EINSTEIN’S MECHANICS
_The mechanical foundation of all the
sciences—Ascending the stream of time—The speed of
light an impassable limit—The addition of speeds
and Fizeau’s experiment—Variability of mass—The
ballistics of electrons—Gravitation and light as
atomic microcosms—Matter and energy—The death of
the sun._
When Baudelaire wrote:
I hate the movement that displaces lines,
he thought only, like the physicists of his time, of the static
deformations which have been known as long as there have been men
to observe them. What we have seen about Einsteinian time and space
has taught us that there must be, in addition to these, kinematic
deformations, to which every material object, however rigid it seems,
is liable.
Movement, therefore, displaces lines much more than Baudelaire
supposed, even the lines of the hardest of marble statues. This kind of
deformation, which is pleasant rather than hateful, since it brings us
nearer to the heart of things, has upset the whole of mechanics.
Mechanics is at the foundation of all the experimental sciences,
because it is the simplest, and because the phenomena it studies are
always present—if not exclusively present—amongst the phenomenal
objects of the other sciences, such as physics, chemistry, and biology.
The converse of this is not true. For instance, there is not a single
phenomenon in chemistry or biology in which one has not to study bodies
in movement, objects endowed with mass and giving out or absorbing
energy. On the other hand, the peculiar aspects of a biological,
chemical, or physical phenomenon, such as the existence of a difference
of potential, an oxidation, or an osmotic pressure, are not always
found in the study of the movements of a ponderable mass and of the
forces which act upon and through it.
Compared with mechanics, the sciences of physics, chemistry, and
biology have, in the order in which we name them, objects of increasing
complexity and generality, or, to put it better, of decreasing
universality. These sciences are mutually dependent in the way that the
trunk, branches, leaves, and flowers of a tree are. They are to some
extent related to each other as are the various parts of the jointed
masts on which military telegraphists fix their antennæ. The lower part
of the mast, the larger part, sustains the whole; but it is the upper
parts which bear the delicate and complicated organs.
The object of the great synthetists in science has always been, and
is, to reduce all phenomena to mechanical phenomena, as Descartes
attempted. Whether these attempts are well-grounded or no, whether
they will some day succeed or are condemned _a priori_ to
failure because physico-biological phenomena involve elements that
are essentially incapable of reduction to mechanical elements, is a
question that has been, and will continue to be, much discussed. But,
however thinkers may differ on that point, they are agreed on this: in
all natural phenomena, in all phenomena that are objects of science,
there is the mechanical element—exclusive in some, the principal
element in others.
All this leads to the conclusion that whatever modifies mechanics,
modifies at the same time the whole structure of ideas founded
thereon—that is to say, the other sciences, the whole of science, our
entire conception of the universe. But we are now going to see that
Einstein’s theory, as a direct effect of what it teaches in regard to
space and time, completely upsets the classical mechanics. It is in
this way, particularly, that it has shaken the rather somnolent frame
of traditional science, and the vibration is not yet over.
In approaching the Einsteinian mechanics we shall have the pleasure of
passing from ideas of time and space that are rather too exclusively
geometrical and psychological to the direct study of material
realities, of _bodies_. Here we can compare theory and reality,
the mathematical premises and the substantial verifications; and we
shall be pleased to see what the facts, given in experience, have to
say on the matter. We shall be able to make our choice, with informed
minds and sound criteria, between the old and the new ideas.
In a word, if I may use this illustration, as long as we were dealing
with ideas of space and time—which are empty frames in themselves,
vases that would interest us chiefly by the liquids they contain—we
were rather like the young men who have to choose a _fiancée_
solely by the description of her which has been given them. We are
now going to see with our own eyes, and see at work the two aspirants
to our affection: classical science and Einstein’s theory. We shall
see both of them take up the paste of facts, and we shall be able to
compare the delicious dishes which they respectively make from it for
the nourishment of the mind.
Theories have no value except as functions of facts. Those which, like
so many in metaphysics, have no real criterion by which we may test
them, are all of the same value. Experience, the sole source of truth
of which Lucretius said long ago:
unde omnia credita pendent,
or the material facts, is going to judge Einstein’s system for us.
* * * * *
The result of the Michelson experiment, the impossibility of proving
any velocity of the earth in relation to the medium in which light is
propagated, amounts to this: we have no means whatever of detecting
a speed higher than that of light. This consequence of the Michelson
experiment will be better understood, perhaps, if we put it in a
tangible form. Here is an illustration that will serve our purpose.
In some astronomical novel an imaginary observer is supposed to recede
from the earth at a speed greater than that of light—at 300,000 miles
a second, let us say—yet to keep his eyes (armed with prodigious
glasses) steadily fixed on this little globe of ours.
