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Title: The A B C of Relativity
Author: Bertrand Russell
Release Date: January 4, 2022 [eBook #67104]
Language: English
Produced by: Tim Lindell and the Online Distributed Proofreading Team at
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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
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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.
[Illustration]
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.
[Illustration]
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=′.
[Illustration]
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.
[Illustration]
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.
[Illustration]
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
when the particles are a mile apart, there will be a quarter as much
attraction when they are two miles apart, a ninth as much when they
are three miles apart, and so on: the attraction diminishes much
faster than the distance increases. Now, of course, Newton, when he
spoke of the distance, meant the distance at a given time: He thought
there could be no ambiguity about time. But we have seen that this
was a mistake. What one observer judges to be the same moment on the
earth and the sun, another will judge to be two different moments.
“Distance at a given moment” is therefore a subjective conception,
which can hardly enter into a cosmic law. Of course, we could make
our law unambiguous by saying that we are going to estimate times as
they are estimated by Greenwich Observatory. But we can hardly believe
that the accidental circumstances of the earth deserve to be taken so
seriously. And the estimate of distance, also, will vary for different
observers. We cannot, therefore, allow that Newton’s form of the law of
gravitation can be quite correct, since it will give different results
according to which of many equally legitimate conventions we adopt.
This is as absurd as it would be if the question whether one man had
murdered another were to depend upon whether they were described by
their Christian names or their surnames. It is obvious that physical
laws must be the same whether distances are measured in miles or in
kilometers, and we are concerned with what is essentially only an
extension of the same principle.
Our measurements are conventional to an even greater extent than
is admitted by the special theory of relativity. Moreover, every
measurement is a physical process carried out with physical material;
the result is certainly an experimental datum, but may not be
susceptible of the simple interpretation which we ordinarily assign to
it. We are, therefore, not going to assume to begin with that we know
how to measure anything. We assume that there is a certain physical
quantity, called “interval,” which is a relation between two events
that are not widely separated; but we do not assume in advance that we
know how to measure it, beyond taking it for granted that it is given
by some generalization of the theorem of Pythagoras such as we spoke of
in the preceding chapter.
We do assume, however, that events have an _order_, and that this order
is four-dimensional. We assume, that is to say, that we know what we
mean by saying that a certain event is nearer to another than to a
third, so that before making accurate measurements we can speak of the
“neighborhood” of an event; and we assume that, in order to assign the
position of an event in space-time, four quantities (co-ordinates) are
necessary—_e.g._ in our former case of an explosion on an airship,
latitude, longitude, altitude and time. But we assume nothing about the
way in which these co-ordinates are assigned, except that neighboring
co-ordinates are assigned to neighboring events.
The way in which these numbers, called co-ordinates, are to be assigned
is neither wholly arbitrary nor a result of careful measurement—it
lies in an intermediate region. While you are making any continuous
journey, your co-ordinates must never alter by sudden jumps. In America
one finds that the houses between (say) Fourteenth Street and Fifteenth
Street are likely to have numbers between 1400 and 1500, while those
between Fifteenth Street and Sixteenth Street have numbers between
1500 and 1600, even if the 1400’s were not used up. This would not do
for our purposes, because there is a sudden jump when we pass from one
block to the next. Or again we might assign the time co-ordinate in the
following way: take the time that elapses between two successive births
of people called Smith; an event occurring between the births of the
3000th and the 3001st Smith known to history shall have a co-ordinate
lying between 3000 and 3001; the fractional part of its co-ordinate
shall be the fraction of a year that has elapsed since the birth of the
3000th Smith. (Obviously there could never be as much as a year between
two successive additions to the Smith family.) This way of assigning
the time co-ordinate is perfectly definite, but it is not admissible
for our purposes, because there will be sudden jumps between events
just before the birth of a Smith and events just after, so that in a
continuous journey your time co-ordinate will not change continuously.
It is assumed that, independently of measurement, we know what a
continuous journey is. And when your position in space-time changes
continuously, each of your four co-ordinates must change continuously.
One, two, or three of them may not change at all; but whatever change
does occur must be smooth, without sudden jumps. This explains what is
_not_ allowable in assigning co-ordinates.
To explain all the changes that are legitimate in your co-ordinates,
suppose you take a large piece of soft india-rubber. While it is in an
unstretched condition, measure little squares on it, each one-tenth
of an inch each way. Put in little tiny pins at the corners of the
squares. We can take as two of the co-ordinates of one of these pins
the number of pins passed in going to the right from a given pin
until we come just below the pin in question, and then the number of
pins we pass on the way up to this pin. In the figure, let =O= be the
pin we start from and =P= the pin to which we are going to assign
co-ordinates. =P= is in the fifth column and the third row, so its
co-ordinates in the plane of the india-rubber are to be 5 and 3.
[Illustration: Fig. 1.]
[Illustration: Fig. 2.]
Now take the india-rubber and stretch it and twist it as much as
you like. Let the pins now be in the shape they have in Fig. 2. The
divisions now no longer represent distances according to our usual
notions, but they will still do just as well as co-ordinates. We may
still take =P= as having the co-ordinates 5 and 3 in the plane of the
india-rubber; and we may still regard the india-rubber as being in a
plane, even if we have twisted it out of what we should ordinarily
call a plane. Such continuous distortions do not matter.
To take another illustration: instead of using a steel measuring rod to
fix our co-ordinates, let us use a live eel, which is wriggling all the
time. The distance from the tail to the head of the eel is to count as
one from the point of view of co-ordinates, whatever shape the creature
may be assuming at the moment. The eel is continuous, and its wriggles
are continuous, so it may be taken as our unit of distance in assigning
co-ordinates. Beyond the requirement of continuity, the method of
assigning co-ordinates is purely conventional, and therefore a live eel
is just as good as a steel rod.
We are apt to think that, for really careful measurements, it is better
to use a steel rod than a live eel. This is a mistake: not because
the eel tells us what the steel rod was thought to tell, but because
the steel rod really tells no more than the eel obviously does. The
point is, not that eels are really rigid, but that steel rods really
wriggle. To an observer in just one possible state of motion, the eel
would appear rigid, while the steel rod would seem to wriggle just
as the eel does to us. For everybody moving differently both from
this observer and ourselves, both the eel and the rod would seem to
wriggle. And there is no saying that one observer is right and another
wrong. In such matters, what is seen does not belong solely to the
physical process observed, but also to the standpoint of the observer.
Measurements of distances and times do not directly reveal properties
of the things measured, but relations of the things to the measurer.
What observation can tell us about the physical world is therefore more
abstract than we have hitherto believed.
It is important to realize that geometry, as taught in schools since
Greek times, ceases to exist as a separate science, and becomes merged
in physics. The two fundamental notions in elementary geometry were
the straight line and the circle. What appears to you as a straight
road, whose parts all exist now, may appear to another observer to
be like the flight of a rocket, some kind of curve whose parts come
into existence successively. The circle depends upon measurement of
distances, since it consists of all the points at a given distance
from its center. And measurement of distances, as we have seen, is
a subjective affair, depending upon the way in which the observer
is moving. The failure of the circle to have objective validity was
demonstrated by the Michelson-Morley experiment, and is thus, in a
sense, the starting point of the whole theory of relativity. Rigid
bodies, which we need for measurement, are only rigid for certain
observers; for others, they will be constantly changing all their
dimensions. It is only our obstinately earth-bound imagination that
makes us suppose a geometry separate from physics to be possible.
That is why we do not trouble to give physical significance to our
co-ordinates from the start. Formerly, the co-ordinates used in physics
were supposed to be carefully measured distances; now we realize
that this care at the start is thrown away. It is at a later stage
that care is required. Our co-ordinates now are hardly more than a
systematic way of cataloguing events. But mathematics provides, in
the method of tensors, such an immensely powerful technique that we
can use co-ordinates assigned in this apparently careless way just
as effectively as if we had applied the whole apparatus of minutely
accurate measurement in arriving at them. The advantage of being
haphazard at the start is that we avoid making surreptitious physical
assumptions, which we can hardly help making, if we suppose that our
co-ordinates have initially some particular physical significance.
We assume that, if two events are close together (but not necessarily
otherwise), there is an interval between them which can be calculated
from the differences between their co-ordinates by some such formula
as we considered in the preceding chapter. That is to say, we take the
squares and products of the differences of co-ordinates, we multiply
them by suitable amounts (which in general will vary from place to
place), and we add the results together. The sum obtained is the
square of the interval. We do not assume in advance that we know the
amounts by which the squares and products must be multiplied; this
is going to be discovered by observing physical phenomena. We know,
however, certain things. We know that the old Newtonian physics is
very nearly accurate when our co-ordinates have been chosen in a
certain way. We know that the special theory of relativity is still
more nearly accurate for suitable co-ordinates. From such facts we can
infer certain things about our new co-ordinates, which, in a logical
deduction, appear as postulates of the new theory.
As such postulates we take:
1. That every body travels in a geodesic in
space-time, except in so far as electromagnetic
forces act upon it.
2. That a light ray travels so that the interval
between two parts of it is zero.
3. That at a great distance from gravitating matter,
we can transform our co-ordinates by mathematical
manipulation so that the interval shall be what it
is in the special theory of relativity; and that
this is approximately true wherever gravitation is
not very powerful.
Each of these postulates requires some explanation.
We saw that a geodesic on a surface is the shortest line that can be
drawn on the surface from one point to another; for example, on the
earth the geodesics are great circles. When we come to space-time,
the mathematics is the same, but the verbal explanations have to be
rather different. In the general theory of relativity, it is only
neighboring events that have a definite interval, independently of
the route by which we travel from one to the other. The interval
between distant events depends upon the route pursued, and has to be
calculated by dividing the route into a number of little bits and
adding up the intervals for the various little bits. If the interval
is space-like, a body cannot travel from one event to the other;
therefore when we are considering the way bodies move, we are confined
to time-like intervals. The interval between neighboring events, when
it is time-like, will appear as the time between them for an observer
who travels from the one event to the other. And so the whole interval
between two events will be judged by a person who travels from one to
the other to be what his clocks show to be the time that he has taken
on the journey. For some routes this time will be longer, for others
shorter; the more slowly the man travels, the longer he will think he
has been on the journey. This must not be taken as a platitude. I am
not saying that if you travel from London to Edinburgh you will take
longer if you travel more slowly. I am saying something much more odd.
I am saying that if you leave London at 10 A.M. and arrive in Edinburgh
at 6.30 P.M. Greenwich time, the more slowly you travel the longer
you will take—if the time is judged by your watch. This is a very
different statement. From the point of view of a person on the earth,
your journey takes eight and a half hours. But if you had been a ray
of light traveling round the solar system, starting from London at 10
A.M., reflected from Jupiter to Saturn, and so on, until at last you
were reflected back to Edinburgh and arrived there at 6.30 P.M., you
would judge that the journey had taken you exactly no time. And if you
had gone by any circuitous route, which enabled you to arrive in time
by traveling fast, the longer your route the less time you would judge
that you had taken; the diminution of time would be continual as your
speed approached that of light. Now I say that when a body travels, if
it is left to itself, it chooses the route which makes the time between
two stages of the journey as long as possible; if it had traveled from
one event to another by any other route, the time, as measured by its
own clocks, would have been shorter. This is a way of saying that
bodies left to themselves do their journeys as slowly as they can; it
is a sort of law of cosmic laziness. Its mathematical expression is
that they travel in geodesics, in which the total interval between any
two events on the journey is _greater_ than by any alternative route.
(The fact that it is greater, not less, is due to the fact that the
sort of interval we are considering is more analogous to time than to
distance.) For example, if a person could leave the earth and travel
about for a time and then return, the time between his departure and
return would be less by his clocks than by those on the earth: the
earth, in its journey round the sun, chooses the route which makes
the time of any bit of its course by its clocks longer than the time
as judged by clocks which move by a different route. This is what is
meant by saying that bodies left to themselves move in geodesics in
space-time.
We assume that the body considered is not acted upon by electromagnetic
forces. We are concerned at present with the law of gravitation, not
with the effects of electromagnetism. These effects have been brought
into the framework of the general theory of relativity by Weyl,[5] but
for the present we will ignore his work. The planets, in any case,
are not subject, as wholes, to appreciable electromagnetic forces; it
is only gravitation that has to be considered in accounting for their
motions, with which we are concerned in this chapter.
[5] See his _Space, Time, Matter_, Methuen, 1922.
Our second postulate, that a light ray travels so that the interval
between two parts of it is zero, has the advantage that it does not
have to be stated only for _small_ distances. If each little bit of
interval is zero, the sum of them all is zero, and so even distant
parts of the same light ray have a zero interval. The course of a light
ray is also a geodesic according to the definition. Thus we now have
two empirical ways of discovering what are the geodesics in space-time,
namely light rays and bodies moving freely. Among freely-moving
bodies are included all which are not subject to constraints or to
electromagnetic forces, that is to say, the sun, stars, planets and
satellites, and also falling bodies on the earth, at least when they
are falling in a vacuum. When you are standing on the earth, you are
subject to electromagnetic forces: the electrons and protons in the
neighborhood of your feet exert a repulsion on your feet which is just
enough to overcome the earth’s gravitation. This is what prevents you
from falling through the earth, which, solid as it looks, is mostly
empty space.
