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In this article, linear transformations of coordinates to a superluminal inertial reference frame are presented. Even if there is no need to use imaginary numbers to maintain c invariant, these functions are just intended as a mathematical curiosity not necessarily having a real physical meaning. Possible applications to our world, if any, are left to the reader.

The special theory of relativity demands, by Einstein's two postulates (i) the principle of relativity and (ii) the constancy of the speed of light in vacuum, that a spherical wave of light in one inertial system transforms, via the Lorentz transformation, into a spherical wave of light (the Lorentz sphere) in another inertial system when the systems are in constant relative rectilinear motion. However, the Lorentz transformation in fact transforms a spherical wave of light into a translated ellipsoidal wave of light even though the speed of light in vacuum is invariant. The special theory of relativity is logically inconsistent and therefore invalid.

All Lorentz four-vectors of Special Relativity (SR) are derived from a basic Lorentz four-position in a Minkowski space. This article explores use of a Lorentz four-displacement, describing translatory motion of particles in a 4-dimensional space irrespective of position, as a basic four-vector. Derivations based on this assumption have striking outcomes and impact: All four-vectors of SR associated with frame invariants are intact and valid. All postulates and predictions of SR are intact and valid, with two exceptions - Length contraction in the boost direction and relativity of simultaneity are both obviated. Notably, these two aspects of SR have never been definitively proven in experiments and can therefore be challenged. We propose an adapted SR theory, based on four-displacement of particles, predicting the following revolutionary and far-reaching consequences: 1. 4D-space is absolute and Euclidean relative to which all particles, mass and massless, are never at rest and perform an equal distance of translatory displacement at a universal displacement rate equivalent to the speed of light. Inertial frames of 3D-space+time remain relative. 2. What appear as Lorentz four-vectors in a Minkowski metric of a relative spacetime are actually Lorentz four-vectors in an absolute Euclidian 4D-space. 3. Special Relativity actually emerges from describing properties of particles in 4 momentum-space, irrespective of position and localization in space. In this form, SR achieves inherent compatibility with Quantum theories, reflecting the particle-wave duality of particle fields. 4. Time is a property of mass particles associated with their displacement in 4D-space, as opposed to a Minkowski-space property. 5. Frames of 3D-space+time represent mixed domain coordinates: 3 spatial coordinates of position-space and a temporal coordinate of momentum-space, unlike the position-space status of 4 spacetime coordinates in conventional SR.

By means of the Lorentz Transformation, Einstein’s Special Theory of Relativity purports invariance of the standard wave equation. Counter-examples, satisfying the Lorentz Transformation, and hence Lorentz Invariance, prove that the Lorentz Transformation does not in fact produce invariance of the standard wave equation. Systems of clock-synchronised stationary observers are Galilean and necessarily transform by the Galilean Transformation. Einstein’s insistence that inertial (i.e. Galilean) systems of clock-synchronised stationary observers transform, not by the Galilean Transformation, but by the non-Galilean Lorentz Transformation, is logically inconsistent. The Special Theory of Relativity is therefore logically inconsistent. Therefore, it is false. The Lorentz Transformation is meaningless.

In this paper the Lorentz Transformation is shown to be merely a set of restricted equations stemmed from the Galileo transformation applied to a particular conversion reflecting the theorized principle of the speed of light invariance implemented in the direction of the relative motion between the inertial reference frames. Consequently, the Lorentz transformation is shown to be restricted to time and longitudinal space coordinate values different from zero. The deduction of the time dilation and length contraction becomes unfeasible under such restrictions. It follows that the Lorentz transformation possesses no other effects than mathematically expressing the speed of light postulate in the relative motion direction; that is, the coordinate of the tip point of a light ray traveling in the direction of the relative motion, given by x=ct in the “stationary” frame, is transformed to x'=ct' with respect to the “traveling” frame, with c being the light speed in empty space. Furthermore, the application of the Lorentz transformation to events having restricted coordinates is shown to result in mathematical contradictions.

