Lorentz Transformation from Symmetry of Reference Principle

The Lorentz Transformation is traditionally derived requiring the Principle of Relativity and light-speed universality. While the latter can be relaxed, the Principle of Relativity is seen as core to the transformation. The present letter relaxes both statements to the weaker, Symmetry of Reference Principle. Thus the resulting Lorentz transformation and its consequences (time dilatation, length contraction) are, in turn, effects of how we manage space and time. Starting with the 1905 paper of A. Einstein in Ann. Phys. [1] the Lorentz Transformation has been traditionally derived based on the Principle of Relativity and light-speed universality. A number of studies [2] have shown that light-speed universality is not needed-the first such publication (1906) being owed to H. Poincaré [3]. Group theory expresses the transitivity property of relativity (C relative to A, if A to B and B to C) in the form of the group closure relation, respectively the product of two group elements being another element of the group. Pure relativity transformations however, cannot form a group on their own, needing rotations to " close " the group. As such, the full group is not immediate from the Principle of Relativity and needs to be specified (in this case a group of transformations invarying the metric: Λ † GΛ = G, where Λ are the transformations and G the metric). Entering the Lorentz group however, is equivalent to admitting light speed invariance. In this sense the Principle of Relativity and (indirectly) light speed invariance are core to the Lorentz Transformation [4]. The present letter shows however, that neither statement is necessary and that the Lorentz transformation stems from the simpler (weaker) Principle of Symmetry of reference systems. Further more, the Minkowsky metric is not unique in defining relativity. There are two possible classes of transformations, one invarying the Minkowsky and the other the Euclidian metric. The ad-hoc terminology of Minkowsky and Euclidian relativities will be thus adopted throughout this letter. Consider two coordinate systems in motion that at some point were at rest relative to each other and were aligned to have the same orientation, offset and (Euclid-ian) space-metric. The transformation between such coordinate systems is: dt d x ′ = scalar 1×1 vector 1×3 vector 3×1 tensor 3×3 Λ dt d x (1) where the dimensions of the objects involved is given by the subscripts. In general the transformation should be an integral, non-linear transformation, however general considerations about …