WEBVTT 00:00:00.000 --> 00:00:01.924 [SQUEAKING] 00:00:01.924 --> 00:00:03.848 [RUSTLING] 00:00:03.848 --> 00:00:06.253 [CLICKING] 00:00:09.817 --> 00:00:11.400 SCOTT HUGHES: So we're just picking up 00:00:11.400 --> 00:00:12.525 where we stopped last time. 00:00:12.525 --> 00:00:15.840 So we are beginning to discuss how 00:00:15.840 --> 00:00:19.710 we are going to sort of do a geometrical approach 00:00:19.710 --> 00:00:24.210 to physics, using a more general set of coordinates now. 00:00:24.210 --> 00:00:27.270 So we began talking about how things change when I discuss 00:00:27.270 --> 00:00:28.860 special relativity, so for the moment 00:00:28.860 --> 00:00:31.080 keeping ourselves just at special relativity. 00:00:31.080 --> 00:00:33.570 We, by the way, are going to begin lifting our assumptions 00:00:33.570 --> 00:00:36.912 that it is simply special relativity fairly soon. 00:00:36.912 --> 00:00:38.370 But to set that up, I need to start 00:00:38.370 --> 00:00:40.890 thinking about how to work in more general coordinate 00:00:40.890 --> 00:00:41.910 systems. 00:00:41.910 --> 00:00:44.620 So we're going to do it in the simplest possible curvilinear 00:00:44.620 --> 00:00:45.120 coordinates. 00:00:45.120 --> 00:00:47.850 So it's basically just going from Cartesian coordinates 00:00:47.850 --> 00:00:51.210 in the spatial sector to plane polar coordinates. 00:00:51.210 --> 00:00:53.460 One of the things which I have emphasized a few times, 00:00:53.460 --> 00:00:55.200 and I'm going to continue to hammer on, 00:00:55.200 --> 00:00:59.640 is that these are a little bit different from the curvilinear 00:00:59.640 --> 00:01:03.000 coordinates that you are used to in your past life. 00:01:03.000 --> 00:01:06.020 In particular, if I write out the displacement, 00:01:06.020 --> 00:01:08.610 the little vector of the displacement element 00:01:08.610 --> 00:01:10.950 in the usual way, I am using what's 00:01:10.950 --> 00:01:14.250 called a "coordinate basis," which means that the vector 00:01:14.250 --> 00:01:17.490 dx is related to the displacement, the differential 00:01:17.490 --> 00:01:19.770 of the coordinates, by just that thing contracted 00:01:19.770 --> 00:01:21.750 with all the basis vectors. 00:01:21.750 --> 00:01:24.480 And so what that means is I have a little displacement in time, 00:01:24.480 --> 00:01:25.620 which looks normal. 00:01:25.620 --> 00:01:28.170 Displacement in radius, which looks normal. 00:01:28.170 --> 00:01:30.420 Displacement in the z direction, which looks normal, 00:01:30.420 --> 00:01:33.248 and a displacement in an angle, which does not. 00:01:33.248 --> 00:01:35.790 In order for this whole thing to be dimensionally consistent, 00:01:35.790 --> 00:01:39.530 that's telling me e phi has to have the dimensions of length. 00:01:39.530 --> 00:01:41.280 And that is a feature, not a bug. 00:01:43.890 --> 00:01:45.750 Last time, we introduced the matrix 00:01:45.750 --> 00:01:48.150 that allows me to convert between one coordinate system 00:01:48.150 --> 00:01:50.452 and another, so just basically the matrix-- 00:01:50.452 --> 00:01:51.660 it's sort of a Jacobi matrix. 00:01:51.660 --> 00:01:54.090 It's a matrix of partials between the two 00:01:54.090 --> 00:01:55.920 coordinate systems. 00:01:55.920 --> 00:01:57.290 And this idea that things are-- 00:01:57.290 --> 00:01:58.350 they look a little weird. 00:01:58.350 --> 00:02:00.725 So the way I did that was I didn't actually write it out, 00:02:00.725 --> 00:02:04.650 but I did the usual mapping between x, y and r and phi, 00:02:04.650 --> 00:02:07.038 worked out all of my derivatives. 00:02:07.038 --> 00:02:08.580 And sure enough, you've got something 00:02:08.580 --> 00:02:10.470 that looks very standard, with the possible exception 00:02:10.470 --> 00:02:12.070 of these r's that are appearing in here. 00:02:12.070 --> 00:02:13.710 So notice the elements of this matrix. 00:02:13.710 --> 00:02:15.690 These do not have consistent units-- 00:02:15.690 --> 00:02:18.030 again, feature, not bug. 00:02:18.030 --> 00:02:20.492 This guy is basically just the inverse of that. 00:02:20.492 --> 00:02:22.450 This is the matrix that affects the coordinates 00:02:22.450 --> 00:02:23.533 in the opposite direction. 00:02:23.533 --> 00:02:26.290 And notice in this case, you have some elements where 00:02:26.290 --> 00:02:29.820 their units are 1 over length. 00:02:29.820 --> 00:02:33.470 So let's just continue to sort of catalog 00:02:33.470 --> 00:02:35.970 what some of the things we are going to be working with look 00:02:35.970 --> 00:02:39.280 like in this new coordinate representation. 00:02:39.280 --> 00:02:41.160 And this will lead us to introduce 00:02:41.160 --> 00:02:42.900 one of the mathematical objects that we 00:02:42.900 --> 00:02:46.850 are going to use extensively as we move forward 00:02:46.850 --> 00:02:48.400 in studying this subject. 00:02:48.400 --> 00:02:50.310 So what I want to do next is look at what 00:02:50.310 --> 00:02:51.650 my basis vectors look like. 00:02:55.090 --> 00:02:57.180 So what I want to do is characterize what 00:02:57.180 --> 00:03:01.310 my e r and my e phi look like. 00:03:01.310 --> 00:03:04.380 And these are going to look very familiar from your intuition, 00:03:04.380 --> 00:03:06.510 from having studied things like E&M 00:03:06.510 --> 00:03:08.550 in non-Cartesian coordinates. 00:03:08.550 --> 00:03:13.590 So your e r is just related to the original Cartesian basis 00:03:13.590 --> 00:03:14.595 vectors, like so. 00:03:18.690 --> 00:03:20.720 And if you like, you can easily read that out 00:03:20.720 --> 00:03:24.200 by performing the following matrix 00:03:24.200 --> 00:03:28.760 multiplication on the original Cartesian basis vectors. 00:03:28.760 --> 00:03:34.110 Your e phi perhaps looks a little wacky. 00:03:48.170 --> 00:03:51.380 So you can see the length coming into play there. 00:03:51.380 --> 00:03:55.520 A good way to think about this is if your intuition 00:03:55.520 --> 00:03:58.710 about basis factors-- 00:03:58.710 --> 00:04:00.710 I have to be careful with this language myself-- 00:04:00.710 --> 00:04:02.127 your intuition about basis vectors 00:04:02.127 --> 00:04:04.010 is typically that they are unit vectors. 00:04:04.010 --> 00:04:05.540 These are not unit vectors. 00:04:05.540 --> 00:04:08.690 They do form a nice basis, but they are not unit vectors. 00:04:08.690 --> 00:04:11.120 In particular, the basic idea we're going to go with here 00:04:11.120 --> 00:04:14.060 is that e phi, it's always going to sort of point 00:04:14.060 --> 00:04:15.710 in the tangential direction. 00:04:15.710 --> 00:04:18.714 But no matter where I put it in radius, 00:04:18.714 --> 00:04:20.089 I want that vector to always sort 00:04:20.089 --> 00:04:23.060 of subtend the same amount of angle. 00:04:23.060 --> 00:04:26.252 In order to do that, its length needs to grow with r. 00:04:26.252 --> 00:04:27.710 So that's where that's a little bit 00:04:27.710 --> 00:04:29.600 different from your intuition. 00:04:29.600 --> 00:04:31.280 And there's a very good reason for this, 00:04:31.280 --> 00:04:33.905 which we will get to, hopefully, well before the end of today's 00:04:33.905 --> 00:04:35.730 class. 00:04:35.730 --> 00:04:40.640 So last time, when we first began 00:04:40.640 --> 00:04:43.130 to talk about tensors a couple of lectures ago, 00:04:43.130 --> 00:04:44.780 the first tensor I gave you-- 00:04:49.430 --> 00:04:52.470 so confining ourselves just to Cartesian coordinates-- 00:04:52.470 --> 00:04:56.910 was the metric, which was originally 00:04:56.910 --> 00:05:00.540 introduced as this mathematical object that 00:05:00.540 --> 00:05:02.958 came out of looking at dot products between basis vectors. 00:05:02.958 --> 00:05:04.500 It's essentially a tensor that allows 00:05:04.500 --> 00:05:07.230 me to feed in two displacements and get 00:05:07.230 --> 00:05:11.340 the invariant interval between those displacements that 00:05:11.340 --> 00:05:12.090 comes out of that. 00:05:16.080 --> 00:05:24.360 I am going to continue to call the dot product of two basis 00:05:24.360 --> 00:05:27.350 vectors the "metric." 00:05:27.350 --> 00:05:30.900 But I'm going to use a slightly different symbol for this. 00:05:30.900 --> 00:05:32.400 I'm going to call this g alpha beta. 00:05:35.340 --> 00:05:38.590 In the coordinate representation that we are using right now, 00:05:38.590 --> 00:05:45.680 so in plane polar coordinates, this becomes-- 00:05:45.680 --> 00:05:49.650 you can work it out from what I've written down right here. 00:05:49.650 --> 00:05:55.980 This is just the diagonal of minus 1, 1 r squared 1. 00:05:58.650 --> 00:05:59.970 So this equals dot here. 00:05:59.970 --> 00:06:03.590 This is-- I'll put PPC for plane polar coordinates under that. 00:06:09.160 --> 00:06:12.190 And then using that, you know that you can always 00:06:12.190 --> 00:06:16.840 work out the invariant displacement between two 00:06:16.840 --> 00:06:18.350 events. 00:06:18.350 --> 00:06:24.660 It's always going to be the metric contracted 00:06:24.660 --> 00:06:30.810 with the differential displacement element. 00:06:30.810 --> 00:06:34.560 And this is going to be minus dt squared 00:06:34.560 --> 00:06:41.220 plus dr squared plus r squared d phi squared plus dz squared. 00:06:41.220 --> 00:06:42.785 That, I hope, makes a lot of sense. 