Title: The theory of relativity and its influence on scientific thought
Author: Eddington, Arthur Stanley, Sir
Language: English
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ITS INFLUENCE ON SCIENTIFIC THOUGHT ***


                        THE ROMANES LECTURE

                                1922



                                _THE_

                        _THEORY OF RELATIVITY_


                              _and its_

                  _Influence on Scientific Thought_



                                  BY

                       ARTHUR STANLEY EDDINGTON

                             M.A., F.R.S.

           _Plumian Professor of Astronomy, Cambridge_
          _President of the Royal Astronomical Society_


                              DELIVERED
                      IN THE SHELDONIAN THEATRE
                             24 MAY, 1922



                                OXFORD
                        AT THE CLARENDON PRESS
                                 1922



Has not a deeper meditation taught certain of every climate and
age, that the where and the when so mysteriously inseparable
from all our thoughts, are but superficial terrestrial adhesions to
thought?

                             CARLYLE, _Sartor Resartus_.



                        Oxford University Press



THE THEORY OF RELATIVITY


In the days before Copernicus the earth was, so it seemed, an immovable
foundation on which the whole structure of the heavens was reared. Man,
favourably situated at the hub of the universe, might well expect that
to him the scheme of nature would unfold itself in its simplest aspect.
But the behaviour of the heavenly bodies was not at all simple; and the
planets literally looped the loop in fantastic curves called epicycles.
The cosmogonist had to fill the skies with spheres revolving upon
spheres to bear the planets in their appointed orbits; and wheels were
added to wheels until the music of the spheres seemed well-nigh drowned
in a discord of whirling machinery. Then came one of the great
revolutions of scientific thought, which swept aside the Ptolemaic
system of spheres and epicycles, and revealed the simple plan of the
solar system which has endured to this day.

The revolution consisted in changing the view-point from which the
phenomena were regarded. As presented to the earth the track of a planet
is an elaborate epicycle; but Copernicus bade us transfer ourselves to
the sun and look again. Instead of a path with loops and nodes, the
orbit is now seen to be one of the most elementary curves--an ellipse.
We have to realize that the little planet on which we stand is of no
great account in the general scheme of nature; to unravel that scheme we
must first disembarrass nature of the distortions arising from the local
point of view from which we observe it. The sun, not the earth, is the
real centre of the scheme of things--at least of those things in which
astronomers at that time had interested themselves--and by transferring
our view-point to the sun the simplicity of the planetary system becomes
apparent. The need for a cumbrous machinery of spheres and wheels has
disappeared.

Every one now admits that the Ptolemaic system, which regarded the earth
as the centre of all things, belongs to the dark ages. But to our dismay
we have discovered that the same _geocentric_ outlook still permeates
modern physics through and through, unsuspected until recently. It has
been left to Einstein to carry forward the revolution begun by
Copernicus--to free our conception of nature from the terrestrial bias
imported into it by the limitations of our earthbound experience. To
achieve a more neutral point of view we have to imagine a visit to some
other heavenly body. That is a theme which has attracted the popular
novelist, and we often smile at his mistakes when sooner or later he
forgets where he is supposed to be and endows his voyagers with some
purely terrestrial appanage impossible on the star they are visiting.
But scientific men, who have not the novelist's licence, have made the
same blunder. When, following Copernicus, they station themselves on the
sun, they do not realize that they must leave behind a certain purely
terrestrial appanage, namely, _the frame of space and time_ in which men
on this earth are accustomed to locate the events that happen. It is
true that the observer on the sun will still locate his experiences in a
frame of space and time, if he uses the same faculties of perception and
the same methods of scientific measurement as on the earth; but the
solar frame of space and time is not precisely the same as the
terrestrial frame, as we shall presently see.

I think you will readily understand what is meant by a _frame_ of space
and time. It is the system of location to which we appeal when we state,
for example, that one event is 100 miles distant from and 10 hours later
than another. The terms space and time have not only a vague descriptive
reference to a boundless void and an ever-rolling stream, but denote an
exact quantitative system of reckoning distances and time-intervals.
Einstein's first great discovery was that there are many such systems of
reckoning--many possible frames of space and time--exactly on all fours
with one another. No one of these can be distinguished as more
fundamental than the rest; no one frame rather than another can be
identified as the scaffolding used in the construction of the world. And
yet one of them does present itself to us as being the actual space and
time of our experience; and we recoil from the other equivalent frames
which seem to us artificial systems in which distance and duration are
mixed up in an extraordinary way. What is the cause of this invidious
selection? It is not determined by anything distinctive in the frame; it
is determined by something distinctive in us--by the fact that our
existence is bound to a particular planet and our motion is the motion
of that planet. _Nature_ offers an infinite choice of frames; we select
the one in which we and our petty terrestrial concerns take the most
distinguished position. Our mischievous geocentric outlook has cropped
out again unsuspected, persuading us to insist on this terrestrial
space-time frame which in the general scheme of nature is in no way
superior to other frames.

The more closely we examine the processes by which events are assigned
to their positions in space and time, the more clearly do we see that
our local circumstances play a considerable part in it. We have no more
right to expect that the space-time frame on the sun will be identical
with our frame on the earth than to expect that the force of gravity
will be the same there as here. If there were no experimental evidence
in support of Einstein's theory, it would nevertheless have made a
notable advance by exposing a fallacy underlying the older mode of
thought--the fallacy of attributing unquestioningly a more than local
significance to our terrestrial reckoning of space and time. But there
is abundant experimental evidence for detecting and determining the
difference between the frames of differently circumstanced observers.
Much of the evidence is too technical to be discussed here, and I can
only refer to the Michelson-Morley experiment. I fear that some of you
must be getting rather tired of the Michelson-Morley experiment; but
those who go to a performance of Hamlet have to put up with the Prince
of Denmark.

