1 00:00:00,000 --> 00:00:04,926 Okay so we're going to take the principle of relativity, and try to understand what 2 00:00:04,926 --> 00:00:09,222 it tells us about physics, and we need a mathematical framework in order to 3 00:00:09,222 --> 00:00:12,316 describe this. And the mathematical framework in which 4 00:00:12,316 --> 00:00:15,840 one discusses relativity is called space time. 5 00:00:15,840 --> 00:00:19,343 This is often introduced as some kind of mystical fourth dimension. 6 00:00:19,343 --> 00:00:23,003 I want to demystify space-time. I want us to get very, very comfortable 7 00:00:23,003 --> 00:00:25,304 with it. So we're going to spend quite a bit of 8 00:00:25,304 --> 00:00:28,023 time getting comfortable with what space-time means. 9 00:00:28,023 --> 00:00:29,906 So, alright, what do we mean by space? 10 00:00:29,906 --> 00:00:33,932 You can get into Kantian philosophy, but what we mean by space is all the 11 00:00:33,932 --> 00:00:37,488 possible places something can be. And we live in a three-dimensional 12 00:00:37,488 --> 00:00:40,102 universe, so all the possible places something can 13 00:00:40,102 --> 00:00:42,821 be are listed as, are labeled by three numbers. 14 00:00:42,821 --> 00:00:45,750 Pick an origin, pick some axes, measure the distance east, 15 00:00:45,750 --> 00:00:50,083 north, and up from that position, and you find all the possible, positions an 16 00:00:50,083 --> 00:00:54,583 object can be, and, the position of an object at a given time is given by these 17 00:00:54,583 --> 00:00:56,750 three numbers. We're studying mechanics. 18 00:00:56,750 --> 00:01:01,028 We're studying the motion of objects. To describe the motion of an object, you 19 00:01:01,028 --> 00:01:05,473 basically let your clock run, and at any given time, you specify where the object 20 00:01:05,473 --> 00:01:09,473 is. So to discuss the motion of an object, you need to compute, or find, or 21 00:01:09,473 --> 00:01:13,585 measure three functions. It's position east and west, it's position north and 22 00:01:13,585 --> 00:01:18,029 south, its position up and down, and, you will remember to the days you were, 23 00:01:18,029 --> 00:01:21,911 doing, high school algebra, that to understand 24 00:01:21,911 --> 00:01:26,680 such functions, you often plot them. You take the T axis. 25 00:01:26,680 --> 00:01:31,892 You plot the function x as a function of t and if you were describing the motion 26 00:01:31,892 --> 00:01:37,233 of an object which started at some point, say x equals zero and moved up the x axis 27 00:01:37,233 --> 00:01:42,316 and then stopped, and then moved a little bit down the x axis and then you stop, 28 00:01:42,316 --> 00:01:46,950 steady okay, then you would have drawn this graph of the function x of t. 29 00:01:46,950 --> 00:01:52,166 This is something we've all done. This taper, this plane in which I've 30 00:01:52,166 --> 00:01:55,805 drawn it, is space time. Space time is the place in which you 31 00:01:55,805 --> 00:01:59,452 clock the motions of objects. Now in this case I was plotting the 32 00:01:59,452 --> 00:02:03,885 motion of an object in one dimension. What would I do if the object were moving 33 00:02:03,885 --> 00:02:07,588 in a two dimensional space? Well then I'd have to specify both the 34 00:02:07,588 --> 00:02:12,487 function x of t and a function y of t. So what I would do, perhaps if I were 35 00:02:12,487 --> 00:02:18,245 graphically talented, which I am not, is I would add an Y axis, and then at any 36 00:02:18,245 --> 00:02:24,154 time, I would plot X by moving my point up and down, Y by moving my point in and 37 00:02:24,154 --> 00:02:27,519 out. And for example, if I study the motion of 38 00:02:27,519 --> 00:02:31,034 that object that was moving in a circle before, 39 00:02:31,034 --> 00:02:36,419 then the motion of the object would be in some circle in this X Y plane. 