00:06:42.785 --> 00:06:44.160 This is exactly what you'd expect 00:06:44.160 --> 00:06:46.680 if I have two events that are separated 00:06:46.680 --> 00:06:51.300 in plane polar coordinates by dt, dr, d phi, dz. 00:06:51.300 --> 00:06:54.660 This is what the distance between them should be. 00:06:54.660 --> 00:06:56.730 So the fact that my basis vectors 00:06:56.730 --> 00:07:01.040 have this slightly annoying form associated with them, 00:07:01.040 --> 00:07:02.790 it all sort of comes out in the wash here. 00:07:02.790 --> 00:07:06.060 Remember at the end of the day, if we think about quantities 00:07:06.060 --> 00:07:07.788 that are representation independent-- 00:07:07.788 --> 00:07:09.330 and that's going to be the key thing. 00:07:09.330 --> 00:07:13.560 When you assemble scalars out of these things, 00:07:13.560 --> 00:07:16.990 the individual tensor components, 00:07:16.990 --> 00:07:19.170 they can be a little bit confusing sometimes. 00:07:19.170 --> 00:07:20.730 They are not things that we measure. 00:07:20.730 --> 00:07:23.250 They are not things that really characterize what we 00:07:23.250 --> 00:07:24.690 are going to be working with. 00:07:24.690 --> 00:07:26.482 And we really want to get into the physics, 00:07:26.482 --> 00:07:28.800 unless we're very careful about it. 00:07:28.800 --> 00:07:30.335 This is something you can measure. 00:07:30.335 --> 00:07:31.710 And so sure enough, it comes out, 00:07:31.710 --> 00:07:33.270 and it's got a good meaning to it. 00:07:33.270 --> 00:07:34.980 Let me just wrap up one last thing 00:07:34.980 --> 00:07:38.230 before I talk about sort of where we're going with this. 00:07:38.230 --> 00:07:43.750 So just for completeness, let me write down the basis one forms. 00:07:43.750 --> 00:07:50.290 Just as the basis vectors had a bit of a funny form associated 00:07:50.290 --> 00:07:53.710 with them, you're going to find the basis one forms likewise 00:07:53.710 --> 00:07:55.710 have a bit of a funny form associated with them. 00:07:55.710 --> 00:07:58.520 And the way I'm going to get these-- 00:07:58.520 --> 00:08:02.560 and so these are going to be the Cartesian basis one forms-- 00:08:02.560 --> 00:08:05.795 basically, I'm not carefully proving all these relations 00:08:05.795 --> 00:08:07.920 at this point, because you all know how to do that. 00:08:07.920 --> 00:08:10.630 I'm just using line up the indices rule. 00:08:10.630 --> 00:08:24.750 And when you do that, you get this. 00:08:24.750 --> 00:08:30.190 And likewise, your basis one form for the axial direction, 00:08:30.190 --> 00:08:39.980 I'll just write down the result. It's going to look like this. 00:08:46.250 --> 00:08:51.745 So the key place where all of this-- so right now, these 00:08:51.745 --> 00:08:53.120 are all just sort of definitions. 00:08:53.120 --> 00:08:56.390 Nothing I've done here should be anything 00:08:56.390 --> 00:08:58.460 that even approaches a surprise, I hope, just 00:08:58.460 --> 00:09:01.105 given the you guys have done-- 00:09:01.105 --> 00:09:02.480 the key thing that's probably new 00:09:02.480 --> 00:09:04.970 is all this garbage associated with coordinate bases, 00:09:04.970 --> 00:09:07.820 this extra factors of r and 1 over r that are popping up. 00:09:07.820 --> 00:09:11.090 But provided you're willing to sort of swallow your discomfort 00:09:11.090 --> 00:09:12.860 and go through the motions, these 00:09:12.860 --> 00:09:15.560 are not difficult calculations. 00:09:15.560 --> 00:09:17.840 The key place where all of this really matters 00:09:17.840 --> 00:09:21.860 is going to be when we calculate derivatives of things. 00:09:36.523 --> 00:09:38.440 It'll turn out there is an important rule when 00:09:38.440 --> 00:09:40.200 we talk about integrals as well a little bit later, 00:09:40.200 --> 00:09:41.200 but let's just say that. 00:09:41.200 --> 00:09:42.985 So for now, we'll focus on derivatives. 00:09:42.985 --> 00:09:45.360 So all the derivatives that we've been looking at so far, 00:09:45.360 --> 00:09:47.850 we have, indeed, done a couple of calculations where we've 00:09:47.850 --> 00:09:51.400 computed the derivatives of various vector valued 00:09:51.400 --> 00:09:53.250 and tensor valued quantities. 00:09:53.250 --> 00:09:56.760 And it was helped by the fact that all the bases, when 00:09:56.760 --> 00:09:59.520 I work in Cartesian coordinates, are constant. 00:09:59.520 --> 00:10:02.130 Well, that's not the case now. 00:10:02.130 --> 00:10:12.150 So now, we need to account for the fact 00:10:12.150 --> 00:10:18.130 that the bases all vary with our coordinates. 00:10:24.020 --> 00:10:26.390 So let me just quickly make a catalog 00:10:26.390 --> 00:10:29.540 of all the non-trivial-- there's basically four. 00:10:29.540 --> 00:10:32.180 In this one, where I'm just doing plane polar coordinates, 00:10:32.180 --> 00:10:34.040 there are four non-trivial derivatives 00:10:34.040 --> 00:10:36.140 we need to worry about. 00:10:36.140 --> 00:10:38.840 One of them actually turns out to be 0. 00:10:38.840 --> 00:10:42.170 So the radial derivative of the radial unit vector is 0. 00:10:44.930 --> 00:10:50.310 But the phi derivative of the phi unit vector is not. 00:10:50.310 --> 00:10:54.300 you go and take the phi derivative of this guy, 00:10:54.300 --> 00:10:56.340 and you basically get-- 00:10:56.340 --> 00:10:58.080 take the phi derivative of this, you're 00:10:58.080 --> 00:11:01.650 going to get this back, modulo factor of radius. 00:11:01.650 --> 00:11:07.920 So I can write d e r d phi as e phi over r. 00:11:13.990 --> 00:11:17.290 If I take the derivative of e phi with respect to r, 00:11:17.290 --> 00:11:20.020 I get e phi back, divided by r. 00:11:20.020 --> 00:11:25.420 So the simplest way to write this is like so. 00:11:31.015 --> 00:11:34.180 And finally, if I take the phi derivative of the phi unit 00:11:34.180 --> 00:11:38.706 vector, I get e r back, with an extra factor of r thrown in. 00:11:44.420 --> 00:11:45.320 And a minus sign. 00:11:52.320 --> 00:11:54.900 So we're going to see a way of doing this that's a little bit 00:11:54.900 --> 00:11:56.400 more systematic later, but I want 00:11:56.400 --> 00:11:59.070 to just keep the simple example, where you can just basically 00:11:59.070 --> 00:12:03.150 by hand calculate all the non-trivial derivatives easily. 00:12:03.150 --> 00:12:05.775 Of course, there's also a t unit vector and a z unit vector. 00:12:05.775 --> 00:12:07.400 But they're constants, so I'm not going 00:12:07.400 --> 00:12:08.525 to bother writing them out. 00:12:08.525 --> 00:12:11.370 All the derivatives associated with them are equal to 0. 00:12:11.370 --> 00:12:17.600 So let's imagine now that I have assembled some vector. 00:12:17.600 --> 00:12:21.250 So I have some vector field that lives in this spacetime. 00:12:21.250 --> 00:12:22.580 And I'm using this basis. 00:12:22.580 --> 00:12:26.830 And so I would write this vector with components v alpha. 00:12:26.830 --> 00:12:27.628 And let's let the-- 00:12:27.628 --> 00:12:29.920 so this is going to be a curvilinear coordinate system, 00:12:29.920 --> 00:12:33.490 so this will be plane polar coordinates being used here, 00:12:33.490 --> 00:12:37.060 plane polar coordinate basis vectors. 00:12:40.030 --> 00:12:43.845 And what I would like to do is assemble the tensor 00:12:43.845 --> 00:12:45.220 that you can think of essentially 00:12:45.220 --> 00:12:47.790 as the gradient of this vector. 00:12:47.790 --> 00:12:51.130 So let's begin by doing this in a sort of abstract notation. 00:12:51.130 --> 00:12:55.030 So the gradient of this guy-- this is sort of ugly notation, 00:12:55.030 --> 00:12:57.860 but live with it. 00:12:57.860 --> 00:13:01.420 Following what we have been doing all along, what 00:13:01.420 --> 00:13:04.150 you would want to do is just take 00:13:04.150 --> 00:13:06.100 the root of this whole thing. 00:13:06.100 --> 00:13:08.560 It's going to have a downstairs component on it. 00:13:08.560 --> 00:13:12.490 So attach to it the basis one form. 00:13:12.490 --> 00:13:15.160 If you prefer, you can write it using the d notation 00:13:15.160 --> 00:13:16.848 like I have there, but I just want 00:13:16.848 --> 00:13:18.640 to stick with the form I wrote in my notes. 00:13:23.810 --> 00:13:27.290 Looking at this the way I've sort of got this right now, 00:13:27.290 --> 00:13:32.600 I can think of, if I don't include the basis 00:13:32.600 --> 00:13:41.290 one forms here, this should be the components of a one form. 00:13:41.290 --> 00:13:43.750 So this should be a kind of object. 00:13:50.490 --> 00:13:52.884 So let's just expand out that derivative. 00:13:55.610 --> 00:13:56.950 Let's write it like this. 00:13:59.540 --> 00:14:03.535 So you just-- we haven't changed calculus. 00:14:03.535 --> 00:14:05.410 So when I do this, I'm going to basically use 00:14:05.410 --> 00:14:08.170 the old-fashioned Leibniz rule for expanding 00:14:08.170 --> 00:14:09.840 the derivative product of two things. 00:14:26.290 --> 00:14:29.020 Here's the key thing which I want to emphasize-- 00:14:29.020 --> 00:14:30.790 in order for this whole thing to be-- 00:14:30.790 --> 00:14:34.780 for this to be a tensorial object, something that I couple 00:14:34.780 --> 00:14:41.310 to this basis one form, the sum of these two objects 00:14:41.310 --> 00:14:44.400 must obey the rules for transforming tensors. 00:14:44.400 --> 00:14:46.680 But the two objects individually will not. 00:15:01.822 --> 00:15:03.280 So this is an important point which 00:15:03.280 --> 00:15:05.650 I'm going to emphasize in slightly different words 00:15:05.650 --> 00:15:07.005 in just a few moments again. 