parallel lines across the drumhead, would keep even all the way. Now
lay a bullet on the center of the drumhead. The rubber sags down and
stretches, most in the middle, least at the edges. The “parallel”
lines are no longer equidistant. The squares are no longer equal. The
lines are no longer of the same length. If now we repeat our worm race
we shall find that those worms following lines close to the weight
have to go down hill and up again and so travel a greater distance to
traverse the same number of squares than those following lines nearer
the edge which lie comparatively flat and are nearly as short as
before. Consequently the worms will be slowed up in proportion to their
nearness to the center and the row of their heads will be swung around
at an angle to their former frontage.
We might “explain” this by assuming that the worms on seeing the bullet
to one side were drawn by their curiosity a little toward it, those
nearest of course being drawn the most. Or if we had got beyond this
crude animistic method of explanation we might assume that the bullet
was attached to the head of each worm by an invisible lariat which
being pulled by the bullet drew the worms more or less to one side, the
shorter the lariat the stronger the pull. Or if we had outgrown this
crude mechanical method of explanation we might assume the existence
of a “force” in the lead which in some mysterious manner attracts the
heads of the worms inversely as the square of their distance. But
instead of inventing a wormhead psychology or an invisible cord or an
incomprehensible force is it not simpler to consider the space between
and to suppose that the lines to be traversed are lengthened in the
neighborhood of the weight?
Now these four successive methods of explanation have been used
to account for gravitation. First it was assumed by the ancient
Babylonians and Hebrews that the sun and stars were living beings, gods
or angels, moving of their own volition around the earth, or at least
that each was guided in its orbit by its particular god or angel. The
later Greeks of Ptolemy’s time supposed the heavenly bodies to be set
in concentric crystal spheres and so revolved; I presume by somebody
turning a crank behind the scenes. Then came Newton and said: “Let’s
discard the Ptolemaic spheres and all mechanical connection and assume
a _force_ of gravitation attracting all bodies in proportion to
their masses and inversely proportional to the squares of the distances
separating them.” Now comes Einstein and says: “Let’s discard this
hypothetical force and simply assume that the field of time and space
traversed by a moving body is altered if there is another body in
the vicinity.” In Einstein’s view gravitation is not a force; it is
a distortion of space and time in the presence of matter. A comet
sweeping past the sun cannot pursue a straight course, as it could in
interstellar space, but follows a curved path about the sun which is
for the comet the shortest way it can go under the circumstances.
So, too, a row of light waves coming from a distant star keeps an
even front as they pass through empty space but as they come close
to the sun they find their paths impeded, or, we may say, stretched.
Those going nearest the sun are slowed up the most; those farthest off
the least. Consequently the wavefront is slued around a bit and the
direction of the ray is slightly altered.
If now light waves have difficulty getting past the sun we should
expect that they would experience like difficulty getting away from the
sun. They would be slowed up a bit by its gravitational pullback. The
frequency would be reduced; the interval of time between wave-crests
lengthened. This means, in the case of sound, lowering the pitch. Touch
your finger to the turntable of your phonograph and you flat the tone.
In the case of light, it means change of color toward the red. This
effect, according to Einstein, should be, but has not been, observed.
“If Einstein’s third prediction is verified,” says Sir Oliver Lodge,
“Einstein’s theory will dominate all higher physics and the next
generation of mathematical physicists will have a terrible time of
it. For university courses and for all practical purposes we shall
have the Galilean and Newtonian dynamics but they will reign as a
limited monarchy and sooner or later the Einstein physics cannot fail
to influence every intelligent man. If these complications are to
come into science we must leave them to the younger men. I hope that
gravitation, now that it has begun to interact with light, will begin
to give up its secrets, but in my time I must be content to get secrets
out dynamically and leave transcendental methods to others.”
One English scientist, Thomas Case, writes to _The Times_ to
protest that it would have been in much better taste for the Royal
Society to have adjourned its discussion “before bringing into question
the reputation of Newton, who was President of the Royal Society for
the last twenty-five years of his life and raised the society to the
acme of its fame.”
