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Relativity: The Special and General I

Contents

Part I — The Theory of Relativity [ edit ]

1 — Physical Meaning of Propositions [ edit ]

In your most of you who read this made acquaintance with the building of Euclid’s geometry, and you — perhaps with more than love — the magnificent on the lofty staircase of which you chased about for uncounted by conscientious teachers. By reason of our experience, you would certainly everyone with disdain who pronounce even the most proposition of this science to be But perhaps this feeling of certainty would leave you if some one were to ask you: then, do you mean by the assertion these propositions are true? Let us to give this question a consideration.

Geometry sets out certain conceptions such as point, and straight line, [ 2 ] which we are able to associate or less definite ideas, and certain simple propositions which, in virtue of these we are inclined to accept as true. on the basis of a logical process, the of which we feel ourselves to admit, all remaining propositions are to follow from those i.e. they are proven. A is then correct (true) it has been derived in the recognized from the axioms. The question of of the individual geometrical propositions is reduced to one of the truth of the axioms. Now it has been known that the question is not only unanswerable by the of geometry, but that it is in itself without meaning. We cannot ask it is true that only one line goes through two We can only say that Euclidean deals with things straight lines, to each of is ascribed the property of being determined by two points situated on it. The true does not tally the assertions of pure geometry, by the word true we are eventually in the of designating always the correspondence a real object; geometry, is not concerned with the relation of the involved in it to objects of experience, but with the logical connection of ideas among themselves.

[ 3 ] It is not to understand why, in spite of we feel constrained to call the of geometry true. Geometrical correspond to more or less objects in nature, and these are undoubtedly the exclusive cause of the of those ideas. Geometry to refrain from such a in order to give to its structure the possible logical unity. The for example, of seeing in a distance two positions on a practically rigid is something which is lodged in our habit of thought. We are accustomed to regard three points as situated on a straight line, if apparent positions can be made to for observation with one eye, suitable choice of our place of

If, in pursuance of our habit of thought, we now the propositions of Euclidean geometry by the proposition that two points on a rigid body always to the same distance (line-interval), of any changes in position to which we may the body, the propositions of Euclidean then resolve themselves propositions on the possible relative of practically rigid bodies. [1] [ 4 ] which has been supplemented in way is then to be treated as a branch of We can now legitimately ask as to the truth of geometrical interpreted in this way, we are justified in asking whether propositions are satisfied for those things we have associated the geometrical ideas. In less terms we can express this by that by the truth of a geometrical in this sense we understand its for a construction with rule and

Of course the conviction of the truth of propositions in this sense is exclusively on rather incomplete For the present we shall assume the of the geometrical propositions, then at a stage (in the general theory of we shall see that this is limited, and we shall consider the of its limitation. [ 5 ]

Section 2 — The of Co-ordinates [ edit ]

On the basis of the interpretation of distance which has indicated, we are also in a position to the distance between two points on a body by means of measurements. For purpose we require a distance S ) which is to be used once and for and which we employ as a standard If, now, A and B are two points on a rigid we can construct the line joining according to the rules of geometry; starting from A . we can mark off the S time after time we reach B . The number of these required is the numerical measure of the AB . This is the basis of all measurement of [2]

Every description of the scene of an or of the position of an object in space is on the specification of the point on a rigid (body of reference) with that event or object [ 6 ] This applies not only to description, but also to everyday If I analyse the place specification Square, London, [3] I arrive at the result. The earth is the rigid to which the specification of place Trafalgar Square, London, is a point, to which a name has assigned, and with which the coincides in space. [4]

This method of place specification only with places on the of rigid bodies, and is dependent on the of points on this surface are distinguishable from each But we can free ourselves from of these limitations without the nature of our specification of position. If, for a cloud is hovering over Square, then we can determine its relative to the surface of the earth by a pole perpendicularly on the Square, so it reaches the cloud. The length of the measured with the standard combined with the specification of the of the foot of the pole, supplies us a complete place specification. On the [ 7 ] of this illustration, we are able to see the in which a refinement of the conception of has been developed.

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

( b ) In locating the position of the we make use of a number (here the of the pole measured with the instead of designated points of

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

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

This of three plane surfaces to each other and rigidly to a rigid [ 8 ] body. Referred to a of co-ordinates, the scene of any event be determined (for the main by the specification of the lengths of the three or co-ordinates ( x, y, z ) which can be dropped the scene of the event to those plane surfaces. The lengths of three perpendiculars can be determined by a of manipulations with rigid performed according to the rules and laid down by Euclidean

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

We thus obtain the result: Every description of in space involves the use of a rigid to which such events to be referred. The resulting relationship for granted that the laws of geometry hold for distances; the being represented physically by of the convention of two marks on a rigid [ 9 ]

Section 3 — Space and in Classical Mechanics [ edit ]

The of mechanics is to describe how bodies their position in space time. I should load my with grave sins the sacred spirit of lucidity I to formulate the aims of mechanics in way, without serious and detailed explanations. Let us proceed to these sins.

