The Great Void
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Relativity: The Special and General Theory


by Albert Einstein

See also Sidelights On Relativity

CONTENTS

Preface

Part I: The Special Theory of Relativity

 

01. Physical Meaning of Geometrical Propositions

02. The System of Co-ordinates

03. Space and Time in Classical Mechanics

04. The Galileian System of Co-ordinates

05. The Principle of Relativity (in the Restricted Sense)

06. The Theorem of the Addition of Velocities employed in

Classical Mechanics

07. The Apparent Incompatability of the Law of Propagation of

Light with the Principle of Relativity

08. On the Idea of Time in Physics

09. The Relativity of Simultaneity

10. On the Relativity of the Conception of Distance

11. The Lorentz Transformation

12. The Behaviour of Measuring-Rods and Clocks in Motion

13. Theorem of the Addition of Velocities. The Experiment of Fizeau

14. The Hueristic Value of the Theory of Relativity

15. General Results of the Theory

16. Expereince and the Special Theory of Relativity

17. Minkowski's Four-dimensial Space

 

 

Part II: The General Theory of Relativity

 

18. Special and General Principle of Relativity

19. The Gravitational Field

20. The Equality of Inertial and Gravitational Mass as an Argument

for the General Postulate of Relativity

21. In What Respects are the Foundations of Classical Mechanics

and of the Special Theory of Relativity Unsatisfactory?

22. A Few Inferences from the General Principle of Relativity

23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of

Reference

24. Euclidean and non-Euclidean Continuum

25. Gaussian Co-ordinates

26. The Space-Time Continuum of the Speical Theory of Relativity

Considered as a Euclidean Continuum

27. The Space-Time Continuum of the General Theory of Relativity

is Not a Eculidean Continuum

28. Exact Formulation of the General Principle of Relativity

29. The Solution of the Problem of Gravitation on the Basis of the

General Principle of Relativity

 

 

Part III: Considerations on the Universe as a Whole

 

30. Cosmological Difficulties of Netwon's Theory

31. The Possibility of a "Finite" and yet "Unbounded" Universe

32. The Structure of Space According to the General Theory of

Relativity

 

 

Appendices:

 

01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)

02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)

03. The Experimental Confirmation of the General Theory of Relativity

04. The Structure of Space According to the General Theory of

Relativity (sup. ch 32)

05. Relativity and the Problem of Space

 

Note: The fifth Appendix was added by Einstein at the time of the

fifteenth re-printing of this book; and as a result is still under

copyright restrictions so cannot be added without the permission of

the publisher.

 

 

 

PREFACE

 

 (December, 1916)

 

The present book is intended, as far as possible, to give an exact

insight into the theory of Relativity to those readers who, from a

general scientific and philosophical point of view, are interested in

the theory, but who are not conversant with the mathematical apparatus

of theoretical physics. The work presumes a standard of education

corresponding to that of a university matriculation examination, and,

despite the shortness of the book, a fair amount of patience and force

of will on the part of the reader. The author has spared himself no

pains in his endeavour to present the main ideas in the simplest and

most intelligible form, and on the whole, in the sequence and

connection in which they actually originated. In the interest of

clearness, it appeared to me inevitable that I should repeat myself

frequently, without paying the slightest attention to the elegance of

the presentation. I adhered scrupulously to the precept of that

brilliant theoretical physicist L. Boltzmann, according to whom

matters of elegance ought to be left to the tailor and to the cobbler.

I make no pretence of having withheld from the reader difficulties

which are inherent to the subject. On the other hand, I have purposely

treated the empirical physical foundations of the theory in a

"step-motherly" fashion, so that readers unfamiliar with physics may

not feel like the wanderer who was unable to see the forest for the

trees. May the book bring some one a few happy hours of suggestive

thought!

 

December, 1916

A. EINSTEIN

 

 

 

PART I

 

THE SPECIAL THEORY OF RELATIVITY

 

PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS

 

 

In your schooldays most of you who read this book made acquaintance

with the noble building of Euclid's geometry, and you remember --

perhaps with more respect than love -- the magnificent structure, on

the lofty staircase of which you were chased about for uncounted hours

by conscientious teachers. By reason of our past experience, you would

certainly regard everyone with disdain who should pronounce even the

most out-of-the-way proposition of this science to be untrue. But

perhaps this feeling of proud certainty would leave you immediately if

some one were to ask you: "What, then, do you mean by the assertion

that these propositions are true?" Let us proceed to give this

question a little consideration.

 

Geometry sets out form certain conceptions such as "plane," "point,"

and "straight line," with which we are able to associate more or less

definite ideas, and from certain simple propositions (axioms) which,

in virtue of these ideas, we are inclined to accept as "true." Then,

on the basis of a logical process, the justification of which we feel

ourselves compelled to admit, all remaining propositions are shown to

follow from those axioms, i.e. they are proven. A proposition is then

correct ("true") when it has been derived in the recognised manner

from the axioms. The question of "truth" of the individual geometrical

propositions is thus reduced to one of the "truth" of the axioms. Now

it has long been known that the last question is not only unanswerable

by the methods of geometry, but that it is in itself entirely without

meaning. We cannot ask whether it is true that only one straight line

goes through two points. We can only say that Euclidean geometry deals

with things called "straight lines," to each of which is ascribed the

property of being uniquely determined by two points situated on it.

The concept "true" does not tally with the assertions of pure

geometry, because by the word "true" we are eventually in the habit of

designating always the correspondence with a "real" object; geometry,

however, is not concerned with the relation of the ideas involved in

it to objects of experience, but only with the logical connection of

these ideas among themselves.

 

It is not difficult to understand why, in spite of this, we feel

constrained to call the propositions of geometry "true." Geometrical

ideas correspond to more or less exact objects in nature, and these

last are undoubtedly the exclusive cause of the genesis of those

ideas. Geometry ought to refrain from such a course, in order to give

to its structure the largest possible logical unity. The practice, for

example, of seeing in a "distance" two marked positions on a

practically rigid body is something which is lodged deeply in our

habit of thought. We are accustomed further to regard three points as

being situated on a straight line, if their apparent positions can be

made to coincide for observation with one eye, under suitable choice

of our place of observation.

 

If, in pursuance of our habit of thought, we now supplement the

propositions of Euclidean geometry by the single proposition that two

points on a practically rigid body always correspond to the same

distance (line-interval), independently of any changes in position to

which we may subject the body, the propositions of Euclidean geometry

then resolve themselves into propositions on the possible relative

position of practically rigid bodies.* Geometry which has been

supplemented in this way is then to be treated as a branch of physics.

We can now legitimately ask as to the "truth" of geometrical

propositions interpreted in this way, since we are justified in asking

whether these propositions are satisfied for those real things we have

associated with the geometrical ideas. In less exact terms we can

express this by saying that by the "truth" of a geometrical

proposition in this sense we understand its validity for a

construction with rule and compasses.

 

Of course the conviction of the "truth" of geometrical propositions in

this sense is founded exclusively on rather incomplete experience. For

the present we shall assume the "truth" of the geometrical

propositions, then at a later stage (in the general theory of

relativity) we shall see that this "truth" is limited, and we shall

consider the extent of its limitation.

 

 

  Notes

 

*) It follows that a natural object is associated also with a

straight line. Three points A, B and C on a rigid body thus lie in a

straight line when the points A and C being given, B is chosen such

that the sum of the distances AB and BC is as short as possible. This

incomplete suggestion will suffice for the present purpose.

 

 

 

THE SYSTEM OF CO-ORDINATES

 

 

On the basis of the physical interpretation of distance which has been

indicated, we are also in a position to establish the distance between

two points on a rigid body by means of measurements. For this purpose

we require a " distance " (rod S) which is to be used once and for

all, and which we employ as a standard measure. If, now, A and B are

two points on a rigid body, we can construct the line joining them

according to the rules of geometry ; then, starting from A, we can

mark off the distance S time after time until we reach B. The number

of these operations required is the numerical measure of the distance

AB. This is the basis of all measurement of length. *

 

Every description of the scene of an event or of the position of an

object in space is based on the specification of the point on a rigid

body (body of reference) with which that event or object coincides.

This applies not only to scientific description, but also to everyday

life. If I analyse the place specification " Times Square, New York,"

**A I arrive at the following result. The earth is the rigid body

to which the specification of place refers; " Times Square, New York,"

is a well-defined point, to which a name has been assigned, and with

which the event coincides in space.**B

 

This primitive method of place specification deals only with places on

the surface of rigid bodies, and is dependent on the existence of

points on this surface which are distinguishable from each other. But

we can free ourselves from both of these limitations without altering

the nature of our specification of position. If, for instance, a cloud

is hovering over Times Square, then we can determine its position

relative to the surface of the earth by erecting a pole

perpendicularly on the Square, so that it reaches the cloud. The

length of the pole measured with the standard measuring-rod, combined

with the specification of the position of the foot of the pole,

supplies us with a complete place specification. On the basis of this

illustration, we are able to see the manner in which a refinement of

the conception of position has been developed.

 

(a) We imagine the rigid body, to which the place specification is

referred, supplemented in such a manner that the object whose position

we require is reached by. the completed rigid body.

 

(b) In locating the position of the object, we make use of a number

(here the length of the pole measured with the measuring-rod) instead

of designated points of reference.

 

(c) We speak of the height of the cloud even when the pole which

reaches the cloud has not been erected. By means of optical

observations of the cloud from different positions on the ground, and

taking into account the properties of the propagation of light, we

determine the length of the pole we should have required in order to

reach the cloud.

 

From this consideration we see that it will be advantageous if, in the

description of position, it should be possible by means of numerical

measures to make ourselves independent of the existence of marked

positions (possessing names) on the rigid body of reference. In the

physics of measurement this is attained by the application of the

Cartesian system of co-ordinates.

 

This consists of three plane surfaces perpendicular to each other and

rigidly attached to a rigid body. Referred to a system of

co-ordinates, the scene of any event will be determined (for the main

part) by the specification of the lengths of the three perpendiculars

or co-ordinates (x, y, z) which can be dropped from the scene of the

event to those three plane surfaces. The lengths of these three

perpendiculars can be determined by a series of manipulations with

rigid measuring-rods performed according to the rules and methods laid

down by Euclidean geometry.

 

In practice, the rigid surfaces which constitute the system of

co-ordinates are generally not available ; furthermore, the magnitudes

of the co-ordinates are not actually determined by constructions with

rigid rods, but by indirect means. If the results of physics and

astronomy are to maintain their clearness, the physical meaning of

specifications of position must always be sought in accordance with

the above considerations. ***

 

We thus obtain the following result: Every description of events in

space involves the use of a rigid body to which such events have to be

referred. The resulting relationship takes for granted that the laws

of Euclidean geometry hold for "distances;" the "distance" being

represented physically by means of the convention of two marks on a

rigid body.

 

 

  Notes

 

* Here we have assumed that there is nothing left over i.e. that

the measurement gives a whole number. This difficulty is got over by

the use of divided measuring-rods, the introduction of which does not

demand any fundamentally new method.

 

**A Einstein used "Potsdamer Platz, Berlin" in the original text.

In the authorised translation this was supplemented with "Tranfalgar

Square, London". We have changed this to "Times Square, New York", as

this is the most well known/identifiable location to English speakers

in the present day. [Note by the janitor.]

 

**B It is not necessary here to investigate further the significance

of the expression "coincidence in space." This conception is

sufficiently obvious to ensure that differences of opinion are

scarcely likely to arise as to its applicability in practice.

 

*** A refinement and modification of these views does not become

necessary until we come to deal with the general theory of relativity,

treated in the second part of this book.

 

 

 

SPACE AND TIME IN CLASSICAL MECHANICS

 

 

The purpose of mechanics is to describe how bodies change their

position in space with "time." I should load my conscience with grave

sins against the sacred spirit of lucidity were I to formulate the

aims of mechanics in this way, without serious reflection and detailed

explanations. Let us proceed to disclose these sins.

 

It is not clear what is to be understood here by "position" and

"space." I stand at the window of a railway carriage which is

travelling uniformly, and drop a stone on the embankment, without

throwing it. Then, disregarding the influence of the air resistance, I

see the stone descend in a straight line. A pedestrian who observes

the misdeed from the footpath notices that the stone falls to earth in

a parabolic curve. I now ask: Do the "positions" traversed by the

stone lie "in reality" on a straight line or on a parabola? Moreover,

what is meant here by motion "in space" ? From the considerations of

the previous section the answer is self-evident. In the first place we

entirely shun the vague word "space," of which, we must honestly

acknowledge, we cannot form the slightest conception, and we replace

it by "motion relative to a practically rigid body of reference." The

positions relative to the body of reference (railway carriage or

embankment) have already been defined in detail in the preceding

section. If instead of " body of reference " we insert " system of

co-ordinates," which is a useful idea for mathematical description, we

are in a position to say : The stone traverses a straight line

relative to a system of co-ordinates rigidly attached to the carriage,

but relative to a system of co-ordinates rigidly attached to the

ground (embankment) it describes a parabola. With the aid of this

example it is clearly seen that there is no such thing as an

independently existing trajectory (lit. "path-curve"*), but only

a trajectory relative to a particular body of reference.

 

In order to have a complete description of the motion, we must specify

how the body alters its position with time ; i.e. for every point on

the trajectory it must be stated at what time the body is situated

there. These data must be supplemented by such a definition of time

that, in virtue of this definition, these time-values can be regarded

essentially as magnitudes (results of measurements) capable of

observation. If we take our stand on the ground of classical

mechanics, we can satisfy this requirement for our illustration in the

following manner. We imagine two clocks of identical construction ;

the man at the railway-carriage window is holding one of them, and the

man on the footpath the other. Each of the observers determines the

position on his own reference-body occupied by the stone at each tick

of the clock he is holding in his hand. In this connection we have not

taken account of the inaccuracy involved by the finiteness of the

velocity of propagation of light. With this and with a second

difficulty prevailing here we shall have to deal in detail later.

 

 

  Notes

 

*) That is, a curve along which the body moves.

 

 

 

THE GALILEIAN SYSTEM OF CO-ORDINATES

 

 

As is well known, the fundamental law of the mechanics of

Galilei-Newton, which is known as the law of inertia, can be stated

thus: A body removed sufficiently far from other bodies continues in a

state of rest or of uniform motion in a straight line. This law not

only says something about the motion of the bodies, but it also

indicates the reference-bodies or systems of coordinates, permissible

in mechanics, which can be used in mechanical description. The visible

fixed stars are bodies for which the law of inertia certainly holds to

a high degree of approximation. Now if we use a system of co-ordinates

which is rigidly attached to the earth, then, relative to this system,

every fixed star describes a circle of immense radius in the course of

an astronomical day, a result which is opposed to the statement of the

law of inertia. So that if we adhere to this law we must refer these

motions only to systems of coordinates relative to which the fixed

stars do not move in a circle. A system of co-ordinates of which the

state of motion is such that the law of inertia holds relative to it

is called a " Galileian system of co-ordinates." The laws of the

mechanics of Galflei-Newton can be regarded as valid only for a

Galileian system of co-ordinates.

 

 

 

THE PRINCIPLE OF RELATIVITY

(IN THE RESTRICTED SENSE)

 

 

In order to attain the greatest possible clearness, let us return to

our example of the railway carriage supposed to be travelling

uniformly. We call its motion a uniform translation ("uniform" because

it is of constant velocity and direction, " translation " because

although the carriage changes its position relative to the embankment

yet it does not rotate in so doing). Let us imagine a raven flying

through the air in such a manner that its motion, as observed from the

embankment, is uniform and in a straight line. If we were to observe

the flying raven from the moving railway carriage. we should find that

the motion of the raven would be one of different velocity and

direction, but that it would still be uniform and in a straight line.

Expressed in an abstract manner we may say : If a mass m is moving

uniformly in a straight line with respect to a co-ordinate system K,

then it will also be moving uniformly and in a straight line relative

to a second co-ordinate system K1 provided that the latter is

executing a uniform translatory motion with respect to K. In

accordance with the discussion contained in the preceding section, it

follows that:

 

If K is a Galileian co-ordinate system. then every other co-ordinate

system K' is a Galileian one, when, in relation to K, it is in a

condition of uniform motion of translation. Relative to K1 the

mechanical laws of Galilei-Newton hold good exactly as they do with

respect to K.

 

We advance a step farther in our generalisation when we express the

tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate

system devoid of rotation, then natural phenomena run their course

with respect to K1 according to exactly the same general laws as with

respect to K. This statement is called the principle of relativity (in

the restricted sense).

 

As long as one was convinced that all natural phenomena were capable

of representation with the help of classical mechanics, there was no

need to doubt the validity of this principle of relativity. But in

view of the more recent development of electrodynamics and optics it

became more and more evident that classical mechanics affords an

insufficient foundation for the physical description of all natural

phenomena. At this juncture the question of the validity of the

principle of relativity became ripe for discussion, and it did not

appear impossible that the answer to this question might be in the

negative.

 

Nevertheless, there are two general facts which at the outset speak

very much in favour of the validity of the principle of relativity.

Even though classical mechanics does not supply us with a sufficiently

broad basis for the theoretical presentation of all physical

phenomena, still we must grant it a considerable measure of " truth,"

since it supplies us with the actual motions of the heavenly bodies

with a delicacy of detail little short of wonderful. The principle of

relativity must therefore apply with great accuracy in the domain of

mechanics. But that a principle of such broad generality should hold

with such exactness in one domain of phenomena, and yet should be

invalid for another, is a priori not very probable.

