as seen by various observers, is also discussed. Some results are generalized for observers arbitrarily moving in flat or curved spacetime. The generally accepted formula for the space line element in a nontime-orthogonal frame is found inappropriate in some cases. Use of Fermi coordinates leads to the result that for any observer the velocity of light is isotropic and is equal to c, providing that it is measured by propagating a light beam in a small neighborhood of the observer.

A free system, considered to be a comparison system, allows for the notion of objective existence and inertial frame. Transformations connecting inertial frames are shown to be either Lorentz or generalized Galilei.