(b) The independence of natural, i.e., physical, laws of their

mathematical mode of expression. Thus, when an equation is written down

as the expression of a physical law it must be satisfied, no matter

what units we choose in order to measure the quantities occurring in

the equation. As our physics teacher used to say "the expression of the

law must have in every term the same dimensions." More than this the

choice of the quantities used to express the law--if there be a choice

open--must have no effect on its correctness. As we were told--"all

physical laws are capable of expression as relations between vectors or

else as relations between magnitudes of the same dimensions." We shall

hope to make this clearer in its proper place in the essay, as its

obvious generalization is Einstein's cardinal principle of relativity.

The measurements which an experimental physicist makes are always the

expression of a coincidence of two points in space at the same time. If

we ask such an experimenter what he means by a point in space he tells

us that, for him, the term has no meaning until he has a material

body with reference to which he can locate the point by measurements;

in general it requires three measurements and he expresses this by

saying that space has three dimensions. He measures his distance,

as a rule, parallel to three mutually perpendicular lines fixed in

the material body--a Cartesian reference-frame so-called. So that a

"point in space" is equivalent to a given material reference-frame

and three numbers or coordinates. If, for any reason, we prefer to

use a new material reference-frame the coordinates or measurements

will change and, if we know the relative positions of the two material

reference-frames, there is a definite relation between the two sets of

three coordinates which is termed a transformation of coordinates. But

which particular material reference-frame shall we use? The first

choice would, we think, be that attached to the earth. But, even yet,

we are in doubt as there are numberless Cartesian frameworks attached

to the earth (as to any material body) and it is here that our idea

(b) begins to function. We say it must be immaterial which of these

Cartesian frames we use. In each frame a vector has three components

and when we change from one frame to another the components change

in such a way that if two vectors have their three components equal

in one framework they will be equal in any other attached to the

same material system. So our idea (b), which says that our physical

equations must be vector equations, is equivalent to saying that the

choice of the framework attached to any given material body can have

no effect on the mode of expression of a natural law.

Shall we carry over our idea (b) to answer the next question: "To

which material body shall we attach our framework?" To this question

Newton gave one answer and Einstein another. We shall first consider

Newton's position and then we may hope to see clearly where the new

theory diverges from the classical or Newtonian mechanics. Newton's

answer was that there is a particular material frame with reference

to which the laws of mechanics have a remarkably simple form commonly

known as "Newton's laws of motion" and so it is preferable to use

this framework which is called an absolute frame.

What is the essential peculiarity of an absolute frame? Newton was

essentially an empiricist of Bacon's school and he observed the

following facts. Let us suppose we have a framework of reference

attached to the earth. Then a small particle of matter under the

gravitational influence of surrounding bodies, including the earth,

takes on a certain acceleration $A_1$. Now suppose the surrounding

bodies removed (since we cannot remove the earth we shall have to

view the experiment as an abstraction), and another set introduced;

the particle, being again at its original position, will begin to

move with an acceleration $A_2$. If both sets of surrounding bodies

are present simultaneously the particle begins to move with an

acceleration which is approximately but not quite the sum of $A_1$

and $A_2$. Newton postulated there there is a certain absolute

reference frame in which the approximation would be an equality;

and so the acceleration, relative to the material frame, furnishes

a convenient measure of the effect of the surrounding bodies--which

effect we call their gravitational force. Notice that if the effect

of the surrounding bodies is small the acceleration is small and

so we obtain as a limiting case, Newton's law of inertia which says

that a body subject to no forces has no acceleration; a law which,

as Poincaré justly observed, can never be subjected to experimental

justification. The natural questions then arise: which is the

absolute and privileged reference-frame and how must the simple laws

be modified when we use a frame more convenient for us--one attached

to the earth let us say? The absolute frame is one attached to the

fixed stars; and to the absolute or real force defined as above,

we must add certain terms, usually called centrifugal forces. These

are referred to as fictitious forces because, as it is explained,

they are due to the motion of the reference-frame with respect to

the absolute frame and in no way depend on the distribution of the

surrounding bodies. Gravitational force and centrifugal forces have

in common the remarkable property that they depend in no way on the

material of the attracted body nor on its chemical state; they act

on all matter and are in this way different from other forces met

with in nature, such as magnetic or electric forces. Further Newton

found that he could predict the facts of observation accurately on the

hypothesis that two small particles of matter attracted each other,

in the direction of the line joining them, with a force varying

inversely as the square of the distance between them. This law is an

"action at a distance" law and so is opposed to the idea (a).

We have tacitly supposed that the space in which we make our

measurements is that made familiar to us by the study of Euclid's

elements. The characteristic property of this space is that

stated by the theorem of Pythagoras that the distance between two

points is found by extracting the square root of the sum of the

squares of the differences of the Cartesian coordinates of the two

points. Mathematicians have long recognized the possibility of other

types of space and Einstein has followed their lead. He abandons

the empiricist method and when asked what he means by a point in

space replies that to him a point in space is equivalent to four

numbers how obtained it is unnecessary to know a priori; in certain

special cases they may be the three Cartesian coordinates of the

experimenter (measured with reference to a definite material framework)

together with the time. Accordingly he says his space is of four

dimensions. Between any two "points" we may insert a sequence of sets

of four numbers, varying continuously from the first set to the second,

thus forming what we call a curve joining the two points. Now we define

the "length" of this curve in a manner which involves all the points

on it and stipulate that this length has a physical reality, i.e.,

according to our idea (b) its value is independent of the particular

choice of coordinates we make in describing the space. Among all

the joining curves there will be one with the property of having

the smallest length; this is called a geodesic and corresponds to

the straight line in Euclidean space. We must now, for lack of an

a priori description of the actual significance of our coordinates,

extend the idea of vector so that we may speak of the components of

a vector no matter what our coordinates may actually signify. In

this way are introduced what are known as tensors; if two tensors

are equal, i.e., have all their components equal, in any one set of

coordinates they are equal in any other and the fundamental demand of

the new physics is that all physical equations which are not merely

the expression of equality of magnitudes must state the equality of

tensors. In this way no one system of coordinates is privileged above

any other and the laws of physics are expressed in a form independent

of the actual coordinates chosen; they are written, as we may say,

in an absolute form.

THE GRAVITATIONAL HYPOTHESIS

Einstein flatly denies Newton's hypothesis that there is an absolute

system (and, indeed, many others before him had found it difficult

to admit that so insignificant a part of the universe as our fixed

star system should have such a privileged position as that accorded

to it in the Newtonian Mechanics). In any system, he says, we have no

reason to distinguish between the so-called real gravitational force

and the so-called fictitious centrifugal forces--if we wish so to

express it gravitational force is fictitious force. [10] A particle

moving in the neighborhood of material bodies moves according to a law

of inertia--a physical law expressible, therefore, in a manner quite

independent of the choice of coordinates. The law of inertia is that

a particle left to itself moves along the geodesics or shortest lines

in the space. If the particle is remote from other bodies the space

has the Euclidean character and we have Newton's law of inertia;

otherwise the particle is in a space of a non-Euclidean character

(the space being always the four-dimensional space) and the path of

the particle is along a geodesic in that space. Einstein, in order

to make the theory more concrete, makes a certain stipulation as to

the nature of the gravitational space which stipulation is expressed,

as are all physical laws, by means of a tensor equation--and this is

sometimes called his law of gravitation.

Perhaps it will be well, in exemplification, to explain why light

rays, which pass close to the sun, should be bent according to the new

theory. It is assumed that light rays travel along certain geodesics

known as minimal geodesics. The sun has an intense gravitational field

near it--or, as we now say, the departure of the four-dimensional

space from the Euclidean is very marked for points near the sun--but

for points so remote as the earth this departure is so small as

to be negligible. Hence the form of the geodesics near the sun is

different from that near the earth. If the space surrounding the

sun were Euclidean the actual paths of the light rays would appear

different from geodesics or straight-lines. Hence Einstein speaks

of the curvature of the light rays due to the gravitational field of

the sun; but we must not be misled by a phrase. Light always travels

along geodesics (or straight lines--the only definition we have of a

straight line is that it is a geodesic); but, owing to the "distortion"

of the space they traverse, due to the sun, these geodesics reach us

with a direction different from that they would have if they did not

pass through the markedly non-Euclidean space near the sun.

The consideration of the fundamental four-dimensional space as

being non-Euclidean where matter is present gives a possibility of

an answer to the world old question: Is space finite or infinite? Is

time eternal or finite? The fascinating possibility arises that the

space may be like the two-dimensional surface of a sphere which to a

limited experience seems infinite in extent and flat or Euclidean in

character. A new Columbus now asks us to consider other possibilities

in which we should have a finite universe--finite not only as to space

measurement but as to time (for the space may be such that all of the

four coordinates of its points are bounded in magnitude). However,

although Einstein speaks of the possibility of a finite universe, we

do not, personally, think his argument convincing. Points on a sphere

may be located by the Cartesian coordinates of their stereographic

projections on the equatorial plane and these coordinates, which

might well be those actually measured, are not bounded.

THE SPECIAL RELATIVITY THEORY

In our account of the Einstein theory we have not followed its

historical order of development for two reasons. Firstly, the

earlier Special Relativity Theory properly belongs to a school

of thought diametrically opposed to that furnishing the "General

Theory of Relativity" and, secondly, the latter cannot be obtained

from the former by the process of generalization as commonly

understood. Einstein, when proposing the earlier theory, adopted

the position of the empiricist so that to him the phrase, a point in

space, had no meaning without a material framework of reference in

which to measure space distances. When he came to investigate what

is meant by time and when he asked the question "what is meant by

the statement that two remote events are simultaneous?" it became

evident that some mode of communication between the two places is

necessary; the mode adopted was that by means of light-signals. The

fundamental hypothesis was then made that the velocity of such signals

is independent of the velocity of their source (some hypothesis is

necessary if we wish to compare the time associated with events, when

one material reference-system is used, and the corresponding time when

another in motion relative to the first is adopted). It develops that

time and space measurements are inextricably interwoven; there is no

such thing as the length of a body or the duration of an event but

rather these are relative to the reference-system. [11] Minkowski

introduced the idea of the space of events--of four dimensions--but

this space was supposed Euclidean like the three-dimensional space

of his predecessors. To Einstein belongs the credit of taking from

this representation a purely formal mathematical character and of

insisting that the "real" space--whose distances have a physical

significance--is the four-dimensional space. But we cannot insist

too strongly on the fact that in the gravitational space of the

general theory there is no postulate of the constancy of velocity of a

light-signal and accordingly no method of assigning a time to events

corresponding to that adopted in the special theory. In this latter

theory attention was confined to material systems moving with uniform

velocity with respect to each other and it developed that the velocity

of light was the ultimate velocity faster than which no system could

move--a result surprising and a priori rather repugnant. It is merely

a consequence of our mode of comparing times of events; if some other

method--thought transference, let us say--were possible the velocity

of this would be the "limiting velocity."

