1

Einstein’s Special Relativity (with quantum clocks) and Poincaré’s Special

Relativity (with classical clocks) in Einstein’s General Relativity

Yves Pierseaux

Faculté des Sciences physiques, Université Libre de Bruxelles , ypiersea@ulb.ac.be

Abstract

Einstein‟s Special Relativity (ESR) without ether (June 1905) is based on the principle of relativity and the principle

of invariance of the speed of light (definition of simultaneity). Poincaré‟s SR (PSR) with ether (June 1905) is based

on the principle of relativity and the principle of real contraction of lengths (Lorentz). All the usual formulas and

concepts of SR can be mathematically deduced, of course, from Einstein‟s two principles but also from Poincaré‟s

two principles (Pierseaux PIRT 1998). So the reciprocal contraction of rigid rods can be deduced with LT (Lorentz

transformations) from Einstein‟s definition of simultaneity whereas Poincaré‟s definition of simultaneity can be

deduced with LT from the real contraction of deformable rods. The main problem is situated in the physical

interpretation of relativistic formulas respectively in ESR and PSR. For instance, Poincaré‟s completely relativistic

ether involves a purely classical wavy representation of the light while Einstein‟s suppression of e ther involves a

corpuscular-wavy (quantum) representation of light (Einstein‟s photons).

We showed that the main border (classical-quantum) of the present physics passes between the two SR (PIRT 2000,

late papers). This is true for the classical representation of the electron in PSR and for the existence of the noncausal

zone (necessarily independent events or space-like events) in ESR. This is also true for the geometrical (Pierseaux

PIRT 2000) definition of units of measure within each inertial system respectively in ESR and PSR. Einstein‟s

identical units of time is unthinkable without the use of quantum identical spectra given by identical clocks-atoms

whereas Poincaré‟s duality true time –local time is thoroughly connected with classical astronomical clocks. The

fundamental difference between the two relativistic conceptions of the contraction of bodies, respectively in PSR and

ESR, is not situated in the reciprocity: Poincaré‟s real contraction is as reciprocal as Einstein‟s contraction. The

difference is rather that Poincaré‟s real contraction of deformable body is the effect of a real acceleration. If we

clearly separate the two SR, we obtain one SR with a real acceleration for the definition of units and one SR without

acceleration for the definition of units. Indeed Einstein‟s identical “rigid” rods don‟t undergo any effect from

Einstein‟s adiabatic “acceleration” which is geometrically in ESR a space-like 4-vector (1908, Minkowski‟s

orthonormalized representation of ESR, acceleration cannot be classically connected with the variation in

Minkowski‟s proper time of the velocity of the same body, PIRT 2000).

When Einstein succeeds of introducing in 1916 a relativistic interpretation of acceleration he gives up the rigid bodies

-as Poincaré - for the deformable bodies (he solves Ehrenfest‟s paradox) but - unlike Poincaré - he completely

reverses the situation. He put not only the accelerated systems on the basis of the theory (Einstein‟s General

Relativity, EGR) but also and above all, he identifies acceleration with gravitation (Einstein‟s principle of

equivalence acceleration-gravitation). Einstein‟s acceleration becomes therefore Einstein‟s curvature of space(-time)

and we find again, at the local limit of EGR, the flat space(-time) of ESR without acceleration for the definitions of

units. But are we sure that the SR locally valid in EGR is ESR? Poincaré‟s classical affine SR (PIRT 2000) with

deformable bodies is clearly the infinitesimal limit in Weyl‟s GR and Einstein‟s famous objections to Weyl (1917,

quantum spectral identity in order to define the units) seems to indicate that the local limit in EGR is ESR. However

Einstein reintroduces the ether in 1921 exactly in the words of Poincaré (“ether has no singular state of movement”).

More fundamentally “locally” means also “infinitesimally” which is precisely the main characteristic of PSR based on

the covariance by LT of differential equations of movement of a material point in independent true time (by contrast

with ESR based on the invariance of finite intervals of independents events).

The SR locally valid in EGR is the semi-classical mixture between ESR and PSR. So the clear separation of the

standard mixing SR into its two components (Einstein's SR and Poincaré's SR) may help to solve the delica te

problems which still persist at the interface between quantum theory and general relativity. This splitting of the

standard mixture SR into its two components “the affine classical PSR” and “the projective quantum ESR” is

therefore very important for an new physical approach (non-idealistic and without Platonist supplementary

dimensions of space) towards the quantization of the gravitation.

2

1 Principles of relativity and group structure of LT

2 Poincaré’s principles and Einstein’s principles of SR

3 Poincaré’s use and Einstein’s use of LT

4 Poincaré’s and Einstein’s conventions of synchronisation

4-1 Poincaré's approximate synchronisation (first order) and the duality true time-local time

4-2 Einstein's synchronisation and « the stationary time of a stationary system»

4-3 Poincaré's exact synchronisation (to second order) and implicit real dilation of time

5 Poincaré’s and Einstein’s definitions of units of space and time

5 Einstein’s preparation of identical isolated stationary systems without acceleration ( adiabatical

hypothesis)

7 Einstein’s atomic quantum clocks and the spectral identity of atoms

Conclusion: the standard mixed SR in Einstein’s General relativity

Appendix A Poincaré's convention of synchronisation-1906 ("Cours à la Sorbonne")

Appendix B: Reichenbach’s parameter and Poincaré's -LT

Appendix C: The spectral identity of atoms in Einstein's letter to Weyl (1918)

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1 Principles of relativity and group structure of LT

The first problem here to be considered is to show that there is not only two approaches of

SR but two genuine theories of SR.

An essential element of a theory of SR is naturally the formulation of a principle of

relativity. Lorentz‟s paper of 1904 “Electromagnetic phenomena in a system moving with any

velocity less than that of light”[3] is not based on a principle of relativity1. Contrary to Lorentz‟s

approach [1], Einstein‟s and Poincaré‟s works are based on a principle of relativity:

1Lorentz admitted [1] that his point of view was not relativistic and that Einstein and Poincaré had a relativist ic point

of view. The historical Poincaré‟s mistake (perhaps by excess of modesty) was certainly to have called his own

principle of relativity the “Lorentz‟s principle of relativity” (1911).

Henri Poincaré

“La Dynamique de l‟électron”

[9-9b] (5 June - 23 July 1905)

This impossibility of experimentally

demonstrating the absolute motion of the Earth

appears to be a general law of the Nature; it is

reasonable to assume existence of this law,

which we shall call the relativity postulate, and

to assume that it is universally valid. [9b,

introduction, 1905]

Albert Einstein

“Zur Electrodynamik bewegter Körper” [3] (27

June 1905)

1° The laws by which the states of physical

systems undergo change are not affected,

whether these changes of state

(Zustandänderungen) be referred to the one or

the other systems of two systems of co-ordinates

in uniform translatory motion [3, §2,1905]

Both 1905 works are almost simultaneous and largely independent. We insist on the fact

that, in Poincaré‟s work on SR (from 1900 to 1912) [6 to 12], the invariance of the speed of light

never appears to be a basic principle. So the question is: what is the second Poincaré‟s

principle (see 2)?

Another essential element for a theory of SR is of course Lorentz‟s transformations (LT, in

the following of this paper, I adopt the respective notations of both authors):

Poincaré’s LT

The essential idea of Lorentz consists in that the

equations of the electromagnetic field will not be

altered by a certain transformation (which I shall

further term the Lorentz transformation) of the

following form

x’ = k l (x - t), y’ = l y, z’= l z, t’ = k l (t - x)

where x, y, z, are the co-ordinates and t the time

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before the transformation, and x’, y’, z’, and t',

are the same after the transformation. [9b, §1,

1905]

Einstein’s LT

To any system of values x, y, z, t, which

completely defines the place and the time of an

event in the stationary system K, there belongs a

system of values , determining that

event relatively to the system k, and our task is

now to find the system of equations connecting

this quantities (…).

We obtain: x (x-vt), = y, = z,

= (t - vx/c2) [3, §3, 1905]

The transformations of co-ordinates of an event2 (I don‟t insist here) are deduced by

Einstein in the kinematical part of his article while they are induced by Poincaré from the

covariance of Maxwell-Lorentz‟s equations3.

A third crucial element for a theory of SR is the structure of group of LT. The physicists

credit often Einstein with the discovery of the relativistic law of composition of the speed and

Poincaré with the discovery of the structure of group of LT. But in fact the two elements are in

each approach4 (Einstein‟s § 5 and Poincaré‟s § 4).

The interesting point is not in these polemical questions of priority but in the question to

know if we must delete the ether because the LT form a group. Let us examine this question in

details in Poincaré‟s work.

