Transcriber’s Notes:

Underscores “_” before and after a word or phrase indicate _italics_
in the original text. Equal signs “=” before and after a word or
phrase indicate =bold= in the original text. Small capitals have been
converted to SOLID capitals. Illustrations have been moved so they do
not break up paragraphs. Typographical and punctuation errors have been
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.

CHAPTER III: THE VELOCITY OF LIGHT

Most of the curious things in the theory of relativity are connected
with the velocity of light. If the reader is to grasp the reasons for
such a serious theoretical reconstruction, he must have some idea of
the facts which made the old system break down.

The fact that light is transmitted with a definite velocity was
first established by astronomical observations. Jupiter’s moons are
sometimes eclipsed by Jupiter, and it is easy to calculate the times
when this ought to occur. It was found that when Jupiter was unusually
near the earth an eclipse of one of his moons would be observed a few
minutes earlier than was expected; and when Jupiter was unusually
remote, a few minutes later than was expected. It was found that these
deviations could all be accounted for by assuming that light has a
certain velocity, so that what we observe to be happening in Jupiter
really happened a little while ago—longer ago when Jupiter is distant
than when it is near. Just the same velocity of light was found to
account for similar facts in regard to other parts of the solar system.
It was therefore accepted that light _in vacuo_ always travels at a
certain constant rate, almost exactly 300,000 kilometers a second. (A
kilometer is about five-eighths of a mile.) When it became established
that light consists of waves, this velocity was that of propagation
of waves in the ether—at least they used to be in the ether, but now
the ether has grown somewhat shadowy, though the waves remain. This
same velocity is that of the waves used in wireless telegraphy (which
are like light waves, only longer) and in X-rays (which are like light
waves, only shorter). It is generally held nowadays to be the velocity
with which gravitation is propagated, though Eddington considers this
not yet certain. (It used to be thought that gravitation was propagated
instantaneously, but this view is now abandoned.)

So far, all is plain sailing. But as it became possible to make more
accurate measurements, difficulties began to accumulate. The waves were
supposed to be in the ether, and therefore their velocity ought to
be relative to the ether. Now since the ether (if it exists) clearly
offers no resistance to the motions of the heavenly bodies, it would
seem natural to suppose that it does not share their motion. If the
earth had to push a lot of ether before it, in the sort of way that
a steamer pushes water before it, one would expect a resistance on
the part of the ether analogous to that offered by the water to the
steamer. Therefore the general view was that the ether could pass
through bodies without difficulty, like air through a coarse sieve,
only more so. If this were the case, then the earth in its orbit must
have a velocity relative to the ether. If, at some point of its orbit,
it happened to be moving exactly with the ether, it must at other
points be moving through it all the faster. If you go for a circular
walk on a windy day, you must be walking against the wind part of the
way, whatever wind may be blowing; the principle in this case is the
same. It follows that, if you choose two days six months apart, when
the earth in its orbit is moving in exactly opposite directions, it
must be moving against an ether wind on at least one of these days.

Now if there is an ether wind, it is clear that, relatively to an
observer on the earth, light signals will seem to travel faster with
the wind than across it, and faster across it than against it. This
is what Michelson and Morley set themselves to test by their famous
experiment. They sent out light signals in two directions at right
angles; each was reflected from a mirror, and came back to the place
from which both had been sent out. Now anybody can verify, either by
trial or by a little arithmetic, that it takes longer to row a given
distance on a river upstream and then back again, than it takes to
row the same distance across the stream and back again. Therefore, if
there were an ether wind, one of the two light signals, which consist
of waves in the ether, ought to have traveled to the mirror and back at
a slower average rate than the other. Michelson and Morley tried the
experiment, they tried it in various positions, they tried it again
later. Their apparatus was quite accurate enough to have detected the
expected difference of speed or even a much smaller difference, if
it had existed, but not the smallest difference could be observed.
The result was a surprise to them as to everybody else; but careful
repetitions made doubt impossible. The experiment was first made as
long ago as 1881, and was repeated with more elaboration in 1887. But
it was many years before it could be rightly interpreted.

The supposition that the earth carries the neighboring ether with it
in its motion was found to be impossible, for a number of reasons.
Consequently a logical deadlock seemed to have arisen, from which at
first physicists sought to extricate themselves by very arbitrary
hypotheses. The most important of these was that of Fitzgerald,
developed by Lorentz, and known as the Fitzgerald contraction
hypothesis.

According to this hypothesis, when a body is in motion it becomes
shortened in the direction of motion by a certain proportion depending
upon its velocity. The amount of the contraction was to be just enough
to account for the negative result of the Michelson-Morley experiment.
The journey up stream and down again was to have been really a shorter
journey than the one across the stream, and was to have been just so
much shorter as would enable the slower light wave to traverse it in
the same time. Of course the shortening could never be detected by
measurement, because our measuring rods would share it. A foot rule
placed in the line of the earth’s motion would be shorter than the
same foot rule placed at right angles to the earth’s motion. This
point of view resembles nothing so much as the White Knight’s “plan to
dye my whiskers green, and always use so large a fan that they could
not be seen.” The odd thing was that the plan worked well enough. Later
on, when Einstein propounded his special theory of relativity (1905),
it was found that the theory was in a certain sense correct, but only
in a certain sense. That is to say, the supposed contraction is not
a physical fact, but a result of certain conventions of measurement
which, when once the right point of view has been found, are seen to
be such as we are almost compelled to adopt. But I do not wish yet to
set forth Einstein’s solution of the puzzle. For the present, it is the
nature of the puzzle itself that I want to make clear.

On the face of it, and apart from hypotheses _ad hoc_, the
Michelson-Morley experiment (in conjunction with others) showed that,
relatively to the earth, the velocity of light is the same in all
directions, and that this is equally true at all times of the year,
although the direction of the earth’s motion is always changing as
it goes round the sun. Moreover, it appeared that this is not a
peculiarity of the earth, but is true of all bodies: if a light signal
is sent out from a body, that body will remain at the center of the
waves as they travel outwards, no matter how it may be moving—at
least, that will be the view of observers moving with the body. This
was the plain and natural meaning of the experiments, and Einstein
succeeded in inventing a theory which accepted it. But at first it was
thought logically impossible to accept this plain and natural meaning.

A few illustrations will make it clear how very odd the facts are. When
a shell is fired, it moves faster than sound: the people at whom it is
fired first see the flash, then (if they are lucky) see the shell go
by, and last of all hear the report. It is clear that if you could put
a scientific observer on the shell, he would never hear the report, as
the shell would burst and kill him before the sound had overtaken him.
But if sound worked on the same principles as light, our observer would
hear everything just as if he were at rest. In that case, if a screen,
suitable for producing echoes, were attached to the shell and traveling
with it, say a hundred yards in front of it, our observer would hear
the echo of the report from the screen after just the same interval
of time as if he and the shell were at rest. This, of course, is an
experiment which cannot be performed, but others which can be performed
will show the difference. We might find some place on a railway where
there is an echo from a place further along the railway—say a place
where the railway goes into a tunnel—and when a train is traveling
along the railway, let a man on the bank fire a gun. If the train is
traveling towards the echo, the passengers will hear the echo sooner
than the man on the bank; if it is traveling in the opposite direction,
they will hear it later. But these are not quite the circumstances
of the Michelson-Morley experiment. The mirrors in that experiment
correspond to the echo, and the mirrors are moving with the earth, so
that echo ought to move with the train. Let us suppose that the shot
is fired from the guard’s van, and the echo comes from a screen on the
engine. We will suppose the distance from the guard’s van to the engine
to be the distance that sound can travel in a second (about one-fifth
of a mile), and the speed of the train to be one-twelfth of the speed
of sound (about sixty miles an hour). We now have an experiment which
can be performed by the people in the train. If the train were at rest,
the guard would hear the echo in two seconds; as it is, he will hear it
in 2 and ²/₁₄₃ seconds. From this difference, if he knows the velocity
of sound, he can calculate the velocity of the train, even if it is a
foggy night so that he cannot see the banks. But if sound behaved like
light, he would hear the echo in two seconds however fast the train
might be traveling.

Various other illustrations will help to show how extraordinary—from
the point of view of tradition and common sense—are the facts about
the velocity of light. Every one knows that if you are on an escalator
you reach the top sooner if you walk up than if you stand still. But if
the escalator moved with the velocity of light (which it does not do
even in New York), you would reach the top at exactly the same moment
whether you walked up or stood still. Again: if you are walking along
a road at the rate of four miles an hour, and a motor-car passes you
going in the same direction at the rate of forty miles an hour, if you
and the motor-car both keep going the distance between you after an
hour will be thirty-six miles. But if the motor-car met you, going in
the opposite direction, the distance after an hour would be forty-four
miles. Now if the motor-car were traveling with the velocity of light,
it would make no difference whether it met or passed you: in either
case, it would, after a second, be 186,000 miles from you. It would
also be 186,000 miles from any other motor-car which happened to be
passing or meeting you less rapidly at the previous second. This seems
impossible: how can the car be at the same distance from a number of
different points along the road?

