WEBVTT 00:00:00.500 --> 00:00:03.374 [SQUEAKING][RUSTLING][CLICKING] 00:00:11.590 --> 00:00:13.520 SCOTT HUGHES: All right, welcome to Tuesday. 00:00:13.520 --> 00:00:16.840 So hopefully, you've all saw the brief announcement 00:00:16.840 --> 00:00:18.320 I send to the class. 00:00:18.320 --> 00:00:22.990 I have to introduce a colloquium speaker over in Astrophysics, 00:00:22.990 --> 00:00:26.167 basically at the second this class officially ends, 00:00:26.167 --> 00:00:28.000 so I will be wrapping things up a little bit 00:00:28.000 --> 00:00:32.020 early so that I can take into account the spatial separation 00:00:32.020 --> 00:00:37.000 and get back there in time to actually do the introduction. 00:00:37.000 --> 00:00:39.850 I've already posted the lecture notes of material 00:00:39.850 --> 00:00:43.253 I'm going to be covering today, and it'll probably spill-- 00:00:43.253 --> 00:00:44.920 I hope to wrap it all up today, but it's 00:00:44.920 --> 00:00:46.920 possible it'll spill a little bit into Thursday. 00:00:46.920 --> 00:00:49.582 So if you've already looked at those notes, 00:00:49.582 --> 00:00:51.790 today will essentially just be sort of my guided tour 00:00:51.790 --> 00:00:53.570 through that material. 00:00:53.570 --> 00:00:56.445 So I want to pick it up with where I left things last time. 00:00:56.445 --> 00:00:57.820 So we covered a bunch of material 00:00:57.820 --> 00:01:00.400 that, again, I kind of emphasize what we're doing right now 00:01:00.400 --> 00:01:03.700 is just laying the mathematical foundations in a very 00:01:03.700 --> 00:01:07.180 thorough, almost excessively thorough way in order 00:01:07.180 --> 00:01:10.180 that we have a very strong structure as we begin to move 00:01:10.180 --> 00:01:12.362 into more physically complicated situations 00:01:12.362 --> 00:01:14.820 in the special relativity that we're focusing on right now. 00:01:14.820 --> 00:01:17.260 So we talked about this definition 00:01:17.260 --> 00:01:20.440 of an inner product between two four-vectors, two vectors 00:01:20.440 --> 00:01:21.850 in space time. 00:01:21.850 --> 00:01:26.170 And it looks just like the inner product between two vectors 00:01:26.170 --> 00:01:28.660 that you are used to from your Euclidean three space 00:01:28.660 --> 00:01:29.290 intuition. 00:01:29.290 --> 00:01:31.720 It's just that we have an extra bit there 00:01:31.720 --> 00:01:33.220 that enters with a minus sign having 00:01:33.220 --> 00:01:36.040 to do with the time like components of those two 00:01:36.040 --> 00:01:38.020 four-vectors. 00:01:38.020 --> 00:01:41.170 And then, using the fact that I can write my four vector 00:01:41.170 --> 00:01:46.510 as components contracted onto elements of a set of basis 00:01:46.510 --> 00:01:50.198 vectors, I can use this to define a tensor, which 00:01:50.198 --> 00:01:52.240 I will get to the mathematical definition of more 00:01:52.240 --> 00:01:54.080 precisely in just a moment. 00:01:54.080 --> 00:01:56.080 The dot product of any two basis vectors, 00:01:56.080 --> 00:02:00.360 I will call that the tensor component, eta alpha beta. 00:02:00.360 --> 00:02:04.240 OK, and so this is the metric tensor of special relativity, 00:02:04.240 --> 00:02:06.820 at least in rectilinear coordinates. 00:02:06.820 --> 00:02:09.970 Rectilinear basically just means Cartesian but throwing 00:02:09.970 --> 00:02:11.730 time in there as well. 00:02:11.730 --> 00:02:14.673 OK, when we start talking about special relativity 00:02:14.673 --> 00:02:16.090 and curvilinear coordinates, it'll 00:02:16.090 --> 00:02:17.630 get a little bit more complicated than that, 00:02:17.630 --> 00:02:19.580 and I not use a symbol eta in that case. 00:02:19.580 --> 00:02:25.390 I am going to reserve eta for this particular form 00:02:25.390 --> 00:02:27.490 of the metric in this coordinate system. 00:02:27.490 --> 00:02:30.640 And of course, when you actually write out these components, 00:02:30.640 --> 00:02:33.235 a very compact way of writing this is it 00:02:33.235 --> 00:02:39.130 is just the matrix that has the elements minus one, one, one, 00:02:39.130 --> 00:02:41.940 one down the diagonal and zeros everywhere else. 00:02:41.940 --> 00:02:44.440 So this is a fairly common way of writing a diagonal matrix. 00:02:44.440 --> 00:02:46.773 This just takes into account the fact that there's zeros 00:02:46.773 --> 00:02:48.730 everywhere else. 00:02:48.730 --> 00:02:51.310 So I call this the metric tensor, 00:02:51.310 --> 00:02:53.700 which kind of begs the question, what's a tensor? 00:02:53.700 --> 00:02:55.750 So this is where we concluded things last time. 00:02:55.750 --> 00:02:59.108 I'm going to generally define a tensor of type 0n-- 00:02:59.108 --> 00:03:01.150 we're going to change that zero to something else 00:03:01.150 --> 00:03:02.710 by the end of today's lecture. 00:03:02.710 --> 00:03:04.600 You'll be able to see where I'm going probably from a mile 00:03:04.600 --> 00:03:06.730 away, but let's just leave it like this for now. 00:03:06.730 --> 00:03:09.820 So a tensor of type 0n is-- 00:03:09.820 --> 00:03:12.010 you can think of it as a function or a mapping 00:03:12.010 --> 00:03:15.730 of n vectors into Lorentz invariant scalars, which 00:03:15.730 --> 00:03:19.360 is linear in its n arguments. 00:03:19.360 --> 00:03:24.190 So the inner product clearly falls into this category. 00:03:24.190 --> 00:03:35.050 If I think of a dot b, let's say this inner product is 00:03:35.050 --> 00:03:38.007 some number, lowercase a. 00:03:40.630 --> 00:03:42.527 A will be a Lorentz invariant. 00:03:42.527 --> 00:03:44.110 I forgot to state this, but the reason 00:03:44.110 --> 00:03:45.770 we define the inner product in this way 00:03:45.770 --> 00:03:49.780 is that we are motivated by the invariant interval in space 00:03:49.780 --> 00:03:50.950 and time between two events. 00:03:50.950 --> 00:03:54.340 We wanted to find an inner product that duplicates 00:03:54.340 --> 00:03:56.350 its mathematical structure. 00:03:56.350 --> 00:04:05.050 If I take one of these vectors, multiply it by some scalar, 00:04:05.050 --> 00:04:06.730 linearity is going to hold. 00:04:06.730 --> 00:04:09.970 This will just be eta alpha beta. 00:04:12.577 --> 00:04:13.660 This is terrible notation. 00:04:13.660 --> 00:04:15.868 I'm using the [INAUDIBLE] symbol alpha for both a pre 00:04:15.868 --> 00:04:18.310 factor and an index. 00:04:18.310 --> 00:04:21.430 Minus five points to me. 00:04:21.430 --> 00:04:23.320 Let's call this gamma. 00:04:29.773 --> 00:04:31.190 You can quickly convince yourself. 00:04:31.190 --> 00:04:33.648 That just comes out, and you get a factor of gamma on this. 00:04:33.648 --> 00:04:36.260 You can do the same thing on the second slot. 00:04:36.260 --> 00:04:44.705 If I take the dot product of a with the sum of two vectors-- 00:05:04.230 --> 00:05:05.008 OK, et cetera. 00:05:05.008 --> 00:05:05.800 You can keep going. 00:05:05.800 --> 00:05:07.717 All the rules for linearity are going to hold. 00:05:07.717 --> 00:05:09.470 I'm not going to step through them all. 00:05:09.470 --> 00:05:10.720 You can see where they all go. 00:05:10.720 --> 00:05:13.608 So whenever I'm going to define a tensor, in my head, 00:05:13.608 --> 00:05:15.400 I'm imagining it's got properties like this 00:05:15.400 --> 00:05:17.480 that come along for the ride. 00:05:17.480 --> 00:05:19.180 Now, especially when you see this 00:05:19.180 --> 00:05:22.060 defined in certain textbooks, MTW 00:05:22.060 --> 00:05:23.880 is particularly fond of doing this. 00:05:23.880 --> 00:05:25.672 So we come back to this idea that it's sort 00:05:25.672 --> 00:05:27.130 of a function or a mapping. 00:05:27.130 --> 00:05:33.640 You can almost abstractly define the tensor 00:05:33.640 --> 00:05:37.960 as a mathematical machine that's got two slots in it. 00:05:53.930 --> 00:05:56.270 So several of the recommended textbooks 00:05:56.270 --> 00:05:58.260 will write down equations. 00:05:58.260 --> 00:06:00.210 I'm going to put two lines over the symbol 00:06:00.210 --> 00:06:02.710 to sort of-- if you actually read this, in for instance, MTW 00:06:02.710 --> 00:06:04.085 or Caroll or something like that, 00:06:04.085 --> 00:06:06.060 this will be written as a boldface symbol. 00:06:06.060 --> 00:06:07.477 It's hard to do on the blackboard, 00:06:07.477 --> 00:06:09.830 so I'm just going to write double bars over it. 00:06:09.830 --> 00:06:15.340 So if I imagine this with my slots filled with these, 00:06:15.340 --> 00:06:16.990 it's got two slots associated with it. 00:06:16.990 --> 00:06:19.270 I fill it with those two vectors. 