Albert Einstein`s - Relativity

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Relativity: The Special and General Theory

Albert Einstein

Relativity

The Special and General Theory

Written: 1916 (this revised edition: 1924)

Publisher: Methuen & Co Ltd

First Published: December, 1916

Translated: Robert W. Lawson (Authorised translation)

Transcription/Markup: Brian Basgen

Convertion to PDF: Sjoerd Langkemper

Offline Version: Einstein Reference Archive (marxists.org) 1999

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?

Relativity: The Special and General Theory

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 Realtiivty 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.

Einstein Reference Archive

Relativity: The Special and General Theory

Albert Einstein

Relativity: The Special and General Theory

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

Next: The Physical Meaning of Geometrical Propositions

Relativity: The Special and General Theory

Albert Einstein: Relativity

Part I: The Special Theory of Relativity

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

Relativity: The Special and General Theory

possible relative position of practically rigid bodies.1) 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.

Next: The System of Co−ordinates

Notes

1) 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.

Relativity: The Special and General Theory

Albert Einstein: Relativity

Part I: The Special Theory of Relativity

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. 1)

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.2)

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.

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

Relativity: The Special and General Theory

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. 3)

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.

Next: Space and Time in Classical Mechanics

Notes

1) 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.]

2) 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.

3) 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.

Relativity: The Special and General Theory

Albert Einstein: Relativity

Part I: The Special Theory of Relativity

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" 1)), 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.

Next: The Galilean System of Co−ordinates

Relativity: The Special and General Theory

Notes

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

Relativity: The Special and General Theory

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Relativity:TheSpecialandGeneralTheoryAlbertEinsteinAlbertEinsteinRelativityTheSpecialandGeneralTheoryWritten:1916(thisrevisededition:1924)Source:Relativity:TheSpecialandGeneralTheory©1920Publisher:Methuen&CoLtdFirstPublished:December,1916Translated:RobertW.Lawson(Authorisedtranslation)Transcription/...

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