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 noninertial 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 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 noninertial frame of reference without considering gravitation.VC 2014 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-27.1.173] Resume: Le succes de la relativite restreinte (RR) provient de l’exigence de la covariance de Lorentz pour toutes les equations physiques. L’explication en ce qui concerne la covariance de Lorentz est fondee sur deux hypotheses, a savoir le principe de la relativite et la constance de la vitesse de la lumiere. Cependant, les declarations de principe de la relativite restreinte sont diverses et confuses. La covariance des equations physiques et l’egalite des referentiels inertiels sont melangees. L’egalite des referentiels inertiels est evidente, mais la covariance des equations physiques est une exigence plus avancee. En outre, la facon dont la propriete de propagation de la lumiere dans une position centrale dans la RR a provoque des malentendus vers l’espace-temps, et il y a aussi une circularite logique entre la mesure de la vitesse de la lumiere et la synchronisation des horloges. Celles-ci ont obstrue d’etendre correctement la theorie de l’espace-temps a partir d’un referentiel inertiel a un referentiel non-inertiel. Ce sont les principales raisons pour lesquelles beaucoup de gens critiquent la RR. Dans cet article, les deux hypotheses sont examinees en detail et une nouvelle exigence pour les equations de la physique est proposee. L’exigence est l’Exigence d’Exhaustivite Special, a savoir, les equations physiques utilisees pour decrire la dynamique de la matiere et/ou les champs devraient inclure les descriptions que non seulement la matiere et/ou les champs sont au repos par rapport a un referentiel inertiel, mais aussi ils se deplacent par rapport a ce referentiel. En appuyant sur cette exigence et l’egalite des referentiels inertiels, nous pouvons nous approcher a la RR. Ainsi, la theorie de Lorentz covariance a une base claire et solide. La constance de la vitesse de la lumiere est seulement une deduction, pas une premisse. La covariance de Lorentz est juste une caracteristique des equations Speciales Completes. Les equations de Maxwell satisfont automatiquement les transformations de Lorentz sans aucune modification, tandis que la Loi de Newton sur la gravite ne le fait pas, parce que les equations de cette loi ne sont pas Speciales Completes mais celles de Maxwell les sont. La nouvelle approche a ouvert une route menant vers gordonliu168@gmail.com 0836-1398/2014/27(1)/173/10/$25.00 VC 2014 Physics Essays Publication 173 PHYSICS ESSAYS 27, 1 (2014)

It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamic quantities are derived from the pre-assumption that the equations of electrodynamics are covariant against these---unknown---transformation rules. There are several problems to be raised concerning these derivations. This is, however, not our main concern in this paper. Even if these derivations are regarded as unquestionable, they leave open the following fundamental question: Are the so-obtained transformation rules indeed identical with the true transformation laws of the empirically ascertained electrodynamic quantities? This is of course an empirical question. In this paper, we will answer this question in a purely theoretical framework by applying what J. S. Bell calls "Lorentzian pedagogy"---according to which the laws of physics in any one reference frame account for all physical phenomena, including what a moving observer must see when performs measurement operations with moving measuring devices. We will show that the real transformation laws are indeed identical with the ones obtained by presuming the covariance of the equations of electrodynamics, and that the covariance is indeed satisfied. Beforehand, however, we need to clarify the operational definitions of the fundamental electrodynamic quantities. As we will see, these semantic issues are not as trivial as one might think., Comment: 23 pages, no figure. arXiv admin note: Majority of the text overlaps with arXiv:0912.4388

