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Title: From Newton to Einstein
Changing Conceptions of the Universe
Author: Benjamin Harrow
Release Date: September 9, 2019 [EBook #60271]
Language: English
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From NEWTON
to EINSTEIN
Changing Conceptions of
THE UNIVERSE
BY
BENJAMIN HARROW, Ph.D.
SECOND EDITION, REVISED AND ENLARGED
With Articles by Prof. Einstein, Prof. J. S. Ames
(Johns Hopkins), Sir Frank Dyson (Astronomer
Royal), Prof. A. S. Eddington (Cambridge) and Sir
J. J. Thomson (President of the Royal Society)
Portraits and Illustrations
NEW YORK
D. VAN NOSTRAND COMPANY
Eight Warren Street
1920
PREFACE
Einstein's contributions to our ideas of time and space, and to our
knowledge of the universe in general, are of so momentous a nature,
that they easily take their place among the two or three greatest
achievements of the twentieth century. This little book attempts
to give, in popular form, an account of this work. As, however,
Einstein's work is so largely dependent upon the work of Newton and
Newton's successors, the first two chapters are devoted to the latter.
B. H.
PREFACE TO SECOND EDITION
The preparation of this new edition has made it possible to correct
errors, to further amplify certain portions of the text and to
enlarge the ever-increasing bibliography on the subject. Photographs
of Professors J. J. Thomson, Michelson, Minkowski and Lorentz are
also new features in this edition.
The explanatory notes and articles in the Appendix will, I believe,
present no difficulties to readers who have mastered the contents of
the book. They are in fact "popular expositions" of various phases
of the Einstein theory; but experience has shown that even "popular
expositions" of the theory need further "popular introductions."
I wish to take this opportunity of thanking Prof. Einstein,
Prof. A. A. Michelson of the University of Chicago, Prof. J. S. Ames
of Johns Hopkins University, and Professor G. B. Pegram of Columbia
University for help in various ways which they were good enough to
extend to me. Prof. J. S. Ames and the editor of Science have been
kind enough to allow me to reprint the former's excellent presidential
address on Einstein's theory, delivered before the members of the
American Physical Society.
B. H.
TABLE OF CONTENTS
PAGE
I. Newton 1
II. The Ether and Its Consequences 27
III. Einstein 41
IV. Appendix 81
Time, Space and Gravitation, by Prof. Einstein 88
Einstein's Law of Gravitation, by Prof. J. S. Ames 93
The Deflection of light by Gravitation and the
Einstein Theory of Relativity, by Sir Frank Dyson,
Prof. A. S. Eddington and Sir J. J. Thomson 112
NEWTON
"Newton was the greatest genius that ever existed."--Lagrange, one
of the greatest of French mathematicians.
"The efforts of the great philosopher were always superhuman; the
questions which he did not solve were incapable of solution in his
time."--Arago, famous French astronomer.
EINSTEIN
"This is the most important result obtained in connection with the
theory of gravitation since Newton's day. Einstein's reasoning is
the result of one of the highest achievements of human thought."--Sir
J. J. Thomson, president of the British Royal Society and professor
of physics at the University of Cambridge.
"It surpasses in boldness everything previously suggested in
speculative natural philosophy and even in the philosophical theories
of knowledge. The revolution introduced into the physical conceptions
of the world is only to be compared in extent and depth with that
brought about by the introduction of the Copernican system of the
universe."--Prof. Max Planck, professor of physics at the University
of Berlin and winner of the Nobel Prize.
I
NEWTON
In speaking of Newton we are tempted to paraphrase a line from the
Scriptures: Before Newton the Solar System was without form, and void;
then Newton came and there was light. To have discovered a law not
only applicable to matter on this earth, but to the planets and sun
and stars beyond, is a triumph which places Newton among the super-men.
What Newton's law of gravitation must have meant to the people of his
day can be pictured only if we conceive what the effect upon us would
be if someone--say Marconi--were actually to succeed in getting into
touch with beings on another planet. Newton's law increased confidence
in the universality of earthly laws; and it strengthened belief in
the cosmos as a law-abiding mechanism.
Newton's Law. The attraction between any two bodies is proportional
to their masses and inversely proportional to the square of the
distance that separates them. This is the concentrated form of
Newton's law. If we apply this law to two such bodies as the
sun and the earth, we can state that the sun attracts the earth,
and the earth, the sun. Furthermore, this attractive power will
depend upon the distance between these two bodies. Newton showed
that if the distance between the sun and the earth were doubled the
attractive power would be reduced not to one-half, but to one-fourth;
if trebled, the attractive power would be reduced to one-ninth. If,
on the other hand, the distance were halved, the attractive power
would be not merely twice, but four times as great. And what is true
of the sun and the earth is true of every body in the firmament, and,
as Professor Rutherford has recently shown, even of the bodies which
make up the solar system of the almost infinitesimal atom.
This mysterious attractive power that one body possesses for another is
called "gravitation," and the law which regulates the motion of bodies
when under the spell of gravitation is the law of gravitation. This
law we owe to Newton's genius.
Newton's Predecessors. We can best appreciate Newton's momentous
contribution to astronomy by casting a rapid glance over the state
of the science prior to the seventeenth century--that is, prior to
Newton's day. Ptolemy's conception of the earth as the center of the
universe held undisputed sway throughout the middle ages. In those days
Ptolemy was in astronomy what Aristotle was in all other knowledge:
they were the gods who could not but be right. Did not Aristotle say
that earth, air, fire and water constituted the four elements? Did
not Ptolemy say that the earth was the center around which the sun
revolved? Why, then, question further? Questioning was a sacrilege.
Copernicus (1473-1543), however, did question. He studied much and
thought much. He devoted his whole life to the investigation of the
movements of the heavenly bodies. And he came to the conclusion that
Ptolemy and his followers in succeeding ages had expounded views which
were diametrically opposed to the truth. The sun, said Copernicus,
did not move at all, but the earth did; and far from the earth being
the center of the universe, it was but one of several planets revolving
around the sun.
The influence of the church, coupled with man's inclination to exalt
his own importance, strongly tended against the acceptance of such
heterodox views. Among the many hostile critics of the Copernican
system, Tycho Brahe (1546-1601) stands out pre-eminently. This
conscientious observer bitterly assailed Copernicus for his
suggestion that the earth moved, and developed a scheme of his own
which postulated that the planets revolved around the sun, and planets
and sun in turn revolved around the earth.
The majority applauded Tycho; a small, very small group of insurgents
had faith in Copernicus. The illustrious Galileo (1564-1642)
belonged to the minority. The telescope of his invention unfolded
a view of the universe which belied the assertions of the many, and
strengthened his belief in the Copernican theory. "It (the Copernican
theory) explains to me the cause of many phenomena which under the
generally accepted theory are quite unintelligible. I have collected
arguments for refuting the latter, but I do not venture to bring
them to publication." So wrote Galileo to his friend, Kepler. "I do
not venture to bring them to publication." How significant of the
times--of any time, one ventures to add.
Galileo did overcome his hesitancy and published his views. They
aroused a storm. "Look through my telescope," he pleaded. But the
professors would not; neither would the body of Inquisitors. The
Inquisition condemned him: "The proposition that the sun is in
the center of the earth and immovable from its place is absurd,
philosophically false and formally heretical; because it is expressly
contrary to the Holy Scriptures." And poor Galileo was made to utter
words which were as far removed from his thoughts as his oppressors'
ideas were from the truth: "I abjure, curse and detest the said errors
and heresies."
The truth will out. Others arose who defied the majority and the
powerful Inquisition. Most prominent of all of these was Galileo's
friend, Kepler. Though a student of Tycho, Kepler did not hesitate to
espouse the Copernican system; but his adoption of it did not mean
unqualified approval. Kepler's criticism was particularly directed
against the Copernican theory that the planets revolve in circles. This
was boldness in the extreme. Ever since Aristotle's discourse on the
circle as a perfect figure, it was taken for granted that motion
in space was circular. Nature is perfect; the circle is perfect;
hence, if the sun revolves, it revolves in circles. So strongly
were men imbued with this "perfection," that Copernicus himself fell
victim. The sun no longer moved, but the earth and the planets did,
and they moved in a circle. Radical as Copernicus was, a few atoms
of conservatism remained with him still.
Not so Kepler. Tycho had taught him the importance of careful
observation,--to such good effect, that Kepler came to the conclusion
that the revolution of the earth around the sun takes the form of an
ellipse rather than a circle, the sun being stationed at one of the
foci of the ellipse.
To picture this ellipse, we shall ask the reader to stick two pins
a short distance apart into a piece of cardboard, and to place over
the pins a loop of string. With the point of a pencil draw the loop
taut. As the pencil moves around the two pins the curve so produced
will be an ellipse. The positions of the two pins represent the
two foci.
Kepler's observation of the elliptical rotation of the planets was the
first of three laws, quantitatively expressed, which paved the way
for Newton's law. Why did the planets move in just this way? Kepler
tried to answer this also, but failed. It remained for Newton to
supply the answer to this question.
Newton's Law of Gravitation. The Great Plague of 1666 drove Newton
from Cambridge to his home in Lincolnshire. There, according to the
celebrated legend, the philosopher sitting in his little garden one
fine afternoon, fell into a deep reverie. This was interrupted by the
fall of an apple, and the thinker turned his attention to the apple
and its fall.
It must not be supposed that Newton "discovered" gravity. Apples
had been seen to fall before Newton's time, and the reason for their
return to earth was correctly attributed to this mysterious force of
attraction possessed by the earth, to which the name "gravity" had been
given. Newton's great triumph consisted in showing that this "gravity,"
which was supposed to be a peculiar property residing in the earth,
was a universal property of matter; that it applied to the moon and the
sun as well as to the earth; that, in fact, the motions of the moon
and the planets could be explained on the basis of gravitation. But
his supreme triumph was to give, in one sublime generalization,
quantitative expression to the motion regulating heavenly bodies.
Let us follow Newton in his train of thought. An apple falls from a
tree 50 yards high. It would fall from a tree 500 yards high. It would
fall from the highest mountain top several miles above sea level. It
would probably fall from a height much above the mountain top. Why
not? Probably the further up you go the less does the earth attract
the apple, but at what distance does this attraction stop entirely?
The nearest body in space to the earth is the moon, some 240,000 miles
away. Would an apple reach the earth if thrown from the moon? But
perhaps the moon itself has attractive power? If so, since the apple
would be much nearer the moon than the earth, the probabilities are
that the apple would never reach the earth.
But hold! The apple is not the only object that falls to the
ground. What is true of the apple is true of all other bodies--of all
matter, large and small. Now there is the moon itself, a very large
body. Does the earth exert any gravitational pull on the moon? To
be sure, the moon is many thousands of miles away, but the moon is
a very large body, and perhaps this size is in some way related to
the power of attraction?
But then if the earth attracts the moon, why does not the moon fall
to the earth?
A glance at the accompanying figure will help to answer this
question. We must remember that the moon is not stationary, but
travelling at tremendous speed--so much so, that it circles the entire
earth every month. Now if the earth were absent the path of the moon
would be a straight line, say MB. If, however, the earth exerts
attraction, the moon would be pulled inward. Instead of following
the line MB it would follow the curved path MB'. And again, the moon
having arrived at B', is prevented from following the line B'C, but
rather B'C'. So that the path instead of being a straight line tends
to become curved. From Kepler's researches the probabilities were that
this curve would assume the shape of an ellipse rather than a circle.
The only reason, then, why the moon does not fall to the earth is on
account of its motion. Were it to stop moving even for the fraction
of a second it would come straight down to us, and probably few would
live to tell the tale.
Newton reasoned that what keeps the moon revolving around the earth is
the gravitational pull of the latter. The next important step was to
discover the law regulating this motion. Here Kepler's observations of
the movements of the planets around the sun was of inestimable value;
for from these Newton deduced the hypothesis that attraction varies
inversely as the square of the distance. Making use of this hypothesis,
Newton calculated what the attractive power possessed by the earth must
be in order that the moon may continue in its path. He next compared
this force with the force exerted by the earth in pulling the apple
to the ground, and found the forces to be identical! "I compared,"
he writes, "the force necessary to keep the moon in her orb with the
force of gravity at the surface of the earth, and found them answer
pretty nearly." One and the same force pulls the moon and pulls the
apple--the force of gravity. Further, the hypothesis that the force
of gravity varies inversely as the square of the distance had now
received experimental confirmation.
The next step was perfectly clear. If the moon's motion is controlled
by the earth's gravitational pull, why is it not possible that the
earth's motion, in turn, is controlled by the sun's gravitational
pull? that, in fact, not only the earth's motion, but the motion of
all the planets is regulated by the same means?
Here again Kepler's pioneer work was a foundation comparable to
reinforced concrete. Kepler, as we have seen, had shown that the earth
revolves around the sun in the form of an ellipse, one of the foci of
this ellipse being occupied by the sun. Newton now proved that such
an elliptic path was possible only if the intensity of the attractive
force between sun and planet varied inversely as the square of the
distance--the very same relationship that had been applied with such
success in explaining the motion of the moon around the earth!
Newton showed that the moon, the sun, the planets--every body in space
conformed to this law. The earth attracts the moon; but so does the
moon the earth. If the moon revolves around the earth rather than the
earth around the moon, it is because the earth is a much larger body,
and hence its gravitational pull is stronger. The same is true of
the relationship existing between the earth and the sun.
Further Developments of Newton's Law of Gravitation. When we speak of
the earth attracting the moon, and the moon the earth, what we really
mean is that every one of the myriad particles composing the earth
attracts every one of the myriad particles composing the moon, and
vice versa. If in dealing with the attractive forces existing between
a planet and its satellite, or a planet and the sun, the power exerted
by every one of these myriad particles would have to be considered
separately, then the mathematical task of computing such forces
might well appear hopeless. Newton was able to present the problem
in a very simple form by pointing out that in a sphere such as the
earth or the moon, the entire mass might be considered as residing
in the center of the sphere. For purposes of computation, the earth
can be considered a particle, with its entire mass concentrated at
the center of the particle. This viewpoint enabled Newton to extend
his law of inverse squares to the remotest bodies in the universe.
If this great law of Newton's found such general application beyond our
planet, it served an equally useful purpose in explaining a number of
puzzling features on this planet. The ebb and flow of the tides was one
of these puzzles. Even in ancient times it had been noticed that a full
moon and a high tide went hand in hand, and various mysterious powers,
were attributed to the satellite and the ocean. Newton pointed out that
the height of the water was a direct consequence of the attractive
power of the moon, and, to a lesser extent, because further away,
of the sun.
One of his first explanations, however, dealt with certain
irregularities in the moon's motion around the earth. If the solar
system would consist of the earth and moon alone, then the path of the
moon would be that of an ellipse, with one of the foci of this ellipse
occupied by the earth. Unfortunately for the simplicity of the problem,
there are other bodies relatively near in space, particularly that huge
body, the sun. The sun not only exerts its pull on the earth but also
on the moon. However, as the sun is much further away from the moon
than is the earth, the earth's attraction for its satellite is much
greater, despite the fact that the sun is much huger and weighs far
more than the earth. The greater pull of the earth in one direction,
and a lesser pull of the sun in another, places the poor moon between
the devil and the deep sea. The situation gives rise to a complexity of
forces, the net result of which is that the moon's orbit is not quite
that of an ellipse. Newton was able to account for all the forces that
come into play, and he proved that the actual path of the moon was
a direct consequence of the law of inverse squares in actual operation.
