1 00:00:00,090 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,830 under a Creative Commons license. 3 00:00:03,830 --> 00:00:06,550 Your support will help MIT OpenCourseWare continue 4 00:00:06,550 --> 00:00:10,160 to offer high-quality educational resources for free. 5 00:00:10,160 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,327 at ocw.mit.edu. 8 00:00:22,310 --> 00:00:24,060 PROFESSOR: OK if there are no questions, 9 00:00:24,060 --> 00:00:25,430 we will get back to physics. 10 00:00:25,430 --> 00:00:33,240 What I want to do today, as it suggests on the slide, 11 00:00:33,240 --> 00:00:36,855 is to finish the kinematics of homogeneous expansion 12 00:00:36,855 --> 00:00:38,905 that we were talking about last time. 13 00:00:38,905 --> 00:00:42,176 And the one topic in that category that we have not 14 00:00:42,176 --> 00:00:44,496 discussed yet is the cosmological redshift 15 00:00:44,496 --> 00:00:46,910 So we'll begin by going over that. 16 00:00:46,910 --> 00:00:50,670 And then we'll begin to go on to the next topic altogether, 17 00:00:50,670 --> 00:00:53,645 which is the dynamics of homogeneous expansion -- 18 00:00:53,645 --> 00:00:57,573 how do we understand how gravity affects the expansion 19 00:00:57,573 --> 00:00:58,555 of the universe? 20 00:00:58,555 --> 00:01:01,712 So that will be the main subject of today's lecture, 21 00:01:01,712 --> 00:01:06,457 once we've finished up the issue of the cosmological redshift 22 00:01:06,457 --> 00:01:07,680 shift. 23 00:01:07,680 --> 00:01:11,292 Let me remind you that at end of the last lecture, 24 00:01:11,292 --> 00:01:14,274 we were talking about the synchronization of clocks, 25 00:01:14,274 --> 00:01:16,262 and the coordinate system that we'll 26 00:01:16,262 --> 00:01:18,910 be using to describe the homogeneously expanding 27 00:01:18,910 --> 00:01:22,150 model of the universe. 28 00:01:22,150 --> 00:01:24,980 Remember, we are introducing spatial coordinates 29 00:01:24,980 --> 00:01:28,853 that grow with the universe, so that we're 30 00:01:28,853 --> 00:01:31,436 going to be assuming the fact, literally, that the universe is 31 00:01:31,436 --> 00:01:33,186 perfectly homogeneous and isotropic, which 32 00:01:33,186 --> 00:01:35,864 means that all objects will be literally addressed, 33 00:01:35,864 --> 00:01:37,833 relative to this coordinate system. 34 00:01:37,833 --> 00:01:40,240 If we're talking about the real universe, 35 00:01:40,240 --> 00:01:42,646 then there would be some motion relative 36 00:01:42,646 --> 00:01:45,940 to this coordinate system, because the universe is not 37 00:01:45,940 --> 00:01:47,513 exactly homogeneous. 38 00:01:47,513 --> 00:01:50,013 But we're going to be working for now with the approximation 39 00:01:50,013 --> 00:01:52,713 that our model universe is exactly homogeneous, which 40 00:01:52,713 --> 00:01:54,432 means that all matter is completely 41 00:01:54,432 --> 00:01:58,730 at rest, relative to this expanding coordinate system. 42 00:01:58,730 --> 00:02:00,771 And now we want to talk about how to define time, 43 00:02:00,771 --> 00:02:02,437 or to review what we said last time when 44 00:02:02,437 --> 00:02:04,590 we talked about how to define time. 45 00:02:04,590 --> 00:02:07,625 What we will imagine is that in every location in the universe 46 00:02:07,625 --> 00:02:11,070 at rest, relative to the matter, is a clock. 47 00:02:11,070 --> 00:02:15,835 And each clock ticks off time, and all those clocks 48 00:02:15,835 --> 00:02:19,144 will be acceptable as a clock which 49 00:02:19,144 --> 00:02:22,976 measures the time at relevant positions-- time 50 00:02:22,976 --> 00:02:25,210 is measured locally-- but we still 51 00:02:25,210 --> 00:02:27,780 have to talk about synchronizing those clocks. 52 00:02:27,780 --> 00:02:30,440 And what we said last time is that we can synchronize 53 00:02:30,440 --> 00:02:33,836 the clocks as long as there's some cosmic phenomena that 54 00:02:33,836 --> 00:02:37,098 can be seen everywhere, which has some time evolution. 55 00:02:37,098 --> 00:02:40,237 And we gave two examples-- one is the evolution of the Hubble 56 00:02:40,237 --> 00:02:43,510 expansion rate, which can be measured locally, 57 00:02:43,510 --> 00:02:47,795 and everybody can agree to set their clocks to midnight when 58 00:02:47,795 --> 00:02:51,530 the Hubble expansion rate has a certain value. 59 00:02:51,530 --> 00:02:55,347 And another cosmic variable is the temperature 60 00:02:55,347 --> 00:02:58,784 of the cosmic microwave background radiation. 61 00:02:58,784 --> 00:03:01,239 So, everybody in this model universe 62 00:03:01,239 --> 00:03:04,185 will agree that we'll set the clocks to midnight when 63 00:03:04,185 --> 00:03:06,640 the temperature of the cosmic background radiation 64 00:03:06,640 --> 00:03:10,570 goes to 5 degrees, or any specified number. 65 00:03:10,570 --> 00:03:13,060 So as long as there's a phenomena of that sort, which 66 00:03:13,060 --> 00:03:16,670 there is in our universe, it's possible to synchronize 67 00:03:16,670 --> 00:03:18,349 these clocks in a unique way. 68 00:03:18,349 --> 00:03:19,765 And the important thing to realize 69 00:03:19,765 --> 00:03:22,018 is that once they're synchronized at one time, 70 00:03:22,018 --> 00:03:24,463 they will remain synchronized as a consequence 71 00:03:24,463 --> 00:03:27,397 of our assumption of homogeneity. 72 00:03:27,397 --> 00:03:30,331 That is, if everybody agrees that the cosmic background 73 00:03:30,331 --> 00:03:33,620 radiation has a temperature of 10 degrees at midnight, 74 00:03:33,620 --> 00:03:38,130 if everybody waits for 15 minutes after midnight, 75 00:03:38,130 --> 00:03:40,390 everybody should see the same fall in temperature 76 00:03:40,390 --> 00:03:42,670 during that time interval, otherwise 77 00:03:42,670 --> 00:03:44,736 it would be a violation of this hypothesis 78 00:03:44,736 --> 00:03:46,668 of perfect homogeneity. 79 00:03:46,668 --> 00:03:48,117 Yes, question. 80 00:03:48,117 --> 00:03:51,015 AUDIENCE: Is it verified that temperature 81 00:03:51,015 --> 00:03:55,879 is invariant for all observers-- all Lorentz observers? 82 00:03:55,879 --> 00:03:58,170 PROFESSOR: OK the question is, is temperature invariant 83 00:03:58,170 --> 00:03:59,325 for all observers? 84 00:03:59,325 --> 00:04:04,020 And the question even included all Lorentz observers. 85 00:04:04,020 --> 00:04:07,497 It's not really invariant to different Lorentz observers. 86 00:04:07,497 --> 00:04:10,163 We're talking about a privileged class of observers, all of whom 87 00:04:10,163 --> 00:04:12,570 are at rest, relative to the average matter. 88 00:04:12,570 --> 00:04:14,950 If you move through the cosmic background radiation, 89 00:04:14,950 --> 00:04:18,249 then you don't see uniform thermal distribution any more. 90 00:04:18,249 --> 00:04:20,209 Rather what you see is radiation that's 91 00:04:20,209 --> 00:04:22,066 hotter in the forward direction and colder 92 00:04:22,066 --> 00:04:23,149 in the backward direction. 93 00:04:23,149 --> 00:04:25,599 And we in fact, as I think I have mentioned here, 94 00:04:25,599 --> 00:04:28,049 see that effect in our real universe. 95 00:04:28,049 --> 00:04:31,390 We're apparently moving relative to the cosmic background 96 00:04:31,390 --> 00:04:35,678 radiation, at about 1/1000th of the speed of light. 97 00:04:35,678 --> 00:04:40,929 So it's not invariant with respect to motion. 98 00:04:40,929 --> 00:04:42,720 There's the additional question, though is, 99 00:04:42,720 --> 00:04:46,240 is it the same everywhere in the visible universe? 100 00:04:46,240 --> 00:04:48,650 As far as we can tell, it is. 101 00:04:48,650 --> 00:04:51,750 There is some direct measurement of that, that we'll probably 102 00:04:51,750 --> 00:04:54,310 talk about later in the course, by looking 103 00:04:54,310 --> 00:04:57,350 at certain spectral lines in distant galaxies. 104 00:04:57,350 --> 00:04:59,724 One can effectively measure the temperature 105 00:04:59,724 --> 00:05:01,140 of the cosmic microwave background 106 00:05:01,140 --> 00:05:03,442 radiation in some distant galaxies. 107 00:05:03,442 --> 00:05:05,150 This line cannot be seen in all galaxies, 108 00:05:05,150 --> 00:05:09,027 and the extent that it's been measured in degrees. 109 00:05:09,027 --> 00:05:10,610 So certainly in our model, we're going 110 00:05:10,610 --> 00:05:12,720 to assume complete homogeneity, so everything's the same 111 00:05:12,720 --> 00:05:15,515 everywhere, and there is strong evidence for that homogeneity. 112 00:05:15,515 --> 00:05:17,530 Although it's not exact, but there's 113 00:05:17,530 --> 00:05:20,040 strong evidence for approximate homogeneity 114 00:05:20,040 --> 00:05:22,438 in the real universe. 115 00:05:22,438 --> 00:05:22,938 Yes? 116 00:05:22,938 --> 00:05:24,399 AUDIENCE: If you were really close to the black hole 117 00:05:24,399 --> 00:05:24,899 [INAUDIBLE]. 118 00:05:34,847 --> 00:05:35,430 PROFESSOR: OK. 119 00:05:35,430 --> 00:05:37,820 The question is, suppose we're a little bit more careful, 120 00:05:37,820 --> 00:05:40,210 and talk about the fact that some people might be living 121 00:05:40,210 --> 00:05:42,980 near black holes, and other people are not. 122 00:05:42,980 --> 00:05:45,080 Will that affect the synchronization of clocks 123 00:05:45,080 --> 00:05:47,100 for the people who are living near black holes? 124 00:05:47,100 --> 00:05:49,580 The answer is sure, it will. 125 00:05:49,580 --> 00:05:53,410 We can only synchronize clocks cosmically 126 00:05:53,410 --> 00:05:56,825 if we assume that the universe is absolutely homogeneous. 127 00:05:56,825 --> 00:05:58,740 As soon as you introduce inhomogeneities 128 00:05:58,740 --> 00:06:01,530 like black holes, or even just stars like the sun, 129 00:06:01,530 --> 00:06:04,250 they create small perturbations, which 130 00:06:04,250 --> 00:06:06,450 then make it really impossible to expect clocks 131 00:06:06,450 --> 00:06:09,160 to stay in sync with each other. 132 00:06:09,160 --> 00:06:11,510 So as soon as you have concentrations of mass, 133 00:06:11,510 --> 00:06:13,810 then the fact that what we're talking about now 134 00:06:13,810 --> 00:06:17,220 is an approximation becomes real. 135 00:06:17,220 --> 00:06:19,569 But those deviations are small. 136 00:06:19,569 --> 00:06:20,985 The deviations coming from the sun 137 00:06:20,985 --> 00:06:24,660 are only on the order of a part in a million or so. 138 00:06:24,660 --> 00:06:28,310 So, to a very good approximation, 139 00:06:28,310 --> 00:06:30,970 the universe obeys what we're describing, 140 00:06:30,970 --> 00:06:34,004 although if you went very close to the surface of one 141 00:06:34,004 --> 00:06:35,420 of these super-massive black holes 142 00:06:35,420 --> 00:06:37,290 in the centers of galaxies, or something, 143 00:06:37,290 --> 00:06:39,070 you would in fact find they had a very 144 00:06:39,070 --> 00:06:42,082 significant effect on your clocks. 145 00:06:42,082 --> 00:06:42,915 Any other questions? 146 00:06:45,420 --> 00:06:46,290 OK. 147 00:06:46,290 --> 00:06:48,270 Let me move on now. 148 00:06:48,270 --> 00:06:51,760 The next topic, as I have warned you, 149 00:06:51,760 --> 00:06:54,063 is the cosmological redshift. 150 00:06:54,063 --> 00:06:59,899 Now in the first lecture beyond the overview, 151 00:06:59,899 --> 00:07:02,190 which I guess was a combination of the second and third 152 00:07:02,190 --> 00:07:05,730 lectures in the course, we talked about the Doppler shift 153 00:07:05,730 --> 00:07:08,720 for sound waves, and we talked about the relativistic Doppler 154 00:07:08,720 --> 00:07:10,990 shift for light waves-- that was all 155 00:07:10,990 --> 00:07:13,680 in the context of special relativity. 156 00:07:13,680 --> 00:07:15,470 Now what we're going to face is the fact 157 00:07:15,470 --> 00:07:18,510 that cosmology is not really governed entirely 158 00:07:18,510 --> 00:07:22,700 by special relativity, although special relativity still 159 00:07:22,700 --> 00:07:26,365 holds locally in our cosmology. 160 00:07:26,365 --> 00:07:29,480 But special relativity does not include the effects of gravity, 161 00:07:29,480 --> 00:07:31,440 and on a global scale, the effects of gravity 162 00:07:31,440 --> 00:07:33,660 are very important for cosmology, 163 00:07:33,660 --> 00:07:35,410 and therefore special relativity by itself 164 00:07:35,410 --> 00:07:41,290 is not enough to understand many properties of the universe, 165 00:07:41,290 --> 00:07:44,026 including the cosmological redshift. 166 00:07:44,026 --> 00:07:45,400 It turns out though, that there's 167 00:07:45,400 --> 00:07:47,525 a way of describing the cosmological redshift which 168 00:07:47,525 --> 00:07:50,030 will make it sound even simpler than special relativity. 169 00:07:50,030 --> 00:07:51,680 And I'll describe that first, and then 170 00:07:51,680 --> 00:07:53,138 afterwards, we'll talk a little bit 171 00:07:53,138 --> 00:07:58,210 about how this very simple-looking derivation jives 172 00:07:58,210 --> 00:08:01,160 with the special relativity derivation, which must also 173 00:08:01,160 --> 00:08:04,730 be correct, at least locally. 174 00:08:04,730 --> 00:08:05,620 OK. 175 00:08:05,620 --> 00:08:13,040 So, the question we want to ask ourselves, 176 00:08:13,040 --> 00:08:16,660 is suppose we look at a distant galaxy, 177 00:08:16,660 --> 00:08:19,820 and light is emitted from that galaxy. 178 00:08:19,820 --> 00:08:22,010 How will the frequency of that light 179 00:08:22,010 --> 00:08:26,830 shift between the frequency it had when it was emitted, 180 00:08:26,830 --> 00:08:28,500 and the frequency that we would measure 181 00:08:28,500 --> 00:08:30,690 as we received the light. 182 00:08:30,690 --> 00:08:37,039 So to draw the situation on the blackboard, let's introduce 183 00:08:37,039 --> 00:08:39,450 a coordinate system, x. 184 00:08:39,450 --> 00:08:41,515 And this will be our comoving coordinate system. 185 00:08:41,515 --> 00:08:44,340 X is measured in notches. 186 00:08:44,340 --> 00:08:51,860 We'll put ourselves at the origin-- there is us. 187 00:08:51,860 --> 00:08:55,100 And we'll put our galaxy out here 188 00:08:55,100 --> 00:09:02,470 someplace-- there is the distant galaxy that we 189 00:09:02,470 --> 00:09:04,880 will be observing. 190 00:09:04,880 --> 00:09:09,240 They galaxy will be at some particular coordinate, which 191 00:09:09,240 --> 00:09:13,600 I will call l sub c, c for coordinate distance, 192 00:09:13,600 --> 00:09:18,210 so l sub c is the coordinate distance to the galaxy. 