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.
[Illustration: FIG. 1. FIG. 2. FIG. 3.]
_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 2β_l_, 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
2_l_(1 + ½β^2).
The other ray will reach the mirror _c_ after the latter has moved
forward a distance
β_l_/(1 − β^2)
and on returning find _m_ at _m’_. Hence its path has a length of
roughly 2_l_(1 + β^2). The difference in path of the two rays is β^2_l_
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^2_l_. 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!
[Illustration: FIG. 4.]
The extreme delicacy of the experiment made it desirable to confirm the
result by repeating it. This was done by Michelson and Morley (=34=,
333, 1887) in 1887. In place of a revolving table a massive slab of
stone floating on mercury was used to carry the apparatus. This slab was
kept in constant rotation, the observer following it around. Moreover,
the precision of the experiment was greatly increased by reflecting each
ray back and forth across the slab a number of times between leaving and
returning to the mirror _m_. The accuracy attained was such as to
justify Michelson in declaring that if the effect sought actually
existed it could not be so great as one-twentieth of its calculated
value. In 1905 Morley and Miller[166] repeated the experiment for the
second time and succeeded in increasing the sensitiveness of the
apparatus to a point such that a motion through the ether of one-tenth
of the earth’s orbital velocity could have been detected.
The displacement looked for in the Michelson-Morley experiment is known
as a second-order effect in that it depends upon the square of the ratio
of the velocity of the earth to that of light. Michelson at first
considered that the negative result obtained confirmed a theory proposed
by Stokes in which it was assumed that the ether inside and near its
surface partakes of the motion of the earth, while that at a distance is
practically quiescent. But there are many objections to Stokes’ theory,
one of which was brought out by an experiment of Michelson’s (=3=, 475,
1897) in which he attempted by an interference method to detect a
difference in the velocity of light at different levels above the
earth’s surface. The negative result obtained led him to conclude that
if Stokes’ theory were true the earth’s influence on the ether would
have to extend to a distance above its surface comparable with its
diameter. Meanwhile a more satisfactory explanation was forthcoming. It
has been pointed out that a uniformly convected electric field is
derivable from an electrostatic field by contracting dimensions in the
direction of motion in the ratio
√(1 − β^2) : 1.
Fitzgerald and Lorentz showed independently that if moving matter is
distorted in this same way the result obtained by Michelson would be
just that to be expected. For then the distance of the mirror _c_ from
_m_ would be
l√(1 − β^2)
instead of _l_, and the path of the ray moving parallel to the earth’s
orbit
2_l_(1 + ½β^2),
which is just that of the other ray. Of course when the apparatus is
rotated through 90°, the distance of this mirror from _m_ assumes its
normal value again, and the distance of the other mirror becomes
shortened. As all measurement consists in comparing the object to be
measured with a standard this contraction could never be detected by
experimental methods, for the measuring rod would contract in exactly
the same ratio as the body to be measured.
In computing its electromagnetic mass Abraham had assumed the electron
to be a uniformly charged rigid sphere which keeps its spherical form no
matter how great a velocity it may be given. He found that the mass
increases with the speed at very high velocities, becoming infinite as
the velocity of light is approached, and that its value depends upon the
direction of the applied force. After the Fitzgerald-Lorentz contraction
was seen to be necessary in order to explain Michelson’s result, Lorentz
calculated the electromagnetic mass of a charged sphere which is
deformed into an oblate spheroid when set in motion. For this type of
electron too, the mass approaches infinity for velocities as great as
that of light, and is different for different directions. If a force is
applied in the direction of motion the inertia to be overcome is a
little greater than when the force is applied at right angles to this
direction. Thus we have to distinguish between longitudinal and
transverse masses. But the masses of Lorentz’s electron are not the same
functions of its velocity as those of Abraham’s. Kaufmann and after him
Bucherer tested experimentally the relation between transverse mass and
velocity by observing the deflections produced by electric and magnetic
fields in the paths of high speed beta particles. The latter’s work was
such an ample confirmation of Lorentz’s formula that it may be
considered as proven that a moving electron at least suffers contraction
in the direction of motion in the ratio
√(1 − β^2) : 1.
