1 00:00:00,000 --> 00:00:04,812 Getting a sense for the way space time works in special relativity is very 2 00:00:04,812 --> 00:00:08,470 challenging and one of the ways that traditionally we 3 00:00:08,470 --> 00:00:13,515 Become accustomed to this and help wrap our minds around the weirdness, is to 4 00:00:13,515 --> 00:00:18,429 challenge ourselves with paradoxes. Almost invariably, they end up having to 5 00:00:18,429 --> 00:00:23,474 do with simultaneity because it's the relativity of what right now means that 6 00:00:23,474 --> 00:00:27,733 is really so current to everything that we are used to seeing. 7 00:00:27,733 --> 00:00:33,368 So, in this optional clip, I'm going to take advantage of my time and go through 8 00:00:33,368 --> 00:00:38,478 two of the most famous paradoxes in special relativity and discuss them in 9 00:00:38,478 --> 00:00:42,410 some detail and hopefully that will help clear things up 10 00:00:42,410 --> 00:00:47,586 and I suspect spur a lot of forum discussion because I'm not going to 11 00:00:47,586 --> 00:00:53,029 manage to clear everything up here. So let's start with a, a ladder paradox. 12 00:00:53,029 --> 00:00:57,202 Ladder paradox is if you want urban parking problem. 13 00:00:57,202 --> 00:01:03,102 The idea is you have a long ladder of length L and the short garage of length G 14 00:01:03,102 --> 00:01:08,930 but G is less than L and you want to fit the long ladder into the short garage. 15 00:01:08,930 --> 00:01:13,260 And the idea is very simple. You come running with a ladder at the 16 00:01:13,260 --> 00:01:17,722 garage at a very high speed. Then, therefore the long ladder is going 17 00:01:17,722 --> 00:01:23,431 to be Lorentz contracted to the point if v is high enough where it is shorter than 18 00:01:23,431 --> 00:01:26,186 g. Notice that by making v close enough to 19 00:01:26,186 --> 00:01:30,320 c, I can make the Lorentz contracted ladder as short as I want. 20 00:01:30,320 --> 00:01:35,373 And voila, the ladder fix in the garage. The ladder is fixed in the garage but 21 00:01:35,373 --> 00:01:40,360 note it only fixed in the garage while its moving very rapidly to the right. 22 00:01:40,360 --> 00:01:45,693 An obvious puzzle occurs because this may be true but relativity says I can look 23 00:01:45,693 --> 00:01:50,837 well, view the same situation from the point of view of somebody sitting on top 24 00:01:50,837 --> 00:01:54,329 of the ladder. And from the point of view of the ladder, 25 00:01:54,329 --> 00:01:59,092 the ladder is at rest, that's an inertial frame, the ladder is moving with a 26 00:01:59,092 --> 00:02:04,172 constant velocity to the right v. in the ladder frame the ladder itself is 27 00:02:04,172 --> 00:02:09,569 stationary, while the garage is moving at exactly the same speed v but to the left 28 00:02:09,569 --> 00:02:13,141 approaching the ladder. But from the ladder point of view, the 29 00:02:13,141 --> 00:02:17,253 ladder's length is L, its at rest. And near the hand, the garage is going to 30 00:02:17,253 --> 00:02:20,421 be low ends contracted. So, the garage is going to be even 31 00:02:20,421 --> 00:02:23,977 narrower than it really is. So, in the ladder frame we obtain the 32 00:02:23,977 --> 00:02:27,089 following picture and the two seem to be in conflict. 33 00:02:27,089 --> 00:02:30,146 In the picture on the left, the ladder is in the garage. 34 00:02:30,146 --> 00:02:34,480 In the picture on the right, there is no way the ladder could be in the garage 35 00:02:34,480 --> 00:02:38,815 because the garage is even shorter than its rest length which is as it is shorter 36 00:02:38,815 --> 00:02:43,209 than a ladder as we see at the top. You can make the puzzle sharper by 37 00:02:43,209 --> 00:02:47,436 saying, imagine that at the instant over here on the frame on the left where the 38 00:02:47,436 --> 00:02:49,973 ladder is inside the garage, I closed both doors. 39 00:02:49,973 --> 00:02:52,720 What happens then? Now I have a ladder in the garage. 40 00:02:52,720 --> 00:02:55,310 Well, now it's not ambiguous. The doors are closed. 41 00:02:55,310 --> 00:02:59,574 So, let's resolve this issue. there is, as always no real paradox. 42 00:02:59,574 --> 00:03:03,188 There's only confusion. And the confusion, as I said, is always 43 00:03:03,188 --> 00:03:06,623 to do with simultaneity. And, I'll solve this problem twice. 44 00:03:06,623 --> 00:03:10,059 The first time, algebraically, using Lorentz Transformation. 45 00:03:10,059 --> 00:03:14,560 And then, I'll demonstrate, graphically how this solution works and along the 46 00:03:14,560 --> 00:03:19,299 way, play with space time diagrams, and what we can or cannot read off of them. 47 00:03:19,299 --> 00:03:23,149 But I want to convince you that Lorentz Transformations are useful. 48 00:03:23,149 --> 00:03:27,888 So, to use Lorentz Transformations, what we do is, we have to be very careful that 49 00:03:27,888 --> 00:03:32,294 we're not talking about wishy-washy things, we want to be very 50 00:03:32,294 --> 00:03:35,109 specific about what are the things we're talking about. 51 00:03:35,109 --> 00:03:38,231 And the things we talk about are events, things that happened. 52 00:03:38,231 --> 00:03:41,199 And they happened a particular place, at a particular time. 53 00:03:41,199 --> 00:03:44,883 So what are the things we know? in order to use coordinates, I'm going to 54 00:03:44,883 --> 00:03:48,517 set up my usual space time diagram if you want, or my coordinates. 55 00:03:48,517 --> 00:03:52,457 I'm going to have the black coordinates presenting the garage because I'm 56 00:03:52,457 --> 00:03:56,347 thinking of the garage as at rest. And the latter is moving to the right and 57 00:03:56,347 --> 00:03:59,520 its coordinates as seen by the garage are t prime and x prime. 58 00:03:59,520 --> 00:04:02,386 I have to pick an origin for each of these. 59 00:04:02,386 --> 00:04:07,385 So I will pick the left end of the garage, remember the ladder is moving to 60 00:04:07,385 --> 00:04:10,851 the right. So the left end of the garage is going to 61 00:04:10,851 --> 00:04:13,984 be the origin for coordinates on the garage. 62 00:04:13,984 --> 00:04:20,708 So the garage stretches from x = 0 to x = G, so in the garages coordinates, 63 00:04:20,708 --> 00:04:27,720 the garage itself where this is x and t. The garage will be something like this. 64 00:04:27,720 --> 00:04:33,278 And this will be the point x = g. And the garage is stationary so it just 65 00:04:33,278 --> 00:04:35,850 sits there, the interior is the garage. 66 00:04:35,850 --> 00:04:40,491 I will pick the left end of the ladder to be x prime equals zero. 67 00:04:40,491 --> 00:04:46,346 And so, of course the left, the ladder is moving to the right, so if I set things 68 00:04:46,346 --> 00:04:52,344 up right and I am going to try to do that is in the standard way that I like the 69 00:04:52,344 --> 00:04:57,843 world line at the left end of the garage which is also because it is x prime 70 00:04:57,843 --> 00:05:01,041 equals zero. This is going to be my t prime access. 