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Études poincaréiennes(I)

Poincaré and the Reaction Principle in Electrodynamics

Olivier Darrigol
p. 63-125

Résumés

Quand, dans les années 1890, Henri Poincaré passa en revue les diverses théories électrodynamiques alors en compétition, il exigea leur compatibilité avec deux principes hérités de la mécanique : le principe de réaction et le principe de relativité. Les historiens de la Relativité se sont généralement concentrés sur ce second principe et ils ont négligé ou mal compris son souci de respecter le principe de réaction. En particulier, la plupart d’entre eux ont interprété son article crucial de 1900 sur la théorie de Lorentz et le principe de réaction comme une tentative de sauver ce principe en admettant une quantité de mouvement électromagnétique en sus de la quantité de mouvement de la matière. Le but du présent article est de réfuter cette interprétation et de montrer que Poincaré avait l’intention opposée de mettre en évidence les conséquences paradoxales de la violation du principe de réaction dans la théorie de Lorentz. Ce faisant, il introduisit formellement une quantité que d’autres interprétèrent plus tard comme une véritable quantité de mouvement électromagnétique ; il développa une conception fondamentalement nouvelle des transformations de Lorentz en relation avec le principe de relativité ; et il identifia les paradoxes qu’Albert Einstein résolut quelques années plus tard en admettant l’inertie de l’énergie.

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Texte intégral

I am very thankful to the two reviewers, David Rowe and Scott Walter, who helped me improve this essay. David Rowe offered useful comments on the methodological difficulties encountered by Poincaré scholars. Scott Walter carefully read my entire manuscript and suggested a number of corrections and improvements. I also thank Christian Bracco and Jean-Pierre Provost for stimulating conversations on Poincaré and relativity theory.

  • 1 In conformity with Poincaré’s usage, in the following I will speak of a “violation of the reaction (...)

1In the late nineteenth century, Henri Poincaré judged the competing electromagnetic theories according to their (in)compatibility with a few general principles, especially the principle of relativity and the principle of the equality of action and reaction. As he expected the relativity principle to hold exactly and naturally in the true electrodynamics of the future and as the relativity theories he and Albert Einstein elaborated in 1905 were devised to fulfill this expectation, historians have usually paid more attention to Poincaré’s concern with this principle than to his concern with the reaction principle. Yet, historically, the latter concern came first, and we will see that Poincaré’s worries about violations of the reaction principle in part informed his discussion of the (in)compatibility of Lorentz’s theory with the relativity principle. When, in 1900, Poincaré was asked to contribute to the silver jubilee of Hendrik Lorentz’s doctorate, he decided to write on “Lorentz’s theory and the principle of reaction,” not on “Lorentz’s theory and the principle of relativity.”1

2The purpose of this article is to correct the bias in the historiography of Poincaré’s criticism of electrodynamic theories. This is all be the more useful as the jubilee article marks an essential step toward the theory of relativity, and as the contents of this article have been frequently misrepresented. As Poincaré there wrote a momentum-balance equation in which a modern physicist immediately recognizes the balance between material and electromagnetic momenta, and as Max Abraham indeed based his electron theory of 1902 on this reading of Poincaré’s equation, it has often been claimed that the main purpose of the jubilee article was to save the reaction principle in Lorentz’s theory by introducing a new, electromagnetic kind of momentum in par with the momentum of material particles.

  • 2 Whittaker 1910, p. 353n; Goldberg 1967, p. 941; Cuvaj 1970, pp. 86, 91; Granek 2000, p. 17; Weinst (...)

3Here are a few samples of this misreading in the historical literature. In 1910 already, Edmund Whittaker stated: “The hypothesis that the ether is a storehouse of mechanical momentum, which was first advanced by J. J. Thomson [in 1893], was afterwards developed by H. Poincaré [in 1900], and by M. Abraham [in 1902].” The claim is correct for Thomson and Abraham, not for Poincaré. In 1967, Stanley Goldberg wrote: “Poincaré indicated how one might reconcile Lorentz’s theory with Newtonian mechanics [by introducing a fictitious fluid of density proportional to the electromagnetic energy density].” In 1970, Camillo Cuvaj described Poincaré’s article as “an attempt to save Newton’s third law and the conservation of momentum by generalizing them to include electromagnetic phenomena,” although he also judged that Poincaré’s attitude toward the reaction principle in Lorentz’s theory was “ambiguous and doubtful.” In 2000, Galina Granek wrote: “Poincaré supposed that he could mathematically re-establish the equality of action and reaction in the theory of Lorentz for the phenomena of radiation pressure.” In 2015, Galina Weinstein wrote: “Lorentz’s theory of the electron violated the principle of action and reaction, but Poincaré believed he could mend this violation.” Lastly, in the Wikipedia entry for Henri Poincaré, one can read (in February 2023): “[Poincaré] noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum.”2

  • 3 Miller 1981, pp. 43, 56 (emphasis mine). Howard Stein [2021, pp. 18–19] also holds an ambiguous po (...)

4At first glance, it would seem that Arthur Miller committed the same error in his influential book of 1981 on the history of relativity as he wrote: “In order to rescue the principle of reaction, Poincaré drew from [a momentum-balance equation derived in Lorentz’s theory] the conclusion that the principles of reaction, and for conservation of momentum, were no longer applicable only to matter with mass; rather electromagnetic energy, which he likened to a fluide fictif with mass, had a ‘momentum’.” This statement is ambiguous and could again be read in terms of an attempt to save the reaction principle. However, in his discussion of Abraham’s electron theory Miller writes: “An important ingredient in Abraham’s theory was the term [Π/c2 where Π is the Poynting vector] from Poincaré’s [momentum-balance equation] which Abraham referred to as the ‘electromagnetic momentum density.’ This was a bold move because, unlike Lorentz and Poincaré, Abraham took [Π/c2] to be a physical entity, thereby maintaining conservation of momentum at the expense of Newton’s third law.” We will indeed see that Poincaré did not at all recommend accepting the violation of the reaction principle in Lorentz’s theory, that he dwelt on the nefarious consequences of this violation, and that he still hoped the electrodynamics of the future would respect the reaction principle.3

5How could a number of respectable Poincaré scholars, with Miller’s partial exception, so crudely misrepresent Poincaré’s intentions? Two reasons have already been suggested: the presentist reading of an equation given by Poincaré, and the authority of Abraham’s first reference to Poincaré. A third reason may be that Paul Langevin and Wilhelm Wien gave Poincaré full credit for the introduction of electromagnetic momentum in widely read obituaries. Lastly, there is the difficulty of Poincaré’s style, especially in the jubilee article, which was written under a short deadline with no careful proofreading. Poincaré tended to formulate his arguments in a subtly nuanced manner, rarely asserting a square fact and rather articulating a range of possibilities through variations of the conditional mode. In the jubilee article he did not clearly announce the purpose of the successive sections, and the reader must figure it out by himself. In the end he did not strictly exclude that nature would violate the reaction principle (or any other principle), he rather accumulated arguments to suggest that this violation came at the cost of grave paradoxes.

6A few years ago, I characterized the contents of Poincaré's jubilee article as follows. Firstly, Poincaré derives the equation

\[(1)\quad\sum{M\mathbf{V}} + \int{c^{- 2}\mathbf{\Pi}\text{d}\tau} = \text{constant}\]

  • 4 Darrigol 1995, pp. 23—29.

from Lorentz's field equations and force formula, where Π stands for the Poynting vector (electromagnetic energy flux), ∑MV for the total momentum of matter, and c for the velocity of light in vacuum. In a fiction in which a fluid of mass density w/c2 is associated with the electromagnetic energy density w, he interprets this equation as the conservation of the total momentum of matter and fictitious fluid. He further derives the uniform, rectilinear motion of this system under the artificial condition that latent fluid at rest is created or annihilated whenever electromagnetic energy is produced or absorbed by matter. He never claims c-2Π to be a genuine momentum density on par with the momentum of matter. In the rest of his article, he will use the fictitious fluid and the associated flux c-2Π only as means to quantify the variation of the total momentum of matter during electromagnetic processes, and he will regard any non-zero value of ∫c2Πdτ as a case of violation of the principle of reaction4.

  • 5 Cf. Katzir 2005, p. 272: “An electromagnetic force that exists only in a moving frame of reference (...)

7In his second section, Poincaré shows that in Lorentz’s theory the air surrounding a unidirectional source of electromagnetic radiation cannot be used to save the reaction principle, because the momentum carried by the air could counterbalance only a small fraction of the recoil momentum of the source. In his third section, he recalls that in Newtonian mechanics a violation of Newton’s third law leads to the possibility of perpetual motion, and he shows that the energy principle and the relativity principle together imply the conservation of momentum. Although these ways of reasoning do not smoothly extend to retarded interactions in electrodynamics, he generally assumes an intimate connection between the two principles, and therefore wonders why in Lorentz’s theory the reaction principle seems to be violated without concomitant violation of the relativity principle. This is indeed the central question of his article. In order to answer this question, he carefully considers how Lorentz’s theory succeeds in accounting for the lack of first-order effects of the motion of the earth through the ether, and he thereby develops a first-order relativity theory based on transformations between true quantities measured by observers in the ether frame and “apparent” quantities measured by observers in the earth frame. This essentially new interpretation of Lorentz’s transformations leads him to the major insight that Lorentz’s “local time” is the time measured by terrestrial observers who synchronize their clocks optically. He relies on this reinterpretation of Lorentz’s transformations to show that for a unidirectional emitter of radiation, the recoil force in the moving frame differs from the recoil force in the ether frame by what he calls “the complementary force.” As he also shows, this force is needed to preserve the energy balance in the moving frame. In his eyes, this force is measurable and represents a first-order violation of the relativity principle, responding to his expectation that a violation of the reaction principle should be accompanied with a violation of the relativity principle.5

8At the end of his article, Poincaré acknowledges that there is no easy way to avoid the violation of the reaction principle in electrodynamics (since 1895 he knows it to result from Fizeau’s experiment under fairly general assumptions), but he still hopes that some deep modification of our basic electrodynamic concepts will allow the principle to be saved.

9This deep modification never happened, and instead physicists adopted the concept of electromagnetic momentum and the concomitant violation of the reaction principle. It thereby became clear that this principle could be violated in a theory that strictly respected the relativity principle. As we will see, one essential ingredient of this conciliation of electromagnetic momentum with the relativity principle was the realization that the mass of a material body depends on its energy content.

  • 6 On this last point, see Darrigol 2000b.

10Retrospectively, Poincaré’s worries about Lorentz’s violation of the reaction principle were exaggerated and his hopes for a deep modification in which this violation would not occur were in vain. However, he did write an equation for the momentum balance in Lorentz’s theory and he did introduce a fictitious electromagnetic momentum that inspired Abraham to introduce a genuine electromagnetic momentum. In addition, Poincaré discovered a physical interpretation of Lorentz’s transformations at first order and he pioneered their application to energy and momentum considerations. Lastly, he showed that the violation of the principle of reaction in Lorentz’s theory had paradoxical consequences when comparing the energy balances in two different inertial frames. Paradoxes of this kind were precisely what later prompted Einstein to introduce the inertia of energy.6

11In this article, I will profusely substantiate my past reading of Poincaré’s jubilee article. The first section gives the necessary background information on the electromagnetic theories criticized by Poincaré and others. The second section retraces the history of his and others’ concern with the reaction principle in electromagnetic theory. The third section addresses the contents of the jubilee article by means of careful textual analysis including translations of the most important passages. The fourth and last section is about the reception of this article.

12The contents of this last section may be summarized as follows. Poincaré’s most careful readers, especially Lorentz, Gustav Mie, Samuel Burbury, and Max Planck, properly understood that he raised the violation of the reaction principle as a major objection to Lorentz’s theory, that he did not try to save this principle through a new concept of momentum, and that Abraham was the true inventor of electromagnetic momentum (with an anticipation by J. J. Thomson). In 1904, Poincaré himself clearly indicated that before Kaufmann’s experiments and Abraham’s relevant theory he had not believed in the reality of electromagnetic momentum. In the same year, Abraham ceased to name Poincaré as the discoverer of electromagnetic momentum and instead joined Lorentz in emphasizing that Poincaré’s intention had been to brandish the violation of the reaction principle as an objection to Lorentz’s theory. As for the inertia of energy, no contemporary physicist read Poincaré as attributing a genuine mass to electromagnetic energy in its exchanges with matter. However, in one of his derivations of the energy-dependence of mass, Einstein employed Poincaré’s theorem for the center of mass of matter and fictitious fluid and reinterpreted this theorem by ascribing to the electromagnetic energy a genuine mass subtracted from radiation emitters and added to a radiation absorbers.

13Some physicists or physicist-trained historians may be inclined to think that even though Poincaré did not support a concept of electromagnetic momentum or a concept of exchangeable electromagnetic mass, he did provide the basic formal structure that supports these new concepts and should therefore be credited for their invention. There is some truth to this: when studying the history of a discovery, one should not belittle the role of formal structures that prepare this discovery. Otherwise one would for instance cease to regard Max Planck as a founding father of quantum theory on the grounds that his energy elements were merely formal and did not imply the quantum discontinuity later imagined by Einstein. Or one would no longer name Werner Heisenberg and Erwin Schrödinger as discoverers of quantum mechanics, on the grounds that in 1925–26 they did not have in hand the true conceptual apparatus of the theory and only knew some of its formal skeleton. At the same time, a competent historian must carefully distinguish between the formal skeleton and the conceptual flesh if he or she does not want to betray the actors’ motivations and thus lose the very essence of historical dynamics. In 1900, Poincaré wanted to spell out damaging consequences of the violation of the reaction principle in Lorentz’s theory. Far from offering a way to avoid these consequences, he aggravated them with the hope this would help others finding the way out. It did.

1 Background

Maxwell’s theory

14In the years 1890–94, several electrodynamic theories appeared in the wake of Heinrich Hertz’s groundbreaking experiments on electromagnetic waves. The first of these theories, proposed by Hertz himself in 1890, was a natural extension of Maxwell’s electromagnetic theory to moving bodies. For bodies at rest, Maxwell’s theory is based on a single ether-matter medium whose states are characterized by four field vectors E, H, D, and B. These vectors satisfy the equations

\[(2)\quad \nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}}{\partial t}, \nabla \times \mathbf{H} = \mathbf{j} + \frac{\partial\mathbf{D}}{\partial t}, \]

\[(3)\quad \nabla \cdot \mathbf{D} = \rho, \nabla \cdot \mathbf{B} = 0,\]

  • 7 Rationalized electromagnetic units are used for all theories before Lorentz’s. Mostly anachronisti (...)

where ρ denotes the electric density and j the electric conduction current.7 In a linear isotropic medium, the relations

\[(4)\quad \mathbf{D} = \varepsilon\mathbf{E},\ \mathbf{B} = \mu\mathbf{H},\ \mathbf{j} = \sigma\mathbf{E} \]

also hold. According to Maxwell, the force density f acting on the medium derives from the stress system

\[(5)\quad p_{ij} = \varepsilon E_{i}E_{j} - \frac{1}{2}\delta_{ij}\varepsilon E^{2} + \mu H_{i}H_{j} - \frac{1}{2}\delta_{ij}\varepsilon H^{2}.\]

Cauchy’s relation fi = ∂jpij then leads to

\[(6)\quad \mathbf{f} = (\nabla \cdot \varepsilon\mathbf{E})\mathbf{E} + (\nabla \times \mathbf{E}) \times \varepsilon\mathbf{E} + (\nabla \cdot \mu\mathbf{H})\mathbf{H} + (\nabla \times \mathbf{H}) \times \mu\mathbf{H},\]

or, using the field equations (2–3),

\[(7)\quad \mathbf{f = \rho E + j}\mathbf{\times}\mathbf{B + \partial(D}\mathbf{\times}\mathbf{B)}/\partial t.\]

  • 8 See Darrigol 2000a, chap. 4, and further reference there. Maxwell only had \( \dot{\mathbf{D}} \ti (...)

The two first terms represent the usual electric and electromagnetic forces. The third term will be discussed in a moment.8

The theories of Hertz and Heaviside

15The two circuital equations (2–3) are the infinitesimal version of the integral laws

\[(8)\quad \small{\oint{\mathbf{E} \cdot \text{d}\mathbf{l} =} - \frac{\text{d}}{\text{d}t}\iint{\mathbf{B} \cdot \text{d}\mathbf{S}}\ \text{and} \oint{\mathbf{H} \cdot \text{d}\mathbf{l}} =\frac{\text{d}}{\text{d}t}\iint{\mathbf{D} \cdot \text{d}\mathbf{S}} + \iint{\mathbf{j} \cdot \text{d}\mathbf{S}}},\]

in which the loop integrals are taken over the boundary of the surfaces for the flux integrals. As the various field vectors in these equations denote intrinsic local properties of the medium, it is natural to assume that in a medium moving at the velocity v (which depends on the point of space), the surfaces of integration should follow the motion of the medium at the local velocity v. Under this assumption, in 1890 Hertz obtained the equations

\[(9)\quad \nabla \times \mathbf{E} = - \frac{\mathrm{D}\mathbf{B}}{\mathrm{D}t},\ \nabla \times \mathbf{H} = \mathbf{j} + \frac{\mathrm{D}\mathbf{D}}{\mathrm{D}t},\]

where D/Dt denotes a convective derivative defined by

\[(10)\quad \frac{\mathrm{D}\mathbf{X}}{\mathrm{D}t} = \frac{\partial\mathbf{X}}{\partial t} - \nabla \times (\mathbf{v} \times \mathbf{X}) + \mathbf{v}(\nabla \cdot \mathbf{X}).\]

In particular, the first circuital equation is replaced with

\[(11)\quad \nabla \times (\mathbf{E} - \mathbf{v} \times \mathbf{B}) = - \frac{\partial\mathbf{B}}{\partial t},\]

  • 9 Hertz 1890. See Darrigol 2000a, pp. 255–257, and further reference there.

which implies the motional electromotive force v×B within a portion of the medium moving across magnetic lines of force. As Hertz showed, his equations captured all the phenomena then known in the electrodynamics of moving conductors, dielectrics, and magnets.9

  • 10 Hertz 1890, pp. 389–398; Heaviside 1891-1892. See Darrigol 2000a, pp. 410–411.