The electromagnetic theory of light had proved so successful when
applied to bodies at rest that Lorentz was anxious to extend this theory
to the optics of moving media. His problem was to find a group of
homogeneous linear transformations that would leave the form of the
electrodynamic equations unchanged. The Michelson-Morley experiment had
shown that dimensions in the direction of motion must be contracted in
the moving system, those at right angles remaining unaltered. But
Lorentz soon found that it was also necessary to use a new unit of time
in the moving system, and as this time was found to depend upon the
_position_ of the point at which it is to be determined, he called it
the _local_ time. Lorentz’s transformation is just that of the principle
of relativity, but he did not succeed in expressing the electrodynamic
equations in terms of the new coördinates and time in exactly the same
form as for a system at rest, for the reason that he failed to endow
these new units with sufficient reality to justify him in using them
when it came to transforming the velocity term involved in an electric
current.
_Principle of Relativity._—In 1905 appeared in the Annalen der
Physik[167] a paper destined to alter entirely the point of view from
which problems in light and electromagnetic theory are to be approached.
The author was Albert Einstein, of Berne, Switzerland, a young man of
twenty-six who had already made a number of notable contributions to
theoretical physics.
The principle of relativity proposed by Einstein was by no means new to
students of dynamics. Newton’s first two laws of motion express very
clearly the fact that in mechanics all motion is relative. Force is
proportional to acceleration, and the relation between the two is the
same whether the motion under consideration is referred to fixed axes or
to axes moving with a constant velocity. But in connection with the
phenomena of light and electromagnetism the case seemed to be quite
different. There everything was referred to a fixed ether, and even
though Lorentz had found a set of transformations which left the
electrodymanic equations practically unchanged, he continued to think in
terms of an ether. So physicists were not a little startled when
Einstein postulated that no experiment, practical or ideal, could ever
distinguish between two systems in such a manner as to warrant the
assertion that one of them is at rest and the other in motion. All
motion is relative, and the laws governing physical, chemical and
biological phenomena are the same in terms of the units of one system as
in terms of those of any other.
Einstein next considers some very fundamental questions. What do we mean
when we say that two events, one at A and the other at a point B far
from A, occur at the same time? Obviously the expression has no
significance unless synchronous clocks are stationed at the two points.
But how is it to be determined whether or not these two clocks are
synchronous? If instantaneous communication could be established between
A and B the matter would be simple enough. Since no infinite velocity of
transmission is available, however, let a light wave be sent from A to B
and returned to A immediately upon its arrival. If the time indicated by
the clock at B when the signal is received is half way between that at
which it left A and the time at which it arrives on its return, then the
two clocks may be considered synchronous. Now if it desired to measure
the length of a bar which is moving parallel to the scale with which the
measurement is to be made, it is necessary to note the positions of the
two ends of the bar at the _same_ instant. So even the measurement of
the length of a moving body depends upon the condition of synchronism at
different points in space.
The principle of relativity requires that the velocity of light shall be
the same in one system as in another relative to which the first is in
motion. Hence the definition of synchronism makes it possible to obtain
a set of transformations connecting space and time measurement on one
system with those on another. This group of transformations is exactly
that which Lorentz had found would transform the electrodynamic
equations into themselves. But Einstein’s point of view brought out a
remarkable reciprocity which Lorentz had missed. If two parallel rods MN
and OP are in motion relative to each other in the direction of their
lengths, not only does OP appear shortened to an observer at rest with
respect to MN, but MN appears shorter than normal in the same ratio to
an observer who is moving along with the rod OP.