The "Principia." The law of gravitation, embodying also laws of motion,
which we shall discuss presently, was first published in Newton's
immortal "Principia" (1686). A selection from the preface will disclose
the contents of the book, and, incidentally, the style of the author:
"... We offer this work as mathematical principles of philosophy; for
all the difficulty in philosophy seems to consist in this--from the
phenomena of motions to investigate the forces of nature, and then from
these forces to demonstrate the other phenomena; and to this end the
general propositions in the first and second book are directed. In the
third book we give an example of this in the explication of the system
of the world; for by the propositions mathematically demonstrated
in the first book, we there derive from the celestial phenomena the
forces of gravity with which bodies tend to the sun and the several
planets. Then, from these forces, by other propositions which are
also mathematical, we deduce the motions of the planets, the comets,
the moon and the sea. I wish we could derive the rest of the phenomena
of nature by the same kind of reasoning from mechanical principles;
for I am induced by many reasons to suspect that they may all depend
upon certain forces by which the particles of bodies, by some causes
hitherto unknown, are either mutually impelled towards each other,
and cohere in regular figures, or are repelled and recede from each
other...."
At this point we may state that neither Newton, nor any of Newton's
successors including Einstein, have been able to advance even a
plausible theory as to the nature of this gravitational force. We
know that this force pulls a stone to the ground; we know, thanks
to Newton, the laws regulating the motions due to gravity; but what
this force we call gravity really is we do not know. The mystery is
as deep as the mystery of the origin of life.
"Prof. Einstein," writes Prof. Eddington, "has sought, and has not
reached, any ultimate explanation of its [that is, gravitation]
cause. A certain connection between the gravitational field and the
measurement of space has been postulated, but this throws light rather
on the nature of our measurements than on gravitation itself. The
relativity theory is indifferent to hypotheses as to the nature of
gravitation, just as it is indifferent to hypotheses as to the nature
of light."
Newton's Laws of Motion. In his Principia Newton begins with a series
of simple definitions dealing with matter and force, and these are
followed by his three famous laws of motion. The nature and amount
of the effort required to start a body moving, and the conditions
required to keep a body in motion, are included in these laws. The
Fundamentals, mass, time and space, are exhibited in their various
relationships. Of importance to us particularly is that in these laws,
time and space are considered as definite entities, and as two distinct
and widely separated manifestations. We shall see that in Einstein's
hands a very close relationship between these two is brought about.
Both Newton and Einstein were led to their theory of gravitation
by profound studies of the mathematics of motion, but as Newton's
conception of motion differed from Einstein's, and as, moreover,
important discoveries into the nature of matter and the relationship
of motion to matter were made subsequent to Newton's time, we need not
wonder that the two theories show divergence; that, as we shall see,
Newton's is probably but an approximation of the truth. If we confine
our attention to our own solar system, the deviation from Newton's
law is, as a rule, so small as to be negligible.
Newton's laws of motion are really axioms, like the axioms of Euclid:
they do not admit of direct proof; but there is this difference,
that the axioms of Euclid seem more obviously true. For example,
when Euclid informs us that "things which are equal to the same
thing are equal to one another," we have no hesitation in accepting
this statement, for it seems so self-evident. When, however, we are
told by Newton that "the alteration of motion is ever proportional
to the motive force impressed," we are at first somewhat bewildered
with the phraseology, and then, even when that has been mastered,
the readiness with which we respond will probably depend upon the
amount of scientific training we have received.
"Every body continues in its state of rest or of uniform motion in a
straight line, unless it is compelled to change that state by forces
impressed thereon." So runs Newton's first law of motion. A body does
not move unless something causes it to move; to make the body move
you must overcome the inertia of the body. On the other hand, if a
body is moving, it tends to continue moving, as witness our forward
movement when the train is brought to a standstill. It may be asked,
why does not a bullet continue moving indefinitely once it has left
the barrel of the gun? Because of the resistance of the air which it
has to overcome; and the path of the bullet is not straight because
gravity acts on it and tends to pull it downwards.
We have no definite means of proving that a body once set in motion
would continue moving, for an indefinite time, and along a straight
line. What Newton meant was that a body would continue moving provided
no external force acted on it; but in actual practise such a condition
is unknown.
Newton's first law defines force as that action necessary to change
a state of rest or of uniform motion, and tells us that force alone
changes the motion of a body. His second law deals with the relation
of the force applied and the resulting change of motion of the body;
that is, it shows us how force may be measured. "The alteration of
motion is ever proportional to the motive force impressed, and is made
in the direction of the right line in which that force is impressed."
Newton's third law runs--"To every action there is always opposed
an equal reaction." The very fact that you have to use force means
that you have to overcome something of an opposite nature. The forward
pull of a horse towing a boat equals the backward pull of the tow-rope
connecting boat and horse. "Many people," says Prof. Watson, "find a
difficulty in accepting this statement ... since they think that if the
force exerted by the horse on the rope were not a little greater than
the backward force exerted by the rope on the horse, the boat would not
progress. In this case we must, however, remember that, as far as their
relative positions are concerned, the horse and the boat are at rest,
and form a single body, and the action and reaction between them, due
to the tension on the rope, must be equal and opposite, for otherwise
there would be relative motion, one with respect to the other."
It may well be asked, what bearing have these laws of Newton on the
question of time and space? Simply this, that to measure force the
factors necessary are the masses of the bodies concerned, the time
involved and the space covered; and Newton's equations for measuring
forces assume time and space to be quite independent of one another. As
we shall see, this is in striking contrast to Einstein's view.
Newton's Researches on Light. In 1665, when but 23 years old,
Newton invented the binomial theorem and the infinitesimal calculus,
two phases of pure mathematics which have been the cause of many a
sleepless night to college freshmen. Had Newton done nothing else his
fame would have been secure. But we have already glanced at his law
of inverse squares and the law of gravitation. We now have to turn
to some of Newton's contributions to optics, because here more than
elsewhere we shall find the starting point to a series of researches
which have culminated so brilliantly in the work of Einstein.
Newton turned his attention to optics in 1666 when he proved that
the light from the sun, which appears white to us, is in reality a
mixture of all the colors of the rainbow. This he showed by placing
a prism between the ray of light and a screen. A spectrum showing
all the colors from red to violet appeared on the screen.
Another notable achievement of his was the design of a telescope which
brought objects to a sharp focus and prevented the blurring effects
which had occasioned so much annoyance to Newton and his predecessors
in all their astronomical observations.
These and other discoveries of very great interest were brought
together in a volume on optics which Newton published in 1704. Our
particular concern here is to examine the views advanced by him as
to the nature of light.
That the nature of light should have been a subject for speculation
even to the ancients need not surprise us. If other senses, as
touch, for example, convey impressions of objects, it is true to say
that the sense of sight conveys the most complete impression. Our
conception of the external world is largely based upon the sense
of sight; particularly so when we deal with objects beyond our
reach. In astronomy, therefore, a study of the properties of light
is indispensable. [1]
But what is this light? We open our eyes and we see; we close our eyes
and we fail to see. At night in a dark room we may have our eyes open
and yet we do not see; light, then, must be absent. Evidently, light
does not wholly depend upon whether our eyes are open or closed. This
much is certain: the eye functions and something else functions. What
is this "something else"?
Strangely enough, Plato and Aristotle regarded light as a property of
the eye and the eye alone. Out of the eye tentacles were shot which
intercepted the object and so illuminated it. From what has already
been said, such a view seems highly unlikely. Far more consistent with
their philosophy in other directions would have been the theory that
light has its source in the object and not in the eye, and travels
from object to eye rather than the reverse. How little substance the
Aristotelian contribution possesses is immediately seen when we refer
to the art of photography. Here light rays produce effects which are
independent of any property of the eye. The blind man may click the
camera and produce the impression on the plate.
Newton, of course, could have fallen into no such error as did Plato
and Aristotle. The source of light to him was the luminous body. Such
a body had the power of emitting minute particles at great speed,
and these when coming in contact with the retina produce the sensation
of sight.
This emission or corpuscular theory of Newton's was combated
very strongly by his illustrious Dutch contemporary, Huyghens, who
maintained that light was a wave phenomenon, the disturbance starting
at the luminous body and spreading out in all directions. The wave
motions of the sea offer a certain analogy.
Newton's strongest objection to Huyghens' wave theory was that it
seemed to offer no satisfactory explanation as to why light travelled
in straight lines. He says: "To me the fundamental supposition itself
seems impossible, namely that the waves or vibrations of any fluid can,
like the rays of light, be propagated in straight lines, without a
continual and very extravagant bending and spreading every way into
the quiescent medium, where they are terminated by it. I mistake if
there be not both experiment and demonstration to the contrary."
In the corpuscular theory the particles emitted by the luminous
body were supposed to travel in straight lines. In empty space the
particles travelled in straight lines and spread in all directions. To
explain how light could traverse some types of matter--liquids, for
example--Newton supposed that these light particles travelled in the
spaces between the molecules of the liquid.
Newton's objection to the wave theory was not answered very
convincingly by Huyghens. Today we know that light waves of high
frequency tend to travel in straight lines, but may be prevented from
doing so by gravitational forces of bodies near its path. But this
is Einstein's discovery.
A very famous experiment by Foucault in 1853 proved beyond the shadow
of a doubt that Newton's corpuscular theory was untenable. According
to Newton's theory, the velocity of light must be greater in a denser
medium (such as water) than in a lighter one (such as air). According
to the wave theory the reverse is true. Foucault showed that light
does travel more slowly in water than in air. The facts were against
Newton and in favor of Huyghens; and where facts and theory clash
there is but one thing to do: discard the theory.
Some Facts about Newton. Newton was a Cambridge man, and Newton made
Cambridge famous as a mathematical center. Since Newton's day Cambridge
has boasted of a Clerk Maxwell and a Rayleigh, and her Larmor, her
J. J. Thomson and her Rutherford are still with us. Newton entered
Trinity College when he was 18 and soon threw himself into higher
mathematics. In 1669, when but 27 years old, he became professor of
mathematics at Cambridge, and later represented that seat of learning
in Parliament. When his friend Montague became Chancellor of the
Exchequer, Newton was offered, and accepted, the lucrative position
of Master of the Mint. As president of the Royal Society Newton was
occasionally brought in contact with Queen Anne. She held Newton in
high esteem, and in 1705 she conferred the honor of knighthood on
him. He died in 1727.
"I do not know," wrote Newton, "what I may appear to the world,
but, to myself, I seem to have been only like a boy playing on the
seashore, and directing myself in now and then finding a smoother
pebble or a prettier shell than ordinary, whilst the great ocean of
truth lay all undiscovered before me."
Such was the modesty of one whom many regard as the greatest intellect
of all ages.
REFERENCES
An excellent account of Newton may be found in Sir R. S. Ball's Great
Astronomers (Sir Isaac Pitman and Sons, Ltd., London). Sedgewick and
Tyler's A Short History of Science (Macmillan, 1918) and Cajori's
A History of Mathematics (Macmillan, 1917) may also be consulted to
advantage. The standard biography is that by Brewster.
II
THE ETHER AND ITS CONSEQUENCES
Huyghens' wave theory of light, now so generally accepted, loses its
entire significance if a medium for the propagation of these waves
is left out of consideration. This medium we call the ether. [2]
Huyghens' reasoning may be illustrated in some such way as this:
If a body moves a force pushes or pulls it. That force itself is
exemplified in some kind of matter--say a horse. The horse in pulling
a cart is attached to the cart. The horse in pulling a boat may not be
attached to the boat directly but to a rope, which in turn is attached
to the boat. In common cases where one piece of matter affects another,
there is some direct contact, some go-between.
But cases are known where matter affects matter without affording us
any evidence of contact. Take the case of a magnet's attraction for
a piece of iron. Where is the rope that pulls the iron towards the
magnet? Perhaps you think the attraction due to the air in between the
magnet and iron? But removing the air does not stop the attraction. Yet
how can we conceive of the iron being drawn to the magnet unless there
is some go-between? some medium not readily perceptible to the senses
perhaps, and therefore not strictly a form of matter?
If we can but picture some such medium we can imagine our magnet
giving rise to vibrations in this medium which are carried to the
iron. The magnet may give rise to a disturbance in that portion of the
medium nearest to it; then this portion hands over the disturbance
to its neighbor, the next portion of the medium; and so on, until
the disturbance reaches the iron. You see, we are satisfying our
sense-perception by arguing in favor of action by actual contact
rather than some vague action at a distance; the go-between instead
of being a rope is the medium called the ether.
Foucault's experiment completely shattered the corpuscular theory
of light, and for want of any other more plausible alternative, we
are thrown back on Huyghens' wave theory. It will presently appear
that this wave theory has elements in it which make it an excellent
alternative. In the meantime, if light is to be considered as a wave
motion, then the query immediately arises, what is the medium through
which these waves are propagated? If water is the medium for the waves
of the sea, what is the medium for the waves of light? Again we answer,
the medium is the ether.
What Is This "Ether"? Balloonists find conditions more and more
uncomfortable the higher they ascend, for the density of the air
(and therefore the amount of oxygen in a given volume of air) becomes
less and less. Meteorologists have calculated that traces of the
air we breathe may reach a height of some 200 miles. But what is
beyond? Nothing but the ether, it is claimed. Light from the sun and
stars reaches us via the ether.
But what is this ether? We cannot handle it. We cannot see it. It
fails to fall within the scope of any of our senses, for every attempt
to show its presence has failed. It is spirit-like in the popular
sense. It is Lodge's medium for the souls of the departed.
Helmholtz and Kelvin tried to arrive at some properties of this
hypothetical substance from a careful study of the manner in
which waves were propagated through this ether. If, as the wave
theory teaches us, the ether can be set in motion, then according
to laws of mechanics, the ether has mass. If so it is smaller in
amount than anything which can be detected with our most accurate
balance. Further--and this is a difficulty not easily explained--if
this ether has any mass, why does it offer no detectable resistance
to the velocity of the planets in it? Why is not the velocity of
the planets reduced in time, just as the velocity of a rifle bullet
decreases owing to the resistance of the air?
Lodge, in arguing in favor of an ether, holds that its presence cannot
be detected because it pervades all space and all matter. His favorite
analogy is to point out the extreme unlikelihood of a deep-sea fish
discovering the presence of the water with which it is surrounded on
all sides;--all of which tells us nothing about the ether, but does
try to tell us why we cannot detect it. [3]
In short, answering the query at the head of this paragraph, we may
say that we do not know.
Waves Set up in This Ether. The waves are not all of the same
length. Those that produce the sensation of sight are not the smallest
waves known, yet their length is so small that it would take anywhere
from one to two million of them to cover a yard. Curiously enough,
our eye is not sensitive to wave lengths beyond either side of these
limits; yet much smaller, and much larger waves are known. The smallest
are the famous X-rays, which are scarcely one ten-thousandth the size
of light waves. Waves which have a powerful chemical action--those
which act on a photographic plate, for example--are longer than
X-rays, yet smaller than light waves. Waves larger than light waves
are those which produce the sensation of heat, and those used in
wireless telegraphy. The latter may reach the enormous length of 5,000
yards. X-ray, actinic, or chemically active ray, light ray, heat ray,
wireless ray--they differ in size, yet they all have this in common:
they travel with the same speed (186,000 miles per second).