Indeed in Lorentz‟s conception there are two systems and one of them, where the ether is at

rest in absolute space, is privileged. This is also the case in §1 of Poincaré‟s paper. But what

happens with the ether if there are three systems K, K‟, K”, connected by three LT of the same

form, when Poincaré establishes in his §4 the group structure ? :

It is noteworthy that the Lorentz transformations form a group. For, if we put:

x‟ = l k (x + t), y‟= l y, z‟ = l z, t‟ = k l (t + x)

2 We insist here neither on the formulation of the invariant laws, “les lois du milieu électromagnétique” for Poincaré

and “Die Gesetze diese Zustandsveränderungen” for Einstein, nor on the crucial role of Einstein‟s concept of event. 3 It is however not true that Poincaré‟s SR would be less general that Einstein‟s SR because Poincaré applies LT, e.g.

to the gravitation force.

5

and

x” =l’ k‟(x‟ + 't‟), y‟ = l’ y, z‟ = l’ z, t” =l’ k‟ (t‟+ 'x‟)

we find that

x” = l” k” (x + "t), y”= l” y, z” = l” z, t” = l” k” (t + "x)

with

"

[9b, §4, 1905]

The group structure is of course totally incompatible with the existence of a privileged

system. Indeed, if the three systems are connected (two by two)5 by the three LT, it is logically

impossible to maintain any absolute conception. One could however object that this property of

transitivity is demonstrated by the mathematician Poincaré on the general group of 2 parameters (l,

) where the parameter l has not any physical meaning. But the “mathematician” Poincaré writes at

the end of his §4 that his group with 2 parameters (l, ) must be reduced to a group with one

parameter because the only physical concept in question here is the concept of velocity :

For our purposes, however, we have to consider only certain of the transformations in this group. We must

regard l as being a function of , the function being chosen so that this partial group is itself a group. [9b,

§4]

Poincaré shows that for this subgroup with one parameter , we must have l () =1. He

insists, in the introduction, on the importance of his own demonstration that implies the purely

longitudinal nature of the Lorentz‟s contraction but also, and above all, that is a relative velocity.

The ether is not deleted by Poincaré but only a relative velocity with respect to it has a

physical meaning

The conception according to which “the absolute space physically exists but it is

impossible to measure an absolute speed with respect to it” is not a Poincaré‟s conception but a

Lorentz‟s conception. Lorentz‟s point of view is the starting point of Poincaré (§ 1) but not the

final point (§ 4).

4 Einstein considers only the structure of group in the case of parallel translations (of three systems). The spatial

rotations are completely absent in Einstein's 1905 paper. In the development of SR without ether, spatial rotations

were considered by Planck, Ehrenfest, Silberstein and Thomas (see 20-3). 5 This is clearly the difference with Lorentz‟s theory in which to pass from one material system for another he uses

the Galilean transformation (see Miller, [18])

6

Poincaré explains in “La relativité de l‟espace”(1907)”this fundamental issue of his 1905

work: “Whoever speaks of absolute space uses a word devoid of meaning” [10]. According to

Poincaré the concept of absolute space doesn't have any physical meaning but only a psychological

meaning

In the same text he dissociates clearly the concept of absolute space from ether's one (“I

mean this time not its absolute velocity, which has no sense, but is velocity in relation with the

ether”). But the splitting between concepts of absolute and relative velocity with respect to the

ether is not clearly developed in this text. On the other hand the concept of relative velocity with

respect to the ether is clearly defined, one year later, in “La dynamique de l‟électron”:

Whatever it can be, it is impossible to escape the impression that the principle of Relativity is a general law of nature,

and that we shall never succeed, by any imaginable method in demonstrating any but relative velocities. By this I

mean not only the velocities of bodies in relation to the ether but also the velocities of bodies in relation to each

other. [11, §6, 1908]

According to Poincaré material bodies and ether must be treated exactly of the same

manner with respect to the concept of relative velocity. In the same text, Poincaré explains that

ether can be regarded by definition in (absolute) rest:

It is not a question, of the velocity in relation to absolute space, but the velocity in relation to the ether,

which is regarded by definition, as being in absolute rest.”(in italics in the text).[10]

What can be the meaning of "being in absolute rest" by definition? In which system is the

ether at rest ? in K, K‟, K”? The answer is not that it is “hidden” but that Poincaré‟s ether doesn‟t

have any singular state of motion. In order to study the laws of physics in two different inertial

systems, Poincaré‟s ether can be chosen by definition, for each couple of inertial systems (KK‟,

KK”, K‟K”, at rest in one of the two frames but the other one is then in movement with respect to

the ether).

The difficulty to present Poincaré's SR is mainly logic because on the level of

mathematical physics the problem is (almost) completely solved by Poincaré in 1905.

There are two logical relativistic answers to the negative results of Michelson‟s experiment:

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The first consists of closely associating ether and absolute space in order to delete the two

concepts. This is Einstein‟s well known answer.

The second consists of radically dissociating ether and absolute space in order to delete

the latter and to transform the former into a relativistic ether. This is Poincaré's answer.

2 Poincaré’s principles and Einstein’s principles of SR

We showed (1) that in each theory there are: a principle of relativity and the structure of

group of LT (first part of Poincaré's 1905 work). But if we know Einstein‟s second principle, we

don‟t know yet Poincaré‟s second principle that is developed in the second part of Poincaré‟s 1905

work (§4-§9).

Let us firstly examine the historical situation. In 1904, in his conference on “the principles

of mathematical physics”, just after his first formulation of principle of relativity, Poincaré

underlines the necessity to admit other principles:

Unhappily, that does not suffice, and complementary hypotheses are necessary. It is necessary to admit

that bodies in motion undergo a uniform contraction in the sense of the motion. (my italics) [8, 1904]

If Poincaré put “hypotheses” at the plural, it is not because the hypotheses of uniform

contraction would not be sufficient but because, as early as 1900 [7], Poincaré was looking for a

dynamical force exerted by the ether on the bodies, in order to justify Lorentz‟s hypothesis (LH) of

contraction and to reconcile Lorentz‟s theory with the principle of reaction. In his fundamental

1905 work, he determines this force:

But in the Lorentz hypothesis [LH], also, the agreement between the formulas does not occur just by itself;

it is obtained together with a possible explanation of the compression of the electron under the assumption

that the deformed and compressed electron is subject to constant external pressure, the work done by

which is proportional to the variation of volume of this electron. (my italics) [9b]

The first part of Poincaré‟s 1905 work consists of showing that LT form a structure of

group (§ 4) and the second part (§5, 6, 7) that a complementary force6 must be introduced not only

in order to balance the electrostatic repelling force (as it is generally admitted) 7 but also, and above

6 Poincaré obtains (§6 of the paper) the fundamental equation of relativistic dynamics (by using a principle of least

action taking in account “Poincaré‟s pressure”). 7 The relativistic mechanics of continuous medium is the starting point of Poincaré‟s SR and the final point of

Einstein‟s SR. Laue in particular rediscovers in 1911 (and Fermi ten years later) Poincaré‟s pressure but in a purely

8

all, in order to justify LH (real contraction of the electron). The deformed and compressed electron

is subject to constant external pressure of the relativistic and deformable8 ether, according to the

principle of reaction.

The situation is very odd because Einstein‟s SR denies the existence of an ether while

Poincaré‟ SR proves the existence of a relativistic and deformable ether. There is a real antinomy

in the sense of Kant. We want to transform this philosophical antinomy into a physical opposition

between the two SR, with and without ether.

We want therefore clearly separate our analysis from this one that consists to say that

absolute ether is “hidden” by Poincaré. Poincaré‟s relativistic ether is not a ghost artificially

introduced in Einstein‟s axiomatic. It exerts an (non-electromagnetic) pressure on the electron not

only in order to balance the electrostatic repulsion but also to contract the deformable electron.

According to Poincaré, the principle of relativity and the principle of real contraction are

dynamically complementary9. However Poincaré 1905 work is not presented on an axiomatic basis

as Einstein‟s kinematics. We shall show that there exists, underlying Poincaré‟s relativistic

dynamic, an implicit kinematics that is based on the compatibility between the principle of

relativity and the principle of real contraction (LH). According to our hypothesis of the existence of

a “fine structure” of SR, we must develop the respective logic of two great spirits, Poincaré and

Einstein. So, in respecting the spirit of Poincaré‟s text and Einstein‟s text, the opposition rigid-

deformable doesn‟t have to take to the letter but must be understood in that way:

1) There is an underlying kinematics “of Poincaré‟s deformable rods”, based, like Einstein‟s

kinematics, on “fundamental principles”.

2) The important concept in Einstein‟s kinematics is not the rigidity of rods but the identity of the

rods within different inertial frames.

Let us develop firstly this second point. From a superficial analysis, one might conclude

that Poincaré's presentation is more coherent than Einstein's one because it is well known that

classical rigidity (instantaneous action-at-a-distance, “à la Descartes”) is incompatible with SR

static sense (more exactly: electrostatic meaning) while, in Poincaré‟s text, this pressure has an explicit dynamical

sense (with an implicit kinematics sense we develop here). Poincaré's force “works”. The work of the supplementary

force, in order to put the rest electron in movement, is proportional to the variation of the volume of the elect ron.

