WEBVTT 00:00:00.000 --> 00:00:01.976 [SQUEAKING] 00:00:01.976 --> 00:00:03.952 [RUSTLING] 00:00:03.952 --> 00:00:04.940 [CLICKING] 00:00:10.975 --> 00:00:12.850 SCOTT HUGHES: So let me just do a quick recap 00:00:12.850 --> 00:00:15.560 of what we did last time. 00:00:15.560 --> 00:00:17.480 So today, we're going to move into things that 00:00:17.480 --> 00:00:18.730 are a little bit more physics. 00:00:18.730 --> 00:00:20.950 Last time we were really doing some things that 00:00:20.950 --> 00:00:23.890 allows us to establish some of the critical mathematical 00:00:23.890 --> 00:00:28.150 concepts we need to study the tensors that 00:00:28.150 --> 00:00:31.750 are going to be used for physics on a curved manifold. 00:00:31.750 --> 00:00:34.210 So one of the things that we saw is 00:00:34.210 --> 00:00:36.670 that if I wanted to formulate differential equations 00:00:36.670 --> 00:00:40.510 on a curved manifold, if I just defined my derivative the most 00:00:40.510 --> 00:00:43.240 naive way you might think of, you 00:00:43.240 --> 00:00:45.940 end up with objects that are not tensorial. 00:00:45.940 --> 00:00:48.608 And so mathematically you might say, well, that's fine. 00:00:48.608 --> 00:00:49.900 It's just not a tensor anymore. 00:00:49.900 --> 00:00:52.780 But we really want tensors for our physics 00:00:52.780 --> 00:00:55.390 because we want to be working with quantities 00:00:55.390 --> 00:00:58.100 that have frame-independent geometric meaning to them. 00:00:58.100 --> 00:01:00.880 So that notion of a derivative-- if I just do it the naive way-- 00:01:00.880 --> 00:01:02.555 isn't the best. 00:01:02.555 --> 00:01:04.180 And so I argued that what we need to do 00:01:04.180 --> 00:01:06.940 is to find some kind of a transport operation 00:01:06.940 --> 00:01:12.250 in which there is a linear mapping between things 00:01:12.250 --> 00:01:13.660 like my vector field or my tensor 00:01:13.660 --> 00:01:17.950 field and the displacement, which allows me to cancel out 00:01:17.950 --> 00:01:20.380 the bits of the partial derivative transformation 00:01:20.380 --> 00:01:23.130 laws that are non tensorial. 00:01:23.130 --> 00:01:24.690 There's a lot of freedom to do that. 00:01:24.690 --> 00:01:27.480 One of the ways I suggest we do that is by demanding that when 00:01:27.480 --> 00:01:30.450 I do this, that derivative-- when applied to the metric-- 00:01:30.450 --> 00:01:31.390 give me 0. 00:01:31.390 --> 00:01:33.480 And if we do that, we see right away 00:01:33.480 --> 00:01:35.473 that the transport law that emerges 00:01:35.473 --> 00:01:37.140 gives me the covariant derivative as one 00:01:37.140 --> 00:01:37.930 of my examples. 00:01:37.930 --> 00:01:39.305 Now this shouldn't be a surprise. 00:01:39.305 --> 00:01:40.980 We introduced the covariant derivative 00:01:40.980 --> 00:01:44.340 by thinking about flat spacetime operations, 00:01:44.340 --> 00:01:47.100 but with all my basis objects being functionals. 00:01:47.100 --> 00:01:52.088 And this in some way is sort of a continuation of that notion. 00:01:52.088 --> 00:01:54.630 The other thing which I talked about is telling you we're not 00:01:54.630 --> 00:01:57.180 going to use a tremendous amount here except to motivate one 00:01:57.180 --> 00:02:01.080 very important result. And that is if I define transport 00:02:01.080 --> 00:02:03.750 by basically imagining that I slide my vectors in order 00:02:03.750 --> 00:02:07.020 to make the comparison-- along some specified vector field-- 00:02:07.020 --> 00:02:08.880 I get what's known as the Lie derivative. 00:02:08.880 --> 00:02:14.610 And so this is an example of the Lie derivative of a vector. 00:02:14.610 --> 00:02:17.790 And you get this form that looks like a commutator 00:02:17.790 --> 00:02:19.740 between the vector field you're sliding along 00:02:19.740 --> 00:02:21.930 and the vector field you are differentiating. 00:02:21.930 --> 00:02:23.997 Similar forms-- which are not really 00:02:23.997 --> 00:02:25.830 the form of a commutator-- but similar forms 00:02:25.830 --> 00:02:29.520 can be written down for general tensors. 00:02:29.520 --> 00:02:33.480 The key thing that you should be aware of 00:02:33.480 --> 00:02:36.720 is that it's got a similar form to the covariant derivative, 00:02:36.720 --> 00:02:38.100 in that you have one term-- let's 00:02:38.100 --> 00:02:39.520 focus on the top line for the second-- 00:02:39.520 --> 00:02:40.895 you have one term that looks just 00:02:40.895 --> 00:02:43.740 like the ordinary vector contracted 00:02:43.740 --> 00:02:46.830 onto a partial derivative of your field. 00:02:46.830 --> 00:02:49.980 And then you have terms which correct every free index 00:02:49.980 --> 00:02:50.730 of your field-- 00:02:50.730 --> 00:02:53.700 one free index if it's a vector, one free index 00:02:53.700 --> 00:02:58.140 if it's a one form, and corrections for an end index 00:02:58.140 --> 00:02:59.130 tensor-- 00:02:59.130 --> 00:03:02.250 with the sign doing something opposite to the sign 00:03:02.250 --> 00:03:04.860 that appears in the covariant derivative. 00:03:04.860 --> 00:03:08.228 What's interesting about this is that so defined, 00:03:08.228 --> 00:03:10.020 the Lie derivative is only written in terms 00:03:10.020 --> 00:03:11.790 of partial derivatives. 00:03:11.790 --> 00:03:14.663 But if you just imagine-- you promote 00:03:14.663 --> 00:03:16.830 those partial derivatives to covariant derivatives-- 00:03:16.830 --> 00:03:19.110 you find the exact same result holds 00:03:19.110 --> 00:03:21.762 because all of your Christoffel symbols-- or connections 00:03:21.762 --> 00:03:23.970 as we like to think of them when we're using parallel 00:03:23.970 --> 00:03:27.737 transport-- all the connective objects cancel each other out. 00:03:27.737 --> 00:03:29.320 And this is nice because this tells me 00:03:29.320 --> 00:03:32.170 that even though this object, strictly speaking, 00:03:32.170 --> 00:03:35.230 only involves partial derivatives, what emerges out 00:03:35.230 --> 00:03:36.780 of it is in fact tensorial. 00:03:36.780 --> 00:03:40.110 And it's an object that I can use for a lot of things 00:03:40.110 --> 00:03:41.110 I want to do in physics. 00:03:41.110 --> 00:03:43.402 In particular where we're going to use it the most-- 00:03:43.402 --> 00:03:45.360 and I said you're going to do this on the PSET, 00:03:45.360 --> 00:03:47.030 but I was wrong-- you're going to do something related to one 00:03:47.030 --> 00:03:47.613 of the PSETs-- 00:03:47.613 --> 00:03:50.002 but I'm going to actually-- if all goes well-- 00:03:50.002 --> 00:03:52.210 derive an important result involving these symmetries 00:03:52.210 --> 00:03:54.040 in today's lecture. 00:03:54.040 --> 00:03:55.540 We can use this to understand things 00:03:55.540 --> 00:03:58.255 that are related to conserved quantities in your space time. 00:03:58.255 --> 00:03:59.880 And where this comes from is that there 00:03:59.880 --> 00:04:02.920 is a definition of an object we call the Killing vector, which 00:04:02.920 --> 00:04:07.480 is an object where if your metric is Lie transported 00:04:07.480 --> 00:04:11.620 along some field C, we call C a Killing vector. 00:04:11.620 --> 00:04:16.519 And from the fact that the covariant derivative the metric 00:04:16.519 --> 00:04:21.769 is 0, you can turn the equation governing 00:04:21.769 --> 00:04:24.593 the Lie derivative along C into what I wrote down just there. 00:04:24.593 --> 00:04:25.760 And I should write its name. 00:04:29.810 --> 00:04:31.920 There's the result known as Killing's Equation. 00:04:43.270 --> 00:04:46.960 If a vector has a Killing vector-- 00:04:46.960 --> 00:04:48.910 if a metric has a Killing vector-- 00:04:48.910 --> 00:04:50.830 then you know that your metric is 00:04:50.830 --> 00:04:53.080 independent of some kind of a parameter that 00:04:53.080 --> 00:04:54.790 characterizes that spacetime. 00:04:54.790 --> 00:04:56.290 The converse also holds. 00:04:56.290 --> 00:04:58.270 If your metric is independent of something-- 00:04:58.270 --> 00:04:59.980 like say the time coordinate-- you 00:04:59.980 --> 00:05:02.620 know that there is a Killing vector corresponding 00:05:02.620 --> 00:05:04.230 to that independent thing. 00:05:04.230 --> 00:05:06.682 And so you'll often see this described as a differing 00:05:06.682 --> 00:05:08.140 amorphism of the spacetime-- if you 00:05:08.140 --> 00:05:10.307 want to dig into some of the more advanced textbooks 00:05:10.307 --> 00:05:11.170 on the subject. 00:05:11.170 --> 00:05:14.110 We'll come back to that in a few more details hopefully shortly 00:05:14.110 --> 00:05:15.890 before the end of today's lecture. 00:05:15.890 --> 00:05:17.587 So where I concluded last time, was we 00:05:17.587 --> 00:05:19.420 started talking about these quantities known 00:05:19.420 --> 00:05:21.370 as tensor densities, which are given 00:05:21.370 --> 00:05:24.760 the less-than-helpful definition-- quantities 00:05:24.760 --> 00:05:27.730 that are like tensors, but not quite. 00:05:27.730 --> 00:05:30.280 The example I gave of this-- where we were starting-- 00:05:30.280 --> 00:05:33.160 was the Levi-Civita symbol. 00:05:33.160 --> 00:05:37.180 So let me just write down again what resulted from that. 00:05:37.180 --> 00:05:40.950 So if I have Levi-Civita-- and the tilde here is going 00:05:40.950 --> 00:05:44.680 to reflect the fact that this is not really a tensor-- 00:05:47.580 --> 00:05:54.070 this guy in some prime coordinates 00:05:54.070 --> 00:06:01.020 is related to this guy in the unprime coordinates 00:06:01.020 --> 00:06:02.340 via the following-- 00:06:10.452 --> 00:06:12.160 let's get the primes in the right place-- 00:06:16.440 --> 00:06:19.310 the following mess of quantities. 00:06:21.870 --> 00:06:23.620 So I'm not going to go through this again. 00:06:23.620 --> 00:06:25.630 This is basically a theorem from linear algebra 00:06:25.630 --> 00:06:29.020 that relates the determinant of a matrix-- 00:06:29.020 --> 00:06:31.660 not metric, but matrix-- 00:06:31.660 --> 00:06:34.360 to what you get when you contract a bunch of matrices 00:06:34.360 --> 00:06:37.092 onto the Levi-Civita symbol. 00:06:37.092 --> 00:06:39.550 And so the key thing to note is that if this were not here, 00:06:39.550 --> 00:06:41.850 this would look just like a tensor transformation. 00:06:41.850 --> 00:06:42.890 But that is there. 00:06:42.890 --> 00:06:44.650 So it's not. 00:06:44.650 --> 00:06:48.010 And so we call this a tensor density of weight 1. 