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We solve the problem of Apollonius by applying the Lorentz transformation within the framework of special relativity to go to a frame of simultaneity, where the solutions are the circumcenter. We transform back to get the solutions in a general frame. We show that the solutions in a general frame are the foci of the ellipse going through the centers of the given 3 circles. [ABSTRACT FROM AUTHOR]

Explicit formulae for the 4 × 4 Lorentz transformation matrices corresponding to a pure boost and a pure three-dimensional rotation are very well known. Significantly less well known is the explicit formula for a general Lorentz transformation with arbitrary non-zero boost and rotation parameters. We revisit this more general formula by presenting two different derivations. The first derivation (which is somewhat simpler than previous ones appearing in the literature) evaluates the exponential of a 4 × 4 real matrix A, where A is a product of the diagonal matrix diag (+ 1 , − 1 , − 1 , − 1) and an arbitrary 4 × 4 real antisymmetric matrix. The formula for exp A depends only on the eigenvalues of A and makes use of the Lagrange interpolating polynomial. The second derivation exploits the observation that the spinor product η † σ ¯ μ χ transforms as a Lorentz four-vector, where χ and η are two-component spinors. The advantage of the latter derivation is that the corresponding formula for a general Lorentz transformation Λ reduces to the computation of the trace of a product of 2 × 2 matrices. Both computations are shown to yield equivalent expressions for Λ. [ABSTRACT FROM AUTHOR]

We show that if we assume the existence of a frame in vacuum where the oneway speed of light is c (c is the measured value of the two-way speed of light in vacuum) than for another frame moving with velocity v1 in relation to that frame we can have time dilation, time contraction or no difference of proper times change at all. Therefore, the standard formulation is a result of a misinterpretation of the mathematical expression between the relation of the proper time of the moving frame in relation to the difference of times of Lorentzian clocks, the so-called time dilation. This is an astonishing conflict that standard formulation cannot solve. This result is easily obtained if we assume time dilation in relation to Einstein Frame (EF) the frame where the speed of light is isotropic. [ABSTRACT FROM AUTHOR]

The common opinion that the Lorentz transformation distorts the Minkowski spacetime arises from a mathematically incorrect neglect of its non-diagonal terms. When the non-diagonal Lorentz transformation is properly treated, the time dilation and the space distortion disappear. Consequently, the Lorentz metric produced by the Lorentz transformation is identical to the Minkowski metric. The spacetime distortion in moving inertial frames is properly described by the so-called Doppler metric produced by the Doppler transformation. This transformation assumes the existence of a preferred frame and depicts the Doppler shift of light for a moving light source and/or observer. The Doppler metric is Lorentz invariant and leaves the Maxwell's equations unchanged from their form in the Minkowski spacetime. Moreover, the Doppler metric is experimentally confirmed because it yields the null result in the Michelson–Morley experiment and other interferometric experiments. The suitability of the Doppler metric for describing light properties in Special and General Relativity is supported by recent cosmological observations of the cosmic microwave background. [ABSTRACT FROM AUTHOR]

Abstract Time‐varying metasurfaces have recently emerged as a promising platform to observe exotic wave behaviors in space‐time systems. In this paper, it theoretically reveals invariant susceptibility parameters for perpendicular‐moving metasurface. Duality‐matching condition is derived by utilizing generalized sheet transition conditions (GSTCs) combined with Lorentz transformation (LT). A metasurface that satisfies the duality‐matching condition can hold its susceptibility parameters when it moves perpendicularly toward the sources. The moving‐invariant behavior is further discussed in time‐varying media and additional reflection is found to be inevitable if the constitutive parameters is discrete in time.

