Reading Help Relativity: The Special and General Theory
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 realily there is not the least `
` incompatibilitiy between the principle of relativity and the law of `
` propagation of light, and that by systematically holding fast to both `
` these laws a logically rigid theory could be arrived at. This theory `
` has been called the 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. `
` `
` `
` `
` ON THE IDEA OF TIME IN PHYSICS `
` `
` `
` Lightning has struck the rails on our railway embankment at two places `
` A and B far distant from each other. I make the additional assertion `
` that these two lightning flashes occurred simultaneously. If I ask you `
` whether there is sense in this statement, you will answer my question `
` with a decided "Yes." But if I now approach you with the request to `
` explain to me the sense of the statement more precisely, you find `
` after some consideration that the answer to this question is not so `
` easy as it appears at first sight. `
` `
` After some time perhaps the following answer would occur to you: "The `
` significance of the statement is clear in itself and needs no further `
` explanation; of course it would require some consideration if I were `
` to be commissioned to determine by observations whether in the actual `
` case the two events took place simultaneously or not." I cannot be `
` satisfied with this answer for the following reason. Supposing that as `
` a result of ingenious considerations an able meteorologist were to `
` discover that the lightning must always strike the places A and B `
` simultaneously, then we should be faced with the task of testing `
` whether or not this theoretical result is in accordance with the `
` reality. We encounter the same difficulty with all physical statements `
` in which the conception " simultaneous " plays a part. The concept `
` does not exist for the physicist until he has the possibility of `
` discovering whether or not it is fulfilled in an actual case. We thus `
` require a definition of simultaneity such that this definition `
` supplies us with the method by means of which, in the present case, he `
` can decide by experiment whether or not both the lightning strokes `
` occurred simultaneously. As long as this requirement is not satisfied, `
` I allow myself to be deceived as a physicist (and of course the same `
` applies if I am not a physicist), when I imagine that I am able to `
` attach a meaning to the statement of simultaneity. (I would ask the `
` reader not to proceed farther until he is fully convinced on this `
` point.) `
` `
` After thinking the matter over for some time you then offer the `
` following suggestion with which to test simultaneity. By measuring `
` along the rails, the connecting line AB should be measured up and an `
` observer placed at the mid-point M of the distance AB. This observer `
` should be supplied with an arrangement (e.g. two mirrors inclined at `
` 90^0) which allows him visually to observe both places A and B at the `
` same time. If the observer perceives the two flashes of lightning at `
` the same time, then they are simultaneous. `
` `
` I am very pleased with this suggestion, but for all that I cannot `
` regard the matter as quite settled, because I feel constrained to `
` raise the following objection: `
` `
` "Your definition would certainly be right, if only I knew that the `
` light by means of which the observer at M perceives the lightning `
` flashes travels along the length A arrow M with the same velocity as `
` along the length B arrow M. But an examination of this supposition `
` would only be possible if we already had at our disposal the means of `
` measuring time. It would thus appear as though we were moving here in `
` a logical circle." `
` `
` After further consideration you cast a somewhat disdainful glance at `
` me -- and rightly so -- and you declare: `
` `
` "I maintain my previous definition nevertheless, because in reality it `
` assumes absolutely nothing about light. There is only one demand to be `
` made of the definition of simultaneity, namely, that in every real `
` case it must supply us with an empirical decision as to whether or not `
` the conception that has to be defined is fulfilled. That my definition `
` satisfies this demand is indisputable. That light requires the same `
` time to traverse the path A arrow M as for the path B arrow M is in `
` reality neither a supposition nor a hypothesis about the physical `
` nature of light, but a stipulation which I can make of my own freewill `
` in order to arrive at a definition of simultaneity." `
` `
` It is clear that this definition can be used to give an exact meaning `
` not only to 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). `
` We are thus led also to a definition of " time " in physics. For this `
` purpose we suppose that clocks of identical construction are placed at `
` the points A, B and C of the railway line (co-ordinate system) and `
` that they are set in such a manner that the positions of their `
` pointers are simultaneously (in the above sense) the same. Under these `
` conditions we understand by the " time " of an event the reading `
` (position of the hands) of that one of these clocks which is in the `
` immediate vicinity (in space) of the event. In this manner a `
` time-value is associated with every event which is essentially capable `
` of observation. `
` `
` This stipulation contains a further physical hypothesis, the validity `
` of which will hardly be doubted without empirical evidence to the `
