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Henri Poincaré : A
decisive contribution to Special Relativity The short story by Jacques. Fric (June 2003) Acknowledgements: This article
relies on a paper published by Jules Leveugle [ ref 7]
and a following paper published by Christian Marchal : Poincaré:
une contribution decisive à
la Relativité Introduction
In
April 1994 « la Jaune et la Rouge » published a survey from Jules Leveugle (Ecole Polytechnique) named
«Poincaré et la Relativité » (Réf 7). Evidences of the major
contribution of Henri Poincaré to the Special Relativity theory are presented
in this survey. According to the success
encountered by this article, and the questions about it, Christian Marchal
published an additional paper in order to answer to the questions. H. Poincaré is very well known
as a famous mathematician, even though he never taught mathematics but physics
(electromagnetism) as a professor at the “Ecole Polytechnique”. His work on
Relativity is less known. The purpose of this paper is to demonstrate that he
set up all the basic concepts of Special Relativity, several years before
Einstein did in his the famous paper (1905: Annalen der Physik vol XVII 1905 p
891-921: ref 6). We will review briefly the
controversy between Lorentz and H. Poincaré
about absolute space time (1898-1905). We will compare the very
different approaches of Poincaré (formal) and Einstein (physical), on
Relativity (illustrated by the demonstration, within the theory, of the Lorentz
equations by each of them) . We will end on some questions:
Why the work of Poincaré is so ignored, why Einstein did not mentioned any
references at all, in his 1905 fundamental article, acting as he had invented
everything, even though, at least the Relativity principle as well as the
method for synchronizing clocks are borrowed from Poincaré published papers
(1898-1902). This should explain why Einstein was not awarded the Nobel Prize
for the Relativity, but for the photo-electric effect! For a detailed bibliography of H. Poincaré
see http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html
…. Lorentz
– Poincaré Controversy (1898-1905) At the end of the 19th
century, when the detection (at the second order) by the Michelson Morley
experience of the absolute motion “in the ether” of the Earth failed, the science was under
trouble. In 1895 Lorentz suggested that
moving bodies should experience a “physical” contraction, compute what it would
be in accordance to the Michelson Morley experience and set up a theory on this
basis. In 1899, Henri Poincaré, who
was teaching electromagnetism at the “Ecole Polytechnique” wrote in one of his
course [ref: “Electricité et Optique”published by Carré et Nadaud, 1901, p 536
] about this hypothesis: “ I am not
satisfied with the explanation ( physical contraction) of the negative result
of the Michelson experiment by the Lorentz theory, I would say that the laws of
optics are only depending on the relative motion of the involved bodies”. It was the beginning of the (
friendly) controversy with Lorentz ,
mainly about the concept of absolute Space and absolute time. In his book “La science et
l’hypothèse” (1902), Poincaré devoted a full chapter to the relativity
principle: “There is no absolute uniform motion, no physical experience can
therefore detect any inertial motion (no force felt), there is no absolute
time, saying that two events have the same duration is conventional, as well as
saying they are simultaneous is purely conventional as they occur in different
places.” One can still keep in mind the concept of ether, if it helps for
thinking, but it is a unphysical concept, it is a metaphysical concept. He defines the way to
synchronise all the clocks of an inertial frame, by using light signals (1900:
La théorie de Lorentz et le principe de réaction, published J.Bosscha]. In 1904 at the St Louis
conference, he proposes to add the Relativity principle to the five classical
“universal” principles of the physics. He emphasised that the Lorentz
contraction was an “ad hoc” hypothesis, just made for adjusting the theory to
the experience. Lorentz reported this point of
view of Poincaré, in his theory of electromagnetism (1904-Ref [1]). “Poincaré has objected to the
existing theory of electric and optical phenomena in moving bodies that, in
order to explain Michelson’s negative result, the introduction of a new
hypothesis has been required, and that the same necessity may occur each time
new facts will be brought to light. Surely this course of inventing special
hypotheses for each new experimental result is somewhat artificial. It would be
more satisfactory if it were possible to show by means of certain assumptions
and without neglecting terms of one order of magnitude or another, that many
electromagnetic actions are entirely independent of the motion of the
system…..” He points out that the “form”
of the Lorentz formulae can be demonstrated
from the Relativity principle alone,
( see annex 1) and therefore are implied by this principle alone (with a
parameter to specify, related to “c” for the Special relativity), only one
hypothesis is required, instead of the local time and the associated three new
hypothesis of the article of Lorentz ( 1904). At last but not at least, In
1900, he noticed that the recoil of a
radiation, of energy E, is m = E/c² [ ref oeuvres de Poincaré ,op.cit.t IX p
471] which is nothing else that the famous E
= mc². We can see that in 1905
Poincaré, in different articles, had set up all the basic concepts of the
special Relativity even though he did not summarize all of them in the
framework of a formal theory as Einstein did.