What will happen? Evidently, our observer will see the train of earthly
events in inverse order, because in the course of his voyage he will
catch up in succession the luminous waves which left the earth before
him. The farther away they are, the longer it must be since they left
the earth. After a time our man, or our superman, will witness the
Battle of the Marne. He will first see the field strewn with the
dead. Gradually the dead men will rise and join their regiments, and
presently they will be seen in groups in Gallieni’s taxis, which will
travel backwards at full speed to Paris, arriving in the midst of a
population that is extremely anxious about the issue of the struggle,
and the soldiers will, naturally, be unable to give them any news. In
a word, our observer will, if he recedes from the earth at a speed
greater than that of light, see terrestrial events happening as if he
were _ascending_ the stream of time.
It would be very different if the observer remained stationary, and
the earth receded from him at a speed of 300,000 miles a second. What
would happen then? It is clear that in this case our observer will see
terrestrial events, not in inverse order, but as they are: except that
they would seem to him to take place with majestic slowness, because
the rays of light which leave the earth at the end of some particular
event will take a much longer time to reach him than the rays which
left the earth at the beginning of the event.
In sum, the phenomena observed by him being essentially different in
the two cases, our imaginary observer would be able to say whether it
is he who is receding from the earth or the earth that is receding
from him; to detect the real movement of the event through space. This
means, of course, movement relatively to the medium of the propagation
of light, not necessarily, as we saw, movement in relation to absolute
space.
The experiment we have imagined could not very well be carried out
with the actual resources of our laboratories. We cannot attain these
fantastic speeds, and even if we could the observer would not
distinguish much. But we have chosen a colossal instance, and the
results of it would be colossal, as there would be question of nothing
less than a reversal of the order of time.
If we were to use more modest means, the results will be more modest,
but according to the older theories they ought to be recorded in our
instruments. But the Michelson experiment—a miniature version of what
we have just described—shows that the differences we should expect
are not observed. Therefore the premise we laid down—that there can
be velocities greater than that of light in empty space—does not
harmonise with reality. Hence this velocity of light is a wall, a limit
that cannot be passed.
* * * * *
Now let us see what follows. There is at the base of classical
mechanics, as it was founded by Galileo, Huyghens, and Newton, and as
it is taught everywhere, a principle which is in the long run, like
all the principles of mechanics, grounded upon experience. It is the
principle of the composition of velocities. If a boat, which makes
ten miles an hour in smooth water, sails down a river which flows at
five miles an hour, the speed of the boat in relation to the bank will
be, as we may find by actual measuring, equal to the sum of the two
speeds, or fifteen miles an hour. This is the rule of the addition of
velocities.
In a more general way, if a body starts from a state of rest, and
under the action of some force takes on in a second the velocity
=V=, what will it do if the action of the force is prolonged for
another second? According to classical mechanics it will take on the
velocity =2V=.[7] Let us imagine an observer who is travelling
at the velocity =V=, yet thinks he is at rest. It will seem to
him, at the end of the first second, that the body is at rest (because
it has the same velocity as the observer). In virtue of the Classical
Principle of Relativity, the apparent movement of the body must be the
same for our observer as if the rest were real. This means that at the
end of the second second the relative velocity of the body in reference
to the observer will be =V=, and, as the observer already has the
velocity =V=, the absolute velocity of the body will be =2V=.
In the same way it will be =3V= at the end of three seconds,
=4V= at the end of four seconds, and so on. Could it increase
indefinitely if the force continues to act long enough? Classical
mechanics says “yes.” Einstein says “no,” because there cannot be a
greater velocity than that of light.
[7] As an example of an identical force acting during periods of time
successively equal to 1, 2, or 3, we may take three guns of the same
calibre, but of lengths equal to 1, 2, and 3, and of which the charges,
or rather, their propulsive forces, are identical and constant. It is
found that the initial velocities of the shells are, in relation to
each other, 1, 2, and 3.
We have imagined an observer who has the velocity V relatively to us,
and who believes that he is at rest. For him the body observed was
likewise at rest at the beginning of the second second, because its
velocity was the same as that of the observer. From the fact that the
apparent movement of the body is for the observer, during the second
second, the same as it was for us during the first, classical mechanics
concluded that its velocity doubles during the second second. It did
not know what Einstein has now taught us: that the time and space of
this observer are different from ours.
What is a velocity? It is the space traversed in the course of a
second. But the space thus measured by our moving observer, which he
believes to be of a certain length, is in reality, for us who are
stationary, smaller than he thinks, because the rules he uses are, as
Einstein has shown, shortened by velocity without his perceiving it.