193 00:09:18,210 --> 00:09:21,930 And then the physical distance-- is 194 00:09:21,930 --> 00:09:25,010 what we've been calling l sub p, p for physical, 195 00:09:25,010 --> 00:09:29,330 which depends on time, because there's Hubble expansion. 196 00:09:29,330 --> 00:09:34,255 So l sub p of t, as we've said a number of times already, 197 00:09:34,255 --> 00:09:38,810 is a of t times l sub c. 198 00:09:38,810 --> 00:09:41,690 The scale factor, which depends on time, times 199 00:09:41,690 --> 00:09:44,520 the coordinate distance, which does not depend on time. 200 00:09:44,520 --> 00:09:49,960 So everything just expands with the scale factor a of t. 201 00:09:49,960 --> 00:09:56,950 So this describes the situation, and now 202 00:09:56,950 --> 00:09:58,830 what we want to ask ourselves, is 203 00:09:58,830 --> 00:10:03,860 suppose a wave is being emitted by the galaxy-- 204 00:10:03,860 --> 00:10:06,800 and we'll be trying to track the distance between wave 205 00:10:06,800 --> 00:10:10,800 crests, which determines what the wavelength is. 206 00:10:10,800 --> 00:10:15,220 Since we'll only be interested in wave crests, 207 00:10:15,220 --> 00:10:18,270 we will talk in language where we just 208 00:10:18,270 --> 00:10:19,885 imagine there's a pulse at each crest, 209 00:10:19,885 --> 00:10:21,343 and what happens in between doesn't 210 00:10:21,343 --> 00:10:23,710 matter for what we're talking about. 211 00:10:23,710 --> 00:10:32,210 So we want to track successive pulses emitted by the galaxy. 212 00:10:32,210 --> 00:10:37,330 Now the important feature of our system 213 00:10:37,330 --> 00:10:40,650 is that we have argued that we know 214 00:10:40,650 --> 00:10:47,160 how to track light waves through this kind of coordinate system. 215 00:10:47,160 --> 00:10:52,940 If x is our cosmic coordinate, dx dt, the coordinate velocity 216 00:10:52,940 --> 00:10:57,900 of light, is just equal to the ordinary velocity of light, 217 00:10:57,900 --> 00:11:03,210 c, but rescaled by the scale factor. 218 00:11:03,210 --> 00:11:04,890 And the scale factor here is playing 219 00:11:04,890 --> 00:11:07,660 the role of converting meters to notches. 220 00:11:07,660 --> 00:11:10,370 So c is measured in meters per second. 221 00:11:10,370 --> 00:11:12,570 By dividing by a of t, we get the speed 222 00:11:12,570 --> 00:11:14,840 in notches per second, which is what we want, 223 00:11:14,840 --> 00:11:18,045 because x is measured, not in meters, but in notches. 224 00:11:18,045 --> 00:11:21,870 A notch being the arbitrary coordinate-- the arbitrary unit 225 00:11:21,870 --> 00:11:26,180 that we adopt to describe our comoving coordinate system. 226 00:11:29,330 --> 00:11:31,600 Now the important feature of this equation, 227 00:11:31,600 --> 00:11:36,010 for our current purpose, is that the speed of light, as we're 228 00:11:36,010 --> 00:11:38,300 going to follow these light pulses 229 00:11:38,300 --> 00:11:42,480 through our coordinate system, depends on time, 230 00:11:42,480 --> 00:11:45,140 but it does not depend on x. 231 00:11:45,140 --> 00:11:49,280 Our universe is homogeneous, so all points x are the same. 232 00:11:49,280 --> 00:11:53,140 So two pulses will travel at the same speed at the same time, 233 00:11:53,140 --> 00:11:55,540 no matter where they are. 234 00:11:55,540 --> 00:11:58,350 And that's all we really need, to understand 235 00:11:58,350 --> 00:12:02,680 the fact that if one pulse leaves our galaxy 236 00:12:02,680 --> 00:12:04,868 and is coming towards us-- I should do that 237 00:12:04,868 --> 00:12:07,076 with my right hand, because the second pulse is going 238 00:12:07,076 --> 00:12:11,280 to be my other hand-- as that second pulse follows it, 239 00:12:11,280 --> 00:12:13,347 the second pulse, at any given time-- 240 00:12:13,347 --> 00:12:15,180 even though the speed will change with time, 241 00:12:15,180 --> 00:12:17,650 but at any given time-- the second pulse 242 00:12:17,650 --> 00:12:20,310 will be traveling at the same speed as the first pulse. 243 00:12:20,310 --> 00:12:25,050 And that means that it'll look something like this. 244 00:12:25,050 --> 00:12:26,541 The speed might change with time, 245 00:12:26,541 --> 00:12:28,915 but as long as they both travel at exactly the same speed 246 00:12:28,915 --> 00:12:32,150 at any given time, they will stay exactly the same distance 247 00:12:32,150 --> 00:12:35,440 apart in our comoving coordinate system. 248 00:12:35,440 --> 00:12:39,080 Delta x, the x distance between the two pulses, 249 00:12:39,080 --> 00:12:41,040 will not change with time. 250 00:12:41,040 --> 00:12:44,600 And if the coordinate distance does not change with time-- 251 00:12:44,600 --> 00:12:47,400 the physical distance is always the scale factor 252 00:12:47,400 --> 00:12:49,440 times the coordinate distance-- it 253 00:12:49,440 --> 00:12:52,120 means that the physical wavelength of the light pulse 254 00:12:52,120 --> 00:12:54,820 will simply be stretched with the scale factor, which 255 00:12:54,820 --> 00:12:57,500 means you'll be stretched with the expansion of the universe, 256 00:12:57,500 --> 00:13:01,230 in exactly the same way as any other distance in this model 257 00:13:01,230 --> 00:13:04,990 universe will be stretched as the universe expands. 258 00:13:04,990 --> 00:13:07,300 So that's the key idea, and it's very simple, 259 00:13:07,300 --> 00:13:09,435 and those words really say it all. 260 00:13:12,640 --> 00:13:23,990 Delta x equals constant implies delta l 261 00:13:23,990 --> 00:13:31,770 physical is proportional to a of t, 262 00:13:31,770 --> 00:13:37,240 and that implies that the wavelength of the light, 263 00:13:37,240 --> 00:13:40,890 as a function of t, is proportional to a of t. 264 00:13:43,480 --> 00:13:46,630 Wavelength is actually what I was calling delta l physical, 265 00:13:46,630 --> 00:13:48,771 the distance between these two pulses, 266 00:13:48,771 --> 00:13:51,870 where each pulse represents a crest of the wave. 267 00:13:51,870 --> 00:13:55,310 And lambda is the standard letter of the wavelength. 268 00:14:22,680 --> 00:14:26,380 Now the wavelength is related to the period of a wave 269 00:14:26,380 --> 00:14:28,580 simply by the relationship that lambda 270 00:14:28,580 --> 00:14:32,770 is equal to c times delta t. 271 00:14:32,770 --> 00:14:37,490 Wavelength is just the distance the wave travels in one period. 272 00:14:37,490 --> 00:14:40,480 So if lambda is proportional to a of t, 273 00:14:40,480 --> 00:14:44,360 so is the time interval, delta t, the period of the wave, 274 00:14:44,360 --> 00:14:46,610 going to be proportional to delta t. 275 00:14:46,610 --> 00:14:48,480 So we have been defining the redshift 276 00:14:48,480 --> 00:14:50,500 in terms of the period. 277 00:14:50,500 --> 00:14:59,800 So delta t observed over delta t at the source 278 00:14:59,800 --> 00:15:11,355 is equal to lambda observed over lambda at the source. 279 00:15:14,869 --> 00:15:18,680 Lambda and delta t are proportional to each other. 280 00:15:18,680 --> 00:15:20,650 And-- let me finish and I'll get to you, OK? 281 00:15:20,650 --> 00:15:21,400 AUDIENCE: Yes. 282 00:15:21,400 --> 00:15:23,850 PROFESSOR: This then, the ratio of the lengths, 283 00:15:23,850 --> 00:15:25,350 is just the amount by which universe 284 00:15:25,350 --> 00:15:27,590 has stretched over that time. 285 00:15:27,590 --> 00:15:30,720 So just the ratio of the scale factors at the two times. 286 00:15:30,720 --> 00:15:35,330 So this is equal to just a of the time of observation, which 287 00:15:35,330 --> 00:15:41,785 I'll call t sub o, over a of the time of the source, t sub s. 288 00:15:44,998 --> 00:15:54,590 So this is the scale factor at source, 289 00:15:54,590 --> 00:16:04,175 and the numerator here is the scale factor at observation. 290 00:16:09,040 --> 00:16:11,710 And this ratio of times, or ratio of wavelengths, 291 00:16:11,710 --> 00:16:15,350 or ratio of scale factors, is defined to be 1 plus z, 292 00:16:15,350 --> 00:16:16,620 as we have always done. 293 00:16:16,620 --> 00:16:19,130 The ratio of the time intervals we had defined originally 294 00:16:19,130 --> 00:16:21,920 as 1 plus z, we'll keep that definition, 295 00:16:21,920 --> 00:16:24,909 and that defines the redshift shift, z. 296 00:16:24,909 --> 00:16:25,450 Question now? 297 00:16:25,450 --> 00:16:26,436 Yes. 298 00:16:26,436 --> 00:16:28,420 AUDIENCE: Is that definition of lambda, 299 00:16:28,420 --> 00:16:31,396 does that have anything to do with the Lorentz invariant? 300 00:16:31,396 --> 00:16:34,662 Like, it just kind of struck me as the first term? 301 00:16:38,260 --> 00:16:40,680 PROFESSOR: Not sure what you mean? 302 00:16:40,680 --> 00:16:42,770 What-- Lorentz invariant what? 303 00:16:42,770 --> 00:16:46,410 AUDIENCE: Like the c delta tau squared equals c delta t-- 304 00:16:46,410 --> 00:16:49,270 PROFESSOR: Oh. 305 00:16:49,270 --> 00:16:52,490 Well, the delta t could be put into that formula, 306 00:16:52,490 --> 00:16:54,897 but that's formula could measure any delta t. 307 00:16:54,897 --> 00:16:55,480 AUDIENCE: Yeah 308 00:16:55,480 --> 00:16:57,590 PROFESSOR: So of course Lorentz is a special case, 309 00:16:57,590 --> 00:17:00,450 but any delta t would be a special case of that formula, 310 00:17:00,450 --> 00:17:02,220 so I don't think there's a lot to say 311 00:17:02,220 --> 00:17:03,973 about it being a special case. 312 00:17:03,973 --> 00:17:06,140 AUDIENCE: All right, cool. 313 00:17:06,140 --> 00:17:07,720 PROFESSOR: Any other questions? 314 00:17:07,720 --> 00:17:08,220 Yes? 315 00:17:08,220 --> 00:17:10,178 AUDIENCE: Is this like fundamentally different? 316 00:17:10,178 --> 00:17:11,497 Or is it similar [INAUDIBLE]? 317 00:17:17,750 --> 00:17:20,190 PROFESSOR: [INAUDIBLE] I was going to come to that. 318 00:17:20,190 --> 00:17:23,810 That's the question of how the cosmological redshift relates 319 00:17:23,810 --> 00:17:26,369 to the special relativity redshift 320 00:17:26,369 --> 00:17:30,051 that we derived earlier, and I'm coming to that immediately. 321 00:17:30,051 --> 00:17:32,640 Good question, we're getting there. 322 00:17:32,640 --> 00:17:35,952 Any other questions, though, before I go there? 323 00:17:35,952 --> 00:17:40,415 In my point of view, that's the next topic. 324 00:17:40,415 --> 00:17:40,914 OK. 325 00:17:43,920 --> 00:17:45,930 OK, so let me move on to exactly that question. 326 00:17:45,930 --> 00:17:47,570 How does this relate to what we already 327 00:17:47,570 --> 00:17:50,075 said about the redshift? 328 00:17:57,570 --> 00:18:00,420 This answer-- I would like to quantify things and say 329 00:18:00,420 --> 00:18:02,481 that it differs in two ways from the calculation 330 00:18:02,481 --> 00:18:03,605 that we've done previously. 331 00:18:29,850 --> 00:18:34,330 And the first is-- the reason why it's important 332 00:18:34,330 --> 00:18:37,750 to us-- is that this actually takes into account, effects 333 00:18:37,750 --> 00:18:40,050 which were not taken into account by our earlier 334 00:18:40,050 --> 00:18:41,430 calculation. 335 00:18:41,430 --> 00:18:43,520 In particular, even though we derived this 336 00:18:43,520 --> 00:18:47,920 by a very simple kinematic argument, which 337 00:18:47,920 --> 00:18:50,890 didn't seem to involve much math at all, 338 00:18:50,890 --> 00:18:53,430 it actually is incredibly strong, 339 00:18:53,430 --> 00:18:59,610 in that it encompasses not only special relativity, but also 340 00:18:59,610 --> 00:19:00,680 general relativity. 341 00:19:00,680 --> 00:19:02,609 It includes all the effects of gravity. 342 00:19:02,609 --> 00:19:04,150 If you think about what gravity might 343 00:19:04,150 --> 00:19:06,750 do to what we're talking about, gravity 344 00:19:06,750 --> 00:19:10,320 doesn't change the fact that the speed of light 345 00:19:10,320 --> 00:19:12,135 is going to be c over a of t. 346 00:19:12,135 --> 00:19:14,450 That really is just a unit conversion, 347 00:19:14,450 --> 00:19:16,790 combined with the fundamental physics assumption 348 00:19:16,790 --> 00:19:18,165 that the speed of light is always 349 00:19:18,165 --> 00:19:22,480 measured at c, relative to any observer. 350 00:19:22,480 --> 00:19:25,210 So when we put in gravity, this relationship 351 00:19:25,210 --> 00:19:26,780 continues to hold-- that was really 352 00:19:26,780 --> 00:19:29,329 all we used to drive this-- so gravity is not 353 00:19:29,329 --> 00:19:30,792 going to affect the answer. 354 00:19:30,792 --> 00:19:32,945 If you think about special relativity, 355 00:19:32,945 --> 00:19:34,460 is there something left out? 356 00:19:34,460 --> 00:19:36,153 Everything I said here, Newton would 357 00:19:36,153 --> 00:19:37,236 have understood perfectly. 358 00:19:37,236 --> 00:19:39,111 I didn't have to mention time dilation, which 359 00:19:39,111 --> 00:19:42,490 was crucial to our special relativity calculation 360 00:19:42,490 --> 00:19:43,841 of the redshift shift. 361 00:19:43,841 --> 00:19:44,715 Did I make a mistake? 362 00:19:44,715 --> 00:19:48,760 Is there some place where time dilation should come in here? 363 00:19:48,760 --> 00:19:52,140 The answer, really, is no, if you think about it. 364 00:19:52,140 --> 00:19:54,555 We had two clocks involved in our system, 365 00:19:54,555 --> 00:19:59,850 a clock on the galaxy, and a clock at us, 366 00:19:59,850 --> 00:20:03,020 which we used to measure the period of emission, 367 00:20:03,020 --> 00:20:08,500 and the period of reception, but those clocks are each at rest, 368 00:20:08,500 --> 00:20:10,300 relative to matter in the region-- 369 00:20:10,300 --> 00:20:13,760 even though they're moving with respect to each other-- 370 00:20:13,760 --> 00:20:18,552 so by definition, they do measure cosmic time. 371 00:20:18,552 --> 00:20:20,385 Cosmic time is a very peculiar kind of time, 372 00:20:20,385 --> 00:20:22,120 it's not the time in any inertial frame. 373 00:20:22,120 --> 00:20:24,260 These clocks are moving with respect to each other, 374 00:20:24,260 --> 00:20:27,434 so if you were defining inertial frame time, 375 00:20:27,434 --> 00:20:29,100 their clocks could never be synchronized 376 00:20:29,100 --> 00:20:31,950 and would never agree with each other. 377 00:20:31,950 --> 00:20:34,800 But in this concept of cosmic time, 378 00:20:34,800 --> 00:20:39,100 they do agree with each other, by construction. 379 00:20:39,100 --> 00:20:42,170 And since each clock is at rest, relative to its local matter, 380 00:20:42,170 --> 00:20:44,450 it measures this t that we're talking about, 381 00:20:44,450 --> 00:20:47,180 this cosmic time variable. 