1 00:00:00,500 --> 00:00:02,890 The following content is provided under a Creative 2 00:00:02,890 --> 00:00:04,430 Commons license. 3 00:00:04,430 --> 00:00:06,730 Your support will help MIT OpenCourseWare 4 00:00:06,730 --> 00:00:11,120 continue to offer high quality educational resources for free. 5 00:00:11,120 --> 00:00:13,720 To make a donation, or view additional materials 6 00:00:13,720 --> 00:00:17,680 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,680 --> 00:00:21,190 at ocw.mit.edu. 8 00:00:21,190 --> 00:00:21,820 PROFESSOR: Hi. 9 00:00:21,820 --> 00:00:26,930 Welcome to Excitatory Topics in Physics, lesson 2. 10 00:00:26,930 --> 00:00:30,424 I hope you all had a good week, good 4th of July. 11 00:00:30,424 --> 00:00:31,840 I was asleep for about half of it, 12 00:00:31,840 --> 00:00:33,381 so I don't really know what happened. 13 00:00:33,381 --> 00:00:36,590 But I'm here now, and-- 14 00:00:36,590 --> 00:00:37,090 yeah. 15 00:00:37,090 --> 00:00:39,310 Let's begin. 16 00:00:39,310 --> 00:00:43,600 Last class I started talking about special relativity, which 17 00:00:43,600 --> 00:00:49,430 is relativity where we assume there's negligible gravity. 18 00:00:49,430 --> 00:00:52,870 To deal with effects where gravity is important, 19 00:00:52,870 --> 00:00:54,810 you have to use general relativity, 20 00:00:54,810 --> 00:00:56,740 but special relativity is a lot simpler, 21 00:00:56,740 --> 00:01:00,730 so we'll start with special relativity first. 22 00:01:00,730 --> 00:01:05,440 Special relativity all derives from two basic postulates-- 23 00:01:05,440 --> 00:01:07,990 the two basic postulates of special relativity, 24 00:01:07,990 --> 00:01:11,080 two postulates which are axioms of the theory-- 25 00:01:11,080 --> 00:01:12,250 unprovable axioms. 26 00:01:12,250 --> 00:01:14,207 And I introduced them last class. 27 00:01:14,207 --> 00:01:15,790 The first was that the laws of physics 28 00:01:15,790 --> 00:01:18,132 are the same for all inertial observers, 29 00:01:18,132 --> 00:01:19,840 and the second is that the speed of light 30 00:01:19,840 --> 00:01:22,390 is the same for all inertial observers. 31 00:01:22,390 --> 00:01:27,940 And I defined inertial observer as simply somebody that 32 00:01:27,940 --> 00:01:30,190 makes measurements-- an observer is just somebody that 33 00:01:30,190 --> 00:01:33,040 makes measurements, and an inertial observer 34 00:01:33,040 --> 00:01:35,710 is somebody that makes measurements 35 00:01:35,710 --> 00:01:38,590 while moving at a constant speed-- or constant velocity, 36 00:01:38,590 --> 00:01:40,060 to be more precise. 37 00:01:40,060 --> 00:01:41,880 I'll just keep things simple. 38 00:01:41,880 --> 00:01:44,550 I'll just say speed. 39 00:01:44,550 --> 00:01:46,330 So if you're accelerating, then you'd 40 00:01:46,330 --> 00:01:47,470 be a non inertial observer. 41 00:01:50,510 --> 00:01:54,022 For non-inertial observers to make measurements, 42 00:01:54,022 --> 00:01:55,480 you have to use general relativity, 43 00:01:55,480 --> 00:01:57,460 because there's actually a deep connection between acceleration 44 00:01:57,460 --> 00:01:58,170 and gravity. 45 00:01:58,170 --> 00:02:01,150 But I'll get to that eventually. 46 00:02:01,150 --> 00:02:04,974 Now, to demonstrate to you-- 47 00:02:04,974 --> 00:02:06,890 the first postulate was pretty easy to accept. 48 00:02:06,890 --> 00:02:08,972 It simply says that nature is fair. 49 00:02:08,972 --> 00:02:10,930 The laws of physics are the same for everybody. 50 00:02:10,930 --> 00:02:17,710 The second postulate is actually very surprising and mysterious, 51 00:02:17,710 --> 00:02:20,980 perhaps disturbing, because it goes completely against what 52 00:02:20,980 --> 00:02:23,680 we'd expect to be true. 53 00:02:23,680 --> 00:02:29,140 Last class I did experiment with someone-- a student, Life-- 54 00:02:29,140 --> 00:02:34,450 where we both walked across the class and Life 55 00:02:34,450 --> 00:02:37,032 measured a speed for me and you measured a speed for me. 56 00:02:37,032 --> 00:02:38,740 They happened to be two different speeds, 57 00:02:38,740 --> 00:02:41,980 and there is a simple way of relating them. 58 00:02:41,980 --> 00:02:44,300 There's a simple way of relating all the speeds. 59 00:02:44,300 --> 00:02:46,450 We simply subtracted one from the other. 60 00:02:46,450 --> 00:02:49,240 You measured four miles per hour for me, two for Life, 61 00:02:49,240 --> 00:02:52,060 and he measured two for me and zero for himself. 62 00:02:52,060 --> 00:02:53,230 And it's easy to get to. 63 00:02:53,230 --> 00:02:55,030 You just take two from four. 64 00:02:55,030 --> 00:02:57,430 Simple subtraction of velocities. 65 00:02:57,430 --> 00:03:00,010 Well, I told you at the end of last class 66 00:03:00,010 --> 00:03:03,700 that that rule actually doesn't apply for light-- 67 00:03:03,700 --> 00:03:06,340 for some reason, some mysterious reason. 68 00:03:06,340 --> 00:03:08,980 But we don't attempt to explain that. 69 00:03:08,980 --> 00:03:11,620 We simply take that as a given. 70 00:03:11,620 --> 00:03:15,880 And we wouldn't take that as a given if we knew for a fact 71 00:03:15,880 --> 00:03:16,980 that it was wrong. 72 00:03:16,980 --> 00:03:19,750 But in fact, numerous experiments 73 00:03:19,750 --> 00:03:22,060 have been conducted testing the validity 74 00:03:22,060 --> 00:03:24,940 of the second hypothesis, and they've all confirmed it. 75 00:03:24,940 --> 00:03:26,440 They've all confirmed it to be true. 76 00:03:29,370 --> 00:03:31,470 Just to remind you what it means, 77 00:03:31,470 --> 00:03:33,620 it means that if I pointed a laser-- 78 00:03:33,620 --> 00:03:38,430 if I pointed a light at the wall and started walking, 79 00:03:38,430 --> 00:03:40,440 then I'd measure a certain speed for light. 80 00:03:40,440 --> 00:03:45,090 I'd measure about 186,000 miles per second for light. 81 00:03:45,090 --> 00:03:48,390 And you'd also measure the same speed, 82 00:03:48,390 --> 00:03:49,920 even though I'm walking with it. 83 00:03:49,920 --> 00:03:52,920 You don't add my speed to it. 84 00:03:52,920 --> 00:03:55,790 And that's crazy. 85 00:03:55,790 --> 00:03:58,455 It's completely counter to what we'd expect. 86 00:04:03,430 --> 00:04:07,292 So actually, you might suspect that maybe the subtraction 87 00:04:07,292 --> 00:04:09,000 of velocities that we were doing before-- 88 00:04:09,000 --> 00:04:11,166 the subtraction of speeds that we were doing before, 89 00:04:11,166 --> 00:04:12,600 four minus two-- 90 00:04:12,600 --> 00:04:16,149 maybe that's just an approximation. 91 00:04:16,149 --> 00:04:19,649 And it turns out that it actually is an approximation. 92 00:04:19,649 --> 00:04:22,190 And I'll give you the real formula. 93 00:04:22,190 --> 00:04:23,450 Oh, you have a question? 94 00:04:23,450 --> 00:04:24,907 OK. 95 00:04:24,907 --> 00:04:26,490 I'll give you the real formula to deal 96 00:04:26,490 --> 00:04:28,460 with subtraction of velocities. 97 00:04:28,460 --> 00:04:29,900 I keep saying velocities. 98 00:04:29,900 --> 00:04:31,400 Speeds. 99 00:04:31,400 --> 00:04:34,930 I won't-- I'll just be doing things in one direction. 100 00:04:34,930 --> 00:04:38,130 So if I say velocity, it's either 101 00:04:38,130 --> 00:04:40,400 going to be negative or positive. 102 00:04:40,400 --> 00:04:42,470 So don't worry about that. 103 00:04:42,470 --> 00:04:44,960 Velocity is just speed with direction, 104 00:04:44,960 --> 00:04:50,210 for any of you that aren't familiar with the word. 105 00:04:50,210 --> 00:04:54,014 Now I'm going to show you something. 106 00:04:54,014 --> 00:04:55,930 Where's-- yeah, this is the chalk I was using. 107 00:04:59,994 --> 00:05:01,410 The way that we're used to dealing 108 00:05:01,410 --> 00:05:06,510 with relating different speeds is, we simply 109 00:05:06,510 --> 00:05:07,890 add them or we subtract them. 110 00:05:11,730 --> 00:05:14,790 If I'm on a train and I'm walking, and you're-- 111 00:05:14,790 --> 00:05:16,960 I'm just giving you another example. 112 00:05:16,960 --> 00:05:19,880 If I'm on a train and I'm walking a few miles per hour, 113 00:05:19,880 --> 00:05:22,980 and you observe the train going at 40 miles per hour, 114 00:05:22,980 --> 00:05:27,390 then you expect to find 40 plus three, or 43, miles per hour 115 00:05:27,390 --> 00:05:30,360 for me, if I'm walking in the direction of the train. 116 00:05:30,360 --> 00:05:31,980 If I walk in the opposite direction, 117 00:05:31,980 --> 00:05:34,313 then you'd expect to find 40 miles three miles per hour, 118 00:05:34,313 --> 00:05:36,420 37 miles per hour. 119 00:05:36,420 --> 00:05:40,530 That rule only works for small speeds. 120 00:05:40,530 --> 00:05:46,020 And that was actually known for a long time. 121 00:05:46,020 --> 00:05:49,560 Nowadays we refer to this rule as the Galilean addition 122 00:05:49,560 --> 00:05:51,360 of velocities. 123 00:05:51,360 --> 00:05:54,271 You make a Galilean transformation. 124 00:05:54,271 --> 00:05:56,520 I won't get into the details of these transformations, 125 00:05:56,520 --> 00:06:00,510 but call it Galilean addition of velocities. 126 00:06:00,510 --> 00:06:07,220 And you simply take the two speeds and you add them. 127 00:06:07,220 --> 00:06:08,120 I'll write up here. 128 00:06:11,850 --> 00:06:15,500 Galilean addition of velocities. 129 00:06:27,688 --> 00:06:28,188 Hi. 130 00:06:32,750 --> 00:06:38,030 Suppose I'm traveling at a speed of v relative to you all. 131 00:06:38,030 --> 00:06:41,360 Suppose I'm traveling at a speed of v. Actually, let me-- 132 00:06:41,360 --> 00:06:42,110 let me start over. 133 00:06:42,110 --> 00:06:43,050 Let me start over. 134 00:06:43,050 --> 00:06:44,140 Let's say I'm on a train. 135 00:06:44,140 --> 00:06:47,210 Let's say I'm on a train that's traveling at a speed of v 136 00:06:47,210 --> 00:06:49,400 relative to you. 137 00:06:49,400 --> 00:06:54,830 And relative to the train, I'm traveling at a speed of u. 138 00:06:54,830 --> 00:06:57,420 So u equals-- 139 00:06:57,420 --> 00:07:07,420 I'll start with v. v equals speed of train, 140 00:07:07,420 --> 00:07:15,520 and u equals speed of me relative to the train. 141 00:07:25,420 --> 00:07:30,490 Then what's the speed of me relative to you? 142 00:07:30,490 --> 00:07:33,550 It's just u plus v-- 143 00:07:33,550 --> 00:07:35,830 or so you would think, and so everybody 144 00:07:35,830 --> 00:07:37,310 thought for a long time. 145 00:07:37,310 --> 00:07:40,360 Everybody since Galileo's time thought it would just 146 00:07:40,360 --> 00:07:44,790 be u plus v. So u plus v-- 147 00:07:44,790 --> 00:07:48,340 I'll put this in quotation marks, 148 00:07:48,340 --> 00:07:53,050 jest to encode the meaning of the speed of me 149 00:07:53,050 --> 00:07:55,700 relative to you. 150 00:07:55,700 --> 00:07:58,390 So u plus-- 151 00:07:58,390 --> 00:07:59,480 I could write it in words. 152 00:07:59,480 --> 00:08:00,370 I was going to put it in quotation marks, 153 00:08:00,370 --> 00:08:02,161 but I'll just write it in words so that you 154 00:08:02,161 --> 00:08:03,860 know what I'm talking about. 155 00:08:03,860 --> 00:08:15,970 So u plus v equals the speed of me relative to you. 156 00:08:19,710 --> 00:08:20,976 Or maybe I should say class. 157 00:08:20,976 --> 00:08:21,600 I'll say class. 158 00:08:26,000 --> 00:08:29,300 I should also specify here, too. 159 00:08:29,300 --> 00:08:31,030 Speed of the train relative to the class. 160 00:08:31,030 --> 00:08:31,320 Yes? 161 00:08:31,320 --> 00:08:32,819 AUDIENCE: It's relative to the class 162 00:08:32,819 --> 00:08:34,419 if the class is surrounding me? 163 00:08:34,419 --> 00:08:35,720 PROFESSOR: Oh, OK. 164 00:08:35,720 --> 00:08:38,120 Well, the class isn't moving as a whole 165 00:08:38,120 --> 00:08:40,650 to any appreciable extent. 166 00:08:40,650 --> 00:08:42,189 So-- yeah. 167 00:08:42,189 --> 00:08:43,230 That's important, though. 168 00:08:43,230 --> 00:08:44,480 That's an important point. 169 00:08:44,480 --> 00:08:46,590 It's an important point. 170 00:08:46,590 --> 00:08:47,870 OK. 171 00:08:47,870 --> 00:08:51,530 To use special relativity at all, to use it at all, 172 00:08:51,530 --> 00:08:55,910 we have to consider only inertial observers-- 173 00:08:55,910 --> 00:08:58,844 only observers that are at rest or moving at a constant speed. 174 00:08:58,844 --> 00:09:00,260 If you're accelerating, you simply 175 00:09:00,260 --> 00:09:01,250 can't use special relativity. 176 00:09:01,250 --> 00:09:02,666 Special relativity is not designed 177 00:09:02,666 --> 00:09:06,770 to deal with problems that arise for non-inertial observers. 178 00:09:06,770 --> 00:09:08,540 You need general relativity, and that's 179 00:09:08,540 --> 00:09:10,850 much more difficult to use. 180 00:09:10,850 --> 00:09:13,250 But I'll let you know-- 181 00:09:13,250 --> 00:09:15,600 I'll talk about general relativity-- 182 00:09:15,600 --> 00:09:18,100 maybe at the end of the class, maybe next class. 183 00:09:18,100 --> 00:09:20,570 Well, definitely next class if I don't get to it today. 184 00:09:20,570 --> 00:09:22,700 But that's an important point. 185 00:09:22,700 --> 00:09:28,760 And it's always important to specify what you're measuring-- 186 00:09:28,760 --> 00:09:34,891 well it's always important to specify the observer that-- 187 00:09:34,891 --> 00:09:35,390 OK. 188 00:09:35,390 --> 00:09:37,655 This is a hard sentence to make. 189 00:09:37,655 --> 00:09:39,210 It's a hard sentence to make. 190 00:09:39,210 --> 00:09:41,720 So I'll just demonstrate with this. 191 00:09:41,720 --> 00:09:44,300 Speed of the train relative to you. 192 00:09:44,300 --> 00:09:45,680 Speed of me relative to you. 193 00:09:45,680 --> 00:09:47,430 It's always important to specify-- to make 194 00:09:47,430 --> 00:09:49,610 that specifier, relative to. 195 00:09:49,610 --> 00:09:51,650 Relative to me, relative to you. 196 00:09:51,650 --> 00:09:53,480 There's no single speed. 197 00:09:53,480 --> 00:09:56,000 There's no single speed. 