00:15:07.005 --> 00:15:09.130 This is one of the key things I want you to get out 00:15:09.130 --> 00:15:11.297 of this lecture, is that when I'm taking derivatives 00:15:11.297 --> 00:15:13.278 of things like this, you've got to be 00:15:13.278 --> 00:15:15.820 a little bit careful about what you consider to be components 00:15:15.820 --> 00:15:17.110 of tensors and what is not. 00:15:19.870 --> 00:15:23.050 Now as written like that, this is kind of annoying. 00:15:25.640 --> 00:15:30.100 So my first object has a nice basis vector attached to it. 00:15:30.100 --> 00:15:33.820 My second object involves a derivative of the basis vector. 00:15:33.820 --> 00:15:35.710 However, something we saw over here 00:15:35.710 --> 00:15:37.600 is that derivatives of basis vectors 00:15:37.600 --> 00:15:39.910 are themselves proportional to basis vectors. 00:15:44.858 --> 00:15:47.150 So what I'm going to do is introduce a bit of notation. 00:15:59.440 --> 00:16:01.510 So let me switch notation slightly here. 00:16:01.510 --> 00:16:12.220 So the beta derivative of e alpha can be written as-- 00:16:12.220 --> 00:16:15.970 in general, it can be written as a linear combination 00:16:15.970 --> 00:16:16.900 of basis vectors. 00:16:33.100 --> 00:16:37.350 So what we're going to do is define d-- 00:16:37.350 --> 00:16:39.850 I want to make sure my Greek letters are legible to everyone 00:16:39.850 --> 00:16:40.480 in the room here. 00:16:40.480 --> 00:16:42.063 So let me write this nice and clearly. 00:16:45.610 --> 00:16:55.860 d beta of e alpha, I'm going to write that as capital gamma mu 00:16:55.860 --> 00:17:01.200 beta alpha e mu. 00:17:01.200 --> 00:17:05.670 This gamma that I've just introduced here in this context 00:17:05.670 --> 00:17:08.310 is known as the Christoffel symbol. 00:17:15.138 --> 00:17:17.680 Fact I'm calling this a symbol, it's got three indices on it. 00:17:17.680 --> 00:17:20.129 You might look at it and go, ooh, smells like a tensor. 00:17:20.129 --> 00:17:21.129 Be a little bit careful. 00:17:24.099 --> 00:17:28.420 In much the same way that these two terms are not individually 00:17:28.420 --> 00:17:31.600 components of a tensor, but their sum is, 00:17:31.600 --> 00:17:33.460 this guy individually is actually 00:17:33.460 --> 00:17:35.620 not a component of a tensor, but when 00:17:35.620 --> 00:17:39.340 combined with other things, it allows us to assemble tensors. 00:17:48.220 --> 00:18:10.530 So for our plane polar coordinates, 00:18:10.530 --> 00:18:15.070 there are exactly three non-zero Christoffel symbols. 00:18:15.070 --> 00:18:22.970 So gamma phi r phi is equal to 1 over r, which is also 00:18:22.970 --> 00:18:28.130 equal to gamma phi phi r. 00:18:31.290 --> 00:18:35.820 Gamma r phi phi is minus r. 00:18:35.820 --> 00:18:38.070 And you can basically just read that out of that table 00:18:38.070 --> 00:18:40.080 that I wrote down over there. 00:18:40.080 --> 00:18:41.640 All the others will be equal to 0. 00:18:50.310 --> 00:18:53.226 Now from this example, this is what 00:18:53.226 --> 00:18:55.500 it makes it smell like every time you introduce 00:18:55.500 --> 00:18:59.070 a new coordinate representation. 00:18:59.070 --> 00:19:02.072 You're going to need to sit down for an hour and a half, 00:19:02.072 --> 00:19:03.780 or something like that, and just work out 00:19:03.780 --> 00:19:06.405 all the bloody derivatives, and then go, oh, crap, and read out 00:19:06.405 --> 00:19:08.197 all the different components of this thing, 00:19:08.197 --> 00:19:09.557 and assemble them together. 00:19:09.557 --> 00:19:11.140 There actually is an algorithm that we 00:19:11.140 --> 00:19:12.848 will get to at the end of this class that 00:19:12.848 --> 00:19:17.310 allows you to easily extract the Christoffel symbols provided 00:19:17.310 --> 00:19:18.420 you know the metric. 00:19:18.420 --> 00:19:21.060 But right now, I just want to illustrate this thing 00:19:21.060 --> 00:19:21.660 conceptually. 00:19:21.660 --> 00:19:23.610 The key thing which you should know about it 00:19:23.610 --> 00:19:27.280 is that it is essentially the-- 00:19:27.280 --> 00:19:30.120 I almost said the word "matrix," but it's got three indices. 00:19:30.120 --> 00:19:34.470 It's a table of functions that allows 00:19:34.470 --> 00:19:38.040 me to relate derivatives of basis vectors 00:19:38.040 --> 00:19:38.970 to the basis vectors. 00:19:44.580 --> 00:19:47.690 So before I go on to talk about some of that stuff, 00:19:47.690 --> 00:19:51.440 let's take a look at the derivative a little bit more 00:19:51.440 --> 00:19:52.323 carefully. 00:20:20.255 --> 00:20:21.630 So the derivative of the vector-- 00:20:21.630 --> 00:20:24.230 so let's basically take what I've written out up there. 00:20:24.230 --> 00:20:28.070 I'm going to write this as the beta derivative of vector 00:20:28.070 --> 00:20:35.510 v. And I can write that as the beta derivative of e of v 00:20:35.510 --> 00:20:36.950 alpha-- 00:20:36.950 --> 00:20:39.290 so the first term where the derivative 00:20:39.290 --> 00:20:40.901 hits the vector components. 00:20:43.550 --> 00:20:45.490 And then I've got a second term where 00:20:45.490 --> 00:20:50.300 the derivative hits the basis. 00:20:50.300 --> 00:20:51.760 I'm going to write this like so. 00:20:58.840 --> 00:21:00.040 This is sort of annoying. 00:21:00.040 --> 00:21:02.200 One term is proportional to e alpha, 00:21:02.200 --> 00:21:03.970 one is proportional to e mu. 00:21:03.970 --> 00:21:08.560 But notice, especially in the second term, both alpha and mu 00:21:08.560 --> 00:21:11.740 are dummy indices, so I'm free to relabel them. 00:21:11.740 --> 00:21:13.870 So what I'm going to do is relabel alpha 00:21:13.870 --> 00:21:20.340 and mu by exchanging them. 00:21:23.570 --> 00:21:31.660 As long as I do that consistently, 00:21:31.660 --> 00:21:33.490 that is totally kosher. 00:21:33.490 --> 00:21:46.140 And when I do that, I can factor out an overall factor 00:21:46.140 --> 00:21:47.040 of the basis object. 00:21:49.980 --> 00:21:54.340 This combination that pops up here-- 00:21:57.290 --> 00:21:59.480 so we give this a name. 00:21:59.480 --> 00:22:01.660 And this is a combination which, by the time 00:22:01.660 --> 00:22:05.140 you have finished this semester, if you don't have at least 00:22:05.140 --> 00:22:07.180 one nightmare in which this name appears, 00:22:07.180 --> 00:22:08.960 I will not have done my job properly. 00:22:08.960 --> 00:22:12.580 This shows up a lot at this point. 00:22:12.580 --> 00:22:17.133 This is called the "covariant derivative." 00:22:24.240 --> 00:22:27.090 And it shows up enough that we introduce 00:22:27.090 --> 00:22:30.780 a whole new notation for the derivative 00:22:30.780 --> 00:22:32.550 to take it into account. 00:22:32.550 --> 00:22:34.290 I'm going to call this combination 00:22:34.290 --> 00:22:37.230 of the partial derivative of v and v 00:22:37.230 --> 00:22:40.940 coupled to the Christoffel symbol-- 00:22:43.560 --> 00:22:50.830 I'm going to write this using the, if you're talking LaTeX, 00:22:50.830 --> 00:22:54.460 this would be the nabla operator. 00:22:54.460 --> 00:22:57.530 So I made a point earlier when we 00:22:57.530 --> 00:22:59.530 were talking about derivatives a couple of weeks 00:22:59.530 --> 00:23:05.740 ago that we were reserving the gradient symbol 00:23:05.740 --> 00:23:07.570 for a special purpose later. 00:23:07.570 --> 00:23:08.980 Here it is. 00:23:08.980 --> 00:23:11.320 So whenever I make a derivative that 00:23:11.320 --> 00:23:13.990 involves the gradient symbol like this, 00:23:13.990 --> 00:23:16.000 it is this covariant derivative. 00:23:16.000 --> 00:23:20.890 And the covariant derivative acting on vector components, 00:23:20.890 --> 00:23:23.710 it generates tensor components. 00:23:23.710 --> 00:23:26.177 Partial derivative does not. 00:23:26.177 --> 00:23:28.510 And what I'm going to do, just in the interest of time-- 00:23:28.510 --> 00:23:30.718 it's one of those calculations that's straightforward 00:23:30.718 --> 00:23:31.600 but fairly tedious-- 00:23:31.600 --> 00:23:34.150 I have a set of notes that I meant 00:23:34.150 --> 00:23:36.550 to make live before I headed over here, but I forgot. 00:23:36.550 --> 00:23:39.160 I have a set of notes that I'm going to show on the website 00:23:39.160 --> 00:23:43.810 by this evening which explicitly works out what happens when you 00:23:43.810 --> 00:23:46.360 apply the coordinate transformation using that-- 00:23:46.360 --> 00:23:47.410 it's been erased-- 00:23:47.410 --> 00:23:49.520 when you use that L matrix to construct 00:23:49.520 --> 00:23:52.270 the coordinate transformation between two representations. 00:23:52.270 --> 00:23:55.990 If you try to do it to partial derivatives of vector 00:23:55.990 --> 00:23:57.880 components, basically what you find 00:23:57.880 --> 00:24:00.910 is that there's an extra term that spoils your ability 00:24:00.910 --> 00:24:04.510 to call that-- it spoils the tensor transformation law, 00:24:04.510 --> 00:24:07.360 spoils your ability to call that a tensor component. 00:24:07.360 --> 00:24:09.790 So the partial on its own doesn't let you. 00:24:09.790 --> 00:24:13.840 You get some extra terms that come along and mess it all up. 00:24:13.840 --> 00:24:15.700 On next p set, you guys are going 00:24:15.700 --> 00:24:20.800 to show that if you then try to apply the tensor transformation 00:24:20.800 --> 00:24:23.500 law to the Christoffel symbols, you 00:24:23.500 --> 00:24:25.390 get something that looks tensorial, 00:24:25.390 --> 00:24:27.880 but with an extra term that spoils your ability 00:24:27.880 --> 00:24:29.210 to call it tensorial. 00:24:29.210 --> 00:24:31.660 There's a little bit of extra junk there. 00:24:31.660 --> 00:24:33.670 But the two terms exactly conspire 00:24:33.670 --> 00:24:37.990 to cancel each other out so that the sum is tensorial. 