1 00:00:00,244 --> 00:00:01,544 - [Voiceover] So let's revisit a scenario 2 00:00:01,544 --> 00:00:03,100 that we have seen in several videos, 3 00:00:03,100 --> 00:00:04,285 especially the last video, 4 00:00:04,285 --> 00:00:07,186 where we tried to find this neutral frame of reference. 5 00:00:07,186 --> 00:00:08,626 Let's say we're in spaceship A. 6 00:00:08,626 --> 00:00:10,902 We are in an inertial frame of reference. 7 00:00:10,902 --> 00:00:14,012 And let's say right at time equal zero 8 00:00:14,012 --> 00:00:15,289 in our frame of reference, 9 00:00:15,289 --> 00:00:18,100 spaceship B is exactly where we are, 10 00:00:18,100 --> 00:00:20,560 but it's traveling in the positive x-direction 11 00:00:20,560 --> 00:00:22,766 at eight tenths the speed of light. 12 00:00:22,766 --> 00:00:24,643 We've already seen that we can overlay 13 00:00:24,643 --> 00:00:27,084 the spacetime diagrams for each of these 14 00:00:27,084 --> 00:00:28,501 frames of reference. 15 00:00:28,501 --> 00:00:32,500 If we do our axes, 16 00:00:32,500 --> 00:00:34,769 the x is one direction of spacetime 17 00:00:34,769 --> 00:00:37,161 that we associate with the x-direction of space 18 00:00:37,161 --> 00:00:39,344 and then the ct, this vertical axis, 19 00:00:39,344 --> 00:00:41,132 this is another direction of spacetime 20 00:00:41,132 --> 00:00:44,985 that we perceive as the passage of time. 21 00:00:44,985 --> 00:00:48,655 Well, you can then overlay B's frame of reference, 22 00:00:48,655 --> 00:00:53,655 and you kind of have these skewed axes. 23 00:00:53,670 --> 00:00:56,201 And you could look at a change in time. 24 00:00:56,201 --> 00:00:57,872 So for example, if you were to compare 25 00:00:57,872 --> 00:01:01,146 the event right when B passes us up to 26 00:01:01,146 --> 00:01:04,211 let's say some amount of time later, 27 00:01:04,211 --> 00:01:07,462 some amount of, so let's say this right over there 28 00:01:07,462 --> 00:01:10,179 is the change in ct. 29 00:01:10,179 --> 00:01:11,385 In our frame of reference, 30 00:01:11,385 --> 00:01:14,637 these two events are happening at the exact same place. 31 00:01:14,637 --> 00:01:16,912 They're just separated, in our frame of reference, 32 00:01:16,912 --> 00:01:19,675 we would perceive them as being separated by time. 33 00:01:19,675 --> 00:01:21,440 But if you want to say, well how much time 34 00:01:21,440 --> 00:01:24,807 seems to separate these events in B's frame of reference? 35 00:01:24,807 --> 00:01:29,217 Well then, you would wanna go parallel to the x prime axis. 36 00:01:29,217 --> 00:01:30,718 We're viewing B's frame of reference 37 00:01:30,718 --> 00:01:32,726 as a primed frame of reference 38 00:01:32,726 --> 00:01:35,928 and see where we intersect the ct prime, 39 00:01:35,928 --> 00:01:37,786 the ct prime axis. 40 00:01:37,786 --> 00:01:39,806 So this is our change in ct 41 00:01:39,806 --> 00:01:43,545 while this is our change in ct, 42 00:01:43,545 --> 00:01:46,842 this looks like our change in ct prime. 43 00:01:46,842 --> 00:01:48,792 And it looks longer but we have to remember 44 00:01:48,792 --> 00:01:50,650 that we haven't scaled these things. 45 00:01:50,650 --> 00:01:52,390 And actually the scales change 46 00:01:52,390 --> 00:01:55,851 depending on the relative velocities. 47 00:01:55,851 --> 00:01:58,275 But we can actually verify algebraically 48 00:01:58,275 --> 00:02:00,642 that our change in ct prime is going to be larger 49 00:02:00,642 --> 00:02:02,057 than our change in ct. 50 00:02:02,057 --> 00:02:03,922 And we just have to look at Lorentz transformations 51 00:02:03,922 --> 00:02:05,410 to realize that. 52 00:02:05,410 --> 00:02:09,317 So our change in ct prime is going to be 53 00:02:09,317 --> 00:02:13,590 equal to the Lorentz factor times our change in ct 54 00:02:13,590 --> 00:02:16,887 minus beta times our change in x. 55 00:02:16,887 --> 00:02:18,906 We've seen that multiple times before. 