382 00:20:47,180 --> 00:20:51,970 And when the pulse arrives at us, 383 00:20:51,970 --> 00:20:55,486 when we measure delta t on our clock, 384 00:20:55,486 --> 00:20:57,360 that's exactly the quantity that, in the end, 385 00:20:57,360 --> 00:21:00,642 we want to talk about-- delta t sub observer. 386 00:21:00,642 --> 00:21:02,380 The quantity measured on our clock, 387 00:21:02,380 --> 00:21:05,450 which is a clock which also measures cosmic time. 388 00:21:05,450 --> 00:21:07,784 So there's no place for any time dilation to enter. 389 00:21:07,784 --> 00:21:10,000 It's not that we forgot it, it's not there. 390 00:21:10,000 --> 00:21:12,320 It's not part of this calculation. 391 00:21:12,320 --> 00:21:15,200 So this result, as simple as it looks, 392 00:21:15,200 --> 00:21:17,190 actually fully encompasses the effects 393 00:21:17,190 --> 00:21:20,490 of both special relativity and gravity. 394 00:21:20,490 --> 00:21:22,470 Now let me just mention, it's not obvious 395 00:21:22,470 --> 00:21:23,600 how gravity came in here. 396 00:21:23,600 --> 00:21:26,960 I'm telling you it satisfies-- includes 397 00:21:26,960 --> 00:21:28,260 all the effects of gravity. 398 00:21:28,260 --> 00:21:29,580 Where is gravity hidden? 399 00:21:29,580 --> 00:21:32,160 Let me throw that out as a question. 400 00:21:32,160 --> 00:21:34,450 How does gravity affect this calculation, 401 00:21:34,450 --> 00:21:37,210 even though I didn't have to mention gravity 402 00:21:37,210 --> 00:21:38,971 when I described the calculation? 403 00:21:38,971 --> 00:21:40,294 Yeah, in back. 404 00:21:40,294 --> 00:21:41,620 AUDIENCE: The scale factor? 405 00:21:41,620 --> 00:21:42,570 PROFESSOR: That's right, the scale factor. 406 00:21:42,570 --> 00:21:45,090 We have not yet talked about how a of t evolves. 407 00:21:45,090 --> 00:21:47,880 And the evolution of the a of t will explicitly 408 00:21:47,880 --> 00:21:49,640 involve the effects of gravity. 409 00:21:49,640 --> 00:21:52,405 And that's why this result depends on gravity, 410 00:21:52,405 --> 00:21:56,210 even though we didn't need to use gravity, 411 00:21:56,210 --> 00:22:00,030 or say anything about gravity to get the results. 412 00:22:00,030 --> 00:22:03,200 So this is the first difference. 413 00:22:03,200 --> 00:22:13,690 This calculation includes the effect of gravity. 414 00:22:22,867 --> 00:22:24,960 Which is through a of t. 415 00:22:31,170 --> 00:22:34,380 Now, because this calculation seems to include everything 416 00:22:34,380 --> 00:22:36,400 that the first calculation included and more, 417 00:22:36,400 --> 00:22:37,899 you'd expect to be more complicated, 418 00:22:37,899 --> 00:22:40,382 but it's less complicated. 419 00:22:40,382 --> 00:22:41,840 Could we have saved ourselves a lot 420 00:22:41,840 --> 00:22:43,923 of time last week by just giving this calculation, 421 00:22:43,923 --> 00:22:46,105 and deriving the other answer from it. 422 00:22:46,105 --> 00:22:49,940 The answer is, not easily, it would not have saved time, 423 00:22:49,940 --> 00:22:51,607 one can't, in principle, do it that way. 424 00:22:51,607 --> 00:22:53,023 But the other important difference 425 00:22:53,023 --> 00:22:55,110 between these two calculations is the variables 426 00:22:55,110 --> 00:22:57,930 that you're using to express your answer. 427 00:22:57,930 --> 00:23:01,369 Once you ask a question, if you ask the question vaguely, 428 00:23:01,369 --> 00:23:03,660 there could be many different answers to that question, 429 00:23:03,660 --> 00:23:08,120 depending on what variables are used to express the answer. 430 00:23:08,120 --> 00:23:22,300 So what we're doing here is we're expressing the redshift z 431 00:23:22,300 --> 00:23:29,000 for objects which are in fact at rest in the comoving coordinate 432 00:23:29,000 --> 00:23:29,500 system. 433 00:23:43,880 --> 00:23:56,715 The special relativity calculation-- I think I'm 434 00:23:56,715 --> 00:23:58,090 going to need another blackboard. 435 00:24:02,730 --> 00:24:17,240 The special relativity calculation, on the other hand 436 00:24:17,240 --> 00:24:32,300 gives z as a function of the velocity, 437 00:24:32,300 --> 00:24:37,070 as measured in an inertial coordinate system. 438 00:24:46,316 --> 00:24:47,940 So the answers are just being expressed 439 00:24:47,940 --> 00:24:50,010 in terms of totally different things, 440 00:24:50,010 --> 00:24:51,969 and the answer is so simple here because a of t 441 00:24:51,969 --> 00:24:53,718 already incorporates a lot of information, 442 00:24:53,718 --> 00:24:55,410 and we've just taken advantage of that 443 00:24:55,410 --> 00:24:58,694 to be able to give a very simple answer in terms of a of t, 444 00:24:58,694 --> 00:25:02,520 without yet saying how we're going to calculate a of t. 445 00:25:02,520 --> 00:25:03,426 Yes. 446 00:25:03,426 --> 00:25:05,410 AUDIENCE: [INAUDIBLE] two questions. 447 00:25:05,410 --> 00:25:07,394 One is about that constant time. 448 00:25:07,394 --> 00:25:08,386 PROFESSOR: Yes. 449 00:25:08,386 --> 00:25:12,354 AUDIENCE: How is that different than the Newton or Galilean 450 00:25:12,354 --> 00:25:15,330 idea of absolute time? 451 00:25:15,330 --> 00:25:16,110 PROFESSOR: OK. 452 00:25:16,110 --> 00:25:20,020 The question was how does the notion of cosmic time 453 00:25:20,020 --> 00:25:22,420 differ from Newton's or Galileo's 454 00:25:22,420 --> 00:25:26,940 notion of absolute time? 455 00:25:26,940 --> 00:25:31,040 And the answer is perhaps not much. 456 00:25:31,040 --> 00:25:34,230 Operationally, I think it is pretty much the same, 457 00:25:34,230 --> 00:25:39,865 but the real point is that Newton and Galileo did not 458 00:25:39,865 --> 00:25:43,159 know anything about relative effects like time dilation. 459 00:25:43,159 --> 00:25:45,325 So for them, it was just obvious that all clocks ran 460 00:25:45,325 --> 00:25:48,350 at the same speed, and time was naturally 461 00:25:48,350 --> 00:25:51,319 universal-- naturally absolute. 462 00:25:51,319 --> 00:25:52,860 In this case, we're aware of the fact 463 00:25:52,860 --> 00:25:55,300 that moving clocks run at different speeds. 464 00:25:55,300 --> 00:25:58,705 So if we were to take these clocks between us 465 00:25:58,705 --> 00:26:00,930 and the galaxy, and transport one to the other, 466 00:26:00,930 --> 00:26:03,500 depending on what path we used to transport them on, 467 00:26:03,500 --> 00:26:06,650 in the end, they would probably not agree with each other. 468 00:26:06,650 --> 00:26:09,334 So, we're setting up a definition 469 00:26:09,334 --> 00:26:13,860 of what we're going to define locally as time, recognizing 470 00:26:13,860 --> 00:26:17,770 that what time means here, versus what time means there, 471 00:26:17,770 --> 00:26:20,260 is a consequence of our assumptions about how we define 472 00:26:20,260 --> 00:26:20,760 things. 473 00:26:20,760 --> 00:26:22,800 It is not given automatically by the fact 474 00:26:22,800 --> 00:26:24,686 that all clocks will run at the same speed. 475 00:26:27,610 --> 00:26:28,224 Follow up? 476 00:26:28,224 --> 00:26:28,849 AUDIENCE: Yeah. 477 00:26:28,849 --> 00:26:30,845 An addition, this is slightly different. 478 00:26:30,845 --> 00:26:32,841 So, in the special relativity calculations, 479 00:26:32,841 --> 00:26:36,416 [INAUDIBLE] z could be [INAUDIBLE] 480 00:26:36,416 --> 00:26:37,332 PROFESSOR: Absolutely. 481 00:26:37,332 --> 00:26:39,827 AUDIENCE: So here we're only seeing a red shift, 482 00:26:39,827 --> 00:26:43,819 but we would obtain a blue shift if we allowed a of t 483 00:26:43,819 --> 00:26:44,840 to be decreasing, right? 484 00:26:44,840 --> 00:26:45,840 PROFESSOR: That's right. 485 00:26:45,840 --> 00:26:47,719 If the universe contracted, we would get a blue shift. 486 00:26:47,719 --> 00:26:48,594 AUDIENCE: [INAUDIBLE] 487 00:26:51,580 --> 00:26:52,580 PROFESSOR: That's right. 488 00:26:52,580 --> 00:26:55,127 It would correspond to the special relativity case. 489 00:26:55,127 --> 00:26:57,460 I was going to say a few words about the correspondence, 490 00:26:57,460 --> 00:26:59,100 but I'll answer questions first. 491 00:26:59,100 --> 00:26:59,647 Yes. 492 00:26:59,647 --> 00:27:01,147 AUDIENCE: I'd kind of like to add on 493 00:27:01,147 --> 00:27:04,550 to that question regarding the causal time. 494 00:27:04,550 --> 00:27:05,230 PROFESSOR: Yes. 495 00:27:05,230 --> 00:27:06,730 AUDIENCE: Isn't the fact that you've 496 00:27:06,730 --> 00:27:08,405 scaled the speed of light, that's 497 00:27:08,405 --> 00:27:12,160 what takes care of this discrepancy between the clocks 498 00:27:12,160 --> 00:27:14,064 themselves? 499 00:27:14,064 --> 00:27:16,230 PROFESSOR: The question is, does the fact that we've 500 00:27:16,230 --> 00:27:18,400 rescaled the speed of light take care 501 00:27:18,400 --> 00:27:21,610 of the discrepancy of times? 502 00:27:21,610 --> 00:27:23,840 Well, partially, but it doesn't say anything 503 00:27:23,840 --> 00:27:25,410 about what moving clocks will do. 504 00:27:25,410 --> 00:27:27,410 If you had a clock moving through this universe, 505 00:27:27,410 --> 00:27:31,660 you would have to calculate a time dilation for that clock, 506 00:27:31,660 --> 00:27:32,990 just as in any other case. 507 00:27:32,990 --> 00:27:35,406 AUDIENCE: What about the two end points, say, of the path. 508 00:27:35,406 --> 00:27:39,310 Is that why you're scaling the speed of light? 509 00:27:39,310 --> 00:27:40,972 PROFESSOR: Not really. 510 00:27:40,972 --> 00:27:43,080 The scaling of the speed of light 511 00:27:43,080 --> 00:27:45,120 really comes about through the scaling of space. 512 00:27:45,120 --> 00:27:48,830 This in fact is just the scale factor that we scale space. 513 00:27:48,830 --> 00:27:50,950 Time is measured locally on every clock, 514 00:27:50,950 --> 00:27:53,652 and we don't think of it as being rescaled. 515 00:27:53,652 --> 00:27:55,110 The speed of light looks different, 516 00:27:55,110 --> 00:27:57,740 simply because a notch is changing with time. 517 00:27:57,740 --> 00:28:00,883 And that formula tells you how to convert meters per second, 518 00:28:00,883 --> 00:28:03,470 which will always be the same to the speed of light, 519 00:28:03,470 --> 00:28:05,220 to notches per second, which will change 520 00:28:05,220 --> 00:28:06,613 as the size of a notch changes 521 00:28:06,613 --> 00:28:07,488 AUDIENCE: Right,yeah. 522 00:28:07,488 --> 00:28:07,988 OK. 523 00:28:07,988 --> 00:28:09,484 I understand. 524 00:28:09,484 --> 00:28:10,645 PROFESSOR: Yes. 525 00:28:10,645 --> 00:28:12,478 AUDIENCE: Further in the line of questioning 526 00:28:12,478 --> 00:28:14,973 about cosmological time-- so we expect 527 00:28:14,973 --> 00:28:17,468 that us and that other galaxy have 528 00:28:17,468 --> 00:28:20,961 simultaneous clocks relative to the cosmic time, 529 00:28:20,961 --> 00:28:23,955 and also we expect our own clocks to be simultaneous 530 00:28:23,955 --> 00:28:25,951 with our cosmological clocks, I assume. 531 00:28:25,951 --> 00:28:28,032 So if we-- Is that true? 532 00:28:28,032 --> 00:28:29,240 PROFESSOR: That's right, yes. 533 00:28:29,240 --> 00:28:31,110 Our own clock is just an example of one 534 00:28:31,110 --> 00:28:33,230 of the clocks sitting on the place called us, 535 00:28:33,230 --> 00:28:36,550 and all clocks sitting on that place will behave the same way. 536 00:28:36,550 --> 00:28:40,920 And they define the local definition of cosmic time. 537 00:28:40,920 --> 00:28:43,370 AUDIENCE: So if we take those clocks and move very slowly 538 00:28:43,370 --> 00:28:47,780 across to the other galaxy, in cosmological-- in comoving 539 00:28:47,780 --> 00:28:50,720 coordinates, we wouldn't expect there to be any time dilation, 540 00:28:50,720 --> 00:28:54,640 in the respect that clocks stay simultaneous. 541 00:28:54,640 --> 00:28:56,686 Safe to say, that we would think it 542 00:28:56,686 --> 00:28:58,560 would be simultaneous with us the whole time, 543 00:28:58,560 --> 00:29:00,030 until we got to the other galaxy. 544 00:29:00,030 --> 00:29:02,030 And then it would still be simultaneous. 545 00:29:02,030 --> 00:29:04,496 But, they're moving at a speed relative to us, 546 00:29:04,496 --> 00:29:07,372 so we wouldn't expect [INAUDIBLE]. 547 00:29:07,372 --> 00:29:08,080 PROFESSOR: Right. 548 00:29:08,080 --> 00:29:08,210 OK. 549 00:29:08,210 --> 00:29:09,659 You raise a good question, which I 550 00:29:09,659 --> 00:29:11,200 would have to think about the answer. 551 00:29:11,200 --> 00:29:13,760 If we brought-- if we carried our clock very slowly 552 00:29:13,760 --> 00:29:17,810 to this galaxy, and the limit was infinitely slow, 553 00:29:17,810 --> 00:29:19,400 would it agree when it got there? 554 00:29:19,400 --> 00:29:22,035 Let me think about that, and answer it next time. 555 00:29:22,035 --> 00:29:23,430 I'm not altogether sure. 556 00:29:26,912 --> 00:29:27,745 Any other questions? 557 00:29:31,060 --> 00:29:31,560 OK. 558 00:29:31,560 --> 00:29:33,476 I want to say something about the relationship 559 00:29:33,476 --> 00:29:36,239 between these two calculations. 560 00:29:36,239 --> 00:29:38,280 What would happen if we tried to actually compare 561 00:29:38,280 --> 00:29:41,730 the answers that we got for the relativistic Doppler shift, 562 00:29:41,730 --> 00:29:46,960 and for this answer, for the cosmological redshift. 563 00:29:46,960 --> 00:29:48,887 There's really only one case where 564 00:29:48,887 --> 00:29:52,930 it would be legitimate to compare them. 565 00:29:52,930 --> 00:29:54,580 Since the calculation we just did 566 00:29:54,580 --> 00:29:56,640 was supposed to include the effects of gravity, 567 00:29:56,640 --> 00:29:58,435 and special relativity calculation does not 568 00:29:58,435 --> 00:30:01,400 include the effects of gravity, the only way 569 00:30:01,400 --> 00:30:03,950 we should be able to compare them, and see that they agree, 570 00:30:03,950 --> 00:30:07,290 would be the case where gravity is negligible. 571 00:30:07,290 --> 00:30:09,994 And one can talk about a cosmological model where 572 00:30:09,994 --> 00:30:11,866 gravity is negligible, there's nothing 573 00:30:11,866 --> 00:30:13,740 inconsistent about that. 574 00:30:13,740 --> 00:30:16,170 If gravity were negligible, what would we 575 00:30:16,170 --> 00:30:19,744 expect for the behavior of a of t for this [INAUDIBLE] 576 00:30:19,744 --> 00:30:20,244 question. 577 00:30:24,670 --> 00:30:26,080 I hear a constant. 