From e5afc88238247a6299e695fcf45c5886b9c7069e Mon Sep 17 00:00:00 2001 From: Francis Russell Date: Tue, 29 Nov 2011 10:43:00 +0000 Subject: [PATCH] Add relat10.tex from Project Gutenberg. --- relat10.tex | 4712 +++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 4712 insertions(+) create mode 100644 relat10.tex diff --git a/relat10.tex b/relat10.tex new file mode 100644 index 0000000..dbaedec --- /dev/null +++ b/relat10.tex @@ -0,0 +1,4712 @@ +%The Project Gutenberg EBook of Relativity: The Special and General Theory +%by Albert Einstein +%(#1 in our series by Albert Einstein) +% +%Note: 58 image files are part of this eBook. They include tables, +%equations and figures that could not be represented well as plain text. +% +%Copyright laws are changing all over the world. 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You can also find out about how to make a +%donation to Project Gutenberg, and how to get involved. +% +% +%**Welcome To The World of Free Plain Vanilla Electronic Texts** +% +%**eBooks Readable By Both Humans and By Computers, Since 1971** +% +%*****These eBooks Were Prepared By Thousands of Volunteers!***** +% +% +%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 +%Transcription to text: Gregory B. Newby +%Typeset into LaTeX: Robert Bradshaw +%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 + +% Copyright (C) 2000 Free Software Foundation, Inc. +% 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA +% Everyone is permitted to copy and distribute verbatim copies +% of this license document, but changing it is not allowed. + +% +%0. 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