In conclusion we should remark that the postulated equivalence of

"gravitational" and "centrifugal" forces demands that anything

possessed of inertia will be acted upon by a gravitational field and

this leads to a possible identification of matter and energy. Further

our guiding idea (a) will prompt us to say, following the example

of Faraday in his electrical researches, that the geodesics of a

gravitational space have a physical existence as distinct from a mere

mathematical one. The four-dimensional space we may call the ether, and

so restore this bearer of physical forces to the position it lost when,

as a three-dimensional idea in the Special Relativity Theory, it had to

bear an identical relation to a multitude of relatively moving material

systems. The reason for our seemingly paradoxical title for an essay on

Relativity will be clear when it is remembered that in the new theory

we consider those space-time properties which are absolute or devoid

of reference to any particular material reference-frame. Nevertheless,

although the general characteristics of the theory are thus described,

without reference to experiment, when the theory is to be tested it

is necessary to state what the four coordinates discussed actually

are--how they are determined by measurement. It is our opinion that

much remains to be done to place this portion of the subject on a

satisfactory basis. For example, in the derivation of the nature

of the gravitational space, surrounding a single attracting body,

most of the accounts use Cartesian coordinates as if the space were

Euclidean and step from these to polar coordinates by the formulæ

familiar in Euclidean geometry. But these details are, perhaps,

like matters of elegance, if we shall be allowed to give Einstein's

quotation from Boltzmann, to be left to the "tailor and the cobbler."

XVII

THE PHYSICAL SIDE OF RELATIVITY

The Immediate Contacts Between Einstein's Theories and Current Physics

and Astronomy

BY PROFESSOR WILLIAM H. PICKERING

HARVARD COLLEGE OBSERVATORY,

MANDEVILLE, JAMAICA

The Theory of Relativity will be treated first from the physical

side, leaving the three astronomical tests to which it has been put

to be discussed later. There is one astronomical fact however that

must be mentioned in this connection, and this is the discovery of

the aberration of light by Bradley in 1726. It is found that every

star in the heavens apparently describes a small annual ellipse,

whose major axis is 41'' in length. This Bradley showed to be due to a

combination of the velocity of the earth in its orbit, and the velocity

of light; and it is so explained in all the elementary text-books on

astronomy. It implies a stationary ether through which the earth is

moving. The importance of this statement will appear presently.

The subject is usually illustrated by supposing a man to go out in a

rainstorm carrying a vertical tube. If the rain is falling vertically,

and the man stands still, the sides of the tube will not be wet,

save by an occasional drop, but if the tube is moved, it must then be

inclined forward in order to keep it dry. The angle of inclination,

which corresponds to aberration, will depend on the relative velocity

of the tube, corresponding to the earth, and the rain drops which

correspond to the waves of light.

If three lines are dropped upon a point in space, each line being

perpendicular to the plane containing the other two, we have what

is known as a system of coordinates. Einstein's original theory

of relativity, which he now designates as the "special theory,"

depends on two principles. The first is that "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 is that

"Light in a vacuum has a definite and constant velocity, independent

of the velocity of its source."

These two sentences may be considered as authoritative, being quoted

in Einstein's own words.1 The first of these principles need not

greatly surprise us. The second is not well expressed, because it

is ambiguous. He does not say how the first "velocity" is measured,

whether relatively to the ether or relatively to the observer. In

fact this is the very gist of the whole matter, as we shall presently

see. In the case of sound the velocity is constant with regard to the

medium, the air, in the case of light it is supposed to be constant

with regard to the observer. It reaches him with a constant velocity,

no matter how he moves.

In order to understand this statement clearly let us consider the

appended tabular diagram. On a calm day imagine a source of sound

at $S$ in line a. This may be either a gun or a bell. Imagine an

observer 1,100 feet distant, located at $O$. The velocity of sound in

air is 1,100 feet per second. This velocity we will take as unity,

as indicated in the third column, and the velocity with which the

sound reaches the observer is also 1, as shown in the fourth. It will

reach him in a unit interval of 1 second, as shown in the fifth. If

the bell is struck, it will give its normal pitch or frequency,

which we will also call unity, in the sixth column.

Now imagine case b where the observer is on a train advancing toward

$S$. When he is 1,100 feet distant, the gun is fired, but as he is

advancing toward it, he hears it at $O$ in rather less than a second,

as shown in the fifth column. The velocity of the sound with regard

to him is rather more than unity, as shown in the fourth column. If

the bell is sounded, the pitch, that is the frequency, is raised,

because he receives more sound waves per second than before.

In case c the observer is stationary, but the source of sound is

receding. At a distance of 1,100 feet the gun is fired, and the

observer hears it after an interval of just one second, as in case

a. The velocities with regard to the observer and through the medium

are also unity. If the bell is struck the pitch is lowered, since he

receives fewer sound waves per second, the reverse of case b.

Velocity

Source in to Interval Frequency Observer

Medium Observer

Air

a S 1 1 1 1 O

b S 1 1+ 1- 1+ O

c S 1 1 1 1- O

d S 1 1+ 1- 1 O

Ether

A S 1 1 1 1 O

B S 1- 1 1 1+ O

C S 1 1 1 1- O

D S 1 - 1 1 1 O

In case d imagine the source and the observer 1,100 feet apart, and

advancing on the same train. When the gun is fired, the velocity of

the sound waves will be greater with regard to the observer, and he

will hear the sound in less than a second, as in case b. When the

bell is struck it will have the normal pitch, the same as in case a.

We find therefore that for sound the velocity with regard to

the medium is always unity, while the velocity with regard to the

observer, and the interval elapsed, depend only on the motion of the

observer himself, and are independent of the motion of the source. The

frequency of the vibrations, on the other hand, depends only on the

relative motion of the observer and the source, but is independent of

their common motion in any direction. Further, it makes no difference

whether the source and the observer are moving on a train, or whether

they are stationary, and a uniform wind is blowing past them.

In the case of light waves we shall find a very different state of

affairs, although the rules for frequency are the same as they are for

sound. In case A we have the normal conditions, where both the source

and observers are stationary. In case B we have a representation of

the Michelson-Morley experiment as supplemented by that of Majorana,2

where the source is stationary and the observer advances. Unlike the

case of sound, the interval elapsed, as shown by the experiment, is now

the same as in case A, and since the distance to the observer is less,

the velocity of light with respect to the ether must also be less than

unity. Since the observer is advancing against the light, this will

permit the velocity of light with regard to the observer to remain

unity, in conformity with the second principle of relativity. Compare

with case b for sound. As Jeans expresses it, "The velocity of light

in all directions is the same, whatever the motion of the observer."3

That is to say it appears to be the same to him, however he moves.

Case C represents Einstein's statement, as confirmed by Majorana's

experiment. It does not differ from case c for sound. Case D is

more complex, but accepting the statement above that the velocity

is constant with regard to the observer, we see that the velocity

through the medium must be less, and that the interval elapsed will

be constant, as in case B. Could we use the brighter stars and planets

as sources of light, several of these cases could be further tested.

This brings us at once to statements that contradict our common

sense. For instance, Jeans says "no matter what the velocity of the

observer is, the light surface, as observed by that observer, is

invariably a sphere having that observer as center."3 That is to say

the light surface, or wave front, is a contracting, not an expanding,

sphere. This, if confirmed, would go a long way toward making our

universe a subjective rather than an objective phenomenon. Again

imagine a flash of light, such as an explosion, to occur when an

observer is in a given position. It makes no difference how the

observer may move while the light is approaching him, whether several

miles forward or backward, the light will reach him in exactly the

same time, as is shown by Michelson's experiment. Or if two observers

are at the same spot when the explosion occurs, and one moves forward,

and the other backward, they will both see the explosion at exactly

the same instant.

This sounds ridiculous, but not only is it what Jeans says, but it is

the logical interpretation of Einstein's second principle, if Einstein

means by velocity, velocity with regard to the observer. If he means

velocity with regard to the medium, then the case is exactly the same

as that of sound in air, and Michelson's experiment as well as the

Maxwell-Lorentz theory of light are contradicted. This theory is now

universally accepted, and Michelson's experiment has been carefully

repeated by other observers, and fully confirmed. This is the very

heart of the relativity question.

If we state the matter objectively it comes to this. The velocity

of light with regard to the ether is a variable quantity, depending

merely on where the observer chooses to go. As Eddington well says,

"these relations to the ether have no effect on the phenomena and can

be disregarded--a step which appears to divest the ether of the last

remnants of substantiality."4

The only way of avoiding this apparent absurdity seems to be to

consider that the ether moves with the earth. Michelson's result would

then be fully explained. Of course this can only be true for a few

miles above the earth's surface. Beyond that the ether must either

be stationary or move with the sun. The velocity of light with regard

to the ether would then be a constant, just as the velocity of sound

is constant with regard to the air. This would contradict Einstein's

second principle as it is generally understood. The trouble with

this suggestion is that it fails to account for aberration, which,

as already explained, appears to require that the earth should be

moving through the ether. To meet this emergency would involve some

modification of the undulatory theory of light, which apparently

would not be impossible, but has not yet been made.

In 1915 Einstein brought out an extension of his first principle. This

he calls the "general theory of relativity." It states that in our

choice of coordinate systems we "should not be limited in any way

so far as their state of motion is concerned."1 This leads to the

three astronomical consequences mentioned later in this paper, two

of which have been more or less confirmed, and the third practically

contradicted as far as quantitative measures are concerned.5

As is well known the kinetic energy of a moving body may be

expressed as $e = 1/2mv^2$, but if the body is charged electrically,

the fraction becomes $1/2(m + m')v^2$, where $m'$ is a quantity

dependent on the square of the electrical charge. That is to say,

we have the normal mass of the body, and also what we may call its

electrical mass. If when in this condition a portion of the mass is

electrical, the question at once occurs to us, why may not the whole

mass be electrical, in other words, a form of energy? Although this

has not been satisfactorily proved hitherto, yet such is the general

belief among physicists. As Einstein puts it "inert mass is nothing

else than latent energy."1 The same idea is sometimes expressed as

"the mass of ordinary matter is due to the electromagnetic energy of

its ultimate particles, and electromagnetic energy wherever found

must possess mass, i.e., inertia."6 If that is so, since a ray of

light on the undulatory theory is a form of electromagnetic energy,

it too must possess mass. Since all mass with which we are familiar is

subject to the attraction of gravitation, it seemed likely that a ray

of light would be bent out of its course in passing near the sun, and

this as we have seen was proved to be true at the recent solar eclipse.