Therefore the velocities are not added together in equal proportions
and indefinitely for a given observer, as classical mechanics
maintained.
Under the action of the same force, the old mechanics said, a body
will always experience the same acceleration, whatever be the velocity
already acquired. Under the action of the same force, the new mechanics
says, the motion of the body will be accelerated less and less in
proportion to its velocity.
Take, for instance, some movable object having, relatively to me, a
velocity of 200,000 kilometres a second. Let us place an observer on
this object. The observer will then start, in the same direction and
under the same conditions as we have done, a second movable object,
which will thus have, _relatively to him_, a speed of 200,000
kilometres. The Relativist says that the resultant velocity of the
second object relatively to us will not be, as the classical addition
of velocities would make it, 200,000 + 200,000 = 400,000 kilometres a
second. It will be only 277,000 kilometres a second. What the second
moving observer took to be 200,000 kilometres (because his measuring
rod was shortened owing to velocity) was really only 77,000 of our
kilometres. How is it possible to calculate that? Simply by using the
formula of Lorentz which I gave in Chapter II, which gives us the value
of the contraction due to velocity. We then easily find that, if we
have two velocities, _v_ and _v_₂, and if we call the
resultant _w_, classical mechanics stated that
_w_ = _v_₁ + _v_₂
The Einstein mechanics says that this is not correct, and that what we
really have (C being the velocity of light) is
_v_₁ + _v_₂
_w_ = —————————————
( _v_₁_v_₂ )
(1 + —————————)
( C² )
I apologise for again introducing—it shall be the last time—an
algebraical formula into my work. But it spares me a large number of
words, and it is so simple that every reader who has even a tincture of
elementary mathematics will at once see its great significance and the
consequences of it.
The formula expresses in the first place the fact that the resultant
of the velocities, however great it may be, cannot be greater than
the speed of light. It conveys also that, if one of the component
velocities is that of light, the resultant velocity must have the same
value. It means, in fine, that in the case of the slight velocities
we have to do with in actual life (that is to say, when the component
velocities are much smaller than that of light) the resultant is
very nearly equal to the sum of the two components, as the classical
mechanics says.
The classical mechanics was, we must remember, founded upon experience.
We understand how, in those circumstances, Galileo and his successors,
dealing only with relatively slowly moving bodies, reached a principle
which seemed to be true for them, but is only a first approximation.
For instance, the resultant of two velocities, each equal to a hundred
kilometres a second (which is far higher than any velocities obtainable
by Galileo and Newton), amounts to, not 200 kilometres, but 199·999978
kilometres. The difference is scarcely twenty-two millimetres in 200
kilometres! We can quite understand that the earlier experimenters
could not detect differences even less minute than that.
* * * * *
Amongst the verifications of the new law of composition of velocities
we may quote one, the outcome of an early experiment of the great
Fizeau, which is very striking.
Imagine a pipe full of some liquid, such as water, and a ray of light
travelling along it. We know the speed of light in water: it is much
lower than in air or in empty space. Suppose, further, that the water
is not stationary, but flows through the pipe at a certain speed.
What will be the velocity of the ray of light when it leaves the pipe
after traversing the moving liquid? That was what Fizeau, with many
variations of the conditions of the experiment, tried to ascertain.
The velocity of light in water is about 220,000 kilometres a second.
There is question here of so rapid a propagation that there is a great
difference between the law of addition of the old classical mechanics
and of Einsteinian mechanics. Now the results of Fizeau’s experiment
are in complete harmony with Einstein’s formula, and are not in harmony
with that of the older mechanics. Many observers, including, recently,
the Dutch physicist Zeeman, have repeated Fizeau’s experiment with the
greatest care, but the result was the same.
When Fizeau made the experiment in the last century, attempts were made
to interpret his results in the light of the older theories. This,
however, led to very improbable hypotheses. Fresnel, for instance,
trying to explain Fizeau’s results, had been compelled to admit that
the ether is partially borne along by the water as it flows, and that
this partial displacement varies with the length of the luminous waves
sent through, or that it is not the same for the blue as for the red
waves! A very startling deduction, and one very difficult to admit.
The new law of composition of velocities given to us by Einstein, on
the other hand, immediately and with perfect accuracy explains Fizeau’s
results. They are opposed to the classical law.
The facts, the sovereign judges and criteria, show in this case that
the new mechanics corresponds to reality; the earlier mechanics does
not, at least in its traditional form. Here is something, therefore,
which enables us to see at once the profound truth (scientific truth
being what is verifiable), the beauty, of the doctrine of Einstein:
something which shows us, superbly, how a scientific, a physical,
theory differs from an arbitrary and more or less consistent
philosophical system.
