MICHEL JANSSEN

THE TROUTON EXPERIMENT, E = Me2,

AND A SLICE OF MINKOWSKI SPACE-TIME

1. THE FORGOTTEN PRECURSOR TO THE TROUTON-NOBLE EXPERIMENT

In the Fall of 1900, Frederick T. Trouton started work on an ingenious experiment inhis laboratory at Trinity College in Dublin. The purpose of the experiment was todetect the earth's presumed motion through the ether, the 19th-century mediumthought to carry light waves and electric and magnetic fields. The experiment wasunusual in that, unlike most of these so-called ether drift experiments, it was not anexperiment in optics.I Trouton tried to detect ether drift by charging and discharginga capacitor in a torsion pendulum at its resonance frequency, which he hoped wouldset the system oscillating. The basic idea behind the experiment came from GeorgeFrancis FitzGerald, whose assistant Trouton was at the time. According to FitzGer­ald, a capacitor moving through the ether should experience an impulse, a jolt, uponbeing charged or discharged. Trouton's torsion pendulum was designed to detectthese jolts. Not surprisingly from a modem relativistic point of view, Trouton foundno such effect. FitzGerald died in February 1901 before the experiment was con­cluded. It was thus left to others to try and reconcile Trouton's result with then cur­rent electromagnetic theory.

The first to do so was Joseph Larmor, who not only got closely involved withTrouton's experiment after FitzGerald 's death, but who also became the editor of avolume of FitzGerald's scientific papers published the following year. Trouton'spaper on the experiment suggested by FitzGerald was reprinted in this volumeaccompanied by an interesting four-page editorial note. Larmor, however, devotedonly one short paragraph of his note to Trouton's original experiment, confidentlyasserting that no effect should have been expected in the first place. He was far moreinterested in a new and promising variant of the experiment that Trouton, most likelywith input from Larmor himself, had proposed in his paper. The idea behind this newexperiment was to detect ether drift not through linear impulses upon charging or dis­charging a capacitor in a torsion pendulum, but through a turning couple on a care­fully insulated charged capacitor in a torsion pendulum of a slightly different design.Trouton eventually carried out this experiment at University College in London,where he was appointed professor of physics in 1903. He was assisted by one of hisresearch students, Henry R. Noble.

27

A. Ashtekar et al. ieds.) , Revisiting the Foundations ofRelativistic Physics. 27-54.© 2003 Kluwer Academic Publishers.

28 MICHEL JANSSEN

Figure 1. FitzGerald was given tofiights offa ncy.

This Trouton-Noble experiment continues to intrigue theorists and experimental­ists, in- and outside the physics mainstream, to this day (see for example, Teukolsky1996; Hayd en 1994; Comille, Naud in, and Szam es 1998, 1999). Its precursor, theTrouton experiment, on the other hand, has been all but forgotten . Thi s is unfortun ate ,for there is as much to be leamed from the original experiment as there is from itsbetter-known sequel. In this paper, I will therefore focus on the Trouton experiment. Ihope to present my findings conceming the Trouton-Noble experiment on some otheroccasion (for a preliminary version, see Janssen 1995) . For a more hist orically ori­ented discu ssion of the experiments, including a careful analysis of the role Larmorplayed in Trouton 's work, I refer to Warwick 1995.2 The emphasis in this pap er willbe on conceptual issues.

The full explanation ofthe Trouton experiment requires that one take into acco untthe inertia of energy expressed in Einstein 's famous equation E = mc-, The Troutonexperiment can actually be seen as a practical version of a thought experim ent withwhich Einstein tried to show that E = me? is both necessary and sufficient to ensurethat the center-o f-mass theorem, accord ing to which no process in an isolated systemcan change the state of motion of the system's center of mass , hold s for systemsinvolving both electromagnetic fields and ordinary matter (Einstein 1906).3 As theseobservation s suggest, the analysis of the Trouton experiment pro vides va luableinsights into the transition from classical to relativistic mechanics." On this score, theTrouton and Trouton-Noble experiments have much more to tell us than such famousoptical ether drift experiments as the Michelson-Morley experiment.

That Lannor was so quick to dismiss FitzGerald's original proposal was probablybecause he real ized that the pred icted effect would violate the center-of-m ass theo­rem.' The probl em with Larmor's reductio is that, at the time, it was at best uncl earwhether current electromagnetic theory , based on the notion of a stationa ry or immo­bile ether, was at all compatible with the center-of-mass theorem or with such closelyrelated laws as Newton 's third law and the law of momentum conservation. In 1895,

THE TROUTON EXPERIMENT 29

H. A. Lorentz, the leading proponent ofthe immobile ether theory, had in fact alreadyrenounced Newton's third law as a universal law of nature. In Lorentz 's theory, mat­ter can never set the ether in motion, yet ether can set matter in motion via theLorentz forces that fields in the ether exert on charged particles in matter. This isclearly in violation ofNewton's principle that action equals reaction.

In 1903, Max Abraham introduced the concept of electromagnetic momentum,with the help of which the notion of an immobile ether could at least be reconciledwith the law of momentum conservation for processes involving both fields and mat­ter. In his well-known paper on the electrodynamics of moving bodies of 1904,Lorentz availed himself of this new concept of electromagnetic momentum in hisanalysis of both the Trouton and the Trouton-Noble experiment. In the case of theTrouton experiment, Lorentz's analysis vindicated FitzGerald's prediction that amoving capacitor should experience ajolt upon being charged or discharged. His cal­culations, however, indicated that Trouton's torsion pendulum had not been sensitiveenough to detect the effect.

Ifwe put Larmor's and Lorentz's accounts ofthe Trouton experiment side by side,we arrive at the following dilemma. If the effect predicted by FitzGerald does occur,the center-of-mass theorem is violated (as is the relativity principle, one may add).That is what Larmor tells us. If, however, the effect does not occur, momentum con­servation appears to be violated. That is what Lorentz tells us. It seems that we haveto choose between the center-of-mass theorem and momentum conservation, twolaws that are essentially equivalent in Newtonian mechanics. E = me- allows us toescape this dilemma. Once the inertia of energy is properly taken into account inLorentz's analysis of the Trouton experiment, a negative result in the experiment,even if the sensitivity of the apparatus were greatly improved, is seen to be compati­ble both with the center-of-mass theorem and with momentum conservation .

It is tempting to speculate that the Trouton experiment would have lived on inphysics textbooks as an elegant illustration of E = mc-, if only the connectionbetween the experiment and the equation had been recognized in the early years ofspecial relativity. This in tum raises the question why nobody in fact made this con­nection, which with hindsight appears to be so obvious. In general, such questions arenot very fruitful, but in this case it will direct us to another complication we are facedwith in working out a detailed explanation of the Trouton experiment (and, for thatmatter, the Trouton-Noble experiment). I already referred obliquely to this complica­tion: how does one properly take into account E = met in Lorentz's analysis of theTrouton experiment? The third part of the title of my paper refers to this problem.

Mathematically, the definition of electromagnetic momentum used by Lorentz inhis analysis of the Trouton and Trouton-Noble experiments is an instance of the stan­dard relativistic definition of the momentum of spatially extended systems, a defini­tion that is not Lorentz invariant. Lorentz defined the electromagnetic momentum ofa charged capacitor moving through the ether as a space integral of the momentumdensity of the capacitor 's electromagnetic field. Such space integrals-to borrowsome modem relativistic terminology-are integrals over hyperplanes of simultane­ity in a frame at rest in the ether, regardless of the velocity of the capacitor withrespect to the ether. It follows that the electromagnetic momentum of a capacitor at

30 MICHEL JANSSEN

rest in the ether and the electromagnetic momentum of the same Lorentz contractedcapacitor carrying the same charge in uniform motion through the ether are notrelated to one another via a Lorentz transformation. Under the standard definition­to once again put it in relativistic terms-electromagnetic momentum does notbehave as the spatial part of a four-vector under Lorentz transformations . And, as weshall see, for the explanation of the Trouton experiment based on the inertia of energyto work out in detail, it is crucial that it does.

