# Lorentz ether theory

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 was created between 1892 and 1895 and 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 (including 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.[1] 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 (see the Robertson–Mansouri–Sexl test theory as an example). Because the same mathematical formalism occurs in both, it is not possible to distinguish between LET and SR by experiment. However, in LET the existence of an undetectable aether 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.

## Historical development

### Basic concept

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.[B 1] Lorentz introduced a strict separation between matter (electrons) and aether, whereby in his model the aether 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 aether with the absolute space of Isaac Newton.[B 2] The condition of this aether can be described by the electric field E and the magnetic field H, where these fields represent the "states" of the aether (with no further specification), related to the charges of the electrons. Thus an abstract electromagnetic aether replaces the older mechanistic aether models. Contrary to Clausius, who accepted that the electrons operate by actions at a distance, the electromagnetic field of the aether 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 aether. A fundamental concept of Lorentz's theory in 1895[A 1] was the "theorem of corresponding states" for terms of order v/c. This theorem states that a moving observer with respect to the aether can use the same electrodynamic equations as an observer in the stationary aether system, thus they are making the same observations.

### 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 aether 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 experiment 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 ${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$. 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 1892[A 2] (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 ${\displaystyle v^{2}/(2c^{2})}$ 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[A 1] Lorentz proposed three possible explanations for this relative contraction:[B 3]

• 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. 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 ${\displaystyle l=l_{0}\cdot {\sqrt {1-v^{2}/c^{2}}}}$ (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.[B 4] For further development of this concept, see the section #Lorentz transformation.[A 3]

### Local time

An important part of the theorem of corresponding states in 1892 and 1895 [A 1] was the local time ${\displaystyle t'=t-vx/c^{2}}$, 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 also resting liquids. 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".[A 4] In The Measure of Time he wrote in 1898:[A 5]

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 ${\displaystyle t'=t-vx/c^{2}}$. But because the moving observers don't know anything about their movement, they don't recognize this.[A 6] 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 aether. So according to Darrigol[B 5] 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 synchronization procedure which was called Einstein synchronisation, Darrigol says that Poincaré had the opinion that clocks resting in the aether are showing the true time.[A 4]

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 ${\displaystyle \varepsilon ^{-1/2}}$ or ${\displaystyle (1-(1/2)v^{2}/c^{2})}$". And in 1899[A 7] also Lorentz noted for the frequency of oscillating electrons "that in S the time of vibrations be ${\displaystyle k\varepsilon }$ times as great as in S0", where S0 is the aether frame, S the mathematical-fictitious frame of the moving observer, k is ${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$, and ${\displaystyle \varepsilon }$ is an undetermined factor. [B 6]

### Lorentz 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 transformation. 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[A 7] 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)[A 3] 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 1905,[A 8] 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.[B 7] 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 waves) 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 ${\displaystyle \ell }$ is an arbitrary function of ${\displaystyle \varepsilon }$, which must be set to unity to conserve the group characteristics. He also set the speed of light to unity.)

${\displaystyle x^{\prime }=k\ell \left(x+\varepsilon t\right),\qquad y^{\prime }=\ell y,\qquad z^{\prime }=\ell z,\qquad t^{\prime }=k\ell \left(t+\varepsilon x\right)}$
${\displaystyle k={\frac {1}{\sqrt {1-\varepsilon ^{2}}}}}$

A substantially extended work (the so-called "Palermo paper")[A 9] was submitted by Poincaré on 23 July 1905, but was published in January 1906, because the journal only appeared twice 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 ${\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$ 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 ${\displaystyle ct{\sqrt {-1}}}$ as a fourth imaginary coordinate, and he used an early form of four-vectors. However, Poincaré later said the translation of physics into the language of four-dimensional geometry 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".[B 8]

