[Illustration: _Wide World Photos._
DR. ALBERT EINSTEIN IN HIS STUDY AT BERLIN.]
EASY LESSONS IN
EINSTEIN
A DISCUSSION OF THE MORE
INTELLIGIBLE FEATURES OF
THE THEORY OF RELATIVITY
BY
EDWIN E. SLOSSON, M.S., Ph.D.
_Literary Editor of The Independent, Associate in the Columbia School
of Journalism. Author of “Great American Universities,”
“Major Prophets of To-day,” “Six Major Prophets,”
“Creative Chemistry,” etc._
_With an Article by Albert Einstein
and a Bibliography_
ILLUSTRATED
[Illustration]
NEW YORK
HARCOURT, BRACE AND HOWE
1920
COPYRIGHT, 1920, BY
HARCOURT, BRACE AND HOWE, INC.
THE QUINN & BODEN COMPANY
RAHWAY, N. J.
_Deepest of all illusory Appearances, for hiding Wonder, as for
many other ends, are your two grand fundamental world-enveloping
Appearances, Space and Time._--CARLYLE.
_Henceforth Space in itself and Time in itself sink into mere
shadows and only a kind of union of the two can be maintained as
self-existent._--MINKOWSKI.
A PREFATORIAL DIALOGUE
(The Purpose of which is to Prevent the Prospective Reader from buying
the Book under False Pretenses)
SCENE: A street car in uniform movement of translation in any
direction.
TIME: The present.
_The Reader_: (looking over the top of a morning paper):
Here’s something queer--a whole page taken with a new discovery
in physics--“Eclipse Observations Confirm Einstein’s Theory of
Relativity.” Anything about it in your paper?
_The Author_: Yes. Here’s a cartoon on it by McCutcheon.
_The Reader_: Must be something to it then. McCutcheon always
knows what’s news. (Reads on with audible fragments) “Most sensational
discovery in the history of science”--“Greatest achievement of the
human intellect”--“Upsets Galileo, Newton, and Euclid”--“Revolution
in philosophy and theology.” It looks as though I ought to know
something about this, doesn’t it?
_The Author_: I think you will have to sometime. And you might as
well do it now and get it over with.
_The Reader_: (running down the column and hitting the high
spots): “Parallel lines meet”--“a man moving with the speed of light
never grows old”--“gravitation due to a warp in space”--“length of a
measuring stick depends upon direction of its motion”--“mass is latent
energy”--“time as a fourth dimension”--why, the man is crazy, isn’t he?
_The Author_: Well, definitions of insanity are so uncertain that
it is not safe to say who is crazy. But it seems there’s method in his
madness--otherwise how could he have hit upon the exact extent of the
sun’s attraction on light?
_The Reader_: (Picks up his paper and reads aloud with
concentrated attention) “Postulate I. Every law of nature which holds
good with respect to a coördinate system K must also hold good for
any other system K′, provided that K and K′ are in uniform movement of
translation.” Say, do you know anything about this business?
_The Author_: Well, yes, a little. I have followed the
controversy--at a safe distance--for a number of years.
_The Reader_: Can you tell me in plain language what it is all
about?
_The Author_: Yes. Just that. I can tell you what it is
_about_, though I can’t tell you what it _is_. Einstein says
that there are only twelve men in the world capable of understanding
his latest paper.
_The Reader_: Are you one of the twelve?
_The Author_: No, nor the thirteenth. But without plunging into
the mathematics of it, we might talk over some of the interesting
aspects of the theory of relativity and in the end I could put you on
track of the twelve so you could read up on the subject if you liked.
_The Reader_: All right. That’s fair. This is a slow car anyhow.
Go ahead.
_The Author_: (See following pages)--
EASY LESSONS IN EINSTEIN
“_A warp in nature has been found,_
_No line is straight, no circle round;_
_For Isaac Newton had unsound_
_Ideas of gravitation._”
Why is it that our newspapers are sending out their reporters to
interview astronomers as well as actresses and devoting pages to
speculations on the nature of space and time as well as on the state
of the market? It is--to get at the bottom of it--merely because a few
photographs taken during the eclipse of the sun on May 29, 1919, by
two telescopes, one at Sobral in northern Brazil and the other on the
island of Principe off the west coast of Africa, showed an abnormal
shift of less than one-324,000th of a right angle in the position of
the stars. When these photograph films were laid over films taken
before the eclipse it was found that the star-images about the darkened
disk of the sun did not exactly coincide with the images when the sun
was not in their midst. Measured with a micrometer the displacement of
the stars from their ordinary positions was found to be 1.60 seconds of
arc on the African plates and 1.98 seconds on the Brazilian plates.
Average these two observations and you get 1.79. This is extremely
close to the 1.73 predicted by Professor Einstein of Berlin and twice
as large as the deflection calculated according to Newton’s law of
gravitation which would be .87 of a second.
