True Transformations of Spacetime Lengths and Apparent Transformations of Spatial and Temporal Distances. I. the Theory

It is shown in this paper that the difference between the two forms of relativity - the ''true transformation (TT) relativity'' and - the ''apparent transformation (AT) relativity'' is essentially caused by the difference in the concept of \emph{sameness} of a physical system, i.e., of a physical quantity, for different, relatively moving, observers. In the ''TT relativity'' the same quantity for different inertial frames of reference is covariantly defined four-dimensional (4D) tensor quantity, which transforms according to the Lorentz transformation as the TT. In the ''AT relativity'' parts of a 4D tensor quantity are often considered as the same quantity for different observers, although they correspond to different quantities in 4D spacetime, and they are not connected by the Lorentz transformation than by the AT. Then the true transformations of a spacetime length and the apparent transformations of a spatial distance (the Lorentz contraction) and of a temporal distance (the usual dilatation of time) are examined in detail. It is proved that only the true transformations of the spacetime length are in agreement with the special relativity as the theory of a 4D spacetime with the pseudo-Euclidean geometry.