1 00:00:01,948 --> 00:00:02,909 [SQUEAKING] 2 00:00:02,909 --> 00:00:03,409 [RUSTLE] 3 00:00:03,409 --> 00:00:04,870 [CLICKING] 4 00:00:19,510 --> 00:00:21,250 PROFESSOR: Welcome back to 8.20. 5 00:00:21,250 --> 00:00:25,270 In this section, we're going to talk about time, timekeeping, 6 00:00:25,270 --> 00:00:29,222 and how to relate time between two different reference point. 7 00:00:29,222 --> 00:00:32,400 Now, let me start with a quote by Albert Einstein. 8 00:00:32,400 --> 00:00:37,390 "Everything should be as simple as possible, but not simpler." 9 00:00:37,390 --> 00:00:39,240 So let's start with this in mind. 10 00:00:39,240 --> 00:00:43,080 And recall that we ended the last section 11 00:00:43,080 --> 00:00:46,620 by finding that the wave aether model doesn't really 12 00:00:46,620 --> 00:00:50,160 describe electromagnetic waves very well. 13 00:00:50,160 --> 00:00:54,240 We see that there is a problem between the experiments, 14 00:00:54,240 --> 00:00:57,480 specifically the one by Michelson-Morley, 15 00:00:57,480 --> 00:01:01,420 and the theoretical picture people had in mind. 16 00:01:01,420 --> 00:01:07,160 So Einstein approached this in an interesting way. 17 00:01:07,160 --> 00:01:10,960 He simply postulated the things he thought need to be true. 18 00:01:10,960 --> 00:01:13,810 He said, "The same law of electrodynamics 19 00:01:13,810 --> 00:01:16,690 will be valid for all reference frames 20 00:01:16,690 --> 00:01:20,350 where all laws of mechanics hold good. 21 00:01:20,350 --> 00:01:24,790 This is the principle of relativity." 22 00:01:24,790 --> 00:01:28,420 The second postulate is that "Light is always propagated 23 00:01:28,420 --> 00:01:31,810 in empty space at the velocity 'c,' independent of the state 24 00:01:31,810 --> 00:01:35,020 of motion of the emitting body." 25 00:01:35,020 --> 00:01:37,870 So with these two postulates, we will now 26 00:01:37,870 --> 00:01:40,580 derive the theory of special relativity. 27 00:01:40,580 --> 00:01:43,270 And again, we'll start by talking about time. 28 00:01:45,920 --> 00:01:47,710 So time is suspect. 29 00:01:47,710 --> 00:01:49,180 And I alluded to this already when 30 00:01:49,180 --> 00:01:51,100 we looked at Galilean transformation, 31 00:01:51,100 --> 00:01:53,890 where it simply, out of our intuition, 32 00:01:53,890 --> 00:01:57,410 assumed that time is invariant. 33 00:01:57,410 --> 00:02:02,320 Now, when we now talk about time, 34 00:02:02,320 --> 00:02:04,490 the viewpoint I would like you to have 35 00:02:04,490 --> 00:02:07,040 is that we want to look at clocks from different reference 36 00:02:07,040 --> 00:02:07,970 frames. 37 00:02:07,970 --> 00:02:10,639 We want to investigate whether or not events 38 00:02:10,639 --> 00:02:13,130 happen simultaneously or not. 39 00:02:13,130 --> 00:02:14,360 What does it mean? 40 00:02:14,360 --> 00:02:17,990 When we make a statement like a train arrives at 7 o'clock, 41 00:02:17,990 --> 00:02:21,950 what we mean is that there is a simultaneous-- two 42 00:02:21,950 --> 00:02:23,340 simultaneous events happen. 43 00:02:23,340 --> 00:02:29,770 One is that this little clock here shows 44 00:02:29,770 --> 00:02:32,500 to point at seven and 12, meaning 45 00:02:32,500 --> 00:02:35,590 that it indicates to us that-- this event indicates 46 00:02:35,590 --> 00:02:37,480 to us that it's 7 o'clock. 47 00:02:37,480 --> 00:02:39,700 And the second event is that the train actually 48 00:02:39,700 --> 00:02:41,200 arrives at the station. 49 00:02:41,200 --> 00:02:44,800 So those two events happen simultaneously. 50 00:02:44,800 --> 00:02:46,930 The question now is whether or not 51 00:02:46,930 --> 00:02:50,560 two observers, one stationary and one moving, 52 00:02:50,560 --> 00:02:52,900 agree with this observation. 