00:24:37.990 --> 00:24:39.610 So part one of this will be notes 00:24:39.610 --> 00:24:42.490 that I post to the website no later than this evening. 00:24:42.490 --> 00:24:44.265 Part two, you guys will do on the p set. 00:25:02.880 --> 00:25:06.270 So just saying in math what I just 00:25:06.270 --> 00:25:10.290 said in words, if I do this, like I said, 00:25:10.290 --> 00:25:13.530 you basically will eventually reach the point 00:25:13.530 --> 00:25:17.910 where what I am writing out right now 00:25:17.910 --> 00:25:24.200 will become so automatic it will haunt your dreams. 00:25:24.200 --> 00:25:25.580 Wait a minute, I screwed that up. 00:25:25.580 --> 00:25:27.622 It's so automatic I can't even write it properly. 00:25:35.050 --> 00:25:38.482 Anyhow, something like that will-- modulo my typo-- 00:25:38.482 --> 00:25:39.690 that should become automatic. 00:25:39.690 --> 00:25:41.273 And the key thing which I want to note 00:25:41.273 --> 00:25:51.840 is that if I take these guys, and I attach 00:25:51.840 --> 00:25:58.590 the appropriate basis objects to them, 00:25:58.590 --> 00:26:00.390 this is an honest-to-god tensor. 00:26:00.390 --> 00:26:03.820 And so this derivative is itself an honest-to-god tensor. 00:26:03.820 --> 00:26:05.850 A typical application of this, so one 00:26:05.850 --> 00:26:09.180 that will come up a fair bit, is how do you 00:26:09.180 --> 00:26:12.600 compute a spacetime divergence in each coordinate system? 00:26:19.940 --> 00:26:38.148 So suppose I take the divergence of some vector field v. 00:26:38.148 --> 00:26:40.350 So you're going to have four terms that 00:26:40.350 --> 00:26:42.930 are just the usual, like when you guys learned 00:26:42.930 --> 00:26:46.840 how to do divergence in freshman E&M in Cartesian coordinates. 00:26:46.840 --> 00:26:51.840 You get one term that's just dv x dx, dv y dy, et cetera. 00:26:51.840 --> 00:26:54.000 So you got one term that looks just like that, 00:26:54.000 --> 00:26:55.917 and you're going to have something that brings 00:26:55.917 --> 00:26:57.240 in all of Christoffel symbols. 00:26:57.240 --> 00:27:01.090 Notice the symmetry that we have on this one. 00:27:01.090 --> 00:27:03.090 Actually, there is Einstein summation convention 00:27:03.090 --> 00:27:04.620 being imposed here. 00:27:04.620 --> 00:27:06.240 But when we look at this, there's 00:27:06.240 --> 00:27:09.210 actually only one Christoffel symbol 00:27:09.210 --> 00:27:12.520 that has repeated indices in that first position. 00:27:12.520 --> 00:27:31.827 So when I put all this together, you 00:27:31.827 --> 00:27:33.660 wind up with something that looks like this. 00:27:38.760 --> 00:27:42.323 So go back and check your copy of Jackson, or Purcell, 00:27:42.323 --> 00:27:44.490 or Griffith, whatever your favorite E&M textbook is. 00:27:44.490 --> 00:27:46.930 And you'll see when you work in cylindrical coordinates, 00:27:46.930 --> 00:27:49.430 you indeed find that there's a correction to the radial term 00:27:49.430 --> 00:27:50.660 that involves 1 over r. 00:27:50.660 --> 00:27:53.030 That's popped out exactly like you think it should. 00:27:53.030 --> 00:27:56.090 You have a bit of a wacky looking thing with your phi 00:27:56.090 --> 00:27:59.270 component, of course. 00:27:59.270 --> 00:28:03.000 And let me just spend a second or two making sure. 00:28:03.000 --> 00:28:05.300 It's often, especially while we're developing intuition 00:28:05.300 --> 00:28:09.380 about working in a coordinate basis, 00:28:09.380 --> 00:28:12.680 it's not a bad idea to do a little sanity check. 00:28:17.660 --> 00:28:21.740 So here's a sanity check that I would do with this. 00:28:21.740 --> 00:28:23.960 If I take the divergence, I take a derivative 00:28:23.960 --> 00:28:26.660 of a vector field, the final object that comes out 00:28:26.660 --> 00:28:29.420 of that should have the dimensions of that vector 00:28:29.420 --> 00:28:31.640 divided by length. 00:28:31.640 --> 00:28:33.930 Remembering c equals 1, that will clearly have 00:28:33.930 --> 00:28:35.180 that vector divided by length. 00:28:35.180 --> 00:28:36.597 That will clearly have that vector 00:28:36.597 --> 00:28:39.200 divided by length, vector divided by length, explicitly 00:28:39.200 --> 00:28:40.368 vector divided by length. 00:28:40.368 --> 00:28:40.910 That's weird. 00:28:44.670 --> 00:28:58.930 But remember, the basis objects themselves are a little weird. 00:29:04.870 --> 00:29:07.600 One of the things we saw was that e phi 00:29:07.600 --> 00:29:08.910 has the dimensions of length. 00:29:15.980 --> 00:29:19.010 In order for the vector to itself be consistent, 00:29:19.010 --> 00:29:26.910 v phi must have the dimensions of v divided by length. 00:29:26.910 --> 00:29:28.910 So in fact, when I just take its phi derivative, 00:29:28.910 --> 00:29:30.327 I get something that looks exactly 00:29:30.327 --> 00:29:32.493 like it should if it is to be a divergence. 00:29:45.250 --> 00:29:47.290 Let's move on and think about how 00:29:47.290 --> 00:29:50.620 I take a covariant derivative of other kinds 00:29:50.620 --> 00:29:53.200 of tensorial objects. 00:29:53.200 --> 00:29:56.440 This is all you need to know if you are worried about taking 00:29:56.440 --> 00:29:58.252 derivatives of vectors. 00:29:58.252 --> 00:30:00.460 But we're going to work with a lot of different kinds 00:30:00.460 --> 00:30:01.330 of tensor objects. 00:30:07.753 --> 00:30:09.170 One of the most important lectures 00:30:09.170 --> 00:30:11.360 we're going to do in about a month 00:30:11.360 --> 00:30:14.600 actually involves looking at a bunch of covariant derivatives 00:30:14.600 --> 00:30:19.308 of some four-indexed objects, so it gets messy. 00:30:19.308 --> 00:30:20.350 Let's walk our way there. 00:30:23.600 --> 00:30:27.620 So suppose I want to take the derivative of a scalar. 00:30:27.620 --> 00:30:29.990 Scalar have no basis object attached to them. 00:30:38.380 --> 00:30:39.490 There's no basis object. 00:30:39.490 --> 00:30:40.540 When I take the derivative, I don't 00:30:40.540 --> 00:30:42.415 have to worry about anything wiggling around. 00:30:42.415 --> 00:30:46.040 No Christoffel symbols come in. 00:30:46.040 --> 00:30:50.440 If I want to take the covariant derivative of some field phi, 00:30:50.440 --> 00:30:53.230 it is nothing more than the partial derivative 00:30:53.230 --> 00:30:54.370 of that field phi-- 00:30:54.370 --> 00:30:55.840 boom. 00:30:55.840 --> 00:30:56.410 Happy days. 00:30:59.125 --> 00:31:00.000 How about a one form? 00:31:06.540 --> 00:31:13.030 The long way to do this would be to essentially say, well, 00:31:13.030 --> 00:31:14.860 the way I started this was by looking 00:31:14.860 --> 00:31:18.570 at how my basis vectors varied as I took their derivatives. 00:31:18.570 --> 00:31:23.650 Let's do the same thing for the basis one forms, 00:31:23.650 --> 00:31:28.285 assemble my table, do a lot of math, blah, blah, blah. 00:31:28.285 --> 00:31:30.410 Knock yourselves out if that's what you want to do. 00:31:30.410 --> 00:31:33.280 There's a shortcut. 00:31:33.280 --> 00:31:39.730 Let's use the fact that when I contract a one form 00:31:39.730 --> 00:31:41.150 on a vector, I get a scalar. 00:31:55.200 --> 00:32:04.260 So let's say I am looking at the beta covariant derivative of p 00:32:04.260 --> 00:32:06.990 alpha on a alpha. 00:32:06.990 --> 00:32:07.890 That's a scalar. 00:32:07.890 --> 00:32:18.155 So this is just the partial derivative. 00:32:27.380 --> 00:32:30.160 And a partial derivative of the product of something I 00:32:30.160 --> 00:32:31.500 can expand out really easily. 00:32:44.550 --> 00:32:46.900 So using the fact that this just becomes the partial, 00:32:46.900 --> 00:32:54.900 I can write this as a alpha d beta p 00:32:54.900 --> 00:32:58.350 alpha plus p downstairs alpha. 00:33:03.580 --> 00:33:06.860 So now what? 00:33:06.860 --> 00:33:13.260 Well, let's rewrite this using the covariant derivative. 00:33:18.862 --> 00:33:21.070 Pardon me a second while I get caught up in my notes. 00:33:21.070 --> 00:33:24.230 Here we are. 00:33:24.230 --> 00:33:31.210 I can write this as the covariant derivative 00:33:31.210 --> 00:33:34.210 minus the correction that comes from that Christoffel symbol. 00:33:45.595 --> 00:33:46.980 Pardon me just a second. 00:33:46.980 --> 00:33:49.480 There's a couple lines here I want to write out very neatly. 00:33:55.740 --> 00:33:57.120 So when I put this in-- 00:34:22.480 --> 00:34:23.170 oops typo. 00:34:29.600 --> 00:34:31.880 That last typo is important, because I'm now going 00:34:31.880 --> 00:34:34.491 to do the relabeling trick. 00:34:34.491 --> 00:34:36.199 So what I'm going to do is take advantage 00:34:36.199 --> 00:34:40.280 of the fact that in this last term, alpha and mu 00:34:40.280 --> 00:34:42.050 are both dummy indices. 00:34:42.050 --> 00:34:47.030 So on this last term that I have written down here, 00:34:47.030 --> 00:34:48.560 I'm going to swap out alpha and mu. 00:35:23.770 --> 00:35:29.570 When I do that, notice that the first term and the last term 00:35:29.570 --> 00:35:32.060 will both be proportional to the component a alpha. 00:35:52.480 --> 00:35:57.130 Now, let's require that the covariant derivative 00:35:57.130 --> 00:36:00.220 when it acts on two things that are multiplied together, 00:36:00.220 --> 00:36:04.430 it's a derivative, so it should do what derivatives ordinarily 00:36:04.430 --> 00:36:04.930 do. 00:36:04.930 --> 00:36:09.680 So what we're going to do is require 00:36:09.680 --> 00:36:15.230 that when I take this covariant derivative, 00:36:15.230 --> 00:36:30.040 I should be able to write the result like so. 00:36:30.040 --> 00:36:35.750 It's a healthy thing that any derivative should do. 00:36:35.750 --> 00:36:37.680 So comparing, I look at that, and go, 00:36:37.680 --> 00:36:40.830 oh, I've got the covariant derivative of my one form 00:36:40.830 --> 00:36:41.330 there. 