In view of numerous experimental results reported in the past decades on the observation of faster-than-light electromagnetic signals, we analyze the structure of relativistic kinematics, where such superluminal signals are allowed. As the first step, we suggest replacing the Einstein postulates with the general relativity principle (the possibility of describing any phenomenon in any frame of reference achievable in nature) applied to an inertial motion in an empty space. Then, as in common relativistic kinematics, we also arrive at the Lorentz transformations between inertial reference frames, where a superluminal motion of massless entities is not prohibited (in particular, for perturbations of bound electromagnetic field). However, for any objects with a finite rest mass, the limited velocity remains always less than the light velocity c, and in such a way we avoid the tachyonic-type theories in their common meaning. We show that the application of superluminal electromagnetic signals to synchronization of distant clocks yields the common expressions for the relativity of the simultaneity of events for different inertial observers. This result confirms the validity of the Lorentz transformations in generalized relativistic kinematics, though along with superluminal signals. Hence we arrive at the invariance of the space-time interval, as in common relativistic kinematics, where, however, the superluminal motion of massless entities is allowed. Even so, no further changes emerge in relativistic dynamics and other common relativistic implications. Finally, we consider causal paradoxes related to the propagation and exchange of superluminal signals between inertial observers and provide their resolution

Assuming that the Universe consists of absolute space, or more properly aether-space, (as evidenced by several well-documented experiments, and supported by the DSSU* theory) the transformation of the coordinates from one moving inertial frame to another frame while retaining the information of their absolute motion (with respect to aether-space) requires the application of the Lorentz transformation, not once, but twice. The combined transformations are detailed. The resulting expressions for space-and-time coordinates are then used to redefine: absolute velocity, apparent velocity, length contraction, momentum, mass, and kinetic energy —all redefined in terms of absolute inertial motion. Each of the resulting expressions are validated by showing that each reduces, under the specified condition, to the Einstein Special Relativity form (as each must) and also reduces, under specified speed restrictions, to the familiar Newtonian-Galilean form (again, as each must). In effect, DSSU relativity (the relativity of absolute motion) encompasses both Special relativity and Classical relativity. * DSSU is the acronym for Dynamic Steady State Universe. It is a model based on the premise that all things are processes.

This paper provides for the categories of “primary stationary system” and “conditional stationary system” for inertial frames of reference, which Einstein believed were all equivalent. The “primary stationary system” is a coordinate system in which all clocks are in sync, in absolute terms, even if clocks were not synchronized as proposed by Einstein. The “conditional coordinate system,” on the other hand, is a coordinate system in which an observer in this coordinate system, in order to be able to state that light propagates isotropically from a light source in his coordinate system, must use the clock synchronization method proposed by Einstein to synchronize with the time of the clock in his coordinate system. This paper considered the coordinate system of rod 2, moving at constant velocity relative to a “conditional stationary system” (Einstein’s inertial frame of reference), as considered in this paper as equivalent to inertial frame of reference of the special theory of relativity. Then, it was shown that, the actual time adjustment made by an observer in this coordinate system to synchronize two clocks at each end of a rod by using Einstein’s method is not the same as the value predicted by the special theory of relativity. The reason for the discrepancy between experimental values and theoretical values is due to the unknown velocity v which is related to “conditional stationary systems,” the

This paper explores the possibility of detecting a difference in the Doppler Shift phenomenon when generated in different inertial frames, as a means of testing Einstein’s Principle of Relativity. Specifically, the Doppler shifts generated alternatively by movement of the observer and movement of the light source are examined and show an observable difference that is inconsistent with the Relativity Principle.

The theory of reference frames is a new theory of relativity that represents the scientific evolution of Einstein’s relativity. In this essay the following points are examined: considerations about the theory of relativity of Albert Einstein, general notes on the theory of reference frames, the Michelson–Morley experiment, the new equations of space-time transformation, the kinematic principles of the theory of reference frames, the transformations of Maxwell’s equations for inertial systems, the Doppler effect, and the mystery of supernovas. The new theory responds to numerous and recent experimental results that are not confirmed by previous theories. In particular, the interpretation of the luminous phenomenon that causes supernovas is at variance with theoretical predictions of Einstein’s relativity but is in accordance with those of the theory of reference frames. The most important theoretical concept in this essay is the inertial time, which is a common time valid for all inertial reference frames that replaces Newtonian– Galilean absolute time and Einstein’s relativistic time.

