Two Theories of Special Relativity ?

Abstract: Recently, [3], it was shown that Special Relativity is in fact based on one single physical axiom which is that of Reciprocity. Originally, Einstein, [1], established Special Relativity on two physical axioms, namely, the Galilean Relativity and the Constancy of the Speed of Light in inertial reference frames. Soon after, [4,5], it was shown that the Galilean Relativity alone is sufficient for Special Relativity. Here it is important to note that, implicitly, three more assumptions have been used on space-time coordinate transformations, namely, the homogeneity of space-time, the isotropy of space, and a mathematical condition of smoothness type. In [3], a boundedness condition on space-time coordinate transformations is used instead of a usual mathematical smoothness type condition. In this paper it is shown that the respective boundedness condition is closely related to a Principle of Transformation Increment Ratio Limitation, or in short, PTIRL, which has an obvious physical meaning. It is also shown that PTIRL is {\it not} a stronger assumption than that of the mentioned boundedness in [3]. Of interest is the fact that, by formulating PTIRL as a physical axiom, the possibility is opened up for the acceptance, or on the contrary, rejection of this physical axiom PTIRL, thus leading to {\it two} possible theories of Special Relativity. And to add further likelihood to such a possibility, the rejection of PTIRL leads easily to effects which involve unlimited time and/or space intervals, thus are not accessible to usual experimentation for the verification of their validity, or otherwise.