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In vector calculus the addition of two vectors means no more than the addition of their components. According to Special Relativity Theory (SRT) the addition of two velocities signifies more than that. Apart from adding the components we have to divide their sum by a coefficient 1+(v/c)(u/c), which presence indicates that the velocity is not a vector in SRT. If the velocities in SRT were vectors they should be added up as vectors. The ratio of velocities v/c should not be considered as the hyperbolic tangent of an angel. The coefficient mentioned above appeared as the result of the baseless introduction of hyperbolic functions to the SRT formulae. On that basis the formulae although fully consistent are evidently wrong. We have shown that they can easily be reduced to the correct formulae of the Galilean Transform. Also we have shown that in the case of a 3D space there are three different coefficients and three corresponding different times for one moving object. Therefore there is a choice to be made. On the one hand by dismissing the hyperbolic functions from SRT we annihilate SRT and on the other, by accepting them we reject (commonly accepted) the rules of vector calculus and obtain in 3D case, three different times instead of one.

In this paper we show that with the use of hyperbolic functions calculus the Einstein formula for velocity addition and the Lorentz transform formulae can be both derived from the Minkowski space-time formula. This simply means that the formulae are fully consistent, although it says nothing about the physical meaning of the symbols used. We claim that two different versions of physical interpretation of the formulae are possible. In the Special Relativity Theory moving objects are considered in two different inertial frames of reference. Except for the Minkowski proper time, other physical quantities are considered as relative. It is believed that even the simultaneity is relative. We propose something quite different, a notion in which we have adopted: the Minkowski formulae as the definition of a local time, proper time as the universal time, relative distance as the absolute distance, and relative time as the local time. In the Minkowski space-time (one frame of reference only) we consider the following: two observers A and B (moving or stationary), their distances from the origin of coordinates and resulting local times. When the distance remains unchanged, i.e. the object or the observer do not move, the difference between the indication of local time and the indication of universal time is constant. With the change of distance (the object or the person moves) the local time depends on an absolute velocity of that movement. In the theory of local time there is no relativity of simultaneity. When comparing the two possible versions of interpretation it is evident that the theory of local time is at least as believable as the Special Relativity Theory.

There are two versions of Maxwell's equations. The classical version uses the Coulomb gauge and the relativistic version has electrodynamic potentials fulfilling the non-homogeneous equations for waves. With the use of the Lorenz gauge such equations have been obtained from the fourth Maxwell's equation and the equation for potentials of vector of electric intensity E. We show that the last equation after differentiation in time and multiplication by dielectric permittivity is similar to the fourth Maxwell's equation. We also show that after obvious corrections the equations are equivalent. We claim that in the fourth Maxwell's equation, instead of the vector of electric current density j, its irrotational component should only be applied. We also claim that in the equation for potentials of vector  E instead of vector A there should be only its solenoidal component. It is shown that the corrected form of these two equations is fully consistent with the rest of Maxwell's equations. As the result of replacing vectors j and A by their proper components the problem of different gauges disappeared. Only one gauge i.e. that of Lorenz with a changed sign is shown to be necessary. As the result of the proposed modification in the case of waves, the wave equations for the potentials - as well as for the field vectors - are homogeneous. It is not so in the relativistic version of Maxwell's equations. We claim that the classical Maxwell's equations in the proposed version are fully consistent, satisfactory and do not fit the Special Relativity Theory.

It is shown that in particular case of photons, the Lorentz transform formulae for distance and time are simpler and take the same form as that of - corrected for relativity - Doppler formulae for the length and period of the wave. It means that considering light as particles and using Lorentz transform we obtained an unexpected result indicating that light is an electromagnetic wave which obeys the Doppler formulae. According to Doppler the length and period of the wave are relative and transform in such a way that their ratio does not change. The Doppler formulae show that the phase speed of the wave defined as the ratio of the length of the wave to its period is absolute. Should light be considered as a wave then the second Einstein postulate would automatically be given by Doppler formulae whether corrected for relativity or not. Light considered as a wave would have at least two speeds: the absolute phase speed and the relative speed of the wave front. Particles (photons) do not have two different speeds but the waves do. Instead of considering the light as photons and introducing the second Einstein postulate, we propose to accept the idea of particle-wave duality of light. In the case of waves it would automatically assure the existence of the absolute phase speed and would provide the relative speed of the light wave front. It is argued that introducing corrections for relativity into the Doppler formulae for electromagnetic waves we should also correct Doppler formulae for elastic waves. Otherwise the first Einstein postulate is violated.

The paper presents the correspondence between physical quantities used in electrodynamics on the one hand and on the other those applied in elasticity and hydrodynamics having taken into account the SI units of physical quantities and assumed an analogy between mass and electric charge. Due to such correspondence almost every equation of electrodynamics is accompanied by an equation used in theory of elastic waves or hydrodynamics. Consistency of the analogy is clear and evidenced in the four tables (P waves, S waves, irrotational direct currents and rotational currents). Some changes concerning the form or the understanding of several equations are proposed. It has been shown that the Coulomb gauge and the first Helmholtz equation used in hydrodynamics have exactly the same physical meaning. Lorentz gauge with its sign changed is recognized as corresponding to one of the relations between stress and strain used in the theory of pressure elastic waves. The formula for the energy volume density of electromagnetic waves was modified in order to obtain the consistency with the corresponding formula for elastic S waves. Then the formula for the volume density of elastic force corresponding to that of the Lorentz force is proposed. The analogy between the Ohm's law for non-rotational direct electric current and the Hagen and Poiseuille's law for non-rotational flows is drawn.

The extension of the idea of local time used on the Earth to the entire solar system is proposed. It is shown how the relative local time in the solar system depends on the observer velocity. The corresponding relative local velocity is defined. The product of such a velocity and a local time interval is required to provide the classical relative distance. Presented formulae for the planetary Local Time Theory (LTT) are compared to those of Special Relativity Theory (SRT). It is shown that the proper time for a moving object defined by Minkowski amounts to the geometrical mean of two local times for two objects moving with velocities of two different orientations. It is shown that in SRT the relative distance between two observers A and B moving with different velocities depends on, which one of two is formally considered as an observer. It is proposed to consider LTT as an alternative to SRT.

The Poincar?-Lorentz approach to relativity differs from the Einstein-Minkowski approach because it includes an aether, and so admits an observer velocity relative to that aether. With that additional parameter, Poincar?-Lorentz theory can produce both time dilation and time contraction.