Abstracts
The socalled principle of relativity is able to fix a general coordinate transformation which differs from the standard Lorentzian form only by an unknown speed which cannot in principle be identified with the light speed. Based on a reanalysis of the MichelsonMorley experiment using this extended transformation we show that such unknown speed is analytically determined regardless of the Maxwell equations and conceptual issues related to synchronization procedures, time and causality definitions. Such a result demonstrates in a pedagogical manner that the constancy of the speed of light does not need to be assumed as a basic postulate of the special relativity theory since it can be directly deduced from an optical experiment in combination with the principle of relativity. The approach presented here provides a simple and insightful derivation of the Lorentz transformations appropriated for an introductory special relativity theory course.
Keywords:
MichelsonMorley experiment; Lorentz transformations; principle of relativity
O chamado princípio da relatividade é capaz de determinar uma transformação geral de coordenadas que difere da forma lorentziana padrão por um velocidade desconhecida que não pode, em princípio, ser identificada com a velocidade da luz. Com base em uma nova análise do experimento de MichelsonMorley, usando esta transformação estendida, mostramos que tal velocidade invariante é determinada analíticamente, sem qualquer referência às equações de Maxwell e questões conceituais relacionadas a procedimentos de sincronização, definição de tempo e causalidade. Tal resultado demonstra de uma maneira pedagógica que a constância da velocidade da luz não precisa ser suposta como um postulado básico da teoria da relatividade especial, uma vez que ela pode ser deduzida diretamente a partir de um experimento óptico em combinação com o princípio da relatividade. O método apresentado aqui resulta numa dedução das transformações de Lorentz que é simples e elucidativa, apropriada para um curso introdutório sobre teoria da relatividade especial.
Palavraschave:
Experimento de MichelsonMorley; transformações de Lorentz; princípio da relatividade
1. Introduction
In the standard lore of special relativity theory (SRT), the pillars of the theory rest on two postulates originally introduced by Einstein [^{1}[1] A. Einstein, Ann. Phys. 17, 891 (1905).], namely: (i) the principle of relativity, and (ii) the principle that states that the speed of light is independent of the velocity of the source (see ref. [^{2}[2] Ralph Baierlein, Am. J. Phys. 74, 193 (2006).] for an explanation of why this is not the same as “the constancy of the speed of light”). These two postulates were explicitly used by him for obtaining the socalled Lorentz transformations. Nevertheless, since the first decade after Einstein’s seminal paper [^{1}[1] A. Einstein, Ann. Phys. 17, 891 (1905).], many authors have tried to show that the second postulate is not necessary. The first attempt was made by Ignatowski [^{3}[3] W. von Ignatowski, Phys. Z. 11, 972 (1910), Arch. Math. Phys. 17, 1 (1911), Arch. Math. Phys. 18, 17 (1911).] in 1910. He replaced the second Einstein postulate by the assumption of isotropy and homogeneity of space, which implies linearity of the transformation equations and the reciprocity of the coordinates transformation  which means that two inertial observers must agree with the numerical value of their relative velocities. In 1911, Frank and Rothe [^{4}[4] P. Frank and H. Rothe, Ann. Phys. 34, 825 (1911).] derived the Lorentz transformations by assuming that they form a homogeneous linear group, the validity of reciprocity principle and the dependency of the length contraction only on the relative velocity. In 1921, Pars [^{5}[5] L.A. Pars, Philos. Mag. 42, 429 (1921).] derived the Lorentz transformation, assuming homogeneity of spacetime, isotropy of space and the reciprocity principle. In fact, it was shown by Berzi and Gorini [^{6}[6] V. Berzi and V. Gorini, J. Math. Phys. 10, 1518 (1969).] that the principle of relativity and spatial isotropy imply reciprocity. LevyLeblond [^{7}[7] J.M. LevyLeblond, Am. J. Phys. 44, 271 (1976).] has shown that the additional hypotheses of group law and causality are necessary. We refer to [^{6}[6] V. Berzi and V. Gorini, J. Math. Phys. 10, 1518 (1969).,^{8}[8] Vittorio Gorini and Antonio Zecca, J. Math. Phys. 11, 2226 (1970).–^{10}[10] Harvey R. Brown, Physical Relativity (Oxford University Press, Oxford, 2005), p. 110.] for discussions of the necessary hypotheses, and also to Miller [^{11}[11] Arthur I. Miller, Albert Einstein's Special Theory of Relativity (Springer, New York, 1998), p. 233.], for a complete historical account of Ignatovski’s work. Pedagogical derivations of Lorentz transformations without the second postulate can be found, for example, in [^{7}[7] J.M. LevyLeblond, Am. J. Phys. 44, 271 (1976).,^{12}[12] Robert Weinstock, Am. J. Phys. 33, 640 (1965).–^{27}[27] A. Drory, Stud. Hist. Philos. Mod. Phys. 51, 57 (2015).]. An additional list of references regarding derivations of this kind is given by Sonego and Pin [^{28}[28] Sebastiano Sonego and Massimo Pin, J. Math. Phys. 50, 042902 (2009)].
All these derivations arrive at formulas for the Lorentz transformations containing an unknown and invariant (constant) limiting speed. However, its identification with the speed of light usually requires the invariance of electrodynamics [^{29}[29] W. Pauli, Theory of Relativity (Dover Publications, Inc., New York, 1981).] or some dynamical effect [^{30}[30] D.H. Frisch and J.H. Smith, Am. J. Phys. 31, 342 (1963).]. The main reason for so many derivations is that some authors have different opinions about what are the most fundamental assumptions, while others present derivations that look pedagogically simpler (see the work of Llosa [^{31}[31] J. Llosa, arXiv:1401.6568 (2014).] for a comprehensive review).