In classical electrodynamics, by motion, it always means a relative movement of two observers in inertia reference frames, so that the covariance of the Maxwell’s equations is preserved respectively in two spaces under Lorentz transformation. The energy is thus conservative for the electromagnetic system. The theory for describing the electromagnetic behavior of the charged particles in vacuum space can be well described using the special relativity because of the invariance of the speed of light in vacuum. However, for engineering applications, the media have shapes and sizes and may move with acceleration, and a system may have multiple moving objects that may be correlated or independently under external mechanical triggering. This paper presents the theory for describing the electromagnetic phenomena in this electro-magnetic-mechano system. We mainly introduce the Maxwell’s equations for a mechano-driven media system (MEs-f-MDMS) under low-speed approximation (v << c). We concluded that the MEs-f-MDMS are required for describing the electrodynamics inside a moving object that moves not only with accelerated translation motion but also has rotation motion. The classical Maxwell’s equations are to describe the electrodynamics in the region where there is no local medium movement. The full solutions of the two regions satisfy the boundary conditions, so that the rotation of the object affects the electromagnetic field at vicinity. The theoretical approaches for solving the MEs-f-MDMS are also presented.

We study partial orders defined on the set of points of space-time that are invariant under Lorentz transformations. Kirszbraun's Theorem allows to show that world lines of particles evolving in space-time are precisely the maximal chains in the causality order. We show that the causality order is well behaved in the sense that it is gradable and level sets under various gradings are precisely the anti-chain cutsets. We also show that the causality orders corresponding to different light speed parameters c are essentially the only partial orders invariant under Lorentz transformations and under some other, more obvious affine transformations of space-time. We characterize optical lines and hyperplanes, inertial lines and planes, and separation lines as well, in terms of the causality order and use these characterizations to provide a variant proof of the Alexandrov-Zeeman Theorem. [ABSTRACT FROM AUTHOR]

The Lorentz transformation of space and time between two inertial frames of reference is one of the pillars of the special theory of relativity. Until now, the Lorentz transformation equations have been considered on the base of one-dimensional motion between the inertial frames. The goal of this article is to extend the ordinary one-dimensional Lorentz transformation to motion along X-, Y-, and Z-directions, i.e., to achieve a space-time coordinate transformation in the three-dimensional (3D) space. We particularly discovered the modified Lorentz transformation equations along 3 directions, and this helped us to analyze the space contraction phenomena of a cuboid due to the relative motion between the inertial frames in the 3D space. As a final point, this study concluded that all length, breadth and height of a cuboid appeared to be shortened to the observer if there is the relative motion between the cuboid and an observer in the 3D space.

This paper points out that the transformation equations for the spatial and temporal coordinates between two frames of reference in the existing generally accepted version of the Lorentz transformation are deficient, since transformation equations are based on one dimensional motion between inertial frames. Therefore, all possible space-time coordinate transformation equations between moving and stationary frames by prolonging Lorentz transformation in a two-dimensional plane are thoroughly proposed in this article. If denotes the relative velocity between stationary frame (x, y) and moving frame (x', y'), then the transformation equations along X and Y-axis under two-dimensional Lorentz are given, respectively, by the formulas (see full text) . In this work we conclude that length and breadth of a rectangle appears to be shortened to the observer when there is the relative velocity between the rectangle and observer along both X and Y-axis. In particular, we present a concise and carefully reasoned account of a new aspect of Lorentz transformation which decently allows for the determination of space-time coordinates transformation equations in two dimensions of space.

Reciprocal length contraction and time dilation, as enshrined in the Lorentz transformation, are the bulwark of the special relativity theory. Reciprocal time dilation is typically discussed (in the context of its apparent contradiction) and usually explained away as the result of differing synchronization in inertial reference frames and the relativity of simultaneity. Further, it is generally suggested that time is an abstract concept, and its perceptions may differ with context. However, we suggest that regarding length measurements no such abstraction can be envisaged. In this manuscript, we examine the reciprocal length contraction in detail to assess whether it leads to any contradiction. [ABSTRACT FROM AUTHOR]

Explicit formulae for the 4×4 Lorentz transformation matrices corresponding to a pure boost and a pure three-dimensional rotation are very well known. Significantly less well known is the explicit formula for a general Lorentz transformation with arbitrary non-zero boost and rotation parameters. We revisit this more general formula by presenting two different derivations. The first derivation (which is somewhat simpler than previous ones appearing in the literature) evaluates the exponential of a 4×4 real matrix A, where A is a product of the diagonal matrix diag(+1,−1,−1,−1) and an arbitrary 4×4 real antisymmetric matrix. The formula for expA depends only on the eigenvalues of A and makes use of the Lagrange interpolating polynomial. The second derivation exploits the observation that the spinor product η†σ¯μχ transforms as a Lorentz four-vector, where χ and η are two-component spinors. The advantage of the latter derivation is that the corresponding formula for a general Lorentz transformation Λ reduces to the computation of the trace of a product of 2×2 matrices. Both computations are shown to yield equivalent expressions for Λ.