When the announcement of this result was made at the meeting of the
Royal Society of London on November 6 all eyes were turned toward Sir
Oliver Lodge, for last February he had been rash enough to express the
hope, if not the prediction, that the results of the eclipse expedition
would support Newton rather than Einstein. But instead of taking part
in the discussion Sir Oliver got up and walked out. It was suspected
that he had “gone off mad,” as we Americans would put it, because the
starlight would not follow his preferred path. But he put a stop to
any such rumors by a letter to _The Times_ in which he explains
that his departure was not due to any dissatisfaction with the universe
but to the necessity of catching the 6 o’clock train. He frankly
acknowledges that “the eclipse result is a great victory for Einstein;
the quantitative agreement is too close to allow much room for doubt”
but he adds “a caution against a strengthening of great and complicated
generalizations concerning space and time on the strength of this
splendid result: I trust that it may be accounted for, with reasonable
simplicity in terms of the ether of space.”
This caution is wise, but we cannot hold our breath till 1922, when
the next eclipse comes, to see if these observations are verified and
we may in the meantime consider some of the implications of Einstein’s
theory of relativity.
Sir Joseph Thomson, President of the Royal Society, in making the
momentous announcement in the session of the Society, said:
If his theory is right, it makes us take an entirely new view of
gravitation. If it is sustained that Einstein’s reasoning holds
good--and it has sustained two very severe tests in connection with
the perihelion of Mercury and the present eclipse--then it is the
result of one of the highest achievements of human thought. The weak
point in the theory is the great difficulty in expressing it. It would
seem that no one can understand the new law of gravitation without a
thorough knowledge of the theory of invariants and of the calculus of
variations.
What is this theory of relativity and why is it so important? The
mathematics of it are too much for most of us, but we can get some
notion of it by a familiar illustration.
Suppose you wake up some morning in a Pullman berth and look out of the
window to see where you are. You find your view blocked by a passing
train on the next track. Now if you do not feel any jar of your car
and cannot catch sight of the landscape beyond the other train you
cannot tell whether (1) your train is moving forward and the other
train is standing still, or (2) your train is standing still and the
other train is moving backward, or (3) whether both trains are moving
in opposite directions, or (4) whether both trains are moving in the
same direction, but your train faster. It is obvious that the trains
are getting past one another. You can measure their speed of parting
as accurately as you please. But all you can perceive is the relative
motion of the two trains. You begin to wonder whether there is any such
thing as absolute motion; whether there is any real difference between
rest and motion. Is there any possible way of telling whether your
train is in motion or not if all you can see out of the window is some
object that itself be moving? Suppose the windows were all curtained,
how could you find out whether you were moving forward or backward or
standing still?
You discuss this curious question with your fellow passengers at the
breakfast table and one of them makes the brilliant suggestion that
it might be possible to determine the absolute motion of the car by
reference to the air. If the car is moving forward the air would
stream from front to rear and the reverse if it were moving backward.
“Suppose,” says the ingenious experimentalist, “that you stand at
one end of the car and I at the other. We will shout at each other
alternately and time the passage of the sound with our stop watches.
Since sound is carried by air waves it will take longer for the shout
to go against the air current than with it, and from that measurement
it might be possible for us not only to determine which way the car
is moving but how to calculate how fast it travels, assuming, of
course, that there is no wind blowing.” That strikes you as a crucial
experiment, but you point out one possible difficulty, that the doors
at the ends of the car may be closed and the air inside is being
carried along with the car, so no difference would be observable in
the speed of the sound even though the car were moving. “All right,”
replies your scientific friend, “we will make a preliminary test to see
if the enclosed air is carried along with the car, and if we find that
it is not then we will try the second experiment with the sound signals
to see which way the air current is moving. These two experiments must
settle it, for either the air is moving with the car or it is moving
through the car. Can you conceive of any other possibility than these
two?” No, you cannot, so you proceed to try the two experiments. First
you visit both ends of the car and find both doors open; the air then
is not being carried along with the car. You turn then with confidence
to the second experiment and you find, of course, that there is a
difference in the speed of sound whether it moves with the air drift or
against it.
There might, I admit, be practical difficulties in the way of carrying
out such a delicate experiment on a moving train, but we need not
bother with them, for probably the current of air through the car would
be so strong as to blow your hat out of the back door and that would
settle the question to your satisfaction--or at least it would settle
the question in the affirmative.
But imagine your amazement if this second experiment should give
negative results like the first one; if you could detect no difference
in time whether the sound was sent forward or back or across the car.
You would then have proved by experiment (1) that the air did not
move with the car and (2) that the air did not move through the car.
You might suppose from this that your car is at rest, but suppose the
people on the other train passing yours tried the same experiments and
got the same result, namely, that they, too, were at rest as regards
the air. You would then be in a quandary, for your two indisputable
experiments had apparently given contradictory results. You might get
out of it by saying that there was no air, but if not what carried the
sound waves--and the hat?
