1 00:00:00,840 --> 00:00:04,628 You may have loved the algebraic derivation of the Lorentz 2 00:00:04,628 --> 00:00:08,091 transformations. You may not have enjoyed the algebra. 3 00:00:08,091 --> 00:00:12,860 Either way, let's remember the logic. What we did is we assumed four facts. 4 00:00:12,860 --> 00:00:17,673 One, that if a blue observer is moving with speed V relative to a black 5 00:00:17,673 --> 00:00:20,926 observer. Then, the black observer is observed by 6 00:00:20,926 --> 00:00:23,841 the blue one to be moving with velocity minus v. 7 00:00:23,841 --> 00:00:28,383 And so, relativity works in that way. That was assumption number one. 8 00:00:28,383 --> 00:00:33,863 Assumption number two was that motion that is observed by the black observer to 9 00:00:33,863 --> 00:00:36,965 be inertial. In other words, to have a straight world 10 00:00:36,965 --> 00:00:41,021 line is observed by the blue observer to have a straight world line. 11 00:00:41,021 --> 00:00:45,257 In other words to be inertial. Newton's laws hold in both frames in the 12 00:00:45,257 --> 00:00:50,217 sense that if an object is observed to be experiencing no force in one frame. 13 00:00:50,217 --> 00:00:54,058 It's observed in the other frame to also be experiencing no force. 14 00:00:54,058 --> 00:00:58,307 That was assumption number two. Assumption number three was the statement 15 00:00:58,307 --> 00:01:01,392 that the velocity of light is the same in all frames. 16 00:01:01,392 --> 00:01:05,466 And putting all those assumptions together, we were ineluctably led to 17 00:01:05,466 --> 00:01:09,889 these Lorentz transformations. Which remember, relate the position and 18 00:01:09,889 --> 00:01:13,091 time of any event. Something happening, a birthday party 19 00:01:13,091 --> 00:01:15,652 starting. As measured by one observer to the 20 00:01:15,652 --> 00:01:19,610 position and time of the same event as measured by another observer. 21 00:01:19,610 --> 00:01:24,315 Two observations about this before we go on, to derive the consequence of this, 22 00:01:24,315 --> 00:01:29,482 which are the meat of special relativity. two things to note right off the bat. 23 00:01:29,482 --> 00:01:33,623 observation number one is if you look at this denominator. 24 00:01:33,623 --> 00:01:37,833 You notice that if v^22 over c^22, is bigger or equal than to one. 25 00:01:37,833 --> 00:01:43,332 Then this denominator is either zero, or you get an infinity or it's negative and 26 00:01:43,332 --> 00:01:46,597 the radical is imaginary. What we conclude, is that if you put 27 00:01:46,597 --> 00:01:50,693 together are, moderest assumptions which we're going to assume as part of physics, 28 00:01:50,693 --> 00:01:54,435 along with the idea of two observers moving relative to each other with a 29 00:01:54,435 --> 00:01:57,166 speed larger than the speed of light you get nonsense. 30 00:01:57,166 --> 00:02:01,060 Conclusion, since we're not going to abandon relativaty, two observers can not 31 00:02:01,060 --> 00:02:04,650 move relative to each other with speeds larger than the speed of light. 32 00:02:04,650 --> 00:02:08,443 The speed V in all of are Lorentz transformations will forever be less than 33 00:02:08,443 --> 00:02:11,067 the speed of light. Therefore, when we talk about 34 00:02:11,067 --> 00:02:15,269 superluminal motion, you cannot accelerate yourself in any way, and we 35 00:02:15,269 --> 00:02:20,020 will see how physics stops you from accelerating yourself to speeds faster 36 00:02:20,020 --> 00:02:21,430 than light. So, 37 00:02:21,430 --> 00:02:23,615 comment one, v^22 is always less than c^2.2. 38 00:02:23,615 --> 00:02:26,698 Comment number two is I worked in a one-dimensional universe. 39 00:02:26,698 --> 00:02:29,303 We don't really live in a one dimensional universe. 40 00:02:29,303 --> 00:02:33,556 we live in a three-dimensional universe so you can ask what happens to the y and 41 00:02:33,556 --> 00:02:35,151 z directions. It's the motions. 42 00:02:35,151 --> 00:02:39,031 I can always pick my x direction. The point along the direction of motion, 43 00:02:39,031 --> 00:02:42,540 what happens in the other directions? Well, in the other directions, 44 00:02:42,540 --> 00:02:46,893 life is much simpler. y prime and z prime are simply given by y 45 00:02:46,893 --> 00:02:49,927 and z. Nothing happens that takes not too much 46 00:02:49,927 --> 00:02:54,611 calculation, so all of the action is indeed in one dimension, the direct, 47 00:02:54,611 --> 00:02:58,570 along the direction of motion. Let's see what we, what happens. 