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In order to describe physical reality, we need a special (gravity-free) relativity (SR) that is founded upon general non-uniform motions as they occur in our environment, and holds for the non-inertial reference frame of our laboratory. Such a generalized form of SR can be built upon a version of relativistic kinematics valid for non-uniform motions. From the view-point of Einstein's principle of reciprocity, the qualitative analysis of coordinate transformations allows the derivation of not only the reciprocal Lorentz transformation (RLT), involving inertial motions, but also a non-reciprocal transformation (N-RLT), called Co-Lorentz transformation (Co-LT), valid for non-uniform motions. Consequently, relativistic kinematics is a double faced theory: it assumes the RLT when the motion is inertial, and the Co-LT when the motion is non-inertial. The complementarity of RLT and Co-LT implies the complementarity of the reciprocal Einstein' special relativity (RESR) and that of the non-reciprocal ESR (N - ESR) called Co-Einstein' special relativity (Co-ESR). By neglecting the gravitational effects a relativistic electrodynamics, founded on Co-ESR, is elaborated. Some properties, related to the non-reciprocity of Co-LT, including the violation of time-reversal symmetry, are discussed. Adding to the classical RLT and RESR their corresponding complementary versions Co-LT and Co-ESR, produces a complete view of the relativity of physical reality.

Motto: Extended Special Relativity is like the Moon which shows us only one of her faces: it is Einstein's SR. The hidden face is Hertz's SR.

The qualitative analysis of coordinate transformations, from the view-point of the reciprocity principle, allows the derivation of not only the Lorentz's transformation (LT), involving inertial motions, but also of a non-reciprocal transformation (N-LT), here called the Co-Lorentz transformation (Co-LT), valid for non-uniform motions. Consequently, relativistic kinematics is a double faced theory: it assumes either the LT, when the motion is inertial, or the Co-LT when the motion is non-inertial. The complementarity of LT and Co-LT implies the complementarity of the corresponding Einstein special relativity (ESR) and a non-reciprocal counter-part (N - ESR), here called Co-Einstein special relativity (Co-ESR). By neglecting gravitational effects, a relativistic electrodynamics, founded on Co-ESR is elaborated. Adding to the classical LT and ESR their corresponding complementary versions Co-LT and Co-ESR, a complete view of the special relativity of physical reality is obtained.

Motto: Extended Special Relativity is like the Moon which shows us only one of her faces: it is Einstein's SR. The hidden face is Hertz's SR.

In this work, Maxwell-Hertz electrodynamics (MHE), valid for non-uniform motions as they occur in the physical reality and hold for the non-inertial reference frame of our laboratory but at small velocities only, is extended at relativistic velocities.  The new theory, called Hertz's relativistic electrodynamics only, is completely independent and built-up in a completely different way as regards Einstein's special relativity (ESR).  HRE, a coordinate-free formulation, does not need postulates, but confirms the constancy principle of speed of light in vacuum.  All experiments of first and second order in v²/c² are correctly interpreted.  To this theory a Hertzian kinematics and dynamics are associated.  HRE with its corresponding mechanics from Hertz's special relativity (HSR), as a complementary theory to ESR.  According to the principle of complementary and neglecting the gravitational effects, the Extended Special Relativity (ExSR) is a double faced theory which becomes either ESR when the motion is inertial or HSR when the motion is non-inertial.  The complementarity of both theories assumes that the two descriptions cannot be employed for the same motion, being mutually exclusive.  Consequently, to every statement of one of the ExSR a complementary statement of the other ExSR corresponds.  The completeness of ESR with HSR ensures an extended view over the relativity in our physical world.

As a consequence of this identity, all experiments on the electrodynamics, correctly interpreted within the frame of Minkowski theory, are explained in the same way within the framework of HRE.  Among these, we quote the experiments of Wilson-Wilson (Roentgen-Eichenwald) of effects of electric (magnetic) type.  Optical experiments, such as those of Michelson-Morley and Sagnac-Harrens are interpreted in the same way.  The first-order Fresnel drag-law predicted by SR is predicted here also.

Einstein's Special Relativity (ESR) in its original formulation, is limited to inertial motions only, while an insight into real-world shows that the motions are non-inertial. It is hard to accept that the launching of a rocket implies a succession of inertial motions. Similarly, during the re-entry of a satellite in the Earth's atmosphere, its continously decaying angular momentum and the violations of the Lorentz - Poincare symmetries, are then evident. Form the viewpoint of kinematics, defined as the science of pure motion, apart from causes, the motion may be either inertial of non-inertial. Since a motion cannot be, at the same time, either a uniform translation or a non-uniform motion, then they form a pair of complementary kinematic concepts, mutually exclusive. If the gravitational effects are neglected, ESR assumes uniform translations, while under similar circumstances (regarding the neglecting of the space curvature) to the best our our knowledge, nothing is known in the literature to take into consideration non-inertial motions. As a consequence, we shall denote by Special Relativity that branch of physics where by neglecting the gravitational effects, the motion may be either inertial or non-inertial, called in this paper permissible motions. As a consequence, the following chart of complementarities:

The identification of the Thomas rotation angle of Cartesian coordinates with the angle between relativistic composite velocities leads to a conflict with the concept of intertial motion. As a consequence Thomas rotation is unable to extend the 1 + 1 Lorentz transformation to 1 + 3 dimensions.

In order to extend the 1 + 1 Lorentz transformation to one with 1 + 3 dimensions, the so-called Thomas rotation is inevitably involved. This, in turn, provides the relativistic interpretations of the non-commutative and non-associative composition laws of non-collinear velocities. When dealing with two successive Lorentz transformations involving non-collinear velocities, two peculiarities are revealed. The first is related to the vector-scaler pair (J, p) and the second to the vector pair (E, B); neither implies the Thomas rotation. Ungar's attempt to solve these difficulties by applying the Thomas rotation if invalidated by the conservation law of the electromagnetic field. The result is a paradox revealing an internal contradiction of the Special Theory of Relativity.