1 00:00:00,500 --> 00:00:03,374 [SQUEAKING][RUSTLING][CLICKING] 2 00:00:11,590 --> 00:00:13,520 SCOTT HUGHES: All right, welcome to Tuesday. 3 00:00:13,520 --> 00:00:16,840 So hopefully, you've all saw the brief announcement 4 00:00:16,840 --> 00:00:18,320 I send to the class. 5 00:00:18,320 --> 00:00:22,990 I have to introduce a colloquium speaker over in Astrophysics, 6 00:00:22,990 --> 00:00:26,167 basically at the second this class officially ends, 7 00:00:26,167 --> 00:00:28,000 so I will be wrapping things up a little bit 8 00:00:28,000 --> 00:00:32,020 early so that I can take into account the spatial separation 9 00:00:32,020 --> 00:00:37,000 and get back there in time to actually do the introduction. 10 00:00:37,000 --> 00:00:39,850 I've already posted the lecture notes of material 11 00:00:39,850 --> 00:00:43,253 I'm going to be covering today, and it'll probably spill-- 12 00:00:43,253 --> 00:00:44,920 I hope to wrap it all up today, but it's 13 00:00:44,920 --> 00:00:46,920 possible it'll spill a little bit into Thursday. 14 00:00:46,920 --> 00:00:49,582 So if you've already looked at those notes, 15 00:00:49,582 --> 00:00:51,790 today will essentially just be sort of my guided tour 16 00:00:51,790 --> 00:00:53,570 through that material. 17 00:00:53,570 --> 00:00:56,445 So I want to pick it up with where I left things last time. 18 00:00:56,445 --> 00:00:57,820 So we covered a bunch of material 19 00:00:57,820 --> 00:01:00,400 that, again, I kind of emphasize what we're doing right now 20 00:01:00,400 --> 00:01:03,700 is just laying the mathematical foundations in a very 21 00:01:03,700 --> 00:01:07,180 thorough, almost excessively thorough way in order 22 00:01:07,180 --> 00:01:10,180 that we have a very strong structure as we begin to move 23 00:01:10,180 --> 00:01:12,362 into more physically complicated situations 24 00:01:12,362 --> 00:01:14,820 in the special relativity that we're focusing on right now. 25 00:01:14,820 --> 00:01:17,260 So we talked about this definition 26 00:01:17,260 --> 00:01:20,440 of an inner product between two four-vectors, two vectors 27 00:01:20,440 --> 00:01:21,850 in space time. 28 00:01:21,850 --> 00:01:26,170 And it looks just like the inner product between two vectors 29 00:01:26,170 --> 00:01:28,660 that you are used to from your Euclidean three space 30 00:01:28,660 --> 00:01:29,290 intuition. 31 00:01:29,290 --> 00:01:31,720 It's just that we have an extra bit there 32 00:01:31,720 --> 00:01:33,220 that enters with a minus sign having 33 00:01:33,220 --> 00:01:36,040 to do with the time like components of those two 34 00:01:36,040 --> 00:01:38,020 four-vectors. 35 00:01:38,020 --> 00:01:41,170 And then, using the fact that I can write my four vector 36 00:01:41,170 --> 00:01:46,510 as components contracted onto elements of a set of basis 37 00:01:46,510 --> 00:01:50,198 vectors, I can use this to define a tensor, which 38 00:01:50,198 --> 00:01:52,240 I will get to the mathematical definition of more 39 00:01:52,240 --> 00:01:54,080 precisely in just a moment. 40 00:01:54,080 --> 00:01:56,080 The dot product of any two basis vectors, 41 00:01:56,080 --> 00:02:00,360 I will call that the tensor component, eta alpha beta. 42 00:02:00,360 --> 00:02:04,240 OK, and so this is the metric tensor of special relativity, 43 00:02:04,240 --> 00:02:06,820 at least in rectilinear coordinates. 44 00:02:06,820 --> 00:02:09,970 Rectilinear basically just means Cartesian but throwing 45 00:02:09,970 --> 00:02:11,730 time in there as well. 46 00:02:11,730 --> 00:02:14,673 OK, when we start talking about special relativity 47 00:02:14,673 --> 00:02:16,090 and curvilinear coordinates, it'll 48 00:02:16,090 --> 00:02:17,630 get a little bit more complicated than that, 49 00:02:17,630 --> 00:02:19,580 and I not use a symbol eta in that case. 50 00:02:19,580 --> 00:02:25,390 I am going to reserve eta for this particular form 51 00:02:25,390 --> 00:02:27,490 of the metric in this coordinate system. 52 00:02:27,490 --> 00:02:30,640 And of course, when you actually write out these components, 53 00:02:30,640 --> 00:02:33,235 a very compact way of writing this is it 54 00:02:33,235 --> 00:02:39,130 is just the matrix that has the elements minus one, one, one, 55 00:02:39,130 --> 00:02:41,940 one down the diagonal and zeros everywhere else. 56 00:02:41,940 --> 00:02:44,440 So this is a fairly common way of writing a diagonal matrix. 57 00:02:44,440 --> 00:02:46,773 This just takes into account the fact that there's zeros 58 00:02:46,773 --> 00:02:48,730 everywhere else. 59 00:02:48,730 --> 00:02:51,310 So I call this the metric tensor, 60 00:02:51,310 --> 00:02:53,700 which kind of begs the question, what's a tensor? 61 00:02:53,700 --> 00:02:55,750 So this is where we concluded things last time. 62 00:02:55,750 --> 00:02:59,108 I'm going to generally define a tensor of type 0n-- 63 00:02:59,108 --> 00:03:01,150 we're going to change that zero to something else 64 00:03:01,150 --> 00:03:02,710 by the end of today's lecture. 65 00:03:02,710 --> 00:03:04,600 You'll be able to see where I'm going probably from a mile 66 00:03:04,600 --> 00:03:06,730 away, but let's just leave it like this for now. 67 00:03:06,730 --> 00:03:09,820 So a tensor of type 0n is-- 68 00:03:09,820 --> 00:03:12,010 you can think of it as a function or a mapping 69 00:03:12,010 --> 00:03:15,730 of n vectors into Lorentz invariant scalars, which 70 00:03:15,730 --> 00:03:19,360 is linear in its n arguments. 71 00:03:19,360 --> 00:03:24,190 So the inner product clearly falls into this category. 72 00:03:24,190 --> 00:03:35,050 If I think of a dot b, let's say this inner product is 73 00:03:35,050 --> 00:03:38,007 some number, lowercase a. 74 00:03:40,630 --> 00:03:42,527 A will be a Lorentz invariant. 75 00:03:42,527 --> 00:03:44,110 I forgot to state this, but the reason 76 00:03:44,110 --> 00:03:45,770 we define the inner product in this way 77 00:03:45,770 --> 00:03:49,780 is that we are motivated by the invariant interval in space 78 00:03:49,780 --> 00:03:50,950 and time between two events. 79 00:03:50,950 --> 00:03:54,340 We wanted to find an inner product that duplicates 80 00:03:54,340 --> 00:03:56,350 its mathematical structure. 81 00:03:56,350 --> 00:04:05,050 If I take one of these vectors, multiply it by some scalar, 82 00:04:05,050 --> 00:04:06,730 linearity is going to hold. 83 00:04:06,730 --> 00:04:09,970 This will just be eta alpha beta. 84 00:04:12,577 --> 00:04:13,660 This is terrible notation. 85 00:04:13,660 --> 00:04:15,868 I'm using the [INAUDIBLE] symbol alpha for both a pre 86 00:04:15,868 --> 00:04:18,310 factor and an index. 87 00:04:18,310 --> 00:04:21,430 Minus five points to me. 88 00:04:21,430 --> 00:04:23,320 Let's call this gamma. 89 00:04:29,773 --> 00:04:31,190 You can quickly convince yourself. 90 00:04:31,190 --> 00:04:33,648 That just comes out, and you get a factor of gamma on this. 91 00:04:33,648 --> 00:04:36,260 You can do the same thing on the second slot. 92 00:04:36,260 --> 00:04:44,705 If I take the dot product of a with the sum of two vectors-- 93 00:05:04,230 --> 00:05:05,008 OK, et cetera. 94 00:05:05,008 --> 00:05:05,800 You can keep going. 95 00:05:05,800 --> 00:05:07,717 All the rules for linearity are going to hold. 96 00:05:07,717 --> 00:05:09,470 I'm not going to step through them all. 97 00:05:09,470 --> 00:05:10,720 You can see where they all go. 98 00:05:10,720 --> 00:05:13,608 So whenever I'm going to define a tensor, in my head, 99 00:05:13,608 --> 00:05:15,400 I'm imagining it's got properties like this 100 00:05:15,400 --> 00:05:17,480 that come along for the ride. 101 00:05:17,480 --> 00:05:19,180 Now, especially when you see this 102 00:05:19,180 --> 00:05:22,060 defined in certain textbooks, MTW 103 00:05:22,060 --> 00:05:23,880 is particularly fond of doing this. 104 00:05:23,880 --> 00:05:25,672 So we come back to this idea that it's sort 105 00:05:25,672 --> 00:05:27,130 of a function or a mapping. 106 00:05:27,130 --> 00:05:33,640 You can almost abstractly define the tensor 107 00:05:33,640 --> 00:05:37,960 as a mathematical machine that's got two slots in it. 108 00:05:53,930 --> 00:05:56,270 So several of the recommended textbooks 109 00:05:56,270 --> 00:05:58,260 will write down equations. 110 00:05:58,260 --> 00:06:00,210 I'm going to put two lines over the symbol 111 00:06:00,210 --> 00:06:02,710 to sort of-- if you actually read this, in for instance, MTW 112 00:06:02,710 --> 00:06:04,085 or Caroll or something like that, 113 00:06:04,085 --> 00:06:06,060 this will be written as a boldface symbol. 114 00:06:06,060 --> 00:06:07,477 It's hard to do on the blackboard, 115 00:06:07,477 --> 00:06:09,830 so I'm just going to write double bars over it. 116 00:06:09,830 --> 00:06:15,340 So if I imagine this with my slots filled with these, 117 00:06:15,340 --> 00:06:16,990 it's got two slots associated with it. 118 00:06:16,990 --> 00:06:19,270 I fill it with those two vectors. 119 00:06:19,270 --> 00:06:24,160 That is equivalent to saying a dot b, 120 00:06:24,160 --> 00:06:29,357 which is equivalent to it in this component notation, 121 00:06:29,357 --> 00:06:30,190 something like this. 122 00:06:33,450 --> 00:06:38,910 So we have repeatedly in the little bit 123 00:06:38,910 --> 00:06:40,160 of time we've spent together-- 124 00:06:40,160 --> 00:06:42,535 I should say I have repeatedly, in the little bit of time 125 00:06:42,535 --> 00:06:47,120 we've spent together, emphasized the distinction between frame 126 00:06:47,120 --> 00:06:49,070 independent geometric objects, things 127 00:06:49,070 --> 00:06:51,200 that reside in the manifold and have 128 00:06:51,200 --> 00:06:55,130 kind of an integral, physical, geometric sensibility 129 00:06:55,130 --> 00:06:59,210 of their own and their representations. 