71 00:05:01,041 --> 00:05:05,600 This is, describes the motion, the straight world line, describes the motion 72 00:05:05,600 --> 00:05:09,789 of the left end of the ladder. And at some point, which I am going to 73 00:05:09,789 --> 00:05:13,732 call by convention, t equals zero and also t prime equal to zero. 74 00:05:13,732 --> 00:05:18,661 I will call that, pick as that event, event number one, or event number zero if 75 00:05:18,661 --> 00:05:20,510 you want. The instant when the. 76 00:05:20,510 --> 00:05:25,600 left end of the ladder enters the garage. So at this point, the ladder is all the 77 00:05:25,600 --> 00:05:30,016 way to the left of the garage. what about the right end of the ladder? 78 00:05:30,016 --> 00:05:35,106 Well, the right end of the ladder will be by con, by construction a distance l as 79 00:05:35,106 --> 00:05:39,644 seen in the primed coordinates to the right of the left end of the garage. 80 00:05:39,644 --> 00:05:44,428 So at coordinate x prime, of the right side of the garage is equal to l, and it 81 00:05:44,428 --> 00:05:48,108 too of course will be moving at speed v, at the same speed v. 82 00:05:48,108 --> 00:05:51,420 So its world line is going to be a parallel line here. 83 00:05:52,480 --> 00:05:58,370 this is the world line spiritually drawn, of the right end of the garage and as 84 00:05:58,370 --> 00:06:02,988 you've, as I've drawn it you can definitely see that at t equals zero, 85 00:06:02,988 --> 00:06:08,209 when the left end of the garage of the ladder enters the garage, the right end 86 00:06:08,209 --> 00:06:13,296 of the ladder has not yet exited the garage, and therefore at this point and 87 00:06:13,296 --> 00:06:17,380 in fact throughout the time that I am going to mark out here. 88 00:06:18,740 --> 00:06:24,822 The, this whole region is a region where the ladder is contained in the garage and 89 00:06:24,822 --> 00:06:30,534 then at this time, the right end emerges first and later the left end emerges. 90 00:06:30,534 --> 00:06:36,765 So the fact that I've drawn this to exist is the statement that while G is less 91 00:06:36,765 --> 00:06:41,290 than L G is bigger than the square root of 1 - v^2 / c^2 L. 92 00:06:42,700 --> 00:06:49,154 I have made the velocity high enough that the Lorentz Contracted Ladder does fit in 93 00:06:49,154 --> 00:06:53,282 the garage. Okay so this is event, the event that 94 00:06:53,282 --> 00:06:59,990 I'll call event number one and this here is event number two, event number two is 95 00:06:59,990 --> 00:07:06,367 the instant when the right end of the ladder is about emerge from the garage 96 00:07:06,367 --> 00:07:12,910 and that occurs of course at the position x^2 is equal to G as seen by the, 97 00:07:12,910 --> 00:07:18,232 by the garage frame and of course at that same, at that same instant, at that 98 00:07:18,232 --> 00:07:23,764 event, is the event where the world line of x prime equals L which is this blue 99 00:07:23,764 --> 00:07:29,296 line, the motion of the right side of the garage intersect x2 = g. 100 00:07:29,296 --> 00:07:34,058 So, that's event number two. The intersection of this green line with 101 00:07:34,058 --> 00:07:37,910 that blue line and I can compute it. In a moment I will. 102 00:07:37,910 --> 00:07:42,301 And then, the question of whether the ladder is inside the garage at any point 103 00:07:42,301 --> 00:07:45,059 or not, is the question of which event came first. 104 00:07:45,059 --> 00:07:49,676 If event number one, the left end of the ladder coming into the garage precedes 105 00:07:49,676 --> 00:07:54,123 event number two, the right end of the ladder emerging from the garage, then the 106 00:07:54,123 --> 00:07:57,783 ladder's in the garage. And in this picture it clearly is as seen 107 00:07:57,783 --> 00:08:01,949 in the frame of the garage. but the question is if on the other hand 108 00:08:01,949 --> 00:08:06,171 event number two precedes event number one, that means the right end of the 109 00:08:06,171 --> 00:08:09,549 ladder protrudes from the garage before the left end was in. 110 00:08:09,549 --> 00:08:14,190 The ladder was never in the garage. So let's figure out the order of these 111 00:08:14,190 --> 00:08:18,092 two events and again we will use Lorentz Transformations. 112 00:08:18,092 --> 00:08:23,775 So I've written out here the full set of Lorentz Transformations front and back 113 00:08:23,775 --> 00:08:29,390 forwards and backwards as you note they only differ by flipping the sign of'v'. 114 00:08:29,390 --> 00:08:34,337 And then. I am going to ask what is the time as 115 00:08:34,337 --> 00:08:40,230 seen by the garage frame at which this world line of the right side of the 116 00:08:40,230 --> 00:08:44,945 garage meets the world line of the right side of the ladder. 117 00:08:44,945 --> 00:08:51,310 Remember, this was the instant when the right side of the ladder pokes out of the 118 00:08:51,310 --> 00:08:57,675 right side of the garage, and so I look at this equation and I say x prime two is 119 00:08:57,675 --> 00:09:03,857 equal to L, so I say L over here. And that is going to be equal to'x', 'x' 120 00:09:03,857 --> 00:09:11,567 is'g' because it's at the right side of the garage and then minus'v' times't2' 121 00:09:11,567 --> 00:09:18,620 and then divided by the square root'1' minus'v' squared of over'c' squared. 122 00:09:19,820 --> 00:09:26,709 And okay from this I can solve, everything here is known, except for T2. 123 00:09:26,709 --> 00:09:34,165 So I multiply by the square root as I'm always doing, move the square root over 124 00:09:34,165 --> 00:09:38,616 to here. Notice I get the foreshortened length 125 00:09:38,616 --> 00:09:45,231 here, and what I find moving things around is that vt2 is G - L * the square 126 00:09:45,231 --> 00:09:52,823 root of 1 - v^2 / c^2 by construction we said that this is less than G so vt2 is 127 00:09:52,823 --> 00:09:57,936 positive because I was running fast enough. This number is small enough, the 128 00:09:57,936 --> 00:10:01,095 Lorentz constructed ladder fits into the garage. 129 00:10:01,095 --> 00:10:06,472 That means its right end will emerge from the garage after the left end entered. 130 00:10:06,472 --> 00:10:11,849 perfect that's what we set up in the garage's frame, the ladder fits into the 131 00:10:11,849 --> 00:10:14,974 garage. How does this look in the ladder's frame? 