This famous experiment is a simple test whether light travels at the
same speed in two different directions. For this purpose an apparatus is
constructed with two equal arms at right angles, providing two equal
tracks for the light. A beam of light is divided into two parts so that
one part travels along one arm and back, and the other along the other
arm and back. The two rays then re-unite, and by delicate interference
tests it is possible to tell if one has been delayed more than the
other; a delay of less than a thousand-billionth of a second could be
detected. The experiment is simply a race between two light-rays with
equal tracks, but pointing in different directions; the result turns out
to be a dead-heat. At first sight this is just what would be expected;
and one almost wonders why it should have been thought worth while to
try the experiment. But Michelson, like a good Copernican, had stationed
himself on the sun to watch the race; accordingly he realized that the
apparatus was being borne along by the earth's orbital motion with a
speed of 20 miles a second. Consequently the light does not travel
exactly the double length of the arm; starting at one end it has to go
to the turning-mark at the other end which has moved on a little in the
meantime; then it returns to the place which the starting-mark has
travelled to whilst the race is in progress. That does not add up to
exactly the double-length of the arm. Making the calculations we easily
find that, although the two arms are equal, the two light-journeys are
unequal; the competitor whose track lies in the line of the earth's
motion has the longer journey, and is at a disadvantage. And yet
according to the experiment he does not suffer the expected delay. From
our standpoint on the sun, the experiment seems to have gone wrong;
Copernicus has met with a rebuff, and Ptolemy is triumphant.

But that is because we have not admitted the full consequences of
transferring our standpoint to the sun. We have all the while been
keeping one foot on earth. Of course, the whole experiment turns on the
two arms having been first adjusted to perfect equality. This could only
be ascertained by experiment; and the test applied was to rotate the
apparatus through a right angle, so that if, for example, the journey in
the line of the earth's motion had had the advantage of the shorter arm
on one occasion, the transverse journey would have had it on the
repetition. That is a perfectly satisfactory test for a terrestrial
observer; to turn a rod from one direction to another is the simple and
direct way of marking out equal lengths. But the test is not
satisfactory to an observer on the sun; he would not think of attempting
to partition equal lengths of space by means of rods travelling at 20
miles a second. His frame of space--the space not only of refined
measurement, but also of the cruder measurements made with the
sense-organs of his body which determine his perception of space--is
partitioned by appliances at rest relatively to him, e.g. his own eyes
and limbs. Lengths of objects carried on the earth must be judged by him
according to the room they occupy in his own frame. In the space of the
terrestrial observer the two arms of the apparatus were adjusted to
equal length; but in the re-partitioned space of the solar observer they
may quite well occupy unequal lengths, and when we take the view-point
of an observer on the sun we must not overlook this inequality. This
inequality is not so much an hypothesis proposed to account for
Michelson's result as a direct deduction from it. The two light-journeys
were found to occupy equal times; this clearly shows that the arm in the
less favoured direction is shorter than the other so as to
counterbalance the handicap to which I have referred.[1]

When the apparatus is turned through a right angle, the experiment still
gives the same result. It does not matter which of the two arms we place
in the line of the earth's motion; that arm must be shorter than the
other. In other words each arm must automatically contract when it is
turned from the transverse to the longitudinal position with respect to
its line of motion. This is the famous FitzGerald contraction of a
moving rod. It is of the same amount whatever the material of the rod,
and depends only on the speed of its motion. For the earth's orbital
motion the contraction amounts to one part in 200 million; in fact the
earth's diameter in the direction of its motion is always shortened by
2½ inches, the transverse diameter being unaffected.

This contraction of a moving material object was first revealed to us by
the Michelson-Morley experiment; but it is not at all disagreeable to
theoretical anticipations. We have to remember that a rod consists of a
large number of molecules kept in position by their mutual forces. The
chief force is the force of cohesion, and there is little doubt that
this is of electrical nature. But when the rod is set in motion, the
electrical forces inside it must change. For example, each electric
charge when put in motion becomes an _electric current_; and the
currents will exert magnetic attractions on each other which did not
occur in the system at rest. Under the new system of forces the
molecules will have to find new positions of equilibrium; they become
differently spaced; and it is therefore not surprising that the form of
the rod changes. Without going beyond the classical laws of Maxwell we
can anticipate theoretically what will be the new equilibrium state of
the rod, and it turns out to be contracted to the exact amount required
by the Michelson-Morley result.

The contraction of the moving rod ought not to surprise us; it would be
much more surprising if the rod were to maintain the same form in spite
of the alteration of the electrical forces which determine the spacing
of the molecules. But the remarkable thing is that the contraction is
only apparent according to the outlook of the solar observer; and we on
the earth, who travel with the rod, cannot appreciate it. The fact that
the contraction happens to be very small is irrelevant. For convenience
suppose that the earth's velocity is 8,000 times faster, so that the
contraction amounts to something like a half the original, length. We
should still fail to notice it in everyday life. Let us say that the
direction of the earth's motion is vertically upwards.

