Part 1 — CHAPTER XII THE EQUATIONS OF ELECTROMAGNETICS AND LORENTZ’S THEORY

WHEN amber is rubbed, it develops the peculiar property of attracting small bodies, such as bits of paper, particles of dust and the like. This phenomenon, which was known to the Greeks, was described by saying that the amber had become electrified (electron meaning amber in Greek). To-day, however, we should say that it had received an electric charge. Soon it was found that electrified bodies did not always attract one another, but that in many instances they appeared to exert a repulsive action. For this reason it was assumed that there existed two different species of electricity, the positive and the negative; and the laws governing the phenomena involved were compressed into the statement that like charges repelled whereas opposite charges attracted. Similar conditions were found to endure between magnetic poles, so that the existence of two different types of magnetism was also assumed. But magnetism and electricity remained entirely distinct; no reciprocal action appeared to exist between an electric charge and a magnetic pole.

We may illustrate these phenomena of attraction and repulsion in a more concrete way by assuming that invisible fields of electric and magnetic forces surround electrified bodies and magnetic poles. To these fields we may ascribe the mechanical actions of attraction and repulsion detected by experiment; the electric fields act on electrified bodies, and the magnetic ones on magnetic poles. It was not until the early years of the nineteenth century that Coulomb submitted the mutual attractions and repulsions of charged bodies to a quantitative test. He found them to be expressed by a law (Coulomb’s law) which was of the inverse-square variety; the same as Newton’s law of gravitation, except that the electric charges took the place of masses, and that the actions could be either attractive or repulsive.

But with the invention of the electric cell by Volta a new manifestation of electricity presented itself for study and experiment, in the form of the electric current. Oersted discovered that an electric current was able to deflect a magnetized needle placed in its vicinity. This was of great significance, as showing that electricity and magnetism were closely allied phenomena. The nature of their relationship became better understood when, by placing iron filings round a wire, it was shown that an electric current was surrounded by a magnetic field. The deviation of the magnet in Oersted’s experiment was accordingly ascribed to the existence of the magnetic field surrounding the current. In the following year Ampère gave an exact quantitative formulation of the laws involved.

Now the fact that an electric current generated a magnetic field rendered it legitimate to suspect that conversely a varying magnetic field should generate an electric current. If this were the case, a magnet displaced near a closed wire should generate or induce an electric current in the wire. This most important phenomenon, known as electromagnetic induction, was discovered by Henry and Faraday, and as a result the connection between magnetism and electricity became still more pronounced. Our present-day dynamos and generators are nothing but machines constructed with a view to utilising this phenomenon of induction to generate electric currents on a commercial basis.

Faraday was the first scientist to realise the enormous importance of the electromagnetic field. He saw in it a reality of a new category differing from matter. It was capable of transmitting effects from place to place, and was not to be likened to a mere mathematical fiction such as the gravitational field was then assumed to be. In his opinion, the phenomena of electricity and magnetism should be approached via the field rather than via the charged bodies and currents. In other words, according to Faraday, when a current was flowing along a wire, the most important aspect of the phenomenon lay not in the current itself but in the fields of electric and magnetic force distributed throughout space in the current’s vicinity. It is this elevation of the field to a position of pre-eminence that is often called the pure physics of the field. Faraday was not a mathematician and was unable to co-ordinate the phenomena he foresaw in a mathematical way, and derive the full benefit from his ideas. Before dying, however, he entrusted this task to his colleague Maxwell; and one of the most astounding theories of science, eclipsed only in recent years by Einstein’s theory of relativity, was the outcome.

In order to appreciate the nature of Maxwell’s contributions, let us recall how matters stood in his day. If we confine-our attention to regions of space where no electric currents, no charged bodies or magnets are present, so that solely the electric and magnetic fields need be considered, the fundamental law of electromagnetism is Faraday’s law of induction. This law states that a variable magnetic field generates an electric field.

