WEBVTT 1 00:00:00.160 --> 00:00:08.240 Section ten on the relativity of the conception of distance. Let us consider two 2 00:00:08.320 --> 00:00:13.160 particular points on the train, for example, the middle of the first and 3 00:00:13.320 --> 00:00:17.960 of the one hundredth carriage, traveling along the embankment with the velocity V, 4 00:00:18.559 --> 00:00:23.839 and the enquire as to their distance apart. We already know that it is 5 00:00:23.879 --> 00:00:28.079 necessary to have a body of reference for the measurement of a distance. With 6 00:00:28.160 --> 00:00:32.280 respect to each body, the distance can be measured up. It is the 7 00:00:32.320 --> 00:00:38.479 simplest plan to use the train itself as reference body or co ordinate system. 8 00:00:38.640 --> 00:00:43.520 An observer in the train measures the interval by marking off his measuring rod in 9 00:00:43.560 --> 00:00:47.840 a straight line, for example, along the floor of the carriage, as 10 00:00:47.920 --> 00:00:51.960 many times as is necessary to take him from the one marked point to the 11 00:00:52.039 --> 00:00:56.399 other. Then the number which tells us how often the rod has to be 12 00:00:56.479 --> 00:01:03.039 laid down is the required distance. It is a different matter when the distance 13 00:01:03.119 --> 00:01:07.079 has to be judged from the railway line. Here, the following method suggests 14 00:01:07.120 --> 00:01:12.640 itself. If we call A prime and B prime the two points on the 15 00:01:12.680 --> 00:01:19.159 train whose distance apart is required, then both of these points are moving with 16 00:01:19.239 --> 00:01:23.439 the velocity V along the embankment. In the first place, we require to 17 00:01:23.519 --> 00:01:30.280 determine the points A and B of the embankment, which are just being passed 18 00:01:30.319 --> 00:01:36.200 by the two points A prime and B prime at a particular time T judged 19 00:01:36.239 --> 00:01:42.079 from the embankment. These points A and B on the embankment can be determined 20 00:01:42.120 --> 00:01:48.840 by applying the definition of time given in section eight. The distance between these 21 00:01:48.879 --> 00:01:53.480 points A and B is then measured by repeated application of the measuring rod along 22 00:01:53.519 --> 00:02:00.280 the embankment a priori. It is by no means certain that this last measurement 23 00:02:00.280 --> 00:02:06.280 will supply us with the same result as the first. Thus, the length 24 00:02:06.319 --> 00:02:10.280 of the train as measured from the embankment may be different from that obtained by 25 00:02:10.319 --> 00:02:16.719 measuring in the train itself. This circumstance leads us to a second objection which 26 00:02:16.800 --> 00:02:23.599 must be raised against the apparently obvious consideration of section six. Namely, if 27 00:02:23.639 --> 00:02:30.120 the man in the carriage covers the distance W in a unit of time measured 28 00:02:30.159 --> 00:02:36.719 from the train, then this distance as measured from the embankment is not necessarily 29 00:02:36.879 --> 00:02:46.560 also equal to W. Section eleven the Lorentz transformation. The results of the 30 00:02:46.639 --> 00:02:53.319 last three sections show that the apparent incompatibility of the law of propagation of light 31 00:02:53.639 --> 00:02:59.319 with the principle of relativity. Section seven has been derived by means of a 32 00:02:59.360 --> 00:03:07.240 considerable which borrowed two unjustifiable hypotheses from classical mechanics. These are as follows one, 33 00:03:07.759 --> 00:03:14.759 the time interval or time between two events is independent of the condition of 34 00:03:14.879 --> 00:03:22.319 motion of the body of reference, and the space interval or distance between two 35 00:03:22.439 --> 00:03:27.159 points of a rigid body is independent of the condition of motion of the body 36 00:03:27.159 --> 00:03:34.280 of reference. If we drop these hypotheses, then the dilemma of section seven 37 00:03:34.479 --> 00:03:39.840 disappears, because the theorem of the addition of velocities derived in section six becomes 38 00:03:40.000 --> 00:03:46.199 invalid. The possibility presents itself that the law of the propagation of light in 39 00:03:46.400 --> 00:03:53.680 vacuo may be compatible with the principle of relativity, and the question