40 00:02:36,419 --> 00:02:41,813 But as I plot it, as a function of T, that would be like the motion of the 41 00:02:41,813 --> 00:02:45,134 object that we studied before was up and down the X axis. 42 00:02:45,134 --> 00:02:49,330 When I plot it as a function of T, the object would say, start here at T equals 43 00:02:49,330 --> 00:02:52,476 zero, and then move along the circle as a function of T, 44 00:02:52,476 --> 00:02:56,322 and I would be plotting, if I could draw, the sort of helical object. 45 00:02:56,322 --> 00:03:00,926 This is, would be called the world line of an object that is moving in a circle 46 00:03:00,926 --> 00:03:04,072 in the X Y plane. Now, my graphical skills are not that 47 00:03:04,072 --> 00:03:08,210 great, so let's do a slightly better drawing and see what it teaches us. 48 00:03:08,210 --> 00:03:12,312 Here we have our friend and this time a green sphere. 49 00:03:12,312 --> 00:03:17,737 And it's going to be moving at a constant speed around this blue circle. 50 00:03:17,737 --> 00:03:21,075 And what I've done is I've embedded this circle, 51 00:03:21,075 --> 00:03:25,763 which is a two-dimensional space. Motion is in two dimensions, and I've 52 00:03:25,763 --> 00:03:31,004 embedded it in a three dimensional cube where the vertical direction is going to 53 00:03:31,004 --> 00:03:34,498 denote time. It is indeed a curiosity of drawing space 54 00:03:34,498 --> 00:03:39,674 time diagrams that rather than having the T access horizontal, it's our habit to 55 00:03:39,674 --> 00:03:44,344 have the T access be vertical. And so if I plot the motion of this 56 00:03:44,344 --> 00:03:47,407 particle is going to be given by the green dot, 57 00:03:47,407 --> 00:03:50,536 there's going to be a red dot that will move, 58 00:03:50,536 --> 00:03:53,600 that will, at any time, at any vertical position, 59 00:03:53,600 --> 00:03:58,685 be dropped at the place along the circle where the particle is at the time 60 00:03:58,685 --> 00:04:02,596 corresponding to that slice. And so, as I plot this, the green 61 00:04:02,596 --> 00:04:07,506 particle moves around the circle. The red dot tracks it, and the red dot 62 00:04:07,506 --> 00:04:13,056 also is placed at any time, at the vertical heights corresponding to that 63 00:04:13,056 --> 00:04:18,758 time, and the net result is that I construct the helix that I was trying so 64 00:04:18,758 --> 00:04:23,396 unsuccessfully to draw. All this is, is the motion, if you want 65 00:04:23,396 --> 00:04:26,893 the storyboard of the motion of this particle. 66 00:04:26,893 --> 00:04:33,380 Space-time is the place in which you draw graphs of X and Y as a function of T. 67 00:04:33,380 --> 00:04:37,880 So. Let's study some simpler motions in space 68 00:04:37,880 --> 00:04:41,472 time. What we're going to have two dots moving 69 00:04:41,472 --> 00:04:44,009 around in our 2-dimensional space. 70 00:04:44,009 --> 00:04:48,121 One is going to be moving up the x axis and the other up the y axis. 71 00:04:48,121 --> 00:04:52,475 The yellow dot moves slowly and the red dot, as you can see, moves faster. 72 00:04:52,475 --> 00:04:56,043 What do the space time plots of these motions look like? 73 00:04:56,043 --> 00:04:58,644 So again, I plot x and y as a function of t. 74 00:04:58,644 --> 00:05:03,240 I'll have a blue trace for the yellow dot and a green trace for the red dot. 75 00:05:03,240 --> 00:05:08,609 And what we see when we look at the world lines that we have drawn is, of course, 76 00:05:08,609 --> 00:05:12,555 they are plots of where the dots where as a function of time. 77 00:05:12,555 --> 00:05:17,665 And when we tilt it we see that the slow moving yellow dot generated a steeper 78 00:05:17,665 --> 00:05:22,642 trace then the more rapidly moving. Red dot, this is reasonable because over 79 00:05:22,642 --> 00:05:27,457 any interval of time, as between each time we plotted these spheres, the 80 00:05:27,457 --> 00:05:33,085 speedier red dot moved farther along the Y axis than the slower yellow dot managed 81 00:05:33,085 --> 00:05:37,426 to move along the X axis. And we see that the traces, these world 82 00:05:37,426 --> 00:05:42,580 lines, these graphs, for both of these red and yellow dots are straight lines. 