Maxwell, however, considered that this law, standing alone, lacked symmetry; so he formulated the hypothesis that conversely a variable electric field should generate a magnetic one, and proceeded to construct his theory with this idea in view.[40] In the particular case of free space in which only fields but no charges or currents are present, Maxwell’s equations of electromagnetics, termed field-equations (since they describe the state of the electromagnetic field), can be written: where represents electric field intensity, magnetic field intensity, denotes time, and a most important constant about which we shall have much to say later. As for “div” and “curl,” they represent mathematical operations of a peculiar sort. At any rate, the reader need not be alarmed at these equations, for it is not necessary that we understand them. Suffice it to say that (2) represents Faraday’s law of induction, while (4) constitutes the hypothetical law postulated by Maxwell in order to ensure symmetry. Prior to Maxwell’s investigations the fourth equation would have been written “”; hence it is the adjunction of the term which constitutes the originality of the theory.

From his field equations, Maxwell by purely mathematical means was able to deduce the two additional ones: where represents a particular species of mathematical operation. In ordinary language, these last two equations connote that varying electric and magnetic intensities will be propagated through the ether in wave form with a velocity , where is a magnitude yet to be determined. This discovery removed all possibility of action at a distance, since the field perturbations now appeared to be propagated from place to place with a finite velocity. It was, of course, of interest to determine the precise value of . The most obvious means would have been to create electromagnetic disturbances and then measure their speed of propagation directly. Physicists, however, were unable, in Maxwell’s day, to devise a means of performing such delicate experiments. But there existed another way of determining the value of . Maxwell remarked that it would be given by the ratio of the magnitude of any electric charge, measured successively in terms of electrostatic units (based on electricity) and then of electromagnetic units (based on magnetism).[41] Precise measurements conducted on electrified bodies and magnets then proved that the value of this ratio was about 186,000 miles per second; whence it became necessary to assume that periodic perturbations in the strains and stresses of the field would be propagated in the form of waves moving through the ether with this particular speed. But this velocity was precisely that of light waves propagated through the luminiferous ether.

The conclusion was obvious. Unless we were to assume that this extraordinary coincidence in the values of these two characteristic velocities, that of electromagnetic induction and that of light waves, was due to blind chance, there was no other alternative but to recognise that what we commonly called a ray of light was nothing else than a series of oscillations in the electromagnetic field, propagated from point to point. Electromagnetic waves and luminous waves were thus all one. Henceforth the two ethers, the electrical ether and the luminiferous ether, were seen to be the same continuum; electricity, magnetism and optics were merged into one single science.

Now, although experimenters had attempted by various means to submit Maxwell’s views to a test, the technical difficulties were so great that no success had been achieved. It appeared clearly from Maxwell’s equations that no appreciable effects could be anticipated unless was very great; and this meant that the electric intensity would have to vary with extreme rapidity. The simplest means of obtaining a result of this kind would be to produce an oscillating field of electric intensity in which the oscillations were extraordinarily rapid, several billions a second. But no mechanical contrivance could yield such results, and apart from mechanical contrivances no other methods suggested themselves. For these reasons a number of years passed before Maxwell’s views could be put to the test.

In 1885 Helmholtz directed the attention of his pupil, Hertz, to the problem. Hertz was one of the most remarkable experimenters of the nineteenth century; he succeeded at last in vanquishing the technical difficulties and in generating by purely electrical means an oscillating electric field of extremely high frequency. Electromagnetic waves of sufficient intensity were thus produced; and after having been sidetracked for a time by a secondary phenomenon whose nature was elucidated by Poincaré, Hertz verified the fact that the waves advanced with the speed of light and indeed possessed all the essential properties of light waves other than those of visibility to the human eye. Thus, as a result of Hertz’s experiments, the foundations were laid for the commercial use of wireless and radio; but, more important still, Maxwell’s electromagnetic theory of light establishing the intimate connection between electricity and optics had at last been vindicated.