130 00:06:59,210 --> 00:07:01,460 I have emphasized quite strongly that you 131 00:07:01,460 --> 00:07:04,550 should think of the vectors a and b 132 00:07:04,550 --> 00:07:06,060 as being geometric objects. 133 00:07:06,060 --> 00:07:08,420 This is some thing that is pointing in space time. 134 00:07:08,420 --> 00:07:10,160 We all agree that this points, if that's 135 00:07:10,160 --> 00:07:14,560 a displacement vector, it points from event one to event two. 136 00:07:14,560 --> 00:07:19,130 OK, everyone agrees on that geometric reality of this. 137 00:07:19,130 --> 00:07:20,900 Different observers may represent 138 00:07:20,900 --> 00:07:22,495 it using different components. 139 00:07:22,495 --> 00:07:23,870 That's just because they're using 140 00:07:23,870 --> 00:07:25,860 different coordinate systems. 141 00:07:25,860 --> 00:07:28,880 So when I write down something like this-- 142 00:07:28,880 --> 00:07:30,590 so let's go back to where I wrote before. 143 00:07:30,590 --> 00:07:33,690 This is going to turn into some frame independent Lorentz 144 00:07:33,690 --> 00:07:34,550 scalar a. 145 00:07:38,460 --> 00:07:41,730 So this is what I like to call a frame independent object. 146 00:07:49,340 --> 00:07:50,980 Frame or Lorentz independent geometric 147 00:07:50,980 --> 00:07:52,585 object, as is this scalar. 148 00:07:55,180 --> 00:07:58,090 And so therefore, the tensor must be a frame 149 00:07:58,090 --> 00:08:00,070 independent geometric object as well. 150 00:08:54,087 --> 00:08:55,920 That's a lot of words around the blackboard, 151 00:08:55,920 --> 00:08:57,780 but I really want to nail that point home. 152 00:08:57,780 --> 00:09:00,840 So tensors just like vectors. 153 00:09:00,840 --> 00:09:03,240 Think of them as geometric objects 154 00:09:03,240 --> 00:09:06,990 that have an intrinsic geometric meaning associated with them 155 00:09:06,990 --> 00:09:08,850 that lives in spacetime. 156 00:09:08,850 --> 00:09:11,100 We will talk about certain examples of them. 157 00:09:11,100 --> 00:09:13,230 OK, you guys all have some intuition about vectors, 158 00:09:13,230 --> 00:09:14,640 because you've been doing vectors 159 00:09:14,640 --> 00:09:16,980 ever since you took your first kindergarten physics course, 160 00:09:16,980 --> 00:09:18,570 and so you know that there's some kind of an object that 161 00:09:18,570 --> 00:09:20,230 points in a certain direction. 162 00:09:20,230 --> 00:09:23,200 Tensors are a little bit more challenging in many cases 163 00:09:23,200 --> 00:09:24,960 to develop an intuition for. 164 00:09:24,960 --> 00:09:26,970 Some of them really do have a fairly simple 165 00:09:26,970 --> 00:09:28,050 geometric interpretation. 166 00:09:28,050 --> 00:09:29,600 You can kind of think of them as-- 167 00:09:29,600 --> 00:09:31,933 for instance, we're going to introduce one a little bit, 168 00:09:31,933 --> 00:09:34,140 which describes the flow of energy and momentum 169 00:09:34,140 --> 00:09:35,340 in space time. 170 00:09:35,340 --> 00:09:37,890 And so it'll have two indices associated with it, 171 00:09:37,890 --> 00:09:40,110 and those indices tell me about what 172 00:09:40,110 --> 00:09:42,360 component of energy or momentum is flowing 173 00:09:42,360 --> 00:09:43,500 in a particular direction. 174 00:09:43,500 --> 00:09:45,090 Really easy to interpret that. 175 00:09:45,090 --> 00:09:47,520 Some of the others, not so much. 176 00:09:47,520 --> 00:09:50,460 Nonetheless, they do have this geometric meaning 177 00:09:50,460 --> 00:09:53,780 underneath the hood. 178 00:09:53,780 --> 00:09:56,160 And that's bound up in the fact that if I 179 00:09:56,160 --> 00:09:59,910 put frame independent geometric objects into all the slots, 180 00:09:59,910 --> 00:10:02,705 I get a Lorentz invariant number out of it. 181 00:10:02,705 --> 00:10:05,910 It's the only way that that can sort of work. 182 00:10:05,910 --> 00:10:07,620 But one reason why I'm going through this 183 00:10:07,620 --> 00:10:11,100 is that just like with vectors, different observers, 184 00:10:11,100 --> 00:10:13,950 different frames will get different components 185 00:10:13,950 --> 00:10:14,790 in general. 186 00:10:14,790 --> 00:10:19,980 So there will be different representations according 187 00:10:19,980 --> 00:10:21,455 to different observers. 188 00:10:21,455 --> 00:10:23,080 I'm going to write down the same thing. 189 00:10:32,430 --> 00:10:33,490 Representations. 190 00:10:52,410 --> 00:10:55,700 So if you want to get a particular observer's 191 00:10:55,700 --> 00:10:57,890 components out of the tensor, there's 192 00:10:57,890 --> 00:10:59,720 actually a very simple recipe for this. 193 00:11:23,970 --> 00:11:29,250 All you do is, you take your tensor, and into its slots, 194 00:11:29,250 --> 00:11:32,910 you plug in the basis vectors that that observer uses. 195 00:11:49,880 --> 00:11:53,300 So if I want to get the components-- 196 00:11:53,300 --> 00:11:55,570 and this, unfortunately, is a fairly stupid example, 197 00:11:55,570 --> 00:11:57,862 but it's the only one we've got at the moment, so let's 198 00:11:57,862 --> 00:12:00,060 just work with it. 199 00:12:00,060 --> 00:12:07,770 If I take the tensor eta, the special relativity metric, 200 00:12:07,770 --> 00:12:10,560 I plug in some observer's basis factors 201 00:12:10,560 --> 00:12:16,590 into this thing, this by definition is eta alpha beta. 202 00:12:16,590 --> 00:12:19,515 Suppose I have a different observer who comes along, 203 00:12:19,515 --> 00:12:21,390 someone who-- we're doing special relativity, 204 00:12:21,390 --> 00:12:22,950 so someone who's dashing through the room at three 205 00:12:22,950 --> 00:12:24,690 quarters of the speed of light. 206 00:12:24,690 --> 00:12:29,490 I want to know what their components would be. 207 00:12:29,490 --> 00:12:34,600 Well, what I do is, I just plug into the slots-- 208 00:12:34,600 --> 00:12:37,020 let's put bars on the complements 209 00:12:37,020 --> 00:12:38,430 to denote this other observer. 210 00:12:46,590 --> 00:12:50,850 Do this operation for this other observers set of basis vectors, 211 00:12:50,850 --> 00:12:53,520 and you will get the components that they will measure. 212 00:12:53,520 --> 00:12:58,640 Now, one of the reasons why I'm going through this 213 00:12:58,640 --> 00:13:00,440 is that last time, we talked about how 214 00:13:00,440 --> 00:13:03,590 to transform basis vectors between different reference 215 00:13:03,590 --> 00:13:04,460 frames. 216 00:13:04,460 --> 00:13:07,850 We know that these guys are just related to one another 217 00:13:07,850 --> 00:13:10,045 by a Lorentz transformation matrix. 218 00:13:10,045 --> 00:13:11,670 So let's just take this a step further. 219 00:13:20,200 --> 00:13:23,530 So this is telling me that the components 220 00:13:23,530 --> 00:13:28,100 of the metric in this barred frame, 221 00:13:28,100 --> 00:13:30,600 they're going to be what I get when I put it into the slots. 222 00:13:45,330 --> 00:13:48,060 Those are the basis vectors in the barred frame using 223 00:13:48,060 --> 00:13:49,560 the Lorentz transformation matrix 224 00:13:49,560 --> 00:13:51,930 to go from the unbarred frame to the barred frame. 225 00:13:55,470 --> 00:13:58,590 Now, remember again-- this is one of those places where 226 00:13:58,590 --> 00:14:02,100 if you're sort of just becoming comfortable with the index 227 00:14:02,100 --> 00:14:04,740 notation, you're temptation at this stage is always to go, 228 00:14:04,740 --> 00:14:06,090 these are matrices. 229 00:14:06,090 --> 00:14:09,030 I should start doing matrix multiplication. 230 00:14:09,030 --> 00:14:14,280 If you set that urge within you aside, you go, no, no, no, no. 231 00:14:14,280 --> 00:14:16,740 Those are just a set of 16 numbers. 232 00:14:16,740 --> 00:14:18,990 For any particular set of complements, 233 00:14:18,990 --> 00:14:20,910 I can just pull them out. 234 00:14:20,910 --> 00:14:24,390 So because of the linearity of all these slots 235 00:14:24,390 --> 00:14:40,380 with this thing, this just becomes those two Lorentz 236 00:14:40,380 --> 00:14:45,390 transformation matrices acting on the abstract metric tensor 237 00:14:45,390 --> 00:14:50,070 with the unbarred basis vectors in the slots. 238 00:14:50,070 --> 00:14:51,562 And this we already know. 239 00:14:51,562 --> 00:14:53,520 This is, by definition, this is just eta mu nu. 240 00:15:12,850 --> 00:15:15,490 Repeat this exercise for any tensor 241 00:15:15,490 --> 00:15:19,120 you care to write down, any 0n tensor you care to write down. 242 00:15:19,120 --> 00:15:21,760 Go through all this manipulation, 243 00:15:21,760 --> 00:15:24,430 and you will always find that there's a very simple algorithm 244 00:15:24,430 --> 00:15:27,840 forgetting the components in, let's 245 00:15:27,840 --> 00:15:29,470 call it the barred observers frame, 246 00:15:29,470 --> 00:15:31,773 as converted from the unbarred observers frame. 