578 00:30:26,080 --> 00:30:27,750 Constant is certainly a possibility, 579 00:30:27,750 --> 00:30:30,110 but it's not the only possibility, 580 00:30:30,110 --> 00:30:32,319 so try to think a little harder, and ask 581 00:30:32,319 --> 00:30:33,786 if there are other possibilities. 582 00:30:38,670 --> 00:30:39,170 Yes. 583 00:30:39,170 --> 00:30:41,190 AUDIENCE: I'm sorry, could you rephrase the question? 584 00:30:41,190 --> 00:30:42,565 PROFESSOR: Rephrase the question. 585 00:30:42,565 --> 00:30:44,830 The question is, if gravity were negligible, 586 00:30:44,830 --> 00:30:48,850 what would we expect for the behavior of the scale factor 587 00:30:48,850 --> 00:30:49,509 a of t? 588 00:30:49,509 --> 00:30:52,008 And so far, it's been suggested that it could be a constant, 589 00:30:52,008 --> 00:30:54,412 and that's true, but that's not the most general answer. 590 00:30:54,412 --> 00:30:54,912 Yes. 591 00:30:54,912 --> 00:30:56,203 AUDIENCE: It could be negative. 592 00:30:56,203 --> 00:30:57,916 PROFESSOR: Could be negative? 593 00:30:57,916 --> 00:30:59,892 I don't know what would mean, actually. 594 00:30:59,892 --> 00:31:00,850 AUDIENCE: What do you-- 595 00:31:00,850 --> 00:31:03,058 PROFESSOR: It would mean the universe was inside out. 596 00:31:03,058 --> 00:31:03,663 AUDIENCE: Oh. 597 00:31:03,663 --> 00:31:04,530 PROFESSOR: It would really Just mean 598 00:31:04,530 --> 00:31:05,990 that you've reversed your coordinates. 599 00:31:05,990 --> 00:31:07,581 I don't think it would have any significance. 600 00:31:07,581 --> 00:31:08,523 AUDIENCE: Oh, the expansion would actually 601 00:31:08,523 --> 00:31:09,940 be a contraction? 602 00:31:09,940 --> 00:31:10,770 PROFESSOR: Oh, well it could decrease 603 00:31:10,770 --> 00:31:12,616 with time, that's not the same as being negative. 604 00:31:12,616 --> 00:31:13,330 AUDIENCE: Oh, I'm sorry 605 00:31:13,330 --> 00:31:15,870 PROFESSOR: It could always increase or decrease with time, 606 00:31:15,870 --> 00:31:17,610 whether gravity is present or not. 607 00:31:17,610 --> 00:31:19,590 For our universe it's increasing with time, 608 00:31:19,590 --> 00:31:22,340 but one could imagine a contracting universe. 609 00:31:22,340 --> 00:31:23,996 Yes, Aviv. 610 00:31:23,996 --> 00:31:24,704 AUDIENCE: Linear? 611 00:31:24,704 --> 00:31:25,454 PROFESSOR: Linear. 612 00:31:25,454 --> 00:31:26,350 That's right. 613 00:31:26,350 --> 00:31:29,975 If there's no gravity, a of t should be a constant times t. 614 00:31:29,975 --> 00:31:32,779 The constant could be zero, and then a of t 615 00:31:32,779 --> 00:31:34,320 is-- and maybe I should say it should 616 00:31:34,320 --> 00:31:37,274 be a constant plus a constant times t, 617 00:31:37,274 --> 00:31:40,630 and then in a special case it could just be a constant. 618 00:31:40,630 --> 00:31:42,880 But it should vary linearly with time. 619 00:31:42,880 --> 00:31:46,174 And that simply means that all velocities are constant. 620 00:31:46,174 --> 00:31:51,840 If all velocities are constant, then a of t is varied linearly 621 00:31:51,840 --> 00:31:54,270 with time, so that the distance -- 622 00:31:54,270 --> 00:31:58,158 the famous relationship is the distance of a of t times l sub 623 00:31:58,158 --> 00:31:59,130 c. 624 00:31:59,130 --> 00:32:01,322 If this distance were growing linearly with time, 625 00:32:01,322 --> 00:32:03,425 it could just be a constant velocity, which is certainly 626 00:32:03,425 --> 00:32:04,841 allowed in the absence of gravity. 627 00:32:04,841 --> 00:32:07,420 It would mean that a of t was growing linearly in time. 628 00:32:07,420 --> 00:32:10,705 So that would be the special case of absence of gravity, a 629 00:32:10,705 --> 00:32:13,820 of t growing linearly in time. 630 00:32:13,820 --> 00:32:16,100 And one can always set the constant 631 00:32:16,100 --> 00:32:19,130 that would be added to the linear to be zero, 632 00:32:19,130 --> 00:32:22,380 just by choosing zero of time to be the time at which a of t 633 00:32:22,380 --> 00:32:24,140 is zero. 634 00:32:24,140 --> 00:32:27,300 So, in the absence of gravity, one can say that a of t 635 00:32:27,300 --> 00:32:29,118 should just be proportional to t. 636 00:32:49,140 --> 00:32:51,580 So for that special case, these two calculations 637 00:32:51,580 --> 00:32:53,340 should really agree. 638 00:32:53,340 --> 00:32:56,670 And it will be, I'm pretty sure, an extra-credit homework 639 00:32:56,670 --> 00:32:58,918 problem coming up soon, in which you'll get a chance 640 00:32:58,918 --> 00:33:00,910 to calculate that. 641 00:33:00,910 --> 00:33:02,600 It's not easy, which is why it will 642 00:33:02,600 --> 00:33:06,170 be an extra-credit problem, probably, not required problem, 643 00:33:06,170 --> 00:33:08,870 because it involves understanding the relationship 644 00:33:08,870 --> 00:33:10,920 between these two coordinate systems. 645 00:33:10,920 --> 00:33:13,570 The special relativity answer is given 646 00:33:13,570 --> 00:33:17,450 in inertial coordinate system which, when gravity is present, 647 00:33:17,450 --> 00:33:18,331 doesn't exist at all. 648 00:33:18,331 --> 00:33:19,705 In the presence of gravity, there 649 00:33:19,705 --> 00:33:23,130 is no global inertial coordinate system. 650 00:33:23,130 --> 00:33:25,260 But without its action, there is. 651 00:33:25,260 --> 00:33:27,450 But it's related to this coordinate system, where 652 00:33:27,450 --> 00:33:30,490 everything's expending in a complicated way, 653 00:33:30,490 --> 00:33:32,913 because of the various time dilations and Lorentz 654 00:33:32,913 --> 00:33:35,131 contractions associated with the motions that 655 00:33:35,131 --> 00:33:40,320 are taking place in our expanding universe. 656 00:33:40,320 --> 00:33:42,270 So what you'll need to do is to figure out 657 00:33:42,270 --> 00:33:44,880 the relationship between these two coordinate systems. 658 00:33:44,880 --> 00:33:47,390 And when you do, and actually compare the answers, 659 00:33:47,390 --> 00:33:49,725 is you find that they actually do agree exactly. 660 00:33:49,725 --> 00:33:52,578 This is all perfectly consistent with special relativity, 661 00:33:52,578 --> 00:33:56,550 but the special case where there is no gravity. 662 00:33:56,550 --> 00:33:57,500 OK. 663 00:33:57,500 --> 00:34:00,945 Ready to leave cosmological redshift altogether, 664 00:34:00,945 --> 00:34:02,570 unless there are any further questions? 665 00:34:08,860 --> 00:34:09,630 OK. 666 00:34:09,630 --> 00:34:13,620 In that case, Onward to the next major topic. 667 00:34:13,620 --> 00:34:15,749 We've now finished what I wanted to say 668 00:34:15,749 --> 00:34:19,360 about the kinematics of homogeneously expanding 669 00:34:19,360 --> 00:34:23,553 universes, and now we're ready to talk about the dynamics. 670 00:34:23,553 --> 00:34:25,719 What happens when we try to think about what gravity 671 00:34:25,719 --> 00:34:28,250 is going to do to this universe, to be 672 00:34:28,250 --> 00:34:30,995 able to calculate how a of t is going to vary with time. 673 00:34:30,995 --> 00:34:32,907 That will be the only goal, to understand 674 00:34:32,907 --> 00:34:35,670 the behavior of a of t. 675 00:34:35,670 --> 00:34:42,000 Now this problem, in a way, goes back to Isaac Newton. 676 00:34:42,000 --> 00:34:46,199 And I might just give a little aside here, 677 00:34:46,199 --> 00:34:49,380 and mention that one of the fun things about cosmology, 678 00:34:49,380 --> 00:34:51,800 actually, is that if one looks back 679 00:34:51,800 --> 00:34:55,650 at the history of cosmology, many great physicists have made 680 00:34:55,650 --> 00:34:59,340 great blunders in trying to analyze cosmological questions. 681 00:34:59,340 --> 00:35:01,470 And in the discussion today, we'll 682 00:35:01,470 --> 00:35:03,750 be discussing one of Newton's blunders. 683 00:35:03,750 --> 00:35:07,450 And to me, it's very consoling to know that even 684 00:35:07,450 --> 00:35:11,600 physicists as great as Newton can make stupid mistakes. 685 00:35:11,600 --> 00:35:13,450 And he actually did make a stupid mistake, 686 00:35:13,450 --> 00:35:18,000 in terms of analyzing the cosmological effect 687 00:35:18,000 --> 00:35:20,850 of his own theory of gravity. 688 00:35:20,850 --> 00:35:25,390 At issue was Newton's view of the universe, 689 00:35:25,390 --> 00:35:31,440 and Newton, like everybody, really, until Hubble, 690 00:35:31,440 --> 00:35:33,770 believed that the universe was static. 691 00:35:33,770 --> 00:35:36,120 He imagined the universe as a static distribution 692 00:35:36,120 --> 00:35:40,220 of stars scattered through space. 693 00:35:40,220 --> 00:35:44,450 And early in his career, from what I understand 694 00:35:44,450 --> 00:35:48,420 of the history, he assumed that this distribution of stars 695 00:35:48,420 --> 00:35:53,480 was finite, and an infinite background space. 696 00:35:53,480 --> 00:35:57,730 But he realized at some point that if you 697 00:35:57,730 --> 00:36:04,160 had a finite distribution of mass, in otherwise empty space, 698 00:36:04,160 --> 00:36:06,540 that everything would attract everything else, 699 00:36:06,540 --> 00:36:09,720 with his one over r squared force of gravity-- which 700 00:36:09,720 --> 00:36:13,170 he knew about, he invented it-- and the result would 701 00:36:13,170 --> 00:36:16,880 be everything would collapse to a point. 702 00:36:16,880 --> 00:36:19,717 So he decided that would not work, 703 00:36:19,717 --> 00:36:21,550 but he was still sure everything was static. 704 00:36:21,550 --> 00:36:22,925 Because everything looked static, 705 00:36:22,925 --> 00:36:26,130 stars don't seem to move very much. 706 00:36:26,130 --> 00:36:29,360 So he asked what he could change, 707 00:36:29,360 --> 00:36:32,470 and decided that instead of assuming that the stars made up 708 00:36:32,470 --> 00:36:34,680 a finite distribution, he could assume 709 00:36:34,680 --> 00:36:37,350 that they were an infinite distribution, sharing 710 00:36:37,350 --> 00:36:39,200 all of space. 711 00:36:39,200 --> 00:36:42,060 And he reasoned-- and this is really 712 00:36:42,060 --> 00:36:45,420 where the fallacy showed up-- but he 713 00:36:45,420 --> 00:36:52,195 reasoned that if the stars filled the infinite space 714 00:36:52,195 --> 00:36:54,570 that, even though they would all be tugging on each other 715 00:36:54,570 --> 00:36:57,030 through the force of gravity, they 716 00:36:57,030 --> 00:36:58,590 wouldn't know which way to go. 717 00:36:58,590 --> 00:37:00,464 And since they wouldn't know which way to go, 718 00:37:00,464 --> 00:37:02,260 because they'd be tugged in all directions, 719 00:37:02,260 --> 00:37:04,750 they would stand still. 720 00:37:04,750 --> 00:37:09,700 So he believed that an infinite, uniform, distribution of mass 721 00:37:09,700 --> 00:37:12,510 would be stable-- that there'd be no gravitational forces 722 00:37:12,510 --> 00:37:16,654 resulting from the masses in this infinite distribution. 723 00:37:16,654 --> 00:37:18,070 And I have some quotes here, which 724 00:37:18,070 --> 00:37:20,560 I think are kind of cute, so I'll show them to you. 725 00:37:20,560 --> 00:37:23,120 Newton had a long discussion about these issues 726 00:37:23,120 --> 00:37:26,350 with Richard Bentley, the theologian. 727 00:37:26,350 --> 00:37:29,360 And we get to read about it, because all these letters have 728 00:37:29,360 --> 00:37:30,160 been preserved. 729 00:37:30,160 --> 00:37:33,110 In fact, I'm told that the original letters are actually 730 00:37:33,110 --> 00:37:35,900 still in existence at Trinity College in Cambridge 731 00:37:35,900 --> 00:37:37,167 University. 732 00:37:37,167 --> 00:37:38,750 And you can find them on the web even, 733 00:37:38,750 --> 00:37:41,660 I'll give you a web reference for the text of these letters, 734 00:37:41,660 --> 00:37:44,820 and they're in books and various places. 735 00:37:44,820 --> 00:37:45,987 So let me read this to you. 736 00:37:45,987 --> 00:37:48,490 I think it's a cute quotation. 737 00:37:48,490 --> 00:37:50,080 "As to your first query"-- by the way, 738 00:37:50,080 --> 00:37:52,329 I think we don't have the letters that Richard Bentley 739 00:37:52,329 --> 00:37:55,570 sent to Newton, only the responses. 740 00:37:55,570 --> 00:37:58,511 But Newton fortunately responded in a way 741 00:37:58,511 --> 00:38:00,052 that made the questions pretty clear, 742 00:38:00,052 --> 00:38:02,385 so it's not an important problem in understanding what's 743 00:38:02,385 --> 00:38:04,330 going on. 744 00:38:04,330 --> 00:38:07,230 Newton says, "It seems to me that if the matter of our sun 745 00:38:07,230 --> 00:38:10,490 and planets and all the matter of the universe were evenly 746 00:38:10,490 --> 00:38:12,695 scattered throughout all the heavens, 747 00:38:12,695 --> 00:38:16,560 and every particle had an innate gravity toward all the rest, 748 00:38:16,560 --> 00:38:19,840 and the whole space throughout which this matter was scattered 749 00:38:19,840 --> 00:38:24,200 was but finite, the matter on the outside space would, 750 00:38:24,200 --> 00:38:27,610 by its gravity, tend toward all the matter on the inside"-- 751 00:38:27,610 --> 00:38:30,874 this is a finite universe he's talking about -- and he says, 752 00:38:30,874 --> 00:38:33,040 "that by its consequence, everything would fall down 753 00:38:33,040 --> 00:38:35,090 into the middle of the whole space, 754 00:38:35,090 --> 00:38:39,260 and there compose one great spherical mass." 755 00:38:39,260 --> 00:38:41,430 So, there he's describing how it would not 756 00:38:41,430 --> 00:38:47,130 work if you had a finite collection of matter. 757 00:38:47,130 --> 00:38:49,290 But, he says, "If the matter was evenly 758 00:38:49,290 --> 00:38:52,210 disposed throughout an infinite space, 759 00:38:52,210 --> 00:38:57,130 it could never convene into one mass, but some of it 760 00:38:57,130 --> 00:38:58,800 would convene into one mass, and some 761 00:38:58,800 --> 00:39:01,515 into another, to as to make an infinite number 762 00:39:01,515 --> 00:39:04,790 of great masses, scattered at great distances 763 00:39:04,790 --> 00:39:08,077 from one to another, throughout all that infinite space." 764 00:39:08,077 --> 00:39:10,660 So he thought there'd be local coagulation, which of course is 765 00:39:10,660 --> 00:39:11,950 what we see in our real world. 766 00:39:11,950 --> 00:39:14,060 We see stars that have formed, and now we 767 00:39:14,060 --> 00:39:15,495 know about galaxies, which Newton 768 00:39:15,495 --> 00:39:16,680 had no way of knowing about. 769 00:39:16,680 --> 00:39:22,020 That's the kind of coagulation process that he's discussing. 