Let us take another illustration. When a fly touches the surface of
a stagnant pool, it causes ripples which move outwards in widening
circles. The center of the circle at any moment is the point of the
pool touched by the fly. If the fly moves about over the surface of
the pool, it does not remain at the center of the ripples. But if the
ripples were waves of light, and the fly were a skilled physicist,
it would find that it always remained at the center of the ripples,
however it might move. Meanwhile a skilled physicist sitting beside the
pool would judge, as in the case of ordinary ripples, that the center
was not the fly, but the point of the pool touched by the fly. And if
another fly had touched the water at the same spot at the same moment,
it also would find that it remained at the center of the ripples, even
if it separated itself widely from the first fly. This is exactly
analogous to the Michelson-Morley experiment. The pool corresponds to
the ether; the fly corresponds to the earth; the contact of the fly and
the pool corresponds to the light signal which Messrs. Michelson and
Morley send out; and the ripples correspond to the light waves.

Such a state of affairs seems, at first sight, quite impossible. It
is no wonder that, although the Michelson-Morley experiment was made
in 1881, it was not rightly interpreted until 1905. Let us see what,
exactly, we have been saying. Take the man walking along a road and
passed by a motor-car. Suppose there are a number of people at the same
point of the road, some walking, some in motor-cars; suppose they are
going at varying rates, some in one direction and some in another. I
say that if, at this moment, a light flash is sent out from the place
where they all are, the light waves will be 186,000 miles from each
one of them after a second by his watch, although the travelers will
not any longer be all in the same place. At the end of a second by your
watch it will be 186,000 miles from you, and it will also be 186,000
miles from a person who met you when it was sent out, but was moving in
the opposite direction, after a second by his watch—assuming both to
be perfect watches. How can this be?

There is only one way of explaining such facts, and that is, to assume
that watches and clocks are affected by motion. I do not mean that
they are affected in ways that could be remedied by greater accuracy
in construction; I mean something much more fundamental. I mean that,
if you say an hour has elapsed between two events, and if you base
this assertion upon ideally careful measurements with ideally accurate
chronometers, another equally precise person, who has been moving
rapidly relatively to you, may judge that the time was more or less
than an hour. You cannot say that one is right and the other wrong,
any more than you could if one used a clock showing Greenwich time and
another a clock showing New York time. How this comes about, I shall
explain in the next chapter.

There are other curious things about the velocity of light. One is,
that no material body can ever travel as fast as light, however great
may be the force to which it is exposed, and however long the force
may act. An illustration may help to make this clear. At exhibitions
one sometimes sees a series of moving platforms, going round and round
in a circle. The outside one goes at four miles an hour; the next
goes four miles an hour faster than the first; and so on. You can
step across from each to the next; until you find yourself going at a
tremendous pace. Now you might think that, if the first platform does
four miles an hour, and the second does four miles an hour relatively
to the first, then the second does eight miles an hour relatively to
the ground. This is an error; it does a little less, though so little
less that not even the most careful measurements could detect the
difference. I want to make quite clear what it is that I mean. I will
suppose that, in the morning, when the apparatus is just about to
start, three men with ideally accurate chronometers stand in a row, one
on the ground, one on the first platform, and one on the second. The
first platform moves at the rate of four miles an hour with respect
to the ground. Four miles an hour is 352 feet in a minute. The man on
the ground, after a minute by his watch, notes the place on the ground
opposite the man on the first platform, who has been standing still
while the platform carried him along. The man on the ground measures
the distance on the ground from himself to the point opposite the
man on the first platform, and finds it is 352 feet. The man on the
first platform, after a minute by his watch, notes the point on his
platform opposite to the man on the second platform. The man on the
first platform measures the distance from himself to the point opposite
the man on the second platform; it is again 352 feet. Problem: how far
will the man on the ground judge that the man on the second platform
has traveled in a minute? That is to say, if the man on the ground,
after a minute by his watch, notes the place on the ground opposite
the man on the second platform, how far will this be from the man on
the ground? You would say, twice 352 feet, that is to say, 704 feet.
But in fact it will be a little less, though so little less as to
be inappreciable. The discrepancy is owing to the fact that the two
watches do not keep perfect time, in spite of the fact that each is
accurate from its owner’s point of view. If you had a long series of
such moving platforms, each moving four miles an hour relatively to the
one before it, you would never reach a point where the last was moving
with the velocity of light relatively to the ground, not even if you
had millions of them. The discrepancy, which is very small for small
velocities, becomes greater as the velocity increases, and makes the
velocity of light an unattainable limit. How all this happens, is the
next topic with which we must deal.

    _Note._ The negative result of the
    Michelson-Morley experiment has recently been called
    in question by Professor Dayton C. Miller, as a
    result of observations by what is said to be an
    improved method. His claim is set forth by Professor
    Silberstein in _Nature_, May 23, 1925, and
    discussed unfavorably by Eddington in the issue of
    June 6. The matter is _sub judice_, but it seems
    highly questionable whether the results bear out the
    interpretation which is put upon them.

CHAPTER IV: CLOCKS AND FOOT RULES

Until the advent of the special theory of relativity, no one had
thought that there could be any ambiguity in the statement that
two events in different places happened at the same time. It might
be admitted that, if the places were very far apart, there might
be difficulty in finding out for certain whether the events were
simultaneous, but every one thought the meaning of the question
perfectly definite. It turned out, however, that this was a mistake.
Two events in distant places may appear simultaneous to one observer
who has taken all due precautions to insure accuracy (and, in
particular, has allowed for the velocity of light), while another
equally careful observer may judge that the first event preceded
the second, and still another may judge that the second preceded
the first. This would happen if the three observers were all moving
rapidly relatively to each other. It would not be the case that one
of them would be right and the other two wrong: they would all be
equally right. The time order of events is in part dependent upon the
observer; it is not always and altogether an intrinsic relation between
the events themselves. Einstein has shown, not only that this view
accounts for the phenomena, but also that it is the one which ought
to have resulted from careful reasoning based upon the old data. In
actual fact, however, no one noticed the logical basis of the theory
of relativity until the odd results of experiment had given a jog to
people’s reasoning powers.

How should we naturally decide whether two events in different places
were simultaneous? One would naturally say: they are simultaneous
if they are seen simultaneously by a person who is exactly half-way
between them. (There is no difficulty about the simultaneity of two
events in the _same_ place, such, for example, as seeing a light
and hearing a noise.) Suppose two flashes of lightning fall in two
different places, say Greenwich Observatory and Kew Observatory.
Suppose that St. Paul’s is half-way between them, and that the flashes
appear simultaneous to an observer on the dome of St. Paul’s. In that
case, a man at Kew will see the Kew flash first, and a man at Greenwich
will see the Greenwich flash first, because of the time taken by
light to travel over the intervening distance. But all three, if they
are ideally accurate observers, will judge that the two flashes were
simultaneous, because they will make the necessary allowance for the
time of transmission of the light. (I am assuming a degree of accuracy
far beyond human powers.) Thus, so far as observers on the earth are
concerned, the definition of simultaneity will work well enough, so
long as we are dealing with events on the surface of the earth. It
gives results which are consistent with each other, and can be used for
terrestrial physics in all problems in which we can ignore the fact
that the earth moves.

But our definition is no longer so satisfactory when we have two sets
of observers in rapid motion relatively to each other. Suppose we see
what would happen if we substitute sound for light, and defined two
occurrences as simultaneous when they are heard simultaneously by a
man half-way between them. This alters nothing in the principle, but
makes the matter easier owing to the much slower velocity of sound.
Let us suppose that on a foggy night two men belonging to a gang of
brigands shoot the guard and engine driver of a train. The guard is at
the end of the train; the brigands are on the line, and shoot their
victims at close quarters. An old gentleman who is exactly in the
middle of the train hears the two shots simultaneously. You would say,
therefore, that the two shots were simultaneous. But a station master
who is exactly half-way between the two brigands hears the shot which
kills the guard first. An Australian millionaire uncle of the guard
and the engine driver (who are cousins) has left his whole fortune to
the guard, or, should he die first, to the engine driver. Vast sums
are involved in the question of which died first. The case goes to the
House of Lords, and the lawyers on both sides, having been educated at
Oxford, are agreed that either the old gentleman or the station master
must have been mistaken. In fact, both may perfectly well be right. The
train travels away from the shot at the guard, and towards the shot at
the engine driver; therefore the noise of the shot at the guard has
farther to go before reaching the old gentleman than the shot at the
engine driver has. Therefore if the old gentleman is right in saying
that he heard the two reports simultaneously, the station master must
be right in saying that he heard the shot at the guard first.

We, who live on the earth, would naturally, in such a case, prefer
the view of simultaneity obtained from a person at rest on the earth
to the view of a person traveling in a train. But in theoretical
physics no such parochial prejudices are permissible. A physicist on a
comet, if there were one, would have just as good a right to his view
of simultaneity as an earthly physicist has to his, but the results
would differ, in just the same sort of way as in our illustration of
the train and the shots. The train is not any more “really” in motion
than the earth; there is no “really” about it. You might imagine a
rabbit and a hippopotamus arguing as to whether man is “really” a large
animal; each would think his own point of view the natural one, and
the other a pure flight of fancy. There is just as little substance
in an argument as to whether the earth or the train is “really” in
motion. And, therefore, when we are defining simultaneity between
distant events, we have no right to pick and choose among different
bodies to be used in defining the point half-way between the events.
All bodies have an equal right to be chosen. But if, for one body, the
two events are simultaneous according to the definition, there will
be other bodies for which the first precedes the second, and still
others for which the second precedes the first. We cannot therefore
say unambiguously that two events in distant places are simultaneous.
Such a statement only acquires a definite meaning in relation to a
definite observer. It belongs to the subjective part of our observation
of physical phenomena, not to the objective part which is to enter into
physical laws.