That portion of the mass of a body due to its electrical charge can

be readily shown experimentally to vary with the velocity of the

body. Einstein has shown the same to be true of the normal mass,

as is illustrated in the advance of the perihelion of the orbit

of Mercury. He has also pointed out that gravitation, inertia and

centrifugal force are all closely related, and obey similar laws. Thus

if we rise from the earth with accelerated velocity, we apparently

increase our weight. Again if the velocity of rotation of the earth on

its axis should be increased, our weight would be diminished. These

facts are suggestive when we come to consider the ultimate cause

of gravitation.

Another fact which must be rather startling to the older school

of scientists is that momentum is no longer simply $mv$, mass times

velocity, but that the velocity of light $c$, comes into the question,

and the formula for momentum now assumes the form of

$$\frac{m v}{\sqrt{1 - \frac{v^2}{c^2}}}$$

For ordinary velocities this correction is extremely small, but it

has been shown to be necessary, both theoretically and experimentally,

when dealing with the high velocities with which we are now familiar.

The theory of relativity is so widespread in its application that

several other theories have become more or less intimately combined

with it, for which Einstein is in no way responsible. One of these is

known as the Fitzgerald-Lorentz theory, that all bodies are subject to

a contraction in the direction of their motions through space. This was

first suggested in order to explain the Michelson-Morley experiment,

but has proved inadequate to do so, particularly when the observer is

receding from the source. This contraction is expressed by the same

factor used in the denominator of the revised expression for momentum,

given above. Again the quantity $c$ is so enormous, that even for

large bodies at planetary velocities the contraction amounts to very

little. Thus the earth moving at a speed of eighteen miles per second

in its orbit, is flattened only 1/200,000,000, or 2.5 inches. On the

other hand for high velocities of many thousand miles per second,

such as we have become familiar with in the case of the radioactive

substances, the flattening is a very considerable fraction of the

diameter of the moving body, one-half or more, and in the case of

the corpuscles of light, if that theory were adopted, this flattening

becomes equal to the diameter, and their thickness is reduced to zero.

When we view Einstein's theories from the astronomical standpoint,

the earliest fact bearing on relativity that we need consider was

the discovery of aberration, by Bradley, in 1726, as seen above. In

1872 Airy observed the star g Draconis through a telescope filled

with water. Since the velocity of light is less in water than in

air, we should naturally expect to find the aberration appreciably

increased. It was found, on the other hand, however, to be unaffected.

In 1887 the results of the famous Michelson-Morley experiment were

published.7 In this experiment the velocity of light was measured

in various directions with regard to the motion of the earth in its

orbit. If the ether were stationary, and the earth moving through it,

different velocities should be obtained in different directions. Such

was not the case however, and the experiment indicated that the ether

moved with the earth. It thus flatly contradicted the conclusions

founded on aberration.

Einstein's Special Theory of Relativity, of 1905, as we have seen,

resolves this contradiction. But as we shall presently see, it is the

General Theory, of 1915, that leads to astronomical applications of

broad scope. It indicates, for instance, that there is no essential

difference between gravitation and inertia. This idea may be crudely

illustrated by our feelings of increased weight when an elevator

starts rapidly upwards. A man while falling freely in space ceases

to feel the pull of gravitation.

But we must not as yet conceive of the theory of relativity as a

universally accepted and unquestioned truth of science. Eddington is

its leading English exponent, and he is supported by such men as Jeans,

Larmor, and Jeffreys. On the other hand, the theory has been severely

criticised by Lodge, Fowler, Silberstein, and Sampson. Few American

scientists have expressed any opinions in print on the subject,

and the recent eclipse observations, to which we shall refer later,

are to be repeated with more suitable instruments for verification

in 1922, in the hope of obtaining more accurate and accordant results.8

An appurtenance of the Einstein theories which bears much the

same relation to them as does the Lorentz-Fitzgerald contraction,

mentioned above, is the idea, first clearly stated by Minkowski,

that time is a kind of space--a fourth dimension. This the reader

will doubtless find to be the most difficult portion of the theory

to picture in his own mind. It is entirely unsupported by experiment

or observation, necessarily so, and is based wholly on mathematical

and philosophical conceptions. Our distinction between space and time

seems to be that the direction in which we progress without effort

is time; the other directions, in which we have to make an exertion

to move ourselves, or in which we are carried, are space. How many

dimensions empty space may have, we really have no means of knowing,

because we can neither see nor feel it. Matter we know has three,

length, breadth, and thickness, also that it lies remote from us in

three corresponding directions. These facts may have given us the

erroneous impression that space too has only three dimensions. Now

it is claimed that time is a fourth, and that there are also others.

In order to illustrate this, Eddington asks us to imagine a movie film

taken of a man or of any moving object. Let the separate pictures be

cut apart and piled on one another. This would form a sort of pictorial

history of the individual for a brief interval in his life, in the form

of a cube. If we attempt to pick it up, it falls apart, thus clearly

showing the difference between time and space. But suppose it now

all glued together in one solid cube, so that it is no easier to cut

a section in one direction than in another. That is Minkowski's idea

of space and time, and further, that the direction in which we should

cut it depends merely on the velocity with which we are moving through

space. I should cut it parallel to the films, but a man on a rapidly

moving star, in order to separate it into space and time, would cut

it in an inclined direction. That is a thing which may be true, but it

is one which we believe no mortal man can clearly picture to himself.

On the other hand Turner has recently made a very interesting

point,9 namely, that the fourth dimension as actually treated

by the mathematicians is not time itself, but time multiplied

by a constant--the velocity of light.10 Without affecting the

astronomical proofs of relativity at all, this simplifies our

conceptions enormously. In ordinary everyday life time and space

cannot be identical, any more than a yard can be identical with a

quart. On what is known to physicists as the centimeter-gram-second

system, distance is represented by $l$, mass by $m$, and time by

$t$. Velocity is then distance divided by time, $l/t$, or as we say

in English units, so many feet per second, and the fourth dimension

may be expressed as time multiplied by velocity, $t \times l/t =

l$. That is to say, it is simply distance, just like the other three

dimensions. To say that time is the fourth dimension from this point

of view, appears to us just as ridiculous as it would be to attempt

to measure the velocity of a train in quarts. It is quite correct,

however, although unusual, to speak of a given train as moving at a

speed of 10 quarts per square inch per second, $l^3/l^2 t = l/t$. This

would be equivalent to a velocity of 33 miles per hour.

If I wish to give a complete dimensional description of myself in my

four dimensions, I must give my length, my breadth, and my thickness,

ever since I came into being, and also the course I have traversed

through space since that time. This latter distance will be expressed

in terms of a unit whose length is 186,000 miles, the distance

traversed by light in one second. The distance which I travel through

space annually is enormous, and very complex as to direction. It

involves not merely my own motions as I cross the room, or take a

train or steamer, but also those due to the rotation of the earth on

its axis, its revolution round the sun, and the motion of the latter

through the heavens. In general I travel, or in other words increase

my length in the fourth dimension, by over 4,000 units a year. The

fourth dimension accordingly, if this view is accepted, is simply a

distance like the other three, and perfectly easy to understand.

We now come to the three actual tests by which the theory has been

tried. The planets as is well known revolve about the sun in ellipses,

with the sun in one of the foci. That is to say, the sun is not in

the center, but a little on one side of it. The end of the ellipse

where the planet comes nearest to the sun is called the perihelion,

and here the planet is moving most rapidly. The other end is called

the aphelion, and here the motion is slowest. According to Newton's

theory of gravitation, if a spherical sun possesses a single planet

or companion, its orbit will be permanently fixed in space unless

perturbed by some other body. If a second planet exist, it will

cause the perihelion of the first slowly to advance. According to

Einstein the mass of a planet depends in part on its velocity. It

will therefore be less at aphelion where it is moving slowly than at

perihelion where it is moving rapidly, consequently in addition to

the Newtonian attraction we have another one which increases as we

approach the sun. The effect of this will be to cause the perihelion

of the orbit to advance, whether there is a second planet or not.

Among the larger planets Mercury has the most eccentric orbit, and it

also moves most rapidly, so that it is particularly well adapted to

test the relativity theory. The observed advance of its perihelion is

574'' per century, instead of the theoretical figure 532'', due to the

other planets--a difference of 42''.11 This has long been a puzzling

discrepancy between observation and the law of gravitation. Prior to

Einstein, attempts were made to eliminate it by assuming a certain

oblateness of the solar disk. If the equatorial diameter exceeded the

polar by only 0''.5 the whole advance would be accounted for, but not

only has this ellipticity failed of detection, but if it existed,

it should produce a very noticeable and inadmissible change in the

inclination of Mercury's orbit, amounting to about 3'' per century,

as has been demonstrated by both Herzer and Newcomb.12

Einstein from computations alone, without introducing any new

constants or hypotheses whatever, showed, if the theory of relativity

be accepted, that the sun should produce an acceleration of 43''

per century, thus entirely accounting for the observed discrepancy,

far within the limits of accuracy of the observations. The only other

planet whose orbit has a large eccentricity, and that is suitable

for investigation, is the planet Mars. Here the discrepancy between

observation and theory is very slight, only 4'', and a portion of

that may be due to the attraction of the asteroids. This deviation

is so slight that it may well be due entirely to accidental errors

of observation, but however that may be, Einstein's theory reduces

it to 2''.7.

This all seems very satisfactory and complete, but the trouble

with it is that the coincidence for Mercury is rather too good. It

is based on the assumption that the sun is a perfect sphere, and

that the density of its surface is uniform from the equator to the

poles. This would doubtless be true if the sun did not revolve on

its axis. In point of fact it does revolve, in a period in general of

about 26 days. Consequently an object on its equator must experience

a certain amount of centrifugal force. Therefore if its surface were

of uniform density the shape of the sun would be an oblate spheroid.