WHO IS EINSTEIN?
Albert Einstein was born in Germany in 1874. He early showed the bent
of his genius and at the age of twelve, when his fellow pupils were
plodding along with their daily tasks, he was plunging through works
of higher mathematics borrowed from his teacher. He was only eighteen
when he conceived the outlines of his theory and ten years later it
was ready to give to the world. He left Germany for Switzerland at
the age of sixteen and became naturalized as a Swiss citizen. His
first academic position was the Professorship of Mathematical Physics
at the Zürich Polytechnic. Then the founding of the Kaiser Wilhelm
Academy for Research at Berlin gave him opportunity to work out his
theories undisturbed by other duties. Shortly before the war he was
called to Berlin to succeed the famous Dutch physicist, Professor
van’t Hoff in the Academy. The object of this institution was the
same as Carnegie had when he founded his institution for scientific
research at Washington, which was to seek out the exceptional man
wherever he may be found and set him at his peculiar tasks. At Berlin
Einstein receives a salary of $4,500 and has nothing to do but sit and
think. This he continued to do all through the five years of war and
revolution as quietly and persistently as Kant at Königsberg during
the wars and revolutions of a century before. Or as Archimedes at the
siege of Syracuse who was absorbed in drawing geometrical figures in
the sand--his blackboard--when a Roman soldier ran him through with a
spear. On two occasions he took part in the world-struggle going on
about his study, both actions greatly to his credit. In the beginning
he refused to sign the manifesto of the German men of science denying
all the charges against Germany, and at the time of the armistice he
signed an appeal in favor of the revolution. He is an ardent Zionist
and has promised to aid the Hebrew university which is to be founded at
Jerusalem.
According to tradition, Isaac Newton was led to his theory of
gravitation by observing an apple falling from a tree in his garden.
The newspaper correspondents start a similar tradition by reporting
that Einstein got his theory of gravitation by observing a man falling
from the roof of a building in Berlin. Now a man has the advantage
of an apple in that he is able to tell his sensations. When Dr.
Einstein, who had seen the accident from his library window in the
top story of a neighboring apartment house, reached the spot he found
the man had hit upon a pile of soft rubbish and had escaped almost
without injury. Asked how it felt to fall he told Dr. Einstein that
he had no sensation of downward pull at all. This led Dr. Einstein to
consider whether the relativity theory, which he had applied only to
the case of uniform motion in a straight line, could not be extended
to difform or accelerated motion by gravitation. So the special
relativity theory which he had enunciated in 1905 developed ten years
later into a generalized relativity theory (_Verallgemeinerten
Relativitätstheorie_).
HOW TO LOSE WEIGHT
A man falling out of an airplane is obeying a natural impulse, namely,
the force of gravitation. So long as he does not resist he is free
as air, light as a feather, and altogether comfortable. He can look
down with complacency and contempt on the poor mortals below him who
are trying to stand up against this natural impulse and laboriously
dragging one foot after another as they crawl about the earth when they
might be flying through space without effort as he is. It is only when
he tries to stop his free fall by bumping against the ground that he
gets into trouble on account of gravitation. It was in this way that
the Calvinists, who were a sort of mathematical theologians, conceived
of the fall of man. The sinner is simply obeying the force of natural
depravity, namely, moral gravity, and so long as he is conscienceless
and does not consider his inevitable end he has no knowledge of the
moral law and is quite happy in his downfall.
A person falling freely loses all his weight. His hat does not press
down on his head. His feet do not press down on his shoes. If he lets
go of his walking-stick it does not “fall down” at his feet. It stands
upright and simply travels along with him. For, as Galileo showed when
he dropped his big and little cannon ball off the Leaning Tower of
Pisa, all bodies fall with the same speed.
If he were in a falling elevator with an opaque door he would not
know he were falling unless he surmised it from the _absence of
gravitation_ as evidenced by his own feeling of lost weight and the
queer behavior of the objects in the car. He might fall all his life
and never find it out. The law of gravitation is like criminal law; you
don’t feel it till you come into conflict with it.
