 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 aspired 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 interests 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 pretense 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 stepmotherly fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the force for the trees. May the book bring some one a few happy hours of suggestive thought. Section 1. Physical Meaning of Geometrical Propositions In your school days, 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 someone 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 plain, point, and straight line with which we are able to associate more or less definite ideas. And from certain simple propositions or 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. Our proposition is then correct or true when it has been derived in the recognized manner from the axioms. The question of truth of the individual geometrical proposition 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 its structure the largest possible logical unity. The practice, for example, of seeing in a distance two market points 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. Begin footnote. 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 point A and C being given, B is chosen such that the sum of the distances A, B and B, C is as short as possible. This incomplete suggestion will suffice for the present purpose. End of footnote. 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 construction with the ruler and compasses. Of course the conviction of the truth of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the truth of the geometrical propositions, then at a later stage, in the general theory of relativity, we shall see that this truth is limited, and we shall consider the extent of its limitation. End of section 1. Section 2. The System of Coordinates. 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, broad 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. Begin footnote. 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 rod, the introduction of which does not demand any fundamentally new method. End footnote. 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 implies not only to scientific description but also to everyday life. If I analyze the place specification, Times Square New York, begin footnote. Einstein used Putzdamerplatz Berlin in the original text. In the authorised translation this was supplemented with Trafalgar Square in 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. End footnote. 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. Begin footnote. It is not necessary here to investigate further the significance of the expression coincide in space. This conception is sufficiently obvious to ensure that the differences of opinion are scarcely likely to arise as to applicability in practice. End footnote. 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 the surface or 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. 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 completed 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 measure to make ourselves independent of the existence of marked positions, those possessing names on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of coordinates. This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of coordinates the scene of any event will be determined for the main part by the specification of the length of the three perpendiculars of coordinates 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 a system of coordinates are generally not available. Furthermore the magnitudes of the coordinates are not actually determined by the constructions with rigid rods but by indirect means. If the result 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. Begin footnote. A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity treated in the second part of this book. End footnote. 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. End of Section 2. Section 3. 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 where I de-formulate the aims of mechanics in this way without serious reflection and detailed explanation. Let us proceed to disclose these sins. It is not clear what is to be understood here by physician in space. I stand at the window of a railway carriage which is traveling uniformly and drop a stone on an 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 physicians traverse by the stone fly in reality on a straight line or on a parabola? Moreover, what is meant here by motion in space? From the consideration 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 preceding section. If instead of body of reference we insert system of coordinates, 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 coordinates rigidly attached to the carriage, but relative to a system of coordinates rigidly attached to the ground embankment it describes the parabola. With the aid of this example, it is clearly seen that there is no such thing as an independently existing trajectory. Literally, path curve. Begin footnote, that is a curve along which the body moves. End footnote, 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, 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 its second difficulty prevailing here, we shall have to deal in detail later. End of section three. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Linda Lu. Relativity, the special and general theory by Albert Einstein. Continuing part one, sections four through six. Section four, the Galilean system of coordinates. As is well known, the fundamental law of mechanics of Galilei Newton, which is known as a 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 coordinates, 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 coordinates which the state of motion is such that the law of inertia holds relative to it is called a Galilean system of coordinates. The laws of the mechanics of Galilean Newton can be regarded as valid only for a Galilean system of coordinates. End of section four. Section five, 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 traveling 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. We were to observe the flying raven from the moving railway carriage. We should find that the motion of the raven will 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 coordinate system K, then it will also be moving uniformly and in a straight line relative to a second coordinate system, K prime, 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 Galilean coordinate system, then every other coordinate system K prime is a Galilean one. When in relation to K, it is in a condition of uniform motion of translation. Relative to K prime, mechanical laws of Galilean Newton hold good exactly as they do with respect to K. We advance a step farther in our generalization when we express the tenets thus. If relative to K, K prime is a uniformly moving coordinate system devoid of rotation, the natural phenomenon run their course with respect to K prime, according to exactly the same general laws as it is respect to K. This statement is called the principle of relativity in a 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 right 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 outside speak very much in favor 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 its great accuracy in the domain of mechanics, but that a principle of such broad generality 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 Galilean coordinate systems k, k prime, k double prime, 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 unconditioned that from amongst all possible Galilean coordinate systems, we should have chosen one, k sub zero 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 and calling the system absolutely at rest and all other Galilean systems k in motion. If, for instance, our embankment with a system k sub zero, then our railway carriage would be a system k relative to which less simple loss would hold than with respect to k sub zero. This diminished simplicity would be due to the fact that the carriage k would be in motion, i.e., really, with respect to k sub zero. 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 organ pipe placed with its axis parallel to the direction of travel would be different from that emitted if the axis of the pipe were placed perpendicular to this direction. Now in virtue of its motion in an orbit around the sun, our Earth is comparable with a railway carriage traveling with a velocity of about 30 kilometers 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 and their behavior would be dependent on the orientation in space with respect to the Earth. For owing to the alteration and 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 sub zero throughout the whole year. However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e., a physical non-equivalence of different directions. This is very powerful argument in favor of the principle of relativity. End of section five. Section six, the theorem of the addition of velocities employed in classical mechanics. Let us oppose our old friend the railway carriage to be traveling 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 capital w does a 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 a second, the distance w being numerically equal to the velocity with which he is walking. Thus in total, he covers the distance capital w equals v plus w relative to the embankment in a 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. End of section six. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Peter Eastman, July 30th, 2006. Relativity, the special and general theory by Albert Einstein. Continuing part one. Section seven, 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 equals 300,000 kilometers per second. At all events, we know with great exactness that this velocity is the same for all colors because if this were not the case, the minimum of emission would not be observed simultaneously for different colors during the eclipse of a fixed star by a stark neighbor. By means of similar considerations, based on observations of double stars, the Dutch astronomer De Zetter 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, coordinate 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 traveling 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 equals c minus v. The velocity of propagation of array of light relative to the carriage thus comes out smaller than c. But this result comes into conflict with the principle of relativity set forth in section 5. For like every other general law of nature, the law of the transmission of light in vacuo must, according to the principle of relativity, be the same for the railway carriage as reference body, as when the rails are the body of reference. But from our above consideration, this would appear to be impossible. If every ray of light is propagated relative to the embankment with the velocity c, then for this reason it would appear that another law of propagation of light must necessarily hold with respect to the carriage, 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. Lorenz 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 therefore more inclined to reject the principle of relativity in spite of the fact that no empirical data had been found which were contradictory to this principle. At this juncture, the theory of relativity entered the arena. As a result of an analysis of the physical conceptions of time and space, it became evident that in reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, and that by systematically holding fast to both these laws, a logically-rigid theory could be arrived at. This theory has been called the 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. End of section 7. Section 8. On the idea of time in physics. Lightning has struck the rails on a 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 the statement, you will answer my question with a decided yes. But, if I now approach you with a 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 simultaneously or not. I cannot be satisfied with this answer for the following reason. Supposing that as a result of ingenious considerations and able meteorologists were to discover that the lightning must always strike the places A and B simultaneously, then we should be faced with a task of testing whether or not this theoretical result is in accordance with the reality. We encounter the same difficulty 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 are referred 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 A-B should be measured up, and an observer placed at the midpoint M of the distance A-B. This observer should be supplied with an arrangement, e.g. two mirrors inclined at 90 degrees, 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 the 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 to M with the same velocity as along the length B to M, but an examination of this opposition 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 to M as for the path B to 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 free will in order to arrive at a definition of simultaneity. It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the body of reference, here the railway embankment. Footnote 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 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. End of footnote 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 coordinate 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 and space of the event. In this manner a time value is associated with every event which is essentially capable of observation. This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly when two clocks arranged at rest in different places of a reference body are set in such a manner that a particular position of the pointers of the one clock is simultaneous in the above sense with the same position of the pointers of the clock, then identical settings are always simultaneous in the sense of the above definition. End of section 8 Section 9 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 velocity v and in the direction indicated in figure 1. People travelling in this train will with a vantage view the train as a rigid reference body coordinate system. They regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of Simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence however, the following question arises. Are two events e.g. the two strokes of lightning A and B which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to the embankment we mean the rays of light emitted at the places A and B where the lightning occurs meet each other at the midpoint M of the length A to B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M' be the midpoint of the distance A to B on the travelling train. Just when the flashes judged from the embankment of lightning occur this point M' naturally coincides with the point M but it moves toward the right in the diagram with a velocity V of the train. If an observer sitting at 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 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 the reference body must therefore come to the conclusion that the lightning flash B earlier than the lightning flash A. We thus arrive at an 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 coordinate 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 passively been assumed in physics that the statement of time had an absolute significance 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 7 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 per second relative to the carriage traverses the same distance also with respect to the embankment in each second of time. But according to the foregoing considerations the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment as reference body. Hence it cannot be contended that the man in walking travels the distance W relative to the railway line in a time which is equal to one second as judged from the embankment. Moreover, the considerations of section 6 are based on yet a second assumption which in the light of a strict consideration appears to be arbitrary although it was always passively made even before the introduction of the theory of relativity. End of section 9 All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org Relativity Continuing part one the special theory of relativity sections 10 to 12 section 10 on the Relativity of the Conception of Distance Let us consider two particular points on the train. example the middle of the first and of the 100th carriage, travelling along the embankment with the velocity v, and then choir us 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, or coordinate system. An observer in the train measures the interval by marking off his measuring rod in a straight line, for example, 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 on 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 the 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 measured from the train, then this distance as measured from the embankment is not necessarily also equal to w. Section 11. 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. 1. The time interval, or time, between two events is independent of the condition of motion of the body of reference. And 2. The space interval, or distance, between two points of a rigid body is independent of the condition of motion of the body of reference. If we drop these hypotheses, then the dilemma of section 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 needs 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 spacetime magnitudes of an event when changing over from one body of reference to another. Before we deal with this we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in section 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 coordinate planes or coordinate system. A coordinate system k then corresponds to the embankment, and a coordinate 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 and z on the coordinate planes and with regard to time by a time value t. Relative to k' the same event would be fixed in respect to space and time by corresponding values x' y' z' and t', which of course are not identical with x, y, z and 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' and t' of an event with respect to k' when the magnitudes x, y, z and 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 coordinate systems indicated in the diagram this problem is solved by means of the equations x' equals x minus vt over the square root of i minus v squared over c squared, y prime equals y, z prime equals z and t' equals t minus v over c squared times x over the square root of i minus v squared over c squared. This system of equations is known as the Lorentz transformation. A simple derivation of the Lorentz transformation is given in appendix 1. 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 x' equals x minus vt, y prime equals y, z prime equals z, t prime equals t. 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 prime. A light signal is sent along the positive x-axis and this light stimulus advances in accordance with the equation x equals ct 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 prime and t prime. In point of fact if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation we obtain x prime equals c minus v times t over the square root of i minus v squared over c squared and t prime equals i minus v over c multiplied by t over the square root of i minus v squared over c squared from which by division the expression x prime equals ct prime immediately follows. If referred to the system k prime the propagation of light takes place according to this equation. We thus see that the velocity of transmission relative to the reference body k prime is also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever. Of course this is not surprising since the equations of the Lorentz transformation were derived conformably to this point of view. The behavior of measuring rods and clocks in motion. Place a meter rod in the x prime axis of k prime in such a manner that one end, the beginning, coincides with the point x prime equals zero while the other end, the end of the rod, coincides with the point x prime equals i. What is the length of the meter 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 equals zero can be shown to be x beginning of rod equals zero over the square root of i minus v squared over c squared. x end of rod equals i over the square root of i minus v squared over c squared. The distance between the points being the square root of i minus v squared over c squared. But the meter rod is moving with the velocity v relative to k. It therefore follows that the length of a rigid meter rod moving in the direction of its length with a velocity v is the square root of i minus v squared over c squared of a meter. 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 equals c we should have the square root of i minus v squared over c squared equals zero and for still greater velocities the square root becomes imaginary. For this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity which can neither be reached nor exceeded by any real body. Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation. For these become meaningless if we choose values of v greater than c. If on the contrary we had considered a meter 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 prime would have been the square root of i minus v squared over c squared. This is quite in accordance with the principle of relativity which forms the basis of our considerations. A priori it is quite clear that we must be able to learn something about the physical behavior of measuring rods and clocks from the equations of transformation for the magnitudes z, y, x and 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 Galilean 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 prime equals zero of k prime. t prime equals zero and t prime equals i are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks t equals zero and t prime equals i divided by the square root of i minus v squared over c squared. 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 i divided by the square root of i minus v squared over c squared seconds 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. Correction for this chapter in mathematical formulae instead of i here one. This is a LibreVox recording. All LibreVox recordings are in the public domain. For more information or to volunteer please visit LibreVox.org. Recording by Linda Lu. Relativity the special and general theory by Albert Einstein. Continuing part one, section 13 through 15. Section 13. Theorem of the addition of velocities the experiment of Esau. Now in practice we can move clocks and measuring rods only with velocities that are small compared to 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 derive the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics. This theorem can also be deduced readily from the Galilei transformation. Section 11. In place of the man walking inside the carriage we introduce a point moving relatively to the coordinate system k' in accordance with the equation x' equals 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 equals parentheses v plus w under parentheses 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 capital W and we then obtain as in section 6 capital W equals v plus w equation a. But we can carry out this consideration just as well on the basis of the theory of relativity in the equation x' equals wt'. 