 

We now proceed to the second argument, to which, moreover, we shall

return later. If the principle of relativity (in the restricted sense)

does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,

which are moving uniformly relative to each other, will not be

equivalent for the description of natural phenomena. In this case we

should be constrained to believe that natural laws are capable of

being formulated in a particularly simple manner, and of course only

on condition that, from amongst all possible Galileian co-ordinate

systems, we should have chosen one (K[0]) of a particular state of

motion as our body of reference. We should then be justified (because

of its merits for the description of natural phenomena) in calling

this system " absolutely at rest," and all other Galileian systems K "

in motion." If, for instance, our embankment were the system K[0] then

our railway carriage would be a system K, relative to which less

simple laws would hold than with respect to K[0]. This diminished

simplicity would be due to the fact that the carriage K would be in

motion (i.e."really")with respect to K[0]. In the general laws of

nature which have been formulated with reference to K, the magnitude

and direction of the velocity of the carriage would necessarily play a

part. We should expect, for instance, that the note emitted by an

organpipe placed with its axis parallel to the direction of travel

would be different from that emitted if the axis of the pipe were

placed perpendicular to this direction.

 

Now in virtue of its motion in an orbit round the sun, our earth is

comparable with a railway carriage travelling with a velocity of about

30 kilometres per second. If the principle of relativity were not

valid we should therefore expect that the direction of motion of the

earth at any moment would enter into the laws of nature, and also that

physical systems in their behaviour would be dependent on the

orientation in space with respect to the earth. For owing to the

alteration in direction of the velocity of revolution of the earth in

the course of a year, the earth cannot be at rest relative to the

hypothetical system K[0] throughout the whole year. However, the most

careful observations have never revealed such anisotropic properties

in terrestrial physical space, i.e. a physical non-equivalence of

different directions. This is very powerful argument in favour of the

principle of relativity.

 

 

 

THE THEOREM OF THE

ADDITION OF VELOCITIES

EMPLOYED IN CLASSICAL MECHANICS

 

 

Let us suppose our old friend the railway carriage to be travelling

along the rails with a constant velocity v, and that a man traverses

the length of the carriage in the direction of travel with a velocity

w. How quickly or, in other words, with what velocity W does the man

advance relative to the embankment during the process ? The only

possible answer seems to result from the following consideration: If

the man were to stand still for a second, he would advance relative to

the embankment through a distance v equal numerically to the velocity

of the carriage. As a consequence of his walking, however, he

traverses an additional distance w relative to the carriage, and hence

also relative to the embankment, in this second, the distance w being

numerically equal to the velocity with which he is walking. Thus in

total be covers the distance W=v+w relative to the embankment in the

second considered. We shall see later that this result, which

expresses the theorem of the addition of velocities employed in

classical mechanics, cannot be maintained ; in other words, the law

that we have just written down does not hold in reality. For the time

being, however, we shall assume its correctness.

 

 

 

THE APPARENT INCOMPATIBILITY OF THE

LAW OF PROPAGATION OF LIGHT WITH THE

PRINCIPLE OF RELATIVITY

 

 

There is hardly a simpler law in physics than that according to which

light is propagated in empty space. Every child at school knows, or

believes he knows, that this propagation takes place in straight lines

with a velocity c= 300,000 km./sec. At all events we know with great

exactness that this velocity is the same for all colours, because if

this were not the case, the minimum of emission would not be observed

simultaneously for different colours during the eclipse of a fixed

star by its dark neighbour. By means of similar considerations based

on observa- tions of double stars, the Dutch astronomer De Sitter was

also able to show that the velocity of propagation of light cannot

depend on the velocity of motion of the body emitting the light. The

assumption that this velocity of propagation is dependent on the

direction "in space" is in itself improbable.

 

In short, let us assume that the simple law of the constancy of the

velocity of light c (in vacuum) is justifiably believed by the child

at school. Who would imagine that this simple law has plunged the

conscientiously thoughtful physicist into the greatest intellectual

difficulties? Let us consider how these difficulties arise.

 

Of course we must refer the process of the propagation of light (and

indeed every other process) to a rigid reference-body (co-ordinate

system). As such a system let us again choose our embankment. We shall

imagine the air above it to have been removed. If a ray of light be

sent along the embankment, we see from the above that the tip of the

ray will be transmitted with the velocity c relative to the

embankment. Now let us suppose that our railway carriage is again

travelling along the railway lines with the velocity v, and that its

direction is the same as that of the ray of light, but its velocity of

course much less. Let us inquire about the velocity of propagation of

the ray of light relative to the carriage. It is obvious that we can

here apply the consideration of the previous section, since the ray of

light plays the part of the man walking along relatively to the

carriage. The velocity w of the man relative to the embankment is here

replaced by the velocity of light relative to the embankment. w is the

required velocity of light with respect to the carriage, and we have

 

                               w = c-v.

 

The velocity of propagation ot a ray of light relative to the carriage

thus comes cut smaller than c.

 

But this result comes into conflict with the principle of relativity

set forth in Section V. For, like every other general law of

nature, the law of the transmission of light in vacuo [in vacuum]

must, according to the principle of relativity, be the same for the

railway carriage as reference-body as when the rails are the body of

reference. But, from our above consideration, this would appear to be

impossible. If every ray of light is propagated relative to the

embankment with the velocity c, then for this reason it would appear

that another law of propagation of light must necessarily hold with

respect to the carriage -- a result contradictory to the principle of

relativity.

 

In view of this dilemma there appears to be nothing else for it than

to abandon either the principle of relativity or the simple law of the

propagation of light in vacuo. Those of you who have carefully

followed the preceding discussion are almost sure to expect that we

should retain the principle of relativity, which appeals so

convincingly to the intellect because it is so natural and simple. The

law of the propagation of light in vacuo would then have to be

replaced by a more complicated law conformable to the principle of

relativity. The development of theoretical physics shows, however,

that we cannot pursue this course. The epoch-making theoretical

investigations of H. A. Lorentz on the electrodynamical and optical

phenomena connected with moving bodies show that experience in this

domain leads conclusively to a theory of electromagnetic phenomena, of

which the law of the constancy of the velocity of light in vacuo is a

necessary consequence. Prominent theoretical physicists were theref

ore more inclined to reject the principle of relativity, in spite of

the fact that no empirical data had been found which were

contradictory to this principle.

 

At this juncture the theory of relativity entered the arena. As a

result of an analysis of the physical conceptions of time and space,

it became evident that in realily there is not the least

incompatibilitiy between the principle of relativity and the law of

propagation of light, and that by systematically holding fast to both

these laws a logically rigid theory could be arrived at. This theory

has been called the special theory of relativity to distinguish it

from the extended theory, with which we shall deal later. In the

following pages we shall present the fundamental ideas of the special

theory of relativity.

 

 

 

ON THE IDEA OF TIME IN PHYSICS

 

 

Lightning has struck the rails on our railway embankment at two places

A and B far distant from each other. I make the additional assertion

that these two lightning flashes occurred simultaneously. If I ask you

whether there is sense in this statement, you will answer my question

with a decided "Yes." But if I now approach you with the request to

explain to me the sense of the statement more precisely, you find

after some consideration that the answer to this question is not so

easy as it appears at first sight.

 

After some time perhaps the following answer would occur to you: "The

significance of the statement is clear in itself and needs no further

explanation; of course it would require some consideration if I were

to be commissioned to determine by observations whether in the actual

case the two events took place simultaneously or not." I cannot be

satisfied with this answer for the following reason. Supposing that as

a result of ingenious considerations an able meteorologist were to

discover that the lightning must always strike the places A and B

simultaneously, then we should be faced with the task of testing

whether or not this theoretical result is in accordance with the

reality. We encounter the same difficulty with all physical statements

in which the conception " simultaneous " plays a part. The concept

does not exist for the physicist until he has the possibility of

discovering whether or not it is fulfilled in an actual case. We thus

require a definition of simultaneity such that this definition

supplies us with the method by means of which, in the present case, he

can decide by experiment whether or not both the lightning strokes

occurred simultaneously. As long as this requirement is not satisfied,

I allow myself to be deceived as a physicist (and of course the same

applies if I am not a physicist), when I imagine that I am able to

attach a meaning to the statement of simultaneity. (I would ask the

reader not to proceed farther until he is fully convinced on this

point.)

 

After thinking the matter over for some time you then offer the

following suggestion with which to test simultaneity. By measuring

along the rails, the connecting line AB should be measured up and an

observer placed at the mid-point M of the distance AB. This observer

should be supplied with an arrangement (e.g. two mirrors inclined at

90^0) which allows him visually to observe both places A and B at the

same time. If the observer perceives the two flashes of lightning at

the same time, then they are simultaneous.

 

I am very pleased with this suggestion, but for all that I cannot

regard the matter as quite settled, because I feel constrained to

raise the following objection:

 

"Your definition would certainly be right, if only I knew that the

light by means of which the observer at M perceives the lightning

flashes travels along the length A arrow M with the same velocity as

along the length B arrow M. But an examination of this supposition

would only be possible if we already had at our disposal the means of

measuring time. It would thus appear as though we were moving here in

a logical circle."

 

After further consideration you cast a somewhat disdainful glance at

me -- and rightly so -- and you declare:

 

"I maintain my previous definition nevertheless, because in reality it

assumes absolutely nothing about light. There is only one demand to be

made of the definition of simultaneity, namely, that in every real

case it must supply us with an empirical decision as to whether or not

the conception that has to be defined is fulfilled. That my definition

satisfies this demand is indisputable. That light requires the same

time to traverse the path A arrow M as for the path B arrow M is in

reality neither a supposition nor a hypothesis about the physical

nature of light, but a stipulation which I can make of my own freewill

in order to arrive at a definition of simultaneity."

 

It is clear that this definition can be used to give an exact meaning

not only to two events, but to as many events as we care to choose,

and independently of the positions of the scenes of the events with

respect to the body of reference * (here the railway embankment).

We are thus led also to a definition of " time " in physics. For this

purpose we suppose that clocks of identical construction are placed at

the points A, B and C of the railway line (co-ordinate system) and

that they are set in such a manner that the positions of their

pointers are simultaneously (in the above sense) the same. Under these

conditions we understand by the " time " of an event the reading

(position of the hands) of that one of these clocks which is in the

immediate vicinity (in space) of the event. In this manner a

time-value is associated with every event which is essentially capable

of observation.

 

This stipulation contains a further physical hypothesis, the validity

of which will hardly be doubted without empirical evidence to the

contrary. It has been assumed that all these clocks go at the same

rate if they are of identical construction. Stated more exactly: When

two clocks arranged at rest in different places of a reference-body

are set in such a manner that a particular position of the pointers of

the one clock is simultaneous (in the above sense) with the same

position, of the pointers of the other clock, then identical "

settings " are always simultaneous (in the sense of the above

definition).

 

 

  Notes

 

*) We suppose further, that, when three events A, B and C occur in

different places in such a manner that A is simultaneous with B and B

is simultaneous with C (simultaneous in the sense of the above

definition), then the criterion for the simultaneity of the pair of

events A, C is also satisfied. This assumption is a physical

hypothesis about the the of propagation of light: it must certainly be

fulfilled if we are to maintain the law of the constancy of the

velocity of light in vacuo.

 

 

 

THE RELATIVITY OF SIMULATNEITY

 

 

Up to now our considerations have been referred to a particular body

of reference, which we have styled a " railway embankment." We suppose

a very long train travelling along the rails with the constant

velocity v and in the direction indicated in Fig 1. People travelling

in this train will with a vantage view the train as a rigid

reference-body (co-ordinate system); they regard all events in

 

                      

 

 

reference to the train. Then every event which takes place along the

line also takes place at a particular point of the train. Also the

definition of simultaneity can be given relative to the train in

exactly the same way as with respect to the embankment. As a natural

consequence, however, the following question arises :

 

Are two events (e.g. the two strokes of lightning A and B) which are

simultaneous with reference to the railway embankment also

simultaneous relatively to the train? We shall show directly that the

answer must be in the negative.

 

When we say that the lightning strokes A and B are simultaneous with

respect to be embankment, we mean: the rays of light emitted at the

places A and B, where the lightning occurs, meet each other at the

mid-point M of the length A arrow B of the embankment. But the events

A and B also correspond to positions A and B on the train. Let M1 be

the mid-point of the distance A arrow B on the travelling train. Just

when the flashes (as judged from the embankment) of lightning occur,

this point M1 naturally coincides with the point M but it moves

towards the right in the diagram with the velocity v of the train. If

an observer sitting in the position M1 in the train did not possess

this velocity, then he would remain permanently at M, and the light

rays emitted by the flashes of lightning A and B would reach him

simultaneously, i.e. they would meet just where he is situated. Now in

reality (considered with reference to the railway embankment) he is

hastening towards the beam of light coming from B, whilst he is riding

on ahead of the beam of light coming from A. Hence the observer will

see the beam of light emitted from B earlier than he will see that

emitted from A. Observers who take the railway train as their

reference-body must therefore come to the conclusion that the

lightning flash B took place earlier than the lightning flash A. We

thus arrive at the important result:

 

Events which are simultaneous with reference to the embankment are not

simultaneous with respect to the train, and vice versa (relativity of

simultaneity). Every reference-body (co-ordinate system) has its own

particular time ; unless we are told the reference-body to which the

statement of time refers, there is no meaning in a statement of the

time of an event.

 

Now before the advent of the theory of relativity it had always

tacitly been assumed in physics that the statement of time had an

absolute significance, i.e. that it is independent of the state of

motion of the body of reference. But we have just seen that this

assumption is incompatible with the most natural definition of

simultaneity; if we discard this assumption, then the conflict between

the law of the propagation of light in vacuo and the principle of

relativity (developed in Section 7) disappears.

 

We were led to that conflict by the considerations of Section 6,

which are now no longer tenable. In that section we concluded that the

man in the carriage, who traverses the distance w per second relative

to the carriage, traverses the same distance also with respect to the

embankment in each second of time. But, according to the foregoing

considerations, the time required by a particular occurrence with

respect to the carriage must not be considered equal to the duration

of the same occurrence as judged from the embankment (as

reference-body). Hence it cannot be contended that the man in walking

travels the distance w relative to the railway line in a time which is

equal to one second as judged from the embankment.

 

Moreover, the considerations of Section 6 are based on yet a second

assumption, which, in the light of a strict consideration, appears to

be arbitrary, although it was always tacitly made even before the

introduction of the theory of relativity.

 

 

 

ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE

 

 

Let us consider two particular points on the train * travelling

along the embankment with the velocity v, and inquire as to their

distance apart. We already know that it is necessary to have a body of

reference for the measurement of a distance, with respect to which

body the distance can be measured up. It is the simplest plan to use

the train itself as reference-body (co-ordinate system). An observer

in the train measures the interval by marking off his measuring-rod in

a straight line (e.g. along the floor of the carriage) as many times

as is necessary to take him from the one marked point to the other.

Then the number which tells us how often the rod has to be laid down

is the required distance.

 

It is a different matter when the distance has to be judged from the

railway line. Here the following method suggests itself. If we call

A^1 and B^1 the two points on the train whose distance apart is

required, then both of these points are moving with the velocity v

along the embankment. In the first place we require to determine the

points A and B of the embankment which are just being passed by the

two points A^1 and B^1 at a particular time t -- judged from the

embankment. These points A and B of the embankment can be determined

by applying the definition of time given in Section 8. The distance

between these points A and B is then measured by repeated application

of thee measuring-rod along the embankment.

 

A priori it is by no means certain that this last measurement will

supply us with the same result as the first. Thus the length of the

train as measured from the embankment may be different from that

obtained by measuring in the train itself. This circumstance leads us

to a second objection which must be raised against the apparently

obvious consideration of Section 6. Namely, if the man in the

carriage covers the distance w in a unit of time -- measured from the

train, -- then this distance -- as measured from the embankment -- is

not necessarily also equal to w.

 

 

  Notes

 

*) e.g. the middle of the first and of the hundredth carriage.

 

 

 

THE LORENTZ TRANSFORMATION

 

 

The results of the last three sections show that the apparent

incompatibility of the law of propagation of light with the principle

of relativity (Section 7) has been derived by means of a

consideration which borrowed two unjustifiable hypotheses from

classical mechanics; these are as follows:

 

(1) The time-interval (time) between two events is independent of the

condition of motion of the body of reference.

 

(2) The space-interval (distance) between two points of a rigid body

is independent of the condition of motion of the body of reference.

 

If we drop these hypotheses, then the dilemma of Section 7

disappears, because the theorem of the addition of velocities derived

in Section 6 becomes invalid. The possibility presents itself that

the law of the propagation of light in vacuo may be compatible with

the principle of relativity, and the question arises: How have we to

modify the considerations of Section 6 in order to remove the

apparent disagreement between these two fundamental results of

experience? This question leads to a general one. In the discussion of

Section 6 we have to do with places and times relative both to the

train and to the embankment. How are we to find the place and time of

an event in relation to the train, when we know the place and time of

the event with respect to the railway embankment ? Is there a

thinkable answer to this question of such a nature that the law of

transmission of light in vacuo does not contradict the principle of

relativity ? In other words : Can we conceive of a relation between

place and time of the individual events relative to both

reference-bodies, such that every ray of light possesses the velocity

of transmission c relative to the embankment and relative to the train

? This question leads to a quite definite positive answer, and to a

perfectly definite transformation law for the space-time magnitudes of

an event when changing over from one body of reference to another.

 

Before we deal with this, we shall introduce the following incidental

consideration. Up to the present we have only considered events taking

place along the embankment, which had mathematically to assume the

function of a straight line. In the manner indicated in Section 2

we can imagine this reference-body supplemented laterally and in a

vertical direction by means of a framework of rods, so that an event

which takes place anywhere can be localised with reference to this

framework.

