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The Einstein Theory of Relativity

A Concise Statement

by

Prof. H.A. Lorentz of the University of Leyden

NOTE

Whether it is true or not that not more than twelve persons in all the

world are able to understand Einstein's Theory, it is nevertheless

a fact that there is a constant demand for information about this

much-debated topic of relativity. The books published on the subject

are so technical that only a person trained in pure physics and

higher mathematics is able to fully understand them. In order to

make a popular explanation of this far-reaching theory available,

the present book is published.

Professor Lorentz is credited by Einstein with sharing the development

of his theory. He is doubtless better able than any other man - except

the author himself - to explain this scientific discovery.

The publishers wish to acknowledge their indebtedness to the New

York Times, The Review of Reviews and The Athenaeum for courteous

permission to reprint articles from their pages. Professor Lorentz's

article appeared originally in The Nieuwe Rotterdamsche Courant of

November 19, 1919.

INTRODUCTION

The action of the Royal Society at its meeting in London on November

6, in recognizing Dr. Albert Einstein's "theory of relativity"

has caused a great stir in scientific circles on both sides of the

Atlantic. Dr. Einstein propounded his theory nearly fifteen years

ago. The present revival of interest in it is due to the remarkable

confirmation which it received in the report of the observations

made during the sun's eclipse of last May to determine whether rays

of light passing close to the sun are deflected from their course.

The actual deflection of the rays that was discovered by the

astronomers was precisely what had been predicted theoretically by

Einstein many years since. This striking confirmation has led certain

German scientists to assert that no scientific discovery of such

importance has been made since Newton's theory of gravitation was

promulgated. This suggestion, however, was put aside by Dr. Einstein

himself when he was interviewed by a correspondent of the New York

Times at his home in Berlin. To this correspondent he expressed the

difference between his conception and the law of gravitation in the

following terms:

"Please imagine the earth removed, and in its place suspended a box as

big as a room or a whole house, and inside a man naturally floating

in the center, there being no force whatever pulling him. Imagine,

further, this box being, by a rope or other contrivance, suddenly

jerked to one side, which is scientifically termed 'difform motion',

as opposed to 'uniform motion.' The person would then naturally reach

bottom on the opposite side. The result would consequently be the

same as if he obeyed Newton's law of gravitation, while, in fact,

there is no gravitation exerted whatever, which proves that difform

motion will in every case produce the same effects as gravitation.

"I have applied this new idea to every kind of difform motion and

have thus developed mathematical formulas which I am convinced give

more precise results than those based on Newton's theory. Newton's

formulas, however, are such close approximations that it was difficult

to find by observation any obvious disagreement with experience."

Dr. Einstein, it must be remembered, is a physicist and not an

astronomer. He developed his theory as a mathematical formula. The

confirmation of it came from the astronomers. As he himself says, the

crucial test was supplied by the last total solar eclipse. Observations

then proved that the rays of fixed stars, having to pass close to

the sun to reach the earth, were deflected the exact amount demanded

by Einstein's formulas. The deflection was also in the direction

predicted by him.

The question must have occurred to many, what has all this to do with

relativity? When this query was propounded by the Times correspondent

to Dr. Einstein he replied as follows:

"The term relativity refers to time and space. According to Galileo and

Newton, time and space were absolute entities, and the moving systems

of the universe were dependent on this absolute time and space. On

this conception was built the science of mechanics. The resulting

formulas sufficed for all motions of a slow nature; it was found,

however, that they would not conform to the rapid motions apparent

in electrodynamics.

"This led the Dutch professor, Lorentz, and myself to develop

the theory of special relativity. Briefly, it discards absolute

time and space and makes them in every instance relative to moving

systems. By this theory all phenomena in electrodynamics, as well as

mechanics, hitherto irreducible by the old formulae - and there are

multitudes - were satisfactorily explained.

"Till now it was believed that time and space existed by themselves,

even if there was nothing else - no sun, no earth, no stars - while

now we know that time and space are not the vessel for the universe,

but could not exist at all if there were no contents, namely, no sun,

earth and other celestial bodies.

"This special relativity, forming the first part of my theory,

relates to all systems moving with uniform motion; that is, moving

in a straight line with equal velocity.

"Gradually I was led to the idea, seeming a very paradox in science,

that it might apply equally to all moving systems, even of difform

motion, and thus I developed the conception of general relativity

which forms the second part of my theory."

As summarized by an American astronomer, Professor Henry Norris

Russell, of Princeton, in the Scientific American for November 29,

Einstein's contribution amounts to this:

"The central fact which has been proved - and which is of great interest

and importance - is that the natural phenomena involving gravitation

and inertia (such as the motions of the planets) and the phenomena

involving electricity and magnetism (including the motion of light)

are not independent of one another, but are intimately related, so

that both sets of phenomena should be regarded as parts of one vast

system, embracing all Nature. The relation of the two is, however, of

such a character that it is perceptible only in a very few instances,

and then only to refined observations."

