1 00:00:00,213 --> 00:00:01,072 - [Voiceover] In the last video, 2 00:00:01,072 --> 00:00:04,021 we started to construct a space-time diagram 3 00:00:04,021 --> 00:00:05,994 for my frame of reference 4 00:00:05,994 --> 00:00:08,061 and I'm just drifting through space 5 00:00:08,061 --> 00:00:10,569 and I'm assuming that I'm in an intertial frame of reference 6 00:00:10,569 --> 00:00:12,496 which means I'm moving at a constant velocity 7 00:00:12,496 --> 00:00:15,027 relative to all other frames of reference 8 00:00:15,027 --> 00:00:17,348 and we set up a situation where 9 00:00:17,348 --> 00:00:19,995 I emitted a photon right at time zero, 10 00:00:19,995 --> 00:00:21,714 so after one second, it would have moved 11 00:00:21,714 --> 00:00:23,409 3 times 10 to the 8th meters. 12 00:00:23,409 --> 00:00:24,570 After two seconds, it would have moved 13 00:00:24,570 --> 00:00:26,543 6 times 10 to the 8th meters, 14 00:00:26,543 --> 00:00:27,913 and then we had a, 15 00:00:27,913 --> 00:00:30,305 we added a little bit of flavor 16 00:00:30,305 --> 00:00:31,930 to our little scenario 17 00:00:31,930 --> 00:00:34,647 where right at time zero, a friend passes me up 18 00:00:34,647 --> 00:00:36,713 in her spaceship and she is traveling 19 00:00:36,713 --> 00:00:39,732 relative to me in the positive-x direction 20 00:00:39,732 --> 00:00:41,728 at half the speed of light, 21 00:00:41,728 --> 00:00:44,352 and so we plotted her path right at time zero. 22 00:00:44,352 --> 00:00:45,606 Her spaceship is right there. 23 00:00:45,606 --> 00:00:46,906 I could draw the spaceship. 24 00:00:46,906 --> 00:00:48,207 It's at the origin. 25 00:00:48,207 --> 00:00:49,901 Then after one, 26 00:00:49,901 --> 00:00:51,341 after one second, 27 00:00:51,341 --> 00:00:53,937 she would have traveled 28 00:00:53,937 --> 00:00:56,168 1 1/2 times 10 to the 8th meters. 29 00:00:56,168 --> 00:00:57,654 After two seconds, she would have traveled 30 00:00:57,654 --> 00:00:59,465 3 times 10 to the 8th meters, 31 00:00:59,465 --> 00:01:00,881 so this blue line in the last video 32 00:01:00,881 --> 00:01:02,414 was her path. 33 00:01:02,414 --> 00:01:03,993 But now I want to make it even more interesting. 34 00:01:03,993 --> 00:01:05,873 Let's assume that we have actually a whole train 35 00:01:05,873 --> 00:01:09,496 of spaceships, all traveling in the positive-x direction 36 00:01:09,496 --> 00:01:11,887 at the same velocity as her spaceship. 37 00:01:11,887 --> 00:01:13,884 So this is her spaceship right over here, 38 00:01:13,884 --> 00:01:16,067 at time equals zero, 39 00:01:16,067 --> 00:01:18,389 she is exactly where I am, 40 00:01:18,389 --> 00:01:21,105 but let's say that 3 times 10 to the 8th meters in front 41 00:01:21,105 --> 00:01:23,218 of her, there's another spaceship, 42 00:01:23,218 --> 00:01:26,051 traveling at exactly 43 00:01:26,051 --> 00:01:29,952 1 1/2 times 10 to the 8th meters per second, 44 00:01:29,952 --> 00:01:34,085 so they're traveling at the same relative velocity to me, 45 00:01:34,085 --> 00:01:35,849 but if you think about it, from each of their point of view, 46 00:01:35,849 --> 00:01:37,591 they would seem to be stationary, 47 00:01:37,591 --> 00:01:39,077 because the distance between them, 48 00:01:39,077 --> 00:01:41,213 in this x-direction is going to stay the same. 49 00:01:41,213 --> 00:01:43,372 So this person, at time zero, is going to be 50 00:01:43,372 --> 