The third postulate, which relates the general to the special theory,
is very useful. It is not necessary for the application of the special
theory to a limited region that there should be no gravitation in the
region; it is enough if the intensity of gravitation is practically the
same throughout the region. This enables us to apply the special theory
within any small region. How small it will have to be, depends upon the
neighborhood. On the surface of the earth, it would have to be small
enough for the curvature of the earth to be negligible. In the spaces
between the planets, it need only be small enough for the attraction
of the sun and the planets to be sensibly constant throughout the
region. In the spaces between the stars it might be enormous—say half
the distance from one star to the next—without introducing measurable
inaccuracies.
At a great distance from gravitating matter, we can so choose our
co-ordinates as to obtain a Euclidean space; this is really only
another way of saying that the special theory of relativity applies. In
the neighborhood of matter, although we can make our space Euclidean
in any small region, we cannot do so throughout any region within
which gravitation varies sensibly—at least, if we do, we shall have
to abandon the view that bodies move in geodesics. In the neighborhood
of a piece of matter, there is, as it were, a hill in space-time;
this hill grows steeper and steeper as it gets nearer the top, like
the neck of a champagne bottle. It ends in a sheer precipice. Now by
the law of cosmic laziness which we mentioned earlier, a body coming
into the neighborhood of the hill will not attempt to go straight
over the top, but will go round. This is the essence of Einstein’s
view of gravitation. What a body does, it does because of the nature
of space-time in its own neighborhood, not because of some mysterious
force emanating from a distant body.
An analogy will serve to make the point clear. Suppose that on a dark
night a number of men with lanterns were walking in various directions
across a huge plain, and suppose that in one part of the plain there
was a hill with a flaring beacon on the top. Our hill is to be such
as we have described, growing steeper as it goes up, and ending in a
precipice. I shall suppose that there are villages dotted about the
plain, and the men with lanterns are walking to and from these various
villages. Paths have been made showing the easiest way from any one
village to any other. These paths will all be more or less curved, to
avoid going too far up the hill; they will be more sharply curved when
they pass near the top of the hill than when they keep some way off
from it. Now suppose that you are observing all this, as best you can,
from a place high up in a balloon, so that you cannot see the ground,
but only the lanterns and the beacon. You will not know that there is a
hill, or that the beacon is at the top of it. You will see that people
turn out of the straight course when they approach the beacon, and
that the nearer they come the more they turn aside. You will naturally
attribute this to an effect of the beacon; you may think that it is
very hot and people are afraid of getting burnt. But if you wait for
daylight you will see the hill, and you will find that the beacon
merely marks the top of the hill and does not influence the people with
lanterns in any way.
Now in this analogy the beacon corresponds to the sun, the people with
lanterns correspond to the planets and comets, the paths correspond
to their orbits, and the coming of daylight corresponds to the coming
of Einstein. Einstein says that the sun is at the top of a hill, only
the hill is in space-time, not in space. (I advise the reader not to
try to picture this, because it is impossible.) Each body, at each
moment, adopts the easiest course open to it, but owing to the hill the
easiest course is not a straight line. Each little bit of matter is at
the top of its own little hill, like the cock on his own dung-heap.
What we call a big bit of matter is a bit which is at the top of a big
hill. The hill is what we know about; the bit of matter at the top is
assumed for convenience. Perhaps there is really no need to assume it,
and we could do with the hill alone, for we can never get to the top of
any one else’s hill, any more than the pugnacious cock can fight the
peculiarly irritating bird that he sees in the looking glass.
I have given only a qualitative description of Einstein’s law of
gravitation; to give its exact quantitative formulation is impossible
without more mathematics than I am permitting myself. The most
interesting point about it is that it makes the law no longer the
result of action at a distance: the sun exerts no force on the planets
whatever. Just as geometry has become physics, so, in a sense, physics
has become geometry. The law of gravitation has become the geometrical
law that every body pursues the easiest course from place to place, but
this course is affected by the hills and valleys that are encountered
on the road.
CHAPTER IX: PROOFS OF EINSTEIN’S LAW OF GRAVITATION
The reasons for accepting Einstein’s law of gravitation rather than
Newton’s are partly empirical, partly logical. We will begin with the
former.
Einstein’s law of gravitation gives very nearly the same results
as Newton’s, when applied to the calculation of the orbits of the
planets and their satellites. If it did not, it could not be true,
since the consequences deduced from Newton’s law have been found to be
almost exactly verified by observation. When, in 1915, Einstein first
published his new law, there was only one empirical fact to which he
could point to show that his theory was better than Newton’s. This was
what is called the “motion of the perihelion of Mercury.”
The planet Mercury, like the other planets, moves round the sun in
an ellipse, with the sun in one of the foci. At some points of its
orbit it is nearer to the sun than at other points. The point where
it is nearest to the sun is called its “perihelion.” Now it was found
by observation that, from one occasion when Mercury is nearest to the
sun until the next, Mercury does not go exactly once round the sun,
but a little bit more. The discrepancy is very small; it amounts to
an angle of forty-two seconds in a century. That is to say, in each
year the planet has to move rather less than half a second of angle
after it has finished a complete revolution from the last perihelion
before it reaches the next perihelion. This very minute discrepancy
from Newtonian theory had puzzled astronomers. There was a calculated
effect due to perturbations caused by the other planets, but this small
discrepancy was the residue after allowing for these perturbations.
Einstein’s theory accounted for this residue, as well as for its
absence in the case of the other planets. (In them it exists, but is
too small to be observed.) This was, at first, his only empirical
advantage over Newton.
His second success was more sensational. According to orthodox
opinion, light in a vacuum ought always to travel in straight lines.
Not being composed of material particles, it ought to be unaffected
by gravitation. However, it was possible, without any serious breach
with old ideas, to admit that, in passing near the sun, light might be
deflected out of the straight path as much as if it were composed of
material particles. Einstein, however, maintained, as a deduction from
his law of gravitation, that light would be deflected twice as much as
this. That is to say, if the light of a star passed very near the sun,
Einstein maintained that the ray from the star would be turned through
an angle of just under one and three-quarters seconds. His opponents
were willing to concede half of this amount. Now it is not every day
that a star almost in line with the sun can be seen. This is only
possible during a total eclipse, and not always then, because there may
be no bright stars in the right position. Eddington points out that,
from this point of view, the best day of the year is May 29, because
then there are a number of bright stars close to the sun. It happened
by incredible good fortune that there was a total eclipse of the sun
on May 29, 1919—the first year after the armistice. Two British
expeditions photographed the stars near the sun during the eclipse,
and the results confirmed Einstein’s prediction. Some astronomers
who remained doubtful whether sufficient precautions had been taken
to insure accuracy were convinced when their own observations in a
subsequent eclipse gave exactly the same result. Einstein’s estimate of
the amount of the deflection of light by gravitation is therefore now
universally accepted.
The third experimental test is on the whole favorable to Einstein,
though the quantities concerned are so small that it is only just
possible to measure them, and the result is therefore not decisive. But
successive investigations have made it more and more probable that the
small effect predicted by Einstein really occurs. Before explaining the
effect in question, a few preliminary explanations are necessary. The
spectrum of an element consists of certain lines of various shades of
light, separated by a prism, and emitted by the element when it glows.
They are the same (to a very close approximation) whether the element
is in the earth or the sun or a star. Each line is of some definite
shade of color, with some definite wave length. Longer wave lengths are
towards the red end of the spectrum, shorter ones towards the violet
end. When the source of light is moving towards you, the apparent wave
lengths grow shorter, just as waves at sea come quicker when you are
traveling against the wind. When the source of light is moving away
from you, the apparent wave lengths grow longer, for the same reason.
This enables us to know whether the stars are moving towards us or away
from us. If they are moving towards us, all the lines in the spectrum
of an element are moved a little toward violet; if away from us, toward
red. You may notice the analogous effect in sound any day. If you are
in a station and an express comes through whistling, the note of the
whistle seems much more shrill while the train is approaching you than
when it has passed. Probably many people think the note has “really”
changed, but in fact the change in what you hear is only due to the
fact that the train was first approaching and then receding. To people
in the train, there was no change of note. This is _not_ the effect
with which Einstein is concerned. The distance of the sun from the
earth does not change much; for our present purposes, we may regard
it as constant. Einstein deduces from his law of gravitation that
any periodic process which takes place in an atom in the sun (whose
gravitation is very intense) must, as measured by our clocks, take
place at a slightly slower rate than it would in a similar atom on the
earth. The “interval” involved will be the same in the sun and on the
earth, but the same interval in different regions does not correspond
to exactly the same time; this is due to the “hilly” character of
space-time which constitutes gravitation. Consequently any given line
in the spectrum ought, when the light comes from the sun, to seem to
us a little nearer the red end of the spectrum than if the light came
from a source on the earth. The effect to be expected is very small—so
small that there is still some slight uncertainty as to whether it
exists or not. But it now seems highly probable that it exists.
No other measurable differences between the consequences of Einstein’s
law and those of Newton’s have hitherto been discovered. But the above
experimental tests are quite sufficient to convince astronomers that,
where Newton and Einstein differ as to the motions of the heavenly
bodies, it is Einstein’s law that gives the right results. Even if
the empirical grounds in favor of Einstein stood alone, they would be
conclusive. Whether his law represents the exact truth or not, it is
certainly more nearly exact than Newton’s, though the inaccuracies in
Newton’s were all exceedingly minute.
But the considerations which originally led Einstein to his law were
not of this detailed kind. Even the consequence about the perihelion of
Mercury, which could be verified at once from previous observations,
could only be deduced after the theory was complete, and could not
form any part of the original grounds for inventing such a theory.
These grounds were of a more abstract logical character. I do not
mean that they were not based upon observed facts, and I do not mean
that they were _à priori_ fantasies such as philosophers indulged in
formerly. What I mean is that they were derived from certain general
characteristics of physical experience, which showed that Newton _must_
be wrong and that something like Einstein’s law _must_ be substituted.
The arguments in favor of the relativity of motion are, as we saw in
earlier chapters, quite conclusive. In daily life, when we say that
something moves, we mean that it moves relatively to the earth. In
dealing with the motions of the planets, we consider them as moving
relatively to the sun, or to the center of mass of the solar system.
When we say that the solar system itself is moving, we mean that it is
moving relatively to the stars. There is no physical occurrence which
can be called “absolute motion.” Consequently the laws of physics must
be concerned with relative motions, since these are the only kind that
occur.
We now take the relativity of motion in conjunction with the
experimental fact that the velocity of light is the same relatively
to one body as relatively to another, however the two may be moving.
This leads us to the relativity of distances and times. This in turn
shows that there is no objective physical fact which can be called “the
distance between two bodies at a given time,” since the time and the
distance will both depend on the observer. Therefore Newton’s law of
gravitation is logically untenable, since it makes use of “distance at
a given time.”
This shows that we cannot rest content with Newton, but it does not
show what we are to put in his place. Here several considerations
enter in. We have in the first place what is called “the equality
of gravitational and inertial mass.” What this means is as follows:
When you apply a given force[6] to a heavy body, you do not give it
as much acceleration as you would to a light body. What is called the
“inertial” mass of a body is measured by the amount of force required
to produce a given acceleration. At a given point of the earth’s
surface, the “mass” is proportional to the “weight.” What is measured
by scales is rather the mass than the weight: the weight is defined as
the force with which the earth attracts the body. Now this force is
greater at the poles than at the equator, because at the equator the
rotation of the earth produces a “centrifugal force” which partially
counteracts gravitation. The force of the earth’s attraction is also
greater on the surface of the earth than it is at a great height or at
the bottom of a very deep mine. None of these variations are shown by
scales, because they affect the weights used just as much as the body
weighed; but they are shown if we use a spring balance. The mass does
not vary in the course of these changes of weight.
[6] Although “force” is no longer to be regarded as one of the
fundamental concepts of dynamics, but only as a convenient way of
speaking, it can still be employed, like “sunrise” and “sunset,”
provided we realize what we mean. Often it would require very
roundabout expressions to avoid the term “force.”
The “gravitational” mass is differently defined. It is capable of two
meanings. We may mean (1), the way a body responds in a situation
where gravitation has a known intensity, for example, on the surface
of the earth, or on the surface of the sun; or (2), the intensity of
the gravitational force produced by the body, as, for example, the sun
produces stronger gravitational forces than the earth does. Newton
says that the force of gravitation between two bodies is proportional
to the product of their masses. Now let us consider the attraction of
different bodies to one and the same body, say the sun. Then different
bodies are attracted by forces which are proportional to their masses,
and which, therefore, produce exactly the same acceleration in all of
them. Thus if we mean “gravitational mass” in sense (1), that is to
say, the way a body responds to gravitation, we find that “the equality
of inertial and gravitational mass,” which sounds formidable, reduces
to this: that in a given gravitational situation, all bodies behave
exactly alike. As regards the surface of the earth, this was one of
the first discoveries of Galileo. Aristotle thought that heavy bodies
fall faster than light ones; Galileo showed that this is not the case,
when the resistance of the air is eliminated. In a vacuum, a feather
falls as fast as a lump of lead. As regards the planets, it was Newton
who established the corresponding facts. At a given distance from the
sun, a comet, which has a very small mass, experiences exactly the
same acceleration towards the sun as a planet experiences at the same
distance. Thus the way in which gravitation affects a body depends only
upon where the body is, and in no degree upon the nature of the body.