132 00:10:14,974 --> 00:10:20,219 Well and indeed there is this whole time as I said between the time the left end 133 00:10:20,219 --> 00:10:25,270 enters and the time that the right end exits that the ladder is completely 134 00:10:25,270 --> 00:10:27,601 contained in the garage. Good for us. 135 00:10:27,601 --> 00:10:32,911 On the other hand, what happens if I look at the same thing from the point of view 136 00:10:32,911 --> 00:10:38,091 of the ladder, what I want to know is at what time did as measured by the ladder, 137 00:10:38,091 --> 00:10:43,271 did the right end of the ladder exit the garage, so I want't' prime at this same 138 00:10:43,271 --> 00:10:47,009 event. That turns out to be easier to extract 139 00:10:47,009 --> 00:10:51,990 from the inverse transformations. So again x is equal to g. 140 00:10:51,990 --> 00:10:56,800 So I put it g over here. Over here x prime is equal to l. 141 00:10:57,920 --> 00:11:01,915 And I want T prime, so again I can solve for T prime. 142 00:11:01,915 --> 00:11:07,140 And I find that V T prime again multiplying through the square root. 143 00:11:09,600 --> 00:11:15,713 I find that'vt' prime is one over, the square root of one minus'v' squared 144 00:11:15,713 --> 00:11:21,588 over'c' squared times'g' minus'l'. Now, G is less than L, this is a number 145 00:11:21,588 --> 00:11:24,264 less than one. T2 prime is negative. 146 00:11:24,264 --> 00:11:29,553 remember that T2, T prime = 0 and also, Tzero, = 0 was the time when the left end 147 00:11:29,553 --> 00:11:34,768 of the garage entered the garage. But here, the right end of the garage 148 00:11:34,768 --> 00:11:38,492 emerged before. So indeed, in the ladder's frame of 149 00:11:38,492 --> 00:11:41,844 reference, the ladder was never in the garage. 150 00:11:41,844 --> 00:11:46,463 You couldn't close both doors. Because, as far as the ladder is 151 00:11:46,463 --> 00:11:52,199 concerned, before the tail end of the ladder entered the garage, the front end 152 00:11:52,199 --> 00:11:57,438 of the ladder was protruding. So what happens you ask if the garage 153 00:11:57,438 --> 00:12:03,287 owner in his frame when the ladder is completely as he sees it contained in the 154 00:12:03,287 --> 00:12:08,478 garage, swiftly closes both doors. Well, lets follow that in a graphic 155 00:12:08,478 --> 00:12:14,255 representation which will give us a chance to play with space time diagrams 156 00:12:14,255 --> 00:12:17,764 and see some of their strengths and limitations. 157 00:12:17,764 --> 00:12:23,540 We could also do the calculation this way but I want to demonstrate a graphic 158 00:12:23,540 --> 00:12:26,100 method of solution. So hopefully. 159 00:12:26,100 --> 00:12:30,734 What we, what did we get from here? We got from here that the source of the 160 00:12:30,734 --> 00:12:34,415 paradox is the simultaneity issue in the Garage's frame. 161 00:12:34,415 --> 00:12:37,193 Indeed the ladder is contained in the garage. 162 00:12:37,193 --> 00:12:42,131 In the ladder's frame, this never happens and the reason is that simultaneity is 163 00:12:42,131 --> 00:12:46,637 relative, and because the ladder is moving, the relativity of simultaneity 164 00:12:46,637 --> 00:12:51,390 means that at the same time, that say, the left hand entered the garage, in the 165 00:12:51,390 --> 00:12:54,230 ladder's frame the right hand was already out, 166 00:12:54,230 --> 00:12:57,073 okay? So, what's our graphic representation? 167 00:12:57,073 --> 00:13:01,406 Well here's our graphic representation, this is already familiar. 168 00:13:01,406 --> 00:13:06,686 I was here careful to remind myself that I'm plotting CT, so the red line that 169 00:13:06,686 --> 00:13:10,952 goes off at 45 degrees is the world line of a light beam. 170 00:13:10,952 --> 00:13:16,232 You don't need a light beam here, but it helps me to draw the t prime, CT prime 171 00:13:16,232 --> 00:13:19,820 and x prime axis so that the light cone bisects them. 172 00:13:19,820 --> 00:13:24,560 And what I have here, these two black lines to represent the garage 173 00:13:24,560 --> 00:13:28,506 These two blue lines represent the world lines. 174 00:13:28,506 --> 00:13:32,285 X prime equal to zero is the T prime axis. 175 00:13:32,285 --> 00:13:39,711 And x prime equals to L is the bold line of the right side of the garage and so 176 00:13:39,711 --> 00:13:43,630 this point here is X prime equal to L and. 177 00:13:43,630 --> 00:13:47,080 Both sides of the garage are moving, of the ladder are moving. 178 00:13:47,080 --> 00:13:51,492 Both sides of the garage are stationary. I may have called the ladder a garage, 179 00:13:51,492 --> 00:13:55,734 but hopefully you understand what I am saying by now, and you're used to my 180 00:13:55,734 --> 00:14:00,446 ambiguity. A word on scales in these diagrams and a 181 00:14:00,446 --> 00:14:06,920 very important point. So, suppose that I were to draw a grid of 182 00:14:06,920 --> 00:14:13,409 lines so these are lines of constant t, these are lines of constant x and some 183 00:14:13,409 --> 00:14:19,566 units so these are, and, and I've picked the unit so they are compatible. 184 00:14:19,566 --> 00:14:27,137 In other words, that if in some units this is 1, 2, 3, 4 so my garage in this 185 00:14:27,137 --> 00:14:34,126 case is precisely four units long. Then CT is also 1, 2, 3, 4 and you can 186 00:14:34,126 --> 00:14:41,940 see that because these these things intersect the 45 degree. 187 00:14:41,940 --> 00:14:46,220 Light count. And so I've drawn these. 188 00:14:46,220 --> 00:14:49,399 How do I draw the point, t prime = to one? 189 00:14:49,399 --> 00:14:53,215 Where along the t prime axis is t prime = to one? 190 00:14:53,215 --> 00:14:59,098 and one way to think about it. One, naive way, is to just measure this 191 00:14:59,098 --> 00:15:03,232 same length along this axis. And because it is, 192 00:15:03,232 --> 00:15:08,400 The axis is, is, not vertical. That would lead you to some point. 193 00:15:08,400 --> 00:15:13,689 Around here and you'd maybe try to set that as t prime equals the one and that 194 00:15:13,689 --> 00:15:17,774 would be incorrect. The point is that the distance along this 195 00:15:17,774 --> 00:15:22,528 axis, in general, distances in this 2-dimensional plot are completely 196 00:15:22,528 --> 00:15:25,809 meaningless. well, how do you measure distances? 197 00:15:25,809 --> 00:15:29,090 Well, the distance is this, between any two points. 198 00:15:29,090 --> 00:15:34,720 Is the square root of the difference in x squared, the horizontal difference, plus 199 00:15:34,720 --> 00:15:39,517 the vertical difference squared. That's, the Pythagorean Theorem which 200 00:15:39,517 --> 00:15:43,480 applies very well to the Euclidean surface of this 201 00:15:43,480 --> 00:15:49,104 A tablet upon which I am drawing it or to this Euclidean surface of this slide, but 202 00:15:49,104 --> 00:15:53,983 this is not a physical quantity. In particular, there is no reason why it 203 00:15:53,983 --> 00:15:58,591 isn't all related to x prime squared plus c^2 t prime squared. 