WEBVTT 1 00:00:00.400 --> 00:00:07.200 Section twenty one, in what respects are the foundations of classical mechanics and of 2 00:00:07.240 --> 00:00:14.919 the special theory of relativity unsatisfactory? We have already stated several times that classical 3 00:00:14.919 --> 00:00:21.120 mechanics starts out from the following law. Material particles sufficiently far removed from other 4 00:00:21.199 --> 00:00:27.399 material particles continue to move uniformly in a straight line, or continue in a 5 00:00:27.440 --> 00:00:32.960 state of rest. We have also repeatedly emphasized that this fundamental law can only 6 00:00:33.000 --> 00:00:39.280 be valid for bodies of reference K, which possess certain unique states of motion, 7 00:00:39.759 --> 00:00:46.200 and which are in uniform translational motion relative to each other. Relative to 8 00:00:46.240 --> 00:00:52.600 other reference bodies K, the law is not valid both in classical mechanics and 9 00:00:52.759 --> 00:00:58.479 in the special theory of relativity. We therefore differentiate between reference bodies K, 10 00:00:59.399 --> 00:01:03.680 relative to which the recognized laws of nature can be said to hold, and 11 00:01:03.920 --> 00:01:10.799 reference bodies K, relative to which these laws do not hold. But no 12 00:01:10.959 --> 00:01:15.599 person whose mode of thought is logical can rest satisfied with this condition of things. 13 00:01:17.159 --> 00:01:22.680 He asks, how does it come that certain reference bodies or their states 14 00:01:22.680 --> 00:01:27.000 of motion are given priority over other reference bodies or their states of motion? 15 00:01:27.959 --> 00:01:34.079 What is the reason for this preference? In order to show clearly what I 16 00:01:34.120 --> 00:01:38.560 mean by this question. I shall make use of a comparison. I am 17 00:01:38.599 --> 00:01:44.319 standing in front of a gas range. Standing alongside of each other. On 18 00:01:44.400 --> 00:01:49.159 the range are two pans, so much alike that one may be mistaken for 19 00:01:49.239 --> 00:01:53.879 the other. Both are half full of water. I notice that steam is 20 00:01:53.879 --> 00:01:59.959 being emitted continuously from one pan, but not from the other. I'm surprised, 21 00:02:00.040 --> 00:02:04.120 rised at this, even if I have never seen either a gas range 22 00:02:04.280 --> 00:02:08.800 or a pan before. But if I now notice a luminous something of bluish 23 00:02:08.800 --> 00:02:14.719 color under the first pan, but not under the other, I cease to 24 00:02:14.759 --> 00:02:19.840 be astonished, even if I have never before seen a gas flame. For 25 00:02:19.960 --> 00:02:24.199 I can only say that this bluish something will cause the emission of the steam, 26 00:02:25.000 --> 00:02:30.280 or at least possibly it may do so. If, however, I 27 00:02:30.400 --> 00:02:35.120 notice the bluish something in neither case, and if I observe that the one 28 00:02:35.199 --> 00:02:40.319 continuously emits steam whilst the other does not, then I shall remain astonished and 29 00:02:40.400 --> 00:02:46.919 dissatisfied until I have discovered some circumstance to which I can attribute the different behavior 30 00:02:47.120 --> 00:02:53.080 of the two pans. Analogously, I seek in vain for a real something 31 00:02:53.240 --> 00:03:00.479 in classical mechanics or in the special theory of relativity to which I can tribute 32 00:03:00.520 --> 00:03:07.039 the different behavior of bodies considered with respect to the reference systems K and K 33 00:03:07.280 --> 00:03:15.240 prime begin footnote. The objection is of importance more especially when the state of 34 00:03:15.360 --> 00:03:19.479 motion of the reference body is of such a nature that it does not require 35 00:03:19.879 --> 00:03:24.280 any external agency for its maintenance, for example, in the case when the 36 00:03:24.360 --> 00:03:34.400 reference body is rotating uniformly. End footnote. Newton saw this objection and