00:06:19.270 --> 00:06:24.160 That is equivalent to saying a dot b, 00:06:24.160 --> 00:06:29.357 which is equivalent to it in this component notation, 00:06:29.357 --> 00:06:30.190 something like this. 00:06:33.450 --> 00:06:38.910 So we have repeatedly in the little bit 00:06:38.910 --> 00:06:40.160 of time we've spent together-- 00:06:40.160 --> 00:06:42.535 I should say I have repeatedly, in the little bit of time 00:06:42.535 --> 00:06:47.120 we've spent together, emphasized the distinction between frame 00:06:47.120 --> 00:06:49.070 independent geometric objects, things 00:06:49.070 --> 00:06:51.200 that reside in the manifold and have 00:06:51.200 --> 00:06:55.130 kind of an integral, physical, geometric sensibility 00:06:55.130 --> 00:06:59.210 of their own and their representations. 00:06:59.210 --> 00:07:01.460 I have emphasized quite strongly that you 00:07:01.460 --> 00:07:04.550 should think of the vectors a and b 00:07:04.550 --> 00:07:06.060 as being geometric objects. 00:07:06.060 --> 00:07:08.420 This is some thing that is pointing in space time. 00:07:08.420 --> 00:07:10.160 We all agree that this points, if that's 00:07:10.160 --> 00:07:14.560 a displacement vector, it points from event one to event two. 00:07:14.560 --> 00:07:19.130 OK, everyone agrees on that geometric reality of this. 00:07:19.130 --> 00:07:20.900 Different observers may represent 00:07:20.900 --> 00:07:22.495 it using different components. 00:07:22.495 --> 00:07:23.870 That's just because they're using 00:07:23.870 --> 00:07:25.860 different coordinate systems. 00:07:25.860 --> 00:07:28.880 So when I write down something like this-- 00:07:28.880 --> 00:07:30.590 so let's go back to where I wrote before. 00:07:30.590 --> 00:07:33.690 This is going to turn into some frame independent Lorentz 00:07:33.690 --> 00:07:34.550 scalar a. 00:07:38.460 --> 00:07:41.730 So this is what I like to call a frame independent object. 00:07:49.340 --> 00:07:50.980 Frame or Lorentz independent geometric 00:07:50.980 --> 00:07:52.585 object, as is this scalar. 00:07:55.180 --> 00:07:58.090 And so therefore, the tensor must be a frame 00:07:58.090 --> 00:08:00.070 independent geometric object as well. 00:08:54.087 --> 00:08:55.920 That's a lot of words around the blackboard, 00:08:55.920 --> 00:08:57.780 but I really want to nail that point home. 00:08:57.780 --> 00:09:00.840 So tensors just like vectors. 00:09:00.840 --> 00:09:03.240 Think of them as geometric objects 00:09:03.240 --> 00:09:06.990 that have an intrinsic geometric meaning associated with them 00:09:06.990 --> 00:09:08.850 that lives in spacetime. 00:09:08.850 --> 00:09:11.100 We will talk about certain examples of them. 00:09:11.100 --> 00:09:13.230 OK, you guys all have some intuition about vectors, 00:09:13.230 --> 00:09:14.640 because you've been doing vectors 00:09:14.640 --> 00:09:16.980 ever since you took your first kindergarten physics course, 00:09:16.980 --> 00:09:18.570 and so you know that there's some kind of an object that 00:09:18.570 --> 00:09:20.230 points in a certain direction. 00:09:20.230 --> 00:09:23.200 Tensors are a little bit more challenging in many cases 00:09:23.200 --> 00:09:24.960 to develop an intuition for. 00:09:24.960 --> 00:09:26.970 Some of them really do have a fairly simple 00:09:26.970 --> 00:09:28.050 geometric interpretation. 00:09:28.050 --> 00:09:29.600 You can kind of think of them as-- 00:09:29.600 --> 00:09:31.933 for instance, we're going to introduce one a little bit, 00:09:31.933 --> 00:09:34.140 which describes the flow of energy and momentum 00:09:34.140 --> 00:09:35.340 in space time. 00:09:35.340 --> 00:09:37.890 And so it'll have two indices associated with it, 00:09:37.890 --> 00:09:40.110 and those indices tell me about what 00:09:40.110 --> 00:09:42.360 component of energy or momentum is flowing 00:09:42.360 --> 00:09:43.500 in a particular direction. 00:09:43.500 --> 00:09:45.090 Really easy to interpret that. 00:09:45.090 --> 00:09:47.520 Some of the others, not so much. 00:09:47.520 --> 00:09:50.460 Nonetheless, they do have this geometric meaning 00:09:50.460 --> 00:09:53.780 underneath the hood. 00:09:53.780 --> 00:09:56.160 And that's bound up in the fact that if I 00:09:56.160 --> 00:09:59.910 put frame independent geometric objects into all the slots, 00:09:59.910 --> 00:10:02.705 I get a Lorentz invariant number out of it. 00:10:02.705 --> 00:10:05.910 It's the only way that that can sort of work. 00:10:05.910 --> 00:10:07.620 But one reason why I'm going through this 00:10:07.620 --> 00:10:11.100 is that just like with vectors, different observers, 00:10:11.100 --> 00:10:13.950 different frames will get different components 00:10:13.950 --> 00:10:14.790 in general. 00:10:14.790 --> 00:10:19.980 So there will be different representations according 00:10:19.980 --> 00:10:21.455 to different observers. 00:10:21.455 --> 00:10:23.080 I'm going to write down the same thing. 00:10:32.430 --> 00:10:33.490 Representations. 00:10:52.410 --> 00:10:55.700 So if you want to get a particular observer's 00:10:55.700 --> 00:10:57.890 components out of the tensor, there's 00:10:57.890 --> 00:10:59.720 actually a very simple recipe for this. 00:11:23.970 --> 00:11:29.250 All you do is, you take your tensor, and into its slots, 00:11:29.250 --> 00:11:32.910 you plug in the basis vectors that that observer uses. 00:11:49.880 --> 00:11:53.300 So if I want to get the components-- 00:11:53.300 --> 00:11:55.570 and this, unfortunately, is a fairly stupid example, 00:11:55.570 --> 00:11:57.862 but it's the only one we've got at the moment, so let's 00:11:57.862 --> 00:12:00.060 just work with it. 00:12:00.060 --> 00:12:07.770 If I take the tensor eta, the special relativity metric, 00:12:07.770 --> 00:12:10.560 I plug in some observer's basis factors 00:12:10.560 --> 00:12:16.590 into this thing, this by definition is eta alpha beta. 00:12:16.590 --> 00:12:19.515 Suppose I have a different observer who comes along, 00:12:19.515 --> 00:12:21.390 someone who-- we're doing special relativity, 00:12:21.390 --> 00:12:22.950 so someone who's dashing through the room at three 00:12:22.950 --> 00:12:24.690 quarters of the speed of light. 00:12:24.690 --> 00:12:29.490 I want to know what their components would be. 00:12:29.490 --> 00:12:34.600 Well, what I do is, I just plug into the slots-- 00:12:34.600 --> 00:12:37.020 let's put bars on the complements 00:12:37.020 --> 00:12:38.430 to denote this other observer. 00:12:46.590 --> 00:12:50.850 Do this operation for this other observers set of basis vectors, 00:12:50.850 --> 00:12:53.520 and you will get the components that they will measure. 00:12:53.520 --> 00:12:58.640 Now, one of the reasons why I'm going through this 00:12:58.640 --> 00:13:00.440 is that last time, we talked about how 00:13:00.440 --> 00:13:03.590 to transform basis vectors between different reference 00:13:03.590 --> 00:13:04.460 frames. 00:13:04.460 --> 00:13:07.850 We know that these guys are just related to one another 00:13:07.850 --> 00:13:10.045 by a Lorentz transformation matrix. 00:13:10.045 --> 00:13:11.670 So let's just take this a step further. 00:13:20.200 --> 00:13:23.530 So this is telling me that the components 00:13:23.530 --> 00:13:28.100 of the metric in this barred frame, 00:13:28.100 --> 00:13:30.600 they're going to be what I get when I put it into the slots. 00:13:45.330 --> 00:13:48.060 Those are the basis vectors in the barred frame using 00:13:48.060 --> 00:13:49.560 the Lorentz transformation matrix 00:13:49.560 --> 00:13:51.930 to go from the unbarred frame to the barred frame. 00:13:55.470 --> 00:13:58.590 Now, remember again-- this is one of those places where 00:13:58.590 --> 00:14:02.100 if you're sort of just becoming comfortable with the index 00:14:02.100 --> 00:14:04.740 notation, you're temptation at this stage is always to go, 00:14:04.740 --> 00:14:06.090 these are matrices. 00:14:06.090 --> 00:14:09.030 I should start doing matrix multiplication. 00:14:09.030 --> 00:14:14.280 If you set that urge within you aside, you go, no, no, no, no. 00:14:14.280 --> 00:14:16.740 Those are just a set of 16 numbers. 00:14:16.740 --> 00:14:18.990 For any particular set of complements, 00:14:18.990 --> 00:14:20.910 I can just pull them out. 00:14:20.910 --> 00:14:24.390 So because of the linearity of all these slots 00:14:24.390 --> 00:14:40.380 with this thing, this just becomes those two Lorentz 00:14:40.380 --> 00:14:45.390 transformation matrices acting on the abstract metric tensor 00:14:45.390 --> 00:14:50.070 with the unbarred basis vectors in the slots. 00:14:50.070 --> 00:14:51.562 And this we already know. 00:14:51.562 --> 00:14:53.520 This is, by definition, this is just eta mu nu. 00:15:12.850 --> 00:15:15.490 Repeat this exercise for any tensor 00:15:15.490 --> 00:15:19.120 you care to write down, any 0n tensor you care to write down. 00:15:19.120 --> 00:15:21.760 Go through all this manipulation, 00:15:21.760 --> 00:15:24.430 and you will always find that there's a very simple algorithm 00:15:24.430 --> 00:15:27.840 forgetting the components in, let's 00:15:27.840 --> 00:15:29.470 call it the barred observers frame, 00:15:29.470 --> 00:15:31.773 as converted from the unbarred observers frame. 