198 00:09:56,000 --> 00:09:57,320 There's no absolute speed. 199 00:09:57,320 --> 00:09:59,510 There's no correct speed. 200 00:09:59,510 --> 00:10:01,520 There's no best speed. 201 00:10:01,520 --> 00:10:02,725 They're all equal. 202 00:10:02,725 --> 00:10:05,330 We're all created equal. 203 00:10:05,330 --> 00:10:07,760 All inertial observers are equal. 204 00:10:07,760 --> 00:10:09,260 But some are more equal than others. 205 00:10:12,680 --> 00:10:18,280 Speed of train relative to the class. 206 00:10:21,160 --> 00:10:24,931 But yeah, this is really relativity at its simplest, 207 00:10:24,931 --> 00:10:25,430 though-- 208 00:10:25,430 --> 00:10:26,780 the relativity of speed. 209 00:10:26,780 --> 00:10:28,954 Some people measure one speed, other people 210 00:10:28,954 --> 00:10:29,870 measure another speed. 211 00:10:33,050 --> 00:10:34,350 They're all equally good. 212 00:10:34,350 --> 00:10:36,820 One's not better than the other. 213 00:10:36,820 --> 00:10:37,820 OK. 214 00:10:37,820 --> 00:10:40,760 So this is the Galilean addition of velocities. 215 00:10:40,760 --> 00:10:43,844 You simply take the speed of me relative to you, 216 00:10:43,844 --> 00:10:46,010 and you take the speed of the train relative to you, 217 00:10:46,010 --> 00:10:48,900 and you add them. u plus v. Simple as that. 218 00:10:48,900 --> 00:10:50,821 But as I said, that doesn't work for light. 219 00:10:50,821 --> 00:10:51,320 Oh, yes. 220 00:10:51,320 --> 00:10:51,820 Question? 221 00:10:51,820 --> 00:10:53,240 AUDIENCE: But are you moving? 222 00:10:53,240 --> 00:10:54,200 Are you on the train? 223 00:10:54,200 --> 00:10:54,620 PROFESSOR: Oh, yeah. 224 00:10:54,620 --> 00:10:55,328 I'm on the train. 225 00:10:55,328 --> 00:10:56,176 I'm on the train. 226 00:10:56,176 --> 00:10:58,550 I'm on the train, and I'm walking at some constant speed, 227 00:10:58,550 --> 00:11:00,090 or running at some constant speed. 228 00:11:00,090 --> 00:11:01,730 I'm being chased by somebody and I'm 229 00:11:01,730 --> 00:11:04,074 running at some constant speed. 230 00:11:04,074 --> 00:11:05,990 But forget about the person that's chasing me. 231 00:11:05,990 --> 00:11:08,860 Say he's invisible and he's undetectable, 232 00:11:08,860 --> 00:11:09,610 because I'm crazy. 233 00:11:14,087 --> 00:11:19,040 The u plus v is this totally familiar formula. 234 00:11:19,040 --> 00:11:22,710 Now, as I said, it's incorrect. 235 00:11:22,710 --> 00:11:24,794 So I'll give you the correct formula, 236 00:11:24,794 --> 00:11:26,960 which is much more complicated than that but, you'll 237 00:11:26,960 --> 00:11:29,210 be able to appreciate it, because-- 238 00:11:29,210 --> 00:11:30,155 well, you'll see. 239 00:11:30,155 --> 00:11:31,280 Here's the correct formula. 240 00:11:33,639 --> 00:11:35,180 I'm just telling you this so that you 241 00:11:35,180 --> 00:11:39,860 can have some power in doing some calculations-- 242 00:11:39,860 --> 00:11:42,584 in doing some correct calculations. 243 00:11:48,400 --> 00:11:50,390 Relativistic addition of velocities. 244 00:11:54,300 --> 00:11:58,660 Or the Einsteinian addition of velocities. 245 00:12:04,540 --> 00:12:06,595 I'll use the same variables that I used up there, 246 00:12:06,595 --> 00:12:11,110 u and v. And now the speed of me relative to the class-- 247 00:12:23,220 --> 00:12:24,840 now the speed of me relative the class 248 00:12:24,840 --> 00:12:38,910 is actually u plus v divided by 1 plus uv over c squared, 249 00:12:38,910 --> 00:12:41,450 where c is the speed of light. 250 00:12:46,620 --> 00:12:48,770 And if you'd like NKS-- 251 00:12:48,770 --> 00:12:53,870 if you like SI units, using meters, kilograms, and seconds, 252 00:12:53,870 --> 00:13:05,090 then that's actually 299,792,458 meters per second-- 253 00:13:05,090 --> 00:13:07,730 exactly, actually. 254 00:13:07,730 --> 00:13:11,997 The meter is actually defined so that the speed of light-- 255 00:13:11,997 --> 00:13:12,830 I won't get to this. 256 00:13:12,830 --> 00:13:15,372 But this is actually the exact speed of light, 257 00:13:15,372 --> 00:13:17,330 and there's a little bit of interesting history 258 00:13:17,330 --> 00:13:18,620 that has to go with that. 259 00:13:18,620 --> 00:13:19,520 And I can tell you-- 260 00:13:19,520 --> 00:13:21,228 if any of you have questions after class, 261 00:13:21,228 --> 00:13:22,170 I can talk about it. 262 00:13:22,170 --> 00:13:22,670 OK. 263 00:13:22,670 --> 00:13:25,010 So c is the speed of light. 264 00:13:25,010 --> 00:13:27,490 And-- yeah. 265 00:13:27,490 --> 00:13:29,250 This is the correct formula to use. 266 00:13:33,920 --> 00:13:38,750 It's actually approximately equal to this formula up there 267 00:13:38,750 --> 00:13:40,470 for small speeds-- 268 00:13:40,470 --> 00:13:42,620 so if u and v are both much, much less 269 00:13:42,620 --> 00:13:44,820 than the speed of light. 270 00:13:44,820 --> 00:13:47,492 So if u and v are everyday speeds-- 271 00:13:47,492 --> 00:13:49,950 certainly, if I'm walking, that would be an everyday speed. 272 00:13:49,950 --> 00:13:53,050 But even if I'm running or even if I'm in a car, 273 00:13:53,050 --> 00:13:56,194 even if I'm on an airplane, even if I'm on a rocket, 274 00:13:56,194 --> 00:13:58,610 all those speeds are so much less than the speed of light. 275 00:13:58,610 --> 00:14:02,210 The speed of light is immensely huge. 276 00:14:02,210 --> 00:14:04,640 It's immensely huge. 277 00:14:04,640 --> 00:14:09,380 And so pretty much all speeds that we can think of, 278 00:14:09,380 --> 00:14:12,310 all speeds that we can get to-- 279 00:14:12,310 --> 00:14:15,855 at least travel ourselves-- all those speeds are much, 280 00:14:15,855 --> 00:14:16,730 much less than light. 281 00:14:16,730 --> 00:14:22,961 So you can use that formula with relative confidence. 282 00:14:22,961 --> 00:14:23,460 Yes? 283 00:14:23,460 --> 00:14:26,956 AUDIENCE: Do you know how fast a speed man has ever traveled? 284 00:14:26,956 --> 00:14:29,080 PROFESSOR: The fastest speed man has ever traveled? 285 00:14:29,080 --> 00:14:29,780 AUDIENCE: Yeah. 286 00:14:29,780 --> 00:14:30,110 PROFESSOR: Oh. 287 00:14:30,110 --> 00:14:32,151 AUDIENCE: Whether it be in an airplane or rocket. 288 00:14:32,151 --> 00:14:34,910 PROFESSOR: Right, right. 289 00:14:34,910 --> 00:14:35,630 I was going to-- 290 00:14:35,630 --> 00:14:38,204 I was actually going to look it up before class, 291 00:14:38,204 --> 00:14:40,370 because I was going to do a simple calculation later 292 00:14:40,370 --> 00:14:41,240 on in the class. 293 00:14:41,240 --> 00:14:45,330 But it's definitely-- definitely less than 10% 294 00:14:45,330 --> 00:14:46,430 of the speed of light. 295 00:14:46,430 --> 00:14:48,388 I'm pretty sure-- yeah, definitely less than 1% 296 00:14:48,388 --> 00:14:50,660 of the speed of light. 297 00:14:50,660 --> 00:14:56,520 Probably something like-- maybe a 1/10,000 the speed of light, 298 00:14:56,520 --> 00:14:57,770 or 1/1,000 the speed of light. 299 00:14:57,770 --> 00:15:00,770 Does anybody actually know what it is? 300 00:15:00,770 --> 00:15:02,730 Well, I'll look it up, and I'll send you 301 00:15:02,730 --> 00:15:04,730 all email about that, because I was 302 00:15:04,730 --> 00:15:08,699 going to do a calculation later on where I do use that speed. 303 00:15:08,699 --> 00:15:10,240 But I'm not going to that calculation 304 00:15:10,240 --> 00:15:14,250 now, because I forgot to look it up. 305 00:15:14,250 --> 00:15:15,929 But we can appreciate it. 306 00:15:15,929 --> 00:15:17,720 But it's much, much-- yeah, much, much less 307 00:15:17,720 --> 00:15:18,720 than the speed of light. 308 00:15:18,720 --> 00:15:21,770 But we haven't actually-- 309 00:15:21,770 --> 00:15:24,140 we haven't accelerated ourselves to that fast. 310 00:15:24,140 --> 00:15:29,930 But we've accelerated particles very, very fast-- 311 00:15:29,930 --> 00:15:34,280 something like 0.99999 times the speed of light. 312 00:15:34,280 --> 00:15:37,160 99.999 times speed of light we've actually accelerated 313 00:15:37,160 --> 00:15:39,012 particles to. 314 00:15:39,012 --> 00:15:40,970 And it's much easier to do that with particles, 315 00:15:40,970 --> 00:15:43,010 because they're much lighter than we are, 316 00:15:43,010 --> 00:15:45,130 so it doesn't require as much energy. 317 00:15:45,130 --> 00:15:46,088 Do you have a question? 318 00:15:46,088 --> 00:15:49,370 AUDIENCE: Do we know why that equation is correct? 319 00:15:49,370 --> 00:15:50,450 PROFESSOR: Oh, yeah. 320 00:15:50,450 --> 00:15:51,280 Oh, yeah, yeah. 321 00:15:51,280 --> 00:15:52,408 This equation-- 322 00:15:52,408 --> 00:15:54,117 AUDIENCE: Can you repeat question? 323 00:15:54,117 --> 00:15:55,700 PROFESSOR: The question is, do we know 324 00:15:55,700 --> 00:15:57,620 why this equation is correct? 325 00:16:00,345 --> 00:16:01,220 Yeah, we do know why. 326 00:16:01,220 --> 00:16:03,350 You can actually derive this equation 327 00:16:03,350 --> 00:16:06,710 with the principles-- with the two postulates of relativity, 328 00:16:06,710 --> 00:16:10,450 along with the definition of velocity. 329 00:16:10,450 --> 00:16:14,540 But I don't plan to go through mathematical derivations 330 00:16:14,540 --> 00:16:17,005 in this class, because I told you from the beginning-- 331 00:16:17,005 --> 00:16:18,380 I promised you from the beginning 332 00:16:18,380 --> 00:16:20,213 that there wasn't going to be a lot of math. 333 00:16:22,511 --> 00:16:24,135 It's pretty simple to derive it though. 334 00:16:24,135 --> 00:16:26,450 And I can show you how to do it, if you're interested. 335 00:16:26,450 --> 00:16:29,180 If anyone else is interested, I can show you how to derive it. 336 00:16:29,180 --> 00:16:35,630 But it certainly agrees with experiment. 337 00:16:35,630 --> 00:16:39,500 I could say that somebody noticed it happens 338 00:16:39,500 --> 00:16:40,770 to agree with experiment. 339 00:16:40,770 --> 00:16:42,320 But no, you can actually derive it. 340 00:16:42,320 --> 00:16:44,319 You can actually prove it, prove that it's true. 341 00:16:47,560 --> 00:16:48,060 Dot. 342 00:16:48,060 --> 00:16:48,926 Relative. 343 00:16:53,720 --> 00:16:57,690 I basically put that there for you to enjoy. 344 00:16:57,690 --> 00:16:59,880 Put in some numbers, have fun with it. 345 00:17:02,680 --> 00:17:04,740 Something that's fun to do with this equation 346 00:17:04,740 --> 00:17:06,210 is to use it wrong. 347 00:17:06,210 --> 00:17:07,710 And that's probably not the best way 348 00:17:07,710 --> 00:17:10,810 to show how to use an equation, using it wrong, 349 00:17:10,810 --> 00:17:13,278 but there's something funny-- there's something 350 00:17:13,278 --> 00:17:14,819 that I think is pretty funny that you 351 00:17:14,819 --> 00:17:16,200 can do with this equation. 352 00:17:16,200 --> 00:17:18,390 You can misinterpret it. 353 00:17:18,390 --> 00:17:19,440 This is probably a-- 354 00:17:19,440 --> 00:17:23,880 well, you can misinterpret this equation, but it's-- 355 00:17:23,880 --> 00:17:25,980 you'll see why it's wrong. 356 00:17:25,980 --> 00:17:27,880 I just want to show you something 357 00:17:27,880 --> 00:17:31,080 that I notice which is kind of funny about the equation. 358 00:17:31,080 --> 00:17:34,940 It allows you to prove, for example, that 1 plus 1 359 00:17:34,940 --> 00:17:36,220 isn't equal to 2. 360 00:17:36,220 --> 00:17:37,470 It's approximately equal to 2. 361 00:17:37,470 --> 00:17:39,480 It's actually equal to 1.99. 362 00:17:39,480 --> 00:17:48,120 And I'll show you how you can get at that conclusion. 363 00:17:48,120 --> 00:17:49,530 Let's say I'm walking-- 364 00:17:49,530 --> 00:17:50,490 well I'll show you-- 365 00:17:50,490 --> 00:17:51,120 OK. 366 00:17:51,120 --> 00:17:54,970 Let's say I'm walking on this train, which is moving-- 367 00:17:54,970 --> 00:17:59,670 let's say the train is moving at 1 miles per hour for you. 368 00:17:59,670 --> 00:18:02,760 And I'm walking on the train 1 miles per hour 369 00:18:02,760 --> 00:18:05,190 relative to the train. 370 00:18:05,190 --> 00:18:09,019 So I can now ask you, how fast am I moving relative to you? 371 00:18:09,019 --> 00:18:10,560 I told you this is the wrong formula, 372 00:18:10,560 --> 00:18:12,900 but it worked pretty well for small speeds. 373 00:18:12,900 --> 00:18:16,290 But this is the correct formula, which works for all speeds. 374 00:18:16,290 --> 00:18:17,920 And so I like to use this formula 375 00:18:17,920 --> 00:18:19,530 in getting at the speed. 376 00:18:19,530 --> 00:18:20,160 Yes? 377 00:18:20,160 --> 00:18:23,500 AUDIENCE: If you do it with the same speed 378 00:18:23,500 --> 00:18:26,126 and you put it into both equations, will-- 379 00:18:26,126 --> 00:18:29,220 if it's a small speed, would it come out the same? 380 00:18:29,220 --> 00:18:30,290 PROFESSOR: The same as-- 381 00:18:30,290 --> 00:18:30,873 the question-- 382 00:18:30,873 --> 00:18:32,650 AUDIENCE: A small speed like 40 miles 383 00:18:32,650 --> 00:18:34,470 per hour the train is moving. 384 00:18:34,470 --> 00:18:35,874 [INAUDIBLE] be the same? 385 00:18:35,874 --> 00:18:37,290 PROFESSOR: The question is, if you 386 00:18:37,290 --> 00:18:39,240 put in small speeds for this equation-- 387 00:18:39,240 --> 00:18:41,340 if you let u and v both be really small, 388 00:18:41,340 --> 00:18:43,745 then is the result the same as this result? 389 00:18:43,745 --> 00:18:44,370 AUDIENCE: Yeah. 390 00:18:44,370 --> 00:18:44,780 PROFESSOR: Yeah. 391 00:18:44,780 --> 00:18:46,100 It's approximately the same. 392 00:18:46,100 --> 00:18:46,600 Yeah. 393 00:18:46,600 --> 00:18:48,390 It's very, very close. 394 00:18:48,390 --> 00:18:53,250 And you can prove-- for those of you who are more mathematically 395 00:18:53,250 --> 00:18:56,400 minded, you can show that if you make the approximation that u 396 00:18:56,400 --> 00:18:59,305 and v are much less than c, much less than the speed of light, 397 00:18:59,305 --> 00:19:01,680 then you can show that is approximately the same as that, 398 00:19:01,680 --> 00:19:05,190 because u times v is both then much 399 00:19:05,190 --> 00:19:07,800 less than the speed of light. 