56 00:02:18,906 --> 00:02:20,463 Well our change in x, 57 00:02:20,463 --> 00:02:22,180 our change in x is zero. 58 00:02:22,180 --> 00:02:23,946 It looks stationary in our frame of reference 59 00:02:23,946 --> 00:02:25,570 so that term is zero. 60 00:02:25,570 --> 00:02:29,100 So our change in ct prime is going to be equal 61 00:02:29,100 --> 00:02:31,422 to the Lorentz factor times, 62 00:02:31,422 --> 00:02:33,210 I don't have to use this parentheses anymore, 63 00:02:33,210 --> 00:02:37,063 times our change in ct. 64 00:02:37,063 --> 00:02:39,410 And our Lorentz factor is going to be greater than one. 65 00:02:39,410 --> 00:02:41,196 I could actually calculate that, let's do it. 66 00:02:41,196 --> 00:02:42,846 So the Lorentz factor, 67 00:02:42,846 --> 00:02:45,707 the Lorentz factor here. 68 00:02:45,707 --> 00:02:47,722 So gamma is going to be equal to one 69 00:02:47,722 --> 00:02:51,694 over the square root of one minus, 70 00:02:51,694 --> 00:02:56,694 well B's relative velocity so 0.8c 71 00:02:57,753 --> 00:03:00,841 over the speed of light squared. 72 00:03:00,841 --> 00:03:02,629 Well what is this going to simplify to? 73 00:03:02,629 --> 00:03:03,697 The c's cancel out. 74 00:03:03,697 --> 00:03:06,274 0.8 squared is 0.64. 75 00:03:06,274 --> 00:03:09,268 One minus that is 0.36. 76 00:03:09,268 --> 00:03:10,198 This is going to be, 77 00:03:10,198 --> 00:03:11,381 and then you take the square root of that, 78 00:03:11,381 --> 00:03:14,842 that's going to be one over 0.6 79 00:03:14,842 --> 00:03:16,839 which is equal to one over six tenths 80 00:03:16,839 --> 00:03:19,578 which is the same thing as 10 over six 81 00:03:19,578 --> 00:03:23,657 which is equal to the same thing as five thirds 82 00:03:23,657 --> 00:03:26,767 which is equal to one and two thirds. 83 00:03:26,767 --> 00:03:28,625 So you can see our change in ct prime 84 00:03:28,625 --> 00:03:30,157 is going to be one and two thirds 85 00:03:30,157 --> 00:03:32,387 times the change in ct. 86 00:03:32,387 --> 00:03:33,362 Now you might wanna just say, 87 00:03:33,362 --> 00:03:35,684 well do these two look like one and two thirds? 88 00:03:35,684 --> 00:03:37,263 And it might look a little bit like that 89 00:03:37,263 --> 00:03:38,702 the way I draw it, but you can't just 90 00:03:38,702 --> 00:03:40,514 look at it purely on, 91 00:03:40,514 --> 00:03:42,302 you can't just take a ruler for this length 92 00:03:42,302 --> 00:03:43,370 and a ruler for this length 93 00:03:43,370 --> 00:03:44,924 because the scales are different 94 00:03:44,924 --> 00:03:46,689 and I haven't marked off the scales. 95 00:03:46,689 --> 00:03:48,709 Well this at least helps us visualize. 96 00:03:48,709 --> 00:03:52,263 But let's think about the other way around. 97 00:03:52,263 --> 00:03:55,977 Let's imagine the change in ct prime 98 00:03:55,977 --> 00:03:58,590 between right where the spaceships pass by 99 00:03:58,590 --> 00:03:59,925 and a little bit later. 100 00:03:59,925 --> 00:04:01,318 Now, I'll do this in a different color 101 00:04:01,318 --> 00:04:04,615 'cause this is actually a different event in spacetime 102 00:04:04,615 --> 00:04:07,657 than the one that we were focusing on right now. 103 00:04:07,657 --> 00:04:10,188 And we're gonna be viewing it from B's frame of reference 104 00:04:10,188 --> 00:04:13,369 so that's our change in ct prime. 105 00:04:13,369 --> 00:04:16,130 But what is going to be our change in ct? 106 00:04:16,130 --> 00:04:17,152 Well, to think about it, 107 00:04:17,153 --> 00:04:20,451 we could go parallel to the x-axis, 108 00:04:20,451 --> 00:04:23,701 the x-axis right over there, so you'll parallel, 109 00:04:23,701 --> 00:04:27,904 parallel to the x-axis and you get right over there 110 00:04:27,904 --> 00:04:30,226 and so our change in ct 111 00:04:30,226 --> 00:04:31,921 looks like it's more. 112 00:04:31,921 --> 00:04:34,057 And once again, we can algebraically verify it 113 00:04:34,057 --> 00:04:35,797 by really doing the same thing. 114 00:04:35,797 --> 00:04:40,650 Our change in ct is going to be equal to gamma 115 00:04:40,650 --> 00:04:43,692 times our change in ct prime 116 00:04:43,692 --> 00:04:48,127 minus beta times delta x prime. 