There is no contradiction with the principle of inertia because the moving electron is also in static equilibrium but its

form is a flattened ellipsoid. According to the principle of reaction (reaction of inertia), Poincaré obtains also a non -

electromagnetic part in the mass of the electron. 8 The non relativistic ether is of course perfectly rigid. Poincaré‟s relativistic ether, directly comes from the

covariance of all the Maxwell-Lorentz equations (transversal waves), is deformable. The non relativistic notion of

rigidity is irrelevant as well for Poincaré‟s ether as for Einstein‟s rods.

9

(Einstein‟s SR and Poincaré‟s SR). But the important concept, according the spirit of the text, is

not the rigidity but the identity:

Let there be given a stationary rigid rod; and let its length be L as measured by a measuring-rod which is

also stationary. (…)

In accordance with the principle of relativity (…) « the length of the rod in the moving system » - must be

equal to « the length L of the stationary rod. » (...)

The length to be discovered [by LT] we will call « the length of the (moving) rod in the stationary

system». This we shall determine on the basis of our two principles, and we shall find that it differs from

L. (My italics but Einstein‟s quotes, 3, §2,1905).

Max Born, who was a specialist of rigidity in Einstein‟s special relativity [14-2], wrote in

1921 in his book on relativity that Einstein introduces a tacit assumption:

A fixed rod that is at rest in the system S and is of length 1 cm, will, of course, also have the length 1 cm,

when it is at rest in the system S‟, provided that the remaining physical conditions are the same in S‟ as in

S. Exactly the same would be postulated of the clocks. We may call this tacit assumption of Einstein‟s

theory the “principle of the physical identity of the units of measure”. [14-1, my italics, p252]

Einstein's principle of the physical identity of the units of measure is not a third

hypothesis because Einstein deduces directly the identity of his (rigid)10

rods from his relativity

principle. The important thing, in the spirit of the young Einstein‟s text, is the postulate of the

existence in Nature of processes giving units of length and time.

Each SR is based on its own system of axioms11

:

9 According to A. Pais [16], there would exist a “third” hypothesis, in Poincaré's texts, that would prove that Poincaré

has not understood the SR. Pais didn‟t try to penetrate the logic of Poincaré‟s SR. 10 The problem doesn't consist only in the incompatibility of prerelativistic rigidity with the existence of a limited

speed of interaction (this is obvious for both SR). The genuine problem of rigidity in Einstein's SR is more serious

and is discussed in the papers of Ehrenfest [15ter], von Laue [24bis] and Born [14-2] between 1909 and 1911 (not

translated in English). Born's concept of proper length of rigid rod (1910) is similar to Minkowski's concept of proper

time of clock (1908) [20-3]. Born's concept of infinitesimal rigidity and Einstein's concept of identity of units are

deeply connected (Einstein's letter to Weyl, 5bis). 11 What we call here Einstein's second principle is here already a result of Einstein's synthesis between his

first principle and his initial formulation of the second principle. Einstein's synthesis implies the deletion of the ether

because the speed of light is not only independent of the speed of the source relative to the ether but also of the speed

of a moving system relative to the ether.

Poincaré’s principles of SR (implicit kinematics of deformable rods)

1 principle of relativity (compensation)

10

2 principle of real contraction (lengths and

units, LH).

Einstein’s principles of SR

(explicit kinematics of rigid rods)

1 principle of relativity (identity)

2 principle of the invariance of the (one way)

speed of light

3 Poincaré’s use and Einstein’s use of LT

In the Lorentz point of view, it is well known that the null-results of the Michelson’s

experiment is explained by a real contraction of lengths.

We don’t want to discuss here the origin of length contraction or more exactly “the

hypothesis of the FitGerald-Lorentz deformation” which is a rather difficult problem. It is often

claimed that this original contraction is purely longitudinal and that it is an “ad hoc” conjecture.

Lorentz's demonstration l=1 for the scale factor was however very debatable and the nature of the

Lorentz's conjecture has deep physical foundations in the atomic structure in Lorentz’s theory [15-

3]. We want here to concentrate the attention on Poincaré representation of LH.

Firstly, if in 1900 Poincaré underlines the “ad hoc” nature of LH it is only because the

local time and the real contraction are independent hypothesis in Lorentz’s theory.

Secondly, if Poincaré insists on his own contribution (the demonstration l () =1) it is

clearly because he considers a purely longitudinal Lorentz‟s contraction [21] in order to show its

compatibility with the principle of relativity [20-2]:

So Lorentz hypothesis [LH] is the only one that is compatible with the impossibility of demonstrating the

absolute motion [RP]; if we admit this impossibility, we must admit that moving electrons are contracted

in such a way to become revolution ellipsoids whose two axis remain constant. 9b, §7

According to Poincaré, LH is the only one compatible with RP (unlike e.g., Langevin's and

Abraham's hypothesis). This last sentence is particularly interesting because LH is not a

consequence of principle of relativity but it is an independent hypothesis. Poincaré admits (like

Lorentz) that the contraction is real but (unlike Lorentz) he raises LH to a status of a postulate.

In the same way that Einstein’s two principles are compatible, we must understand why

Poincaré’s two principles are compatible. Poincaré writes, in his § 6 of his 1905 work, on LH:

11

In accordance with LH, moving electrons are deformed in such a manner that the real electron becomes an

ellipsoid, while the ideal electron at rest is always a sphere of radius r (…) The LT replaces thus a moving

real electron by a motionless ideal electron. [9b, §6]

Poincaré’s compatibility LH and RP implies specifically use of LT. In order to illustrate

this, we can use Tonnelat’s diagram [23] (fig 1, we adopt Poincaré’s and Einstein’s respective

notations in the following, see former respective quotations about LT):

In fig 1, in Einstein’s standard SR, the contraction of the moving rod -1L is not real

(dashed line) but is the reciprocal result of a comparison of measurement made on identical rods L

(continuous lines) from one system to the other with the well known use of LT (dashed lines).

In fig2, in Poincaré‟s SR the contraction of the moving rod k-1L is in principle (LH) real

(continuous lines) in K‟. By the use of LT the length of the rod in K‟ (for observers in K‟) seems

be equal to L (dashed lines). Reciprocally, we can of course reverse the role of K and K‟ (where

ether is now chosen at rest) and reverse the continuous lines and dashed lines. Contrary to

Lorentz‟s point of view, the contraction is reciprocal because of course we can always choose the

system in which the ether is at rest.

The calculation with the LT is also very easy. Suppose the ether is chosen by definition at

rest in K. The real length of the rod placed in the moving system K’ is thus k-1 L. The first LT x’ =

k (x – t) “replaces” (in Poincaré’s words) in any time t (see below) the length moving real rod k-

1L by a motionless rod L.

L in K

L in k L in K

k-1L in K‟

Fig 1

The dashed lines, in Einstein‟s words: “The

length to be discovered [with LT]“the length of

the (moving) rod in the stationary system”

Fig 2

The dashed lines in Poincaré‟s words: “LT

replaces thus a moving real electron [rod] by

a motionless ideal electron [rod]”

12

Poincaré’s principles are compatible12

but the historical difference with respect to Einstein

is that Poincaré has never developed explicitly his way of using of LT on a basic example (a

deformable rod). According to Poincaré’s implicit kinematics, the real differences are compensated

by a “good use” of LT. According to Einstein‟s explicit kinematics, the identical processes seem to

be different by another “good use” of LT.

The principle of contraction and the principle of inertia (and therefore relativity) seem in

the first sight incompatible but we see now that there is no problem in Poincaré's kinematics

because the LT rules the connection between the two inertial or Galilean systems K and K‟. The

contraction of the deformable (and non-charged) rods is purely kinematics exactly in the same way

that in Einstein's logic the reciprocal contraction of identical rods is purely kinematics. In

Einstein‟s logic there is waves without (acceleration in) ether and in Poincaré‟s logic there is a

reaction of the ether without force (Poincaré‟s pression applies only for charged electron). So we

can say that in Poincaré‟s logic the principle of reaction has a (local) kinematics meaning

(Poincaré‟s force of contraction doesn‟t modify the velocity of the rod).

This is a very important result: the relativistic Poincaré‟s ether supposes a relativistic

conception of the principle of reaction.

Another fundamental question is the definition of simultaneity in Poincaré‟s kinematics.

In Einstein’s use of LT the reciprocal contraction of rigid rods are deduced from the

definition of simultaneity. In Poincaré’s use of LT the simultaneity is deduced from the real

contraction of deformable rods.

Indeed the real length of a moving rod (velocity ) relative to K is k-1 L. Suppose now two

observers, 1 and 2, fixed in K are separated by the distance x = k-1

L: The observer 2 notes the

time t2 when the head of the rod is in x2 and the observer 1 notes the time t1 when the tail of the rod

is in x1. According to Poincaré’s use of LT, we have x’1 = k (x1 – t1) and x’2 = k (x2 – t2) so we

have x’ = k k-1

L – k (t2 – t1). The length of the rod x’ is L (see above) and so we have k (t2

- t1) = 0 and then t2 = t1.

