Part 60

Now an ether, if it is to explain anything at all, must have at least some of the simpler properties of material media. The most fundamental of these, perhaps, is position in space. As a first approximation in explaining optical phenomena on the earth’s surface, the earth might be supposed to be at rest relative to the ether. But the establishment of the Copernican system made the sun the center of the solar system and gave the earth an orbital speed of eighteen miles a second. It may be remarked parenthetically that the speed of a point on the equator due to the earth’s diurnal rotation is quite insignificant compared to its orbital velocity. Hence as a second approximation the sun might be considered at rest relative to the ether and the earth as moving through this unresisting medium.

The first indication of this motion lay in the discovery of aberration by the British astronomer Bradley in 1728. Bradley noticed that stars near the pole of the ecliptic describe small circles during the course of a year, while those in the plane of the ecliptic vibrate back and forth in straight lines, stars in intermediate positions describing ellipses. The surprising thing, however, was that the time taken to complete one of these small orbits is in all cases exactly a year. Bradley concluded that the phenomenon is in some way dependent on the earth’s motion around the sun, and he was not long in reaching the correct explanation. For suppose the earth to be at rest. Then in observing a star at the pole of the ecliptic it would be necessary to keep the axis of the telescope exactly at right angles to the plane of the earth’s orbit. However, as the earth is in motion, the telescope must be pointed a little forward, just as in walking rapidly through the rain an umbrella must be inclined forward so as to intercept the raindrops which would otherwise fall on the spot to be occupied at the end of the next step. The angle through which the telescope has to be tilted is known as the angle of aberration, and the tangent of this angle may easily be shown to be equal to the ratio of the velocity of the earth to the velocity of light. Knowing the velocity of the earth, the velocity of light can then be calculated. This method was one of the first of obtaining the value of this important quantity.

More recently, terrestrial methods of great precision have been devised for measuring the velocity of light. The most accurate of these is that employed by the French physicist Foucault in 1862. A ray of light is reflected by a rotating mirror to a fixed mirror placed at some distance, which in turn reflects the ray back to the moving mirror. The latter, however, has turned through a small angle during the time elapsed since the first reflection, and consequently the direction of the ray on returning to the source is not quite opposite to that in which it had started out. This deviation in direction is determined from the displacement of the image formed by the returning light, and from it the velocity of light is calculated. In order to make the deflection appreciable the distance between the two mirrors should be very great. As originally arranged by Foucault, it was found impractical to make this distance greater than twenty meters, and consequently the displacement of the image was less than a millimeter. Such a small deflection limited the accuracy of the experiment to one percent. In 1879, however, Michelson (=18=, 390, 1879), then a master in the United States Navy, improved Foucault’s optical arrangements to such an extent that he was able to use a distance of nearly seven hundred meters between the two mirrors. With a rate of two hundred and fifty-seven revolutions a second for the rotating mirror, the displacement obtained was over thirteen centimeters. This experiment gave 299,910 kilometers a second for the velocity of light, with a probable error of one part in ten thousand. Later investigations by Newcomb and Michelson (=31=, 62, 1886) gave substantially the same result. So great has been the accuracy of these terrestrial determinations that recent practice has been to calculate from them and the angle of aberration the earth’s orbital velocity, and hence the distance of the earth from the sun. This indirect method of measuring the astronomical unit has a probable error no greater than the best parallax methods of the astronomer. (J. Lovering, =36=, 161, 1863.)

Aberration is a first order effect, i. e., it depends upon the first power of the ratio of the velocity of the earth to the velocity of light, and at first sight it seemed to prove conclusively that the earth must be in motion relative to the luminiferous medium. Other questions had to be settled, however, and one of these was whether or not light coming from a star would be refracted differently when passing through optical instruments from light which had a terrestial origin. Arago subjected the matter to experiment, and concluded that in every respect the light from a star behaved as if the earth were at rest and the star actually occupied the position which it appears to occupy on account of aberration. Finally optical experiments with terrestrial sources seemed to be in no way affected by the motion of the earth through the ether.

In order to account for these facts Fresnel advanced the following theory. To explain the refraction that takes place when light enters a transparent body, it is necessary to assume that light waves travel more slowly through matter than in free ether. Now the velocity of sound is known to vary inversely with the square root of the density of the material medium through which it passes. Hence it is natural to assume that ether is condensed inside material objects to such an extent that this same relation connects its density with the velocity of light traveling through it. But when a lens or prism is set in motion, Fresnel supposed it to carry along only the excess ether which it contains, ether of the normal density remaining behind. This assumption suffices to explain Arago’s results, and yet fits in with the phenomenon of aberration. It gives for light traveling in the direction of motion through a moving material medium of index of refraction n an absolute velocity greater than that when the medium is at rest by an amount

