text:
\section{Introduction}
Sometimes Woldemar Voigt, a German physicist, is considered as ``Relativity's
forgotten figure'' \cite{I-1}. Although his contribution was not quite
forgotten and cited ``by those who have taken the trouble to read the original
literature'' \cite{I-2}, it is a matter of fact that this contribution had no
influence at all on the development of special relativity. Therefore, in our
opinion, it is more appropriate in this case to talk about missed
opportunities in the sense of Dyson \cite{I-3}, than about forgotten
contribution.

Nearly two decades before the vigorous development of special relativity has
started, in 1887 Woldemar Voigt published an article on the Doppler effect in
which some fundamental principles underlying the relativity theory were
anticipated. Namely, he was the first who used Einstein's second postulate
(universal speed of light) and the restricted form of the first postulate
(invariance of the wave equation  when changing the inertial reference system)
to show that the Doppler shift of frequency was incompatible with Newtonian
absolute time and required a relative time identical with the Lorentz's local
time introduced later.

The modern reader may wonder why this paper published by Voigt remained
unnoticed and played no role in the emergence and development of special
relativity. The simple answer is that we are reading too much in too little:
what we see in Voigt's paper today is affected by our modern scientific
background and it doesn't coincide with the readings of Voigt himself and his
contemporaries.

This relativity of perception of scientific papers is valid, of course, for
Poincar\'{e}'s far more important contributions too and, in our opinion, is in
the base of the pointless Einstein-Poincar\'{e} priority dispute.

\section{Voigt --- a missing link in the Einstein-Poincar\'{e}
mysterious connection?}
It is unfortunate that Einstein didn't cite anybody in his seminal paper
on the foundations of special relativity \cite{1}. On one hand, this
circumstance can induce a wrong impression that the creation of special
relativity was a miraculous event, a single strike of a young genius, at that
time a 26-year-old junior clerk at the Swiss patent office in Bern who
worked it out in his spare time within a few months in complete scientific
isolation.

Einstein's own account \cite{2} seems to confirm this simplified version
of how the special relativity was created. He recalls that after reading
the Lorentz's 1895 monograph {\it Treatise on a Theory of Electrical and
Optical Phenomena in Moving Bodies} he spent almost a year in vain trying to
resolve a difficult problem of how to reconcile the invariance of the velocity
of light, implied by the validity of Maxwell equations in the reference frame
of the moving body, to the addition rule of velocities used in mechanics.
Then suddenly during a conservation with his friend Michele Besso a solution
came to him: ``An analysis of the concept of time was my solution. Time cannot
be absolutely defined, and there is an inseparable relation between time and
signal velocity. With this new concept, I could resolve all the difficulties
completely for the first time. Within five weeks the special theory of
relativity was completed'' \cite{2}.

We have no reason not to believe Einstein. Rindler argues that ``contrary
claims notwithstanding, apparently no one before Einstein in 1905 ventured the
idea that all clocks in motion might go slow'' \cite{3}. It seems very natural
that this startling new idea psychologically played a role of a trigger for
Einstein to develop his version of special relativity. This version turned out
very influential and determined the route of special relativity for years
ahead.

Nonetheless, historically the genesis of the special relativity theory was
a long process involving at least two more key players: Lorentz and
Poincar\'{e}. Besides, ``the construction of the special theory of relativity
did not end with Einstein's papers of 1905. Some features that today's
physicists judge essential were added only later'' \cite{4}. Many prominent
physicists participated in this process of shaping the special relativity to
its modern form, most notably Hermann Minkowski, Arnold Sommerfeld, Alfred
Robb, Max Planck and Max von Laue, to name a few.

Einstein's paper \cite{1} is undoubtedly a strike of a genius. It gives an
extraordinary clear presentation of fundamentally new concepts about space
and time which shattered the ideas which existed before. However, there is
a mystery here. As observed by Dyson, ``when the great innovation appears,
it will almost certainly be in a muddled, incomplete and confusing form.
To the discoverer himself it will be only half-understood; to everybody else
it will be a mystery'' \cite{5}. How can we reconcile the extreme clarity
of Einstein's presentation with Dyson's observation?

First of all, Einstein's great discovery has not  appeared out of nowhere.
As Einstein himself wrote in 1955 ``there is no doubt, that the
\underline{special} theory of relativity, if we regard its development
in retrospect, was ripe for discovery in 1905. Lorentz had already observed
that for the analysis of Maxwell's equations the transformations which later
were known by his name are essential, and Poincar\'{e} had even penetrated
deeper into these connections'' \cite{6}. Although, as the known historical
documents witness \cite{6,7}, Einstein's version of special relativity is
solely his own long-sought creature, barely influenced by whatever background
reading he had at that time; undoubtedly Einstein benefited from his scientific
background in this process, at least unconsciously: Einstein received good
education for his time. Although ``the scientific courses offered to him in
Z\"{u}rich soon seemed insufficient and inadequate, so that he habitually cut
his classes. His development as a scientist did not suffer thereby. With
a veritable mania for reading, day and night, he went through the works
of the great physicists --- Kirchhoff, Hertz, Helmholtz, F\"{o}ppl'' \cite{6}.

As Einstein cites nobody in his paper \cite{1}, we don't know for sure what
contemporary papers, if any, influenced him while writing his masterwork
\cite{1} and whether his development of special relativity was eased by his
readings. Letters of Einstein, written between 1998 and 1902, confirm \cite{8}
his later recollections that he was engaged in the problems of electrodynamics
of moving bodies many years before writing his epoch-making article \cite{1}.
However, surviving correspondence sheds very little light on what happened in
the crucial years between 1902 and 1905. ``Whatever reading and writing he may
have done at this time, Einstein published nothing on the subject of optics
and electrodynamics of moving bodies for 3.5 years'' \cite{6,8}.

When Einstein was asked in 1955 whether Poincar\'{e} had had any influence
on his development of special relativity, he answered \cite{6}:
``Concerning myself, I knew only Lorentz's important work of 1895 ---
'La th\'{e}orie \'{e}lectromagn\'{e}tique de Maxwell' and 'Versuch einer
Theorie der elektrischen und Optischen Erscheinungenin bewegten K\"{o}rpern'
--- but not Lorentz's later work, nor the consecutive investigations by
Poincar\'{e}. In this sense my work of 1905 was independent. The new feature
of it was the realization of the fact that the bearing of the Lorentz
transformation transcended its connection with Maxwell's equations and was
concerned with the nature of space and time in general. A further new result
was that the 'Lorentz invariance' is a general condition for any theory''.

Einstein is not very accurate here. The universal character of Lorentz
invariance, the fact that it is a general requirement for all laws of
physics, not merely a property of  Maxwell's equations, was, no doubt,
anticipated by Poincar\'{e} \cite{4,9}. Moreover, there is still another
mystery. From the modern perspective, it may appear that the Einstein's
synchronization procedure via light signals and not absolute nature of
simultaneity were also Poincar\'{e}'s inventions. Already in 1898,
``Poincar\'{e} had presented exactly the same light signaling and clock
synchronization thought experiment that would later be found in Einstein's
1905 relativity paper'' \cite{10}, although Poincar\'{e}'s presentation is
without any mention of the relativity principle and Lorentz's local time.

Two years later in his lecture ``Lorentz's theory and the principle of
reaction'' Poincar\'{e} used his light signaling and clock  synchronization
thought experiment to explain the physical meaning of the Lorentz's local
time \cite{10}. It is true however that Poincar\'{e} never abandoned the
concept of {\ae}ther and considered the above mentioned relativity of
simultaneity as only an apparent, not genuine, physical effect related to
the fact that we have synchronized the clocks in the moving frame as if the
light velocity were the same in all directions, like what happens in the
preferred {\ae}ther frame. For Poincar\'{e}, only clocks so synchronized in the
{\ae}ther frame show the real time, despite the fact that he was fully aware
that, because of the relativity principle, it was impossible to experimentally
find out that the local time defined in the moving frame was not the real time.
In 1902 letter to the Nobel committee to nominate Lorentz for the Nobel prize
in Physics, which he indeed was awarded, Poincar\'{e} praises very highly
Lorentz's ``most ingenious invention'' of ``local time'' and writes:
``Two phenomena happening in two different places can appear simultaneous even
though they are not: everything happens as if the clock in one of these places
were late with respect to that of the other, and as if no conceivable
experiment could show evidence of this discordance'' \cite{10}.

Could Poincar\'{e}'s insights about simultaneity and clock synchronization
somehow influenced Einstein? This is not an easy question.
We know from Einstein himself that he had read Lorentz's 1895 treatise and
``was therefore aware of the local time and the role it served in preserving
the form of the Maxwell-Lorentz equations to first order'' \cite{4}. We also
know that he and his friends from Olympia Academy were fascinated by
Poincar\'{e}'s {\it La science et l'hypoth\'{e}se}, which contained the
relativity principle and a brief critical discussion of simultaneity. ``He may
also have known the exact form of the Lorentz transformations, for in 1904
several German theorists commented on them in journals that he regularly read''
\cite{4}. For Torretti this is enough to conclude that ``there is no doubt
that Einstein could have drawn inspiration and support for his first work on
relativity from the writings of Poincar\'{e}'' \cite{11}. However matters are
not as simple, because Einstein denies such an influence and we have no reason
not to believe him. Knowing the paradoxical nature of human memory and
psychology, we cannot exclude that the true story was hidden for Einstein too.

``It is certain that sometime before 17 May 1906, Einstein did read
Poincar\'{e}'s 1900 paper --- on that day Einstein submitted a paper of his
own that explicitly used the contents of Poincar\'{e}'s article (though not
local time)'' \cite{12}. This was the first and last occasion Einstein ever
cited Poincar\'{e} in the context of relativity (in connection with the
$E=mc^2$ formula).

Similar ideas about clock synchronization and physical definition of the local
time can be also found (in German) in Emil Cohn's November 1904 article
``Toward the electrodynamics of moving systems'' \cite{12}. Cohn's views are
even closer to Einstein's than to  Poincar\'{e}'s, as he rejected the
{\ae}ther, preferring ``the vacuum''. We know that sometime before
25 September 1907 Einstein did read the Cohn's paper --- at that day he sent
to the editor of a journal his review paper on relativity which contains
a footnote-compliment: ``The pertinent studies of E.~Cohn also enter into
consideration, but I did not make use of them here'' \cite{12}. However, again
we have no evidence that Einstein knew the Cohn's work prior his founding 1905
paper. How can we then explain a strange fact that ``Einstein's reasoning in
1905 regarding synchronization of clocks using light signals is incredibly
similar to Poincar\'{e}'s'' \cite{13}? Probably Chavan or Solovine (members of
the Olympia Academy), who were fluent in French, reported about Poincar\'{e}'s
1898 paper (in French) ``the measurement of time'' at an Olympia Academy
session \cite{12}. Even more interesting guess about what could have happened
then is suggested in \cite{10}.

Norton argues that it is quite possible that thoughts of clocks
synchronization by light signals played no essential role in Einstein's
discovery of the relativity of simultaneity: ``A plausible scenario is that
Einstein was compelled to the Lorentz transformation for space and time as
a formal result, but needed some way to make its use of local time physically
comprehensible. Thoughts of light signals and clock synchronization would then
briefly play their role. It is also entirely possible that these thoughts
entered only after Einstein had become convinced of the relativity of
simultaneity; that is, they were introduced as an effective means of conveying
the result to readers of his 1905 paper and convincing them of it. In both
cases, thoughts about the light signals and clock synchronization most likely
played a role only at one brief moment, some five to six weeks prior to the
completion of the paper, at the time that Einstein brought his struggle with
him to a celebrated meeting with his friend Michele Besso'' \cite{14}. Indeed,
five to six weeks before completing his relativity paper, Einstein wrote to
his friend Conrad Habicht: ``The fourth paper is only a rough draft at this
point, and is an electrodynamics of moving bodies which employs a modification
of the theory of space and time'' \cite{10}. It seems that at that time
Einstein already discovered the relativity of simultaneity, as he speaks of
modification of the theory of space and time, but yet had not found a good way
how to present his new discoveries. Then there was a ``celebrated meeting''
with his friend Michele Besso. At the end of his relativity paper, Einstein
writes ``In conclusion I wish to say that in working at the problem here dealt
with I have had the loyal assistance of my friend and colleague M. Besso, and
that I am indebted to him for several valuable suggestions'' \cite{1}. What
was Besso's role in completing Einstein's relativity paper? Granek suggests
``that in the Einstein-Besso 'celebrated meeting' Besso indirectly and
unnoticeably conveyed Poincar\'{e}'s light signaling and clock synchronization
thought experiment, without necessarily knowing that this thought experiment
originated in Poincar\'{e}'s works'' \cite{10}. Probably Besso was told about
this light signaling and clock synchronization thought experiment by another
member of Olympia Academy who read Poincar\'{e}'s papers (as was already
mentioned, it is possible that Solovine read Poincar\'{e}'s 1898 paper). Then
he conveyed to Einstein a loose idea about the light signaling and clock
synchronization intermingled with other ideas \cite{10}. No wonder that
Einstein afterwords refused any connection with Poincar\'{e}, because he was
convinced that he came to the idea by himself, with the help of his friend
Besso, and had not realized that in fact this was Poincar\'{e}'s idea which
came to him in such a bizarre way.

We think this is the best we can do on the question whether Poincar\'{e}'s
or Cohn's papers influenced Einstein's reasoning. ``The limits of
reconstruction are evident, and would be even if it were written in stone that
Einstein had seen one of these papers'' \cite{12}.

However, the real enigma is not the absence of Poincar\'{e}'s name in
Einstein's 1905 paper. As we have seen above, we can imagine a plausible
explanation of this fact. The real enigma is why Poincar\'{e}'s contribution
to relativity was downplayed by his contemporaries.

From the modern perspective, Poincar\'{e}'s treatment of the Lorentz group in
his last pre-Einstein papers \cite{15,16} is extraordinary advanced and modern
\cite{9,17}. In some sense, Poincar\'{e} was far ahead of his time. Perhaps
very few physicists (if any) at that time had enough mathematical background
to duly appreciate Poincar\'{e}'s achievements. Besides, Poincar\'{e}'s 1906
Rendiconti paper \cite{16} was both ``rather long (47 pages) and technically
quite complex, especially in the final section'' where ``many of the key new
results of Poincar\'{e} on Relativity are contained'' \cite{18}. However,
Hermann Minkowski was undoubtedly among those who were capable to appreciate
Poincar\'{e}'s writings. And a blatant fact, which leaves us almost speechless,
is that Poincar\'{e}'s name was never mentioned in Minkowski's famous Cologne
lecture ``Raum und Zeit''.

Minkowski's September 21st, 1908, lecture ``Space and Time'' was a crucial
event in the history of relativity. In a sense, the appearance of the
Einstein's paper \cite{1} was an unexpected and peculiar event which changed
the natural flow of ideas of contemporary physics.
Suppose Einstein had not existed. How would our understanding of space-time
physics have developed then? In an interesting essay \cite{19}
L\'{e}vy-Leblond tried to answer this question. His conclusion is that ``apart
from a few details perhaps, the formalism, that is, the notations and
equations, would be very similar to ours today. But the language and,
underneath, the words and the ideas we would use could  be rather different.
Many familiar terms, beginning with 'relativity' itself, might be absent from
our vocabulary'' \cite{19}. In this respect, Minkowski's Cologne lecture
constitutes the shift of focus from reference frames and relativity of lengths
and time intervals towards the modern concepts of symmetry and associated
space-time geometry. After the Cologne lecture, relativity was gradually but
firmly embarked  on the royal way leading to the modern understanding of
special relativity as the Minkowskian chronogeometry of the 4-dimensional
flat space-time.

Initially Einstein was skeptical about Minkowski's innovations and called
them ``superfluous learnedness'' \cite{20}. According to Sommerfeld's
recollections, he said  on one occasion: ``Since the mathematicians have
invaded the relativity theory, I do not understand it myself any more''
\cite{21}. However, as early as 1912 Einstein fully appreciated the strength
of Minkowski's 4-dimensional approach. At that year he wrote to
Sommerfeld: ``I am now occupied exclusively with the gravitational problem,
and believe that I can overcome all difficulties with the help of a local
mathematician friend [Marcel Grossmann]. But one thing is certain, never before
in my life have I troubled myself over anything so much, and that I have
gained great respect for mathematics, whose more subtle parts I considered
until now, in my ignorance, as pure luxury! Compared with this problem, the
original theory of relativity is childish'' \cite{21}.

The influence of the Cologne lecture was enormous. Its published version
``sparked an explosion of publications in relativity theory, with the number
of papers on relativity tripling between 1908 (32 papers) and 1910
(95 papers)'' \cite{22}. The response to the Minkowski's lecture was
overwhelmingly positive on the part of mathematicians, and more mixed
on the part of physicists --- only in the 1950s their attitude began to
converge toward Minkowski's space-time view \cite{22}.

Due to the importance of the Cologne lecture and to the impact it had on the
historical development of special relativity, a sad omission of Poincar\'{e}'s
name from it played, in our opinion, a crucial role in the subsequent
downplaying of Poincar\'{e}'s contribution to relativity by contemporary
physicists for a long time.

Some leading scientists remarked the absence of Poincar\'{e}'s name in the
Cologne lecture and even tried somehow to correct injustice. In particular,
``in the notes he added to a 1913 reprint of this lecture, Sommerfeld
attempted to right the wrong by making it clear that a Lorentz-covariant law
of gravitation and the idea of a 4-vector had both been proposed earlier by
Poincar\'{e}'' \cite{23}. Though Sommerfeld acknowledges Poincar\'{e}'s
contribution only partially, the young Pauli does ``a better job'' \cite{18}
in giving Poincar\'{e} a due credit in his famous review article on the theory
of relativity \cite{24}. In fact, this happened under Felix Klein's
direct influence. In a letter of acknowledgment of the receipt of the
manuscript Klein advises Pauli to pay great attention to historical facts
and in particular emphasizes Poincar\'{e}'s priority before Einstein in
recognizing the group properties of the Lorentz transformations \cite{25}.
However, these efforts were insufficient. As the citation record shows
\cite{26}, Poincar\'{e}'s papers \cite{15,16} were rarely cited during the
founding period of relativity 1905-1912 and thus had virtually no effect
on the development of special relativity (in this time period Einstein's
paper \cite{1} was cited about five times more frequently than Poincar\'{e}'s
papers \cite{15,16}).