Already before the war, Einstein had immense fame among physicists,

and among all who are interested in the philosophy of science,

because of his principle of relativity.

Clerk Maxwell had shown that light is electro-magnetic, and had reduced

the whole theory of electro-magnetism to a small number of equations,

which are fundamental in all subsequent work. But these equations

were entangled with the hypothesis of the ether, and with the notion

of motion relative to the ether. Since the ether was supposed to be

at rest, such motion was indistinguishable from absolute motion. The

motion of the earth relatively to the ether should have been different

at different points of its orbit, and measurable phenomena should

have resulted from this difference. But none did, and all attempts to

detect effects of motions relative to the ether failed. The theory of

relativity succeeded in accounting for this fact. But it was necessary

incidentally to throw over the one universal time, and substitute

local times attached to moving bodies and varying according to their

motion. The equations on which the theory of relativity is based are

due to Lorentz, but Einstein connected them with his general principle,

namely, that there must be nothing, in observable phenomena, which

could be attributed to absolute motion of the observer.

In orthodox Newtonian dynamics the principle of relativity had a

simpler form, which did not require the substitution of local time

for general time. But it now appeared that Newtonian dynamics is only

valid when we confine ourselves to velocities much less than that

of light. The whole Galileo-Newton system thus sank to the level

of a first approximation, becoming progressively less exact as the

velocities concerned approached that of light.

Einstein's extension of his principle so as to account for gravitation

was made during the war, and for a considerable period our astronomers

were unable to become acquainted with it, owing to the difficulty

of obtaining German printed matter. However, copies of his work

ultimately reached the outside world and enabled people to learn more

about it. Gravitation, ever since Newton, had remained isolated from

other forces in nature; various attempts had been made to account

for it, but without success. The immense unification effected by

electro-magnetism apparently left gravitation out of its scope. It

seemed that nature had presented a challenge to the physicists which

none of them were able to meet.

At this point Einstein intervened with a hypothesis which, apart

altogether from subsequent verification, deserves to rank as one

of the great monuments of human genius. After correcting Newton,

it remained to correct Euclid, and it was in terms of non-Euclidean

geometry that he stated his new theory. Non-Euclidean geometry is

a study of which the primary motive was logical and philosophical;

few of its promoters ever dreamed that it would come to be applied

in physics. Some of Euclid's axioms were felt to be not "necessary

truths," but mere empirical laws; in order to establish this view,

self-consistent geometries were constructed upon assumptions other

than those of Euclid. In these geometries the sum of the angles of

a triangle is not two right angles, and the departure from two right

angles increases as the size of the triangle increases. It is often

said that in non-Euclidean geometry space has a curvature, but this

way of stating the matter is misleading, since it seems to imply a

fourth dimension, which is not implied by these systems.

Einstein supposes that space is Euclidean where it is sufficiently

remote from matter, but that the presence of matter causes it

to become slightly non-Euclidean - the more matter there is in the

neighborhood, the more space will depart from Euclid. By the help of

this hypothesis, together with his previous theory of relativity, he

deduces gravitation - very approximately, but not exactly, according

to the Newtonian law of the inverse square. The minute differences

between the effects deduced from his theory and those deduced from

Newton are measurable in certain cases. There are, so far, three

crucial tests of the relative accuracy of the new theory and the old.

(1) The perihelion of Mercury shows a discrepancy which has long

puzzled astronomers. This discrepancy is fully accounted for by

Einstein. At the time when he published his theory, this was its only

experimental verification.

(2) Modern physicists were willing to suppose that light might be

subject to gravitation - i.e., that a ray of light passing near a

great mass like the sun might be deflected to the extent to which a

particle moving with the same velocity would be deflected according

to the orthodox theory of gravitation. But Einstein's theory required

that the light should be deflected just twice as much as this. The

matter could only be tested during an eclipse among a number of

bright stars. Fortunately a peculiarly favourable eclipse occurred

last year. The results of the observations have now been published,

and are found to verify Einstein's prediction. The verification is not,

of course, quite exact; with such delicate observations that was not to

be expected. In some cases the departure is considerable. But taking

the average of the best series of observations, the deflection at

the sun's limb is found to be 1.98'', with a probable error of about

6 per cent., whereas the deflection calculated by Einstein's theory

should be 1.75''. It will be noticed that Einstein's theory gave a

deflection twice as large as that predicted by the orthodox theory,

and that the observed deflection is slightly larger than Einstein

predicted. The discrepancy is well within what might be expected in

view of the minuteness of the measurements. It is therefore generally

acknowledged by astronomers that the outcome is a triumph for Einstein.

