WEBVTT 00:00:00.000 --> 00:00:01.944 [SQUEAKING] 00:00:01.944 --> 00:00:03.888 [RUSTLING] 00:00:03.888 --> 00:00:06.804 [CLICKING] 00:00:09.982 --> 00:00:10.690 SCOTT HUGHES: OK. 00:00:10.690 --> 00:00:12.160 So let's just dive in. 00:00:12.160 --> 00:00:15.580 And I'll just quickly recap where we were last time. 00:00:15.580 --> 00:00:18.160 So what I did last time was, we are now 00:00:18.160 --> 00:00:20.290 beginning the adventure, so to speak, 00:00:20.290 --> 00:00:24.730 of looking at how one examines-- 00:00:24.730 --> 00:00:29.140 how one finds and then studies the properties of solutions 00:00:29.140 --> 00:00:30.550 to the Einstein field equations. 00:00:30.550 --> 00:00:32.710 In other words, given a particular source, 00:00:32.710 --> 00:00:35.423 what is the space-time that emerges? 00:00:35.423 --> 00:00:37.840 What is the space-time that is consistent with that source 00:00:37.840 --> 00:00:41.562 of gravity in Einstein's relativistic theory here? 00:00:41.562 --> 00:00:43.270 So we begin-- we're going to look at this 00:00:43.270 --> 00:00:44.650 in a couple of different ways. 00:00:44.650 --> 00:00:47.110 And the first one is where we're going 00:00:47.110 --> 00:00:49.490 to-- what we do-- linearize the Einstein equation. 00:00:49.490 --> 00:00:51.198 So we're going to imagine that space-time 00:00:51.198 --> 00:00:53.770 is a small deviation from the flat space-time that 00:00:53.770 --> 00:00:55.930 contains no gravity. 00:00:55.930 --> 00:00:59.590 And we are going to expand the Einstein field equations, 00:00:59.590 --> 00:01:04.720 discarding all terms that are non-linear in the perturbation 00:01:04.720 --> 00:01:06.710 away from flat space-time. 00:01:06.710 --> 00:01:11.410 So last time we did this, we developed our field equations, 00:01:11.410 --> 00:01:13.480 we recast them in a form that's particularly 00:01:13.480 --> 00:01:16.840 convenient for making solutions, and we 00:01:16.840 --> 00:01:19.450 noted that the coordinate degrees of freedom 00:01:19.450 --> 00:01:21.610 can be thought of as a kind of gauge freedom 00:01:21.610 --> 00:01:25.690 that, in some ways, it's very useful for us, 00:01:25.690 --> 00:01:28.780 because it allows us to write the equations in a form that 00:01:28.780 --> 00:01:30.820 is particularly amenable to solving, 00:01:30.820 --> 00:01:32.500 but can also be a little bit dangerous, 00:01:32.500 --> 00:01:34.420 because you can sometimes wind up 00:01:34.420 --> 00:01:37.630 with a solution whose physical character is masked 00:01:37.630 --> 00:01:40.720 by the nature of that gauge. 00:01:40.720 --> 00:01:42.580 I went through a somewhat advanced topic 00:01:42.580 --> 00:01:44.770 for the purposes of 8.962, and I do not 00:01:44.770 --> 00:01:47.860 expect students to follow this in detail. 00:01:47.860 --> 00:01:50.140 But it's good to at least survey this and understand 00:01:50.140 --> 00:01:51.307 what was going on with this. 00:01:51.307 --> 00:01:56.740 What I did was I decomposed the perturbation to flat space-time 00:01:56.740 --> 00:01:58.660 into irreducible degrees of freedom. 00:01:58.660 --> 00:02:00.850 I sort of chose a time coordinate. 00:02:00.850 --> 00:02:04.270 I then imagined the other three coordinates in our space, 00:02:04.270 --> 00:02:07.740 and I looked at how, having chosen time, 00:02:07.740 --> 00:02:10.259 the different degrees of freedom decompose 00:02:10.259 --> 00:02:14.440 in two scalars with respect to coordinate rotations, 00:02:14.440 --> 00:02:16.540 vectors with respect to coordinate rotations, 00:02:16.540 --> 00:02:18.010 and two-index tensors with respect 00:02:18.010 --> 00:02:20.370 to coordinate rotations. 