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 a we then obtain the equation capital W equals the sum v plus w over the sum i plus v w over c squared equation b 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 a better in accord with experience. On this point we're enlightened by a most important experiment which the brilliant physicist Faisal 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. 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 3. When the liquid above mentioned is flowing through the tube with a velocity v. In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid. Whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known and we require the velocity of light relative to the tube. It is clear that we have a problem in section 6 again before us. The tube plays a part of the railway embankment or the coordinate system k. The liquid plays a part of the carriage or the coordinate system k' and finally the light plays a part of the man walking along the carriage or the moving point in the present section. If we denote the velocity of the light relative to the tube by capital W then this is the given by the equation a or b. According as Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment decides in favor of equation b derived from the theory of relativity and the agreement is indeed very exact. Footnote. Phase O bound capital W equals w plus v. Open parenthesis i minus i over n squared close parenthesis where n equals c over w is the index of refraction of the liquid. On the other hand owing to the smallness of vw over c squared as compared with i we can replace v in the first place by capital W equals open parenthesis w plus v close parenthesis open parenthesis i minus the fraction vw over c squared close parenthesis or to the same order of approximation by w plus v open parenthesis i minus i over n squared close parenthesis which agrees with phase O's result. End footnote. According to recent and most excellent measurements by Siemens the influence of the velocity flow v on the propagation of light is represented by formula b within 1%. 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 favor of the theory of relativity for the electrodynamics of Maxwell-Lorentz in which the original theory was based in no way opposes the theory of relativity. Rather has a ladder been developed from electrodynamics as an astoundingly simple combination and generalization of the hypotheses formally independent of each other in which electrodynamics was built. End of section 13. Section 14. The heuristic value of the theory of relativity. Our train of thought in the foregoing pages can be epitomized in the following manner. Experience is 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 Bacchwell has to be considered equal to a constant C. By uniting these two postulates we obtain the law of transformation for the rectangular coordinates x, y, c 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 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 a 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 spacetime variables x, y, z, t of the original coordinate system k we introduce new spacetime variables x prime, y prime, z prime, t prime of your coordinate system k prime. 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 transformation. 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 viable 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 of the latter theory as hitherto evinced. End of section 14. Section 15. 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 hypotheses forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible that the latter would have been generally accepted by physicists even if it was an experiment and decided less unequivocally in its favor. Classical mechanics required to be modified before it could come in 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 a 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 mv squared over 2 but by the expression mc squared over the square root of the difference i minus the fraction v squared over c squared. This expression approaches infinity as a velocity v approaches a 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 squared plus mv squared over 2 plus 3ac mv to the fourth over c squared plus etc. When v squared over c squared 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 squared does not contain the velocity and requires no consideration if we are only dealing with a 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 recognized two conservation laws of fundamental importance namely the law of conservation of energy and the law of the conservation of mass. These two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about and what meaning is to be attached to it. The principle of relativity requires that the law of the conservation of energy should hold not only with reference to a coordinate system k but also with respect to every coordinate system k prime which is in a state of uniform motion of translation relative to k or briefly relative to every Galilean system of coordinates. 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 conclusions from these premises in conjunction with the fundamental equations of the electrodynamics of Maxwell, a body moving with a velocity v which absorbs footnote one. E sum zero is the energy taken up as judged from a coordinate system moving with a body. And footnote an amount of energy E sum zero in the form of radiation without suffering an alteration in velocity in the process has as a consequence its energy increased by an amount E sum zero over the square root of the difference i minus v squared over c squared. In consideration of the expression given above for the kinetic energy of the body the required energy of the body comes out to be the sum m plus E sum zero over c squared times c squared over the square root of the difference i minus v squared over c squared. Thus the body has the same energy as a body of mass m plus E sum zero over c squared moving with a velocity v hence we can say if a body takes up an amount of energy E sub zero then its inertial mass increases by an amount E sum zero over c squared. 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 the sum mc squared plus E sub zero over the square root of the difference i minus v squared over c squared. We see that the term mc squared which is the two attracted our attention is nothing else than the energy possessed by the body. Footnote 2 as judged from a coordinate system moving with a body. End footnote before it absorbs the energy E sum zero. A direct comparison of this relation with experiment is not possible at the present time. Note the equation E equals mc squared has been thoroughly proved time and again since this time. End note owing to the fact that the changes in energy E zero 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. E zero over c squared is too small in comparison with the mass in which was present before the alteration of the energy. It is owing to this circumstance that classical mechanics was able to establish successfully 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 a velocity of light always takes a 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 two we shall see in what way this result becomes modified in the general theory relativity. End of section 15. Correction for this chapter in mathematical formulae instead of i here one. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Recording by Kelly Boucher. Relativity the Special and General Theory by Alfred Einstein. Continuing part one. Section 16 and 17. Section 16. 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 the zoo. The Special Theory of Relativity has crystallized out from the Maxwell-Lorenz 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 mentioned 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 from 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 color of the light reaching us from them. The latter effect manifests itself in a slightest placement 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 favor of the Maxwell-Lorenz theory, which are at the same time arguments in favor of the theory of relativity, are too numerous to be said for theory. In reality they limit the theoretical possibilities to such an extent that no other theory than that of Maxwell and Lorenz has been able to hold it to one tested by experience. But there are two classes of experimental facts hitherto obtained which can be represented in the Maxwell-Lorenz theory only by the introduction of an auxiliary hypothesis which in itself, i.e. without making use of the theory of relativity, appears extraneous. It is known that cathode rays in the so-called beta rays emitted by radioactive substances consists 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. Footnoted. The general theory of relativity renders it likely that electrical masses of an electron are held together by gravitational forces. And footnoted. 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. Lorenz was the first to introduce the hypothesis that the particles constituting the electronics experience a contraction in the direction of motion and consequence of that motion. The amount of this contraction being proportional to the expression the square root of the difference i minus the fraction v squared over c squared. This hypothesis, which is not justifiable by any electrodynamical facts, supplies us 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 behavior of the electron. We arrived at a similar conclusion in section 13 in connection with the experiment of the zoo, 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 referenced 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 at this nature led to a negative result. Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and space did not allow any doubt to arise as to the prime importance of the Galilei transformation for changing over from one body of reference to another. Now, assuming that Maxwell-Lorentz equations hold for a reference body k, we then find that they do not hold for a reference body k-prime moving uniformly with respect to k. If we assume that the relations of the Galilean transformation exist between the coordinates of k and k-prime, it thus appears that of all Galilean coordinate 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 ether of space. On the other hand, all coordinate systems k-prime moving relatively to k were to be regarded as in motion with respect to the ether. To this motion of k-prime against the ether, ether drift relative to k-prime, were assigned the more complicated laws which were supposed to hold relative to k-prime, strictly seeking such an ether drift not 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 ether 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 back again, if the whole system be at rest with respect to the ether. It is found by calculation, however, that a slight different time t-prime is required for this process if the body together with the mirrors be moving relatively to the ether, and yet another point it is shown by calculation that for a given velocity v with reference to the ether, this time t-prime 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 result should have been clearly detectable, but the experiment gave a negative result, a fact very perplexing to physicists. Lorenz and Fitzgerald rescued this theory from this difficulty by assuming that the motion of the body relative to the ether produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in section 12 shows us that from the standpoint also of the theory of relativity, the 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 favored, unique coordinate system to occasion the introduction of the ether idea, and hence there can be no ether 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 coordinate system moving with the earth, the mirror system of Michelson and Morley is not shortened, but it is shortened for a coordinate system which is at rest relatively to the sun. And of section 16, section 17, Mankowski's four-dimensional space. The non-mathematician is seized by mysterious shuttering when he hears of four-dimensional things, by feeling not unlike that awakened by thoughts of the occult, and yet there is no more commonplace 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, or coordinates, x, y, z, and that there is an indefinite number of points in the neighborhood of this one, the position of which can be described by coordinates such as x sub one, y sub one, z sub one, which may be as near as we choose the respective values of the coordinate x, y, z, of the first point. In virtue of the latter property we speak of a continuum, knowing to the fact that there are three coordinates we speak of it as being three-dimensional. Similarly, the world of physical phenomena which was briefly called world by Mankowski 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 coordinates x, y, z, and a time coordinate, the time value t. The world is, in this sense, also a continuum, for to every event there are as many neighboring events, realized or at least thinkable as we care to choose, the coordinate x sub one, y sub one, z sub one, t sub one, 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 state's 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, i.e. it is independent of the position and the condition of the motion of the system of coordinates. We see this expressed in the last equation of the Galilean transformation, t prime equals 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 prime equals the fraction in which the numerator is t minus the fraction vx over c squared, and the denominator is the square root of the difference i minus the fraction v squared over c squared. Moreover, according to this equation the time difference delta t prime of two events with respect to k prime does not in general vanish, even when the time difference of delta t of the same events with reference to k vanishes. Here, a space distance of two events with respect to k results in time distance of the same events with respect to k prime. But the discovery of Mankowski, which was of importance in 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 and its most essential formal properties shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. Begin footnote. Compare the somewhat more detailed discussion in appendix two. End footnote. In order to give due prominence to this relationship, however, we must replace the usual time coordinate t by an imaginary magnitude, square root of negative 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 coordinate plays exactly the same role as the three-space coordinates. Formally these four coordinates correspond exactly to the three-space coordinates 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 clear lists and no mean measure. These inadequate remarks can give the reader only a vague notion of the important idea contributed by Mankowski. 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 close. Mankowski's work is doubtless difficult of access to anyone in inexperience in mathematics, but since it is not necessary to have a very exact grasp of this work in order to understand the fundamental ideas of either the special or the general theory of relativity, I shall at present leave it here, and shall revert to it only towards the end of part two. End of section 17. End of part one. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Relativity, The Special and General Theory by Albert Einstein. Recorded by Laurie Ann Walden. Part two, The General Theory of Relativity. Sections 18 through 20. Section 18, 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 us once more analyze 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. A. The carriage is in motion relative to the embankment. B. The embankment is in motion relative to the carriage. N. A. The embankment. N. 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, A. The embankment as reference body. B. The railway carriage as reference body. Then these general laws of nature, 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, literally specially marked out, as compared with the other. Unlike the first, this latter statement need not of necessity hold apriari. It is not contained in the conceptions of motion and reference body and arrivalable 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 lines. In the first place, we started out from the assumption that there exists a reference body K, whose condition of motion is such that the Galilean 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, Galilean 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 Galilean reference bodies. The validity of the principle of relativity was assumed only for these reference bodies, but not for others, 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, or 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 generalization 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 traveling at a uniform rate. As long as it is moving uniformly, the occupant of the carriage is not sensible of its motion, and it is for this reason that he can, 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 behavior of bodies relative to the person in the railway carriage. The mechanical behavior 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 Galilean law does not hold with respect to the non-uniformly moving carriage. Because of this we feel compelled at the present juncture to grant the kind of absolute physical reality to non-uniform motion in opposition to the general principle of relativity. But in what follows we shall soon see that this conclusion cannot be maintained. Section 19, The Gravitational Field. If we pick up a stone and then let it go, why does it fall to the ground? The usual answer to this question is because it is attracted by the earth. Modern physics formulates the answer rather differently for the following reason. As a result of the more careful study of electromagnetic phenomena, we have come to regard action at a distance as a process impossible without the intervention of some intermediary medium. If, for instance, a magnet attracts a piece of iron, we cannot be content to regard this as meaning that the magnet acts directly on the iron through the intermediate empty space, but we are constrained to imagine, 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 surroundings a gravitational field which acts on the stone and produces its motion of fall. As we know from experience, the intensity of the action on a body diminishes according to a quite definite law as we proceed farther and farther away from the earth. From our point of view this means the law governing the properties of the gravitational field in space must be a perfectly definite one in order correctly to represent the denonution of gravitational action with the distance from operative bodies. It is something like this. The body, e.g. the earth, produces a field in its immediate neighborhood directly. The intensity and direction of the field at points farther removed from the body are then 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 force equals inertial mass times acceleration, where the inertial mass is a characteristic constant of the accelerated body. If now gravitation is the cause of the acceleration, we then have force equals gravitational mass times intensity of the gravitational field, where the gravitational mass is likewise a characteristic constant for the body. From these two relations follows acceleration equals the fraction gravitational mass over inertial mass times 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 mass. 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 recognize the following fact. The same quality of a body manifests itself according to the circumstances as inertia or as weight, literally 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. Section 20. 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 Galilean 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 toward 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 house on our earth. If he release a body which he previously had in his hand, the acceleration of the chest will no longer be transmitted to this body, and for this reason the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that the acceleration of the body towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment. Relying on his knowledge of the gravitational field, as it was discussed in the preceding section, the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time. Of course, he will be puzzled for a moment as to why the chest does not fall in this gravitational field. Just then, however, he discovers the hook in the middle of the lid of the chest and the rope which is attached to it, and he consequently comes to the conclusion that the chest is suspended at rest in the gravitational field. ought we to smile at the man and say that he errs in his conclusion? I do not believe we ought 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 Galilean 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 generalized 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 behavior of the bodies around him on the supposition of a gravitational field, and he would not be justified on the grounds of experience in supposing his reference body to be at rest. Suppose that the man in the chest fixes a rope to the inner side of the lid, and that he attaches a body to the free end of the rope. The result of this will be to stretch the rope so that it will hang vertically downwards. If we ask for an opinion of the cause of tension in the rope, the man in the chest will say, the suspended body experiences a downward force in the gravitational field, and this is neutralized 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 affect 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 law 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 coordinate system first chosen. Now we might easily suppose that the existence of a gravitational field is always only an apparent one. We might also think that regardless of the kind of gravitational field which may be present, we could always choose another reference body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth, in its entirety, vanishes. We can now appreciate why that argument is not convincing, which we brought forward against the general principle of relativity at the end of section 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 recognizes in this the non-uniformity of motion or retardation of the carriage. But he is compelled by nobody to refer this jerk to a real acceleration or 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. End of section 20. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recorded by Annie Coleman, www.annecolman.com. Relativity, the Special and General Theory, by Albert Einstein. Continuing Part 2. Sections 21 through 23. Section 21. 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 emphasized 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 recognized 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 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 color 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 behavior 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 behavior of bodies considered with respect to the reference systems K and K prime. Begin footnote. The objection is of importance more especially when the state of motion of the reference body is of such a nature that it does not require any external agency for its maintenance. For example, in the case when the reference body is rotating uniformly. End footnote. Newton saw this objection and attempted to invalidate it, but without success. But E. Mock recognized 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. Section 22. 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 spacetime course for any natural process whatsoever as regards the manner in which it takes place in the Galilean domain relative to a Galilean body of reference k. By means of purely theoretical operations, i.e. simply by calculation, we are then able to find how this known natural process appears as seen from a reference body k prime which is accelerated relatively to k. But since the gravitational field exists with respect to this new body of reference k prime, 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 prime chest. This acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to k prime. 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 Galilean reference body k, such a ray of light is transmitted rectilinearly with a 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 prime. 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 is 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 neighborhood 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. Again footnote. By means of the star photographs of two expeditions equipped by a joint committee of the Royal and Royal Astronomical Societies the existence of the deflection of light demanded by theory was first confirmed during the solar eclipse of 29th May 1919. End footnote. 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 unlimited domain of validity. Its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena, for example 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 realized in which the electrical masses are quite at rest relatively to each other and to the coordinate system. Should we be justified in saying that for this reason electrostatics is overthrown by the field equations of Maxwell in electrodynamics? Not in the least electrostatics is contained in electrodynamics as a limiting case. The laws of the latter lead directly to those of the former for the case in which the fields are invariable with regard to time. No fairer destiny could be allotted to any physical theory than that it should of itself point out the way to the introduction of a more comprehensive theory in which it lives on as a limiting case. In the example of the transmission of light just dealt with we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes. The laws of which are already known when a gravitational field is absent but the most attractive problem to the solution of which the general theory of relativity supplies the key concerns the investigation of the laws satisfied by the gravitational field itself. Let us consider this for a moment. We are acquainted with spacetime domains which behave approximately in a Galilean fashion under suitable choice of reference body 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. Begin footnote. This follows from a generalization of the discussion in section 20. End footnote. The character of this field will of course depend on the motion chosen for k' according to the general theory of relativity the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. Even though by no means all gravitational fields can be produced in this way yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. This hope has been realized in the most beautiful manner but between the clear vision of this goal and its actual realization 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 spacetime continuum still farther. Section 23. Behavior of clocks and measuring rods on a rotating body of reference. Hitherto I have purposefully refrained from speaking about the physical interpretation of space and time data in the case of this 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 spacetime 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 Galilean reference body as regards the domain considered and the results of the special theory of relativity hold relative to k. Let us suppose the same domain referred to a second body of reference k prime which is rotating uniformly with respect to k. In order to fix our ideas we shall imagine k prime to be in the form of a plain circular disc which rotates uniformly in its own plane about its center. An observer who is sitting eccentrically on the disc k prime is sensible of a force which acts outward 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. Begin footnote. The field disappears at the center of the disc and increases proportionally to the distance from the center as we proceed outwards. And footnote. 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 significance of time and space data with reference to the circular disc k prime. 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 center 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 Galilean reference body k. As judged from this body the clock at the center 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 center of the circular disc 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 center 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 coordinates 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 Galilean 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 measuring rod will not experience a shortening in length as judged from k if it is applied to the disc in the direction of the radius. If then the observer first measures the circumference of the disc with his measuring rod and then the diameter of the disc on dividing the one by the other he will not obtain as quotient the familiar number pi equals 3.14 etc but a larger number. Begin footnote. Throughout this consideration we have to use the Galilean 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 prime a gravitational field prevails. End footnote. But a larger number 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 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 coordinates x, y, z relative to the disc by means of the method used in discussing the special theory and as long as the coordinates 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. End of sections 21 to 23 read by Annie Coleman in St. Louis, Missouri on August 13th 2006. Correction for this chapter in mathematical formulae. Instead of i here one. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Recording by Meredith Hughes, Cambridge, Massachusetts. Relativity the special and general theory by Albert Einstein. Continuing part two the general theory of relativity sections 24 through 26. Section 24 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 neighboring one and repeating this process a large number of times or in other words by going from point to point without executing jumps. I'm sure the reader will appreciate with sufficient clearness what I mean here by neighboring and by jumps if he is not too pedantic. We express this property at 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 to the dimensions of the marble slab. When I say they are of equal length I mean that one can be laid on any other one 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 in a special favour of the marble slab and of the little rods about which I can only be thankfully surprised. We must needs experience many such surprises if the construction is to be successful. If everything has really gone smoothly then I say that the points of the marble slab constitute a Euclidean continuum with respect to the little rod which has been used as a distance, line interval. By choosing one corner of a square as origin I can characterize 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 coordinates of this corner with reference to the Cartesian coordinate system which is determined by the arrangement of little rods. By making use of the following modification of this abstract experiment we recognize that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods expand by an amount proportional to the increase of temperature. We heat the central part of the marble slab but not the periphery in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating because the little rods on the central region of the table expand whereas those on the outer part do not. With reference to our little rods 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 coordinates 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 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 behavior 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. Begin footnote. Mathematicians have been confronted with our problem in the following form. If we are given a surface 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 plain 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 plain geometry. The surface is not a Euclidean continuum with respect to the rods and we cannot define Cartesian coordinates in the surface. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface and thus pointed out the way to the method of Riemann of treating multi-dimensional 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. End footnote. The reader will notice that the situation depicted here corresponds to the one brought about by the general postulate of relativity. Section 23. End of section 24. Section 25. Gaussian coordinates. 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, C figure 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 equals 1, U equals 2, and U equals 3 are drawn in the diagram. Between the curves U equals 1 and U equals 2, we must imagine an infinitely large number to be drawn, all of which correspond to the real numbers lying between 1 and 2. We have then a system of U curves and this infinitely dense system covers the whole surface of the table. These U curves must not intersect each other and through each point of the surface, one and only one curve must pass. Thus a perfectly definite value of U belongs to every point on the surface of the marble slab. In like manner, we imagine a system of V curves drawn on the surface. These satisfy the same conditions as the U curves. They are provided with numbers in a corresponding manner and they may likewise be of arbitrary shape. It follows that a value of U and a value of V belong to every point on the surface of the table. We call these two numbers the coordinates of the surface of the table, Gaussian coordinates. For example, the point capital P in the diagram has the Gaussian coordinates U equals 3, V equals 1. Two neighboring points, capital P and capital P prime on the surface, then correspond to the coordinates capital P colon U comma V, capital P prime colon U plus du comma V plus dV, where dU and dV signify very small numbers. In a similar manner, we may indicate the distance line interval between capital P and capital P prime as measured with a little rod by means of the very small number dS. Then, according to Gauss we have dF squared equals g sub 1 1 du squared plus 2 g sub 1 2 du dV plus g sub 2 2 dV squared, where g sub 1 1 g sub 1 2 g sub 2 2 are magnitudes which depend in a perfectly definite way on U and V. The magnitudes g sub 1 1 g sub 1 2 and g sub 2 2 determine the behavior 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 squared equals du squared plus dV squared. 