Henri Poincaré : a decisive contribution to Relativity



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Henri Poincaré : a decisive contribution to Relativity

C. Marchal

General Scientific Direction, ONERA, BP 72, 92322 Châtillon cedex, FRANCE


Abstract
Maxwell’s electromagnetism equations and the old notions of Newtonian absolute time and Euclidean absolute space were contradictory with the impossibility of detection of Earth’s absolute motion.

Henri Poincaré and Hendrik Antoon Lorentz exchanged many friendly letters on these subjects over more than ten years. They improved their analysis step by step and this situation led Henri Poincaré to consider that the notions of absolute time and space as well as the corresponding "ether" are artificial and do not really exist. The modifications of inertial references frames do not follow the Galilean rules but those of the Lorentz transformation that can be deduced from Poincaré’s relativity principle presented in 1904 at the world scientific congress of Saint Louis (Missouri).

Poincaré’s later works of June and July 1905 have led to the full development of Special Relativity with its links to electromagnetism and the first mention of gravitational waves.

Unfortunately Henri Poincaré did not survive much to his gigantic pioneering work, he got cancer in 1909 and died in 1912.
Key words : Special Relativity, History of Science, Henri Poincaré.


  1. Introduction

The theory of Relativity is the result of a very long maturation of the knowledge and reflection of mankind confronted with the properties of matter, energy, space and time.

Let us begin with the state of this confrontation in the second half of the nineteenth century. The five main elements were then the following :


  1. Galilean Relativity.

For centuries it was believed that force was proportional to velocity : you push on an object and it moves, you stop pushing and it stops…It requires difficult observations and reflections on rubbings to understand that in absence of force the motion is rectilinear and uniform (Galileo) and that force is proportional to acceleration (Newton).

The real motive of Galileo was the understanding of Earth’s orbital motion. He needed what we now call the Galilean relativity : the Earth does not lose its atmosphere and its oceans along its orbit !… “A mechanical experiment gives the same results in a fixed laboratory and in a laboratory with a rectilinear and uniform motion” i.e. in practical terms, you can drink your coffee as usual as long as your airplane is flying in a rectilinear and uniform motion and is not buffeted by the wind. _


For this principle and several other fundamental philosophical reflections Galileo is considered by the scientists as one of the founding fathers of modern science while he is known among the public essentially because of his trial of 1633.

However notice the irony and the historical chance : it is because he was under house arrest in his country home of Arcetri near Florence that he found the time necessary for philosophical reflections. Without his condemnation he would have probably remained the busy professor and the burning, violent and sometimes unfair polemist he had been all his life. _____

2) The Earth’s motion.


Copernicus and Galileo had no physical proof of Earth’s motion, so Copernicus presented his work as an hypothesis even if Galileo was more adventurous. Fortunately by the middle of the nineteenth century this motion was firmly established on its three classical proofs : A) The aberration of stars (Bradley 1727). B) The parallax of stars (Bessel 1840) and C) The Foucault pendulum (1851).
3) Absolute or “Newtonian” time.

“Tempus absolutum, verum et mathematicum…”

“True mathematical absolute time is, by its very nature, independent of all other variables. It flows uniformly and shall be called duration.

Apparent, relative and common time is the more or less accurate and subjective measure of the duration by the motions of celestial bodies as is usually done, instead of the true time, with the hours, days, months, years.” (Newton. Philosophia Naturalis Principia Mathematica. Second edition, 1713).

In Newton’s days, and even two centuries later, no clock was able to reveal the small differences related to the relativistic effects. It was therefore very natural to assume the existence of “absolute time” , this essential parameter of so many physical laws, and Newton’s definition of this time appeared then essentially as a warning : “Let us be careful, Earth’s rotation is perhaps slightly irregular”.



  1. Absolute Euclidean space and the notion of force.

The law of inertia:

Acceleration = Force / Mass

is valid only in the “Galilean reference frames” or “inertial reference frames” that have no rotation and the relative motion of which are rectilinear and uniform.

In the second half of the nineteenth century the non-euclidean geometries of Lobatchevsky, Bolyai and Riemann were considered as mathematical curiosities without much interest and everybody considered the physical space as Euclidean.

