Lorentz ether theory
Encyclopedia
What is now often called Lorentz Ether theory ("LET") has its roots in Hendrik Lorentz
's "Theory of electrons", which was the final point in the development of the classical aether theories
at the end of the 19th and at the beginning of the 20th century.
Lorentz's initial theory created in 1892 and 1895 was based on a completely motionless aether. It explained the failure of the negative aether drift experiments to first order in v/c by introducing an auxiliary variable called "local time" for connecting systems at rest and in motion in the aether. In addition, the negative result of the Michelson-Morley experiment
led to the introduction of the hypothesis of length contraction
in 1892. However, other experiments also produced negative results and (guided by Henri Poincaré
's principle of relativity
) Lorentz tried in 1899 and 1904 to expand his theory to all orders in v/c by introducing the Lorentz transformation
. In addition, he assumed that also non-electromagnetic forces (if they exist) transform like electric forces. However, Lorentz's expression for charge density and current were incorrect, so his theory did not fully exclude the possibility of detecting the aether. Eventually, it was Henri Poincaré
who in 1905 corrected the errors in Lorentz's paper and actually incorporated non-electromagnetic forces (incl. Gravitation
) within the theory, which he called "The New Mechanics". Many aspects of Lorentz's theory were incorporated into special relativity
(SR) with the works of Albert Einstein
and Hermann Minkowski
.
Today LET is often treated as some sort of "Lorentzian" or "neo-Lorentzian" interpretation of special relativity. The introduction of length contraction
and time dilation
for all phenomena in a "preferred" frame of reference
(which plays the role of Lorentz's immobile aether), leads to the complete Lorentz transformation. Because of the same mathematical formalism it is not possible to distinguish between LET and SR by experiment. However, in LET the existence of an undetectable ether is assumed and the validity of the relativity principle seems to be only coincidental, which is one reason why SR is commonly preferred over LET. Another important reason for preferring SR is that the new understanding of space and time was also fundamental for the development of general relativity
.
This theory, which was developed mainly between 1892 and 1906 by Lorentz and Poincaré, was based on the aether theory of Augustin-Jean Fresnel
, Maxwell's equations
and the electron theory of Rudolf Clausius
.Whittaker (1951), 386ff Lorentz introduced a strict separation between matter (electrons) and ether, whereby in his model the ether is completely motionless, and it won't be set in motion in the neighborhood of ponderable matter. As Max Born
later said, it was natural (though not logically necessary) for scientists of that time to identify the rest frame of the Lorentz ether with the absolute space of Isaac Newton
.Born (1964), 172ff The condition of this ether can be described by the electric field
E and the magnetic field
H, where these fields represent the "states" of the ether (with no further specification), related to the charges of the electrons. Thus an abstract electromagnetic ether replaces the older mechanistic ether models. Contrary to Clausius, who accepted that the electrons operate by actions at a distance
, the electromagnetic field of the ether appears as a mediator between the electrons, and changes in this field can propagate not faster than the speed of light
. Lorentz theoretically explained the Zeeman effect
on the basis of his theory, for which he received the Nobel Prize in Physics
in 1902. Joseph Larmor
found a similar theory simultaneously, but his concept was based on a mechanical ether. A fundamental concept of Lorentz's theory in 1895Lorentz (1895) was the "theorem of corresponding states" for terms of order v/c. This theorem states that a moving observer makes the same observations as an observer in the stationary system.
and the effect of aberration disproved that model.
A possible solution came in sight, when in 1889 Oliver Heaviside
derived from the Maxwell's equations
that the magnetic vector potential field around a moving body is altered by a factor of
. Based on that result and to bring the hypothesis of an immobile ether in accordance with the Michelson–Morley experiment, George FitzGerald
in 1889 (qualitatively) and independently of him Lorentz in 1892Lorentz (1892) (already quantitatively) suggested that not only the electrostatic fields, but also the molecular forces are affected in such a way that the dimension of a body in the line of motion is less by the value
than the dimension perpendicularly to the line of motion. However, an observer co-moving with the earth would not notice this contraction, because all other instruments contract at the same ratio. In 1895 Lorentz proposed three possible explanation for this relative contraction:Brown (2001)
Although the possible connection between electrostatic and intermolecular forces was used by Lorentz as a plausibility argument, the contraction hypothesis was soon considered as purely ad hoc
. It is also important that this contraction only affected the space between the electron but not the electrons themselves, therefore the name "intermolecular hypotheses" was sometimes used of this effect. The so called Length contraction
without expansion perpendicularly to the line of motion and by the precise value
(where l0 is the length at rest in the ether) was given by Larmor in 1897 and by Lorentz in 1904. In the same year Lorentz also argued that also electrons themselves are affected by this contraction.Miller (1981), 70-75, For further development of this concept, see the section #Lorentz transformation.Lorentz (1904b)
, where t is the time coordinate for an observer resting in the ether, and t is the time coordinate for an observer moving in the ether. (Woldemar Voigt
had previously used the same expression for local time in 1887 in connection with the Doppler effect
and an incompressible medium.) With the help of this concept Lorentz could explain the aberration of light
, the Doppler effect and the Fizeau experiment
(i.e. measurements of the Fresnel drag coefficient
) by Hippolyte Fizeau
in moving and resting liquids as well. While for Lorentz length contraction was a real physical effect, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation to simplify the calculation from the resting to a "fictitious" moving system. Contrary to Lorentz, Poincaré saw more than a mathematical trick in the definition of local time, which he called Lorentz's "most ingenious idea".Poincaré (1904); Poincaré (1905a), Ch. 8 In The Measure of Time he wrote in 1898:Poincaré (1898); Poincaré (1905a), Ch. 2
In 1900 Poincaré interpreted local time as the result of a synchronization procedure based on light signals. He assumed that 2 observers A and B which are moving in the ether, synchronize their clocks by optical signals. Since they believe to be at rest they must consider only the transmission time of the signals and then crossing their observations to examine whether their clocks are synchronous. However, from the point of view of an observer at rest in the ether the clocks are not synchronous and indicate the local time
. But because the moving observers don't know anything about their movement, they don't recognize this.Poincaré (1900b) In 1904 he illustrated the same procedure in the following way: A sends a signal at the time 0 to B, which arrives at the time t. B also sends a signal at the time 0 to A, which arrives at the time t. If in both cases t has the same value the clocks are synchronous, but only in the system in which the clocks are at rest in the ether. So according to DarrigolDarrigol (2005), 10-11 Poincaré understood local time as a physical effect just like length contraction - in contrast to Lorentz, who used the same interpretation not before 1906. However, contrary to Einstein, who later used a similar synchronisation procedure which was called Einstein synchronisation
, he still was the opinion that only clocks resting in the ether are showing the „true“ time.
However, at the beginning it was unknown that local time includes what is now known as time dilation
. This effect was first noticed by Larmor (1897), who wrote that "individual electrons describe corresponding parts of their orbits in times shorter for the [ether] system in the ratio
or 
". And in 1899Lorentz (1899) also Lorentz noted for the frequency of oscillating electrons "that in S the time of vibrations be 
times as great as in S0", where S0 is the ether frame, S the mathematical-fictitious frame of the moving observer, k is 
, and 
is an undetermined factor. Janssen (1995), Chap. 3.5.4
. It was Voigt in 1887 who already derived a similar set of equations (however, with a different scale factor). Afterwards, Larmor in 1897 and Lorentz in 1899 derived equations in an algebraically equivalent form to those, which are used up to this day (however, Lorentz used an undetermined factor l in his transformation). In his paper Electromagnetic phenomena in a system moving with any velocity smaller than that of light (1904) Lorentz attempted to create such a theory, according to which all forces between the molecules are affected by the Lorentz transformation (in which Lorentz set the factor l to unity) in the same manner as electrostatic forces. In other words, Lorentz attempted to create a theory in which the relative motion of earth and aether is (nearly or fully) undetectable. Therefore he generalized the contraction hypothesis and argued that not only the forces between the electrons, but also the electrons themselves are contracted in the line of motion. However, Max Abraham
(1904) quickly noted a defect of that theory: Within a purely electromagnetic theory the contracted electron-configuration is unstable and one has to introduce non-electromagnetic force to stabilize the electrons - Abraham himself questioned the possibility of including such forces within the theory of Lorentz.