Just as Hertz had proved that electromagnetic waves were of the same nature as light waves, so now, conversely, Lebedew succeeded in showing that light waves were of the same nature as electromagnetic ones. It is, indeed, a necessary consequence of Maxwell’s theory that electromagnetic waves should exert a definite pressure on bodies upon which they impinge, and this pressure was verified and measured by Lebedew and again by Nichols and Hull in the case of light waves falling on matter.[42]

Yet it must be realised that marvellous as were Maxwell’s equations, so far as the field phenomena in empty space were concerned, they led to conflict with experiment in the presence of matter. The conflict was notably apparent when the phenomenon of dispersion was considered; and by dispersion we mean the splitting up of white light by a prism into a rainbow of colors. Maxwell’s equations proved that on entering a transparent dielectric such as glass, the electromagnetic vibrations which constitute light would be slowed down; and this slowing down, as had long been known, would account for refraction. But dispersion demands that the angle of refraction, hence the slowing down, should depend on the frequency, hence on the colour of the light vibrations; and this is what Maxwell’s theory failed to account for. His equations showed that the slowing down would be the same for all frequencies. A similar difficulty was encountered when the phenomenon of the magnetic rotation of the plane of polarisation was considered. Faraday had discovered the existence of this phenomenon, and any theory of electromagnetics would have to account for it. Nevertheless, it will be seen that all these difficulties which beset Maxwell’s theory arose solely when experiments involving matter were taken into consideration. It appeared, therefore, that the fault resided not in the field equations themselves, but in the too crude assumptions that had been made with respect to the constitution of matter.

Furthermore, Maxwell had considered more especially the case of matter at rest in the ether, and it was necessary to investigate the general case of matter in motion. This extension of Maxwell’s theory, known as the electrodynamics of moving bodies, was attempted by Hertz, who assumed that matter in motion through the ether would drag the ether along with it in its interior—a hypothesis not so radical, but somewhat in line with that of Stokes. So long as conductors were contemplated, Hertz’s hypothesis seemed to be in accord with experiment; but when we came to consider dielectrics in motion, Hertz’s anticipations were contradicted. Fizeau’s experiment in particular, proving that the velocity of running water did not compound with the velocity of the light in stagnant water, was incompatible with Hertz’s hypothesis of a total drag. Also, in an experiment of a totally different nature, dealing with the motion of a dielectric through a magnetic field, Wilson proved the existence of the same partial drag, given once again by Fresnel’s convection coefficient. It appeared, therefore, that Fresnel’s convection coefficient was a fact which would have to be fitted into any theoretical scheme of the electrodynamics of moving dielectrics.

Now, Fresnel’s partial drag hypothesis, though seemingly confirmed by experiment, had yet to be accounted for theoretically, and this appeared impossible so long as we remained in such utter ignorance of the ultimate constitution of dielectrics and of matter in general. At this stage Lorentz started his theoretical investigations which were to lead him to a refinement of Maxwell’s equations, extending their application to cases where matter was present, either at rest or in motion through the ether. As we have mentioned, Maxwell’s equations had proved themselves incapable of accounting for dispersion. It appeared necessary to conceive of some structure for dielectrics which would act selectively, imposing different degrees of retardation on light waves of different frequencies. Lorentz achieved this result by assuming that electricity was atomic and that matter was constituted by more or less complicated groupings of these electric atoms or electrons.

In the case of dielectrics or non-conductors such as glass, the electrons were assumed to be tied down to their atoms by elastic forces. When disturbed they would vibrate with some characteristic frequency depending on the magnitude of the elastic forces which held them in place. If, then, a wave of light were transmitted through the transparent dielectric, the electrons would be set into forced vibration and the retarding effect these vibrations would impose on the incident light would depend on the relative frequencies of the incident light and of the natural frequencies of the electrons. In this way it was possible to obtain a mathematical interpretation of dispersion.

Further phenomena were accounted for by taking into consideration the frictional resistances that would interfere with rapid vibrations of the electrons. When these frictional resistances were weak, oscillatory disturbances, such as rays of light, could be propagated through the dielectric, which was then termed transparent (glass). When these frictional forces were considerable, the light ray was unable to set the electrons into vibration; its energy was consumed in the attempt, and as a result it could not proceed; the dielectric was then opaque (ebonite, sulphur).