247 00:15:31,773 --> 00:15:33,940 Essentially, you just hit it with a bunch of Lorentz 248 00:15:33,940 --> 00:15:37,420 transform matrices, and as an old professor of mine 249 00:15:37,420 --> 00:15:39,640 liked to say at this point, line up the indices. 250 00:15:39,640 --> 00:15:40,660 That's really all we're doing, is 251 00:15:40,660 --> 00:15:42,118 we're just going to line up the mus 252 00:15:42,118 --> 00:15:43,480 to convert them to alpha bars. 253 00:15:43,480 --> 00:15:46,150 Alpha bar here, mu there, put this guy here. 254 00:15:46,150 --> 00:15:49,540 I want to convert my new into a beta bar. 255 00:15:49,540 --> 00:15:50,670 I put my matrix there. 256 00:15:50,670 --> 00:15:51,170 Boom. 257 00:15:51,170 --> 00:15:53,820 Just line up the indices, and we're done. 258 00:15:53,820 --> 00:15:57,420 Now, this is, as I kind of emphasized, 259 00:15:57,420 --> 00:16:06,940 a fairly stupid example, because if you 260 00:16:06,940 --> 00:16:11,110 take the diagonal of minus one, one, one, one 261 00:16:11,110 --> 00:16:13,417 and you apply the most God awful immense Lorentz 262 00:16:13,417 --> 00:16:15,250 transformation you care to write down to it, 263 00:16:15,250 --> 00:16:17,410 you do all the matrix manipulation 264 00:16:17,410 --> 00:16:22,090 and you line it all up, what you'll end up finding 265 00:16:22,090 --> 00:16:28,510 is that this is the diagonal of minus one, one, one, one 266 00:16:28,510 --> 00:16:29,500 in all frames. 267 00:16:33,045 --> 00:16:35,170 That's actually one of the defining characteristics 268 00:16:35,170 --> 00:16:36,920 of the metric of special relativity. 269 00:16:36,920 --> 00:16:38,680 If you're working curvilinear coordinates, 270 00:16:38,680 --> 00:16:44,110 the metric is always minus one, one, one, one to all observers. 271 00:16:44,110 --> 00:16:48,220 So the recipe holds in general. 272 00:16:48,220 --> 00:16:50,530 This will hold whenever we are studying tensors 273 00:16:50,530 --> 00:16:51,922 from now and from henceforth. 274 00:16:51,922 --> 00:16:54,130 Just so happens that this first example we were given 275 00:16:54,130 --> 00:16:55,047 is kind of a dumb one. 276 00:16:59,850 --> 00:17:05,220 Nonetheless, learn the lesson and overlook the example, 277 00:17:05,220 --> 00:17:08,109 and wisdom shall be yours. 278 00:17:08,109 --> 00:17:10,050 I want to spend a few moments talking 279 00:17:10,050 --> 00:17:19,650 about a particular subset of tensors, of the 0n tensors, 280 00:17:19,650 --> 00:17:21,260 where n equals one. 281 00:17:26,560 --> 00:17:28,660 This is a subset of tensions in general 282 00:17:28,660 --> 00:17:31,780 that is known in many textbooks as one forms. 283 00:17:40,740 --> 00:17:42,990 For reasons that I will elaborate on in probably 284 00:17:42,990 --> 00:17:47,295 10 or so minutes, these are also sometimes called dual vectors. 285 00:17:53,660 --> 00:17:57,200 And just if that's sort of making some neurons light up 286 00:17:57,200 --> 00:18:00,080 in your head, set that aside for a moment. 287 00:18:00,080 --> 00:18:02,510 I want to carefully go through them 288 00:18:02,510 --> 00:18:05,660 before I indicate the manner in which there is a duality that 289 00:18:05,660 --> 00:18:07,890 is being applied here. 290 00:18:07,890 --> 00:18:12,810 So if we go back to the definition of a tensor, 291 00:18:12,810 --> 00:18:22,180 this tells us that a one form is a mapping 292 00:18:22,180 --> 00:18:31,760 from a single vector to the Lorentz invariant scalar. 293 00:18:34,940 --> 00:18:37,693 So using some notation that I will probably only 294 00:18:37,693 --> 00:18:39,110 use in this lecture, because we're 295 00:18:39,110 --> 00:18:41,810 going to move past this notation pretty soon, 296 00:18:41,810 --> 00:18:43,350 let's say a one form-- 297 00:18:43,350 --> 00:18:46,440 I'm going to denote this with an over tilde on it. 298 00:18:46,440 --> 00:18:51,140 So let's say p is a one form. 299 00:18:51,140 --> 00:18:54,260 It will have in this sort of abstract notation 300 00:18:54,260 --> 00:18:58,840 a single slot, so I put the vector a into it. 301 00:18:58,840 --> 00:19:03,280 And this then gives me some scalar out. 302 00:19:30,490 --> 00:19:33,130 So in my notes, I go through some stuff indicating 303 00:19:33,130 --> 00:19:35,090 that this guy is-- it's a linear operation, 304 00:19:35,090 --> 00:19:37,090 but that's obvious, because it just inherits all 305 00:19:37,090 --> 00:19:38,560 the properties from tensors, so I'm not 306 00:19:38,560 --> 00:19:39,280 going to go through that. 307 00:19:39,280 --> 00:19:41,305 If you want to double check some of the details, 308 00:19:41,305 --> 00:19:44,140 they're in the notes that have been posted. 309 00:19:44,140 --> 00:19:46,570 Just like with the tensors, I extract 310 00:19:46,570 --> 00:19:58,500 components from this thing by putting my basis vectors 311 00:19:58,500 --> 00:19:59,000 inside. 312 00:20:05,430 --> 00:20:12,720 So if I take my one form p and I put it in my alpha basis 313 00:20:12,720 --> 00:20:18,150 vector, this gives me the alpha component of the one form. 314 00:20:18,150 --> 00:20:20,980 Notice, it's in the downstairs position. 315 00:20:20,980 --> 00:20:28,610 So one of the reasons why I want to go through this step 316 00:20:28,610 --> 00:20:32,370 is it gives me a way to think about what's 317 00:20:32,370 --> 00:20:34,870 going on at this scalar that I wrote down on the top board 318 00:20:34,870 --> 00:20:35,370 here. 319 00:20:35,370 --> 00:20:37,203 So what is this scalar that I get by putting 320 00:20:37,203 --> 00:20:38,370 a vector into my one form? 321 00:20:41,350 --> 00:20:44,160 So I take this, put my vector in here. 322 00:20:47,300 --> 00:20:52,440 I use the fact that I can write my vector 323 00:20:52,440 --> 00:20:56,910 using its components and the basis vector. 324 00:20:56,910 --> 00:20:58,860 I use linearity to pull this guy out. 325 00:21:19,920 --> 00:21:22,240 So the scalar that I get by doing this 326 00:21:22,240 --> 00:21:27,480 is just the scalar that one gets by contracting the upstairs 327 00:21:27,480 --> 00:21:29,850 components that I use to set up my vector 328 00:21:29,850 --> 00:21:33,760 with the downstairs components I use to set up my one form. 329 00:21:33,760 --> 00:21:39,190 This is an operation that is called contraction, for reasons 330 00:21:39,190 --> 00:21:41,320 that I hope are fairly obvious. 331 00:21:41,320 --> 00:21:45,250 So let me define a few other characteristics of this thing, 332 00:21:45,250 --> 00:21:48,730 and in just a few moments, we'll see what this is good for. 333 00:21:59,310 --> 00:22:03,800 OK, so one of things I'm going to want to do 334 00:22:03,800 --> 00:22:05,722 is change the representation of those things. 335 00:22:05,722 --> 00:22:08,180 So I'm going to want to know how these components transform 336 00:22:08,180 --> 00:22:10,283 between different frames of reference. 337 00:22:10,283 --> 00:22:11,450 But we've already done that. 338 00:22:11,450 --> 00:22:14,390 We did this using tensors, and this is just a tensor. 339 00:22:14,390 --> 00:22:16,700 So if I change reference frames, if I 340 00:22:16,700 --> 00:22:19,320 want to know what the components are according 341 00:22:19,320 --> 00:22:22,100 to some barred observer, I will step 342 00:22:22,100 --> 00:22:25,010 through the algebra in my notes, but I 343 00:22:25,010 --> 00:22:28,670 think you know where I'm going to go with this. 344 00:22:28,670 --> 00:22:32,000 You just take the components in the unbarred frame, 345 00:22:32,000 --> 00:22:35,390 line it up, contract it with the correct setup of my Lorentz 346 00:22:35,390 --> 00:22:36,650 transformation matrix, boom. 347 00:22:36,650 --> 00:22:40,290 Line at the indices, and we've got it there. 348 00:22:40,290 --> 00:22:44,180 So the last thing which I want to do with this before I talk 349 00:22:44,180 --> 00:22:49,260 a little bit about what this is really good for is say, 350 00:22:49,260 --> 00:22:52,460 you know, I've got these basis vectors that 351 00:22:52,460 --> 00:22:57,323 allow me to relate the components of my vectors 352 00:22:57,323 --> 00:22:59,240 to the geometric object in a way where I don't 353 00:22:59,240 --> 00:23:00,407 use a represented bi-symbol. 354 00:23:00,407 --> 00:23:03,320 I actually have an honest to God equal sign. 355 00:23:03,320 --> 00:23:06,635 Can I define a similar set of basis one forms? 356 00:23:13,760 --> 00:23:28,530 What I want to do is define a family of geometric objects, 357 00:23:28,530 --> 00:23:42,070 and I will denote them with an omega and a tilde such 358 00:23:42,070 --> 00:23:49,390 that any one form can be written as its components attached 359 00:23:49,390 --> 00:23:52,970 to these little basis vectors. 360 00:23:52,970 --> 00:23:54,920 Well, the way I'm going to do this 361 00:23:54,920 --> 00:23:58,310 is, I'm going to exploit the fact that I already 362 00:23:58,310 --> 00:24:01,250 know what basis vectors are. 363 00:24:09,260 --> 00:24:12,800 So I know, for instance, that if I take my one form 364 00:24:12,800 --> 00:24:14,885 and I plug in a basis factor, I get this. 365 00:24:18,090 --> 00:24:20,820 And so what I want to do is combine this thing 366 00:24:20,820 --> 00:24:33,020 which I would like to do with the defining 367 00:24:33,020 --> 00:24:34,870 operation of contractions. 