We show that relativistic rotation transformations represent transfer maps between the laboratory system and a local observer on an observer manifold, rather than an event manifold, in the spirit of C-equivalence. Rotation is, therefore, not a parameterised motion on a background space or spacetime, but is determined by a particular sequence of tetrads related by specific special Lorentz transformations or boosts. Because such Lorentz boosts do not form a group, these tetrads represent distinct observers that cannot put together their local descriptions into a manifold in the usual sense. The choice of observer manifold depends on the dynamical situation under consideration, and is not solely determined by the kinematics. Three examples are given: Franklin's rotation transformation for uniform plane rotation, the Thomas precession of a vector attached to an electron, and the motion of a charged particle in an electromagnetic field. In each case, at each point of its trajectory, there is a distinguished tetrad and a special Lorentz transformation that maps Minkowski space to the spacetime of the local observer on the curve. [ABSTRACT FROM AUTHOR]

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The paper demonstrates the incompleteness of the relativistic space contraction formula for space-time coordinates transformation between inertial frames of reference in the existing normally acknowledged version of the theory of special relativity. The reason is the lack of a formula expressing the space contraction in terms of the cosines of the direction of an object’s movement. Thus a modified version of the special relativity theory has been presented through a natural extension of the Lorentz transformation to three dimensions of space. The new transformations are based on the same mathematical operation as the Lorentz transformation, and provide mathematical information similar to that of a single-axis movement. An expression for the new transformation between inertial frames, when there was simultaneous relative motion in three directions, was introduced. The presence of relative motion in three directions between a parallelepiped and an observer (were introduced) made it possible to obtain the formulas of the directions of the ends of the parallelepiped. In fact, the Lorentz transformation was interpreted, revealing transformation of space-time coordinates in three dimensions. This offers ample scope for finding the space contraction within moving frame of reference.

The importance of synchronization of spatially separated clocks in observing event coordinates cannot be overemphasized. We illustrate this with a real-world example. The results show the dependence of both length contraction and time dilation on the adopted synchronization convention. It is illustrated that a change in synchronization convention completely alters the appearance of the physical world. This is to be viewed in the context of the stated position in physics that synchronization of spatially separated clocks in any inertial frame depended on the adoption of a convention for its resolution. The preferred method of synchronization of spatially separated clocks is the standard convention recommended by the theory of relativity; however, it still remains a convention. [ABSTRACT FROM AUTHOR]

Maxwell’s equations and the Dirac equation are the first-order differential relativistic wave equation for electromagnetic waves and electronic waves respectively. Hence, there is a notable similarity between these two wave equations, which has been widely researched since the Dirac equation was proposed. In this paper, we show that the Maxwell equations can be written in an exact form of the Dirac equation by representing the four Dirac operators with $8\times8$ matrices. Unlike the ordinary $4\times4$ Dirac equation, both spin–1/2 and spin–1 operators can be derived from the $8\times8$ Dirac equation, manifesting that the $8\times8$ Dirac equation is able to describe both electrons and photons. As a result of the restrictions that the electromagnetic wave is a transverse wave, the photon is a spin–1 particle. The four–current in the Maxwell equations and the mass in the electronic Dirac equation also force the electromagnetic field to transform differently to the electronic field. We use this $8\times8$ representation to find that the Zitterbewegung of the photon is actually the oscillatory part of the Poynting vector, often neglected upon time averaging.