400 00:19:07,800 --> 00:19:10,290 A small number over a really big number is approximately 0, 401 00:19:10,290 --> 00:19:12,650 and then 1 plus approximately 0 is 1. 402 00:19:12,650 --> 00:19:14,750 And so then u plus v of approximately 1 403 00:19:14,750 --> 00:19:16,950 is approximately u plus v. Yes? 404 00:19:16,950 --> 00:19:18,845 AUDIENCE: Why is light involved? 405 00:19:18,845 --> 00:19:20,220 PROFESSOR: Why is light involved? 406 00:19:20,220 --> 00:19:22,970 The question is, why is light involved? 407 00:19:22,970 --> 00:19:26,190 Well, I told you there are two postulates 408 00:19:26,190 --> 00:19:27,216 to special relativity. 409 00:19:27,216 --> 00:19:28,590 So you might suspect that light's 410 00:19:28,590 --> 00:19:32,070 involved in all these equations, because it's 411 00:19:32,070 --> 00:19:33,540 right there in the postulates. 412 00:19:37,380 --> 00:19:40,110 I can't tell you precisely why it's involved, because I 413 00:19:40,110 --> 00:19:42,080 would have to get mathematic-- 414 00:19:42,080 --> 00:19:43,680 I would have to get more mathematical. 415 00:19:47,301 --> 00:19:51,650 But I can give a plausibility argument for why it's involved, 416 00:19:51,650 --> 00:19:52,844 for why this isn't-- 417 00:19:52,844 --> 00:19:54,260 I can give a plausibility argument 418 00:19:54,260 --> 00:19:57,450 for why this equation is not totally wrong, because-- 419 00:19:57,450 --> 00:19:59,290 it's actually because of what I said before. 420 00:19:59,290 --> 00:20:00,831 If u and v are really small, then you 421 00:20:00,831 --> 00:20:02,490 expect this to be true, because we 422 00:20:02,490 --> 00:20:04,590 find this to be true in our everyday life 423 00:20:04,590 --> 00:20:07,200 to an incredible accuracy. 424 00:20:07,200 --> 00:20:09,900 And so since the speed of light is 425 00:20:09,900 --> 00:20:13,290 much faster than everyday speeds, 426 00:20:13,290 --> 00:20:15,180 this is a reasonable formula, at least. 427 00:20:15,180 --> 00:20:17,085 It's a reasonable formula. 428 00:20:17,085 --> 00:20:17,820 OK. 429 00:20:17,820 --> 00:20:21,392 Now I want to do that calculation. 430 00:20:21,392 --> 00:20:23,600 The train is moving 1 miles per hour relative to you, 431 00:20:23,600 --> 00:20:25,849 and I'm moving 1 miles per hour relative to the train, 432 00:20:25,849 --> 00:20:27,735 so how fast am I moving relative to you? 433 00:20:27,735 --> 00:20:29,700 AUDIENCE: Two. 434 00:20:29,700 --> 00:20:32,040 PROFESSOR: Approximately two, because I'm 435 00:20:32,040 --> 00:20:35,490 moving much less than the speed of light, but not exactly 2. 436 00:20:35,490 --> 00:20:38,241 And if any of you have a calculator, 437 00:20:38,241 --> 00:20:41,200 you can go with me as I do this. 438 00:20:41,200 --> 00:20:48,420 So u equals-- actually, if you're actually 439 00:20:48,420 --> 00:20:51,060 going to calculate it, then we have to use consistent units. 440 00:20:51,060 --> 00:20:53,307 And I was talking about miles per hour, 441 00:20:53,307 --> 00:20:55,140 and I don't remember what the speed of light 442 00:20:55,140 --> 00:20:56,350 is in miles per hour. 443 00:20:56,350 --> 00:20:57,330 So let's say that I'm just moving 444 00:20:57,330 --> 00:20:58,705 at 1 miles-- let's say I'm moving 445 00:20:58,705 --> 00:21:00,885 at 1 meters per second relative to the train, 446 00:21:00,885 --> 00:21:03,450 and the train is moving 1 meters per second relative to you. 447 00:21:03,450 --> 00:21:06,900 So let's say u equals 1 meters per second, 448 00:21:06,900 --> 00:21:10,400 and v equals 1 meters per second. 449 00:21:10,400 --> 00:21:18,150 Then u plus v is 1 meters per second 450 00:21:18,150 --> 00:21:21,761 plus 1 meters per second. 451 00:21:21,761 --> 00:21:27,590 And then the denominator is 1 plus 1 meters per second times 452 00:21:27,590 --> 00:21:33,930 1 meters per second over approximately 300 million 453 00:21:33,930 --> 00:21:38,197 meters per second squared. 454 00:21:38,197 --> 00:21:39,780 And those of you who have a calculator 455 00:21:39,780 --> 00:21:41,670 can plug in these numbers, and you'll 456 00:21:41,670 --> 00:21:44,870 find it's something like-- 457 00:21:44,870 --> 00:21:46,680 if you're actually doing it, then you 458 00:21:46,680 --> 00:21:48,138 can tell me in a second what it is. 459 00:21:48,138 --> 00:21:53,322 But it's something like 1.99999-- 460 00:21:53,322 --> 00:21:55,980 give me another nine-- 461 00:21:55,980 --> 00:21:57,580 meters per second. 462 00:21:57,580 --> 00:22:02,230 So we've just shown-- 463 00:22:02,230 --> 00:22:05,810 wink-- you've just shown that 1 meters per second 464 00:22:05,810 --> 00:22:16,000 plus 1 meters per second equals 1.99999 meters per second. 465 00:22:16,000 --> 00:22:21,630 So if you divide by meters per second, then you get 1 plus 1 466 00:22:21,630 --> 00:22:27,295 equals 1.99999. 467 00:22:27,295 --> 00:22:28,670 So it's approximately equal to 2. 468 00:22:34,270 --> 00:22:35,790 Yes? 469 00:22:35,790 --> 00:22:38,600 AUDIENCE: Isn't technically an infinite nine. 470 00:22:38,600 --> 00:22:39,767 The same as-- 471 00:22:39,767 --> 00:22:40,350 PROFESSOR: OK. 472 00:22:40,350 --> 00:22:42,582 It's not an infinite nine. 473 00:22:42,582 --> 00:22:44,290 I don't know if anyone is calculating it, 474 00:22:44,290 --> 00:22:46,342 but actually it's 999-- 475 00:22:46,342 --> 00:22:48,550 there was something like six or seven nines something 476 00:22:48,550 --> 00:22:52,746 like that, and then you get a four and a three. 477 00:22:52,746 --> 00:22:53,620 How many nines is it? 478 00:22:57,204 --> 00:22:58,120 AUDIENCE: Eight nines. 479 00:22:58,120 --> 00:22:58,710 PROFESSOR: Eight nines. 480 00:22:58,710 --> 00:22:59,490 OK. 481 00:22:59,490 --> 00:23:01,180 I'll fix it. 482 00:23:01,180 --> 00:23:05,680 One, two, three, four, five, six, seven, eight. 483 00:23:05,680 --> 00:23:08,280 AUDIENCE: But how do you get a three [INAUDIBLE] 484 00:23:08,280 --> 00:23:10,040 PROFESSOR: Oh, how do you get a three? 485 00:23:10,040 --> 00:23:11,220 It's just arithmetic. 486 00:23:11,220 --> 00:23:12,678 The calculator is doing it for you. 487 00:23:15,640 --> 00:23:18,330 I'll fix this. 488 00:23:18,330 --> 00:23:28,530 1.999999993-- sorry, 3. 489 00:23:28,530 --> 00:23:32,650 Therefore 1 plus 1 is approximately 2, 490 00:23:32,650 --> 00:23:34,438 and that's wonderful. 491 00:23:34,438 --> 00:23:36,610 AUDIENCE: Yay. 492 00:23:36,610 --> 00:23:38,750 PROFESSOR: Of course-- of course, 493 00:23:38,750 --> 00:23:40,240 I haven't really proven anything, 494 00:23:40,240 --> 00:23:43,900 because I'm not really saying 1 meters per second 495 00:23:43,900 --> 00:23:46,690 plus 1 meters per second. 496 00:23:46,690 --> 00:23:49,610 That's the interpretation of what I'm doing, kind of, 497 00:23:49,610 --> 00:23:51,610 but it's not precisely the interpretation of it. 498 00:23:54,179 --> 00:23:55,720 I'm actually not dealing with numbers 499 00:23:55,720 --> 00:23:58,270 when I say the speed of-- 500 00:23:58,270 --> 00:24:00,670 well, I'm not dealing purely with numbers 501 00:24:00,670 --> 00:24:04,160 when I say the speed of me relative to you. 502 00:24:04,160 --> 00:24:11,030 And that's why this is wrong. 503 00:24:11,030 --> 00:24:12,560 Of course, it is true that 1 plus 1 504 00:24:12,560 --> 00:24:15,760 is approximately equal to 2, because 1 plus 1 is equal to 2, 505 00:24:15,760 --> 00:24:16,850 and 2 is approximately 2. 506 00:24:19,999 --> 00:24:22,040 So you get the right answer for the wrong reason. 507 00:24:22,040 --> 00:24:23,081 I love when that happens. 508 00:24:23,081 --> 00:24:25,770 I love when that happens. 509 00:24:25,770 --> 00:24:28,070 OK. 510 00:24:28,070 --> 00:24:32,990 No, let's-- I'll leave this here. 511 00:24:32,990 --> 00:24:33,490 OK. 512 00:24:38,450 --> 00:24:41,990 This demonstrates that speed is relative, 513 00:24:41,990 --> 00:24:45,050 and you know for every day experiment-- 514 00:24:45,050 --> 00:24:46,064 everyday experiments. 515 00:24:46,064 --> 00:24:47,480 Maybe you do experiments everyday. 516 00:24:47,480 --> 00:24:49,370 But you know from everyday experience 517 00:24:49,370 --> 00:24:52,190 that speed is relative. 518 00:24:52,190 --> 00:24:54,854 Everybody notices this, that speed is relative. 519 00:24:54,854 --> 00:24:56,270 But there are some things that you 520 00:24:56,270 --> 00:24:58,880 wouldn't expect to be relative. 521 00:24:58,880 --> 00:25:01,550 One of the most striking relative things 522 00:25:01,550 --> 00:25:06,260 is simultaneity, and by that I mean that-- 523 00:25:06,260 --> 00:25:11,780 by that I mean that if I observe two events to be simultaneous-- 524 00:25:11,780 --> 00:25:14,982 two events occurring at the same time-- 525 00:25:14,982 --> 00:25:17,780 if I measure them as being simultaneous events, 526 00:25:17,780 --> 00:25:19,340 then that doesn't necessarily mean 527 00:25:19,340 --> 00:25:25,490 that they'll be simultaneous for you, and that's unexpected. 528 00:25:25,490 --> 00:25:32,290 And I can show you very simply why simultaneity is relative. 529 00:25:32,290 --> 00:25:35,560 Again, let's say that I'm on a train, because I like trains. 530 00:25:35,560 --> 00:25:36,560 Say I'm on a train. 531 00:25:40,790 --> 00:25:42,432 Let's say I'm on a train. 532 00:25:42,432 --> 00:25:43,640 I'm sitting inside the train. 533 00:25:43,640 --> 00:25:46,940 I'm sitting in the center of the train. 534 00:25:46,940 --> 00:25:53,100 And I turn on a flashlight. 535 00:25:53,100 --> 00:25:55,184 This is supposed to be a train. 536 00:25:55,184 --> 00:25:58,136 [LAUGHTER] 537 00:26:02,300 --> 00:26:03,260 Yeah. 538 00:26:03,260 --> 00:26:05,960 I say train because in Einstein's time, 539 00:26:05,960 --> 00:26:11,380 that was pretty much the fastest vehicle they had, I guess. 540 00:26:11,380 --> 00:26:14,900 All of his experiments-- well, all of his thought experiments, 541 00:26:14,900 --> 00:26:16,810 because he did thought experiments-- all 542 00:26:16,810 --> 00:26:20,120 of his thought experiments involve trains. 543 00:26:20,120 --> 00:26:23,918 And so in his tradition, I will continue to talk about trains. 544 00:26:26,424 --> 00:26:28,340 Let's say I'm sitting in the middle of a train 545 00:26:28,340 --> 00:26:30,410 and I turn on a flashlight. 546 00:26:30,410 --> 00:26:34,850 And the light rays go in all directions. 547 00:26:34,850 --> 00:26:36,507 They go in all directions. 548 00:26:36,507 --> 00:26:38,590 And eventually they'll hit the walls of the train. 549 00:26:38,590 --> 00:26:42,415 They'll hit-- they'll hit this point. 550 00:26:42,415 --> 00:26:45,040 One ray of light we'll hit this point, and another ray of light 551 00:26:45,040 --> 00:26:48,440 will hit this point. 552 00:26:48,440 --> 00:26:50,750 And if I'm sitting in the center of the train, 553 00:26:50,750 --> 00:26:55,490 then since both of these points are equidistant from me, 554 00:26:55,490 --> 00:26:57,576 I would measure them as-- 555 00:26:57,576 --> 00:26:59,450 I would measure the light hitting both points 556 00:26:59,450 --> 00:27:00,630 at the same time. 557 00:27:00,630 --> 00:27:05,660 So I would observe the two events to be simultaneous. 558 00:27:05,660 --> 00:27:07,310 Let me be-- 559 00:27:07,310 --> 00:27:09,506 I keep talking about events. 560 00:27:09,506 --> 00:27:12,200 By event I just mean something that happens. 561 00:27:12,200 --> 00:27:16,631 Lightning strikes here, I punch a guy here. 562 00:27:16,631 --> 00:27:18,380 I don't know why that was the second thing 563 00:27:18,380 --> 00:27:19,171 to come to my mind. 564 00:27:21,560 --> 00:27:23,060 Events are just things that happen-- 565 00:27:23,060 --> 00:27:26,630 things that happen at a specific place, a specific location, 566 00:27:26,630 --> 00:27:28,290 and a specific time. 567 00:27:28,290 --> 00:27:30,460 So you actually need four numbers. 568 00:27:30,460 --> 00:27:32,520 You need four numbers to specify an event. 569 00:27:32,520 --> 00:27:36,860 You need the location, which is three numbers, because there 570 00:27:36,860 --> 00:27:39,890 are three dimensions of space-- 571 00:27:39,890 --> 00:27:42,380 this direction, this direction, and this direction. 572 00:27:42,380 --> 00:27:44,610 And you measure the time of the events. 573 00:27:44,610 --> 00:27:46,950 So events are specified by four numbers. 574 00:27:46,950 --> 00:27:49,220 And it's for this reason that light-- 575 00:27:49,220 --> 00:27:49,740 not light. 576 00:27:49,740 --> 00:27:52,490 It's for this reason that time is often considered 577 00:27:52,490 --> 00:27:54,650 as the fourth dimension, because you simply 578 00:27:54,650 --> 00:27:57,390 need it to specify events. 579 00:27:57,390 --> 00:27:59,880 And there are other reasons too, much deeper reasons, 580 00:27:59,880 --> 00:28:02,366 but I don't know if I'll get to those. 581 00:28:02,366 --> 00:28:03,740 Anyway, getting back getting back 582 00:28:03,740 --> 00:28:07,114 to the train, which I love. 583 00:28:07,114 --> 00:28:08,780 I'm sitting in the center, and I observe 584 00:28:08,780 --> 00:28:10,820 these two events to be simultaneous, 585 00:28:10,820 --> 00:28:14,360 and it's as simple as that. 586 00:28:14,360 --> 00:28:19,550 Let's say now that I ask you, which 587 00:28:19,550 --> 00:28:21,595 do you observe to occur first? 588 00:28:21,595 --> 00:28:25,850 Do you observe the light hitting at this edge to occur first, 589 00:28:25,850 --> 00:28:29,181 or the light hitting this edge to occur first? 590 00:28:29,181 --> 00:28:30,680 But before you answer that, you have 591 00:28:30,680 --> 00:28:32,388 to know which way the train is traveling. 592 00:28:32,388 --> 00:28:37,931 So I'll tell you that the train is traveling in this direction. 593 00:28:37,931 --> 00:28:38,430 Yeah. 594 00:28:38,430 --> 00:28:41,120 The train is traveling in this direction. 