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+%The Project Gutenberg EBook of Relativity: The Special and General Theory
+%by Albert Einstein
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+%Title: Relativity: The Special and General Theory
+%
+%Author: Albert Einstein
+%
+%Release Date: February, 2004 [EBook #5001]
+%[Yes, we are more than one year ahead of schedule]
+%[This file was first posted on April 1, 2002]
+%
+%Edition: 10
+%
+%Language: English
+%
+%Character set encoding: ASCII
+%
+%*** START OF THE PROJECT GUTENBERG EBOOK, RELATIVITY ***
+%
+%
+%
+%
+%ALBERT EINSTEIN REFERENCE ARCHIVE
+%
+%RELATIVITY: THE SPECIAL AND GENERAL THEORY
+%
+%BY ALBERT EINSTEIN
+%
+%
+%Written: 1916 (this revised edition: 1924)
+%Source: Relativity: The Special and General Theory (1920)
+%Publisher: Methuen & Co Ltd
+%First Published: December, 1916
+%Translated: Robert W. Lawson (Authorised translation)
+%Transcription/Markup: Brian Basgen <brian@marxists.org>
+%Transcription to text: Gregory B. Newby <gbnewby@petascale.org>
+%Typeset into LaTeX: Robert Bradshaw <rwb00@myrealbox.com>
+%Copyleft: Einstein Reference Archive (marxists.org) 1999, 2002.
+%Permission is granted to copy and/or distribute this document under
+%the terms of the GNU Free Documentation License (end of this file)
+%The Einstein Reference Archive is online at:
+%http://www.marxists.org/reference/archive/einstein/index.htm
+
+
+
+%\input gutenberg-simple.tex
+\documentclass[11pt]{report}
+
+\renewcommand{\labelenumi}{\alph{enumi}.}
+\renewcommand{\thesection}{\alph{section}.}
+%\renewcommand{\thesection}{}
+
+
+\begin{document}
+
+%\gtitle{Relativity: The Special and General Theory}
+\title{Relativity: The Special and General Theory}
+\date{1916}
+
+%\gauthor{Albert Einstein}
+
+\author{Albert Einstein}
+
+%\frontmatter
+
+\maketitle
+
+\tableofcontents
+
+\newpage
+
+%CONTENTS
+
+%Preface
+
+%Part I: The Special Theory of Relativity
+
+%01. Physical Meaning of Geometrical Propositions
+%02. The System of Co-ordinates
+%03. Space and Time in Classical Mechanics
+%04. The Galileian System of Co-ordinates
+%05. The Principle of Relativity (in the Restricted Sense)
+%06. The Theorem of the Addition of Velocities employed in
+%Classical Mechanics
+%07. The Apparent Incompatability of the Law of Propagation of
+%Light with the Principle of Relativity
+%08. On the Idea of Time in Physics
+%09. The Relativity of Simultaneity
+%10. On the Relativity of the Conception of Distance
+%11. The Lorentz Transformation
+%12. The Behaviour of Measuring-Rods and Clocks in Motion
+%13. Theorem of the Addition of Velocities. The Experiment of Fizeau
+%14. The Hueristic Value of the Theory of Relativity
+%15. General Results of the Theory
+%16. Expereince and the Special Theory of Relativity
+%17. Minkowski's Four-dimensial Space
+
+
+%Part II: The General Theory of Relativity
+
+%18. Special and General Principle of Relativity
+%19. The Gravitational Field
+%20. The Equality of Inertial and Gravitational Mass as an Argument
+%for the General Postulate of Relativity
+%21. In What Respects are the Foundations of Classical Mechanics
+%and of the Special Theory of Relativity Unsatisfactory?
+%22. A Few Inferences from the General Principle of Relativity
+%23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of
+%Reference
+%24. Euclidean and non-Euclidean Continuum
+%25. Gaussian Co-ordinates
+%26. The Space-Time Continuum of the Speical Theory of Relativity
+%Considered as a Euclidean Continuum
+%27. The Space-Time Continuum of the General Theory of Relativity
+%is Not a Euclidean Continuum
+%28. Exact Formulation of the General Principle of Relativity
+%29. The Solution of the Problem of Gravitation on the Basis of the
+%General Principle of Relativity
+
+
+%Part III: Considerations on the Universe as a Whole
+
+%30. Cosmological Difficulties of Netwon's Theory
+%31. The Possibility of a ``Finite" and yet ``Unbounded" Universe
+%32. The Structure of Space According to the General Theory of
+%Relativity
+
+
+%Appendices:
+
+%01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
+%02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
+%03. The Experimental Confirmation of the General Theory of Relativity
+%04. The Structure of Space According to the General Theory of
+%Relativity (sup. ch 32)
+%05. Relativity and the Problem of Space
+
+%Note: The fifth Appendix was added by Einstein at the time of the
+%fifteenth re-printing of this book; and as a result is still under
+%copyright restrictions so cannot be added without the permission of
+%the publisher.
+
+
+%\chapter{Preface}
+
+ (December, 1916)
+
+ ~
+
+The present book is intended, as far as possible, to give an exact
+insight into the theory of Relativity to those readers who, from a
+general scientific and philosophical point of view, are interested in
+the theory, but who are not conversant with the mathematical apparatus
+of theoretical physics. The work presumes a standard of education
+corresponding to that of a university matriculation examination, and,
+despite the shortness of the book, a fair amount of patience and force
+of will on the part of the reader. The author has spared himself no
+pains in his endeavour to present the main ideas in the simplest and
+most intelligible form, and on the whole, in the sequence and
+connection in which they actually originated. In the interest of
+clearness, it appeared to me inevitable that I should repeat myself
+frequently, without paying the slightest attention to the elegance of
+the presentation. I adhered scrupulously to the precept of that
+brilliant theoretical physicist L. Boltzmann, according to whom
+matters of elegance ought to be left to the tailor and to the cobbler.
+I make no pretence of having withheld from the reader difficulties
+which are inherent to the subject. On the other hand, I have purposely
+treated the empirical physical foundations of the theory in a
+"step-motherly" fashion, so that readers unfamiliar with physics may
+not feel like the wanderer who was unable to see the forest for the
+trees. May the book bring some one a few happy hours of suggestive
+thought!
+
+~
+
+December, 1916
+
+A. EINSTEIN
+
+%\mainmatter
+
+\part{The Special Theory of Relativity}
+
+\chapter{Physical Meaning of Geometrical Propositions}
+
+In your schooldays most of you who read this book made acquaintance
+with the noble building of Euclid's geometry, and you remember---perhaps
+with more respect than love---the magnificent structure, on
+the lofty staircase of which you were chased about for uncounted hours
+by conscientious teachers. By reason of our past experience, you would
+certainly regard everyone with disdain who should pronounce even the
+most out-of-the-way proposition of this science to be untrue. But
+perhaps this feeling of proud certainty would leave you immediately if
+some one were to ask you: ``What, then, do you mean by the assertion
+that these propositions are true?" Let us proceed to give this
+question a little consideration.
+
+Geometry sets out form certain conceptions such as ``plane,'' ``point,"
+and ``straight line," with which we are able to associate more or less
+definite ideas, and from certain simple propositions (axioms) which,
+in virtue of these ideas, we are inclined to accept as ``true." Then,
+on the basis of a logical process, the justification of which we feel
+ourselves compelled to admit, all remaining propositions are shown to
+follow from those axioms, {\it i.e.} they are proven. A proposition is then
+correct (``true") when it has been derived in the recognised manner
+from the axioms. The question of ``truth" of the individual geometrical
+propositions is thus reduced to one of the ``truth" of the axioms. Now
+it has long been known that the last question is not only unanswerable
+by the methods of geometry, but that it is in itself entirely without
+meaning. We cannot ask whether it is true that only one straight line
+goes through two points. We can only say that Euclidean geometry deals
+with things called ``straight lines," to each of which is ascribed the
+property of being uniquely determined by two points situated on it.
+The concept ``true" does not tally with the assertions of pure
+geometry, because by the word ``true" we are eventually in the habit of
+designating always the correspondence with a ``real" object; geometry,
+however, is not concerned with the relation of the ideas involved in
+it to objects of experience, but only with the logical connection of
+these ideas among themselves.
+
+It is not difficult to understand why, in spite of this, we feel
+constrained to call the propositions of geometry ``true." Geometrical
+ideas correspond to more or less exact objects in nature, and these
+last are undoubtedly the exclusive cause of the genesis of those
+ideas. Geometry ought to refrain from such a course, in order to give
+to its structure the largest possible logical unity. The practice, for
+example, of seeing in a ``distance" two marked positions on a
+practically rigid body is something which is lodged deeply in our
+habit of thought. We are accustomed further to regard three points as
+being situated on a straight line, if their apparent positions can be
+made to coincide for observation with one eye, under suitable choice
+of our place of observation.
+
+If, in pursuance of our habit of thought, we now supplement the
+propositions of Euclidean geometry by the single proposition that two
+points on a practically rigid body always correspond to the same
+distance (line-interval), independently of any changes in position to
+which we may subject the body, the propositions of Euclidean geometry
+then resolve themselves into propositions on the possible relative
+position of practically rigid bodies.\footnotemark\ Geometry which has been
+supplemented in this way is then to be treated as a branch of physics.
+We can now legitimately ask as to the ``truth" of geometrical
+propositions interpreted in this way, since we are justified in asking
+whether these propositions are satisfied for those real things we have
+associated with the geometrical ideas. In less exact terms we can
+express this by saying that by the ``truth" of a geometrical
+proposition in this sense we understand its validity for a
+construction with rule and compasses.
+
+Of course the conviction of the ``truth" of geometrical propositions in
+this sense is founded exclusively on rather incomplete experience. For
+the present we shall assume the ``truth" of the geometrical
+propositions, then at a later stage (in the general theory of
+relativity) we shall see that this ``truth" is limited, and we shall
+consider the extent of its limitation.
+
+
+% Notes
+
+\footnotetext[1]{It follows that a natural object is associated also with a
+straight line. Three points A, B and C on a rigid body thus lie in a
+straight line when the points A and C being given, B is chosen such
+that the sum of the distances AB and BC is as short as possible. This
+incomplete suggestion will suffice for the present purpose.}
+
+
+
+\chapter{The System of Co-ordinates}
+
+
+On the basis of the physical interpretation of distance which has been
+indicated, we are also in a position to establish the distance between
+two points on a rigid body by means of measurements. For this purpose
+we require a ``distance'' (rod S) which is to be used once and for
+all, and which we employ as a standard measure. If, now, A and B are
+two points on a rigid body, we can construct the line joining them
+according to the rules of geometry; then, starting from A, we can
+mark off the distance S time after time until we reach B. The number
+of these operations required is the numerical measure of the distance
+AB. This is the basis of all measurement of length.\footnotemark
+
+Every description of the scene of an event or of the position of an
+object in space is based on the specification of the point on a rigid
+body (body of reference) with which that event or object coincides.
+This applies not only to scientific description, but also to everyday
+life. If I analyse the place specification ``Times Square, New York,"\footnotemark
+I arrive at the following result. The earth is the rigid body
+to which the specification of place refers; ``Times Square, New York,"
+is a well-defined point, to which a name has been assigned, and with
+which the event coincides in space.\footnotemark
+
+This primitive method of place specification deals only with places on
+the surface of rigid bodies, and is dependent on the existence of
+points on this surface which are distinguishable from each other. But
+we can free ourselves from both of these limitations without altering
+the nature of our specification of position. If, for instance, a cloud
+is hovering over Times Square, then we can determine its position
+relative to the surface of the earth by erecting a pole
+perpendicularly on the Square, so that it reaches the cloud. The
+length of the pole measured with the standard measuring-rod, combined
+with the specification of the position of the foot of the pole,
+supplies us with a complete place specification. On the basis of this
+illustration, we are able to see the manner in which a refinement of
+the conception of position has been developed.
+
+\begin{enumerate}
+\item 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.
+
+\item 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.
+
+\item 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.
+\end{enumerate}
+
+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. \footnotemark
+
+We thus obtain the following result: Every description of events in
+space involves the use of a rigid body to which such events have to be
+referred. The resulting relationship takes for granted that the laws
+of Euclidean geometry hold for ``distances;" the ``distance" being
+represented physically by means of the convention of two marks on a
+rigid body.
+
+
+% Notes
+
+\footnotetext[1]{Here we have assumed that there is nothing left over {\it 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.}
+
+\footnotetext[2]{Einstein used ``Potsdamer Platz, Berlin" in the original text.
+In the authorised translation this was supplemented with ``Tranfalgar
+Square, London". We have changed this to ``Times Square, New York", as
+this is the most well known/identifiable location to English speakers
+in the present day. [Note by the janitor.]}
+
+\footnotetext[3]{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.}
+
+\footnotetext[4]{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.}
+
+
+
+\chapter{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"\footnotemark), but only
+a trajectory relative to a particular body of reference.
+
+In order to have a complete description of the motion, we must specify
+how the body alters its position with time; {\it i.e.} for every point on
+the trajectory it must be stated at what time the body is situated
+there. These data must be supplemented by such a definition of time
+that, in virtue of this definition, these time-values can be regarded
+essentially as magnitudes (results of measurements) capable of
+observation. If we take our stand on the ground of classical
+mechanics, we can satisfy this requirement for our illustration in the
+following manner. We imagine two clocks of identical construction;
+the man at the railway-carriage window is holding one of them, and the
+man on the footpath the other. Each of the observers determines the
+position on his own reference-body occupied by the stone at each tick
+of the clock he is holding in his hand. In this connection we have not
+taken account of the inaccuracy involved by the finiteness of the
+velocity of propagation of light. With this and with a second
+difficulty prevailing here we shall have to deal in detail later.
+
+
+% Notes
+
+\footnotetext[1]{That is, a curve along which the body moves.}
+
+
+\chapter{The Galilean System of Co-ordinates}
+
+
+As is well known, the fundamental law of the mechanics of
+Galilei-Newton, which is known as the law of inertia, can be stated
+thus: A body removed sufficiently far from other bodies continues in a
+state of rest or of uniform motion in a straight line. This law not
+only says something about the motion of the bodies, but it also
+indicates the reference-bodies or systems of coordinates, permissible
+in mechanics, which can be used in mechanical description. The visible
+fixed stars are bodies for which the law of inertia certainly holds to
+a high degree of approximation. Now if we use a system of co-ordinates
+which is rigidly attached to the earth, then, relative to this system,
+every fixed star describes a circle of immense radius in the course of
+an astronomical day, a result which is opposed to the statement of the
+law of inertia. So that if we adhere to this law we must refer these
+motions only to systems of coordinates relative to which the fixed
+stars do not move in a circle. A system of co-ordinates of which the
+state of motion is such that the law of inertia holds relative to it
+is called a ``Galileian system of co-ordinates." The laws of the
+mechanics of Galflei-Newton can be regarded as valid only for a
+Galileian system of co-ordinates.
+
+
+\chapter{The Principle of Relativity in the Restricted Sense}
+
+In order to attain the greatest possible clearness, let us return to
+our example of the railway carriage supposed to be travelling
+uniformly. We call its motion a uniform translation (``uniform" because
+it is of constant velocity and direction, ``translation'' because
+although the carriage changes its position relative to the embankment
+yet it does not rotate in so doing). Let us imagine a raven flying
+through the air in such a manner that its motion, as observed from the
+embankment, is uniform and in a straight line. If we were to observe
+the flying raven from the moving railway carriage. we should find that
+the motion of the raven would be one of different velocity and
+direction, but that it would still be uniform and in a straight line.
+Expressed in an abstract manner we may say: If a mass m is moving
+uniformly in a straight line with respect to a co-ordinate system $K$,
+then it will also be moving uniformly and in a straight line relative
+to a second co-ordinate system $K'$ 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
+mechanics. But that a principle of such broad generality should hold
+with such exactness in one domain of phenomena, and yet should be
+invalid for another, is a priori not very probable.
+
+We now proceed to the second argument, to which, moreover, we shall
+return later. If the principle of relativity (in the restricted sense)
+does not hold, then the Galileian co-ordinate systems $K$, $K'$, $K''$, etc.,
+which are moving uniformly relative to each other, will not be
+equivalent for the description of natural phenomena. In this case we
+should be constrained to believe that natural laws are capable of
+being formulated in a particularly simple manner, and of course only
+on condition that, from amongst all possible Galileian co-ordinate
+systems, we should have chosen \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 ({\it i.e.} ``really") with respect to $K_0$. In the general laws of
+nature which have been formulated with reference to $K$, the magnitude
+and direction of the velocity of the carriage would necessarily play a
+part. We should expect, for instance, that the note emitted by an
+organpipe placed with its axis parallel to the direction of travel
+would be different from that emitted if the axis of the pipe were
+placed perpendicular to this direction.
+
+Now in virtue of its motion in an orbit round the sun, our earth is
+comparable with a railway carriage travelling with a velocity of about
+30 kilometres per second. If the principle of relativity were not
+valid we should therefore expect that the direction of motion of the
+earth at any moment would enter into the laws of nature, and also that
+physical systems in their behaviour would be dependent on the
+orientation in space with respect to the earth. For owing to the
+alteration in direction of the velocity of revolution of the earth in
+the course of a year, the earth cannot be at rest relative to the
+hypothetical system $K_0$ throughout the whole year. However, the most
+careful observations have never revealed such anisotropic properties
+in terrestrial physical space, {\it i.e.} a physical non-equivalence of
+different directions. This is very powerful argument in favour of the
+principle of relativity.