In special relativity, the easiest way out is to use an alternative Lorentz-invariantdefinition of the four-momentum of spatially extended systems. This alternative defi­nition was first proposed by Fermi (1922) and made popular by Rohrlich (1960,1965).6 Under this new definition, the electromagnetic momentum of a capacitor isthe integral of the electromagnetic energy-momentum density over a hyperplane ofsimultaneity in the capacitor's rest frame, regardless of the frame of reference inwhich the momentum is calculated. Defined in this way, the electromagnetic momen­tum of the capacitor does transform as the spatial part of a four-vector under Lorentztransformations . In special relativity, it is ultimately a matter of convention whetherone chooses the standard definition or the definition of Fermi, Rohrlich, et al.' InLorentz's ether theory, however, the hyperplanes of simultaneity in the frames at restin the ether are privileged, which commits the proponents of the theory to a definitionthat mathematically is a special case of the standard definition. Hence, the easy wayout is not available to the ether theorist.

Under the standard definition, only the four-momentum of a closed system trans­forms as a four-vector under Lorentz transformations, not the four-momentum of itsopen sub-systems. This result was first announced in 1911 by Max Laue for the spe­cial case of static closed systems." The electromagnetic field of a capacitor is not aclosed system and its four-momentum (under the standard definition) does not trans­form as a four-vector. A satisfactory account of the Trouton experiment in Lorentz'stheory therefore can not be given without considering the material part of the capaci­tor as well.

From the point of view of classical mechanics, this is a completely unexpectedcomplication . Classical mechanics, of course, already tells us that there will bestresses in the material part of a charged capacitor that prevent the capacitor from col­lapsing under the Coulomb attraction between its plates. But relativistic mechanics(at least under the standard definition of the four-momentum of spatially extendedsystems) predicts the entirely new effect that stresses in a system's rest frame giverise to momentum in a frame in which the system is moving. It is only the sum of thismomentum and the electromagnetic momentum that transforms as the spatial part ofa four-vector under Lorentz transformations as is required for the detailed explana­tion of the Trouton experiment. Laue, unaware of the conventional element in his def­inition of momentum, put great emphasis on this relation between stresses andmomentum. He presented it as on par with the relation between energy current andmomentum density, which, as was first noted by Planck (1908), can be seen as anexpression ofthe inertia of energy. For a modem relativist Laue's effect is just an arti­fact of a definition that the relativist is under no obligation to adopt. The ether theorydoes not allow such leeway, and Laue's peculiar effect is another element that has to

T HE T ROUTON EXPERIMENT 31

be taken over from relativistic mechanics, along with the inertia of energy, to arrive ata satisfactory account of the Trouton experiment.

So, the full explanation of the negative result of the Trouton experiment, be it inspecial rclativityor in Lorentz's ether theory, involves three clements completely for­eign to classical mechani cs that were introduc ed in the decade followin g the experi­ment: electromagnetic momentum (Abraham 1903), the inertia of energy (Einste in1905b , 1906), and the effect that stresses give rise to momentum (Laue 1911a).? Inspecial relativity , the third element can be replaced by the later insight that there is acert ain freedom in defining the four-momentum of spatially extended systems. Inview of all this , I would say that the Trouton experiment fully deserves to be rescuedfrom oblivion.

2. LARMOR, TH E TROUTO N EXPERIMENT, AND THE CENTER OFMASS THEORE M

veloci ty withrespect to the ether

Figure 2. The basic idea behind the Trouton experiment.

Figure 2 illustrates the basic idea behind the Trouton exper iment. A battery is used tocharge a capac itor. If the power is switched on, an electromagnetic field is producedlargely confined to the volum e between the plates of the capacitor. If the system is atrest in the ether, the charges will only produce an electric field; if the system is mov­ing, the charges will also produce a magnetic field. As Trouton wrote in his paper onthe experiment:

The question then naturally arises as to the source supplying the energy required to pro­duce this magnetic field . If we attribute it to the electric generator, say a battery, there isno difficulty [...] FitzGerald 's view, however, was that it would be found to be suppliedthrough there being a mechanical drag on the condenser itself at the moment of charging(Trouton 1902, 557- 558)

In other words, FitzGerald thought that the energy for the magnetic field would comefrom the capacitor's kinetic energy. Elementary Newton ian mechanics tells us that inthat case a moving capacitor upon being charged should experience a jolt in the direc­tion oppo site to its direction of mot ion. In figure 2, the effect is illustrated for a capac-

32 MICHEL JANSSEN

itor suspended on a wire from the ceiling of the laboratory with its plates parallel tothe direction of motion.

The actual arrangement, shown in figure 3 taken from Trouton's paper, was a littlemore subtle. At FitzGerald' s suggestion, Trouton made the capacitor part of a torsionpendulum . The capacitor was "charged and discharged continuously by means of aclock-work, at the intervals corresponding to the free period of swing of the appara­tus. In this way any effect produced would cumulate and be made easier of observa­tion" (Trouton 1902, 560).10

+

CONDENSER

Figure 3. Trouton s torsion pendulum with capacitor (from: Trouton 1902, 560) .

Trouton did not find any effect. He concluded: "it is evident that some othersource for the energy or some countervailing effect must clearly be looked for"(Trouton 1902, 562). The passage quoted at the beginning of this section suggest sthat Trouton , probably under Larmor's influence, preferred the first option . FitzGer­ald, as Trouton reports , was leaning toward the second:

On the last opportunity I had of discussing the matter with Professor FitzGerald, prelim­inary experiments had been made, giving as far as they went negative results: the finalresults not being completed till after Science had to deplore the grievous loss it sustainedat his death, FitzGcrald, on that occasion, made a remark which, as well as I remember,was to the effect that should the negative results then obtained be sustained by furtherwork, he would attribute the non-occurrence of any observable effect to the same generalcause as produced the negative results in Michelson and Morley's experiments on the rel­ative motion of the Earth and the ether by means of the interference of light. (Trouton1902, 562)

THE TROUTON EXPERIMENT 33

As Trouton goes on to explain, "the same general cause" refers to the Lorentz­FitzGerald contraction, or, as Trouton and Larmor preferred, the FitzGerald-Lorentz con­traction. In the next paragraph, Trouton elaborates on FitzGerald's suggestion: "Fromsome such cause a diminution ofthe electrostatic energy might be brought about [. .. ] justsufficient in amount to provide the energy required for the magnetic field" (Trouton1902, 562-563). It is not clear whether this elaboration is Trouton's or FitzGerald's.However,no matter whose idea it was, it is hard to see how it could be made to work.

Trouton acknowledged the help of Larmor in his paper. And Larmor includedTrouton's paper in his edition of FitzGerald's scientific papers, adding an interestingnote himself. For the most part Larmor's note deals with the proposal in Trouton'spaper to look for a turning couple on a charged capacitor rather than for impulsesupon charging and discharging a capacitor. Anticipating the negative result of theTrouton-Noble experiment aimed at detecting this turning couple, Larmor argued thata negative result could be accounted for on the basis of the Lorentz-FitzGerald con­traction .II The original Trouton experiment is disposed of in the final short paragraphof Larmor's note :

If the condenser [...J is held absolutely fixe d while it is being charged, any impulsivetorque there might be could do no work; yet the condenser gets its energy. This seems byitself sufficient to negative the suggestion that the energies of charge and discharge [. .. Jhave to do directly with mechanical forces (Larmor 1902, 569; italics in the original) .

I take it that by 'absolutely fixed' Larmor meant ' fixed with respect to the laboratoryrather than suspended on a torsion wire as in the actual experiment.' The alternativereading 'fixed with respect to the ether' does not seem to make sense, since the prob­lem only arises for a moving capacitor. At first glance, Larmor's argument is a grossnon-sequitur. It is perfectly consistent to maintain that if the capacitor is held fixedwith respect to the laboratory the energy of the magnetic field comes from an ever soslight decrease in the kinetic energy of the earth as a whole . It is hard to believe thatLarmor could have overlooked such an obvious point. It is more likely that his reduc­tio is, in fact, to the absurdity of the notion that the earth's kinetic energy coulddecrease in this experiment. But why would Larmor have thought this to be absurd?