This is a very important result: In Poincaré’s relativistic kinematics the simultaneity of

two “events” is deduced from the use of the first LT on a moving rod really contracted. The

simultaneity is therefore completely relative in Poincaré‟s kinematics. In others words, without the

real contraction, the simultaneity would be absolute.

12 Poincaré doesn't demonstrate only the compatibility between electromagnetic Lorentz theory and the principle of

relativity but also the incompatibility of the other electromagnetic with the principle of relativity.

13

To sum up, we have:

Einstein identity first LT contraction of length

(dilation of time)

simultaneity contraction of length

Poincaré contraction first LT compensation

contraction of length simultaneity

It is easy to see that the difficult problem in Poincaré's SR is to deduce from the real

contraction the dilation of time (see 4-3). Until now we have only use the first LT. What happens

about the fourth LT?

4 Poincaré’s and Einstein’s conventions of synchronisation

If it is true that Poincaré‟s second principle has the same status as Einstein‟s second

principle, we must show that Poincaré‟s conception of time can be deduced from his second

principle of real contraction exactly in the same manner that Einstein‟s conception of time is

deduced from his second principle of invariance of the light velocity.

In 1911 Poincaré was aware that the German relativistic school (Einstein, Planck, Laue,

Sommerfeld, Minkowksi…) has adopted another convention for the SR (“une autre convention”):

Today some physicists want to adopt a new convention. This is not that they have to do it; they consider

that this convention is easier, that‟s all; and those who have another opinion may legitimately keep the old

assumption in order not to disturb their old habits. [12, my italics]

Poincaré and Einstein use the same method of distant clock synchronisation (exchange -

forth and back - of signals of light) but we shall show that Poincaré‟s convention is not the same as

Einstein‟s convention.

The synchronisation method by exchange of signals of light is developed by Poincaré in

1900 in a paper [7, Appendix A] on the reaction principle in the Lorentz theory. Poincaré explains

that when Lorentz‟s local time t‟ = x + vx/c2 is used in the system K‟ moving with respect to the

ether, the observers remark no difference (to first order) between the forth travel time and the back

travel time of the light. For the second order Poincaré envisages already in 1900 that the hypothesis

of Lorentz is necessary [Appendix A].

14

Once again the purely mathematical Lorentz conception of local time (t‟ = t + x) is the

starting point for Poincaré13

(1900) but not the final point. Poincaré's aims are to find not only a

physical interpretation (synchronisation of clocks) of Lorentz's local time but also a harmonious

connection between the two independent Lorentz hypotheses (real contraction and local time)

(1901).

This is Poincaré‟s “tour de force” to have shown that the principle of real contraction

implies a (real) dilation of time and therefore for the local time the expression: t‟ = k (t + x). But it

is also his weakness because his complete historical demonstration in his lectures in La Sorbonne,

based on lengthened light ellipsoids, is complicated. Let us examine in detail Poincaré's analysis

(see Appendix A).

4-1 Poincaré's approximate 1900 synchronisation (first order) and the duality true time-local

time

Let us examine first Poincaré's physical interpretation of Lorentz local time (to first order).

In his talk on “The principles of the mathematical physics”, Poincaré explains (we break down his

argumentation into two parts):

1- (Two observers are at rest relative to ether, System K)

The most ingenious idea has been that of local time. Imagine two observers [A and B] who wish to adjust

their watches by optical signals; they exchange signals, (…) And in fact, they [The clocks of A and B]

mark the same hour at the same physical instant, but on one condition, namely, that the stations are fixed.

2- (Two observers are moving relative to ether, System K’)

In the contrary case the duration of the transmission will not be the same in the two senses, since the

station A, for example, moves forward to meet the optical perturbation emanating from B, while the

station B flies away before the perturbation emanating from A.

The watches adjusted in that manner do not mark, therefore the true time; they mark the local time, so

that one of them goes slow on the other (de telle manière que l‟une retarde sur l‟autre). It matters little,

since we have no means of perceiving it. So, as the principle of relativity would have it, he will have no

means of knowing whether he is at rest or in absolute motion. Unhappily, that does not suffice, and

complementary hypotheses are necessary. It is necessary to admit that bodies in motion undergo a

uniform contraction in the sense of the motion. [my italics, 7]

Poincaré‟s synchronisation (to first order and the second order) is clearly based for his

second system K‟ on the duality of the true time t and the local time t’. According to Poincaré, in

order to obtain the exact compensation (second order) for the synchronisation, we must add LH

13 Lorentz’s local time is a necessary mathematical change of variable (t is the universel time) in order to obtain the

same wave equation in the absolute ether and in the moving system [1]).

15

(see below and Appendix A). Poincaré develops explicitly his synchronisation to the first order (t' =

t + x, p218) and to the second order (t' = k (t + x), lengthened light ellipsoïds, p219-220) in his

1906 “cours à la Sorbonne “[9c, see Appendix A].

4-2 Einstein's 1905 synchronisation and « the stationary time of a stationary system»

We haven‟t yet answered the fundamental question: why are both conventions of

synchronisation (Poincaré and Einstein) deeply different?

Indeed Poincaré adopts in his system K, where the ether is chosen at rest, exactly the same

convention (assumption) as Einstein's one in his stationary system K. Let us now examine in detail

Einstein's convention.

1- (Einstein’s «stationary time of a stationary system K»)

The young Einstein writes in the famous §1 of his 1905 paper:

But it is not possible without further assumption to compare, in respect to time, an event at A with an

event at B. We have so far defined only an “A time” and a “B time”. We have not defined a common

“time” for A and B, for the latter cannot be defined at all unless to establish by definition that the time it

required by light to travel from A to B equals the time it requires to travel from B to A.

Let a ray light start at the “A time tA” from A towards B, let it at the “B time” tB be reflected at B in the

direction of A, and arrive again at A at the A time t‟A. In accordance with definition the two clocks

synchronize if

tB - tA = t‟A - tB.

(…) It is essential to have time defined by means of stationary clocks in stationary system (…) 3, my

italics, §1, 1905]

Both authors speak about a convention (an assumption). It is reasonable to think that

Poincaré knew what he said when he insisted in 1911 (see above) on the fact that the conventions

were not the same. Indeed what happens in Einstein‟s second system k?

2- (Einstein’s «stationary time of a stationary system k »)

The repetition of the concept stationary is essential because in his §3, Einstein notices [10]

about his second system k ():

16

To do this14 [deduce LT] we have to express in equations that is nothing else than the set of data of

clocks at rest in system k, which have been synchronized according to the rule given in paragraph 1.3,

§3, 1905]

Einstein‟s synchronisation of identical clocks within his second system k is exactly the

same as Einstein‟s synchronisation within his first system because the speed of light is of course

exactly the same.

Einstein‟s radical elimination of ether implies that his two systems, K(t) and k(), are

prepared in internal identical states of synchronisation. This is the reason why the duality true time-

local time have no sense in Einstein‟s logic. Poincaré‟s relativistic ether is always at rest in a given

inertial frame (his relativistic ether doesn't have a particular state of movement), but the other one

is then moving relative to the ether.

It is crucial to notice that former Einstein's convention of synchronisation is not only to

first order as Poincaré's one (see Appendix A). Indeed Einstein admits the principle of identity of

the rods while Poincaré admits the real contraction of the rod in order to have the exact

synchronisation of clocks (cf. 3).

Einstein's convention seems "easier" than Poincaré's one but in fact this assessment is very

subjective. If we consider that the real contraction is a "supplementary" hypotheses for the

synchronisation of clocks, we must therefore also consider that the identity of clocks is also

"supplementary" hypotheses because Einstein's concept of identity is based on a quantum

conception of the clocks (see §6).

4-3 Poincaré's 1906 exact synchronisation (to second order) and implicit real dilation of time

The subtle difference between the two conventions is situated in Einstein‟s preparation of

stationary systems where the k-clocks (like K-clocks) are synchronised which each other15

-

without the use of local time and LH – provides a truly identical rhythm or rate of the clocks. We

14 Einstein admits a priori the same relation of synchronisation within the two systems in order to deduce LT of course

without the introduction of a contracted rod in the moving system k [3, §3]. 15 In Poincaré‟s relativistic logic, the clocks in K‟ are adjusted (t' = t + x, to first order) in such a way that the real

difference of duration in true time is (exactly with LH) compensated afterwards by using the local time. Poincaré

notices in 1904 (we repeat the former quotation): "The watches adjusted in that manner do not mark, therefore the

true time; they mark the local time, so that one of them goes slow on the other (de telle manière que l‟une retarde sur

l‟autre). It matters little, since we have no means of perceiving it." [7] English translation (“goes slow”) is perhaps

ambiguous: Poincaré means that K‟ clocks are not put in the same origin of time. There are adjusted ( t' = t- x, to

first order) with respect to the true time but they are not adjusted a priori with each other.

17

can now better understand why Poincaré never speaks of duration given by identical clocks or a

fortiori never speaks of dilation of such a duration.

Let us show now [20, p154] that there exists an implicit Poincaré’s dilation of duration

in Poincaré‟s lengthened (elongated) light ellipsoids (1906 & 1908). We shall give, in this

paragraph, extracts of the English translation of Poincaré's 1908 paper and in the Appendix A the

completed development of Poincaré‟s 1906 “Cours à la Sorbonne”.