` contrary. It has been assumed that all these clocks go 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 other clock, then identical " `
` settings " are always simultaneous (in the sense of the above `
` definition). `
` `
` `
` Notes `
` `
` *) We suppose further, that, when three events A, B and C occur in `
` different places in such a manner that A is simultaneous with B and B `
` is simultaneous with C (simultaneous in the sense of the above `
` definition), then the criterion for the simultaneity of the pair of `
` events A, C is also satisfied. This assumption is a physical `
` hypothesis about the the of propagation of light: it must certainly be `
` fulfilled if we are to maintain the law of the constancy of the `
` velocity of light in vacuo. `
` `
` `
` `
` THE RELATIVITY OF SIMULATNEITY `
` `
` `
` Up to now our considerations have been referred to a particular body `
` of reference, which we have styled a " railway embankment." We suppose `
` a very long train travelling along the rails with the constant `
` velocity v and in the direction indicated in Fig 1. People travelling `
` in this train will with a vantage view the train as a rigid `
` reference-body (co-ordinate system); they regard all events in `
` `
` Fig. 01: file fig01.gif `
` `
` `
` 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 be embankment, we mean: the rays of light emitted at the `
` places A and B, where the lightning occurs, meet each other at the `
` mid-point M of the length A arrow B of the embankment. But the events `
` A and B also correspond to positions A and B on the train. Let M1 be `
` the mid-point of the distance A arrow B on the travelling train. Just `
` when the flashes (as judged from the embankment) of lightning occur, `
` this point M1 naturally coincides with the point M but it moves `
` towards the right in the diagram with the velocity v of the train. If `
` an observer sitting in the position M1 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 their `
` reference-body must therefore come to the conclusion that the `
` lightning flash B took place earlier than the lightning flash A. We `
` thus arrive at the important result: `
` `
` Events which are simultaneous with reference to the embankment are not `
` simultaneous with respect to the train, and vice versa (relativity of `
` simultaneity). Every reference-body (co-ordinate system) has its own `
` particular time ; unless we are told the reference-body to which the `
` statement of time refers, there is no meaning in a statement of the `
` time of an event. `
` `
` Now before the advent of the theory of relativity it had always `
` tacitly been assumed in physics that the statement of time had an `
` absolute significance, 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 tacitly made even before the `
`
` result of an analysis of the physical conceptions of time and space, `
` it became evident that in realily there is not the least `
` incompatibilitiy between the principle of relativity and the law of `
` propagation of light, and that by systematically holding fast to both `
` these laws a logically rigid theory could be arrived at. This theory `
` has been called the 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. `
` `
` `
` `
` ON THE IDEA OF TIME IN PHYSICS `
` `
` `
` Lightning has struck the rails on our railway embankment at two places `
` A and B far distant from each other. I make the additional assertion `
` that these two lightning flashes occurred simultaneously. If I ask you `
` whether there is sense in this statement, you will answer my question `
` with a decided "Yes." But if I now approach you with the request to `
` explain to me the sense of the statement more precisely, you find `
` after some consideration that the answer to this question is not so `
` easy as it appears at first sight. `
` `
` After some time perhaps the following answer would occur to you: "The `
` significance of the statement is clear in itself and needs no further `
` explanation; of course it would require some consideration if I were `
` to be commissioned to determine by observations whether in the actual `
` case the two events took place simultaneously or not." I cannot be `
` satisfied with this answer for the following reason. Supposing that as `
` a result of ingenious considerations an able meteorologist were to `
` discover that the lightning must always strike the places A and B `
` simultaneously, then we should be faced with the task of testing `
` whether or not this theoretical result is in accordance with the `
` reality. We encounter the same difficulty with all physical statements `
` in which the conception " simultaneous " plays a part. The concept `
` does not exist for the physicist until he has the possibility of `
` discovering whether or not it is fulfilled in an actual case. We thus `
` require a definition of simultaneity such that this definition `
` supplies us with the method by means of which, in the present case, he `
` can decide by experiment whether or not both the lightning strokes `
` occurred simultaneously. As long as this requirement is not satisfied, `
` I allow myself to be deceived as a physicist (and of course the same `
` applies if I am not a physicist), when I imagine that I am able to `
` attach a meaning to the statement of simultaneity. (I would ask the `
` reader not to proceed farther until he is fully convinced on this `
` point.) `
` `
` After thinking the matter over for some time you then offer the `
` following suggestion with which to test simultaneity. By measuring `
` along the rails, the connecting line AB should be measured up and an `
` observer placed at the mid-point M of the distance AB. This observer `
` should be supplied with an arrangement (e.g. two mirrors inclined at `
` 90^0) which allows him visually to observe both places A and B at the `
` same time. If the observer perceives the two flashes of lightning at `
` the same time, then they are simultaneous. `
` `
` I am very pleased with this suggestion, but for all that I cannot `
` regard the matter as quite settled, because I feel constrained to `
` raise the following objection: `
` `
` "Your definition would certainly be right, if only I knew that the `
` light by means of which the observer at M perceives the lightning `
` flashes travels along the length A arrow M with the same velocity as `
` along the length B arrow M. But an examination of this supposition `
` would only be possible if we already had at our disposal the means of `
` measuring time. It would thus appear as though we were moving here in `
` a logical circle." `
` `
` After further consideration you cast a somewhat disdainful glance at `
` me -- and rightly so -- and you declare: `
` `
` "I maintain my previous definition nevertheless, because in reality it `
` assumes absolutely nothing about light. There is only one demand to be `
` made of the definition of simultaneity, namely, that in every real `
` case it must supply us with an empirical decision as to whether or not `
` the conception that has to be defined is fulfilled. That my definition `
` satisfies this demand is indisputable. That light requires the same `
` time to traverse the path A arrow M as for the path B arrow M is in `
` reality neither a supposition nor a hypothesis about the physical `
` nature of light, but a stipulation which I can make of my own freewill `
` in order to arrive at a definition of simultaneity." `
` `
` It is clear that this definition can be used to give an exact meaning `
` not only to 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). `
` We are thus led also to a definition of " time " in physics. For this `
` purpose we suppose that clocks of identical construction are placed at `
` the points A, B and C of the railway line (co-ordinate system) and `
` that they are set in such a manner that the positions of their `
` pointers are simultaneously (in the above sense) the same. Under these `
` conditions we understand by the " time " of an event the reading `
` (position of the hands) of that one of these clocks which is in the `
` immediate vicinity (in space) of the event. In this manner a `
` time-value is associated with every event which is essentially capable `
` of observation. `
` `
` This stipulation contains a further physical hypothesis, the validity `
` of which will hardly be doubted without empirical evidence to the `
` contrary. It has been assumed that all these clocks go 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 other clock, then identical " `
` settings " are always simultaneous (in the sense of the above `
` definition). `
` `
` `
` Notes `
` `
` *) We suppose further, that, when three events A, B and C occur in `
` different places in such a manner that A is simultaneous with B and B `
` is simultaneous with C (simultaneous in the sense of the above `
` definition), then the criterion for the simultaneity of the pair of `
` events A, C is also satisfied. This assumption is a physical `
` hypothesis about the the of propagation of light: it must certainly be `
` fulfilled if we are to maintain the law of the constancy of the `
` velocity of light in vacuo. `
` `
` `
` `
` THE RELATIVITY OF SIMULATNEITY `
` `
` `
` Up to now our considerations have been referred to a particular body `
` of reference, which we have styled a " railway embankment." We suppose `
` a very long train travelling along the rails with the constant `
` velocity v and in the direction indicated in Fig 1. People travelling `
` in this train will with a vantage view the train as a rigid `
` reference-body (co-ordinate system); they regard all events in `
` `
` Fig. 01: file fig01.gif `
` `
` `
` 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 be embankment, we mean: the rays of light emitted at the `
` places A and B, where the lightning occurs, meet each other at the `
` mid-point M of the length A arrow B of the embankment. But the events `
` A and B also correspond to positions A and B on the train. Let M1 be `
` the mid-point of the distance A arrow B on the travelling train. Just `
` when the flashes (as judged from the embankment) of lightning occur, `
` this point M1 naturally coincides with the point M but it moves `
` towards the right in the diagram with the velocity v of the train. If `
` an observer sitting in the position M1 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 their `
` reference-body must therefore come to the conclusion that the `
` lightning flash B took place earlier than the lightning flash A. We `
` thus arrive at the important result: `
` `
` Events which are simultaneous with reference to the embankment are not `
` simultaneous with respect to the train, and vice versa (relativity of `
` simultaneity). Every reference-body (co-ordinate system) has its own `
` particular time ; unless we are told the reference-body to which the `
` statement of time refers, there is no meaning in a statement of the `
` time of an event. `
` `
` Now before the advent of the theory of relativity it had always `
` tacitly been assumed in physics that the statement of time had an `
` absolute significance, 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 tacitly made even before the `
`