00:02:20.370 --> 00:02:23.320 I then took those, particularly the vectors and the tensors, 00:02:23.320 --> 00:02:26.230 and I further broke them up into degrees of freedom that 00:02:26.230 --> 00:02:34.660 are either divergence-free vector fields or the gradients 00:02:34.660 --> 00:02:37.390 of scalar fields. 00:02:37.390 --> 00:02:39.580 And in doing this, what we found was 00:02:39.580 --> 00:02:43.050 that the 10 degrees of freedom that are specified-- excuse me, 00:02:43.050 --> 00:02:46.700 the 10 degrees of freedom that are present in the perturbation 00:02:46.700 --> 00:02:48.730 the space time, can be broken down 00:02:48.730 --> 00:02:54.130 into six functions that are gauge-invariant plus four gauge 00:02:54.130 --> 00:02:56.050 degrees of freedom. 00:02:56.050 --> 00:02:59.890 Applying a similar decomposition to the source, 00:02:59.890 --> 00:03:03.610 we found that those six gauge invariant degrees of freedom 00:03:03.610 --> 00:03:06.900 are governed by four Poisson type equations. 00:03:06.900 --> 00:03:09.870 And I will refer you to the previous lecture 00:03:09.870 --> 00:03:11.620 for exact definitions of these things, 00:03:11.620 --> 00:03:14.620 but there is a scalar field phi that 00:03:14.620 --> 00:03:22.385 is related to the mean density of energy in your space-time 00:03:22.385 --> 00:03:24.010 and pressures in your space-time-- sort 00:03:24.010 --> 00:03:26.410 of isotropic stresses. 00:03:26.410 --> 00:03:29.500 This DTS is an additional degree of freedom 00:03:29.500 --> 00:03:32.500 that is defined in that previous lecture. 00:03:32.500 --> 00:03:35.050 You find that a Poisson equation for a scalar 00:03:35.050 --> 00:03:40.330 field I called theta is directly related to the energy density. 00:03:40.330 --> 00:03:46.450 There are two degrees of freedom associated 00:03:46.450 --> 00:03:49.960 with sort of off diagonal terms in the metric-- the time-space 00:03:49.960 --> 00:03:51.130 terms in the metric. 00:03:51.130 --> 00:03:53.980 And they are related to the momentum flow. 00:03:53.980 --> 00:03:56.020 These are only two degrees of freedom, 00:03:56.020 --> 00:04:00.910 because the psi is a divergence-free field. 00:04:00.910 --> 00:04:03.430 And finally, what we found is if you 00:04:03.430 --> 00:04:06.940 take the spatial piece of the space-time metric, 00:04:06.940 --> 00:04:09.460 you project out the bits that are 00:04:09.460 --> 00:04:12.080 transverse-- in other words, it's divergence-free. 00:04:12.080 --> 00:04:14.860 We will come to a more physical meaning of this a little bit 00:04:14.860 --> 00:04:16.329 later in today's lecture. 00:04:16.329 --> 00:04:19.089 And you require that this be traceless-- 00:04:19.089 --> 00:04:24.340 then those components of the spatial space-time perturbation 00:04:24.340 --> 00:04:27.340 are already gauge invariant, and they're 00:04:27.340 --> 00:04:31.050 related to sort of anisotropic stresses in our source. 00:04:31.050 --> 00:04:33.100 They are governed by a wave equation. 00:04:33.100 --> 00:04:36.940 And as such, these represent a radiative degree of freedom. 00:04:36.940 --> 00:04:38.710 Because this is [coughs] (excuse me!)-- 00:04:38.710 --> 00:04:42.640 because this is trace-free and divergence-right, 00:04:42.640 --> 00:04:45.940 naively you would think that this has six numbers in it. 00:04:45.940 --> 00:04:49.120 A symmetric 3 by 3 matrix is how you would represent it. 00:04:49.120 --> 00:04:52.210 But because it is trace-free and because it is divergenceless, 00:04:52.210 --> 00:04:53.590 there are four constraints. 