204 00:15:58,591 --> 00:16:04,013 Its not a physically invariant object, Lorentz transformations change it and 205 00:16:04,013 --> 00:16:09,163 indeed this distance means nothing physical and reading it off the graph 206 00:16:09,163 --> 00:16:14,042 produces no useful information. So, it is not true that I can measure off 207 00:16:14,042 --> 00:16:17,970 distances. They see and on the same scale as I did 208 00:16:17,970 --> 00:16:24,957 over there and thus, we figure out where the t prime equal to two save point is. 209 00:16:24,957 --> 00:16:28,950 So, how do I do it? Well, if I really want to plot where 210 00:16:28,950 --> 00:16:34,866 along the t prime axis, the point say t prime equals to 1 is, what I need to do 211 00:16:34,866 --> 00:16:40,930 is the following construction: remember there is something that is preserved in 212 00:16:40,930 --> 00:16:47,289 this weird and is physically meaningful in this low ends relativistic world, and 213 00:16:47,289 --> 00:16:58,314 that is that c^2 t^2 - x^2 is the same as c^2 sorry for the color violation here, 214 00:16:58,314 --> 00:17:02,720 minus x prime squared. And along the 215 00:17:02,720 --> 00:17:07,746 t prime axis well, that's the point where x prime = zero. 216 00:17:07,746 --> 00:17:12,860 So I can erase that. Furthermore, along the t prime axis, x = 217 00:17:12,860 --> 00:17:19,650 vt because this describes the motion of the left side of the ladder which is 218 00:17:19,650 --> 00:17:23,090 moving with speed v. So I get that c^2 - v^2. 219 00:17:23,090 --> 00:17:27,495 T squared is equal to equal to c squared t prime squared. 220 00:17:27,495 --> 00:17:33,525 I am going to eventually figure out exactly the time dilation, relativistic 221 00:17:33,525 --> 00:17:39,708 time dilation formula which is that t prime is t times square root of one minus 222 00:17:39,708 --> 00:17:43,187 v squared over c squared. What does that mean? 223 00:17:43,187 --> 00:17:48,520 That means that the time here, t prime at this point is less than one. 224 00:17:48,520 --> 00:17:53,814 If I draw the point t prime equal to one. It will be somewhere over here. 225 00:17:53,814 --> 00:17:56,898 And the point t'= two is somewhere over there. 226 00:17:56,898 --> 00:18:00,530 Notice that the scales of the two axis are different. 227 00:18:00,530 --> 00:18:05,971 This is confusing and troubling. It has to do with the fact that what is 228 00:18:05,971 --> 00:18:12,245 physically meaningful is this Lewenstein Distant, distance if you want this sort 229 00:18:12,245 --> 00:18:16,780 of weird distance with the minus sign rather than the 230 00:18:16,780 --> 00:18:23,005 Euclidean distance that were. Is the distance that we observe in the 231 00:18:23,005 --> 00:18:27,565 graph. And you can perform with negative value 232 00:18:27,565 --> 00:18:35,018 for the interval a similar calculation. And you will find that the scale of the 233 00:18:35,018 --> 00:18:41,770 x-axes also is extended. And in general in any space-time diagram 234 00:18:41,770 --> 00:18:45,320 the scale is smallest. And the Observers. 235 00:18:45,320 --> 00:18:52,070 For the observer that is addressed. In other words if I draw here the lines 236 00:18:52,070 --> 00:18:59,360 of constant x, the constant t prime and the lines of constant x prime then I find 237 00:18:59,360 --> 00:19:05,710 indeed that if I drew. X equals 1, 2, 3 and CT is 1, 2, 3. 238 00:19:05,710 --> 00:19:11,755 And I then want to draw the lines of x prime equal 1 2, 3. 239 00:19:11,755 --> 00:19:15,540 Then I need to draw them approximately here. 240 00:19:15,540 --> 00:19:20,381 And then x prime equals three would lie, lay, lie way over there. 241 00:19:20,381 --> 00:19:26,836 And likewise, t prime equals to 1 would be way up here, and 2 and 3 and so on. 242 00:19:26,836 --> 00:19:30,294 So the scales in this diagram are misleading. 243 00:19:30,294 --> 00:19:36,288 If you really want to lay scales on it, you need to use the invariant interval 244 00:19:36,288 --> 00:19:41,130 or, the appropriate hyperbolas to draw these distances. 245 00:19:41,130 --> 00:19:45,838 This is one of the reasons why doing calculations from these graphs can be 246 00:19:45,838 --> 00:19:50,547 confusing the thing that is always meaningful in these graphs is the order 247 00:19:50,547 --> 00:19:53,686 of things. In other words, what is true is that the 248 00:19:53,686 --> 00:19:58,270 entire region above the'x' axis is positive't' and that an event that is. 249 00:19:58,270 --> 00:20:04,392 further up than another is later as viewed by this observer and similarly the 250 00:20:04,392 --> 00:20:11,271 entire region above in this sense, the x prime axis is at +t prime and the region 251 00:20:11,271 --> 00:20:17,470 below it in this direction is at -t prime and the order of the events 252 00:20:17,470 --> 00:20:22,817 can be read off from this grid, you draw the grid and if you have some 253 00:20:22,817 --> 00:20:28,383 random event over here the infamous birthday party that we spoke of, over 254 00:20:28,383 --> 00:20:31,680 here. Then to read off its X and T we drop. 255 00:20:31,680 --> 00:20:38,337 Parallel to the x and t axis and we see that this happened at x a little larger 256 00:20:38,337 --> 00:20:44,579 than two and t getting close to four. If you want to plot this in x prime, t 257 00:20:44,579 --> 00:20:49,870 prime co-ordinates we. Draw lines parallel to the axis because 258 00:20:49,870 --> 00:20:54,110 remember this line is a line of constant T prime. 259 00:20:54,110 --> 00:20:58,490 It's parallel to the X prime axis, and this is a line of constant X prime. 260 00:20:58,490 --> 00:21:03,230 And we see that this happened at X prime a little less then one and T prime 261 00:21:03,230 --> 00:21:07,370 somewhere between two and three is when this birthday party happened. 262 00:21:07,370 --> 00:21:12,350 So this is how we read coordinates off of these space time diagrams, I hope that 263 00:21:12,350 --> 00:21:15,825 was helpful. And now I am going to erase all these 264 00:21:15,825 --> 00:21:20,756 grids which are just confusing. And note that I have marked here our 265 00:21:20,756 --> 00:21:25,035 favorite two events. Event number 1 is the instant t equal to 266 00:21:25,035 --> 00:21:30,401 zero when the left end of the ladder enters the garage and notice that as 267 00:21:30,401 --> 00:21:37,352 anticipated because I made this point which we know is L root 1 - v^2 over c^2 268 00:21:37,352 --> 00:21:43,650 less than G then there, the entire ladder at this point is enclosed in the garage 269 00:21:43,650 --> 00:21:49,718 and this goes on until the time of this event labelled two, event number two is 270 00:21:49,718 --> 00:21:55,862 the instant where the right end of the ladder starts protruding from the garage 271 00:21:55,862 --> 00:22:02,140 and so as I suggested there, there is this whole region here from t1 to t2 272 00:22:02,140 --> 00:22:06,437 where the entire ladder's inside the garage, 273 00:22:06,437 --> 00:22:10,185 okay? So, now we can ask what happens if I 274 00:22:10,185 --> 00:22:15,305 close the door. Well if I close the right door and oh, 275 00:22:15,305 --> 00:22:22,162 sorry and I forgot to note on the other hand we can see directly from this 276 00:22:22,162 --> 00:22:29,042 diagram that at event number two. While in the garage frame event number 277 00:22:29,042 --> 00:22:34,548 two happened after event number one and indeed for this whole time the ladder was 278 00:22:34,548 --> 00:22:38,242 inside the garage. In the ladder frame, event number two 279 00:22:38,242 --> 00:22:43,211 occurs before event number one, remember these are all the events that are 280 00:22:43,211 --> 00:22:46,502 simultaneous with the ladder entering the garage. 