00:06:56.290 --> 00:06:58.810 So the other one-- which I hinted at the end of the last 00:06:58.810 --> 00:07:02.530 lecture, but did not have time to get into-- 00:07:02.530 --> 00:07:05.060 is suppose we look at the metric. 00:07:05.060 --> 00:07:08.427 Now, the metric-- no ifs, ands, or buts about it-- 00:07:08.427 --> 00:07:09.010 it's a tensor. 00:07:15.081 --> 00:07:17.870 And it's actually the first tensor 00:07:17.870 --> 00:07:22.890 we've started talking about back in our toddler 00:07:22.890 --> 00:07:26.470 years of studying flat spacetime, which by the way, 00:07:26.470 --> 00:07:29.367 was about three weeks ago. 00:07:29.367 --> 00:07:30.450 Obviously that's a tensor. 00:07:30.450 --> 00:07:32.100 It's a simple tensor relationship. 00:07:32.100 --> 00:07:34.230 Let's take the determinant of both sides of this. 00:07:47.293 --> 00:07:49.710 You might look at this and go, why do you want to do that? 00:07:49.710 --> 00:07:52.730 Well when I do this, I'm going to call 00:07:52.730 --> 00:07:55.710 the determinant of the metric in the primed representation. 00:07:55.710 --> 00:07:58.700 Let's call that G prime. 00:07:58.700 --> 00:08:00.050 I get 2 powers-- 00:08:03.530 --> 00:08:07.200 2 powers of this Jacobian matrix is determinant. 00:08:11.810 --> 00:08:16.190 And I get the determinant in my original representation. 00:08:16.190 --> 00:08:18.080 Now I want to write this in a way that's 00:08:18.080 --> 00:08:19.747 similar to the way I wrote it over here. 00:08:19.747 --> 00:08:21.500 Notice I have all my primed objects 00:08:21.500 --> 00:08:22.880 over here on the left-hand side. 00:08:22.880 --> 00:08:26.900 And my factor of this determinant relates-- 00:08:26.900 --> 00:08:29.360 it's got primed indices in the upstairs position, unprimed 00:08:29.360 --> 00:08:30.830 in the downstairs. 00:08:30.830 --> 00:08:34.850 But the determinant of 1 over a metric is just 1 00:08:34.850 --> 00:08:36.289 over the determinant of-- 00:08:36.289 --> 00:08:39.799 the determinant of the inverse of a matrix is just 1 00:08:39.799 --> 00:08:42.080 over the determinant of that matrix. 00:08:42.080 --> 00:08:52.790 And so I can really simply just say this looks like so. 00:08:52.790 --> 00:09:00.440 So the determinant of the metric is a tensor density 00:09:00.440 --> 00:09:01.790 of weight minus 2. 00:09:23.550 --> 00:09:28.170 What this basically tells us is I now have two of these things. 00:09:28.170 --> 00:09:30.120 I've been arguing basically this entire course 00:09:30.120 --> 00:09:32.670 that we want to use tensors because of the fact 00:09:32.670 --> 00:09:37.590 that they give me a covariant way of encoding 00:09:37.590 --> 00:09:40.110 geometric concepts. 00:09:40.110 --> 00:09:43.350 I've got these two things that are not quite tensors. 00:09:43.350 --> 00:09:46.560 I can put them together and get a tensor out of this. 00:09:46.560 --> 00:09:52.440 So what this tells me now is I can convert any tensor density 00:09:52.440 --> 00:09:53.625 into a proper tensor. 00:09:56.540 --> 00:09:58.820 So suppose I have a tensor density of weight 00:09:58.820 --> 00:10:11.160 W. I can convert this into a proper tensor 00:10:11.160 --> 00:10:21.430 by multiplying by a power of that G. So multiply it 00:10:21.430 --> 00:10:26.940 by G to the W over 2. 00:10:26.940 --> 00:10:31.140 One slight subtlety here, when we work in spacetime-- 00:10:31.140 --> 00:10:35.620 let's just stop for a second and think about special relativity. 00:10:35.620 --> 00:10:41.020 In special relativity in an inertial reference frame, 00:10:41.020 --> 00:10:44.410 my metric is minus 1 1 1 1 on the diagonal. 00:10:44.410 --> 00:10:46.880 So its determinant is negative 1. 00:10:46.880 --> 00:10:51.610 And when I take negative 1 to some power that involves 00:10:51.610 --> 00:10:54.190 a square root, I get sad. 00:10:54.190 --> 00:10:56.480 We all know how to work with complex numbers. 00:10:56.480 --> 00:10:58.330 You might think that's all OK. 00:10:58.330 --> 00:10:59.260 It's not in this case. 00:10:59.260 --> 00:11:03.370 But the way I can fix that is that equation's still true 00:11:03.370 --> 00:11:06.353 if I multiply both sides by minus 1. 00:11:06.353 --> 00:11:08.020 I want this to be a positive number when 00:11:08.020 --> 00:11:08.978 I take the square root. 00:11:08.978 --> 00:11:13.240 So I'm allowed just to take the absolute value. 00:11:13.240 --> 00:11:16.240 So we take the absolute value to clear out the fact 00:11:16.240 --> 00:11:19.820 that in spacetime, we tend to have an indeterminate metric, 00:11:19.820 --> 00:11:22.210 where the sign depends on the interval. 00:11:33.753 --> 00:11:35.670 So remember the only reason we're doing this-- 00:11:35.670 --> 00:11:36.520 this is just a-- 00:11:36.520 --> 00:11:38.640 I don't want to say it's a trick. 00:11:38.640 --> 00:11:42.320 But it's not that far off from a trick. 00:11:42.320 --> 00:11:44.810 I'm just combining two tensor densities in order 00:11:44.810 --> 00:11:46.250 to get a tensor out of it. 00:11:46.250 --> 00:11:49.940 And minus a tensor density is still a tensor density. 00:11:49.940 --> 00:11:50.990 So I'm OK to do that. 00:11:50.990 --> 00:11:52.907 And I'm just doing this so that my square root 00:11:52.907 --> 00:11:55.310 doesn't go haywire on me. 00:11:55.310 --> 00:11:56.630 So a particular example-- 00:11:56.630 --> 00:11:59.960 in fact the one that in my career has come up the most-- 00:12:02.850 --> 00:12:08.550 is making a proper volume element 00:12:08.550 --> 00:12:16.677 converting my Levi-Civita symbol into a tensor that 00:12:16.677 --> 00:12:17.760 gives me a volume element. 00:12:22.570 --> 00:12:25.930 So my Levi-Civita symbol has a tensor density of weight 1. 00:12:25.930 --> 00:12:28.630 If I want to make that into a proper tensor, 00:12:28.630 --> 00:12:33.010 I multiply by the square root of the determinant of the metric. 00:12:33.010 --> 00:12:36.040 So now I will no longer have that tilde 00:12:36.040 --> 00:12:38.260 on there, which was meant to be a signpost that this 00:12:38.260 --> 00:12:39.760 as a quantity is a little bit goofy. 00:12:46.990 --> 00:12:49.570 You wind up with something like this. 00:12:49.570 --> 00:12:51.670 When you go-- and by the way, sometimes 00:12:51.670 --> 00:12:52.510 when you're working with this, you 00:12:52.510 --> 00:12:54.370 need to have this thing with indices in the upstairs 00:12:54.370 --> 00:12:55.150 position. 00:12:55.150 --> 00:12:56.260 You have to be a little bit careful. 00:12:56.260 --> 00:12:57.718 But I'll just give you one example. 00:12:57.718 --> 00:13:02.440 If you raise all four of the indices, what 00:13:02.440 --> 00:13:04.025 you find when everything goes through, 00:13:04.025 --> 00:13:06.400 this one is not that hard to see because you're basically 00:13:06.400 --> 00:13:08.317 playing with a similar relationship to the one 00:13:08.317 --> 00:13:09.850 that I wrote down over here-- 00:13:09.850 --> 00:13:13.008 just a short homework exercise to demonstrate this. 00:13:13.008 --> 00:13:14.800 And then you end up with the tensor density 00:13:14.800 --> 00:13:16.260 of the opposite sign. 00:13:16.260 --> 00:13:18.760 Weight minus 1, you wind up with a 1 over square root there. 00:13:23.430 --> 00:13:25.020 So like I said one of the reasons 00:13:25.020 --> 00:13:30.030 why this is an important example is that we use it to form 00:13:30.030 --> 00:13:32.088 covariant volume operators. 00:13:55.520 --> 00:13:59.440 So in four-dimensional space-- 00:13:59.440 --> 00:14:06.860 so imagine here's my basis direction for spatial direction 00:14:06.860 --> 00:14:10.610 1, spatial direction 2, spatial direction 3-- 00:14:10.610 --> 00:14:12.860 you guys can figure out how to write spatial direction 00:14:12.860 --> 00:14:17.060 0 on your own time-- 00:14:17.060 --> 00:14:20.790 I would define a covariant 4 volume-- 00:14:20.790 --> 00:14:22.430 4 volume element from this. 00:14:34.755 --> 00:14:36.250 It'll look like this. 00:14:36.250 --> 00:14:40.080 And if this is an orthogonal basis, 00:14:40.080 --> 00:14:52.930 this simply turns into something like-- 00:14:55.745 --> 00:14:57.650 these are meant to be superscripts 00:14:57.650 --> 00:15:00.265 because these are coordinates. 00:15:00.265 --> 00:15:02.480 So it just turns into something like this 00:15:02.480 --> 00:15:04.210 if I'm working in an orthogonal basis. 00:15:09.210 --> 00:15:12.470 And again for intuition, I suggest go down 00:15:12.470 --> 00:15:16.685 to 3-dimensional spherical coordinates. 00:15:16.685 --> 00:15:17.810 And I wrote this last time. 00:15:17.810 --> 00:15:19.490 But let me just quickly write it up. 00:15:19.490 --> 00:15:23.750 I mean everything I did here, I tend to-- since this is 00:15:23.750 --> 00:15:25.670 a course on spacetime-- 00:15:25.670 --> 00:15:28.250 by default I write down all my formulas 00:15:28.250 --> 00:15:32.415 for three space dimensions, one time dimension. 00:15:32.415 --> 00:15:35.180 But it's perfectly good in 3 spatial dimensions, 00:15:35.180 --> 00:15:39.703 2 spatial dimensions, 17 spatial dimensions-- whatever crazy 00:15:39.703 --> 00:15:41.870 spacetime your physics want you to put yourself in-- 00:15:41.870 --> 00:15:43.620 or space your physics wants to put you in. 00:15:48.460 --> 00:15:49.620 So I'll just remind you-- 00:16:03.610 --> 00:16:05.360 that when you do this, you've got yourself 00:16:05.360 --> 00:16:12.650 a metric across the diagonal of 1 r 00:16:12.650 --> 00:16:17.690 squared r squared sine squared theta. 00:16:17.690 --> 00:16:19.560 And just to be consistent-- 00:16:19.560 --> 00:16:24.572 I usually use Latin letters for only spatial things. 00:16:24.572 --> 00:16:25.280 So let's do that. 00:16:35.640 --> 00:16:39.685 This would be how I would then write my volume element. 00:16:39.685 --> 00:16:40.560 Did I miss something? 00:16:40.560 --> 00:16:41.185 AUDIENCE: Yeah. 00:16:41.185 --> 00:16:42.270 [INAUDIBLE] 00:16:42.270 --> 00:16:43.340 SCOTT HUGHES: Absolutely. 00:16:43.340 --> 00:16:45.720 Yeah. 00:16:45.720 --> 00:16:46.767 Thank you. 00:16:46.767 --> 00:16:47.600 I'm writing quickly. 00:16:47.600 --> 00:16:48.100 Yeah? 00:16:48.100 --> 00:16:51.827 AUDIENCE: Is there [INAUDIBLE]? 00:16:51.827 --> 00:16:52.910 SCOTT HUGHES: This is dx-- 00:16:52.910 --> 00:16:53.940 oh, bugger. 00:16:53.940 --> 00:16:55.912 Yep. 00:16:55.912 --> 00:16:57.370 I'm trying to get to something new. 00:16:57.370 --> 00:16:59.037 And I'm afraid I'm rushing a little bit. 00:16:59.037 --> 00:17:01.053 So thank you for catching this. 00:17:01.053 --> 00:17:03.220 And so with this, take the determiner of this thing. 