(3) In the excitement of this sensational verification, there has

been a tendency to overlook the third experimental test to which

Einstein's theory was to be subjected. If his theory is correct as it

stands, there ought, in a gravitational field, to be a displacement

of the lines of the spectrum towards the red. No such effect has

been discovered. Spectroscopists maintain that, so far as can be

seen at present, there is no way of accounting for this failure if

Einstein's theory in its present form is assumed. They admit that some

compensating cause may be discovered to explain the discrepancy, but

they think it far more probable that Einstein's theory requires some

essential modification. Meanwhile, a certain suspense of judgment

is called for. The new law has been so amazingly successful in two

of the three tests that there must be some thing valid about it,

even if it is not exactly right as yet.

Einstein's theory has the very highest degree of aesthetic merit:

every lover of the beautiful must wish it to be true. It gives a

vast unified survey of the operations of nature, with a technical

simplicity in the critical assumptions which makes the wealth of

deductions astonishing. It is a case of an advance arrived at by

pure theory: the whole effect of Einstein's work is to make physics

more philosophical (in a good sense), and to restore some of that

intellectual unity which belonged to the great scientific systems of

the seventeenth and eighteenth centuries, but which was lost through

increasing specialization and the overwhelming mass of detailed

knowledge. In some ways our age is not a good one to live in, but

for those who are interested in physics there are great compensations.

THE EINSTEIN THEORY OF RELATIVITY

A Concise Statement by Prof. H. A. Lorentz, of the University of Leyden

The total eclipse of the sun of May 29, resulted in a striking

confirmation of the new theory of the universal attractive power

of gravitation developed by Albert Einstein, and thus reinforced

the conviction that the defining of this theory is one of the most

important steps ever taken in the domain of natural science. In

response to a request by the editor, I will attempt to contribute

something to its general appreciation in the following lines.

For centuries Newton's doctrine of the attraction of gravitation has

been the most prominent example of a theory of natural science. Through

the simplicity of its basic idea, an attraction between two bodies

proportionate to their mass and also proportionate to the square

of the distance; through the completeness with which it explained

so many of the peculiarities in the movement of the bodies making

up the solar system; and, finally, through its universal validity,

even in the case of the far-distant planetary systems, it compelled

the admiration of all.

But, while the skill of the mathematicians was devoted to making

more exact calculations of the consequences to which it led, no

real progress was made in the science of gravitation. It is true

that the inquiry was transferred to the field of physics, following

Cavendish's success in demonstrating the common attraction between

bodies with which laboratory work can be done, but it always was

evident that natural philosophy had no grip on the universal power

of attraction. While in electric effects an influence exercised

by the matter placed between bodies was speedily observed - the

starting-point of a new and fertile doctrine of electricity - in

the case of gravitation not a trace of an influence exercised by

intermediate matter could ever be discovered. It was, and remained,

inaccessible and unchangeable, without any connection, apparently,

with other phenomena of natural philosophy.

Einstein has put an end to this isolation; it is now well established

that gravitation affects not only matter, but also light. Thus

strengthened in the faith that his theory already has inspired,

we may assume with him that there is not a single physical or

chemical phenomenon - which does not feel, although very probably in

an unnoticeable degree, the influence of gravitation, and that, on the

other side, the attraction exercised by a body is limited in the first

place by the quantity of matter it contains and also, to some degree,

by motion and by the physical and chemical condition in which it moves.

It is comprehensible that a person could not have arrived at such a

far-reaching change of view by continuing to follow the old beaten

paths, but only by introducing some sort of new idea. Indeed,

Einstein arrived at his theory through a train of thought of great

originality. Let me try to restate it in concise terms.

THE EARTH AS A MOVING CAR

Everyone knows that a person may be sitting in any kind of a vehicle

without noticing its progress, so long as the movement does not vary

in direction or speed; in a car of a fast express train objects fall

in just the same way as in a coach that is standing still. Only when

we look at objects outside the train, or when the air can enter the

car, do we notice indications of the motion. We may compare the earth

with such a moving vehicle, which in its course around the sun has

a remarkable speed, of which the direction and velocity during a

considerable period of time may be regarded as constant. In place

of the air now comes, so it was reasoned formerly, the ether which

fills the spaces of the universe and is the carrier of light and of

electro-magnetic phenomena; there were good reasons to assume that the

earth was entirely permeable for the ether and could travel through it

without setting it in motion. So here was a case comparable with that

of a railroad coach open on all sides. There certainly should have

been a powerful "ether wind" blowing through the earth and all our

instruments, and it was to have been expected that some signs of it

would be noticed in connection with some experiment or other. Every

attempt along that line, however, has remained fruitless; all the

phenomena examined were evidently independent of the motion of the

earth. That this is the way they do function was brought to the front

by Einstein in his first or "special" theory of relativity. For him

the ether does not function and in the sketch that he draws of natural

phenomena there is no mention of that intermediate matter.