770 00:39:22,020 --> 00:39:23,130 And he-- oops, sorry. 771 00:39:27,310 --> 00:39:29,475 "And thus might the sun and the fixed stars 772 00:39:29,475 --> 00:39:33,664 be formed, supposing the matter were of a lucid nature." 773 00:39:33,664 --> 00:39:35,640 That's a cute phrase. 774 00:39:35,640 --> 00:39:38,130 I can tell you what it means, it may not be obvious. 775 00:39:38,130 --> 00:39:40,700 But at this point, nobody had any idea 776 00:39:40,700 --> 00:39:42,305 what the sun was made out of, and why 777 00:39:42,305 --> 00:39:43,846 the sun was different from the earth. 778 00:39:43,846 --> 00:39:46,220 In fact nobody really had much of a real idea what 779 00:39:46,220 --> 00:39:49,750 the earth was made out of either, here. 780 00:39:49,750 --> 00:39:53,300 Chemistry wasn't really invented yet. 781 00:39:53,300 --> 00:39:55,870 So the assumption was that there were 782 00:39:55,870 --> 00:40:00,550 two kinds of matter, lucid matter and opaque matter. 783 00:40:00,550 --> 00:40:03,305 Where lucid matter is the stuff the sun is made out of, 784 00:40:03,305 --> 00:40:07,230 and the stars, that glows, and is fundamentally different 785 00:40:07,230 --> 00:40:09,917 in some way, that was of course not understood at all, 786 00:40:09,917 --> 00:40:12,375 from opaque matter, which is what the Earth is made out of. 787 00:40:12,375 --> 00:40:15,860 You can't see through it, and it doesn't, obviously, glow. 788 00:40:15,860 --> 00:40:19,910 So here, when he's talking about matter 789 00:40:19,910 --> 00:40:22,310 forming the stars and the sun, he 790 00:40:22,310 --> 00:40:25,119 says if the matter was lucid, if it 791 00:40:25,119 --> 00:40:26,535 was the kind of matter that glows. 792 00:40:31,090 --> 00:40:35,800 Going on-- so far, what he said sounds pretty good. 793 00:40:35,800 --> 00:40:37,740 Going on, he goes on now to talk more 794 00:40:37,740 --> 00:40:39,842 about this lucid versus opaque business. 795 00:40:39,842 --> 00:40:41,107 And I think it's cute. 796 00:40:41,107 --> 00:40:42,690 I don't know where exactly it's going, 797 00:40:42,690 --> 00:40:45,023 but it shows something about Newton's personality, which 798 00:40:45,023 --> 00:40:47,839 one might not have known otherwise. 799 00:40:47,839 --> 00:40:49,630 "But how the matter should divide itself"-- 800 00:40:49,630 --> 00:40:53,470 I should also warn you, all of this is one sentence. 801 00:40:53,470 --> 00:40:56,210 If you think sometimes my sentences sound convoluted, 802 00:40:56,210 --> 00:40:58,710 just think how lucky you are that you don't have Newton here 803 00:40:58,710 --> 00:40:59,550 as your lecturer. 804 00:40:59,550 --> 00:41:01,390 This is just impossible. 805 00:41:01,390 --> 00:41:05,770 So, "But how the matter should divide itself into two sorts"-- 806 00:41:05,770 --> 00:41:08,630 how we'd have lucid and opaque matter in the right places-- 807 00:41:08,630 --> 00:41:11,480 "and that part of it which is to comprise a shining body should 808 00:41:11,480 --> 00:41:15,080 fall down into one mass, and make a sun, and the rest, 809 00:41:15,080 --> 00:41:17,210 of which is fit to compose an opaque body, 810 00:41:17,210 --> 00:41:20,826 should coalesce not into one great body, like shiny matter, 811 00:41:20,826 --> 00:41:23,200 but into many little ones"-- somehow he's forgotten about 812 00:41:23,200 --> 00:41:25,158 the stars here, when he's talking about the sun 813 00:41:25,158 --> 00:41:28,240 and the planets, many planets and one sun. 814 00:41:28,240 --> 00:41:37,190 So he says that, "how the opaque matter should fall instead 815 00:41:37,190 --> 00:41:40,802 into many little masses"-- and then 816 00:41:40,802 --> 00:41:42,260 he talks about other possibilities. 817 00:41:42,260 --> 00:41:44,900 It's wonderful the way he lists all the possibilities. 818 00:41:44,900 --> 00:41:48,040 "Or," he says, "if the sun were at first an opaque body, 819 00:41:48,040 --> 00:41:51,875 like the planets, or if the planets were lucid bodies like 820 00:41:51,875 --> 00:41:55,345 the sun, how he alone"-- he being the sun, 821 00:41:55,345 --> 00:41:59,080 if you track everything back, "how the sun alone should be 822 00:41:59,080 --> 00:42:04,620 changed into a shiny body, while all the"-- lost track -- 823 00:42:04,620 --> 00:42:07,150 "where all they"-- of the planets-- 824 00:42:07,150 --> 00:42:10,280 "continue to be opaque, or"-- he's considering all 825 00:42:10,280 --> 00:42:13,070 possibilities-- "or they all be changed to opaque ones, 826 00:42:13,070 --> 00:42:17,560 while he,"-- the sun-- "remains unchanged as a lucid one." 827 00:42:17,560 --> 00:42:20,435 He does not know how to explain all that, is what he's saying. 828 00:42:20,435 --> 00:42:22,310 Bottom line of the sentence is, I don't know, 829 00:42:22,310 --> 00:42:23,752 I don't have a clue. 830 00:42:23,752 --> 00:42:26,210 And he says, "I don't think it's explicable by mere natural 831 00:42:26,210 --> 00:42:29,080 causes, but am forced," Newton says, 832 00:42:29,080 --> 00:42:31,830 "to ascribe it to the council and contrivance 833 00:42:31,830 --> 00:42:34,850 of a voluntary agent." 834 00:42:34,850 --> 00:42:37,180 So the theory of intelligence design, 835 00:42:37,180 --> 00:42:38,900 as well the theory of gravity, actually 836 00:42:38,900 --> 00:42:41,600 both go back to Newton, it turns out. 837 00:42:41,600 --> 00:42:43,104 Newton was a very religious person, 838 00:42:43,104 --> 00:42:44,520 and in certain aspects of physics, 839 00:42:44,520 --> 00:42:48,255 he was happy to ascribe to a voluntary agent, 840 00:42:48,255 --> 00:42:50,440 as he calls it. 841 00:42:50,440 --> 00:42:52,052 I have some references here, and I'll 842 00:42:52,052 --> 00:42:53,760 be posting this so you'll be able to read 843 00:42:53,760 --> 00:42:55,718 those references and type them in, if you want. 844 00:42:58,362 --> 00:43:04,140 Now Newton decided that you could not 845 00:43:04,140 --> 00:43:07,471 have a finite distribution, because it would collapse. 846 00:43:07,471 --> 00:43:08,970 If you had an infinite distribution, 847 00:43:08,970 --> 00:43:11,970 he thought it would be stable, but he apparently 848 00:43:11,970 --> 00:43:16,180 had heard different arguments to that same conclusion. 849 00:43:16,180 --> 00:43:18,930 And one argument that you might give 850 00:43:18,930 --> 00:43:21,305 for saying that the infinite distribution would be stable 851 00:43:21,305 --> 00:43:25,040 would be the argument that if you look at the force 852 00:43:25,040 --> 00:43:27,780 one any one particle, there is an infinite force pulling it 853 00:43:27,780 --> 00:43:31,100 to the right-- my right, your left-- and an infinite 854 00:43:31,100 --> 00:43:34,060 force pulling it to my left, your right, 855 00:43:34,060 --> 00:43:37,210 and since they're both infinite, they would cancel each other. 856 00:43:37,210 --> 00:43:39,700 Newton did not accept that argument. 857 00:43:39,700 --> 00:43:43,160 He was sophisticated enough to realize that infinity minus 858 00:43:43,160 --> 00:43:45,620 infinity isn't necessarily zero. 859 00:43:45,620 --> 00:43:47,916 And he has a bit of a tirade on that, 860 00:43:47,916 --> 00:43:49,290 that I thought was worth quoting. 861 00:43:49,290 --> 00:43:53,380 And this is a second letter to the same Richard Bentley. 862 00:43:53,380 --> 00:43:55,296 I guess it was Bentley who made this argument, 863 00:43:55,296 --> 00:43:57,190 and Newton rejected it. 864 00:43:57,190 --> 00:44:00,230 Infinity minus infinity, Newton realized, is ambiguous. 865 00:44:00,230 --> 00:44:02,220 It's not something that we should necessarily 866 00:44:02,220 --> 00:44:03,710 think of zero. 867 00:44:03,710 --> 00:44:06,460 "But you argue in the next paragraph of your letter that 868 00:44:06,460 --> 00:44:09,310 every particle of the matter in the infinite space has 869 00:44:09,310 --> 00:44:11,810 an infinite quantity of matter on all sides, 870 00:44:11,810 --> 00:44:15,470 and by consequence, an infinite attraction every way, 871 00:44:15,470 --> 00:44:18,450 and therefore must rest in equilibrio, 872 00:44:18,450 --> 00:44:20,190 because all infinities are equal:"-- 873 00:44:20,190 --> 00:44:23,530 he's summarizing Richard Bentley's argument-- 874 00:44:23,530 --> 00:44:28,057 "yet you suspect a parologism"-- that means logical error, 875 00:44:28,057 --> 00:44:33,240 I think-- "in this argument: and I can see the paralogism lies 876 00:44:33,240 --> 00:44:36,700 in the position that all infinities are equal. 877 00:44:36,700 --> 00:44:40,400 The generality of mankind consider infinities no other 878 00:44:40,400 --> 00:44:43,510 ways than indefinitely"-- and in this sentence they said all 879 00:44:43,510 --> 00:44:47,380 infinities are equal-- "though they would speak more truly 880 00:44:47,380 --> 00:44:51,010 if they should say that they are neither equal nor unequal, 881 00:44:51,010 --> 00:44:54,230 nor have any certain difference or proportion, one to another." 882 00:44:54,230 --> 00:44:57,302 So he realizes that the ratio of infinity could be anything, 883 00:44:57,302 --> 00:45:00,210 and infinity minus infinity could be anything, all of which 884 00:45:00,210 --> 00:45:01,960 is consistent with our modern view of how 885 00:45:01,960 --> 00:45:03,920 to do the mathematics. 886 00:45:03,920 --> 00:45:05,780 "In this sense, therefore, no conclusions 887 00:45:05,780 --> 00:45:09,090 can be drawn from them about the equality, proportions 888 00:45:09,090 --> 00:45:12,870 or differences of things, and they that attempt to do so 889 00:45:12,870 --> 00:45:17,552 usually fall into paralogisms." 890 00:45:17,552 --> 00:45:20,610 He goes on, now I just have one more Newton quote-- 891 00:45:20,610 --> 00:45:22,232 I like Newton quotes-- 892 00:45:22,232 --> 00:45:23,690 I have one more Newton quote, again 893 00:45:23,690 --> 00:45:25,580 from the same series of letters. 894 00:45:25,580 --> 00:45:27,850 These are all from 1692 and 1693, 895 00:45:27,850 --> 00:45:30,800 I believe, where he gives an example-- 896 00:45:30,800 --> 00:45:33,110 I think this follows the quotes of the previous slide 897 00:45:33,110 --> 00:45:35,575 immediately-- where he gives an example of a false argument 898 00:45:35,575 --> 00:45:37,700 that you get into-- and apparently it's an argument 899 00:45:37,700 --> 00:45:39,530 that he had heard from other people-- 900 00:45:39,530 --> 00:45:42,290 if you think all infinities are equal. 901 00:45:42,290 --> 00:45:45,296 What he says is, "So when men"-- he doesn't say who men are, 902 00:45:45,296 --> 00:45:46,796 and I don't really know the history. 903 00:45:46,796 --> 00:45:49,300 He may referring to some particular philosophers 904 00:45:49,300 --> 00:45:53,480 at the time-- "when men argue that the infinite divisibility 905 00:45:53,480 --> 00:45:57,730 of magnitude by saying that an inch may be divided 906 00:45:57,730 --> 00:46:01,310 into an infinite number of parts, the sum of those parts 907 00:46:01,310 --> 00:46:04,250 would be an inch-- and a foot can 908 00:46:04,250 --> 00:46:06,600 be divided into an infinite number of parts, 909 00:46:06,600 --> 00:46:09,330 the sum of those parts must be a foot-- 910 00:46:09,330 --> 00:46:12,000 and therefore, since all infinities are equal, 911 00:46:12,000 --> 00:46:14,302 these sums must be equal." 912 00:46:14,302 --> 00:46:15,510 Understand the argument here. 913 00:46:15,510 --> 00:46:16,690 He's saying that if you divide and inch 914 00:46:16,690 --> 00:46:18,106 into an infinite number of parts-- 915 00:46:18,106 --> 00:46:19,950 this is all you've been given as a foil. 916 00:46:19,950 --> 00:46:22,020 He's not claiming the argument is right, 917 00:46:22,020 --> 00:46:23,950 he's claiming it's wrong-- that argument is that if you divide 918 00:46:23,950 --> 00:46:25,355 an inch into an infinite number of parts, 919 00:46:25,355 --> 00:46:27,020 you get an infinite number of points, 920 00:46:27,020 --> 00:46:29,195 if you put them together, you get an inch. 921 00:46:29,195 --> 00:46:31,445 If you divide a foot into an infinite number of parts, 922 00:46:31,445 --> 00:46:33,170 you get an infinite number of points, 923 00:46:33,170 --> 00:46:35,760 and if you put them together, you should get a foot. 924 00:46:35,760 --> 00:46:37,700 But they're both an infinite number 925 00:46:37,700 --> 00:46:39,460 of points in the description. 926 00:46:39,460 --> 00:46:41,255 So if you think all infinities are equal, 927 00:46:41,255 --> 00:46:43,000 the infinite number of points that make an inch 928 00:46:43,000 --> 00:46:45,100 should be the same as the infinite number of points that 929 00:46:45,100 --> 00:46:47,225 make a foot, therefore a foot should equal an inch, 930 00:46:47,225 --> 00:46:48,150 obviously. 931 00:46:48,150 --> 00:46:49,142 Right. 932 00:46:49,142 --> 00:46:51,110 Not right, he know. 933 00:46:51,110 --> 00:46:54,630 So he says that the falseness of the conclusion 934 00:46:54,630 --> 00:46:57,170 shows an error in the premises, and the error lies 935 00:46:57,170 --> 00:47:00,640 in the position that all infinities are equal. 936 00:47:00,640 --> 00:47:02,990 So Newton has given us a very nice example 937 00:47:02,990 --> 00:47:05,280 of how you can convince yourself that you 938 00:47:05,280 --> 00:47:07,970 get into logical paradoxes if you pretend 939 00:47:07,970 --> 00:47:11,240 that all infinities are equal. 940 00:47:11,240 --> 00:47:12,939 But, this does not change the fact 941 00:47:12,939 --> 00:47:15,480 that Newton was still convinced that an infinite distribution 942 00:47:15,480 --> 00:47:18,170 of mass would be stable. 943 00:47:18,170 --> 00:47:20,670 The argument that convinced him was not the infinity 944 00:47:20,670 --> 00:47:23,690 on each side, but rather the symmetry. 945 00:47:23,690 --> 00:47:25,736 Newton's argument, the one he believed, 946 00:47:25,736 --> 00:47:27,110 was that if you look at any point 947 00:47:27,110 --> 00:47:30,590 in this infinite distribution, if you look around that point, 948 00:47:30,590 --> 00:47:32,965 all directions would look exactly the same, with matter 949 00:47:32,965 --> 00:47:34,964 extending off to infinity, and therefore there'd 950 00:47:34,964 --> 00:47:36,940 be no direction that the force should 951 00:47:36,940 --> 00:47:39,229 point on any given particle. 952 00:47:39,229 --> 00:47:41,520 And if there's no direction in that force at the point, 953 00:47:41,520 --> 00:47:42,730 it must be zero. 954 00:47:42,730 --> 00:47:45,930 That was the argument Newton believed. 