This suggests that the gravitational effect is a characteristic of the
locality, which is what Einstein makes it.
As for the gravitational mass in sense (2), _i.e._, the intensity of
the force produced by a body, this is no longer _exactly_ proportional
to its inertial mass. The question involves some rather complicated
mathematics, and I shall not go into it.[7]
[7] See Eddington, _The Mathematical Theory of Relativity_, Cambridge
University Press, 2d edition, p. 128.
We have another indication as to what sort of thing the law of
gravitation _must_ be, if it is to be a characteristic of a
neighborhood, as we have seen reason to suppose that it is. It must
be expressed in some law which is unchanged when we adopt a different
kind of co-ordinates. We saw that we must not, to begin with, regard
our co-ordinates as having any physical significance: they are merely
systematic ways of naming different parts of space-time. Being
conventional, they cannot enter into physical laws. That means to say
that, if we have expressed a law correctly in terms of one set of
co-ordinates, it must be expressed by the same formula in terms of
another set of co-ordinates. Or, more exactly, it must be possible
to find a formula which expresses the law, and which is unchanged
however we change the co-ordinates. It is the business of the theory
of tensors to deal with such formulæ. And the theory of tensors shows
that there is one formula which obviously suggests itself as being
possibly the law of gravitation. When this possibility is examined,
it is found to give the right results; it is here that the empirical
confirmations come in. But if Einstein’s law had not been found to
agree with experience, we could not have gone back to Newton’s law. We
should have been compelled by logic to seek some law expressed in terms
of “tensors,” and therefore independent of our choice of co-ordinates.
It is impossible without mathematics to explain the theory of
tensors; the non-mathematician must be content to know that it is the
technical method by which we eliminate the conventional element from
our measurements and laws, and thus arrive at physical laws which are
independent of the observer’s point of view. Of this method, Einstein’s
law of gravitation is the most splendid example.
CHAPTER X: MASS, MOMENTUM, ENERGY AND ACTION
The pursuit of quantitative precision is as arduous as it is important.
Physical measurements are made with extraordinary exactitude; if
they were made less carefully, such minute discrepancies as form
the experimental data for the theory of relativity could never be
revealed. Mathematical physics, before the coming of relativity, used
a set of conceptions which were supposed to be as precise as physical
measurements, but it has turned out that they were logically defective,
and that this defectiveness showed itself in very small deviations from
expectations based upon calculation. In this chapter I want to show how
the fundamental ideas of pre-relativity physics are affected, and what
modifications they have had to undergo.
We have already had occasion to speak of mass. For purposes of
daily life, mass is much the same as weight; the usual measures of
weight—ounces, grams, etc.—are really measures of mass. But as
soon as we begin to make accurate measurements, we are compelled to
distinguish between mass and weight. Two different methods of weighing
are in common use, one, that of scales, the other that of the spring
balance. When you go a journey and your luggage is weighed, it is not
put on scales, but on a spring; the weight depresses the spring a
certain amount, and the result is indicated by a needle on a dial. The
same principle is used in automatic machines for finding your weight.
The spring balance shows weight, but scales show _mass_. So long as
you stay in one part of the world, the difference does not matter;
but if you test two weighing machines of different kinds in a number
of different places, you will find, if they are accurate, that their
results do not always agree. Scales will give the same result anywhere,
but a spring balance will not. That is to say, if you have a lump of
lead weighing ten pounds by the scales, it will also weigh ten pounds
by scales in any other part of the world. But if it weighs ten pounds
by a spring balance in London, it will weigh more at the North Pole,
less at the equator, less high up in an aeroplane, and less at the
bottom of a coal mine, if it is weighed in all those places on the same
spring balance. The fact is that the two instruments measure quite
different quantities. The scales measure what may be called (apart from
refinements which will concern us presently) “quantity of matter.”
There is the same “quantity of matter” in a pound of feathers as in a
pound of lead. Standard “weights,” which are really standard “masses,”
will measure the amount of mass in any substance put into the opposite
scales. But “weight” is a properly due to the earth’s gravitation: It
is the amount of the force by which the earth attracts a body. This
force varies from place to place. In the first place, anywhere outside
the earth the attraction varies inversely as the square of the distance
from the center of the earth; it is therefore less at great heights.
In the second place, when you go down a coal mine, part of the earth
is above you, and attracts matter upwards instead of downwards, so
that the net attraction downwards is less than on the surface of the
earth. In the third place, owing to the rotation of the earth, there is
what is called a “centrifugal force,” which acts against gravitation.
This is greatest at the equator, because there the rotation of the
earth involves the fastest motion; at the poles it does not exist,
because they are on the axis of rotation. For all these reasons, the
force with which a given body is attracted to the earth is measureably
different at different places. It is this force that is measured by a
spring balance; that is why a spring balance gives different results
in different places. In the case of scales, the standard “weights” are
altered just as much as the body to be weighed, so that the result is
the same everywhere; but the result is the “mass,” not the “weight.”
A standard “weight” has the same mass everywhere, but not the same
“weight”; it is in fact a unit of mass, not of weight. For theoretical
purposes, mass, which is almost invariable for a given body, is much
more important than weight, which varies according to circumstances.
Mass may be regarded, to begin with, as “quantity of matter”; we shall
see that this view is not strictly correct, but it will serve as a
starting point for subsequent refinements.
For theoretical purposes, a mass is defined as being determined by the
amount of force required to produce a given acceleration: The more
massive a body is, the greater will be the force required to alter its
velocity by a given amount in a given time. It takes a more powerful
engine to make a long train attain a speed of ten miles an hour at the
end of the first half-minute, than it does to make a short train do so.
Or we may have circumstances where the force is the same for a number
of different bodies; in that case, if we can measure the accelerations
produced in them, we can tell the ratios of their masses: the greater
the mass, the smaller the acceleration. We may take, in illustration
of this method, an example which is important in connection with
relativity. Radio-active bodies emit beta-particles (electrons) with
enormous velocities. We can observe their path by making them travel
through water vapor and form a cloud as they go. We can at the same
time subject them to known electric and magnetic forces, and observe
how much they are bent out of a straight line by these forces. This
makes it possible to compare their masses. It is found that the faster
they travel, the greater is their mass, as measured by the stationary
observer; the increase is greatest as applied to their mass as measured
by the effect of a force in the line of motion. In regard to forces at
right angles to the line of motion, there is a change of mass with
velocity in the same proportion as the changes of length and time. It
is known otherwise that, apart from the effect of motion, all electrons
have the same mass.
All this was known before the theory of relativity was invented, but
it showed that the traditional conception of mass had not quite the
definiteness that had been ascribed to it. Mass used to be regarded as
“quantity of matter,” and supposed to be quite invariable. Now mass was
found to be relative to the observer, like length and time, and to be
altered by motion in exactly the same proportion. However, this could
be remedied. We could take the “proper mass,” the mass as measured by
an observer who shares the motion of the body. This was easily inferred
from the measured mass, by taking the same proportion as in the case of
lengths and times.
But there is a more curious fact, and that is, that after we have
made this correction we still have not obtained a quantity which is
at all times exactly the same for the same body. When a body absorbs
energy—for example, by growing hotter—its “proper mass” increases
slightly. The increase is very slight, since it is measured by
dividing the increase of energy by the square of the velocity of
light. On the other hand, when a body parts with energy it loses mass.
The most notable case of this is that four hydrogen atoms can come
together to make one helium atom, but a helium atom has rather less
than four times the mass of one hydrogen atom.
We have thus two kinds of mass, neither of which quite fulfils the old
ideal. The mass as measured by an observer who is in motion relative
to the body in question is a relative quantity, and has no physical
significance as a property of the body. The “proper mass” is a genuine
property of the body, not dependent upon the observer; but it, also,
is not strictly constant. As we shall see shortly, the notion of mass
becomes absorbed into the notion of energy; it represents, so to speak,
the energy which the body expends internally, as opposed to that which
it displays to the outer world.
Conservation of mass, conservation of momentum, and conservation of
energy were the great principles of classical mechanics. Let us next
consider conservation of momentum.
The momentum of a body in a given direction is its velocity in that
direction multiplied by its mass. Thus a heavy body moving slowly may
have the same momentum as a light body moving fast. When a number of
bodies interact in any way, for instance by collisions, or by mutual
gravitation, so long as no outside influences come in, the total
momentum of all the bodies in any direction remains unchanged. This law
remains true in the theory of relativity. For different observers, the
mass will be different, but so will the velocity; these two differences
neutralize each other, and it turns out that the principle still
remains true.
The momentum of a body is different in different directions. The
ordinary way of measuring it is to take the velocity in a given
direction (as measured by the observer) and multiply it by the mass (as
measured by the observer). Now the velocity in a given direction is
the distance traveled in that direction in unit time. Suppose we take
instead the distance traveled in that direction while the body moves
through unit “interval.” (In ordinary cases, this is only a very slight
change, because, for velocities considerably less than that of light,
interval is very nearly equal to lapse of time.) And suppose that
instead of the mass as measured by the observer we take the proper
mass. These two changes increase the velocity and diminish the mass,
both in, the same proportion. Thus the momentum remains the same, but
the quantities that vary according to the observer have been replaced
by quantities which are fixed independently of the observer—with the
exception of the distance traveled by the body in the given direction.
When we substitute space-time for time, we find that the measured
mass (as opposed to the proper mass) is a quantity of the same kind
as the momentum in a given direction; it might be called the momentum
in the time direction. The measured mass is obtained by multiplying
the invariant mass by the _time_ traversed in traveling through unit
interval; the momentum is obtained by multiplying the same invariant
mass by the _distance_ traversed (in the given direction) in traveling
through unit interval. From a space-time point of view, these naturally
belong together.
Although the measured mass of a body depends upon the way the observer
is moving relatively to the body, it is none the less a very important
quantity. For any given observer, the measured mass of the whole
physical universe is constant.[8] The proper mass of all the bodies
in the world is not necessarily the same at one time as at another,
so that in this respect the measured mass has an advantage. The
conservation of measured mass is the same thing as the conservation of
energy. This may seem surprising, since at first sight mass and energy
are very different things. But it has turned out that energy is the
same thing as measured mass. To explain how this comes about is not
easy; nevertheless we will make the attempt.
[8] This is subject to the explanations given below as regards
conservation of energy.
In popular talk, “mass” and “energy” do not mean at all the same thing.
We associate “mass” with the idea of a fat man in a chair, very slow to
move, while “energy” suggests a thin person full of hustle and “pep.”
Popular talk associates “mass” and “inertia,” but its view of inertia
is one-sided: it includes slowness in beginning to move, but not
slowness in stopping, which is equally involved. All these terms have
technical meanings in physics, which are only more or less analogous
to the meanings of the terms in popular talk. For the present, we are
concerned with the technical meaning of “energy.”
Throughout the latter half of the nineteenth century, a great deal was
made of the “conservation of energy,” or the “persistence of force,”
as Herbert Spencer preferred to call it. This principle was not easy
to state in a simple way, because of the different forms of energy;
but the essential point was that energy is never created or destroyed,
though it can be transformed from one kind into another. The principle
acquired its position through Joule’s discovery of “the mechanical
equivalent of heat,” which showed that there was a constant proportion
between the work required to produce a given amount of heat and the
work required to raise a given weight through a given height: in fact,
the same sort of work could be utilized for either purpose according to
the mechanism. When heat was found to consist in motion of molecules,
it was seen to be natural that it should be analogous to other forms of
energy. Broadly speaking, by the help of a certain amount of theory,
all forms of energy were reduced to two, which were called respectively
“kinetic” and “potential.” These were defined as follows:
The kinetic energy of a particle is half the mass multiplied by the
square of the velocity. The kinetic energy of a number of particles is
the sum of the kinetic energies of the separate particles.
The potential energy is more difficult to define. It represents any
state of strain, which can only be preserved by the application of
force. To take the easiest case: If a weight is lifted to a height and
kept suspended, it has potential energy, because, if left to itself, it
will fall. Its potential energy is equal to the kinetic energy which it
would acquire in falling through the same distance through which it was
lifted. Similarly when a comet goes round the sun in a very eccentric
orbit, it moves much faster when it is near the sun than when it is far
from it, so that its kinetic energy is much greater when it is near the
sun. On the other hand, its potential energy is greatest when it is
farthest from the sun, because it is then like the stone which has been
lifted to a height. The sum of the kinetic and potential energies of
the comet is constant, unless it suffers collisions or loses matter by
forming a tail. We can determine accurately the _change_ of potential
energy in passing from one position to another, but the total amount of
it is to a certain extent arbitrary, since we can fix the zero level
where we like. For example, the potential energy of our stone may be
taken to be the kinetic energy it would acquire in falling to the
surface of the earth, or what it would acquire in falling down a well
to the center of the earth, or any assigned lesser distance. It does
not matter which we take, so long as we stick to our decision. We are
concerned with a profit-and-loss account, which is unaffected by the
amount of the assets with which we start.