368 00:24:41,870 --> 00:24:46,610 So I know p alpha a alpha is what 369 00:24:46,610 --> 00:24:50,900 I get when I've got my one form and I plug into its slot 370 00:24:50,900 --> 00:24:51,811 the vector a. 371 00:25:14,900 --> 00:25:26,580 OK, so let's insist that when I do this, 372 00:25:26,580 --> 00:25:34,125 I can write this as p beta omega beta. 373 00:25:34,125 --> 00:25:35,750 Now remember, these are the components. 374 00:25:35,750 --> 00:25:37,550 This is the actual basis one form. 375 00:25:37,550 --> 00:25:42,450 So I'm going to stick into my basis one form 376 00:25:42,450 --> 00:25:43,650 this form of the vector. 377 00:25:58,930 --> 00:26:07,000 I can then use the linearity of the tensor nature 378 00:26:07,000 --> 00:26:09,040 to pull out that component of a. 379 00:26:12,410 --> 00:26:14,920 So what this tells me is this is exactly what I want, 380 00:26:14,920 --> 00:26:19,460 provided whatever this geometric object is, 381 00:26:19,460 --> 00:26:33,420 it obeys the rule that when I plug basis vectors into it, 382 00:26:33,420 --> 00:26:34,890 I get the Kronecker delta back. 383 00:26:37,590 --> 00:26:39,930 Now, this may all seem really, really trivial right now. 384 00:26:39,930 --> 00:26:41,310 And indeed, if you think about this, 385 00:26:41,310 --> 00:26:43,435 in terms of just running down mathematical symbols, 386 00:26:43,435 --> 00:26:45,042 this is fairly trivial. 387 00:26:45,042 --> 00:26:46,500 One thing which I want to emphasize 388 00:26:46,500 --> 00:26:48,480 is that if you go through and say, well, 389 00:26:48,480 --> 00:26:50,550 if I'm working in a basis where this 390 00:26:50,550 --> 00:26:53,160 has a time-like component that I'll say is one, 391 00:26:53,160 --> 00:26:56,550 the time-like direction is zero everywhere else. 392 00:26:56,550 --> 00:26:58,480 Remember to set the points in the x direction, 393 00:26:58,480 --> 00:27:02,250 so it's zero, one along x, zero everywhere else. 394 00:27:02,250 --> 00:27:03,720 This then leads to-- 395 00:27:03,720 --> 00:27:13,880 so as an example, a set of basis objects 396 00:27:13,880 --> 00:27:26,590 that a particular observer would write just like so. 397 00:27:26,590 --> 00:27:29,710 I won't write out the two and three components. 398 00:27:29,710 --> 00:27:33,550 And again, you look at this and you think to yourself, dude, 399 00:27:33,550 --> 00:27:35,310 you're just repeating basis vectors. 400 00:27:35,310 --> 00:27:37,332 What's the big deal here? 401 00:27:37,332 --> 00:27:39,790 Now, I'm going to explain the fact that these are sometimes 402 00:27:39,790 --> 00:27:41,373 called dual vectors. 403 00:27:41,373 --> 00:27:43,540 So if we want to think about this in a language that 404 00:27:43,540 --> 00:27:46,390 is reminiscent of linear algebra, 405 00:27:46,390 --> 00:27:51,160 if you think of the basis vectors as column vectors, 406 00:27:51,160 --> 00:27:54,940 then my basis one forms are essentially row vectors. 407 00:28:03,630 --> 00:28:07,520 So these look a lot like my basis vectors. 408 00:28:18,070 --> 00:28:19,800 They enter in a dual way. 409 00:28:36,150 --> 00:28:39,960 And so they're going to play an important role 410 00:28:39,960 --> 00:28:41,460 in helping us to-- 411 00:28:41,460 --> 00:28:43,440 whenever I contract two objects together 412 00:28:43,440 --> 00:28:46,530 to make some kind of a Lorentz invariant scalar, 413 00:28:46,530 --> 00:28:48,382 I'm going to want to only combine objects 414 00:28:48,382 --> 00:28:49,590 that have a dual nature like. 415 00:28:49,590 --> 00:28:52,048 That's the only way I can get something sensible out of it. 416 00:28:52,048 --> 00:28:53,340 So let me give you an example. 417 00:28:53,340 --> 00:29:09,960 Mathematically, this is an equation that I can write down. 418 00:29:09,960 --> 00:29:10,500 No question. 419 00:29:10,500 --> 00:29:11,640 If I'm in a particular frame, I've 420 00:29:11,640 --> 00:29:13,170 got the components of vector a, I've 421 00:29:13,170 --> 00:29:15,000 got the complements of vector b, I 422 00:29:15,000 --> 00:29:17,400 can multiply their complements together, sum them, 423 00:29:17,400 --> 00:29:19,650 and square them. 424 00:29:19,650 --> 00:29:23,970 So this is mathematically well-defined but plays no role 425 00:29:23,970 --> 00:29:33,340 in the physics we are going to talk about this term, 426 00:29:33,340 --> 00:29:36,490 because this is not related to the underlying invariant 427 00:29:36,490 --> 00:29:39,910 structure of the manifold that we are working with. 428 00:29:39,910 --> 00:29:42,970 So remember I talked about how a manifold is essentially 429 00:29:42,970 --> 00:29:46,360 a sufficiently smooth set of points endowed with a metric? 430 00:29:46,360 --> 00:29:51,160 Well, the metric is what tells us that this is mathematically 431 00:29:51,160 --> 00:29:51,660 short. 432 00:29:51,660 --> 00:29:54,850 Write it down, but it means nothing. 433 00:29:54,850 --> 00:30:08,070 By contrast, of course this has frame independent meaning. 434 00:30:08,070 --> 00:30:09,539 This is important. 435 00:30:15,770 --> 00:30:21,130 So that's the sense in which these one forms are often 436 00:30:21,130 --> 00:30:23,170 called duels actors. 437 00:30:23,170 --> 00:30:25,197 It is when they are combined with vectors, 438 00:30:25,197 --> 00:30:27,280 they are duel to it in the sense that when they're 439 00:30:27,280 --> 00:30:30,070 combined in this appropriate way, 440 00:30:30,070 --> 00:30:34,290 we find that they describe the physics, 441 00:30:34,290 --> 00:30:36,040 they capture the invariant characteristics 442 00:30:36,040 --> 00:30:38,050 of the physics that are important in the theory 443 00:30:38,050 --> 00:30:39,250 that we are describing. 444 00:30:39,250 --> 00:30:43,780 If I may give one more example that 445 00:30:43,780 --> 00:30:45,590 is from a completely different field 446 00:30:45,590 --> 00:30:48,590 but I think helps to sort of illustrate a useful analogy 447 00:30:48,590 --> 00:30:50,500 to think about these things, suppose 448 00:30:50,500 --> 00:30:53,260 you are doing quantum mechanics and I give you 449 00:30:53,260 --> 00:30:54,760 two wave functions. 450 00:30:57,910 --> 00:31:04,520 So suppose you have a wave function 451 00:31:04,520 --> 00:31:10,020 psi of x and another wave function phi of x. 452 00:31:10,020 --> 00:31:17,080 If you wanted to, you could multiply them together 453 00:31:17,080 --> 00:31:20,077 and integrate overall space. 454 00:31:20,077 --> 00:31:22,410 I don't know what you would do with that, but you could. 455 00:31:24,528 --> 00:31:26,820 On the other hand, you could take the complex conjugate 456 00:31:26,820 --> 00:31:31,040 of one of them, multiply it by the other one, 457 00:31:31,040 --> 00:31:34,952 multiply it overall space, and in the notation 458 00:31:34,952 --> 00:31:36,660 that you learn about, this is, of course, 459 00:31:36,660 --> 00:31:39,780 just the inner product of wave function psi with wave function 460 00:31:39,780 --> 00:31:41,250 phi. 461 00:31:41,250 --> 00:31:43,290 Forming the one form, using a one 462 00:31:43,290 --> 00:31:47,720 form is akin to selecting an object that has-- 463 00:31:47,720 --> 00:31:50,550 it allows us to make a mathematical construction 464 00:31:50,550 --> 00:31:52,710 similar to the quantum mechanical enterprise 465 00:31:52,710 --> 00:31:54,270 that we use here. 466 00:31:54,270 --> 00:31:56,170 And as we'll see in just a couple of minutes, 467 00:31:56,170 --> 00:31:58,440 it's actually really easy to flip back and forth 468 00:31:58,440 --> 00:32:03,760 between one forms and vectors. 469 00:32:03,760 --> 00:32:05,892 OK, so a better way to move forward 470 00:32:05,892 --> 00:32:07,600 with this is rather than talking in terms 471 00:32:07,600 --> 00:32:12,700 of these more abstract things, let me you a good example. 472 00:32:12,700 --> 00:32:19,407 So imagine I have some trajectory through spacetime. 473 00:32:36,310 --> 00:32:38,370 So let's let the t axis go up. 474 00:32:41,410 --> 00:32:42,740 There's my x and y-axes. 475 00:32:42,740 --> 00:32:44,880 You all can imagine the z-axis. 476 00:32:44,880 --> 00:32:49,790 And some observer moves through spacetime like so. 477 00:32:58,680 --> 00:33:00,660 Last lecture, we talked about a couple 478 00:33:00,660 --> 00:33:03,090 of important examples of four-vectors. 479 00:33:03,090 --> 00:33:05,730 And so one which is germane to the situation 480 00:33:05,730 --> 00:33:08,730 is the four velocity of this observer. 481 00:33:08,730 --> 00:33:11,610 That just expresses the rate of change 482 00:33:11,610 --> 00:33:17,390 of its position in spacetime per unit proper time. 483 00:33:17,390 --> 00:33:23,870 And I'll remind you, tau is time as 484 00:33:23,870 --> 00:33:26,270 measured along this observer's trajectory. 485 00:33:26,270 --> 00:33:28,490 So roughly speaking, not even roughly speaking, 486 00:33:28,490 --> 00:33:31,730 exactly speaking, tau is the time 487 00:33:31,730 --> 00:33:34,670 that is measured by the by the watch of the person moving 488 00:33:34,670 --> 00:33:36,352 along there. 489 00:33:36,352 --> 00:33:40,040 So suppose in addition to this person sort of trundling along 490 00:33:40,040 --> 00:33:48,120 through spacetime here, suppose spacetime 491 00:33:48,120 --> 00:34:00,390 is filled with some field, phi, which depends on all 492 00:34:00,390 --> 00:34:03,870 of my spacetime coordinates. 493 00:34:03,870 --> 00:34:05,340 Question I want to ask is, what is 494 00:34:05,340 --> 00:34:09,704 the rate of change of phi along this observer's trajectory? 