955 00:47:45,930 --> 00:47:46,430 OK. 956 00:47:46,430 --> 00:47:49,870 What I want to do now is to talk about this in a little bit 957 00:47:49,870 --> 00:47:53,050 more detail, and try to understand 958 00:47:53,050 --> 00:47:57,157 how modern folks would look at the argument. 959 00:47:57,157 --> 00:47:59,240 And by the way, I might just add a little bit more 960 00:47:59,240 --> 00:48:00,675 about the history first. 961 00:48:00,675 --> 00:48:02,570 Newton's argument, as far as I know, 962 00:48:02,570 --> 00:48:06,505 was not questioned by anybody for hundreds of years, 963 00:48:06,505 --> 00:48:08,750 until the time of Albert Einstein. 964 00:48:08,750 --> 00:48:12,330 Albert Einstein, in trying to describe cosmology using 965 00:48:12,330 --> 00:48:15,006 his new theory of general relativity, 966 00:48:15,006 --> 00:48:16,380 was the first person, as far as I 967 00:48:16,380 --> 00:48:18,720 know, to realize that even if you 968 00:48:18,720 --> 00:48:22,170 had an infinite distribution of mass, it would collapse-- 969 00:48:22,170 --> 00:48:24,085 and we'll talk about why. 970 00:48:24,085 --> 00:48:26,460 And Einstein did realize that the same thing would happen 971 00:48:26,460 --> 00:48:28,110 with Newtonian physics, it's not really 972 00:48:28,110 --> 00:48:29,810 a special feature of general relativity, 973 00:48:29,810 --> 00:48:32,220 it just somehow historically took 974 00:48:32,220 --> 00:48:34,740 the invention of general relativity 975 00:48:34,740 --> 00:48:37,690 to cause people to rethink these ideas 976 00:48:37,690 --> 00:48:42,470 and realize that Newton had been wrong. 977 00:48:42,470 --> 00:48:43,670 So, what's going on. 978 00:49:04,400 --> 00:49:09,860 The difficulty in trying to analyze things the way 979 00:49:09,860 --> 00:49:14,270 in which Newton did is that Newton was thinking of gravity, 980 00:49:14,270 --> 00:49:17,357 in the language that he first proposed it, as a force 981 00:49:17,357 --> 00:49:17,940 at a distance. 982 00:49:17,940 --> 00:49:21,745 If you have two objects in space, the distance r apart, 983 00:49:21,745 --> 00:49:24,320 they will exert a force on each other proportional to one 984 00:49:24,320 --> 00:49:27,020 over r squared. 985 00:49:27,020 --> 00:49:29,120 Since the time of Newton, other ways 986 00:49:29,120 --> 00:49:31,557 of describing Newtonian gravity itself 987 00:49:31,557 --> 00:49:33,890 have been invented, which make it much more clear what's 988 00:49:33,890 --> 00:49:35,050 going on. 989 00:49:35,050 --> 00:49:36,870 The difficulty in using Newton's method-- 990 00:49:36,870 --> 00:49:39,619 we'll talk about in more detail in a few minutes-- 991 00:49:39,619 --> 00:49:42,160 but it's simply that we try to add up all of these one over r 992 00:49:42,160 --> 00:49:44,635 squared forces, you get divergent sums 993 00:49:44,635 --> 00:49:48,330 that you have to figure out how to interpret. 994 00:49:48,330 --> 00:49:50,610 But to understand that Newton couldn't possibly 995 00:49:50,610 --> 00:49:53,230 have been right, the easiest thing to do 996 00:49:53,230 --> 00:49:56,870 is to look at other formulations of Newton's gravity. 997 00:49:56,870 --> 00:49:59,180 And I'll describe two of them, both of which 998 00:49:59,180 --> 00:50:03,202 will probably have some familiarity to you. 999 00:50:03,202 --> 00:50:04,950 The first one I'm quite sure will. 1000 00:50:04,950 --> 00:50:09,920 And I'm going to describe it by analogy with Coulomb's law, 1001 00:50:09,920 --> 00:50:12,920 because 802 goes a little further with Coulomb's law 1002 00:50:12,920 --> 00:50:15,170 than any course you are likely to have taken 1003 00:50:15,170 --> 00:50:16,874 has gone with gravity. 1004 00:50:16,874 --> 00:50:18,915 But Coulomb's law is really the same as the force 1005 00:50:18,915 --> 00:50:19,840 law of gravity. 1006 00:50:19,840 --> 00:50:21,720 So Coulomb's law says that any charged 1007 00:50:21,720 --> 00:50:24,560 particle will create an electric field, which 1008 00:50:24,560 --> 00:50:27,530 is the charge divided by the distance squared times the unit 1009 00:50:27,530 --> 00:50:29,842 vector pointing radially outward. 1010 00:50:29,842 --> 00:50:31,730 That's Coulomb's law. 1011 00:50:31,730 --> 00:50:33,944 People can-- sometimes there's constants in here, 1012 00:50:33,944 --> 00:50:35,720 depending on what units you measure q in, 1013 00:50:35,720 --> 00:50:38,010 but that won't be important for us. 1014 00:50:38,010 --> 00:50:39,627 So I'm going to assume we're using 1015 00:50:39,627 --> 00:50:40,960 this where that constant is one. 1016 00:50:44,820 --> 00:50:49,000 You know that Coulomb's law can be reformulated 1017 00:50:49,000 --> 00:50:51,232 in terms of what we call Gauss's law. 1018 00:50:51,232 --> 00:50:53,850 If Coulomb's law is true, you can make a definite statement 1019 00:50:53,850 --> 00:50:55,391 about what happens when you integrate 1020 00:50:55,391 --> 00:50:58,060 the flux of the electric field over any surface. 1021 00:50:58,060 --> 00:51:02,380 It's proportionate to the total amount of charge inside. 1022 00:51:02,380 --> 00:51:07,800 So Coulomb's law implies Gauss's law, 1023 00:51:07,800 --> 00:51:14,692 which says that the integral over any closed surface of E 1024 00:51:14,692 --> 00:51:24,310 dotted into da is equal to 4 pi times the total enclosed 1025 00:51:24,310 --> 00:51:26,690 charge. 1026 00:51:26,690 --> 00:51:31,340 q encloses the total amount of charge inside that volume. 1027 00:51:31,340 --> 00:51:33,500 And what constants appear depends on what constants 1028 00:51:33,500 --> 00:51:36,090 appear here, which depends on what units you're using, 1029 00:51:36,090 --> 00:51:38,700 but these equations are consistent. 1030 00:51:38,700 --> 00:51:41,520 Those are the correct constants, if you 1031 00:51:41,520 --> 00:51:44,880 measure charge in a way which makes the electric field be 1032 00:51:44,880 --> 00:51:46,745 given by that simple formula. 1033 00:51:46,745 --> 00:51:49,286 OK, so I'm going to assume you know this, that you learned it 1034 00:51:49,286 --> 00:51:51,880 in 802 or elsewhere. 1035 00:51:51,880 --> 00:51:53,720 If this is true, then, since this 1036 00:51:53,720 --> 00:51:56,560 is the same inverse square law, if we write down 1037 00:51:56,560 --> 00:51:58,269 Newton's law of gravity, almost as Newton 1038 00:51:58,269 --> 00:51:59,934 would have written it, we can express it 1039 00:51:59,934 --> 00:52:01,790 as the acceleration of gravity at a given 1040 00:52:01,790 --> 00:52:03,762 distance from an object. 1041 00:52:03,762 --> 00:52:05,470 So we could write Newton's law of gravity 1042 00:52:05,470 --> 00:52:07,440 by saying the acceleration of gravity 1043 00:52:07,440 --> 00:52:11,320 is equal to minus Newton's constant times the mass 1044 00:52:11,320 --> 00:52:15,090 of the object, the analog of the charge up there, 1045 00:52:15,090 --> 00:52:18,942 divided by r squared times r hat. 1046 00:52:18,942 --> 00:52:20,525 Again, it's the inverse r squared law, 1047 00:52:20,525 --> 00:52:23,030 and the point radiating outward is just like Coulomb's law, 1048 00:52:23,030 --> 00:52:26,187 except for the constant out front. 1049 00:52:26,187 --> 00:52:28,020 The constant actually has the opposite sign, 1050 00:52:28,020 --> 00:52:29,478 which is important for some issues, 1051 00:52:29,478 --> 00:52:31,780 but not for what we're saying now. 1052 00:52:31,780 --> 00:52:34,160 The important point is that this can also 1053 00:52:34,160 --> 00:52:37,390 be recast as a Gauss's law, and it's 1054 00:52:37,390 --> 00:52:40,107 called Gauss's law of gravity. 1055 00:52:40,107 --> 00:52:42,440 And the only thing that differs is a constant out front, 1056 00:52:42,440 --> 00:52:45,920 so it's a trivial transformation. 1057 00:52:45,920 --> 00:52:48,980 The integral over any closed surface 1058 00:52:48,980 --> 00:52:52,440 of the gravitational acceleration vector, little g 1059 00:52:52,440 --> 00:52:59,918 dotted into da is equal to minus 4 pi 1060 00:52:59,918 --> 00:53:03,020 g times the total mass enclosed. 1061 00:53:06,918 --> 00:53:09,126 The only difference is the minus sign, and the factor 1062 00:53:09,126 --> 00:53:14,152 of g, which follow from the difference of the minus sign 1063 00:53:14,152 --> 00:53:16,110 and the factor of g in the formula on the left. 1064 00:53:18,894 --> 00:53:21,319 OK, does everybody believe that? 1065 00:53:21,319 --> 00:53:23,610 OK, now let's think about this homogeneous distribution 1066 00:53:23,610 --> 00:53:26,027 of mass that Newton was trying to think about. 1067 00:53:26,027 --> 00:53:27,590 Newton's claim was that you could 1068 00:53:27,590 --> 00:53:29,740 have a homogeneous distribution of mass filling 1069 00:53:29,740 --> 00:53:33,260 all of the infinite space, and that would be static, that is, 1070 00:53:33,260 --> 00:53:35,180 there would be no acceleration. 1071 00:53:35,180 --> 00:53:37,700 No acceleration means Newton is claiming in this language 1072 00:53:37,700 --> 00:53:40,820 that little g could be zero everywhere. 1073 00:53:40,820 --> 00:53:44,000 But if you look at this formula, if little g is zero everywhere, 1074 00:53:44,000 --> 00:53:46,890 then the integral of g over any surface is going to zero, 1075 00:53:46,890 --> 00:53:49,640 and therefore the total mass enclosed had better be zero. 1076 00:53:49,640 --> 00:53:51,760 But if we have a uniform distribution of mass, 1077 00:53:51,760 --> 00:53:53,590 the total mass enclosed will certainly not 1078 00:53:53,590 --> 00:53:56,870 be zero for anything with non-zero volume. 1079 00:53:56,870 --> 00:54:00,590 So clearly this assertion that the system would be static 1080 00:54:00,590 --> 00:54:05,420 was in direct contradiction with the Gauss's law formulation 1081 00:54:05,420 --> 00:54:06,550 of Newton's law of gravity. 1082 00:54:12,759 --> 00:54:14,300 Just for the fun of it, I'll give you 1083 00:54:14,300 --> 00:54:24,620 another similar argument using another more modern formulation 1084 00:54:24,620 --> 00:54:25,535 of Newtonian gravity. 1085 00:54:35,035 --> 00:54:37,160 Another way of formulating Newtonian gravity, which 1086 00:54:37,160 --> 00:54:39,580 you may or may not have seen-- and if you haven't seen it, 1087 00:54:39,580 --> 00:54:41,100 don't understand what I'm saying, don't worry about it, 1088 00:54:41,100 --> 00:54:42,540 it's not that important. 1089 00:54:42,540 --> 00:54:44,190 But for those of you who have seen it, 1090 00:54:44,190 --> 00:54:46,950 I'll give you this argument. 1091 00:54:46,950 --> 00:54:49,030 Another way of formulating Newtonian gravity 1092 00:54:49,030 --> 00:54:52,620 is to introduce the gravitational potential. 1093 00:54:52,620 --> 00:54:54,620 So I'm going to use the letter phi 1094 00:54:54,620 --> 00:54:56,870 for the gravitational potential. 1095 00:54:56,870 --> 00:54:59,726 I'll tell you in a second how that relates to gravity-- well, 1096 00:54:59,726 --> 00:55:01,120 I guess I'll tell you now. 1097 00:55:01,120 --> 00:55:03,800 It's related to the gravitational acceleration 1098 00:55:03,800 --> 00:55:11,140 by g is equal to minus the gradient of phi, 1099 00:55:11,140 --> 00:55:16,010 and gradient of phi is something that you probably all 1100 00:55:16,010 --> 00:55:21,160 learned in 802, but I'll write down the formula anyway. 1101 00:55:21,160 --> 00:55:24,250 It's equal to i hat, a unit vector in the x direction, 1102 00:55:24,250 --> 00:55:26,560 times the derivative of phi with respect 1103 00:55:26,560 --> 00:55:31,410 to x, plus j hat, a unit vector in the y direction, 1104 00:55:31,410 --> 00:55:34,624 times the partial of phi with respect to y, 1105 00:55:34,624 --> 00:55:39,564 plus k hat times the partial of phi with respect to z. 1106 00:55:42,530 --> 00:55:46,500 And once one defines this gravitational potential, 1107 00:55:46,500 --> 00:55:48,420 one can write down the differential form 1108 00:55:48,420 --> 00:55:50,710 of the Gauss's law, which becomes 1109 00:55:50,710 --> 00:55:52,668 what's called Laplace's equation. 1110 00:55:52,668 --> 00:56:00,500 And it says the del squared phi is 1111 00:56:00,500 --> 00:56:08,531 equal to 4 pi times Newton's constant times rho, 1112 00:56:08,531 --> 00:56:12,475 where rho is the mass density. 1113 00:56:21,870 --> 00:56:24,192 And this is called Laplace's equation, 1114 00:56:24,192 --> 00:56:26,330 and if you're given the mass density, 1115 00:56:26,330 --> 00:56:29,197 it allows you to find the gravitational potential, 1116 00:56:29,197 --> 00:56:30,655 and then you can take its gradient, 1117 00:56:30,655 --> 00:56:33,510 and that determines what g is. 1118 00:56:33,510 --> 00:56:37,110 And it's equivalent to the other formulations of gravity. 1119 00:56:37,110 --> 00:56:40,570 But it gives us another test of Newton's claim 1120 00:56:40,570 --> 00:56:43,460 that you could have a homogeneous distribution 1121 00:56:43,460 --> 00:56:46,750 of matter, and no gravitational forces. 1122 00:56:46,750 --> 00:56:49,580 If there are no gravitational forces, 1123 00:56:49,580 --> 00:56:53,200 then g would have to be zero, as we said a minute ago, 1124 00:56:53,200 --> 00:56:55,090 and this formulation of g is zero, 1125 00:56:55,090 --> 00:56:58,660 that implies the gradient of phi is zero. 1126 00:56:58,660 --> 00:57:01,120 If we look at the formula for the gradient, it's a vector. 1127 00:57:01,120 --> 00:57:03,411 For the vector to be zero, each of the three components 1128 00:57:03,411 --> 00:57:05,630 has to be zero, and therefore the derivative 1129 00:57:05,630 --> 00:57:07,450 of phi with respect to x has to vanish, 1130 00:57:07,450 --> 00:57:09,700 the derivative of phi with respect to y has to vanish, 1131 00:57:09,700 --> 00:57:12,330 the derivative of phi with respect to z has to vanish, 1132 00:57:12,330 --> 00:57:13,780 that means phi has to be constant everywhere, 1133 00:57:13,780 --> 00:57:15,154 it has no derivative with respect 1134 00:57:15,154 --> 00:57:17,310 to any spacial coordinate. 1135 00:57:17,310 --> 00:57:20,640 So if g vanishes, the gradient of phi 1136 00:57:20,640 --> 00:57:23,770 vanishes, and phi is a constant throughout space. 1137 00:57:23,770 --> 00:57:25,990 And if phi is a constant throughout space, 1138 00:57:25,990 --> 00:57:27,380 now we can look at this formula-- 1139 00:57:27,380 --> 00:57:30,990 and I forgot to write down the definition of del squared. 1140 00:57:30,990 --> 00:57:36,020 Del squared phi is defined to be the second derivative of phi 1141 00:57:36,020 --> 00:57:41,990 with respect to x squared, plus the second derivative of phi 1142 00:57:41,990 --> 00:57:48,500 with respect to y squared, plus the second derivative of phi 1143 00:57:48,500 --> 00:57:51,560 with respect to z squared. 