CONTRADICTORY EXPERIMENTS
Now this is the quandary in which physicists have been in for the last
thirty-three years. Is there any way of _discovering_ absolute
motion among the heavenly bodies? We can observe and measure with
great accuracy their relative motion. The sun is seen to pass across
the sky from east to west and man at first assumed that the earth was
still and the sun went around it. This is the natural and instinctive
assumption, for when you first glance out of your Pullman window you
get the impression that the other train is the moving one. But for
the last three hundred years it has been the fashion to assume the
earth was moving and not the sun. That assumption has the advantage of
simplifying the calculations of the astronomers, though I never could
see why we should have to give up our simple notions of sunrise and
sunset to save them a little trouble figuring.
The earth moves--if it does move--so quietly and silently that we feel
no jar or engine-beat to tell us of its motion. If the earth were
perpetually shrouded by clouds could we find out its motion through
space or even its rotation? And do we actually get any proof on this
point from observation of the heavenly bodies? We see them moving about
relatively to each other and we can represent their movements most
easily by supposing that the moon goes around the earth and that the
earth and the rest of the planets go around the sun. But is this whole
solar system in motion? So it seems when we compare it with the stars.
But who knows if the solar system and all the visible stars are not
altogether moving off through space at the rate of a mile or a thousand
miles a second? How can we tell unless we have something that is still
and fixed to measure the motion by?
It seemed until recently that we had such a fixture, the ether. We know
of the sun and stars only from the light that comes from them to us.
Light, as we can prove by simple experiments, consists of wave motion.
Now, can you have wave motion without something to wave? Sound waves
are conveyed by air but there is no air between the earth and the sun.
So as nothing could be found to fill this empty space scientists had
to invent something to satisfy their sense of the fitness of things.
The ether was the product of their excogitations. It was a British
invention, devised in the Royal Institution, whence have come so many
useful theories and discoveries.
The ether, as Salisbury said, is simply the nominative of the verb “to
undulate.” It was conceived of as a sort of transparent jelly filling
all space, more rigid than any solid, more frictionless than any fluid,
more easily penetrated than any gas. It must be more elastic than steel
and yet so rarefied that ordinary matter passes through it without the
slightest effort. The ether is supposed to slip between the particles
of the rushing earth as the wind blows through the branches of a tree.
For many years after its invention the ether had nothing to do
except to carry light about from one place to another. But when
the electro-magnetic waves of the wireless telegraph were produced
something was needed also to carry them and this new task was laid upon
the shoulders of the uncomplaining ether. When Röntgen discovered the
X-rays, whose waves are 10,000 times shorter than the shortest light
waves, these were turned over to the ether to run. In fact, it got so
that whenever a physicist found any action that he could not explain by
ordinary matter he said: “Let the ether do it,” and that hypothetical
substance apparently answered every purpose until it came to this
question of relative motion.
Now whatever we may think about the ether it would seem that if there
is any such thing filling all “empty” space we might use it for
measuring the motion of the earth through it as we did the air current
in the car. If the earth is really revolving around the sun the ether
must be whizzing through its pores at the rate of about nineteen miles
a second.
But wait--there is the possibility that the earth carries along with it
in its flight through space a sort of atmosphere of ether as it does
of air. We must first get rid of this possibility by a preliminary
experiment to see if a swiftly moving mass of matter does catch up
and carry along with it a little of the ether. This would cause a
sort of an eddy or disturbance in the ether in the neighborhood of
the moving mass as a boat disturbs the water. For instance, a ray of
light passing close to a rapidly revolving wheel would be a little
deflected and show a distorted image. Sir Oliver Lodge tried this
experiment and got negative results. That is, moving matter does not
disturb or carry with it the ether. Consequently, it would seem, we
are left to the only other logical alternative, that the ether drifts
through matter and we should expect to detect this drift by measuring
the speed of light in the direction of the earth’s motion. It ought
to take longer for light to travel from one point to another if the
earth meantime is moving away from the first point and it ought to take
less time if the earth is moving toward it. Well, Michelson and Morley
tried this experiment--and also got negative results! It did not make
any difference whether the ray of light was sent in the direction of
the earth’s movement or the reverse or across the line, it traveled
invariably at the same speed, 186,000 miles a second. Here then were
two unquestionable experiments apparently contradicting each other. One
proved that the ether did not travel with the earth. The other proved
that the ether did not stand still while the earth traveled through it.
Now when we get contradictory answers to the questions we put to Nature
we must assume--unless Nature is nonsensical--that we are asking
nonsensical questions. If in the trial of a pickpocket one witness
swears that the thief did not run up the street and another witness
that he did not run down the street the lawyer does not necessarily
say that one of them must be a liar. He meditates a moment and then it
occurs to him that possibly the pickpocket did not move or that perhaps
he disappeared into the third dimension by climbing up a fire-escape or
dropping into a coalhole.
So with our ether quandary. If the ether does not move and does not
stand still perhaps there isn’t any ether or perhaps there is a fourth
dimension. These are two conceivable ways out of the dilemma though
they are not easy to accept, either of them. If there is no ether
what carries the light waves? If there is a fourth dimension in what
direction does it lie? But it is no harder to believe in or conceive
of a fourth dimension than it is the ether, and if the physicist finds
that he needs it in his business he will have to have it. Einstein says
that he needs a fourth dimension for his formulas.
