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\title{Relativity: The Special and General Theory}


\author{Albert Einstein\\
{\normalsize Translated by Robert W. Lawson}}

\maketitle
\vspace*{\stretch{1}}

\noindent This public-domain (U.S.) text was originally published
by Henry Holt and Company, 1920. The present edition was created from
material archived at bartleby.com. The text was converted to \LaTeX{}
by Ron Burkey, and modified (both for formatting reasons, and to correct
obvious errors). All mathematical expressions which had been represented
as scanned graphics have been translated to pure \LaTeX{}. All scanned
figures have been completely redrawn, except for Fig.\ 5 (in which
only the text has been redrawn). Revision D differs from revision
C in Appendix I, where the definition of the symbol $b$ and the left-hand
side of equation 14 have been corrected thanks to a report by Charles
Cameron. D1 has the further change that GIF images are changed to
JPEG for greater portability of the \LaTeX{}. Report other problems
to info@sandroid.org.

\noindent \begin{center}Revision: D1\\
Date: 01/25/2008\end{center}

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\frontmatter

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{\small \tableofcontents{}}{\small \par}

\newpage


\chapter*{\raggedright{}Preface}

\addcontentsline{toc}{chapter}{Preface}

\markboth{Relativity}{Preface}

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%
\footnote{The mathematical fundaments of the special theory of relativity are
to be found in the original papers of H.\ A.\ Lorentz, A.\ Einstein,
H.\ Minkowski published under the title \emph{Das Relativitäts-prinzip}
(\emph{The Principle of Relativity}) in B.\ G.\ Teubner's collection
of monographs \emph{Fortschritte der mathematischen Wissenschaften}
(\emph{Advances in the Mathematical Sciences}), also in M.\ Laue's
exhaustive book \emph{Das Relativitäts prinzip}\mdsh{---}published
by Friedr.\ Vieweg \& Son, Braunschweig. The general theory of relativity,
together with the necessary parts of the theory of invariants, is
dealt with in the author's book \emph{Die Grundlagen der allgemeinen
Relativitätstheorie} (\emph{The Foundations of the General Theory
of Relativity})\mdsh{---}Joh.\ Ambr.\ Barth, 1916; this book assumes
some familiarity with the special theory of relativity.%
} 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 trees. May the book bring some
one a few happy hours of suggestive thought! 

\noindent \begin{center}A\noun{. Einstein}\\
\noun{December, 1916}\end{center}


\section*{NOTE TO THE THIRD EDITION}

In the present year (1918) an excellent and detailed manual on the
general theory of relativity, written by H.\ Weyl, was published
by the firm Julius Springer (Berlin). This book, entitled \emph{Raum\mdsh{---}Zeit\mdsh{---}Materie}
(\emph{Space\mdsh{---}Time\mdsh{---}Matter}), may be warmly recommended
to mathematicians and physicists.


\chapter*{\raggedright{}Biographical Note}

\addcontentsline{toc}{chapter}{Biographical Note}

\markboth{Relativity}{Biographical Note}

Albert Einstein is the son of German-Jewish parents. He was born in
1879 in the town of Ulm, Würtemberg, Germany. His schooldays were
spent in Munich, where he attended the Gymnasium until his sixteenth
year. After leaving school at Munich, he accompanied his parents to
Milan, whence he proceeded to Switzerland six months later to continue
his studies.

From 1896 to 1900 Albert Einstein studied mathematics and physics
at the Technical High School in Zurich, as he intended becoming a
secondary school (\emph{Gymnasium}) teacher. For some time afterwards
he was a private tutor, and having meanwhile become naturalised, he
obtained a post as engineer in the Swiss Patent Office in 1902, which
position he occupied till 1909. The main ideas involved in the most
important of Einstein's theories date back to this period. Amongst
these may be mentioned: The Special Theory of Relativity, Inertia
of Energy, Theory of the Brownian Movement, and the Quantum-Law of
the Emission and Absorption of Light (1905). These were followed some
years later by the Theory of the Specific Heat of Solid Bodies, and
the fundamental idea of the General Theory of Relativity.

During the intervel 1909 to 1911 he occupied the post of Professor
Extraordinarius at the University of Prague, Bohemia, where he remained
as Professor Ordinarius until 1912. In the latter year Professor Einstein
accepted a similar chair at the Polytechnikum, Zurich, and continued
his activities there until 1914, when he received a call to the Prussian
Academy of Science, Berlin, as successor to Van't Hoff. Professor
Einstein is able to devote himself freely to his studies at the Berlin
Academy, and it was here that he succeeded in completing his work
on the General Theory of Relativity (1915-17). Professor Einstein
also lectures on various special branches of physics at the University
of Berlin, and, in addition, he is Director of the Institute for Physical
Research of the Kaiser Wilhelm Gesellschaft.

Professor Einstein has been twice married. His first wife, whom he
married at Berne in 1903, was a fellow-student from Serbia. There
were two sons of this marriage, both of whom are living in Zurich,
the elder being sixteen years of age. Recently Professor Einstein
married a widowed cousin, with whom he is now living in Berlin. 

\noindent \begin{center}\noun{R. W. L.}\end{center}


\chapter*{\raggedright{}Translator's Note}

\addcontentsline{toc}{chapter}{Translator's Note}

\markboth{Relativity}{Translator's Note}

In presenting this translation to the English-reading public, it is
hardly necessary for me to enlarge on the Author's prefatory remarks,
except to draw attention to those additions to the book which do not
appear in the original.

At my request, Professor Einstein kindly supplied me with a portrait
of himself, by one of Germany's most celebrated artists. Appendix
III, on {}``The Experimental Confirmation of the General Theory of
Relativity,'' has been written specially for this translation. Apart
from these valuable additions to the book, I have included a biographical
note on the Author, and, at the end of the book, an Index and a list
of English references to the subject. This list, which is more suggestive
than exhaustive, is intended as a guide to those readers who wish
to pursue the subject farther.

I desire to tender my best thanks to my colleagues Professor S.\ R.\ Milner,
D.Sc., and Mr.\ W.\ E.\ Curtis, A.R.C.Sc., F.R.A.S., also to my
friend Dr.\ Arthur Holmes, A.R.C.SC., F.G.S., of the Imperial College,
for their kindness in reading through the manuscript, for helpful
criticism, and for numerous suggestions. I owe an expression of thanks
also to Messrs.\ Methuen for their ready counsel and advice, and
for the care they have bestowed on the work during the course of its
publication. 

\noindent \begin{center}\noun{Robert W.\ Lawson }\\
\noun{The Physics Laboratory}\\
\noun{The University of Sheffield}\\
\noun{June 12, 1920}\end{center}

\mainmatter


\part*{Part I: The Special Theory of Relativity}

\addcontentsline{toc}{part}{Part I: The Special Theory\\
of Relativity}


\chapter*{\raggedright{}I. Physical Meaning of Geometrical Propositions}

\addcontentsline{toc}{chapter}{Physical Meaning of Geometrical\\
Propositions}

\markboth{Relativity}{Physical Meaning of Geometry}

In your schooldays most of you who read this book made acquaintance
with the noble building of Euclid's geometry, and you remember\mdsh{---}perhaps
with more respect than love\mdsh{---}the magnificent structure, on
the lofty staircase of which you were chased about for uncounted hours
by conscientious teachers. By reason of your past experience, you
would certainly regard every one 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 from 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, \emph{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 the {}``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
line,'' 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.%
\footnote{It follows that a natural object is associated also with a straight
line. Three points \emph{$A$}, \emph{$B$} and \emph{$C$} on a rigid
body thus lie in a straight when, the points \emph{$A$} and \emph{$C$}
being given, \emph{$B$} is chosen such that the sum of the distances
\emph{$AB$} and \emph{$BC$} is as short as possible. This incomplete
suggestion will suffice for our present purpose.%
} 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 ruler 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.