Exactly in the same way as in his 1904 talk in Saint Louis, Poincaré writes in his 1908

paper that the approximate compensation neglects the square of aberration (2):

It follows from this that the compensation is easy to explain so long as we neglect the square of

aberration ...[11, principle of relativity]

Poincaré‟s reasoning about lengthened light ellipsoids (to second order) is very subtle

because the spherical waves of the a moving source appear for the observers at rest relative to the

source as ellipsoidal waves (real contraction of units of length):

A body that is spherical when in repose will thus assume the form of a flattened ellipsoid of revolution

when it is in motion. But the observer will always believe it to be spherical, because he had himself

undergoes an analogous deformation, as well as all the objects that serve him as points of reference. On

the contrary, the surfaces of the waves of light, which have remained exactly spherical, will appear to him

as elongated ellipsoids.

The null results of Michelson's experiment implies that the round trip speed of light is

invariant. This is precisely Poincaré's 1906-1908 interpretation16

of the invariance of the quadratic

form (x2+ y2 +

z2

- t2 =

x'2+ y'2

+

z'2 - t'2). If Poincaré poses “c =1” in the Maxwell-Lorentz

equations and in the LT it doesn't mean that Poincare raises -like Einstein- the famous constant in

Maxwell equation to a status of a principle (Einstein‟s second principle). In Poincaré's logic c is

only the constant in the Maxwell equation, which are invariant in virtue of the first principle.

According to Poincaré "we can choose the units of time and space in such a way that c equals one".

But the metre (unit of length) are really contracted (Poincaré‟s second principle) and the spherical

wave surfaces become elongated ellipsoids:

16 This is the only essential development after Poincaré's fundamental 1905 work where the invariance of

the quadratic form is not connected with the propagation of the light.

18

What will happen then? Imagine an observer and a source involved together in the transposition. The

wave surfaces emanating for the source will be spheres, having as centre the successive positions of the

source. The distance of this centre from the present position (la position actuelle) F of the source will be

proportional to the time elapsed since the emission - that is to say, to the radius of the sphere. All these

spheres are accordingly homothetic one to the other, in the relation to the present position (la position

actuelle) of the source. But for our observer, on account of the contraction, all these spheres appears will

appear as elongated ellipsoids, and all these ellipsoids will still be homothetic in relation to the point F.

This time the compensation is exact, and this is explained by Michelson's experiments [my italics and my

underlining, 11, principle of relativity].

The contraction (k-1

L) of the lengths or the elongation (kR, R is the radius of the light

sphere) of the light waves are real by principle for the observers of K.

But now what is the meaning, according to Poincaré, of the elongation of the ellipsoid?

("The radius is proportional to the time elapsed since the emission"). The elongation of the ellipsoid

(kR) is directly translated by Poincaré as a dilation of time (kT). We give the complete calculation

in the Appendix A. We propose to call the real dilation "Poincaré's real dilation of time" and the

expression that results immediately for the local time, t' = k (t +x), "Poincaré's local time" or

perhaps better "Poincaré's apparent time" (see Appendix A), not in the sense of illusion, but in the

sense that the compensation is perfect for the observers of K'. The factor k in Poincaré‟s expression

of local time, t' = k (t + x), is therefore a consequence of Poincaré‟s second principle LH [20,

p154].

This is another "tour de force" of Poincaré, with respect to Lorentz, to have interpreted the

null result of Michelson's experiment directly with the time and the round trip invariance of the

speed of light:

This hypothesis of Lorentz and FitzGerald will appear most extraordinary at first sight. All that can be

said in its favour is that it is merely the immediate interpretation of Michelson's experimental result if we

define distances by the time taken by light to traverse them (in italic in the text).[11, principle of

relativity]

What is the time taken by light to traverse the distance? Is it the same concept as Einstein's

concept of one-way speed of light? Not only the context but also the explicit calculation (see

Appendix A) shows that the round trip speed of light is the basis concept in Poincaré's SR. In

Poincaré's explicit calculation there is not only a source A but the two usual stations A and B

(fixed in K'). This is the reason why Poincaré speaks, in the previous quotation (p17) about "the

time elapsed since the emission". Poincaré's evaluates the mean time forth-back in the exchange of

19

light signals between two stations A and B and Poincaré's dilation is of course a dilation of this

mean time.

For those who are sceptical by principle we propose a verification of the compatibility of

real contraction and real dilation with LT (Appendix B). But the best prove of the coherence of

Poincaré's real dilation is that Poincaré's formula of lengthened ellipsoid are mathematically

exactly the same as Einstein's relativistic formula of Doppler effect (see 20 and Appendix A).

In summary: the local time t‟ is defined by the fourth LT as a function of the independent

true time t (we can reverse the roles of course). Independent doesn‟t mean absolute – as in

Lorentz‟s theory - but only that t is a parameter. Poincaré‟s temporality is as physical (i.e.

compatible with experiments that measure the dilation of time) as Einstein‟s one but it belongs

clearly to classical physics (“La Mecanique Nouvelle”, with his relativistic kinematics and

relativistic dynamics).

Einstein identity first LT contraction of length

fourth LT dilation of time

Poincaré real contraction first LT compensation

real dilation fourth LT compensation

elongated ellipsoid in true time t sphere for K' in local time t‟=k (t- x)

5 Poincaré’s and Einstein’s definitions of units of space and time

So let us return to our main problem: the definition of units. What is finally the

quintessence of Poincaré's light lengthened ellipsoids? There exists a fundamental relation between

the speed of light, the unit of time and the unit of length. Poincaré's covariance of the speed of light

implies that the variation of unit of time is inversely proportional to the unit of length.

x t = k-1

x k t = x' t'

Poincaré‟s definition of units in relationship with his conception

of the covariance of the speed of light

20

This is consistent with Poincaré's homothetic choice of units because, with the use of

fourth LT, we have also x'/t'=1 in the primed system. So, thanks to the use of LT we have the

same, primed or not primed, units:

I shall choose the units of length and of time in such a way that the velocity of light is equal to the unity (c=1).

It is well known that we have only spherical waves in Einstein's SR:

At the time t 0, when the origin of the two coordinates is common to the two systems, let a spherical

wave be emitted therefrom, and be propagated -with the velocity c in system K. If x, y, z be a point just

attained by this wave, then

x2 +y2 +z2 = c2t2

Transforming this equation of our equations of transformation we obtain after a simple calculation

c

The wave under consideration is therefore no less a spherical wave with velocity of propagation c when

view in the moving system. This shows that our two fundamental principles are compatible.[5, §3]

Einstein doesn't say in which system, K or k, the source is at rest. The speed of light

(without ether!) depends neither on the speed the source (at rest in K or in k) nor on the speed of

the system (respectively k and K).

The relation x2 - c

2t2 =

- c

= 0 can be understood as an equality (from one system to

the other) but also, and above all as an identity (within each system we have an identical process

that give identical units of measure). Einstein's invariance of the speed of light implies that the ratio

of unit of length and unit of time is a priori identical c = x/ t = within each system.

x2 - c

2t

2 =

- c

= 0

Einstein's definition of units in relationship with his conception of invariance

of the speed of light

In other words Poincaré’s fundamental relativistic invariant (i.e. for the definitions of

units) is the four volume and Einstein-Minkowski’s fundamental invariant is the interval between

two events (see Pierseaux, PIRT 2000, 20-3). We have the freedom of choice between Poincaré’s

and Einstein’s definitions of units.

21

5 Einstein’s preparation of identical isolated stationary systems without acceleration

(adiabatical hypothesis)

There is another way to prove the existence of a “structure fine” of SR. Indeed, the young

Einstein distinguishes, in original 1905 and 190717

articles two stages in his preparation of inertial

frames in his deduction of LT:

first stage. The preparation of the two systems in state of rest:

“Let us consider K et k two equivalent systems of reference; we may say that the systems have measuring-

rod of same length and clocks giving the same indications, the comparison between this objects being

made when they are in state of relative rest (im Zustande relative Ruhe miteinander) ” [4, §1-1907]

Second stage18

: the “launching of the boost”:

“Now let a constant velocity v (Es werde nun dem Anfangspunkte … erteilt) to the origin of one of two

systems (k)” [3, §1-1905]

The young specialist of statistical thermodynamics Einstein formulates explicitly in 1907

the hypothesis (I discuss Einstein‟s adiabatical hypothesis19

in my book, 20) that his identical

clocks and his “rigid” rods are not modified by the passage from the velocity 0 to the velocity v.

Many authors think that the problem of rigidity in Einstein‟s 1905 text can only be solved

in general relativity (GR) because the concept of rigidity is deeply connected with Euclidean

geometry.

