1 00:00:00,500 --> 00:00:01,924 [SQUEAKING] 2 00:00:01,924 --> 00:00:03,367 [RUSTLING] 3 00:00:03,367 --> 00:00:04,810 [CLICKING] 4 00:00:10,467 --> 00:00:11,050 PROFESSOR: OK. 5 00:00:11,050 --> 00:00:11,950 Great. 6 00:00:11,950 --> 00:00:15,640 So at this point, we basically have 7 00:00:15,640 --> 00:00:17,850 all of the most important-- well, 8 00:00:17,850 --> 00:00:19,300 we now have the full understanding 9 00:00:19,300 --> 00:00:20,560 of how to work with tensors. 10 00:00:20,560 --> 00:00:22,893 I haven't really done too much physics with them at this 11 00:00:22,893 --> 00:00:25,420 point, but we've very carefully-- 12 00:00:25,420 --> 00:00:28,330 one might even argue excessively carefully-- 13 00:00:28,330 --> 00:00:30,190 laid out this mathematical structure 14 00:00:30,190 --> 00:00:31,710 that is going to be-- we're going 15 00:00:31,710 --> 00:00:34,450 to use it to contain the geometric objects that 16 00:00:34,450 --> 00:00:35,950 will describe the physics that we're 17 00:00:35,950 --> 00:00:39,070 going to study over the course of the entire semester. 18 00:00:39,070 --> 00:00:43,720 So there was a lot of twiddly detail in the preceding 19 00:00:43,720 --> 00:00:46,350 lecture, but I would say the two most important things I want 20 00:00:46,350 --> 00:00:50,550 you to take out of this is the idea that we now have tensors 21 00:00:50,550 --> 00:00:55,180 as a general class of geometric objects which map one-forms-- 22 00:00:55,180 --> 00:00:57,520 or dual vectors, if you prefer-- 23 00:00:57,520 --> 00:01:00,260 a combination of one-forms and vectors to the Lorentz 24 00:01:00,260 --> 00:01:01,930 invariant real numbers. 25 00:01:01,930 --> 00:01:05,200 And further, the distinction between one-forms, and vectors, 26 00:01:05,200 --> 00:01:10,660 and what is being mapped to what is not so important, because I 27 00:01:10,660 --> 00:01:13,930 can always use the metric to either raise or lower 28 00:01:13,930 --> 00:01:16,810 indices on tensors or vectors or one forms 29 00:01:16,810 --> 00:01:18,650 in order to convert from one to another. 30 00:01:18,650 --> 00:01:20,890 So if I raise an index-- if I have a tensor that's 31 00:01:20,890 --> 00:01:22,930 got m indices in the upstairs position 32 00:01:22,930 --> 00:01:24,790 and n in the downstairs, I raise an index, 33 00:01:24,790 --> 00:01:27,760 and I've got m plus 1 upstairs, n minus 1 downstairs. 34 00:01:27,760 --> 00:01:29,928 And likewise, I can lower index-- 35 00:01:29,928 --> 00:01:31,720 put one of those ones in the upstairs, down 36 00:01:31,720 --> 00:01:34,168 to the downstairs. 37 00:01:34,168 --> 00:01:36,460 We wanted to go through all that stuff with great care, 38 00:01:36,460 --> 00:01:38,585 because we're going to need the foundations of that 39 00:01:38,585 --> 00:01:41,170 to make a lot of what we talk about later rigorous. 40 00:01:41,170 --> 00:01:44,410 It is really overkill for where we're starting, 41 00:01:44,410 --> 00:01:47,980 but it's worthwhile to have that scaffolding in place. 42 00:01:47,980 --> 00:01:49,972 I'm sick of it, though, so today I'm 43 00:01:49,972 --> 00:01:51,430 going to try to do some things that 44 00:01:51,430 --> 00:01:55,005 are more physics, because, well, not all of you 45 00:01:55,005 --> 00:01:56,380 are necessarily physics students, 46 00:01:56,380 --> 00:01:58,210 but I'm a physics professor and I'm 47 00:01:58,210 --> 00:01:59,830 a little tired of doing math. 48 00:01:59,830 --> 00:02:02,240 So let's think about-- 49 00:02:02,240 --> 00:02:03,960 when we're studying physics, generally we 50 00:02:03,960 --> 00:02:08,857 are interested in the behavior of bodies and fields. 51 00:02:08,857 --> 00:02:10,690 Let's move this to the relativity framework. 52 00:02:10,690 --> 00:02:14,080 We're looking at the behavior of bodies and fields in spacetime. 53 00:02:14,080 --> 00:02:17,320 So far, if I think about the quantities 54 00:02:17,320 --> 00:02:19,030 that we have introduced-- 55 00:02:19,030 --> 00:02:21,740 and by the way, I apologize for the blackboards. 56 00:02:21,740 --> 00:02:24,760 A previous lecturer today used chalk that cannot be erased 57 00:02:24,760 --> 00:02:28,030 very well, so they're sort of grayish boards today, 58 00:02:28,030 --> 00:02:28,620 unfortunately. 59 00:02:28,620 --> 00:02:30,840 But we'll manage. 60 00:02:30,840 --> 00:02:41,870 Anyhow, so far, the quantities that we have introduced 61 00:02:41,870 --> 00:02:45,840 that have solid physics content are 62 00:02:45,840 --> 00:02:49,860 good for describing particles-- 63 00:02:49,860 --> 00:02:53,010 individual particles-- maybe a handful of them at once. 64 00:02:59,240 --> 00:03:02,420 So the two that I want to focus on in particular are 65 00:03:02,420 --> 00:03:04,470 the 4-velocity-- 66 00:03:04,470 --> 00:03:08,830 so if I have a body with a 4-velocity u, 67 00:03:08,830 --> 00:03:11,670 a particular Lorentz observer will describe that as having 68 00:03:11,670 --> 00:03:13,950 a timelike component gamma-- 69 00:03:13,950 --> 00:03:19,980 special relativistic gamma-- and a spacelike component gamma v. 70 00:03:19,980 --> 00:03:26,280 This is defined such that u dot u equals minus 1. 71 00:03:26,280 --> 00:03:28,070 It is great for describing-- it is 72 00:03:28,070 --> 00:03:32,280 the tool we use to describe the trajectory of a material body 73 00:03:32,280 --> 00:03:35,310 that follows a timelike trajectory through spacetime. 74 00:03:35,310 --> 00:03:38,170 Notice that its norm is minus 1. 75 00:03:38,170 --> 00:03:40,230 The fact that it's 1 means that it's normalized, 76 00:03:40,230 --> 00:03:42,150 and the minus tells us that it is timelike. 77 00:03:51,100 --> 00:03:56,030 By the way, because this is timelike, 78 00:03:56,030 --> 00:03:58,720 the 4-velocity is not going to be a useful tool for us 79 00:03:58,720 --> 00:04:01,102 when we want to describe the behavior of light. 80 00:04:01,102 --> 00:04:02,560 If we want to talk about the motion 81 00:04:02,560 --> 00:04:06,460 of a photon in our spacetime, we can't use a 4-velocity for it. 82 00:04:09,250 --> 00:04:12,580 The trajectory of a photon is null. 83 00:04:12,580 --> 00:04:15,330 Whatever quantity that is going to describe its trajectory-- 84 00:04:15,330 --> 00:04:18,160 I take its dot product with itself, I have to get 0. 85 00:04:18,160 --> 00:04:19,657 And you can intuitively get a sense 86 00:04:19,657 --> 00:04:20,740 what's going on with that. 87 00:04:20,740 --> 00:04:24,460 If I take the way a particular observer interprets 88 00:04:24,460 --> 00:04:27,640 these things and says, what would 89 00:04:27,640 --> 00:04:30,130 this turn into if v equals c, well, you 90 00:04:30,130 --> 00:04:32,760 have infinity and infinity, because your gammas 91 00:04:32,760 --> 00:04:33,640 are diverging there. 92 00:04:33,640 --> 00:04:34,810 So it's a singular limit. 93 00:04:34,810 --> 00:04:36,640 It doesn't behave well. 94 00:04:36,640 --> 00:04:38,724 We will overcome this difficulty. 95 00:04:43,370 --> 00:04:45,375 The other quantity that we have used-- 96 00:04:45,375 --> 00:04:47,500 let me see if I can clean this a little bit better. 97 00:04:56,430 --> 00:04:56,930 All right. 98 00:04:56,930 --> 00:04:58,690 The other quantity that we have defined 99 00:04:58,690 --> 00:05:02,530 which is good for particles is the 4-momentum, 100 00:05:02,530 --> 00:05:12,390 which is just that 4-velocity multiplied by a rest mass. 101 00:05:12,390 --> 00:05:13,960 So this is defined-- you can clearly, 102 00:05:13,960 --> 00:05:17,020 by trimming together a bunch of definitions-- 103 00:05:17,020 --> 00:05:21,102 you have p dot p equals minus m squared. 104 00:05:21,102 --> 00:05:23,060 We're actually going to interpret this, though. 105 00:05:23,060 --> 00:05:24,850 So if a particular Lorentz observer 106 00:05:24,850 --> 00:05:26,710 makes a measurement of this, they 107 00:05:26,710 --> 00:05:33,850 will call the timelike component E and the spatial component p. 108 00:05:33,850 --> 00:05:35,650 And so you put all these things together, 109 00:05:35,650 --> 00:05:42,360 and this tells me that E squared minus magnitude of p 110 00:05:42,360 --> 00:05:46,240 squared equals m squared. 111 00:05:46,240 --> 00:05:48,550 Notice this is perfectly well-behaved 112 00:05:48,550 --> 00:05:49,700 for a massless particle. 113 00:05:49,700 --> 00:05:51,700 This is actually going to be the trick by which, 114 00:05:51,700 --> 00:05:55,180 when I want to describe the trajectory of photons 115 00:05:55,180 --> 00:05:57,280 or of light in a spacetime-- 116 00:05:57,280 --> 00:05:58,690 I can't use 4-velocity. 117 00:05:58,690 --> 00:06:00,150 I can use 4-momentum. 118 00:06:00,150 --> 00:06:02,170 It works perfectly well. 119 00:06:02,170 --> 00:06:06,528 It's 0 times a 4-velocity, but if I use this intuition that 120 00:06:06,528 --> 00:06:09,070 this thing is diverging when I take the speed of light limit, 121 00:06:09,070 --> 00:06:10,135 I sort of-- 122 00:06:10,135 --> 00:06:11,560 I guess I cheat, but I'm basically 123 00:06:11,560 --> 00:06:13,720 getting 0 times infinity. 124 00:06:13,720 --> 00:06:15,280 And it behaves. 125 00:06:15,280 --> 00:06:16,780 We're going to do it a little bit more rigorously than that, 126 00:06:16,780 --> 00:06:18,530 but if you want a little bit of intuition, 127 00:06:18,530 --> 00:06:19,955 that's essentially why it works. 128 00:06:19,955 --> 00:06:21,580 So the last thing which I will say here 129 00:06:21,580 --> 00:06:27,000 is that this is good for m equals 0. 130 00:06:27,000 --> 00:06:27,580 OK. 131 00:06:27,580 --> 00:06:33,340 And when you do have m equals 0, it's 132 00:06:33,340 --> 00:06:38,020 often convenient to write this as h bar omega. 133 00:06:38,020 --> 00:06:41,610 So use the usual formula for the energy of a photon-- 134 00:06:41,610 --> 00:06:45,190 let me move this so that you can see-- 135 00:06:45,190 --> 00:06:48,650 and then I just put a unit vector 136 00:06:48,650 --> 00:06:52,300 into the spatial direction that defines the direction 137 00:06:52,300 --> 00:06:53,300 in which this is moving. 138 00:07:09,800 --> 00:07:10,300 OK. 139 00:07:12,940 --> 00:07:18,500 So this is all fine as far as it goes, 140 00:07:18,500 --> 00:07:20,290 but it's kind of restrictive. 141 00:07:20,290 --> 00:07:23,180 We want to a little bit more with our physics 142 00:07:23,180 --> 00:07:24,910 than just worry about the kinetics 143 00:07:24,910 --> 00:07:26,150 of individual particles. 144 00:07:26,150 --> 00:07:28,030 There's a lot we can do with that, 145 00:07:28,030 --> 00:07:29,770 but one of the things that we like 146 00:07:29,770 --> 00:07:31,990 to do when we're studying gravity 147 00:07:31,990 --> 00:07:34,930 is to have a description of continual matter, 148 00:07:34,930 --> 00:07:36,560 like things that gravitate. 