Or if our illustration requires too tall a skyscraper, let us imagine
that a comet as it flies by knocks a chip off the earth with a group
of people on it. This terrestrial fragment, cast loose in space, gets
caught by the attractive force of some gigantic and distant star and
falls toward it with ever-increasing velocity for thousands of years.
The inhabitants of this errant orb would never know it from their own
feelings or any observations they could make _on their own little
world_. Does that seem incredible to you? Then tell me how do you
know but this our world is such a planet and together with the solar
system has been falling for thousands of years toward some center of
attraction? Astronomers, indeed, say that we are moving at tremendous
speed toward Canis Major, in other words that the world is going to the
dogs.
All this means that uniformly accelerated motion, such as gravitation
imparts to a freely falling body, is, like uniform translatory motion,
a question of relativity and cannot be discovered by an observer
carried along by such movement.
The idea that uniform translation, like the moving train we have
considered, is merely relative motion, is an old idea and not hard
to understand or accept. But when we try to extend the principle of
relativity to acceleration, that is, to a rate of motion that is
continuously increased or retarded, we get a new and revolutionary
conception of the universe and are drawn into some very startling
conclusions. Einstein took this step five years ago and that is what
has caused the present excitement. For Einstein when he once gets
hold of an idea follows it wherever it leads him with the undaunted
determination of a Nantucket sailor towed by a harpooned whale. It was
a whale of an idea that he harpooned in 1915 and it carried him into
strange waters. It led directly to a contradiction or correction of
one of the two fundamental postulates which he had laid down as the
foundation of his theory of the universe in 1905, namely, that the
velocity of light in space is a constant. But he promptly abandoned
this idea with cheerful nonchalance in favor of the new notion that the
velocity of light is affected by gravitation.
A SUBSTITUTE FOR GRAVITY
Let us then follow Einstein and apply his Principle of Equivalence to
accelerated motion and see what it leads to. Imagine yourself shut up
inside a closed chamber, like an elevator car, somewhere out in space
away from the gravitational forces of the earth or sun. Suppose this
chamber to be rising with a constantly increasing velocity. We can,
if we want to be definite about it, assume that the chamber is a big
shell pulled up by a cable coiling around a conical windlass that hauls
it up faster all the time. Or we can assume that it is propelled from
behind by the continuous backfire of explosives, like the rocket which
Professor Goddard proposes to send to the moon. All we need is some
force, not gravitation, capable of giving the chamber every second an
additional velocity of thirty-two feet a second. Now the point is that
if you were in such an upward-moving chamber you would not know but
what you were resting on the earth. Everything would behave exactly
the same. If you now weigh one hundred and fifty pounds on the scales,
that is, if your shoe soles press down with that force, the floor of
the rising chamber would press upward with that same force and you
would not know the difference. If you let loose a ball from your hand
the floor would rise up to meet it and it would appear to fall. If you
threw the ball upward with a velocity greater than the velocity of the
chamber at the moment, the ball would rise, but since the velocity of
the chamber was constantly increasing the floor would gain on the ball
and catch up with it. This would look to you just the same as when on
earth you threw a ball into the air and it fell back to the ground,
drawn, as you are accustomed to think, by “the force of gravitation.”
But here we have no “force,” but merely a mode of motion.
Under such circumstances it would seem that all Nature conspired to
keep you in the dark. You appeal to the ether, that supposedly stable
and stationary medium that fills all space, but that also fails you.
You try the Michelson-Morley experiment to see if you are moving
through the ether or at rest on the earth but your apparatus expands or
contracts just enough to deceive you.
You now try observing horizontal rays of light but they seem to bend;
that is, a beam of sunshine entering a pinhole on one side of your
_camera obscura_ will not strike the wall at a spot exactly
opposite but a little below it, if you have instruments sufficiently
delicate to show this. You try vertical rays of light in this fashion:
You examine with the spectroscope rays of light coming from two sources
below (behind) your instrument, one at a distance and the other nearer.
