WEBVTT 00:00:00.000 --> 00:00:04.920 [SQUEAKING] [RUSTLING] [CLICKING] 00:00:11.320 --> 00:00:14.710 SCOTT HUGHES: So this is unfortunately a rather sad 00:00:14.710 --> 00:00:17.350 lecture, [LAUGHS] since this will be the last one that 00:00:17.350 --> 00:00:19.650 I'm giving to a live audience here, 00:00:19.650 --> 00:00:21.400 although any of you who are grad students, 00:00:21.400 --> 00:00:24.670 if you want to come and keep me company while I'm recording 00:00:24.670 --> 00:00:28.200 my next couple of lectures-- 00:00:28.200 --> 00:00:29.200 I'm actually not joking. 00:00:29.200 --> 00:00:30.992 It's really weird talking to an empty room, 00:00:30.992 --> 00:00:33.640 and so it might be kind of nice to at least have a face 00:00:33.640 --> 00:00:34.810 to react to. 00:00:34.810 --> 00:00:38.920 So I will be recording these, and making them available. 00:00:38.920 --> 00:00:41.920 And per the announcements that I sent out today, 00:00:41.920 --> 00:00:44.710 there are some additional assignments 00:00:44.710 --> 00:00:45.785 that I've begun posting. 00:00:45.785 --> 00:00:47.410 I've begun putting things like that up. 00:00:47.410 --> 00:00:50.333 But for the time being, ignore due dates. 00:00:50.333 --> 00:00:52.000 In fact, I would like to say, especially 00:00:52.000 --> 00:00:54.880 if you are getting ready to leave town, just 00:00:54.880 --> 00:00:56.990 ignore the assignments. 00:00:56.990 --> 00:01:00.550 There will be updates posted fairly soon. 00:01:00.550 --> 00:01:03.430 You know, [CHUCKLES] sometimes-- there's 00:01:03.430 --> 00:01:05.820 a saying that Martin Schmidt, our provost, 00:01:05.820 --> 00:01:10.060 said one time in respect to a particular initiative 00:01:10.060 --> 00:01:12.460 as it was unwinding, and I feel like it applies here: 00:01:12.460 --> 00:01:15.055 we're basically sewing together the parachute after we've 00:01:15.055 --> 00:01:16.180 jumped out of the airplane. 00:01:16.180 --> 00:01:19.570 [CHUCKLES] And so that's kind of where we are right now, 00:01:19.570 --> 00:01:22.930 and hopefully it'll be assembled before we hit the ground. 00:01:22.930 --> 00:01:25.220 But we're all kind of figuring out the way this goes. 00:01:25.220 --> 00:01:27.652 So there will be updates with respect to assignments 00:01:27.652 --> 00:01:29.110 and things like that, and you know, 00:01:29.110 --> 00:01:30.870 we're just trying to figure out-- 00:01:30.870 --> 00:01:32.320 I'll just be blunt-- how the hell 00:01:32.320 --> 00:01:34.906 we're going to actually pull this off. 00:01:34.906 --> 00:01:39.210 [LAUGHS] It's not curved enough. 00:01:39.210 --> 00:01:41.790 All right, so let me get back to where we were. 00:01:41.790 --> 00:01:42.630 I want to begin-- 00:01:42.630 --> 00:01:44.520 so what we worked with yesterday was 00:01:44.520 --> 00:01:47.220 we derived this mathematical object called the Riemann 00:01:47.220 --> 00:01:48.480 curvature tensor. 00:01:48.480 --> 00:01:52.080 And the way we discussed it was by thinking about some vector 00:01:52.080 --> 00:01:55.290 that I parallel transported around a closed loop. 00:01:55.290 --> 00:01:58.410 I chose a parallelogram because it allowed me to sort of nicely 00:01:58.410 --> 00:02:00.450 formulate the mathematics. 00:02:00.450 --> 00:02:02.353 So if I imagine a parallelogram who's 00:02:02.353 --> 00:02:04.020 got sides that are sort of close enough, 00:02:04.020 --> 00:02:06.600 that parallel is a well-defined term, 00:02:06.600 --> 00:02:10.169 and the sides are in two different directions, 00:02:10.169 --> 00:02:12.270 one's of length delta x in the lambda direction, 00:02:12.270 --> 00:02:14.910 one is a delta x in the sigma direction, 00:02:14.910 --> 00:02:17.130 when I take this vector all the way around the loop, 00:02:17.130 --> 00:02:18.940 I find the vector has changed. 00:02:18.940 --> 00:02:21.320 It's pointing in a different direction, 00:02:21.320 --> 00:02:25.130 and this describes the way in which it has sort of moved 00:02:25.130 --> 00:02:27.230 as I go around that loop. 00:02:27.230 --> 00:02:30.070 That object is-- even though it's created-- 00:02:30.070 --> 00:02:33.118 it's assembled from the Christoffel symbols, which 00:02:33.118 --> 00:02:34.910 are not tensor, but it's done in such a way 00:02:34.910 --> 00:02:37.970 that their nontensorialness cancels out, 00:02:37.970 --> 00:02:41.270 and this four-index object R is a tensor. 00:02:41.270 --> 00:02:42.710 As you can see, it involves terms 00:02:42.710 --> 00:02:45.740 like derivatives of the Christoffels and Christoffel 00:02:45.740 --> 00:02:46.550 times Christoffel. 00:02:46.550 --> 00:02:50.150 So that's where nonlinearity is going to enter. 00:02:50.150 --> 00:02:52.155 I want to sort of fairly quickly-- 00:02:52.155 --> 00:02:54.530 because I do want to get into some other stuff that we do 00:02:54.530 --> 00:02:55.310 with this-- 00:02:55.310 --> 00:02:58.493 talk a bit about the symmetry of this thing. 00:02:58.493 --> 00:03:00.410 So I discussed last time how you look at this, 00:03:00.410 --> 00:03:02.120 and naively, you think you've got-- 00:03:02.120 --> 00:03:04.328 in four dimensions, four dimensions, four [INAUDIBLE] 00:03:04.328 --> 00:03:04.828 states. 00:03:04.828 --> 00:03:06.500 It looks like you have 256 components. 00:03:10.130 --> 00:03:12.793 But in fact, there are quite a few symmetries 00:03:12.793 --> 00:03:14.960 associated with this which reduces it significantly. 00:03:14.960 --> 00:03:16.640 I kind of gave you the answer last time, 00:03:16.640 --> 00:03:18.920 but let me talk about where that comes from. 00:03:18.920 --> 00:03:22.340 I'm going to describe a way to see these symmetries, which 00:03:22.340 --> 00:03:23.840 there's a couple ways you can do it, 00:03:23.840 --> 00:03:25.010 and I'm going to pick the one that's 00:03:25.010 --> 00:03:26.052 essentially the simplest. 00:03:26.052 --> 00:03:27.710 It's perhaps not as-- 00:03:27.710 --> 00:03:31.670 there's a few ways to do it that a purist might prefer, 00:03:31.670 --> 00:03:33.170 but in the interest of time, I think 00:03:33.170 --> 00:03:34.520 this is sort of the best one. 00:03:34.520 --> 00:03:39.320 So the first thing is that just by inspecting this answer, 00:03:39.320 --> 00:03:45.950 you should see that this object is antisymmetric if you 00:03:45.950 --> 00:03:49.150 exchange the final two indices. 00:03:49.150 --> 00:03:52.880 OK, just look at that formula, switch the indices 00:03:52.880 --> 00:03:55.440 sigma and lambda, and you get a minus sign. 00:03:55.440 --> 00:03:57.440 Physically, that actually hopefully makes sense, 00:03:57.440 --> 00:03:59.120 because what that is basically telling you-- remember the way 00:03:59.120 --> 00:04:01.460 we did this, this operation of parallel transporting 00:04:01.460 --> 00:04:02.420 around a parallelogram. 00:04:02.420 --> 00:04:05.660 We sort of went along, say, the lambda direction, then 00:04:05.660 --> 00:04:08.015 the sigma, then the lambda, than the sigma. 00:04:08.015 --> 00:04:09.735 If I exchange those indices, it's 00:04:09.735 --> 00:04:12.110 kind of like I actually go in the opposite direction, OK? 00:04:12.110 --> 00:04:13.580 I go in the other direction first. 00:04:13.580 --> 00:04:15.650 I go on the sigma first, then the lambda. 00:04:15.650 --> 00:04:19.700 And so this corresponds to doing this operation, which 00:04:19.700 --> 00:04:21.510 I'll remind you is called a holonomy. 00:04:21.510 --> 00:04:23.978 It corresponds to reversing direction. 00:04:39.914 --> 00:04:40.650 OK? 00:04:40.650 --> 00:04:42.990 So that's a fairly easy one to say. 00:04:42.990 --> 00:04:46.950 The others are a little bit tougher, and like I said, 00:04:46.950 --> 00:04:49.920 what I'm going to do is follow kind of a shortcut to do this. 00:04:49.920 --> 00:04:53.550 One can do it using sort of the full glory of the Riemann 00:04:53.550 --> 00:04:56.490 tensor here, but life gets a little bit easier for us 00:04:56.490 --> 00:04:57.885 if we do the following. 00:05:03.010 --> 00:05:07.005 So what you want to do first is lower the first index. 00:05:11.100 --> 00:05:14.400 So what I'm going to do is look at R, 00:05:14.400 --> 00:05:19.740 alpha in the downstairs, mu lambda sigma. 00:05:19.740 --> 00:05:27.520 This is what I get when I contract, like so. 00:05:27.520 --> 00:05:29.020 And then what I'm going to do is I'm 00:05:29.020 --> 00:05:32.330 going to do all of my analysis that follows in a local Lorentz 00:05:32.330 --> 00:05:32.830 frame. 00:05:39.500 --> 00:05:41.720 When I go into the local Lorentz frame, 00:05:41.720 --> 00:05:44.840 the metric at a particular point is 00:05:44.840 --> 00:05:46.250 the metric of flat spacetime. 00:05:46.250 --> 00:05:47.570 It's the A to mu nu. 00:05:56.830 --> 00:06:02.333 My Christoffel symbols all vanish, 00:06:02.333 --> 00:06:04.000 but you have to be a little bit careful, 00:06:04.000 --> 00:06:06.490 because the derivatives of the Christoffel do not vanish, 00:06:06.490 --> 00:06:06.990 right? 00:06:06.990 --> 00:06:08.532 So one of the key points is that when 00:06:08.532 --> 00:06:10.680 you go into those local Lorentz frame, 00:06:10.680 --> 00:06:13.840 there is a little bit of curvature associated with it. 00:06:13.840 --> 00:06:15.060 So the derivatives do not-- 00:06:24.810 --> 00:06:26.780 OK, so the reason I'm doing this-- like I said, 00:06:26.780 --> 00:06:29.570 you actually could do the little counting 00:06:29.570 --> 00:06:32.600 exercise I'm about to go through with this whole thing. 