495 00:34:34,909 --> 00:34:42,630 So if you are working in ordinary Euclidean space, 496 00:34:42,630 --> 00:34:44,380 you would basically say, ah, this is easy. 497 00:34:52,600 --> 00:34:54,139 So you're three space intuition. 498 00:34:54,139 --> 00:34:56,120 You don't have this proper time to worry about. 499 00:34:56,120 --> 00:35:06,000 So you would just say that d phi dt along this trajectory 500 00:35:06,000 --> 00:35:08,800 is just what I get when I calculate dx dt. 501 00:35:11,305 --> 00:35:12,680 And then look at the x derivative 502 00:35:12,680 --> 00:35:17,160 of my field phi plus dy dt. 503 00:35:34,615 --> 00:35:35,990 This is one of those places where 504 00:35:35,990 --> 00:35:37,657 getting the difference between a partial 505 00:35:37,657 --> 00:35:39,810 and a total derivative right is important. 506 00:35:39,810 --> 00:35:43,040 So if you see me do that again, if I don't correct it, 507 00:35:43,040 --> 00:35:43,850 yell at me. 508 00:35:43,850 --> 00:35:45,850 OK, so you get this. 509 00:35:45,850 --> 00:35:48,770 And then you say, ah, this is nothing more 510 00:35:48,770 --> 00:35:51,470 than that particle's velocity dotted 511 00:35:51,470 --> 00:35:54,430 into the gradient of the field phi. 512 00:35:54,430 --> 00:35:58,250 It's a directional derivative along the velocity 513 00:35:58,250 --> 00:36:00,110 of this trajectory. 514 00:36:00,110 --> 00:36:03,078 So generalizing this to spacetime, 515 00:36:03,078 --> 00:36:04,870 you basically have the same thing going on. 516 00:36:27,740 --> 00:36:30,160 Only now, time is a coordinate. 517 00:36:30,160 --> 00:36:32,762 So we don't treat time as the independent parameter 518 00:36:32,762 --> 00:36:35,095 that describes the ticking of clocks as I move along it. 519 00:36:35,095 --> 00:36:37,480 I use the proper time of the observer 520 00:36:37,480 --> 00:36:39,022 as my independent parameter. 521 00:36:41,620 --> 00:36:47,880 So what I will say is, the rate at which 522 00:36:47,880 --> 00:36:50,070 the field changes per unit of this guy's 523 00:36:50,070 --> 00:37:12,620 proper time, every one of these is a component of the four 524 00:37:12,620 --> 00:37:13,650 velocity. 525 00:37:23,650 --> 00:37:25,750 So now, we introduce a little bit of notation. 526 00:37:25,750 --> 00:37:47,750 So this derivative is what I get when I contract the four 527 00:37:47,750 --> 00:37:50,720 velocity against a quantity that's defined by taking 528 00:37:50,720 --> 00:37:52,325 the derivative of this field. 529 00:37:55,280 --> 00:37:58,070 We're going to be taking derivatives like this a lot, 530 00:37:58,070 --> 00:38:00,470 so a little bit of notation being introduced 531 00:38:00,470 --> 00:38:02,030 to save us some writing. 532 00:38:06,520 --> 00:38:10,300 This is the directional derivative along the trajectory 533 00:38:10,300 --> 00:38:10,990 of this body. 534 00:38:17,140 --> 00:38:19,270 Now, this is a frame independent scalar. 535 00:38:19,270 --> 00:38:22,240 This is a quality that all observers will agree on. 536 00:38:22,240 --> 00:38:24,347 This is a four velocity. 537 00:38:24,347 --> 00:38:25,680 We know this is a four velocity. 538 00:38:25,680 --> 00:38:28,300 These are the components of a four-vector, 539 00:38:28,300 --> 00:38:32,780 so these are the components of a one form. 540 00:38:32,780 --> 00:38:35,800 So the generalization of a gradient 541 00:38:35,800 --> 00:38:37,420 is an example of a one form. 542 00:39:16,390 --> 00:39:22,590 So re-using that abstract notation that I gave earlier, 543 00:39:22,590 --> 00:39:25,920 I can write-- so the way you will sometimes see this 544 00:39:25,920 --> 00:39:33,090 is, the abstract gradient one form of the field phi 545 00:39:33,090 --> 00:39:41,080 is represented by the components, the alpha phi. 546 00:39:41,080 --> 00:39:43,570 You will also in some cases-- 547 00:39:43,570 --> 00:39:47,800 and I actually have a few lines about this in my notes-- 548 00:39:47,800 --> 00:39:52,120 sometimes, people will recycle the notation of a gradient 549 00:39:52,120 --> 00:40:04,130 that you guys learn about in undergraduate E&M. 550 00:40:04,130 --> 00:40:07,820 I urge a little bit of caution with this notation, 551 00:40:07,820 --> 00:40:10,930 because we are going to use this symbol for a derivative 552 00:40:10,930 --> 00:40:13,010 to mean something a little bit different 553 00:40:13,010 --> 00:40:15,510 in just a couple of lectures. 554 00:40:15,510 --> 00:40:17,703 It turns out that the something different reduces 555 00:40:17,703 --> 00:40:19,370 to this in the special relativity limit, 556 00:40:19,370 --> 00:40:21,440 so there's no harm being done. 557 00:40:21,440 --> 00:40:23,630 But just bear in mind this particular notion 558 00:40:23,630 --> 00:40:26,300 of a derivative here is going to change its meaning 559 00:40:26,300 --> 00:40:28,400 in a little bit. 560 00:40:28,400 --> 00:40:32,060 Another little bit of notation that is sometimes used here, 561 00:40:32,060 --> 00:40:33,710 and this is one more unfortunately, 562 00:40:33,710 --> 00:40:40,950 I think I'm stuck with this notation, one often says-- 563 00:40:40,950 --> 00:40:44,710 so this idea of taking a derivative 564 00:40:44,710 --> 00:40:50,680 along a particular four velocity, it comes up a lot. 565 00:40:50,680 --> 00:40:56,150 So sometimes, what people then do 566 00:40:56,150 --> 00:40:59,810 is, they define it as the directional gradient 567 00:40:59,810 --> 00:41:02,490 along u of the field phi. 568 00:41:02,490 --> 00:41:04,430 So don't worry about this too much. 569 00:41:04,430 --> 00:41:06,050 We'll come back to us a little bit more appropriate. 570 00:41:06,050 --> 00:41:07,040 I just want you to be aware, especially 571 00:41:07,040 --> 00:41:08,873 for those of you who might be reading ahead, 572 00:41:08,873 --> 00:41:09,920 when you see this. 573 00:41:09,920 --> 00:41:12,320 So just think of this as what you get when I am taking 574 00:41:12,320 --> 00:41:13,900 a gradient along a velocity u. 575 00:41:13,900 --> 00:41:17,690 It basically refers to the gradient one form contracted 576 00:41:17,690 --> 00:41:21,620 with the four-vector u. 577 00:41:21,620 --> 00:41:32,860 So the last thing I want to do as I talk about this 578 00:41:32,860 --> 00:41:35,740 is revisit this notion of one forms 579 00:41:35,740 --> 00:41:37,870 as being dual to vectors for just a moment. 580 00:41:45,960 --> 00:41:48,590 So we've just introduced the gradient 581 00:41:48,590 --> 00:41:50,750 as our first example of a one form. 582 00:42:03,400 --> 00:42:08,740 So the notion of the gradient as a one form, this 583 00:42:08,740 --> 00:42:17,660 gives us a nice way to think about what the basis one 584 00:42:17,660 --> 00:42:18,180 forms mean. 585 00:42:24,270 --> 00:42:26,580 So when I introduced basis one forms a few moments ago, 586 00:42:26,580 --> 00:42:28,290 it was a purely mathematical definition. 587 00:42:28,290 --> 00:42:30,330 I just wanted to have objects such 588 00:42:30,330 --> 00:42:32,550 that when I popped in the basis vectors, 589 00:42:32,550 --> 00:42:34,740 I got the Kronecker delta back. 590 00:42:34,740 --> 00:42:37,622 And after belaboring the obvious, perhaps 591 00:42:37,622 --> 00:42:39,330 for a little bit too long, we essentially 592 00:42:39,330 --> 00:42:42,920 got a bunch of ones and zeros out of it. 593 00:42:42,920 --> 00:42:55,530 Now, putting in math all of those words, 594 00:42:55,530 --> 00:42:58,730 I did a lot of junk to get that. 595 00:42:58,730 --> 00:43:08,750 But we also know that if I just take 596 00:43:08,750 --> 00:43:13,580 the derivative of my coordinate with my coordinate, 597 00:43:13,580 --> 00:43:18,470 I'm going to get the Kronecker deltas. 598 00:43:21,560 --> 00:43:24,960 This and this, these are the same thing. 599 00:43:24,960 --> 00:43:29,720 So I can think of this operation here as telling me 600 00:43:29,720 --> 00:43:41,240 if I regard this as this kind of abstract form of the gradient 601 00:43:41,240 --> 00:43:47,250 applied to the coordinate itself, 602 00:43:47,250 --> 00:43:51,360 this is nothing more than my basis one form. 603 00:43:51,360 --> 00:43:54,690 So my basis one forms are kind of like gradients 604 00:43:54,690 --> 00:43:56,248 of my coordinates. 605 00:43:56,248 --> 00:43:57,790 OK, you're sitting here thinking, OK, 606 00:43:57,790 --> 00:43:59,832 what the hell does this have to do with anything? 607 00:44:11,220 --> 00:44:13,490 So when we combine-- 608 00:44:13,490 --> 00:44:17,550 set that aside for just a second-- 609 00:44:17,550 --> 00:44:20,225 and remind you of some pretty important intuition 610 00:44:20,225 --> 00:44:22,100 that you probably learned the very first time 611 00:44:22,100 --> 00:44:23,392 you learned about the gradient. 612 00:44:30,100 --> 00:44:41,540 So imagine I just draw level surfaces of some function in-- 613 00:44:41,540 --> 00:44:43,060 I'll just do two dimensional space. 614 00:44:48,490 --> 00:44:55,300 OK, so I have some function h of x and y. 615 00:44:55,300 --> 00:44:57,920 This represents, like for instance, a height field. 