So it was Poincaré (1905) on 5 June 1905Poincaré (1905b) who introduced the so called "Poincaré stresses" to solve that problem. Those stresses were interpreted by him as an external, non-electromagnetic pressure, which stabilize the electrons and also served as an explanation for length contraction.Janssen/Mecklenburg (2007) Although he argued that Lorentz succeeded in creating a theory which complies to the postulate of relativity, he showed that Lorentz's equations of electrodynamics were not fully Lorentz covariant
. So by pointing out the group characteristics of the transformation Poincaré demonstrated the Lorentz covariance of the Maxwell-Lorentz equations and corrected Lorentz's transformation formulae for charge density
and current density
. He went on to sketch a model of gravitation (incl. gravitational wave
s) which might be compatible with the transformations. Poincaré used for the first time the term "Lorentz transformation", and he gave them a form which is used up to this day. (Where
is an arbitrary function of 
, which must be set to unity to conserve the group characteristics. He also set the speed of light to unity.)
A substantially extended work (the so called „Palermo paper“)Poincaré (1906) was submitted by Poincaré on 23 July 1905, but was published on January 1906, because the journal only appeared two times in a year. He spoke literally of "the postulate of relativity", he showed that the transformations are a consequence of the principle of least action
; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group
, and he showed that the combination
is invariant. While elaborating his gravitational theory he noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing 
as a fourth imaginary coordinate, and he used an early form of four-vector
s. However, Poincaré later said the translation of physics into the language of four-dimensional metry would entail too much effort for limited profit, and therefore he refused to work out the consequences of this notion. This was later done by Minkowski, see "The shift to relativity".Walter (2007), Kap. 1
(1881) and others noticed, that electromagnetic energy contributes to the mass of charged bodies by the amount
, which was called electromagnetic or "apparent mass". Another derivation of some sort of electromagnetic mass was conducted by Poincaré (1900). By using the momentum
of electromagnetic fields, he concluded that these fields contribute a mass of
to all bodies, which is necessary to save the center of mass
theorem.
As noted by Thomson and others, this mass increases also with velocity. Thus in 1899, Lorentz calculated that the ratio of the electron's mass in the moving frame and that of the ether frame is
parallel to the direction of motion, and 
perpendicular to the direction of motion, where 
and 
is an undetermined factor. And in 1904, he set 
, arriving at the expressions for the masses in different directions (longitudinal and transverse):
,
where
Many scienists now believed, that the entire mass and all forms of forces are electromagnetic in nature. This idea had to be given up, however, in the course of the development of relativistic mechanics. Abraham (1904) argued (as described in the preceding section #Lorentz transformation), that non-electrical binding forces were necessary within Lorentz's electrons model. But Abraham also noted that different results occurred, dependent on whether the em-mass is calculated from the energy or from the momentum. To solve those problems, Poincaré in 1905 and 1906 introduced some sort of pressure of non-electrical nature, which contributes the amount
to the energy of the bodies, and therefore explains the 4/3-factor in the expression for the electromagnetic mass-energy relation. However, while Poincaré's expression for the energy of the electrons was correct, he erroneously stated that only the em-energy contributes to the mass of the bodies.Janssen/Mecklenburg (2007)
The concept of electromagnetic mass is not considered anymore as the cause of mass per se, because the entire mass (not only the electromagnetic part) is proportional to energy, and can be converted into different forms of energy, which is explained by Einstein's mass–energy equivalence.Miller (1981), 359-360
and argued that there possibly exists a universal radiation field, consisting of very penetrating em-radiation, and exerting a uniform pressure on every body. Lorentz showed that an attractive force between charged particles would indeed arise, if it is assumed that the incident energy is entirely absorbed. This was the same fundamental problem which had afflicted the other Le Sage models, because the radiation must vanish somehow and any absorption must lead to an enormous heating. Therefore Lorentz abandoned this model.
In the same paper, he assumed like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner
that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity
is that of light. This leads to a conflict with the law of gravitation by Isaac Newton, in which it was shown by Pierre Simon Laplace that a finite speed of gravity leads to some sort of aberration and therefore makes the orbits unstable. However, Lorentz showed that the theory is not concerned by Laplace's critique, because due to the structure of the Maxwell equations only effects in the order v2/c2 arise. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low. He wrote:
In 1908Poincaré (1908a); Poincaré (1908b) Book 3, Ch. 3 Poincaré examined the gravitational theory of Lorentz and classified it as compatible with the relativity principle, but (like Lorentz) he criticized the inaccurate indication of the perihelion advance of Mercury. Contrary to Poincaré, Lorentz in 1914 considered his own theory as incompatible with the relativity principle and rejected it.Lorentz (1914) primary sources
However, in 1905 and 1906 Poincaré pointed out the possibility of a gravitational theory, in which changes propagate with the speed of light and which is Lorentz covariant. He pointed out that in such a theory the gravitational force not only depends on the masses and their mutual distance, but also on their velocities and their position due to the finite propagation time of interaction. On that occasion Poincaré introduced four-vectors. Following Poincaré, also Minkowski (1908) and Arnold Sommerfeld
(1910) tried to establish a lorentz-invariant gravitational law.Walter (2007) However, these attempts were superseded because of Einstein's theory of general relativity
, see "The shift to relativity".
" with the help of hypotheses like local time, but he confessed that this venture was possible only by an accumulation of hypotheses. And he defined the principle in this way (according to MillerMiller (1981), 79 based on Lorentz's theorem of corresponding states): "The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion."
Referring to the critique of Poincaré from 1900, Lorentz wrote in his famous paper in 1904, where he extended his theorem of corresponding states: "Surely, the 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 fundamental 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."
One of the first assessments of Lorentz's paper was by Paul Langevin
in May 1905. According to him, this extension of the electron theories of Lorentz and Larmor led to "the physical impossibility to demonstrate the translational motion of the earth". However, Poincaré noticed in 1905 that Lorentz's theory of 1904 was not perfectly "Lorentz invariant" in a few equations such as Lorentz's expression for current density (it was admitted by Lorentz in 1921 that these were defects). As this required just minor modifications of Lorentz's work, also Poincaré asserted that Lorentz had succeeded in harmonizing his theory with the principle of relativity: "It appears that this impossibility of demonstrating the absolute motion of the earth is a general law of nature. [..] Lorentz tried to complete and modify his hypothesis in order to harmonize it with the postulate of complete impossibility of determining absolute motion. He has succeeded in doing so in his article [Lorentz, 1904b]."French original: Il semble que cette impossibilité de démontrer le mouvement absolu soit une loi générale de la nature [..] Lorentz a cherché à more compléter et à more modifier son hypothèse de façon à la mettre en concordance avec le postulate de l' impossibilité complète de la détermination du mouvement absolu. C'est ce qu'il a réussi dans son article intitulé [Lorentz, 1904b]
In his Palermo paper (1906), Poincaré called this "the postulate of relativity“, and although he stated that it was possible this principle might be disproved at some point (and in fact he mentioned at the paper's end that the discovery of magneto-cathode ray
s by Paul Ulrich Villard
(1904) seems to threaten itWalter (2007), Chap. 1), he believed it was interesting to consider the consequences if we were to assume the postulate of relativity was valid without restriction. This would imply that all forces of nature (not just electromagnetism) must be invariant under the Lorentz transformation. In 1921 Lorentz credited Poincaré for establishing the principle and postulate of relativity and wrote:Lorentz (1921), pp. 247-261 "I have not established the principle of relativity as rigorously and universally true. Poincaré, on the other hand, has obtained a perfect invariance of the electro-magnetic equations, and he has formulated 'the postulate of relativity', terms which he was the first to employ."French original: je n'ai pas établi le principe de relativité comme rigoureusement et universellement vrai. Poincaré, au contraire, a obtenu une invariance parfaite des équations de l’électrodynamique, et il a formule le « postulat de relativité » , termes qu’il a été le premier a employer.