attempted 37 00:03:34.439 --> 00:03:39.000 to invalidate it, but without success. But E. Mok recognized it most 38 00:03:39.039 --> 00:03:45.400 clearly of all, and because of this objection he claimed that mechanics must be 39 00:03:45.479 --> 00:03:50.599 placed on a new basis. It can only be got rid of by means 40 00:03:50.719 --> 00:03:55.240 of a physics which is conformable to the general principle of relativity, since the 41 00:03:55.319 --> 00:04:00.639 equations of such a theory hold for every body of reference, whatever may be 42 00:04:00.879 --> 00:04:10.680 its state of motion. Section twenty two a few inferences from the general principle 43 00:04:10.800 --> 00:04:17.120 of relativity. The considerations of section twenty show that the general principle of relativity 44 00:04:17.600 --> 00:04:23.560 puts us in a position to derive properties of the gravitational field in a purely 45 00:04:23.680 --> 00:04:28.360 theoretical manner. Let us suppose, for instance, that we know the space 46 00:04:28.439 --> 00:04:33.759 time course for any natural process whatsoever as regards the manner in which it takes 47 00:04:33.759 --> 00:04:41.639 place in the Galilean domain relative to a Galilean body of reference k by means 48 00:04:41.680 --> 00:04:47.160 of purely theoretical operations by e simply by calculation. We are then able to 49 00:04:47.199 --> 00:04:54.879 find how this known natural process appears as seen from a reference body K prime, 50 00:04:55.399 --> 00:05:01.439 which is accelerated relatively to K. But the gravitational field exists with respect 51 00:05:01.439 --> 00:05:06.879 to this new body of reference k prime. Our consideration also teaches us how 52 00:05:06.920 --> 00:05:14.800 the gravitational field influences the process studied. For example, we learn that a 53 00:05:14.839 --> 00:05:18.560 body which is in a state of uniform rectilinear motion with respect to K, 54 00:05:19.240 --> 00:05:25.800 in accordance with the law of Galiley, is executing an accelerated and in general 55 00:05:26.120 --> 00:05:33.360 curvilinear motion with respect to the accelerated reference body k prime chest. This acceleration 56 00:05:33.839 --> 00:05:40.680 or curvature corresponds to the influence on the moving body of the gravitational field prevailing 57 00:05:40.759 --> 00:05:46.560 relatively to K prime. It is known that a gravitational field influences the movement 58 00:05:46.639 --> 00:05:51.199 of bodies in this way, so that our consideration supplies us with nothing essentially 59 00:05:51.279 --> 00:05:58.279 new. However, we obtain a new result of fundamental importance when we carry 60 00:05:58.279 --> 00:06:03.399 out the analogous consider duration for a ray of light with respect to the Galilean 61 00:06:03.519 --> 00:06:10.439 reference body K. Such a ray of light is transmitted rectilinearly with the velocity 62 00:06:10.639 --> 00:06:15.079 C. It can easily be shown that the path of the same ray of 63 00:06:15.199 --> 00:06:18.839 light is no longer a straight line when we consider it with reference to the 64 00:06:18.920 --> 00:06:27.480 accelerated chest reference body K. Prime. From this we conclude that in general, 65 00:06:27.920 --> 00:06:33.959 rays of light are propagated curvilinearly in gravitational fields. In two respects. 66 00:06:33.959 --> 00:06:39.240 This result is of great importance. In the first place, it can be 67 00:06:39.319 --> 00:06:44.839 compared with the reality. Although a detailed examination of the question shows that the 68 00:06:44.879 --> 00:06:49.199 curvature of light ray is required by the general theory of relativity, is only 69 00:06:49.240 --> 00:06:55.839 exceedingly small for the gravitational fields at our disposal. In practice, its estimated 70 00:06:55.879 --> 00:07:01.319 magnitude for light rays passing the Sun at grazing incidents is nevertheless one point seven 71 00:07:01.439 --> 00:07:08.240 seconds of arc. This ought to manifest itself