83 00:05:42,580 --> 00:05:46,533 They were moving in a straight line and their world lines are straight lines. 84 00:05:46,533 --> 00:05:50,127 That's not exactly the same thing. In this diagram, what we're going to is 85 00:05:50,127 --> 00:05:53,722 we're going to have the same red dot, it's going to move along the y axis in 86 00:05:53,722 --> 00:05:56,392 the same straight line, but it's going to accelerate. 87 00:05:56,392 --> 00:05:58,599 What is that going to do to its world line? 88 00:05:58,599 --> 00:06:02,502 Well, when it's moving slow, its world line will be close to the vertical, and 89 00:06:02,502 --> 00:06:06,712 as it speeds up, its world line will curl away from the vertical, so that the graph 90 00:06:06,712 --> 00:06:11,021 that we draw is no longer a straight line, despite the fact that the motion of 91 00:06:11,021 --> 00:06:15,167 the object was in a straight line. Because it was not moving at a constant 92 00:06:15,167 --> 00:06:17,797 speed, its work line curved. So this is space time. 93 00:06:17,797 --> 00:06:20,135 Space time is the place where you plot R of T. 94 00:06:20,135 --> 00:06:22,624 Now, of course, I plotted a one dimensional motion, 95 00:06:22,624 --> 00:06:26,638 and whenever I'm plotting anything, it'll only be motion in one dimension, because 96 00:06:26,638 --> 00:06:29,840 I can barely plot that. The computer managed to do a decent job 97 00:06:29,840 --> 00:06:34,108 manning, plotting two dimensional motion. If you were really going to plot motion 98 00:06:34,108 --> 00:06:37,614 in a three-dimensional space, you would need a four-dimensional graph. 99 00:06:37,614 --> 00:06:41,120 We don't really have the technology to draw a four dimensional graph 100 00:06:41,120 --> 00:06:43,000 effectively. We can't even imagine it, 101 00:06:43,000 --> 00:06:47,863 but we can mathematically describe it. And really this fourth dimension is 102 00:06:47,863 --> 00:06:52,333 basically the statement that we are drawing three functions at once. 103 00:06:52,333 --> 00:06:56,999 There is nothing fancier about that. And so, the points in space time. 104 00:06:56,999 --> 00:06:59,760 The points in this four dimensional graph. 105 00:06:59,760 --> 00:07:03,598 what are they? The points in space rather places you can 106 00:07:03,598 --> 00:07:06,200 be. The points in space time are all the 107 00:07:06,200 --> 00:07:10,883 places you can be at a given time. These are all the places all the ways 108 00:07:10,883 --> 00:07:15,957 something can happen, what we call given position in space in given time is an 109 00:07:15,957 --> 00:07:18,980 event. It's a more specific statement than a, a 110 00:07:18,980 --> 00:07:23,221 position. So for example Durham at this particular 111 00:07:23,221 --> 00:07:26,503 time is an event. Durham is a location. 112 00:07:26,503 --> 00:07:32,826 Right now my world line passes through Durham, because at this point I am in 113 00:07:32,826 --> 00:07:37,868 Durham at this event. but Durham in 1985 is also the same 114 00:07:37,868 --> 00:07:42,030 location. It's a different event and I was not there. 115 00:07:42,030 --> 00:07:45,369 And so a point in space time is called an event. 116 00:07:45,369 --> 00:07:49,195 It specifies something happening or not happening, 117 00:07:49,195 --> 00:07:54,482 and through all of the possible places and times, you draw a one-dimensional 118 00:07:54,482 --> 00:07:58,643 line. That is the, trajectory of a particle. 