00:01:46,437 3 times 10 to the 8th meters in the positive-x direction 51 00:01:46,437 --> 00:01:47,343 from me. 52 00:01:47,343 --> 00:01:49,990 Now, if I were to wait two seconds, 53 00:01:49,990 --> 00:01:51,197 they are going to be, 54 00:01:51,197 --> 00:01:54,192 so if I were to wait two seconds, 55 00:01:54,192 --> 00:01:55,214 they are going to be, 56 00:01:55,214 --> 00:01:59,161 they are going to be 6 times 10 to the 8th meters 57 00:01:59,161 --> 00:02:01,413 away from me, because they're going half 58 00:02:01,413 --> 00:02:03,271 the speed of light, 59 00:02:03,271 --> 00:02:05,059 and so I could draw their path 60 00:02:05,059 --> 00:02:07,845 and I'm going to do it in a slightly more muted, 61 00:02:07,845 --> 00:02:10,562 thinner color, so that we, 62 00:02:10,562 --> 00:02:12,558 what I'm essentially going to be doing here 63 00:02:12,558 --> 00:02:15,484 is I'm setting up gridlines for 64 00:02:15,484 --> 00:02:19,431 my friend's alternate frame of reference. 65 00:02:19,431 --> 00:02:22,241 I'm going to put that on top of my frame of reference 66 00:02:22,241 --> 00:02:23,727 and to be clear, 67 00:02:23,727 --> 00:02:26,025 I'm not assuming special relativity. 68 00:02:26,025 --> 00:02:27,813 I'm assuming a Newtonian world, 69 00:02:27,813 --> 00:02:29,183 classical mechanics. 70 00:02:29,183 --> 00:02:31,157 Just to get familiar with these ideas 71 00:02:31,157 --> 00:02:34,825 and to see where the Newtonian world is going to break down 72 00:02:34,825 --> 00:02:36,288 but let's say that's not the only ship, 73 00:02:36,288 --> 00:02:37,356 let's say there was another ship, 74 00:02:37,356 --> 00:02:40,491 that at time equals zero is 6 times 10 to the 8th meters 75 00:02:40,491 --> 00:02:42,697 away from me in the positive x-direction. 76 00:02:42,697 --> 00:02:44,415 Well, where are they going to be? 77 00:02:44,415 --> 00:02:47,782 Where are they going to be after 78 00:02:47,782 --> 00:02:49,105 two seconds? 79 00:02:49,105 --> 00:02:50,104 Well, after, 80 00:02:50,104 --> 00:02:51,566 let's see, after one second, 81 00:02:51,566 --> 00:02:54,469 they're going to be 1 1/2 times 10 to the 8th meters 82 00:02:54,469 --> 00:02:55,723 further. 83 00:02:55,723 --> 00:03:00,499 Then after two seconds, they're going to be 84 00:03:00,499 --> 00:03:03,616 3 times 10 to the 8th meters further, 85 00:03:03,616 --> 00:03:05,891 and then after three seconds, 86 00:03:05,891 --> 00:03:08,724 after three seconds, so they're going to be 87 00:03:08,724 --> 00:03:12,718 1 1/2 times 10 to the 8th meters further than that, 88 00:03:12,718 --> 00:03:15,481 so their path is going to look like, 89 00:03:15,481 --> 00:03:17,710 so let me draw this. 90 00:03:17,710 --> 00:03:21,146 I'm really just doing these to show what stationary 91 00:03:21,146 --> 00:03:23,584 objects in my friend's frame of reference, 92 00:03:23,584 --> 00:03:25,442 how they actually look to me, 93 00:03:25,442 --> 00:03:27,647 and these should be parallel lines. 94 00:03:27,647 --> 00:03:29,157 These should be parallel lines, 95 00:03:29,157 --> 00:03:31,780 so that's a pretty good attempt, 96 00:03:31,780 --> 00:03:32,407 and so I'll, 97 00:03:32,407 --> 00:03:35,054 let me, and so let me actually just set up 98 00:03:35,054 --> 00:03:37,028 that alternate frame of reference, 99 00:03:37,028 --> 00:03:40,209 so this is going to be not only my x-axis, 100 00:03:40,209 --> 00:03:42,693 I'm also going to call this my x-prime axis 101 00:03:42,693 --> 00:03:44,644 and I'll do it in a second of thinking about 102 00:03:44,644 --> 00:03:46,455 how we can read this. 103 00:03:46,455 --> 00:03:49,589 So, I can color this in with the blue color. 