In the case of conductors such as metals, the electrons were assumed to be very loosely held to their atoms so that the slightest difference of potential would tear them away and cause them to rush in the same direction, thereby producing an electric current. It was precisely because electrons in conductors were not tied down to fixed positions by elastic forces that they were incapable of vibrating; and so conductors were necessarily opaque to electromagnetic vibrations or to light. Conversely, it was because the electrons were all tied down to fixed positions in dielectrics, that they could not rush along in one direction. As a result dielectrics were opaque to currents, and hence were non-conductors. According to these views of Lorentz, an electric current passing through matter was nothing but a rush of electrons. If this were the case, we should expect the motion of a charged body to generate the magnetic field of a current. Rowland had established the existence of this effect many years previously, so that Lorentz’s theory conformed to facts in this respect. This same effect is illustrated in a more vivid way by the magnetic field which surrounds cathode rays. These are constituted by streams of electrons travelling with enormous velocities through a partial vacuum; and for this reason the cathode rays have much in common with a current passing through a metal wire.

In perfectly empty space no electrons were assumed to be present; and the propagation of electromagnetic disturbances through space was credited entirely to the oscillations of the field which stood out as a manifestation of energy differing radically from the substantial electrons.

In contradistinction to Hertz, Lorentz assumed that the ether was never disturbed by the motion of matter; and just as in Maxwell’s theory, a luminous source in motion through the ether would never communicate its velocity to the light waves it emitted. With the constitution of matter postulated in this way, Lorentz set out to extend Maxwell’s theory of electromagnetics in the free ether to cases where matter was present, either at rest or in motion. As already mentioned, he succeeded in giving a theory of dispersion, one of the phenomena which Maxwell had been unable to explain. Furthermore, such highly complicated phenomena as abnormal dispersion, absorption, metallic reflection, selective reflection, body colour and many others were accounted for by the theory with great precision.

The apparent partial drag of the ether verified in Fizeau’s experiment and also in Wilson’s (Fresnel’s convection coefficient) was proved to be a necessary consequence of the theory. It had nothing to do with a partial dragging of the ether, since the ether was in no wise disturbed by the motion of the matter; the apparent drag was accounted for solely in terms of the electronic constitution of the moving matter. In this way Fresnel’s convection coefficient received the rational explanation which it had thus far lacked. The most refined experiments confirmed Lorentz’s theory in a number of minute details; and when the electron was finally isolated and studied, all doubts appeared to be removed, at least as to the general correctness of the doctrine.

Theories, however, are especially convincing, not so much when they account for what has been observed but has not been explained, as when they allow us to foretell phenomena which no one has anticipated. Triumphs of this sort were soon forthcoming in the case of Lorentz’s theory. For instance, the theory required that light waves be produced by electrons vibrating in the interior of the atom. Inasmuch as the presence of a magnetic or an electric field affects the motion of electrified bodies, the vibrations of the electrons in the luminous atom should be modified by the presence of a strong magnetic field. Calculation then showed Lorentz that a magnetic field would cause an atom normally emitting a monochromatic light to emit two or three separate lights, according to the relative inclination of the magnetic field to the line of sight of the observer. This totally unexpected effect was soon verified by Zeeman, one of Lorentz’s colleagues, and is now known as the Zeeman effect. It has since been observed in the light emitted by sunspots, proving that very intense magnetic fields must there be present.

It is quite unnecessary to dwell on a number of further verifications of Lorentz’s theory. All that we have wished to show is that the empirical discoveries that lend weight to Lorentz’s electronic theory are so numerous that it can certainly not be cast aside lightly. Although no achievement in theoretical physics can ever claim to be permanent, yet in view of the wonderful accuracy of Lorentz’s previsions it would require some reason of a truly imperative nature for us to feel justified in tampering with it in any essential detail.

Lorentz’s electronic theory is also perfectly consistent with the negative experiments we have heretofore mentioned, since it explains the existence of the Fresnel convection coefficient; and we have seen that this coefficient in turn accounts for the negative results of experiments conducted to the first order of approximation in dielectric media moving through the ether. But Lorentz’s theory as it now stands suggests that experiments conducted to a higher order of precision, to the second order, for example, would cease to yield negative results; so that with experiments of this sort our motion through the ether would be detectible.[43]

Now, the Michelson and Morley experiment is precisely of this type. It is a second-order experiment. It is assumed that the reader is sufficiently familiar with it to render its detailed description unnecessary. Stated briefly, the experiment proves that if waves of light leave the centre of a sphere simultaneously, they will return in perfect unison to the centre of the sphere after having been reflected against the sphere’s inner surface, no matter in what direction through the ether, or with what constant velocity, the sphere attached to the earth may be moving. An explanation was suggested by both FitzGerald and Lorentz, and was to the effect that the sphere had contracted in a definite degree in the direction of its motion through the ether. The sphere accordingly became an ellipsoid, and Michelson’s negative result was explained.