119 00:07:58,643 --> 00:08:01,280 So, we do not move in space time. 120 00:08:01,280 --> 00:08:06,548 We do not move in time. these are all ideas that are confusing, 121 00:08:06,548 --> 00:08:11,817 that come from treating time in the same way that we treat space. 122 00:08:11,817 --> 00:08:18,382 What the extra dimension is that we add to space-time is basically the slices of 123 00:08:18,382 --> 00:08:24,866 the movie of our life where you prod-, use the variable T to plot your position 124 00:08:24,866 --> 00:08:28,919 as a function of time. Hopefully that helps demystify this 125 00:08:28,919 --> 00:08:31,630 story. we've learned something about the 126 00:08:31,630 --> 00:08:36,421 geometry of space-time. We've learned that if your trajectory in space is 127 00:08:36,421 --> 00:08:41,339 curved, then your worldline will curve. The worldline is this graph of all the 128 00:08:41,339 --> 00:08:45,689 positions that you've occupied over the course of some time interval. 129 00:08:45,689 --> 00:08:50,559 So, if the object is changing direction of motion in space, then it's worldline 130 00:08:50,559 --> 00:08:54,050 tracking this change will change it's direction as well. 131 00:08:54,050 --> 00:08:58,901 If an object is moving in a straight line, then its worldline will not have to 132 00:08:58,901 --> 00:09:03,575 change direction, but if the object accelerates, the worldline curves towards 133 00:09:03,575 --> 00:09:07,540 or away from the vertical, as the object is slowing or accelerating. 134 00:09:07,540 --> 00:09:11,246 What does that tell us about an object whose world line is straight? 135 00:09:11,246 --> 00:09:15,498 Well an object whose world line is a straight line in space time is an object 136 00:09:15,498 --> 00:09:18,496 that is moving in the fixed direction at a fixed speed. 137 00:09:18,496 --> 00:09:20,895 Aha. What we have is a geometric description 138 00:09:20,895 --> 00:09:24,874 of Newton's first law in space-time. Space-time gives us a nice geometric 139 00:09:24,874 --> 00:09:28,145 description of objects. The, the world line of an object upon 140 00:09:28,145 --> 00:09:31,034 which no force acts is a straight line in space-time. 141 00:09:31,034 --> 00:09:35,449 And conversely, an object whose world line is a straight line in space-time is 142 00:09:35,449 --> 00:09:38,776 an object upon which no force has acted. So space-time is useful. 143 00:09:38,776 --> 00:09:44,544 File that away, we'll come back to it. So, here is that, version of space-time 144 00:09:44,544 --> 00:09:48,827 that I am able to draw. We imagine a one dimensional universe, 145 00:09:48,827 --> 00:09:53,812 objects are free to move to the left or to the right along this X axis. 146 00:09:53,812 --> 00:09:57,182 We plot their motion along X as a function of T. 147 00:09:57,182 --> 00:10:02,799 As usual we have the T axis vertical and so there are some names that you might 148 00:10:02,799 --> 00:10:06,872 give to all things. So all of the things, all of the points 149 00:10:06,872 --> 00:10:12,278 in this graph that are, I've picked some time T=0, so the coordinates of a point, 150 00:10:12,278 --> 00:10:15,545 a event. In this graph is given by its X 151 00:10:15,545 --> 00:10:20,277 coordinate and its T coordinate. X gives you the distance from some 152 00:10:20,277 --> 00:10:24,448 particular point, which I've called X equals zero, at which this, event 153 00:10:24,448 --> 00:10:29,025 occurred, and T gives you the time after some particular time that I decided to 154 00:10:29,025 --> 00:10:33,196 call T equals zero, when this event occurred, so give me an X and a T, and 155 00:10:33,196 --> 00:10:37,077 that tells you if something happened at that position at that time. 156 00:10:37,077 --> 