The tremendous success of Newton’s theory of universal attraction, with the accurate description of planetary motions and the discovery of planet Neptune (1846) after the long computations of Leverrier and Adams, emphasizes the validity of all above notions.

By 1850 all laws of Mechanics satisfied the Galilean relativity, being conserved by the usual Galilean transformations of reference frames, for instance by the classical expression :

│ x1 = x  Vt : constant velocity V of the second set of axes with respect to the first.│

│ y1 = y │

│ z1 = z │(1)

│ t1 = t : absolute time. │


  1. Maxwell’s electromagnetism equations (1864).

Maxwell’s equations represent a major progress in the knowledge of matter, perhaps a progress of the order of magnitude of Newton’s law of universal attraction. But they are also the source of the trouble : they are not conserved in Galilean transformations of reference frames.

Let us consider their simplest expression in vacuum. The electric field E and the magnetic induction B are related by the four following equations :

div E = 0 ; curl E =  B /t │

div B = 0 ; curl B = µoE /t │ (2)

with :


µo = magnetic permeability of vacuum = 4 henry per meter.

= permittivity of vacuum = 8.854 187 81810 farad per meter.


The simplest solutions are the plane waves, for instance those propagating in the Ox direction :

u = x ct ; with c = [µo0. 5 299 792 458 meter per second │



E = [ 0 ; cF(u) ; cG(u) ] ; B = [ 0 ; G(u) ; F(u) ] │ (3) F(u) and G(u) are two arbitrary continuously differentiable functions │
Thus , with respect to the suitable Maxwell’s reference frame Oxyzt in which the equations (2) are valid, the plane waves are moving with the velocity c, the velocity of electromagnetic waves. This velocity was also recognized as the velocity of light after the Hertz experiment on the similarities of electromagnetism and light.

Unfortunately the Galilean transformation (1) does not conserve the velocity c and thus we must choose between the two following possibilities :

Either :


  1. Maxwell’s equations are valid with respect to a particular frame of reference Oxyzt and are only approximate for slowly moving frames of reference (as those of our usual experiments on Earth).

Or :
  1. Maxwell’s equations are valid for all inertial reference frames and the relativity can be extended from Mechanics to electromagnetism and optics. But there is a price to pay : the notions of absolute time and absolute space must be forsaken since they are contradictory with the invariance of the velocity of light.

The Newtonian absolute time seemed so obvious and so natural that the hypothesis A was immediately adopted. The hypothetical reference frame Oxyzt took a concrete form with “ether” or “aether”, a medium assumed to be very light and subtle that was considered as playing for light and electromagnetism the role of air for sound. Of course this ether offers a vivid picture of absolute space.

The next step was obviously the research of the properties of ether and also the determination of the “absolute” Earth’s motion by suitable optic or electromagnetic experiments.

The Fizeau experiment (measurement of the velocity of light in a stream of water ; 1851) and the Airy experiment (measurement of the angle of aberration in a telescope full of water ; 1871) seem to show a “partial ether drag” by transparent media.

Using all kinds of ideas and equipments many experimenters (Trouton and Noble, Lodge, Kennedy and Thorndyke etc…) tried to study the ether and to determine the Earth’s absolute motion but without success.

The most famous experimenters were Michelson and Morley. Their experiment was unable to detect an anisotropy of the velocity of light in spite of an accuracy that would theoretically enable the detection of an Earth’s motion of a few km/s only.

It is very fortunate that the Earth’s orbital motion was firmly established in the mind of the scientists of this century. Two centuries earlier the simplest explanation would have been : “The Earth is not moving…”

Whilst these experiments were carried out, the theorists obtained some interesting results.

Lorentz and Fitzgerald noticed that a suitable contraction by the “wind of ether” could explain the apparent isotropy given by the Michelson and Morley experiment.

In 1887 Voigt obtained a transformation of coordinates that preserves Maxwell’s plane and spherical waves.

In 1895 Lorentz noticed that the first-order of Voigt transformation preserves the first-order of Maxwell’s equations.

Larmor gave the second order a little later.