in the following way. As 72 00:07:08.240 --> 00:07:12.839 seen from the Earth, certain fixed stars appear to be in the neighborhood of 73 00:07:12.879 --> 00:07:17.240 the Sun and are thus capable of observation during a total eclipse of the Sun. 74 00:07:18.160 --> 00:07:23.680 At such times, these stars ought to appear to be displaced outwards from 75 00:07:23.720 --> 00:07:29.240 the Sun by an amount indicated above, as compared with their apparent position in 76 00:07:29.240 --> 00:07:32.800 the sky when the Sun is situated at another part of the heavens. The 77 00:07:32.920 --> 00:07:39.519 examination of the correctness or otherwise of this deduction is a problem of the greatest 78 00:07:39.560 --> 00:07:46.040 importance, the early solution of which is to be expected of astronomers. Begin 79 00:07:46.160 --> 00:07:51.360 footnote. By means of the star photographs of two expeditions equipped by a joint 80 00:07:51.360 --> 00:07:57.560 committee of the Royal and Royal Astronomical Societies. The existence of the deflection of 81 00:07:57.639 --> 00:08:03.399 light demanded by theory was first confirmed during the solar eclipse of twenty ninth May 82 00:08:03.519 --> 00:08:11.759 nineteen nineteen. End footnote. In the second place, our result shows that, 83 00:08:11.959 --> 00:08:16.000 according to the general theory of relativity, the law of the constancy of 84 00:08:16.040 --> 00:08:22.480 the velocity of light in vacuo, which constitutes one of the two fundamental assumptions 85 00:08:22.480 --> 00:08:28.120 in the special theory of relativity, and to which we have already frequently referred, 86 00:08:28.680 --> 00:08:33.639 cannot claim any unlimited validity. A curvature of rays of light can only 87 00:08:33.679 --> 00:08:39.360 take place when the velocity of propagation of light varies with position. Now, 88 00:08:39.440 --> 00:08:43.799 we might think that as a consequence of this, the special theory of relativity 89 00:08:45.240 --> 00:08:48.639 and with it the whole theory of relativity, would be laid in the dust. 90 00:08:48.360 --> 00:08:52.360 But in reality this is not the case. We can only conclude that 91 00:08:52.440 --> 00:08:58.919 the special theory of relativity cannot claim an unlimited domain of validity. Its results 92 00:09:00.159 --> 00:09:05.440 hold only so long as we are able to disregard the influences of gravitational fields 93 00:09:05.440 --> 00:09:09.240 on the phenomena, for example, of light. Since it has often been 94 00:09:09.240 --> 00:09:16.039 contended by opponents of the theory of relativity that the special theory of relativity is 95 00:09:16.120 --> 00:09:20.240 overthrown by the general theory of relativity, it is perhaps advisable to make the 96 00:09:20.240 --> 00:09:26.720 facts of the case clearer by means of an appropriate comparison. Before the development 97 00:09:26.799 --> 00:09:33.639 of electrodynamics, the laws of electrostatics were looked upon as the laws of electricity. 98 00:09:33.559 --> 00:09:37.200 At the present time, we know that electric fields can be derived correctly 99 00:09:37.240 --> 00:09:43.360 from electrostatic considerations only for the case which is never strictly realized, in which 100 00:09:43.360 --> 00:09:48.840 the electrical masses are quite at rest relatively to each other and to the coordinate 101 00:09:48.879 --> 00:09:54.200 system. Should we be justified in saying that for this reason, electrostatics is 102 00:09:54.240 --> 00:10:01.720 overthrown by the field equations of Maxwell and electrodynamics not in the least. Electrostatics 103 00:10:01.840 --> 00:10:07.679 is contained in electrodynamics as a limiting case. The laws of the latter lead 104 00:10:07.759 --> 00:10:11.639 directly to those of the former. For the case in which the fields are 105 00:10:11.679 --> 00:10:18.919 invariable with regard to time. No fairer destiny could be allotted to any physical 106 00:10:18.960 --> 00:10:22.960 theory than that it should