00:15:31.773 --> 00:15:33.940 Essentially, you just hit it with a bunch of Lorentz 00:15:33.940 --> 00:15:37.420 transform matrices, and as an old professor of mine 00:15:37.420 --> 00:15:39.640 liked to say at this point, line up the indices. 00:15:39.640 --> 00:15:40.660 That's really all we're doing, is 00:15:40.660 --> 00:15:42.118 we're just going to line up the mus 00:15:42.118 --> 00:15:43.480 to convert them to alpha bars. 00:15:43.480 --> 00:15:46.150 Alpha bar here, mu there, put this guy here. 00:15:46.150 --> 00:15:49.540 I want to convert my new into a beta bar. 00:15:49.540 --> 00:15:50.670 I put my matrix there. 00:15:50.670 --> 00:15:51.170 Boom. 00:15:51.170 --> 00:15:53.820 Just line up the indices, and we're done. 00:15:53.820 --> 00:15:57.420 Now, this is, as I kind of emphasized, 00:15:57.420 --> 00:16:06.940 a fairly stupid example, because if you 00:16:06.940 --> 00:16:11.110 take the diagonal of minus one, one, one, one 00:16:11.110 --> 00:16:13.417 and you apply the most God awful immense Lorentz 00:16:13.417 --> 00:16:15.250 transformation you care to write down to it, 00:16:15.250 --> 00:16:17.410 you do all the matrix manipulation 00:16:17.410 --> 00:16:22.090 and you line it all up, what you'll end up finding 00:16:22.090 --> 00:16:28.510 is that this is the diagonal of minus one, one, one, one 00:16:28.510 --> 00:16:29.500 in all frames. 00:16:33.045 --> 00:16:35.170 That's actually one of the defining characteristics 00:16:35.170 --> 00:16:36.920 of the metric of special relativity. 00:16:36.920 --> 00:16:38.680 If you're working curvilinear coordinates, 00:16:38.680 --> 00:16:44.110 the metric is always minus one, one, one, one to all observers. 00:16:44.110 --> 00:16:48.220 So the recipe holds in general. 00:16:48.220 --> 00:16:50.530 This will hold whenever we are studying tensors 00:16:50.530 --> 00:16:51.922 from now and from henceforth. 00:16:51.922 --> 00:16:54.130 Just so happens that this first example we were given 00:16:54.130 --> 00:16:55.047 is kind of a dumb one. 00:16:59.850 --> 00:17:05.220 Nonetheless, learn the lesson and overlook the example, 00:17:05.220 --> 00:17:08.109 and wisdom shall be yours. 00:17:08.109 --> 00:17:10.050 I want to spend a few moments talking 00:17:10.050 --> 00:17:19.650 about a particular subset of tensors, of the 0n tensors, 00:17:19.650 --> 00:17:21.260 where n equals one. 00:17:26.560 --> 00:17:28.660 This is a subset of tensions in general 00:17:28.660 --> 00:17:31.780 that is known in many textbooks as one forms. 00:17:40.740 --> 00:17:42.990 For reasons that I will elaborate on in probably 00:17:42.990 --> 00:17:47.295 10 or so minutes, these are also sometimes called dual vectors. 00:17:53.660 --> 00:17:57.200 And just if that's sort of making some neurons light up 00:17:57.200 --> 00:18:00.080 in your head, set that aside for a moment. 00:18:00.080 --> 00:18:02.510 I want to carefully go through them 00:18:02.510 --> 00:18:05.660 before I indicate the manner in which there is a duality that 00:18:05.660 --> 00:18:07.890 is being applied here. 00:18:07.890 --> 00:18:12.810 So if we go back to the definition of a tensor, 00:18:12.810 --> 00:18:22.180 this tells us that a one form is a mapping 00:18:22.180 --> 00:18:31.760 from a single vector to the Lorentz invariant scalar. 00:18:34.940 --> 00:18:37.693 So using some notation that I will probably only 00:18:37.693 --> 00:18:39.110 use in this lecture, because we're 00:18:39.110 --> 00:18:41.810 going to move past this notation pretty soon, 00:18:41.810 --> 00:18:43.350 let's say a one form-- 00:18:43.350 --> 00:18:46.440 I'm going to denote this with an over tilde on it. 00:18:46.440 --> 00:18:51.140 So let's say p is a one form. 00:18:51.140 --> 00:18:54.260 It will have in this sort of abstract notation 00:18:54.260 --> 00:18:58.840 a single slot, so I put the vector a into it. 00:18:58.840 --> 00:19:03.280 And this then gives me some scalar out. 00:19:30.490 --> 00:19:33.130 So in my notes, I go through some stuff indicating 00:19:33.130 --> 00:19:35.090 that this guy is-- it's a linear operation, 00:19:35.090 --> 00:19:37.090 but that's obvious, because it just inherits all 00:19:37.090 --> 00:19:38.560 the properties from tensors, so I'm not 00:19:38.560 --> 00:19:39.280 going to go through that. 00:19:39.280 --> 00:19:41.305 If you want to double check some of the details, 00:19:41.305 --> 00:19:44.140 they're in the notes that have been posted. 00:19:44.140 --> 00:19:46.570 Just like with the tensors, I extract 00:19:46.570 --> 00:19:58.500 components from this thing by putting my basis vectors 00:19:58.500 --> 00:19:59.000 inside. 00:20:05.430 --> 00:20:12.720 So if I take my one form p and I put it in my alpha basis 00:20:12.720 --> 00:20:18.150 vector, this gives me the alpha component of the one form. 00:20:18.150 --> 00:20:20.980 Notice, it's in the downstairs position. 00:20:20.980 --> 00:20:28.610 So one of the reasons why I want to go through this step 00:20:28.610 --> 00:20:32.370 is it gives me a way to think about what's 00:20:32.370 --> 00:20:34.870 going on at this scalar that I wrote down on the top board 00:20:34.870 --> 00:20:35.370 here. 00:20:35.370 --> 00:20:37.203 So what is this scalar that I get by putting 00:20:37.203 --> 00:20:38.370 a vector into my one form? 00:20:41.350 --> 00:20:44.160 So I take this, put my vector in here. 00:20:47.300 --> 00:20:52.440 I use the fact that I can write my vector 00:20:52.440 --> 00:20:56.910 using its components and the basis vector. 00:20:56.910 --> 00:20:58.860 I use linearity to pull this guy out. 00:21:19.920 --> 00:21:22.240 So the scalar that I get by doing this 00:21:22.240 --> 00:21:27.480 is just the scalar that one gets by contracting the upstairs 00:21:27.480 --> 00:21:29.850 components that I use to set up my vector 00:21:29.850 --> 00:21:33.760 with the downstairs components I use to set up my one form. 00:21:33.760 --> 00:21:39.190 This is an operation that is called contraction, for reasons 00:21:39.190 --> 00:21:41.320 that I hope are fairly obvious. 00:21:41.320 --> 00:21:45.250 So let me define a few other characteristics of this thing, 00:21:45.250 --> 00:21:48.730 and in just a few moments, we'll see what this is good for. 00:21:59.310 --> 00:22:03.800 OK, so one of things I'm going to want to do 00:22:03.800 --> 00:22:05.722 is change the representation of those things. 00:22:05.722 --> 00:22:08.180 So I'm going to want to know how these components transform 00:22:08.180 --> 00:22:10.283 between different frames of reference. 00:22:10.283 --> 00:22:11.450 But we've already done that. 00:22:11.450 --> 00:22:14.390 We did this using tensors, and this is just a tensor. 00:22:14.390 --> 00:22:16.700 So if I change reference frames, if I 00:22:16.700 --> 00:22:19.320 want to know what the components are according 00:22:19.320 --> 00:22:22.100 to some barred observer, I will step 00:22:22.100 --> 00:22:25.010 through the algebra in my notes, but I 00:22:25.010 --> 00:22:28.670 think you know where I'm going to go with this. 00:22:28.670 --> 00:22:32.000 You just take the components in the unbarred frame, 00:22:32.000 --> 00:22:35.390 line it up, contract it with the correct setup of my Lorentz 00:22:35.390 --> 00:22:36.650 transformation matrix, boom. 00:22:36.650 --> 00:22:40.290 Line at the indices, and we've got it there. 00:22:40.290 --> 00:22:44.180 So the last thing which I want to do with this before I talk 00:22:44.180 --> 00:22:49.260 a little bit about what this is really good for is say, 00:22:49.260 --> 00:22:52.460 you know, I've got these basis vectors that 00:22:52.460 --> 00:22:57.323 allow me to relate the components of my vectors 00:22:57.323 --> 00:22:59.240 to the geometric object in a way where I don't 00:22:59.240 --> 00:23:00.407 use a represented bi-symbol. 00:23:00.407 --> 00:23:03.320 I actually have an honest to God equal sign. 00:23:03.320 --> 00:23:06.635 Can I define a similar set of basis one forms? 00:23:13.760 --> 00:23:28.530 What I want to do is define a family of geometric objects, 00:23:28.530 --> 00:23:42.070 and I will denote them with an omega and a tilde such 00:23:42.070 --> 00:23:49.390 that any one form can be written as its components attached 00:23:49.390 --> 00:23:52.970 to these little basis vectors. 00:23:52.970 --> 00:23:54.920 Well, the way I'm going to do this 00:23:54.920 --> 00:23:58.310 is, I'm going to exploit the fact that I already 00:23:58.310 --> 00:24:01.250 know what basis vectors are. 00:24:09.260 --> 00:24:12.800 So I know, for instance, that if I take my one form 00:24:12.800 --> 00:24:14.885 and I plug in a basis factor, I get this. 00:24:18.090 --> 00:24:20.820 And so what I want to do is combine this thing 00:24:20.820 --> 00:24:33.020 which I would like to do with the defining 00:24:33.020 --> 00:24:34.870 operation of contractions. 