\chapter*{\raggedright{}II. The System of Co-ordinates}

\addcontentsline{toc}{chapter}{The System of Co-ordinates}

\markboth{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 \emph{$S$}) which is to
be used once and for all, and which we employ as a standard measure.
If, now, \emph{$A$} and \emph{$B$} are two points on a rigid body,
we can construct the line joining them according to the rules of geometry;
then, starting from \emph{$A$}, we can mark off the distance \emph{$S$}
time after time until we reach \emph{$B$}. The number of these operations
required is the numerical measure of the distance \emph{$AB$}. This
is the basis of all measurement of length.%
\footnote{Here we have assumed that there is nothing left over, \emph{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.%
}

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 {}``Trafalgar Square,
London,''%
\footnote{I have chosen this as being more familiar to the English reader than
the {}``Potsdamer Platz, Berlin,'' which is referred to in the original.
(R. W. L.)%
} I arrive at the following result. The earth is the rigid body to
which the specification of place refers; {}``Trafalgar Square, London''
is a well-defined point, to which a name has been assigned, and with
which the event coincides in space.%
\footnote{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.%
}

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 Trafalgar 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.%
\footnote{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.%
}

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.


\chapter*{\raggedright{}III. Space and Time in Classical Mechanics}

\addcontentsline{toc}{chapter}{Space and Time in Classical Mechanics}

\markboth{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''%
\footnote{That is, a curve along which the body moves.%
}), but only a trajectory relative to a particular body of reference.

In order to have a \emph{complete} description of the motion, we must
specify how the body alters its position \emph{with time}; \emph{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.


\chapter*{\raggedright{}IV. The Galileian System of Co-ordinates}

\addcontentsline{toc}{chapter}{The Galileian System of Co-ordinates}

\markboth{Relativity}{The Galileian System of Co-ordinates}

As is well known, the fundamental law of the mechanics of Galilei-Newton,
which is known as the \emph{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 co-ordinates, 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 co-ordinates 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 Galilei-Newton can be regarded as valid only for
a Galileian system of co-ordinates.


\chapter*{\raggedright{}V. The Principle of Relativity (In the Restricted
Sense)}

\addcontentsline{toc}{chapter}{The Principle of Relativity (In the\\
Restricted Sense)}

\markboth{Relativity}{The Restricted Principle of Relativity}

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 $K'$, 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 $K'$
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 $K$, $K'$ is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course
with respect to $K'$ according to exactly the same general laws as
with respect to $K$. This statement is called the \emph{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 \emph{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 \emph{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 $K$, $K'$,
$K''$, \emph{etc.}, which are moving uniformly relative to each other,
will not be \emph{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 \emph{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 (\emph{i.e.}\ {}``really'')
with respect to $K_{0}$. In the general laws of natural phenomena
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
organ-pipe 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 rotation 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,
\emph{i.e.}\ a physical non-equivalence of different directions.
This is a very powerful argument in favour of the principle of relativity.


\chapter*{\raggedright{}VI. The Theorem of the Addition of Velocities Employed
in Classical Mechanics}

\addcontentsline{toc}{chapter}{The Theorem of the Addition of \\
Velocities Employed in Classical\\
Mechanics}

\markboth{Relativity}{Classical Addition of Velocities}

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 he 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.


\chapter*{\raggedright{}VII. The Apparent Incompatibility of the Law of Propagation
of Light with the Principle of Relativity}

\addcontentsline{toc}{chapter}{The Apparent Incompatibility of the
\\
Law of Propagation of Light with \\
the Principle of Relativity}

\markboth{Relativity}{The Apparent Incompatibility}

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 $c=300000$ 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 observations 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 of a ray of light relative to the carriage
thus comes out 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 \emph{in vacuo} 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\mdsh{---}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 \emph{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 \emph{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 \emph{in vacuo} is a necessary
consequence. Prominent theoretical physicists were therefore 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 \emph{in reality there is not the least incompatibility
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 \emph{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.


\chapter*{\raggedright{}VIII. On the Idea of Time in Physics}

\addcontentsline{toc}{chapter}{On the Idea of Time in Physics}

\markboth{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 now 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 (\emph{e.g.}\ two mirrors
inclined at 90°) 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 I only knew that the light by means of which the observer
at $M$ perceives the lightning flashes travels along the length $A\rightarrow M$
with the same velocity as along the length $B\rightarrow M$. 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\mdsh{---}and rightly so\mdsh{---}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 $A\rightarrow M$ as for the path $B\rightarrow M$
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%
\footnote{We suppose further that, when three events $A$, $B$ and $C$ take
place in different places in such a manner that, if $A$ 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 law of propagation of light; it
must certainly be fulfilled if we are to maintain the law of the constancy
of the velocity of light \emph{in vacuo}. %
} (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).


\chapter*{\raggedright{}IX. The Relativity of Simultaneity}

\addcontentsline{toc}{chapter}{The Relativity of Simultaneity}

\markboth{Relativity}{The Relativity of Simultaneity}

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 advantage use 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 (\emph{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. 

\noindent \begin{center}\includegraphics[%
  width=1\columnwidth]{EinsteinRelativity-f01.jpg}\end{center}

\noindent \begin{center}\noun{Fig. 1.}\end{center}

When we say that the lightning strokes $A$ and $B$ are simultaneous
with respect to the 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 $A\rightarrow B$ of the
embankment. But the events $A$ and $B$ also correspond to positions
$A$ and $B$ on the train. Let $M'$ be the mid-point of the distance
$A\rightarrow B$ on the travelling train. Just when the flashes%
\footnote{As judged from the embankment. %
} of lightning occur, this point $M'$ 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
$M'$ 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, \emph{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, \emph{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 \emph{in vacuo} and the principle of relativity
(developed in Section VII) disappears.

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


\chapter*{\raggedright{}X. On the Relativity of the Conception of Distance}

\addcontentsline{toc}{chapter}{On the Relativity of the Conception\\
of Distance}

\markboth{Relativity}{Relativity of the Conception of Distance}

Let us consider two particular points on the train%
\footnote{\emph{E.g.}\ the middle of the first and of the hundredth carriage.%
} 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 the reference-body (co-ordinate system).
An observer in the train measures the interval by marking off his
measuring-rod in a straight line (\emph{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
$A'$ and $B'$ 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 $A'$ and $B'$ at a particular time $t$\mdsh{---}judged
from the embankment. These points $A$ and $B$ of the embankment
can be determined by applying the definition of time given in Section
VIII. The distance between these points $A$ and $B$ is then measured
by repeated application of the measuring-rod along the embankment.

\emph{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 VI. Namely, if the man in the carriage
covers the distance $w$ in a unit of time\mdsh{---}measured from
the train,\mdsh{---}then this distance\mdsh{---}as measured from
the embankment\mdsh{---}is not necessarily also equal to $w$.