of itself point out the way to the introduction of 107 00:10:22.960 --> 00:10:28.120 a more comprehensive theory in which it lives on as a limiting case. In 108 00:10:28.200 --> 00:10:33.159 the example of the transmission of light just dealt with, we have seen that 109 00:10:33.200 --> 00:10:39.480 the general theory of relativity enables us to derive theoretically the influence of a gravitational 110 00:10:39.519 --> 00:10:43.200 field on the course of natural processes, the laws of which are already known 111 00:10:43.440 --> 00:10:48.679 when a gravitational field is absent. But the most attractive problem to the solution 112 00:10:48.720 --> 00:10:54.120 of which the general theory of relativity supplies the key, concerns the investigation of 113 00:10:54.159 --> 00:11:00.440 the laws satisfied by the gravitational field itself. Let us consider this for a 114 00:11:00.519 --> 00:11:05.399 moment. We are acquainted with space, time domains which behave approximately in a 115 00:11:05.600 --> 00:11:13.480 Galileyan fashion under suitable choice of reference body, ie, domains in which gravitational 116 00:11:13.519 --> 00:11:18.399 fields are absent. If we now refer such a domain to a reference body 117 00:11:18.519 --> 00:11:22.600 k prime possessing any kind of motion, then relative to k prime, there 118 00:11:22.679 --> 00:11:30.399 exists a gravitational field which is variable with respect to space and time. Begin 119 00:11:30.519 --> 00:11:37.279 footnote. This follows from a generalization of the discussion in section twenty end footnote. 120 00:11:37.919 --> 00:11:41.840 The character of this field will of course depend on the motion chosen for 121 00:11:41.919 --> 00:11:48.360 k prime. According to the general theory of relativity, the general law of 122 00:11:48.399 --> 00:11:52.919 the gravitational field must be satisfied for all gravitational fields obtainable in this way. 123 00:11:54.000 --> 00:11:58.879 Even though by no means all gravitational fields can be produced in this way, 124 00:12:00.039 --> 00:12:05.039 yet we may entertain the hope that the general law of gravitation will be derivable 125 00:12:05.159 --> 00:12:09.759 from such gravitational fields of a special kind. This hope has been realized in 126 00:12:09.799 --> 00:12:16.600 the most beautiful manner. But between the clear vision of this goal and its 127 00:12:16.639 --> 00:12:22.840 actual realization it was necessary to surmount a serious difficulty. And as this lies 128 00:12:22.919 --> 00:12:26.720 deep at the root of things, I dare not withhold it from the reader. 129 00:12:26.200 --> 00:12:33.720 We require to extend our ideas of the space time continuum still farther Section 130 00:12:33.879 --> 00:12:41.480 twenty three behavior of clocks and measuring rods on a rotating body of reference. 131 00:12:43.000 --> 00:12:48.159 Hitherto, I have purposefully refrained from speaking about the physical interpretation of space and 132 00:12:48.320 --> 00:12:54.080 time data in the case of this general theory of relativity. As a consequence, 133 00:12:54.080 --> 00:12:58.080 I am guilty of a certain slovenliness of treatment, which, as we 134 00:12:58.200 --> 00:13:03.519 know from the special theory of relativetivity, is far from being unimportant and pardonable. 135 00:13:03.399 --> 00:13:07.000 It is now high time that we remedy this defect, But I would 136 00:13:07.039 --> 00:13:11.399 mention at the outset that this matter lays no small claims on the patients and 137 00:13:11.519 --> 00:13:16.200 on the power of abstraction of the reader. We start off again from quite 138 00:13:16.240 --> 00:13:22.840 special cases which we have frequently used before. Let us consider a space time 139 00:13:22.919 --> 00:13:28.080 domain in which no gravitational field exists relative to a reference body K, whose 140 00:13:28.080 --> 00:13:33.480 state of motion has been suitably chosen. K is then a galiley in reference 141 00:13:33.519 --> 