 

 

 

 

 

 

 Similarly, we can imagine the train travelling with

the velocity v to be continued across the whole of space, so that

every event, no matter how far off it may be, could also be localised

with respect to the second framework. Without committing any

fundamental error, we can disregard the fact that in reality these

frameworks would continually interfere with each other, owing to the

impenetrability of solid bodies. In every such framework we imagine

three surfaces perpendicular to each other marked out, and designated

as " co-ordinate planes " (" co-ordinate system "). A co-ordinate

system K then corresponds to the embankment, and a co-ordinate system

K' to the train. An event, wherever it may have taken place, would be

fixed in space with respect to K by the three perpendiculars x, y, z

on the co-ordinate planes, and with regard to time by a time value t.

Relative to K1, the same event would be fixed in respect of space and

time by corresponding values x1, y1, z1, t1, which of course are not

identical with x, y, z, t. It has already been set forth in detail how

these magnitudes are to be regarded as results of physical

measurements.

 

Obviously our problem can be exactly formulated in the following

manner. What are the values x1, y1, z1, t1, of an event with respect

to K1, when the magnitudes x, y, z, t, of the same event with respect

to K are given ? The relations must be so chosen that the law of the

transmission of light in vacuo is satisfied for one and the same ray

of light (and of course for every ray) with respect to K and K1. For

the relative orientation in space of the co-ordinate systems indicated

in the diagram ([7]Fig. 2), this problem is solved by means of the

equations :

 

                        

 

                                y1 = y

                                z1 = z

 

                        

 

This system of equations is known as the " Lorentz transformation." *

 

If in place of the law of transmission of light we had taken as our

basis the tacit assumptions of the older mechanics as to the absolute

character of times and lengths, then instead of the above we should

have obtained the following equations:

 

                             x1 = x - vt

                                y1 = y

                                z1 = z

                                t1 = t

 

This system of equations is often termed the " Galilei

transformation." The Galilei transformation can be obtained from the

Lorentz transformation by substituting an infinitely large value for

the velocity of light c in the latter transformation.

 

Aided by the following illustration, we can readily see that, in

accordance with the Lorentz transformation, the law of the

transmission of light in vacuo is satisfied both for the

reference-body K and for the reference-body K1. A light-signal is sent

along the positive x-axis, and this light-stimulus advances in

accordance with the equation

 

                               x = ct,

 

i.e. with the velocity c. According to the equations of the Lorentz

transformation, this simple relation between x and t involves a

relation between x1 and t1. In point of fact, if we substitute for x

the value ct in the first and fourth equations of the Lorentz

transformation, we obtain:

 

                        

 

 

                        

 

from which, by division, the expression

 

                               x1 = ct1

 

immediately follows. If referred to the system K1, the propagation of

light takes place according to this equation. We thus see that the

velocity of transmission relative to the reference-body K1 is also

equal to c. The same result is obtained for rays of light advancing in

any other direction whatsoever. Of cause this is not surprising, since

the equations of the Lorentz transformation were derived conformably

to this point of view.

 

 

  Notes

 

*) A simple derivation of the Lorentz transformation is given in

Appendix I.

 

 

 

THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION

 

 

Place a metre-rod in the x1-axis of K1 in such a manner that one end

(the beginning) coincides with the point x1=0 whilst the other end

(the end of the rod) coincides with the point x1=I. What is the length

of the metre-rod relatively to the system K? In order to learn this,

we need only ask where the beginning of the rod and the end of the rod

lie with respect to K at a particular time t of the system K. By means

of the first equation of the Lorentz transformation the values of

these two points at the time t = 0 can be shown to be

 

 

 

 

the distance between the points being  

 

But the metre-rod is moving with the velocity v relative to K. It

therefore follows that the length of a rigid metre-rod moving in the

direction of its length with a velocity v is  of a metre.

 

The rigid rod is thus shorter when in motion than when at rest, and

the more quickly it is moving, the shorter is the rod. For the

velocity v=c we should have  ,

 

and for stiII greater velocities the square-root becomes imaginary.

From this we conclude that in the theory of relativity the velocity c

plays the part of a limiting velocity, which can neither be reached

nor exceeded by any real body.

 

Of course this feature of the velocity c as a limiting velocity also

clearly follows from the equations of the Lorentz transformation, for

these became meaningless if we choose values of v greater than c.

 

If, on the contrary, we had considered a metre-rod at rest in the

x-axis with respect to K, then we should have found that the length of

the rod as judged from K1 would have been  ;

 

this is quite in accordance with the principle of relativity which

forms the basis of our considerations.

 

A Priori it is quite clear that we must be able to learn something

about the physical behaviour of measuring-rods and clocks from the

equations of transformation, for the magnitudes z, y, x, t, are

nothing more nor less than the results of measurements obtainable by

means of measuring-rods and clocks. If we had based our considerations

on the Galileian transformation we should not have obtained a

contraction of the rod as a consequence of its motion.

 

Let us now consider a seconds-clock which is permanently situated at

the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of

this clock. The first and fourth equations of the Lorentz

transformation give for these two ticks :

 

                                t = 0

 

and

 

                       

 

As judged from K, the clock is moving with the velocity v; as judged

from this reference-body, the time which elapses between two strokes

of the clock is not one second, but

 

                       

 

seconds, i.e. a somewhat larger time. As a consequence of its motion

the clock goes more slowly than when at rest. Here also the velocity c

plays the part of an unattainable limiting velocity.

 

 

 

 

 

THEOREM OF THE ADDITION OF VELOCITIES.

THE EXPERIMENT OF FIZEAU

 

 

Now in practice we can move clocks and measuring-rods only with

velocities that are small compared with the velocity of light; hence

we shall hardly be able to compare the results of the previous section

directly with the reality. But, on the other hand, these results must

strike you as being very singular, and for that reason I shall now

draw another conclusion from the theory, one which can easily be

derived from the foregoing considerations, and which has been most

elegantly confirmed by experiment.

 

In Section 6 we derived the theorem of the addition of velocities

in one direction in the form which also results from the hypotheses of

classical mechanics- This theorem can also be deduced readily horn the

Galilei transformation (Section 11). In place of the man walking

inside the carriage, we introduce a point moving relatively to the

co-ordinate system K1 in accordance with the equation

 

                               x1 = wt1

 

By means of the first and fourth equations of the Galilei

transformation we can express x1 and t1 in terms of x and t, and we

then obtain

 

                             x = (v + w)t

 

This equation expresses nothing else than the law of motion of the

point with reference to the system K (of the man with reference to the

embankment). We denote this velocity by the symbol W, and we then

obtain, as in Section 6,

 

                           W=v+w         A)

 

But we can carry out this consideration just as well on the basis of

the theory of relativity. In the equation

 

                         x1 = wt1         B)

 

we must then express x1and t1 in terms of x and t, making use of the

first and fourth equations of the Lorentz transformation. Instead of

the equation (A) we then obtain the equation

 

                       

 

 

which corresponds to the theorem of addition for velocities in one

direction according to the theory of relativity. The question now

arises as to which of these two theorems is the better in accord with

experience. On this point we axe enlightened by a most important

experiment which the brilliant physicist Fizeau performed more than

half a century ago, and which has been repeated since then by some of

the best experimental physicists, so that there can be no doubt about

its result. The experiment is concerned with the following question.

Light travels in a motionless liquid with a particular velocity w. How

quickly does it travel in the direction of the arrow in the tube T

 

 when the liquid above

mentioned is flowing through the tube with a velocity v ?

 

In accordance with the principle of relativity we shall certainly have

to take for granted that the propagation of light always takes place

with the same velocity w with respect to the liquid, whether the

latter is in motion with reference to other bodies or not. The

velocity of light relative to the liquid and the velocity of the

latter relative to the tube are thus known, and we require the

velocity of light relative to the tube.

 

It is clear that we have the problem of Section 6 again before us. The

tube plays the part of the railway embankment or of the co-ordinate

system K, the liquid plays the part of the carriage or of the

co-ordinate system K1, and finally, the light plays the part of the

man walking along the carriage, or of the moving point in the present

section. If we denote the velocity of the light relative to the tube

by W, then this is given by the equation (A) or (B), according as the

Galilei transformation or the Lorentz transformation corresponds to

the facts. Experiment * decides in favour of equation (B) derived

from the theory of relativity, and the agreement is, indeed, very

exact. According to recent and most excellent measurements by Zeeman,

the influence of the velocity of flow v on the propagation of light is

represented by formula (B) to within one per cent.

 

Nevertheless we must now draw attention to the fact that a theory of

this phenomenon was given by H. A. Lorentz long before the statement

of the theory of relativity. This theory was of a purely

electrodynamical nature, and was obtained by the use of particular

hypotheses as to the electromagnetic structure of matter. This

circumstance, however, does not in the least diminish the

conclusiveness of the experiment as a crucial test in favour of the

theory of relativity, for the electrodynamics of Maxwell-Lorentz, on

which the original theory was based, in no way opposes the theory of

relativity. Rather has the latter been developed trom electrodynamics

as an astoundingly simple combination and generalisation of the

hypotheses, formerly independent of each other, on which

electrodynamics was built.

 

 

  Notes

 

*) Fizeau found  ,

 

 where  

 

is the index of refraction of the liquid. On the other hand, owing to

the smallness of  as compared with I,

 

we can replace (B) in the first place by  , or to the same order

of approximation by

 

 , which agrees with Fizeau's result.

 

 

 

THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY

 

 

Our train of thought in the foregoing pages can be epitomised in the

following manner. Experience has led to the conviction that, on the

one hand, the principle of relativity holds true and that on the other

hand the velocity of transmission of light in vacuo has to be

considered equal to a constant c. By uniting these two postulates we

obtained the law of transformation for the rectangular co-ordinates x,

y, z and the time t of the events which constitute the processes of

nature. In this connection we did not obtain the Galilei

transformation, but, differing from classical mechanics, the Lorentz

transformation.

 

The law of transmission of light, the acceptance of which is justified

by our actual knowledge, played an important part in this process of

thought. Once in possession of the Lorentz transformation, however, we

can combine this with the principle of relativity, and sum up the

theory thus:

 

Every general law of nature must be so constituted that it is

transformed into a law of exactly the same form when, instead of the

space-time variables x, y, z, t of the original coordinate system K,

we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate

system K1. In this connection the relation between the ordinary and

the accented magnitudes is given by the Lorentz transformation. Or in

brief : General laws of nature are co-variant with respect to Lorentz

transformations.

 

This is a definite mathematical condition that the theory of

relativity demands of a natural law, and in virtue of this, the theory

becomes a valuable heuristic aid in the search for general laws of

nature. If a general law of nature were to be found which did not

satisfy this condition, then at least one of the two fundamental

assumptions of the theory would have been disproved. Let us now

examine what general results the latter theory has hitherto evinced.

 

 

 

GENERAL RESULTS OF THE THEORY

 

 

It is clear from our previous considerations that the (special) theory

of relativity has grown out of electrodynamics and optics. In these

fields it has not appreciably altered the predictions of theory, but

it has considerably simplified the theoretical structure, i.e. the

derivation of laws, and -- what is incomparably more important -- it

has considerably reduced the number of independent hypothese forming

the basis of theory. The special theory of relativity has rendered the

Maxwell-Lorentz theory so plausible, that the latter would have been

generally accepted by physicists even if experiment had decided less

unequivocally in its favour.

 

Classical mechanics required to be modified before it could come into

line with the demands of the special theory of relativity. For the

main part, however, this modification affects only the laws for rapid

motions, in which the velocities of matter v are not very small as

compared with the velocity of light. We have experience of such rapid

motions only in the case of electrons and ions; for other motions the

variations from the laws of classical mechanics are too small to make

themselves evident in practice. We shall not consider the motion of

stars until we come to speak of the general theory of relativity. In

accordance with the theory of relativity the kinetic energy of a

material point of mass m is no longer given by the well-known

expression

 

                       

but by the expression

 

                       

 

This expression approaches infinity as the velocity v approaches the

velocity of light c. The velocity must therefore always remain less

than c, however great may be the energies used to produce the

acceleration. If we develop the expression for the kinetic energy in

the form of a series, we obtain

 

                       

 

When  is small compared with unity, the third of these terms is

always small in comparison with the second,

 

which last is alone considered in classical mechanics. The first term

mc^2 does not contain the velocity, and requires no consideration if

we are only dealing with the question as to how the energy of a

point-mass; depends on the velocity. We shall speak of its essential

significance later.

 

The most important result of a general character to which the special

theory of relativity has led is concerned with the conception of mass.

Before the advent of relativity, physics recognised two conservation

laws of fundamental importance, namely, the law of the canservation of

energy and the law of the conservation of mass these two fundamental

laws appeared to be quite independent of each other. By means of the

theory of relativity they have been united into one law. We shall now

briefly consider how this unification came about, and what meaning is

to be attached to it.

 

The principle of relativity requires that the law of the concervation

of energy should hold not only with reference to a co-ordinate system

K, but also with respect to every co-ordinate system K1 which is in a

state of uniform motion of translation relative to K, or, briefly,

relative to every " Galileian " system of co-ordinates. In contrast to

classical mechanics; the Lorentz transformation is the deciding factor

in the transition from one such system to another.

 

By means of comparatively simple considerations we are led to draw the

following conclusion from these premises, in conjunction with the

fundamental equations of the electrodynamics of Maxwell: A body moving

with the velocity v, which absorbs * an amount of energy E[0] in

the form of radiation without suffering an alteration in velocity in

the process, has, as a consequence, its energy increased by an amount

 

                       

 

In consideration of the expression given above for the kinetic energy

of the body, the required energy of the body comes out to be

 

                       

 

Thus the body has the same energy as a body of mass

 

                         

 

moving with the velocity v. Hence we can say: If a body takes up an

amount of energy E[0], then its inertial mass increases by an amount

 

                       

 

the inertial mass of a body is not a constant but varies according to

the change in the energy of the body. The inertial mass of a system of

bodies can even be regarded as a measure of its energy. The law of the

conservation of the mass of a system becomes identical with the law of

the conservation of energy, and is only valid provided that the system

neither takes up nor sends out energy. Writing the expression for the

energy in the form

 

                       

 

we see that the term mc^2, which has hitherto attracted our attention,

is nothing else than the energy possessed by the body ** before it

absorbed the energy E[0].

 

A direct comparison of this relation with experiment is not possible

at the present time (1920; see *** Note, p. 48), owing to the fact that

the changes in energy E[0] to which we can Subject a system are not

large enough to make themselves perceptible as a change in the

inertial mass of the system.

 

                               

 

is too small in comparison with the mass m, which was present before

the alteration of the energy. It is owing to this circumstance that

classical mechanics was able to establish successfully the

conservation of mass as a law of independent validity.

 

Let me add a final remark of a fundamental nature. The success of the

Faraday-Maxwell interpretation of electromagnetic action at a distance

resulted in physicists becoming convinced that there are no such

things as instantaneous actions at a distance (not involving an

intermediary medium) of the type of Newton's law of gravitation.

According to the theory of relativity, action at a distance with the

velocity of light always takes the place of instantaneous action at a

distance or of action at a distance with an infinite velocity of

transmission. This is connected with the fact that the velocity c

plays a fundamental role in this theory. In Part II we shall see in

what way this result becomes modified in the general theory of

relativity.

 

 

  Notes

 

*) E[0] is the energy taken up, as judged from a co-ordinate system

moving with the body.

 

**) As judged from a co-ordinate system moving with the body.

 

***[Note] The equation E = mc^2 has been thoroughly proved time and

again since this time.

 

 

 

EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY

 

 

To what extent is the special theory of relativity supported by

experience?  This question is not easily answered for the reason

already mentioned in connection with the fundamental experiment of

Fizeau. The special theory of relativity has crystallised out from the

Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of

experience which support the electromagnetic theory also support the

theory of relativity. As being of particular importance, I mention

here the fact that the theory of relativity enables us to predict the

effects produced on the light reaching us from the fixed stars. These

results are obtained in an exceedingly simple manner, and the effects

indicated, which are due to the relative motion of the earth with

reference to those fixed stars are found to be in accord with

experience. We refer to the yearly movement of the apparent position

of the fixed stars resulting from the motion of the earth round the

sun (aberration), and to the influence of the radial components of the

relative motions of the fixed stars with respect to the earth on the

colour of the light reaching us from them. The latter effect manifests

itself in a slight displacement of the spectral lines of the light

transmitted to us from a fixed star, as compared with the position of

the same spectral lines when they are produced by a terrestrial source

of light (Doppler principle). The experimental arguments in favour of

the Maxwell-Lorentz theory, which are at the same time arguments in

favour of the theory of relativity, are too numerous to be set forth

here. In reality they limit the theoretical possibilities to such an

extent, that no other theory than that of Maxwell and Lorentz has been

able to hold its own when tested by experience.

 

But there are two classes of experimental facts hitherto obtained

which can be represented in the Maxwell-Lorentz theory only by the

introduction of an auxiliary hypothesis, which in itself -- i.e.

without making use of the theory of relativity -- appears extraneous.

 

It is known that cathode rays and the so-called b-rays emitted by

radioactive substances consist of negatively electrified particles

(electrons) of very small inertia and large velocity. By examining the

deflection of these rays under the influence of electric and magnetic

fields, we can study the law of motion of these particles very

exactly.

 

In the theoretical treatment of these electrons, we are faced with the

difficulty that electrodynamic theory of itself is unable to give an

account of their nature. For since electrical masses of one sign repel

each other, the negative electrical masses constituting the electron

would necessarily be scattered under the influence of their mutual

repulsions, unless there are forces of another kind operating between

them, the nature of which has hitherto remained obscure to us.*   If

we now assume that the relative distances between the electrical

masses constituting the electron remain unchanged during the motion of

the electron (rigid connection in the sense of classical mechanics),

we arrive at a law of motion of the electron which does not agree with

experience. Guided by purely formal points of view, H. A. Lorentz was

the first to introduce the hypothesis that the form of the electron

experiences a contraction in the direction of motion in consequence of

that motion. the contracted length being proportional to the

expression

 

                       

 

This, hypothesis, which is not justifiable by any electrodynamical

facts, supplies us then with that particular law of motion which has

been confirmed with great precision in recent years.