This question of time in different places is perhaps, for the
imagination, the most difficult aspect of the theory of relativity. We
are accustomed to the idea that everything can be dated. Historians
make use of the fact that there was an eclipse of the sun visible in
China on August 29 in the year 776 B. C.[1] No doubt astronomers could
tell the exact hour and minute when the eclipse began to be total at
any given spot in North China. And it seems obvious that we can speak
of the positions of the planets at a given instant. The Newtonian
theory enables us to calculate the distance between the earth and (say)
Jupiter at a given time by the Greenwich clocks; this enables us to
know how long light takes at that time to travel from Jupiter to the
earth—say half an hour; this enables us to infer that half an hour ago
Jupiter was where we see it now. All this seems obvious. But in fact it
only works in practice because the relative velocities of the planets
are very small compared with the velocity of light. When we judge that
an event on the earth and an event on Jupiter have happened at the
same time—for example, that Jupiter eclipsed one of his moons when
the Greenwich clocks showed twelve midnight—a person moving rapidly
relatively to the earth would judge differently, assuming that both he
and we had made the proper allowance for the velocity of light. And
naturally the disagreement about simultaneity involves a disagreement
about periods of time. If we judged that two events on Jupiter were
separated by twenty-four hours, another person might judge that they
were separated by a longer time, if he were moving rapidly relatively
to Jupiter and the earth.

[1] A contemporary Chinese ode, after giving the day of the year
correctly, proceeds:

    “For the moon to be eclipsed
     Is but an ordinary matter.
     Now that the sun has been eclipsed,
     How bad it is.”

The universal cosmic time which used to be taken for granted is thus no
longer admissible. For each body, there is a definite time order for
the events in its neighborhood; this may be called the “proper” time
for that body. Our own experience is governed by the proper time for
our own body. As we all remain very nearly stationary on the earth,
the proper times of different human beings agree, and can be lumped
together as terrestrial time. But this is only the time appropriate to
_large_ bodies on the earth. For Beta-particles in laboratories, quite
different times would be wanted; it is because we insist upon using
our own time that these particles seem to increase in mass with rapid
motion. From their own point of view, their mass remains constant,
and it is we who suddenly grow thin or corpulent. The history of a
physicist as observed by a Beta-particle would resemble Gulliver’s
travels.

The question now arises: what really is measured by a clock? When we
speak of a clock in the theory of relativity, we do not mean only
clocks made by human hands: we mean anything which goes through some
regular periodic performance. The earth is a clock, because it rotates
once in every twenty-three hours and fifty-six minutes. An atom is a
clock, because the electrons go round the nucleus a certain number of
times in a second; its properties as a clock are exhibited to us in
its spectrum, which is due to light waves of various frequencies. The
world is full of periodic occurrences, and fundamental mechanisms,
such as atoms, show an extraordinary similarity in different parts of
the universe. Any one of these periodic occurrences may be used for
measuring time; the only advantage of humanly manufactured clocks is
that they are specially easy to observe. One question is: If cosmic
time is abandoned, what is really measured by a clock in the wide sense
that we have just given to the term?

Each clock gives a correct measure of its own “proper” time, which,
as we shall see presently, is an important physical quantity. But it
does not give an accurate measure of any physical quantity connected
with events on bodies that are moving rapidly in relation to it. It
gives one datum towards the discovery of a physical quantity connected
with such events, but another datum is required, and this has to be
derived from measurement of distances in space. Distances in space,
like periods of time, are in general not objective physical facts, but
partly dependent upon the observer. How this comes about must now be
explained.

First of all, we have to think of the distance between two events, not
between two bodies. This follows at once from what we have found as
regards time. If two bodies are moving relatively to each other—and
this is really always the case—the distance between them will be
continually changing, so that we can only speak of the distance
between them at a given time. If you are in a train traveling towards
Edinburgh, we can speak of your distance from Edinburgh at a given
time. But, as we said, different observers will judge differently as
to what is the “same” time for an event in the train and an event in
Edinburgh. This makes the measurement of distances relative, in just
the same way as the measurement of times has been found to be relative.
We commonly think that there are two separate kinds of interval between
two events, an interval in space and an interval in time: between your
departure from London and your arrival in Edinburgh, there are 400
miles and ten hours. We have already seen that another observer will
judge the time differently; it is even more obvious that he will judge
the distance differently. An observer in the sun will think the motion
of the train quite trivial, and will judge that you have traveled the
distance traveled by the earth in its orbit and its diurnal rotation.
On the other hand, a flea in the railway carriage will judge that you
have not moved at all in space, but have afforded him a period of
pleasure which he will measure by his “proper” time, not by Greenwich
Observatory. It cannot be said that you or the sun dweller or the
flea are mistaken: each is equally justified, and is only wrong if he
ascribes an objective validity to his subjective measures. The distance
in space between two events is, therefore, not in itself a physical
fact. But, as we shall see, there is a physical fact which can be
inferred from the distance in time together with the distance in space.
This is what is called the “interval” in space-time.

Taking any two events in the universe, there are two different
possibilities as to the relation between them. It may be physically
possible for a body to travel so as to be present at both events, or it
may not. This depends upon the fact that no body can travel as fast as
light. Suppose, for example, that it were possible to send out a flash
of light from the earth and have it reflected back from the moon. The
time between the sending of the flash and the return of the reflection
would be about two and a half seconds. No body could travel so fast
as to be present on the earth during any part of those two and a half
seconds and also present on the moon at the moment of the arrival of
the flash, because in order to do so the body would have to travel
faster than light. But theoretically a body could be present on the
earth at any time before or after those two and a half seconds and also
present on the moon at the time when the flash arrived. When it is
physically impossible for a body to travel so as to be present at both
events, we shall say that the interval[2] between the two events is
“space-like”; when it is physically possible for a body to be present
at both events, we shall say that the interval between the two events
is “time-like.” When the interval is “space-like,” it is possible for
a body to move in such a way that an observer on the body will judge
the two events to be simultaneous. In that case, the “interval” between
the two events is what such an observer will judge to be the distance
in space between them. When the interval is “time-like,” a body can
be present at both events; in that case, the “interval” between the
two events is what an observer on the body will judge to be the time
between them, that is to say, it is his “proper” time between the two
events. There is a limiting case between the two, when the two events
are parts of one light flash—or, as we might say, when the one event
is the seeing of the other. In that case, the interval between the two
events is zero.

[2] I shall define “interval” in a moment.

There are thus three cases. (1) It may be possible for a ray of light
to be present at both events; this happens whenever one of them is the
seeing of the other. In this case the interval between the two events
is zero. (2) It may happen that no body can travel from one event to
the other, because in order to do so it would have to travel faster
than light. In that case, it is always physically possible for a body
to travel in such a way that an observer on the body would judge the
two events to be simultaneous. The interval is what he would judge to
be the distance in space between the two events. Such an interval is
called “space-like.” (3) It may be physically possible for a body to
travel so as to be present at both events; in that case, the interval
between them is what an observer on such a body will judge to be the
time between them. Such an interval is called “time-like.”

The interval between two events is a physical fact about them, not
dependent upon the particular circumstances of the observer.

There are two forms of the theory of relativity, the special and the
general. The former is in general only approximate, but is exact at
great distances from gravitating matter. When the special theory can
be applied, the interval can be calculated when we know the distance
in space and the distance in time between the two events, estimated by
any observer. If the distance in space is greater than the distance
that light would have traveled in the time, the separation is
space-like. Then the following construction gives the interval between
the two events: Draw a line =AB= as long as the distance that light
would travel in the time; round =A= describe a circle whose radius is
the distance in space between the two events; through =B= draw =BC=
perpendicular to =AB=, meeting the circle in =C=. Then =BC= is the
length of the interval between the two events.

When the distance is time-like, use the same figure, but let =AC= be
now the distance that light would travel in the time, while =AB= is the
distance in space between the two events. The interval between them is
now the time that light would take to travel the distance =BC=.

Although =AB= and =AC= are different for different observers, =BC= is
the same length for all observers, subject to corrections made by the
general theory. It represents the one interval in “space-time” which
replaces the two intervals in space and time of the older physics. So
far, this notion of interval may appear somewhat mysterious, but as we
proceed it will grow less so, and its reason in the nature of things
will gradually emerge.

CHAPTER V: SPACE-TIME

Everybody who has ever heard of relativity knows the phrase
“space-time,” and knows that the correct thing is to use this phrase
when formerly we should have said “space _and_ time.” But very few
people who are not mathematicians have any clear idea of what is meant
by this change of phraseology. Before dealing further with the special
theory of relativity, I want to try to convey to the reader what is
involved in the new phrase “space-time,” because that is, from a
philosophical and imaginative point of view, perhaps the most important
of all the novelties that Einstein has introduced.