00:17:03.220 --> 00:17:09.130 And sure enough you get r squared sine theta d r d theta 00:17:09.130 --> 00:17:11.170 d phi. 00:17:11.170 --> 00:17:12.640 So this is the main thing that we 00:17:12.640 --> 00:17:15.460 are going to use this result for-- this thing with tensor 00:17:15.460 --> 00:17:16.480 densities. 00:17:16.480 --> 00:17:18.760 I want to go on a brief aside, which 00:17:18.760 --> 00:17:21.790 is relevant to the problem that I delayed on 00:17:21.790 --> 00:17:22.890 this week's problem 7. 00:17:26.640 --> 00:17:29.370 So there are three parts of problem 7 00:17:29.370 --> 00:17:41.940 that I moved from PSET 3 to PSET 4 because they rely on a result 00:17:41.940 --> 00:17:43.230 that I want to talk about now. 00:17:46.010 --> 00:17:49.000 So the main thing that we use the determinant of the metric 00:17:49.000 --> 00:17:51.880 for in a formal way is this-- that it's a tensor 00:17:51.880 --> 00:17:53.950 density of weight minus 2. 00:17:53.950 --> 00:17:55.750 And so it's a really useful quantity 00:17:55.750 --> 00:17:59.023 for converting tensor densities into proper tensors. 00:17:59.023 --> 00:18:00.940 And really the most common application of this 00:18:00.940 --> 00:18:02.842 tends to be to volume elements. 00:18:02.842 --> 00:18:04.300 But it turns out that it's actually 00:18:04.300 --> 00:18:07.930 really useful for what a former professor of mine 00:18:07.930 --> 00:18:10.450 used to like to call party tricks. 00:18:10.450 --> 00:18:15.760 There's some really-- it offers a really nice shortcut 00:18:15.760 --> 00:18:18.610 to computing certain Christoffel symbols. 00:18:18.610 --> 00:18:21.190 So in honor of Saul Teukolsky let's call this a party trick. 00:18:26.480 --> 00:18:36.660 So we're using the determinant of the metric 00:18:36.660 --> 00:18:38.470 to compute certain Christoffels. 00:18:47.590 --> 00:18:49.340 So this is going to rely on the following. 00:18:49.340 --> 00:18:55.100 So suppose I calculate the Christoffel symbol, 00:18:55.100 --> 00:18:58.243 but I'm going to sum on the raised index. 00:18:58.243 --> 00:19:00.410 And bearing in mind it's symmetric in the lower one, 00:19:00.410 --> 00:19:02.630 I'm going to do a contraction of the raised index with one 00:19:02.630 --> 00:19:03.505 of the lower indices. 00:19:06.260 --> 00:19:10.630 So let's just throw in a couple of definitions. 00:19:10.630 --> 00:19:17.280 This is equivalent to the following. 00:19:17.280 --> 00:19:21.150 And so throwing in the definition of the Christoffel 00:19:21.150 --> 00:19:23.417 with all the indices in the downstairs position-- 00:19:43.820 --> 00:19:45.470 this formula, by the way, is something 00:19:45.470 --> 00:19:50.240 that I've been writing down now for about 27 years. 00:19:50.240 --> 00:19:51.830 And I have to look it up every time. 00:19:51.830 --> 00:19:54.447 Usually by the end of a semester of teaching 8.962, 00:19:54.447 --> 00:19:55.280 I have it memorized. 00:19:55.280 --> 00:19:56.660 But it decays by then. 00:19:56.660 --> 00:19:59.135 So if you're wondering how to go from here to here-- 00:19:59.135 --> 00:20:00.260 this is the kind of thing-- 00:20:00.260 --> 00:20:02.340 just look it up. 00:20:02.340 --> 00:20:04.320 So let's pause for a second. 00:20:04.320 --> 00:20:08.090 Remember that the metric is-- 00:20:08.090 --> 00:20:10.400 it's itself symmetric. 00:20:10.400 --> 00:20:12.640 So in keeping with that, I'm going 00:20:12.640 --> 00:20:19.860 to flip the indices on this last term, which-- hang on a second. 00:20:19.860 --> 00:20:21.420 That was stupid. 00:20:21.420 --> 00:20:21.920 Wait. 00:20:24.960 --> 00:20:25.460 Pardon me. 00:20:32.055 --> 00:20:33.930 This is the term I want to switch indices on. 00:20:33.930 --> 00:20:36.693 My apologies. 00:20:36.693 --> 00:20:38.110 So the reason I did that is I want 00:20:38.110 --> 00:20:39.610 to have both of these guys ending 00:20:39.610 --> 00:20:43.312 with the alpha because notice this and this-- 00:20:43.312 --> 00:20:44.020 they're the same. 00:20:44.020 --> 00:20:48.490 But I have interchanged the beta and the mu. 00:20:48.490 --> 00:20:49.850 So these two terms-- 00:20:49.850 --> 00:20:51.970 the first term and the third term-- 00:20:51.970 --> 00:20:56.970 are anti symmetric upon exchange of beta and mu. 00:20:56.970 --> 00:20:58.690 They are contracted with the metric, 00:20:58.690 --> 00:21:02.380 which is symmetric upon exchange of beta and mu. 00:21:02.380 --> 00:21:02.880 Question? 00:21:02.880 --> 00:21:05.080 AUDIENCE: Does the metric have to be symmetric? 00:21:05.080 --> 00:21:07.482 SCOTT HUGHES: The metric has to be symmetric. 00:21:07.482 --> 00:21:16.330 [LAUGHS] I don't want to get into that right now, but yes 00:21:16.330 --> 00:21:17.620 [LAUGHS]. 00:21:17.620 --> 00:21:19.780 So these guys are anti symmetric. 00:21:19.780 --> 00:21:21.080 This guy is symmetric. 00:21:21.080 --> 00:21:23.050 And remember the rule. 00:21:23.050 --> 00:21:25.720 Whenever you contract some kind of a symmetric object 00:21:25.720 --> 00:21:28.900 with an anti symmetric m you get 0. 00:21:28.900 --> 00:21:32.050 So that means this term and this term die. 00:21:47.751 --> 00:21:51.590 And what we are left with is gamma mu 00:21:51.590 --> 00:22:02.520 mu alpha is 1/2 g u beta, and the alpha derivative of g u 00:22:02.520 --> 00:22:04.960 beta. 00:22:04.960 --> 00:22:07.240 There is the way it is contracting 00:22:07.240 --> 00:22:09.460 the indices in the 2 metric with the other one. 00:22:09.460 --> 00:22:11.918 Well here's a theorem that I'm going to prove in a second-- 00:22:11.918 --> 00:22:16.100 or at least motivate-- that it's going to rely on a result 00:22:16.100 --> 00:22:17.680 that I will pull out of thin air, 00:22:17.680 --> 00:22:22.320 but can be found in most linear algebra textbooks. 00:22:22.320 --> 00:22:26.820 It's not too hard to show that this can be further written 00:22:26.820 --> 00:22:34.390 as 1 over square root of the determinant times the partial 00:22:34.390 --> 00:22:44.400 derivative of the square root of the determinant, 00:22:44.400 --> 00:22:47.090 which is sometimes-- depending on your applications-- 00:22:47.090 --> 00:22:50.700 this can be written very nicely as the derivative 00:22:50.700 --> 00:22:54.030 of the logarithm of the absolute value of the-- 00:22:54.030 --> 00:22:56.489 the square root of the absolute value of the determinant. 00:23:01.650 --> 00:23:05.500 So before I go on and actually demonstrate this, 00:23:05.500 --> 00:23:08.770 you can see why this is actually a pretty-- 00:23:08.770 --> 00:23:10.150 so this actually comes up. 00:23:10.150 --> 00:23:11.650 I'm going to show a few applications 00:23:11.650 --> 00:23:13.880 as to why this particular combination of Christoffel 00:23:13.880 --> 00:23:16.480 symbols shows up more often than you might guess. 00:23:16.480 --> 00:23:19.190 It's really important for certain important calculations. 00:23:19.190 --> 00:23:20.920 And this is telling me that I can get it 00:23:20.920 --> 00:23:23.380 by just taking one partial derivative of a scalar 00:23:23.380 --> 00:23:24.900 function. 00:23:24.900 --> 00:23:27.916 And if you know your metric, that's easy. 00:23:27.916 --> 00:23:31.000 So this becomes really easy thing to calculate. 00:23:31.000 --> 00:23:31.750 So let's prove it. 00:23:41.170 --> 00:23:45.107 So the proof of this relies on a few results 00:23:45.107 --> 00:23:45.940 from linear algebra. 00:24:03.370 --> 00:24:05.650 So let's not think about tensors for a second. 00:24:05.650 --> 00:24:07.150 And let's just think about matrices. 00:24:07.150 --> 00:24:09.850 So imagine I've got some matrix m. 00:24:12.910 --> 00:24:15.410 I'm going to be agnostic about the dimensions of this thing. 00:24:18.860 --> 00:24:21.040 And suppose I look at the following variation 00:24:21.040 --> 00:24:21.665 of this matrix. 00:24:31.240 --> 00:24:34.340 So suppose I imagine doing a little variation. 00:24:34.340 --> 00:24:37.210 So suppose every element of m is a function. 00:24:37.210 --> 00:24:43.390 And I look at a little variation of the log of the determinant 00:24:43.390 --> 00:24:45.750 of that matrix. 00:24:45.750 --> 00:24:58.280 Well this can be written as log is basically 00:24:58.280 --> 00:25:06.250 a definition of this. 00:25:08.850 --> 00:25:13.260 Now, if I exploit properties of logarithms, 00:25:13.260 --> 00:25:20.440 this can be written as the log of the determinant-- 00:25:20.440 --> 00:25:23.240 m plus delta m-- 00:25:23.240 --> 00:25:26.010 divided by the determinant of m. 00:25:33.060 --> 00:25:34.920 Now I'm going to use the fact that 1 00:25:34.920 --> 00:25:38.317 over the determinant of m is the determinant 00:25:38.317 --> 00:25:39.150 of the inverse of m. 00:26:00.850 --> 00:26:04.700 So taking advantage of that, I can further write this guy 00:26:04.700 --> 00:26:20.000 as something like this. 00:26:20.000 --> 00:26:23.495 Now I'm going to invoke an identity, which 00:26:23.495 --> 00:26:26.120 I believe you can find proven in many linear algebra textbooks. 00:26:26.120 --> 00:26:27.770 It just occurred to me as I'm thinking about this, 00:26:27.770 --> 00:26:30.620 I don't know if I've ever seen it explicitly proven myself. 00:26:30.620 --> 00:26:32.870 But it's something that's very easy to demonstrate 00:26:32.870 --> 00:26:35.200 with just a quick calculation. 00:26:35.200 --> 00:26:36.463 You can just do-- 00:26:36.463 --> 00:26:37.130 I'm a physicist. 00:26:37.130 --> 00:26:38.280 So for me I'll use Mathematica. 00:26:38.280 --> 00:26:40.060 I'll look at six or seven examples and go, 00:26:40.060 --> 00:26:41.130 it seems right. 00:26:41.130 --> 00:26:42.890 And so I've definitely done that. 00:26:42.890 --> 00:26:44.557 But I believe this is something that you 00:26:44.557 --> 00:26:46.100 can find proven explicitly-- 00:26:46.100 --> 00:26:47.930 like I said-- in most books. 00:26:47.930 --> 00:26:49.920 So remember these are all matrices. 00:26:49.920 --> 00:26:51.360 So this isn't the number 1. 00:26:51.360 --> 00:26:53.360 We want to think of this as the identity matrix. 00:26:59.900 --> 00:27:02.090 Oh and I'm also going to regard this variation 00:27:02.090 --> 00:27:04.580 as a small quantity. 00:27:04.580 --> 00:27:15.400 So if I regard epsilon as a small matrix-- 00:27:15.400 --> 00:27:17.290 this can be made formal by defining something 00:27:17.290 --> 00:27:19.290 like condition number associated with the matrix 00:27:19.290 --> 00:27:20.080 or something like that. 00:27:20.080 --> 00:27:21.480 But generally what I want to mean 00:27:21.480 --> 00:27:23.220 by that is if I take this epsilon 00:27:23.220 --> 00:27:26.640 and I add it to 1, all of this-- 00:27:26.640 --> 00:27:29.010 so my identity is 1 on the diagonal-- 00:27:29.010 --> 00:27:31.380 0s everywhere else-- all the things 00:27:31.380 --> 00:27:33.480 that are put into the sum of 1 plus epsilon 00:27:33.480 --> 00:27:36.390 are much, much smaller than that 1 that's on the diagonal. 