48 00:02:58,570 --> 00:03:03,761 Let's follow through, go to the consequences of adopting these 49 00:03:03,761 --> 00:03:08,654 Lorentz transformations. By far the most confusing one is the idea 50 00:03:08,654 --> 00:03:10,360 of simultaneity. Notice, 51 00:03:10,360 --> 00:03:18,105 we are not shocked that when we draw the now and the here of the black observer 52 00:03:18,105 --> 00:03:21,106 and then we draw the here of the blue observer. 53 00:03:21,106 --> 00:03:23,980 This describes a tilted world line. Well, good. 54 00:03:23,980 --> 00:03:28,515 That's not shocking. We expect this describes the fact, that the blue 55 00:03:28,515 --> 00:03:32,219 observer is moving. What is shocking and very confusing, is 56 00:03:32,219 --> 00:03:36,498 that if you look at the Lorentz transformations and we look for the 57 00:03:36,498 --> 00:03:41,608 collection of all events that, such that they are characterized by t prime equal 58 00:03:41,608 --> 00:03:46,398 to zero, you find that, that is not equivalent to t equal to zero, but rather 59 00:03:46,398 --> 00:03:51,761 to t equal vx / c^2 so that the, blue observers now differs from the black 60 00:03:51,761 --> 00:03:55,355 observers now. They agree and when I say now, these are 61 00:03:55,355 --> 00:04:00,681 all of the events in the history of the universe that each observer imagines to 62 00:04:00,681 --> 00:04:04,742 have occurred simultaneously with in this case, their meeting. 63 00:04:04,742 --> 00:04:08,536 Because that's how he said, t as t prime is zero. 64 00:04:08,536 --> 00:04:12,930 Is, they both started their watches at the moment when they met. 65 00:04:12,930 --> 00:04:18,388 And so this violation of simultaneity is the cause of much, much, much confusion, 66 00:04:18,388 --> 00:04:23,174 and many paradoxes. And to, to, to, get a sense of it for, 67 00:04:23,174 --> 00:04:29,321 for first how much should we worry. Let's measure the magnitude of this effect. 68 00:04:29,321 --> 00:04:32,634 So for example, when I am driving in my car. 69 00:04:32,634 --> 00:04:39,379 Things that I consider simultaneously are not the same as things that a bystander 70 00:04:39,379 --> 00:04:45,211 on the sidewalk considers simultaneous if you move far enough out in the direction 71 00:04:45,211 --> 00:04:48,724 of my motion. It's a happy coincidence that if I'm 72 00:04:48,724 --> 00:04:53,853 moving, say, at ten meters per second, a reasonably fast clip then I can 73 00:04:53,853 --> 00:04:58,432 estimate that our time our, our, our space axes or our definitions of 74 00:04:58,432 --> 00:05:02,508 simultaneity will differ by about a second for objects that are a light year 75 00:05:02,508 --> 00:05:04,995 away. So if something is happening a light year 76 00:05:04,995 --> 00:05:07,324 away, and I measure when I think it happened. 77 00:05:07,324 --> 00:05:11,241 And I'm driving in my car towards it. And someone standing on the sidewalk 78 00:05:11,241 --> 00:05:14,258 measures that time. We will disagree by a second over the 79 00:05:14,258 --> 00:05:17,381 distance of a light year. But how are you going to know what's 80 00:05:17,381 --> 00:05:19,975 going on right now, at a distance of a light year? 81 00:05:19,975 --> 00:05:24,210 That's a subject of our next question. So we'll, we'll address that in a second. 82 00:05:24,210 --> 00:05:29,583 Nonetheless when you are moving not at ten meters per second but at relativistic 83 00:05:29,583 --> 00:05:33,933 velocities, this idea of simultaneity meaning something different, is 84 00:05:33,933 --> 00:05:38,602 significant and understanding it correctly, or learning to wrap your mind 85 00:05:38,602 --> 00:05:43,530 around the weirdness that it is, is the, the, the, the main obstacle to 86 00:05:43,530 --> 00:05:48,420 feel comfortable with relativity. And that comes from working problems. 87 00:05:48,420 --> 00:05:54,315 We'll probably have an optional clip where I will work paradoxes in relativity 88 00:05:54,315 --> 00:05:59,741 to give you a sense for how that works. you'll note that when I've drawn these 89 00:05:59,741 --> 00:06:05,636 diagrams, I am often drawing the red lines which are the world lines. Remember 90 00:06:05,636 --> 00:06:08,718 of the two light beams at 45 degree angles. 