More generally, any body, a yardstick, for example, when lying in the direction of the earth’s motion through the ether, would be contracted and become shorter than when placed perpendicularly to the ether hurricane. However, we should never have any means of detecting this contraction, for all things, the human body included, would participate with it, whence observationally, at least, nothing would be changed.

This Lorentz-FitzGerald contraction hypothesis was a hypothesis formulated ad hoc for the sole purpose of explaining the null result of Michelson’s experiment; and like all such hypotheses, it could be accepted only provided it was justified by a wide variety of experiments. At the time this hypothesis was suggested, scientists were busying themselves, with the study of the electron when in motion through the ether. Abraham assumed that an electron moving at high speed through the ether would remain spherical and rigid; and he worked out mathematically the path a rigid electron would pursue when acted upon by electric and magnetic forces. Lorentz, holding to his view of the FitzGerald contraction, attacked the same problem, assuming the electron to contract more and more as its speed through the ether increased. Experimenters were called upon to decide between the hypotheses of the rigid and the contractile electron. Accordingly, highly refined experiments were conducted on the electrons emitted by radium ( rays). Kaufmann, by photographing the paths followed by the electrons, came to the conclusion that Abraham’s rigid electron corresponded to reality. But, still more precise experiments conducted by Bucherer reversed the verdict, and little doubt remained that the electron suffered the FitzGerald contraction in the precise degree demanded by Lorentz.

Inasmuch as matter was assumed to be electronically constituted, these experiments of Bucherer lent weight to the FitzGerald contraction of matter, hence to the Lorentz-FitzGerald explanation of Michelson’s null result. This contraction being seemingly established, attempts were made by experimenters to detect it in bodies which were attached to the earth. Could this result be achieved, we should have a means of ascertaining the earth’s velocity through the ether; and it was confidently expected that this time success would attend us.

Now, as we have mentioned, if in our frame of reference, on the earth’s surface, for instance, all things, including our measuring rods and our own bodies, contracted in exactly the same way owing to the motion of our planet through the ether, it would, of course, be idle for us to expect this contraction to be observable by direct measurement. But although not observable in a direct way, it was argued that this contraction of a body should modify its optical and electrical properties so that a study of the variations in these properties when the body was placed successively in different directions should yield us the earth’s motion through the ether. In particular the contraction of a solid transparent body, owing to its motion through the ether, should give rise to a phenomenon of double refraction, much as would follow from an ordinary physical compression of the body. The experiment of Rayleigh and of Brace was based on this idea, but the result was a complete disappointment. The FitzGerald contraction gave rise to no such effect, and a negative result was registered. The experiment of Trouton and Rankine aimed to detect a variation in the electrical conductivity of a strip of metal as a result of the FitzGerald contraction. Once more a negative result was the outcome. These experiments having failed, still another was suggested and carried out by Trouton and Noble. The essence of this experiment was based on the well-known phenomenon verified by Rowland, that an electrified body in motion produces all the effects of an electric current. If, therefore, two electrified bodies were placed side by side, the motion of the earth through the ether should produce two parallel electric currents; and since parallel electric currents attract one another, a very sensitive torsion balance should be able to detect this attraction. Again, the result of this experiment was negative; the velocity of the earth through the ether evaded us completely.

Now, it should be noticed that the last two experiments pertain purely to electricity and electrodynamics; there is nothing of an optical nature about them. Hence, it would appear that all electromagnetic phenomena behave in exactly the same way, regardless of the velocity of the ether hurricane. In other words, they, too, yield negative results. But there is yet another aspect to these experiments.