philosophy in 1889: Poincaré (1889); Poincaré (1902), Ch. 12 "Whether the ether exists or not matters little - let us leave that to the metaphysicians; what is essential for us is, that everything happens as if it existed, and that this hypothesis is found to be suitable for the explanation of phenomena. After all, have we any other reason for believing in the existence of material objects? That, too, is only a convenient hypothesis; only, it will never cease to be so, while some day, no doubt, the ether will be thrown aside as useless."
He also denied the existence of absolute space and time by saying in 1901:Poincaré (1901a); Poincaré (1902), Ch. 6 "1. There is no absolute space, and we only conceive of relative motion ; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred. 2. There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention. 3. Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events occurring in two different places. I have explained this in an article entitled "Mesure du Temps" [1898]. 4. Finally, is not our Euclidean geometry in itself only a kind of convention of language?"
However, Poincaré himself never abandoned the ether hypothesis and stated in 1900: "Does our ether actually exist ? We know the origin of our belief in the ether. If light takes several years to reach us from a distant star, it is no longer on the star, nor is it on the earth. It must be somewhere, and supported, so to speak, by some material agency." And referring to the Fizeau experiment
, he even wrote: "The ether is all but in our grasp." He also said the ether is necessary to harmonize Lorentz's theory with Newton's third law. Even in 1912 in a paper called "The Quantum theory", Poincaré ten times used the word "ether", and described light as "luminous vibrations of the ether".Poincaré 1912; Poincaré 1913, Ch. 6
And although he admitted the relative and conventional character of space and time, he believed that the classical convention is more "convenient" and continued to distinguish between "true" time in the ether and "apparent" time in moving systems. Addressing the question if a new convention of space and time is needed he wrote in 1912:Poincaré (1913), Ch. 2 "Shall we be obliged to modify our conclusions? Certainly not; we had adopted a convention because it seemed convenient and we had said that nothing could constrain us to abandon it. Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one in order not to disturb their old habits, I believe, just between us, that this is what they shall do for a long time to come."
Also Lorentz argued during his lifetime that in all frames of reference this one has to be preferred, in which the ether is at rest. Clocks in this frame are showing the "real“ time and simultaneity is not relative. However, if the correctness of the relativity principle is accepted, it is impossible to find this system by experiment.Lorentz (1913), p. 75
published his paper on what is now called special relativity
.Einstein (1905a) In this paper, by examining the fundamental meanings of the space and time coordinates used in physical theories, Einstein showed that the "effective" coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. From this followed all of the physically observable consequences of LET, along with others, all without the need to postulate an unobservable entity (the ether). Einstein identified two fundamental principles, each founded on experience, from which all of Lorentz's electrodynamics follows:
1. The laws by which physical processes occur are the same with respect to any system of inertial coordinates (the principle of relativity
)
2. In empty space light propagates at an absolute speed c in any system of inertial coordinates (the principle of the constancy of light)
Taken together (along with a few other tacit assumptions such as isotropy and homogeneity of space), these two postulates lead uniquely to the mathematics of special relativity. Lorentz and Poincaré had also adopted these same principles, as necessary to achieve their final results, but didn't recognize that they were also sufficient, and hence that they obviated all the other assumptions underlying Lorentz's initial derivations (many of which later turned out to be incorrect The three best known examples are (1) the assumption of Maxwell's equations, and (2) the assumptions about finite structure of the electron, and (3) the assumption that all mass was of electromagnetic origin. Maxwell's equations were subsequently found to be invalid and were replaced with quantum electrodynamics, although one particular feature of Maxwell's equations, the invariance of a characteristic speed, has remained. The electron's mass is now regarded as a pointlike particle, and Poincaré already showed in 1905 that it is not possible for all the mass of the electron to be electromagnetic in origin. This is how relativity invalidated the 19th century hopes for basing all of physics on electromagnetism.). Therefore, special relativity very quickly gained wide acceptance among physicists, and the 19th century concept of a luminiferous ether was no longer considered useful.Darrigol (2005), 15-18Janssen (1995), Kap. 4
Einstein's 1905 presentation of special relativity was soon supplemented, in 1907, by Hermann Minkowski
, who showed that the relations had a very natural interpretationSee Whittaker's History of the Aether, in which he writes "the great advances made by Minkowski were connected with his formulation of physics in terms of a four-dimensional manifold... in order to represent natural phenomena without introducing contingent elements, it is necessary to abandon the customary three-dimensional system of coordinates and to operate in four dimensions". See also Pais's Subtle is the Lord, in which it says of Minkowski's interpretation "Thus began the enormous simplification of special relativity". See also Miller's "Albert Einstein's Special Theory of Relativity" in which it says "Minkowski's results led to a deeper understanding of relativity theory". in terms of a unified four-dimensional "spacetime
" in which absolute intervals are seen to be given by an extension of the Pythagorean theorem. (Already in 1906 Poincaré anticipated some of Minkowski's ideas, see the section "Lorentz-transformation").Walter (1999) The utility and naturalness of the representations by Einstein and Minkowski contributed to the rapid acceptance of special relativity, and to the corresponding loss of interest in Lorentz's ether theory.
In 1907 Einstein criticized the "ad hoc
" character of Lorentz's contraction hypothesis in his theory of electrons, because according to him it was only invented to rescue the hypothesis of an immobile ether. Einstein thought it necessary to replace Lorentz's theory of electrons by assuming that Lorentz's "local time" can simply be called "time", and he stated that the immobile ether as the theoretical fundament of electrodynamics was unsatisfactory.Einstein (1907) And in 1910Einstein (1909) and 1912Einstein (1912) Einstein explained that he borrowed the principle of the constancy of light from Lorentz's immobile ether, but he recognized that this principle together with the principle of relativity makes the ether useless and leads to special relativity. Minkowski ironically said that for Lorentz the contraction hypothesis is only a "gift from above". And although Lorentz's hypothesis is "completely equivalent with the new concept of space and time", Minkowski held that it becomes much more comprehensible in the framework of the new spacetime physics. However, Lorentz disagreed that it was "ad-hoc" and he argued in 1913 that there is little difference between his theory and the negation of a preferred reference frame, as in the theory of Einstein and Minkowski, so that it is a matter of taste which theory one prefers..
, but in contrast to Poincaré's 1900-paper, Einstein recognized that matter itself loses or gains mass during the emission or absorption.Einstein (1905b) So the mass of any form of matter is equal to a certain amount of energy, which can be converted into and re-converted from other forms of energy. This is the mass–energy equivalence, represented by 
. So Einstein didn't have to introduce "fictitious" masses and also avoided the perpetual motion
problem, because according to DarrigolDarrigol (2005), 18-21
, Poincaré's radiation paradox can simply be solved by applying Einstein's equivalence. If the light source loses mass during the emission by
, the contradiction in the momentum law vanishes without the need of any compensating effect in the ether.