Einstein’s theory makes the velocity of light the maximum speed with
which a signal can be transmitted. This leads to his celebrated addition
theorem. Consider three observers A, B and C. Let B be moving relative
to A with a velocity of nine-tenths the velocity of light, and C in the
same direction with an equal velocity relative to B. In terms of
old-fashioned notions of time and space, the velocity of C relative to A
would be computed as one and eight-tenths the velocity of light. But the
relativity theory gives it as ninety-nine hundredths the velocity of
light. For the velocity of light can never be surpassed by that of any
material object. This deduction from theory is most strikingly confirmed
by the fact that although beta particles have been observed with
velocities as high as ninety-nine hundredths that of light, the velocity
of light is never quite equalled. It may be remarked in passing that the
principle of relativity requires that the masses of all material bodies
shall vary with the velocity in the same manner as Lorentz found to be
the case for the electromagnetic mass of the deformable electron. In
this connection Bumstead (=26=, 498, 1908) has devised an elegant method
of deducing the ratio of longitudinal to transverse mass.
The close connection between electrodynamics and the principle of
relativity is obvious from the fact that both lead to the same time and
space transformations. Furthermore L. Page (=37=, 169, 1914) has shown
that the electrodynamic equations can be derived exactly and in their
entirety from nothing more than the kinematics of relativity and the
assumption that every element of charge is a center of uniformly
diverging lines of force. Hence it may safely be asserted that no purely
electromagnetic phenomenon can ever come into contradiction with this
principle. The simplicity thus introduced into the solution of a certain
class of problems is enormous. As an example consider the question as to
whether a moving star is retarded by the reaction of its own radiation.
This purely electrodynamical problem is of such complexity that attempts
to solve it have led to some controversy among mathematical physicists.
The principle of relativity tells us without recourse to analysis that
no retardation can exist.
Throughout the nineteenth century the ether has played a fundamental
part in all important physical theories of light and electromagnetism.
But if it is not possible for experiment to detect even the state of
motion of the ether, why postulate the existence of such a medium? If it
does not possess the most fundamental characteristic of matter, how can
it possess such derived properties as density and elasticity,—properties
which any conceivable _mechanical_ medium must have in order to transmit
transverse vibrations? The relativist does not deny the existence of an
ether. To him the question has no more meaning than if he were asked to
express an opinion as to the reality of parallels of latitude on the
earth’s surface. As a convenient medium of expression in describing
certain phenomena the ether has justified much of the use which has been
made of it. But to attribute to it a degree of substantiality for which
there is no warrant in experiment, is to change it from an aid into an
obstacle to the progress of science. From the relativist point of view
the distinction is very sharp between those motions of charged particles
which are experimentally observable, and such geometrical conventions as
electromagnetic fields, or analytical symbols as electric and magnetic
intensities. These modes of representation have been and still are of
the greatest use and importance, but their value in scientific
description must not lead to lack of appreciation of their purely
speculative character.
Finally attention must be drawn to the fact that the discoveries of
inductive science, embodied in the great generalization we have just
been discussing, have led to a more intimate knowledge of the nature of
time and space than twenty centuries of introspection on the part of
professional philosophers. Minskowski, whose promise of greater
achievement was cut off by an untimely death, has shown that four
dimensional geometry makes possible the representation with beautiful
simplicity of the time and space relationships of this theory. The one
time and three space dimensions merge in such a manner as to form a
single whole with not a vestige of differentiation between these
fundamental quantities. Wilson and Lewis[168] have made this
representation familiar to American readers through their admirable
translation of Minskowski’s work into the notation of Gibbs’s vector
analysis.
Aberration, the Doppler effect, anomalous dispersion, —indeed all known
phenomena,—are found to be in accord with the principle of relativity.
It must be borne in mind, however, that this principle applies only to
systems moving relative to one another in straight lines with constant
velocities. That there is something absolute about rotation has been
recognized since Foucault performed his famous pendulum experiment in
1851. This experiment (C. S. Lyman, =12=, 251 and 398, 1851) consisted
in setting a pendulum composed of a heavy-brass ball suspended by a long
wire into oscillation in such a way as to avoid appreciable ellipticity
in its motion. Observation of the rate at which the ground rotates
relative to the plane of vibration of the pendulum furnished a method of
measuring the rotation of the earth about its axis _without reference to
celestial bodies_. The gyroscopic compass in use to-day provides yet
another terrestrial method of detecting this rotation.