Why did Minkowski not mention  Poincar\'{e}? Minkowski was certainly aware of
Poincar\'{e}'s papers. One year before the celebrated lecture in Cologne,
in his lecture to  the G\"{o}ttinger Mathematischen Gesellschaft, the text of
which was published posthumously in 1915 through the efforts of Sommerfeld,
Minkowski frequently and positively cites  Poincar\'{e}. In fact Poincar\'{e}
was the third most cited author (cited six times), after Planck (cited eleven
times) and Lorentz (cited ten times), in this lecture \cite{18}. For
comparison, Einstein was cited only twice. How can we then explain the
complete disregard of Poincar\'{e}'s contribution only one year later? In
absence of any historical document resolving this conundrum, only speculations
on Minkowski's motivations may be entertained, as was done in \cite{18} and
\cite{23}. Maybe the following circumstances played a psychological role.

According to Max Born's recollections, later after the Cologne lecture
Minkowski told him that ``it came to him as a great shock when Einstein
published his paper in which the equivalence of the different local times of
observers moving relative to each other was pronounced; for he had reached the
same conclusions independently but did not publish them because he wished
first to work out the mathematical structure in all its splendor'' \cite{21}.
We may suppose that the discovery of the 4-dimensional formalism was also
a result of this process of working out the mathematical structure behind the
Lorentz transformations and was made by Minkowski independently of
Poincar\'{e}'s 1905 papers. ``If that reconstruction is correct, he must have
been all the more eager, when he later realized that he had been preceded by
Poincar\'{e}, to find reasons for downplaying Poincar\'{e}'s work'' \cite{18}.
So, perhaps, in 1907 Minkowski had only a background reading of Poincar\'{e}'s
papers. Then, as he prepared for the Cologne lecture, he more thoroughly
studied Poincar\'{e}'s papers and realized that if he ``had chosen to include
some mention of Poincar\'{e}'s work, his own contribution may have appeared
derivative'' \cite{23}. However, in our opinion, to make the decision to
exclude Poincar\'{e}'s name from the Cologne lecture Minkowski needed some
serious reason to psychologically justify such an unfair omission. Maybe
another oddity of the Cologne lecture gives a clue what was this reason.

In the Cologne lecture Minkowski never mentions Lorentz transformations (so
named by Poincar\'{e} in \cite{15,16}), instead he refers to transformations
of the group $G_c$. Even more surprisingly, on December 21, 1907,
Minkowski talked to the G\"{o}ttingen scientific society and subsequently in
April 1908 published its contents in {\it G\"{o}ttinger Nachrichten} as a
technical paper {\it Die Grundgleichungen f\"{u}r die electromagnetischen
Vorg\"{a}nge in bewegten K\"{o}rpern} where one can find all the results of
the Cologne lecture presented with great details \cite{18,21}. In this paper
Minkowski quotes Poincar\'{e} only twice and ``in a rather derogatory
manner'' \cite{18}. In particular,  at the beginning, he mentions Poincar\'{e}
only as an originator of the name ``Lorentz transformations''. In fact,
probably, this talk of Minkowski was a beginning of his downplaying of the
Poincar\'{e}'s contribution, as witnessed by the following facts \cite{26A}.
In it Minkowski gives Lorentz the credit of having found the ``purely
mathematical fact'' of the Lorentz covariance of the Maxwell equations and
of having ``created the relativity-postulate''. None of these claims is true
--- it is Poincar\'{e} who should be credited for these discoveries as
was explicitly stated by Lorentz himself. The priority in clarifying the link
between the Lorentz transformations and a new concept of time is fully given
to Einstein without mentioning Poincar\'{e}'s important insights in this
question. Overall impression that the reader could infer from this
historically important publication might be that Minkowski  is ``suggesting
that the main (if not the only) contribution by Poincar\'{e} is to have given
the Lorentz transformations \ldots their name'' \cite{26A}.

The reason for suppression of the name ``Lorentz transformations'' in the
Cologne lecture is unknown, but very probably \cite{23} it was linked to
Minkowski's discovery that essential application of the central role the
Lorentz symmetry plays in optics goes back to Woldemar Voigt's 1887 paper
\cite{27}.

Thanks to a letter sent by Minkowski to David Hilbert in 1889, we know that
Minkowski was studying Voigt's treatment of elasticity \cite{28A}. However,
it is not known whether Minkowski read Voigt's 1887 paper on the Doppler
principle at that time. Probably he did. At least such a suggestion seems
tenable \cite{28}. Even if he read the paper, it seems he had not recognized
the potential of Voigt's transformations. In his letter of 1889 to Hilbert
he mentions that ``abstract speculation was not sufficient for physical
understanding'' and that ``it was inconceivable to him that anyone would
develop mathematical equations only with the hope that someone might later
demonstrate their utility'' \cite{28A}.

However, it is quite possible and even more plausible that Minkowski became
aware of the Voigt's 1887 Doppler principle paper only later and his above
given quotations concern to another paper by Voigt {\it Theoretische
Studien \"{u}ber die Elasticit\"{a}tsverh\"{a}ltnisse der Krystalle} published
also in 1887 \cite{28B}. Interestingly, Walter gives a slightly different
translation of Minkowski's quotation: ``how anyone can apply odd calculations
in the hope that later, perhaps, someone will be found who can get something
out of it'' \cite{28B}.

Although Voigt and Lorentz were in correspondence since 1883,  Lorentz was
unaware of Voigt's Doppler principle paper and it was not until 1908 that
Voigt sent him a reprint of this paper. In a letter of acknowledgment of the
receipt of the preprint from July, 1908, Lorentz writes ``I would like to
thank you very much for sending me your paper on Doppler's principle
together with your enclosed remarks. I really regret that your paper has
escaped my notice. I can only explain it by the fact, that many lectures kept
me back from reading everything, while I was already glad to be able to work
a little bit. Of course I will not miss the first opportunity to mention, that
the concerned transformation and the introduction of a local time has been
your idea'' \cite{28}.

Voigt and Minkowski were friends at G\"{o}ttingen. So it is possible that
Voigt informed Minkowski about his correspondence with Lorentz in around July,
1908 and probably that's how Minkowski became aware of the Voigt's 1887
Doppler principle paper \cite{28}.

In confirmation of this suggestion Minkowski cites (incorrectly) Voigt in the
Cologne lecture as the discoverer of the Lorentz symmetry. Namely, he says
``Now the impulse and true motivation for assuming the group $G_{c}$ came from
the fact that the differential equation for the propagation of light waves in
empty space possesses the group $G_{c}$'' and adds in the footnote that
an essential application of this fact can already be found in Voigt's 1887
paper \cite{28C}.

Two days after the Cologne lecture, in a discussion that followed the
Alfred Bucherer's lecture {\it Measurements of Becquerel rays. The Experimental
Confirmation of the Lorentz-Einstein Theory}, Minkowski again tried to promote
Voigt as the discoverer of Lorentz transformations. He said: ``Historically, I
want to add that the transformation, which play the main role in the relativity
principle, were first mathematically discussed by Voigt in the year 1887.
Already then, Voigt drew some consequences with their aid, in respect to the
principle of Doppler'' \cite{28D}.

Convincing himself that it was Voigt who should be credited for apprehension
of the central role of the Lorentz symmetry, perhaps, it was psychologically
more easy for Minkowski to decide to omit Poincar\'{e}'s name from the Cologne
lecture. Very likely, this pernicious decision was further eased by
Poincar\'{e}'s style of writing who ``habitually wrote in a self-effacing
manner. He named many of his discoveries after other people, and expounded many
important and original ideas in writings that were ostensibly just reviewing
the works of others, with 'minor amplifications and corrections'.
Poincar\'{e}'s style of writing, especially on topics in physics, always gave
the impression that he was just reviewing someone else's work'' \cite{28E}.

Curiously, Voigt's name resurfaces, and in a rather mysterious way, in
relation with Einstein's paper \cite{1} too (\cite{29}). In \cite{1} Einstein
doesn't use Lorentz's and Poincar\'{e}'s nomenclature $(x, y, z, t)
\leftrightarrow (x^\prime, y^\prime, z^\prime, t^\prime)$ while writing the
final form of Lorentz transformations, but rather the one used in Voigt's
paper \cite{27}: $(x, y, z, t)\leftrightarrow (\xi, \eta, \zeta, \tau)$.
We don't know the reason of this weird coincidence. Did Einstein read Voigt's
paper? Or was it reported on the Olympia Academy session? Probably
not and all this is just a coincidence. After all
$(\xi_1, \eta_1, \zeta_1)$, $(\xi_2, \eta_2, \zeta_2)$, etc. were used by
Lorentz to denote locations of the molecules in his 1895 treatise which
Einstein was well aware of. Poincar\'{e} also uses $(\xi, \eta, \zeta )$,
though to denote velocity components, in \cite{15,16}.

If you still consider such a coincidence as improbable, one can remember
another amazing coincidence of this kind. The following two identities,
expressing parametric solutions for representing a forth power as a sum of
five  forth powers, can be found in Ramanujan's third notebook \cite{29A}:
\begin{equation}
(8s^2+40st-24t^2)^4+(6s^2-44st-18t^2)^4+(14s^2-4st-42t^2)^4+
(9s^2+27t^2)^4+(4s^2+12t^2)^4=(15s^2+45t^2)^4,
\label{eqR1}
\end{equation}
and
\begin{equation}
(4m^2-12n^2)^4+(3m^2+9n^2)^4+(2m^2-12mn-6n^2)^4+(4m^2+12n^2)^4+
(2m^2+12mn-6n^2)^4=(5m^2+15n^2)^4.
\label{eqR2}
\end{equation}
What is amazing about this identities is that the first identity was published
by Haldeman in 1904 and Ramanujan uses the same notations and precisely the
same order of the terms as Haldeman. ``One might conclude that Ramanujan saw
Haldeman's paper or a secondary source quoting it. However, this seems highly
unlikely as Ramanujan had access to very few journals in India and,
moreover, Haldeman's paper was published in a very obscure journal. It is also
quite possible that Ramanujan made his discovery before Haldeman did. Thus, in
conclusion, the identical notation must be an amazing coincidence''
\cite{29A}.

The second identity (\ref{eqR2}) is also contained in that Haldeman's paper.
However, in this case Ramanujan uses not the same notation as Haldeman.
His notation (but not order of the terms) coincides to what was used by
A.~Martin, another mathematician who published his version of the proof of
this identity in the same volume in which Haldeman's paper appeared.
``Again, this must be an astonishing coincidence'' \cite{29A}. Probably,
the same can be said about Einstein-Voigt notation coincidence.

\section{Voigt transformations}
Now let us take a closer look at the Voigt's 1887 Doppler principle paper
\cite{28,29,30,31,32}. The first impression is that this paper \cite{27} looks
more like a technical note with messy calculations than an ordinary
research paper: there is no abstract present nor any explanation of
the idea and purpose of the paper, and, like Einstein's \cite{1}, it does not
contain references \cite{31}. Voigt's objective was to derive the Doppler
effect formula from the invariance of the wave equation
\begin{equation}
\Box\, \Phi(x,y,z,t)=\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-
\Delta \right) \Phi(x,y,z,t)=0,
\label{eq1}
\end{equation}
under the change of variables
\begin{equation}
\begin{array}{l}
\xi=m_1(V)\,x+n_1(V)\,y+p_1(V)\,z-\alpha_0(V)\, t, \\
\eta=m_2(V)\,x+n_2(V)\,y+p_2(V)\,z-\beta_0(V)\, t, \\
\zeta=m_3(V)\,x+n_3(V)\,y+p_3(V)\,z-\gamma_0(V)\, t, \\
\tau=t-\left [a_0(V)\,x+b_0(V)\,y+c_0(V)\,z\right ],
\end{array}
\label{eq2}
\end{equation}
which corresponds to the transition from the {\ae}ther rest frame to an
other frame moving with a constant velocity $\vec{V}$. Voigt doesn't explain
why such invariance (which implies that the light velocity is the same in
both frames) should take place, he simply says that the wave equation in the
new frame has the form
\begin{equation}
\left(\frac{1}{c^2}\frac{\partial^2}{\partial \tau^2}-
\frac{\partial^2}{\partial \xi^2}-\frac{\partial^2}{\partial \eta^2}-
\frac{\partial^2}{\partial \zeta^2}\right) \Phi(\xi,\eta,\zeta,\tau)=0,
\label{eq3}
\end{equation}
``as it must be'' \cite{31}. Neither he provides any rationale behind the
coordinate transformation (\ref{eq2}).

Comparing (\ref{eq1}) and (\ref{eq3}), Voigt obtains a system of nine
equations for fifteen unknown constants
$$m_1,\,n_1,\,p_1,\,m_2,\,n_2,\,p_2,\,m_3,\,n_3,\,p_3,a_0,\, b_0,\, c_0,\,
\alpha_0,\,\beta_0,\,\gamma_0.$$
Therefore, he needs some assumptions to solve this troublesome system.
First of all he notices that in the moving frame $S^\prime$ the origin
($x=0,y=0,z=0$) of the stationary frame $S$ moves with the velocity
$-(\alpha_0,\beta_0,\gamma_0)$. Therefore, $(\alpha_0,\beta_0,\gamma_0)$ can
be identified with $\vec{V}$ and assuming the standard configuration
($S^\prime$ moves along the $x$-axis with respect to $S$), we get
$\alpha_0=V,\,\beta_0=0,\,\gamma_0=0$. The following can then be calculated
from (\ref{eq2}):
\begin{eqnarray} &&
\frac{\partial}{\partial x}=m_1\,\frac{\partial}{\partial \xi}+
m_2\,\frac{\partial}{\partial \eta}+m_3\,\frac{\partial}{\partial \zeta}-
a_0\,\frac{\partial}{\partial \tau}, \;\;\;\;
\frac{\partial}{\partial y}=n_1\,\frac{\partial}{\partial \xi}+
n_2\,\frac{\partial}{\partial \eta}+n_3\,\frac{\partial}{\partial \zeta}-
b_0\,\frac{\partial}{\partial \tau}, \nonumber \\ &&
\frac{\partial}{\partial z}=\;p_1\,\frac{\partial}{\partial \xi}+
\;p_2\;\frac{\partial}{\partial \eta}+\;p_3\;\frac{\partial}{\partial \zeta}-
\;c_0\;\frac{\partial}{\partial \tau}, \;\;\;\;
\frac{\partial}{\partial t}=\frac{\partial}{\partial \tau}-V\,\frac{\partial}
{\partial \xi}.
\label{eq4}
\end{eqnarray}
The invariance of the wave equation implies that
\begin{equation}
\Box^\prime=\frac{1}{c^2}\frac{\partial^2}{\partial \tau^2}-
\frac{\partial^2}{\partial \xi^2}-\frac{\partial^2}{\partial \eta^2}-
\frac{\partial^2}{\partial \zeta^2}=\gamma^2(V)\,\Box=\gamma^2(V)\left (
\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}
{\partial x^2}-\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial z^2}
\right ),
\label{eq5}
\end{equation}
where $\gamma^2(V)$ is some constant. In combination with (\ref{eq4}) this
condition gives ten equations which we organize into three groups (starting
from this point we somewhat modify Voigt's original derivation in order to
make it more transparent). Equations of the first group correspond to the
diagonal elements of the d'Alembert operator and they have the form
\begin{eqnarray} &&
\gamma^2\left [1-(a_0^2+b_0^2+c_0^2)c^2\right ]=1,\;\;\;\;
\gamma^2\left [(m_1^2+n_1^2+p_1^2)-\frac{V^2}{c^2}\right ]=1, \nonumber \\ &&
\gamma^2\left [m_2^2+n_2^2+p_2^2\right ]=1,\hspace*{20mm}
\gamma^2\left [m_3^2+n_3^2+p_3^2\right ]=1.
\label{eq6}
\end{eqnarray}
The second group corresponds to the equations that ensure the vanishing of the
mixed derivatives $\frac{\partial^2}{\partial \tau \partial \xi}$,
$\frac{\partial^2}{\partial \tau \partial \eta}$ and
$\frac{\partial^2}{\partial \tau \partial \zeta}$. These equations are
\begin{equation}
m_1\,a_0+n_1\,b_0+p_1\,c_0-\frac{V}{c^2}=0,\;\;
m_2\,a_0+n_2\,b_0+p_2\,c_0=0,\;\;m_3\,a_0+n_3\,b_0+p_3\,c_0=0.
\label{eq7}
\end{equation}
The third group of equations ensures the vanishing of the mixed
derivatives $\frac{\partial^2}{\partial \xi \partial \eta}$,
$\frac{\partial^2}{\partial \xi \partial \zeta}$ and
$\frac{\partial^2}{\partial \eta \partial \zeta}$. They have the form
\begin{equation}
m_1\,m_2+n_1\,n_2+p_1\,p_2=0,\;\;m_1\,m_3+n_1\,n_3+p_1\,p_3=0,\;\;
m_2\,m_3+n_2\,n_3+p_2\,p_3=0.
\label{eq8}
\end{equation}
As we see, we have thirteen unknowns (including $\gamma^2$) and ten equations.
So, Voigt concludes that three of them can be chosen arbitrarily. The natural
choice is $m_1=1,\,n_1=0,\,p_1=0$, which makes the transformation law for
$\xi$ identical to the Galilean transformation. Then equations (\ref{eq8})
imply $m_2=m_3=0$ and
\begin{equation}
n_2\,n_3=-p_2\,p_3,
\label{eq9}
\end{equation}
while from equations (\ref{eq6}) we get
\begin{equation}
\gamma^{-2}=1-\frac{V^2}{c^2},
\label{eq10}
\end{equation}
and
\begin{equation}
n_2^2+p_2^2=n_3^2+p_3^2=\gamma^{-2}.
\label{eq11}
\end{equation}
At last, equations (\ref{eq7}) will give us in this case
\begin{equation}
a_0=\frac{V}{c^2},
\label{eq12}
\end{equation}
and
\begin{equation}
n_2\,b_0=-p_2\,c_0,\;\;\;n_3\,b_0=-p_3\,c_0.
\label{eq13}
\end{equation}
Equations (\ref{eq9}) and (\ref{eq13}) can be combined to produce the
following relations
\begin{equation}
(n_2^2+p_2^2)p_3\,c_0=0,\;\;\;(n_3^2+p_3^2)n_2\,b_0=0,
\label{eq14}
\end{equation}
which in light of (\ref{eq11}) imply
\begin{equation}
p_3\,c_0=0,\;\;\;n_2\,b_0=0.
\label{eq15}
\end{equation}
However, in the limit $V=0$ we should have $n_2(0)=1$ and $p_3(0)=1$.
Therefore, $n_2$ and $p_3$ can't be identically zero and the only way to
satisfy (\ref{eq15}) is to take $b_0=c_0=0$.