104 00:03:49,589 --> 00:03:51,308 Actually, let me do it in that light blue color 105 00:03:51,308 --> 00:03:53,955 that we're going to be using for my friend's 106 00:03:53,955 --> 00:03:54,907 frame of reference, 107 00:03:54,907 --> 00:03:56,323 so this is also, 108 00:03:56,323 --> 00:04:00,224 this is also going to be our x-prime axis overlaid, 109 00:04:00,224 --> 00:04:03,428 and this one, let me call this the t-prime axis, 110 00:04:03,428 --> 00:04:04,891 so my friend's frame of reference, 111 00:04:04,891 --> 00:04:06,702 I'm going to call that the prime frame of reference, 112 00:04:06,702 --> 00:04:07,607 or if I called, 113 00:04:07,607 --> 00:04:10,881 if I called my frame of reference the S frame of reference, 114 00:04:10,881 --> 00:04:13,551 I can call my friend's frame of reference 115 00:04:13,551 --> 00:04:18,310 the S-prime frame of reference, 116 00:04:18,310 --> 00:04:21,073 and so, let me draw horizontal lines to show the 117 00:04:21,074 --> 00:04:23,327 passage of each second, 118 00:04:23,327 --> 00:04:26,995 so, that's one second, 119 00:04:26,995 --> 00:04:29,271 two seconds, 120 00:04:29,271 --> 00:04:31,314 three seconds, 121 00:04:31,314 --> 00:04:33,543 so on and so forth. 122 00:04:33,543 --> 00:04:36,352 So let's make sure we're comfortable with reading 123 00:04:36,352 --> 00:04:38,373 what's going on here. 124 00:04:38,373 --> 00:04:40,137 So, a point, 125 00:04:40,137 --> 00:04:41,693 let's say this point on our diagram. 126 00:04:41,693 --> 00:04:42,436 Let me just see. 127 00:04:42,436 --> 00:04:45,361 This point on our diagram, 128 00:04:45,361 --> 00:04:48,705 from my frame of reference, in the S frame of reference, 129 00:04:48,705 --> 00:04:51,514 this is the point time is, 130 00:04:51,514 --> 00:04:52,745 or space, 131 00:04:52,745 --> 00:04:54,045 or my distance, 132 00:04:54,045 --> 00:04:57,342 my distance from the origin in the positive-x direction 133 00:04:57,342 --> 00:04:59,107 is going to be, 134 00:04:59,107 --> 00:05:00,570 I just drop a, 135 00:05:00,570 --> 00:05:02,961 I just go parallel to my vertical axis, 136 00:05:02,961 --> 00:05:04,308 so I'm just going to do that, 137 00:05:04,308 --> 00:05:08,604 and I get 4.5 times 10 to the 8th meters, 138 00:05:08,604 --> 00:05:11,715 so 4.5 times 10 to the 8th meters, 139 00:05:11,715 --> 00:05:13,364 and what is our, 140 00:05:13,364 --> 00:05:15,128 so I can write the units there if I want, 141 00:05:15,128 --> 00:05:17,636 and our time is at one second, 142 00:05:17,636 --> 00:05:18,286 one second. 143 00:05:18,286 --> 00:05:20,190 So this is our coordinates from, 144 00:05:20,190 --> 00:05:22,883 this is the coordinates from my frame of reference, 145 00:05:22,883 --> 00:05:24,346 from the S frame of reference, 146 00:05:24,346 --> 00:05:25,670 I guess S for Sal, 147 00:05:25,670 --> 00:05:27,202 and so, what would be the frame? 148 00:05:27,202 --> 00:05:28,549 What would be those same coordinates 149 00:05:28,549 --> 00:05:30,313 from my friend's frame of reference? 150 00:05:30,313 --> 00:05:33,587 Let's say her name is Sally, or S-prime. 