THE CONUNDRUM OF THE AGES
For twenty-four hundred years philosophic thought has been concerned
with the problem of the relation of space and time. Drop into any of
the scientific societies of today and you will find them discussing
whether space is finite or infinite, whether there is any difference
between rest and motion, whether length is absolute or relative,
whether time and space have real existence, which are the very
questions discussed by Pythagoras and Zeno in the Greek cities of Asia
Minor. Now the time spent in these speculations has not been wasted,
although it has led to no definite conclusion, for out of it have
grown our mathematics and physics. The Wandering Jew, who is the only
mortal having the privilege of attending the schools of the Eleatics
and those of the present day, would observe one difference, that modern
scientists try to put their theories to the test of experiment wherever
possible, while the ancients were content with thinking them out.
Of all the guesses that have been given to this riddle of the universe
none has been more bold and revolutionary than that contained in a
paper of four or five pages contributed in 1905 to the _Annalen der
Physik_ by Albert Einstein. The controversy it precipitated has not
altogether been confined to the realm of pure reason, for scientists
are but human and as such are not entirely uninfluenced by patriotic
prejudice.
In this brief paper he proposed a new theory of the universe based
upon two postulates. The first was the principle of relativity; that
all _motion is relative_. This means, for instance, that we would
never know the motion of a smoothly moving train if the windows were
darkened and that we could never discover the forward movement of the
earth if we could not see the heavenly bodies.
Einstein’s second postulate was that _the velocity of light is
independent of the motion of the source_. This is a hard one for
our reason to swallow, for it means that nothing can travel faster
than light, 186,000 miles a second, and that you cannot make light
travel faster than that by giving it a swift send-off. It is the same
as saying that if a man standing on the cowcatcher of an engine threw a
ball forward, it would not make any difference with the velocity of the
ball whether the train was running at full speed forward or backward
or standing still. But the experiments of the American physicists,
Michelson and Morley, who measured the speed of light and found it the
same whether the earth was moving toward the source of the ray or away
from it, or at right angles to its direction, confirm Einstein’s second
assumption.
If we accept Einstein’s two primary postulates and his later “Principle
of Equivalence” his theory clears up this ether-drift difficulty as
well as various other riddles of the universe. It explains the shifting
of the orbit of Mercury that Newton’s theory could never account for.
It foretold the deflection of light by the sun’s gravitation that
the observations on the eclipse of last May confirmed. A third test,
the shifting of the lines of the solar spectrum toward the red end
in a gravitational field, has not been met. Such technical points
concern only physicists and astronomers, but Einstein’s relativity
theory, which two out of the three experiments support, carries with
it certain speculations as to time and space that are upsetting to
current conceptions.
PARADOXES OF RELATIVITY
All three of Newton’s laws of motion are now questioned and the
world is called upon to unlearn the lesson which Euclid taught it
that parallel lines never meet. According to Einstein they may meet.
According to Newton the action of gravitation is instantaneous
throughout all space. According to Einstein no action can exceed the
velocity of light. If the theory of relativity is right there can be
no such thing as absolute time or way of finding whether clocks in
different places are synchronous. Our yardsticks may vary according to
how we hold them and the weight of a body may depend upon its velocity.
The shortest distance between two points may not be a straight line.
These are a few of the startling implications of Einstein’s theory of
relativity. If he had put it forward as a mere metaphysical fancy, as a
possible but unverifiable hypothesis, it would have aroused mere idle
curiosity. But he deduced from it mathematical laws governing physical
phenomena which could be put to the test of experiment. They have been
tested in these two crucial cases and prove to be true.
In the preceding pages we have discussed the question of the relativity
of motion and seen how impossible it is to tell, for instance, whether
a train or a ship you are on is moving or not unless you can compare it
with something that you are “sure” is stationary. But what are you sure
is stationary? Nothing on earth surely, for the earth compared with the
“fixed” stars is spinning around at the rate of about a thousand miles
an hour and rushing around the sun at the rate of nearly 70,000 miles
an hour. But are we sure the stars are fixed since we have nothing else
to compare them with? You may remember Herbert Spencer’s illustration
of the sea captain who was walking west on the deck of a ship sailing
east at the same rate. Is he moving or not? If you are in the same
boat, you say he is. If you are on shore when the ship is passing you
say he is standing still and “marking time.” It all depends on the
point of view.
Now you may readily admit that all motion is relative, not absolute,
and yet you may balk at the idea that space and time are also relative,
not absolute. But motion is merely simultaneous change of position in
space and time, and why should we feel so certain about space and time
when we have never seen either?
You may say, for instance, that you are sure your desk is _so_
long. But if I ask you _how_ long you have to say as long as
something else. You may say it is a yard long. But how long is a yard?
It is as long as some tape or stick marked “one yard,” and this in turn
has been taken from some other yardstick, until you get back to the
brass rod in London that is just as long as the distance from the tip
of the nose of King Henry I to the end of his royal thumb. But such a
standard of absolute measurement is unsatisfactory to everyone except
an absolute monarchist. But apart from the difficulty of the present
inaccessibility of King Henry’s nose and thumb, can we be confident
that our yardstick keeps the same length while we are measuring with
it? We must admit indeed that it is longer on a summer day than on a
winter day, but can we be sure that it does not alter in length when we
hold it upright or lay it horizontally? Or, rather, could we tell if it
did change in length as it is changed in direction?
