RELATIVITY
* FOR ALL *
HERBERT DINGLE
UU-NRLF
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o
RELATIVITY FOR ALL
RELATIVITY FOR ALL
BY
HERBERT PINGLE, B.Sc.
LECTURER ON ASTROPHYSICS AT THE
IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY
WITH TWO DIAGRAMS
BOSTON
LITTLE, BROWN, AND COMPANY
1922
PREFACE
AN apology is needed for the production of
a popular work on Relativity after Einstein
himself has undertaken such a task. As a
matter of fact, our purpose here is somewhat
different from Einstein's. This little book is
written, not so much for those who wish to under-
stand a physical theory from the point of view of
a physicist, as for the large body of intelligent
men and women who look on physics as one of
many avenues into the secret of the Universe,
and wish to know its windings in their relation
to the larger field of human inquiry.
The dominant aim throughout the book has
been to make the ideas definite and intelligible
to the ordinary mind. All other considerations
strict philosophical phraseology, literary graces,
conventional forms of presentation ; everything,
in fact, but truth have been subordinated to
vi RELATIVITY FOR ALL
this end. Between the Chary bdis of inaccuracy
and the Scylla of abstruseness, the course is narrow
and the sea is rough. The vessel will hardly
escape bufferings from either side. It is hoped,
nevertheless, that in the present voyage a passage
will be made without fatal mishap.
Those who wish to pursue the subject more
deeply, from either the philosophical or the
scientific standpoint, are recommended to the
works of Professor A. N. Whitehead, F.R.S. The
author is glad to acknowledge his deep indebted-
ness to Professor Whitehead for invaluable help
and unwearying kindness in unveiling the mysteries
of a difficult subject.
H. D
IMPERIAL COLLEGE OF SCIENCE
AND TECHNOLOGY
July 1921
CONTENTS
PART I
THE FOUNDATIONS OF SCIENCE
CHAP. PAGB
I. How THE THEORY AROSE . . i
II. SPACE, TIME, AND MATTER . . .10
III. THE " FOUR-DIMENSIONAL CONTINUUM" . 22
IV. THE VELOCITY OF LIGHT . . .32
PART II
THE LAWS OF NATURE
V. WHAT is A NATURAL LAW?. . . 39
VI. THE WORK OF NEWTON . . .45
VII. RELATIVITY AND THE MOVEMENTS OF
BODIES . . . . .53
VIII. SOME PROBLEMS OF RELATIVITY . . 63
INDEX . . . . .71
RELATIVITY FOR ALL
PART I
THE FOUNDATIONS OF SCIENCE
CHAPTER I
HOW THE THEORY AROSE
"Space is thought's, and the wonders thereof, and the
secret of space;
Is thought not more than the thunders and lightnings ?
shall thought give place ?
Tune, father of life, and more great than the life it begat
and began,
Earth's keeper and heaven's and their fate, lives, thinks
and hath substance in man."
WHEN Swinburne wrote these words, he
was thinking what a wonderful being
he was. That they would ever come to
be a poetical expression of cold, scientific ideas
about matter, time, and space, was probably the
thought farthest from his inaccessible mind. Yet
so it is. The new doctrine of Relativity entails
a complete uprooting of the conceptions that
I
y FOR ALL
have formerly been held to lie inviolable at the
foundations of thought and experience. The
theory is not merely a metaphysical speculation.
It has arisen in order to explain certain facts of
observation, which seem to point to it as the most
probable statement of the nature of the Universe
which we perceive.
Let us think for a moment of the way in which
we are accustomed tacitly, almost subcon-
sciously to regard the physical world. We think
of it as a number of pieces of matter. These
pieces of matter exist in space. We do not gener-
ally take the trouble to define to ourselves exactly
what we mean by " space," but we understand
one another quite well when we refer to it in
conversation. It is a sort of receptacle, without
limit in any direction, in which the material of the
world exists and moves about. When we say a
certain object is " there," we have a clear idea of
what we mean, and we feel confident that, pro-
vided the object does not move, it will always be
" there," no matter what we do ourselves. If we
could take the wings of the morning, and dwell
in the uttermost parts of the sea, the object would
still be " there." Then we have also an idea of
time. We do not define this either, but we know
what it means. When something has happened,
it belongs to the past. Nothing can ever bring
the same happening into the present or the future
again. It happened at some definite time, and
HOW THE THEORY AROSE 3
every person in the Universe who observed it
would agree with every other person as to what
that time was, supposing every one had an accurate
clock.
These three " things " matter, space, and time
are the three independent, immovable founda-
tion-stones of the World, as we are accustomed to
regard it, and Science has hitherto adopted them
as the only possible data in terms of which to
express its discoveries. For instance, the law of
gravitation expresses the way in which matter
will move near other matter, i.e. it describes how
the position of matter in space changes as time
advances. All other physical laws have been
essentially of the same kind.
But recently, scientists have had reason to
question whether space, time, and matter are really
the absolute and fundamental things we have
supposed. The doubt arises in the following
way. As the result of a considerable accumula-
tion of experience, it has been impressed upon
physicists that space is not empty, but is filled,
in every nook and cranny of its infinite extent,
with a kind of invisible, intangible super-matter,
which has been called the " ether." The first
and still one of the most convincing of the
indications of this substance came from the study
of the propagation of light through space. Cer-
tain remarkable laboratory experiments seemed
to assert that light could travel from a luminous
4 RELATIVITY FOR ALL
body to the eye in no other form than that of
a train of waves, like the ripples spreading out
in a pool of water when a stone is thrown into it.
The facts pointed unanimously in this direction,
and their combined force was almost irresistible.
But waves are unthinkable without some medium
in which they exist. There must obviously be
something through which light waves travel :
what is that something ? It is not the air, because
light reaches us from stars millions of miles away,
and there is, to the best of our knowledge, no air
or matter of any kind reaching all the way from
the stars to us. It must be something of whose
existence we have not previously been aware ;
something that fills all space, for light comes to
us from all directions and from unimaginable
distances. It must, moreover, penetrate the
pores and secret places of matter itself, for does
not light pass through some bodies, which are
said to be " transparent " ? Scientists, then,
were led to the idea of a space filled with this
infinite, all-permeating ether.
Now there is nothing in all this to challenge
our common sense. The conception of an omni-
present ether offers no difficulties to the imagina-
tion. Space might as well be full as empty, so
far as mere possibility is concerned. Neverthe-
less, we should feel more satisfied on the matter
if we had some direct sign of the ether's existence.
An experiment giving immediate evidence of it
HOW THE THEORY AROSE 5
would be more convincing than its appearance
as the last link in a chain of reasoning. This
was recognized by physicists, and many attempts
were made to betray the ether into a declaration
of its reality. One of the most promising of these
was the search for the velocity of the earth as it
travels through the ether. Since tte ether filled
all space, it had to be regarded as being at rest
as a whole. It could not move bodily because
it was infinite and there was nothing for it to move
into. Consequently, the velocity of the earth
through the ether could be looked upon as its
" absolute" velocity something more funda-
mental than its velocity of revolution round the \
Sun, which ignores any possible motion of the
Sun itself.
To understand the most famous of all experi-
ments made to measure this absolute velocity,
we must picture the earth swimming through the
ether at some speed which we are to find out.
Suppose that, on the earth's surface, and travelling
with it, there are two objects a lamp and a
mirror (A), represented in Fig. i. Suppose also
that these objects he in the line of absolute motion
of the earth whatever that may be the mirror
in advance of the lamp. Let the lamp be uncovered
for an instant, so that it sends a beam towards
the mirror (A). Now light travels with a definite
velocity (186,000 miles a second). It moves at
this speed through the ether towards the mirror (A).
6 RELATIVITY FOR ALL
But the mirror (A) is running away from the beam,
since it is fixed to the earth, and the earth is moving
through the ether. Consequently, the light should
take longer to reach the mirror (A) than it would
if the earth were not moving. On reaching the
mirror (A), the light is reflected back to the lamp.
But now the lamp is moving to meet it, so that
Mirror (B)
< 7J> Lamp
Mirror (A)
__Direction of Motion of
Earbh through Ether
FIG. i.
the light will make the return journey more quickly
than it would have done if the earth had been
at rest. It is a very simple matter to calculate
what the time should be for the total journey to
and fro, in terms of the unknown absolute velocity
of the earth. But now suppose that, at the same
time as the beam of light left the lamp, another
HOW THE THEORY AROSE 7
beam left the same point at right angles to tlve
first, towards the mirror (B). This beam would
move across the line of motion of the earth, and
the time it would take to perform its complete
journey to the mirror (B) and back, can be cal-
culated also. On making the calculations, we
find that the second beam should return to the
lamp before the first, and we can tell exactly how
much sooner it should arrive in terms, of course,
of the unknown velocity. The interval should
vary throughout the year, owing to the change
in the direction and rate of motion of the earth.
Now an experiment on these lines the famous
Michelson-Morley experiment was actually per-
formed in 1887. The apparatus used was so
delicate that it was capable of detecting a far
smaller quantity than that which it was expected
to measure. Every one awaited the result with
confidence. But when the experiment was made,
it was found that the two beams arrived back
at the same time. The apparatus was turned
round, so that the mirrors were in different positions
relative to the lamp ; the experiment was repeated
at different times of the year ; but always the
result was the same the two beams took precisely
the same time for their respective journeys.