I turn my arm from horizontal to vertical and it contracts to half its
length. No, you cannot convince me I am wrong; I am not afraid of a
yard-measure. Bring one and measure my arm; first horizontally, the
result is 30 inches; now vertically, the result is 30--half-inches!
Because you must remember that you have turned the scale into the line
of the earth's motion so that each inch-division contracts to half an
inch. 'But we can see that your arm does not contract. Are we not to
trust our eyes?' Certainly not, unless you first correct your visual
impressions for the contraction of the retina in the vertical direction,
and for the effect of our rapid motion on the apparent direction of
propagation of the waves of light. You will find, when you calculate
these corrections, that they just conceal the contraction. 'But if the
contraction takes place, ought one not to feel it happening to the arm?'
Not necessarily; I am an observer on the earth, and my feelings like
other sense-impressions belong to the geocentric outlook on nature,
which Copernicus has persuaded us to abandon.

Take a pair of compasses and twiddle them on a sheet of paper. Is the
resulting curve a circle or an ellipse? Copernicus from his standpoint
on the sun declares that owing to the FitzGerald contraction the two
points drew nearer together when turned in the direction of the earth's
orbital motion; hence the curve is flattened into an ellipse. But here I
think Ptolemy has a right to be heard; he points out that from the
beginning of geometry circles have always been drawn with compasses in
this way, and that when the word 'circle' is mentioned every intelligent
person understands that this is the curve meant. The same pencil line is
in fact a circle in the space of the terrestrial observer and an ellipse
in the space of a solar observer. It is at the same time a moving
ellipse and a stationary circle. I think that illustrates as well as
possible what we mean by _the relativity of space_.

It is sometimes complained that Einstein's conclusion that the frame of
space and time is different for observers with different motions tends
to make a mystery of a phenomenon which is not after all intrinsically
strange. We have seen that it depends on a contraction of moving objects
which turns out to be quite in accordance with Maxwell's classical
theory. But even if we have succeeded in explaining it to ourselves
intelligibly, that does not make the statement any the less true! A new
result may often be expressed in various ways; one mode of statement may
sound less mysterious; but another mode may show more clearly what will
be the consequences in amending and extending our knowledge. It is for
the latter reason that we emphasize the relativity of space--that
lengths and distances differ according to the observer implied. Distance
and duration are the most fundamental terms in physics; velocity,
acceleration, force, energy, and so on, all depend on them; and we can
scarcely make any statement in physics without direct or indirect
reference to them. Surely then we can best indicate the revolutionary
consequences of what we have learnt by the statement that distance and
duration, and all the physical quantities derived from them, do not as
hitherto supposed refer to anything absolute in the external world, but
are relative quantities which alter when we pass from one observer to
another with different motion. The consequence in physics of the
discovery that a yard is not an absolute chunk of space, and that what
is a yard for one observer may be eighteen inches for another observer,
may be compared with the consequences in economics of the discovery that
a pound sterling is not an absolute quantity of wealth, and in certain
circumstances may 'really' be seven and sixpence. The theorist may
complain that this last statement tends to make a mystery of phenomena
of currency which have really an intelligible explanation; but it is a
statement which commends itself to the man who has an eye to the
practical applications of currency.

Ptolemy on the earth and Copernicus on the sun are both contemplating
the same external universe. But their experiences are different, and it
is in the process of experiencing events that they become fitted into
the frame of space and time--the frame being different according to the
local circumstances of the observer who is experiencing them. That, I
take it, is Kant's doctrine, 'Space and time are forms of experience.'
The frame then is not in the world; it is supplied by the observer and
depends on him. And those relations of simplicity, which we seek when we
try to obtain a comprehension of how the universe functions, must lie in
the events themselves before they have been arbitrarily fitted into the
frame. The most we can hope for from any frame is that it will not have
distorted the simplicity which was originally present; whilst an
ill-chosen frame may play havoc with the natural simplicity of things.
We have seen that the simplicity of planetary motions was obscured in
Ptolemy's frame, and became apparent in Copernicus's frame. But for
ordinary terrestrial phenomena the position is reversed and Ptolemy's
frame allows their natural simplicity to become apparent. In
Copernicus's frame the most simple phenomena are brought about by highly
complicated processes which mutually cancel one another. Ordinary
objects contract and expand as they are moved about, and the changes are
concealed by an elaborate conspiracy in which all the quantities of
nature--electrical, optical, mechanical, gravitational--have joined. In
Copernicus's frame we have a great complication of description which has
no counterpart in anything occurring in the external world; because the
terms of our description refer to the irrelevant process of fitting into
the selected frame of space and time. This elaborate Copernican scheme
rather reminds one of the schemes of the White Knight--


    But I was thinking of a plan
      To dye one's whiskers green,
    And always use so large a fan
      That they could not be seen.


We do not deny the subtlety and the remarkable efficiency of the plan;
but we may be allowed to question whether it is the simplest
interpretation of the drab monotony of the face of nature presented to
us. The simple fact is that a terrestrial or Ptolemaic frame fits
naturally the terrestrial phenomena, and a solar or Copernican frame
fits the phenomena of the solar system; but we cannot make one frame
serve for both without introducing irrelevant, complications.

We go beyond Copernicus nowadays, and are not content with a visit to
the sun. Why choose the sun rather than some other star in order to
obtain an undistorted view of things? The astronomer now places himself
so as to travel with the centre of gravity of the stellar universe, and
is not even then quite satisfied. The physicist dreams of a land of
Weissnichtwo, which shall be truly at rest in the ether. We realize the
distortion imported into the world of nature by the parochial standpoint
from which we observe it, and we try to place ourselves so as to
eliminate this distortion--so as to observe that which actually is. But
it is a vain pursuit. Wherever we pitch our camera, the photograph is
necessarily a two-dimensional picture distorted according to the laws of
perspective; it is never a true semblance of the building itself.