In his great memorandum of May 1904, “Magnetic phenomena in a system moving with any velocity smaller than that of light” (Ref. 1), Lorentz gave an extension of Voigt transformation which preserve Maxwell’s equations in vacuum.

The greatest progress was achieved by the mathematician, physicist and philosopher Henri Poincaré who was a friend of Lorentz. They exchanged many scientific letters dating from 1895 and improved their analysis step by step.

Poincaré’s successive main ideas and improvements are the following.

A) In the book “La science et l’hypothèse” (1902) in pages 111, 245, 246 (Ref. 2). “There is no absolute space and we only conceive relative motions”.

There is no absolute time, the equality of two durations can be defined only by convention and we have no intuition of the simultaneity of events occurring in different places”.

“Is the hypothesis of ether true ? This is only a question for metaphysicians. This hypothesis is convenient for the explanation of phenomena but it can be sacrified. Usually this hypothesis is conserved for clarity, but this reason is the only one”.

B) The world scientific congress of Saint-Louis (Missouri, September 1904, published in November 1904).

Henri Poincaré was asked to present a general lecture on : “The present state and the future of mathematical Physics”.

He boldly added to the five classical principles of Physics the “Principle of Relativity”: “The laws of physical phenomena must be the same for a fixed observer and for an observer in rectilinear and uniform motion so that we have no possibility of perceiving whether or not we are dragged in such a motion”. (Ref. 11, page 306). .



Surprisingly this very first expression of the Relativity Principle at its true level is not mentioned in the very interesting and generally well documented ref 12. I don’t find it either in ref 13 in spite of its presence in ref 14 and also in the famous Encyklopädie der Mathematischen Wissenchaften (Ref. 15). .

This principle was of course essentially based on the negative results of all ether experiments of this time and most of the lecture was devoted to the defense of the new principle.

The conclusion of the lecture :

“Thus the Relativity Principle was these last years valiantly defended, but the energy of defense proves how serious was the attack…We will perhaps be forced to built a complete new Mechanics in which the inertia increasing with velocity, the velocity of light would become an unsurpassable limit”.(Ref. 11, page 324)

C) The note to the French Academy of Science (June 5, 1905, published June 9, 1905, Ref. 3).

Poincaré states the Relativity Principle once again and analyses the “change of variables” presented by Lorentz in his 1904 memorandum. He simplifies its presentation and gives its present name : “The essential point, established by Lorentz, is that Maxwell’s equations are preserved by a transformation to which I will give the name of Lorentz” (later, in 1914, Lorentz will correct that statement : “My demonstration was not general, only that of Poincaré and later those of Einstein and Minkowsky were complete. (Ref . 10, page 249)).

Poincaré notices that the Lorentz and associated transformations are the element of a group (today the Poincaré group) and this allows him to give the value of the coefficient L used by Lorentz in his transformation : this coefficient must be identical to one.

Finally, after a short development about the transformation of electromagnetic equations, he notices that this relativity theory implies the existence of “gravitational waves” moving at the velocity of light, but his later researches in this direction will remain unsuccessful.

We must remark that the second time t’ appearing in the Lorentz transformation is as physical as the first because of the non-existence of ether and absolute time and because of the perfect symmetry of the transformation. Poincaré had already given a physical meaning to this time t’ by the synchronization between distant clocks with light signals, and with the invariance of the velocity of light (Ref. 4).

It is essential to notice that the Lorentz transformation is a direct consequence of the Relativity Principle, without need for an invariant velocity of light (see appendix).

D)The final fundamental work of Poincaré on relativity is his paper “Sur la dynamique de l’électron” in which he develops and demonstrates the ideas of his note to the Academy (Ref. 5, July 1905, published in January 1906).

The expression of the transformation of the electromagnetic field is particularly impressive : the electromagnetism appears as the association between electrostatic and relativity.

Poincaré also develops the analysis of his group and find its invariant : the quantity (L2 c2 T2) where L represents the space interval (length or distance) and T the time interval. A few years later Minkowski will present that same invariant under the famous differential form :

c2 dt2 dx2 dy2 dz2 = c2 ds2

The parameter s is the “proper time” given by the onboard clocks of the space vehicle of interest. That parameter, being a physical parameter, must of course have the same value in all referentials.