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 coordinates 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 slightly from each other are associated with neighboring 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 sub 1, x sub 2, x sub 3, x sub 4, which are known as coordinates. Adjacent points correspond to adjacent values of the coordinates. If a distance dF is associated with the adjacent points capital P and capital P prime, this distance being measurable and well-defined from a physical point of view, then the following formula holds. dF squared equals g sub 1, 1 dx sub 1 squared plus 2g sub 1, 2 dx sub 1 dx sub 2 dot dot dot plus g sub 4, 4 dx sub 4 squared, where the magnitudes g sub 1, 1 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 coordinates x sub 1 to x sub 4 with the points of the continuum, so that we have simply dS squared equals dx sub 1 squared plus dx sub 2 squared plus dx sub 3 squared plus dx sub 4 squared. 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 squared, 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 continuum. For example, this obviously holds in the case of the marble slab of the table and local variation of the temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behavior 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 continuum in general, in which size relations, distances between neighboring 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 generalization of the Cartesian coordinate system. It is also applicable to non-Euclidean continuum, but only one 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. End of section 25. Section 26, the spacetime 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 17. In accordance with the special theory of relativity, certain coordinate systems are given preference for the description of the four-dimensional spacetime continuum. We called these Galilean coordinate systems. For these systems, the four coordinates 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 Galilean 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 Galilean systems of reference. Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighboring events, the relative position of which in the four-dimensional continuum is given with respect to a Galilean reference body capital K by the space coordinate differences dx, dy, dz, and the time difference dt. With reference to a second Galilean system, we shall suppose that the corresponding differences for these two events are dx prime, dy prime, dz prime, dt prime. Then these magnitudes always fulfill the condition begin footnote, cf appendices one and two. The relations which are derived there for the coordinates themselves are valid also for coordinate differences, and thus also for coordinate differentials, indefinitely small differences, end footnote. dx squared plus dy squared plus dz squared minus c squared dt squared equals dx prime squared plus dy prime squared plus dz prime squared minus c squared dt prime squared. The validity of the Lorentz transformation follows from this condition. We can express this as follows. The magnitude ds squared equals dx squared plus dy squared plus dz squared minus c squared dt squared, which belongs to two adjacent points of the four-dimensional spacetime continuum has the same value for all selected Galilean reference bodies. If we replace x, y, z square root of quantity minus one and quantity ct by x sub one, x sub two, x sub three, x sub four, we also obtain the result that ds squared equals dx sub one squared plus dx sub two squared plus dx sub three squared plus dx sub four squared 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 square root quantity minus one and quantity ct, instead of the real quantity t, we can regard the spacetime continuum in accordance with the special theory of relativity as a Euclidean four-dimensional continuum, a result which follows from the considerations of the preceding section. End of section 26. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Relativity, the special and general theory by Albert Einstein. Continuing part two, the general theory of relativity. Sections 27 to 29. Section 27. The spacetime 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 spacetime coordinates which allowed of a simple and direct physical interpretation and which according to section 26 can be regarded as four-dimensional Cartesian coordinates. This was possible on the basis of the law of the constancy of the velocity of light. 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 coordinates 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 time which led us to our objective in the special theory of relativity. In view of the results of these considerations we are led to the conviction that according to the general principle of relativity the spacetime continuum cannot be regarded as a Euclidean one but that here we have the general case corresponding to the Marbles lab 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 coordinate system from equal rods so here it is impossible to build up a system or reference body from rigid bodies and clocks which shall be of such a nature that measuring rods and clocks arranged rigidly with respect to one another shall indicate position and time directly. Such was the essence of the difficulty with which we were confronted in section 23 but the considerations of section 25 and 26 show us the way to surmount this difficulty. We refer the four-dimensional spacetime continuum in an arbitrary manner to Gauss coordinates. We assign to every point of the continuum or event four numbers x sub 1, x sub 2, x sub 3 and x sub 4 coordinates 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 kinds that we must regard x sub 1, x sub 2 and x sub 3 as space coordinates and x sub 4 as a time coordinate. 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 coordinates x sub 1, x sub 2, x sub 3 and x sub 4 if in themselves these coordinates have no significance? More careful consideration shows however that this anxiety is unfounded. Let us consider for instance a material point with any kind of motion. If this point had only a momentary existence without duration then it would be described in space time by a single system of values x sub 1, x sub 2, x sub 3 and x sub 4. Thus its permanent existence must be characterized by an infinitely large number of such systems of values the coordinate values of which are so close together as to give continuity corresponding to the material point we thus have a unidimensional 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 coordinate values x sub 1, x sub 2, x sub 3 and x sub 4 in common. After mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements. When we were describing the motion of a material point relative to a body of reference we stated nothing more than the encounters of this point with particular points of the reference body. We can also determine the corresponding values of the time by the observation of encounters of the body with clocks in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials. It is just the same in the case of space measurements by means of measuring rods as a little consideration will show. The following statements hold generally. Every physical description resolves itself into a number of statements each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian coordinates every such statement is expressed by the agreement of their four coordinates x sub 1, x sub 2, x sub 3 and x sub 4. Thus in reality the description of the time-space continuum by means of Gauss coordinates 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. Section 28 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 18 by an exact formulation. The form they're used quote all bodies of reference k k prime etc are equivalent for the description of natural phenomena or formulation of the general laws of nature whatever may be their state of motion unquote 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 coordinate 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 coordinate 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 and t of a Galilean reference body k by the space-time variables x prime y prime z prime and t prime of a new reference body k prime. According to the general theory of relativity on the other hand by application of arbitrary substitutions of the Gauss variables x sub 1 x sub 2 x sub 3 and x sub 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 coordinate system into another. If we desire to adhere to our old time three-dimensional view of things then we can characterize 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 Galilean domains i.e. to those in which no gravitational field exists. In this connection a Galilean reference body serves as body of reference i.e. a rigid body the state of motion of which is so chosen that the Galilean law of the uniform rectilinear motion of isolated material points holds relatively to it. Certain considerations suggest that we should refer the same Galilean domains to non-Galilean 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 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 mollusk is in the main equivalent to a Gaussian four-dimensional coordinate system chosen arbitrarily. That which gives the mollusk a certain comprehensibility as compared with the Gauss coordinate system is the really unjustified formal retention of the separate existence of the space coordinates as opposed to the time coordinate. Every point on the mollusk is treated as a space point and every material point which is at rest relatively to it is at rest so long as the mollusk is considered as reference body. The general principle of relativity requires that all these mollux can be used as reference bodies with equal right and equal success in the formulation of the general laws of nature. The laws themselves must be quite independent of the choice of mollusk. The great power possessed by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence of what we have seen above. Section 29 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 Galilean domain i.e a domain in which there is no gravitational field relative to the Galilean reference body k. The behavior of measuring rods and clocks with reference to k is known from the special theory of relativity. Likewise the behavior 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 mollusk as reference body k prime. Then with respect to k prime 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 prime simply by mathematical transformation. We interpret this behavior as the behavior of measuring rods clocks and material points under the influence of the gravitational field g. Hereupon we introduce a hypothesis that the influence of the gravitational field on measuring rods clocks and freely moving material points continues to take place according to the same laws even in the case where the prevailing gravitational field is not derivable from the Galilean special case simply by means of a transformation of coordinates. The next step is to investigate the space-time behavior of the gravitational field g which was derived from the Galilean special case simply by transformation of the coordinates. This behavior is formulated in a law which is always valid no matter how the reference body or mollusk 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 generalization of the law as found above. This can be obtained without caprice however by taking into consideration the following demands. A. The required generalization must likewise satisfy the general postulate of relativity. B. 