+
+
+
+\chapter{The Theorem of the Addition of Velocities Employed in
+Classical Mechanics}
+
+
+Let us suppose our old friend the railway carriage to be travelling
+along the rails with a constant velocity $v$, and that a man traverses
+the length of the carriage in the direction of travel with a velocity
+$w$. How quickly or, in other words, with what velocity $W$ does the man
+advance relative to the embankment during the process? The only
+possible answer seems to result from the following consideration: If
+the man were to stand still for a second, he would advance relative to
+the embankment through a distance $v$ equal numerically to the velocity
+of the carriage. As a consequence of his walking, however, he
+traverses an additional distance $w$ relative to the carriage, and hence
+also relative to the embankment, in this second, the distance w being
+numerically equal to the velocity with which he is walking. Thus in
+total be covers the distance $W=v+w$ relative to the embankment in the
+second considered. We shall see later that this result, which
+expresses the theorem of the addition of velocities employed in
+classical mechanics, cannot be maintained; in other words, the law
+that we have just written down does not hold in reality. For the time
+being, however, we shall assume its correctness.
+
+
+
+\chapter{The Apparent Incompatability of the Law of Propagation of Light
+with the Principle of Relativity}
+
+
+There is hardly a simpler law in physics than that according to which
+light is propagated in empty space. Every child at school knows, or
+believes he knows, that this propagation takes place in straight lines
+with a velocity $c= 300,000$ km./sec. At all events we know with great
+exactness that this velocity is the same for all colours, because if
+this were not the case, the minimum of emission would not be observed
+simultaneously for different colours during the eclipse of a fixed
+star by its dark neighbour. By means of similar considerations based
+on 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 ot a ray of light relative to the carriage
+thus comes cut smaller than $c$.
+
+But this result comes into conflict with the principle of relativity
+set forth in Section V. For, like every other general law of
+nature, the law of the transmission of light in vacuo [in vacuum]
+must, according to the principle of relativity, be the same for the
+railway carriage as reference-body as when the rails are the body of
+reference. But, from our above consideration, this would appear to be
+impossible. If every ray of light is propagated relative to the
+embankment with the velocity $c$, then for this reason it would appear
+that another law of propagation of light must necessarily hold with
+respect to the carriage---a result contradictory to the principle of
+relativity.
+
+In view of this dilemma there appears to be nothing else for it than
+to abandon either the principle of relativity or the simple law of the
+propagation of light in vacuo. Those of you who have carefully
+followed the preceding discussion are almost sure to expect that we
+should retain the principle of relativity, which appeals so
+convincingly to the intellect because it is so natural and simple. The
+law of the propagation of light in vacuo would then have to be
+replaced by a more complicated law conformable to the principle of
+relativity. The development of theoretical physics shows, however,
+that we cannot pursue this course. The epoch-making theoretical
+investigations of H. A. Lorentz on the electrodynamical and optical
+phenomena connected with moving bodies show that experience in this
+domain leads conclusively to a theory of electromagnetic phenomena, of
+which the law of the constancy of the velocity of light in vacuo is a
+necessary consequence. Prominent theoretical physicists were theref
+ore more inclined to reject the principle of relativity, in spite of
+the fact that no empirical data had been found which were
+contradictory to this principle.
+
+At this juncture the theory of relativity entered the arena. As a
+result of an analysis of the physical conceptions of time and space,
+it became evident that \emph{in realily there is not the least
+incompatibilitiy between the principle of relativity and the law of
+propagation of light}, and that by systematically holding fast to both
+these laws a logically rigid theory could be arrived at. This theory
+has been called the \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{On the Idea of Time in Physics}
+
+Lightning has struck the rails on our railway embankment at two places
+A and B far distant from each other. I make the additional assertion
+that these two lightning flashes occurred simultaneously. If I ask you
+whether there is sense in this statement, you will answer my question
+with a decided ``Yes." But if I now approach you with the request to
+explain to me the sense of the statement more precisely, you find
+after some consideration that the answer to this question is not so
+easy as it appears at first sight.
+
+After some time perhaps the following answer would occur to you: ``The
+significance of the statement is clear in itself and needs no further
+explanation; of course it would require some consideration if I were
+to be commissioned to determine by observations whether in the actual
+case the two events took place simultaneously or not." I cannot be
+satisfied with this answer for the following reason. Supposing that as
+a result of ingenious considerations an able meteorologist were to
+discover that the lightning must always strike the places A and B
+simultaneously, then we should be faced with the task of testing
+whether or not this theoretical result is in accordance with the
+reality. We encounter the same difficulty with all physical statements
+in which the conception ``simultaneous'' plays a part. The concept
+does not exist for the physicist until he has the possibility of
+discovering whether or not it is fulfilled in an actual case. We thus
+require a definition of simultaneity such that this definition
+supplies us with the method by means of which, in the present case, he
+can decide by experiment whether or not both the lightning strokes
+occurred simultaneously. As long as this requirement is not satisfied,
+I allow myself to be deceived as a physicist (and of course the same
+applies if I am not a physicist), when I imagine that I am able to
+attach a meaning to the statement of simultaneity. (I would ask the
+reader not to proceed farther until he is fully convinced on this
+point.)
+
+After thinking the matter over for some time you then offer the
+following suggestion with which to test simultaneity. By measuring
+along the rails, the connecting line AB should be measured up and an
+observer placed at the mid-point M of the distance AB. This observer
+should be supplied with an arrangement ({\it e.g.} two mirrors inclined at
+$90^\circ$) which allows him visually to observe both places A and B at the
+same time. If the observer perceives the two flashes of lightning at
+the same time, then they are simultaneous.
+
+I am very pleased with this suggestion, but for all that I cannot
+regard the matter as quite settled, because I feel constrained to
+raise the following objection:
+
+"Your definition would certainly be right, if only I knew that the
+light by means of which the observer at M perceives the lightning
+flashes travels along the length A~$\longrightarrow$~M with the same velocity as
+along the length B~$\longrightarrow$~M. But an examination of this supposition
+would only be possible if we already had at our disposal the means of
+measuring time. It would thus appear as though we were moving here in
+a logical circle."
+
+After further consideration you cast a somewhat disdainful glance at
+me---and rightly so---and you declare:
+
+ ``I maintain my previous definition nevertheless, because in reality it
+assumes absolutely nothing about light. There is only one demand to be
+made of the definition of simultaneity, namely, that in every real
+case it must supply us with an empirical decision as to whether or not
+the conception that has to be defined is fulfilled. That my definition
+satisfies this demand is indisputable. That light requires the same
+time to traverse the path A~$\longrightarrow$~M as for the path B~$\longrightarrow$~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 \emph{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\footnotemark[1] (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 \emph{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 \emph{particular} position of the pointers of
+the one clock is \emph{simultaneous} (in the above sense) with the same
+position, of the pointers of the other clock, then identical ``settings''
+are always simultaneous (in the sense of the above
+definition).
+
+
+% Notes
+
+\footnotetext[1]{We suppose further, that, when three events A, B, and C occur in
+different places in such a manner that A is simultaneous with B and B
+is simultaneous with C (simultaneous in the sense of the above
+definition), then the criterion for the simultaneity of the pair of
+events A, C is also satisfied. This assumption is a physical
+hypothesis about the the of propagation of light: it must certainly be
+fulfilled if we are to maintain the law of the constancy of the
+velocity of light in vacuo.}
+
+
+
+\chapter{The Relativity of Simulatneity}
+
+
+Up to now our considerations have been referred to a particular body
+of reference, which we have styled a ``railway embankment." We suppose
+a very long train travelling along the rails with the constant
+velocity v and in the direction indicated in Fig. \ref{fig:1}. People travelling
+in this train will with a vantage view the train as a rigid
+reference-body (co-ordinate system); they regard all events in
+reference to the train. Then every event which takes place along the
+line also takes place at a particular point of the train. Also the
+definition of simultaneity can be given relative to the train in
+exactly the same way as with respect to the embankment. As a natural
+consequence, however, the following question arises:
+
+% Fig. 01:
+%
+% v ---> M' -----> v ---> Train
+% -------|------------|------------|----/
+% ---------|------------|------------|-------
+% A M B Embankment
+%
+%
+
+\begin{figure}[hbtp]
+
+\centering
+\caption{}
+\label{fig:1}
+
+\begin{picture}(250,100)(0,0)
+\thicklines
+\put(0,40){\line(1,0){250}}
+\put(230,30){Embankment}
+\put(15,50){\line(1,0){210}}
+\put(225,50){\line(1,1){10}}
+\put(235,60){Train}
+\put(40,35){\line(0,1){20}}
+\put(37,23){A}
+\put(125,48){\line(0,1){7}}
+\put(125,35){\line(0,1){7}}
+\put(121,23){M}
+\put(210,35){\line(0,1){20}}
+\put(207,23){B}
+\thinlines
+\put(15,60){$v$}
+\put(22,62){\vector(1,0){25}}
+\put(195,60){$v$}
+\put(202,62){\vector(1,0){25}}
+\put(105,65){M$'$}
+\put(122,67){\vector(1,0){25}}
+\end{picture}
+
+\end{figure}
+
+Are two events ({\it e.g.} the two strokes of lightning A and B) which are
+simultaneous \emph{with reference to the railway embankment} also
+simultaneous \emph{relatively to the train}? We shall show directly that the
+answer must be in the negative.
+
+When we say that the lightning strokes A and B are simultaneous with
+respect to be embankment, we mean: the rays of light emitted at the
+places A and B, where the lightning occurs, meet each other at the
+mid-point M of the length A $\longrightarrow$ 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 $\longrightarrow$ B on the travelling train. Just
+when the flashes (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, {\it 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, {\it i.e.} that it is independent of the state of
+motion of the body of reference. But we have just seen that this
+assumption is incompatible with the most natural definition of
+simultaneity; if we discard this assumption, then the conflict between
+the law of the propagation of light in vacuo and the principle of
+relativity (developed in Section 6) disappears.
+
+We were led to that conflict by the considerations of Section 6,
+which are now no longer tenable. In that section we concluded that the
+man in the carriage, who traverses the distance $w$ \emph{per second} relative
+to the carriage, traverses the same distance also with respect to the
+embankment \emph{in each second} of time. But, according to the foregoing
+considerations, the time required by a particular occurrence with
+respect to the carriage must not be considered equal to the duration
+of the same occurrence as judged from the embankment (as
+reference-body). Hence it cannot be contended that the man in walking
+travels the distance $w$ relative to the railway line in a time which is
+equal to one second as judged from the embankment.
+
+Moreover, the considerations of Section 6 are based on yet a second
+assumption, which, in the light of a strict consideration, appears to
+be arbitrary, although it was always tacitly made even before the
+introduction of the theory of relativity.
+
+
+
+\chapter{On the Relativity of the Conception of Distance}
+
+
+Let us consider two particular points on the train\footnotemark travelling
+along the embankment with the velocity $v$, and inquire as to their
+distance apart. We already know that it is necessary to have a body of
+reference for the measurement of a distance, with respect to which
+body the distance can be measured up. It is the simplest plan to use
+the train itself as reference-body (co-ordinate system). An observer
+in the train measures the interval by marking off his measuring-rod in
+a straight line ({\it 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 $t$---judged from the
+embankment. These points A and B of the embankment can be determined
+by applying the definition of time given in Section 8. The distance
+between these points A and B is then measured by repeated application
+of thee measuring-rod along the embankment.
+
+A priori it is by no means certain that this last measurement will
+supply us with the same result as the first. Thus the length of the
+train as measured from the embankment may be different from that
+obtained by measuring in the train itself. This circumstance leads us
+to a second objection which must be raised against the apparently
+obvious consideration of Section 6. Namely, if the man in the
+carriage covers the distance $w$ in a unit of time---\emph{measured from the
+train},---then this distance--\emph{as measured from the embankment}---is
+not necessarily also equal to $w$.
+
+
+% Notes
+
+\footnotetext{{\it e.g.} the middle of the first and of the hundredth carriage.}
+
+
+\chapter{The Lorentz Transformation}
+
+
+The results of the last three sections show that the apparent
+incompatibility of the law of propagation of light with the principle
+of relativity (Section 7) has been derived by means of a
+consideration which borrowed two unjustifiable hypotheses from
+classical mechanics; these are as follows:
+
+\begin{enumerate}
+\item The time-interval (time) between two events is independent of the
+condition of motion of the body of reference.
+\item The space-interval (distance) between two points of a rigid body
+is independent of the condition of motion of the body of reference.
+\end{enumerate}
+
+If we drop these hypotheses, then the dilemma of Section 7
+disappears, because the theorem of the addition of velocities derived
+in Section 6 becomes invalid. The possibility presents itself that
+the law of the propagation of light in vacuo may be compatible with
+the principle of relativity, and the question arises: How have we to
+modify the considerations of Section 6 in order to remove the
+apparent disagreement between these two fundamental results of
+experience? This question leads to a general one. In the discussion of
+Section 6 we have to do with places and times relative both to the
+train and to the embankment. How are we to find the place and time of
+an event in relation to the train, when we know the place and time of
+the event with respect to the railway embankment? Is there a
+thinkable answer to this question of such a nature that the law of
+transmission of light in vacuo does not contradict the principle of
+relativity? In other words: Can we conceive of a relation between
+place and time of the individual events relative to both
+reference-bodies, such that every ray of light possesses the velocity
+of transmission $c$ relative to the embankment and relative to the train?
+This question leads to a quite definite positive answer, and to a
+perfectly definite transformation law for the space-time magnitudes of
+an event when changing over from one body of reference to another.
+
+% Figure 2
+
+% z'
+% | --->
+% z | y'
+% | | / --->
+% | y | / v
+% | / | / --->
+% | / |/______________x'
+% | / K'
+% |/______________x
+% K
+
+\begin{figure}[hbtp]
+
+\centering
+\caption{}
+\label{fig:2}
+
+\begin{picture}(200,220)(0,0)
+\thicklines
+\put(15,10){$K$}
+\put(20,20){\line(1,0){125}}
+\put(149,17){$x$}
+\put(20,20){\line(0,1){125}}
+\put(17,150){$z$}
+\put(20,20){\line(1,2){40}}
+\put(55,105){$y$}
+
+\put(85,25){$K'$}
+\put(90,35){\line(1,0){125}}
+\put(219,32){$x'$}
+\put(90,35){\line(0,1){125}}
+\put(87,165){$z'$}
+\put(90,35){\line(1,2){40}}
+\put(125,120){$y'$}
+
+\thinlines
+\put(95,155){\vector(1,0){35}}
+\put(135,110){\vector(1,0){35}}
+\put(110,40){\vector(1,0){35}}
+\end{picture}
+
+\end{figure}
+
+Before we deal with this, we shall introduce the following incidental
+consideration. Up to the present we have only considered events taking
+place along the embankment, which had mathematically to assume the
+function of a straight line. In the manner indicated in Section 2
+we can imagine this reference-body supplemented laterally and in a
+vertical direction by means of a framework of rods, so that an event
+which takes place anywhere can be localised with reference to this
+framework. Similarly, we can imagine the train travelling with
+the velocity $v$ to be continued across the whole of space, so that
+every event, no matter how far off it may be, could also be localised
+with respect to the second framework. Without committing any
+fundamental error, we can disregard the fact that in reality these
+frameworks would continually interfere with each other, owing to the
+impenetrability of solid bodies. In every such framework we imagine
+three surfaces perpendicular to each other marked out, and designated
+as ``co-ordinate planes" (``co-ordinate system"). A co-ordinate
+system $K$ then corresponds to the embankment, and a co-ordinate system
+$K'$ to the train. An event, wherever it may have taken place, would be
+fixed in space with respect to $K$ by the three perpendiculars $x, y, z$
+on the co-ordinate planes, and with regard to time by a time value $t$.
+Relative to $K'$, 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 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 \ref{fig:2}), this problem is solved by means of the
+equations:
+
+\begin{eqnarray*}
+x' &=& \frac{x-vt}{\sqrt{I-\frac{v^2}{c^2}}} \\
+y' &=& y \\
+z' &=& z \\
+t' &=& \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}} \\
+\end{eqnarray*}
+
+\noindent This system of equations is known as the ``Lorentz transformation."\footnotemark
+
+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*}
+x' &=& x - vt \\
+y' &=& y \\
+z' &=& z \\
+t' &=& t \\
+\end{eqnarray*}
+
+\noindent This system of equations is often termed the ``Galilei
+transformation." The Galilei transformation can be obtained from the
+Lorentz transformation by substituting an infinitely large value for
+the velocity of light $c$ in the latter transformation.