 

The theory of relativity leads to the same law of motion, without

requiring any special hypothesis whatsoever as to the structure and

the behaviour of the electron. We arrived at a similar conclusion in

Section 13 in connection with the experiment of Fizeau, the result

of which is foretold by the theory of relativity without the necessity

of drawing on hypotheses as to the physical nature of the liquid.

 

The second class of facts to which we have alluded has reference to

the question whether or not the motion of the earth in space can be

made perceptible in terrestrial experiments. We have already remarked

in Section 5 that all attempts of this nature led to a negative

result. Before the theory of relativity was put forward, it was

difficult to become reconciled to this negative result, for reasons

now to be discussed. The inherited prejudices about time and space did

not allow any doubt to arise as to the prime importance of the

Galileian transformation for changing over from one body of reference

to another. Now assuming that the Maxwell-Lorentz equations hold for a

reference-body K, we then find that they do not hold for a

reference-body K1 moving uniformly with respect to K, if we assume

that the relations of the Galileian transformstion exist between the

co-ordinates of K and K1. It thus appears that, of all Galileian

co-ordinate systems, one (K) corresponding to a particular state of

motion is physically unique. This result was interpreted physically by

regarding K as at rest with respect to a hypothetical æther of space.

On the other hand, all coordinate systems K1 moving relatively to K

were to be regarded as in motion with respect to the æther. To this

motion of K1 against the æther ("æther-drift " relative to K1) were

attributed the more complicated laws which were supposed to hold

relative to K1. Strictly speaking, such an æther-drift ought also to

be assumed relative to the earth, and for a long time the efforts of

physicists were devoted to attempts to detect the existence of an

æther-drift at the earth's surface.

 

In one of the most notable of these attempts Michelson devised a

method which appears as though it must be decisive. Imagine two

mirrors so arranged on a rigid body that the reflecting surfaces face

each other. A ray of light requires a perfectly definite time T to

pass from one mirror to the other and back again, if the whole system

be at rest with respect to the æther. It is found by calculation,

however, that a slightly different time T1 is required for this

process, if the body, together with the mirrors, be moving relatively

to the æther. And yet another point: it is shown by calculation that

for a given velocity v with reference to the æther, this time T1 is

different when the body is moving perpendicularly to the planes of the

mirrors from that resulting when the motion is parallel to these

planes. Although the estimated difference between these two times is

exceedingly small, Michelson and Morley performed an experiment

involving interference in which this difference should have been

clearly detectable. But the experiment gave a negative result -- a

fact very perplexing to physicists. Lorentz and FitzGerald rescued the

theory from this difficulty by assuming that the motion of the body

relative to the æther produces a contraction of the body in the

direction of motion, the amount of contraction being just sufficient

to compensate for the differeace in time mentioned above. Comparison

with the discussion in Section 11 shows that also from the

standpoint of the theory of relativity this solution of the difficulty

was the right one. But on the basis of the theory of relativity the

method of interpretation is incomparably more satisfactory. According

to this theory there is no such thing as a " specially favoured "

(unique) co-ordinate system to occasion the introduction of the

æther-idea, and hence there can be no æther-drift, nor any experiment

with which to demonstrate it. Here the contraction of moving bodies

follows from the two fundamental principles of the theory, without the

introduction of particular hypotheses ; and as the prime factor

involved in this contraction we find, not the motion in itself, to

which we cannot attach any meaning, but the motion with respect to the

body of reference chosen in the particular case in point. Thus for a

co-ordinate system moving with the earth the mirror system of

Michelson and Morley is not shortened, but it is shortened for a

co-ordinate system which is at rest relatively to the sun.

 

 

  Notes

 

*) The general theory of relativity renders it likely that the

electrical masses of an electron are held together by gravitational

forces.

 

 

 

MINKOWSKI'S FOUR-DIMENSIONAL SPACE

 

 

The non-mathematician is seized by a mysterious shuddering when he

hears of "four-dimensional" things, by a feeling not unlike that

awakened by thoughts of the occult. And yet there is no more

common-place statement than that the world in which we live is a

four-dimensional space-time continuum.

 

Space is a three-dimensional continuum. By this we mean that it is

possible to describe the position of a point (at rest) by means of

three numbers (co-ordinales) x, y, z, and that there is an indefinite

number of points in the neighbourhood of this one, the position of

which can be described by co-ordinates such as x[1], y[1], z[1], which

may be as near as we choose to the respective values of the

co-ordinates x, y, z, of the first point. In virtue of the latter

property we speak of a " continuum," and owing to the fact that there

are three co-ordinates we speak of it as being " three-dimensional."

 

Similarly, the world of physical phenomena which was briefly called "

world " by Minkowski is naturally four dimensional in the space-time

sense. For it is composed of individual events, each of which is

described by four numbers, namely, three space co-ordinates x, y, z,

and a time co-ordinate, the time value t. The" world" is in this sense

also a continuum; for to every event there are as many "neighbouring"

events (realised or at least thinkable) as we care to choose, the

co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely

small amount from those of the event x, y, z, t originally considered.

That we have not been accustomed to regard the world in this sense as

a four-dimensional continuum is due to the fact that in physics,

before the advent of the theory of relativity, time played a different

and more independent role, as compared with the space coordinates. It

is for this reason that we have been in the habit of treating time as

an independent continuum. As a matter of fact, according to classical

mechanics, time is absolute, i.e. it is independent of the position

and the condition of motion of the system of co-ordinates. We see this

expressed in the last equation of the Galileian transformation (t1 =

t)

 

The four-dimensional mode of consideration of the "world" is natural

on the theory of relativity, since according to this theory time is

robbed of its independence. This is shown by the fourth equation of

the Lorentz transformation:

 

                       

 

Moreover, according to this equation the time difference Dt1 of two

events with respect to K1 does not in general vanish, even when the

time difference Dt1 of the same events with reference to K vanishes.

Pure " space-distance " of two events with respect to K results in "

time-distance " of the same events with respect to K. But the

discovery of Minkowski, which was of importance for the formal

development of the theory of relativity, does not lie here. It is to

be found rather in the fact of his recognition that the

four-dimensional space-time continuum of the theory of relativity, in

its most essential formal properties, shows a pronounced relationship

to the three-dimensional continuum of Euclidean geometrical

space.*  In order to give due prominence to this relationship,

however, we must replace the usual time co-ordinate t by an imaginary

magnitude eq. 25 proportional to it. Under these conditions, the

natural laws satisfying the demands of the (special) theory of

relativity assume mathematical forms, in which the time co-ordinate

plays exactly the same role as the three space co-ordinates. Formally,

these four co-ordinates correspond exactly to the three space

co-ordinates in Euclidean geometry. It must be clear even to the

non-mathematician that, as a consequence of this purely formal

addition to our knowledge, the theory perforce gained clearness in no

mean measure.

 

These inadequate remarks can give the reader only a vague notion of

the important idea contributed by Minkowski. Without it the general

theory of relativity, of which the fundamental ideas are developed in

the following pages, would perhaps have got no farther than its long

clothes. Minkowski's work is doubtless difficult of access to anyone

inexperienced in mathematics, but since it is not necessary to have a

very exact grasp of this work in order to understand the fundamental

ideas of either the special or the general theory of relativity, I

shall leave it here at present, and revert to it only towards the end

of Part 2.

 

 

  Notes

 

*) Cf. the somewhat more detailed discussion in Appendix II.

 

 

 

 

PART II

 

THE GENERAL THEORY OF RELATIVITY

 

 

SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY

 

 

The basal principle, which was the pivot of all our previous

considerations, was the special principle of relativity, i.e. the

principle of the physical relativity of all uniform motion. Let as

once more analyse its meaning carefully.

 

It was at all times clear that, from the point of view of the idea it

conveys to us, every motion must be considered only as a relative

motion. Returning to the illustration we have frequently used of the

embankment and the railway carriage, we can express the fact of the

motion here taking place in the following two forms, both of which are

equally justifiable :

 

(a) The carriage is in motion relative to the embankment,

(b) The embankment is in motion relative to the carriage.

 

In (a) the embankment, in (b) the carriage, serves as the body of

reference in our statement of the motion taking place. If it is simply

a question of detecting or of describing the motion involved, it is in

principle immaterial to what reference-body we refer the motion. As

already mentioned, this is self-evident, but it must not be confused

with the much more comprehensive statement called "the principle of

relativity," which we have taken as the basis of our investigations.

 

The principle we have made use of not only maintains that we may

equally well choose the carriage or the embankment as our

reference-body for the description of any event (for this, too, is

self-evident). Our principle rather asserts what follows : If we

formulate the general laws of nature as they are obtained from

experience, by making use of

 

(a) the embankment as reference-body,

(b) the railway carriage as reference-body,

 

then these general laws of nature (e.g. the laws of mechanics or the

law of the propagation of light in vacuo) have exactly the same form

in both cases. This can also be expressed as follows : For the

physical description of natural processes, neither of the reference

bodies K, K1 is unique (lit. " specially marked out ") as compared

with the other. Unlike the first, this latter statement need not of

necessity hold a priori; it is not contained in the conceptions of "

motion" and " reference-body " and derivable from them; only

experience can decide as to its correctness or incorrectness.

 

Up to the present, however, we have by no means maintained the

equivalence of all bodies of reference K in connection with the

formulation of natural laws. Our course was more on the following

Iines. In the first place, we started out from the assumption that

there exists a reference-body K, whose condition of motion is such

that the Galileian law holds with respect to it : A particle left to

itself and sufficiently far removed from all other particles moves

uniformly in a straight line. With reference to K (Galileian

reference-body) the laws of nature were to be as simple as possible.

But in addition to K, all bodies of reference K1 should be given

preference in this sense, and they should be exactly equivalent to K

for the formulation of natural laws, provided that they are in a state

of uniform rectilinear and non-rotary motion with respect to K ; all

these bodies of reference are to be regarded as Galileian

reference-bodies. The validity of the principle of relativity was

assumed only for these reference-bodies, but not for others (e.g.

those possessing motion of a different kind). In this sense we speak

of the special principle of relativity, or special theory of

relativity.

 

In contrast to this we wish to understand by the "general principle of

relativity" the following statement : All bodies of reference K, K1,

etc., are equivalent for the description of natural phenomena

(formulation of the general laws of nature), whatever may be their

state of motion. But before proceeding farther, it ought to be pointed

out that this formulation must be replaced later by a more abstract

one, for reasons which will become evident at a later stage.

 

Since the introduction of the special principle of relativity has been

justified, every intellect which strives after generalisation must

feel the temptation to venture the step towards the general principle

of relativity. But a simple and apparently quite reliable

consideration seems to suggest that, for the present at any rate,

there is little hope of success in such an attempt; Let us imagine

ourselves transferred to our old friend the railway carriage, which is

travelling at a uniform rate. As long as it is moving unifromly, the

occupant of the carriage is not sensible of its motion, and it is for

this reason that he can without reluctance interpret the facts of the

case as indicating that the carriage is at rest, but the embankment in

motion. Moreover, according to the special principle of relativity,

this interpretation is quite justified also from a physical point of

view.

 

If the motion of the carriage is now changed into a non-uniform

motion, as for instance by a powerful application of the brakes, then

the occupant of the carriage experiences a correspondingly powerful

jerk forwards. The retarded motion is manifested in the mechanical

behaviour of bodies relative to the person in the railway carriage.

The mechanical behaviour is different from that of the case previously

considered, and for this reason it would appear to be impossible that

the same mechanical laws hold relatively to the non-uniformly moving

carriage, as hold with reference to the carriage when at rest or in

uniform motion. At all events it is clear that the Galileian law does

not hold with respect to the non-uniformly moving carriage. Because of

this, we feel compelled at the present juncture to grant a kind of

absolute physical reality to non-uniform motion, in opposition to the

general principle of relatvity. But in what follows we shall soon see

that this conclusion cannot be maintained.

 

 

 

THE GRAVITATIONAL FIELD

 

 

"If we pick up a stone and then let it go, why does it fall to the

ground ?" The usual answer to this question is: "Because it is

attracted by the earth." Modern physics formulates the answer rather

differently for the following reason. As a result of the more careful

study of electromagnetic phenomena, we have come to regard action at a

distance as a process impossible without the intervention of some

intermediary medium. If, for instance, a magnet attracts a piece of

iron, we cannot be content to regard this as meaning that the magnet

acts directly on the iron through the intermediate empty space, but we

are constrained to imagine -- after the manner of Faraday -- that the

magnet always calls into being something physically real in the space

around it, that something being what we call a "magnetic field." In

its turn this magnetic field operates on the piece of iron, so that

the latter strives to move towards the magnet. We shall not discuss

here the justification for this incidental conception, which is indeed

a somewhat arbitrary one. We shall only mention that with its aid

electromagnetic phenomena can be theoretically represented much more

satisfactorily than without it, and this applies particularly to the

transmission of electromagnetic waves. The effects of gravitation also

are regarded in an analogous manner.

 

The action of the earth on the stone takes place indirectly. The earth

produces in its surrounding a gravitational field, which acts on the

stone and produces its motion of fall. As we know from experience, the

intensity of the action on a body dimishes according to a quite

definite law, as we proceed farther and farther away from the earth.

From our point of view this means : The law governing the properties

of the gravitational field in space must be a perfectly definite one,

in order correctly to represent the diminution of gravitational action

with the distance from operative bodies. It is something like this:

The body (e.g. the earth) produces a field in its immediate

neighbourhood directly; the intensity and direction of the field at

points farther removed from the body are thence determined by the law

which governs the properties in space of the gravitational fields

themselves.

 

In contrast to electric and magnetic fields, the gravitational field

exhibits a most remarkable property, which is of fundamental

importance for what follows. Bodies which are moving under the sole

influence of a gravitational field receive an acceleration, which does

not in the least depend either on the material or on the physical

state of the body. For instance, a piece of lead and a piece of wood

fall in exactly the same manner in a gravitational field (in vacuo),

when they start off from rest or with the same initial velocity. This

law, which holds most accurately, can be expressed in a different form

in the light of the following consideration.

 

According to Newton's law of motion, we have

 

(Force) = (inertial mass) x (acceleration),

 

where the "inertial mass" is a characteristic constant of the

accelerated body. If now gravitation is the cause of the acceleration,

we then have

 

(Force) = (gravitational mass) x (intensity of the gravitational

field),

 

where the "gravitational mass" is likewise a characteristic constant

for the body. From these two relations follows:

 

                               

 

If now, as we find from experience, the acceleration is to be

independent of the nature and the condition of the body and always the

same for a given gravitational field, then the ratio of the

gravitational to the inertial mass must likewise be the same for all

bodies. By a suitable choice of units we can thus make this ratio

equal to unity. We then have the following law: The gravitational mass

of a body is equal to its inertial law.

 

It is true that this important law had hitherto been recorded in

mechanics, but it had not been interpreted. A satisfactory

interpretation can be obtained only if we recognise the following fact

: The same quality of a body manifests itself according to

circumstances as " inertia " or as " weight " (lit. " heaviness '). In

the following section we shall show to what extent this is actually

the case, and how this question is connected with the general

postulate of relativity.

 

 

 

 

THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS

AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY

 

 

We imagine a large portion of empty space, so far removed from stars

and other appreciable masses, that we have before us approximately the

conditions required by the fundamental law of Galilei. It is then

possible to choose a Galileian reference-body for this part of space

(world), relative to which points at rest remain at rest and points in

motion continue permanently in uniform rectilinear motion. As

reference-body let us imagine a spacious chest resembling a room with

an observer inside who is equipped with apparatus. Gravitation

naturally does not exist for this observer. He must fasten himself

with strings to the floor, otherwise the slightest impact against the

floor will cause him to rise slowly towards the ceiling of the room.

 

To the middle of the lid of the chest is fixed externally a hook with

rope attached, and now a " being " (what kind of a being is immaterial

to us) begins pulling at this with a constant force. The chest

together with the observer then begin to move "upwards" with a

uniformly accelerated motion. In course of time their velocity will

reach unheard-of values -- provided that we are viewing all this from

another reference-body which is not being pulled with a rope.

 

But how does the man in the chest regard the Process ? The

acceleration of the chest will be transmitted to him by the reaction

of the floor of the chest. He must therefore take up this pressure by

means of his legs if he does not wish to be laid out full length on

the floor. He is then standing in the chest in exactly the same way as

anyone stands in a room of a home on our earth. If he releases a body

which he previously had in his land, the accelertion of the chest will

no longer be transmitted to this body, and for this reason the body

will approach the floor of the chest with an accelerated relative

motion. The observer will further convince himself that the

acceleration of the body towards the floor of the chest is always of

the same magnitude, whatever kind of body he may happen to use for the

experiment.

 

Relying on his knowledge of the gravitational field (as it was

discussed in the preceding section), the man in the chest will thus

come to the conclusion that he and the chest are in a gravitational

field which is constant with regard to time. Of course he will be

puzzled for a moment as to why the chest does not fall in this

gravitational field. just then, however, he discovers the hook in the

middle of the lid of the chest and the rope which is attached to it,

and he consequently comes to the conclusion that the chest is

suspended at rest in the gravitational field.

 

Ought we to smile at the man and say that he errs in his conclusion ?

I do not believe we ought to if we wish to remain consistent ; we must

rather admit that his mode of grasping the situation violates neither

reason nor known mechanical laws. Even though it is being accelerated

with respect to the "Galileian space" first considered, we can

nevertheless regard the chest as being at rest. We have thus good

grounds for extending the principle of relativity to include bodies of

reference which are accelerated with respect to each other, and as a

result we have gained a powerful argument for a generalised postulate

of relativity.