149 00:07:36,560 --> 00:07:39,820 You want to be able to build a description of something like 150 00:07:39,820 --> 00:07:43,300 a star, or an exotic star-- a neutron star-- 151 00:07:43,300 --> 00:07:44,900 something like that. 152 00:07:44,900 --> 00:07:47,928 And so just having the behavior of individual particles 153 00:07:47,928 --> 00:07:48,720 is not good enough. 154 00:07:48,720 --> 00:07:49,960 I want to be able to describe things like fluids. 155 00:07:49,960 --> 00:07:52,210 I want to be able to have a continuum that 156 00:07:52,210 --> 00:07:53,240 describes things. 157 00:07:53,240 --> 00:07:54,850 So what I'm going to start doing today 158 00:07:54,850 --> 00:07:57,280 is begin to introduce the mathematical tools 159 00:07:57,280 --> 00:07:58,990 that we will use. 160 00:07:58,990 --> 00:07:59,490 Pardon me. 161 00:07:59,490 --> 00:08:03,020 I seem to have twisted this around myself in a crazy way. 162 00:08:03,020 --> 00:08:05,650 We're going to describe the mathematical tools that 163 00:08:05,650 --> 00:08:10,480 are useful for dealing with continuous matter, 164 00:08:10,480 --> 00:08:11,998 rather than just particles. 165 00:08:14,810 --> 00:08:18,280 So let's call this more interesting matter. 166 00:08:22,800 --> 00:08:26,940 So the simplest continuum form of matter which 167 00:08:26,940 --> 00:08:38,030 we're going to talk about is-- 168 00:08:38,030 --> 00:08:39,980 we're going to call it dust. 169 00:08:39,980 --> 00:08:40,961 OK. 170 00:08:40,961 --> 00:08:42,419 When you're here at the chalkboard, 171 00:08:42,419 --> 00:08:43,836 you can't get away with it, so you 172 00:08:43,836 --> 00:08:45,090 have an idea of what it means. 173 00:08:45,090 --> 00:08:48,630 Physically, what I mean by dust is 174 00:08:48,630 --> 00:08:51,690 that it is a collection of particles that do not 175 00:08:51,690 --> 00:08:52,950 interact with each other. 176 00:08:52,950 --> 00:08:54,910 So they pass through each other. 177 00:08:54,910 --> 00:08:57,070 There's no pressure that is generated. 178 00:08:57,070 --> 00:09:10,440 So they have energy density associated with them, 179 00:09:10,440 --> 00:09:12,980 but no interaction. 180 00:09:12,980 --> 00:09:15,188 So it's the most boring kind of matter that you like. 181 00:09:15,188 --> 00:09:17,272 You can think of it as essentially just particles, 182 00:09:17,272 --> 00:09:18,800 but it's a ton of them, and they're 183 00:09:18,800 --> 00:09:22,222 smearing out into a continuum. 184 00:09:22,222 --> 00:09:23,930 In particular, one thing I'm going to say 185 00:09:23,930 --> 00:09:29,800 is, so I can imagine, if I take my two erasers here, 186 00:09:29,800 --> 00:09:32,570 I have a nice dust field in front of me 187 00:09:32,570 --> 00:09:35,600 now, which I helpfully just inhaled. 188 00:09:35,600 --> 00:09:38,900 And you can think of this thing as a field of little dust 189 00:09:38,900 --> 00:09:41,030 elements that are all moving around. 190 00:09:41,030 --> 00:09:44,090 If I go in and I track an individual element 191 00:09:44,090 --> 00:09:44,780 of that dust-- 192 00:09:44,780 --> 00:09:46,520 I go in and I make a little cube that's 193 00:09:46,520 --> 00:09:48,590 a nanometer on each side-- 194 00:09:48,590 --> 00:09:52,560 every little element has its own rest frame. 195 00:09:52,560 --> 00:09:54,560 In that cloud-- I'm not going to do that again-- 196 00:09:54,560 --> 00:09:56,900 but as I made that cloud, different elements 197 00:09:56,900 --> 00:09:59,270 within the cloud have different rest frames, 198 00:09:59,270 --> 00:10:02,840 but for each individual element, I can define a rest frame. 199 00:10:06,820 --> 00:10:08,320 So we're going to use the properties 200 00:10:08,320 --> 00:10:11,350 of the dust in that rest frame as a way of beginning 201 00:10:11,350 --> 00:10:13,742 to normalize and get a grasp on how 202 00:10:13,742 --> 00:10:14,950 we're going to describe this. 203 00:10:26,180 --> 00:10:35,000 But I just want to emphasize again-- in a given cloud, 204 00:10:35,000 --> 00:10:42,528 different elements may have different rest frames. 205 00:10:50,210 --> 00:10:50,710 OK. 206 00:10:50,710 --> 00:10:53,680 But let's suppose that I clap my erasers, 207 00:10:53,680 --> 00:10:55,720 I make my little cloud, and I go and I zoom in 208 00:10:55,720 --> 00:10:59,260 on one particular nanometer by nanometer 209 00:10:59,260 --> 00:11:00,890 by nanometer chunk of that thing, 210 00:11:00,890 --> 00:11:03,640 and I go into the rest frame of that particular little bit 211 00:11:03,640 --> 00:11:05,290 of dust. 212 00:11:05,290 --> 00:11:07,870 So how am I going to characterize that? 213 00:11:16,880 --> 00:11:20,480 So presumably, each particle has its own mass 214 00:11:20,480 --> 00:11:21,830 associated with it. 215 00:11:21,830 --> 00:11:24,752 Each little bit of dust that's in that thing has its own mass. 216 00:11:24,752 --> 00:11:26,210 We're going to worry about how that 217 00:11:26,210 --> 00:11:28,100 comes into the picture a little bit later. 218 00:11:28,100 --> 00:11:29,715 Let's just begin by saying, one of the things it's 219 00:11:29,715 --> 00:11:31,090 going to be interested in knowing 220 00:11:31,090 --> 00:11:34,700 is, how much dust is there in that little cubic nanometer 221 00:11:34,700 --> 00:11:37,380 of dust in front of me there? 222 00:11:37,380 --> 00:11:44,220 So the first thing we're going to focus on 223 00:11:44,220 --> 00:11:55,532 is just counting how many bits of dust are in this element. 224 00:11:55,532 --> 00:11:57,990 In particular, I'm going to want to know, how many elements 225 00:11:57,990 --> 00:11:59,160 per unit volume? 226 00:12:06,922 --> 00:12:08,380 In other words, the number density. 227 00:12:11,957 --> 00:12:14,290 So I claim that's a good thing for us to get started on. 228 00:12:18,710 --> 00:12:20,232 And it looks like they haven't quite 229 00:12:20,232 --> 00:12:21,440 poisoned this board as badly. 230 00:12:21,440 --> 00:12:22,280 Good. 231 00:12:22,280 --> 00:12:31,310 So let's call n sub 0 the number density 232 00:12:31,310 --> 00:12:34,160 in the rest frame of that element. 233 00:12:43,230 --> 00:12:43,730 OK. 234 00:12:43,730 --> 00:12:46,860 So this will just be some number per unit volume. 235 00:12:46,860 --> 00:12:47,360 OK. 236 00:12:47,360 --> 00:12:51,540 So it's a quantity that has dimensions 1 over volume. 237 00:12:51,540 --> 00:12:54,340 Now, in the rest frame, like I said, 238 00:12:54,340 --> 00:12:58,492 we're going to talk about how I treat the mass associated 239 00:12:58,492 --> 00:13:00,200 with each element, and how much energy is 240 00:13:00,200 --> 00:13:01,930 in that thing, a little bit later. 241 00:13:01,930 --> 00:13:04,880 There's a few other tools I want to introduce first 242 00:13:04,880 --> 00:13:07,730 before I get to there. 243 00:13:07,730 --> 00:13:09,220 But in the rest frame, if I'm not 244 00:13:09,220 --> 00:13:11,345 worrying about the mass and the energy, this is it. 245 00:13:11,345 --> 00:13:13,680 This is really the only thing I can say about this. 246 00:13:13,680 --> 00:13:16,580 So what do I'm going to do now is say, OK, well, if I 247 00:13:16,580 --> 00:13:18,777 have a general cloud here-- 248 00:13:18,777 --> 00:13:21,110 as I described, when we made this little turbulent cloud 249 00:13:21,110 --> 00:13:23,420 of dust in front of us, it was all swirling around 250 00:13:23,420 --> 00:13:25,370 and doing its own thing. 251 00:13:25,370 --> 00:13:27,010 You're generally not in the rest frame. 252 00:13:27,010 --> 00:13:27,510 OK. 253 00:13:27,510 --> 00:13:31,760 So you also want to know how to characterize the dust out 254 00:13:31,760 --> 00:13:33,000 of the rest frame. 255 00:13:33,000 --> 00:13:35,161 So let's move out of the rest frame. 256 00:13:42,040 --> 00:13:43,720 When I do this, two things happen. 257 00:13:54,820 --> 00:13:58,790 So first of all, let's continue to focus on-- 258 00:13:58,790 --> 00:14:01,750 so we've got a particular nanometer cubed of dust 259 00:14:01,750 --> 00:14:03,230 that we are looking at here. 260 00:14:03,230 --> 00:14:05,760 So we're very attached to that particular thing. 261 00:14:05,760 --> 00:14:07,135 So I now boost into a frame where 262 00:14:07,135 --> 00:14:08,980 I'm going 3/4 of the speed of light or something like that, 263 00:14:08,980 --> 00:14:11,110 but I'm going to pay attention to that-- 264 00:14:11,110 --> 00:14:13,240 I'm very attached to the dust in that little cube, 265 00:14:13,240 --> 00:14:16,150 so I'm going to keep myself focused on that. 266 00:14:16,150 --> 00:14:19,070 Boosting into frame can't change the total number, 267 00:14:19,070 --> 00:14:22,420 so the amount of dust has to be the same. 268 00:14:22,420 --> 00:14:25,310 But of course, there's going to be a Lorentz contraction, 269 00:14:25,310 --> 00:14:27,565 and so the volume will get smaller. 270 00:14:33,370 --> 00:14:35,680 So the number in a particular volume 271 00:14:35,680 --> 00:14:44,435 stays the same while the volume Lorentz contracts. 272 00:14:50,260 --> 00:14:57,770 And so let's call n the number density in this new frame. 273 00:15:02,890 --> 00:15:06,770 This is just going to be the Lorentz gamma times n0. 274 00:15:14,090 --> 00:15:17,230 So again, I'm assuming you're all perfectly 275 00:15:17,230 --> 00:15:18,530 fluent with special relativity. 276 00:15:18,530 --> 00:15:20,530 So all I did there was say it's going to contract along 277 00:15:20,530 --> 00:15:21,220 that direction. 278 00:15:21,220 --> 00:15:23,650 The lengths are smaller by a factor of gamma, 279 00:15:23,650 --> 00:15:26,170 therefore the volume is larger by a factor of gamma. 280 00:15:26,170 --> 00:15:28,170 But there's a second thing that happens as well. 281 00:15:28,170 --> 00:15:29,890 When I'm in this new frame, there 282 00:15:29,890 --> 00:15:33,730 is now a flow associated with the dust. 283 00:15:49,030 --> 00:15:52,790 So this will now be flowing through space. 284 00:15:52,790 --> 00:15:54,370 In particular, what you can do is 285 00:15:54,370 --> 00:16:00,280 define a flux that describes the number of particles crossing 286 00:16:00,280 --> 00:16:04,130 unit area in unit time. 287 00:16:04,130 --> 00:16:24,787 So let's define n 3-vector to be this. 288 00:16:24,787 --> 00:16:26,370 And you use a little bit of intuition, 289 00:16:26,370 --> 00:16:29,370 and just think about the flow of these particles as they 290 00:16:29,370 --> 00:16:32,070 kind of go translating past you. 291 00:16:32,070 --> 00:16:37,200 This can only be the number density 292 00:16:37,200 --> 00:16:44,300 that you measure in that frame times the 3-velocity. 293 00:16:44,300 --> 00:16:45,030 OK? 294 00:16:45,030 --> 00:16:45,780 Think about that a little bit. 295 00:16:45,780 --> 00:16:47,280 It's very much like when you learn 296 00:16:47,280 --> 00:16:53,520 about current density in basic E&M. Notice it's n and not n0 297 00:16:53,520 --> 00:16:55,390 here. 298 00:16:55,390 --> 00:16:58,850 So when you look at these two things for a moment or two-- 299 00:16:58,850 --> 00:17:01,140 let me just write down one more step. 300 00:17:01,140 --> 00:17:09,930 I can write this as gamma n0 v. 301 00:17:09,930 --> 00:17:13,440 These are screaming to be put together to make a 4-vector. 