WEBVTT 1 00:00:00.400 --> 00:00:08.560 Section thirty cosmological difficulties of Newton's theory. A far from the difficulty discussed in 2 00:00:08.599 --> 00:00:16.000 section twenty one, there is a second fundamental difficulty attending classical celestial mechanics, 3 00:00:16.079 --> 00:00:20.199 which, to the best of my knowledge, was first discussed in detail by 4 00:00:20.199 --> 00:00:25.359 the astronomers Selucre. If we ponder over the question as to how the universe, 5 00:00:25.800 --> 00:00:29.719 considered as a whole, is to be regarded, the first answer that 6 00:00:29.800 --> 00:00:35.880 suggestsself to us is surely this, as reguard space and time, universe is 7 00:00:35.920 --> 00:00:41.880 infinite. There are stars everywhere, so that the density of matter, although 8 00:00:42.039 --> 00:00:46.960 very variable in detail, is nevertheless, on the average everywhere the same. 9 00:00:47.600 --> 00:00:52.600 In other words, however far we might travel through space, we should find 10 00:00:52.719 --> 00:00:59.399 everywhere in attenuated swarm of fixed stars of approximately the same kind and density. 11 00:01:00.600 --> 00:01:03.599 This view is not in harmony with the theory of Newton, but the latter 12 00:01:03.680 --> 00:01:08.280 theory rather requires that the universe should have a kind of center in which the 13 00:01:08.359 --> 00:01:14.439 density of the stars is a maximum, and that as we proceed outwards from 14 00:01:14.439 --> 00:01:19.480 the center, the group density of the stars should diminish, until finally, 15 00:01:19.959 --> 00:01:26.439 at great distances it is succeeded by an infinite region of emptiness. The stellar 16 00:01:26.519 --> 00:01:32.439 universe ought to be a finite island in the infinite ocean of space. Begin 17 00:01:32.640 --> 00:01:38.840 fine proof. According to the theory of Newton, the number of lines of 18 00:01:38.840 --> 00:01:45.439 force which come from infinity and terminate in a mass M is proportional to the 19 00:01:45.519 --> 00:01:52.519 mass M. If on the average, the mass density rose sum zero is 20 00:01:52.599 --> 00:01:57.480 constant throughout the universe, then a sphere of volume V will enclose the average 21 00:01:57.519 --> 00:02:04.439 man rose zero V. Thus, the number of lines of force passing through 22 00:02:04.480 --> 00:02:10.000 the surface F of the sphere into its interior is proportional to row subzero V 23 00:02:12.199 --> 00:02:15.000 or unit area of the surface of the sphere. The number of lines of 24 00:02:15.120 --> 00:02:23.479 force which enters the sphere is thus proportional to row subzero V over F or 25 00:02:23.639 --> 00:02:30.159 two row subzero R. Hence, the intensity of the field at the surface 26 00:02:30.840 --> 00:02:38.000 would ultimately become infinite with increasing radius R of the sphere, which is impossible 27 00:02:39.560 --> 00:02:46.360 n footnote this conception is in itself not very satisfactory. It is still less 28 00:02:46.360 --> 00:02:53.319 satisfactory because it leads to the result that the light emitted by the stars and 29 00:02:53.479 --> 00:02:59.439 also individual stars of the stellar system are perpetually passing out into infinite space, 30 00:03:00.080 --> 00:03:05.479 never to return and without ever again coming into interaction with other objects of nature, 31 00:03:06.319 --> 00:03:13.400 Such a finite material universe would be destined to become gradually but systematically impoverished. 32 00:03:14.400 --> 00:03:19.719 In order to escape this dilemma, Seeliger suggested a modification of Newton's law, 33 00:03:20.639 --> 00:03:24.360 in which he assumes that for great distances, the force of attraction between 34 00:03:24.439 --> 00:03:31.639 two masses diminishes more rapidly than would result from the inverse square law. In 35 00:03:31.680 --> 00:03:38.000 this way, it is possible for the mean density of matter to be constant 36 00:03:38.199 --> 00:03:46.360 everywhere, even to infinity, without infinitely large gravitational fields being produced. We 37 00:03:46.439 --> 00:03:52.719 thus free ourselves from the distasteful conception that the material universe ought to