In an article in this journal [Verheest, Am. J. Phys. 90, 425–429 (2022)], Frank Verheest presented a proof for the linearity of the Lorentz transformation. We fill in some gaps in his derivation and analyze the role of the light postulate that some physicists, including Verheest, have criticized as a necessary hypothesis for formulating the theory of relativity.
I. INTRODUCTION
In Ref. 1, Verheest derived the linearity property of the Lorentz transformation. The reasons for Lorentz transformations must be linear often mentioned only in passing and without due rigor. Although that attitude is justified from a heuristic viewpoint, a rigorous derivation employing only elementary mathematical tools can be educationally fruitful.
On the other hand, homogeneity and isotropy of space and homogeneity of time imply that linearity is often mentioned without giving further references or comments, leaving the impression that it is a trivial implication.^{5,6}
Rigorous proofs of linearity can be given using different approaches and techniques.^{7–12} Some authors^{8,10,11} prove linearity only from spacetime homogeneity, leaving out isotropy. Whether a detailed proof is necessary or a heuristic justification suffices, as well as the method employed, is a matter of personal taste and philosophical attitude towards a rigorous formulation of the foundations of physics.
In any case, Verheest's approach is a valid contribution for those interested in the foundations of relativity. Although his derivation is generally correct, there are some issues that we consider omissions or gaps in the reasoning rather than mistakes. We explain that problem in Sec. III. However, first, in Sec. II, we delve into another fundamental question that Verheest brought about.
In addition to the linearity issue, the author of Ref. 1 expressed a longstanding concern that some physicists have pointed out regarding the central role that the speed of light seems to play in the principles of relativity theory.^{12–15} That uneasiness is justified since relativity constitutes a central pillar in the theories of modern physics. Notwithstanding the importance of electromagnetic theory, it seems odd that a particular type of phenomenon should play such a central role. In the following section, we explain that the crucial role that light purportedly plays in Einstein's formulation is only apparent and is owed to historical and practical reasons.
II. THE LIGHT PRINCIPLE
As observed in Ref. 1, Einstein based his special theory of relativity on two principles (i) the laws of physics are invariant in all inertial frames of reference and (ii) light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.^{16}
Principle (i) is an extension of the equivalence of inertial reference frames from mechanics to all physical phenomena,^{17} while (ii) is also known as the light principle.
It is relevant to note that (i) and (ii) together imply that the speed of light has the same value for every inertial observer. In modern literature, postulate (ii) is sometimes replaced by stating straightly that light speed is the same in all inertial frames.^{18,19}
In 1905, only two fundamental interactions were known, gravitational and electromagnetic. Newtonian gravity is described by an action at a distance law, i.e., instantaneous interaction. On the other hand, light was known to be an electromagnetic phenomenon with a finite speed, while all attempts to find evidence of a lightcarrying medium had failed. That historical prospect explains why Einstein gave light such a central role, notwithstanding that principle (i) encompasses all physical laws.
The tradition of teaching relativity through the light principle continues to this day. As Verheest has observed, from a conceptual viewpoint, it is more compelling to derive the Lorentz transformations without mentioning the speed of light at all. The first to do that was Vladimir Ignatowski, as early as 1910.^{13} Then, many such formulations followed using different approaches and techniques.^{10,12,14,15,20}
Regarding the role of light in the formulation of the theory of relativity, it is relevant to observe the following:

The light principle can be replaced by the more general principle, (ii′) the principle of finiteness of the speed of propagation of interactions.^{5,21}

As a result of (i) and (ii′), we obtain that interactions taking place as a consequence of the fundamental physical laws, such as electromagnetic, gravitational, weak, and strong or any eventual not yet known law, must take place at the same speed which is, therefore, a universal constant. So, this approach explains why, according to relativity all fundamental interactions, and not just light (electromagnetic), occurs at speed c.
Since arbitrary interactions must occur as a consequence of the fundamental ones, the universal speed represents the upper limit for the speed transmission of any other influence. A typical example is that one hears the thunder much later than one sees the lightning, with light being a fundamental interaction while sound is not.
Principles (i) and (ii′), plus the homogeneity and isotropy assumptions, can lead us to Lorentz transformations through the usual derivations replacing light speed with a finite universal limiting speed based on the exclusion of unobservable instantaneous interactions.
Also, as done by Verheest, we can hold only to principle (i), which puts Galilean and Lorentz transformations on the same basis. Ironically, such an approach has the conceptual advantage of making more evident the essential difference between Galilean and Einstein's relativity, namely, the existence of a universal finite speed limit and the exclusion of instantaneous interactions.
Thus, when we assume there is no limit to the speed of the transmission of interactions, Newton's absolute time is not optional but a necessary imposition.
III. LINEARITY
A. First issue
The first issue arises after equation (8*). There Verheest asserts, “This implies that F is a function of the combined argument x – vt as well as of v” without further explanation. Note that v enters the equation as a parameter, F being a function of the variables x and t.
It is clear that if F has the functional form $ F ( x \u2212 v t ; v )$, (17) is satisfied. However, the former argument constitutes only a sufficient condition, and Verheest's derivation requires F to have that functional form necessarily.
B. Second issue
IV. CONCLUSIONS
We have complemented Verheet's linearity proof with two observations that may ease its detailed understanding for the reader. However, the foundationally relevant points were discussed in Sec. II. From a conceptual viewpoint, we have stressed that it is better to base the derivation of Lorentz transformation using axioms (i) and (ii′), replacing the light principle with a more physically compelling one. In this respect, we highlight two authoritative references, Landau and Lifshitz^{21} and Jackson;^{5} both postulate (ii′) instead of the light principle. In particular, Jackson explicitly spells out, Because special relativity applies to everything, not just light, it is desirable to express the second postulate in terms that convey its generality:
In every inertial frame, there is a finite universal limiting speed C for physical entities.
Thus, by using postulate (ii′) instead of the light principle, we gain physical insight regarding the finite character of the speed of interactions avoiding any particular reference to light.
On the other hand, if to avoid any reference to light, we try to derive the spacetime transformations using only postulate (i), it becomes hard to motivate the puzzling abandonment of the absolute character of time and we leave unanswered Verheest justified question:
How do we incorporate the transformation of time from one inertial observer to the next? Of course, we all know the Lorentz transformation, but how to get there?
The abovementioned question was duly responded to in Sec. II. The form of incorporating time into the transformation is necessarily subject to our assumption about the existence of instantaneous interactions or their impossibility. Instantaneous interactions necessarily imply absolute Newtonian time (12). Hence, the time transformation cannot include spatial variables.
On the contrary, the existence of a universal finite limiting speed for physical interactions requires abandoning absolute time (13), which demands that time enter the transformation as a fourth coordinate depending on the spatial variables.
Finally, we observe that instantaneous interactions can be considered a special case of finite speed interactions by setting the universal constant $ c = \u221e$.
Thus, the mathematical reduction of the Lorentz transformations to the Galilean case when $ c \u2192 \u221e$ (in practice, when $ v / c \u226a 1$) is naturally justified. This is according to “The Hierarchy of Theories.” When an old theory (Newtonian physics) is replaced by a new one (special relativity), the new one does not disprove the former. What actually happened was that the old one continued to provide the correct predictions, but only for a limited set of phenomena.^{23}
ACKNOWLEDGMENTS
The author thanks the reviewers for significantly improving the article.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.