00:36:41.330 --> 00:36:42.140 Just compare forms. 00:37:06.840 --> 00:37:11.850 Very, very similar, but notice the minus sign. 00:37:11.850 --> 00:37:14.530 There's a minus sign that's been introduced there, 00:37:14.530 --> 00:37:19.470 and that minus sign guarantees, if you actually expand out 00:37:19.470 --> 00:37:22.290 that combination of covariant derivatives 00:37:22.290 --> 00:37:24.660 I have on the previous line, there's a nice cancellation 00:37:24.660 --> 00:37:28.050 so that the scalar that I get when I contract p 00:37:28.050 --> 00:37:31.740 on a, in fact, doesn't have anything special going on when 00:37:31.740 --> 00:37:34.033 I do the covariant derivative. 00:37:41.483 --> 00:37:43.150 So I'm going to generalize this further, 00:37:43.150 --> 00:37:45.130 but let me just make a quick comment here. 00:37:45.130 --> 00:37:48.820 I began this little calculation by saying, 00:37:48.820 --> 00:37:53.260 given how we started our calculation 00:37:53.260 --> 00:37:55.240 of the covariant derivative of a vector, 00:37:55.240 --> 00:37:56.920 we could have begun by just taking 00:37:56.920 --> 00:37:59.620 lots of derivatives of the basis one forms, and assembling 00:37:59.620 --> 00:38:01.840 all these various tables, and things like that. 00:38:01.840 --> 00:38:10.960 If you had done this, it's simple to find, 00:38:10.960 --> 00:38:22.050 based on an analysis like this, that if you 00:38:22.050 --> 00:38:34.640 take a partial derivative of a one form, 00:38:34.640 --> 00:38:37.740 that you get sort of a linear combination of one forms back. 00:38:37.740 --> 00:38:39.167 Looks just like what you got when 00:38:39.167 --> 00:38:41.250 you took a partial derivative of the basis vector, 00:38:41.250 --> 00:38:42.560 but with a minus sign. 00:38:42.560 --> 00:38:48.670 And what that minus sign does is it enforces, if you go back 00:38:48.670 --> 00:38:51.730 to a lecture from ages ago, when I first introduced basis one 00:38:51.730 --> 00:39:04.480 forms, it enforces the idea that when I combine basis one forms 00:39:04.480 --> 00:39:08.110 with basis vectors, I get an identity object out of this, 00:39:08.110 --> 00:39:09.910 which is itself a constant. 00:39:09.910 --> 00:39:12.190 If you are the kind of person who 00:39:12.190 --> 00:39:14.320 likes that sort of mathematical rigor, 00:39:14.320 --> 00:39:16.090 some textbooks will start with this, 00:39:16.090 --> 00:39:18.790 and then derive other things from that-- 00:39:18.790 --> 00:39:21.310 sort of six of one, half a dozen of the other. 00:39:21.310 --> 00:39:24.460 So we could go on at this point. 00:39:24.460 --> 00:39:27.310 And I could say, how do I do this 00:39:27.310 --> 00:39:30.070 with a tensor that has two indices in the upstairs 00:39:30.070 --> 00:39:30.890 position? 00:39:30.890 --> 00:39:32.350 How do I do this with a tensor that 00:39:32.350 --> 00:39:34.373 has two indices in the downstairs position? 00:39:34.373 --> 00:39:35.790 How do I do it for a tensor that's 00:39:35.790 --> 00:39:37.960 got 17 indices in the upstairs position 00:39:37.960 --> 00:39:40.490 and 38 in the downstairs position? 00:39:40.490 --> 00:39:43.480 The answer is easily deduced from doing 00:39:43.480 --> 00:39:44.970 these kinds of rules, so I'm just 00:39:44.970 --> 00:39:46.900 going to write down a couple of examples and state 00:39:46.900 --> 00:39:47.900 what it turns out to be. 00:39:56.660 --> 00:40:01.390 So basically, imagine I want to take the covariant derivative-- 00:40:01.390 --> 00:40:03.470 let's do the stress energy tensor-- 00:40:03.470 --> 00:40:06.747 covariant derivative of T mu nu. 00:40:06.747 --> 00:40:09.080 So remember, the way that the Christoffel got into there 00:40:09.080 --> 00:40:11.420 is that when I looked at the derivative of a vector, 00:40:11.420 --> 00:40:15.000 I was looking at derivatives of basis objects. 00:40:15.000 --> 00:40:16.750 Well, now I'm going to look at derivatives 00:40:16.750 --> 00:40:18.870 of two different basis objects. 00:40:18.870 --> 00:40:22.140 So I'm going to wind up with two Christoffel symbols. 00:40:29.100 --> 00:40:42.543 You can kind of think of it as coming along and correcting 00:40:42.543 --> 00:40:43.460 each of these indices. 00:40:46.440 --> 00:40:50.520 I can do this with the indices in the downstairs position. 00:40:57.290 --> 00:40:58.250 Guess what? 00:40:58.250 --> 00:41:07.510 Comes along and corrects all them with minus signs. 00:41:29.860 --> 00:41:31.930 Just for completeness, let me just 00:41:31.930 --> 00:41:35.250 write down the general rule. 00:41:35.250 --> 00:41:40.330 If I am looking at the covariant derivative of a tensor 00:41:40.330 --> 00:41:48.330 with a gajillion upstairs indices 00:41:48.330 --> 00:41:51.990 and a gajillion downstairs indices, 00:41:51.990 --> 00:41:59.690 you get one term that's just a partial derivative of that guy, 00:41:59.690 --> 00:42:10.800 and you get a Christoffel coupling 00:42:10.800 --> 00:42:13.270 for every one of these. 00:42:13.270 --> 00:42:32.020 Plus sign for all the upstairs, minus sign 00:42:32.020 --> 00:42:35.820 for all the downstairs. 00:42:43.380 --> 00:42:44.500 That was a little tedious. 00:42:44.500 --> 00:42:46.450 You basically just, when I give you a tensor like that, 00:42:46.450 --> 00:42:47.950 you just kind of have to go through. 00:42:47.950 --> 00:42:49.723 And it becomes sort of almost monkey work. 00:42:49.723 --> 00:42:51.640 You just have to rotely go through and correct 00:42:51.640 --> 00:42:54.500 every one of the indices using an algorithm that kind of looks 00:42:54.500 --> 00:42:55.000 like this. 00:42:59.800 --> 00:43:02.300 Oh, jeez, there's absolutely a minus sign on the second one. 00:43:02.300 --> 00:43:05.085 Thank you. 00:43:05.085 --> 00:43:06.040 I appreciate that. 00:43:14.160 --> 00:43:19.687 So the way that we have done things so far, 00:43:19.687 --> 00:43:22.020 and I kind of emphasized, it sort of smells like the way 00:43:22.020 --> 00:43:26.190 to do this is you pick your new coordinate representation, 00:43:26.190 --> 00:43:33.420 you throw together all of your various basis objects, 00:43:33.420 --> 00:43:35.467 and then you just start going whee, 00:43:35.467 --> 00:43:37.050 let's start taking derivatives and see 00:43:37.050 --> 00:43:39.380 how all these things vary with respect to each other, 00:43:39.380 --> 00:43:42.780 assemble my table of the gammas, and then do 00:43:42.780 --> 00:43:45.550 my covariant derivative. 00:43:45.550 --> 00:43:48.220 If that were, in fact, the way we did it, 00:43:48.220 --> 00:43:50.260 I would not have chosen my research career 00:43:50.260 --> 00:43:51.480 to focus on this field. 00:43:51.480 --> 00:43:52.990 That would suck. 00:43:52.990 --> 00:43:58.030 Certainly prior to Odin providing us with Mathematica, 00:43:58.030 --> 00:44:01.750 it would have been absolutely undoable. 00:44:01.750 --> 00:44:05.290 Even with it, though, it would be incredibly tedious. 00:44:05.290 --> 00:44:28.890 So there is a better way to do this, 00:44:28.890 --> 00:44:33.170 and it comes via the metric. 00:44:33.170 --> 00:44:39.800 Before I derive what the algorithm actually is, 00:44:39.800 --> 00:44:43.250 I want to introduce an extremely important property of tensor 00:44:43.250 --> 00:44:46.430 relationships that we are going to come back to and use 00:44:46.430 --> 00:44:47.750 quite a bit in this course. 00:44:57.923 --> 00:45:00.340 So this is something that we have actually kind of alluded 00:45:00.340 --> 00:45:03.280 to repeatedly, but I want to make it a little more formal 00:45:03.280 --> 00:45:04.742 and just clearly state it. 00:45:12.180 --> 00:45:14.370 So this relationship that I'm going to use 00:45:14.370 --> 00:45:27.350 is some kind of a tensor equation, a tensorial equation 00:45:27.350 --> 00:45:37.620 that holds in one representation must 00:45:37.620 --> 00:45:42.182 hold in all representations. 00:45:46.250 --> 00:45:48.350 Come back to the intuition when I first 00:45:48.350 --> 00:45:50.780 began describing physics in terms of geometric objects 00:45:50.780 --> 00:45:51.460 in spacetime. 00:45:51.460 --> 00:45:54.560 One of the key points I tried to really emphasize 00:45:54.560 --> 00:45:57.410 the difference of is that I can have different-- 00:45:57.410 --> 00:46:00.457 let's say my arm is a particular vector in spacetime. 00:46:00.457 --> 00:46:02.540 Someone running through the room at three-quarters 00:46:02.540 --> 00:46:05.030 the speed of light will use different representations 00:46:05.030 --> 00:46:06.013 to describe my arm. 00:46:06.013 --> 00:46:07.430 They will see length contractions. 00:46:07.430 --> 00:46:09.722 They will see things sort of spanning different things. 00:46:09.722 --> 00:46:11.450 But the geometric object, the thing 00:46:11.450 --> 00:46:16.400 which goes between two events in spacetime, that does not 00:46:16.400 --> 00:46:18.920 change, even though the representation of those events 00:46:18.920 --> 00:46:20.138 might. 00:46:20.138 --> 00:46:22.430 This remains true not just for Lorentz transformations, 00:46:22.430 --> 00:46:25.070 but for all classes of transformations 00:46:25.070 --> 00:46:28.850 that we might care to use in our analysis. 00:46:28.850 --> 00:46:32.560 Changing the representation cannot change the equation. 