53 00:02:52,900 --> 00:02:55,690 And I take it away-- the answer is no. 54 00:02:55,690 --> 00:02:59,320 There is a relativity of simultaneity, 55 00:02:59,320 --> 00:03:02,380 meaning that two observers can very much agree 56 00:03:02,380 --> 00:03:04,430 on the description of two events, 57 00:03:04,430 --> 00:03:06,460 but not necessarily that those two 58 00:03:06,460 --> 00:03:11,000 events happen simultaneously. 59 00:03:11,000 --> 00:03:11,950 So let's investigate. 60 00:03:11,950 --> 00:03:14,320 And we use our two friends, Alice and Bob, in order 61 00:03:14,320 --> 00:03:16,736 to have this discussion. 62 00:03:16,736 --> 00:03:19,440 All right, so we start from a situation 63 00:03:19,440 --> 00:03:22,110 where Alice and Bob are both stationary. 64 00:03:22,110 --> 00:03:24,360 Alice is on her spacecraft, and she 65 00:03:24,360 --> 00:03:28,395 has a device on her spacecraft which shoots light or paint 66 00:03:28,395 --> 00:03:30,540 balls towards two clocks. 67 00:03:30,540 --> 00:03:35,030 And each time this happens, the clock ticks, right? 68 00:03:35,030 --> 00:03:36,750 And we just look at one situation. 69 00:03:36,750 --> 00:03:40,250 So she has a clock on the left and a clock on the right. 70 00:03:40,250 --> 00:03:42,830 Bob observes Alice's clock. 71 00:03:42,830 --> 00:03:46,610 And he can compare this observation of Alice's clock 72 00:03:46,610 --> 00:03:49,530 with his own. 73 00:03:49,530 --> 00:03:51,290 So in this station, as to duration, 74 00:03:51,290 --> 00:03:54,290 there's a TA and a TB. 75 00:03:54,290 --> 00:03:56,810 Those are the times of Alice and Bob. 76 00:03:56,810 --> 00:03:57,470 Both are 0. 77 00:03:57,470 --> 00:03:59,660 This is when the situation starts. 78 00:03:59,660 --> 00:04:04,460 And the capital T indicates for Alice and for Bob 79 00:04:04,460 --> 00:04:07,880 when they observe that the clock has been hit. 80 00:04:10,460 --> 00:04:13,770 I should add here that when we talk about observation 81 00:04:13,770 --> 00:04:17,779 in this entire class, unless I make a very explicit exception 82 00:04:17,779 --> 00:04:22,220 to this, we don't consider the fact that observing actually 83 00:04:22,220 --> 00:04:25,610 means that light has to be emitted from the clock 84 00:04:25,610 --> 00:04:28,100 and enters Bob's eye in order for him 85 00:04:28,100 --> 00:04:30,770 to conclude that there was something happening. 86 00:04:30,770 --> 00:04:37,920 The observation is like taking an instantaneous picture. 87 00:04:37,920 --> 00:04:39,750 OK, so we have to keep this in mind. 88 00:04:39,750 --> 00:04:43,230 But in this simple situation, nothing is moving. 89 00:04:43,230 --> 00:04:48,000 We can hopefully agree that the times being 90 00:04:48,000 --> 00:04:51,600 read for Alice and Bob on the left and the right clock 91 00:04:51,600 --> 00:04:54,460 are all the same. 92 00:04:54,460 --> 00:04:56,190 Now, we go in the second situation, 93 00:04:56,190 --> 00:04:59,910 where we use the same device but with a paintball. 94 00:04:59,910 --> 00:05:03,750 So now, Alice moves and Bob is observing her. 95 00:05:03,750 --> 00:05:06,600 She moves with a relative velocity, v, 96 00:05:06,600 --> 00:05:09,930 and shoots off the paint balls with a velocity, u. 97 00:05:09,930 --> 00:05:16,260 The velocities will add, meaning that the answers to clocks 98 00:05:16,260 --> 00:05:17,550 are initially synchronized. 99 00:05:17,550 --> 00:05:23,910 So there is a small tA equals small tB equals 0. 100 00:05:23,910 --> 00:05:26,400 Once the clock hits, you can hopefully 101 00:05:26,400 --> 00:05:29,430 agree that Alice and Bob will agree 102 00:05:29,430 --> 00:05:33,150 that the times when the left clock and the right clock hit 103 00:05:33,150 --> 00:05:36,270 are the same, right? 