Now it must be recognized at once that this
was a most extraordinary thing. Here was an
experiment, performed with every care and
apparently with full understanding of what was
8 RELATIVITY FOR ALL
being done, which completely failed to give the
result that common sense would have thought
inevitable. For what the experiment seems to
imply is this. We know that, if a bird flies from
one end of a train to the other, he will complete
the journey sooner if the train is moving towards
him than he would do if it were at rest. The
experiment suggests that, if he only moves as
quickly as light, he will appear to the engine-
driver to reach the end in the same time, no matter
whether the train is at rest, or moving towards
him, or moving away from him. It seems im-
possible, but experience shows it to be true.
If any explanation is to be given, therefore, it
must necessarily involve something revolutionary.
Various suggestions were offered, but, in the light
of future investigations at any rate, none of them
was so satisfactory or far-reaching as the most
revolutionary of all the principle of relativity.
Let us ask ourselves why common sense says
that the bird cannot reach the end of the train
in the same time, when the train is moving, as he
does when it is at rest. We reply that he has
to travel different distances in space in the two
cases. If, during his flight, the train has moved,
its far end, when he reaches it, will be at a point in
space different from that which it occupied at the
beginning. The times taken by the two journeys
will therefore be different. But, in saying this,
we are assuming that " space " and " time "
HOW THE THEORY AROSE 9
mean the same things for the engine-driver at
rest as they do for the engine-driver in motion.
What if they are different ? In that case, of course,
we shall not know what to expect. If what one
man calls an hour, another calls a minute, and
what the first pronounces a yard, the second asserts
to be a mile and if there is no possible criterion
for testing their statements, so that both are equally
right or equally wrong then it will not be surprising
if results are obtained which would otherwise
be considered impossible. This is, in essence,
just what the principle of relativity says. It\i
declares that the conceptions of space and time, I -
and, as will subsequently appear, of matter I
also are not absolute and independent, but are I
relative to the observer. What do we mean by *
this ? We will try to explain it in the next chapter.
CHAPTER II
SPACE, TIME, AND MATTER
SUPPOSE a being, endowed with full human
intelligence, but without any experience or
knowledge of the world, were suddenly
created and placed, say, on Hampstead Heath :
what would he perceive ? The answer we should
naturally give to this question is contained in
the first chapter he would perceive material
things in space and time. The answer of the
relativist, however, is different. According to
him, the man would perceive a number of happen-
ings, occurrences, events. Their interpretation as
material objects in space and time would come
later, and would be the result of his intelligent
ordering of the events among themselves.
Let us take an example. Suppose our visitor
sees a wasp alight on a flower. That is an event.
Next, suppose the wasp alights on his hand.
That is another event. We have here, then, two
events, and to the man they would at first be
merely two events and nothing more. But now,
suppose he begins to use his intelligence, and tries
to impose some order or arrangement on the
SPACE, TIME, AND MATTER 11
circumstances in which he finds himself. He
notices that there is something common to the
two events, and also to a number of intermediate
events, with which we need not concern ourselves.
He has an impression of an "object" with black
and yellow bands, which characterizes the whole
series of events from the wasp on the flower to
the wasp on the hand. This " character " of the
events he calls " matter/' and the particular
example with which we are dealing, a "wasp."
He has now the first of the three entities which
we supposed were his original perceptions matter.
But that is not enough. If he confines himself
to what is common to the two events, he will
not be able to distinguish them, one from the
other. He must construct some other relation
between them. He does this by saying that they
are " in different places " : the flower is in one
" place," and the hand in another. In this way
he forms an idea of place, and by extending the
same relation to other events which he perceives,
he becomes conscious of " infinite space." Matter,
space two types of relation between events
have arisen as conceptions derived from a common
source the events themselves.
Are these conceptions sufficient to enable our
observer to think clearly and to comprehend the
world around him ? Not quite : matter and
space will not relate all the events which he per-
ceives. Consider a third event : suppose he feels
12 RELATIVITY FOR ALL
a stinging sensation in his hand. How can he
relate this to the second of the events we have
already considered the arrival of the wasp on
his hand ? The space relations are the same
if we make the legitimate assumption that there
is no movement between the two events. The
material relations are the same the wasp and
the hand. He must find a third type of relation.
He therefore says that one of the events occurs
. " before " the other. By generalizing this rela-
^ tion, he forms the conception of " time."
Matter, space, and time, then, according to the
relativist, are types of relation between events.
Together they appear to be capable of relating
the whole of inanimate Nature in a consistent
and orderly way. Our visitor employs them for
purposes of thought ; he hands them down to
his successors, generation after generation, until,
ultimately, they come to be regarded as the funda-
mental perceptions of the human mind, and the
poor event, the legitimate father of them all, sinks
to the rank of a dependent.
This idea of the derivative character of matter,
\ space, and time lies at the heart of the modern
j principle of relativity. It deserves particular
emphasis, for, if it is once firmly grasped, the
greater part of the difficulty of the subject dis-
appears. It is the event that is the immediate
^ entity of perception ; Nature is the sum-total
V of events, and every instrument of thought that
SPACE, TIME, AND MATTER 13
our minds employ can be traced back to its ultimate
origin in events. Two observers of Nature see,
not necessarily the same matter, but the same r"
events, because events finally constitute the ex-
ternal physical world. What about the spatial,
temporal, and material relations the observers
impose on the events : will they be the same ?
Evidently it is not necessary that they should be.
We have no right to say, without experimental
test, that a man on the Earth and a man on Mars,
say, who are moving relatively to one another,
will both declare Regent Street to be half a mile
long. What they may both be immediately
aware of are the two events which are the existence
of the two ends of the street during their perception
of them. If one man relates them spatially by
saying that they are half a mile apart, there is
no fundamental necessity, so far as we know, for
the other man to do the same. It is essentially
a matter for experiment.
Let us illustrate this point, which is of basic
importance, by an example which, however,
must not be pressed too close. Consider two
events : first, a young man sees a young maiden ;
second, he shows signs of agitation. Consider,
further, two observers of these events the young
man himself and another young maiden. Each of
them relates the events in a certain way. The
young man calls his relation, "love" ; the second
young maiden supposing she is honest with her-
14 RELATIVITY FOR ALL "
self calls her relation " jealousy." Here, then,
we have a type of relation between events which
we definitely recognize as relative* Why is this
so ? We do not know : it is a complete, mystery.
But we certainly do know that the events are
related differently by different persons. May not,
then, the space and time relations also be relative ?
" Ah ! " yo^i exclaim, " but the tyo' observers in
your example are in different circumstances.
They have different predispositions, different
histories, different emotional states. Conse-
quently, their emotional relations between events
will inevitably be different. But 'space' and time
are independent of our predispositions, our
histories, our emotional states. Tittiey dre on an
entirely different footing." That i^ quite true :
space and time do not change with dyr emotions.
But it does not follow that they do not change
with anything. Emotions are relative because
they depend on our emotional state. Might not
space and time depend on our spatio-temporal
state ? Might they not be modified by motion,
for example, i.e., by a change of our position in
space as our position in time advances ? It seems
to be possible, at any rate. If I am moving
relatively to you, it does not seem to be imperative
that my spatial and my temporal relations,
between events that are observed by both of us,
shall be the same as yours. "But," you reply,
"it is idle to talk of what seems to be possible.
SPACE, TIME, AND MATTER 15
Is it not a fact that there is no such difference
between us ? If my watch does not keep mean
solar time when I am on an express train, have '
I not legitimate ground of complaint against my
watchmaker ? If the road becomes longer or
shorter when I am travelling along it on a bus,
shall not my habits justly be open to suspicion ?
You may be right with your possibilities, but
experience shows that they are not actual."
But then, after all, experience has its limitations.
Perhaps, in a journey from London to Manchester,
your watch keeps " perfect " time, so far as you
can judge : it may yet have varied by an amount
too small for you to detect. If you could travel
at the same rate for 100,000 years, or if you
could move at a speed of 100,000 miles a second,
might not your extended experience show that
the former conclusion was too hasty ? Experience
is certainly the final judge in the case before us,
but it gives a verdict strictly according to the
facts in its possession. If we wish to get at the
truth, we must elicit all the facts.
The question, then, comes down to this : Can
we make an experiment that will decide definitely ,
' whether space and time are different for observers
in relative motion ? Such an experiment is not
inconceivable, but it would be one of colossal
difficulty. At present, all we can hope to do is
to see if there are any facts which receive a simple
explanation if we assume the relativity of time
16 RELATIVITY FOR ALL
and space, but which can be interpreted only
with difficulty, or not at all, if those relations are
absolute. Now facts of this kind are presented
to us in the Michelson-Morley experiment, and in
other attempts to observe the drift of matter
through ether. They are to be found also in
certain astronomical observations, to which we
shall refer later. The theory of relativity, in fact,
. has passed with honours every test it has so far
been found possible to apply. It is only the
absence of direct experimental confirmation that
^prevents it from being recognized as a proven law
of the Universe.
We shall consider the precise nature of the
relativity in the next chapter, but it will be useful
to state at once the kind of effect the relativist
requires. The table opposite shows the loss of
a watch during one day, and the shortening of a
i-foot rule in the direction of motion, as they would
appear to an observer moving relatively to them
at different speeds, who tests them by standard
instruments moving with him. 1
1 As we have said, the principle from which these
results are obtained will be stated in the next chapter.
The reader, however, may, at this point, be somewhat
curious as to its nature. We will, therefore, say in
advance that the theory of relativity assumes that it is
1 impossible for any observer ever to obtain experimental
evidence of relative motion between matter and ether.