91 00:06:08,718 --> 00:06:13,502 Now, the tilt of the world line from the vertical should correspond to the 92 00:06:13,502 --> 00:06:17,343 velocity or the, the speed with which something is moving. 93 00:06:17,343 --> 00:06:21,056 So essentially I'm saying that light moves with speed one, 94 00:06:21,056 --> 00:06:25,857 the way that's typically implemented is that you plot not t, but ct, on your 95 00:06:25,857 --> 00:06:28,865 diagram. So you're effectively working in units 96 00:06:28,865 --> 00:06:33,680 such that the speed of light is one. In my field we do that all the time 97 00:06:33,680 --> 00:06:37,585 anyway and so I have to remind myself to put the c's in. 98 00:06:37,585 --> 00:06:42,884 The c's will be in all my equations. I do not promise to always remember the 99 00:06:42,884 --> 00:06:47,696 to plot ct on this plot. But I am indeed plotting ct and if you 100 00:06:47,696 --> 00:06:53,553 remember that what you are plotting is ct then you can see that the geometry of the 101 00:06:53,553 --> 00:06:57,040 situation is as depicted here the 45 degree 102 00:06:57,040 --> 00:07:03,874 world line of a light beam disects the angle between the time and space axis for 103 00:07:03,874 --> 00:07:09,977 any observer so if you have a green observer who is moving faster than the 104 00:07:09,977 --> 00:07:17,614 blue observer that means their t double prime access will be tilted farther away 105 00:07:17,614 --> 00:07:23,693 from the vertical. Then, so well, there x double prime axis 106 00:07:23,693 --> 00:07:30,834 be tilted farther from the horizontal. This is the geometry of the Lorentz 107 00:07:30,834 --> 00:07:37,700 transformations that we wrote up. So, how do we know what's going on? 108 00:07:37,700 --> 00:07:41,679 A light year away right now? What do we actually mean by right now? 109 00:07:41,679 --> 00:07:46,383 Well now that we understand that the speed of light is a constant we can give 110 00:07:46,383 --> 00:07:50,664 a technical measurable operative meaning to what is going on right now. 111 00:07:50,664 --> 00:07:55,488 if for example 6,000 light years away on the crab nebula something is happening 112 00:07:55,488 --> 00:07:58,286 now. Well, I don't know it yet but in exactly 113 00:07:58,286 --> 00:08:01,481 6,000 years the light from there will reach Earth. 114 00:08:01,481 --> 00:08:06,402 So I define what's going on now at the crab nebula to be what we will see on 115 00:08:06,402 --> 00:08:10,428 Earth in 6,000 years. If I make that definition that now I have 116 00:08:10,428 --> 00:08:15,030 an operational meaning of what the present is at any distance things 117 00:08:15,030 --> 00:08:20,014 happening now are the things from which we will hear at a time related to its 118 00:08:20,014 --> 00:08:24,552 distance from us by this expression. And so if you use that operational 119 00:08:24,552 --> 00:08:27,620 definition of what you mean by the present, then. 120 00:08:27,620 --> 00:08:30,725 Following through the behavior of light beams. 121 00:08:30,725 --> 00:08:34,100 Drawing them onto that diagram that I drew before. 122 00:08:34,100 --> 00:08:39,027 You find that this relativity of simultaneity is obviously a fact is a, a 123 00:08:39,027 --> 00:08:43,752 property of the fact that light moves at the same speed in all frames. 124 00:08:43,752 --> 00:08:46,117 Okay, we'll swallow simultaneity, will work 125 00:08:46,117 --> 00:08:48,877 through it. it's only, it only matters at large 126 00:08:48,877 --> 00:08:52,932 distances or high speed, so the fact that it conflicts with our everyday 127 00:08:52,932 --> 00:08:57,212 experience, well, we don't have everyday experience of moving at the speed of 128 00:08:57,212 --> 00:08:59,577 light. Well what other weird things do the 129 00:08:59,577 --> 00:09:02,450 Lorentz transformations teach us? Well, for starters, 130 00:09:02,450 --> 00:09:06,013 that teaches about something called length contraction which is a very 131 00:09:06,013 --> 00:09:09,778 puzzling a property and we need to understand it so let's see what it does 132 00:09:09,778 --> 00:09:11,937 how do you measure the length of something? 133 00:09:11,937 --> 00:09:15,049 Well, here's the idea. suppose that the moving blue observer is 134 00:09:15,049 --> 00:09:17,760 holding a ruler this ruler has nominal length l prime. 