This paper examines the possibility of detecting a difference in the stellar aberration phenomenon when generated in different inertial frames, as a means of testing Einstein’s principle of relativity. Specifically, stellar aberration resulting alternately from movement of Earth and movement of the light source are examined and show an observable difference that is inconsistent with the relativity principle.

The definition of frame of reference in classical space–time bundles is investigated, in the general case of bundles whose fibres are not euclidean and of frames without the rigidity requirement. The enriching of the geometry of the bundle following from the presence of a frame of reference, viewed as a particular time–like vector field, is analyzed. Space–like tensor fields, unless totally contravariant, are proved to be correctly defined only if a frame of reference H is assigned. The connection associated to H is introduced and the selection of special covariant and time derivatives associated to H is soundly motivated. The usual kinematic quantities relative to H are defined. The relationships between pairs of frames are examinated and the concepts of relative rigidity, relative spin and relative acceleration are defined and investigated. The main results of Relative Kinematics are shown to fit readily into this geometric context.

The Lorentz transformation has some limitations: (a) it is not associative, (b) it is not isotropic (in the sense that the triangular law of addition of velocities is valid for velocities in some directions and not valid for velocities in other directions), and (c) it does not have the group property without rotation. In this paper we have applied mixed-number algebra to develop the mixed-number Lorentz trans-formation, which is free of these limitations. Mixed-number Lorentz transforma-tions for electric and magnetic fields are also presented. Key words: associative, isotropic, group property 1. INTRODUCTION Let us consider two inertial frames of reference S and S¢, where the frame S is at rest and the frame S¢ is moving along the X axis with velocity V with respect to the S frame. The space and time coordinates of S and S¢ are ( x , y , z , t ) and ( x ¢, y ¢, z ¢, t ¢), respectively. The relation between the coordinates of S and S¢, which is called the special Lorentz transformation, can be written as

Einstein's argument on the relativity of simultaneity itself is the first result of the special theory of relativity. This argument is reflected in the structure and functioning of the physical world. The arbitrary nature of the decision regarding the particular inertial reference frame from which the argument on the relativity of simultaneity begins is discussed. It is this arbitrary, or freely made, decision that is the basis for the significance of the argument of the relativity of simultaneity itself on the structure and functioning of the physical world. The same concrete circumstances in the physical world can support either direction in the argument on the relativity of simultaneity, as well as other results, in the special theory (i.e., with either of two inertial reference frames in uniform translational motion relative to one another considered "stationary" while the other inertial reference frame is considered "moving"). There are different sets of results in the physical world associated with the choice in direction that is made, for example as concerns the temporal durations of occurrences or the spatial lengths of existents in these inertial reference frames. Experiments testing the equations derived in the special theory that relate the temporal durations of occurrences or the spatial lengths of physical existents in inertial reference frames in uniform motion relative to one another generally have not tested predictions with both of the two inertial reference frames considered, in separate scenarios, the "stationary" reference frame. It is proposed that empirical tests in two directions be developed and conducted. If empirical results from such tests support the predictions of the special theory, these empirical results would support the thesis that there is a cognitive factor in Einstein's argument on the relativity of simultaneity that affects results derived in the special theory concerning the physical world. One such test is proposed.

The ratio of the wave functions of an accelerating frame of reference and an inertial frame of reference is determined. The velocity of points of a frame of reference with respect to an inertial frame of reference is then allowed to depend on space and time. The curl of this velocity is shown to be zero. Resume: Nous determinons le rapport des fonctions d’onde d’un referentiel en acceleration et d’un referentiel inertiel. Nous laissons la vitesse des points d’un referentiel par rapport a un referentiel inertiel etre une fonction de l’espace et du temps. Nous montrons que le rotationnel de cette vitesse est nul.

We consider the detection of a superluminal signal. We show that there exists a particular inertial frame in which the superluminal signal is not uniquely localizable and that it is detected everywhere at the same instant within the inertial frame. This leads to a novel paradox to which we attribute the relativity of simultaneity. keywords: superluminal, simultaneity, paradox, tachyons, localization.