302 00:17:13,440 --> 00:17:15,750 They're looking at you and saying-- 303 00:17:15,750 --> 00:17:19,170 this bit's kind of like, I'm a timelike piece of a 4-vector. 304 00:17:19,170 --> 00:17:20,490 Spatial piece, I love you. 305 00:17:20,490 --> 00:17:21,900 Let's get together. 306 00:17:21,900 --> 00:17:24,245 They clearly are just screaming to go together. 307 00:17:24,245 --> 00:17:27,060 And when you do that, you say, hmm, 308 00:17:27,060 --> 00:17:28,820 let's see what happens here. 309 00:17:28,820 --> 00:17:31,330 Two great tastes that taste great together. 310 00:17:34,980 --> 00:17:35,730 OK. 311 00:17:35,730 --> 00:17:38,100 That's nice. 312 00:17:38,100 --> 00:17:40,620 Let's take advantage of the fact that both of these 313 00:17:40,620 --> 00:17:42,420 have a really simple form in terms 314 00:17:42,420 --> 00:17:45,060 of the density in the dust's own rest frame. 315 00:17:49,090 --> 00:17:52,080 And you look at this and go, holy moly. 316 00:17:52,080 --> 00:17:53,670 That is nothing more than the number 317 00:17:53,670 --> 00:17:56,220 density times a 4-velocity. 318 00:18:03,460 --> 00:18:05,550 So that's pretty cool. 319 00:18:05,550 --> 00:18:07,140 If I have this stuff in front of me-- 320 00:18:07,140 --> 00:18:08,520 you know, again, what the heck. 321 00:18:08,520 --> 00:18:10,980 This time I'm going to actually do my clap again. 322 00:18:10,980 --> 00:18:11,760 So I do this. 323 00:18:11,760 --> 00:18:13,230 Every little element-- 324 00:18:13,230 --> 00:18:16,320 I can think of it as having some trajectory through spacetime. 325 00:18:16,320 --> 00:18:19,360 I can attach a 4-velocity to that. 326 00:18:19,360 --> 00:18:23,310 And if I know what the density of dust is in that rest frame, 327 00:18:23,310 --> 00:18:25,830 I put it together-- now I've got a geometric object that 328 00:18:25,830 --> 00:18:27,280 describes this thing. 329 00:18:27,280 --> 00:18:31,980 And one of the reasons why this is powerful is, 330 00:18:31,980 --> 00:18:35,130 geometric objects-- as I have emphasized repeatedly-- 331 00:18:35,130 --> 00:18:39,630 have a geometric meaning that transcends 332 00:18:39,630 --> 00:18:42,120 a particular coordinate representation. 333 00:18:42,120 --> 00:18:45,840 Another way to think about this is that all observers-- 334 00:18:45,840 --> 00:18:48,330 I don't care whether you're at rest in this classroom, 335 00:18:48,330 --> 00:18:49,997 or you've had too much coffee and you're 336 00:18:49,997 --> 00:18:51,940 dashing through at 3/4 the speed of light. 337 00:18:51,940 --> 00:18:54,750 We all agree on N-- 338 00:18:54,750 --> 00:18:59,610 capital N vector-- but we choose different ways of splitting 339 00:18:59,610 --> 00:19:01,990 spacetime into space and time. 340 00:19:01,990 --> 00:19:04,170 And so I will have a different n from you, 341 00:19:04,170 --> 00:19:06,690 I will have a different nv from you. 342 00:19:06,690 --> 00:19:09,312 We all have different ways of splitting this spacetime 343 00:19:09,312 --> 00:19:10,770 into space and time, and so we have 344 00:19:10,770 --> 00:19:12,870 different ways of splitting these two things up. 345 00:19:12,870 --> 00:19:15,323 But this is a geometric object we all agree on. 346 00:19:15,323 --> 00:19:17,490 It's something that we can hang a lot of our physics 347 00:19:17,490 --> 00:19:24,120 on, and build frame-independent powerful geometric 348 00:19:24,120 --> 00:19:25,870 representations. 349 00:19:25,870 --> 00:19:29,520 So before going on to do a few things with this, 350 00:19:29,520 --> 00:19:31,770 let's just explore a couple of the properties of this. 351 00:19:50,250 --> 00:19:51,130 All right. 352 00:19:51,130 --> 00:19:55,465 So whenever you've got a 4-vector, at a certain point, 353 00:19:55,465 --> 00:19:56,840 you just sort of say to yourself, 354 00:19:56,840 --> 00:19:57,935 well, I've got this thing. 355 00:19:57,935 --> 00:19:59,560 Let's take the dot product with itself. 356 00:19:59,560 --> 00:20:00,060 Why not? 357 00:20:02,920 --> 00:20:07,960 So in this case, N dot N equals minus n0 squared. 358 00:20:07,960 --> 00:20:11,620 Or, turning this around, this tells you 359 00:20:11,620 --> 00:20:14,590 that if someone comes buy and goes, psst, here's 360 00:20:14,590 --> 00:20:18,370 a number 4-vector, you can figure out 361 00:20:18,370 --> 00:20:23,030 density in the rest frame associated with each element 362 00:20:23,030 --> 00:20:25,195 by the following operations. 363 00:20:25,195 --> 00:20:26,020 So that's nice. 364 00:20:26,020 --> 00:20:26,520 OK. 365 00:20:26,520 --> 00:20:29,230 In other words, the number density, in a sense, 366 00:20:29,230 --> 00:20:31,360 tells you what the normalization of this vector is. 367 00:20:36,490 --> 00:20:39,340 A few moments ago, I defined N as being 368 00:20:39,340 --> 00:20:41,270 kind of a flux of these things. 369 00:20:41,270 --> 00:20:46,900 It's a flux in the spatial directions. 370 00:20:46,900 --> 00:20:50,780 We want to think about flux in a more general kind of way. 371 00:20:50,780 --> 00:20:54,070 And so another thing which we're going to do-- 372 00:20:54,070 --> 00:20:57,290 and this is where 1-forms are going to play an important role 373 00:20:57,290 --> 00:20:57,790 for us-- 374 00:21:03,140 --> 00:21:10,185 is a nice systematic way to pick out a flux across a surface. 375 00:21:15,333 --> 00:21:17,500 So here's the bit where, when I was talking about it 376 00:21:17,500 --> 00:21:20,050 in Tuesday's lecture, I was getting the blank stares. 377 00:21:20,050 --> 00:21:22,400 And sorry, I'm coming right back to this again, 378 00:21:22,400 --> 00:21:25,720 so we're going to harp on this concept a little bit more. 379 00:21:25,720 --> 00:21:29,290 So recall we talked about the fact 380 00:21:29,290 --> 00:21:31,520 that if I have a particular coordinate system-- 381 00:21:31,520 --> 00:21:33,550 I have a particular set of coordinates x alpha 382 00:21:33,550 --> 00:21:36,430 that I use to represent spacetime. 383 00:21:36,430 --> 00:21:41,000 If I make 1-forms that are the gradients of those coordinates, 384 00:21:41,000 --> 00:21:51,300 I can represent them as essentially, 385 00:21:51,300 --> 00:22:03,100 level surfaces at unit ticks of the coordinate x alpha. 386 00:22:03,100 --> 00:22:03,840 OK. 387 00:22:03,840 --> 00:22:05,465 And basically, the way to think of it-- 388 00:22:05,465 --> 00:22:09,020 I kept emphasizing that the basis 1-forms are 389 00:22:09,020 --> 00:22:11,157 dual to basis vectors. 390 00:22:11,157 --> 00:22:13,490 And if you have this intuition that your basis vector is 391 00:22:13,490 --> 00:22:16,040 a little arrow pointing like so, then the thing 392 00:22:16,040 --> 00:22:20,090 that is dual to it is a surface that is everywhere but pointing 393 00:22:20,090 --> 00:22:20,873 along like so. 394 00:22:20,873 --> 00:22:23,040 That's kind of the way I want you to think about it. 395 00:22:23,040 --> 00:22:26,593 And those graphics that are scans from MTW I put on the web 396 00:22:26,593 --> 00:22:28,760 give better pictures of that than I am able to draw. 397 00:22:33,180 --> 00:22:52,970 So putting these concepts together, 398 00:22:52,970 --> 00:22:54,980 it basically tells me that I can use 399 00:22:54,980 --> 00:23:00,830 these basis 1-forms as a way of defining in an abstract form. 400 00:23:00,830 --> 00:23:13,060 So if I want to know the flux of N in the-- 401 00:23:13,060 --> 00:23:14,320 let's write it this way-- 402 00:23:14,320 --> 00:23:24,710 in the x alpha direction. 403 00:23:24,710 --> 00:23:28,510 So remember, the basis 1-forms-- 404 00:23:28,510 --> 00:23:31,480 the alpha in that case is not like a coordinate index. 405 00:23:31,480 --> 00:23:34,070 It's labeling a particular member of a set. 406 00:23:34,070 --> 00:23:37,070 And so what you would do is, you would construct this by saying, 407 00:23:37,070 --> 00:23:46,300 OK, let's take the beta component of a basis 408 00:23:46,300 --> 00:23:49,610 1-form alpha. 409 00:23:49,610 --> 00:23:50,830 Contract it like so. 410 00:24:00,750 --> 00:24:03,470 And that tells me about the flux-- how much of this stuff 411 00:24:03,470 --> 00:24:07,180 is flowing in the direction associated with alpha. 412 00:24:07,180 --> 00:24:09,680 So this is a tool that we're going to use from time to time. 413 00:24:09,680 --> 00:24:11,180 One little thing which I want to emphasize-- actually, 414 00:24:11,180 --> 00:24:12,630 two things I want to emphasize. 415 00:24:12,630 --> 00:24:15,440 So if I'm working in an intelligent coordinate system, 416 00:24:15,440 --> 00:24:23,860 remember, this is basically just the identity matrix, right? 417 00:24:23,860 --> 00:24:26,560 And so this ends up being a very simple thing that I actually 418 00:24:26,560 --> 00:24:27,060 work out. 419 00:24:27,060 --> 00:24:29,420 And it says, if I want to know the timelike component, 420 00:24:29,420 --> 00:24:32,990 then this is 1 in the timelike direction, 0 everywhere else, 421 00:24:32,990 --> 00:24:34,960 and I just pick out N sub t-- 422 00:24:34,960 --> 00:24:37,980 in other words, the density itself in that frame. 423 00:24:37,980 --> 00:24:40,210 That is the other thing which I want to emphasize. 424 00:24:40,210 --> 00:24:50,990 So when you do this, if I pick out the timelike piece of this, 425 00:24:50,990 --> 00:24:54,080 this just gives me the 0 component 426 00:24:54,080 --> 00:24:56,480 of this thing, which is the density 427 00:24:56,480 --> 00:24:58,610 that I measure in this frame. 428 00:24:58,610 --> 00:25:00,860 This is very, very simple in terms of the calculation, 429 00:25:00,860 --> 00:25:04,100 but I want you to stop and think about what it means physically. 430 00:25:04,100 --> 00:25:06,665 So flux in the timelike direction-- what 431 00:25:06,665 --> 00:25:09,290 does it mean for a flux to be in the timelike direction, right? 432 00:25:09,290 --> 00:25:11,057 If i take my water-- 433 00:25:11,057 --> 00:25:13,640 actually, I'm kind of thirsty-- so I want a little bit of flux 434 00:25:13,640 --> 00:25:16,547 of water going down my throat. 435 00:25:16,547 --> 00:25:18,380 You have an intuition about what that means. 436 00:25:18,380 --> 00:25:21,710 The water's actually flowing in a particular direction. 437 00:25:21,710 --> 00:25:24,970 If it's flowing through time-- 438 00:25:24,970 --> 00:25:26,835 there it goes. 439 00:25:26,835 --> 00:25:28,710 That basically means it's just sitting there. 440 00:25:28,710 --> 00:25:30,860 It's not doing anything, but just moving in time, 441 00:25:30,860 --> 00:25:33,470 as we all are doing. 