135 00:09:17,760 --> 00:09:21,863 Bought it at a store, it's a one meter ruler, so L prime may be one meter. 136 00:09:21,863 --> 00:09:26,423 He's holding it and so it's left end is in his hand at X prime equals zero, it's 137 00:09:26,423 --> 00:09:29,957 left end is a distance one meter, L prime, to the right of that. 138 00:09:29,957 --> 00:09:34,517 And of course it's moving with him and he measures its distance and he says the 139 00:09:34,517 --> 00:09:38,621 distance is L prime, very good. We want to know what happens if the black 140 00:09:38,621 --> 00:09:41,300 observer measures the length of the same ruler? 141 00:09:41,300 --> 00:09:46,131 So how do we do that we turn our friends the Lorentz transformations which contain 142 00:09:46,131 --> 00:09:50,848 all of the information and we use these transformations to understand where the 143 00:09:50,848 --> 00:09:55,452 black observer sees the ends of the ruler so not too difficult let's see what we 144 00:09:55,452 --> 00:09:58,586 get. the left end of the ruler is at all time 145 00:09:58,586 --> 00:10:03,259 at x prime equal to zero. It's clear from setting this that x prime 146 00:10:03,259 --> 00:10:06,397 equals 0 corresponds to x equals vt. So, good. 147 00:10:06,397 --> 00:10:11,871 The left end of the ruler is in the hands of the blue observer, and as seen by the 148 00:10:11,871 --> 00:10:16,210 black observer, it's moving to the left, to the right with speed v. 149 00:10:16,210 --> 00:10:19,080 Good. What about the right end of the ruler? 150 00:10:19,080 --> 00:10:23,820 I set x prime equal to l prime, and I get an equation that says l prime. 151 00:10:23,820 --> 00:10:34,275 is x2 - vt divide it, at any time by the square root of 1 - v^2 / c^2. 152 00:10:34,275 --> 00:10:38,780 So I can solve this. I multiply through by the square root. 153 00:10:38,780 --> 00:10:48,360 Erase it here, put it here. And I then move the vt over to the other 154 00:10:48,360 --> 00:10:55,226 side and I find that x2 is vt + l prime route 1 - v^2 / c^2. 155 00:10:55,226 --> 00:10:57,136 Okay, let's write that out cleanly. 156 00:10:57,136 --> 00:10:58,372 x1 is this, x2 is this. 157 00:10:58,372 --> 00:11:01,237 Well, good. Both ends of the ruler are moving to the 158 00:11:01,237 --> 00:11:05,226 right at speed v because it's at rest in the moving frame, and the distance 159 00:11:05,226 --> 00:11:06,911 between them is constant. Good. 160 00:11:06,911 --> 00:11:10,731 The ruler is not wiggling. What is the distance between the two ends 161 00:11:10,731 --> 00:11:13,934 at any given time? At any given time, this is the distance 162 00:11:13,934 --> 00:11:18,372 between the two ends of the ruler, so this is what you'd call the length of the 163 00:11:18,372 --> 00:11:23,368 ruler, and so I conclude that the length of the ruler is this number, and since 164 00:11:23,368 --> 00:11:29,143 this multiplier here this square root is smaller than one, the ruler shrunk. 165 00:11:29,143 --> 00:11:33,875 I see a shorter ruler than the person moving with the ruler sees. 166 00:11:33,875 --> 00:11:38,260 In other words when you move something fast its length shrinks. 167 00:11:38,260 --> 00:11:41,553 Along the direction of motion. This is a real effect. 168 00:11:41,553 --> 00:11:46,494 Now, of course, relativity says that if I were holding, if the black observer is 169 00:11:46,494 --> 00:11:51,245 holding a ruler of length L, and you ask what length does the blue observer 170 00:11:51,245 --> 00:11:55,553 measure for that same ruler? Well, same expression, flip the sign of v 171 00:11:55,553 --> 00:11:58,912 makes no difference. The blue observer thinks the black 172 00:11:58,912 --> 00:12:03,516 observer's ruler is shrinking and of course these equations are inconsistent 173 00:12:03,516 --> 00:12:08,358 and so what you have to keep in mind is that they mean two very different things. 174 00:12:08,358 --> 00:12:12,902 This is the length as measured by the black observer of a ruler of length L 175 00:12:12,902 --> 00:12:16,788 prime at rest in the blue frame, so moving with the blue observer. 176 00:12:16,788 --> 00:12:21,571 This is the length as measured by the blue observer of a ruler of length L that 177 00:12:21,571 --> 00:12:26,115 is moving with the black observer. Anything when it moves relative to us and 178 00:12:26,115 --> 00:12:30,360 I don't need an observer for it, the ruler is an observer all by itself. 