We propose another direction of research. In this paragraph Einstein introduces his active

interpretation of the LT (the boost). Thanks to the adiabatical hypothesis the active and the passive

interpretation are perfectly coherent in Einstein‟s logic. After Einstein‟s original preparation of his

17 Einstein‟s formulation of 1905 is of course the same: “Let each system be provided with a rigid-measuring-rod and

a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects

alike” 18 There is of course a well known third stage in Einstein‟s logic(his paragraph 4): when the system k is in moving

relative to the system K, it is impossible to compare directly the rods (or the clocks) in rest in k with the rods (and the

clocks) in rest in K. We must use LT to rely the two systems. The dilation of time as contraction of length is not real

but is the (reciprocal) result of a comparison of measurement made from one system to the other. 19 Van der Waerden reminds the crucial importance of the adiabatical hypothesis formulated explicitly by Ehrenfest

in the development of quantum theory. Two important heuristic principles have guided quantum physicists during the

period 1913-1925. Ehrenfest‟s adiabatic hypothesis and Bohr‟s principle of correspondance. The adiabatic

hypothesis, first formulated by Ehrenfest in 1913 (“A theorem of Boltzmann and its connection with the theory of

quanta”): if a system be affected in a reversible adiabatic way, allowed motions are transformed into allowed mot ions.

The name adiabatic hypothesis is due to Einstein as Ehrenfest states in his paper. Finally, Ehrenfest shows that the

22

isolated stationary frames, the internal state of the second system remains identical when it returns

to another non-accelerated state.

Poincaré‟s temporality and Einstein‟s temporality are both relativistic but the time is not a

parameter in Einstein‟s SR. In Einstein‟s own words, time is a “set of data of identical clocks

(Gleichbesshaffene Uhren)” that beat time with identical rhythm. A finite time is necessary to

synchronise the two distant clocks when they are at rest in both systems (Einstein's first stage).

With identical rods and with the (one-way) speed of light c numerically identical within each

inertial system, we have of course the same internal duration. Einstein‟s concept of “synchronous

clocks” implies that the repetition of the processes of synchronisation (sometimes called Einstein‟s

ideal light clock, for Einstein's real atomic clock, see §6) provides an identical rhythm (unit of

time) for the clocks: Einstein’s synchronisation and Einstein’s identity of units are exactly the

same concept.

We can introduce a genuine quantum unit of time (an identical unit within both frames) in

Einstein‟s logic. This is a quantum solution of the problem of rigidity in Einstein‟s SR (without

GR). The idea of use light signals to define distances in Einstein‟s SR (Bondi H., the factor k) is

necessary but not sufficient. We will show now that we must absolutely introduce a quantum of

time (identical unit within the two systems).

7 Einstein’s atomic quantum clocks and the spectral identity of atoms

The young Einstein identifies explicitly “clock and atom” in his second fundamental

synthesis on SR in 1907:

Since the oscillatory process that corresponds to a spectral line is to be considered as a intra-atomic

process, whose frequency is determined by the ion alone, we can consider such an ion as a clock of

definite frequency 0; this frequency is given, for example, by the light emitted by identically constituted

ions at rest with respect to the observer. [4, §3].

The young specialist of statistical thermodynamics is frighteningly clear-sighted. His

intuition implies not only that the atoms (“producers of spectral lines” in Einstein‟s own terms) of

a same nature are identical but also that this identity permit us to known the frequency in his own

adiabatic hypothesis is closely connected with the second law of thermodynamics. I showed in my book the crucial

importance of the invariance of entropy in Einstein‟s SR.

23

system (after Minkowski, we say proper frequency) if the atom (the clock) is moving relative to us

(the observers) .

There exists therefore in Nature physical process giving identical units of time within each

inertial system. In the framework of classical mechanics, the spectral identity of atoms is not

comprehensible. Weiskopf underlines this essential point – often forgotten – of the quantum

conception:

The main idea of quantum theory, I said, there is idea of identity. (…) Understanding the idea of identity,

there is the understanding the concept of quantum state established by Bohr in the first period of his

scientific activity. 24-1

Within the framework of prequantum concepts two objects could not be identical in every respect since, in

principle prequantum physics requires an infinite set of indications for the full description of an object. It

could always differ in some very small detail. The orbit of an electron around the nucleus differs by some

amount. Indeed it would be extremely improbable to find two atoms with exactly the same electrons orbit.

Therefore a new conceptual framework was needed in which the state of a system is fully define in all his

qualities by a finite set of indicators. This new framework was quantum mechanics and its leading concept

is quantum state.24-2

In order to have his identical units of measure, the young Einstein requires not only the

classical concept of identity (in the sense of orbital identity in classical statistical mechanic or in

the sense of minimum scale of classical chemistry) but the quantum concept of spectral identity.

The identical processes must be associated to identical units of time (or a frequency of course).

Our argument is not only an argument based on historical foundations. Indeed, as the unit

of time is identified by Einstein () to the (inverse of) frequency (of the spectrum), we

absolutely need a purely monochromatic wave of a determined frequency. Einstein‟s intuition of

identity () can only be justified in the framework of quantum theory because the emission of

monochromatic radiation is closely connected to the concept of quantum state of the atom (E = h

)20

.

We notice that Einstein uses the same argument (existence of spectral lines) about a (rigid)

element of length of a rod in his letter to H. Weyl in 1918 (see concl. and 15bis).

20 Moreover the most radical conception that identifies thermodynamically a monochromatic radiation with a set of

independent quanta of light is of course his own conception: "From this we further conclude that monochromatic

radiation of low density behaves thermodynamically as if it consisted of mutually independent energy quanta" 2, §6

We showed that independent quanta are closely connected to independent events in Einstein's special relativity.

24

Now if Einstein‟s clocks are atomic clocks, what are Poincaré‟s clocks? In “La mesure du

temps” [2], The great specialist of Celestial Mechanics writes in 1998:

In fact the best clocks have to be corrected from time to time, and corrections are made with the

astronomical observations; (…) In other terms, it is the sidereal day or the duration of rotation of the

Earth, that is the constant unit of time. [6]

It is sometimes claimed that Poincaré‟s analysis of distant simultaneity (1898) is the same

as Einstein‟s one, seven years later. But it is completely wrong because Poincaré, in “La Mesure

du temps”, underlines the conventionality of simultaneity21. In the same text Poincaré writes:

When we use the pendulum to measure time, which is the postulate that we admit implicitly? It is that

duration of two identical phenomena is the same. Watch out one moment (Prenons-y garde un instant). Is

it possible that experiment denies our postulate.? If experiment made us the observers of such a spectacle

our postulate would be contradicted. [6]

In a paper in 1910 on his SR Einstein affirms explicitly his postulate of identity:

Thus, we postulate that two identical phenomena are of the same duration. The perfect clock so defined

plays a role in the measurement of time that is analogous to the role played by the perfect solid in the

measurement of lengths.[5]

We can therefore formulate the next conjecture: the borderline classical-quantum passes

between the two SR and the existence of a “fine structure” of SR (two very close but not merging

theories) could be established on foundations that are non only metaphysical (Poincaré's approach

of identity is clearly Leibnitzian) but physical. In other words, there is a SR with Einstein‟s

"quantum" clocks and SR with Poincaré’s classical clocks.

In order to take a well known Galileo‟s expression we might say that there is a SR from

the messenger of the atoms who is Einstein and a SR from the messenger of the stars who is

Poincaré.

21 Poincaré‟s analysis of conventionality of simultaneity is (at the starting point) the same as Reichenbach‟s analysis

23 years later. We show in Appendix B the connection between Reichenbach‟s parameter and Poincaré‟s

synchronisation. In a sense Poincaré's SR is more "general" than Einstein's one because Reichenbach‟s parameter is

not =1/2 within each inertial system (Einstein's invariance of the one-way speed of light).

25

The difference between Poincaré and Minkowski is also very clear. In Poincaré‟s

kinematics dt and dt‟ are given in the LT. In Minkowski‟s representation of Einstein‟s kinematics,

d is the “invariant” of LT (ds = c d). In fact Minkowski considers not only the proper time in

stationary systems k (like Einstein) but also the element d of “eigenzeit” of any accelerated system

k (the acceleration is therefore reduces to a series stationary systems k1, k2, … kn with different

successive velocities relative to K, the inertial system).

The only possible clocks in Einstein-Minkowski‟s SR are atomic-quantum clocks. The

classical mechanical clocks (not only pendulum or the rotation of the earth but all harmonic

classical oscillations) are incompatible with Einstein‟s axioms because they are deformable and

undergo a real acceleration in Poincaré‟s sense. More fundamentally, in order to obtain the

“eigenzeit” (in the same place), Einstein‟s clocks cannot depend of a movement in space. The

“eigenvalues” of the “eigenzeit” must be given by the frequencies associated with the changes of

state of energy (quantum harmonic oscillator) and not by a velocity associated to the changes of

state in space (classical harmonic oscillator).

The geometrical difference between Poincaré‟s local time dt‟ given by LT and

Minkowski‟s d (recall that it is not a total differential) by the invariant of LT are deeply rooted in

physics. ESR is a theory of time entirely based on measure: Newton‟s absolute time or Poincaré‟s

apparent time are radically eliminated. ESR is based on a radical identification between time and

clock. The proper time implies a point clock i.e. a clock that gives the time (i.e. the frequency)

without movement in space. This clock can only be a quantum clock.