 

We must note carefully that the possibility of this mode of

interpretation rests on the fundamental property of the gravitational

field of giving all bodies the same acceleration, or, what comes to

the same thing, on the law of the equality of inertial and

gravitational mass. If this natural law did not exist, the man in the

accelerated chest would not be able to interpret the behaviour of the

bodies around him on the supposition of a gravitational field, and he

would not be justified on the grounds of experience in supposing his

reference-body to be " at rest."

 

Suppose that the man in the chest fixes a rope to the inner side of

the lid, and that he attaches a body to the free end of the rope. The

result of this will be to strech the rope so that it will hang "

vertically " downwards. If we ask for an opinion of the cause of

tension in the rope, the man in the chest will say: "The suspended

body experiences a downward force in the gravitational field, and this

is neutralised by the tension of the rope ; what determines the

magnitude of the tension of the rope is the gravitational mass of the

suspended body." On the other hand, an observer who is poised freely

in space will interpret the condition of things thus : " The rope must

perforce take part in the accelerated motion of the chest, and it

transmits this motion to the body attached to it. The tension of the

rope is just large enough to effect the acceleration of the body. That

which determines the magnitude of the tension of the rope is the

inertial mass of the body." Guided by this example, we see that our

extension of the principle of relativity implies the necessity of the

law of the equality of inertial and gravitational mass. Thus we have

obtained a physical interpretation of this law.

 

From our consideration of the accelerated chest we see that a general

theory of relativity must yield important results on the laws of

gravitation. In point of fact, the systematic pursuit of the general

idea of relativity has supplied the laws satisfied by the

gravitational field. Before proceeding farther, however, I must warn

the reader against a misconception suggested by these considerations.

A gravitational field exists for the man in the chest, despite the

fact that there was no such field for the co-ordinate system first

chosen. Now we might easily suppose that the existence of a

gravitational field is always only an apparent one. We might also

think that, regardless of the kind of gravitational field which may be

present, we could always choose another reference-body such that no

gravitational field exists with reference to it. This is by no means

true for all gravitational fields, but only for those of quite special

form. It is, for instance, impossible to choose a body of reference

such that, as judged from it, the gravitational field of the earth (in

its entirety) vanishes.

 

We can now appreciate why that argument is not convincing, which we

brought forward against the general principle of relativity at theend

of Section 18. It is certainly true that the observer in the

railway carriage experiences a jerk forwards as a result of the

application of the brake, and that he recognises, in this the

non-uniformity of motion (retardation) of the carriage. But he is

compelled by nobody to refer this jerk to a " real " acceleration

(retardation) of the carriage. He might also interpret his experience

thus: " My body of reference (the carriage) remains permanently at

rest. With reference to it, however, there exists (during the period

of application of the brakes) a gravitational field which is directed

forwards and which is variable with respect to time. Under the

influence of this field, the embankment together with the earth moves

non-uniformly in such a manner that their original velocity in the

backwards direction is continuously reduced."

 

 

 

IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE

SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?

 

 

We have already stated several times that classical mechanics starts

out from the following law: Material particles sufficiently far

removed from other material particles continue to move uniformly in a

straight line or continue in a state of rest. We have also repeatedly

emphasised that this fundamental law can only be valid for bodies of

reference K which possess certain unique states of motion, and which

are in uniform translational motion relative to each other. Relative

to other reference-bodies K the law is not valid. Both in classical

mechanics and in the special theory of relativity we therefore

differentiate between reference-bodies K relative to which the

recognised " laws of nature " can be said to hold, and

reference-bodies K relative to which these laws do not hold.

 

But no person whose mode of thought is logical can rest satisfied with

this condition of things. He asks : " How does it come that certain

reference-bodies (or their states of motion) are given priority over

other reference-bodies (or their states of motion) ? What is the

reason for this Preference? In order to show clearly what I mean by

this question, I shall make use of a comparison.

 

I am standing in front of a gas range. Standing alongside of each

other on the range are two pans so much alike that one may be mistaken

for the other. Both are half full of water. I notice that steam is

being emitted continuously from the one pan, but not from the other. I

am surprised at this, even if I have never seen either a gas range or

a pan before. But if I now notice a luminous something of bluish

colour under the first pan but not under the other, I cease to be

astonished, even if I have never before seen a gas flame. For I can

only say that this bluish something will cause the emission of the

steam, or at least possibly it may do so. If, however, I notice the

bluish something in neither case, and if I observe that the one

continuously emits steam whilst the other does not, then I shall

remain astonished and dissatisfied until I have discovered some

circumstance to which I can attribute the different behaviour of the

two pans.

 

Analogously, I seek in vain for a real something in classical

mechanics (or in the special theory of relativity) to which I can

attribute the different behaviour of bodies considered with respect to

the reference systems K and K1.*  Newton saw this objection and

attempted to invalidate it, but without success. But E. Mach recognsed

it most clearly of all, and because of this objection he claimed that

mechanics must be placed on a new basis. It can only be got rid of by

means of a physics which is conformable to the general principle of

relativity, since the equations of such a theory hold for every body

of reference, whatever may be its state of motion.

 

 

  Notes

 

*) The objection is of importance more especially when the state of

motion of the reference-body is of such a nature that it does not

require any external agency for its maintenance, e.g. in the case when

the reference-body is rotating uniformly.

 

 

 

A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY

 

 

The considerations of Section 20 show that the general principle of

relativity puts us in a position to derive properties of the

gravitational field in a purely theoretical manner. Let us suppose,

for instance, that we know the space-time " course " for any natural

process whatsoever, as regards the manner in which it takes place in

the Galileian domain relative to a Galileian body of reference K. By

means of purely theoretical operations (i.e. simply by calculation) we

are then able to find how this known natural process appears, as seen

from a reference-body K1 which is accelerated relatively to K. But

since a gravitational field exists with respect to this new body of

reference K1, our consideration also teaches us how the gravitational

field influences the process studied.

 

For example, we learn that a body which is in a state of uniform

rectilinear motion with respect to K (in accordance with the law of

Galilei) is executing an accelerated and in general curvilinear motion

with respect to the accelerated reference-body K1 (chest). This

acceleration or curvature corresponds to the influence on the moving

body of the gravitational field prevailing relatively to K. It is

known that a gravitational field influences the movement of bodies in

this way, so that our consideration supplies us with nothing

essentially new.

 

However, we obtain a new result of fundamental importance when we

carry out the analogous consideration for a ray of light. With respect

to the Galileian reference-body K, such a ray of light is transmitted

rectilinearly with the velocity c. It can easily be shown that the

path of the same ray of light is no longer a straight line when we

consider it with reference to the accelerated chest (reference-body

K1). From this we conclude, that, in general, rays of light are

propagated curvilinearly in gravitational fields. In two respects this

result is of great importance.

 

In the first place, it can be compared with the reality. Although a

detailed examination of the question shows that the curvature of light

rays required by the general theory of relativity is only exceedingly

small for the gravitational fields at our disposal in practice, its

estimated magnitude for light rays passing the sun at grazing

incidence is nevertheless 1.7 seconds of arc. This ought to manifest

itself in the following way. As seen from the earth, certain fixed

stars appear to be in the neighbourhood of the sun, and are thus

capable of observation during a total eclipse of the sun. At such

times, these stars ought to appear to be displaced outwards from the

sun by an amount indicated above, as compared with their apparent

position in the sky when the sun is situated at another part of the

heavens. The examination of the correctness or otherwise of this

deduction is a problem of the greatest importance, the early solution

of which is to be expected of astronomers.[2]*

 

In the second place our result shows that, according to the general

theory of relativity, the law of the constancy of the velocity of

light in vacuo, which constitutes one of the two fundamental

assumptions in the special theory of relativity and to which we have

already frequently referred, cannot claim any unlimited validity. A

curvature of rays of light can only take place when the velocity of

propagation of light varies with position. Now we might think that as

a consequence of this, the special theory of relativity and with it

the whole theory of relativity would be laid in the dust. But in

reality this is not the case. We can only conclude that the special

theory of relativity cannot claim an unlinlited domain of validity ;

its results hold only so long as we are able to disregard the

influences of gravitational fields on the phenomena (e.g. of light).

 

Since it has often been contended by opponents of the theory of

relativity that the special theory of relativity is overthrown by the

general theory of relativity, it is perhaps advisable to make the

facts of the case clearer by means of an appropriate comparison.

Before the development of electrodynamics the laws of electrostatics

were looked upon as the laws of electricity. At the present time we

know that electric fields can be derived correctly from electrostatic

considerations only for the case, which is never strictly realised, in

which the electrical masses are quite at rest relatively to each

other, and to the co-ordinate system. Should we be justified in saying

that for this reason electrostatics is overthrown by the

field-equations of Maxwell in electrodynamics ? Not in the least.

Electrostatics is contained in electrodynamics as a limiting case ;

the laws of the latter lead directly to those of the former for the

case in which the fields are invariable with regard to time. No fairer

destiny could be allotted to any physical theory, than that it should

of itself point out the way to the introduction of a more

comprehensive theory, in which it lives on as a limiting case.

 

In the example of the transmission of light just dealt with, we have

seen that the general theory of relativity enables us to derive

theoretically the influence of a gravitational field on the course of

natural processes, the Iaws of which are already known when a

gravitational field is absent. But the most attractive problem, to the

solution of which the general theory of relativity supplies the key,

concerns the investigation of the laws satisfied by the gravitational

field itself. Let us consider this for a moment.

 

We are acquainted with space-time domains which behave (approximately)

in a " Galileian " fashion under suitable choice of reference-body,

i.e. domains in which gravitational fields are absent. If we now refer

such a domain to a reference-body K1 possessing any kind of motion,

then relative to K1 there exists a gravitational field which is

variable with respect to space and time.[3]**  The character of this

field will of course depend on the motion chosen for K1. According to

the general theory of relativity, the general law of the gravitational

field must be satisfied for all gravitational fields obtainable in

this way. Even though by no means all gravitationial fields can be

produced in this way, yet we may entertain the hope that the general

law of gravitation will be derivable from such gravitational fields of

a special kind. This hope has been realised in the most beautiful

manner. But between the clear vision of this goal and its actual

realisation it was necessary to surmount a serious difficulty, and as

this lies deep at the root of things, I dare not withhold it from the

reader. We require to extend our ideas of the space-time continuum

still farther.

 

 

  Notes

 

*) By means of the star photographs of two expeditions equipped by

a Joint Committee of the Royal and Royal Astronomical Societies, the

existence of the deflection of light demanded by theory was first

confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix

III.)

 

**) This follows from a generalisation of the discussion in

Section 20

 

 

 

BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE

 

 

Hitherto I have purposely refrained from speaking about the physical

interpretation of space- and time-data in the case of the general

theory of relativity. As a consequence, I am guilty of a certain

slovenliness of treatment, which, as we know from the special theory

of relativity, is far from being unimportant and pardonable. It is now

high time that we remedy this defect; but I would mention at the

outset, that this matter lays no small claims on the patience and on

the power of abstraction of the reader.

 

We start off again from quite special cases, which we have frequently

used before. Let us consider a space time domain in which no

gravitational field exists relative to a reference-body K whose state

of motion has been suitably chosen. K is then a Galileian

reference-body as regards the domain considered, and the results of

the special theory of relativity hold relative to K. Let us supposse

the same domain referred to a second body of reference K1, which is

rotating uniformly with respect to K. In order to fix our ideas, we

shall imagine K1 to be in the form of a plane circular disc, which

rotates uniformly in its own plane about its centre. An observer who

is sitting eccentrically on the disc K1 is sensible of a force which

acts outwards in a radial direction, and which would be interpreted as

an effect of inertia (centrifugal force) by an observer who was at

rest with respect to the original reference-body K. But the observer

on the disc may regard his disc as a reference-body which is " at rest

" ; on the basis of the general principle of relativity he is

justified in doing this. The force acting on himself, and in fact on

all other bodies which are at rest relative to the disc, he regards as

the effect of a gravitational field. Nevertheless, the

space-distribution of this gravitational field is of a kind that would

not be possible on Newton's theory of gravitation.* But since the

observer believes in the general theory of relativity, this does not

disturb him; he is quite in the right when he believes that a general

law of gravitation can be formulated- a law which not only explains

the motion of the stars correctly, but also the field of force

experienced by himself.

 

The observer performs experiments on his circular disc with clocks and

measuring-rods. In doing so, it is his intention to arrive at exact

definitions for the signification of time- and space-data with

reference to the circular disc K1, these definitions being based on

his observations. What will be his experience in this enterprise ?

 

To start with, he places one of two identically constructed clocks at

the centre of the circular disc, and the other on the edge of the

disc, so that they are at rest relative to it. We now ask ourselves

whether both clocks go at the same rate from the standpoint of the

non-rotating Galileian reference-body K. As judged from this body, the

clock at the centre of the disc has no velocity, whereas the clock at

the edge of the disc is in motion relative to K in consequence of the

rotation. According to a result obtained in Section 12, it follows

that the latter clock goes at a rate permanently slower than that of

the clock at the centre of the circular disc, i.e. as observed from K.

It is obvious that the same effect would be noted by an observer whom

we will imagine sitting alongside his clock at the centre of the

circular disc. Thus on our circular disc, or, to make the case more

general, in every gravitational field, a clock will go more quickly or

less quickly, according to the position in which the clock is situated

(at rest). For this reason it is not possible to obtain a reasonable

definition of time with the aid of clocks which are arranged at rest

with respect to the body of reference. A similar difficulty presents

itself when we attempt to apply our earlier definition of simultaneity

in such a case, but I do not wish to go any farther into this

question.

 

Moreover, at this stage the definition of the space co-ordinates also

presents insurmountable difficulties. If the observer applies his

standard measuring-rod (a rod which is short as compared with the

radius of the disc) tangentially to the edge of the disc, then, as

judged from the Galileian system, the length of this rod will be less

than I, since, according to Section 12, moving bodies suffer a

shortening in the direction of the motion. On the other hand, the

measaring-rod will not experience a shortening in length, as judged

from K, if it is applied to the disc in the direction of the radius.

If, then, the observer first measures the circumference of the disc

with his measuring-rod and then the diameter of the disc, on dividing

the one by the other, he will not obtain as quotient the familiar

number p = 3.14 . . ., but a larger number,[4]** whereas of course,

for a disc which is at rest with respect to K, this operation would

yield p exactly. This proves that the propositions of Euclidean

geometry cannot hold exactly on the rotating disc, nor in general in a

gravitational field, at least if we attribute the length I to the rod

in all positions and in every orientation. Hence the idea of a

straight line also loses its meaning. We are therefore not in a

position to define exactly the co-ordinates x, y, z relative to the

disc by means of the method used in discussing the special theory, and

as long as the co- ordinates and times of events have not been

defined, we cannot assign an exact meaning to the natural laws in

which these occur.

 

Thus all our previous conclusions based on general relativity would

appear to be called in question. In reality we must make a subtle

detour in order to be able to apply the postulate of general

relativity exactly. I shall prepare the reader for this in the

following paragraphs.

 

 

  Notes

 

*) The field disappears at the centre of the disc and increases

proportionally to the distance from the centre as we proceed outwards.

 

**) Throughout this consideration we have to use the Galileian

(non-rotating) system K as reference-body, since we may only assume

the validity of the results of the special theory of relativity

relative to K (relative to K1 a gravitational field prevails).

 

 

 

EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM

 

 

The surface of a marble table is spread out in front of me. I can get

from any one point on this table to any other point by passing

continuously from one point to a " neighbouring " one, and repeating

this process a (large) number of times, or, in other words, by going

from point to point without executing "jumps." I am sure the reader

will appreciate with sufficient clearness what I mean here by "

neighbouring " and by " jumps " (if he is not too pedantic). We

express this property of the surface by describing the latter as a

continuum.

 

Let us now imagine that a large number of little rods of equal length

have been made, their lengths being small compared with the dimensions

of the marble slab. When I say they are of equal length, I mean that

one can be laid on any other without the ends overlapping. We next lay

four of these little rods on the marble slab so that they constitute a

quadrilateral figure (a square), the diagonals of which are equally

long. To ensure the equality of the diagonals, we make use of a little

testing-rod. To this square we add similar ones, each of which has one

rod in common with the first. We proceed in like manner with each of

these squares until finally the whole marble slab is laid out with

squares. The arrangement is such, that each side of a square belongs

to two squares and each corner to four squares.

 

It is a veritable wonder that we can carry out this business without

getting into the greatest difficulties. We only need to think of the

following. If at any moment three squares meet at a corner, then two

sides of the fourth square are already laid, and, as a consequence,

the arrangement of the remaining two sides of the square is already

completely determined. But I am now no longer able to adjust the

quadrilateral so that its diagonals may be equal. If they are equal of

their own accord, then this is an especial favour of the marble slab

and of the little rods, about which I can only be thankfully

surprised. We must experience many such surprises if the construction

is to be successful.

 

If everything has really gone smoothly, then I say that the points of

the marble slab constitute a Euclidean continuum with respect to the

little rod, which has been used as a " distance " (line-interval). By

choosing one corner of a square as " origin" I can characterise every

other corner of a square with reference to this origin by means of two

numbers. I only need state how many rods I must pass over when,

starting from the origin, I proceed towards the " right " and then "

upwards," in order to arrive at the corner of the square under

consideration. These two numbers are then the " Cartesian co-ordinates

" of this corner with reference to the " Cartesian co-ordinate system"

which is determined by the arrangement of little rods.

 

By making use of the following modification of this abstract

experiment, we recognise that there must also be cases in which the

experiment would be unsuccessful. We shall suppose that the rods "

expand " by in amount proportional to the increase of temperature. We

heat the central part of the marble slab, but not the periphery, in

which case two of our little rods can still be brought into

coincidence at every position on the table. But our construction of

squares must necessarily come into disorder during the heating,

because the little rods on the central region of the table expand,

whereas those on the outer part do not.

