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in the original text. Equal signs “=” before and after a word or
phrase indicate =bold= in the original text. Small capitals have been
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silently corrected.
HARPER’S MODERN SCIENCE SERIES
THE A B C OF RELATIVITY
BY
BERTRAND RUSSELL
AUTHOR OF
“THE PRINCIPLES OF MATHEMATICS”
“PROPOSED ROADS TO FREEDOM”
AND “WHY MEN FIGHT”
PUBLISHERS
HARPER & BROTHERS
NEW YORK AND LONDON
THE A B C OF RELATIVITY
Copyright, 1925, by Harper & Brothers
Printed in the United States of America
_Contents_
CHAPTER PAGE
I. TOUCH AND SIGHT: THE EARTH AND THE HEAVENS 1
II. WHAT HAPPENS AND WHAT IS OBSERVED 14
III. THE VELOCITY OF LIGHT 28
IV. CLOCKS AND FOOT RULES 43
V. SPACE-TIME 58
VI. THE SPECIAL THEORY OF RELATIVITY 71
VII. INTERVALS IN SPACE-TIME 91
VIII. EINSTEIN’S LAW OF GRAVITATION 111
IX. PROOFS OF EINSTEIN’S LAW OF GRAVITATION 131
X. MASS, MOMENTUM, ENERGY AND ACTION 144
XI. IS THE UNIVERSE FINITE? 163
XII. CONVENTIONS AND NATURAL LAWS 177
XIII. THE ABOLITION OF “FORCE” 192
XIV. WHAT IS MATTER? 206
XV. PHILOSOPHICAL CONSEQUENCES 219
THE A B C OF RELATIVITY
CHAPTER ONE: TOUCH AND SIGHT: THE EARTH AND THE HEAVENS
Everybody knows that Einstein has done something astonishing, but
very few people know exactly what it is that he has done. It is
generally recognized that he has revolutionized our conception of the
physical world, but his new conceptions are wrapped up in mathematical
technicalities. It is true that there are innumerable popular
accounts of the theory of relativity, but they generally cease to
be intelligible just at the point where they begin to say something
important. The authors are hardly to blame for this. Many of the new
ideas can be expressed in non-mathematical language, but they are none
the less difficult on that account. What is demanded is a change in
our imaginative picture of the world—a picture which has been handed
down from remote, perhaps pre-human, ancestors, and has been learned
by each one of us in early childhood. A change in our imagination is
always difficult, especially when we are no longer young. The same sort
of change was demanded by Copernicus, when he taught that the earth
is not stationary and the heavens do not revolve about it once a day.
To us now there is no difficulty in this idea, because we learned it
before our mental habits had become fixed. Einstein’s ideas, similarly,
will seem easy to a generation which has grown up with them; but for
our generation a certain effort of imaginative reconstruction is
unavoidable.
In exploring the surface of the earth, we make use of all our senses,
more particularly of the senses of touch and sight. In measuring
lengths, parts of the human body are employed in pre-scientific
ages: a “foot,” a “cubit,” a “span” are defined in this way. For
longer distances, we think of the time it takes to walk from one
place to another. We gradually learn to judge distances roughly by
the eye, but we rely upon touch for accuracy. Moreover it is touch
that gives us our sense of “reality.” Some things cannot be touched:
rainbows, reflections in looking-glasses, and so on. These things
puzzle children, whose metaphysical speculations are arrested by the
information that what is in the looking glass is not “real.” Macbeth’s
dagger was unreal because it was not “sensible to feeling as to sight.”
Not only our geometry and physics, but our whole conception of what
exists outside us, is based upon the sense of touch. We carry this even
into our metaphors: a good speech is “solid,“ a bad speech is “gas,”
because we feel that a gas is not quite “real.”
In studying the heavens, we are debarred from all senses except sight.
We cannot touch the sun, or travel to it; we cannot walk round the
moon, or apply a foot rule to the Pleiades. Nevertheless, astronomers
have unhesitatingly applied the geometry and physics which they found
serviceable on the surface of the earth, and which they had based
upon touch and travel. In doing so, they brought down trouble on
their heads, which it has been left for Einstein to clear up. It has
turned out that much of what we learned from the sense of touch was
unscientific prejudice, which must be rejected if we are to have a true
picture of the world.