179 00:12:30,360 --> 00:12:33,551 When a ruler moves along its length, it appears shorter. 180 00:12:33,551 --> 00:12:37,207 Again, unless the speeds are relativistic, this is not important. 181 00:12:37,207 --> 00:12:41,270 When the speeds are relativistic, this is important, and it does happen. 182 00:12:41,270 --> 00:12:44,870 So, moving fast messes with your idea of length. 183 00:12:44,870 --> 00:12:48,689 It also messes with your idea of time. We know about rulers. 184 00:12:48,689 --> 00:12:52,832 Let's think about clocks. So imagine that the moving observer is 185 00:12:52,832 --> 00:12:56,134 holding a clock in their hand. And that clock ticks. 186 00:12:56,134 --> 00:13:01,054 And we'll select two specific ticks. One at the time of the historic meeting 187 00:13:01,054 --> 00:13:04,873 between the two observers at x prime = t prime = 0x= = x = t = 0. 188 00:13:04,873 --> 00:13:08,175 And then on the one sometime later say a second. 189 00:13:08,175 --> 00:13:13,095 So the moving observer measures the tick of the clock at t0, = 0, t prime = 0/ And 190 00:13:13,095 --> 00:13:17,151 then at t prime = 1 or capital T prime. I want to know, what does the black 191 00:13:17,151 --> 00:13:21,379 observer think about this moving clock? In this case, it's easier to use the 192 00:13:21,379 --> 00:13:25,324 inverse Lorentz transformations. The ones that give me x and t and 193 00:13:25,324 --> 00:13:29,270 functions of x prime t prime. Remember they're no harder to write down 194 00:13:29,270 --> 00:13:32,982 than the other ones. You just flip the sign of v and so we're 195 00:13:32,982 --> 00:13:35,774 working at x prime = to zero in this case. 196 00:13:35,774 --> 00:13:39,450 So that's easy. Both terms with x prime are irrelevant 197 00:13:39,450 --> 00:13:44,488 for all of this process because the clock is always at x prime = to 0. 198 00:13:44,488 --> 00:13:49,322 And so I set t prime equal to 0. And I find directly that t1 is zero. 199 00:13:49,322 --> 00:13:53,430 Okay, that's good. We agree that the clock was ticking at 200 00:13:53,430 --> 00:13:57,516 the same event, which was the meeting of the observers. 201 00:13:57,516 --> 00:14:02,055 Excellent, we agree about that. Events are unambiguous things. 202 00:14:02,055 --> 00:14:05,082 But then, what about t2? Well t2 is equal, 203 00:14:05,082 --> 00:14:10,151 t prime is now blue t prime. Pardon me for not respecting the color 204 00:14:10,151 --> 00:14:11,967 scheme. But so it goes divided by 12. 205 00:14:11,967 --> 00:14:15,119 - v^2 / c^22. and indeed, if you ask the black 206 00:14:15,119 --> 00:14:17,358 observer. When did the clock tick? 207 00:14:17,358 --> 00:14:20,886 This is the time when it ticked for the second time. 208 00:14:20,886 --> 00:14:24,685 This when it ticked for the first time. Write those down. 209 00:14:24,685 --> 00:14:29,841 It is clear that the time interval between those two is a lot longer than t 210 00:14:29,841 --> 00:14:32,623 prime. It is t prime divided this quantity less 211 00:14:32,623 --> 00:14:35,743 than one. The black observer observes the blue 212 00:14:35,743 --> 00:14:39,685 clock to be running slow. Of course, by the same relativistic 213 00:14:39,685 --> 00:14:43,954 argument we're used to by now, if the blue observer looks at a clock that is 214 00:14:43,954 --> 00:14:48,316 stationary in the black frame, if you observe that the black blocks are moving 215 00:14:48,316 --> 00:14:50,713 slow. And again, the fact that these two are 216 00:14:50,713 --> 00:14:55,006 consistent just like the fact that the other two length contraction equation 217 00:14:55,006 --> 00:14:59,466 expressions are consistent is associated with being confused about simultaneity. 218 00:14:59,466 --> 00:15:02,867 What you mean by simultaneity in the two frames is different. 219 00:15:02,867 --> 00:15:06,993 And that makes these, it possible for each observer to observe the other's 220 00:15:06,993 --> 00:15:10,171 clocks moving slow. We'll deal with that in the paradoxes 221 00:15:10,171 --> 00:15:14,520 section and we'll soon discuss about, something about why we know this is true. 222 00:15:14,520 --> 00:15:18,600 First, let's follow one important astronomical consequence of this. 223 00:15:18,600 --> 00:15:22,951 We have Doppler shifts a we describe Doppler shifts in the traditional sense 224 00:15:22,951 --> 00:15:27,529 as having to do with radial motion if an object is moving away from me then each 225 00:15:27,529 --> 00:15:31,994 subsequent wave front that it emits has a longer distance to go so I get a red 226 00:15:31,994 --> 00:15:34,424 shift that's moving at negative blue shift. 