Suppose you wish to say where and when some event has occurred—say
an explosion on an airship—you will have to mention four quantities,
say the latitude and longitude, the height above the ground, and the
time. According to the traditional view, the first three of these
give the position in space, while the fourth gives the position in
time. The three quantities that give the position in space may be
assigned in all sorts of ways. You might, for instance, take the
plane of the equator, the plane of the meridian of Greenwich, and the
plane of the ninetieth meridian, and say how far the airship was from
each of these planes; these three distances would be what are called
“Cartesian co-ordinates,” after Descartes. You might take any other
three planes all at right angles to each other, and you would still
have Cartesian co-ordinates. Or you might take the distance from London
to a point vertically below the airship, the direction of this distance
(northeast, west-southwest, or whatever it might be), and the height of
the airship above the ground. There are an infinite number of such ways
of fixing the position in space, all equally legitimate; the choice
between them is merely one of convenience.

When people said that space had three dimensions, they meant just this:
that three quantities were necessary in order to specify the position
of a point in space, but that the method of assigning these quantities
was wholly arbitrary.

With regard to time, the matter was thought to be quite different. The
only arbitrary elements in the reckoning of time were the unit, and
the point of time from which the reckoning started. One could reckon
in Greenwich time, or in Paris time, or in New York time; that made a
difference as to the point of departure. One could reckon in seconds,
minutes, hours, days, or years; that was a difference of unit. Both
these were obvious and trivial matters. There was nothing corresponding
to the liberty of choice as to the method of fixing position in space.
And, in particular, it was thought that the method of fixing position
in space and the method of fixing position in time could be made wholly
independent of each other. For these reasons, people regarded time and
space as quite distinct.

The theory of relativity has changed this. There are now a number of
different ways of fixing position in time, which do not differ merely
as to the unit and the starting point. Indeed, as we have seen, if one
event is simultaneous with another in one reckoning, it will precede
it in another, and follow it in a third. Moreover, the space and time
reckonings are no longer independent of each other. If you alter the
way of reckoning position in space, you may also alter the time
interval between two events. If you alter the way of reckoning time,
you may also alter the distance in space between two events. Thus space
and time are no longer independent, any more than the three dimensions
of space are. We still need four quantities to determine the position
of an event, but we cannot, as before, divide off one of the four as
quite independent of the other three.

It is not quite true to say that there is no longer any distinction
between time and space. As we have seen, there are time-like intervals
and space-like intervals. But the distinction is of a different sort
from that which was formerly assumed. There is no longer a universal
time which can be applied without ambiguity to any part of the
universe; there are only the various “proper” times of the various
bodies in the universe, which agree approximately for two bodies which
are not in rapid relative motion, but never agree exactly except for
two bodies which are at rest relatively to each other.

The picture of the world which is required for this new state of
affairs is as follows: Suppose an event =E= occurs to me, and
simultaneously a flash of light goes out from me in all directions.
Anything that happens to any body after the light from the flash has
reached it is definitely after the event =E= in any system of reckoning
time. Any event anywhere which I could have seen before the event =E=
occurred to me is definitely before the event =E= in any system of
reckoning time. But any event which happened in the intervening time
is not definitely either before or after the event =E=. To make the
matter definite: suppose I could observe a person in Sirius, and he
could observe me. Anything which he does, and which I see before the
event =E= occurs to me, is definitely before =E=; anything he does
after he has seen the event =E= is definitely after =E=. But anything
that he does before he sees the event =E=, but so that I see it after
the event =E= has happened, is not definitely before or after =E=.
Since light takes many years to travel from Sirius to the earth, this
gives a period of twice as many years in Sirius which may be called
“contemporary” with =E=, since these years are not definitely before or
after =E=.

Dr. A. A. Robb, in his _Theory of Time and Space_, suggests a point
of view which may or may not be philosophically fundamental, but is
at any rate a help in understanding the state of affairs we have
been describing. He maintains that one event can only be said to be
definitely _before_ another if it can influence that other in some
way. Now influences spread from a center at varying rates. Newspapers
exercise an influence emanating from London at an average rate of about
twenty miles an hour—rather more for long distances. Anything a man
does because of what he reads in the newspaper is clearly subsequent
to the printing of the newspaper. Sounds travel much faster: it would
be possible to arrange a series of loud speakers along the main roads,
and have newspapers shouted from each to the next. But telegraphing is
quicker, and wireless telegraphy travels with the velocity of light,
so that nothing quicker can ever be hoped for. Now what a man does in
consequence of receiving a wireless message he does _after_ the message
was sent; the meaning here is quite independent of conventions as to
the measurement of time. But anything that he does while the message
is on its way cannot be influenced by the sending of the message, and
cannot influence the sender until some little time after he sent the
message. That is to say, if two bodies are widely separated, neither
can influence the other except after a certain lapse of time; what
happens before that time has elapsed cannot affect the distant body.
Suppose, for instance, that some notable event happens on the sun:
there is a period of sixteen minutes on the earth during which no event
on the earth can have influenced or been influenced by the said notable
event on the sun. This gives a substantial ground for regarding that
period of sixteen minutes on the earth as neither before nor after the
event on the sun.

The paradoxes of the special theory of relativity are only paradoxes
because we are unaccustomed to the point of view, and in the habit
of taking things for granted when we have no right to do so. This is
especially true as regards the measurement of lengths. In daily life,
our way of measuring lengths is to apply a foot rule or some other
measure. At the moment when the foot rule is applied, it is at rest
relatively to the body which is being measured. Consequently the length
that we arrive at by measurement is the “proper” length, that is to
say, the length as estimated by an observer who shares the motion of
the body. We never, in ordinary life, have to tackle the problem of
measuring a body which is in continual motion. And even if we did, the
velocities of visible bodies on the earth are so small relatively to
the earth that the anomalies dealt with by the theory of relativity
would not appear. But in astronomy, or in the investigation of atomic
structure, we are faced with problems which cannot be tackled in this
way. Not being Joshua, we cannot make the sun stand still while we
measure it; if we are to estimate its size, we must do so while it is
in motion relatively to us. And similarly if you want to estimate the
size of an electron, you have to do so while it is in rapid motion,
because it never stands still for a moment. This is the sort of problem
with which the theory of relativity is concerned. Measurement with a
foot rule, when it is possible, gives always the same result, because
it gives the “proper” length of a body. But when this method is not
possible, we find that curious things happen, particularly if the
body to be measured is moving very fast relatively to the observer. A
figure like the one at the end of the previous chapter will help us to
understand the state of affairs.

Let us suppose that the body on which we wish to measure lengths
is moving relatively to ourselves, and that in one second it moves
the distance =OM=. Let us draw a circle round =O= whose radius is
the distance that light travels in a second. Through =M= draw =MP=
perpendicular to =OM=, meeting the circle in =P=. Thus =OP= is the
distance that light travels in a second. The ratio of =OP= to =OM=
is the ratio of the velocity of light to the velocity of the body.
The ratio of =OP= to =MP= is the ratio in which apparent lengths are
altered by the motion. That is to say, if the observer judges that
two points in the line of motion on the moving body are at a distance
from each other represented by =MP=, a person moving with the body
would judge that they were at a distance represented (on the same
scale) by =OP=. Distances on the moving body at right angles to the
line of motion are not affected by the motion. The whole thing is
reciprocal; that is to say, if an observer moving with the body were to
measure lengths on the previous observer’s body, they would be altered
in just the same proportion. When two bodies are moving relatively
to each other, lengths on either appear shorter to the other than
to themselves. This is the Fitzgerald contraction, which was first
invented to account for the result of the Michelson-Morley experiment.
But it now emerges naturally from the fact that the two observers do
not make the same judgment of simultaneity.

The way in which simultaneity comes in is this: We say that two points
on a body are a foot apart when we can _simultaneously_ apply one end
of a foot rule to the one and the other end to the other. If, now, two
people disagree about simultaneity, and the body is in motion, they
will obviously get different results from their measurements. Thus the
trouble about time is at the bottom of the trouble about distance.

The ratio of =OP= to =MP= is the essential thing in all these matters.
Times and lengths and masses are all altered in this proportion when
the body concerned is in motion relatively to the observer. It will
be seen that, if =OM= is very much smaller than =OP=, that is to say,
if the body is moving very much more slowly than light, =MP= and =OP=
are very nearly equal, so that the alterations produced by the motion
are very small. But if =OM= is nearly as large as =OP=, that is to
say, if the body is moving nearly as fast as light, =MP= becomes very
small compared to =OP=, and the effects become very great. The apparent
increase of mass in swiftly moving particles had been observed,
and the right formula had been found, before Einstein invented his
special theory of relativity. In fact, Lorentz had arrived at the
formulæ called the “Lorentz transformation,” which embody the whole
mathematical essence of the special theory of relativity. But it was
Einstein who showed that the whole thing was what we ought to have
expected, and not a set of makeshift devices to account for surprising
experimental results. Nevertheless, it must not be forgotten that
experimental results were the original motive of the whole theory,
and have remained the ground for undertaking the tremendous logical
reconstruction involved in Einstein’s theories.