+
+Aided by the following illustration, we can readily see that, in
+accordance with the Lorentz transformation, the law of the
+transmission of light in vacuo is satisfied both for the
+reference-body $K$ and for the reference-body $K'$. A light-signal is sent
+along the positive $x$-axis, and this light-stimulus advances in
+accordance with the equation
+
+ $$x = ct,$$
+
+\noindent {\it 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{I-\frac{v^2}{c^2}}} \\
+t' &=& \frac{(I-\frac{v}{c})t}{\sqrt{I-\frac{v^2}{c^2}}}
+\end{eqnarray*}
+
+\noindent from which, by division, the expression
+
+ $$x' = ct'$$
+
+\noindent 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 cause this is not surprising, since
+the equations of the Lorentz transformation were derived conformably
+to this point of view.
+
+
+% Notes
+
+\footnotetext{A simple derivation of the Lorentz transformation is given in
+Appendix I.}
+
+
+
+\chapter{The Behaviour of Measuring-Rods and Clocks in Motion}
+
+
+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 $x'=I$. What is the length
+of the metre-rod relatively to the system $K$? In order to learn this,
+we need only ask where the beginning of the rod and the end of the rod
+lie with respect to $K$ at a particular time $t$ of the system $K$. By means
+of the first equation of the Lorentz transformation the values of
+these two points at the time $t = 0$ can be shown to be
+
+\begin{eqnarray*}
+x_{\mbox{(begining of rod)}} &=& 0 \overline{\sqrt{I-\frac{v^2}{c^2}}} \\
+x_{\mbox{(end of rod)}} &=& 1 \overline{\sqrt{I-\frac{v^2}{c^2}}}
+\end{eqnarray*}
+~
+
+\noindent the distance between the points being $\sqrt{I-v^2/c^2}$.
+
+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 $\sqrt{I-v^2/c^2}$ 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 $\sqrt{I-v^2/c^2} = 0$,
+and for stiII greater velocities the square-root becomes imaginary.
+From this we conclude that in the theory of relativity the velocity $c$
+plays the part of a limiting velocity, which can neither be reached
+nor exceeded by any real body.
+
+Of course this feature of the velocity $c$ as a limiting velocity also
+clearly follows from the equations of the Lorentz transformation, for
+these became meaningless if we choose values of $v$ greater than $c$.
+
+If, on the contrary, we had considered a metre-rod at rest in the
+$x$-axis with respect to $K$, then we should have found that the length of
+the rod as judged from $K'$ would have been $\sqrt{I-v^2/c^2}$;
+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 $z, y, x, t$, are
+nothing more nor less than the results of measurements obtainable by
+means of measuring-rods and clocks. If we had based our considerations
+on the Galileian transformation we should not have obtained a
+contraction of the rod as a consequence of its motion.
+
+Let us now consider a seconds-clock which is permanently situated at
+the origin ($x'=0$) of $K'$. $t'=0$ and $t'=I$ 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
+
+$$t' = \frac{I}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+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
+
+$$\frac{I}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+\noindent seconds, {\it 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{Theorem of the Addition of Velocities.\\
+The Experiment of Fizeau}
+
+
+Now in practice we can move clocks and measuring-rods only with
+velocities that are small compared with the velocity of light; hence
+we shall hardly be able to compare the results of the previous section
+directly with the reality. But, on the other hand, these results must
+strike you as being very singular, and for that reason I shall now
+draw another conclusion from the theory, one which can easily be
+derived from the foregoing considerations, and which has been most
+elegantly confirmed by experiment.
+
+In Section 6 we derived the theorem of the addition of velocities
+in one direction in the form which also results from the hypotheses of
+classical mechanics---This theorem can also be deduced readily horn the
+Galilei transformation (Section 11). In place of the man walking
+inside the carriage, we introduce a point moving relatively to the
+co-ordinate system $K'$ 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 6,
+\begin{equation}
+W=v+w
+\label{eqnA}
+\end{equation}
+
+But we can carry out this consideration just as well on the basis of
+the theory of relativity. In the equation
+\begin{equation}
+x'=wt'
+\label{eqnB}
+\end{equation}
+
+\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{eqnA} we then obtain the equation
+
+$$W = \frac{v+w}{I+\frac{vw}{c^2}}$$
+~
+
+\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 axe enlightened by a most important
+experiment which the brilliant physicist Fizeau performed more than
+half a century ago, and which has been repeated since then by some of
+the best experimental physicists, so that there can be no doubt about
+its result. The experiment is concerned with the following question.
+Light travels in a motionless liquid with a particular velocity $w$. How
+quickly does it travel in the direction of the arrow in the tube T
+(see the accompanying diagram, Figure \ref{fig:3}) when the liquid above
+mentioned is flowing through the tube with a velocity $v$?
+
+% Figure 3
+%
+% T
+% /
+% --------------------------------------
+% v --------->
+% --------------------------------------
+%
+
+
+\begin{figure}[hbtp]
+
+\centering
+\caption{}
+\label{fig:3}
+
+\begin{picture}(200,75)(0,0)
+\thicklines
+\put(0,15){\line(1,0){200}}
+\put(0,35){\line(1,0){200}}
+\put(100,35){\line(1,3){5}}
+\put(107,52){T}
+
+\thinlines
+\put(40,25){\vector(1,0){50}}
+\put(60,26){$v$}
+\end{picture}
+
+\end{figure}
+
+
+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 \emph{with respect to the liquid}, whether the
+latter is in motion with reference to other bodies or not. The
+velocity of light relative to the liquid and the velocity of the
+latter relative to the tube are thus known, and we require the
+velocity of light relative to the tube.
+
+It is clear that we have the problem of Section 6 again before us. The
+tube plays the part of the railway embankment or of the co-ordinate
+system $K$, the liquid plays the part of the carriage or of the
+co-ordinate system $K'$, 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{eqnA} or \ref{eqnB}, according as the
+Galilei transformation or the Lorentz transformation corresponds to
+the facts. Experiment\footnotemark decides in favour of equation \ref{eqnB} 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{eqnB} to within one per cent.
+
+Nevertheless we must now draw attention to the fact that a theory of
+this phenomenon was given by H. A. Lorentz long before the statement
+of the theory of relativity. This theory was of a purely
+electrodynamical nature, and was obtained by the use of particular
+hypotheses as to the electromagnetic structure of matter. This
+circumstance, however, does not in the least diminish the
+conclusiveness of the experiment as a crucial test in favour of the
+theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
+which the original theory was based, in no way opposes the theory of
+relativity. Rather has the latter been developed trom electrodynamics
+as an astoundingly simple combination and generalisation of the
+hypotheses, formerly independent of each other, on which
+electrodynamics was built.
+
+
+% Notes
+
+\footnotetext{Fizeau found $W=w+v\left(I-\frac{I}{n^2}\right)$, where $n=\frac{c}{w}$
+is the index of refraction of the liquid. On the other hand, owing to
+the smallness of $\frac{vw}{c^2}$ as compared with $I$,
+we can replace (B) in the first place by $W=(w+v)\left(I-\frac{vw}{c^2}\right)$, or to the same order
+of approximation by
+$w+v\left(I-\frac{I}{n^2}\right)$, which agrees with Fizeau's result.}
+
+
+
+\chapter{The Heuristic Value of the Theory of Relativity}
+
+
+Our train of thought in the foregoing pages can be epitomised in the
+following manner. Experience has led to the conviction that, on the
+one hand, the principle of relativity holds true and that on the other
+hand the velocity of transmission of light in vacuo has to be
+considered equal to a constant $c$. By uniting these two postulates we
+obtained the law of transformation for the rectangular co-ordinates $x,
+y, z$ and the time $t$ of the events which constitute the processes of
+nature. In this connection we did not obtain the Galilei
+transformation, but, differing from classical mechanics, the \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 coordinate 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{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, {\it i.e.} the
+derivation of laws, and---what is incomparably more important---it
+has considerably reduced the number of independent hypothese forming
+the basis of theory. The special theory of relativity has rendered the
+Maxwell-Lorentz theory so plausible, that the latter would have been
+generally accepted by physicists even if experiment had decided less
+unequivocally in its favour.
+
+Classical mechanics required to be modified before it could come into
+line with the demands of the special theory of relativity. For the
+main part, however, this modification affects only the laws for rapid
+motions, in which the velocities of matter $v$ are not very small as
+compared with the velocity of light. We have experience of such rapid
+motions only in the case of electrons and ions; for other motions the
+variations from the laws of classical mechanics are too small to make
+themselves evident in practice. We shall not consider the motion of
+stars until we come to speak of the general theory of relativity. In
+accordance with the theory of relativity the kinetic energy of a
+material point of mass m is no longer given by the well-known
+expression
+
+$$m\frac{v^2}{2}$$
+
+\noindent but by the expression
+
+$$\frac{mc^2}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+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, we obtain
+
+$$mc^2 + m\frac{v^2}{2} + \frac{3}{8} m \frac{v^4}{c^2} + \cdots$$
+~
+
+When $v^2/c^2$ is small compared with unity, the third of these terms is
+always small in comparison with the second,
+which last is alone considered in classical mechanics. The first term
+$mc^2$ does not contain the velocity, and requires no consideration if
+we are only dealing with the question as to how the energy of a
+point-mass; depends on the velocity. We shall speak of its essential
+significance later.
+
+The most important result of a general character to which the special
+theory of relativity has led is concerned with the conception of mass.
+Before the advent of relativity, physics recognised two conservation
+laws of fundamental importance, namely, the law of the canservation of
+energy and the law of the conservation of mass these two fundamental
+laws appeared to be quite independent of each other. By means of the
+theory of relativity they have been united into one law. We shall now
+briefly consider how this unification came about, and what meaning is
+to be attached to it.
+
+The principle of relativity requires that the law of the concervation
+of energy should hold not only with reference to a co-ordinate system
+$K$, but also with respect to every co-ordinate system $K'$ 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\footnotemark\ 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
+
+$$\frac{E_0}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+In consideration of the expression given above for the kinetic energy
+of the body, the required energy of the body comes out to be
+
+$$\frac{\left(m+\frac{E_0}{c^2}\right)c^2}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+\noindent Thus the body has the same energy as a body of mass
+
+$$\left(m+\frac{E_0}{c^2}\right)$$
+~
+
+\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
+
+$$\frac{mc^2+E_0}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+\noindent we see that the term $mc^2$, which has hitherto attracted our attention,
+is nothing else than the energy possessed by the body\footnotemark\ before it
+absorbed the energy $E_0$.
+
+A direct comparison of this relation with experiment is not possible
+at the present time (1920; see\footnotemark\ Note, p. 48), owing to the fact that
+the changes in energy E[0] to which we can Subject a system are not
+large enough to make themselves perceptible as a change in the
+inertial mass of the system.
+
+$$\frac{E_0}{c^2}$$
+~
+
+\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 role in this theory. In Part II we shall see in
+what way this result becomes modified in the general theory of
+relativity.
+
+
+% Notes
+
+\footnotetext[1]{$E_0$ is the energy taken up, as judged from a co-ordinate system
+moving with the body.}
+
+\footnotetext[2]{As judged from a co-ordinate system moving with the body.}
+
+\footnotetext[3]{The equation $E = mc^2$ has been thoroughly proved time and
+again since this time.}
+
+
+
+\chapter{Experience and the Special Theory of Relativity}
+
+
+To what extent is the special theory of relativity supported by
+experience? This question is not easily answered for the reason
+already mentioned in connection with the fundamental experiment of
+Fizeau. The special theory of relativity has crystallised out from the
+Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
+experience which support the electromagnetic theory also support the
+theory of relativity. As being of particular importance, I mention
+here the fact that the theory of relativity enables us to predict the
+effects produced on the light reaching us from the fixed stars. These
+results are obtained in an exceedingly simple manner, and the effects
+indicated, which are due to the relative motion of the earth with
+reference to those fixed stars are found to be in accord with
+experience. We refer to the yearly movement of the apparent position
+of the fixed stars resulting from the motion of the earth round the
+sun (aberration), and to the influence of the radial components of the
+relative motions of the fixed stars with respect to the earth on the
+colour of the light reaching us from them. The latter effect manifests
+itself in a slight displacement of the spectral lines of the light
+transmitted to us from a fixed star, as compared with the position of
+the same spectral lines when they are produced by a terrestrial source
+of light (Doppler principle). The experimental arguments in favour of
+the Maxwell-Lorentz theory, which are at the same time arguments in
+favour of the theory of relativity, are too numerous to be set forth
+here. In reality they limit the theoretical possibilities to such an
+extent, that no other theory than that of Maxwell and Lorentz has been
+able to hold its own when tested by experience.
+
+But there are two classes of experimental facts hitherto obtained
+which can be represented in the Maxwell-Lorentz theory only by the
+introduction of an auxiliary hypothesis, which in itself---{\it i.e.}
+without making use of the theory of relativity---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.\footnotemark\ If
+we now assume that the relative distances between the electrical
+masses constituting the electron remain unchanged during the motion of
+the electron (rigid connection in the sense of classical mechanics),
+we arrive at a law of motion of the electron which does not agree with
+experience. Guided by purely formal points of view, H. A. Lorentz was
+the first to introduce the hypothesis that the form of the electron
+experiences a contraction in the direction of motion in consequence of
+that motion. the contracted length being proportional to the
+expression
+
+$$\overline{\sqrt{I-\frac{v^2}{c^2}}}.$$
+
+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 13 in connection with the experiment of Fizeau, the result
+of which is foretold by the theory of relativity without the necessity
+of drawing on hypotheses as to the physical nature of the liquid.
+
+The second class of facts to which we have alluded has reference to
+the question whether or not the motion of the earth in space can be
+made perceptible in terrestrial experiments. We have already remarked
+in Section 5 that all attempts of this nature led to a negative
+result. Before the theory of relativity was put forward, it was
+difficult to become reconciled to this negative result, for reasons
+now to be discussed. The inherited prejudices about time and space did
+not allow any doubt to arise as to the prime importance of the
+Galileian transformation for changing over from one body of reference
+to another. Now assuming that the Maxwell-Lorentz equations hold for a
+reference-body $K$, we then find that they do not hold for a
+reference-body $K'$ moving uniformly with respect to $K$, if we assume
+that the relations of the Galileian transformstion 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 coordinate 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 {\ae}ther (``{\ae}ther-drift'' relative to $K'$) were
+attributed the more complicated laws which were supposed to hold
+relative to $K'$. Strictly speaking, such an {\ae}ther-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
+{\ae}ther-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 {\ae}ther. And yet another point: it is shown by calculation that
+for a given velocity v with reference to the {\ae}ther, 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---a
+fact very perplexing to physicists. Lorentz and FitzGerald rescued the
+theory from this difficulty by assuming that the motion of the body
+relative to the æther produces a contraction of the body in the
+direction of motion, the amount of contraction being just sufficient
+to compensate for the differeace in time mentioned above. Comparison
+with the discussion in Section 11 shows that also from the
+standpoint of the theory of relativity this solution of the difficulty
+was the right one. But on the basis of the theory of relativity the
+method of interpretation is incomparably more satisfactory. According
+to this theory there is no such thing as a ``specially favoured''
+(unique) co-ordinate system to occasion the introduction of the
+æther-idea, and hence there can be no æther-drift, nor any experiment
+with which to demonstrate it. Here the contraction of moving bodies
+follows from the two fundamental principles of the theory, without the
+introduction of particular hypotheses; and as the prime factor
+involved in this contraction we find, not the motion in itself, to
+which we cannot attach any meaning, but the motion with respect to the
+body of reference chosen in the particular case in point. Thus for a
+co-ordinate system moving with the earth the mirror system of
+Michelson and Morley is not shortened, but it is shortened for a
+co-ordinate system which is at rest relatively to the sun.
+
+
+% Notes
+
+\footnotetext{The general theory of relativity renders it likely that the
+electrical masses of an electron are held together by gravitational
+forces.}
+
+
+
+\chapter{Minkowski's Four-Dimensional Space}
+
+
+The non-mathematician is seized by a mysterious shuddering when he
+hears of ``four-dimensional" things, by a feeling not unlike that
+awakened by thoughts of the occult. And yet there is no more
+common-place statement than that the world in which we live is a
+four-dimensional space-time continuum.
+
+Space is a three-dimensional continuum. By this we mean that it is
+possible to describe the position of a point (at rest) by means of
+three numbers (co-ordinales) $x, y, z$, and that there is an indefinite
+number of points in the neighbourhood of this one, the position of
+which can be described by co-ordinates such as $x_1, y_1, z_1$, which
+may be as near as we choose to the respective values of the
+co-ordinates $x, y, z$, of the first point. In virtue of the latter
+property we speak of a ``continuum," and owing to the fact that there
+are three co-ordinates we speak of it as being ``three-dimensional."
+
+Similarly, the world of physical phenomena which was briefly called
+ ``world" by Minkowski is naturally four dimensional in the space-time
+sense. For it is composed of individual events, each of which is
+described by four numbers, namely, three space co-ordinates $x, y, z$,
+and a time co-ordinate, the time value $t$. The ``world" is in this sense
+also a continuum; for to every event there are as many ``neighbouring"
+events (realised or at least thinkable) as we care to choose, the
+co-ordinates $x_1, y_1, z_1, t_1$ of which differ by an indefinitely
+small amount from those of the event $x, y, z, t$ originally considered.
