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.

1.

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.

2.
J.
Aharoni
,
The Special Theory of Relativity
, 2nd ed. (
Clarendon Press
,
Oxford, England
,
1965
).
3.
A. P.
French
,
Special Relativity
(
W.W. Norton & Company
,
New York
,
1968
).
4.
N. David
Mermin
,
Space and Time in Special Relativity
(
McGraw-Hill
,
New York
,
1968
).
5.
R. D.
Sard
,
Relativistic Mechanics
(
W. A. Benjamin
,
New York
,
1970
).
6.
Edward M.
Purcell
,
Electricity and Magnetism, Berkeley Physics Course
Vol.
2
, 2nd ed. (
McGraw-Hill
,
Hew York
,
1985
).
7.
Enders A.
Robinson
,
Einstein's Relativity in Metaphor and Mathematics
(
Prentice-Hall
,
Englewood Cliffs
,
1990
).
8.
Robert
Resnick
and
David
,
Basic Concepts in Relativity and Early Quantum Theory
, 2nd ed. (
Macmillan
,
New York
,
1992
).
9.
Edwin F.
Taylor
and
John Archibald
Wheeler
,
Spacetime Physics: Introduction to Special Relativity
, 2nd ed. (
W. H. Freeman
,
New York
,
1992
).
10.
Thomas
Moore
,
A Traveler's Guide to Spacetime: An Introduction to the Special Theory of Relativity
(
McGraw-Hill
,
New York
,
1995
).
11.
John David
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
New York
,
1999
).
12.
N.
David Mermin
,
(
Princeton U. P.
,
Princeton
,
2005
)
13.
Daniel F.
Styer
,
Relativity for the Questioning Mind
(
The Johns Hopkins U. P
.,
Baltimore
,
2011
).
14.
Andrew M.
Steane
,
(
Oxford U. P
.,
Oxford, England
,
2012
).
15.

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.

16.
M. P.
Hobson
,
G. P.
Efstathiou
, and
A. N.
Lasenby
,
General Relativity: An Introduction for Physicists
(
Cambridge U. P
.,
Cambridge, UK
,
2006
)
17.
Leonid J.
Eisenberg
, “
Necessity of the linearity of relativistic transformations between inertial systems
,”
Am. J. Phys.
35
(
7
),
649
651
(
1967
).
18.

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.

19.
W.
von Ignatowsky
, “
Einige allgemeine Bemerkungen über das Relativitätsprinzip
,”
Phys. Z.
11
,
972
976
(
1910
);
W.
von Ignatowsky
Arch. Math. Phys.
17
,
1
24
(
1911
);
W.
von Ignatowsky
Arch. Math. Phys.
18
,
17
40
(
1911
). These papers can be found by using “W. von Ignatowsky” as the search string on Wikipedia. A translation of the first can be found at <https://en.wikisource.org/wiki/Translation:Some_General_Remarks_on_the_Relativity_Principle>. A brief discussion of Ignatowsky's work and some additional references can be found in Ref. 28. We note that Ignatowsky was known under several names, as noted in the German National Library at <https://portal.dnb.de/opac.htm?method=simpleSearch&cqlMode=true&query=idn%3D142420689>. We also note a little known fact of historical interest, which is movingly related on p. 296 of Vol. 2 of 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.
20.
P.
Frank
and
H.
Rothe
, “
Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme
,”
Ann. Phys.
34
(
5
),
825
855
(
1911
).
21.
H. M.
Schwartz
, “
Axiomatic deduction of the general Lorentz transformations
,”
Am. J. Phys.
30
(
10
),
697
707
(
1963
);
H. M.
Schwartz
Erratum: Axiomatic deduction of the general Lorentz transformation
,”
Am. J. Phys.
31
(
2
),
140
(
1963
).
22.
Yakov P.
Terletskii
,
Paradoxes in the Theory of Relativity
(
Plenum Press
,
New York
,
1968
), Sec. 7.
23.
A. R.
Lee
and
T. M.
Kalotas
, “
Lorentz transformations from the first postulate
,”
Am. J. Phys.
43
(
5
),
434
437
(
1975
).
24.
Jean-Marc Lévy-Leblond.
One more derivation of the Lorentz transformation
,”
Am. J. Phys.
44
(
3
),
271
277
(
1976
).
25.
H. M.
Schwartz
, “
Deduction of the general Lorentz transformations from a set of necessary assumptions
,”
Am. J. Phys.
52
(
4
),
346
350
(
1984
).
26.
H. M.
Schwartz
, “
A simple new approach to the deduction of the Lorentz transformations
,”
Am. J. Phys.
53
(
10
),
1007
1008
(
1985
).
27.
Sardar
Singh
, “
Lorentz transformations in Mermin's relativity without light
,”
Am. J. Phys.
54
(
2
),
183
(
1986
).
28.
Brian
Coleman
, “
A dual first-postulate basis for special relativity
,”
Eur. J. Phys.
24
(
3
),
301
313
(
2003
).
29.
Palash B.
Pal
, “
Nothing but relativity
,”
Eur. J. Phys.
24
(
3
),
315
319
(
2003
).
30.
Andrea
Pelissetto
and
Massimo
Testa
, “
Getting the Lorentz transformation without requiring an invariant speed
,”
Am. J. Phys.
83
(
4
),
338
340
(
2015
).
31.
Albert
Einstein
, “
Zur Elektrodynamik bewegter Körper
,”
Ann. Phys.
17
,
891
921
(
1905
). There are two English translations: “On the electrodynamics of moving bodies,” by Megh Nad Saha (1920), which can be found at <https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)>; “On the electrodynamics of moving bodies,” by George Barker Jeffery and Wilfrid Perrett, published in 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
H. M.
Schwartz
, “
Einstein's first paper on relativity
Am. J. Phys.
45
(
1
),
18
25
(
1977
).
32.