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 effect in exciting a field. C. Gravitational field and matter together must satisfy the law of the conservation of energy and of impulse. Finally the general principle of relativity permits us to determine the influence of the gravitational field on the course of all those processes which take place according to known laws when a gravitational field is absent 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 can find 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 around 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 Laverier it has been known that the eclipse corresponding to the orbit of Mercury after it has been corrected for the influences mentioned above is not stationary with respect to the fixed stars but that it rotates exceedingly slowly in the plane of the orbit and in the sense of the orbital motion. The value obtained for this rotary movement of the orbital ellipse was 43 seconds of arc per century an amount ensured to be correct to within a few seconds of arc. This effect can be explained by means of classical mechanics only on the assumption of hypotheses which have little probability and which were devised solely for this purpose. On the basis of the general theory of relativity it is found that the ellipse of every planet around 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 first observed by Eddington and others in 1919 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 i.e by the same kind of atom established by Adams in 1924 these two deductions from the theory have both been confirmed. End of section 29. This is a LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org. Recording by Linda Lu. Relativity the special and general theory by Albert Einstein. Part three considerations on the universe as a whole. Sections 30, 31 and 32. Section 30 cosmological difficulties of Newton's theory. Apart from the difficulty discussed in section 21 there's a second fundamental difficulty attending classical celestial mechanics which to the best of my knowledge was first discussed in detail by the astronomer Seeliker. 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 we guard space and time universe is infinite. There are stars everywhere so that the density and 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 in attenuated swarm of fixed stars of approximately the same kind and density. This view is not in harmony with the theory of Newton. The latter theory rather requires that the universe should have a kind of center in which the density of the stars is a maximum and that as we proceed outwards from the center the group density of the stars should diminish until finally at great distances it has succeeded by an infinite region of emptiness. The stellar universe found to be a finite island in the infinite ocean of space. Begin but not prove. According to the theory of Newton the number of lines of force which come from infinity and terminate in a mass in is proportional to the mass in if on the average the mass density rho sum serum is constant throughout the universe then a sphere of line v will enclose the average man rho sub zero v thus the number of lines of force passing through the surface at of the sphere into its interior is proportional to rho sub zero v or unit area of the surface of the sphere the number of lines of force which enters the sphere is thus proportional to rho sub zero v over f or to rho sub zero r hence the intensity of the field at the surface would ultimately become infinite with increasing radius r of the sphere which is impossible and but no this conception isn't 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 if a night material universe would be destined to become gradually but systematically impoverished in order to escape this dilemma silica 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 then 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 center of course we purchase our emancipation from the fundamental difficulties mentioned at the cost of a modification and complication of newton's law which is neither empirical nor theoretical foundation we can imagine innumerable laws which would serve the same purpose without our being able to state the 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 end of section 30 section 31 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 out on the infiniteness of our space without coming into conflict with the laws of thought or with experience reyman helmholtz these questions have already been treated in detail and with unsurpassable acidity by helmholtz and boncaret 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 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 for example 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 i.e its volume surface is infinite if these beings say their universe is quote plane unquote 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 the surface 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 with all its realization of distance they cannot do this for if they attempt to realize a straight line they will obtain a curve which we three-dimensional beings designate as a great circle 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 great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is for night 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 three 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 equation 27 pi times sine parentheses little r over big r and parentheses divided by parentheses little r over big r and parentheses i.e smaller value than pi the difference being more considerable greater as radius of the circle in comparison with the radius r of the world sphere by means of this relation spherical beings can determine the radius of their universe quote world unquote even when only a relatively small part of their world sphere is available for their measurements but if this part is very small indeed they will no longer be able to demonstrate that they are on a spherical world and not on a Euclidean plane for a small part of a spherical surface differs only slightly from a piece of a plane of the same size thus the spherical surface beings are living on a planet 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 an infinite universe because a piece of universe to which they have access is in both cases practically play or Euclidean it follows directly from this discussion that for our sphere beings circumference of a circle first increases with the radius till the circumference of the universe is reached and that it bends forward 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 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 justifications in the fact that of all closed surfaces 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 for a given value of r it is the same for all points of the world sphere in other words the world sphere is a surface of constant curvature to this two-dimensional sphere universe there is a three-dimensional analogy namely the three-dimensional spherical space which was discovered by Riemann its points are likewise all equivalent it possesses a finite volume which is determined by its radius two pi squared r cubed 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 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 the measuring run all the free ends of these lengths lie on a spherical surface we can specially measure up the area f of the surface by means of a square made up of measuring rods if the universe is euclidean then f equals four pi r squared if it is spherical then f is always less than four pi r squared 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 quote counterpoint unquote 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 i.e of the night volume and has no balance it may be mentioned that there is yet another kind of curved space quote elliptical space unquote it can be regarded as a curved space in which the two counterpoints 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 spherical space and the elliptical excels in its simplicity since all points on it are equivalent as a result of this discussion the most interesting question arises for astronomers and physicists and that is whether the universe in which we live is infinite whether it is finite in the manner of the spherical universe our experience is far from being sufficient to enable us to answer this question but the general theory of relativity permits of our answering it with a moderate degree of certainty and in this connection the difficulty mentioned in section 30 finds solution. End of section 31 section 32 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 coordinate system the velocities of the stars are small as compared with a 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 behavior of measuring rods and clocks is influenced by gravitational fields i.e by the distribution of matter this in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe but it is conceivable that our universe differs only slightly from a Euclidean one and this notion seems all the more probable since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun we might imagine that as regards geometry our universe behaves analogously to a surface which is irregularly curved in its individual parts but which nowhere departs appreciably from a plane something like the rippled surface of a lake such a universe might fittingly be called a quasi-euclidean universe as regards its space it would be infinite but calculation shows that in a quasi-euclidean universe the average density of matter would necessarily be nil thus such a universe could not be inhabited by matter everywhere it would present to us that unsatisfactory picture which we portrayed in section 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 i.e the universe will be quasi-spherical but it will be necessarily for ninth in fact the theory supplies us with a simple connection begin footnote for the radius r of the universe we obtain the equation r squared equals two over kappa rho the use of the cgs system in this equation gives two over k equals one to the eighth power times 10 to the 27th power p is the average density of the matter and k is a constant connected with the Newtonian constant of gravitation end footnote between the space expanse of the universe and the average density of matter in it end section 32 end of part three and of relativity special in general theory by albert einstein correction for this chapter in mathematical formula instead of i here one this is a liberbox recording all liberbox recordings are in the public domain for more information or to volunteer please visit liberbox.org recording by ml cohen cleveland ohio march 2007 relativity the special and general theory by albert einstein appendix