00:06:32.600 --> 00:06:33.828 It's just messy, OK? 00:06:33.828 --> 00:06:35.870 And so this is sort of a quick way to see the way 00:06:35.870 --> 00:06:36.578 it all comes out. 00:06:36.578 --> 00:06:39.625 Because this is a tensor relationship, 00:06:39.625 --> 00:06:41.000 you are guaranteed that something 00:06:41.000 --> 00:06:43.430 you conclude in a convenient reference frame 00:06:43.430 --> 00:06:45.050 will hold in all of them, OK? 00:06:45.050 --> 00:06:47.055 So it's a nice way to do this. 00:06:47.055 --> 00:06:48.680 Those of you who are sort of, you know, 00:06:48.680 --> 00:06:52.238 particularly purist, knock yourselves out 00:06:52.238 --> 00:06:54.530 trying to figure out how to do this sort of in general. 00:06:54.530 --> 00:06:58.290 This is adequate for our purposes here. 00:06:58.290 --> 00:07:01.800 So when I go and I do this, here's 00:07:01.800 --> 00:07:06.300 what my all downstairs Riemann tensor turns into. 00:07:08.980 --> 00:07:11.800 I'm going to lose the Christoffel squared 00:07:11.800 --> 00:07:15.820 terms because Christoffel vanishes, 00:07:15.820 --> 00:07:19.930 and the only thing that is left are 00:07:19.930 --> 00:07:24.628 the terms that involve the derivative of the Christoffel, 00:07:24.628 --> 00:07:26.910 OK? 00:07:26.910 --> 00:07:28.840 Let's do something a little bit further. 00:07:28.840 --> 00:07:34.680 So let's now plug in the definition of the Christoffel, 00:07:34.680 --> 00:07:38.580 and we'll use the metric before we actually go into the frame, 00:07:38.580 --> 00:07:40.897 so that we can sort of get the form that we 00:07:40.897 --> 00:07:41.730 would get with this. 00:08:04.760 --> 00:08:08.700 I should have made myself be a little bit more careful here. 00:08:08.700 --> 00:08:12.560 So I'm going to put LLF on this, to make it clear that I'm doing 00:08:12.560 --> 00:08:14.890 this in a local Lorentz frame. 00:08:18.080 --> 00:08:23.540 So I do this in a local Lorentz frame, 00:08:23.540 --> 00:08:26.450 and what this turns into-- so there's a lot of algebra 00:08:26.450 --> 00:08:27.710 that I'm not writing out. 00:08:37.580 --> 00:08:49.490 This is an alpha, g sigma mu minus d sigma d mu g alpha 00:08:49.490 --> 00:08:59.370 lambda plus d sigma d alpha g lambda mu, OK? 00:08:59.370 --> 00:09:02.445 So notice, it is only the second derivative of the metric 00:09:02.445 --> 00:09:03.570 that is appearing here, OK? 00:09:03.570 --> 00:09:05.958 If I go into the local Lorentz frame, that's 00:09:05.958 --> 00:09:07.500 the one degree of freedom we were not 00:09:07.500 --> 00:09:09.522 able to transform away. 00:09:09.522 --> 00:09:11.730 Anything involving the metric in the first derivative 00:09:11.730 --> 00:09:13.105 dies in this particular frame. 00:09:13.105 --> 00:09:14.730 But this is good enough for us to begin 00:09:14.730 --> 00:09:17.820 to count up symmetries, and to see what 00:09:17.820 --> 00:09:19.420 things are going to look like. 00:09:19.420 --> 00:09:24.328 And so I don't have any great guidance for how to do this. 00:09:24.328 --> 00:09:26.370 This is one of those things where, literally, you 00:09:26.370 --> 00:09:28.203 just sort of stare at it for a little while, 00:09:28.203 --> 00:09:32.010 and you see what happens when you play around with exchanging 00:09:32.010 --> 00:09:35.012 various pairs of indices. 00:09:35.012 --> 00:09:36.220 So stare at this for a while. 00:09:39.260 --> 00:09:41.080 OK, so you can see right away-- 00:09:41.080 --> 00:09:43.680 just for counting purposes, let me write down 00:09:43.680 --> 00:09:47.010 the one that I argued was kind of obvious even 00:09:47.010 --> 00:09:48.360 in the full form. 00:09:48.360 --> 00:09:51.000 It's the one you get when you exchange 00:09:51.000 --> 00:09:52.530 the last pairs of indices. 00:09:52.530 --> 00:09:54.120 We'll call that symmetry one. 00:09:57.380 --> 00:10:00.380 Oh, bugger, thank you. 00:10:00.380 --> 00:10:05.030 [CHUCKLES] 00:10:05.030 --> 00:10:05.550 Yeah. 00:10:05.550 --> 00:10:07.100 [CHUCKLES] That's a good way to-- that's 00:10:07.100 --> 00:10:09.183 a very complicated way of writing zero, otherwise. 00:10:09.183 --> 00:10:10.670 [LAUGHING] 00:10:10.670 --> 00:10:13.905 All right, if you exchange the first ones, 00:10:13.905 --> 00:10:15.530 you also-- if you look at this formula, 00:10:15.530 --> 00:10:17.380 you can see when you exchange alpha and mu, you 00:10:17.380 --> 00:10:18.140 get a minus sign. 00:10:29.940 --> 00:10:33.190 So one that might be-- 00:10:33.190 --> 00:10:35.690 this one takes a little bit of creativity to sort of see it, 00:10:35.690 --> 00:10:36.640 where it comes out. 00:10:42.250 --> 00:10:45.900 Suppose what I do is I exchange the first two indices 00:10:45.900 --> 00:10:48.190 for the second two indices. 00:10:48.190 --> 00:10:50.140 Well, it turns out, if you do that, you just 00:10:50.140 --> 00:10:51.140 get the expression back. 00:10:55.116 --> 00:10:56.110 OK? 00:10:56.110 --> 00:10:59.842 So if I just swap alpha mu, make them my last two, 00:10:59.842 --> 00:11:01.800 make lambda sigma my first two, [CLICKS TONGUE] 00:11:01.800 --> 00:11:03.450 I get Riemann right back. 00:11:03.450 --> 00:11:04.590 So that's another symmetry. 00:11:10.680 --> 00:11:15.980 And then finally, someday, when I've 00:11:15.980 --> 00:11:19.393 got a little bit more bandwidth in my class to do this, 00:11:19.393 --> 00:11:20.810 I'd really like to sort of justify 00:11:20.810 --> 00:11:23.268 where you can derive this one a little bit more rigorously. 00:11:23.268 --> 00:11:25.100 But for now, it's sufficient just 00:11:25.100 --> 00:11:27.210 to sort of say, look at that formula, 00:11:27.210 --> 00:11:29.720 and you'll see that it's true. 00:11:29.720 --> 00:11:34.490 If you take-- and then you cyclically 00:11:34.490 --> 00:11:54.180 permute the second, third, and fourth indices, 00:11:54.180 --> 00:11:59.090 you get zero, OK? 00:11:59.090 --> 00:12:02.680 There is a variant of this particular symmetry, which 00:12:02.680 --> 00:12:09.260 is that if you take this and you antisymmetrize on these things, 00:12:09.260 --> 00:12:10.340 you get zero, OK? 00:12:10.340 --> 00:12:11.960 And I've got some words in the notes 00:12:11.960 --> 00:12:13.900 that, in the interest of time, I'm 00:12:13.900 --> 00:12:15.993 not going to go through them, but they are posted, 00:12:15.993 --> 00:12:17.660 demonstrating why those two are actually 00:12:17.660 --> 00:12:18.980 equivalent to one another. 00:12:18.980 --> 00:12:22.688 What it boils down to is, expand out this antisymmetrization-- 00:12:22.688 --> 00:12:24.980 and I'm going to do that again for a three-index object 00:12:24.980 --> 00:12:26.840 a little bit later in this lecture-- 00:12:26.840 --> 00:12:30.800 and then occasionally invoke one of these other ones, 00:12:30.800 --> 00:12:34.220 and you'll see that these two, four and four prime, are 00:12:34.220 --> 00:12:37.110 saying the same thing. 00:12:37.110 --> 00:12:41.360 So go through all of these different symmetries, 00:12:41.360 --> 00:12:45.860 and you'll find n to the 4 independent components 00:12:45.860 --> 00:12:53.042 goes over to n squared times n squared minus 1 over 12, OK? 00:12:53.042 --> 00:12:54.250 Not terribly hard to do that. 00:12:54.250 --> 00:12:57.650 It's a little exercise in combinatorics. 00:12:57.650 --> 00:13:01.490 And when you do that for n equals 20, as I mentioned, 00:13:01.490 --> 00:13:04.031 you end up with-- 00:13:04.031 --> 00:13:05.505 excuse me, not n equals 20. 00:13:05.505 --> 00:13:12.980 When you do this with n equals 4, you end up with 20. 00:13:16.190 --> 00:13:18.680 And just for completeness, let me actually write out 00:13:18.680 --> 00:13:27.430 that if you actually-- if you go into the local Lorentz frame, 00:13:27.430 --> 00:13:36.130 your spacetime metric is in fact minus 1, 1, 1, 00:13:36.130 --> 00:13:38.060 1 on the diagonal. 00:13:38.060 --> 00:13:41.680 But you, of course, have these quadratic corrections, 00:13:41.680 --> 00:13:44.503 and one can in fact write them out explicitly. 00:13:49.820 --> 00:13:52.950 So that is what the time piece ends up looking like. 00:13:56.440 --> 00:13:59.700 There is an off-diagonal piece, which 00:13:59.700 --> 00:14:02.370 only enters at quadratic order. 00:14:07.463 --> 00:14:10.780 It looks like this, and move this down a little bit lower. 00:14:28.640 --> 00:14:30.750 That's what your space-based piece will look like. 00:14:30.750 --> 00:14:34.010 So this explicitly constructs the coordinate system 00:14:34.010 --> 00:14:38.990 used in a freely falling frame, including these second order 00:14:38.990 --> 00:14:39.920 corrections. 00:14:39.920 --> 00:14:45.077 So this particular form is known as Riemann normal coordinates. 00:14:50.810 --> 00:14:52.993 So if you are-- 00:14:52.993 --> 00:14:55.160 this is discussed in a little bit more detail in one 00:14:55.160 --> 00:15:00.620 of the optional readings in the textbook 00:15:00.620 --> 00:15:04.520 by Eric Poisson, A Relativist's Toolkit. 00:15:04.520 --> 00:15:05.875 Very nice discussion. 00:15:14.800 --> 00:15:29.750 All right, so now that we have the curvature tensor in hand, 00:15:29.750 --> 00:15:32.900 we really have essentially every tool 00:15:32.900 --> 00:15:35.713 that matters for 8.962, OK? 00:15:35.713 --> 00:15:37.130 There's a little bit more analysis 00:15:37.130 --> 00:15:44.210 we need to do with this guy, but we now have all the pieces. 