(1 − (1/n^2))v,

which is only a fraction of the velocity v which would have to be added if convected matter carried along all the ether which resides within it. This expression was tested directly, first by Fizeau in 1851, and later by Michelson and Morley (=31=, 377, 1886) in this country. The experiment consists in bifurcating a beam of light, passing one-half in one direction and the other in the opposite direction through a stream of running water. On reuniting the two rays the usual interference fringes are produced. Reversing the direction of motion of the water causes the fringes to shift, and from the amount of this shift the velocity imparted to the light by the motion of the stream is computed. The divergence between the experimental value of this quantity and that calculated from Fresnel’s coefficient of entrainment was found by Michelson and Morley to be less than one percent, which was about their experimental error. Thus Fresnel’s expression for the velocity of light in a moving medium is entirely confirmed by experiment. The derivation of it accepted to-day, however, is very different from his original deduction.

It has been noted that the phenomena of polarization led Newton to reject the wave theory of light. The only type of wave known to him was the longitudinal wave, in which the vibrations of the particles of the medium are in the same direction as that of propagation of the wave, and it was impossible to suppose that such a wave could have different properties in different directions at right angles to the line in which it is advancing. But in 1817 Young suggested that this inconsistency between the wave theory and the facts of polarization could be removed by supposing the vibrations constituting light to be executed at right angles to the direction of propagation. Thus in ordinary light the vibrations are to be conceived as taking place haphazard in all directions in the plane perpendicular to the ray, while in plane polarized light these vibrations are confined to a single direction. This supposition explained so many of the puzzling results of experiment, that it was accepted at once and led to the complete vindication of the undulatory theory.

Elastic Solid Theory.—Shortly afterwards Poisson succeeded in solving the differential equation which determines the motion of a wave through an elastic medium. His solution shows that such a medium is capable of transmitting two types of wave—one longitudinal, the other transverse. If κ denotes the volume elasticity, η the rigidity and ρ the density of the medium, the velocities of the two waves are respectively

√((κ + (⁴⁄₃)η)/ρ) and √(η/ρ)

Now a solid has both compressibility and rigidity, and transmits in general both types of wave. A fluid, on the other hand, on account of its lack of rigidity, cannot support a transverse vibration. Hence it was natural that Green, in searching for a dynamical explanation of the ether, should have proposed in a paper read before the Cambridge Philosophical Society in 1837 that the ether has the elastic properties of a solid. One great difficulty presented itself; disturbances inside an elastic solid must give rise to compressional as well as to transverse waves. But no such thing as a compressional wave had been found in the experimental study of light. Green attempted to overcome this difficulty by attributing an infinite volume elasticity to the ether. The expression above shows that longitudinal waves originating in such an incompressible medium would be carried away with an infinite velocity, and it may be shown that the energy associated with them would be infinitesimal in amount. The next step was to calculate the coefficients of transmission and reflection for light passing from one material medium to another. Here the elastic solid theory is not altogether successful. If the ether is supposed to have different densities in the two media, as in Fresnel’s theory, but the same rigidity, certain of these coefficients fail to give the values demanded by experiment, while if the densities are assumed the same but the rigidities different, other of the coefficients have discordant values. In connection with the phenomena of double refraction even more serious difficulties are encountered.

Electromagnetic Theory.—It was beginning to be felt that an ether must explain more than the phenomena of light, for Faraday’s conception of electromagnetic action as carried on through the agency of a medium had added greatly to its functions. Finally Maxwell’s demonstration that electromagnetic waves are propagated with the velocity of light made the theory of light into a subdivision of electrodynamics. Maxwell himself did not apply electromagnetic theory to the explanation of reflection and refraction. This deficiency, however, was remedied by Lorentz in 1875. The results obtained, as well as those for double refraction (J. W. Gibbs, =23=, 262, 1882 et seq.), and metallic reflection (L. P. Wheeler, =32=, 85, 1911), provided a complete vindication of the electromagnetic theory of light. This is all the more significant when the extreme precision obtainable in optical experiments is taken into account. For instance, Hastings (=35=, 60, 1888) has tested Huygens’ construction for double refraction in Iceland spar and found that “the difference between a measured index of refraction ... at an angle of 30° with the crystalline axis, and the index calculated from Huygens’ law and the measured principal indices of refraction” is a matter of only 4–5 units in the sixth decimal place. Since Maxwell’s time the gamut of electromagnetic waves has been steadily extended. The shortest Hertzian waves merge almost imperceptibly into the longest heat waves of the infra-red, and from there the known spectrum runs continuously through the visible region to the short waves of the extreme ultra-violet recently disclosed by Lyman. Here there is a short gap until soft X-rays are reached, and finally the domain of radiation comes to an end with gamma rays a billionth of a centimeter in length.