This means (cf. the Michelson-Morley experiment) that
all observers, whatever their state of relative motion, will
SPACE, TIME, AND MATTER 1.7
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38 RELATIVITY FOR ALL
It is clear now why the relativity of space and time,
if it is true, did not declare itself long ago. The effect
is so small for speeds which we ordinarily use that
it is quite impossible to detect it. High velocities,
or observations extending over long periods, are
necessary before its importance begins to appear.
We have still to consider the nature of matter.
Is that relative also ? If two of the primary rela-
tions between events vary with the observer, it
seems probable that the third will do so as well.
As a matter of fact, the principle of relativity
requires that there shall be a change in matter
arising from motion. The quantity of matter in
a body (not its " size," which is an attribute of
space, but the actual amount of matter in it ;
what we call its " mass," and usually measure by
weighing the body) should be different for different
observers moving relatively to one another. Or,
obtain the same measure of the velocity of light. Such
a result can only occur if the spaces and times used by
the observers are related in one particular way, which,
granted the principle, is readily determined by mathe-
matical calculation. In this manner the foregoing table has
been constructed. Thus, to a hypothetical observer on
Arcturus, a beam of light would travel i^oop- inch per
foot less than the same beam to an observer on the Earth.
But the former observer would find that it did so in a
period of time less than that measured by the Earth man
by -g^th second per day. These numbers are such that
both observers would obtain the same value for the velocity
of the beam, and are the only ones that are consistent with
the application of the same principle to all possible relative
velocities.
SPACE, TIME, AND MATTER 19
in other words, if a body moved with gradually
increasing velocity relative to us, we should find
that its mass, supposing we could measure it,
would grow continuously. 1 U
Now this, again, seems at first to be contrary
to experience. From the time of Lavoisier, in
the eighteenth century, at least, the invariability
of mass has been one of the cardinal doctrines of
chemistry and physics, and experience has tended
consistently to confirm it. Can it conceivably be
an illusion ? Here, as before, we must understand
clearly what experience shows. All that we can
fairly deduce from the experiments which support
the law of invariability of mass is that, under the
particular conditions of those experiments, the
, mass of a body does not vary by any amount that
we can measure. The case might be totally
different if we had instruments of much greater
1 The mass of a body, as we shall see in Chapter VI, is
the resistance the body offers to change of velocity. If f/
this resistance remained constant, then any force capable
of increasing the velocity at all would, if it continued
acting, go on increasing it indefinitely. From the point
of view of relativity, however, this is impossible, because,
as in the Michelson-Morley experiment, space and time
adjust themselves so as to make the velocity of light the
highest relative velocity possible between two bodies.
(See also Chapter IV.) Consequently, resistance to
change of velocity (i.e. mass) must increase as speed
increases, in some way which will allow of its becoming
infinite when the velocity of light is reached . The calcula-
tion of the necessary change is a simple piece of mathe-
matics, and gives the results embodied in Table 2.
20
RELATIVITY FOR ALL
precision, or if we could compare the mass of a
body at rest with its mass when moving at an
extremely high speed. As with time and space,
the demands of relativity are so humble at
ordinary velocities that they might be granted in
full without giving us the slightest suspicion that
they are made. As the body moves faster and
faster, the theory gathers confidence, and asks for
more and more. Table 2 shows, for the same
velocities as before, the increase in mass of a body
containing i Ib. of matter when it is at rest with
respect to the observer.
TABLE 2.
Speed.
Increase of Mass.
60 miles per hour
67,000 miles per hour
700,000 miles per hour
93,000 miles per second .
161,000 miles per second .
186,000 miles per second .
250.000,000,000,000
Ib.
1,730,000
lh
IU *
Ib.
i Ib.
Infinitely great.
Can we test this by experiment ? It is ex-
ceedingly difficult. We must rely again on
indirect evidence. Nature has provided us with
extremely small electrified particles which move
SPACE, TIME, AND MATTER 21
with enormous speed. It has been found possible
to measure their mass at different rates of motion.
The result is that they actually do show a change'
of mass with speed, of just the amount required
by the theory. Moreover, theoretical results
deduced from the assumption of such a change
when these bodies move inside material atoms,
have been verified with almost incredible accuracy.
The theory again has been successful in every
test to which it has been subjected. But we
must remark that these are highly special cases.
The particles are electrically charged they are
believed, in fact, to be particles of electricity
itself and the effects observed can be explained
equally well on the ordinary electro-magnetic theory,
without reference to relativity. We cannot, there-
fore, put them forward as unequivocal evidence for
relativity. But we can say that, so far as experi-
ment has yet gone, there is nothing that has put
the theory in the slightest difficulty, or necessitated
any modification of its fundamental principles.
Let us see, then, just where we stand. We have
said that, since the only element in Nature is the
event, space, time, and matter may be relative to
the observer. We have considered experiments
which seem to make it very probable that they
are relative. It remains for us now to investigate
the principles governing the magnitude of their
variation with change of speed. It is to this
question that we turn in the next chapter.
CHAPTER III
THE "FOUR-DIMENSIONAL CON-
TINUUM"
IT has already been pointed out and it is
so important that it will bear repetition
that the principle of relativity is a de-
duction from facts of observation. It is emphati-
cally not an arm-chair doctrine, proceeding from
the inner recesses of the brain without reference
to the results of experience. When the modern
relativist says that space, time, and matter are
different ideas for different observers, he does so
because he believes that the interpretation of
experimental facts which thereby becomes pos-
sible is the simplest and, on the whole, the most
plausible that he can devise. Consequently, he
is not content with the mere statement that these
relations change with one's state of motion. He
must say exactly by how much they change. As
we implied in the first chapter, if a yard and an
hour to one observer become a quite indiscrimi-
nate length and time to another, anything might
happen. But, actually, " anything " does not
happen : particular things happen for example,
" FOUR-DIMENSIONAL CONTINUUM " 28
the Michelson-Morley experiment. The magnitude
of the yard and the hour to the second observer
must be such as to explain those particular
things, and cannot be anything else.
The numerical side of the theory of relativity is
derived from the failure of all attempts to detect ^
the relative motion of matter and ether. The
relativist assumes that, in the nature of things, it
is impossible to observe such a motion ; in other
words, that space and time change with motion
in such a way as always to make the measure of
the velocity of matter through ether equal to
nothing. If this is granted, the calculation of
the necessary change becomes a simple piece of
mathematics.
We are endeavouring in this book to avoid the
use of general mathematical formulae. We shall,
therefore, not give the theoretical expressions
for the dependence of the time, space, and mass
units on velocity. The reader who is interested
in this side of the question may find them in
any of the more technical expositions of the prin-
ciple. We call attention rather to Tables i and 2
in Chapter II, from which most of the essential
facts may be understood quite as well as from the
general expressions. It will be observed at once
that, until the relative velocity reaches a very
high value, the change is almost infinitesimally
small. As the velocity grows, however, the '
change increases at a more rapid rate, until, at
24 RELATIVITY FOR ALL
the velocity of light, the state of affairs would
appear to be inconceivable.
We shall return to this point in the next
chapter. For the moment we will direct our
attention to a very simple relation between
the space and time used by any one observer
a relation which summarizes the quantitative
requirements of the principle. It is not brought
out in the tables, and, as it is of very great im-
portance, we will deal with it at length.
All who have tried to get at the meaning of
relativity have, at one time or another, come
across the blessed phrase, " the four-dimensional
continuum." What does it mean ? Let us .be
quite clear, first of all, as to the meaning of the
word " dimension." Suppose we have a room,
enclosed by four square, vertical walls, and a
square floor and ceiling. Suppose an electric
lamp globe is suspended somewhere in its interior.
How can we describe to an architect, say the
exact position of the globe in the room ? (We
are not concerning ourselves now with the question
of what we mean by " position," which occupied
us in the last chapter. We are using the word in
its ordinary, everyday sense, just as we might
have done if we had had no reason to doubt the
absolute nature of space.) We should tell him
something about it if we said the globe was 7 feet
above the ground. But that would not be enough.
If that were all, the globe might be anywhere in a
" FOUR-DIMENSIONAL CONTINUUM " 25
movable floor placed at that height. Suppose,
then, we say, further, that it is 6 feet from the
wall containing the door. That is better, but
still he is not satisfied. There may be a whole
line of objects on our imaginary movable floor
which satisfy this condition, and the globe might
be at any point in it. But let us now say that
it is 5 feet from the adjacent wall containing the
window. We have then determined its position
completely. There is only one point in the line
that is 5 feet from the wall containing the window :
or, in other words, there is only one point in the
room that is 7 feet above the ground, 6 feet from
one specified wall, and 5 feet from another. We
have definitely fixed the position of the globe by
these measurements.
Now there are other ways in which we could
have done this, but, in all of them, three inde-
pendent measurements are necessary and sufficient.
This is expressed, in technical language, by saying
that space has three dimensions. Another aspect
of the same property of space is embodied in the
statement that three independent measurements
are necessary to calculate the spatial volume
occupied by a body e.g. its length, breadth, and
height.
Three independent measurements, we say, will
define the position of a body in relation to a *"'
given structure (e.g. a room), and these three
measurements may be made in different ways.
26
RELATIVITY FOR ALL
We must now point out that, in whatever way
they are made, there is a certain relation between
them that is always satisfied. Let the continuous
lines in Fig. 2 indicate the floor and the two walls
of the room with which we are dealing, and let the
globe be suspended in its defined position, A. It
1 l-l l-l I
1 ' 1 1 1 1
FIG. 2.
will be 7, 6, and 5 feet from the respective planes
of reference. It is easily proved that the distance
from the globe, A, to the corner, B, in which these
three planes meet, is obtained by adding together
the three quantities, 7 2 , 6 2 , 5 2 , and finding the
square root of the sum. Thus, 7 2 + 6 2 + 5 2 = iio,
and the square root of no is nearly loj, so that
the globe is nearly loj feet from the corner, B.