An illustration may help us to understand how much is impossible to
the astronomer as compared to the man who is interested in things on
the surface of the earth. Let us suppose that a drug is administered
to you which makes you temporarily unconscious, and that when you
wake you have lost your memory but not your reasoning powers. Let us
suppose further that while you were unconscious you were carried into
a balloon, which, when you come to, is sailing with the wind in a dark
night—the night of the fifth of November if you are in England, or of
the fourth of July if you are in America. You can see fireworks which
are being sent off from the ground, from trains, and from aeroplanes
traveling in all directions, but you cannot see the ground or the
trains or the aeroplanes be cause of the darkness. What sort of picture
of the world will you form? You will think that nothing is permanent:
there are only brief flashes of light, which, during their short
existence, travel through the void in the most various and bizarre
curves. You cannot touch these flashes of light, you can only see them.
Obviously your geometry and your physics and your metaphysics will be
quite different from those of ordinary mortals. If an ordinary mortal
is with you in the balloon, you will find his speech unintelligible.
But if Einstein is with you, you will understand him more easily than
the ordinary mortal would, because you will be free from a host of
preconceptions which prevent most people from understanding him.
The theory of relativity depends, to a considerable extent, upon
getting rid of notions which are useful in ordinary life but not to
our drugged balloonist. Circumstances on the surface of the earth,
for various more or less accidental reasons, suggest conceptions
which turn out to be inaccurate, although they have come to seem like
necessities of thought. The most important of these circumstances is
that most objects on the earth’s surface are fairly persistent and
nearly stationary from a terrestrial point of view. If this were not
the case, the idea of going a journey would not seem so definite as it
does. If you want to travel from King’s Cross to Edinburgh, you know
that you will find King’s Cross where it always has been, that the
railway line will take the course that it did when you last made the
journey, and that Waverley Station in Edinburgh will not have walked up
to the Castle. You therefore say and think that you have traveled to
Edinburgh, not that Edinburgh has traveled to you, though the latter
statement would be just as accurate. The success of this common sense
point of view depends upon a number of things which are really of the
nature of luck. Suppose all the houses in London were perpetually
moving about, like a swarm of bees; suppose railways moved and changed
their shapes like avalanches; and finally suppose that material objects
were perpetually being formed and dissolved like clouds. There is
nothing impossible in these suppositions: something like them must have
been verified when the earth was hotter than it is now. But obviously
what we call a journey to Edinburgh would have no meaning in such a
world. You would begin, no doubt, by asking the taxi-driver: “Where
is King’s Cross this morning?“ At the station you would have to ask a
similar question about Edinburgh, but the booking-office clerk would
reply: “What part of Edinburgh do you mean, Sir? Prince’s Street has
gone to Glasgow, the Castle has moved up into the Highlands, and
Waverley Station is under water in the middle of the Firth of Forth.”
And on the journey the stations would not be staying quiet, but some
would be travelling north, some south, some east or west, perhaps much
faster than the train. Under these conditions you could not say where
you were at any moment. Indeed the whole notion that one is always in
some definite “place” is due to the fortunate immovability of most of
the large objects on the earth’s surface. The idea of “place” is only
a rough practical approximation: there is nothing logically necessary
about it, and it cannot be made precise.
If we were not much larger than an electron, we should not have this
impression of stability, which is only due to the grossness of our
senses. King’s Cross, which to us looks solid, would be too vast to
be conceived except by a few eccentric mathematicians. The bits of it
that we could see would consist of little tiny points of matter, never
coming into contact with each other, but perpetually whizzing round
each other in an inconceivably rapid ballet-dance. The world of our
experience would be quite as mad as the one in which the different
parts of Edinburgh go for walks in different directions. If—to take
the opposite extreme—you were as large as the sun and lived as long,
with a corresponding slowness of perception, you would again find
a higgledy-piggledy universe without permanence—stars and planets
would come and go like morning mists, and nothing would remain in a
fixed position relatively to anything else. The notion of comparative
stability which forms part of our ordinary outlook is thus due to the
fact that we are about the size we are, and live on a planet of which
the surface is no longer very hot. If this were not the case, we should
not find pre-relativity physics intellectually satisfying. Indeed, we
should never have invented such theories. We should have had to arrive
at relativity physics at one bound, or remain ignorant of scientific
laws. It is fortunate for us that we were not faced with this
alternative, since it is almost inconceivable that one man could have
done the work of Euclid, Galileo, Newton, and Einstein. Yet without
such an incredible genius physics could hardly have been discovered
in a world where the universal flux was obvious to non-scientific
observation.