If we regard the FitzGerald contraction as physically real—and we must do so in order to explain Michelson’s negative result—we must recognise that it is certainly compensated in a miraculous way; since no experiments, whether optical or electrical, appear to be capable of revealing it. Furthermore, in the last experiment, that of Trouton and Noble, we are not dealing primarily with the FitzGerald contraction; we are here in the presence of an experiment of electrodynamics, and we are endeavouring to detect a purely mechanical force arising as a result of the motion of our electrified bodies through the ether. The laws of electrodynamics require that such a force be generated, and yet no force can be observed; so we must assume that the mysterious compensations that conceal our motion through the ether are of so perfectly general a nature that they affect mechanics as well as electromagnetics and optics.

True, none of the experiments we had been able to execute exceeded the second order in precision; and there was nothing in Lorentz’s theory, even when the FitzGerald contraction was taken into consideration, to suggest that all experiments, whatever their nature and however precise they might be, would fail to detect our velocity through the ether. A very simple hypothetical experiment will make this point clear. Suppose, for instance, that the earth were rushing through the ether with a speed approximating that of light; and suppose that we suddenly stepped in front of a mirror. All that the FitzGerald contraction hypothesis would ensure would be that the time required by the light rays to leave our body and be reflected back to our eyes should be independent of the orientation of the mirror with respect to the earth’s motion. But the actual duration which the light rays would require, hence the time it would take for our image to appear to us, would depend essentially on the earth’s speed through the ether; so that theoretically, at least, a measure of duration should reveal this mysterious speed.

However, in view of the fact that the highly precise experiments that had been performed were incapable of detecting this velocity through the ether, it appeared legitimate to extrapolate, to assume that something deeper was at stake, and that in all probability this velocity would remain undetected, however far we might extend the precision of our electrodynamic or optical experiments, and however much we might vary their nature. Lorentz accepted what appeared to him as the inevitable, and asserted that the time had come to recognise that nature seemed to have entered into some giant conspiracy to defraud us of a knowledge of our velocity through the ether. Accordingly, he laid down his celebrated principle of correlation, according to which adjustments were so regulated in nature that the velocity of our planet through the ether could never be detected, however precise our experiments.

Lorentz applied his mathematical talents to the discovery of the necessary adjustments which would have to exist in nature for this correlation to be satisfied completely so far as electrodynamics and optics were concerned. In order to understand the significance of his investigations, we must recall what we have said of the laws of nature and of the space and time transformations.

We remember that Newtonian relativity expressed the fact that mechanical phenomena, hence the laws of mechanics regulating these mechanical phenomena, remained unchanged when we passed from one Galilean frame to another. The significance of this statement is as follows:

To begin with, when we wish to fix the position of a body in our frame, it is also necessary to specify the instant of time at which the body occupies this position. We thus have four co-ordinates to take into consideration: three for space, , , ; and one for time, . If, now, we change our frame of reference, taking our stand in some other Galilean system, which is in motion with respect to the first, these co-ordinates will change in value or will be transformed. For instance, if a body remains at rest at the meeting point of the three axes of our first frame, its co-ordinates will always be , , in this frame for all values of the time . But computed with respect to our second frame, the new co-ordinates represented by , , will vary with time, since with respect to this second frame the body is moving—i.e., is adopting successive positions in time.

The classical Galilean transformations had for their object to tell us exactly how these co-ordinates would vary when we passed from one Galilean frame to another. As it was always assumed by classical science that time was absolute, the same for all systems, the only co-ordinates susceptible of variation as we changed frames were the space co-ordinates , , . As for , it had the same value in all frames, and this was expressed by writing . Now the laws or equations which govern phenomena contain in their expression, among other quantities, these co-ordinates, , , , . Hence, in the general case it is only to be expected that the laws will change in form when we refer them first to one Galilean frame, then to another; or again, when we replace , , , by , , , . When we say that the laws suffer no change in form or are transformed into themselves we mean that this substitution of the new co-ordinates for the old does not modify the form of the equations or laws.