\chapter*{\raggedright{}XI. The Lorentz Transformation}

\addcontentsline{toc}{chapter}{The Lorentz Transformation}

\markboth{Relativity}{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 VII) has been derived by means of a consideration which borrowed
two unjustifiable hypotheses from classical mechanics; these are as
follows: The time-interval (time) between two events is independent
of the condition of motion of the body of reference. 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 VII disappears,
because the theorem of the addition of velocities derived in Section
VI becomes invalid. The possibility presents itself that the law of
the propagation of light \emph{in vacuo} may be compatible with the
principle of relativity, and the question arises: How have we to modify
the considerations of Section VI 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 VI 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 \emph{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 II
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 $K'$, the same event would be fixed
in respect of space and time by corresponding values $x'$, $y'$,
$z'$, $t'$, 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 $x'$, $y'$, $z'$, $t'$ of an event with respect
to $K'$, 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 \emph{in vacuo} is satisfied
for one and the same ray of light (and of course for every ray) with
respect to $K$ and $K'$. For the relative orientation in space of
the co-ordinate systems indicated in the diagram (Fig. 2), this problem
is solved by means of the equations: \begin{eqnarray*}
x' & = & \frac{x-vt}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\
y' & = & y\\
z' & = & z\\
t' & = & \frac{t-\frac{v}{c^{2}}x}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.\end{eqnarray*}


This system of equations is known as the {}``Lorentz transformation.''%
\footnote{A simple derivation of the Lorentz transformation is given in Appendix
I.%
}

\noindent \begin{center}\includegraphics[%
  scale=0.5]{EinsteinRelativity-f02.jpg}\end{center}

\noindent \begin{center}\noun{Fig. 2.}\end{center}

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: \begin{eqnarray*}
\begin{array}{c}
x'\end{array} & = & x-vt\\
y' & = & y\\
z' & = & z\\
t' & = & t.\end{eqnarray*}


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
\emph{in vacuo} is satisfied both for the reference-body $K$ and
for the reference-body $K'$. A light-signal is sent along the positive
x-axis, and this light-stimulus advances in accordance with the equation\[
x=ct;\]


\noindent \emph{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 $x'$ and $t'$. In point of fact,
if we substitute for $x$ the value ct in the first and fourth equations
of the Lorentz transformation, we obtain

\begin{eqnarray*}
x' & = & \frac{(c-v)t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\
t' & = & \frac{(1-\frac{v}{c})t}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\end{eqnarray*}


\noindent from which, by division, the expression \[
x'=ct'\]
immediately follows. If referred to the system $K'$, the propagation
of light takes place according to this equation. We thus see that
the velocity of transmission relative to the reference-body $K'$
is also equal to $c$. The same result is obtained for rays of light
advancing in any other direction whatsoever. Of course this is not
surprising, since the equations of the Lorentz transformation were
derived conformably to this point of view.


\chapter*{\raggedright{}XII. The Behaviour of Measuring-Rods and Clocks in
Motion}

\addcontentsline{toc}{chapter}{The Behaviour of Measuring-Rods\\
and Clocks in Motion}

\markboth{Relativity}{Moving Measuring-Rods and Clocks}

I place a metre-rod in the $x'$-axis of $K'$ in such a manner that
one end (the beginning) coincides with the point $x'=0$, whilst the
other end (the end of the rod) coincides with the point $x'=1$. 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\begin{eqnarray*}
x_{(beginning\, of\, rod)} & = & 0\cdot\sqrt{1-\frac{v^{2}}{c^{2}}}\\
x_{(end\, of\, rod)} & = & 1\cdot\sqrt{1-\frac{v^{2}}{c^{2}}};\end{eqnarray*}


\noindent the distance between the points being

\[
\sqrt{1-\frac{v^{2}}{c^{2}}}.\]


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\[
\sqrt{1-v^{2}/c^{2}}\]


\noindent 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

\[
\sqrt{1-v^{2}/c^{2}}=0,\]


\noindent and for still 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 become 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 $K'$ would have been\[
\sqrt{1-v^{2}/c^{2}};\]


\noindent this is quite in accordance with the principle of relativity
which forms the basis of our considerations.

\emph{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 $x$, $y$, $z$,
$t$, 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 Galilei 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 ($x'=0$) of $K'$. $t'=0$ and $t'=1$ are two successive
ticks of this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks: \[
t=0\]


\noindent and

\[
t=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.\]


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\[
\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\]


\noindent seconds, \emph{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.


\chapter*{\raggedright{}XIII. Theorem of the Addition of Velocities. The Experiment
of Fizeau}

\addcontentsline{toc}{chapter}{Theorem of the Addition of Velocities.\\
The Experiment of Fizeau}

\markboth{Relativity}{Theorem of the Addition of Velocities}

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 VI 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 from
the Galilei transformation (Section XI). In place of the man walking
inside the carriage, we introduce a point moving relatively to the
co-ordinate system $K'$ in accordance with the equation\[
x'=wt'.\]


By means of the first and fourth equations of the Galilei transformation
we can express $x'$ and $t'$ 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 VI,\begin{equation}
W=v+w.\label{eq:A}\end{equation}


But we can carry out this consideration just as well on the basis
of the theory of relativity. In the equation\[
x'=wt',\]


\noindent we must then express $x'$ and $t'$ in terms of $x$ and
$t$, making use of the first and fourth equations of the Lorentz
transformation. Instead of the equation (\ref{eq:A}) we then obtain
the equation

\begin{equation}
W=\frac{v+w}{1+\frac{vw}{c^{2}}},\label{eq:B}\end{equation}


\noindent 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 are 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
$T$ (see the accompanying diagram, Fig. 3) when the liquid above
mentioned is flowing through the tube with a velocity $v$? 

\noindent \begin{center}\includegraphics[%
  width=1\columnwidth]{EinsteinRelativity-f03.jpg}\end{center}

\noindent \begin{center}\noun{Fig. 3.}\end{center}

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 VI 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 $K'$, 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 (\ref{eq:A}) or (\ref{eq:B}),
according as the Galilei transformation or the Lorentz transformation
corresponds to the facts. Experiment%
\footnote{Fizeau found\[
W=w+v(1-\frac{1}{n^{2}}),\]


\noindent where 

\[
n=\frac{c}{w}\]


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

\[
\frac{vw}{c^{2}}\]


\noindent as compared with 1, we can replace (\ref{eq:B}) in the
first place by

\[
W=(w+v)(1-\frac{vw}{c^{2}}),\]


\noindent or to the same order of approximation by 

\[
w+v(1-\frac{1}{n^{2}}),\]


\noindent which agrees with Fizeau's result.%
} decides in favour of equation (\ref{eq: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 (\ref{eq: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 from electrodynamics as an astoundingly
simple combination and generalisation of the hypotheses, formerly
independent of each other, on which electrodynamics was built.


\chapter*{\raggedright{}XIV. The Heuristic Value of the Theory of Relativity}

\addcontentsline{toc}{chapter}{The Heuristic Value of the Theory
of\\
Relativity}

\markboth{Relativity}{The Heuristic Value 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 \emph{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
\emph{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 co-ordinate system
$K$, we introduce new space-time variables $x'$, $y'$, $z'$, $t'$
of a co-ordinate system $K'$. 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.