117 00:04:48,127 --> 00:04:49,242 And if you were to actually, 118 00:04:49,242 --> 00:04:51,517 if we actually did have a change in x prime here, 119 00:04:51,517 --> 00:04:53,815 the beta, the velocity now has a different direction 120 00:04:53,815 --> 00:04:55,163 so it would all be the negative 121 00:04:55,163 --> 00:04:57,763 but our change in x prime from B's point of view, 122 00:04:57,763 --> 00:05:00,318 these two events are happening in the same place 123 00:05:00,318 --> 00:05:02,663 so our change in x prime is zero. 124 00:05:02,663 --> 00:05:07,663 So you have change in ct is equal to gamma times change, 125 00:05:08,072 --> 00:05:09,628 once again I don't need my parentheses, 126 00:05:09,628 --> 00:05:12,600 times change in ct prime. 127 00:05:12,600 --> 00:05:14,458 And it's going to be the same gamma 128 00:05:14,458 --> 00:05:16,431 because remember, we're taking v over c 129 00:05:16,431 --> 00:05:18,080 and so whether it's either v or negative v, 130 00:05:18,080 --> 00:05:20,170 when you square it, it gets the same value 131 00:05:20,170 --> 00:05:23,837 so once again, gamma is going to be one and two thirds. 132 00:05:23,837 --> 00:05:26,670 So it's going to be one and two thirds. 133 00:05:26,670 --> 00:05:28,574 So it seems a little bit strange. 134 00:05:28,574 --> 00:05:30,177 You know, I have some passage of time 135 00:05:30,177 --> 00:05:31,895 in my frame of reference where it looks, 136 00:05:31,895 --> 00:05:33,381 you know, it's something that looks stationary, 137 00:05:33,381 --> 00:05:34,542 two events that look like they're happening 138 00:05:34,542 --> 00:05:37,026 in the same place but one after another. 139 00:05:37,026 --> 00:05:38,722 It looks like their change in time, 140 00:05:38,722 --> 00:05:41,461 it takes longer for those two events to happen 141 00:05:41,461 --> 00:05:44,920 in the ct prime in the moving frame of reference. 142 00:05:44,920 --> 00:05:47,574 But then, if we have some event 143 00:05:47,574 --> 00:05:49,704 that looks stationary in B's frame of reference 144 00:05:49,704 --> 00:05:52,211 and they're separated by a change in ct prime, 145 00:05:52,211 --> 00:05:55,625 it looks like the change in ct between those two events 146 00:05:55,625 --> 00:05:57,134 is even larger. 147 00:05:57,134 --> 00:06:00,503 So it looks like this somewhat bizarre phenomenon. 148 00:06:00,503 --> 00:06:02,382 And to help us reconcile these 149 00:06:02,382 --> 00:06:03,890 and to visualize a little bit better, 150 00:06:03,890 --> 00:06:07,746 actually even to be able to put ct and ct prime 151 00:06:07,746 --> 00:06:10,020 on the same scale, 152 00:06:10,020 --> 00:06:12,575 we can look for this neutral frame of reference 153 00:06:12,575 --> 00:06:15,361 which we did in actually the last video I made. 154 00:06:15,361 --> 00:06:17,149 I don't know if it's the last video you've seen. 155 00:06:17,149 --> 00:06:19,262 Or we could say look, if A and B 156 00:06:19,262 --> 00:06:22,256 are travelling with a relative velocity of 0.8 157 00:06:22,256 --> 00:06:23,906 times of the speed of light relative to each other, 158 00:06:23,906 --> 00:06:26,832 B is travelling 0.8c in the positive x direction 159 00:06:26,832 --> 00:06:28,968 relative to a stationary A 160 00:06:28,968 --> 00:06:33,937 or A is travelling 0.8c in the negative x direction 161 00:06:33,937 --> 00:06:35,608 relative to a stationary B. 162 00:06:35,608 --> 00:06:38,092 You can find a frame of reference 163 00:06:38,092 --> 00:06:40,693 where A and B were in that frame of reference 164 00:06:40,693 --> 00:06:42,667 to a stationary observer in that frame of reference, 165 00:06:42,667 --> 00:06:44,896 A and B are both travelling outwards 166 00:06:44,896 --> 00:06:46,614 at half the speed of light. 167 00:06:46,614 --> 00:06:47,797 And we figured that out where we 168 00:06:47,797 --> 00:06:50,560 did those videos on a neutral frame of reference. 