If the spaces of the universe are filled with an ether, let us suppose

with a substance, in which, aside from eventual vibrations and other

slight movements, there is never any crowding or flowing of one part

alongside of another, then we can imagine fixed points existing in it;

for example, points in a straight line, located one meter apart, points

in a level plain, like the angles or squares on a chess board extending

out into infinity, and finally, points in space as they are obtained

by repeatedly shifting that level spot a distance of a meter in the

direction perpendicular to it. If, consequently, one of the points

is chosen as an "original point" we can, proceeding from that point,

reach any other point through three steps in the common perpendicular

directions in which the points are arranged. The figures showing how

many meters are comprized in each of the steps may serve to indicate

the place reached and to distinguish it from any other; these are, as

is said, the "co-ordinates" of these places, comparable, for example,

with the numbers on a map giving the longitude and latitude. Let

us imagine that each point has noted upon it the three numbers that

give its position, then we have something comparable with a measure

with numbered subdivisions; only we now have to do, one might say,

with a good many imaginary measures in three common perpendicular

directions. In this "system of co-ordinates" the numbers that fix

the position of one or the other of the bodies may now be read off

at any moment.

This is the means which the astronomers and their mathematical

assistants have always used in dealing with the movement of the

heavenly bodies. At a determined moment the position of each body

is fixed by its three co-ordinates. If these are given, then one

knows also the common distances, as well as the angles formed by the

connecting lines, and the movement of a planet is to be known as soon

as one knows how its co-ordinates are changing from one moment to

the other. Thus the picture that one forms of the phenomena stands

there as if it were sketched on the canvas of the motionless ether.

EINSTEIN'S DEPARTURE

Since Einstein has cut loose from the ether, he lacks this canvas, and

therewith, at the first glance, also loses the possibility of fixing

the positions of the heavenly bodies and mathematically describing

their movement - i.e., by giving comparisons that define the positions

at every moment. How Einstein has overcome this difficulty may be

somewhat elucidated through a simple illustration.

On the surface of the earth the attraction of gravitation causes

all bodies to fall along vertical lines, and, indeed, when one omits

the resistance of the air, with an equally accelerated movement; the

velocity increases in equal degrees in equal consecutive divisions of

time at a rate that in this country gives the velocity attained at

the end of a second as 981 centimeters (32.2 feet) per second. The

number 981 defines the "acceleration in the field of gravitation,"

and this field is fully characterized by that single number; with its

help we can also calculate the movement of an object hurled out in an

arbitrary direction. In order to measure the acceleration we let the

body drop alongside of a vertical measure set solidly on the ground;

on this scale we read at every moment the figure that indicates the

height, the only co-ordinate that is of importance in this rectilinear

movement. Now we ask what would we be able to see if the measure were

not bound solidly to the earth, if it, let us suppose, moved down or

up with the place where it is located and where we are ourselves. If

in this case the speed were constant, then, and this is in accord with

the special theory of relativity, there would be no motion observed at

all; we should again find an acceleration of 981 for a falling body. It

would be different if the measure moved with changeable velocity.

If it went down with a constant acceleration of 981 itself, then an

object could remain permanently at the same point on the measure,

or could move up or down itself alongside of it, with constant

speed. The relative movement of the body with regard to the measure

should be without acceleration, and if we had to judge only by what

we observed in the spot where we were and which was falling itself,

then we should get the impression that there was no gravitation at

all. If the measure goes down with an acceleration equal to a half

or a third of what it just was, then the relative motion of the body

will, of course, be accelerated, but we should find the increase

in velocity per second one-half or two-thirds of 981. If, finally,

we let the measure rise with a uniformly accelerated movement, then

we shall find a greater acceleration than 981 for the body itself.

Thus we see that we, also when the measure is not attached to the

earth, disregarding its displacement, may describe the motion of the

body in respect to the measure always in the same way - i.e., as one

uniformly accelerated, as we ascribe now and again a fixed value to

the acceleration of the sphere of gravitation, in a particular case

the value of zero.

Of course, in the case here under consideration the use of a measure

fixed immovably upon the earth should merit all recommendation. But

in the spaces of the solar system we have, now that we have abandoned

the ether, no such support. We can no longer establish a system of

co-ordinates, like the one just mentioned, in a universal intermediate

matter, and if we were to arrive in one way or another at a definite

system of lines crossing each other in three directions, then we should

be able to use just as well another similar system that in respect to

the first moves this or that way. We should also be able to remodel the

system of co-ordinates in all kinds of ways, for example by extension

or compression. That in all these cases for fixed bodies that do not

The Einstein Theory of Relativity

A Concise Statement

by

Prof. H.A. Lorentz of the University of Leyden

NOTE

Whether it is true or not that not more than twelve persons in all the

world are able to understand Einstein's Theory, it is nevertheless

a fact that there is a constant demand for information about this

much-debated topic of relativity. The books published on the subject

are so technical that only a person trained in pure physics and

higher mathematics is able to fully understand them. In order to

make a popular explanation of this far-reaching theory available,

the present book is published.