Special
Relativity : Poincaré versus Einstein On one hand, Poincaré as a
mathematician had a clear formal approach of the Relativity principle. Whether
we have a look at his demonstration (annex 1), we see that he states first that
inertial frames are homogenous allowing to express each coordinate (x,t) of
one frame as a linear function of the coordinates of the other frame (x’,t’)
and vice versa , he had previously
rotated one axis by 180 ° in order to have a full symmetrical situation.
This results in four relations with eight unknown parameters. The symmetry of
the situation reduces them to four and using relations leaves the equations
with only one undetermined parameter, function of the relative speed. The form
of this last parameter can be computed taking into account that into a group,
by definition, operation by two elements is an element of the group. He considered three inertial frames (1), (2),
(3) and applied the equations according (1) ->(3) = (1)->(2)->(3): he
got: . x'=(x-Vt) (1-V²/K)-1/2
; y'=y ; z'=z t'=[(t-(Vx/K)] (1-V²/K)-1/2
(1) Note
that in this demonstration, he never uses the (second) postulate ( constancy of the speed of
light). So it is more general. Where V is the relative
velocity of the two frames and K is a parameter having the dimension of a
square of a speed which, is the maximum possible value of V (to keep the
equation real) . Whether K = c², we recognize the Lorentz equations, whether K
= infinity, it is Newton. The interesting thing is that each
value of K (continuous parameter) defines a continuous family (with a group
structure) of inertial frames. The least we can say is that
this demonstration is very simple. Poincaré understood that such Relativity principle reflects symmetry
properties of the physical laws (prefiguring Noether theorem: 1915):
invariance of physical laws under a group of transformations : the Poincaré
group, [ Ref 3] including three spatial rotations, three hyperbolic rotations
(boosts) and four translations,
conferring to the Minkowski space ( The special Relativity spacetime), the
maximum possible number of symmetries : n( n+1)/2 = 10. This is obviously, the modern
approach. . On the other hand, Einstein,
more intuitive and sticking to physical principles, first states the two
postulates: Relativity principle ( borrowed to Poincaré) and constancy of light
speed in all Galilean frames. Then he studies the methods for synchronizing
clocks in a frame ( borrowed to Poincaré) , allowing also to define space
milestones ( with the help of the constant light speed). Then, after defining exactly
the terminology, he considers a moving frame and writes the relations between
the coordinates of the two frames by considering the clock synchronisation
equation in the two different frames derived from the light speed constancy.
See equation (5) in annex 2. In this step by step
demonstration, we discover the expected result only at the end.. The
last part is quite classical. The paper of Einstein [ ref 6]
is a more extensive treaty about Relativity, as in addition to the
demonstration of the Lorentz formula, he considers also the cinematic, and the
dynamic of the electron, it is a first synthesis about the theory of special
relativity and its implications. ( E = mc² will come later in a second paper
[Annalen der Physic, 1905 vol XVIII, p 639-641]) That’s the facts. So,
why the work of Poincaré in Special Relativity is so ignored ? One point is that Poincaré did
not realize the revolution implied by this new theory, as he had many others
topics of interest especially in mathematics. His formal (maybe too
speculative) approach did not reflect the “physical reality” implied by the
theory. His position on the ether remains quite ambiguous, as he did not reject
definitely the concept as Einstein did (at the beginning, as later Einstein had
a more ambiguous position too, ether being a “hard to kill “concept). In some
following papers, he expresses some doubts about the universality of the
Relativity principle. In addition, he never published a so extensive treaty
than Einstein. Einstein approach more
physical, more practical, was closer to the physical reality, and may be seen
more striking than the Poincaré approach. In 1905 Poincaré, who was in
his fifty’s, was known as a shy person (For instance, he did most of the work
for the Fuchsian functions, but let the paternity to Fuchs). He was recognised
as a famous mathematician, his comfortable situation and his reputation can
explain his cautious and tedious attitude. Soon later, he got seriously ill. He