It is not clear is to be understood here by position and I stand at the window of a railway which is travelling uniformly, and a stone on the embankment, without it. Then, disregarding the influence of the air I see the stone descend in a straight A pedestrian who observes the misdeed the footpath notices that the falls to earth in a parabolic I now ask: do the positions traversed by the lie in reality on a straight line or on a Moreover, what is meant by motion in space? From the of the previous section the answer is In the first place we entirely the vague word space, [ 10 ] of we must honestly acknowledge, we form the slightest conception, and we it by motion relative to a practically body of reference. The positions to the body of reference (railway or embankment) have already defined in detail in the preceding If instead of body of reference we system of co-ordinates, which is a idea for mathematical description, we are in a to say: The stone traverses a line relative to a system of rigidly attached to the carriage, but to a system of co-ordinates rigidly to the ground (embankment) it describes a With the aid of this example it is seen that there is no thing as an independently existing (lit. path-curve [6] ), but only a relative to a particular body of

In order to have a complete of the motion, we must specify how the alters its position with ; i.e. for every point on the it must be stated at what the body is situated there. data must be supplemented by a definition of time that, in of this definition, these can be regarded essentially as magnitudes of measurements) capable of observation. If we our stand on the ground of classical [ 11 ] we can satisfy this requirement for our in the following manner. We imagine two of identical construction; the man at the railway-carriage is holding one of them, and the man on the footpath the Each of the observers determines the on his own reference-body occupied by the stone at tick of the clock he is holding in his In this connection we have not account of the inaccuracy involved by the of the velocity of propagation of light. this and with a second prevailing here we shall to deal in detail later. [ 12 ]

4 — The Galileian System of [ edit ]

As is well known, the law of the mechanics of Galilei-Newton, which is as the law of inertia . can be stated thus: a removed sufficiently far from bodies continues in a state of or of uniform motion in a straight This law not only says about the motion of the bodies, but it indicates the reference-bodies or systems of permissible in mechanics, which can be in mechanical description. The visible stars are bodies for which the law of certainly holds to a high of approximation. Now if we use a system of co-ordinates is rigidly attached to the earth, relative to this system, fixed star describes a of immense radius in the course of an day, a result which is to the statement of the law of inertia. So that if we to this law we must refer motions only to systems of relative to which the fixed do not move in a circle. A system of of [ 13 ] which the state of motion is that the law of inertia holds to it is called a Galileian system of The laws of the mechanics of Galilei-Newton can be as valid only for a Galileian of co-ordinates. [ 14 ]

Section 5 — The of Relativity (In the Restricted Sense) [ ]

In order to attain the greatest clearness, let us return to our example of the carriage supposed to be travelling We call its motion a uniform (uniform because it is of constant and direction, translation because the carriage changes its position to the embankment yet it does not rotate in so Let us imagine a raven flying the air in such a manner that its as observed from the embankment, is and in a straight line. If we were to the flying raven from the railway carriage. we should that the motion of the raven be one of different velocity and direction, but it would still be uniform and in a line. Expressed in an abstract we may say: If a mass is moving in a straight line with to a co-ordinate system. then it also be moving uniformly and in a line relative to a second system provided that [ 15 ] the is executing a uniform translatory with respect to. In accordance the discussion contained in the preceding it follows that:

If is a Galileian system. then every co-ordinate system is a Galileian when, in relation to. it is in a condition of motion of translation. Relative to the laws of Galilei-Newton hold exactly as they do with to .

We advance a step farther in our when we express the tenet If, relative to. is a uniformly moving system devoid of rotation, natural phenomena run their with respect to according to the same general laws as respect to. This statement is the principle of relativity (in the restricted

As long as one was convinced that all phenomena were capable of with the help of classical there was no need to doubt the of this principle of relativity. But in of the more recent development of and optics it became more and evident that classical affords an insufficient foundation for the description of all natural phenomena. At juncture the question of the validity of the of relativity became ripe for and it did not appear [ 16 ] impossible that the to this question might be in the

Nevertheless, there are two general which at the outset speak much in favour of the validity of the of relativity. Even though mechanics does not supply us a sufficiently broad basis for the presentation of all physical phenomena, we must grant it a considerable of truth, since it supplies us the actual motions of the heavenly with a delicacy of detail short of wonderful. The principle of must therefore apply great accuracy in the domain of . But that a principle of such generality should hold such exactness in one domain of and yet should be invalid for another, is a not very probable.


We now proceed to the argument, to which, moreover, we return later. If the principle of (in the restricted sense) does not then the Galileian co-ordinate etc. which are moving relative to each other, not be equivalent for the description of natural In this case we should be to believe that natural are capable of being formulated in a simple manner, and of course on condition that, from all possible Galileian [ 17 ] co-ordinate we should have chosen one () of a state of motion as our body of We should then be justified of its merits for the description of natural in calling this system at rest, and all other Galileian in motion. If, for instance, our embankment the system then our railway would be a system. relative to less simple laws hold than with to. This diminished simplicity be due to the fact that the carriage be in motion ( i.e .really) with to. In the general laws of nature have been formulated reference to. the magnitude and direction of the of the carriage would necessarily a part. We should expect, for that the note emitted by an placed with its axis to the direction of travel would be from that emitted if the of the pipe were placed to this direction. Now in virtue of its in an orbit round the sun, our is comparable with a railway travelling with a velocity of 30 kilometres per second. If the principle of were not valid we should expect that the direction of of the earth at any moment would into the laws of nature, and that physical systems in behaviour would be dependent on the in space [ 18 ] with respect to the For owing to the alteration in direction of the of revolution of the earth in the course of a the earth cannot be at rest to the hypothetical system throughout the year. However, the most observations have never such anisotropic properties in physical space, i.e. a non-equivalence of different directions. is a very powerful argument in of the principle of relativity. [ 19 ]

Section 6 The Theorem of the Addition of Velocities in Classical Mechanics [ edit ]

