# 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 `

` `

` `

`