possess something 38 00:03:52.719 --> 00:03:57.680 of the nature of a center. Of course, we purchase our emancipation from 39 00:03:57.680 --> 00:04:03.199 the fundamental difficulties mentioned at the cost of a modification and complication of Newton's law, 40 00:04:03.680 --> 00:04:11.000 which is neither empirical nor theoretical foundation. We can imagine innumerable laws which 41 00:04:11.000 --> 00:04:15.479 would serve the same purpose without our being able to state a reason why one 42 00:04:15.520 --> 00:04:18.360 of them is to be preferred to the others. For any one of these 43 00:04:18.480 --> 00:04:25.240 laws would be founded just as little on more general theoretical principles as is the 44 00:04:25.360 --> 00:04:33.240 law of Newton end a Section thirty Section thirty one the possibility of a finite 45 00:04:33.279 --> 00:04:41.639 and yet unbounded universe. But speculations on the structure of the universe also move 46 00:04:41.680 --> 00:04:47.160 in quite another direction. The development of non Euclidean geometry led to the recognition 47 00:04:47.199 --> 00:04:53.720 of the fact that we can cast out on the infiniteness of our space without 48 00:04:53.720 --> 00:04:59.920 coming into conflict with the laws of thought or with experience Raymond Home whole. 49 00:05:00.279 --> 00:05:06.160 These questions have already been treated in detail and with unsurpassedbu lucidity by Helmwoltz and 50 00:05:06.279 --> 00:05:12.879 Poincare, whereas I can only touch on them briefly. Here. In the 51 00:05:12.920 --> 00:05:17.759 first place, we imagine in existence in two dimensional space flat beings with flat 52 00:05:17.839 --> 00:05:24.360 implements, and in particular flat rigid measuring rods, are free to move in 53 00:05:24.439 --> 00:05:30.480 a plane. For them, nothing exists outside of this plane. That which 54 00:05:30.480 --> 00:05:34.720 they observe to happen to themselves and their flat things is the all inclusive reality 55 00:05:35.199 --> 00:05:43.000 of their plane. In particular, the constructions of plain Euclidean geometry can be 56 00:05:43.079 --> 00:05:48.600 carried out by means of the rods. For example, the lattice construction considered 57 00:05:48.600 --> 00:05:55.199 in Section twenty four. In contrast to ours. The universe of these beings 58 00:05:55.600 --> 00:06:01.000 is two dimensional, but like ours, it extends to infinity. In their 59 00:06:01.120 --> 00:06:06.480 universe there is room for an infinite number of identical squares made up of rods 60 00:06:06.759 --> 00:06:15.720 ie its volume surface is infinite. If these beings say their universe is quote 61 00:06:15.079 --> 00:06:20.319 plain, there is sense in the statement, because they mean that they can 62 00:06:20.360 --> 00:06:27.639 perform the constructions of plain Euclidean geometry with their rods. In this connection, 63 00:06:27.959 --> 00:06:34.000 the individual rods always represent the same distance, independently of their position. Let 64 00:06:34.079 --> 00:06:40.560 us consider now a second two dimensional existence, but this time on a spherical 65 00:06:40.600 --> 00:06:45.399 surface. Instead of auto play. The flat beings, with their measuring rods 66 00:06:45.639 --> 00:06:49.480 and other objects, fit exactly on the surface. They are unable to leave 67 00:06:49.519 --> 00:06:57.519 it. Their holy universe of observation extends exclusively over the surface of the sphere. 68 00:06:57.560 --> 00:07:01.199 Are these beings able to regard the geometry of their universe as being plain 69 00:07:01.319 --> 00:07:08.639 geometry and the rods with all as a realization of distance. They cannot do 70 00:07:08.680 --> 00:07:13.800 this, for if they attempt to realize a straight line, they will obtain 71 00:07:13.839 --> 00:07:20.040 a curve, which we three dimensional beings designate as a great circle, i 72 00:07:20.279 --> 00:07:27.240 e. A self contained line of definite phenite length which can be measured up 73 00:07:27.560 --> 00:07:32.560 by means of a measuring rod. Similarly, this universe has a phenite area 74 00:07:32.720 --> 00:07:39.920 that can be compared with the area of a square constructed