If we introduce two complex numbers $z_1=n_2+ip_2$ and $z_2=p_3+in_3$, the
equations (\ref{eq9}) and (\ref{eq11}) indicate that $|z_1|=|z_2|=\gamma^{-1}$
and that $z_1z_2$ is a real number. Therefore, $z_2=\pm z_1^*$. Only the plus
sign is acceptable, because $n_2(0)=p_3(0)$. Thus, the most general solution
for $z_1$ and $z_2$ has the form
\begin{eqnarray} &&
n_2=\gamma^{-1}\cos{\phi(V)},\;\;\;\;\;p_2=\gamma^{-1}\sin{\phi(V)},
\nonumber \\ &&
n_3=-\gamma^{-1}\sin{\phi(V)},\;\;\;p_3=\gamma^{-1}\cos{\phi(V)},
\label{eq16}
\end{eqnarray}
where $\phi(V)$ is any function of the velocity $V$ such that $\phi(0)=0$.
The presence of an angle $\phi(V)$ reflects the obvious fact that
a transformation from the {\ae}ther frame to the moving frame can always be
superimposed by a rotation around the axis parallel to $\vec{V}$ by an angle
$\phi(V)$. Excluding this trivial possibility of an additional rotation,
we see that the invariance of the wave equation under Voigt's assumptions
essentially uniquely leads to Voigt transformations:
\begin{equation}
\begin{array}{l}
\xi=x-V\, t, \\
\eta=\gamma^{-1}\,y, \\
\zeta=\gamma^{-1}\,z, \\
\tau=t-\frac{V}{c^2}\,x.
\end{array}
\label{eq17}
\end{equation}
The inverse transformations have the form
\begin{equation}
\begin{array}{l}
x=\gamma^2\left(\xi+V\, \tau\right), \\
y=\gamma\,\eta, \\
z=\gamma\,\zeta, \\
t=\gamma^2\left(\tau+\frac{V}{c^2}\,\xi\right).
\end{array}
\label{eq18}
\end{equation}
For a modern reader it is obvious that Voigt was very close to discovering
special relativity. His transformations reveal the crucial fact that the
invariance of the wave equation is inconsistent with Newtonian absolute time.
In fact, Voigt was the first who discovered ``that a 'natural' clock would
alter its rate on motion'' \cite{33}. More precisely, to be fair, Voigt missed
this opportunity because he apparently did not realized a great conceptual
novelty of his transformations. His main objective was the Doppler effect.
Using his transformations he concludes that the vibration period experiences
time dilation with the (wrong) factor $\gamma^2$. However, the conclusion that
all moving clocks would actually behave like this, was not ventured and to
a great extent ``his paper appears to be a mathematical exercise'' without
giving any conceptual reason for developing this particular set of
transformations \cite{33}.

Voigt transformations differ from Lorentz transformations only by a scale
factor. Many physicists, including Minkowski and Lorentz, considered
this difference insignificant and credited Voigt with the discovery of Lorentz
transformations \cite{31}. For example, Lorentz wrote in 1914: ``These
considerations published by myself in 1904, have stimulated Poincar\'{e} to
write his article on the dynamics of electron where he has given my name to
the just mentioned transformation. I have to note as regards this that
a similar transformation has been already given in an article by Voigt
published in 1887 and I have not taken all possible benefit from it''
\cite{9}. It seems Voigt himself finally shared this opinion. Voigt's 1887
paper was reprinted in {\it Physikalische Zeitschrift} on occasion of the
tenth anniversary of the principle of relativity in 1915. In the reprinted
version Voigt included additional comments and in particular when referring to
his transformations he wrote: ``This is, except for the factor $q$ [$\gamma$]
which is irrelevant for the application, exactly the Lorentz transformation of
the year 1904'' \cite{31}.

Are Voigt transformations and Lorentz transformations indeed equivalent? The
answer depends on the reading of Voigt transformations: modern reader can
see quite different contexts for them compared to Voigt's contemporaries.
At first sight, the scale factor ``indicates only a uniform magnification of
the scales of space and time, or what is the same thing, a change of units.
It does not introduce any essential modification'' \cite{34}. The same
argument is also given in \cite{35}.

Indeed, let us assume that to define a time unit the observer in the moving
frame uses a standard atomic clock of the {\ae}ther frame. As this clock moves
with respect to the observer, his inferred time unit will be $\gamma$ times
larger than if he would use, as tacitly assumed in usual Lorentz
transformations, an atomic clock at rest in his own frame as the standard
clock. Modern definition of the length unit relates it through the light
velocity $c$ to the time unit. Therefore, the length unit will be also
$\gamma$ times larger, and to express the Lorentz transformations in these new
units of the moving frame we should divide primed coordinates by $\gamma$.
As a result, we get precisely the Voigt transformations (\ref{eq17}).

Therefore, the relativistic reading of Voigt transformations is certainly
possible and the resulting theory will be completely equivalent (although
perhaps less convenient) to special relativity. Note that the introduction of
the preferred {\ae}ther frame in this case is purely formal: we can choose any
inertial reference frame instead of the {\ae}ther frame and use its
standard clocks to define time units in other inertial reference frames.

However, surprisingly enough, a non-relativistic Galilean
reading of Voigt transformations (\ref{eq17}) is also possible. Let us
consider a hypothetical Newtonian world with the truly {\ae}ther frame $S$
which is the only inertial frame in which light propagates isotropically with
velocity $c$. Transition to an other inertial frame $S^\prime$ is given
by Galilean transformations
\begin{equation}
\begin{array}{l}
\xi=x-V\, t, \\
\eta=y, \\
\zeta=z, \\
\tau=t.
\end{array}
\label{eq19}
\end{equation}
This form of Galilean transformations assumes Newtonian absolute time.
However, if the velocity $V$ of $S^\prime$ with respect to the {\ae}ther frame
$S$ is smaller than light velocity $c$, it is possible to perform
Poincar\'{e}-Einstein synchronization of clocks in $S^\prime$ and as a result
we get another parametrization of the Galilean space-time in $S^\prime$.
Let us find this  parametrization \cite{36}.

Suppose a light signal is sent from the origin $O^\prime$ of the inertial
reference frame $S^\prime$ at absolute time $\tau_1$, it arrives to some
point $P^\prime(\xi,\eta,\zeta)$ at absolute time $\tau_2$, is instantaneously
reflected by a mirror at $P^\prime$ and returns to $O^\prime$  at absolute
time $\tau_3$. According to Poincar\'{e}-Einstein synchronization convention,
to synchronize the clock at $P^\prime$ with the clock at $O^\prime$,
one should set its reading to $t^\prime_2=(\tau_1+\tau_3)/2$ at an instant of
signal reflection. Taking
\begin{equation}
\tau_3=\tau,\;\;\; \tau_1=\tau-\frac{r^\prime}{c_1^\prime}-
\frac{r^\prime}{c_2^\prime},
\label{eq20}
\end{equation}
where $r^\prime=\sqrt{\xi^2+\eta^2+\zeta^2}$, and $c_1^\prime,c_2^\prime$
are light velocities in the frame $S^\prime$ when it moves from $O^\prime$
to $P^\prime$ and back respectively, we get
\begin{equation}
t^\prime_2=\tau-\frac{1}{2}\left (\frac{r^\prime}{c_1^\prime}+
\frac{r^\prime}{c_2^\prime}\right ).
\label{eq21}
\end{equation}
Therefore, at the time $\tau_3=\tau$ at $O^\prime$, the Poincar\'{e}-Einstein
synchronized clock at $P^\prime$ will show
\begin{equation}
t^\prime=t^\prime_2+\frac{r^\prime}{c_2^\prime}=\tau-\frac{1}{2}\left (
\frac{r^\prime}{c_1^\prime}-\frac{r^\prime}{c_2^\prime}\right )=
\tau-\frac{r^\prime}{2}\,\frac{c_2^\prime-c_1^\prime}{c_1^\prime c_2^\prime}.
\label{eq22}
\end{equation}
On the other hand, according to the Galilean velocity addition formula,
\begin{equation}
c^2=(\vec{c}_1^\prime+\vec{V})^2=c_1^{\prime\,2}+2c_1^\prime\,V
\cos{\theta^\prime}+V^2,\;\;
c^2=(\vec{c}_2^\prime+\vec{V})^2=c_2^{\prime\,2}-2c_2^\prime\,V
\cos{\theta^\prime}+V^2,
\label{eq23}
\end{equation}
where $\theta^\prime$ is the angle between the radius-vector $\vec{r}^\prime$
and the $x^\prime$-axis. The positive solutions of (\ref{eq23}) are
\begin{equation}
c_1^\prime=-V\cos{\theta^\prime}+\sqrt{V^2\cos^2{\theta^\prime}+c^2-V^2},\;\;
c_2^\prime=V\cos{\theta^\prime}+\sqrt{V^2\cos^2{\theta^\prime}+c^2-V^2}.
\label{eq24}
\end{equation}
Substituting these velocities into (\ref{eq22}), we get (instead of
$(\xi,\eta,\zeta)$, we have returned to the more traditional
$(x^\prime,y^\prime, z^\prime)$ notation)
\begin{equation}
t^\prime=\tau-\frac{r^\prime\, V\,\cos{\theta^\prime}}{c^2-V^2}=
\tau-\gamma^2\,\frac{V}{c^2}\,x^\prime.
\label{eq25}
\end{equation}
Therefore, if the Poincar\'{e}-Einstein synchronization convention is adopted,
instead of Galilean transformations we will get Zahar transformations
\cite{36}:
\begin{eqnarray} &&
x^\prime=x-V\,t, \nonumber \\ &&
y^\prime=y,\nonumber \\ &&
z^\prime=z,\nonumber \\ &&
t^\prime=t-\gamma^2\,\frac{V}{c^2}\,x^\prime=\gamma^2\left (t-\frac{V}{c^2}x
\right).
\label{eq26}
\end{eqnarray}
Let us note that, contrary to a prevailing belief, the non-relativistic limit
of Lorentz transformations, when $\beta=V/c\ll 1$, is not the Galilean
transformations (\ref{eq19}) but the Zahar transformations (\ref{eq26})
in which $\beta^2$ terms are neglected \cite{37A,37,38}, so that
$\gamma^2\approx 1$. The Galilean limit additionally requires
\begin{equation}
\beta\ll\frac{ct}{x},\;\;\;\frac{x}{ct}\sim\beta,
\label{eq27}
\end{equation}
which is not necessarily true if the spatial separation $x$ between two events
is comparable or larger than temporal separation $ct$. It seems that this
simple fact is not widely known as witnessed by its rediscovery in
\cite{39,40}. The reason for this mismatch of non relativistic limits is the
use of different synchronization conventions in Lorentz transformations and
Galilean transformations: for sufficiently distant events in Minkowski world
one cannot ignore not absolute nature of distant simultaneity and only when
Poincar\'{e}-Einstein synchronization convention is adopted in Galilean world
too the resulting Zahar transformations share the common non relativistic limit
with the Lorentz transformations \cite{39}.

Now let us assume that inhabitants of the frame $S^\prime$ decided to use
an atomic clock of the frame $S$ as the time standard (and let us assume that
clocks when in motion behave as prescribed by Newtonian physics of Galilean
space-time). Then (\ref{eq26}) shows that the time unit so defined in
$S^\prime$ will be $\gamma^2$ times larger than the time unit inferred from
the stationary atomic clock in $S^\prime$, as assumed in (\ref{eq26}).
Length units, as usually, are determined from the time unit and light
velocity. However we should be careful here because the light velocity is not
isotropic in $S^\prime$ in Galilean world even under the
Poincar\'{e}-Einstein synchronization convention, if the ``natural'' units,
as in (\ref{eq26}), are used. The velocity addition rule that follows from
(\ref{eq26}) has the form
\begin{equation}
V_x^\prime=\frac{V_x-V}{\gamma^2\left(1-\frac{V}{c^2}V_x\right)},\;\;
V_y^\prime=\frac{V_y}{\gamma^2\left(1-\frac{V}{c^2}V_x\right)},\;\;
V_z^\prime=\frac{V_z}{\gamma^2\left(1-\frac{V}{c^2}V_x\right)}.
\label{eq28}
\end{equation}
Suppose light signal moves in $S^\prime$ along the $x$-axis with velocity
$c_\parallel=V_x^\prime$. Then (\ref{eq28}) shows that, in the frame $S$,
$V_y=V_z=0$. Hence $V_x=c$ and
\begin{equation}
c_\parallel=\frac{c-V}{\gamma^2\left(1-\frac{V}{c}\right)}=\frac{c}{\gamma^2}.
\label{eq29}
\end{equation}
If the light impulse in $S^\prime$ moves along the $y$-axis with velocity
$c_\perp=V_y^\prime$, then (\ref{eq28}) shows that $V_x=V,\,V_z=0$, and hence
$V_y=\sqrt{c^2-V^2}$. In this case (\ref{eq28}) will give
\begin{equation}
c_\perp=\frac{\sqrt{c^2-V^2}}{\gamma^2\left(1-\frac{V^2}{c^2}\right)}=
\frac{c}{\gamma}.
\label{eq30}
\end{equation}
Therefore, when the time unit is increased $\gamma^2$ times, the length unit
in the $x$-direction (parallel to the $S^\prime$ velocity) will not change
at all and in the transverse $y$- and $z$-directions will be increased
$\gamma$ times. To write Zahar transformations (\ref{eq26}) in these new
units, we must therefore divide $t^\prime$ by $\gamma^2$, $y^\prime$ and
$z^\prime$ by $\gamma$, and leave $x^\prime$ unchanged. As a result we again
obtain Voigt transformations (\ref{eq17}).

As we see, Voigt transformations by themselves are empirically empty
without detailing how coordinates involved in these transformations are
related to the behavior of real clocks and rulers. Voigt transformations
require an additional input about the nature of real clocks not to be just
a mathematical exercise but an instrument to answer questions about the physics
of the real world . Let us demonstrate this by considering a natural question
whether the Poincar\'{e}-Einstein clock synchronization is equivalent
to another synchronization method by slow clock transport.

Let us consider an inertial reference frame $S^\prime$ moving with respect to
{\ae}ther frame $S$ (which is singled out by Voigt transformations) with
velocity $V$. Clocks in $S^\prime$ are synchronized by Poincar\'{e}-Einstein
procedure. Let one of the so synchronized clocks be transported slowly in
$S^\prime$ from one point $A$ to an another point $B$ with the velocity
$u^\prime\ll c$ along the $x$-axis (for simplicity). If this process takes
time $\tau$ according to the clocks in $S^\prime$, then $\Delta \xi=u^\prime
\tau$ and the inverse Voigt transformations (\ref{eq18}) will give
\begin{equation}
\Delta x=\gamma_V^2(u^\prime+V)\tau,\;\;\;\Delta t=\gamma_V^2\left [1+
\frac{u^\prime V}{c^2}
\right ] \tau.
\label{eq31}
\end{equation}
The Voigt transformations (\ref{eq17}) imply just the same velocity addition
rule as the Lorentz transformations. Therefore, with respect to $S$, the clock
moving in $S^\prime$ has the velocity
\begin{equation}
u=\frac{u^\prime+V}{1+\frac{V}{c^2}u^\prime}\approx (u^\prime+V)\left (
1-\frac{V}{c^2}u^\prime\right )\approx V+\frac{u^\prime}{\gamma_V^2}.
\label{eq32}
\end{equation}
Hence, according to (\ref{eq17}) and (\ref{eq31}), while moving from $A$ to
$B$, the clock will measure the proper time
\begin{equation}
\tau^\prime=\Delta t-\frac{u}{c^2}\Delta x\approx\gamma_V^2\tau\left [1-
\frac{V^2}{c^2}-\frac{u^\prime V}{c^2\gamma_V^2}\right ]=
\left (1-\frac{u^\prime V}{c^2}\right )\tau.
\label{eq33}
\end{equation}
However, this time interval is measured in time units of the reference frame
moving with velocity $u$, not $V$. Therefore, we need to rescale it in
the $S^\prime$ time units in order to compare to the reading of the clock
at rest in  $S^\prime$ at point $B$. This rescaling cannot be done on
the base of Voigt transformations alone and this is the point where an extra
empirical input about the behavior of real clocks is needed. In particular
we must remember how time units for Voigt transformations were defined if
we stick the interpretation of these transformations given above with either
Lorentz or Zahar transformations as the primary source describing the
behavior of real clocks. Having all this in mind, we write the rescaling as
follows
\begin{equation}
\tilde \tau=\tau^\prime\,\frac{\gamma_u^n}{\gamma_V^n},
\label{eq34}
\end{equation}
where $n=1$, if the real clocks behave as prescribed by special relativity,
and $n=2$, if the real clocks are Galilean. But from (\ref{eq32}) it is easy
to find that
\begin{equation}
\gamma_u\approx \gamma_V\left( 1+\frac{u^\prime V}{c^2}\right ).
\label{eq35}
\end{equation}
Therefore, from (\ref{eq33}) and (\ref{eq35}), we get
\begin{equation}
\tilde \tau= \tau\left (1-\frac{u^\prime V}{c^2}\right )
\left (1+n\frac{u^\prime V}{c^2}\right )\approx \tau\left [1+(n-1)
\frac{u^\prime V}{c^2}\right ].
\label{eq36}
\end{equation}
As we see, $\tilde \tau=\tau$ if and only if $n=1$. That is only in the
Minkowski world the Poincar\'{e}-Einstein synchronization and
synchronization by slow clock transport agree with each other.
In Galilean world these two synchronization methods will give different
results.