" FOUR-DIMENSIONAL CONTINUUM " 27
Now suppose as the result of an earthquake,
say the room is twisted into the position shown
by the dotted lines in the figure, in such a way
that the points A and B remain in exactly the same
positions as before. The distance AB will then
be unaltered about loj feet. But the distances
of the lamp from the walls and floor will now be
quite different. Yet and this is the point we
are trying to illustrate provided the walls and
floor still remain at right angles to one another,
those distances must be such that the sum of their
squares is equal to no. Thus, if the lamp is 9 feet
and 5 feet from the two walls, it must be 2 feet
from the floor, for 9 2 + 5 2 + 2 2 = no. It cannot
possibly be at any other distance, however the
room is twisted within the restrictions we have
mentioned.
Now we need not have had the earthquake to
alter the position of the room. We could have
imagined the walls and floor to be anywhere we
liked, and defined the position of the globe relative
to the point B by reference to the imaginary room.
Or, without supposing the room twisted at all,
we could have fixed the position by three other
measurements of a different kind. The essential
point is that, however we do it, there is a definite
way of combining the measurements so as to give
the number no, and the measurements are bound
to be related among themselves so that the com-
bination will give this number.
28 RELATIVITY FOR ALL
To summarize, then, the statement that space
has three dimensions implies two things : first,
three independent measurements are necessary to
fix one point relative to another ; second, what-
ever three measurements we select for this purpose,
there is a certain combination of them that is the
same for all selections.
Bearing this in mind, we are now in a position
to understand what a four-dimensional continuum
is. We repeat that, in all that we have said about
the three dimensions of space, we have been speak-
ing in terms of absolute space. We have supposed
that the distances 7, 6, 5 feet and the number no
have the same value for all observers. From the
point of view of relativity, as we know, this is only
true as long as we are at rest in the room. If
we begin to move, the distances change, and our
measurement of time changes also. In view of the
constancy of the number no, whatever alterations
took place in the separate measurements in our
supposed absolute space, it is perhaps natural to
inquire whether there is any way of combining
our new space and new time measurements, what-
ever they may be, in such a way as to obtain the
same result as that given by the corresponding
combination of our old space and old time measure-
ments. As a matter of fact, there is such a com-
bination.
To illustrate this, we must choose two events
instead of two points, A and B, for, to deal with
" FOUR-DIMENSIONAL CONTINUUM " 29
time, we must introduce something containing the
temporal quality. Actually, the perceived ex-
istence of the corner and the lamp are events, for
perception itself takes time. It will simplify
matters, however, if our data are more readily
recognized as events. Let us consider the total
interval (which we usually analyse into space in-
terval and time interval) between the lighting
of the lamp, A, and the arrival of a spider at
the corner, B. Suppose, first, that we are at
rest in the room. Then the square of the spatial
distance between these two events is, as we have
seen, f + 6 2 + 5 2 , i.e. no. Suppose the time inter-
val is 10 units ; l so that its square is 100. The
difference between the squares of the space and
time intervals (which, we shall soon see, is an
important quantity) will then be 7 2 + 6 2 + 5 2 - io 2 =
no - 100 = 10. Now consider how the events
would appear to us if we were moving relatively
to the room. As we know, both the space
interval and the time interval would be different.
Suppose our speed to be such as to give us
a space interval of 9 feet between the events.
What would the time interval be ? It is found
that it must be just large enough for its square to
differ from the square of the space interval by
exactly the same amount (namely, 10) as in the
former case. It must therefore be nearly 8J units,
1 The magnitude of a time unit for this purpose is one
thousand-millionth of a second.
30 RELATIVITY FOR ALL
for 9 2 -(8J) 2 =io, very nearly. And, however we
move, this condition must always be satisfied. 1
And that is all that the four-dimensional con-
tinuum means. There is nothing essentially
mysterious about it. The mental panic that it
sometimes creates seems to arise from the complete
illusion that there is actually a fourth dimension
in space a sort of additional direction of spatial
extension, of which we could obtain experience if
we possessed an additional sense. In reality, the
fourth dimension does not exist, except as an
academic expression of our familiar experience of
time. Nothing exists fundamentally but events.
It is not true to say that they take place in a four-
dimensional continuum. Strictly speaking, they
do not take place at all : they simply exist in them-
selves. We do not say that Nature " takes place,"
and Nature is simply the aggregate of events. The
four-dimensional continuum is merely the mathe-
matician's shorthand way of saying, first, that
four measurements are necessary to define the
complete interval between two events, and second,
1 Strictly speaking, this particular combination (space
interval) 2 - (time interval) 2 is invariant only in free
space. In the neighbourhood of heavy bodies or, to
use scientific language, in strong gravitational fields a
slightly different combination must be taken. This will
be dealt with in a later chapter. In all circumstances,
the general statement that there is a particular combina-
tion of the four measurements that is constant for all
observers is strictly true.
" FOUR-DIMENSIONAL CONTINUUM " 31
that a certain combination of these four measure-
ments is constant for all observers, whatever their
spaces and times might be. It has no other
physical meaning." It derives its name simply by
analogy to the familiar three dimensions of space.
We have dwelt in detail on this point, because a
great deal of the terror inspired by the idea of
relativity is due to preliminary misconceptions.
The theory is approached as if it were something
quite outside the scope of ordinary intelligence and
everyday experience. It cannot be said too often
that this is a complete mistake. The only reason
why the practical effects of the principle (supposing
it to be true) were not recognized long ago is that
they are exceedingly minute not at all that they
require new organs of sense and intelligence. If
this is once thoroughly understood, the subject
will lose its esoteric appearance and begin to be
instructive.
CHAPTER IV
THE VELOCITY OF LIGHT
WE have already called attention to the
remarkable properties which seem to be
possessed by the velocity of light. Suppose
we have two observers, A and B, moving relatively
to one another with this velocity. Tables i
and 2 show us what to expect. It would appear]
that events separated by a finite time interval;
to A would be simultaneous to B. For since,
under these conditions, B's watch would lose
twenty-four hours in one of A's days (see Table
i), time would appear to stand still for B, and the
whole of A's world past, present, and future
would be concentrated for B's perception in a
moment of time.. The relations of A and B are,
of course, reciprocal, for it is just as true to say
that A is moving relatively to B as to say that B
is moving relatively to A. Hence, B's world is
presented instantaneously to A also. Our ob-
servers, then, would appear each to inhabit a
double world of events : one world is perceived
instantaneously, and the other stretches through
time.
32
THE VELOCITY OF LIGHT 38
It is very interesting to think that the whole
panorama of the world's history from the dawn
of created things to the last sunset of time
might be conjured up for our inspection, if only
we could move past it quickly enough. Like
every good thing, however, the realization of such
a prospect entails some compensating conditions.
We say nothing for the moment about the difficulty
of attaining the requisite relative speed, or of
perceiving terrestrial events clearly when it is
attained. But it should be pointed out that the
perception, supposing it were possible, would
occupy but an instant of our time a duration so
minute that our sluggish intelligences would be
powerless to apprehend it. Moreover, the picture
would present a very different aspect from that
to which we are accustomed. According to Table
I, the dimension of every material object in the
direction of relative motion would be nothing.
The instantaneous world would consist of a collec-
tion of plane objects, having no thickness at all.
The historian, then, has no great incentive to
study the production of high velocities. But
the economist appears to be in better case. It
seems from Table 2 that if matter moves past
him with sufficient speed, it can increase its
content to any desired amount. The widow
no longer needs an Elijah to conserve her supplies ;
if she agrees to the conditions, Maskelyne can
do it. But here again, the juggling fiends of
3
34 RELATIVITY FOR ALL
relativity are not to be believed. They may keep
the word of promise to our ear ; they will certainly
break it to our hope. The " mass " that grows
with velocity is not " size " ; the size of the
matter, in fact, actually diminishes, owing to the
shortening in the direction of motion. What
mass really means (see Chapter VI) is inertia,
or resistance of matter to change of motion. So
that, all that Table 2 implies is that the faster a
body moves past an observer, the more difficult
does it become to increase the relative speed :
the body itself gets smaller and smaller all the
time.
From the practical point of view, then, any
anticipations of increased wealth or novel orders
of experience that the theory might have aroused,
are likely to meet with disappointment. It
might, however, be some consolation to the in-
tellect to know that it is not called upon to
conceive the possibility of such anticipations.
But we have yet to indicate another property of
this remarkable velocity of light : it is the highest
velocity that one body can have relative to an-
other a natural maximum of speed, to exceed
which is for ever impossible. Let us look once
more at Table 2. At the velocity of light we
see that the mass of a body is " infinitely great."
Remembering that " mass " means resistance to
change of motion, it follows that there is an in-
finitely great resistance to the change of speed of
THE VELOCITY OF LIGHT 35
a body moving with this critical velocity. Not
only can a body never exceed this speed. If it
once reaches it, it can never begin to move more
slowly, for the resistance to change of motion is \
as great in one direction as in the other. The
relative velocity of the body is fixed*
Now there appears to be an obvious objection
to this. Suppose a body is moving past me with
the velocity of light. If I begin to move in the
same direction, do I not decrease our relative
velocity ? Or, if I move in the opposite direction,
do I not increase it ? The answer in each case
is, No. When I begin to move, my space and
time alter, and my new measure of the relative
velocity, with my new space and time, is exactly
the same as the old measure, with the old space
and time. At whatever speed I move, the result
is the same. This is, in fact, the essence of the\
Michelson-Morley experiment, in which light
moved relatively to the observer with exactly
the same speed, in whatever direction it was
travelling, or however the earth was moving.