442 00:25:33,470 --> 00:25:35,770 One time I remember my wife saying about our daughter-- 443 00:25:35,770 --> 00:25:37,700 she's like, she's growing up so fast. 444 00:25:37,700 --> 00:25:40,130 I was like, eh, she's growing up at 1 second per second. 445 00:25:40,130 --> 00:25:42,180 And that's just the way things go. 446 00:25:42,180 --> 00:25:44,180 It's just sitting there and it's living its life 447 00:25:44,180 --> 00:25:47,360 at 1 second per second, as we all do. 448 00:25:47,360 --> 00:25:48,980 That is density. 449 00:25:48,980 --> 00:25:51,350 So we described the flux of a thing in time 450 00:25:51,350 --> 00:25:52,460 as just being its density. 451 00:25:52,460 --> 00:25:55,220 That's another way-- when we do a lot of our calculations, 452 00:25:55,220 --> 00:25:57,057 and we talk about the flow of things 453 00:25:57,057 --> 00:25:58,640 in the timelike direction-- that tends 454 00:25:58,640 --> 00:26:03,273 to be just the simple density associated with stuff. 455 00:26:03,273 --> 00:26:04,940 And then if I did this-- if I pulled out 456 00:26:04,940 --> 00:26:08,015 the x direction of this thing, I would pull out the x component 457 00:26:08,015 --> 00:26:09,022 of velocity times that. 458 00:26:09,022 --> 00:26:11,480 And that's sort of the flux of something in the x direction 459 00:26:11,480 --> 00:26:15,170 like you are probably used to from other classes. 460 00:26:15,170 --> 00:26:17,190 Now, a place where this turns out-- 461 00:26:17,190 --> 00:26:20,720 so doing it when I'm just using the basis 1-forms 462 00:26:20,720 --> 00:26:33,910 is kind of trivial, [INAUDIBLE] sucks. 463 00:26:36,920 --> 00:26:38,630 More generally, what you can do is 464 00:26:38,630 --> 00:26:53,220 define a surface as being the solution of some scalar 465 00:26:53,220 --> 00:26:55,519 function in spacetime. 466 00:27:09,510 --> 00:27:14,390 So let's say I do something like psi of t comma x comma y 467 00:27:14,390 --> 00:27:17,010 equals a constant. 468 00:27:17,010 --> 00:27:17,510 OK. 469 00:27:17,510 --> 00:27:20,390 This is very abstract, so let me just make it a little bit more 470 00:27:20,390 --> 00:27:21,260 concrete. 471 00:27:21,260 --> 00:27:25,790 Suppose my function psi were square root 472 00:27:25,790 --> 00:27:28,980 of x squared plus y squared plus z squared, 473 00:27:28,980 --> 00:27:33,060 and my constant was 5. 474 00:27:33,060 --> 00:27:33,560 OK. 475 00:27:33,560 --> 00:27:35,518 Well then, my scalar field would be picking out 476 00:27:35,518 --> 00:27:37,080 a sphere of radius 5. 477 00:27:37,080 --> 00:27:37,580 OK. 478 00:27:37,580 --> 00:27:39,870 You can make a little bit more complicated than that, 479 00:27:39,870 --> 00:27:41,080 and people often do. 480 00:27:47,040 --> 00:27:50,620 You can define the unit 1-form that is associated 481 00:27:50,620 --> 00:27:51,870 with the normal to this thing. 482 00:27:54,560 --> 00:27:57,780 Or rather, it's the 1-form associated with the surface. 483 00:27:57,780 --> 00:27:59,527 If you translate this to a vector-- 484 00:27:59,527 --> 00:28:01,110 you raise the index-- with the vector, 485 00:28:01,110 --> 00:28:02,985 it would be the vector normal to the surface. 486 00:28:11,300 --> 00:28:12,980 You might need to normalize it. 487 00:28:12,980 --> 00:28:15,950 Let's imagine that we normalize this thing. 488 00:28:15,950 --> 00:28:23,990 And then we would just need to contract it along this, 489 00:28:23,990 --> 00:28:26,120 and that tells me about the flux through 490 00:28:26,120 --> 00:28:27,280 this particular surface. 491 00:28:27,280 --> 00:28:27,780 OK? 492 00:28:31,160 --> 00:28:33,540 This is one of the things that-- this idea 493 00:28:33,540 --> 00:28:34,915 of 1-forms being level surfaces-- 494 00:28:34,915 --> 00:28:36,707 it tends to be useful for things like that. 495 00:28:36,707 --> 00:28:38,915 We're going to come back to a similar sort of picture 496 00:28:38,915 --> 00:28:40,790 in just a moment, because we'll have to start 497 00:28:40,790 --> 00:28:41,965 talking about integration. 498 00:28:49,100 --> 00:28:50,610 OK. 499 00:28:50,610 --> 00:28:55,220 In prep for that discussion, now that I'm 500 00:28:55,220 --> 00:28:56,810 talking about things a little bit 501 00:28:56,810 --> 00:29:05,330 more complicated than just the kinematics of simple particles, 502 00:29:05,330 --> 00:29:08,930 we're going to want to have some laws. 503 00:29:08,930 --> 00:29:11,900 And we want to have geometric ways of describing 504 00:29:11,900 --> 00:29:16,100 those laws that essentially act as constraints 505 00:29:16,100 --> 00:29:18,700 on what those particles can do. 506 00:29:18,700 --> 00:29:19,200 OK. 507 00:29:19,200 --> 00:29:21,550 So the form in which we're going to express them-- we're 508 00:29:21,550 --> 00:29:24,175 going to tend to put things into the form of conservation laws. 509 00:29:32,440 --> 00:29:41,772 So suppose, here's my little element that's got dust in it, 510 00:29:41,772 --> 00:29:43,230 and it's embedded in an environment 511 00:29:43,230 --> 00:29:44,940 with a bunch of dust around it. 512 00:29:44,940 --> 00:29:47,610 And over some time interval, some dust flows in, 513 00:29:47,610 --> 00:29:49,132 some does flows out. 514 00:29:49,132 --> 00:29:51,090 The density can go up, the density can go down. 515 00:29:51,090 --> 00:29:53,333 The total number-- it may vary depending 516 00:29:53,333 --> 00:29:54,375 on how things are flying. 517 00:29:59,310 --> 00:30:10,000 The spatial flux out of the sides must come-- 518 00:30:10,000 --> 00:30:11,750 and I'm going to say it's the flux out-- 519 00:30:11,750 --> 00:30:22,820 so let's say that that comes at the expense of the density 520 00:30:22,820 --> 00:30:23,860 of dust already there. 521 00:30:28,140 --> 00:30:28,640 OK. 522 00:30:28,640 --> 00:30:30,348 And so if you were to just to intuitively 523 00:30:30,348 --> 00:30:32,210 write down what kind of conservation law 524 00:30:32,210 --> 00:30:36,440 you would expect to see, you would write it as something 525 00:30:36,440 --> 00:30:40,487 like this, based on simple Euclidean intuition. 526 00:30:42,683 --> 00:30:44,850 What's kind of nice-- you look at that for a second. 527 00:30:44,850 --> 00:30:48,400 You go, ooh, if I think about this as being the time 528 00:30:48,400 --> 00:30:49,980 complement of my 4-vector. 529 00:30:49,980 --> 00:30:53,370 This is the space component of my 4-vector. 530 00:30:53,370 --> 00:30:56,490 This has a very obvious form when I write it 531 00:30:56,490 --> 00:30:59,280 in a geometric framework. 532 00:30:59,280 --> 00:31:05,230 This whole thing can be rewritten as a conservation 533 00:31:05,230 --> 00:31:07,570 law that looks like this. 534 00:31:07,570 --> 00:31:09,750 OK. 535 00:31:09,750 --> 00:31:18,460 So I'll remind you, d sub alpha equals d by dx alpha. 536 00:31:18,460 --> 00:31:21,240 I'm going to talk a little bit about some of the derivatives 537 00:31:21,240 --> 00:31:23,250 a little bit later, because there are a few subtle points 538 00:31:23,250 --> 00:31:24,250 that can get introduced. 539 00:31:24,250 --> 00:31:27,090 But for now, we just know that d downstairs 540 00:31:27,090 --> 00:31:31,380 t is d by dt. d downstairs x is d by dx. d by downstairs y, 541 00:31:31,380 --> 00:31:31,880 et cetera. 542 00:31:34,900 --> 00:31:35,950 So this is really nice. 543 00:31:35,950 --> 00:31:37,783 One thing which I want to emphasize-- again, 544 00:31:37,783 --> 00:31:41,280 coming back to what I said over there just a moment ago. 545 00:31:41,280 --> 00:31:44,070 When I write down this conservation law, 546 00:31:44,070 --> 00:31:46,620 I'm assuming that someone has defined 547 00:31:46,620 --> 00:31:50,520 what time means and someone has defined what space means. 548 00:31:50,520 --> 00:31:53,100 This is a form that's covariant, right? 549 00:31:53,100 --> 00:31:55,560 All observers agree that this goes together. 550 00:31:55,560 --> 00:31:57,938 When they actually make their own coordinate systems, 551 00:31:57,938 --> 00:32:00,480 they're going to have their own time coordinate and their own 552 00:32:00,480 --> 00:32:03,540 x-coordinate, but they're all going to be different flavors-- 553 00:32:03,540 --> 00:32:07,910 just different ways of instantiating what this is. 554 00:32:07,910 --> 00:32:15,860 Now, pardon me for just one moment. 555 00:32:19,340 --> 00:32:20,855 Depending on the pace of the course, 556 00:32:20,855 --> 00:32:23,480 we're about to switch over to a different set of lecture notes, 557 00:32:23,480 --> 00:32:24,620 and I want to make sure I smoothly 558 00:32:24,620 --> 00:32:25,700 go from one to the other. 559 00:32:25,700 --> 00:32:26,660 OK. 560 00:32:26,660 --> 00:32:30,050 So in many of your physics classes, you have learned, 561 00:32:30,050 --> 00:32:32,960 when you get a conservation law, both a differential 562 00:32:32,960 --> 00:32:34,160 form like this-- 563 00:32:34,160 --> 00:32:37,490 the rate of change of N in a box is related to the amount of N 564 00:32:37,490 --> 00:32:39,740 flowing out of that box and its sides. 565 00:32:39,740 --> 00:32:42,290 And you also learn an integral form of the conservation law. 566 00:33:08,790 --> 00:33:11,760 So without proof, let me just say 567 00:33:11,760 --> 00:33:14,610 that it should be intuitively clear 568 00:33:14,610 --> 00:33:17,640 that what I've written down over here is equivalent to-- 569 00:33:43,730 --> 00:33:45,457 OK, so it looks like this. 570 00:33:45,457 --> 00:33:46,040 You know what? 571 00:33:46,040 --> 00:33:48,922 Let me just fix up my notation a little bit. 572 00:33:48,922 --> 00:33:50,130 Let me call this lowercase n. 573 00:33:56,940 --> 00:33:58,140 OK. 574 00:33:58,140 --> 00:33:59,890 A few symbols that I've introduced here-- 575 00:33:59,890 --> 00:34:10,090 so V3 is some volume in three-dimensional space, 576 00:34:10,090 --> 00:34:12,760 and dV like this-- this is a symbol 577 00:34:12,760 --> 00:34:15,760 that means the boundary of that 3-volume. 578 00:34:29,915 --> 00:34:32,130 And that's basically-- it's a form of Gauss's law. 579 00:34:32,130 --> 00:34:33,080 That's what I've written down there. 580 00:34:33,080 --> 00:34:33,580 OK? 581 00:34:33,580 --> 00:34:35,409 So you've all seen things like that. 582 00:34:35,409 --> 00:34:38,199 Again, let me emphasize that when I write down 583 00:34:38,199 --> 00:34:42,400 a formula like that, I can only do that having assumed 584 00:34:42,400 --> 00:34:44,320 a particular Lorentz frame. 585 00:34:44,320 --> 00:34:47,770 That t is the t of some observer. 586 00:34:47,770 --> 00:34:50,650 That volume is the volume of that particular observer 587 00:34:50,650 --> 00:34:52,179 who is using t. 588 00:34:52,179 --> 00:34:54,389 You jump into a different Lorentz frame, 589 00:34:54,389 --> 00:34:55,855 their volumes will not be the same. 590 00:34:55,855 --> 00:34:57,230 Their times will not be the same. 591 00:34:57,230 --> 00:34:58,813 I'm going to make some coordinates up. 592 00:34:58,813 --> 00:35:02,230 So an integral form like this, as I've written it there, only 593 00:35:02,230 --> 00:35:04,870 works in one given Lorentz frame. 