We may now recapitulate the reasons which have made it necessary to
substitute “space-time” for space and time. The old separation of
space and time rested upon the belief that there was no ambiguity in
saying that two events in distant places happened at the same time;
consequently it was thought that we could describe the topography of
the universe at a given instant in purely spatial terms. But now that
simultaneity has become relative to a particular observer, this is
no longer possible. What is, for one observer, a description of the
state of the world at a given instant, is, for another observer, a
series of events at various different times, whose relations are not
merely spatial but also temporal. For the same reason, we are concerned
with _events_, rather than with _bodies_. In the old theory, it was
possible to consider a number of bodies all at the same instant, and
since the time was the same for all of them it could be ignored. But
now we cannot do that if we are to obtain an objective account of
physical occurrences. We must mention the date at which a body is to be
considered, and thus we arrive at an “event,” that is to say, something
which happens at a given time. When we know the time and place of an
event in one observer’s system of reckoning, we can calculate its time
and place according to another observer. But we must know the time as
well as the place, because we can no longer ask what is its place for
the new observer at the “same” time as for the old observer. There is
no such thing as the “same” time for different observers, unless they
are at rest relatively to each other. We need four measurements to
fix a position, and four measurements fix the position of an event in
space-time, not merely of a body in space. Three measurements are not
enough to fix any position. That is the essence of what is meant by the
substitution of space-time for space and time.

CHAPTER VI: THE SPECIAL THEORY OF RELATIVITY

The special theory of relativity arose as a way of accounting for the
facts of electromagnetism. We have here a somewhat curious history. In
the eighteenth and early nineteenth centuries the theory of electricity
was wholly dominated by the Newtonian analogy. Two electric charges
attract each other if they are of different kinds, one positive and
one negative, but repel each other if they are of the same kind; in
each case, the force varies as the inverse square of the distance,
as in the case of gravitation. This force was conceived as an action
at a distance, until Faraday, by a number of remarkable experiments,
demonstrated the effect of the intervening medium. Faraday was no
mathematician; Clerk Maxwell first gave a mathematical form to the
results suggested by Faraday’s experiments. Moreover Clerk Maxwell gave
grounds for thinking that light is an electromagnetic phenomenon,
consisting of electromagnetic waves. The medium for the transmission of
electromagnetic effects could therefore be taken to be the ether, which
had long been assumed for the transmission of light. The correctness
of Maxwell’s theory of light was proved by the experiments of Hertz in
manufacturing electromagnetic waves; these experiments afforded the
basis for wireless telegraphy. So far, we have a record of triumphant
progress, in which theory and experiment alternately assume the leading
role. At the time of Hertz’s experiments, the ether seemed securely
established, and in just as strong a position as any other scientific
hypothesis not capable of direct verification. But a new set of facts
began to be discovered, and gradually the whole picture was changed.

The movement which culminated with Hertz was a movement for making
everything continuous. The ether was continuous, the waves in it were
continuous, and it was hoped that matter would be found to consist
of some continuous structure in the ether. Then came the discovery
of the electron, a small finite unit of negative electricity, and
the proton, a small finite unit of positive electricity. The most
modern view is that electricity is never found except in the form of
electrons and protons; all electrons have the same amount of negative
electricity, and all protons have an exactly equal and opposite amount
of positive electricity. It appeared that an electric current, which
had been thought of as a continuous phenomenon, consists of electrons
traveling one way and positive ions traveling the other way; it is no
more strictly continuous than the stream of people going up and down
an escalator. Then came the discovery of quanta, which seems to show
a fundamental discontinuity in all such natural processes as can be
measured with sufficient precision. Thus physics has had to digest new
facts and face new problems.

But the problems raised by the electron and the quantum are not those
that the theory of relativity can solve, at any rate at present; as
yet, it throws no light upon the discontinuities which exist in nature.
The problems solved by the special theory of relativity are typified by
the Michelson-Morley experiment. Assuming the correctness of Maxwell’s
theory of electromagnetism, there should have been certain discoverable
effects of motion through the ether; in fact, there were none. Then
there was the observed fact that a body in very rapid motion appears
to increase its mass; the increase is in the ratio of =OP= to =MP=
in the figure in the preceding chapter. Facts of this sort gradually
accumulated, until it became imperative to find some theory which would
account for them all.

Maxwell’s theory reduced itself to certain equations, known as
“Maxwell’s equations.” Through all the revolutions which physics has
undergone in the last fifty years, these equations have remained
standing; indeed they have continually grown in importance as well as
in certainty—for Maxwell’s arguments in their favor were so shaky that
the correctness of his results must almost be ascribed to intuition.
Now these equations were, of course, obtained from experiments in
terrestrial laboratories, but there was a tacit assumption that the
motion of the earth through the ether could be ignored. In certain
cases, such as the Michelson-Morley experiment, this ought not to have
been possible without measurable error; but it turned out to be always
possible. Physicists were faced with the odd difficulty that Maxwell’s
equations were more accurate than they should be. A very similar
difficulty was explained by Galileo at the very beginning of modern
physics. Most people think that if you let a weight drop it will fall
vertically. But if you try the experiment in the cabin of a moving
ship, the weight falls, in relation to the cabin, just as if the ship
were at rest; for instance, if it starts from the middle of the ceiling
it will drop onto the middle of the floor. That is to say, from the
point of view of an observer on the shore it does not fall vertically,
since it shares the motion of the ship. So long as the ship’s motion
is steady, everything goes on inside the ship as if the ship were not
moving. Galileo explained how this happens, to the great indignation
of the disciples of Aristotle. In orthodox physics, which is derived
from Galileo, a uniform motion in a straight line has no discoverable
effects. This was, in its day, as astonishing a form of relativity
as that of Einstein is to us. Einstein, in the special theory of
relativity, set to work to show how electromagnetic phenomena could be
unaffected by uniform motion through the ether if there be an ether.
This was a more difficult problem, which could not be solved by merely
adhering to the principles of Galileo.

The really difficult effort required for solving this problem was in
regard to time. It was necessary to introduce the notion of “proper”
time which we have already considered, and to abandon the old belief in
one universal time. The quantitative laws of electromagnetic phenomena
are expressed in Maxwell’s equations, and these equations are found
to be true for any observer, however he may be moving.[3] It is a
straight-forward mathematical problem to find out what differences
there must be between the measures applied by one observer and the
measures applied by another, if, in spite of their relative motion,
they are to find the same equations verified. The answer is contained
in the “Lorentz transformation,” found as a formula by Lorentz, but
interpreted and made intelligible by Einstein.

The Lorentz transformation tells us what estimate of distances and
periods of time will be made by an observer whose relative motion is
known, when we are given those of another observer. We may suppose that
you are in a train on a railway which travels due east. You have been
traveling for a time which, by the clocks at the station from which
you started, is _t_. At a distance _x_ from your starting point, as
measured by the people on the line, an event occurs at this moment—say
the line is struck by lightning. You have been traveling all the time
with a uniform velocity _v_. The question is: How far from you will you
judge that this event has taken place, and how long after you started
will it be by your watch, assuming that your watch is correct from the
point of view of an observer on the train?

[3] So long as he has no considerable acceleration. The treatment of
acceleration belongs to the _general_ theory of relativity.

Our solution of this problem has to satisfy certain conditions. It has
to bring out the result that the velocity of light is the same for all
observers, however they may be moving. And it has to make physical
phenomena—in particular, those of electromagnetism—obey the same
laws for different observers, however they may find their measures of
distances and times affected by their motion. And it has to make all
such effects on measurement reciprocal. That is to say, if you are in
a train and your motion affects your estimate of distances outside the
train, there must be an exactly similar change in the estimate which
people outside the train make of distances inside it. These conditions
are sufficient to determine the solution of the problem, but the
method of obtaining the solution cannot be explained without more
mathematics than is possible in the present work.

Before dealing with the matter in general terms, let us take an
example. Let us suppose that you are in a train on a long straight
railway, and that you are traveling at three-fifths of the velocity
of light. Suppose that you measure the length of your train, and find
that it is a hundred yards. Suppose that the people who catch a glimpse
of you as you pass succeed, by skilful scientific methods, in taking
observations which enable them to calculate the length of your train.
If they do their work correctly, they will find that it is eighty
yards long. Everything in the train will seem to them shorter in the
direction of the train than it does to you. Dinner plates, which you
see as ordinary circular plates, will look to the outsider as if they
were oval: they will seem only four-fifths as broad in the direction
in which the train is moving as in the direction of the breadth of the
train. And all this is reciprocal. Suppose you see out of the window a
man carrying a fishing rod which, by his measurement, is fifteen feet
long. If he is holding it upright, you will see it as he does; so you
will if he is holding it horizontally at right angles to the railway.
But if he is pointing it along the railway, it will seem to you to
be only twelve feet long. All lengths in the direction of motion are
diminished by twenty per cent, both for those who look into the train
from outside and for those who look out of the train from inside.