Professor Lorentz is credited by Einstein with sharing the development

of his theory. He is doubtless better able than any other man - except

the author himself - to explain this scientific discovery.

The publishers wish to acknowledge their indebtedness to the New

York Times, The Review of Reviews and The Athenaeum for courteous

permission to reprint articles from their pages. Professor Lorentz's

article appeared originally in The Nieuwe Rotterdamsche Courant of

November 19, 1919.

INTRODUCTION

The action of the Royal Society at its meeting in London on November

6, in recognizing Dr. Albert Einstein's "theory of relativity"

has caused a great stir in scientific circles on both sides of the

Atlantic. Dr. Einstein propounded his theory nearly fifteen years

ago. The present revival of interest in it is due to the remarkable

confirmation which it received in the report of the observations

made during the sun's eclipse of last May to determine whether rays

of light passing close to the sun are deflected from their course.

The actual deflection of the rays that was discovered by the

astronomers was precisely what had been predicted theoretically by

Einstein many years since. This striking confirmation has led certain

German scientists to assert that no scientific discovery of such

importance has been made since Newton's theory of gravitation was

promulgated. This suggestion, however, was put aside by Dr. Einstein

himself when he was interviewed by a correspondent of the New York

Times at his home in Berlin. To this correspondent he expressed the

difference between his conception and the law of gravitation in the

following terms:

"Please imagine the earth removed, and in its place suspended a box as

big as a room or a whole house, and inside a man naturally floating

in the center, there being no force whatever pulling him. Imagine,

further, this box being, by a rope or other contrivance, suddenly

jerked to one side, which is scientifically termed 'difform motion',

as opposed to 'uniform motion.' The person would then naturally reach

bottom on the opposite side. The result would consequently be the

same as if he obeyed Newton's law of gravitation, while, in fact,

there is no gravitation exerted whatever, which proves that difform

motion will in every case produce the same effects as gravitation.

"I have applied this new idea to every kind of difform motion and

have thus developed mathematical formulas which I am convinced give

more precise results than those based on Newton's theory. Newton's

formulas, however, are such close approximations that it was difficult

to find by observation any obvious disagreement with experience."

Dr. Einstein, it must be remembered, is a physicist and not an

astronomer. He developed his theory as a mathematical formula. The

confirmation of it came from the astronomers. As he himself says, the

crucial test was supplied by the last total solar eclipse. Observations

then proved that the rays of fixed stars, having to pass close to

the sun to reach the earth, were deflected the exact amount demanded

by Einstein's formulas. The deflection was also in the direction

predicted by him.

The question must have occurred to many, what has all this to do with

relativity? When this query was propounded by the Times correspondent

to Dr. Einstein he replied as follows:

"The term relativity refers to time and space. According to Galileo and

Newton, time and space were absolute entities, and the moving systems

of the universe were dependent on this absolute time and space. On

this conception was built the science of mechanics. The resulting

formulas sufficed for all motions of a slow nature; it was found,

however, that they would not conform to the rapid motions apparent

in electrodynamics.

"This led the Dutch professor, Lorentz, and myself to develop

the theory of special relativity. Briefly, it discards absolute

time and space and makes them in every instance relative to moving

systems. By this theory all phenomena in electrodynamics, as well as

mechanics, hitherto irreducible by the old formulae - and there are

multitudes - were satisfactorily explained.

"Till now it was believed that time and space existed by themselves,

even if there was nothing else - no sun, no earth, no stars - while

now we know that time and space are not the vessel for the universe,

but could not exist at all if there were no contents, namely, no sun,

earth and other celestial bodies.

"This special relativity, forming the first part of my theory,

relates to all systems moving with uniform motion; that is, moving

in a straight line with equal velocity.

"Gradually I was led to the idea, seeming a very paradox in science,

that it might apply equally to all moving systems, even of difform

motion, and thus I developed the conception of general relativity

which forms the second part of my theory."

As summarized by an American astronomer, Professor Henry Norris

Russell, of Princeton, in the Scientific American for November 29,

Einstein's contribution amounts to this:

"The central fact which has been proved - and which is of great interest

and importance - is that the natural phenomena involving gravitation

and inertia (such as the motions of the planets) and the phenomena

involving electricity and magnetism (including the motion of light)

are not independent of one another, but are intimately related, so

that both sets of phenomena should be regarded as parts of one vast

system, embracing all Nature. The relation of the two is, however, of

such a character that it is perceptible only in a very few instances,

and then only to refined observations."