1144 00:57:51,560 --> 00:57:53,440 So if phi is a constant everywhere, 1145 00:57:53,440 --> 00:57:56,610 as it would have to be if there were no gravitational forces, 1146 00:57:56,610 --> 00:57:58,649 then one can see immediately from this equation 1147 00:57:58,649 --> 00:58:00,440 that del squared phi would have to be zero, 1148 00:58:00,440 --> 00:58:01,640 and one can see from this equation 1149 00:58:01,640 --> 00:58:03,181 that rho would have to be zero, there 1150 00:58:03,181 --> 00:58:04,867 would have to be no mass density. 1151 00:58:04,867 --> 00:58:06,575 But Newton wanted to have a non-zero mass 1152 00:58:06,575 --> 00:58:08,830 density, the matter of the universe spread 1153 00:58:08,830 --> 00:58:11,060 out uniformly over an infinite space. 1154 00:58:11,060 --> 00:58:14,730 So this is another demonstration that Newton's argument 1155 00:58:14,730 --> 00:58:17,500 was inconsistent. 1156 00:58:17,500 --> 00:58:18,000 Yes. 1157 00:58:18,000 --> 00:58:21,074 AUDIENCE: I'm sorry, what does phi represent? 1158 00:58:21,074 --> 00:58:23,240 PROFESSOR: Phi is really defined by these equations, 1159 00:58:23,240 --> 00:58:25,130 it's defined, really, by this equation. 1160 00:58:25,130 --> 00:58:29,615 The name is that it's the gravitational potential. 1161 00:58:29,615 --> 00:58:30,448 AUDIENCE: Potential. 1162 00:58:37,782 --> 00:58:39,240 PROFESSOR: And its physical meaning 1163 00:58:39,240 --> 00:58:41,450 is simply that it gives you another way 1164 00:58:41,450 --> 00:58:43,762 of writing what g is. 1165 00:58:43,762 --> 00:58:44,387 AUDIENCE: Yeah. 1166 00:58:49,249 --> 00:58:50,540 PROFESSOR: Any other questions? 1167 00:58:55,240 --> 00:58:59,210 OK, so the conclusion seems to be that Newton has not 1168 00:58:59,210 --> 00:59:01,770 gotten the right answer, here, but we still 1169 00:59:01,770 --> 00:59:03,653 have to analyze Newton's argument a little bi 1170 00:59:03,653 --> 00:59:07,730 more carefully, to see exactly where he went wrong. 1171 00:59:07,730 --> 00:59:11,230 So, the next thing I want to talk about 1172 00:59:11,230 --> 00:59:16,120 is the ambiguity associated with trying to add up the Newtonian 1173 00:59:16,120 --> 00:59:18,970 gravitational forces, as Newton was 1174 00:59:18,970 --> 00:59:22,220 thinking, for an infinite universe. 1175 00:59:22,220 --> 00:59:24,380 I mentioned that the real problem with Newton's 1176 00:59:24,380 --> 00:59:28,820 calculation is that the quantum he was calculating actually 1177 00:59:28,820 --> 00:59:31,330 diverges, and you have to be more careful about trying 1178 00:59:31,330 --> 00:59:33,440 to calculate it in a reliable way. 1179 00:59:50,250 --> 00:59:52,050 So to make this clear, I want to begin 1180 00:59:52,050 --> 00:59:57,060 by giving an example of this general notion of integrals 1181 00:59:57,060 --> 01:00:00,190 that give ambiguous values. 1182 01:00:00,190 --> 01:00:03,530 And I want to define just a couple of mathematical terms. 1183 01:00:03,530 --> 01:00:06,673 I want to consider just-- again, starting 1184 01:00:06,673 --> 01:00:09,172 talking about general functions, and when integrals are well 1185 01:00:09,172 --> 01:00:10,770 defined and when they're not. 1186 01:00:10,770 --> 01:00:14,920 I want to imagine that we just have some arbitrary function 1187 01:00:14,920 --> 01:00:18,141 f of x where x would not just be one variable. 1188 01:00:18,141 --> 01:00:20,140 We'll generalize this to three dimensions, which 1189 01:00:20,140 --> 01:00:22,450 is the case that we'll be interested in, 1190 01:00:22,450 --> 01:00:24,835 but we'll start by talking in terms of one variable. 1191 01:00:24,835 --> 01:00:28,080 If we have a function f of x, we can 1192 01:00:28,080 --> 01:00:31,620 discuss what I'll call I sub 1, which 1193 01:00:31,620 --> 01:00:35,279 is the integral, from minus infinity to infinity, 1194 01:00:35,279 --> 01:00:38,237 of f of x dx. 1195 01:00:38,237 --> 01:00:39,845 This is exactly the kind of integral 1196 01:00:39,845 --> 01:00:41,440 that you're thinking of when we wanted 1197 01:00:41,440 --> 01:00:44,670 to-- thinking about adding up all the gravitational forces 1198 01:00:44,670 --> 01:00:46,100 acting on a given body. 1199 01:00:52,610 --> 01:01:05,076 Now I want to consider the case where I1 is finite. 1200 01:01:05,076 --> 01:01:05,937 I'm sorry. 1201 01:01:05,937 --> 01:01:08,270 I need to first define more carefully what I mean by I1. 1202 01:01:14,294 --> 01:01:17,020 OK, to even define what you mean by this minus v to infinity, 1203 01:01:17,020 --> 01:01:19,740 you should say something a little bit more precise. 1204 01:01:19,740 --> 01:01:22,640 So we could define I1 a little bit more precisely, 1205 01:01:22,640 --> 01:01:26,247 and I'll call this I1 prime, for clarity. 1206 01:01:26,247 --> 01:01:27,830 This will really just be a clearer way 1207 01:01:27,830 --> 01:01:29,507 of describing what one probably meant 1208 01:01:29,507 --> 01:01:31,854 when one wrote the first line. 1209 01:01:31,854 --> 01:01:33,770 We can define the integral from minus infinity 1210 01:01:33,770 --> 01:01:39,130 to infinity as the limit, as some quantity L goes 1211 01:01:39,130 --> 01:01:46,152 to infinity, of the integral from minus L to L of f of x dx. 1212 01:01:49,140 --> 01:01:52,380 So this says to do the integral from minus L to L, 1213 01:01:52,380 --> 01:01:54,340 and if we assume f of x is itself finite, 1214 01:01:54,340 --> 01:01:55,400 this is always finite. 1215 01:01:55,400 --> 01:01:58,490 I will assume f of x itself is finite, 1216 01:01:58,490 --> 01:02:01,390 we'll only worry about the convergence of the integral. 1217 01:02:01,390 --> 01:02:04,800 So for any given L, this is a number, then you can ask, 1218 01:02:04,800 --> 01:02:08,562 does this number approach a limit as L goes to infinity? 1219 01:02:08,562 --> 01:02:11,170 And if it does, you say that's the value of this integral. 1220 01:02:11,170 --> 01:02:13,094 That just defines what we mean by the integral 1221 01:02:13,094 --> 01:02:15,980 from minus infinity to infinity. 1222 01:02:15,980 --> 01:02:18,172 I want to now consider the case where that exists. 1223 01:02:26,930 --> 01:02:29,740 So consider the case where I1 prime is-- I'll write 1224 01:02:29,740 --> 01:02:33,520 is less than infinity, meaning it has some finite value. 1225 01:02:33,520 --> 01:02:35,655 The limit as L goes to infinity exists. 1226 01:02:38,240 --> 01:02:42,888 But now, I want to also consider-- 1227 01:02:42,888 --> 01:02:44,930 and I'll move on to the next blackboard-- 1228 01:02:44,930 --> 01:03:01,450 to consider this-- consider an integral that I'll call I2, 1229 01:03:01,450 --> 01:03:03,900 for future reference, which is just 1230 01:03:03,900 --> 01:03:08,400 defined to be the integral from minus infinity to infinity. 1231 01:03:08,400 --> 01:03:10,750 Defined as the same kind of limit that we used here, 1232 01:03:10,750 --> 01:03:12,205 but I won't rewrite it. 1233 01:03:12,205 --> 01:03:14,455 I'll just assume that the integral from minus infinity 1234 01:03:14,455 --> 01:03:16,689 to infinity means that limit. 1235 01:03:16,689 --> 01:03:18,980 But I want to consider the integral from minus infinity 1236 01:03:18,980 --> 01:03:24,120 to infinity of the absolute value of f of x dx. 1237 01:03:28,960 --> 01:03:31,600 And now I want to introduce some terminology. 1238 01:03:34,690 --> 01:03:43,900 If I2 is less than infinity, if it converges, 1239 01:03:43,900 --> 01:03:49,740 then I1 is called absolutely convergent. 1240 01:04:02,700 --> 01:04:05,130 So absolutely convergent means that it would converge, 1241 01:04:05,130 --> 01:04:06,680 even if you had absolute value signs. 1242 01:04:10,784 --> 01:04:16,770 Conversely, this I2 is divergent-- 1243 01:04:16,770 --> 01:04:19,470 and I'll just write that as I2 equals infinity, 1244 01:04:19,470 --> 01:04:24,856 if that limit does not exist, if its a divergent integral. 1245 01:04:24,856 --> 01:04:28,350 But remember, we assumed I1 did exist, 1246 01:04:28,350 --> 01:04:30,767 so I1 still converges, but it's called conditionally 1247 01:04:30,767 --> 01:04:31,267 convergent. 1248 01:04:49,616 --> 01:04:51,490 So if an integral converges, but the integral 1249 01:04:51,490 --> 01:04:54,210 of the absolute value of that same client does not converge, 1250 01:04:54,210 --> 01:04:56,829 that's the case that's called conditional convergence. 1251 01:04:56,829 --> 01:04:58,370 And the moral of the story, that I'll 1252 01:04:58,370 --> 01:05:02,800 be beginning to tell you now, is that conditionally convergent 1253 01:05:02,800 --> 01:05:04,714 integrals are very dangerous. 1254 01:05:04,714 --> 01:05:06,880 What makes them dangerous is that they're not really 1255 01:05:06,880 --> 01:05:07,970 well defined. 1256 01:05:07,970 --> 01:05:11,180 You can get any value you want by adding up 1257 01:05:11,180 --> 01:05:14,520 the integrand in different orders. 1258 01:05:14,520 --> 01:05:16,760 As long as you stick to a particular order, which 1259 01:05:16,760 --> 01:05:20,310 is how we define the symbol, you will get a unique answer, 1260 01:05:20,310 --> 01:05:22,750 but if, for example, you just shift your origin, 1261 01:05:22,750 --> 01:05:25,457 you can get a different answer, which 1262 01:05:25,457 --> 01:05:27,040 is something you don't usually expect. 1263 01:05:27,040 --> 01:05:28,680 You usually think of just integrating over 1264 01:05:28,680 --> 01:05:30,263 the whole real line, it doesn't matter 1265 01:05:30,263 --> 01:05:32,770 what you took to be the center of the line. 1266 01:05:32,770 --> 01:05:34,565 So things become much less well defined 1267 01:05:34,565 --> 01:05:37,680 when one is discussing conditionally 1268 01:05:37,680 --> 01:05:40,030 convergent integrals. 1269 01:05:40,030 --> 01:05:42,277 And before we get to the particular integral 1270 01:05:42,277 --> 01:05:43,860 that we're really interested in, which 1271 01:05:43,860 --> 01:05:45,818 is trying to add up the gravitational forces of 1272 01:05:45,818 --> 01:05:48,355 and infinite distribution of matter, which I'll get to, 1273 01:05:48,355 --> 01:05:50,980 I'm going to give you an example of a very simple function that 1274 01:05:50,980 --> 01:05:54,560 just illustrates this ambiguity, that the integral converges, 1275 01:05:54,560 --> 01:05:55,990 but is not absolutely convergent. 1276 01:05:55,990 --> 01:05:58,857 You can get any answer you want by adding it 1277 01:05:58,857 --> 01:06:01,930 in different orders-- adding up the pieces of the integral 1278 01:06:01,930 --> 01:06:03,950 in different orders. 1279 01:06:03,950 --> 01:06:11,810 So let me consider an example-- and this is just 1280 01:06:11,810 --> 01:06:16,230 to illustrate the ambiguity-- the example I'll consider 1281 01:06:16,230 --> 01:06:25,550 will be a function f of x, which is defined to equal plus 1 if x 1282 01:06:25,550 --> 01:06:32,176 is greater than zero, and minus one if x is less than zero. 1283 01:06:35,040 --> 01:06:37,000 And I have neglected to specify what 1284 01:06:37,000 --> 01:06:39,437 happens if x is exactly equal to zero, 1285 01:06:39,437 --> 01:06:41,270 but when you integrate, that doesn't matter. 1286 01:06:41,270 --> 01:06:42,810 A single point never matters. 1287 01:06:42,810 --> 01:06:44,226 So you could measure it's anything 1288 01:06:44,226 --> 01:06:46,485 you want at x equals zero, it won't change anything 1289 01:06:46,485 --> 01:06:47,568 you're going to be saying. 1290 01:06:49,880 --> 01:06:51,600 Let me draw a graph of this. 1291 01:06:57,866 --> 01:07:00,758 f of x versus x. 1292 01:07:00,758 --> 01:07:06,210 I'll put plus 1 there, and minus 1 there. 1293 01:07:06,210 --> 01:07:08,047 The function is plus 1. 1294 01:07:08,047 --> 01:07:09,797 Maybe I have a little bit of colored chalk 1295 01:07:09,797 --> 01:07:12,060 here to draw the function. 1296 01:07:12,060 --> 01:07:17,207 The function is plus 1 for all positive values of x, 1297 01:07:17,207 --> 01:07:20,470 and minus one for all negative values of x. 1298 01:07:20,470 --> 01:07:22,470 And there's the function. 1299 01:07:22,470 --> 01:07:24,750 And if we integrate it symmetrically, 1300 01:07:24,750 --> 01:07:26,680 following this definition of what 1301 01:07:26,680 --> 01:07:29,786 we mean by integrating from minus infinity to infinity, 1302 01:07:29,786 --> 01:07:31,160 we do get a perfect cancellation. 1303 01:07:31,160 --> 01:07:33,060 When you integrate from minus L to L, 1304 01:07:33,060 --> 01:07:35,720 we get zero, because you get a perfect cancellation 1305 01:07:35,720 --> 01:07:38,330 between the negative parts and the positive parts. 1306 01:07:38,330 --> 01:07:41,250 And then if you take the limit as L goes to infinity, 1307 01:07:41,250 --> 01:07:44,438 the limit of zero is zero. 1308 01:07:44,438 --> 01:07:47,180 There's not really any ambiguity to that statement. 1309 01:07:47,180 --> 01:07:50,400 So in the order specified, this has unique integral, 1310 01:07:50,400 --> 01:07:52,230 which is zero. 1311 01:07:52,230 --> 01:07:57,610 But, it depends on how you've chosen to add things up. 1312 01:07:57,610 --> 01:07:59,980 In particular, if you just change your origin, 1313 01:07:59,980 --> 01:08:03,380 and integrate starting moving outward from the new origin, 1314 01:08:03,380 --> 01:08:05,900 you'll get a different answer, and that's 1315 01:08:05,900 --> 01:08:08,040 what I want to illustrate next. 1316 01:08:08,040 --> 01:08:36,020 Suppose-- suppose we consider the limit 1317 01:08:36,020 --> 01:08:39,810 as L goes to infinity, we'll pick the limit the same way, 1318 01:08:39,810 --> 01:08:42,229 but instead of integrating from minus L to L, 1319 01:08:42,229 --> 01:08:51,668 we can integrate from a minus L to a plus L of f of x dx. 1320 01:08:56,648 --> 01:08:58,189 Now this is really the same integral, 1321 01:08:58,189 --> 01:08:59,813 we've just basically changed our origin 1322 01:08:59,813 --> 01:09:01,110 by integrating from a outwards. 1323 01:09:01,110 --> 01:09:03,020 In the special case a equals zero, 1324 01:09:03,020 --> 01:09:05,140 it's exactly the same as what we did before, 1325 01:09:05,140 --> 01:09:07,439 but if a is non-zero, it means that our integral 1326 01:09:07,439 --> 01:09:10,350 is centered about x equals a, instead of centered 1327 01:09:10,350 --> 01:09:12,600 about x equals zero. 1328 01:09:12,600 --> 01:09:16,890 So we can draw that on the blackboard. 