+That we have not been accustomed to regard the world in this sense as
+a four-dimensional continuum is due to the fact that in physics,
+before the advent of the theory of relativity, time played a different
+and more independent role, as compared with the space coordinates. It
+is for this reason that we have been in the habit of treating time as
+an independent continuum. As a matter of fact, according to classical
+mechanics, time is absolute, {\it 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
+on the theory of relativity, since according to this theory time is
+robbed of its independence. This is shown by the fourth equation of
+the Lorentz transformation:
+
+$$t' = \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}}$$
+~
+
+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.\footnotemark \ In order to give due prominence to this relationship,
+however, we must replace the usual time co-ordinate $t$ by an imaginary
+magnitude $\sqrt{-I}ct$ 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 role as the three space co-ordinates. Formally,
+these four co-ordinates correspond exactly to the three space
+co-ordinates in Euclidean geometry. It must be clear even to the
+non-mathematician that, as a consequence of this purely formal
+addition to our knowledge, the theory perforce gained clearness in no
+mean measure.
+
+These inadequate remarks can give the reader only a vague notion of
+the important idea contributed by Minkowski. Without it the general
+theory of relativity, of which the fundamental ideas are developed in
+the following pages, would perhaps have got no farther than its long
+clothes. Minkowski's work is doubtless difficult of access to anyone
+inexperienced in mathematics, but since it is not necessary to have a
+very exact grasp of this work in order to understand the fundamental
+ideas of either the special or the general theory of relativity, I
+shall leave it here at present, and revert to it only towards the end
+of Part 2.
+
+
+% Notes
+
+\footnotetext{Cf. the somewhat more detailed discussion in Appendix II.}
+
+
+
+
+%PART II
+
+\part{Then General Theory of Relativity}
+
+\chapter{Special and General Principle of Relativity}
+
+The basal principle, which was the pivot of all our previous
+considerations, was the special principle of relativity, \emph{{\it i.e.}} the
+principle of the physical relativity of all \emph{uniform} motion. Let as
+once more analyse its meaning carefully.
+
+It was at all times clear that, from the point of view of the idea it
+conveys to us, every motion must be considered only as a relative
+motion. Returning to the illustration we have frequently used of the
+embankment and the railway carriage, we can express the fact of the
+motion here taking place in the following two forms, both of which are
+equally justifiable:
+
+\begin{enumerate}
+\item The carriage is in motion relative to the embankment,
+\item The embankment is in motion relative to the carriage.
+\end{enumerate}
+
+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
+
+\begin{enumerate}
+\item the embankment as reference-body,
+\item the railway carriage as reference-body,
+\end{enumerate}
+
+\noindent then these general laws of nature ({\it e.g.} the laws of mechanics or the
+law of the propagation of light in vacuo) have exactly the same form
+in both cases. This can also be expressed as follows: For the
+physical description of natural processes, neither of the reference
+bodies $K$, $K'$ is unique (lit. ``specially marked out'') as compared
+with the other. Unlike the first, this latter statement need not of
+necessity hold a priori; it is not contained in the conceptions of
+``motion" and ``reference-body'' and derivable from them; only
+experience can decide as to its correctness or incorrectness.
+
+Up to the present, however, we have by no means maintained the
+equivalence of all bodies of reference $K$ in connection with the
+formulation of natural laws. Our course was more on the following
+Iines. In the first place, we started out from the assumption that
+there exists a reference-body $K$, whose condition of motion is such
+that the Galileian law holds with respect to it: A particle left to
+itself and sufficiently far removed from all other particles moves
+uniformly in a straight line. With reference to $K$ (Galileian
+reference-body) the laws of nature were to be as simple as possible.
+But in addition to $K$, all bodies of reference $K'$ 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 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 ({\it e.g.}
+those possessing motion of a different kind). In this sense we speak
+of the special principle of relativity, or special theory of
+relativity.
+
+In contrast to this we wish to understand by the ``general principle of
+relativity" the following statement: All bodies of reference $K$, $K'$,
+etc., are equivalent for the description of natural phenomena
+(formulation of the general laws of nature), whatever may be their
+state of motion. But before proceeding farther, it ought to be pointed
+out that this formulation must be replaced later by a more abstract
+one, for reasons which will become evident at a later stage.
+
+Since the introduction of the special principle of relativity has been
+justified, every intellect which strives after generalisation must
+feel the temptation to venture the step towards the general principle
+of relativity. But a simple and apparently quite reliable
+consideration seems to suggest that, for the present at any rate,
+there is little hope of success in such an attempt; Let us imagine
+ourselves transferred to our old friend the railway carriage, which is
+travelling at a uniform rate. As long as it is moving unifromly, the
+occupant of the carriage is not sensible of its motion, and it is for
+this reason that he can without reluctance interpret the facts of the
+case as indicating that the carriage is at rest, but the embankment in
+motion. Moreover, according to the special principle of relativity,
+this interpretation is quite justified also from a physical point of
+view.
+
+If the motion of the carriage is now changed into a non-uniform
+motion, as for instance by a powerful application of the brakes, then
+the occupant of the carriage experiences a correspondingly powerful
+jerk forwards. The retarded motion is manifested in the mechanical
+behaviour of bodies relative to the person in the railway carriage.
+The mechanical behaviour is different from that of the case previously
+considered, and for this reason it would appear to be impossible that
+the same mechanical laws hold relatively to the non-uniformly moving
+carriage, as hold with reference to the carriage when at rest or in
+uniform motion. At all events it is clear that the Galileian law does
+not hold with respect to the non-uniformly moving carriage. Because of
+this, we feel compelled at the present juncture to grant a kind of
+absolute physical reality to non-uniform motion, in opposition to the
+general principle of relatvity. But in what follows we shall soon see
+that this conclusion cannot be maintained.
+
+
+\chapter{The Gravitational Field}
+
+``f we pick up a stone and then let it go, why does it fall to the
+ground?" The usual answer to this question is: ``Because it is
+attracted by the earth." Modern physics formulates the answer rather
+differently for the following reason. As a result of the more careful
+study of electromagnetic phenomena, we have come to regard action at a
+distance as a process impossible without the intervention of some
+intermediary medium. If, for instance, a magnet attracts a piece of
+iron, we cannot be content to regard this as meaning that the magnet
+acts directly on the iron through the intermediate empty space, but we
+are constrained to imagine---after the manner of Faraday---that the
+magnet always calls into being something physically real in the space
+around it, that something being what we call a ``magnetic field." In
+its turn this magnetic field operates on the piece of iron, so that
+the latter strives to move towards the magnet. We shall not discuss
+here the justification for this incidental conception, which is indeed
+a somewhat arbitrary one. We shall only mention that with its aid
+electromagnetic phenomena can be theoretically represented much more
+satisfactorily than without it, and this applies particularly to the
+transmission of electromagnetic waves. The effects of gravitation also
+are regarded in an analogous manner.
+
+The action of the earth on the stone takes place indirectly. The earth
+produces in its surrounding a gravitational field, which acts on the
+stone and produces its motion of fall. As we know from experience, the
+intensity of the action on a body dimishes according to a quite
+definite law, as we proceed farther and farther away from the earth.
+From our point of view this means: The law governing the properties
+of the gravitational field in space must be a perfectly definite one,
+in order correctly to represent the diminution of gravitational action
+with the distance from operative bodies. It is something like this:
+The body ({\it e.g.} the earth) produces a field in its immediate
+neighbourhood directly; the intensity and direction of the field at
+points farther removed from the body are thence determined by the law
+which governs the properties in space of the gravitational fields
+themselves.
+
+In contrast to electric and magnetic fields, the gravitational field
+exhibits a most remarkable property, which is of fundamental
+importance for what follows. Bodies which are moving under the sole
+influence of a gravitational field receive an acceleration, which does
+not in the least depend either on the material or on the physical
+state of the body. For instance, a piece of lead and a piece of wood
+fall in exactly the same manner in a gravitational field (in vacuo),
+when they start off from rest or with the same initial velocity. This
+law, which holds most accurately, can be expressed in a different form
+in the light of the following consideration.
+
+According to Newton's law of motion, we have
+
+\begin{center}
+(Force) = (inertial mass) $\times$ (acceleration),
+\end{center}
+
+\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
+
+\begin{center}
+(Force) = (gravitational mass) $\times$ (intensity of the gravitational
+field),
+\end{center}
+
+\noindent where the ``gravitational mass" is likewise a characteristic constant
+for the body. From these two relations follows:
+
+$$\mbox{(acceleration)} = \frac{\mbox{gravitational mass}}{\mbox{inertial mass}}
+ \times \mbox{intensity of the gravitational field}$$
+~
+
+If now, as we find from experience, the acceleration is to be
+independent of the nature and the condition of the body and always the
+same for a given gravitational field, then the ratio of the
+gravitational to the inertial mass must likewise be the same for all
+bodies. By a suitable choice of units we can thus make this ratio
+equal to unity. We then have the following law: The gravitational mass
+of a body is equal to its inertial law.
+
+It is true that this important law had hitherto been recorded in
+mechanics, but it had not been interpreted. A satisfactory
+interpretation can be obtained only if we recognise the following fact:
+The same quality of a body manifests itself according to
+circumstances as ``inertia'' or as ``weight'' (lit. ``heaviness''). In
+the following section we shall show to what extent this is actually
+the case, and how this question is connected with the general
+postulate of relativity.
+
+
+
+\chapter{The Equality of Inertial and Gravitational Mass
+as an Argument for the General Postule of Relativity}
+
+We imagine a large portion of empty space, so far removed from stars
+and other appreciable masses, that we have before us approximately the
+conditions required by the fundamental law of Galilei. It is then
+possible to choose a Galileian reference-body for this part of space
+(world), relative to which points at rest remain at rest and points in
+motion continue permanently in uniform rectilinear motion. As
+reference-body let us imagine a spacious chest resembling a room with
+an observer inside who is equipped with apparatus. Gravitation
+naturally does not exist for this observer. He must fasten himself
+with strings to the floor, otherwise the slightest impact against the
+floor will cause him to rise slowly towards the ceiling of the room.
+
+To the middle of the lid of the chest is fixed externally a hook with
+rope attached, and now a ``being'' (what kind of a being is immaterial
+to us) begins pulling at this with a constant force. The chest
+together with the observer then begin to move ``upwards" with a
+uniformly accelerated motion. In course of time their velocity will
+reach unheard-of values---provided that we are viewing all this from
+another reference-body which is not being pulled with a rope.
+
+But how does the man in the chest regard the Process? The
+acceleration of the chest will be transmitted to him by the reaction
+of the floor of the chest. He must therefore take up this pressure by
+means of his legs if he does not wish to be laid out full length on
+the floor. He is then standing in the chest in exactly the same way as
+anyone stands in a room of a home on our earth. If he releases a body
+which he previously had in his land, the accelertion of the chest will
+no longer be transmitted to this body, and for this reason the body
+will approach the floor of the chest with an accelerated relative
+motion. The observer will further convince himself that the
+acceleration of the body towards the floor of the chest is always of
+the same magnitude, whatever kind of body he may happen to use for the
+experiment.
+
+Relying on his knowledge of the gravitational field (as it was
+discussed in the preceding section), the man in the chest will thus
+come to the conclusion that he and the chest are in a gravitational
+field which is constant with regard to time. Of course he will be
+puzzled for a moment as to why the chest does not fall in this
+gravitational field. just then, however, he discovers the hook in the
+middle of the lid of the chest and the rope which is attached to it,
+and he consequently comes to the conclusion that the chest is
+suspended at rest in the gravitational field.
+
+Ought we to smile at the man and say that he errs in his conclusion?
+I do not believe we ought to if we wish to remain consistent; we must
+rather admit that his mode of grasping the situation violates neither
+reason nor known mechanical laws. Even though it is being accelerated
+with respect to the ``Galileian space" first considered, we can
+nevertheless regard the chest as being at rest. We have thus good
+grounds for extending the principle of relativity to include bodies of
+reference which are accelerated with respect to each other, and as a
+result we have gained a powerful argument for a generalised postulate
+of relativity.
+
+We must note carefully that the possibility of this mode of
+interpretation rests on the fundamental property of the gravitational
+field of giving all bodies the same acceleration, or, what comes to
+the same thing, on the law of the equality of inertial and
+gravitational mass. If this natural law did not exist, the man in the
+accelerated chest would not be able to interpret the behaviour of the
+bodies around him on the supposition of a gravitational field, and he
+would not be justified on the grounds of experience in supposing his
+reference-body to be ``at rest."
+
+Suppose that the man in the chest fixes a rope to the inner side of
+the lid, and that he attaches a body to the free end of the rope. The
+result of this will be to strech the rope so that it will hang
+``vertically'' downwards. If we ask for an opinion of the cause of
+tension in the rope, the man in the chest will say: ``The suspended
+body experiences a downward force in the gravitational field, and this
+is neutralised by the tension of the rope; what determines the
+magnitude of the tension of the rope is the gravitational mass of the
+suspended body." On the other hand, an observer who is poised freely
+in space will interpret the condition of things thus: ``The rope must
+perforce take part in the accelerated motion of the chest, and it
+transmits this motion to the body attached to it. The tension of the
+rope is just large enough to effect the acceleration of the body. That
+which determines the magnitude of the tension of the rope is the
+inertial mass of the body." Guided by this example, we see that our
+extension of the principle of relativity implies the necessity of the
+law of the equality of inertial and gravitational mass. Thus we have
+obtained a physical interpretation of this law.
+
+From our consideration of the accelerated chest we see that a general
+theory of relativity must yield important results on the laws of
+gravitation. In point of fact, the systematic pursuit of the general
+idea of relativity has supplied the laws satisfied by the
+gravitational field. Before proceeding farther, however, I must warn
+the reader against a misconception suggested by these considerations.
+A gravitational field exists for the man in the chest, despite the
+fact that there was no such field for the co-ordinate system first
+chosen. Now we might easily suppose that the existence of a
+gravitational field is always only an apparent one. We might also
+think that, regardless of the kind of gravitational field which may be
+present, we could always choose another reference-body such that no
+gravitational field exists with reference to it. This is by no means
+true for all gravitational fields, but only for those of quite special
+form. It is, for instance, impossible to choose a body of reference
+such that, as judged from it, the gravitational field of the earth (in
+its entirety) vanishes.
+
+We can now appreciate why that argument is not convincing, which we
+brought forward against the general principle of relativity at theend
+of Section 18. It is certainly true that the observer in the
+railway carriage experiences a jerk forwards as a result of the
+application of the brake, and that he recognises, in this the
+non-uniformity of motion (retardation) of the carriage. But he is
+compelled by nobody to refer this jerk to a ``real ``acceleration
+(retardation) of the carriage. He might also interpret his experience
+thus: ``My body of reference (the carriage) remains permanently at
+rest. With reference to it, however, there exists (during the period
+of application of the brakes) a gravitational field which is directed
+forwards and which is variable with respect to time. Under the
+influence of this field, the embankment together with the earth moves
+non-uniformly in such a manner that their original velocity in the
+backwards direction is continuously reduced."
+
+
+
+\chapter{In What Respects Are the Foundations of Classical Mechanics and of the
+Special Theory of Relativity Unsatisfactory?}
+
+
+We have already stated several times that classical mechanics starts
+out from the following law: Material particles sufficiently far
+removed from other material particles continue to move uniformly in a
+straight line or continue in a state of rest. We have also repeatedly
+emphasised that this fundamental law can only be valid for bodies of
+reference $K$ which possess certain unique states of motion, and which
+are in uniform translational motion relative to each other. Relative
+to other reference-bodies $K$ the law is not valid. Both in classical
+mechanics and in the special theory of relativity we therefore
+differentiate between reference-bodies $K$ relative to which the
+recognised ``laws of nature'' can be said to hold, and
+reference-bodies $K$ relative to which these laws do not hold.
+
+But no person whose mode of thought is logical can rest satisfied with
+this condition of things. He asks: ``How does it come that certain
+reference-bodies (or their states of motion) are given priority over
+other reference-bodies (or their states of motion)? What is the
+reason for this Preference?'' In order to show clearly what I mean by
+this question, I shall make use of a comparison.
+
+I am standing in front of a gas range. Standing alongside of each
+other on the range are two pans so much alike that one may be mistaken
+for the other. Both are half full of water. I notice that steam is
+being emitted continuously from the one pan, but not from the other. I
+am surprised at this, even if I have never seen either a gas range or
+a pan before. But if I now notice a luminous something of bluish
+colour under the first pan but not under the other, I cease to be
+astonished, even if I have never before seen a gas flame. For I can
+only say that this bluish something will cause the emission of the
+steam, or at least possibly it may do so. If, however, I notice the
+bluish something in neither case, and if I observe that the one
+continuously emits steam whilst the other does not, then I shall
+remain astonished and dissatisfied until I have discovered some
+circumstance to which I can attribute the different behaviour of the
+two pans.
+
+Analogously, I seek in vain for a real something in classical
+mechanics (or in the special theory of relativity) to which I can
+attribute the different behaviour of bodies considered with respect to
+the reference systems $K$ and $K$.\footnotemark\ Newton saw this objection and
+attempted to invalidate it, but without success. But E. Mach recognsed
+it most clearly of all, and because of this objection he claimed that
+mechanics must be placed on a new basis. It can only be got rid of by
+means of a physics which is conformable to the general principle of
+relativity, since the equations of such a theory hold for every body
+of reference, whatever may be its state of motion.