281 00:22:46,502 --> 00:22:52,082 Event number two corresponds. To some negative value of t prime down 282 00:22:52,082 --> 00:22:57,432 here and indeed it preceded as seen by the ladder event number one there as seen 283 00:22:57,432 --> 00:23:02,915 by the ladder the right end poked out of the garage before the left end got in and 284 00:23:02,915 --> 00:23:07,670 the ladder was never in the garage. So now you can ask, alright well, what 285 00:23:07,670 --> 00:23:11,954 happens if I close the door? Well if I close the door, then let me 286 00:23:11,954 --> 00:23:16,944 decide that the front door of the gara, the, the, the the door of the garage at g 287 00:23:16,944 --> 00:23:20,778 is going to be closed. What that means is that the right end of 288 00:23:20,778 --> 00:23:25,464 the ladder travels, at event number two it hits the garage door and from that 289 00:23:25,464 --> 00:23:30,211 moment on the left end, the right end of the ladder is at rest and moves along 290 00:23:30,211 --> 00:23:32,970 this green trajectory. Now. 291 00:23:32,970 --> 00:23:37,478 The way we're used to thinking about it, is a ladder is a ladder. 292 00:23:37,478 --> 00:23:42,278 If at this point in time. The ladder stopped, then the entire 293 00:23:42,278 --> 00:23:47,967 ladder stopped and if voila, we have the entire ladder contained in the garage as 294 00:23:47,967 --> 00:23:52,322 seen by the garage frame. But, this is confusing because in the 295 00:23:52,322 --> 00:23:57,520 ladder frame, remember, at when this event happened, in the ladder frame, the 296 00:23:57,520 --> 00:24:00,892 left end of the garage wasn't in the garage yet. 297 00:24:00,892 --> 00:24:04,544 The left end of the ladder wasn't in the garage yet. 298 00:24:04,544 --> 00:24:08,791 So what is this saying? Well, the answer to this is that we have 299 00:24:08,791 --> 00:24:13,924 made an unwarranted assumption here. Which is that when you stop the front end 300 00:24:13,924 --> 00:24:18,464 of the garage, of the ladder, you also stop the left end of the ladder. 301 00:24:18,464 --> 00:24:23,464 Remember, that at this point the right end of the ladder smashes into a wall, 302 00:24:23,464 --> 00:24:26,874 and it stops. Now, what is going to happen is the back 303 00:24:26,874 --> 00:24:31,983 end of the ladder just like our slinky, when I dropped it has not yet heard that 304 00:24:31,983 --> 00:24:35,495 the front end of the ladder has crashed into something. 305 00:24:35,495 --> 00:24:40,604 The left end of the ladder continues to come at speed v until somebody tells it. 306 00:24:40,604 --> 00:24:43,798 What tells it? Well, a sound wave, of course will be 307 00:24:43,798 --> 00:24:48,970 propagating down this ladder as the front end of it crumples, propagating back to 308 00:24:48,970 --> 00:24:54,080 tell the back end of the ladder, wait, there is an extra force, we need to stop. 309 00:24:54,080 --> 00:24:58,517 How fast does the sound wave travel? Well that depends on what the ladder is 310 00:24:58,517 --> 00:25:02,371 made of, but I know a limit. The sound wave will not travel back to 311 00:25:02,371 --> 00:25:07,159 the ladder in it's own frame, or in any other frame that's happily an invariant 312 00:25:07,159 --> 00:25:11,246 faster then the speed of light. So at this event when the front end of 313 00:25:11,246 --> 00:25:16,034 the ladder crashes into the door, you can imagine a sound wave propagating back up 314 00:25:16,034 --> 00:25:19,480 the ladder to the left, trying to tell the left end to stop. 315 00:25:19,480 --> 00:25:24,258 So, how fast can this signal propagate? Well, this is the fastest it can 316 00:25:24,258 --> 00:25:27,466 propagate. The fastest that the rear end of the 317 00:25:27,466 --> 00:25:31,835 ladder can hear, but the front has stopped, is the speed of light. 318 00:25:31,835 --> 00:25:37,364 So I've drawn a light speed, 45 degree propagation of the signal from the back 319 00:25:37,364 --> 00:25:42,415 of the ladder, from the front of the ladder to the back telling it to stop. 320 00:25:42,415 --> 00:25:47,262 And if this were the case. you notice that by the time this light 321 00:25:47,262 --> 00:25:53,334 being reaches the back end of the ladder, back end of the ladder is well within the 322 00:25:53,334 --> 00:25:59,260 garage because even in the ladder frame, we are now talking about a positive time. 323 00:25:59,260 --> 00:26:01,076 Right? And so, time has 324 00:26:01,076 --> 00:26:05,486 The, the, remember, the left end of the ladder entered the garage here. 325 00:26:05,486 --> 00:26:09,118 By now, the left end of the ladder is well in the garage. 326 00:26:09,118 --> 00:26:13,723 If you stop the front end of the ladder, the back end would keep coming. 327 00:26:13,723 --> 00:26:17,809 You would not, be able to stop it until it was in the garage. 328 00:26:17,809 --> 00:26:21,939 Now, what happens at the moment. To the actual ladder when the soundwave) 329 00:26:21,939 --> 00:26:26,237 propagates and reaches the back. Well the energies involved are such that 330 00:26:26,237 --> 00:26:30,593 most ladders would probably blow themselves up to smithereenes or crumble 331 00:26:30,593 --> 00:26:33,596 to atoms. But in any event by the time the rear end 332 00:26:33,596 --> 00:26:36,716 of the ladder appears that the front end has stopped. 333 00:26:36,716 --> 00:26:41,543 And maybe we have a very powerful ladder and it's elastic and it springs back out 334 00:26:41,543 --> 00:26:45,606 and re-expands by the time all that and crashes into the garage door. 335 00:26:45,606 --> 00:26:47,784 But the time, by the time all of that, 336 00:26:47,784 --> 00:26:51,140 any of that happens the ladder is well within the garage. 337 00:26:51,140 --> 00:26:55,040 And so the fact that you managed to close the doors. 338 00:26:55,040 --> 00:26:59,370 Is not an interruption when you close the doors. 339 00:26:59,370 --> 00:27:05,285 at this event, the ladder, had, the left end of the ladder was already in the 340 00:27:05,285 --> 00:27:10,707 garage, this door stopped the front end but the left hand kept, as seen by the 341 00:27:10,707 --> 00:27:16,130 ladder this left hand kept coming. So, there is no contradiction, there is a 342 00:27:16,130 --> 00:27:21,060 lot of kinetic energy to be lost and there might be a great explosion. 343 00:27:21,060 --> 00:27:27,185 And, of course, if the latter doesn't blow itself to smitherines, then of 344 00:27:27,185 --> 00:27:33,807 course there will be an elastic rebound. It will, this, the back end of the latter 345 00:27:33,807 --> 00:27:37,780 will now pop back to the right at high speed. 