595 00:28:41,120 --> 00:28:45,200 So who thinks that event are still 596 00:28:45,200 --> 00:28:49,160 simultaneous for you guys? 597 00:28:49,160 --> 00:28:52,422 One, two, three, four. 598 00:28:52,422 --> 00:28:54,170 Anybody else? 599 00:28:54,170 --> 00:28:54,670 Five. 600 00:28:54,670 --> 00:28:58,254 OK, some of you think-- six, seven. 601 00:28:58,254 --> 00:28:59,150 OK. 602 00:28:59,150 --> 00:28:59,930 Seven. 603 00:28:59,930 --> 00:29:00,430 OK. 604 00:29:00,430 --> 00:29:02,204 Who thinks-- I guess the rest of you 605 00:29:02,204 --> 00:29:03,370 think it's not simultaneous. 606 00:29:03,370 --> 00:29:04,780 I'll assume that. 607 00:29:04,780 --> 00:29:08,200 Who thinks that-- well, if they're not simultaneous, then 608 00:29:08,200 --> 00:29:09,910 obviously one occurs before the other. 609 00:29:09,910 --> 00:29:11,750 One event occurs before the other. 610 00:29:11,750 --> 00:29:14,440 Now, who thinks that in the event of light 611 00:29:14,440 --> 00:29:17,770 hitting this wall occurs before the event of light 612 00:29:17,770 --> 00:29:18,680 hitting this wall? 613 00:29:22,120 --> 00:29:23,750 Several of you. 614 00:29:23,750 --> 00:29:29,290 And I guess the rest of you think it hits this wall first. 615 00:29:29,290 --> 00:29:29,790 Well-- 616 00:29:29,790 --> 00:29:31,542 AUDIENCE: [INAUDIBLE] 617 00:29:31,542 --> 00:29:32,500 PROFESSOR: What's that? 618 00:29:32,500 --> 00:29:34,210 AUDIENCE: Where are we relative to? 619 00:29:34,210 --> 00:29:36,130 PROFESSOR: Oh, you're just sitting here. 620 00:29:36,130 --> 00:29:36,790 Where are you relative to? 621 00:29:36,790 --> 00:29:37,290 The train. 622 00:29:37,290 --> 00:29:38,540 You're just sitting here. 623 00:29:38,540 --> 00:29:42,820 And I'm sit-- suppose I'm sitting on a train, 624 00:29:42,820 --> 00:29:47,164 and I hold out a flashlight, and it's going in all directions. 625 00:29:47,164 --> 00:29:48,330 The train is going this why. 626 00:29:48,330 --> 00:29:49,630 The train is going this way. 627 00:29:49,630 --> 00:29:51,762 I'm sitting down in the middle of it, 628 00:29:51,762 --> 00:29:53,470 and I turn on a flashlight, and the light 629 00:29:53,470 --> 00:29:55,332 goes in all directions. 630 00:29:55,332 --> 00:29:57,647 AUDIENCE: Yeah, but what way are you pointing? 631 00:29:57,647 --> 00:29:58,730 PROFESSOR: Oh, which way-- 632 00:29:58,730 --> 00:29:59,230 OK. 633 00:29:59,230 --> 00:30:01,570 I'm-- OK. 634 00:30:01,570 --> 00:30:03,730 Maybe a candle would be better than a flashlight. 635 00:30:03,730 --> 00:30:06,150 I went to the light to go in all directions. 636 00:30:06,150 --> 00:30:08,330 The light to go in all directions. 637 00:30:08,330 --> 00:30:11,170 So the light will hit-- 638 00:30:11,170 --> 00:30:13,140 the light will hit the top walls too, 639 00:30:13,140 --> 00:30:14,890 and it'll hit a lot of walls, but I'm just 640 00:30:14,890 --> 00:30:18,720 focusing on the events of light hitting these two walls. 641 00:30:18,720 --> 00:30:20,590 OK? 642 00:30:20,590 --> 00:30:24,750 Now, the answer is that the events 643 00:30:24,750 --> 00:30:27,430 aren't simultaneous to you, even though they 644 00:30:27,430 --> 00:30:28,420 are simultaneous to me. 645 00:30:28,420 --> 00:30:30,419 And I kind of hinted at that, because I told you 646 00:30:30,419 --> 00:30:33,119 from the outset that simultaneity is relative, 647 00:30:33,119 --> 00:30:34,660 which is a wonderful phrase, I think. 648 00:30:34,660 --> 00:30:36,610 Simultaneity is relative. 649 00:30:36,610 --> 00:30:38,320 I like saying it. 650 00:30:38,320 --> 00:30:39,115 OK. 651 00:30:39,115 --> 00:30:41,240 Now, how could we determine which one occurs first, 652 00:30:41,240 --> 00:30:42,840 which event occurs first? 653 00:30:42,840 --> 00:30:46,150 Well, we've got these two postulates here. 654 00:30:46,150 --> 00:30:48,390 The first one doesn't really seem to be useful. 655 00:30:48,390 --> 00:30:50,764 OK, yeah, the laws of physics are the same for everybody. 656 00:30:50,764 --> 00:30:51,470 cool. 657 00:30:51,470 --> 00:30:54,440 The speed of light is the same for everybody. 658 00:30:54,440 --> 00:30:57,190 Well, that's actually the useful one, 659 00:30:57,190 --> 00:30:59,486 because you'll observe the same speed of light 660 00:30:59,486 --> 00:31:01,360 that I'll observe while sitting on the train. 661 00:31:03,940 --> 00:31:08,630 Light has to travel from here to here and from here to here. 662 00:31:08,630 --> 00:31:13,480 And if the train is moving, then actually, the distance 663 00:31:13,480 --> 00:31:17,850 the light has to travel when going from here to here 664 00:31:17,850 --> 00:31:19,510 is less than the distance the light 665 00:31:19,510 --> 00:31:21,760 has to travel when going from here to here, 666 00:31:21,760 --> 00:31:25,770 because the train is coming this way 667 00:31:25,770 --> 00:31:27,310 and so the wall gets closer. 668 00:31:27,310 --> 00:31:30,010 The wall gets closer to the light coming to it, 669 00:31:30,010 --> 00:31:32,840 so the light actually travels a shorter distance 670 00:31:32,840 --> 00:31:37,690 than it has to travel when going to this wall. 671 00:31:37,690 --> 00:31:41,620 As a result, since the speed of light is just a number-- 672 00:31:41,620 --> 00:31:45,304 since the speed of light is a constant, 673 00:31:45,304 --> 00:31:46,720 you can get the time that it takes 674 00:31:46,720 --> 00:31:50,890 to get from the center of the train to the walls 675 00:31:50,890 --> 00:31:53,890 just by using the simple formula distance equals 676 00:31:53,890 --> 00:31:54,570 rate times time. 677 00:31:54,570 --> 00:31:57,520 You can say distance divided by speed is time. 678 00:31:57,520 --> 00:32:01,210 And you get that the time interval 679 00:32:01,210 --> 00:32:04,300 between when I turn on the light and when 680 00:32:04,300 --> 00:32:06,610 it hits here, the time interval that 681 00:32:06,610 --> 00:32:10,180 is less than the time interval between the two 682 00:32:10,180 --> 00:32:12,730 events of my turning on the light and the light 683 00:32:12,730 --> 00:32:14,370 reaching this wall. 684 00:32:14,370 --> 00:32:17,860 And so you all will observe the events 685 00:32:17,860 --> 00:32:19,300 to occur at a different order. 686 00:32:21,980 --> 00:32:25,840 That's the simple proof that simultaneity is relative. 687 00:32:25,840 --> 00:32:29,710 And it's really interesting to think about, 688 00:32:29,710 --> 00:32:35,650 because we often think of things happening at the same time. 689 00:32:35,650 --> 00:32:36,520 We often think of-- 690 00:32:40,672 --> 00:32:42,700 you're on the phone talking to somebody, 691 00:32:42,700 --> 00:32:47,410 and you both see something happen, say, outside, 692 00:32:47,410 --> 00:32:48,760 like there's lightning. 693 00:32:48,760 --> 00:32:50,300 And you'd expect them-- 694 00:32:50,300 --> 00:32:53,320 well, if someone says, hey, I just saw lightning, 695 00:32:53,320 --> 00:32:55,600 and you say, yeah, I saw it too, you'd 696 00:32:55,600 --> 00:32:58,480 expect to think that they happened at the same time. 697 00:32:58,480 --> 00:33:05,530 And, well, just as before, for everyday phenomena 698 00:33:05,530 --> 00:33:11,560 it is approximately true that simultaneity is not relative. 699 00:33:11,560 --> 00:33:13,840 For everyday speeds-- if the train was traveling 700 00:33:13,840 --> 00:33:18,910 at a speed much less than light, then the two events 701 00:33:18,910 --> 00:33:21,730 would be approximately simultaneous. 702 00:33:21,730 --> 00:33:23,710 So the relativity of simultaneity 703 00:33:23,710 --> 00:33:27,820 is only apparent at high speeds, again. 704 00:33:27,820 --> 00:33:31,955 Special relativity reduces, once again, 705 00:33:31,955 --> 00:33:35,950 to simple Newtonian mechanics for low speeds. 706 00:33:35,950 --> 00:33:39,250 And for that reason, Newton wasn't completely wrong. 707 00:33:39,250 --> 00:33:40,930 He wasn't completely wrong. 708 00:33:40,930 --> 00:33:43,150 He was just incomplete. 709 00:33:43,150 --> 00:33:47,110 Newtonian mechanics just deals correctly 710 00:33:47,110 --> 00:33:50,110 with speeds that are much less than light. 711 00:33:50,110 --> 00:33:51,970 And you can actually prove that rigorously. 712 00:33:51,970 --> 00:33:58,040 You can prove that rigorously using the modified laws 713 00:33:58,040 --> 00:34:01,134 and mechanics that you get from the two postulates 714 00:34:01,134 --> 00:34:02,050 of special relativity. 715 00:34:02,050 --> 00:34:05,860 You can prove rigorously that Newtonian mechanics reduces 716 00:34:05,860 --> 00:34:09,562 as a limiting case of special relativity. 717 00:34:12,107 --> 00:34:13,940 Now we know of two things that are relative. 718 00:34:13,940 --> 00:34:16,106 We know about at least two things that are relative. 719 00:34:16,106 --> 00:34:18,199 We know simultaneity is relative and we 720 00:34:18,199 --> 00:34:22,050 know that speed is relative. 721 00:34:22,050 --> 00:34:24,206 We know that the speed of light's not relative. 722 00:34:24,206 --> 00:34:25,800 The speed of light is invariant. 723 00:34:25,800 --> 00:34:29,420 That's the word that we use, invariant. 724 00:34:29,420 --> 00:34:30,630 Not relative. 725 00:34:30,630 --> 00:34:33,654 Oh, by the way, I just remembered-- 726 00:34:33,654 --> 00:34:35,570 you're all good with the word relative by now, 727 00:34:35,570 --> 00:34:37,486 I'm sure, because I've been using a whole lot. 728 00:34:37,486 --> 00:34:40,489 But I realized I didn't really know what the word relative 729 00:34:40,489 --> 00:34:45,110 meant until I started reading some books on relativity. 730 00:34:45,110 --> 00:34:48,409 I just had never-- 731 00:34:48,409 --> 00:34:53,730 I never used it in the context of speed relative to this, 732 00:34:53,730 --> 00:34:58,999 or measurements relative to this observer. 733 00:34:58,999 --> 00:34:59,540 I don't know. 734 00:34:59,540 --> 00:35:02,870 I guess-- maybe I was so illiterate or something, 735 00:35:02,870 --> 00:35:06,060 or maybe I still am really illiterate. 736 00:35:06,060 --> 00:35:08,930 But if any of you've found yourself thinking 737 00:35:08,930 --> 00:35:11,870 that that's a strange word, well, I 738 00:35:11,870 --> 00:35:14,200 was in the same situation. 739 00:35:14,200 --> 00:35:15,565 And so I learned-- 740 00:35:15,565 --> 00:35:18,290 I learned what the word relative meant by reading 741 00:35:18,290 --> 00:35:20,008 popular books on relativity. 742 00:35:22,756 --> 00:35:24,700 I also learned with the word dumbbell meant. 743 00:35:24,700 --> 00:35:25,647 You know, a dumbbell? 744 00:35:25,647 --> 00:35:27,230 I learned what the word dumbbell meant 745 00:35:27,230 --> 00:35:30,060 by learning about p orbitals. 746 00:35:30,060 --> 00:35:31,421 [LAUGHTER] 747 00:35:31,421 --> 00:35:31,920 Yeah. 748 00:35:31,920 --> 00:35:35,990 I-- I had no idea what a dumbbell was before I 749 00:35:35,990 --> 00:35:37,260 learned about p orbitals. 750 00:35:37,260 --> 00:35:40,300 For those of you who haven't heard about p orbitals-- 751 00:35:40,300 --> 00:35:42,770 I'm just giving you more evidence to my illiteracy. 752 00:35:42,770 --> 00:35:43,270 OK? 753 00:35:55,272 --> 00:35:58,240 Now that you can see that simultaneity is relative, 754 00:35:58,240 --> 00:36:01,200 you're probably-- or hopefully you're 755 00:36:01,200 --> 00:36:03,870 wondering a bit about time. 756 00:36:03,870 --> 00:36:05,835 What is it about time that-- 757 00:36:05,835 --> 00:36:10,050 well, there might be something about time that's 758 00:36:10,050 --> 00:36:12,540 a little bit different than what we expected, 759 00:36:12,540 --> 00:36:16,488 something a little bit different from what we're used to. 760 00:36:16,488 --> 00:36:18,070 And it's true. 761 00:36:18,070 --> 00:36:23,280 Time is actually very mysterious once you 762 00:36:23,280 --> 00:36:25,620 invoke special relativity. 763 00:36:25,620 --> 00:36:28,080 And the meaning of space and time, 764 00:36:28,080 --> 00:36:30,810 once Einstein introduced his relativity, 765 00:36:30,810 --> 00:36:37,440 was completely contrary to what Newton 766 00:36:37,440 --> 00:36:43,320 had conceived of the meaning of space and time. 767 00:36:43,320 --> 00:36:45,780 According to Newton, everybody-- 768 00:36:45,780 --> 00:36:52,920 every observer measures the same time for events, 769 00:36:52,920 --> 00:36:57,030 and they measure the same coordinates of events. 770 00:36:57,030 --> 00:36:59,700 Well, they measure the same lengths of-- 771 00:36:59,700 --> 00:37:02,340 actually, it's probably best if I don't get to space yet, 772 00:37:02,340 --> 00:37:05,820 because I haven't told you of the strange things about space 773 00:37:05,820 --> 00:37:07,720 yet. 774 00:37:07,720 --> 00:37:08,550 I'll leave-- OK. 775 00:37:08,550 --> 00:37:09,450 I'll leave this in a second. 776 00:37:09,450 --> 00:37:10,350 I'll talk about time. 777 00:37:10,350 --> 00:37:11,474 I'll talk about time first. 778 00:37:11,474 --> 00:37:13,060 Remind me to get to space. 779 00:37:13,060 --> 00:37:16,200 Well, I won't forget, but-- 780 00:37:16,200 --> 00:37:18,496 let me just talk a little bit about time first. 781 00:37:21,125 --> 00:37:23,150 For the simultaneity of-- 782 00:37:23,150 --> 00:37:25,500 from the relativity of simultaneity, 783 00:37:25,500 --> 00:37:27,990 we suspect something is mysterious about time, 784 00:37:27,990 --> 00:37:29,520 and that's in fact true. 785 00:37:29,520 --> 00:37:34,280 And relativity of simultaneity in fact isn't-- 786 00:37:34,280 --> 00:37:36,140 well, it's shocking, but I don't think 787 00:37:36,140 --> 00:37:40,350 it's as shocking as the relativity of time intervals. 788 00:37:40,350 --> 00:37:41,440 Let me explain this now. 789 00:37:45,945 --> 00:37:48,750 It turns out that-- 790 00:37:48,750 --> 00:37:50,580 OK, well-- sorry. 791 00:37:50,580 --> 00:37:51,570 I want to be-- sorry. 792 00:37:51,570 --> 00:37:54,030 I want to be precise. 793 00:37:54,030 --> 00:37:55,262 They're related. 