00:15:44.210 --> 00:15:47.840 OK, there's no major new mathematical object 00:15:47.840 --> 00:15:50.210 or no object described in geometry 00:15:50.210 --> 00:15:51.922 that I need to build in order for us 00:15:51.922 --> 00:15:53.630 to make a relativistic theory of gravity. 00:15:55.845 --> 00:15:58.220 I do want to talk about the curvature tensor a little bit 00:15:58.220 --> 00:16:01.430 more, because there are a couple of properties about it 00:16:01.430 --> 00:16:03.905 that are very important for us. 00:16:03.905 --> 00:16:05.780 And I think we'll have just enough time today 00:16:05.780 --> 00:16:06.738 to sort of get to them. 00:16:06.738 --> 00:16:10.220 That will allow us to set up, and in the first lecture 00:16:10.220 --> 00:16:11.900 that I will record in an empty room, 00:16:11.900 --> 00:16:15.990 actually derive a relativistic gravity equation. 00:16:15.990 --> 00:16:18.568 So let me talk about a few variants 00:16:18.568 --> 00:16:19.610 of this curvature tensor. 00:16:25.570 --> 00:16:35.140 So first, suppose you take the trace on some pair of indices. 00:16:35.140 --> 00:16:38.280 So what I mean by the trace is, essentially, 00:16:38.280 --> 00:16:41.040 imagine all the indices are in the downstairs position, 00:16:41.040 --> 00:16:43.110 and I contract it with the metric, 00:16:43.110 --> 00:16:44.852 so that I'm summing over them. 00:16:44.852 --> 00:16:46.560 Well, first of all, you'll note that when 00:16:46.560 --> 00:16:50.130 you do this, if you were to take the trace on indices 1 and 2, 00:16:50.130 --> 00:16:51.660 they are antisymmetric. 00:16:51.660 --> 00:16:55.710 The metric is symmetric, so contracting metric with indices 00:16:55.710 --> 00:16:57.720 1 and 2 is going to give me 0. 00:16:57.720 --> 00:17:02.580 If I can track symmetric with indices 3 and 4, I get 0, OK? 00:17:02.580 --> 00:17:05.099 So the only trace that makes sense 00:17:05.099 --> 00:17:09.359 is do it on either indices 1 and 3 or on indices 2 and 4. 00:17:09.359 --> 00:17:10.920 Let's do it on indices 1 and 3. 00:17:21.300 --> 00:17:28.740 So what I'm going to do is evaluate R, alpha mu alpha nu-- 00:17:28.740 --> 00:17:35.922 which if you like, you can write this as R alpha beta of R. 00:17:35.922 --> 00:17:37.610 Here, I'll turn it like this. 00:17:37.610 --> 00:17:40.500 Beta mu alpha nu-- 00:17:40.500 --> 00:17:48.410 we're going to define this as R with two indices, OK? 00:17:48.410 --> 00:17:50.042 Just little r and mu, nu. 00:17:50.042 --> 00:17:52.250 This actually shows up enough that it's given a name. 00:17:52.250 --> 00:17:55.900 This is called the Ricci curvature tensor. 00:18:13.370 --> 00:18:16.580 If you were to like, I said, if you do your trace on indices 1 00:18:16.580 --> 00:18:18.630 and 2, you get 0. 00:18:18.630 --> 00:18:24.120 If you do it on 3 and 4, you get 0. 00:18:24.120 --> 00:18:26.810 If you do it on 2 and 4, it's exactly the same 00:18:26.810 --> 00:18:29.810 as doing it on 1 and 3, because of all these other symmetries. 00:18:29.810 --> 00:18:31.970 So in fact, this ends up, when you count up 00:18:31.970 --> 00:18:33.680 all your different symmetries, this 00:18:33.680 --> 00:18:36.108 is the only trace that is meaningful, OK? 00:18:36.108 --> 00:18:38.150 There are a few other pairs where you get a minus 00:18:38.150 --> 00:18:40.860 sign, but still the same thing. 00:18:40.860 --> 00:18:44.040 So this is the only trace that ends up being meaningful. 00:18:44.040 --> 00:18:47.120 This is actually, in fact, it's not 00:18:47.120 --> 00:18:48.925 too hard to show that this is symmetric. 00:18:48.925 --> 00:18:50.300 So even though, you know, Riemann 00:18:50.300 --> 00:18:52.910 had all these crazy antisymmetries and symmetries, 00:18:52.910 --> 00:18:54.440 this one is simpler, OK? 00:19:07.070 --> 00:19:10.280 One way to do this is just to take the exact expression 00:19:10.280 --> 00:19:14.390 that we wrote down for the Riemann tensor 00:19:14.390 --> 00:19:16.730 in terms of derivatives of Christoffel and Christoffel 00:19:16.730 --> 00:19:20.930 squared, and just write it out in this traced over form. 00:19:20.930 --> 00:19:29.570 So when you do this, you get this. 00:19:29.570 --> 00:19:33.050 This guy is symmetric on the bottom two indices. 00:19:40.680 --> 00:19:43.210 Now, this isn't obviously symmetric, 00:19:43.210 --> 00:19:45.340 but recall, a lecture or two ago, I 00:19:45.340 --> 00:19:47.650 worked through a couple of identities that involved 00:19:47.650 --> 00:19:49.465 the determinant of the metric. 00:19:49.465 --> 00:19:52.420 It turns out, if you go back and you look up these identities, 00:19:52.420 --> 00:19:59.550 you can rewrite this term as d nu of d mu of the log of j-- 00:19:59.550 --> 00:20:02.230 excuse me, log of square root of j. 00:20:02.230 --> 00:20:04.298 Symmetric when you exchange mu and nu. 00:20:07.440 --> 00:20:21.710 And then you get two terms that look like this. 00:20:21.710 --> 00:20:23.600 OK, this one, pretty obviously symmetric 00:20:23.600 --> 00:20:25.268 on exchange of mu and nu. 00:20:25.268 --> 00:20:27.560 When you do this one, you might think to yourself, hmm, 00:20:27.560 --> 00:20:29.143 that one doesn't quite look symmetric. 00:20:29.143 --> 00:20:32.760 But remember, alpha and beta are both dummy indices. 00:20:32.760 --> 00:20:35.120 And so in fact, when you exchange mu and nu, 00:20:35.120 --> 00:20:37.948 this last term just turns back into itself. 00:20:37.948 --> 00:20:39.740 So that's kind of interesting, because R mu 00:20:39.740 --> 00:20:42.020 nu-- the reason I went through that little exercise-- 00:20:42.020 --> 00:20:47.360 this is a symmetric tensor, and we're 00:20:47.360 --> 00:20:49.620 going to work in four dimensions of spacetime here. 00:20:49.620 --> 00:20:56.180 So symmetric 4 by 4 has 10 independent components. 00:21:01.710 --> 00:21:02.880 Riemann had 20. 00:21:02.880 --> 00:21:06.510 Somehow, this Ricci has-- 00:21:06.510 --> 00:21:09.570 in some sense, you can of it as having 10 of-- 00:21:09.570 --> 00:21:11.880 the 20 components associated with Riemann 00:21:11.880 --> 00:21:15.230 are encoded in this guy. 00:21:15.230 --> 00:21:16.340 Where are the other 10? 00:21:16.340 --> 00:21:18.548 I'm going to talk about that in just a moment or two. 00:21:35.520 --> 00:21:37.290 Before I do that, I will just note 00:21:37.290 --> 00:21:39.060 that we are going to want to also know 00:21:39.060 --> 00:21:43.440 about the trace of the Ricci tensor. 00:21:43.440 --> 00:21:56.090 So if you compute this, this is often just abbreviated R 00:21:56.090 --> 00:21:59.000 with no indices whatsoever. 00:21:59.000 --> 00:22:01.660 This is called the Ricci scalar or the curvature scalar. 00:22:16.510 --> 00:22:19.930 So there's another variant of curvature which we're not 00:22:19.930 --> 00:22:22.490 going to use very much, but I wanted to just talk about very, 00:22:22.490 --> 00:22:25.400 very briefly. 00:22:25.400 --> 00:22:29.173 The derivation of this is highly non-obvious, 00:22:29.173 --> 00:22:30.673 so let me just write it out, and I'm 00:22:30.673 --> 00:22:32.840 going to talk about it briefly. 00:22:32.840 --> 00:22:37.880 So suppose I define a four-index tensor, 00:22:37.880 --> 00:22:45.240 C alpha mu lambda sigma, to be the Riemann tensor-- 00:22:45.240 --> 00:22:50.780 minus-- so working in n dimensions-- 00:22:50.780 --> 00:22:56.492 2 over n minus 2, g alpha-- 00:22:56.492 --> 00:22:59.620 I'm going to antisymmetrize here. 00:22:59.620 --> 00:23:01.910 So antisymmetrizing on lambda and sigma-- 00:23:33.340 --> 00:23:42.460 OK, so in the famous words of Rabi, who ordered that? 00:23:42.460 --> 00:23:47.050 So the way that this has been constructed, if you go through 00:23:47.050 --> 00:23:48.940 and you carefully look at the way this thing 00:23:48.940 --> 00:23:52.000 behaves under exchange of any pairs of indices, 00:23:52.000 --> 00:23:54.790 it has exactly the same symmetries as Riemann. 00:24:06.510 --> 00:24:11.160 But if you take the trace of this, turns out to be zero. 00:24:20.512 --> 00:24:22.470 In fact, when you go through, when you count up 00:24:22.470 --> 00:24:24.330 how many independent components it has, 00:24:24.330 --> 00:24:29.291 because it has no trace, it has 10 independent components. 00:24:48.440 --> 00:24:50.500 So this tensor has a name. 00:24:50.500 --> 00:24:55.710 It was first formulated by the mathematician Hermann Weyl. 00:24:55.710 --> 00:24:57.220 And so although it's a German name 00:24:57.220 --> 00:24:59.860 and it's not spelled that way, it is appropriately the "vile" 00:24:59.860 --> 00:25:00.360 tensor. 00:25:00.360 --> 00:25:02.710 It is a pretty vile thing to look at, 00:25:02.710 --> 00:25:04.190 but it plays an important role. 00:25:14.570 --> 00:25:16.270 So in keeping with the idea that it's 00:25:16.270 --> 00:25:18.670 got 10 independent components, Ricci 00:25:18.670 --> 00:25:22.720 has 10 independent components, Riemann has 20. 00:25:22.720 --> 00:25:28.390 Heuristically, you can sort of think and put big quotes 00:25:28.390 --> 00:25:29.560 around all these objects. 00:25:32.330 --> 00:25:33.720 This doesn't mean approximately. 00:25:33.720 --> 00:25:34.960 Let's just put it this way. 00:25:34.960 --> 00:25:45.020 This is, got all the same information as Ricci plus Weyl, 00:25:45.020 --> 00:25:45.610 OK? 00:25:45.610 --> 00:25:48.692 One can, in fact-- by bringing in appropriate powers 00:25:48.692 --> 00:25:50.150 in the metric and things like that, 00:25:50.150 --> 00:25:52.370 one can actually write down a real equation relating these. 