594 00:35:04,870 --> 00:35:07,722 Nonetheless, we are going to find it useful, 595 00:35:07,722 --> 00:35:09,430 even though in some sense, when you do it 596 00:35:09,430 --> 00:35:11,650 in an integral form like this, you're 597 00:35:11,650 --> 00:35:14,928 saying things in the framework of some particular observer. 598 00:35:14,928 --> 00:35:16,720 Sometimes you want to know in the framework 599 00:35:16,720 --> 00:35:17,887 of some particular observer. 600 00:35:17,887 --> 00:35:18,890 It could be you, right? 601 00:35:18,890 --> 00:35:21,310 And you might care about these sorts of things. 602 00:35:21,310 --> 00:35:25,090 And that's good, but the way I've written it here-- 603 00:35:25,090 --> 00:35:29,373 first of all, it's in language that-- 604 00:35:29,373 --> 00:35:30,790 it's in a mathematical formulation 605 00:35:30,790 --> 00:35:32,832 that's not easy to generalize as I take things up 606 00:35:32,832 --> 00:35:34,790 to higher dimensions. 607 00:35:34,790 --> 00:35:37,360 And so what I would like to do is 608 00:35:37,360 --> 00:35:42,610 think about how to step up a formulation like this 609 00:35:42,610 --> 00:35:49,080 in such a way that things are put into as frame-independent 610 00:35:49,080 --> 00:35:52,863 a language as is possible, and that will generalize forward 611 00:35:52,863 --> 00:35:55,030 when we start looking at more complicated geometries 612 00:35:55,030 --> 00:35:58,430 than just geometry of special relativity. 613 00:35:58,430 --> 00:36:00,310 So I want to spend the next roughly 10 614 00:36:00,310 --> 00:36:05,780 or 15 minutes talking about volumes and volume integrals. 615 00:36:09,700 --> 00:36:11,600 And my goal here is to try to-- 616 00:36:11,600 --> 00:36:13,600 I'm going to start by just doing stuff that 617 00:36:13,600 --> 00:36:14,860 comes from the journal of duh. 618 00:36:14,860 --> 00:36:17,240 It's stuff you have seen over and over and over again, 619 00:36:17,240 --> 00:36:22,090 but I want to re-express it using mathematical formulation 620 00:36:22,090 --> 00:36:24,190 that maybe-- 621 00:36:24,190 --> 00:36:26,330 you have seen all the symbols, but perhaps not used 622 00:36:26,330 --> 00:36:28,180 in quite this way. 623 00:36:28,180 --> 00:36:31,420 And then it'll carry forward in a framework that generalizes 624 00:36:31,420 --> 00:36:33,590 in a very useful way for us. 625 00:36:33,590 --> 00:36:38,960 So let's begin with just simple 3D space. 626 00:36:38,960 --> 00:36:41,150 So I'm going to begin in 3D, and I'm 627 00:36:41,150 --> 00:36:50,580 going to consider a parallelepiped-- 628 00:36:50,580 --> 00:36:52,786 parallelepiped. 629 00:36:52,786 --> 00:36:53,286 Ha-ha. 630 00:36:53,286 --> 00:36:58,905 Got it right-- whose sides are a set of vectors. 631 00:37:03,120 --> 00:37:07,150 So there are three vectors, A, B, and C. OK. 632 00:37:14,600 --> 00:37:18,890 So here is a vector A. This one going into the board 633 00:37:18,890 --> 00:37:23,790 is vector B. And this one going up here is vector C. OK. 634 00:37:23,790 --> 00:37:26,040 So those are my three vectors. 635 00:37:26,040 --> 00:37:30,600 And if I go draw the ghost legs associated with these things-- 636 00:37:35,330 --> 00:37:35,830 OK. 637 00:37:39,540 --> 00:37:41,050 That's a little bit better. 638 00:37:41,050 --> 00:37:44,340 So these three vectors define a particular volume. 639 00:37:44,340 --> 00:37:46,080 And you guys have probably all seen-- 640 00:37:46,080 --> 00:37:47,040 you know you have three vectors. 641 00:37:47,040 --> 00:37:48,915 You can define a volume associated with this. 642 00:37:52,200 --> 00:37:54,130 A really easy way to get that volume 643 00:37:54,130 --> 00:37:56,390 given those three vectors is to take A 644 00:37:56,390 --> 00:38:03,460 and dot it into the cross product of B and C. 645 00:38:03,460 --> 00:38:07,310 This is a quantity which is cyclic, so if you prefer, 646 00:38:07,310 --> 00:38:13,720 you can write it as B dotted into C cross A, 647 00:38:13,720 --> 00:38:26,160 or C dotted into A cross B. 648 00:38:26,160 --> 00:38:29,730 That can be expressed as a determinant, or-- 649 00:38:33,873 --> 00:38:36,540 you guys can look at my notes to see the determinant written out 650 00:38:36,540 --> 00:38:38,332 if you like, but it's not that interesting, 651 00:38:38,332 --> 00:38:41,750 so I'm not going to use it very much. 652 00:38:41,750 --> 00:38:43,920 An equivalent way of writing all that 653 00:38:43,920 --> 00:38:46,110 is to use the Levi-Civita symbol. 654 00:39:06,110 --> 00:39:18,780 So that 3-volume is given by epsilon ijk Ai Bj Ck, where-- 655 00:39:18,780 --> 00:39:29,940 I'll remind you-- epsilon ijk equals plus 1 if i equals 1, 656 00:39:29,940 --> 00:39:38,600 j equals 2, k equals 3, and even permutations. 657 00:39:43,040 --> 00:39:45,730 Even permutations means I swap two pairs of indices. 658 00:39:45,730 --> 00:39:50,090 So 123, 231, 312-- 659 00:39:50,090 --> 00:39:51,260 those all give me plus 1. 660 00:39:54,320 --> 00:39:58,280 It gives me minus 1 for any odd permutations of those. 661 00:40:04,230 --> 00:40:08,940 So 132, 231, et cetera. 662 00:40:08,940 --> 00:40:11,700 Those will all give me minus 1. 663 00:40:11,700 --> 00:40:14,280 And it's 0 if any index is repeated. 664 00:40:28,410 --> 00:40:30,660 So you probably have all seen things like this before. 665 00:40:30,660 --> 00:40:32,575 This is fairly basic vector geometry. 666 00:40:35,950 --> 00:40:39,850 We are going to regard the Levi-Civita symbol 667 00:40:39,850 --> 00:40:44,140 as the components of a 0, 3 tensor. 668 00:40:49,040 --> 00:40:49,540 OK. 669 00:40:49,540 --> 00:40:52,000 Bear in mind for just a moment here I'm working only 670 00:40:52,000 --> 00:40:54,460 in-- sorry, just ran out of good chalk-- 671 00:40:54,460 --> 00:40:58,240 I'm only working in 3-space, so my tensor definition 672 00:40:58,240 --> 00:40:59,260 is slightly different. 673 00:40:59,260 --> 00:41:01,990 It's not going to be a set of things that maps to Lorentz 674 00:41:01,990 --> 00:41:04,360 invariants, but it's going to be invariant with respect 675 00:41:04,360 --> 00:41:06,280 to things like rotations and translations 676 00:41:06,280 --> 00:41:08,960 in three-dimensional space. 677 00:41:08,960 --> 00:41:22,160 So I'm going to regard these as the components of a 0, 3 678 00:41:22,160 --> 00:41:29,180 tensor that basically takes in vectors and spits 679 00:41:29,180 --> 00:41:35,180 out the volume associated with the element whose edges are 680 00:41:35,180 --> 00:41:36,350 bounded by those vectors. 681 00:42:03,090 --> 00:42:03,880 OK. 682 00:42:03,880 --> 00:42:09,650 So I could say, in this abstract form I wrote down earlier, 683 00:42:09,650 --> 00:42:12,740 imagine a boldfaced epsilon which is my volume tensor. 684 00:42:16,410 --> 00:42:19,880 I put these slots into it, and voila. 685 00:42:19,880 --> 00:42:22,550 I get the volume out of it. 686 00:42:22,550 --> 00:42:28,520 Now, with that in mind, remember some of the games 687 00:42:28,520 --> 00:42:32,340 that we played with tensors in the previous lecture. 688 00:42:32,340 --> 00:42:36,050 So when I was talking about spacetime tensors, 689 00:42:36,050 --> 00:42:41,240 if I filled up all of their slots, 690 00:42:41,240 --> 00:42:42,890 I got out a Lorentz invariant number. 691 00:42:42,890 --> 00:42:45,265 In this case, I'm in 3-space, so I fill up all its slots, 692 00:42:45,265 --> 00:42:48,530 I get an invariant number in this 3-space. 693 00:42:48,530 --> 00:42:50,180 Suppose I only put in two vectors. 694 00:42:59,100 --> 00:43:04,200 So suppose I do something like, I plug in-- 695 00:43:04,200 --> 00:43:06,920 let's leave the first slot blank-- 696 00:43:06,920 --> 00:43:12,650 and I put in vectors B and C. OK, well, 697 00:43:12,650 --> 00:43:16,070 writing this out in component form, 698 00:43:16,070 --> 00:43:21,800 I know this is epsilon ijk Bj Ck. 699 00:43:21,800 --> 00:43:25,200 That's just B cross C, right? 700 00:43:25,200 --> 00:43:28,130 That's the area spanned by the side that 701 00:43:28,130 --> 00:43:31,220 is B cross C. And you guys have learned in other classes 702 00:43:31,220 --> 00:43:33,000 that you have an extra index left over, 703 00:43:33,000 --> 00:43:35,840 so it's a vector that has a direction associated to it. 704 00:43:35,840 --> 00:43:38,810 So it's sort of an oriented surface. 705 00:43:38,810 --> 00:43:40,798 We put the index in the downstairs position, 706 00:43:40,798 --> 00:43:42,590 so we're actually going to think about this 707 00:43:42,590 --> 00:43:47,180 as a 1-form corresponding to the side whose 708 00:43:47,180 --> 00:43:54,200 edges are B and C. So let's call this side 1-form sigma. 709 00:43:58,700 --> 00:44:15,060 This is a 1-form whose magnitude is the area of the side spanned 710 00:44:15,060 --> 00:44:20,400 by the vectors B and C. 711 00:44:20,400 --> 00:44:24,300 And although I can still tell it hasn't quite gelled yet, 712 00:44:24,300 --> 00:44:26,580 it's useful to think of 1-forms as being 713 00:44:26,580 --> 00:44:28,530 associated with surfaces. 714 00:44:28,530 --> 00:44:29,030 Guess what? 715 00:44:29,030 --> 00:44:30,750 This is the side of a parallelepiped. 716 00:44:30,750 --> 00:44:32,730 That's a surface. 717 00:44:32,730 --> 00:44:34,500 So it actually holds together. 718 00:44:39,350 --> 00:44:40,460 All right. 719 00:44:40,460 --> 00:44:50,050 So using all of this, if I wanted to write down 720 00:44:50,050 --> 00:45:31,133 how to do something like Gauss's theorem in this kind 721 00:45:31,133 --> 00:45:33,300 of geometric language-- and again, we emphasize this 722 00:45:33,300 --> 00:45:37,290 is very much in the spirit right now of mosquito 723 00:45:37,290 --> 00:45:38,280 with a sledgehammer. 724 00:45:38,280 --> 00:45:39,610 We don't need all this sort of stuff, 725 00:45:39,610 --> 00:45:41,235 but we're about to step up to something 726 00:45:41,235 --> 00:45:42,640 a little bit more complicated. 727 00:45:42,640 --> 00:45:46,960 So what you would do is say, OK, well, I know Gauss's theorem. 728 00:45:46,960 --> 00:45:50,190 I pick a particular 3-volume. 729 00:45:50,190 --> 00:45:52,560 I say the divergence of some vector field integrated 730 00:45:52,560 --> 00:45:56,520 over that volume is given by integrating 731 00:45:56,520 --> 00:45:59,560 the flux of that vector over the surface of this thing. 732 00:46:03,005 --> 00:46:04,380 So what you might want to do then 733 00:46:04,380 --> 00:46:06,780 at this point is say, oh, OK, well, what I'm going to do, 734 00:46:06,780 --> 00:46:09,460 then, is say that my volume-- 735 00:46:09,460 --> 00:46:11,130 oops, pardon me a second. 736 00:46:11,130 --> 00:46:16,070 First thing I'll do is define a differential triple. 