151 00:05:33,587 --> 00:05:35,515 Well, from her frame of reference, 152 00:05:35,515 --> 00:05:37,581 our time is still one second. 153 00:05:37,581 --> 00:05:39,044 We see the horizontal line there. 154 00:05:39,044 --> 00:05:42,016 We're just going parallel to our horizontal axis 155 00:05:42,016 --> 00:05:43,084 or our x-axis 156 00:05:43,084 --> 00:05:46,172 so the time is still one second, 157 00:05:46,172 --> 00:05:47,263 but her, 158 00:05:47,263 --> 00:05:49,632 the distance that the, 159 00:05:49,632 --> 00:05:53,765 or the x-coordinate is 3 times 10 to the 8th meters, 160 00:05:53,765 --> 00:05:57,131 3 times 10 to the 8th meters, 161 00:05:57,131 --> 00:05:59,453 and I really want you to sit and think about this 162 00:05:59,453 --> 00:06:01,752 and even take out a ruler and draw this on your 163 00:06:01,752 --> 00:06:04,167 own paper and feel good about this. 164 00:06:04,167 --> 00:06:05,978 Now, why does this make sense? 165 00:06:05,978 --> 00:06:09,182 Well, this, you could view this point as where 166 00:06:09,182 --> 00:06:10,598 that second spaceship was 167 00:06:10,598 --> 00:06:14,453 after a second, and from Sally's point of view, 168 00:06:14,453 --> 00:06:16,241 from the S-prime frame of reference, 169 00:06:16,241 --> 00:06:17,564 at time equals zero, 170 00:06:17,564 --> 00:06:20,379 that person, that ship was 3 times 10 to the 8th meters 171 00:06:20,379 --> 00:06:23,351 in front of her, and after a second, that spaceship is still 172 00:06:23,351 --> 00:06:25,603 3 times 10 to the 8th meters in front of her. 173 00:06:25,603 --> 00:06:26,764 After two seconds, it's still 174 00:06:26,764 --> 00:06:29,806 3 times 10 to the 8th meters in front of her, 175 00:06:29,806 --> 00:06:32,522 and so from her point of view and her frame of reference, 176 00:06:32,522 --> 00:06:34,542 it is stationary, 177 00:06:34,542 --> 00:06:36,539 so the way to read this for any point, 178 00:06:36,539 --> 00:06:38,234 if you want my frame of reference, 179 00:06:38,234 --> 00:06:40,092 you, the time is easy, 180 00:06:40,092 --> 00:06:42,158 you just go horizontally to either one, 181 00:06:42,158 --> 00:06:45,246 so on her axis, we've essentially just taken the coordinate 182 00:06:45,246 --> 00:06:47,545 axis and sheared it and skewed it. 183 00:06:47,545 --> 00:06:48,961 This is still one second, 184 00:06:48,961 --> 00:06:51,097 this is two seconds, this is three seconds, 185 00:06:51,097 --> 00:06:53,768 but it turned all of these kind of a grid 186 00:06:53,768 --> 00:06:57,459 of squares if I were to draw a grid for my coordinates 187 00:06:57,459 --> 00:06:58,969 into a bunch of, 188 00:06:58,969 --> 00:07:02,243 into a bunch of parallelograms 189 00:07:02,243 --> 00:07:04,054 which we see right over here, 190 00:07:04,054 --> 00:07:06,840 and so for any given point in, 191 00:07:06,840 --> 00:07:09,255 remember we're thinking of Newtonian space 192 00:07:09,255 --> 00:07:11,089 and time, 193 00:07:11,089 --> 00:07:13,504 to read it, we go horizontally for time, 194 00:07:13,504 --> 00:07:14,990 in either case, 195 00:07:14,990 --> 00:07:17,149 but for me, you would go straight down 196 00:07:17,149 --> 00:07:19,239 because my time axis is vertical. 