It can be readily shown that the theoretical excess of the equatorial

over the polar diameter, due to the centrifugal force, should amount to

only 0''.04, an amount which could hardly be detected by observation,

and might readily be concealed by a slight excess of equatorial over

polar density. Any reasonable excess of density at the center would

diminish this result but slightly. The molecular weight of the central

material13 is probably about 2. This computed equatorial excess is

one-twelfth of the amount necessary to cause the observed advance,

and should therefore cause an advance of the perihelion of about 3''.5

per century, reducing the difference between the observed advance

and that caused by gravitation to 38''.5. According to Einstein the

advance due to relativity should be, as we saw, 43'', a discrepancy

of 4''.5 per century, or 10 per cent. Jeffreys has remarked that

any discrepancy such as 10'' "would be fatal to a theory such as

Einstein's, which contains no arbitrary constituent capable of

adjustment to suit empirical facts."14 It must be pointed out here

however, that so far as known, this small correction to the motion

of Mercury's perihelion has not previously been suggested, so that

there has been no opportunity hitherto for its criticism by others.

It was due largely to the success with Mercury that it was decided to

put the relativity theory to another test. According to the Newtonian

theory, as stated by Newton himself, corpuscles as well as planets

have mass, and must therefore be attracted by the sun. According to

Einstein, owing to their high velocity, this attraction must be twice

as great as it would be according to the theory of gravitation. If the

ray of light proceeding from a star were to pass nearly tangent to the

sun's limb it should be deflected 0''.87 according to Newton. According

to the theory of relativity it should be deflected 1''.75. Stars

of course cannot usually be observed near the sun. It is therefore

necessary to take advantage of a total solar eclipse, when the sun is

completely hidden by the moon, in order to secure these observations.

Two expeditions, one to Africa, and one to South America, observed

successfully the total eclipse of May 29, 1919. The former was

located on the Island of Principe in the Gulf of Guinea. The latter

was located at Sobral, Brazil. Their equipment and results are shown

in the following table, where the successive columns give the location,

the aperture in inches of the telescopes employed, their focus in feet,

the number of plates secured, the number of stars measured, their

mean deduced deflection from their true positions by the attraction

of the sun, and the deviations from the theoretical results.15 In

the first and last line of the table shown herewith, this

Location Aperture Focus Plates Stars Defl. Dev.

Principe 13 11 2 5 1''.60 -0''.15

Sobral 13 11 19 12 0 .93 (+0 .06)

,,   4 19 8 7 1 .98 +0 .23

deviation is taken from Einstein's computed value of 1''.75. In the

second line the difference shown is from the value required by the

Newtonian theory, 0''.87. The results obtained with this telescope

were rejected however, although they were much the most numerous,

because it was found that for some reason, supposed to be the heating

of the mirror by the sun before the eclipse, the star images were

slightly out of focus, and were therefore considered unreliable. The

results with the two other telescopes were not very accordant, but the

4-inch had the longer focus, secured the greater number of plates,

and showed the greater number of stars. The results obtained with

it therefore appear to have been the more reliable. They differ from

Einstein's prediction by 13 per cent. In future expeditions to test

this question, the mirror in front of the telescope will be eliminated.

We now come to the final test which has been applied to Einstein's

theory. Einstein showed that in the intense gravitational field of the

sun, the theory of relativity required that all of the spectrum lines

should be shifted slightly toward the red end. The shift however is

exceedingly small, and can only be detected and measured with the

most powerful modern instruments. Moreover only certain lines can

be used, because owing to varying pressure in the solar atmosphere,

which affects many lines, as well as to rapid motion in the line of

sight, which may affect all of them, still larger displacements are

liable to occur.

According to the theory of relativity the displacement of the lines

should be $+0.0080 A$. St. John at Mt. Wilson found a displacement

for the cyanogen lines of only $+0.0018 A$.16 Evershed at Kodaikanal

found +0.0060 at the north pole of the sun, and +0.0080 at the south

pole. These latter values however were only for the stronger lines. The

weaker lines give much smaller shifts, as do those of calcium and

magnesium.17 According to Einstein all lines should give nearly the

same shift, an amount proportional to the wave length. It therefore

appears that we must conclude by saying that Einstein's theory of

relativity has been partially, but not completely, verified.

The reference numbers in the above text have nothing to do with the

numbers used in other parts of this volume to acknowledge the work

of the various contestants; they refer to Dr. Pickering's sources,

as follows:

1 Journ. Brit. Astron. Assoc., 1919, 30, 76.

2 Comptes Rendus, 165, 424, and 167, 71.

3 Monthly Notices R. A. S., 1919, 80, 104.

4 Monthly Notices R. A. S., 1917, 77, 379.

5 Astro-Physical Journal, 1917, 46, 249. Journ. Brit. Astron. Assoc.,

1920, 30, 276.

6 Monthly Notices, R. A. S., 1917, 77, 377.

7 Amer. Journ. Sci., 34, 333.

8 Monthly Notices, R. A. S., 1920, 80, 628.

9 The Observatory 1920, April. From an Oxford Note Book.

10 Monthly Notices, R. A. S., 1917. 78, 3 De Sitter, 1919, 80, 121,

Jeans, 80, 146 Jeffreys.

11 "Gravitation and the Principle of Relativity," Eddington. Royal

Institution of Great Britain, 1918.

12 Journ. Brit. Astron. Assoc., 1920, 30, 125.

13 "The Interior of a Star," Eddington. Scientia, 1918, 23, 15.

14 Monthly Notices, R. A. S., 1919, 80, 138.

15 Monthly Notices. R. A. S., 1920, 80,

415. Journ. Brit. Astron. Assoc., 1919, 30, 46.

16 Astro-Physical Journ., 1917, 46, 249.

17 Journ. Brit. Astron. Assoc., 1920, 30, 276.

XVIII

THE PRACTICAL SIGNIFICANCE OF RELATIVITY

The Best Discussion of the Special Theory Among All the Competing

Essays

BY PROFESSOR HENRY NORRIS RUSSELL, PRINCETON UNIVERSITY

Can a small child catch a baseball moving sixty miles an hour without

getting hurt? We should probably answer "No"--but suppose that the boy

and his father were sitting side by side in an express train, and the

ball was tossed lightly from one to the other. Then there would be

no trouble about it, whether the train was standing still, or going

at full speed. Only the relative motion of ball and boy would count.

This every-day experience is a good illustration of the much discussed

Principle of Relativity, in its simplest form. If there were no

jolting, the motion of the train, straight ahead at a uniform speed,

would have no effect at all upon the relative motions of objects

inside it, nor on the forces required to produce or change these

motions. Indeed, the motion of the earth in its orbit, which is free

from all jar, but a thousand times faster, does not influence even

the most delicate apparatus. We are quite unconscious of it, and

would not know that the earth was moving, if we could not see other

bodies outside it. This sort of relativity has been recognized for

more than two centuries and lies at the bottom of all our ordinary

dynamical reasoning, upon which both science and engineering are based.

But there are other things in nature besides moving bodies,--above

all, light, which is intimately related to electricity and magnetism,

and can travel through empty space, between the stars. It moves at

the enormous speed of 186,000 miles per second, and behaves exactly

like a series of vibrations or "waves." We naturally think of it as

travelling through some medium, and call this thing, which carries

the light, the "ether."

Can we tell whether we are moving through this ether, even though all

parts of our apparatus move together, and at the same rate? Suppose

that we have two mirrors, $M$ and $N$, at equal distances, $d$, from a

point $O$, but in directions at right angles to one another, and send

out a flash of light from $O$. If everything is at rest, the reflected

flashes will evidently come back to $O$ at the same instant, and the

elapsed time will be $2d/c$ seconds if $c$ is the velocity of light.

But suppose that $O$, $M$, and $N$ are fastened to a rigid frame work,

and all moving in the direction $O M$, with velocity $V$. The light

which goes from $O$ toward $M$, at the speed $c$, will overtake it

with the difference of their speeds, $c-v$, taking $d/(c-v)$ seconds to

reach $M$. On the way back, $O$ will be advancing to meet it, and the

return trip will occupy $d/(c+v)$ seconds. The elapsed time for the

round trip comes out $2cd/(c^2-v^2)$ seconds, which is longer than

when the system was at rest--the loss of time in the "stern chase"

exceeding the saving on the return.

The light which is reflected from N has a different history. When

it starts, $O$ and $N$ have certain positions in the ether, $O_1$

and $N_1$. By the time it reaches the mirror, this is at $N_2$, and

$O$ is at $O_2$, and when it returns, it finds $O$ at $O_3$. The

distances for the outward and inward journeys are now equal, but

(as is obvious from the figure), each of them is greater than $d$,

or $O_2N_2$, and the time for the round trip will be correspondingly

increased. A simple calculation shows that it is $2d/\sqrt{c^2-v^2}$.

The increase above the time required when the system was at rest

is less in this case than the preceding. Hence, if the apparatus

is moving through the ether, the flashes reflected from $M$ and $N$

will not return at the same instant.

For such velocities as are attainable--even the 18 miles per second of

the earth in its orbit--the difference is less than a hundred-millionth

of the elapsed time. Nevertheless, Michelson and Morley tried to

detect it in their famous experiment.

A beam of light was allowed to fall obliquely upon a clear glass

mirror (placed at $O$ in the diagram) which reflected part of it

toward the mirror, $M$, and let the rest pass through to the mirror

$N$. By reuniting the beams after their round trips, it was possible

to tell whether one had gained upon the other by even a small fraction

of the time of vibration of a single light wave. The apparatus was

so sensitive that the predicted difference, though amounting to less

than a millionth part of a billionth of a second, could easily have

been measured; but they actually found no difference at all--though

the earth is certainly in motion.

Other optical experiments, more intricate, and even more delicate,

were attempted, with the same object of detecting the motion of the

earth through the ether; and they all failed.

THE SPECIAL THEORY AND ITS SURPRISING CONSEQUENCES

It was upon these facts that Einstein based his original, or "special"

theory of Relativity. He assumed boldly that the universe is so

constituted that uniform straight-ahead motion of an observer and all

his apparatus will not produce any difference whatever in the result

of any physical process or experiment of any kind. Granting this,

it follows that if all objects in the visible universe were moving

uniformly together in any direction, no matter how fast, we could

not find this out at all. We cannot determine whether the universe,

as a whole, is at rest or in motion, and may as well make one guess

as another. Only the relative motions of its parts can be detected

or studied.

This seems simple and easy enough to understand. But the consequences

which follow from it are extraordinary, and at first acquaintance

seem almost absurd.