In astronomy, although the sun, moon, and stars continue to exist year
after year, yet in other respects the world we have to deal with is
very different from that of everyday life. As already observed, we
depend exclusively on sight: the heavenly bodies cannot be touched,
heard, smelt or tasted. Everything in the heavens is moving relatively
to everything else. The earth is going round the sun, the sun is
moving, very much faster than an express train, towards a point in the
constellation “Hercules,” the “fixed” stars are scurrying hither and
thither like a lot of frightened hens. There are no well-marked places
in the sky, like King’s Cross and Edinburgh. When you travel from place
to place on the earth, you say the train moves and not the stations,
because the stations preserve their topographical relations to each
other and the surrounding country. But in astronomy it is arbitrary
which you call the train and which the station: the question is to be
decided purely by convenience and as a matter of convention.
In this respect, it is interesting to contrast Einstein and Copernicus.
Before Copernicus, people thought that the earth stood still and the
heavens revolved about it once a day. Copernicus taught that “really”
the earth rotates once a day, and the daily revolution of sun and stars
is only “apparent.” Galileo and Newton endorsed this view, and many
things were thought to prove it—for example, the flattening of the
earth at the poles, and the fact that bodies are heavier there than at
the equator. But in the modern theory the question between Copernicus
and his predecessors is merely one of convenience; all motion is
relative, and there is no difference between the two statements: “the
earth rotates once a day” and “the heavens revolve about the earth
once a day.” The two mean exactly the same thing, just as it means the
same thing if I say that a certain length is six feet or two yards.
Astronomy is easier if we take the sun as fixed than if we take the
earth, just as accounts are easier in a decimal coinage. But to say
more for Copernicus is to assume absolute motion, which is a fiction.
All motion is relative, and it is a mere convention to take one body as
at rest. All such conventions are equally legitimate, though not all
are equally convenient.
There is another matter of great importance, in which astronomy
differs from terrestrial physics because of its exclusive dependence
upon sight. Both popular thought and old-fashioned physics used the
notion of “force,” which seemed intelligible because it was associated
with familiar sensations. When we are walking, we have sensations
connected with our muscles which we do not have when we are sitting
still. In the days before mechanical traction, although people could
travel by sitting in their carriages, they could see the horses
exerting themselves and evidently putting out “force” in the same
way as human beings do. Everybody knew from experience what it is to
push or pull, or to be pushed or pulled. These very familiar facts
made “force” seem a natural basis for dynamics. But Newton’s law of
gravitation introduced a difficulty. The force between two billiard
balls appeared intelligible, because we know what it feels like to bump
into another person; but the force between the earth and the sun, which
are ninety-three million miles apart, was mysterious. Newton himself
regarded this “action at a distance” as impossible, and believed that
there was some hitherto undiscovered mechanism by which the sun’s
influence was transmitted to the planets. However, no such mechanism
was discovered, and gravitation remained a puzzle. The fact is that the
whole conception of “force” is a mistake. The sun does not exert any
force on the planets; in Einstein’s law of gravitation, the planet only
pays attention to what it finds in its own neighborhood. The way in
which this works will be explained in a later chapter; for the present
we are only concerned with the necessity of abandoning the notion of
“force,” which was due to misleading conceptions derived from the sense
of touch.
As physics has advanced, it has appeared more and more that sight is
less misleading than touch as a source of fundamental notions about
matter. The apparent simplicity in the collision of billiard balls is
quite illusory. As a matter of fact, the two billiard balls never touch
at all; what really happens is inconceivably complicated, but is more
analogous to what happens when a comet penetrates the solar system and
goes away again than to what common sense supposes to happen.