 

With reference to our little rods -- defined as unit lengths -- the

marble slab is no longer a Euclidean continuum, and we are also no

longer in the position of defining Cartesian co-ordinates directly

with their aid, since the above construction can no longer be carried

out. But since there are other things which are not influenced in a

similar manner to the little rods (or perhaps not at all) by the

temperature of the table, it is possible quite naturally to maintain

the point of view that the marble slab is a " Euclidean continuum."

This can be done in a satisfactory manner by making a more subtle

stipulation about the measurement or the comparison of lengths.

 

But if rods of every kind (i.e. of every material) were to behave in

the same way as regards the influence of temperature when they are on

the variably heated marble slab, and if we had no other means of

detecting the effect of temperature than the geometrical behaviour of

our rods in experiments analogous to the one described above, then our

best plan would be to assign the distance one to two points on the

slab, provided that the ends of one of our rods could be made to

coincide with these two points ; for how else should we define the

distance without our proceeding being in the highest measure grossly

arbitrary ? The method of Cartesian coordinates must then be

discarded, and replaced by another which does not assume the validity

of Euclidean geometry for rigid bodies.*  The reader will notice

that the situation depicted here corresponds to the one brought about

by the general postitlate of relativity (Section 23).

 

 

  Notes

 

*) Mathematicians have been confronted with our problem in the

following form. If we are given a surface (e.g. an ellipsoid) in

Euclidean three-dimensional space, then there exists for this surface

a two-dimensional geometry, just as much as for a plane surface. Gauss

undertook the task of treating this two-dimensional geometry from

first principles, without making use of the fact that the surface

belongs to a Euclidean continuum of three dimensions. If we imagine

constructions to be made with rigid rods in the surface (similar to

that above with the marble slab), we should find that different laws

hold for these from those resulting on the basis of Euclidean plane

geometry. The surface is not a Euclidean continuum with respect to the

rods, and we cannot define Cartesian co-ordinates in the surface.

Gauss indicated the principles according to which we can treat the

geometrical relationships in the surface, and thus pointed out the way

to the method of Riemman of treating multi-dimensional, non-Euclidean

continuum. Thus it is that mathematicians long ago solved the formal

problems to which we are led by the general postulate of relativity.

 

 

 

GAUSSIAN CO-ORDINATES

 

According to Gauss, this combined analytical and geometrical mode of

handling the problem can be arrived at in the following way. We

imagine a system of arbitrary curves (see Fig. 4) drawn on the surface

of the table. These we designate as u-curves, and we indicate each of

them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in

the diagram. Between the curves u= 1 and u= 2 we must imagine an

infinitely large number to be drawn, all of which correspond to real

numbers lying between 1 and 2. fig. 04 We have then a system of

u-curves, and this "infinitely dense" system covers the whole surface

of the table. These u-curves must not intersect each other, and

through each point of the surface one and only one curve must pass.

Thus a perfectly definite value of u belongs to every point on the

surface of the marble slab. In like manner we imagine a system of

v-curves drawn on the surface. These satisfy the same conditions as

the u-curves, they are provided with numbers in a corresponding

manner, and they may likewise be of arbitrary shape. It follows that a

value of u and a value of v belong to every point on the surface of

the table. We call these two numbers the co-ordinates of the surface

of the table (Gaussian co-ordinates). For example, the point P in the

diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring

points P and P1 on the surface then correspond to the co-ordinates

 

                       P:       u,v

 

                       P1:     u + du, v + dv,

 

where du and dv signify very small numbers. In a similar manner we may

indicate the distance (line-interval) between P and P1, as measured

with a little rod, by means of the very small number ds. Then

according to Gauss we have

 

                ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2

 

where g[11], g[12], g[22], are magnitudes which depend in a perfectly

definite way on u and v. The magnitudes g[11], g[12] and g[22],

determine the behaviour of the rods relative to the u-curves and

v-curves, and thus also relative to the surface of the table. For the

case in which the points of the surface considered form a Euclidean

continuum with reference to the measuring-rods, but only in this case,

it is possible to draw the u-curves and v-curves and to attach numbers

to them, in such a manner, that we simply have :

 

                           ds2 = du2 + dv2

 

Under these conditions, the u-curves and v-curves are straight lines

in the sense of Euclidean geometry, and they are perpendicular to each

other. Here the Gaussian coordinates are samply Cartesian ones. It is

clear that Gauss co-ordinates are nothing more than an association of

two sets of numbers with the points of the surface considered, of such

a nature that numerical values differing very slightly from each other

are associated with neighbouring points " in space."

 

So far, these considerations hold for a continuum of two dimensions.

But the Gaussian method can be applied also to a continuum of three,

four or more dimensions. If, for instance, a continuum of four

dimensions be supposed available, we may represent it in the following

way. With every point of the continuum, we associate arbitrarily four

numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."

Adjacent points correspond to adjacent values of the coordinates. If a

distance ds is associated with the adjacent points P and P1, this

distance being measurable and well defined from a physical point of

view, then the following formula holds:

 

ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,

 

where the magnitudes g[11], etc., have values which vary with the

position in the continuum. Only when the continuum is a Euclidean one

is it possible to associate the co-ordinates x[1] . . x[4]. with the

points of the continuum so that we have simply

 

ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

 

In this case relations hold in the four-dimensional continuum which

are analogous to those holding in our three-dimensional measurements.

 

However, the Gauss treatment for ds2 which we have given above is not

always possible. It is only possible when sufficiently small regions

of the continuum under consideration may be regarded as Euclidean

continua. For example, this obviously holds in the case of the marble

slab of the table and local variation of temperature. The temperature

is practically constant for a small part of the slab, and thus the

geometrical behaviour of the rods is almost as it ought to be

according to the rules of Euclidean geometry. Hence the imperfections

of the construction of squares in the previous section do not show

themselves clearly until this construction is extended over a

considerable portion of the surface of the table.

 

We can sum this up as follows: Gauss invented a method for the

mathematical treatment of continua in general, in which "

size-relations " (" distances " between neighbouring points) are

defined. To every point of a continuum are assigned as many numbers

(Gaussian coordinates) as the continuum has dimensions. This is done

in such a way, that only one meaning can be attached to the

assignment, and that numbers (Gaussian coordinates) which differ by an

indefinitely small amount are assigned to adjacent points. The

Gaussian coordinate system is a logical generalisation of the

Cartesian co-ordinate system. It is also applicable to non-Euclidean

continua, but only when, with respect to the defined "size" or

"distance," small parts of the continuum under consideration behave

more nearly like a Euclidean system, the smaller the part of the

continuum under our notice.

 

 

 

THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A

EUCLIDEAN CONTINUUM

 

 

We are now in a position to formulate more exactly the idea of

Minkowski, which was only vaguely indicated in Section 17. In

accordance with the special theory of relativity, certain co-ordinate

systems are given preference for the description of the

four-dimensional, space-time continuum. We called these " Galileian

co-ordinate systems." For these systems, the four co-ordinates x, y,

z, t, which determine an event or -- in other words, a point of the

four-dimensional continuum -- are defined physically in a simple

manner, as set forth in detail in the first part of this book. For the

transition from one Galileian system to another, which is moving

uniformly with reference to the first, the equations of the Lorentz

transformation are valid. These last form the basis for the derivation

of deductions from the special theory of relativity, and in themselves

they are nothing more than the expression of the universal validity of

the law of transmission of light for all Galileian systems of

reference.

 

Minkowski found that the Lorentz transformations satisfy the following

simple conditions. Let us consider two neighbouring events, the

relative position of which in the four-dimensional continuum is given

with respect to a Galileian reference-body K by the space co-ordinate

differences dx, dy, dz and the time-difference dt. With reference to a

second Galileian system we shall suppose that the corresponding

differences for these two events are dx1, dy1, dz1, dt1. Then these

magnitudes always fulfil the condition*

 

     dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.

 

The validity of the Lorentz transformation follows from this

condition. We can express this as follows: The magnitude

 

                   ds2 = dx2 + dy2 + dz2 - c^2dt2,

 

which belongs to two adjacent points of the four-dimensional

space-time continuum, has the same value for all selected (Galileian)

reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],

x[2], x[3], x[4], we also obtaill the result that

 

             ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

 

is independent of the choice of the body of reference. We call the

magnitude ds the " distance " apart of the two events or

four-dimensional points.

 

Thus, if we choose as time-variable the imaginary variable sq. rt. -I

. ct instead of the real quantity t, we can regard the space-time

contintium -- accordance with the special theory of relativity -- as a

", Euclidean " four-dimensional continuum, a result which follows from

the considerations of the preceding section.

 

 

  Notes

 

*) Cf. Appendixes I and 2. The relations which are derived

there for the co-ordlnates themselves are valid also for co-ordinate

differences, and thus also for co-ordinate differentials (indefinitely

small differences).

 

 

 

THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A

ECULIDEAN CONTINUUM

 

 

In the first part of this book we were able to make use of space-time

co-ordinates which allowed of a simple and direct physical

interpretation, and which, according to Section 26, can be regarded

as four-dimensional Cartesian co-ordinates. This was possible on the

basis of the law of the constancy of the velocity of tight. But

according to Section 21 the general theory of relativity cannot

retain this law. On the contrary, we arrived at the result that

according to this latter theory the velocity of light must always

depend on the co-ordinates when a gravitational field is present. In

connection with a specific illustration in Section 23, we found

that the presence of a gravitational field invalidates the definition

of the coordinates and the ifine, which led us to our objective in the

special theory of relativity.

 

In view of the resuIts of these considerations we are led to the

conviction that, according to the general principle of relativity, the

space-time continuum cannot be regarded as a Euclidean one, but that

here we have the general case, corresponding to the marble slab with

local variations of temperature, and with which we made acquaintance

as an example of a two-dimensional continuum. Just as it was there

impossible to construct a Cartesian co-ordinate system from equal

rods, so here it is impossible to build up a system (reference-body)

from rigid bodies and clocks, which shall be of such a nature that

measuring-rods and clocks, arranged rigidly with respect to one

another, shaIll indicate position and time directly. Such was the

essence of the difficulty with which we were confronted in Section

23.

 

But the considerations of Sections 25 and 26 show us the way to

surmount this difficulty. We refer the fourdimensional space-time

continuum in an arbitrary manner to Gauss co-ordinates. We assign to

every point of the continuum (event) four numbers, x[1], x[2], x[3],

x[4] (co-ordinates), which have not the least direct physical

significance, but only serve the purpose of numbering the points of

the continuum in a definite but arbitrary manner. This arrangement

does not even need to be of such a kind that we must regard x[1],

x[2], x[3], as "space" co-ordinates and x[4], as a " time "

co-ordinate.

 

The reader may think that such a description of the world would be

quite inadequate. What does it mean to assign to an event the

particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these

co-ordinates have no significance ? More careful consideration shows,

however, that this anxiety is unfounded. Let us consider, for

instance, a material point with any kind of motion. If this point had

only a momentary existence without duration, then it would to

described in space-time by a single system of values x[1], x[2], x[3],

x[4]. Thus its permanent existence must be characterised by an

infinitely large number of such systems of values, the co-ordinate

values of which are so close together as to give continuity;

corresponding to the material point, we thus have a (uni-dimensional)

line in the four-dimensional continuum. In the same way, any such

lines in our continuum correspond to many points in motion. The only

statements having regard to these points which can claim a physical

existence are in reality the statements about their encounters. In our

mathematical treatment, such an encounter is expressed in the fact

that the two lines which represent the motions of the points in

question have a particular system of co-ordinate values, x[1], x[2],

x[3], x[4], in common. After mature consideration the reader will

doubtless admit that in reality such encounters constitute the only

actual evidence of a time-space nature with which we meet in physical

statements.

 

When we were describing the motion of a material point relative to a

body of reference, we stated nothing more than the encounters of this

point with particular points of the reference-body. We can also

determine the corresponding values of the time by the observation of

encounters of the body with clocks, in conjunction with the

observation of the encounter of the hands of clocks with particular

points on the dials. It is just the same in the case of

space-measurements by means of measuring-rods, as a litttle

consideration will show.

 

The following statements hold generally : Every physical description

resolves itself into a number of statements, each of which refers to

the space-time coincidence of two events A and B. In terms of Gaussian

co-ordinates, every such statement is expressed by the agreement of

their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the

description of the time-space continuum by means of Gauss co-ordinates

completely replaces the description with the aid of a body of

reference, without suffering from the defects of the latter mode of

description; it is not tied down to the Euclidean character of the

continuum which has to be represented.

 

 

 

EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY

 

 

We are now in a position to replace the pro. visional formulation of

the general principle of relativity given in Section 18 by an exact

formulation. The form there used, "All bodies of reference K, K1,

etc., are equivalent for the description of natural phenomena

(formulation of the general laws of nature), whatever may be their

state of motion," cannot be maintained, because the use of rigid

reference-bodies, in the sense of the method followed in the special

theory of relativity, is in general not possible in space-time

description. The Gauss co-ordinate system has to take the place of the

body of reference. The following statement corresponds to the

fundamental idea of the general principle of relativity: "All Gaussian

co-ordinate systems are essentially equivalent for the formulation of

the general laws of nature."

 

We can state this general principle of relativity in still another

form, which renders it yet more clearly intelligible than it is when

in the form of the natural extension of the special principle of

relativity. According to the special theory of relativity, the

equations which express the general laws of nature pass over into

equations of the same form when, by making use of the Lorentz

transformation, we replace the space-time variables x, y, z, t, of a

(Galileian) reference-body K by the space-time variables x1, y1, z1,

t1, of a new reference-body K1. According to the general theory of

relativity, on the other hand, by application of arbitrary

substitutions of the Gauss variables x[1], x[2], x[3], x[4], the

equations must pass over into equations of the same form; for every

transformation (not only the Lorentz transformation) corresponds to

the transition of one Gauss co-ordinate system into another.

 

If we desire to adhere to our "old-time" three-dimensional view of

things, then we can characterise the development which is being

undergone by the fundamental idea of the general theory of relativity

as follows : The special theory of relativity has reference to

Galileian domains, i.e. to those in which no gravitational field

exists. In this connection a Galileian reference-body serves as body

of reference, i.e. a rigid body the state of motion of which is so

chosen that the Galileian law of the uniform rectilinear motion of

"isolated" material points holds relatively to it.

 

Certain considerations suggest that we should refer the same Galileian

domains to non-Galileian reference-bodies also. A gravitational field

of a special kind is then present with respect to these bodies (cf.

Sections 20 and 23).

 

In gravitational fields there are no such things as rigid bodies with

Euclidean properties; thus the fictitious rigid body of reference is

of no avail in the general theory of relativity. The motion of clocks

is also influenced by gravitational fields, and in such a way that a

physical definition of time which is made directly with the aid of

clocks has by no means the same degree of plausibility as in the

special theory of relativity.

 

For this reason non-rigid reference-bodies are used, which are as a

whole not only moving in any way whatsoever, but which also suffer

alterations in form ad lib. during their motion. Clocks, for which the

law of motion is of any kind, however irregular, serve for the

definition of time. We have to imagine each of these clocks fixed at a

point on the non-rigid reference-body. These clocks satisfy only the

one condition, that the "readings" which are observed simultaneously

on adjacent clocks (in space) differ from each other by an

indefinitely small amount. This non-rigid reference-body, which might

appropriately be termed a "reference-mollusc", is in the main

equivalent to a Gaussian four-dimensional co-ordinate system chosen

arbitrarily. That which gives the "mollusc" a certain

comprehensibility as compared with the Gauss co-ordinate system is the

(really unjustified) formal retention of the separate existence of the

space co-ordinates as opposed to the time co-ordinate. Every point on

the mollusc is treated as a space-point, and every material point

which is at rest relatively to it as at rest, so long as the mollusc

is considered as reference-body. The general principle of relativity

requires that all these molluscs can be used as reference-bodies with

equal right and equal success in the formulation of the general laws

of nature; the laws themselves must be quite independent of the choice

of mollusc.

 

The great power possessed by the general principle of relativity lies

in the comprehensive limitation which is imposed on the laws of nature

in consequence of what we have seen above.

 

 

 

THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL

PRINCIPLE OF RELATIVITY

 

 

If the reader has followed all our previous considerations, he will

have no further difficulty in understanding the methods leading to the

solution of the problem of gravitation.

 

We start off on a consideration of a Galileian domain, i.e. a domain

in which there is no gravitational field relative to the Galileian

reference-body K. The behaviour of measuring-rods and clocks with

reference to K is known from the special theory of relativity,

likewise the behaviour of "isolated" material points; the latter move

uniformly and in straight lines.

 

Now let us refer this domain to a random Gauss coordinate system or to

a "mollusc" as reference-body K1. Then with respect to K1 there is a

gravitational field G (of a particular kind). We learn the behaviour

of measuring-rods and clocks and also of freely-moving material points

with reference to K1 simply by mathematical transformation. We

interpret this behaviour as the behaviour of measuring-rods, docks and

material points tinder the influence of the gravitational field G.

Hereupon we introduce a hypothesis: that the influence of the

gravitational field on measuringrods, clocks and freely-moving

material points continues to take place according to the same laws,

even in the case where the prevailing gravitational field is not

derivable from the Galfleian special care, simply by means of a

transformation of co-ordinates.

 

The next step is to investigate the space-time behaviour of the

gravitational field G, which was derived from the Galileian special

case simply by transformation of the coordinates. This behaviour is

formulated in a law, which is always valid, no matter how the

reference-body (mollusc) used in the description may be chosen.

 

This law is not yet the general law of the gravitational field, since

the gravitational field under consideration is of a special kind. In

order to find out the general law-of-field of gravitation we still

require to obtain a generalisation of the law as found above. This can

be obtained without caprice, however, by taking into consideration the

following demands:

 

(a) The required generalisation must likewise satisfy the general

postulate of relativity.