ARE YOU SURE OF YOUR SHAPE?
If you have ever been in any of those funny places at the amusement
parks you will have noticed the convex mirrors there and how ridiculous
they make other people look. If you cannot afford the nickel necessary
for the study of optics in such an establishment you can contemplate
your reflection in the side of a shiny tin cup or can. In a plane
mirror you see a man who looks as you suppose yourself to be except
that somehow you seem to have become left-handed. But when you look
into a convex cylindrical mirror set upright you see a man thinner than
you “really are.” Look into the same mirror set horizontal and you see
a man shorter than you “really are.” You grin at the sight of such
queer-looking creatures, but you notice that they are equally amused
at your shape. Now how are you going to prove to the men in the curved
glasses that they are mere caricatures and that you are not really
built on the plan of either of these images? You naturally resort to
measurement, as a scientist should. You cannot get into the mirror
world to measure the tall man who pretends to represent you, but you
can explain to him in the sign language what you want him to do and he
instantly complies. You stand up a measuring rod at your side and show
him that you are exactly 72 inches tall. He also sets up a rod and that
also reads 72 inches. Never mind, let him use any kind of measure he
likes, you will catch him when it comes to measurement of width with
the same stick. You hold your rule across your shoulders and it reads
18 inches, that is, one-fourth your height. But he also measures his
width with his rule and makes it just the same, 18 inches, although as
you see him he looks at least six times as high as he is broad.
[Illustration: THE MEASURE OF A MAN
When the man in the middle looks at himself in a curved mirror he
sees what he regards as a distorted image. The image on the right is
thinner and seems taller because it is reflected from a cylindrical
surface set upright. The image on the left is shorter and seems broader
because it is reflected from a cylindrical surface set horizontally.
But if the man and his image are measured by scales in the real world
and the mirror world they come out the same. So, too, it would be
impossible for us to find out if everything in the world were expanded
or contracted in all directions. In other words, all measurements are
relative. According to Einstein any body in movement is shortened in
the direction of the line of motion while the transverse dimension
remains the same. If, then, a man is being carried headlong through
space with a velocity approaching the speed of light he would be
shortened like the man on the left. If he were moving sideways he would
be like the man on the right.
The man’s image in a plane mirror seems to him symmetrical but
reversed. His right hand has somehow got over on his left side and
vice versa. Such a transformation as the mirror seems to effect cannot
be actually accomplished in ordinary space, but would conceivably
be possible in a space of four dimensions.]
Now you are sure he is cheating--must have some sort of telescoping
rod that contracts and expands according to the way he holds it. You
point out to him that his measure is unreliable, but to your surprise
his gestures seem intended to convince you that you instead are using
the elastic rule. You shake your fist in his face--to which he responds
with equal indignation--and then you turn to the squatty chap in
the other mirror, hoping he will be amenable to reason. But he also
measures himself as 72 inches high and 18 inches wide by his own rule.
If you try the still queerer-looking fellow in the concavo-convex
mirror who is distorted in all sorts of ways you will find that his
rule lengthens and shortens and bends just enough to make him as
symmetrical a man as yourself. And how can he be otherwise since he is
the image of yourself?
You are therefore driven to doubt the invariableness of your own
yardsticks. Suppose when you wake up tomorrow everything, including
all means of measuring, is twice as big as it is today. Could you
tell the difference? Would it make any difference? Would there be any
difference? Is there any such thing as absolute distance? Are not all
measurements relative?
Such questions had from the earliest times occupied the attention
of speculative philosophers, but they passed from the realm of
metaphysics to the realm of physics in 1886 when Michelson and
Morley made their famous experiment on the speed of light in various
directions. Their object was to find out if the ether, the hypothetical
medium carrying the light waves, was stationary and drifted back
through the earth as the earth moved onward. They devised an instrument
of such delicacy that the stamp of a foot a hundred yards off would
be noticeable. A ray of light was divided into two parts; one half
was sent forward and back in the direction toward which that part of
the earth where the experiment was made was moving at the time; the
other half was sent back and forth across the line of this motion.
But the two rays of light following different routes came back at
the same instant and matched up exactly. In order to correct for any
inequality in the instrument, Michelson and Morley turned it around so
the arm that formerly pointed across the line of motion now pointed
in the direction of that motion and the other arm pointed across, but
that made no difference. The light traveled with the same velocity
regardless of the motion of the earth.
