Lorentz Transformations from Intrinsic Symmetries

Sheng Chao

doi:10.3390/sym8090094

Abstract

We reveal the frame-exchange space-inversion (FESI) symmetry and the frame-exchange time-inversion (FETI) symmetry in the Lorentz transformation and propose a symmetry principle stating that the space-time transformation between two inertial frames is invariant under the FESI or the FETI transformation. In combination with the principle of relativity and the presumed nature of Euclidean space and time, the symmetry principle is employed to derive the proper orthochronous Lorentz transformation without assuming the constancy of the speed of light and specific mathematical requirements (such as group property) a priori. We explicitly demonstrate that the constancy of the speed of light in all inertial frames can be derived using the velocity reciprocity property, which is a deductive consequence of the space–time homogeneity and the space isotropy. The FESI or the FETI symmetry remains to be preserved in the Galilean transformation at the non-relativistic limit. Other similar symmetry operations result in either trivial transformations or improper and/or non-orthochronous Lorentz transformations, which do not form groups.

Highlights

The importance of the Lorentz transformation (LT) in the special theory of relativity can hardly be overemphasized
We show that either the frame-exchange space-inversion (FESI) or the frame-exchange time-inversion (FETI) can lead to the proper orthochronous LT, and both are preserved for the Galilean transformation (GT)
Among the possible discrete-type space–time symmetry operations similar to the LT, we have found that only the satisfies the following two studied which leave the coordinate transformation between two inertial frames formally similar requirements: (1)

Summary

Introduction

The importance of the Lorentz transformation (LT) in the special theory of relativity can hardly be overemphasized. It is known that the second postulate is not a necessary ingredient in the axiomatic development of the theory It has been shown, as far back as 1910s [2,3], that the LT can be derived using the velocity reciprocity property for the relative velocity of two inertial frames and a mathematical requirement of the transformation to be a one-parameter linear group [4,5,6,7]. Relaxing some presumed postulates can lead to a “Very Special Relativity” proposed by Cohen and Glashow [18,19,20] or an extension of special relativity by Hill and Cox [21] that is applicable to relative velocities greater than the speed of light These extensions largely widen our scope of exploring the more fundamental side of Lorentz symmetry and give impetus to further experimental research. The necessary condition for physical causality is shown to be a deductive consequence of this symmetry principle

Derivation of the Lorentz Transformation

Discussion

Conclusions