We show that Special and General Relativity lead to the introduction of an observer manifold, in addition to the usual event manifold. Axiomatics that require that manifolds be Hausdorff are not appropriate for the observer manifold. We propose an alternative axiomatics and show how most of the usual local computations extend to this framework without difficulty. The derivation of the Lorentz transformation takes a new meaning in this context, enabling the identification of the representations of several observers and hence reducing the observer manifold to the event manifold. However, we show in an example relevant to the radiation of accelerated electrons that this identification is not always correct. This appears to be relevant in any situation where gravitational fields in remote locations have to be measured on Earth, such as the detection of gravitational waves, or when high accelerations are involved, such as in electron radiation or laser cooling. [ABSTRACT FROM AUTHOR]

The twin paradox, a classical puzzle in Special Relativity (SR), is typically resolved by acknowledging the asymmetry introduced by the acceleration of one twin's frame, associated with a rocket's motion through space. A similar paradox arises without any accelerated system, involving three or more inertial systems. This is commonly resolved by employing the relativistic concepts of clock synchronization and simultaneity. Here, we reformulate the paradox for two free-falling systems, where the twins traverse identical circular orbits in opposite directions around a central mass with a spherically symmetric gravitational field. This redefined version of the paradox eliminates asymmetry inherent in the original problem. Since both frames are free-falling, they can be viewed as locally inertial according to Einstein's Equivalence Principle. Additionally, no synchronization with clocks in other frames is needed. We explore a potential resolution to the paradox and highlight that there may be a misinterpretation of the Lorentz transformation in SR. • The twin paradox for free-falling frames in a gravitational field is examined. • The new formulation of the paradox removes asymmetry and synchronization issues. • Resolving the paradox requires revising the physical interpretation of the Lorentz transformation. [ABSTRACT FROM AUTHOR]

We show that relativistic rotation transformations represent transfer maps between the laboratory system and a local observer on an observer manifold, rather than an event manifold, in the spirit of C-equivalence. Rotation is, therefore, not a parameterised motion on a background space or spacetime, but is determined by a particular sequence of tetrads related by specific special Lorentz transformations or boosts. Because such Lorentz boosts do not form a group, these tetrads represent distinct observers that cannot put together their local descriptions into a manifold in the usual sense. The choice of observer manifold depends on the dynamical situation under consideration, and is not solely determined by the kinematics. Three examples are given: Franklin’s rotation transformation for uniform plane rotation, the Thomas precession of a vector attached to an electron, and the motion of a charged particle in an electromagnetic field. In each case, at each point of its trajectory, there is a distinguished tetrad and a special Lorentz transformation that maps Minkowski space to the spacetime of the local observer on the curve.

We use the method of field decomposition, a widely used technique in relativistic magnetohydrodynamics, to study the small velocity approximation (SVA) of the Lorentz transformation in Maxwell equations for slowly moving media. The "deformed" Maxwell equations derived using SVA in the lab frame can be put into the conventional form of Maxwell equations in the medium's co-moving frame. Our results show that the Lorentz transformation in the SVA of up to O (v / c) (v is the speed of the medium and c is the speed of light in a vacuum) is essential to derive these equations: the time and charge density must also change when transforming to a different frame, even in the SVA, not just the position and current density, as in the Galilean transformation. This marks the essential difference between the Lorentz transformation and the Galilean one. We show that the integral forms of Faraday and Ampere equations for slowly moving surfaces are consistent with Maxwell equations. We also present Faraday equation in the covariant integral form, in which the electromotive force can be defined as a Lorentz scalar that is independent of the observer's frame. No evidence exists to support an extension or modification of Maxwell equations. [ABSTRACT FROM AUTHOR]

Abstract This study deals with the evaluation of the longitudinal and transverse coupling impedance of a charge travelling in a drift tube with a perfectly electric conductive angular strip. The problem is formulated in the particle frame as dual series equations, and efficiently solved through the representation of the unknown in terms of Neumann series. Then, the electric parameters are obtained in the pipe frame through the Lorentz transforms. The presented solution can be adopted as the general methodology in case of angular discontinuities in particle accelerators.