00:04:53.590 --> 00:04:57.400 And so those six independent numbers are reduced to two. 00:04:57.400 --> 00:04:59.110 And as we'll see in this lecture, 00:04:59.110 --> 00:05:01.990 those two degrees of freedom correspond to the polarizations 00:05:01.990 --> 00:05:04.210 of gravitational waves. 00:05:04.210 --> 00:05:07.780 Today's lecture is all about understanding these guys. 00:05:07.780 --> 00:05:10.120 Given our kind of funny schedule that we're on today, 00:05:10.120 --> 00:05:12.370 I got up early and have not had time to take a shower, 00:05:12.370 --> 00:05:16.510 so I am wearing my LIGO hat. 00:05:16.510 --> 00:05:18.580 And what we're going to be doing today 00:05:18.580 --> 00:05:22.630 is the foundations of how one calculates this 00:05:22.630 --> 00:05:25.510 and why one builds an observatory like LIGO 00:05:25.510 --> 00:05:28.090 to measure these things. 00:05:28.090 --> 00:05:28.700 All right. 00:05:28.700 --> 00:05:29.950 So let me just write that out. 00:05:29.950 --> 00:05:32.195 So our goal for today-- 00:05:32.195 --> 00:05:33.820 I should say our goal for this lecture, 00:05:33.820 --> 00:05:35.610 because this is the first of three lectures 00:05:35.610 --> 00:05:36.693 I'm going to record today. 00:05:39.190 --> 00:05:49.310 Our goal is to understand hij TT in terms of quantities 00:05:49.310 --> 00:05:59.680 that we can observe and also to understand 00:05:59.680 --> 00:06:04.120 how to compute this field-- 00:06:04.120 --> 00:06:15.015 this hij TT given us source. 00:06:19.590 --> 00:06:20.960 So let's begin by just assuming. 00:06:20.960 --> 00:06:23.720 Let's begin by focusing on how we can understand 00:06:23.720 --> 00:06:27.342 what hij TT actually means-- how we can go and measure this. 00:06:27.342 --> 00:06:29.300 So what I'm going to do is just we'll start out 00:06:29.300 --> 00:06:33.290 by let's just hand ourselves an hij TT 00:06:33.290 --> 00:06:35.117 and study its properties. 00:06:37.980 --> 00:06:49.290 So what I'm going to do is begin with a space-time perturbation 00:06:49.290 --> 00:06:53.580 that has everything except hij TT equal to 0. 00:06:53.580 --> 00:07:01.860 So this is something that has phi equals 0, theta equals 0, 00:07:01.860 --> 00:07:04.710 psi equal to 0. 00:07:04.710 --> 00:07:07.410 I am going to pick hij TT. 00:07:11.736 --> 00:07:14.885 If we have a form such that it only depends-- 00:07:17.530 --> 00:07:19.180 it's a function that depends only 00:07:19.180 --> 00:07:21.670 on the combination T minus z. 00:07:21.670 --> 00:07:25.520 So this represents something propagating-- 00:07:25.520 --> 00:07:27.340 as we see, it actually describes radiation. 00:07:37.860 --> 00:07:41.440 in the z-direction. 00:07:41.440 --> 00:07:52.626 So this assumed form can be represented 00:07:52.626 --> 00:07:54.115 by the following matrix. 00:08:21.120 --> 00:08:23.810 So if you take a look at this, I'm 00:08:23.810 --> 00:08:26.547 going to justify a few details of this later. 00:08:26.547 --> 00:08:27.630 There's a few constraints. 00:08:27.630 --> 00:08:29.910 So it looks like I have four numbers here. 00:08:29.910 --> 00:08:37.440 Because my space-time metric is always symmetric under-- 00:08:37.440 --> 00:08:39.150 we'll just say symmetry-- 00:08:39.150 --> 00:08:51.000 under exchange of indices, h xy is the same thing as h yx. 00:08:51.000 --> 00:08:53.790 So these two things are the same. 00:08:53.790 --> 00:09:00.240 And from the requirement that this construction be 00:09:00.240 --> 00:09:01.530 trace-free-- 00:09:01.530 --> 00:09:04.500 so from the way I've written this here, 00:09:04.500 --> 00:09:13.640 that tells me that h xx plus hyy h yy must equal 0. 00:09:19.350 --> 00:09:22.110 So these two quantities on the diagonal are not independent. 