737 00:46:22,140 --> 00:46:27,440 I'll define some x1 that points along one direction 738 00:46:27,440 --> 00:46:33,800 I care about, and x2, and an x3. 739 00:46:33,800 --> 00:46:47,510 And then I will say dV equals epsilon ijk dx1i dx2j dx3k. 740 00:46:47,510 --> 00:46:50,630 I can likewise define a 1-form associated 741 00:46:50,630 --> 00:46:52,970 with my area element, as I have done over here. 742 00:47:00,082 --> 00:47:01,790 I'm not going to actually write this out. 743 00:47:01,790 --> 00:47:02,720 The key thing which I want to say 744 00:47:02,720 --> 00:47:03,887 is you have all the pieces-- 745 00:47:03,887 --> 00:47:07,040 you put all these things together, 746 00:47:07,040 --> 00:47:08,690 and you can define this thing. 747 00:47:08,690 --> 00:47:11,820 It's now very easy for you to prove Gauss's theorem using 748 00:47:11,820 --> 00:47:13,970 this kind of ingredients. 749 00:47:13,970 --> 00:47:15,740 What I want to move onto-- 750 00:47:15,740 --> 00:47:17,520 there's a few more details in my notes. 751 00:47:17,520 --> 00:47:20,720 It's not super difficult or interesting to go through this. 752 00:47:20,720 --> 00:47:24,380 What I want to now start doing is generalize 753 00:47:24,380 --> 00:47:27,080 all of these ideas to the way we're 754 00:47:27,080 --> 00:47:29,450 going to approach them in spacetime. 755 00:47:48,040 --> 00:47:52,210 Basically, we're going to do exactly the same kind 756 00:47:52,210 --> 00:47:54,130 of operations that I just did in space-- 757 00:47:54,130 --> 00:47:55,810 three-dimensional space-- 758 00:47:55,810 --> 00:47:57,760 but I'm going to put an extra index on things, 759 00:47:57,760 --> 00:48:01,130 and I'm going to do all of my quantities in spacetime. 760 00:48:01,130 --> 00:48:01,630 OK. 761 00:48:01,630 --> 00:48:22,260 So imagine a parallelepiped with sides A, B, C, 762 00:48:22,260 --> 00:48:27,810 and D. Four dimensions, so it's going to point along four 763 00:48:27,810 --> 00:48:30,022 different-- these can be mutually orthogonal. 764 00:48:33,330 --> 00:48:36,240 I'm going to define the invariant 4-volume associated 765 00:48:36,240 --> 00:48:49,180 with these things like so. 766 00:48:49,180 --> 00:48:53,150 Where now my four-index Levi-Civita is defined such 767 00:48:53,150 --> 00:49:02,030 that epsilon 0123 equals plus 1. 768 00:49:02,030 --> 00:49:03,770 If I do an odd permutation of those-- 769 00:49:03,770 --> 00:49:07,450 I exchange one pair of them-- epsilon 1023-- 770 00:49:07,450 --> 00:49:09,110 this equals minus 1. 771 00:49:09,110 --> 00:49:11,600 If I repeat any index, I get 0. 772 00:49:16,270 --> 00:49:18,580 And likewise, all even permutations of this 773 00:49:18,580 --> 00:49:19,300 give me plus 1. 774 00:49:19,300 --> 00:49:22,630 All odd permutations of this give me minus 1. 775 00:49:22,630 --> 00:49:27,400 Or likewise, just do even permutations of this one. 776 00:49:27,400 --> 00:49:29,500 So that is how I'm going to generalize 777 00:49:29,500 --> 00:49:31,875 my Levi-Civita symbol. 778 00:49:31,875 --> 00:49:33,250 As they say on The Simpsons, it's 779 00:49:33,250 --> 00:49:34,840 a perfectly cromulent object. 780 00:49:41,300 --> 00:49:44,600 I'm going to need to talk about the area associated 781 00:49:44,600 --> 00:49:47,100 with the faces of each of these things. 782 00:49:47,100 --> 00:49:50,975 So what is the area of the face of a 4-volume? 783 00:49:50,975 --> 00:49:52,469 A 3-volume. 784 00:50:30,960 --> 00:50:36,580 So you can define a 1-form that tells me about the-- 785 00:50:36,580 --> 00:50:38,880 you can either call it the 3-volume or the 4-area. 786 00:50:38,880 --> 00:50:41,965 Knock yourselves out as to how you want to call it. 787 00:50:41,965 --> 00:50:44,400 And the obvious generalization-- let's 788 00:50:44,400 --> 00:50:52,590 say I leave off edge A. I want to get something like this. 789 00:50:52,590 --> 00:50:56,340 I do my similar exercise of defining a-- in this case, 790 00:50:56,340 --> 00:50:59,320 it'll be a differential quartet associated with directions 0, 791 00:50:59,320 --> 00:51:01,610 1, 2, and 3. 792 00:51:01,610 --> 00:51:05,250 And so by going through a procedure very similar to this, 793 00:51:05,250 --> 00:51:07,440 you get a generalization of Gauss's theorem. 794 00:51:18,894 --> 00:51:28,710 It says that if I integrate the spacetime divergence 795 00:51:28,710 --> 00:51:38,680 of some 4-vector over a four-dimensional volume, 796 00:51:38,680 --> 00:51:44,590 it looks like what I get when I sum up 797 00:51:44,590 --> 00:51:50,730 the flux of that guy over all of the little faces. 798 00:51:55,275 --> 00:51:57,400 So I'm not going to step through the proof of that. 799 00:51:57,400 --> 00:52:01,360 It's fairly elementary, and basically it's 800 00:52:01,360 --> 00:52:05,590 just like proofs of Gauss's law that you have seen elsewhere, 801 00:52:05,590 --> 00:52:07,750 but there's an extra dimension attached to it. 802 00:52:07,750 --> 00:52:10,180 Really nothing new that's going on here. 803 00:52:10,180 --> 00:52:12,910 The thing which is new is, this is now 804 00:52:12,910 --> 00:52:15,880 being done in an additional dimension, 805 00:52:15,880 --> 00:52:17,710 and where this tends to be useful 806 00:52:17,710 --> 00:52:20,530 is when there is some kind of a conservation law 807 00:52:20,530 --> 00:52:23,650 that tells you something about this left-hand side here. 808 00:52:28,800 --> 00:52:32,120 So the whole starting point of this discussion 809 00:52:32,120 --> 00:52:36,800 was I wrote down, on intuitive grounds, 810 00:52:36,800 --> 00:52:43,450 that the rate of change of the total amount of-- so 811 00:52:43,450 --> 00:52:47,060 what I did was I had an integral of dust density 812 00:52:47,060 --> 00:52:49,580 over a 3-volume, and I said, d by dt of that 813 00:52:49,580 --> 00:52:52,250 was balanced by the flux through the surfaces 814 00:52:52,250 --> 00:52:53,540 on the edge of that 3-volume. 815 00:52:56,420 --> 00:52:58,760 As I argued, that's an observer-dependent statement, 816 00:52:58,760 --> 00:53:00,660 because you have chosen a particular time. 817 00:53:00,660 --> 00:53:03,035 You have chosen a particular space to make those volumes. 818 00:53:05,510 --> 00:53:07,550 This, on the other hand-- 819 00:53:07,550 --> 00:53:11,630 so let's switch my general vector field 820 00:53:11,630 --> 00:53:15,460 V to be N that we started this discussion off with. 821 00:53:15,460 --> 00:53:24,370 So this guy-- oops. 822 00:53:24,370 --> 00:53:26,660 Let me write it the way I wrote it over there. 823 00:53:26,660 --> 00:53:28,727 d4x over some 4-volume. 824 00:53:32,310 --> 00:53:35,440 This must be 0, because we are going 825 00:53:35,440 --> 00:53:37,230 to require that this thing be-- 826 00:53:37,230 --> 00:53:41,730 so if my number density is conserved, this must be 0. 827 00:53:48,110 --> 00:53:56,170 So this tells me that when I do this flux integral, 828 00:53:56,170 --> 00:53:57,760 it's going to have to be 0. 829 00:53:57,760 --> 00:54:00,550 Let me now break this integral up, and actually 830 00:54:00,550 --> 00:54:04,630 write this into a form that is a little closer to the way 831 00:54:04,630 --> 00:54:06,970 that you may have seen something like this before. 832 00:54:17,230 --> 00:54:21,690 So what I want to do is actually zoom in and think 833 00:54:21,690 --> 00:54:23,220 about the four-dimensional volume 834 00:54:23,220 --> 00:54:25,445 that I'm doing this integral over. 835 00:54:25,445 --> 00:54:27,570 So I'm just going to do a two-dimensional cut of it 836 00:54:27,570 --> 00:54:28,770 here on the blackboard. 837 00:54:28,770 --> 00:54:32,310 Let's let the time axis go up, and let's 838 00:54:32,310 --> 00:54:36,210 define the edges of my volumes to be t1 and t2. 839 00:54:36,210 --> 00:54:38,820 x-axis going across here. 840 00:54:38,820 --> 00:54:40,365 Boundaries are x1 and x2. 841 00:54:56,330 --> 00:54:57,090 OK. 842 00:54:57,090 --> 00:55:00,520 So here is my V4. 843 00:55:04,820 --> 00:55:14,060 And every face here is an example of my boundary of d4. 844 00:55:14,060 --> 00:55:14,560 OK. 845 00:55:14,560 --> 00:55:19,560 Of course there's one over here as well, and over here. 846 00:55:22,138 --> 00:55:23,680 So if I were to do the top line here, 847 00:55:23,680 --> 00:55:25,170 I know that's got to get me 0, so I'm 848 00:55:25,170 --> 00:55:27,295 going to take advantage of this and say, let's just 849 00:55:27,295 --> 00:55:29,110 look at what happens when I do the integral 850 00:55:29,110 --> 00:55:34,690 of the flux of this thing across the many different faces. 851 00:55:34,690 --> 00:55:38,520 So in a four-dimensional parallelepiped-- 852 00:55:38,520 --> 00:55:41,590 an n-dimensional parallelepiped has 2n phases, 853 00:55:41,590 --> 00:55:43,730 so there will be eight integrals we need to do. 854 00:55:43,730 --> 00:55:44,700 They're pretty obvious though, so I'm just 855 00:55:44,700 --> 00:55:46,283 going to write down a handful of them. 856 00:56:07,920 --> 00:56:08,490 OK. 857 00:56:08,490 --> 00:56:15,160 So over all those faces-- 858 00:56:15,160 --> 00:56:18,300 N alpha sigma alpha. 859 00:56:18,300 --> 00:56:19,980 So what I'm going to do now is say, OK, 860 00:56:19,980 --> 00:56:27,560 let's evaluate this on the face that is at t equals t2. 861 00:56:27,560 --> 00:56:34,220 So when I do this, I'm going to get N0 dx dy dz. 862 00:56:37,820 --> 00:56:40,610 Let's do next the contribution at moment-- 863 00:56:40,610 --> 00:56:42,360 actually, let me move this over, because I 864 00:56:42,360 --> 00:56:43,527 want a little bit more room. 865 00:56:56,830 --> 00:56:59,680 I also do one on the slice t equals t1. 866 00:57:06,190 --> 00:57:08,680 When I do this, though, in the same way 867 00:57:08,680 --> 00:57:13,030 that when you guys do fluxes in three-dimensional space, 868 00:57:13,030 --> 00:57:16,060 you get a sign associated with the orientation 869 00:57:16,060 --> 00:57:19,180 of these things, because the Levi-Civita symbol here 870 00:57:19,180 --> 00:57:23,050 has sort of a right-hand rule built into it. 871 00:57:23,050 --> 00:57:25,510 And so when I do it for the side that 872 00:57:25,510 --> 00:57:28,090 is on the future side of the box, 873 00:57:28,090 --> 00:57:29,740 I'm going to get a plus sign. 874 00:57:29,740 --> 00:57:32,830 Do all the analysis carefully on the side 875 00:57:32,830 --> 00:57:35,580 that's on the past side the box, you get a minus sign. 876 00:57:39,290 --> 00:57:41,870 Then I'm going to do it-- 877 00:57:41,870 --> 00:57:46,710 I'll pick out the component N1, and I'm 878 00:57:46,710 --> 00:57:49,170 going to do this along the face that's at x equals x1. 879 00:57:56,060 --> 00:57:56,560 Sorry. 880 00:57:56,560 --> 00:57:57,518 This is going to be x2. 