+
+
+% Notes
+
+\footnotetext{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, {\it e.g.} in the case when
+the reference-body is rotating uniformly.}
+
+
+
+\chapter{A Few Inferences from the General Principle of Relativity}
+
+The considerations of Section 20 show that the general principle of
+relativity puts us in a position to derive properties of the
+gravitational field in a purely theoretical manner. Let us suppose,
+for instance, that we know the space-time ``course'' for any natural
+process whatsoever, as regards the manner in which it takes place in
+the Galileian domain relative to a Galileian body of reference $K$. By
+means of purely theoretical operations ({\it 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.\footnotemark
+
+In the second place our result shows that, according to the general
+theory of relativity, the law of the constancy of the velocity of
+light in vacuo, which constitutes one of the two fundamental
+assumptions in the special theory of relativity and to which we have
+already frequently referred, cannot claim any unlimited validity. A
+curvature of rays of light can only take place when the velocity of
+propagation of light varies with position. Now we might think that as
+a consequence of this, the special theory of relativity and with it
+the whole theory of relativity would be laid in the dust. But in
+reality this is not the case. We can only conclude that the special
+theory of relativity cannot claim an unlinlited domain of validity;
+its results hold only so long as we are able to disregard the
+influences of gravitational fields on the phenomena ({\it e.g.} of light).
+
+Since it has often been contended by opponents of the theory of
+relativity that the special theory of relativity is overthrown by the
+general theory of relativity, it is perhaps advisable to make the
+facts of the case clearer by means of an appropriate comparison.
+Before the development of electrodynamics the laws of electrostatics
+were looked upon as the laws of electricity. At the present time we
+know that electric fields can be derived correctly from electrostatic
+considerations only for the case, which is never strictly realised, in
+which the electrical masses are quite at rest relatively to each
+other, and to the co-ordinate system. Should we be justified in saying
+that for this reason electrostatics is overthrown by the
+field-equations of Maxwell in electrodynamics? Not in the least.
+Electrostatics is contained in electrodynamics as a limiting case;
+the laws of the latter lead directly to those of the former for the
+case in which the fields are invariable with regard to time. No fairer
+destiny could be allotted to any physical theory, than that it should
+of itself point out the way to the introduction of a more
+comprehensive theory, in which it lives on as a limiting case.
+
+In the example of the transmission of light just dealt with, we have
+seen that the general theory of relativity enables us to derive
+theoretically the influence of a gravitational field on the course of
+natural processes, the Iaws of which are already known when a
+gravitational field is absent. But the most attractive problem, to the
+solution of which the general theory of relativity supplies the key,
+concerns the investigation of the laws satisfied by the gravitational
+field itself. Let us consider this for a moment.
+
+We are acquainted with space-time domains which behave (approximately)
+in a ``Galileian'' fashion under suitable choice of reference-body,
+{\it 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.\footnotemark\ 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 gravitationial fields can be
+produced in this way, yet we may entertain the hope that the general
+law of gravitation will be derivable from such gravitational fields of
+a special kind. This hope has been realised in the most beautiful
+manner. But between the clear vision of this goal and its actual
+realisation it was necessary to surmount a serious difficulty, and as
+this lies deep at the root of things, I dare not withhold it from the
+reader. We require to extend our ideas of the space-time continuum
+still farther.
+
+
+% Notes
+
+\footnotetext[1]{By means of the star photographs of two expeditions equipped by
+a Joint Committee of the Royal and Royal Astronomical Societies, the
+existence of the deflection of light demanded by theory was first
+confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
+III.)}
+
+\footnotetext{This follows from a generalisation of the discussion in
+Section 20}
+
+
+\chapter{Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference}
+
+Hitherto I have purposely refrained from speaking about the physical
+interpretation of space- and time-data in the case of the general
+theory of relativity. As a consequence, I am guilty of a certain
+slovenliness of treatment, which, as we know from the special theory
+of relativity, is far from being unimportant and pardonable. It is now
+high time that we remedy this defect; but I would mention at the
+outset, that this matter lays no small claims on the patience and on
+the power of abstraction of the reader.
+
+We start off again from quite special cases, which we have frequently
+used before. Let us consider a space time domain in which no
+gravitational field exists relative to a reference-body $K$ whose state
+of motion has been suitably chosen. $K$ is then a Galileian
+reference-body as regards the domain considered, and the results of
+the special theory of relativity hold relative to $K$. Let us supposse
+the same domain referred to a second body of reference $K'$, 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.\footnotemark\ But since the
+observer believes in the general theory of relativity, this does not
+disturb him; he is quite in the right when he believes that a general
+law of gravitation can be formulated---a law which not only explains
+the motion of the stars correctly, but also the field of force
+experienced by himself.
+
+The observer performs experiments on his circular disc with clocks and
+measuring-rods. In doing so, it is his intention to arrive at exact
+definitions for the signification of time- and space-data with
+reference to the circular disc $K'$, 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 12, it follows
+that the latter clock goes at a rate permanently slower than that of
+the clock at the centre of the circular disc, {\it i.e.} as observed from $K$.
+It is obvious that the same effect would be noted by an observer whom
+we will imagine sitting alongside his clock at the centre of the
+circular disc. Thus on our circular disc, or, to make the case more
+general, in every gravitational field, a clock will go more quickly or
+less quickly, according to the position in which the clock is situated
+(at rest). For this reason it is not possible to obtain a reasonable
+definition of time with the aid of clocks which are arranged at rest
+with respect to the body of reference. A similar difficulty presents
+itself when we attempt to apply our earlier definition of simultaneity
+in such a case, but I do not wish to go any farther into this
+question.
+
+Moreover, at this stage the definition of the space co-ordinates also
+presents insurmountable difficulties. If the observer applies his
+standard measuring-rod (a rod which is short as compared with the
+radius of the disc) tangentially to the edge of the disc, then, as
+judged from the Galileian system, the length of this rod will be less
+than I, since, according to Section 12, moving bodies suffer a
+shortening in the direction of the motion. On the other hand, the
+measaring-rod will not experience a shortening in length, as judged
+from $K$, if it is applied to the disc in the direction of the radius.
+If, then, the observer first measures the circumference of the disc
+with his measuring-rod and then the diameter of the disc, on dividing
+the one by the other, he will not obtain as quotient the familiar
+number $\pi$ = 3.14 . . ., but a larger number,\footnotemark\ 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 I to the rod
+in all positions and in every orientation. Hence the idea of a
+straight line also loses its meaning. We are therefore not in a
+position to define exactly the co-ordinates $x, y, z$ relative to the
+disc by means of the method used in discussing the special theory, and
+as long as the co-ordinates and times of events have not been
+defined, we cannot assign an exact meaning to the natural laws in
+which these occur.
+
+Thus all our previous conclusions based on general relativity would
+appear to be called in question. In reality we must make a subtle
+detour in order to be able to apply the postulate of general
+relativity exactly. I shall prepare the reader for this in the
+following paragraphs.
+
+
+% Notes
+
+\footnotetext[1]{The field disappears at the centre of the disc and increases
+proportionally to the distance from the centre as we proceed outwards.}
+
+\footnotetext[2]{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).}
+
+
+\chapter{Euclidean and Non-Euclidean Continuum}
+
+
+
+The surface of a marble table is spread out in front of me. I can get
+from any one point on this table to any other point by passing
+continuously from one point to a ``neighbouring'' one, and repeating
+this process a (large) number of times, or, in other words, by going
+from point to point without executing ``jumps." I am sure the reader
+will appreciate with sufficient clearness what I mean here by
+``neighbouring'' and by ``jumps'' (if he is not too pedantic). We
+express this property of the surface by describing the latter as a
+continuum.
+
+Let us now imagine that a large number of little rods of equal length
+have been made, their lengths being small compared with the dimensions
+of the marble slab. When I say they are of equal length, I mean that
+one can be laid on any other without the ends overlapping. We next lay
+four of these little rods on the marble slab so that they constitute a
+quadrilateral figure (a square), the diagonals of which are equally
+long. To ensure the equality of the diagonals, we make use of a little
+testing-rod. To this square we add similar ones, each of which has one
+rod in common with the first. We proceed in like manner with each of
+these squares until finally the whole marble slab is laid out with
+squares. The arrangement is such, that each side of a square belongs
+to two squares and each corner to four squares.
+
+It is a veritable wonder that we can carry out this business without
+getting into the greatest difficulties. We only need to think of the
+following. If at any moment three squares meet at a corner, then two
+sides of the fourth square are already laid, and, as a consequence,
+the arrangement of the remaining two sides of the square is already
+completely determined. But I am now no longer able to adjust the
+quadrilateral so that its diagonals may be equal. If they are equal of
+their own accord, then this is an especial favour of the marble slab
+and of the little rods, about which I can only be thankfully
+surprised. We must experience many such surprises if the construction
+is to be successful.
+
+If everything has really gone smoothly, then I say that the points of
+the marble slab constitute a Euclidean continuum with respect to the
+little rod, which has been used as a ``distance'' (line-interval). By
+choosing one corner of a square as ``origin" I can characterise every
+other corner of a square with reference to this origin by means of two
+numbers. I only need state how many rods I must pass over when,
+starting from the origin, I proceed towards the ``right'' and then
+ ``upwards," in order to arrive at the corner of the square under
+consideration. These two numbers are then the ``Cartesian co-ordinates"
+of this corner with reference to the ``Cartesian co-ordinate system"
+which is determined by the arrangement of little rods.
+
+By making use of the following modification of this abstract
+experiment, we recognise that there must also be cases in which the
+experiment would be unsuccessful. We shall suppose that the rods
+``expand'' by in amount proportional to the increase of temperature. We
+heat the central part of the marble slab, but not the periphery, in
+which case two of our little rods can still be brought into
+coincidence at every position on the table. But our construction of
+squares must necessarily come into disorder during the heating,
+because the little rods on the central region of the table expand,
+whereas those on the outer part do not.
+
+With reference to our little rods---defined as unit lengths---the
+marble slab is no longer a Euclidean continuum, and we are also no
+longer in the position of defining Cartesian co-ordinates directly
+with their aid, since the above construction can no longer be carried
+out. But since there are other things which are not influenced in a
+similar manner to the little rods (or perhaps not at all) by the
+temperature of the table, it is possible quite naturally to maintain
+the point of view that the marble slab is a ``Euclidean continuum."
+This can be done in a satisfactory manner by making a more subtle
+stipulation about the measurement or the comparison of lengths.
+
+But if rods of every kind ({\it i.e.} of every material) were to behave in
+the same way as regards the influence of temperature when they are on
+the variably heated marble slab, and if we had no other means of
+detecting the effect of temperature than the geometrical behaviour of
+our rods in experiments analogous to the one described above, then our
+best plan would be to assign the distance one to two points on the
+slab, provided that the ends of one of our rods could be made to
+coincide with these two points; for how else should we define the
+distance without our proceeding being in the highest measure grossly
+arbitrary? The method of Cartesian coordinates must then be
+discarded, and replaced by another which does not assume the validity
+of Euclidean geometry for rigid bodies.\footnotemark\ The reader will notice
+that the situation depicted here corresponds to the one brought about
+by the general postitlate of relativity (Section 23).
+
+
+% Notes
+
+\footnotetext{Mathematicians have been confronted with our problem in the
+following form. If we are given a surface ({\it e.g.} an ellipsoid) in
+Euclidean three-dimensional space, then there exists for this surface
+a two-dimensional geometry, just as much as for a plane surface. Gauss
+undertook the task of treating this two-dimensional geometry from
+first principles, without making use of the fact that the surface
+belongs to a Euclidean continuum of three dimensions. If we imagine
+constructions to be made with rigid rods in the surface (similar to
+that above with the marble slab), we should find that different laws
+hold for these from those resulting on the basis of Euclidean plane
+geometry. The surface is not a Euclidean continuum with respect to the
+rods, and we cannot define Cartesian co-ordinates in the surface.
+Gauss indicated the principles according to which we can treat the
+geometrical relationships in the surface, and thus pointed out the way
+to the method of Riemman of treating multi-dimensional, non-Euclidean
+continuum. Thus it is that mathematicians long ago solved the formal
+problems to which we are led by the general postulate of relativity.}
+
+
+
+\chapter{Gaussian Co-Ordinates}
+
+% Figure 4
+%
+% __ u=1
+% _/
+% ____/ ___ u=2
+% / \___/ ___ u=3
+% __/_ _/ \___/
+% / \/__ _/ \_
+%__/__/ \/__ \
+% / \__/ \_ v=3
+%/ / /\__ \
+% / / \ v=2
+% v=1
+
+\begin{figure}[bthp]
+
+\centering
+\caption{}
+\label{fig:4}
+
+%Created by jPicEdt 1.x
+%Standard LaTeX format (emulated lines)
+%Thu Aug 25 17:20:05 PDT 2005
+\unitlength 1mm
+\begin{picture}(75.00,55.00)(0,12)
+
+\qbezier(20.00,20.00)(35.00,50.00)(65.00,50.00)
+\qbezier(15.00,25.00)(25.00,55.00)(45.00,60.00)
+\qbezier(30.00,20.00)(45.00,40.00)(70.00,40.00)
+
+\qbezier(50.00,15.00)(40.00,30.00)(15.00,35.00)
+\qbezier(55.00,25.00)(45.00,40.00)(20.00,45.00)
+\qbezier(65.00,30.00)(55.00,50.00)(25.00,55.00)
+
+\put(34.00,16.00){\makebox(0,0)[cc]{P}}
+
+\put(34.00,19.00){\vector(1,3){2}}
+\put(36.50,27.50){\circle*{1.50}}
+
+\put(54.00,12.500){\makebox(0,0)[cc]{$v=1$}}
+\put(59.00,22.50){\makebox(0,0)[cc]{$v=2$}}
+\put(69.00,27.50){\makebox(0,0)[cc]{$v=3$}}
+\put(52.00,61.00){\makebox(0,0)[cc]{$u=1$}}
+\put(72.00,51.00){\makebox(0,0)[cc]{$u=2$}}
+\put(77.00,41.00){\makebox(0,0)[cc]{$u=3$}}
+
+\end{picture}
+
+\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. \ref{fig:4}) drawn on the surface
+of the table. These we designate as $u$-curves, and we indicate each of
+them by means of a number. The curves $u=1$, $u=2$ and $u=3$ are drawn in
+the diagram. Between the curves $u=1$ and $u=2$ we must imagine an
+infinitely large number to be drawn, all of which correspond to real
+numbers lying between 1 and 2. fig. 04 We have then a system of
+u-curves, and this ``infinitely dense" system covers the whole surface
+of the table. These u-curves must not intersect each other, and
+through each point of the surface one and only one curve must pass.
+Thus a perfectly definite value of u belongs to every point on the
+surface of the marble slab. In like manner we imagine a system of
+v-curves drawn on the surface. These satisfy the same conditions as
+the u-curves, they are provided with numbers in a corresponding
+manner, and they may likewise be of arbitrary shape. It follows that a
+value of u and a value of v belong to every point on the surface of
+the table. We call these two numbers the co-ordinates of the surface
+of the table (Gaussian co-ordinates). For example, the point $P$ in the
+diagram has the Gaussian co-ordinates $u=3$, $v=1$. Two neighbouring
+points $P$ and $P_1$ on the surface then correspond to the co-ordinates
+\begin{eqnarray*}
+P: & u ~~,~~v \\
+P': & u + du , v + dv
+\end{eqnarray*}
+where $du$ and $dv$ signify very small numbers. In a similar manner we may
+indicate the distance (line-interval) between $P$ and $P_1$, as measured
+with a little rod, by means of the very small number $ds$. Then
+according to Gauss we have
+
+ $$ds_2 = g_{11}du^2 + 2g_{12}dudv = g_{22}dv^2$$
+
+\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 coordinates 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 coordinates. If a
+distance $ds$ is associated with the adjacent points $P$ and $P_1$, this
+distance being measurable and well defined from a physical point of
+view, then the following formula holds:
+
+$$ds^2 = g_{11}dx_1^2 + 2g_{12}dx_1dx_2 . . . . g_{44}dx_4^2$$
+
+\noindent where the magnitudes g[11], etc., have values which vary with the
+position in the continuum. Only when the continuum is a Euclidean one
+is it possible to associate the co-ordinates $x_1 \ldots x_4$. with the
+points of the continuum so that we have simply
+
+$$ds2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$$
+
+In this case relations hold in the four-dimensional continuum which
+are analogous to those holding in our three-dimensional measurements.
+
+However, the Gauss treatment for $ds^2$ 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 coordinates) as the continuum has dimensions. This is done
+in such a way, that only one meaning can be attached to the
+assignment, and that numbers (Gaussian coordinates) which differ by an
+indefinitely small amount are assigned to adjacent points. The
+Gaussian coordinate system is a logical generalisation of the
+Cartesian co-ordinate system. It is also applicable to non-Euclidean
+continua, but only when, with respect to the defined ``size'' or
+``distance,'' small parts of the continuum under consideration behave
+more nearly like a Euclidean system, the smaller the part of the
+continuum under our notice.