All this follows, as we have said more than once,
from the experimental failure to detect any change
at all in the measured velocity of light, arising
from motion of the observer, and the assumption
that no such change exists. It is grounded in
actual physical experience. Einstein has gone a
step further, and has attempted to deduce from
it the meaning of the term ''simultaneity." He
36 RELATIVITY FOR ALL
points out that when we say that two things
happen " at the same time," we may have a
general idea of what we mean, but we should be
in difficulties if we were asked to give a rigid
explanation of the phrase. In the light of the
theory of relativity, he suggests a definition on the
following lines. Suppose we have two events,
and an observer situated midway in his spatial
distance between them. Suppose light signals
reach the observer from the two events at the
same time. Then, according to Einstein, the two
events themselves were simultaneous. To an-
other observer, in motion relative to the first,
the events would not necessarily be simultaneous,
because the same light signals might reach him,
at the mid-point of his spatial distance, at different
times. This, it should be noted, is not a thing to
be proved. It is a definition, not a proposition.
If we were to try to test it, we should require
some means of determining whether the two
events actually were simultaneous, and we could
not do this without knowing what the simul-
taneity of events means, i.e., without falling back
on the statement itself. Also, it is not a complete
philosophical definition of simultaneity. It can
only be applied to events separated in space from
the observer, and it assumes, moreover, that the
observer knows what he means when he says the
light signals reach him " at the same time."
This book is concerned solely with experimental
THE VELOCITY OF LIGHT 37
Science and its lessons. We shall, therefore,
not pursue the present line of thought, which
enters the borderland of metaphysics. Never-
theless, for the sake of completeness, it should
perhaps be stated that the choice of light signals
for the purpose of the definition goes slightly
beyond experimental justification. The choice
is made, of course, in order that the application
of the definition shall give the peculiar properties
of the velocity of light that the theory of relativity
requires. But we cannot be quite sure whether
those properties belong to the actual velocity
of light or to another velocity whose magnitude
is too close to it to be distinguished by the experi-
ments on which the theory is founded. The
latter possibility is favoured by Whitehead. His
reasons are, in the main, philosophical, and there-
fore do not fall within our scope. For general
description, it is sufficient to indicate that there
is a peculiar velocity, very close to that of light,
which cannot possibly be exceeded by any body
relative to any other.
PART II
THE LAWS OF NATURE
CHAPTER V
WHAT IS A NATURAL LAW?
AT the very beginning of scientific inquiry
lies the assumption that Nature works in
an orderly way. The aim of the scientist
is to express, in as simple a statement as possible,
the principles underlying the order and arrange-
ment of phenomena. To do this, he has to observe
what Nature does. He can provoke her in various
ways, and from her response he can draw certain
conclusions as to her character. He states those
conclusions in terms which he believes will be
understood by all in terms, hitherto, of matter,
space, and time.
There was a period in scientific history when
it was thought that the whole world of possible
experience could be described without going
beyond these three fundamental ideas. It was
hoped that one all-embracing law, expressing the
relation of matter to time and space (or, in other
40 RELATIVITY FOR ALL
words, the movements of matter), would be the
complete and ultimate reward of the physicist
a law from which the whole of physical history,
throughout the infinity of space, would issue and
run its inevitable course from age to age.
That hope passed away. There were actions
of Nature that would not be pressed within the
limits of such a law. An ether seemed inevitable :
light, electricity these were not to be made
subordinate to matter and motion ; they demanded
a status of their own. Apart altogether from
spiritual facts (with which we are not concerned
in any part of this book), a strict materialism was
found to be untenable. As a matter of fact,
there never was a time when it could be said to
have triumphed : its hope, even when brightest,
was for the future. The Newtonian law of gravita-
tion, from which it sprang, demanded something
other than matter, space, and time; namely,
gravitation. It presumed a force, which modified
the movements of matter. It is true that the
universal scope of this force made it appear very
likely that it belonged, in some way, to the essential
qualities of matter itself. But, until this was
proved, it could not be said that matter and
motion had established their sway over the whole
physical universe.
It is not without regret that one sees the failure
of a simple explanation of things. To the mind
that aims at the unification of phenomena, the
WHAT IS A NATURAL LAW ? 41
discovery of a new element in Nature brings
disappointment as well as elation. But facts are
invincible. The progress of physics demanded
'the admission that there were other physical
existences besides matter, time, and space. Never-
theless, there was no need to modify ideas as to
what was a law of Nature. The new entities could
manifest themselves only by their effect on matter
in space and time. Electricity, for example, was,
in itself, merely a hypothesis though, apparently,
a necessary one. All that was observed was a
peculiar kind of material movement. A natural
law was still a statement of the way in which
matter moved in space and time, though, to make
the statement simple, it was necessary to intro-
duce other conceptions.
The reader will be prepared, by the first part
of this book, for a new conception of natural law.
Relativity gives the death-blow to whatever
might remain of the old form of materialism.
Not only does it make it impossible to reduce
Nature to matter and motion : it makes the
description of the course of Nature in these terms
an incomplete, and therefore a false one. What
has hitherto been called a law of Nature becomes
a law of our particular aspect of Nature, which
is only one of an infinite number of aspects. Space,
time, and matter are seen to be an inadequate
alphabet for a universal language. They may,
or may not, be capable of forming all the words
42 RELATIVITY FOR ALL
we need, but there are tongues whose sounds
they certainly will not fit, and these tongues, as
well as ours, belong to Nature. We must go
back to the primitive sounds for the expression of
natural laws. We can afterwards spell them out
in our own characters for our own special use,
but the laws themselves must be universal.
In other words, the only possible terms for the
statement of a law of Nature are events. Any
materia-spatio-temporal statement that is peculiar
to a particular observer, and has not its exact
equivalent in the relations of other observers, is
not a natural law.
Relativity demands, therefore, a review of
existing laws. Only those which can be generalized
in the way we have suggested can survive ; the
others must be restated. Since it is the inevit-
able custom of physicists to express their con-
clusions in mathematical form, the test must be
a mathematical one. We therefore pass over its
details, which are of interest only to the specialist.
The general idea, however, we hope has been
made clear.
The new point of view is of especial interest
because it suggests the possibility of a more
complete unification of Nature than any previously
imagined. With one hand relativity destroys
the throne of matter and motion : with the other
it erects an altar to the event. Matter, space,
and time, even if they could have explained
WHAT IS A NATURAL LAW ? 43
everything in Nature, were, after all, three in-
dependent things : the event is one thing. But,
as we have seen, the three things failed : can the
one thing succeed ? It has a better chance, for
the following reason. Matter, space, and time,
when they were thought to be fundamental,
were, on that very account, incapable of modifica-
tion to meet new discoveries. If they did the
unexpected, it was not they, but something else,
that was responsible. They were absolute, wrapped
in immutability as in an impenetrable garment.
Force had to be invented ; electricity, magnetism
were postulated all because matter, space, and
time were held to be above caprice. But if we
start with the event, there is only one deity on
our Olympus. Matter, space, and time are his
broken lights, and there is no sacrilege in suppos-
ing them liable to change. Consequently, what
we formerly attributed to force might perhaps be
derived from a modification of space or time or
matter in which case, force can be dispensed
with. Possibly, also, the other extraneous entities
that we have called into being might be treated
in the same way.
It should particularly be noted that it is not
sufficient merely to conceive an idea of this kind.
It must be something more than a philosophical
possibility before it can apply for recognition as
a scientific hypothesis. A particular modifica-
tion, that will tally exactly with experiment, must
44 RELATIVITY FOR ALL
be found, and the new laws, expressed in terms
of the modified relations, must be capable of
generalization so as to include the experience of
all observers, in the way we have already pointed
out. If these conditions cannot be fulfilled, the
particular unification suggested must be abandoned.
We may say at once that it has been found
possible to describe the phenomena of gravita-
tion by a certain modification of space. Electro-
magnetism is being treated in the same way, with
every prospect of success. We choose the case
of gravitation to illustrate the argument we have
been trying to develop in this chapter. General
statements are sometimes difficult to follow, and
are liable to be misunderstood. The next chapter,
then, is devoted to the work of Newton on the
movements of bodies, while the succeeding one
attempts to explain the attitude of the relativist
to the same facts.
CHAPTER VI
THE WORK OF NEWTON
WHEN Newton began his work, the time
was ripe for a great generalization.
Galileo had studied the manner in
which bodies on the earth's surface fell to the
ground, and had shown how the distances fallen
through increased with the time of flight. Kepler
had succeeded, after many failures, in expressing,
in three famous laws, how the planets moved round
the Sun. The manner in which different bodies
moved was known with great exactitude. What
was lacking was a co-ordination between the fall
of a body to the Earth and the journey of a planet
round the Sun.
It was not obvious to all that such a co-ordina-
tion was necessary. One could regard the elliptic
motion of the Earth and the linear motion of a
falling stone as distinct phenomena. It was
natural, said some, for the Earth to move in an
ellipse, and it was natural for the stone to fall in
a straight line. To attempt to get behind these
facts was absurd. They were ultimate facts of
Nature.
45
46 RELATIVITY FOR ALL
Newton, however, regarded them in a different
light. Matter was matter, and if it moved in a
certain way in one set of circumstances, and in a
different way in another set, the reason must be
sought in the circumstances, and not placed to
the account of the moving bodies. We must
remember that Newton thought in terms of
absolute space, time, and matter.