This negative result was just as astonishing as if you should stand
at a certain spot on the bank of a river half a mile wide and should
send out two boats, one to go up the river half a mile against the
current and then back with the current and the other boat to go across
the river and back. If both boats should return at the same moment
you would be puzzled to account for it. One way of accounting for it
would be that your measurement of the half-mile course upstream had
been a little short. This was the explanation of the Michelson-Morley
experiment given by the Dutch physicist, Lorentz. He suggested that the
arm of the instrument shortened a trifle as it was turned from across
the line of the earth’s motion to the direction of that motion. The
amount of shrinkage necessary to compensate for the ether drift would
be exceedingly small. Besides how could you measure the change in the
length of the arm if the rule you laid alongside of it altered in the
same proportion? Lorentz’s explanation could not be disproved, yet it
was so upsetting to our ordinary ideas of the stability of matter that
it was hard to accept.
Einstein took Lorentz’s idea and made it one of the fundamental
principles of his new theory of the universe and then deduced from
this theory sundry very startling conclusions, some of which could
be--and have been--confirmed by experiment. According to Einstein the
size and shape of any body depends upon the rate and direction of its
movement. For ordinary speeds the alteration is very slight, but it
becomes considerable at rates approaching the speed of light, 186,000
miles a second. If, for instance, you could shoot an arrow from a bow
with a velocity of 160,000 miles a second, it would shrink to about
half its length, as measured by a man remaining still on earth. A man
traveling along with the arrow could discover no change. No force
could bring the arrow or even the smallest particle of matter to a
motion greater than the speed of light, and the nearer it comes to this
limit the greater the force required to move it faster. This means
that the mass of a body, instead of being absolute and unalterable as
we have supposed, increases with the speed of its movement. Newton’s
laws of dynamics are therefore valid only for matter in motion at such
moderate speeds as we have to deal with in our experiments on earth and
in our observations of the heavenly bodies. When we come to consider
velocities approximating that of light the ordinary laws of physics are
subject to an increasing correction.
If a person calculates that he is attaining a speed faster than light
he will seem to another observer to be moving the other way. That
is, any motion above the speed of light is negative motion. Just as
a tourist traveling more than 12,000 miles away from home in any
direction will really be getting nearer home the farther he goes.
Such speculations would not have bothered anybody twenty years ago,
for then the physicist did not have to handle any cases of such high
speeds. But when radium was discovered it was found that this metal
was continuously throwing off particles of negative electricity with
approximately the speed of light. Now if these electrons are not matter
they are at any rate the material of which matter is made. They can be
detected and counted and tracked and deflected and speeded and weighed.
They are very real things, perhaps the ultimate reality of all things,
yet their extreme velocity carries them out of Newton’s world and into
Einstein’s.
INTRODUCING THE FOURTH DIMENSION
Now Einstein’s world, as I said before, differs from the world in which
we are accustomed to live in many particulars. It has four dimensions
instead of three. One of these dimensions may be time. Time, too, must
be relative, not absolute. This is even harder to imagine than the
relativity of space.
As some schoolboy said: “If there were no matter in the universe the
law of gravitation would fall to the ground.” Quite so.
WHAT IS MEANT BY DIMENSIONS
No dimensions:
A mathematical point.
Has position but no size.
Represented by a dot. Like this .
One dimension:
Has length but no breadth.
Made by moving a point along straight in any direction.
Represented by a line. Like this ----
Two dimensions:
A plane surface like this page.
Has length and breadth but no thickness.
Made by moving a line in a direction perpendicular to its
length (that is, into the second dimension).
Represented by two straight lines of indefinite length perpendicular
to each other.
The lines are called axes and are labeled _x_ and _y_.
The point where they meet, the origin, is marked O.
Like this [Illustration]
Three dimensions:
A solid like a cube.
Has length, breadth and thickness.
Made by moving a plane in a direction perpendicular to the
other two (that is, into the third dimension).
Cannot be pictured on paper, but is indicated by three axes,
_x_, _y_, and _z_, of which _x_ and _y_ are on the plane of the
page and _z_ is supposed to be stuck up at right-angles to
the other two. Stick a pin into the paper at the point
O and you will have the third or _z_ axis.
Like this [Illustration]
Four dimensions:
Has length, breadth, thickness and extension into a fourth
dimension, say time.
Made by moving a cube in a direction perpendicular to the
other three (that is, into the fourth dimension).
Cannot be pictured on paper, but may be indicated by four
axes, _x_, _y_, _z_ and _t_ (or _u_), each at right-angles to the
other three.
Like this [Illustration]
More dimensions:
Any desired number of dimensions can be worked out
mathematically but with increasing difficulty because of
the impracticability of diagrammatical representation. We
can generalize the idea by speaking of a “geometry of _n_
dimensions” where _n_ may stand for any number whatever
from zero to infinity.
A line of a given length contains an infinite number of points.
A square of a given size contains an infinite number of lines.
A cube of a given size contains an infinite number of plane squares.
A tesseract (four-dimensional cuboid) of a given size contains an infinite
number of solid cubes.
And what would there be left of space if you took everything out of it,
and what would become of time if nothing ever happened? In other words
are not space and time merely forms of thought, the framework of ideas,
and if so cannot we fix them over to suit our need of new conceptions?