794 00:37:55,262 --> 00:37:57,720 Relativity of simultaneity and relativity of time intervals 795 00:37:57,720 --> 00:38:02,200 are related, because you measure some time interval, 796 00:38:02,200 --> 00:38:04,040 and I measure a different time interval. 797 00:38:04,040 --> 00:38:08,870 They couldn't be the same if you measured something different. 798 00:38:08,870 --> 00:38:10,830 But now I'd like to just be a bit more 799 00:38:10,830 --> 00:38:15,180 precise about this relativity of time intervals. 800 00:38:15,180 --> 00:38:16,555 OK. 801 00:38:16,555 --> 00:38:20,620 Let's say I'm on a train again, because trains 802 00:38:20,620 --> 00:38:23,070 are so much fun to go on. 803 00:38:30,260 --> 00:38:33,850 Let's say measure the time interval between two events. 804 00:38:33,850 --> 00:38:40,622 Let's say I take out a watch, and I write down the time 805 00:38:40,622 --> 00:38:41,830 of the first event happening. 806 00:38:41,830 --> 00:38:44,500 And then I write down the time of the second event happening, 807 00:38:44,500 --> 00:38:47,240 and then I just subtract the two times to get the time interval. 808 00:38:47,240 --> 00:38:49,540 So I measure the time interval between two events 809 00:38:49,540 --> 00:38:50,710 while I'm on the train. 810 00:38:50,710 --> 00:38:53,140 So I write down some time interval. 811 00:38:53,140 --> 00:38:55,450 You could do the same thing. 812 00:38:55,450 --> 00:38:59,650 Everybody observes events happening. 813 00:38:59,650 --> 00:39:01,445 Reality isn't relative. 814 00:39:04,420 --> 00:39:05,770 Reality isn't relative. 815 00:39:05,770 --> 00:39:09,760 The same-- at least according to special relativity, reality 816 00:39:09,760 --> 00:39:10,960 isn't relative. 817 00:39:10,960 --> 00:39:13,540 Events will happen regardless of-- the same events 818 00:39:13,540 --> 00:39:17,454 will happen regardless of who's observing the events happening. 819 00:39:17,454 --> 00:39:19,870 It's just the numbers that we associate to the events that 820 00:39:19,870 --> 00:39:20,700 are relative. 821 00:39:20,700 --> 00:39:22,650 But the same events will still happen, 822 00:39:22,650 --> 00:39:25,060 so in that sense reality is not relative. 823 00:39:25,060 --> 00:39:29,193 Reality is invariant, special relativistically invariant. 824 00:39:29,193 --> 00:39:29,692 Whew. 825 00:39:32,920 --> 00:39:34,960 So you'll see the same events happening, 826 00:39:34,960 --> 00:39:40,230 and you can measure a time interval between the two events 827 00:39:40,230 --> 00:39:42,030 by taking out your watch and writing down 828 00:39:42,030 --> 00:39:43,960 the time of the first event happening, 829 00:39:43,960 --> 00:39:46,293 and then writing the time of the second event happening, 830 00:39:46,293 --> 00:39:49,540 and then subtracting the two times to get a time interval. 831 00:39:49,540 --> 00:39:51,880 And as I've just told you before, 832 00:39:51,880 --> 00:39:55,180 that time interval isn't the same as the time interval 833 00:39:55,180 --> 00:39:59,920 that I get, and that's crazy. 834 00:39:59,920 --> 00:40:02,920 It's crazy that the time intervals should be different. 835 00:40:02,920 --> 00:40:05,370 I think it's crazy. 836 00:40:05,370 --> 00:40:11,660 You'd expect for time to be the same. 837 00:40:11,660 --> 00:40:14,830 This effect is called time dilation, 838 00:40:14,830 --> 00:40:20,690 and it's called time dilation because actually, 839 00:40:20,690 --> 00:40:23,550 as you can derive from special relativity, 840 00:40:23,550 --> 00:40:28,120 it turns out that clocks that are moving relative to you 841 00:40:28,120 --> 00:40:33,580 actually tick at a slower pace than the clock 842 00:40:33,580 --> 00:40:35,470 that you're holding, that's stationary. 843 00:40:35,470 --> 00:40:37,456 So moving clocks runs slow. 844 00:40:37,456 --> 00:40:39,580 That's a phrase that's commonly used to encapsulate 845 00:40:39,580 --> 00:40:41,440 the meaning of time dilation. 846 00:40:41,440 --> 00:40:44,200 Moving clocks run slow. 847 00:40:44,200 --> 00:40:45,550 So that means that if I-- 848 00:40:50,680 --> 00:40:52,390 if I measure a certain-- 849 00:40:52,390 --> 00:40:56,590 well, I could be a little bit-- 850 00:40:56,590 --> 00:40:59,470 I was about to be more quantitative in my explanation, 851 00:40:59,470 --> 00:41:01,720 but I think I'll just write down a formula for you, 852 00:41:01,720 --> 00:41:05,410 because this'll a very simple formula 853 00:41:05,410 --> 00:41:06,700 and you can appreciate it. 854 00:41:06,700 --> 00:41:10,370 And you'll have so much fun with it, believe me. 855 00:41:10,370 --> 00:41:13,330 And this is the-- 856 00:41:13,330 --> 00:41:15,130 I just want to make sure I don't forget it, 857 00:41:15,130 --> 00:41:19,790 because it's so important. 858 00:41:19,790 --> 00:41:23,210 Let's say I measure-- 859 00:41:23,210 --> 00:41:26,157 let's say I measure a time interval of-- 860 00:41:26,157 --> 00:41:27,470 sorry. 861 00:41:27,470 --> 00:41:29,890 Let's say-- sorry, sorry. 862 00:41:29,890 --> 00:41:31,690 Let's say that you measure-- 863 00:41:31,690 --> 00:41:35,170 see, we have to be careful when we say who measures things, 864 00:41:35,170 --> 00:41:38,182 because so many things are relative. 865 00:41:38,182 --> 00:41:39,640 So many things are relative, so you 866 00:41:39,640 --> 00:41:41,780 have to be careful in how you phrase things. 867 00:41:41,780 --> 00:41:44,230 And that's-- it gets kind of confusing. 868 00:41:44,230 --> 00:41:46,120 It gets kind of confusing. 869 00:41:46,120 --> 00:41:46,870 So let's say now-- 870 00:41:50,470 --> 00:41:54,490 let's say that you measure a certain time 871 00:41:54,490 --> 00:41:55,930 interval between two events. 872 00:41:55,930 --> 00:41:57,790 Let's say that you're stationary. 873 00:41:57,790 --> 00:42:00,910 You're holding a clock, and you measure some time interval 874 00:42:00,910 --> 00:42:01,800 between two events. 875 00:42:01,800 --> 00:42:04,420 And let's let that time interval be T0. 876 00:42:07,965 --> 00:42:11,220 T0 just because you're moving at zero speed. 877 00:42:11,220 --> 00:42:13,630 We just want to have some reference number. 878 00:42:13,630 --> 00:42:25,520 T0 is time interval that a stationary observer-- 879 00:42:25,520 --> 00:42:30,062 time interval that a stationary observer measures. 880 00:42:35,474 --> 00:42:49,250 A stationary-- OK. 881 00:42:49,250 --> 00:42:54,280 Now let's say that T is the time that an observer in motion 882 00:42:54,280 --> 00:42:55,440 measures. 883 00:42:55,440 --> 00:42:57,440 So I'm on the train, and I take out my watch, 884 00:42:57,440 --> 00:43:02,840 and I measure the time interval between two events. 885 00:43:02,840 --> 00:43:12,500 Let's call it T. Time interval that a moving 886 00:43:12,500 --> 00:43:15,300 observer measures. 887 00:43:15,300 --> 00:43:19,460 It turns out that you can relate the two in a very easy way, 888 00:43:19,460 --> 00:43:21,710 and in a very fascinating way. 889 00:43:21,710 --> 00:43:32,990 It turns out that T equals T0 divided by the square root of 1 890 00:43:32,990 --> 00:43:36,330 minus V squared over c squared. 891 00:43:39,010 --> 00:43:40,850 I forgot to tell you what V is. 892 00:43:40,850 --> 00:43:43,965 V is the relative speed-- 893 00:43:49,080 --> 00:43:52,941 the relative speed of the two observers. 894 00:43:56,710 --> 00:43:58,240 Observers. 895 00:43:58,240 --> 00:44:04,360 And, of course, c is the speed of light, as usual. 896 00:44:04,360 --> 00:44:10,780 Approximately 3 times time to the eighth meters per second. 897 00:44:10,780 --> 00:44:11,530 Can you read this? 898 00:44:11,530 --> 00:44:12,280 AUDIENCE: Yes. 899 00:44:12,280 --> 00:44:13,250 PROFESSOR: OK. 900 00:44:13,250 --> 00:44:14,060 I'll make it a little bit nicer. 901 00:44:14,060 --> 00:44:14,559 Sorry. 902 00:44:18,660 --> 00:44:19,390 OK. 903 00:44:19,390 --> 00:44:22,860 That's supposed to be T equals T0 over the square root of 1 904 00:44:22,860 --> 00:44:25,521 minus V squared over c squared. 905 00:44:25,521 --> 00:44:26,020 Yes? 906 00:44:26,020 --> 00:44:28,800 AUDIENCE: Does c always stand for the speed of light? 907 00:44:28,800 --> 00:44:29,960 PROFESSOR: c will always stand for the speed of light 908 00:44:29,960 --> 00:44:31,070 when I write it down. 909 00:44:31,070 --> 00:44:32,050 AUDIENCE: OK. 910 00:44:32,050 --> 00:44:35,127 PROFESSOR: Even when I write my name, N-I-C-K. 911 00:44:35,127 --> 00:44:36,710 I want for c to be the speed of light, 912 00:44:36,710 --> 00:44:39,130 but people don't understand that. 913 00:44:39,130 --> 00:44:39,928 Yes? 914 00:44:39,928 --> 00:44:41,920 AUDIENCE: Can you read the [INAUDIBLE] again? 915 00:44:41,920 --> 00:44:42,630 PROFESSOR: What's that? 916 00:44:42,630 --> 00:44:44,025 AUDIENCE: Can you read the formula out again? 917 00:44:44,025 --> 00:44:44,858 PROFESSOR: Oh, yeah. 918 00:44:44,858 --> 00:44:46,630 I'll read the formula again. 919 00:44:46,630 --> 00:44:49,510 Oh, I'm going to write down a couple of more formulas today, 920 00:44:49,510 --> 00:44:52,735 and I'll send them all to you with instructions 921 00:44:52,735 --> 00:44:55,210 on how to use them, so that you can all play with them 922 00:44:55,210 --> 00:44:57,520 and so that you can get them all done correctly. 923 00:44:57,520 --> 00:44:59,020 If you're not writing them down now, 924 00:44:59,020 --> 00:45:00,520 it's OK, because I'll send them all to you. 925 00:45:00,520 --> 00:45:02,144 So don't worry about writing them down. 926 00:45:02,144 --> 00:45:03,580 I'll read it out again. 927 00:45:03,580 --> 00:45:07,690 It says that T-- which is the time interval that a moving 928 00:45:07,690 --> 00:45:09,480 observer measures-- 929 00:45:09,480 --> 00:45:12,070 T equals T0 over the square root of 1 930 00:45:12,070 --> 00:45:15,640 minus V squared over c squared. 931 00:45:15,640 --> 00:45:16,140 Yes? 932 00:45:16,140 --> 00:45:18,824 AUDIENCE: What is the speed at which these effects become 933 00:45:18,824 --> 00:45:19,556 noticeable? 934 00:45:19,556 --> 00:45:21,392 PROFESSOR: OK. 935 00:45:21,392 --> 00:45:23,850 The speed at which all of these effects that I'm discussing 936 00:45:23,850 --> 00:45:25,890 becomes noticeable is a speed that's 937 00:45:25,890 --> 00:45:27,250 close to the speed of light. 938 00:45:27,250 --> 00:45:30,000 You won't notice any of these effects 939 00:45:30,000 --> 00:45:33,690 if you're moving much smaller than speed of light, even 1% 940 00:45:33,690 --> 00:45:34,680 the speed of light. 941 00:45:34,680 --> 00:45:36,054 Probably 1% of the speed of light 942 00:45:36,054 --> 00:45:37,570 is when they start to be noticeable. 943 00:45:37,570 --> 00:45:43,830 So you'll never have any dream of observing these effects when 944 00:45:43,830 --> 00:45:46,130 you're walking your dog. 945 00:45:46,130 --> 00:45:48,097 Unless it's a space dog or something. 946 00:45:48,097 --> 00:45:48,680 AUDIENCE: Yes? 947 00:45:48,680 --> 00:45:51,560 Well, why does the moving clock go slower? 948 00:45:51,560 --> 00:45:54,020 PROFESSOR: Why does it go slower? 949 00:45:54,020 --> 00:45:57,690 I could derive this for you. 950 00:45:57,690 --> 00:45:59,380 It's as simple-- what's that? 951 00:45:59,380 --> 00:46:00,330 AUDIENCE: Please do. 952 00:46:00,330 --> 00:46:02,070 PROFESSOR: Please do. 953 00:46:02,070 --> 00:46:06,110 Would you be OK if I get did simple geome-- 954 00:46:06,110 --> 00:46:08,640 I guess I could. 955 00:46:08,640 --> 00:46:10,560 The question was, can you derive this for me? 956 00:46:13,930 --> 00:46:14,430 Sure. 957 00:46:14,430 --> 00:46:17,061 I-- I won't go through all the details, 958 00:46:17,061 --> 00:46:18,810 but I'll go through the essential details. 959 00:46:20,912 --> 00:46:22,745 I'm going to derive this first, and then you 960 00:46:22,745 --> 00:46:25,240 can ask the question. 961 00:46:25,240 --> 00:46:28,830 And I'm kind of going against the promise I made to the class 962 00:46:28,830 --> 00:46:31,590 about not being mathematical, but if you guys want 963 00:46:31,590 --> 00:46:33,840 to see how you get this, then I'll show it to you. 964 00:46:37,361 --> 00:46:37,860 OK. 965 00:46:37,860 --> 00:46:42,517 So let's say-- can you guys guess what I'm drawing? 966 00:46:42,517 --> 00:46:43,680 AUDIENCE: Train. 967 00:46:43,680 --> 00:46:45,055 PROFESSOR: No, it's a school bus. 968 00:46:48,790 --> 00:46:51,120 Let's say that's the-- 969 00:46:51,120 --> 00:46:51,620 what? 970 00:46:55,010 --> 00:46:59,540 Let's say that the school bus is traveling at a speed of V 971 00:46:59,540 --> 00:47:01,870 relative to you. 972 00:47:01,870 --> 00:47:04,430 Say it's traveling at a speed of V relative to you. 973 00:47:04,430 --> 00:47:09,860 And let's say it has a height of h. 974 00:47:09,860 --> 00:47:12,168 Crazy h. 975 00:47:12,168 --> 00:47:14,870 It has a height of h. 976 00:47:14,870 --> 00:47:20,180 And let's say I'm inside of it. 977 00:47:20,180 --> 00:47:24,570 So I'm stationary relative to the train. 978 00:47:24,570 --> 00:47:29,240 Let's say-- let's say I'm hanging at the top. 979 00:47:29,240 --> 00:47:30,420 I'm at the top of the train. 980 00:47:30,420 --> 00:47:31,628 I'm hanging at the top of it. 981 00:47:34,080 --> 00:47:36,470 And I shine a flashlight going downwards. 982 00:47:36,470 --> 00:47:40,250 I shine a flash light going downwards, going down. 983 00:47:43,100 --> 00:47:46,520 This is the path that the light makes. 984 00:47:46,520 --> 00:47:49,350 I'll draw it again. 985 00:47:49,350 --> 00:47:52,370 A little squiggly path. 986 00:47:52,370 --> 00:47:53,840 This is the path the light makes. 987 00:47:53,840 --> 00:47:55,700 It goes down like this. 988 00:47:55,700 --> 00:47:58,370 Now, I can measure the time interval 989 00:47:58,370 --> 00:48:01,970 between the two events of the emission of light 990 00:48:01,970 --> 00:48:07,015 and then when the light hits the bottom floor. 991 00:48:07,015 --> 00:48:10,810 I could calculate how long that takes. 