# Reading Help Relativity: The Special and General Theory

`

` eq. 05b: file eq05b.gif `

` `

` `

` the distance between the points being eq. 06 . `

` `

` But the metre-rod is moving with the velocity v relative to K. It `

` therefore follows that the length of a rigid metre-rod moving in the `

` direction of its length with a velocity v is eq. 06 of a metre. `

` `

` The rigid rod is thus shorter when in motion than when at rest, and `

` the more quickly it is moving, the shorter is the rod. For the `

` velocity v=c we should have eq. 06a , `

` `

` and for stiII greater velocities the square-root becomes imaginary. `

` From this we conclude that in the theory of relativity the velocity c `

` plays the part of a limiting velocity, which can neither be reached `

` nor exceeded by any real body. `

` `

` Of course this feature of the velocity c as a limiting velocity also `

` clearly follows from the equations of the Lorentz transformation, for `

` these became meaningless if we choose values of v greater than c. `

` `

` If, on the contrary, we had considered a metre-rod at rest in the `

` x-axis with respect to K, then we should have found that the length of `

` the rod as judged from K1 would have been eq. 06 ; `

` `

` this is quite in accordance with the principle of relativity which `

` forms the basis of our considerations. `

` `

` A Priori it is quite clear that we must be able to learn something `

` about the physical behaviour of measuring-rods and clocks from the `

` equations of transformation, for the magnitudes z, y, x, t, are `

` nothing more nor less than the results of measurements obtainable by `

` means of measuring-rods and clocks. If we had based our considerations `

` on the Galileian transformation we should not have obtained a `

` contraction of the rod as a consequence of its motion. `

` `

` Let us now consider a seconds-clock which is permanently situated at `

` the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of `

` this clock. The first and fourth equations of the Lorentz `

` transformation give for these two ticks : `

` `

` t = 0 `

` `

` and `

` `

` eq. 07: file eq07.gif `

` `

` As judged from K, the clock is moving with the velocity v; as judged `

` from this reference-body, the time which elapses between two strokes `

` of the clock is not one second, but `

` `

` eq. 08: file eq08.gif `

` `

` seconds, i.e. a somewhat larger time. As a consequence of its motion `

` the clock goes more slowly than when at rest. Here also the velocity c `

` plays the part of an unattainable limiting velocity. `

` `

` `

` `

` THEOREM OF THE ADDITION OF VELOCITIES. `

` THE EXPERIMENT OF FIZEAU `

` `

` `

` Now in practice we can move clocks and measuring-rods only with `

` velocities that are small compared with the velocity of light; hence `

` we shall hardly be able to compare the results of the previous section `

` directly with the reality. But, on the other hand, these results must `

` strike you as being very singular, and for that reason I shall now `

` draw another conclusion from the theory, one which can easily be `

` derived from the foregoing considerations, and which has been most `

` elegantly confirmed by experiment. `

` `

` In Section 6 we derived the theorem of the addition of velocities `

` in one direction in the form which also results from the hypotheses of `

` classical mechanics- This theorem can also be deduced readily horn the `

` Galilei transformation (Section 11). In place of the man walking `

` inside the carriage, we introduce a point moving relatively to the `

` co-ordinate system K1 in accordance with the equation `

` `

` x1 = wt1 `

` `

` By means of the first and fourth equations of the Galilei `

` transformation we can express x1 and t1 in terms of x and t, and we `

` then obtain `

` `

` x = (v + w)t `

` `

` This equation expresses nothing else than the law of motion of the `

` point with reference to the system K (of the man with reference to the `

` embankment). We denote this velocity by the symbol W, and we then `

` obtain, as in Section 6, `

` `

` W=v+w A) `

` `

` But we can carry out this consideration just as well on the basis of `

` the theory of relativity. In the equation `

` `

` x1 = wt1 B) `

` `

` we must then express x1and t1 in terms of x and t, making use of the `

` first and fourth equations of the Lorentz transformation. Instead of `

` the equation (A) we then obtain the equation `

` `

` eq. 09: file eq09.gif `

` `

` `

` which corresponds to the theorem of addition for velocities in one `

` direction according to the theory of relativity. The question now `

` arises as to which of these two theorems is the better in accord with `