00:25:52.370 --> 00:25:53.080 It's a bit of a mess. 00:25:53.080 --> 00:25:54.070 It's not that interesting. 00:25:54.070 --> 00:25:55.695 The key thing I want you to be aware of 00:25:55.695 --> 00:25:59.470 is that all of the curvature content of Riemann 00:25:59.470 --> 00:26:02.137 is sort of split into Ricci and Weyl. 00:26:02.137 --> 00:26:04.720 Looking ahead a little bit-- and because I'm going to be doing 00:26:04.720 --> 00:26:07.960 this in an empty room, I just want to make this point now-- 00:26:07.960 --> 00:26:14.590 we are soon going to see that Ricci is very closely 00:26:14.590 --> 00:26:23.015 related to sources of gravity. 00:26:27.470 --> 00:26:30.650 So we're going to-- when we formulate a theory of gravity, 00:26:30.650 --> 00:26:32.893 working with things, we're going to find 00:26:32.893 --> 00:26:34.310 that there's a tensor that is just 00:26:34.310 --> 00:26:37.010 a slight modification of the Ricci tensor, that 00:26:37.010 --> 00:26:38.510 is equal to the stress energy tensor 00:26:38.510 --> 00:26:39.920 that we use as our source. 00:26:39.920 --> 00:26:42.830 Stress energy tensor is a symmetric 4 by 4 object. 00:26:42.830 --> 00:26:46.550 Those 10 degrees of freedom in the stress energy tensor 00:26:46.550 --> 00:26:48.530 essentially determine the 10 curvature degrees 00:26:48.530 --> 00:26:50.002 of freedom encoded in Ricci. 00:26:50.002 --> 00:26:51.710 But what that tells you is that if you're 00:26:51.710 --> 00:26:56.630 in a region of vacuum, where there is no stress energy, 00:26:56.630 --> 00:26:59.160 there's no Ricci, OK? 00:26:59.160 --> 00:27:00.750 But there is curvature. 00:27:00.750 --> 00:27:01.770 We measure tides. 00:27:01.770 --> 00:27:03.810 We see gravitational effects. 00:27:03.810 --> 00:27:07.260 Weyl ends up being the quantity that describes behavior 00:27:07.260 --> 00:27:11.130 of gravity in vacuum regions. 00:27:11.130 --> 00:27:13.860 Ricci ends up very closely related to how it describes it 00:27:13.860 --> 00:27:15.277 in regions with matter. 00:27:15.277 --> 00:27:17.235 One reason I mentioned this-- so there's only-- 00:27:17.235 --> 00:27:19.140 [CHUCKLES] only one LIGO student here now. 00:27:19.140 --> 00:27:20.193 Hi, Sylvia. 00:27:20.193 --> 00:27:23.010 [CHUCKLES] But in fact, when one is describing 00:27:23.010 --> 00:27:25.830 gravitational radiation, the behavior of the Weyl tensor 00:27:25.830 --> 00:27:28.320 ends up being very important for characterizing 00:27:28.320 --> 00:27:30.390 the degrees of freedom associated with radiation 00:27:30.390 --> 00:27:31.960 and general relativity. 00:27:31.960 --> 00:27:33.173 So that's a little bit ahead. 00:27:33.173 --> 00:27:34.590 There's a few other things you can 00:27:34.590 --> 00:27:36.548 do with it which are related to what are called 00:27:36.548 --> 00:27:38.140 conformal transformations. 00:27:38.140 --> 00:27:40.042 I have a few notes on them, but they're not 00:27:40.042 --> 00:27:41.250 that important for our class. 00:27:41.250 --> 00:27:42.875 It's discussed a little bit in Carroll, 00:27:42.875 --> 00:27:46.333 so I'll leave that as a reading if you are interested, 00:27:46.333 --> 00:27:48.000 but we don't need to go through it here. 00:27:50.753 --> 00:27:52.170 So there are two other things that 00:27:52.170 --> 00:27:54.420 are much more important for us to discuss with respect 00:27:54.420 --> 00:27:57.870 to Riemann first, and I'm going to focus 00:27:57.870 --> 00:28:01.830 the time that we have on them. 00:28:01.830 --> 00:28:04.880 So one of the really important aspects 00:28:04.880 --> 00:28:07.100 of curvature that I've emphasized a few times now 00:28:07.100 --> 00:28:12.320 is the idea that initially parallel geodesics 00:28:12.320 --> 00:28:14.690 become no longer parallel when they're moving 00:28:14.690 --> 00:28:16.790 on a manifold that is curved. 00:28:16.790 --> 00:28:18.740 We're going to use the Riemann curvature 00:28:18.740 --> 00:28:22.640 tensor to quantify what the breakdown of parallelism 00:28:22.640 --> 00:28:23.450 actually means. 00:28:37.130 --> 00:28:39.800 There's a couple of different discussions of this 00:28:39.800 --> 00:28:42.450 that you'll see in various places. 00:28:42.450 --> 00:28:46.460 Carroll's discussion is very brief. 00:28:46.460 --> 00:28:47.210 It's rigorous. 00:28:47.210 --> 00:28:47.930 It's very brief. 00:28:47.930 --> 00:28:51.823 I'm doing something that's a little bit-- 00:28:51.823 --> 00:28:53.240 to my mind, it's a little bit more 00:28:53.240 --> 00:28:54.740 physically motivated OK so I'm going 00:28:54.740 --> 00:28:56.782 to do it in a slightly different way from the way 00:28:56.782 --> 00:28:57.870 it's done in Carroll. 00:28:57.870 --> 00:29:02.930 It's a little close to the way it's done in Schutz's textbook. 00:29:02.930 --> 00:29:05.930 So what we want to do is imagine we have initially 00:29:05.930 --> 00:29:15.830 parallel geodesics, and what we want to do 00:29:15.830 --> 00:29:22.700 is characterize how they become nonparallel. 00:29:30.775 --> 00:29:33.960 Well, we'll put it this way, how they deviate 00:29:33.960 --> 00:29:37.658 as one moves along their world lines-- 00:29:37.658 --> 00:29:38.700 around these world lines. 00:29:41.380 --> 00:29:43.740 OK, so here's what I want to do. 00:29:43.740 --> 00:29:46.270 That's the idea of the calculation that I want to do. 00:29:46.270 --> 00:29:48.570 So what I want to start out with is-- 00:29:48.570 --> 00:29:53.580 let's consider two nearby geodesics. 00:30:01.140 --> 00:30:03.140 And what I'm going to mean by nearby 00:30:03.140 --> 00:30:06.080 is that they are close enough that they're essentially 00:30:06.080 --> 00:30:08.540 in the same local Lorentz frame, OK? 00:30:08.540 --> 00:30:10.260 So they're going to have the same metric. 00:30:10.260 --> 00:30:12.010 I'm going to be able to choose coordinates 00:30:12.010 --> 00:30:14.480 such that the Christoffel symbol is zero for both of them. 00:30:14.480 --> 00:30:16.420 I will not be able to get rid of the second derivative, 00:30:16.420 --> 00:30:17.210 though, OK? 00:30:17.210 --> 00:30:21.010 So that will be where a bit of a difference begins to enter. 00:30:21.010 --> 00:30:25.310 Two nearby geodesics-- and I'm going 00:30:25.310 --> 00:30:28.010 to use just lambda as the affine parameter along them. 00:30:37.610 --> 00:30:38.980 So here's my first one. 00:30:43.660 --> 00:30:45.910 I'm going to make a few definitions on the next board. 00:30:45.910 --> 00:30:53.920 I'm going to call this gamma sub v. And here's my next one. 00:30:53.920 --> 00:30:56.800 I'm going to call it gamma sub u. 00:31:01.740 --> 00:31:02.840 Define two points here. 00:31:05.500 --> 00:31:08.200 OK, I'm going to make a couple of definitions 00:31:08.200 --> 00:31:08.950 on the next board. 00:31:23.310 --> 00:31:25.560 OK, so definitions-- 00:31:25.560 --> 00:31:31.690 I'm going to call u the tangent vector to the curve gamma sub 00:31:31.690 --> 00:31:32.190 u. 00:31:37.760 --> 00:31:38.260 OK? 00:31:38.260 --> 00:31:43.720 So this is equal to dx d lambda on that curve. 00:31:43.720 --> 00:31:52.505 v is the tangent vector to gamma sub 00:31:52.505 --> 00:32:05.660 v. The point A is at lambda 0 on curve gamma sub u. 00:32:08.810 --> 00:32:21.160 A prime is at lambda sub 0 on gamma v. 00:32:21.160 --> 00:32:24.910 So they're both parameterized by a parameter lambda, 00:32:24.910 --> 00:32:26.310 and I'm going to set the-- 00:32:26.310 --> 00:32:29.250 they're basically both synchronizing their clocks 00:32:29.250 --> 00:32:30.160 at the same time. 00:32:30.160 --> 00:32:32.320 Like, at those points, they're going 00:32:32.320 --> 00:32:34.990 to find their starting points as A and A prime. 00:32:34.990 --> 00:32:36.700 What I'm going to do now is I'm going 00:32:36.700 --> 00:32:39.640 to define what's called a geodesic displacement 00:32:39.640 --> 00:32:43.780 factor that points from lambda on the u curve 00:32:43.780 --> 00:32:47.405 to lambda on the v curve. 00:32:47.405 --> 00:32:50.930 And I'm going to use my favorite Greek letter, xi. 00:32:54.570 --> 00:33:00.730 So this points from lambda on gamma u 00:33:00.730 --> 00:33:04.050 to lambda that is basically to the same value. 00:33:07.350 --> 00:33:09.450 Let me make this a little bit more precise. 00:33:09.450 --> 00:33:19.410 Points from the event at lambda on gamma u 00:33:19.410 --> 00:33:23.840 to the event at lambda on gamma v-- 00:33:23.840 --> 00:33:24.340 OK? 00:33:24.340 --> 00:33:27.540 Apologies for being somewhat didactic there, 00:33:27.540 --> 00:33:29.880 but we need to define things carefully. 00:33:29.880 --> 00:33:34.810 So on this-- so here's my initial-- 00:33:38.400 --> 00:33:43.440 so that's what xi looks like at parameter lambda 0. 00:33:43.440 --> 00:33:47.340 And what we want to do is examine 00:33:47.340 --> 00:33:50.340 how xi evolves as one moves along these geodesics. 00:33:50.340 --> 00:33:53.365 That's going to be our goal. 00:33:53.365 --> 00:33:55.740 So let's make things a little bit more quantitative here. 00:34:12.340 --> 00:34:20.650 Xi is going to be equal to x gamma v at lambda 00:34:20.650 --> 00:34:26.290 minus x on gamma u at lambda. 00:34:26.290 --> 00:34:29.817 Finally, I'm going to assume that the curves begin parallel 00:34:29.817 --> 00:34:30.400 to each other. 00:34:38.790 --> 00:34:54.409 That's a statement that u of lambda 0 equals v of lambda 0. 