The Electromagnetic Theory of Light. Powerful support to the conception
that space is pervaded by ether was given when Maxwell discovered
light to be an electromagnetic phenomenon. From purely theoretical
considerations this gifted English physicist was led to the view
that waves could be set up as a result of electrical disturbances. He
proved that such waves would travel with the same velocity as light
waves. As air is not needed to transmit electrical phenomena--for you
can pump all air out of a system and produce a vacuum, and electrical
phenomena will continue--Maxwell was forced to the conclusion that
the waves set up by electrical disturbances and transmitted with the
same velocity as light, were enabled to do so with the help of the
same medium as light, namely, the ether.
It was now but a step for Maxwell to formulate the theory that light
itself is nothing but an electrical phenomenon, the sensation of light
being due to the passage of electric waves through the ether. This
theory met with considerable opposition at first. Physicists had been
brought up in a school which had taught that light and electricity
were two entirely unrelated phenomena, and it was difficult for
them to loosen the shackles that bound them to the older school. But
two startling discoveries helped to fasten attention upon Maxwell's
theory. One was an experimental confirmation of Maxwell's theoretical
deduction. Hertz, a pupil of Helmholtz, showed how the discharge from
a Leyden jar set up oscillations, which in turn gave rise to waves in
the ether, comparable, in so far as velocity is concerned, to light
waves, but differing from the latter in wave length, the Hertzian
waves being much longer. At a later date these waves were further
investigated by Marconi, with the result that wireless messages soon
began to be flashed from one place to another.
Just as there is a close connection between light and electricity,
so there is between light and magnetism. The first to point out such
a relationship was the illustrious Michael Faraday, but we owe to
Zeeman the most extensive investigations in this field.
If we throw some common salt into a flame, and, with the help of a
spectroscope, examine the spectrum produced, we are struck by two
bright lines which stand out very prominently. These lines, yellow
in color, are known as the D-lines and serve to identify even minute
traces of sodium. What is true of sodium is true of other elements:
they all produce very characteristic spectra. Now Zeeman found that if
the flame is placed between a powerful magnet, and then some common
salt thrown into the flame, the two yellow lines give place to ten
yellow lines. Such is one of the results of the effect of a magnetic
field on light.
The Electron. The "Zeeman effect" led to several theories regarding
its nature. The most successful of these was one proposed by Larmor and
more fully treated by Lorentz. It has already been pointed out that the
only difference between wireless and light waves is that the former
are much "longer," and, we may now add, their vibrations are much
slower. Light and wireless waves bear a relationship to one another
comparable to the relationship born by high and low-pitched sounds. To
produce wireless waves we allow a charge of electricity to oscillate
to and fro. These oscillations, or oscillating charges, are the cause
of such waves. What charges give rise to light waves? Lorentz, from
a study of the Zeeman effect, ascribed them to minute particles of
matter, smaller than the chemical atom, to which the name "electron"
was given.
The unit of electricity is the electron. Electrons in motion give rise
to electricity, and electrons in vibration, to light. The Zeeman effect
gave Lorentz enough data to calculate the mass of such electrons. He
then showed that these electrons in a magnetic field would be disturbed
by precisely the amount to which Zeeman's observations pointed. In
other words, the assumption of the electron fitted in most admirably
with Zeeman's experiments on magnetism and light.
In the meantime, a study of the discharge of electricity through
gases, and, later, the discovery of radium, led, among other things,
to a study of beta or cathode rays--negatively charged particles
of electricity. Through a series of strikingly original experiments
J. J. Thomson ascertained the mass of such particles or corpuscles, and
then the very striking fact was brought out that Thomson's corpuscle
weighed the same as Lorentz's electron. The electron was not merely
the unit of electricity but the smallest particle of matter.
The Nature of Matter. All matter is made up of some eighty-odd
elements. Oxygen, copper, lead are examples of such elements. Each
element in turn consists of an innumerable number of atoms, of a size
so small, that 300 million of them could be placed alongside of one
another without their total length exceeding one inch.
John Dalton more than a hundred years ago postulated a theory, now
known as the atomic theory, to explain one of the fundamental laws
in chemistry. This theory started out with an old Greek assumption
that matter cannot be divided indefinitely, but that, by continued
subdivision, a point would be reached beyond which no further breaking
up would be possible. The particles at this stage Dalton called atoms.
Dalton's atomic hypothesis became one of the pillars upon which the
whole superstructure of chemistry rested, and this because it explained
a number of perplexing difficulties so much more satisfactorily than
any other hypothesis.
For nearly a century Dalton stood as firm as a rock. But early
in the nineties some epoch-making experiments on the discharge
of electricity through gases were begun by a group of physicists,
particularly Crookes, Rutherford, Lenard, Roentgen, Becquerel, and,
above all, J. J. Thomson, which pointed very clearly to the fact that
the atoms are not the smallest particles of matter at all; that,
in fact, they could be broken up into electrons, of a diameter one
one-hundred-thousandth that of an atom.
It remained for the illustrious Madame Curie to confirm this beyond
all doubt by her isolation of radium. Here, as Madame Curie showed,
was an element whose atoms were actually breaking up under one's very
eyes, so to speak.
So far have we advanced since Dalton's day, that Dalton's unit,
the atom, is now pictured as a complex particle patterned after our
solar system, with a nucleus of positive electricity in the center,
and particles of negative electricity, or electrons, surrounding
the nucleus.
All this leads to one inevitable conclusion: matter is electrical in
nature. But now if matter and light have the same origin, and matter
is subject to gravitation, why not light also? So reasoned Einstein.
Summary. Newton's studies of matter in motion led to his theory of
gravitation, and, incidentally, to his conception of time and space
as definite entities. As we shall see, Einstein in his theory of
gravitation based it upon discoveries belonging to the post-Newtonian
period. One of these is Minkowski's theory of time and space as one
and inseparable. This theory we shall discuss at some length in the
next chapter.
Other important discoveries which led up to Einstein's work are the
researches which culminated in the electron theory of matter. The
origin of this theory may be traced to studies dealing with the nature
of light.
Here again Newton appears as a pioneer. Newton's corpuscular theory,
however, proved wholly untenable when Foucault showed that the
velocity of light in water is less than in air, which is the very
reverse of what the corpuscular theory demands, but which does agree
with Huyghens' wave theory.
But Huyghens' wave theory postulated some medium in which the waves
can act. To this medium the name "ether" was given. However, all
attempts to show the presence of such an ether failed. Naturally
enough, some began to doubt the existence of an ether altogether.
Huyghens' wave theory received a new lease of life with Maxwell's
discovery that light is an electromagnetic phenomenon; that the waves
set up by a source of light were comparable to waves set up by an
electrical disturbance.
Zeeman next showed that magnetism was also, closely related to light.
A study of Zeeman's experiments led Lorentz to the conclusion that
electrical phenomena are due to the motion of charged particles called
"electrons," and that the vibrations of these electrons give rise
to light.
The conception of the electron as the very fundamental of matter was
arrived at in an entirely different way: from studies dealing with
the discharge of electricity through gases and the breaking up of
the atoms of radium.
If matter and light have the same origin, and if matter is subject
to gravitation, why not light also?
REFERENCES
For the general subject of light the reader must be referred to a
rather technical work, but one of the best in the English language:
Edwin Edser, Light for Students (Macmillan, 1907).
The nature of matter and electricity is excellently discussed in
several books of a popular variety. The very best and most complete
of its kind that has come to the author's attention is Comstock and
Troland's The Nature of Matter and Electricity (D. Van Nostrand Co.,
1919). Two other very readable books are Soddy's Matter and Energy
(Henry Holt and Co.) and Crehore's The Mystery of Matter and Energy
(D. Van Nostrand Co., 1917).
III
EINSTEIN
"This is the most important result obtained in connection with the
theory of gravitation since Newton's day. Einstein's reasoning is
the result of one of the highest achievements of human thought."
These words were uttered by Sir J. J. Thomson, the president of the
Royal Society, at a meeting of that body held on November 6, 1919,
to discuss the results of the Eclipse Expedition.
Einstein another Newton--and this from the lips of J. J. Thomson,
England's most illustrious physicist! If ever man weighed words
carefully it is this Cambridge professor, whose own researches have
assured him immortality for all time.
What has this Albert Einstein done to merit such extraordinary
praise? With the world in turmoil, with classes and races in a death
struggle, with millions suffering and starving, why do we find time
to turn our attention to this Jew? His ideas have no bearing on
Europe's calamity. They will not add one bushel of wheat to starving
populations.
The answer is not hard to find. Men come and men go, but the mystery of
the universe remains. It is Einstein's glory to have given us a deeper
insight into the universe. Our scientists are Huxley's agnostics:
they do not deny activities beyond our planet; they merely center
their attention on the knowable on this earth. Our philosophers,
on the other hand, go far afield. Some of them soar so high that,
like one poet's opinion of Shelley, the bubble bursts. Einstein,
using the tools of the scientist--the experimentalist--builded a
skyscraper which ultimately reached the philosophical school. His rôle
is the rôle of alcohol in causing water and ether (the anæsthetic)
to mix. Ether and water will mix no better than oil and water, without
the presence of alcohol; in its presence a uniform mixture is obtained.
The Object of the Eclipse Expedition. Einstein prophesied that a
ray of light passing near the sun would be pulled out of its course,
due to the action of gravity. He went even further. He predicted how
much out of its course the ray would be deflected. This prediction
was based on a theory of gravitation which Einstein had developed in
great mathematical detail. The object of the British Eclipse Expedition
was either to prove or disprove Einstein's assumption.
The Result of the Expedition. Einstein's prophecy was fulfilled almost
to the letter.
The Significance of the Result. Since Einstein's theory of gravitation
is intimately associated with certain revolutionary ideas concerning
time and space, and, therefore, with Fundamentals of the Universe,
the net result of the expedition was to strengthen our belief in the
validity of his new view of the universe.
It is our intention in the following pages to discuss the expedition
and the larger aspects of Einstein's theory that follow from it. But
before we do so we must have a clear idea of our solar system.
Our Solar System. In the center of our system is the sun, a flaming
mass of fire, much bigger than our own earth, and very, very far
away. The sun has its family of eight planets--of which the earth is
one--which travel around the sun; and around some of the planets there
travel satellites, or moons. The earth has such a satellite, the moon.
Now we have good reasons for believing that every star which twinkles
in the sky is a sun comparable to our own, having also its own planets
and its own moons. These stars, or suns, are so much further away
from us than our own sun, that but a speck of their light reaches us,
and then only at night, when, as the poets would say, our sun has
gone to its resting place.
The distances between bodies in the solar system is so immense that,
like the number of dollars spent in the Great War, the number of miles
conveys little, or no impression. But picture yourself in an express
train going at the average rate of 30 miles an hour. If you start
from New York and travel continuously you would reach San Francisco
in 4 days. If you could continue your journey around the earth at
the same rate you would complete it in 35 days. If now you could
travel into space and to the moon, still with the same velocity,
you would reach it in 350 days. Having reached the moon, you could
circumscribe it with the same express train in 8 days, as compared to
the 35 days it would take you to circumscribe the earth. If instead
of travelling to the moon you would book your passage for the sun
you, or rather your descendants, would get there in 350 years, and
it would then take them 10 additional years to travel around the sun.
Immense as these distances are, they are small as compared to the
distances that separate us from the stars. It takes light which,
instead of travelling 30 miles an hour, travels 186,000 miles a second,
about 8 minutes to get to us from the sun, and a little over 4 years
to reach us from the nearest star. The light from some of the other
stars do not reach us for several hundred years.
The Eclipse of the Sun. Now to return to an infinitesimal part of
the universe--our solar system. We have seen that the earth travels
around the sun, and the moon around the earth. At some time in the
course of these revolutions the moon must come directly between the
earth and the sun. Then we get the eclipse of the sun. As the moon
is smaller than the earth, only a portion of the earth's surface
will be cut off from the sun's rays. That portion which is so cut
off suffers a total eclipse. This explains why the eclipse of May,
1919, which was a total one for Brazil, was but a partial one for us.
Einstein's Assertion Re-stated. Einstein claimed that a ray of light
from one of the stars, if passing near enough to the surface of the
sun, would be appreciably deflected from its course; and he calculated
the exact amount of this deflection. To begin with, why should Einstein
suppose that the path of a ray of light would be affected by the son?
Newton's law of gravitation made it clear that bodies which have
mass attract one another. If light has mass--and very recent work
tends to show that it has--there is no reason why light should not be
attracted by the sun, or any other planetary body. The question that
agitated scientists was not so much whether a ray of light would be
deviated from its path, but to what extent this deviation would take
place. Would Einstein's figures be confirmed?
Of the bodies within our solar system the sun is by far the largest,
and therefore it would exert a far greater pull than any of the planets
on light rays coming from the stars. Under ordinary conditions,
however, the sun itself shines with such brilliancy, that objects
around it, including rays of light passing near its surface, are wholly
dimmed. Hence the necessity of putting our theory to the test only when
the moon covers up the sun--when there is a total eclipse of the sun.
A Graphical Representation. Imagine a star A, so selected that as
its light comes to us the ray just grazes the sun. If the path of
the ray is straight--if the sun has no influence on it--then the path
can be represented by the line AB. If, however, the sun does exert a
gravitational pull, then its real path will be AB', and to an observer
on the earth the star will have appeared to shift from A to A'.
What the Eclipse Expedition Set Out to Do. Photographs of stars
around the sun were to be taken during the eclipse, and these
photographs compared with others of the same region taken at night,
with the sun absent. Any apparent shifting of the stars could be
determined by measuring the distances between the stars as shown on
the photographic plates.
Three Possibilities Anticipated. According to Newton's assumption,
light consists of corpuscles, or minute particles, emitted from
the source of light. If this be true these particles, having mass,
should be affected by the gravitational pull of the sun. If we apply
Newton's theory of gravitation and make use of his formula, it can be
shown that such a gravitational pull would displace the ray of light
by an average amount equal to 0.75 (seconds of angular distance.) [4]
On the other hand, where light is regarded as waves set in motion in
the "ether" of space (the wave theory of light), and where weight is
denied light altogether, no deviation need be expected. Finally there
is a third alternative: Einstein's. Light, says Einstein, has mass,
and therefore probably weight. Mass is the matter light contains;
weight represents pull by gravity. Light rays will be attracted by
the sun, but according to Einstein's theory of gravitation the sun's
gravitational pull will displace the rays by an average amount equal
to 1.75 (seconds of angular distance).
The Expeditions. That science is highly international, despite many
recent examples to the contrary, is evidenced by this British Eclipse
Expedition. Here was a theory propounded by one who had accepted a
chair of physics in the university of Berlin, and across the English
Channel were Germany's mortal enemies making elaborate preparations
to test the validity of the Berlin professor's theory.
The British Astronomical Society began to plan the eclipse expedition
even before the outbreak of the Great War. During the years that
followed, despite the destinies of nations which hung on threads from
day to day, despite the darkest hours in the history of the British
people, our English astronomers continued to give attention to the
details of the proposed expedition. When the day of the eclipse came
all was in readiness.