As Andrew Warwick (1995, 317) has emphasized, Larmor believed that it wasimpossible to extract energy from an object's motion through the ether, except maybe avery small amount, proportional to some higher power of vic, the ratio of the object'svelocity with respect to the ether and the velocity of light. If it were possible to extractmore, the whole universe, or so Larmor believed, would have come to rest in the ethera long time ago. If FitzGerald were right, the energy extracted from the motion of thecapacitor and the earth through the ether in the Trouton experiment would be of order(v 2/c 2 ) . Perhaps this then is the absurdity Larmor sensed in FitzGerald's proposal.His attitude toward the new experiment Trouton proposed would seem to support thisinterpretation . Referring to the latter experiment, Larmor writes: "Thus the energy ofmotion of the Earth through the rether is available for mechanical work to an unlimitedextent, unless [...] the FitzGerald-Lorentz shrinkage of moving bodies is a fact" (Lar­mor 1902, 568). Larmor appears to have been quite confident that the result of theexperiment would be negative. The importance of the experiment for him was that it

34 MICHEL JANSSEN

would provide further evidence for the contraction hypothesis. There is an importantdifference though between Larmor's attitude toward the original experiment and hisattitude toward Trouton's new proposal. In the latter case, Larmor offered a detailedexplanation as to why it would be impossible to extract energy from the earth's motionthrough the ether in this fashion. In the case ofthe original Trouton experiment, he sim­ply dismissed FitzGerald's proposal out of hand. This suggests that Larmor had someother reason for believing FitzGerald's idea to be absurd.

John Stachel (see note 5) has suggested such a reason . Ifthe fully isolated systemof earth and capacitor were to change its velocity upon charging or discharging thecapacitor, we would have a blatant violation of the center-of-mass theorem. Onemight object that the earth-capacitor system is not fully isolated since it interacts withthe ether. However, since Larmor assumed the ether to be immobile, the center-of­mass theorem is also violated if we consider the earth-capacitor-ether system. Itseems very plausible to me that this violation of the center-of-mass theorem is indeedthe absurdity that Larmor sensed in FitzGerald's proposal. In that case, it has to besaid that Larmor's intuition was right on the mark. The connection between the Trou­ton experiment and the center-of-mass theorem, however, turns out to be much morecomplicated than Larmor, given the extreme brevity of his remarks, can possibly haverealized at the time .

3. THE ACTION-EQUALS-REACTION PRINCIPLE AND THEINTRODUCTION OF ELECTROMAGNETIC MOMENTUM

The center-of-mass theorem is closely related to Newton's third law, the principlethat action equals reaction, which, in tum, is closely related to the law of momentumconservation. When Larmor wrote his comment on the Trouton experiment, the statusof momentum conservation in its various guises in theories positing an immobileether had been the subject of some serious debate , notably between Lorentz andPoincare .'? In 1902, the situation was unclear at best.

Newton 's principle of the equality of action and reaction is hard to reconcile withthe notion of an ether that can set matter in motion (through the Lorentz forces of elec­tromagnetic fields on charged particles), yet can itself never be set in motion by matter.Lorentz clearly stated this obvious difficulty in his widely read monograph of 1895.After discussing the problem of how to make sense of forces acting on an immobileether and concluding that the easiest way to solve the problem would be never to applythe notion of force to the ether at all, Lorentz wrote, in an often quoted passage:

It is true that this conception would violate the principle of the equality of action andreaction-because we do have grounds for saying that the ether exerts forces on ponder­able matter-but nothing , as far as I can see, forces us to elevate that principle to the rankofa fundamental law of unlimited validity. (Lorentz 1895,28; italics in the original.)

Poincare strongly objected to this aspect of Lorentz 's theory, especially to the viola­tions of the center-of-mass theorem it entails . In fact, he made this the topic of hiscontribution to a Festschrifton the occasion of the 25th anniversary of Lorentz's doc­torate (Poincare 1900b) .

THE TROUTON EXPERIMENT 35

Poincare illustrated his objection with the example of a mirror recoiling upon thereflection of light (Poincare 1900b, 273). He used this same example in an importantlecture during the International Congress ofArts and Sciences in St. Louis in 1904:

Imagine, for example, a Hertzian oscillator, like those used in wireless telegraphy ; itsends out energy in every direction; but we can provide it with a parabolic mirror, asHertz did with his smallest oscillators, so as to send all the energy produced in a singledirection. What happens then according to the theory? The apparatus recoils, as if it werea cannon and the projected energy a ball; and that is contrary to the [action equals reac­tion] principle of Newton, since our projectile here has no mass, it is not matter, it isenergy" (Poincare 1904, 101; my italics) .

The italicized final remark , which is not to be found in Poincare's more detailed dis­cussion of the example in 1900, shows how tantalizingly close he came to the resolu­tion of the problem through E = mc 2. 13

In a letter to Poincare in response to the latter's contribution to his Festschrift ,Lorentz reiterated that any theory based on an immobile ether-or, as Lorentz put it,"any theory that can explain Fizeau's experiment," which was generalIy understoodto provide strong evidence for the notion of an immobile ether (see Janssen andStachel I999)-will violate the action-equals-reaction principle and thereby the cen­ter-of-mass theorem . He made it clear that he did not see this as a serious problem forhis theory. 14

From the point of view of classical mechanics, Poincare's recoiling mirror exam­ple also violates momentum conservation. This can be avoided by ascribing momen­tum to the electromagnetic field. The concept of electromagnetic momentum wasintroduced by Abraham in 1903. Today we are so accustomed to a concept ofmomentum that is broader than mechanical momentum that it is easy to forget thatthis was by no means obvious at the beginning of the century," It may not be inap­propriate therefore to illustrate this point with the folIowing extensive quotation froma paper by Planck. The paper is based on a lecture delivered during the annual meet­ing ofthe Versammlung Deutscher Naturforscher und Ifrzte in Cologne on September23, 1908, two days after Minkowski's famous lecture on "Space and Time"(Minkowski 1909). Planck's paper, entitled "Comments on the Principle of Actionand Reaction in General Dynamics," contains a vivid description of the difficultiessurrounding the action-equals-reaction principle around the tum of the century:

As is well-known, the real content of the Newtonian principle ofthe equality of action andreaction is the theorem of the constancy of the quantity of motion or of the momentum ofmotion; I therefore want to talk about this principle only in the sense of that theorem, and,more specifically, about its relevance for general dynamics, which not only includesmechanics in a more restricted sense, but also electrodynamics and thermodynamics.

Many of us will still recall the stir it caused, when H. A. Lorentz , in laying the founda­tions of an atomistic electrodynamics on the basis of a stationary ether, denied Newton'sthird axiom absolute validity, and inevitably this circumstance was turned into a seriousobjection against Lorentz's theory, as was done, for instance, by H. Poincare . A calmnessof sorts [eine Art Beruhigung] only returned when it became clear, especially through theinvestigations of M. Abraham, that the reaction principle could be saved after all, in itsfull generality at that, if only one introduces , besides the mechanical quantity of motion ,the only kind known at that point , a new quantity of motion , the electromagnetic kind .

36 MICHEL JANSSEN

Abraham made this notion even more plausible by a comparison between the conserva­tion of the quantity of motion and the conservation of energy. Just as the energy principleis violated if one docs not take electromagnetic energy into account and satisfied if onedocs introduce this form of energy, so is the reaction principle violated if one only con­siders the mechanical quantity of motion but satisfied as soon as one also takes intoaccount the electromagnetic quantity of motion.

However, this comparison , incontestable in and of itself, leaves one essential differenceuntouched. In the case of energy, we already knew a whole series of different kinds­kinetic energy, gravitation [sic], clastic energy of deformation, heat, chemical cnergy­so it docs not constitute a fundamental innovation if one adds electromagnetic energy tothese different forms of energy as yet another form. In the case of the quantity of motion,however, we only knew one kind so far: the mechanical kind. Whereas energy wasalready a universal physical concept, the quantity of motion had so far been a typicallymechanical concept and the reaction principle had been a typically mechanical theorem.Consequently, its generalization, while recognized to be necessary, was bound to beexperienced as a revolution of a fundamental nature, through which the up to that pointrelatively simple and uniform concept of the quantity of motion acquired a considerablymore complicated character (Planck 1908, 828-829).

Planck may have exaggerated the difficulties physicists were experiencing with thenotion ofelectromagnetic momentum somewhat for rhetorical purposes (he goes on toshow that the idea ofputting energy and momentum on equal footing is a very naturalone in relativity theory), but this passage does make it clear that the introduction ofelectromagnetic momentum was indeed, as Planck says, a "fundamental innovation."