Robertson \cite{41} and later Mansouri and Sexl \cite{42} developed a general
framework for test theories of special relativity which can be considered
as a generalization of Voigt, Lorentz and Zahar transformations. These
generalized transformations can be written in the form \cite{43}
\begin{eqnarray} &&
x^\prime=\lambda\,\frac{c_\parallel}{c_\perp}\,\gamma(x-V\,t), \nonumber \\ &&
y^\prime=\lambda y,\nonumber \\ &&
z^\prime=\lambda z,\nonumber \\ &&
t^\prime=\lambda\,\frac{c}{c_\perp}\,\gamma\left (t-\frac{V}{c^2}x
\right),
\label{eq37}
\end{eqnarray}
where $c_\parallel$ and $c_\perp$ are two-way  light speeds in the moving
frame $S^\prime$ in the parallel and perpendicular directions to the
the velocity $\vec{V}$ of $S^\prime$ with respect to the preferred
({\ae}ther) frame $S$, in which light propagates isotropically with the speed
$c$. The parameters $c_\parallel/c$, $c_\perp/c$ and the conformal factor
$\lambda$ are, in general, functions of $\gamma$. For example, if
$\lambda=1,\,c_\parallel=c_\perp=c$, we get from (\ref{eq37}) Lorentz
transformations; if $\lambda=1,\,c_\parallel=c/\gamma^2,\,c_\perp=c/\gamma$,
we get Zahar transformations, and, finally, if $\lambda=\gamma^{-1},\,
c_\parallel=c_\perp=c$, we get Voigt transformations. It
can be shown using (\ref{eq37}) that ``clock synchronization by clock
transport and by the Einstein procedure agree if and only if the time
dilatation factor is given exactly by the special relativistic value''
\cite{42}, that is if $\lambda\,c/c_\perp=1$.

We hope that our above discussions of the Voigt transformations convinced you
that a modern reader can find a lot of relativistic content in these
transformations. However, all of them are later time readings and we suspect
that none were actually possible at Voigt's time. It would be not true to
say that Voigt was at the origin of relativistic revolution. Alas, he missed
this opportunity.

\section{Another derivation of Voigt transformations}
One of the reasons why Voigt's 1887 paper remained unnoticed was its clumsy
and unwieldy derivation of Voigt's transformations. In this section we will try
to give more modern and succinct derivation which will reveal basic physical
inputs behind it. Inspiration for this derivation and its subsequent
generalization comes mostly from one postulate derivation of Lorentz
transformations originated by von Ignatowsky \cite{44} and further developed
by Frank and Rothe \cite{44A}. More precisely, we have in mind the idea
presented in \cite{45}.

Under the assumption that measuring rods do not change their length when
gently set into a state of uniform motion, the Galilean transformations
(\ref{eq19}) are simply a statement that the length of a finite interval is
an additive quantity: the length of a union of two intervals is equal to the
sum of their lengths. However, it seems not too fantastic to imagine that Voigt
in 1887 could question whether the measuring rods really do not change their
lengths when moving through {\ae}ther. At least in 1888 Oliver Heaviside
showed that the electric field of a charge in motion relative to the {\ae}ther
is no longer spherically symmetric and becomes distorted in the the
longitudinal direction. Influenced by the Heaviside's result, FitzGerald and
later Lorentz came out with a hypothesis that Heaviside distortion might be
applied to the molecular forces too and thus rigid bodies when moving through
the {\ae}ther should experience deformations. In particular, in 1889
FitzGerald published in then little known American journal {\it Science}
a letter in which he wrote:

``I have read with much interest Messrs. Michelson and Morley's wonderfully
delicate experiment attempting to decide the important question as to how far
the {\ae}ther is carried along by the earth. Their result seems opposed to
other experiments showing that the {\ae}ther in the air can be carried along
only to an inappreciable extent. I would suggest that almost the only
hypothesis that can reconcile this opposition is that the length of material
bodies changes, according as they are moving through the {\ae}ther or across
it, by an amount depending on the square of the ratio of their velocities to
that of light. We know that electric forces are affected by the motion of the
electrified bodies relative to the {\ae}ther, and it seems a not improbable
supposition that the molecular forces are affected by the motion, and that the
size of a body alters consequently'' \cite{46}.

Interestingly, this FitzGerald's publication sank into oblivion and was
virtually unknown until 1967 when it was unearthed by Brush \cite{46}.
Paradoxically, FitzGerald himself was not aware whether his letter to
{\it Science} was published or not: ``FitzGerald did publish the contraction
hypothesis before Lorentz, but no one involved in later research on the subject
knew about this publication, including FitzGerald himself'' \cite{46}. He did
promote his deformation idea mostly in lectures and was relieved to know that
in 1892 Lorentz came out with essentially the same idea. In 1894 FitzGerald
wrote to Lorentz: ``I am particularly delighted to hear that you agree with me,
for I have been rather laughed at for my view over here'' \cite{46}.

If we accept the possibility that measuring rods can change their length when
in motion, instead of $x=x^\prime+Vt$ we should write $x=k_1(V)x^\prime+Vt$,
where $k_1(V)$ accounts for a possible change of the measuring rod's length
in the moving frame $S^\prime$. But from the point of view of an observer in
$S^\prime$, the reference frame $S$ moves with velocity $-V$ (although there
are some subtleties in this reciprocity principle \cite{47,48}, we assume that
in 1887 Voigt probably would not question it). Therefore we can analogously
write $x^\prime=k_2(-V)x-Vt^\prime$. If we assume the validity of the
relativity principle, then $k_1(V)=k_2(V)$, and if we assume the spatial
isotropy, then $k_1(-V)=k_1(V)$. However, for Voigt, a true believer in
{\ae}ther, relativity principle was not evident, so we do not
assume its validity for a while, neither the spatial isotropy. Hence, we have
\begin{equation}
x^\prime=\frac{1}{k_1(V)}\left (x-Vt\right ),\;\;\;
x=\frac{1}{k_2(-V)}\left (x^\prime+Vt^\prime\right ).
\label{eq38}
\end{equation}
From these equations we can express $t^\prime$ in terms of $x$ and $t$,
or $t$ in terms of $x^\prime$ and $t^\prime$:
\begin{equation}
t^\prime=\frac{1}{k_1(V)}\left [t-\frac{1-K(V)}{V}\,x\right],\;\;\;
t=\frac{1}{k_2(-V)}\left [t^\prime+\frac{1-K(V)}{V}\,x^\prime\right],
\label{eq39}
\end{equation}
where $K(V)=k_1(V)k_2(-V)$. Therefore, we get the transformation
\begin{eqnarray} &&
x^\prime=\frac{1}{k_1(V)}\,(x-V\,t), \nonumber \\ &&
y^\prime=y,\nonumber \\ &&
z^\prime=z,\nonumber \\ &&
t^\prime=\frac{1}{k_1(V)}\,\left [t-\frac{1-K(V)}{V}\,x\right ],
\label{eq40}
\end{eqnarray}
and its inverse
\begin{eqnarray} &&
x=\frac{1}{k_2(-V)}\,(x^\prime+V\,t^\prime), \nonumber \\ &&
y=y^\prime,\nonumber \\ &&
z=z^\prime,\nonumber \\ &&
t=\frac{1}{k_2(-V)}\,\left [t^\prime+\frac{1-K(V)}{V}\,x^\prime\right ].
\label{eq41}
\end{eqnarray}
We have assumed that the transverse coordinates do not change. This is not
the most general possibility but probably the one which Voigt's contemporaries
would infer from  Heaviside's result on the moving Coulomb field distortions.

Voigt was interested in wave equation (\ref{eq1}) which is obviously invariant
under the scale transformations. Considering all coordinates that differ only
by a scale change as equivalent, we can choose new coordinates
\begin{equation}
X^\prime=k_1(V)x^\prime, \;\;Y^\prime=k_1(V)y^\prime,\;\;
Z^\prime=k_1(V)z^\prime, \;\;T^\prime=k_1(V)t^\prime,
\label{eq42}
\end{equation}
to regain the Galilean relation $x=X^\prime+Vt$. In new coordinates
(\ref{eq40}) takes the form
\begin{eqnarray} &&
X^\prime=x-V\,t, \nonumber \\ &&
Y^\prime=k_1(V)\,y,\nonumber \\ &&
Z^\prime=k_1(V)\,z,\nonumber \\ &&
T^\prime=t-\frac{1-K(V)}{V}\,x,
\label{eq43}
\end{eqnarray}
while its inverse (\ref{eq41}) transforms into
\begin{eqnarray} &&
x=\frac{1}{K(V)}\left[X^\prime+V\,T^\prime\right],, \nonumber \\ &&
y=k^{-1}_1(V)\,Y^\prime,\nonumber \\ &&
Z=k^{-1}_1(V)\,Z^\prime,\nonumber \\ &&
t=\frac{1}{K(V)}\left[T^\prime+\frac{1-K(V)}{V}\,X^\prime \right ].
\label{eq44}
\end{eqnarray}
The velocity addition rule following from (\ref{eq44}) has the form
\begin{equation}
v_x=\frac{v^\prime_x+V}{1+\frac{1-K(V)}{V}\,v^\prime_x},\;\;\;
v_y=\frac{K(V)\,v^\prime_y}{k_1(V)\left[1+\frac{1-K(V)}{V}\,v^\prime_x\right]},
\;\;\;
v_z=\frac{K(V)\,v^\prime_z}{k_1(V)\left[1+\frac{1-K(V)}{V}\,v^\prime_x\right]}.
\label{eq45}
\end{equation}
If we now require the universality of the light velocity as it follows from
the assumed invariance of the wave equation, and assume $v^\prime_x=c$,
$v^\prime_y=v^\prime_z=0$, then we get from (\ref{eq45}) $v_y=v_z=0$,
\begin{equation}
c=v_x=\sqrt{c^2-v_y^2+v_z^2}=\frac{c+V}{1+\frac{1-K(V)}{V}\,c},
\label{eq46}
\end{equation}
and therefore,
\begin{equation}
K(V)=1-\frac{V^2}{c^2}=\frac{1}{\gamma^2}.
\label{eq47}
\end{equation}
On the other hand, if we take  $v^\prime_y=c$, $v^\prime_x=v^\prime_z=0$, from
(\ref{eq45}) we obtain
\begin{equation}
v_x=V,\;\;\;v_y=\frac{K(V)\,c}{k_1(V)}=\frac{c}{\gamma^2\,k_1(V)},\;\;
v_z=0,
\label{eq48}
\end{equation}
which together with the condition $v_x^2+v_y^2+v_z^2=c^2$ then determines
$k_1(V)$:
\begin{equation}
k_1(V)=\frac{1}{\gamma}.
\label{eq49}
\end{equation}
Substituting (\ref{eq47}) and (\ref{eq49}) into (\ref{eq43}) we get indeed
the Voigt transformations (\ref{eq17}).

Now we will show that a little generalization of this derivation provides an
interesting perspective on the relativity principle. Let us return to the
original coordinates before rescaling (\ref{eq42}), but this time we will
not assume that the transverse coordinates remain unchanged. So we modify
(\ref{eq40}) and (\ref{eq41}) into
\begin{eqnarray} &&
x^\prime=\frac{1}{k_1(V)}\,(x-V\,t), \nonumber \\ &&
y^\prime=\lambda(V)y,\nonumber \\ &&
z^\prime=\lambda(V)z,\nonumber \\ &&
t^\prime=\frac{1}{k_1(V)}\,\left [t-\frac{1-K(V)}{V}\,x\right ]
\label{eq50}
\end{eqnarray}
and
\begin{eqnarray} &&
x=\frac{1}{k_2(-V)}\,(x^\prime+V\,t^\prime), \nonumber \\ &&
y=\lambda^{-1}(V)y^\prime,\nonumber \\ &&
z=\lambda^{-1}(V)z^\prime,\nonumber \\ &&
t=\frac{1}{k_2(-V)}\,\left [t^\prime+\frac{1-K(V)}{V}\,x^\prime\right ]
\label{eq51}
\end{eqnarray}
respectively. From (\ref{eq51}) we get the following velocity addition rule:
\begin{equation}
v_x=\frac{v^\prime_x+V}{1+\frac{1-K(V)}{V}\,v^\prime_x}=F(v^\prime_x,V),\;\;\;
v_y=\frac{k_2(-V)\,v^\prime_y}{\lambda(V)\left[1+\frac{1-K(V)}{V}\,v^\prime_x
\right]},\;\;\;
v_z=\frac{k_2(-V)\,v^\prime_z}{\lambda(V)\left[1+\frac{1-K(V)}{V}\,v^\prime_x
\right]}.
\label{eq52}
\end{equation}
It is clear from the above discussion that then the universality of the light
velocity will demand (\ref{eq47}) and
\begin{equation}
\frac{k_2(-V)}{\lambda(V)}=\frac{1}{\gamma}.
\label{eq53}
\end{equation}
From (\ref{eq47}) and (\ref{eq53}) we can get
\begin{equation}
\lambda(V)=\frac{k_2(-V)}{\sqrt{K(V)}}=\sqrt{\frac{k_2(-V)}{k_1(V)}},
\label{eq54}
\end{equation}
and
\begin{equation}
\frac{1}{k_1(V)}=\sqrt{\frac{k_2(-V)}{k_1(V)}}\,\frac{1}{\sqrt{k_1(V)k_2(-V)}}
=\frac{\lambda(V)}{\sqrt{K(V)}}=\lambda(V)\gamma.
\label{eq55}
\end{equation}
Therefore, (\ref{eq50}) takes the form
\begin{eqnarray} &&
x^\prime=\lambda(V)\gamma\left(x-V\,t\right ), \nonumber \\ &&
y^\prime=\lambda(V)y,\nonumber \\ &&
z^\prime=\lambda(V)z,\nonumber \\ &&
t^\prime=\lambda(V)\gamma\,\left (t-\frac{V}{c^2}\,x\right ).
\label{eq56}
\end{eqnarray}
Both Einstein \cite{1} and Poincar\'{e} \cite{15,16} got these
$\lambda$-Lorentz transformations and then both argued that $\lambda(V)=1$.
The arguments of Einstein were more physical as he correctly identified the
physical meaning of $\lambda(V)$: it corresponds to a contraction of the
length of a moving rod which is perpendicular to the direction of its motion.
''He makes then the clever statement that, because the rod is perpendicular to
the direction of motion, this possible contraction can well depend on the
relative velocity, but not on its sign!'' \cite{17}. Therefore, $\lambda(-V)=
\lambda(V)$. On the other hand, if we adopt the relativity principle so that
the {\ae}ther frame loses its privileged role, then transformations
(\ref{eq56}) should form a group which implies
\begin{equation}
\lambda(V_1\oplus V_2)=\lambda(V_1)\lambda(V_2),
\label{eq57}
\end{equation}
where $V_1\oplus V_2$ denotes the relativistic sum of velocities. In
particular, if we take $V_1=-V_2=V$, then we get $\lambda(V)\lambda(-V)=
\lambda(0)=1$ which in combination with $\lambda(-V)=\lambda(V)$ and
positivity of $\lambda(V)$ implies $\lambda(V)=1$.

Poincar\'{e}'s reasoning was more formal and mathematical. He makes a complete
analysis of the Lorentz's group including not only the boosts, but also the
spatial rotations and gets $\lambda(V)=1$ from the group theory. Note that
the multiplication law (\ref{eq57}) first appeared in Poincar\'{e}'s May 1905
letter to Lorentz \cite{49}. Einstein in \cite{1} doesn't provide this
multiplication law in its full generality --- he considers only mutually
inverse transformations with $V_1=-V_2$.

In fact, both derivations of $\lambda(V)=1$ implicitly assume spatial
isotropy. Although natural, this is not the most general possibility. So let
us go ahead without this assumption and see what gems were thrown overboard
by Einstein and Poincar\'{e} when insisting on it.

First of all, Poincar\'{e} missed an opportunity to discover that Maxwell
equations and wave equation are invariant not only with respect to the Lorentz
group but also with respect to much wider class of conformal transformations
(we think, for Einstein this feat was unrealistic in 1905). This fact was
established several years later by Cunningham and Bateman \cite{50} (their
road to this discovery is described in \cite{50A}). Conformal group is
a fifteen parameter Lie group. It contains as a subgroup the ten parameter
Poincar\'{e} group (inhomogeneous Lorentz transformations). A general
Poincar\'{e} transformation is a combination of a rotation (three parameters)
with a boost (three parameters), followed by a translation in space-time
(four  parameters). Additional five parameters of the conformal group are:
$\lambda$ of the scale transformation, promoted to an independent parameter
representing dilatation, and four parameters of the so called special
conformal transformation ($x_0=x^0=ct,x_1=-x^1=-x,x_2=-x^2=-y,x_3=-x^3=-z$)
\begin{equation}
x^{\prime\,\mu}=\frac{x^\mu-\alpha^\mu \,x_\nu x^\nu}{1-2\alpha_\mu x^\mu+
\alpha_\mu \alpha^\mu\,x_\nu x^\nu}.
\label{eq58}
\end{equation}
Taking $\alpha^\mu=(0,\vec{a})$, in the non-relativistic limit $c\to\infty,\,
|\vec{a}|\to 0,\,c^2\vec{a}\to\vec{g}/2$, where $\vec{g}$ is some constant
3-vector, we get
\begin{equation}
t^\prime=t,\;\;\;\vec{r}{\,^\prime}=\vec{r}-\frac{\vec{g}\,t^2}{2}.
\label{eq59}
\end{equation}
Therefore, in the non-relativistic limit the transformation (\ref{eq58})
describes the transition to a non-inertial frame moving with a constant
acceleration $\vec{g}$.