Similar to Poincaré, Einstein concluded in 1906 that the inertia of (electromagnetic) energy is a necessary condition for the center of mass theorem to hold in systems, in which electromagnetic fields and matter are acting on each other. Based on the mass–energy equivalence he showed that emission and absorption of em-radiation and therefore the transport of inertia solves all problems. On that occasion, Einstein referred to Poincaré's 1900-paper and wrote:Einstein (1906)
Also Poincaré's rejection of the reaction principle due to the violation of the mass conservation law can be avoided through Einstein's
, because mass conservation appears as a special case of the energy conservation law.
) to formulate a theory of gravitation, were superseded by Einstein's theory of general relativity
.Walter 2007 This theory is based on principles like the equivalence principle
, the general principle of relativity
, the principle of general covariance
, geodesic
motion, local Lorentz covariance (the laws of special relativity apply locally for all inertial observers), and that spacetime curvature is created by stress-energy within the spacetime.
In 1920 Einstein compared Lorentz's ether with the "gravitational ether" of general relativity. He said that immobility is the only mechanical property of which the ether has not been deprived by Lorentz, but contrary to the luminiferous and Lorentz's ether the ether of general relativity has no mechanical property, not even immobility:Einstein (1922)
.
Subsequent to the advent of special relativity, only a small number of individuals have advocated the Lorentzian approach to physics. Many of these, such as Herbert E. Ives
(who, along with G. R. Stilwell, performed the first experimental confirmation of time dilation) have been motivated by the belief that special relativity is logically inconsistent, and so some other conceptual framework is needed to reconcile the relativistic phenomena. For example, Ives wrote "The 'principle' of the constancy of the velocity of light is not merely 'ununderstandable', it is not supported by 'objective matters of fact'; it is untenable..."Herbert E. Ives, "Revisions of the Lorentz Transformations", October 27, 1950. However, the logical consistency of special relativity (as well as its empirical success) is well established, so the views of such individuals are considered unfounded within the mainstream scientific community.
John Stewart Bell
advocated teaching special relativity first from the viewpoint of a single Lorentz inertial frame, then showing that Poincare invariance of the laws of physics such as Maxwell's equations is equivalent to the frame-changing arguments often used in teaching special relativity. Because a single Lorentz inertial frame is one of a preferred class of frames, he called this approach Lorentzian in spirit J. Bell, How to Teach Special Relativity.
|volume=25
|pages=363–552}}
|place=Leipzig & Berlin
|publisher=B.G. Teubner}}
|place=Leipzig & Berlin
|publisher=B.G. Teubner}}
Preface partly reprinted in "Science and Hypothesis", Ch. 12.
. Reprinted in Poincaré, Oeuvres, tome IX, pp. 395–413
. Reprinted in "Science and Hypothesis", Ch. 9-10.
. See also the English translation.
. Reprinted in "Science and Hypothesis", Ch. 6-7.
|publisher=Gauthier-Villars
|place=Paris}}
|place=London and Newcastle-on-Cyne (1905)
|publisher=The Walter Scott publishing Co.}}
(Wikisource translation)
(Wikisource translation)
Reprinted in Poincaré 1913, Ch. 6.
|place=New York
|publisher=Dover Publication (1963)}}
. See also: English translation.
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
's "Theory of electrons", which was the final point in the development of the classical aether theories
Aether theories
Aether theories in early modern physics proposed the existence of a medium, the aether , a space-filling substance or field, thought to be necessary as a transmission medium for the propagation of electromagnetic waves...
at the end of the 19th and at the beginning of the 20th century.
Lorentz's initial theory created in 1892 and 1895 was based on a completely motionless aether. It explained the failure of the negative aether drift experiments to first order in v/c by introducing an auxiliary variable called "local time" for connecting systems at rest and in motion in the aether. In addition, the negative result of the Michelson-Morley experiment
Michelson-Morley experiment
The Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University in Cleveland, Ohio. Its results are generally considered to be the first strong evidence against the theory of a luminiferous ether and in favor of special...
led to the introduction of the hypothesis of length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
in 1892. However, other experiments also produced negative results and (guided by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
's principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
) Lorentz tried in 1899 and 1904 to expand his theory to all orders in v/c by introducing the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
. In addition, he assumed that also non-electromagnetic forces (if they exist) transform like electric forces. However, Lorentz's expression for charge density and current were incorrect, so his theory did not fully exclude the possibility of detecting the aether. Eventually, it was Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
who in 1905 corrected the errors in Lorentz's paper and actually incorporated non-electromagnetic forces (incl. Gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
) within the theory, which he called "The New Mechanics". Many aspects of Lorentz's theory were incorporated into special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
(SR) with the works of Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
and Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
.
Today LET is often treated as some sort of "Lorentzian" or "neo-Lorentzian" interpretation of special relativity. The introduction of length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
and time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
for all phenomena in a "preferred" frame of reference
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
(which plays the role of Lorentz's immobile aether), leads to the complete Lorentz transformation. Because of the same mathematical formalism it is not possible to distinguish between LET and SR by experiment. However, in LET the existence of an undetectable ether is assumed and the validity of the relativity principle seems to be only coincidental, which is one reason why SR is commonly preferred over LET. Another important reason for preferring SR is that the new understanding of space and time was also fundamental for the development of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
Basic concept
Augustin-Jean Fresnel
Augustin-Jean Fresnel , was a French engineer who contributed significantly to the establishment of the theory of wave optics. Fresnel studied the behaviour of light both theoretically and experimentally....
, Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
and the electron theory of Rudolf Clausius
Rudolf Clausius
Rudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...
.Whittaker (1951), 386ff Lorentz introduced a strict separation between matter (electrons) and ether, whereby in his model the ether is completely motionless, and it won't be set in motion in the neighborhood of ponderable matter. As Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
later said, it was natural (though not logically necessary) for scientists of that time to identify the rest frame of the Lorentz ether with the absolute space of Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
.Born (1964), 172ff The condition of this ether can be described by the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
E and the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
H, where these fields represent the "states" of the ether (with no further specification), related to the charges of the electrons. Thus an abstract electromagnetic ether replaces the older mechanistic ether models. Contrary to Clausius, who accepted that the electrons operate by actions at a distance
Action at a distance (physics)
In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant...
, the electromagnetic field of the ether appears as a mediator between the electrons, and changes in this field can propagate not faster than the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. Lorentz theoretically explained the Zeeman effect
Zeeman effect
The Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field...
on the basis of his theory, for which he received the Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...
in 1902. Joseph Larmor
Joseph Larmor
Sir Joseph Larmor , a physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter...
found a similar theory simultaneously, but his concept was based on a mechanical ether. A fundamental concept of Lorentz's theory in 1895Lorentz (1895) was the "theorem of corresponding states" for terms of order v/c. This theorem states that a moving observer makes the same observations as an observer in the stationary system.
Length contraction
A big challenge for this theory was the Michelson–Morley experiment in 1887. According to the theories of Fresnel and Lorentz a relative motion to an immobile ether had to be determined by this experiment, however, the result was negative. Michelson himself thought that the result confirmed the aether drag hypothesis, in which the aether is fully dragged by matter. However, other experiments like the Fizeau experimentFizeau experiment
The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Albert Einstein later pointed out the importance of the experiment for special relativity...
and the effect of aberration disproved that model.