got a cancer in 1909 and died in 1912. Einstein was young, bold,
unknown at that time, and therefore took over the whole thing. Einstein has always denied to
have known Poincaré publications. It’s hard to believe as his friends Maurice Solovine
and Carl Seelig, report Einstein had read the Poincaré book “La Science et l'hypothèse”
(no absolute time, no absolute space, no ether ... ) around
1902-1904. This book was commented at their reading commitee « Académie
Olympia » during several weeks (ref. 8, pages 129 et 139 ; ref. 9, page VIII
and ref. 17, page 30). His position at
the Swiss office patent in charge of “electromagnetism” implied that part of
his job was to read and summarize the main publications on this topic (he
summarized several papers from the French “Academie des sciences”). At the end of his life,
Einstein wrote in 1955 in a letter to Carl Seelig: «There is no doubt, if we look
back to the development of the Relativity theory, special Relativity was about to
be discovered in 1905. Lorentz already noticed that the transformations (named
Lorentz transformations) were essential in the Maxwell theory and Poincaré had
gone even further. At that time I only knew
Lorentz work of 1895, but I knew neither
Lorentz nor Poincaré further work. This why I can say that my work of 1905 was independent » (ref 8, page 11). Anyway, Einstein had a
different approach, he popularised the Special Relativity, and developed “
alone” the General Relativity ( even though Hilbert derived the field equation
of GR a few days before Einstein, but by using Einstein work so the equation
was rightly attributed to Einstein) . When the Nobel Committee decided to award
Einstein the price, Lorentz, who was a member, objected that it was Poincaré
who had found the full group of transformations of the SR, and it would be
unfair not to associate him. But as Poincaré was dead at that time, Einstein
was awarded the Nobel price for his work on the photo electric effect: That’s diplomacy… . Références 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. Edition Flammarion, Paris, 1902. 3 Poincaré H. Sur la dynamique de l'électron. Comptes rendus Acad. Sci. Paris, 140, pages 1504-1508, 5 Juin 1905. 4 Poincaré H. La mesure du temps. Revue de métaphysique et de morale. 6, pages 371384, 1898. 5 Poincaré H. Sur la dynamique de l'électron. Rendiconti del Circolo Matematico di Palermo, 21, pages 129-175, reçu le 23 Juillet 1905, published in January 1906. 6 Einstein A. Zur Elektrodynamik der bewegten Körper. Annalen der Physik, 17, pages 891-921, reçu le 30 Juin 1905, publié le 26 Septembre 1905. 7 Leveugle J. Poincaré et la relativité. La Jaune et la Rouge, pages 3 1-5 1, Avril 1994. 8
Miller A.I. Albert Einstein's Special Theory of Relativity. Ed.
Addison-Wesley Publishing Company Inc. Reading Mass., 1981. 9 Solovine M. Lettres à Maurice Solovine. Ed. Gauthier-Villars, Paris, 1956. 10 Lorentz H.A. Deux mémoires de Henri Poincaré dans la Physique mathématique. Acta Mathematica, 38, pages 293-308, 1921. 1 1 Poincaré H. L'état actuel et l'avenir de la physique mathématique. Bulletin des Sciences Mathématiques, 28, 2° série (réorganisé 39-1), pages 302-324, 1904. 12 Tonnelat
M.A. Histoire du principe de relativité. Ed.
Flammarion, Paris, 1971. 13
Ginzburg V.L. On the theory of relativity. Ed. Nauka,
Moscow, 1979. 14
Bol'shaia Sovetskaia Entsiklopedia. Great Soviet Encyclopedia-A
translation of the third edition. Volume 18, Macmillan
Inc., New-York, Collier Macmillan Publishers. Relativity, Theory of, page
653, 1974. 15
Pauli W., Kottler F. Encyclopädie der mathematsichen
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 langue scientifique moderne. Publication ONERA 2000-1, pages 1-48, 2000. 1 7 Merleau-Ponty J. Einstein. Ed Flammarion ISBN, page 30, 1993. 1 8 Einstein A.Beiblâtter zu der Annalen der Physik. 29, N' 18, pages 952-953, 1905. 1 9 Einstein A. L'éther et la théorie de la relativité. Conférence faîte à Leyde (Pays-Bas) le 5 Mai 1920. Traduction en français par Maurice Solovine et M.A. Tonnelat dans: Albert Einstein, Réflexions sur l'électrodynamique, l'éther , la géométrie et la relativité. Collection « Discours de la méthode », nouvelle édition, Gauthier-Villars éd. 55 Quai des Grands Augustins, Paris 6è, page 74,1972. 20
Darrigol 0. Henri Poincaré
's criticism of Fin de Siècle electrodynamics Studies in History and Philosophy
of modem Physics, pages 1-4, April 1995. References 3, 5 et 10 appear also in "Oeuvres de Henri Poincaré", respectively tome 9, pages 489-493 ; tome 9, pages 494-550 et tome 11, page 247-261; Gauthier-Villars éditeur, Paris, 1956.