00:24:41.870 --> 00:24:46.610 So I know p alpha a alpha is what 00:24:46.610 --> 00:24:50.900 I get when I've got my one form and I plug into its slot 00:24:50.900 --> 00:24:51.811 the vector a. 00:25:14.900 --> 00:25:26.580 OK, so let's insist that when I do this, 00:25:26.580 --> 00:25:34.125 I can write this as p beta omega beta. 00:25:34.125 --> 00:25:35.750 Now remember, these are the components. 00:25:35.750 --> 00:25:37.550 This is the actual basis one form. 00:25:37.550 --> 00:25:42.450 So I'm going to stick into my basis one form 00:25:42.450 --> 00:25:43.650 this form of the vector. 00:25:58.930 --> 00:26:07.000 I can then use the linearity of the tensor nature 00:26:07.000 --> 00:26:09.040 to pull out that component of a. 00:26:12.410 --> 00:26:14.920 So what this tells me is this is exactly what I want, 00:26:14.920 --> 00:26:19.460 provided whatever this geometric object is, 00:26:19.460 --> 00:26:33.420 it obeys the rule that when I plug basis vectors into it, 00:26:33.420 --> 00:26:34.890 I get the Kronecker delta back. 00:26:37.590 --> 00:26:39.930 Now, this may all seem really, really trivial right now. 00:26:39.930 --> 00:26:41.310 And indeed, if you think about this, 00:26:41.310 --> 00:26:43.435 in terms of just running down mathematical symbols, 00:26:43.435 --> 00:26:45.042 this is fairly trivial. 00:26:45.042 --> 00:26:46.500 One thing which I want to emphasize 00:26:46.500 --> 00:26:48.480 is that if you go through and say, well, 00:26:48.480 --> 00:26:50.550 if I'm working in a basis where this 00:26:50.550 --> 00:26:53.160 has a time-like component that I'll say is one, 00:26:53.160 --> 00:26:56.550 the time-like direction is zero everywhere else. 00:26:56.550 --> 00:26:58.480 Remember to set the points in the x direction, 00:26:58.480 --> 00:27:02.250 so it's zero, one along x, zero everywhere else. 00:27:02.250 --> 00:27:03.720 This then leads to-- 00:27:03.720 --> 00:27:13.880 so as an example, a set of basis objects 00:27:13.880 --> 00:27:26.590 that a particular observer would write just like so. 00:27:26.590 --> 00:27:29.710 I won't write out the two and three components. 00:27:29.710 --> 00:27:33.550 And again, you look at this and you think to yourself, dude, 00:27:33.550 --> 00:27:35.310 you're just repeating basis vectors. 00:27:35.310 --> 00:27:37.332 What's the big deal here? 00:27:37.332 --> 00:27:39.790 Now, I'm going to explain the fact that these are sometimes 00:27:39.790 --> 00:27:41.373 called dual vectors. 00:27:41.373 --> 00:27:43.540 So if we want to think about this in a language that 00:27:43.540 --> 00:27:46.390 is reminiscent of linear algebra, 00:27:46.390 --> 00:27:51.160 if you think of the basis vectors as column vectors, 00:27:51.160 --> 00:27:54.940 then my basis one forms are essentially row vectors. 00:28:03.630 --> 00:28:07.520 So these look a lot like my basis vectors. 00:28:18.070 --> 00:28:19.800 They enter in a dual way. 00:28:36.150 --> 00:28:39.960 And so they're going to play an important role 00:28:39.960 --> 00:28:41.460 in helping us to-- 00:28:41.460 --> 00:28:43.440 whenever I contract two objects together 00:28:43.440 --> 00:28:46.530 to make some kind of a Lorentz invariant scalar, 00:28:46.530 --> 00:28:48.382 I'm going to want to only combine objects 00:28:48.382 --> 00:28:49.590 that have a dual nature like. 00:28:49.590 --> 00:28:52.048 That's the only way I can get something sensible out of it. 00:28:52.048 --> 00:28:53.340 So let me give you an example. 00:28:53.340 --> 00:29:09.960 Mathematically, this is an equation that I can write down. 00:29:09.960 --> 00:29:10.500 No question. 00:29:10.500 --> 00:29:11.640 If I'm in a particular frame, I've 00:29:11.640 --> 00:29:13.170 got the components of vector a, I've 00:29:13.170 --> 00:29:15.000 got the complements of vector b, I 00:29:15.000 --> 00:29:17.400 can multiply their complements together, sum them, 00:29:17.400 --> 00:29:19.650 and square them. 00:29:19.650 --> 00:29:23.970 So this is mathematically well-defined but plays no role 00:29:23.970 --> 00:29:33.340 in the physics we are going to talk about this term, 00:29:33.340 --> 00:29:36.490 because this is not related to the underlying invariant 00:29:36.490 --> 00:29:39.910 structure of the manifold that we are working with. 00:29:39.910 --> 00:29:42.970 So remember I talked about how a manifold is essentially 00:29:42.970 --> 00:29:46.360 a sufficiently smooth set of points endowed with a metric? 00:29:46.360 --> 00:29:51.160 Well, the metric is what tells us that this is mathematically 00:29:51.160 --> 00:29:51.660 short. 00:29:51.660 --> 00:29:54.850 Write it down, but it means nothing. 00:29:54.850 --> 00:30:08.070 By contrast, of course this has frame independent meaning. 00:30:08.070 --> 00:30:09.539 This is important. 00:30:15.770 --> 00:30:21.130 So that's the sense in which these one forms are often 00:30:21.130 --> 00:30:23.170 called duels actors. 00:30:23.170 --> 00:30:25.197 It is when they are combined with vectors, 00:30:25.197 --> 00:30:27.280 they are duel to it in the sense that when they're 00:30:27.280 --> 00:30:30.070 combined in this appropriate way, 00:30:30.070 --> 00:30:34.290 we find that they describe the physics, 00:30:34.290 --> 00:30:36.040 they capture the invariant characteristics 00:30:36.040 --> 00:30:38.050 of the physics that are important in the theory 00:30:38.050 --> 00:30:39.250 that we are describing. 00:30:39.250 --> 00:30:43.780 If I may give one more example that 00:30:43.780 --> 00:30:45.590 is from a completely different field 00:30:45.590 --> 00:30:48.590 but I think helps to sort of illustrate a useful analogy 00:30:48.590 --> 00:30:50.500 to think about these things, suppose 00:30:50.500 --> 00:30:53.260 you are doing quantum mechanics and I give you 00:30:53.260 --> 00:30:54.760 two wave functions. 00:30:57.910 --> 00:31:04.520 So suppose you have a wave function 00:31:04.520 --> 00:31:10.020 psi of x and another wave function phi of x. 00:31:10.020 --> 00:31:17.080 If you wanted to, you could multiply them together 00:31:17.080 --> 00:31:20.077 and integrate overall space. 00:31:20.077 --> 00:31:22.410 I don't know what you would do with that, but you could. 00:31:24.528 --> 00:31:26.820 On the other hand, you could take the complex conjugate 00:31:26.820 --> 00:31:31.040 of one of them, multiply it by the other one, 00:31:31.040 --> 00:31:34.952 multiply it overall space, and in the notation 00:31:34.952 --> 00:31:36.660 that you learn about, this is, of course, 00:31:36.660 --> 00:31:39.780 just the inner product of wave function psi with wave function 00:31:39.780 --> 00:31:41.250 phi. 00:31:41.250 --> 00:31:43.290 Forming the one form, using a one 00:31:43.290 --> 00:31:47.720 form is akin to selecting an object that has-- 00:31:47.720 --> 00:31:50.550 it allows us to make a mathematical construction 00:31:50.550 --> 00:31:52.710 similar to the quantum mechanical enterprise 00:31:52.710 --> 00:31:54.270 that we use here. 00:31:54.270 --> 00:31:56.170 And as we'll see in just a couple of minutes, 00:31:56.170 --> 00:31:58.440 it's actually really easy to flip back and forth 00:31:58.440 --> 00:32:03.760 between one forms and vectors. 00:32:03.760 --> 00:32:05.892 OK, so a better way to move forward 00:32:05.892 --> 00:32:07.600 with this is rather than talking in terms 00:32:07.600 --> 00:32:12.700 of these more abstract things, let me you a good example. 00:32:12.700 --> 00:32:19.407 So imagine I have some trajectory through spacetime. 00:32:36.310 --> 00:32:38.370 So let's let the t axis go up. 00:32:41.410 --> 00:32:42.740 There's my x and y-axes. 00:32:42.740 --> 00:32:44.880 You all can imagine the z-axis. 00:32:44.880 --> 00:32:49.790 And some observer moves through spacetime like so. 00:32:58.680 --> 00:33:00.660 Last lecture, we talked about a couple 00:33:00.660 --> 00:33:03.090 of important examples of four-vectors. 00:33:03.090 --> 00:33:05.730 And so one which is germane to the situation 00:33:05.730 --> 00:33:08.730 is the four velocity of this observer. 00:33:08.730 --> 00:33:11.610 That just expresses the rate of change 00:33:11.610 --> 00:33:17.390 of its position in spacetime per unit proper time. 00:33:17.390 --> 00:33:23.870 And I'll remind you, tau is time as 00:33:23.870 --> 00:33:26.270 measured along this observer's trajectory. 00:33:26.270 --> 00:33:28.490 So roughly speaking, not even roughly speaking, 00:33:28.490 --> 00:33:31.730 exactly speaking, tau is the time 00:33:31.730 --> 00:33:34.670 that is measured by the by the watch of the person moving 00:33:34.670 --> 00:33:36.352 along there. 00:33:36.352 --> 00:33:40.040 So suppose in addition to this person sort of trundling along 00:33:40.040 --> 00:33:48.120 through spacetime here, suppose spacetime 00:33:48.120 --> 00:34:00.390 is filled with some field, phi, which depends on all 00:34:00.390 --> 00:34:03.870 of my spacetime coordinates. 00:34:03.870 --> 00:34:05.340 Question I want to ask is, what is 00:34:05.340 --> 00:34:09.704 the rate of change of phi along this observer's trajectory? 