104 00:05:36,270 --> 00:05:39,180 But now, we want to enter the situation where we use light. 105 00:05:39,180 --> 00:05:41,950 So we use a phaser in order to do the very same. 106 00:05:41,950 --> 00:05:44,850 So Einstein just postulated that the speed of light 107 00:05:44,850 --> 00:05:46,260 is constant, is c. 108 00:05:46,260 --> 00:05:48,300 And it's the same in all reference frames. 109 00:05:48,300 --> 00:05:50,490 And it's independent of the emitter, 110 00:05:50,490 --> 00:05:54,420 which means that we cannot add the velocities anymore. 111 00:05:54,420 --> 00:05:59,340 So the velocity, as seen by Alice, of light is c. 112 00:05:59,340 --> 00:06:04,350 The velocity of the same light by the moving observer, Bob, 113 00:06:04,350 --> 00:06:05,620 is also c. 114 00:06:05,620 --> 00:06:10,290 So here, we can conclude that the times for Alice 115 00:06:10,290 --> 00:06:14,130 for where the situation is stationary, both clocks 116 00:06:14,130 --> 00:06:16,900 will hit at the very same time. 117 00:06:16,900 --> 00:06:19,660 Those two events, clock one and clock two 118 00:06:19,660 --> 00:06:22,150 are hit are simultaneous. 119 00:06:22,150 --> 00:06:25,860 Why, for Bob, this is clearly not the case. 120 00:06:25,860 --> 00:06:33,730 You can see here that this lagging clock 121 00:06:33,730 --> 00:06:36,790 is being hit first, while the leading clock is 122 00:06:36,790 --> 00:06:39,710 hit a little while after. 123 00:06:39,710 --> 00:06:44,120 So if Bob and Alice now meet and they discuss whether or not 124 00:06:44,120 --> 00:06:46,370 those two events happened simultaneously, 125 00:06:46,370 --> 00:06:47,810 they will disagree. 126 00:06:47,810 --> 00:06:51,800 For Alice, those two clocks were hit simultaneously-- 127 00:06:51,800 --> 00:06:53,330 at the same time for her. 128 00:06:53,330 --> 00:06:57,530 But for Bob, the first clock was hit first 129 00:06:57,530 --> 00:07:01,900 and the leading clock was hit second. 130 00:07:01,900 --> 00:07:05,800 All right, we can conclude the two events 131 00:07:05,800 --> 00:07:08,680 can be simultaneously to one observer 132 00:07:08,680 --> 00:07:11,020 but not to another one. 133 00:07:11,020 --> 00:07:13,360 This is rather confusing. 134 00:07:13,360 --> 00:07:15,980 And we will see and use this fact 135 00:07:15,980 --> 00:07:20,110 a few times later on when we discuss the famous paradoxes 136 00:07:20,110 --> 00:07:23,570 of special relativity. 137 00:07:23,570 --> 00:07:26,500 So let's look at this in a concept question 138 00:07:26,500 --> 00:07:28,810 to just make sure that we're all on the same page. 139 00:07:28,810 --> 00:07:31,340 Again, we discuss your diagram three. 140 00:07:31,340 --> 00:07:32,980 Alice move to the right. 141 00:07:32,980 --> 00:07:34,420 Bob is the observer. 142 00:07:34,420 --> 00:07:40,150 Alice's fires her phaser at times equal 0. 143 00:07:40,150 --> 00:07:42,220 Then the situation unfolds. 144 00:07:42,220 --> 00:07:44,950 At time TA, capital TA, Alice observes 145 00:07:44,950 --> 00:07:46,720 that both blocks are hit. 146 00:07:46,720 --> 00:07:50,230 At time TB1, Bob observes that the left clock is hit. 147 00:07:50,230 --> 00:07:53,200 At time TB2, it's the right clock. 148 00:07:53,200 --> 00:07:56,060 Which of the following answers is correct? 149 00:07:56,060 --> 00:07:58,000 So here, you want to stop the video 150 00:07:58,000 --> 00:08:01,510 and think about which of the answers is correct. 151 00:08:01,510 --> 00:08:04,240 So moving forward, the correct answer 152 00:08:04,240 --> 00:08:08,290 is number three, where TB1 is smaller 153 00:08:08,290 --> 00:08:11,380 than TA is smaller than TB2. 