 

(b) If there is any matter in the domain under consideration, only its

inertial mass, and thus according to Section 15 only its energy is

of importance for its etfect in exciting a field.

 

(c) Gravitational field and matter together must satisfy the law of

the conservation of energy (and of impulse).

 

Finally, the general principle of relativity permits us to determine

the influence of the gravitational field on the course of all those

processes which take place according to known laws when a

gravitational field is absent i.e. which have already been fitted into

the frame of the special theory of relativity. In this connection we

proceed in principle according to the method which has already been

explained for measuring-rods, clocks and freely moving material

points.

 

The theory of gravitation derived in this way from the general

postulate of relativity excels not only in its beauty ; nor in

removing the defect attaching to classical mechanics which was brought

to light in Section 21; nor in interpreting the empirical law of

the equality of inertial and gravitational mass ; but it has also

already explained a result of observation in astronomy, against which

classical mechanics is powerless.

 

If we confine the application of the theory to the case where the

gravitational fields can be regarded as being weak, and in which all

masses move with respect to the coordinate system with velocities

which are small compared with the velocity of light, we then obtain as

a first approximation the Newtonian theory. Thus the latter theory is

obtained here without any particular assumption, whereas Newton had to

introduce the hypothesis that the force of attraction between mutually

attracting material points is inversely proportional to the square of

the distance between them. If we increase the accuracy of the

calculation, deviations from the theory of Newton make their

appearance, practically all of which must nevertheless escape the test

of observation owing to their smallness.

 

We must draw attention here to one of these deviations. According to

Newton's theory, a planet moves round the sun in an ellipse, which

would permanently maintain its position with respect to the fixed

stars, if we could disregard the motion of the fixed stars themselves

and the action of the other planets under consideration. Thus, if we

correct the observed motion of the planets for these two influences,

and if Newton's theory be strictly correct, we ought to obtain for the

orbit of the planet an ellipse, which is fixed with reference to the

fixed stars. This deduction, which can be tested with great accuracy,

has been confirmed for all the planets save one, with the precision

that is capable of being obtained by the delicacy of observation

attainable at the present time. The sole exception is Mercury, the

planet which lies nearest the sun. Since the time of Leverrier, it has

been known that the ellipse corresponding to the orbit of Mercury,

after it has been corrected for the influences mentioned above, is not

stationary with respect to the fixed stars, but that it rotates

exceedingly slowly in the plane of the orbit and in the sense of the

orbital motion. The value obtained for this rotary movement of the

orbital ellipse was 43 seconds of arc per century, an amount ensured

to be correct to within a few seconds of arc. This effect can be

explained by means of classical mechanics only on the assumption of

hypotheses which have little probability, and which were devised

solely for this purponse.

 

On the basis of the general theory of relativity, it is found that the

ellipse of every planet round the sun must necessarily rotate in the

manner indicated above ; that for all the planets, with the exception

of Mercury, this rotation is too small to be detected with the

delicacy of observation possible at the present time ; but that in the

case of Mercury it must amount to 43 seconds of arc per century, a

result which is strictly in agreement with observation.

 

Apart from this one, it has hitherto been possible to make only two

deductions from the theory which admit of being tested by observation,

to wit, the curvature of light rays by the gravitational field of the

sun,*x and a displacement of the spectral lines of light reaching

us from large stars, as compared with the corresponding lines for

light produced in an analogous manner terrestrially (i.e. by the same

kind of atom).**  These two deductions from the theory have both

been confirmed.

 

 

  Notes

 

*) First observed by Eddington and others in 1919. (Cf. Appendix

III, pp. 126-129).

 

**) Established by Adams in 1924. (Cf. p. 132)

 

 

 

 

PART III

 

CONSIDERATIONS ON THE UNIVERSE AS A WHOLE

 

 

COSMOLOGICAL DIFFICULTIES OF NEWTON'S THEORY

 

 

Part from the difficulty discussed in Section 21, there is a second

fundamental difficulty attending classical celestial mechanics, which,

to the best of my knowledge, was first discussed in detail by the

astronomer Seeliger. If we ponder over the question as to how the

universe, considered as a whole, is to be regarded, the first answer

that suggests itself to us is surely this: As regards space (and time)

the universe is infinite. There are stars everywhere, so that the

density of matter, although very variable in detail, is nevertheless

on the average everywhere the same. In other words: However far we

might travel through space, we should find everywhere an attenuated

swarm of fixed stars of approrimately the same kind and density.

 

This view is not in harmony with the theory of Newton. The latter

theory rather requires that the universe should have a kind of centre

in which the density of the stars is a maximum, and that as we proceed

outwards from this centre the group-density of the stars should

diminish, until finally, at great distances, it is succeeded by an

infinite region of emptiness. The stellar universe ought to be a

finite island in the infinite ocean of space.*

 

This conception is in itself not very satisfactory. It is still less

satisfactory because it leads to the result that the light emitted by

the stars and also individual stars of the stellar system are

perpetually passing out into infinite space, never to return, and

without ever again coming into interaction with other objects of

nature. Such a finite material universe would be destined to become

gradually but systematically impoverished.

 

In order to escape this dilemma, Seeliger suggested a modification of

Newton's law, in which he assumes that for great distances the force

of attraction between two masses diminishes more rapidly than would

result from the inverse square law. In this way it is possible for the

mean density of matter to be constant everywhere, even to infinity,

without infinitely large gravitational fields being produced. We thus

free ourselves from the distasteful conception that the material

universe ought to possess something of the nature of a centre. Of

course we purchase our emancipation from the fundamental difficulties

mentioned, at the cost of a modification and complication of Newton's

law which has neither empirical nor theoretical foundation. We can

imagine innumerable laws which would serve the same purpose, without

our being able to state a reason why one of them is to be preferred to

the others ; for any one of these laws would be founded just as little

on more general theoretical principles as is the law of Newton.

 

 

  Notes

 

*) Proof -- According to the theory of Newton, the number of "lines

of force" which come from infinity and terminate in a mass m is

proportional to the mass m. If, on the average, the Mass density p[0]

is constant throughout tithe universe, then a sphere of volume V will

enclose the average man p[0]V. Thus the number of lines of force

passing through the surface F of the sphere into its interior is

proportional to p[0] V. For unit area of the surface of the sphere the

number of lines of force which enters the sphere is thus proportional

to p[0] V/F or to p[0]R. Hence the intensity of the field at the

surface would ultimately become infinite with increasing radius R of

the sphere, which is impossible.

 

 

 

THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE

 

 

But speculations on the structure of the universe also move in quite

another direction. The development of non-Euclidean geometry led to

the recognition of the fact, that we can cast doubt on the

infiniteness of our space without coming into conflict with the laws

of thought or with experience (Riemann, Helmholtz). These questions

have already been treated in detail and with unsurpassable lucidity by

Helmholtz and Poincaré, whereas I can only touch on them briefly here.

 

In the first place, we imagine an existence in two dimensional space.

Flat beings with flat implements, and in particular flat rigid

measuring-rods, are free to move in a plane. For them nothing exists

outside of this plane: that which they observe to happen to themselves

and to their flat " things " is the all-inclusive reality of their

plane. In particular, the constructions of plane Euclidean geometry

can be carried out by means of the rods e.g. the lattice construction,

considered in Section 24. In contrast to ours, the universe of

these beings is two-dimensional; but, like ours, it extends to

infinity. In their universe there is room for an infinite number of

identical squares made up of rods, i.e. its volume (surface) is

infinite. If these beings say their universe is " plane," there is

sense in the statement, because they mean that they can perform the

constructions of plane Euclidean geometry with their rods. In this

connection the individual rods always represent the same distance,

independently of their position.

 

Let us consider now a second two-dimensional existence, but this time

on a spherical surface instead of on a plane. The flat beings with

their measuring-rods and other objects fit exactly on this surface and

they are unable to leave it. Their whole universe of observation

extends exclusively over the surface of the sphere. Are these beings

able to regard the geometry of their universe as being plane geometry

and their rods withal as the realisation of " distance " ? They cannot

do this. For if they attempt to realise a straight line, they will

obtain a curve, which we " three-dimensional beings " designate as a

great circle, i.e. a self-contained line of definite finite length,

which can be measured up by means of a measuring-rod. Similarly, this

universe has a finite area that can be compared with the area, of a

square constructed with rods. The great charm resulting from this

consideration lies in the recognition of the fact that the universe of

these beings is finite and yet has no limits.

 

But the spherical-surface beings do not need to go on a world-tour in

order to perceive that they are not living in a Euclidean universe.

They can convince themselves of this on every part of their " world,"

provided they do not use too small a piece of it. Starting from a

point, they draw " straight lines " (arcs of circles as judged in

three dimensional space) of equal length in all directions. They will

call the line joining the free ends of these lines a " circle." For a

plane surface, the ratio of the circumference of a circle to its

diameter, both lengths being measured with the same rod, is, according

to Euclidean geometry of the plane, equal to a constant value p, which

is independent of the diameter of the circle. On their spherical

surface our flat beings would find for this ratio the value

 

                        

 

i.e. a smaller value than p, the difference being the more

considerable, the greater is the radius of the circle in comparison

with the radius R of the " world-sphere." By means of this relation

the spherical beings can determine the radius of their universe ("

world "), even when only a relatively small part of their worldsphere

is available for their measurements. But if this part is very small

indeed, they will no longer be able to demonstrate that they are on a

spherical " world " and not on a Euclidean plane, for a small part of

a spherical surface differs only slightly from a piece of a plane of

the same size.

 

Thus if the spherical surface beings are living on a planet of which

the solar system occupies only a negligibly small part of the

spherical universe, they have no means of determining whether they are

living in a finite or in an infinite universe, because the " piece of

universe " to which they have access is in both cases practically

plane, or Euclidean. It follows directly from this discussion, that

for our sphere-beings the circumference of a circle first increases

with the radius until the " circumference of the universe " is

reached, and that it thenceforward gradually decreases to zero for

still further increasing values of the radius. During this process the

area of the circle continues to increase more and more, until finally

it becomes equal to the total area of the whole " world-sphere."

 

Perhaps the reader will wonder why we have placed our " beings " on a

sphere rather than on another closed surface. But this choice has its

justification in the fact that, of all closed surfaces, the sphere is

unique in possessing the property that all points on it are

equivalent. I admit that the ratio of the circumference c of a circle

to its radius r depends on r, but for a given value of r it is the

same for all points of the " worldsphere "; in other words, the "

world-sphere " is a " surface of constant curvature."

 

To this two-dimensional sphere-universe there is a three-dimensional

analogy, namely, the three-dimensional spherical space which was

discovered by Riemann. its points are likewise all equivalent. It

possesses a finite volume, which is determined by its "radius"

(2p2R3). Is it possible to imagine a spherical space? To imagine a

space means nothing else than that we imagine an epitome of our "

space " experience, i.e. of experience that we can have in the

movement of " rigid " bodies. In this sense we can imagine a spherical

space.

 

Suppose we draw lines or stretch strings in all directions from a

point, and mark off from each of these the distance r with a

measuring-rod. All the free end-points of these lengths lie on a

spherical surface. We can specially measure up the area (F) of this

surface by means of a square made up of measuring-rods. If the

universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is

always less than 4pR2. With increasing values of r, F increases from

zero up to a maximum value which is determined by the " world-radius,"

but for still further increasing values of r, the area gradually

diminishes to zero. At first, the straight lines which radiate from

the starting point diverge farther and farther from one another, but

later they approach each other, and finally they run together again at

a "counter-point" to the starting point. Under such conditions they

have traversed the whole spherical space. It is easily seen that the

three-dimensional spherical space is quite analogous to the

two-dimensional spherical surface. It is finite (i.e. of finite

volume), and has no bounds.

 

It may be mentioned that there is yet another kind of curved space: "

elliptical space." It can be regarded as a curved space in which the

two " counter-points " are identical (indistinguishable from each

other). An elliptical universe can thus be considered to some extent

as a curved universe possessing central symmetry.

 

It follows from what has been said, that closed spaces without limits

are conceivable. From amongst these, the spherical space (and the

elliptical) excels in its simplicity, since all points on it are

equivalent. As a result of this discussion, a most interesting

question arises for astronomers and physicists, and that is whether

the universe in which we live is infinite, or whether it is finite in

the manner of the spherical universe. Our experience is far from being

sufficient to enable us to answer this question. But the general

theory of relativity permits of our answering it with a moduate degree

of certainty, and in this connection the difficulty mentioned in

Section 30 finds its solution.

 

 

 

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

 

 

According to the general theory of relativity, the geometrical

properties of space are not independent, but they are determined by

matter. Thus we can draw conclusions about the geometrical structure

of the universe only if we base our considerations on the state of the

matter as being something that is known. We know from experience that,

for a suitably chosen co-ordinate system, the velocities of the stars

are small as compared with the velocity of transmission of light. We

can thus as a rough approximation arrive at a conclusion as to the

nature of the universe as a whole, if we treat the matter as being at

rest.

 

We already know from our previous discussion that the behaviour of

measuring-rods and clocks is influenced by gravitational fields, i.e.

by the distribution of matter. This in itself is sufficient to exclude

the possibility of the exact validity of Euclidean geometry in our

universe. But it is conceivable that our universe differs only

slightly from a Euclidean one, and this notion seems all the more

probable, since calculations show that the metrics of surrounding

space is influenced only to an exceedingly small extent by masses even

of the magnitude of our sun. We might imagine that, as regards

geometry, our universe behaves analogously to a surface which is

irregularly curved in its individual parts, but which nowhere departs

appreciably from a plane: something like the rippled surface of a

lake. Such a universe might fittingly be called a quasi-Euclidean

universe. As regards its space it would be infinite. But calculation

shows that in a quasi-Euclidean universe the average density of matter

would necessarily be nil. Thus such a universe could not be inhabited

by matter everywhere ; it would present to us that unsatisfactory

picture which we portrayed in Section 30.

 

If we are to have in the universe an average density of matter which

differs from zero, however small may be that difference, then the

universe cannot be quasi-Euclidean. On the contrary, the results of

calculation indicate that if matter be distributed uniformly, the

universe would necessarily be spherical (or elliptical). Since in

reality the detailed distribution of matter is not uniform, the real

universe will deviate in individual parts from the spherical, i.e. the

universe will be quasi-spherical. But it will be necessarily finite.

In fact, the theory supplies us with a simple connection *  between

the space-expanse of the universe and the average density of matter in

it.

 

 

  Notes

 

*) For the radius R of the universe we obtain the equation

 

                       

 

The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27;

p is the average density of the matter and k is a constant connected

with the Newtonian constant of gravitation.

 

 

 

APPENDIX I

 

SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION

(SUPPLEMENTARY TO SECTION 11)

 

 

For the relative orientation of the co-ordinate systems indicated in

Fig. 2, the x-axes of both systems pernumently coincide. In the

present case we can divide the problem into parts by considering first

only events which are localised on the x-axis. Any such event is

represented with respect to the co-ordinate system K by the abscissa x

and the time t, and with respect to the system K1 by the abscissa x'

and the time t'. We require to find x' and t' when x and t are given.

 

A light-signal, which is proceeding along the positive axis of x, is

transmitted according to the equation

 

                                x = ct

 

or

 

                 x - ct = 0     .     .     .    (1).

 

Since the same light-signal has to be transmitted relative to K1 with

the velocity c, the propagation relative to the system K1 will be

represented by the analogous formula

 

                x' - ct' = O     .     .     .    (2)

 

Those space-time points (events) which satisfy (x) must also satisfy

(2). Obviously this will be the case when the relation

 

          (x' - ct') = l (x - ct)     .     .     .    (3).

 

is fulfilled in general, where l indicates a constant ; for, according

to (3), the disappearance of (x - ct) involves the disappearance of

(x' - ct').

 

If we apply quite similar considerations to light rays which are being

transmitted along the negative x-axis, we obtain the condition

 

           (x' + ct') = µ(x + ct)    .     .     .    (4).

 

By adding (or subtracting) equations (3) and (4), and introducing for

convenience the constants a and b in place of the constants l and µ,

where

 

                       

 

and

 

                       

 

we obtain the equations

 

                       

 

We should thus have the solution of our problem, if the constants a

and b were known. These result from the following discussion.

 

For the origin of K1 we have permanently x' = 0, and hence according

to the first of the equations (5)

 

                       

 

If we call v the velocity with which the origin of K1 is moving

relative to K, we then have

 

                       

 

The same value v can be obtained from equations (5), if we calculate

the velocity of another point of K1 relative to K, or the velocity

(directed towards the negative x-axis) of a point of K with respect to

K'. In short, we can designate v as the relative velocity of the two

systems.

 

Furthermore, the principle of relativity teaches us that, as judged

from K, the length of a unit measuring-rod which is at rest with

reference to K1 must be exactly the same as the length, as judged from

K', of a unit measuring-rod which is at rest relative to K. In order

to see how the points of the x-axis appear as viewed from K, we only

require to take a " snapshot " of K1 from K; this means that we have

to insert a particular value of t (time of K), e.g. t = 0. For this

value of t we then obtain from the first of the equations (5)

 

                               x' = ax

 

Two points of the x'-axis which are separated by the distance Dx' = I

when measured in the K1 system are thus separated in our instantaneous

photograph by the distance

 

                       

 

But if the snapshot be taken from K'(t' = 0), and if we eliminate t

from the equations (5), taking into account the expression (6), we

obtain

 

                       

 

From this we conclude that two points on the x-axis separated by the

distance I (relative to K) will be represented on our snapshot by the

distance

 

                       

 

But from what has been said, the two snapshots must be identical;

hence Dx in (7) must be equal to Dx' in (7a), so that we obtain

 

                       

 

The equations (6) and (7b) determine the constants a and b. By

inserting the values of these constants in (5), we obtain the first

and the fourth of the equations given in Section 11.