A possible solution came in sight, when in 1889 Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...
derived from the Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
that the magnetic vector potential field around a moving body is altered by a factor of
. Based on that result and to bring the hypothesis of an immobile ether in accordance with the Michelson–Morley experiment, George FitzGeraldGeorge FitzGerald
George Francis FitzGerald was an Irish professor of "natural and experimental philosophy" at Trinity College in Dublin, Ireland, during the last quarter of the 19th century....
in 1889 (qualitatively) and independently of him Lorentz in 1892Lorentz (1892) (already quantitatively) suggested that not only the electrostatic fields, but also the molecular forces are affected in such a way that the dimension of a body in the line of motion is less by the value
than the dimension perpendicularly to the line of motion. However, an observer co-moving with the earth would not notice this contraction, because all other instruments contract at the same ratio. In 1895 Lorentz proposed three possible explanation for this relative contraction:Brown (2001)
- The body contracts in the line of motion and preserves its dimension perpendicularly to it.
- The dimension of the body remains the same in the line of motion, but it expands perpendicularly to it.
- The body contracts in the line of motion, and expands at the same time perpendicularly to it.
Although the possible connection between electrostatic and intermolecular forces was used by Lorentz as a plausibility argument, the contraction hypothesis was soon considered as purely ad hoc
Ad hoc
Ad hoc is a Latin phrase meaning "for this". It generally signifies a solution designed for a specific problem or task, non-generalizable, and not intended to be able to be adapted to other purposes. Compare A priori....
. It is also important that this contraction only affected the space between the electron but not the electrons themselves, therefore the name "intermolecular hypotheses" was sometimes used of this effect. The so called Length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
without expansion perpendicularly to the line of motion and by the precise value
(where l0 is the length at rest in the ether) was given by Larmor in 1897 and by Lorentz in 1904. In the same year Lorentz also argued that also electrons themselves are affected by this contraction.Miller (1981), 70-75, For further development of this concept, see the section #Lorentz transformation.Lorentz (1904b)Local time
An important part of the theorem of corresponding states in 1892 and 1895 was the local timeRelativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...
, where t is the time coordinate for an observer resting in the ether, and t is the time coordinate for an observer moving in the ether. (Woldemar VoigtWoldemar Voigt
Woldemar Voigt was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 1914 by Peter Debye, who took charge of the theoretical department of the Physical Institute...
had previously used the same expression for local time in 1887 in connection with the Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...
and an incompressible medium.) With the help of this concept Lorentz could explain the aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...
, the Doppler effect and the Fizeau experiment
Fizeau experiment
The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Albert Einstein later pointed out the importance of the experiment for special relativity...
(i.e. measurements of the Fresnel drag coefficient
Aether drag hypothesis
In the 19th century, the theory of the luminiferous aether as the hypothetical medium for the propagation of light was widely discussed. An important part of this discussion was the question concerning the state of motion of Earth with respect to this medium. The aether drag hypothesis dealt with...
) by Hippolyte Fizeau
Hippolyte Fizeau
Armand Hippolyte Louis Fizeau was a French physicist.-Biography:Fizeau was born in Paris. His earliest work was concerned with improvements in photographic processes. Following suggestions by François Arago, Léon Foucault and Fizeau collaborated in a series of investigations on the interference of...
in moving and resting liquids as well. While for Lorentz length contraction was a real physical effect, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation to simplify the calculation from the resting to a "fictitious" moving system. Contrary to Lorentz, Poincaré saw more than a mathematical trick in the definition of local time, which he called Lorentz's "most ingenious idea".Poincaré (1904); Poincaré (1905a), Ch. 8 In The Measure of Time he wrote in 1898:Poincaré (1898); Poincaré (1905a), Ch. 2
In 1900 Poincaré interpreted local time as the result of a synchronization procedure based on light signals. He assumed that 2 observers A and B which are moving in the ether, synchronize their clocks by optical signals. Since they believe to be at rest they must consider only the transmission time of the signals and then crossing their observations to examine whether their clocks are synchronous. However, from the point of view of an observer at rest in the ether the clocks are not synchronous and indicate the local time
. But because the moving observers don't know anything about their movement, they don't recognize this.Poincaré (1900b) In 1904 he illustrated the same procedure in the following way: A sends a signal at the time 0 to B, which arrives at the time t. B also sends a signal at the time 0 to A, which arrives at the time t. If in both cases t has the same value the clocks are synchronous, but only in the system in which the clocks are at rest in the ether. So according to DarrigolDarrigol (2005), 10-11 Poincaré understood local time as a physical effect just like length contraction - in contrast to Lorentz, who used the same interpretation not before 1906. However, contrary to Einstein, who later used a similar synchronisation procedure which was called Einstein synchronisationEinstein synchronisation
Einstein synchronisation is a convention for synchronising clocks at different places by means of signal exchanges. This synchronisation method was used already by telegraphers in the middle 19th century, but was popularized by H. Poincaré and A...
, he still was the opinion that only clocks resting in the ether are showing the „true“ time.
However, at the beginning it was unknown that local time includes what is now known as time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
. This effect was first noticed by Larmor (1897), who wrote that "individual electrons describe corresponding parts of their orbits in times shorter for the [ether] system in the ratio
or ". And in 1899Lorentz (1899) also Lorentz noted for the frequency of oscillating electrons "that in S the time of vibrations be times as great as in S0", where S0 is the ether frame, S the mathematical-fictitious frame of the moving observer, k is , and is an undetermined factor. Janssen (1995), Chap. 3.5.4Lorentz transformation
While local time could explain the negative aether drift experiments to first order to v/c, it was necessary – due to other unsuccessful ether drift experiments like the Trouton–Noble experiment – to modify the hypothesis to include second order effects. The mathematical tool for that is the so called Lorentz transformationLorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
. It was Voigt in 1887 who already derived a similar set of equations (however, with a different scale factor). Afterwards, Larmor in 1897 and Lorentz in 1899 derived equations in an algebraically equivalent form to those, which are used up to this day (however, Lorentz used an undetermined factor l in his transformation). In his paper Electromagnetic phenomena in a system moving with any velocity smaller than that of light (1904) Lorentz attempted to create such a theory, according to which all forces between the molecules are affected by the Lorentz transformation (in which Lorentz set the factor l to unity) in the same manner as electrostatic forces. In other words, Lorentz attempted to create a theory in which the relative motion of earth and aether is (nearly or fully) undetectable. Therefore he generalized the contraction hypothesis and argued that not only the forces between the electrons, but also the electrons themselves are contracted in the line of motion. However, Max Abraham
Max Abraham
Max Abraham was a German physicist.Abraham was born in Danzig, Imperial Germany to a family of Jewish merchants. His father was Moritz Abraham and his mother was Selma Moritzsohn. Attending the University of Berlin, he studied under Max Planck. He graduated in 1897...
(1904) quickly noted a defect of that theory: Within a purely electromagnetic theory the contracted electron-configuration is unstable and one has to introduce non-electromagnetic force to stabilize the electrons - Abraham himself questioned the possibility of including such forces within the theory of Lorentz.
So it was Poincaré (1905) on 5 June 1905Poincaré (1905b) who introduced the so called "Poincaré stresses" to solve that problem. Those stresses were interpreted by him as an external, non-electromagnetic pressure, which stabilize the electrons and also served as an explanation for length contraction.Janssen/Mecklenburg (2007) Although he argued that Lorentz succeeded in creating a theory which complies to the postulate of relativity, he showed that Lorentz's equations of electrodynamics were not fully Lorentz covariant
Lorentz covariance
In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space"...