Let us our old friend the railway carriage to be along the rails with a velocity. and that a man traverses the of the carriage in the direction of travel a velocity. How quickly or, in other with what velocity the man advance relative to the embankment the process? The only possible seems to result from the consideration: If the man were to stand for a second, he would advance to the embankment through a distance numerically to the velocity of the carriage. As a of his walking, however, he traverses an distance relative to the carriage, and also relative to the embankment, in second, the distance being equal to the velocity with he is walking. Thus in total he the distance relative to the embankment in the considered. We shall see later this result, which the theorem of the addition [ 20 ] of velocities in classical mechanics, cannot be in other words, the law that we just written down not hold in reality. For the time however, we shall assume its [ 21 ]

Section 7 — The Apparent of the Law of Propagation of Light with the of Relativity [ edit ]

There is a simpler law in physics than according to which light is in empty space. Every at school knows, or believes he that this propagation place in straight lines a velocity = 300,000 km./sec. At all we know with great that this velocity is the for all colours, because if this not the case, the minimum of emission not be observed simultaneously for different during the eclipse of a fixed by its dark neighbour. By means of considerations based on observations of stars, the Dutch astronomer De was also able to show the velocity of propagation of light depend on the velocity of motion of the emitting the light. The assumption this velocity of propagation is on the direction in space is in itself

In short, let us assume that the law of the constancy of the velocity of light (in [ 22 ] is justifiably believed by the child at Who would imagine that simple law has plunged the conscientiously physicist into the greatest difficulties? Let us consider how these arise.

Of course we must the process of the propagation of light indeed every other to a rigid reference-body (co-ordinate As such a system let us again our embankment. We shall imagine the air it to have been removed. If a ray of be sent along the embankment, we see the above that the tip of the ray will be with the velocity relative to the Now let us suppose that our railway is again travelling along the lines with the velocity. and its direction is the same as that of the ray of but its velocity of course much Let us inquire about the velocity of of the ray of light relative to the carriage. It is that we can here apply the of the previous section, since the ray of plays the part of the man walking relatively to the carriage. The velocity of the man to the embankment is here replaced by the of light relative to the embankment. is the velocity of light with to the carriage, and we have

[ 23 ] The velocity of of a ray of light relative to the carriage comes out smaller than .

But result comes into with the principle of relativity set in Section 5. For, like other general law of nature, the law of the of light in vacuo must, to the principle of relativity, be the same for the carriage as reference-body as when the are the body of reference. But, our above consideration, this appear to be impossible. If every ray of is propagated relative to the embankment the velocity. then for this it would appear that law of propagation of light must hold with respect to the — a result contradictory to the principle of

In view of this dilemma appears to be nothing else for it to abandon either the principle of or the simple law of the propagation of light in . Those of you who have carefully the preceding discussion are almost to expect that we should the principle of relativity, which so convincingly to the intellect because it is so and simple. The law of the propagation of light in would then have to be by a more complicated law conformable to the of relativity. The development of theoretical [ 24 ] shows, however, that we pursue this course. The theoretical investigations of H. A. Lorentz on the and optical phenomena connected moving bodies show experience in this domain conclusively to a theory of electromagnetic of which the law of the constancy of the velocity of in vacuo is a necessary consequence. theoretical physicists were more inclined to reject the of relativity, in spite of the fact no empirical data had been which were contradictory to principle.

At this juncture the of relativity entered the arena. As a of an analysis of the physical conceptions of and space, it became evident in reality there is not the least between the principle of relativity and the law of of light . and that by systematically fast to both these a logically rigid theory be arrived at. This theory has called the special theory of to distinguish it from the extended with which we shall later. In the following pages we present the fundamental ideas of the theory of relativity. [ 25 ]

Section 8 On the Idea of Time in Physics [ ]

Lightning has struck the rails on our embankment at two places and far distant each other. I make the assertion that these two flashes occurred simultaneously. If I ask you there is sense in this you will answer my question a decided Yes. But if I now approach you the request to explain to me the sense of the more precisely, you find some consideration that the to this question is not so easy as it at first sight.

After time perhaps the following would occur to you: The of the statement is clear in itself and no further explanation; of course it require some consideration if I to be commissioned to determine by observations in the actual case the two events place simultaneously or not. I be satisfied with this for the following reason. Supposing as a result of ingenious considerations an meteorologist were to discover [ 26 ] the lightning must always the places and simultaneously, then we be faced with the task of whether or not this theoretical is in accordance with the reality. We the same difficulty with all statements in which the conception plays a part. The concept not exist for the physicist until he has the of discovering whether or not it is fulfilled in an case. We thus require a of simultaneity such that definition supplies us with the by means of which, in the present he can decide by experiment whether or not the lightning strokes occurred As long as this requirement is not I allow myself to be deceived as a (and of course the same if I am not a physicist), when I imagine I am able to attach a meaning to the of simultaneity. (I would ask the reader not to farther until he is fully on this point.)

After the matter over for some you then offer the following with which to test By measuring along the rails, the line should be measured up and an placed at the mid-point of the distance. observer should be supplied an arrangement ( e.g. two mirrors at 90°) which allows him to observe [ 27 ] both places and at the time. If the observer perceives the two of lightning at the same time, they are simultaneous.

I am very with this suggestion, but for all I cannot regard the matter as settled, because I feel to raise the following objection: definition would certainly be if only I knew that the by means of which the observer at the lightning flashes travels the length with the same as along the length. But an examination of supposition would only be if we already had at our disposal the means of time. It would thus as though we were moving in a logical circle.