Experience, the supreme judge, decides in favour of the Einsteinian
mechanics against the older mechanics. We shall see further examples;
and we shall not find a single case in which the verdict is the other
way.
* * * * *
Let us turn now to a different matter. The new law of composition
of velocities and the resistance of a velocity-limit equal to that
of light may be expressed in a different language from that we have
hitherto used. Up to this we have spoken only of velocities and
movements. Let us see how these things look when we at the same time
examine the particular qualities of the moving objects, of bodies, of
matter.
Everybody knows that the characteristic feature of matter is what we
call inertia. If matter is at rest, a force is needed to set it in
motion. If it is in motion, it needs a force to stop it. It needs
one to accelerate the movement and one to alter the direction. This
resistance which matter offers to the forces which tend to modify its
condition of rest or movement is what we call _inertia_. But
different bodies may offer a different degree of resistance to these
forces. If a force is applied to an object, it will give it a certain
acceleration. But the same force applied to another object will, as a
rule, give it a different acceleration. A race-horse making a supreme
effort will get along much more quickly under a small jockey than under
a man of fifteen stone. A draught-horse will run more quickly if the
cart it draws is empty than if it is full of goods. You can start a
perambulator with a push that would be useless in the case of a heavy
truck.
When a locomotive with a few coaches suddenly starts, the velocity
imparted to the train during the first second is what we call its
acceleration. If the same locomotive starts, in the same conditions,
with a much longer train, we see that the acceleration is less. Hence
the idea, introduced into science by Newton, of the _mass_ of
bodies, which is the measure of their inertia.
If in our example the locomotive produces in the second case an
acceleration only half as great, we express this by saying that the
mass of the second train is double that of the first. If we find that
the acceleration produced by the locomotive is the same for three
trucks loaded with wheat as for a single truck loaded with metal, we
see that the two trains are equal in mass.
In a word, the masses of bodies are conventional data defined by the
fact that they are proportional to the accelerations caused by one
and the same force. To put it differently, the mass of a body is the
quotient of the force which acts upon it by the acceleration given to
it. Poincaré used to say picturesquely: “Masses are coefficients which
it is convenient to use in calculations.”
If there is one property of bodies which comes within the range of our
senses, a property of which every man has some sort of instinct or
intuition, it is _mass_. Yet careful analysis shows us that we are
unable to define it otherwise than by disguised conventions. Poincaré’s
definition seems paradoxical in its admission of powerlessness. But it
is correct. Mass is only a “coefficient,” a conventional outcome of our
weakness!
Nevertheless, something remained upon which we thought we could
base, if not our craving for certainty—genuine men of science gave
up the idea of certainty long ago—at least our desire for accuracy
of deduction in our classification of phenomena. We believed in
the constancy of mass, of this convenient and so clearly defined
_coefficient_.
Here again, unfortunately, we have to recant—or, perhaps, we should
say fortunately, as there is no pleasure like that of novelty.
The older mechanics taught us that mass is constant in one and the
same body, and is therefore independent of the velocity which the body
acquires. From which it followed, as we have already explained, that,
if a force continues to act, the velocity acquired at the end of a
second will be doubled at the end of two seconds, tripled at the end of
three seconds, and so on indefinitely.
But we have just seen that the velocity increases less during
the second second than during the first, and so on, continuously
diminishing until, when the velocity of light is attained, that of the
moving body can increase no further, whatever force may act upon it.
What does that mean? If the velocity of a body increases less during
the second second, it must be because it offers an increasing
resistance to the accelerating force. Everything happens as if its
inertia, its mass, had changed! Which amounts to saying that _the
mass of bodies is not constant: it depends upon their velocity, and
increases with an increase of velocity_.
In the case of feeble velocities this influence is imperceptible.
It was because the founders of classical mechanics, an experimental
science, had experience only of relatively feeble velocities that they
found that mass was _perceptibly_ constant, and believed they
might conclude that it was _absolutely_ constant. In the case of
greater velocities that is not so.
Similarly, in the case of feeble velocities, in the new mechanics as
well as the old, bodies perceptibly oppose the same resistance of
inertia to the forces which tend to accelerate their movement as to
those which tend to alter the direction, to give a curve to their
trajectories. In the case of great velocities that is not so.
Mass, therefore, increases rapidly with velocity. It becomes infinite
when the velocity equals that of light. No body whatever can attain or
surpass the velocity of light, because, in order to pass that limit, it
would need to overcome an infinite resistance.
In order to make it quite clear, let us give certain figures which
show how mass varies with velocity. The calculation is easy, thanks to
the formula which we have previously seen, giving the values of the
Fitzgerald-Lorentz construction.