. So by pointing out the group characteristics of the transformation Poincaré demonstrated the Lorentz covariance of the Maxwell-Lorentz equations and corrected Lorentz's transformation formulae for charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
and current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
. He went on to sketch a model of gravitation (incl. gravitational wave
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...
s) which might be compatible with the transformations. Poincaré used for the first time the term "Lorentz transformation", and he gave them a form which is used up to this day. (Where
is an arbitrary function of , which must be set to unity to conserve the group characteristics. He also set the speed of light to unity.)A substantially extended work (the so called „Palermo paper“)Poincaré (1906) was submitted by Poincaré on 23 July 1905, but was published on January 1906, because the journal only appeared two times in a year. He spoke literally of "the postulate of relativity", he showed that the transformations are a consequence of the principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
, and he showed that the combination
is invariant. While elaborating his gravitational theory he noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing as a fourth imaginary coordinate, and he used an early form of four-vectorFour-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
s. However, Poincaré later said the translation of physics into the language of four-dimensional metry would entail too much effort for limited profit, and therefore he refused to work out the consequences of this notion. This was later done by Minkowski, see "The shift to relativity".Walter (2007), Kap. 1
Electromagnetic mass
J. J. ThomsonJ. J. Thomson
Sir Joseph John "J. J." Thomson, OM, FRS was a British physicist and Nobel laureate. He is credited for the discovery of the electron and of isotopes, and the invention of the mass spectrometer...
(1881) and others noticed, that electromagnetic energy contributes to the mass of charged bodies by the amount
, which was called electromagnetic or "apparent mass". Another derivation of some sort of electromagnetic mass was conducted by Poincaré (1900). By using the momentumMomentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
of electromagnetic fields, he concluded that these fields contribute a mass of
to all bodies, which is necessary to save the center of massCenter of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
theorem.
As noted by Thomson and others, this mass increases also with velocity. Thus in 1899, Lorentz calculated that the ratio of the electron's mass in the moving frame and that of the ether frame is
parallel to the direction of motion, and perpendicular to the direction of motion, where and is an undetermined factor. And in 1904, he set , arriving at the expressions for the masses in different directions (longitudinal and transverse):,where
Many scienists now believed, that the entire mass and all forms of forces are electromagnetic in nature. This idea had to be given up, however, in the course of the development of relativistic mechanics. Abraham (1904) argued (as described in the preceding section #Lorentz transformation), that non-electrical binding forces were necessary within Lorentz's electrons model. But Abraham also noted that different results occurred, dependent on whether the em-mass is calculated from the energy or from the momentum. To solve those problems, Poincaré in 1905 and 1906 introduced some sort of pressure of non-electrical nature, which contributes the amount
to the energy of the bodies, and therefore explains the 4/3-factor in the expression for the electromagnetic mass-energy relation. However, while Poincaré's expression for the energy of the electrons was correct, he erroneously stated that only the em-energy contributes to the mass of the bodies.Janssen/Mecklenburg (2007)The concept of electromagnetic mass is not considered anymore as the cause of mass per se, because the entire mass (not only the electromagnetic part) is proportional to energy, and can be converted into different forms of energy, which is explained by Einstein's mass–energy equivalence.Miller (1981), 359-360
Lorentz's theories
In 1900Lorentz (1900) Lorentz tried to explain gravity on the basis of the Maxwell equations. He first considered a Le Sage type modelLe Sage's theory of gravitation
Le Sage's theory of gravitation is a kinetic theory of gravity originally proposed by Nicolas Fatio de Duillier in 1690 and later by Georges-Louis Le Sage in 1748. The theory proposed a mechanical explanation for Newton's gravitational force in terms of streams of tiny unseen particles impacting...
and argued that there possibly exists a universal radiation field, consisting of very penetrating em-radiation, and exerting a uniform pressure on every body. Lorentz showed that an attractive force between charged particles would indeed arise, if it is assumed that the incident energy is entirely absorbed. This was the same fundamental problem which had afflicted the other Le Sage models, because the radiation must vanish somehow and any absorption must lead to an enormous heating. Therefore Lorentz abandoned this model.
In the same paper, he assumed like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner
Johann Karl Friedrich Zöllner
Johann Karl Friedrich Zöllner was a German astrophysicist who studied optical illusions. He invented the Zöllner illusion where lines that are parallel appear diagonal. The lunar Zöllner crater is named in his honor...
that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...
is that of light. This leads to a conflict with the law of gravitation by Isaac Newton, in which it was shown by Pierre Simon Laplace that a finite speed of gravity leads to some sort of aberration and therefore makes the orbits unstable. However, Lorentz showed that the theory is not concerned by Laplace's critique, because due to the structure of the Maxwell equations only effects in the order v2/c2 arise. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low. He wrote:
In 1908Poincaré (1908a); Poincaré (1908b) Book 3, Ch. 3 Poincaré examined the gravitational theory of Lorentz and classified it as compatible with the relativity principle, but (like Lorentz) he criticized the inaccurate indication of the perihelion advance of Mercury. Contrary to Poincaré, Lorentz in 1914 considered his own theory as incompatible with the relativity principle and rejected it.Lorentz (1914) primary sources
Lorentz-invariant gravitational law
Poincaré argued in 1904 that a propagation speed of gravity which is greater than c is contradicting the concept of local time and the relativity principle. He wrote:However, in 1905 and 1906 Poincaré pointed out the possibility of a gravitational theory, in which changes propagate with the speed of light and which is Lorentz covariant. He pointed out that in such a theory the gravitational force not only depends on the masses and their mutual distance, but also on their velocities and their position due to the finite propagation time of interaction. On that occasion Poincaré introduced four-vectors. Following Poincaré, also Minkowski (1908) and Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
(1910) tried to establish a lorentz-invariant gravitational law.Walter (2007) However, these attempts were superseded because of Einstein's theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, see "The shift to relativity".
Constancy of light
Already in his philosophical writing on time measurements (1898) Poincaré wrote that astronomers like Ole Rømer, in determining the speed of light, simply assume that light has a constant speed, and that this speed is the same in all directions. Without this postulate it would not be possible to infer the speed of light from astronomical observations, as Rømer did based on observations of the moons of Jupiter. Poincaré went on to note that Rømer also had to assume that Jupiter's moons obey Newton's laws, including the law of gravitation, whereas it would be possible to reconcile a different speed of light with the same observations if we assumed some different (probably more complicated) laws of motion. According to Poincaré, this illustrates that we adopt for the speed of light a value that makes the laws of mechanics as simple as possible. (This is an example of Poincaré's conventionalist philosophy.) Poincaré also noted that the propagation speed of light can be (and in practice often is) used to define simultaneity between spatially separate events. However, in that paper he did not go on to discuss the consequences of applying these "conventions" to multiple relatively moving systems of reference. This next step was done by Poincaré in 1900, when he recognized that synchronization by light signals in earth's reference frame leads to Lorentz's local time.Galison (2002)Miller (1981), 186-189 (See the section on "local time" above). And in 1904 Poincaré wrote:Principle of relativity
In 1895Poincaré (1895)Katzir (2005), 275-288 Poincaré argued that experiments like that of Michelson-Morley show that it seems to be impossible to detect the absolute motion of matter or the relative motion of matter in relation to the ether. And although most physicists had other views, Poincaré in 1900Poincaré (1900a); Poincaré (1902), Ch. 9-10 stood to his opinion and alternately used the expressions "principle of relative motion" and "relativity of space". He criticized Lorentz by saying, that it would be better to create a more fundamental theory, which explains the absence of any ether drift, than to create one hypothesis after the other. In 1902Poincaré (1902), Ch. 13 he used for the first time the expression "principle of relativity". In 1904 he appreciated the work of the mathematicians, who saved what he now called the "principle of relativityPrinciple of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
" with the help of hypotheses like local time, but he confessed that this venture was possible only by an accumulation of hypotheses. And he defined the principle in this way (according to MillerMiller (1981), 79 based on Lorentz's theorem of corresponding states): "The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion."