00:34:34.909 --> 00:34:42.630 So if you are working in ordinary Euclidean space, 00:34:42.630 --> 00:34:44.380 you would basically say, ah, this is easy. 00:34:52.600 --> 00:34:54.139 So you're three space intuition. 00:34:54.139 --> 00:34:56.120 You don't have this proper time to worry about. 00:34:56.120 --> 00:35:06.000 So you would just say that d phi dt along this trajectory 00:35:06.000 --> 00:35:08.800 is just what I get when I calculate dx dt. 00:35:11.305 --> 00:35:12.680 And then look at the x derivative 00:35:12.680 --> 00:35:17.160 of my field phi plus dy dt. 00:35:34.615 --> 00:35:35.990 This is one of those places where 00:35:35.990 --> 00:35:37.657 getting the difference between a partial 00:35:37.657 --> 00:35:39.810 and a total derivative right is important. 00:35:39.810 --> 00:35:43.040 So if you see me do that again, if I don't correct it, 00:35:43.040 --> 00:35:43.850 yell at me. 00:35:43.850 --> 00:35:45.850 OK, so you get this. 00:35:45.850 --> 00:35:48.770 And then you say, ah, this is nothing more 00:35:48.770 --> 00:35:51.470 than that particle's velocity dotted 00:35:51.470 --> 00:35:54.430 into the gradient of the field phi. 00:35:54.430 --> 00:35:58.250 It's a directional derivative along the velocity 00:35:58.250 --> 00:36:00.110 of this trajectory. 00:36:00.110 --> 00:36:03.078 So generalizing this to spacetime, 00:36:03.078 --> 00:36:04.870 you basically have the same thing going on. 00:36:27.740 --> 00:36:30.160 Only now, time is a coordinate. 00:36:30.160 --> 00:36:32.762 So we don't treat time as the independent parameter 00:36:32.762 --> 00:36:35.095 that describes the ticking of clocks as I move along it. 00:36:35.095 --> 00:36:37.480 I use the proper time of the observer 00:36:37.480 --> 00:36:39.022 as my independent parameter. 00:36:41.620 --> 00:36:47.880 So what I will say is, the rate at which 00:36:47.880 --> 00:36:50.070 the field changes per unit of this guy's 00:36:50.070 --> 00:37:12.620 proper time, every one of these is a component of the four 00:37:12.620 --> 00:37:13.650 velocity. 00:37:23.650 --> 00:37:25.750 So now, we introduce a little bit of notation. 00:37:25.750 --> 00:37:47.750 So this derivative is what I get when I contract the four 00:37:47.750 --> 00:37:50.720 velocity against a quantity that's defined by taking 00:37:50.720 --> 00:37:52.325 the derivative of this field. 00:37:55.280 --> 00:37:58.070 We're going to be taking derivatives like this a lot, 00:37:58.070 --> 00:38:00.470 so a little bit of notation being introduced 00:38:00.470 --> 00:38:02.030 to save us some writing. 00:38:06.520 --> 00:38:10.300 This is the directional derivative along the trajectory 00:38:10.300 --> 00:38:10.990 of this body. 00:38:17.140 --> 00:38:19.270 Now, this is a frame independent scalar. 00:38:19.270 --> 00:38:22.240 This is a quality that all observers will agree on. 00:38:22.240 --> 00:38:24.347 This is a four velocity. 00:38:24.347 --> 00:38:25.680 We know this is a four velocity. 00:38:25.680 --> 00:38:28.300 These are the components of a four-vector, 00:38:28.300 --> 00:38:32.780 so these are the components of a one form. 00:38:32.780 --> 00:38:35.800 So the generalization of a gradient 00:38:35.800 --> 00:38:37.420 is an example of a one form. 00:39:16.390 --> 00:39:22.590 So re-using that abstract notation that I gave earlier, 00:39:22.590 --> 00:39:25.920 I can write-- so the way you will sometimes see this 00:39:25.920 --> 00:39:33.090 is, the abstract gradient one form of the field phi 00:39:33.090 --> 00:39:41.080 is represented by the components, the alpha phi. 00:39:41.080 --> 00:39:43.570 You will also in some cases-- 00:39:43.570 --> 00:39:47.800 and I actually have a few lines about this in my notes-- 00:39:47.800 --> 00:39:52.120 sometimes, people will recycle the notation of a gradient 00:39:52.120 --> 00:40:04.130 that you guys learn about in undergraduate E&M. 00:40:04.130 --> 00:40:07.820 I urge a little bit of caution with this notation, 00:40:07.820 --> 00:40:10.930 because we are going to use this symbol for a derivative 00:40:10.930 --> 00:40:13.010 to mean something a little bit different 00:40:13.010 --> 00:40:15.510 in just a couple of lectures. 00:40:15.510 --> 00:40:17.703 It turns out that the something different reduces 00:40:17.703 --> 00:40:19.370 to this in the special relativity limit, 00:40:19.370 --> 00:40:21.440 so there's no harm being done. 00:40:21.440 --> 00:40:23.630 But just bear in mind this particular notion 00:40:23.630 --> 00:40:26.300 of a derivative here is going to change its meaning 00:40:26.300 --> 00:40:28.400 in a little bit. 00:40:28.400 --> 00:40:32.060 Another little bit of notation that is sometimes used here, 00:40:32.060 --> 00:40:33.710 and this is one more unfortunately, 00:40:33.710 --> 00:40:40.950 I think I'm stuck with this notation, one often says-- 00:40:40.950 --> 00:40:44.710 so this idea of taking a derivative 00:40:44.710 --> 00:40:50.680 along a particular four velocity, it comes up a lot. 00:40:50.680 --> 00:40:56.150 So sometimes, what people then do 00:40:56.150 --> 00:40:59.810 is, they define it as the directional gradient 00:40:59.810 --> 00:41:02.490 along u of the field phi. 00:41:02.490 --> 00:41:04.430 So don't worry about this too much. 00:41:04.430 --> 00:41:06.050 We'll come back to us a little bit more appropriate. 00:41:06.050 --> 00:41:07.040 I just want you to be aware, especially 00:41:07.040 --> 00:41:08.873 for those of you who might be reading ahead, 00:41:08.873 --> 00:41:09.920 when you see this. 00:41:09.920 --> 00:41:12.320 So just think of this as what you get when I am taking 00:41:12.320 --> 00:41:13.900 a gradient along a velocity u. 00:41:13.900 --> 00:41:17.690 It basically refers to the gradient one form contracted 00:41:17.690 --> 00:41:21.620 with the four-vector u. 00:41:21.620 --> 00:41:32.860 So the last thing I want to do as I talk about this 00:41:32.860 --> 00:41:35.740 is revisit this notion of one forms 00:41:35.740 --> 00:41:37.870 as being dual to vectors for just a moment. 00:41:45.960 --> 00:41:48.590 So we've just introduced the gradient 00:41:48.590 --> 00:41:50.750 as our first example of a one form. 00:42:03.400 --> 00:42:08.740 So the notion of the gradient as a one form, this 00:42:08.740 --> 00:42:17.660 gives us a nice way to think about what the basis one 00:42:17.660 --> 00:42:18.180 forms mean. 00:42:24.270 --> 00:42:26.580 So when I introduced basis one forms a few moments ago, 00:42:26.580 --> 00:42:28.290 it was a purely mathematical definition. 00:42:28.290 --> 00:42:30.330 I just wanted to have objects such 00:42:30.330 --> 00:42:32.550 that when I popped in the basis vectors, 00:42:32.550 --> 00:42:34.740 I got the Kronecker delta back. 00:42:34.740 --> 00:42:37.622 And after belaboring the obvious, perhaps 00:42:37.622 --> 00:42:39.330 for a little bit too long, we essentially 00:42:39.330 --> 00:42:42.920 got a bunch of ones and zeros out of it. 00:42:42.920 --> 00:42:55.530 Now, putting in math all of those words, 00:42:55.530 --> 00:42:58.730 I did a lot of junk to get that. 00:42:58.730 --> 00:43:08.750 But we also know that if I just take 00:43:08.750 --> 00:43:13.580 the derivative of my coordinate with my coordinate, 00:43:13.580 --> 00:43:18.470 I'm going to get the Kronecker deltas. 00:43:21.560 --> 00:43:24.960 This and this, these are the same thing. 00:43:24.960 --> 00:43:29.720 So I can think of this operation here as telling me 00:43:29.720 --> 00:43:41.240 if I regard this as this kind of abstract form of the gradient 00:43:41.240 --> 00:43:47.250 applied to the coordinate itself, 00:43:47.250 --> 00:43:51.360 this is nothing more than my basis one form. 00:43:51.360 --> 00:43:54.690 So my basis one forms are kind of like gradients 00:43:54.690 --> 00:43:56.248 of my coordinates. 00:43:56.248 --> 00:43:57.790 OK, you're sitting here thinking, OK, 00:43:57.790 --> 00:43:59.832 what the hell does this have to do with anything? 00:44:11.220 --> 00:44:13.490 So when we combine-- 00:44:13.490 --> 00:44:17.550 set that aside for just a second-- 00:44:17.550 --> 00:44:20.225 and remind you of some pretty important intuition 00:44:20.225 --> 00:44:22.100 that you probably learned the very first time 00:44:22.100 --> 00:44:23.392 you learned about the gradient. 00:44:30.100 --> 00:44:41.540 So imagine I just draw level surfaces of some function in-- 00:44:41.540 --> 00:44:43.060 I'll just do two dimensional space. 00:44:48.490 --> 00:44:55.300 OK, so I have some function h of x and y. 00:44:55.300 --> 00:44:57.920 This represents, like for instance, a height field. 