The Lorentz
Transformation ( Poincaré) The basic idea is that coordinates transformations form a group. We will
derive it, by using the symmetries implied by the Relativity principle alone. A- First general
considerations Let O'x' be one inertial
frame sliding on Ox, an other inertial frame at a constant speed V. ___________________________________________O'___________________________>
x'
______________________O______OO'= V_____________________________________>
x
In order to have a full
symmetrical configuration, let’s rotate O’x’ by 180°, as shown below. x'<_________________________________________O'________________ ________________________O_____________________________________> x Homogeneity would imply a
linear transformation. If we set t = t' = 0 in Ox and O'x’ frames, when O and O' coincide : transformations (x, t) ® (x', t') and (x',
t')® (x, t) will be as follow, with eight constants A to
D' (4)
x' = Ax +
Bt
t' = Cx + Dt x = A'x' + B't' t = C'x' + D't' Relativity principle and
symmetry imply: (5)
A = A' B =
B' C =
C' D = D' In addition, for O' we have x'
= 0 and x = Vt, therefore x' = Ax + Bt implies AV +
B = 0, as well as x = Ax' + Bt' and t = Cx' + Dt' implies B
= DV, and therefore D = -A. Finally, consistency implies: (6)
x = Ax' + Bt' = A(Ax + Bt) + B(Cx + Dt) = (D' + CDV)X t = Cx' + Dt' = C(Ax +
Bt) + D(Cx + Dt) ='(D' + CDV) t, therefore D^2 + CDV = 1, that
is: C = (1 – D^2) / DV. The transformation
(x, t) -> (x', t') becomes (7) x'
= -Dx + DVt,
t' = [(l – D^2) / DV]x + Dt B- The only unknown
parameter, D, is obviously a function of the speed V. It will be
determined by considering that the “product” of two
transformations is a transformation of the group and computing it in two
different ways : Now we resume O'x' initial
position and let consider three axes Ox, O'x' et O"x" oriented in the
same way .
__O"____________________________>x"
OO"=V"t, O'O"
=
V't
____________________________________O'__________________________________>x'
OO' = Vt __________________________O_____________________________________________>x The relation (7) becomes with opposite sign for x' (8)
x' = D(x - Vt) ,
t'= [(l – D^2) / DV]x + Dt If we do the same for D' with V' and D" with V "(This new D' is
not the same than the one used in (4)-(5), no longer used after (5)) we get : (9)
x"= D'(x'- V't'); t" = [(l - D'^2)/D'V']x' + D't' (10)
x"= D"(x - V"t); t" = [(l - D"^2)/
D"V"]x + D"t Plugging x' and t' from (8)
into (9), we get an other expression
of (10) (11)
x"={DD'+[D'V'(D^2
- 1) / DV] }x - DD'(V + V')t t"={[(D - DD'^2) / D'V'] + [(D'- D^2. D') / DV]} x + { DD'+ [DV(D'^2 - 1) /D'V']} t
Identifying (10) and (11) leads to the
four equations described below: (12)
D " = DD' + [D'V'(D^2
-1) / DV] (13)
D"V "= DD'(V + V') (14)
(1 - D"^2 ) / D"V"= [(D - DD'^2) /D'V']
+ [(D'- D^2.D') / DV] (15)
D"= DD' + [DV(D'^2 - 1) / D'V'] Therefore with (12) and (15): 16)
D"- DD' = D'V'(D^2
- 1) / DV = DV(D'^2 - 1) /
D'V' C- This last equation allows
us to define a constant parameter K as: (17)
K=D^2.