As a matter of fact we do. We have constructed by the aid of Euclid
and his successors a geometry of three dimensions that works perfectly
for all ordinary requirements and if we need a fourth dimension to
accommodate these new astronomical and physical phenomena we will build
on the necessary addition to our conception of space. There was no use
having a fourth dimension so long as we had nothing to put in it. For
ordinary earth measurements (geometry) such as laying out a town lot we
only use two dimensions, length and breadth. We speak of “flat ground”
and “water-level” regardless of the fact that all our “straight” lines
on the earth’s surface are really curves that come back to us after
going 25,000 miles or less. It is only when measuring mile lengths
that we have to correct for the curvature of the earth in the third
dimension. So if, as seems probable, we shall have to make allowance
in astronomical measurements for the curvature of the universe in a
fourth dimension it will merely mean a little labor to the astronomers
and it will relieve their minds of some of their perplexities.
There is nothing more mystical or mysterious or “psychical” about a
fourth dimension than about the other three. A dimension is simply
a measurable direction and we can use five dimensions or _n_
dimensions if we need to.
It does not matter that we cannot “see” a figure in four dimensions
even with our mind’s eye. Actually we cannot see any figure of more or
less than two dimensions: we have to take the others on faith. Nobody
can see the mathematician’s point because it has no dimensions, no size
at all. The schoolboy says: “Let that be the point A,” and we let it
be although what he is pointing at with his stick is not a point but
a vast irregular splotch of white chalk on the blackboard. So, too,
we cannot see a mathematical line because it has only one dimension,
length and no breadth. But set four lines at right angles to one
another and we get a square. This we can really see if the enclosed
surface is of a different color such as a shadow or black print. Set
six squares together at right angles and we get a cube. This we cannot
see in its entirety at one time. All that we see when we look squarely
at a cube is a square. If we look at it from an angle we see what
looks like a square with a couple of lozenges on the sides. The retina
of the eye is practically a plane surface, so all we can get is a
two-dimensional projection of a solid.
[Illustration: HOW TO DRAW A FOUR-DIMENSIONAL FIGURE
The best way to get an idea of the construction of a cubical solid in
four dimensions is to draw a diagram yourself and trace out in turn
each of the eight cubes that inclose it. I am indebted to K. W. Lamson
of Barnard College for the following sketch and directions:
Draw the four coördinate axes _OX_, _OY_, _OZ_,
_OU_.
Lay off the unit _a_{1}a_{4}_ on the _X_ axis,
_a_{1}a_{2}_ on the _Y_ axis, _a_{1}a_{7}_ on the
_Z_ axis and _a_{1}b_{1}_ on the _U_ axis.
Draw the cube _a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}_ on the
three axes _XYZ_.
Draw parallel to this the cube
_b_{1}b_{2}b_{3}b_{4}b_{5}b_{6}b_{7}b_{8}_ on the _U_ axis.
Draw the cube _a_{1}a_{2}a_{3}a_{4}b_{1}b_{2}b_{3}b_{4}_ on the
three axes _XYU_. This is partly drawn already.
Draw parallel to this the cube
_a_{7}a_{8}a_{5}a_{6}b_{7}b_{8}b_{5}b_{6}_ on the three axes
_XYU_.
This completes the figure.
There are four other cubes in the figure besides those described above:
The cube on the _XZU_ axes
_a_{1}a_{4}b_{1}b_{4}a_{7}a_{8}b_{7}b_{8}_ and its opposite
_a_{2}a_{3}b_{2}b_{3}a_{5}a_{6}b_{5}b_{6}_.
The cube on the _YZU_ axes
_a_{5}a_{2}a_{7}a_{1}b_{5}b_{2}b_{7}b_{1}_ and its opposite
_a_{6}a_{3}a_{8}a_{4}b_{6}b_{3}b_{8}b_{4}_.
The figure has: 16 corners, 32 edges, 24 bounding squares, 8 bounding
cubes.
The heavy line _a_{1}b_{6}_ might be called the principal diagonal
and makes an angle of 60 degrees with each of the four axes. It is
foreshortened in the sketch, but its real length is twice that of one
edge of the cube. Every line except this is on the outside of the
four-dimensional figure.]
[Illustration: THE TESSERACT
A four-dimensional cube-like solid if transparent and looked at
with one eye would appear something like this. But it is obviously
impossible to depict a four-dimensional figure on a two-dimensional
surface like this page.--From “The Fourth Dimension Simply Explained,”
Munn and Company, N. Y.]
Since our two eyes present us slightly different pictures of an object
we infer from these its size, shape and distance, but this is guesswork.
Still we have a pretty clear idea of a cube although we have never
seen it in its solidity. But the attempt to visualize the hypercube,
the four-dimensional figure corresponding to the cube, strains our
imagination to the breaking point. Some mathematicians endowed with
constructive imaginations of high power claim to have got by long hard
thinking some sort of a shadowy and fleeting perception of it, but
their visions--if they are not imaginary--do not help out us ordinary
folks. But if we cannot imagine--that is, image--the hypercube we know
all about it, even its name. It is called the “tesseract,” and it is
bounded by eight cubes just as the cube is bounded by six squares and
the square by four lines. The tesseract has 24 square faces, 32 edges
and 16 rightangular corners.
