We present a comprehensive discussion of the formulation of the kinematics of special relativity, i.e., the Lorentz transformation. We begin with a concise new proof that the principle of relativity implies that the transformation of event coordinates between inertial reference frames is linear. We then give a clear derivation of the pre-Lorentz transformation, which follows from the principle of relativity. We then show that the pre-Lorentz transformation and the inertial invariance of the speed of light together result in the Lorentz transformation. This, of course, is essentially the traditional formulation. We next present two additional formulations, one using Lorentz–Fitzgerald contraction and one using time dilation, instead of inertial invariance. This is reasonable since Lorentz–Fitzgerald contraction and time dilation are about as well established as and are arguably less abstract than inertial invariance, and thus may profitably be used instead of inertial invariance to complete the formulation. We then present a complete proof that the pre-Lorentz transformation and the requirement of closure upon composition together imply that the transformation is either a Galilean transformation or a generalized Lorentz transformation. This is noteworthy in that it gets ever so close to the Lorentz transformation without invoking light. In the course of this, we obtain a generalized velocity addition rule, which reduces to the velocity addition rule of special relativity. We next show that the generalized Lorentz transformation, together with inertial invariance, Lorentz–Fitzgerald contraction, and time dilation, used one at a time, yields three more formulations. We then show that the unspecified, nonzero, constant speed in the generalized Lorentz transformation can be determined without any reference to light, thereby obtaining a seventh formulation. Light plays no explicit role in the four formulations employing Lorentz–Fitzgerald contraction and time dilation and plays no role whatsoever in the seventh formulation. Thus, and this is a fact which should be strongly emphasized, the formulation of special relativity in no way depends upon the nature of electromagnetic radiation. We conclude by briefly discussing these seven formulations of the kinematics of special relativity and some associated implications.

## References

It appears that of all the subfields of physics, the one that has commanded the most attention, as far as books published, is special relativity. Accordingly, we cannot give even a representative selection of the published books on special relativity. Rather, we note a baker's dozen of the books that we have found most accessible and useful.

One might argue that the principle of relativity does not imply that physical space is three-dimensional, but does imply that it is Euclidean. Nevertheless, we consider it preferable to explicitly state that we assume that physical space is three-dimensional and Euclidean.

In this regard, we note the statement on p. 452 of Ref. 6 …there appears a universal speed… whose value must be found by experiment. Sometimes added as a second postulate is the statement that a measurement of the velocity of light in any frame of reference gives the same result …One may regard this as a statement about the nature of light rather than as an independent postulate.

*The Gulag Archipelago*, by Aleksandr Solzhenitsyn, namely, that Russian scientist, Wladimir Ignatowsky, was murdered, in 1942, by the NKVD, in Leningrad, which at the time was under siege by the Wehrmacht.

**(**

**)**,

*The Principle of Relativity*(Methuen and Company, Ltd./Dover Publications, Inc., London/Mineola, NY, 1923/1952). For a more recent translation of a portion of Einstein's paper, see

A list of acronyms used in this paper is given under the heading of NOMENCLATURE just before the Acknowledgments.

Our point here is that, as stated, the POR implies the homogeneity and isotropy of space and the homogeneity of time. The matter of whether or not the homogeneity and isotropy of space are equivalent or imply one another may be of intrinsic interest, but is irrelevant here. This point and the homogeneity of time are discussed in some detail in Ref. 24.

See Chap. 4, particularly pp. 94–105 of Ref. 3.

See Sec. 2.7 of Ref. 5.

See Ref. 6, Appendix A, particularly p. 455.

See PROBLEMS 1–11, pp. 23–24, and Sec. 3.8 of Ref. 9.

See Ref. 12, p. 65.

These requirements limit our considerations to the *special* Lorentz transformations, i.e., to those which involve spatial coordinate frames with their respective Cartesian axes parallel to one another and their relative velocity along one pair of coordinate axes. For the general case, where *S* and $S\u2032$ do not necessarily have their respective axes parallel and the relative velocity is in a general direction, see Sec. 2-3 of Ref. 5 and Sec. 11.3 of Ref. 11. We note that Eqs. (11.19) of Ref. 11 are correct even when the respective axes of *K* and $K\u2032$ (*S* and $S\u2032$ in our notation) are not parallel.

There is a sign error in Eq. (34) of Ref. 30. Essentially the same sign error occurs in Ref. 2, beginning with the second line before Eq. (2.14) on p. 9; apparently this error, which amounts to taking the velocity of $S\u2032$ relative to *S* to be $\u2212x\u0302w$ (velocity of $K\u2032$ relative to *K* to be $\u2212x\u0302v$, in the notation of Ref. 2), propagates throughout the book.

Section 3 of Ref. 28 contains a brief discussion of what was probably the first observation implying IISL. The argument presented amounts to inferring that observations of starlight indicate that the speed of starlight, and presumably all light, is an inertial invariant. That is, the author takes IISL as an observed fact, rather than a postulate, which nevertheless results in a formulation of KSR that depends on IISL.

Logically, the GT follows from α = 0. It is true that if one uses Eqs. (66) and (73) to write the transformation as $t\u2032=\gamma \xaf(w)(t\u2212wc\xaf2x),\u2003x\u2032=\gamma \xaf(w)(\u2212wt+x),\u2003y\u2032=y,\u2003z\u2032=z,$ then letting $c\xaf\u2192\u221e$ in these equations and Eq. (72) does formally yield the GT. Of course, one is then left with the question of the physical significance of $c\xaf\u2192\u221e$. Observation readily reveals that it certainly does not apply to electromagnetic radiation. Moreover, it is not clear why any other choice for $c\xaf$ should be possible. Consequently, this approach to obtaining the GT appears to be dubious and confusing at best.

Except for the use of $c\xaf$ and *u* instead of *λ* and *c*, respectively, Eq. (79) is the same as the last equation of Ref. 43. As a check of Eq. (79), we note the following: from $w=D/t$ and *D* = *uT*, it follows that $uT/t=w$; from one of the other formulations of special relativity, we have $t\u2032/t=1/\gamma (w)$; it immediately follows that $c\xaf=c$.

*chronocity*, which is “…a factor corresponding to distance-rated time ‘displacement’.”

^{28}Chronocity is also discussed in Part 1 of a recent book by

In Ref. 30, the assertion is made that …the Lorentz transformations are not directly related to the properties of electromagnetic radiation. Electromagnetism is only relevant, if present within the theory, as a way to fix the arbitrary velocity scale, which is then identified with the speed of light

Cosmology is a rapidly developing field which has commanded a great deal of attention in the last 30 years or so. Accordingly, we note only a handful of the most recent books on cosmology.

*American Journal of Physics*and

*The Physics Teacher*as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.