154 00:08:11,380 --> 00:08:14,150 So again, the leading clock lags. 155 00:08:14,150 --> 00:08:16,810 The leading clock has a larger TB, 156 00:08:16,810 --> 00:08:21,740 which means that clock ticks a little slower. 157 00:08:21,740 --> 00:08:24,620 And again, the two events, they can be simultaneously 158 00:08:24,620 --> 00:08:25,580 to one observer-- 159 00:08:25,580 --> 00:08:28,310 Alice, in this case-- but not to another, Bob. 160 00:08:31,160 --> 00:08:33,380 All right, let's look at clocks a little bit more 161 00:08:33,380 --> 00:08:36,510 and design an optical clock. 162 00:08:36,510 --> 00:08:39,429 So here, the situation is as follows. 163 00:08:39,429 --> 00:08:42,240 We have two mirrors in which we inject light. 164 00:08:42,240 --> 00:08:44,910 The light travels up and travels down. 165 00:08:44,910 --> 00:08:47,130 And that's what we call one clock 166 00:08:47,130 --> 00:08:49,440 tick of this optical clock. 167 00:08:49,440 --> 00:08:51,930 The length between the two mirrors is L. 168 00:08:51,930 --> 00:08:54,570 So for Alice, she has this clock in her hand. 169 00:08:54,570 --> 00:09:00,450 And she can happily observe the ticking of the clock. 170 00:09:00,450 --> 00:09:03,000 OK, Bob observes Alice's clock and compares it 171 00:09:03,000 --> 00:09:05,430 with his own identical clock. 172 00:09:05,430 --> 00:09:07,650 There's a relative speed between Alice and Bob, 173 00:09:07,650 --> 00:09:11,230 and that's v in x direction. 174 00:09:11,230 --> 00:09:15,210 Now, the task for you is to relate the clock ticks which 175 00:09:15,210 --> 00:09:18,570 are observed by Bob and the ones which are observed 176 00:09:18,570 --> 00:09:21,480 by Alice in Alice's clock. 177 00:09:21,480 --> 00:09:25,800 So again, stop the video and work out the algebra. 178 00:09:25,800 --> 00:09:28,380 The answer is going to be, again, surprising. 179 00:09:30,980 --> 00:09:36,230 So if you do this now, we find this picture. 180 00:09:36,230 --> 00:09:38,660 So we calculate how long does a clock tick take. 181 00:09:38,660 --> 00:09:42,020 The light has to travel to L with a velocity c. 182 00:09:42,020 --> 00:09:44,360 So the clock tick is 2L over c. 183 00:09:44,360 --> 00:09:49,640 The length can be expressed as c times t delta tA over 2. 184 00:09:49,640 --> 00:09:52,220 For Bob's, the situation is a little bit more complicated. 185 00:09:52,220 --> 00:09:57,290 And we have to use Pythagoras in order to calculate the length. 186 00:09:57,290 --> 00:09:59,570 So we define that the length the light has to travel 187 00:09:59,570 --> 00:10:02,870 is D, then the delta tB as Bob observes 188 00:10:02,870 --> 00:10:06,080 this is 2 times D over c. 189 00:10:06,080 --> 00:10:08,390 Again, for Bob, the light travels 190 00:10:08,390 --> 00:10:09,740 with the speed of light. 191 00:10:09,740 --> 00:10:12,020 Einstein just postulated it. 192 00:10:12,020 --> 00:10:15,080 And then we find the length as expressed to the time 193 00:10:15,080 --> 00:10:18,430 as c times delta tB over 2. 194 00:10:18,430 --> 00:10:22,750 The length in x is simply given by the relative velocity, v, 195 00:10:22,750 --> 00:10:25,690 times the time it takes for the clock to tick-- 196 00:10:25,690 --> 00:10:27,640 v times delta tB. 197 00:10:27,640 --> 00:10:33,010 So then we can express D square via L squared plus x squared 198 00:10:33,010 --> 00:10:36,380 over 4 and use those expressions here. 199 00:10:36,380 --> 00:10:40,370 So we just use this for L, this for D, and this 4x, 200 00:10:40,370 --> 00:10:42,970 we find this expression here. 201 00:10:42,970 --> 00:10:43,780 All right. 