We must try another plan. I do not think we can ever eliminate
altogether the human element in our conception of nature; but we can
eliminate a particular human element, namely, this framework of space
and time. If our thought must be anthropocentric, it need not be
geocentric. Nor are we permanently better off if we merely substitute
the space-time frame of some other star or standard of motion. We must
leave the frame entirely indeterminate. When we do that, we find that
the world common to all observers--in which each observer traces a
different space-time frame according to his own outlook--is a world of
four dimensions. When we look at any object, say a chair, the impression
on our eyes is a two-dimensional picture depending on the position from
which we are looking; but we have no difficulty in conceiving of the
chair as a solid object, not to be identified with any one of our
two-dimensional pictures of it, but giving rise them all as the position
of the observer is varied. We must now realize that this solid chair in
three dimensions is itself only an appearance, which changes according
to the motion of the observer, and that there is a super-object in
four-dimensions, not to be identified with the three-dimensional chair
in Ptolemy's scheme, or the same chair in Copernicus's scheme, but
giving rise to both these appearances. The synthesis of a
three-dimensional chair from a number of flat pictures is easy to us
because we are accustomed to assume different positions in rapid
succession; indeed our two eyes give us slightly different points of
view simultaneously. By sheer necessity our brains have been forced to
construct the conception of the solid chair to combine these changing
appearances. But we do not vary our motion to any appreciable extent and
our brains have not hitherto been called upon to combine the appearances
for different motions; thus the effort which we now ask the brain to
make is a novel one. That explains why the result seems to transcend our
ordinary mode of thought.

The discovery, or one should rather say the rediscovery, of the world of
four dimensions is due to Minkowski. Einstein had worked out fully the
relations between the frames of space and time for observers with
different motions. To the genius of Minkowski we owe the realization
that these frames are merely systems of partitions arbitrarily drawn
across a four-dimensional world which is common to all observers.

There is a strange delusion that the fourth dimension must be something
wholly beyond the conception of the ordinary man, and that only the
mathematician can be initiated into its mysteries. It is true that the
mathematician has the advantage of understanding the technical machinery
for solving the problems which may arise in studying the world of four
dimensions; but as regards the conception of the four dimensions of the
world his point of view is the same as that of anybody else. Is it
supposed that by intense thought he throws himself into some state of
trance in which he perceives some hitherto unsuspected direction
stretching away at right angles to length, breadth, and thickness? That
would not be much use. The world of four dimensions, of which we are now
speaking, is perfectly familiar to everybody. It is obvious to every
one--even to the mathematician--that the world of solid and permanent
_objects_ has three dimensions and no more; that objects are arranged in
a threefold order, which for any particular individual may be analysed
into right-and-left, backwards-and-forwards, up-and-down. But it is no
less obvious to every one that the world of _events_ is of four
dimensions; that events are arranged in a fourfold order, which in the
experience of any particular individual will be analysed into
right-and-left, backwards-and-forwards, up-and-down, _sooner-and-later_.
The subject of our study is external nature, which is a world of events,
common to all observers but represented by them differently in their
parochial frames of space and time; it is obvious to the most
commonplace experience that this absolute world contains a fourfold
order.[2]

The news that the events around us form a world of four dimensions is as
stale as the news that Queen Anne is dead. The reason why the relativist
resurrects this ancient truism is because it is only in this undissected
combination of four dimensions that the experiences of all observers
meet. In our own experience one dimension is sharply separated from the
other three and is distinguished as time; but our experience is solely
terrestrial, and if we insist on building the scheme of nature on purely
terrestrial experience we are limiting ourselves to the mediaeval
geocentric system of the world.

We have been accustomed to regard the enduring world as composed of a
continuous succession of instantaneous states, as though the world of
events were _stratified_. Each event is supposed to lie in a definite
instant or stratum, and the orderly succession of these strata makes up
the whole of reality. The instant 'now' represents one such stratum
running throughout the universe. Indeed we are accustomed to extend it
beyond the universe, and we even use the word 'now' with reference to
the existence of those who have passed away from the material world. The
investigations of the relativity theory show incontrovertibly that this
supposed stratification is an illusion; there is not the slightest
evidence for such a view of world-structure. The instantaneous state,
which we have hitherto taken to be a natural stratum in the
four-dimensional world of events, is merely an arbitrary partition
created by ourselves to correspond with our geocentric outlook. We can
take a differently inclined partition,[3] that is to say, a section
which includes on the one side of us events which happened a little
while ago and on the other side of us events which have not yet
happened; such a farcical combination is in every way equivalent to our
so-called instantaneous state, and indeed it _is_ an instantaneous state
according to the outlook of some non-terrestrial observer with suitably
assigned motion.

It is so contrary to our natural prejudices to recognize that the
world-wide instant now is created by ourselves and has no existence
apart from our geocentric outlook, that I will spend a few moments
trying to show its artificiality. When I say that I am conscious of an
instant now, I am only conscious of it in so far as it is HERE--inside
me. What then has led me to imagine that there exists a continuation of
it outside me? It is because I look out on the world and see various
events happening 'now', so that I jump to the conclusion that this
instant of which I am conscious has to be extended to include
them. But that idea is another inheritance from the dark ages,
overthrown by Römer in 1675. It is not the events themselves but the
sense-impressions to which they give rise which are happening in the
instant now. So my justification for placing the events outside me in
the instants of which I am conscious has entirely disappeared.
Unfortunately, however, the crude outlook was not abolished, but patched
up; it was found that the immediate difficulties could be met by
locating the external events not in the instant of our visual perception
of them but in an instant which we had experienced a little time
back--allowing, as we say, for the time of propagation of light. Thus
our instants were still made to extend through space; but they were
carried like partitions among the events by an artificial process of
computation, and no longer by immediate intuition. The relativity theory
recognizes these _worldwide instants_ for what they are--artificial
partitions constructed for purposes of calculation. I may add that it in
no way tampers with the _local instants_ which form the stream of our
consciousness; it fully recognizes that the chain of events in such a
time-succession is a series of an entirely distinctive character from
the succession of points along a line in space. Those who suspect that
Einstein's theory is playing unjustifiable tricks with time should
realize that it leaves entirely untouched that time-succession of which
we have intuitive knowledge, and confines itself to overhauling the
artificial scheme of time which Römer first introduced into physics.

