Relativity: The Special and General Theory — Inside the Classic

Relativity: The Special and General Theory

Authorised Translation by Robert W. Lawson

ALBERT EINSTEIN REFERENCE ARCHIVE RELATIVITY: THE SPECIAL AND GENERAL THEORY BY ALBERT EINSTEIN

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

Part I: The Special Theory of Relativity I. Physical Meaning of Geometrical Propositions II. The System of Co-ordinates III. Space and Time in Classical Mechanics IV. The Galileian System of Co-ordinates V. The Principle of Relativity (in the Restricted Sense) VI. The Theorem of the Addition of Velocities employed in Classical Mechanics VII. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity VIII. On the Idea of Time in Physics IX. The Relativity of Simultaneity X. On the Relativity of the Conception of Distance XI. The Lorentz Transformation XII. The Behaviour of Measuring-Rods and Clocks in Motion XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau XIV. The Heuristic Value of the Theory of Relativity XV. General Results of the Theory XVI. Experience and the Special Theory of Relativity XVII. Minkowski’s Four-dimensional Space

Part II: The General Theory of Relativity XVIII. Special and General Principle of Relativity XIX. The Gravitational Field XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity XXI. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory? XXII. A Few Inferences from the General Principle of Relativity XXIII. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference XXIV. Euclidean and non-Euclidean Continuum XXV. Gaussian Co-ordinates XXVI. The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum XXVII. The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum XXVIII. Exact Formulation of the General Principle of Relativity XXIX. 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 XXX. Cosmological Difficulties of Newton’s Theory XXXI. The Possibility of a “Finite” and yet “Unbounded” Universe XXXII. The Structure of Space According to the General Theory of Relativity

Appendices: I. Simple Derivation of the Lorentz Transformation (supplementary to section XI) II. Minkowski’s Four-Dimensional Space (“World”) (supplementary to section XVII) III. The Experimental Confirmation of the General Theory of Relativity IV. The Structure of Space According to the General Theory of Relativity (supplementary to section XXXII) V. 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.

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!

PART I: THE SPECIAL THEORY OF RELATIVITY

I. 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 from certain conceptions such as “plane,” “point,” and “straight line,” with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as “true.” Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, _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.[1] Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the “truth” of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the “truth” of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses.

[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.

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.

II. 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.[2]

[2] Here we have assumed that there is nothing left over _i.e._ that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.

Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place specification “Trafalgar Square, London”[3] I arrive at the following result. The earth is the rigid body to which the specification of place refers; “Trafalgar Square, London” is a well-defined point, to which a name has been assigned, and with which the event coincides in space.[4]

I have chosen this as being more familiar to the English reader than the “Potzdammer Platz, Berlin,” which is referred to in the original. (R. W. L.)

[4] It is not necessary here to investigate further the significance of the expression “coincidence in space.” This conception is sufficiently obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice.

This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other. But we can free ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering over Trafalgar Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed.

(_a_) We imagine the rigid body, to which the place specification is referred, supplemented in such a manner that the object whose position we require is reached by the completed rigid body.

(_b_) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.

(_c_) We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud.

From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates.

This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (_x, y, z_) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry.

In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be sought in accordance with the above considerations.[5]

[5] A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book.

We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distances;” the “distance” being represented physically by means of the convention of two marks on a rigid body.

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”[6], but only a trajectory relative to a particular body of reference.

[6] That is, a curve along which the body moves.

In order to have a _complete_ description of the motion, we must specify how the body alters its position _with time; i.e._ for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction; the man at the railway-carriage window is holding one of them, and the man on the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.

IV. THE GALILEIAN 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 Galilei-Newton can be regarded as valid only for a Galileian system of co-ordinates.

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_.