992 00:48:10,810 --> 00:48:15,453 It's simply the height of the-- 993 00:48:15,453 --> 00:48:18,361 I almost said train, but I said this is the school bus. 994 00:48:18,361 --> 00:48:20,110 The height of the school bus divided by c. 995 00:48:23,360 --> 00:48:25,541 Let's say that-- 996 00:48:25,541 --> 00:48:26,040 OK. 997 00:48:26,040 --> 00:48:31,580 So T-- T as I've defined it there. 998 00:48:31,580 --> 00:48:36,320 T equals h over c. 999 00:48:36,320 --> 00:48:38,780 Now, I ask-- sorry. 1000 00:48:38,780 --> 00:48:40,640 I don't have that good handwriting. 1001 00:48:40,640 --> 00:48:42,240 I'm trying really hard. 1002 00:48:42,240 --> 00:48:44,570 T equals h over c. 1003 00:48:44,570 --> 00:48:46,670 And now I ask, how long-- 1004 00:48:46,670 --> 00:48:49,460 what's the time interval between these two events for you guys, 1005 00:48:49,460 --> 00:48:51,680 with the train traveling at some speed? 1006 00:48:51,680 --> 00:48:52,370 Well, for you-- 1007 00:48:55,460 --> 00:48:56,990 I should make another drawing. 1008 00:48:56,990 --> 00:48:58,760 For you, it doesn't look like this. 1009 00:48:58,760 --> 00:49:01,070 For you, the light doesn't travel straight down. 1010 00:49:01,070 --> 00:49:04,542 It actually travels on a diagonal path. 1011 00:49:04,542 --> 00:49:06,500 It travels on a diagonal path because the train 1012 00:49:06,500 --> 00:49:07,291 is moving this way. 1013 00:49:13,020 --> 00:49:15,316 Oh, I said train, didn't I? 1014 00:49:15,316 --> 00:49:15,940 AUDIENCE: Yeah. 1015 00:49:15,940 --> 00:49:17,064 PROFESSOR: OK it's a train. 1016 00:49:17,064 --> 00:49:17,710 It's a train. 1017 00:49:22,960 --> 00:49:25,909 I'll remove this label now, because this actually is only 1018 00:49:25,909 --> 00:49:27,950 what the path of the light looks like to somebody 1019 00:49:27,950 --> 00:49:29,330 inside the train. 1020 00:49:29,330 --> 00:49:32,770 It's not what it looks like to you. 1021 00:49:32,770 --> 00:49:33,830 OK. 1022 00:49:33,830 --> 00:49:39,680 So this is for an observer in the train. 1023 00:49:39,680 --> 00:49:45,080 Observer in the train. 1024 00:49:45,080 --> 00:49:54,090 And this is for the observer watching the train. 1025 00:50:00,756 --> 00:50:02,880 For the observers watching the train, for you guys, 1026 00:50:02,880 --> 00:50:07,730 the light actually will travel in a diagonal path, 1027 00:50:07,730 --> 00:50:10,440 and then it'll hit the floor. 1028 00:50:10,440 --> 00:50:14,040 The height of the train is still h, 1029 00:50:14,040 --> 00:50:16,470 although you should be a little bit skeptical 1030 00:50:16,470 --> 00:50:20,250 because I told you that there was 1031 00:50:20,250 --> 00:50:22,590 something fishy about space. 1032 00:50:22,590 --> 00:50:29,030 And there actually is an effect of the relativity of length. 1033 00:50:29,030 --> 00:50:30,960 But it turns out that it's still h. 1034 00:50:30,960 --> 00:50:31,930 It's still h, OK? 1035 00:50:31,930 --> 00:50:34,360 It's still h. 1036 00:50:34,360 --> 00:50:37,500 The height of the train is still h. 1037 00:50:37,500 --> 00:50:41,400 Now, this time the light doesn't travel h over c. 1038 00:50:41,400 --> 00:50:42,270 It doesn't-- sorry. 1039 00:50:42,270 --> 00:50:46,370 This time the light doesn't travel a distance of h. 1040 00:50:46,370 --> 00:50:48,210 It actually travels a slightly longer-- 1041 00:50:51,280 --> 00:50:54,810 slightly longer path, because now it's got to go down 1042 00:50:54,810 --> 00:50:55,920 and it's got to go across. 1043 00:50:58,560 --> 00:51:02,445 It's got to go vertical and horizontal. 1044 00:51:02,445 --> 00:51:05,790 So the path that actually has to travel is now-- 1045 00:51:05,790 --> 00:51:09,060 well, let me make a little diagram, a little triangle. 1046 00:51:13,790 --> 00:51:15,740 OK. 1047 00:51:15,740 --> 00:51:17,120 This is h. 1048 00:51:17,120 --> 00:51:18,866 This is a right triangle. 1049 00:51:18,866 --> 00:51:20,780 h is just the height of the train. 1050 00:51:20,780 --> 00:51:22,940 It's always going to be the height of the train. 1051 00:51:22,940 --> 00:51:23,930 I can ask now-- 1052 00:51:23,930 --> 00:51:24,612 what's that? 1053 00:51:24,612 --> 00:51:25,507 AUDIENCE: The bus. 1054 00:51:25,507 --> 00:51:26,465 PROFESSOR: What's that? 1055 00:51:26,465 --> 00:51:28,210 AUDIENCE: It's a bus. 1056 00:51:28,210 --> 00:51:29,210 PROFESSOR: I changed it. 1057 00:51:29,210 --> 00:51:30,540 I couldn't help myself. 1058 00:51:30,540 --> 00:51:31,940 It's a train again. 1059 00:51:31,940 --> 00:51:33,370 It's always going to be a train. 1060 00:51:33,370 --> 00:51:35,745 You're never going to hear me mention the word bus again, 1061 00:51:35,745 --> 00:51:36,770 probably, at least not-- 1062 00:51:39,660 --> 00:51:40,550 I have to really try. 1063 00:51:40,550 --> 00:51:41,049 OK. 1064 00:51:41,049 --> 00:51:42,560 It's a train. 1065 00:51:42,560 --> 00:51:45,350 The height here is h, and-- 1066 00:51:45,350 --> 00:51:49,580 does anybody know what this distance is? 1067 00:51:49,580 --> 00:51:50,307 Yes? 1068 00:51:50,307 --> 00:51:56,400 AUDIENCE: [INAUDIBLE] 1069 00:51:56,400 --> 00:51:59,517 PROFESSOR: No, this distance is just how far the train's moved. 1070 00:51:59,517 --> 00:52:00,350 Forget about length. 1071 00:52:00,350 --> 00:52:02,060 This is just how far the train's moved. 1072 00:52:02,060 --> 00:52:03,870 And the train has actually-- 1073 00:52:03,870 --> 00:52:06,350 you can get that distance just by saying, well, 1074 00:52:06,350 --> 00:52:08,480 it's the speed of the train times 1075 00:52:08,480 --> 00:52:12,200 how long it takes for the light to get down. 1076 00:52:12,200 --> 00:52:14,450 [INAUDIBLE] to make that diagonal path, 1077 00:52:14,450 --> 00:52:18,130 and that time is T0. 1078 00:52:18,130 --> 00:52:20,000 Same way we've defined it before. 1079 00:52:20,000 --> 00:52:24,560 This distance is V times T0. 1080 00:52:27,210 --> 00:52:28,110 Rate times time. 1081 00:52:28,110 --> 00:52:30,000 You get a distance. 1082 00:52:30,000 --> 00:52:34,720 And so the total length the light travels is therefore-- 1083 00:52:34,720 --> 00:52:40,940 using our little theorem, using the theorem all of you know-- 1084 00:52:40,940 --> 00:52:47,260 it's just h squared plus V times T0 squared. 1085 00:52:47,260 --> 00:52:51,830 That's a T. So that's the total distance the light travels. 1086 00:52:51,830 --> 00:52:55,760 The total time-- the total amount 1087 00:52:55,760 --> 00:52:57,860 of time that it'll take for the light to travel 1088 00:52:57,860 --> 00:53:01,520 is therefore this distance divided by the speed of light. 1089 00:53:01,520 --> 00:53:06,700 And that distance is-- 1090 00:53:06,700 --> 00:53:07,700 OK. 1091 00:53:07,700 --> 00:53:11,480 That time-- I told you from the very beginning that time is T0. 1092 00:53:11,480 --> 00:53:13,260 So we already know how long it is, 1093 00:53:13,260 --> 00:53:16,880 but now we'll get an equation in which we can solve for T0. 1094 00:53:16,880 --> 00:53:19,910 So that equation is now going to be 1095 00:53:19,910 --> 00:53:26,944 T0 equals the square root of h squared plus V 1096 00:53:26,944 --> 00:53:31,130 T0 squared over c. 1097 00:53:31,130 --> 00:53:33,320 And you can solve for T0 now. 1098 00:53:33,320 --> 00:53:35,900 You can solve for T0 to get an expression for T0. 1099 00:53:35,900 --> 00:53:39,470 And then you can relate T0 to T, and then you 1100 00:53:39,470 --> 00:53:41,640 can get this equation. 1101 00:53:41,640 --> 00:53:42,140 Voila. 1102 00:53:46,600 --> 00:53:49,030 This equation is kind of complicated to solve. 1103 00:53:49,030 --> 00:53:50,980 It's just quadratic, I guess. 1104 00:53:50,980 --> 00:53:53,170 Yeah, you can do that if you want. 1105 00:53:53,170 --> 00:53:59,090 But-- you don't really gain anything by solving it. 1106 00:53:59,090 --> 00:54:00,580 I mean, you know how to solve it. 1107 00:54:00,580 --> 00:54:02,060 You don't really gain anything. 1108 00:54:02,060 --> 00:54:03,815 So that's where it comes from. 1109 00:54:03,815 --> 00:54:04,940 That's where it comes from. 1110 00:54:08,640 --> 00:54:09,870 And this whole effect-- 1111 00:54:09,870 --> 00:54:12,720 this effect of relativity of time intervals 1112 00:54:12,720 --> 00:54:14,420 is called time dilation. 1113 00:54:14,420 --> 00:54:15,642 I'll put this right here. 1114 00:54:23,160 --> 00:54:24,810 I also learned what the word dilate 1115 00:54:24,810 --> 00:54:27,240 means by learning about time dilation. 1116 00:54:27,240 --> 00:54:30,870 Time dilation-- they call it dilation because it's a time 1117 00:54:30,870 --> 00:54:34,455 interval that dilates, like your eyes dilate-- 1118 00:54:34,455 --> 00:54:35,880 no, your pupils dilate. 1119 00:54:35,880 --> 00:54:38,831 I learned about that after I learned about time dilation. 1120 00:54:38,831 --> 00:54:39,330 Yes? 1121 00:54:39,330 --> 00:54:41,620 AUDIENCE: Why do all of these equations-- 1122 00:54:41,620 --> 00:54:46,320 I guess except for that one-- use the speed of light squared? 1123 00:54:46,320 --> 00:54:47,190 Except for that one. 1124 00:54:47,190 --> 00:54:48,920 PROFESSOR: Oh, except for this one? 1125 00:54:48,920 --> 00:54:49,420 Oh-- 1126 00:54:49,420 --> 00:54:50,586 AUDIENCE: Why does it have-- 1127 00:54:50,586 --> 00:54:53,060 does it have to be-- why is it a square? 1128 00:54:53,060 --> 00:54:54,390 PROFESSOR: [INAUDIBLE] square? 1129 00:54:54,390 --> 00:54:56,427 OK. 1130 00:54:56,427 --> 00:54:58,260 The question is, why does the speed of light 1131 00:54:58,260 --> 00:55:01,502 appear as a square in these equations? 1132 00:55:01,502 --> 00:55:03,210 Well, you can see that pretty simply when 1133 00:55:03,210 --> 00:55:04,501 you try to solve this equation. 1134 00:55:04,501 --> 00:55:07,869 To solve this equation, you first square both sides 1135 00:55:07,869 --> 00:55:09,660 so that you can get rid of the square root, 1136 00:55:09,660 --> 00:55:10,909 and then you have a c squared. 1137 00:55:10,909 --> 00:55:13,890 And then it travels around the equations 1138 00:55:13,890 --> 00:55:15,810 and it ends up right there. 1139 00:55:15,810 --> 00:55:20,700 And something similar happens when you get that equation. 1140 00:55:20,700 --> 00:55:23,230 Something similar happens. 1141 00:55:23,230 --> 00:55:26,910 And then once again, this effect is only noticeable 1142 00:55:26,910 --> 00:55:32,680 for very high speeds, because if V small, then this ratio, 1143 00:55:32,680 --> 00:55:35,340 V squared over c squared is very small. 1144 00:55:35,340 --> 00:55:38,830 Then 1 minus a very small number is approximately 1, 1145 00:55:38,830 --> 00:55:40,710 and then you get T is approximately T0. 1146 00:55:40,710 --> 00:55:43,620 So the two intervals are approximately the same 1147 00:55:43,620 --> 00:55:45,723 for everyday phenomena. 1148 00:55:45,723 --> 00:55:46,223 Yes? 1149 00:55:46,223 --> 00:55:47,550 AUDIENCE: But it's still there? 1150 00:55:47,550 --> 00:55:48,216 PROFESSOR: Yeah. 1151 00:55:48,216 --> 00:55:49,740 The phenomenon is still there. 1152 00:55:49,740 --> 00:55:53,185 Time dilation is always occurring, just to a very-- 1153 00:55:56,790 --> 00:55:59,250 totally imperceptible extent for humans. 1154 00:55:59,250 --> 00:56:03,786 But this effect is actually measurable. 1155 00:56:03,786 --> 00:56:04,660 It's not just theory. 1156 00:56:04,660 --> 00:56:07,830 We've actually measured this in lots of particle physics 1157 00:56:07,830 --> 00:56:08,948 experiments. 1158 00:56:13,830 --> 00:56:16,950 One characteristic of particles that you can measure 1159 00:56:16,950 --> 00:56:20,880 is the half-life of a particle. 1160 00:56:20,880 --> 00:56:25,860 Well-- a single particle doesn't really have a half-life. 1161 00:56:25,860 --> 00:56:29,520 Half-life is really a statistical property. 1162 00:56:29,520 --> 00:56:31,950 What I mean by half-life is, if you 1163 00:56:31,950 --> 00:56:35,250 have many of these identical particles, 1164 00:56:35,250 --> 00:56:39,360 then after the half-life time, after the half-life, 1165 00:56:39,360 --> 00:56:41,640 approximately half of the particles will have decayed. 1166 00:56:45,360 --> 00:56:49,308 And you can actually notice-- 1167 00:56:49,308 --> 00:56:51,360 you can notice time dilation for these particles 1168 00:56:51,360 --> 00:56:53,401 because you're able to accelerate these particles 1169 00:56:53,401 --> 00:56:56,220 to very high fractions of c, very high fractions 1170 00:56:56,220 --> 00:56:58,020 of the speed of light. 1171 00:56:58,020 --> 00:57:02,340 For instance, it might be that usually-- 1172 00:57:02,340 --> 00:57:05,520 it might be that if you have a particle at rest, 1173 00:57:05,520 --> 00:57:07,890 then it has a certain half-life-- 1174 00:57:07,890 --> 00:57:11,610 which might be a microsecond, like one microsecond. 1175 00:57:11,610 --> 00:57:16,560 And if I make that particle travel at a very high speed, 1176 00:57:16,560 --> 00:57:19,300 then if I go into that-- 1177 00:57:21,697 --> 00:57:23,530 I were traveling right next to that particle 1178 00:57:23,530 --> 00:57:26,360 at that very high speed, then the half-life 1179 00:57:26,360 --> 00:57:27,360 would still be the same. 1180 00:57:27,360 --> 00:57:30,120 But if I'm at rest, sitting in the lab 1181 00:57:30,120 --> 00:57:34,060 at rest, than that half-life will actually be dilated. 1182 00:57:34,060 --> 00:57:36,690 That half-life will actually be much longer. 1183 00:57:36,690 --> 00:57:40,950 And that's a direct confirmation of time dilation, 1184 00:57:40,950 --> 00:57:44,460 which is pretty cool. 1185 00:57:44,460 --> 00:57:45,050 Let's see. 1186 00:57:45,050 --> 00:57:46,680 It's-- whoa. 1187 00:57:46,680 --> 00:57:49,510 An hour has already passed. 1188 00:57:49,510 --> 00:57:50,760 Does everybody else have 2:30? 1189 00:57:50,760 --> 00:57:51,661 AUDIENCE: Yeah. 1190 00:57:51,661 --> 00:57:52,410 PROFESSOR: Oh, OK. 1191 00:57:52,410 --> 00:57:53,430 Wow. 1192 00:57:53,430 --> 00:57:55,456 Let's take a short break. 