three 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 observation 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 catalog it is as it were a purely empirical enterprise but this point of view by no means embraces the whole of the actual process where 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 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 deduction from the theories which are capable of being tested the agreement between the theories may be so complete that it becomes difficult to find such deductions in which the two theories differ from each other as an example a case of general interest is available in the province of biology in the Darwinian theory of the development of species by selection in the struggle for existence and in the theory of development which is based on the hypothesis of the hereditary transmission of acquired characters we have another instance of far-reaching agreement between the deductions from two theories and Newtonian mechanics on the one hand and the general theory of relativity on the other this agreement goes so far that up to the present we have been able to find only a few deductions from the general theory of relativity which are capable of investigation and to which the physics of pre- relativity days does not also lead and this despite a profound difference in the fundamental assumption 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 a motion of the perihelion of mercury according to Newtonian mechanics and Newton's law of gravitation a planet which is revolving around the sun would describe an ellipse around the latter or more correctly around the common center of gravity of the sun and the planet in such a system the sun or the common center 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 parentheses from perihelion closest proximity to the sun to perihelion and parentheses would differ from 360 degrees 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 that is between the circle of least and the circle of greatest distance of the planet from the sun according also to the general theory of relativity which differs of course from the theory of newton a small variation from the newton kepler motion of a planet in its orbit should take place and in such a way that the angle described by the radius sun planet between one perihelion and the next should exceed that corresponding to one complete revolution by an amount given by the formula plus 24 pi cubed a squared divided by t squared c squared times the quantity one minus e squared nb one complete revolution corresponds to the angle two to the pi power in the absolute angle room measure customary in physics and the above expression gives the amount by which the radius sun planet exceeds this angle during an interval between one perihelion and the next close print 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 around the sun in the same sense as the orbital motion of the planet theory requires that this rotation should amount to 43 seconds of arc per century for the planet mercury but for the other planets of our solar system its magnitude should be so small that it would necessarily escape detection footnote especially since the next planet venus has an orbit that is almost an exact circle which makes it more difficult to locate the perihelion with precision and footnote 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 brentzies levrier 1859 and newcom 1895 close brentzies then an unexplained perihelion movement of the orbit of mercury remained over the amount of which does not differ sensibly from the above mentioned plus 43 seconds of arc per century the uncertainty of the empirical results amounts to a few seconds only be 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 center the angle of deflection parentheses alpha close print should amount to alpha equals 1.7 seconds of arc divided by delta it may be added that according to the theory half of this deflection is produced by the Newtonian field of attraction of the sun and the other half by the geometrical modification parentheses quote curvature end quote end parentheses of space caused by the sun 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 of 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 reader's annotation figure five the earth is shown as a dot at the bottom of the diagram a straight line proceeding from there labeled d sub one proceeds upward and slightly to the right passing the sun at a tangent sun being represented by a circle a second line labeled d sub two starts at the earth proceeds at a relatively smaller angle which results in its passing the sun at a greater distance than the initial line d one which is signified by the symbol delta after passing the sun the line becomes parallel to d sub one and a reader's annotation if the sun parentheses s and parentheses were not present a star which is practically infinitely distant would be seen in the direction d sub one as observed from the earth but as a consequence of the deflection of light from the star by the sun the star will be seen in the direction d sub two that is at a somewhat greater distance from the center 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 that is a few months earlier or later as compared with the standard photograph the positions of the stars on the eclipse photograph ought to appear displaced radially outwards parentheses away from the center of the sun close friends by an amount corresponding to the angle a we are indebted to the royal society and to royal astronomical society for the investigation of this important deduction undaunted by the war and by difficulties of both a material and a psychological nature aroused by the war these societies equip two expeditions to sobral brazil and to the island of prunseep west africa and sent several of britain's most celebrated astronomers parentheses eddington coddingham chromolin davidson and prince in order to obtain photographs of the solar eclipse of 29th may 1919 the relative discrepancies to be expected between the stellar photographs of zane during the eclipse and a comparison photographs amounted to a few hundredths of a millimeter only thus great accuracy was necessary in making the adjustments required for taking of the photographs and in their subsequent measurement the results of the measurements confirm the theory in a thoroughly satisfactory manner the rectangular components of the observed and of the calculated deviation of the stars parentheses and seconds of an arc and parentheses are set forth in the following table of results reader's annotation the table consists of measurements on seven stars which are then tabulated in four additional columns which are entitled first coordinate and second coordinate and then for each of those the observed and calculated measurements are given and reader's annotation number of the star 11 first coordinate observed minus 0.19 calculated minus 0.22 second coordinate observed plus 0.16 calculated plus 0.02 star numbered five first coordinate observed plus 0.29 calculated plus 0.31 second coordinate observed negative 0.46 calculated minus 0.43 star number four observed 0.11 calculated 0.10 second coordinate observed 0.83 calculated plus 0.74 star number three observed plus 0.20 calculated plus 0.12 second coordinate observed plus 1.00 calculated plus 0.87 star number six observed at the first coordinate plus 0.10 calculated plus 0.04 second coordinate observed plus 0.57 calculated plus 0.40 number the star 10 observed minus 0.08 calculated plus 0.09 second coordinate observed plus 0.35 calculated plus 0.32 number of the star two observed plus 0.95 calculated plus 0.85 and at the second coordinate observed minus 0.27 calculated minus 0.09 c displacement of the spectral line towards the red in section 23 it has been shown that in a system k prime which is in rotation with regard to a Galilean 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 position of the clocks. We shall now examine this dependence quantitatively. A clock which is situated at a distance r from the center of the disk has a velocity relative to k which is given by v equals omega r where omega represents the angle of velocity of rotation of the disk k prime with respect to k. If v sub zero represents the number of ticks of a clock per unit time parentheses quote rate and quote of the clock close friends relative to k when the clock is at rest then the quote rate and quote of the clock parentheses v close friends when it is moving relative to k with a velocity v but at rest with respect to the disk will in accordance with section 12 be given by v equals v sub zero times the square root of 1 minus v squared over c squared or what's efficient accuracy by v equals v zero times the quantity one minus one half v squared over c squared. This expression may be also stated in the following form v equals v sub zero times the quantity one minus one over c squared times omega squared r squared over two. If we represent the difference of potential of the centrifugal force between the position of the clock and the center of the disk by phi that is 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 disk to the center of the disk then we have phi equals minus omega squared r squared divided by two. From this it follows that v equals v sub zero times the quantity one plus phi over c squared. In the first place we see from this expression that the two clocks of identical construction will go at different rates when situated at different distances from the center of the disk. This result is also valid from the standpoint of an observer who is rotating with the disk. Now as judged from the disk the latter is in a gravitational field of potential phi hence the result we have obtained will hold quite generally for gravitational fields. Furthermore we can regard an atom which is emitting spectral lines as a clock so that the following statement will hold. 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 parentheses or on the surface of a smaller celestial body closed parentheses period. Now phi equals minus k times m over r where k is Newton's constant of gravitation and m is the mass of the heavenly body. Thus the 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 v sub zero minus v divided by v sub zero equals k over c squared times m over r. For the Sun the displacement towards the red predicted by theory amounts to about two millions of the wavelength. A trustworthy calculation is not possible in the case of the stars because in general neither the mass m nor the radius r is known. It is an open question whether or not this effect exists and at the present time astronomers are working with great zeal towards the solution. Owing to the smallness of the effect in the case of the Sun it is difficult to form an opinion as to its existence whereas greb and bachem parentheses van and prens as a result of their own measurements and those of evershed and schwarzchild on the cyanogen vans have placed the existence of the effect almost beyond doubt. Other investigators particularly saint john have been led to the opposite opinion in consequence of their measurements. Mean displacements of lines towards the less-refrangeable end of the spectrum are certainly revealed by statistical investigation of the thick 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 feindlich entitled ser profunder a la mingan revitatsiari prensi's dianitio zonshaft in 1919 number 35 page 520 julia springer berlin close prens period. 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. end of appendix three