_The Future of Physics._—At times during the history of physics it has
seemed as if the fundamental laws of this science had been so completely
formulated that nothing remained to future generations beyond the
routine of deducing to the full the consequences of these laws, and
increasing the precision of the methods used to measure the constants
appearing in them. That Laplace held this view has already been pointed
out, and Maxwell, in his introductory lecture at the opening of the
Cavendish laboratory in 1871, said, “This characteristic of modern
experiments—that they consist principally of measurements—is so
prominent, that the opinion seems to have gotten abroad that in a few
years all the great physical constants will have been approximately
estimated, and that the only occupation which will then be left to men
of science will be to carry on these measurements to another place of
decimals.” That he himself did not entertain this view is made evident
by a succeeding paragraph. “But we have no right to think thus of the
unsearchable riches of creation, or of the untried fertility of those
fresh minds into which these riches will continue to be poured. It may
possibly be true that, in some of those fields of discovery which lie
open to such rough observations as can be made without artificial
methods, the great explorers of former times have appropriated most of
what is valuable, and that the gleanings which remain are sought after
rather for their abstruseness than for their intrinsic worth. But the
history of science shows that even during that phase of her progress in
which she devotes herself to improving the accuracy of the numerical
measurement of quantities with which she has long been familiar, she is
preparing the materials for the subjugation of new regions, which would
have remained unknown if she had been contented with the rough methods
of her early pioneers....”
That Maxwell’s forecast of the prospects of his science was no
overestimate will be granted by those who have followed the progress of
physics during the last twenty years. Yet the work accomplished in the
past appears small compared to that which is left to the future. Many of
the unsolved problems are matters of fitting together puzzling details,
but there is at least one whose solution appears to demand a radical
modification in our fundamental physical conceptions. This is the
formulation of the laws which govern the motions of electrons and
positively charged particles inside the atom.
_Black Radiation._—The significance of the problem was first brought to
light through the study of black radiation. By a black body is meant one
whose distinguishing characteristic is that it emits and absorbs
radiation of all frequencies, and black radiation is that which will
exist in thermal equilibrium with such a body. The interest of this type
of radiation lies in the fact, demonstrated by Kirchhoff, that its
nature depends only upon the temperature of the black body with which it
is in equilibrium, and on none of this body’s physical or chemical
characteristics. Thus we may speak of the “temperature” of the radiation
itself, meaning by this the temperature of the material body with which
it would be in equilibrium.
The problem of black radiation is to find the distribution of energy
among the waves of different frequencies at any given temperature. The
first step toward a solution was made when Stefan showed experimentally,
and Boltzmann as a deduction from thermodynamics and electrodynamics,
that the total energy density summed up over all wave lengths varies
with the fourth power of the absolute temperature. If the energy density
is plotted as ordinate against the wave length as abscissa, the
experimental curve for any one temperature rises from the axis of
abscissas at the origin, reaches a maximum, and falls to zero again as
the wave length becomes infinitely great. Now Wien’s displacement law,
the second important step toward the determination of the form of this
curve, shows that as the temperature is raised the wave length to which
its highest point corresponds becomes shorter,—in fact this particular
wave length varies inversely with the absolute temperature. This
theoretical conclusion is entirely confirmed by experiment. (J. W.
Draper, =4=, 388, 1847.)
Farther than this general thermodynamical principles are unable to go.
Statistical mechanics, however, asserts that when a large number of like
elements are in thermal equilibrium, the average kinetic energy
associated with each degree of freedom is equal to a universal constant
multiplied by the absolute temperature. This “principle of
equi-partition of energy” has been applied in various ways to obtain a
radiation law. The most straightforward method is based on the
equilibrium which must ensue between radiation field and material
oscillators when the latter emit, on the average, as much energy as they
absorb. From whatever aspect the problem is treated, however, the
radiation law obtained from the application of the equi-partition
principle is the same. And while this law agrees well with the
experimental curve for long wave lengths, it shows an energy density
that becomes indefinitely great for extremely short waves, which is not
only at variance with the facts, but actually leads to an _infinite_
value of this quantity when integrated over the entire spectrum.