Both the kinetic and the potential energies of a given set of bodies
will be different for different observers. In classical dynamics,
the kinetic energy differed according to the state of motion of the
observer, but only by a constant amount; the potential energy did not
differ at all. Consequently, for each observer, the total energy was
constant—assuming always that the observers concerned were moving
in straight lines with uniform velocities, or, if not, were able to
refer their motions to bodies which were so moving. But in relativity
dynamics the matter becomes more complicated. We cannot profitably
adapt the idea of potential energy to the theory of relativity, and
therefore the conservation of energy, in a strict sense, cannot
be maintained. But we obtain a property, closely analogous to
conservation, which applies to kinetic energy alone. As Eddington
puts it: the kinetic energy is not always strictly conserved, and the
classical theory therefore introduces a supplementary quantity, the
potential energy, so that the sum of the two is strictly conserved. The
relativity treatment, on the other hand, discovers another formula,
analogous to the one expressing conservation, which holds always for
the kinetic energy. “The relativity treatment adheres to the physical
quantity and modifies the law; the classical treatment adheres to
the law and modifies the physical quantity.” The new formula, he
continues, may be spoken of “as the law of conservation of energy and
momentum, because, though it is not formally a law of conservation, it
expresses exactly the phenomena which classical mechanics attributes to
conservation.”[9] It is only in this modified and less rigorous sense
that the conservation of energy remains true.
[9] _Mathematical Theory of Relativity_, p. 135.
What is meant by “conservation” in practice is not exactly what it
means in theory. In theory we say that a quantity is conserved when the
amount of it in the world is the same at any one time as at any other.
But in practice we cannot survey the whole world, so we have to mean
something more manageable. We mean that, taking any given region, if
the amount of the quantity in the region has changed, it is because
some of the quantity has passed across the boundary of the region. If
there were no births and deaths, population would be conserved; in that
case the population of a country could only change by emigration or
immigration, that is to say, by passing across the boundaries. We might
be unable to take an accurate census of China or Central Africa, and,
therefore, we might not be able to ascertain the total population of
the world. But we should be justified in assuming it to be constant if,
wherever statistics were possible, the population never changed except
through people crossing the frontiers. In fact, of course, population
is not conserved. A physiologist of my acquaintance once put four mice
into a thermos. Some hours later, when he went to take them out, there
were eleven of them. But mass is not subject to these fluctuations:
the mass of the eleven mice at the end of the time was no greater than
the mass of the four at the beginning.
This brings us back to the problem for the sake of which we have been
discussing energy. We stated that, in relativity theory, measured mass
and energy are regarded as the same thing, and we undertook to explain
why. It is now time to embark upon this explanation. But here, as at
the end of Chapter VI, the totally unmathematical reader will do well
to skip, and begin again at the following paragraph.
Let us take the velocity of light as the unit of velocity; this is
always convenient in relativity theory. Let _m_ be the proper mass of a
particle, _v_ its velocity relative to the observer. Then its measured
mass will be
_m_
——————————
√(1 - _v²_)
while its kinetic energy, according to the usual formula, will be
½ _mv²_
As we saw before, energy only occurs in a profit-and-loss account,
so that we can add any constant quantity to it that we like. We may
therefore take the energy to be
_m_ + ½(_mv²_).
Now if _v_ is a small fraction of the velocity of light,
_m_ + ½ _mv²_
is almost exactly equal to
_m_
—————————
√(1 - _v²_).
Consequently, for velocities such as large bodies have, the energy and
the measured mass turn out to be indistinguishable within the limits of
accuracy attainable. In fact, it is better to alter our definition of
energy, and take it to be
_m_
——————————
√(1 - _v²_),
because this is the quantity for which the law analogous to
conservation holds. And when the velocity is very great, it gives a
better measure of energy than the traditional formula. The traditional
formula must therefore be regarded as an approximation, of which the
new formula gives the exact version. In this way, energy and measured
mass become identified.
I come now to the notion of “action,” which is less familiar to
the general public than energy, but has become more important in
relativity physics, as well as in the theory of quanta.[10] (The
quantum is a small amount of action.) The word “action” is used to
denote energy multiplied by time. That is to say, if there is one unit
of energy in a system, it will exert one unit of action in a second,
100 units of action in 100 seconds, and so on; a system which has
100 units of energy will exert 100 units of action in a second, and
10,000 in 100 seconds, and so on. “Action” is thus, in a loose sense,
a measure of how much has been accomplished: it is increased both by
displaying more energy and by working for a longer time. Since energy
is the same thing as measured mass, we may also take action to be
measured mass multiplied by time. In classical mechanics, the “density”
of matter in any region is the mass divided by the volume; that is
to say, if you know the density in a small region, you discover the
total amount of matter by multiplying the density by the volume of the
small region. In relativity mechanics, we always want to substitute
space-time for space; therefore a “region” must no longer be taken to
be merely a volume, but a volume lasting for a time; a small region
will be a small volume lasting for a small time. It follows that, given
the density, a small region in the new sense contains, not a small mass
merely, but a small mass multiplied by a small time, that is to say, a
small amount of “action.” This explains why it is to be expected that
“action” will prove of fundamental importance in relativity mechanics.
And so in fact it is.
[10] On this subject, see the present author’s _A.B.C. of Atoms_,
chaps. VI and XIII.
All the laws of dynamics have been put together into one principle,
called “The Principle of Least Action.” This states that, in passing
from one state to another, a body chooses a route involving less action
than any slightly different route—again a law of cosmic laziness. The
principle is subject to certain limitations, which have been pointed
out by Eddington,[11] but it remains one of the most comprehensive
ways of stating the purely formal part of mechanics. The fact that
the quantum is a unit of action seems to show that action is also
fundamental in the empirical structure of the world. But at present
there is no bridge connecting the quantum with the theory of relativity.
[11] _Op. cit._ § 60.
CHAPTER XI: IS THE UNIVERSE FINITE?
We have been dealing hitherto with matters that must be regarded as
acquired scientific results—not that they will never be found to need
improvement, but that further progress must be built upon them, as
Einstein is built upon Newton. Science does not aim at establishing
immutable truths and eternal dogmas: its aim is to approach truth by
successive approximations, without claiming that at any stage final and
complete accuracy has been achieved. There is a difference, however,
between results which are pretty certainly in the line of advance, and
speculations which may or may not prove to be well founded. Some very
interesting speculations are connected with the theory of relativity,
and we shall consider certain of them. But it must not be supposed that
we are dealing with theories having the same solidity as those with
which we have been concerned hitherto.
One of the most fascinating of the speculations to which I have been
alluding is the suggestion that the universe may be of finite extent.
Two somewhat different finite universes have been constructed, one by
Einstein, the other by De Sitter. Before considering their differences,
we will discuss what they have in common.
There are, to begin with, certain reasons for thinking that the total
amount of matter in the universe is limited. If this were not the
case, the gravitational effects of enormously distant matter would
make the kind of world in which we live impossible. We must therefore
suppose that there is some definite number of electrons and protons in
the world: theoretically, a complete census would be possible. These
are all contained within a certain finite region; whatever space lies
outside that region is, so to speak, waste, like unfurnished rooms in a
house too large for its inhabitants. This seems futile, but in former
days no one knew of any alternative possibility. It was obviously
impossible to conceive of an edge to space, and therefore, it was
thought, space must be infinite.
Non-Euclidean geometry, however, showed other possibilities. The
surface of a sphere has no boundary, yet it is not infinite. In
traveling round the earth, we never reach “the edge of the world,” and
yet the earth is not infinite. The surface of the earth is contained
in three-dimensional space, but there is no reason in logic why
three-dimensional space should not be constructed on an analogous plan.
What we imagine to be straight lines going on for ever will then be
like great circles on a sphere: they will ultimately return to their
starting point. There will not be in the universe anything straighter
than these great circles; the Euclidean straight line may remain as
a beautiful dream, but not as a possibility in the actual world. In
particular, light rays in empty space will travel in what are really
great circles. If we could make measurements with sufficient accuracy,
we should be able to infer this state of affairs even from a small part
of space, because the sum of the angles of a triangle would always be
greater than two right angles, and the excess would be proportional to
the size of the triangle. The suggestion we have to consider is the
suggestion that our universe may be spherical in this sense.
The reader must not confuse this suggestion with the non-Euclidean
character of space upon which the new law of gravitation depends. The
latter is concerned with small regions such as the solar system. The
departures from flatness which it notices are like hills and valleys
on the surface of the earth, local irregularities, not characteristics
of the whole. We are now concerned with the possible curvature of the
universe as a whole, not with the occasional ups and downs due to the
sun and the stars. It is suggested that on the average, and in regions
remote from matter, the universe is not quite flat, but has a slight
curvature, analogous, in three dimensions, to the curvature of a sphere
in two dimensions.
It is important to realize, in the first place, that there is not the
slightest reason _à priori_ why this should not be the case. People
unaccustomed to non-Euclidean geometry may feel that, even if such a
thing be _logically_ possible, the world simply _cannot_ be so odd
as all that. We all have a tendency to think that the world must
conform to our prejudices. The opposite view involves some effort of
thought, and most people would die sooner than think—in fact, they
do so. But the fact that a spherical universe seems odd to people
who have been brought up on Euclidean prejudices is no evidence that
it is impossible. There is no law of nature to the effect that what
is taught at school must be true. We cannot therefore dismiss the
hypothesis of a spherical universe as in any degree less worthy of
examination than any other. We have to ask ourselves the same two
questions as we should in any other case, namely: (1) Are the facts
consistent with this hypothesis? (2) Is this hypothesis the only one
with which the facts are consistent?
With regard to the first question, the answer is undoubtedly in the
affirmative. All the known facts are perfectly consistent with the
hypothesis of a spherical universe. A very slight modification of the
law of gravitation—a modification suggested by Einstein himself—leads
to a spherical space, without producing any measurable differences in a
small region such as the solar system. The known stars are all within
a certain distance from us. There is nothing whatever in the stellar
universe as we know it to show that space must be infinite. There can
therefore be no doubt whatever that, so far as our present knowledge
goes, the hypothesis of a finite universe _may_ be true.
But when we ask whether the hypothesis of a finite universe _must_
be true, the answer is different. It is obvious, on general grounds,
that we cannot, from what we know, draw conclusive inferences as to
the totality of things. A very slight change in the Newtonian formula
for gravitation would prevent masses beyond the limits of the visible
universe from having appreciable effects if they existed, and would
therefore destroy our reason for supposing that they do not exist.
All arguments as to regions which are too distant to be observed
depend upon extending to them the laws which hold in our part of
the world, and upon assuming that there is not, in these laws, some
inaccuracy which is inappreciable for observable distances, but fatal
to inferences in which very much greater distances are involved. We
cannot, therefore, say that the universe _must_ be finite. We can say
that it may be, and we can even say a little more than this. We can say
that a finite universe fits in better with the laws that hold in the
part we know, and that awkward adjustments of the laws have to be made
in order to allow the universe to be infinite. From the point of view
of choosing the best framework into which to fit what we know—best, I
mean, from a logico-æsthetic point of view—there is no doubt that the
hypothesis of a finite universe is preferable. This, I think, is the
extent of what can be said in its favor.
Let us now see what the two finite universes are like. The difference
between them is that in Einstein’s world it is only space that
is queer, whereas in De Sitter’s time is queer too. Consequently
Einstein’s world is less puzzling, and we will describe it first.
In Einstein’s world, light travels round the whole universe in a time
which is supposed to be something like a thousand million years. The
odd thing is that all the rays of light which start (say) from the sun
will meet again, after their enormous journey, in the place where the
sun was when they started. The case is exactly analogous to that of a
number of travelers who set out from London to go round the world in
great circles, all traveling at the same rate in different aeroplanes.
One starts due north, passes the North Pole, then the South Pole, and
finally comes home. Another starts due south, reaches the South Pole
first and then the North Pole. Another starts westward, but he must not
continue to travel due west, because then he would not be traveling on
a great circle. Another starts eastward, and so on. They all meet in
the antipodes of London, and then they all meet again in London. Now
if instead of aeronauts going round the earth you take rays of light
going round the universe, the same sort of thing happens: they all meet
first at the antipodes of their starting point, and then meet again at
their starting point. That means to say that a person who is near the
antipodes of the place where the sun was about five hundred million
years ago will see what is apparently a body as bright as the sun then
was (except for the small amount of light that has been stopped on the
way by opaque bodies), and having the same shape and size. And a person
who is near where the sun was a thousand million years ago will see
what is apparently a body just like what the sun was a thousand million
years ago. And the same applies to the antipodes of the sun fifteen
hundred million years ago, and to the place of the sun two thousand
million years ago, and so on. This series only ends when it carries us
back to a time before the sun existed.
But all these suns are only ghosts; that is to say, you could pass
through them without experiencing resistance, and they do not exert
gravitation. They are, in fact, like images in a mirror: they exist
only for the sense of sight, not for any other sense. It is rather
disturbing to reflect that, if this theory is true, any number of the
objects we see in the heavens may be merely ghosts. They are like
ghosts in their habit of revisiting the scenes of their past life.