In the first place, if an observer measures the velocity of light,

he must always get the same result, no matter how fast he and his

apparatus are moving, or in what direction (so long as the motion is

uniform and rectilinear). This sounds harmless; but let us go back to

the Michelson-Morley experiment where the light came back in exactly

the same time from the two mirrors. If the observer supposes himself

to be at rest, he will say that the distances $O M$ and $O N$ were

equal. But if he fancies that the whole universe is moving in the

direction $O M$, he will conclude that $M$ is nearer to $O$ than $N$

is--for if they were equidistant, the round-trip would take longer in

the first case, as we have proved. If once more he fancies that the

universe is moving in the direction $O N$, he will conclude that N is

nearer to $O$ than $M$ is. His answer to the question which of the

two distances, $O M$ or $O N$, is the greater will therefore depend

on his assumption about the motion of the universe as a whole.

Similar complications arise in the measurement of time. Suppose that

we have two observers, $A$ and $B$, provided with clocks which run

with perfect uniformity, and mirrors to reflect light signals to one

another. At noon exactly by his clock, $A$ sends a flash of light

towards $B$. $B$ sees it come in at 12:01 by his clock. The flash

reflected from $B$'s mirror reaches $A$ at 12:02 by $A$'s clock. They

communicate these observations to one another.

If $A$ and $B$ regard themselves as being at rest, they will agree that

the light took as long to go out as it did to come back, and therefore

that it reached $B$ at just 12:01 by $A$'s clock, and that the two

clocks are synchronized. But they may, if they please, suppose that

they (and the whole universe) are moving in the direction from $A$

towards $B$, with half the speed of light. They will then say that

the light had a "stern chase" to reach $B$, and took three times as

long to go out as to come back. This means that it got to $B$ at 1

1/2 minutes past noon by $A$'s clock, and that $B$'s clock is slow

compared with $A$'s. If they should assume that they were moving with

the same speed in the opposite direction, they would conclude that

$B$'s clock is half a minute fast.

Hence their answer to the question whether two events at different

places happen at the same time, or at different times, will depend

on their assumption about the motion of the universe as a whole.

Once more, let us suppose that $A$ and $B$, with their clocks and

mirrors, are in relative motion, with half the speed of light, and pass

one another at noon by both clocks. At 12:02 by $A$'s clock, he sends a

flash of light, which reaches $B$ at 12:04 by his clock, is reflected,

and gets back to $A$'s clock at 12:06. They signal these results to

each other, and sit down to work them out. $A$ thinks that he is at

rest, and $B$ moving. He therefore concludes that the light had the

same distance to go out as to return to him and took two seconds each

way, reaching $B$ at 12:04 by $A$'s clock, and that the two clocks,

which agreed then, as well as at noon, are running at the same rate.

$B$, on the contrary, thinks that he is at rest and $A$ in motion. He

then concludes that $A$ was much nearer when he sent out the flash

than when he got it back, and that the light had three times as far to

travel on the return journey. This means that it was 12:03 by $A$'s

clock at the instant when the light reached $B$ and $B$'s clock read

12:04. Hence $A$'s clock is running slow, compared with $B$'s.

Hence the answer to the question whether two intervals of time,

measured by observers who are in motion relative to one another, are

of the same or of different durations, depends upon their assumptions

about the motion of the universe as a whole.

Now we must remember that one assumption about the motion of the

universe as a whole is exactly as good--or bad--as another. No

possible experiment can distinguish between them. Hence on the

Principle of Relativity, we have left no absolute measurement of

time or space. Whether two distances in different directions are to

be called equal or not--whether two events in different places are

to be called simultaneous or not--depends on our arbitrary choice of

such an assumption, or "frame of reference." All the various schemes

of measurement corresponding to these assumptions will, when applied

to any imaginable experiment, predict exactly the same phenomena. But,

in certain important cases, these predictions differ from those of the

old familiar theory, and, every time that such experiments have been

tried, the result has agreed with the new theory, and not with the old.

We are therefore driven to accept the theory of relativity, strange

as it is, as being more nearly "true to nature" than our older

ideas. Fortunately, the difference between the results of the two

become important only when we assume that the whole visible universe is

moving together much faster than any of its parts are moving relatively

to one another. Unless we make such an unwarranted assumption, the

differences are so small that it takes the most ingenious and precise

experiments to reveal them.

THE GENERALIZATION

Not content with all this, Einstein proceeded, a few years ago, to

develop a "general" theory of relativity, which includes the effects

of gravitation.

To make this idea clear, let us imagine two observers, each, with

his measuring instruments, in a large and perfectly impervious box,

which forms his "closed system."

The first observer, with his box and its contents, alone in space,

is entirely at rest.

The second observer, with his box and its contents, is, it may be

imagined, near the earth or the sun or some star, and falling freely

under the influence of its gravitation.

This second box and its contents, including the observer, will then

fall under the gravitational force, that is, get up an ever-increasing

speed, but at exactly the same rate, so that there will be no tendency

for their relative positions to be altered.

According to Newton's principles, this will make not the slightest

difference in the motions of the physical objects comprising the system

or their attractions on one another, so that no dynamical experiment

can distinguish between the condition of the freely falling observer

in the second box and the observer at rest in the first.

But once more the question arises: What could be done by an optical

experiment?

Einstein assumed that the principle of relativity still applied in

this case, so that it would be impossible to distinguish between the

conditions of the observers in the two boxes by any optical experiment.

It can easily be seen that it follows from this new generalized

relativity that light cannot travel in a straight line in a

gravitational field.

Imagine that the first observer sets up three slits, all in a straight

line. A ray of light which passes through the first and second will

obviously pass exactly through the third.

Suppose the observer in the freely falling system attempts the same

experiment, having his slits $P$, $Q$, $R$, equally spaced, and placing

them at right angles to the direction in which he is falling. When

the light passes through $P$, the slits will be in certain position

$P_1 Q_1 R_1$ (Figure). By the time it reaches $Q$, they will have

fallen to a lower level, $P_2 Q_2 R_2$, and when it reaches $R$,

they will be still lower, $P_3 Q_3 R_3$. The times which the light

takes to move from $P$ to $Q$ and $Q$ to $R$ will be the same: but,

since the system is falling ever faster and faster the distance $R_2

R_3$ will be greater than $Q_1 Q_2$. Hence, if the light which has

passed through $P$ and $Q$ moves in a straight line, it will strike

above $R$, as is illustrated by the straight line in the figure. But,

on Einstein's assumption, the light must go through the third slit,

as it would do in the system at rest, and must therefore move in a

curved line, like the curved line in the figure, and bend downward

in the direction of the gravitational force.

THE TESTS

Calculation shows that the deviation of light by the moon or planets

would be too small to detect. But for a ray which had passed near

the sun, the deflection comes out 1.7'', which the modern astronomer

regards as a large quantity, easy to measure. Observations to test

this can be made only at a total eclipse, when we can photograph stars

near the sun, on a nearly dark sky. A very fine chance came in May,

1919, and two English expeditions were sent to Brazil and the African

coast. These photographs were measured with extreme care, and they

show that the stars actually appear to be shifted, in almost exactly

the way predicted by Einstein's theory.

Another consequence of "general relativity" is that Newton's law of

gravitation needs a minute correction. This is so small that there is

but a single case in which it can be tested. On Newton's theory, the

line joining the sun to the nearest point upon a planet's orbit (its

perihelion) should remain fixed in direction, (barring certain effects

of the attraction of the other planets, which can be allowed for). On

Einstein's theory it should move slowly forward. It has been known for

years that the perihelion of Mercury was actually moving forward, and

all explanations had failed. But Einstein's theory not only predicts

the direction of the motion, but exactly the observed amount.

Einstein also predicts that the lines of any element in the solar

spectrum should be slightly shifted towards the red, as compared

with those produced in our laboratories. Different observers have

investigated this, and so far they disagree. The trouble is that

there are several other influences which may shift the lines, such

as pressure in the sun's atmosphere, motion of currents on the sun's

surface, etc., and it is very hard to disentangle this Gordian knot. At

present, the results of these observations can neither be counted

for or against the theory, while those in the other two cases are

decisively favorable.

The mathematical expression of this general relativity is intricate

and difficult. Mathematicians--who are used to conceptions which

are unfamiliar, if not incomprehensible, to most of us--find that

these expressions may be described (to the trained student) in terms

of space of four dimensions and of the non-Euclidean geometry. We

therefore hear such phrases as "time as a sort of fourth dimension,"

"curvature of space" and others. But these are simply attempts--not

altogether successful--to put mathematical relationships into ordinary

language, instead of algebraic equations.

More important to the general reader are the physical bearings of

the new theory, and these are far easier to understand.

Various assumptions which we may make about the motion of the universe

as a whole, though they do not influence the observed facts of nature,

will lead us to different ways of interpreting our observations as

measurements of space and time.

Theoretically, one of these assumptions is as good as any other. Hence

we no longer believe in absolute space and time. This is of great

interest philosophically. Practically, it is unimportant, for,

unless our choice of an assumption is very wild, our conclusions and

measurements will agree substantially with those which are already

familiar.

Finally, the "general" relativity shows that gravitation and

electro-magnetic phenomena--(including light) do not form two

independent sides of nature, as we once supposed, but influence one

another (though slightly) and are parts of one greater whole.

XIX

EINSTEIN'S THEORY OF RELATIVITY

A Simple Explanation of His Postulates and Their Consequences

BY T. ROYDS, KODAIKANAL OBSERVATORY, INDIA

Einstein's theory of relativity seeks to represent to us the world as

it really is instead of the world of appearances which may be deceiving

us. When I was in town last week to buy 5 yards of calico I watched

the draper very carefully as he measured the cloth to make sure I was

not cheated. Yet experiment can demonstrate, and Einstein's theory

can explain, that the draper's yardstick became longer or shorter

according to the direction in which it was held. The length of the

yardstick did not appear to me to change simply because everything else

in the same direction, the store, the draper, the cloth, the retina

of my eye, changed length in the same ratio. Einstein's theory points

out not only this, but every case where appearances are deceptive,

and tries to show us the world of reality.

Einstein's theory is based on the principle of relativity and

before we try to follow his reasoning we must spend a little time

in understanding what he means by "relativity" and in grasping how

the idea arises. Suppose I wish to define my motion as I travel

along in an automobile. I may be moving at the rate of 25 miles an

hour relative to objects fixed on the roadside, but relative to a

fellow-passenger I am not moving at all; relative to the sun I am

moving with a speed of 18 1/2 miles per second in an elliptical orbit,

and again relative to the stars I am moving in the direction of the

star Vega at a speed of 12 miles per second. Thus motion can only be

defined relative to some object or point of reference. Now this is

not satisfactory to the exact scientist. Scientists are not content

with knowing, for example, that the temperature of boiling water is

+100° C. relative to the temperature of freezing; they have set out

to determine absolute temperatures and have found that water boils at

373° C. above absolute zero. Why should I not, therefore, determine

the absolute motion of the automobile, not its motion relative to

the road, earth, sun or stars, but relative to absolute rest?