881 00:58:09,090 --> 00:58:09,890 OK. 882 00:58:09,890 --> 00:58:12,470 You can write down four other integrals 883 00:58:12,470 --> 00:58:15,560 according to when you do the y side and the z side 884 00:58:15,560 --> 00:58:16,490 that I've not drawn. 885 00:58:36,908 --> 00:58:37,430 OK. 886 00:58:37,430 --> 00:58:40,680 What I'm going to do now is imagine 887 00:58:40,680 --> 00:58:44,590 that this box gets very small in the timelike direction. 888 00:58:44,590 --> 00:58:55,470 So let's let t2 go to t1 plus dt. 889 00:58:55,470 --> 00:58:57,480 And I'm going to do this and then rearrange 890 00:58:57,480 --> 00:58:58,313 things a little bit. 891 00:59:06,040 --> 00:59:06,820 Again, apologies. 892 00:59:06,820 --> 00:59:07,737 I'm eating chalk here. 893 00:59:14,790 --> 00:59:16,440 So I'm going to rearrange this integral 894 00:59:16,440 --> 00:59:18,610 so that it's of the form-- 895 00:59:18,610 --> 00:59:21,500 this side's being evaluated at t1 plus t2. 896 00:59:40,160 --> 00:59:44,860 So I'm going to rearrange stuff so that I can then write-- 897 00:59:44,860 --> 00:59:45,760 just one moment. 898 00:59:54,990 --> 00:59:55,490 Oh. 899 01:00:00,750 --> 01:00:01,890 Apologies. 900 01:00:01,890 --> 01:00:05,162 This whole thing, of course, equals 0. 901 01:00:05,162 --> 01:00:06,870 I'm just looking over my notes and went-- 902 01:00:11,180 --> 01:00:12,580 there was a magic sign flip. 903 01:00:12,580 --> 01:00:13,580 I was just looking at it and going, where 904 01:00:13,580 --> 01:00:14,820 the hell did that come from? 905 01:00:14,820 --> 01:00:15,320 OK. 906 01:00:15,320 --> 01:00:17,190 So I left off my equals 0 there. 907 01:00:17,190 --> 01:00:20,900 I'm basically moving a bunch of terms to the other side. 908 01:00:20,900 --> 01:00:24,640 So now I'm integrating this over the face at x2. 909 01:00:31,570 --> 01:00:34,725 And I do an integral over the face at x1. 910 01:00:37,810 --> 01:00:40,510 And then I'm not drawing in a bunch of faces 911 01:00:40,510 --> 01:00:43,390 along y2 and y1, z1 and z2. 912 01:00:46,940 --> 01:00:50,030 Divide both sides by dt. 913 01:00:50,030 --> 01:00:53,180 Take the limit of dt going to 0. 914 01:00:53,180 --> 01:00:54,230 What is this? 915 01:00:54,230 --> 01:00:57,900 This is the derivative of the volume integral of N0. 916 01:01:25,720 --> 01:01:28,150 So let's just write that out explicitly. 917 01:01:28,150 --> 01:01:39,390 So if I've got integral t1 plus dt N0 dx dy dz 918 01:01:39,390 --> 01:01:41,295 minus the integral at t1-- 919 01:01:46,570 --> 01:01:49,780 all this divided by dt. 920 01:01:49,780 --> 01:01:51,640 Take the limit as dt goes to 0. 921 01:02:00,480 --> 01:02:02,200 That becomes this. 922 01:02:02,200 --> 01:02:04,400 That i is a bad erasure. 923 01:02:04,400 --> 01:02:05,430 My apologies. 924 01:02:10,930 --> 01:02:12,510 What do I get for the other term? 925 01:02:12,510 --> 01:02:16,230 Well, when I divide out that dt, I 926 01:02:16,230 --> 01:02:21,780 am just left with the flux of the spatial component 927 01:02:21,780 --> 01:02:29,280 of the number vector N through all six sides at moment t, 928 01:02:29,280 --> 01:02:29,910 or t1. 929 01:02:54,780 --> 01:02:58,060 And at last-- so I will wrap things up in just a moment 930 01:02:58,060 --> 01:02:59,380 here. 931 01:02:59,380 --> 01:03:04,450 What this finally leads us to is that this fully covariant 932 01:03:04,450 --> 01:03:06,340 form that I have at the top there-- 933 01:03:16,890 --> 01:03:44,253 this becomes-- so in the language 934 01:03:44,253 --> 01:03:46,420 that you probably learned about in an earlier class, 935 01:03:46,420 --> 01:03:49,700 you can think of this as the area element of each side. 936 01:03:49,700 --> 01:03:52,450 These are completely equivalent to one another 937 01:03:52,450 --> 01:03:55,090 after you have chosen a particular Lorentz frame 938 01:03:55,090 --> 01:03:57,370 and invoked this four-dimensional analog 939 01:03:57,370 --> 01:03:58,240 of Gauss's law. 940 01:04:18,070 --> 01:04:21,740 So that was a lot to take, but I wanted to do it 941 01:04:21,740 --> 01:04:24,230 for this one particularly-- 942 01:04:24,230 --> 01:04:27,170 to be blunt, this was a particularly simple example, 943 01:04:27,170 --> 01:04:28,580 because-- 944 01:04:28,580 --> 01:04:30,380 so right now, the only matter that I've 945 01:04:30,380 --> 01:04:33,350 introduced beyond particle kinematics is dust. 946 01:04:33,350 --> 01:04:35,900 And dust is actually surprisingly important. 947 01:04:35,900 --> 01:04:38,580 When we actually get to cosmology, 948 01:04:38,580 --> 01:04:41,120 there are essentially two forms of matter 949 01:04:41,120 --> 01:04:44,240 that we consider when we study cosmology, and one of them 950 01:04:44,240 --> 01:04:46,110 is dust. 951 01:04:46,110 --> 01:04:49,460 In cosmological situations, a dust particle 952 01:04:49,460 --> 01:04:52,738 is basically a galaxy, or even a cluster of galaxies. 953 01:04:52,738 --> 01:04:53,530 We're thinking big. 954 01:04:56,100 --> 01:05:00,247 So it's not trivial to do this, but we are soon 955 01:05:00,247 --> 01:05:02,330 going to need to introduce mathematical tools that 956 01:05:02,330 --> 01:05:05,150 are a little better for describing things like fluids, 957 01:05:05,150 --> 01:05:06,350 and stuff like that. 958 01:05:06,350 --> 01:05:08,270 And when we do that, we are at last 959 01:05:08,270 --> 01:05:13,970 going to have things like conservation of energy 960 01:05:13,970 --> 01:05:16,260 and conservation of momentum. 961 01:05:16,260 --> 01:05:20,180 It will be really easy to write down 962 01:05:20,180 --> 01:05:24,230 a differential conservation law that describes conservation 963 01:05:24,230 --> 01:05:27,590 of both energy and momentum using that mathematical object. 964 01:05:27,590 --> 01:05:31,580 Via a process very similar to what I just did here, 965 01:05:31,580 --> 01:05:33,890 we can then turn this into integrals 966 01:05:33,890 --> 01:05:36,307 that describe how energy is conserved 967 01:05:36,307 --> 01:05:38,390 in a particular volume where things may be flowing 968 01:05:38,390 --> 01:05:41,930 into or flowing out of it, and how momentum is conserved 969 01:05:41,930 --> 01:05:44,810 per unit volume as things flow into it and flow out of it. 970 01:05:44,810 --> 01:05:46,580 I'm not going to go through that in detail 971 01:05:46,580 --> 01:05:49,540 on the board, but having done this-- 972 01:05:49,540 --> 01:05:52,040 I have some notes that I'm going to post to the website that 973 01:05:52,040 --> 01:05:56,950 describe this, and that will be the way we communicate this. 974 01:05:56,950 --> 01:05:58,950 And we're going to take advantage-- to be blunt, 975 01:05:58,950 --> 01:06:00,908 we will mostly just use this differential form. 976 01:06:00,908 --> 01:06:03,650 We'll use the fact there is a particular mathematical object 977 01:06:03,650 --> 01:06:07,560 whose divergence-like derivative there is equal to 0. 978 01:06:07,560 --> 01:06:08,060 OK. 979 01:06:08,060 --> 01:06:10,970 We have a few minutes left and we've ended early a few times, 980 01:06:10,970 --> 01:06:12,512 so I'd like to take advantage of this 981 01:06:12,512 --> 01:06:14,450 to switch gears a little bit and talk 982 01:06:14,450 --> 01:06:19,160 about another important 4-vector that 983 01:06:19,160 --> 01:06:21,920 plays some role in physics, and allows me to introduce 984 01:06:21,920 --> 01:06:26,197 a few other very useful tricks that we will often 985 01:06:26,197 --> 01:06:28,280 take advantage of at various points in this class. 986 01:06:33,020 --> 01:06:35,110 I forgot there's a straw. 987 01:06:35,110 --> 01:06:37,180 When you tip it, the straw doesn't work. 988 01:06:37,180 --> 01:06:41,970 Anyway, so the next example of stuff 989 01:06:41,970 --> 01:06:46,140 that we will occasionally talk about is an electric current. 990 01:06:56,220 --> 01:06:59,090 So switching gears very, very much now. 991 01:07:20,770 --> 01:07:24,680 So we will describe this as a 4-vector whose 992 01:07:24,680 --> 01:07:28,130 timelike component is the charge density as seen 993 01:07:28,130 --> 01:07:32,420 by some observer, and whose spatial component 994 01:07:32,420 --> 01:07:34,940 is the current density as seen by that observer. 995 01:07:34,940 --> 01:07:36,470 Bear in mind, you might look at this 996 01:07:36,470 --> 01:07:38,690 and twitch a little bit because the units look wrong. 997 01:07:38,690 --> 01:07:41,180 Don't forget c equals 1. 998 01:07:41,180 --> 01:07:43,280 So if you use this in other systems of units, 999 01:07:43,280 --> 01:07:48,320 sometimes we call this component charge density times c. 1000 01:07:48,320 --> 01:07:52,322 So you guys have all-- 1001 01:07:52,322 --> 01:07:54,530 you know what, let me just go ahead and write it out. 1002 01:07:54,530 --> 01:07:57,080 So a couple properties about this 1003 01:07:57,080 --> 01:07:58,730 are important and interesting. 1004 01:07:58,730 --> 01:08:03,080 One is that the current and the charge density 1005 01:08:03,080 --> 01:08:07,130 obey a continuity equation. 1006 01:08:07,130 --> 01:08:15,600 We can think of it as a conservation of charge. 1007 01:08:15,600 --> 01:08:21,198 It's expressed-- so in elementary E&M, 1008 01:08:21,198 --> 01:08:23,490 you guys all presumably learned that the rate of change 1009 01:08:23,490 --> 01:08:25,380 of charge density is related to the divergence 1010 01:08:25,380 --> 01:08:26,338 of the current density. 1011 01:08:29,930 --> 01:08:33,260 Well, this is exactly the same as saying 1012 01:08:33,260 --> 01:08:37,640 that the spacetime divergence of the current 4-vector 1013 01:08:37,640 --> 01:08:40,149 is equal to 0. 1014 01:08:40,149 --> 01:08:42,819 This is really useful for us. 1015 01:08:42,819 --> 01:08:45,370 And indeed, bearing this in mind, 1016 01:08:45,370 --> 01:08:48,729 we find that if we want to express Maxwell's equations 1017 01:08:48,729 --> 01:08:51,729 in a covariant way, we can do so such that this 1018 01:08:51,729 --> 01:08:55,229 is built in automatically. 1019 01:08:55,229 --> 01:09:04,389 So skipping over a few sets of things in my notes, 1020 01:09:04,389 --> 01:09:05,264 we're going to find-- 1021 01:09:08,250 --> 01:09:09,250 we're not going to find. 1022 01:09:09,250 --> 01:09:14,590 We have found that electric fields and magnetic fields 1023 01:09:14,590 --> 01:09:19,270 are inconvenient objects to describe 1024 01:09:19,270 --> 01:09:22,750 using geometric objects that are appropriate for spacetime. 1025 01:09:22,750 --> 01:09:24,130 If I want a geometric object that 1026 01:09:24,130 --> 01:09:27,250 is appropriate for spacetime, the first thing you think of 1027 01:09:27,250 --> 01:09:30,130 is a 4-vector. 1028 01:09:30,130 --> 01:09:33,420 A 4-vector has four components. 1029 01:09:33,420 --> 01:09:34,899 E-fields and B-fields have a total 1030 01:09:34,899 --> 01:09:36,790 of six components among them. 1031 01:09:36,790 --> 01:09:40,479 So what you going to do, have two 4-vectors and just ignore 1032 01:09:40,479 --> 01:09:42,430 two of the components? 