Suppose a star had exploded at a certain place, as stars sometimes
will. Every thousand million years its ghost would return to the scene
of the disaster and explode again in the same place. There is, however,
considerable doubt whether rays of light could perform the journey with
sufficient accuracy to produce a clear image. Some would be stopped by
matter on the way, some would be turned out of the straight course by
passing near heavy bodies, as in the eclipse observations described in
Chapter IX, and for one reason or another their return would not be
punctual and exact.
There are various reasons for doubting whether Einstein’s universe can
be quite right.[12] Some of these are rather complicated. But there
is one objection which is easily appreciated: in Einstein’s theory,
absolute space and time re-enter by another door. The ghostly sun
is formed in the “place” where it was a thousand million years ago.
Both the “place” and the period of time are in a sense absolute. We
saw as early as Chapter I that “place” is a vague and popular notion,
incapable of scientific precision. It seems hardly worth while to go
through such a vast intellectual labor if the errors we set out to
correct are to reappear at the end.
[12] See Eddington, _Space, Time and Gravitation_, p. 162ff.
De Sitter’s world is even odder than Einstein’s, because time goes
mad as well as space. I despair of explaining, in non-mathematical
language, the particular form of lunacy with which time is afflicted,
but some of its manifestations can be described. An observer in this
world, if he observes a number of clocks, each of which is perfectly
accurate from its own point of view, will think that distant clocks
are going slow as compared with those in his neighborhood. They will
seem to go slower and slower, until, at a distance of one quarter of
the circumference of the universe, they will seem to have stopped
altogether. That region will seem to our observer a sort of lotus
land, where nothing is ever done. He will not be able to have any
cognizance of things farther off, because no light waves can get across
the boundary. Not that there is any real boundary: the people who live
in what our observer takes to be lotus land live just as bustling a
life as he does, but get the impression that he is eternally standing
still. As a matter of fact, you would never become aware of the lotus
land, because it would take an infinite time for light to travel from
it to you. You could become aware of places just short of it, but it
would remain itself always just beyond your ken. There will not be the
ghostly suns of Einstein’s world, because light cannot travel so far.
One of the oddest things about this state of affairs is that empirical
evidence for or against it is possible, and that there is actually
some slight evidence in its favor. If all “clocks” are slowed down at
a great distance from the observer, this will apply to the periodic
motions of atoms, and therefore to the light which they emit.
Consequently all rays of light emitted by distant objects ought, when
they reach us, to look rather more red or less violet than when they
started. This can be tested by the spectroscope. We can compare a
known line, as it appears in the spectrum of a spiral nebula, with
the same line as it appears in a terrestrial laboratory. We find, as a
matter of fact, that in a large majority of spiral nebulæ there is a
considerable displacement of spectral lines towards the red. The spiral
nebulæ are the most distant objects we can see: Eddington states that
their distances “may perhaps be of the order of a million light-years.”
(A light-year is the distance light travels in a year.) The usual
interpretation of a shifting of spectral lines towards the red is that
it is a “Doppler effect,” due to the fact that the source of light is
moving away from us. But one would expect to find the nebulæ just as
often moving towards us as moving away from us, if nothing operated but
the law of chances. If the world is such as De Sitter says it is, the
spectral lines of the spiral nebulæ will be displaced towards the red
owing to the slowing down of distant clocks, even if in fact they are
not moving away from us. This, for what it is worth, is an argument in
favor of De Sitter.
The same facts afford another argument in favor of De Sitter, for
another reason. If, at a given moment, a body is at rest relatively to
the observer, and at a distance from him, it will (in the absence of
counteracting causes) not remain at rest from his point of view, but
will begin to move away from him, and will continue to move away faster
and faster; the further it is from him, the more its retreat will be
accelerated. For bodies which are not too distant from each other,
gravitation may overcome this tendency; but as this tendency increases
with the distance, while gravitation diminishes, we should expect
to find very distant bodies receding from us if De Sitter’s theory
is right. Thus we have two reasons for the displacement of spectral
lines in spiral nebulæ: one, the slowing down of time; the other, the
movement away from us which we should expect at distances too great
for gravitation to be sensible. However, it cannot be said that the
argument, on either ground, is very strong. Eddington gives a list
of forty-one spiral nebulæ, of which five have their spectral lines
shifted towards the violet, not towards the red. Thus the material is
neither very copious nor quite harmonious.
Einstein’s and De Sitter’s hypotheses do not exhaust the possibilities
of a finite world: they are merely the two simplest forms of such a
world. There are arguments against each, and it hardly seems probable
that either is quite true. But it does seem probable that something
more or less analogous is true. If the universe is finite, it is
theoretically conceivable that there should be a complete inventory
of it. We may be coming to the end of what physics can do in the way
of stretching the imagination and systematizing the world. The period
since Galileo has been essentially the period of physics, as the age of
the Greeks was the period of geometry. It may be that physics will lose
its attractions through success: if the fundamental laws of physics
come to be fully known, adventurous and inquiring intellects will turn
to other fields. This may alter profoundly the whole texture of human
life, since our present absorption in machinery and industrialism is
the reflection in the practical world of the theorist’s interest in
physical laws. But such speculations are even more rash than those of
De Sitter, and I do not wish to lay any stress upon them.
CHAPTER XII: CONVENTIONS AND NATURAL LAWS
One of the most difficult matters in all controversy is to distinguish
disputes about words from disputes about facts: it ought not to be
difficult, but in practice it is. This is quite as true in physics as
in other subjects. In the seventeenth century there was a terrific
debate as to what “force” is; to us now, it was obviously a debate
as to how the word “force” should be defined, but at the time it was
thought to be much more. One of the purposes of the method of tensors,
which is employed in the mathematics of relativity, is to eliminate
what is purely verbal (in an extended sense) in physical laws. It is
of course obvious that what depends on the choice of co-ordinates is
“verbal” in the sense concerned. A man punting walks along the boat,
but keeps a constant position with reference to the river bed so
long as he does not pick up his pole. The Lilliputians might debate
endlessly whether he is walking or standing still: the debate would
be as to words, not as to facts. If we choose co-ordinates fixed
relatively to the boat, he is walking; if we choose co-ordinates
fixed relatively to the river bed, he is standing still. We want to
express physical laws in such a way that it shall be obvious when we
are expressing the same law by reference to two different systems
of co-ordinates, so that we shall not be misled into supposing we
have different laws when we only have one law in different words.
This is accomplished by the method of tensors. Some laws which seem
plausible in one language cannot be translated into another; these are
impossible as laws of nature. The laws that can be translated into
_any_ co-ordinate language have certain characteristics: this is a
substantial help in looking for such laws of nature as the theory of
relativity can admit to be possible. Combined with what we know of the
actual motions of bodies, it enables us to decide what must be the
correct expression of the law of gravitation: logic and experience
combine in equal proportions in obtaining this expression.
But the problem of arriving at genuine laws of nature is not to be
solved by the method of tensors alone; a good, deal of careful thought
is wanted in addition. Some of this has been done, especially by
Eddington; much remains to be done.
To take a simple illustration: Suppose, as in the hypothesis of the
Fitzgerald contraction, that lengths in one direction were shorter than
in another. Let us assume that a foot rule pointing north is only half
as long as the same foot rule pointing east, and that this is equally
true of all other bodies. Does such an hypothesis have any meaning?
If you have a fishing rod fifteen feet long when it is pointing west,
and you then turn it to the north, it will still measure fifteen feet,
because your foot rule will have shrunk too. It won’t “look” any
shorter, because your eye will have been affected in the same way. If
you are to find out the change, it cannot be by ordinary measurement;
it must be by some such method as the Michelson-Morley experiment, in
which the velocity of light is used to measure lengths. Then you still
have to decide whether it is simpler to suppose a change of length
or a change in the velocity of light. The experimental fact would be
that light takes longer to traverse what your foot rule declares to
be a given distance in one direction than in another—or, as in the
Michelson-Morley experiment, that it ought to take longer but doesn’t.
You can adjust your measures to such a fact in various ways; in any
way you choose to adopt, there will be an element of convention. This
element of convention survives in the laws that you arrive at after
you have made your decision as to measures, and often it takes subtle
and elusive forms. To eliminate the element of convention is, in fact,
extraordinarily difficult; the more the subject is studied, the greater
the difficulty is seen to be.
A more important example is the question of the size and shape of the
electron. We find experimentally that all electrons are the same size,
and that they are symmetrical in all directions. How far is this a
genuine fact ascertained by experiment, and how far is it a result of
our conventions of measurement? We have here a number of different
comparisons to make: (1) between different directions in regard to one
electron at one time; (2) in regard to one electron at different times;
(3) in regard to two electrons at the same time. We can then arrive
at the comparison of two electrons at different times, by combining
(2) and (3). We may dismiss any hypothesis which would affect all
electrons equally; for example, it would be useless to suppose that in
one region of space-time they were all larger than in another. Such a
change would affect our measuring appliances just as much as the things
measured, and would therefore produce no discoverable phenomena. This
is as much as to say that it would be no change at all. But the fact
that two electrons have the same mass, for instance, cannot be regarded
as purely conventional. Given sufficient minuteness and accuracy, we
could compare the effects of two different electrons upon a third;
if they were equal under like circumstances, we should be able to
infer equality in a not purely conventional sense. The question of
the symmetry of the forces exerted by an electron—_i.e._, that these
forces depend only upon the distance from the electron, and not upon
the direction—is more complicated. Eddington finally comes to the
conclusion that this, too, is a matter of convention. The argument
is difficult and I have not fully understood it; but I feel some
hesitation in accepting it as valid.
Eddington describes the process concerned in the more advanced portions
of the theory of relativity as “world-building.” The structure to be
built is the physical world as we know it; the economical architect
tries to construct it with the smallest possible amount of material.
This is a question for logic and mathematics. The greater our technical
skill in these two subjects, the more real building we shall do, and
the less we shall be content with mere heaps of stones. But before we
can use in our building the stones that nature provides, we have to
hew them into the right shapes: this is all part of the process of
budding. In order that this may be possible, the raw material must
have _some_ structure (which we may conceive as analogous to the
grain in timber), but almost any structure will do. By successive
mathematical refinements, we whittle away our initial requirements
until they amount to very little. Given this necessary minimum of
structure in the raw material, we find that we can construct from it a
mathematical expression which will have the properties that are needed
for describing the world we perceive—in particular, the properties
of conservation which are characteristic of momentum and energy (or
mass). Our raw material consisted merely of events; but when we find
that we can build out of it something which, as measured, will seem
to be never created or destroyed, it seems not surprising that we
should come to believe in “bodies.” These are really mere mathematical
constructions out of events, but owing to their permanence they are
practically important, and our senses (which were presumably developed
by biological needs) are adapted for noticing them, rather than the
crude continuum of events which is theoretically more fundamental. From
this point of view, it is astonishing how little of the real world is
revealed by physical science: our knowledge is limited, not only by the
conventional element, but also by the selectiveness of our perceptual
apparatus.
We assume that there is an “interval” between two events, in the
sense explained in Chapter VII, but we no longer assume that we can
unambiguously compare the length of an interval in one region with the
length of an interval in another. It is assumed by Weyl, who introduced
this limitation, that we can compare a number of small intervals which
all start from the same point; also that, in a very small journey,
our measuring rod will not alter its length much, so that there will
only be a small error if we compare lengths in neighboring places by
the usual methods. Weyl found that, by diminishing our assumptions as
to interval in this way, it was possible to bring electromagnetism
and gravitation into one system. The mathematics of Weyl’s theory is
complicated, and I shall not attempt to explain it. For the present,
I am concerned with a different consequence of his theory. If lengths
in different regions cannot be compared directly, there is an element
of convention in the indirect comparisons which we actually make. This
element will be at first unrecognized, but will be such as to simplify
to the utmost the expression of the laws of nature. In particular,
conditions of symmetry may be entirely created by conventions as to
measurement, and there is no reason to suppose that they represent any
property of the real world. The law of gravitation itself, according to
Eddington, may be regarded as expressing conventions of measurement.
“The conventions of measurement,” he says, “introduce an isotropy[13]
and homogeneity into measured space which need not originally have any
counterpart in the relation-structure which is being surveyed. This
isotropy and homogeneity is exactly expressed by Einstein’s law of
gravitation.”[14]
[13] “Isotropy” means being similar in all directions—_e.g._, that a
foot rule is as long when it points north as when it points east.
[14] _Mathematical Theory of Relativity_, p. 238.
The limitations of knowledge introduced by the selectiveness of our
perceptual apparatus may be illustrated by the indestructibility
of matter. This has been gradually discovered by experiment, and
seemed a well-founded empirical law of nature. Now it turns out
that, from our original space-time continuum, we can construct a
mathematical expression which will have properties causing it to appear
indestructible. The statement that matter is indestructible then ceases
to be a proposition of physics, and becomes instead a proposition
of linguistics and psychology. As a proposition of linguistics:
“Matter” is the name of the mathematical expression in question. As a
proposition of psychology: Our senses are such that we notice what is
roughly the mathematical expression in question, and we are led nearer
and nearer to it as we refine upon our crude perceptions by scientific
observation. This is much less than physicists used to think they knew
about matter.