We can now understand what mathematical condition will have to be satisfied for Lorentz’s principle of correlation to hold. The principle asserts that all things are so adjusted in nature that absolute velocity through the ether must ever elude us. In other words, all electrodynamic experiments will yield negative results. But this obviously makes it necessary that the laws of electrodynamics which govern all electrodynamic phenomena, hence all electrodynamic and all optical experiments, should remain unchanged in form when we pass from one Galilean system to another. In mathematical language, the laws of electrodynamics will remain invariant when we submit them to the space and time transformations.

Now we have said that when submitted to the classical space and time transformations, the invariance of the equations of electrodynamics does, not hold. It appeared then that, contrary to the negative results of experiment and contrary to Lorentz’s principle of correlation, velocity through the ether should be detectible by electromagnetic experiments. Science was thus placed in an exceedingly difficult position; the equations of electrodynamics had been so strongly confirmed by extremely accurate experiment that we could scarcely question their accuracy. There remained, but one alternative, namely, to suspect the accuracy of the classical transformations.

This was indeed the course followed by Lorentz. He succeeded in establishing the transformations which would be in harmony with the invariance of the electrodynamic equations, and he found these to differ perceptibly from the classical ones; although reducing to the latter in the case of low velocities. The new transformations constitute the celebrated Lorentz transformations.[44]

Translated into ordinary language, these transformations expressed the existence of two separate phenomena: first, the FitzGerald contraction of bodies moving through the ether, a phenomenon with which we are already acquainted; and, secondly, a new phenomenon consisting in the slowing down (with increase of velocity through the ether) of the rate of time-flow as applying to electromagnetic processes. Thus, contrary to the classical transformations, where time was absolute (), we see that in the Lorentz transformations this absoluteness of time begins to be questioned. For psychological reasons, however, imbued as he was with the spirit of classical science, Lorentz was unable to realise the importance of his discovery; and he never, succeeded in ridding himself of his belief in the absoluteness of time. For him, as also for Larmor, who contributed to these discoveries, this new species of variable duration, depending as it did on the motion through the ether of the Galilean frame, was not real time. It was a species of “local time”—a distortion of real time—and it was not assumed that this local time corresponded in any way to the time which the observer in the moving frame would live and sense.

And so Lorentz assumed that these new transformations applied only to purely electromagnetic quantities, and no reference was made to their being applicable to mechanical phenomena as well. Though, as a result of these transformations, the velocity of light proved to have always the same invariant value through all Galilean frames when measured by the observer in the frame, no suspicion was cast on the classical formula for the composition of the velocities of material bodies; and this in spite of the fact that these two circumstances were mutually incompatible.

It appeared then as if Lorentz still considered the classical transformation-formulæ to be the real ones, the new transformations being merely of limited application and referring solely to the electromagnetic quantities. When, therefore, in such experiments as that of Trouton and Noble, mechanical and purely electromagnetic effects were indissolubly connected, a large measure of obscurity was involved. Lorentz, however, succeeded in extricating himself from his difficulties and in accounting for all the negative experiments; but he was compelled to appeal to additional hypotheses, such as a modification in those elastic forces which cause the electrons to vibrate in transparent bodies (Rayleigh and Brace experiments). In a similar way it was possible to explain the negative result of the Trouton and Noble experiment. Nevertheless, we can understand the numerous difficulties into which Lorentz’s theory was leading us. We knew very little about the constitution of matter, and here, in Lorentz’s theory, we were compelled to account for negative experiments by taking this unknown constitution of matter into consideration.

Taking Lorentz’s theory as it stands, one cannot help but recognise that this accumulation of hypotheses postulated ad hoc makes it painfully artificial. According to the theory, the ether must be regarded as stagnant and the earth as moving through it with some definite velocity. This velocity must, however, remain eternally unknowable to us. Its elusiveness arises from the slowing down of time and the FitzGerald contraction, in the case of the simplest electromagnetic experiments; in the case of experiments dealing with dielectrics and conductors, we have to appeal to still more complicated adjustments pertaining to the constitution of matter. But although forever indiscernible, our velocity through the ether was a reality; it was there, and had it not been for all these compensating effects, it would have been observed. All the forces of the cosmos appeared to have conspired to deprive us of a knowledge we were so anxious to obtain, namely, the velocity of our planet through the ether.