The study of the four-dimensional world of events gives us a new insight
into the processes of nature because it removes the irrelevant
stratification in a particular direction--the instantaneous
states--which we have so unnecessarily introduced in our customary
outlook. When this stratification is ignored we are enabled to see the
processes in their simplest aspect, though not, of course, in their most
familiar aspect. We must distinguish between simplicity and familiarity;
a pig may be most familiar to us in the form of rashers, but the
_unstratified_ pig is a simpler object of study to the biologist who
wishes to understand how the animal functions.

I will conclude this part of the argument with an experimental
application which illustrates the power of Einstein's method. Much study
has of late been given to electrons moving with very high speeds; for
example, the β particles shot off from radioactive substances are
negative electrons which sometimes attain speeds of 100,000 miles a
second. It is found by experiment that the rapid motion produces an
increase of mass of these particles. I want to show that the theory of
relativity gives a very simple explanation of just how this increase of
mass occurs. But I must first remark that an explanation had been
previously given which had generally been accepted as satisfactory. The
phenomenon was actually predicted by J. J. Thomson before relativity was
thought of; because, assuming that the mass of a particle is of
electrical origin, an application of Maxwell's equations shows that it
ought to increase with velocity. But the precise law of increase cannot
be predicted on this basis, since various plausible assumptions lead to
slightly different results. Moreover, Maxwell's equations are after all
only empirical laws, with a mystery of their own; it was a notable
advance to connect the change of mass at high speeds with other
phenomena whose strangeness has disappeared by long familiarity, but
there is still scope for a more far-reaching explanation. Einstein takes
us straight to the root of the mystery, and he clears up one point which
was misleading, if not actually wrong, in the older explanation. The
change of mass does not in any way depend on whether the mass is of
electrical origin or not; it arises simply from the fact that mass is a
_relative_ quantity, depending by its definition on the relative
quantities length and time. Let us look at the β particle from its own
point of view; it is just an ordinary electron in no way different from
any other. 'But it is travelling unusually rapidly?' 'That', says the
electron, 'is a matter of opinion. So far as I am aware I am at rest, if
the word "rest" has any meaning. In fact I was just contemplating with
amazement _your_ extraordinary speed of 100,000 miles a second with
which you are shooting past me.' Of course our motion is of no
particular concern to the electron, and it will not modify its
constitution on our account; so it keeps its mass, radius, electric
field, &c., equal to the standard constants applying to electrons in
general. These terms are relative, and refer therefore to some
particular frame of space and time--clearly the frame appropriate to an
electron in self-contemplation, viz. the one with, respect to which it
is at rest. But this frame is not the usual geocentric frame to which
_we_ refer quantities such as length, time, and mass; there is a
difference of 100,000 miles a second between our station of observation
and that of the β particle in self-contemplation. It is a mere matter
of geometry to discover what the β particle's lengths and times become
when referred to the partitions which we have drawn across the world.
But when we calculate the consequential change of mass resulting from
the changes of length and time, we find that it should be increased in
precisely the proportion indicated by the most refined experiments.

The point is that every electron, at rest or in motion, is a perfectly
constant structure; but we distort it by fitting it into the space-time
frame appropriate to our own motion with which the electron has no
concern. The greater our motion with respect to the electron, the
greater will be the distortion. The distortion is not produced by any
physical agency at work in the electron; it is a purely subjective
distortion depending on our transformation of the reference frame of
space and time. This distortion involves a change in our physical
description of the electron in terms of mass, shape, size; and in
particular the change of mass agrees precisely with that found
experimentally.

You see that it is not altogether idle discussing the natural space-time
frames for observers moving with huge velocities. We know of no animate
observers with these speeds; but we do know of inanimate material
objects. Their common resemblance is obscured when we refer them
indiscriminately to our irrelevant geocentric frame; we think they have
altered their properties, varied in mass, and so on; but the resemblance
is restored when we refer each individual to the frame appropriate to
it, and so describe them all in comparable terms.

Our measurements of distance in space are found to be subject to certain
laws--the laws of geometry. But it has now become impossible to regard
the subject of space-geometry as complete in itself. Consider a triangle
formed by three points (or events) in the four-dimensional world; if we
happen to have drawn our instantaneous strata so that the three points
lie in one stratum, then the triangle is a space-triangle and its
properties fall within the scope of our classical geometry. But another
observer will draw his strata in a different direction, and for him the
triangle would be partly in space and partly in time, so that it would
not be a fit subject for space-geometry. The subject of geometry is in a
desperate condition, because Copernicus and Ptolemy not merely disagree
as to the geometry of a configuration; they even disagree as to whether
a given configuration is one to which space-geometry is applicable. It
is clear that to save it we must extend our geometry so as to include
time as well as space. Let me give an illustration of this extension.
The terrestrial observer can have a space-triangle (formed by three
points or events at the same instant) whose sides he can measure with
scales; he can also have a 'time-triangle', formed by three events on
different dates, whose sides he must measure with _clocks_.[4] You all
know the law of the space-triangle--that if you measure with a scale
from _A_ to _B_ and from _B_ to _C_ the sum of the readings is always
_greater_ than the measure from _A_ to _C_. It is not so well known that
there is a precisely analogous law for the time-triangle--that if you
measure with a clock from _A_ to _B_ and from _B_ to _C_ the sum of the
readings is always _less_ than the reading of a clock measuring directly
from _A_ to _C_. In the space-triangle any two sides are together
_greater_ than the third side; in the time-triangle two sides are
together _less_ than the third side.[5] Both these laws must be combined
in our general geometry of four dimensions, so that it will not be quite
so simple a geometry as that to which we are accustomed.[6]