But the effects in regard to time are even more strange. This matter
has been explained with almost ideal lucidity by Eddington in _Space,
Time and Gravitation_. He supposes an aviator traveling, relatively to
the earth, at a speed of 161,000 miles a second, and he says:

“If we observed the aviator carefully we should infer that he was
unusually slow in his movements; and events in the conveyance moving
with him would be similarly retarded—as though time had forgotten to
go on. His cigar lasts twice as long as one of ours. I said ‘infer’
deliberately; we should _see_ a still more extravagant slowing down
of time; but that is easily explained, because the aviator is rapidly
increasing his distance from us and the light impressions take longer
and longer to reach us. The more moderate retardation referred to
remains after we have allowed for the time of transmission of light.
But here again reciprocity comes in, because in the aviator’s opinion
it is we who are traveling at 161,000 miles a second past him; and when
he has made all allowances, he finds that it is we who are sluggish.
Our cigar lasts twice as long as his.”

What a situation for envy! Each man thinks that the other’s cigar lasts
twice as long as his own. It may, however, be some consolation to
reflect that the other man’s visits to the dentist also last twice as
long.

This question of time is rather intricate, owing to the fact that
events which one man judges to be simultaneous another considers to be
separated by a lapse of time. In order to try to make clear how time
is affected, I shall revert to our railway train traveling due east at
a rate three-fifths of that of light. For the sake of illustration, I
assume that the earth is large and flat, instead of small and round.

If we take events which happen at a fixed point on the earth, and ask
ourselves how long after the beginning of the journey they will seem to
be to the traveler, the answer is that there will be that retardation
that Eddington speaks of, which means in this case that what seems an
hour in the life of the stationary person is judged to be an hour and a
quarter by the man who observes him from the train. Reciprocally, what
seems an hour in the life of the person in the train is judged by the
man observing him from outside to be an hour and a quarter. Each makes
periods of time observed in the life of the other a quarter as long
again as they are to the person who lives through them. The proportion
is the same in regard to times as in regard to lengths.

But when, instead of comparing events at the same place on the earth,
we compare events at widely separated places, the results are still
more odd. Let us now take all the events along the railway which, from
the point of view of a person who is stationary on the earth, happen
at a given instant, say the instant when the observer in the train
passes the stationary person. Of these events, those which occur at
points towards which the train is moving will seem to the traveler to
have already happened, while those which occur at points behind the
train will, for him, be still in the future. When I say that events
in the forward direction will seem to have already happened, I am
saying something not strictly accurate, because he will not yet have
seen them; but when he does see them, he will, after allowing for the
velocity of light, come to the conclusion that they must have happened
before the moment in question. An event which happens in the forward
direction along the railway, and which the stationary observer judges
to be now (or rather, will judge to have been now when he comes to know
of it), if it occurs at a distance along the line which light could
travel in a second, will be judged by the traveler to have occurred
three-quarters of a second ago. If it occurs at a distance from the two
observers which the man on the earth judges that light could travel
in a year, the traveler will judge (when he comes to know of it) that
it occurred nine months earlier than the moment when he passed the
earth dweller. And generally, he will ante-date events in the forward
direction along the railway by three-quarters of the time that it would
take light to travel from them to the man on the earth whom he is just
passing, and who holds that these events are happening now—or rather,
will hold that they happened now when the light from them reaches him.
Events happening on the railway behind the train will be post-dated by
an exactly equal amount.

We have thus a two-fold correction to make in the date of an event when
we pass from the terrestrial observer to the traveler. We must first
take five-fourths of the time as estimated by the earth dweller, and
then subtract three-fourths of the time that it would take light to
travel from the event in question to the earth dweller.

Take some event in a distant part of the universe, which becomes
visible to the earth dweller and the traveler just as they pass each
other. The earth dweller, if he knows how far off the event occurred,
can judge how long ago it occurred, since he knows the speed of light.
If the event occurred in the direction towards which the traveler is
moving, the traveler will infer that it happened twice as long ago as
the earth dweller thinks. But if it occurred in the direction from
which he has come, he will argue that it happened only half as long
ago as the earth dweller thinks. If the traveler moves at a different
speed, these proportions will be different.

Suppose now that (as sometimes occurs) two new stars have suddenly
flared up, and have just become visible to the traveler and to the
earth dweller whom he is passing. Let one of them be in the direction
towards which the train is traveling, the other in the direction from
which it has come. Suppose that the earth dweller is able, in some way,
to estimate the distance of the two stars, and to infer that light
takes fifty years to reach him from the one in the direction towards
which the traveler is moving, and one hundred years to reach him from
the other. He will then argue that the explosion which produced the
new star in the forward direction occurred fifty years ago, while the
explosion which produced the other new star occurred a hundred years
ago. The traveler will exactly reverse these figures: he will infer
that the forward explosion occurred a hundred years ago, and the
backward one fifty years ago. I assume that both argue correctly on
correct physical data. In fact, both are right, unless they imagine
that the other must be wrong. It should be noted that both will have
the same estimate of the velocity of light, because their estimates
of the distances of the two new stars will vary in exactly the same
proportion as their estimates of the times since the explosions.
Indeed, one of the main motives of this whole theory is to secure that
the velocity of light shall be the same for all observers, however they
may be moving. This fact, established by experiment, was incompatible
with the old theories, and made it absolutely necessary to admit
something startling. The theory of relativity is just as little
startling as is compatible with the facts. Indeed, after a time, it
ceases to seem startling at all.

There is another feature of very great importance in the theory we
have been considering, and that is that, although distances and times
vary for different observers, we can derive from them the quantity
called “interval,” which is the same for all observers. The “interval,”
in the special theory of relativity, is obtained as follows: Take
the square of the distance between two events, and the square of the
distance traveled by light in the time between the two events; subtract
the lesser of these from the greater, and the result is defined as
the square of the interval between the events. The interval is the
same for all observers, and represents a genuine physical relation
between the two events, which the time and the distance do not. We
have already given a geometrical construction for the interval at the
end of Chapter IV; this gives the same result as the above rule. The
interval is “time-like” when the time between the events is longer than
light would take to travel from the place of the one to the place
of the other; in the contrary case it is “space-like.” When the time
between the two events is exactly equal to the time taken by light to
travel from one to the other, the interval is zero; the two events are
then situated on parts of one light ray, unless no light happens to be
passing that way.

When we come to the general theory of relativity, we shall have to
generalize the notion of interval. The more deeply we penetrate into
the structure of the world, the more important this concept becomes;
we are tempted to say that it is the reality of which distances and
periods of time are confused representations. The theory of relativity
has altered our view of the fundamental structure of the world; that is
the source both of its difficulty and of its importance.

The remainder of this chapter may be omitted by readers who have not
even the most elementary acquaintance with geometry or algebra. But for
the benefit of those whose education has not been _entirely_ neglected,
I will add a few explanations of the general formula of which I have
hitherto given only particular examples. The general formula in
question is the “Lorentz transformation,” which tells, when one body
is moving in a given manner relatively to another, how to infer the
measures of lengths and times appropriate to the one body from those
appropriate to the other. Before giving the algebraical formulæ, I
will give a geometrical construction. As before, we will suppose that
there are two observers, whom we will call =O= and =O=′, one of whom is
stationary on the earth while the other is traveling at a uniform speed
along a straight railway. At the beginning of the time considered, the
two observers were at the same point of the railway, but now they are
separated by a certain distance. A flash of lightning strikes a point
=X= on the railway, and =O= judges that at the moment when the flash
takes place the observer in the train has reached the point =O=′. The
problem is: how far will =O=′ judge that he is from the flash, and
how long after the beginning of the journey (when he was at =O=) will
he judge that the flash took place? We are supposed to know =O=′s
estimates, and we want to calculate those of =O=′.

In the time that, according to =O=, has elapsed since the beginning of
the journey, let =OC= be the distance that light would have traveled
along the railway. Describe a circle about =O=, with =OC= as radius,
and through =O′= draw a perpendicular to the railway, meeting the
circle in =D=. On =OD= take a point =Y= such that =OY= is equal to =OX=
(=X= is the point of the railway where the lightning strikes). Draw
=YM= perpendicular to the railway, and =OS= perpendicular to =OD=. Let
=YM= and =OS= meet in =S=. Also let =DO′= produced and =OS= produced
meet in =R=. Through =X= and =C= draw perpendiculars to the railway
meeting =OS= produced in =Q= and =Z= respectively. Then =RQ= (as
measured by =O=) is the distance at which =O′= will believe himself to
be from the flash, not =O′X= as it would be according to the old view.
And whereas =O= thinks that, in the time from the beginning of the
journey to the flash, light would travel a distance =OC=, =O′= thinks
that the time elapsed is that required for light to travel the distance
=SZ= (as measured by =O=). The interval as measured by =O= is got by
subtracting the square on =OX= from the square on =OC=; the interval
as measured by =O′= is got by subtracting the square on =RQ= from the
square on =SZ=. A little very elementary geometry shows that these are
equal.

The algebraical formulæ embodied in the above construction are as
follows: From the point of view of =O=, let an event occur at a
distance _x_ along the railway, and at a time _t_ after the beginning
of the journey (when =O′= was at =O=). From the point of view of =O′=,
let the same event occur at a distance _x′_ along the railway, and at a
time _t′_ after the beginning of the journey. Let _c_ be the velocity
of light, and _v_ the velocity of =O′= relative to =O=. Put

                _c_
    β =    ————————————
           √(_c_² - _v_²)

Then

    _x′_ = β(_x_ - _vt_)

                 ( _vx_ )
    _t′_ = β(_t_ - —————)
                 ( _c²_ )

This is the Lorentz transformation, from which everything in this
chapter can be deduced.