00:13:39.519 body as regards the domain considered, and the results of the special theory of 142 00:13:39.559 --> 00:13:46.000 relativity hold relative to K. Let Us suppose the same domain referred to a 143 00:13:46.080 --> 00:13:50.519 second body of reference K prime, which is rotating uniformly with respect to K. 144 00:13:52.639 --> 00:13:56.360 In order to fix our ideas, we shall imagine K prime to be 145 00:13:56.399 --> 00:14:01.360 in the form of a plain circular disk which rotates uniformly in its own plane 146 00:14:01.399 --> 00:14:07.919 about its center. An observer who is sitting eccentrically on the disk k prime 147 00:14:07.399 --> 00:14:13.240 is sensible of a force which acts outward in a radial direction, and which 148 00:14:13.240 --> 00:14:18.000 would be interpreted as an effect of inertia centrifugal force by an observer who was 149 00:14:18.080 --> 00:14:24.519 at rest with respect to the original reference body K. But the observer on 150 00:14:24.559 --> 00:14:28.559 the disk may regard his disk as a reference body which is at rest on 151 00:14:28.600 --> 00:14:33.080 the basis of the general principle of relativity. He is justified in doing this. 152 00:14:33.960 --> 00:14:39.159 The force acting on himself, and in fact on all other bodies which 153 00:14:39.159 --> 00:14:43.240 are at rest relative to the disk, he regards as the effect of a 154 00:14:43.279 --> 00:14:50.120 gravitational field. Nevertheless, the space distribution of this gravitational field is of a 155 00:14:50.200 --> 00:14:56.039 kind that would not be possible on Newton's theory of gravitation. Begin footnote. 156 00:14:56.960 --> 00:15:03.279 The field disappears at the center of the disk and increases proportionally to the distance 157 00:15:03.360 --> 00:15:09.799 from the center as we proceed atwards and footnote. But since the observer believes 158 00:15:09.840 --> 00:15:15.320 in the general theory of relativity. This does not disturb him. He is 159 00:15:15.399 --> 00:15:18.279 quite in the right when he believes that a general law of gravitation can be 160 00:15:18.320 --> 00:15:24.000 formulated, a law which not only explains the motion of the stars correctly, 161 00:15:24.519 --> 00:15:30.759 but also the field of force experienced by himself. The observer performs experiments on 162 00:15:30.840 --> 00:15:35.559 his circular disc with clocks and measuring rods. In doing so, it is 163 00:15:35.600 --> 00:15:41.320 his intention to arrive at exact definitions for the significance of time and space data 164 00:15:41.679 --> 00:15:46.919 with reference to the circular disc k prime, these definitions being based on his 165 00:15:46.039 --> 00:15:52.120 observations, what will be his experience in the enterprise. To start with, 166 00:15:52.320 --> 00:15:58.879 he places one of two identically constructed clocks at the center of the circular disc 167 00:15:58.360 --> 00:16:03.000 and the other on the edge of the disk, so that they are at 168 00:16:03.039 --> 00:16:07.799 rest relative to it. We now ask ourselves whether both clocks go at the 169 00:16:07.879 --> 00:16:15.480 same rate from the standpoint of the non rotating Galilean reference body K. As 170 00:16:15.600 --> 00:16:18.519 judged from this body, the clock at the center of the disc has no 171 00:16:18.639 --> 00:16:23.159 velocity, whereas the clock at the edge of the disk is in motion relative 172 00:16:23.200 --> 00:16:29.919 to K in consequence of the rotation. According to a result obtained in section 173 00:16:30.039 --> 00:16:34.559 twelve. It follows that the latter clock goes at a rate permanently slower than 174 00:16:34.600 --> 00:16:40.159 that of the clock at the center of the circular disk, ie as observed 175 00:16:40.240 --> 00:16:44.840 from k It is obvious that the same effect would be noted by an observer, 176 00:16:45.240 --> 00:16:48.679 whom we will imagine sitting alongside his clock at the center of the circular 177 00:16:48.720 --> 