An allegorical thought experiment occurring in a pseudo Huxleyean world in the future is conducted, in which "Euclidean" geometry has been forgotten and can only be retrieved by traversing backward in time, physically and virtually by two protagonists, Alpha and Beta, inspired by the challenge issued by their teacher Aleph 1. In the year 136 A.H (After Huxley, or late twenty-first century of the Common Era), all instruction occurs virtually. Mathematics is the exclusive purview of 0.01% of society (the alphas). The thought experiment leads to the discovery and realization from the two protagonists that Euclidean geometry is possible using sensorimotor functions with tools called a straight edge and compass or simply a rope with knots. As an ode to Aldous Huxley (1894–1963) this article is deliberately in the form of a satirical provocation that examines the interaction of technology with the learning of mathematics in a dystopian future. A prolog explaining the context of the thought experiment along with a coda with implications for the immediate future in the learning of mathematics is made. A glossary with terms unfamiliar to the reader is also provided. [ABSTRACT FROM AUTHOR]

Abstract Four‐dimensional (4D) relativistic scattering of electromagnetic waves from an arbitrary collection of uniformly translational moving lossy dielectric spheres is discussed. Two reference frames, four 4D coordinate systems and Lorentz transformation are used to obtain the scattered electromagnetic fields. The direct scattering of the spheres and their interactions are considered with a novel approach. The introduced method is straightforward and the analytical relations for the fields are achieved. To check the validity of the proposed method, different examples for both stationary and moving scatterers are investigated. The effects of key parameters such as the size, material, velocity, number, position of the spheres and also the frequency of the incident wave are discussed. The derived scattered fields are valid for low, medium and high velocities but according to practical applications low and moderate velocities are highlighted in numerical results.

We show that the comment by Franklin (Int. J. Mod. Phys. A 36, 2175001 (2021)) on our paper "Lorentz transformation of a charge-current density and "relativistic polarization" of a moving current loop" (Int. J. Mod. Phys. A 36, 2175001 (2021)), like the previous publication of this author entitled "Complete Lorentz transformation of a charge-current density" (Int. J. Mod. Phys. A 35, 2050061 (2020)), contains fundamental errors in the physical interpretation of Lorentz transformations, which are once again disclosed with some particular examples. [ABSTRACT FROM AUTHOR]

We show that Special and General Relativity lead to the introduction of an observer manifold, in addition to the usual event manifold. Axiomatics that require that manifolds be Hausdorff are not appropriate for the observer manifold. We propose an alternative axiomatics and show how most of the usual local computations extend to this framework without difficulty. The derivation of the Lorentz transformation takes a new meaning in this context, enabling the identification of the representations of several observers and hence reducing the observer manifold to the event manifold. However, we show in an example relevant to the radiation of accelerated electrons that this identification is not always correct. This appears to be relevant in any situation where gravitational fields in remote locations have to be measured on Earth, such as the detection of gravitational waves, or when high accelerations are involved, such as in electron radiation or laser cooling.

The radar echo signal of a high-speed target is simulated in this letter. Using the Lorentz transformation, the linear frequency modulation pulse radar signal is transformed from the laboratory coordinate system into a moving system. The near-zone scattered fields and the far-zone echo signals are solved by the finite-difference time-domain (FDTD) algorithm in the moving system that is kept the same velocity as the target. Subsequently, transform the echo signal back to the laboratory system. Meanwhile, this letter combines the graphic processing unit parallel technique with the Lorentz–FDTD method to realize the simulation of a high-speed target excited by a large time-width signal and extracts the one-dimensional(1-D) range profile of the target. The validity of the proposed method is verified by the analytical results of a high-speed target excited by arbitrary directions incident waves. The numerical examples show that the motion's effects on the range information of the radar signal are nonnegligible for high-speed objects. [ABSTRACT FROM AUTHOR]

In the new mathematical description of special relativity in terms of the relativistic velocity space, many physical aspects acquire new geometric meanings. Performing conformal deformations upon the 2-dimensional relativistic velocity space for the (2+1)-dimensional special relativity, we find that these conformal deformations correspond to the generalized Lorentz transformations, which are akin to the ordinary Lorentz transformation, but are morphed by a global rescaling of the polar angle and correspondingly characterized by a topological integral index. The generalized Lorentz transformations keep the two fundamental principles of special relativity intact, suggesting that the indexed generalization may be related to the Bondi–Metzner–Sachs (BMS) group of the asymptotic symmetries of the spacetime metric. Furthermore, we investigate the Doppler effect of light, the Planck photon rocket, and the Thomas precession, affirming that they all remain in the same forms of the standard special relativity under the generalized Lorentz transformation. Additionally, we obtain the general formula of the Thomas precession, which gives a clear geometric meaning from the perspective of the gauge field theory in the relativistic velocity space.