00:46:39.480 --> 00:46:43.350 Written that way, it sounds like, well, duh, 00:46:43.350 --> 00:46:47.741 but as we'll see, it's got important consequences. 00:47:01.980 --> 00:47:06.440 So as a warm-up exercise of how we might want to use this, 00:47:06.440 --> 00:47:09.710 let's think about the double gradient of a scalar. 00:47:40.540 --> 00:47:47.710 So let's define-- let's just say that this is the object that I 00:47:47.710 --> 00:47:50.320 want to compute. 00:47:50.320 --> 00:47:54.044 Let's first do this in a Cartesian representation. 00:47:57.920 --> 00:47:59.710 In a Cartesian representation, I just 00:47:59.710 --> 00:48:02.875 take two partial derivatives. 00:48:05.490 --> 00:48:07.790 I've got a couple basis one forms for this. 00:48:17.520 --> 00:48:20.102 So I've got something like this. 00:48:20.102 --> 00:48:21.560 The thing which I want to emphasize 00:48:21.560 --> 00:48:25.220 is that as written, in Cartesian coordinates, d 00:48:25.220 --> 00:48:27.350 alpha d beta of phi-- 00:48:27.350 --> 00:48:32.564 those are the components of a tensor in this representation. 00:48:40.150 --> 00:48:42.880 And the key thing is that they are obviously 00:48:42.880 --> 00:48:48.690 symmetric on exchange of the indices alpha and beta. 00:48:53.795 --> 00:48:55.420 If I'm just taking partial derivatives, 00:48:55.420 --> 00:48:57.400 doesn't matter what order I take them in. 00:48:57.400 --> 00:48:58.794 That's got to be symmetric. 00:49:02.970 --> 00:49:05.780 Let's now look at the double gradient 00:49:05.780 --> 00:49:08.374 of a scalar in a more general representation. 00:49:31.630 --> 00:49:37.430 So in a general representation, I'm 00:49:37.430 --> 00:49:40.220 going to require these two derivatives 00:49:40.220 --> 00:49:42.870 to be covariant derivatives. 00:49:42.870 --> 00:49:47.690 Now, we know one of them can be very trivially replaced with 00:49:47.690 --> 00:49:50.090 a partial, but the other one cannot. 00:49:50.090 --> 00:49:51.590 Hold that thought for just a second. 00:49:55.610 --> 00:50:01.220 If this thing is symmetric in the Cartesian representation, 00:50:01.220 --> 00:50:04.822 I claim it must also be true in a general representation. 00:50:17.370 --> 00:50:27.890 In other words, exchanging the order of covariant derivatives 00:50:27.890 --> 00:50:31.853 when they act on a scalar should give me the same thing back. 00:50:31.853 --> 00:50:33.020 Let's see what this implies. 00:51:02.540 --> 00:51:11.725 So if I require the following to be true, that's saying-- 00:51:18.630 --> 00:51:19.130 oops. 00:51:29.230 --> 00:51:31.030 So let's expand this out one more level. 00:51:37.680 --> 00:51:48.700 So now, I'm correcting that downstairs index and over here. 00:52:04.640 --> 00:52:08.740 So the terms involving nothing but partials, 00:52:08.740 --> 00:52:10.940 they obviously cancel. 00:52:10.940 --> 00:52:13.247 I have a common factor of d mu of phi. 00:52:13.247 --> 00:52:15.330 So let's move one of these over to the other side. 00:52:18.240 --> 00:52:20.360 What we've learned is that this requirement, 00:52:20.360 --> 00:52:33.097 that this combination of derivatives be symmetric, 00:52:33.097 --> 00:52:35.430 tells me something about the symmetry of the Christoffel 00:52:35.430 --> 00:52:36.140 symbols itself. 00:52:43.640 --> 00:52:46.880 If you go back to that little table 00:52:46.880 --> 00:52:49.250 that I wrote down for plane polar coordinates, 00:52:49.250 --> 00:52:50.750 that was one where I just calculated 00:52:50.750 --> 00:52:53.720 only three non-trivial components, 00:52:53.720 --> 00:52:55.202 but there was a symmetry in there. 00:52:55.202 --> 00:52:56.660 And if you go and you check it, you 00:52:56.660 --> 00:53:00.673 will see it's consistent with what I just found right here. 00:53:00.673 --> 00:53:01.840 Pardon me for just a second. 00:53:01.840 --> 00:53:06.600 I want to organize a few of my notes. 00:53:06.600 --> 00:53:08.320 These have gotten all out of order. 00:53:15.960 --> 00:53:16.460 Here it is. 00:53:49.897 --> 00:53:51.605 So let me just use this as an opportunity 00:53:51.605 --> 00:53:52.897 to introduce a bit of notation. 00:53:56.210 --> 00:54:00.770 Whenever I give you a tensor that's got two indices, 00:54:00.770 --> 00:54:10.870 if I write parentheses around those indices, 00:54:10.870 --> 00:54:14.110 this is going to mean that I do what 00:54:14.110 --> 00:54:15.931 is called "symmetrization." 00:54:19.020 --> 00:54:21.090 We're going to use this from time to time. 00:54:21.090 --> 00:54:29.310 If I write square braces, this is what we call 00:54:29.310 --> 00:54:31.057 "anti-symmetrization." 00:54:33.800 --> 00:54:41.220 And so what we just learned is that gamma mu alpha beta 00:54:41.220 --> 00:54:45.450 is equal to gamma mu alpha beta with symmetrization 00:54:45.450 --> 00:54:48.630 on those last two indices. 00:54:48.630 --> 00:54:53.430 We have likewise learned that if I contract 00:54:53.430 --> 00:55:03.780 this against some object, if these were anti-symmetric, 00:55:03.780 --> 00:55:07.027 I must get a 0 out of it. 00:55:07.027 --> 00:55:09.360 So that's a brief aside, but these are important things, 00:55:09.360 --> 00:55:11.610 and I want to make sure you have a chance to see them. 00:55:14.640 --> 00:55:19.710 So trying to make a decision here about where we're going to 00:55:19.710 --> 00:55:22.410 want to carry things forward. 00:55:22.410 --> 00:55:25.140 We're approaching the end of one set of notes. 00:55:25.140 --> 00:55:27.810 There's still one more thing I want to do. 00:55:27.810 --> 00:55:29.760 So I set this whole thing up by saying 00:55:29.760 --> 00:55:32.760 that I wanted to give you guys an algorithm for how 00:55:32.760 --> 00:55:34.640 to generate the Christoffel symbols. 00:55:51.230 --> 00:56:02.200 The way I'm going to do this is by examining 00:56:02.200 --> 00:56:11.300 the gradient of the metric. 00:56:11.300 --> 00:56:16.140 So suppose I want to compute the following tensor 00:56:16.140 --> 00:56:18.390 quantity-- let's say is g the metric tensor, 00:56:18.390 --> 00:56:20.890 written here in the fairly abstract notation. 00:56:20.890 --> 00:56:26.530 And this is my full-on tensor gradient of this thing. 00:56:32.500 --> 00:56:36.380 So if you want to write this out in its full glory, 00:56:36.380 --> 00:56:38.150 I might write this as something like this. 00:56:44.030 --> 00:56:47.320 But if you stop and think about this for just a second, 00:56:47.320 --> 00:56:53.370 let's go back to this principle. 00:56:53.370 --> 00:56:55.830 An equation that is tensorial in one representation 00:56:55.830 --> 00:56:57.990 must be tensorial in all. 00:56:57.990 --> 00:57:03.600 Suppose I choose the Cartesian representation of this thing. 00:57:11.330 --> 00:57:26.870 Well, then here's what it looks like there. 00:57:26.870 --> 00:57:29.970 But this is a constant. 00:57:29.970 --> 00:57:36.030 So if I do this in Cartesian coordinates, it has to be 0. 00:57:36.030 --> 00:57:39.840 The only way that I can make this sort of comport 00:57:39.840 --> 00:57:43.170 with this principle that an equation that 00:57:43.170 --> 00:57:45.480 is tensorial in one representation 00:57:45.480 --> 00:57:48.370 holds in all representations-- 00:57:48.370 --> 00:57:57.150 this leads me to say, I need to require 00:57:57.150 --> 00:58:02.970 that the covariant derivative of the metric be equal to 0. 00:58:06.410 --> 00:58:07.460 We're going to use this. 00:58:07.460 --> 00:58:09.710 And I think this will be the last detailed calculation 00:58:09.710 --> 00:58:11.250 I do in today's lecture. 00:58:11.250 --> 00:58:12.830 We're going to use this to find a way 00:58:12.830 --> 00:58:16.027 to get the Christoffel symbol from partial derivatives 00:58:16.027 --> 00:58:16.610 of the metric. 00:58:45.860 --> 00:58:48.050 There's a lot of terms here and there's 00:58:48.050 --> 00:58:49.260 a lot of little indices. 00:58:49.260 --> 00:58:51.552 So I'm going to do my best to make my handwriting neat. 00:58:54.137 --> 00:58:55.720 I'm going to write down a relationship 00:58:55.720 --> 00:58:57.100 that I call "Roman numeral I." 00:58:59.740 --> 00:59:03.750 The covariant derivative in the gamma direction, G alpha beta-- 00:59:15.298 --> 00:59:17.590 you know what, let me put this down a little bit lower, 00:59:17.590 --> 00:59:19.507 so I can get these two terms on the same line. 00:59:29.140 --> 00:59:34.020 So I get this thing that involves two Christoffel 00:59:34.020 --> 00:59:38.790 symbols correcting those two indices. 00:59:43.418 --> 00:59:44.460 This is going to equal 0. 00:59:49.490 --> 00:59:52.680 I don't really seem to have gotten very far. 00:59:52.680 --> 00:59:57.120 This is true, but I now have two bloody Christoffel 00:59:57.120 --> 00:59:59.760 symbols that I've somehow managed to work into this. 00:59:59.760 --> 01:00:01.410 What I'm trying to do is find a way 01:00:01.410 --> 01:00:04.470 to get one, and equate it to things involving derivatives 01:00:04.470 --> 01:00:05.520 of the metric. 01:00:05.520 --> 01:00:09.670 So this is sort of a ruh-roh kind of moment. 01:00:09.670 --> 01:00:11.770 But there's nothing special about this order 01:00:11.770 --> 01:00:14.030 of the indices. 01:00:14.030 --> 01:00:18.790 So with the audacity that only comes 01:00:18.790 --> 01:00:21.790 from knowing the answer in advance, what I'm going to do 01:00:21.790 --> 01:00:23.500 is permute the indices. 