An invariant formulation of special relativity, or the "true transformations relativity," and comparison with experiments

Different formulations of special relativity (SR) are briefly theoretically discussed. In the first formulation SR is understood as the theory of a four-dimensional (4D) spacetime with the pseudo-Euclidean geometry. It is an invariant formulation of SR, which we call the "true transformations (TT) relativity." There a physical quantity in the 4D spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant formulation is compared with the usual covariant formulation, which mainly deals with the basis components of tensors in a specific, i.e., Einstein's coordinatization of the chosen inertial frame of reference. The third formulation is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from "3+1" space and time, e.g., the synchronously determined spatial length. This formulation is called the "apparent transformations (AT) relativity." Some of the well-known experiments: the "muon" experiment, the Michelson-Morley type experiments, the Kennedy-Thorndike type experiments, and the Ives-Stilwell type experiments are analyzed using the nonrelativistic theory and the mentioned different formulations of SR. It is shown that all the experiments (when they are complete from the "TT relativity" viewpoint) are in agreement with the "TT relativity"' but not always with the "AT relativity."