00:09:22.110 --> 00:09:24.250 They must be of opposite sign. 00:09:24.250 --> 00:09:26.400 So there are my two degrees of freedom. 00:09:26.400 --> 00:09:29.250 Why have I set the z part equal to 0? 00:09:29.250 --> 00:09:32.880 We'll look at that in a little bit more detail a little later 00:09:32.880 --> 00:09:33.580 in the lecture. 00:09:33.580 --> 00:09:35.205 But essentially, it comes from the fact 00:09:35.205 --> 00:09:38.130 that I'm making it propagate in the z-direction. 00:09:38.130 --> 00:09:41.100 What we're going to see is that if-- 00:09:41.100 --> 00:09:44.310 whatever direction the wave is propagating in, 00:09:44.310 --> 00:09:46.530 my field has to be orthogonal to it. 00:09:46.530 --> 00:09:50.350 And we'll prove that a little bit more carefully later. 00:09:50.350 --> 00:09:53.740 So, for now let's just take this as handed to us. 00:09:53.740 --> 00:09:56.620 We can see this is at least consistent with the properties 00:09:56.620 --> 00:09:58.450 I've written down here, from the fact 00:09:58.450 --> 00:10:01.360 that it depends on T minus z, it's 00:10:01.360 --> 00:10:04.350 not hard to show that this thing will satisfy the wave equation. 00:10:10.810 --> 00:10:15.690 So we have a good source for exploration in front of us-- 00:10:15.690 --> 00:10:18.340 excuse me-- a good space-time for us 00:10:18.340 --> 00:10:20.540 to explore the properties of. 00:10:20.540 --> 00:10:22.940 So let's begin by saying great. 00:10:22.940 --> 00:10:26.290 One of the most important things we've learned how to do 00:10:26.290 --> 00:10:30.010 is, given a space-time, I can compute 00:10:30.010 --> 00:10:34.420 how a freely falling body moves in that space-time. 00:10:34.420 --> 00:10:40.690 So let's consider what happens if I have freely falling bodies 00:10:40.690 --> 00:11:01.780 in a space-time, in a space-time such that this-- 00:11:01.780 --> 00:11:05.467 in a space-time that is this-- 00:11:05.467 --> 00:11:07.050 so I am not saying this very fluently. 00:11:12.480 --> 00:11:13.730 So this is what we want to do. 00:11:13.730 --> 00:11:16.272 We're going to consider freely falling bodies in a space-time 00:11:16.272 --> 00:11:17.905 that involves this h mu nu. 00:11:17.905 --> 00:11:19.280 And the way I'm going to do this, 00:11:19.280 --> 00:11:20.923 my first pass is to sort of say, OK, 00:11:20.923 --> 00:11:23.090 I know all about the motion of freely falling bodies 00:11:23.090 --> 00:11:24.080 in space-time. 00:11:24.080 --> 00:11:25.280 They are geodesics. 00:11:36.270 --> 00:11:38.020 Geodesics describe such bodies. 00:11:38.020 --> 00:11:48.360 So let's go ahead and look at the geodesic equation 00:11:48.360 --> 00:11:49.900 for this setup. 00:11:49.900 --> 00:11:51.408 Now I wanted to-- 00:11:51.408 --> 00:11:53.200 let's give ourselves a couple of conditions 00:11:53.200 --> 00:11:54.620 to try and understand this. 00:11:54.620 --> 00:11:57.580 Let's imagine that these are-- we're initially 00:11:57.580 --> 00:12:02.071 in a space-time that is just flat space-time. 00:12:07.150 --> 00:12:12.039 So we're going to imagine that h mu nu is 0 initially. 00:12:16.930 --> 00:12:19.290 And perhaps it then-- 00:12:19.290 --> 00:12:21.520 we're sort of imagining a space-time in which there's 00:12:21.520 --> 00:12:24.440 just you falling freely in empty space. 00:12:24.440 --> 00:12:26.320 Absolutely nothing going on here. 00:12:26.320 --> 00:12:29.670 And this h mu nu field passes over you. 00:12:29.670 --> 00:12:33.247 A wave packet passes over your freely falling particles. 00:12:38.980 --> 00:12:43.590 So h mu nu, not equal 0, comes along, 00:12:43.590 --> 00:12:45.554 and passes over your particles. 