00:44:57.920 --> 00:44:59.420 If I'm looking at a topographic map, 00:44:59.420 --> 00:45:01.910 this might tell me about where things are high 00:45:01.910 --> 00:45:04.250 and where things are low. 00:45:04.250 --> 00:45:08.530 So h of xy, there might be level surfaces on my map. 00:45:08.530 --> 00:45:11.240 It'd kind of look like this. 00:45:11.240 --> 00:45:14.455 And there'd be another one that kind of looks like this. 00:45:14.455 --> 00:45:16.525 And maybe it'd have something like this, 00:45:16.525 --> 00:45:21.170 and then something kind of right here. 00:45:21.170 --> 00:45:24.940 So we know looking at this thing that the gradient is very 00:45:24.940 --> 00:45:27.760 low here and very high here. 00:45:30.280 --> 00:45:32.020 Let's put this into the language of what 00:45:32.020 --> 00:45:34.170 we are looking at right now. 00:45:34.170 --> 00:45:39.700 OK, let's let delta x be a displacement vector in the xy 00:45:39.700 --> 00:45:55.474 plane, and dh is going to be my one form of my [] function. 00:46:04.870 --> 00:46:06.850 How do I get the change in height 00:46:06.850 --> 00:46:10.060 as I move along my displacement vector? 00:46:10.060 --> 00:46:30.140 Well, take my one form, plop into its slot delta x, 00:46:30.140 --> 00:46:31.450 I get something like this. 00:46:35.950 --> 00:46:38.360 The thing which I kind of want to emphasize here 00:46:38.360 --> 00:46:43.700 is, we have a lot of geometric intuition about vectors. 00:46:43.700 --> 00:46:45.500 So if I have a delta x-- 00:46:45.500 --> 00:46:46.940 let' say this is my delta x. 00:46:46.940 --> 00:46:51.880 It lasts for about this long over here. 00:46:51.880 --> 00:46:56.090 I take the exact same delta x, and I apply it over here. 00:46:56.090 --> 00:46:58.460 I get a very different result, because that 00:46:58.460 --> 00:47:02.450 goes through many more contours on the left side 00:47:02.450 --> 00:47:05.120 than it does on the right side. 00:47:05.120 --> 00:47:08.158 And the thing-- this is where this duality kind of comes 00:47:08.158 --> 00:47:10.700 in, and I'm going to put up a couple of graphics illustrating 00:47:10.700 --> 00:47:11.930 this here-- 00:47:11.930 --> 00:47:18.670 is that you should think of the one form 00:47:18.670 --> 00:47:22.480 as essentially that set of level surfaces. 00:47:22.480 --> 00:47:23.480 It's a little confusing. 00:47:23.480 --> 00:47:24.230 I'm not going to-- 00:47:24.230 --> 00:47:26.750 I mean, I can see a couple blank looks. 00:47:26.750 --> 00:47:28.250 Maybe even the majority of you have 00:47:28.250 --> 00:47:30.290 kind of blank looks on your faces here. 00:47:30.290 --> 00:47:32.810 And that's fine. 00:47:32.810 --> 00:47:34.850 So what I want you to regard is that when 00:47:34.850 --> 00:47:36.410 I'm talking about basis one forms 00:47:36.410 --> 00:47:38.660 and one forms of functions, they have a very different 00:47:38.660 --> 00:47:40.790 geometric interpretation, even though you're kind of used 00:47:40.790 --> 00:47:42.290 to gradient as telling you something 00:47:42.290 --> 00:47:45.670 about the direction along which something is changing. 00:47:45.670 --> 00:47:47.180 Actually, you define that direction. 00:47:47.180 --> 00:47:48.200 The thing that you're worried about 00:47:48.200 --> 00:47:49.867 is sort of how close the different level 00:47:49.867 --> 00:47:51.840 surfaces are of things. 00:47:51.840 --> 00:47:57.380 OK, so coming back to this idea that my basis one form 00:47:57.380 --> 00:47:59.600 that I use are essentially just the gradients 00:47:59.600 --> 00:48:00.800 of the coordinates. 00:48:00.800 --> 00:48:02.300 So I'm going to put some graphics up 00:48:02.300 --> 00:48:04.790 on the website, which I have actually 00:48:04.790 --> 00:48:09.840 scanned out of the textbook by Misner, Thorne, and Wheeler. 00:48:09.840 --> 00:48:16.460 And what they basically show is, let's 00:48:16.460 --> 00:48:19.180 say this is the time direction. 00:48:19.180 --> 00:48:22.030 Let's say this is my x-axis. 00:48:22.030 --> 00:48:24.350 And this is my y-axis. 00:48:24.350 --> 00:48:27.080 Your intuition is that the x basis 00:48:27.080 --> 00:48:30.450 vector will be a little arrow pointing along x. 00:48:33.020 --> 00:48:35.480 Well, what your intuition should be like for the x basis 00:48:35.480 --> 00:48:41.990 one form is a series of sheets normal to the x-axis that 00:48:41.990 --> 00:48:43.750 fill all of space. 00:48:43.750 --> 00:48:47.910 OK, spaced one unit apart, filling 00:48:47.910 --> 00:48:51.120 all of space kind of like that. 00:48:51.120 --> 00:48:59.100 So here's one example of one of those sheets. 00:49:17.500 --> 00:49:21.195 And notice, the x-axis pierces every one of those things. 00:49:21.195 --> 00:49:22.570 That's another way in which these 00:49:22.570 --> 00:49:24.490 are sort of a set of dual functions 00:49:24.490 --> 00:49:28.810 to the vectors themselves. 00:49:28.810 --> 00:49:31.540 This notion-- so I'm going to turn 00:49:31.540 --> 00:49:33.970 to something which is perhaps a little less 00:49:33.970 --> 00:49:36.405 weird in about 30 seconds. 00:49:36.405 --> 00:49:37.780 But the one thing which I kind of 00:49:37.780 --> 00:49:39.430 want to emphasize with this-- and again, I'm 00:49:39.430 --> 00:49:40.750 going to put a couple graphics up that 00:49:40.750 --> 00:49:41.560 help to illustrate this. 00:49:41.560 --> 00:49:43.602 And there's some really nice discussions of this. 00:49:43.602 --> 00:49:46.780 MTW is particularly good for this discussion. 00:49:46.780 --> 00:49:48.910 This ends up being a really useful notion 00:49:48.910 --> 00:49:51.850 for capturing how we are going to compute fluxes 00:49:51.850 --> 00:49:53.882 through particular directions. 00:49:53.882 --> 00:49:55.840 Because if I want to know the flux of something 00:49:55.840 --> 00:49:59.110 in the x direction, well, my x basis one form 00:49:59.110 --> 00:50:01.270 is actually like a sheet that captures everything 00:50:01.270 --> 00:50:02.767 that flows in the x direction. 00:50:02.767 --> 00:50:04.600 So it's sort of a mathematical object that's 00:50:04.600 --> 00:50:06.085 designed for catching fluxes. 00:50:11.530 --> 00:50:15.128 If isn't quite gelling with you, that's fine. 00:50:15.128 --> 00:50:17.170 This is without a doubt one of the goofier things 00:50:17.170 --> 00:50:19.720 we're going to talk about in this first introductory period 00:50:19.720 --> 00:50:20.990 of stuff. 00:50:20.990 --> 00:50:22.480 This is one of those places where 00:50:22.480 --> 00:50:24.370 I think it's sort of fair to say, if you're 00:50:24.370 --> 00:50:26.320 not quite getting what's going on, 00:50:26.320 --> 00:50:28.703 shut up and calculate works well enough. 00:50:28.703 --> 00:50:31.120 It's kind of like you know the Feynman's mantra on quantum 00:50:31.120 --> 00:50:31.620 mechanics. 00:50:31.620 --> 00:50:34.410 Sometimes, you've just got to say, OK, whatever. 00:50:34.410 --> 00:50:36.208 Learn the way it goes, and it's kind 00:50:36.208 --> 00:50:37.750 of like playing a musical instrument. 00:50:37.750 --> 00:50:39.280 You sort of strum it and practice it for a while, 00:50:39.280 --> 00:50:41.072 and it becomes second nature after a while. 00:50:47.630 --> 00:50:54.130 All right, so to wrap this up, what 00:50:54.130 --> 00:50:58.460 I want to do for the last major thing today is, 00:50:58.460 --> 00:51:02.960 I hope I can kind of put a bow on our discussion of tensors. 00:51:02.960 --> 00:51:05.720 Let's come back to the metric as our original example 00:51:05.720 --> 00:51:07.110 of a tensor here. 00:51:07.110 --> 00:51:14.790 So when I give you the metric as an abstract tensor, 00:51:14.790 --> 00:51:26.710 and I imagine I have filled both slots, 00:51:26.710 --> 00:51:28.870 I get a Lorentz invariant number. 00:51:34.830 --> 00:51:38.220 A dot b is what I get when I take this tensor 00:51:38.220 --> 00:51:42.400 and I put a and b into it slots. 00:51:42.400 --> 00:51:44.680 Suppose I only fill one of its slots. 00:51:51.100 --> 00:51:56.080 Well, if I take this, I plug in the vector 00:51:56.080 --> 00:52:02.770 a but I leave the other slot blank, well, what I've got 00:52:02.770 --> 00:52:07.790 is a mathematical object that will take a vector 00:52:07.790 --> 00:52:09.690 and produce a Lorentz invariance number. 00:52:27.770 --> 00:52:28.520 That's a one form. 00:52:41.740 --> 00:52:44.240 So let's do this very carefully and abstractly for a moment. 00:52:50.065 --> 00:52:53.630 But at this point, we basically have almost all the pieces 00:52:53.630 --> 00:52:54.750 in place. 00:52:54.750 --> 00:52:57.620 And so I'm going to kind of tone down some of the formality 00:52:57.620 --> 00:52:58.680 fairly soon. 00:52:58.680 --> 00:53:05.430 So let's define a one form, an object that 00:53:05.430 --> 00:53:09.030 takes a single vector inside of it 00:53:09.030 --> 00:53:14.960 as what I get when I take the metric 00:53:14.960 --> 00:53:16.620 and put that vector in there. 00:53:16.620 --> 00:53:19.520 If I want to get its components out, 00:53:19.520 --> 00:53:22.190 well, I know the way I do that is I put the basis 00:53:22.190 --> 00:53:22.880 vector in there. 00:53:39.350 --> 00:53:43.520 This guy is symmetric, so I can flip the order of the metric. 