4. LORENTZ, THE TROUTON EXPERIMENT,AND MOMENTUM CONSERVATION

Both the Trouton and the Trouton-Noble experiment are discussed in the paper thatforms the crowning achievement of Lorentz's work on the electrodynamics of mov­ing bodies before the advent of special relativity (Lorentz I904a) . Lorentz discussedboth experiments in terms of electromagnetic momentum. This does not mean that heaccepted the interpretation ofthis quantity as a new kind of momentum . In the case ofthe Trouton-Noble experiment, Lorentz made it clear that he only used the phraseelectromagnetic momentum to describe the result of a derivation that is completelyindependent of the quantity's interpretation as a form of momentum. 16 Lorentz's cau­tion illustrates the physics community's somewhat reluctant acceptance of electro­magnetic momentum as momentum sui generis in the early years of this century (cf.the quotation from Planck 1908 above).'? In his discussion of the Trouton experi­ment, as we shall see shortly, Lorentz uncharacteristically did invoke the interpreta­tion of electromagnetic momentum as a form of momentum . It cannot be ruled out,however, that he simply suppressed a derivation that he felt would justify his argu­ment without relying on this interpretation. It is important to keep in mind in this con­text that the Trouton experiment plays a rather modest role in Lorentz's paper. TheTrouton-Noble experiment is prominently discussed in the introduction of the paper,where it is presented as one of two new ether drift experiments that partly motivatedthe paper. Discussion ofthe Trouton experiment is relegated to the final section of thepaper, a section that has the character of an appendix .

T HE TROUTON EXPERIMENT 37

After these cautionary remarks, let us see what Lorentz actually had to say aboutthe Trouton experiment:

I take this opportunity for mentioning an experiment that has been made by Trouton atthe suggestion of FitzGerald, and in which it was tried to observe the existence of a sud­den impulse acting on a condenser at the moment of charging or discharging; for thispurpose the condenser was suspended by a torsion-balance, with its plates parallel to theearth 's motion. POl' forming an estimate of the effect that may be expected, it will sufficeto cons ider a condenser with aether as dielectricum. Now if the apparatus is chargedthere will be [...Jan electromagnet ic momentum

(Terms of the third and higher orders arc here neglected). This momentum being pro­duced at the moment of charging and disappearing at that of discharging, the condens ermust experience in the first case an impulse - (8 and at the second an impulse + (8.

However Trouton has not been able to observe these jerks.I believe it may be shown (though his calculations have led him to a different conclusion)that the sensibility of the apparatus was far from sufficient for the objec t Trouton had inview (Lorentz 1904a, 829-830).

mechanical mom entum

---~

velocity withrespect to the ether

... ele ctromagnetic momentum

Figure 4. Lorentz 's analysis a/ the Trouton experiment.

Figure 4 illustrates Lorentz's analysis in this passage. As the moving capacitor ischarged, a certain amount of electromagnetic momentum is created in the direction ofmotion (or rather, to first order in (v / c), in the direction parallel to the plates of thecapaci tor," which Lorentz, following Trouton, assumed to coincid e with the directionof motion). Implicitly invoking momentum conservation, Lorentz concludes that thecapacitor's gain in electromagnetic momentum must be compensated by a loss ofordinary mechanic al momentum. Hence, the capacitor should experience a jerk back­wards upon being charged, just as FitzGerald had originally predicted.

Here we have the dilemma mentioned in the introduction . Momentum conversa­tion seems to require a positive result if the Trouton experime nt were repeated withincreased accuracy, whereas the center-of-mass theorem seems to require a strictly

38 MICHEL JANSSEN

negative result. As we saw in the preceding section, Lorentz was quite willing to giveup the center-of-mass theorem, so for him there was no real dilemma.

Of course, a positive result would violate not only the center-of-mass theorem butalso the relativity principle. In ether-theoretic terms, a positive result in the Troutonexperiment would have provided evidence of the earth's motion through the ether. In1904, this would have been no more troubling for Lorentz than a possible violation ofthe center-of-mass theorem . Citing Poincare's scathing criticism of the way in whichhe had introduced the contraction hypothesis to explain the negative result of theMichelson-Morley experiment, Lorentz described the task he had set himself in his1904 paper as showing "by means of certain fundamental assumptions, and withoutneglecting terms of one order of magnitude or another, that many electromagneticactions are entirely independent of the motion of the system" (Lorentz 1904a, 8 I I;my emphasis). In fact, the reason that the Trouton-Noble experiment played such animportant role in the paper was that the theory developed in the paper provided an ele­gant explanation of the experiment's negative result. However, as the statement justquoted shows, Lorentz did not claim that no experiment could ever detect ether drift.

After 1905 the situation changed . Borrowing some insights from special relativ­ity, in particular the notion that moving observers will actually measure the "fictive"Lorentz transformed quantities of Lorentz's theorem of corresponding states insteadof the "real" Galilean transformed ones, Lorentz perfected his theory in such a waythat it predicted in full generality that no experiment could ever detect ether drift.Lorentz put great emphasis on the fact that his theory was therefore empirically indis­tinguishable from special relativity.-" Yet, as far as I know, he never returned to theTrouton experiment to explain what was wrong with his analysis of 1904.

5. E = MC2: HOW THE CENTER-Of-MASS THEOREM AND MOMENTUMCONSERVATION CAN BOTH HOLD IN THE TROUTON EXPERIMENT

The dilemma that we arrived at in the preceding section is easily resolved once we real­ize that energy has mass. Qualitatively, the argument runs as follows. If energy has mass,a transfer of energy from the battery to the capacitor means a transfer of mass, and, in aframe of reference in which battery and capacitor are moving, a transfer of momentum.So, figure 4 showing the momentum of the capacitor in the Trouton experiment beforeand after it is charged should be replaced by figure 5 below showing the momentum ofboth the capacitor and the battery before and after the capacitor is charged. When themoving capacitor is charged, it gains a certain amount of energy, mass, and momentum,while the moving battery loses that same amount of energy, mass, and momentum. Thetotal amount ofmomentum is conserved. Contrary to what Lorentz thought in 1904, thisdoes not require the capacitor to change its velocity. The increase in the capacitor'smomentum corresponds to a change in the capacitor's mass, not to a change in its veloc­ity. Hence, there is no violation of the center-of-mass theorem. Once the inertia ofenergy is taken into account, a strictly negative result of the Trouton experiment is thusseen to be compatible both with momentum conservation and with the center-of-masstheorem.

THE TROUTON EXPERIMENT 39

+

Figure 5. Transfer ofmomentum in the Trouton experiment.

All that is neede d for this simple argume nt to work out quantitatively as well, isthat energy and momentum transform as the components of a four-vector underLorentz transformations. Let the transfer of four-momen tum from the battery to thecapaci tor in a frame with coordinates x ' ~ = (et', x ', y' , z' ) in which battery andcapacitor arc at rest, be:

A ,~ _ A'~ (~E' )up Capacitor - -up Battery = c' 0, 0, 0 . (I)

In a frame of reference in which the laboratory is moving with a velocity v in the x­direction-a frame with coordinates X ~l = i\~vx'v 21-the transfer of four-momen­tum is:22

, (~E') ( ~E' (~EI) )~p~l = A~v~p v = i\~o c = Y-c' Y 7 v, 0, 0 . (2)

The spat ial components of this four-vector give the transfer of ordinary three­momentum:

(~E')~p = Y~ v. (3)

Since ~PCapacitor = -~PBattery , momen tum is conserve d. Yet, there is no violation ofthe center-of-mass theorem: ~P corresponds to a change in mass (~m = y~E' / e2 ) ,

not to a change in velocity (~v = 0).Abou t a year afte r he first introduced the inertia of energy (Einstein 1905b) , Ein­

stein published a paper, entitled "The Principle of the Conservation of Motion ofthe Cen ter of Grav ity and the Inertia of Energy," in which he showed that E = me?is necessary and sufficie nt to ensure that the center-of-mass theorem holds for sys­tems in which "not only mecha nica l, but also electromagnetic processes take place"

40 MICHEL JANSSEN

(Einstein 1906, 627). As Einstein acknowledges, his paper is similar to Poincare'scontribution to the Lorentz Fest schrift (Poincare 1900b). Einstein showed that inorder to avoid the kind of violations of the center-of-mass theorem discussed byPoincare, one has to assume that energy has inertia. Instead of Poincare's recoilingmirror, Einstein cons idered the thought experiment illustrated in figure 6.