### Electromagnetic mass

J. J. Thomson (1881) and others noticed, that electromagnetic energy contributes to the mass of charged bodies by the amount ${\displaystyle m=(4/3)E/c^{2}}$, 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 ${\displaystyle E_{em}/c^{2}}$ 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 ${\displaystyle k^{3}\varepsilon }$ parallel to the direction of motion, and ${\displaystyle k\varepsilon }$ perpendicular to the direction of motion, where ${\displaystyle k={\sqrt {1-v^{2}/c^{2}}}}$ and ${\displaystyle \varepsilon }$ is an undetermined factor.[A 7] And in 1904, he set ${\displaystyle \varepsilon =1}$, arriving at the expressions for the masses in different directions (longitudinal and transverse):[A 3]

${\displaystyle m_{L}={\frac {m_{0}}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{3}}},\quad m_{T}={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},}$

where

${\displaystyle m_{0}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}}$

Many scientists now believed that the entire mass and all forms of forces were 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[A 8] and 1906[A 9] introduced some sort of pressure of non-electrical nature, which contributes the amount ${\displaystyle -(1/3)E/c^{2}}$ 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.[B 9]

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.[B 10]

### Gravitation

#### Lorentz's theories

In 1900[A 10] Lorentz tried to explain gravity on the basis of the Maxwell equations. He first considered a Le Sage type model 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 1908[A 11] 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.[A 12]

#### 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: [A 4]

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.[A 8] Following Poincaré, also Minkowski (1908) and Arnold Sommerfeld (1910) tried to establish a Lorentz-invariant gravitational law.[B 11] However, these attempts were superseded because of Einstein's theory of general relativity, see "The shift to relativity".

## Principles and conventions

Henri Poincaré

### Constancy of light

Already in his philosophical writing on time measurements (1898),[A 5] 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,[A 6] when he recognized that synchronization by light signals in earth's reference frame leads to Lorentz's local time.[B 12][B 13] (See the section on "local time" above). And in 1904 Poincaré wrote:[A 4]

### Principle of relativity

In 1895[A 13][B 14] 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 1900[A 14] 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 1902[A 15] he used for the first time the expression "principle of relativity". In 1904[A 4] he appreciated the work of the mathematicians, who saved what he now called the "principle of 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 Miller[B 15] 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:[A 3] "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 [A 8] 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. It is what he has succeeded in doing in his article entitled Electromagnetic phenomena in a system moving with any velocity smaller than that of light [Lorentz, 1904b]."[C 2]

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 rays by Paul Ulrich Villard (1904) seems to threaten it[B 16]), 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.[A 9] In 1921 Lorentz credited Poincaré for establishing the principle and postulate of relativity and wrote:[A 16] "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."[C 3]

### Aether

Poincaré wrote in the sense of his conventionalist philosophy in 1889: [A 17] "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:[A 18] "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: [A 14] "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".[A 19]

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:[A 20] "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.[A 21]

## The shift to relativity

Albert Einstein

### Special relativity

In 1905, Albert Einstein published his paper on what is now called special relativity.[A 22] 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 [C 4]). Therefore, special relativity very quickly gained wide acceptance among physicists, and the 19th century concept of a luminiferous ether was no longer considered useful.[B 17][B 18]

Einstein's 1905 presentation of special relativity was soon supplemented, in 1907, by Hermann Minkowski, who showed that the relations had a very natural interpretation[C 5] 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").[B 19] 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 1909[A 23] and 1912[A 24] Einstein explained:[B 20]

In 1907 Einstein criticized the "ad hoc" character of Lorentz's contraction hypothesis in his theory of electrons, because according to him it was an artificial assumption to make the Michelson–Morley experiment conform to Lorentz's stationary aether and the relativity principle.[A 25] Einstein argued that Lorentz's "local time" can simply be called "time", and he stated that the immobile ether as the theoretical foundation of electrodynamics was unsatisfactory.[A 26] He wrote in 1920:[A 27]