Michelson and Morley set out in their famous experiment to measure the

absolute velocity of their laboratory, which was, of course, fixed

on the earth. The experiment consisted of timing two rays of light

over two equal tracks at right angles to each other. When one track

was situated in the direction of the earth's motion they expected to

get the same result as when two scullers of equal prowess are racing

in a river, one up and down the stream and the other across and back;

the winner will be the sculler rowing across the stream, as working

out an example will convince. Even if the earth had been stationary

at the time of one experiment, the earth's motion round the sun would

have been reversed 6 months later and would then have given double the

effect. They found, however, that the two rays of light arrived always

an exact dead heat. All experimenters who have tried since have arrived

at the same result and found it impossible to detect absolute motion.

The principle of relativity has its foundation in fact on these

failures to detect absolute motion. This principle states that the

only motion we can ever know about is relative motion. If we devise

an experiment which ought to reveal absolute motion, nature will

enter into a conspiracy to defeat us. In the Michelson and Morley

experiment the conspiracy was that the track in the direction of the

earth's absolute motion should contract its length by just so much

as would allow the ray of light along it to arrive up to time.

We see, therefore, that according to the principle of relativity

motion must always remain a relative term, in much the same way as

vertical and horizontal, right and left, are relative terms having

only meaning when referred to some observer. We do not expect to

find an absolute vertical and are wise enough not to attempt it;

in seeking to find absolute motion physicists were not so wise and

only found themselves baffled.

The principle that all motion is relative now requires to be worked

out to all its consequences, as has been done by Einstein, and we

have his theory of relativity. Einstein conceives a world of four

dimensions built up of the three dimensions of space, namely up and

down, backwards and forwards, right and left, with time as the fourth

dimension. This is an unusual conception to most of us, so let us

simplify it into something which we can more easily picture but which

will still allow us to grasp Einstein's ideas. Let us confine ourselves

for the present to events which happen on this sheet of paper, i.e.,

to space of two dimensions only and take time as our third dimension

at right angles to the plane of the paper. We have thus built up a

three dimensional world of space-time which is every bit as useful

to us as a four dimensional representation so long as we only need

study objects moving over the sheet of paper.

Suppose a fly is crawling over this sheet of paper and let us make

a movie record of it. If we cut up the strip of movie film into the

individual pictures and cement them together one above another in

their proper order, we shall build up a solid block of film which

will be a model of our simplified world of space-time and in which

there will be a series of dots representing the motion of the fly

over the paper. Just as I can state the exact position of an object

in my room by defining its height above the floor, its distance

from the north wall and its distance from the east wall, so we can

reduce the positions of the dots to figures for use in calculations

by measuring their distances from the three faces intersecting in the

lines $OX$, $OY$, and $OT$, where $OXAYTBCD$ represents the block of

film. The mathematician would call the three lines $OX$, $OY$, $OT$

the coordinate axes. Measuring all the dots in this way we shall

obtain the motion of the fly relative to the coordinate axes $OX$,

$OY$, $OT$. If we add a block $OTDYEFGH$ of plain film we can use

$EX$, $EH$, $EF$ as coordinate axes and again obtain the motion of

the fly relative to these new axes; or we can add block after block

so as to keep the axes moving. We can conceive of other changes of

axes. The operator making the movie record might have taken the fly

for the hero of the piece and moved the camera about so as to keep the

fly more or less central in the picture; or he might, by turning the

handle first fast and then slow and by moving the camera, have made

the fly appear to be doing stunts. Moving the camera would change the

axes of $x$ and $y$, and turning the handle at different speeds would

change the axis of time. Again, we might change the axes by pushing

the block out of shape or by distorting it into a state of strain.

Whatever change of axes we make, any dot in the block of film will

signify a coincidence of the fly with a certain point of the paper

at a certain time, and the series of dots will, in every case, be a

representation of the motion of the fly. Maybe the representation

will be a distorted one, but who is to say which is the absolutely

undistorted representation? The principle of relativity which we laid

down before says that no one set of coordinates will give the absolute

motion of the fly, so that one set is as good as another. The principle

that all motion is relative means, therefore, that no matter how we

change our coordinates of space-time, the laws of motion which we

deduce must be the same for all changes.

To use an analogy, the sculptured head of Shakespeare on my table may

appear to have hollow cheeks when I admit light from the east window

only, or to have sunken eyes with light from the skylight in the roof,

but the true shape of the head remains the same in all lights.

Hence, if with reference to two consecutive dots in our block of

film a mathematical quantity can be found which will not change no

matter how we changes our axes of coordinates, that quantity must

be an expression of the true law of motion of the fly between the

two points of the paper and the two times represented by these two

dots. Einstein has worked out such a quantity remaining constant for

all changes of coordinates of the four dimensional world of space-time.

In passing we may notice a feature of Einstein's world of space-time

which we shall doubtless find it difficult to conceive, namely,

that there is no essential difference between a time and a distance

in space. Since one set of coordinates is as good as another, we

can transform time into space and space into time according as we

choose our axes. For example if we change $OX$, $OT$, the axes of $x$

and time in Fig. 2, into $OX'$, $OT'$ by a simple rotation, the new

time represented by $OT'$ consists partly of $OA$ in the old time

and partly of $OB in the old $x$ direction. Referring to our block

of movie film again, it means that although I might separate the

block into space and time by slicing it into the original pictures,

I can just as readily slice it in any direction I choose and still

get individual pictures representing the motion of the fly but with,

of course, new time and space. So whilst I may be believe that a liner

has travelled 3,000 miles in 4 days, an observer on a star who knows

nothing of my particular axes in space-time may say, with equal truth,

that it went 2,000 miles in 7 days. Thus, time and space are not two

separate identities in Einstein's view; there only exists a world of

four dimensions which we can split up into time and space as we choose.

Let us see now how Einstein explains gravitation. When a body is not

acted on by any forces (except gravitation) the quantity which remains

constant for all changes of coordinates implies that the body will

follow that path in the space of an outside observer which takes the

least time. It is an observed fact that one body attracts another by

gravitation; that is, the path of one body is bent from its course by

the presence of another. Now we can bend the path of the fly in our

block of film by straining the block in some way. Suppose, therefore,

that I strain the world so as to bend the path of a body exactly as

the gravitation due to some other body bends it; i.e., by a change

of coordinates I have obtained the same effect as that produced

by gravitation. Einstein's theory, therefore, explains gravitation

as a distortion of the world of space-time due to the presence of

matter. Suppose first that a body is moving with no other bodies near;

according to Einstein it will take the path in space which requires the

least time, i.e., a straight line as agrees with our experience. If now

the world be strained by the presence of another body or by a change

of coordinates it will still pursue the path of least time, but this

path is now distorted from the straight line, just as in a similar way

the path on a globe requiring the least time to travel follows a great

circle. So, on Einstein's view of gravitation, the earth moves in an

elliptical path around the sun not because a force is acting on it,

but because the world of space-time is so distorted by the presence of

the sun that the path of least time through space is the elliptical

path observed. There is, therefore, no need to introduce any idea of

"force" of gravitation. Einstein's theory explains gravitation only

in the sense that he has explained it away as a force of nature and

makes it a property of space-time, namely, a distortion not different

from an appropriate change of coordinates. He does not, however,

explain how or why a body can distort space-time. It is noteworthy

that whilst the law of gravitation and the law of uniform motion in

a straight line when no force is acting were separate and independent

laws under Newton, Einstein finds one explanation for both under the

principle of relativity. [12]

XX

EINSTEIN'S THEORY OF GRAVITATION

The Discussion of the General Theory and Its Most Important

Application, from the Essay by

PROF. W. F. G. SWANN, UNIVERSITY OF MINNESOTA, MINNEAPOLIS

Newton's great discovery regarding the motion of the planets consisted

in his showing that these could all be summed up in the following

statement: consider any planet in its relation to all particles in the

universe. Write down, for the planet, in the line joining it to any

particle, an acceleration proportional to the mass of the particle and

to the inverse square of its distance from the planet. Then calculate

the planet's resultant acceleration by combining all the accelerations

thus obtained.

We have here purposely avoided the use of the word "force," for

Newton's law is complete as a practical statement of fact without it;

and this word adds nothing to the law by way of enhancing its power

in actual use. Nevertheless, the fact that the acceleration is made

up as it were of non-interfering contributions from each particle

in the line joining it to the planet strongly suggests to the mind

something of the nature of an elastic pull for which the particle is

responsible, and to which the planet's departure from a straight-line

motion is due. The mind likes to think of the elastic; ever since the

time of Newton people have sought to devise some mechanism by which

these pulls might be visualized as responsible for the phenomena in

the same way as one pictures an elastic thread as controlling the

motion of a stone which swings around at its end.

This search has been always without success; and now Einstein has

found a rather different law which fits the facts better than Newton's

law. It is of such a type that it does not lend itself conveniently

to expression in terms of force; the mind would gain nothing by

trying to picture such forces as are necessary. It compensates for

this, however, in being capable of visualization in terms of what is

ultimately a much simpler concept.

In order to appreciate the fundamental ideas involved, suppose for

a moment that gravitation could be annihilated, completely, and

suppose I find myself upon this earth in empty space. You shall be

seated at some point in space and shall watch my doings. If I am in

the condition of mind of the people of the reign of King Henry VIII,

I shall believe that the earth does not rotate. If I let go a stone,

there being no gravity, I shall find that it flies away from me with

an acceleration. You will know, however, that the stone really moves

in a straight line with constant velocity, and that the apparent

acceleration which I perceive is due to the earth's rotation. If I

have argued that acceleration is due to force, I shall say that the

earth repels the stone, and shall try to find the law governing the

variation of this force with distance. I may go farther, and try to

imagine some reason for the force, some pushing action transmitted from

the earth to the stone through a surrounding medium; and, you will

pity me for all this wasted labor, and particularly for my attempt

to find a mechanism to account for the force, since you know that if

I would only accept your measurements all would appear so simple.