616 00:44:57,920 --> 00:44:59,420 If I'm looking at a topographic map, 617 00:44:59,420 --> 00:45:01,910 this might tell me about where things are high 618 00:45:01,910 --> 00:45:04,250 and where things are low. 619 00:45:04,250 --> 00:45:08,530 So h of xy, there might be level surfaces on my map. 620 00:45:08,530 --> 00:45:11,240 It'd kind of look like this. 621 00:45:11,240 --> 00:45:14,455 And there'd be another one that kind of looks like this. 622 00:45:14,455 --> 00:45:16,525 And maybe it'd have something like this, 623 00:45:16,525 --> 00:45:21,170 and then something kind of right here. 624 00:45:21,170 --> 00:45:24,940 So we know looking at this thing that the gradient is very 625 00:45:24,940 --> 00:45:27,760 low here and very high here. 626 00:45:30,280 --> 00:45:32,020 Let's put this into the language of what 627 00:45:32,020 --> 00:45:34,170 we are looking at right now. 628 00:45:34,170 --> 00:45:39,700 OK, let's let delta x be a displacement vector in the xy 629 00:45:39,700 --> 00:45:55,474 plane, and dh is going to be my one form of my [] function. 630 00:46:04,870 --> 00:46:06,850 How do I get the change in height 631 00:46:06,850 --> 00:46:10,060 as I move along my displacement vector? 632 00:46:10,060 --> 00:46:30,140 Well, take my one form, plop into its slot delta x, 633 00:46:30,140 --> 00:46:31,450 I get something like this. 634 00:46:35,950 --> 00:46:38,360 The thing which I kind of want to emphasize here 635 00:46:38,360 --> 00:46:43,700 is, we have a lot of geometric intuition about vectors. 636 00:46:43,700 --> 00:46:45,500 So if I have a delta x-- 637 00:46:45,500 --> 00:46:46,940 let' say this is my delta x. 638 00:46:46,940 --> 00:46:51,880 It lasts for about this long over here. 639 00:46:51,880 --> 00:46:56,090 I take the exact same delta x, and I apply it over here. 640 00:46:56,090 --> 00:46:58,460 I get a very different result, because that 641 00:46:58,460 --> 00:47:02,450 goes through many more contours on the left side 642 00:47:02,450 --> 00:47:05,120 than it does on the right side. 643 00:47:05,120 --> 00:47:08,158 And the thing-- this is where this duality kind of comes 644 00:47:08,158 --> 00:47:10,700 in, and I'm going to put up a couple of graphics illustrating 645 00:47:10,700 --> 00:47:11,930 this here-- 646 00:47:11,930 --> 00:47:18,670 is that you should think of the one form 647 00:47:18,670 --> 00:47:22,480 as essentially that set of level surfaces. 648 00:47:22,480 --> 00:47:23,480 It's a little confusing. 649 00:47:23,480 --> 00:47:24,230 I'm not going to-- 650 00:47:24,230 --> 00:47:26,750 I mean, I can see a couple blank looks. 651 00:47:26,750 --> 00:47:28,250 Maybe even the majority of you have 652 00:47:28,250 --> 00:47:30,290 kind of blank looks on your faces here. 653 00:47:30,290 --> 00:47:32,810 And that's fine. 654 00:47:32,810 --> 00:47:34,850 So what I want you to regard is that when 655 00:47:34,850 --> 00:47:36,410 I'm talking about basis one forms 656 00:47:36,410 --> 00:47:38,660 and one forms of functions, they have a very different 657 00:47:38,660 --> 00:47:40,790 geometric interpretation, even though you're kind of used 658 00:47:40,790 --> 00:47:42,290 to gradient as telling you something 659 00:47:42,290 --> 00:47:45,670 about the direction along which something is changing. 660 00:47:45,670 --> 00:47:47,180 Actually, you define that direction. 661 00:47:47,180 --> 00:47:48,200 The thing that you're worried about 662 00:47:48,200 --> 00:47:49,867 is sort of how close the different level 663 00:47:49,867 --> 00:47:51,840 surfaces are of things. 664 00:47:51,840 --> 00:47:57,380 OK, so coming back to this idea that my basis one form 665 00:47:57,380 --> 00:47:59,600 that I use are essentially just the gradients 666 00:47:59,600 --> 00:48:00,800 of the coordinates. 667 00:48:00,800 --> 00:48:02,300 So I'm going to put some graphics up 668 00:48:02,300 --> 00:48:04,790 on the website, which I have actually 669 00:48:04,790 --> 00:48:09,840 scanned out of the textbook by Misner, Thorne, and Wheeler. 670 00:48:09,840 --> 00:48:16,460 And what they basically show is, let's 671 00:48:16,460 --> 00:48:19,180 say this is the time direction. 672 00:48:19,180 --> 00:48:22,030 Let's say this is my x-axis. 673 00:48:22,030 --> 00:48:24,350 And this is my y-axis. 674 00:48:24,350 --> 00:48:27,080 Your intuition is that the x basis 675 00:48:27,080 --> 00:48:30,450 vector will be a little arrow pointing along x. 676 00:48:33,020 --> 00:48:35,480 Well, what your intuition should be like for the x basis 677 00:48:35,480 --> 00:48:41,990 one form is a series of sheets normal to the x-axis that 678 00:48:41,990 --> 00:48:43,750 fill all of space. 679 00:48:43,750 --> 00:48:47,910 OK, spaced one unit apart, filling 680 00:48:47,910 --> 00:48:51,120 all of space kind of like that. 681 00:48:51,120 --> 00:48:59,100 So here's one example of one of those sheets. 682 00:49:17,500 --> 00:49:21,195 And notice, the x-axis pierces every one of those things. 683 00:49:21,195 --> 00:49:22,570 That's another way in which these 684 00:49:22,570 --> 00:49:24,490 are sort of a set of dual functions 685 00:49:24,490 --> 00:49:28,810 to the vectors themselves. 686 00:49:28,810 --> 00:49:31,540 This notion-- so I'm going to turn 687 00:49:31,540 --> 00:49:33,970 to something which is perhaps a little less 688 00:49:33,970 --> 00:49:36,405 weird in about 30 seconds. 689 00:49:36,405 --> 00:49:37,780 But the one thing which I kind of 690 00:49:37,780 --> 00:49:39,430 want to emphasize with this-- and again, I'm 691 00:49:39,430 --> 00:49:40,750 going to put a couple graphics up that 692 00:49:40,750 --> 00:49:41,560 help to illustrate this. 693 00:49:41,560 --> 00:49:43,602 And there's some really nice discussions of this. 694 00:49:43,602 --> 00:49:46,780 MTW is particularly good for this discussion. 695 00:49:46,780 --> 00:49:48,910 This ends up being a really useful notion 696 00:49:48,910 --> 00:49:51,850 for capturing how we are going to compute fluxes 697 00:49:51,850 --> 00:49:53,882 through particular directions. 698 00:49:53,882 --> 00:49:55,840 Because if I want to know the flux of something 699 00:49:55,840 --> 00:49:59,110 in the x direction, well, my x basis one form 700 00:49:59,110 --> 00:50:01,270 is actually like a sheet that captures everything 701 00:50:01,270 --> 00:50:02,767 that flows in the x direction. 702 00:50:02,767 --> 00:50:04,600 So it's sort of a mathematical object that's 703 00:50:04,600 --> 00:50:06,085 designed for catching fluxes. 704 00:50:11,530 --> 00:50:15,128 If isn't quite gelling with you, that's fine. 705 00:50:15,128 --> 00:50:17,170 This is without a doubt one of the goofier things 706 00:50:17,170 --> 00:50:19,720 we're going to talk about in this first introductory period 707 00:50:19,720 --> 00:50:20,990 of stuff. 708 00:50:20,990 --> 00:50:22,480 This is one of those places where 709 00:50:22,480 --> 00:50:24,370 I think it's sort of fair to say, if you're 710 00:50:24,370 --> 00:50:26,320 not quite getting what's going on, 711 00:50:26,320 --> 00:50:28,703 shut up and calculate works well enough. 712 00:50:28,703 --> 00:50:31,120 It's kind of like you know the Feynman's mantra on quantum 713 00:50:31,120 --> 00:50:31,620 mechanics. 714 00:50:31,620 --> 00:50:34,410 Sometimes, you've just got to say, OK, whatever. 715 00:50:34,410 --> 00:50:36,208 Learn the way it goes, and it's kind 716 00:50:36,208 --> 00:50:37,750 of like playing a musical instrument. 717 00:50:37,750 --> 00:50:39,280 You sort of strum it and practice it for a while, 718 00:50:39,280 --> 00:50:41,072 and it becomes second nature after a while. 719 00:50:47,630 --> 00:50:54,130 All right, so to wrap this up, what 720 00:50:54,130 --> 00:50:58,460 I want to do for the last major thing today is, 721 00:50:58,460 --> 00:51:02,960 I hope I can kind of put a bow on our discussion of tensors. 722 00:51:02,960 --> 00:51:05,720 Let's come back to the metric as our original example 723 00:51:05,720 --> 00:51:07,110 of a tensor here. 724 00:51:07,110 --> 00:51:14,790 So when I give you the metric as an abstract tensor, 725 00:51:14,790 --> 00:51:26,710 and I imagine I have filled both slots, 726 00:51:26,710 --> 00:51:28,870 I get a Lorentz invariant number. 727 00:51:34,830 --> 00:51:38,220 A dot b is what I get when I take this tensor 728 00:51:38,220 --> 00:51:42,400 and I put a and b into it slots. 729 00:51:42,400 --> 00:51:44,680 Suppose I only fill one of its slots. 730 00:51:51,100 --> 00:51:56,080 Well, if I take this, I plug in the vector 731 00:51:56,080 --> 00:52:02,770 a but I leave the other slot blank, well, what I've got 732 00:52:02,770 --> 00:52:07,790 is a mathematical object that will take a vector 733 00:52:07,790 --> 00:52:09,690 and produce a Lorentz invariance number. 734 00:52:27,770 --> 00:52:28,520 That's a one form. 735 00:52:41,740 --> 00:52:44,240 So let's do this very carefully and abstractly for a moment. 736 00:52:50,065 --> 00:52:53,630 But at this point, we basically have almost all the pieces 737 00:52:53,630 --> 00:52:54,750 in place. 738 00:52:54,750 --> 00:52:57,620 And so I'm going to kind of tone down some of the formality 739 00:52:57,620 --> 00:52:58,680 fairly soon. 740 00:52:58,680 --> 00:53:05,430 So let's define a one form, an object that 741 00:53:05,430 --> 00:53:09,030 takes a single vector inside of it 742 00:53:09,030 --> 00:53:14,960 as what I get when I take the metric 743 00:53:14,960 --> 00:53:16,620 and put that vector in there. 744 00:53:16,620 --> 00:53:19,520 If I want to get its components out, 745 00:53:19,520 --> 00:53:22,190 well, I know the way I do that is I put the basis 746 00:53:22,190 --> 00:53:22,880 vector in there. 747 00:53:39,350 --> 00:53:43,520 This guy is symmetric, so I can flip the order of the metric. 