00:10:41,364 You've specified where and when it happened, so all of the X axis here are 157 00:10:41,364 --> 00:10:44,724 all of the events that happened at the time T equals zero. 158 00:10:44,724 --> 00:10:47,100 By convention, I'm going to call that now. 159 00:10:47,100 --> 00:10:51,801 what are all the events that occurred at the position x equals zero? 160 00:10:51,801 --> 00:10:55,840 Those are all the things that naturally would be called here. 161 00:10:55,840 --> 00:11:00,211 So, the t axis might be called here. The x axis might be called now. 162 00:11:00,211 --> 00:11:05,574 And what we saw is that the world lines of objects that are moving at a constant 163 00:11:05,574 --> 00:11:10,475 velocity are going to be straight lines. Your instantaneous velocity is 164 00:11:10,475 --> 00:11:15,270 essentially the slope of your world line. But the slope is measured from the t 165 00:11:15,270 --> 00:11:17,857 axis. So it's a sort of backward slope because 166 00:11:17,857 --> 00:11:21,569 we've put t vertically. So if you take this red world line, that 167 00:11:21,569 --> 00:11:26,124 describes an object moving to the right. It starts at this negative time way over 168 00:11:26,124 --> 00:11:28,823 here to the left and then it moves to the right. 169 00:11:28,823 --> 00:11:33,266 And if you look at this green world line, it describes an object which moves 170 00:11:33,266 --> 00:11:36,037 faster. You could measure your velocity by taking 171 00:11:36,037 --> 00:11:39,502 some time interval. Notice that I've arranged and will often 172 00:11:39,502 --> 00:11:43,314 do that, for both objects to be at the same place at the same time. 173 00:11:43,314 --> 00:11:45,047 At t0, equals zero, they were both at X equals zero. 174 00:11:45,047 --> 00:11:49,436 If these were material objects, they would have crashed into each other with 175 00:11:49,436 --> 00:11:53,132 the green object overtaking the red object and slamming into it. 176 00:11:53,132 --> 00:11:57,463 But they, in this case, pass right through each other and continue moving at 177 00:11:57,463 --> 00:12:00,582 a straight line. And if I take some time, delta t later. 178 00:12:00,582 --> 00:12:03,123 And I measure where these objects have gone. 179 00:12:03,123 --> 00:12:06,820 You can see that in the time, delta t, the red object has gone. 180 00:12:06,820 --> 00:12:11,586 This distance, delta x red, and the green object has gone the larger distance over 181 00:12:11,586 --> 00:12:15,293 here, delta x green. this is telling me that the green object 182 00:12:15,293 --> 00:12:19,001 is moving faster than the red object. This is what we're saying. 183 00:12:19,001 --> 00:12:22,649 What does it mean for an object to have a vertical world line. 184 00:12:22,649 --> 00:12:26,239 A vertical world line means x is a function of t as constant. 185 00:12:26,239 --> 00:12:31,005 The blue object is a stationary object that sits at the same location throughout 186 00:12:31,005 --> 00:12:33,334 time. What does it mean to have horizontal 187 00:12:33,334 --> 00:12:35,589 world? Nobody has a horizontal world line. 188 00:12:35,589 --> 00:12:39,878 A horizontal world line is a collection of all the event that happened at the 189 00:12:39,878 --> 00:12:44,332 same time so just as the X axis is what I call now, the orange axis might be what 190 00:12:44,332 --> 00:12:48,622 you call a little bit later, these are a collection of all of the events along 191 00:12:48,622 --> 00:12:52,636 this line are all the things that happen at a particular time so at this 192 00:12:52,636 --> 00:12:56,815 particular time indicated by the orange line, the blue particle was where it 193 00:12:56,815 --> 00:13:01,215 always was, the red particle was at this position, the green particle was at that 194 00:13:01,215 --> 00:13:05,078 position and so on. I