Referring to the critique of Poincaré from 1900, Lorentz wrote in his famous paper in 1904, where he extended his theorem of corresponding states: "Surely, the 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 fundamental 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."
One of the first assessments of Lorentz's paper was by Paul Langevin
Paul Langevin
Paul Langevin was a prominent French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the Comité de vigilance des intellectuels antifascistes, an antifascist organization created in the wake of the 6 February 1934 far right riots...
in May 1905. According to him, this extension of the electron theories of Lorentz and Larmor led to "the physical impossibility to demonstrate the translational motion of the earth". However, Poincaré noticed in 1905 that Lorentz's theory of 1904 was not perfectly "Lorentz invariant" in a few equations such as Lorentz's expression for current density (it was admitted by Lorentz in 1921 that these were defects). As this required just minor modifications of Lorentz's work, also Poincaré asserted that Lorentz had succeeded in harmonizing his theory with the principle of relativity: "It appears that this impossibility of demonstrating the absolute motion of the earth is a general law of nature. [..] Lorentz tried to complete and modify his hypothesis in order to harmonize it with the postulate of complete impossibility of determining absolute motion. He has succeeded in doing so in his article [Lorentz, 1904b]."French original: Il semble que cette impossibilité de démontrer le mouvement absolu soit une loi générale de la nature [..] Lorentz a cherché à more compléter et à more modifier son hypothèse de façon à la mettre en concordance avec le postulate de l' impossibilité complète de la détermination du mouvement absolu. C'est ce qu'il a réussi dans son article intitulé [Lorentz, 1904b]
In his Palermo paper (1906), Poincaré called this "the postulate of relativity“, and although he stated that it was possible this principle might be disproved at some point (and in fact he mentioned at the paper's end that the discovery of magneto-cathode ray
Cathode ray
Cathode rays are streams of electrons observed in vacuum tubes. If an evacuated glass tube is equipped with two electrodes and a voltage is applied, the glass opposite of the negative electrode is observed to glow, due to electrons emitted from and travelling perpendicular to the cathode Cathode...
s by Paul Ulrich Villard
Paul Ulrich Villard
Paul Ulrich Villard was a French chemist and physicist, born in Saint-Germain-au-Mont-d'Or, Rhône, 28 September 1860...
(1904) seems to threaten itWalter (2007), Chap. 1), he believed it was interesting to consider the consequences if we were to assume the postulate of relativity was valid without restriction. This would imply that all forces of nature (not just electromagnetism) must be invariant under the Lorentz transformation. In 1921 Lorentz credited Poincaré for establishing the principle and postulate of relativity and wrote:Lorentz (1921), pp. 247-261 "I have not established the principle of relativity as rigorously and universally true. Poincaré, on the other hand, has obtained a perfect invariance of the electro-magnetic equations, and he has formulated 'the postulate of relativity', terms which he was the first to employ."French original: je n'ai pas établi le principe de relativité comme rigoureusement et universellement vrai. Poincaré, au contraire, a obtenu une invariance parfaite des équations de l’électrodynamique, et il a formule le « postulat de relativité » , termes qu’il a été le premier a employer.
Ether
Poincaré wrote in the sense of his conventionalistConventionalism
Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on agreements in society, rather than on external reality...
philosophy in 1889: Poincaré (1889); Poincaré (1902), Ch. 12 "Whether the ether exists or not matters little - let us leave that to the metaphysicians; what is essential for us is, that everything happens as if it existed, and that this hypothesis is found to be suitable for the explanation of phenomena. After all, have we any other reason for believing in the existence of material objects? That, too, is only a convenient hypothesis; only, it will never cease to be so, while some day, no doubt, the ether will be thrown aside as useless."
He also denied the existence of absolute space and time by saying in 1901:Poincaré (1901a); Poincaré (1902), Ch. 6 "1. There is no absolute space, and we only conceive of relative motion ; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred. 2. There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention. 3. Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events occurring in two different places. I have explained this in an article entitled "Mesure du Temps" [1898]. 4. Finally, is not our Euclidean geometry in itself only a kind of convention of language?"
However, Poincaré himself never abandoned the ether hypothesis and stated in 1900: "Does our ether actually exist ? We know the origin of our belief in the ether. If light takes several years to reach us from a distant star, it is no longer on the star, nor is it on the earth. It must be somewhere, and supported, so to speak, by some material agency." And referring to the Fizeau experiment
Fizeau experiment
The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Albert Einstein later pointed out the importance of the experiment for special relativity...
, he even wrote: "The ether is all but in our grasp." He also said the ether is necessary to harmonize Lorentz's theory with Newton's third law. Even in 1912 in a paper called "The Quantum theory", Poincaré ten times used the word "ether", and described light as "luminous vibrations of the ether".Poincaré 1912; Poincaré 1913, Ch. 6
And although he admitted the relative and conventional character of space and time, he believed that the classical convention is more "convenient" and continued to distinguish between "true" time in the ether and "apparent" time in moving systems. Addressing the question if a new convention of space and time is needed he wrote in 1912:Poincaré (1913), Ch. 2 "Shall we be obliged to modify our conclusions? Certainly not; we had adopted a convention because it seemed convenient and we had said that nothing could constrain us to abandon it. Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one in order not to disturb their old habits, I believe, just between us, that this is what they shall do for a long time to come."
Also Lorentz argued during his lifetime that in all frames of reference this one has to be preferred, in which the ether is at rest. Clocks in this frame are showing the "real“ time and simultaneity is not relative. However, if the correctness of the relativity principle is accepted, it is impossible to find this system by experiment.Lorentz (1913), p. 75
Special relativity
In 1905, Albert EinsteinAlbert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
published his paper on what is now called special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
.Einstein (1905a) In this paper, by examining the fundamental meanings of the space and time coordinates used in physical theories, Einstein showed that the "effective" coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. From this followed all of the physically observable consequences of LET, along with others, all without the need to postulate an unobservable entity (the ether). Einstein identified two fundamental principles, each founded on experience, from which all of Lorentz's electrodynamics follows:
1. The laws by which physical processes occur are the same with respect to any system of inertial coordinates (the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
)
2. In empty space light propagates at an absolute speed c in any system of inertial coordinates (the principle of the constancy of light)
Taken together (along with a few other tacit assumptions such as isotropy and homogeneity of space), these two postulates lead uniquely to the mathematics of special relativity. Lorentz and Poincaré had also adopted these same principles, as necessary to achieve their final results, but didn't recognize that they were also sufficient, and hence that they obviated all the other assumptions underlying Lorentz's initial derivations (many of which later turned out to be incorrect The three best known examples are (1) the assumption of Maxwell's equations, and (2) the assumptions about finite structure of the electron, and (3) the assumption that all mass was of electromagnetic origin. Maxwell's equations were subsequently found to be invalid and were replaced with quantum electrodynamics, although one particular feature of Maxwell's equations, the invariance of a characteristic speed, has remained. The electron's mass is now regarded as a pointlike particle, and Poincaré already showed in 1905 that it is not possible for all the mass of the electron to be electromagnetic in origin. This is how relativity invalidated the 19th century hopes for basing all of physics on electromagnetism.). Therefore, special relativity very quickly gained wide acceptance among physicists, and the 19th century concept of a luminiferous ether was no longer considered useful.Darrigol (2005), 15-18Janssen (1995), Kap. 4
Einstein's 1905 presentation of special relativity was soon supplemented, in 1907, by Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
, who showed that the relations had a very natural interpretationSee Whittaker's History of the Aether, in which he writes "the great advances made by Minkowski were connected with his formulation of physics in terms of a four-dimensional manifold... in order to represent natural phenomena without introducing contingent elements, it is necessary to abandon the customary three-dimensional system of coordinates and to operate in four dimensions". See also Pais's Subtle is the Lord, in which it says of Minkowski's interpretation "Thus began the enormous simplification of special relativity". See also Miller's "Albert Einstein's Special Theory of Relativity" in which it says "Minkowski's results led to a deeper understanding of relativity theory". in terms of a unified four-dimensional "spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
" in which absolute intervals are seen to be given by an extension of the Pythagorean theorem. (Already in 1906 Poincaré anticipated some of Minkowski's ideas, see the section "Lorentz-transformation").Walter (1999) The utility and naturalness of the representations by Einstein and Minkowski contributed to the rapid acceptance of special relativity, and to the corresponding loss of interest in Lorentz's ether theory.