00:12:52.480 --> 00:12:56.270 I'm going to imagine that these bodies are initially at rest. 00:12:56.270 --> 00:13:00.070 So when I'm in this regime where h mu nu is equal to 0, 00:13:00.070 --> 00:13:02.290 these guys are just sitting stationary with respect 00:13:02.290 --> 00:13:04.515 to whoever is making these measurements. 00:13:15.273 --> 00:13:16.440 So we're going to say great. 00:13:16.440 --> 00:13:18.660 I've got these bodies sitting at rest in what 00:13:18.660 --> 00:13:20.610 is initially empty space-time. 00:13:20.610 --> 00:13:22.540 A wave passes along. 00:13:22.540 --> 00:13:23.870 What happens to these bodies? 00:13:30.630 --> 00:13:33.920 So if these guys are at rest, then I know that, 00:13:33.920 --> 00:13:36.860 at initial times-- so let's just say that this comes along, 00:13:36.860 --> 00:13:41.360 and it becomes h mu nu not equal to 0 at some time-- 00:13:41.360 --> 00:13:43.040 what is defined as the origin of time. 00:13:46.460 --> 00:13:51.100 So t equals 0 is when the non-zero wave comes along. 00:13:57.840 --> 00:14:02.340 So my space-time is just eta mu nu for all earlier times. 00:14:02.340 --> 00:14:04.920 My particles are sitting here at rest at all earlier times, 00:14:04.920 --> 00:14:07.290 then h mu nu comes along. 00:14:07.290 --> 00:14:10.320 So what we want to do, then, is look 00:14:10.320 --> 00:14:16.410 at how does this guy behave at t equals 0. 00:14:26.050 --> 00:14:31.540 Now, initially, the only component that will get picked 00:14:31.540 --> 00:14:34.661 out of this is the 00 component. 00:14:38.990 --> 00:14:43.205 And, working to linear order, I can write this. 00:14:49.878 --> 00:14:51.670 Just wanted to write one more step in here. 00:14:51.670 --> 00:14:58.160 Let's say that this, I'm going to write as eta alpha beta beta 00:14:58.160 --> 00:14:58.748 00. 00:14:58.748 --> 00:15:00.290 Go back to your earlier lecture, when 00:15:00.290 --> 00:15:02.000 we defined the Christoffel symbol to look up 00:15:02.000 --> 00:15:03.125 the definition of this guy. 00:15:15.860 --> 00:15:18.100 So these are all the only components 00:15:18.100 --> 00:15:22.150 of the space-time that enter into this Christoffel symbol. 00:15:22.150 --> 00:15:24.490 Every single one-- let's go back and take a look 00:15:24.490 --> 00:15:26.750 at the space-time that we're looking down here. 00:15:26.750 --> 00:15:32.500 Notice this involves beta 0, 0 beta, and 00. 00:15:32.500 --> 00:15:36.130 That is the space-time parts, all of which are 0. 00:15:36.130 --> 00:15:38.530 The other space-time parts, all of which are 0. 00:15:38.530 --> 00:15:42.140 And the time-time part, which is 0. 00:15:42.140 --> 00:15:46.100 So the initial acceleration is 0. 00:15:46.100 --> 00:15:53.550 That means du alpha d tau is equal to 0. 00:15:59.620 --> 00:16:02.320 Which means that my particle begins on a geodesic, 00:16:02.320 --> 00:16:04.510 and it stays on the geodesic. 00:16:04.510 --> 00:16:07.210 In these coordinates, it does not move at all. 00:16:26.880 --> 00:16:30.030 This seems to suggest that gravitational waves have 00:16:30.030 --> 00:16:31.290 no effect. 00:16:31.290 --> 00:16:32.850 I have a small body here. 00:16:32.850 --> 00:16:35.780 Gravitational waves come along. 00:16:35.780 --> 00:16:36.943 It's unaccelerated. 00:16:43.898 --> 00:16:44.690 So what's this for? 00:16:47.840 --> 00:16:51.620 Well, if you think about this for a second, what you should 00:16:51.620 --> 00:16:54.308 convince yourself is that the calculation I just did, 00:16:54.308 --> 00:16:55.850 it's probably the first one you would 00:16:55.850 --> 00:16:57.225 think of doing when you're trying 00:16:57.225 --> 00:16:59.360 to ascertain how does a body move 00:16:59.360 --> 00:17:01.040 in a particular space-time. 00:17:01.040 --> 00:17:04.890 But, in some ways, it was kind of a dumb calculation to do. 00:17:04.890 --> 00:17:12.750 So let's bear in mind, what this is doing 00:17:12.750 --> 00:17:17.339 is tell me the geodesic equation describes motion 00:17:17.339 --> 00:17:20.910 with respect to some given coordinate system. 