00:27:36.390 --> 00:27:38.260 That will be sufficient. 00:27:38.260 --> 00:27:46.230 So if epsilon is a small matrix, then the determinant 00:27:46.230 --> 00:27:55.540 of 1 plus epsilon is approximately equal to 1 00:27:55.540 --> 00:27:59.130 plus the trace of epsilon. 00:28:05.840 --> 00:28:08.000 What that approximately refers to is-- of course 00:28:08.000 --> 00:28:09.180 you can take that further. 00:28:09.180 --> 00:28:10.370 And you'll get additional corrections 00:28:10.370 --> 00:28:12.860 that involve epsilon times epsilon, epsilon times epsilon, 00:28:12.860 --> 00:28:14.570 times epsilon. 00:28:14.570 --> 00:28:17.150 I believe when you do that, the coefficient is 00:28:17.150 --> 00:28:18.020 no longer universal. 00:28:18.020 --> 00:28:20.990 But it depends upon the dimensions of the matrix. 00:28:20.990 --> 00:28:23.000 But leading order it's independent of dimensions 00:28:23.000 --> 00:28:23.600 of the matrix. 00:28:23.600 --> 00:28:24.500 And that's something that you can 00:28:24.500 --> 00:28:26.460 you can play with a little bit yourself. 00:28:26.460 --> 00:28:28.960 Like I said, this is sufficient for what we want to do here. 00:28:35.070 --> 00:28:42.460 So I'm going to think of my small matrix 00:28:42.460 --> 00:28:50.350 as the matrix of inverse m times a variation of m. 00:28:50.350 --> 00:28:51.940 This is our epsilon. 00:28:55.010 --> 00:28:58.490 So we're going to apply it to the line that I have up here. 00:28:58.490 --> 00:29:04.180 And this tells me that my delta on the log of the derivative 00:29:04.180 --> 00:29:17.690 of m is the log of 1 plus the trace of m to the minus 1 00:29:17.690 --> 00:29:20.380 on the matrix m. 00:29:20.380 --> 00:29:23.130 Log of 1 plus a small number is that small number. 00:29:33.000 --> 00:29:34.440 Now the application. 00:29:34.440 --> 00:29:36.900 So this is-- like I said, this the theorem that you 00:29:36.900 --> 00:29:41.730 can find in books that I don't know about but truly exist. 00:29:41.730 --> 00:29:45.690 This is something I've seen documented in a lot of places. 00:29:45.690 --> 00:29:50.115 Let's treat our m as the metric of spacetime. 00:29:55.700 --> 00:29:58.300 So my m will be g alpha beta. 00:29:58.300 --> 00:30:01.010 My m inverse will be g in the upstairs position. 00:30:06.030 --> 00:30:09.070 And I will write this something like so. 00:30:09.070 --> 00:30:12.600 And I'm going to apply this by looking 00:30:12.600 --> 00:30:14.070 at variations in my metric. 00:30:50.510 --> 00:30:54.070 So delta log-- 00:30:54.070 --> 00:30:56.330 I'm going to throw my absolute values in here. 00:30:56.330 --> 00:31:00.320 That's perfectly allowed to go ahead and put that into there. 00:31:00.320 --> 00:31:03.350 Applying this to what I've got, this 00:31:03.350 --> 00:31:15.338 is going to be the trace of g mu beta times the variation of g 00:31:15.338 --> 00:31:16.935 beta gamma. 00:31:16.935 --> 00:31:19.310 And I forgot to say, how do I take the trace of a matrix? 00:31:33.482 --> 00:31:35.190 So the trace that we're going to use-- we 00:31:35.190 --> 00:31:37.710 want it to be something that has geometric meaning 00:31:37.710 --> 00:31:40.240 and has a tensorial meaning to it. 00:31:40.240 --> 00:31:48.200 So we're going to call the trace of this thing g alpha 00:31:48.200 --> 00:31:50.360 beta epsilon alpha beta. 00:31:53.217 --> 00:31:54.800 If you think about what this is doing, 00:31:54.800 --> 00:31:57.380 you're essentially going to take your-- 00:31:57.380 --> 00:31:59.915 let's say I apply this to the metric itself. 00:31:59.915 --> 00:32:01.790 I put one index in the upstairs position, one 00:32:01.790 --> 00:32:05.120 the downstairs position, and then I am summing along 00:32:05.120 --> 00:32:08.390 the diagonal when I do this. 00:32:08.390 --> 00:32:15.990 You will sometimes see this written as something like that. 00:32:21.300 --> 00:32:25.790 So in this case, when I'm taking the trace of this guy here, 00:32:25.790 --> 00:32:28.810 that is going to force me to-- 00:32:28.810 --> 00:32:31.810 let's see. 00:32:31.810 --> 00:32:33.560 So this gives me a quantity where 00:32:33.560 --> 00:32:35.910 I'm summing over my betas. 00:32:35.910 --> 00:32:38.630 And then I'm just going to sum over the diagonal indices. 00:32:38.630 --> 00:32:42.780 I'm forcing my two remaining indices to be the same. 00:32:42.780 --> 00:32:46.880 So putting this together, this tells me-- 00:32:59.160 --> 00:33:02.630 so now what I'm going to do is say, 00:33:02.630 --> 00:33:06.070 I basically have part of a partial derivative here. 00:33:06.070 --> 00:33:18.050 All I need to do is now divide by a variation in my coordinate 00:33:18.050 --> 00:33:19.010 and take the limit. 00:33:39.340 --> 00:33:43.633 So it comes out of this as the partial derivative-- 00:33:53.310 --> 00:33:54.670 looks like this. 00:33:54.670 --> 00:33:56.940 Now let's trace it back to our Christoffel symbol. 00:34:02.060 --> 00:34:05.270 My Christoffel symbol-- the thing 00:34:05.270 --> 00:34:07.100 which I'm trying to compute-- is one half 00:34:07.100 --> 00:34:08.110 of this right-hand side. 00:34:15.630 --> 00:34:17.922 So it's one half of the left-hand side. 00:34:17.922 --> 00:34:20.464 And I can take that one half, march it through my derivative, 00:34:20.464 --> 00:34:23.300 and use the fact that 1/2 the log of x 00:34:23.300 --> 00:34:24.830 is the log of the square root of x. 00:34:40.909 --> 00:34:43.260 Check. 00:34:43.260 --> 00:34:46.860 So like I said, this is what an old mentor of mine 00:34:46.860 --> 00:34:48.570 used to like to call a party trick. 00:34:48.570 --> 00:34:52.440 It is a really useful party trick for certain calculations. 00:34:52.440 --> 00:34:54.949 So I want to make sure you saw where that comes from. 00:34:54.949 --> 00:34:57.360 This is something you will now use on the problem 00:34:57.360 --> 00:35:00.087 that I just moved from PSET 3 to PSET 4. 00:35:00.087 --> 00:35:02.170 It's useful for you to know where this comes from. 00:35:02.170 --> 00:35:03.630 You're certainly not going to need to go through this 00:35:03.630 --> 00:35:04.130 yourself. 00:35:04.130 --> 00:35:06.720 But this is a good type of calculation 00:35:06.720 --> 00:35:10.440 to be comfortable with. 00:35:10.440 --> 00:35:12.960 Those of you who are more rigorous in your math than me, 00:35:12.960 --> 00:35:15.660 you might want to run off and verify 00:35:15.660 --> 00:35:17.910 a couple of these identities that I used. 00:35:17.910 --> 00:35:24.600 But this is very nice for physics level rigor-- 00:35:24.600 --> 00:35:26.730 at least astrophysicists level rigor. 00:35:26.730 --> 00:35:29.070 So let me talk about one of the places where this 00:35:29.070 --> 00:35:32.550 shows up and is quite useful. 00:35:39.310 --> 00:35:42.000 So a place where I've seen this show up the most 00:35:42.000 --> 00:35:44.190 is when you're looking at the spacetime divergence 00:35:44.190 --> 00:35:44.970 of a vector field. 00:35:55.320 --> 00:35:58.550 So when you're calculating the covariant derivative 00:35:58.550 --> 00:36:00.980 of alpha contracting on the indices-- 00:36:00.980 --> 00:36:01.940 let's just throw in-- 00:36:05.140 --> 00:36:07.030 expand out the full definition of things-- 00:36:13.920 --> 00:36:16.080 all you've gotta do is correct one index. 00:36:16.080 --> 00:36:21.090 And voila, this is exactly the things that change-- 00:36:21.090 --> 00:36:23.190 where is it-- change my alpha to a mu-- that's 00:36:23.190 --> 00:36:26.610 exactly what I've got before. 00:36:26.610 --> 00:36:32.680 And so-- hang on just one moment. 00:36:32.680 --> 00:36:34.440 I know what I'm doing. 00:36:34.440 --> 00:36:36.750 These are all dummy indices. 00:36:36.750 --> 00:36:39.610 So in order to keep things from getting crossed, 00:36:39.610 --> 00:36:41.300 I'm going to relabel these over here. 00:36:47.660 --> 00:36:51.340 So I can take advantage of this identity and write this. 00:37:08.150 --> 00:37:17.260 So stare at this for a second. 00:37:17.260 --> 00:37:21.200 And you'll see that the whole thing can be rewritten 00:37:21.200 --> 00:37:22.560 in a very simple form. 00:37:47.560 --> 00:37:49.660 Ta-da. 00:37:49.660 --> 00:37:52.360 You haven't done as much work with covariant in your lives 00:37:52.360 --> 00:37:53.350 as I have. 00:37:53.350 --> 00:37:56.205 So let me just emphasize that ordinarily when 00:37:56.205 --> 00:37:57.580 you see an expression like you've 00:37:57.580 --> 00:37:59.950 got up there on the top line, you look at that, 00:37:59.950 --> 00:38:02.388 and you kind of go [GROANS] because you look at that, 00:38:02.388 --> 00:38:04.180 and the first thing that comes to your mind 00:38:04.180 --> 00:38:07.360 is you've got to work out every one of those Christoffel 00:38:07.360 --> 00:38:10.470 symbols and sum it up to get those things. 00:38:10.470 --> 00:38:13.870 And in a general spacetime, there will be 40 of them. 00:38:13.870 --> 00:38:16.960 And before Odin gave us Mathematica, 00:38:16.960 --> 00:38:18.310 that was a fair amount of labor. 00:38:18.310 --> 00:38:20.870 Even with Mathematica it's not necessarily trivial 00:38:20.870 --> 00:38:23.520 because it's really easy to screw things up. 00:38:23.520 --> 00:38:26.350 With this you calculate the determinant of the metric, 00:38:26.350 --> 00:38:28.990 you take its square root, you multiply your guy, 00:38:28.990 --> 00:38:31.330 and you take a partial derivative, and you divide. 00:38:31.330 --> 00:38:33.580 That is something that most of us 00:38:33.580 --> 00:38:35.410 learned how to do quite a long time ago. 00:38:35.410 --> 00:38:38.892 It cleans the hell out of this up. 00:38:38.892 --> 00:38:41.350 So the fact that this gives us something that only involves 00:38:41.350 --> 00:38:43.794 partial derivatives is awesome. 00:38:55.660 --> 00:38:59.590 This also-- it turns out-- so when you have things like this, 00:38:59.590 --> 00:39:03.730 it gives us a nice way to express Gauss's theorem 00:39:03.730 --> 00:39:04.740 in a curved manifold. 00:39:08.830 --> 00:39:11.225 So Gauss's theorem-- if I just look at the integrals 00:39:11.225 --> 00:39:13.600 that-- or rather the integral for Gauss's theorem-- let's 00:39:13.600 --> 00:39:15.070 put it that way. 00:39:15.070 --> 00:39:17.560 Let's say a Gauss's-type integral. 