+
+
+
+\chapter{The Space-Time Continuum of the Speical Theory of Relativity Considered as a
+Euclidean Continuum}
+
+
+We are now in a position to formulate more exactly the idea of
+Minkowski, which was only vaguely indicated in Section 17. In
+accordance with the special theory of relativity, certain co-ordinate
+systems are given preference for the description of the
+four-dimensional, space-time continuum. We called these ``Galileian
+co-ordinate systems." For these systems, the four co-ordinates $x, y,
+z, t$, which determine an event or---in other words, a point of the
+four-dimensional continuum---are defined physically in a simple
+manner, as set forth in detail in the first part of this book. For the
+transition from one Galileian system to another, which is moving
+uniformly with reference to the first, the equations of the Lorentz
+transformation are valid. These last form the basis for the derivation
+of deductions from the special theory of relativity, and in themselves
+they are nothing more than the expression of the universal validity of
+the law of transmission of light for all Galileian systems of
+reference.
+
+Minkowski found that the Lorentz transformations satisfy the following
+simple conditions. Let us consider two neighbouring events, the
+relative position of which in the four-dimensional continuum is given
+with respect to a Galileian reference-body $K$ by the space co-ordinate
+differences $dx, dy, dz$ and the time-difference $dt$. With reference to a
+second Galileian system we shall suppose that the corresponding
+differences for these two events are $dx', dy', dz', dt'$. Then these
+magnitudes always fulfil the condition\footnotemark.
+
+ $$dx^2 + dy^2 + dz^2 - c^2dt^2 = dx' 2 + dy' 2 + dz' 2 - c^2dt'^2$$
+
+The validity of the Lorentz transformation follows from this
+condition. We can express this as follows: The magnitude
+
+ $$ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2$$
+
+\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 $x, y, z$, $\sqrt{-I} \cdot ct$ , by $x_1,
+x_2, x_3, x_4$, we also obtaill 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 $\sqrt{-I} \cdot ct$
+instead of the real quantity $t$, we can regard the space-time
+contintium---accordance with the special theory of relativity---as a
+``Euclidean'' four-dimensional continuum, a result which follows from
+the considerations of the preceding section.
+
+
+% Notes
+
+\footnotetext{Cf. Appendixes I and 2. The relations which are derived
+there for the co-ordlnates themselves are valid also for co-ordinate
+differences, and thus also for co-ordinate differentials (indefinitely
+small differences).}
+
+
+
+\chapter{The Space-Time Continuum of the General Theory of Relativity is Not a
+Euclidean Continuum}
+
+
+In the first part of this book we were able to make use of space-time
+co-ordinates which allowed of a simple and direct physical
+interpretation, and which, according to Section 26, can be regarded
+as four-dimensional Cartesian co-ordinates. This was possible on the
+basis of the law of the constancy of the velocity of tight. But
+according to Section 21 the general theory of relativity cannot
+retain this law. On the contrary, we arrived at the result that
+according to this latter theory the velocity of light must always
+depend on the co-ordinates when a gravitational field is present. In
+connection with a specific illustration in Section 23, we found
+that the presence of a gravitational field invalidates the definition
+of the coordinates and the ifine, which led us to our objective in the
+special theory of relativity.
+
+In view of the resuIts of these considerations we are led to the
+conviction that, according to the general principle of relativity, the
+space-time continuum cannot be regarded as a Euclidean one, but that
+here we have the general case, corresponding to the marble slab with
+local variations of temperature, and with which we made acquaintance
+as an example of a two-dimensional continuum. Just as it was there
+impossible to construct a Cartesian co-ordinate system from equal
+rods, so here it is impossible to build up a system (reference-body)
+from rigid bodies and clocks, which shall be of such a nature that
+measuring-rods and clocks, arranged rigidly with respect to one
+another, shaIll indicate position and time directly. Such was the
+essence of the difficulty with which we were confronted in Section
+23.
+
+But the considerations of Sections 25 and 26 show us the way to
+surmount this difficulty. We refer the fourdimensional space-time
+continuum in an arbitrary manner to Gauss co-ordinates. We assign to
+every point of the continuum (event) four numbers, $x_1, x_2, x_3,
+x_4$ (co-ordinates), which have not the least direct physical
+significance, but only serve the purpose of numbering the points of
+the continuum in a definite but arbitrary manner. This arrangement
+does not even need to be of such a kind that we must regard $x_1,
+x_2, x_3$, as ``space" co-ordinates and $x_4$, as a ``time''
+co-ordinate.
+
+The reader may think that such a description of the world would be
+quite inadequate. What does it mean to assign to an event the
+particular co-ordinates $x_1, x_2, x_3, x_4$, if in themselves these
+co-ordinates have no significance? More careful consideration shows,
+however, that this anxiety is unfounded. Let us consider, for
+instance, a material point with any kind of motion. If this point had
+only a momentary existence without duration, then it would to
+described in space-time by a single system of values $x_1, x_2, x_3,
+x_4$. Thus its permanent existence must be characterised by an
+infinitely large number of such systems of values, the co-ordinate
+values of which are so close together as to give continuity;
+corresponding to the material point, we thus have a (uni-dimensional)
+line in the four-dimensional continuum. In the same way, any such
+lines in our continuum correspond to many points in motion. The only
+statements having regard to these points which can claim a physical
+existence are in reality the statements about their encounters. In our
+mathematical treatment, such an encounter is expressed in the fact
+that the two lines which represent the motions of the points in
+question have a particular system of co-ordinate values, $x_1, x_2,
+x_3, x_4$, in common. After mature consideration the reader will
+doubtless admit that in reality such encounters constitute the only
+actual evidence of a time-space nature with which we meet in physical
+statements.
+
+When we were describing the motion of a material point relative to a
+body of reference, we stated nothing more than the encounters of this
+point with particular points of the reference-body. We can also
+determine the corresponding values of the time by the observation of
+encounters of the body with clocks, in conjunction with the
+observation of the encounter of the hands of clocks with particular
+points on the dials. It is just the same in the case of
+space-measurements by means of measuring-rods, as a litttle
+consideration will show.
+
+The following statements hold generally: Every physical description
+resolves itself into a number of statements, each of which refers to
+the space-time coincidence of two events A and B. In terms of Gaussian
+co-ordinates, every such statement is expressed by the agreement of
+their four co-ordinates $x_1, x_2, x_3, x_4$. Thus in reality, the
+description of the time-space continuum by means of Gauss co-ordinates
+completely replaces the description with the aid of a body of
+reference, without suffering from the defects of the latter mode of
+description; it is not tied down to the Euclidean character of the
+continuum which has to be represented.
+
+
+
+\chapter{Exact Formulation of the General Principle of Relativity}
+
+
+We are now in a position to replace the pro. visional formulation of
+the general principle of relativity given in Section 18 by an exact
+formulation. The form there used, ``All bodies of reference $K, K^1,$
+etc., are equivalent for the description of natural phenomena
+(formulation of the general laws of nature), whatever may be their
+state of motion," cannot be maintained, because the use of rigid
+reference-bodies, in the sense of the method followed in the special
+theory of relativity, is in general not possible in space-time
+description. The Gauss co-ordinate system has to take the place of the
+body of reference. The following statement corresponds to the
+fundamental idea of the general principle of relativity: ``All Gaussian
+co-ordinate systems are essentially equivalent for the formulation of
+the general laws of nature."
+
+We can state this general principle of relativity in still another
+form, which renders it yet more clearly intelligible than it is when
+in the form of the natural extension of the special principle of
+relativity. According to the special theory of relativity, the
+equations which express the general laws of nature pass over into
+equations of the same form when, by making use of the Lorentz
+transformation, we replace the space-time variables $x, y, z, t$, of a
+(Galileian) reference-body $K$ by the space-time variables $x^1, y^1, z^1,
+t^1$, of a new reference-body $K^1$. According to the general theory of
+relativity, on the other hand, by application of arbitrary
+substitutions of the Gauss variables $x_1, x_2, x_3, x_4$, the
+equations must pass over into equations of the same form; for every
+transformation (not only the Lorentz transformation) corresponds to
+the transition of one Gauss co-ordinate system into another.
+
+If we desire to adhere to our ``old-time" three-dimensional view of
+things, then we can characterise the development which is being
+undergone by the fundamental idea of the general theory of relativity
+as follows: The special theory of relativity has reference to
+Galileian domains, {\it i.e.} to those in which no gravitational field
+exists. In this connection a Galileian reference-body serves as body
+of reference, {\it i.e.} a rigid body the state of motion of which is so
+chosen that the Galileian law of the uniform rectilinear motion of
+``isolated" material points holds relatively to it.
+
+Certain considerations suggest that we should refer the same Galileian
+domains to non-Galileian reference-bodies also. A gravitational field
+of a special kind is then present with respect to these bodies (cf.
+Sections 20 and 23).
+
+In gravitational fields there are no such things as rigid bodies with
+Euclidean properties; thus the fictitious rigid body of reference is
+of no avail in the general theory of relativity. The motion of clocks
+is also influenced by gravitational fields, and in such a way that a
+physical definition of time which is made directly with the aid of
+clocks has by no means the same degree of plausibility as in the
+special theory of relativity.
+
+For this reason non-rigid reference-bodies are used, which are as a
+whole not only moving in any way whatsoever, but which also suffer
+alterations in form {\it ad lib.} during their motion. Clocks, for which the
+law of motion is of any kind, however irregular, serve for the
+definition of time. We have to imagine each of these clocks fixed at a
+point on the non-rigid reference-body. These clocks satisfy only the
+one condition, that the ``readings" which are observed simultaneously
+on adjacent clocks (in space) differ from each other by an
+indefinitely small amount. This non-rigid reference-body, which might
+appropriately be termed a ``reference-mollusc", is in the main
+equivalent to a Gaussian four-dimensional co-ordinate system chosen
+arbitrarily. That which gives the ``mollusc" a certain
+comprehensibility as compared with the Gauss co-ordinate system is the
+(really unjustified) formal retention of the separate existence of the
+space co-ordinates as opposed to the time co-ordinate. Every point on
+the mollusc is treated as a space-point, and every material point
+which is at rest relatively to it as at rest, so long as the mollusc
+is considered as reference-body. The general principle of relativity
+requires that all these molluscs can be used as reference-bodies with
+equal right and equal success in the formulation of the general laws
+of nature; the laws themselves must be quite independent of the choice
+of mollusc.
+
+The great power possessed by the general principle of relativity lies
+in the comprehensive limitation which is imposed on the laws of nature
+in consequence of what we have seen above.
+
+
+
+\chapter{The Solution of the Problem of Gravitation on the Basis of the General
+Principle of Relativity}
+
+
+If the reader has followed all our previous considerations, he will
+have no further difficulty in understanding the methods leading to the
+solution of the problem of gravitation.
+
+We start off on a consideration of a Galileian domain, {\it i.e.} a domain
+in which there is no gravitational field relative to the Galileian
+reference-body $K$. The behaviour of measuring-rods and clocks with
+reference to K is known from the special theory of relativity,
+likewise the behaviour of ``isolated" material points; the latter move
+uniformly and in straight lines.
+
+Now let us refer this domain to a random Gauss coordinate system or to
+a ``mollusc" as reference-body $K^1$. Then with respect to $K^1$ there is a
+gravitational field $G$ (of a particular kind). We learn the behavior
+of measuring-rods and clocks and also of freely-moving material points
+with reference to $K^1$ simply by mathematical transformation. We
+interpret this behaviour as the behaviour of measuring-rods, docks and
+material points tinder the influence of the gravitational field $G$.
+Hereupon we introduce a hypothesis: that the influence of the
+gravitational field on measuringrods, clocks and freely-moving
+material points continues to take place according to the same laws,
+even in the case where the prevailing gravitational field is not
+derivable from the Galfleian special care, simply by means of a
+transformation of co-ordinates.
+
+The next step is to investigate the space-time behaviour of the
+gravitational field $G$, which was derived from the Galileian special
+case simply by transformation of the coordinates. This behaviour is
+formulated in a law, which is always valid, no matter how the
+reference-body (mollusc) used in the description may be chosen.
+
+This law is not yet the general law of the gravitational field, since
+the gravitational field under consideration is of a special kind. In
+order to find out the general law-of-field of gravitation we still
+require to obtain a generalisation of the law as found above. This can
+be obtained without caprice, however, by taking into consideration the
+following demands:
+
+\begin{enumerate}
+\item The required generalisation must likewise satisfy the general
+postulate of relativity.
+\item If there is any matter in the domain under consideration, only its
+inertial mass, and thus according to Section 15 only its energy is
+of importance for its etfect in exciting a field.
+\item Gravitational field and matter together must satisfy the law of
+the conservation of energy (and of impulse).
+\end{enumerate}
+
+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 {\it i.e.} which have already been fitted into
+the frame of the special theory of relativity. In this connection we
+proceed in principle according to the method which has already been
+explained for measuring-rods, clocks and freely moving material
+points.
+
+The theory of gravitation derived in this way from the general
+postulate of relativity excels not only in its beauty; nor in
+removing the defect attaching to classical mechanics which was brought
+to light in Section 21; nor in interpreting the empirical law of
+the equality of inertial and gravitational mass; but it has also
+already explained a result of observation in astronomy, against which
+classical mechanics is powerless.
+
+If we confine the application of the theory to the case where the
+gravitational fields can be regarded as being weak, and in which all
+masses move with respect to the coordinate system with velocities
+which are small compared with the velocity of light, we then obtain as
+a first approximation the Newtonian theory. Thus the latter theory is
+obtained here without any particular assumption, whereas Newton had to
+introduce the hypothesis that the force of attraction between mutually
+attracting material points is inversely proportional to the square of
+the distance between them. If we increase the accuracy of the
+calculation, deviations from the theory of Newton make their
+appearance, practically all of which must nevertheless escape the test
+of observation owing to their smallness.
+
+We must draw attention here to one of these deviations. According to
+Newton's theory, a planet moves round the sun in an ellipse, which
+would permanently maintain its position with respect to the fixed
+stars, if we could disregard the motion of the fixed stars themselves
+and the action of the other planets under consideration. Thus, if we
+correct the observed motion of the planets for these two influences,
+and if Newton's theory be strictly correct, we ought to obtain for the
+orbit of the planet an ellipse, which is fixed with reference to the
+fixed stars. This deduction, which can be tested with great accuracy,
+has been confirmed for all the planets save one, with the precision
+that is capable of being obtained by the delicacy of observation
+attainable at the present time. The sole exception is Mercury, the
+planet which lies nearest the sun. Since the time of Leverrier, it has
+been known that the ellipse corresponding to the orbit of Mercury,
+after it has been corrected for the influences mentioned above, is not
+stationary with respect to the fixed stars, but that it rotates
+exceedingly slowly in the plane of the orbit and in the sense of the
+orbital motion. The value obtained for this rotary movement of the
+orbital ellipse was 43 seconds of arc per century, an amount ensured
+to be correct to within a few seconds of arc. This effect can be
+explained by means of classical mechanics only on the assumption of
+hypotheses which have little probability, and which were devised
+solely for this purponse.
+
+On the basis of the general theory of relativity, it is found that the
+ellipse of every planet round the sun must necessarily rotate in the
+manner indicated above; that for all the planets, with the exception
+of Mercury, this rotation is too small to be detected with the
+delicacy of observation possible at the present time; but that in the
+case of Mercury it must amount to 43 seconds of arc per century, a
+result which is strictly in agreement with observation.
+
+Apart from this one, it has hitherto been possible to make only two
+deductions from the theory which admit of being tested by observation,
+to wit, the curvature of light rays by the gravitational field of the
+sun\footnotemark[1], 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 ({\it i.e.} by the same
+kind of atom)\footnotemark[2]. These two deductions from the theory have both
+been confirmed.
+
+
+% Notes
+
+\footnotetext[1]{First observed by Eddington and others in 1919. (Cf. Appendix
+III, pp. 126-129).}
+
+\footnotetext[2]{Established by Adams in 1924. (Cf. p. 132)}
+
+
+
+
+%PART III
+
+\part{Considerations on the Universe as a Whole}
+
+
+\chapter{Cosmological Difficulties of Newton's Theory}
+
+
+Part from the difficulty discussed in Section 21, there is a second
+fundamental difficulty attending classical celestial mechanics, which,
+to the best of my knowledge, was first discussed in detail by the
+astronomer Seeliger. If we ponder over the question as to how the
+universe, considered as a whole, is to be regarded, the first answer
+that suggests itself to us is surely this: As regards space (and time)
+the universe is infinite. There are stars everywhere, so that the
+density of matter, although very variable in detail, is nevertheless
+on the average everywhere the same. In other words: However far we
+might travel through space, we should find everywhere an attenuated
+swarm of fixed stars of approrimately the same kind and density.
+
+This view is not in harmony with the theory of Newton. The latter
+theory rather requires that the universe should have a kind of centre
+in which the density of the stars is a maximum, and that as we proceed
+outwards from this centre the group-density of the stars should
+diminish, until finally, at great distances, it is succeeded by an
+infinite region of emptiness. The stellar universe ought to be a
+finite island in the infinite ocean of space\footnotemark.
+
+This conception is in itself not very satisfactory. It is still less
+satisfactory because it leads to the result that the light emitted by
+the stars and also individual stars of the stellar system are
+perpetually passing out into infinite space, never to return, and
+without ever again coming into interaction with other objects of
+nature. Such a finite material universe would be destined to become
+gradually but systematically impoverished.