\chapter*{\raggedright{}XV. General Results of the Theory}

\addcontentsline{toc}{chapter}{General Results of the Theory}

\markboth{Relativity}{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, \emph{i.e.}\ the
derivation of laws, and\mdsh{---}what is incomparably more important\mdsh{---}it
has considerably reduced the number of independent hypotheses 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\[
m\frac{v^{2}}{2},\]


\noindent but by the expression

\[
\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.\]


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,\[
mc^{2}+m\frac{v^{2}}{2}+\frac{3}{8}m\frac{v^{4}}{c^{2}}+\ldots.\]


When\[
\frac{v^{2}}{c^{2}}\]


\noindent 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 conservation
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 conservation
of energy should hold not only with reference to a co-ordinate system
$K$, but also with respect to every co-ordinate system $K'$ 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%
\footnote{$E_{0}$ is the energy taken up, as judged from a co-ordinate system
moving with the body.%
} 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\[
\frac{E_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.\]


In consideration of the expression given above for the kinetic energy
of the body, the required energy of the body comes out to be\[
\frac{(m+\frac{E_{0}}{c^{2}})c^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.\]


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

\[
(m+\frac{E_{0}}{c^{2}})\]


\noindent 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 

\[
\frac{E_{0}}{c^{2}},\]


\noindent 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

\[
\frac{mc^{2}+E_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\]


\noindent we see that the term $mc^{2}$, which has hitherto attracted
our attention, is nothing else than the energy possessed by the body%
\footnote{As judged from a co-ordinate system moving with the body.%
} before it absorbed the energy $E_{0}$.

A direct comparison of this relation with experiment is not possible
at the present time, 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.\[
\frac{E_{0}}{c^{2}}\]


\noindent 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 rôle in this theory. In Part II we shall see in
what way this result becomes modified in the general theory of relativity. 


\chapter*{\raggedright{}XVI. Experience and the Special Theory of Relativity}

\addcontentsline{toc}{chapter}{Experience and the Special Theory
of\\
Relativity}

\markboth{Relativity}{Experience and Special 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\mdsh{---}\emph{i.e.}\ without
making use of the theory of relativity\mdsh{---}appears extraneous.

It is known that cathode rays and the so-called $\beta$-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.%
\footnote{The general theory of relativity renders it likely that the electrical
masses of an electron are held together by gravitational forces.%
} 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 particles constituting
the electron experience a contraction in the direction of motion in
consequence of that motion, the amount of this contraction being proportional
to the expression\[
\sqrt{1-\frac{v^{2}}{c^{2}}}\]


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 XIII in connection with the experiment of Fizeau, the result
of which is fore-told 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 V 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 Galilei 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 $K'$ moving
uniformly with respect to $K$, if we assume that the relations of
the Galileian transformation exist between the co-ordinates of $K$
and $K'$. 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 co-ordinate systems $K'$ moving relatively to $K$ were
to be regarded as in motion with respect to the æther. To this motion
of $K'$ against the æther ({}``æther-drift'' relative to $K'$)
were assigned the more complicated laws which were supposed to hold
relative to $K'$. 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 $T'$ 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 $T'$
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\mdsh{---}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 difference in time mentioned above. Comparison
with the discussion in Section XII shows that from the standpoint
also 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.


\chapter*{\raggedright{}XVII. Minkowski's Four-Dimensional Space}

\addcontentsline{toc}{chapter}{Minkowski's Four-Dimensional Space}

\markboth{Relativity}{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-ordinates) $x$, $y$, $z$, 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
rôle, as compared with the space co-ordinates. 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, \emph{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
($t'=t$). 

The four-dimensional mode of consideration of the {}``world'' is
natural in 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:\[
t'=\frac{t-\frac{v}{c^{2}}x}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\]


Moreover, according to this equation the time difference $\Delta t'$
of two events with respect to $K'$ does not in general vanish, even
when the time difference $\Delta t$ 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.%
\footnote{\emph{Cf.}\ the somewhat more detailed discussion in Appendix II.%
} In order to give due prominence to this relationship, however, we
must replace the usual time co-ordinate $t$ by an imaginary magnitude
$\sqrt{-1}ct$ 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 rôle 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 at present leave it here, and shall revert to it only towards
the end of Part II.


\part*{Part II: The General Theory of Relativity}

\addcontentsline{toc}{part}{Part II: The General Theory\\
of Relativity}


\chapter*{\raggedright{}XVIII. Special and General Principle of Relativity}

\addcontentsline{toc}{chapter}{Special and General Principle of\\
Relativity}

\markboth{Relativity}{Special and General Relativity}

The basal principle, which was the pivot of all our previous considerations,
was the \emph{special} principle of relativity, \emph{i.e.}\ the
principle of the physical relativity of all \emph{uniform} motion.
Let us 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 only be considered 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: The carriage is in motion relative to the
embankment. 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 the embankment as reference-body, the railway carriage as reference-body,
then these general laws of nature (\emph{e.g}.\ the laws of mechanics
or the law of the propagation of light \emph{in vacuo}) have exactly
the same form in both cases. This can also be expressed as follows:
For the \emph{physical} description of natural processes, neither
of the reference-bodies $K$, $K'$ is unique (lit.\ {}``specially
marked out'') as compared with the other. Unlike the first, this
latter statement need not of necessity hold \emph{a priori}; it is
not contained in the conceptions of {}``motion'' and {}``reference-body''
and derivable from them; only \emph{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 lines. 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 $K'$ 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
\emph{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 (\emph{e.g.}\ those possessing
motion of a different kind). In this sense we speak of the \emph{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
$K$, $K'$, \emph{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 uniformly, the occupant of the carriage
is not sensible of its motion, and it is for this reason that he can
un-reluctantly 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 relativity. But in what follows we shall
soon see that this conclusion cannot be maintained.


\chapter*{\raggedright{}IX. The Gravitational Field}

\addcontentsline{toc}{chapter}{The Gravitational Field}

\markboth{Relativity}{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\mdsh{---}after the manner
of Faraday\mdsh{---}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 surroundings 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 diminishes 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 (\emph{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, \emph{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 (\emph{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\[
(Force)=(inertial\, mass)×(acceleration),\]


\noindent where the {}``inertial mass'' is a characteristic constant
of the accelerated body. If now gravitation is the cause of the acceleration,
we then have%
\footnote{For editing reasons, in the following two expressions the phrase {}``intensity
of the gravitational field'' has been replaced by the phrase {}``gravitational
intensity''\mdsh{---}R.\ Burkey.%
} 

\begin{eqnarray*}
(Force) & = & (gravitational\, mass)×\\
 &  & (gravitational\, intensity),\end{eqnarray*}


\noindent where the {}``gravitational mass'' is likewise a characteristic
constant for the body. From these two relations follows: 

\begin{eqnarray*}
(acceleration) & = & \frac{(gravitational\, mass)}{(inertial\, mass)}\times\\
 &  & (gravitational\, intensity).\end{eqnarray*}


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 \emph{gravitational} mass of a body is
equal to its \emph{inertial} mass. 

It is true that this important law had hitherto been recorded in mechanics,
but it had not been \emph{interpreted}. A satisfactory interpretation
can be obtained only if we recognise the following fact: The \emph{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. 


\chapter*{\raggedright{}XX. The Equality of Inertial and Gravitational Mass
as an Argument for the General Postulate of Relativity}

\addcontentsline{toc}{chapter}{The Equality of Inertial and Gravitational\\
Mass as an Argument for the General\\
Postulate of Relativity}

\markboth{Relativity}{Inertial and Gravitational Mass}

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\mdsh{---}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 house on our earth. If he release a body which
he previously had in his hand, the acceleration 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 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 stretch 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 the
end of Section XVIII. 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.''


\chapter*{\raggedright{}XXI. In What Respects Are the Foundations of Classical
Mechanics and of the Special Theory of Relativity Unsatisfactory?}

\addcontentsline{toc}{chapter}{In What Respects Are the Foundations
of\\
Classical Mechanics and of the Special\\
Theory of Relativity Unsatisfactory?}

\markboth{Relativity}{Unsatisfactory Foundations}

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 $K'$.%
\footnote{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, \emph{e.g.}\ in the case
when the reference-body is rotating uniformly.%
} Newton saw this objection and attempted to invalidate it, but without
success. But E.\ Mach recognised 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.