One expedition under Dr. Crommelin was sent to Sobral, Brazil; another,
under Prof. Eddington, to Principe, an island off the west coast of
Africa. In both these places a total eclipse was anticipated.
The eclipse occurred on May 29, 1919. It lasted for six to eight
minutes. Some 15 photographs, with an average exposure of five to six
seconds, were taken. Two months later another series of photographs
of the same region were taken, but this time the sun was no longer
in the midst of these stars.
The photographs were brought to the famous Greenwich Observatory,
near London, and the astronomers and mathematicians began their
laborious measurements and calculations.
On November 6, at the meeting of the Royal Society, the result
was announced. The Sobral expedition reported 1.98; the Principe
expedition 1.62. The average was 1.8. Einstein had predicted 1.75,
Newton might have predicted 0.75, and the orthodox scientists would
have predicted 0. There could now no longer be any question as to which
of the three theories rested on a sure foundation. To quote Sir Frank
Dyson, the Astronomer Royal: "After a careful study of the plates I am
prepared to say that there can be no doubt that they confirm Einstein's
prediction. A very definite result has been obtained that light is
deflected in accordance with Einstein's law of gravitation." [5]
Where Did Einstein Get His Idea of Gravitation? In 1905 Einstein
published the first of a series of papers supporting and extending a
theory of time and space to which the name "the theory of relativity"
had been given. These views as expounded by Einstein came into direct
conflict with Newton's ideas of time and space, and also with Newton's
law of gravitation. Since Einstein had more faith in his theory of
relativity than in Newton's theory of gravitation, Einstein so changed
the latter as to make it harmonize with the former. More will be said
on this subject.
Let not the reader misunderstand. Newton was not wholly in the wrong;
he was only approximately right. With the knowledge existing in
Newton's day Newton could have done no more than he did; no mortal
could have done more. But since Newton's day physics--and science in
general--has advanced in great strides, and Einstein can interpret
present-day knowledge in the same masterful fashion that Newton
could in his day. With more facts to build upon, Einstein's law
of gravitation is more universal than Newton's; it really includes
Newton's.
But now we must turn our attention very briefly to the theory of
relativity--the theory that led up to Einstein's law of gravitation.
The Theory of Relativity. The story goes that Einstein was led to
his ideas by watching a man fall from a roof. This story bears a
striking similarity to Newton and the apple. Perhaps one is as true
as the other. [6]
However that may be, the principle of relativity is as old as
philosophical thought, for it denies the possibility of measuring
absolute time, or absolute space. All things are relative. We say
that it takes a "long time" to get from New York to Albany; long as
compared to what? long, perhaps, as compared to the time it takes to
go from New York City to Brooklyn. We say the White House is "large";
large when compared to a room in an apartment. But we can just as
well reverse our ideas of time and distance. The time it takes to go
from New York to Albany is "short" when compared to the time it takes
to go from New York to San Francisco. The size of the White House is
"small" when compared to the size of the city of Washington.
Let us take another illustration. Every time the earth turns on its
axis we mark down as a day. With this as a basis, we say that it
takes a little over 365 days for the earth to complete its revolution
around the sun, and our 365 days we call a year. But now consider some
of our other planets. With our time as a basis, it takes Jupiter or
Saturn 10 hours to turn on its axis, as compared to the 24 hours it
takes the earth to turn. Saturn's day is less than one-half our day,
and our day is more than twice Saturn's--that is, according to the
calculations of the inhabitants of the earth. Mercury completes her
circuit around the sun in 88 days; Neptune, in 164 years. Mercury's
year is but one-fourth ours, Neptune's, 164 times ours. And observers
at Mercury and Neptune would regard us from their standard of time,
which differs from our standard.
You may say, why not take our standard of time as the standard,
and measure everything by it? But why should you? Such a selection
would be quite arbitrary. It would not be based on anything absolute,
but would merely depend on our velocity around the sun.
These ideas are old enough in metaphysics. Einstein's improvement
of them consists not merely in speculating about them, but in giving
them mathematical form.
The Origin of the Theory of Relativity. A train moves with reference
to the earth. The earth moves with reference to the sun. We say the
sun is stationary and the earth moves around the sun. But how do we
know that the sun itself does not move with reference to some other
body? How do we know that our planetary system, and the stars, and
the cosmos as a whole is not in motion?
There is no way of answering such a question unless we could get a
point of reference which is fixed--fixed absolutely in space.
We have already alluded to our view of the nature of light, known as
the wave theory of light. This theory postulates the existence of an
all-pervading "ether" in space. Light sets up wave disturbances in
this ether, and is thus propagated. If the ocean were the ether, the
waves of the ocean would compare with the waves set up by the ether.
But what is this ether? It cannot be seen. It defies weight. It
permeates all space. It permeates all matter. So say the exponents of
this ether. To the layman this sounds very much like another name
for the Deity. To Sir Oliver Lodge it represents the spirits of
the departed.
To us, of importance is the conception that this ether is absolutely
stationary. Such a conception is logical if the various developments in
optics and electricity are considered. But if absolutely stationary,
then the ether is the long-sought-for point of reference, the guide
to determine the motion of all bodies in the universe.
The Famous Experiment Performed by Prof. Michelson. If there is an
ether, and a stationary ether, and if the earth moves with reference to
this ether, the earth, in moving, must set up ether "currents"--just
as when a train moves it sets up air currents. So reasoned Michelson,
a young Annapolis graduate at the time. And forthwith he devised a
crucial experiment the explanation of which we can simplify by the
following analogy:
Which is the quicker, to swim up stream a certain length, say a
hundred yards, and back again, or across stream the same length and
back again? The swimmer will answer that the up-and-down journey is
longer. [7]
Our river is the ether. The earth, if moving in this ether, will
set up an ether stream, the up stream being parallel to the earth's
motion. Now suppose we send a beam of light a certain distance up
this ether stream and back, and note the time; and then turn the
beam of light at right angles and send it an equal distance across
the stream and back, and note the time. How will the time taken for
light to travel in these two directions compare? Reasoning by analogy,
the up-and-down stream should take longer.
Michelson's results did not accord with analogy. No difference in time
could be detected between the beam of light travelling up-and-down,
and across-and-back.
But this was contrary to all reason if the postulate of an ether was
sound. Must we then revise our ideas of an ether? Perhaps after all
there is no ether.
But if no ether, how are we to explain the propagation of light in
space, and various electrical phenomena connected with it, such as
the Hertzian, or wireless waves?
There was another alternative, one suggested by Larmor in England
and Lorentz in Holland,--that matter is contracted in the direction
of its motion through the ether current. To say that bodies are
actually shortened in the direction of their motion--by an amount
which increases as the velocity of these bodies approaches that of
light--is so revolutionary an idea that Larmor and Lorentz would
hardly have adopted such a viewpoint but for the fact that recent
investigations into the nature of matter gave basis for such belief.
Matter, it has been shown, is electrical in nature. The forces which
hold the particles together are electrical. Lorentz showed that
mathematical formulas for electrical forces could be developed which
would inevitably lead to the view that material bodies contract in
the direction of their motion. [8]
"But this is ridiculous," you say; "if I am shorter in one direction
than in another I would notice it." You would if some things were
shortened and others were not. But if all things pointing in a certain
direction are shortened to an equal extent, how are you going to notice
it? Will you apply the yard stick? That has been shortened. Will you
pass judgment with the help of your eyes? But your retina has also
contracted. In brief, if all things contract to the same amount it
is as if there were no contraction at all.
Lorentz's Plausible Explanation Really Deepens the Mystery. The
startling ideas just outlined have opened up several new vistas,
but they have left unanswered the two problems we set out to solve:
whether there is an ether, and if so, what is the velocity of the earth
in reference to this ether? Lorentz maintains that there is an ether,
but the velocities of bodies relative to it must forever remain a
mystery. As you change your position your distances change; you change;
everything about you changes accordingly; and all basis for comparison
is lost. Nature has entered into a conspiracy to keep you ignorant.
Einstein Comes upon the Scene. Einstein starts with the assumption
that there is no possible way of identifying this ether. Suppose we
ignore the ether altogether, what then? [9]
If we do ignore the ether we no longer have any absolute point of
reference; for if the ether is considered stationary the velocity of
all bodies within the ether may be referred to it; any point in space
may be considered a fixed point. If, however, there is no ether, or if
we are to ignore it, how are we to get the velocity of bodies in space?
The Principle of Relativity. If we are to believe in the "causal
relationship between only such things as lie within the realm of
observation," then observation teaches us that bodies move only
relative to one another, and that the idea of absolute motion of a
body in space is meaningless. Einstein, therefore, postulates that
there is no such thing as absolute motion, and that all we can discuss
is the relative motion of one body with respect to another. This
is just as logical a deduction from Michelson's experiment as the
attempt to explain Michelson's anomalous results in the light of an
all-pervading ether.
Consider for a moment Newton's scheme. This great pioneer pictured
an absolute standard of position in space relative to which all
velocities are measured. Velocities were measured by noting the
distance covered and dividing the result by the time taken to cover
the distance. Space was a definite entity; and so was time. "Time,"
said Newton, "flows evenly on," independent of aught else. To Newton
time and space were entirely different, in no way to be confounded.
Just as Newton conceived of absolute space, so he conceived of absolute
time. From the latter standard of reference the idea of a "simultaneity
of events" at different places arose. But now if there is no standard
of reference, if the ether does not exist or does not function, if
two points A and B cannot be referred to a third, and fixed point C,
how can we talk of "simultaneity of events" at A and B? [10]
In fact, Einstein shows that if all you can speak about is relative
motion, then one event which takes say one minute on one planet would
not take one minute on another. For consider two bodies in space,
say the planets Venus and the earth, with an observer B on Venus
and another A on the earth. B notes the time taken for a ray of
light to travel from B to the distance M. A on the earth has means
of observing the same event. B records one minute. A is puzzled,
for his watch records a little more than one minute. What is the
explanation? Granting that the two clocks register the same time
to start with, and assuming further Einstein's hypothesis that the
velocity of light is independent of its source, the difference in
time is due to the fact that the planet Venus moves with reference to
the observer on the earth; so that A in reality does not measure the
path BM and MB, but BM' and M'B', where BB' represents the distance
Venus itself has moved in the interval. And if you put yourself in B's
position on Venus the situation is exactly reversed. All of which is
simply another way of saying that what is a certain time on one body
in space is another time on another body in space. There is nothing
definite in time.
Prof. Cohen's Illustration. Further bewildering possibilities are
clearly outlined in this apt illustration: "If when you are going away
on a long and continuous journey you write home at regular intervals,
you should not be surprised that with the best possible mail service
your letters will reach home at progressively longer intervals,
since each letter will have a greater distance to travel than its
predecessor. If you were armed with instruments to hear the home
clock ticking, you would find that as your distance from home keeps
on increasing, the intervals between the successive ticks (that is,
its seconds) grow longer, so that if you travelled with the velocity
of sound the home clock would seem to slow down to a standstill--you
would never hear the next tick.
"Precisely the same is true if you substitute light rays for sound
waves. If with the naked eye or with a telescope you watch a clock
moving away from you, you will find that its minute hand takes a longer
time to cover its five-minute intervals than does the chronometer in
your hand, and if the clock travelled with the velocity of light you
would forever see the minute hand at precisely the same point. That
which is true of the clock is, of course, also true of all time
intervals which it measures, so that if you moved away from the earth
with the velocity of light everything on it would appear as still as
on a painted canvas."
Your time has apparently come to a standstill in one position and
is moving in another! All this seems absurd enough, but it does show
that time alone has little meaning.
Minkowski's Conclusion. The relativity theory requires that we
thoroughly reorganise our method of measuring time. But this is
intimately associated with our method of measuring space, the distance
between two points. As we proceed we find that space without time has
little meaning, and vice versa. This leads Minkowski to the conclusion
that "time by itself and space by itself are mere shadows; they are
only two aspects of a single and indivisible manner of coordinating
the facts of the physical world." Einstein incorporated this time-space
idea in his theory of relativity.
How We Measure a Point in Space. Suppose I say to you that the chemical
laboratory of Columbia University faces Broadway; would that locate
the laboratory? Hardly, for any building along Broadway would face
Broadway. But suppose I add that it is situated at Broadway and 117th
Street, south-east? there could be little doubt then. But if, further,
this laboratory would occupy but part of the building, say the third
floor; then the situation would be specified by naming Broadway,
117th Street S. E., third floor. If Broadway represents length, 117th
Street width, and third floor height, we can see what is meant when we
say that three dimensions are required to locate a position in space.
The Fourth Dimension. A point on a line may be located by one
dimension; a point on a wall requires two dimensions; a point in the
room, like the chemical laboratory above ground, needs three. The
layman cannot grasp the meaning of a fourth dimension; yet the
mathematician does imagine it, and plays with it in mathematical
terms. Minkowski and Einstein picture time as the fourth dimension. To
them time occupies no more important position than length, breadth,
or thickness, and is as intimately related to these three as the three
are to one another. H. G. Wells, the novelist, has beautifully caught
this spirit when in his novel, "The Time Machine," he makes his hero
travel backwards and forwards along time just as a man might go north
or south. When the man with his time machine goes forward he is in
the future; when he goes backwards he is in the past.
In reality, if we stop to think a minute, there is no valid
reason for the non-existence of a fourth dimension. If one, two
and three dimensions, why not four--and five and six, for that
matter? Theoretically at least there is no reason why the limit
should be set at three. However, our minds become sluggish when we
attempt to picture dimensions beyond three; just as an extraordinary
effort on our part is needed to follow Einstein when he "juggles"
with space and time.
Our difficulty in imagining four dimensions may be likened to the
difficulty two-dimensioned beings would experience in imagining
us, beings of the conventional three dimensions. Suppose these
two-dimensional beings were living on the surface of the earth;
what could they see? They could see nothing below and nothing above
the surface. They would see shifting surfaces as we walked about,
but being sensitive to length and breadth only, and not to height,
they could gain no notion whatsoever of what we really look like. It
is thus with us when we attempt to picture four-dimensional space.
Perhaps the analogy of the motion picture may help us somewhat. As
everybody knows, these motion pictures consist of a series of
photographs which are shown in rapid succession on the screen. Each
photograph by itself conveys a sensation of space, that is, of three
dimensions; but one photograph rapidly following another conveys
the sensation of space and time--four dimensions. Space and time
are interlinked.
The Time-space Idea Further Developed. We have already alluded to the
fact that objects in space moving with different velocities build up
different time intervals. Thus the velocity of the star Arcturus, if
compared with reference to the earth, moves at the rate of 200 miles a
second. Its motion through space is different from ours. Objects which,
according to Lorentz, contract in the direction of their motion to an
extent proportional to their velocity, will contract differently on
the surface of Arcturus than on the earth. Our space is not Arcturus'
space; neither is Arcturus' time our time. And what is true of the
discrepancies existing between the space and time conceptions of the
earth and Arcturus is true of any other two bodies in space moving
at different velocities.