_The Energy Quantum._—Now the principle of equi-partition of energy
rests securely on most general dynamical principles. That these
dynamical laws are inexact to any such extent as the divergence between
theory and experiment would indicate, is inconceivable; that they are
_insufficient_ when applied to motions of electrons in such intense
fields as occur within the atom seems no longer open to doubt. In order
to obtain a radiation formula in accord with experiment Planck has found
it necessary to extend the atomic idea to energy, which he conceives to
exist in multiples of a fundamental quantum _h_ν, ν being the frequency
and _h_ Planck’s constant. That some such hypothesis of discontinuity is
essential in order to obtain any law that will even approximately fit
the experimental facts has been proved by Poincaré. But the precise spot
at which the quantum is introduced differs for every new derivation of
Planck’s law. As deduced most recently by Planck himself, the quantum
shows itself in connection with the emission of energy by the material
oscillators with which the radiation field is in equilibrium. These
oscillators are supposed to act quite normally in every respect except
emission; here the radiation demanded by the electrodynamic equations is
cast aside, and an oscillator is supposed to emit at once all its energy
after it has accumulated an amount equal to some integral multiple of
_h_ν. A form of the theory which does not contain this improbable
contradiction of the firmly established facts of electrodynamics
introduces the quantum into the specification of the energy of vibration
which is permitted to each oscillator. Here both emission and absorption
follow the classical theory, but the motion of an emitting and absorbing
linear oscillator of frequency ν is supposed to be stable only for those
amplitudes for which the energy of its oscillations is an integral
multiple of _h_ν. In order to maintain the energy at these particular
values, the oscillator may draw energy from, or deposit surplus energy
with, other degrees of freedom which partake neither in emission nor
absorption, but act merely as storehouses.
_Photoelectric Effect._—When investigating the production of
electromagnetic waves, Hertz had noticed that a spark passed more
readily between the terminals of his oscillator when the negative
electrode was illuminated by light from another spark. Further
investigation by Hallwachs, Elster and Geitel, and others showed that
this effect was due to the emission of electrons by a metal exposed to
the influence of ultra-violet light. Lenard discovered that the energy
with which a negatively charged particle is ejected is entirely
independent of the intensity of the light, and further investigation
showed it to depend only on the frequency. Einstein suggested that the
electrons appearing in this so-called photo-electric effect start from
within the metal with an initial energy _h_ν. In passing through the
surface a resistance is encountered, however, so he concluded that the
energy with which the fastest moving electrons appear outside the metal
should be equal to _h_ν less the work done in overcoming this
resistance. Recent experiments not only confirm this relation, but
provide a most satisfactory method of determining the value of _h_.
Millikan[169] finds it to be 6·57(10)^{–27} ergs sec., which gives the
quantum for yellow light a value sixty times as great as the heat energy
of a monatomic gas molecule at O°C. That this large amount of energy can
be transferred from the incident light to the ejected electron is quite
out of the question; it must come from within the atom. In this way some
indication is obtained of how vast intra-atomic energies must be.
_Structure of the Atom._—The generally accepted model of the atom is
that due chiefly to Rutherford.[170] He considers it to be constituted
of electrons revolving about a positive nucleus either singly or grouped
in concentric rings, in much the same manner as the planets revolve
around the sun. Experiments on the scattering of alpha rays, however,
show that the nucleus, while it must have a positive charge sufficient
to neutralize the charges of all the electrons moving around it, cannot
have a volume of an order of magnitude greater than that of the
electron. The number of unit charges residing on it, except in the case
of hydrogen, which is supposed to consist of a singly charged nucleus
and only one electron, is found to be approximately half the atomic
weight. Thus helium, with an atomic weight of about four, has a doubly
charged nucleus with two electrons revolving about it, and lithium a
triply charged nucleus and three electrons. The number of unit charges
on the nucleus is supposed to correspond with the atomic number used by
Moseley in interpreting the results of his experiment on the X-ray
spectra of the elements.
Now the electron which is revolving around the positive nucleus of a
hydrogen atom, must, according to electrodynamic laws, radiate energy.