TIME AS THE FOURTH DIMENSION
Although we find it hard to conceive of a fourth dimension in space we
have no such difficulty in case the fourth dimension is time. In fact,
we use this idea all the while and could not get along without it. To
fix the position of any event requires four dimensions. For instance,
a man is shot. Where? At the corner of 7th Avenue and 42d Street, New
York. This fixes the place by two coördinates crossing at right angles
in a plane. But was it above or below this, on the twentieth floor
of the Times Building or in the Subway? Knowing this fixes the third
dimension, but we have still to fix its position in a fourth dimension,
time. Was it today or last week and what hour? If then we find out all
four we can distinguish this shooting from any that may have occurred
in other places at the same time or at other times in the same place.
Or consider this simple illustration: Cut a strip of motion picture
film into its separate scenes and pile them up in order till it is as
high as it is broad. You have then a cubical event. Two dimensions of
the cube are spatial; the third dimension is essentially temporal,
although in a spatial form. If one of the films from the middle of the
pack represents the present then the films below represent the past and
those above the future. The people on the picture you picked out know
only of the scene there depicted though they may have a fading memory
of the past and a dim anticipation of the future. But to you who are
outside of the film pack all the scenes are equally visible. They are
all present to you. This is the way most Christians have conceived of
God, as one to whom past and future form one eternal present, so he
sees simultaneously all things that have been, are or will be.
If our pile of film were made up of snapshots taken one a day
throughout a man’s life we should see at one glance his growth from
babyhood to boyhood, to maturity and old age. We could turn the leaves
of his life backward or forward as we will. Some day perhaps we shall
have stereo-movies, scenes in three dimensions with time as the fourth.
This idea of time as a fourth dimension is not a new one. In 1754
d’Alembert, defining “dimension” in the Encyclopedia, wrote: “A
brilliant man of my acquaintance believes that one may regard duration
as a fourth dimension.” In 1903 Minkowski worked out the idea in
mathematical form. H. G. Wells, always quick to catch up a new
scientific theory to use as a plot for a story, wrote in 1895 of “The
Time Machine,” a vehicle by which a man could travel back and forth in
time as he can travel east and west in a motor car. In this he visits
the future and finds mankind split into two species, a subterranean
working class living on--literally--a pleasure-loving leisure class.
In “The Plattner Case” Wells tells of a chemical professor who was by
an explosion knocked into--not the middle of next week as we commonly
say--but into the fourth dimension of space. Ten days later he was
knocked back again into our world but the only evidence of the truth
of his story was that his heart beat on the right side and he was left
handed and otherwise reversed in a way that would be impossible in a
space of three dimensions.
[Illustration: In space of three dimensions we cannot make a right hand
glove and a left hand glove look the same no matter how we turn them
around. But if we turn one glove inside out it will match the other
except that the lining now appears on the outside.]
[Illustration: Our two hands cannot be turned inside out so as to look
the same in three dimensions, though they might in four dimensions.]
We can turn a glove inside out in three dimensions and so make it just
like its mate of the other hand, but we cannot turn a solid inside out
except in four-dimensional space.
In another of his “Thirty Strange Stories” Wells tells “The Story of
Davidson’s Eyes.” While Davidson was working in his London laboratory
a lightning shock so affected his eyesight that he could not see the
familiar objects about him which he could feel but looked instead at
a South Sea island on the opposite side of the globe. This might be
possible in a curved space of four dimensions although Wells professes
to pooh-pooh such an absurd suggestion while he ingeniously insinuates
it. George Macdonald in his fantastic romance “Lilith” also introduces
the fourth dimension.
Points that are far apart if measured in three dimensions may be close
together in the fourth. We can readily understand this if time is the
fourth dimension, for events can happen at the same instant though
thousands of miles apart. But it is not impossible to conceive of the
fourth dimension as spatial instead of temporal if we approach the
problem from a simpler standpoint. Let us think of ourselves as living
in a “Flatland” of two dimensions with no thought of a third. There yet
survive in enlightened America individuals who believe that “the sun do
move” and who deny that the earth is “round like a ball.” That is, they
do not recognize the curvature of the earth in the third dimension. But
if such an individual were to travel in a “straight” line westward over
the “level” land and water he would, much to his surprise, come back to
his starting point which he had left 25,000 miles behind him.
[Illustration: By movement in one dimension we cannot make the lines
_AB_ and _B′A′_ coincide for if we drag _B′A′_ straight
on to _AB_ the ends will not match. But if we swing _B′A′_
around through the second dimension we bring it on _AB_ so the
letters correspond.]
[Illustration: In space of two dimensions, such as a table top, we
cannot bring these two triangles into the same position. If we drag one
straight over on to the other (movement in one dimension) they will
not fit together. If we swing one triangle around (movement in two
dimensions) they still do not fit. But if we take one triangle off the
table and turn it over (movement in the third dimension) we can then
lay it by the side of the other and they will match perfectly.]
A WORM’S-EYE VIEW OF THE WORLD
[Illustration]