After consideration you cast a somewhat glance at me — and rightly so — and you declare: I my previous definition nevertheless, in reality it assumes absolutely about light. There is one demand to be made of the definition of namely, that in every case it must supply us an empirical decision as to whether or not the that has to be defined is fulfilled. my definition satisfies this is indisputable. That light the same time to traverse the as for the path is in reality neither a nor a hypothesis about the physical of light, [ 28 ] but a stipulation which I can of my own freewill in order to arrive at a of simultaneity.

It is clear that definition can be used to give an meaning not only to two events, but to as events as we care to choose, and of the positions of the scenes of the events respect to the body of reference [7] the railway embankment). We are thus led to a definition of time in physics. For purpose we suppose that of identical construction are placed at the and of the railway line (co-ordinate and that they are set in such a that the positions of their are simultaneously (in the above sense) the Under these conditions we by the time of an event the reading of the hands) of that one of these which is in the immediate vicinity (in of the event. In this manner a is associated with every which is essentially capable of

This stipulation contains a physical [ 29 ] hypothesis, the validity of will hardly be doubted empirical evidence to the contrary. It has assumed that all these go at the same rate if they are of construction. Stated more When two clocks arranged at in different places of a reference-body are set in a manner that a particular of the pointers of the one clock is simultaneous (in the sense) with the same of the pointers of the other clock, identical settings are always (in the sense of the above definition). [ 30 ]

9 — The Relativity of Simultaneity [ ]

Up to now our considerations have been to a particular body of reference, we have styled a railway We suppose a very long travelling along the rails the constant velocity and in the direction in Fig 1. People travelling in this will with a vantage the train as a rigid reference-body system); they regard all in

reference to the train. Then event which takes along the line also place at a particular point of the Also the definition of simultaneity can be relative to the train in exactly the way as with respect to the embankment. As a consequence, however, the following arises:

Are two events ( e.g. the two of lightning and ) which are simultaneous reference to [ 31 ] the railway embankment simultaneous relatively to the train . We show directly that the must be in the negative.

When we say the lightning strokes and are simultaneous respect to the embankment, we mean: the of light emitted at the places where the lightning occurs, each other at the mid-point of the of the embankment. But the events and also to positions and on the train. Let be the mid-point of the on the travelling train. Just the flashes [8] of lightning occur, point naturally coincides the point but it moves towards the in the diagram with the velocity of the If an observer sitting in the position in the did not possess this velocity, he would remain permanently at. and the rays emitted by the flashes of and would reach him simultaneously, they would meet where he is situated. Now in reality with reference to the railway he is hastening towards the beam of coming from. whilst he is on ahead of the beam of light from. Hence the observer see the beam of light emitted earlier than he will see emitted from. Observers who the railway train as their [ 32 ] must therefore come to the that the lightning flash place earlier than the flash. We thus arrive at the result:

Events which are with reference to the embankment are not with respect to the train, and versa (relativity of simultaneity). reference-body (co-ordinate system) has its own time; unless we are told the to which the statement of time there is no meaning in a statement of the of an event.

Now before the advent of the of relativity it had always tacitly assumed in physics that the of time had an absolute significance, that it is independent of the state of of the body of reference. But we have seen that this is incompatible with the most definition of simultaneity; if we discard assumption, then the conflict the law of the propagation of light in vacuo and the of relativity (developed in Section 7 )

We were led to that conflict by the of Section 6. which are now no longer In that section we concluded the man in the carriage, who traverses the distance w per relative to the carriage, traverses the distance also with to the embankment [ 33 ] in each second of But, according to the foregoing the time required by a particular with respect to the carriage not be considered equal to the duration of the occurrence as judged from the (as reference-body). Hence it cannot be that the man in walking travels the w relative to the railway line in a which is equal to one second as from the embankment.

Moreover, the of Section 6 are based on yet a second which, in the light of a strict appears to be arbitrary, although it was tacitly made even the introduction of the theory of relativity. [ 34 ]

10 — On the Relativity of the Conception of [ edit ]

Let us consider two particular on the train [9] travelling along the with the velocity. and inquire as to distance apart. We already that it is necessary to have a of reference for the measurement of a distance, respect to which body the can be measured up. It is the simplest plan to use the itself as reference-body (co-ordinate An observer in the train measures the by marking off his measuring-rod in a straight ( e.g. along the floor of the as many times as is necessary to him from the one marked point to the Then the number which us how often the rod has to be laid down is the distance.

It is a different matter the distance has to be judged from the line. Here the following suggests itself. If we call and the two on the train whose distance is required, then both of points are [ 35 ] moving with the along the embankment. In the first we require to determine the points and of the which are just being by the two points and at a particular time — from the embankment. These and of the embankment can be determined by applying the of time given in Section 8. The between these points and is measured by repeated application of the along the embankment.

A priori it is by no certain that this measurement will supply us the same result as the first. the length of the train as measured the embankment may be different from obtained by measuring in the train This circumstance leads us to a objection which must be against the apparently obvious of Section 6. Namely, if the man in the carriage the distance in a unit of time — from the train . — then distance — as measured from the — is not necessarily also equal to. [ 36 ]

11 — The Lorentz Transformation [ ]

The results of the last three show that the apparent of the law of propagation of light with the of relativity (Section 7 ) has been by means of a consideration which two unjustifiable hypotheses from mechanics; these are as follows:

(1) The (time) between two events is of the condition of motion of the body of

(2) The space-interval (distance) between two of a rigid body is independent of the of motion of the body of reference.