A mass of 1,000 grammes will weigh an additional two grammes at the
velocity of 1,000 kilometres a second. It will weigh 1,060 grammes
at the velocity of 100,000 kilometres a second; 1,341 grammes at the
velocity of 200,000 kilometres a second; 2,000 grammes (or double)
at the velocity of 259,806 kilometres a second; 3,905 grammes at the
velocity of 290,000 kilometres a second.
* * * * *
That is what the new theory tells us. But how can we verify it?
It would have been impossible only fifty years ago, when the only
velocities known were those of our vehicles and projectiles, which
then did not rise, even in the case of shells, above one kilometre a
second. The planets themselves are far too slow for the purpose of
verification. Mercury, for instance, the swiftest of them, travels at a
speed of only a hundred kilometres a second, which is not enough.
If we had at our disposal no higher velocities than these, we should
have no means of settling which was right, the classical mechanics
with its constancy of mass or the new mechanics with its assertion of
variability.
It is the cathode rays and the Beta rays of radium which have provided
us with velocities great enough for the purpose of verification. These
rays consist of an uninterrupted bombardment by small and very rapid
projectiles, each of a mass less than the two-thousandth part that of
an atom of hydrogen, and charged with negative electricity. They are
the _electrons_.
The cathode tubes of radium give out a continuous bombardment of these
minute projectiles, charged, not with melinite, but electricity:
far smaller than the shells of our artillery, but animated with
infinitely greater initial speeds. The velocity of “Bertha’s” shells is
contemptible in comparison.
But how was it possible to measure the speed of these projectiles?
We know that electrified bodies act upon each other. They attract or
repel each other. Now our electrons are charged with electricity.
If, therefore, we put them in an electric field, between two plates
connected at the edges by an electrical machine or an induction coil,
they will be subjected to a force that will cause them to change
their direction. The cathode rays, in other words, will change their
direction under the influence of an electric field. The amount of
diversion will depend upon the speed of the projectiles and upon their
mass; that is to say, upon the resistance of inertia which the mass
opposes to the causes which tend to divert it.
But this is not all. The electric charges borne by the projectiles are
in movement, even rapid movement. Now, electricity in movement is an
electric current, and we know that currents are diverted by magnets or
magnetic fields. Therefore the cathode rays will be diverted by the
magnet. This diversion will, like the former, depend upon the velocity
and the mass of the projectile; but not quite in the same way. Other
things being equal, the magnetic diversion will be greater than the
electrical diversion, if the velocity is high. As a matter of fact, the
magnetic diversion is due to the action of the magnet on the current.
It will be greater in proportion to the intensity of the current; and
the current will be more intense in proportion to the height of the
velocity, since it is the movement of the projectile which causes the
current. On the other hand, the trajectory of our little projectiles
will be less influenced by the electrical attraction in proportion as
the velocity of the projectile is great.
Hence it is easy to see that when we subject a cathode ray to the
action of an electric field, then to that of a magnetic field, we may,
by comparing the two deviations, measure at one and the same time the
velocity of the projectile and its mass (related to the known electric
charge of the electron).
In this way we find enormous velocities, rising from a few tens of
kilometres to 150,000 kilometres a second, and even more. As to the
Beta rays of radium, they are still more rapid. In cases they attain
velocities not far short of that of light, and higher than 290,000
kilometres a second. Here are just the velocities we need in order to
test whether or no mass increases with them.
* * * * *
In order to understand clearly the progress of the experiments, it
remains to say a few words about the curious phenomenon of electrical
inertia which is called _self-induction_. When we want to set up
an electric current, we find a certain initial resistance which ceases
as soon as the current begins. If afterwards we want to break the
current, it tends to maintain itself, and we have just the same trouble
to stop it as to stop a vehicle in motion. It is a matter of daily
experience. Sometimes the trolley of a tramcar leaves for a moment the
wire which conducts the current, and we then see sparks. Why? There
was a current passing from the wire to the trolley, and if the trolley
breaks away from the wire for a moment, leaving an interval of air
which obstructs the passage of electricity, the current will not stop.
It has been set going, as it were, and it leaps the obstacle in the
form of a spark. This phenomenon is what we call self-induction.
Self-induction—or “self” as the electrical workers call it—is a real
inertia. The surrounding medium offers resistance to the force which
tends to establish an electric current, and to that which tends to stop
a current already set up; just as matter resists the force which tends
to cause it to pass from rest to movement, or from movement to rest.
There is, therefore, a real electrical inertia as well as mechanical
inertia.