arises, 40 00:03:53.360 --> 00:04:00.039 how have we to modify the considerations of section six in order to remove the 41 00:04:00.080 --> 00:04:06.479 apparent disagreement between these two fundamental results of experience. This question leads to a 42 00:04:06.520 --> 00:04:11.639 general one. In the discussion of section six, we have to do with 43 00:04:11.759 --> 00:04:17.000 places and times relative both to the train and to the embankment. How are 44 00:04:17.040 --> 00:04:20.879 we to find the place and time of an event in relation to the train 45 00:04:21.519 --> 00:04:26.639 When we know the place and time of the event with respect to the railway 46 00:04:26.639 --> 00:04:31.680 embankment? Is there a thinkable answer to this question of such a nature that 47 00:04:31.720 --> 00:04:38.600 the law of transmission of light in vacuo does not contradict the principle of relativity? 48 00:04:39.560 --> 00:04:44.279 In other words, can we conceive of a relation between place and time 49 00:04:44.399 --> 00:04:49.399 of the individual events relative to both reference bodies, such that every ray of 50 00:04:49.480 --> 00:04:57.040 light possesses the velocity of transmissions ce relative to the embankment and relative to the 51 00:04:57.079 --> 00:05:02.680 train. This question leads to a quite definite positive answer, and to a 52 00:05:02.759 --> 00:05:10.639 perfectly definite transformation law for the space time magnitudes of an event when changing over 53 00:05:10.680 --> 00:05:15.879 from one body of reference to another. Before we deal with this, we 54 00:05:15.920 --> 00:05:20.839 shall introduce the following incidental consideration. Up to the present, we have only 55 00:05:20.879 --> 00:05:28.120 considered events taking place along the embankment, which had mathematically to assume the function 56 00:05:28.160 --> 00:05:32.600 of a straight line in the manner indicated in section two. We can imagine 57 00:05:32.600 --> 00:05:39.639 this reference body supplemented laterally and in a vertical direction by means of a framework 58 00:05:39.639 --> 00:05:44.680 of rods, so that an event which takes place anywhere can be localized with 59 00:05:44.800 --> 00:05:49.800 reference to this framework. Similarly, we can imagine the train traveling with the 60 00:05:49.879 --> 00:05:56.759 velocity v to be continued across the whole of space, so that every event, 61 00:05:57.160 --> 00:06:00.279 no matter how far off it may be, could all be localized with 62 00:06:00.319 --> 00:06:08.839 respect to the second framework without committing any fundamental error. We can disregard the 63 00:06:08.879 --> 00:06:14.600 fact that in reality these frameworks would continually interfere with each other, owing to 64 00:06:14.680 --> 00:06:20.360 the impenetrability of solid bodies. In every such framework, we imagine three surfaces 65 00:06:20.360 --> 00:06:29.759 perpendicular to each other, marked out and designated as coordinate planes or coordinate system. 66 00:06:30.199 --> 00:06:35.399 A coordinate system k then corresponds to the embankment, and a coordinate system 67 00:06:35.519 --> 00:06:42.120 K prime to the train. An event, wherever it may have taken place, 68 00:06:42.560 --> 00:06:47.120 would be fixed in space with respect to K by the three perpendiculars x, 69 00:06:47.519 --> 00:06:53.240 y, and z on the coordinate planes, and with regard to time 70 00:06:53.519 --> 00:07:00.199 by a time value T relative to K prime. The same event would be 71 00:07:00.240 --> 00:07:05.560 fixed in respect of space and time by corresponding values x prime, y prime, 72 00:07:05.839 --> 00:07:11.560 z prime, and T prime, which of course are not identical with 73 00:07:11.879 --> 00:07:15.879 x, y, z and T. It has already been set forth in 74 00:07:15.920 --> 00:07:23.160 detail how these magnitudes are to be regarded as results of physical measurements. Obviously, 75 00:07:23.360 --> 00:07:27.879 our problem can be exactly formulated in the following manner. What are the 76 00:07:27.959 --> 00:07:31.759 values x prime, y prime, z prime, and t prime of an 77 00:07:31.759 --> 00:07:36.399 event with respect to k prime. When the magnitudes x, y, z, 78 00:07:36.680 --> 00:07:43.240 and t of the same event with respect to k are given. The 79 00:07:43.319 --> 00:07:47.240 relations must