00:39:23.330 --> 00:39:25.520 So go back to when you're talking 00:39:25.520 --> 00:39:26.750 about conservation laws. 00:39:30.210 --> 00:39:38.400 If I imagine I'm integrating the divergence of some vector field 00:39:38.400 --> 00:39:41.130 over a four-dimensional volume, look at that, 00:39:41.130 --> 00:39:44.820 I get a nice cancellation. 00:39:44.820 --> 00:39:57.180 So this turns into an integral of that nice, clean derivative 00:39:57.180 --> 00:40:00.780 over my four coordinates. 00:40:00.780 --> 00:40:03.570 And then you can take advantage of the actual content 00:40:03.570 --> 00:40:05.977 of Gauss's Theorem to turn that into an integral 00:40:05.977 --> 00:40:08.310 over the three-dimensional surface that bounds that four 00:40:08.310 --> 00:40:08.810 volume. 00:40:13.330 --> 00:40:14.320 It's a good point-- 00:40:14.320 --> 00:40:15.825 so you're emboldened by this. 00:40:15.825 --> 00:40:16.950 You say, yay, look at that. 00:40:16.950 --> 00:40:19.230 We can do all this awesome stuff with this identity. 00:40:19.230 --> 00:40:20.875 It gives me a great way to express some 00:40:20.875 --> 00:40:23.400 of these conservation laws. 00:40:23.400 --> 00:40:24.767 You might think to yourself-- 00:40:24.767 --> 00:40:26.850 and I realized as I was looking over these notes-- 00:40:26.850 --> 00:40:30.390 I'm about to I think give away a part of one of the problems 00:40:30.390 --> 00:40:32.510 on the PSET-- but c'est la vie. 00:40:32.510 --> 00:40:35.510 It's an important point. 00:40:35.510 --> 00:40:38.432 Can we do something similar for tensors? 00:40:38.432 --> 00:40:40.640 So this is great that you have this form for vectors. 00:40:43.220 --> 00:40:46.287 The divergence of a vector is a mathematical notion that 00:40:46.287 --> 00:40:47.495 comes up in various contexts. 00:40:47.495 --> 00:40:49.320 So this is important. 00:40:49.320 --> 00:40:51.410 But we've already talked about the fact 00:40:51.410 --> 00:40:53.150 that things like energy and momentum 00:40:53.150 --> 00:40:55.670 are described by a stress energy tensor. 00:40:55.670 --> 00:40:57.080 So can we do this for tensors? 00:41:05.190 --> 00:41:09.000 Well the answer turns out to be no, 00:41:09.000 --> 00:41:10.245 except in a handful of cases. 00:41:13.070 --> 00:41:17.260 And I have a comment about those handful of cases. 00:41:17.260 --> 00:41:19.510 So suppose I take this-- 00:41:19.510 --> 00:41:21.040 and I'm taking the divergence on say 00:41:21.040 --> 00:41:22.360 the first index of this guy-- 00:41:27.150 --> 00:41:31.540 so there's the bit involves my partial derivative-- 00:41:31.540 --> 00:41:36.450 I'm going to have a bit that involves 00:41:36.450 --> 00:41:38.640 correcting the first index. 00:41:51.600 --> 00:41:58.160 So the first correction is it's of a form that 00:41:58.160 --> 00:41:59.720 does in fact involve this guy we just 00:41:59.720 --> 00:42:00.963 worked out this identity for. 00:42:00.963 --> 00:42:02.630 And in principle we could take advantage 00:42:02.630 --> 00:42:05.480 of that to massage this and use this identity. 00:42:05.480 --> 00:42:08.508 But the second one there's nothing to do with that. 00:42:08.508 --> 00:42:10.050 This you just have to go and work out 00:42:10.050 --> 00:42:12.840 all of your 40 different Christoffel symbols 00:42:12.840 --> 00:42:14.310 and sit down and slog through it. 00:42:14.310 --> 00:42:15.935 This spoils your ability to do anything 00:42:15.935 --> 00:42:21.270 with it, with one exception. 00:42:21.270 --> 00:42:25.102 What if a is an anti-symmetric tensor? 00:42:25.102 --> 00:42:26.630 If a is an anti-symmetric tensor, 00:42:26.630 --> 00:42:29.870 you've got symmetry, anti symmetry, and it dies. 00:42:29.870 --> 00:42:32.420 So that is one example of where you can actually apply it. 00:42:32.420 --> 00:42:35.840 And I had you guys play with that a little bit on the PSET. 00:42:35.840 --> 00:42:38.780 It's worth noting though that the main reason why one often 00:42:38.780 --> 00:42:41.300 finds this to be a useful thing to do 00:42:41.300 --> 00:42:44.810 is that when you take the divergence of something 00:42:44.810 --> 00:42:47.570 like a vector, you get a scalar out. 00:42:47.570 --> 00:42:50.570 You get a quantity that is-- 00:42:50.570 --> 00:42:52.490 really its transformation properties 00:42:52.490 --> 00:42:55.400 between different inertial frames or freely-falling frames 00:42:55.400 --> 00:42:55.940 is simple. 00:43:06.230 --> 00:43:09.290 So even when you can do this and take advantage of this thing, 00:43:09.290 --> 00:43:13.028 working with the divergence of a tensor-- 00:43:13.028 --> 00:43:15.320 exploiting a trick like this turns out to generally not 00:43:15.320 --> 00:43:16.240 be all that useful. 00:43:16.240 --> 00:43:20.000 And I'll use the example of the stress energy tensor. 00:43:20.000 --> 00:43:24.390 So conservation of stress energy in special relativity-- 00:43:24.390 --> 00:43:27.802 it was the partial derivative-- 00:43:27.802 --> 00:43:29.260 the divergence of the stress energy 00:43:29.260 --> 00:43:31.093 tensor expressed with the partial derivative 00:43:31.093 --> 00:43:32.200 was equal to 0. 00:43:32.200 --> 00:43:34.555 We're going to take this over to covariant derivative 00:43:34.555 --> 00:43:36.430 of the stress energy tensor being equal to 0. 00:43:36.430 --> 00:43:37.810 That's what the equivalence principle tells us 00:43:37.810 --> 00:43:38.435 that we can do. 00:43:40.960 --> 00:43:43.600 Now when I take the divergence of something like the stress 00:43:43.600 --> 00:43:45.820 energy tensor, I get a 4 vector. 00:43:48.580 --> 00:43:51.130 Every 4 vector always has implicitly 00:43:51.130 --> 00:43:54.138 a set of basis objects attached to it. 00:43:54.138 --> 00:43:55.930 When I've got basis objects attached to it, 00:43:55.930 --> 00:43:59.008 those are defined with respect to the tangent space 00:43:59.008 --> 00:44:00.550 at a particular point in the manifold 00:44:00.550 --> 00:44:02.260 where you are currently working. 00:44:02.260 --> 00:44:05.680 And so if I want to try to do something like an integral like 00:44:05.680 --> 00:44:09.190 this-- where I add up the four vector I get by taking 00:44:09.190 --> 00:44:12.220 the divergence of stress energy and integrate it over 00:44:12.220 --> 00:44:13.330 a volume-- 00:44:13.330 --> 00:44:16.180 I'm going to get nonsense because what's going on 00:44:16.180 --> 00:44:18.880 is I'm combining vector fields that are defined 00:44:18.880 --> 00:44:22.060 in different tangent spaces that can't be properly compared 00:44:22.060 --> 00:44:23.710 to one another. 00:44:23.710 --> 00:44:25.400 In order to do that kind of comparison, 00:44:25.400 --> 00:44:28.520 you have to introduce a transport law. 00:44:28.520 --> 00:44:30.160 And when you start doing transports 00:44:30.160 --> 00:44:34.060 over macroscopic regions, you run into trouble. 00:44:34.060 --> 00:44:36.370 They turn out to be path dependent. 00:44:36.370 --> 00:44:38.430 And this is where we run into ambiguities 00:44:38.430 --> 00:44:40.180 that have to do with the curvature content 00:44:40.180 --> 00:44:41.560 of your manifold. 00:44:41.560 --> 00:44:44.440 We'll discuss where that comes into our calculations 00:44:44.440 --> 00:44:45.610 a little bit later. 00:44:45.610 --> 00:44:49.270 But what it basically boils down to is 00:44:49.270 --> 00:44:51.850 if I use a stress energy tensor as an example, 00:44:51.850 --> 00:44:56.050 this equation tells me about local conservation 00:44:56.050 --> 00:44:59.520 of energy and momentum. 00:44:59.520 --> 00:45:03.720 In general relativity I cannot take the local conservation 00:45:03.720 --> 00:45:06.120 of energy and momentum and promote it to a global 00:45:06.120 --> 00:45:08.130 conservation of energy and momentum. 00:45:08.130 --> 00:45:10.510 It's ambiguous. 00:45:10.510 --> 00:45:15.910 We'll deal with that and the conceptual difficulties 00:45:15.910 --> 00:45:18.100 that that presents a little bit later in the course. 00:45:18.100 --> 00:45:21.850 But it's a good see the plant at this point. 00:45:21.850 --> 00:45:23.350 So let's switch gears. 00:45:23.350 --> 00:45:25.570 We have a new set of mathematical tools. 00:45:30.340 --> 00:45:35.860 I want to take a detour away from thinking about some more 00:45:35.860 --> 00:45:39.070 abstract mathematical notions and start thinking about how 00:45:39.070 --> 00:45:41.300 we actually do some physics. 00:45:41.300 --> 00:45:44.200 So what I want to do is talk today about how 00:45:44.200 --> 00:45:47.980 do we formulate the kinematics of a body 00:45:47.980 --> 00:45:49.698 moving in curved spacetime? 00:46:08.427 --> 00:46:10.010 So I've already hinted at this in some 00:46:10.010 --> 00:46:11.000 of my previous lectures. 00:46:11.000 --> 00:46:12.375 And what I want to do now is just 00:46:12.375 --> 00:46:14.990 basically fill in some of the gaps. 00:46:14.990 --> 00:46:18.470 The way that we do this really just 00:46:18.470 --> 00:46:21.940 builds on Einstein's insight about what 00:46:21.940 --> 00:46:24.500 the weak equivalence principle means. 00:46:24.500 --> 00:46:27.320 So go into a freely falling frame. 00:46:34.910 --> 00:46:36.870 Go in that freely-falling frame. 00:46:36.870 --> 00:46:42.160 Put things into locally Lorentz coordinates. 00:46:42.160 --> 00:46:44.538 In other words perform that little calculation 00:46:44.538 --> 00:46:46.330 that make spacetime look like the spacetime 00:46:46.330 --> 00:46:49.219 of special relativity of the curvature corrections. 00:46:59.210 --> 00:47:01.490 And to start with, let's consider what we always 00:47:01.490 --> 00:47:04.440 do in physics, is we'll look at the simplest body first. 00:47:04.440 --> 00:47:06.440 We're going to look at what we call a test body. 00:47:15.410 --> 00:47:22.150 So this is the body that has no charge, no spatial extent, 00:47:22.150 --> 00:47:29.920 it's of zero dimensional point, no spin-- 00:47:29.920 --> 00:47:33.580 nothing interesting, except a mass. 00:47:42.350 --> 00:47:44.360 So if you want to think about this-- 00:47:44.360 --> 00:47:46.430 I use a way that I find to think about this 00:47:46.430 --> 00:47:51.200 is all these various aspects to it, you're adding additional-- 00:47:51.200 --> 00:47:54.800 either charges to it or additional multipolar structure 00:47:54.800 --> 00:47:56.090 to this body. 00:47:56.090 --> 00:47:59.090 I'm thinking of this-- this is sort of like a pure monopole. 