But is there no relationship existing between the space and time
of one body in the universe as compared to the space and time of
another? Can we not find something which holds good for all bodies
in the universe? We can. We can express it mathematically. It is the
concept of time and space interlinked; of time as the fourth dimension,
length, breadth and thickness being the other three; of time as
one of four co-ordinates and at right angles to the other three
(a situation which requires a terrific stretch of the imagination
to visualize). The four dimensions are sufficient to co-ordinate
the time-space relationships of all bodies in the cosmos, and hence
have a universality which is totally lacking when time and space
are used independently of one another. The four components of our
time-space are up-and-down, right-and-left, backwards-and-forwards,
and sooner-and-later.
"Strain" and "Distortion" in Space. The four-dimensional unit has been
given the name "world-line," for the "world-line" of any particle in
space is in reality a complete history of that particle as it moves
about in space. Particles, we know, attract one another. If each
particle is represented by a world-line these world-lines will be
deflected from their course owing to such attraction.
Imagine a bladder representing the universe, with lines on it
representing world-lines. Now squeeze the bladder. The world-lines are
bent in various directions; they are "distorted." This illustrates the
influence of gravity on these world-lines; it is the "strain" brought
about due to the force of attraction. The distorted bladder illustrates
even more, for it is a true representation of the real world.
How Einstein's Conception of Time and Space Led to a New View of
Gravitation. In our conventional language we speak of the sun as
exerting a "force" on the earth. We have seen, however, that this
force brings about a "distortion" or "strain" in world-lines; or,
what amounts to the same thing, a "distortion" or "strain" of time
and space. The sun's "force," the "force" of any body in space, is the
"force" due to gravity; and these "forces" may now be treated in terms
of the laws of time and space. "The earth," Prof. Eddington tells us,
"moves in a curved orbit, not because the sun exerts any direct pull,
but because the earth is trying to find the shortest way through a
space and time which have been tangled up by an influence radiating
from the sun." [11]
At this point Newton's conceptions fail, for his views and his laws
do not include "strains" in space. Newton's law of gravitation must
be supplanted by one which does include such distortions. It is
Einstein's great glory to have supplied us with this new law.
Einstein's Law of Gravitation. This appears to be the only law
which meets all requirements. It includes Newton's law, and cannot be
distinguished from it if our experiments are confined to the earth and
deal with relatively small velocities. But when we betake ourselves
to some orbits in space, with a gravitational pull much greater than
the earth's, and when we deal with velocities comparable to that of
light, the differences become marked.
Einstein's Theory Scores Its First Great Victory. In the beginning
of this chapter we referred to the elaborate eclipse expedition sent
by the British to test the validity of Einstein's new theory of
gravitation. The British scientists would hardly have expended so
much time and energy on this theory of Einstein's but for the fact
that Einstein had already scored one great victory. What was it?
Imagine but a single planet revolving about the sun. According to
Newton's law of gravitation, the planet's path would be that of an
ellipse--that is, oval--and the planet would travel indefinitely along
this path. According to Einstein the path would also be elliptical,
but before a revolution would be quite completed, the planet would
start along a slightly advanced line, forming a new ellipse slightly in
advance of the first. The elliptic orbit slowly turns in the direction
in which the planet is moving. After many years--centuries--the orbit
will be in a different direction.
The rapidity of the orbit's change of direction depends on the velocity
of the planet. Mercury moving at the rate of 30 miles a second is the
fastest among the planets. It has the further advantage over Venus or
the earth in that its orbit, as we have said, is an ellipse, whereas
the orbits of Venus and the earth are nearly circular; and how are
you going to tell in which direction a circle is pointing?
Observation tells us that the orbit of Mercury is advancing at the
rate of 574 seconds (of arc) per century. We can calculate how much
of this is due to the gravitational influence of other planets. It
amounts to 532 seconds per century. What of the remaining 42 seconds?
You might be inclined to attribute this shortcoming to experimental
error. But when all such possibilities are allowed for our
mathematicians assure us that the discrepancy is 30 times greater
than any possible experimental error.
This discrepancy between theory and observation remained one
of the great puzzles in astronomy until Einstein cleared up the
mystery. According to Einstein's theory the mathematics of the
situation is simply this: in one revolution of the planet the orbit
will advance by a fraction of a revolution equal to three times the
square of the ratio of the velocity of the planet to the velocity
of light. When we allow mathematicians to work this out we get the
figure 43, which is certainly close enough to 42 to be called identical
with it.
Still Another Victory? Einstein's third prediction--the shifting of
spectral lines toward the red end of the spectrum in the case of light
coming to us from the stars of appreciable mass--seems to have been
confirmed recently (March, 1920). "The young physicists in Bonn,"
writes Prof. Einstein to a friend, "have now as good as certainty
(so gut wie sicher) proved the red displacement of the spectral lines
and have cleared up the grounds of a previous disappointment."
Summary. Velocity, or movement in space, is at the basis of Einstein's
work, as it was at the basis of Newton's. But time and space no
longer have the distinct meanings that they had when examined with
the help of Newton's equations. Time and space are not independent but
interdependent. They are meaningless when treated as separate entities,
giving results which may hold for one body in the universe but do not
hold for any other body. To get general laws which are applicable to
the cosmos as a whole the Fundamentals of Mechanics must be united.
Einstein's great achievement consists in applying this
revised conception of space and time to elucidate cosmical
problems. "World-lines," representing the progress of particles in
space, consisting of space-time combinations (the four dimensions),
are "strained" or "distorted" in space due to the attraction that
bodies exhibit for one another (the force of gravitation). On
the other hand, gravitation itself--more universal than anything
else in the universe--may be interpreted in terms of strains on
world-lines, or, what amounts to the same thing, strains of space-time
combinations. This brings gravitation within the field of Einstein's
conception of time and space.
That Einstein's conception of the universe is an improvement upon that
of Newton's is evidenced by the fact that Einstein's law explains all
that Newton's law does, and also other facts which Newton's law is
incapable of explaining. Among these may be mentioned the distortion
of the oval orbits of planets round the sun (confirmed in the case of
the planet Mercury), and the deviation of light rays in a gravitational
field (confirmed by the English Solar Eclipse Expedition).
Einstein's Theories and the Inferences to be Drawn from
Them. Einstein's theories, supported as they are by very convincing
experiments, will probably profoundly influence philosophic and
perhaps religious thought, but they can hardly be said to be of
immediate consequence to the man in the street. As I have said
elsewhere, Einstein's theories are not going to add one bushel of
wheat to war-torn and devastated Europe, but in their conception of
a cosmos decidedly at variance with anything yet conceived by any
school of philosophy, they will attract the universal attention of
thinking men in all countries. The scientist is immediately struck
by the way Einstein has utilized the various achievements in physics
and mathematics to build up a co-ordinated system showing connecting
links where heretofore none were perceived. The philosopher is equally
fascinated with a theory, which, in detail extremely complex, shows
a singular beauty of unity in design when viewed as a whole. The
revolutionary ideas propounded regarding time and space, the brilliant
way in which the most universal property of matter, gravitation,
is for the first time linked up with other properties of matter,
and, above all, the experimental confirmation of several of his
more startling predictions--always the finest test of scientific
merit--stamps Einstein as one of those super-men who from time to
time are sent to us to give us a peep into the beyond.
Some Facts about Einstein Himself. Albert Einstein was born in Germany
some 45 years ago. At first he was engaged at the Patent Bureau in
Berne, and later became professor at the Zürich Polytechnic. After
a short stay at Prague University he accepted one of those tempting
"Akademiker" professorships at the university of Berlin--professorships
which insure a comfortable income to the recipient of one of them,
little university work beyond, perhaps, one lecture a week, and
splendid facilities for research. A similar inducement enticed the
chemical philosopher, Van 't Hoff, to leave his Amsterdam, and the
Swedes came perilously near losing their most illustrious scientist,
Arrhenius.
Einstein published his first paper on relativity in 1905, when not
more than 30 years old. Of this paper Planck, the Nobel Laureate
in physics this year, has offered this opinion: "It surpasses in
boldness everything previously suggested in speculative natural
philosophy and even in the philosophical theories of knowledge. The
revolution introduced into the physical conceptions of the world is
only to be compared in extent and depth with that brought about by
the introduction of the Copernican system of the universe."
Einstein published a full exposition of the relativity theory in 1916.
During the momentous years of 1914-19, Einstein quietly pursued his
labors. There seems to be some foundation for the belief that the ways
of the German High Command found little favor in his eyes. At any
rate, he was not one of the forty professors who signed the famous
manifesto extolling Germany's aims. "We know for a fact," writes
Dr. O. A. Rankine, of the Imperial College of Science and Technology,
London, "that Einstein never was employed on war work. Whatever may
have been Germany's mistakes in other directions, she left her men
of science severely alone. In fact, they were encouraged to continue
in their normal occupations. Einstein undoubtedly received a large
measure of support from the Imperial Government, even when the German
armies were being driven back across Belgium."
Quite recently (June, 1920) the Barnard Medal of Columbia University
was conferred on him "in recognition of his highly original
and fruitful development of the fundamental concepts of physics
through application of mathematics." In acknowledging the honor,
Prof. Einstein wrote to President Butler that "... quite apart from
the personal satisfaction, I believe I may regard your decision [to
confer the medal upon him] as a harbinger of a better time in which
a sense of international solidarity will once more unite scholars of
the various countries."
REFERENCES
For those lacking all astronomical knowledge, an excellent plan would
be to read the first 40 pages of W. H. Snyder's Everyday Science
(Allyn and Bacon), in which may be found a clear and simple account
of the solar system. This could be followed with Bertrand Russell's
chapter on The Nature of Matter in his little volume, The Problems of
Philosophy (Henry Holt and Co.). Here the reader will be introduced
to the purely philosophical side of the question--quite a necessary
equipment for the understanding of Einstein's theory.
Of the non-mathematical articles which have appeared, those by
Prof. A. S. Eddington (Nature, volume 101, pages 15 and 34, 1918)
and Prof. M. R. Cohen (The New Republic, Jan. 21, 1920) are the best
which have come to the author's notice. Other articles on Einstein's
theory, some easily comprehensible, others somewhat confusing, and
still others full of noise and rather empty, are by H. A. Lorentz,
The New York Times, Dec. 21, 1919 (since reprinted in book form by
Brentano's, New York, 1920); J. Q. Stewart, Scientific American,
Jan. 3, 1920; E. Cunningham, Nature, volume 104, pages 354 and 374,
1919; F. H. Loring, Chemical News, volume 112, pages 226, 236, 248,
and 260, 1915; E. B. Wilson, Scientific Monthly, volume 10, page 217,
1920; J. S. Ames, Science, volume 51, page 253, 1920 [12]; L. A. Bauer,
Science, volume 51, page 301 (1920), and volume 51, page 581 (1920);
Sir Oliver Lodge, Scientific Monthly, volume 10, page 378, 1920;
E. E. Slosson, Independent, Nov. 29, Dec. 13, Dec. 20, Dec. 27,
1919 (since collected and published in book form by Harcourt,
Brace and Howe); Isabel M. Lewis, Electrical Experimenter, Jan.,
1920; A. J. Lotka, Harper's Magazine, March, 1920, page 477; and
R. D. Carmichael, New York Times, March 28, 1920. Einstein himself
is responsible for a brief article in English which first appeared
in the London Times, and was later reprinted in Science, volume 51,
page 8, 1920 (see the Appendix).
A number of books deal with the subject, and all of them are more
or less mathematical. However, in every one of these volumes certain
chapters, or portions of chapters, may be read with profit even by the
non-mathematical reader. Some of these books are: Erwin Freundlich,
The Foundations of Einstein's Theory of Gravitation (University Press,
Cambridge, 1920). (A very complete list of references--up to Feb.,
1920--is also given); A. S. Eddington, Report on the Relativity Theory
of Gravitation for the Physical Society of London (Fleetway Press,
Ltd., London, 1920); R. C. Tolman, Theory of the Relativity of Motion
(University of California Press, 1917); E. Cunningham, Relativity and
the Electron Theory (Longmans, Green and Co., 1915); R. D. Carmichael,
The Theory of Relativity (John Wiley and Sons, 1913); L. Silberstein,
The Theory of Relativity (Macmillan, 1914); and E. Cunningham, The
Principle of Relativity (University Press, Cambridge, England, 1914).
To those familiar with the German language Einstein's book, Über die
spezielle und die allgemeine Relativitätstheorie (Friedr. Vieweg und
Sohn, Braunschweig, 1920), may be recommended. [13]
The mathematical student may be referred to a volume incorporating
the more important papers of Einstein, Minkowski and Lorentz: Das
Relativitätsprinzip, (B. G. Teubner, Berlin, 1913).
Einstein's papers have appeared in the Annalen der Physik, Leipzig,
volume 17, page 132, 1905, volume 49, page 760, 1916, and volume 55,
page 241, 1918.
APPENDIX
Note 1 (page 21)
"On this earth there is indeed a tiny corner of the universe accessible
to other senses [than the sense of sight]: but feeling and taste act
only at those minute distances which separate particles of matter
when 'in contact:' smell ranges over, at the utmost, a mile or two,
and the greatest distance which sound is ever known to have traveled
(when Krakatoa exploded in 1883) is but a few thousand miles--a mere
fraction of the earth's girdle."--Prof. H. H. Turner of Oxford.
Note 2 (page 27)
Huyghens and Leibniz both objected to Newton's inverse square law
because it postulated "action at a distance,"--for example, the
attractive force of the sun and the earth. This desire for "continuity"
in physical laws led to the supposition of an "ether." We may here
anticipate and state that the reason which prompted Huyghens to object
to Newton's law led Einstein in our own day to raise objections to
the "ether" theory. "In the formulation of physical laws, only those
things were to be regarded as being in causal connection which were
capable of being actually observed." And the "ether" has not been
"actually observed."
The idea of "continuity" implies distances between adjacent points
that are infinitesimal in extent; hence the idea of "continuity"
comes in direct opposition with the finite distances of Newton.
The statement relating to causal connection--the refusal to accept an
"ether" as an absolute base of reference--leads to the principle of
the relativity of motion.
Note 3 (page 30)
Sir Oliver Lodge goes to the extreme of pinning his faith in the
reality of this ether rather than in that of matter. Witness the
following statement he made recently before a New York audience:
"To my mind the ether of space is a substantial reality with
extraordinarily perfect properties, with an immense amount of energy
stored up in it, with a constitution which we must discover, but a
substantial reality far more impressive than that of matter. Empty
space, as we call it, is full of ether, but it makes no appeal to
our senses. The appearance is as if it were nothing. It is the most
important thing in the material universe. I believe that matter
is a modification of ether, a very porous substance, a thing more
analogous to a cobweb or the Milky Way or something very slight and
unsubstantial, as compared to ether."
And again:
"The properties of ether seem to be perfect. Matter is less so; it has
friction and elasticity. No imperfection has been discovered in the
ether space. It doesn't wear out; there is no dissipation of energy;
there is no friction. Ether is material, yet it is not matter; both
are substantial realities in physics, but it is the ether of space
that holds things together and acts as a cement. My business is to
call attention to the whole world of etherealness of things, and I
have made it a subject of thirty years' study, but we must admit that
there is no getting hold of ether except indirectly."
"I consider the ether of space," says Lodge, in conclusion, "the one
substantial thing in the universe." And Lodge is certainly entitled
to his opinion.