This study deals with the evaluation of the longitudinal and transverse coupling impedance of a charge travelling in a drift tube with a perfectly electric conductive angular strip. The problem is formulated in the particle frame as dual series equations, and efficiently solved through the representation of the unknown in terms of Neumann series. Then, the electric parameters are obtained in the pipe frame through the Lorentz transforms. The presented solution can be adopted as the general methodology in case of angular discontinuities in particle accelerators. [ABSTRACT FROM AUTHOR]

The simultaneity framework describes the relativistic interaction of time with space. The two major proposed simultaneity frameworks are differential simultaneity, in which time is offset with distance in "moving" or rotating frames for each "stationary" observer, and absolute simultaneity, in which time is not offset with distance. We use the Mansouri and Sexl test theory to analyze the simultaneity framework in rotating frames in the absence of spacetime curvature. The Mansouri and Sexl test theory has four parameters. Three parameters describe relativistic effects. The fourth parameter, 𝜖(v), was described as a convention on clock synchronization. We show that 𝜖(v) is not a convention, but is instead a descriptor of the simultaneity framework whose value can be determined from the extent of anisotropy in the unidirectional one-way speed of light. In rotating frames, one-way light speed anisotropy is described by the Sagnac effect equation. We show that four published Sagnac equations form a relativistic series based on relativistic kinematics and simultaneity framework. Only the conventional Sagnac effect equation, and its associated isotropic two-way speed of light, is found to match high-resolution optical data. Using the conventional Sagnac effect equation, we show that 𝜖(v) has a null value in rotating frames, which implies absolute simultaneity. Introducing the empirical Mansouri and Sexl parameter values into the test theory equations generates the rotational form of the absolute Lorentz transformation, implying that this transformation accurately describes rotational relativistic effects. [ABSTRACT FROM AUTHOR]

Rotational transformations describe relativistic effects in rotating frames. There are four major kinematic rotational transformations: the Langevin metric; Post transformation; Franklin transformation; and the rotational form of the absolute Lorentz transformation. The four transformations exhibit different combinations of relativistic effects and simultaneity frameworks, and generate different predictions for relativistic phenomena. Here, the predictions of the four rotational transformations are compared with recent optical data that has sufficient resolution to distinguish the transformations. We show that the rotational absolute Lorentz transformation matches diverse relativistic optical and nonoptical rotational data. These include experimental observations of length contraction, directional time dilation, anisotropic one-way speed of light, isotropic two-way speed of light, and the conventional Sagnac effect. In contrast, the other three transformations do not match the full range of rotating-frame relativistic observations. [ABSTRACT FROM AUTHOR]

The time transformation governing the operation of the global positioning system (GPS) clocks and the time corrections required to ensure sustained clock synchronization are determined to be the Selleri time transformation and not the Lorentz time transformation. This condition, as well as GPS-demonstrated light speed anisotropy, falsifies the Lorentz transformations and special relativity. [ABSTRACT FROM AUTHOR]

We use the method of field decomposition, a widely used technique in relativistic magnetohydrodynamics, to study the small velocity approximation (SVA) of the Lorentz transformation in Maxwell equations for slowly moving media. The “deformed” Maxwell equations derived using SVA in the lab frame can be put into the conventional form of Maxwell equations in the medium’s co-moving frame. Our results show that the Lorentz transformation in the SVA of up to O(v/c) (v is the speed of the medium and c is the speed of light in a vacuum) is essential to derive these equations: the time and charge density must also change when transforming to a different frame, even in the SVA, not just the position and current density, as in the Galilean transformation. This marks the essential difference between the Lorentz transformation and the Galilean one. We show that the integral forms of Faraday and Ampere equations for slowly moving surfaces are consistent with Maxwell equations. We also present Faraday equation in the covariant integral form, in which the electromotive force can be defined as a Lorentz scalar that is independent of the observer’s frame. No evidence exists to support an extension or modification of Maxwell equations.