Special conformal transformation is tightly related to uniformly accelerated
motion in relativistic case too. Uniformly accelerated motion is defined as
having a zero jerk (time derivative of the acceleration 3-vector) in
instantaneous rest frame \cite{51}. Differentiating velocity 4-vector
\begin{equation}
u^\mu=\gamma\,(c,\,\vec{V}),
\label{eq60}
\end{equation}
we get successively the acceleration 4-vector
\begin{equation}
\dot u^\mu=\frac{du^\mu}{d\tau}=\gamma\,\frac{du^\mu}{dt}=
\gamma^4\,\left(\vec{\beta}\cdot\vec{a},\,
\frac{\vec{a}}{\gamma^2}+(\vec{\beta}\cdot\vec{a})\vec{\beta}\right )=
\gamma^4\,\left(\vec{\beta}\cdot\vec{a},\,\vec{a}+\vec{\beta}\times
(\vec{\beta}\times\vec{a})\right ),
\label{eq61}
\end{equation}
and the naive jerk 4-vector
\begin{equation}
\ddot u^\mu=\frac{d\dot u^\mu}{d\tau}=
\gamma^5\left(\frac{\vec{a}\cdot\vec{a}}{c}+\vec{\beta}\cdot\vec{b}+
\frac{4\gamma^2}{c}\,(\vec{\beta}\cdot\vec{a})^2,\;\frac{\vec{b}}{\gamma^2}+
3\,\frac{(\vec{\beta}\cdot\vec{a})}{c}\,\vec{a}+\left[\vec{\beta}\cdot\vec{b}+
\frac{\vec{a}\cdot\vec{a}}{c}+4\,\frac{\gamma^2}{c}\,
(\vec{\beta}\cdot\vec{a})^2\right]\vec{\beta}\right ),
\label{eq62}
\end{equation}
where $\vec{a}=\frac{d\vec{V}}{dt}$ and $\vec{b}=\frac{d\vec{a}}{dt}$.
In the instantaneous rest frame
\begin{equation}
\ddot u^\mu_{(0)}=\left(\frac{\vec{a}_{(0)}\cdot\vec{a}_{(0)}}{c},\;
\vec{b}_{(0)}\right),
\label{eq63}
\end{equation}
which indicates that $\ddot u^\mu$ is not necessarily spacelike and could be
nonzero even if 3-jerk satisfies $\vec{b}_{(0)}=0$. Due to these properties
$\ddot u^\mu$ can not be really considered as a relativistic jerk 4-vector
\cite{51A}. To find a better candidate for the role of the relativistic jerk
4-vector let us note that
\begin{equation}
u_\mu\ddot u^\mu=\vec{a}_{(0)}\cdot\vec{a}_{(0)}=-\frac{\dot u_\nu\dot u^\nu}
{c^2}\,u_\mu u^\mu,
\label{eq64}
\end{equation}
which implies
\begin{equation}
u_\mu\left (\ddot u^\mu+\frac{\dot u_\nu\dot u^\nu}{c^2}\,u^\mu\right )=0.
\label{eq65}
\end{equation}
Therefore, the 4-vector
\begin{equation}
\Gamma^\mu=\ddot u^\mu+\frac{\dot u_\nu\dot u^\nu}{c^2}\,u^\mu
\label{eq66}
\end{equation}
is spacelike because it is orthogonal to the timelike 4-vector $u^\mu$.
Besides, in the instantaneous rest frame $\Gamma^\mu_{(0)}=(0,\;\vec{b}_{(0)}
)$. Hence, it makes more sense to define the relativistic jerk to be
$\Gamma^\mu$ \cite{51A}. Then the covariant condition of uniformly accelerated
motion is the vanishing of jerk 4-vector $\Gamma^\mu$ \cite{51} (in \cite{51}
$\Gamma^\mu$ is called the Abraham 4-vector due to its relation to the
Lorentz-Abraham radiation reaction force). Using
\begin{equation}
\dot u_\nu\dot u^\nu=-\gamma^4\left [\vec{a}\cdot\vec{a}+\gamma^2(\vec{\beta}
\cdot\vec{a})^2\right],
\label{eq67}
\end{equation}
which follows from (\ref{eq61}), and
\begin{equation}
\vec{b}+3\,\frac{\gamma^2}{c}\,(\vec{\beta}\cdot\vec{a})\,\vec{a}=
\frac{1}{\gamma^3}\,\frac{d}{dt}(\gamma^3\,\vec{a}),
\label{eq68}
\end{equation}
we get after some algebra
\begin{equation}
\Gamma^\mu=\gamma^2\left(\vec{\beta}\cdot\frac{d}{dt}(\gamma^3\,\vec{a}),\;
\frac{1}{\gamma^2}\,\frac{d}{dt}(\gamma^3\,\vec{a})+\left (\vec{\beta}\cdot
\frac{d}{dt}(\gamma^3\,\vec{a})\right )\,\vec{\beta}\right )=
\gamma^2\left(\vec{\beta}\cdot\frac{d}{dt}(\gamma^3\,\vec{a}),\;
\frac{d}{dt}(\gamma^3\,\vec{a})+\vec{\beta}\times\left(\vec{\beta}\times
\frac{d}{dt}(\gamma^3\,\vec{a})\right )\right  ).
\label{eq69}
\end{equation}
Therefore, a uniformly accelerated motion is characterized by the condition
\cite{51}
\begin{equation}
\frac{1}{\gamma^3}\,\frac{d}{dt}(\gamma^3\,\vec{a})=
\vec{b}+3\,\frac{\gamma^2}{c}\,(\vec{\beta}\cdot\vec{a})\,\vec{a}=0,
\label{eq70}
\end{equation}
which is equivalent to:
\begin{equation}
\gamma^3\,\vec{a}=\vec{g}
\label{eq71}
\end{equation}
with some constant vector $\vec{g}$.

Suppose we want to interpret special conformal transformation (\ref{eq58})
as a transition to a new reference frame. How does the origin
$\vec{r}^{\,^\prime}=0$ of this frame move? When $\vec{r}^{\,^\prime}=0$,
we will have from (\ref{eq58})
\begin{equation}
\vec{r}=\vec{\alpha}\,x\cdot x,
\;\;\;x\cdot x=x_\mu\,x^\mu=c^2t^2-\vec{r}\cdot\vec{r}.
\label{eq72}
\end{equation}
Differentiating this relation we find
\begin{equation}
\vec{V}=2\vec{\alpha}\,(c^2t-\vec{r}\cdot\vec{V}),\;\;
\vec{a}=2\vec{\alpha}\,(c^2-V^2-\vec{r}\cdot\vec{a})=\frac{2\vec{\alpha}}
{\gamma^2}(c^2-\gamma^2\,\vec{r}\cdot\vec{a}).
\label{eq73}
\end{equation}
On the other hand, taking the scalar products of (\ref{eq73}) and (\ref{eq72}),
we get the relations
\begin{equation}
\vec{r}\cdot\vec{V}=2\,\vec{\alpha}\cdot\vec{\alpha}\,x\cdot x\,
(c^2t-\vec{r}\cdot\vec{V}),\;\;\vec{r}\cdot\vec{a}=2\,\vec{\alpha}\cdot
\vec{\alpha}\,x\cdot x\left(\frac{c^2}{\gamma^2}-\vec{r}\cdot\vec{a}\right ).
\label{eq74}
\end{equation}
From these equations $\vec{r}\cdot\vec{V}$ and $\vec{r}\cdot\vec{a}$ can be
determined, and substituting the results back in (\ref{eq73}) we get
\begin{equation}
\vec{V}=\frac{2\vec{\alpha}c^2t}{1+2\,\vec{\alpha}\cdot\vec{\alpha}\;
x\cdot x},\;\;\;
\vec{a}=\frac{2\vec{\alpha}c^2}{\gamma^2}\,\frac{1}{1+2\,\vec{\alpha}\cdot
\vec{\alpha}\;x\cdot x}.
\label{eq75}
\end{equation}
Therefore,
\begin{equation}
1-\beta^2=1-\frac{4\,\vec{\alpha}\cdot\vec{\alpha}\,c^2t^2}{(1+2\,\vec{\alpha}
\cdot\vec{\alpha}\;x\cdot x)^2}=\frac{1}{(1+2\,\vec{\alpha}\cdot\vec{\alpha}
\;x\cdot x)^2},
\label{eq76}
\end{equation}
where we have taken into account that in light of (\ref{eq72})
\begin{equation}
c^2t^2=x\cdot x+\vec{r}\cdot\vec{r}=x\cdot x+\vec{\alpha}\cdot\vec{\alpha}\;
(x\cdot x)^2.
\label{eq77}
\end{equation}
As we see,
\begin{equation}
\gamma=1+2\,\vec{\alpha}\cdot\vec{\alpha}\;x\cdot x,
\label{eq78}
\end{equation}
and then the second equation in (\ref{eq75}) gives
$\gamma^3\,\vec{a}=2c^2\vec{\alpha}$ what implies that the point
$\vec{r}^{\,\prime}=0$ experiences uniformly accelerated motion.

It seems therefore tempting to interpret the special conformal transformation
(\ref{eq58}) as a transition to a uniformly accelerated reference frame.
There were long-lasting efforts \cite{51,52,53,54} to make sense of this
interpretation however they were not very successful, and the modern
consensus is that such interpretation is untenable \cite{55,56}. Passive
interpretation (as a change of the reference frame) is not always
advantageous whereas an active interpretation of transformations and the
corresponding notion of symmetries is often more fruitful.

The structure of the conformal group (more precisely, the commutation
relations of its Lie algebra) indicates that ``a Poincar\'{e} invariant
theory that is also conformally invariant necessarily enjoys scale
invariance'', but the contrary is not true: ``group theory does not establish
conformal invariance from scale and Poincar\'{e} invariance'' \cite{57}.
This fact probably explains why Poincar\'{e} missed an opportunity to
discover conformal invariance of the Maxwell equations. Interestingly,
if the dimensionality of space-time is greater than four, the Maxwell
equations are no longer conformal invariant even though they remain scale
invariant \cite{57}.

The second postulate about the universality of light speed in inertial
reference frames plays the role of founding principle in Einstein's version
of special relativity. One may wonder what is so special  about light that
makes its speed a fundamental constant. In the Standard Model there is no
compelling theoretical reason to assume that the photon mass is strictly zero
\cite{58}. A tiny photon mass can be generated by means of the St\"{u}ckelberg
mechanism that retains gauge invariance, unitarity and renormalizability
\cite{59}. Of course, if the Standard Model is a low energy remnant of a grand
unification model based on a simple nonabelian group like $SU(5)$, $SO(10)$,
or $E_6$, then the photon will be necessarily truly massless because one
cannot generalize St\"{u}ckelberg mechanism to a non-Abelian gauge symmetry.
However, even in this case the masslessness of the photon originates from the
particular pattern of the electroweak symmetry breaking. One may suspect
therefore that the fundamental constant $c$ appearing in the Lorentz
transformations is not necessarily light speed in its essence. This is indeed
the case as the following simple argument shows \cite{45}.

Let us return to the velocity addition rule (\ref{eq52}). If we assume the
validity of the relativity principle, then $k_1(V)=k_2(V)$ and hence, $K(-V)=
K(V)$ so that the function $F(V_1,V_2)$ in (\ref{eq52}) is an odd function:
\begin{equation}
F(-V_1,-V_2)=-F(V_1,V_2).
\label{eq79}
\end{equation}
Let us combine this oddness of $F(V_1,V_2)$ with the reciprocity principle
$V_{AB}=-V_{BA}$ for the velocity $V_{AB}$ of an object $A$ relative to
an object $B$. We will have
\begin{equation}
V_{AB}=F(V_{AC},V_{CB})=-V_{BA}=-F(V_{BC},V_{CA})=-F(-V_{CB},-V_{AC})=
F(V_{CB},V_{AC}).
\label{eq80}
\end{equation}
Therefore, $F(V_1,V_2)$ is a symmetric function of its arguments what in light
of its definition in (\ref{eq52}) implies
\begin{equation}
\frac{1-K(V_2)}{V_2}\,V_1=\frac{1-K(V_1)}{V_1}\,V_2.
\label{eq81}
\end{equation}
This is only possible if
\begin{equation}
\frac{1-K(V)}{V^2}=k=\mathrm{const}
\label{eq82}
\end{equation}
If $k<0$, then we will face a number of strange features like positive
velocities that sum up in a negative velocity. Most importantly, in this
case it will be difficult if not impossible to define a causal structure
on the corresponding space-time \cite{45,47,60}.

If $k\ge 0$, we can take $k=1/c^2$, where $c$ is some constant with a
dimensionality of velocity. Nothing however indicates in such a derivation
that $c$ is the light velocity in vacuum. In this case $K(V)$ is given by
(\ref{eq47}) and the first equation in (\ref{eq52}) coincides to the
relativistic addition law for parallel velocities. To promote $c$ to the
invariant velocity for perpendicular velocities too, we will need
(\ref{eq53}). Then (\ref{eq52}) will completely coincide with the
special-relativistic velocity-addition law. $k=0$ corresponds to the
$c\to \infty$ limit and gives Galilean velocity-addition law. Whether the
light velocity in vacuum is the invariant velocity $c$ is the question about
the properties of light and it has nothing to do with the logical structure of
special relativity. However, experimental limits on the nonzero photon mass
are very tough: the photon Compton wavelength is already known to be
at least comparable in dimension to the Earth-Sun distance \cite{61}. So
for all laboratory-scale purposes the photon mass safely may be taken as zero
and therefore $c$ may be considered as light velocity  for all practical
purposes.

\section{anisotropic special relativity and Lalan-Alway-Bogoslovsky
transformations}
It was demonstrated in the previous section that $\lambda$-Lorentz
transformations (\ref{eq56}) with $\lambda(V)$ satisfying the functional
equation (\ref{eq57}) are the most general transformations  that follow
from Einstein's two postulates under some natural assumptions. To solve
the functional equation (\ref{eq57}) it is beneficial to realize that
the natural parameter for special Lorentz transformations is not the velocity
$V$ but the rapidity $\psi$ \cite{62} defined through
\begin{equation}
\tanh{\psi}=\beta=\frac{V}{c}.
\label{eq83}
\end{equation}
Rapidities are additive for the special Lorentz transformations, and the
functional equation (\ref{eq57}) takes the form of the Cauchy exponential
functional equation
\begin{equation}
\lambda(\psi_1+\psi_2)=\lambda(\psi_1)\,\lambda(\psi_2).
\label{eq84}
\end{equation}
Although there exist infinitely many wildly discontinuous solutions of
(\ref{eq84}), the continuous solutions, which are the only ones acceptable in
the context of $\lambda$-Lorentz transformations, all have the form \cite{63}
\begin{equation}
\lambda(\psi)=e^{-b\psi}=\left(\frac{1-\beta}{1+\beta}\right )^{b/2}
\label{eq85}
\end{equation}
with some real constant $b$. Therefore, $\lambda$-Lorentz transformations
(\ref{eq56}) take the form
\begin{eqnarray} &&
x^\prime=\left(\frac{1-\beta}{1+\beta}\right )^{b/2}
\gamma\left(x-V\,t\right ), \nonumber \\ &&
y^\prime=\left(\frac{1-\beta}{1+\beta}\right )^{b/2}y,\nonumber \\ &&
z^\prime=\left(\frac{1-\beta}{1+\beta}\right )^{b/2}z,\nonumber \\ &&
t^\prime=\left(\frac{1-\beta}{1+\beta}\right )^{b/2}
\gamma\,\left (t-\frac{V}{c^2}\,x\right ).
\label{eq86}
\end{eqnarray}
Let us introduce light-cone coordinates \cite{64}
\begin{equation}
u=ct+x,\;\;\;v=ct-x.
\label{eq87}
\end{equation}
Then we get from (\ref{eq86})
\begin{equation}
u^\prime=e^{-(1+b)\psi}\,u,\;\;\;v^\prime=e^{(1-b)\psi}\,v,\;\;\;
y^\prime=e^{-b\psi}\,y,\;\;\;z^\prime=e^{-b\psi}\,z.
\label{eq88}
\end{equation}
Using
\begin{equation}
\frac{u^\prime}{v^\prime}=e^{-2\psi}\,\frac{u}{v},\;\;\;
u^\prime\,v^\prime=e^{-2b\psi}u\,v,
\label{eq89}
\end{equation}
we can easily find invariant quantities
\begin{equation}
\left(\frac{v^\prime}{u^\prime}\right )^b u^\prime v^\prime=
\left(\frac{v}{u}\right )^b u v,\;\;
\left(\frac{v^\prime}{u^\prime}\right )^b y^{\,\prime\,2}=
\left(\frac{v}{u}\right )^b y^2,\;\;
\left(\frac{v^\prime}{u^\prime}\right )^b z^{\,\prime\,2}=
\left(\frac{v}{u}\right )^b z^2,\;\;\frac{u^\prime v^\prime}{y^{\,\prime\,2}}
=\frac{u v}{y^{2}},\;\;\frac{u^\prime v^\prime}{z^{\,\prime\,2}}
=\frac{u v}{z^{2}}.
\label{eq90}
\end{equation}
Therefore, the following quantity, which can be considered as a generalization
of the relativistic interval, is invariant under $\lambda$-Lorentz
transformations (\ref{eq56}):
\begin{equation}
s^2=\left(\frac{v}{u}\right )^b\left (uv-y^2-z^2\right)\left(\frac{uv}
{uv-y^2-z^2}\right)^b=v^{2b}(uv-y^2-z^2)^{1-b}=\left(ct-x\right)^{2b}
\left(c^2t^2-x^2-y^2-z^2\right)^{1-b}.
\label{eq91}
\end{equation}
Since the  $\lambda$-Lorentz transformations (\ref{eq56}) are linear, the
differentials of coordinates are transformed as the coordinates themselves.
Consequently, the metric of the space-time, invariant with respect to
the $\lambda$-Lorentz transformations (\ref{eq56}) has the form
\begin{equation}
ds^2=\left(c\,dt-dx\right)^{2b}
\left(c^2\,dt^2-dx^2-dy^2-dz^2\right)^{1-b}=\left(n_\nu dx^\nu\right)^{2b}
\left(dx_\mu dx^\mu\right)^{1-b},
\label{eq92}
\end{equation}
where $n^\mu=(1,1,0,0)=(1,\vec{n}),\;\vec{n}^{\,2}=1$, is the fixed null-vector
defining a preferred null-direction in the space-time.

As we see, in general, if the spatial isotropy is not assumed, Einstein's two
postulates do not lead to Lorentz transformations and Minkowski metric.
Instead they lead to more general $\lambda$-Lorentz transformations and
Finslerian metric (\ref{eq92}) (for applications of Finsler geometry in
physics see, for example, \cite{65,66}).

