In order to explain the Lorentz transformation, it is advantageous to use its eigencoordinates rather than Cartesian coordinates. In this manner, the matrix can be diagonalized. Moreover, the diagonal entries have a direct physical meaning—permitting their determination without the need of algebraic equations. This yields a proof that is easier to remember and reproduce.
REFERENCES
1.
Einstein's original derivation is in the famous 1905 paper:
A.
Einstein
, “
Zur Elektrodynamik bewegter Körper
,” Ann. Phys.
322
(10
), 891
–921
(1905
).English translation by
George Barker
Jeffery
and
Wilfrid
Perrett
, “
On the electrodynamics of moving bodies
,” in The Principle of Relativity
(
Dover Publications
,
New York
, 1952
). Chap. III.2.
One attempt at a simple derivation—by Einstein himself—may be found in:
Albert
Einstein
, Relativity, The Special and the General Theory, A Popular Exposition
(
Crown Publishers
,
New York
, 1961
), Appendix I.3.
Minkowski's landmark 1908 address, introducing spacetime, may be found here:
Hermann
Minkowski
, “
Space and time
,” translated by George Barker
Jeffery
and
Wilfrid
Perrett
, in The Principle of Relativity
(
Dover Publications
,
New York
, 1952
), Chap. V.4.
Pais′ incomparable biography of Einstein contains a representative history of special relativity:
Abraham
Pais
, Subtle is the Lord: The Science and the Life of Albert Einstein
(
Oxford U. P.
,
Oxford
, 1982
).5.
H. R. Brown's monograph is an invaluable historical resource, itself containing a vast bibliography of many sources not directly needed in the context of this paper:
Harvey R.
Brown
, Physical Relativity: Space-Time Structure from a Dynamical Perspective
(
Clarendon Press
,
Oxford
, 2005
).6.
Any student of relativity ought to explore the works of Wolfgang Rindler. A wonderfully precise author, his books on the subject are full of insight and contain beautiful exercises for the reader to enhance his/her understanding of the subject. These books include:
Special Relativity
(
Interscience Publishers
,
New York
, 1960
);Relativity: Special, General, and Cosmological
, 2nd ed. (
Oxford U. P
.,
Oxford
, 2006
).In addition, the following paper is of historical interest
W.
Rindler
, “
Einstein's priority in recognizing time dilation physically
,” Am. J. Phys.
38
, 1111
–1115
(1970
).7.
This book is one of a kind, a remarkable and creative introduction to relativity:
Edwin F.
Taylor
and
John Archibald
Wheeler
, Spacetime Physics: Introduction to Special Relativity
, 2nd ed. (
W. H. Freeman
,
New York
, 1992
). It contains a wealth of exercises, with solutions provided for odd-numbered problems.8.
Lest anyone take the reciprocity principle for granted, see
P.
Moylan
, “
Velocity reciprocity and the relativity principle
,” Am. J. Phys.
90
(2
), 126
–134
(2022
).9.
Compare
L. J.
Eisenberg
, “
Necessity of the linearity of relativistic transformations between inertial systems
,” Am. J. Phys.
35
(7
), 649
–649
(1967
);also see Lévy–Leblond below. The differentiability assumption is made only for brevity. For a proof without this assumption, see
V.
Berzi
and
V.
Gorini
, “
Reciprocity principle and the Lorentz transformations
,” J. Math. Phys.
10
, 1518
–1524
(1969
).10.
Seminal works by
W.
von Ignatowsky
include: “
Der starre Körper und das Relativitätsprinzip
,” Ann. Phys.
338
(13
), 607
–630
(1910
);“
Einige allgemeine Bemerkungen über das Relativitätsprinzip
,” Phys. Z.
11
, 972
–976
(1910
);“
Eine Bemerkung zu meiner Arbeit: ‘Einige allgemeine Bemerkungen zum Relativitätsprinzip
,’” Phys. Z.
12
, 779
–779
(1911
).Compare
P.
Frank
and
H.
Rothe
, “
Uber die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme
,” Ann. Phys.
339
(5
), 825
–855
(1911
).Also see
Vladimir
Fock
, Theory of Space Time and Gravitation
, 2nd ed. (
Pergamon Press
,
New York
, 1964
), Appendix A.More modern papers include:
A. R.
Lee
and
T. M.
Kalotas
, “
Lorentz transformations from the first postulate
,” Am. J. Phys.
43
, 434
–437
(1975
);J. M.
Lévy-Leblond
, “
One more derivation of the Lorentz transformation
,” Am. J. Phys.
44
, 271
–277
(1976
);R. J.
Cook
, “
Comment on “Self-inverse form of the Lorentz transformation
,” Am. J. Phys.
47
, 117
–118
(1979
);W. N.
Mathews
, Jr., “
Seven formulations of the kinematics of special relativity
,” Am. J. Phys.
88
, 269
–278
(2020
).11.
Thus, is proportional to .
12.
L.
Parker
and
G. M.
Schmieg
, “
Special relativity and diagonal transformations
,” Am. J. Phys.
38
(2
), 218
–222
(1970
);S.
Chao
, “
Lorentz transformations from intrinsic symmetries
,” Symmetry
8
, 94
(2016
).13.
Malcolm S.
Longair
, Theoretical Concepts in Physics
(
Cambridge U. P
.,
Cambridge
, 1984
), pp. 262
–264
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2024
Author(s)
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