with rods. Great 75 00:07:40.000 --> 00:07:46.079 charm resulting from this consideration lies in the recognition of the fact that the universe 76 00:07:46.120 --> 00:07:54.199 of these beings is phenite and yet has no limits. But the spherical surface 77 00:07:54.279 --> 00:07:59.000 beings do not need to go on a world tour in order to perceive that 78 00:07:59.040 --> 00:08:03.839 they are not living any Euclidean universe. They can convince themselves of this on 79 00:08:05.000 --> 00:08:09.319 every part of their world, provided they do not use too small a piece 80 00:08:09.360 --> 00:08:13.480 of it. Starting from a point, they draw straight lines. Arcs of 81 00:08:13.560 --> 00:08:20.360 circles is judged in three dimensional space of equal length in all directions. They 82 00:08:20.399 --> 00:08:26.839 will call the line joining the free ends of these lines a circle. For 83 00:08:26.000 --> 00:08:31.720 a plane surface, the ratio of the circumference of a circle to its diameter, 84 00:08:31.279 --> 00:08:37.960 both lengths being measured with the same round is, according to Euclidean geometry 85 00:08:37.960 --> 00:08:43.759 of the plane, equal to a constant value pie, which is independent of 86 00:08:43.799 --> 00:08:48.480 the diameter of the circle. On their spherical surface. Our flat beings would 87 00:08:48.519 --> 00:08:58.840 find for this ratio the value equation twenty seven Pie times sign parentheses little R 88 00:09:00.080 --> 00:09:11.039 over big r unparentheses divided by parentheses little r over big r unparentheses i e. 89 00:09:11.720 --> 00:09:18.200 Smaller value than pie, the difference being the more considerable greater is the 90 00:09:18.320 --> 00:09:22.440 radius of the circle in comparison with the radius R of the world's sphere. 91 00:09:24.519 --> 00:09:28.519 By means of this relation, spherical beings can determine the radius of their universe 92 00:09:30.840 --> 00:09:37.320 quote world unquote, even when only relatively small part of their world's sphere it's 93 00:09:37.360 --> 00:09:41.360 available for their measurements. But if this part is very small, indeed, 94 00:09:41.840 --> 00:09:46.240 they will no longer be able to demonstrate that they are on a spherical world 95 00:09:46.799 --> 00:09:52.759 and not on a Euclidean plane, For a small part of a spherical surface 96 00:09:52.320 --> 00:09:58.200 differs only slightly from a piece of a plane of the same size. Thus, 97 00:09:58.279 --> 00:10:03.639 of the spherical surface beings are living on a planet of which the Solar 98 00:10:03.679 --> 00:10:09.200 System occupies only a negligibly small part of the spherical universe. They am no 99 00:10:09.279 --> 00:10:15.919 means of determining whether they are living in a finite or an infinite universe, 100 00:10:16.559 --> 00:10:22.360 because a piece of universe to which they have access is in both cases practically 101 00:10:22.480 --> 00:10:30.000 play more Euclidean. It follows directly from this discussion that for our sphere beings, 102 00:10:30.320 --> 00:10:35.639 the circumference of a circle first increases with the radius till the circumference of 103 00:10:35.639 --> 00:10:41.279 the universe is reached, and that it thenceforward gradually decreases to cyril for still 104 00:10:41.399 --> 00:10:48.240 further increasing values of the radius. During this process, the area of the 105 00:10:48.320 --> 00:10:54.639 circle continues to increase more and more, until finally becomes equal to the total 106 00:10:54.759 --> 00:11:00.159 area of the whole world's sphere. Perhaps the reader will wonder why we have 107 00:11:00.240 --> 00:11:05.000 placed our beings on a sphere rather than on another closed surface. But this 108 00:11:05.120 --> 00:11:11.679 choice has its justifications in the fact that of all closed surfaces, sphere is 109 00:11:11.799 --> 00:11:16.799 unique in possessing the property that all points on it are equivalent. I admit 110 00:11:16.840 --> 00:11:22.480 that the ratio of the circumference of a circle to its radius R depends on 111 00:11:22.840 --> 00:11:26.879 R, but for a given value of R, it is the same for 112 00:11:26.960 --> 