But the point to which I would especially direct attention is this.
Evidently the proposition which I have given you about time-triangles
cannot be dissociated from the corresponding proposition about
space-triangles. When we give up the mediaeval geocentric standpoint, we
must recognize that they belong to one geometry, of which our ordinary
space-geometry is only a part or projection. But if you examine the
proposition about time-triangles, you will see that it is a statement
about the behaviour of clocks when they move about, a subject which
obviously comes under the heading of mechanics. When we deal with the
four-dimensional world we can no longer distinguish between geometry and
mechanics. They become the same subject. When we have completely
mastered the geometry of the world of events, we shall have inevitably
learnt the mechanics of it. That is why Einstein, studying the geometry
of the world and discovering that it was strictly non-Euclidean, found
that he was at the same time studying the mechanical force of
gravitation. And when he had made up his mind which of the possible
varieties of non-Euclidean geometry was obeyed, and so settled the laws
of the new geometry, the same decision settled the law of gravitation--a
law approximating to, but not identical with, the law which Newton had
given.

Here a wide vista opens before us. We see that two great divisions of
mathematical physics, viz. geometry and mechanics, have met in the
four-dimensional world. It is not merely that mechanical problems can be
treated by formulae originally belonging to pure geometry; that device
has long been in use. Experimental geometry and mechanics actually
relate to the same subject-matter; and the young student who discovers
experimental laws with ruler and compasses and cardboard figures, and
later goes on to pendulums and spring-balances, is developing a single
subject which cannot be divided any more than the subject of magnetism
can be divided from electricity.

It is through this unification of geometry and mechanics that I should
like to approach the problem of gravitation, showing that a field of
force is a manifestation of the geometry of space and time. But I fear
that that would be too technical; so we will approach it from a
different angle.

We have shown that the contemplation of the world from the standpoint
of a single observer is liable to distort its simplicity, and we have
tried to obtain a juster idea by taking into account and combining
other points of view. The more standpoints the better. Let us now
consider another point of view, which we have not previously thought
about--the point of view of an observer who has tumbled out of an
aeroplane and is falling headlong. In many respects his is an ideal
situation--temporarily. Unfortunately on _terra firma_ we are
continually subjected to a very disturbing influence; we undergo a
terrific bombardment by the molecules of the ground, which are hammering
on the soles of our boots with a total force of some ten stone weight
pressing us upwards. Now our bodies are the scientific appliances which
we use to make our common observations of the world. I am sure that no
physicist would permit any one to enter his laboratory and hammer on his
clocks and galvanometers whilst he was observing with them; at any rate
he would think it necessary to apply some corrections for the effect of
the disturbance. Let us then allow ourselves to fall freely _in vacuo_;
then we shall be free from this disturbing bombardment and able to take
a much more natural view of what is going on around us.

Whilst falling, we perform the experiment of letting go an apple held in
the hand. The apple is now free, but it cannot fall any more than it was
falling already; consequently it remains poised in contact with our
hand. In our new outlook--in our new frame of space and time--an apple
does not drop. There is no mysterious force accelerating it. And
remember that this new frame of space and time is the natural frame of a
free observer; whereas the old frame, in which the mysterious
accelerating force occurred, was the frame of a very much disturbed
observer. It is true that when we look down at the earth we see trees
and houses rushing up to meet us; but there is no mystery about that.
There is an obvious cause for it; plainly they are being propelled
upwards from below by that molecular bombardment which I have mentioned.
You see that the apple's view of things is simpler than Newton's. Newton
had to invent a mysterious force dragging the apple down; the apple
observes only a familiar physical agency propelling Newton up.

It is not my purpose to emphasize unduly the superiority of the apple's
view over Newton's, but rather to regard both on an equal footing. I
have perhaps been a little unfair to Newton. His position on the surface
of the earth was unfortunate, but he would have been perfectly content
to be at the centre of the earth, where he could have remained without
support, i.e. without disturbance by molecular bombardment. From there
he would still have observed the well-known acceleration of the apple;
and the apple would have observed a corresponding acceleration of Newton
without any molecular bombardment causing it. From either point of view
there is a mysterious agent at work. How shall we picture to ourselves
this agent? Shall we picture it as a force--a tug of some kind? But if
so, to which of them is the tug applied? If we take the standpoint of
Newton the tug is applied to the apple, if the standpoint of the apple
the tug is applied to Newton; so that in our synthesis of all
standpoints we cannot decide which is being tugged, and the picture of
gravitation as a tugging agent becomes impossible. Einstein replaces it
by a different picture, which we shall perhaps better understand if we
compare it with a very similar revolution of scientific thought which
occurred long ago.