01:00:54.620 --> 01:00:57.800 Then go, oh, let's permute the indices once more. 01:01:25.665 --> 01:01:27.790 So I'll give you guys a moment to catch up with me. 01:01:27.790 --> 01:01:29.520 Don't forget, these notes will be scanned and added 01:01:29.520 --> 01:01:30.187 to the web page. 01:01:30.187 --> 01:01:33.240 So if you don't want to follow along writing down 01:01:33.240 --> 01:01:35.760 every little detail, I understand, 01:01:35.760 --> 01:01:38.568 although personally, I find that these things gel 01:01:38.568 --> 01:01:40.110 a little bit better when you actually 01:01:40.110 --> 01:01:41.110 write them out yourself. 01:01:45.780 --> 01:01:52.260 So those are three ways that I can assert that the metric has 01:01:52.260 --> 01:01:54.125 no covariant derivative. 01:01:54.125 --> 01:01:55.500 They all are basically expressing 01:01:55.500 --> 01:01:56.790 that same physical fact. 01:01:56.790 --> 01:01:59.730 I'm just permuting the indices. 01:01:59.730 --> 01:02:03.080 Now there's no better way to describe this 01:02:03.080 --> 01:02:06.240 than you sort of just stare at this for a few moments, 01:02:06.240 --> 01:02:10.694 and then go, gee, I wonder what would happen if-- 01:02:10.694 --> 01:02:12.760 so stare at this for a little while. 01:02:12.760 --> 01:02:15.850 And then construct-- you know I have three things that 01:02:15.850 --> 01:02:17.320 are equal to 0. 01:02:17.320 --> 01:02:20.020 So I can add them together, I can subtract one 01:02:20.020 --> 01:02:20.830 from the other. 01:02:20.830 --> 01:02:24.760 I can add two and subtract one, whatever. 01:02:24.760 --> 01:02:26.800 They should all give me 0. 01:02:26.800 --> 01:02:33.160 And the particular combination I want to look at 01:02:33.160 --> 01:02:36.530 is what I get when I take relationship one 01:02:36.530 --> 01:02:40.840 and I subtract from it two and three. 01:02:44.760 --> 01:02:48.470 So I'm going to get one term that 01:02:48.470 --> 01:02:54.210 are just these three combinations of derivatives, 01:02:54.210 --> 01:02:54.710 gamma. 01:03:03.020 --> 01:03:05.040 And I get something that looks like-- 01:03:22.570 --> 01:03:25.490 let me write this out and then pause and make a comment. 01:03:52.470 --> 01:03:54.690 So I sort of made some lame jokes a few moments ago 01:03:54.690 --> 01:03:56.250 that essentially, the only reason I was able to get 01:03:56.250 --> 01:03:57.930 this was by knowing the answer in the back of the book, 01:03:57.930 --> 01:03:58.890 essentially. 01:03:58.890 --> 01:04:01.560 And to be perfectly blunt, for me personally, that's 01:04:01.560 --> 01:04:02.220 probably true. 01:04:02.220 --> 01:04:04.157 When I first wrote this down, I probably 01:04:04.157 --> 01:04:05.490 did need to follow an algorithm. 01:04:05.490 --> 01:04:07.823 But if I was doing this ab initio, if I was sitting down 01:04:07.823 --> 01:04:09.840 to first do this, what's really going on here 01:04:09.840 --> 01:04:12.600 is the reason I wrote out all these different combinations 01:04:12.600 --> 01:04:16.620 of things is that I was trying to gather terms together 01:04:16.620 --> 01:04:20.010 in such a way that I could take advantage of that symmetry. 01:04:20.010 --> 01:04:24.330 A few moments ago, we showed that the Christoffel symbols 01:04:24.330 --> 01:04:28.210 are symmetric on the lower two indices. 01:04:28.210 --> 01:04:30.630 And so by putting out all these different combinations 01:04:30.630 --> 01:04:33.660 of things, I was then able to combine them in such a way 01:04:33.660 --> 01:04:35.510 that certain terms-- look at this 01:04:35.510 --> 01:04:38.610 and go, ah, symmetry on alpha and gamma means 01:04:38.610 --> 01:04:41.490 this whole term dies. 01:04:41.490 --> 01:04:45.620 Symmetry on beta and gamma means this whole term dies. 01:04:45.620 --> 01:04:49.020 Symmetry on alpha and beta means these two guys combine, 01:04:49.020 --> 01:04:50.100 and I get a factor of 2. 01:05:05.150 --> 01:05:06.400 So let's clean up our algebra. 01:05:16.200 --> 01:05:19.060 Move a bunch of our terms to the other side equation, 01:05:19.060 --> 01:05:22.255 since it's a blah, blah, blah equals 0. 01:05:22.255 --> 01:05:27.010 And what we get when we do this is g mu downstairs 01:05:27.010 --> 01:05:32.140 gamma is equal to 1/2. 01:05:50.450 --> 01:05:52.770 What we're going to do now is we will define everything 01:05:52.770 --> 01:05:53.770 on the right-hand side-- 01:05:53.770 --> 01:05:58.380 I've kind of emphasized earlier that the Christoffels are not 01:05:58.380 --> 01:06:01.202 themselves tensors, but we're going to imagine that we can 01:06:01.202 --> 01:06:02.910 nonetheless-- we're not going to imagine, 01:06:02.910 --> 01:06:04.010 we're just going to define-- 01:06:04.010 --> 01:06:06.240 we're going to say that we're allowed to raise and lower 01:06:06.240 --> 01:06:08.782 their indices using the metric, in the same way you guys been 01:06:08.782 --> 01:06:11.640 doing with vectors and one forms and other kinds of tensors. 01:06:11.640 --> 01:06:14.580 So let's call everything on the right-hand side 01:06:14.580 --> 01:06:21.890 here gamma with all the indices in the downstairs position, 01:06:21.890 --> 01:06:24.240 gamma sub gamma alpha beta. 01:06:24.240 --> 01:06:29.500 And then this is simply what I get 01:06:29.500 --> 01:06:37.370 when I click all of these things together like so. 01:06:41.870 --> 01:06:44.390 If you go and you look up the formulas 01:06:44.390 --> 01:06:46.040 for this in various textbooks that 01:06:46.040 --> 01:06:49.070 give these different kinds of formulas, 01:06:49.070 --> 01:06:53.520 you will typically see it written as 1/2 g upstairs 01:06:53.520 --> 01:06:57.590 indices, and then all this stuff in parentheses after that. 01:06:57.590 --> 01:06:59.780 When you look things up, this is the typical formula 01:06:59.780 --> 01:07:01.133 that is given in these books. 01:07:01.133 --> 01:07:02.300 This is where it comes from. 01:07:08.750 --> 01:07:15.850 So I need to check one thing because it appears my notes are 01:07:15.850 --> 01:07:18.530 a little bit out of order here. 01:07:18.530 --> 01:07:22.210 But nonetheless, this is a good point, 01:07:22.210 --> 01:07:29.490 since we've just finished a pretty long calculation, 01:07:29.490 --> 01:07:30.900 this is a good point to introduce 01:07:30.900 --> 01:07:32.175 an important physical point. 01:07:32.175 --> 01:07:33.550 We're going to come back to this. 01:07:33.550 --> 01:07:35.835 We're going to start this on Thursday. 01:07:38.940 --> 01:07:41.460 But I want to begin making some physical points that 01:07:41.460 --> 01:07:43.800 are going to take us from special relativity 01:07:43.800 --> 01:07:45.200 to general relativity. 01:07:48.433 --> 01:07:50.100 So despite the fact that I've introduced 01:07:50.100 --> 01:07:54.810 this new mathematical framework, everything 01:07:54.810 --> 01:08:04.050 that I have done so far is in the context 01:08:04.050 --> 01:08:06.810 of special relativity. 01:08:06.810 --> 01:08:08.880 I'm going to make a more precise definition 01:08:08.880 --> 01:08:10.490 of special relativity right now. 01:08:17.390 --> 01:08:18.890 So special relativity-- we are going 01:08:18.890 --> 01:08:32.990 to think of this moving forward as the theory which allows us 01:08:32.990 --> 01:08:50.270 to cover the entire spacetime manifold 01:08:50.270 --> 01:08:52.520 using inertial reference frames. 01:09:00.865 --> 01:09:02.990 So we use inertial reference frames or essentially, 01:09:02.990 --> 01:09:05.660 Lorenz reference frames, and saying that Lorentz coordinates 01:09:05.660 --> 01:09:07.220 are good everywhere. 01:09:07.220 --> 01:09:09.890 We know we can go between different Lorentz reference 01:09:09.890 --> 01:09:13.550 frames using Lorentz transformations. 01:09:13.550 --> 01:09:15.800 But the key thing is that if special relativity were 01:09:15.800 --> 01:09:19.100 correct, the entire universe would be accurately described 01:09:19.100 --> 01:09:23.720 by any inertial reference frame you care to write down. 01:09:23.720 --> 01:09:28.160 And I will probably only be able to do about half of this right 01:09:28.160 --> 01:09:29.330 now. 01:09:29.330 --> 01:09:32.109 We'll pick it up next time, if I cannot finish this. 01:09:32.109 --> 01:09:33.800 The key thing which I want to emphasize 01:09:33.800 --> 01:09:38.000 is, gravity breaks this. 01:09:45.899 --> 01:09:51.660 As soon as you put gravity into your theory of relativity, 01:09:51.660 --> 01:09:53.220 you cannot have-- 01:09:53.220 --> 01:09:58.820 so we will call this a global inertial frame, 01:09:58.820 --> 01:10:03.950 an inertial frame that is good everywhere, 01:10:03.950 --> 01:10:06.080 so "global" in the mathematical sense, 01:10:06.080 --> 01:10:08.330 not "global" in the geographic sense-- not just earth, 01:10:08.330 --> 01:10:11.200 the whole universe, essentially. 01:10:11.200 --> 01:10:13.580 As soon as we put in gravity, we no longer 01:10:13.580 --> 01:10:16.580 have global reference frames and global inertial reference 01:10:16.580 --> 01:10:18.230 frames. 01:10:18.230 --> 01:10:19.977 That word "inertial" is important. 01:10:29.730 --> 01:10:34.650 But we are going to be allowed to have local inertial frames. 01:10:34.650 --> 01:10:36.570 I have not precisely defined the difference 01:10:36.570 --> 01:10:40.035 what "local" means in this case, and I won't for a few lectures. 01:10:48.440 --> 01:10:52.998 But to give you a preview as to what that means, 01:10:52.998 --> 01:10:54.540 it's essentially going to say that we 01:10:54.540 --> 01:10:56.100 can define an inertial coordinate 01:10:56.100 --> 01:11:01.200 system that is good over a particular region of spacetime. 