16As Hertz and Oliver Heaviside proved, the generalized field equations imply the relation10

\[(12)\quad\begin{split}\frac{\partial w}{\partial t} + \nabla \cdot (\mathbf{\Pi} + w\mathbf{v}) + \mathbf{j} \cdot \mathbf{E} - p_{ij}\partial_{i}\upsilon_{j} = 0\\ \text{with}\ w = \frac{1}{2}(\varepsilon E^{2} + \mu H^{2})\ \text{and}\ \mathbf{\Pi} = \mathbf{E} \times \mathbf{H}.\end{split}\]

In the simplest case of a linear, isotropic medium in which the effects of the motion of the medium on the value of the ε and μ parameters are neglected, the stress system pij is still given by Maxwell’s Eq. (5). In words, Eq. (12) means that the variation of the electromagnetic energy of density w within a fixed volume element balances the energy flux across the boundary of this element (as given by the Poynting vector Π and the convective flux wv), the Joule heat, and the work of the stresses. This reasoning provides an independent justification of the Maxwell stresses. Again, the resulting force density fi=∂jpij contains the force ∂(D×B)/∂t. As Hertz noted, this force is too small to imply an observable motion of a material medium, even for air at very low density. In a strict vacuum, however, this force still has the non-vanishing value \(\dot{\mathbf{\Pi}}/c^{2}\) (since εμc2 = 1). Hertz commented:

[In this case] the derived pressures should move the interior of the ether—which we have explicitly assumed to be movable—with velocities which we could compute if only we had an indication of the mass of the ether. This result seems to have little intrinsic probability.

  • 11 Hertz 1890, p. 284; 1892, p. 295; Heaviside 1891-1892. All translations are mine. Helmholtz later (...)

In a later footnote, Hertz further mentioned that in his theory a radiating body could lose momentum even in a vacuum, thus violating the conservation of momentum unless the ether carried the missing momentum. He gave the example of a magnetized steel sphere rotating around an axis different from the axis of magnetization. The emitted radiation causes the sphere to lose angular momentum. He commented: “Such consequences seem improbable, but in this domain we have no right to judge from probabilities only, so complete is our ignorance of eventual motions of the ether.”11

  • 12 Hertz 1890, p. 370; Fizeau 1851, 1859. On the Fizeau experiment, see Darrigol 2022, pp. 105–107, a (...)
  • 13 Hertz 1890, p. 399.

17At any rate, Hertz regarded the assumption of a single ether-matter medium as a working hypothesis that could not possibly agree with the few clues physicists then had on the relation between ether and matter. At the beginning of his text, he alluded to experiments indicating that ether and matter move independently of each other. He was certainly aware of Hippolyte Fizeau’s experiment of 1851, in which the phase shift of a wave traveling through running water is measured.12 If the ether were fully dragged by the water, the velocity of light in the water would be c/n in a frame moving together with the water (n being the optical index of water), and the velocity in the laboratory frame would then be c/n + υ in the direction of motion of the water, if υ denotes the velocity of the water in the direction of light propagation. In reality, the phase shift measured by Fizeau agreed with Fresnel’s formula, according to which the net velocity should be c/n+υ(1−1/n2). If this result was correct (Michelson and Morley confirmed it in 1886), ether and matter had to move separately. At the end of his article, Hertz wrote:13

This theory shows how the electromagnetic phenomena in moving bodies can be treated under certain arbitrary limitations. There is little chance that nature should comply with these limitations. The true theory is much more likely to be one that distinguishes the states of the ether from the states of the embedded matter at every point. However, it seems to me that a theory building on this intuition would require more numerous and more arbitrary assumptions than the theory I have developed.

18In 1892 Heaviside perfected Hertz’s electrodynamics of moving bodies and gave his own comments on the force F≡∂(D×B)/∂t:

The vector D×B, or the flux of energy divided by the square of the speed of propagation, is, therefore, the momentum [...] provided the force F is the complete force from all causes acting [...]. But have we any right to safely write F=m∂v/∂t where m is the density of the ether? To do so is to assume that F is the only force acting, and, therefore, equivalent to the time-variation of the momentum of a moving particle.

  • 14 Heaviside 1891-1892, p. 524 (2 citation), 557–558 (1 citation)..

Heaviside goes on to explain that in a dynamical medium one should expect other forces, internal pressures of various kinds, to be acting in addition to the force F. The linear momentum of the medium therefore has little chance to reach the value D×B. In the present theory, the equations for the bulk motion of the medium remain unknown. Fortunately, this bulk motion does not matter much for electrodynamic phenomena as long as the linear velocity of the medium remains negligible compared to the velocity of propagation of disturbances through the medium. This is no longer the case in the optics of moving bodies, because the phase-shift caused by a motion of the transparent medium becomes measurable. As Heaviside knows, Fizeau’s experiment indicates a partial drag of the ether by running water, against the Hertz-Heaviside assumption of a compound ether-matter medium. This is why Heaviside agrees with Hertz that this assumption could only be tentative:14

Now, a really difficult and highly speculative question, at present, is the connection between matter (in the ordinary sense) and ether. When the medium transmitting the electrical disturbances consists of ether and matter, do they move together or does the matter only partially carry forward the ether which immediately surrounds it? Optical reasons may lead us to conclude, though only tentatively, that the latter may be the case; but at present, for the purpose of fixing the data, and in the pursuit of investigations not having especially optical bearing, it is convenient to assume that the matter and ether in contact move together. This is the hypothesis made by Hertz in his recent treatment of the electrodynamics of moving bodies.

To sum up, the electrodynamics of Hertz and Heaviside implied the force ∂(D×B)/∂t acting on dielectrics. Hertz believed this force to be imparting the momentum D×B on the dielectric, but he doubted this would still be true in the vacuum limit for which the dielectric is reduced to the ether. He must have felt that the ether was too immaterial to carry momentum. Heaviside had no such reluctance, for he shared Maxwell’s inclination to treat the ether as a dynamical medium on par with ordinary matter. He did not hesitate to call D×B a momentum, although he suspected that other internal, non-electromagnetic forces, prevented this momentum to appear as a translational motion of the medium.

J. J. Thomson’s moving tubes of force

  • 15 Thomson 1893, pp. 6–21 (p. 21 for the electromagnetic momentum of a moving charge). See Buchwald 1 (...)

19In 1893, J. J. Thomson published his Recent researches in which he attempted to explain electromagnetic phenomena through the motion of electric tubes of force in vacuum. In order to take into account the discrete character of electric charge (as observed in electrolysis and in electric gas discharge), he requires his tubes to carry a unit of electric charge (namely, the flux of the microscopic displacement d across a section of a tube is equated to the electrolytic quantum). The average D in a small, macroscopic portion of space is derived from the flux of the included (quasi-parallel) tubes of force. An energy transfer in the field implies a motion of the tubes at the global velocity u, and this motion in turn implies the average magnetic force H = u × D. By generalized dynamics, the magnetic energy 1/2μ(u×D)2dτ of this motion implies the momentum D×B through derivation with respect to u. This momentum, unlike the one introduced by Hertz, is not to be associated with a translational motion of the ether. There is no mechanical ether in Thomson’s theory. The tubes of force are primitive physical entities endowed with their own dynamics. Their momentum has direct physical significance in studying the motion of electrified bodies. In particular, Thomson computes the electromagnetic momentum of the field associated with a moving electrified sphere and finds it to be proportional to the velocity of the sphere, thus revealing an electromagnetic contribution to the mass of the sphere.15

Lorentz’s theory

20Hertz, Heaviside, and J. J. Thomson were mostly concerned with electrodynamics per se, although they understood that Maxwell’s theory and derived theories should ultimately include optics as a part of electrodynamics. They knew that various optical phenomena, including dispersion, magneto-optics, and the optics of moving bodies eluded Maxwell’s original theory. Indeed, Maxwell’s approach to electromagnetism was essentially phenomenological and macroscopic: it treated matter as continuous and it oversimplified the relation between ether and matter by representing them through a single continuous medium. The first theorist who successfully overcame these limitations of Maxwell’s theory was Hendrik Lorentz. In his theory of 1892, Lorentz represents the electrically active components of matter as ions carrying the electrolytic quantum of charge. He assumes the ether between (and within) the ions to be strictly immobile, and he applies Maxwell’s equations to the electromagnetic field in the ether at rest between the ions. The form of the coupling between the field and the ions is dictated by charge conservation and dynamical consistency. The resulting field equations in the ether frame read

\[(13)\quad \nabla \times \mathbf{e} = - \frac{1}{c}\frac{\partial\mathbf{b}}{\partial t},\ \nabla \times \mathbf{b} = \frac{1}{c}\left( \rho\mathbf{v} + \frac{\partial\mathbf{e}}{\partial t}\right),\ \nabla \cdot \mathbf{e} = \rho,\ \nabla \cdot \mathbf{b} = 0, \]

for the microscopic fields e and b in the presence of ions carrying the charge density ρ and moving with the velocity v. The force density f acting within the ions is given by

\[(14)\quad \mathbf{f} = \rho(\mathbf{e} +{c}^{\mathbf{-}\mathbf{1}}\mathbf{v}\mathbf{\times}\mathbf{b)}.\]

  • 16 Lorentz 1892b. See Darrigol 2000a, pp. 322–332, and further reference there. I use Hertz’s units, (...)

Lorentz derives macroscopic field equations by summing the microscopic equations over all the ions in a microscopic volume element. The ions are free to move within conductors (with viscous resistance), and they are elastically bound to the molecules to which they belong in dielectrics. In moving bodies, molecules and ions follow the bulk motion. In this framework, Lorentz managed to account for all electromagnetic and optical phenomena known at that time. In particular, he derived the partial drag of light waves by moving transparent matter, in agreement with Fresnel’s old prediction and with Fizeau’s experiment of 1851.16

  • 17 See Darrigol 2022, pp. 102–104, and further reference there.
  • 18 Lorentz 1892a, 1895.

21Fresnel originally introduced the partial drag of light waves in order to explain the lack of effect of the motion of the earth on the laws of refraction.17 The Fresnel drag also explains the failure of most other attempts to detect the motion of the earth, through the ether, at first order in the ratio of the velocity of the earth to the velocity of light. In 1892, Lorentz further accounted for Albert Michelson’s failure to detect a second-order effect of the motion of the earth, through the famous contraction hypothesis. In the Versuch published in 1895, he greatly improved his theory through a more direct derivation of the lack of effects of the earth’s motion at first and second order. The idea is to write the field equations in the ether frame for macroscopic bodies moving together at the velocity of the earth, and then to perform a change of variables through which the equations retrieve the form they would have if the earth were at rest in the ether. This invariance indirectly explains the absence of observable optical effects of the earth’s motion. It involves (at first order) the “local time” t−ux/c2, with a shift depending on the abscissa x along the motion of the earth at the velocity u. It should be emphasized that for Lorentz the “corresponding states” resulting from this change of variables do not have any observable meaning, they refer to a fictitious system of bodies at rest.18

  • 19 Reiff 1893. See Darrigol 2000a, p. 322.

22In 1894, the Heilbronn Gymnasium professor Richard Reiff gave another electromagnetic derivation of the Fresnel drag, independently of Lorentz, based on Helmholtz’s special reinterpretation of Maxwell’s equations (1870) and on Helmholtz’s dispersion theory (1893). In this theory, there are two kinds of electric charges, the charges of ions in matter (conductors and dielectrics), and neutral doublets belonging to an infinitely polarizable ether. The interaction between the primitive charges and their currents occurs directly and instantaneously at a distance, but the effective interaction between the ions is retarded owing to the polarizability of the ether. Material polarization and ethereal polarization being of an essentially different nature in this theory, Reiff assumes the former to be dragged by a moving dielectric and the latter not. The resulting macroscopic field equations are the same as in Lorentz’s theory, and the Fresnel drag follows, despite the discrepancy in the basic ideas on which the theory is founded.19

  • 20 Larmor 1894, 1895, 1897. See Darrigol 2000a, pp. 332–343, and further reference there.

23The theories of J. J. Thomson, Lorentz, and Helmholtz-Reiff shared the idea of accommodating the atomistic, ionic structure of matter in Maxwell’s theory, although Thomson and Lorentz were far more concerned with atomistic processes than Helmholtz was. Another influential proponent of the integration of discrete matter and discrete electricity in an electromagnetic field theory was Joseph Larmor. The Irish theorist started with the best quasi-mechanical model of the ether, James MacCullagh’s rotational ether, and tried to graft Kelvin’s vortex atoms on this ether. After this attempt failed, in the Spring of 1894, he introduced electric “monads” or “electrons” as centers of radial twist in the rotational ether. In the end, this ether and the immersed electrons obeyed equations very similar to those of Lorentz’s theory. After encountering Lorentz’s theory, Larmor duplicated most of Lorentz’s results.20

2 The principle of reaction in electromagnetic theories

Poincaré’s criticism in 1895

24Poincaré grew deeply conversant with Maxwell’s theory, with the electromagnetic theory of light, and with Hertz’s experiments and theory as he taught them at the Sorbonne during the years 1887–88, 1891–1892, and 1892–93. At some point, he learned about Larmor’s first theory (of 1893) and about Lorentz’s theory in the version of 1892. This prompted him, in the spring of 1895, to publish a critical review of the competing electrodynamic theories in L’éclairage électrique under the misleadingly narrow title “À propos de la théorie de M. Larmor.” Of the section devoted to Larmor, it will be sufficient to retain the remark that one would vainly seek to detect a flow of the ether and thereby determine its density. As Poincaré knew, Larmor’s theory implies a flow of the ether in a magnetic field, with a velocity inversely proportional to the density of the ether. Oliver Lodge had tried and failed to detect an optical effect of this motion near a powerful magnet. To this result Poincaré commented:

Thus, this motion [of the ether] was so slow that Mr. Lodge’s experiments, although they were very precise, were still not precise enough to detect it. To say all my thoughts, had these experiments been hundred or thousand times more precise, I believe the result would have still been negative. In support of this opinion, I have only reasons of sentiment to offer. Had the result been positive, we would have measured the density of the ether, and—may the reader pardon the vulgarity of this expression—it repugns me to imagine the ether so arrivé.

  • 21 Poincaré 1895, pp. 381–382. On Poincaré’s lectures and on his criticism, see Darrigol 1995. On Poi (...)

The remark is important, as it confirms Poincaré’s reluctance (already expressed in his lectures on optics) to convey truly mechanical attributes to the ether.21

  • 22 In reality, Reiff’s system of equations is incomplete and it can easily be completed to comply wit (...)
  • 23 Poincaré 1895, pp. 391–392.

25Poincaré next considers Hertz’s theory and deplores its incompatibility with the Fresnel-drag. He salutes the success of the Helmholtz-Reiff theory in this regard, but still condemns this theory for violating the conservation of the electric charge.22 Lorentz’s theory evidently does not have this defect, since the permanence of ionic charges is inscribed in its foundation. Poincaré also notes that this theory accounts for the Fresnel drag. He goes on:23

Unfortunately, there remains a grave difficulty: the equality of action and reaction no longer holds.

In order to understand this without entering detailed calculations, a simple example shall suffice. Consider a small conductor A, positively charged and surrounded with ether. Suppose that an electromagnetic wave travels through the ether and that at a certain time this wave reaches A, the electric force caused by this perturbation acts on the charge of A and produces a ponderomotive force acting on the body A. In regards to the principle of action and reaction, this ponderomotive force will not be counterbalanced by any other force acting on ponderable matter, because all other ponderable bodies can be assumed to be very far outside the region in which the ether is disturbed.

One would avoid this difficulty by saying that there is a reaction of the body A on the ether. Yet, it remains true that one could perform or at least conceive an experiment in which the principle of reaction would seem to be failing since the experimenter can only act on ponderable bodies and could not reach the ether. This conclusion seems difficult to admit.

  • 24 Poincaré 1895, pp. 392.
  • 25 This is only a guess. Lorentz will prove it in 1900: see below, section on Poincaré’s jubilee arti (...)
  • 26 See above, section on Hertz’s theory.

This is Poincaré’s first statement of his conviction that the principle of reaction should hold for matter alone, without counting the ether as one of the bodies involved in the balance of forces. In Lorentz’s theory, the process described in this extract implies an action without reaction. Poincaré goes on to remark that in Hertz’s theory the imagined process does not violate the principle of reaction because even the smallest amount of air would provide the needed reaction, since in this theory the air and the ether move together as a single body.24 In Lorentz’s theory, Poincaré observes that the ether is strictly immovable and asserts that the air can only provide a small part of the reaction to the motion of the body A.25 Note that Hertz, unlike Poincaré, conceived a perfect vacuum in which there is pure, matter-less ether only, and therefore deplored a violation the principle of reaction in his own theory.26

  • 27 Poincaré 1895, p. 392 (citation), 395–402 (Hertz’s theory), 403–409 (last section).

26In Poincaré’s reasoning, there is an evident correlation between the assumption of an immovable ether (in Lorentz’s theory) and the difficulty with the principle of reaction. Remember that this assumption played an essential role in Lorentz’s explanation of the partial drag of waves by moving matter. This is why Poincaré remarks: “Thinking about this point [the violation of the reaction principle], one sees that the difficulty is not particular to Lorentz’s theory and that one can hardly explain the partial drag of waves [as Lorentz does] without violating the principle of action and reaction.” Poincaré develops this remark later in his article, after a precise discussion of Hertz’s theory and the resulting force density. He verifies that this force density should be the divergence of the Maxwell stress tensor. The space integral of the divergence of a quantity vanishes if this quantity vanishes at infinity. Therefore, the space integral of the force density vanishes identically, in conformity with the principle of reaction. In a later section, Poincaré proves that in the limit of small and slowly variable fields and velocities, Hertz’s theory is the only field theory that satisfies the principle of reaction and the conservation of electricity and magnetism. Since Hertz’s theory is incompatible with the Fresnel drag and since the conservation of electricity and magnetism cannot be doubted, it turns out to be impossible to explain the Fresnel drag without violating the principle of reaction. Poincaré’s demonstration goes as follows.27

27Any alternative to Hertz’s field equations can be obtained by adding complementary terms X and Y in these equations, so that

\[(15)\quad \nabla \times \mathbf{E} = - \frac{\mathrm{D}\mathbf{B}}{\mathrm{D}t} + \mathbf{X},\ \nabla \times \mathbf{H} = \mathbf{j} + \frac{\mathrm{D}\mathbf{D}}{\mathrm{D}t} + \mathbf{Y}.\]

  • 28 For a density ρ, the relevant convective derivative is Dρ/Dt=∂ρ/∂t+∇⋅(ρv).