169 00:06:50,560 --> 00:06:51,721 And what's neat about that 170 00:06:51,721 --> 00:06:55,786 is if you make what looks like a Minkowski spacetime diagram 171 00:06:55,786 --> 00:06:57,741 in that neutral frame of reference 172 00:06:57,741 --> 00:07:01,267 then the ct and ct prime, A and B's frame of references, 173 00:07:01,267 --> 00:07:04,637 get equally skewed to the, you know, 174 00:07:04,637 --> 00:07:07,093 if you think about the time axis to the left and the right. 175 00:07:07,093 --> 00:07:09,252 And since they're equally skewed, 176 00:07:09,252 --> 00:07:12,805 the time dilation relative to this rest frame 177 00:07:12,805 --> 00:07:15,545 is the scaling is going to be the same. 178 00:07:15,545 --> 00:07:18,261 So we could put both of these on, 179 00:07:18,261 --> 00:07:20,374 both of these on the same scale. 180 00:07:20,374 --> 00:07:23,672 And you could see that this neutral frame of reference, 181 00:07:23,672 --> 00:07:26,319 this ct prime prime frame of reference, 182 00:07:26,319 --> 00:07:27,503 I drew them over here, 183 00:07:27,503 --> 00:07:30,127 that if you look at either of A or B's frame of reference, 184 00:07:30,127 --> 00:07:31,658 they're going to be in between 185 00:07:31,658 --> 00:07:33,121 A and B's frame of reference. 186 00:07:33,121 --> 00:07:34,422 But here it's the neutral one. 187 00:07:34,422 --> 00:07:36,244 It's the one where we're drawing the time 188 00:07:36,244 --> 00:07:38,392 or we're drawing the two axes being perpendicular 189 00:07:38,392 --> 00:07:39,391 to each other. 190 00:07:39,391 --> 00:07:41,434 And now if you look at these two events, 191 00:07:41,434 --> 00:07:43,035 so if you look at this first event, 192 00:07:43,035 --> 00:07:45,103 where you have our delta ct, 193 00:07:45,103 --> 00:07:47,750 so you look at this first event 194 00:07:47,750 --> 00:07:50,722 where you have a delta in ct between 195 00:07:50,722 --> 00:07:53,137 right when the spaceships pass and right over there. 196 00:07:53,137 --> 00:07:54,739 You see that if you were to look at that, 197 00:07:54,739 --> 00:07:57,710 if you were to look at the ct prime coordinate 198 00:07:57,710 --> 00:08:01,750 for that event, when you go parallel to the x prime axis, 199 00:08:01,750 --> 00:08:03,654 you get right over there. 200 00:08:03,654 --> 00:08:08,182 So this is, that is your change in ct prime. 201 00:08:08,182 --> 00:08:11,223 And likewise, if you have that other event 202 00:08:11,223 --> 00:08:14,475 that I drew in that blue-green color, right over here. 203 00:08:14,475 --> 00:08:17,191 And this is your change in ct prime. 204 00:08:17,191 --> 00:08:20,000 This is your change in ct prime, 205 00:08:20,000 --> 00:08:23,181 well then if you think of it from A's frame of reference, 206 00:08:23,181 --> 00:08:25,736 well we just follow the x-axis not the x prime. 207 00:08:25,736 --> 00:08:27,106 We go parallel to it. 208 00:08:27,106 --> 00:08:29,312 We end up right over there. 209 00:08:29,312 --> 00:08:31,493 Now what's really interesting about this, 210 00:08:31,493 --> 00:08:33,385 what's really interesting about this 211 00:08:33,385 --> 00:08:36,951 is from A's frame of reference, 212 00:08:36,951 --> 00:08:40,038 the yellow event happens before the blue event. 213 00:08:40,038 --> 00:08:43,499 But in B's frame of reference, the blue event 214 00:08:43,499 --> 00:08:45,728 happens before the yellow event. 215 00:08:45,728 --> 00:08:48,467 So really, really, really fascinating things going on 216 00:08:48,467 --> 00:08:50,255 but what I really like about this diagram 217 00:08:50,255 --> 00:08:52,623 is that A and B's frame of reference 218 00:08:52,623 --> 00:08:53,993 are going to have the same scale 219 00:08:53,993 --> 00:08:56,902 since they're both equally skewed to the left and the right 220 00:08:56,902 --> 00:08:58,382 if we're thinking about the time axis. 221 00:08:58,382 --> 00:09:01,492 And this type of diagram is called a Loedel diagram. 222 00:09:01,492 --> 00:09:04,790 Loedel, Loedel diagram which is really 223 00:09:04,790 --> 00:09:07,461 a variation of a Minkowski diagram 224 00:09:07,461 --> 00:09:10,339 but it lets us really appreciate the symmetry 225 00:09:10,339 --> 00:00:00,000 between these frames of reference.