1193 00:57:55,456 --> 00:57:57,080 We'll take a short break for right now. 1194 00:57:57,080 --> 00:57:58,129 When we come back-- 1195 00:57:58,129 --> 00:58:00,420 when we come back, we'll talk about length contraction. 1196 00:58:00,420 --> 00:58:03,670 It turns out the length is actually relative as well. 1197 00:58:03,670 --> 00:58:05,340 OK. 1198 00:58:05,340 --> 00:58:07,830 We saw there's something fishy about time. 1199 00:58:07,830 --> 00:58:11,262 We saw there something fishy about simultaneity. 1200 00:58:11,262 --> 00:58:13,095 You saw there's something fishy about speed. 1201 00:58:15,750 --> 00:58:18,960 Everything that we held so dear, things 1202 00:58:18,960 --> 00:58:22,080 that we used to rely on-- 1203 00:58:22,080 --> 00:58:26,370 time, speed, everything we used to rely on so dearly 1204 00:58:26,370 --> 00:58:30,720 turned out to be just a misconception. 1205 00:58:30,720 --> 00:58:34,071 And it turns out that there's something fishy about length, 1206 00:58:34,071 --> 00:58:34,570 too. 1207 00:58:34,570 --> 00:58:36,861 And by something fishy, I mean that length is relative. 1208 00:58:40,462 --> 00:58:41,670 I mean that something-- yeah. 1209 00:58:41,670 --> 00:58:43,800 Length is relative. 1210 00:58:43,800 --> 00:58:45,830 Now, what exactly do I mean by length? 1211 00:58:45,830 --> 00:58:47,400 Well, suppose you took an object-- 1212 00:58:47,400 --> 00:58:51,700 suppose you took an object, you looked at the two ends of it 1213 00:58:51,700 --> 00:58:52,659 at the same exact time. 1214 00:58:52,659 --> 00:58:55,158 You wrote down the coordinates of this end, your coordinates 1215 00:58:55,158 --> 00:58:57,870 of this end, and then you simply found the difference of the two 1216 00:58:57,870 --> 00:59:00,290 coordinates an then took the absolute value. 1217 00:59:00,290 --> 00:59:01,269 That's the length. 1218 00:59:01,269 --> 00:59:02,560 That's the length of an object. 1219 00:59:02,560 --> 00:59:04,860 You have to measure the coordinates of the endpoints 1220 00:59:04,860 --> 00:59:07,230 at the same time, because if you don't measure them 1221 00:59:07,230 --> 00:59:08,690 at the same time, then-- 1222 00:59:11,820 --> 00:59:14,690 of course that's not length. 1223 00:59:14,690 --> 00:59:17,700 if I have a water bottle traveling-- well, 1224 00:59:17,700 --> 00:59:21,890 if I have a water bottle here, I measure the coordinate of this, 1225 00:59:21,890 --> 00:59:23,622 then a minute later it's here and I 1226 00:59:23,622 --> 00:59:25,580 measure the coordinate of this, well, of course 1227 00:59:25,580 --> 00:59:27,871 this coordinate minus this coordinate's not the length. 1228 00:59:27,871 --> 00:59:30,665 So you have to do it at the same time. 1229 00:59:30,665 --> 00:59:32,810 It turns out that length is relative. 1230 00:59:32,810 --> 00:59:34,400 And there's a very simple formula 1231 00:59:34,400 --> 00:59:38,270 for the relation of lengths that you observe 1232 00:59:38,270 --> 00:59:41,850 and the lengths that the train observer observes. 1233 00:59:41,850 --> 00:59:44,270 Again, let's say I'm on the train. 1234 00:59:44,270 --> 00:59:48,110 I hold out a pole, and I measure the length of it 1235 00:59:48,110 --> 00:59:49,610 and you measure the length of it. 1236 00:59:49,610 --> 00:59:51,470 And we say, how do these two lengths-- 1237 00:59:51,470 --> 00:59:53,178 we ask, how do these two lengths compare? 1238 00:59:56,280 --> 00:59:59,120 Let's say that the length of-- 1239 00:59:59,120 --> 01:00:06,290 let's say that the length of the pole that I'm holding is L0. 1240 01:00:06,290 --> 01:00:09,050 Well, the length of the pole at rest. 1241 01:00:09,050 --> 01:00:13,130 Say the length of the pole at rest is L0. 1242 01:00:13,130 --> 01:00:15,964 And we call this-- 1243 01:00:15,964 --> 01:00:17,380 we call it the proper length, just 1244 01:00:17,380 --> 01:00:19,040 as we call T0 the proper time. 1245 01:00:19,040 --> 01:00:20,640 Those are just words. 1246 01:00:20,640 --> 01:00:24,260 L0 is-- I'll be more generic, and I'll 1247 01:00:24,260 --> 01:00:33,660 say L0 is the length of object at rest. 1248 01:00:38,510 --> 01:00:40,120 Now I can ask, is that length going 1249 01:00:40,120 --> 01:00:43,840 to be the same when the object starts moving? 1250 01:00:43,840 --> 01:00:46,402 And it turns out the answer is no. 1251 01:00:46,402 --> 01:00:47,110 The answer is no. 1252 01:00:49,996 --> 01:00:52,120 And I'll call the length of the object 1253 01:00:52,120 --> 01:00:55,820 when it's moving L, without the 0. 1254 01:00:55,820 --> 01:01:00,720 L is the length of an object in motion. 1255 01:01:04,290 --> 01:01:09,990 In motion moving at speed V. And it turns out 1256 01:01:09,990 --> 01:01:15,100 that these two lengths are related by a simple formula-- 1257 01:01:15,100 --> 01:01:20,220 namely that L equals L0 times the square root of 1 1258 01:01:20,220 --> 01:01:22,557 minus V squared over c squared. 1259 01:01:22,557 --> 01:01:24,390 There's that quantity again, the square root 1260 01:01:24,390 --> 01:01:25,940 of 1 minus V square over c squared. 1261 01:01:30,450 --> 01:01:32,190 I could derive this one again, similarly 1262 01:01:32,190 --> 01:01:33,440 to the way I derived that one. 1263 01:01:33,440 --> 01:01:35,100 But I think this is kind of a longer 1264 01:01:35,100 --> 01:01:36,630 derivation than the previous one, 1265 01:01:36,630 --> 01:01:39,240 and we only have a little bit of time left in the class, 1266 01:01:39,240 --> 01:01:41,751 so I'll just give this to you. 1267 01:01:41,751 --> 01:01:43,500 However, I will-- I'll send you some notes 1268 01:01:43,500 --> 01:01:44,220 where I do derive it. 1269 01:01:44,220 --> 01:01:46,386 I'll send you some notes where I derive the addition 1270 01:01:46,386 --> 01:01:47,946 of velocities formula, too. 1271 01:01:47,946 --> 01:01:50,410 I'll fix the square root. 1272 01:01:50,410 --> 01:01:53,040 OK. 1273 01:01:53,040 --> 01:01:56,740 Let's say that you measure-- 1274 01:01:56,740 --> 01:02:01,680 well, let's say that we measure the length of some stick 1275 01:02:01,680 --> 01:02:02,820 to be-- 1276 01:02:02,820 --> 01:02:07,230 some pole to be 10 feet, just to plug in some numbers. 1277 01:02:07,230 --> 01:02:08,820 Let's say that we let the pole be-- 1278 01:02:11,740 --> 01:02:13,630 I'll say 10 meters, or-- 1279 01:02:13,630 --> 01:02:15,880 10 meters is pretty big, but let's just say 10 meters. 1280 01:02:19,000 --> 01:02:21,240 It's safe to use meters. 1281 01:02:21,240 --> 01:02:24,540 It's safe to use meters because I 1282 01:02:24,540 --> 01:02:26,940 know the speed of light in meters, 1283 01:02:26,940 --> 01:02:31,290 and we should be consistent with our units. 1284 01:02:31,290 --> 01:02:33,120 So we'll say L0 is 10 meters. 1285 01:02:33,120 --> 01:02:37,260 And let's say that V-- 1286 01:02:37,260 --> 01:02:38,970 well, let's actually say V over c. 1287 01:02:38,970 --> 01:02:44,130 We'll say that's the fraction of the speed of light that V is. 1288 01:02:44,130 --> 01:02:47,720 Let's say it's 0.6. 1289 01:02:47,720 --> 01:02:51,601 So V is actually 60% the speed of light, which is very fast. 1290 01:02:51,601 --> 01:02:53,100 That's how fast this pole is moving, 1291 01:02:53,100 --> 01:02:55,470 60% of the speed of light. 1292 01:02:55,470 --> 01:02:58,635 And I ask, how fast does it-- 1293 01:02:58,635 --> 01:03:01,470 I ask, how long is this pole? 1294 01:03:01,470 --> 01:03:03,990 How long is this pole while it's in motion? 1295 01:03:03,990 --> 01:03:06,330 And I will have this formula, and I just 1296 01:03:06,330 --> 01:03:08,610 plug in these numbers. 1297 01:03:08,610 --> 01:03:13,860 The pole is then 10 meters times the square root of 1 1298 01:03:13,860 --> 01:03:20,640 minus 0.6 square, which is 0.36. 1299 01:03:20,640 --> 01:03:21,370 Square roots. 1300 01:03:21,370 --> 01:03:23,530 1 minus 0.36 is 0.64. 1301 01:03:23,530 --> 01:03:26,430 The square root of 0.64 is 0.8. 1302 01:03:26,430 --> 01:03:29,830 So the length-- let me fix this. 1303 01:03:29,830 --> 01:03:32,670 So the length of the pole while it in motion 1304 01:03:32,670 --> 01:03:36,330 is actually 8 meters. 1305 01:03:36,330 --> 01:03:38,310 So it contracted. 1306 01:03:38,310 --> 01:03:39,909 This is called length contraction. 1307 01:03:51,160 --> 01:03:52,307 Question? 1308 01:03:52,307 --> 01:03:52,890 AUDIENCE: Yes. 1309 01:03:52,890 --> 01:03:59,280 The original equation, L equals L0 times the square root 1310 01:03:59,280 --> 01:04:02,185 of 1 minus V squared over c squared-- 1311 01:04:02,185 --> 01:04:05,563 can't you simplify the square root of 1 minus V squared 1312 01:04:05,563 --> 01:04:08,810 over c squared to 1 minus V over c? 1313 01:04:08,810 --> 01:04:10,060 PROFESSOR: You can say-- sure. 1314 01:04:10,060 --> 01:04:13,810 You can simplify this question by writing 1 minus V squared 1315 01:04:13,810 --> 01:04:14,890 over c. 1316 01:04:14,890 --> 01:04:16,480 This is a difference of two squares, 1317 01:04:16,480 --> 01:04:17,930 so you could simplify it if you wanted to. 1318 01:04:17,930 --> 01:04:19,721 There's no reason to, but if you wanted to, 1319 01:04:19,721 --> 01:04:22,120 you could say that 1 minus V squared equals 1 plus V 1320 01:04:22,120 --> 01:04:25,360 over c times 1 minus V over c. 1321 01:04:25,360 --> 01:04:26,300 But we don't need to. 1322 01:04:26,300 --> 01:04:27,900 We're just plugging in numbers. 1323 01:04:27,900 --> 01:04:29,555 But you could if you want. 1324 01:04:29,555 --> 01:04:30,054 Question? 1325 01:04:30,054 --> 01:04:32,467 AUDIENCE: [INAUDIBLE] 1326 01:04:32,467 --> 01:04:33,550 PROFESSOR: Does it matter? 1327 01:04:33,550 --> 01:04:34,049 OK. 1328 01:04:34,049 --> 01:04:35,445 Good question, good question. 1329 01:04:35,445 --> 01:04:36,600 AUDIENCE: Can you repeat the question? 1330 01:04:36,600 --> 01:04:39,220 PROFESSOR: The question is, does it matter which direction 1331 01:04:39,220 --> 01:04:42,350 that the pole is moving and which direction the pole 1332 01:04:42,350 --> 01:04:44,195 is oriented? 1333 01:04:44,195 --> 01:04:45,650 Yeah, it does matter, actually. 1334 01:04:45,650 --> 01:04:47,000 It does matter. 1335 01:04:47,000 --> 01:04:52,130 Actually-- if I have a three-dimensional object-- 1336 01:04:52,130 --> 01:04:56,240 if I have a three-dimensional object, like a box, 1337 01:04:56,240 --> 01:04:58,580 that's moving at some speed relative to me, 1338 01:04:58,580 --> 01:05:00,410 then it turns out-- 1339 01:05:00,410 --> 01:05:01,610 I won't prove it for you. 1340 01:05:01,610 --> 01:05:02,776 You'll have to take my word. 1341 01:05:02,776 --> 01:05:05,210 Well, I'll prove it in some notes that I'll send to you. 1342 01:05:05,210 --> 01:05:09,620 It turns out that the only direction-- 1343 01:05:09,620 --> 01:05:13,190 the only direction that the length is contracted 1344 01:05:13,190 --> 01:05:16,370 is the direction of the box that's 1345 01:05:16,370 --> 01:05:19,280 the same direction as the motion of the box. 1346 01:05:19,280 --> 01:05:22,830 So if the box is moving this way, 1347 01:05:22,830 --> 01:05:27,900 then this dimension is going to be contracted. 1348 01:05:27,900 --> 01:05:32,780 It'll get shorter this way, but the two perpendicular 1349 01:05:32,780 --> 01:05:35,255 directions actually are unaffected. 1350 01:05:35,255 --> 01:05:39,020 They're unaffected, it turns out. 1351 01:05:39,020 --> 01:05:41,540 It's pretty interesting. 1352 01:05:41,540 --> 01:05:45,395 And this whole effect is called length contraction. 1353 01:05:45,395 --> 01:05:46,520 Was there another question? 1354 01:05:46,520 --> 01:05:47,020 Yes? 1355 01:05:47,020 --> 01:05:51,830 AUDIENCE: [INAUDIBLE] 1356 01:05:51,830 --> 01:05:52,620 PROFESSOR: Oh, OK. 1357 01:05:52,620 --> 01:05:53,820 So the question is, what happens as you 1358 01:05:53,820 --> 01:05:54,986 approach the speed of light? 1359 01:05:58,592 --> 01:05:59,550 That's a good question. 1360 01:05:59,550 --> 01:06:00,300 The question is, what happens as you 1361 01:06:00,300 --> 01:06:01,720 approach the speed of light? 1362 01:06:01,720 --> 01:06:05,480 And the answer is that the closer 1363 01:06:05,480 --> 01:06:07,200 that this object in motion approaches-- 1364 01:06:07,200 --> 01:06:09,720 the closer this object gets to the speed of light, 1365 01:06:09,720 --> 01:06:10,890 the shorter it will get. 1366 01:06:10,890 --> 01:06:13,470 So it actually approaches 0 length 1367 01:06:13,470 --> 01:06:15,480 as it approaches the speed of light. 1368 01:06:15,480 --> 01:06:19,860 And similarly, with the time dilation formula, 1369 01:06:19,860 --> 01:06:21,900 as you approach speed of light, you 1370 01:06:21,900 --> 01:06:25,230 have some number over a very, very small number, 1371 01:06:25,230 --> 01:06:26,870 because now V squared over c squared 1372 01:06:26,870 --> 01:06:30,570 is a very large number that's close to 1, very close to 1, 1373 01:06:30,570 --> 01:06:33,390 so 1 minus this number is approximately 0, 1374 01:06:33,390 --> 01:06:35,730 and some number over a number that's approximately 0 1375 01:06:35,730 --> 01:06:37,271 is going to be a really large number. 1376 01:06:40,360 --> 01:06:43,590 So the time interval actually gets very, very dilated. 1377 01:06:43,590 --> 01:06:45,350 It's much, much longer. 1378 01:06:45,350 --> 01:06:48,300 The half-lives of the particles I mentioned 1379 01:06:48,300 --> 01:06:51,630 get much longer than the half-lives of those particles 1380 01:06:51,630 --> 01:06:52,260 at rest. 1381 01:07:01,910 --> 01:07:05,160 So there's something about space and time-- 1382 01:07:05,160 --> 01:07:06,382 yes? 1383 01:07:06,382 --> 01:07:09,230 AUDIENCE: Is there any way to test length contraction? 1384 01:07:09,230 --> 01:07:09,980 PROFESSOR: Length? 1385 01:07:09,980 --> 01:07:10,896 OK. 1386 01:07:10,896 --> 01:07:12,730 AUDIENCE: [INAUDIBLE] 1387 01:07:12,730 --> 01:07:15,710 PROFESSOR: The question is, are there ways of testing length 1388 01:07:15,710 --> 01:07:17,870 contraction? 