00:53:43.520 --> 00:53:46.520 It's symmetric, so I can flip the order of these guys. 00:53:46.520 --> 00:53:55.380 And what this tells me is that my one form component of a 00:53:55.380 --> 00:53:58.260 is just the vector component of a hit 00:53:58.260 --> 00:54:01.290 by the components of the metric. 00:54:01.290 --> 00:54:14.490 In other words, the metric converts vectors into one forms 00:54:14.490 --> 00:54:19.220 by lowering the indices. 00:54:22.870 --> 00:54:25.580 This is an invertebrate procedure as well. 00:54:25.580 --> 00:54:34.960 So this metric, I can define eta with indices in the upstairs 00:54:34.960 --> 00:54:46.390 position by requiring that eta alpha beta contracted 00:54:46.390 --> 00:54:48.530 with it in the downstairs position 00:54:48.530 --> 00:54:53.360 gives me the identity back. 00:54:53.360 --> 00:54:55.700 Incidentally, when you do this, you again 00:54:55.700 --> 00:54:59.370 find it's got exactly the same matrix representation. 00:54:59.370 --> 00:55:02.300 So this thing with its indices in the upstairs position 00:55:02.300 --> 00:55:05.360 is just minus one, one, one, one. 00:55:05.360 --> 00:55:08.580 It will not always be that way, though. 00:55:08.580 --> 00:55:10.910 Again, this is just because special relativity 00:55:10.910 --> 00:55:14.030 in rectilinear coordinates is simple. 00:55:18.470 --> 00:55:23.445 So I will often call that the inverse metric. 00:55:26.953 --> 00:55:28.370 And then, you shouldn't have a lot 00:55:28.370 --> 00:55:33.380 of trouble convincing yourself that if I've got a one form, 00:55:33.380 --> 00:55:47.926 I can make a vector out of it by a contraction operation. 00:55:54.560 --> 00:55:58.797 That now tells me that I have about 16 gagillian-- well, 00:55:58.797 --> 00:56:00.630 actually, it's a countable and finite thing. 00:56:00.630 --> 00:56:07.130 But I have many ways that I can write down the inner product 00:56:07.130 --> 00:56:13.620 between two vectors. 00:56:13.620 --> 00:56:25.530 This guy is-- if you like, you can now 00:56:25.530 --> 00:56:28.890 regard a vector as being a sort of a function 00:56:28.890 --> 00:56:30.560 that you put one forms into. 00:56:40.020 --> 00:56:41.920 These are all the same. 00:56:51.820 --> 00:56:54.520 These are all equivalents to one another. 00:56:54.520 --> 00:56:56.800 And actually making a distinction 00:56:56.800 --> 00:56:58.690 between vector in one form and all that, 00:56:58.690 --> 00:57:02.980 it's just kind of gotten stupid at this point. 00:57:02.980 --> 00:57:05.300 So the distinction among these different objects, 00:57:05.300 --> 00:57:07.950 the different names, kind of doesn't matter. 00:57:29.498 --> 00:57:31.040 And indeed, you sort of look at this. 00:57:31.040 --> 00:57:32.880 Up until now, we've regarded tensors 00:57:32.880 --> 00:57:35.660 as being these sort of things that operate on vectors. 00:57:35.660 --> 00:57:37.670 OK, but why not regard vectors as things 00:57:37.670 --> 00:57:39.740 that operate on one forms? 00:57:39.740 --> 00:57:43.220 What this sort of tells you is that this whole notion 00:57:43.220 --> 00:57:45.980 of tensors being separate from vectors that I talked 00:57:45.980 --> 00:57:51.330 about before is kind of silly. 00:57:51.330 --> 00:57:54.080 So I'm going to revisit the definition of a tensor 00:57:54.080 --> 00:58:05.970 that I started the lecture off with like so. 00:58:05.970 --> 00:58:08.790 So a new and more complete definition. 00:58:15.990 --> 00:58:40.780 A tensor of type mn is a linear mapping of m1 forms and n 00:58:40.780 --> 00:58:47.085 vectors to the Lorentz scalars. 00:58:50.240 --> 00:58:53.635 In this definition-- so we've already introduced zero, 00:58:53.635 --> 00:58:54.980 one tensors. 00:58:54.980 --> 00:58:56.090 Those are one forms. 00:58:59.600 --> 00:59:01.400 One zero tensors are vectors. 00:59:11.540 --> 00:59:13.040 Furthermore, as I kind of emphasized 00:59:13.040 --> 00:59:15.650 when I wrote this sentence here, the distinction 00:59:15.650 --> 00:59:18.980 between the slots that operate on vectors and the slots that 00:59:18.980 --> 00:59:21.800 operate on one forms, it's nice for getting some 00:59:21.800 --> 00:59:23.965 of the basic foundations laid. 00:59:23.965 --> 00:59:25.340 This is one of those places where 00:59:25.340 --> 00:59:27.170 now that the scaffolding is in place-- 00:59:27.170 --> 00:59:29.128 we've had the scaffolding in place for a while, 00:59:29.128 --> 00:59:31.140 but this wall of our edifice is pretty steady, 00:59:31.140 --> 00:59:33.200 so we can kick the scaffolding away. 00:59:33.200 --> 00:59:35.099 We can sort of lose this distinction. 00:59:45.860 --> 00:59:47.590 The metric lets us convert the nature 00:59:47.590 --> 00:59:51.880 of the slots on a tensor. 00:59:51.880 --> 01:00:05.710 So if I have an mn tensor and I use the metric to lower, 01:00:05.710 --> 01:00:10.760 that means I make it an m minus one n plus one tensor. 01:00:10.760 --> 01:00:22.185 So an example would be, if I have a tensor 01:00:22.185 --> 01:00:36.573 r mu beta gamma delta and I lower that first index, 01:00:36.573 --> 01:00:38.615 so this went from something that operates on one, 01:00:38.615 --> 01:00:40.220 one form and three vectors. 01:00:40.220 --> 01:00:42.170 Now, it's one it operates on four-vectors. 01:00:47.180 --> 01:00:52.190 Likewise using the inverse metric, you can raise-- 01:00:58.660 --> 01:01:00.660 and just for completeness, let's write that out. 01:01:11.990 --> 01:01:18.640 So an example of this would be if I have some tensor su beta 01:01:18.640 --> 01:01:27.735 gamma, and let's say I raised that first index 01:01:27.735 --> 01:01:28.860 to get something like this. 01:01:32.700 --> 01:01:35.687 OK, so let's see. 01:01:35.687 --> 01:01:37.270 In my last couple of minutes-- recall, 01:01:37.270 --> 01:01:38.900 I do have to leave a little bit early today, because I 01:01:38.900 --> 01:01:40.067 need to introduce a speaker. 01:01:40.067 --> 01:01:44.800 But I just want to wrap up one thing kind of quickly. 01:01:44.800 --> 01:01:52.560 I've spent a bunch of time talking about basis objects. 01:02:01.670 --> 01:02:03.810 And I'm going to go through this fairly quickly. 01:02:03.810 --> 01:02:05.880 The notes, if you want to see a few more details, 01:02:05.880 --> 01:02:07.713 you're welcome to download and look at them. 01:02:07.713 --> 01:02:10.860 They're not really tricky or super critical. 01:02:10.860 --> 01:02:12.782 We know we have basis objects for vectors, 01:02:12.782 --> 01:02:14.990 which hopefully you have pretty good intuition about. 01:02:14.990 --> 01:02:18.080 We have basis objects for one forms, where your intuition is 01:02:18.080 --> 01:02:19.580 perhaps a little bit more befuddled, 01:02:19.580 --> 01:02:21.440 but it'll come with time. 01:02:21.440 --> 01:02:24.890 So you might think, uh, now I've got two index tenors. 01:02:24.890 --> 01:02:26.200 I've got three index tensors. 01:02:26.200 --> 01:02:28.212 There's a four index tensor on the board. 01:02:28.212 --> 01:02:30.170 Scott's probably going to write 17 index tensor 01:02:30.170 --> 01:02:31.295 on the board at some point. 01:02:31.295 --> 01:02:34.443 Do I need a basis object for every one of those? 01:02:34.443 --> 01:02:35.735 So in other words, do we need-- 01:02:45.552 --> 01:02:46.760 glad that's caught on video-- 01:02:49.340 --> 01:02:51.110 do I need to be able to say something 01:02:51.110 --> 01:02:56.210 like the abstract metric tensor is 01:02:56.210 --> 01:03:02.055 these components times some kind of a two index basis object? 01:03:16.420 --> 01:03:18.940 Do I need to do something like this? 01:03:18.940 --> 01:03:21.640 Cutting to the chase, the answer is no. 01:03:27.120 --> 01:03:31.097 Basis one forms and vectors are sufficient. 01:03:38.890 --> 01:03:41.610 So what we're going to do is, abstractly 01:03:41.610 --> 01:03:44.340 just imagine that if I do have a tensor, 01:03:44.340 --> 01:03:46.860 I kind of have an outer product. 01:03:46.860 --> 01:03:51.000 I have both of the basis objects attached to this thing, 01:03:51.000 --> 01:03:54.000 and each one is just attached to those two slots. 01:03:54.000 --> 01:04:06.012 OK, so in this particular case, my two index basis two-form 01:04:06.012 --> 01:04:07.470 that would go with this thing here, 01:04:07.470 --> 01:04:10.020 the thing with two indices on it, 01:04:10.020 --> 01:04:11.595 I'm just going to regard this as-- 01:04:11.595 --> 01:04:16.160 I'll abstractly write this as just an outer product 01:04:16.160 --> 01:04:17.340 on the basis one forms. 01:04:21.860 --> 01:04:27.380 If I ever need a basis object for a tensor like this, 01:04:27.380 --> 01:04:33.370 I will just regard this as an outer product of these two 01:04:33.370 --> 01:04:33.950 things. 01:04:33.950 --> 01:04:35.533 So what's going on with that notation? 