Nothing but Relativity, Redux

A previous paper demonstrated that spacetime transformations consistent with the principle of relativity can be derived without assuming explicitly the constancy of the speed of light. Here we correct an error in the earlier paper while showing that this derivation can be done under weaker assumptions, in particular, without an implicit assumption of differentiability or even continuity for the spacetime mapping. Hence, these historic results could have been derived centuries ago, even before the advent of calculus. Relativity—the notion that the laws of physics remain the same in all inertial frames—is a cornerstone of Einstein’s Special Theory of Relativity [1] as well as a cornerstone of physics in general. The other cornerstone of Einstein’s Relativity is the notion that the speed of light remains constant in all inertial frames. Einstein postulated the constancy of the speed of light and showed how that brought electrodynamic phenomena under the purview of the principle of relativity. In his thought-provoking paper titled Nothing but Relativity, Palash B. Pal showed how spacetime transformations consistent with the Einsteinian (Lorentz) transformation can be derived without assuming explicitly the constancy of the speed of light and without invoking gedanken experiments involving light rays [2]. This observation, which may seem remarkable to many, has been noted by numerous authors dating as far back as 1910. Papers by Ignatowsky [3, 4] and Frank and Rothe [5] are notable early examples. 1 Pal [2] lists authors who published variations on this theme going back to the mid-1960s, while Berzi and Gorini’s 1969 paper [6] lists many earlier references. Some physics textbooks reflect this “relativity only” observation [7], while others [8] follow Einstein’s seminal paper [1] by basing the derivation on an assumed constancy of the speed of light. The straightforward mathematical derivation given by Pal [2] was based on the principle of relativity, the isotropy of space, and the homogeneity of space and time. Pal invoked the principle of relativity only insofar as needed to assert the reciprocity property of the inertial spacetime transformation, which Berzi and Gorini [6] showed was a nontrivial consequence of the principle of relativity and the isotropy of space. We point out here that a less restrictive but adequate set of assumptions for deriving these results can be † Eur. J. Phys. 28 (2007) 1145-1150. 1 Ignatowsky’s paper in vol. 17 of Archiv Der Mathematik Und Physik [4] has been erroneously referenced as published in 1910, while his vol. 18 paper in the same journal has been referenced as published in 1911. In fact, both vols. 17 and 18 were published in 1911 (vol. 16 covered 1910 and vol. 19 covered 1912).