` experience. On this point we axe enlightened by a most important `

` experiment which the brilliant physicist Fizeau performed more than `

` half a century ago, and which has been repeated since then by some of `

` the best experimental physicists, so that there can be no doubt about `

` its result. The experiment is concerned with the following question. `

` Light travels in a motionless liquid with a particular velocity w. How `

` quickly does it travel in the direction of the arrow in the tube T `

` (see the accompanying diagram, Fig. 3) when the liquid above `

` mentioned is flowing through the tube with a velocity v ? `

` `

` In accordance with the principle of relativity we shall certainly have `

` to take for granted that the propagation of light always takes place `

` with the same velocity w with respect to the liquid, whether the `

` latter is in motion with reference to other bodies or not. The `

` velocity of light relative to the liquid and the velocity of the `

` latter relative to the tube are thus known, and we require the `

` velocity of light relative to the tube. `

` `

` It is clear that we have the problem of Section 6 again before us. The `

` tube plays the part of the railway embankment or of the co-ordinate `

` system K, the liquid plays the part of the carriage or of the `

` co-ordinate system K1, and finally, the light plays the part of the `

` `

` Figure 03: file fig03.gif `

` `

` `

` man walking along the carriage, or of the moving point in the present `

` section. If we denote the velocity of the light relative to the tube `

` by W, then this is given by the equation (A) or (B), according as the `

` Galilei transformation or the Lorentz transformation corresponds to `

` the facts. Experiment * decides in favour of equation (B) derived `

` from the theory of relativity, and the agreement is, indeed, very `

` exact. According to recent and most excellent measurements by Zeeman, `

` the influence of the velocity of flow v on the propagation of light is `

` represented by formula (B) to within one per cent. `

` `

` Nevertheless we must now draw attention to the fact that a theory of `

` this phenomenon was given by H. A. Lorentz long before the statement `

` of the theory of relativity. This theory was of a purely `

` electrodynamical nature, and was obtained by the use of particular `

` hypotheses as to the electromagnetic structure of matter. This `

` circumstance, however, does not in the least diminish the `

` conclusiveness of the experiment as a crucial test in favour of the `

` theory of relativity, for the electrodynamics of Maxwell-Lorentz, on `

` which the original theory was based, in no way opposes the theory of `

` relativity. Rather has the latter been developed trom electrodynamics `

` as an astoundingly simple combination and generalisation of the `

` hypotheses, formerly independent of each other, on which `

` electrodynamics was built. `

` `

` `

` Notes `

` `

` *) Fizeau found eq. 10 , where eq. 11 `

` `

` is the index of refraction of the liquid. On the other hand, owing to `

` the smallness of eq. 12 as compared with I, `

` `

` we can replace (B) in the first place by eq. 13 , or to the same order `

` of approximation by `

` `

` eq. 14 , which agrees with Fizeau's result. `

` `

` `

` `

` THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY `

` `

` `

` Our train of thought in the foregoing pages can be epitomised in the `

` following manner. Experience has led to the conviction that, on the `

` one hand, the principle of relativity holds true and that on the other `

` hand the velocity of transmission of light in vacuo has to be `

` considered equal to a constant c. By uniting these two postulates we `

` obtained the law of transformation for the rectangular co-ordinates x, `

` y, z and the time t of the events which constitute the processes of `

` nature. In this connection we did not obtain the Galilei `

` transformation, but, differing from classical mechanics, the Lorentz `

` transformation. `

` `

` The law of transmission of light, the acceptance of which is justified `

` by our actual knowledge, played an important part in this process of `

` thought. Once in possession of the Lorentz transformation, however, we `

` can combine this with the principle of relativity, and sum up the `

` theory thus: `

` `

` Every general law of nature must be so constituted that it is `

` transformed into a law of exactly the same form when, instead of the `

` space-time variables x, y, z, t of the original coordinate system K, `

` we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate `

` system K1. In this connection the relation between the ordinary and `