227 00:15:34,424 --> 00:15:39,068 If it's moving transversely there is no Doppler shift but now there is see, the 228 00:15:39,068 --> 00:15:43,490 reason for their, their there's a Doppler shift that's imagined that an object that 229 00:15:43,490 --> 00:15:47,751 is moving has some atom in it that is vibrating the period in which that atom 230 00:15:47,751 --> 00:15:52,066 vibrates is determined by the physics of the atom and that physics is invariant 231 00:15:52,066 --> 00:15:55,302 because of relativity. And so that atom addressed in its own 232 00:15:55,302 --> 00:15:59,293 frame is vibrating at exactly the same frequency that the same atom would 233 00:15:59,293 --> 00:16:02,637 vibrate when I held it in my hand here addressed in my frame. 234 00:16:02,637 --> 00:16:06,901 But when moving, it vibrates with the same rate relative 235 00:16:06,901 --> 00:16:12,453 to the clock that I observed to run slow. Thusly, the vibrations of the atom are 236 00:16:12,453 --> 00:16:15,476 slowed down by any motion so there's a redshift. 237 00:16:15,476 --> 00:16:18,814 There's always a redshift. The frequency is decreased. 238 00:16:18,814 --> 00:16:23,600 the speed of light is constant. The frequency times the speed of light is 239 00:16:23,600 --> 00:16:26,623 always. the frequency times the wavelength is 240 00:16:26,623 --> 00:16:30,843 always the speed of light. So there is a relativistic redshift that 241 00:16:30,843 --> 00:16:34,559 occurs even if in the case when there is transverse motion. 242 00:16:34,559 --> 00:16:39,220 There's no radial motion whatsoever. And this has to do with time dilation. 243 00:16:39,220 --> 00:16:43,362 So there's always a red shift. what happens if the object happens to 244 00:16:43,362 --> 00:16:47,219 actually be moving radially? Well, then you have this time delation 245 00:16:47,219 --> 00:16:50,350 effect but in, that happens independent of the direction. 246 00:16:50,350 --> 00:16:54,654 But in, in addition to this, you have the geometric fact of how far each light 247 00:16:54,654 --> 00:16:57,897 frame has to go. So you basically multiply the two effects 248 00:16:57,897 --> 00:17:01,251 and you find the relativistic formula for the Doppler shift. 249 00:17:01,251 --> 00:17:05,866 notice I'm using the astronomical convention here where the positive sign 250 00:17:05,866 --> 00:17:11,327 of v means the object is receding from us so we've now converted completely to the 251 00:17:11,327 --> 00:17:16,854 astronomical radial velocity component so this is for a radial or longitudinal I 252 00:17:16,854 --> 00:17:21,790 perhaps should have called it radial because that's what we have called it. 253 00:17:21,790 --> 00:17:26,200 And transverse is tangential when looking at a star. 254 00:17:26,200 --> 00:17:31,458 So all observations are have Doppler shift. 255 00:17:31,458 --> 00:17:35,869 When an object is moving at us sufficiently rapidly. 256 00:17:35,869 --> 00:17:43,587 the the, the blue shift from negative v here can overcome the red shift from 257 00:17:43,587 --> 00:17:47,403 this. But this is the relativistic expression 258 00:17:47,403 --> 00:17:51,032 for a Doppler shift. Okay, that's a lot to swallow. 259 00:17:51,032 --> 00:17:53,297 Is this for real? What is going on? 260 00:17:53,297 --> 00:17:56,895 The answer is yes. We have a lot, a ton of experimental 261 00:17:56,895 --> 00:18:00,960 evidence for each and every one of these phenomenon. 262 00:18:00,960 --> 00:18:06,278 we observe for example, length contraction, when we collide large ions 263 00:18:06,278 --> 00:18:12,022 in accelerator experiments, we accelerate a gold nuclei to high energies, to 264 00:18:12,022 --> 00:18:17,766 velocities close to the speed of light and these are large objects remember they 265 00:18:17,766 --> 00:18:23,581 are hundreds of ferometer across and when they collide the collision happens much 266 00:18:23,581 --> 00:18:26,963 too fast. For relative to the size of these nuclei 267 00:18:26,963 --> 00:18:31,517 and the reason is that as they observe each other well because of their large 268 00:18:31,517 --> 00:18:35,720 relative velocity, these things are colliding as essentially pancakes. 