In 1907 Einstein criticized the "ad hoc
Ad hoc
Ad hoc is a Latin phrase meaning "for this". It generally signifies a solution designed for a specific problem or task, non-generalizable, and not intended to be able to be adapted to other purposes. Compare A priori....
" character of Lorentz's contraction hypothesis in his theory of electrons, because according to him it was only invented to rescue the hypothesis of an immobile ether. Einstein thought it necessary to replace Lorentz's theory of electrons by assuming that Lorentz's "local time" can simply be called "time", and he stated that the immobile ether as the theoretical fundament of electrodynamics was unsatisfactory.Einstein (1907) And in 1910Einstein (1909) and 1912Einstein (1912) Einstein explained that he borrowed the principle of the constancy of light from Lorentz's immobile ether, but he recognized that this principle together with the principle of relativity makes the ether useless and leads to special relativity. Minkowski ironically said that for Lorentz the contraction hypothesis is only a "gift from above". And although Lorentz's hypothesis is "completely equivalent with the new concept of space and time", Minkowski held that it becomes much more comprehensible in the framework of the new spacetime physics. However, Lorentz disagreed that it was "ad-hoc" and he argued in 1913 that there is little difference between his theory and the negation of a preferred reference frame, as in the theory of Einstein and Minkowski, so that it is a matter of taste which theory one prefers..
Mass–energy equivalence
It was derived by Einstein (1905) as a consequence of the relativity principle, that inertia of energy is actually represented by, but in contrast to Poincaré's 1900-paper, Einstein recognized that matter itself loses or gains mass during the emission or absorption.Einstein (1905b) So the mass of any form of matter is equal to a certain amount of energy, which can be converted into and re-converted from other forms of energy. This is the mass–energy equivalence, represented by . So Einstein didn't have to introduce "fictitious" masses and also avoided the perpetual motionPerpetual motion
Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not....
problem, because according to DarrigolDarrigol (2005), 18-21
, Poincaré's radiation paradox can simply be solved by applying Einstein's equivalence. If the light source loses mass during the emission by
, the contradiction in the momentum law vanishes without the need of any compensating effect in the ether.Similar to Poincaré, Einstein concluded in 1906 that the inertia of (electromagnetic) energy is a necessary condition for the center of mass theorem to hold in systems, in which electromagnetic fields and matter are acting on each other. Based on the mass–energy equivalence he showed that emission and absorption of em-radiation and therefore the transport of inertia solves all problems. On that occasion, Einstein referred to Poincaré's 1900-paper and wrote:Einstein (1906)
Also Poincaré's rejection of the reaction principle due to the violation of the mass conservation law can be avoided through Einstein's
, because mass conservation appears as a special case of the energy conservation law.General relativity
The attempts of Lorentz and Poincaré (and other attempts like those of Abraham and Gunnar NordströmGunnar Nordström
Gunnar Nordström was a Finnish theoretical physicist best remembered for his theory of gravitation, which was an early competitor of general relativity...
) to formulate a theory of gravitation, were superseded by Einstein's theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.Walter 2007 This theory is based on principles like the equivalence principle
Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
, the general principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
, the principle of general covariance
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
, geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
motion, local Lorentz covariance (the laws of special relativity apply locally for all inertial observers), and that spacetime curvature is created by stress-energy within the spacetime.
In 1920 Einstein compared Lorentz's ether with the "gravitational ether" of general relativity. He said that immobility is the only mechanical property of which the ether has not been deprived by Lorentz, but contrary to the luminiferous and Lorentz's ether the ether of general relativity has no mechanical property, not even immobility:Einstein (1922)
Priority
Some claim that Poincaré and Lorentz are the true founders of special relativity, but not Einstein. For more Details see the article on Relativity priority disputeRelativity priority dispute
Albert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most...
.
Later activity and Current Status
Viewed as a theory of elementary particles, Lorentz's electron/ether theory was superseded during the first few decades of the 20th century, first by quantum mechanics and then by quantum field theory. As a general theory of dynamics, Lorentz and Poincare had already (by about 1905) found it necessary to invoke the principle of relativity itself in order to make the theory match all the available empirical data. By this point, the last vestiges of a substantial ether had been eliminated from Lorentz's "ether" theory, and it became both empirically and deductively equivalent to special relativity. The only difference was the metaphysicalLorentz commented that the difference between his and Einstein's view was purely metaphysical, and could be left to the metaphysicians. postulate of a unique absolute rest frame, which was empirically undetectable and played no role in the physical predictions of the theory. As a result, the term "Lorentz ether theory" is sometimes used today to refer to a neo-Lorentzian interpretation of special relativity. The prefix "neo" is used in recognition of the fact that the interpretation must now be applied to physical entities and processes (such as the standard model of quantum field theory) that were unknown in Lorentz's day.Subsequent to the advent of special relativity, only a small number of individuals have advocated the Lorentzian approach to physics. Many of these, such as Herbert E. Ives
Herbert E. Ives
Herbert Eugene Ives was a scientist and engineer who headed the development of facsimile and television systems at AT&T in the first half of the twentieth century. He was also a critic of the special theory of relativity, and attempted to disprove the theory by means of logical arguments and...
(who, along with G. R. Stilwell, performed the first experimental confirmation of time dilation) have been motivated by the belief that special relativity is logically inconsistent, and so some other conceptual framework is needed to reconcile the relativistic phenomena. For example, Ives wrote "The 'principle' of the constancy of the velocity of light is not merely 'ununderstandable', it is not supported by 'objective matters of fact'; it is untenable..."Herbert E. Ives, "Revisions of the Lorentz Transformations", October 27, 1950. However, the logical consistency of special relativity (as well as its empirical success) is well established, so the views of such individuals are considered unfounded within the mainstream scientific community.
John Stewart Bell
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...
advocated teaching special relativity first from the viewpoint of a single Lorentz inertial frame, then showing that Poincare invariance of the laws of physics such as Maxwell's equations is equivalent to the frame-changing arguments often used in teaching special relativity. Because a single Lorentz inertial frame is one of a preferred class of frames, he called this approach Lorentzian in spirit J. Bell, How to Teach Special Relativity.
Works of Lorentz, Poincaré, Einstein
|journal=Archives néerlandaises des sciences exactes et naturelles|volume=25
|pages=363–552}}
|place=Leipzig & Berlin
|publisher=B.G. Teubner}}
|place=Leipzig & Berlin
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Preface partly reprinted in "Science and Hypothesis", Ch. 12.
. Reprinted in Poincaré, Oeuvres, tome IX, pp. 395–413
. Reprinted in "Science and Hypothesis", Ch. 9-10.
. See also the English translation.
. Reprinted in "Science and Hypothesis", Ch. 6-7.
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|place=Paris}}
|place=London and Newcastle-on-Cyne (1905)
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(Wikisource translation)
(Wikisource translation)
Reprinted in Poincaré 1913, Ch. 6.
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. See also: English translation.