In this sense, the existence of a “fine structure” of SR means that the borderline classical-

quantum, the main cut of the physics of XX century, passes between the two SR.

The starting point of S. Reynaud‟s and T. Jaekel‟s research [16] on time operators and on

the statute of acceleration in SR (without GR) is also the quantum interpretation of Einstein‟s

synchronisation that is:

“The physical observables describing space-time positions cannot be confused with a classical co-ordinate parameter

(…). Their definition has to reach limits associated with quantum nature of the physical world.”

It is very easy to see that the famous time operator is in fact in ESR clearly separated of

the continuous differential PSR. Poincaré‟s temporality and Einstein‟s temporality are both

relativistic but time is not a parameter in Einstein‟s SR. In Einstein‟s own words, time (t or ) is a

set of data from identical clocks (Gleichbesshaffene Uhren), that beat time with an identical

rhythm that is infinite but countable (numerable). Poincaré‟s set of points of true time is also

26

infinite but non-countable [20, p262]. The development of both logics lead us to a very interesting

mathematical contrast between “countable (numerable) set” and “non-countable (unnumerable)

set” 22

.

The measurement of values of time supposes the definition of a unit of time. Einstein‟s “set

of data of clocks” represents a set of values or a spectrum of values of the observable time

Einstein‟s internal time called proper time –eigenzeit- by Minkowski in 1908. This eigenzeit is

essentially a duration of time (Zeitteilchen, Zeitelement in Einstein‟s 1905 words). The skeleton of

identical units in hard SR is in fact the spectrum of eigenvalues of the observable time.

Conclusion: the standard mixed SR in Einstein’s General relativity

The idea to try to underscore the quantum features of the metrical structure of space-time

is of course not a new idea. Anandan follows a suggestion of Penrose and argues for the quantum

mechanical nature of clocks in their fondamental role as hodometers of the metrical structure of

space-time [15-1, §IX]

In this paper I didn‟t analyse the geometrical concept of Minkowski‟s metric and I only

showed that Penrose‟s suggestion (quantum nature of clocks) is true for one of the two components

of the mixing: Einstein‟s SR. If both theories, Einstein’s one and Poincaré’s, are already not very

easy to distinguish, it is still much more difficult to distinguish Minkowski’s and Poincaré’s

geometrical representations of space-time.

We note that Einstein's crucial connection between spectral lines and units of measure

doesn't concern only SR but also GR. Cao writes:

Weyl's concept of the non integrability of the transference of length, derived from the local definition of

length unit, invited many criticisms. Most famous among them was that of Einstein, who pointed out its

contradiction with the observed frequencies of spectral lines. (… ) Einstein's pointed out that Weyl's

concept [gauge system of standard units] meant that spectral lines with definite frequencies could not

exist.[my italics, 15bis]

In others words, in GR, Einstein gives up completely the concept of rigidity (the Galilean

frames are replaced by the Gauss frames) but not the quantum definitions of units. Einstein‟s

22 We arrive at the same conclusion from an analysis of deeply discontinuous Einstein‟s concept of

independent events [17 and 20].

27

objection could let think that the SR locally valid in GR is his own ESR. In this case Weyl‟s affine

point of view in GR would be the “generalisation” of Poincaré‟s point of view in PSR.

However Einstein, in 1921, reintroduces the ether (the space-time in GR has physical

properties) exactly in the words of Poincaré (“relativistic ether without singular state of

movement”). The main difference between the space-time in ESR and space-time in GSR is that the

latter “acts and reacts” on the matter. This is also the main characteristic of Poincaré‟s space-time

or Poincaré “relativistic ether”: Action (pressure of the ether) and Reaction (contraction of the

volume of the electron) or conversely of course.

The space-time in ESR is a space-time of (independent) events (fundamentally collisions),

completely indifferent at the presence of matter.

Now EGR absolutely needs that a mass placed in space-time deforms this space-time. So it

is only possible if the flat space-time is deformable and it is only possible in PSR with relativistic

ether (Poincaré 1905- Einstein 1921).

Poincaré‟s space-time is not a space-time of events. It is the space-time in which the

material points (the centres of gravity of the classical electrons) are moving. In Poincaré‟s own

words the deformable electron is a hole in ether which exerts a (non electromagnetic) pressure in

order to balance the electrostatic repulsion. Of course Poincaré calculates not the curvature of the

ether but the work exerted by the ether on electron23

. Laue showed that the energy-momentum

tensor is very easy to construct with Poincaré‟s pressure. Poincaré‟s starting point for PSR is the

relativistic theory of continuous medium. This is also a very important element in EGR.

The SR locally valid in EGR is therefore a semi-classical or pseudo-classical mixture

between ESR and PSR: the definition of units comes from ESR and the deformable space-time

comes from PSR. So the clear separation of the standard mixing SR into its two components

(Einstein's SR and Poincaré's SR) may help to solve the delicate problems which still persist at the

interface between quantum theory and general relativity. This splitting of the standard mixture SR

into its two components “the affine classical PSR” and “the projective quantum ESR”24

is therefore

very important for an new physical approach (non-idealistic and without Platonist supplementary

dimensions of space) towards the quantization of the gravitation.

23

It is very curious that Einstein wrote in 1919 that the pressure of Poincaré‟s is perhaps the force of gravity.

“The general theory of relativity renders it likely that the electrical masses of an electron are held together by

gravitational forces” Einstein, “The special and general theory of relativity”, Routledge, p51. 24 We showed [20-3, pirt 2000] that it is possible to separate Poincaré‟s four-dimensional and Minkowski‟s four-

dimensional representation and thus to give a geometrical sense to Einstein‟s principle of identity of units of measure.

28

Appendix A Poincaré's convention of synchronisation-1906 ("Cours à la Sorbonne")

First order

"Let us consider two stations A and B separated by a distance D; (…) In the forward travel, the light takes

the time D/c =; and in the backward travel = D/c.. A notices the time + +, and take the average

1/2 ( ) as correction of his chronometer. This is true if the two stations are fixed.

A B

D

v But let us assume now a transtation of AB with a velocity v in the direction AB. We have:

- = D

c - v + = D

c + v

If we take as correction 1/2 ( + +) in place of , the error is 1/2 (+).

= -

+

2 = D c

c2 - v2 =

--

+

2 = D v

c2 - v2

(…)The error for the adjusting of the watch at the square of the aberration v2/c2 is

= --

+

2 = D v

c2

Thus the watch A will be the fixed quantity Dv/c2 fast with respect to the watch B. So adjusted the watch

A gives the reduced time This is true only if we neglect the square of the aberration.

Second order spherical waves

Let a source be in motion ; at the instant zero it is in O; at the instant it is in E; at the instant 2 it is in

F. The wave emanating from the point O, at the instant 2, will be on a sphere of centre O et of radius 2

c; the wave emanating from O at the instant will be at the instant 2 on a sphere of centre E and of

radius c. The surfaces will be non concentric, homothetic spheres with respect to the point B, the present

position of the source."

E FO

2cc

Lorentz's Hypothesis (exact compensation)

"Let us introduce the Lorentz hypothesis. Apparently, the waves surface are becoming homothetic

elongated ellipsoids with respect to the point F. If we call k-1 the coefficient of the contraction in the sense

of the velocity (and l = in the perpendicular sense), both axes of the ellipsoid are k c = a and c = b

29

O A (F) P

B (M)

Moreover OF = vbeing the present position of the source, v/c being the eccentricity of the ellipse and the

point F (centre of homothetie) the common focus of the meridian sections of the ellipsoids.

Then the compensation is rigorous

Indeed, let F and M be the two stations A and B previously considered, they are both involved in a

common transposition with velocity v. FM is the apparent distance of the two points. We have

supposing

c ( 1 - e2) = p

Thus the time is given by

+ = FMp

+ ep

FP

If the two stations exchange light signal, for the return from M to F, we will have

- = FMp

- ep

FP

therefore

= +

-

2 = e

p FP = v

c2

1

( 1- v2

c2)

FP

Therefore the difference of times is rigorously proportional to the differences of the abscissa, et that

without neglecting the square of the aberration We have with this Lorentz's deformation no means to put

in an obvious the absolute movement of the Earth with respect to the ether."(end of the quotation)

M = ++

-

2

M = ABp

= k ABc

(MEAN REAL DILATION of TIME )

1 - v2

c2

= M

DOPPLER MAIN REAL CONTRACTION OF FREQUENCY

c + FP e = c + ( 1 - e2)

(1 + e cos ) = + ( 1 - e2)

If A sends a signal each secondes:

= + (1 + v

c cos )

( 1- (vc)2)

FM FPe a e c e ( ) ( )1 12 2

30

We have the particular cases of longitudinal Doppler = 0 and transversal Doppler:

= + 1 + v

c

1 - vc

= + 1

( 1- (vc)2)

Appendix B: Reichenbach’s parameter and Poincaré's -LT

Let us thus examine Poincaré‟s convention of synchronisation. Consider two systems of reference

K (ether is by definition at rest) and K‟ where there are two stations A and B placed in the direction of the

movement (velocity v).

t(1) et t(3) t(2)

K K’

y y ’

x

x’A B

z z’

v

Let start an electromagnetic wave with the velocity c (in the ether, = v/c) from A towards B (1), let it be

reflected at B in the direction of A (2), and arrive again at A. The length really contracted AB of the

deformable rod is k-1 L.