$\lambda$-Lorentz transformations (\ref{eq56}) correspond to the inertial
frame which moves along the preferred direction $\vec{n}$. What about other
inertial frames? Let us consider an inertial reference frame $S^\prime$
which moves with the velocity $\vec{V}$. If for a moment we consider
$n^\mu$ as a normal 4-vector, not the fixed one, we may write
\begin{equation}
\left(n_\nu dx^\nu\right)^{2b}\left(dx_\mu dx^\mu\right)^{1-b}=
\left(n^\prime_\nu dx^{\prime\nu}\right)^{2b}\left(dx^\prime_\mu
dx^{\prime\mu}\right)^{1-b},
\label{eq93}
\end{equation}
where
\begin{equation}
n_0^\prime=\gamma\left(n_0-\beta\,\frac{\vec{n}\cdot\vec{V}}{V}\right)=
\gamma(1-\vec{n}\cdot\vec{\beta}),
\label{eq94}
\end{equation}
and
\begin{equation}
\vec{n}^\prime=\frac{\vec{V}}{V}\,\gamma\left(\frac{\vec{n}\cdot\vec{V}}{V}-
\beta n_0\right)+\left(\vec{n}-\frac{\vec{n}\cdot\vec{V}}{V}\,\frac{\vec{V}}
{V}\right)=\vec{n}+\frac{\vec{V}}{V}\left[(\gamma-1)\frac{\vec{n}\cdot\vec{V}}
{V}-\gamma\beta\right].
\label{eq95}
\end{equation}
Using
\begin{equation}
\gamma-1=\frac{\gamma^2-1}{\gamma+1}=\frac{\gamma^2\beta^2}{\gamma+1},
\label{eq96}
\end{equation}
we can rewrite (\ref{eq95}) as follows
\begin{equation}
\vec{n}^\prime=\vec{n}-\gamma\vec{\beta}\left[1-\frac{\gamma}{\gamma+1}\,
\vec{n}\cdot\vec{\beta}\right]=\vec{n}-\frac{\gamma\vec{\beta}}{\gamma+1}
\left[1+\gamma(1-\vec{n}\cdot\vec{\beta})\right].
\label{eq97}
\end{equation}
If $\vec{V}$ is not parallel to $\vec{n}$, $\vec{n}^\prime$ will be not
parallel to $\vec{n}$ neither. If $\alpha$ is an angle between $\vec{n}
^\prime$ and $\vec{n}$, we will have
\begin{equation}
\sin{\alpha}=\frac{|\vec{n}\times\vec{n}^\prime|}{n_0^\prime}=
\frac{|\vec{n}\times\vec{\beta}|[1+\gamma(1-\vec{n}\cdot\vec{\beta})]}
{(\gamma+1)(1-\vec{n}\cdot\vec{\beta})},
\label{eq98}
\end{equation}
where we have used that $|\vec{n}^\prime|=n_0^\prime$ because the Lorentz
transformation leaves $n_\mu^\prime$ lightlike. Alternatively, we can write
\begin{equation}
\cos{\alpha}=\frac{\vec{n}\cdot\vec{n}^\prime}{n_0^\prime}=
\frac{1-\gamma(\vec{n}\cdot\vec{\beta}-1+1)+\frac{\gamma^2}{\gamma+1}\,
(\vec{n}\cdot\vec{\beta})^2}{\gamma(1-\vec{n}\cdot\vec{\beta})}=
1+\frac{1-\gamma+\frac{1}{\beta^2}\,(\gamma-1)(\vec{n}\cdot\vec{\beta})^2}
{\gamma(1-\vec{n}\cdot\vec{\beta})}=1-\frac{\gamma-1}{\gamma\beta^2}\,
\frac{\beta^2-(\vec{n}\cdot\vec{\beta})^2}{1-\vec{n}\cdot\vec{\beta}}.
\label{eq99}
\end{equation}
But $\beta^2-(\vec{n}\cdot\vec{\beta})^2=[\vec{n}\times\vec{\beta}]^2$, and
finally
\begin{equation}
\cos{\alpha}=1-\frac{\gamma-1}{\gamma\beta^2}\,\frac{[\vec{n}\times
\vec{\beta}]^2}{1-\vec{n}\cdot\vec{\beta}}=1-\frac{\gamma}{\gamma+1}\,
\frac{[\vec{n}\times\vec{\beta}]^2}{1-\vec{n}\cdot\vec{\beta}}.
\label{eq100}
\end{equation}
We can make $\vec{n}^\prime$ again parallel to $\vec{n}$ by an additional
rotation around the axis $\vec{n}^\prime\times\vec{n}\parallel \vec{n}\times
\vec{\beta}$ by certain angle $\alpha$. For the radius vector $\vec{r}$ the
result of this additional rotation is given by the Euler-Rodrigues formula
\cite{67} (for proofs of this formula see, for example, \cite{68,69})
\begin{equation}
\vec{r}^{\,\prime\prime}=\vec{r}^{\,\prime}+(\vec{m}\times\vec{r}^{\,\prime})
\sin{\alpha}+[\vec{m}\times(\vec{m}\times\vec{r}^{\,\prime})](1-\cos{\alpha}),
\label{eq101}
\end{equation}
where
\begin{equation}
\vec{m}=\frac{\vec{n}\times\vec{\beta}}{|\vec{n}\times\vec{\beta}|}
\label{eq102}
\end{equation}
is the unit vector along the axis of the rotation and
\begin{equation}
\vec{r}^{\,\prime}=\vec{r}+\frac{\vec{\beta}}{\beta}\left[(\gamma-1)\,\frac{
\vec{r}\cdot\vec{\beta}}{\beta}-\gamma\beta\,x_0\right]
\label{eq103}
\end{equation}
is the result of the preceding Lorentz transformation on the spatial part
of the $x^\mu$ 4-vector. Using (\ref{eq98}) and (\ref{eq100}) along with
\begin{equation}
\vec{m}\times\vec{r}^{\,\prime}=\frac{(\vec{n}\times\vec{\beta})\times
\vec{r}^{\,\prime}}{|\vec{n}\times\vec{\beta}|}=\frac{\vec{\beta}(\vec{n}
\cdot\vec{r}^{\,\prime})-\vec{n}(\vec{\beta}\cdot\vec{r}^{\,\prime})}
{|\vec{n}\times\vec{\beta}|}
\label{eq104}
\end{equation}
and
\begin{equation}
\vec{m}\times(\vec{m}\times\vec{r}^{\,\prime})=\frac{(\vec{n}\times
\vec{\beta})\times[\vec{\beta}(\vec{n}\cdot\vec{r}^{\,\prime})-\vec{n}
(\vec{\beta}\cdot\vec{r}^{\,\prime})]}{[\vec{n}\times\vec{\beta}]^2}=
\frac{\vec{\beta}\,[(\vec{n}\cdot\vec{r}^{\,\prime})(\vec{n}\cdot\vec{\beta})-
\vec{\beta}\cdot\vec{r}^{\,\prime}]+\vec{n}\,[(\vec{\beta}\cdot\vec{r}^{\,
\prime})(\vec{\beta}\cdot\vec{n})-\beta^2(\vec{n}\cdot\vec{r}^{\,\prime})]}
{[\vec{n}\times\vec{\beta}]^2},
\label{eq105}
\end{equation}
we get
\begin{equation}
\vec{r}^{\,\prime\prime}=\vec{r}^{\,\prime}+\frac{\vec{\beta}}{1-\vec{n}\cdot
\vec{\beta}}\left[\vec{n}\cdot\vec{r}^{\,\prime}-\frac{\gamma}{\gamma+1}\,
\vec{\beta}\cdot\vec{r}^{\,\prime}\right]+\frac{\vec{n}}{1-\vec{n}\cdot
\vec{\beta}}\left[(\vec{\beta}\cdot\vec{r}^{\,\prime})\left(\frac{2\gamma}
{\gamma+1}\,\vec{\beta}\cdot\vec{n}-1\right)-\frac{\gamma-1}{\gamma}\,
\vec{n}\cdot\vec{r}^{\,\prime}\right].
\label{eq106}
\end{equation}
This result coincides with the one that follows from Eq.34 of \cite{70}. As an
other check, if we take $\vec{r}^{\,\prime}=\vec{n}^\prime$ and use the
relations
\begin{equation}
\vec{\beta}\cdot\vec{n}^\prime=\gamma[\vec{\beta}\cdot\vec{n}-\beta^2],\;\;\;
\vec{n}\cdot\vec{n}^\prime=1-\gamma\,\vec{\beta}\cdot\vec{n}+\frac{\gamma^2}
{\gamma+1}\,(\vec{\beta}\cdot\vec{n})^2
\label{eq107}
\end{equation}
that follow from (\ref{eq97}), we get after some calculations the correct
result
\begin{equation}
\vec{n}^{\prime\prime}=\gamma(1-\vec{\beta}\cdot\vec{n})\,\vec{n}.
\label{eq108}
\end{equation}
Consequently, after the combined transformations
\begin{equation}
x^{\prime\prime\mu}=R^\mu_{\;\nu}(\vec{m};\alpha)\,L^\nu_{\;\sigma}(\vec{V})
x^\sigma,\;\;\;n^{\prime\prime\mu}=R^\mu_{\;\nu}(\vec{m};\alpha)\,L^\nu_{
\;\sigma}(\vec{V})n^\sigma,
\label{eq109}
\end{equation}
we will obtain
\begin{equation}
\left(n_\nu dx^\nu\right)^{2b}\left(dx_\mu dx^\mu\right)^{1-b}=
\left(n^{\prime\prime}_\nu dx^{\prime\prime\nu}\right)^{2b}
\left(dx^{\prime\prime}_\mu dx^{\prime\prime\mu}\right)^{1-b}=
\left[\gamma(1-\vec{\beta}\cdot\vec{n})\right]^{2b}\left(n_\nu
dx^{\prime\prime\nu}\right)^{2b}\left(dx^{\prime\prime}_\mu
dx^{\prime\prime\mu}\right)^{1-b}.
\label{eq110}
\end{equation}
The unwanted conformal factor $\left[\gamma(1-\vec{\beta}\cdot\vec{n})
\right]^{2b}$ can be compensated by final dilatation
\begin{equation}
x^{\prime\prime\prime\mu}=D(\lambda)x^{\prime\prime\mu}=
\lambda\,x^{\prime\prime\mu},
\label{eq111}
\end{equation}
with the suitably chosen scale-factor
\begin{equation}
\lambda=\left[\gamma(1-\vec{\beta}\cdot\vec{n})\right]^b,
\label{eq112}
\end{equation}
and we finally get the desired invariance
\begin{equation}
\left(n_\nu dx^\nu\right)^{2b}\left(dx_\mu dx^\mu\right)^{1-b}=
\left(n_\nu dx^{\prime\prime\prime\nu}\right)^{2b}\left(
dx^{\prime\prime\prime}_\mu dx^{\prime\prime\prime\mu}\right)^{1-b}.
\label{eq113}
\end{equation}
To summarize, the Finsler metric (\ref{eq92}) is invariant under the
generalized Lorentz transformations \cite{70}
\begin{equation}
x^{\prime\mu}=D(\lambda)R^\mu_{\;\nu}(\vec{m};\alpha)\,L^\nu_{\;\sigma}
(\vec{V})x^\sigma.
\label{eq114}
\end{equation}
Using (\ref{eq103}) and (\ref{eq106}) we can get an explicit form of these
generalized Lorentz transformations:
\begin{eqnarray} &&
x_0^\prime=\left[\gamma(1-\vec{\beta}\cdot\vec{n})\right]^b\gamma\,(x_0-
\vec{\beta}\cdot\vec{r}), \nonumber \\ &&
\vec{r}^{\,\prime}=\left[\gamma(1-\vec{\beta}\cdot\vec{n})\right]^b\left\{
\vec{r}-\frac{\vec{\beta}\,(x_0-\vec{n}\cdot\vec{r})}{1-\vec{\beta}\cdot
\vec{n}}-\vec{n}\left[\gamma\,\vec{\beta}\cdot\vec{r}+\frac{\gamma-1}{\gamma}
\,\frac{\vec{n}\cdot\vec{r}}{1-\vec{\beta}\cdot\vec{n}}+\frac{(\gamma-1)
\vec{\beta}\cdot\vec{n}-\gamma\beta^2}{1-\vec{\beta}\cdot\vec{n}}\,x_0
\right]\right\}.
\label{eq115}
\end{eqnarray}
We have checked that these expressions are equivalent to  Eq.29 of \cite{70}.
If $\vec{\beta}$ and $\vec{n}$ are parallel and the $x$-axis is along
$\vec{\beta}$, (\ref{eq115}) are reduced to the $\lambda$-Lorentz
transformations (\ref{eq86}). Another interesting limiting case is when
$\vec{\beta}$ and $\vec{n}$ are perpendicular to each other. Assuming that
the $x$ and $y$ axes are respectively along $\vec{\beta}$ and $\vec{n}$,
we get from (\ref{eq115})
\begin{eqnarray} &&
x^\prime=\gamma^b\left[x-Vt+\beta\,y\right],\nonumber \\ &&
y^\prime=\gamma^{1+b}\left[(1-\beta^2)\,y-\beta\,(x-Vt)\right ],
\nonumber \\ &&
z^\prime=\gamma^b\,z,\nonumber \\ &&
t^\prime=\gamma^{1+b}\left[t-\frac{V}{c^2}\,x\right ].
\label{eq116}
\end{eqnarray}
It may appear that transformations (\ref{eq116}) violate the reciprocity
principle because according to (\ref{eq116}) the origin $x=0,y=0,z=0$ of
the frame $S$ moves in the frame $S^\prime$ with velocity $V^\prime_x=
-V/\gamma,\,V^\prime_y=\beta\,V,\,V^\prime_z=0$, not with $V^\prime_x=-V,\,
V^\prime_y=0,\,V^\prime_z=0$ as one might expect. However, remember that
the generalized Lorentz transformations (\ref{eq114}) contain the rotation
$R(\vec{m};\alpha)$. When passively interpreted, this rotation implies in
the particular case of (\ref{eq116}) that the $x$ and $y$-axes of $S^\prime$
are rotated around the $z$-axis with an angle
$\alpha$ such that $\sin{\alpha}=\beta,\,\cos{\alpha}=1/\gamma$ according to
(\ref{eq98}) and (\ref{eq100}). No wonder that the vector $-\vec{V}$  has the
coordinates $V^\prime_x=-V\cos{\alpha}=-V/\gamma$ and $V^\prime_y=
V\sin{\alpha}=\beta\,V$ in $S^\prime$.

To our best knowledge, it was Lalan \cite{71,72,73} who first derived
$(1+1)$-dimensional $\lambda$-Lorentz transformations and  demonstrated that
the metric invariant under such transformations was of pseudo-Finslerian type
rather than of the Minkowski pseudo-Riemannian. $(1+3)$-dimensional
generalized Lorentz transformations (\ref{eq115}) were first discovered by
Alway \cite{74}. He gives no details on how the transformations were obtained
but it seems that the spirit of the derivation was somewhat similar to the one
presented above as he cites Pars \cite{75} and mentions that one of the Pars'
assumptions, namely isotropic behavior of clocks, is not guaranteed in general.
Pars like von Ignatowsky \cite{44} provides the derivation of
the Lorentz transformations without Einstein's second  postulate. He doesn't
cite neither von Ignatowsky nor anybody else in his paper.  The fate of Pars's
and von Ignatowsky's innovative derivations of Lorentz transformations were
similar: ``like numerous others that followed, these have gone largely
unnoticed'' \cite{76}.  Alway's generalized Lorentz transformations and
the corresponding Finslerian space-time metric discovered by him also fell
into oblivion immediately after it was recognized that experimental
observations severely constrain spatial anisotropy essentially making the
parameter $b$ zero \cite{77}. Fortunately, generalized Lorentz transformations
(\ref{eq115}) were soon rediscovered by Bogoslovsky \cite{70,78} who then
thoroughly investigated their physical consequences \cite{79,80,81}.
$(1+1)$-dimensional $\lambda$-Lorentz transformations for the case $b=1/2$
was independently rediscovered by Brown \cite{82} and then generalized to any
value of $b$ by Budden \cite{83}. Apparently, initially they were unaware of
Bogoslovsky's pioneering contributions. Later they cite Bogoslovsky in
\cite{84,85} and \cite{86}, but neither  Lalan nor Alway are mentioned
in their work. Slightly different perspective on Finslerian relativity is
given by Asanov \cite{87}.

As was already mentioned, observations severely constrain the Finslerian
parameter $b$. Some {\ae}ther-drift experiments imply $|b|<10^{-10}$
while the Hughes-Drever type limits on the anisotropy of inertia can
potentially lower the limit on the Finslerian parameter $b$ up to
$|b|<10^{-26}$, though in a model-dependent way \cite{88}. It may appear
therefore that Strnad's conclusion \cite{77} that the special relativity is
valid for all practical purposes, and the Alway's extension of Lorentz
transformations \cite{74} is superfluous, is well justified. However,
notwithstanding our firm belief that special relativity is indeed valid for
all practical purposes, its Finslerian extension described above, in our
opinion, has a very significant conceptual value. Let us explain why.

Besides the generalized pure Lorentz boosts (\ref{eq115}), the Finsler metric
(\ref{eq92}) is left invariant by rotations about preferred spatial direction
$\vec{n}$ and by space-time translations $x^{\prime\mu}=x^\mu+a^\mu$. Together
these transformations constitute an 8-parameter group of isometries of the
Finsler space-time with metric (\ref{eq92}). This group is given a fancy name
DISIM$_b$(2) (deformed ISIM(2), $b$ being the deformation parameter) in
\cite{88}. Both DISIM$_b$(2) and the 10-parameter Poincar\'{e} group are
subgroups of the 11-parameter Weyl group (Poincar\'{e} transformations
augmented with arbitrary dilations). It may appear therefore that DISIM$_b$(2)
plays the role similar to the Poincar\'{e} group but it doesn't. The role
analogues to the Poincar\'{e} group is played by ISIM(2) which is the
$b\to 0$ limit (or better to say contraction in the sense of In\"{o}n\"{u} and
Wigner \cite{89}) of DISIM$_b$(2). Although the Finsler metric (\ref{eq92})
is reduced to the Minkowski metric in the $b\to 0$ limit, the generalized
Lorentz transformations (\ref{eq115}) are not reduced to the ordinary Lorentz
transformations. Instead in this limit the transformations (\ref{eq115})
augmented with  rotations about the preferred direction $\vec{n}$ constitute
a 4-parameter subgroup of the Lorentz group called SIM(2) (similitude group of
the plane consisting of dilations, translations and rotations of the plane).
If we further add space-time translations, we get a 8-parameter subgroup
of the Poincar\'{e} group ISIM(2) which is a semi-direct product of SIM(2)
with the group of space-time translations. Very special relativity developed
by Cohen and Glashow \cite{90} assumes that the exact symmetry group of nature
may be not the  Poincar\'{e} group but its subgroup ISIM(2).