00:34:54.409 --> 00:34:57.995 And it also tells me that I can use-- 00:35:07.280 --> 00:35:11.190 that this must equal 0 at the initial point. 00:35:11.190 --> 00:35:12.960 Not going to equal 0 everywhere. 00:35:12.960 --> 00:35:13.710 In fact, it won't. 00:35:17.230 --> 00:35:19.230 But I want to use this as a boundary condition 00:35:19.230 --> 00:35:20.688 in the calculation I'm about to do. 00:35:31.840 --> 00:35:32.340 OK. 00:35:40.470 --> 00:35:44.000 So what we're going to do is essentially say, you know, 00:35:44.000 --> 00:35:46.670 as we move along these two curves, these are geodesics. 00:35:46.670 --> 00:35:48.800 We're going to use the geodesic equation to slide 00:35:48.800 --> 00:35:50.030 along these two curves. 00:35:50.030 --> 00:35:52.520 We know what the equation is that governs 00:35:52.520 --> 00:35:54.230 u as it moves along gamma u. 00:35:54.230 --> 00:35:56.030 We know what the equation is that governs 00:35:56.030 --> 00:35:58.110 v it moves along gamma v. We're just 00:35:58.110 --> 00:35:59.860 going to take the difference between them, 00:35:59.860 --> 00:36:03.800 and we're going to kind of use that to develop an acceleration 00:36:03.800 --> 00:36:08.757 equation that governs that displacement xi. 00:36:08.757 --> 00:36:11.090 Here's the bit where I differ a little bit from Carroll. 00:36:11.090 --> 00:36:13.940 So Carroll, like I said, Carroll has a very brief discussion, 00:36:13.940 --> 00:36:15.300 which is-- 00:36:15.300 --> 00:36:16.873 it's absolutely right. 00:36:16.873 --> 00:36:19.040 But I want to give a little bit of physical insight, 00:36:19.040 --> 00:36:23.435 and I think you can do that by choosing to work 00:36:23.435 --> 00:36:24.560 in the local Lorentz frame. 00:36:48.510 --> 00:36:51.170 So we're going to work in the local Lorentz frame, 00:36:51.170 --> 00:36:52.920 and I'm going to center this local Lorentz 00:36:52.920 --> 00:37:05.170 frame on the event A. The reason why I'm doing that is this then 00:37:05.170 --> 00:37:12.990 allows me to say, g mu nu at the event is A mu nu. 00:37:23.140 --> 00:37:24.290 Pardon me a second. 00:37:24.290 --> 00:37:31.130 My Christoffel symbols at event A are all going to be zero. 00:37:36.480 --> 00:37:43.702 g mu nu at A prime is also going to be A mu nu. 00:37:43.702 --> 00:37:45.160 We have to be a little bit careful. 00:37:45.160 --> 00:37:49.950 Our Christoffel symbols do not vanish at point A prime 00:37:49.950 --> 00:37:51.700 because there's a little bit of curvature. 00:37:51.700 --> 00:38:06.010 They are in fact related to the fact that I'm sort of slightly 00:38:06.010 --> 00:38:08.708 far away, and I'm picking up that second order correction, 00:38:08.708 --> 00:38:11.150 OK? 00:38:11.150 --> 00:38:16.082 This is sort of where all the important bits of the analysis 00:38:16.082 --> 00:38:17.040 are going to come from. 00:38:17.040 --> 00:38:19.590 This is the fact that they're close to each other. 00:38:19.590 --> 00:38:21.260 I can set up a local Lorentz frame, 00:38:21.260 --> 00:38:23.660 but it's got that little bit of sort of schmutz 00:38:23.660 --> 00:38:25.280 at second order that's coming in, 00:38:25.280 --> 00:38:27.610 and kind of pushing me away from a simple local Lorentz 00:38:27.610 --> 00:38:28.800 frame there-- 00:38:28.800 --> 00:38:31.380 the simple mathematical form right there. 00:38:31.380 --> 00:38:34.430 OK, let's put that up. 00:38:37.010 --> 00:38:37.510 OK. 00:38:51.600 --> 00:38:55.320 So let's look at the equations that govern 00:38:55.320 --> 00:38:56.820 motion along these two curves. 00:39:02.730 --> 00:39:09.160 So the geodesic equation along curve gamma u-- 00:39:09.160 --> 00:39:17.360 and we'll just look at it at A. It's a second order-- 00:39:17.360 --> 00:39:20.630 I'm going to write it in terms of the coordinates. 00:39:20.630 --> 00:39:22.790 It looks like this. 00:39:22.790 --> 00:39:27.620 We're evaluating this at A. At A, my Christoffel symbols are 00:39:27.620 --> 00:39:30.740 all zero, so this is just zero. 00:39:38.830 --> 00:39:46.760 Let's look at it along the other curve, at A prime. 00:40:16.888 --> 00:40:17.388 OK? 00:40:28.280 --> 00:40:32.765 So in just a second, I am going to-- 00:40:37.210 --> 00:40:40.270 wait a second. 00:40:40.270 --> 00:40:45.340 I'm going to substitute in that derivative of Christoffel 00:40:45.340 --> 00:40:46.630 that's going to go in there. 00:40:46.630 --> 00:40:48.400 Before I do that-- so notice, this also 00:40:48.400 --> 00:40:50.800 depends on the velocity as I move along at that point, 00:40:50.800 --> 00:40:51.300 right? 00:40:51.300 --> 00:40:52.697 Typical geodesic-- 00:40:52.697 --> 00:40:54.280 So I'm going to substitute in the fact 00:40:54.280 --> 00:40:59.740 that both the x mu and the x nu velocity here, 00:40:59.740 --> 00:41:04.516 this is defined as v mu. 00:41:04.516 --> 00:41:06.910 But remember, these guys are defined 00:41:06.910 --> 00:41:08.350 as being initially parallel. 00:41:14.655 --> 00:41:16.120 OK? 00:41:16.120 --> 00:41:18.805 So what this tells me is-- 00:41:25.238 --> 00:41:26.420 OK, that's an alpha. 00:41:43.032 --> 00:41:43.990 So this is interesting. 00:41:43.990 --> 00:41:46.780 What I'm seeing is I end up with an equation 00:41:46.780 --> 00:41:50.200 where the derivative of the Christoffel symbol 00:41:50.200 --> 00:41:56.170 is coupling to sort of my motion along that world 00:41:56.170 --> 00:41:58.810 line and the displacement. 00:41:58.810 --> 00:42:01.480 Now, let me just remind you, what I really wanted to do 00:42:01.480 --> 00:42:05.350 was get an equation that governs how this guy changes, OK? 00:42:05.350 --> 00:42:08.560 But this guy is just defined as the difference 00:42:08.560 --> 00:42:13.600 between the position along curve v minus the position 00:42:13.600 --> 00:42:16.758 along curve u. 00:42:16.758 --> 00:42:18.800 So if I take the difference of these two things-- 00:42:27.980 --> 00:42:32.230 pardon me, forgot to label which curve this one is on. 00:42:32.230 --> 00:42:40.570 So I'm doing this at A prime, doing this one at A. 00:42:40.570 --> 00:42:48.560 This is an equation governing the acceleration 00:42:48.560 --> 00:42:49.960 of this displacement. 00:43:00.460 --> 00:43:01.960 OK? 00:43:01.960 --> 00:43:03.390 So this is interesting, OK? 00:43:03.390 --> 00:43:06.260 So what you're sort of seeing is that the geodesic displacement 00:43:06.260 --> 00:43:10.310 looks something like two derivatives 00:43:10.310 --> 00:43:13.820 of the metric coupling in the forward velocity 00:43:13.820 --> 00:43:16.370 and the displacement itself. 00:43:16.370 --> 00:43:22.630 Now, as written, this equation is fine if all you are doing 00:43:22.630 --> 00:43:24.830 is living life in the local Lorentz frame, OK? 00:43:24.830 --> 00:43:26.830 We want to do a little better than that, though. 00:43:26.830 --> 00:43:30.520 So I'm going to do a little bit more massaging of this, 00:43:30.520 --> 00:43:33.610 but I kind of want to emphasize that this already brings out 00:43:33.610 --> 00:43:35.228 the key physical point, OK? 00:43:35.228 --> 00:43:37.520 I can't yet-- you know, you're sort of looking at this, 00:43:37.520 --> 00:43:39.062 and you're thinking to yourself, that 00:43:39.062 --> 00:43:41.620 looks like a piece of Riemann, but it ain't Riemann, right? 00:43:41.620 --> 00:43:43.370 It's a derivative of a Christoffel symbol, 00:43:43.370 --> 00:43:44.370 and that's not a tensor. 00:43:44.370 --> 00:43:46.453 So this isn't quite the kind of thing we want yet. 00:43:46.453 --> 00:43:48.050 We need to do a little bit more work. 00:43:52.430 --> 00:43:55.035 And so what I would sort of say is, 00:43:55.035 --> 00:43:56.410 everything I did up to here, this 00:43:56.410 --> 00:43:58.032 is like the key important physics. 00:43:58.032 --> 00:43:59.740 Now I'm going to do a little bit of stuff 00:43:59.740 --> 00:44:03.140 to sort of put the suit and tie that a tensor is supposed 00:44:03.140 --> 00:44:03.640 to wear. 00:44:03.640 --> 00:44:05.348 I'm going to dress it up a little bit, so 00:44:05.348 --> 00:44:07.462 that it's wearing the clothes that all quantities 00:44:07.462 --> 00:44:08.920 in this class are supposed to wear. 00:44:19.070 --> 00:44:21.580 So we want to make this tensorial. 00:44:30.860 --> 00:44:34.175 And so for guidance of that, these time derivatives 00:44:34.175 --> 00:44:36.050 or derivatives with respect to the parameter, 00:44:36.050 --> 00:44:38.510 it's not really nicely formulated, OK? 00:44:38.510 --> 00:44:41.300 The thing which we should note is, 00:44:41.300 --> 00:44:45.710 d by d lambda, that's what I get when 00:44:45.710 --> 00:44:48.890 I contract the forward velocity with a partial derivative. 00:44:48.890 --> 00:44:52.400 What we should do is replace this with something 00:44:52.400 --> 00:45:03.580 like what I'm going to call capital D by d lambda, which 00:45:03.580 --> 00:45:06.920 is what I'm going to get when I do derivatives using-- 00:45:06.920 --> 00:45:08.630 when I use forward velocity contracted 00:45:08.630 --> 00:45:12.270 on a covariant derivative, OK? 00:45:12.270 --> 00:45:12.770 So-- 00:45:21.250 --> 00:45:22.590 Suppose I just-- 00:45:22.590 --> 00:45:24.510 I'm agnostic about what xi actually is. 00:45:24.510 --> 00:45:26.400 I just know it's a vector field, and I 00:45:26.400 --> 00:45:27.660 want to compute this thing. 