346 00:27:37,780 --> 00:27:42,032 Whoops. High speed but less than the speed of 347 00:27:42,032 --> 00:27:47,078 light crashing to the garage door and then whatever happens will happen 348 00:27:47,078 --> 00:27:53,507 but [COUGH] you can indeed enclose the ladder in the garage, at least in the 349 00:27:53,507 --> 00:27:56,550 instant. And the error that we were making when we 350 00:27:56,550 --> 00:28:01,663 were trying to compare the two frames and say you stock the ladder at the front was 351 00:28:01,663 --> 00:28:04,889 the assumption was that the ladder is a rigid object. 352 00:28:04,889 --> 00:28:09,880 One of the things we learned here is that the objects with a given shape and size. 353 00:28:09,880 --> 00:28:15,307 Do not make relativistic sense.An object whose pen has a size and the shape in a 354 00:28:15,307 --> 00:28:20,041 sense that when I move this side of it. And I drop it, the other side moves as 355 00:28:20,041 --> 00:28:22,554 well. This could not realistically be true, in 356 00:28:22,554 --> 00:28:25,011 fact, if you look very closely, it's not true. 357 00:28:25,011 --> 00:28:28,026 When I move this side, actually the pen bends a little. 358 00:28:28,026 --> 00:28:32,102 A sound wave goes travelling down the pen, and if it were a slinky with a 359 00:28:32,102 --> 00:28:34,447 slower sound wave, you'd be able to see it. 360 00:28:34,447 --> 00:28:38,635 And when the sound wave reaches here, then and only then, does the other end 361 00:28:38,635 --> 00:28:41,762 start rising. when you're dealing in realistic speeds, 362 00:28:41,762 --> 00:28:44,442 this little time delay makes all the difference. 363 00:28:44,442 --> 00:28:48,853 So thinking of an object as having a given size in realistic terms is a really 364 00:28:48,853 --> 00:28:51,310 bad idea. So hopefully we've learned some 365 00:28:51,310 --> 00:28:58,362 results of some of the ways to think about relativistic kinematics and 366 00:28:58,362 --> 00:29:03,554 hopefully you understand the resolution of the ladder diagram though as I said, 367 00:29:03,554 --> 00:29:08,812 it takes seeing the solution and then thinking about it a lot and arguing and I 368 00:29:08,812 --> 00:29:13,410 hope the forums will be lively. A perhaps even more famous paradox of 369 00:29:13,410 --> 00:29:17,781 special relativity is the twin paradox. And the twin paradox I'm going to 370 00:29:17,781 --> 00:29:21,902 demonstrate graphically. We won't do too many calculations there's 371 00:29:21,902 --> 00:29:25,274 not much to compute, so what's the story with the twin 372 00:29:25,274 --> 00:29:28,521 paradox? We have two twin brothers or sisters and 373 00:29:28,521 --> 00:29:33,329 they live until some age on Earth. And then, at some point, one of them gets 374 00:29:33,329 --> 00:29:37,013 into a spaceship traveling at high speed in some direction. 375 00:29:37,013 --> 00:29:40,760 close to relativistic speed, or traveling for a long time. 376 00:29:40,760 --> 00:29:44,372 And for a long time this twin travels off into space. 377 00:29:44,372 --> 00:29:49,080 So, he's going moving to the right. And then after have achieving, achieved a 378 00:29:49,080 --> 00:29:53,628 certain distance, he hops on another spaceship moving at the same speed but in 379 00:29:53,628 --> 00:29:57,710 the opposite direction. And, after a while returns to Earth and 380 00:29:57,710 --> 00:30:02,347 gets reunited with his twin. Now of course, when they left they were 381 00:30:02,347 --> 00:30:06,071 the same age. So the question is when twin number two 382 00:30:06,071 --> 00:30:11,412 returns from his voyage, who is younger? Are they the same age and the answer is 383 00:30:11,412 --> 00:30:14,714 no. They are not the same age and it's easy 384 00:30:14,714 --> 00:30:19,000 to make that calculation but then its puzzling to understand. 385 00:30:19,000 --> 00:30:23,778 Wait a minute, so let's first start with the question who is younger. 386 00:30:23,778 --> 00:30:26,963 Well. We have here the usual two events, event 387 00:30:26,963 --> 00:30:33,264 number zero and event number one and the Coordinates of these events, again I have 388 00:30:33,264 --> 00:30:39,455 my usual axis x prime, t prime t double prime here is the a new time axis which 389 00:30:39,455 --> 00:30:44,942 is the world line of the twin number two on his return but I'm going to start by 390 00:30:44,942 --> 00:30:50,500 dealing with this segment, it is sort of clear by symmetry that everything that 391 00:30:50,500 --> 00:30:55,283 happens along this segment happens symmetrically along this segment. 392 00:30:55,283 --> 00:31:00,023 And, so lets figure out what happened at event number one. 393 00:31:00,023 --> 00:31:04,159 So let's say that the twin on earth measured a time T1. 394 00:31:04,159 --> 00:31:10,249 And therefore twin on the, the twin that was traveling measured a time T prime 395 00:31:10,249 --> 00:31:13,528 one. Now where did this event occurred. 396 00:31:13,528 --> 00:31:22,601 Well this event occurred at the position. x1 which is equal to vt1 because the twin 397 00:31:22,601 --> 00:31:29,080 was moving with speed v, but it also occurred at position x prime one. 398 00:31:29,080 --> 00:31:34,505 And we know what x prime one is. X prime one is zero because in his own 399 00:31:34,505 --> 00:31:40,618 frame, the moving twin was addressed and therefore we could now set, the point is 400 00:31:40,618 --> 00:31:46,732 that we can now compute the invariant interval between here and here in the two 401 00:31:46,732 --> 00:31:50,017 different frames and set it to be the same. 402 00:31:50,017 --> 00:31:55,596 So, that tells us, and again I'm not going to maintain all of the colors, that 403 00:31:55,596 --> 00:32:02,594 c squared, t1 squared minus x1 squared. Has to be equal to'c' squared't1' prime 404 00:32:02,594 --> 00:32:09,111 squared minus'x1' prime squared. But we know some things, we know'x1' 405 00:32:09,111 --> 00:32:17,765 prime is zero, we know that'x1' is'vt1'. So, c squared t1 squared minus v squared, 406 00:32:17,765 --> 00:32:27,595 t1 squared is c squared t1 prime squared and we do the usual calculation and we 407 00:32:27,595 --> 00:32:33,740 find that t1 prime is t1 times the square root. 408 00:32:33,740 --> 00:32:41,193 Of one minus V squared over C squared. Or, written otherwise, t1 is t1 prime / 409 00:32:41,193 --> 00:32:47,861 by the square root of one - v^2 over c^2. So that, what we are saying is that this 410 00:32:47,861 --> 00:32:54,968 event appears to the stationary twin to have, happened at a later time than it 411 00:32:54,968 --> 00:33:00,144 appears to the moving twin. Why do I know it's a later time? 412 00:33:00,144 --> 00:33:04,530 Because I'm dividing by a number smaller than one. 413 00:33:04,530 --> 00:33:10,321 Well, yeah, you say, of course. But on the other hand, I can draw this 414 00:33:10,321 --> 00:33:15,483 line. And say that is true, but