From this point of view he was faced with a
double problem. He had, first of all, to determine
what was the natural tendency of matter in no
circumstances at all, i.e., when it was entirely
uninfluenced by anything outside itself ; secondly,
he had to consider the effect on this tendency of
the various circumstances occurring in Nature.
For the second task he was guided by observed
facts : the effect must be such as to produce the
phenomena we actually find. But for the first
he had to fall back on his own inspired imagina-
tion. There was no experience to guide him,
because he could never be quite sure that any
movements he observed were free from external
influence.
In making his fundamental assumption, Newton
took as his starting-point the essential deadness,
or " inertia," of matter. He put forward the
hypothesis that matter by itself could do nothing
to change its state of motion. If it was at rest,
it would remain at rest until something moved it.
If it was moving, it would continue to move, in
THE WORK OF NEWTON 47
exactly the same direction, and with exactly the
'same speed, until it was disturbed by outside
agencies. This he declared to be the natural con-
dition of matter, and any departure of a body
from either of these states of rest or of uniform
motion in a straight line was evidence that
something was interfering with it. To this some-
thing, whatever change it produced, he gave the
name " force."
We should understand quite clearly that
"force," in the Newtonian sense, is not a thing
observed : it is a hvpo^fcesis^ What we observe
is a change of motion. According to Newton's
assumption, this change implies a cause,* and
force is created to act the part. Newton did not
discover force ; he invented it. He was thus at
liberty to deal with it as he liked. He could define
its magnitude in whatever fashion best suited him
so long as its calculated effects were consistent
with the facts. He set himself, then, to define
force in such a way that it would be possible to
find one particular force capable of explaining, at
the same time, the movements of bodies on the
Earth and in the Heavens.
Now since force is the supposed cause of change
of motion of a body, StS^OJ55?^ffl:4,fe| ?9I ne
^Y-teJ^L^L! 1 !. J^Zb : jLprMuces_thfiLjchange.
Newton, still with his one object in view, tried,
first of all, the simplest possible definition. He
assumed force to be actually equal to the rate at
48 RELATIVITY FOR ALL
which it changes motion. But he recognized that
he would probably not achieve much success with
a definition of this kind, unless he understood by
motion something more than mere velocity. The
idea that the agency producing a given change of
velocity was of the same magnitude, no matter
what was the bulk of the moving body, did not
appear very promising as the basis of a universal
law. He therefore defined the quantity of motion
a body as the product of its "mass" and its
velocity. It followed that the change of velocity
produced in a body by a given force was greater,
the smaller the mass of the body, the change of
motion being the same in both cases.
It was in this way that the idea of mass first
became definite in physics. From its derivation,
its meaning is simply inertia, or resistance to
change of velocity. Since inertia istaken tp_j2-
the fundamental property of matterTtEe mass of
a body can also be interpreted as the quantity
of matter in it. The motion of the body, then,
according to Newton, arises from the quantity of
matter it contains and the velocity with which it
is moving. A change in either of these things will
alter the motion and reveal the existence of a
force.
Later experiments showed that, to the degree
of accuracy attainable, the mass of a body never
varied. If disintegration took place, the surr^
of the masses of the various parts was always
THE WORK OF NEWTON 49
exactly equal to the mass of the original body,
whatever treatment the body or its parts received.
In this way the idea of matter became absolute.
Change of motion was due entirely to change of'
velocity, the mass remaining constant all the
time. Newton's law of force, therefore, amounted
to a statement that the force acting on a body was
equal to the product of the mass and the change
of velocity (or, the " acceleration ") which it
produced.
Newton had now provided himself with general
laws expressing the possible motions of matter.
He had next to apply them to the facts of Nature,
and see if it were possible to devise a single force
that would give rise to the varied motions actually
observed. The problem was no easy one ; it
required a Newton to solve it. A stone fell in a
constant direction, but with varying speed ; the
Earth revolved with almost uniform speed, but
with continually changing direction. Moreover,
a heavy body fell from a given height to the Earth
in the same time as a light one. All these experi-
mental facts had to be the inevitable result of
one simple hypothesis.
As every one knows, Newton was almost com-
pletely successful. He assumed that, between
every two pieces of matter in the Universe
or, at any rate, in our own Solar System there
existed a force of attraction (gravitation) which
was proportional to the product of the masses of
4
50 RELATIVITY FOR ALL
the two bodies divided by the square of the dis-
tance between them. This force acted on each
of the bodies, pulling them towards one another,
with accelerations which, of course, depended on
their masses. The force pulling the stone to the
Earth also pulled the Earth to the stone, but it
produced a far greater acceleration in the stone
than it did in the Earth, because of the great
difference between the masses of the two bodies.
All bodies on the Earth's surface, however, would
fall with the same acceleration. For, suppose one
body had twice the mass of another. The force
pulling it towards the Earth would then be twice
the force pulling the second body towards the
Earth. But the resistance to the force (i.e. the
mass) would also be twice as great for the first
body as for the second. Consequently, the same
acceleration would be produced in both bodies.
To explain why the planets did not fall into the
Sun if they were attracted by it, it was necessary
only to suppose that they had some motion of their
own, independent of that produced by gravitation.
The attractive force would then fulfil its function
by constantly changing the direction of motion
and, to a slight extent, the speed.
Newton's laws of motion and gravitation have
been the basis of physics for more than two
hundred years. Their success in explaining and
predicting new phenomena has been almost
complete. It is true that not every material
THE WORK OF NEWTON 51
movement can be said to come within the scope
of gravitation. Electricity, magnetism, radia-
tion all have had to be recognized as origins of
force. Nevertheless, observations have been con-
sistent with the idea that the forces they produce
are in accordance with Newton's definition.
Almost every observed change of motion in
Nature can be explained as the result of a New-
tonian force, arising from particular physical
conditions. But there are one or two that have
defied such explanation. They are exceedingly
small so small, in fact, that one might be in-
clined at first to neglect them. But astronomical
observations are very exact, and they leave no
doubt at all that there are motions in the Solar ^
System that so far it has not been possible
to bring under the sway of Newton's laws. One
of the most important of these is exhibited by
the planet Mercury the nearest to the Sun of all /
the planets yet discovered. Mercury, like all the
Sun's satellites, revolves round its primary in an
ellipse. There is one point in its orbit (its
" perihelion ") which is nearer to the Sun than
any other. Now Mercury is, of course, attracted
by the other planets as well as by the Sun, and
calculation shows that, as a result, its perihelion
should gradually change its position in space
(the absolute space of the Newtonian system).
Mercury has been under observation for many
years, and it is found that the position of the
52 RELATIVITY FOR ALL
perihelion does change, but npt by quite the same
amount as the calculations require. The differ-
ence in one century amounts only to the apparent
length of a i-foot rule placed one mile away, but
this is much greater than the possible errors of
observation. It must have some cause not yet
revealed, or else the Newtonian laws are not quite
exact. Until the advent of the theory of rela-
tivity, it can be said that there was no explana-
tion of this phenomenon.
CHAPTER VII
RELATIVITY AND THE MOVEMENTS
OF BODIES
THE attitude of the relativist to Newton's
laws of motion and gravitation is not
exactly that of criticism. The more he
studies them, the more are their wonderful,
almost magical, beauty and simplicity brought
home to him. He looks on them as on Prospero's
fairy visions perfect in their kind, but springing
from such stuff as dreams are made on. It is
the tacit assumptions underlying the laws that
are the objects of his attack.
Newton, as we have said, presupposed absolute
space, time, and matter. If these things are
relative, the laws become, not false, but meaning-
less. A body left to itself moves in a straight
line. But what is a straight line ? A line which
is straight in A's space may be curved in B's.
Again, force is measured by the rate of change of
motion of a body. But what is the " rate " of
change ? In whose time system must it be
measured ? How, indeed, are we to know whether
force exists or not ? A, with his space and time,
53
54 RELATIVITY FOR ALL
finds a change of motion : B, with his space
and time, finds none. A asserts a force ; B denies
it : which is right ? Once more, two bodies
attract one another with a force proportional
to the product of their masses divided by the
square of the distance between them. But what
are their masses ? Is A to measure them, or B ?
Clearly, we must start afresh if the relativist
is right. We must go behind the motions of
bodies, which we observe from our own particular
standpoint, and think in terms of events, which are
common to all. Let us take an example. On
8th February 1921, the Moon was between the
Earth and the Sun. On 22nd February 1921, it
was in the opposite direction from that of the
Sun, as seen from the Earth. Newton gave laws
to account for the elliptic motion of the Moon
from the first of these positions to the second, as
we observe it. The relativist looks for the con-
nection between the series of events which we
speak of as the successive positions of the Moon
between the dates mentioned, but which another
observer might regard differently. He takes a
standpoint beyond the Moon, the Sun, the dates,
and studies the events from which they spring.
He asks why those particular events are what they
are, and not something else. Afterwards, when
he has found the answer to his question, he descends
to Mother Earth again, and translates it into our
language of space, time, and matter.
THE MOVEMENTS OF BODIES 55
Now, in dealing with events, we must make
use of the only relation among them known to
us so far ; namely, that their complete separation
what we have called, for convenience, the in-
terval between them in the four-dimensional
continuum is constant for all observers. This
separation, as we have seen, is obtained by sub-
tracting the square of the time interval from the
square of the space interval as measured by the
same observer, and finding the square root of
the difference.