01:04:35.533 --> 01:04:36.592 What does this mean? 01:04:36.592 --> 01:04:38.050 Don't lose too much sleep about it. 01:04:38.050 --> 01:04:40.600 It's basically saying that I have separate objects attached 01:04:40.600 --> 01:04:43.420 to all of my different indices, and they're 01:04:43.420 --> 01:04:46.000 kind of coming along here and giving a sense of direction 01:04:46.000 --> 01:04:47.060 to these things. 01:04:47.060 --> 01:04:49.233 So for instance, if I have some kind of a tensor-- 01:04:49.233 --> 01:04:51.400 I mean, a great one, which we're going to talk about 01:04:51.400 --> 01:04:52.300 in just a little bit. 01:04:56.820 --> 01:04:58.570 It's a quantity known as the stress energy 01:04:58.570 --> 01:05:20.340 tensor, in which I can abstractly 01:05:20.340 --> 01:05:23.925 think of the tensor as having-- 01:05:27.060 --> 01:05:28.020 not just a [INAUDIBLE]. 01:05:28.020 --> 01:05:28.990 It will have two components. 01:05:28.990 --> 01:05:30.490 And I can think of it as essentially 01:05:30.490 --> 01:05:35.497 pointing in two different directions at once. 01:05:35.497 --> 01:05:37.830 Now, we're going to talk about this in a lot more detail 01:05:37.830 --> 01:05:39.588 in a couple of weeks. 01:05:39.588 --> 01:05:41.130 Actually, not even a couple of weeks. 01:05:41.130 --> 01:05:42.420 A couple of lectures. 01:05:42.420 --> 01:05:45.140 What I'm going to teach you is that the alpha beta component 01:05:45.140 --> 01:05:47.250 of the stress energy tensor tells me 01:05:47.250 --> 01:05:52.620 about the flux of form momentum component alpha 01:05:52.620 --> 01:05:55.128 in the beta direction. 01:05:55.128 --> 01:05:56.670 And so this is basically just saying, 01:05:56.670 --> 01:05:59.040 when I think of it as the actual object, not 01:05:59.040 --> 01:06:02.010 just the representation according to some observer, 01:06:02.010 --> 01:06:06.530 here is the thing that gives me the direction of my four 01:06:06.530 --> 01:06:08.250 momentum, and here is the direction 01:06:08.250 --> 01:06:10.350 in which it is flowing. 01:06:10.350 --> 01:06:13.170 And sometimes, we will make more complicated objects. 01:06:13.170 --> 01:06:16.560 And so you might need to imagine-- 01:06:16.560 --> 01:06:20.280 here's one which I actually wrote down in my notes here-- 01:06:20.280 --> 01:06:21.780 there will be times when we're going 01:06:21.780 --> 01:06:39.750 to care about a tensor which at least abstractly, we might need 01:06:39.750 --> 01:06:44.175 to regard as having this whole set of sort of vomitous basis 01:06:44.175 --> 01:06:45.800 vectors coming along for the ride here. 01:06:48.370 --> 01:06:50.770 And it is actually fairly important 01:06:50.770 --> 01:06:54.645 to have all these things that are in place. 01:06:54.645 --> 01:06:56.020 Where I will just conclude things 01:06:56.020 --> 01:06:59.170 for today is that the place where it is particularly 01:06:59.170 --> 01:07:02.260 important to remember that we kind of sometimes almost just 01:07:02.260 --> 01:07:05.230 implicitly have these things coming along 01:07:05.230 --> 01:07:06.070 for the ride here-- 01:07:17.990 --> 01:07:21.037 it's important when we calculate derivatives. 01:07:27.580 --> 01:07:35.010 So I gave you guys an example of a directional derivative 01:07:35.010 --> 01:07:37.310 for a scalar field that filled all its spacetime. 01:07:37.310 --> 01:07:39.750 I imagine that there was some trajectory of an observer 01:07:39.750 --> 01:07:41.160 moving through this. 01:07:41.160 --> 01:07:46.780 Now, imagine it isn't a scalar field that fills all of, 01:07:46.780 --> 01:07:48.330 spacetime but it's a tensor field. 01:08:02.160 --> 01:08:03.862 Here's t. 01:08:03.862 --> 01:08:05.550 We'll say this is the y direction. 01:08:05.550 --> 01:08:07.440 This is the x direction. 01:08:07.440 --> 01:08:10.440 Here's my observer moving through all this thing. 01:08:10.440 --> 01:08:12.990 And again, I'm going to say that this trajectory is 01:08:12.990 --> 01:08:17.380 characterized by a four velocity, dxt tau, 01:08:17.380 --> 01:08:28.270 and I'm going to imagine that there is some tensor field that 01:08:28.270 --> 01:08:35.580 fills all of spacetime when I go and calculate 01:08:35.580 --> 01:08:39.300 the derivative of this thing, when 01:08:39.300 --> 01:08:41.160 we're working in special relativity 01:08:41.160 --> 01:08:44.410 where we are right now, these guys are going to be constant, 01:08:44.410 --> 01:08:45.611 so it doesn't really matter. 01:08:45.611 --> 01:08:47.069 But soon, we're going to generalize 01:08:47.069 --> 01:08:51.420 to more complicated geometries, more complicated spacetimes, 01:08:51.420 --> 01:08:53.310 and the basis objects will themselves 01:08:53.310 --> 01:08:55.620 vary as I move along the trajectory. 01:08:55.620 --> 01:08:59.279 And I will need to-- in order to have 01:08:59.279 --> 01:09:01.350 a notion of a derivative that is a properly 01:09:01.350 --> 01:09:02.948 formed geometric object, I'm going 01:09:02.948 --> 01:09:05.490 to have to worry about how the basis objects change as I move 01:09:05.490 --> 01:09:07.585 along this trajectory as well. 01:09:07.585 --> 01:09:09.210 So that tends to just make the analysis 01:09:09.210 --> 01:09:11.140 a little bit more complicated. 01:09:11.140 --> 01:09:13.545 I have a few notes about this that I will put up 01:09:13.545 --> 01:09:16.170 on to the web page, but I don't want to go into too much detail 01:09:16.170 --> 01:09:17.640 beyond that until we actually get 01:09:17.640 --> 01:09:19.620 into some of the details of these derivatives. 01:09:19.620 --> 01:09:23.430 So I'm just going to leave it at that for now. 01:09:23.430 --> 01:09:31.240 So yeah, I'll just say, the derivative in principle 01:09:31.240 --> 01:09:43.500 and quite often in practice, it will 01:09:43.500 --> 01:09:48.640 depend on how these guys vary in space and time. 01:09:58.060 --> 01:10:00.400 And let me just say, you guys kind of already know that. 01:10:00.400 --> 01:10:02.720 Because when you studied E&M, you 01:10:02.720 --> 01:10:04.540 got these somewhat complicated formulas 01:10:04.540 --> 01:10:07.487 for things like the divergence and curl and stuff like that. 01:10:07.487 --> 01:10:09.070 And those are essential, because those 01:10:09.070 --> 01:10:11.320 are notions of derivative where you are taking 01:10:11.320 --> 01:10:13.030 into account the fact that when you're in a curvilinear 01:10:13.030 --> 01:10:15.405 coordinate system, your basis vectors are shifting as you 01:10:15.405 --> 01:10:17.008 move from point to point. 01:10:17.008 --> 01:10:18.550 The stuff we're going to get out this 01:10:18.550 --> 01:10:20.140 will look a little bit different, 01:10:20.140 --> 01:10:22.780 and it comes to the fact, as I emphasized in my last lecture, 01:10:22.780 --> 01:10:24.700 that we are going to tend to use what we call 01:10:24.700 --> 01:10:28.220 a coordinate basis, whereas when you guys learn stuff in E&M, 01:10:28.220 --> 01:10:30.460 you were using what's known as an orthonormal basis. 01:10:30.460 --> 01:10:32.085 And it does lead to slight differences. 01:10:32.085 --> 01:10:33.730 There's a mapping between them. 01:10:33.730 --> 01:10:35.530 Not that hard to figure it out, but we 01:10:35.530 --> 01:10:37.870 don't need to get into those weeds just now. 01:10:37.870 --> 01:10:42.400 All right, so I will pick it up from there on Thursday. 01:10:42.400 --> 01:10:44.980 So I'll begin with a brief recap of everything we did. 01:10:44.980 --> 01:10:48.010 The primary thing which I really want to emphasize more 01:10:48.010 --> 01:10:52.300 than anything is this board plus this idea 01:10:52.300 --> 01:10:54.670 that the metric can be used to raise and lower 01:10:54.670 --> 01:10:56.440 the indices of a tensor. 01:10:56.440 --> 01:10:58.720 At this point, talking about vectors, 01:10:58.720 --> 01:11:02.860 talking about one forms, many of you in a math class 01:11:02.860 --> 01:11:05.170 probably learned about contravariant vector components 01:11:05.170 --> 01:11:07.210 and covariant vector components. 01:11:07.210 --> 01:11:10.450 Once you've got a metric, it's kind of like, who cares? 01:11:10.450 --> 01:11:12.350 You can just go from one to the other. 01:11:12.350 --> 01:11:15.940 And that's why I tend to, almost religiously, I 01:11:15.940 --> 01:11:17.780 avoid the terms covariant and contravariant, 01:11:17.780 --> 01:11:19.832 and I just say, upstairs and downstairs. 01:11:19.832 --> 01:11:21.790 Because I can flip back and forth between them, 01:11:21.790 --> 01:11:24.377 and there's really no physical meaning in them. 01:11:24.377 --> 01:11:25.960 You have to think carefully about what 01:11:25.960 --> 01:11:27.730 is physically measurable, and it has 01:11:27.730 --> 01:11:30.940 nothing to do with whether it's covariant or convariant, 01:11:30.940 --> 01:11:32.562 upstairs or downstairs. 01:11:32.562 --> 01:11:34.270 All right, I will end there, since I have 01:11:34.270 --> 01:11:36.390 to go and introduce someone.