The Lorentz-Poincaré theory then lead to the relative character of physical space and physical time, it is in agreement with the Relativity Principle, with Maxwell’s equations not only in vacuum but in all cases, with the ether drag experiments (Fizeau, Airy, Michelson…) and with the classical results of electromagnetism as discovered by the pioneers : Coulomb, Ampère, Volta, Laplace, Gauss, Oersted, Faraday…The theory of special Relativity was thus complete.

Meanwhile Einstein prepared and published his first and most famous paper on Relativity : “Zür Elektrodynamik der bewegten Körper” (Ref. 6). This paper is presented without reference and is considered for that reason by some people as a compilation of former works (Ref. 7).

The basic idea of Einstein is the invariance of the velocity of light (which forbid the slightest mass for the photon).

Einstein was led to the Relativity Principle. He obtained all the same results as Poincaré. He mentioned that the Lorentz and associated transformations are the elements of a mathematical group but made no use of this property.

Was Einstein aware of Poincaré’s work ? This is a difficult question.

On one hand he wrote to his friend Carl Seelig in 1955 :

“There is no doubt, that the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905. Lorentz has already observed that for the analysis of Maxwell’s equations the transformations which later were known by his name are essential, and Poincaré had even penetrated deeper into these connections. Concerning myself, I knew only Lorentz’ important work of 1895 La théorie électromagnétique de Maxwell and Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern but not Lorentz later work, nor the consecutive work of Poincaré. In this sense my work of 1905 was independent”. (Ref. 8, page 11).

On the other hand :

The 1905 Einstein’s paper presents all the same results as Poincaré and even the group property of the elements of the Lorentz and associated transformations.

This notion of a mathematical group was then very new and practically ignored among physicists. Einstein makes no use of it. Einstein had of course no opportunity to read the July 1905 Poincaré’s paper while writing his own paper, but the June 1905 Note to the Academy arrived in Bern, in time, by June 12 or 13, and it was a part of Einstein’s job to read it. Furthermore several analyses by Einstein of interesting physical 1905 notes to the French Academy of Sciences have recently been noticed (see for instance in ref. 18 the analysis of the work of Mr Ponsot, C.R. 140, S pages 1176-1179, 1905 ).

Finally, last but not least, according to his friends Maurice Solovine and Carl Seelig, Einstein read Poincaré’s book La Science et l’Hypothèse (no absolute time, no absolute space, no ether…) during the years 1902-1904. This book was discussed at their lecture circle “Olympia Academy” over several weeks (Ref. 8, pages 129 and 139 ; ref. 9 page VIII and ref. 17 page 30).

It is sometimes considered that a significant part of scientific and technical progress have been made at the same time by several people (in other words Poincaré and Einstein could have obtained the same results independently). The best example is this of Bell and Gray putting down the same day their patent on the telephone, but even if this is sometimes true for ordinary research it never happen for the major transformations of Science that immediately raise up opposition. Copernicus was alone, Kepler was alone, Galileo was alone, Darwin and Wallace were alone, Pasteur was alone, Freud was alone and all of them were either ignored or even fought and persecuted.

Nevertheless, even if the Principle of Relativity could be called Poincaré’s Principle and even if Einstein was not the first, we owe him not only the General Relativity of 1916 but also a magnificent popularization of Special Relativity. This is very fortunate because the health of Poincaré was bad and he didn’t survive much to his gigantic work (he got cancer in 1909 and died in 1912 at the age of 58).

The bad health of Poincaré and the absence of reference in Einstein’s 1905 paper are of course not the only reasons for which Poincaré is so little known and Einstein so famous.

If a great physicist such as Paul Langevin (who discussed Physics with Poincaré, his former professor, during the weeks of their trip to Saint-Louis in 1904), if Paul Langevin had supported Poincaré the evidence would have been recognized immediately.

If Poincaré had had the possibility of publishing in a famous physics publication, such as Einstein’s paper in the Annalen der Physik, he would have a large audience. But he only found the Rendiconti del Circolo Matematico di Palermo for his founding July 1905 paper…a small journal of mathematics which was not known among physicists. (Today, of course, Italians are very proud that the founding paper of Relativity was published in Italy, and in all Italian universities you can see the letter sent by Henri Poincaré to the director of the Rendiconti del circolo Matematico di Palermo and published by this journal for its centennial in 1984).