The ancients believed that the earth was flat. The small portion of its
surface with which they were chiefly concerned could be represented
without serious distortion on a flat map. As more distant countries were
added, it would be natural to think that they also could be included in
the flat map. You have all seen such maps of the world, e. g. Mercator's
projection, and you will remember how Greenland appears enormously
exaggerated in size. Now those who adhered to the flat-earth theory must
hold that the flat map gives the _true_ size of Greenland. How then
would they explain that travellers in that country reported that the
distances were much shorter? They would, I suppose, invent a theory that
a demon resided in that country who helped travellers on their way,
making the journeys appear much shorter than they 'really' were. No
doubt the scientists would preserve their self-respect by using some
Graeco-Latin polysyllable instead of the word 'demon', but that must not
disguise from us the fact that they were appealing to a _deus ex
machina_. The name demon is rather suitable, however, because he has the
impish characteristic that we cannot pin him down to any particular
locality. We might equally well start our flat map with its centre in
Greenland; then it would be found that journeys there were quite normal,
and that the activities of the demon were disturbing travellers in
Europe. We now recognize that the true explanation is that the earth's
surface is curved; and the demoniacal complications appeared because we
were forcing the earth's surface into an inappropriate flat frame which
distorts the simplicity of things.

What has happened in the case of the earth has happened also in the case
of the world, and a similar revolution of thought is needed. An
observer, say at the centre of the earth, finds that there is a frame of
space and time--a flat or Euclidean frame--in which he can locate things
happening in his neighbourhood without distorting their natural
simplicity. There is no gravitation, no tendency of bodies to fall, so
long as the observer confines his observations to his immediate
neighbourhood. He extends this frame of space and time to greater
distances, and ultimately to the earth's surface where he encounters the
phenomenon of falling apples. This new phenomenon must be accounted for,
so he invents a _deus ex machina_ which he calls gravitation to whose
activities the disturbance is attributed. But we have seen that we may
just as well start with the falling apple. It has a flat frame of space
and time into which phenomena in its neighbourhood fit without
distortion; and from its point of view bodies near it do not undergo any
acceleration. But when it extends this frame farther afield, the
simplicity is lost; and it too has to postulate the demon force of
gravitation existing in distant parts, and for example causing
undisturbed objects at the centre of the earth to fall towards it. As we
change from one observer to another--from one flat space-time frame to
another--so we have to change the region of activity of this demon. Is
not the solution now apparent? The demon is simply the complication
which arises when we force the world into a flat Euclidean space-time
frame into which it does not fit without distortion. It does not fit the
frame, because _it is not a Euclidean or flat world_. Admit a curvature
of the world and the mysterious disturbance disappears. Einstein has
exorcized the demon.

Einstein, recognizing that in the phenomena of gravitation he was not
dealing with a 'tug' but with a curvature of the world, had to
reconsider the law of gravitation. He could not make any possible law of
curvature correspond exactly with the previously assumed law of tugging.
Thus he was led to propound a new law of gravitation--a law which in
most practical cases differs very little from that of Newton, although
it has an essentially different foundation. I need not here dwell on the
very remarkable way in which Einstein's emendation of the law of
gravitation has been confirmed both by the anomalous secular change in
the orbit of the planet Mercury, and by the observed displacement of the
stars near the sun during the total eclipse of 1919. I might, however,
remind you that in the latter observation the point at issue between
Newton's and Einstein's theory was not the _existence_ of a deflexion of
light-rays passing near the sun but the amount of the deflexion,
Einstein predicting twice the deflexion possible on the Newtonian
theory. The larger deflexion was quantitatively confirmed by the eclipse
observations. Einstein's main achievement is a new law, not a new
explanation, of gravitation. He attributes the gravitation of massive
bodies to a curvature of the world in the region surrounding them and so
throws a flood of light on the whole problem; but he is not primarily
concerned to explain how material bodies produce (or are associated
with) this curvature of the world around them, nor how this curvature is
made subject to a law. Although it would be an entire misunderstanding
of Einstein's attitude in propounding the general relativity theory to
regard it as a search for an explanation of gravitation, nevertheless I
think that the further following up of his ideas has led to a genuine
explanation as complete as could be desired. But I am not going to give
you the explanation in this lecture; sometimes an explanation requires a
great deal of explaining.[7]

I think that we can without mathematics form a general idea of why
Einstein found it necessary to amend Newton's law of gravitation. Let us
return to the illustration of the pig, and imagine that we wish to
discover the law governing the distribution of fat and lean in the
animal. From the breakfast-table standpoint a plausible type of law
would be that half of each rasher is fat and the other half lean; and if
this turned out to be confirmed very approximately by observation we
might well imagine that we had discovered the exact law of porcine
structure. But the case is altered if we give up the breakfast-table
standpoint and contemplate the animal in a more general way, remembering
that he has not been designed with any particular reference to the
series of rashers into which our grocer has chosen to slice him. We must
now look for a different type of law altogether. Two possibilities may
arise. We may find that our proposed law, although expressed in
breakfast-table parlance, is nevertheless equivalent to a possible
biological law; it may be immediately capable of translation into a more
general statement which makes no reference to a particular
stratification. But on the other hand, it may happen that the suggested
law cannot be freed from this reference to a particular system of
slicing. In that case we can only regard it as approximate, perhaps
holding fairly well for the slices of which we have most experience but
becoming less and less accurate in the more tortuous parts of the
animal. Both these cases are illustrated in Einstein's modifications of
classical theory. Newton's law of gravitation explicitly refers to a
space-time frame and therefore to a world stratified into instantaneous
states. It proves to be impossible to free it from this reference to a
particular stratification without modifying it. In fact if the crucial
astronomical observations had shown that Newton's law and not Einstein's
was the exact law of gravitation, this would have been evidence of a
real stratification of the structure of the world--a stratification
revealed by no other phenomena. Einstein's law is the simpler law
because it is consistent with what we now know of the general plan of
world-structure; Newton's law could only be made possible by introducing
a novel and specialized feature--a stratified arrangement of
structure--which is not revealed in any other phenomena.