\chapter*{\raggedright{}XXII. A Few Inferences from the General Theory of
Relativity}

\addcontentsline{toc}{chapter}{A Few Inferences from the General\\
Theory of Relativity}

\markboth{Relativity}{Inferences from General Relativity}

The considerations of Section XX show that the general theory 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 (\emph{i.e.}\ simply by calculation) we are
then able to find how this known natural process appears, as seen
from a reference-body $K'$ which is accelerated relatively to $K$.
But since a gravitational field exists with respect to this new body
of reference $K'$, 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 $K'$ (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
$K'$). 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.%
\footnote{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 confirmed
during the solar eclipse of 29th May, 1919. (\emph{Cf}.\ Appendix
III.)%
}

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 \emph{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 unlimited domain of validity;
its result hold only so long as we are able to disregard the influences
of gravitational fields on the phenomena (\emph{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 and the
laws of electricity were regarded indiscriminately. 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 laws 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,
\emph{i.e.}\ domains in which gravitational fields are absent. If
we now refer such a domain to a reference-body $K'$ possessing any
kind of motion, then relative to $K'$ there exists a gravitational
field which is variable with respect to space and time.%
\footnote{This follows from a generalisation of the discussion in Section XX.%
} The character of this field will of course depend on the motion chosen
for $K'$. 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 gravitational
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. 


\chapter*{\raggedright{}XXIII. Behaviour of Clocks and Measuring Rods on a
Rotating Body of Reference}

\addcontentsline{toc}{chapter}{Behaviour of Clocks and Measuring
Rods\\
on a Rotating Body of Reference}

\markboth{Relativity}{Rotating Clocks and Measuring Rods}

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
fields 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 suppose the same domain
referred to a second body of reference $K'$, which is rotating uniformly
with respect to $K$. In order to fix our ideas, we shall imagine
$K'$ 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 $K'$ 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.%
\footnote{The field disappears at the centre of the disc and increases proportionally
to the distance from the centre as we proceed outwards.%
} 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\mdsh{---}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 $K'$, 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 XII, it
follows that the latter clock goes at a rate permanently slower than
that of the clock at the centre of the circular disc, \emph{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 simultaneously 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 unsurmountable 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 1, since, according to Section XII, moving bodies suffer a shortening
in the direction of the motion. On the other hand, the measuring-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 $\pi=3.14\ldots$,
but a larger number,%
\footnote{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 $K'$ a gravitational field prevails).%
} whereas of course, for a disc which is at rest with respect to $K$,
this operation would yield $\pi$ 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 1 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. 


\chapter*{\raggedright{}XXIV. Euclidean and Non-Euclidean Continuum}

\addcontentsline{toc}{chapter}{Euclidean and Non-Euclidean\\
Continuum}

\markboth{Relativity}{Euclidean \& 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 our 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 needs 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 an 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\mdsh{---}defined as unit lengths\mdsh{---}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 (\emph{i.e.}\ of every material) were to
behave \emph{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 co-ordinates must then
be discarded, and replaced by another which does not assume the validity
of Euclidean geometry for rigid bodies.%
\footnote{Mathematicians have been confronted with our problem in the following
form. If we are given a surface (\emph{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.%
} The reader will notice that the situation depicted here corresponds
to the one brought about by the general postulate of relativity (Section
XXIII). 

\noun{Note}:\mdsh{---}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
\emph{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 \emph{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 Riemann of
treating multi-dimensional, \emph{non-Euclidean continua}. Thus it
is that mathematicians long ago solved the formal problems to which
we are led by the general postulate of relativity. 


\chapter*{\raggedright{}XXV. Gaussian Co-ordinates}

\addcontentsline{toc}{chapter}{Gaussian Co-ordinates}

\markboth{Relativity}{Gaussian Co-ordinates}

%
\begin{figure}[htbp]
\noindent \begin{center}\includegraphics[%
  scale=0.5]{EinsteinRelativity-f04.jpg}\end{center}

\noindent \begin{center}\noun{Fig. 4.}\end{center}
\end{figure}
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. 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 $P'$ on the surface then correspond
to the co-ordinates \begin{eqnarray*}
P & : & u,v\\
P' & : & u+du,v+dv\end{eqnarray*}


\noindent where $du$ and $dv$ signify very small numbers. In a similar
manner we may indicate the distance (line-interval) between $P$ and
$P'$, as measured with a little rod, by means of the very small number
$ds$. Then according to Gauss we have

\[
ds^{2}=g_{11}+2g_{12}du\, dv+g_{22}dv^{2},\]


\noindent 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: 

\[
ds^{2}=du^{2}+dv^{2}.\]


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 co-ordinates are simply 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 co-ordinates.
If a distance $ds$ is associated with the adjacent points $P$ and
$P'$, this distance being measurable and well-defined from a physical
point of view, then the following formula holds:\[
ds^{2}=g_{11}dx_{1}^{2}+2g_{12}dx_{1}dx_{2}+\ldots+g_{44}dx_{4}^{2},\]


\noindent where the magnitudes $g_{11}$ \emph{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 $x_{1}\ldots x_{4}$
with the points of the continuum so that we have simply

\[
ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}.\]


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 $ds^{2}$ 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 co-ordinates)
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 co-ordinates) which differ by an indefinitely small amount
are assigned to adjacent points. The Gaussian co-ordinate 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.


\chapter*{\raggedright{}XXVI. The Space-Time Continuum of the Special Theory
of Relativity Considered as a Euclidean Continuum}

\addcontentsline{toc}{chapter}{The Space-Time Continuum of the Special\\
Theory of Relativity Considered as a\\
Euclidean Continuum}

\markboth{Relativity}{Euclidean Special Relativity}

We are now in a position to formulate more exactly the idea of Minkowski,
which was only vaguely indicated in Section XVII. 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\mdsh{---}in other words\mdsh{---}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 $dx',dy',dz',dt'$. Then these
magnitudes always fulfil the condition%
\footnote{\emph{Cf.}\ Appendices I and II. The relations which are derived
there for the co-ordinates themselves are valid also for co-ordinate
\emph{differences}, and thus also for co-ordinate differentials (indefinitely
small differences).%
}\[
dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}=dx'^{2}+dy'^{2}+dz'^{2}-c^{2}dt'^{2}.\]


The validity of the Lorentz transformation follows from this condition.
We can express this as follows: The magnitude\[
ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2},\]


\noindent 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 $x,y,z,\sqrt{-1}ct$, by $x_{1},x_{2},x_{3},x_{4}$,
we also obtain the result that 

\[
ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}\]


\noindent 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 $\sqrt{-1}ct$
instead of the real quantity $t$, we can regard the space-time continuum\mdsh{---}in
accordance with the special theory of relativity\mdsh{---}as a {}``Euclidean''
four-dimensional continuum, a result which follows from the considerations
of the preceding section. 


\chapter*{\raggedright{}XXVII. The Space-Time Continuum of the General Theory
of Relativity is not a Euclidean Continuum}

\addcontentsline{toc}{chapter}{The Space-Time Continuum of the\\
General Theory of Relativity is not\\
a Euclidean Continuum}

\markboth{Relativity}{Non-Euclidean General Relativity}

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 XXVI, 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 light. But according to Section
XXI, 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 coordinates when a
gravitational field is present. In connection with a specific illustration
in Section XXIII, we found that the presence of a gravitational field
invalidates the definition of the co-ordinates and the time, which
led us to our objective in the special theory of relativity. 