If we these hypotheses, then the of Section 7 disappears, because the of the addition of velocities derived in 6 becomes invalid. The possibility itself that the law of the propagation of in vacuo may be compatible with the of relativity, and the question arises: How we to modify the considerations of Section 6 in to remove [ 37 ] the apparent disagreement these two fundamental results of This question leads to a one. In the discussion of Section 6 we to do with places and times both to the train and to the embankment. How are we to the place and time of an event in to the train, when we know the and time of the event with to the railway embankment? Is there a answer to this question of a nature that the law of transmission of in vacuo does not contradict the of relativity? In other words: Can we of a relation between place and of the individual events relative to reference-bodies, such that ray of light possesses the velocity of relative to the embankment and relative to the This question leads to a definite positive answer, and to a definite transformation law for the space-time of an event when changing from one body of reference to

Before we deal with we shall introduce the following consideration. Up to the present we have considered events taking along the embankment, which had to assume the function of a straight In the manner indicated in Section 2 we can this reference-body supplemented and in a vertical direction by means of a [ 38 ] of rods, so that an event takes place anywhere can be with reference to this Similarly, we can imagine the train with the velocity to be continued the whole of space, so that event, no matter how far off it may be, could be localised with respect to the framework. Without committing any error, we can disregard the fact in reality these frameworks continually interfere with other, owing to the impenetrability of bodies. In every such we imagine three surfaces to each other marked and designated as co-ordinate planes system). A co-ordinate system corresponds to the embankment, and a co-ordinate to the train. An event, wherever it may taken place, would be in space with respect to by the perpendiculars. on the co-ordinate planes, and regard to time by a time Relative to. the same event be fixed in respect of space and by corresponding values. which of are not identical with. It has already set forth in detail how these are to be regarded as results of physical

Obviously our problem can be exactly in the following manner. What are the [ 39 ] of an event with respect to. the magnitudes. of the same event respect to are given? The relations be so chosen that the law of the transmission of in vacuo is satisfied for one and the same ray of (and of course for every with respect to and. For the orientation in space of the co-ordinate indicated in the diagram ([7]Fig. 2), problem is solved by means of the :

This system of equations is as the Lorentz transformation. [10]

If in of the law of transmission of light we had taken as our the tacit assumptions of the older as to the absolute character [ 40 ] of times and then instead of the above we have obtained the following

This system of equations is termed the Galilei transformation. The transformation can be obtained from the transformation by substituting an infinitely value for the velocity of light in the transformation.

Aided by the following illustration, we can see that, in accordance with the transformation, the law of the transmission of light in is satisfied both for the reference-body and for the A light-signal is sent along the -axis, and this light-stimulus in accordance with the equation

with the velocity. According to the of the Lorentz transformation, this relation between and involves a between and. In point of if we substitute for the value in the first and equations of the Lorentz transformation, we

[ 41 ] from which, by division, the

immediately follows. If referred to the the propagation of light takes according to this equation. We see that the velocity of transmission to the reference-body is also equal to. The result is obtained for rays of advancing in any other direction Of cause this is not surprising, the equations of the Lorentz transformation derived conformably to this of view. [ 42 ]

Section 12 — The of Measuring-Rods and Clocks in Motion [ ]

Place a metre-rod in the -axis of in a manner that one end (the coincides with the point the other end (the end of the rod) with the point. What is the of the metre-rod relatively to the system. In to learn this, we need ask where the beginning of the rod and the end of the rod lie with to at a particular time of the system. By of the first equation of the Lorentz the values of these two points at the can be shown to be

the distance between the being. But the metre-rod is moving the velocity relative to. It therefore that the length of a rigid moving in the direction of its length a velocity is of a metre. The rigid rod is shorter when in motion [ 43 ] when at rest, and the more it is moving, the shorter is the rod. For the we should have. and for still velocities the square-root becomes From this we conclude in the theory of relativity the velocity the part of a limiting velocity, can neither be reached nor exceeded by any body.

Of course this of the velocity as a limiting velocity clearly follows from the of the Lorentz transformation, for these meaningless if we choose values of than .

If, on the contrary, we had considered a at rest in the -axis with to. then we should have that the length of the rod as judged would have been ; is quite in accordance with the of relativity which forms the of our considerations.

A priori it is quite that we must be able to something about the physical of measuring-rods and clocks from the of transformation, for the magnitudes. are nothing nor less than the results of obtainable by means of measuring-rods and If we had based our considerations on the Galileian we should not have obtained a of the rod as a consequence of its motion.

[ 44 ] Let us now consider a which is permanently situated at the () of. and are two successive ticks of this The first and fourth equations of the transformation give for these two

and

As judged from. the clock is with the velocity ; as judged this reference-body, the time elapses between two strokes of the is not one second, but seconds, i.e. a larger time. As a consequence of its the clock goes more than when at rest. also the velocity plays the of an unattainable limiting velocity. [ 45 ]

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13 — Theorem of the Addition of The Experiment of Fizeau [ edit ]

Now in we can move clocks and measuring-rods with velocities that are compared with the velocity of hence we shall hardly be to compare the results of the previous directly with the reality. on the other hand, these must strike you as being singular, and for that reason I now draw another conclusion the theory, one which can easily be from the foregoing considerations, and has been most elegantly by experiment.