Already before the war, Einstein had immense fame among physicists,

and among all who are interested in the philosophy of science,

because of his principle of relativity.

Clerk Maxwell had shown that light is electro-magnetic, and had reduced

the whole theory of electro-magnetism to a small number of equations,

which are fundamental in all subsequent work. But these equations

were entangled with the hypothesis of the ether, and with the notion

of motion relative to the ether. Since the ether was supposed to be

at rest, such motion was indistinguishable from absolute motion. The

motion of the earth relatively to the ether should have been different

at different points of its orbit, and measurable phenomena should

have resulted from this difference. But none did, and all attempts to

detect effects of motions relative to the ether failed. The theory of

relativity succeeded in accounting for this fact. But it was necessary

incidentally to throw over the one universal time, and substitute

local times attached to moving bodies and varying according to their

motion. The equations on which the theory of relativity is based are

due to Lorentz, but Einstein connected them with his general principle,

namely, that there must be nothing, in observable phenomena, which

could be attributed to absolute motion of the observer.

In orthodox Newtonian dynamics the principle of relativity had a

simpler form, which did not require the substitution of local time

for general time. But it now appeared that Newtonian dynamics is only

valid when we confine ourselves to velocities much less than that

of light. The whole Galileo-Newton system thus sank to the level

of a first approximation, becoming progressively less exact as the

velocities concerned approached that of light.

Einstein's extension of his principle so as to account for gravitation

was made during the war, and for a considerable period our astronomers

were unable to become acquainted with it, owing to the difficulty

of obtaining German printed matter. However, copies of his work

ultimately reached the outside world and enabled people to learn more

about it. Gravitation, ever since Newton, had remained isolated from

other forces in nature; various attempts had been made to account

for it, but without success. The immense unification effected by

electro-magnetism apparently left gravitation out of its scope. It

seemed that nature had presented a challenge to the physicists which

none of them were able to meet.

At this point Einstein intervened with a hypothesis which, apart

altogether from subsequent verification, deserves to rank as one

of the great monuments of human genius. After correcting Newton,

it remained to correct Euclid, and it was in terms of non-Euclidean

geometry that he stated his new theory. Non-Euclidean geometry is

a study of which the primary motive was logical and philosophical;

few of its promoters ever dreamed that it would come to be applied

in physics. Some of Euclid's axioms were felt to be not "necessary

truths," but mere empirical laws; in order to establish this view,

self-consistent geometries were constructed upon assumptions other

than those of Euclid. In these geometries the sum of the angles of

a triangle is not two right angles, and the departure from two right

angles increases as the size of the triangle increases. It is often

said that in non-Euclidean geometry space has a curvature, but this

way of stating the matter is misleading, since it seems to imply a

fourth dimension, which is not implied by these systems.

Einstein supposes that space is Euclidean where it is sufficiently

remote from matter, but that the presence of matter causes it

to become slightly non-Euclidean - the more matter there is in the

neighborhood, the more space will depart from Euclid. By the help of

this hypothesis, together with his previous theory of relativity, he

deduces gravitation - very approximately, but not exactly, according

to the Newtonian law of the inverse square. The minute differences

between the effects deduced from his theory and those deduced from

Newton are measurable in certain cases. There are, so far, three

crucial tests of the relative accuracy of the new theory and the old.

(1) The perihelion of Mercury shows a discrepancy which has long

puzzled astronomers. This discrepancy is fully accounted for by

Einstein. At the time when he published his theory, this was its only

experimental verification.

(2) Modern physicists were willing to suppose that light might be

subject to gravitation - i.e., that a ray of light passing near a

great mass like the sun might be deflected to the extent to which a

particle moving with the same velocity would be deflected according

to the orthodox theory of gravitation. But Einstein's theory required

that the light should be deflected just twice as much as this. The

matter could only be tested during an eclipse among a number of

bright stars. Fortunately a peculiarly favourable eclipse occurred

last year. The results of the observations have now been published,

and are found to verify Einstein's prediction. The verification is not,

of course, quite exact; with such delicate observations that was not to

be expected. In some cases the departure is considerable. But taking

the average of the best series of observations, the deflection at

the sun's limb is found to be 1.98'', with a probable error of about

6 per cent., whereas the deflection calculated by Einstein's theory

should be 1.75''. It will be noticed that Einstein's theory gave a

deflection twice as large as that predicted by the orthodox theory,

and that the observed deflection is slightly larger than Einstein

predicted. The discrepancy is well within what might be expected in

view of the minuteness of the measurements. It is therefore generally

acknowledged by astronomers that the outcome is a triumph for Einstein.