202 00:10:43,780 --> 00:10:46,780 And then we solve this for delta tB. 203 00:10:46,780 --> 00:10:50,740 And we find the relation between delta tB and delta tA 204 00:10:50,740 --> 00:10:54,010 and can find that it's 1 over square root 205 00:10:54,010 --> 00:10:56,470 1 minus v squared over c squared, 206 00:10:56,470 --> 00:10:58,930 which is the Lorentz factor. 207 00:10:58,930 --> 00:11:02,590 So we just used a simple clock and Einstein's postulate 208 00:11:02,590 --> 00:11:05,410 derived time dilation. 209 00:11:05,410 --> 00:11:11,020 We find that for Bob, Alice's moving clock moves slower. 210 00:11:14,250 --> 00:11:15,000 Great. 211 00:11:15,000 --> 00:11:19,230 So again, gamma is 1 over square root 1 212 00:11:19,230 --> 00:11:20,790 minus v squared over c squared. 213 00:11:20,790 --> 00:11:25,500 We often use, in short, beta as a relativistic velocity. 214 00:11:25,500 --> 00:11:29,670 It's unitless and defined as v over c. 215 00:11:29,670 --> 00:11:32,580 gamma is always greater or equal to 1. 216 00:11:32,580 --> 00:11:34,260 And it's mostly one for everything 217 00:11:34,260 --> 00:11:35,950 we observe in nature. 218 00:11:35,950 --> 00:11:39,600 So in one of the p sets and also here, 219 00:11:39,600 --> 00:11:42,960 I invite you to simply calculate values 220 00:11:42,960 --> 00:11:47,040 for gamma for things you might think are fast-moving objects. 221 00:11:47,040 --> 00:11:49,140 So we start with a fighter jet. 222 00:11:49,140 --> 00:11:51,480 We look at the International Space Station, 223 00:11:51,480 --> 00:11:54,550 the Earth around the sun, the particle which almost moves 224 00:11:54,550 --> 00:11:57,480 with the speed of light, and the proton at the Large Hadron 225 00:11:57,480 --> 00:12:00,660 Collider, which is only 3 meters per second slower 226 00:12:00,660 --> 00:12:02,860 than the speed of light. 227 00:12:02,860 --> 00:12:06,360 So again, stop the video and work out those numbers. 228 00:12:06,360 --> 00:12:09,770 You will need a calculator for that. 229 00:12:09,770 --> 00:12:12,950 So if I do this, I find for this very, 230 00:12:12,950 --> 00:12:17,830 very fast F15 fighter jet, which moves with speeds of 2,680 231 00:12:17,830 --> 00:12:21,130 kilometers per hour, that the number for gamma is 232 00:12:21,130 --> 00:12:31,210 1.00000000000, which is 11 zeros, 3. 233 00:12:31,210 --> 00:12:34,660 So we find this very, very small number or number 234 00:12:34,660 --> 00:12:37,090 which is very, very close to 1. 235 00:12:37,090 --> 00:12:39,460 The duration for the International Space Station 236 00:12:39,460 --> 00:12:40,480 changes a bit-- 237 00:12:40,480 --> 00:12:42,185 only 9 zeros. 238 00:12:42,185 --> 00:12:43,810 For the Earth around the sun, the Earth 239 00:12:43,810 --> 00:12:46,210 is really, really fast, travels a long distance. 240 00:12:46,210 --> 00:12:48,880 Every year, we travel once around the sun. 241 00:12:48,880 --> 00:12:51,440 And you know, every year you get older. 242 00:12:51,440 --> 00:12:53,830 You have a lot of mileage on your back. 243 00:12:53,830 --> 00:12:55,750 Here, you have eight zeros. 244 00:12:55,750 --> 00:12:58,960 Particle which moves with 0.9 times the speed of light, 245 00:12:58,960 --> 00:13:02,410 here the gamma factor is very different from one. 246 00:13:02,410 --> 00:13:03,580 It's 2.3. 247 00:13:03,580 --> 00:13:05,710 And the protons we have at the LHC, 248 00:13:05,710 --> 00:13:07,930 they have a gamma factor of 7,000. 249 00:13:07,930 --> 00:13:10,570 So you see, once you get close to the speed of light, 250 00:13:10,570 --> 00:13:14,562 the gamma factor approaches large numbers. 251 00:13:14,562 --> 00:13:17,080 And that's where our relativistic effects really 252 00:13:17,080 --> 00:13:18,750 are visible.