The conservation of magnetism and the conservation of electricity read28

\[(16)\quad \frac{\mathrm{D}}{\mathrm{D}t}\nabla \cdot \mathbf{B} = 0,\ \nabla \cdot \mathbf{j} + \frac{\mathrm{D}}{\mathrm{D}t}\nabla \cdot \mathbf{D} = 0.\]

They will be satisfied if and only if ∇⋅= 0 and ∇⋅= 0. We may therefore rewrite the modified Hertz equations under the form

\[(17)\quad\begin{split}\nabla \times (\mathbf{E} - \mathbf{e}) = - \frac{\mathrm{D}\mathbf{B}}{\mathrm{D}t},\ \nabla \times (\mathbf{H} - \mathbf{h}) = \mathbf{j} + \frac{\mathrm{D}\mathbf{D}}{Dt},\\\ \text{with}\ \mathbf{X} = \nabla \times \mathbf{e}\ \text{and}\ \mathbf{Y} = \nabla \times \mathbf{h}.\end{split}\]

  • 29 Poincaré does not write this local balance. Instead he obtains and uses the global balance given b (...)

The resulting equation for the local energy balance reads29

\[(18)\quad \frac{\partial w}{\partial t} + \nabla \cdot (\mathbf{\Pi} + w\mathbf{v}) + \mathbf{j} \cdot \mathbf{E} - p_{ij}\partial_{i}\upsilon_{j} + \mathbf{E} \cdot (\nabla \times \mathbf{h}) + \mathbf{H} \cdot (\nabla \times \mathbf{e}) = 0.\]

Consequently, the integral

\[(19)\quad W = \int{\lbrack\mathbf{E} \cdot (\nabla \times \mathbf{h}) + \mathbf{H} \cdot (\nabla \times \mathbf{e})\rbrack}\mathrm{d}\tau\]

represents the work of a complementary force density acting on the medium. The principle of reaction will be satisfied if and only if this work vanishes for a constant, uniform value of the velocity v, and for any fields E and H vanishing at infinity. In the limit considered by Poincaré, e and h can only be linear functions of E, H, and v, and they cannot involve derivatives. The medium being isotropic, their form is restricted to

\[(20)\quad \mathbf{e} = \alpha\mathbf{v} \times \mathbf{E} + \beta\mathbf{v} \times \mathbf{H},\ \mathbf{h} = \gamma\mathbf{v} \times \mathbf{E} + \delta\mathbf{v} \times \mathbf{H},\]

  • 30 I have simplified Poincaré’s ingenious proof by using the isotropy of the medium at an early stage (...)

Where α, β, γ, δ are four scalar fields.30 Integrating W by parts and using these expressions of e and h, we arrive at the condition that the integral

\[(21)\quad \int{\lbrack(\nabla \times \mathbf{E})} \times (\gamma\mathbf{E} + \delta\mathbf{H}) + (\nabla \times \mathbf{H}) \times (\alpha\mathbf{E} + \beta\mathbf{H})\rbrack \mathrm{d}\tau\]

must vanish identically. The special choices E = 0 and H = 0 lead to β = 0 and γ = 0 respectively. Then the special choices ∇×E = 0 and ∇×0 lead to α = 0 and δ = 0 respectively. This ends the proof that in the limit of small, slowly varying fields and velocity Hertz’s theory is the only electromagnetic field theory that satisfies the principle or reaction (and the conservation of electricity and magnetism).

28Consequently, there is no electromagnetic field theory for which the Fresnel drag can be accounted for without violating the principle of reaction:

The preceding considerations imply that no theory can satisfy the three conditions [Fresnel drag, conservation of electricity, equality of action and reaction] because Hertz’s theory is the only one that satisfies the two last conditions and it does not satisfy the first.

Consequently, we will not be able to avoid this difficulty without deeply altering the generally accepted ideas. Besides, it is hard to see in which direction this alteration should be sought for.

We should therefore give up developing a perfectly satisfactory theory and we should provisionally hold on the least defective theory, which seems to be Lorentz’s.

  • 31 Poincaré 1895, pp. 409, 412.

Poincaré goes on to sketch a way of developing Lorentz’s theory at the macroscopic level, and further comments: 31

It goes without saying that this theory, even though it can serve some of our purposes by giving some shape to our ideas, cannot fully satisfy us, cannot be regarded as definitive.

I find it difficult to admit that the principle of reaction should be violated, even in an apparent manner, and that it would no longer hold when we consider only the actions on ponderable matter and if we leave aside the reaction of this matter on the ether.

Consequently, we will someday have to alter our ideas regarding some important point and break the frame in which we try to place both optical and electrical phenomena.

29At this point, Poincaré remarks that even in the narrower domain of optics, the existing explanations of the Fresnel drag remain unsatisfactory. He of course knows that this drag is adjusted so that the laws of refraction on earth would not be affected by the motion of the earth through the ether. In addition, he is aware of the failure of many attempts to detect effects of the motion of the earth on optical phenomena, including the most recent attempt by Albert Michelson and Edward Morley (1887). This explains why he suddenly shifts from the theoretical difficulties with the principle of reaction to the theoretical difficulties with the relativity of optical phenomena:

Experience has revealed a great number of facts that can be summarized in the following formula: it is impossible to detect the absolute motion of matter—better, the relative motion of ponderable matter with respect to the ether. All we can detect is the motion of ponderable matter with respect to ponderable matter.

Various theories from Fresnel to Lorentz, Poincaré’s goes on, account for this law as long as we neglect quantities proportional to the square of the ratio of the velocity of the earth to the velocity of light. Yet the law seems to remain true at second order according to Michelson and Morley.

We here have a lacuna that does not seem unrelated to the lacuna to which this paragraph is devoted. Indeed, the impossibility of detecting the motion of matter with respect to the ether, and the equality which probably holds between action and reaction without considering the action of matter on the ether, are two facts whose connexity seems evident.

Perhaps the two lacunas will be corrected at the same time.

Poincaré here expresses, for the first time, three convictions that will direct his thinking on electrodynamic theories for a few years:

  • Motion with respect to the ether should forever remain undetectable.
  • The principle of reaction should hold for matter alone, without including the ether.
  • Violations of the first principle should be intimately related to violations of the second.
  • 32 Poincaré 1895, pp. 412–413. In the foreword to his first Sorbonne course, Poincaré [1889, p. II] h (...)

No one in the 1890s would have shared these convictions. The source of Poincaré’s originality may be found in his earlier opinion that the ether was a disposable, formal construct. In his eyes, the ether was not a movable thing, and it could be regarded neither as a concrete reference of motion nor as a concrete momentum carrier. As Poincaré had already written in reaction to Lodge’s ether-whirling experiment, the ether could not be so arrivé.32

Lorentz in 1895

30Poincaré had not seen Lorentz’s Versuch of 1895 when he published his article in L’éclairage électrique. In the first section of this treatise, Lorentz derives the energy balance

\[(22)\quad \rho\mathbf{v} \cdot \mathbf{e} + \frac{\partial w}{\partial t} + \nabla \cdot \mathbf{\Pi} = 0,\ \text{with}\ w = \frac{1}{2}(\mathbf{e}^{2} + \mathbf{b}^{2})\ \text{and}\ \mathbf{\Pi} = c(\mathbf{e} \times \mathbf{b}),\]

simply by multiplying his first field equation by b, the second by e, and adding the two resulting equations. In this balance, the first term yields the work of the Lorentz force in a unit volume element, the second the variation of the field energy in this element, and the third the energy flux across the surface of this element. In addition, Lorentz derives the force formula

\[(23)\quad f_{i} = \partial_{j}p_{ij} - \frac{\partial}{\partial t}\frac{\Pi_{i}}{c^{2}},\ \text{with}\ p_{ij} = e_{i}e_{j} - \frac{1}{2}\delta_{ij}e^{2} + b_{i}b_{j} - \frac{1}{2}\delta_{ij}b^{2},\]

by injecting ρ = ∇⋅e and c−1ρv = ∇×bc−1e/∂t into the Lorentz force f = ρe + c1ρv × b.

31In a stationary field for which the Poynting vector Π  does not depend on time, the Lorenz force is simply given by the divergence of the Maxwell stresses. Lorentz finds it convenient to use these stresses in such cases. However, he notes that in general these stresses would imply a force density c−2Π/∂t acting on the ether in vacuum. One could, he goes on, try to discuss the motion of the ether under these forces, as Helmholtz had done in his last years. But Lorentz prefers to regard the ether as strictly immobile and to refrain from speaking of an action of matter on the ether:

In fact, having already assumed that the ether does not move, why would we speak of a force acting on this medium? The simplest would well be to assume that no force ever acts on a volume element of the ether [...], or not even to apply the concept of force on such an immovable element. Admittedly, this view would run against the principle of the equality of action and reaction—since we indeed have reason to say that the ether exerts forces on ponderable matter—but, as far as I can see, nothing compels us to regard this principle as a fundamental principle of unlimited validity.

32In this case, Lorentz goes one, one should not speak of a stress in the ether, since this stress would represent a force between two contiguous portions of the ether. The Maxwell stresses can still be used as a convenient fiction, for instance in the calculation of radiation pressure. Note that Lorentz, unlike Poincaré, does not directly discuss a violation of the equality of action and reaction for matter alone. The violation he admits concerns the ether/matter interaction. It is nonetheless evident that Eq. (23) implies

\[(24)\quad \int{\mathbf{f}d\tau} = - \frac{\mathrm{d}}{\mathrm{d}t}\int{\frac{\mathbf{\Pi}}{c^{2}}d\tau},\]

  • 33 Lorentz 1895, p. 28.

which means that the sum of the forces acting on all the ions of the system does not vanish. In Poincaré’s words, the principle of action and reaction does not apply to matter alone.33

Liénard in 1898

33Having read Poincaré’s critical review of 1895 as well as the previous remarks by Lorentz, the engineering professor Alfred Liénard felt compelled to discuss the principle of reaction in Lorentz’s theory. In a valuable analysis of this theory published in March 1898 in L’éclairage électrique, he rederives Lorentz’s expression (23) for the Lorentz force and integrates it over the entire space to get Eq. (24). He comments:

[The total force acting on matter] will vanish only if the integral (e × b)dτ has a constant value. This is not true in general, and the principle of the equality of action and reaction is not satisfied.

This result should not at all surprise us: as soon as we reject the theory of action at a distance and instead assume that forces take a certain time to propagate through the ether, one cannot have the equality of action and reaction at every instant; action and reaction do not occur at the same time. All we can demand is that the resultant of all forces be zero on average, and this is indeed the case in Lorentz’s theory according to [Eq. (24)] [...].

Consequently, Lorentz’s theory satisfies the principle of reaction as much as is possible when one rejects the theories of instantaneous action at a distance.

  • 34 Liénard 1898a, pp. 457–458. On Liénard’s theoretical works, see Nio 2020, chap. 12.

Liénard goes on to argue, pace Poincaré, that the principle of reaction is also violated in Hertz’s theory because this theory includes the force \(\dot{\mathbf{D}} \times \mathbf{B}\) acting on the ether in vacuum. He adds that the violation is worse in Hertz’s theory, because the time-average of this force does not vanish. The latter remark is erroneous, for it overlooks the force \(\mathbf{D} \times \dot{\mathbf{B}}\) that hould be added to the former force in Hertz’s theory.34

Wien in 1898

  • 35 Wien 1898, p. 52.

34In September 1898, Wilhelm Wien reported on the problem of ether motion at the Naturforscherversammlung in Düsseldorf. Although, like Poincaré and a growing number of experts, he regarded Lorentz’s theory as the best available, he denounced “a fundamental difficulty in the consequent development of this theory”:35

This difficulty is closely related to the fact that variable electromagnetic states exert forces that would set the ether in motion if it were movable. Consider a body in the free ether, shaped like a thin plate and having a different emissivity for thermal radiation on both sides. Since according to Maxwell’s theory the emitted radiation exerts a pressure on the surface, this pressure would be stronger on the side of higher emissivity and it would set the body in motion. We would here have a case in which a body can set its center of mass in motion through its own internal energy. If the ether is regarded as unmovable, this represents a violation of the general principle of the center of mass. In contrast, the assumption of a movable ether endowed with inertia would elude this objection.

At any rate, this point is to be kept in mind in the future development of the theory.

35At the same meeting, Lorentz answered Wien’s and similar objections at the very end of his own report:

I still have to consider a frequent objection to the theory of the stationary ether, an objection also formulated by Mr. Wien in his report: this theory contradicts the principle of action and reaction. First, I must remark that this principle, it seems to me, need not apply to all the elementary actions to which we reduce phenomena. I shall not deny that it brings us some satisfaction when a law holding uniformly in macroscopic phenomena can already be recognized in these elementary actions. Yet, we will perhaps have to renounce this satisfaction in order to gain another that appears to us more valuable.

  • 36 Lorentz 1898, pp. 64–65.

Lorentz then offered two possibilities: one could either regard the ether as strictly immovable and deny any reaction of matter on the ether, as he did in the Versuch, or one could endow the ether with a very high inertia so that the motions induced by the action of matter on it would remain undetectable. In either case it is difficult to understand why the principle of reaction normally holds at the macroscale even though it does not apply to matter alone at the microscale. Lorentz hoped that “this defect might be mended in a future simplification and elucidation of the theoretical considerations.”36

The situation in 1898

36To sum up, by 1898 there had been much discussion of the principle of reaction in Lorentz’s theory. Poincaré, Lorentz, Wien, and Liénard all agreed that the principle was violated in the sense that the total force acting on matter does not vanish in this theory. Wien brought out the connection with radiation pressure through a thought experiment. Poincaré and Liénard both underlined that this violation was to be expected in a theory in which actions are retarded through a non-material medium. Liénard remarked that the principle was still satisfied on average, and thus unwittingly answered Lorentz’s Düsseldorf call for an explanation of why the principle generally held at the macro-level despite being violated at the micro-level.

37There was much disagreement on how tolerable the violation was for the elementary actions in Lorentz’s and in Hertz’s theory. Heaviside believed that the ether was sufficiently material to provide the missing reaction, partly through additional ethereal forces, partly through bulk motion. Poincaré believed that Hertz’s medium, being movable and quasi-material, could always carry the missing momentum. Lorentz was willing to consider a similar reaction in his theory, if only the density of the ether were large enough to prevent a detectable ether flow. However, he and Liénard had no qualms about a fundamental violation of the principle: being a principle of mechanics, the equality of action and reaction did not have to extend to electromagnetic actions. Poincaré and Hertz (for a vacuum) were both reluctant to admit this violation. They believed that the principle should apply to matter alone, presumably because in their mind the ether (in a vacuum) was too immaterial to provide the missing reaction. Also, both of them believed in a physics of principles in which one should require new theories to satisfy well-established principles as long as possible. Or else it could be that they believed any violation of the reaction principle would have paradoxical consequences. We will see in a moment that Poincaré later said as much.

Poincaré’s lectures of 1899

38Poincaré returned to Lorentz’s theory in 1897 in a discussion of the Zeeman effect, and he taught it in 1899 together with Hertz’s theory in a sequel to his Sorbonne lectures on electricity and optics. His chapter on Hertz’s theory ends with the statement that the principle of reaction is satisfied in this theory. The proof there given differs from the proof of 1895, which was based on the fact that the ponderomotive forces in Hertz’s theory derive from stresses. In 1899, Poincaré no longer relies on stresses and directly obtains expressions for the ponderomotive force f by manipulating the time derivative of the field energy

\[(25)\quad J = \int{\frac{1}{2}(\varepsilon E^{2} + \mu H^{2})\mathrm{d}\tau}.\]

These expressions do not involve the velocity field v of the medium. The form of the field equations remains unchanged when they are referred to a frame moving at the constant velocity u with respect to the original frame (this is an obvious consequence of their being built on convective derivatives). Therefore, the energy balance should have the same form in both frames, except that v should be replaced with v − u in the new frame. Hence, we have both

\[(26-27) \quad {\small \dot{J} + \int{\mathbf{j} \cdot \mathbf{E}\mathrm{d}\tau} + \int{\mathbf{f} \cdot \mathbf{v}\mathrm{d}\tau} = 0\,\text{and}\,\dot{J} + \int{\mathbf{j} \cdot \mathbf{E}\mathrm{d}\tau} + \int{\mathbf{f} \cdot (\mathbf{v} - \mathbf{u})\mathrm{d}\tau}\,=\,0},\]

  • 37 Poincaré 1901, pp. 420–421. Boltzmann’s assistant Ignaz Schütz (1897) had earlier established the (...)

which requires ∫fdτ 0. The net force acting on the medium vanishes, and the reaction principle is satisfied. Poincaré thus exhibits a direct connection between the (Galilean) relativity principle, the energy principle, and the reaction principle.37

39In his next chapter, Poincaré addresses Lorentz’s theory, “which explains fairly well the optical phenomena that eluded Hertz’s theory but which, unfortunately, does not satisfy the principle of the equality of action and reaction.” In the section devoted to this violation, Lorentz first repeats the argument he has given in 1895 regarding the effect of a traveling disturbance on a charged particle. He then derives Eq. (24) for the total force acting on matter, \(\int{\mathbf{f}{d}\tau} = - \int{c^{- 2}\dot{\mathbf{\Pi}}\mathrm{d}\tau}\), and concludes that Lorentz’s theory in general does not satisfy the reaction principle. He next cites Liénard’s argument that the principle of reaction is still met on average because the time average of the time derivative of a bounded quantity (the Poynting vector) goes to zero when the integration time goes to infinity. This argument is “insufficient” because during the emission of radiation by an electric system, the momentum of this system changes by the amount

\[(28)\quad \Delta\mathbf{P} = - \Delta\int{c^{- 2}\mathbf{\Pi}\mathrm{d}\tau},\]

  • 38 Poincaré 1901, p. 422 (citation), pp. 448–451, p. 453 (citation).

the integral being taken over a sphere including all the emitted radiation. As Poincaré notes, this integral is proportional to the total energy emitted by the radiator. Consequently, “if the system producing the electromagnetic energy emits this energy in one direction only, it should recoil as a piece of artillery would do.” If three million joules are emitted and if the mass of the emitter is one kilogram, Poincaré reckons, the recoil velocity takes the “non-negligible” value of 1 cm per second.38

40Lastly, Poincaré addresses one way (already suggested by Lorentz) of saving the principle of reaction:

One could still say that if the principle of reaction seems to be violated, this is perhaps because we did not take the motion of the ether into account. Let us admit this objection and let us see where it takes us. In order that the principle of reaction should not be violated, the [...] velocity of the ether must be represented by [e × b]. This is the radiant vector, up to a constant [factor]. We must conclude: if the field is doubled, the velocity of the ether is quadrupled. This is evidently not satisfactory.