00:11:31.000 all points of the world's sphere. In other words, the world's sphere is 113 00:11:31.000 --> 00:11:37.720 a surface of constant curvature. To this two dimensional sphere universe, there is 114 00:11:37.720 --> 00:11:45.440 a three dimensional analogy, namely the three dimensional spherical space, which was discovered 115 00:11:45.440 --> 00:11:50.519 by Ramond. Its points are likewise all equivalent. It possesses a finite volume, 116 00:11:50.919 --> 00:11:58.159 which is determined by its radius two pie squared are cubed. Is it 117 00:11:58.200 --> 00:12:03.120 possible to imagine a spherical space. To imagine a space means nothing else than 118 00:12:03.159 --> 00:12:09.000 that we imagine an epitome of our space experience, i e. Of experience 119 00:12:09.000 --> 00:12:13.720 that we can have in the movement of rigid bodies. In this sense, 120 00:12:13.879 --> 00:12:20.759 we can imagine a spherical space. Suppose we draw lines or stretch strings in 121 00:12:20.840 --> 00:12:24.480 all directions from a point, and mark off from each of these the distance 122 00:12:24.639 --> 00:12:31.360 are the measuring ron. All the free ends of these lengths lie on a 123 00:12:31.399 --> 00:12:37.759 spherical surface. We can specially measure up the area F of the surface by 124 00:12:37.799 --> 00:12:43.120 means of a square made up of measuring rods. If the universe is Euclidean, 125 00:12:43.559 --> 00:12:48.879 then F equals four pi are squared. If it is spherical, then 126 00:12:48.919 --> 00:12:56.200 F is always less than four pi are squared. With increasing values of R, 127 00:12:56.960 --> 00:13:01.879 f increases from zero up to a maximum value, which is determined by 128 00:13:01.919 --> 00:13:07.519 the world radius. But for still further increasing values of R, the area 129 00:13:07.600 --> 00:13:13.399 gradually diminishes to zero. At first, the straight lines which radiate from the 130 00:13:13.399 --> 00:13:18.039 starting point diverge farther and farther from one another, but later they approach each 131 00:13:18.080 --> 00:13:26.000 other, and finally they run together again at a quote counterpoint unquote to the 132 00:13:26.039 --> 00:13:31.480 starting point. Under such conditions, they have traversed the whole spherical space. 133 00:13:31.200 --> 00:13:37.159 It is easily seen that the three dimensional spherical space is quite analogous to the 134 00:13:37.200 --> 00:13:41.879 two dimensional spherical surface. It is phinite i e. A phenite volume, 135 00:13:43.480 --> 00:13:48.639 and has no bounds. It may be mentioned that there is yet another kind 136 00:13:48.720 --> 00:13:56.679 of curved space quote elliptical space. It can be regarded as a curved space 137 00:13:56.759 --> 00:14:03.919 in which the two counterpoints are identical indistinguishable from each other. An elliptical universe 138 00:14:03.960 --> 00:14:11.320 can thus be considered to some extent as a curved universe possessing central symmetry. 139 00:14:11.240 --> 00:14:16.919 It follows from what has been said that closed spaces without limits are conceivable. 140 00:14:18.559 --> 00:14:24.120 From amongst these, spherical space and the elliptical excels in its simplicity, since 141 00:14:24.159 --> 00:14:30.480 all points on it are equivalent. As a result of this discussion, a 142 00:14:30.600 --> 00:14:35.320 most interesting question arises for astronomers and physicists, and that is whether the universe 143 00:14:35.360 --> 00:14:41.279 in which we live is infinite or whether it is finite in the manner of 144 00:14:41.279 --> 00:14:46.480 the spherical universe. Our experience is far from being sufficient to enable us to 145 00:14:46.519 --> 00:14:50.879 answer this question, But the general theory of relativity permits of our answering it 146 00:14:52.279 --> 00:14:58.000 with a moderate degree of certainty, and in this connection the difficulty mentioned in 147 00:14:58.080 --> 00:15:07.600 section thirty finds solution. End of section thirty one. Section thirty two structure 148 00:15:07.639 --> 00:15:13.679 a space according to the general theory of relativity. According to the general theory 149 00:15:13.679 --> 00:15:20.399 of relativity, the geometrical properties of space are not independent, but they are 150 00:15:20.480 --> 00:15:24.519 determined by matter. Thus, we can draw conclusions about the geometrical structure of 151 00:15:24.519 --> 00:15:31.120 the universe only if we base our considerations on the state of the matter as 152 00:15:31.159 --> 00:15:37.000 being something that is known. We know from experience that for a suitably chosen 153 00:15:37.080 --> 00:15:41.279 coordinates system, the velocities of the stars are small as compared with the velocity 154 00:15:41.279 --> 00:15:46.799 of transmission of light. We can, thus, as a rough approximation, 155 00:15:46.440 --> 00:15:50.559 arrive at a conclusion as to the nature of the universe as a whole if 156 00:15:50.639 --> 00:15:56.320 we treat the matter as being at rest. We already know from our previous 157 00:15:56.320 --> 00:16:02.320 discussion that the behavior of measuring rods in clocks is influenced by gravitational fields, 158 00:16:03.200 --> 00:16:07.200 i e. By the distribution of matter. This in itself is sufficient to 159 00:16:07.240 --> 00:16:14.639 exclude the possibility of the exact validity of Euclidean geometry in our universe. But 160 00:16:14.759 --> 00:16:19.159 it is conceivable that our universe differs only slightly from a Euclidean one, and 161 00:16:19.320 --> 00:16:25.799 this notion seems all the more probable since calculations show that the metrics of surrounding 162 00:16:25.840 --> 00:16:30.720 space is influenced only to an exceedingly small extent by masses, even of the 163 00:16:30.759 --> 00:16:37.559 magnitude of our sun. We might imagine that, as regards geometry, our 164 00:16:37.679 --> 00:16:42.399 universe behaves analogously to a surface which is irregularly curved in its individual parts, 165 00:16:44.000 --> 00:16:48.120 but which nowhere departs appreciably from a plane, something like the rippled surface of 166 00:16:48.120 --> 00:16:53.759 a lake. Such a universe might fittingly be called a quasi Euclidean universe. 167 00:16:55.639 --> 00:17:00.360 As regards its space, it would be infinite. But calculation shows that in 168 00:17:00.399 --> 00:17:06.839 a quasi Euclidian universe the average density of matter would necessarily be nil. Thus 169 00:17:06.920 --> 00:17:11.039 such a universe could not be inhabited by matter everywhere. It would present to 170 00:17:11.200 --> 00:17:18.000 us that unsatisfactory picture which we portrayed in section thirty. If we are to 171 00:17:18.039 --> 00:17:22.359 have in the universe an average density of matter which differs from zero, however 172 00:17:22.400 --> 00:17:26.519 small may be, that difference, then the universe cannot be quasi euclidian. 173 00:17:27.160 --> 00:17:33.319 On the contrary, the results of calculation indicate that if matter be distributed uniformly, 174 00:17:33.680 --> 00:17:40.839 the universe would necessarily be spherical or elliptical. Since in reality, the 175 00:17:40.960 --> 00:17:45.559 detailed distribution of matter is not uniform, the real universe will deviate in individual 176 00:17:45.599 --> 00:17:51.400 parts from the spherical ie. The universe will be quasi spherical, but it 177 00:17:51.440 --> 00:17:56.200 will be necessarily finite. In fact, the theory supplies us with a simple 178 00:17:56.279 --> 00:18:03.160 connection. Begin footnote. For the radius R of the universe, we obtain 179 00:18:03.200 --> 00:18:10.160 the equation R squared equals two over a capa row. The use of the 180 00:18:10.200 --> 00:18:15.279 C G S system in this equation gives two over a K equals one to 181 00:18:15.359 --> 00:18:19.640 the eighth power times ten to the twenty seventh power. P is the average 182 00:18:19.680 --> 00:18:26.240 density of the matter, and K is a constant connected with the Newtonian constant 183 00:18:26.240 --> 00:18:33.240 of gravitation. End footnote between the space expanse of the universe and the average 184 00:18:33.279 --> 00:18:41.079 density of matter in it. End Section thirty two, end of Part three. 185 00:18:42.000 --> 00:18:47.680 End of relativity special in general theory by Albert Einstein.