But our cathodic projectiles, our electrons, are charged. When they
begin to move, they start an electric current; when they come to
rest, the current ceases. Besides mechanical inertia, then, they
must also have electrical inertia. _They have, so to speak, two
inertias; that is to say, two inert masses, a real and mechanical
mass, and an apparent mass due to the phenomena of electro-magnetic
self-induction._ By studying the two deviations, electric and
magnetic, of the Beta rays of radium or of the cathode rays, it is
possible to determine the respective parts of each of these masses in
the total mass of the electron. The electro-magnetic mass due to the
causes which we have explained varies with the velocity, according to
certain laws which we gather from the theory of electricity. Hence, by
observing the relation between the total mass and the velocity, we can
see what part belongs to the real and invariable mass and what to the
apparent mass of electro-magnetic origin.
The experiment has been made repeatedly by physicists of distinction.
The result of it is surprising: the real mass is _nil_, and
the whole mass of the particle is of electro-magnetic origin. Here
is something that is calculated to modify entirely our ideas of the
essence of what we call matter. But that is another story.
Physicists then asked themselves—this is what we were coming to, after
clearing the way of various difficulties—whether the relation between
the mass and the velocity of the cathodic projectiles was the same as
that which we found in virtue of the Principle of Relativity.
The result of the experiments is absolutely clear and consistent, and
some of them have dealt with Beta rays corresponding to a mass-value
ten times greater than the original mass. This result is: mass
varies with velocity, and in exact accord with the numerical laws of
Einstein’s dynamics.
Here is a new and valuable experimental confirmation. This in
turn tends to show that classical mechanics was merely a rough
approximation, valid at the most only for the comparatively slight
velocities with which we have to deal in the very restricted course of
daily life.
Thus the mass of bodies, the Newtonian property which was believed
to be the very symbol of constancy, the equivalent of what loyalty
to treaties is in the moral order of things, is now merely a small
coefficient, variable, undulating, and relative to the point of view.
In virtue of the reciprocity which we have described, when there is
question of contraction due to velocity, the mass of an object
increases in the same way, not only if the object is displaced, but if
the observer is displaced, and without any other observer, connected
with the object, being able to detect the difference.
For instance, a measuring rod that moves at a velocity of about
260,000 kilometres a second will not only have its length shortened
by one-half, but will have its mass doubled at the same time. Hence
its density, which is the relation of its mass to its volume, will be
quadrupled.
The physical ideas which were believed to be most solidly established,
most constant, most unshakeable, have been uprooted by the storm of
the new mechanics. They have become soft and plastic things moulded by
velocity.
* * * * *
Further confirmations of the new formula, quite independent of the one
we have just described, have recently been provided by physicists. One
of the most astonishing of these is given in spectroscopy.
As is well known, when we cause a ray of sunlight, admitted through
a narrow slit, to pass through the edge of a glass prism, the ray
expands, as it issues from the prism, like a beautiful fan, the
successive blades of which consist of the different colours of the
rainbow. When we examine closely this coloured fan, we notice certain
fine discontinuities, narrow lines or gaps, in which there is no light.
They look like cuts made with a pair of scissors in our polychrome fan.
They are the dark lines of the solar spectrum. Each of these lines,
or each group of them, corresponds to a special chemical element, and
serves to identify this, whether in our laboratories or in the sun and
the stars.
It was explained long ago that these lines are due to electrons which
revolve rapidly round the nuclei of the atoms. Their sudden changes of
velocity give rise to a wave (like those caused in water when you drop
a pebble into it) in the surrounding medium, and this is one of the
characteristic luminous waves of the atom. It reveals itself in one
of the lines of the spectrum. The Danish physicist Bohr has recently
developed this theory in detail, and has shown that it accurately
explains the various spectral lines of the different chemical elements.
These, I may note, differ from each other in the number and arrangement
of the electrons which revolve within their atoms.
Now Sommerfeld has argued as follows. The electrons which gravitate
near the centre of an atom must have a higher velocity than those
which revolve in its outer part; just as the smaller planets, Mercury
and Venus, revolve round the sun far more rapidly than the larger
planets, Jupiter and Saturn. It follows if Lorentz and Einstein are
right that the mass of the interior electrons of the atoms must be
greater than that of the exterior electrons: appreciably greater, as
the former revolve with enormous velocities. We can calculate that,
in those conditions, each line in the spectrum of a chemical element
must in reality consist of a number of fine lines joined together. This
is precisely what Paschen afterwards (1916) found. He discovered that
the structure of the fine lines is strictly such as Sommerfeld had
predicted. It was an astonishing confirmation of an hypothesis: a proof
of the soundness of the new mechanics.