In Section 6 we derived the of the addition of velocities in one direction in the which also results the hypotheses of classical mechanics. theorem can also be deduced from the Galilei transformation 11 ). In place of the man walking inside the we introduce a point moving to the co-ordinate system in accordance the equation

By means of the first and equations of the [ 46 ] Galilei transformation we can and in terms of and. and we then

This equation expresses else than the law of motion of the with reference to the system (of the man reference to the embankment). We denote velocity by the symbol. and we then as in Section 6 ,

which corresponds to the of addition for velocities in one direction to the theory of relativity. The question now as to which of these two theorems is the in accord with experience. On point we are enlightened by a most experiment which the brilliant Fizeau performed more half a century ago, and has been repeated since [ 47 ] by some of the best experimental so that there can be no doubt its result. The experiment is concerned the following question. Light in a motionless liquid with a velocity. How quickly does it in the direction of the arrow in the tube T the accompanying diagram, Fig. 3) the liquid above mentioned is through the tube with a ?

In accordance with the principle of we shall certainly have to for granted that the propagation of always takes place the same velocity with to the liquid . whether the latter is in with reference to other or not. The velocity of light to the liquid and the velocity of the latter to the tube are thus known, and we the velocity of light relative to the

It is clear that we have the of Section 6 again before us. The plays the part of the railway or of the co-ordinate system. the liquid the part of the carriage or of the co-ordinate and finally, the light plays the of the man walking along the carriage, or of the point in the present [ 48 ] section. If we the velocity of the light relative to the by. then this is given by the (A) or (B), according as the Galilei or the Lorentz transformation corresponds to the Experiment [11] decides in of equation (B) derived from the of relativity, and the agreement is, indeed, exact. According to recent and excellent measurements by Zeeman, the of the velocity of flow on the propagation of is represented by formula (B) to within one per

Nevertheless we must now draw to the fact that a theory of phenomenon was given by H. A. Lorentz before the statement of the theory of This theory was of a purely nature, and was obtained by the use of particular as to the electromagnetic structure of matter. circumstance, however, does not in the diminish the conclusiveness of the experiment as a test in favour of the theory of for the [ 49 ] electrodynamics of Maxwell-Lorentz, on which the theory was based, in no way opposes the of relativity. Rather has the latter developed from electrodynamics as an simple combination and generalisation of the formerly independent of each on which electrodynamics was built. [ 50 ]

14 — The Heuristic Value of the of Relativity [ edit ]

Our train of in the foregoing pages can be epitomised in the manner. Experience has led to the conviction on the one hand, the principle of relativity true and that on the other the velocity of transmission of light in has to be considered equal to a constant. By these two postulates we obtained the law of for the rectangular co-ordinates. and the time of the which constitute the processes of In this connection we did not obtain the transformation, but, differing classical mechanics, the Lorentz .

The law of transmission of light, the acceptance of is justified by our actual knowledge, an important part in this of thought. Once in possession of the transformation, however, we can combine with the principle of relativity, and sum up the thus:

Every general law of must be so constituted that it is into a law of exactly the same when, instead of the space-time [ 51 ] of the original coordinate system. we new space-time variables. of a co-ordinate In this connection the relation the ordinary and the accented magnitudes is by the Lorentz transformation. Or in brief: laws of nature are co-variant respect to Lorentz transformations.

is a definite mathematical condition the theory of relativity demands of a law, and in virtue of this, the becomes a valuable heuristic aid in the for general laws of nature. If a law of nature were to be found did not satisfy this condition, at least one of the two fundamental assumptions of the would have been Let us now examine what general the latter theory has hitherto [ 52 ]

Section 15 — General of the Theory [ edit ]

It is clear from our previous that the (special) theory of has grown out of electrodynamics and optics. In fields it has not appreciably altered the of theory, but it has considerably simplified the structure, i.e. the derivation of and — what is incomparably more — it has considerably reduced the number of hypotheses forming the basis of The special theory of relativity has the Maxwell-Lorentz theory so plausible, the latter would have generally accepted by physicists if experiment had decided less in its favour.

Classical mechanics to be modified before it could into line with the of the special theory of relativity. For the part, however, this affects only the laws for motions, in which the velocities of are not very small as compared the velocity of light. We have of such rapid motions in the case of electrons and [ 53 ] ions; for motions the variations from the of classical mechanics are too small to themselves evident in practice. We not consider the motion of stars we come to speak of the general of relativity. In accordance with the of relativity the kinetic energy of a point of mass is no longer by the well-known expression

but by the expression

expression approaches infinity as the approaches the velocity of light. The must therefore always less than. however may be the energies used to produce the If we develop the expression for the kinetic in the form of a series, we obtain

is small compared with the third of these terms is small in comparison with the which last is alone in classical mechanics. The first does not contain the velocity, and no consideration if we [ 54 ] are only dealing the question as to how the energy of a point-mass on the velocity. We shall speak of its significance later.

The most result of a general character to the special theory of relativity has led is with the conception of mass. the advent of relativity, physics two conservation laws of fundamental namely, the law of the conservation of energy and the law of the of mass these two fundamental appeared to be quite independent of other. By means of the theory of they have been into one law. We shall now consider how this unification about, and what meaning is to be to it.

The principle of relativity requires the law of the conservation of energy should not only with reference to a system. but also with to every co-ordinate system is in a state of uniform motion of relative to. or, briefly, relative to Galileian system of co-ordinates. In to classical mechanics, the Lorentz is the deciding factor in the transition one such system to another.