In view of the results 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, shall indicate
position and time directly. Such was the essence of the difficulty
with which we were confronted in Section XXIII. 

But the considerations of Sections XXV and XXVI show us the way to
surmount this difficulty. We refer the four-dimensional 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 be 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 little 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.


\chapter*{\raggedright{}XXVIII. Exact Formulation of the General Principle
of Relativity}

\addcontentsline{toc}{chapter}{Exact Formulation of the\\
General Principle of Relativity}

\markboth{Relativity}{Exact Formulation of General Relativity}

We are now in a position to replace the provisional formulation of
the general principle of relativity given in Section XVIII by an exact
formulation. The form there used, {}``All bodies of reference $K$,
$K'$, \emph{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: {}``\emph{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 $x',y',z',t'$, of a new reference-body
$K'$. According to the general theory of relativity, on the other
hand, by application of \emph{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, \emph{i.e.}\ to those in which no gravitational field exists.
In this connection a Galileian reference-body serves as body of reference,
\emph{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 (\emph{cf.}\ Sections
XX and XXIII). 

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 \emph{ad lib}.\ during their motion. Clocks,
for which the law of motion is 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-mollusk,'' is in the
main equivalent to a Gaussian four-dimensional co-ordinate system
chosen arbitrarily. That which gives the {}``mollusk'' a certain
comprehensibleness as compared with the Gauss co-ordinate system is
the (really unqualified) formal retention of the separate existence
of the space co-ordinate. Every point on the mollusk is treated as
a space-point, and every material point which is at rest relatively
to it as at rest, so long as the mollusk is considered as reference-body.
The general principle of relativity requires that all these mollusks
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 mollusk. 

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.


\chapter*{\raggedright{}XXIX. The Solution of the Problem of Gravitation on
the Basis of the General Principle of Relativity}

\addcontentsline{toc}{chapter}{The Solution of the Problem of\\
Gravitation on the Basis of the General\\
Principle of Relativity}

\markboth{Relativity}{Solution of the Problem of Gravitation}

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 from a consideration of a Galileian domain, \emph{i.e.}\ a
domain in which there is no gravitational field relative to the Galileian
reference-body \emph{K}. The behaviour of measuring-rods and clocks
with reference to \emph{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 co-ordinate system
or to a {}``mollusk'' as reference-body $K'$. Then with respect
to $K'$ 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 $K'$ simply by mathematical transformation.
We interpret this behaviour as the behaviour of measuring-rods, clocks
and material points under the influence of the gravitational field
$G$. Hereupon we introduce a hypothesis: that the influence of the
gravitational field on measuring-rods, clocks and freely-moving material
points continues to take place according to the same laws, even in
the case when the prevailing gravitational field is \emph{not} derivable
from the Galileian special case, 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 co-ordinates. This behaviour is formulated
in a law, which is always valid, no matter how the reference-body
(mollusk) used in the description may be chosen. 

This law is not yet the \emph{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: The required generalisation must likewise satisfy
the general postulate of relativity. If there is any matter in the
domain under consideration, only its inertial mass, and thus according
to Section XV only its energy is of importance for its effect in exciting
a field. 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, \emph{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
XXI; 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 co-ordinate 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 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
purpose. 

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,%
\footnote{Observed by Eddington and others in 1919. (\emph{Cf.}\ Appendix III.)%
} 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 (\emph{i.e.}\ by the same kind
of molecule). I do not doubt that these deductions from the theory
will be confirmed also. 


\part*{Part III: Considerations on the Universe as a Whole}

\addcontentsline{toc}{part}{Part III: Considerations on the Universe
as a Whole}


\chapter*{\raggedright{}XXX. Cosmological Difficulties of Newton's Theory}

\addcontentsline{toc}{chapter}{Cosmological Difficulties of Newton's\\
Theory}

\markboth{Relativity}{Cosmological Difficulties of Newton}

Apart from the difficulty discussed in Section XXI, 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 approximately 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.%
\footnote{\emph{Proof}.\mdsh{---}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 the universe, then
a sphere of volume $V$ will enclose the average mass $P_{0}V$. 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 $P_{0}\cdot V/F$ or $P_{0}R$.
Hence the intensity of the field at the surface would ultimately become
infinite with increasing radius $R$ of the sphere, which is impossible. %
}

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 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. 


\chapter*{\raggedright{}XXXI. The Possibility of a {}``Finite'' and Yet
{}``Unbounded'' Universe}

\addcontentsline{toc}{chapter}{The Possibility of a {}``Finite''
and Yet\\
{}``Unbounded'' Universe}

\markboth{Relativity}{{}``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 \emph{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, \emph{e.g.}\ the lattice construction,
considered in Section XXIV. 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, \emph{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, \emph{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 \emph{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 $\pi$,
which is independent of the diameter of the circle. On their spherical
surface our flat beings would find for this ratio the value\[
\pi\frac{\sin(\frac{r}{R})}{(\frac{r}{R})},\]


\noindent \emph{i.e}.\ a smaller value than $\pi$, 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 world-sphere 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 {}``world-sphere'';
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'' ($2\pi^{2}R^{3}$).
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, \emph{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 ($F=4\pi r^{2}$) of this surface
by means of a square made up of measuring-rods. If the universe is
Euclidean, then $F=4\pi r^{2}$; if it is spherical, then $F$ is
always less than $4\pi r^{2}$. 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 (\emph{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 moderate degree of certainty, and
in this connection the difficulty mentioned in Section XXX finds its
solution.


\chapter*{\raggedright{}XXXII. The Structure of Space According to the General
Theory of Relativity}

\addcontentsline{toc}{chapter}{The Structure of Space According to
the\\
General Theory of Relativity}

\markboth{Relativity}{The Structure of Space}

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, \emph{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
XXX. 

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, \emph{i.e.}\ the
universe will be quasi-spherical. But it will be necessarily finite.
In fact, the theory supplies us with a simple connection%
\footnote{For the {}``radius'' $R$ of the universe we obtain the equation\[
R^{2}=\frac{2}{\kappa\rho}\]


\noindent The use of the C.G.S. system in this equation gives is the
average density of the matter.%
} between the space-expanse of the universe and the average density
of matter in it. 


\part*{Appendices}

\addcontentsline{toc}{part}{Appendices}


\chapter*{\raggedright{}I. Simple Derivation of the Lorentz Transformation}

\addcontentsline{toc}{chapter}{Simple Derivation of the Lorentz\\
Transformation}

\markboth{Relativity}{Derivation of Lorentz Transformation}

For the relative orientation of the co-ordinate systems indicated
in Fig.\ 2, the $x$-axes of both systems permanently 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 $K'$
by the abscissa $x'$ and the time $t'$, when $x$ and $t$ are given. 