197 00:07:19,239 --> 00:07:21,120 You go straight down and you read where it intersects 198 00:07:21,120 --> 00:07:23,442 the x-axis to figure out our position 199 00:07:23,442 --> 00:07:24,835 but from her frame of reference 200 00:07:24,835 --> 00:07:28,294 we move parallel to her t-axis, 201 00:07:28,294 --> 00:07:30,036 or her t-prime axis, I should say, 202 00:07:30,036 --> 00:07:32,334 to get what her x, 203 00:07:32,334 --> 00:07:35,144 I guess we could say what her x-prime coordinates 204 00:07:35,144 --> 00:07:38,464 are going to be, so this is x-prime, t-prime. 205 00:07:38,464 --> 00:07:41,552 This is x and this is t. 206 00:07:41,552 --> 00:07:43,619 Now, what's clear, at any given point, 207 00:07:43,619 --> 00:07:45,244 at any given point, 208 00:07:45,244 --> 00:07:49,424 t is going to be equal to t-prime, 209 00:07:49,424 --> 00:07:52,419 or we could say t-prime is equal to t, 210 00:07:52,419 --> 00:07:53,324 either way, 211 00:07:53,324 --> 00:07:56,830 but what's the relationship between x-prime and x? 212 00:07:56,830 --> 00:07:58,851 Well, let's think about that for a second, 213 00:07:58,851 --> 00:08:03,285 so when x is 4.5 times 10 to the 8th meters, 214 00:08:03,285 --> 00:08:07,047 x-prime is 3 times 10 to the 8th meters, 215 00:08:07,047 --> 00:08:08,626 and so it looks like, 216 00:08:08,626 --> 00:08:12,712 it looks like x-prime is going to be equal to 217 00:08:12,712 --> 00:08:16,311 x minus, well what's it going to be minus? 218 00:08:16,311 --> 00:08:17,867 It's going to be minus 219 00:08:17,867 --> 00:08:22,867 1.5 times 10 to the 8th meters 220 00:08:24,182 --> 00:08:28,641 but that difference is going to grow as time goes on, 221 00:08:28,641 --> 00:08:31,729 meters per second times, 222 00:08:31,729 --> 00:08:34,956 times the amount of time that has passed by, 223 00:08:34,956 --> 00:08:36,071 and you should see that 224 00:08:36,071 --> 00:08:37,951 because after one second, 225 00:08:37,951 --> 00:08:42,061 after one second, the difference is 1.5 times 10 to the 8th 226 00:08:42,061 --> 00:08:44,592 meters, after two seconds, 227 00:08:44,592 --> 00:08:46,403 after two seconds, so let's look at this point 228 00:08:46,403 --> 00:08:47,703 right over here, 229 00:08:47,703 --> 00:08:51,372 in my frame of reference, that person is sitting at 230 00:08:51,372 --> 00:08:55,691 6 times 10 to the 8th meters in the positive-x direction, 231 00:08:55,691 --> 00:08:57,455 but in Sally's frame of reference, 232 00:08:57,455 --> 00:08:59,104 in the S-prime frame of reference, 233 00:08:59,104 --> 00:09:03,051 they're still at 3 times 10 to the 8th meters 234 00:09:03,051 --> 00:09:04,816 in the positive-x direction, 235 00:09:04,816 --> 00:09:06,929 so we'd multiply two seconds 236 00:09:06,929 --> 00:09:09,436 times 1.5 times 10 to the 8th 237 00:09:09,436 --> 00:09:12,664 so we have 3 times 10 to the 8th meters, 238 00:09:12,664 --> 00:09:14,777 so we take 6 times 10 to the 8th 239 00:09:14,777 --> 00:09:15,798 which would be that. 240 00:09:15,798 --> 00:09:18,097 We subtract out 3 times 10 to the 8th 241 00:09:18,097 --> 00:09:20,581 and we get the coordinates in S-prime, 242 00:09:20,581 --> 00:09:22,346 so this might get a little bit confusing 243 00:09:22,346 --> 00:09:25,550 and the key here is really try to draw this yourself 244 00:09:25,550 --> 00:09:27,083 and really try to plot points 245 00:09:27,083 --> 00:09:29,335 and think about how they are different coordinates 246 00:09:29,335 --> 00:09:32,980 in the different frames of reference, 247 00:09:32,980 --> 00:09:34,513 or different time-space coordinates in the different 248 00:09:34,513 --> 00:00:00,000 frames of reference.