1033 01:09:42,430 --> 01:09:43,896 That seems sketchy. 1034 01:09:43,896 --> 01:09:45,229 So you think, eh, you know what? 1035 01:09:45,229 --> 01:09:47,200 Why don't we make a tensor. 1036 01:09:47,200 --> 01:09:49,930 Ah crap, a tensor has 16 components. 1037 01:09:49,930 --> 01:09:50,930 That doesn't seem right. 1038 01:09:50,930 --> 01:09:53,060 Then you go, ooh, I can make it symmetric. 1039 01:09:53,060 --> 01:10:06,310 If you have a symmetric 4-by-4 tensor, 1040 01:10:06,310 --> 01:10:08,130 well, that basically means that the number 1041 01:10:08,130 --> 01:10:10,470 of independent numbers that go into this thing-- you 1042 01:10:10,470 --> 01:10:13,340 have four down the diagonal. 1043 01:10:13,340 --> 01:10:15,090 And then you count the number that are off 1044 01:10:15,090 --> 01:10:16,700 the diagonal-- you have six. 1045 01:10:16,700 --> 01:10:19,200 There's 16 in total, but the ones that are off the diagonal 1046 01:10:19,200 --> 01:10:26,140 are equal to one another, so you have four on diagonal, six off. 1047 01:10:26,140 --> 01:10:26,640 Too many. 1048 01:10:31,940 --> 01:10:35,350 So you say, well, what about antisymmetric? 1049 01:10:41,720 --> 01:10:44,170 If I have an antisymmetric object, 1050 01:10:44,170 --> 01:10:46,020 that means that component F alpha 1051 01:10:46,020 --> 01:10:51,060 beta is the negative of component F beta alpha. 1052 01:10:51,060 --> 01:10:54,120 When you do that, that forces you the conclusion 1053 01:10:54,120 --> 01:10:55,710 that there are, in fact, actually 1054 01:10:55,710 --> 01:10:58,410 only six independent numbers in that thing. 1055 01:10:58,410 --> 01:11:02,040 The diagonal has to be 0, because set beta equal 1056 01:11:02,040 --> 01:11:02,970 to alpha here-- 1057 01:11:02,970 --> 01:11:05,280 F alpha alpha equals minus F alpha alpha. 1058 01:11:05,280 --> 01:11:07,770 That only works if that component is equal to 0. 1059 01:11:07,770 --> 01:11:19,420 So the diagonal becomes zero, and only the six off-diagonals 1060 01:11:19,420 --> 01:11:19,920 survive. 1061 01:11:25,210 --> 01:11:27,050 And then you go, holy crap. 1062 01:11:27,050 --> 01:11:27,740 Six. 1063 01:11:27,740 --> 01:11:30,830 That's exactly what I need to have a geometric object that 1064 01:11:30,830 --> 01:11:33,620 cleanly holds the three independent electric field 1065 01:11:33,620 --> 01:11:36,050 components and the three independent magnetic field 1066 01:11:36,050 --> 01:11:37,970 components. 1067 01:11:37,970 --> 01:11:41,450 So many of you have seen all this. 1068 01:11:41,450 --> 01:11:43,260 If this isn't familiar to you, take a look 1069 01:11:43,260 --> 01:11:45,260 at a book like Griffiths or something like that. 1070 01:11:45,260 --> 01:11:46,940 It goes through this. 1071 01:11:46,940 --> 01:11:52,820 The punchline is that what you find 1072 01:11:52,820 --> 01:11:57,200 is that the electric and magnetic field is very nicely 1073 01:11:57,200 --> 01:12:03,410 represented by this antisymmetric 2-index object, 1074 01:12:03,410 --> 01:12:07,220 whose components, in the units that we are using, 1075 01:12:07,220 --> 01:12:19,700 are filled with the E and the B like so. 1076 01:12:24,330 --> 01:12:24,830 OK. 1077 01:12:24,830 --> 01:12:26,490 Last semester I had these memorized, 1078 01:12:26,490 --> 01:12:27,907 but I have totally forgotten them. 1079 01:12:41,650 --> 01:12:42,150 OK. 1080 01:12:42,150 --> 01:12:45,840 So this is a geometric object that-- whoops. 1081 01:12:52,878 --> 01:12:54,670 Anybody know why I just said whoops and had 1082 01:12:54,670 --> 01:12:56,450 to put a dot on that there? 1083 01:12:56,450 --> 01:13:00,010 The point is that electric and magnetic fields 1084 01:13:00,010 --> 01:13:01,900 look different to different observers. 1085 01:13:01,900 --> 01:13:04,150 This is its representation according to one 1086 01:13:04,150 --> 01:13:06,250 particular Lorentz observer. 1087 01:13:06,250 --> 01:13:07,750 So it's important to get that right. 1088 01:13:18,950 --> 01:13:21,810 So in terms of this-- 1089 01:13:21,810 --> 01:13:25,029 you're all familiar with the four Maxwell's equations. 1090 01:13:34,020 --> 01:13:36,840 They turn out to be equivalent to-- 1091 01:13:36,840 --> 01:13:46,688 so if you take a divergence of this F, it is-- 1092 01:13:46,688 --> 01:13:48,230 depending on your units, so you might 1093 01:13:48,230 --> 01:13:50,840 want to put u0's in there, and things like that. 1094 01:13:50,840 --> 01:13:53,820 Basically, I'm setting everything 1095 01:13:53,820 --> 01:13:56,480 that I can never remember to 1. 1096 01:13:56,480 --> 01:13:58,230 The divergence of that thing-- by the way, 1097 01:13:58,230 --> 01:14:00,050 actually divergence on the second index-- 1098 01:14:00,050 --> 01:14:02,990 becomes the current density. 1099 01:14:02,990 --> 01:14:04,820 This will actually only give you half 1100 01:14:04,820 --> 01:14:07,340 of the Maxwell's equations. 1101 01:14:07,340 --> 01:14:16,000 The other half-- what you do is you lower these two indices, 1102 01:14:16,000 --> 01:14:24,327 and there's this cyclic permutation of derivatives 1103 01:14:24,327 --> 01:14:25,160 that give you these. 1104 01:14:25,160 --> 01:14:27,020 So you put these two things together, 1105 01:14:27,020 --> 01:14:29,930 you apply it to this form I've written out here, 1106 01:14:29,930 --> 01:14:31,993 and you will reproduce Maxwell's equations 1107 01:14:31,993 --> 01:14:33,410 as they are presented in textbooks 1108 01:14:33,410 --> 01:14:36,530 like Purcell and Griffiths. 1109 01:14:36,530 --> 01:14:38,510 The thing which I want to emphasize here 1110 01:14:38,510 --> 01:14:44,180 is, this form that we've got is written in such a way 1111 01:14:44,180 --> 01:14:49,680 that the conservation of source is built into it. 1112 01:14:49,680 --> 01:14:56,750 This geometric language requires that J mu have no divergence. 1113 01:14:56,750 --> 01:14:58,520 And let me just show you can do this. 1114 01:14:58,520 --> 01:15:01,993 All you need to know is that F is an antisymmetric tensor. 1115 01:15:01,993 --> 01:15:04,160 You don't need to know anything about the properties 1116 01:15:04,160 --> 01:15:05,420 of the E- and the B-field. 1117 01:15:05,420 --> 01:15:06,295 So let's just try it. 1118 01:15:25,160 --> 01:15:27,720 Let's look at the divergence of the current. 1119 01:15:27,720 --> 01:15:34,760 So I'm going to do 4pi d mu J mu. 1120 01:15:34,760 --> 01:15:43,740 So I'm going to take a mu derivative of this. 1121 01:15:46,500 --> 01:15:48,980 Now bear in mind, mu and nu-- 1122 01:15:48,980 --> 01:15:50,790 I'm using Einstein's summation convention-- 1123 01:15:50,790 --> 01:15:54,420 they are dummy indices, so I can relabel them. 1124 01:15:54,420 --> 01:15:55,230 I can change them. 1125 01:15:55,230 --> 01:15:58,590 I can change mu to an alpha, nu to a beta, 1126 01:15:58,590 --> 01:16:01,770 or I can just change mu to nu, nu to mu. 1127 01:16:01,770 --> 01:16:10,240 So this-- as long as I do it consistently on all objects 1128 01:16:10,240 --> 01:16:13,020 that have those things-- this is the exact same thing. 1129 01:16:20,260 --> 01:16:22,900 But I also know that this tensor is antisymmetric. 1130 01:16:29,420 --> 01:16:35,550 So if I switch these guys back, I get a minus sign. 1131 01:16:35,550 --> 01:16:36,810 OK? 1132 01:16:36,810 --> 01:16:37,830 Antisymmetry. 1133 01:16:44,700 --> 01:16:49,240 What happens if I switch the order of the derivatives? 1134 01:16:49,240 --> 01:16:52,113 Does it matter whether I take the x derivative first and then 1135 01:16:52,113 --> 01:16:54,030 the y derivative, or the y derivative and then 1136 01:16:54,030 --> 01:16:55,740 the x derivative? 1137 01:16:55,740 --> 01:16:58,200 Partial derivatives commute with each other, right? 1138 01:16:58,200 --> 01:17:00,150 They are perfectly symmetric. 1139 01:17:00,150 --> 01:17:15,680 So this-- so what I've got is d mu d nu of F mu nu 1140 01:17:15,680 --> 01:17:20,120 is minus d mu d nu of F mu nu, and that only 1141 01:17:20,120 --> 01:17:23,090 works if the whole thing is 0. 1142 01:17:26,640 --> 01:17:33,860 What I just did is, I actually just invoked a trick 1143 01:17:33,860 --> 01:17:38,120 that we are going to use many times. 1144 01:17:38,120 --> 01:17:41,930 And the one reason why-- this is a bit of a tangent. 1145 01:17:41,930 --> 01:17:45,050 Or more than a tangent-- 1146 01:17:45,050 --> 01:17:47,960 150-degree turn from what I've been discussing 1147 01:17:47,960 --> 01:17:51,680 before-- but I wanted to make sure you saw this little trick. 1148 01:17:51,680 --> 01:18:04,840 Whenever I have an object that is antisymmetric in its indices 1149 01:18:04,840 --> 01:18:07,840 and I contract it with an object that 1150 01:18:07,840 --> 01:18:18,260 is symmetric in its indices, you just get 0. 1151 01:18:18,260 --> 01:18:19,850 You can go through the little exercise 1152 01:18:19,850 --> 01:18:22,730 if you don't feel fluent in this yet. 1153 01:18:22,730 --> 01:18:26,240 You can do the exercise I just did up there again over. 1154 01:18:26,240 --> 01:18:28,580 Take advantage of dummy indices, swap one. 1155 01:18:28,580 --> 01:18:30,620 Work in the antisymmetry, swap the other. 1156 01:18:30,620 --> 01:18:32,360 Work in the symmetry-- boom. 1157 01:18:32,360 --> 01:18:34,617 You will necessarily prove that you've got 0. 1158 01:18:34,617 --> 01:18:36,200 So I lay this out here because there's 1159 01:18:36,200 --> 01:18:38,660 going to be several times later in the course, where 1160 01:18:38,660 --> 01:18:40,648 I'm going to get to a particular calculation 1161 01:18:40,648 --> 01:18:42,440 and there's going to be some godawful mess. 1162 01:18:42,440 --> 01:18:44,150 We're going to look at it and go, oh, this is horrible, 1163 01:18:44,150 --> 01:18:46,490 and then go, wait, symmetry-antisymmetry, boom. 1164 01:18:46,490 --> 01:18:49,560 We just killed 13 terms. 1165 01:18:49,560 --> 01:18:52,170 Tricks are fun. 1166 01:18:52,170 --> 01:18:55,660 So I am going to stop there for today. 1167 01:18:55,660 --> 01:18:58,530 There is another symmetry-antisymmetry thing 1168 01:18:58,530 --> 01:19:00,240 which allows you to-- 1169 01:19:00,240 --> 01:19:02,100 if you apply it to the equation of motion 1170 01:19:02,100 --> 01:19:04,770 of a charge in an electromagnetic field, 1171 01:19:04,770 --> 01:19:07,620 it just shows you that that equation of motion 1172 01:19:07,620 --> 01:19:11,760 builds in the fact that in spacetime, the acceleration is 1173 01:19:11,760 --> 01:19:14,490 always orthogonal to the 4-velocity. 1174 01:19:14,490 --> 01:19:15,760 I won't do it in class. 1175 01:19:15,760 --> 01:19:18,710 It is on page 6 of the notes that I'm about to post up. 1176 01:19:18,710 --> 01:19:19,950 We will pick up next time. 1177 01:19:19,950 --> 01:19:23,510 We'll begin by talking about the stress energy tensor.