748 00:53:43,520 --> 00:53:46,520 It's symmetric, so I can flip the order of these guys. 749 00:53:46,520 --> 00:53:55,380 And what this tells me is that my one form component of a 750 00:53:55,380 --> 00:53:58,260 is just the vector component of a hit 751 00:53:58,260 --> 00:54:01,290 by the components of the metric. 752 00:54:01,290 --> 00:54:14,490 In other words, the metric converts vectors into one forms 753 00:54:14,490 --> 00:54:19,220 by lowering the indices. 754 00:54:22,870 --> 00:54:25,580 This is an invertebrate procedure as well. 755 00:54:25,580 --> 00:54:34,960 So this metric, I can define eta with indices in the upstairs 756 00:54:34,960 --> 00:54:46,390 position by requiring that eta alpha beta contracted 757 00:54:46,390 --> 00:54:48,530 with it in the downstairs position 758 00:54:48,530 --> 00:54:53,360 gives me the identity back. 759 00:54:53,360 --> 00:54:55,700 Incidentally, when you do this, you again 760 00:54:55,700 --> 00:54:59,370 find it's got exactly the same matrix representation. 761 00:54:59,370 --> 00:55:02,300 So this thing with its indices in the upstairs position 762 00:55:02,300 --> 00:55:05,360 is just minus one, one, one, one. 763 00:55:05,360 --> 00:55:08,580 It will not always be that way, though. 764 00:55:08,580 --> 00:55:10,910 Again, this is just because special relativity 765 00:55:10,910 --> 00:55:14,030 in rectilinear coordinates is simple. 766 00:55:18,470 --> 00:55:23,445 So I will often call that the inverse metric. 767 00:55:26,953 --> 00:55:28,370 And then, you shouldn't have a lot 768 00:55:28,370 --> 00:55:33,380 of trouble convincing yourself that if I've got a one form, 769 00:55:33,380 --> 00:55:47,926 I can make a vector out of it by a contraction operation. 770 00:55:54,560 --> 00:55:58,797 That now tells me that I have about 16 gagillian-- well, 771 00:55:58,797 --> 00:56:00,630 actually, it's a countable and finite thing. 772 00:56:00,630 --> 00:56:07,130 But I have many ways that I can write down the inner product 773 00:56:07,130 --> 00:56:13,620 between two vectors. 774 00:56:13,620 --> 00:56:25,530 This guy is-- if you like, you can now 775 00:56:25,530 --> 00:56:28,890 regard a vector as being a sort of a function 776 00:56:28,890 --> 00:56:30,560 that you put one forms into. 777 00:56:40,020 --> 00:56:41,920 These are all the same. 778 00:56:51,820 --> 00:56:54,520 These are all equivalents to one another. 779 00:56:54,520 --> 00:56:56,800 And actually making a distinction 780 00:56:56,800 --> 00:56:58,690 between vector in one form and all that, 781 00:56:58,690 --> 00:57:02,980 it's just kind of gotten stupid at this point. 782 00:57:02,980 --> 00:57:05,300 So the distinction among these different objects, 783 00:57:05,300 --> 00:57:07,950 the different names, kind of doesn't matter. 784 00:57:29,498 --> 00:57:31,040 And indeed, you sort of look at this. 785 00:57:31,040 --> 00:57:32,880 Up until now, we've regarded tensors 786 00:57:32,880 --> 00:57:35,660 as being these sort of things that operate on vectors. 787 00:57:35,660 --> 00:57:37,670 OK, but why not regard vectors as things 788 00:57:37,670 --> 00:57:39,740 that operate on one forms? 789 00:57:39,740 --> 00:57:43,220 What this sort of tells you is that this whole notion 790 00:57:43,220 --> 00:57:45,980 of tensors being separate from vectors that I talked 791 00:57:45,980 --> 00:57:51,330 about before is kind of silly. 792 00:57:51,330 --> 00:57:54,080 So I'm going to revisit the definition of a tensor 793 00:57:54,080 --> 00:58:05,970 that I started the lecture off with like so. 794 00:58:05,970 --> 00:58:08,790 So a new and more complete definition. 795 00:58:15,990 --> 00:58:40,780 A tensor of type mn is a linear mapping of m1 forms and n 796 00:58:40,780 --> 00:58:47,085 vectors to the Lorentz scalars. 797 00:58:50,240 --> 00:58:53,635 In this definition-- so we've already introduced zero, 798 00:58:53,635 --> 00:58:54,980 one tensors. 799 00:58:54,980 --> 00:58:56,090 Those are one forms. 800 00:58:59,600 --> 00:59:01,400 One zero tensors are vectors. 801 00:59:11,540 --> 00:59:13,040 Furthermore, as I kind of emphasized 802 00:59:13,040 --> 00:59:15,650 when I wrote this sentence here, the distinction 803 00:59:15,650 --> 00:59:18,980 between the slots that operate on vectors and the slots that 804 00:59:18,980 --> 00:59:21,800 operate on one forms, it's nice for getting some 805 00:59:21,800 --> 00:59:23,965 of the basic foundations laid. 806 00:59:23,965 --> 00:59:25,340 This is one of those places where 807 00:59:25,340 --> 00:59:27,170 now that the scaffolding is in place-- 808 00:59:27,170 --> 00:59:29,128 we've had the scaffolding in place for a while, 809 00:59:29,128 --> 00:59:31,140 but this wall of our edifice is pretty steady, 810 00:59:31,140 --> 00:59:33,200 so we can kick the scaffolding away. 811 00:59:33,200 --> 00:59:35,099 We can sort of lose this distinction. 812 00:59:45,860 --> 00:59:47,590 The metric lets us convert the nature 813 00:59:47,590 --> 00:59:51,880 of the slots on a tensor. 814 00:59:51,880 --> 01:00:05,710 So if I have an mn tensor and I use the metric to lower, 815 01:00:05,710 --> 01:00:10,760 that means I make it an m minus one n plus one tensor. 816 01:00:10,760 --> 01:00:22,185 So an example would be, if I have a tensor 817 01:00:22,185 --> 01:00:36,573 r mu beta gamma delta and I lower that first index, 818 01:00:36,573 --> 01:00:38,615 so this went from something that operates on one, 819 01:00:38,615 --> 01:00:40,220 one form and three vectors. 820 01:00:40,220 --> 01:00:42,170 Now, it's one it operates on four-vectors. 821 01:00:47,180 --> 01:00:52,190 Likewise using the inverse metric, you can raise-- 822 01:00:58,660 --> 01:01:00,660 and just for completeness, let's write that out. 823 01:01:11,990 --> 01:01:18,640 So an example of this would be if I have some tensor su beta 824 01:01:18,640 --> 01:01:27,735 gamma, and let's say I raised that first index 825 01:01:27,735 --> 01:01:28,860 to get something like this. 826 01:01:32,700 --> 01:01:35,687 OK, so let's see. 827 01:01:35,687 --> 01:01:37,270 In my last couple of minutes-- recall, 828 01:01:37,270 --> 01:01:38,900 I do have to leave a little bit early today, because I 829 01:01:38,900 --> 01:01:40,067 need to introduce a speaker. 830 01:01:40,067 --> 01:01:44,800 But I just want to wrap up one thing kind of quickly. 831 01:01:44,800 --> 01:01:52,560 I've spent a bunch of time talking about basis objects. 832 01:02:01,670 --> 01:02:03,810 And I'm going to go through this fairly quickly. 833 01:02:03,810 --> 01:02:05,880 The notes, if you want to see a few more details, 834 01:02:05,880 --> 01:02:07,713 you're welcome to download and look at them. 835 01:02:07,713 --> 01:02:10,860 They're not really tricky or super critical. 836 01:02:10,860 --> 01:02:12,782 We know we have basis objects for vectors, 837 01:02:12,782 --> 01:02:14,990 which hopefully you have pretty good intuition about. 838 01:02:14,990 --> 01:02:18,080 We have basis objects for one forms, where your intuition is 839 01:02:18,080 --> 01:02:19,580 perhaps a little bit more befuddled, 840 01:02:19,580 --> 01:02:21,440 but it'll come with time. 841 01:02:21,440 --> 01:02:24,890 So you might think, uh, now I've got two index tenors. 842 01:02:24,890 --> 01:02:26,200 I've got three index tensors. 843 01:02:26,200 --> 01:02:28,212 There's a four index tensor on the board. 844 01:02:28,212 --> 01:02:30,170 Scott's probably going to write 17 index tensor 845 01:02:30,170 --> 01:02:31,295 on the board at some point. 846 01:02:31,295 --> 01:02:34,443 Do I need a basis object for every one of those? 847 01:02:34,443 --> 01:02:35,735 So in other words, do we need-- 848 01:02:45,552 --> 01:02:46,760 glad that's caught on video-- 849 01:02:49,340 --> 01:02:51,110 do I need to be able to say something 850 01:02:51,110 --> 01:02:56,210 like the abstract metric tensor is 851 01:02:56,210 --> 01:03:02,055 these components times some kind of a two index basis object? 852 01:03:16,420 --> 01:03:18,940 Do I need to do something like this? 853 01:03:18,940 --> 01:03:21,640 Cutting to the chase, the answer is no. 854 01:03:27,120 --> 01:03:31,097 Basis one forms and vectors are sufficient. 855 01:03:38,890 --> 01:03:41,610 So what we're going to do is, abstractly 856 01:03:41,610 --> 01:03:44,340 just imagine that if I do have a tensor, 857 01:03:44,340 --> 01:03:46,860 I kind of have an outer product. 858 01:03:46,860 --> 01:03:51,000 I have both of the basis objects attached to this thing, 859 01:03:51,000 --> 01:03:54,000 and each one is just attached to those two slots. 860 01:03:54,000 --> 01:04:06,012 OK, so in this particular case, my two index basis two-form 861 01:04:06,012 --> 01:04:07,470 that would go with this thing here, 862 01:04:07,470 --> 01:04:10,020 the thing with two indices on it, 863 01:04:10,020 --> 01:04:11,595 I'm just going to regard this as-- 864 01:04:11,595 --> 01:04:16,160 I'll abstractly write this as just an outer product 865 01:04:16,160 --> 01:04:17,340 on the basis one forms. 866 01:04:21,860 --> 01:04:27,380 If I ever need a basis object for a tensor like this, 867 01:04:27,380 --> 01:04:33,370 I will just regard this as an outer product of these two 868 01:04:33,370 --> 01:04:33,950 things. 869 01:04:33,950 --> 01:04:35,533 So what's going on with that notation? 