00:45:27.660 --> 00:45:30.230 Well, we learned how to do that a couple lectures ago. 00:45:34.960 --> 00:45:36.320 I'm going to work this guy out. 00:45:44.280 --> 00:45:55.070 And-- [EXCLAIMS] I couple in a term that looks like this. 00:45:55.070 --> 00:45:56.570 You might be tempted at this point 00:45:56.570 --> 00:45:58.400 to go, oh, I'm in a local Lorentz frame. 00:45:58.400 --> 00:46:00.230 I can get rid of that Christoffel. 00:46:00.230 --> 00:46:02.180 Don't do that quite yet, because we're 00:46:02.180 --> 00:46:04.190 going to want to take one more derivative. 00:46:04.190 --> 00:46:06.725 Christoffel does vanish, but its derivative does not. 00:46:06.725 --> 00:46:08.600 Wait till you've done all of your derivatives 00:46:08.600 --> 00:46:11.587 before you insert that relationship. 00:46:30.100 --> 00:46:31.600 Just getting another piece of chalk. 00:46:40.150 --> 00:46:43.420 So my covariant derivative along the trajectory 00:46:43.420 --> 00:46:51.375 with my parameter is the usual total derivative plus-- 00:46:57.831 --> 00:46:59.630 that, OK? 00:46:59.630 --> 00:47:02.079 Now, I'm going to want to take one more derivative. 00:47:29.524 --> 00:47:31.060 OK? 00:47:31.060 --> 00:47:34.990 So there's a lot of junk in here. 00:47:34.990 --> 00:47:36.100 I'm going to get one term. 00:47:40.830 --> 00:47:43.110 It just looks like this. 00:47:43.110 --> 00:47:45.600 But now I'm going to get a whole bunch of other terms 00:47:45.600 --> 00:47:46.738 that involve-- 00:47:46.738 --> 00:47:48.780 so this is what I get when I expand this guy out, 00:47:48.780 --> 00:47:49.590 and I just have-- 00:47:51.957 --> 00:47:53.290 well, I should hang on a second. 00:47:53.290 --> 00:47:54.748 So first, I'm going to get one that 00:47:54.748 --> 00:47:59.340 involves essentially the first term, u on the partial, 00:47:59.340 --> 00:48:01.000 hitting both of these terms. 00:48:01.000 --> 00:48:02.700 OK, so when I do that-- 00:48:06.210 --> 00:48:07.610 pardon me just one moment. 00:48:13.430 --> 00:48:14.290 OK, never mind. 00:48:19.232 --> 00:48:20.690 Let me just write it down, and I'll 00:48:20.690 --> 00:48:21.898 describe where it comes from. 00:48:26.450 --> 00:48:32.360 OK, so this first term, this is basically the connection term 00:48:32.360 --> 00:48:33.950 coupling to this, OK? 00:48:33.950 --> 00:48:36.077 Associated with this covariant derivative. 00:48:36.077 --> 00:48:37.910 Now I'm going to have a whole bunch of terms 00:48:37.910 --> 00:48:43.570 that involve this guy operating on these terms over here. 00:48:43.570 --> 00:48:46.652 And my apologies that the board is not clearing as well as I 00:48:46.652 --> 00:48:47.360 would like today. 00:48:55.980 --> 00:48:56.740 And you know what? 00:48:56.740 --> 00:48:59.670 I'm going to just write it out like this, 00:48:59.670 --> 00:49:01.590 and move to a different board in a second. 00:49:09.430 --> 00:49:09.930 OK? 00:49:16.023 --> 00:49:18.440 All right, so I'm going to put this over on the other side 00:49:18.440 --> 00:49:20.390 here. 00:49:20.390 --> 00:49:22.610 What I have there-- so the first term, like I said, 00:49:22.610 --> 00:49:25.395 I'm basically just operating that thing 00:49:25.395 --> 00:49:26.520 on the two different terms. 00:49:26.520 --> 00:49:29.450 So it's a little easy when I first hit it on the d xi d 00:49:29.450 --> 00:49:31.617 lambda, and it's going to be messy when I expand out 00:49:31.617 --> 00:49:33.575 all the derivatives that operate on here, which 00:49:33.575 --> 00:49:35.480 is why I'm writing it with some care, 00:49:35.480 --> 00:49:38.390 because we're about to make a lot of mess on the board. 00:49:49.288 --> 00:49:49.788 OK. 00:50:19.250 --> 00:50:25.842 OK, so first, I'm going to do what happens when this guy hits 00:50:25.842 --> 00:50:26.800 the Christoffel symbol. 00:50:46.290 --> 00:50:48.030 Then I'm going to get a term that 00:50:48.030 --> 00:50:56.680 involves this combination of guys 00:50:56.680 --> 00:50:58.668 acting on the four velocity. 00:51:13.330 --> 00:51:18.440 And then I get these guys acting on my displacement. 00:51:18.440 --> 00:51:21.220 OK, [BLOWS AIR] now I'm ready to simplify. 00:51:21.220 --> 00:51:24.520 OK, so let me just emphasize, everything I have here, 00:51:24.520 --> 00:51:27.850 all I did was expand out those derivatives 00:51:27.850 --> 00:51:30.430 and write them in a form where I want to basically call out 00:51:30.430 --> 00:51:33.550 different terms and see how they behave. 00:51:33.550 --> 00:51:35.590 Two things to bear in mind, I am going 00:51:35.590 --> 00:51:38.590 to do this in the local Lorenz frame 00:51:38.590 --> 00:51:52.080 and in the vicinity of the point A. 00:51:52.080 --> 00:51:55.970 That's going to allow me to get rid of a lot of terms. 00:51:58.968 --> 00:52:00.760 I'll use the-- here's an open box of chalk. 00:52:06.550 --> 00:52:10.780 OK, so earlier, I didn't want to get rid of my Christoffel 00:52:10.780 --> 00:52:13.360 symbols because I was going to take another derivative. 00:52:13.360 --> 00:52:14.780 I'm done with that. 00:52:14.780 --> 00:52:17.890 So anything that's just a Christoffel on its own, 00:52:17.890 --> 00:52:19.510 I go into local Lorentz form-- 00:52:19.510 --> 00:52:22.060 Christoffel, you die! 00:52:22.060 --> 00:52:24.348 Die! 00:52:24.348 --> 00:52:25.640 Can't really do much with this. 00:52:25.640 --> 00:52:27.150 Let's set this aside. 00:52:27.150 --> 00:52:29.730 Let's look at this guy. 00:52:29.730 --> 00:52:34.400 What do we know about the forward vector u? 00:52:34.400 --> 00:52:35.550 It's a geodesic. 00:52:35.550 --> 00:52:37.200 It obeys the geodesic equation. 00:52:37.200 --> 00:52:40.950 The geodesic equation is, this thing in parentheses equals 0. 00:52:40.950 --> 00:52:42.570 You die! 00:52:42.570 --> 00:52:46.260 Finally, I have this condition here at the beginning. 00:52:46.260 --> 00:52:54.050 That is essentially the velocity at point A 00:52:54.050 --> 00:52:58.070 minus the velocity at point B. Our initial condition 00:52:58.070 --> 00:53:02.420 is that these things start out parallel to each other. 00:53:02.420 --> 00:53:05.495 These die because these are initially parallel. 00:53:09.040 --> 00:53:13.900 So initially parallel geodesic-- 00:53:13.900 --> 00:53:15.910 local Lorentz frame-- 00:53:15.910 --> 00:53:18.160 So the only derivative I'm going to need to expand out 00:53:18.160 --> 00:53:21.820 is this guy, and because I'm working in a local Lorentz 00:53:21.820 --> 00:53:24.490 frame, that's actually not that bad. 00:53:37.960 --> 00:53:42.030 So what I finally get after all the smoke clears 00:53:42.030 --> 00:53:45.590 is that my covariant acceleration 00:53:45.590 --> 00:53:48.450 of this displacement vector is related 00:53:48.450 --> 00:53:55.040 to my noncovariant acceleration, with a term 00:53:55.040 --> 00:53:57.980 that basically ends up just being-- 00:53:57.980 --> 00:54:05.540 oops-- a derivative of the Christoffel symbol. 00:54:05.540 --> 00:54:08.940 It couples in two powers of the four 00:54:08.940 --> 00:54:15.210 velocity and the displacement. 00:54:15.210 --> 00:54:16.980 But we actually already have-- 00:54:16.980 --> 00:54:22.830 we worked out earlier what this guy was, OK? 00:54:22.830 --> 00:54:27.960 So my noncovariant acceleration was a different derivative 00:54:27.960 --> 00:54:29.920 of the Christoffel symbol. 00:54:29.920 --> 00:54:31.740 So if I go and I plug this in-- 00:54:55.560 --> 00:54:56.880 My apologies for pausing here. 00:54:56.880 --> 00:54:58.960 There are a lot of indices on this page. 00:55:03.880 --> 00:55:06.670 Look at this, I've got a derivative of Christoffel 00:55:06.670 --> 00:55:09.550 minus a derivative of a Christoffel. 00:55:09.550 --> 00:55:11.800 That is exactly what the Riemann curvature 00:55:11.800 --> 00:55:14.957 tensor looks like in the local Lorentz frame. 00:55:14.957 --> 00:55:17.540 Now, if you actually go back and you look up your definitions, 00:55:17.540 --> 00:55:19.390 you'll see it's not quite right. 00:55:19.390 --> 00:55:23.350 There's a couple of indices that are sort of a little bit off, 00:55:23.350 --> 00:55:25.840 but they turn out to all be dummy indices. 00:55:25.840 --> 00:55:27.730 So what you should do at this point 00:55:27.730 --> 00:55:31.900 is just relabel a couple of your dummy indices. 00:55:40.000 --> 00:55:57.300 So if on the second term, you take beta to mu, mu to gamma, 00:55:57.300 --> 00:56:23.190 and nu to beta, what you finally wind up with is-- 00:56:32.470 --> 00:56:34.470 We'll write it first in the local Lorentz frame. 00:56:50.740 --> 00:56:54.080 This is nothing but the Riemann tensor in the local Lorentz 00:56:54.080 --> 00:56:54.580 frame. 00:56:54.580 --> 00:56:57.130 And from the principle that a tensor equation that 00:56:57.130 --> 00:57:00.670 holds in one frame must hold in all, 00:57:00.670 --> 00:57:15.560 we can deduce from this that this 00:57:15.560 --> 00:57:18.320 is the way the displacement behaves. 00:57:18.320 --> 00:57:21.190 So I start out with two geodesics 00:57:21.190 --> 00:57:23.830 that are perfectly parallel to one another. 00:57:23.830 --> 00:57:25.940 If they are in a spacetime that is curved, 00:57:25.940 --> 00:57:28.970 the Riemann curvature tensor tells how those initially 00:57:28.970 --> 00:57:34.370 parallel trajectories evolve as I move along those geodesics. 00:57:34.370 --> 00:57:36.450 So I want to make two remarks. 