01:11:01.200 --> 01:11:03.120 And we're going to have to discuss and come up 01:11:03.120 --> 01:11:05.738 with ways of understanding what the boundaries of that region 01:11:05.738 --> 01:11:07.155 are, and how to make this precise. 01:11:11.160 --> 01:11:13.020 So the statement that gravity breaks 01:11:13.020 --> 01:11:16.990 the existence of global Lorentz frames, like I said, 01:11:16.990 --> 01:11:18.000 it's a two-part thing. 01:11:24.070 --> 01:11:30.730 I'm going to use a very handwavy argument which 01:11:30.730 --> 01:11:32.620 can be made quite rigorous later, 01:11:32.620 --> 01:11:35.380 but I want to keep it to this handwaving level, because first 01:11:35.380 --> 01:11:37.030 of all, it actually was first done 01:11:37.030 --> 01:11:41.205 by a very high-level mathematical physicist named 01:11:41.205 --> 01:11:43.705 Alfred Schild, who worked in the early theory of relativity. 01:11:43.705 --> 01:11:45.413 It's sort of like he was so mathematical, 01:11:45.413 --> 01:11:48.550 if it was good enough for him, that's good enough for me. 01:11:48.550 --> 01:11:51.190 And I think even though it's a little handwavy, and kind 01:11:51.190 --> 01:11:53.890 of goofy in at least one place, it 01:11:53.890 --> 01:11:57.220 gives a good physical sense as to why it is gravity 01:11:57.220 --> 01:11:58.990 begins to mess things up. 01:11:58.990 --> 01:12:04.988 So part one is the fact that there exists 01:12:04.988 --> 01:12:06.030 a gravitational redshift. 01:12:20.020 --> 01:12:25.760 So here's where I'm going to be particularly silly, 01:12:25.760 --> 01:12:28.753 but I will back up my silliness by the fact that everything 01:12:28.753 --> 01:12:30.170 silly that I say here has actually 01:12:30.170 --> 01:12:33.100 been experimentally verified, or at least 01:12:33.100 --> 01:12:34.800 the key physical output of this. 01:12:34.800 --> 01:12:38.840 So imagine you are on top of a tower 01:12:38.840 --> 01:12:49.110 and you drop a rock of rest mass m off the top of this tower. 01:12:59.484 --> 01:13:00.400 So here you are. 01:13:04.112 --> 01:13:04.820 Here's your rock. 01:13:08.060 --> 01:13:08.720 The rock falls. 01:13:15.770 --> 01:13:22.810 There's a wonderful device down here which I label with a p. 01:13:22.810 --> 01:13:24.301 It's called a photonulater. 01:13:27.070 --> 01:13:34.820 And what the photonulater does, it 01:13:34.820 --> 01:13:44.370 converts the rock into a single photon, 01:13:44.370 --> 01:13:46.680 and it does so conserving energy. 01:13:53.790 --> 01:13:56.710 So when this rock falls, the instant 01:13:56.710 --> 01:14:05.880 before it goes into your photonulater, 01:14:05.880 --> 01:14:10.590 just use Newtonian physics plus the notion of rest energy. 01:14:10.590 --> 01:14:15.510 So it's got an energy of m-- mC squared, if you prefer, 01:14:15.510 --> 01:14:17.100 its rest energy-- 01:14:17.100 --> 01:14:19.948 plus what it acquired after falling-- 01:14:19.948 --> 01:14:21.990 pardon me, I forgot to give you a distance here-- 01:14:21.990 --> 01:14:23.450 after falling a distance h. 01:14:30.330 --> 01:14:32.432 So that means that the photon that I shoot up 01:14:32.432 --> 01:14:33.140 from this thing-- 01:14:36.090 --> 01:14:39.300 let me put a few things on this board. 01:14:49.860 --> 01:15:00.350 So the instant that I create this photon, 01:15:00.350 --> 01:15:02.730 this thing goes out, and it's going 01:15:02.730 --> 01:15:05.882 to have a frequency omega bottom, which is simply 01:15:05.882 --> 01:15:06.840 related to that energy. 01:15:15.320 --> 01:15:17.880 This photon immediately is shot back up to the top, 01:15:17.880 --> 01:15:20.760 where clever you, you happen to have 01:15:20.760 --> 01:15:23.507 in your hands a rerockulater. 01:15:26.310 --> 01:15:29.430 The rerockulater, as the name obviously implies, 01:15:29.430 --> 01:15:31.080 converts the photon back into a rock. 01:15:52.500 --> 01:15:54.930 Now, suppose it does so-- both the photonulater 01:15:54.930 --> 01:15:57.660 and the rerockulater are fine MIT engineering. 01:15:57.660 --> 01:16:00.930 There are no losses anywhere in this thing. 01:16:00.930 --> 01:16:02.160 So there's no friction. 01:16:02.160 --> 01:16:03.510 There's no extra heat generated. 01:16:03.510 --> 01:16:07.830 It does it conserving energy, 100%. 01:16:07.830 --> 01:16:09.620 What is the energy at the top? 01:16:18.450 --> 01:16:21.717 Well, you might naively say, ah, it's just 01:16:21.717 --> 01:16:22.800 going to go up to the top. 01:16:22.800 --> 01:16:24.300 It's going to have that same energy. 01:16:24.300 --> 01:16:27.780 It might just have it in the form of a photon and omega b. 01:16:30.740 --> 01:16:34.850 There will be some frequency at the top. 01:16:34.850 --> 01:16:37.103 And your initial guess might be it's 01:16:37.103 --> 01:16:39.270 going to be the same as the frequency at the bottom. 01:16:39.270 --> 01:16:41.478 But if you do that, you're going to suddenly discover 01:16:41.478 --> 01:16:43.850 that your rock has more energy than it started out with, 01:16:43.850 --> 01:16:45.890 and you can redirect it back down, send it back up. 01:16:45.890 --> 01:16:47.480 Next thing you know, you've got yourself a perpetual motion 01:16:47.480 --> 01:16:48.770 machine. 01:16:48.770 --> 01:16:50.750 So all you need to do is get your photonulater 01:16:50.750 --> 01:16:52.375 and your rerockulater going, and you've 01:16:52.375 --> 01:16:54.630 got yourself a perpetual motion machine here. 01:16:54.630 --> 01:16:56.463 I will grant that's probably not the weakest 01:16:56.463 --> 01:16:58.550 part of this argument. 01:16:58.550 --> 01:16:59.500 Suppose you had this. 01:16:59.500 --> 01:17:00.542 I mean, you look at this. 01:17:00.542 --> 01:17:05.210 If technology allowed you to make these goofy devices, 01:17:05.210 --> 01:17:08.000 you would instantly look at this and say, look, 01:17:08.000 --> 01:17:09.542 if I am not to have-- 01:17:09.542 --> 01:17:11.000 let's just say I live in a universe 01:17:11.000 --> 01:17:12.417 where I'm fine with photonulaters. 01:17:12.417 --> 01:17:14.780 I'm fine with rerockulaters, but damn it, 01:17:14.780 --> 01:17:16.100 energy has to be conserved. 01:17:16.100 --> 01:17:19.100 I am not fine with perpetual motion machines. 01:17:19.100 --> 01:17:26.200 If that's the case, we always fight perpetual motion. 01:17:31.990 --> 01:17:35.485 We must have that the energy at the top 01:17:35.485 --> 01:17:37.360 is equal to the energy this guy started with. 01:17:37.360 --> 01:17:38.980 When it sort of gets back into-- 01:17:38.980 --> 01:17:41.140 imagine that your rerockulater is shaped 01:17:41.140 --> 01:17:42.460 like a baseball catcher's mitt. 01:17:42.460 --> 01:17:45.100 You want that thing to just land gently in your mitt, 01:17:45.100 --> 01:17:48.170 and just be a perfectly gentle, little landing there. 01:17:48.170 --> 01:18:07.290 And when you put all this together, going through this, 01:18:07.290 --> 01:18:10.223 taking advantage of the fact that if you work in units where 01:18:10.223 --> 01:18:11.640 you've put your c's back in, there 01:18:11.640 --> 01:18:21.860 will be a factor of g h over c squared appearing in here, what 01:18:21.860 --> 01:18:25.340 you find is that the frequency at the top 01:18:25.340 --> 01:18:30.000 is less than the frequency at the bottom. 01:18:30.000 --> 01:18:33.340 In other words, the light has gotten a little bit redder. 01:18:33.340 --> 01:18:36.630 now I fully confess, I did this via the silliest argument 01:18:36.630 --> 01:18:37.650 possible. 01:18:37.650 --> 01:18:39.720 But I want to emphasize that this 01:18:39.720 --> 01:18:45.060 is one of the most precisely verified predictions of gravity 01:18:45.060 --> 01:18:47.250 and relativity theory. 01:18:47.250 --> 01:18:51.030 This was first done, actually, up the street at Harvard, 01:18:51.030 --> 01:18:54.740 by what's called the Pound-Rebka experiment. 01:18:54.740 --> 01:18:57.240 And the basic principles of what is going on with this right 01:18:57.240 --> 01:18:57.958 now-- 01:18:57.958 --> 01:19:00.000 I just took this out to make sure my alarm is not 01:19:00.000 --> 01:19:01.667 about to go off, but I want to emphasize 01:19:01.667 --> 01:19:03.208 it's actually built into the workings 01:19:03.208 --> 01:19:04.630 of the global positioning system. 01:19:04.630 --> 01:19:07.170 Because this fact that light signals may travel out 01:19:07.170 --> 01:19:09.960 of a gravitational potential, they 01:19:09.960 --> 01:19:12.390 get redshifted, needs to be taken into account in order 01:19:12.390 --> 01:19:15.210 to do the precise metrology that GPS allows. 01:19:15.210 --> 01:19:21.630 Now, this is part one, this idea that light 01:19:21.630 --> 01:19:24.720 gets redder as it climbs out of a gravitational field. 01:19:24.720 --> 01:19:27.230 Part two, which I will do on Thursday, 01:19:27.230 --> 01:19:31.500 is to show that if there is a global inertial frame, 01:19:31.500 --> 01:19:34.830 there is no way for light to get redder as it climbs out 01:19:34.830 --> 01:19:38.080 of a gravitational potential. 01:19:38.080 --> 01:19:42.280 You cannot have both gravity and a global inertial reference 01:19:42.280 --> 01:19:42.898 frame. 01:19:42.898 --> 01:19:44.690 That's where I will pick it up on Thursday. 01:19:44.690 --> 01:19:45.700 So we'll do that. 01:19:45.700 --> 01:19:49.000 And we will then begin talking about how 01:19:49.000 --> 01:19:52.510 we can try to put the principles of relativity and gravity 01:19:52.510 --> 01:19:53.695 together. 01:19:53.695 --> 01:19:56.320 And in some sense, this is when our study of general relativity 01:19:56.320 --> 01:19:58.060 will begin in earnest. 01:19:58.060 --> 01:20:01.110 All right, so let us stop there.