+
+In order to escape this dilemma, Seeliger suggested a modification of
+Newton's law, in which he assumes that for great distances the force
+of attraction between two masses diminishes more rapidly than would
+result from the inverse square law. In this way it is possible for the
+mean density of matter to be constant everywhere, even to infinity,
+without infinitely large gravitational fields being produced. We thus
+free ourselves from the distasteful conception that the material
+universe ought to possess something of the nature of a centre. Of
+course we purchase our emancipation from the fundamental difficulties
+mentioned, at the cost of a modification and complication of Newton's
+law which has neither empirical nor theoretical foundation. We can
+imagine innumerable laws which would serve the same purpose, without
+our being able to state a reason why one of them is to be preferred to
+the others; for any one of these laws would be founded just as little
+on more general theoretical principles as is the law of Newton.
+
+
+% Notes
+
+\footnotetext[1]{Proof---According to the theory of Newton, the number of ``lines
+of force" which come from infinity and terminate in a mass $m$ is
+proportional to the mass $m$. If, on the average, the Mass density $p_0$
+is constant throughout tithe universe, then a sphere of volume $V$ will
+enclose the average mass $p_0V$. Thus the number of lines of force
+passing through the surface $F$ of the sphere into its interior is
+proportional to $p_0 V$. For unit area of the surface of the sphere the
+number of lines of force which enters the sphere is thus proportional
+to $p_0 V/F$ or to $p_0R$. Hence the intensity of the field at the
+surface would ultimately become infinite with increasing radius $R$ of
+the sphere, which is impossible.}
+
+
+
+\chapter{The Possibility of a ``Finite" and yet ``Unbounded" Universe}
+
+
+But speculations on the structure of the universe also move in quite
+another direction. The development of non-Euclidean geometry led to
+the recognition of the fact, that we can cast doubt on the
+infiniteness of our space without coming into conflict with the laws
+of thought or with experience (Riemann, Helmholtz). These questions
+have already been treated in detail and with unsurpassable lucidity by
+Helmholtz and Poincar\'{e}, 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 {\it e.g.} the lattice construction,
+considered in Section 24. In contrast to ours, the universe of
+these beings is two-dimensional; but, like ours, it extends to
+infinity. In their universe there is room for an infinite number of
+identical squares made up of rods, {\it 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, {\it i.e.} a self-contained line of definite finite length,
+which can be measured up by means of a measuring-rod. Similarly, this
+universe has a finite area that can be compared with the area, of a
+square constructed with rods. The great charm resulting from this
+consideration lies in the recognition of the fact that the universe of
+these beings is finite and yet has no limits.
+
+But the spherical-surface beings do not need to go on a world-tour in
+order to perceive that they are not living in a Euclidean universe.
+They can convince themselves of this on every part of their ``world,"
+provided they do not use too small a piece of it. Starting from a
+point, they draw ``straight lines'' (arcs of circles as judged in
+three dimensional space) of equal length in all directions. They will
+call the line joining the free ends of these lines a ``circle." For a
+plane surface, the ratio of the circumference of a circle to its
+diameter, both lengths being measured with the same rod, is, according
+to Euclidean geometry of the plane, equal to a constant value $\pi$, which
+is independent of the diameter of the circle. On their spherical
+surface our flat beings would find for this ratio the value
+
+ $$\pi \frac{\sin \frac{r}{R}}{\frac{r}{R}}$$
+{\it 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 worldsphere
+is available for their measurements. But if this part is very small
+indeed, they will no longer be able to demonstrate that they are on a
+spherical ``world'' and not on a Euclidean plane, for a small part of
+a spherical surface differs only slightly from a piece of a plane of
+the same size.
+
+Thus if the spherical surface beings are living on a planet of which
+the solar system occupies only a negligibly small part of the
+spherical universe, they have no means of determining whether they are
+living in a finite or in an infinite universe, because the ``piece of
+universe'' to which they have access is in both cases practically
+plane, or Euclidean. It follows directly from this discussion, that
+for our sphere-beings the circumference of a circle first increases
+with the radius until the ``circumference of the universe'' is
+reached, and that it thenceforward gradually decreases to zero for
+still further increasing values of the radius. During this process the
+area of the circle continues to increase more and more, until finally
+it becomes equal to the total area of the whole ``world-sphere."
+
+Perhaps the reader will wonder why we have placed our ``beings ``on a
+sphere rather than on another closed surface. But this choice has its
+justification in the fact that, of all closed surfaces, the sphere is
+unique in possessing the property that all points on it are
+equivalent. I admit that the ratio of the circumference $c$ of a circle
+to its radius $r$ depends on $r$, but for a given value of $r$ it is the
+same for all points of the ``worldsphere''; in other words, the ``
+world-sphere'' is a ``surface of constant curvature."
+
+To this two-dimensional sphere-universe there is a three-dimensional
+analogy, namely, the three-dimensional spherical space which was
+discovered by Riemann. its points are likewise all equivalent. It
+possesses a finite volume, which is determined by its ``radius"
+($2\pi^2R^3$). 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, {\it i.e.} of experience that we can have in the
+movement of ``rigid'' bodies. In this sense we can imagine a spherical
+space.
+
+Suppose we draw lines or stretch strings in all directions from a
+point, and mark off from each of these the distance r with a
+measuring-rod. All the free end-points of these lengths lie on a
+spherical surface. We can specially measure up the area ($F$) of this
+surface by means of a square made up of measuring-rods. If the
+universe is Euclidean, then $F = 4\pi R^2$; 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 ({\it i.e.} of finite
+volume), and has no bounds.
+
+It may be mentioned that there is yet another kind of curved space:
+``elliptical space." It can be regarded as a curved space in which the
+two ``counter-points'' are identical (indistinguishable from each
+other). An elliptical universe can thus be considered to some extent
+as a curved universe possessing central symmetry.
+
+It follows from what has been said, that closed spaces without limits
+are conceivable. From amongst these, the spherical space (and the
+elliptical) excels in its simplicity, since all points on it are
+equivalent. As a result of this discussion, a most interesting
+question arises for astronomers and physicists, and that is whether
+the universe in which we live is infinite, or whether it is finite in
+the manner of the spherical universe. Our experience is far from being
+sufficient to enable us to answer this question. But the general
+theory of relativity permits of our answering it with a moduate degree
+of certainty, and in this connection the difficulty mentioned in
+Section 30 finds its solution.
+
+
+
+\chapter{The Structure of Space According to the General Theory of Relativity}
+
+
+According to the general theory of relativity, the geometrical
+properties of space are not independent, but they are determined by
+matter. Thus we can draw conclusions about the geometrical structure
+of the universe only if we base our considerations on the state of the
+matter as being something that is known. We know from experience that,
+for a suitably chosen co-ordinate system, the velocities of the stars
+are small as compared with the velocity of transmission of light. We
+can thus as a rough approximation arrive at a conclusion as to the
+nature of the universe as a whole, if we treat the matter as being at
+rest.
+
+We already know from our previous discussion that the behaviour of
+measuring-rods and clocks is influenced by gravitational fields, {\it 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 {\it nil}. Thus such a universe could not be inhabited
+by matter everywhere; it would present to us that unsatisfactory
+picture which we portrayed in Section 30.
+
+If we are to have in the universe an average density of matter which
+differs from zero, however small may be that difference, then the
+universe cannot be quasi-Euclidean. On the contrary, the results of
+calculation indicate that if matter be distributed uniformly, the
+universe would necessarily be spherical (or elliptical). Since in
+reality the detailed distribution of matter is not uniform, the real
+universe will deviate in individual parts from the spherical, {\it i.e.} the
+universe will be quasi-spherical. But it will be necessarily finite.
+In fact, the theory supplies us with a simple connection\footnotemark between
+the space-expanse of the universe and the average density of matter in
+it.
+
+
+% Notes
+
+\footnotetext{For the radius R of the universe we obtain the equation
+
+ $$R^2=\frac{2}{\kappa p}$$
+
+The use of the C.G.S. system in this equation gives $2/k = 1^.08 \cdot 10^{27}$;
+$p$ is the average density of the matter and $k$ is a constant connected
+with the Newtonian constant of gravitation.}
+
+
+
+%APPENDIX I
+
+\appendix
+
+\chapter{Simple Derivation of the Lorentz Transformation
+(Supplementary to Section 11)}
+
+
+For the relative orientation of the co-ordinate systems indicated in
+Fig. 2, the x-axes of both systems pernumently coincide. In the
+present case we can divide the problem into parts by considering first
+only events which are localised on the $x$-axis. Any such event is
+represented with respect to the co-ordinate system $K$ by the abscissa $x$
+and the time $t$, and with respect to the system $K^1$ by the abscissa $x'$
+and the time $t'$. We require to find $x'$ and $t'$ when $x$ and $t$ are given.
+
+A light-signal, which is proceeding along the positive axis of $x$, is
+transmitted according to the equation
+
+ $$x = ct$$
+or
+\begin{equation}
+\label{eqn:a1}
+ x - ct = 0
+\end{equation}
+
+Since the same light-signal has to be transmitted relative to $K^1$ with
+the velocity $c$, the propagation relative to the system $K^1$ will be
+represented by the analogous formula
+
+\begin{equation}
+\label{eqn:a2}
+ x' - ct' = 0
+\end{equation}
+
+Those space-time points (events) which satisfy (\ref{eqn:a1}) must also satisfy
+(\ref{eqn:a2}). Obviously this will be the case when the relation
+
+\begin{equation}
+\label{eqn:a3}
+ (x' - ct') = \lambda (x - ct)
+\end{equation}
+
+\noindent is fulfilled in general, where $\lambda$ indicates a constant; for, according
+to (\ref{eqn:a3}), the disappearance of $(x - ct)$ involves the disappearance of
+$(x' - ct')$.
+
+If we apply quite similar considerations to light rays which are being
+transmitted along the negative x-axis, we obtain the condition
+
+\begin{equation}
+\label{eqn:a4}
+ (x' + ct') = \mu (x + ct)
+\end{equation}
+
+By adding (or subtracting) equations (\ref{eqn:a3}) and (\ref{eqn:a4}), 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
+
+ $$a = \frac{\lambda-\mu}{2}$$ % ??
+
+\noindent we obtain the equations
+
+\begin{equation}
+\label{eqn:a5}
+ \left. \begin{array}{rcl} x' &=& ax-bct \\ ct' &=& act-bx \end{array} \right\}
+\end{equation}
+
+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 $K^1$ we have permanently $x' = 0$, and hence according
+to the first of the equations (\ref{eqn:a5})
+
+ $$x = \frac{bc}{a}t$$
+
+If we call $v$ the velocity with which the origin of $K^1$ is moving
+relative to $K$, we then have
+
+\begin{equation}
+\label{eqn:a6}
+ v=\frac{bc}{a}
+\end{equation}
+
+The same value $v$ can be obtained from equations (\ref{eqn:a5}), if we calculate
+the velocity of another point of $K^1$ 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 $K^1$ 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 $K^1$ from $K$; this means that we have
+to insert a particular value of $t$ (time of $K$), {\it e.g.} $t = 0$. For this
+value of $t$ we then obtain from the first of the equations (5)
+
+ $$x' = ax$$
+
+Two points of the $x'$-axis which are separated by the distance $\Delta x' = I$
+when measured in the $K^1$ system are thus separated in our instantaneous
+photograph by the distance
+
+\begin{equation}
+\label{eqn:a7}
+ \Delta x = \frac{I}{a}
+\end{equation}
+
+\noindent But if the snapshot be taken from $K'(t' = 0)$, and if we eliminate $t$
+from the equations (\ref{eqn:a5}), taking into account the expression (\ref{eqn:a6}), we
+obtain
+
+ $$x' = a \left( I - \frac{v^2}{c^2} \right) x$$
+
+\noindent From this we conclude that two points on the $x$-axis separated by the
+distance $I$ (relative to $K$) will be represented on our snapshot by the
+distance
+
+ $$\Delta x' = a \left( I - \frac{v^2}{c^2} \right) \quad . \quad . \quad . \quad \mbox{(7a)}$$
+
+But from what has been said, the two snapshots must be identical;
+hence $\Delta x$ in (7) must be equal to $\Delta x'$ in (7a), so that we obtain
+
+ $$a = \frac{I}{I-\frac{v^2}{c^2}} \quad . \quad . \quad . \quad \mbox{(7b)} $$
+
+The equations (\ref{eqn:a6}) and (7b) determine the constants $a$ and $b$. By
+inserting the values of these constants in (\ref{eqn:a5}), we obtain the first
+and the fourth of the equations given in Section 11.
+
+\begin{equation}
+\label{eqn:a8}
+ \left. \begin{array}{rcl}
+ x' &=& \frac{x-vt}{\sqrt{I-\frac{v^2}{c^2}}} \\
+ ~ \\
+ t' &=& \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}} \end{array} \right\}
+\end{equation}
+
+Thus we have obtained the Lorentz transformation for events on the
+$x$-axis. It satisfies the condition
+
+ $$x'^2 - c^2t'^2 = x^2 - c^2t^2 \quad . \quad . \quad . \quad \mbox{(8a)} $$
+
+The extension of this result, to include events which take place
+outside the $x$-axis, is obtained by retaining equations (\ref{eqn:a8}) and
+supplementing them by the relations
+
+\begin{equation}
+\label{eqn:a9}
+ \left. \begin{array}{rcl} y' &=& y \\ z' &=& z \end{array} \right\}
+\end{equation}
+
+In this way we satisfy the postulate of the constancy of the velocity
+of light in vacuo for rays of light of arbitrary direction, both for
+the system $K$ and for the system $K'$. This may be shown in the following
+manner.
+
+We suppose a light-signal sent out from the origin of $K$ at the time $t
+= 0$. It will be propagated according to the equation
+
+ $$r = \sqrt{x^2+y^2+z^2} = ct$$
+
+\noindent or, if we square this equation, according to the equation
+
+\begin{equation}
+\label{eqn:a10}
+ x^2 + y^2 + z^2 = c^2t^2 = 0
+\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---as judged from $K^1$---in accordance with
+the corresponding formula
+
+ $$r' = ct'$$
+
+\noindent or,
+
+ $$x'^2 + y'^2 + z'^2 - c^2t'^2 = 0 \quad . \quad . \quad . \quad \mbox{(10a)} $$
+
+In order that equation (10a) may be a consequence of equation (\ref{eqn:a10}), we
+must have
+
+\begin{equation}
+\label{eqn:a11}
+ x'^2 + y'^2 + z'^2 - c^2t'^2 = \sigma (x^2 + y^2 + z^2 - c^2t^2)
+\end{equation}
+
+
+Since equation (8a) must hold for points on the $x$-axis, we thus have $\sigma
+= I$. It is easily seen that the Lorentz transformation really
+satisfies equation (\ref{eqn:a11}) for $\sigma = I$; for (\ref{eqn:a11}) is a consequence of (8a)
+and (\ref{eqn:a9}), and hence also of (\ref{eqn:a8}) and (\ref{eqn:a9}). We have thus derived the
+Lorentz transformation.
+
+The Lorentz transformation represented by (\ref{eqn:a8}) and (\ref{eqn:a9}) still requires
+to be generalised. Obviously it is immaterial whether the axes of $K^1$
+be chosen so that they are spatially parallel to those of $K$. It is
+also not essential that the velocity of translation of $K^1$ 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, {\it viz.} from Lorentz
+transformations in the special sense and from purely spatial
+transformations. which corresponds to the replacement of the
+rectangular co-ordinate system by a new system with its axes pointing
+in other directions.
+
+Mathematically, we can characterise the generalised Lorentz
+transformation thus:
+
+It expresses $x', y', x', t'$, in terms of linear homogeneous functions
+of $x, y, x, t$, of such a kind that the relation
+
+ $$x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2 \quad . \quad . \quad . \quad \mbox{(11a)} $$
+
+\noindent is satisficd identically. That is to say: If we substitute their
+expressions in $x, y, x, t$, in place of $x', y', x', t'$, on the
+left-hand side, then the left-hand side of (11a) agrees with the
+right-hand side.
+
+
+
+% APPENDIX II
+
+\chapter{MINKOWSKI'S FOUR-DIMENSIONAL SPACE (``WORLD")
+(SUPPLEMENTARY TO SECTION 17)}
+
+
+We can characterise the Lorentz transformation still more simply if we
+introduce the imaginary $\sqrt{-I} \cdot ct$ 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{-I} \cdot ct
+\end{eqnarray*}
+and similarly for the accented system $K^1$, then the condition which is
+identically satisfied by the transformation can be expressed thus:
+
+$${x'_1}^2 + {x'}_2^2 + {x'}_3^2 + {x'}_4^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2 \quad . \quad . \quad . \quad \mbox{(12)}.$$
+
+\noindent That is, by the afore-mentioned choice of ``coordinates," (11a) [see
+the end of Appendix II] is transformed into this equation.
+
+We see from (12) that the imaginary time co-ordinate $x_4$, enters into
+the condition of transformation in exactly the same way as the space
+co-ordinates $x_1, x_2, x_3$. It is due to this fact that, according
+to the theory of relativity, the ``time'' $x_4$, enters into natural
+laws in the same form as the space co ordinates $x_1, x_2, x_3$.