Minkowski argued that Lorentz's introduction of the contraction hypothesis "sounds rather fantastical", since it is not the product of resistance in the aether but a "gift from above". He said that this hypothesis is "completely equivalent with the new concept of space and time", though it becomes much more comprehensible in the framework of the new spacetime geometry.[A 28] 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.[A 21]

### Mass–energy equivalence

It was derived by Einstein (1905) as a consequence of the relativity principle, that inertia of energy is actually represented by ${\displaystyle E/c^{2}}$, but in contrast to Poincaré's 1900-paper, Einstein recognized that matter itself loses or gains mass during the emission or absorption.[A 29] 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 ${\displaystyle E=mc^{2}}$. So Einstein didn't have to introduce "fictitious" masses and also avoided the perpetual motion problem, because according to Darrigol[B 21] , Poincaré's radiation paradox can simply be solved by applying Einstein's equivalence. If the light source loses mass during the emission by ${\displaystyle E/c^{2}}$, 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:[A 30]

Also Poincaré's rejection of the reaction principle due to the violation of the mass conservation law can be avoided through Einstein's ${\displaystyle E=mc^{2}}$, 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 Nordström) to formulate a theory of gravitation were superseded by Einstein's theory of general relativity.[B 22] 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:[A 27]

### Priority

Some claim that Poincaré and Lorentz are the true founders of special relativity, not Einstein. For more details see the article on this dispute.

## Later activity

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, most 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 main difference was the metaphysical postulate of a unique absolute rest frame, which was empirically undetectable and played no role in the physical predictions of the theory, as Lorentz wrote in 1909,[C 7] 1910 (published 1913),[C 8] 1913 (published 1914),[C 9] or in 1912 (published 1922).[C 10]

As a result, the term "Lorentz ether theory" is sometimes used today to refer to a neo-Lorentzian interpretation of special relativity.[B 23] 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 (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...".[C 11] 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.[B 24]

Also some test theories of special relativity use some sort of Lorentzian framework. For instance, the Robertson–Mansouri–Sexl test theory introduces a preferred aether frame and includes parameters indicating different combinations of length and times changes. If time dilation and length contraction of bodies moving in the aether have their exact relativistic values, the complete Lorentz transformation can be derived and the aether is hidden from any observation, which makes it kinematically indistinguishable from the predictions of special relativity. Using this model, the Michelson–Morley experiment, Kennedy–Thorndike experiment, and Ives–Stilwell experiment put sharp constraints on violations of Lorentz invariance.

## References

For a more complete list with sources of many other authors, see History of special relativity#References.