Let us probe this matter a little farther, however, from the

stand-point of myself. I must believe in the reality of the force,

since I have to be tied to my chair to prevent my departure from

the earth. I might wonder how this field of force would affect the

propagation of light, chemical action and so forth. For, even though I

had discovered that, by using your measures, I could transform away the

apparent effects of my field of force as far as concerned its power

to hurl stones about, I could still regard this as a mathematical

accident, and believe that the force was really there. Although I

might suspect that the same transformation of view-point that would

annul the field's effect as regards the stones would also annul its

effect as regards light, etc., I should not be sure of this, as you

would be; and my conscience would hardly allow me to do more than

look upon the assumption of complete equivalence between the apparent

field and a change in the system of measurement as a hypothesis. I

should be strongly tempted to make the hypothesis, however.

Now the question raised by Einstein is whether the force of gravity,

which we experience as a very real thing, may be put upon a footing

which is in some way analogous to that of the obviously fictitious

centrifugal force cited above: whether gravitation may be regarded as

a figment of our imagination engendered by the way in which we measure

things. He found that it could be so regarded. He went still farther,

and in his Principle of Equivalence, he postulated that the apparent

effects of gravitation in all phenomena could be attributed to the same

change in the system of our measurements that would account for the

ordinary phenomena of gravitation. On the basis of this hypothesis

he was able to deduce for subsequent experimental verification,

the effects of gravitation on light. He did not limit himself to

such simple changes in our measurements as were sufficient to serve

the purpose of the problem of centrifugal force cited above; but,

emboldened by the assumptions, in the older theory of relativity,

of change in standards of length and time on account of motion,

he went even farther than this, and considered the possibility of

change of our measures due to mere proximity to matter.

His problem amounted to an attempt to find some way in which it was

possible to conceive our scales and clocks as altered, relatively to

some more fundamental set, so as to allow of the planetary motions

being uniform and rectilinear with respect to these fundamental

measures, although they appear as they do to us. If we allow our

imaginations perfect freedom as to how the scales may be altered, we

shall not balk at assuming alterations varying in any way we please,

Einstein does, however, introduce restrictions for reasons which we

will now discern.

If we imagine our whole universe, with its observers, planetary

orbits, instruments, and everything else, embedded in a jelly, and

then distort the jelly and contents in any way, the numbers at which

our planetary orbits (or rather their telescopic images) intersect

our scales will be unaltered. Moreover, we could vary, in any manner,

the times at which all objects (including the clock hands) occupied

their distorted positions, and the hand of some clock near the point

where the planetary image crossed the scale would record for this

occurrence the same dial reading as before. An inhabitant of this

distorted universe would be absolutely unconscious of the change. Now

the General Theory of Relativity which expresses itself in slightly

varied forms, amounts to satisfying a certain philosophical craving

of the mind, by asserting that the laws of nature which control our

universe ought to be such that another universe like the above, whose

inhabitants would be unconscious of their change, would also satisfy

these laws, not merely from the standpoint of its own inhabitants,

but also from the standpoint of our measurements. In other words,

this second universe ought to appear possible to us as well as to

its inhabitants.

Einstein decides to make his theory conform to this philosophical

desire, and this greatly limits the modifications of clocks and

scales which he permits himself for the purpose of representing

gravitation. Further, if we express the alterations of the measures

as functions of proximity to matter, velocity and so forth, our

expressions for these alterations will include, as a particular case,

that where matter is absent, although the scales and observer may still

remain. Our alteration of the scales and clocks with velocity must thus

revert, for this case, to that corresponding to the older theory of

relativity, in order to avoid predicting that two observers, in uniform

motion relative to each other in empty space, will measure different

values for the velocity of light. In this way, the velocity of light

comes to play a part in expressing the alterations of the measures.

Even with these restrictions, Einstein was able to do the equivalent

of finding an alteration of scales and clocks in the presence of

matter which would account for our finding that the planetary motions

take place very nearly in accordance with Newton's law. The new law

has accounted with surprising accuracy for certain astronomical

irregularities for which Newton's law failed to account, and has

predicted at least one previously unknown phenomenon which was

immediately verified.

In conclusion, it may be of interest to state how the new law describes

the motion of a particle in the vicinity of a body like the earth. The

law amounts to stating that, if we measure a short distance, radially

as regards the earth's center, we must allow for the peculiarity of

our units by dividing by

$$\sqrt{1-\frac{2mG}{c^2r}}$$

where $r$ is the distance from the earth's center, $m$ the mass

of the earth, $c$ the velocity of light, and $G$ the Newtonian

gravitational constant. Tangential measurements require no correction,

but intervals of time as measured by our clocks must be multiplied,

for each particular place, by the above factor. Then, in terms of

the corrected measures so obtained, the particle will be found to

describe a straight line with constant velocity although, in terms

of our actual measures, it appears to fall with an acceleration.

XXI

THE EQUIVALENCE HYPOTHESIS

The Discussion of This, With Its Difficulties and the Manner in Which

Einstein Has Resolved Them, from the Essay by

PROF. E. N. DA C. ANDRADE, ORDNANCE COLLEGE, WOOLWICH, ENGLAND

Having shown that, of several systems all moving with reference to one

another with uniform motion, no one is entitled to any preference over

the others, and having deduced the laws for such systems, Einstein

was confronted with a difficulty which had long been felt. A body

rotating, which is a special case of an accelerated body, can be

distinguished from one at rest, without looking outside it, by the

existence of the so-called centrifugal forces.

This circumstance, which gives certain bodies an absolute or

preferential motion, is unpalatable to the relativist; he would

like there to be no difference as regards forces [13] between the

case when the earth rotates with reference to outside bodies (the

stars) considered as fixed, and the case when the earth is considered

fixed and all outside bodies rotate around it. This point cannot be

investigated by direct experiment; we can spin a top but we cannot

keep a top at rest and spin the world round it, to see if the forces

are same.

In considering the problem of how to devise laws which should make all

rotations relative, Einstein conceived the brilliant yet simple idea

that gravitation could be brought into the scheme as an acceleration

effect, since both ordinary accelerational forces and gravitational

forces are proportional to the same thing, the mass of a body. The

impossibility of separating the two kinds of effect can be easily

seen by considering the starting of an elevator. When the elevator

is quickly accelerated upwards we feel a downward pull, just as if

the gravitational pull had been increased, and if the acceleration

continued to be uniform, bodies tested with a spring balance would

all weigh more in the elevator than they did on firm ground. In a

similar way the whole of the gravitational pull may be considered to

be an accelerational effect, the difficulty being to devise laws of

motion which will give the effects that we find by actual observation.

But it is obvious that we cannot, by ordinary mechanics, consider the

earth as being accelerated in all directions, which we should have to

do, apparently, to account for the fact that the gravitational pull

is always toward the center. [It is obvious that we cannot explain

gravitation by assuming that the earth's surface is continually moving

outward with an accelerated velocity.]227 So Einstein found that, as

long as we treat the problem by Euclid's geometry, we cannot reach

a satisfactory solution. But he found that to the four-dimensional

space made up of the three ordinary dimensions of space, together with

the time-dimension which we have already mentioned in discussing the

special theory, may be attributed a peculiar geometry, the nature of

which departs more and more from Euclidean geometry as we approach a

gravitational body, and the net result of which is to make possible

the universal correspondence of gravitation and acceleration.

This modification of the geometry of space is often spoken of as the

"curvature of space," an expression which is puzzling, especially as

the space which is "curved" is four-dimensional time-space. But we

can get an idea of what is meant by considering figures, triangles

say, drawn on the surface, of a sphere. These triangles, although

drawn on a surface, will not have the same properties as triangles

drawn on flat paper--their three angles will not together equal right

angles. They will be non-Euclidean. This is only a rough analogy, but

we can see that the curvature of the surface causes a departure from

Euclidean geometry for plane figures, and consequently the departure

from Euclidean laws extended to four dimensions may be referred to

as caused by "curvature of space."

It is difficult to imagine a lump of matter affecting the geometry of

the space round it. Once more we must use a rough illustration. Imagine

a very hot body, and that, knowing nothing of its properties, we

have to measure up the space round it with metal measuring-rods. The

nearer we are to the body, the longer the rods will become, owing to

the expansion of the metal. When we measure out a square, one side

of which is nearer the body than the opposite side, its angles will

not be right angles. If we knew nothing of the laws of heat we should

say that the body had made the space round it non-Euclidean.

Einstein found, then, that by taking the properties of space, as given

by measurement, to be modified in the neighborhood of masses of matter,

he could devise general laws according to which gravitational effects

would be produced, and there would be no absolute rotation. All

forces will be the same whether a body rotates with everything

outside it fixed, or the body is fixed, and everything rotates

round it. All motion is then relative, and the theory is one of

"general relativity." The velocity of light is, however, no longer

constant, and its path is not a straight line, if it is passing near

gravitating matter. This does not contradict the special theory,

which did not allow for gravitation. Rather, the special theory is

a particular case to which the generalised theory reduces when there

is no matter about, just as the Newtonian dynamics is a special case

of the special theory, which we obtain when all velocities are small

compared to that of light.

XXII

THE GENERAL THEORY

Fragments of Particular Merit on This Phase of the Subject

BY VARIOUS CONTRIBUTORS

When Dorothy was carried by the cyclone from her home in Kansas to the

land of Oz, together with her uncle's house and her little dog Toto,

she neglected to lower the trap door over the hole in the floor which

formerly led to the cyclone cellar and Toto stepped through. Dorothy

rushed to the opening expecting to see him dashed onto the rocks below

but found him floating just below the floor. She drew him back into

the room and closed the trap.

The author of the chronicle of Dorothy's adventures explains that the

same force which held up the house held up Toto but this explanation

is not necessary. Dorothy was now floating through space and house

and dog were subject to the same forces of gravitation which gave

them identical motions. Dorothy must have pushed the dog down onto the

floor and in doing so must herself have floated to the ceiling whence

she might have pushed herself back to the floor. In fact gravitation

was apparently suspended and Dorothy was in a position to have tried

certain experiments which Einstein has never tried because he was

never in Dorothy's unique position.]188

The Principle of Equivalence, of which Einstein's suspended cage

experiment is the usual illustration, and upon which the generalized

theory of relativity is built, is thus stated by Prof. Eddington: "A

gravitational field of force is precisely equivalent to an artificial

field of force, so that in any small region it is impossible, by any

conceivable experiment, to distinguish between them. In other words,

force is purely relative."