00:57:36.450 --> 00:57:41.240 One is just sort of a way of thinking about the calculation 00:57:41.240 --> 00:57:42.080 I just did. 00:57:42.080 --> 00:57:43.580 It may not-- you might be slightly 00:57:43.580 --> 00:57:44.700 dissatisfied with the fact that I 00:57:44.700 --> 00:57:46.220 decided to do the whole calculation 00:57:46.220 --> 00:57:48.110 in a whole special frame. 00:57:48.110 --> 00:57:52.940 If that's-- you would prefer to see something a little bit more 00:57:52.940 --> 00:57:56.480 rigorous, feel free to explore some of the other textbooks 00:57:56.480 --> 00:57:58.460 that we have for this class. 00:57:58.460 --> 00:58:01.215 They do treat this a little bit more rigorously than this. 00:58:01.215 --> 00:58:02.840 One reason why I wanted to do this is I 00:58:02.840 --> 00:58:04.880 wanted to really emphasize this idea 00:58:04.880 --> 00:58:06.680 that where this effect comes from is 00:58:06.680 --> 00:58:10.040 that if I think about the freely falling frame describing two 00:58:10.040 --> 00:58:12.645 nearby geodesics, it is that derivative, 00:58:12.645 --> 00:58:15.020 it's that secondary derivative of the metric, that really 00:58:15.020 --> 00:58:18.320 plays a fundamental role in driving these two things apart. 00:58:18.320 --> 00:58:22.010 What we get out of it is a completely frame-independent 00:58:22.010 --> 00:58:24.890 equation, and this ends up really 00:58:24.890 --> 00:58:27.620 being the key equation describing the behavior 00:58:27.620 --> 00:58:30.590 of tides in general relativity. 00:58:30.590 --> 00:58:33.943 We're going to-- when you go into a freely falling frame, 00:58:33.943 --> 00:58:35.360 we've basically at that point said 00:58:35.360 --> 00:58:38.960 there's no longer such a thing as gravitational acceleration. 00:58:38.960 --> 00:58:40.960 And if there's no gravitational acceleration 00:58:40.960 --> 00:58:43.310 that's coming, well, what the hell does gravity do? 00:58:43.310 --> 00:58:46.580 General relativity, tides become the key thing 00:58:46.580 --> 00:58:50.330 that really defines what the action of gravity is. 00:58:50.330 --> 00:58:52.880 One of the ones I am personally very interested in applying 00:58:52.880 --> 00:58:54.710 this to is that this equation gives you 00:58:54.710 --> 00:58:56.870 a very rigorous and frame-independent way 00:58:56.870 --> 00:59:00.530 to describe how a gravitational wave detector responds 00:59:00.530 --> 00:59:02.747 to the impact of a gravitational wave. 00:59:02.747 --> 00:59:04.580 If you imagine you have two test masses that 00:59:04.580 --> 00:59:09.590 are moving through spacetime, they just follow geodesics, OK? 00:59:09.590 --> 00:59:10.730 They feel gravity. 00:59:10.730 --> 00:59:12.170 They are free falling. 00:59:12.170 --> 00:59:13.415 They don't do anything. 00:59:13.415 --> 00:59:15.457 If you look at them, if you're falling with them, 00:59:15.457 --> 00:59:18.770 you don't see any effect whatsoever. 00:59:18.770 --> 00:59:21.590 But if they're sufficiently far away from one another 00:59:21.590 --> 00:59:24.860 that their displacement kind of takes the curvature 00:59:24.860 --> 00:59:28.550 to that freely falling frame, then this equation 00:59:28.550 --> 00:59:30.890 governs how the separation between the two of them 00:59:30.890 --> 00:59:32.687 actually evolves with time. 00:59:32.687 --> 00:59:34.520 And so for instance, if you're a student who 00:59:34.520 --> 00:59:36.853 works in LIGO, where some of the many important analyses 00:59:36.853 --> 00:59:40.220 describing how gravitational wave detectors respond 00:59:40.220 --> 00:59:41.820 to an [INAUDIBLE] radiation field, 00:59:41.820 --> 00:59:44.120 they are based on this equation. 00:59:44.120 --> 00:59:45.950 If you're interested in understanding 00:59:45.950 --> 00:59:49.340 how an extended object is tidally 00:59:49.340 --> 00:59:52.040 distorted due to the fact that it's sort of big enough 00:59:52.040 --> 00:59:54.410 that it doesn't all sort of live at a single point, 00:59:54.410 --> 00:59:57.830 but its shape spills across the local Lorentz frame, 00:59:57.830 --> 00:59:59.390 this equation governs-- it allows 00:59:59.390 --> 01:00:01.015 you to set up some of the stresses that 01:00:01.015 --> 01:00:04.700 act upon that body and change it from a simple point mass. 01:00:04.700 --> 01:00:07.250 So I'm waxing slightly rhapsodic here, 01:00:07.250 --> 01:00:10.250 because this is one of the more important physical equations 01:00:10.250 --> 01:00:13.390 that come out of the class at this point. 01:00:13.390 --> 01:00:21.940 All right, in our last 12 or 13 minutes, 01:00:21.940 --> 01:00:26.020 I would like to do one other little identity which 01:00:26.020 --> 01:00:40.510 is extremely important, but rather messy. 01:00:40.510 --> 01:00:43.427 In fact, I think I'm going to just-- if I have the time-- 01:00:43.427 --> 01:00:44.260 yeah, you know what? 01:00:44.260 --> 01:00:45.302 I think I will have time. 01:00:45.302 --> 01:00:51.050 I'm going to set up the first page of the next lecture. 01:00:51.050 --> 01:00:54.310 So let's move away from-- oh, by the way, 01:00:54.310 --> 01:00:56.990 I forgot to give the name of this. 01:00:56.990 --> 01:01:01.200 This is the equation of geodesic deviation. 01:01:20.270 --> 01:01:25.810 OK, so recall when we first talked about the Riemann 01:01:25.810 --> 01:01:26.310 tensor. 01:01:35.940 --> 01:01:37.612 There's a very mathematical action 01:01:37.612 --> 01:01:38.820 that the Riemann tensor does. 01:01:38.820 --> 01:01:40.278 You can think of it as what happens 01:01:40.278 --> 01:01:43.260 when you have a commutator of covariant derivatives acting 01:01:43.260 --> 01:01:43.830 on a vector. 01:01:58.630 --> 01:02:02.790 So if I evaluate the commentator, sigma lambda-- 01:02:02.790 --> 01:02:09.270 excuse me-- covariant of lambda covariant with sigma 01:02:09.270 --> 01:02:18.460 acting on v alpha, OK, it ends up looking like this. 01:02:18.460 --> 01:02:21.900 And I argue that this definition is kind of the differential 01:02:21.900 --> 01:02:24.870 version of that little holonomy operation by which I derived 01:02:24.870 --> 01:02:26.550 Riemann in the first place. 01:02:26.550 --> 01:02:29.370 A more generalized form of this is, 01:02:29.370 --> 01:02:38.420 suppose I act on some object with two indices, one 01:02:38.420 --> 01:02:40.100 in the upstairs, one in the downstairs. 01:02:50.446 --> 01:02:50.946 Oops. 01:03:02.310 --> 01:03:03.750 OK? 01:03:03.750 --> 01:03:06.720 So these are just some reminders of a couple definitions. 01:03:10.560 --> 01:03:11.700 I want to use this. 01:03:11.700 --> 01:03:15.465 I'm going to apply these things to two relationships 01:03:15.465 --> 01:03:17.990 that I'm going to write down, and I'm 01:03:17.990 --> 01:03:20.840 going to do something kind of crazy, 01:03:20.840 --> 01:03:24.230 then I'm going to do something a little bit crazier, 01:03:24.230 --> 01:03:27.950 and something kind of cool is going to emerge from this. 01:03:27.950 --> 01:03:30.650 And this is something that deserves a clean board. 01:03:50.050 --> 01:03:55.570 So I'm going to write down relationship A. That's 01:03:55.570 --> 01:03:58.160 what I get when I do the commutator 01:03:58.160 --> 01:04:06.000 of the alpha beta derivatives on the gamma derivative 01:04:06.000 --> 01:04:08.850 of some one form, OK? 01:04:08.850 --> 01:04:12.480 Now, if I apply those little definitions I worked out, 01:04:12.480 --> 01:04:17.940 this is Riemann mu gamma alpha beta-- 01:04:32.975 --> 01:04:33.970 OK? 01:04:33.970 --> 01:04:36.760 So this is relationship A. 01:04:36.760 --> 01:04:42.750 Relationship B, I'm going to take 01:04:42.750 --> 01:04:54.540 the covariant derivative along alpha, of the beta gamma 01:04:54.540 --> 01:04:59.550 commutator on p delta, OK? 01:04:59.550 --> 01:05:00.800 So notice the difference here. 01:05:00.800 --> 01:05:02.630 One, I'm doing the commutator acting 01:05:02.630 --> 01:05:04.613 on this particular derivative. 01:05:04.613 --> 01:05:06.030 Next one, I'm doing the derivative 01:05:06.030 --> 01:05:07.310 of commutator acting on this. 01:05:10.720 --> 01:05:14.480 And so I'm going to skip a line or two that are in my notes, 01:05:14.480 --> 01:05:19.460 but this ends up turning into minus p mu-- 01:05:25.870 --> 01:05:36.145 beta alpha delta minus R mu and delta gamma. 01:05:39.260 --> 01:05:40.150 OK? 01:05:40.150 --> 01:05:42.880 So the way I went to get this line, so the two lines 01:05:42.880 --> 01:05:47.190 that I skipped over, one is I just straightforwardly applied 01:05:47.190 --> 01:05:49.690 the commutator derivative, and then took a derivative. 01:05:49.690 --> 01:05:51.640 And then I took advantage of the fact 01:05:51.640 --> 01:06:00.370 that I can commute the metric with covariant derivatives, 01:06:00.370 --> 01:06:03.110 and sort of raising the next, move things-- 01:06:03.110 --> 01:06:05.573 move an index up on one side and down on the other, 01:06:05.573 --> 01:06:07.740 and take advantage of some symmetries of the Riemann 01:06:07.740 --> 01:06:11.420 tensor to slough my indices around. 01:06:11.420 --> 01:06:13.575 OK, so this board's kind of a disaster, 01:06:13.575 --> 01:06:15.120 so I'm going to go to a cleaner board 01:06:15.120 --> 01:06:17.360 here for what I want to do next. 01:06:17.360 --> 01:06:19.110 I'm just going to clean this so you're not 01:06:19.110 --> 01:06:20.193 distracted by its content. 01:06:23.720 --> 01:06:28.060 So two equations, what the hell do they 01:06:28.060 --> 01:06:30.440 have to do with each other? 