be so chosen that the law of the transmission of light in vacuum 80 00:07:47.759 --> 00:07:53.079 is satisfied for one and the same ray of light, and of course for 81 00:07:53.199 --> 00:07:59.720 every ray with respect to k and k prime. For the relative orientation in 82 00:08:00.079 --> 00:08:05.399 base of the coordinate systems indicated in the diagram. This problem is solved by 83 00:08:05.519 --> 00:08:13.600 means of the equations x prime equals x vt over the square root of i 84 00:08:15.720 --> 00:08:22.279 v squared over C squared, why prime equals y, z prime equals z, 85 00:08:24.399 --> 00:08:35.240 and t prime equals t v overc squared times x over the square root 86 00:08:35.279 --> 00:08:41.720 of i v squared over C squared. This system of equations is known as 87 00:08:41.759 --> 00:08:50.039 the Lorentz transformation. Footnote. A simple derivation of the Lorentz transformation is given 88 00:08:50.120 --> 00:08:54.840 in Appendix one. If, in place of the law of transmission of light, 89 00:08:56.360 --> 00:09:01.000 we had taken as our basis the tacit assumptions of the older mechanics as 90 00:09:01.039 --> 00:09:05.559 to the absolute character of times and lengths, then instead of the above, 91 00:09:05.679 --> 00:09:13.000 we should have obtained the following equations. X prime equals x vt, y 92 00:09:13.080 --> 00:09:20.200 prime equals y, z prime equals z, t prime equals t. This 93 00:09:20.240 --> 00:09:28.039 system of equations is often termed the Galileay transformation. The Galileay transformation can be 94 00:09:28.080 --> 00:09:35.679 obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of 95 00:09:35.759 --> 00:09:43.559 light c in the latter transformation. Aided by the following illustration, we can 96 00:09:43.600 --> 00:09:48.559 readily see that, in accordance with the Lorentz transformation, the law of the 97 00:09:48.559 --> 00:09:54.519 transmission of light in vacuo is satisfied both for the reference body K and for 98 00:09:54.559 --> 00:10:00.360 the reference body k prime. A light signal is sent along the positive x 99 00:10:00.399 --> 00:10:05.759 axis, and this light stimulus advances in accordance with the equation x equals c 100 00:10:07.080 --> 00:10:13.759 t i e with the velocity c. According to the equations of the Lorentz 101 00:10:13.759 --> 00:10:20.120 transformation. This simple relation between x and t involves a relation between x prime 102 00:10:20.159 --> 00:10:26.000 and t prime. In point of fact, if we substitute for x the 103 00:10:26.080 --> 00:10:31.039 value ct in the first and fourth equations of the Lorentz transformation, we obtain 104 00:10:33.080 --> 00:10:41.600 x prime equals c v times t. Over the square root of i v 105 00:10:41.799 --> 00:10:50.960 squared overse squared, and T prime equals I minus v overse multiplied by t 106 00:10:52.240 --> 00:11:01.080 over the square root of i v squared oversea squared, from which by division 107 00:11:01.360 --> 00:11:09.279 the expression x prime equals ct prime immediately follows. If referred to the system 108 00:11:09.440 --> 00:11:13.000 k prime, the propagation of light takes place. According to this equation. 109 00:11:13.799 --> 00:11:20.279 We thus see that the velocity of transmission relative to the reference body K prime 110 00:11:20.679 --> 00:11:24.679 is also equal to c. The same result is obtained for rays of light 111 00:11:26.159 --> 00:11:31.559 advancing in any other direction whatsoever. Of course, this is not surprising, 112 00:11:31.000 --> 00:11:37.080 since the equations of the Lorentz transformation were derived conformably to this point of view. 113 00:11:39.799 --> 00:11:48.840 Section twelve The behavior of measuring rods and clocks in motion. Place a 114 00:11:48.000 --> 00:11:54.399 meter rod in the x prime axis of K prime in such a manner. 