It seems incredible that Poincaré had almost no possibility of publishing in a journal of Physics, but the physicists of this time refused to consider that this prodigious mathematician was also one of them. Even today some physicists still consider that the physical character of the variables x’ and t’ of the Lorentz transformation was not sufficiently emphasized by the father of the Relativity Principle ! (Ref. 20).

All along the twentieth century many physicists will blame Poincaré for having not ruled out more strongly the notion of ether. But this was not so obvious and even much later, in May 1920, in the conclusion of his Leyde conference, Einstein will say : “ Finally we can say : according to the General Relativity theory, space has physical properties ; therefore in this meaning an ether exists. According to the General Relativity theory a space without ether cannot be conceived, for not only the propagation of light would be impossible, but there would be no possibility of existence for rulers and clocks, and thus consequently for space-time distances in the meaning of Physics. However this ether must not be conceived as having the most common property of material media, that is with parts that can be followed in time, the notion of motion must not be applied to ether”. (Ref. 19).

We must recognize that Poincaré had been unlucky. His work Sur la dynamique de l’électron is practically not known before the thirties and meanwhile Science and the scientific vocabulary have drastically changed. While from one translation to the next the Einstein’s text is constantly updated…Hence the comparison of the two texts is apparently edifying. Poincaré’s text is difficult to read and even some readers conclude that Poincaré had not really understood Relativity…Fortunately a Russian physicist, the Academician Anatoly Logunov has recently translated Sur la dynamique de l’électron into modern scientific language (Ref. 16). Then everything become clear, it becomes impossible to claim that Poincaré had not understood Relativity or was unaware of the importance of his work.

Above all this a typically French story. Most professors of the French universities were politically from the left during this time of hard political confrontation (Dreyfus affair, separation of Church and State). They refused to support Henri Poincaré who was identified with his cousin Raymond Poincaré, leader of the right and future president of the French Republic. This is perhaps the true reason for the silence of Paul Langevin.

Poincaré was not a self-publicist. He had given to Lorentz more than his share of credit, which was fair-mindedly refused by Lorentz. He had called Fuchsian functions, functions of Professor Fuchs, some functions for which he had completed two thirds of the work…

Fortunately the friendship of Lorentz saved him. In 1921, after the triumph of the solar eclipse of 1919, the Nobel committee met and initially decided : “We must give the Nobel prize to Einstein for Relativity”. But Lorentz, Nobel prize 1902 for Physics, protested : “ This is unfair !” and he published the notice on Poincaré’s life he had written in 1914: “ I have not established the Relativity Principle as rigorously and universally true. On the contrary Poincaré has obtained a perfect invariance (of Maxwell’s equations) and has expressed the Postulate of Relativity (i.e. the Relativity Principle), expression that he was the first to use…”(Ref. 10, page 252).The embarrassed Nobel committee decide to think the subject over carefully and a few months later finally gave the Nobel prize to Einstein, not for the Relativity…but for the photoelectric effect !

Thus, in spite of his modesty and his shyness, Henri Poincaré must be considered not only as an excellent philosopher of science and one of the greatest mathematicians. He is also a major physicist (electromagnetism and radio, optics, fluorescence, kinetic theory of gas, quanta, etc.), the father of the Relativity Principle and the founder of Special Relativity.

References


  1. Lorentz H.A. Electromagnetic phenomena in a system moving with any velocity less than that of light. Proc. Royal Acad. Amsterdam. 6, page 809, 1904.

  2. Poincaré H. La Science et l’Hypothèse. Flammarion ed. Paris . 1902.

  3. Poincaré H. Sur la dynamique de l’électron. Compte rendus de l’Académie des Sciences de Paris, 140, pages 1504-1508. June 5, 1905.

  4. Poincaré H. La mesure du temps. Revue de métaphysique et de morale, 6, pages 371-384. 1898.

  5. Poincaré H. Sur la dynamique de l’électron. Rendiconti del circolo matematico di Palermo, 21, pages 129-175, received July 23, 1905, published in January 1906.