  • 39 Poincaré 1901, pp. 453–454, 631–632.

Thus, the only way of saving the principle of reaction that Poincaré can imagine is to endow the ether with a linear momentum in the mechanical sense of a mass multiplied by a velocity. His following argument against this kind of ether momentum is asserted with great confidence. But why is it so “evident” to Poincaré that the velocity of the ether could not be quadrupled when the field is doubled? Maybe he has in mind that in the best quasi-mechanical representation of Lorentz’s ether, given by Larmor, the magnetic field gives the velocity of the ether. Or maybe he believes that any motion of the ether should be induced by the electric and magnetic fields regarded as forces, and therefore could only be proportional to these fields. The first guess seems to be confirmed by Poincaré’s remark, at the very end of his lectures, that the motion of Larmor’s ether, being proportional to the magnetic field could not possibly provide a momentum proportional to the Poynting vector, as would be needed to save the reaction principle. Poincaré further notes that Lodge’s ether-whirling experiment requires a very high ether density, in which case the ether wind on earth would imply strong and never seen magnetic effects.39

  • 40 Poincaré 1901, chap. 6. On p. 535 Poincaré notes that the local time shift, being non-negligible c (...)

41Poincaré devotes an entire chapter of his course to the optics of moving bodies. He recalls that nearly all attempts to detect optical effects of the motion of the earth through the ether have failed. He then derives the Fresnel drag in Lorentz’s first manner, that is, from the plane-wave solutions of the macroscopic field equations in a moving body. The Fresnel drag indirectly explains the absence of most effects of the earth’s motion at first order in u/c. Poincaré next gives Lorentz’s general proof of 1895 that optical phenomena are not affected in his theory at this order, based on the local time t – u ⋅ r/c2. There is a slight difference: whereas Lorentz first derived the (macroscopic) field equation in the earth frame by means of a Galilean transformation and then retrieved the original form of the field equations (for a body at rest in the ether) by referring them to the local time, Poincaré directly considers what we would now call a first-order Lorentz transformation, involving a change of both the space and time coordinates as well as a concomitant field transformation. He verifies the first-order invariance of the field equations through these transformations, and connects this invariance to the absence of effects of the motion of the earth by noting, as Lorentz has already done, that the field transformations do not affect the patterns of darkness and brightness observed in optical experiments and that the space-dependence of the local time is too small (of the order of 10−9 second per km) to affect the observed phenomena.40

42This reasoning works only at first order. At second order, the Michelson-Morley experiment of 1887 still gives a negative result, which Lorentz has explained through the Lorentz contraction of the longitudinal arm of the Michelson interferometer. About this assumption, Poincaré writes:

This strange property would appear to be a true “nudge” [coup de pouce] given by nature to prevent detection of the absolute motion of the earth through optical phenomena. This could not satisfy me and I think I should tell you what I feel: I regard it as very probable that optical phenomena depend only on the relative motion of the implied material bodies, sources and optical apparatus, and this not only by neglecting third- or fourth-order quantities [...] but rigorously. The more exact the experiments will become, the more precisely this principle will be verified.

Will we need a new nudge, a new hypothesis, at each approximation? Obviously not: a well-built theory should allow us to demonstrate the principle in one stroke in all rigor. Lorentz’s theory does not do that yet. Of all theories that have been proposed, Lorentz’s is the one which is closest to do so. We may therefore hope to make it perfectly satisfactory in this respect without modifying it too deeply.

  • 41 Poincaré 1901, p. 536.

Here again we meet with Poincaré’s conviction that the principle of relativity (as he will later call it) should hold exactly in the future improved theory.41

  • 42 Poincaré considers only the case v = u. The ρ and the v in these equations refer to macroscopic av (...)

43In his next chapter, Poincaré investigates whether electrodynamic phenomena satisfy the relativity principle. For this purpose, he verifies that Lorentz’s macroscopic field equations for a system of conductors in a medium of negligible polarizability,42

\[(29)\quad\begin{split}c\nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}}{\partial t},\ c\nabla\times\mathbf{H} = \mathbf{j} + \rho\mathbf{v} + \frac{\partial\mathbf{E}}{\partial t},\ \nabla \cdot \mathbf{E} = \rho,\\ \nabla \cdot \mathbf{B} = 0,\ \mathbf{j} = \sigma(\mathbf{E} + c^{- 1}\mathbf{v} \times \mathbf{B}),\end{split}\]

are invariant through the first-order Lorentz transformations,

\[(30)\quad \small{\mathbf{r}' = \mathbf{r} - \mathbf{u}t,\ t' = t - \mathbf{u} \cdot \mathbf{r}/c^{2},\ \mathbf{E}' = \mathbf{E} + c^{- 1}\mathbf{u} \times \mathbf{B},\ \mathbf{B}' = \mathbf{B} - c^{- 1}\mathbf{u} \times \mathbf{E}},\]

provided that the electric density and the conduction current transform as

\[(31)\quad \rho' = \rho - \mathbf{j} \cdot \mathbf{u}/c^{2},\ \mathbf{j}' = \mathbf{j}.\]

He then computes the transformed force f' = ρ'(E' + c−1 v' × B') with v' = u to first order in u, and finds

\[(32)\quad \mathbf{f}' = \mathbf{f} - c^{- 2}(\mathbf{j} \cdot \mathbf{E})\mathbf{u},\]

  • 43 Liénard 1898b, pp. 323–324. Liénard also shows that the same result can be obtained by transformin (...)
  • 44 Poincaré 1901, p. 543.

a result already obtained by Liénard the previous year by similar reasoning.43 As Poincaré reports, Liénard has estimated that for the force acting on a resistor fed by a dynamo, the complementary force  −∫c2 (jE)udτ would be only of the order of a hundredth of a milligram for a dissipated energy of the order of 10joule per second. Poincaré concludes: “To summarize, we see that in the case of electrostatic [sic] phenomena, even though there are terms of first order, there is no effect that could be experimentally exhibited. —Lorentz’s theory therefore remains compatible with experimental facts.” This statement clearly shows that for Poincaré, the complementary force contradicts the relativity principle at first order. However, the small value of this force makes it impossible to experimentally confirm the contradiction. Neither Poincaré nor Liénard indicates how the force could be measured. They just give the order of magnitude.44

Poincaré’s keynote lecture in August 1900

44On August 7, 1900, Poincaré discoursed on “The relations between experimental physics and mathematical physics” in his keynote lecture for the first international congress of physics, organized by the Société française de physique in the context of the universal exhibition in Paris. In a section devoted to mechanical explanations, Poincaré raises the question: “And our ether, does it really exist?” In answer, he first recalls the main reason why physicists introduced the ether: the light from a distant body takes time to reach us, and this could not happen without some intermediate body unless we violate the laws of mechanics. In addition, Fizeau’s experiment seems to require two interpenetrating media, ether and matter, able to move separately: “We feel like we are touching the ether with our finger.” Poincaré goes on:

We may however conceive experiments in which we would touch the ether even closer. Suppose that Newton’s principle of the equality of action and reaction is no longer true when we apply it to matter alone and suppose we confirm this. The geometric sum of all forces applied to material molecules would no longer vanish. Unless we are willing to change the mechanics entirely, we would have to introduce the ether in order that this apparent action on matter would be counterbalanced by the reaction of matter on something.

Or else, suppose we find that optical and electrical phenomena are influenced by the motion of the earth. We would be driven to conclude that these phenomena could reveal not only the relative motion of material bodies, but also what would seem to be their absolute motion. Again, an ether would have to exist so that this alleged absolute motion would not be a displacement with respect to a vacuum but a displacement with respect to something concrete.

Will we ever come to that point? I do not have this hope—I will say why in a moment—and yet it is not so absurd since others have had it.

For example, if Lorentz’s theory were true, Newton’s principle [of reaction] would no longer apply to matter alone and the difference would not be far from being experimentally accessible.

Besides, there have been many investigations of the influence of the motion of the earth. The results have always been negative. But we undertook these experiments only because we were not sure the result would be negative. On the contrary, according to the reigning theories the compensation would only be approximative, and more precise methods [of measurement] would be expected to yield a positive result.

I believe this hope is illusory. It was nonetheless interesting to show that a success of this kind would, as it were, open a new world to us.

  • 45 Poincaré 1900b, pp. 21–22.

Poincaré justifies his belief in the relativity principle as he has done in his lectures of the previous year, by noting that the successive compensations imagined by Lorentz for the effects of the earth’s motion at successive orders of approximation could not have worked by mere chance. There should be a global, rigorous explanation of the lack of effects of the motion of the earth.45

  • 46 Pyotr Nikolaevich Lebedev gave a first account of his confirmation of radiation pressure at the Pa (...)

45In this discussion, Poincaré addresses the violations of the principle of relativity and of the principle reaction in parallel, as he had started to do in 1895. In his mind, a violation of these principles (for matter alone) would compel us to regard the ether as a concrete body, able to carry momentum and able to affect the phenomena in systems globally moving through it. There are yet no experiments implying a violation of these principles.46 On the contrary, every attempt to detect effects of the motion of the earth through the ether has failed. Lorentz’s theory violates the reaction principle, and it saves the relativity principle only in an approximative manner, at the price of accumulated hypotheses. Poincaré bets that in the final theory the two principles will be satisfied exactly and naturally. Later in his talk, he returns to Lorentz’s theory and the reaction principle in the following words:

The most satisfactory theory we have is Lorentz’s theory. Undoubtedly, this theory is the one that best accounts for known facts; it is the one that sheds light on the greatest number of true relations; it is the one that will leave the strongest imprint on the final construction. Nonetheless, Lorentz’s theory still has a grave defect, which I have mentioned earlier: it contradicts Newton’s principle of the equality of action and reaction, or, rather, in Lorentz’s eyes this principle would not be applicable to matter alone. For this principle to be true, one would have to consider the actions of ether on matter, and the reaction of matter on the ether. Yet, until further notice, things don’t happen that way.

  • 47 Poincaré 1900b, pp. 24–25.

Here Poincaré reasserts in the clearest manner his reluctance to save the reaction principle of reaction by enabling a reaction of the ether. He then repeats his remark that the reaction is strictly impossible in Larmor’s theory because this theory specifies the motion of the ether in a manner incompatible with the form of the missing momentum.47

3 The jubilee offering

Introduction

  • 48 Heike Kamerlingh Onnes launched the invitations in May 1900 (see, e.g., his letter to Wien of 5 Ma (...)

46Later in the same year 1900, Poincaré contributed to a volume presented to Lorentz on 11 December 1900 for the silver jubilee of his doctorate.48 Somewhat oddly, he decided to write on “Lorentz’s theory and the principle of reaction,” that is, he focused on what he had long regarded as the main defect of Lorentz’s theory. The introduction reads as follows:

It will probably seem strange that in a monument to Lorentz’s glory I would return to considerations which I formerly presented as an objection to his theory. I could say that the following pages tend to attenuate rather than aggravate this objection.

But I disdain this excuse because I have a thousand-times better one. Good theories are flexible. The theories that have a rigid form and cannot shed this form without collapsing very much lack vitality.

If a theory reveals us certain true relations, it can be dressed in thousand different manners, it will resist all assaults and what makes its essence will not change. This is what I recently explained at the Congress of Physics.

Good theories get the better of all objections. Merely specious objections cause no harm to them, and they triumph even over serious objections, although they do so by transforming themselves.

Objections therefore serve them, far from hurting them, since they allow them to develop all their latent virtue. Yes, Lorentz’s theory belongs to this kind, and this is the only excuse I shall invoke.

I will therefore not apologize to the reader for [presenting objections], but only for spending so much time on ideas of so little originality.

  • 49 Poincaré 1900a, p. 252.

In this preamble, Poincaré makes clear that he is going to dwell on objections to Lorentz’s theory, and his chief excuse is that, in conformity with his philosophy of knowledge, he believes that something essential, the true relations [rapports vrais] will remain of a good theory such as Lorentz’s, provided this theory is modified to answer the objections. In his second sentence, there also is the fleeting suggestion that what follows might attenuate Poincaré’s older objections. We will later try to see to what extent this is the case.49

Section 1

47In the first section of his article, Poincaré rederives Lorentz’s Eq. (23)

\[f_{i} = \partial_{j}p_{ij} - \frac{\partial}{\partial t}\frac{\Pi_{i}}{c^{2}},\ \text{with}\ p_{ij} = e_{i}e_{j} - \frac{1}{2}\delta_{ij}e^{2} + b_{i}b_{j} - \frac{1}{2}\delta_{ij}b^{2}\]

and integrates it (as Liénard has already done) to get Eq. (24)

\[\int{\mathbf{f}\mathrm{d}\tau} + \int{c^{- 2}\dot{\mathbf{\Pi}}\mathrm{d}\tau} = \mathbf{0}.\]

As he has already done in his Sorbonne lectures, he applies the Newtonian acceleration law to macroscopic portions of the system of mass M and velocity V. He writes: “If the principle of reaction could be applied, one would have ∑MV = constant Instead we have

\[(33)\quad \sum{M\mathbf{V} + \int{c^{- 2}\mathbf{\Pi}\mathrm{d}\tau} = \text{constant}},”\]

  • 50 Poincaré 1900a, pp. 252–255.

which is a variant of the earlier Eq. (28).50

48Poincaré then recalls Poynting’s Eq. (22) (in Lorentz’s version),

\[\rho\mathbf{v} \cdot \mathbf{e} + \frac{\partial w}{\partial t} + \nabla \cdot \mathbf{\Pi} = 0,\ \text{with}\ w = \frac{1}{2}(\mathbf{e}^{2} + \mathbf{b}^{2})\ \text{and}\ \mathbf{\Pi} = c(\mathbf{e} \times \mathbf{b}),\]

and goes on to introduce the newer idea of the energy as a fluid:

We can regard the electromagnetic energy as a fictitious fluid of density w/c2 moving in space according to Poynting laws. However, we must assume that this fluid is not indestructible and that in the volume element dτ the quantity c2 ρevdτ is destroyed [...]. This is what prevents us, in our reasoning, to fully equate our fictitious fluid to a real fluid.

Poincaré adjusts the velocity U of the fluid so that its flux wU  agrees with the Poynting vector Π, and rewrites Eq. (33) as

\[(34)\quad \sum{M\mathbf{V} + \int{c^{- 2}w\mathbf{U}\mathrm{d}\tau} = \text{constant}},\]

  • 51 Poincaré 1900a, pp. 256 (citation), 258 (Eq. (35)).

which means that the momentum of matter properly speaking plus the momentum of the fictitious fluid is a constant vector. As Poincaré knows, momentum conservation is traditionally used to derive the theorem of the center of mass, according to which the motion of the center of mass of a closed mechanical system is rectilinear and uniform. This theorem requires mass to be conserved, which is not the case for the fictitious fluid. The Poynting relation and the conservation of the total momentum of matter and fluid together lead to51

\[(35)\quad \frac{\mathrm{d}}{\mathrm{d}t}\left( \sum{M\mathbf{R}} + \int{c^{- 2}w\mathbf{r}\mathrm{d}\tau} \right) + \int{c^{- 2}\rho(\mathbf{v} \cdot \mathbf{E})\mathbf{r}\mathrm{d}\tau = \text{constant}}.\]

  • 52 Poincaré 1900a, p. 258.

The expression in parentheses gives the position of the center mass multiplied by the total mass of matter and fluid. As long as no electromagnetic energy is created or absorbed at any point of space, the motion of this center of mass is rectilinear and uniform. In order to save the center-of-mass theorem in the general case, Poincaré imagines a reservoir of latent fluid at rest at every point of space, and assumes the creation of electromagnetic energy at a given point to correspond to the liberation of a portion of latent fluid at this point, with a sudden change of velocity from 0 to U, and vice versa in the case of absorption. With this convention, the former equation implies the rectilinear, uniform motion of the center of mass of matter plus free fluid plus latent fluid. For those who would be baffled by this unreal convention, Poincaré writes: “There is nothing shocking in this convention, since we are dealing only with a mathematical fiction.”52

  • 53 Poincaré 1900a, pp. 259–260. The conservation of the total angular momentum follows from the local (...)