The reader may say: What then is left of physics? What do we really
know about the world of matter? Here we may distinguish three
departments of physics. There is first what is included within the
theory of relativity, generalized as widely as possible. Next, there
are laws which cannot be brought within the scope of relativity.
Thirdly, there is what may be called geography. Let us consider each of
these in turn.
The theory of relativity, apart from convention, tells us that the
events in the universe have a four-dimensional order, and that,
between any two events which are near together in this order, there
is a relation called “interval,” which is capable of being measured
if suitable precautions are taken. We make also an assumption as to
what happens when a little measuring rod is carried round a closed
circuit in a certain manner; the consequences of this assumption are
such as to make it highly probable that it is true. Beyond this, there
is little in the theory of relativity that can be regarded as physical
laws. There is a great deal of mathematics, showing that certain
mathematically-constructed quantities must behave like the things we
perceive; and there is a suggestion of a bridge between psychology and
physics in the theory that these mathematically-constructed quantities
are what our senses are adapted for perceiving. But neither of these
things is physics in the strict sense.
The part of physics which cannot, at present, be brought within
the scope of relativity is large and important. There is nothing
in relativity to show why there should be electrons and protons;
relativity cannot give any reason why matter should exist in little
lumps. With this goes the whole theory of the structure of the atom.
The theory of quanta also is quite outside the scope of relativity.
Relativity is, in a sense, the most extreme application of what may
be called next-to-next methods. Gravitation is no longer regarded
as due to the effect of the sun upon a planet, but as expressing
characteristics of the region in which the planet happens to be.
Distance, which used to be thought to have a definite meaning however
far apart two points might be, is now only definite for neighboring
points. The distance between widely separated places depends upon the
route chosen. We may, it is true, define _the_ distance as the geodesic
distance, but that can only be estimated by adding up little bits,
that is to say, by the method we use in estimating the length of a
curve. What applies to distance applies equally to the straight line.
There is nothing in the actual world having exactly the properties
that straight lines were supposed to have; the nearest approach is the
track of a light ray. Straight lines have to be replaced by geodesics,
which are defined by what they do at each point, not all at once,
like Euclidean straight lines. Measurement, in Weyl’s theory, suffers
the same fate. We can only use a measuring rod to give lengths in one
place: when we move it to another region, there is no knowing how it
will alter. We do assume, however, that, if it alters, it alters bit
by bit, gradually, continuously, and not by sudden jumps. Perhaps
this assumption is unjustified. It belongs to the general outlook of
relativity, which is that of continuity. No doubt it is owing to this
outlook that relativity is unable to account for the discontinuities in
physics, such as quanta, electrons and protons. Perhaps relativity will
conquer these domains when it learns to dispense with the assumption of
continuity.
Finally we come to geography, in which I include history. The
separation of history from geography rests upon the separation of time
from space; when we amalgamate the two in space-time, we need one word
to describe the combination of geography and history. For the sake of
simplicity, I shall use the one word geography in this extended sense.
Geography, in this sense, includes everything that, as a matter of
crude fact, distinguishes one part of space-time from another. One
part is occupied by the sun, one by the earth; the intermediate
regions contain light waves, but no matter (apart from a very little
here and there). There is a certain degree of theoretical connection
between different geographical facts; to establish this is the purpose
of physical laws. It is thought that a sufficient knowledge of the
geographical facts of the solar system throughout any finite time,
however short, would enable an ideally competent physicist to predict
the future of the solar system so long as it remained remote from other
stars. We are already in a position to calculate the large facts about
the solar system backwards and forwards for vast periods of time. But
in all such calculations we need a basis of crude fact. The facts are
interconnected, but facts can only be inferred from other facts, not
from general laws alone. Thus the facts of geography have a certain
independent status in physics. No amount of physical laws will enable
us to infer a physical fact unless we know other facts as data for our
inference. And here when I speak of “facts” I am thinking of particular
facts of geography, in the extended sense in which I am using the term.
In the theory of relativity, we are concerned with _structure_, not
with the material of which the structure is composed. In geography,
on the other hand, the material is relevant. If there is to be any
difference between one place and another, there must either be
differences between the material in one place and that in another, or
places where there is material and places where there is none. The
former of these alternatives seems the more satisfactory. We might
try to say: There are electrons and protons, and the rest is empty.
But in the “empty” regions there are light waves, so that we cannot
say nothing happens in them. Some people maintain that the light
waves take place in the ether, others are content to say simply that
they take place; but in any case events are occurring where there are
light waves. That is all that we can really say for the places where
there is matter, since matter has turned out to be a mathematical
construction built out of events. We may say, therefore, that there
are events everywhere in space-time, but they must be of a somewhat
different kind according as we are dealing with a region where there is
an electron or proton or with the sort of region we should ordinarily
call empty. But as to the intrinsic nature of these events we can know
nothing, except when they happen to be events in our own lives. Our own
perceptions and feelings must be part of the crude material of events
which physics arranges into a pattern—or rather, which physics finds
to be arranged in a pattern. As regards events which do not form part
of our own lives, physics tells us the pattern of them, but is quite
unable to tell us what they are like in themselves. Nor does it seem
possible that this should be discovered by any other method.
CHAPTER XIII: THE ABOLITION OF “FORCE”
In the Newtonian system, bodies under the action of no forces move in
straight lines with uniform velocity; when bodies do not move in this
way, their change of motion is ascribed to a “force.” Some forces seem
intelligible to our imagination: those exerted by a rope or string,
by bodies colliding, or by any kind of obvious pushing or pulling. As
explained in an earlier chapter, our apparent imaginative understanding
of these processes is quite fallacious; all that it really means is
that past experience enables us to foresee more or less what is going
to happen without the need of mathematical calculations. But the
“forces” involved in gravitation and in the less familiar forms of
electrical action do not seem in this way “natural” to our imagination.
It seems odd that the earth can float in the void: the natural thing
to suppose is that it must fall. That is why it has to be supported on
an elephant, and the elephant on a tortoise, according to some early
speculators. The Newtonian theory, in addition to action at a distance,
introduced two other imaginative novelties. The first was, that
gravitation is not always and essentially directed what we should call
“downwards,” _i.e._, towards the center of the earth. The second was,
that a body going round and round in a circle with uniform velocity is
not “moving uniformly” in the sense in which that phrase is applied to
the motion of bodies under no forces, but is perpetually being turned
out of the straight course towards the center of the circle, which
requires a force pulling it in that direction. Hence Newton arrived at
the view that the planets are attracted to the sun by a force, which is
called gravitation.
This whole point of view, as we have seen, is superseded by relativity.
There are no longer such things as “straight lines” in the old
geometrical sense. There are “straightest lines,” or geodesics, but
these involve time as well as space. A light ray passing through
the solar system does not describe the same orbit as a comet, from
a geometrical point of view; nevertheless each moves in a geodesic.
The whole imaginative picture is changed. A poet might say that water
runs down hill because it is attracted to the sea, but a physicist or
an ordinary mortal would say that it moves as it does, at each point,
because of the nature of the ground at that point, without regard to
what lies ahead of it. Just as the sea does not cause the water to run
towards it, so the sun does not cause the planets to move round it. The
planets move round the sun because that is the easiest thing to do—in
the technical sense of “least action.” It is the easiest thing to do
because of the nature of the region in which they are, not because of
an influence emanating from the sun.
The supposed necessity of attributing gravitation to a “force”
attracting the planets towards the sun has arisen from the
determination to preserve Euclidean geometry at all costs. If we
suppose that our space is Euclidean, when in fact it is not, we shall
have to call in physics to rectify the errors of our geometry. We shall
find bodies not moving in what we insist upon regarding as straight
lines, and we shall demand a cause for this behavior. Eddington has
stated this matter with admirable lucidity. He supposes a physicist
who has assumed the formula for interval which is used in the special
theory of relativity—a formula which still supposes that the
observer’s space is Euclidean. He continues:
Since intervals can be compared by experimental
methods, he ought soon to discover that his
(formula for the interval) cannot be reconciled
with observational results, and so realize his
mistake. But the mind does not so readily get rid of
an obsession. It is more likely that our observer
will continue in his opinion, and attribute the
discrepancy of the observations to some influence
which is present and affects the behavior of his
test-bodies. He will, so to speak, introduce a
supernatural agency which he can blame for the
consequences of his mistake.... The name given to
any agency which causes deviation from uniform
motion in a straight line is _force_ according
to the Newtonian definition of force. Hence the
agency invoked through our observer’s mistake is
described as a “field of force.”... _A field of
force represents the discrepancy between the natural
geometry of a co-ordinate system and the abstract
geometry arbitrarily ascribed to it._[15]
[15] _Mathematical Theory of Relativity_, pp. 37-38. Italics in the
original.
If people were to learn to conceive the world in the new way, without
the old notion of “force,” it would alter not only their physical
imagination, but probably also their morals and politics. The latter
effect would be quite illogical, but is none the less probable on that
account. In Newton’s theory of the solar system, the sun seems like a
monarch whose behests the planets have to obey. In Einstein’s world
there is more individualism and less government than in Newton’s.
There is also far less hustle: we have seen that laziness is the
fundamental law of Einstein’s universe. The word “dynamic” has come to
mean, in newspaper language, “energetic and forceful”; but if it meant
“illustrating the principles of dynamics,” it ought to be applied to
the people in hot climates who sit under banana trees waiting for the
fruit to drop into their mouths. I hope that journalists, in future,
when they speak of a “dynamic personality,” will mean a person who
does what is least trouble at the moment, without thinking of remote
consequences. If I can contribute to this result, I shall not have
written in vain.
It has been customary for people to draw arguments from the laws of
nature as to what we ought to do. Such arguments seem to me a mistake:
to imitate nature may be merely slavish. But if nature, as portrayed by
Einstein, is to be our model, it would seem that the anarchists will
have the best of the argument. The physical universe is orderly, not
because there is a central government, but because every body minds
its own business. No two particles of matter ever come into contact;
when they get too close, they both move off. If a man were had up
for knocking another man down, he would be scientifically correct in
pleading that he had never touched him. What happened was that there
was a hill in space-time in the region of the other man’s nose, and it
fell down the hill.
The abolition of “force” seems to be connected with the substitution
of sight for touch as the source of physical ideas, as explained in
Chapter I. When an image in a looking glass moves, we do not think that
something has pushed it. In places where there are two large mirrors
opposite to each other, you may see innumerable reflections of the
same object. Suppose a gentleman in a top-hat is standing between the
mirrors, there may be twenty or thirty top-hats in the reflections.
Suppose now somebody comes and knocks off the gentleman’s hat with a
stick: all the other twenty or thirty top-hats will tumble down at the
same moment. We think that a force is needed to knock off the “real”
top-hat, but we think the remaining twenty or thirty tumble off, so to
speak, of themselves, or out of a mere passion for imitation. Let us
try to think out this matter a little more seriously.
Obviously something happens when an image in a looking glass moves.
From the point of view of sight, the event seems just as real as if it
were not in a mirror. But nothing has happened from the point of view
of touch or hearing. When the “real” top-hat falls, it makes a noise;
the twenty or thirty reflections fall without a sound. If it falls on
your toe, you feel it; but we believe that the twenty or thirty people
in the mirrors feel nothing, though top-hats fall on their toes too.
But all this is equally true of the astronomical world. It makes no
noise, because sound cannot travel across a vacuum. So far as we know,
it causes no “feelings,” because there is no one on the spot to “feel”
it. The astronomical world, therefore, seems hardly more “real” or
“solid” than the world in the looking glass, and has just as little
need of “force” to make it move.
The reader may feel that I am indulging in idle sophistry. “After all,”
he may say, “the image in the mirror is the reflection of something
solid, and the top-hat in the mirror only falls off because of the
force applied to the real top-hat. The top-hat in the mirror cannot
indulge in behavior of its own; it has to copy the real one. This shows
how different the image is from the sun and the planets, because _they_
are not obliged to be perpetually imitating a prototype. So you had
better give up pretending that an image is just as real as one of the
heavenly bodies.”
There is, of course, some truth in this; the point is to discover
exactly _what_ truth. In the first place, images are not “imaginary.”
When you see an image, certain perfectly real light waves reach your
eye; and if you hang a cloth over the mirror, these light waves cease
to exist. There is, however, a purely optical difference between an
“image” and a “real” thing. The optical difference is bound up with
this question of imitation. When you hang a cloth over the mirror,
it makes no difference to the “real” object; but when you move the
“real” object away, the image vanishes also. This makes us say that the
light rays which make the image are only reflected at the surface of
the mirror, and do not really come from a point behind it, but from
the “real” object. We have here an example of a general principle of
great importance. Most of the events in the world are not isolated
occurrences, but members of groups of more or less similar events,
which are such that each group is connected in an assignable manner
with a certain small region of space-time. This is the case with the
light rays which make us see both the object and its reflection in the
mirror: they all emanate from the object as a center. If you put an
opaque globe round the object at a certain distance, the object and
its reflection are invisible at any point outside the globe. We have
seen that gravitation, although no longer regarded as an action at a
distance, is still connected with a center: there is, so to speak, a
hill symmetrically arranged about its summit, and the summit is the
place where we conceive the body to be which is connected with the
gravitational field we are considering. For simplicity, common sense
lumps together all the events which form one group in the above sense.