This may be otherwise stated by going back to our idea of a

four-dimensional world, the points of which represent the positions

and times of events. If we mark in such a space-time the successive

positions of an object we get a line, or curve, which represents the

whole history of the object, inasmuch as it shows us the position

of the object at every time. The reader may imagine that all events

happen in one plane, so that only two perpendicular dimensions

are needed to fix positions in space, with a third perpendicular

dimension for time. He may then conceive, if he may not picture,

an analogous process for four-dimensional space-time. These lines,

"tracks of objects through space-time," were called by Minkowski

"world-lines." We may now say that all the events we observe are the

intersections of world-lines. The temperature at noon was 70°. This

means that if I plot the world-line of the top of the mercury column

and the world-line of a certain mark on the glass they intersect in

a certain point of space-time. All that we know are intersections of

these world-lines. Suppose now we have a large number of them drawn

in our four-dimensional world, satisfying all known intersections,

and let us suppose the whole embedded in a jelly. We may distort

this jelly in any way, changing our coordinates as we please, but we

shall neither destroy nor create intersections of world-lines. It

may be proved that a change from one system of reference, to which

observations are referred, to any other system, moving in any way

with respect to the first system, may be pictured as a distortion of

the four-dimensional jelly. The laws of nature, therefore, being laws

that describe intersections, must be expressible in a form independent

of the reference system chosen.

From these postulates, Einstein was able to show such a formulation

possible. His law may be stated very simply:--All bodies move through

space-time in the straightest possible tracks.

The fact that an easy non-mathematical explanation can not be given,

of how this law is reached, or of just why the straightest track of

Mercury through space-time will give us an ellipse in space after we

have split space-time up into space and time, is no valid objection to

the theory. Newton's law that bodies attract with a force proportional

to their masses and inversely proportional to the square of the

distance is simple, but no one has ever given an easy non-mathematical

proof of how that law requires the path of Mercury to be an ellipse,

with the sun at a focus, instead of some other curve.]182

One of the grave difficulties we have in gaining a satisfactory

comprehension of Einstein's conceptions, is that they do not readily

relate themselves to our modes of geometrical thought. Within limits

we may choose our own geometry, but it may be at the cost of unwieldy

complication. If we think with Newton in terms of Euclidean geometry

and consider the earth as revolving around the sun, the motions of

our solar system can be stated in comparatively simple terms. If,

on the other hand, we should persist in stating them, as Ptolemy

would have done, from the earth as a relatively stationary center,

our formulas will become complicated beyond ready comprehension. For

this reason it is much simpler in applying the theory of relativity,

and in considering and describing what actually happens in the physical

universe, to use geometrical conceptions to which the actual conditions

can be easily related. We find such an instrument in non-Euclidean

geometry, wherein space will appear as though it were projected

from a slightly concave mirror. It is in this sense that some speak

of space as curved. The analogy is so suggestive it tempts one to

linger over it. Unless there were material objects within the range

of the mirror, its conformation would be immaterial; the thought of

the space which the mirror, as it were, circumscribes, is dependent

upon the presence of such material objects. The lines of light and of

all other movement will not be quite "straight" from the view-point

of Euclidean geometry. A line drawn in a universe of such a nature

must inevitably return upon itself. Nothing therefore, can ever pass

out of this unlimitedly great but yet finite cosmos. But even now,

since our imaginary mirror is only very slightly concave, it follows

that for limited regions like the earth or even the solar system, our

conception of geometry may well be rectilinear and Euclidean. Newton's

law of gravitation will be quite accurate with only a theoretical

modification drawn from the theory of relativity.]82

The way in which a curvature of space might appear to us as a force

is made plainer by an example. Suppose that in a certain room a

marble dropped anywhere on the floor always rolled to the center of

the room; suppose the same thing happened to a baseball, a billiard

ball, and a tennis ball. These results could be explained in two

ways; we might assume that a mysterious force of attraction existed

at the center of the floor, which affected all kinds of balls alike;

or we might assume that the floor was curved. We naturally prefer the

latter explanation. But when we find that in the neighborhood of a

large material body all other bodies move toward it in exactly the

same manner, regardless of their nature or their condition, we are

accustomed to postulate a mysterious attractive force (gravitation);

Einstein, on the contrary, adopts the other alternative, that the

space around the body is curved.]223

In the ordinary "analytical geometry," the position and motion of

all the points considered is referred to a rigid "body" or "frame of

reference." This usually consists of an imaginary room of suitable

size. The position of any point is then given by three numbers, i.e.,

its distances from one side wall and from the back wall and its height

above the floor. These three numbers can only give one point, every

other point having at least one number different. In four-dimensional

geometry a fourth wall may be vaguely imagined as perpendicular to

all three walls, and a fourth number added, giving the distance of

the "point" from this wall also. Since "rigid" bodies do not exist in

gravitational fields the "frame of reference" must be "non-rigid." The

frame of reference in the Gaussian system need not be rigid, it can

be of any shape and moving in any manner, in fact a kind of jelly. A

"point" or "event" in the four dimensioned world is still given by four

numbers but these numbers do not represent distances from anywhere;

all that is necessary is that no two events shall have exactly the

same four numbers to represent them, and that two events which are

very close together shall be represented by numbers which differ only

slightly from one another. This system assumes so little that it will

be seen to be very wide in its scope; although to the ordinary mind,

what is gained in scope seems more than that lost in concreteness. This

does not concern the mathematician, however, and by using this system

he gains his object, proving that the general laws of nature remain

the same when expressed in any Gaussian coordinate system whatever.]220

Einstein enunciates a general principle that it is possible to find

a transformation of coordinate axes which is exactly equivalent

to any force, and in particular one which is equivalent to the

force of gravitation. That is by concentrating our attention on

the transformation which is a purely mathematical operation we can

afford to neglect the force completely. To get a better idea of this

principle of equivalence as it is called, let us consider a relatively

simple example (which actually has nothing to do with gravitation,

but which will serve to make our notions clearer.) A person on the

earth unconsciously refers all his experiences, i.e., the motions

of the objects around him to a set of axes fixed in the earth on

which he stands. However, we know that the earth is rotating about

its axis, and his axes of reference are also rotating with respect to

the space about him. From the point of view of general relativity it

is exactly because we do refer motions on the surface of the earth

to axes rotating with the earth that we experience the so-called

centrifugal force of the earth's rotation, with which everyone is

familiar. If we could find it convenient to transform from moving axes

to fixed axes, the force would vanish, since it is exactly equivalent

to the transformation from one set of axes to the other. However,

we find it unnatural to refer daily experiences to axes that are not

placed where we happen to be, and so we prefer to take the force and

rotating axes instead of no force and fixed axes.]272

We seem to have a direct experience of force in our muscular

sensations. By pushing or pulling we can set bodies in motion. It

is natural to assume, that something similar occurs, when Nature set

bodies in motion. But is this not a relic of animism? The savage and

the ancients peopled all the woods and skies with Gods and demons, who

carries on the activities of nature by their own bodily efforts. Today

we have dispossessed the demons, but the ghost of a muscular pull

still holds the planets in place.]141

The general theory is an extension of the special theory which enables

the law of gravitation to be deduced. Not in Newton's form, it is true,

but in a better form, that is, one that accounts for two important

facts otherwise not explained. But it is a far more general theory

that indicated above. It is a complete study of the relations between

laws expressed by means of any four coordinates (of which three space

and one time is a special case), and the same laws expressed in the

four coordinates of a system having any motion whatever with respect

to the first system. By restricting this general study in accordance

with certain postulates about the nature of the universe we live in,

we arrive at a number of conclusions which fit more closely with

observed facts that the conclusions drawn from Newton's theory.]221

NOTES

[1] It will be noted that Mr. Bolton pronounces the geometry of

space to be Euclidean in the absence of gravitational fields, not

that of space-time. This is in accord with what was pointed out on

page 161.--Editor.

[2] The author here comes perilously close to ascribing to this

"contraction" the sort of physical reality which it does not

possess. See page 96.--Editor.

[3] Dr. Davis went rather fully into the algebra of the

Michelson-Morley experiment. But Dr. Russell has covered the same

ground in a form somewhat more advantageous from the typographical

viewpoint, and the point is not one which it is profitable to discuss

twice; so we eliminate this part of Dr. Davis' text.--Editor.

[4] This statement is objectionable, as explained in Chapter

IV.--Editor.

[5] A. Einstein: Die Grundlage der allgemeinen

Relativitätstheorie. Ann. d. Physik. 4, vol. 49, page 776.

[6] At this point we have again used the blue pencil on Dr. Davis'

text, his discussion of the three observational tests of the General

Theory adding nothing to Dr. Pickering's.--The Editor.

[7] Commander McHardy uses the term "event" in a sense somewhat

different from that seen in a majority of the essays. He reserves

for the four-dimensional element--the instant of time at a point in

space--the name "point-event"; and the term "event" he applies to a

collection of these forming, together, an observable whole. An actual

physical happening, like a railroad wreck or a laboratory experiment,

it will be realized is of the latter sort, occupying an appreciable

region of space rather than a single point, and an appreciable interval

of time rather than a single second. To the element, the "point-event"

of Commander McHardy's essay, this bears the same relation that the

geometer's solid bears to his point. This comment is in no sense

to be taken as criticism of Commander McHardy's terminology, which

rather appeals to us; we make it merely to guard against confusion

in the reader's mind.--Editor.

[8] This paragraph is the result of an editorial revision

of the author's text, designed to retain the substance of his

presentation, while tying up what he has to say more definitely with

the preceding essays, and eliminating the distinction between finite

and infinitesimal intervals, which we believe to be out of place in

an essay of this character. We will not apologize to our mathematical

readers for having used finite and differential notation in the same

equation, in violation of mathematical convention.--Editor.

[9] Although gravitational force in a small region can be imitated

or annulled by accelerating motion, there remains the disturbing

influence of gravitational matter already referred to and expressed in

the fabric curvature. It is this that defines how unique tracks run,

or rather, how bodies progress.--Author.

[10] Not all gravitational fields may be transformed away by a

proper choice of coordinates. If this were so, the space, whose

nature is independent of any choice of coordinates, would always

be Euclidean.--Author.

[11] Thus when it it said that a body contracts or that a clock runs

slow when it is put in motion no actual physical change is implied. The

judgment of different observers--one at rest with respect to the body

and one not--are different.--Author.

[12] The balance of Dr. Royds' essay is given to a discussion of

the phenomena of Mercury's perihelial advance, the deflection of

light under the gravitational field of the sun, and the shift in

spectral lines, in connection with which alone Einstein's theory

makes predictions which are sufficiently at variance with those

of Newtonian science to be of value in checking up the theory

observationally. In the interest of space conservation and in the

presence of Dr. Pickering's very complete discussion of these matters

we omit Dr. Royds' statement.--Editor.

[13] There is, on any view, no difference as regards observation of

position only.--Author.