### Works of Lorentz, Poincaré, Einstein, Minkowski (group A)

1. ^ a b c Lorentz (1895)
2. ^ Lorentz (1892)
3. ^ a b c d Lorentz (1904b)
4. Poincaré (1904); Poincaré (1905a), Ch. 8
5. ^ a b Poincaré (1898); Poincaré (1905a), Ch. 2
6. ^ a b Poincaré (1900b)
7. ^ a b c Lorentz (1899)
8. ^ a b c d Poincaré (1905b)
9. ^ a b c Poincaré (1906)
10. ^ Lorentz (1900)
11. ^ Poincaré (1908a); Poincaré (1908b) Book 3, Ch. 3
12. ^ Lorentz (1914) primary sources
13. ^ Poincaré (1895)
14. ^ a b Poincaré (1900a); Poincaré (1902), Ch. 9–10
15. ^ Poincaré (1902), Ch. 13
16. ^ Lorentz (1921), pp. 247–261
17. ^ Poincaré (1889); Poincaré (1902), Ch. 12
18. ^ Poincaré (1901a); Poincaré (1902), Ch. 6
19. ^ Poincaré 1912; Poincaré 1913, Ch. 6
20. ^ Poincaré (1913), Ch. 2
21. ^ a b Lorentz (1913), p. 75
22. ^ Einstein (1905a)
23. ^ Einstein (1909)
24. ^ a b Einstein (1912)
25. ^ Einstein (1908a)
26. ^ Einstein (1907)
27. ^ a b Einstein (1922)
28. ^ Minkowski (1908)
29. ^ Einstein (1905b)
30. ^ Einstein (1906)
• Lorentz, Hendrik Antoon (1886), "De l'influence du mouvement de la terre sur les phénomènes lumineux", Archives néerlandaises des sciences exactes et naturelles, 21: 103–176
• Lorentz, Hendrik Antoon (1892a), "La Théorie electromagnétique de Maxwell et son application aux corps mouvants", Archives néerlandaises des sciences exactes et naturelles, 25: 363–552
• Lorentz, Hendrik Antoon (1909), The theory of electrons and its applications to the phenomena of light and radiant heat, Leipzig & Berlin: B.G. Teubner
• Lorentz, Hendrik Antoon; Einstein, Albert & Minkowski, Hermann (1913), Das Relativitätsprinzip. Eine Sammlung von Abhandlungen, Leipzig & Berlin: B.G. Teubner
• Lorentz, Hendrik Antoon (1931) [1922], Lecture on theoretical physics, Vol.3 (Lectures held between 1910–1912, first published in Dutch in 1922, English translation in 1931), London: MacMillan
• Lorentz, Hendrik Antoon; Lorentz, H. A.; Miller, D. C.; Kennedy, R. J.; Hedrick, E. R.; Epstein, P. S. (1928), "Conference on the Michelson–Morley Experiment", The Astrophysical Journal, 68: 345–351, Bibcode:1928ApJ....68..341M, doi:10.1086/143148
• Poincaré, Henri (1889), Théorie mathématique de la lumière, 1, Paris: G. Carré & C. Naud Preface partly reprinted in "Science and Hypothesis", Ch. 12.
• Poincaré, Henri (1895), "A propos de la Théorie de M. Larmor", L'éclairage électrique, 5: 5–14. Reprinted in Poincaré, Oeuvres, tome IX, pp. 395–413
• Poincaré, Henri (1913) [1898], "The Measure of Time", The foundations of science, New York: Science Press, pp. 222–234
• Poincaré, Henri (1901a), "Sur les principes de la mécanique", Bibliothèque du Congrès international de philosophie: 457–494. Reprinted in "Science and Hypothesis", Ch. 6–7.
• Poincaré, Henri (1906a) [1904], "The Principles of Mathematical Physics", Congress of arts and science, universal exposition, St. Louis, 1904, 1, Boston and New York: Houghton, Mifflin and Company, pp. 604–622
• Poincaré, Henri (1913) [1908], "The New Mechanics", The foundations of science (Science and Method), New York: Science Press, pp. 486–522
• Poincaré, Henri (1910) [1909], "La Mécanique nouvelle (Göttingen)", Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und mathematischen Physik, Leipzig und Berlin: B.G.Teubner, pp. 41–47
• Poincaré, Henri (1912), "L'hypothèse des quanta", Revue scientifique, 17: 225–232 Reprinted in Poincaré 1913, Ch. 6.
• Minkowski, Hermann (1909) [1908], "Space and Time", Physikalische Zeitschrift, 10: 75–88