01:06:30.440 --> 01:06:43.130 Here is where I'm going to do something which you might-- 01:06:43.130 --> 01:06:44.790 let's do the other board. 01:06:44.790 --> 01:06:47.940 I'm going through something that you might legitimately 01:06:47.940 --> 01:06:49.320 think is crazy. 01:06:49.320 --> 01:06:52.680 What I want to do is take equation A-- 01:06:56.450 --> 01:06:58.800 actually, I'm going to look at both of these equations. 01:06:58.800 --> 01:07:05.350 And I am going to antisymmetrize on the indices alpha, beta, 01:07:05.350 --> 01:07:05.850 and gamma. 01:07:24.622 --> 01:07:25.970 OK? 01:07:25.970 --> 01:07:27.220 So just bear with me a second. 01:07:30.050 --> 01:07:32.447 So the way I'm going to do this-- 01:07:32.447 --> 01:07:33.530 here's what I'm doing now. 01:07:36.140 --> 01:07:37.593 Here's my commutator. 01:07:49.930 --> 01:07:51.730 So if I look at equation A-- 01:07:56.340 --> 01:07:56.840 OK? 01:07:56.840 --> 01:07:58.520 So I need to expand this guy out. 01:08:02.260 --> 01:08:05.681 Let me just write out a step of this analysis. 01:08:10.980 --> 01:08:20.581 The way you do this is you add up every even permutation 01:08:20.581 --> 01:08:21.289 of those indices. 01:08:39.370 --> 01:08:40.360 That's a beta. 01:08:46.340 --> 01:08:51.016 And then you subtract off every odd permutation 01:08:51.016 --> 01:08:51.724 of these indices. 01:09:21.300 --> 01:09:23.790 Oh, and don't forget-- 01:09:23.790 --> 01:09:26.180 exactly on that one form, OK? 01:09:29.680 --> 01:09:32.210 So now we can-- 01:09:32.210 --> 01:09:39.170 if you expand these guys out, and gather terms exactly 01:09:39.170 --> 01:09:45.130 correctly, it's not hard to show that this turns out 01:09:45.130 --> 01:09:58.620 to be what you get if you just slide those commutators over 01:09:58.620 --> 01:09:59.190 by one. 01:10:41.435 --> 01:10:43.423 OK. 01:10:43.423 --> 01:10:47.290 You know, if you're really feeling motivated, just try it, 01:10:47.290 --> 01:10:50.020 OK? 01:10:50.020 --> 01:10:55.360 What this is is, the right hand side-- 01:10:55.360 --> 01:10:58.870 sorry, this is the left hand side-- 01:10:58.870 --> 01:11:02.140 we'll put this way. 01:11:02.140 --> 01:11:09.870 So when I apply this to the left hand side of equation A, 01:11:09.870 --> 01:11:13.440 what emerges is the antisymmetrized left hand 01:11:13.440 --> 01:11:38.160 side of equation B. 01:11:38.160 --> 01:11:43.590 So when I antisymmetrize on the indices alpha, beta, gamma, 01:11:43.590 --> 01:11:45.210 these two relationships become-- 01:11:45.210 --> 01:11:46.440 they say the same thing. 01:12:03.430 --> 01:12:08.170 So that means the right hand side must be the same as well. 01:12:08.170 --> 01:12:10.270 So let's apply this and see what happens 01:12:10.270 --> 01:12:14.440 if I require the antisymmetrized right hand side of A 01:12:14.440 --> 01:12:30.600 to equal the antisymmetrized right hand side of B. 01:12:30.600 --> 01:12:34.650 OK, so first, if I remind myself how I order these equations-- 01:12:34.650 --> 01:12:35.150 OK, yeah. 01:12:35.150 --> 01:13:32.580 So let's do-- here's A, and here is B. 01:13:32.580 --> 01:13:36.830 OK, so a few things to notice. 01:13:36.830 --> 01:13:38.900 This term, we actually showed when 01:13:38.900 --> 01:13:42.410 we looked at the different symmetries of the Riemann 01:13:42.410 --> 01:13:43.900 tensor. 01:13:43.900 --> 01:13:46.400 This is-- I didn't actually show this, but it's in the notes 01:13:46.400 --> 01:13:47.670 and I stated it. 01:13:47.670 --> 01:13:49.610 This is one of those symmetries. 01:13:49.610 --> 01:13:51.560 If I take this thing, and I just add up 01:13:51.560 --> 01:13:56.150 what I get when I permute the three final indices, 01:13:56.150 --> 01:13:57.622 I get zero. 01:13:57.622 --> 01:13:59.330 And that's equivalent to antisymmetrizing 01:13:59.330 --> 01:14:00.690 on these things. 01:14:00.690 --> 01:14:02.810 So this is zero by Riemann symmetry. 01:14:11.190 --> 01:14:16.890 This term here and this term here are exactly the same, OK? 01:14:16.890 --> 01:14:19.055 Because the only indices that are 01:14:19.055 --> 01:14:21.430 in a slightly different order are alpha, beta, and gamma. 01:14:21.430 --> 01:14:22.870 This one goes alpha, beta, gamma. 01:14:22.870 --> 01:14:25.245 Where I wrote it here, it just became beta, gamma, alpha. 01:14:25.245 --> 01:14:27.420 That's a cyclic permutation I've antisymmetrized. 01:14:27.420 --> 01:14:29.323 They are exactly the same. 01:14:29.323 --> 01:14:31.490 So these are on the same side-- or on opposite sides 01:14:31.490 --> 01:14:32.610 of the equation. 01:14:32.610 --> 01:14:35.350 So they cancel each other out. 01:14:35.350 --> 01:14:40.370 And so what arises from all this is p mu-- 01:14:44.700 --> 01:14:46.678 pardon me, I missed the derivative. 01:15:04.990 --> 01:15:09.880 I've set no properties on p, so the only way 01:15:09.880 --> 01:15:22.090 this can always hold is if in fact, this is equal to zero. 01:15:24.640 --> 01:15:29.562 This is a result known as the Bianchi identity. 01:15:37.756 --> 01:15:40.250 Let me just write another form of it, 01:15:40.250 --> 01:15:42.920 and then I'm fairly pressed for time, 01:15:42.920 --> 01:15:44.630 so I think I'm just going to write down 01:15:44.630 --> 01:15:47.060 the result of the next thing, and I 01:15:47.060 --> 01:15:51.010 will put the details of this into 01:15:51.010 --> 01:15:52.490 the first prerecorded lecture. 01:16:01.310 --> 01:16:04.400 If you expand out that antisymmetry 01:16:04.400 --> 01:16:11.330 and take advantage of the Riemann symmetry-- 01:16:11.330 --> 01:16:23.630 the various Riemann symmetry relationships, 01:16:23.630 --> 01:16:24.820 this is an equivalent form. 01:16:30.170 --> 01:16:33.760 So these two things both are an important geometric 01:16:33.760 --> 01:16:37.870 relationship the curvature tensor must go by. 01:16:37.870 --> 01:16:44.270 Now, in some notes that I am going to-- well, 01:16:44.270 --> 01:16:49.030 they're actually already scanned and on the web. 01:16:49.030 --> 01:16:53.980 What I do is take the Bianchi identity and contract 01:16:53.980 --> 01:17:02.610 on some of the indices to convert my Riemann tensor-- 01:17:02.610 --> 01:17:05.040 my Riemann curvature, into a Ricci tensor-- 01:17:05.040 --> 01:17:06.215 Ricci curvature. 01:17:17.860 --> 01:17:20.965 So in particular, if I use this second form of it, 01:17:20.965 --> 01:17:22.590 it's just covariant derivatives, right? 01:17:22.590 --> 01:17:24.780 And the metric commutes with covariant derivatives, 01:17:24.780 --> 01:17:27.750 so I can just sort of walk it through. 01:17:27.750 --> 01:17:32.940 What you find when you do this is that you can write 01:17:32.940 --> 01:17:40.020 this relationship in the following form, 01:17:40.020 --> 01:17:42.920 and the derivation of this will be scanned and posted, 01:17:42.920 --> 01:17:46.040 and I will step through it in the first recorded lecture. 01:17:48.660 --> 01:17:51.390 The divergence of a particular combination 01:17:51.390 --> 01:17:56.040 of the Ricci curvature and the Ricci scalar is equal to zero. 01:17:56.040 --> 01:18:00.060 This is a sufficiently interesting tensor 01:18:00.060 --> 01:18:01.920 that it is now given a name. 01:18:05.350 --> 01:18:12.550 We write this G, and we call this the Einstein curvature 01:18:12.550 --> 01:18:13.050 tensor. 01:18:26.180 --> 01:18:28.520 I sort of mentioned that one of our governing principles 01:18:28.520 --> 01:18:31.670 here is, we're going to change the source of gravitation 01:18:31.670 --> 01:18:34.850 from just matter density to the stress energy tensor. 01:18:34.850 --> 01:18:38.720 That is a two-index divergence-free tensor. 01:18:38.720 --> 01:18:40.393 We want our gravitational-- 01:18:40.393 --> 01:18:41.810 the left hand side of the equation 01:18:41.810 --> 01:18:43.280 to look like two derivatives in the metric, which 01:18:43.280 --> 01:18:43.905 is a curvature. 01:18:43.905 --> 01:18:47.760 So we need a divergence-free two-index curvature. 01:18:47.760 --> 01:18:49.227 Ta-da! 01:18:49.227 --> 01:18:50.810 It's got Einstein's name on it, so you 01:18:50.810 --> 01:18:52.520 know it's got to be good. 01:18:52.520 --> 01:18:57.030 All right, so that is where, unfortunately, the COVID virus 01:18:57.030 --> 01:18:59.390 is requiring us to stop for the semester, 01:18:59.390 --> 01:19:01.680 at least as far as in-person goes. 01:19:01.680 --> 01:19:07.100 So look for notes to be posted, videos to be posted, 01:19:07.100 --> 01:19:11.300 and I, of course, will be in contact, 01:19:11.300 --> 01:19:12.860 behind a sick wall or something. 01:19:12.860 --> 01:19:14.960 But anyway, [CHUCKLES] certainly, 01:19:14.960 --> 01:19:16.730 I am not going out of contact. 01:19:16.730 --> 01:19:24.170 And so those of you who are scattering to parts unknown-- 01:19:24.170 --> 01:19:26.570 known to you, that is, unknown to me-- 01:19:26.570 --> 01:19:29.630 good luck with your travels and getting yourselves settled. 01:19:29.630 --> 01:19:31.793 I really feel-- everyone feels terrible 01:19:31.793 --> 01:19:33.710 that you're going through this crap right now, 01:19:33.710 --> 01:19:38.330 but hopefully, this will flatten the curve, as they say, 01:19:38.330 --> 01:19:40.610 and it will be the right thing to do. 01:19:40.610 --> 01:19:42.020 But stay in touch, OK? 01:19:42.020 --> 01:19:45.590 This campus is going to be weird and sad without the students 01:19:45.590 --> 01:19:49.090 here, and so we want to hear from you.