+
+A four-dimensional continuum described by the ``co-ordinates" $x_1,
+x_2, x_3, x_4$, was called ``world" by Minkowski, who also termed a
+point-event a ``world-point." From a ``happening'' in three-dimensional
+space, physics becomes, as it were, an ``existence ``in the
+four-dimensional ``world."
+
+This four-dimensional ``world'' bears a close similarity to the
+three-dimensional ``space'' of (Euclidean) analytical geometry. If we
+introduce into the latter a new Cartesian co-ordinate system ($x'_1,
+x'_2, x'_3$) with the same origin, then $x'_1, x'_2, x'_3$, are
+linear homogeneous functions of $x_1, x_2, x_3$ which identically
+satisfy the equation
+
+ $${x'}_1^2 + {x'}_2^2 + {x'}_3^2 = x_1^2 + x_2^2 + x_3^2$$
+
+The analogy with (12) is a complete one. We can regard Minkowski's ``world''
+in a formal manner as a four-dimensional Euclidean space (with
+an imaginary time coordinate); the Lorentz transformation corresponds
+to a ``rotation'' of the co-ordinate system in the four-dimensional
+``world."
+
+
+
+%APPENDIX III
+
+\chapter{The Experimental Confirmation of the General Theory of Relativity}
+
+
+From a systematic theoretical point of view, we may imagine the
+process of evolution of an empirical science to be a continuous
+process of induction. Theories are evolved and are expressed in short
+compass as statements of a large number of individual observations in
+the form of empirical laws, from which the general laws can be
+ascertained by comparison. Regarded in this way, the development of a
+science bears some resemblance to the compilation of a classified
+catalogue. It is, as it were, a purely empirical enterprise.
+
+But this point of view by no means embraces the whole of the actual
+process; for it slurs over the important part played by intuition and
+deductive thought in the development of an exact science. As soon as a
+science has emerged from its initial stages, theoretical advances are
+no longer achieved merely by a process of arrangement. Guided by
+empirical data, the investigator rather develops a system of thought
+which, in general, is built up logically from a small number of
+fundamental assumptions, the so-called axioms. We call such a system
+of thought a {\it theory}. The theory finds the justification for its
+existence in the fact that it correlates a large number of single
+observations, and it is just here that the ``truth'' of the theory
+lies.
+
+Corresponding to the same complex of empirical data, there may be
+several theories, which differ from one another to a considerable
+extent. But as regards the deductions from the theories which are
+capable of being tested, the agreement between the theories may be so
+complete that it becomes difficult to find any deductions in which the
+two theories differ from each other. As an example, a case of general
+interest is available in the province of biology, in the Darwinian
+theory of the development of species by selection in the struggle for
+existence, and in the theory of development which is based on the
+hypothesis of the hereditary transmission of acquired characters.
+
+We have another instance of far-reaching agreement between the
+deductions from two theories in Newtonian mechanics on the one hand,
+and the general theory of relativity on the other. This agreement goes
+so far, that up to the preseat we have been able to find only a few
+deductions from the general theory of relativity which are capable of
+investigation, and to which the physics of pre-relativity days does
+not also lead, and this despite the profound difference in the
+fundamental assumptions of the two theories. In what follows, we shall
+again consider these important deductions, and we shall also discuss
+the empirical evidence appertaining to them which has hitherto been
+obtained.
+
+\section{Motion of the Perihelion of Mercury}
+
+According to Newtonian mechanics and Newton's law of gravitation, a
+planet which is revolving round the sun would describe an ellipse
+round the latter, or, more correctly, round the common centre of
+gravity of the sun and the planet. In such a system, the sun, or the
+common centre of gravity, lies in one of the foci of the orbital
+ellipse in such a manner that, in the course of a planet-year, the
+distance sun-planet grows from a minimum to a maximum, and then
+decreases again to a minimum. If instead of Newton's law we insert a
+somewhat different law of attraction into the calculation, we find
+that, according to this new law, the motion would still take place in
+such a manner that the distance sun-planet exhibits periodic
+variations; but in this case the angle described by the line joining
+sun and planet during such a period (from perihelion--closest
+proximity to the sun--to perihelion) would differ from $360^\circ$. The line
+of the orbit would not then be a closed one but in the course of time
+it would fill up an annular part of the orbital plane, viz. between
+the circle of least and the circle of greatest distance of the planet
+from the sun.
+
+According also to the general theory of relativity, which differs of
+course from the theory of Newton, a small variation from the
+Newton-Kepler motion of a planet in its orbit should take place, and
+in such away, that the angle described by the radius sun-planet
+between one perhelion and the next should exceed that corresponding to
+one complete revolution by an amount given by
+
+ $$+ \frac{24\pi^3a^2}{T^2e^2(I-e^2)}$$
+
+\noindent (N.B. -- One complete revolution corresponds to the angle $2\pi$ in the
+absolute angular measure customary in physics, and the above
+expression giver the amount by which the radius sun-planet exceeds
+this angle during the interval between one perihelion and the next.)
+In this expression $a$ represents the major semi-axis of the ellipse, $e$
+its eccentricity, $c$ the velocity of light, and $T$ the period of
+revolution of the planet. Our result may also be stated as follows:
+According to the general theory of relativity, the major axis of the
+ellipse rotates round the sun in the same sense as the orbital motion
+of the planet. Theory requires that this rotation should amount to 43
+seconds of arc per century for the planet Mercury, but for the other
+Planets of our solar system its magnitude should be so small that it
+would necessarily escape detection.\footnotemark
+
+In point of fact, astronomers have found that the theory of Newton
+does not suffice to calculate the observed motion of Mercury with an
+exactness corresponding to that of the delicacy of observation
+attainable at the present time. After taking account of all the
+disturbing influences exerted on Mercury by the remaining planets, it
+was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
+perihelial movement of the orbit of Mercury remained over, the amount
+of which does not differ sensibly from the above mentioned +43 seconds
+of arc per century. The uncertainty of the empirical result amounts to
+a few seconds only.
+
+\section{Deflection of Light by a Gravitational Field}
+
+In Section 22 it has been already mentioned that according to the
+general theory of relativity, a ray of light will experience a
+curvature of its path when passing through a gravitational field, this
+curvature being similar to that experienced by the path of a body
+which is projected through a gravitational field. As a result of this
+theory, we should expect that a ray of light which is passing close to
+a heavenly body would be deviated towards the latter. For a ray of
+light which passes the sun at a distance of $\Delta$ sun-radii from its
+centre, the angle of deflection (a) should amount to
+
+ $$a = \frac{1.7 \mbox{seconds of arc}}{\Delta}$$
+
+It may be added that, according to the theory, half of Figure 05 this
+deflection is produced by the Newtonian field of attraction of the
+sun, and the other half by the geometrical modification (``curvature")
+of space caused by the sun.
+
+\begin{figure}[hbtp]
+
+\centering
+\caption{}
+\label{fig:5}
+
+% / D1
+% /
+% / /
+% / /
+% / /
+% /D /
+% S( )/--/
+% / /
+% D1 / / D2
+% //
+% /
+% _/
+%
+
+
+\begin{picture}(110,250)(0,30)
+\thicklines
+\put(38,138){\circle{15}}
+\put(22,135){S}
+
+\multiput(5,45)(15,15){2}{\line(1,1){10}}
+\multiput(30,70)(5,20){6}{\line(1,4){3}}
+\multiput(30,70)(10,20){4}{\line(1,2){5}}
+\multiput(70,150)(5,20){6}{\line(1,4){3}}
+
+\put(40,90){\vector(1,2){5}}
+\put(35,90){\vector(1,4){3}}
+\put(90,230){\vector(1,4){3}}
+
+\put(15,100){$D_1$}
+\put(50,90){$D_2$}
+\put(100,230){$D_1$}
+
+\put(45,135){\line(3,-1){15}}
+\put(50,137){$\Delta$}
+
+\end{picture}
+
+\end{figure}
+
+
+This result admits of an experimental test by means of the
+photographic registration of stars during a total eclipse of the sun.
+The only reason why we must wait for a total eclipse is because at
+every other time the atmosphere is so strongly illuminated by the
+light from the sun that the stars situated near the sun's disc are
+invisible. The predicted effect can be seen clearly from the
+accompanying diagram. If the sun (S) were not present, a star which is
+practically infinitely distant would be seen in the direction $D_1$, as
+observed front the earth. But as a consequence of the deflection of
+light from the star by the sun, the star will be seen in the direction
+$D_2$, {\it 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 neighborhood 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, {\it i.e.} a few months
+earlier or later. As compared whh the standard photograph, the
+positions of the stars on the eclipse-photograph ought to appear
+displaced radially outwards (away from the centre of the sun) by an
+amount corresponding to the angle a.
+
+We are indebted to the [British] Royal Society and to the Royal
+Astronomical Society for the investigation of this important
+deduction. Undaunted by the [first world] war and by difficulties of
+both a material and a psychological nature aroused by the war, these
+societies equipped two expeditions---to Sobral (Brazil), and to the
+island of Principe (West Africa)---and sent several of Britain's most
+celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),
+in order to obtain photographs of the solar eclipse of 29th May, 1919.
+The relative discrepancies to be expected between the stellar
+photographs obtained during the eclipse and the comparison photographs
+amounted to a few hundredths of a millimetre only. Thus great accuracy
+was necessary in making the adjustments required for the taking of the
+photographs, and in their subsequent measurement.
+
+The results of the measurements confirmed the theory in a thoroughly
+satisfactory manner. The rectangular components of the observed and of
+the calculated deviations of the stars (in seconds of arc) are set
+forth in the following table of results:
+
+% Table 01:
+$$
+\begin{array}{r|rr|rr}
+\mbox{Number of the Star} & \mbox{First} & \mbox{Co-ordinate~~} & \mbox{Second} & \mbox{Co-ordinate~~} \\
+\hline
+ & \mbox{Observed} & \mbox{Calculated} & \mbox{Observed} & \mbox{Calculated} \\
+11 & -0'19 & -0'22 & +0'16 & +0'02 \\
+5 & +0'29 & +0'31 & -0'46 & -0'43 \\
+4 & +0'11 & +0'10 & +0'83 & +0'73 \\
+3 & +0'22 & +0'12 & +1'00 & +0'87 \\
+6 & +0'10 & +0'04 & +0'57 & +0'40 \\
+10 & -0'08 & +0'09 & +0'35 & +0'32 \\
+2 & +'095 & +0'85 & -0'27 & -0'09
+\end{array}
+$$
+
+\section{Displacement of Spectral Lines Towards the Red}
+
+In Section 23 it has been shown that in a system $K^1$ 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 angular velocity of rotation of the disc $K^1$
+with respect to $K$. If $v_0$, represents the number of ticks of the
+clock per unit time (``rate'' of the clock) relative to $K$ when the
+clock is at rest, then the ``rate'' of the clock ($v$) when it is moving
+relative to $K$ with a velocity $V$, but at rest with respect to the disc,
+will, in accordance with Section 12, be given by
+
+ $$v = v_2\sqrt{I-\frac{v^2}{c^2}}$$
+
+\noindent or with sufficient accuracy by
+
+ $$v = v_0 \left( I-\frac{1}{2} \frac{v^2}{c^2} \right)$$
+
+\noindent This expression may also be stated in the following form:
+
+ $$v = v_0 \left( I-\frac{1}{c^2} \frac{\omega^2r^2}{2} \right)$$
+
+If we represent the difference of potential of the centrifugal force
+between the position of the clock and the centre of the disc by $\phi$,
+{\it 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
+
+ $$\phi = \frac{\omega^2r^2}{2}$$
+
+\noindent From this it follows that
+
+ $$v = v_0 \left( I + \frac{\phi}{c^2} \right)$$
+
+In the first place, we see from this expression that two clocks of
+identical construction will go at different rates when situated at
+different distances from the centre of the disc. This result is aiso
+valid from the standpoint of an observer who is rotating with the
+disc.
+
+Now, as judged from the disc, the latter is in a gravititional field
+of potential $\phi$, 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:
+
+{\it An atom absorbs or emits light of a frequency which is dependent on
+the potential of the gravitational field in which it is situated.}
+
+The frequency of an atom situated on the surface of a heavenly body
+will be somewhat less than the frequency of an atom of the same
+element which is situated in free space (or on the surface of a
+smaller celestial body).
+
+Now $\phi = - K (M/r)$, where $K$ is Newton's constant of gravitation, and $M$
+is the mass of the heavenly body. Thus a displacement towards the red
+ought to take place for spectral lines produced at the surface of
+stars as compared with the spectral lines of the same element produced
+at the surface of the earth, the amount of this displacement being
+
+ $$\frac{v_0-v}{v_0} = \frac{K}{c^2} \frac{M}{r}$$
+
+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 $r$ are known.
+
+It is an open question whether or not this effect exists, and at the
+present time (1920) astronomers are working with great zeal towards
+the solution. Owing to the smallness of the effect in the case of the
+sun, it is difficult to form an opinion as to its existence. Whereas
+Grebe and Bachem (Bonn), as a result of their own measurements and
+those of Evershed and Schwarzschild on the cyanogen bands, have placed
+the existence of the effect almost beyond doubt, while other
+investigators, particularly St. John, have been led to the opposite
+opinion in consequence of their measurements.
+
+Mean displacements of lines towards the less refrangible end of the
+spectrum are certainly revealed by statistical investigations of the
+fixed stars; but up to the present the examination of the available
+data does not allow of any definite decision being arrived at, as to
+whether or not these displacements are to be referred in reality to
+the effect of gravitation. The results of observation have been
+collected together, and discussed in detail from the standpoint of the
+question which has been engaging our attention here, in a paper by E.
+Freundlich entitled ``Zur Prüfung der allgemeinen
+Relativit\"ats-Theorie" ({\it 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. \footnotemark
+
+
+% Notes
+
+\footnotetext[1]{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.}
+
+\footnotetext[2]{The displacentent of spectral lines towards the red end of the
+spectrum was definitely established by Adams in 1924, by observations
+on the dense companion of Sirius, for which the effect is about thirty
+times greater than for the Sun. R.W.L. -- translator}
+
+
+
+%APPENDIX IV
+\chapter{The Structure of Space According to the General Theory of Relativity
+(Supplementary to Section 32)}
+
+Since the publication of the first edition of this little book, our
+knowledge about the structure of space in the large (``cosmological
+problem'') has had an important development, which ought to be
+mentioned even in a popular presentation of the subject.
+
+My original considerations on the subject were based on two
+hypotheses:
+
+\begin{enumerate}
+\item There exists an average density of matter in the whole of space
+which is everywhere the same and different from zero.
+
+\item The magnitude (``radius'') of space is independent of time.
+\end{enumerate}
+
+Both these hypotheses proved to be consistent, according to the
+general theory of relativity, but only after a hypothetical term was
+added to the field equations, a term which was not required by the
+theory as such nor did it seem natural from a theoretical point of
+view (``cosmological term of the field equations'').
+
+Hypothesis (2) appeared unavoidable to me at the time, since I thought
+that one would get into bottomless speculations if one departed from
+it.
+
+However, already in the 'twenties, the Russian mathematician Friedman
+showed that a different hypothesis was natural from a purely
+theoretical point of view. He realized that it was possible to
+preserve hypothesis (1) without introducing the less natural
+cosmological term into the field equations of gravitation, if one was
+ready to drop hypothesis (2). Namely, the original field equations
+admit a solution in which the ``world radius'' depends on time
+(expanding space). In that sense one can say, according to Friedman,
+that the theory demands an expansion of space.
+
+A few years later Hubble showed, by a special investigation of the
+extra-galactic nebulae (``milky ways''), that the spectral lines
+emitted showed a red shift which increased regularly with the distance
+of the nebulae. This can be interpreted in regard to our present
+knowledge only in the sense of Doppler's principle, as an expansive
+motion of the system of stars in the large---as required, according
+to Friedman, by the field equations of gravitation. Hubble's discovery
+can, therefore, be considered to some extent as a confirmation of the
+theory.
+
+There does arise, however, a strange difficulty. The interpretation of
+the galactic line-shift discovered by Hubble as an expansion (which
+can hardly be doubted from a theoretical point of view), leads to an
+origin of this expansion which lies ``only'' about $10^9$ years ago,
+while physical astronomy makes it appear likely that the development
+of individual stars and systems of stars takes considerably longer. It
+is in no way known how this incongruity is to be overcome.
+
+I further want to remark that the theory of expanding space, together
+with the empirical data of astronomy, permit no decision to be reached
+about the finite or infinite character of (three-dimensional) space,
+while the original ``static'' hypothesis of space yielded the closure
+(finiteness) of space.
+
+\newpage
+
+~\\
+$K$ = co-ordinate system \\
+$x, y$ = two-dimensional co-ordinates \\
+$x, y, z$ = three-dimensional co-ordinates \\
+$x, y, z, t$ = four-dimensional co-ordinates \\
+
+~\\
+$t$ = time \\
+$I$ = distance \\
+$v$ = velocity \\
+
+~\\
+$F$ = force \\
+$G$ = gravitational field
+
+
+%
+% GNU Free Documentation License
+% Version 1.1, March 2000
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