A light-signal, which is proceeding along the positive axis of $x$,
is transmitted according to the equation\[
x=ct\]


\noindent or

\begin{equation}
x-ct=0.\label{eq:1}\end{equation}


\noindent Since the same light-signal has to be transmitted relative
to $K'$ with the velocity $c$, the propagation relative to the system
$K'$ will be represented by the analogous formula

\begin{equation}
x'-ct'=0.\label{eq:2}\end{equation}


\noindent Those space-time points (events) which satisfy (\ref{eq:1})
must also satisfy (\ref{eq:2}). Obviously this will be the case when
the relation

\begin{equation}
(x'-ct')=\lambda\cdot(x-ct)\label{eq:3}\end{equation}


\noindent is fulfilled in general, where $\lambda$ indicates a constant;
for, according to (\ref{eq: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 \emph{x}-axis, we obtain the condition\begin{equation}
(x'+ct')=\mu\cdot(x+ct).\label{eq:4}\end{equation}


By adding (or subtracting) equations (\ref{eq:3}) and (\ref{eq:4}),
and introducing for convenience the constants $a$ and $b$ in place
of the constants $\lambda$ and $\mu$ where\[
a=\frac{\lambda+\mu}{2}\]


\noindent and

\[
b=\frac{\lambda-\mu}{2}\]


\noindent we obtain the equations

\begin{eqnarray}
x' & = & ax-bct\label{eq:5a}\\
ct' & = & act-bx.\label{eq:5b}\end{eqnarray}


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 $K'$ we have permanently $x'=0$, and hence according
to the first of the equations (\ref{eq:5a}) and (\ref{eq:5b})\[
x=\frac{bc}{a}t.\]


If we call $v$ the velocity with which the origin of $K'$ is moving
relative to $K$, we then have\begin{equation}
v=\frac{bc}{a}.\label{eq:6}\end{equation}


The same value $v$ can be obtained from equations (\ref{eq:5a})
and (\ref{eq:5b}), if we calculate the velocity of another point
of $K'$ 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 $K'$ 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 $K'$ from
$K$; this means that we have to insert a particular value of $t$
(time of $K$), \emph{e.g.}\ $t=0$. For this value of $t$ we then
obtain from the equation (\ref{eq:5a})\[
x'=ax.\]


Two points of the $x'$-axis which are separated by the distance $x'=1$
when measured in the $K'$ system are thus separated in our instantaneous
photograph by the distance\begin{equation}
\Delta x=\frac{1}{a}.\label{eq:7}\end{equation}


But if the snapshot be taken from $K'$ ($t'=0$), and if we eliminate
$t$ from the equations (\ref{eq:5a}) and (\ref{eq:5b}), taking
into account the expression (\ref{eq:6}), we obtain\[
x'=a\cdot(1-\frac{v^{2}}{c^{2}})\cdot x.\]


From this we conclude that two points on the $x$-axis and separated
by the distance 1 (relative to $K$) will be represented on our snapshot
by the distance\begin{equation}
\Delta x'=a\cdot(1-\frac{v^{2}}{c^{2}}).\label{eq:7a}\end{equation}


But from what has been said, the two snapshots must be identical;
hence $\Delta x$ in (\ref{eq:7}) must be equal to $\Delta x'$ in
(\ref{eq:7a}), so that we obtain\begin{equation}
a^{2}=\frac{1}{1-\frac{v^{2}}{c^{2}}}.\label{eq:7b}\end{equation}


The equations (\ref{eq:6}) and (\ref{eq:7b}) determine the constants
$a$ and $b$. By inserting the values of these constants in (\ref{eq:5a})
and (\ref{eq:5b}), we obtain the first and the fourth of the equations
given in Section XI.\begin{eqnarray}
x' & = & \frac{x-vt}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\label{eq:8}\\
t' & = & \frac{t-\frac{v}{c^{2}}x}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.\label{eq:8b}\end{eqnarray}


Thus we have obtained the Lorentz transformation for events on the
$x$-axis. It satisfies the condition\begin{equation}
x'^{2}-c^{2}t'^{2}=x^{2}-c^{2}t^{2}\label{eq:8a}\end{equation}


The extension of this result, to include events which take place outside
the $x$-axis, is obtained by retaining equations (\ref{eq:8}) and
(\ref{eq:8b}) and supplementing them by the relations\begin{equation}
\left.\begin{array}{rcl}
y' & = & y\\
z' & = & z.\end{array}\right\} \label{eq:9}\end{equation}


\noindent In this way we satisfy the postulate of the constancy of
the velocity of light \emph{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\[
r=\sqrt{x^{2}+y^{2}+z^{2}}=ct,\]


\noindent or, if we square this equation, according to the equation

\begin{equation}
x^{2}+y^{2}+z^{2}-c^{2}t^{2}=0.\label{eq:10}\end{equation}


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\mdsh{---}as judged from $K'$\mdsh{---}in
accordance with the corresponding formula\[
r'=ct'\]


\noindent or, 

\begin{equation}
x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=0.\label{eq:10a}\end{equation}


\noindent In order that equation (\ref{eq:10a}) may be a consequence
of equation (\ref{eq:10}), we must have

\begin{equation}
x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=\sigma\cdot(x^{2}+y^{2}+z^{2}-c^{2}t^{2}).\label{eq:11}\end{equation}


Since equation (\ref{eq:8a}) must hold for points on the $x$-axis,
we thus have $\sigma=1$; for (\ref{eq:11}) is a consequence of (\ref{eq:8a})
and (\ref{eq:9}), and hence also of (\ref{eq:8}) and (\ref{eq:9}).
We have thus derived the Lorentz transformation. 

The Lorentz transformation represented by (\ref{eq:8}) and (\ref{eq:9})
still requires to be generalised. Obviously it is immaterial whether
the axes of $K'$ be chosen so that they are spatially parallel to
those of $K$. It is also not essential that the velocity of translation
of $K'$ 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,
\emph{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 $x',y',z',t'$, in terms of linear homogeneous
functions of $x,y,z,t$, of such a kind that the relation\begin{equation}
x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=x^{2}+y^{2}+z^{2}-c^{2}t^{2}\label{eq:11a}\end{equation}


\noindent is satisfied identically. That is to say: If we substitute
their expressions in $x,y,z,t$, in place of $x',y',z',t'$, on the
left-hand side, then the left-hand side of (\ref{eq:11a}) agrees
with the right-hand side.


\chapter*{\raggedright{}II. Minkowski's Four-Dimensional Space ({}``World'')}

\addcontentsline{toc}{chapter}{Minkowski's Four-Dimensional Space\\
({}``World'')}

\markboth{Relativity}{Minkowski's Four-Dimensional Space}

We can characterise the Lorentz transformation still more simply if
we introduce the imaginary $\sqrt{-1}ct$ in place of $t$, as time-variable.
If, in accordance with this, we insert\begin{eqnarray*}
x_{1} & = & x\\
x_{2} & = & y\\
x_{3} & = & z\\
x_{4} & = & \sqrt{-1}ct\end{eqnarray*}


\noindent and similarly for the accented system $K'$, then the condition
which is identically satisfied by the transformation can be expressed
thus:

\begin{equation}
x_{1}'^{2}+x_{2}'^{2}+x_{3}'^{2}+x_{4}'^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}.\label{eq:12}\end{equation}


That is, by the afore-mentioned choice of {}``co-ordinates'' (\ref{eq:11a})
is transformed into this equation. 

We see from (\ref{eq: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\[
x_{1}'^{2}+x_{2}'^{2}+x_{3}'^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}.\]


\noindent The analogy with (\ref{eq:12}) is a complete one. We can
regard Minkowski's {}``world'' in a formal manner as a four-dimensional
Euclidean space (with imaginary time co-ordinate); the Lorentz transformation
corresponds to a {}``rotation'' of the co-ordinate system in the
four-dimensional {}``world.''


\chapter*{\raggedright{}III. The Experimental Confirmation of the General
Theory of Relativity}

\addcontentsline{toc}{chapter}{The Experimental Confirmation of the\\
General Theory of Relativity}

\markboth{Relativity}{The Experimental Confirmation}

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 \emph{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 such 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 present 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. 