It is known \cite{91,92} that the Lorentz group has no 5-parameter subgroup
and only one 4-parameter subgroup SIM(2) up to isomorphism. As a
result, very special relativity breaks Lorentz symmetry in a very mild and
minimal way. For amplitudes satisfying appropriate analyticity properties,
$CPT$ discrete symmetry follows from SIM(2), but SIM(2) violates $P$ and
$T$ discrete symmetries \cite{90}. However, the incorporation of either $P$,
$T$ or $CP$ discrete symmetries enlarges SIM(2) subgroup to the full Lorentz
group and thus, for particle physics theories conserving any
one of these three discrete symmetries Lorentz-violating effects in very
special relativity are absent \cite{90}. The existence of the preferred
light-like direction $n^\mu$ can be interpreted as the existence of
light-like {\ae}ther. However, this peculiar form of {\ae}ther is very
difficult to detect because it doesn't single out any preferred inertial
reference frame and, since $CP$ violating effects are small, Lorentz-violating
effects in very special relativity are expected to be also small.

Although the $b\to 0$ limits of (\ref{eq92}) and (\ref{eq115}) lead at the
first sight naturally to the very special relativity, it is not so. When
$b=0$, or when the space is isotropic, we have no reason to introduce
the preferred light-like direction $n^\mu$. Of course, we can do this
artificially and consequently arrive to the generalized Lorentz
transformations (\ref{eq115}) instead of usual  Lorentz transformations.
However, in this case $\vec{n}$ has no physical meaning and just serves to
calibrate the orientations of space axes of the inertial frames of reference
in such way that if in one such frame of reference the beam of light has the
direction $\vec{n}$, it will have the same direction in all inertial reference
frames \cite{70}. The resulting theory will be not the very special relativity
but the ordinary special relativity under the indicated special convention
about the spatial axes orientations. This is true if we can choose $n^\mu$
arbitrarily and all such choices are equivalent. In fact, it is sufficient to
require that the choices $\vec{n}$ and  $-\vec{n}$ are equivalent. Indeed,
when  $\vec{n}\to -\vec{n}$, then $\vec{m}\to -\vec{m}$ in (\ref{eq102}) and,
since $R^\mu_{\;\lambda}(-\vec{m},\alpha)R^\lambda_{\;\nu}(\vec{m},\alpha)=
\delta^\mu_\nu$, if we require the symmetry under rotations about $-\vec{m}$
and if $b=0$, we can easily transform the generalized Lorentz transformations
(\ref{eq115}) into the ordinary Lorentz transformations by an additional
rotation $x^{\prime\prime\,\mu}=R^\mu_{\;\nu}(-\vec{m},\alpha)x^{\prime\,\nu}$.

Therefore, if the space-time is Minkowskian, the very special relativity
(ISIM(2) symmetry) is not natural in light of relativity principle and will
indicate its breaking, although very subtle, if it is indeed realized in
nature. In our opinion far more natural possibility which completely
respects the relativity principle, is that the space-time is Finslerian with
the Lalan-Alway-Bogoslovsky metric (\ref{eq92}) but the parameter $b$ of this
metric is very small. This conclusion is reinforced by the following analogy
with the cosmological constant.

A surprising fact about Minkowski's ``Raum und Zeit'' lecture is that it
never mentions Klein's Erlangen program of defining a geometry by its
symmetry group \cite{18}. A link between Minkowski's presentation of special
relativity and Erlangen program was immediately recognized by Felix Klein
himself \cite{93} who remarked: ``What the modern physicists call the theory
of relativity is the theory of invariants of the 4-dimensional space-time
region $x,y,z,t$ (the Minkowski 'world') under a certain group of
collineations, namely, the 'Lorentz group' ''. Untimely death of Minkowski
presumably hindered the appreciation of this important fact by physicists.
Only in 1954 Fantappi\'{e} rediscovered the connection and put forward an idea
which he himself calls ``an Erlangen program for physics'': a classification
of possible physical theories through their group of symmetries \cite{94,95}.

Interestingly, we have one more would-be ``missed opportunities'' story here
\cite{I-3}. Fantappi\'{e} discovered that the ``final relativity'' group is
not the Poincar\'{e} group but the De Sitter group, the Poincar\'{e} group
being just a limit of the latter when the radius of curvature of the
De Sitter space-time goes to infinity, much like the Galilei group being
a limit of the Lorentz group when the speed of light goes to infinity. In fact,
the full story of possible kinematic groups and their interconnections under
different limits was given later by H.~Bacry and J.~L\'{e}vy-Leblond
\cite{96}. This picture seems quite natural in light of the Erlangen program
and of ideas of group contractions and deformations put forward by
In\"{o}n\"{u} and Wigner \cite{89} and Irving Segal \cite{97} about the same
time Fantappi\'{e} found his ``final relativity''. Dyson argues \cite{I-3}
that Minkowski, or any other pure mathematician, were in a position to
conjecture already in 1908 that the true invariance group of the universe
should be the de Sitter group rather than the Poincar\'{e} group and thus
suggest a cosmological space expansion much earlier than Hubble discovered it,
if only they would have brought Minkowski's argument to its logical conclusion.
 However, ``to be honest, it would be rather anachronistic to expect an
approach like the one of Fantappi\'{e} at the beginning of the 20th century,
as Dyson advocates'' \cite{95}. Even Fantappi\'{e} himself ``missed the chance
to anticipate the theory of Lie group contractions and deformations''
\cite{95}.

Segal's principle  \cite{97,98,99} states that a true physical theory should
be stable against small deformations of its underlying algebraic (group)
structure (a nice informal review of Lie algebra contractions and deformations
can be found in \cite{100}). Lie algebra of inhomogeneous Galilei group is not
stable and its deformation leads to Lie algebra of  the Poincar\'{e} group.
Consequently, the relativity theory based on the Poincar\'{e} group has a
greater range of validity than Galilean relativity. However, the Poincar\'{e}
Lie algebra is by itself unstable and its deformation leads to either de Sitter
or anti-de Sitter Lie algebras \cite{96}. Therefore, it is not surprising
that the cosmological constant turned out to be not zero and correspondingly,
the asymptotic vacuum space-time is not Minkowski but de Sitter space-time.
What is really surprising is why the cosmological constant is so small that
makes special relativity valid for all practical purposes.

Lie algebra of the very special relativity symmetry group ISIM(2) is not
stable against small deformations of its structure and a physically relevant
deformation DISIM$_b$(2) of it does exist \cite{88}. Therefore, in light of
Segal's principle, we expect that the very special relativity cannot be a true
symmetry of nature and should be replaced by DISIM$_b$(2) and the
corresponding Finslerian space-time.  Drawing an analogy with the cosmological
constant, it can be argued that $b$ is really not zero but very small. In
this case, to detect the effects of Finslerian nature of  space-time in
laboratory experiments will be almost impossible. Nevertheless, the question
of the true value of the parameter $b$ has the same fundamental significance
as the question why the cosmological constant is so small \cite{88}. Perhaps,
both questions are just different parts of the same mystery.

\section{Concluding remarks}
In parallel to the advance in modern physics, in the middle of the twentieth
century it became increasingly evident that Poincar\'{e}'s contribution to
relativity was unjustly downplayed. As a result, some attempts to restore the
justice followed. Unfortunately it was forgotten in the majority of these
attempts that ``injustice cannot be corrected by committing another
injustice'' \cite{101} and recurrent attempts to deny Einstein the discovery
of special relativity took place \cite{102}. A great amount of these attempts
don't deserve even to be mentioned. However, there are some of them that are
worth our attention and which are desirable to be clarified.

Renowned English mathematician Edmund Whittaker wrote a book {\em A History
of the Theories of Aether and Electricity}, first edition of which was
published in 1910. This is an  excellent and very detailed book describing
the development of field theories from Descartes up to the year 1900. In the
final chapter of the book there is some survey of the work done after 1900.
It includes a note about Minkowski's {\em Raum und Zeit} paper. Then little
known Einstein is mentioned twice in footnotes \cite{103}. First footnote
says that Einstein completed Lorentz's work on Lorentz transformations. Then
the footnote cites Voigt and says that ``It should be added that the
transformation in question had been applied to the equation of vibratory
motions many years before by Voigt'' \cite{103} that is not quite correct as
we have explained earlier. The next footnote correctly attributes to
Einstein the first clear expression of the opinion that all inertial frames
are equivalent and no one is granted a primacy by having an absolute relation
to the {\ae}ther. Poincar\'{e} is mentioned three times, also in footnotes,
but never in the context of relativity.

Forty-three years later, in 1953, Whittaker published a revised edition of his
book, which now included  a second volume covering the years from 1900 to
1926 \cite{103A}. The chapter devoted to relativity was called ``The
relativity theory of Poincar\'{e} and Lorentz'' and its content left little
doubt that Whittaker attributes the whole credit for the discovery of special
relativity  almost exclusively to Poincar\'{e} and Lorentz. About Einstein's
groundbreaking 1905 paper it was said that ``Einstein published a paper which
set forth the relativity theory of Poincare and Lorentz with some
amplifications, and which attracted much attention''. Why such injustice to
Einstein? Maybe the following long excerpt from Max Born's 26 September, 1953
letter to Einstein can shed some light on this difficult question.

``Very often I feel the need to write to you, but I usually suppress it to
spare you the trouble of replying. Today, though, I have a definite reason
--- that Whittaker, the old mathematician, who lives here as Professor
Emeritus and is a good friend of mine, has written a new edition of his old
book History of the Theory of the Ether, of which the second volume has
already been published. Among other things it contains a history of the theory
of relativity which is peculiar in that Lorentz and Poincar\'{e} are credited
with its discovery while your papers are treated as less important. Although
the book originated in Edinburgh, I am not really afraid you will think that
I could be behind it. As a matter of fact I have done everything I could
during the last three years to dissuade Whittaker from carrying out his plan,
which he had already cherished for a long time and loved to talk about.
I re-read the originals of some of the old papers, particularly some rather
off-beat ones by Poincar\'{e}, and have given Whittaker translations of German
papers (for example, I translated many pages of Pauli's Encyclopedia article
into English with the help of my lecturer, Dr. Schlapp, in order to make it
easier for Whittaker to form an opinion). But all in vain. He insisted that
everything of importance had already been said by Poincar\'{e}, and that
Lorentz quite plainly had the physical interpretation. As it happens, I know
quite well how skeptical Lorentz was and how long it took him to become
a relativist. I have told Whittaker all this, but without success''
\cite{104}.

Born's role in this affair is more ambivalent that it can appear from this
letter \cite{105}. At first sight Born was in a good position to be
a competent judge in the relativity priority dispute. He recollects \cite{106}
that at the time when Einstein's  1905 paper appeared he was ``in Gottingen
and well acquainted with the difficulties and puzzles encountered in the study
of electromagnetic and optical phenomena in moving bodies, which we thoroughly
discussed in a seminar held by Hilbert and Minkowski. We studied the recent
papers by Lorentz and Poincar\'{e}, we discussed the contraction hypothesis
brought forward by Lorentz and Fitzgerald, and we knew the transformations now
known under Lorentz's name. Minkowski was already working on his
4-dimensional representation of space and time, published in 1907, which
became later the standard method in fundamental physics'' and all this was
before Born became aware of Einstein's 1905 paper. Therefore, Born was well
versed in pre-Einstein relativity of Lorentz and Poincar\'{e}. In spite of
this, Einstein's paper greatly impressed Born. In his book {\em Physics in my
generation} \cite{106} he wrote:

``A long time before I read Einstein's famous 1905 paper, I knew the formal
mathematical side of the special theory of relativity through my teacher
Hermann Minkowski. Even so, Einstein's paper was a revelation to me which had
a stronger influence on my thinking than any other scientific experience\ldots
Einstein's simple consideration, by which he disclosed the epistemological
root of the problem\ldots made an enormous impression, and I think it right
that the principle of relativity is connected with his name, though Lorentz
and Poincar\'{e} should not be forgotten'' \cite{105, 106}.

Born's letter to Einstein leaves an impression that ``Born's attention was
drawn to Poincar\'{e}'s papers by Whittaker, rather than the other way around,
in contradiction to Born's recollection of having read Poincar\'{e}'s papers
in his student days before reading Einstein's paper'' \cite{105}. We think the
contradiction is only apparent here. There is little doubt that Poincar\'{e}'s
papers were discussed at Minkowski's seminars and they were well known to
Born. The only explanation why he needed to re-read them in fifties to discover
that several of Poincar\'{e}'s pre-1905 statements sound very much like
special relativity is that in 1905 or before Poincar\'{e}'s papers have not
impressed him as much as Einstein's 1905 paper which was ``a revelation'' to
him. Whittaker also was not aware of the importance of Poincar\'{e}'s papers
in 1910. Only in fifties he was in a position to acknowledge their importance.

It is not surprising that modern physicists like, for example, Logunov
\cite{9}, find Poincar\'{e}'s papers important and rise doubts about
Poincar\'{e}'s priority in discovery of special relativity. Even Born,
somewhat contrary to what he writes in the letter to Einstein, hesitated in
giving the priority: ``Does this mean that Poincar\'{e} knew all this before
Einstein? It is possible, but the strange thing is that this lecture
[Poincar\'{e}'s 1909 lecture {\em La m\'{e}canique nouvelle}]
definitely gives you the impression that he is recording Lorentz's work\ldots
On the other hand, Lorentz himself never claimed to be the author of the
principle of relativity'' \cite{105, 106}. Further he writes about Einstein's
1905 paper: ``The striking point is that it contains not a single reference to
previous literature. It gives you the impression of quite a new venture. But
that is, of course, as I have tried to explain, not true'' \cite{106}. And
this is, of course, ``precisely the conclusion that Whittaker reached\ldots
after long discussions of the subject with Born'' \cite{105}.

However, Whittaker's conclusion, implicitly shared by many modern physicists
who had taken a trouble to indeed read Poincar\'{e}'s papers, is wrong.
This conclusion is based on the retrospective reading of Poincar\'{e}'s papers.
However, in retrospective reading of really deep papers, and there is no doubt
that Poincar\'{e}'s relativity papers are really very deep, the reader can see
much more than it was possible to see by contemporaries, including the author,
who were bound by concrete historical context and prejudices of those days.
If we inspect the physical literature of the founding period of
relativity (1905-1918), we will clearly see that the scientific community
never hesitated to give Einstein a due credit, and there was no such a thing
as Poincar\'{e}'s version of relativity ``accessible and perceived as such by
physicists of the times'' \cite{26}. It is a historical fact that the
relativistic revolution, as seen in the contemporary physical literature
of those days, rightly or wrongly is dominated by only one name, the
Einstein's papers playing the major role, while Poincar\'{e}'s ones being
left virtually unnoticed \cite{26}. The importance of these papers was
recognized only later. In this respect, Poincar\'{e}-Einstein mystery
``is an artefact of projecting backward a particular reading of scientific
papers that does not correspond to what the actors of the time saw in them''
\cite{26}. We cannot rewrite the history to restore justice towards
Poincar\'{e}. This is not needed indeed. A scientific rehabilitation of
Poincar\'{e}'s contribution to relativity is an accomplished fact now. This
contribution, being far ahead of his time, is still alive and modern scholars
can still find inspiration in it. Isn't this a wonderful miracle?

Interestingly, it seems like Born in his letter to Einstein is trying to
provoke Einstein over the priority issue \cite{105}. He writes: ``I am annoyed
about this, for he is considered a great authority in the English speaking
countries and many people are going to believe him'' \cite{104}. Einstein's
response in his 12 October, 1953 letter was short and priceless in its wisdom:

``Don't lose any sleep over your friend's book. Everybody does what he
considers right or, in deterministic terms, what he has to do. If he manages
to convince others, that is their own affair. I myself have certainly found
satisfaction in my efforts, but I would not consider it sensible to defend
the results of my work as being my own 'property', as some old miser might
defend the few coppers he had laboriously scraped together. I do not hold
anything against him, nor of course against you. After all, I do not need to
read the thing'' \cite{104}.

As witnessed by Oppenheimer, Einstein in his last years `was
a twentieth-century Ecclesiastes, saying with unrelenting and indomitable
cheerfulness, 'Vanity of vanities, all is vanity' '' \cite{107}.

Of course we can wonder why Einstein, and not then much famous and educated
Poincar\'{e}, was at the origin of relativistic revolution. Undoubtedly some
peculiar personal character traits of Einstein played the role. ``He was
almost wholly without sophistication and wholly without worldliness. I think
that in England people would have said that he did not have much
'background' and in America that he lacked 'education'. This may throw some
light on how these words are used. I think that this simplicity, this lack of
clutter and this lack of cant, had a lot to do with his preservation
throughout of a certain pure, rather Spinoza-like, philosophical monism, which
of course is hard to maintain if you have been 'educated' and have a
'background'. There was always with him a wonderful purity at once childlike
and profoundly stubborn'' \cite{107}.

From the scientific point of view, the fundamental difference between the
conceptual foundations of the Lorentz-Poincar\'{e} {\ae}ther based theory
from one side, and Einstein's special relativity from another, played the
crucial role. Most succinctly this difference was expressed by Lorentz
himself: `the chief difference being that  Einstein simply
postulates what we have deduced, with some difficulty, and not altogether
satisfactorily, from the fundamental equations of the electromagnetic field''
\cite{108}. It is clear that in contrary to Einstein,  Lorentz and
Poincar\'{e} considered special relativity as a derivative, not fundamental
theory and tried to base it on a more general premises. Although very modern
in its spirit, such an approach had no chance in 1905. Even now we have little
clue how special relativity can be considered as an emergent and not a
fundamental phenomenon.

Whittaker was not the only prominent mathematician who ill treated Einstein's
contribution to relativity. Outstanding Russian mathematician Vladimir
Arnold also gives a quite distorted picture:

``Minkowski, being a teacher of  Einstein and a friend of Poincar\'{e}, had
early suggested to Einstein that he should study Poincar\'{e}'s theory, and
Einstein did (though never referring to this until a 1945 article)''
\cite{109}.