00:17:37.950 --> 00:17:40.110 Satisfying this equation is just saying, 00:17:40.110 --> 00:17:43.020 yup, you're following a geodesic. 00:17:43.020 --> 00:17:44.400 And guess what? 00:17:44.400 --> 00:17:48.330 These are-- this radiation is a form of gravity. 00:17:48.330 --> 00:17:51.990 Even if the body responds to it, it is still 00:17:51.990 --> 00:17:53.700 going to follow a geodesic. 00:17:53.700 --> 00:17:55.988 It's going to be in free fall. 00:17:55.988 --> 00:17:57.780 The gravitational field may be a little bit 00:17:57.780 --> 00:17:59.238 different from the ones that you're 00:17:59.238 --> 00:18:02.220 more used to from your previous experiences with gravity, 00:18:02.220 --> 00:18:06.090 but it is still just following the free fall trajectory 00:18:06.090 --> 00:18:07.470 that gravity demands. 00:18:15.230 --> 00:18:22.920 So free fall in these coordinates 00:18:22.920 --> 00:18:27.099 means the body is remaining fixed in these coordinates. 00:18:37.860 --> 00:18:40.620 This is one of the most important lessons 00:18:40.620 --> 00:18:41.550 in general relativity. 00:18:41.550 --> 00:18:44.340 When you ask a statement that depends 00:18:44.340 --> 00:18:47.100 on the coordinate system you've written down, 00:18:47.100 --> 00:18:48.870 you have to understand your coordinates. 00:18:48.870 --> 00:18:50.040 And I would submit we don't really 00:18:50.040 --> 00:18:52.680 understand the coordinates that we wrote down here with this. 00:18:52.680 --> 00:19:02.970 These coordinates essentially follows the body. 00:19:02.970 --> 00:19:08.470 Even if the body is moving, they essentially 00:19:08.470 --> 00:19:10.750 wiggle right along with the body itself. 00:19:26.570 --> 00:19:28.420 So if you want to understand what 00:19:28.420 --> 00:19:32.480 the impact of gravitational waves is on things, 00:19:32.480 --> 00:19:35.680 you have to try to formulate what you're 00:19:35.680 --> 00:19:38.860 able to measure using coordinate-independent language 00:19:38.860 --> 00:19:41.440 as much as possible. 00:19:41.440 --> 00:19:43.840 At the very least, you need to understand 00:19:43.840 --> 00:19:47.185 your coordinate system a lot better than I did in setting up 00:19:47.185 --> 00:19:48.310 this calculation over here. 00:19:59.400 --> 00:20:03.500 So let's reframe the way we do this analysis. 00:20:03.500 --> 00:20:05.750 Let's imagine-- so one of the things 00:20:05.750 --> 00:20:09.380 that we learned in thinking about how 00:20:09.380 --> 00:20:13.310 gravity works in general relativity is I can always, 00:20:13.310 --> 00:20:17.210 if I'm just focusing on a single body, a single point mass, 00:20:17.210 --> 00:20:20.060 I can always go into freely falling coordinates. 00:20:20.060 --> 00:20:22.910 And space-time is essentially the space-time 00:20:22.910 --> 00:20:24.810 of special relativity. 00:20:24.810 --> 00:20:27.530 I can find a representation so that everything looks just 00:20:27.530 --> 00:20:29.330 like special relativity near there. 00:20:29.330 --> 00:20:35.490 Gravity is basically completely thrown away, 00:20:35.490 --> 00:20:37.830 at least as far as the mathematical representation 00:20:37.830 --> 00:20:39.600 is concerned. 00:20:39.600 --> 00:20:40.700 I'm in free fall. 00:20:40.700 --> 00:20:41.700 I don't really see much. 00:20:41.700 --> 00:20:45.840 If I wanted to see the impact of these waves, what I need to do 00:20:45.840 --> 00:20:50.370 is think about how things behave over some finite region. 