Now since you are moving away with increasing speed, the light from
the farther source will have to take longer strides to catch up. Or
in other words, its frequency will be reduced and it will be shoved
toward the red end of the spectrum where the longer waves are. You will
have noticed that when a whistling train rushes past the train you are
on, the whistle as it comes toward you is raised in pitch (decreased
wave-length) and as it recedes from you is lowered in pitch (increased
wave-length).
Now, says Einstein to himself, if my Principle of Equivalence is
correct and there is no difference between (1) weight and (2) the
accelerated upward movement of an observer, then all the optical
effects that I have thought out in the second case must apply to the
first, that is, to gravitation. It must follow that a ray of light
passing through a gravitational field will be bent out of its course
as though it were attracted by the heavy body. This prediction has
been verified. It must further follow that light proceeding from a
heavy body like the sun or a star will be held back or slowed up by the
attraction of gravitation, and the spectral lines will be displaced
toward the left as compared with the same lines in the spectrum of
an earthly light. Now such displacement has been observed in stellar
spectra but it does not seem to be of the right value to satisfy
Einstein’s equation and it has not been observed in sunlight.
The remarkable thing about it is that Einstein, by following a line of
reasoning somewhat like that which I have crudely outlined, not merely
supplied an explanation for phenomena that had been observed but could
not be explained (such as the discrepancy in the orbit of Mercury) but
he provided in advance the explanation for phenomena that had never
been observed until he directed attention to it (such as the deflection
of starlight by the sun). Sir Oliver Lodge says of this:[5]
Before Einstein’s prediction nothing of the kind had been seen,
nothing of the kind had been looked for, nor, so far as it is known,
had such an amount of deflection been suspected.
Whatever may ultimately be thought of the validity of Einstein’s views
as a whole it is evident that he has worked out a mathematical method
of unprecedented power and wide usefulness.
Professor Bumstead of Yale says:
Einstein’s theory is important in that it exemplifies a method which
is in many respects new in theoretical physics and which may prove to
be a very powerful method for advancing scientific knowledge. There
was no idea that the prediction of the bending of light would fix
up Mercury’s perihelion and incidentally explain a two-century old
astronomical difficulty. That came straight out of a blue sky.
MECHANICAL VERSUS MATHEMATICAL MINDS
We sometimes hear it said that “Einstein has overthrown Newton’s theory
of gravitation.” That is impossible because Newton did not have any
theory of gravitation. He merely laid down the law of gravitation. He
told how bodies behaved toward their neighbors; he did not tell why.
Newton was not content with the idea of action at a distance through
empty space and he tried to explain gravitation by the pressure of the
ether on material bodies but he was not satisfied with the results
and did not publish them. In the 234 years since many men have tried
their hands at devising some sort of machinery that will “explain”
gravitation. For human beings are like Toddie of “Helen’s Babies”
and want to have the watch opened so they can “see the wheels go
wound.” At least Anglo-Saxons have that desire. Poincaré, the French
physicist, said this is the distinction between the Anglo-Saxon and
Latin minds; the former are uneasy until they can imagine a mechanical
model to represent natural phenomena, the latter are satisfied with
a mathematical formula expressing the action. The ether, which was
invented to explain light, also required “explanation.” Lord Kelvin
imagined it to consist of spinning tops which have a sort of mobile
stability. Sir Oliver Lodge has filled it with a complicated structure
of interlocking geared wheels to account for electro-magnetic action.
These are typical Anglo-Saxon modes of thinking. On the other hand,
Einstein, who, in spite of his Hebrew blood and German training, has
preëminently what Poincaré claims as the Latin temperament, does not
have any use for the ether and does not care at all whether he can
“picture” the fourth dimensions on paper or not.