that's because 415 00:33:15,483 --> 00:33:23,660 I am measuring the time at event one. I can invent an event number two here. 416 00:33:23,660 --> 00:33:34,342 Which is simultaneous with the, the, the moving twins decision to return and that 417 00:33:34,342 --> 00:33:40,520 event number two clearly occurs at a time. 418 00:33:40,520 --> 00:33:46,720 Earlier than T one. And if I compute I will find that in fact 419 00:33:46,720 --> 00:33:49,976 T two. Is exactly't1' prime times you can 420 00:33:49,976 --> 00:33:54,362 compute this. It's again these two happened at the same 421 00:33:54,362 --> 00:34:00,411 value of't' prime but one happened at the'x' equals zero and one happened at'x' 422 00:34:00,411 --> 00:34:04,570 prime equals zero. And the symmetry between these is of 423 00:34:04,570 --> 00:34:10,846 course the usual understanding that we have the stationary twin sees time 424 00:34:10,846 --> 00:34:16,895 dilation, sees the moving twin's clock running slow and the moving twin sees the 425 00:34:16,895 --> 00:34:19,920 stationary twin's clock running slow. So. 426 00:34:19,920 --> 00:34:24,321 The twin whose coordinates are black thinks that his brother is aging slowly. 427 00:34:24,321 --> 00:34:29,200 The twin whose coordinates are blue thinks his brother is aging slowly. 428 00:34:29,200 --> 00:34:35,857 So how is it that upon, but on the other hand, it is clear that since from at this 429 00:34:35,857 --> 00:34:42,761 point t1 is less, is clearly more than t1 prime and since this is t1 prime is how 430 00:34:42,761 --> 00:34:48,185 much the moving twin aged, t1 is how much the stationary twin aged. 431 00:34:48,185 --> 00:34:54,760 Since the same thing is now repeated on their return, it's clear that if this is 432 00:34:54,760 --> 00:35:01,089 true, then 2t1 = 2t1 prime / that and upon their return the stationary twin 433 00:35:01,089 --> 00:35:05,140 will be older. How can that be? 434 00:35:05,140 --> 00:35:10,522 We just computed that on the one hand I, we I thought we had this symmetry 435 00:35:10,522 --> 00:35:13,685 relation. Relativity tells us that if the black 436 00:35:13,685 --> 00:35:19,068 twin sees the blue clock running slow so does the blue twin see the black clock 437 00:35:19,068 --> 00:35:21,760 running slow. What ruins relativity here? 438 00:35:21,760 --> 00:35:26,621 Well, when you think about it. It's pretty clear that these two twins 439 00:35:26,621 --> 00:35:29,770 are not related by Lorentz Transformation. 440 00:35:29,770 --> 00:35:34,837 They were related by Lorentz Transformation, as long as twin number 1, 441 00:35:34,837 --> 00:35:38,466 2 was moving to the right at this constant speed, v. 442 00:35:38,466 --> 00:35:42,163 But there is a big difference in their life histories. 443 00:35:42,163 --> 00:35:46,340 And the big difference occurs, of course, here, at this instant. 444 00:35:46,340 --> 00:35:50,200 Twin Number two jumped from one spaceship to the other. 445 00:35:50,200 --> 00:35:54,464 He was not inertial. At that instant twin number two was 446 00:35:54,464 --> 00:35:58,679 actually accelerating. Not only that but as I drew it, since his 447 00:35:58,679 --> 00:36:03,355 velocity changed instantaneously the rate of change of his velocity was infinite. 448 00:36:03,355 --> 00:36:05,953 He was experiencing an infinite acceleration. 449 00:36:05,953 --> 00:36:09,705 So that's not very realistic, but it makes the calculation simple. 450 00:36:09,705 --> 00:36:12,992 But this is why the. Life history of the two twins is 451 00:36:12,992 --> 00:36:18,290 different, it's different and it's not, you don't expect symmetry if blue thinks 452 00:36:18,290 --> 00:36:23,394 black clock runs slow then black thinks blue clock runs slow, the difference is 453 00:36:23,394 --> 00:36:28,692 that one of them accelerated and the one who accelerated will be younger and this 454 00:36:28,692 --> 00:36:33,990 is brought out by drawing some auxiliary world lines here, so what I've done is at 455 00:36:33,990 --> 00:36:39,178 this event, event number one. I took, I drew before the t double prime 456 00:36:39,178 --> 00:36:43,655 axis the world line of the moving twin as he returns. 457 00:36:43,655 --> 00:36:50,243 I can now draw the auxiliary light cone and therefore the x double prime axis. 458 00:36:50,243 --> 00:36:56,324 And here we see the resolution to the mystery because remember before I 459 00:36:56,324 --> 00:37:01,730 suggested that we introduce event number two over here which is. 460 00:37:01,730 --> 00:37:06,026 What is this? This event is from the point of view of 461 00:37:06,026 --> 00:37:12,267 the blue twin of the, the moving twin as long as he is moving away from earth, 462 00:37:12,267 --> 00:37:16,968 event number two is simultaneous with event number one. 463 00:37:16,968 --> 00:37:23,940 So if these guys were exchanging messages then indeed when he decided to change 464 00:37:23,940 --> 00:37:27,921 The, to, to jump ships. The moving twin was, would have been 465 00:37:27,921 --> 00:37:31,483 convinced that his brother was younger than he was. 466 00:37:31,483 --> 00:37:37,630 And in just as the stationary twin would have been convinced that his brother was 467 00:37:37,630 --> 00:37:41,682 younger than he was. Until that point, everybody's inertial, 468 00:37:41,682 --> 00:37:45,873 and life is symmetric. But at the moment that he jumps ships. 469 00:37:45,873 --> 00:37:51,042 What happens is that, now, simul-, what he considers, when he's moving to the 470 00:37:51,042 --> 00:37:56,490 left, to be simultaneous with this same event #one is actually event #three over 471 00:37:56,490 --> 00:38:00,454 here. Is line of simultaneity changed because 472 00:38:00,454 --> 00:38:07,592 you changed velocities instantaneously. So in particular over the instantaneous 473 00:38:07,592 --> 00:38:12,110 jump from ship to ship, his brother aged this much. 474 00:38:12,110 --> 00:38:19,429 All of this aging took place essentially instantaneously during the instant that 475 00:38:19,429 --> 00:38:25,017 the moving twin jumped chips. This is the resolution.This is where the 476 00:38:25,017 --> 00:38:28,820 symmetry is broken, indeed this segment. Is less. 477 00:38:28,820 --> 00:38:33,012 The time, the, the twin number, The twin on Earth aged during this 478 00:38:33,012 --> 00:38:37,454 segment is, indeed, less than the time that, the moving twin aged here. 479 00:38:37,454 --> 00:38:40,082 And the same bi-symmetry is true up there. 480 00:38:40,082 --> 00:38:45,150 But when you add this extra segment from jumping ships, it's unambiguous the twin 481 00:38:45,150 --> 00:38:48,215 on Earth is older. This is not a contradiction to 482 00:38:48,215 --> 00:38:51,093 relativity. There is a difference between them. 483 00:38:51,093 --> 00:38:54,660 You could measure the difference. One of them accelerated. 