By of comparatively simple considerations we are led to the following conclusion from premises, in conjunction with the [ 55 ] equations of the electrodynamics of Maxwell: A moving with the velocity. absorbs [12] an amount of in the form of radiation without an alteration in velocity in the process, as a consequence, its energy increased by an

In consideration of the expression given for the kinetic energy of the body, the energy of the body comes out to be

the body has the same energy as a of mass moving with the Hence we can say: If a body up an amount of energy. then its mass increases by an amount ; the mass of a body is not a constant but according to the change in the energy of the The inertial mass of a system of can even be regarded as a measure [ 56 ] of its The law of the conservation of the mass of a system identical with the law of the conservation of and is only valid provided the system neither takes up nor out energy. Writing the expression for the in the form

we see that the term. has hitherto attracted our attention, is else than the energy by the body [13] before it the energy .

A direct comparison of relation with experiment is not at the present time, owing to the that the changes in energy to we can subject a system are not large to make themselves perceptible as a in the inertial mass of the system. is too in comparison with the mass. was present before the alteration of the It is owing to this circumstance classical mechanics was able to successfully the conservation of mass as a law of validity.

Let me add a final remark of a nature. The success of the Faraday-Maxwell [ 57 ] of electromagnetic action at a distance in physicists becoming convinced there are no such things as actions at a distance (not an intermediary medium) of the type of law of gravitation. According to the theory of action at a distance with the of light always takes the of instantaneous action at a distance or of at a distance with an infinite of transmission. This is connected the fact that the velocity a fundamental role in this In Part II we shall see in what way result becomes modified in the theory of relativity. [ 58 ]

Section 16 Experience and the Special Theory of [ edit ]

To what extent is the theory of relativity supported by This question is not easily for the reason already mentioned in with the fundamental experiment of The special theory of relativity has out from the Maxwell-Lorentz theory of phenomena. Thus all facts of which support the electromagnetic also support the theory of As being of particular importance, I here the fact that the of relativity enables us to predict the produced on the light reaching us the fixed stars. These are obtained in an exceedingly simple and the effects indicated, which are due to the motion of the earth with to those fixed stars are to be in accord with experience. We to the yearly movement of the apparent of the fixed stars resulting the motion of the earth round the sun and to the influence of the radial [ 59 ] components of the motions of the fixed stars respect to the earth on the colour of the reaching us from them. The effect manifests itself in a displacement of the spectral lines of the transmitted to us from a fixed as compared with the position of the spectral lines when are produced by a terrestrial source of (Doppler principle). The experimental in favour of the Maxwell-Lorentz theory, are at the same time arguments in of the theory of relativity, are too numerous to be set here. In reality they the theoretical possibilities to such an that no other theory that of Maxwell and Lorentz has able to hold its own when by experience.

But there are two classes of facts hitherto obtained can be represented in the Maxwell-Lorentz theory by the introduction of an auxiliary hypothesis, in itself — i.e. without use of the theory of relativity — appears

It is known that cathode and the so-called β -rays emitted by substances consist of negatively particles (electrons) of very inertia and large velocity. By the deflection of these rays the influence of electric and magnetic we can study the law of motion of these very exactly.

[ 60 ] In the theoretical of these electrons, we are faced the difficulty that electrodynamic of itself is unable to give an of their nature. For since masses of one sign repel other, the negative electrical constituting the electron would be scattered under the influence of mutual repulsions, unless are forces of another kind between them, the nature of has hitherto remained obscure to us. If we now assume that the relative between the electrical masses the electron remain unchanged the motion of the electron (rigid in the sense of classical mechanics), we at a law of motion of the electron which not agree with experience. by purely formal points of H. A. Lorentz was the first to introduce the that the form of the electron a contraction in the direction of motion in of that motion. the contracted being proportional to the expression. hypothesis, which is not justifiable by any facts, supplies us then that particular law of motion has been confirmed with precision in recent years.

[ 61 ] The of relativity leads to the same law of without requiring any special whatsoever as to the structure and the behaviour of the We arrived at a similar conclusion in 13 in connection with the experiment of the result of which is foretold by the of relativity without the necessity of on hypotheses as to the physical nature of the

The second class of facts to we have alluded has reference to the whether or not the motion of the earth in can be made perceptible in terrestrial We have already remarked in 5 that all attempts of this led to a negative result. Before the of relativity was put forward, it was difficult to reconciled to this negative for reasons now to be discussed. The inherited about time and space did not any doubt to arise as to the prime of the Galileian transformation for changing from one body of reference to Now assuming that the Maxwell-Lorentz hold for a reference-body. we then that they do not hold for a moving uniformly with to. if we assume that the relations of the transformation exist between the of and. It thus appears of all Galileian co-ordinate [ 62 ] systems one () to a particular state of motion is unique. This result was physically by regarding as at rest respect to a hypothetical æther of On the other hand, all coordinate moving relatively to were to be as in motion with respect to the To this motion of against the (æther-drift relative to ) were the more complicated laws were supposed to hold to. Strictly speaking, such an ought also to be assumed to the earth, and for a long time the of physicists were devoted to to detect the existence of an æther-drift at the surface.

In one of the most notable of these Michelson devised a method appears as though it must be Imagine two mirrors so arranged on a body that the reflecting face each other. A ray of requires a perfectly definite to pass from one mirror to the and back again, if the whole be at rest with respect to the It is found by calculation, however, a slightly different time is for this process, if the body, with the mirrors, be moving to the æther. And yet another point: it is by calculation that for a given with reference to the æther, time is different [ 63 ] when the is moving perpendicularly to the planes of the from that resulting the motion is parallel to these Although the estimated difference these two times is exceedingly Michelson and Morley performed an involving interference in which difference should have clearly detectable. But the experiment a negative result — a fact perplexing to physicists. Lorentz and rescued the theory from difficulty by assuming that the of the body relative to the æther a contraction of the body in the direction of the amount of contraction being sufficient to compensate for the difference in mentioned above. Comparison the discussion in Section 11 shows also from the standpoint of the of relativity this solution of the was the right one. But on the basis of the of relativity the method of interpretation is more satisfactory. According to theory there is no such as a specially favoured (unique) system to occasion the introduction of the and hence there can be no æther-drift, nor any with which to demonstrate it. the contraction of moving bodies from the two fundamental principles of the without the introduction of particular and as the [ 64 ] prime factor involved in contraction we find, not the motion in to which we cannot attach any but the motion with respect to the of reference chosen in the particular in point. Thus for a co-ordinate moving with the earth the system of Michelson and Morley is not but it is shortened for a co-ordinate system is at rest relatively to the sun. [ 65 ]

17 — Minkowski’s Four-Dimensional [ edit ]

The non-mathematician is seized by a shuddering when he hears of things, by a feeling not unlike awakened by thoughts of the occult. And yet is no more common-place statement that the world in which we is a four-dimensional space-time continuum.

is a three-dimensional continuum. By this we that it is possible to describe the of a point (at rest) by means of numbers (co-ordinates). and that is an indefinite number of points in the of this one, the position of can be described by co-ordinates such as. may be as near as we choose to the respective of the co-ordinates. of the first point. In of the latter property we speak of a and owing to the fact that are three co-ordinates we speak of it as three-dimensional.

Similarly, the world of phenomena which was briefly world by Minkowski [ 66 ] is naturally dimensional in the space-time sense. For it is of individual events, each of is described by four numbers, three space co-ordinates. and a co-ordinate, the time value. The is in this sense also a for to every event there are as neighbouring events (realised or at thinkable) as we care to choose, the of which differ by an indefinitely amount from those of the originally considered. That we not been accustomed to regard the in this sense as a four-dimensional is due to the fact that in physics, the advent of the theory of relativity, played a different and more role, as compared with the coordinates. It is for this reason we have been in the habit of time as an independent continuum. As a of fact, according to classical time is absolute, i.e . it is independent of the and the condition of motion of the system of We see this expressed in the last of the Galileian transformation ().

The four-dimensional of consideration of the world is natural on the of relativity, since according to theory time is robbed of its This is shown by the fourth of the Lorentz transformation: [ 67 ]

Moreover, to this equation the time of two events with respect to not in general vanish, even the time difference of the same with reference to vanishes. space-distance of two events with to results in time-distance of the same with respect to. But the discovery of which was of importance for the formal of the theory of relativity, does not lie It is to be found rather in the fact of his that the four-dimensional space-time of the theory of relativity, in its most formal properties, shows a relationship to the three-dimensional continuum of geometrical space. [15] In to give due prominence to this however, we must replace the time co-ordinate by an imaginary proportional to it. Under these the natural laws satisfying the of the (special) theory of relativity mathematical forms, in which the co-ordinate plays exactly the role as the three space Formally, these four correspond exactly to the three co-ordinates in Euclidean geometry. It be clear even to the non-mathematician as a consequence of this purely addition to our knowledge, the theory gained clearness in no mean

These inadequate remarks can the reader only a vague of the important idea contributed by Without it the general theory of of which the fundamental ideas are in the following pages, would have got no farther than its clothes. Minkowski’s work is difficult of access to anyone in mathematics, but since it is not necessary to a very exact grasp of work in order to understand the ideas of either the special or the theory of relativity, I shall it here at present, and revert to it towards the end of Part 2.

↑ It follows a natural object is associated with a straight line. points A, B and C on a rigid body lie in a straight line when the A and C being given, B is chosen that the sum of the distances AB and BC is as short as This incomplete suggestion suffice for the present purpose. ↑ we have assumed that is nothing left over, that the measurement gives a number. This difficulty is got by the use of divided measuring-rods, the introduction of does not demand any fundamentally new ↑ I have chosen this as more familiar to the English than the Potsdamer Platz, which is referred to in the original. (R. W. L.) ↑ It is not here to investigate further the of the expression coincidence in space. conception is sufficiently obvious to that differences of opinion are likely to arise as to its applicability in ↑ A refinement and modification of these does not become necessary we come to deal with the theory of relativity, treated in the part of this book. ↑ is, a curve along which the moves. ↑ We suppose further, when three events. and in different places in such a that is simultaneous with and is with (simultaneous in the sense of the definition), then the criterion for the of the pair of events. is also This assumption is a physical about the propagation of light: it certainly be fulfilled if we are to maintain the law of the of the velocity of light in vacuo . ↑ As from the embankment ↑ e.g. the of the first and of the hundredth carriage. ↑ A derivation of the Lorentz transformation is in Appendix I ↑ Fizeau found. is the index of refraction of the liquid. On the hand, owing to the smallness of as with 1, we can replace (B) in the first by. or to the same order of approximation by. agrees with Fizeau’s ↑ is the energy taken up, as judged a co-ordinate system moving the body. ↑ As judged from a system moving with the ↑ The general theory of relativity it likely that the electrical of an electron are held together by forces. ↑ Cf. the somewhat more discussion in Appendix II .

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