Most of what we have said hitherto was already recognized by physicists
before Einstein invented the theory of relativity. “Force” was known
to be merely a mathematical fiction, and it was generally held that
motion is a merely relative phenomenon—that is to say, when two
bodies are changing their relative position, we cannot say that one is
moving while the other is at rest, since the occurrence is merely a
change in their relation to each other. But a great labor was required
in order to bring the actual procedure of physics into harmony with
these new convictions. Newton believed in force and in absolute space
and time; he embodied these beliefs in his technical methods, and his
methods remained those of later physicists. Einstein invented a new
technique, free from Newton’s assumptions. But in order to do so he
had to change fundamentally the old ideas of space and time, which had
been unchallenged from time immemorial. This is what makes both the
difficulty and the interest of his theory. But before explaining it
there are some preliminaries which are indispensable. These will occupy
the next two chapters.
CHAPTER II: WHAT HAPPENS AND WHAT IS OBSERVED
A certain type of superior person is fond of asserting that “everything
is relative.” This is, of course, nonsense, because, if _everything_
were relative, there would be nothing for it to be relative to.
However, without falling into metaphysical absurdities it is possible
to maintain that everything in the physical world is relative to
an observer. This view, true or not, is _not_ that adopted by the
“theory of relativity.” Perhaps the name is unfortunate; certainly
it has led philosophers and uneducated people into confusions. They
imagine that the new theory proves _everything_ in the physical world
to be relative, whereas, on the contrary, it is wholly concerned to
exclude what is relative and arrive at a statement of physical laws
that shall in no way depend upon the circumstances of the observer. It
is true that these circumstances have been found to have more effect
upon what appears to the observer than they were formerly thought to
have, but at the same time Einstein showed how to discount this effect
completely. This was the source of almost everything that is surprising
in his theory.
When two observers perceive what is regarded as one occurrence, there
are certain similarities, and also certain differences, between their
perceptions. The differences are obscured by the requirements of
daily life, because from a business point of view they are as a rule
unimportant. But both psychology and physics, from their different
angles, are compelled to emphasize the respects in which one man’s
perception of a given occurrence differs from another man’s. Some of
these differences are due to differences in the brains or minds of
the observers, some to differences in their sense organs, some to
differences of physical situation: these three kinds may be called
respectively psychological, physiological, and physical. A remark made
in a language we know will be heard, whereas an equally loud remark
in an unknown language may pass entirely unnoticed. Of two men in the
Alps, one will perceive the beauty of the scenery while the other will
notice the waterfalls with a view to obtaining power from them. Such
differences are psychological. The difference between a long-sighted
and a short-sighted man, or between a deaf man and a man who hears
well, are physiological. Neither of these kinds concerns us, and I have
mentioned them only in order to exclude them. The kind that concerns us
is the purely physical kind. Physical differences between two observers
will be preserved when the observers are replaced by cameras or
phonographs, and can be reproduced on the movies or the gramophone. If
two men both listen to a third man speaking, and one of them is nearer
to the speaker than the other is, the nearer one will hear louder and
slightly earlier sounds than are heard by the other. If two men both
watch a tree falling, they see it from different angles. Both these
differences would be shown equally by recording instruments: they are
in no way due to idiosyncrasies in the observers, but are part of the
ordinary course of physical nature as we experience it.
The physicist, like the plain man, believes that his perceptions give
him knowledge about what is really occurring in the physical world,
and not only about his private experiences. Professionally, he regards
the physical world as “real,” not merely as something which human
beings dream. An eclipse of the sun, for instance, can be observed
by any person who is suitably situated, and is also observed by the
photographic plates that are exposed for the purpose. The physicist
is persuaded that something has really happened over and above the
experiences of those who have looked at the sun or at photographs of
it. I have emphasized this point, which might seem a trifle obvious,
because some people imagine that Einstein has made a difference in this
respect. In fact he has made none.
But if the physicist is justified in this belief that a number of
people can observe the “same” physical occurrence, then clearly the
physicist must be concerned with those features which the occurrence
has in common for all observers, for the others cannot be regarded
as belonging to the occurrence itself. At least, the physicist must
confine himself to the features which are common to all “equally
good” observers. The observer who uses a microscope or a telescope is
preferred to one who does not, because he sees all that the latter sees
and more too. A sensitive photographic plate may “see” still more,
and is then preferred to any eye. But such things as differences of
perspective, or differences of apparent size due to difference of
distance, are obviously not attributable to the object; they belong
solely to the point of view of the spectator. Common sense eliminates
these in judging of objects; physics has to carry the same process much
further, but the principle is the same.