But that is not all. We know that the X-rays are vibrations analogous
to light, the same in origin, but consisting of much shorter waves, or
waves with a far higher frequency. Hence, while light comes from the
external electrons of the miniature solar system which we call an atom,
the X-rays come from the most rapid electrons—those nearest to the
centre. It follows that the special structure of the fine lines, due
to the variation of the mass of the electron with its velocity, must
be much more marked in the case of the X-rays than in the case of the
spectral lines of light. This, again, was confirmed by experiment.
The figures expressing the observed facts correspond exactly with the
calculations of the new mechanics, as regards the predicted variation
of mass with velocity.
It is therefore settled that the phenomena which take place in the
microcosm of each atom are subject to the laws of the new mechanics,
not the old, and that, in particular, masses in motion vary as the new
mechanics demands.
Experience, “sole source of truth,” has given its verdict.
We are now very far from the ideas which were once prevalent. Lavoisier
taught us that matter can neither be created nor destroyed. It remains
always the same. What he meant was that mass is invariable, as he
proved by means of scales. Now it appears that, perhaps, bodies have no
mass at all—if it is entirely of electro-magnetic origin—and that, in
any case, mass is not invariable. This does not mean that Lavoisier’s
law has now no meaning. There remains something that corresponds to
mass at low velocities. Our idea of matter is, however, revolutionised.
By matter we particularly meant mass, which seemed to us to be at once
the most tangible and most enduring of its properties. Now this “mass”
has no more reality than the time and space in which we thought we
located it! Our solid realities were but phantoms.
The reader must pardon me for whatever difficulties he finds in this
exposition. The new mechanics opens out to us such strange new horizons
that it is worth far more than a rapid and superficial glance. If
you want to see a vast prospect in an unexplored world, you must not
hesitate to do some rough climbing, however breathless it may leave you
for the time.
* * * * *
There is, in fine, another fundamental idea of mechanics, that of
_energy_, which takes on a new aspect in the light of Einstein’s
theory: an aspect which, in turn, is largely justified by experiment.
We saw that a body charged with electricity and in motion makes a
certain resistance to interference, on account of the electrical
inertia which is known as self-induction. Calculation and experiment
show that, if we reduce the dimensions of a body that is charged with
a certain quantity of electricity, without altering the charge, the
electrical inertia increases. As a matter of fact, in our hypotheses,
and if the inertia is entirely electro-magnetic in origin, the
electrons are now merely a sort of electric trails moving in the
propagating medium of electrical and luminous waves which we call ether.
The electrons are no longer anything in themselves. They are merely,
in the words of Poincaré, a sort of “holes in ether,” round which the
ether presses much as a lake makes eddies which check the progress of a
boat.
In that case, however, the smaller the holes in the ether are, the
more important will be the agitation of the ether round them; and,
consequently, the greater will be the inertia of the “hole in ether”
which represents the corpuscle under investigation. What will
follow? We know from measurements we have made that the mass of the
tiny sun of each atom, the _positive nucleus_, round which the
planet-electrons revolve, is greater than that of an electron. If this
mass and the corresponding inertia are electro-magnetic in origin, it
follows that the positive nucleus of the atom is much smaller than the
electron.
Let us consider the atom of hydrogen, the lightest and simplest of
the gases. We know that it consists of one planet only, one single
negative electron revolving round the minute central sun, the positive
nucleus. We know also that the mass of the electron is two thousand
times as small as that of the hydrogen atom. It follows, as we can
calculate, that the _positive nucleus_ must have a radius two
thousand times smaller than that of the electron. Now, the experiments
of the English physicists have proved that the large Alpha particles of
the radium emanation can pass through hundreds of thousands of atoms
without being appreciably diverted by the positive nucleus. We conclude
that the latter is in reality much smaller than the electron, as theory
predicted.
All this irresistibly compels us to think that the inertia of the
various component parts of atoms—that is to say, of all matter—is
exclusively electro-magnetic in origin. There is now no matter. There
is only electrical energy, which, by the reactions of the surrounding
medium upon it, leads us to the fallacious belief in the existence of
this substantial and massive something which hundreds of generations
have been wont to call “matter.”
And from all this it also follows, by calculation and by the simple and
elegant reasoning of Einstein, of which I here convey only the faintest
adumbration, that mass and energy are the same thing, or are at least
the two different sides of one and the same coin. There is, then, no
longer a material mass. There is nothing but energy in the external
universe. A strange—in a sense, an almost spiritual—turn for modern
physics to take!
According to all this the greater part of the “mass” of bodies must be
due to a considerable and concealed internal energy. It is this energy
which we find gradually dissipated in radio-active bodies, the only
reservoirs of atomic energy which have as yet opened externally.