00:47:59.090 --> 00:48:00.920 It's nothing but mass concentrated 00:48:00.920 --> 00:48:03.680 in a single zero size point. 00:48:03.680 --> 00:48:05.299 Obviously it's an idealization. 00:48:05.299 --> 00:48:06.716 But you've got to start somewhere. 00:48:14.810 --> 00:48:18.920 So since it's got no charge, no spatial extent, 00:48:18.920 --> 00:48:22.803 it's got nothing but mass, nothing's 00:48:22.803 --> 00:48:23.720 going to couple to it. 00:48:23.720 --> 00:48:26.685 It's not going to basically do anything but freefall. 00:48:37.040 --> 00:48:49.906 In this frame the body moves on a purely inertial trajectory. 00:49:04.262 --> 00:49:06.470 And what does a purely inertial trajectory look like? 00:49:06.470 --> 00:49:11.360 Well you take whatever your initial conditions are. 00:49:15.330 --> 00:49:17.490 And you move in a straight line with respect 00:49:17.490 --> 00:49:20.560 to time as measured on your own clock. 00:49:20.560 --> 00:49:23.670 Simplest, stupidest possible motion that you can. 00:49:37.320 --> 00:49:39.890 So we would obviously call that a straight line with respect 00:49:39.890 --> 00:49:41.390 to the parameterization that's being 00:49:41.390 --> 00:49:43.740 used in this representation. 00:49:43.740 --> 00:49:46.640 So what does that mean in a more general sense 00:49:46.640 --> 00:49:47.730 of the representation? 00:49:47.730 --> 00:49:50.300 So if we think about this a little bit more geometrically, 00:49:50.300 --> 00:49:53.780 when a body is moving in a straight line, that basically 00:49:53.780 --> 00:49:58.250 means that whatever the tangent vector to its world line is, 00:49:58.250 --> 00:50:02.540 it's essentially moving such that the tangent vector at time 00:50:02.540 --> 00:50:08.550 T1 is parallel to the tangent vector at T1 plus delta T1, 00:50:08.550 --> 00:50:10.160 provided that's actually small enough 00:50:10.160 --> 00:50:13.010 that they're sort of within the same local Lorentz frame. 00:50:16.950 --> 00:50:20.120 So a more geometric way of thinking about this motion 00:50:20.120 --> 00:50:22.110 is that it's parallel transporting its tangent 00:50:22.110 --> 00:50:22.610 vector. 00:51:13.858 --> 00:51:15.650 Let's make this a little bit more rigorous. 00:51:15.650 --> 00:51:17.150 So let's imagine this body's moving 00:51:17.150 --> 00:51:19.983 on a particular trajectory through spacetime. 00:51:25.780 --> 00:51:28.840 So it's a trajectory parameterized. 00:51:28.840 --> 00:51:33.250 I will define its parameterization a little bit 00:51:33.250 --> 00:51:34.440 more carefully very soon. 00:51:38.660 --> 00:51:42.090 So for now, just think of lambda as some kind of a quantity. 00:51:42.090 --> 00:51:44.420 It's a scale that just accumulates uniformly as it 00:51:44.420 --> 00:51:48.000 moves along the world line. 00:51:48.000 --> 00:51:50.620 So I'm going to say the small body has 00:51:50.620 --> 00:51:55.900 a path through spacetime, given by u x of lambda. 00:51:55.900 --> 00:52:08.190 Its tangent is given by this. 00:52:08.190 --> 00:52:10.590 And if it is parallel transporting its own tangent 00:52:10.590 --> 00:52:24.753 vector, that is-- 00:52:24.753 --> 00:52:26.170 I'll remind you that the condition 00:52:26.170 --> 00:52:31.950 for parallel transport was that you 00:52:31.950 --> 00:52:34.080 take the covariant derivative your field. 00:52:34.080 --> 00:52:38.670 And as you are moving along, you contract it 00:52:38.670 --> 00:52:41.370 along the tangent vector of the trajectory you're moving on. 00:52:41.370 --> 00:52:43.260 And you get 0. 00:52:43.260 --> 00:52:48.600 So in my notes, there's a couple of equivalent ways 00:52:48.600 --> 00:52:49.280 of writing this. 00:52:49.280 --> 00:52:51.000 So you will sometimes see this written 00:52:51.000 --> 00:52:55.320 as the gradient along u of u. 00:52:59.090 --> 00:53:06.970 And you'll sometimes see this written as capital u 00:53:06.970 --> 00:53:08.108 u lambda equals 0. 00:53:08.108 --> 00:53:08.900 So these are just-- 00:53:08.900 --> 00:53:09.790 I just throw that out because these 00:53:09.790 --> 00:53:11.915 are different forms that are common in the notation 00:53:11.915 --> 00:53:12.842 that you will see. 00:53:12.842 --> 00:53:14.050 So let's expand this guy out. 00:53:39.018 --> 00:53:40.060 It's something like this. 00:53:40.060 --> 00:53:43.870 So what we're going to do-- so remember this is dx d lambda. 00:53:43.870 --> 00:53:46.438 This is d by dx. 00:53:46.438 --> 00:53:48.730 That's a total derivative with respect to the parameter 00:53:48.730 --> 00:53:49.230 lambda. 00:53:51.712 --> 00:53:52.420 So this becomes-- 00:54:05.500 --> 00:54:07.590 I'm going to write it in two forms. 00:54:07.590 --> 00:54:11.070 This is often written expanding out the u into a second order 00:54:11.070 --> 00:54:11.570 form. 00:54:18.890 --> 00:54:21.230 This is obvious but sufficiently important. 00:54:21.230 --> 00:54:26.900 It's worth calling it out. 00:54:26.900 --> 00:54:30.410 And this has earned itself a box. 00:54:39.570 --> 00:54:42.930 This result is known as the geodesic equation. 00:55:03.020 --> 00:55:05.040 The trajectories which solve these equations 00:55:05.040 --> 00:55:06.520 are known as geodesics. 00:55:22.083 --> 00:55:25.580 One of the reasons why I highlight this is it's-- 00:55:29.790 --> 00:55:32.610 I'm trying to keep a straight face with the comment I 00:55:32.610 --> 00:55:33.600 want to make. 00:55:33.600 --> 00:55:37.350 A tremendous amount of research in general relativity 00:55:37.350 --> 00:55:41.670 is based around doing solutions of this equation 00:55:41.670 --> 00:55:45.060 for various spacetimes that go in to make the Christoffel 00:55:45.060 --> 00:55:47.370 symbols. 00:55:47.370 --> 00:55:54.240 My career-- [LAUGHS] it's probably not false to say that 00:55:54.240 --> 00:56:00.720 about 65% of my papers have this equation at its centerpiece 00:56:00.720 --> 00:56:03.540 at some point with the thing that goes into making 00:56:03.540 --> 00:56:04.860 my gammas-- 00:56:04.860 --> 00:56:08.190 things related to black hole spacetimes. 00:56:08.190 --> 00:56:12.450 This is really important because this gives me the motion 00:56:12.450 --> 00:56:14.260 of a freely-falling frame. 00:56:14.260 --> 00:56:16.380 What does a freely-falling frame describe? 00:56:16.380 --> 00:56:18.610 Somebody who's moving under gravity. 00:56:18.610 --> 00:56:21.420 So when you're doing things like describing orbits, for example, 00:56:21.420 --> 00:56:23.700 this is your tool. 00:56:23.700 --> 00:56:25.350 A tremendous number of applications 00:56:25.350 --> 00:56:27.840 where if what you care about is the motion 00:56:27.840 --> 00:56:30.450 of a body due to relativistic gravity, 00:56:30.450 --> 00:56:32.580 this gives you a leading solution. 00:56:32.580 --> 00:56:35.190 Now bear in mind when I did this, 00:56:35.190 --> 00:56:36.900 this is the motion of a test body. 00:56:36.900 --> 00:56:38.880 This is an object with no charge, 00:56:38.880 --> 00:56:41.730 no spatial extent, no spin-- 00:56:41.730 --> 00:56:45.570 that describes no object. 00:56:45.570 --> 00:56:49.080 So it should be borne in mind that this is 00:56:49.080 --> 00:56:52.110 the leading solution to things. 00:56:52.110 --> 00:56:54.170 Suppose the body is charged. 00:56:54.170 --> 00:56:56.130 And there is an electromagnetic field 00:56:56.130 --> 00:56:58.410 that this body is interacting with. 00:56:58.410 --> 00:57:01.170 Then what you do is you are no longer going 00:57:01.170 --> 00:57:03.900 to be parallel transporting this tangent factor. 00:57:03.900 --> 00:57:05.733 It will be pushed away-- we like to say-- 00:57:05.733 --> 00:57:06.900 from the parallel transport. 00:57:06.900 --> 00:57:09.180 And you'll replace the 0 on the right hand side 00:57:09.180 --> 00:57:11.550 here with a properly-constructed force 00:57:11.550 --> 00:57:14.730 that describes the interactions of those charges 00:57:14.730 --> 00:57:17.280 with the fields. 00:57:17.280 --> 00:57:19.770 Suppose the body has some size. 00:57:19.770 --> 00:57:22.650 Well then what ends up happening is that the body actually 00:57:22.650 --> 00:57:25.020 doesn't just couple to a single-- remember what's 00:57:25.020 --> 00:57:29.190 going on here is that in the freely falling frame, 00:57:29.190 --> 00:57:32.490 I'm imagining that spacetime is flat at some point. 00:57:32.490 --> 00:57:35.183 And in a decent enough vicinity of that point, 00:57:35.183 --> 00:57:36.600 the first order corrections are 0. 00:57:36.600 --> 00:57:38.610 But there might be second order corrections. 00:57:38.610 --> 00:57:41.040 Well imagine a body is so big that it fills 00:57:41.040 --> 00:57:42.800 that freely-falling frame. 00:57:42.800 --> 00:57:46.368 And it actually tastes those second order corrections. 00:57:46.368 --> 00:57:47.910 Then what's going to happen is you're 00:57:47.910 --> 00:57:50.320 going to get additional terms on this equation, which 00:57:50.320 --> 00:57:53.200 have to do with the coupling of the spatial extent of that body 00:57:53.200 --> 00:57:55.780 to the curvature of the spacetime. 00:57:55.780 --> 00:58:00.580 That is where-- so for people who study astrophysical systems 00:58:00.580 --> 00:58:03.970 involving binaries, when you have spinning bodies, 00:58:03.970 --> 00:58:06.400 that ends up actually-- you cannot describe a body 00:58:06.400 --> 00:58:08.950 that's spinning without it having some spatial extent. 00:58:08.950 --> 00:58:11.830 And you find terms here that involve coupling of those 00:58:11.830 --> 00:58:14.720 spins to the curvature of the spacetime. 00:58:14.720 --> 00:58:18.910 So this is the leading piece of the motion of a body moving 00:58:18.910 --> 00:58:20.680 in the current spacetime. 00:58:20.680 --> 00:58:22.460 And it's enough to do a tremendous amount. 00:58:22.460 --> 00:58:27.470 Basically because gravity is just so bloody strong that all 00:58:27.470 --> 00:58:28.792 of these various things-- 00:58:28.792 --> 00:58:30.250 it's the weakest fundamental force. 00:58:30.250 --> 00:58:33.340 But it adds up because it's only got one sine. 00:58:33.340 --> 00:58:36.190 And when you're dealing with some of these things, 00:58:36.190 --> 00:58:38.770 it really ends up being the coupling to the monopole-- 00:58:38.770 --> 00:58:40.540 the most important thing. 00:58:40.540 --> 00:58:43.210 So all these other terms that come in and correct this 00:58:43.210 --> 00:58:45.430 are small enough that we can add them in. 00:58:45.430 --> 00:58:48.610 And that, to be blunt, is modern research. 