Maxwell's laws of electromagnetism afford an example of the other type.
These, it is true, are stated as relating to the particular slices of
the world of events, which are served up to us like rashers instant by
instant. But they can be restated, without alteration of effect, in a
form making no reference to slices. This is a very remarkable property
of Maxwell's equations which was quite unknown at the time they were
first put forward. It was brought to light much later by the researches
of Larmor and Lorentz. In consequence of this Einstein is able to take
over the whole classical theory of electromagnetism unaltered; he
restates it so as to show how it applies generally and is not bound up
with the purely terrestrial point of view, but he does not amend the
laws. He metes out different treatment to the gravitational laws and
electromagnetic laws, because he finds the latter already adapted to his
scheme.

If I have succeeded in my object, you will have realized that the
present revolution of scientific thought follows in natural sequence on
the great revolutions at earlier epochs in the history of science.
Einstein's special theory of relativity, which explains the
indeterminateness of the frame of space and time, crowns the work of
Copernicus who first led us to give up our insistence on a geocentric
outlook on nature; Einstein's general theory of relativity, which
reveals the curvature or non-Euclidean geometry of space and time,
carries forward the rudimentary thought of those earlier astronomers who
first contemplated the possibility that their existence lay on something
which was not flat. These earlier revolutions are still a source of
perplexity in childhood, which we soon outgrow; and a time will come
when Einstein's amazing revelations have likewise sunk into the
commonplaces of educated thought.

To free our thought from the fetters of space and time is an aspiration
of the poet and the mystic, viewed somewhat coldly by the scientist who
has too good reason to fear the confusion of loose ideas likely to
ensue. If others have had a suspicion of the end to be desired, it has
been left to Einstein to show the way to rid ourselves of these
'terrestrial adhesions to thought'. And in removing our fetters he
leaves us, not (as might have been feared) vague generalities for the
ecstatic contemplation of the mystic, but a precise scheme of
world-structure to engage the mathematical physicist.


[Footnote 1: The only alternative is that (relatively to a solar
observer) the velocity of light differs in different directions, at
least in the region where the experiment is conducted. This would
presumably be due to some influence of the moving earth on the
propagation of light (convection of the ether). This explanation was at
one time favoured, but it could not be reconciled with the observed
phenomena of the aberration of light.]

[Footnote 2: The relativity theory does not suggest that there is such
a thing in nature as a four-dimensional space. The whole object of the
recognition of the four-dimensional world is to eliminate the harassing
frame of space.]

[Footnote 3: The inclination must not exceed a certain limit. This
limiting angle may be regarded as a fundamental constant of the
world-structure, and owing to its fundamental character it appears in
many kinds of phenomena; for example, it determines the velocity of
propagation of light. The instant on the sun which is simultaneous with
a given instant on the earth is indeterminate (varying according to the
space and time frame employed) but only within a range of 16 minutes.
Any event on the sun happening before this 16 minutes is _absolutely_ in
the past, all observers agreeing on this point; in fact it would be
possible for us to have already received a wireless message announcing
its occurrence. Events after the 16 minutes are in the _absolute_
future. The neutral zone which is (absolutely) neither past nor future
becomes proportionately wider as the distance increases; at the nearest
fixed star it extends to 8 years, and at the most distant stars yet
known it reaches 400,000 years.]

[Footnote 4: The three events must not be at the same place since that
would give a time-_line_ not a triangle. The clock must move so that the
two events whose time-distance is to be determined both happen where it
is, just as the scale must be directed so that the two points fall on
it. You are not allowed to 'bend' the clock, i. e. apply force so as to
make it move with other than uniform velocity, any more than you are
allowed to bend the scale by applying force.]

[Footnote 5: Of course, it is not true that _any_ two sides are less
than the third side. A clock, unlike a scale, can only measure in one
direction, viz. from past to future, so that the sides _AB_ + _BC_ and
_AC_ can be chosen in only one way.]

[Footnote 6: This involves only a comparatively trifling generalization
of Euclidean geometry, not to be confused with the 'non-Euclidean'
geometry introduced later in the lecture.]

[Footnote 7: The following brief outline will give a hint of the nature
of the explanation. Einstein's law of gravitation is usually expressed
as a set of ten very lengthy differential equations; these equations are
exactly equivalent to the geometrical statement that 'the radius of
spherical curvature of any 3-dimensional section of the 4-dimensional
world is a universal constant length, the same for all points of the
world and for all directions of the section'. The law therefore implies
that the world has a certain type of homogeneity and isotropy (not,
however, the _complete_ homogeneity and isotropy of a sphere). To
explain the law of gravitation and the phenomena governed by it, we have
to explain how this isotropy and homogeneity is secured. Our explanation
is that the homogeneity and isotropy is not initially in the external
world, but _in the measurements which we make of it_. It is introduced
in all our operations of measurement, because the appliances which we
use for measurement are themselves part of the world. In the earlier
part of this lecture we saw that the contraction of the arm turned from
horizontal to vertical is not detected by measurements made with a
yard-measure which shares the contraction; in the same way any
anisotropy of the world does not appear in measurements of it by
appliances which, being part of the world, share the same anisotropy.
The law of gravitation therefore arises from the fact that a certain
type of non-homogeneity and non-isotropy of the world cannot come into
observational experience, because it is necessarily eliminated in all
observations and measurements made with material appliances. The orderly
phenomena of gravitation are due to the _absence_ of certain conceivable
effects. We have been trying to find a key to the mystery; but the
secret of the lock lies not in the key but in the wards.]




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