When two people see the same object, two different events occur, but
they are events belonging to one group and connected with the same
center. Just the same applies when two people (as we say) hear the
same noise. And so the reflection in a mirror is less “real” than the
object reflected, even from an optical point of view, because light
rays do not spread in _all_ directions from the place where the image
seems to be, but only in directions in front of the mirror, and only so
long as the object reflected remains in position. This illustrates the
usefulness of grouping connected events about a center in the way we
have been considering.
When we examine the changes in such a group of objects, we find that
they are of two kinds: there are those which affect only some member
of the group, and those which make connected alterations in all the
members of the group. If you put a candle in front of a mirror, and
then hang black cloth over the mirror, you alter only the reflection
of the candle as seen from various places. If you shut your eyes,
you alter its appearance to you, but not its appearance elsewhere.
If you put a red globe round it at a distance of a foot, you alter
its appearance at any distance greater than a foot, but not at any
distance less than a foot. In all these cases, you do not regard the
candle itself as having changed; in fact, in all of them, you find that
there are groups of changes connected with a different center or with
a number of different centers. When you shut your eyes, for instance,
your eyes, not the candle, look different to any other observer: the
center of the changes that occur is in your eyes. But when you blow
out the candle, its appearance _everywhere_ is changed; in this case
you say that the change has happened to the candle. The changes that
happen to an object are those that affect the whole group of events
which center about the object. All this is only an interpretation of
common sense, and an attempt to explain what we mean by saying that the
image of the candle in the mirror is less “real” than the candle. There
is no connected group of events situated all round the place where the
image seems to be, and changes in the image center about the candle,
not about a point behind the mirror. This gives a perfectly verifiable
meaning to the statement that the image is “only” a reflection. And at
the same time it enables us to regard the heavenly bodies, although
we can only see and not touch them, as more “real” than an image in a
looking glass.
We can now begin to interpret the common sense notion of one body
having an “effect” upon another, which we must do if we are really to
understand what is meant by the abolition of “force.” Suppose you come
into a dark room and switch on the electric light: the appearance of
everything in the room is changed. Since everything in the room is
visible because it reflects the electric light, this case is really
analogous to that of the image in the mirror; the electric light is the
center from which all the changes emanate. In this case, the “effect”
is explained by what we have already said. The more important case is
when the effect is a movement. Suppose you let loose a tiger in the
middle of a Bank Holiday crowd: they would all move, and the tiger
would be the center of their various movements. A person who could
see the people but not the tiger would infer that there was something
repulsive at that point. We say in this case that the tiger has an
effect upon the people, and we might describe the tiger’s action upon
them as of the nature of a repulsive force. We know, however, that they
fly because of something which happens to _them_, not merely because
the tiger is where he is. They fly because they can see and hear him,
that is to say, because certain waves reach their eyes and ears. If
these waves could be made to reach them without there being any tiger,
they would fly just as fast, because the neighborhood would seem to
them just as unpleasant.
Let us now apply similar considerations to the sun’s gravitation. The
“force” exerted by the sun only differs from that exerted by the tiger
in being attractive instead of repulsive. Instead of acting through
waves of light or sound, the sun acquires its apparent power through
the fact that there are modifications of space-time all round the sun.
Like the noise of the tiger, they are more intense near their source;
as we travel away they grow less and less. To say that the sun “causes”
these modifications of space-time is to add nothing to our knowledge.
What we know is that the modifications proceed according to a certain
rule, and that they are grouped symmetrically about the sun as center.
The language of cause and effect adds only a number of quite irrelevant
imaginings, connected with will, muscular tension, and such matters.
What we can more or less ascertain is merely the formula according to
which space-time is modified by the presence of gravitating matter.
More correctly: we can ascertain what kind of space-time _is_ the
presence of gravitating matter. When space-time is not accurately
Euclidean in a certain region, but has a non-Euclidean character which
grows more and more marked as we approach a certain center, and when,
further, the departure from Euclid obeys a certain law, we describe
this state of affairs briefly by saying that there is gravitating
matter at the center. But this is only a compendious account of what
we know. What we know is about the places where the gravitating matter
is _not_, not about the place where it is. The language of cause
and effect (of which “force” is a particular case) is thus merely
a convenient shorthand for certain purposes; it does not represent
anything that is genuinely to be found in the physical world.
And how about matter? Is matter also no more than a convenient
shorthand? This question, however, being a large one, demands a
separate chapter.
CHAPTER XIV: WHAT IS MATTER?
The question “What is matter?” is of the kind that is asked by
metaphysicians, and answered in vast books of incredible obscurity.
But I am not asking the question as metaphysician: I am asking it as a
person who wants to find out what is the moral of modern physics, and
more especially of the theory of relativity. It is obvious from what we
have learned of that theory that matter cannot be conceived quite as it
used to be. I think we can now say more or less what the new conception
must be.
There were two traditional conceptions of matter, both of which have
had advocates ever since scientific speculation began. There were
the atomists, who thought that matter consisted of tiny lumps which
could never be divided; these were supposed to hit each other and then
bounce off in various ways. After Newton, they were no longer supposed
actually to come into contact with each other, but to attract and
repel each other, and move in orbits round each other. Then there
were those who thought that there is matter of some kind everywhere,
and that a true vacuum is impossible. Descartes held this view, and
attributed the motions of the planets to vortices in the ether. The
Newtonian theory of gravitation caused the view that there is matter
everywhere to fall into discredit, the more so as light was thought by
Newton and his disciples to be due to actual particles traveling from
the source of the light. But when this view of light was disproved, and
it was shown that light consisted of waves, the ether was revived so
that there should be something to undulate. The ether became still more
respectable when it was found to play the same part in electromagnetic
phenomena as in the propagation of light. It was even hoped that atoms
might turn out to be a mode of motion of the ether. At this stage, the
atomic view of matter was, on the whole, getting the worst of it.
Leaving relativity aside for the moment, modern physics has provided
proof of the atomic structure of ordinary matter, while not disproving
the arguments in favor of the ether, to which no such structure is
attributed. The result was a sort of compromise between the two views,
the one applying to what was called “gross” matter, the other to the
ether. There can be no doubt about electrons and protons, though, as we
shall see shortly, they need not be conceived as atoms were conceived
traditionally. As for the ether, its status is very curious: many
physicists still maintain that, without it, the propagation of light
and other electromagnetic waves would be inconceivable, but except in
this way it is difficult to see what purpose it serves. The truth is,
I think, that relativity demands the abandonment of the old conception
of “matter,” which is infected by the metaphysics associated with
“substance,” and represents a point of view not really necessary in
dealing with phenomena. This is what we must now investigate.
In the old view, a piece of matter was something which survived all
through time, while never being at more than one place at a given time.
This way of looking at things is obviously connected with the complete
separation of space and time in which people formerly believed. When we
substitute space-time for space and time, we shall naturally expect to
derive the physical world from constituents which are as limited in
time as in space. Such constituents are what we call “events.” An event
does not persist and move, like the traditional piece of matter; it
merely exists for its little moment and then ceases. A piece of matter
will thus be resolved into a series of events. Just as, in the old
view, an extended body was composed of a number of particles, so, now,
each particle, being extended in time, must be regarded as composed
of what we may call “event-particles.” The whole series of these
events makes up the whole history of the particle, and the particle is
regarded as _being_ its history, not some metaphysical entity to which
the events happen. This view is rendered necessary by the fact that
relativity compels us to place time and space more on a level than they
were in the older physics.
This abstract requirement must be brought into relation with the known
facts of the physical world. Now what are the known facts? Let us
take it as conceded that light consists of waves traveling with the
received velocity. We then know a great deal about what goes on in
the parts of space-time where there is no matter; we know, that is to
say, that there are periodic occurrences (light waves) obeying certain
laws. These light waves start from atoms, and the modern theory of
the structure of the atoms enables us to know a great deal about the
circumstances under which they start, and the reasons which determine
their wave lengths. We can find out not only how one light wave
travels, but how its source moves relatively to ourselves. But when I
say this I am assuming that we can recognise a source of light as the
same at two slightly different times. This is, however, the very thing
which had to be investigated.
We saw, in the preceding chapter, how a group of connected events can
be formed, all related to each other by a law, and all ranged about a
center in space-time. Such a group of events will be the arrival, at
various places, of the light waves emitted by a brief flash of light.
We do not need to suppose that anything particular is happening at the
center; certainly we do not need to suppose that we know _what_ is
happening there. What we know is that, as a matter of geometry, the
group of events in question are ranged about a center, like widening
ripples on a pool when a fly has touched it. We can hypothetically
invent an occurrence which is to have happened at the center, and set
forth the laws by which the consequent disturbance is transmitted. This
hypothetical occurrence will then appear to common sense as the “cause”
of the disturbance. It will also count as one event in the biography of
the particle of matter which is supposed to occupy the center of the
disturbance.
Now we find not only that one light wave travels outward from a center
according to a certain law, but also that, in general, it is followed
by other closely similar light waves. The sun, for example, does not
change its appearance suddenly; even if a cloud passes across it during
a high wind, the transition is gradual, though swift. In this way a
group of occurrences connected with a center at one point of space-time
is brought into relation with other very similar groups whose centers
are at neighboring points of space-time. For each of these other groups
common sense invents similar hypothetical occurrences to occupy their
centers, and says that all these hypothetical occurrences are part of
one history; that is to say, it invents a hypothetical “particle” to
which the hypothetical occurrences are to have occurred. It is only by
this double use of hypothesis, perfectly unnecessary in each case, that
we arrive at anything that can be called “matter” in the old sense of
the word.
If we are to avoid unnecessary hypotheses, we shall say that an
electron at a given moment is the various disturbances in the
surrounding medium which, in ordinary language, would be said to be
“caused” by it. But we shall not take these disturbances at what is,
for us, the moment in question, since that would make them depend upon
the observer; we shall instead travel outward from the electron with
the velocity of light, and take the disturbance we find in each place
as we reach it. The closely similar set of disturbances, with very
nearly the same center, which is found existing slightly earlier or
slightly later, will be defined as _being_ the electron at a slightly
earlier or slightly later moment. In this way, we preserve all the
laws of physics, without having recourse to unnecessary hypotheses or
inferred entities, and we remain in harmony with the general principle
of economy which has enabled the theory of relativity to clear away so
much useless lumber.
Common sense imagines that when it sees a table it sees a table. This
is a gross delusion. When common sense sees a table, certain light
waves reach its eyes, and these are of a sort which, in its previous
experience, has been associated with certain sensations of touch, as
well as with other people’s testimony that they also saw the table.
But none of this ever brought us to the table itself. The light waves
caused occurrences in our eyes, and these caused occurrences in the
optic nerve, and these in turn caused occurrences in the brain. Any one
of these, happening without the usual preliminaries, would have caused
us to have the sensations we call “seeing the table,” even if there had
been no table. (Of course, if matter in general is to be interpreted
as a group of occurrences, this must apply also to the eye, the optic
nerve, and the brain.) As to the sense of touch when we press the table
with our fingers, that is an electric disturbance in the electrons and
protons of our finger tips, produced, according to modern physics, by
the proximity of the electrons and protons in the table. If the same
disturbance in our finger tips arose in any other way, we should have
the same sensations, in spite of there being no table. The testimony
of others is obviously a second-hand affair. A witness in a law court,
if asked whether he had seen some occurrence, would not be allowed to
reply that he believed so because of the testimony of others to that
effect. In any case, testimony consists of sound waves and demands
psychological as well as physical interpretation; its connection with
the object is therefore very indirect. For all these reasons, when
we say that a man “sees a table,” we use a highly abbreviated form
of expression, concealing complicated and difficult inferences, the
validity of which may well be open to question.
But we are in danger of becoming entangled in psychological questions,
which we must avoid if we can. Let us therefore return to the purely
physical point of view.
What I wish to suggest may be put as follows. Everything that occurs
elsewhere, owing to the existence of an electron, can be explored
experimentally, at least in theory, unless it occurs in certain
concealed ways. But what occurs within the electron (if anything occurs
there) it is absolutely impossible to know: there is no conceivable
apparatus by which we could obtain even a glimpse of it. An electron is
known by its “effects.” But the word “effects” belongs to a view of
causation which will not fit modern physics, and in particular will
not fit relativity. All that we have a right to say is that certain
groups of occurrences happen together, that is to say, in neighboring
parts of space-time. A given observer will regard one member of the
group as earlier than the other, but another observer may judge the
time order differently. And even when the time order is the same for
all observers, all that we really have is a connection between the two
events, which works equally backwards and forwards. It is not true that
the past determines the future in some sense other than that in which
the future determines the past: the apparent difference is only due to
our ignorance, because we know less about the future than about the
past. This is a mere accident: there might be beings who would remember
the future and have to infer the past. The feelings of such beings
in these matters would be the exact opposite of our own, but no more
fallacious.
The moral of this is that, if an electron is only known by its
“effects,” there is no reason to suppose that anything exists except
the “effects.” In so far as these “effects” consist of light waves
and other electromagnetic disturbances, we may say that what is
called “empty space” consists of regions where these disturbances are
propagated freely. Every such disturbance, we find, has a center, and
when we get very near the center (though still at a finite distance
from it) we find that the law of propagation of the disturbance ceases
to be valid. This region within which the law does not hold is called
“matter”; it will be an electron or proton according to circumstances.