49In this picture, any variation of the total momentum of matter is counterbalanced by a variation in the total momentum of the fictitious fluid. Of course, the latent fluid does not contribute to the latter momentum, since by definition it is permanently at rest. But it is needed to preserve the constancy of the total fluid mass and the rectilinear uniform motion of the center of mass of the global system. Eq. (23) then means that the electromagnetic force acting on the matter within a volume element is the resultant of the Maxwell stresses on the surface of this element plus the inertial force of the fictitious fluid within this element. This force results from the momentum variation of the fictitious fluid, and this variation involves both a velocity variation of the free fluid, and the sudden acquisition (or loss) of this velocity by the latent fluid when electromagnetic energy is produced (or destroyed) in this element. Integrating over the entire space, the net electromagnetic force acting on all matter equals the total inertial force of the fictitious fluid. At the same time, the net electromagnetic force must be balanced by the net inertial force of matter since any non-electromagnetic (ordinary mechanical) force should satisfy the principle of action and reaction. Poincaré concludes:53

The two kinds of inertial force [for the fictitious fluid and for matter] must balance each other, with a double consequence:

  1. The principle of [...] the conservation of momentum applies to the system of matter and fictitious fluid, in conformity with [Eq. (34)].
  2. The principle of the conservation of angular momentum [...] applies to the system of matter and fluid [...].

Electromagnetic energy therefore behaves, from the point of view we are dealing with, like a fluid endowed with inertia, and we must conclude that if an apparatus, having produced electromagnetic energy, radiates it in a given direction, this apparatus must recoil as would a cannon after sending a projectile.

50Poincaré estimates that a radiator with a mass of 1 kg would recoil at a velocity of 1 cm/s after emitting 3 million joules. For a constant radiating power of 3000 watts, a force of one dyne [10−5 newton] must be applied to the radiator in order to prevent its recoil. Poincaré goes on:

Such a weak force would evidently be impossible to detect experimentally. However, if it could be confirmed by a sufficiently sensitive instrument, one might imagine that one would thus demonstrate that the principle of reaction does not apply to matter alone; this would confirm Lorentz’s theory and condemn the other theories.

This is not at all the case, because Hertz’s theory and all the other theories predict the same recoil as Lorentz’s theory.

  • 54 Poincaré 1900a, p. 260.

Indeed, the recoil force is easily seen to be equal to the resultant of the Maxwell stresses on the surface of the emitter, and the Maxwell stresses are the same in all electromagnetic field theories after Maxwell. The emitter could be a Hertz resonator placed at the focus of a parabolic mirror, as Poincaré first imagined, but one could instead consider the reflection of a normal plane wave on the surface of a mirror as Maxwell did in his treatise. The Maxwell stresses imply a radiation pressure in both cases.54

  • 55 Poincaré’s expression “the momentum of the electromagnetic energy,” found in Section 2 (p. 264), s (...)

51From Poincaré’s earlier comments on a possible momentum of the ether, we know that Poincaré, like all his contemporaries, conceived momentum mechanically, as the product of mass by velocity. Being reluctant to regard the ether as a genuine mechanical body and being alert to the absurdity of the ether motions that the mechanical concept would imply, he could not possibly interpret  c-2Π as the momentum density of the ether. Instead he observed that this vector, being proportional to the energy flux, could be regarded as the flux of a fictitious fluid of density proportional to the energy density w: c-2Π = c-2wU, with UΠ/w. This fluid, its mass, its velocity, and its momentum could only be fictitious in Poincaré’s eyes, since it could be created or absorbed without any change in the material masses. Poincaré introduced it only as a convenient illustration of the term ∫c2 Πdτ in Eq. (33) for the momentum balance. Nowhere in his article does he suggest regarding this term as a genuine momentum on par with the momentum of matter.55 On the contrary, his central concern is the violation of the principle of reaction (or of momentum conservation) for matter alone.

Section 2

  • 56 Poincaré 1900a, p. 261.

52In his second section, Poincaré addresses the following difficulty: why is it that some field theories, in particular Hertz’s, imply the recoil force without violating the principle of reaction? The answer is simple, and it is already found in Poincaré’s critical study of 1895: in Hertz’s theory the medium is not pure ether, it is a comoving combination of matter and ether. Even the faintest trace of air will be enough for the medium to carry the missing momentum. In Poincaré’s words:56

In Lorentz’s theory and in Hertz’s theory, the apparatus that produces the energy and sends it in one direction recoils, but the radiated energy propagates through a medium, the air for instance.

In Lorentz’s theory, when the air receives the radiated energy, it does not undergo any mechanical action; it does not either when this energy leaves it after passing through it. On the contrary, in Hertz’s theory the air is pushed forward when it receives the [radiated] energy, and recoils when this energy leaves it. With respect to the principle of reaction, the motions of the air through which the energy passes thus compensate for those of the apparatus producing this energy.

  • 57 Poincaré 1900a, p. 269.

53In 1895, Poincaré had claimed, somewhat more accurately, that in Lorenz’s theory the dielectric (the air) could only carry a small portion of the missing momentum. The proof of this statement occupies most of the second section of the jubilee article. After computations implying Lorentz’s picture of dielectrics through elastically bound ions, Lorentz’s fundamental equations, and the fictitious fluid introduced in the first section, Poincaré finds that the dielectric provides only the fraction (n2–1)/(n2+1) of the missing momentum, while the rest goes to the fictitious fluid. In the case of a unidirectional emitter in air, the recoil of the emitter is very far from being compensated by the motion of the air. Poincaré next provides a new derivation of the fact that “in Hertz’s theory the principle of reaction is not violated and applies to matter alone,” which has the following consequence:57

In order to prove experimentally that the principle of reaction is violated in reality as it is in Lorentz’s theory, it would not be sufficient to show that the energy emitters recoil—although this would be difficult enough—it would be necessary to show that this recoil is not compensated by the motion of dielectrics, in particular of the air through which the electromagnetic waves travel. Evidently, this would be far more difficult.

54To summarize, Lorentz’s theory violates the principle of reaction when applied to radiation processes. However, in order to experimentally verify this violation, it is not enough to measure the recoil of the emitter, one would also have to make sure that the missing momentum does not go into residual matter around the emitter. If the question cannot be decided experimentally, are there a priori reasons why the principle of reaction should be satisfied? This is in substance the question with which Poincaré launches his next section.

Section 3

55Poincaré begins:

Why does the principle of reaction impose itself on our minds? It is important to know why in order to judge whether the preceding paradoxes can be truly considered as an objection to Lorentz’s theory.

In most cases this principle imposes itself because its negation would lead to perpetual motion. Is this the case in the present situation?

Consider to bodies A and B acting on each other but isolated from any external action. If the action of one of the bodies did not equate the reaction of the other, one could bind them through a rod of invariable length so that they behave like a single rigid body. The forces applied to this body being out of equilibrium, the system would start to move and this motion would keep accelerating, provided the following condition is satisfied : the mutual action of the two bodies depends only on their relative motion and on their relative velocity and does not depend on their absolute position and their absolute velocity.

  • 58 Poincaré 1900a, p. 270; Newton 1687, pp. 23–24.

Note that Poincaré here characterizes the contents of the earlier sections of his article as “paradoxes.” Namely, he regards the established violation of the reaction principle in Lorentz’s theory as a paradox. He is very far from suggesting that the paradox is solved by introducing an electromagnetic momentum. The argument that a violation of the principle of reaction leads to the possibility of perpetual motion belongs to Newton, as a scholium to his third law. The only difference is that Poincaré emphasizes the condition that the interaction should satisfy a relativity condition.58

56Poincaré next proves that for a mechanical system in which forces derive from the potential U, the “principle of relative motion” and the conservation of energy together imply the principle of reaction. Indeed, if m and r denote the mass and position of the various material points of the system, the conservation of energy reads

\[(36)\quad \sum\frac{1}{2}m{\dot{\mathbf{r}}}^{2} + U = \text{constant}\]

in a given inertial frame. According to the principle of relative motion, the potential U can only depend on the relative positions of the material points, and the conservation of energy in a reference frame moving at the velocity u with respect of the former frame reads

\[(37)\quad \sum\frac{1}{2}m({\dot{\mathbf{r}}} + \mathbf{u})^2 + U = \text{constant}.\]

Subtracting this equation from the former equation, Poincaré gets

\[(38-39)\quad \small{\mathbf{u} \cdot \sum{m\dot{\mathbf{r}}} + \frac{1}{2}\mathbf{u}^{2}\sum m = \text{constant}\ \text{for any}\ \mathbf{u},\ \text{so that}\ \sum{m\dot{\mathbf{r}}} = \mathbf{0}}.\]

  • 59 Poincaré 1900a, pp. 270–271.

The latter equation is “the analytical expression of the principle of reaction.” The reasoning is similar to the one Poincaré had given a few months earlier in his Sorbonne lectures to prove that the principle of reaction is satisfied in Hertz’s theory.59

57Poincaré goes on:

The principle of reaction thus appears to be a consequence of the energy principle together with the principle of relative motion. The latter principle strongly imposes itself on our mind when it is applied to an isolated system.

However, in the case at hand we are not dealing with an isolated system, since we consider only ordinary matter, in addition to which there still is the ether. If all material objects are dragged in a common translation, for instance the translation of the earth, the phenomena might differ from what they would be if this translation did not exist because the ether does not have to be dragged by the earth. The principle of relative motion understood in this way and applied to matter is so far from compelling the mind that some have conducted experiments designed to reveal the translation of the earth. While it is true that these experiments have shown negative results, this outcome was rather surprising.

One question remains. These experiments, I just said, have given negative results, and Lorentz’s theory explains these negative results. It seems that the principle of relative motion, which did not impose itself a priori, is verified a posteriori and that the principle of reaction should follow; and yet this is not the case. How is that possible?

  • 60 Poincaré 1900a, pp. 271–272.

Since 1895, Poincaré suspected a connection between the violation of the principle of reaction in Lorentz’s theory and the imperfect, convoluted way in which this theory accounted for the relativity of optical phenomena. He now had an argument to suggest that in general the two principles should stand or fall together. Note that this argument applies only to direct, instantaneous action at a distance, represented by the potential U. The same limitation applies to Newton’s justification of his third law. However, Poincaré strongly suspected that the connection was general. He therefore expected that the violation of the principle of reaction in Lorentz’s theory should imply a concomitant violation of the relativity principle. Yet, Lorentz’s theory had been able to account for the relativity of optical phenomena, at least at first order. This strange anomaly may have motivated his entire essay: the two first sections confirm and sharpen the violation of the reaction principle in Lorentz’s theory, and the third section examines how Lorentz’s theory can save the relativity of optical phenomena and yet violate the principle of reaction.60

  • 61 Poincaré 1900a, pp. 272–273.

58Poincaré’s reply to this central question begins as follows:61

The reason is that in reality what we have called the principle of relative motion has been verified only in an imperfect manner as is shown in Lorentz’s theory. This verification results from a compensation of effects but:

  1. This compensation holds only when neglecting u2, unless we introduce a complementary hypothesis I will not discuss for the moment.
    This does not matter for our purpose, because if we neglect u2, [Eq. (38)] will directly give [Eq. (39)], that is, the principle of reaction.
  2. For the compensation to succeed, phenomena should be referred not to the true time t but to a certain local time t' defined in the following manner.
    Let two observers at different points of space set their watches by means of light signals. Let them correct these signals by the transmission time, thereby ignoring their translational motion [owing to the motion of the earth]. As they believe that the signals travel at equal speed in the two opposite directions, they just cross their observations, by sending a signal from A to B, then another signal from B to A. The local time t’ is the time given by the watches adjusted in this manner.
    If [c] denotes the velocity of light, and [u] the translation of the earth along the Ox axis, we have [t' = t − ux/c2].
  3. The apparent energy propagates in the relative motion according to the same laws as the real energy does in the absolute motion, but the apparent energy is not exactly equal to the corresponding real energy.
  4. In the relative motion, the bodies producing electromagnetic energy are subjected to an apparent complementary force which does not exist in the absolute motion.
  • 62 For details, see Darrigol 2022, pp. 167–169. According to Peter Galison [2003, chap. 4], Poincaré’ (...)

59As Poincaré has already explained in his Sorbonne lectures, Lorentz’s demonstration of the lack of first-order effects relies on “corresponding states” satisfying Lorentz’s equations. For Lorentz and also for Poincaré in his lectures, the corresponding states referred to the states the system would take if it were globally at rest in the ether. In contrast, in the cited extract the “apparent states” refer to the states observed and measured in the earth frame. In particular, the local time is the time measured by moving observers when they synchronize their clocks by optical means and thereby ignore their common motion. This is an essential insight, following which the Lorentz transformations relate the observable field states in the ether frame to the observable field states in the earth frame, and the Lorentz invariance directly implies the invariance of optical phenomena.62

  • 63 Poincaré [1900a, p. 274] observes that for the Lorentz-transformed fields the “apparent propagatio (...)

60One may wonder why Poincaré arrived at this insight while investigating Lorentz’s violation of the reaction principle, and not any earlier when he first encountered Lorentz’s local time. As we are about to see, his general idea of an interconnection between the energy principle, the relativity principle and the reaction principle brought him to discuss the recoil of a radiator in two different frames. The comparison prompted him to consider the fields measured in the moving frame. He thereby noticed that Lorentz’s corresponding fields truly were the field measured by terrestrial observers (if e and b are the fields in the ether frame, the electric field measured by a moving observer through a comoving test charge is e + c−1 × b), except for the shift in the time variable. At this point, he gathered that the time shift was not only small (as he had argued in his lectures), it was strictly unobservable if the terrestrial observers synchronized their clocks optically. Indeed, as Poincaré could not fail to notice, the local time is the time with respect to which the apparent velocity of light is the same on earth as it is in the ether to first order in u/c :63

\[(40)\quad x'^{2} - c^2 t'^{2} = x^{2} - c^{2}t^{2} + O(u^{2})\ \text{if}\ x' = x - ut\ \text{and}\ t' = t - ux/c^{2}.\]

61Another remark: the four conditions of compensation listed in the cited extract are quite heterogeneous. The first condition concerns the second order only and therefore can be ignored in reasoning confined to first order, as Poincaré notes. The second condition concerns the first-order Lorentz transformations and their physical interpretation as the transformations connecting the fields measured in two different frames. The third and fourth conditions truly are consequences of the second, as they can be derived from the Lorentz transformations applied to the fields included in the electromagnetic energy and force formulas. Yet, as we will see, there is an essential difference between these two conditions. The third does not imply any violation of the relativity principle, because energies naturally depend on the reference frame even in a Galilean context. In contrast, we will see that for Poincaré the fourth condition implies a violation of the relativity principle because he assumes forces to be measurable in an absolute manner.

62One may then wonder why Poincaré displays his four conditions on the same footing, as ways to “verify” the relativity principle. There are two reasons for this peculiarity. The first is that, as was just said, the Lorentz transformations (at first and second order) are the common formal core of the four conditions. The second, surely the most important one, is that the three last conditions will serve Poincaré to save the energy balance in the moving frame. However, as we will see in a moment, the last condition in itself implies a violation of the relativity principle.

  • 64 Poincaré 1900a, p. 273.

63Let us return to Poincaré’s text. Poincaré proceeds to show that Conditions 2, 3, 4 “solve the contradiction just mentioned” between the theory’s seeming compatibility with the relativity principle and the theory’s violation of the reaction principle. For this purpose, Poincaré considers the unidirectional emission of radiation (by a Hertz resonator placed at the focus of a parabolic mirror) in two different frames.64

  • 65 Poincaré 1900a, p. 273.

64If the emitter is originally at rest (in the ether) and starts to emit radiation, the energy spent by the emitter must equal the energy of the emitted radiation after a small time has elapsed. At a later time, the emitter has acquired a recoil velocity. Poincaré introduces what we would now call the tangent frame at this later time, namely, the inertial frame that moves at the velocity of the emitter at that time. In this frame, the emitter is at rest and the recoil force does not work. Therefore, the apparent energy of the emitted radiation must equal the energy spent by the emitter (per unit time). In the ether frame, the recoil force performs a work proportional to the velocity of the emitter and thus increases the translational energy of the emitter. The energy spent by the emitter now serves not only to produce radiation but also to increase the translational energy of the emitter. Therefore, the true radiated energy is smaller than the apparent radiated energy, in conformity with Condition 3.65

65In a subsequent, more precise reasoning, Poincaré considers an emitter moving at the constant velocity v in the ether frame in the direction of emission. In a given amount of time, the radiator emits a plane wave train of length L in the ether frame. In a frame moving at the velocity u with respect to the ether, the apparent fields are

\[(41)\quad \mathbf{E}' = \mathbf{E} + c^{- 1}\mathbf{u} \times \mathbf{B},\ \mathbf{B}' = \mathbf{B} - c^{- 1}\mathbf{u} \times \mathbf{E},\]

and the apparent field energy density therefore is

\[(42)\quad w' = w(1 - 2u/c)\]

  • 66 In a modernized proof, the intensity of the wave train in the ether frame is 
    χ (kx − kct) , where (...)

at first order in u/c. The apparent length L' of this wave train differs from the real length L because the events corresponding to the beginning and end of the wave train in the ether frame are not synchronous in the moving frame. The expression t' = t −ux/c2  of the apparent time implies66

\[(43)\quad L' = L(1 + u/c).\]

The product wL being proportional to the energy J emitted in a unit time, we have

\[(44)\quad J' = J(1 - u/c)\]

at first order in u/c. In the ether frame, the energy D spent by the emitter in a unit time must equal the radiated energy plus the work of the recoil force. Poincaré derives this force from the amount −J/c of fictitious fluid generated during the emission, and thus reaches the balance

\[(45)\quad D = J - J\upsilon/c.\]

In order that energy be conserved in the moving frame, one would like to have

\[(46)\quad D = J' - J'(\upsilon - u)/c,\]

because the expenditure D is evidently invariant (it corresponds to an intrinsic alteration of the source). In reality, we have

\[(47)\quad \small{J' - J'(\upsilon - u)/c = J(1 - \upsilon/c) + (Ju/c^{2})(\upsilon - u) = D + (Ju/c^{2})(\upsilon - u)}.\]

  • 67 Poincaré 1900a, pp. 274–276.

In order to save energy conservation, one must add to the recoil force −J'/c the complementary force −Ju/c2 performing the additional work 
–(Ju/c2)(v – u) on the emitter.67

  • 68 Poincaré 1900a, p. 277.