Consider a box of mass M and length L. Suppose some energy E is stored on theinside of the left wall of the box, and suppose that at time t = 0 this energy is some­how converted into electromagnetic radiation travelling to the other side of the box.The radiation is absorbed at the other end of the box, where the energy is convertedback to its original form . According to standard electromagnetic theory, the box willrecoil upon emission of the radiation, and it will recoil again upon re-absorption ofthe radiation, bringing the box back to rest. Standard electromagnetic theory tells usthat the radiation will have momentum (E I c). Momentum conservation requiresthat the box will recoil with that same momentum in the opposite direction. So, whatthis thought experiment shows is that by moving energy from one side of the box tothe other, the completely isolated system of box plus energy E can move itself. If theenergy E has no mass, this is in blatant violation of the center-of-mass theorem. Withthe help of figure 6, it can easily be shown that the only way to avoid this conse­quence is to ascribe mass m = EI c2 to the energy E.

t < 0

o

Lc I~

Box of mass M

Strip of mass m

Strip converted into

electromagnetic energy E,

momentum E l c

8 = displacement of Box

8' = displacement of center

of mass of Box and Strip

Figure 6: Einstein 's thought experiment to show that E = mc2

is necessary and suffi cientf or the center-of-mass theorem.

Let the energy E initially be contained in a strip of as yet unknown mass m « M stuckagainst the inside of the left wall of the box. This means that the center of mass ofbox plus strip will be slightl y to the left of the middle of the box. The energy is then

THE TROUTON EXPERIMENT 41

converted into electromagnetic radiation and, a short time later, reconverted into astrip of mass m stuck against the inside of the right wall of the box. The center ofmass of box plus strip is now slightly to the right of the middle of the box. The dis­placement D' of the center of mass can be calculated from the following conditionthat determines where a wedge supporting the system should be placed so that thesystem is perfectly balanced:

(D')_ (L -D')M"2 - m -2- ' (4)

It follows that, to a very good approximation, the displacement of the center of massis given by:

(5)

The center-of-mass theorem is satisfied if and only if the displacement D' of the cen­ter of mass to the right is equal to the distance D that the box travels to the left duringthe time it takes for the radiation to move from one end of the box to the other. To avery good approximation, the time that the box is in flight can be set equal to (Lie),and the velocity of the box can be set equal to its momentum EIc divided by itsmass M. Hence, to a very good approximation, the distance travelled by the box isgiven by:

Comparing eqs. (5) and (6), one sees that indeed

D= D' ~ E = mc-,

(6)

(7)

The conclusion is that E = me? is the necessary and sufficient condition for the cen ­ter-of-mass theorem to hold in systems in which processes involving both electro­magnetic fields and ordinary matter occur. Strictly speaking, there should of coursebe approximately-equal signs rather than equal signs in eq . (7), just as in eqs. (5) and(6). In other words, the thought experiment only yields the conclusion to a very goodapproximation. Einstein was happy to leave it at that (Einstein 1906,629).23

The Trouton experiment can be seen as a practical version of Einstein 's thoughtexperiment. In the case of the Trouton experiment it is the conversion of chemicalenergy of the battery into the energy of the electromagnetic field between the platesof the capacitor that would lead to a violation of the center-of-mass theorem were itnot for the inertia of energy expressed in E = mct . 24

42 MICHEL JANSSEN

6. THE TROUTON EXPERIMENT FORGOTTEN

Why did the Trouton experiment not live on in textbooks on relativity as a practicalversion of Einstein's thought experiment of 1906? Something along these lines, afterall, did happen to the Trouton-Noble experiment. Ever since Max Laue (1911b) firstmade the connection between the Trouton-Noble experiment and a thought experi­ment involving a turning couple on a moving right-angled lever (Lewis and Tolman1909), the Trouton-Noble experiment has frequently been cited as a physical instanti­ation of this thought experiment (see for example, Pauli 1921, 128). The Troutonexperiment, however, seems to have been forgotten before anybody could make theconnection between the experiment and E = mc- ,

The main reason for this sad fate is undoubtedly that the Trouton-Noble experi­ment completely overshadowed the Trouton experiment. We have seen this in thework of Larmor and Lorentz. Larmor was interested in the Trouton-Noble experimentas a test of FitzGerald's contraction hypothesis, but saw no merit in the reasoningbehind FitzGerald 's prediction of the effect sought after in the Trouton experiment.And FitzGerald was no longer there to defend or elucidate his argument. Lorentztook the Trouton experiment more seriously, but the fact remains that the experimentdoes not play the prominent role that the Trouton-Noble experiment plays in theexposition of the 1904 version of his theory for the electrodynamics of moving bod­ies. To make matters worse, the section on the Trouton experiment was omitted whenLorentz's paper was reprinted as part of The Principle ofRelativity, an anthology stillpopular today (Lorentz et al. 1913, 1922). The omission was probably not just to savespace. The editors may have found it more than a little awkward for a collection ofpapers on the relativity principle to include a passage implying that a more accuraterepetition of some experiment should produce a violation of this principle."

The far greater importance that both Larmor and Lorentz attached to the Trouton­Noble experiment appears to have sealed the fate of the Trouton experiment. The onlysubsequent mention of the experiment in the physics literature that I am aware ofoccurs in a review article on the experimental evidence in support of the relativityprinciple written by Jakob Laub, one of Einstein 's early collaborators (Laub 1910,428--430).26 Laub listed the experiment as one offour first-order electrodynamical (asopposed to optical) ether drift experiments with a negative result. The Trouton-Nobleexperiment is also listed, as one of two second-order electrodynamical ether driftexperiments with a negative result. The classifications 'first order' and 'second order'refer to the fact that the experiments aimed at detecting effects of first and secondorder in (v / c) , respectively.

The fact that the Trouton-Noble experiment was a second-order experiment wasan important factor favoring it over the first-order Trouton experiment. In the yearsbefore the advent of special relativity, it was widely felt that Lorentz's theory gave asatisfactory account of almost all first-order experiments and that the main problemwas how to extend this theory to cover second-order experiments as well. This isreflected both in the statement of purpose in Lorentz 1904a, quoted in section 4, andin Poincare's criticism of the theory presented in Lorentz 1895 to which Lorentz wasin part responding in 1904. And it is precisely for this reason that the Trouton-Noble

THE TROUTON EXPERIMENT 43

experiment was so important for Lorentz in 1904.Laub's discussion of the Trouton experiment nicely illustrates another factor that

may have been partly responsible for the neglect of the Trouton experiment after1905. Recall that Einstein , following the example of thermodynamics, presented spe­cial relativity in his 1905 paper in the form of what he would later call a "theory ofprinciple" (Einstein 1919).27 One can thus take the attitude that in special relativitythe negative result of a particular ether drift experiment needs as little explanation asthe impossibility of some alleged perpetuum mobile in thermodynamics. This indeedappears to have been Laub's attitude. He reported that according to Lorentz's theorythere should have been a small effect in the Trouton experiment but that the apparatushad not been sensitive enough to detect it. He then noted that the negative result oftheexperiment is in accordance with the relativity principle and left it at that. He madeno attempt to pinpoint what was wrong with Lorentz 's analysis.

The analogy between special relativity and thermodynamics obviously cannotfully explain the lack of interest in producing a detailed account of the Trouton exper­iment, for in the case of the Trouton-Noble experiment there was no such lack ofinterest. It also does not explain why after 1905 Lorentz did not work out a detailedaccount of the Trouton experiment in his own "constructive theory" (to use the termi­nology of Einstein 1919 again), which was now supposed to predict null results forall conceivable ether drift experiments. Lorentz was in an ideal position, it seems, toproduce such an account. He was familiar with all relevant papers . Trouton 1902 andLarmor 1902 are cited in Lorentz 1904a. And he must have read Einstein 1906 withspecial interest given its relation to Poincare 1900b, a contribution to a Festschrift inhis honor. In fact, in his lectures on relativity at the University of Leiden in 1910-12,Lorentz used the thought experiment of Einstein 1906 in explaining E = mc 2

(Lorentz 1922, 242-243). Yet, despite all this, he apparently never made the connec­tion between the Trouton experiment and Einstein's thought experiment. It is tempt­ing to ask why not.