It is true that Minkowski was Einstein's former mathematics professor at the
Federal Polytechnic School of Z\"{u}rich. However, at that time Einstein was
less interested in abstract mathematics and often skipped Minkowski's classes
during his studies at the Polytechnic \cite{21}. Needless to say, it is simply
not true that Minkowski ever supervised Einstein in his scientific research.
According to Arnold Sommerfeld's recollections ``Strangely enough no personal
contacts resulted between his teacher of mathematics, Hermann Minkowski, and
Einstein'' \cite{21}.

At first sight it seems difficult to explain why Poincar\'{e} and Einstein
almost did not refer to each other when writing about relativity. First we
consider Poincar\'{e}'s silence which is, perhaps, easier to explain. As was
mentioned above, Poincar\'{e}'s objective was much more ambitious than
Einstein's as he wanted to derive special relativity as an emergent phenomenon.
It is quite possible therefore that Poincar\'{e} simply considered Einstein's
contribution as being too trivial in light of this bigger goal.
``To Poincar\'{e}, Einstein's theory must have been seen as a poor attempt
to explain a small part of the phenomena embraced by the Lorentz theory''
\cite{110}. Perhaps, later in his life Poincar\'{e} became aware of the
importance of  Einstein's work. In November of 1911, one year before his
unexpected death, when he was asked to write a recommendation letter for
Einstein who was looking to become a professor of theoretical physics
at Swiss Federal Institute of Technology in Zurich, Poincar\'{e} wrote:

``Monsieur Einstein is one of the most original minds I have known; in spite
of his youth he already occupies a very honorable position among the leading
scholars of his time. We must especially admire in him the ease with which he
adapts himself to new concepts and his ability to infer all the consequences
from them. He does not remain attached to the classical principles and, faced
with a physics problem, promptly envisages all possibilities. This is
translated immediately in his mind into an anticipation of new phenomena,
susceptible some day to experimental verification. I would not say that all
his expectations will resist experimental check when such checks will become
possible. Since he is probing in all directions, one should anticipate, on the
contrary, that most of the roads he is following will lead to dead ends; but,
at the same time, one must hope that one of the directions he has indicated
will be a good one; and that suffice'' \cite{20}.

Einstein's silence is more difficult to explain. It is reasonable to assume
that young Einstein, like most other physicists of those times, just lacked
the proper mathematical education to dully appreciate Poincar\'{e}'s work.
``More surprisingly, Einstein continued to ignore Poincar\'{e}'s contribution
in all his later writings on special relativity and in his autobiographical
notes'' \cite{111}. For late Einstein it is impossible to assume that he
was not capable to recognize the importance of Poincar\'{e}'s work. How can we
then explain Einstein's silence? We think he simply had not read
Poincar\'{e}'s key papers on relativity until perhaps later times.
Contrary to what can be expected, most scholars do not have a habit to read
the papers  of their fellow scientists attentively. Empirical studies of
misprint distributions in citations indicate that only about 20\% of
scientists read the original that they cite \cite{112}. In case of Einstein
we have his own confession to Ehrenfest: ``My dear friend, do you think that
I make a habit of reading papers written by others?'' \cite{113}. In the
early 1950s Abraham Pais asked Einstein how Poincar\'{e}'s Palermo paper had
affected his thinking. Einstein answered that he had never read the
paper. Then Pais lent to Einstein his own copy of the paper which afterwords
vanished \cite{20}. ``Perhaps he did read it. In 1953 Einstein received an
invitation to attend the forthcoming Bern celebration of the fiftieth
anniversary of special relativity. Einstein wrote back that his health did
not permit him to plan such a trip. In this letter Einstein mentions for
the first time (as far as I know) Poincar\'{e}'s role in regard to the special
theory: 'I hope that one will also take care on that occasion to honor
suitably the merits of H.~A.~Lorentz and H.~Poincar\'{e}' '' \cite{20}.
Besides, two months before his death, Einstein wrote to his biographer Carl
Seelig that Poincar\'{e} further deepened Lorentz's insight about the
essential role of Lorentz's transformations for the analysis of Maxwell's
equations \cite{111}.

There is still another aspect which makes Einstein-Poincar\'{e} priority
dispute pointless. Modern understanding of relativity is significantly
different from the one that was cultivated at the beginning of the twentieth
century. Two examples are the notions of {\ae}ther and relativity of
simultaneity which are often used in the priority dispute.

Einstein proponents stress that Poincar\'{e} never abandoned the {\ae}ther.
Indeed, in 1912, a few months before his death, he gave a talk at
a conference  at the University of London in which he said: ``Everything
happens as if time were a fourth dimension of space, and as if
4-dimensional space resulting from the combination of ordinary space and
of time could rotate not only around an axis of ordinary space in such a way
that time were not altered, but around any axis whatever. For the comparison
to be mathematically accurate, it would be necessary to assign purely
imaginary values to this fourth coordinate of space'' \cite{114,115}. Taken
out of context, it is tempting to consider this statement as an evidence
that Poincar\'{e} was accepting the Einstein-Minkowski concept of space-time.
Especially when he continues: ``the essential thing is to notice that in the
new conception space and time are no longer two entirely distinct entities
which can be considered separately, but two parts of the same whole, two parts
which are so closely knit that they cannot be easily separated'' \cite{114,
115}. However, it is clear from the full text of his talk that  he is not at
all ready to accept this new conception, as he further continues with the
conclusion: ``What shall be our position in view of these new conceptions?
Shall we be obliged to modify our conclusions? Certainly not; we had adopted
a convention because it seemed convenient and we had said that nothing could
constrain us to abandon it. Today some physicists want to adopt a new
convention. It is not that they are constrained to do so; they consider this
new convention more convenient; that is all. And those who are not of this
opinion can legitimately retain the old one in order not to disturb their old
habits. I believe, just between us, that this is what they shall do for
a long time to come'' \cite{114,115}.

Therefore, clearly, ``Poincar\'{e} never believed in the physical relevance
of the conceptual revolution brought by Einstein in the concept of time
(and extended by Minkowski to a revolutionary view of the physical meaning of
space-time)'' \cite{115}. In particular, for Poincar\'{e} the {\ae}ther, even
unobservable by physical experiments, remained an important conceptual element.

Usually this stubbornness of Poincar\'{e} with respect to the {\ae}ther is
considered as his weak point, as an evidence that he didn't really understand
relativity. It is historically true that the abolishment of the {\ae}ther
by Einstein played a crucial role and revolutionized physics. However, frankly
speaking, in retrospect, when this revolution came to its logical end in
modern physics, we can equally well consider Poincar\'{e}'s attitude as
prophetical.

As modern physics has progressed in the twentieth century, it became
increasingly evident that the vacuum, the basic  state of quantum field
theory, is anything but empty space. In fact, at present an {\ae}ther,
``renamed and thinly disguised, dominates the accepted laws of physics''
\cite{116}. It is clear that only ``intellectual inertia'' \cite{117}
prevents us from using historically venerable word ``{\ae}ther'' instead of
``vacuum state'' when referring to the states with such complex physical
properties as vacuum states of modern quantum theories.

However, the {\ae}ther of modern physics is Lorentz invariant and hardly the
one which Poincar\'{e} had in mind. Interestingly both in classical mechanical
systems \cite{118} and in condensed matter physics \cite{119} an effective
Lorentz symmetry may arise from non-relativistic background physics. This is
reminiscent of Lorentz-Poincar\'{e} program of emergent Lorentz symmetry.
This program was never completely abandoned. For most notable attempts in
this direction see the books of Brown \cite{86} and J\'{a}nossy \cite{120}.
Such attempts had, and still have, little chance of complete success. It is
not excluded however that they will get a better chance when our knowledge of
the Planck scale physics progresses. Therefore, in this respect Poincar\'{e}'s
idea of the {\ae}ther in which an effective Lorentz symmetry emerges in some
degrees of freedom at low energy limit is still alive and modern.

Now about relativity of simultaneity. Poincar\'{e} proponents in the priority
dispute argue that Einstein synchronization, which Einstein himself considered
as the crucial element of special relativity, has in fact originated from
Poincar\'{e}'s work. Above we had already commented on this difficult issue.
Now we would like to consider another aspect of it. The fact is that in light
of the modern view on relativity, Einstein synchronization can no longer be
considered as a crucial element. What is really essential is Minkowski
geometry (or slightly Finslerian) of space-time in the absence of gravity (and
cosmological constant). Einstein synchronization is just a convention, or as
Einstein himself put it, a ``stipulation'' which allows to introduce
a convenient coordinate chart in Minkowski space-time. However, an
introduction of any other coordinate charts, although probably less
convenient, is equally possible. For example, we can use 'everyday' clock
synchronization when clocks are  adjusted by using time signal broadcasted by
some radio-station telling that, for example,  ``At the six stroke, the time
will be 12 o'clock exactly'' \cite{121}. All coordinate-dependent quantities,
including the Lorentz transformations too, can experience drastic changes
under the change of synchronization convention. For example, such once thought
intrinsic feature of relativity as relativity of simultaneity is no longer
true when 'everyday' clock synchronization is used \cite{121}. Under such
synchronization  the ``Lorentz'' transformations relating the two 'everyday'
observers $S$ and $S^\prime$ with relative velocity $V$ (as measured in the
frame $S$) along the $x$-axis have the following form \cite{121}
\begin{equation}
\begin{array}{l}
x^\prime=\frac{x-Vt}{\sqrt{1+2\beta}}, \\
y^\prime=y, \\
z^\prime=z, \\
t^\prime=\sqrt{1+2\beta}\,t.
\end{array}
\label{eq117}
\end{equation}
where $\beta=V/c$. It is clear from these transformations that simultaneity
under 'everyday' synchronization is absolute: if $\Delta t=0$ in some system
$S$ then in all other inertial systems $S^\prime$ we also shall have
$\Delta t^\prime=0$.

Conventionality of simultaneity was systematically investigated by Reichenbach
(see, for example, \cite{122}). Suppose that two distant clocks $A$ and $B$
are motionless in a common inertial frame $S$. If a light signal is sent from
$A$ at time $t_A$, is instantaneously reflected by $B$ at time $t_B$,
and arrives back at $A$ at time $t_A^\prime$, then $A$ and $B$ are
Poincar\'{e}-Einstein synchronized if $t_B-t_A=t_A^\prime-t_B$, or
\begin{equation}
t_B=\frac{1}{2}(t_A+t_A^\prime)=t_A+\frac{1}{2}(t_A^\prime-t_A).
\label{eq118}
\end{equation}
Reichenbach modifies this definition of synchronization as follows
\begin{equation}
t_B=t_A+\epsilon (t_A^\prime-t_A),
\label{eq119}
\end{equation}
where now $\epsilon$ is some real parameter and $0\le\epsilon\le 1$ (because
for causality reasons we need $t_B\ge t_A$ and $t_B\le t_A^\prime$), the
equality corresponding to the infinite one-way velocity of the light signal.
In fact, these limiting cases were excluded by Reichenbach).

The modification of Lorentz transformations needed to meet this general
$\epsilon$-synchronization, was considered by Winnie \cite{123}. If
$\epsilon$-synchronization is adopted in the reference frame $S$ while
$\epsilon^\prime$-synchronization is adopted in the reference frame
$S^\prime$, Winnie transformations have the form \cite{123}
\begin{equation}
\begin{array}{l}
x^\prime=\frac{x-Vt}{\alpha}, \\
y^\prime=y, \\
z^\prime=z, \\
t^\prime=\frac{1}{\alpha}\left\{\left[1+2\beta(1-\epsilon-\epsilon^\prime)
\right]t-\left[2(\epsilon-\epsilon^\prime)+4\beta\epsilon(1-\epsilon)\right]
\frac{x}{c}\right\},
\end{array}
\label{eq120}
\end{equation}
where
\begin{equation}
\alpha=\sqrt{[1-(2\epsilon-1)\beta]^2-\beta^2}.
\label{eq121}
\end{equation}
If $\epsilon=\epsilon^\prime=1/2$, we recover the ordinary Lorentz
transformations, and if $\epsilon=\epsilon^\prime=0$ we get the
transformations (\ref{eq117}) which correspond to 'everyday' synchronization.

It is interesting to note that the 'everyday' synchronization is not the only
choice that corresponds to absolute simultaneity. Historically Frank Robert
Tangherlini was the first to realize that absolute simultaneity doesn't
contradict special relativity in his 1958 PhD dissertation supervised by
Sidney Drell and Donald Yennie (at the initial stage of the work)
\cite{124,125,126}. Tangherlini's transformations have the form
\begin{equation}
\begin{array}{l}
x^\prime=\gamma(x-Vt), \\
y^\prime=y, \\
z^\prime=z, \\
t^\prime=\gamma^{-1}t,
\end{array}
\label{eq122}
\end{equation}
and they correspond to the special case of the Winnie transformations
(\ref{eq117}) when $\epsilon=1/2$ and $\epsilon^\prime=(1+\beta)/2$.
External synchronization provides a possible realization of the Tangherlini's
simultaneity \cite{42,126A}. First clocks in the ``{\ae}ther'' frame $S$ are
Poincar\'{e}-Einstein synchronized with $\epsilon=1/2$. Then these clocks
are used to synchronize nearby clocks in the moving frame $S^\prime$ by
adjusting clocks from $S^\prime$ to $t^\prime=0$ whenever they fly past
a clock in $S$ which shows $t=0$.

Note that the Tangherlini transformations (\ref{eq122}) were obtained
twenty years before his thesis by English mathematician Albert Eagle,
although from erroneous anti-relativistic premises \cite{126A, 126B} ---
one more example that correct mathematics doesn't guarantee a correct
physical interpretation.

Some other simultaneity conventions, not necessarily based on light signals
and their two-way universal velocity $c$,  are also feasible. For examples,
inhabitants of the homogeneous isotropic ocean can find convenient to use
for synchronization light signals with two-way velocity $c^\prime<c$
\cite{127}, and dolphins can even prefer sound waves for this goal
\cite{128}. Of course, coordinate-independent quantities are not affected
by synchronization conventions. Various synchronization conventions are useful
to separate the effects due to synchronization and the real contraction of
moving bodies and slowing down of moving clocks. Such investigations allow
us to come to the conclusion that ``the results of relativistic experiments
have their origin in the length contraction and time dilatation effects which
are so real as a change of the length of a rod caused by the change of
temperature'' \cite{129,130}.

Rosen's $c^\prime$-relativity gives an interesting perspective on superluminal
objects \cite{127}. Effective $c^\prime$-relativity (as far as the
electromagnetic phenomena are concerned) precludes charged particles in the
Rosen's ocean to acquire speeds greater than or equal to $c^\prime$. However,
one might have an energetic cosmic muon enter the ocean from outside with
a speed greater than $c^\prime$. The behavior of such a particle could not
be described in the framework of $c^\prime$-relativity. For example, the
inhabitants of the ocean will observe  ``vacuum'' Cherenkov radiation emanated
from such a particle --- a phenomenon they certainly think impossible on
the base of $c^\prime$-relativity. It was speculated in \cite{131} that
likewise $c$-relativity might also not embrace some hidden sectors weakly
coupled to our visible sector of the world. Then superluminal objects called
elvisebrions in \cite{131} can enter the visible sector and their behavior
could not be described in the framework of special relativity.

Tangherlini was guided by the logic of general relativity when undertaking
his investigation of absolute simultaneity \cite{126}. According to this
logic, equations (\ref{eq122}) represent just a coordinate transformation
which makes the metric tensor non-diagonal. Indeed, it is easy to get from
(\ref{eq122}) that
\begin{equation}
dx=\gamma^{-1}dx^\prime+\gamma V\,dt^\prime,\;\;\;dt=\gamma \,dt^\prime,
\label{eq123}
\end{equation}
and the relativistic interval (line element) takes the form
\begin{equation}
ds^2=c^2\,dt^{\prime\,2}-2V\,dx^\prime dt^\prime -\gamma^{-2}dx^{\prime\,2}-
dy^{\prime\,2}-dz^{\prime\,2}.
\label{eq124}
\end{equation}
From the point of view of general relativity, these coordinates are as good
as any others because of general covariance. However, as was already shown by
Kretschmann in 1917, any space-time theory can be expressed in
general covariant manner, not only general relativity. Therefore, the
coordinates (\ref{eq122}) are as good as any others in special relativity
too. It is just an other expression of ``intellectual inertia'' that
standard textbook presentations of special relativity are based solely on
inertial observers and diagonal metric. Another misbelief, which still
prevails, is that accelerated observers cannot be treated by special
relativity, and that one needs general relativity for their incorporation.
In fact, there is no principal difficulty to consider Minkowski space-time
from the point of view of non-inertial observers at the expense of more
sophisticated mathematics \cite{133}.

We hope that the above given examples show clearly how different the present
day interpretation of key features of special relativity is from what had in
mind founding fathers of this magnificent theory. And the story of special
relativity is not finished yet. For example, the debate about conventionality
of simultaneity still continues and seems to be far from being settled
(see,  for instance, \cite{134} and references therein). The initial
observer-centered presentation of special relativity was changed to the modern
geometry-centered presentation. However, it seems that the full potential of
the notion of an observer as ontologically prior to either space or space-time
is not yet exhausted \cite{135, 136} and the initial presentation of special
relativity, although renewed and enriched by our experience with space-time
picture, can still strike back.

In light of this immense and still continuing progress of modern physics,
attempts to retrospectively induce an artificial  Poincar\'{e}-Einstein
priority dispute and rewrite the history seem minute. We will be happy if
this arid and futile dispute will come to its end. There is nothing
scientific in it and its presence only emphasizes hideous traits of human
nature. However, there is no single thing we can do, except writing this
article, to help to end this unfortunate dispute. It remains, like aging
Einstein, to repeat after  Ecclesiastes 'Vanity of vanities, all is vanity'.

\section*{Acknowledgments}
An occasional remarks of A.I. Milstein and V.~I.~Telnov about  Vladimir
Arnold's account of Poincar'{e}'s role in the development of special
relativity triggered our interest and led to this investigation.
The work of Z.K.S. is supported by the Ministry of Education and Science of
the Russian Federation.

redpajama_set_name: RedPajamaArXiv