CHAPTER VII: INTERVALS IN SPACE-TIME

The special theory of relativity, which we have been considering
hitherto, solved completely a certain definite problem: to account for
the experimental fact that, when two bodies are in uniform relative
motion, all the laws of physics, both those of ordinary dynamics and
those connected with electricity and magnetism, are exactly the same
for the two bodies. “Uniform” motion, here, means motion in a straight
line with constant velocity. But although one problem was solved by
the special theory, another was immediately suggested: what if the
motion of the two bodies is not uniform? Suppose, for instance, that
one is the earth while the other is a falling stone. The stone has
an accelerated motion: it is continually falling faster and faster.
Nothing in the special theory enables us to say that the laws of
physical phenomena will be the same for an observer on the stone as for
one on the earth. This is particularly awkward, as the earth itself
is, in an extended sense, a falling body: It has at every moment
an acceleration[4] towards the sun, which makes it go round the sun
instead of moving in a straight line. As our knowledge of physics is
derived from experiments on the earth, we cannot rest satisfied with
a theory in which the observer is supposed to have no acceleration.
The general theory of relativity removes this restriction, and allows
the observer to be moving in any way, straight or crooked, uniformly
or with an acceleration. In the course of removing the restriction,
Einstein was led to his new law of gravitation, which we shall consider
presently. The work was extraordinarily difficult, and occupied him for
ten years. The special theory dates from 1905, the general theory from
1915.

[4] This does not mean that its velocity is increasing, but that it
is changing its direction. The only sort of motion which is called
“unaccelerated” is motion with uniform velocity _in a straight line_.

It is obvious from experiences with which we are all familiar that an
accelerated motion is much more difficult to deal with than a uniform
one. When you are in a train which is traveling steadily, the motion
is not noticeable so long as you do not look out of the window; but
when the brakes are applied suddenly you are precipitated forwards,
and you become aware that something is happening without having to
notice anything outside the train. Similarly in a lift everything
seems ordinary while it is moving steadily, but at starting and
stopping, when its motion is accelerated, you have odd sensations
in the pit of the stomach. (We call a motion “accelerated” when it
is getting slower as well as when it is getting quicker; when it is
getting slower the acceleration is negative.) The same thing applies
to dropping a weight in the cabin of a ship. So long as the ship is
moving uniformly, the weight will behave, relatively to the cabin,
just as if the ship were at rest: if it starts from the middle of
the ceiling, it will hit the middle of the floor. But if there is an
acceleration everything is changed. If the boat is increasing its
speed very rapidly, the weight will seem to an observer in the cabin
to fall in a curve directed towards the stern; if the speed is being
rapidly diminished, the curve will be directed towards the bow. All
these facts are familiar, and they led Galileo and Newton to regard an
accelerated motion as something radically different, in its own nature,
from a uniform motion. But this distinction could only be maintained by
regarding motion as absolute, not relative. If all motion is relative,
the earth is accelerated relatively to the lift just as truly as the
lift relatively to the earth. Yet the people on the ground have no
sensations in the pits of their stomachs when the lift starts to go
up. This illustrates the difficulty of our problem. In fact, though
few physicists in modern times have believed in absolute motion, the
technique of mathematical physics still embodied Newton’s belief in it,
and a revolution in method was required to obtain a technique free from
this assumption. This revolution was accomplished in Einstein’s general
theory of relativity.

It is somewhat optional where we begin in explaining the new ideas
which Einstein introduced, but perhaps we shall do best by taking the
conception of “interval.” This conception, as it appears in the special
theory of relativity, is already a generalization of the traditional
notion of distance in space and time; but it is necessary to generalize
it still further. However, it is necessary first to explain a certain
amount of history, and for this purpose we must go back as far as
Pythagoras.

Pythagoras, like many of the greatest characters in history, perhaps
never existed: he is a semi-mythical character, who combined
mathematics and priestcraft in uncertain proportions. I shall, however,
assume that he existed, and that he discovered the theorem attributed
to him. He was roughly a contemporary of Confucius and Buddha; he
founded a religious sect, which thought it wicked to eat beans,
and a school of mathematicians, who took a particular interest in
right-angled triangles. The theorem of Pythagoras (the forty-seventh
proposition of Euclid) states that the sum of the squares on the two
shorter sides of a right-angled triangle is equal to the square on
the side opposite the right angle. No proposition in the whole of
mathematics has had such a distinguished history. We all learned to
“prove” it in youth. It is true that the “proof” proved nothing, and
that the only way to prove it is by experiment. It is also the case
that the proposition is not _quite_ true—it is only approximately
true. But everything in geometry, and subsequently in physics, has been
derived from it by successive generalizations. The latest of these
generalizations is the general theory of relativity.

The theorem of Pythagoras was itself, in all probability, a
generalization of an Egyptian rule of thumb. In Egypt, it had been
known for ages that a triangle whose sides are 3, 4, and 5 units of
length is a right-angled triangle; the Egyptians used this knowledge
practically in measuring their fields. Now if the sides of a triangle
are 3, 4, and 5 inches, the squares on these sides will contain
respectively 9, 16, and 25 square inches; and 9 and 16 added together
make 25. Three times three is written “3²”; four times four, “4²”; five
times five, “5².” So that we have

    3² + 4² = 5².

It is supposed that Pythagoras noticed this fact, after he had learned
from the Egyptians that a triangle whose sides are 3, 4 and 5 has a
right angle. He found that this could be generalized, and so arrived
at his famous theorem: In a right-angled triangle, the square on the
side opposite the right angle is equal to the sum of the squares on the
other two sides.

Similarly in three dimensions: if you take a right-angled solid block,
the square on the diagonal (the dotted line in the figure) is equal to
the sum of the squares on the three sides.

This is as far as the ancients got in this matter.

The next step of importance is due to Descartes, who made the theorem
of Pythagoras the basis of his method of analytical geometry. Suppose
you wish to map out systematically all the places on a plain—we will
suppose it small enough to make it possible to ignore the fact that
the earth is round. We will suppose that you live in the middle of the
plain. One of the simplest ways of describing the position of a place
is to say: starting from my house, go first such and such a distance
east, then such and such a distance north (or it may be west in the
first case, and south in the second). This tells you exactly where
the place is. In the rectangular cities of America, it is the natural
method to adopt: in New York you will be told to go so many blocks east
(or west) and then so many blocks north (or south). The distance you
have to go east is called _x_, and the distance you have to go north
is called _y_. (If you have to go west, _x_ is negative; if you have
to go south, _y_ is negative.) Let =O= be your starting point (the
“origin”); let =OM= be the distance you go east, and =MP= the distance
you go north. How far are you from home in a direct line when you reach
=P=? The theorem of Pythagoras gives the answer. The square on =OP= is
the sum of the squares on =OM= and =MP=. If =OM= is four miles, and
=MP= is three miles, =OP= is 5 miles. If =OM= is 12 miles and =MP= is 5
miles, =OP= is 13 miles, because 12² + 5² = 13². So that if you adopt
Descartes’ method of mapping, the theorem of Pythagoras is essential in
giving you the distance from place to place. In three dimensions the
thing is exactly analogous. Suppose that, instead of wanting merely
to fix positions on the plain, you want to fix stations for captive
balloons above it, you will then have to add a third quantity, the
height at which the balloon is to be. If you call the height _z_, and
if _r_ is the direct distance from =O= to the balloon, you will have

    _r_² = _x_² + _y_² + _z_²,

and from this you can calculate _r_ when you know _x_, _y_, and _z_.
For example, if you can get to the balloon by going 12 miles east, 4
miles north, and then 3 miles up, your distance from the balloon in a
straight line is 13 miles, because 12 × 12 = 144, 4 × 4 = 16, 3 × 3 =
9, 144 + 16 + 9 = 169 = 13 × 13.

But now suppose that, instead of taking a small piece of the earth’s
surface which can be regarded as flat, you consider making a map of
the world. An accurate map of the world on flat paper is impossible.
A globe can be accurate, in the sense that everything is produced
to scale, but a flat map cannot be. I am not talking of practical
difficulties, I am talking of a theoretical impossibility. For example:
the northern halves of the meridian of Greenwich and the ninetieth
meridian of west longitude, together with the piece of the equator
between them, make a triangle whose sides are all equal and whose
angles are all right angles. On a flat surface, a triangle of that sort
would be impossible. On the other hand, it is possible to make a square
on a flat surface, but on a sphere it is impossible. Suppose you try on
the earth: walk 100 miles west, then 100 miles north, then 100 miles
east, then 100 miles south. You might think this would make a square,
but it wouldn’t, because you would not at the end have come back to
your starting point. If you have time, you may convince yourself of
this by experiment. If not, you can easily see that it must be so. When
you are nearer the pole, 100 miles takes you through more longitude
than when you are nearer the equator, so that in doing your 100 miles
east (if you are in the northern hemisphere) you get to a point further
east than that from which you started. As you walk due south after
this, you remain further east than your starting point, and end up at a
different place from that in which you began. Suppose, to take another
illustration, that you start on the equator 4,000 miles east of the
Greenwich meridian; you travel till you reach the meridian, then you
travel northwards along it for 4,000 miles, through Greenwich and up
to the neighborhood of the Shetland Islands; then you travel eastward
for 4,000 miles, and then 4,000 miles south. This will take you to the
equator at a point 4,000 miles further east than the point from which
you started.