The complete separation, we say, is constant
for all observers. But that does not tell us any-
thing about the course of Nature. It does not
tell us why (speaking in our own terms, for brevity)
the Moon should travel from its position on the
8th February to its position on the 22nd February
in an ellipse, as seen from the Earth, and not in
a straight line or a circle. If it moved in either
of these paths, the complete separation between
the two events which we describe as its positions
on the dates given, would still have the same
value for all observers, though a different value
from that which it actually has. Clearly, to obtain
a law of Nature we must make some hypothesis
as to the actual value of the interval between
events.
Einstein assumed Nature to be such that the
total four-dimensional interval between any two
events, when computed from event to event
56 RELATIVITY FOR ALL
along the actual succession, has a maximum value.
That is to say to refer to our example again
if the Moon moved in any path slightly different
from that which it chooses, then the total interval
between the two events which we call its positions
on the 8th February and the 22nd February
would be smaller than it is. This will probably
appear very surprising at first, because, accustomed
as we are to intervals in space, it seems incon-
ceivable that there can be a path which has not
a slightly greater neighbouring one. But we
must note that the interval with which we are
dealing is to be calculated in the hypothetical
four-dimensional continuum, and not in space.
Referring to Chapter III, we see that it depends
on the difference between the squares of the space
and time intervals. This difference can be in-
creased by diminishing the time interval as well
as by augmenting the space interval, so that we
shall be getting nearer to the idea of Einstein
if we think of the actual path of the Moon as
being that in which it can cover the greatest
spatial distance in the shortest time.
It must be recognized that Einstein's hypothesis
was a guess, though an inspired one. It depended
for its justification on its ability to explain the
facts of observation. It is very like Newton's
guess about the movement of matter in the free
state. Newton assumed that free matter would
move in a straight line, i.e. that it would take
THE MOVEMENTS OF BODIES 57
the minimum spatial distance between any two
points in its path. Einstein assumed that an
actual event would be separated from its neigh-
bour by the maximum four-dimensional distance.
There is this important difference, however,
between the two assumptions. Newton was
thinking of an ideal case, which hardly occurs
in Nature, for matter is never quite free. To
account for actual motions he had to introduce
force. But Einstein dealt with actual events.
If his assumption was successful, he would there-
fore have no need of force or any agency at all
outside the events themselves.
Now this assumption of Einstein's can be
tested, for, from our knowledge of the Moon's
path, for instance, we can calculate the four-
dimensional interval between any two events in
it, and compare the result with the interval
between the same two events, supposing the
Moon had travelled in a slightly different path ;
i.e. supposing the series of events we call the
Moon's successive positions in space had been
slightly different from what they are. This has
been done, using the geometry of Euclid in the
calculation. The result shows that the actual
path does not give the maximum four-dimensional
interval.
We are faced, then, with two possibilities.
Either Einstein's assumption is contrary to Nature,
or else the definitions and axioms of Euclid are
58 RELATIVITY FOR ALL
not relevant to the space in which the members
of the Solar System travel. Or in other words
Einstein's assumption must be either abandoned
altogether, or modified by the employment of a
different type of combination of the four measure-
ments (see Chapter III) for the purpose of de-
fining the four-dimensional interval in the neigh-
bourhood of material bodies. Which of these
alternatives can we adopt ? The natural impulse
is, of course, towards the former. It seems to be
impossible to cast doubt on Euclid. But, once
more, the matter is not to be decided by prejudice :
it must submit to experiment. If we assume
Einstein to be on the right lines, then space must
be non-Euclidean, and geometrical measurements
in it should show results contrary to Euclid's
assertions. On the other hand, if experiment
shows space to be completely Euclidean, then
Einstein's assumption falls to the ground, and
some other hypothesis becomes necessary. Ex-
periment, as always, is the final court of appeal.
Let us pause for a moment to see what we mean
by space being " non-Euclidean." There is
nothing occult about it ; it is essentially a state-
ment about actual physical fact to be tested
by ordinary experience. It simply means that
the assumptions that Euclid made about space
are not applicable to the actual space which we
use as a relation between events. The non-
Euclidean character of space, if it is actual, would
THE MOVEMENTS OF BODIES 59
lead us to expect some, at least, of the propositions
of Euclid to be falsified by exact measurements /,
in space. For example, the sum of the three
angles of a triangle might not be exactly equal
to two right angles.
The test seems an easy one, but, as before, we
are baffled by the extreme minuteness of the
crucial effect. The differences between practi-
cable measurements in the space of Euclid and in
the space which must be assumed in order to
justify the hypothesis of Einstein, are beyond
the power of existing instruments to detect. Our
only hope at present, at least is to assume
Einstein's hypothesis, and see if the consequences
agree with fact better than those of any other
assumption.
The result of investigations of this kind has
been all in favour of Einstein. There are two
points on which the Einstein and the Newtonian
theories give definitely conflicting results. The
first is connected with the orbit of the planet ^
Mercury. The Newtonian laws, as we have seen, J\
leave a small movement of the perihelion of this
planet unexplained. Einstein's modified assump-
tion gives a motion equal to that observed, within
the limits of experimental error. It does not need
any additional modification for this explanation,
nor was the hypothesis constructed for the pur-
pose of explaining the motion. The result follows
directly from the assumption that Mercury moves
60 RELATIVITY FOR ALL
in the maximum four-dimensional path, and the
consequent supposition of non-Euclidean space.
This is the only explanation of the phenomenon
that has not been devised ad hoc and found
inapplicable to, or conflicting with, other ob-
servations.
The second point at issue between the Einstein
^Vand Newtonian theories involves observations
/ that had not been made previous to the formula-
tion of the principle of relativity. A ray of light,
passing close to a heavy body, should, on Einstein's
assumption, suffer a slight change of direction, as
if it were pulled towards the body. According
to Newton's principles, there seems to be no
reason why the light should be bent at all. It is
possible, however, that light possesses the equi-
valent of weight in a material body, and, if so,
the gravitational force should cause a bending
similar to that predicted by the theory of rela-
tivity, but of only about half the amount. The
two theories are therefore definitely at variance
in their predictions, and an experimental test
becomes possible. This test was made at the
total solar eclipse of 2Qth May 1919. The heavy
body chosen was the Sun, and the light examined
was that emitted by stars which were almost
directly behind the Sun as seen from the Earth.
During the eclipse the sunlight was extinguished,
and the stars became visible, apparently very
close to the obscured Sun. Now these stars
THE MOVEMENTS OF BODIES 61
necessarily appeared to be in the directions of
their own light, by which they were seen. If,
therefore, that light was bent, they would appear
to be displaced from their normal positions in
the sky, which were known with great exactitude.
The amount of the displacement would be a
measure of the bending of the light. The result
was that bending of the light did occur, of just
the amount (within the limits of experimental
error) required by the Einstein hypothesis. Once
more, experiment justified the assumption that
the space of experience is non-Euclidean.
There is a third possible consequence of the
theory that is not predicted by Newton's laws ;
namely, that the colour of the light emitted by a
glowing substance in a very massive body, such
as the Sun, should be slightly different from that
of light from the same kind of material on the
Earth. It is not quite certain, however, that this
conclusion necessarily follows from the theory.
Einstein himself considers that it does, but there
are distinguished mathematicians who hold the
opposite view. The question is one of great
difficulty. Experimental tests have been made,
but the colour of the light may be influenced in
so many ways, and the results are so complicated,
that no certain conclusions can yet be drawn
from them.
We are left, then, with these facts. The theory
of relativity, requiring that space does not con-
62 RELATIVITY FOR ALL
>form to the definitions and axioms of Euclid,
explains all the movements of bodies that are
accounted for by the Newtonian law of gravitation.
In addition, it explains a movement of the planet
Mercury that stands outside the Newtonian law,
and it has predicted the true path of a ray of
light, which no other theory seems able to do.
It requires the supposition of no imaginary exist-
ences, such as force, but proceeds entirely and
completely from a single hypothesis as to the
association of events. There is no known pheno-
menon with which it is at variance.
CHAPTER VIII
SOME PROBLEMS OF RELATIVITY
A GREAT idea invariably creates as many
problems as it solves : that is a sign of
its greatness. The thoughtful student
will not be baffled by its novelty or lose himself
in its details. He will patiently probe it to the
core, and lay bare to his mind its inner meaning
and its relation to the world of experience. And
in so doing he will meet with difficulties not,
perhaps, the difficulties that are dealt with in
books, for they were the authors', and may not
be his own. The attainment of a new point of
view is hindered, not so much by the roughness
of the road as by the tendency to return to the
old standpoint. It is our prepossessions that hold
us back, drawing us, like a magnet, to themselves.
Each of us has his own standpoint and prejudices ;
each will have his own difficulties.
It may be said of the principle of relativity that,
for the most part, it offers the same problems to
all plain men. Its point of view is so remote
from that to which most of us are accustomed
that we are relatively together, and approach it
63
ALL
along parallel roads. This, our concluding chapter,
is devoted to a few general comments on some of the
more prominent difficulties that are our common
lot. It makes no claim to be exhaustive or
final ; its sole purpose is to help.
It is important that we should recognize that
the principle of relativity is not a complex, fantastic
theory a sort of last hope, called in to save the
human mind from defeat by the manoeuvres of
phenomena. It is, on the contrary, a straight-
forward attack on the problems of Nature, an
attempt to see them as they are. It is a quest
after the simple. It is inevitable that it should
pursue its ends at the expense of plausibility.