V^2 / (D^2 -1)
= D'^2.V'^2 / (D'^2 - 1)
The
parameter K has the same value for two arbitrary speeds (
with their respective D). It is therefore the same for all speeds. In addition, V = 0
gives x = x' and t = t', therefore D = 1 in (8), we must choose the
positive solution of equation (17) (18) D=
1/ sqrt(1-V^2/K) That’s the famous relation we
were looking for… Plugging this in equation (8),
gives the transformation (x, t)®(x',
t). Poincaré generalises this easily to
the general transformation (x, y, z, t) ® (x', y', z', t'). (19) x'=(x-Vt)
/sqrt(1-V^2/K) ;
y'=y ;
z'=z
t'=[(t-(Vx/K)] / sqrt(1-V^2/K) K looks to be a
free parameter giving the Galilée transformation if infinite and Lorentz
transformation if K
= c^2. Obviously these transformations are very close when V/c
<< 1. Parameter K can’t
be negative (it should be possible to travel backwards in time) and it square
root looks to be a maximum speed for V. This is confirmed
by the square root (1-V^2/K) and also by the speed combinaison equation
deducted from (12) and (13): (19)
V" = (V+V')/ [1 + (VV'/ K)] if we pose sqrt(K)
= k (19’)
(k-V")/(k+V") = [(k-V)/(k+V)].[(k-V')/(k+v')] therefore êV ê and êV' ê < k imply êV" ê < k. Poincaré and Lorentz obviously
selected K = c^2, in accordance with light speed invariance and conservation of
the Maxwell equations in inertial frames. Annexe 2 : Einstein method ( Zur
Elektrodynamik der bewegten Körper. Annalen der Physik.) Einstein first states the two ( independent and
non contradictory) postulates of the
Special Relativity. -
Relativity principle:
all the physical laws are the same in all inertial frames. -
Speed of light is the
same in all inertial frame and is equal to “c” Then, Einstein analyses the measurement methods
for deriving his equations. We use the original Einstein notations but
equations have been labeled He defines how to synchronise all the clocks of
a frame ( Poincaré method) with obvious notation ( Signal emitted in A at T0,
received and reflected at B at T1, received back in A at T2) T1 = 1/2(T0+T2).
(1) This measurement is performed in a frame (R1)
sliding at constant speed "v
" along "x" axis of frame R0 ( supposed here at rest, arbitrary
choice). Let’s try to find the value of « T1 » in frame R0 . For a master ruler of length L in R1, it is easy to show that in R0 (
coordinate "t" ), we have: t1-t0 = L/(V-v) et t2 - t1 = L/(V+v) with V = light speed.
(2) . Let introduce
x' = x-vt, which maps a point
at rest in R1 (coordinates in greek letters) to a fix point of R0 ( coordinates
in latin letters). Let’s write Equation (1) in R1 1/2(t0 + t2) = t1
(3) As t= t(x,y,z,t) , with the relevant values of the arguments we get (4) 1/2 [t (0,0,0,t) + t (0,0,0,{t
+x'/(V-v) + x'/(V+v)})] = t
(x',0,0,t+x'/(V-v))
(5) if
x' is supposed to be infinitesimal, we get: (1/2)dt (2,0)= dt (1) that is (1/2) (¶t/¶t) dt = (¶t/¶x'). dx' + (¶t/¶t) . dt
(6) Applying to (5) and plugging values of dt in
the two members of the equation dt =
dx' [( 1/(V-v) + 1 /(V+v)] on the left, and dt = dx' ( 1/(V-v)) on the right , we get: 1/2
( [1/(V-v)] + [1/(V+v)] )¶t/¶t = (¶t/¶x') +
[1/(V-v)]. ¶t/¶t that is : ¶t/¶x' +
[v/(V²-v²)] ¶t/¶t = 0
(7) Equation easy to solve whether we suppose that t is a linear function of t,x' t = a ( t - v.x’/(V²-v²))
(8) where a is a function of v, Now let’s replace x' by its value ( x-vt)
in (8) and et let’s simplify, we get t = j (v).b.( t - vx/V²)
(9) where j(v) is a function of v with b = (1- v²/V²)-1/2
(10) Let’s apply equation (9) first for + v and then
for -v ( we should the initial coordinates again ) to find : j (v). j (-v) = j² (v) = 1: j (v) = j (-v) according to the symmetry of the problem.
(11) therefore t = b.( t - vx/V²)
(12) Computing space coordinate x from t ( corresponding to à x) is an easy game, just apply the fact
the the speed of light is constant x = Vt ( we know t ). The other space coordinates are identical
in both frames, in our example. One can check constancy of speed light by x²+y²+z² = V²t² = V²t² =x²+h²+z² The following
part of the article is related to the general equations of Lorentz, the metric
and the Maxwell equation invariance under these transformations. |