269 00:18:35,720 --> 00:18:40,332 And therefore they pass through each other in a much faster time than would 270 00:18:40,332 --> 00:18:44,769 the nuclei given their size and the speeds that we know because we know how 271 00:18:44,769 --> 00:18:48,564 fast we've accelerated them. time delation is something that's 272 00:18:48,564 --> 00:18:53,039 measured all the time if you want to give we, we take unstable particles. 273 00:18:53,039 --> 00:18:55,543 Unstable particles decay within some time. 274 00:18:55,543 --> 00:19:00,195 We put them in an accelerator, speed them up to a large fraction, a substantial 275 00:19:00,195 --> 00:19:03,356 fraction of the speed of light and they last forever. 276 00:19:03,356 --> 00:19:05,622 Why? Because their clocks are retarded. 277 00:19:05,622 --> 00:19:10,453 Our time their time is dilated relative to ours and we can keep you unstable for 278 00:19:10,453 --> 00:19:14,868 hours by moving them fast enough. there are more low speed measurement of 279 00:19:14,868 --> 00:19:17,030 this. In fact, the lowest speed, I think at 280 00:19:17,030 --> 00:19:21,514 which this has been successfully measured has been obtained by taking really high 281 00:19:21,514 --> 00:19:25,628 precision atomic clocks, putting one of them on the runway, putting another one 282 00:19:25,628 --> 00:19:29,901 on a jet that circumnavigated the globe a couple of times and then noticing when 283 00:19:29,901 --> 00:19:34,226 they landed that the one that was moving fast was in fact, behind the atomic clock 284 00:19:34,226 --> 00:19:37,813 that had been left on the ground. The effects of jet speeds are tiny. 285 00:19:37,813 --> 00:19:40,240 So, you need very high precision atomic clocks. 286 00:19:40,240 --> 00:19:42,508 There is no doubt that these are all facts. 287 00:19:42,508 --> 00:19:45,702 They're counter intuitive, too bad. We have to change our intuition. 288 00:19:45,702 --> 00:19:48,800 now how does it happen? And the other question is also how you 289 00:19:48,800 --> 00:19:51,306 take this ruler. What happened to the rest of the ruler? 290 00:19:51,306 --> 00:19:54,677 You got a one meter ruler, you're now moving at some large fraction of the 291 00:19:54,677 --> 00:19:57,684 speed of, you move it at some large fraction of the speed of light. 292 00:19:57,684 --> 00:20:00,873 The ruler is 90 centimeters length, in length, what happened to the other ten 293 00:20:00,873 --> 00:20:03,406 centimeters? Well the answer is what determined the 294 00:20:03,406 --> 00:20:07,174 length of the ruler in the first place. Well the ruler is held together by 295 00:20:07,174 --> 00:20:10,281 electromagnetic forces. Electromagnetic interactions, remember 296 00:20:10,281 --> 00:20:13,693 these are Maxwell equations, are invariant in the Lorentz transformations, 297 00:20:13,693 --> 00:20:15,680 which means that they give the same physics. 298 00:20:15,680 --> 00:20:19,590 If you do Lorentz transformations electromagnetic interactions if you were 299 00:20:19,590 --> 00:20:23,154 able to make the count detailed calculation of what determines the size 300 00:20:23,154 --> 00:20:26,916 of this ruler would actually predict that it shortens when you move it fast. 301 00:20:26,916 --> 00:20:30,530 In fact Lorentz, when he first derived the Lorentz contraction, the length 302 00:20:30,530 --> 00:20:33,747 contraction expression was not trying to do relativistic physics. 303 00:20:33,747 --> 00:20:37,707 He had not adopted Einstein's principle of relativity he was doing calculations 304 00:20:37,707 --> 00:20:41,420 with electromagnetic fields and he saw that the electromagnetic field of a 305 00:20:41,420 --> 00:20:45,083 moving charge appears compressed along the direction of its motion he was 306 00:20:45,083 --> 00:20:48,350 observing that Maxwell's equations know about Lorentz contraction. 307 00:20:48,350 --> 00:20:51,835 Which is not surprising since Lorentz contraction follows from the invariance 308 00:20:51,835 --> 00:20:54,740 of the speed of light which is a property of Maxwell's equations. 