\section*{\raggedright{}(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\mdsh{---}closest proximity to the
sun\mdsh{---}to perihelion) would differ from 360°. 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, \emph{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 a way,
that the angle described by the radius sun-planet between one perihelion
and the next should exceed that corresponding to one complete revolution
by an amount given by\[
+\frac{24\pi^{3}a^{2}}{T^{2}c^{2}(1-e^{2})}.\]


(\emph{N.B}.\mdsh{---}One complete revolution corresponds to the
angle $2\pi$ in the absolute angular measure customary in physics,
and the above expression gives 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.%
\footnote{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. %
}

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\mdsh{---}1859\mdsh{---}and Newcomb\mdsh{---}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. 


\section*{\raggedright{}(b) DEFLECTION OF LIGHT BY A GRAVITATIONAL FIELD}

In Section XXII 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 $\Delta$ sun-radii from its centre, the
angle of deflection ($\alpha$) should amount to\[
\alpha=\frac{1.7\, seconds\, of\, arc}{\Delta}.\]


\noindent It may be added that, according to the theory, half of 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.

\noindent \begin{center}\includegraphics[%
  scale=0.5]{EinsteinRelativity-f05.jpg}\end{center}

\noindent \begin{center}\noun{Fig. 5.}\end{center}

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 from
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 $D_{2}$,
\emph{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, \emph{i.e.}\ a few
months earlier or later. As compared with 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 $\alpha$. 

We are indebted to the Royal Society and to Royal Astronomical Society
for the investigation of this important deduction. Undaunted by the
war and by difficulties of both a material and a psychological nature
aroused by the war, these societies equipped two expeditions\mdsh{---}to
Sobral (Brazil) and to the island of Principe (West Africa)\mdsh{---}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 descrepancies 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:%
\footnote{For editorial reasons, the format of the table has been slightly modified\mdsh{---}Ron
Burkey.%
} 

\noindent \begin{center}\begin{tabular}{|c|c|c|c|c|}
\hline 
&
\multicolumn{2}{c|}{\emph{1st Co-ordinate}}&
\multicolumn{2}{c|}{\emph{2nd Co-ordinate}}\tabularnewline
\cline{2-3} \cline{4-5} 
\hline 
\textbf{Star}&
\textbf{Obs.}&
\textbf{Calc.}&
\textbf{Obs.}&
\textbf{Calc.}\tabularnewline
\hline
\hline 
\#11&
-0.19 &
-0.22 &
+0.16 &
+0.02\tabularnewline
\hline 
\#5&
+0.29 &
+0.31&
-0.46 &
-0.43\tabularnewline
\hline 
\#4&
+0.11&
0.10&
+0.83 &
+0.74\tabularnewline
\hline 
\#3&
+0.20&
+0.12&
+1.00&
+0.87\tabularnewline
\hline 
\#6&
+0.10&
+0.04&
+0.57&
+0.40\tabularnewline
\hline 
\#10&
-0.08&
+0.09&
+0.35&
+0.32\tabularnewline
\hline 
\#2&
+0.95&
+0.85&
-0.27&
-0.09\tabularnewline
\hline
\end{tabular}\end{center}


\section*{\raggedright{}(c) DISPLACEMENT OF SPECTRAL LINES TOWARDS THE RED}

In Section XXIII it has been shown that in a system $K'$ 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\[
v=\omega r,\]


\noindent where $\omega$ represents the velocity of rotation of the
disc $K'$ 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
XII, be given by

\[
v=v_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}\]


\noindent or with sufficient accuracy by

\[
v=v_{0}(1-\frac{1}{2}\frac{v^{2}}{c^{2}}).\]


\noindent This expression may also be stated in the following form: 

\[
v=v_{0}(1-\frac{1}{c^{2}}\frac{\omega^{2}r^{2}}{2}).\]


If we represent the difference of potential of the centrifugal force
between the position of the clock and the centre of the disc by $\phi$,
\emph{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\[
\phi=-\frac{\omega^{2}r^{2}}{2}.\]


\noindent From this it follows that

\[
v=v_{0}(1+\frac{\phi}{c^{2}}).\]


\noindent 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
also valid from the standpoint of an observer who is rotating with
the disc. 

Now, as judged from the disc, the latter is in a gravitational field
of potential $\phi$, 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: 

\begin{quote}
An atom absorbs or emits light of a frequency which is dependent on
the potential of the gravitational field in which it is situated. 
\end{quote}
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 $\phi=-K\cdot M/r$ 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\[
\frac{v_{0}-v}{v_{0}}=\frac{K}{c^{2}}\frac{M}{r}.\]


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 $r$ is known. 

It is an open question whether or not this effect exists, and at the
present time 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, 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 Prüfungder allgemeinen Relativitäts-Theorie''
(\emph{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. 


\chapter*{\raggedright{}Bibliography}

\addcontentsline{toc}{chapter}{Bibliography}

\markboth{Relativity}{Bibliography}


\section*{INTRODUCTORY}

{}

\emph{\indent{}The Foundations of Einstein's Theory of Gravitation}:
Erwin Freundlich (translation by H. L. Brose). Camb. Univ. Press,
1920. 

\emph{Space and Time in Contemporary Physics}: Moritz Schlick (translation
by H. L. Brose). Clarendon Press, Oxford, 1920. 


\section*{THE SPECIAL THEORY}

{}

\emph{\indent{}The Principle of Relativity}: E. Cunningham. Camb.
Univ. Press. 

\emph{Relativity and the Electron Theory}: E. Cunningham, Monographs
on Physics. Longmans, Green \& Co. 

\emph{The Theory of Relativity}: L. Silberstein. Macmillan \& Co. 

{}``The Space-Time Manifold of Relativity'': E. B. Wilson and G.
N. Lewis, \emph{Proc. Amer. Soc. Arts \& Science}, vol. xlviii., No.
11, 1912. 


\section*{THE GENERAL THEORY}

{}

\emph{\indent{}Report on the Relativity Theory of Gravitation}: A.
S. Eddington. Fleetway Press Ltd., Fleet Street, London. 

{}``On Einstein's Theory of Gravitation and its Astronomical Consequences'':
W. de Sitter, \emph{M. N. Roy. Astron. Soc.}, lxxvi. p. 699, 1916;
lxxvii. p. 155, 1916; lxxviii. p. 3, 1917. 

{}``On Einstein's Theory of Gravitation'': H. A. Lorentz, \emph{Proc.
Amsterdans Acad.}, vol. xix. p. 1341, 1917. 

\emph{Space, Time and Gravitation}: W. de Sitter: The Observatory,
No. 505, p. 412. Taylor \& Francis, Fleet Street, London. 

{}``The Total Eclipse of 29th May 1919, and the Influence of Gravitation
on Light'': A. S. Eddington, ibid., March, 1919. 

{}``Discussion on the Theory of Relativity'': \emph{M. N. Roy. Astron.
Soc.}, vol. lxxx., No. 2., p. 96, December 1919. 

{}``The Displacement of Spectrum Lines and the Equivalence Hypothesis'':
W. G. Duffield, \emph{M. N. Roy. Astron. Soc.}, vol. lxxx.; No. 3,
p. 262, 1920. 

\emph{Space, Time and Gravitation}: A. S. Eddington. Camb. Univ. Press,
1920. 


\section*{ALSO, CHAPTERS IN}

{}

\emph{\indent{}The Mathematical Theory of Electricity and Magnetism}:
J. H. Jeans (4th edition). Camb. Univ. Press, 1920. 

\emph{The Electron Theory of Matter}: O. W. Richardson. Camb. Univ.
Press.
\end{document}