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

33.

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.

34.
Alan
McDonald
, “
Derivation of the Lorentz transformation
,”
Am. J. Phys.
49
(
5
),
493
(
1981
).
35.
Achin
Sen
, “
How Galileo could have derived the special theory of relativity
,”
Am. J. Phys.
62
,
157
162
(
1994
).
36.
R. S.
Shankland
,
S. W.
McCuskey
,
F. C.
Leone
, and
G.
Kuerti
, “
New analysis of the interferometer observations of Dayton C. Miller
,”
Rev. Mod. Phys.
27
(
2
),
167
178
(
1955
).
37.
David
Appell
, “
The invisibility of length contraction
,”
Phys. World
32
(
8
),
41
45
(
2019
).
38.

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

39.

See Sec. 2.7 of Ref. 5.

40.

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

41.

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

42.

See Ref. 12, p. 65.

43.
Brian
Coleman
, “
An elementary first-postulate measurement of the cosmic limit speed
,”
Eur. J. Phys.
25
(
3
),
L31
L32
(
2004
).
44.

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 $S′$ 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 $K′$ (S and $S′$ in our notation) are not parallel.

45.

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 $S′$ relative to S to be $−x̂w$ (velocity of $K′$ relative to K to be $−x̂v$, in the notation of Ref. 2), propagates throughout the book.

46.

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.

47.

The authors of Ref. 30 choose Eq. (61) without indicating why the other sign for the square root is incorrect.

48.

Logically, the GT follows from α = 0. It is true that if one uses Eqs. (66) and (73) to write the transformation as $t′=γ¯(w)(t−wc¯2x), x′=γ¯(w)(−wt+x), y′=y, z′=z,$ then letting $c¯→∞$ 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 $c¯→∞$. Observation readily reveals that it certainly does not apply to electromagnetic radiation. Moreover, it is not clear why any other choice for $c¯$ should be possible. Consequently, this approach to obtaining the GT appears to be dubious and confusing at best.

49.

This velocity addition rule is not derived in Ref. 30. A derivation of it without invoking light is given in Refs. 20, 21, 23, 24, 27–29, and the following reference.

50.
N. David
Mermin
, “
Relativity without light
,”
Am. J. Phys.
52
(
2
),
119
124
(
1984
).
51.

Except for the use of $c¯$ 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 $t′/t=1/γ(w)$; it immediately follows that $c¯=c$.

52.
We note in passing that Refs. 28 and 43 employ a little known and generally unappreciated quantity, namely, chronocity, which is “…a factor corresponding to distance-rated time ‘displacement’.”28 Chronocity is also discussed in Part 1 of a recent book by
Brian
Coleman
,
Spacetime Fundamentals Intelligibly (Re)Learnt: Special Relativity's Cosmographicum
(
BCS, VelChronos, Moyard
,
County Galway, Ireland
,
2017
). In Parts 1 and 2 of this book, Coleman also discusses the role of electromagnetic radiation in the formulation of KSR.
53.

Calculus is used only in Sec. II, in the proof that POR implies TIL, in Eqs. (30) and (32) of Sec. III, and in the consideration of the vacuum electromagnetic wave equation in Secs. IV and VI.

54.

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

55.

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.

56.
Andrew
Liddle
,
An Introduction to Modern Cosmology
, 3rd ed. (
Wiley
,
New York
,
2015
).
57.
Christian
G.
,
Böhmer
,
Introduction to General Relativity and Cosmology
(
World Scientific
,
London
,
2016
).
58.
Barbara
Ryden
,
Introduction to Cosmology
, 2nd ed. (
Cambridge U. P
.,
Cambridge, UK
,
2016
).
59.
Stuart
Clark
,
The Unknown Universe: A New Exploration of Time, Space, and Modern Cosmology
(
Pegasus Books
,
New York
,
2017
).
60.
Delia
Perlov
and
Alex
Vilenkin
,
Cosmology for the Curious
(
Springer
,
New York
,
2017
).
61.
For the LIGO detection papers, see <https://wow.ligo.caltech.edu/page/detection-companion-papers>. For the LIGO Scientific Collaboration and Virgo Collaboration publications, see <https://www.lsc-group.phys.uwm.edu/ppcomm/Papers.html>.
62.
Francis E.
Low
,
Classical Field Theory: Electromagnetism and Gravitation
(
John Wiley & Sons
,
New York
,
1997
).