This radiation will act as a resistance to its motion, causing its orbit
to become smaller and its frequency to increase. Hence luminous hydrogen
would be expected to give off a continuous spectrum. The very fine lines
actually found seem inexplicable on the classical dynamical and
electrodynamical theories. These lines, and those of many other spectra,
may even be grouped into series, and the relations between them
expressed in mathematical form. Formulæ have been proposed by Balmer,
Rydberg, Ritz and others, all of which contain a universal constant _N_
as well as certain parameters which must be varied by unity in passing
from one line of a series to the next.
In 1913 Bohr[171] proposed anatomic theory which brings to light a
remarkable numerical relationship between this quantity _N_ and Planck’s
constant _h_. He postulated that the electron in the hydrogen atom, for
instance, cannot revolve in a circle of any arbitrary radius, but is
confined to those orbits for which its kinetic energy is an integral
multiple of ½_hn_, _n_ being its orbital frequency. Now at times this
electron is supposed to jump from an outer to an inner orbit, when the
excess energy of the first orbit over the second is radiated away. But
the energy emitted is also taken to be equal to _h_ν, where ν is the
frequency of the radiation. Hence ν can be determined, and the
expression obtained for it is exactly that given long before by Balmer
as an empirical law. The most remarkable thing about it, however, is
that Bohr’s result contains a constant involving _h_ and the electronic
charge and mass which has precisely the value of the universal constant
_N_ of Balmer’s and Rydberg’s formulæ. In all, the theory accounts for
three series of hydrogen, and yields satisfactory results for helium
atoms which have lost an electron, or lithium atoms which have a double
positive charge. But for atoms which retain more than a single electron
it seems no longer to hold.
The three mentioned are only the most clearly defined of a growing group
of phenomena in which the quantum manifests itself. Its significance and
the alteration in our fundamental conceptions to which it seems to be
leading is for the future to make clear. That it presents the most
important and interesting problem as yet unsolved few physicists would
deny.
_American Physicists._—In attempting to cover the progress of physics
during the last hundred years in the space of a few pages, many
important developments of the subject have of necessity remained
untouched, and the treatment of many others has been entirely
inadequate. Among those appearing in the Journal of which no mention has
been made are LeConte’s (=25=, 62, 1858) discovery of the sensitive
flame and Rood’s (=46=, 173, 1893) invention of the flicker photometer.
However, enough has been recounted to indicate the preeminent position
in the history of physics in America occupied by four men: Joseph Henry,
of the Albany Academy, Princeton, and the Smithsonian Institution; Henry
Augustus Rowland, of Johns Hopkins University; Josiah Willard Gibbs, of
Yale; and Albert Abraham Michelson, of the United States Naval Academy,
Case School of Applied Science, Clark University, and the University of
Chicago. Of these, the last named has the distinction of being the only
American physicist to have received the Nobel prize, though there is
little doubt that the other three would have been similarly honored had
not their important work been published prior to the institution of this
award. All four occupy high places in the ranks of the world’s great men
of science, and the investigations carried out by them and their fellow
workers in America have given to their country a position in the annals
of physics which is by no means insignificant.
_The Journal’s Part in Meteorology._
The meteorological investigations published in the early numbers of the
Journal have played an important role in establishing a correct theory
of storms. Before the origin of the United States Signal Service in 1871
no systematic weather reports were issued by any governmental agency in
this country, and consequently the work of collecting as well as
interpreting meteorological data rested entirely in the hands of
interested individuals and institutions. The earliest important studies
of storms to appear in the Journal were contributed by Redfield of New
York, whose first paper (=20=, 17, 1831) treated in considerable detail
a violent storm which passed over Long Island, Connecticut and
Massachusetts in 1821. He concluded that “the direction of the wind at a
particular place, forms no part of the essential character of a storm,
but is only incidental to that particular portion ... of the track of
the storm which may chance to become the point of observation, ... the
direction of the wind being, in all cases, compounded of both the
rotative and progressive velocities of the storm.” A few years later,
analyses of twelve “gales and hurricanes of the Western Atlantic” (=31=,
115, 1837) led to the statement that the phenomena involved “are to be
ascribed mainly to the mechanical gravitation of the atmosphere, as
connected with the rotative and orbital movements of the earth’s
surface.” In this paper is emphasized the fact that the wind may blow in
diametrically opposite directions at points near the storm center.
