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Relativity: The Special and General Theory

Albert Einstein

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Section 1 of 12
ALBERT EINSTEIN REFERENCE ARCHIVE

RELATIVITY: THE SPECIAL AND GENERAL THEORY

BY ALBERT EINSTEIN


Written: 1916 (this revised edition: 1924)
Source: Relativity: The Special and General Theory (1920)
Publisher: Methuen & Co Ltd
First Published: December, 1916
Translated: Robert W. Lawson (Authorised translation)
Transcription/Markup: Brian Basgen 
Transcription to text: Gregory B. Newby 
Thanks to: Einstein Reference Archive (marxists.org)
The Einstein Reference Archive is online at:
http://www.marxists.org/reference/archive/einstein/index.htm

Transcriber note: This file is a plain text rendition of HTML.
Because many equations cannot be presented effectively in plain text,
images are supplied for many equations and for all figures and tables.


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.
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