66Poincaré now believes to have “dissipated the aforementioned contradiction” (violation of the reaction principle without concomitant violation of the energy principle). He explains:68

The reason why in Lorentz’s theory the recoil can exist without violating the energy principle, is that the apparent energy for an observer attached to the moving frame is not equal to the real energy. Suppose that our radiator recoils and that the observer follows this motion (u = υ < 0), the emitter will appear to be at rest for this observer and it will seem to be radiating as much energy as it would if it were at rest. In reality, it will radiate less, and this is what compensates for the recoil work.

  • 69 The effect is opposite to the one derived in the previous reasoning because the emitter now moves (...)

Indeed, in the ether frame the energy balance reads D = J + Jυ/c (since the recoil force now acts in the direction of motion), and this agrees with the balance D=J' in the comoving frame if and only if J' = J(1 + υ/c): the true radiated energy J is smaller than the apparent radiated energy J'.69 Poincaré could have derived the expression of the apparent energy in this manner, as he implicitly did in his preliminary reasoning. He did not do so for two reasons: he wanted to relate this expression to the Lorentz transformations, and he wanted to introduce the complementary force. As he asserts, the necessity of this force (at first-order in u/c) appears only when a frame moving at a velocity different from the velocity of the emitter is used.

  • 70 Poincaré 1900a, pp. 277–278. Poincaré does not clearly distinguish between the recoil momentum and (...)
  • 71 Poincaré 1900a, p. 278.

67Poincaré then briefly indicates a simpler way to derive the complementary force.70 The recoil momentum, as computed in the ether frame, is –(J/c) per unit time. This momentum increment should be the same in the moving frame, since (for Poincaré) the mass of the emitter is invariable (and invariant) and since, in a sufficient approximation, the velocity increment is the same in the two frames (despite the different time variables). In the moving frame, the recoil force is –(J'/c) and the force  J'/c – J/c must be added to this force in order to produce the recoil momentum –(J/c) per unit time. Using J' J(1 – u/c), this force is the complementary force –Ju/c2. Poincaré comments on this result as follows:71

The existence of the apparent complementary force is therefore a necessary consequence of the recoil phenomenon.

Thus, in Lorentz’s theory the principle of reaction should not be applied to matter alone; the principle of relative motion does not either apply to matter alone. The important thing to note is that there is an intimate and necessary connection between these two facts.

It would therefore be sufficient to experimentally establish one of these facts in order that the other fact be established ipso facto. It would probably be less difficult to demonstrate the second fact. But this is already nearly impossible since for example Mr. Liénard has computed that for a machine of 100 kW the apparent complementary force would only be 1/600 dyne.

  • 72 From the fact that the complementary force can be obtained through the Lorentz transformations, on (...)
  • 73 Poincaré 1900a, pp. 271–272 (fuller quote above, note 60): “The principle of relative motion under (...)
  • 74 Also, the quantities in Eq. (46) are all apparent, measurable quantities and the fact that this re (...)

As Poincaré indicates in the last part of this extract, the complementary force agrees with the “supplementary force” Lenard had obtained two years earlier by applying a Lorentz transformation to the fields in the Lorentz force formula.72 Indeed, if in Liénard’s expression of this force, −c2 ujEdτ, the field E represents the emf of the generator that feeds the Hertz resonator (minus the resistive cemf), the integral jEdτ gives the energy D  spent by the resonator in a unit time, which equals the emitted energy when v = 0. In the first sentence of the extract, Poincaré tells us that the existence of the recoil force entails the existence of the complementary force. In his eyes, the recoil force implies a violation of the principle of reaction (applied to matter alone) and the existence of the Liénard force entails a violation of “the principle of relative motion applied to matter alone,” which he has earlier defined as the impossibility of experimentally detecting effects of the motion of a system through the ether.73 Hence comes the second sentence of the extract, in which Poincaré states that the reaction principle and the relativity principle are both violated and that the two violations are deeply interconnected. From the last sentence of the extract, it is clear that Poincaré regards the complementary force as principally measurable. Since the expression of this force involves the velocity of the moving frame, an observer in this frame could detect his motion in the ether by measuring the complementary force. For instance, for a terrestrial radiator, terrestrial observers could determine the velocity of the earth in the ether by measuring the complementary force.74

  • 75 Lorentz 1899.

68In today’s theory of relativity, the Liénard force does not constitute a violation of the relativity principle. As Lorentz already noted in 1899 (unknown to Poincaré), a force is measured by balancing it with another (known) force, and this balance will not be altered in the moving frame if all forces transform (linearly) according to the same rule.75 The simplest balancing force one can imagine is the inertial force of a body moving under the effect of the measured force. In the case of the Hertz resonator, Poincaré would write this balance as

\[(48-49)\quad - J/c = M\dot{\upsilon},\ - J'/c - Ju/c^{2} = M\dot{\upsilon}\]

in the ether frame and in the moving frame respectively, in a first approximation neglecting terms in υ2, u2, and υu. The motion of the resonator would therefore offer a means to measure the complementary force. Nowadays, we would invoke the variation of the mass of the radiator and replace these two equations with

\[(50-51)\quad - J/c = M\dot{\upsilon} + \dot{M}\upsilon,\ - J'/c + Ju/c^{2} = M\dot{\upsilon} + \dot{M}(\upsilon - u) \]

(in the same approximation), and we would compensate for the effect of the Liénard force by demanding \(\dot{M} = - J/c^{2}\). Poincaré was of course unaware of this escape, and he therefore regarded the complementary force as a violation of the relativity principle. In his mind, this violation brought the desired counterpart to the undesired violation of the reaction principle in Lorentz’s theory.

  • 76 Poincaré 1900a, 278.

69Poincaré ends his article with the following words:76

From this correlation between the two facts [violation of the reaction principle, violation of the relativity principle] there follows an important consequence: Fizeau’s experiment is already in itself contrary to the principle of reaction. If indeed, as this experiment shows, the dragging of waves is only partial, then the relative propagation of waves in a moving medium does not follow the same laws as the propagation in a medium at rest. That is to say, the principle of relative motion does not apply to matter alone and we must correct it in at least one way [Condition 2 above]: everything must be referred to the “local time.” If this correction is not compensated by others, we will have to conclude that the principle of reaction is not true either for matter alone.

All theories respecting this principle would thus be globally condemned, unless we consent to deeply modify all our ideas on electrodynamics. This is an idea that I have developed at length in an earlier article [Poincaré 1895].

  • 77 It would if there were non-optical means to synchronize moving clocks so that they give the true t (...)

Remember that in 1895, Poincaré has proven that Hertz’s theory is the only field theory satisfying the energy principle, the conservation of electricity and magnetism, and the reaction principle, with the corollary that Fizeau’s experiment cannot be explained in any field theory compatible with the reaction principle (since Hertz’s theory implies a complete ether drag). He now draws the same conclusion from the simpler remark that the Fresnel drag, being incompatible with the principle of relative motion, must also be incompatible with the principle of reaction. Clearly, Poincaré means that the Galilean law for the addition of velocities is incompatible with the partial drag, and that this law must be replaced with the law obtained by referring processes in the moving frame to Lorentz’s local time. It should be noted, however, that the Fresnel drag, unlike the Liénard force (according to Poincaré) does not offer a means to detect absolute motion (in the ether).77 Rather confusingly, Poincaré here uses the phrase “principle of relative motion applied to matter alone” to refer to Galilean relativity (based on the usual composition of velocities), whereas in the rest of his article he has used this expression to refer to the impossibility of experimentally detecting motion through the ether.

70At any rate, Poincaré does not doubt that Lorentz’s theory violates the reaction principle, and since 1895 he has not ceased to believe that this violation is unavoidable in electromagnetic field theories that account for the Fresnel drag without violating other basic principles. Hence comes his conclusion that all theories satisfying the reaction principle (for matter alone) would be condemned, “unless we consent to deeply modify all our ideas on electrodynamics.” In simpler words, we will have to deeply modify our ideas if we want to preserve the principle of reaction. The emphasis on “unless we consent…” and the earlier arguments that any violation of the reaction principle has paradoxical consequences (including an undesired violation of the relativity principle) strongly suggest that Poincaré was in favor of the “deep modifications.”

  • 78 As we will see in a moment, Lorentz indeed read this remark as attenuating Poincaré’s objection.

71We may now return to Poincaré’s introduction and to his suggestion that “the following considerations [might be] of a kind that attenuates rather than aggravates this objection.” There is no doubt that the objection is still there: Poincaré persists in regarding the violation of the reaction principle as a major defect of Lorentz’s theory, and he persists in hoping that the theory will be modified so as to satisfy the principle—although he does not exclude that the violation might be real. Moreover, he now believes that the violation of the reaction principle implies a first-order violation of the relativity principle, which should make it even more harmful in his eyes. One could perhaps speculate that he regards the fictitious fluid as a means to alleviate the violation: in a formal sense, the theorem of the center of mass applies to matter plus fluid. The speculation would be legitimate if only he anticipated or wished that someday the momentum of this fluid or at least the symbolic quantity c-2Π would be regarded as a genuine momentum. But there is not even the smallest hint that he does so. On the contrary, in the rest of his article he uses the fluid as a means to quantify the violations of the reaction principle. We are left with three possibilities: the attenuation claim is mere politeness, or it refers to the fact that the violation of the reaction principle is not a difficulty specific to Lorentz’s theory, or else Poincaré believes that the newly established connection between the violation of the reaction principle and a violation of the relativity principle brings one closer to a better theory. The second option rests on the remark that Fizeau’s result can hardly be explained without violating the reaction principle.78 But this remark dates from 1895, and all Poincaré adds to it in 1900 is a link with the general correlation between the relativity and reaction principles. This brings us to the third option: Poincaré was somewhat relieved to see that the violations of the two principles, undesirable though they were, were mutually consistent. Better understood violations might bring us closer to a better principled theory. This is the gist of Poincaré’s liminary remark that objections “serve to develop the latent virtue of theories.”

4 Reception

Lorentz’s thank-you letter

72In a letter of 20 January, 1901, Lorentz thanked Poincaré for his contribution to the jubilee volume:

Allow me to thank you most sincerely for agreeing to take part in the Recueil offered to me [...]. Since your judgment is of the greatest value to me, you have particularly obliged me by the choice of your topic and by the words preceding your article. I have studied your considerations with all the attention they deserve, and I feel the force of your remarks. I must admit I find it impossible to modify the theory in a way that would eliminate the difficulty you point out. It even seems improbable to me that anyone will succeed in this task. Instead I believe—this is also the result toward which your remarks tend—that the violation of the principle of reaction is necessary in all the theories that can explain Fizeau’s result. Yet, should we worry about that?

  • 79 Poincaré to Lorentz, 20 January 1901, in Walter, Bolmont et al. 2007, pp. 252– 254.

Lorentz goes on to express his dislike of theories, such as Helmholtz’s, admitting a movable ether, and his preference for a strictly immobile ether on which the electrons cannot react and in which the Maxwell stresses are mere fictions. He then writes:79

As for the principle of reaction, I do not think it should be regarded as a fundamental principle of physics. Admittedly, in every case in which a body acquires a certain momentum a, our mind will not rest until we can point to a simultaneous change in some other body and until we know that, in all phenomena in which the ether is not involved, this change consists in the acquisition of the momentum −a. But I think one could be equally satisfied if this simultaneous change were not itself the production of a motion. You derived the beautiful formula

\[\sum{M\mathbf{V} + \int{c^{- 2}\mathbf{\Pi}\mathrm{d}\tau} = \text{constant}}.\]

It seems to me that we could just regard the quantity ∫c2 Πdτ  as a quantity depending on the state of the ether and so to say “equivalent” to a quantity of motion. For any change in the momentum of matter, your theorem yields a simultaneous change of the equivalent quantity. I think we could make do with this. [...] What I just called “equivalence” could well someday appear to be an “identity.” This could happen if we succeed in regarding ponderable matter as a modification of the ether itself.

  • 80 On the latter point, see Darrigol 2004.

73Clearly, Lorentz understands that Poincaré finds it difficult to admit a violation of the principle of reaction and that Poincaré hopes for a “modified” theory that would save the principle. Lorentz also understands that by writing Eq. (33) for the momentum balance, Poincaré does not mean to interpret c-2Π as a genuine momentum density (for the ether). That this quantity could be treated as equivalent to a momentum, or even as a genuine momentum in an electromagnetic theory of matter, is Lorentz’s own suggestion. Alas, Lorentz has not paid attention to another aspect of Poincaré’s article: the essentially new conception of the Lorentz transformations, and in his important memoir of 1904 with the fuller Lorentz transformations he will maintain his old concept of corresponding states. No one before 1905—except Samuel Burbury in print and perhaps Einstein in private—seems to have noticed the crucial difference between Lorentz’s corresponding states and Poincaré’s apparent states, which involve a time defined through optically synchronized clocks.80

Mie’s abstract

  • 81 Mie 1901.

74Around the same time, the competent field theorist Gustav Mie reviewed Poincaré’s jubilee article for the Beiblätter zu den Annalen der Physik (the German abstracts of the time). The report includes: Poincaré’s derivation of the momentum-balance equation; the interpretation of c-2Π as the momentum of a fictitious fluid of density c-2w ; the need to include this fluid to formally save the reaction principle, with the conclusion that “for real matter the principle does not hold”; the impossibility of compensating this violation by a motion of the dielectric in Lorentz’s theory (in contrast with Hertz’s); the conviction that the energy principle and the relativity principle [Prinzip der Relativität der Bewegungen] together imply the reaction principle; that consequently the relativity principle cannot hold in Lorentz’s theory; the deduction, for a moving radiator, of a complementary recoil force that indeed violates this principle; the quasi-impossibility of detecting this force owing to its smallness; the idea that Fizeau’s result in itself implies a violation of the relativity principle. So far, everything Mie writes is accurate, in fact much more accurate than what most historians have written. However, he errs at the very end of his review by having Poincaré say that we will have to renounce all theories respecting the relativity principle unless we deeply modify our ideas on electrodynamics. Also, Mie did not pay attention to Poincaré’s groundbreaking reinterpretation of the Lorentz transformations.81

Burbury’s abstract

  • 82 Burbury 1901.

75Samuel Burbury, a British mathematician and expert on statistical mechanics and electrodynamics, reviewed Poincaré’s article for the Science abstracts of 1901. He begins with: “The author explains in his first section how the law of equality of action and reaction is not satisfied in Lorentz’s theory, if that law is to be applied to ordinary matter only.” He gives the momentum-balance equation and the attached interpretation through the fictitious “Poynting fluid.” He indicates how Poincaré uses this fluid to derive the recoil of a unidirectional radiator. He understands that in his second section, Poincaré wants to explain why the same recoil is predicted by Hertz’s theory even though this theory satisfies the principle of reaction. The answer, according to Burbury, is that in Hertz’s theory the dielectric “takes the whole recoil” whereas in Lorentz’s theory the dielectric takes only a small fraction of the recoil (this infelicitous wording suggests that part of the recoil momentum goes to the radiator, part to the dielectric, whereas Poincaré means that the recoil of the resonator is partly (entirely in Hertz’s case) compensated by an opposite motion of the dielectric). In a sketchy summary of the third section, Burbury tells us that for Poincaré the principle of reaction is “axiomatic” owing to its “intimate connection” with the principle of relative motion. Lastly, he recounts how Poincaré verifies this connection in the case of a unidirectional radiator: firstly, Poincaré interprets the Lorentz’s local time as the time measured by moving observers who synchronize their clocks by optical signaling and thereby ignore their motion in the ether; secondly he derives the difference between the true and apparent energy of the emitted radiation; thirdly, he shows the necessity of the complementary force by comparing the energy balances in two different reference frames. Burbury concludes: “This is the explanation of the fact that in Lorentz’s theory the principle of action and reaction cannot apply to matter alone, and the same is true of the principle of relative motion.” Altogether, we see that Burbury has captured the essence of Poincaré’s reasoning, and he probably is the first reader of the jubilee article to have publicly remarked it contained a new interpretation of Lorentz’s local time.82

Abraham’s momentum

  • 83 Wien 1900; Lorentz 1900-1901; Thomson 1893; Abraham 1902. See Darrigol 2022, pp. 170–171, and furt (...)

76As Lorentz knew, J. J. Thomson, Heaviside, Wien and others had shown that the electromagnetic field of a charged particle implies an additional inertia of this particle in proportion to the electrostatic self-energy. In 1900 Wien and Lorentz had both speculated that the mass of an electron could be entirely electromagnetic, in which case it should significantly depend on the velocity of the electron. In this picture, the momentum of an electron derives entirely from the attached electromagnetic field, and it becomes natural to conceive an electromagnetic momentum of which every momentum in nature would result. This is the gist of Lorentz’s private suggestion that c-2Π may someday become a genuine momentum. The first implementation of this idea belongs to J. J. Thomson and dates from 1893, as was earlier mentioned. The second is found in an article of 1902 by Max Abraham, motivated by Walter Kaufmann’s recent success in experimentally demonstrating a velocity-dependence of the mass of the electron.83

77In order to compute both the longitudinal and the transversal mass of a fully electromagnetic electron (the electromagnetic energy gives only the longitudinal mass), Kaufmann found it convenient to compute the total momentum ∫c2 Πdτ  of the accompanying field. For the origin of this idea, he referred to Poincaré’s article of 1900 in the following words:

Mr. Poincaré has discussed the position of Lorentz’s theory regarding Newton’s third axiom with special thoroughness. He emphasized that the theorem of momentum conservation remains valid when a proper momentum is ascribed to the electromagnetic field. This electromagnetic momentum is given by the integral [∫c2 Πdτ ] over the entire field [...]. Accordingly, the generalized theorem of the center of mass reads: the time derivative of the sum of the electromagnetic and mechanical momenta equals the external force.

Abraham introduced this electromagnetic momentum in parallel with the electromagnetic energy: just as Maxwell had generalized the energy principle by adding a field energy to the energy of matter, one could add a field momentum to the momentum of matter: “The theorem of the center of mass behaves like the energy principle [with respect to electromagnetic interactions].” A few months later, Abraham similarly wrote:

It will be shown [...] that from Lorentz’s theory one can derive not only an electromagnetic energy but also an electromagnetic momentum. Poincaré was first to bring this out. He showed that by introducing this momentum one can extend the theorem of the center of mass to systems of electrons and he alleged the same for the conservation of angular momentum. The existence of an electromagnetic momentum is of fundamental import for the dynamics of electrons.