Note 4 (page 51)
For the benefit of those readers who wish to gain a deeper insight
into the relativity principle, we shall here discuss it very briefly.
Newton and Galileo had developed a relativity principle in mechanics
which may be stated as follows: If one system of reference is in
uniform rectilinear motion with respect to another system of reference,
then whatever physical laws are deduced from the first system hold
true for the second system. The two systems are equivalent. If the
two systems be represented by $xyz$ and $x'y'z'$, and if they move
with the velocity of v along the x-axis with respect to one another,
then the two systems are mathematically related thus:
$$x' = x - vt, y' = y, z' = z, t' = t,$$
and this immediately provides us with a means of transforming the
laws of one system to those of another.
With the development of electrodynamics (which we may call electricity
in motion) difficulties arose which equations in mechanics of type
(1) could no longer solve. These difficulties merely increased when
Maxwell showed that light must be regarded as an electromagnetic
phenomenon. For suppose we wish to investigate the motion of a source
of light (which may be the equivalent of the motion of the earth
with reference to the sun) with respect to the velocity of the light
it emits--a typical example of the study of moving systems--how are
we to coordinate the electrodynamical and mechanical elements? Or,
again, suppose we wish to investigate the velocity of electrons shot
out from radium with a speed comparable to that of light, how are we
to coordinate the two branches in tracing the course of these negative
particles of electricity?
It was difficulties such as these that led to the Lorentz-Einstein
modifications of the Newton-Galileo relativity equations (1). The
Lorentz-Einstein equations are expressed in the form:
$$x' = \frac{x-vt}{\sqrt{1-\frac{v^2}{c}}}, y' = y, z' = z, t' =
\frac{t-\frac{v}{c^2}\cdot x}{\sqrt{1-\frac{v^2}{c^2}}},$$
c denoting the velocity of light in vacuo (which, according to all
observations, is the same, irrespective of the observer's state of
motion). Here, you see, electrodynamical systems (light and therefore
"ray" velocities such as those due to electrons) are brought into play.
This gives us Einstein's special theory of relativity. From it Einstein
deduced some startling conceptions of time and space.
Note 5 (page 55)
The velocity (v) of an object in one system will have a different
velocity (v') if referred to another system in uniform motion relative
to the first. It had been supposed that only a "something" endowed
with infinite velocity would show the same velocity in all systems,
irrespective of the motions of the latter. Michelson and Morley's
results actually point to the velocity of light as showing the
properties of the imaginary "infinite velocity." The velocity of
light possesses universal significance; and this is the basis for
much of Einstein's earlier work.
Note 6 (page 56)
"Euclid assumes that parallel lines never meet, which they cannot
do of course if they be defined as equidistant. But are there such
lines? And if not, why not assume that all lines drawn through a
point outside a given line will eventually intersect it? Such an
assumption leads to a geometry in which all lines are conceived as
being drawn on the surface of a sphere or an ellipse, and in it the
three angles of a triangle are never quite equal to two right angles,
nor the circumference of a circle quite [pi] times its diameter. But
that is precisely what the contraction effect due to motion requires."
(Dr. Walker)
Note 7 (page 57)
Einstein had become tired of assumptions. He had no particular
objection to the "ether" theory beyond the fact that this "ether"
did not come within the range of our senses; it could not be
"observed." "The consistent fulfilment of the two postulates--'action
by contact' and causal relationship between only such things
as lie within the realm of observation [see Note 2] combined
together is, I believe, the mainspring of Einstein's method of
investigation...." (Prof. Freundlich).
Note 8 (page 59)
That the conception of the "simultaneity" of events is devoid of
meaning can be deduced from equation (2) [see Note 4]. We owe the proof
to Einstein. "It is possible to select a suitable time-coordinate
in such a way that a time-measurement enters into physical laws
in exactly the same manner as regards its significance as a space
measurement (that is, they are fully equivalent symbolically), and
has likewise a definite coordinate direction.... It never occurred to
anyone that the use of a light-signal as a means of connection between
the moving-body and the observer, which is necessary in practice in
order to determine simultaneity, might affect the final result, i.e.,
of time measurements in different systems." (Freundlich). But that
is just what Einstein shows, because time-measurements are based on
"simultaneity of events," and this, as pointed out above, is devoid
of meaning.
Had the older masters the occasion to study enormous velocities, such
as the velocity of light, rather than relatively small ones--and even
the velocity of the earth around the sun is small as compared to the
velocity of light--discrepancies between theory and experiment would
have become apparent.
Note 9 (page 67)
How the special theory of relativity (see Note 4) led to the general
theory of relativity (which included gravitation) may now be briefly
traced.
When we speak of electrons, or negative particles of electricity,
in motion, we are speaking of energy in motion. Now these electrons
when in motion exhibit properties that are very similar to matter in
motion. Whatever deviations there are are due to the enormous velocity
of these electrons, and this velocity, as has already been pointed
out, is comparable to that of light; whereas before the advent of the
electron, the velocity of no particles comparable to that of light
had ever been measured.
According to present views "all inertia of matter consists only of the
inertia of the latent energy in it; ... everything that we know of the
inertia of energy holds without exception for the inertia of matter."
Now it is on the assumption that inertial mass and gravitational
"pull" are equivalent that the mass of a body is determined by its
weight. What is true of matter should be true of energy.
The special theory of relativity, however, takes into account only
inertia ("inertial mass") but not gravitation (gravitational pull or
weight) of energy. When a body absorbs energy equation 2 (see Note 4)
will record a gain in inertia but not in weight--which is contrary
to one of the fundamental facts in mechanics.
This means that a more general theory of relativity is required to
include gravitational phenomena. Hence Einstein's General Theory of
Relativity. Hence the approach to a new theory of gravitation. Hence
"the setting up of a differential equation which comprises the
motion of a body under the influence of both inertia and gravity,
and which symbolically expresses the relativity of motions.... The
differential law must always preserve the same form, irrespective of
the system of coordinates to which it is referred, so that no system of
coordinates enjoys a preference to any other." (For the general form
of the equation and for an excellent discussion of its significance,
see Freundlich's monograph, pages 27-33.)
TIME, SPACE, AND GRAVITATION [14]
By Prof. Albert Einstein
There are several kinds of theory in physics. Most of them are
constructive. These attempt to build a picture of complex phenomena out
of some relatively simple proposition. The kinetic theory of gases,
for instance, attempts to refer to molecular movement the mechanical
thermal, and diffusional properties of gases. When we say that we
understand a group of natural phenomena, we mean that we have found
a constructive theory which embraces them.
Theories of Principle.--But in addition to this most weighty group of
theories, there is another group consisting of what I call theories of
principle. These employ the analytic, not the synthetic method. Their
starting-point and foundation are not hypothetical constituents, but
empirically observed general properties of phenomena, principles from
which mathematical formulæ are deduced of such a kind that they apply
to every case which presents itself. Thermodynamics, for instance,
starting from the fact that perpetual motion never occurs in ordinary
experience, attempts to deduce from this, by analytic processes,
a theory which will apply in every case. The merit of constructive
theories is their comprehensiveness, adaptability, and clarity,
that of the theories of principle, their logical perfection, and the
security of their foundation.
The theory of relativity is a theory of principle. To understand it,
the principles on which it rests must be grasped. But before stating
these it is necessary to point out that the theory of relativity is
like a house with two separate stories, the special relativity theory
and the general theory of relativity.
Since the time of the ancient Greeks it has been well known that in
describing the motion of a body we must refer to another body. The
motion of a railway train is described with reference to the ground,
of a planet with reference to the total assemblage of visible fixed
stars. In physics the bodies to which motions are spatially referred
are termed systems of coordinates. The laws of mechanics of Galileo
and Newton can be formulated only by using a system of coordinates.
The state of motion of a system of coordinates can not be chosen
arbitrarily if the laws of mechanics are to hold good (it must be
free from twisting and from acceleration). The system of coordinates
employed in mechanics is called an inertia-system. The state of
motion of an inertia-system, so far as mechanics are concerned,
is not restricted by nature to one condition. The condition in the
following proposition suffices; a system of coordinates moving in the
same direction and at the same rate as a system of inertia is itself
a system of inertia. The special relativity theory is therefore the
application of the following proposition to any natural process:
"Every law of nature which holds good with respect to a coordinate
system K must also hold good for any other system K' provided that
K and K' are in uniform movement of translation."
The second principle on which the special relativity theory rests is
that of the constancy of the velocity of light in a vacuum. Light
in a vacuum has a definite and constant velocity, independent of
the velocity of its source. Physicists owe their confidence in this
proposition to the Maxwell-Lorentz theory of electro-dynamics.
The two principles which I have mentioned have received strong
experimental confirmation, but do not seem to be logically
compatible. The special relativity theory achieved their logical
reconciliation by making a change in kinematics, that is to say,
in the doctrine of the physical laws of space and time. It became
evident that a statement of the coincidence of two events could have
a meaning only in connection with a system of coordinates, that the
mass of bodies and the rate of movement of clocks must depend on
their state of motion with regard to the coordinates.
The Older Physics.--But the older physics, including the laws of
motion of Galileo and Newton, clashed with the relativistic kinematics
that I have indicated. The latter gave origin to certain generalized
mathematical conditions with which the laws of nature would have to
conform if the two fundamental principles were compatible. Physics had
to be modified. The most notable change was a new law of motion for
(very rapidly) moving mass-points, and this soon came to be verified
in the case of electrically-laden particles. The most important result
of the special relativity system concerned the inert mass of a material
system. It became evident that the inertia of such a system must depend
on its energy-content, so that we were driven to the conception that
inert mass was nothing else than latent energy. The doctrine of the
conservation of mass lost its independence and became merged in the
doctrine of conservation of energy.
The special relativity theory which was simply a systematic extension
of the electro-dynamics of Maxwell and Lorentz, had consequences which
reached beyond itself. Must the independence of physical laws with
regard to a system of coordinates be limited to systems of coordinates
in uniform movement of translation with regard to one another? What has
nature to do with the coordinate systems that we propose and with their
motions? Although it may be necessary for our descriptions of nature
to employ systems of coordinates that we have selected arbitrarily,
the choice should not be limited in any way so far as their state of
motion is concerned. (General theory of relativity.) The application
of this general theory of relativity was found to be in conflict
with a well-known experiment, according to which it appeared that
the weight and the inertia of a body depended on the same constants
(identity of inert and heavy masses). Consider the case of a system of
coordinates which is conceived as being in stable rotation relative
to a system of inertia in the Newtonian sense. The forces which,
relatively to this system, are centrifugal must, in the Newtonian
sense, be attributed to inertia. But these centrifugal forces are,
like gravitation, proportional to the mass of the bodies. Is it not,
then, possible to regard the system of coordinates as at rest, and
the centrifugal forces as gravitational? The interpretation seemed
obvious, but classical mechanics forbade it.
This slight sketch indicates how a generalized theory of relativity
must include the laws of gravitation, and actual pursuit of the
conception has justified the hope. But the way was harder than
was expected, because it contradicted Euclidian geometry. In other
words, the laws according to which material bodies are arranged in
space do not exactly agree with the laws of space prescribed by the
Euclidian geometry of solids. This is what is meant by the phrase
"a warp in space." The fundamental concepts "straight," "plane,"
etc., accordingly lose their exact meaning in physics.
In the generalized theory of relativity, the doctrine of space and
time, kinematics, is no longer one of the absolute foundations of
general physics. The geometrical states of bodies and the rates
of clocks depend in the first place on their gravitational fields,
which again are produced by the material system concerned.
Thus the new theory of gravitation diverges widely from that of Newton
with respect to its basal principle. But in practical application
the two agree so closely that it has been difficult to find cases in
which the actual differences could be subjected to observation. As
yet only the following have been suggested:
1. The distortion of the oval orbits of planets round the sun
(confirmed in the case of the planet Mercury).
2. The deviation of light-rays in a gravitational field (confirmed
by the English Solar Eclipse expedition).
3. The shifting of spectral lines towards the red end of the spectrum
in the case of light coming to us from stars of appreciable mass
(not yet confirmed).
The great attraction of the theory is its logical consistency. If
any deduction from it should prove untenable, it must be given up. A
modification of it seems impossible without destruction of the whole.
No one must think that Newton's great creation can be overthrown in
any real sense by this or by any other theory. His clear and wide
ideas will for ever retain their significance as the foundation on
which our modern conceptions of physics have been built.
EINSTEIN'S LAW OF GRAVITATION [15]
By Prof. J. S. Ames
Johns Hopkins University
... In the treatment of Maxwell's equations of the electromagnetic
field, several investigators realized the importance of deducing the
form of the equations when applied to a system moving with a uniform
velocity. One object of such an investigation would be to determine
such a set of transformation formulæ as would leave the mathematical
form of the equations unaltered. The necessary relations between
the new space-coordinates, those applying to the moving system,
and the original set were of course obvious; and elementary methods
led to the deduction of a new variable which should replace the time
coordinate. This step was taken by Lorentz and also, I believe, by
Larmor and by Voigt. The mathematical deductions and applications
in the hands of these men were extremely beautiful, and are probably
well known to you all.