Maxwell’s ether was not a dynamical ether in the sense of Green’s elastic solid medium. In spite of the fact that Maxwell was always active in devising mechanical analogues to illustrate the phenomena of electromagnetism, he was never enthusiastic over the speculations of the advocates of a dynamical ether. The electrodynamic equations provided an accurate representation of the electric and magnetic fields, and beyond that he felt it was needless to go. That Gibbs (=23=, 475, 1882) held the same view is made evident by the closing paragraphs of a paper in which he shows that the electromagnetic theory of light accounts in minutest detail for the intricate phenomena accompanying the passage of light through circularly polarizing media. He says:

“The laws of the propagation of light in plane waves, which have thus been derived from the single hypothesis that the disturbance by which light is transmitted consists of solenoidal electrical fluxes, ... are essentially those which are received as embodying the results of experiment. In no particular, so far as the writer is aware, do they conflict with the results of experiment, or require the aid of auxiliary and forced hypotheses to bring them into harmony therewith.

In this respect the electromagnetic theory of light stands in marked contrast with that theory in which the properties of an elastic solid are attributed to the ether,—a contrast which was very distinct in Maxwell’s derivation of Fresnel’s laws from electrical principles, but becomes more striking as we follow the subject farther into its details, and take account of the want of absolute homogeneity in the medium, so as to embrace the phenomena of the dispersion of colors and circular and elliptical polarization.”

Further Dynamical Theories.—Kelvin, however, was not satisfied with this type of ether. To him dynamics was the foundation of all physical phenomena, and nothing could be said to be explained until a mechanical model was provided. So he returned to the elastic solid theory, and developed the consequences of the assumption, already made use of by Cauchy, that the ether has a negative volume elasticity of such a value as to make the velocity of the compressional wave zero. In order to prevent such an ether from collapsing it is necessary to assume that it is rigidly attached at its boundaries and that cavities cannot be formed at any point in its interior. Now Gibbs (=37=, 129, 1889) has pointed out the remarkable fact that the equations describing the motion of Kelvin’s quasi-labile ether are of exactly the same form as the electromagnetic equations. Electric displacement is represented by an actual displacement of the ether, magnetic intensity by a rotation. Hence everything which can be explained by the electrodynamic equations finds an analogue in terms of Kelvin’s ether. Still another type of dynamic ether which fits the known facts was proposed by McCullagh and perfected by Larmor. In this ether a rotational elasticity is premised, such as would exist if each particle of the medium consisted of three rigidly connected gyrostats with mutually perpendicular axes. In this ether electrical displacements correspond to rotations, and magnetic strains to etherial displacement.

A New Point of View.—While the dynamical school was still dominant in England, another point of view was developing on the continent. Kirchhoff denied that it was the province of science to provide mechanical explanations of the ether and electrodynamic phenomena such as Kelvin conceived to be necessary in order to make these phenomena intelligible. Kirchhoff’s contention was that the object of science is purely descriptive,—phenomena must be observed, classified, and mutual connections described by the fewest number of differential equations possible. Mach expressed the same idea somewhat more concisely when he asserted that the aim of science is “economy of thought.” For instance, in the time of Newton, planetary motions could be described quite satisfactorily by means of the three laws of Kepler. The motion of falling bodies on the earth’s surface had been described with a fair degree of accuracy by Galileo. The value of Newton’s law of gravitation, however, lay in the fact that this great generalization made it possible to describe these and many other types of motion by a single simple formula, instead of leaving each to be governed by a number of separate and apparently unrelated laws. The importance of such a generalization is measured by the economy of thought which it introduces.

Electron Theory.—The electron theory was leading to a reversal of Kelvin’s idea that dynamical principles must underlie electrodynamics. Lorentz had shown that a rigorous solution of the electrodynamic equations did away entirely with Maxwell’s displacement current, but made the electromagnetic field at a point in space depend not upon the distribution of charges and currents at the same instant, but at a time earlier sufficient to allow the effect to travel with the velocity of light from the charges and currents producing the field to the point at which the electric and magnetic intensities are to be found. The position of a charge or current element at this earlier time he denoted its “effective position.” The effective distribution, then, is that actually seen by an observer stationed at the point under consideration at the instant for which the intensity of the electromagnetic field is to be determined. This solution of the electrodynamic equations led in turn to rigorous expressions for the electric and magnetic intensities produced by a very small charged particle, such as an electron. Fig. 1 shows the electrostatic field produced by a charged particle at rest. The lines of force spread out radially and uniformly in all directions. In fig. 2 the electron is supposed to have a velocity v horizontally to the right of an amount smaller than, though comparable with, the velocity of light c. It is seen that the lines of electric force still diverge radially from the charge, but are crowded in the equatorial plane and spread apart in the polar regions. The dissymmetry grows as the velocity increases until if the velocity of light should be reached the field would be entirely concentrated in a plane at right angles to the direction of motion. Now it may be shown that fig. 2 is obtainable from fig. 1 by reducing dimensions in the direction of motion in the ratio of

√(1 − β^2) : 1, where β ≡ v/c.