`

` eq. 05b: file eq05b.gif `

` `

` `

` the distance between the points being eq. 06 . `

` `

` But the metre-rod is moving with the velocity v relative to K. It `

` therefore follows that the length of a rigid metre-rod moving in the `

` direction of its length with a velocity v is eq. 06 of a metre. `

` `

` The rigid rod is thus shorter when in motion than when at rest, and `

` the more quickly it is moving, the shorter is the rod. For the `

` velocity v=c we should have eq. 06a , `

` `

` and for stiII greater velocities the square-root becomes imaginary. `

` From this we conclude that in the theory of relativity the velocity c `

` plays the part of a limiting velocity, which can neither be reached `

` nor exceeded by any real body. `

` `

` Of course this feature of the velocity c as a limiting velocity also `

` clearly follows from the equations of the Lorentz transformation, for `

` these became meaningless if we choose values of v greater than c. `

` `

` If, on the contrary, we had considered a metre-rod at rest in the `

` x-axis with respect to K, then we should have found that the length of `

` the rod as judged from K1 would have been eq. 06 ; `

` `

` this is quite in accordance with the principle of relativity which `

` forms the basis of our considerations. `

` `

` A Priori it is quite clear that we must be able to learn something `

` about the physical behaviour of measuring-rods and clocks from the `

` equations of transformation, for the magnitudes z, y, x, t, are `

` nothing more nor less than the results of measurements obtainable by `

` means of measuring-rods and clocks. If we had based our considerations `

` on the Galileian transformation we should not have obtained a `

` contraction of the rod as a consequence of its motion. `

` `

` Let us now consider a seconds-clock which is permanently situated at `

` the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of `

` this clock. The first and fourth equations of the Lorentz `

` transformation give for these two ticks : `

` `

` t = 0 `

` `

` and `

` `

` eq. 07: file eq07.gif `

` `

` As judged from K, the clock is moving with the velocity v; as judged `

` from this reference-body, the time which elapses between two strokes `

` of the clock is not one second, but `

` `

` eq. 08: file eq08.gif `

` `

` seconds, i.e. a somewhat larger time. As a consequence of its motion `

` the clock goes more slowly than when at rest. Here also the velocity c `

` plays the part of an unattainable limiting velocity. `

` `

` `

` `

` THEOREM OF THE ADDITION OF VELOCITIES. `

` THE EXPERIMENT OF FIZEAU `

` `

` `

` Now in practice we can move clocks and measuring-rods only with `

` velocities that are small compared with the velocity of light; hence `

` we shall hardly be able to compare the results of the previous section `

` directly with the reality. But, on the other hand, these results must `

` strike you as being very singular, and for that reason I shall now `

` draw another conclusion from the theory, one which can easily be `

` derived from the foregoing considerations, and which has been most `

` elegantly confirmed by experiment. `

` `

` In Section 6 we derived the theorem of the addition of velocities `

` in one direction in the form which also results from the hypotheses of `

` classical mechanics- This theorem can also be deduced readily horn the `

` Galilei transformation (Section 11). In place of the man walking `

` inside the carriage, we introduce a point moving relatively to the `

` co-ordinate system K1 in accordance with the equation `

` `

` x1 = wt1 `

` `

` By means of the first and fourth equations of the Galilei `

` transformation we can express x1 and t1 in terms of x and t, and we `

` then obtain `

` `

` x = (v + w)t `

` `

` This equation expresses nothing else than the law of motion of the `

` point with reference to the system K (of the man with reference to the `

` embankment). We denote this velocity by the symbol W, and we then `

` obtain, as in Section 6, `

` `

` W=v+w A) `

` `

` But we can carry out this consideration just as well on the basis of `

` the theory of relativity. In the equation `

` `

` x1 = wt1 B) `

` `

` we must then express x1and t1 in terms of x and t, making use of the `

` first and fourth equations of the Lorentz transformation. Instead of `

` the equation (A) we then obtain the equation `

` `

` eq. 09: file eq09.gif `

` `

` `

` which corresponds to the theorem of addition for velocities in one `

` direction according to the theory of relativity. The question now `

` arises as to which of these two theorems is the better in accord with `