We are accustomed to the complex. We think in
terms of three things matter, space, and time
and we are so much at home with them that we
do not, perhaps, give our race full credit for the
consistency and success with which it has applied
them to the interpretation of the Universe. Jug-
gling with three balls is not an easy matter, and
we have done it almost to perfection. It is not
surprising that, when we are left with one, we are
at a loss to know how to perform our tricks. But
let u? once clearly realize the conditions; let us
suitably arrange our mirrors to make up for the
lost balls, and we shall find our repertoire augmented
and our own effort simplified. That is, in essence,
the central meaning of relativity. It takes us
SOME PROBEEMS OF RELATIVITY. 65
to the view-point of the Gods, from which we see
things as they are, unmodified by reflection in
matter, space, and time. It is a step towards
truth, and truth is simple to the simple-minded.
The question of the existence of the ether in the
light of relativity has aroused much discussion.
Apparently there are still differences of opinion
on this point. It would be unwise, therefore, to
make any definite pronouncement. We shall
merely offer a few suggestions from our own point
of view. There seems to be nothing in the theory
of relativity that is incompatible with the ether.
On the other hand, the theory has no need of the
ether's existence. There has probably been a
little misunderstanding in some quarters as to
what relativity really implies. The theory is
based on the assumption that it is inconceivable
that we shall ever detect relative motion between
matter and ether. But that does not necessarily
mean that there is no ether. The relative motion
is hidden because the velocity with which the
ether (if it exists) transmits waves is almost,' if
not quite, identical with the peculiar velocity
that takes part in the relativity formulae for
change of the time, space, and matter units with
motion. If the changes of these so-called funda-
mental units are granted, then experiment appears
to decide neither for nor against the ether's exist-
ence. There is still the possibility that the ether
66 .. i i. itELOTjtVITY. FOR ALL
, j
or
possesses some physical property, other than the
power to transmit waves, for which no compensa-
tion is made by the relativity transformations.
The essence of relativity is the universal character
the event, and the subordination of space, time,
and matter. A subordinate ether in addition
would appear not to be inconsistent with this.
The ether may, at any rate, have an existence as
real as that of matter, and that is all that is
demanded of it by the physical facts which called
it into existence.
We have already dealt more than once with the
idea that relativity is not a physical theory, but
a metaphysical one. This book has failed in its
purpose if it has not made it clear that the entire
theory is built up with the one object of accounting
for actual physical facts, and stands or falls at the
dictate of experiment. The theory of relativity
is no more metaphysical than the wave theory of
light, or Newton's laws of motion. It partakes
of their nature, and is vulnerable to the same
weapons. It appears, perhaps, at first blush to be
metaphysical, because it deals with the nature of
matter, space, and time, which are part of the
playground of metaphysics. But it is with these
things as objective entities that relativity is
concerned. Its assertions about them are sus-
ceptible to actual physical tests with clocks,
scales, and balances. They may be considered
SOME PROBLEMS OF RELATIVITY 67
metaphysically, it is true, but relativity does
not so consider them. Love is fair game for
the metaphysician, but we do not fall in love
metaphysically. It is essentially a matter of
experience.
But perhaps the greatest difficulty of relativity
is presented to the imagination. The conse-
quences of the theory are so extraordinary that
we cannot picture them. It seems impossible,
for instance, that the order of events in time can
be different for different persons. We must re-
member, however, that relativity does not entail
anything that is contrary to experience. It
would have no right to exist if that were so.
Its predictions, that appear to us so strange, are
related to matters as yet outside our experience,
about which we can only form conjectures.
Everything that we come across in our everyday
life is left untouched. Space, time, and matter
have an absolute meaning for observers relatively
at rest. For them there is a real and definite
meaning in the statements that a man is 6 feet
high, that is it nine o'clock, and that sugar is 8d.
a pound. With all terrestrial movements, even,
the statements are, for practical purposes, exact.
They are quite as true as the statement that, in
sunlight, grass is green. It is only when we
get into conditions that are far beyond our common
experience that the reasoned effects belie our
68 RELATIVITY FOR ALL
anticipations. With a Sun composed of sodium,
grass would no longer be green, and space, time,
and matter, in appropriate circumstances, suffer
a corresponding change. It is not true to say
that relativity is revolting to our experience. All
we can say is that we did not expect it. But,
after all, Nature has nothing to do with our
expectations.
It is well with Science when reason and imagina-
tion go hand in hand. One assists the other,
and the mind is satisfied. But the history of
Science shows that it is not in this way that the
greatest advances have been made. The imagina-
tion of Faraday saw the electric field threaded
by "lines of force," to which bodies were harnessed
and in obedience to which they moved in their
courses. But the mind was baffled : reason
could find no lines of force until Clerk-Maxwell
brought them within its light. In the words of
Helmholtz : " Now that the mathematical inter-
pretation of Faraday's conceptions regarding the
nature of electric and magnetic forces has been
given by Clerk-Maxwell, we see how great a
degree of exactness and precision was really
hidden behind the words, which to Faraday's
contemporaries appeared either vague or ob-
scure. ... I confess that many times I have
myself sat hopelessly looking upon some para-
graph of Faraday's descriptions of lines of force,
or of the galvanic current being an axis of power."
SOME PROBLEMS OF RELATIVITY 69
Would it not have been unwise to restrain the
imagination of Faraday because reason could not
follow it ?
Or to take an example more closely allied to
the present discussion think of the dawn of the
idea that the Earth was round and rotated on an
axis. Reason demanded it; there was no Bother
explanation of facts. But where was the imagina-
tion ? Was it conceivable that we were whirling
through space with breathless speed, and yet were
insensible of the motion ? Could one really be-
lieve that there were people on the Earth who
were upside down and yet did not fall off ? It
was impossible ; common sense scorned the sugges-
tion. Yet, would progress have been possible if
we had listened to common sense and silenced the
voice of reason ?
We are in much the same position to-day.
Can we not take heart from the experience of the
past ? To-day the dullest schoolboy knows our
place in the Solar System, and finds it not hard
to accept. Surely it is not impossible that the
paradox of relativity will one day become a part
of our common knowledge and fashion our view
of Nature. Reason calls for it : if it is true, the
imagination will not be left far behind.
Meanwhile, reason will march along and explore
fresh country, and we can at least meditate on its
conquests. Whatever our attitude towards them
may be, we must recognize that the world is a far
70 RELATIVITY FOR ALL
more wonderful thing than we have ever imagined.
Conceptions which we thought were universal ap-
pear as merely one set of an infinite number of
possible conceptions. Our idea of Nature con-
sistent though it has been, and, therefore, to
some extent, true is yet not the whole truth.
We have " swayed about upon a rocking-horse,
and thought it Pegasus." The theory of relativity
does not countenance the bombast of Swinburne,
with which we opened : it should make us very
humble.
Finally, we must not imagine that the theory of
relativity is complete and self-sufficient. Rather
is it the beginning of a new chapter in Science.
Some pages of that chapter have already been
written, and the work is even now in progress.
Electro-magnetism is being examined ; the nature
of space, and its extent, are being subjected to
searching inquiry. We have tried, in this book,
to indicate the view-point ; the view we must
/eave till our eyes are adjusted to the new dis-
tances. But the view is there, and the reward of
our labours will be limited only by the quality
of our vision.
INDEX
Acceleration, 49
Astronomy, 16, 51
Atoms, 21
Chemistry, 19
Clerk-Maxwell, 68
Conservation of mass, 19, 48
Dimension, 24
Earth, the, 5, 45, 49, 50, 54,
55, 60, 6 1, 69
Eclipse, solar, 60
Economics, 33
Einstein, v, 35, 36, 55 et seq.
Electricity, 21, 40, 41, 43, 51,
68
Electro-magnetism, 21, 44,
70
Emotions, 14
Ether, the, 3, 40, 65
Euclid, 57 et seq., 62
Events, 10, 28, 30, 42, 54, 57,
62,66
Faraday, 68, 69
Force, 40, 43, 47 et seq., 62
" Four - Dimensional Con-
tinuum," 22, 24, 28, 30, 55,
56
Galileo, 45
Geometry, 57
Gravitation, 3, 30, 40, 44, 49,
60
Helmholtz. 68
History, 33
Imagination, 67
Inertia, 34, 46, 48
Interval, total, 29, 30, 55,
58, 60
Kepler, 45
Lavoisier, 19
Laws, natural, 3, 19, 39, 49,
Light, 3, 5, 1 8, 32, 36, 37, 40,
60, 61, 62, 66
Lines of force, 68
Magnetism, 43, 51, 68
Mass, 18, 19, 34, 48
Materialism, 40, 41
Mathematics, 18, 19, 23, 42,
61
Matter, i, 10, n, 18, 22, 39,
4i 46, 53, 57, 64
Mercury, 51, 59, 62
Metaphysics, 2, 37, 66
Michelson - Morley experi-
ment, 5, 7, 16, 19, 23, 35
Moon, the, 54, 55, 57
Motion, 8, 1 6, 18, 22, 29, 34,
40, 41, 45, 46, 66
Newton, 40, 44, 45 et seq., 53,
54. 56, 59, 62, 66
Non-Euclidean space, 58, 61
72
RELATIVITY FOR ALL
Physics, v, 19, 40, 42, 48, 50,
66
Planets, 45, 50, 51
Radiation, 51
Reason, 68
Simultaneity, 35
Solar System, 49, 51, 58, 69
Space, i, 2, 8, 10, n, 22,
25, 28, 39, 46, 53, 58, 64,
70
Stars, 4, 60
Sim, the, 5, 45, 50, 51, 54,
60, 61
Swinburne, i, 70
Time, i, 2, 8, 10, 12, 22, 28,
39, 4 6 > 53, 64
Velocity, 5, 18, 19, 23, 32, 37,
48
Whitehead, vi, 37
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