“While one vessel has been lying-to in a heavy gale of wind, another,
not more than thirty leagues distant, has at the very same time been in
another gale equally heavy, and lying-to with the wind in quite an
opposite direction.” From an accompanying sketch showing wind
directions, the reader would infer that, at this time, Redfield believed
the motion of the air to be very nearly in circles about the storm
center. The same idea is conveyed by a later paper (=42=, 112, 1842).
Espy (=39=, 120, 1840) of Philadelphia, however, claimed that
observation showed rather that the wind blew inwards toward a central
point, if the storm were round in shape, or toward a central line, if it
were oblong. This view Redfield (=42=, 112, 1842) contested, and brought
forth much evidence to prove its falsity. A later statement (=1=, 1,
1846) of his own theory is as follows: “I have never been able to
conceive, that the wind in violent storms moves only in _circles_. On
the contrary, a vortical movement ... appears to be an essential element
of their violent and long-continued action, of their increased energy
towards the center or axis, and of the accompanying rain.... The
_degree_ of vorticular inclination in violent storms must be subject,
locally, to great variations; but it is not probable that, on an average
of the different sides, it ever comes near to forty-five degrees from
the tangent of a circle,—and that such average inclination ever exceeds
two points of the compass, may well be doubted.” A qualitative
explanation of the effect of the earth’s rotation on the direction of
the wind near the storm center had already been given by Tracy (=45=,
65, 1843), and this was followed some years later by Ferrel’s (=31=, 27,
1861) very thorough quantitative investigation of the dynamics of the
atmosphere.
A number of individuals kept systematic records of meteorological
observations, among whom was Loomis, whose storm analyses did much to
settle the merits of the rival theories of Redfield and Espy. In
studying the storm of 1836 (=40=, 34, 1841) he had drawn on the map
lines through those points in the track of the storm where the
barometer, at any given hour, is lowest. While this method revealed the
general direction in which the storm was progressing, it failed to give
much indication of its size or shape. In discussing the two tornadoes of
February, 1842, one of which had already been described in the Journal
(=43=, 278, 1842), he adopted a new and more illuminating graphical
method. Instead of connecting points of lowest pressure, he drew a curve
through all points where the barometer stood at its normal level, then
one through those points at which the pressure was ²⁄₁₀ of an inch below
normal, and so on. Temperature he treated in much the same way, and the
strength and direction of the wind were indicated by arrows. This
innovation gave to his storm analyses a significance which had been
entirely lacking in those of his predecessors, and led to the familiar
systems of isobars and isotherms in use on the daily charts issued by
the Weather Bureau at the present time. Loomis advocated careful
observations for one year at stations 50 miles apart all over the United
States, so that sufficient data might be obtained to settle once for all
the law of storms. His efforts, seconded by those of Henry, Bache,
Pierce, Abbe, and Lapham, led eventually to the establishment of the
Signal Service, and the publication of daily weather maps according to
the plan advocated thirty years before. These maps afforded a basis for
further analyses of storms, which he published in numerous
“Contributions to Meteorology” (=8=, 1, 1874, _et seq._) between 1874
and his death in 1890.
In addition to his work on storms, Loomis made a careful study of the
earth’s magnetism (=34=, 290, 1838 _et seq._), and of the aurora
borealis (=28=, 385, 1859 _et seq._). That a connection existed between
sunspots, aurora, and terrestrial magnetism was already recognized.
Loomis (=50=, 153, 1870 _et seq._), however, showed that the periodicity
of the aurora borealis, as well as of excessive disturbances in the
earth’s magnetic field, corresponds very closely with that of sunspots.
_Notes._
Footnote 154:
J. W. Gibbs, Trans. Conn. Acad. Arts and Sci., =3=, 108 and 343.
Abstract by the author, the Journal, =16=, 441, 1878.
Footnote 155:
H. K. Onnes, Nature, =93=, 481, 1914.
Footnote 156: