Reading Help Relativity: The Special and General Theory
be arbitrary, although it was always tacitly made even before the `
` introduction of the theory of relativity. `
` `
` `
` `
` ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE `
` `
` `
` Let us consider two particular points on the train * travelling `
` along the embankment with the velocity v, and inquire as to their `
` distance apart. We already know that it is necessary to have a body of `
` reference for the measurement of a distance, with respect to which `
` body the distance can be measured up. It is the simplest plan to use `
` the train itself as reference-body (co-ordinate system). An observer `
` in the train measures the interval by marking off his measuring-rod in `
` a straight line (e.g. 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^1 and B^1 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^1 and B^1 at a particular time t -- judged from the `
` embankment. These points A and B of 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 thee 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. `
` `
` `
` Notes `
` `
` *) e.g. the middle of the first and of the hundredth carriage. `
` `
` `
` `
` 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 (time) between two events is independent of the `
` condition of motion of the body of reference. `
` `
` (2) The space-interval (distance) between two points of a rigid body `
` is independent of 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 leads 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 space-time 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. Fig. 2 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 " co-ordinate planes " (" co-ordinate system "). A co-ordinate `
` system K then corresponds to the embankment, and a co-ordinate 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, z `
` on the co-ordinate planes, and with regard to time by a time value t. `
` Relative to K1, the same event would be fixed in respect of space and `
` time by corresponding values x1, y1, z1, t1, which of course are not `
` identical with x, y, z, 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 x1, y1, z1, t1, of an event with respect `
` to K1, when the magnitudes x, y, z, 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 K1. For `
` the relative orientation in space of the co-ordinate systems indicated `
` in the diagram ([7]Fig. 2), this problem is solved by means of the `
` equations : `
` `
` eq. 1: file eq01.gif `
` `
` y1 = y `
` z1 = z `
` `
` eq. 2: file eq02.gif `
` `
` This system of equations is known as the " Lorentz transformation." * `
` `
` 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: `
` `
` x1 = x - vt `
` y1 = y `
` z1 = z `
` t1 = 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 K1. A light-signal is sent `
` along the positive x-axis, and this light-stimulus advances in `
` accordance with the equation `
` `
` x = 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 x1 and t1. In point of fact, if we substitute for x `
` the value ct in the first and fourth equations of the Lorentz `
` transformation, we obtain: `
` `
` eq. 3: file eq03.gif `
` `
` `
` eq. 4: file eq04.gif `
` `
` from which, by division, the expression `
` `
` x1 = ct1 `
` `
` immediately follows. If referred to the system K1, the propagation of `
` light takes place according to this equation. We thus see that the `
` velocity of transmission relative to the reference-body K1 is also `
` equal to c. The same result is obtained for rays of light advancing in `
` any other direction whatsoever. Of cause this is not surprising, since `
` the equations of the Lorentz transformation were derived conformably `
` to this point of view. `
` `
` `
` Notes `
` `
` *) A simple derivation of the Lorentz transformation is given in `
` Appendix I. `
` `
` `
` `
` THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION `
` `
` `
` Place a metre-rod in the x1-axis of K1 in such a manner that one end `
` (the beginning) coincides with the point x1=0 whilst the other end `
` (the end of the rod) coincides with the point x1=I. What is the length `
` of the metre-rod relatively to the system K? In order to learn this, `
` we need only ask where the beginning of the rod and the end of the rod `
` lie with respect to K at a particular time t of the system K. By means `
` of the first equation of the Lorentz transformation the values of `
` these two points at the time t = 0 can be shown to be `
` `
` eq. 05a: file eq05a.gif `
` `
` `
`
` introduction of the theory of relativity. `
` `
` `
` `
` ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE `
` `
` `
` Let us consider two particular points on the train * travelling `
` along the embankment with the velocity v, and inquire as to their `
` distance apart. We already know that it is necessary to have a body of `
` reference for the measurement of a distance, with respect to which `
` body the distance can be measured up. It is the simplest plan to use `
` the train itself as reference-body (co-ordinate system). An observer `
` in the train measures the interval by marking off his measuring-rod in `
` a straight line (e.g. 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^1 and B^1 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^1 and B^1 at a particular time t -- judged from the `
` embankment. These points A and B of 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 thee 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. `
` `
` `
` Notes `
` `
` *) e.g. the middle of the first and of the hundredth carriage. `
` `
` `
` `
` 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 (time) between two events is independent of the `
` condition of motion of the body of reference. `
` `
` (2) The space-interval (distance) between two points of a rigid body `
` is independent of 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 leads 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 space-time 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. Fig. 2 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 " co-ordinate planes " (" co-ordinate system "). A co-ordinate `
` system K then corresponds to the embankment, and a co-ordinate 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, z `
` on the co-ordinate planes, and with regard to time by a time value t. `
` Relative to K1, the same event would be fixed in respect of space and `
` time by corresponding values x1, y1, z1, t1, which of course are not `
` identical with x, y, z, 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 x1, y1, z1, t1, of an event with respect `
` to K1, when the magnitudes x, y, z, 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 K1. For `
` the relative orientation in space of the co-ordinate systems indicated `
` in the diagram ([7]Fig. 2), this problem is solved by means of the `
` equations : `
` `
` eq. 1: file eq01.gif `
` `
` y1 = y `
` z1 = z `
` `
` eq. 2: file eq02.gif `
` `
` This system of equations is known as the " Lorentz transformation." * `
` `
` 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: `
` `
` x1 = x - vt `
` y1 = y `
` z1 = z `
` t1 = 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 K1. A light-signal is sent `
` along the positive x-axis, and this light-stimulus advances in `
` accordance with the equation `
` `
` x = 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 x1 and t1. In point of fact, if we substitute for x `
` the value ct in the first and fourth equations of the Lorentz `
` transformation, we obtain: `
` `
` eq. 3: file eq03.gif `
` `
` `
` eq. 4: file eq04.gif `
` `
` from which, by division, the expression `
` `
` x1 = ct1 `
` `
` immediately follows. If referred to the system K1, the propagation of `
` light takes place according to this equation. We thus see that the `
` velocity of transmission relative to the reference-body K1 is also `
` equal to c. The same result is obtained for rays of light advancing in `
` any other direction whatsoever. Of cause this is not surprising, since `
` the equations of the Lorentz transformation were derived conformably `
` to this point of view. `
` `
` `
` Notes `
` `
` *) A simple derivation of the Lorentz transformation is given in `
` Appendix I. `
` `
` `
` `
` THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION `
` `
` `
` Place a metre-rod in the x1-axis of K1 in such a manner that one end `
` (the beginning) coincides with the point x1=0 whilst the other end `
` (the end of the rod) coincides with the point x1=I. What is the length `
` of the metre-rod relatively to the system K? In order to learn this, `
` we need only ask where the beginning of the rod and the end of the rod `
` lie with respect to K at a particular time t of the system K. By means `
` of the first equation of the Lorentz transformation the values of `
` these two points at the time t = 0 can be shown to be `
` `
` eq. 05a: file eq05a.gif `
` `
` `
`