One modern relevance of this result lies in the fact that to study the consequences of Lorentz symmetry breaking one has to abandon or modify the principle of relativity [^{32}[32] David Mattingly, Living Rev. Relativity 8, 5 (2005), available at http://www.livingreviews.org/lrr20055.
http://www.livingreviews.org/lrr20055...
–^{34}[34] S. Liberati, Class. Quantum Grav. 8, 1 (2013).]. Also, when taking in account theories for varying speed of light, it is important to know the origin of the terms containing the speed of light in the equations [^{35}[35] George F.R. Ellis and JeanPhilippe Uzan, Am. J. Phys.73, 240 (2005).].
In this work, we show how the identification of this constant speed with the speed of light could have been made in the early years of the theory of special relativity, by applying the derived general transformations to the null results obtained in the MichelsonMorley experiment [^{36}[36] A.A. Michelson and E.W. Morley, Am. J. Sci. 34, 333 (1887).]. To obtain the usual Lorentz transformations, we replace the second postulate by a careful interpretation of the empirical (null) result of that optical experiment. The approach discussed here not only establishes the speed of light as the limiting speed to be used at Lorentz transformations but also shows explicitly that the hypothesis of the existence of a luminiferous aether does not interfere with the result since it becomes irrelevant as a consequence of the null MichelsonMorley experiment.
2. Lorentz transformations without the second postulate
Let us now suppose that Cartesian coordinates
We choose the work of LevyLeblond [^{7}[7] J.M. LevyLeblond, Am. J. Phys. 44, 271 (1976).] for its elegance and generality^{1} 1 Llosa [31] presents a similar derivation that does not require the counting of parameters of the transformation group. . Assuming validity of the principle of relativity plus the hypotheses of homogeneity of spacetime, the linearity of inertial transformations, isotropy of space and the group law, he derived a set of coordinate transformations between two inertial frames slightly more general than that proposed by Lorentz and Einstein.
Using the standard configuration coordinates of the inertial frames
where
and
where
It is also worth mentioning that the above transformations are a particular case of the more general set assumed by Robertson [^{37}[37] H.P. Robertson, Rev. Mod. Phys. 21, 378 (1949).], from which he concluded that the three secondorder optical experiments taken together, namely: MichelsonMorley [^{36}[36] A.A. Michelson and E.W. Morley, Am. J. Sci. 34, 333 (1887).] (1887), KennedyThorndike [^{38}[38] R.J. Kennedy and E.M. Thorndike, Phys. Rev. 42, 400 (1932).] (1932), and IvesStilwell [^{39}[39] H.E. Ives and G.R. Stilwell, J. Opt. Soc. Am. 28, 215 (1938).,^{40}[40] H.E. Ives and G.R. Stilwell, J. Opt. Soc. Am. 31, 369 (1941).] (1938, 1941) are sufficient to single out the Lorentz transformations.
In contrast, we have found that a suitable analysis of any of the cited optical experiments is enough to obtain the relativistic result, that is,
3. MichelsonMorley experiment and the limiting invariant speed
The MichelsonMorley experiment consists of an optical interferometer assembled on a platform that can horizontally be rotated. A simplified diagram is shown in Figure 1.
We suppose that a beam of light coming from a source
Let us suppose that the whole interferometer is at rest in the lab frame
We suppose that the apparatus is configured in such a way that the arm 1 is aligned to this constant relative speed
Now, by using the
and
where
Let us denote by
where
By using equation (12) we also find that the time
The interference pattern, between the light beam coming out of the two optical paths is determined by the time delay
When the interferometer is rotated clockwise by
while the corresponding time delay reads:
Due to the rotation of the apparatus, there is a net difference in the time delays associated to each angular configuration:
The expected fringe shift after rotation of the apparatus can be written as the ratio:
where
In the original experiment, using multiple reflections, the total length of the arms (
For an arbitrary relative speed
We remark that the MichelsonMorley experiment measures only the isotropy of the twoway speed of light. Therefore, its result does not depend on the synchronization procedure [^{42}[42] Clifford M. Will, Phys. Rev. D 45, 403 (1982).].
4. Conclusions
Standard derivations of the generalized Lorentz transformations given by equations (1–5) show that the unique freeparameter to be determined is an invariant (and unknown) maximum speed,
The step from the generalized Lorentz transformations to the usual ones presented here has a methodological and also a clear pedagogical advantage for undergraduate teaching. In particular, it does not require from fresh undergraduate students a previous knowledge of Maxwell’s equations. More puzzling kinematic concepts, like the relativity of simultaneity and synchronization procedures [^{44}[44] Reza Mansouri and Roman U. Sexl, Gen. Relativ. Gravit. 8, 497 (1977).–^{46}[46] Alberto A. Martínez, Am. J. Phys. 73, 452 (2005).] can be postponed for a second study of the Lorentz transformations.
Another interesting pedagogical aspect of our complete derivation without the second postulate is that the existence of the luminiferous aether was explicitly assumed from the very beginning, but its possible effects on the light propagation work only to provide the expected identification of the invariant undetermined speed, namely:

1
Llosa [^{31}[31] J. Llosa, arXiv:1401.6568 (2014).] presents a similar derivation that does not require the counting of parameters of the transformation group.
Acknowledgments
J. A. S. Lima is partially supported by CNPq and FAPESP (Brazilian Research Agencies).
References

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Publication Dates

Publication in this collection
2017
History

Received
19 Nov 2016 
Accepted
11 Jan 2017