said that space time is the right place 195 00:13:05,078 --> 00:13:09,238 to discuss relativity. Let's see what relativity looks like in 196 00:13:09,238 --> 00:13:13,197 space time. So what I have here is space time as seen 197 00:13:13,197 --> 00:13:17,186 by some observer. Their axis are drawn in black, I will use 198 00:13:17,186 --> 00:13:21,962 a green pen to denote their observations because I don't have a black pen. 199 00:13:21,962 --> 00:13:26,351 So this is the black observers now this is the black observers here. 200 00:13:26,351 --> 00:13:31,320 And I have here a world line that is tilted to the right, that means it's the 201 00:13:31,320 --> 00:13:36,613 world line of something moving from left to right at some constant velocity along 202 00:13:36,613 --> 00:13:40,098 this diagram. And what this is going to be is the node 203 00:13:40,098 --> 00:13:42,809 or the origin as seen by another observer. 204 00:13:42,809 --> 00:13:46,940 The blue observer thinks that this is here, what he thinks of as. 205 00:13:46,940 --> 00:13:51,855 Something that is stationary is going to be something that is moving relative the 206 00:13:51,855 --> 00:13:54,973 first observer. This is precisely the situation that 207 00:13:54,973 --> 00:13:59,730 relativity describes. The world is observed by an observer in 208 00:13:59,730 --> 00:14:04,831 the black purse axes, and the world is observed by someone else who is moving at 209 00:14:04,831 --> 00:14:07,560 a constant velocity relative to the first one. 210 00:14:07,560 --> 00:14:11,679 And the statement is that the laws of physics as observed in the blue system 211 00:14:11,679 --> 00:14:16,119 should be the same as the laws of physics as observed in the black system with the 212 00:14:16,119 --> 00:14:20,207 green notations. And so we have arranged things so that at 213 00:14:20,207 --> 00:14:24,917 the time that the black axis s-, select as T equals zero. 214 00:14:24,917 --> 00:14:30,496 We've picked that time to be the exact instant where along their motion from 215 00:14:30,496 --> 00:14:36,148 left to right the blue observer passed the location of the gr-, black observer. 216 00:14:36,148 --> 00:14:41,944 And moreover, we have set it up, so that at that same time both observers started 217 00:14:41,944 --> 00:14:45,277 their clocks. What this means is that the blue 218 00:14:45,277 --> 00:14:49,309 observer. Considers T equals zero to be the same as 219 00:14:49,309 --> 00:14:55,103 what the black observer considers. So they agree on what now is, and then 220 00:14:55,103 --> 00:15:01,299 they measure time with synchronized precise clocks from that moment on. 221 00:15:01,299 --> 00:15:06,646 And what happens then is that if there's some event going on. 222 00:15:06,646 --> 00:15:12,607 Somewhere over here an event occurs, a birthday party takes place. 223 00:15:12,607 --> 00:15:20,128 Well, then the black observer, measures when and where the event occurred, and he 224 00:15:20,128 --> 00:15:23,980 declares that it occurred at a distance x. 225 00:15:23,980 --> 00:15:28,720 From the place from the origin and at a time t. 226 00:15:28,720 --> 00:15:34,260 What does the, blue observer measure? Well, it's clear that the blue observer 227 00:15:34,260 --> 00:15:40,226 measures a different distance, since, the blue observer is at a different position 228 00:15:40,226 --> 00:15:45,198 at the time the party occurred. He measures the party to occur at this 229 00:15:45,198 --> 00:15:50,100 distance, to the right of his origin, and he also measures the time of. 230 00:15:50,100 --> 00:15:54,284 At which the party occurred. And since they synchronize their clocks, 231 00:15:54,284 --> 00:15:57,300 they agree on the measure of, measurement of time. 232 00:15:57,300 --> 00:16:01,792 So, they measure, for the same birthday party, the same event, the same time. 233 00:16:01,792 --> 00:16:05,054 But they disagree on where the birthday party occurs. 