Lorentz' paper on this subject appeared in the Proceedings of the
Amsterdam Academy in 1904. In the following year there was published
in the Annalen der Physik a paper by Einstein, written without any
knowledge of the work of Lorentz, in which he arrived at the same
transformation equations as did the latter, but with an entirely
different and fundamentally new interpretation. Einstein called
attention in his paper to the lack of definiteness in the concepts
of time and space, as ordinarily stated and used. He analyzed clearly
the definitions and postulates which were necessary before one could
speak with exactness of a length or of an interval of time. He disposed
forever of the propriety of speaking of the "true" length of a rod or
of the "true" duration of time, showing, in fact, that the numerical
values which we attach to lengths or intervals of time depend upon the
definitions and postulates which we adopt. The words "absolute" space
or time intervals are devoid of meaning. As an illustration of what
is meant Einstein discussed two possible ways of measuring the length
of a rod when it is moving in the direction of its own length with
a uniform velocity, that is, after having adopted a scale of length,
two ways of assigning a number to the length of the rod concerned. One
method is to imagine the observer moving with the rod, applying along
its length the measuring scale, and reading off the positions of the
ends of the rod. Another method would be to have two observers at rest
on the body with reference to which the rod has the uniform velocity,
so stationed along the line of motion of the rod that as the rod
moves past them they can note simultaneously on a stationary measuring
scale the positions of the two ends of the rod. Einstein showed that,
accepting two postulates which need no defense at this time, the two
methods of measurements would lead to different numerical values, and,
further, that the divergence of the two results would increase as the
velocity of the rod was increased. In assigning a number, therefore,
to the length of a moving rod, one must make a choice of the method to
be used in measuring it. Obviously the preferable method is to agree
that the observer shall move with the rod, carrying his measuring
instrument with him. This disposes of the problem of measuring space
relations. The observed fact that, if we measure the length of the rod
on different days, or when the rod is lying in different positions,
we always obtain the same value offers no information concerning the
"real" length of the rod. It may have changed, or it may not. It
must always be remembered that measurement of the length of a
rod is simply a process of comparison between it and an arbitrary
standard, e.g., a meter-rod or yard-stick. In regard to the problem
of assigning numbers to intervals of time, it must be borne in mind
that, strictly speaking, we do not "measure" such intervals, i.e.,
that we do not select a unit interval of time and find how many times
it is contained in the interval in question. (Similarly, we do not
"measure" the pitch of a sound or the temperature of a room.) Our
practical instruments for assigning numbers to time-intervals depend
in the main upon our agreeing to believe that a pendulum swings in
a perfectly uniform manner, each vibration taking the same time as
the next one. Of course we cannot prove that this is true, it is,
strictly speaking, a definition of what we mean by equal intervals
of time; and it is not a particularly good definition at that. Its
limitations are sufficiently obvious. The best way to proceed is
to consider the concept of uniform velocity, and then, using the
idea of some entity having such a uniform velocity, to define equal
intervals of time as such intervals as are required for the entity
to traverse equal lengths. These last we have already defined. What
is required in addition is to adopt some moving entity as giving our
definition of uniform velocity. Considering our known universe it
is self-evident that we should choose in our definition of uniform
velocity the velocity of light, since this selection could be made by
an observer anywhere in our universe. Having agreed then to illustrate
by the words "uniform velocity" that of light, our definition of equal
intervals of time is complete. This implies, of course, that there is
no uncertainty on our part as to the fact that the velocity of light
always has the same value at any one point in the universe to any
observer, quite regardless of the source of light. In other words,
the postulate that this is true underlies our definition. Following
this method Einstein developed a system of measuring both space and
time intervals. As a matter of fact his system is identically that
which we use in daily life with reference to events here on the
earth. He further showed that if a man were to measure the length
of a rod, for instance, on the earth and then were able to carry the
rod and his measuring apparatus to Mars, the sun, or to Arcturus he
would obtain the same numerical value for the length in all places
and at all times. This doesn't mean that any statement is implied
as to whether the length of the rod has remained unchanged or not;
such words do not have any meaning--remember that we can not speak of
true length. It is thus clear that an observer living on the earth
would have a definite system of units in terms of which to express
space and time intervals, i.e., he would have a definite system
of space coordinates (x, y, z) and a definite time coordinate (t);
and similarly an observer living on Mars would have his system of
coordinates (x', y', z', t'). Provided that one observer has a definite
uniform velocity with reference to the other, it is a comparatively
simple matter to deduce the mathematical relations between the two
sets of coordinates. When Einstein did this, he arrived at the same
transformation formulæ as those used by Lorentz in his development of
Maxwell's equations. The latter had shown that, using these formulæ,
the form of the laws for all electromagnetic phenomena maintained
the same form; so Einstein's method proves that using his system of
measurement an observer, anywhere in the universe, would as the result
of his own investigation of electromagnetic phenomena arrive at the
same mathematical statement of them as any other observer, provided
only that the relative-velocity of the two observers was uniform.
Einstein discussed many other most important questions at this time;
but it is not necessary to refer to them in connection with the
present subject. So far as this is concerned, the next important
step to note is that taken in the famous address of Minkowski, in
1908, on the subject of "Space and Time." It would be difficult to
overstate the importance of the concepts advanced by Minkowski. They
marked the beginning of a new period in the philosophy of physics. I
shall not attempt to explain his ideas in detail, but shall confine
myself to a few general statements. His point of view and his line of
development of the theme are absolutely different from those of Lorentz
or of Einstein; but in the end he makes use of the same transformation
formulæ. His great contribution consists in giving us a new geometrical
picture of their meaning. It is scarcely fair to call Minkowski's
development a picture; for to us a picture can never have more than
three dimensions, our senses limit us; while his picture calls for
perception of four dimensions. It is this fact that renders any even
semi-popular discussion of Minkowski's work so impossible. We can all
see that for us to describe any event a knowledge of four coordinates
is necessary, three for the space specification and one for the time. A
complete picture could be given then by a point in four dimensions. All
four coordinates are necessary: we never observe an event except at
a certain time, and we never observe an instant of time except with
reference to space. Discussing the laws of electromagnetic phenomena,
Minkowski showed how in a space of four dimensions, by a suitable
definition of axes, the mathematical transformation of Lorentz and
Einstein could be described by a rotation of the set of axes. We are
all accustomed to a rotation of our ordinary cartesian set of axes
describing the position of a point. We ordinarily choose our axes at
any location on the earth as follows: one vertical, one east and west,
one north and south. So if we move from any one laboratory to another,
we change our axes; they are always orthogonal, but in moving from
place to place there is a rotation. Similarly, Minkowski showed that
if we choose four orthogonal axes at any point on the earth, according
to his method, to represent a space-time point using the method of
measuring space and time intervals as outlined by Einstein; and, if
an observer on Arcturus used a similar set of axes and the method of
measurement which he naturally would, the set of axes of the latter
could be obtained from those of the observer on the earth by a pure
rotation (and naturally a transfer of the origin). This is a beautiful
geometrical result. To complete my statement of the method, I must
add that instead of using as his fourth axis one along which numerical
values of time are laid off, Minkowski defined his fourth coordinate
as the product of time and the imaginary constant, the square root of
minus one. This introduction of imaginary quantities might be expected,
possibly, to introduce difficulties; but, in reality, it is the very
essence of the simplicity of the geometrical description just given
of the rotation of the sets of axes. It thus appears that different
observers situated at different points in the universe would each have
their own set of axes, all different, yet all connected by the fact
that any one can be rotated so as to coincide with any other. This
means that there is no one direction in the four-dimensional space
that corresponds to time for all observers. Just as with reference to
the earth there is no direction which can be called vertical for all
observers living on the earth. In the sense of an absolute meaning
the words "up and down," "before and after," "sooner or later,"
are entirely meaningless.
This concept of Minkowski's may be made clearer, perhaps, by the
following process of thought. If we take a section through our
three-dimensional space, we have a plane, i.e., a two-dimensional
space. Similarly, if a section is made through a four-dimensional
space, one of three dimensions is obtained. Thus, for an observer on
the earth a definite section of Minkowski's four-dimensional space will
give us our ordinary three-dimensional one; so that this section will,
as it were, break up Minkowski's space into our space and give us our
ordinary time. Similarly, a different section would have to be used
to the observer on Arcturus; but by a suitable selection he would
get his own familiar three-dimensional space and his own time. Thus
the space defined by Minkowski is completely isotropic in reference
to measured lengths and times, there is absolutely no difference
between any two directions in an absolute sense; for any particular
observer, of course, a particular section will cause the space to
fall apart so as to suit his habits of measurement; any section,
however, taken at random will do the same thing for some observer
somewhere. From another point of view, that of Lorentz and Einstein,
it is obvious that, since this four-dimensional space is isotropic,
the expression of the laws of electromagnetic phenomena take identical
mathematical forms when expressed by any observer.
The question of course must be raised as to what can be said in regard
to phenomena which so far as we know do not have an electromagnetic
origin. In particular what can be done with respect to gravitational
phenomena? Before, however, showing how this problem was attacked by
Einstein; and the fact that the subject of my address is Einstein's
work on gravitation shows that ultimately I shall explain this, I
must emphasize another feature of Minkowski's geometry. To describe
the space-time characteristics of any event a point, defined by its
four coordinates, is sufficient; so, if one observes the life-history
of any entity, e.g., a particle of matter, a light-wave, etc., he
observes a sequence of points in the space-time continuum; that is,
the life-history of any entity is described fully by a line in this
space. Such a line was called by Minkowski a "world-line." Further,
from a different point of view, all of our observations of nature
are in reality observations of coincidences, e.g., if one reads
a thermometer, what he does is to note the coincidence of the
end of the column of mercury with a certain scale division on
the thermometer tube. In other words, thinking of the world-line
of the end of the mercury column and the world-line of the scale
division, what we have observed was the intersection or crossing of
these lines. In a similar manner any observation may be analyzed;
and remembering that light rays, a point on the retina of the eye,
etc., all have their world-lines, it will be recognized that it is
a perfectly accurate statement to say that every observation is the
perception of the intersection of world-lines. Further, since all we
know of a world-line is the result of observations, it is evident
that we do not know a world-line as a continuous series of points,
but simply as a series of discontinuous points, each point being where
the particular world-line in question is crossed by another world-line.
It is clear, moreover, that for the description of a world-line
we are not limited to the particular set of four orthogonal axes
adopted by Minkowski. We can choose any set of four-dimensional
axes we wish. It is further evident that the mathematical expression
for the coincidence of two points is absolutely independent of our
selection of reference axes. If we change our axes, we will change
the coordinates of both points simultaneously, so that the question
of axes ceases to be of interest. But our so-called laws of nature are
nothing but descriptions in mathematical language of our observations;
we observe only coincidences; a sequence of coincidences when put in
mathematical terms takes a form which is independent of the selection
of reference axes; therefore the mathematical expression of our laws
of nature, of every character, must be such that their form does
not change if we make a transformation of axes. This is a simple but
far-reaching deduction.
There is a geometrical method of picturing the effect of a change
of axes of reference, i.e., of a mathematical transformation. To a
man in a railway coach the path of a drop of water does not appear
vertical, i.e., it is not parallel to the edge of the window; still
less so does it appear vertical to a man performing manoeuvres in
an airplane. This means that whereas with reference to axes fixed
to the earth the path of the drop is vertical; with reference to
other axes, the path is not. Or, stating the conclusion in general
language, changing the axes of reference (or effecting a mathematical
transformation) in general changes the shape of any line. If one
imagines the line forming a part of the space, it is evident that
if the space is deformed by compression or expansion the shape of
the line is changed, and if sufficient care is taken it is clearly
possible, by deforming the space, to make the line take any shape
desired, or better stated, any shape specified by the previous change
of axes. It is thus possible to picture a mathematical transformation
as a deformation of space. Thus I can draw a line on a sheet of paper
or of rubber and by bending and stretching the sheet, I can make the
line assume a great variety of shapes; each of these new shapes is
a picture of a suitable transformation.
Now, consider world-lines in our four-dimensional space. The
complete record of all our knowledge is a series of sequences of
intersections of such lines. By analogy I can draw in ordinary space
a great number of intersecting lines on a sheet of rubber; I can
then bend and deform the sheet to please myself; by so doing I do
not introduce any new intersections nor do I alter in the least the
sequence of intersections. So in the space of our world-lines, the
space may be deformed in any imaginable manner without introducing
any new intersections or changing the sequence of the existing
intersections. It is this sequence which gives us the mathematical
expression of our so-called experimental laws; a deformation of
our space is equivalent mathematically to a transformation of axes,
consequently we see why it is that the form of our laws must be the
same when referred to any and all sets of axes, that is, must remain
unaltered by any mathematical transformation.
Now, at last we come to gravitation. We can not imagine any world-line
simpler than that of a particle of matter left to itself; we shall
therefore call it a "straight" line. Our experience is that two
particles of matter attract one another. Expressed in terms of
world-lines, this means that, if the world-lines of two isolated
particles come near each other, the lines, instead of being straight,
will be deflected or bent in towards each other. The world-line of
any one particle is therefore deformed; and we have just seen that a
deformation is the equivalent of a mathematical transformation. In
other words, for any one particle it is possible to replace the
effect of a gravitational field at any instant by a mathematical
transformation of axes. The statement that this is always possible
for any particle at any instant is Einstein's famous "Principle
of Equivalence."
Let us rest for a moment, while I call attention to a most interesting
coincidence, not to be thought of as an intersection of world-lines. It
is said that Newton's thoughts were directed to the observation of
gravitational phenomena by an apple falling on his head; from this
striking event he passed by natural steps to a consideration of the
universality of gravitation. Einstein in describing his mental process
in the evolution of his law of gravitation says that his attention
was called to a new point of view by discussing his experiences with
a man whose fall from a high building he had just witnessed. The man
fortunately suffered no serious injuries and assured Einstein that in
the course of his fall he had not been conscious in the least of any
pull downward on his body. In mathematical language, with reference to
axes moving with the man the force of gravity had disappeared. This is
a case where by the transfer of the axes from the earth itself to the
man, the force of the gravitational field is annulled. The converse
change of axes from the falling man to a point on the earth could be
considered as introducing the force of gravity into the equations of
motion. Another illustration of the introduction into our equations of
a force by a means of a change of axes is furnished by the ordinary
treatment of a body in uniform rotation about an axis. For instance,
in the case of a so-called conical pendulum, that is, the motion of a
bob suspended from a fixed point by string, which is so set in motion
that the bob describes a horizontal circle and the string therefore
describes a circular cone, if we transfer our axes from the earth and
have them rotate around the vertical line through the fixed point with
the same angular velocity as the bob, it is necessary to introduce into
our equations of motion a fictitious "force" called the centrifugal
force. No one ever thinks of this force other than as a mathematical
quantity introduced into the equations for the sake of simplicity of
treatment; no physical meaning is attached to it. Why should there
be to any other so-called "force," which like centrifugal force,
is independent of the nature of the matter? Again, here on the earth
our sensation of weight is interpreted mathematically by combining
expressions for centrifugal force and gravity; we have no distinct
sensation for either separately. Why then is there any difference in
the essence of the two? Why not consider them both as brought into
our equations by the agency of mathematical transformations? This is
Einstein's point of view.
Granting, then, the principle of equivalence, we can so choose axes at
any point at any instant that the gravitational field will disappear;
these axes are therefore of what Eddington calls the "Galilean"
type, the simplest possible. Consider, that is, an observer in a
box, or compartment, which is falling with the acceleration of the
gravitational field at that point. He would not be conscious of the
field. If there were a projectile fired off in this compartment,
the observer would describe its path as being straight. In this space
the infinitesimal interval between two space-time points would then
be given by the formula
$$ds^2 = dx^2_1 + dx2_2 + dx^2_3 + dx2_4,$$
where ds is the interval and $x_1, x_2, x_3, x_4$ are coordinates. If
we make a mathematical transformation, i.e., use another set of axes,
this interval would obviously take the form
$$ds^2 = g_{11}dx^2_{33} + g_{22}dx^2_2 + g_{33}dx^2_3 +
g_{44}dx2_4 + 2g_{12}dx_1dx_2 + \rm{etc.},$$
where $x_1, x_2, x_3$ and $x_4$ are now coordinates referring to the
new axes. This relation involves ten coefficients, the coefficients
defining the transformation.
But of course a certain dynamical value is also attached to the
g's, because by the transfer of our axes from the Galilean type
we have made a change which is equivalent to the introduction of
a gravitational field; and the g's must specify the field. That
is, these g's are the expressions of our experiences, and hence
their values can not depend upon the use of any special axes; the
values must be the same for all selections. In other words, whatever
function of the coordinates any one g is for one set of axes, if other
axes are chosen, this g must still be the same function of the new
coordinates. There are ten g's defined by differential equations;
so we have ten covariant equations. Einstein showed how these g's
could be regarded as generalized potentials of the field. Our own
experiments and observations upon gravitation have given us a certain
knowledge concerning its potential; that is, we know a value for it
which must be so near the truth that we can properly call it at least
a first approximation. Or, stated differently, if Einstein succeeds in
deducing the rigid value for the gravitational potential in any field,
it must degenerate to the Newtonian value for the great majority of
cases with which we have actual experience. Einstein's method, then,
was to investigate the functions (or equations) which would satisfy
the mathematical conditions just described. A transformation from
the axes used by the observer in the following box may be made so as
to introduce into the equations the gravitational field recognized
by an observer on the earth near the box; but this, obviously, would
not be the general gravitational field, because the field changes as
one moves over the surface of the earth. A solution found, therefore,
as just indicated, would not be the one sought for the general field;
and another must be found which is less stringent than the former
but reduces to it as a special case. He found himself at liberty to
make a selection from among several possibilities, and for several
reasons chose the simplest solution. He then tested this decision
by seeing if his formulæ would degenerate to Newton's law for the
limiting case of velocities small when compared with that of light,
because this condition is satisfied in those cases to which Newton's
law applies. His formulæ satisfied this test, and he therefore was
able to announce a "law of gravitation," of which Newton's was a
special form for a simple case.