If this is true, if energy and mass are synonymous, if mass is merely
energy, it follows that free energy must possess the property of mass.
As a matter of fact, light, for instance, has mass. Careful experiments
have shown that when a ray of light strikes a material object, it
exerts upon it a pressure which has been measured. Light has mass;
therefore it has weight, like all masses. When we come to consider the
new form given by Einstein to the problem of gravitation, we shall see
a further and beautiful proof that light has weight.
We can calculate that the light received from the sun by the earth in
the space of a year is rather more than 58,000 tons. It seems very
little when one thinks of the formidable weight of coal that would be
needed to maintain our globe at the temperature at which the sun keeps
it—in the event of a sudden extinction of our luminary.
The reason for the difference is that, when we produce heat from a
certain amount of coal, we use only a small proportion of its total
energy, its chemical energy. Its intra-atomic energy is inaccessible
to us. It is a pity, as otherwise we should need only a few ounces of
coal to supply heat for a whole year to all the towns and workshops
of England! How many problems that would simplify! When humanity
emerges from the ignorance and the clumsy barbarism in which it lives
to-day—that is to say, in some hundreds of centuries—this will be
accomplished. Yes, it will one day be done. It will be a glorious
spectacle, one in which we may justly rejoice in advance.
Meantime, our sun, like all the other stars, like every incandescent
body, loses its weight in proportion as it radiates. But this happens
so slowly that we need not fear to see it disappear at some early date,
like the ephemeral things which die because they gave themselves too
freely.
* * * * *
To finish with Einstein’s mechanics, let me reproduce a very suggestive
application of these ideas about the identity of energy and mass.
There is in chemistry a well-known elementary law which is called
“Prout’s Law.” It states that the atomic masses of all the elements
must be whole multiples of the mass of hydrogen. Since hydrogen has the
lightest atoms amongst all known bodies Prout’s Law started from the
hypothesis that all the atoms are built up of a fundamental element,
the atom of hydrogen. This supposed unity of matter seems to be more
and more confirmed by the facts. On the one hand, it is proved that the
electrons which come from different chemical elements are identical. On
the other hand, in the transformation of radio-active bodies we find
heavy atoms simplifying themselves by successively emitting atoms of
helium gas. Lastly, the great British physicist Sir Ernest Rutherford
showed in 1919 that by bombarding the atoms of nitrogen gas, in
certain circumstances, by means of radium emanation, we can detach
hydrogen atoms from them. This experiment, the importance of which has
not been fully realised—it is the first instance of transmutation
really effected by man—also tends to prove the soundness of Prout’s
hypothesis.
Yet, when we accurately measure and compare the atomic masses of the
various chemical elements, we find that they do not strictly conform to
Prout’s Law. For instance, while the atomic mass of hydrogen is 1, that
of chlorium is 35·46, which is not a whole multiple of 1.
But we can calculate that, if the formation of complex atoms from
hydrogen upwards is accompanied, as is probable, by variations of
internal energy, as a consequence of the radiation of a certain amount
of energy during the combination, it necessarily follows (since the
lost energy has weight) that there will be variations in the mass of
the body composed, and these will explain the known departures from
Prout’s Law.
* * * * *
In our somewhat hurried and informal excursion into the bush of the new
facts which confirm the mechanics outlined by Lorentz and completed
by Einstein our progress has been rather difficult. It is because,
since we could not use terminology and technical formulæ which would
be unsuitable in this work, we have had to be content with bold and
rapid moves into the districts we wished to reconnoitre. Perhaps they
have sufficed to enable the reader to understand what a revolution
in the very bases of science, what an explosion amidst its age-old
foundations, the brilliant synthesis of Einstein has caused. New light
now streams upon all who slowly climb the slopes of knowledge: upon all
who, wisely renouncing the desire to know “why,” would at least learn
the “how” in many things.
A little before his death, foreseeing, with the intuition of genius,
that a new era opened in mechanics, Poincaré advised professors not to
teach the new truths to the young until they were steeped to the very
marrow of their bones in the older mechanics.
“It is,” he added, “with ordinary mechanics that their life is
concerned: it is that alone that they will ever have to apply. Whatever
speed our motor-cars may attain, they will never reach a speed at
which the old mechanics ceases to be true. The new is a luxury, and
we must think of luxuries only when it can be done without injury to
necessaries.”
I would appeal from Poincaré’s text to the man himself. For him this
luxury, the truth, was a necessary. On the day in question, it is true,
he thought of the young. But do men ever cease to be children? To that
the master, too early taken from us, would have replied, in his grave,
smiling manner: “Yes—at all events, it is better to suppose so.”