There is a plausible answer to this question. As the reader may have noticed, thereappears to be a discrepancy between Lorentz's expression for the momentum gener­ated upon charging a moving capacitor and the expression given in the explanation ofthe Trouton experiment on the basis of E = me? in section 5. According to Lorentz ,the momentum gained by the capacitor (to first order in vic) is 2( UI c2)v, whereasour relativistic analysis suggests that the gain is only half that amount (cf. eq. 3). Aswe shall see in the next section, it is a non-trivial task to find out where the extra fac­tor 2 is coming from. In fact, this was not fully clarified until Laue's work on relativ­istic mechanics in 1911. Suppose that Lorentz had realized sometime between 1905and 1911 that the transfer of energy from battery to capacitor in a laboratory movingthrough the ether is accompanied by a transfer of momentum . This would still onlyaccount for half the momentum gained by the capacitor. Hence, momentum conser­vation would still require the capacitor to experience a jolt upon being charged .

By the time that Laue had provided the tools with which Lorentz's analysis ofthe Trouton experiment can fully be reconciled with the simple relativistic analysisgiven in section 5, the experiment seems to have been thoroughly forgotten . Yearslater, Laue-by now Max von Laue-ehose the inertia of energy as the topic for his

44 MICHEL JANSSEN

contribution to the volume in The Library ofLiving Philosophers devoted to Ein­stein (Laue 1949).28Von Laue discussed Einstein's thought experiment of 1906 andhis own analysis of the Trouton-Noble experiment in the same section of this essay.He did not mention the Trouton experiment.

7. THE TROUTON EXPERIMENT AND DEFINING THE FOUR-MOMENTUMOF SPATIALLY EXTENDED SYSTEMS.

The account of the Trouton experiment in section 5 was based on the assumption thatenergy and momentum transform as the components of a four-vector under Lorentztransformations. As a matter of fact, the energy and momentum of the electromag­netic field of a charged capacitor do not transform in this manner, at least not underthe definitions of these quantities that were used by Lorentz, nor for that matter underthe closely related definitions still standard in special relativity today. If we want toretain these definitions, we have to revise the simple account of the Trouton experi­ment. If we want to retain this simple account, we have to adopt new definitions.

In modern notation.s? Lorentz, following Abraham, defined the electromagneticmomentum G of an electric field E and a magnetic field B produced by a charge dis­tribution moving through the ether as

(8)

where £0 is the dielectric constant of the ether. In Lorentz's theory, the integral is tobe taken in a coordinate system moving with the charge distribution. For Lorentz thiscoordinate system is related to a coordinate system at rest in the ether through aGalilean transformation. So, in relativistic terms, the integral is to be taken over ahyperplane of simultaneity in a frame that, from Lorentz's point of view, would be atrest in the ether. I will call such frames 'ether frames ' for short.

To first order in (v / c), the field of a charged capacitor moving at velocity v withrespect to the ether, its plates tilted at an angle tJ with v, has momentum

(9)

pointing in the direction of the plates." In this equation, V'is the energy of the elec­tromagnetic field of the same capacitor at rest in the ether, or-what in special rela­tivity comes to the same thing-the energy in the moving capacitor's rest frame . Forthe special case that the plates are parallel to the direction of motion (tJ = 0), werecover the expression for the electromagnetic momentum given in Lorentz's 1904discussion of the Trouton experiment (see the quotation from Lorentz I904a in sec­tion 4 above) . As I mentioned at the end of section 6, even in this special case and tofirst order in (v / c ),

THE TROUTON EX PERIMENT 45

(10)

as would be required for the simpl e explanation of the Trouton experiment based onE = m e? given in section 5 (cf. eqs. 1-3). In other words, the electromagneticmomentum of the capacitor does not transform as the spatial part of a four-vectorunder Lorentz transformation s. In general, it is not even in the direction of mot ion.

With the standard definiti on in special relativity of the four-momentum of spa­tially extended systems,

(11)

where TllV is the system 's energy-mom entum tensor, we run into the exact sameproblem. In fact, the quantity G as defined in eq. (8) is just the spatia l part of thequantity Pll as defined in eq. (11), iffor Tvv one takes the energy-momentum tensorof the electromagnetic field of a movin g charge distribution in an ether frame .'!

How is the four-momentum of the electromagnetic field of the moving capacitorrelated to the four-momentum of the electromagnetic field of that same capa citor atrest, or- what in special relativity comes to the same thing-to the four-momentumof the electromagnetic field of the moving capacitor in its rest frame? 1 will answerthis question along the lines of a seminal paper by Max Laue (19 11a), in which theenergy-momentum tens or was for the first time put at the center of relativisticmechanics."

With the help of the transformation law for the energy-momentum tensor, eq. (11)can be rewritten as

( 12)

where T'IlV the energy-momentum tensor for the field ofthe capacitor at rest. Since acharged capac itor is a static system, the iO- and Oi-components (i = 1, 2,3 ) of T'llvvanish, and the volum e integral in the xll- frame can be set equal to l / y times thevolume integral in the x 'll -frame.P Using in addition that A00 = y, we can rewrit eeq. ( 12) for this speci al case as

(13)

The first term on the right-hand side is equal to Allo(U' / c ) . This is just the Lorentztransform of the four-momentum P'v = (U' / c ,0, 0, 0) of the system at rest (cf. eqs .1- 2). It is the second term that is responsible for the fact that Pv does not transformas a four-vector.

Thi s term depends on the electromagnetic stresses T'U. Relativistic mechanics

46 MICHEL JANSSEN

thus seems to predict a rather peculiar effect. Stresses in a system 's rest frame giverise to energy and momentum in a frame in which the system is in motion . In general ,such momentum will not be in the direction of motion. This means that it will giverise to a turning couple." This is just the type of effect sought after in the Trouton­Noble experiment. The electromagnetic stresses in a charged capacitor, however, willbe exactly balanced by stresses in the material part of the capacitor that prevent theplates from collapsing onto one another under the influence of their mutual Coulombattraction. These material stresses will also give rise to energy and momentum in aframe in which the system is in motion. The energy and momentum coming from thematerial stresses will be equal and opposite to the energy and momentum comingfrom the electrom agnetic stresses. The material stresses will thus give rise to a turn­ing couple compensating the turning couple coming from the electromagneticstresses. This, essentially, is Laue's explanation of the negative result of the Trouton­Noble experiment, which has become the standard explanation of the experiment inspecial relativity (sec for example, Pauli 1921, 129).

Laue was able to show that

(14)

for the total energy-momentum tensor of any so-called "complete static system"(Vollstiindiges statisehes System, Laue 1911a, 539), where 'co mplete' means ' closed'(i.e., dvTfi V = 0 ).35 So, for a complete static system, the second term on the right­hand side of eq. (13) vanishes. Therefore, the total four-momentum of a completestatic system, unlike the four-momentum of its various constituents, transforms as afour-vector under Lorentz transformations." One of the examples Laue gives of acomplete static system is that of a charged capacitor and its electromagnetic field.

The upshot of these considerations based on Laue' s work of 1911 is that the sim­ple explanation of the Trouton experiment of section 5 based on E = me- works ifwe consider both the electromagnetic field of the capacitor and the stresses in thematerial part of the capacitor, but that it does not work if we only consider the electro­magnetic field, as Lorentz did in 1904. Nothing in classical mechanics could haveprepared Lorentz for Laue 's strange effect of stresses producing momentum. So, evenif Lorentz had immediately recognized the connection between the thought experi­ment used to derive E = me2 in Einstein's 1906 paper and the Trouton experiment,he would have been in no position to correct his 1904 analysis of the Trouton experi­ment before Laue's work of 1911.

From a modem point of view, Laue's analysis of the condition s under which Pfitransforms as a four-vector is rather clumsy. Most importantly, the restriction tostatic systems that Laue was forced to impose is completely unnecessary. The totalfour-momentum of any closed system, static or not, transform s as a four-vectorunder Lorentz transformations. This can easily be shown with the help of the obvi­ous generali zation from 3-dimensional space to 4-dimen sional space-time of thestandard theorem of Gauss that says that the integral of any vector field over a closedsurface is equal to the integral of the divergence of that vector field over the volume

THE TROUTON EXPERIMENT 47

enclosed by that surface. It will still be worth our while to take a somewhat closerlook at this derivation, for it will reveal that special relativity allows us a certain free­dom in defining the four-momentum of spatially extended systems. Taking advan­tage of this freedom, we can simplify the explanation of the Trouton and Trouton­Noble experiments considerably.