 

                       

 

Thus we have obtained the Lorentz transformation for events on the

x-axis. It satisfies the condition

 

         x'2 - c^2t'2 = x2 - c^2t2    .     .     .    (8a).

 

The extension of this result, to include events which take place

outside the x-axis, is obtained by retaining equations (8) and

supplementing them by the relations

 

                       

 

In this way we satisfy the postulate of the constancy of the velocity

of light in vacuo for rays of light of arbitrary direction, both for

the system K and for the system K'. This may be shown in the following

manner.

 

We suppose a light-signal sent out from the origin of K at the time t

= 0. It will be propagated according to the equation

 

                       

 

or, if we square this equation, according to the equation

 

          x2 + y2 + z2 = c^2t2 = 0    .     .     .    (10).

 

It is required by the law of propagation of light, in conjunction with

the postulate of relativity, that the transmission of the signal in

question should take place -- as judged from K1 -- in accordance with

the corresponding formula

 

                               r' = ct'

 

or,

 

       x'2 + y'2 + z'2 - c^2t'2 = 0    .     .     .    (10a).

 

In order that equation (10a) may be a consequence of equation (10), we

must have

 

   x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2)       (11).

 

Since equation (8a) must hold for points on the x-axis, we thus have s

= I. It is easily seen that the Lorentz transformation really

satisfies equation (11) for s = I; for (11) is a consequence of (8a)

and (9), and hence also of (8) and (9). We have thus derived the

Lorentz transformation.

 

The Lorentz transformation represented by (8) and (9) still requires

to be generalised. Obviously it is immaterial whether the axes of K1

be chosen so that they are spatially parallel to those of K. It is

also not essential that the velocity of translation of K1 with respect

to K should be in the direction of the x-axis. A simple consideration

shows that we are able to construct the Lorentz transformation in this

general sense from two kinds of transformations, viz. from Lorentz

transformations in the special sense and from purely spatial

transformations. which corresponds to the replacement of the

rectangular co-ordinate system by a new system with its axes pointing

in other directions.

 

Mathematically, we can characterise the generalised Lorentz

transformation thus :

 

It expresses x', y', x', t', in terms of linear homogeneous functions

of x, y, x, t, of such a kind that the relation

 

     x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2       (11a).

 

is satisficd identically. That is to say: If we substitute their

expressions in x, y, x, t, in place of x', y', x', t', on the

left-hand side, then the left-hand side of (11a) agrees with the

right-hand side.

 

 

 

APPENDIX II

 

MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")

(SUPPLEMENTARY TO SECTION 17)

 

 

We can characterise the Lorentz transformation still more simply if we

introduce the imaginary eq. 25 in place of t, as time-variable. If, in

accordance with this, we insert

 

                              x[1] = x

                              x[2] = y

                              x[3] = z

                              x[4] = …….eq. 25

 

and similarly for the accented system K1, then the condition which is

identically satisfied by the transformation can be expressed thus :

 

x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2

    (12).

 

That is, by the afore-mentioned choice of " coordinates," (11a) [see

the end of Appendix II] is transformed into this equation.

 

We see from (12) that the imaginary time co-ordinate x[4], enters into

the condition of transformation in exactly the same way as the space

co-ordinates x[1], x[2], x[3]. It is due to this fact that, according

to the theory of relativity, the " time "x[4], enters into natural

laws in the same form as the space co ordinates x[1], x[2], x[3].

 

A four-dimensional continuum described by the "co-ordinates" x[1],

x[2], x[3], x[4], was called "world" by Minkowski, who also termed a

point-event a " world-point." From a "happening" in three-dimensional

space, physics becomes, as it were, an " existence " in the

four-dimensional " world."

 

This four-dimensional " world " bears a close similarity to the

three-dimensional " space " of (Euclidean) analytical geometry. If we

introduce into the latter a new Cartesian co-ordinate system (x'[1],

x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are

linear homogeneous functions of x[1], x[2], x[3] which identically

satisfy the equation

 

        x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2

 

The analogy with (12) is a complete one. We can regard Minkowski's "

world " in a formal manner as a four-dimensional Euclidean space (with

an imaginary time coordinate) ; the Lorentz transformation corresponds

to a " rotation " of the co-ordinate system in the fourdimensional "

world."

 

 

 

APPENDIX III

 

THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY

 

 

From a systematic theoretical point of view, we may imagine the

process of evolution of an empirical science to be a continuous

process of induction. Theories are evolved and are expressed in short

compass as statements of a large number of individual observations in

the form of empirical laws, from which the general laws can be

ascertained by comparison. Regarded in this way, the development of a

science bears some resemblance to the compilation of a classified

catalogue. It is, as it were, a purely empirical enterprise.

 

But this point of view by no means embraces the whole of the actual

process ; for it slurs over the important part played by intuition and

deductive thought in the development of an exact science. As soon as a

science has emerged from its initial stages, theoretical advances are

no longer achieved merely by a process of arrangement. Guided by

empirical data, the investigator rather develops a system of thought

which, in general, is built up logically from a small number of

fundamental assumptions, the so-called axioms. We call such a system

of thought a theory. The theory finds the justification for its

existence in the fact that it correlates a large number of single

observations, and it is just here that the " truth " of the theory

lies.

 

Corresponding to the same complex of empirical data, there may be

several theories, which differ from one another to a considerable

extent. But as regards the deductions from the theories which are

capable of being tested, the agreement between the theories may be so

complete that it becomes difficult to find any deductions in which the

two theories differ from each other. As an example, a case of general

interest is available in the province of biology, in the Darwinian

theory of the development of species by selection in the struggle for

existence, and in the theory of development which is based on the

hypothesis of the hereditary transmission of acquired characters.

 

We have another instance of far-reaching agreement between the

deductions from two theories in Newtonian mechanics on the one hand,

and the general theory of relativity on the other. This agreement goes

so far, that up to the preseat we have been able to find only a few

deductions from the general theory of relativity which are capable of

investigation, and to which the physics of pre-relativity days does

not also lead, and this despite the profound difference in the

fundamental assumptions of the two theories. In what follows, we shall

again consider these important deductions, and we shall also discuss

the empirical evidence appertaining to them which has hitherto been

obtained.

 

 (a) Motion of the Perihelion of Mercury

 

According to Newtonian mechanics and Newton's law of gravitation, a

planet which is revolving round the sun would describe an ellipse

round the latter, or, more correctly, round the common centre of

gravity of the sun and the planet. In such a system, the sun, or the

common centre of gravity, lies in one of the foci of the orbital

ellipse in such a manner that, in the course of a planet-year, the

distance sun-planet grows from a minimum to a maximum, and then

decreases again to a minimum. If instead of Newton's law we insert a

somewhat different law of attraction into the calculation, we find

that, according to this new law, the motion would still take place in

such a manner that the distance sun-planet exhibits periodic

variations; but in this case the angle described by the line joining

sun and planet during such a period (from perihelion--closest

proximity to the sun--to perihelion) would differ from 360^0. The line

of the orbit would not then be a closed one but in the course of time

it would fill up an annular part of the orbital plane, viz. between

the circle of least and the circle of greatest distance of the planet

from the sun.

 

According also to the general theory of relativity, which differs of

course from the theory of Newton, a small variation from the

Newton-Kepler motion of a planet in its orbit should take place, and

in such away, that the angle described by the radius sun-planet

between one perhelion and the next should exceed that corresponding to

one complete revolution by an amount given by

 

                       

 

(N.B. -- One complete revolution corresponds to the angle 2p in the

absolute angular measure customary in physics, and the above

expression giver the amount by which the radius sun-planet exceeds

this angle during the interval between one perihelion and the next.)

In this expression a represents the major semi-axis of the ellipse, e

its eccentricity, c the velocity of light, and T the period of

revolution of the planet. Our result may also be stated as follows :

According to the general theory of relativity, the major axis of the

ellipse rotates round the sun in the same sense as the orbital motion

of the planet. Theory requires that this rotation should amount to 43

seconds of arc per century for the planet Mercury, but for the other

Planets of our solar system its magnitude should be so small that it

would necessarily escape detection. *

 

In point of fact, astronomers have found that the theory of Newton

does not suffice to calculate the observed motion of Mercury with an

exactness corresponding to that of the delicacy of observation

attainable at the present time. After taking account of all the

disturbing influences exerted on Mercury by the remaining planets, it

was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained

perihelial movement of the orbit of Mercury remained over, the amount

of which does not differ sensibly from the above mentioned +43 seconds

of arc per century. The uncertainty of the empirical result amounts to

a few seconds only.

 

 (b) Deflection of Light by a Gravitational Field

 

In Section 22 it has been already mentioned that according to the

general theory of relativity, a ray of light will experience a

curvature of its path when passing through a gravitational field, this

curvature being similar to that experienced by the path of a body

which is projected through a gravitational field. As a result of this

theory, we should expect that a ray of light which is passing close to

a heavenly body would be deviated towards the latter. For a ray of

light which passes the sun at a distance of D sun-radii from its

centre, the angle of deflection (a) should amount to

 

 

                       

 

 

 

 

It may be added that, according to the theory, half of Figure 05 this

deflection is produced by the Newtonian field of attraction of the

sun, and the other half by the geometrical modification (" curvature

") of space caused by the sun.

 

 

This result admits of an experimental test by means of the

photographic registration of stars during a total eclipse of the sun.

The only reason why we must wait for a total eclipse is because at

every other time the atmosphere is so strongly illuminated by the

light from the sun that the stars situated near the sun's disc are

invisible. The predicted effect can be seen clearly from the

accompanying diagram. If the sun (S) were not present, a star which is

practically infinitely distant would be seen in the direction D[1], as

observed front the earth. But as a consequence of the deflection of

light from the star by the sun, the star will be seen in the direction

D[2], i.e. at a somewhat greater distance from the centre of the sun

than corresponds to its real position.

 

In practice, the question is tested in the following way. The stars in

the neighbourhood of the sun are photographed during a solar eclipse.

In addition, a second photograph of the same stars is taken when the

sun is situated at another position in the sky, i.e. a few months

earlier or later. As compared whh the standard photograph, the

positions of the stars on the eclipse-photograph ought to appear

displaced radially outwards (away from the centre of the sun) by an

amount corresponding to the angle a.

 

We are indebted to the [British] Royal Society and to the Royal

Astronomical Society for the investigation of this important

deduction. Undaunted by the [first world] war and by difficulties of

both a material and a psychological nature aroused by the war, these

societies equipped two expeditions -- to Sobral (Brazil), and to the

island of Principe (West Africa) -- and sent several of Britain's most

celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),

in order to obtain photographs of the solar eclipse of 29th May, 1919.

The relative discrepancies to be expected between the stellar

photographs obtained during the eclipse and the comparison photographs

amounted to a few hundredths of a millimetre only. Thus great accuracy

was necessary in making the adjustments required for the taking of the

photographs, and in their subsequent measurement.

 

The results of the measurements confirmed the theory in a thoroughly

satisfactory manner. The rectangular components of the observed and of

the calculated deviations of the stars (in seconds of arc) are set

forth in the following table of results :

 

                     

 

 (c) Displacement of Spectral Lines Towards the Red

 

In Section 23 it has been shown that in a system K1 which is in

rotation with regard to a Galileian system K, clocks of identical

construction, and which are considered at rest with respect to the

rotating reference-body, go at rates which are dependent on the

positions of the clocks. We shall now examine this dependence

quantitatively. A clock, which is situated at a distance r from the

centre of the disc, has a velocity relative to K which is given by

 

                                V = wr

 

where w represents the angular velocity of rotation of the disc K1

with respect to K. If v[0], represents the number of ticks of the

clock per unit time (" rate " of the clock) relative to K when the

clock is at rest, then the " rate " of the clock (v) when it is moving

relative to K with a velocity V, but at rest with respect to the disc,

will, in accordance with Section 12, be given by

 

                       

 

or with sufficient accuracy by

 

                       

 

This expression may also be stated in the following form:

 

                       

 

If we represent the difference of potential of the centrifugal force

between the position of the clock and the centre of the disc by f,

i.e. the work, considered negatively, which must be performed on the

unit of mass against the centrifugal force in order to transport it

from the position of the clock on the rotating disc to the centre of

the disc, then we have

 

                       

 

From this it follows that

 

                        

 

In the first place, we see from this expression that two clocks of

identical construction will go at different rates when situated at

different distances from the centre of the disc. This result is aiso

valid from the standpoint of an observer who is rotating with the

disc.

 

Now, as judged from the disc, the latter is in a gravititional field

of potential f, hence the result we have obtained will hold quite

generally for gravitational fields. Furthermore, we can regard an atom

which is emitting spectral lines as a clock, so that the following

statement will hold:

 

An atom absorbs or emits light of a frequency which is dependent on

the potential of the gravitational field in which it is situated.

 

The frequency of an atom situated on the surface of a heavenly body

will be somewhat less than the frequency of an atom of the same

element which is situated in free space (or on the surface of a

smaller celestial body).

 

Now f = - K (M/r), where K is Newton's constant of gravitation, and M

is the mass of the heavenly body. Thus a displacement towards the red

ought to take place for spectral lines produced at the surface of

stars as compared with the spectral lines of the same element produced

at the surface of the earth, the amount of this displacement being

 

                       

 

For the sun, the displacement towards the red predicted by theory

amounts to about two millionths of the wave-length. A trustworthy

calculation is not possible in the case of the stars, because in

general neither the mass M nor the radius r are known.

 

It is an open question whether or not this effect exists, and at the

present time (1920) astronomers are working with great zeal towards

the solution. Owing to the smallness of the effect in the case of the

sun, it is difficult to form an opinion as to its existence. Whereas

Grebe and Bachem (Bonn), as a result of their own measurements and

those of Evershed and Schwarzschild on the cyanogen bands, have placed

the existence of the effect almost beyond doubt, while other

investigators, particularly St. John, have been led to the opposite

opinion in consequence of their measurements.

 

Mean displacements of lines towards the less refrangible end of the

spectrum are certainly revealed by statistical investigations of the

fixed stars ; but up to the present the examination of the available

data does not allow of any definite decision being arrived at, as to

whether or not these displacements are to be referred in reality to

the effect of gravitation. The results of observation have been

collected together, and discussed in detail from the standpoint of the

question which has been engaging our attention here, in a paper by E.

Freundlich entitled "Zur Prüfung der allgemeinen

Relativit&umlaut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35,

p. 520: Julius Springer, Berlin).

 

At all events, a definite decision will be reached during the next few

years. If the displacement of spectral lines towards the red by the

gravitational potential does not exist, then the general theory of

relativity will be untenable. On the other hand, if the cause of the

displacement of spectral lines be definitely traced to the

gravitational potential, then the study of this displacement will

furnish us with important information as to the mass of the heavenly

bodies. [5][A]

 

 

  Notes

 

*) Especially since the next planet Venus has an orbit that is

almost an exact circle, which makes it more difficult to locate the

perihelion with precision.

 

The displacentent of spectral lines towards the red end of the

spectrum was definitely established by Adams in 1924, by observations

on the dense companion of Sirius, for which the effect is about thirty

times greater than for the Sun. R.W.L. -- translator

 

 

 

APPENDIX IV

 

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

(SUPPLEMENTARY TO SECTION 32)

 

 

Since the publication of the first edition of this little book, our

knowledge about the structure of space in the large (" cosmological

problem ") has had an important development, which ought to be

mentioned even in a popular presentation of the subject.

 

My original considerations on the subject were based on two

hypotheses:

 

(1) There exists an average density of matter in the whole of space

which is everywhere the same and different from zero.

 

(2) The magnitude (" radius ") of space is independent of time.

 

Both these hypotheses proved to be consistent, according to the

general theory of relativity, but only after a hypothetical term was

added to the field equations, a term which was not required by the

theory as such nor did it seem natural from a theoretical point of

view (" cosmological term of the field equations ").

 

Hypothesis (2) appeared unavoidable to me at the time, since I thought

that one would get into bottomless speculations if one departed from

it.

 

However, already in the 'twenties, the Russian mathematician Friedman

showed that a different hypothesis was natural from a purely

theoretical point of view. He realized that it was possible to

preserve hypothesis (1) without introducing the less natural

cosmological term into the field equations of gravitation, if one was

ready to drop hypothesis (2). Namely, the original field equations

admit a solution in which the " world radius " depends on time

(expanding space). In that sense one can say, according to Friedman,

that the theory demands an expansion of space.

 

A few years later Hubble showed, by a special investigation of the

extra-galactic nebulae (" milky ways "), that the spectral lines

emitted showed a red shift which increased regularly with the distance

of the nebulae. This can be interpreted in regard to our present

knowledge only in the sense of Doppler's principle, as an expansive

motion of the system of stars in the large -- as required, according

to Friedman, by the field equations of gravitation. Hubble's discovery

can, therefore, be considered to some extent as a confirmation of the

theory.

 

There does arise, however, a strange difficulty. The interpretation of

the galactic line-shift discovered by Hubble as an expansion (which

can hardly be doubted from a theoretical point of view), leads to an

origin of this expansion which lies " only " about 10^9 years ago,

while physical astronomy makes it appear likely that the development

of individual stars and systems of stars takes considerably longer. It

is in no way known how this incongruity is to be overcome.

 

I further want to rernark that the theory of expanding space, together

with the empirical data of astronomy, permit no decision to be reached

about the finite or infinite character of (three-dimensional) space,

while the original " static " hypothesis of space yielded the closure

(finiteness) of space.

 

 

K = co-ordinate system

x, y = two-dimensional co-ordinates

x, y, z = three-dimensional co-ordinates

x, y, z, t = four-dimensional co-ordinates

 

t = time

I = distance

v = velocity

 

F = force

G = gravitational field