(3) In the excitement of this sensational verification, there has

been a tendency to overlook the third experimental test to which

Einstein's theory was to be subjected. If his theory is correct as it

stands, there ought, in a gravitational field, to be a displacement

of the lines of the spectrum towards the red. No such effect has

been discovered. Spectroscopists maintain that, so far as can be

seen at present, there is no way of accounting for this failure if

Einstein's theory in its present form is assumed. They admit that some

compensating cause may be discovered to explain the discrepancy, but

they think it far more probable that Einstein's theory requires some

essential modification. Meanwhile, a certain suspense of judgment

is called for. The new law has been so amazingly successful in two

of the three tests that there must be some thing valid about it,

even if it is not exactly right as yet.

Einstein's theory has the very highest degree of aesthetic merit:

every lover of the beautiful must wish it to be true. It gives a

vast unified survey of the operations of nature, with a technical

simplicity in the critical assumptions which makes the wealth of

deductions astonishing. It is a case of an advance arrived at by

pure theory: the whole effect of Einstein's work is to make physics

more philosophical (in a good sense), and to restore some of that

intellectual unity which belonged to the great scientific systems of

the seventeenth and eighteenth centuries, but which was lost through

increasing specialization and the overwhelming mass of detailed

knowledge. In some ways our age is not a good one to live in, but

for those who are interested in physics there are great compensations.

THE EINSTEIN THEORY OF RELATIVITY

A Concise Statement by Prof. H. A. Lorentz, of the University of Leyden

The total eclipse of the sun of May 29, resulted in a striking

confirmation of the new theory of the universal attractive power

of gravitation developed by Albert Einstein, and thus reinforced

the conviction that the defining of this theory is one of the most

important steps ever taken in the domain of natural science. In

response to a request by the editor, I will attempt to contribute

something to its general appreciation in the following lines.

For centuries Newton's doctrine of the attraction of gravitation has

been the most prominent example of a theory of natural science. Through

the simplicity of its basic idea, an attraction between two bodies

proportionate to their mass and also proportionate to the square

of the distance; through the completeness with which it explained

so many of the peculiarities in the movement of the bodies making

up the solar system; and, finally, through its universal validity,

even in the case of the far-distant planetary systems, it compelled

the admiration of all.

But, while the skill of the mathematicians was devoted to making

more exact calculations of the consequences to which it led, no

real progress was made in the science of gravitation. It is true

that the inquiry was transferred to the field of physics, following

Cavendish's success in demonstrating the common attraction between

bodies with which laboratory work can be done, but it always was

evident that natural philosophy had no grip on the universal power

of attraction. While in electric effects an influence exercised

by the matter placed between bodies was speedily observed - the

starting-point of a new and fertile doctrine of electricity - in

the case of gravitation not a trace of an influence exercised by

intermediate matter could ever be discovered. It was, and remained,

inaccessible and unchangeable, without any connection, apparently,

with other phenomena of natural philosophy.

Einstein has put an end to this isolation; it is now well established

that gravitation affects not only matter, but also light. Thus

strengthened in the faith that his theory already has inspired,

we may assume with him that there is not a single physical or

chemical phenomenon - which does not feel, although very probably in

an unnoticeable degree, the influence of gravitation, and that, on the

other side, the attraction exercised by a body is limited in the first

place by the quantity of matter it contains and also, to some degree,

by motion and by the physical and chemical condition in which it moves.

It is comprehensible that a person could not have arrived at such a

far-reaching change of view by continuing to follow the old beaten

paths, but only by introducing some sort of new idea. Indeed,

Einstein arrived at his theory through a train of thought of great

originality. Let me try to restate it in concise terms.

THE EARTH AS A MOVING CAR

Everyone knows that a person may be sitting in any kind of a vehicle

without noticing its progress, so long as the movement does not vary

in direction or speed; in a car of a fast express train objects fall

in just the same way as in a coach that is standing still. Only when

we look at objects outside the train, or when the air can enter the

car, do we notice indications of the motion. We may compare the earth

with such a moving vehicle, which in its course around the sun has

a remarkable speed, of which the direction and velocity during a

considerable period of time may be regarded as constant. In place

of the air now comes, so it was reasoned formerly, the ether which

fills the spaces of the universe and is the carrier of light and of

electro-magnetic phenomena; there were good reasons to assume that the

earth was entirely permeable for the ether and could travel through it

without setting it in motion. So here was a case comparable with that

of a railroad coach open on all sides. There certainly should have

been a powerful "ether wind" blowing through the earth and all our

instruments, and it was to have been expected that some signs of it

would be noticed in connection with some experiment or other. Every

attempt along that line, however, has remained fruitless; all the

phenomena examined were evidently independent of the motion of the

earth. That this is the way they do function was brought to the front

by Einstein in his first or "special" theory of relativity. For him

the ether does not function and in the sketch that he draws of natural

phenomena there is no mention of that intermediate matter.