I want to make it clear that I am not concerned with anything that can
be called inaccuracy. I am concerned with genuine physical differences
between occurrences each of which is a correct record of a certain
event, from its own point of view. When a man fires a gun, people who
are not quite close to him see the flash before they hear the report.
This is not due to any defect in their senses, but to the fact that
sound travels more slowly than light. Light travels so fast that, from
the point of view of phenomena on the surface of the earth, it may
be regarded as instantaneous. Anything that we can see on the earth
happens practically at the moment when we see it. In a second, light
travels 300,000 kilometers (about 186,000 miles). It travels from the
sun to the earth in about eight minutes, and from the stars to us in
anything from three to a thousand years. But of course we cannot place
a clock in the sun, and send out a flash of light from it at 12 noon,
Greenwich Mean Time, and have it received at Greenwich at 12.08 P.M.
Our methods of estimating the speed of light have to be more or less
indirect. The only direct method would be that which we apply to sound
when we use an echo. We could send a flash to a mirror, and observe how
long it took for the reflection to reach us; this would give the time
of the double journey to the mirror and back. On the earth, however,
the time would be so short that a great deal of theoretical physics
has to be utilized if this method is to be employed—more even than is
required for the employment of astronomical data.
The problem of allowing for the spectator’s point of view, we may be
told, is one of which physics has at all times been fully aware; indeed
it has dominated astronomy ever since the time of Copernicus. This is
true. But principles are often acknowledged long before their full
consequences are drawn. Much of traditional physics is incompatible
with the principle, in spite of the fact that it was acknowledged
theoretically by all physicists.
There existed a set of rules which caused uneasiness to the
philosophically minded, but were accepted by physicists because
they worked in practice. Locke had distinguished “secondary”
qualities—colors, noises, tastes, smells, etc.—as subjective, while
allowing “primary” qualities—shapes and positions and sizes—to be
genuine properties of physical objects. The physicist’s rules were
such as would follow from this doctrine. Colors and noises were
allowed to be subjective, but due to waves proceeding with a definite
velocity—that of light or sound as the case may be—from their source
to the eye or ear of the percipient. Apparent shapes vary according to
the laws of perspective, but these laws are simple and make it easy to
infer the “real” shapes from several visual apparent shapes; moreover,
the “real” shapes can be ascertained by touch in the case of bodies in
our neighborhood. The objective time of a physical occurrence can be
inferred from the time when we perceive it by allowing for the velocity
of transmission—of light or sound or nerve currents according to
circumstances. This was the view adopted by physicists in practice,
whatever qualms they may have had in unprofessional moments.
This view worked well enough until physicists became concerned with
much greater velocities than those that are common on the surface of
the earth. An express train travels about a mile in a minute; the
planets travel a few miles in a second. Comets, when they are near
the sun, travel much faster, and behave somewhat oddly; but they were
puzzling in various ways. Practically, the planets were the most
swiftly moving bodies to which dynamics could be adequately applied.
With radio-activity a new range of observations became possible.
Individual electrons can be observed, emanating from radium with a
velocity not far short of that of light. The behavior of bodies moving
with these enormous speeds is not what the old theories would lead
us to expect. For one thing, mass seems to increase with speed in a
perfectly definite manner. When an electron is moving very fast, a
bigger force is required to have a given effect upon it than when it
is moving slowly. Then reasons were found for thinking that the size
of a body is affected by its motion—for example, if you take a cube
and move it very fast, it gets shorter in the direction of its motion,
from the point of view of a person who is not moving with it, though
from its own point of view (_i.e._ for an observer traveling with it)
it remains just as it was. What was still more astonishing was the
discovery that lapse of time depends on motion; that is to say, two
perfectly accurate clocks, one of which is moving very fast relatively
to the other, will not continue to show the same time if they come
together again after a journey. It follows that what we discover by
means of clocks and foot rules, which used to be regarded as the acme
of impersonal science, is really in part dependent upon our private
circumstances, _i.e._ upon the way in which we are moving relatively to
the bodies measured.
This shows that we have to draw a different line from that which is
customary in distinguishing between what belongs to the observer and
what belongs to the occurrence which he is observing. If a man is
wearing blue spectacles he knows that the blue look of everything is
due to his spectacles, and does not belong to what he is observing.