In a sense, what we have just been saying is not quite fair, because,
except on the equator, traveling due east is not the shortest route
from a place to another place due east of it. A ship traveling (say)
from New York to Lisbon, which is nearly due east, will start by going
a certain distance northward. It will sail on a “great circle,” that
is to say, a circle whose centre is the centre of the earth. This
is the nearest approach to a straight line that can be drawn on the
surface of the earth. Meridians of longitude are great circles, and so
is the equator, but the other parallels of latitude are not. We ought,
therefore, to have supposed that, when you reach the Shetland Islands,
you travel 4,000 miles, not due east, but along a great circle which
lands you at a point due east of the Shetland Islands. This, however,
only reinforces our conclusion: you will end at a point even further
east of your starting point than before.

What are the differences between the geometry on a sphere and the
geometry on a plane? If you make a triangle on the earth, whose sides
are great circles, you will not find that the angles of the triangle
add up to two right angles: they will add up to rather more. The amount
by which they exceed two right angles is proportional to the size of
the triangle. On a small triangle such as you could make with strings
on your lawn, or even on a triangle formed by three ships which can
just see each other, the angles will add up to so little more than two
right angles that you will not be able to detect the difference. But
if you take the triangle made by the equator, the Greenwich meridian,
and the ninetieth meridian, the angles add up to _three_ right angles.
And you can get triangles in which the angles add up to anything up to
six right angles. All this you could discover by measurements on the
surface of the earth, without having to take account of anything in the
rest of space.

The theorem of Pythagoras also will fail for distances on a sphere.
From the point of view of a traveler bound to the earth, the distance
between two places is their great circle distance, that is to say, the
shortest journey that a man can make without leaving the surface of
the earth. Now suppose you take three bits of great circles which make
a triangle, and suppose one of them is at right angles to another—to
be definite, let one be the equator and one a bit of the meridian of
Greenwich going northward from the equator. Suppose you go 3,000 miles
along the equator, and then 4,000 miles due north; how far will you
be from your starting point, estimating the distance along a great
circle? If you were on a plane, your distance would be 5,000 miles,
as we saw before. In fact, however, your great circle distance will be
considerably less than this. In a right-angled triangle on a sphere,
the square on the side opposite the right angle is less than the sum of
the squares on the other two sides.

These differences between the geometry on a sphere and the geometry on
a plane are intrinsic differences; that is to say, they enable you to
find out whether the surface on which you live is like a plane or like
a sphere, without requiring that you should take account of anything
outside the surface. Such considerations led to the next step of
importance in our subject, which was taken by Gauss, who flourished a
hundred years ago. He studied the theory of surfaces, and showed how to
develop it by means of measurements on the surfaces themselves, without
going outside them. In order to fix the position of a point in space,
we need three measurements; but in order to fix the position of a point
on a surface we need only two—for example, a point on the earth’s
surface is fixed when we know its latitude and longitude.

Now Gauss found that, whatever system of measurement you adopt,
and whatever the nature of the surface, there is always a way of
calculating the distance between two not very distant points of the
surface, when you know the quantities which fix their positions.
The formula for the distance is a generalization of the formula of
Pythagoras: it tells you the square of the distance in terms of the
squares of the differences between the measure quantities which fix
the points, and also the product of these two quantities. When you
know this formula, you can discover all the intrinsic properties of
the surface, that is to say, all those which do not depend upon its
relations to points outside the surface. You can discover, for example,
whether the angles of a triangle add up to two right angles, or more,
or less, or more in some cases and less in others.

But when we speak of a “triangle,” we must explain what we mean,
because on most surfaces there are no straight lines. On a sphere, we
shall replace straight lines by great circles, which are the nearest
possible approach to straight lines. In general, we shall take,
instead of straight lines, the lines that give the shortest route on
the surface from place to place. Such lines are called “geodesics.”
On the earth, the geodesics are great circles. In general, they are
the shortest way of traveling from point to point if you are unable
to leave the surface. They take the place of straight lines in the
intrinsic geometry of a surface. When we inquire whether the angles of
a triangle add up to two right angles or not, we mean to speak of a
triangle whose sides are geodesics. And when we speak of the distance
between two points, we mean the distance along a geodesic.

The next step in our generalizing process is rather difficult: it is
the transition to non-Euclidean geometry. We live in a world in which
space has three dimensions, and our empirical knowledge of space is
based upon measurement of small distances and of angles. (When I speak
of small distances, I mean distances that are small compared to those
in astronomy; all distances on the earth are small in this sense.) It
was formerly thought that we could be sure _à priori_ that space is
Euclidean—for instance, that the angles of a triangle add up to two
right angles. But it came to be recognized that we could not prove this
by reasoning; if it was to be known, it must be known as the result
of measurements. Before Einstein, it was thought that measurements
confirm Euclidean geometry within the limits of exactitude attainable;
now this is no longer thought. It is still true that we can, by what
may be called a natural artifice, cause Euclidean geometry to _seem_
true throughout a small region, such as the earth; but in explaining
gravitation Einstein is led to the view that over large regions where
there is matter we cannot regard space as Euclidean. The reasons for
this will concern us later. What concerns us now is the way in which
non-Euclidean geometry results from a generalization of the work of
Gauss.

There is no reason why we should not have the same circumstances in
three-dimensional space as we have, for example, on the surface of a
sphere. It might happen that the angles of a triangle would always
add up to more than two right angles, and that the excess would be
proportional to the size of the triangle. It might happen that the
distance between two points would be given by a formula analogous
to what we have on the surface of a sphere, but involving three
quantities instead of two. Whether this does happen or not, can only
be discovered by actual measurements. There are an infinite number of
such possibilities.

This line of argument was developed by Riemann, in his dissertation
“On the hypotheses which underlie geometry” (1854), which applied
Gauss’s work on surfaces to different kinds of three-dimensional
spaces. He showed that all the essential characteristics of a kind
of space could be deduced from the formula for small distances. He
assumed that, from the small distances in three given directions
which would together carry you from one point to another not far from
it, the distances between the two points could be calculated. For
instance, if you know that you can get from one point to another by
first moving a certain distance east, then a certain distance north,
and finally a certain distance straight up in the air, you are to be
able to calculate the distance from the one point to the other. And
the rule for the calculation is to be an extension of the theorem of
Pythagoras, in the sense that you arrive at the square of the required
distance by adding together multiples of the squares of the component
distances, together possibly with multiples of their products. From
certain characteristics in the formula, you can tell what sort of
space you have to deal with. These characteristics do not depend upon
the particular method you have adopted for determining the positions of
points.

In order to arrive at what we want for the theory of relativity, we
now have one more generalization to make: we have to substitute the
“interval” between events for the distance between points. This takes
us to space-time. We have already seen that, in the special theory
of relativity, the square of the interval is found by subtracting
the square of the distance between the events from the square of the
distance that light would travel in the time between them. In the
general theory, we do not assume this special form of interval, except
at a great distance from matter. Elsewhere, we assume to begin with a
general form, like that which Riemann used for distances. Moreover,
like Riemann, Einstein only assumes his formula for _neighboring_
events, that is to say, events which have only a small interval
between them. What goes beyond these initial assumptions depends upon
observation of the actual motion of bodies, in ways which we shall
explain in later chapters.

We may now sum up and re-state the process we have been describing.
In three dimensions, the position of a point relatively to a fixed
point (the “origin”) can be determined by assigning three quantities
(“co-ordinates”). For example, the position of a balloon relatively to
your house is fixed if you know that you will reach it by going first
a given distance due east, then another given distance due north,
then a third given distance straight up. When, as in this case, the
three co-ordinates are three distances all at right angles to each
other, which, taken successively, transport you from the origin to the
point in question, the square of the direct distance to the point in
question is got by adding up the squares of the three co-ordinates. In
all cases, whether in Euclidean or in non-Euclidean spaces, it is got
by adding multiples of the squares and products of the co-ordinates
according to an assignable rule. The co-ordinates may be any quantities
which fix the position of a point, provided that neighboring points
must have neighboring quantities for their co-ordinates. In the general
theory of relativity, we add a fourth co-ordinate to give the time, and
our formula gives “interval” instead of spatial distance; moreover we
assume the accuracy of our formula for small distances only. We assume
further that, at great distances from matter, the formula approximates
more and more closely to the formula for interval which is used in the
special theory.

We are now at last in a position to tackle Einstein’s theory of
gravitation.

CHAPTER VIII: EINSTEIN’S LAW OF GRAVITATION

Before tackling Einstein’s new law, it is as well to convince
ourselves, on logical grounds, that Newton’s law of gravitation cannot
be quite right.

Newton said that between any two particles of matter there is a force
which is proportional to the product of their masses and inversely
proportional to the square of their distance. That is to say, ignoring
for the present the question of mass, if there is a certain attraction