The existence of the FitzGerald contraction, holding in exactly the same degree for all bodies, was an instance of this marvellous conspiracy of nature. Even if we admitted that all matter were electronically constituted, and that the constituent electrons became flattened as a result of their motion through the ether, it was still impossible to conceive of an identical flattening for all bodies, whether rigid or soft, unless we assumed that an appropriate adjustment had also taken place in the other factors entering into the constitution of matter (elastic forces, etc.). The whole theory was unsatisfactory in the extreme. If Nature was blind, by what marvellous coincidence had all things been so adjusted as to conceal a velocity through the ether? And if Nature was wise, she had surely other subjects to attend to, more worthy of her consideration, and would scarcely be interested in hampering our feeble attempts to philosophise. In Lorentz’s theory, Nature, when we read into her system all these extraordinary adjustments ad hoc, is made to appear mischievous; it was exceedingly difficult to reconcile one’s self to finding such human traits in the universal plan.

Possibly more space than was needed has been devoted to a discussion of these difficulties of pre-Einsteinian physics, but the purpose of these lengthy explanations has been to show that science was confronted with an exceedingly difficult situation. Before proceeding to a study of Einstein’s solution, we must mention a tentative hypothesis put forward by the Swiss physicist Ritz.

Ritz remarked that Michelson’s experiment could be explained by assuming that, contrary to the requirements of the electromagnetic theory of light, waves of light were shot out from the luminous source just as bullets are shot out from a gun. With this species of ballistic theory, the speed of the light ray with respect to the observer would be given by the constant speed of the ray with respect to its source, compounded with the relative speed of the observer and source. Now, there is not the slightest doubt that Ritz’s hypothesis affords an explanation of Michelson’s negative result, but, on the other hand, it leads us into serious difficulties when we wish to interpret a number of other experiments, Bucherer’s, for example. This can be understood when we realize that the Ritz hypothesis compels us to deny the existence of the FitzGerald contraction, since were the contraction to subsist in addition to Ritz’s assumption, Michelson’s experiment would no longer yield a negative result. But, as we have said, the FitzGerald contraction permits a very simple explanation of the moving electron’s increase in mass in Bucherer’s experiment. This difficulty would not be insuperable, but still it would constitute an argument adverse to the acceptance of Ritz’s idea. Other experiments which it would be extremely difficult to account for are those connected with the Fresnel convection coefficient, e.g., Fizeau’s experiment.

Furthermore, with Ritz’s conception of the velocity of a light wave through the ether being compounded with the velocity of the source, the frequency of light issuing from a source, moving towards us through the ether, would be increased, whereas the wave length would remain unchanged. Now, it so happens that the angle of refraction of a monochromatic light ray passing through a prism depends on the frequency of the incident light, whereas the dispersion caused by a grating depends on the wave length. It should, therefore, be possible to determine whether frequency and wave length truly behaved like independent variables as demanded by Ritz’s hypothesis. Experiments conducted by Stark in 1910 on canal rays disproved Ritz’s anticipations. Subsequently other experiments led to the same results.

Besides these objections to Ritz’s views, it must be realised that Michelson’s experiment was only one of many. Here it cannot be emphasised too strongly that the gravity of the situation arose from the fact that as a result of a large number of different experiments (many of which we have not mentioned), the entire structure of classical electrodynamics seemed to be crumbling on every side. Ritz’s hypothesis consists in shoving one individual brick back into place. Lorentz, thanks to his mathematical knowledge, realised that something much deeper was at the root of all the trouble and that more radical means would have to be adopted. Accordingly, he proceeded to tackle the very foundations of science, namely, the space and time transformations which had endured for centuries. But Lorentz only proceeded in a half-hearted way and did not have the courage to push his discoveries to their logical conclusion, contenting himself with patching up rather than reconstructing. Einstein, as we shall see, took the bull by the horns, abandoned the classical structure, and proceeded to build up an entirely new edifice, superbly coherent and free from all artificial support and scaffoldings. Thus it was that even before Ritz’s hypothesis was disproved by numerous direct investigations (experiments of Majorana and of Sagnac, and astronomical observations on double stars by de Sitter), scientists had ceased to give it serious consideration.