00:58:48.610 --> 00:58:51.370 So let me make a couple of comments about this. 00:58:54.750 --> 00:58:56.145 A more general form-- 00:58:59.300 --> 00:59:02.930 this will help to clarify what the meaning of that lambda 00:59:02.930 --> 00:59:03.650 actually is. 00:59:14.940 --> 00:59:21.700 Suppose that as my vector is transported along itself-- 00:59:21.700 --> 00:59:24.940 so one way is recall how we derive parallel transport. 00:59:24.940 --> 00:59:28.810 We imagine going into a freely-falling frame 00:59:28.810 --> 00:59:30.580 and a Lorentz representation. 00:59:30.580 --> 00:59:32.782 And we said, in that frame, I'm going 00:59:32.782 --> 00:59:34.240 to imagine moving this thing along, 00:59:34.240 --> 00:59:37.240 holding all the components constants-- that 00:59:37.240 --> 00:59:39.430 defined parallel transport. 00:59:39.430 --> 00:59:43.900 Imagine that I don't keep the components constant, 00:59:43.900 --> 00:59:46.270 but I hold them all in a constant ratio with respect 00:59:46.270 --> 00:59:49.090 to each other, but I allow the overall magnitude 00:59:49.090 --> 00:59:50.380 to expand or contract. 00:59:56.050 --> 01:00:03.420 So suppose we allow the vector's normalization 01:00:03.420 --> 01:00:05.220 to change as it slides along. 01:00:18.740 --> 01:00:20.330 Well the way I would mathematically 01:00:20.330 --> 01:00:24.860 formulate this is I'm going to use a notation that 01:00:24.860 --> 01:00:25.540 looks like this. 01:00:25.540 --> 01:00:27.920 So recall this capital D-- it's a shorthand 01:00:27.920 --> 01:00:38.630 for this combination of the tangent 01:00:38.630 --> 01:00:41.997 and the covariant derivative. 01:00:41.997 --> 01:00:43.580 I'm going to call the parameterization 01:00:43.580 --> 01:00:45.650 I use when I set up like this lambda star, 01:00:45.650 --> 01:00:47.775 for reasons that I hope will be clear in just about 01:00:47.775 --> 01:00:49.100 two minutes. 01:00:49.100 --> 01:00:52.070 So what I'm basically saying is that as I move along, 01:00:52.070 --> 01:00:54.098 I don't keep the components constant. 01:00:54.098 --> 01:00:55.640 But I keep them proportional to where 01:00:55.640 --> 01:00:57.238 they were on the previous step. 01:00:57.238 --> 01:00:59.780 But I allow their magnitude to change by some function, which 01:00:59.780 --> 01:01:00.530 I'll call a kappa. 01:01:08.573 --> 01:01:10.490 So you might look at that and think, you know, 01:01:10.490 --> 01:01:13.220 that's a more general kind of transport law. 01:01:13.220 --> 01:01:16.970 It seems to describe physically a very similar situation here. 01:01:16.970 --> 01:01:20.000 It's kind of annoying that this normalization is changing. 01:01:20.000 --> 01:01:22.940 Is there anything going on with this? 01:01:22.940 --> 01:01:29.930 Well what you guys are going to do as a homework exercise, 01:01:29.930 --> 01:01:32.525 you're going to prove that if this 01:01:32.525 --> 01:01:34.310 is the situation you're in, you've 01:01:34.310 --> 01:01:36.140 chosen a dumb parameterization. 01:01:36.140 --> 01:01:38.780 And you can actually convert this 01:01:38.780 --> 01:01:41.060 to the normal geodesic parameterization 01:01:41.060 --> 01:01:43.520 by just relabeling your lambda. 01:01:51.310 --> 01:01:59.910 So we can always reparameterize this, such 01:01:59.910 --> 01:02:03.480 that the right-hand side is 0. 01:02:03.480 --> 01:02:05.460 And right-hand side being 0 corresponds 01:02:05.460 --> 01:02:08.010 to the transport vector remaining constant 01:02:08.010 --> 01:02:09.670 as it moves along. 01:02:09.670 --> 01:02:11.070 So I'll just quickly sketch-- 01:02:11.070 --> 01:02:18.050 so imagine there exists some different parameterization, 01:02:18.050 --> 01:02:19.790 which I will call lambda. 01:02:29.030 --> 01:02:32.480 So imagine something that gives me my normal parallel transport 01:02:32.480 --> 01:02:35.180 exists. 01:02:35.180 --> 01:02:40.970 And I have a different one that involves the star parameter. 01:02:40.970 --> 01:02:43.700 You can actually show that these two things describe exactly 01:02:43.700 --> 01:02:50.630 the same motion, but with lambda and the dumb parameterization, 01:02:50.630 --> 01:02:56.090 lambda star, related to each other by a particular integral. 01:03:04.680 --> 01:03:07.820 So what this shows us is we can always-- 01:03:07.820 --> 01:03:10.400 as long as I'm talking about motion where I'm in this 01:03:10.400 --> 01:03:12.260 regime-- where there's no forces acting-- 01:03:12.260 --> 01:03:14.510 it's not an extended body-- it's just a test body-- 01:03:14.510 --> 01:03:16.557 I can always put it into a regime 01:03:16.557 --> 01:03:18.890 where it'll [INAUDIBLE] geodesic and the right-hand side 01:03:18.890 --> 01:03:20.030 is equal to 0. 01:03:20.030 --> 01:03:21.860 If I'm finding that's not the case, 01:03:21.860 --> 01:03:25.820 I need to adjust my parameterization. 01:03:25.820 --> 01:03:28.550 When you are, in fact, in a prioritization 01:03:28.550 --> 01:03:33.650 such as the right-hand side is 0, 01:03:33.650 --> 01:03:36.822 you are using what is called an affine parameterization. 01:03:36.822 --> 01:03:38.530 That's a name that's worth knowing about. 01:03:57.550 --> 01:04:01.910 So your intuition is that the affine parameterization-- 01:04:01.910 --> 01:04:03.410 I described this in words last time. 01:04:03.410 --> 01:04:06.035 And this just helps to make it a little bit more mathematically 01:04:06.035 --> 01:04:07.440 precise what those words mean. 01:04:17.210 --> 01:04:25.340 Affine parameters correspond to the tick marks 01:04:25.340 --> 01:04:38.065 on the world line, being uniformly spaced 01:04:38.065 --> 01:04:39.190 in the local Lorentz frame. 01:04:45.550 --> 01:04:47.670 If you are working with time-like trajectories-- 01:04:47.670 --> 01:04:51.420 which if you're a physicist, you will be much of the time-- 01:04:51.420 --> 01:04:54.030 a really good choice of the affine parameter 01:04:54.030 --> 01:04:58.020 is the proper time of a body moving through the spacetime. 01:04:58.020 --> 01:05:01.050 That is something that is uniformly spaced, 01:05:01.050 --> 01:05:02.010 assuming that's-- 01:05:02.010 --> 01:05:03.030 you don't have to assume anything. 01:05:03.030 --> 01:05:04.405 Just by definition it's the thing 01:05:04.405 --> 01:05:07.080 that uniformly measures the time as experienced 01:05:07.080 --> 01:05:07.995 by that observer. 01:05:51.130 --> 01:05:54.130 So this is-- you guys are going to do on PSET 4-- 01:05:54.130 --> 01:05:56.140 this is the exercise you need to do to convert 01:05:56.140 --> 01:05:58.360 a nonaffine parameterized geodesic 01:05:58.360 --> 01:06:01.270 to an affine parameterized one. 01:06:01.270 --> 01:06:03.100 That kind of parameterization is not 01:06:03.100 --> 01:06:09.960 too hard to show that if we adjust the parameterization 01:06:09.960 --> 01:06:10.920 in a linear fashion-- 01:06:20.580 --> 01:06:25.160 so in other words, let's say I go from lambda to some lambda 01:06:25.160 --> 01:06:29.220 prime, which is equal to a lambda plus b, 01:06:29.220 --> 01:06:31.700 where and b are both constants-- 01:06:37.450 --> 01:06:40.195 we get a new affine parameterization. 01:06:43.450 --> 01:06:45.510 But that's the only class of reparamterizations 01:06:45.510 --> 01:06:46.593 that allows me to do that. 01:06:49.288 --> 01:06:50.580 And hopefully that makes sense. 01:06:50.580 --> 01:06:53.100 If you imagine that you're using proper time 01:06:53.100 --> 01:06:55.440 as your reparameterization, this is basically 01:06:55.440 --> 01:06:57.690 saying that you just chose a different origin for when 01:06:57.690 --> 01:06:58.648 you started your clock. 01:06:58.648 --> 01:07:00.690 And this means you changed the units in which you 01:07:00.690 --> 01:07:01.560 are measuring time. 01:07:01.560 --> 01:07:02.060 That's all. 01:07:08.210 --> 01:07:11.780 So I'm going to skip a bunch of the details. 01:07:11.780 --> 01:07:13.940 But I'm going to scan and put up the notes 01:07:13.940 --> 01:07:16.580 corresponding to one other route to getting 01:07:16.580 --> 01:07:20.227 to the geodesic equation, which I think it's definitely worth 01:07:20.227 --> 01:07:20.810 knowing about. 01:07:28.880 --> 01:07:36.328 It connects very nicely to other work in classical mechanics. 01:07:36.328 --> 01:07:38.870 So it's a bit of a shame we're going to need to skip over it. 01:07:38.870 --> 01:07:41.030 But we're a little bit behind pace. 01:07:41.030 --> 01:07:42.590 And this is straightforward enough 01:07:42.590 --> 01:07:45.950 that I feel OK posting the notes that you can read it. 01:07:45.950 --> 01:07:50.910 So there is a second path to geodesics. 01:07:56.040 --> 01:07:59.380 So recall the way that we argued how to get the geodesic 01:07:59.380 --> 01:08:02.205 equation, which we said we're going to go 01:08:02.205 --> 01:08:04.330 into-- it's actually in the board right above where 01:08:04.330 --> 01:08:05.410 I'm writing right now-- 01:08:05.410 --> 01:08:07.120 go into the freely-falling frame. 01:08:07.120 --> 01:08:09.070 I have a body that isn't coupling to anything 01:08:09.070 --> 01:08:10.270 but gravity. 01:08:10.270 --> 01:08:11.870 Therefore in the freely-falling frame, 01:08:11.870 --> 01:08:13.270 it just maintains its momentum. 01:08:13.270 --> 01:08:15.190 It's going to go in a straight line. 01:08:15.190 --> 01:08:17.590 Straight means parallel transporting tangent vector-- 01:08:17.590 --> 01:08:20.830 math, math, math-- and that's how we get all that. 01:08:20.830 --> 01:08:22.569 So what this boiled down to is I was 01:08:22.569 --> 01:08:26.109 trying to make rigorous in a geometric sense what straight 01:08:26.109 --> 01:08:28.620 meant. 01:08:28.620 --> 01:08:30.222 There's another notion of straight 01:08:30.222 --> 01:08:31.930 that one can imagine applying when you're 01:08:31.930 --> 01:08:33.013 working in a curved space. 01:08:35.850 --> 01:08:37.740 So your intuition for-- 01:08:37.740 --> 01:08:40.529 if you're talking about how do I make a straight line 01:08:40.529 --> 01:08:42.960 between two points on a globe-- 01:08:42.960 --> 01:08:45.689 your intuition is you say, oh, well the straightest line 01:08:45.689 --> 01:08:48.330 that I can make is the path that is shortest. 01:09:19.290 --> 01:09:21.460 We're going to formulate-- and I'll 01:09:21.460 --> 01:09:23.165 leave the details and the calculation 01:09:23.165 --> 01:09:24.790 to the notes-- we're going to formulate 01:09:24.790 --> 01:09:26.920 how one can apply a similar thing 01:09:26.920 --> 01:09:28.600 to the notion of geodesics. 