### Secondary sources (group B)

1. ^ Whittaker (1951), 386ff
2. ^ Born (1964), 172ff
3. ^ Brown (2001)
4. ^ Miller (1981), 70–75,
5. ^ Darrigol (2005), 10–11
6. ^ Janssen (1995), Chap. 3.5.4
7. ^ Janssen/Mecklenburg (2007)
8. ^ Walter (2007), Kap. 1
9. ^ Janssen/Mecklenburg (2007)
10. ^ Miller (1981), 359–360
11. ^ Walter (2007)
12. ^ Galison (2002)
13. ^ Miller (1981), 186–189
14. ^ Katzir (2005), 275–288
15. ^ Miller (1981), 79
16. ^ Walter (2007), Chap. 1
17. ^ Darrigol (2005), 15–18
18. ^ Janssen (1995), Kap. 4
19. ^ Walter (1999)
20. ^ Martinez (2009)
21. ^ Darrigol (2005), 18–21
22. ^ Walter 2007
23. ^ Balashov / Janssen, 2002
24. ^ J. Bell, How to Teach Special Relativity
• Born, Max (1964), Einstein's Theory of Relativity, Dover Publications, ISBN 0-486-60769-0
• Darrigol, Olivier (2000), Electrodynamics from Ampére to Einstein, Oxford: Clarendon Press, ISBN 0-19-850594-9
• Galison, Peter (2003), Einstein's Clocks, Poincaré's Maps: Empires of Time, New York: W.W. Norton, ISBN 0-393-32604-7
• Alberto A. Mart́ínez (2009), Kinematics: the lost origins of Einstein's relativity, Johns Hopkins University Press, ISBN 0-8018-9135-3
• Miller, Arthur I. (1981), Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2
In English: Pauli, W. (1981) [1921]. Theory of Relativity. Fundamental Theories of Physics. 165. Dover Publications. ISBN 0-486-64152-X.
• Whittaker, Edmund Taylor (1951), A History of the theories of aether and electricity Vol. 1: The classical theories (2. ed.), London: Nelson