1389 01:07:17,870 --> 01:07:20,452 I haven't heard of any direct experimental confirmation 1390 01:07:20,452 --> 01:07:22,160 for length contraction, just because it's 1391 01:07:22,160 --> 01:07:25,355 so hard to get objects of large size going 1392 01:07:25,355 --> 01:07:27,230 at really large speeds so they could actually 1393 01:07:27,230 --> 01:07:29,405 notice it occurring. 1394 01:07:29,405 --> 01:07:31,530 I don't know of any direct experimental detections. 1395 01:07:31,530 --> 01:07:38,660 But time dilation, there's a lot of experimental support for. 1396 01:07:38,660 --> 01:07:40,260 Did you have a question? 1397 01:07:40,260 --> 01:07:41,640 Oh, no, you're just doing this. 1398 01:07:41,640 --> 01:07:43,372 OK. 1399 01:07:43,372 --> 01:07:48,540 Now, as I was starting to say, time and space 1400 01:07:48,540 --> 01:07:52,590 are completely different from what you would 1401 01:07:52,590 --> 01:07:54,019 you'd expect intuitively. 1402 01:07:54,019 --> 01:07:56,310 They're completely different from how Newton envisioned 1403 01:07:56,310 --> 01:07:57,870 time and space being. 1404 01:07:57,870 --> 01:08:02,250 Newton thought that time intervals and space intervals-- 1405 01:08:02,250 --> 01:08:03,330 which are lengths-- 1406 01:08:03,330 --> 01:08:04,660 are the same for everybody. 1407 01:08:04,660 --> 01:08:06,550 And he didn't question it at all. 1408 01:08:06,550 --> 01:08:08,200 In fact, he said-- 1409 01:08:08,200 --> 01:08:11,580 he said, "I do not define time, space, place, 1410 01:08:11,580 --> 01:08:13,837 and motion, because they are well known to all." 1411 01:08:13,837 --> 01:08:14,670 That's what he said. 1412 01:08:20,430 --> 01:08:24,479 Other people have speculated about the nature of time. 1413 01:08:24,479 --> 01:08:26,020 It's a hard question to ask. 1414 01:08:26,020 --> 01:08:27,479 What is time? 1415 01:08:27,479 --> 01:08:29,470 Philosophers have thought about it. 1416 01:08:29,470 --> 01:08:30,720 Physics have thought about it. 1417 01:08:33,124 --> 01:08:35,290 Probably lots of other people have thought about it. 1418 01:08:35,290 --> 01:08:39,622 And I have some interesting quotes that you might like. 1419 01:08:39,622 --> 01:08:40,574 Let's see. 1420 01:08:42,939 --> 01:08:43,689 Here's a good one. 1421 01:08:46,594 --> 01:08:47,750 Let me see. 1422 01:08:47,750 --> 01:08:48,250 OK. 1423 01:08:48,250 --> 01:08:51,040 "Nothing puzzles me more than time and space, and yet 1424 01:08:51,040 --> 01:08:53,560 nothing troubles me less, as I never think about them." 1425 01:08:56,640 --> 01:08:59,920 "Either this man is dead, or my watch has stopped." 1426 01:09:02,439 --> 01:09:06,097 "It's good to reach 100, because very few people die after 100." 1427 01:09:06,097 --> 01:09:08,180 That doesn't really have anything to do with time, 1428 01:09:08,180 --> 01:09:09,149 does it? 1429 01:09:09,149 --> 01:09:10,779 OK. 1430 01:09:10,779 --> 01:09:11,920 Here's one of my favorites. 1431 01:09:11,920 --> 01:09:14,260 "Time is nature's way to keep everything 1432 01:09:14,260 --> 01:09:15,370 from happening at once." 1433 01:09:19,750 --> 01:09:22,720 We actually don't really define time in physics. 1434 01:09:22,720 --> 01:09:24,580 We simply don't define it, because-- 1435 01:09:24,580 --> 01:09:27,370 well, first of all, it's hard, and second, it might not even 1436 01:09:27,370 --> 01:09:28,532 be meaningful. 1437 01:09:28,532 --> 01:09:30,490 So what we do is, we define time operationally. 1438 01:09:30,490 --> 01:09:32,680 We define time in terms of ticks of clocks, 1439 01:09:32,680 --> 01:09:34,779 as I was mentioning earlier. 1440 01:09:34,779 --> 01:09:39,740 And it seems to be a very successful way of doing things, 1441 01:09:39,740 --> 01:09:43,060 but it makes you wonder, what really is time? 1442 01:09:43,060 --> 01:09:46,420 it's hard to-- how do you define it, though? 1443 01:09:46,420 --> 01:09:47,620 You can wonder for hours. 1444 01:09:47,620 --> 01:09:49,600 What does time mean? 1445 01:09:49,600 --> 01:09:51,399 And you probably won't get-- 1446 01:09:51,399 --> 01:09:51,985 what's that? 1447 01:09:51,985 --> 01:09:53,109 AUDIENCE: You'll lose time. 1448 01:09:53,109 --> 01:09:53,710 PROFESSOR: You'll lose time. 1449 01:09:53,710 --> 01:09:55,418 Yeah, you'll lose time by pondering time. 1450 01:09:55,418 --> 01:09:56,841 Yeah, exactly. 1451 01:09:56,841 --> 01:09:57,340 Exactly. 1452 01:10:00,030 --> 01:10:01,920 OK. 1453 01:10:01,920 --> 01:10:04,660 There are couple of paradoxes I wanted to talk about. 1454 01:10:08,020 --> 01:10:14,080 I can talk about one paradox, one brain-- 1455 01:10:14,080 --> 01:10:14,770 brain what? 1456 01:10:14,770 --> 01:10:16,500 What do paradoxes do to brains? 1457 01:10:16,500 --> 01:10:17,500 They don't-- they twist. 1458 01:10:17,500 --> 01:10:18,570 Yeah, Brain twisting. 1459 01:10:18,570 --> 01:10:19,300 I don't know. 1460 01:10:19,300 --> 01:10:20,860 Brain tweaser? 1461 01:10:20,860 --> 01:10:22,105 Teaser? 1462 01:10:22,105 --> 01:10:23,190 Brain teaser. 1463 01:10:23,190 --> 01:10:23,770 Yeah. 1464 01:10:23,770 --> 01:10:24,270 OK. 1465 01:10:26,770 --> 01:10:29,350 But it's an important paradox, and it 1466 01:10:29,350 --> 01:10:30,980 has to do with time dilation. 1467 01:10:30,980 --> 01:10:31,870 It has to do with time dilation. 1468 01:10:31,870 --> 01:10:33,578 And I'm not going to tell you the answer. 1469 01:10:33,578 --> 01:10:35,665 I want you to think about it afterwards. 1470 01:10:35,665 --> 01:10:41,750 It's known as the twin paradox, and it goes as follows. 1471 01:10:41,750 --> 01:10:46,360 Suppose there's this astronaut who on her 21st-- 1472 01:10:46,360 --> 01:10:50,210 21st birthday decides to go out into space 1473 01:10:50,210 --> 01:10:57,160 in a rocket very fast, very close to the speed of light. 1474 01:10:57,160 --> 01:11:00,450 I'll give you some numbers, because I have them. 1475 01:11:00,450 --> 01:11:02,100 You can check these numbers-- 1476 01:11:02,100 --> 01:11:04,630 let me find it. 1477 01:11:04,630 --> 01:11:06,670 You can check these numbers in a bit. 1478 01:11:06,670 --> 01:11:07,820 Where do I have it? 1479 01:11:07,820 --> 01:11:08,320 Gosh. 1480 01:11:08,320 --> 01:11:10,420 Oh, here it is. 1481 01:11:10,420 --> 01:11:12,980 Suppose that there's an astronaut that travels-- 1482 01:11:12,980 --> 01:11:14,490 there's this astronaut that travels 1483 01:11:14,490 --> 01:11:19,480 into space on her 21st birthday, going at 12/13 1484 01:11:19,480 --> 01:11:22,000 of the speed of light, 12/13 c. 1485 01:11:22,000 --> 01:11:26,230 And she goes-- she leaves on Earth 1486 01:11:26,230 --> 01:11:28,120 and she travels for five years at this speed. 1487 01:11:28,120 --> 01:11:31,570 She travels for five years at this speed on her clock, 1488 01:11:31,570 --> 01:11:33,220 and then she decides to head back. 1489 01:11:33,220 --> 01:11:35,380 So it takes her five years to get back. 1490 01:11:35,380 --> 01:11:41,290 It takes her five years both directions. 1491 01:11:41,290 --> 01:11:43,450 I'll make-- I won't write the things down 1492 01:11:43,450 --> 01:11:45,516 because I'm running out of time, but I'll 1493 01:11:45,516 --> 01:11:46,890 send it in an email all precisely 1494 01:11:46,890 --> 01:11:48,070 so you can think about it. 1495 01:11:48,070 --> 01:11:50,050 So she goes five years this direction, then 1496 01:11:50,050 --> 01:11:51,490 she comes back for five-- 1497 01:11:51,490 --> 01:11:53,600 takes her five years to come back. 1498 01:11:53,600 --> 01:11:57,820 Now, she'll arrive back on her 31st, birthday because 5 plus 5 1499 01:11:57,820 --> 01:12:01,150 is 10, and then we just add 10 to 21. 1500 01:12:01,150 --> 01:12:02,290 Now, she has a twin. 1501 01:12:02,290 --> 01:12:04,930 Suppose she has a twin that lives on Earth. 1502 01:12:04,930 --> 01:12:10,850 And so her twin was also 21 when she left. 1503 01:12:10,850 --> 01:12:17,460 But-- the question is, how old will he be-- 1504 01:12:17,460 --> 01:12:19,420 or she be. 1505 01:12:19,420 --> 01:12:20,350 I'll say he. 1506 01:12:20,350 --> 01:12:22,370 How old will he be when she comes back? 1507 01:12:22,370 --> 01:12:24,370 Well, you can calculate this using this formula. 1508 01:12:24,370 --> 01:12:28,900 Just plug in 12/13 and then multiply by 10 years, 1509 01:12:28,900 --> 01:12:34,390 and you get that while it takes 10 years for her 1510 01:12:34,390 --> 01:12:38,200 to travel out and come back, 26 years have actually 1511 01:12:38,200 --> 01:12:39,870 passed on his watch. 1512 01:12:39,870 --> 01:12:43,030 26 years have actually passed on his watch. 1513 01:12:43,030 --> 01:12:45,880 So when she's 31-- when she comes back, she's 31, 1514 01:12:45,880 --> 01:12:47,500 and he's not going be 31. 1515 01:12:47,500 --> 01:12:50,340 He'll be 47, because 26 plus 21 is 47. 1516 01:12:58,112 --> 01:13:00,320 It looks like there's something wrong here, because-- 1517 01:13:06,410 --> 01:13:08,240 the astronaut twin you can actually 1518 01:13:08,240 --> 01:13:10,040 do the same calculation. 1519 01:13:10,040 --> 01:13:11,810 She can say, well-- 1520 01:13:11,810 --> 01:13:14,270 she can say, well, my twin is moving relative to me-- 1521 01:13:14,270 --> 01:13:16,444 because, remember, all of these observers are equal. 1522 01:13:16,444 --> 01:13:18,110 I mean, there's not one preferred speed. 1523 01:13:18,110 --> 01:13:19,560 They're all as good as any other. 1524 01:13:19,560 --> 01:13:21,740 They're all as good as any other. 1525 01:13:21,740 --> 01:13:24,080 She can say, well, my twin is moving-- 1526 01:13:24,080 --> 01:13:26,560 my twin is moving at 12/13 speed. 1527 01:13:26,560 --> 01:13:27,695 I'm stationary. 1528 01:13:32,260 --> 01:13:37,220 So 26-- sorry, so 10 years should pass for my twin, 1529 01:13:37,220 --> 01:13:39,187 and 26 years should pass for me. 1530 01:13:39,187 --> 01:13:40,520 That's what the twin should say. 1531 01:13:40,520 --> 01:13:42,230 You just run the calculation backwards. 1532 01:13:42,230 --> 01:13:43,580 All of these equations are-- 1533 01:13:43,580 --> 01:13:45,540 I didn't emphasize, but all these equations 1534 01:13:45,540 --> 01:13:46,584 that I've given you-- 1535 01:13:46,584 --> 01:13:47,750 well, these two right here-- 1536 01:13:47,750 --> 01:13:48,950 they're symmetric. 1537 01:13:51,860 --> 01:13:52,885 If I see that-- 1538 01:13:52,885 --> 01:13:54,740 if I say that you're moving relative to me, 1539 01:13:54,740 --> 01:13:58,067 then you can equally say, well, you're moving relative to me. 1540 01:13:58,067 --> 01:14:00,150 Did I just say that, or did I actually reverse it? 1541 01:14:00,150 --> 01:14:01,862 I meant to reverse it. 1542 01:14:01,862 --> 01:14:03,072 I meant to reverse it. 1543 01:14:06,800 --> 01:14:09,950 So the twin will think that, actually-- 1544 01:14:09,950 --> 01:14:11,450 sorry, the astronaut twin will think 1545 01:14:11,450 --> 01:14:15,770 that her Earth counterpart will have aged more than her, 1546 01:14:15,770 --> 01:14:19,190 but only one of them can actually be true. 1547 01:14:19,190 --> 01:14:23,240 Only one of the predictions can actually be true. 1548 01:14:23,240 --> 01:14:26,060 Either the astronaut twin will have aged more than the Earth 1549 01:14:26,060 --> 01:14:29,510 twin, or the Earth one will have aged more 1550 01:14:29,510 --> 01:14:31,340 than the astronaut twin, or they'll 1551 01:14:31,340 --> 01:14:32,810 both have aged the same. 1552 01:14:32,810 --> 01:14:34,160 Those are three possibilities. 1553 01:14:34,160 --> 01:14:37,460 Only one of them can be true, and I'll tell you 1554 01:14:37,460 --> 01:14:38,925 right now which one is true. 1555 01:14:38,925 --> 01:14:41,000 I'll tell you right now which one is true. 1556 01:14:41,000 --> 01:14:42,750 It's not that they both age the same. 1557 01:14:42,750 --> 01:14:44,750 They haven't both aged the same. 1558 01:14:44,750 --> 01:14:49,850 Actually, the astronaut twin ages less than the Earth twin. 1559 01:14:49,850 --> 01:14:54,380 And that seems very crazy, because it's a completely 1560 01:14:54,380 --> 01:14:57,590 symmetric situation-- 1561 01:14:57,590 --> 01:15:00,298 or is it? 1562 01:15:00,298 --> 01:15:01,756 I don't know if that was effective. 1563 01:15:05,240 --> 01:15:08,320 I'll let you ponder that, that twin paradox. 1564 01:15:08,320 --> 01:15:13,040 It has a nice resolution, and I'll let you think about it. 1565 01:15:13,040 --> 01:15:17,670 But it goes against-- 1566 01:15:17,670 --> 01:15:22,470 well, it goes against this perceived symmetry. 1567 01:15:22,470 --> 01:15:24,170 Maybe it's not actually a symmetry. 1568 01:15:24,170 --> 01:15:25,730 Maybe it's not. 1569 01:15:25,730 --> 01:15:28,334 If your hand's up to answer the paradox-- 1570 01:15:28,334 --> 01:15:30,000 if you're trying to resolve the paradox, 1571 01:15:30,000 --> 01:15:33,089 then I'll have to ask-- 1572 01:15:33,089 --> 01:15:34,380 I'll listen to you after class. 1573 01:15:34,380 --> 01:15:35,420 If you have a different question, 1574 01:15:35,420 --> 01:15:36,315 then I'll listen to you. 1575 01:15:36,315 --> 01:15:37,050 Do you have a question? 1576 01:15:37,050 --> 01:15:37,780 AUDIENCE: No. 1577 01:15:37,780 --> 01:15:38,400 PROFESSOR: Oh, you had an answer. 1578 01:15:38,400 --> 01:15:38,870 OK. 1579 01:15:38,870 --> 01:15:39,370 Yeah. 1580 01:15:39,370 --> 01:15:41,850 I want you guys to think about that paradox. 1581 01:15:41,850 --> 01:15:46,120 And I'll send you an email with it stated more precisely 1582 01:15:46,120 --> 01:15:48,960 so that you can really enjoy it, so you can really 1583 01:15:48,960 --> 01:15:51,630 confused yourselves, so that you can really never sleep. 1584 01:15:51,630 --> 01:15:52,860 Because I don't want you guys to sleep at all. 1585 01:15:52,860 --> 01:15:53,943 You shouldn't be sleeping. 1586 01:15:53,943 --> 01:15:54,855 Great summer. 1587 01:15:54,855 --> 01:15:55,980 You should be up all night. 1588 01:15:55,980 --> 01:15:57,750 AUDIENCE: [INAUDIBLE] 1589 01:15:57,750 --> 01:15:58,650 PROFESSOR: Yeah. 1590 01:15:58,650 --> 01:16:01,100 Should be up all night.