01:09:28.600 --> 01:09:36.760 So imagine I've got an event p here and event q up here. 01:09:36.760 --> 01:09:43.750 And I ask myself, what is the accumulated proper time 01:09:43.750 --> 01:09:48.939 experienced by all possible paths that take me 01:09:48.939 --> 01:09:50.368 from event p to event q? 01:09:50.368 --> 01:09:51.910 I'm going to need to restrict myself. 01:09:51.910 --> 01:09:54.285 I want it to be something that an observer can physically 01:09:54.285 --> 01:09:56.730 ride-- so all the time-like trajectories that 01:09:56.730 --> 01:09:59.050 connect event p to event q. 01:09:59.050 --> 01:10:01.660 So I've got one a path that goes like this, 01:10:01.660 --> 01:10:04.735 got a path that goes like this, path that goes like this, path 01:10:04.735 --> 01:10:06.610 goes like this, path goes like-- some of them 01:10:06.610 --> 01:10:08.270 might have just become somewhat space like, 01:10:08.270 --> 01:10:09.460 so I should rule them out. 01:10:09.460 --> 01:10:10.380 But you get the idea. 01:10:10.380 --> 01:10:12.880 Imagine I take all the possible time-like paths 01:10:12.880 --> 01:10:14.230 that connect p and q. 01:10:18.010 --> 01:10:22.600 Some of those paths will involve strong accelerations. 01:10:22.600 --> 01:10:25.690 So they will not be the freefall path. 01:10:25.690 --> 01:10:28.260 Among them there will be one that corresponds exactly 01:10:28.260 --> 01:10:28.760 to freefall. 01:10:43.290 --> 01:10:45.540 So if I were talking about-- imagine I was trying to-- 01:10:45.540 --> 01:10:48.560 and this is something that Muslim astronomers worked out 01:10:48.560 --> 01:10:50.810 long, long ago-- they wanted to know the shortest path 01:10:50.810 --> 01:10:53.660 from some point on earth towards Mecca. 01:10:53.660 --> 01:10:56.680 And so you need to find what the shortest distance was 01:10:56.680 --> 01:10:57.680 for something like that. 01:10:57.680 --> 01:11:00.290 And when you're doing this on the surface of a sphere, 01:11:00.290 --> 01:11:02.137 that's complicated. 01:11:02.137 --> 01:11:03.720 And that's where the qibla arose from, 01:11:03.720 --> 01:11:06.210 was working out the mathematics to know how to do this. 01:11:06.210 --> 01:11:07.730 This is a similar kind of concept. 01:11:07.730 --> 01:11:09.860 I'm trying to define-- 01:11:09.860 --> 01:11:13.740 in this case, it's going to turn out it's not the shortest path, 01:11:13.740 --> 01:11:17.600 but it's the path on which an observer ages the most because 01:11:17.600 --> 01:11:19.400 as soon as you accelerate someone-- 01:11:19.400 --> 01:11:20.000 it's not hard. 01:11:20.000 --> 01:11:21.542 Go back to some of those problem sets 01:11:21.542 --> 01:11:23.840 you guys did where you look at accelerated observers. 01:11:23.840 --> 01:11:26.720 Acceleration tends to decrease the amount of aging you have 01:11:26.720 --> 01:11:29.370 as you move through some interval of spacetime. 01:11:29.370 --> 01:11:33.920 So the path that has no acceleration on it, 01:11:33.920 --> 01:11:36.480 this is going to be the one on which an observer is maximally 01:11:36.480 --> 01:11:36.980 aged. 01:11:47.065 --> 01:11:48.440 Why maximum instead of a minimum? 01:11:48.440 --> 01:11:50.810 Well it comes down to the bloody minus sign 01:11:50.810 --> 01:11:53.780 that enters into the timepiece of an interval 01:11:53.780 --> 01:11:55.052 that we have in relativity. 01:11:55.052 --> 01:11:56.510 And that's all I'll say about that, 01:11:56.510 --> 01:11:58.970 is just boils down to that. 01:11:58.970 --> 01:12:03.350 So what we want to do is say, well along all 01:12:03.350 --> 01:12:06.380 of these trajectories, the amount 01:12:06.380 --> 01:12:09.648 of proper time that's accumulated-- so let's 01:12:09.648 --> 01:12:11.690 just say that every one of these is parameterized 01:12:11.690 --> 01:12:15.440 by some lambda that describes the motion along these things. 01:12:34.480 --> 01:12:35.890 This is the amount of proper time 01:12:35.890 --> 01:12:39.400 that someone accumulates as they move from point p-- 01:12:39.400 --> 01:12:42.688 which is at-- let's say this is defined as lambda equals 0-- 01:12:42.688 --> 01:12:44.980 and it's indeterminate what that top lambda is actually 01:12:44.980 --> 01:12:45.480 going to be. 01:12:45.480 --> 01:12:47.860 It's whatever it takes when you get up to lambda of q. 01:12:53.820 --> 01:12:58.020 So what the notes I'm going to post do, 01:12:58.020 --> 01:13:01.530 is they define an action principle that 01:13:01.530 --> 01:13:05.445 can be applied to understand what the trajectory is 01:13:05.445 --> 01:13:06.570 that allows you to do this. 01:13:23.790 --> 01:13:25.760 So I'll just hit the highlights. 01:13:25.760 --> 01:13:37.060 So in notes to be posted, I show that this delta t-- 01:13:37.060 --> 01:13:38.150 this delta tau rather-- 01:13:43.410 --> 01:13:45.580 this can be used to define an action. 01:14:11.600 --> 01:14:12.410 It looks like this. 01:14:17.370 --> 01:14:19.820 And then if you vary the action-- 01:14:26.970 --> 01:14:29.760 or rather you do a variation of your trajectory-- 01:14:29.760 --> 01:14:33.630 where you require that the action remain stationary under 01:14:33.630 --> 01:14:42.610 that variation-- in other words I require delta i equals 0 as x 01:14:42.610 --> 01:14:43.420 goes over to such-- 01:14:57.820 --> 01:15:08.840 so-- what you wind up with-- 01:15:20.870 --> 01:15:25.715 is delta i equals-- 01:16:10.040 --> 01:16:14.340 Notice what I've got in here. 01:16:14.340 --> 01:16:15.790 This is just a Christoffel symbol. 01:16:21.290 --> 01:16:25.715 So when I do this variation, what I find-- 01:16:25.715 --> 01:16:28.950 and by the way going from essentially that board 01:16:28.950 --> 01:16:33.080 to that board, it's about 2/3 a page of algebra. 01:16:33.080 --> 01:16:34.850 Going down to this one, there's a bunch 01:16:34.850 --> 01:16:36.683 of straightforward but fairly tedious stuff. 01:16:36.683 --> 01:16:38.985 It's one reasons why I'm skipping over the details. 01:16:38.985 --> 01:16:40.610 We've got enough G mu nus on the board. 01:17:00.720 --> 01:17:08.250 So the key point is I am going to require that my action be 01:17:08.250 --> 01:17:11.850 stationary, independent of the nature of the variation 01:17:11.850 --> 01:17:13.150 that I make. 01:17:13.150 --> 01:17:16.110 For that to be true, the quantity in braces 01:17:16.110 --> 01:17:20.057 here must be equal to 0. 01:17:20.057 --> 01:17:21.640 Let me just write that down over here. 01:17:21.640 --> 01:17:23.682 This is a good place to conclude today's lecture. 01:17:42.630 --> 01:17:45.390 So we require this to be 0 for any variation. 01:17:53.040 --> 01:18:00.020 Yet the bracketed term being equal to 0, pull that 01:18:00.020 --> 01:18:07.390 out, and clear out that factor of the metric with an inverse, 01:18:07.390 --> 01:18:09.860 you've got your geodesic equation back. 01:18:09.860 --> 01:18:11.590 So we just quickly wrap this up. 01:18:11.590 --> 01:18:15.080 So it's worth looking over these notes. 01:18:15.080 --> 01:18:17.080 It's not worth going through them in gory detail 01:18:17.080 --> 01:18:18.705 on the board, which is why I'm skipping 01:18:18.705 --> 01:18:20.020 a few pages of these things. 01:18:20.020 --> 01:18:23.500 But what this demonstrates is that geodesics-- 01:18:28.070 --> 01:18:29.780 our original definition is that they 01:18:29.780 --> 01:18:33.380 carry the notion of a straight line in a straightforward way 01:18:33.380 --> 01:18:39.470 from where they are obvious in a locally Lorentz frame 01:18:39.470 --> 01:18:41.240 to a more covariant formulation of that-- 01:18:48.170 --> 01:18:55.170 so a generalized straight line to a curved spacetime. 01:18:55.170 --> 01:19:08.500 And they give the trajectory of extremal aging in other words 01:19:08.500 --> 01:19:12.850 a trajectory along which between two points in spacetime, 01:19:12.850 --> 01:19:15.520 an observer moving from one to the other will 01:19:15.520 --> 01:19:17.060 accumulate the most proper time. 01:19:20.350 --> 01:19:26.380 So I'm going to stop here. 01:19:26.380 --> 01:19:28.748 There's a bit more, which I would like to do, 01:19:28.748 --> 01:19:30.040 but I just don't have the time. 01:19:30.040 --> 01:19:33.940 But I'll tell you the key things that I want to say next. 01:19:33.940 --> 01:19:37.060 Everything that I've done here so far 01:19:37.060 --> 01:19:41.367 is I've really fixated on time-like trajectories. 01:19:41.367 --> 01:19:43.450 I've imagined there's a body with some finite rest 01:19:43.450 --> 01:19:47.500 mass where I can make a sensible notion of proper time. 01:19:47.500 --> 01:19:49.000 We are also going to want to talk 01:19:49.000 --> 01:19:50.750 about the behavior of light. 01:19:50.750 --> 01:19:53.590 Light moves on null trajectories. 01:19:53.590 --> 01:19:58.660 I cannot sensibly define proper time on long such a trajectory. 01:19:58.660 --> 01:19:59.500 They are massless. 01:19:59.500 --> 01:20:01.667 There's all sorts of properties associated with them 01:20:01.667 --> 01:20:03.250 that just make this analysis. 01:20:03.250 --> 01:20:05.260 The way I've done it so far, I'll 01:20:05.260 --> 01:20:08.188 need to tweak things a little bit in order for it to work. 01:20:08.188 --> 01:20:09.230 We will do that tweaking. 01:20:09.230 --> 01:20:11.290 It's actually quite straightforward 01:20:11.290 --> 01:20:14.110 and allows us to also bring in a bit more intuition 01:20:14.110 --> 01:20:16.210 about what affine parameters mean when we do that. 01:20:16.210 --> 01:20:17.680 So that'll be the one thing we do. 01:20:17.680 --> 01:20:19.780 The other-- it's unfortunate I wasn't able to get to it 01:20:19.780 --> 01:20:21.620 today-- but it's a straightforward saying, 01:20:21.620 --> 01:20:24.100 which I think I may include in the notes that I post-- 01:20:24.100 --> 01:20:28.420 is including what happens, if your spacetime-- 01:20:28.420 --> 01:20:31.570 so if the metric you use to generate these Christoffels has 01:20:31.570 --> 01:20:33.860 a Killing factor associated with it, 01:20:33.860 --> 01:20:35.770 you can combine Killing's equation 01:20:35.770 --> 01:20:37.360 with the geodesic equation to prove 01:20:37.360 --> 01:20:40.570 the existence of conserved quantities associated 01:20:40.570 --> 01:20:41.343 with that motion. 01:20:41.343 --> 01:20:42.760 And that's where we start to begin 01:20:42.760 --> 01:20:46.150 to see that if I have a spacetime that 01:20:46.150 --> 01:20:48.850 is independent of time, there's a notion of conserved energy 01:20:48.850 --> 01:20:50.020 associated with it. 01:20:50.020 --> 01:20:52.410 So we will do that on Tuesday.