But if he observes two flashes of lightning, and notes the interval
of time between his observations; if he knows where the flashes took
place, and allows, in each case, for the time the light took to reach
him—in that case, if his chronometer is accurate, he naturally thinks
that he has discovered the actual interval of time between the two
flashes, and not something merely personal to himself. He is confirmed
in this view by the fact that all other careful observers to whom he
has access agree with his estimates. This, however, is only due to the
fact that all these observers are on the earth, and share its motion.
Even two observers in aeroplanes moving in opposite directions would
have at the most a relative velocity of 400 miles an hour, which is
very little in comparison with 186,000 miles a second (the velocity
of light). If an electron shot out from a piece of radium with a
velocity of 170,000 miles a second could observe the time between the
two flashes, it would arrive at a quite different estimate, after
making full allowance for the velocity of light. How do you know this?
the reader may ask. You are not an electron, you cannot move at these
terrific speeds, no man of science has ever made the observations which
would prove the truth of your assertion. Nevertheless, as we shall see
in the sequel, there is good ground for the assertion—ground, first
of all, in experiment, and—what is remarkable—ground in reasonings
which could have been made at any time, but were not made until
experiments had shown that the old reasonings must be wrong.
There is a general principle to which the theory of relativity appeals,
which turns out to be more powerful than anybody would suppose. If
you know that one man is twice as rich as another, this fact must
appear equally whether you estimate the wealth of both in pounds or
dollars or francs or any other currency. The numbers representing their
fortunes will be changed, but one number will always be double the
other. The same sort of thing, in more complicated forms, reappears in
physics. Since all motion is relative, you may take any body you like
as your standard body of reference, and estimate all other motions
with reference to that one. If you are in a train and walking to the
dining-car, you naturally, for the moment, treat the train as fixed
and estimate your motion by relation to it. But when you think of the
journey you are making, you think of the earth as fixed, and say you
are moving at the rate of sixty miles an hour. An astronomer who is
concerned with the solar system takes the sun as fixed, and regards you
as rotating and revolving; in comparison with this motion, that of the
train is so slow that it hardly counts. An astronomer who is interested
in the stellar universe may add the motion of the sun relatively to
the average of the stars. You cannot say that one of these ways of
estimating your motion is more correct than another; each is perfectly
correct as soon as the reference body is assigned. Now just as you can
estimate a man’s fortune in different currencies without altering its
relations to the fortunes of other men, so you can estimate a body’s
motion by means of different reference bodies without altering its
relations to other motions. And as physics is entirely concerned with
relations, it must be possible to express all the laws of physics by
referring all motions to any given body as the standard.
We may put the matter in another way. Physics is intended to give
information about what really occurs in the physical world, and not
only about the private perceptions of separate observers. Physics must,
therefore, be concerned with those features which a physical process
has in common for all observers, since such features alone can be
regarded as belonging to the physical occurrence itself. This requires
that the _laws_ of phenomena should be the same whether the phenomena
are described as they appear to one observer or as they appear to
another. This single principle is the generating motive of the whole
theory of relativity.
Now what we have hitherto regarded as the spatial and temporal
properties of physical occurrences are found to be in large part
dependent upon the observer; only a residue can be attributed to the
occurrences in themselves, and only this residue can be involved in
the formulation of any physical law which is to have an _à priori_
chance of being true. Einstein found ready to his hand an instrument of
pure mathematics, called the theory of tensors, which enabled him to
discover laws expressed in terms of the objective residue and agreeing
approximately with the old laws. Where Einstein’s laws differed from
the old ones, they have hitherto proved more in accord with observation.
If there were no reality in the physical world, but only a number of
dreams dreamed by different people, we should not expect to find any
laws connecting the dreams of one man with the dreams of another. It
is the close connection between the perceptions of one man and the
(roughly) simultaneous perceptions of another that makes us believe in
a common external origin of the different related perceptions. Physics
accounts both for the likenesses and for the differences between
different people’s perceptions of what we call the “same” occurrence.
But in order to do this it is first necessary for the physicist to
find out just what are the likenesses. They are not quite those
traditionally assumed, because neither space nor time separately can
be taken as strictly objective. What is objective is a kind of mixture
of the two called “space-time.” To explain this is not easy, but the
attempt must be made; it will be begun in the next chapter.