870 01:04:35,533 --> 01:04:36,592 What does this mean? 871 01:04:36,592 --> 01:04:38,050 Don't lose too much sleep about it. 872 01:04:38,050 --> 01:04:40,600 It's basically saying that I have separate objects attached 873 01:04:40,600 --> 01:04:43,420 to all of my different indices, and they're 874 01:04:43,420 --> 01:04:46,000 kind of coming along here and giving a sense of direction 875 01:04:46,000 --> 01:04:47,060 to these things. 876 01:04:47,060 --> 01:04:49,233 So for instance, if I have some kind of a tensor-- 877 01:04:49,233 --> 01:04:51,400 I mean, a great one, which we're going to talk about 878 01:04:51,400 --> 01:04:52,300 in just a little bit. 879 01:04:56,820 --> 01:04:58,570 It's a quantity known as the stress energy 880 01:04:58,570 --> 01:05:20,340 tensor, in which I can abstractly 881 01:05:20,340 --> 01:05:23,925 think of the tensor as having-- 882 01:05:27,060 --> 01:05:28,020 not just a [INAUDIBLE]. 883 01:05:28,020 --> 01:05:28,990 It will have two components. 884 01:05:28,990 --> 01:05:30,490 And I can think of it as essentially 885 01:05:30,490 --> 01:05:35,497 pointing in two different directions at once. 886 01:05:35,497 --> 01:05:37,830 Now, we're going to talk about this in a lot more detail 887 01:05:37,830 --> 01:05:39,588 in a couple of weeks. 888 01:05:39,588 --> 01:05:41,130 Actually, not even a couple of weeks. 889 01:05:41,130 --> 01:05:42,420 A couple of lectures. 890 01:05:42,420 --> 01:05:45,140 What I'm going to teach you is that the alpha beta component 891 01:05:45,140 --> 01:05:47,250 of the stress energy tensor tells me 892 01:05:47,250 --> 01:05:52,620 about the flux of form momentum component alpha 893 01:05:52,620 --> 01:05:55,128 in the beta direction. 894 01:05:55,128 --> 01:05:56,670 And so this is basically just saying, 895 01:05:56,670 --> 01:05:59,040 when I think of it as the actual object, not 896 01:05:59,040 --> 01:06:02,010 just the representation according to some observer, 897 01:06:02,010 --> 01:06:06,530 here is the thing that gives me the direction of my four 898 01:06:06,530 --> 01:06:08,250 momentum, and here is the direction 899 01:06:08,250 --> 01:06:10,350 in which it is flowing. 900 01:06:10,350 --> 01:06:13,170 And sometimes, we will make more complicated objects. 901 01:06:13,170 --> 01:06:16,560 And so you might need to imagine-- 902 01:06:16,560 --> 01:06:20,280 here's one which I actually wrote down in my notes here-- 903 01:06:20,280 --> 01:06:21,780 there will be times when we're going 904 01:06:21,780 --> 01:06:39,750 to care about a tensor which at least abstractly, we might need 905 01:06:39,750 --> 01:06:44,175 to regard as having this whole set of sort of vomitous basis 906 01:06:44,175 --> 01:06:45,800 vectors coming along for the ride here. 907 01:06:48,370 --> 01:06:50,770 And it is actually fairly important 908 01:06:50,770 --> 01:06:54,645 to have all these things that are in place. 909 01:06:54,645 --> 01:06:56,020 Where I will just conclude things 910 01:06:56,020 --> 01:06:59,170 for today is that the place where it is particularly 911 01:06:59,170 --> 01:07:02,260 important to remember that we kind of sometimes almost just 912 01:07:02,260 --> 01:07:05,230 implicitly have these things coming along 913 01:07:05,230 --> 01:07:06,070 for the ride here-- 914 01:07:17,990 --> 01:07:21,037 it's important when we calculate derivatives. 915 01:07:27,580 --> 01:07:35,010 So I gave you guys an example of a directional derivative 916 01:07:35,010 --> 01:07:37,310 for a scalar field that filled all its spacetime. 917 01:07:37,310 --> 01:07:39,750 I imagine that there was some trajectory of an observer 918 01:07:39,750 --> 01:07:41,160 moving through this. 919 01:07:41,160 --> 01:07:46,780 Now, imagine it isn't a scalar field that fills all of, 920 01:07:46,780 --> 01:07:48,330 spacetime but it's a tensor field. 921 01:08:02,160 --> 01:08:03,862 Here's t. 922 01:08:03,862 --> 01:08:05,550 We'll say this is the y direction. 923 01:08:05,550 --> 01:08:07,440 This is the x direction. 924 01:08:07,440 --> 01:08:10,440 Here's my observer moving through all this thing. 925 01:08:10,440 --> 01:08:12,990 And again, I'm going to say that this trajectory is 926 01:08:12,990 --> 01:08:17,380 characterized by a four velocity, dxt tau, 927 01:08:17,380 --> 01:08:28,270 and I'm going to imagine that there is some tensor field that 928 01:08:28,270 --> 01:08:35,580 fills all of spacetime when I go and calculate 929 01:08:35,580 --> 01:08:39,300 the derivative of this thing, when 930 01:08:39,300 --> 01:08:41,160 we're working in special relativity 931 01:08:41,160 --> 01:08:44,410 where we are right now, these guys are going to be constant, 932 01:08:44,410 --> 01:08:45,611 so it doesn't really matter. 933 01:08:45,611 --> 01:08:47,069 But soon, we're going to generalize 934 01:08:47,069 --> 01:08:51,420 to more complicated geometries, more complicated spacetimes, 935 01:08:51,420 --> 01:08:53,310 and the basis objects will themselves 936 01:08:53,310 --> 01:08:55,620 vary as I move along the trajectory. 937 01:08:55,620 --> 01:08:59,279 And I will need to-- in order to have 938 01:08:59,279 --> 01:09:01,350 a notion of a derivative that is a properly 939 01:09:01,350 --> 01:09:02,948 formed geometric object, I'm going 940 01:09:02,948 --> 01:09:05,490 to have to worry about how the basis objects change as I move 941 01:09:05,490 --> 01:09:07,585 along this trajectory as well. 942 01:09:07,585 --> 01:09:09,210 So that tends to just make the analysis 943 01:09:09,210 --> 01:09:11,140 a little bit more complicated. 944 01:09:11,140 --> 01:09:13,545 I have a few notes about this that I will put up 945 01:09:13,545 --> 01:09:16,170 on to the web page, but I don't want to go into too much detail 946 01:09:16,170 --> 01:09:17,640 beyond that until we actually get 947 01:09:17,640 --> 01:09:19,620 into some of the details of these derivatives. 948 01:09:19,620 --> 01:09:23,430 So I'm just going to leave it at that for now. 949 01:09:23,430 --> 01:09:31,240 So yeah, I'll just say, the derivative in principle 950 01:09:31,240 --> 01:09:43,500 and quite often in practice, it will 951 01:09:43,500 --> 01:09:48,640 depend on how these guys vary in space and time. 952 01:09:58,060 --> 01:10:00,400 And let me just say, you guys kind of already know that. 953 01:10:00,400 --> 01:10:02,720 Because when you studied E&M, you 954 01:10:02,720 --> 01:10:04,540 got these somewhat complicated formulas 955 01:10:04,540 --> 01:10:07,487 for things like the divergence and curl and stuff like that. 956 01:10:07,487 --> 01:10:09,070 And those are essential, because those 957 01:10:09,070 --> 01:10:11,320 are notions of derivative where you are taking 958 01:10:11,320 --> 01:10:13,030 into account the fact that when you're in a curvilinear 959 01:10:13,030 --> 01:10:15,405 coordinate system, your basis vectors are shifting as you 960 01:10:15,405 --> 01:10:17,008 move from point to point. 961 01:10:17,008 --> 01:10:18,550 The stuff we're going to get out this 962 01:10:18,550 --> 01:10:20,140 will look a little bit different, 963 01:10:20,140 --> 01:10:22,780 and it comes to the fact, as I emphasized in my last lecture, 964 01:10:22,780 --> 01:10:24,700 that we are going to tend to use what we call 965 01:10:24,700 --> 01:10:28,220 a coordinate basis, whereas when you guys learn stuff in E&M, 966 01:10:28,220 --> 01:10:30,460 you were using what's known as an orthonormal basis. 967 01:10:30,460 --> 01:10:32,085 And it does lead to slight differences. 968 01:10:32,085 --> 01:10:33,730 There's a mapping between them. 969 01:10:33,730 --> 01:10:35,530 Not that hard to figure it out, but we 970 01:10:35,530 --> 01:10:37,870 don't need to get into those weeds just now. 971 01:10:37,870 --> 01:10:42,400 All right, so I will pick it up from there on Thursday. 972 01:10:42,400 --> 01:10:44,980 So I'll begin with a brief recap of everything we did. 973 01:10:44,980 --> 01:10:48,010 The primary thing which I really want to emphasize more 974 01:10:48,010 --> 01:10:52,300 than anything is this board plus this idea 975 01:10:52,300 --> 01:10:54,670 that the metric can be used to raise and lower 976 01:10:54,670 --> 01:10:56,440 the indices of a tensor. 977 01:10:56,440 --> 01:10:58,720 At this point, talking about vectors, 978 01:10:58,720 --> 01:11:02,860 talking about one forms, many of you in a math class 979 01:11:02,860 --> 01:11:05,170 probably learned about contravariant vector components 980 01:11:05,170 --> 01:11:07,210 and covariant vector components. 981 01:11:07,210 --> 01:11:10,450 Once you've got a metric, it's kind of like, who cares? 982 01:11:10,450 --> 01:11:12,350 You can just go from one to the other. 983 01:11:12,350 --> 01:11:15,940 And that's why I tend to, almost religiously, I 984 01:11:15,940 --> 01:11:17,780 avoid the terms covariant and contravariant, 985 01:11:17,780 --> 01:11:19,832 and I just say, upstairs and downstairs. 986 01:11:19,832 --> 01:11:21,790 Because I can flip back and forth between them, 987 01:11:21,790 --> 01:11:24,377 and there's really no physical meaning in them. 988 01:11:24,377 --> 01:11:25,960 You have to think carefully about what 989 01:11:25,960 --> 01:11:27,730 is physically measurable, and it has 990 01:11:27,730 --> 01:11:30,940 nothing to do with whether it's covariant or convariant, 991 01:11:30,940 --> 01:11:32,562 upstairs or downstairs. 992 01:11:32,562 --> 01:11:34,270 All right, I will end there, since I have 993 01:11:34,270 --> 01:11:36,390 to go and introduce someone.