  6. Einstein A. Zür Elektrodynamik der bewegten Körper. Annalen der Physik, 17, pages 891-921, received June 30, published September 26, 1905.

  7. Leveugle J. Poincaré et la Relativité. La Jaune et la Rouge, pages 31-51, Avril 1994.

  8. Miller A.I. Albert Einstein’s Special Theory of Relativity. Addison-Wesley Publishing Company Inc., Reading Mass. 1981.

  9. Solovine M. Lettres à Maurice Solovine. Gauthier-Villars ed, Paris, page VIII, 1956.
  10. Lorentz H.A. Deux mémoires de Henri Poincaré dans la Physique mathématique. Acta Matematica, 38, pages 293-308, 1921 ; and also : Œuvres de Henri Poincaré. Gauthier-, Villars ed. Paris, tome 11, pages 247-261,1956.


  11. Poincaré H. L’état actuel et l’avenir de la physique mathématique. Bulletin des Sciences mathématiques, 28, 2nd series (reorganized 39-1), pages 302-324, 1904.

  12. Tonnelat M.A. Histoire du principe de relativité. Flammarion ed. Paris, 1971.

  13. Ginzburg V.L. On the theory of relativity. Nauka ed. Moscow 1979.

14. Bol’shaia Sovietskaïa Entsiklopedia. Great Soviet Encyclopedia-A translation of the

third edition. 18, Macmillan Inc, Collier Macmillan Publishers , New-York. Relativity,

Theory of, page 653, 1974.



  1. Pauli W. , Kottler F. Encyclopädie der mathematischen Wissenchaften. Leipzig Verlag

und Druck von B. G. Teubner . Relativitätstheorie V-2 , pages 545-546 (1904 -1922).

Gravitation und Relativitätstheorie VI-2-2, page 171 (1922-1934).

16. Logunov A. A. On the articles by Henri Poincaré : “On the dynamics of the electron” . Publishing Dept of the Joint Institute for Nuclear Research, Dubna, 1995.



Sur les articles de Henri Poincaré : « Sur la dynamique de l’électron ». Le texte fondateur

de la Relativité en langage scientifique moderne. Publication ONERA 2000-1. 2000.

  1. Merleau-Ponty J. Einstein. Flammarion ed. ISBN, page 30, 1993.
  2. Einstein A. Beiblätter zu der Annalen der Physik. 29, n° 18 , pages 952-953, 1905.


  3. Einstein A . L’éther et la théorie de la Relativité. Leyde conference (Netherland) , May

5, 1920 . French translation by Maurice Solovine and Marie-Antoinette Tonnelat in :

Albert Einstein, Réflexions sur l’électrodynamique , l’éther , la géométrie et la relativité.

Collection « Discours de la méthode » , nouvelle édition , Gauthier-Villars ed. Paris ,

Page 74, 1972.


  1. Darrigol O. Henri Poincaré’s criticism of Fin de Siècle electrodynamics . Studies in

History and Philosophy of modern Physics, pages 1-4, April 1995.

___________


Appendix
The Lorentz transformation
Let us look for the Lorentz transformation along two axes Ox and O’x’ moving along each other with the constant relative velocity V.





OO’ = Vt O’ x’

>

O x

In order to obtain a perfect symmetry between the two referentials we will put O’x’ in the other direction.



<

x’ O’


>

O x

The homogeneity will lead to linear transformations and if we choose t = t’ = 0 when the two origins O and O’ cross each other , the transformations (x, t)  (x’, t’) and (x’, t’)  (x, t) will be given by the following expressions with eight suitable constants from A to D’ :

x’ = Ax + Bt ; t’ = Cx + Dt

x = A’x’ + B’t’ ; t = C’x’ + D’t’ (4)

The Principle of Relativity and the symmetry lead to :

A = A’ ; B = B’ ; C = C’ ; D = D’ (5)

Furthermore we have in the origin O’ : x’ = 0 and x = Vt , hence from x’ = Ax + Bt

we deduce 0 = AV + B , and from x = Ax’ + Bt’ ; t = Cx’ + Dt’ we deduce B = DV, and thus D = A .