234 00:16:05,054 --> 00:16:09,731 It's not that they disagree, but they describe it with different coordinates. 235 00:16:09,731 --> 00:16:14,346 The position to the right of the axis. Or the origin, at which, the moving 236 00:16:14,346 --> 00:16:20,070 observer, measures the party differs from the pos-, distance from the origin that, 237 00:16:20,070 --> 00:16:25,095 black stationary observer measures the party to be at, by a quantity that is V 238 00:16:25,095 --> 00:16:30,040 T, where V T is simply the distance over here, 239 00:16:30,040 --> 00:16:34,868 by which the two origins differ. And it changes the function of t, because 240 00:16:34,868 --> 00:16:39,828 over time, the blue observer is moving to the right, and can invert these 241 00:16:39,828 --> 00:16:45,738 relations, and figure out how to find the description given by the stationary 242 00:16:45,738 --> 00:16:51,746 observer in black, given the data in the measurements of the moving observer in 243 00:16:51,746 --> 00:16:54,046 blue. And it's, you can solve these linear 244 00:16:54,046 --> 00:16:58,083 equations rather easily for x and t as functions of x prime and t prime. 245 00:16:58,083 --> 00:17:01,391 And it's not surprising that these equations look the same, 246 00:17:01,391 --> 00:17:04,476 except the sign of v has changed. What this means is, 247 00:17:04,476 --> 00:17:08,681 the black observer sees the blue observer moving to the right with speed v. 248 00:17:08,681 --> 00:17:12,663 If the blue observer were to describe what the black observer is doing, 249 00:17:12,663 --> 00:17:17,093 they would find that the black observer is moving to the left with exactly the 250 00:17:17,093 --> 00:17:20,155 same speed. There is not a difference in principle 251 00:17:20,155 --> 00:17:23,917 between these two systems. What they measure is the relative 252 00:17:23,917 --> 00:17:26,362 velocity. V is not some absolute object. 253 00:17:26,362 --> 00:17:30,501 It's the relative velocity. The blue observer is moving say to the 254 00:17:30,501 --> 00:17:35,454 right in this picture relative to the black observer, or equivalently the black 255 00:17:35,454 --> 00:17:39,279 observer is moving to the left relative to the blue observer, 256 00:17:39,279 --> 00:17:42,540 and physics does not select who is right about this. 257 00:17:42,540 --> 00:17:46,784 This is all very logical and am, I am indeed belaboring the obvious, but let's 258 00:17:46,784 --> 00:17:50,476 belabor it one more time. Let's understand what this tells us about 259 00:17:50,476 --> 00:17:53,508 some an important topic called velocity addition. 260 00:17:53,508 --> 00:17:57,807 So here's the same collection of two observers with their here's and now's, 261 00:17:57,807 --> 00:18:01,390 and now I've introduced the world line of some other object here, 262 00:18:01,390 --> 00:18:04,945 and this object is also moving to the right in this picture. 263 00:18:04,945 --> 00:18:09,566 It's moving faster, it turns out, than the blue observer so that it's moving to 264 00:18:09,566 --> 00:18:14,069 the right both respect to the black observer and respect to, with respect to 265 00:18:14,069 --> 00:18:17,387 the blue observer. But the blue observer can measure its 266 00:18:17,387 --> 00:18:20,113 velocity. Notice this is a straight world line, 267 00:18:20,113 --> 00:18:23,550 the motion is inertial motion with a constant velocity. 