115 00:11:54.440 --> 00:12:00.720 At one end the beginning coincides with the point x prime equals zero, while 116 00:12:00.759 --> 00:12:05.279 the other end, the end of the rod coincides with the point x prime 117 00:12:05.360 --> 00:12:11.240 equals i. What is the length of the meter rod relatively to the system 118 00:12:11.360 --> 00:12:16.320 K. In order to learn this, we need only ask where the beginning 119 00:12:16.320 --> 00:12:18.679 of the rod and the end of the rod lie with respect to K. 120 00:12:20.120 --> 00:12:24.600 At a particular time T of the system K. By means of the first 121 00:12:24.639 --> 00:12:30.240 equation of the Lorentz transformation, the values of these two points at the time 122 00:12:30.320 --> 00:12:37.960 T equals zero can be shown to be X. Beginning of rod equals zero 123 00:12:39.240 --> 00:12:46.720 over the square root of i v squared over C squared. X end of 124 00:12:46.840 --> 00:12:52.440 rod equals i over the square root of i v squared over C squared, 125 00:12:54.639 --> 00:13:01.000 the distance between the points being the square root of i v square oversea squared. 126 00:13:01.399 --> 00:13:05.879 But the meter rod is moving with the velocity V relative to K. 127 00:13:07.840 --> 00:13:11.320 It therefore follows that the length of a rigid meter rod moving in the direction 128 00:13:11.360 --> 00:13:18.240 of its length with a velocity V is the square root of I v squared 129 00:13:18.279 --> 00:13:26.480 oversea squared of a meter The rigid rod is thus shorter when in motion than 130 00:13:26.559 --> 00:13:30.960 when at rest, and the more quickly it is moving, the shorter is 131 00:13:31.000 --> 00:13:37.480 the rod. For the velocity V equals c, we should have the square 132 00:13:37.559 --> 00:13:45.039 root of i v squared oversea squared equals zero, and for still greater velocities 133 00:13:45.279 --> 00:13:50.600 the square root becomes imaginary. For this, we conclude that in the theory 134 00:13:50.600 --> 00:13:56.240 of relativity, the velocity C plays the part of a limiting velocity, which 135 00:13:56.279 --> 00:14:01.519 can neither be reached nor exceeded by any real body. Of course, this 136 00:14:01.600 --> 00:14:07.840 feature of the velocity C as a limiting velocity also clearly follows from the equations 137 00:14:07.879 --> 00:14:13.440 of the Lorentz transformation, for these become meaningless if we choose values of v 138 00:14:15.000 --> 00:14:20.120 greater than C. If, on the contrary, we had considered a meter 139 00:14:20.279 --> 00:14:24.759 rod at rest in the x axis with respect to K, then we should 140 00:14:24.759 --> 00:14:28.159 have found that the length of the rod, as judged from K prime, 141 00:14:28.399 --> 00:14:33.960 would have been the square root of i v squared over C squared. This 142 00:14:35.120 --> 00:14:39.799 is quite in accordance with the principle of relativity, which forms the basis of 143 00:14:39.840 --> 00:14:46.360 our considerations. A priori. It is quite clear that we must be able 144 00:14:46.360 --> 00:14:50.919 to learn something about the physical behavior of measuring rods and clocks from the equations 145 00:14:50.960 --> 00:14:56.840 of transformation. For the magnitudes z, y, x, and t are 146 00:14:56.919 --> 00:15:03.080 nothing more nor less than the result of measurements obtainable by means of measuring rods 147 00:15:03.080 --> 00:15:09.240 and clocks. If we had based our considerations on the Galilean transformation, we 148 00:15:09.279 --> 00:15:13.720 should not have obtained a contraction of the rod as a consequence of its motion. 149 00:15:15.480 --> 00:15:20.720 Let us now consider a seconds clock which is permanently situated at the origin 150 00:15:20.399 --> 00:15:28.360 x prime equals zero of k prime T prime equals zero and t prime equals 151 00:15:28.440 --> 00:15:33.960 i are two successive ticks of this clock. The first and fourth equations of 152 00:15:35.039 --> 00:15:41.200 the Lorentz transformation give. For these two ticks, t equals zero and t 153 00:15:41.519 --> 00:15:50.240 prime equals i divided by the square root of i v squared over C squared. 154 00:15:50.720 --> 00:15:54.960 As judged from k, the clock is moving with the velocity V. 155 00:15:56.159 --> 00:16:02.320 As judged from this reference body, the time which elapses between two strokes of 156 00:16:02.360 --> 00:16:07.320 the clock is not one second, but i divided by the square root of 157 00:16:07.600 --> 00:16:15.480 i v squared over C squared seconds i e a somewhat larger time. As 158 00:16:15.519 --> 00:16:21.080 a consequence of its motion, the clock goes more slowly than when at rest. 159 00:16:22.159 --> 00:16:26.679 Here also, the velocity C plays the part of an unattainable limiting velocity. 160 00:16:29.879 --> 00:16:30.399 End of section twelve.