00:16:53.679 disk. Thus on our circular disc, or, to make the case more 178 00:16:53.759 --> 00:17:00.399 general, in every gravitational field, a clock will go more quickly or less 179 00:17:00.480 --> 00:17:06.680 quickly according to the position in which the clock is situated at rest. For 180 00:17:06.720 --> 00:17:11.359 this reason, it is not possible to obtain a reasonable definition of time with 181 00:17:11.440 --> 00:17:15.680 the aid of clocks which are arranged at rest with respect to the body of 182 00:17:15.720 --> 00:17:22.319 reference. A similar difficulty presents itself when we attempt to apply our earlier definition 183 00:17:22.400 --> 00:17:26.279 of simultaneity in such a case. But I do not wish to go any 184 00:17:26.319 --> 00:17:30.759 farther into this question. Moreover, at this stage, the definition of the 185 00:17:30.799 --> 00:17:38.039 space coordinates also presents insurmountable difficulties. If the observer applies his standard measuring rod, 186 00:17:38.519 --> 00:17:44.119 a rod which is short as compared with the radius of the disk tangentially 187 00:17:44.240 --> 00:17:48.720 to the edge of the disk. Then, as judged from the Galilean system, 188 00:17:48.839 --> 00:17:52.279 the length of this rod will be less than I, since, according 189 00:17:52.319 --> 00:17:57.400 to Section twelve, moving bodies suffer a shortening in the direction of the motion. 190 00:17:59.200 --> 00:18:02.160 On the other hand, end, the measuring rod will not experience a 191 00:18:02.240 --> 00:18:06.400 shortening in length as judged from K, if it is applied to the disc 192 00:18:06.519 --> 00:18:11.160 in the direction of the radius. If then the observer first measures the circumference 193 00:18:11.160 --> 00:18:15.880 of the disc with his measuring rod, and then the diameter of the disk. 194 00:18:15.480 --> 00:18:19.160 On dividing the one by the other, he will not obtain as quotient 195 00:18:19.279 --> 00:18:25.359 the familiar number Pi equals three point one, four, etc. But a 196 00:18:25.480 --> 00:18:33.920 larger number begin footnote. Throughout this consideration we have to use the Galilean non 197 00:18:33.079 --> 00:18:38.279 rotating system K as reference body, since we may only assume the validity of 198 00:18:38.319 --> 00:18:44.880 the results of the special theory of relativity relative to K. Relative to k 199 00:18:45.039 --> 00:18:52.519 prime a gravitational field prevails end footnote, but a larger number, whereas, 200 00:18:52.680 --> 00:18:56.559 of course, for a disc which is at rest with respect to K, 201 00:18:56.519 --> 00:19:03.559 this operation would yield pi actly. This proves that the propositions of Euclidean geometry 202 00:19:03.880 --> 00:19:10.640 cannot hold exactly on the rotating disk, nor in general in a gravitational field, 203 00:19:11.079 --> 00:19:15.559 at least if we attribute the length I to the rod in all positions 204 00:19:15.559 --> 00:19:21.480 in every orientation. Hence the idea of a straight line also loses its meaning. 205 00:19:22.440 --> 00:19:26.079 We are therefore not in a position to define exactly the coordinates x, 206 00:19:26.519 --> 00:19:32.200 y z relative to the disk by means of the method used in discussing the 207 00:19:32.279 --> 00:19:36.880 special theory, And as long as the coordinates and times of events have not 208 00:19:37.039 --> 00:19:41.880 been defined, we cannot assign an exact meaning to the natural laws in which 209 00:19:41.920 --> 00:19:48.680 these occur. Thus, all our previous conclusions based on general relativity would appear 210 00:19:48.720 --> 00:19:55.000 to be called in question. In reality, we must make a subtle detour 211 00:19:55.200 --> 00:20:00.519 in order to be able to apply the postulate of general relativity. Actually, 212 00:20:00.359 --> 00:20:07.519 I shall prepare the reader for this in the following paragraphs. End of sections 213 00:20:07.599 --> 00:20:08.759 twenty one to twenty three.