268 00:18:23,550 --> 00:18:28,349 And as the blue observer measures it this velocity is given by some number in 269 00:18:28,349 --> 00:18:32,306 meters per second, u prime. Now both observes will agree that this 270 00:18:32,306 --> 00:18:35,901 thing is moving. They will not agree on its speed relative 271 00:18:35,901 --> 00:18:38,999 to them. Let's compute the speed with which the 272 00:18:38,999 --> 00:18:43,709 black observer measures, the speed that the black observer measures for this 273 00:18:43,709 --> 00:18:48,046 green object for which the blue observers, observer measures a speed u 274 00:18:48,046 --> 00:18:49,844 prime. Well how do we do that? 275 00:18:49,844 --> 00:18:54,814 This equation describes the motion, in the, blue observers system and, the 276 00:18:54,814 --> 00:18:59,854 coefficient gives us, the velocity. Note that I have selected, particularly 277 00:18:59,854 --> 00:19:04,893 nice world line for the green object, I have set it up so that the green object 278 00:19:04,893 --> 00:19:08,976 meets both observers at the same time that they meet each other. 279 00:19:08,976 --> 00:19:13,888 You will often discuss such things, discussing an object which makes the same 280 00:19:13,888 --> 00:19:19,247 motion, but, meets the observers a little bit later or a little bit earlier, is not 281 00:19:19,247 --> 00:19:23,396 that much more complicated, adds in a necessary algebra step, so we 282 00:19:23,396 --> 00:19:27,005 won't deal with it. And so, how do we transcribe the motion 283 00:19:27,005 --> 00:19:31,921 in as seen by the moving observer to the motion as scene by the black observer 284 00:19:31,921 --> 00:19:35,515 very easy We know the relation, we wrote it in 285 00:19:35,515 --> 00:19:41,394 general, between, the coordinates as observed by the observer in black, and 286 00:19:41,394 --> 00:19:44,876 those as observed by the observer in, blue, 287 00:19:44,876 --> 00:19:49,517 and given X prime and T prime, you can figure out X and T. 288 00:19:49,517 --> 00:19:54,933 Now we need to plug into this, the information we have, which is that X 289 00:19:54,933 --> 00:19:58,260 prime is U prime T prime. So, I plug that in. 290 00:19:58,260 --> 00:20:04,530 U prime T prime plus V T prime. I collect terms and write that as U prime 291 00:20:04,530 --> 00:20:09,082 plus V, T prime. And then I remember that because they 292 00:20:09,082 --> 00:20:13,707 synchronize their clocks, t and t prime are the same thing, and I 293 00:20:13,707 --> 00:20:16,693 reach the conclusion that x is u prime plus vt. 294 00:20:16,693 --> 00:20:19,806 Oh, good. This is the equation of exactly the same 295 00:20:19,806 --> 00:20:23,554 form but describes an object moving at a constant velocity. 296 00:20:23,554 --> 00:20:25,102 Excellent. Inertial motion. 297 00:20:25,102 --> 00:20:29,093 Remember physics should not match, should not change between frames. 298 00:20:29,093 --> 00:20:33,712 If we met one observer who says no force is acting on the green object, the other 299 00:20:33,712 --> 00:20:36,848 observer agrees. No force is acting on the green object. 300 00:20:36,848 --> 00:20:40,269 Its motion, its world line is a straight line in both frames. 301 00:20:40,269 --> 00:20:44,774 Its motion has a constant velocity, and that constant velocity is just obtained 302 00:20:44,774 --> 00:20:49,394 by adding the velocity as measured from the moving observers viewpoint with the 303 00:20:49,394 --> 00:20:51,960 relative velocity between the two observers. 304 00:20:51,960 --> 00:20:56,001 This is not that, anything. I mean, this is not saying anything that you didn't 305 00:20:56,001 --> 00:20:58,746 already know. If you are standing in a train that is 306 00:20:58,746 --> 00:21:02,477 moving 100 miles an hour, and you are walking at one mile an hour in the 307 00:21:02,477 --> 00:21:06,621 direction the train is moving, relative to the ground you're moving at 101 miles 308 00:21:06,621 --> 00:21:08,072 per hour. This is so obvious. 309 00:21:08,072 --> 00:21:12,113 Why did I take so long to discuss it? We'll see why it's worth discussing in a 310 00:21:12,113 --> 00:21:12,476 minute.