` experience. On this point we axe enlightened by a most important `

` experiment which the brilliant physicist Fizeau performed more than `

` half a century ago, and which has been repeated since then by some of `

` the best experimental physicists, so that there can be no doubt about `

` its result. The experiment is concerned with the following question. `

` Light travels in a motionless liquid with a particular velocity w. How `

` quickly does it travel in the direction of the arrow in the tube T `

` (see the accompanying diagram, Fig. 3) when the liquid above `

` mentioned is flowing through the tube with a velocity v ? `

` `

` In accordance with the principle of relativity we shall certainly have `

` to take for granted that the propagation of light always takes place `

` with the same velocity w with respect to the liquid, whether the `

` latter is in motion with reference to other bodies or not. The `

` velocity of light relative to the liquid and the velocity of the `

` latter relative to the tube are thus known, and we require the `

` velocity of light relative to the tube. `

` `

` It is clear that we have the problem of Section 6 again before us. The `

` tube plays the part of the railway embankment or of the co-ordinate `

` system K, the liquid plays the part of the carriage or of the `

` co-ordinate system K1, and finally, the light plays the part of the `

` `

` Figure 03: file fig03.gif `

` `

` `

` man walking along the carriage, or of the moving point in the present `

` section. If we denote the velocity of the light relative to the tube `

` by W, then this is given by the equation (A) or (B), according as the `

` Galilei transformation or the Lorentz transformation corresponds to `

` the facts. Experiment * decides in favour of equation (B) derived `

` from the theory of relativity, and the agreement is, indeed, very `

` exact. According to recent and most excellent measurements by Zeeman, `

` the influence of the velocity of flow v on the propagation of light is `

` represented by formula (B) to within one per cent. `

` `

` Nevertheless we must now draw attention to the fact that a theory of `

` this phenomenon was given by H. A. Lorentz long before the statement `

` of the theory of relativity. This theory was of a purely `

` electrodynamical nature, and was obtained by the use of particular `

` hypotheses as to the electromagnetic structure of matter. This `

` circumstance, however, does not in the least diminish the `

` conclusiveness of the experiment as a crucial test in favour of the `

` theory of relativity, for the electrodynamics of Maxwell-Lorentz, on `

` which the original theory was based, in no way opposes the theory of `

` relativity. Rather has the latter been developed trom electrodynamics `

` as an astoundingly simple combination and generalisation of the `

` hypotheses, formerly independent of each other, on which `

` electrodynamics was built. `

` `

` `

` Notes `

` `

` *) Fizeau found eq. 10 , where eq. 11 `

` `

` is the index of refraction of the liquid. On the other hand, owing to `

` the smallness of eq. 12 as compared with I, `

` `

` we can replace (B) in the first place by eq. 13 , or to the same order `

` of approximation by `

` `

` eq. 14 , which agrees with Fizeau's result. `

` `

` `

` `

` THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY `

` `

` `

` Our train of thought in the foregoing pages can be epitomised in the `

` following manner. Experience has led to the conviction that, on the `

` one hand, the principle of relativity holds true and that on the other `

` hand the velocity of transmission of light in vacuo has to be `

` considered equal to a constant c. By uniting these two postulates we `

` obtained the law of transformation for the rectangular co-ordinates x, `

` y, z and the time t of the events which constitute the processes of `

` nature. In this connection we did not obtain the Galilei `

` transformation, but, differing from classical mechanics, the Lorentz `

` transformation. `

` `

` The law of transmission of light, the acceptance of which is justified `

` by our actual knowledge, played an important part in this process of `

` thought. Once in possession of the Lorentz transformation, however, we `

` can combine this with the principle of relativity, and sum up the `

` theory thus: `

` `

` Every general law of nature must be so constituted that it is `

` transformed into a law of exactly the same form when, instead of the `

` space-time variables x, y, z, t of the original coordinate system K, `

` we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate `

` system K1. In this connection the relation between the ordinary and `

`