We begin by defining the manifestly Lorentz invariant quantity

r»=f TIlYn elI.L - L Y' (15)

where L is a space-like hyperplane in Minkowski space-time defined in some arbi­trary frame with coordinates x ll through nllxll = cr (nil is the normal to the hyper­plane, ct the hyperplane's distance to the origin of the xll-frame). For arbitrary butfixed hyperplanes L, P~ transforms as a four-vector under Lorentz transformations.It is an example of what Felix Klein called a "free vector" (Klein 1918, 398-399).37

The standard definition of the four-momentum of spatially extended systems isobtained from eq. (15) by stipulating that in every frame we choose L to be a hyper­plane of simultaneity in that frame. In that case, nil = (I, 0, 0, 0) and the right-handside of eq. (15) reduces to the right-hand side of eq. (11). With this convention, four­momentum will , in general , not transform as a four-vector. Let P't(x'U) be the four­momentum in the x'il -frame. So, L(x'll) is a hyperplane of simultaneity in the x'il ­

frame. The Lorentz transform Pt(x'U) = AllyP'~(x'U) in the x ll -frame will still be anintegral over L( x'Il), which is not a hyperplane of simultaneity in the x ll -frame, So,pt(x'U) will in general not give the four-momentum in the x ll -frame. It will only givethe four-momentum in the xll-frame if Pt(x'U) = Pt(xU), i.e., if the values of theintegrals do not depend on which hyperplane of simultaneity is being integrated over.The generalization of Gauss's theorem referred to above tells us that under fairly gen­eral conditions (e.g., that TIlY falls off sufficiently rapidly as we go to infinity):

(P~ independent of L) ¢:} (dyTIlY = °everywhere). (16)

So, under the standard definition, the four-momentum of a spatially extended systemtransforms as a four-vector under arbitrary Lorentz transformations if and only if thesystem is closed."

In 1922, Enrico Fermi proposed an alternative Lorentz invariant definition of thefour-momentum of spatially extended systems . His proposal was forgotten , then inde­pendently rediscovered several times, until Fritz Rohrlich (1960, 1965) made sure thatit would not be forgotten again. I will refer to this alternative definition as the Fermi­Rohrlich definition. It differs from the standard definition in the convention that isadopted for choosing the hyperplane L in eq. (15). The standard convention, as wejust saw, is to pick different families of hyperplanes in different frames of reference,viz. the hyperplanes of simultaneity in .whatever frame we happen to be using. Thisresults in a definition that is not Lorentz invariant. The Fermi-Rohrlich definition stip­ulates that we pick the same family of hyperplanes in all frames of reference, viz. thehyperplanes of simultaneity in the rest frame of the system under consideration.'?

48 MICHEL JANSSEN

Given this definition, all four-momentum transforms as a four-vector under Lorentztransformations. It does not matter whether we are dealing with elosed systems orwith their open subsystems.

The relativist is free to adopt either the standard definition or the Fenni-Rohrlichdefinition . After all, the only difference between the two is a convention about how toslice Minkowski space-time. A proponent of Lorentz's theory, however, is committedto a definition that mathematically is a special case of the standard definition . To con­vince the reader that the proponent of Lorentz's theory is indeed thus committed anddoes not enjoy the same freedom as his relativistic counterpart in this matter, I willonce again resort to describing the state of affairs in Lorentz's theory with the help ofsome relativistic terminology,

After 1905, as I mentioned at the end of section 4, Lorentz came to realize that asystem in motion through the ether will always appear to a co-moving observer as ifit were at rest in the ether (in relativistic terms, the co-moving observer will alwaysmeasure the quantities in the system's rest frame). Even after 1905, however, the truedescription of the system continues to be the description in terms of the quantities ofa coordinate system either at rest in the ether or related to such a coordinate systemby a Galilean transformation,

Consider the specific example of a charged capacitor moving through the ether.To a co-moving observer the capacitor will appear as if it is at rest in the ether." Theco-moving observer will thus measure the four-momentum ofthe capacitor's electro­magnetic field in the capacitor's rest frame. There is no need to specify under whichdefinition of four-momentum because in the capacitor's rest frame the standard defi­nition and the Fenni-Rohrlich definition coincide . It is also no problem, of course, tointroduce the notion of four-momentum into Lorentz's theory for the combination ofenergy and momentum.

Now consider the true description of the system, which, up to a Galilean transfor­mation, is a description in an ether frame. Hence, the true four-momentum ofthe elec­tromagnetic field of a moving capacitor will be an integral over a hyperplane ofsimultaneity in an ether frame. Lorentz's definition of the four-momentum of electro­magnetic fields or other spatially extended systems is thus obtained from eq. (15) bystipulating that p~ always be evaluated in an ether frame and that L be a hyperplaneof simultaneity in such a frame. From a purely mathematical point of view, this makesLorentz's definition a special case of the standard definition in special relativity."

The explanations of the Trouton and Trouton-Noble experiments are much sim­pler under the Fermi-Rohrlich definition of the four-momentum of spatially extend cdsystems than under the standard definition . In the case of the Trouton experiment, theFenni-Rohrlich definition allows us to use the simple explanation of section 5 (1-3)without ever having to worry about stresses in the material part of the capacitor. Thisis because under the Fenni-Rohrlich definition the four-momentum of the capacitor'selectromagnetic field taken by itself will transform as a four-vector under Lorentztransformations, So, for the energy 1:1£' transferred from the battery to the capacitor(see eq. I) we can simply substitute the energy U' of the electric field in the capaci ­tor's rest frame. In the case of the Trouton-Noble experiment, the simplificationbrought about by adopting the Fenni-Rohrlich definition is even greater. Instead of

THE TROUTON EXPERIMENT 49

the delicately balanced turning couples that we find under the standard definition,there will be no turning couples whatsoever under the Fenni-Rohrlich deftnition!? AsI explained above, the Fermi-Rohrlich definition is unacceptable to proponents ofLorentz's theory. In Lorentz's theory one is therefore stuck with the more cumber­some explanations of these experiments on the basis of the standard definition. I thinkthat this circumstance can be turned into a strong argument for preferring special rel­ativity over an empirically equivalent version of Lorentz's theory. But that is a storyfor another paper.

ACKNOWLEDGMENTS

I am greatly indebted to John Stachel for his perceptive comments following a talk Igave on the experiments of Trouton and Noble in a colloquium of the Max-Planck­Institut filr Bildungsforschung in Berlin in the Summer of 1994 (cf. notes 3 and 5). Ialso want to thank Jon Dorling, Tony Duncan, Gordon Fleming, David Hillman, A. J.Kox, John Norton, Jiirgen Renn, and again John Stachel for helpful discussions onmany other occasions.

NOTES

I. For a concise survey of 19th-century ether drift experiments and their theoretical ramifications, withselected references to further literature, sec Janssen and Stachel 1999.

2. Warwick docs not always distinguish carefully, as I think one should , between the effect sought afterin the Trouton experiment and the effect sought after in the Trouton-Noble experiment (sec for exam­ple, Warwick 1995, 318).

3. John Stachel first drew my attention to the connection between the Trouton experiment and this paperby Einstein .

4. This focus on developments at the level ofmechanics in my analysis of the history ofspecial relativitywas partly inspired by Damerow et al. 1992 and by several discussions over the years with JilrgenRenn.

5. John Stachel first suggested this interpretation of Larmor's rather cryptic comments on the Troutonexperiment to me.

6. Both Fermi and Rohrlieh proposed this alternative definition to give a Lorentz-invariant definition ofthe electromagnetic four-momentum of the classical electron model of Lorentz and Poincare . In thismodel , the electron is a physical system very similar to a charged capacitor.

7. One can avoid such arbitrary conventions altogether by accepting that the four-momentum of a spa­tially extended system is an example of a hyperplane-dependent quantity. This position, to which I amvery sympathetic, has been championed by Gordon Fleming (1998) .

8. Laue's proof was not entirely satisfactory even for this special case (see note 33 below) . A satisfactoryproof, without the unnecessary restriction to static systems , was first given by Felix Klein in the con­text of general rather than special relativity (Klein 1918). Klein 's proof was inspired by correspon­dence with Einstein in 1918 (sec Schulmann et al. 1998, in particular doc. 581, note 9). Klein coinedthe name "free vector" for such non-local quantities as the four-momentum of spatially extended sys­tems obtained through integration over some hypersurface in space-t ime.