1329 01:09:16,890 --> 01:09:25,700 If we let a be over here, our integral 1330 01:09:25,700 --> 01:09:32,859 will go from a minus L, and that will 1331 01:09:32,859 --> 01:09:35,620 be to the left of distance L, you 1332 01:09:35,620 --> 01:09:38,220 will extend to a plus L, which will be to the right 1333 01:09:38,220 --> 01:09:44,560 by distance L. The integral defined 1334 01:09:44,560 --> 01:09:48,040 by the equation on the blackboard at the left 1335 01:09:48,040 --> 01:09:50,346 will correspond to that region of integration. 1336 01:09:50,346 --> 01:09:52,970 And the specification is that we should do that interval first, 1337 01:09:52,970 --> 01:09:54,886 and then take the limit as L goes to infinity, 1338 01:09:54,886 --> 01:09:57,160 and see what we get. 1339 01:09:57,160 --> 01:09:59,730 It's easy to see what we will get. 1340 01:09:59,730 --> 01:10:05,180 Once L is bigger than a, you can see 1341 01:10:05,180 --> 01:10:08,710 that the answer won't change any more, as we make L bigger. 1342 01:10:08,710 --> 01:10:10,640 As you make L bigger, we will always 1343 01:10:10,640 --> 01:10:13,460 be adding a certain amount of minus 1 on the left, 1344 01:10:13,460 --> 01:10:16,210 and certain amount-- the same amount of plus 1 on the right, 1345 01:10:16,210 --> 01:10:21,741 and they will cancel each other once L is bigger than a. 1346 01:10:21,741 --> 01:10:23,657 And we don't care about small l, because we're 1347 01:10:23,657 --> 01:10:25,654 only interested in taking the limit of large L, 1348 01:10:25,654 --> 01:10:27,070 but we should look at what happens 1349 01:10:27,070 --> 01:10:30,390 when L equals a, and then from any bigger value of L 1350 01:10:30,390 --> 01:10:32,500 will give us exactly the same number. 1351 01:10:32,500 --> 01:10:34,800 And when L equals a, the integral 1352 01:10:34,800 --> 01:10:41,816 will go from 0 up to 2a-- a plus L which 1353 01:10:41,816 --> 01:10:45,640 is a equals L, so that's 2a. 1354 01:10:45,640 --> 01:10:47,735 So the integral will be only on the positive side, 1355 01:10:47,735 --> 01:10:50,089 and we'll have a length of 2a, and that 1356 01:10:50,089 --> 01:10:52,130 means the integral will be 2a, because we're just 1357 01:10:52,130 --> 01:10:54,940 integrating one from 0 to 2a. 1358 01:10:54,940 --> 01:10:57,190 And that will be what we get for any bigger value of L 1359 01:10:57,190 --> 01:10:59,480 also, because as we increase L, as I said, 1360 01:10:59,480 --> 01:11:02,146 we just get a cancellation between adding more plus 1 1361 01:11:02,146 --> 01:11:06,410 on the right, and adding more minus 1 on the left. 1362 01:11:06,410 --> 01:11:10,280 So this limit has a perfectly well defined value, 1363 01:11:10,280 --> 01:11:11,020 which is 2a. 1364 01:11:13,930 --> 01:11:17,502 And a is just where we chose to start integrating, 1365 01:11:17,502 --> 01:11:18,460 so a could be anything. 1366 01:11:18,460 --> 01:11:19,918 We could choose a to be anything we 1367 01:11:19,918 --> 01:11:23,500 want if we're free to integrate in any order. 1368 01:11:23,500 --> 01:11:24,880 So we can get any answer we want, 1369 01:11:24,880 --> 01:11:26,715 if we're free to integrate in any order, 1370 01:11:26,715 --> 01:11:29,300 to add up the pieces of this integral in the order 1371 01:11:29,300 --> 01:11:31,050 that we choose. 1372 01:11:31,050 --> 01:11:33,230 And that is a fundamental ambiguity 1373 01:11:33,230 --> 01:11:36,300 of conditionally convergent integrals. 1374 01:11:36,300 --> 01:11:37,930 And what we'll see is that trying 1375 01:11:37,930 --> 01:11:41,167 to add up the force on a particle in an infinite mass 1376 01:11:41,167 --> 01:11:42,542 distribution is exactly this kind 1377 01:11:42,542 --> 01:11:45,260 of conditionally convergent integral. 1378 01:11:45,260 --> 01:11:47,500 And that's why you get any answer you want, 1379 01:11:47,500 --> 01:11:49,300 and it doesn't really mean anything 1380 01:11:49,300 --> 01:11:53,740 unless you do things very carefully. 1381 01:11:53,740 --> 01:11:54,830 OK. 1382 01:11:54,830 --> 01:11:56,870 Let's move on. 1383 01:11:56,870 --> 01:11:59,960 We only have a few minutes left, which 1384 01:11:59,960 --> 01:12:04,370 I guess means I will set up this calculation, 1385 01:12:04,370 --> 01:12:08,110 but not quite get the answer, and we'll continue next time. 1386 01:12:19,380 --> 01:12:21,930 I actually have some diagrams here on my slides. 1387 01:12:26,074 --> 01:12:30,840 What I want to do now, is calculate the force 1388 01:12:30,840 --> 01:12:33,394 on some particle in an infinite mass distribution, 1389 01:12:33,394 --> 01:12:35,310 and show you that I can get different answers, 1390 01:12:35,310 --> 01:12:38,022 depending on what order I add things up. 1391 01:12:38,022 --> 01:12:40,610 I will add things up in a definite order at each stage, 1392 01:12:40,610 --> 01:12:43,026 so I will get a definite answer at each stage, though I'll 1393 01:12:43,026 --> 01:12:46,210 get different answers, depending on what ordering I choose. 1394 01:12:46,210 --> 01:12:47,664 So, we're going to start by trying 1395 01:12:47,664 --> 01:12:49,830 to calculate-- and the only thing [? of interest, ?] 1396 01:12:49,830 --> 01:12:52,810 actually, in calculating the gravitational force 1397 01:12:52,810 --> 01:12:56,790 on some point, p in an infinite distribution of mass. 1398 01:12:56,790 --> 01:13:00,740 Mass fills the slide, and everything, out to infinity. 1399 01:13:00,740 --> 01:13:03,690 And we're going to add up that mass in contributions 1400 01:13:03,690 --> 01:13:05,695 that are specified. 1401 01:13:05,695 --> 01:13:07,850 And for our first calculation, we're 1402 01:13:07,850 --> 01:13:11,070 going to add up the forces for masses 1403 01:13:11,070 --> 01:13:13,674 that are defined in concentric shells, where we're 1404 01:13:13,674 --> 01:13:15,340 going to take the innermost shell first, 1405 01:13:15,340 --> 01:13:17,173 then the second shell, then the third shell, 1406 01:13:17,173 --> 01:13:19,710 going outward from the center. 1407 01:13:19,710 --> 01:13:23,630 In that case, it's easy to see that the force on p calculated 1408 01:13:23,630 --> 01:13:27,190 in that order of integration is 0, 1409 01:13:27,190 --> 01:13:30,440 because every shell has p exactly at the center, 1410 01:13:30,440 --> 01:13:33,610 and by symmetry, it has to cancel exactly. 1411 01:13:33,610 --> 01:13:36,240 In fact, we know-- and we'll use this fact shortly-- 1412 01:13:36,240 --> 01:13:40,050 that the gravitational field of a shell, inside the shell, 1413 01:13:40,050 --> 01:13:44,030 is exactly zero-- Newton figured this out-- 1414 01:13:44,030 --> 01:13:47,140 and outside the shell, the gravitational field of a shell 1415 01:13:47,140 --> 01:13:50,010 looks exactly the same as the gravitational field of a point 1416 01:13:50,010 --> 01:13:52,330 mass located at the center of the shell 1417 01:13:52,330 --> 01:13:53,556 with the same total mass. 1418 01:13:53,556 --> 01:13:55,180 So we're going to be using those facts. 1419 01:13:55,180 --> 01:13:58,780 And clearly those facts indicate that, for this case, 1420 01:13:58,780 --> 01:14:01,210 the answer is 0. 1421 01:14:01,210 --> 01:14:04,320 P equals 0. 1422 01:14:04,320 --> 01:14:09,352 Now we're going to consider a more complicated case-- going 1423 01:14:09,352 --> 01:14:12,400 too far, here, don't want to tell you the answer yet-- 1424 01:14:12,400 --> 01:14:15,020 this more complicated case, we're going to still calculate 1425 01:14:15,020 --> 01:14:18,340 the force at the point p, but we're 1426 01:14:18,340 --> 01:14:20,890 going to choose concentric spherical shells 1427 01:14:20,890 --> 01:14:23,706 which are centered around a different point, q. 1428 01:14:23,706 --> 01:14:25,330 So q just defines the shells that we're 1429 01:14:25,330 --> 01:14:26,672 going to use for adding things up, 1430 01:14:26,672 --> 01:14:29,297 and we're still going to add up all the shells out to infinity, 1431 01:14:29,297 --> 01:14:31,810 so we're going to be adding up the force on p 1432 01:14:31,810 --> 01:14:34,140 due to the entire infinite mass distribution, 1433 01:14:34,140 --> 01:14:35,962 but we'll be taking those contributions 1434 01:14:35,962 --> 01:14:37,420 in a different order, because we're 1435 01:14:37,420 --> 01:14:39,980 going to be ordering it according to shells that 1436 01:14:39,980 --> 01:14:42,480 are all centered on q, starting with the innermost, and then 1437 01:14:42,480 --> 01:14:44,990 the second, and then the third, and so on. 1438 01:14:44,990 --> 01:14:48,620 Now in this case, we can first talk about the contribution 1439 01:14:48,620 --> 01:14:53,085 of the shaded region, which are all the shells around q which 1440 01:14:53,085 --> 01:14:57,690 have radii which are less than the distance to p. 1441 01:14:57,690 --> 01:15:03,590 For all of these shells, p lies outside the shell. 1442 01:15:03,590 --> 01:15:05,740 And therefore all of those shells 1443 01:15:05,740 --> 01:15:09,680 act just like a point mass, with the same total mass 1444 01:15:09,680 --> 01:15:13,760 concentrated at q, the center of all those spheres. 1445 01:15:13,760 --> 01:15:16,270 So the mass that's in the shaded region 1446 01:15:16,270 --> 01:15:18,220 will give a contribution to the force 1447 01:15:18,220 --> 01:15:21,840 at p, which is just equal to the force of the mass 1448 01:15:21,840 --> 01:15:26,500 given by the same total mass the point q, located at q. 1449 01:15:26,500 --> 01:15:29,580 On the other hand, all the shells outside 1450 01:15:29,580 --> 01:15:32,260 will be shells for which p is inside. 1451 01:15:32,260 --> 01:15:34,500 P is no longer at the center of those shells, 1452 01:15:34,500 --> 01:15:38,110 but Newton figured out, and I'll assume that we all believe, 1453 01:15:38,110 --> 01:15:39,190 it doesn't matter. 1454 01:15:39,190 --> 01:15:41,560 Inside the spherical shell, the gravitational force 1455 01:15:41,560 --> 01:15:43,330 is zero anywhere, no matter how close 1456 01:15:43,330 --> 01:15:44,765 you are to the boundaries. 1457 01:15:44,765 --> 01:15:46,680 It just cancels out perfectly. 1458 01:15:46,680 --> 01:15:48,140 As you get closer to one boundary, 1459 01:15:48,140 --> 01:15:50,400 you might think you'd be pulled toward that boundary, 1460 01:15:50,400 --> 01:15:53,090 but-- let me just tell you what's happening here-- 1461 01:15:53,090 --> 01:15:54,600 as you get closer to one boundary, 1462 01:15:54,600 --> 01:15:57,377 it is true that the force pulling you 1463 01:15:57,377 --> 01:15:59,960 towards the particular particles at the boundary get stronger, 1464 01:15:59,960 --> 01:16:02,214 because it's 1 over r squared, but as you get close 1465 01:16:02,214 --> 01:16:04,255 to this boundary, there's more mass on this side, 1466 01:16:04,255 --> 01:16:06,270 because all the mass except for a little sliver 1467 01:16:06,270 --> 01:16:07,560 is on the other side. 1468 01:16:07,560 --> 01:16:10,910 And those two effects cancel out exactly. 1469 01:16:10,910 --> 01:16:13,980 So the force on a particle inside a shell 1470 01:16:13,980 --> 01:16:16,970 is exactly zero, as you can prove very easily by the way, 1471 01:16:16,970 --> 01:16:19,772 from the Gauss's law of formulation of gravity. 1472 01:16:19,772 --> 01:16:21,980 And therefore, the outer shells give no contribution. 1473 01:16:21,980 --> 01:16:25,082 So we've completely calculated now the force at p 1474 01:16:25,082 --> 01:16:27,770 is just equal to the force due to the shaded mass. 1475 01:16:36,650 --> 01:16:38,275 It's just given by that simple formula, 1476 01:16:38,275 --> 01:16:40,500 it's g times the total mass, divided by b squared, 1477 01:16:40,500 --> 01:16:43,850 that would be it's distance between q and p. 1478 01:16:43,850 --> 01:16:45,330 And it's non-zero. 1479 01:16:45,330 --> 01:16:48,800 So you get 0 or non-zero answer, depending 1480 01:16:48,800 --> 01:16:50,961 on what ordering you chose for adding up 1481 01:16:50,961 --> 01:16:52,960 the pieces of the mass that are going to make up 1482 01:16:52,960 --> 01:16:54,497 this infinite distribution. 1483 01:16:54,497 --> 01:16:56,830 And furthermore, this answer could be anything you want, 1484 01:16:56,830 --> 01:16:58,538 because I could let b be anything I want. 1485 01:16:58,538 --> 01:17:00,320 And this answer depends on b, and becomes 1486 01:17:00,320 --> 01:17:03,210 arbitrarily large in magnitude as b gets bigger. 1487 01:17:03,210 --> 01:17:05,250 The mass grows like b cubed. 1488 01:17:05,250 --> 01:17:06,860 It might look like it falls with b, 1489 01:17:06,860 --> 01:17:09,050 but actually it grows with b. 1490 01:17:09,050 --> 01:17:10,950 And we could get a point in any direction, 1491 01:17:10,950 --> 01:17:13,890 by choosing q on any side we want of p, so we can get, 1492 01:17:13,890 --> 01:17:16,130 really, any answer we want by using 1493 01:17:16,130 --> 01:17:18,208 this particular way of adding up the masses. 1494 01:17:18,208 --> 01:17:18,725 Yes. 1495 01:17:18,725 --> 01:17:21,016 AUDIENCE: Well, although we can get any answer we want, 1496 01:17:21,016 --> 01:17:24,252 every answer [INAUDIBLE] 1497 01:17:24,252 --> 01:17:25,710 PROFESSOR: Every answer, say again? 1498 01:17:25,710 --> 01:17:27,376 AUDIENCE: Like every single one of those 1499 01:17:27,376 --> 01:17:29,126 answers corresponds to a setup. 1500 01:17:29,126 --> 01:17:33,518 I mean like the g equals 0, [INAUDIBLE] 1501 01:17:36,637 --> 01:17:38,345 PROFESSOR: Well the reason it's a problem 1502 01:17:38,345 --> 01:17:40,140 is that these shells don't really exist. 1503 01:17:40,140 --> 01:17:41,765 We're just thinking about these shells. 1504 01:17:41,765 --> 01:17:43,510 The shells only determine what order 1505 01:17:43,510 --> 01:17:46,800 we are going to use for adding up the different contributions. 1506 01:17:46,800 --> 01:17:48,750 The matter is just uniformly distributed 1507 01:17:48,750 --> 01:17:51,422 and there's no shells present. 1508 01:17:51,422 --> 01:17:53,323 The shells are purely a mental construct, 1509 01:17:53,323 --> 01:17:55,981 which should not affect the answer. 1510 01:17:55,981 --> 01:17:58,160 This is not part of the physical system at all. 1511 01:18:00,885 --> 01:18:02,470 The shells only reflect the order 1512 01:18:02,470 --> 01:18:05,650 that we have used to add up the masses. 1513 01:18:05,650 --> 01:18:06,484 So we'll stop there. 1514 01:18:06,484 --> 01:18:08,566 If anybody has questions, we can talk after class, 1515 01:18:08,566 --> 01:18:11,240 and we can talk more about the question at the beginning 1516 01:18:11,240 --> 01:18:14,870 of the next class, but class is over for now.