The region so defined is found to move relatively to other such
regions, and its movements follow the known laws of dynamics. So far,
this theory provides for electromagnetic phenomena and the motions of
matter; and it does so without assuming that “matter” is anything but
systems of electromagnetic phenomena. In order to carry out the theory
fully, it would no doubt be necessary to introduce many complications.
But it seems fairly clear that all the facts and laws of physics
can be interpreted without assuming that “matter” is anything more
than groups of events, each event being of the sort which we should
naturally regard as “caused” by the matter in question. This does not
involve any change in the symbols or formulæ of physics: it is merely
a question of interpretation of the symbols.
This latitude in interpretation is a characteristic of mathematical
physics. What we know is certain very abstract logical relations,
which we express in mathematical formulæ; we know also that, at
certain points, we arrive at results which are capable of being tested
experimentally. Take, for example, the eclipse observations by which
Einstein’s theory as to the bending of light was established. The
actual observation consisted in the careful measurement of certain
distances on certain photographic plates. The formulæ which were to
be verified were concerned with the course of light in passing near
the sun. Although the part of these formulæ which gives the observed
result must always be interpreted in the same way, the other part of
them may be capable of a great variety of interpretations. The formulæ
giving the motions of the planets are almost exactly the same in
Einstein’s theory as in Newton’s, but the meaning of the formulæ is
quite different. It may be said generally that, in the mathematical
treatment of nature, we can be far more certain that our formulæ are
approximately correct than we can be as to the correctness of this or
that interpretation of them. And so in the case with which this chapter
is concerned: the question as to the nature of an electron or a proton
is by no means answered when we know all that mathematical physics has
to say as to the laws of its motion and the laws of its interaction
with the environment. A definite and conclusive answer to our question
is not possible just because a variety of answers are compatible with
the truth of mathematical physics. Nevertheless some answers are
preferable to others, because some have a greater probability in their
favor. We have been seeking, in this chapter, to define matter so that
there _must_ be such a thing if the formulæ of physics are true. If we
had made our definition such as to secure that a particle of matter
should be what one thinks of as substantial, a hard, definite lump, we
should not have been _sure_ that any such thing exists. That is why
our definition, though it may seem complicated, is preferable from the
point of view of logical economy and scientific caution.
CHAPTER XV: PHILOSOPHICAL CONSEQUENCES
The philosophical consequences of relativity are neither so great nor
so startling as is sometimes thought. It throws very little light on
time-honored controversies, such as that between realism and idealism.
Some people think that it supports Kant’s view that space and time are
“subjective” and are “forms of intuition.” I think such people have
been misled by the way in which writers on relativity speak of “the
observer.” It is natural to suppose that the observer is a human being,
or at least a mind; but he is just as likely to be a photographic
plate or a clock. That is to say, the odd results as to the difference
between one “point of view” and another are concerned with “point of
view” in a sense applicable to physical instruments just as much as to
people with perceptions. The “subjectivity” concerned in the theory of
relativity is a _physical_ subjectivity, which would exist equally if
there were no such things as minds or senses in the world.
Moreover, it is a strictly limited subjectivity. The theory does
not say that _everything_ is relative; on the contrary, it gives a
technique for distinguishing what is relative from what belongs to a
physical occurrence in its own right. If we are going to say that the
theory supports Kant about space and time, we shall have to say that it
refutes him about space-time. In my view, neither statement is correct.
I see no reason why, on such issues, philosophers should not all stick
to the views they previously held. There were no conclusive arguments
on either side before, and there are none now; to hold either view
shows a dogmatic rather than a scientific temper.
Nevertheless, when the ideas involved in Einstein’s work have become
familiar, as they will when they are taught in schools, certain changes
in our habits of thought are likely to result, and to have great
importance in the long run.
One thing which emerges is that physics tells us much less about
the physical world than we thought it did. Almost all the “great
principles” of traditional physics turn out to be like the “great
law” that there are always three feet to a yard; others turn out to
be downright false. The conservation of mass may serve to illustrate
both these misfortunes to which a “law” is liable. Mass used to be
defined as “quantity of matter,” and as far as experiment showed it
was never increased or diminished. But with the greater accuracy of
modern measurements, curious things were found to happen. In the first
place, the mass as measured was found to increase with the velocity;
this kind of mass was found to be really the same thing as energy. This
kind of mass is not constant for a given body, but the total amount of
it in the universe is conserved, or at least obeys a law very closely
analogous to conservation. This law itself, however, is to be regarded
as a truism, of the nature of the “law” that there are three feet to a
yard; it results from our methods of measurement, and does not express
a genuine property of matter. The other kind of mass, which we may call
“proper mass,” is that which is found to be the mass by an observer
moving with the body. This is the ordinary terrestrial case, where
the body we are weighing is not flying through the air. The “proper
mass” of a body is very nearly constant, but not quite, and the total
amount of “proper mass” in the world is not quite constant. One would
suppose that if you have four one-pound weights, and you put them all
together into the scales, they will together weigh four pounds. This is
a fond delusion: they weigh rather less, though not enough less to be
discovered by even the most careful measurements. In the case of four
hydrogen atoms, however, when they are put together to make one helium
atom, the defect is noticeable; the helium atom weighs measurably less
than four separate hydrogen atoms.
Broadly speaking, traditional physics has collapsed into two portions,
truisms and geography. There are, however, newer portions of physics,
such as the theory of quanta, which do not come under this head, but
appear to give genuine knowledge of laws reached by experiment.
The world which the theory of relativity presents to our imagination
is not so much a world of “things” in “motion” as a world of _events_.
It is true that there are still electrons and protons which persist,
but these (as we saw in the preceding chapter) are really to be
conceived as strings of connected events, like the successive notes
of a song. It is _events_ that are the stuff of relativity physics.
Between two events which are not too remote from each other there is,
in the general theory as in the special theory, a measurable relation
called “interval,” which appears to be the physical reality of which
lapse of time and distance in space are two more or less confused
representations. Between two distant events, there is not any one
definite interval. But there is one way of moving from one event to
another which makes the sum of all the little intervals along the route
greater than by any other route. This route is called a “geodesic,” and
it is the route which a body will choose if left to itself.
The whole of relativity physics is a much more step-by-step matter than
the physics and geometry of former days. Euclid’s straight lines have
to be replaced by light rays, which do not quite come up to Euclid’s
standard of straightness when they pass near the sun or any other very
heavy body. The sum of the angles of a triangle is still thought to be
two right angles in very remote regions of empty space, but not where
there is matter in the neighborhood. We, who cannot leave the earth,
are incapable of reaching a place where Euclid is true. Propositions
which used to be proved by reasoning have now become either
conventions, or merely approximate truths verified by observation.
It is a curious fact—of which relativity is not the only
illustration—that, as reasoning improves, its claims to the power of
proving facts grow less and less. Logic used to be thought to teach
us how to draw inferences; now, it teaches us rather how not to draw
inferences. Animals and children are terribly prone to inference: a
horse is surprised beyond measure if you take an unusual turning. When
men began to reason, they tried to justify the inferences that they
had drawn unthinkingly in earlier days. A great deal of bad philosophy
and bad science resulted from this propensity. “Great principles,”
such as the “uniformity of nature,” the “law of universal causation,”
and so on, are attempts to bolster up our belief that what has often
happened before will happen again, which is no better founded than the
horse’s belief that you will take the turning you usually take. It is
not altogether easy to see what is to replace these pseudo-principles
in the practice of science; but perhaps the theory of relativity gives
us a glimpse of the kind of thing we may expect. Causation, in the
old sense, no longer has a place in theoretical physics. There is,
of course, something else which takes its place, but the substitute
appears to have a better empirical foundation than the old principle
which it has superseded.
The collapse of the notion of one all-embracing time, in which all
events throughout the universe can be dated, must in the long run
affect our views as to cause and effect, evolution, and many other
matters. For instance, the question whether, on the whole, there is
progress in the universe, may depend upon our choice of a measure of
time. If we choose one out of a number of equally good clocks, we may
find that the universe is progressing as fast as the most optimistic
American thinks it is; if we choose another equally good clock, we may
find that the universe is going from bad to worse as fast as the most
melancholy Slav could imagine. Thus optimism and pessimism are neither
true nor false, but depend upon the choice of clocks.
The effect of this upon a certain type of emotion is devastating. The
poet speaks of
One far-off divine event
To which the whole creation moves.
But if the event is sufficiently far off, and the creation moves
sufficiently quickly, some parts will judge that the event has already
happened, while others will judge that it is still in the future. This
spoils the poetry. The second line ought to be:
To which some parts of the creation move,
while others move away from it.
But this won’t do. I suggest that an emotion which can be destroyed by
a little mathematics is neither very genuine nor very valuable. But
this line of argument would lead to a criticism of the Victorian Age,
which lies outside my theme.
What we know about the physical world, I repeat, is much more
abstract, than was formerly supposed. Between bodies there are
occurrences, such as light waves; of the _laws_ of these occurrences,
we know something—just so much as can be expressed in mathematical
formulæ—but of their _nature_ we know nothing. Of the bodies
themselves, as we saw in the preceding chapter, we know so little
that we cannot even be sure that they are anything: they _may_ be
merely groups of events in other places, those events which we should
naturally regard as their effects. We naturally interpret the world
pictorially; that is to say, we imagine that what goes on is more or
less like what we see. But in fact this likeness can only extend to
certain formal logical properties expressing structure, so that all we
can know is certain general characteristics of its changes. Perhaps an
illustration may make the matter clear. Between a piece of orchestral
music as played, and the same piece of music as printed in the score,
there is a certain resemblance, which may be described as a resemblance
in structure. The resemblance is of such a sort that, when you know the
rules, you can infer the music from the score or the score from the
music. But suppose you had been stone deaf from birth, but had lived
among musical people. You could understand, if you had learned to speak
and to do lip-reading, that the musical scores represented something
quite different from themselves in intrinsic quality, though similar in
structure.[16] The value of music would be completely unimaginable to
you, but you could infer all its mathematical characteristics, since
they are the same as those of the score. Now our knowledge of nature is
something like this. We can read the scores, and infer just so much as
our stone-deaf person could have inferred about music. But we have not
the advantages which he derived from association with musical people.
We cannot know whether the music represented by the scores is beautiful
or hideous; perhaps, in the last analysis, we cannot be quite sure that
the scores represent anything but themselves. But this is a doubt which
the physicist, in his professional capacity, cannot permit himself to
entertain.
[16] For the definition of “structure,” see the present author’s
_Introduction to Mathematical Philosophy_.
Assuming the utmost that can be claimed for physics, it does not tell
us what it is that changes, or what are its various states; it only
tells us such things as that changes follow each other periodically,
or spread with a certain speed. Even now we are probably not at the
end of the process of stripping away what is merely imagination, in
order to reach the core of true scientific knowledge. The theory of
relativity has accomplished a very great deal in this respect, and in
doing so has taken us nearer and nearer to bare structure, which is
the mathematician’s goal—not because it is the only thing in which he
is interested as a human being, but because it is the only thing that
he can express in mathematical formulæ. But far as we have traveled in
the direction of abstraction, it may be that we shall have to travel
further still.
In the preceding chapter, I suggested what may be called a minimum
definition of matter, that is to say, one in which matter has, so
to speak, as little “substance” as is compatible with the truth of
physics. In adopting a definition of this kind, we are playing for
safety: our tenuous matter will exist, even if something more beefy
also exists. We tried to make our definition of matter, like Isabella’s
gruel in Jane Austen, “thin, but not too thin.” We shall, however, fall
into error if we assert positively that matter is nothing more than
this. Leibniz thought that a piece of matter is really a colony of
souls. There is nothing to show that he was wrong, though there is also
nothing to show that he was right: we know no more about it either way
than we do about the flora and fauna of Mars.
To the non-mathematical mind, the abstract character of our physical
knowledge may seem unsatisfactory. From an artistic or imaginative
point of view, it is perhaps regrettable, but from a practical point
of view it is of no consequence. Abstraction, difficult as it is, is
the source of practical power. A financier, whose dealings with the
world are more abstract than those of any other “practical” man, is
also more powerful than any other practical man. He can deal in wheat
or cotton without needing ever to have seen either: all he needs to
know is whether they will go up or down. This is abstract mathematical
knowledge, at least as compared to the knowledge of the agriculturist.
Similarly the physicist, who knows nothing of matter except certain
laws of its movements, nevertheless knows enough to enable him to
manipulate it. After working through whole strings of equations, in
which the symbols stand for things whose intrinsic nature can never be
known to us, he arrives at last at a result which can be interpreted
in terms of our own perceptions, and utilized to bring about desired
effects in our own lives. What we know about matter, abstract and
schematic as it is, is enough, in principle, to tell us the rules
according to which it produces perceptions and feelings in ourselves;
and it is upon these rules that the _practical_ uses of physics depend.
The final conclusion is that we know very little, and yet it is
astonishing that we know so much, and still more astonishing that so
little knowledge can give us so much power.
THE END
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