9. Readers who think ofEinstein and Lorentz as proponents of competing paradigms, research programs ,or what have you, will probably feel uneasy about this identification of clements common to the ether­theoretic and the relativistic explanations of the Trouton experiment. This is not the place to argue thepoint (sec Janssen 1995, 1997), but I think the difference between the positions of Lorentz and Ein­stein is much more naturally understood as a difference of opinion over the interpretation of theLorentz invariance of a formalism they agreed upon. Although Lorentz formally retained remnants of

50 MICHEL JANSSEN

Newtonian mechanics in his theory, he accepted that physical systems are in fact governed by relativ­istic mechanics (see for example, Lorentz 1922, based on lectures of 1910-12).

10. There is an obvious improvement ofTrouton 's design. As Trouton explains in his paper: "It was origi­nally intended to have two condensers, one at each end of the cross arm, the one to be charged at themoment the other was discharged, not only to double the effect, but also to secure a pure torque actingon the wire . This idea had to be abandoned in the final experiment, owing to all the condensers avail­able breaking down under the excessive voltage employed save only one" (Trouton 1902, 559).

II . See Janssen 1995, section 1.3, for my reconstruction and analysis of Larmor's argument.12. For an insightful discussion of this issue , sec Darrigol 1995.13. In accordance with his infamous attribution of special relativity to Lorentz and Poincare, Sir Edmund

T. Whittaker has Poincare proclaim the inertia of energy in 1900: "In 1900 Poincare suggested thatelectromagnetic energy might possess mass density [...] that is to say E = me2 [ •..] and he remarkedthat if this were so , then a Hertz oscillator, which sends out electromagnetic energy preponderantly inone direction, should recoil as a gun does when it is fired" (Whittaker 195 I- 53, II, 5 I) . The passagequoted above from Poincare 's lecture four years later clearly refutes Whittaker's claim. I am gratefulto Tony Duncan for alerting me to Whittaker's claim.

14. Lorentz to Poincare, January 20,1901. The letter is quoted in full in Miller 1986, 6-7. Lorentz alsodiscussed these issues in his lectures at Columbia University in New York in 1906 (Lorentz 1915,30­33) .

15. Given the initial resistance in the 1920s to the notion of spin as a non-mechanical form of angularmomentum it should perhaps not surprise us that there was some resistance to the notion of electro­magnetic momentum as non-mechanical momentum two decades earlier.

16. Lorentz only sketched this derivation in his paper and referred to one of his contributions to the Eney­clopddie der mathematischen Wissenschaften (Lorentz 1904b) for further details . Using nothing butNewtonian mechanics, the Maxwell-Lorentz equations, and the expression for the Lorentz force ,Lorentz showed that the force F and the turning couple T that a charged system, that is static exceptfor an overall velocity v with respect to the ether, experiences from its self-field are given byF = - d G / d t and T = - v x G , respectively (where G is the self-field's electromagnetic momen­tum). These equations arc just what one would expect on the basis of the interpretation ofG as a formof momentum. They express conservation of momentum and angular momentum, respectively. (SecJanssen 1995, Sees , 1.2, 1.4.2, and 3.4 .2, for reconstructions of Lorentz's derivations in modern nota­tion.)

17. In his lectures at Columbia University two years later, Lorentz was still careful not to commit himselfto the interpretation of electromagnetic momentum as a new form of momentum (see Lorentz 1915,32) .

18. In this equation, U is "the energy of the charged condenser in the state of rest" (Lorentz 1904a, 830­83 I), e is the velocity oflight, and w is the velocity of the capacitor with respect to the ether.

19. Sec section 7, eq. (9), below.20 . For a more detailed account of the development of Lorentz 's theory, see (Janssen I 995-which also

contains a detailed critique of older accounts of this development; Janssen 1997 ; and Janssen andStachel 1999) .

21. Where A~ = [yY~ y: ~ ~] , with ~ == ~ and t ss_1_ .v 0010 e J1=I32

o 0 0 I

22. Notice that it does not matter what the angle between the plates of the capacitor and its velocity is.23 . The argument as it stands cannot be made exact, because it tacitly involves the assumption that the box

in the thought experiment can be treated as a rigid body, a notion incompatible with special relativity.In fact, the disturbance at the left end ofthc box upon emission of the radiation will not even havereached the right side of the box at the time of reabsorption of the radiation at the right end! The prob­lem can be circumvented by modifying the thought experiment. One option (suggested to me by JohnNorton) is to have the energy E transmitted from one end of the box to the othe r in small parcels over

TH E TROUTON EXPERIMENT 51

a long period of time. Another option is to only retain the walls at the two ends of the box without any­thing connecting them.

24. Thinking of the Trouton exper iment in terms of Einstein's thought experiment, we see that thereshould still be a minuscu le effect in the former due to the slight displacement of the center of mass ofthe battery-capacitor system that results from transferring energy from the battery to the capacitor.

25. In terms of distorting the history of special relativity, the omission of this section of Lorentz I904apales in compar ison, ofcourse, to the decision to include only the section on the Michelson-Morleyexperiment of Lorentz 1895 (ef. Miller 1981, 391-392).

26. I am grateful to John Stachel for bringing this paper to my attention. For discussion of the collabora­tion between Einstein and Laub, sec the editorial note, "Einstein and Laub on the Electrodynamics inMoving Media ," in Stachel et al. 1989, 503- 507.

27. Einstein contrasted theories of princip le with constructive theories . Roughly, the distinction is that aconstructi ve theory provides a detailed model of (features of) the physical world, whereas a theory ofprinciple only provides constraints on such modeling, constraints based on empi rically well confirmedregularities.

28. In his response to the papers brought together in this volume, Einstein characterized Von Laue's essayas "An historical investigation of the development of the conservation postulates, which, in my opin­ion, is oflasting value. I think it would be worth while to make this essay easily accessi ble to studentsby way of independent publication" (Einstein 1949,686).

29. I use SI or MKSA units. For conversion to other units, sec for example , Jackson 1975, 817- 819.30. See Janssen 1995, section 1.4, for a reconstruct ion in modem notation of Lorentz's derivat ion of this

result. That G will be in the direction of the plates of the capacitor to first order in vi c can be seenupon inspect ion of the integrand in eq. (8): the E-field will be perpendicular to the plates, while the S­field will be parallel to the plates and perpendicular to the velocity v.

31. Usc TUY", fo(TI«pFU«FPY +~TlUY TI «pTlP"FP"F«P). where

[

0 - e. r « - Eylc - E/C]uY= Exl c 0 - Bz By

F - ,E/c s, 0 - Bx

E/ c - By s, 0

TluY = TluY'" diag( I ,-1,- 1,- 1) , and c = I I M (with Ilo the permeability in vacuo).

32. For further discussion of this work by Laue, see Norton 1992, section 9.33. In general, this will not be true: a volume integral in the XU -frame and a volume integral in the x'u ­

frame arc integrals over different hyperplanes of simultaneity. For static systems, however, the resultsof the two integrals differ only by a factor y. It is unclear whether Laue, when he wrote his 19 11paper, realized that the assumption that the system is static is crucial at this j uncture. For further dis­cussion, sec Janssen 1995, section 2.1.4.

34. Cf. the equation T = - v x G in note 16.35. This result is sometimes called "Laue's theorem" (Miller 1981, 373; unfortunate ly, Miller uses the

phrase "perfectly static system" instead of "comp lete static system").36. It follows that there will never be a net turning couple acting on a complete static system.37. Cf. note 8.38. For a more detailed version of this proof, sec Rohrlich 1965, 89-90, 279-28 I.39. As Gordon Flem ing has pointed out (private communication), it is not clear exactly how to define the

rest frame of an arbitrary spatia lly extended system. For static systems , however, such as a chargedcapacitor and its electromagnetic field, this problem docs not arise.

40. This observer, for instance , will not notice the Lorentz-FitzGera ld contraction that the capacitor expe­riences as a result of its motion through the ether.

41. It follows from Lorentz's definition (cf. the discussion following eq. (15) above) that the four-momen ­tum of the electromagnetic field of a capacitor moving through the ether is not the Lorentz transform

52 M ICHEL JANSSEN

of the four-momentum of the electromagnetic field of the same capacitor at rest in the ether. TheLorent z transform of the four-moment um of the field of the capacitor at rest in the ether will be anintegra l over a hyperp lane of simultaneity in the moving system 's rest frame , where as the (true) four­momentum of the field of the capacitor in motion through the ether will be an integral over a hyper­plane of simultan eity in the ether frame. Since the electromagn etic field of a capacitor is not a closedsystem, these integrals over different hyperplanes will give different results.

42. See Butler 1968, Janssen 1995, Teukolsky 1996.

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