### Other notes and comments (group C)

1. ^ French original: Nous n’avons pas l’intuition directe de la simultanéité, pas plus que celle de l’égalité de deux durées. Si nous croyons avoir cette intuition, c’est une illusion. Nous y suppléons à l’aide de certaines règles que nous appliquons presque toujours sans nous en rendre compte. [...] Nous choisissons donc ces règles, non parce qu’elles sont vraies, mais parce qu’elles sont les plus commodes, et nous pourrions les résumer en disant: « La simultanéité de deux événements, ou l’ordre de leur succession, l’égalité de deux durées, doivent être définies de telle sorte que l’énoncé des lois naturelles soit aussi simple que possible. En d’autres termes, toutes ces règles, toutes ces définitions ne sont que le fruit d’un opportunisme inconscient. »
2. ^ French original: Il semble que cette impossibilité de démontrer le mouvement absolu soit une loi générale de la nature [..] Lorentz a cherché à compléter et à 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 Electromagnetic phenomena in a system moving with any velocity smaller than that of light.
3. ^ 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.
4. ^ 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.
5. ^ See 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".
6. ^ German original: Trotzdem die einfachen formalen Betrachtungen, die zum Nachweis dieser Behauptung durchgeführt werden müssen, in der Hauptsache bereits in einer Arbeit von H. Poincaré enthalten sind [Lorentz-Festschrift, p. 252, 1900], werde ich mich doch der Übersichtlichkeit halber nicht auf jene Arbeit stützen.
7. ^ Lorentz 1909, p. 229: It will be clear by what has been said that the impressions received by the two observers A0 and A would be alike in all respects. It would be impossible to decide which of them moves or stands still with respect to the ether, and there would be no reason for preferring the times and lengths measured by the one to those determined by the other, nor for saying that either of them is in possession of the "true" times or the "true" lengths. This is a point which Einstein has laid particular stress on, in a theory in which he starts from what he calls the principle of relativity, i. e. the principle that the equations by means of which physical phenomena may be described are not altered in form when we change the axes of coordinates for others having a uniform motion of translation relatively to the original system.
I cannot speak here of the many highly interesting applications which Einstein has made of this principle. His results concerning electromagnetic and optical phenomena (...) agree in the main with those which we have obtained in the preceding pages, the chief difference being that Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field. By doing so, he may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh and Brace, not a fortuitous compensation of opposing effects, but the manifestation of a general and fundamental principle.
Yet, I think, something may also be claimed in favour of the form in which I have presented the theory. I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. In this line of thought, it seems natural not to assume at starting that it can never make any difference whether a body moves through the ether or not, and to measure distances and lengths of time by means of rods and clocks having a fixed position relatively to the ether.
It would be unjust not to add that, besides the fascinating boldness of its starting point, Einstein's theory has another marked advantage over mine. Whereas I have not been able to obtain for the equations referred to moving axes exactly the same form as for those which apply to a stationary system, Einstein has accomplished this by means of a system of new variables slightly different from those which I have introduced.
8. ^ Lorentz 1913, p. 75: Provided that there is an aether, then under all systems x, y, z, t, one is preferred by the fact, that the coordinate axes as well as the clocks are resting in the aether. If one connects with this the idea (which I would abandon only reluctantly) that space and time are completely different things, and that there is a "true time" (simultaneity thus would be independent of the location, in agreement with the circumstance that we can have the idea of infinitely great velocities), then it can be easily seen that this true time should be indicated by clocks at rest in the aether. However, if the relativity principle had general validity in nature, one wouldn't be in the position to determine, whether the reference system just used is the preferred one. Then one comes to the same results, as if one (following Einstein and Minkowski) deny the existence of the aether and of true time, and to see all reference systems as equally valid. Which of these two ways of thinking one is following, can surely be left to the individual.
9. ^ Lorentz 1914, p. 23: If the observers want to see the concept of time as something primary, something entirely separated from the concept of space, then they would certainly recognize that there is an absolute simultaneity; though they would leave it undecided, whether simultaneity is indicated by equal values of t, or by equal values of t′, or maybe neither by that or the other.
Einstein said in a nutshell, that all of those mentioned questions have no meaning. Then he arrives at the "abandonment" of the aether. Incidentally, the latter is to a certain extent a quarrel about words: it makes no great difference whether one speaks about the vacuum or the aether. In any case, according to Einstein it has no meaning to speak about motion relative to the aether. He also denies the existence of absolute simultaneity.
It is certainly remarkable that these relativity concepts, also with respect to time, have been incorporated so quickly.
The evaluation of these concepts belongs largely to epistemology to which we can left the judgment, trusting that it can consider the discussed questions with the necessary thoroughness. But it is sure that for a large part it depends on the way of thinking to which one is accustomed, whether one feels attracted to the one view or the other. Regarding to the lecturer himself, he finds a certain satisfaction in the older views, that the aether has at least some substantiality, that space and time can be strictly separated, that one can speak about simultaneity without further specification. Regarding the latter, one can probably refer to the ability that arbitrary great velocities can at least imagined by us. By that, one comes very near to the concept of absolute simultaneity.
10. ^ Lorentz 1922, p. 125: We thus have the choice between two different plans: we can adhere to the concept of an aether or else we can assume a true simultaneity. If one keeps strictly to the relativistic view that all systems are equivalent, one must give up the substantiality of the aether as well as the concept of a true time. The choice of the standpoint depends thus on very fundamental considerations, especially about the time.
Of course, the description of natural phenomena and the testing of what the theory of relativity has to say about them can be carried out independently of what one thinks of the aether and the time. From a physical point of view these questions can be left on one side, and especially the question of the true time can be handed over to the theory of knowledge.
The modern physicists, as Einstein and Minkowski, speak no longer about the aether at all. This, however, is a question of taste and of words. For, whether there is an aether or not, electromagnetic fields certainly exist, and so also does the energy of the electrical oscillations. If we do not like the name of "aether," we must use another word as a peg to hang all these things upon. It is not certain whether "space" can be so extended as to take care not only of the geometrical properties but also of the electric ones.
One cannot deny to the bearer of these properties a certain substantiality, and if so, then one may, in all modesty, call true time the time measured by clocks which are fixed in this medium, and consider simultaneity as a primary concept.
11. ^ Herbert E. Ives, "Revisions of the Lorentz Transformations", October 27, 1950