We shall estimate thus the forth an back travel time (from A to B and from B to A) of the

electromagnetic wave successively in true time t and in local time t‟(3).

1) FORTH AND BACK TIMES TRAVEL IN TRUE TIME

(Poincaré's convention of synchronisation in one dimension)

Suppose that A is the common origin of the two systems at time t(1) = 0.

We have thus xA (1) = 0 et xB (1) = k-1 L.

31

We shall seek the co-ordinates of arrival of the wave at A in true time by the evaluation of the distance

covered by the light in the system K (with the speed c).

Let t(2) the time for the electromagnetic wave to arrive in B. During this time the point B is translated

towards the right of v t(2). The distance covered by the light is thus c t(2) = k-1 L + v t(2) We have ( =

v/c):

c t(2) = L 1 - 2 + v t(2) and thus t (2) (1 - ) = L

c1 - 2

So we obtain the time t(2) of arrival in B:

t(2) = L

c 1 +

1 -

The co-ordinate xB(2) is

xB (2) = v L

c 1 +

1 - + 1 - 2 L = L 1 +

1 - + 1 - 2 L

Now let the time t(3) for the light to return in A. The distance covered by the light, c t(3), is the sum of

the forth distance covered by the light during the time t(2) and of the back distance covered by the light

during the time t(3) - t(2):

c t(3) = [ L 1 - 2 + v t(2) ] + [ L 1 - 2 - v (t(3) - t(2)) ]

We obtain for the time t(3):

t(3) = 2 Lc

1

1 - 2

The co-ordinate xA(3) is:

xA (3) = v t(3) = v 2 Lc

1

1 - 2

= 2 L 1

1 - 2

So we have

FORTH: t(2) = Lc

1 +

1 - BACK: t(3) - t(2) = 2 L

c1

1 - 2

- Lc

1 +

1 -

In Poincaré‟s SR with ether, the equality of forth travel time and back travel time is not postulated in true

time. The true time determines the real state (of synchronization) of the primed system K‟.

The mean time is k L/c and so we have Poincaré's real dilation.

2) FORTH AND BACK TRAVEL TIMES IN LOCAL TIMES OF A AND B

Poincaré's compensation with the local times defined by the fourth LT

In the SR with ether the LT defines the local time (at A or at B):

32

t'A = k (t - v

c2 xA) t'B = k (t - v

c2 xB )

Let t‟A(1), tB‟(2) and tA‟(3) the locals times of the departure, arrival and return on the light‟s wave. We

have

t'A(1) = k (t(1) - v

c2 xA(1) ) t'B(2) = k (t(2) - v

c2 xB(2) ) t'A(3) = k (t(3) - v

c2 xA(3) )

From the first equation, we have t‟A (1) =0.

From the second equation, we replace t(2) et xB(2) by their values and we find:

t'B(2) = k (L

c 1 +

1 - - 2 L

c1 +

1 - - L

c1 - 2 )

t'B(2) = L

c

The forth travel time t‟B(2) - t‟A(1) is thus L/c.

From the third equation, we replace t(3) et xA(3) by their values, and we find:

t'A(3) = k (2 Lc

1

1 - 2

- 2 L 1

1 - 2

)

We obtain:

t'A(3) = 2 L

c

The back travel time, t‟A(3) - t‟B(2) equals also to L/c.

The compensation of dilation k with local times, defined by the fourth LT, is made exactly of the same

way that the compensation of real contraction k-1 L by the first LT.

Reichenbach‟s aim [22] is, in the starting point, the same as Poincaré‟s one in 1898 [6]: the

demonstration of the conventionality of the distant simultaneity or more precisely the comparison of the

travel forth and the travel back of a light signal in a given inertial system.

Reichenbach has introduced a parameter (I adopt the letter for Reichenbach‟s parameter in

place of the standard notation to avoid the confusion with Poincaré‟s notation ) whose value is the

travel-time forth divided by the travel-time forth-back.

It is elementary to evaluate this parameter in both SR.

Einstein’s SR

The “one-way speed of light”[25] is of course an invariant in the relativistic logic of Einstein. It is well

known that this parameter equal to one half within all inertial systems in SR without ether. In particular

we have in each Einstein‟s frame (K and k): = ½ for

33

Poincaré’s SR

For Poincaré‟s system K, = ½ because ether is by definition at rest in this system.

For Poincaré‟s system K‟, we must distinguish the situation in true time and in local time (with LT)

In true time t, we have calculated (see Appendix A)

FORTH: t(2) = Lc

1 +

1 - BACK: t(3) - t(2) = 2 L

c1

1 - 2

- Lc

1 +

1 -

The value of Reichenbach‟s parameter , may be determined in function of the speed relatively

to the ether:

=

1 +

1 -

1 +

1 - + 1 -

1 +

= 1 +

2

The connection between Reichenbach‟s parameter and the relative speed with respect to the

ether is quite natural.

1/2 in K‟ in true time.

We have only = 1/2 when , the speed relative to the ether, equals to zero. This result seems

trivial but precisely we shall show that the elimination of the ether by Einstein (or in other words, the

elimination of Poincaré's hidden variable, the true time) may be interpreted as if the two systems, K and

k, were in the same state (at rest, = 0) of synchronization.

In local time(s) t’ (with LT), we have calculated (see Appendix A):

“The back travel time, t‟A(3) - t‟B(2) equals also to L/c.”

and thus = ½ in K‟ in local time(s)

Everything happens as if when A and B used their local time, the forth travel time was the same as the

back travel time. But the two times are different in the “reality”.

Poincaré‟s convention of synchronisation is therefore the definitive physical answer to the

problem of Reichenbach‟s parameter. Indeed if we consider only one inertial system we can discuss

philosophically endless on the value of Reichenbach‟s parameter. The true physics begins when it is

possible to compare experiments from two systems in uniform translation (this is not only true for

Poincaré and Einstein but also for Galileo). The value of the parameter, in true time (Poincaré's hidden

variable), for Poincaré's second system K‟ is ½ that becomes = 1/2 in local time(s). Poincaré's SR

34

can be considered as a theory with hidden variable but unlike Bell's local theories, the true time is not an

indefinite parameter but the main parameter of classical mechanics and unlike Bohm non local theories,

Poincaré's theory (with his supplementary potential) is of course relativistic.

It is therefore a mistake in a relativistic point of view to introduce this parameter (or another

synchronisation‟s parameter) in LT.

According to the “fine structure” of SR, the -Lorentz transformation is in fact the -Lorentz

transformations of Poincaré.

I showed in my paper “Euclidean Poincaré‟s SR and non Euclidean Einstein-Minkowski‟s SR” that the

genuine physical contrast is situated between Einstein‟s v-LT of independent events and Poincaré‟s -LT

of the co-ordinates of a material point.

Appendix C: The spectral identity of atoms in Einstein's letter to Weyl (1918)

[extracts, Doc 512, 5bis]

Wenn Lichstrahlen das einzige Mittel wären, um die metrischen Verhältnisse [Weyl's concept] in der

Umgebung eines Weltpunktes empirish zu ermitteln, so bliebe in dem Abstand ds (sowie in den g)

allerdings ein Faktor unbestimmt. Diese Unbestimmtheit ist aber nicht vorhanden, wenn man zur

definition von ds Messergebnisse heranzieht, die mit (unendlich kleinen) starren Körpern (Massstäben)

[Born's concept, see note 9] und Uhren zu gewinnen sind. Ein zeitartiges ds kan dann unmittelbar

gemessen werden durch eine Einheitsuhr, deren Weltlinie ds enthält.

Eine derartige Definition des elementaren Abstandes ds würde nur dan illusorisch werden, wenn die

Begriffe Einheitsmassstab und Einheitsuhr auf einer prinzipiell falschen Voraussetzung beruhten; dies

wäre dann der Fall, wenn die Länge eines Einheitsmassstab (bezw. Die Gang-Geschwindigkeit einer

Einheitsuhr) von der Vorgeschichte abhingen. Ware dies in der Natur wirklich so, dann könnte es nicht

chemische Element mit Sprektrallinien von bestimmter Frequenz geben, sonder es müsste die relative

Frequenz zweier (räumlich benachäher) Atom der gleichen Art im allgemeinen verschieden sein. Da dies

nicht der Fall ist, scheint , mir die Grundhypothesis der Theorie leider nicht annehmbar, deren Tiefe und

Kühnheit aber jeden Leser mit Bewunderung erfüllen muss. [5bis, my italics, my bold and my

underlining]

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35

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