For a uniformly convected electric field differs from an electrostatic field only in that the dimensions in the direction of motion are contracted in this particular ratio. Fig. 3 represents the electric field of a charged particle which has a uniform acceleration to the right. Consider Faraday’s analogy between lines of force and stretched elastic bands. The symmetry of the first two figures shows that in neither of these cases would there be a resultant force on the charged particle. But in the third figure it is obvious that a force to the left is exerted on the charge by its own field. Calculation shows this force to be proportional in magnitude to the acceleration. Let it be postulated that the resultant force on a charged particle is always zero. Then if F is the applied force, the force on the particle due to the reaction of its field will be — m f, where f stands for the acceleration and m is a positive constant, and we have the fundamental equation of dynamics

F − m f = 0

Hence, instead of admitting Kelvin’s contention that all physical phenomena must be given a mechanical explanation, it would seem more logical to assert that electrodynamics actually underlies mechanics.

Calculation shows the electromagnetic mass m to vary inversely with the radius of the charged particle. Now Thomson’s experiments made it possible to calculate the mass of an electron. Hence its radius can be computed, and is found to be about 2(10)^{–13} part of a centimeter, or one fifty-thousandth part of the radius of the atom. Since numbers so small convey little meaning, consider the following illustration, due, in part, to Kelvin. Imagine a single drop of water to be magnified until it is as large as the earth. The individual atoms would then have the size of baseballs. Now magnify one of these atoms until it is comparable in size with St. Peter’s cathedral at Rome. The electrons within the atom would appear as a few grains of sand scattered about the nave. This separation between the constituent electrons of the atom,—so great in comparison with their dimensions,—explains how alpha particles can be shot by the billion through thin-walled glass tubing without leaving any holes behind or impairing in the slightest degree the high vacuum within the tube. The much smaller high speed beta particles pass through an average of ten thousand atoms without even coming near enough to one of the component electrons to detach it and form an ion.

Michelson-Morley Experiment.—In 1881 Michelson (=22=, 120, 1881) conceived an ingenious and bold method of measuring the orbital motion of the earth through the luminiferous ether. As the experiment was one involving considerable expense, Bell, the inventor of the telephone receiver, was appealed to successfully for the funds necessary to carry it through. Michelson’s experimental plan was as follows: A beam of light traveling in the direction of the earth’s motion strikes an unsilvered mirror m at an angle of 45°. Part of the light passes through, the rest being reflected at right angles to its original direction. Each ray is returned by a mirror at a distance l from m. On meeting again, the ray whose path has been at right angles to the direction of the earth’s motion passes on through the mirror, while the other ray is reflected so as to bring the two in line and form interference fringes. Now consider the effect of the earth’s motion on the paths of the two rays. In fig. 4 the earth is supposed to be moving to the right. The unsilvered mirror m bifurcates a beam of light coming from a source a. By the time the ray reflected from m has traveled to the mirror b and back, m will have moved forward to m’; a distance 2βl, where the small quantity β is the ratio of the earth’s velocity to the velocity of light. Hence the length of the path traversed by this ray is approximately

2l(1 + ½β^2).

The other ray will reach the mirror c after the latter has moved forward a distance

βl/(1 − β^2)

and on returning find m at m’. Hence its path has a length of roughly 2l(1 + β^2). The difference in path of the two rays is β^2l and consequently they should be a little out of phase on meeting at d. By rotating the apparatus clockwise through 90° the directions of the two rays relative to the earth’s motion are interchanged, and the interference fringes would be expected to shift an amount corresponding to a difference in path of 2β^2l. This quantity is of course small,—β^2 is about one one hundred millionth,—but so sensitive are the methods of interferometry that Michelson felt confident that he would be able to detect the earth’s motion through the ether. The apparatus consisted of a table which could be rotated about a vertical axis in much the same way as a spectrometer table, and provided with arms a meter long to carry the mirrors b and c. With this length of arm the interference fringes from sodium light should shift by an amount corresponding to four hundredths of a wave length when the table is rotated through a right angle. When the experiment was first performed the apparatus was placed on a stone pier in the Physical Institute at Berlin. So sensitive was the instrument to outside vibrations that even after midnight it was found impossible to get consistent readings. Finally a satisfactory foundation was constructed in the cellar of the Astrophysical observatory at Potsdam. But what was the astonishment of the experimenters to find that the expected shift of the interference fringes did not exist!