If the spaces of the universe are filled with an ether, let us suppose

with a substance, in which, aside from eventual vibrations and other

slight movements, there is never any crowding or flowing of one part

alongside of another, then we can imagine fixed points existing in it;

for example, points in a straight line, located one meter apart, points

in a level plain, like the angles or squares on a chess board extending

out into infinity, and finally, points in space as they are obtained

by repeatedly shifting that level spot a distance of a meter in the

direction perpendicular to it. If, consequently, one of the points

is chosen as an "original point" we can, proceeding from that point,

reach any other point through three steps in the common perpendicular

directions in which the points are arranged. The figures showing how

many meters are comprized in each of the steps may serve to indicate

the place reached and to distinguish it from any other; these are, as

is said, the "co-ordinates" of these places, comparable, for example,

with the numbers on a map giving the longitude and latitude. Let

us imagine that each point has noted upon it the three numbers that

give its position, then we have something comparable with a measure

with numbered subdivisions; only we now have to do, one might say,

with a good many imaginary measures in three common perpendicular

directions. In this "system of co-ordinates" the numbers that fix

the position of one or the other of the bodies may now be read off

at any moment.

This is the means which the astronomers and their mathematical

assistants have always used in dealing with the movement of the

heavenly bodies. At a determined moment the position of each body

is fixed by its three co-ordinates. If these are given, then one

knows also the common distances, as well as the angles formed by the

connecting lines, and the movement of a planet is to be known as soon

as one knows how its co-ordinates are changing from one moment to

the other. Thus the picture that one forms of the phenomena stands

there as if it were sketched on the canvas of the motionless ether.

EINSTEIN'S DEPARTURE

Since Einstein has cut loose from the ether, he lacks this canvas, and

therewith, at the first glance, also loses the possibility of fixing

the positions of the heavenly bodies and mathematically describing

their movement - i.e., by giving comparisons that define the positions

at every moment. How Einstein has overcome this difficulty may be

somewhat elucidated through a simple illustration.

On the surface of the earth the attraction of gravitation causes

all bodies to fall along vertical lines, and, indeed, when one omits

the resistance of the air, with an equally accelerated movement; the

velocity increases in equal degrees in equal consecutive divisions of

time at a rate that in this country gives the velocity attained at

the end of a second as 981 centimeters (32.2 feet) per second. The

number 981 defines the "acceleration in the field of gravitation,"

and this field is fully characterized by that single number; with its

help we can also calculate the movement of an object hurled out in an

arbitrary direction. In order to measure the acceleration we let the

body drop alongside of a vertical measure set solidly on the ground;

on this scale we read at every moment the figure that indicates the

height, the only co-ordinate that is of importance in this rectilinear

movement. Now we ask what would we be able to see if the measure were

not bound solidly to the earth, if it, let us suppose, moved down or

up with the place where it is located and where we are ourselves. If

in this case the speed were constant, then, and this is in accord with

the special theory of relativity, there would be no motion observed at

all; we should again find an acceleration of 981 for a falling body. It

would be different if the measure moved with changeable velocity.

If it went down with a constant acceleration of 981 itself, then an

object could remain permanently at the same point on the measure,

or could move up or down itself alongside of it, with constant

speed. The relative movement of the body with regard to the measure

should be without acceleration, and if we had to judge only by what

we observed in the spot where we were and which was falling itself,

then we should get the impression that there was no gravitation at

all. If the measure goes down with an acceleration equal to a half

or a third of what it just was, then the relative motion of the body

will, of course, be accelerated, but we should find the increase

in velocity per second one-half or two-thirds of 981. If, finally,

we let the measure rise with a uniformly accelerated movement, then

we shall find a greater acceleration than 981 for the body itself.

Thus we see that we, also when the measure is not attached to the

earth, disregarding its displacement, may describe the motion of the

body in respect to the measure always in the same way - i.e., as one

uniformly accelerated, as we ascribe now and again a fixed value to

the acceleration of the sphere of gravitation, in a particular case

the value of zero.

Of course, in the case here under consideration the use of a measure

fixed immovably upon the earth should merit all recommendation. But

in the spaces of the solar system we have, now that we have abandoned

the ether, no such support. We can no longer establish a system of

co-ordinates, like the one just mentioned, in a universal intermediate

matter, and if we were to arrive in one way or another at a definite

system of lines crossing each other in three directions, then we should

be able to use just as well another similar system that in respect to

the first moves this or that way. We should also be able to remodel the

system of co-ordinates in all kinds of ways, for example by extension

or compression. That in all these cases for fixed bodies that do not

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