To the ordinary scholar the difficulties surmounted by Einstein in
his investigations appear stupendous. It is not improbable that the
statement which he is alleged to have made to his editor, that only
ten men in the world could understand his treatment of the subject,
is true. I am fully prepared to believe it, and wish to add that I
certainly am not one of the ten. But I can also say that, after a
careful and serious study of his papers, I feel confident that there
is nothing in them which I can not understand, given the time to become
familiar with the special mathematical processes used. The more I work
over Einstein's papers, the more impressed I am, not simply by his
genius in viewing the problem, but also by his great technical skill.
Following the path outlined, Einstein, as just said, arrived at certain
mathematical laws for a gravitational field, laws which reduced
to Newton's form in most cases where observations are possible,
but which led to different conclusions in a few cases, knowledge
concerning which we might obtain by careful observations. I shall
mention a few deductions from Einstein's formulæ.
1. If a heavy particle is put at the center of a circle, and, if
the length of the circumference and the length of the diameter
are measured, it will be found that their ratio is not [pi]
(3.14159). In other words the geometrical properties of space in such
a gravitational field are not those discussed by Euclid; the space is,
then, non-Euclidean. There is no way by which this deduction can be
verified, the difference between the predicted ratio and [pi] is
too minute for us to hope to make our measurements with sufficient
exactness to determine the difference.
2. All the lines in the solar spectrum should with reference to lines
obtained by terrestrial sources be displaced slightly towards longer
wave-lengths. The amount of displacement predicted for lines in the
blue end of the spectrum is about one-hundredth of an Angstrom unit,
a quantity well within experimental limits. Unfortunately, as far
as the testing of this prediction is concerned, there are several
physical causes which are also operating to cause displacement of the
spectrum-lines; and so at present a decision can not be rendered as
to the verification. St. John and other workers at the Mount Wilson
Observatory have the question under investigation.
3. According to Newton's law an isolated planet in its motion around a
central sun would describe, period after period, the same elliptical
orbit; whereas Einstein's laws lead to the prediction that the
successive orbits traversed would not be identically the same. Each
revolution would start the planet off on an orbit very approximately
elliptical, but with the major axis of the ellipse rotated slightly in
the plane of the orbit. When calculations were made for the various
planets in our solar system, it was found that the only one which
was of interest from the standpoint of verification of Einstein's
formulæ was Mercury. It has been known for a long time that there
was actually such a change as just described in the orbit of Mercury,
amounting to 574'' of arc per century; and it has been shown that of
this a rotation of 532'' was due to the direct action of other planets,
thus leaving an unexplained rotation of 42'' per century. Einstein's
formulæ predicted a rotation of 43'', a striking agreement.
4. In accordance with Einstein's formulæ a ray of light passing close
to a heavy piece of matter, the sun, for instance, should experience
a sensible deflection in towards the sun. This might be expected from
"general" consideration of energy in motion; energy and mass are
generally considered to be identical in the sense that an amount
of energy E has the mass $E1c^2$ where c is the velocity of light;
and consequently a ray of light might fall within the province of
gravitation and the amount of deflection to be expected could be
calculated by the ordinary formula for gravitation. Another point
of view is to consider again the observer inside the compartment
falling with the acceleration of the gravitational field. To him the
path of a projectile and a ray of light would both appear straight;
so that, if the projectile had a velocity equal to that of light, it
and the light wave would travel side by side. To an observer outside
the compartment, e.g., to one on the earth, both would then appear
to have the same deflection owing to the sun. But how much would
the path of the projectile be bent? What would be the shape of its
parabola? One might apply Newton's law; but, according to Einstein's
formulæ, Newton's law should be used only for small velocities. In the
case of a ray passing close to the sun it was decided that according
to Einstein's formula there should be a deflection of 1''.75 whereas
Newton's law of gravitation predicted half this amount. Careful
plans were made by various astronomers, to investigate this question
at the solar eclipse last May, and the result announced by Dyson,
Eddington and Crommelin, the leaders of astronomy in England, was
that there was a deflection of 1''.9. Of course the detection of such
a minute deflection was an extraordinarily difficult matter, so many
corrections had to be applied to the original observations; but the
names of the men who record the conclusions are such as to inspire
confidence. Certainly any effect of refraction seems to be excluded.
It is thus seen that the formulæ deduced by Einstein have been
confirmed in a variety of ways and in a most brilliant manner. In
connection with these formulæ one question must arise in the minds of
everyone; by what process, where in the course of the mathematical
development, does the idea of mass reveal itself? It was not in
the equations at the beginning and yet here it is at the end. How
does it appear? As a matter of fact it is first seen as a constant
of integration in the discussion of the problem of the gravitational
field due to a single particle; and the identity of this constant with
mass is proved when one compares Einstein's formulæ with Newton's
law which is simply its degenerated form. This mass, though, is the
mass of which we become aware through our experiences with weight;
and Einstein proceeded to prove that this quantity which entered as
a constant of integration in his ideally simple problem also obeyed
the laws of conservation of mass and conservation of momentum when
he investigated the problems of two and more particles. Therefore
Einstein deduced from his study of gravitational fields the well-known
properties of matter which form the basis of theoretical mechanics. A
further logical consequence of Einstein's development is to show that
energy has mass, a concept with which every one nowadays is familiar.
The description of Einstein's method which I have given so far is
simply the story of one success after another; and it is certainly
fair to ask if we have at last reached finality in our investigation
of nature, if we have attained to truth. Are there no outstanding
difficulties? Is there no possibility of error? Certainly, not
until all the predictions made from Einstein's formulæ have been
investigated can much be said; and further, it must be seen whether
any other lines of argument will lead to the same conclusions. But
without waiting for all this there is at least one difficulty which
is apparent at this time. We have discussed the laws of nature as
independent in their form of reference axes, a concept which appeals
strongly to our philosophy; yet it is not at all clear, at first sight,
that we can be justified in our belief. We can not imagine any way
by which we can become conscious of the translation of the earth in
space; but by means of gyroscopes we can learn a great deal about its
rotation on its axis. We could locate the positions of its two poles,
and by watching a Foucault pendulum or a gyroscope we can obtain a
number which we interpret as the angular velocity of rotation of axes
fixed in the earth; angular velocity with reference to what? Where
is the fundamental set of axes? This is a real difficulty. It can be
surmounted in several ways. Einstein himself has outlined a method
which in the end amounts to assuming the existence on the confines
of space of vast quantities of matter, a proposition which is not
attractive. deSitter has suggested a peculiar quality of the space
to which we refer our space-time coordinates. The consequences of
this are most interesting, but no decision can as yet be made as to
the justification of the hypothesis. In any case we can say that
the difficulty raised is not one that destroys the real value of
Einstein's work.
In conclusion I wish to emphasize the fact, which should be obvious,
that Einstein has not attempted any explanation of gravitation;
he has been occupied with the deduction of its laws. These laws,
together with those of electromagnetic phenomena, comprise our store
of knowledge. There is not the slightest indication of a mechanism,
meaning by that a picture in terms of our senses. In fact what we have
learned has been to realize that our desire to use such mechanisms
is futile.
THE DEFLECTION OF LIGHT BY GRAVITATION AND THE EINSTEIN THEORY OF
RELATIVITY. [16]
Sir Frank Dyson
the Astronomer Royal
The purpose of the expedition was to determine whether any displacement
is caused to a ray of light by the gravitational field of the sun,
and if so, the amount of the displacement. Einstein's theory predicted
a displacement varying inversely as the distance of the ray from
the sun's center, amounting to 1''.75 for a star seen just grazing
the sun....
A study of the conditions of the 1919 eclipse showed that the sun
would be very favorably placed among a group of bright stars--in fact,
it would be in the most favorable possible position. A study of the
conditions at various points on the path of the eclipse, in which
Mr. Hinks helped us, pointed to Sobral, in Brazil, and Principe, an
island off the west coast of Africa, as the most favorable stations....
The Greenwich party, Dr. Crommelin and Mr. Davidson, reached Brazil
in ample time to prepare for the eclipse, and the usual preliminary
focusing by photographing stellar fields was carried out. The day of
the eclipse opened cloudy, but cleared later, and the observations
were carried out with almost complete success. With the astrographic
telescope Mr. Davidson secured 15 out of 18 photographs showing the
required stellar images. Totality lasted 6 minutes, and the average
exposure of the plates was 5 to 6 seconds. Dr. Crommelin with the
other lens had 7 successful plates out of 8. The unsuccessful plates
were spoiled for this purpose by the clouds, but show the remarkable
prominence very well.
When the plates were developed the astrographic images were found to
be out of focus. This is attributed to the effect of the sun's heat
on the coelostat mirror. The images were fuzzy and quite different
from those on the check-plates secured at night before and after the
eclipse. Fortunately the mirror which fed the 4-inch lens was not
affected, and the star images secured with this lens were good and
similar to those got by the night-plates. The observers stayed on in
Brazil until July to secure the field in the night sky at the altitude
of the eclipse epoch and under identical instrumental conditions.
The plates were measured at Greenwich immediately after the observers'
return. Each plate was measured twice over by Messrs. Davidson and
Furner, and I am satisfied that such faults as lie in the results
are in the plates themselves and not in the measures. The figures
obtained may be briefly summarized as follows: The astrographic plates
gave 0''.97 for the displacement at the limb when the scale-value was
determined from the plates themselves, and 1''.40 when the scale-value
was assumed from the check plates. But the much better plates gave
for the displacement at the limb 1''.98, Einstein's predicted value
being 1''.75. Further, for these plates the agreement was all that
could be expected....
After a careful study of the plates I am prepared to say that there
can be no doubt that they confirm Einstein's prediction. A very
definite result has been obtained that light is deflected according
to Einstein's law of gravitation.
Professor A. S. Eddington
Royal Observatory
Mr. Cottingham and I left the other observers at Madeira and arrived
at Principe on April 23.... We soon realized that the prospect of
a clear sky at the end of May was not very good. Not even a heavy
thunderstorm on the morning of the eclipse, three weeks after the end
of the wet season, saved the situation. The sky was completely cloudy
at first contact, but about half an hour before totality we began to
see glimpses of the sun's crescent through the clouds. We carried
out our program exactly as arranged, and the sky must have been
clearer towards the end of totality. Of the 16 plates taken during
the five minutes of totality the first ten showed no stars at all;
of the later plates two showed five stars each, from which a result
could be obtained. Comparing them with the check-plates secured at
Oxford before we went out, we obtained as the final result from the
two plates for the value of the displacement of the limb 1''.6 ±
0.3.... This result supports the figures obtained at Sobral....
I will pass now to a few words on the meaning of the result. It points
to the larger of the two possible values of the deflection. The
simplest interpretation of the bending of the ray is to consider
it as an effect of the weight of light. We know that momentum is
carried along on the path of a beam of light. Gravity in acting
creates momentum in a direction different from that of the path of
the ray and so causes it to bend. For the half-effect we have to
assume that gravity obeys Newton's law; for the full effect which has
been obtained we must assume that gravity obeys the new law proposed
by Einstein. This is one of the most crucial tests between Newton's
law and the proposed new law. Einstein's law had already indicated a
perturbation, causing the orbit of Mercury to revolve. That confirms
it for relatively small velocities. Going to the limit, where the
speed is that of light, the perturbation is increased in such a way
as to double the curvature of the path, and this is now confirmed.
This effect may be taken as proving Einstein's law rather than his
theory. It is not affected by the failure to detect the displacement
of Fraunhofer lines on the sun. If this latter failure is confirmed
it will not affect Einstein's law of gravitation, but it will affect
the views on which the law was arrived at. The law is right, though
the fundamental ideas underlying it may yet be questioned....
One further point must be touched upon. Are we to attribute the
displacement to the gravitational field and not to the refracting
matter around the sun? The refractive index required to produce the
result at a distance of 15' from the sun would be that given by gases
at a pressure of 1/60 to 1/200 of an atmosphere. This is of too great
a density considering the depth through which the light would have
to pass.
Sir J. J. Thomson
President of the Royal Society
... If the results obtained had been only that light was affected by
gravitation, it would have been of the greatest importance. Newton,
did, in fact, suggest this very point in his "Optics," and his
suggestion would presumably have led to the half-value. But this result
is not an isolated one; it is part of a whole continent of scientific
ideas affecting the most fundamental concepts of physics.... This is
the most important result obtained in connection with the theory of
gravitation since Newton's day, and it is fitting that it should be
announced at a meeting of the society so closely connected with him.
The difference between the laws of gravitation of Einstein and Newton
come only in special cases. The real interest of Einstein's theory
lies not so much in his results as in the method by which he gets
them. If his theory is right, it makes us take an entirely new view
of gravitation. If it is sustained that Einstein's reasoning holds
good--and it has survived two very severe tests in connection with the
perihelion of mercury and the present eclipse--then it is the result
of one of the highest achievements of human thought. The weak point
in the theory is the great difficulty in expressing it. It would
seem that no one can understand the new law of gravitation without
a thorough knowledge of the theory of invariants and of the calculus
of variations.
One other point of physical interest arises from the discussion. Light
is deflected in passing near huge bodies of matter. This involves
alterations in the electric and magnetic field. This, again, implies
the existence of electric and magnetic forces outside matter--forces
at present unknown, though some idea of their nature may be got from
the results of this expedition.
NOTES
[1] See Note 1 at the end of the volume.
[2] See Note 2.
[3] See Note 3.
[4] A circle--in our case the horizon--is measured by dividing the
circumference into 360 parts; each part is called a degree. Each
degree is divided into 60 minutes, and each minute into 60 seconds.
[5] See page 113.
[6] See Note 4.
[7] See Note 5.
[8] See Note 6.
[9] See Note 7.
[10] See Note 8.
[11] See Note 9.
[12] See page 93.
[13] This has since been translated into English by Dr. Lawson and
published by Methuen (London).
Since the above has been written two excellent books have been
published. One is by Prof. A. S. Eddington, Space, Time and
Gravitation (Cambridge Univ. Press, 1920). The other, somewhat more
of a philosophical work, is Prof. Moritz Schlick's Space and Time in
Contemporary Physics (Oxford Univ. Press, 1920).
Though published as early as 1897, Bertrand Russell's An Essay on
the Foundations of Geometry (Cambridge Univ. Press, 1897) contains
a fine account of non-Euclidean geometry.
[14] Republished by permission from "Science."
[15] Presidential address delivered at the St. Louis meeting of the
Physical Society, December 30, 1919. Republished by permission from
"Science."
[16] From a report in The Observatory, of the Joint Eclipse Meeting of
the Royal Society and the Royal Astronomical Society, November 6, 1919.
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