Now some of us are excessively Anglo-Saxon in our attitude toward
mathematics. It is with a fellow-feeling for such folks that I have
filled this little volume with such crude and absurd analogies as
trains and elevators and projectiles flying through space and Coney
Island mirrors. To the mathematically minded such illustrations are not
simplifications but complications, not representations but caricatures.
Mathematics is the proper language of physics as the five-barred staff
is the proper language of music. Ask a musician to explain a symphony
in plain everyday English and he cannot do it, though he carry the
Oxford Dictionary in his head. He can have the music played for us
or he can show us the printed score but he could never convey it in
ordinary language however long he might be willing to talk or we to
listen. But we must not do the musician or the mathematician the
injustice to suspect that his notions are hazy or absurd because he
cannot explain (_i.e._ translate) them to us.
Nor should we assume that the new ideas, because they are more
difficult for us to grasp, are necessarily more complicated or
extravagant than the old. A friend of mine who is familiar with both
tells me that Einstein’s papers are easier reading than Newton’s
“Principia.”
The aim of science is simplification through generalization and
this is the widest generalization yet attempted. It promises to
bring gravitation into relationship with other forces. One great
generalization, the law of the conservation of energy worked out by
Joule and others in the forties, brought heat and work and chemical
power all into one simple system. Clerk Maxwell in the seventies
brought together in one beautiful formulation all the diverse phenomena
of light, electricity and magnetism.
But gravitation has always stood out against any such league of natural
forces. It refused to come into the combine. It remained unique,
independent, irreducible, unalterable and inexplicable. Everything
else is correlated and interactive. Heat destroys magnetism, magnetism
produces electricity; electricity dissolves chemical combination;
chemical combination produces heat; heat causes motion; motion makes
magnetism; magnetism produces heat; and so on in endless round, each
affecting all the others. Different substances behave very differently;
one is more easily heated than another; some are readily magnetized or
electrified, others are not so susceptible; certain elements rush into
each others’ arms, others cannot be forced into combination.
But gravitation seemed indifferent to all these things; it showed no
prejudices or preferences. It attracted with equal force all sorts of
substances, no matter whether they were hot or cold, shiny or black,
moving or still, electrified or magnetized or neither. Other forces and
effects too required time for action at a distance. Sound travels at
the rate of 1,100 feet a second in ordinary air. Light travels at the
rate of 186,337 miles a second in a vacuum. But the force of gravity
seemed not to require any time but to be everywhere, acting all the
while, and nothing could shield it off or shut it out or in any way
interfere with it. The substance or mass of a body as measured by its
weight (the gravitational pull of the earth) was always identical with
its mass as measured by its inertia (its resistance to being set in
motion). All the energies are interchangeable. All other forces could
be reduced or increased, annulled or brought into effect at will.
Not so gravitation. Any bodies of a certain mass placed at a certain
distance apart are always drawn by the same attraction. That is,
gravitation is affected by nothing except geometrical relationships.
This naturally leads us to suspect that gravitation is nothing but
a geometrical relationship, that it is somehow a peculiarity of
space itself. If so, our demand of the physicist that he show us
gravitation,--drag out this mysterious force from its hiding-place
and let us see it--is altogether irrational. It is like a blind man
hunting in a dark cellar at midnight for a black cat that isn’t there.
The geometrician tells us that the internal angles of any triangle
are equal to two right angles. Shall we ask him, what is the force
that makes it so? Shall we refuse to ride on a trolley car until the
electrician can answer our persistent question; “but what _is_
electricity?” When we ask such a question we are really asking him to
tell us what electricity _is not_. To show us what electricity is
he can keep his mouth shut and simply point to the dynamo that produces
it, the wire that conveys it and the motor that consumes it. But what
we secretly mean is that he show us a mechanical model that imperfectly
imitates some of the actions of electricity or a mathematical formula
that will calculate its effects.
Now Einstein seems in the way of making gravitation the foundation of a
new system of geometry. Instead of “explaining” gravitation in terms of
something else he will explain other things in terms of gravitation,
or rather of his space-time manifold of which gravitation is one of the
properties.