Abraham now accompanied the reference to Poincaré with the remark: “J. J. Thomson has given a particular derivation of the electromagnetic momentum based on the impulse of moving Faraday tubes.”Abraham 1902, pp. 25–26; Abraham 1903, pp. 110, 110n.

78Clearly, Abraham then regarded the momentum of Poincaré’s fictitious fluid as a genuine momentum, and he credited Poincaré for showing the necessity of this momentum in Lorentz’s theory. In an article of 1904 on the theory of radiation, he more correctly asserted:

The concept of electromagnetic momentum is as fundamental for the theory of light and heat radiation as it is for the theory of electrons. As is well known, this concept is the basis of the theory of the inertial mass of the electron, which is so strikingly confirmed by W. Kaufmann’s experiment on the deviation of Becquerel rays. Thus—whereas H. Poincaré regarded the violation of the momentum law as an objection to the theory of electrons—now the extended momentum law has become the safest basis of this theory. For this reason, the concept of electromagnetic momentum plays a significant role in the electromagnetic foundation of mechanics that is pursued in modern physics.

In his treatise of 1905 on the electromagnetic theory of radiation, Abraham similarly wrote:

That the principle of action and reaction, understood according to Newton’s mechanics, is violated by Lorentz’s theory, has been raised by H. Poincaré as an objection against this theory. This objection holds only if we regard the axioms of mechanics as a priori valid. If, on the contrary, we regard physics as a science whose principles must be adapted to our progressing experience, this objection should not disconcert us. We will rather base the mechanics of the electromagnetic field on the generalized impulse theorem [Eq. (33)] and investigate whether this theorem has consequences compatible with experience. If this is the case, it is not the foundations of electron theory but the axioms of mechanics that we should revise.

  • 84 Abraham 1904, pp. 237–238; 1905, pp. 31 (citation), 27 (credit); 1908, p. 365 (synchronization). I (...)

Regarding the origin of the concept of electromagnetic momentum, Abraham referred to his own article of 1903, and no longer to Poincaré. For compensation, in 1908 he credited Poincaré with the physical interpretation of Lorentz’s local time through optical synchronization, while (as far as I know) no one else in the preceding years (except Burbury) referred to Poincaré in this respect.84

79Presumably, after writing his first articles of 1902–03 on electron dynamics, Abraham read Poincaré’s text more carefully than he had earlier done and understood his true intentions. Also, he had probably seen Lorentz’s encyclopedia article of 1903 on electron theory, in which one can read:

Here [in Eq. (33)] we have a violation of the principle of action and reaction, which Poincaré brought to the fore as a defect of the theory, although he concedes that this defect is hard to avoid. In fact, this deviation from ordinary mechanics is rooted in the foundations of the theory if we assume that the ether exerts forces on charged matter [...] and yet never consider forces acting on the ether. However, it is worth mentioning that every change in the momentum of matter is accompanied with the opposite change of a certain quantity [∫c2 Πdτ ] depending on the state of the ether, although this quantity is not a momentum in the ordinary sense of the word [...]. Owing to this circumstance, we may call [∫c2 Πdτ] the electromagnetic impulse or the electromagnetic momentum.

  • 85 Lorentz 1903, pp. 162–163 (citation), 190–193 (electromagnetic mass).

Lorentz (correctly) credits Abraham for introducing this concept of electromagnetic momentum. He of course does not mention his private letter of 1901 to Poincaré, despite the striking similarity of the present extract with the suggestions given in this letter. In another section of his article, Lorentz presents Abraham’s momentum-based method for determining the electromagnetic inertia of an electron.85

80Despite Lorentz’s authoritative judgment on this matter of priority and despite Abraham’s implicit approval of this judgment, Abraham’s early statement that Poincaré had introduced the concept of electromagnetic momentum in 1900 was often repeated by electron theorists in Göttingen and elsewhere. In his posthumous eulogy of Poincaré the physicist, Paul Langevin wrote:

[Poincaré] spent much time on the following point: Lorentz’s theory contradicts the principle of the equality of action and reaction or of momentum conservation. He first saw in this point a reason to reject Lorentz’s theory, but he soon brought a decisive contribution by showing that the difficulty disappears by introducing what he called the electromagnetic momentum, a new notion which singularly eased and even provoked the ulterior development of electron dynamics.

  • 86 Langevin 1913a, p. 699; Wien 1921, (written in 1914), 290. See also Langevin 1905, p. 265. Langevi (...)

This is doubly incorrect: Poincaré did not use the phrase “electromagnetic momentum” and in 1900 he did not believe the difficulty disappeared by introducing what he truly called “the momentum of the electromagnetic energy [regarded as a fictitious fluid].” In his own praise of Poincaré the physicist, Wien similarly wrote:86

Of very high significance for theoretical physics is a work which was published in 1900 in the Lorentz jubilee volume. There [Poincaré] introduced the electromagnetic momentum through which the contradiction with the principle of reaction is removed, a theory which became very important in the further development of electrodynamics.

Poincaré’s later thoughts

  • 87 Poincaré 1904, pp. 303–312; 312, 316 (all forces transform in the same way).

81In September 1904, Poincaré discoursed on “The present state and the future of mathematical physics” at the St. Louis Congress of Arts and Science in Missouri. He first explained how theoretical physics had evolved from the Laplacian mechanical picture to a physics of more abstract and general principles, namely: the energy principle, the principle of energy dissipation, the principle of reaction, the relativity principle, the principle of mass conservation, and the principle of least action. He then diagnosed a crisis in which all these principles (except the principle of least action) seemed to be under threat. The principle of energy dissipation, Poincaré explains, can only have a statistical validity if Boltzmann and Gibbs are right, and Brownian motion could well represent a visible violation of this principle. The principle of relativity has so far survived any experimental attempt to disprove it by detecting the motion of the earth through the ether, but our best electromagnetic theory, Lorentz’s, satisfies this principle in an artificial and imperfect manner, by accumulating hypotheses. Poincaré now understands that the lack of invariance of forces does not imply a violation of the relativity principle if all forces are altered in the same manner when passing to a moving frame. But he is still dissatisfied with the current state of affairs.87

82Poincaré’s next remarks concern the principle of reaction:

Let us move to Newton’s principle of the equality of action and reaction. This principle is intimately related to the relativity principle and it seems clear that the fall of one principle would precipitate the fall of the other. Thus, we should not be surprised to encounter the same difficulties in this case.

Poincaré rehearses his old argument of 1895 that retarded action necessarily implies a violation of the reaction principle, and he also introduces the unidirectional radiator of 1899. He is now aware that the radiative recoil of a mirror has been observed (by Lebedev in particular), and he argues that the residual air could not possibly compensate for the missing momentum because in this case Fizeau would have observed a complete drag of light waves by moving matter. As for the subterfuge of endowing the ether itself with inertia, he writes:

The principle [thus saved] could explain everything, since, whatever be the visible motions one could always imagine hypothetical motions that compensate for them. But if the principle can explain everything, it cannot predict anything, since it does not allow us to choose among competing hypotheses, since it explains everything in advance. It therefore becomes useless.

Also, Poincaré recalls his argument of 1899 according to which the ether velocity would strangely be quadrupled when the fields are doubled, and concludes:

This is why I have long believed that the consequences of the theory adverse to Newton’s principle should someday be abandoned and yet recent experiments on the motion of electrons from radium rather seem to confirm them.

  • 88 Poincaré 1904, pp. 312, 314.

Here Poincaré clearly indicates that before encountering Kaufmann’s experiments, which proved the electromagnetic nature of the inertia of electrons, he believed that our theories should be modified so as to avoid the violation of the principle of reaction (for matter alone). This indeed agrees with what Poincaré wrote at the end of the jubilee article. In his present view, the electromagnetic inertia of the electrons deeply alters the game since it requires every momentum in the world to be of electromagnetic origin.88

  • 89 Poincaré 1904, pp. 315, 316–317.

83These considerations naturally bring Poincaré to a violation of the principle of mass conservation, since purely electromagnetic electrons have a variable mass: “There is no mass other than electrodynamic inertia. But in this case the mass cannot be a constant, it must depend on velocity.” In fact, even if there are non-electromagnetic masses, they should also depend on velocity like electromagnetic masses in order to preserve the relativity principle. Indeed, as Poincaré has now learned from Lorentz, the non-invariance of forces (manifested in the Liénard force) does not imply any violation of the relativity principle if every force, including inertial forces, are altered in the same manner when passing to a moving frame of reference. Then every mass must vary like an electromagnetic mass. Poincaré comments:89

It goes without saying that the fall of Lavoisier’s principle [of the conservation of mass] precipitates the fall of Newton’s principle [of reaction]. The latter principle stipulates that the center of mass of an isolated system moves [uniformly] on a straight line. But if mass is no longer a constant, there is no center of mass, we do not even know what that is. This is why I earlier said that the experiments on cathode rays [rather: on beta rays] had seemed to justify Lorentz’s doubts regarding Newton’s principle.

Of all these results, if they were confirmed, would emerge an entirely new mechanics that would be mainly characterized by the fact that no velocity could exceed the velocity of light. For an observer unconsciously dragged in a translation [through the ether], no apparent velocity could ever exceed the velocity of light. This would be a contradiction if we did not remember that this observer would not use the same clocks as an observer at rest but instead would use clocks giving the “local time.”

  • 90 See Poincaré to Lorentz [ca. May 1905], in Walter, Bolmont et al. 2007, p. 255: “I have recently s (...)

84To sum up, by 1904 Poincaré has changed his mind about the reaction principle, or at least he is willing to change his mind. He now believes that the principle can be violated, so much so that the electrons’ inertia and momentum could be of electromagnetic origin. This violation, despite his earlier contrary guess, would not contradict the relativity principle as long as all forces including inertial forces would undergo the same alteration in a moving frame of reference. Poincaré is aware of Lorentz’s memoir of 1904,90 which provides a proof that for contractile electrons, the electromagnetic mass varies exactly in the way needed for the inertial force to behave like the Lorentz force under a Lorentz transformation. He is not yet certain that this theory is the correct one and he formulates his prophecies about a new dynamics in the conditional mode. But he now understands that the reaction principle in the old mechanical sense could be dissolved in the electromagnetic worldview without losing the relativity principle.

85In a manuscript document written in 1908 to instruct his campaign for the Nobel prize in physics, Poincaré wrote:

I had to examine various consequences of Lorentz’s theory. I showed it was incompatible with the principle of the equality of action and reaction and how this principle should be modified in order to be reconciled with this theory. This result served Abraham as a starting point for the calculation through which he proved that the mass of electrons if of purely electromagnetic origin and that their transversal mass differs from their longitudinal mass.

This claim flatly contradicts his statement of 1904 that he was not willing to give up the principle of reaction before he heard about Kaufmann’s experiments. It also differs from the description included in the “Analysis of his scientific works” he wrote in 1901 for Gösta Mittag-Leffler:

I reviewed the main electromagnetic theories of light by Maxwell, Helmholtz, Hertz, Lorentz, and Larmor. Lorentz’s was the most satisfactory in my eyes. However, I made the following objection: this theory does not agree with the principle of the action and reaction (applied to matter alone).

[In 1900] I returned in greater details to the relations between Lorentz’s theory and this principle.

  • 91 Poincaré 1908, p. 416; 1921, (written in 1901), p. 124.

There is no hint here at a putative attempt to save the reaction principle through the concept of electromagnetic momentum.91

86Was Poincaré being dishonest by claiming, in 1908, that he had given to Abraham the momentum concept he needed to build his theory of the electron? The claim is misleading insofar as it suggests that in 1900 Poincaré was willing to accept the violation of the principle of reaction (for matter alone) in Lorentz’s theory and was arguing in favor of an electromagnetic momentum of genuine physical import. Yet, one cannot deny that Abraham found inspiration in Poincaré’s article, by misreading it as a call for a new kind of momentum. As already said in the introduction, the history of science is full of episodes in which the formal skeleton of an argument (here the equations Poincaré wrote for his fictitious fluid) get fruitfully reinterpreted by a reader working with a different agenda. The reinterpretation, conscious or not, does not deprive the author of the argument of the merit to have stimulated an important development. In this sense Poincaré deserves to be recognized as an important actor in the genesis of the concept of electromagnetic momentum. At the same time, historians should by no means confuse the original argument and its reinterpretation.

Planck in 1908

87In September 1908, Planck lectured on “the principle of action and reaction in the generalized dynamics” at the Naturforscherversammlung in Cologne. He included the following historical remarks:

Many of us remember the sensation that Lorentz created when he renounced the general validity of Newton’s axiom in his construction of an atomistic electrodynamics based on a stationary ether. Inevitably, this circumstance was asserted, e.g. on the part of H. Poincaré, as a serious objection against Lorentz’s theory. Some sort of relief came only when it became clear, especially in M. Abraham’s investigations, that the reaction principle could be saved in full generality if one admitted a new, electromagnetic momentum in addition to the mechanical momentum that had been the only known kind of momentum. Abraham made this move plausible by comparing momentum conservation to energy conservation: just as the energy principle is violated when one does not take the electromagnetic energy into account but holds when one takes it into account, the momentum principle is violated when one does not take the electromagnetic momentum into account but holds when one takes it into account.

This inherently indisputable comparison leaves a fundamental difference untouched: in the energy case, we know from the start a whole series of different kinds of energy: kinetic, gravitational, elastic, thermal, and chemical, so that one does not fundamentally innovate when he adds to these kinds one more kind: the electromagnetic energy. In contrast, in the case of momentum only one kind of momentum had been known: the mechanical momentum. Whereas energy represented a universal physics concept from the start, momentum had so far been a specifically mechanical concept and the reaction principle had been a specifically mechanical theorem. Therefore, the necessary extension of this concept was perceived as a fundamental upheaval in which the relatively simple and unitary concept of momentum acquired a considerably more complex character.

  • 92 Planck 1908, pp. 828–829.

In this extract, Planck clearly identifies Poincaré as one who used the violation of the reaction principle as an objection to Lorentz’s theory, and Abraham as the one who saved this principle through the electromagnetic momentum. Most insightfully, Planck explains why Abraham’s parallel between electromagnetic energy and electromagnetic momentum did not earlier induce theorists, including Poincaré, to admit this kind of momentum as a physical reality. The reason is that no one before Abraham (save for Lorentz in private, and J. J. Thomson in a singular theory of moving tubes of force) was willing to introduce a momentum that was not mechanically understood as the product of mass by velocity. There is indeed much evidence that Poincaré, before 1904, conceived momentum in this mechanical manner only, and this partially explains why in 1900 he persisted in his hope for a future electromagnetic theory that would spare the principle of reaction (applied to matter alone). As we saw, another reason is that he believed this violation to have paradoxical consequences.92

Einstein in 1905–1906

  • 93 Whittaker [1953, pp. 51–52], and a few others after him, have wrongly attributed to Poincaré the g (...)

88In retrospect, Poincaré’s greatest error, in 1900, was to bet that the future, correct theory of electrodynamics should respect the principle of reaction (applied to matter alone). His two greatest insights were the physical interpretation of Lorentz’s corresponding states, including the local time, as the states measured by terrestrial observers; and the idea that the violation of the reaction principle in Lorentz’s theory led to a paradoxical conflict between the energy principle and the relativity principle. Also important was the strategy he used to display this paradox: consider the same radiation process in two different inertial frames and apply the energy principle in the two frames. By 1904, Poincaré knew how to solve this paradox at the electronic level by assuming the proper velocity dependence of the electron’s mass. The paradox remained intact for macroscopic radiation processes.93

89When, in 1905, Einstein offered his first argument for the inertia of energy, he similarly considered a radiation process in two different frames. The main difference is that the process imagined by Einstein is symmetrical in a frame in which the source is at rest, the same quantity of radiative energy J/2 being emitted in two opposite directions, without recoil because of the symmetry. In this frame, the total emitted energy is J and the energy spent by the radiator is also J. In a frame in which the source moves at the velocity u  along the direction of emission, and with γu = (1 − u2/c2)−1/2, the total emitted energy is

\[(52)\quad J' = \gamma_{u}(1 + u/c)(J/2) + \gamma_{u}(1 - u/c)(J/2)\sim J + (1/2)(J/c^{2})u^{2}\]

  • 94 Einstein 1905.

owing to the relativistic rule for transforming the energy of a light complex. In order to retrieve the value J of the energy spent by the source, Einstein assumes the mass of the source to vary by −J/c2 during the emission process. This variation indeed implies a variation –(J/c2)u2/2 of the kinetic energy of the source (to second order), which exactly compensates for the increase of the emitted energy.94

90This argument would not have been available to Poincaré in 1900, for he then reasoned to first order only. Paul Langevin is reported to have given in 1906 the following first-order argument, based on momentum balance. The process is the same as in Einstein’s argument. In the comoving frame, the total momentum of the emitted radiation vanishes. In the rest frame, the emitted momentum at first order is

\[(53)\quad (1 + u/c)(J/2c) - (1 - u/c)(J/2c) = Ju/c^{2}.\]

  • 95 Langevin 1913b. For the 1906 dating, see Langevin 1913b, p. 414, and Edmond Bauer’s testimony repo (...)

This variation is exactly compensated by the variation –(J/c2)u of the momentum of the source if its mass diminishes by –J/c2.95

91In an article published in the same year [1906], Einstein gave two new arguments in favor of the inertia of energy. In his introduction, he refers to Poincaré’s article of 1900 in the following words:

In this article I will show that [the inertia of energy] is a necessary and sufficient condition for extending the law of the conservation of motion of the center of mass (at least in a first approximation) to systems in which electromagnetic processes occur in addition to the mechanical processes. Although the simple formal considerations that must be completed before proving this assertion are essentially contained in an earlier work by Poincaré, for the sake of transparency I will not rely on this work.

  • 96 Einstein 1906, pp. 627–629. The argument is slightly defective since the absorption of radiation i (...)