309 00:20:54,740 --> 00:20:57,429 Woo-hoo. Similarly, build a clock of any kind and 310 00:20:57,429 --> 00:21:00,287 it will run slow when you move it to high speed. 311 00:21:00,287 --> 00:21:04,939 in fact we just did that calculation for one clock construction, the light clock. 312 00:21:04,939 --> 00:21:09,590 The light clock moving transversely where we assumed that the speed of light was a 313 00:21:09,590 --> 00:21:12,168 constant. We found slowed down by one over the 314 00:21:12,168 --> 00:21:17,555 square root of 1 - v^2 / c^2. What about the longitudinal time clock? 315 00:21:17,555 --> 00:21:22,336 You'll do that in the homework. one last thing before we let go of these 316 00:21:22,336 --> 00:21:27,450 weird and bizarre consequences of Lorentz transformations, we talked about velocity 317 00:21:27,450 --> 00:21:32,009 addition, what happens when you add to velocity, so back to our old picture. 318 00:21:32,009 --> 00:21:35,275 We have the blue observer, we have the black observer. 319 00:21:35,275 --> 00:21:40,456 The blue observer note sees an object and the object is moving at a velocity which 320 00:21:40,456 --> 00:21:44,462 I should properly call u prime, and I ask at what 321 00:21:44,462 --> 00:21:49,487 I will call it u I think, I guess. And I ask, at what velocity will the 322 00:21:49,487 --> 00:21:52,983 black observer observe this object to be moving? 323 00:21:52,983 --> 00:21:59,173 So again, I draw the world line of the moving object as x prime = blue u * t 324 00:21:59,173 --> 00:22:02,523 prime. I use my Lorentz transformations, plug in 325 00:22:02,523 --> 00:22:07,840 this information, and solve for x. So, for example, I can do the calculation 326 00:22:07,840 --> 00:22:18,564 x = ut prime + vt prime / the square root of 1 - v^2 / c^2.2. 327 00:22:18,564 --> 00:22:34,734 On the other hand t = t prime + vx prime, x prime is ut prime / c^2 / square root 328 00:22:34,734 --> 00:22:43,770 of 1 - v^2 / c^2. And, what does that tell me? 329 00:22:43,770 --> 00:22:50,846 Well this is equal to u plus v, just as it was in the Galillean case except for 330 00:22:50,846 --> 00:22:57,433 the funny square root there. This is equal to t prime * 1UV + uv / c^2 331 00:22:57,433 --> 00:23:02,831 divided by the same square root. So indeed x is some number times t, 332 00:23:02,831 --> 00:23:08,953 inertial motion translates to inertial motion and the number is obtained by 333 00:23:08,953 --> 00:23:14,028 dividing this by that. The reason I didn't write out the square 334 00:23:14,028 --> 00:23:20,312 roots is because they happily cancel. I see that x is in fact not uV + v * t as 335 00:23:20,312 --> 00:23:26,595 it was before but uV + / 1UV + uv / c^2. This is the new velocity addition 336 00:23:26,595 --> 00:23:28,930 formula. Let me write it cleanly. 337 00:23:28,930 --> 00:23:32,310 Here's the new velocity addition formula and 338 00:23:32,310 --> 00:23:36,604 the great here I've remembered to put u prime and what is the great property of 339 00:23:36,604 --> 00:23:39,019 this? Well if u and v are both small compared 340 00:23:39,019 --> 00:23:43,260 to the speed of light if you are dealing with slow moving objects then you can 341 00:23:43,260 --> 00:23:47,793 erase this u2 / c * v over c and you're back to Galileo. 342 00:23:47,793 --> 00:23:53,335 Logic makes sense at the small velocities at which we're used to applying it. 343 00:23:53,335 --> 00:23:57,437 The other nice property of this is that if you plug in vC = c. 344 00:23:57,437 --> 00:24:02,762 If you ask, if I add some speed, if I observe something moving at a speed C, 345 00:24:02,762 --> 00:24:06,361 what will you observe? Well if vC = c, then it's an easy 346 00:24:06,361 --> 00:24:11,974 experiment to show that if, sorry not v, v isn't always necessary, if u prime is 347 00:24:11,974 --> 00:24:13,270 equal to c, then uC. = c. 348 00:24:13,270 --> 00:24:16,943 So this has a property that on one hand at low velocity, this is Galileo. 349 00:24:16,943 --> 00:24:21,134 On the other hand, light travels at the speed of light no matter who is measuring 350 00:24:21,134 --> 00:24:23,889 it. These are the consequences of Lorentz. 351 00:24:23,889 --> 00:24:27,494 we can play with, if you'd like, you can go on from here to 352 00:24:27,494 --> 00:24:32,221 play with some simultaneity puzzles and if not, we'll move on to constructing 353 00:24:32,221 --> 00:24:33,462 relativistic physics.