Finally the coherence requires :

x = Ax’ + Bt’ = A(Ax + Bt) + B(Cx + Dt) = (D2 + CDV)x

t = Cx’ + Dt’ = C(Ax + Bt) + D(Cx + Dt) = (D2 + CDV)t (6)

hence we need D2 + CDV = 1 , that is C = (1  D2 ) / DV , and the transformation (x, t)  (x’, t’) becomes :

x’ = Dx + DVt ; t’ = [(1 D2 ) / DV] x + Dt (7)

The only remaining unknown is D that is a priori a suitable function of the velocity V. This function can be obtained through the comparison of several velocities.

Let us reverse again the axis O’x’ and let us consider the three axes Ox, O’x’, O’’x’’ in the same direction.

>

OO’’ = V’’t O’O’’ = V’t’ O’’ x’’

>



OO’ = Vt O’ x’

>

O OO’ = Vt ; O’O’’ = V’t’ ; OO’’ = V’’t x

With the opposite sign for x’ the relation (7) becomes :

x’ = D(x Vt) ; t’ = [(1D2) / DV] x + Dt (8)

and thus similarly, with D’ for V’ and D’’ for V’’ (this new D’ is independent from that of (4)-(5) that is not used after (5) ).

x’’ = D’(x’V’t’) ; t’’ = [(1 D’2) / D’V’] x’ + D’t’ (9)

x’’ = D’’(xV’’t) ; t’’ = [(1D’’2) / D’’V’’] x + D’’t (10)

The elimination of x’ and t’ in (8) and (9) gives another expression of (10) :

x’’ = {DD’ + [D’V’(D21) / DV]}x DD’(V + V’)t

t’’ = {[(DDD’2) / D’V’] + [(D’ D2D’) / DV]}x + {DD’ + [DV(D’2 1) / D’V’]}t (11)

The identification of (10) and (11) leads to the four following equalities :

D’’ = DD’ + [D’V’(D2 1) / DV] (12)

D’’V’’ = DD’ (V + V’) (13)

(1 D’’2) / D’’V’’ = [(DDD’2) / D’V’] + [(D’D2D’) / DV] (14)

D’’ = DD’ + [DV(D’2 1) / D’V’] (15)

Hence with (12) and (15) :

D’’DD’ = D’V’( D21) / DV = DV( D2 1) / D’V’ (16)

The last inequality of (16) allows to define the quantity K by :

K = D2 V2 / ( D2 1) = D’2 V’2 ( D’21) (17)

The quantity K has the same value for two arbitrary velocities V and V’ (and their corresponding D and D’), hence it is a constant for all velocities. On the other hand the case V= 0 gives D = 1 in (8) , hence we must choose the positive solution of (17), that is :

D = 1 / (18)

With (8) we thus reach the transformation (x, t)  (x’, t’) and Poincaré has no difficulty to extend it to the general transformation (x, y, z, t)  (x’, y’, z’, t’) :

x’ = (x Vt) / ; y’ = y ; z’ = z ; t’ = [t (Vx / K)] / (19)

It remains to determine the constant K that gives the Galilean transformation (1) if K = and the ordinary Lorentz transformation if K = c2 . These two transformations are of course very near to each other if the ratio V / c is small.

The constant K cannot be negative (it would become possible to go back in the past) and its square root appears as an unsurpassable limit velocity. This latter point is confirmed by the expression and also by the composition of velocities that is given by (12) and (13) :

V’’ = ( V + V’ ) / [1 (VV’/ K)] (20)

that is, with = k :


k V’’ = (k V) . (k V’)

k + V’’ (k + V) (k + V’) (21)


hence { V< k , and V’< k } implies V’’< k .

Very naturally Poincaré and Lorentz have chosen K = c2 , that agree with the invariance of the velocity of light and with the conservation of Maxwell’s equations in all inertial frames of reference. Let us notice however that, if necessary, it remains possible to assume that the constant K is very slightly larger than c2 . The photons would then have a very small but non zero mass and their velocity, the velocity of light, would be almost and would be an increasing function of their energy.

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End of the appendix.

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