Einstein’s first argument is based on a thought experiment, perhaps in analogy with the perpetual motion Poincaré (and Newton) generate by rigidly connecting two bodies for which the action does not equal the reaction. Einstein imagines an emitter and an absorber facing each other and rigidly connected. The emitter sends a flash of energy J  toward the absorber, which implies the recoil momentum variation –J/c for the rigid system. This system returns to rest when the flash reaches the absorber. The transferred energy J is then brought back to the emitter through a massless carrier. In this cycle, the system globally shifts by –(J/Mc)(L/c) if M denotes the mass of the system and L the distance between the emitter and the absorber. This is a kind of perpetual motion, since the cycle could be indefinitely repeated, without any energy expense, so as to move the system to any distance. The paradox is solved by assuming that the return of the energy J  to the emitter involves the transfer of the mass J/c2. Indeed this transfer implies the shift (c2J/M)L of the rigid system (in a first approximation), which compensates for the shift produced during the first step of the cycle.96

92In his second argument, Einstein rederives Poincaré’s Eq. (35), which is

\[\frac{\mathrm{d}}{\mathrm{d}t}\left( \sum{M\mathbf{R}} + \int{c^{- 2}w\mathbf{r}\mathrm{d}\tau} \right) + \int{c^{- 2}\rho(\mathbf{v} \cdot \mathbf{E})\mathbf{r}\mathrm{d}\tau = \text{constant}}.\]

Assuming that the increase ρ(vE) of the energy density of matter implies the increase ρ(vE)/c2 of its mass density, then the variation of the mass of a particle (obtained by integrating the latter density over the volume of the particle) exactly compensated for the third term in the former equation, and we are left with

\[(54)\quad \tag{54} \frac{\mathrm{d}}{\mathrm{d}t}\left( \sum{M\mathbf{R}} + \int{c^{- 2}w\mathbf{r}\mathrm{d}\tau} \right) = \text{constant,}\]

which expresses the conservation of the motion of the center of mass of the field-matter system, if only the mass density w/c2 is ascribed to the electromagnetic field.

  • 97 Einstein 1906, pp. 629–633. For a comparison between Poincaré’s and Einstein’s approaches to relat (...)

93Even though Poincaré did not anticipate the mass-energy equivalence in Einstein’s sense, he detected the paradoxes that Einstein solved through this equivalence. Evidently, Einstein had seen Poincaré’s article of 1900 when he devised his second argument for the inertia of energy. Poincaré’s article was frequently cited in literature Einstein read, and the Lorentz jubilee volume in which it was published was easily available. It is not known whether he had read this article when he constructed his relativity theory in 1905. At any rate, there are interesting similarities between Einstein’s and Poincaré’s ways of thinking, as well as deep differences in the understanding of relativity theory.97

  • 98 Planck 1908; Einstein to Lorentz, 16 Aug. 1913, in Einstein 1993, p. 552. As Max Laue later showed (...)

94In his earlier cited conference of 1908 on the principle of reaction in generalized dynamics, Planck proposed to extend the relation g = c−2Π between the electromagnetic momentum density g and the electromagnetic energy flux Π to any form of energy, in harmony with the Einsteinian equivalence between mass and energy. Namely, a momentum density can be seen as a mass flux, and Planck’s relation then means that the transfer of the energy E across a surface implies the transfer of the mass E/c2 across this surface. As Einstein put it in 1913, Planck’s relation is “the simplest expression of the equivalence between mass and energy.” Although Planck and Einstein did not refer to Poincaré in this regard, it jumps to the eyes that Poincaré’s interpretation of the vector c-2Π as the momentum density of a fictitious fluid of mass density c2w anticipates Planck’s general intuition. The anticipation is merely formal: in 1900 Poincaré could not possibly imagine that his fictitious mass transfers would someday become reality.98

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Notes

1 In conformity with Poincaré’s usage, in the following I will speak of a “violation of the reaction principle” when the principle does not apply to matter alone. Poincaré sometimes specified “for matter alone,” sometimes not. As David Rowe pointed out to me, the secondary literature’s focus on Poincaré’s relativity principle (instead of the reaction principle) could be related to the Einstein-centered character of this literature.

2 Whittaker 1910, p. 353n; Goldberg 1967, p. 941; Cuvaj 1970, pp. 86, 91; Granek 2000, p. 17; Weinstein 2015, p. 187. In addition, Granek 2000 (p. 22) and Weinstein 2015 (p. 188) both misinterpreted Poincaré’s complementary force (discussed below) as a means to save the reaction principle, whereas in Poincaré’s eyes it was a consequence of the violation of this principle. In a forthcoming critical edition of Poincaré’s main writings regarding relativity theory, Christian Bracco and Jean-Pierre Provost support the reaction-saving narrative. I thank them for numerous exchanges that helped me sharpen my old arguments against this narrative.

3 Miller 1981, pp. 43, 56 (emphasis mine). Howard Stein [2021, pp. 18–19] also holds an ambiguous position: he regards Poincaré’s extension of the conservation of momentum to Lorentz’s theory as his most important result in 1900, but he contrasts this piece of “mathematical artist[ry]” with Abraham’s assertion of the physical existence of electromagnetic momentum.

4 Darrigol 1995, pp. 23—29.

5 Cf. Katzir 2005, p. 272: “An electromagnetic force that exists only in a moving frame of reference [the complementary force] had to be assumed. Lorentz’s theory therefore conflicted with the principle of relativity, and Poincaré once again saw a close connection between that principle and the principle of action and reaction.”

6 On this last point, see Darrigol 2000b.

7 Rationalized electromagnetic units are used for all theories before Lorentz’s. Mostly anachronistic vector and tensor notation is used through the paper. On electrodynamic theories before relativity theory, see Hirosige 1966, 1976; Buchwald 1985; and much other literature cited in Darrigol 2000a.

8 See Darrigol 2000a, chap. 4, and further reference there. Maxwell only had \( \dot{\mathbf{D}} \times \mathbf{B}\) for the third term; he overlooked the Hertz force \(\mathbf{D} \times \dot{\mathbf{B}}\).

9 Hertz 1890. See Darrigol 2000a, pp. 255–257, and further reference there.

10 Hertz 1890, pp. 389–398; Heaviside 1891-1892. See Darrigol 2000a, pp. 410–411.

11 Hertz 1890, p. 284; 1892, p. 295; Heaviside 1891-1892. All translations are mine. Helmholtz later attempted and failed to compensate for the Hertz force with a pressure gradient of the ether [Helmholtz 1894].

12 Hertz 1890, p. 370; Fizeau 1851, 1859. On the Fizeau experiment, see Darrigol 2022, pp. 105–107, and further reference there.

13 Hertz 1890, p. 399.

14 Heaviside 1891-1892, p. 524 (2 citation), 557–558 (1 citation)..

15 Thomson 1893, pp. 6–21 (p. 21 for the electromagnetic momentum of a moving charge). See Buchwald 1985, pp. 275–276; Darrigol 2000a, pp. 295–298.

16 Lorentz 1892b. See Darrigol 2000a, pp. 322–332, and further reference there. I use Hertz’s units, for which the electromagnetic energy is simply given by
½(e2 + b2).

17 See Darrigol 2022, pp. 102–104, and further reference there.

18 Lorentz 1892a, 1895.

19 Reiff 1893. See Darrigol 2000a, p. 322.

20 Larmor 1894, 1895, 1897. See Darrigol 2000a, pp. 332–343, and further reference there.

21 Poincaré 1895, pp. 381–382. On Poincaré’s lectures and on his criticism, see Darrigol 1995. On Poincaré and L’éclairage électrique, see Nio 2020, chap. 10.

22 In reality, Reiff’s system of equations is incomplete and it can easily be completed to comply with the conservation of electricity.

23 Poincaré 1895, pp. 391–392.

24 Poincaré 1895, pp. 392.

25 This is only a guess. Lorentz will prove it in 1900: see below, section on Poincaré’s jubilee article of 1900.

26 See above, section on Hertz’s theory.

27 Poincaré 1895, p. 392 (citation), 395–402 (Hertz’s theory), 403–409 (last section).

28 For a density ρ, the relevant convective derivative is Dρ/Dt=∂ρ/∂t+∇⋅(ρv).

29 Poincaré does not write this local balance. Instead he obtains and uses the global balance given by the space integral of this equation.

30 I have simplified Poincaré’s ingenious proof by using the isotropy of the medium at an early stage of the reasoning. One could also use parity invariance to directly infer α=0 and δ=0.

31 Poincaré 1895, pp. 409, 412.

32 Poincaré 1895, pp. 412–413. In the foreword to his first Sorbonne course, Poincaré [1889, p. II] had written: “It does not matter whether the ether really exists. This a question for the metaphysicians. For us the essential thing is that everything happens as if the ether existed and that this hypothesis is convenient for the explanation of phenomena. After all, do we have any other reason to believe in the existence of material objects? This too is only a convenient hypothesis. However, this hypothesis will never cease to be convenient whereas the day will probably come when the ether will be rejected as useless.”

33 Lorentz 1895, p. 28.

34 Liénard 1898a, pp. 457–458. On Liénard’s theoretical works, see Nio 2020, chap. 12.

35 Wien 1898, p. 52.

36 Lorentz 1898, pp. 64–65.

37 Poincaré 1901, pp. 420–421. Boltzmann’s assistant Ignaz Schütz (1897) had earlier established the same connection between the three principles

38 Poincaré 1901, p. 422 (citation), pp. 448–451, p. 453 (citation).

39 Poincaré 1901, pp. 453–454, 631–632.

40 Poincaré 1901, chap. 6. On p. 535 Poincaré notes that the local time shift, being non-negligible compared to optical periods, would seem to be detectable by interferential means. He then excludes this possibility by noting that phase differences can be observed only for vibrations occurring at the same point

41 Poincaré 1901, p. 536.

42 Poincaré considers only the case v = u. The ρ and the v in these equations refer to macroscopic averages.

43 Liénard 1898b, pp. 323–324. Liénard also shows that the same result can be obtained by transforming the expression f = ρ(e + c1 v × b)  of the Lorentz force at the micro-level.

44 Poincaré 1901, p. 543.

45 Poincaré 1900b, pp. 21–22.

46 Pyotr Nikolaevich Lebedev gave a first account of his confirmation of radiation pressure at the Paris International Congress, soon after Poincaré gave his talk. Even if Poincaré had known and trusted this result before giving his talk, he would not necessarily have seen in it a violation of the reaction principle since, as he soon argued in the jubilee article, one could still imagine that some rarefied matter would provide the missing momentum.

47 Poincaré 1900b, pp. 24–25.

48 Heike Kamerlingh Onnes launched the invitations in May 1900 (see, e.g., his letter to Wien of 5 May 1900, recently auctioned at https://natedsanders.com/heike_kamerlingh_onnes_autograph_letter_signed_fro-lot58883.aspx). Poincaré’s replies (Boerhaave Museum, accessible through http://henripoincarepapers.univ-nantes.fr/) are undated. His text was received on 10 Nov. 1900.

49 Poincaré 1900a, p. 252.

50 Poincaré 1900a, pp. 252–255.

51 Poincaré 1900a, pp. 256 (citation), 258 (Eq. (35)).

52 Poincaré 1900a, p. 258.

53 Poincaré 1900a, pp. 259–260. The conservation of the total angular momentum follows from the local angular momentum balance that can be derived from Eq. (23) and from the symmetry of the stress tensor. The reasoning being quite similar to the one leading from Eq. (23) to the global momentum balance, Poincaré simply omits it.

54 Poincaré 1900a, p. 260.

55 Poincaré’s expression “the momentum of the electromagnetic energy,” found in Section 2 (p. 264), should not be confused with what we now call the electromagnetic momentum. For Poincaré (or anyone at that time), an energy does not have a momentum; what has a momentum is the fictitious fluid associated with the electromagnetic energy, as Poincaré makes clear in his next sentence.

56 Poincaré 1900a, p. 261.

57 Poincaré 1900a, p. 269.

58 Poincaré 1900a, p. 270; Newton 1687, pp. 23–24.

59 Poincaré 1900a, pp. 270–271.

60 Poincaré 1900a, pp. 271–272.

61 Poincaré 1900a, pp. 272–273.

62 For details, see Darrigol 2022, pp. 167–169. According to Peter Galison [2003, chap. 4], Poincaré’s contemporary involvement in practical aspects of the measurement of time would explain his recourse to optical synchronization.

63 Poincaré [1900a, p. 274] observes that for the Lorentz-transformed fields the “apparent propagation velocity” has the value  .

64 Poincaré 1900a, p. 273.

65 Poincaré 1900a, p. 273.

66 In a modernized proof, the intensity of the wave train in the ether frame is 
χ (kx − kct) , where χ is the characteristic function of the interval [0,1] and k = 1\L. The 2-vector (k,k) transforms like (x,ct) under a Lorentz transformation. Therefore,
χ(kx – kct )= χ(k'x' − k'ct'), with k' = k(1 − u/c) and L' = 1/k' = (1 + u/c)L at first order.

67 Poincaré 1900a, pp. 274–276.

68 Poincaré 1900a, p. 277.

69 The effect is opposite to the one derived in the previous reasoning because the emitter now moves in the direction opposite to the direction of emission.

70 Poincaré 1900a, pp. 277–278. Poincaré does not clearly distinguish between the recoil momentum and the force that causes it, which makes his reasoning difficult to follow. The following is a clarification of what he had in mind.

71 Poincaré 1900a, p. 278.

72 From the fact that the complementary force can be obtained through the Lorentz transformations, one should not conclude that it must satisfy the relativity principle. The relativity principle will be satisfied if the field equations are invariant and if the equation of motion of the radiator is also invariant. The latter condition is not satisfied if, as Poincaré implicitly assumes, the inertial force −d/dt of the radiator is (approximately) invariant (while the recoil force is not).

73 Poincaré 1900a, pp. 271–272 (fuller quote above, note 60): “The principle of relative motion understood in this way and applied to matter alone [namely, the principle according to which phenomena depend only on the relative motion of material objects only] is so far from compelling the mind that some have conducted experiments designed to reveal the translation of the earth. While it is true that these experiments have shown negative results, this outcome was rather surprising.”

74 Also, the quantities in Eq. (46) are all apparent, measurable quantities and the fact that this relation is not satisfied therefore represents a violation of the relativity principle.

75 Lorentz 1899.

76 Poincaré 1900a, 278.

77 It would if there were non-optical means to synchronize moving clocks so that they give the true time. Poincaré indeed considered this possibility in [1904, p. 312], as he thought gravitational actions might propagate faster than light (by Laplace’s old argument).

78 As we will see in a moment, Lorentz indeed read this remark as attenuating Poincaré’s objection.

79 Poincaré to Lorentz, 20 January 1901, in Walter, Bolmont et al. 2007, pp. 252– 254.

80 On the latter point, see Darrigol 2004.

81 Mie 1901.

82 Burbury 1901.

83 Wien 1900; Lorentz 1900-1901; Thomson 1893; Abraham 1902. See Darrigol 2022, pp. 170–171, and further reference there.

84 Abraham 1904, pp. 237–238; 1905, pp. 31 (citation), 27 (credit); 1908, p. 365 (synchronization). In 1904–1905, Emil Cohn, Einstein, and Abraham all employed optical synchronization for the time to be used in a moving frame [see Darrigol 2000a, pp. 382], without referring to Poincaré.

85 Lorentz 1903, pp. 162–163 (citation), 190–193 (electromagnetic mass).

86 Langevin 1913a, p. 699; Wien 1921, (written in 1914), 290. See also Langevin 1905, p. 265. Langevin [1913a, p. 700], also attributed to Poincaré the parallel between electromagnetic energy and electromagnetic momentum, which truly belongs to Abraham. Langevin’s opinion may just be reflecting his familiarity with Abraham’s article [1903], which he translated for the collection Ions, électrons, corpuscules he published in 1905 with Henri Abraham for the Société française de physique.

87 Poincaré 1904, pp. 303–312; 312, 316 (all forces transform in the same way).

88 Poincaré 1904, pp. 312, 314.

89 Poincaré 1904, pp. 315, 316–317.

90 See Poincaré to Lorentz [ca. May 1905], in Walter, Bolmont et al. 2007, p. 255: “I have recently studied your memoir [Lorentz 1904] in greater details; this memoir is extremely important, and I had cited its main results in my St. Louis conference.”

91 Poincaré 1908, p. 416; 1921, (written in 1901), p. 124.

92 Planck 1908, pp. 828–829.

93 Whittaker [1953, pp. 51–52], and a few others after him, have wrongly attributed to Poincaré the general idea of the inertia of energy.

94 Einstein 1905.

95 Langevin 1913b. For the 1906 dating, see Langevin 1913b, p. 414, and Edmond Bauer’s testimony reported in A. Langevin 1971, pp. 58–59.

96 Einstein 1906, pp. 627–629. The argument is slightly defective since the absorption of radiation is an irreversible process (the absorber could be replaced by an ideal photovoltaic cell).

97 Einstein 1906, pp. 629–633. For a comparison between Poincaré’s and Einstein’s approaches to relativity theory, see Darrigol 2022, pp. 232–234.

98 Planck 1908; Einstein to Lorentz, 16 Aug. 1913, in Einstein 1993, p. 552. As Max Laue later showed, Planck’s relation derives from the symmetry of the stress-energy tensor in Hermann Minkowski’s four-dimensional formalism. See Darrigol 2022, pp. 220–221, 225. On Planck and relativity, see Liu 1997.

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Olivier Darrigol, « Poincaré and the Reaction Principle in Electrodynamics »Philosophia Scientiæ, 27-2 | 2023, 63-125.

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Olivier Darrigol, « Poincaré and the Reaction Principle in Electrodynamics »Philosophia Scientiæ [En ligne], 27-2 | 2023, mis en ligne le 16 octobre 2023, consulté le 19 mai 2024. URL : http://journals.openedition.org/philosophiascientiae/4084 ; DOI : https://doi.org/10.4000/philosophiascientiae.4084

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Olivier Darrigol

CNRS: UMR SPHere (France)

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