484 00:38:54,660 --> 00:38:57,920 Okay. So we've cleared up the paradox 485 00:38:57,920 --> 00:39:02,344 We understand that the twin that accelerated remained younger and there is 486 00:39:02,344 --> 00:39:07,240 not a violation of relativity because you could measure the difference between the 487 00:39:07,240 --> 00:39:09,482 two. One was inertial, one accelerating. 488 00:39:09,482 --> 00:39:13,966 Okay, but there is an unsatisfactory feeling to this resolution because what 489 00:39:13,966 --> 00:39:18,508 I've essentially done is hidden all my ignorance in this instant of infinite 490 00:39:18,508 --> 00:39:22,520 acceleration where instantaneously as seen by the moving twin is. 491 00:39:22,520 --> 00:39:26,760 Suddenly aged all the difference that makes up the, the, the, for the, for the, 492 00:39:26,760 --> 00:39:30,022 Lorentz, the time dilation that he otherwise would have seen. 493 00:39:30,022 --> 00:39:32,686 This infinity doesn't make sense. Can I do better? 494 00:39:32,686 --> 00:39:36,274 Well of course I can do better. What I really want to do, and in fact, 495 00:39:36,274 --> 00:39:40,080 the only physical thing to do, is you can't have infinite acceleration. 496 00:39:40,080 --> 00:39:44,449 You could apply a finite acceleration and you could round off this corner a little 497 00:39:44,449 --> 00:39:46,625 bit. And now I have a story without infinite 498 00:39:46,625 --> 00:39:48,984 acceleration. You can ask well what happens here? 499 00:39:48,984 --> 00:39:53,062 computing here is a little bit more difficult, because now the moving twin is 500 00:39:53,062 --> 00:39:56,207 not in an inertial frame. The way I cheated before was, that I, he 501 00:39:56,207 --> 00:39:59,008 jumped from one inertial frame to another inertial frame. 502 00:39:59,008 --> 00:40:02,841 And I ignored the acceleration that this implies, now I have to deal with this 503 00:40:02,841 --> 00:40:05,414 acceleration. And that requires a little more 504 00:40:05,414 --> 00:40:09,344 calculation then we have. But we can get a sense for what this 505 00:40:09,344 --> 00:40:13,576 implies, by imagining here's a trick that will cause him to accelerate. 506 00:40:13,576 --> 00:40:18,352 Remember I want him to be moving to the right and then accelerated to the left. 507 00:40:18,352 --> 00:40:21,920 And the way I'm going to do this is I'm going to imagine that 508 00:40:21,920 --> 00:40:27,110 Somewhere out here, I don't know. Little distance not exactly in the 509 00:40:27,110 --> 00:40:30,897 direction it is moving so it isn't crashing there. 510 00:40:30,897 --> 00:40:35,290 But not far from there lies a star a heavy massive object. 511 00:40:35,290 --> 00:40:40,444 And what I am going to imagine is that as this twin approaches, the star is 512 00:40:40,444 --> 00:40:45,730 accelerated and then goes into hyperbolic orbit and is deflected by the star so 513 00:40:45,730 --> 00:40:50,620 that he comes back to the earth. So basically I arrange the angles so that 514 00:40:50,620 --> 00:40:54,736 he overshoots the star. Gets captured affected by its gravity, 515 00:40:54,736 --> 00:40:58,749 loops around and comes back. Okay, now since he was going very fast I 516 00:40:58,749 --> 00:41:03,469 wanted this to be relativistic, he needs quite a bit of acceleration to flip the 517 00:41:03,469 --> 00:41:08,131 direction of that velocity, so he'll to get near the star or the star has to be 518 00:41:08,131 --> 00:41:11,520 very massive whichever way you want to think about it. 519 00:41:11,520 --> 00:41:16,676 And this gives us another point of view on what is going on here and why it is 520 00:41:16,676 --> 00:41:21,963 that in an unambiguous way, not the usual time dilation that is symmetric between 521 00:41:21,963 --> 00:41:25,179 two, Two inertial frames but in an asymmetric 522 00:41:25,179 --> 00:41:30,428 way honestly physically the moving twins clock was running slower than the 523 00:41:30,428 --> 00:41:36,014 stationary twins clock and it somehow has to do with the acceleration and the idea 524 00:41:36,014 --> 00:41:40,119 is imagine braking up the trajectory in sort of three pieces. 525 00:41:40,119 --> 00:41:44,762 Piece number one is the piece over here until he approaches the star. 526 00:41:44,762 --> 00:41:49,810 So, some at this point, we imagine we can ignore the gravity of the star and 527 00:41:49,810 --> 00:41:54,799 everything is inertial and indeed, The stationary twin thinks his brother's 528 00:41:54,799 --> 00:41:57,697 aging slower. The moving twin thinks the stationary 529 00:41:57,697 --> 00:42:01,049 twin's aging slower. And indeed the same thing obtains here. 530 00:42:01,049 --> 00:42:05,764 And all the symmetric Lorentz invariant. Lorentz transformation's back and forth 531 00:42:05,764 --> 00:42:09,798 describe what was going on here. And then there's the region in between 532 00:42:09,798 --> 00:42:12,866 these two events where there's this high acceleration. 533 00:42:12,866 --> 00:42:15,650 And now we have a sense of what is going on here. 534 00:42:15,650 --> 00:42:19,991 We know that during this period the blue clock runs slower than the black clock. 535 00:42:19,991 --> 00:42:24,278 But from our point of view its somewhat clear, the black clock is very far away 536 00:42:24,278 --> 00:42:27,642 from the star, the blue clock is running very near to the star. 537 00:42:27,642 --> 00:42:31,712 The blue clock runs slow during this period because of a gravitational red 538 00:42:31,712 --> 00:42:34,045 shift, and how much gravitational red shift? 539 00:42:34,045 --> 00:42:37,410 Well that depends how mass of the star in is, is and how close. 540 00:42:37,410 --> 00:42:42,499 And that is determined by how massive the star has to be and how close you have to 541 00:42:42,499 --> 00:42:47,405 get to it in order to deflect this velocity V change your velocity from plus 542 00:42:47,405 --> 00:42:50,103 V to minus V, zooming you back towards Earth. 543 00:42:50,103 --> 00:42:54,580 So at least without a calculation, you can do a calculation, but it's some 544 00:42:54,580 --> 00:43:00,553 intuitive level that is clear that indeed The accelerating phase of this moving 545 00:43:00,553 --> 00:43:06,902 twins trajectory is the part the phase of the journey during which unambiguously 546 00:43:06,902 --> 00:43:12,937 the moving twin ages less than the stationary twin in a way in which they 547 00:43:12,937 --> 00:43:16,230 both agree that this is not a question of. 548 00:43:16,230 --> 00:43:20,019 Whose clock is measuring it, but he the literally he is edging slower this is 549 00:43:20,019 --> 00:43:23,857 because we can now see the asymmetry. He is at the bottom of the gravitational 550 00:43:23,857 --> 00:43:27,499 level and his brother at the top. So, thinking about the gravitational time 551 00:43:27,499 --> 00:43:30,993 dilation is a very power tool and in particular, it gives us the way of 552 00:43:30,993 --> 00:43:34,733 reasoning our way through the twin paradox and understanding what there had 553 00:43:34,733 --> 00:43:35,718 been being going on.