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Authorised Translation by Robert W. Lawson
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://w
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PREFACE
PREFACE
The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus 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 par
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I. PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
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 pr
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II. THE SYSTEM OF CO-ORDINATES
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 distan
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III. SPACE AND TIME IN CLASSICAL MECHANICS
III. 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 emb
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IV.THE GALILEIAN SYSTEM OF CO-ORDINATES
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 wh
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V. THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)
V. 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
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VI. THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL MECHANICS
VI. 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 thr
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VII. THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH THE PRINCIPLE OF RELATIVITY
VII. THE APPARENT INCOMPATIBILITY 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 dar
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VIII. ON THE IDEA OF TIME IN PHYSICS
VIII. 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 appear
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IX. THE RELATIVITY OF SIMULTANEITY
IX. THE RELATIVITY OF SIMULTANEITY
Up to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig 1. People travelling in this train will with 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 particula
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X. ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
X. ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
Let us consider two particular points on the train [8] 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 ( e.
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XI. THE LORENTZ TRANSFORMATION
XI. THE LORENTZ TRANSFORMATION
The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: (1) The time-interval (time) between two events is independent of the condition of motion of the body of reference. (2) The space-interval (distance) between two points of a rigid body is independent of
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XII. THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
XII. 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′ = 1. What is the length of the metre-rod relatively to the system K ? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of the system K . By means of the first equation of the Lorentz transformation the values of these two
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XIII. THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU
XIII. 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 expe
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XIV. THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
XIV. 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 conne
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XV. GENERAL RESULTS OF THE THEORY
XV. 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, 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
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XVI. EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
XVI. 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 theo
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XVII. MINKOWSKI’S FOUR-DIMENSIONAL SPACE
XVII. MINKOWSKI’S FOUR-DIMENSIONAL SPACE
The non-mathematician is seized by a mysterious shuddering when he hears of “four-dimensional” things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum. Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (co-ordinates) x, y, z , and that there is an indefinit
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XVIII. SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
XVIII. SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all 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
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XIX. THE GRAVITATIONAL FIELD
XIX. THE GRAVITATIONAL FIELD
“If we pick up a stone and then let it go, why does it fall to the ground?” The usual answer to this question is: “Because it is attracted by the earth.” Modern physics formulates the answer rather differently for the following reason. As a result of the more careful study of electromagnetic phenomena, we have come to regard action at a distance as a process impossible without the intervention of some intermediary medium. If, for instance, a magnet attracts a piece of iron, we cannot be content
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XX. THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE GENERAL POSTULATE OF RELATIVITY
XX. THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE GENERAL POSTULATE 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
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XXI. IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
XXI. 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-b
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XXII. A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
XXII. A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
The considerations of Section XX 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 ( i.e. simply by calculation) we are then able to find how
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XXIII. BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
XXIII. 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 o
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XXIV. EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
XXIV. 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
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XXV. GAUSSIAN CO-ORDINATES
XXV. GAUSSIAN CO-ORDINATES
According to Gauss, this combined analytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig. 4) drawn on the surface of the table. These we designate as u -curves, and we indicate each of them by means of a number. The Curves u = 1, u = 2 and u = 3 are drawn in the diagram. Between the curves u = 1 and u = 2 we must imagine an infinitely large number to be drawn, all of which correspond to real numbers lying b
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XXVI. THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED AS A EUCLIDEAN CONTINUUM
XXVI. THE SPACE-TIME CONTINUUM OF THE SPECIAL 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 XVII. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these “Galileian co-ordinate systems.” For these systems, the four co-ordinates x, y, z, t , which determine an event or—in other words—a point of the four-dimensional continuum, are defined physic
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XXVII. THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A EUCLIDEAN CONTINUUM
XXVII. 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 XXVI, can be regarded as four-dimensional Cartesian co-ordinates. This was possible on the basis of the law of the constancy of the velocity of light. But according to Section XXI the general theory of relativity cannot retain this law. On the contrary, we arrived at the result that according to this latter theory the vel
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XXVIII. EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
XXVIII. EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
We are now in a position to replace the provisional formulation of the general principle of relativity given in Section XVIII by an exact formulation. The form there used, “All bodies of reference K, K′ , 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 gene
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XXIX. THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL PRINCIPLE OF RELATIVITY
XXIX. 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, 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; t
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XXX. COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY
XXX. COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY
Part from the difficulty discussed in Section XXI, there is a second fundamental difficulty attending classical celestial mechanics, which, to the best of my knowledge, was first discussed in detail by the astronomer Seeliger. If we ponder over the question as to how the universe, considered as a whole, is to be regarded, the first answer that suggests itself to us is surely this: As regards space (and time) the universe is infinite. There are stars everywhere, so that the density of matter, alt
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XXXI. THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE
XXXI. 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, whereas I can only touch on them briefly here. In the first place, we ima
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XXXII. THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
XXXII. 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
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APPENDIX I SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION (SUPPLEMENTARY TO SECTION XI)
APPENDIX I SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION (SUPPLEMENTARY TO SECTION XI)
For the relative orientation of the co-ordinate systems indicated in Fig. 2, the x -axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the x -axis. Any such event is represented with respect to the co-ordinate system K by the abscissa x and the time t , and with respect to the system K′ by the abscissa x′ and the time t′ . We require to find x′ and t′ when x and t are given. A light-signal, w
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APPENDIX II MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”) (SUPPLEMENTARY TO SECTION XVII)
APPENDIX II MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”) (SUPPLEMENTARY TO SECTION XVII)
We can characterise the Lorentz transformation still more simply if we introduce the imaginary in place of t , as time-variable. If, in accordance with this, we insert and similarly for the accented system K′ , 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 (12). That is, by the afore-mentioned choice of “coordinates,” (11 a ) [see the end of Appendix II] is transformed into this
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(a) Motion of the Perihelion of Mercury
(a) Motion of the Perihelion of Mercury
( N.B. —One complete revolution corresponds to the angle 2π in the absolute angular measure customary in physics, and the above expression given 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
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(b) Deflection of Light by a Gravitational Field
(b) Deflection of Light by a Gravitational Field
In Section XXII it has been already mentioned that according to the general theory of relativity, a ray of light will experience a curvature of its path when passing through a gravitational field, this curvature being similar to that experienced by the path of a body which is projected through a gravitational field. As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter. For a ray of light which passes the s
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(c) Displacement of Spectral Lines Towards the Red
(c) Displacement of Spectral Lines Towards the Red
In Section XXIII it has been shown that in a system K′ which is in rotation with regard to a Galileian system K , clocks of identical construction, and which are considered at rest with respect to the rotating reference-body, go at rates which are dependent on the positions of the clocks. We shall now examine this dependence quantitatively. A clock, which is situated at a distance r from the centre of the disc, has a velocity relative to K which is given by v = ω r , where ω represents the angul
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APPENDIX IV THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY (SUPPLEMENTARY TO SECTION XXXII)
APPENDIX IV THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY (SUPPLEMENTARY TO SECTION XXXII)
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: (1) There exists an average density of matter in the whole of space which is everywhere the same and different from zero. (2) The magnitude (“radius”) of space is independent
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