00:20:50.370 --> 00:20:56.850 I wanted to put together a setup such that I have two bodies, 00:20:56.850 --> 00:21:00.450 and the relative effect of things between those two bodies 00:21:00.450 --> 00:21:03.190 will allow me to see the effect of space-time curvature 00:21:03.190 --> 00:21:03.690 in action. 00:21:06.810 --> 00:21:09.660 So let's now-- with this in mind, 00:21:09.660 --> 00:21:21.624 let's consider two nearby bodies that each follow geodesics. 00:21:29.210 --> 00:21:34.560 So body A is right here. 00:21:34.560 --> 00:21:36.570 And I'm going to define body A as living 00:21:36.570 --> 00:21:38.462 at the origin of my coordinates. 00:21:42.320 --> 00:21:47.230 And we've got a body B over here. 00:21:47.230 --> 00:21:54.950 And we're going to say that this guy is displaced 00:21:54.950 --> 00:22:04.640 from the origin by a small amount, some epsilon, in what I 00:22:04.640 --> 00:22:07.250 will call the x direction. 00:22:07.250 --> 00:22:10.100 And what I want to do is say, OK, I've got these two guys. 00:22:10.100 --> 00:22:11.870 They are moving through a space-time. 00:22:11.870 --> 00:22:15.112 Let's make this be the space-time that we had just 00:22:15.112 --> 00:22:16.070 discussed a moment ago. 00:22:19.220 --> 00:22:24.950 So they are each moving on geodesics of this space time. 00:22:24.950 --> 00:22:28.250 We know that if I set up my coordinate system in the way 00:22:28.250 --> 00:22:30.740 that I've done it here, I am not going 00:22:30.740 --> 00:22:34.050 to see any acceleration on each of these bodies. 00:22:34.050 --> 00:22:40.070 Let's now try to focus on some kind of a quantity that 00:22:40.070 --> 00:22:43.210 tells me about the separation between these two. 00:22:43.210 --> 00:22:44.733 I'm going to do this in two ways. 00:22:44.733 --> 00:22:46.400 So the first thing which I'm going to do 00:22:46.400 --> 00:22:53.840 is say, let's imagine that I am bouncing a light pulse back 00:22:53.840 --> 00:23:42.730 and forth between A and B. What I'm going to do 00:23:42.730 --> 00:23:48.820 is write the 4-momentum associated with that light. 00:23:48.820 --> 00:23:53.340 So recall I can't really define a 4-velocity for light 00:23:53.340 --> 00:23:55.440 because proper time is not defined. 00:23:55.440 --> 00:24:03.270 But I can introduce some kind of an affine parameter 00:24:03.270 --> 00:24:07.050 such that d by d lambda tells me about motion on the world line 00:24:07.050 --> 00:24:08.582 that this light follows. 00:24:08.582 --> 00:24:11.040 And this is just bouncing back and forth between these two. 00:24:11.040 --> 00:24:14.430 So it moves in the x direction. 00:24:14.430 --> 00:24:23.950 Because this is light, p dot T equals 0. 00:24:45.860 --> 00:24:47.540 So these components, the 4-momentum 00:24:47.540 --> 00:24:50.360 associated with the light, the dt d lambda and the dx d 00:24:50.360 --> 00:24:54.440 lambda, they are related to each other like so. 00:24:54.440 --> 00:24:55.930 And look. 00:24:55.930 --> 00:24:58.190 The space-time metric, the gauge-invariant piece 00:24:58.190 --> 00:25:03.360 of the space-time, hxxtt, it is appearing in here. 00:25:03.360 --> 00:25:06.470 So let's say, OK, what I am interested now 00:25:06.470 --> 00:25:09.050 is computing what-- 00:25:09.050 --> 00:25:12.680 suppose I'm sitting at A, and I want 00:25:12.680 --> 00:25:17.000 to compute the time, according to A's clock, 00:25:17.000 --> 00:25:21.108 that it takes for a pulse of light to travel from A to B. 00:25:21.108 --> 00:25:22.400 Maybe I'll make it bounce back. 00:25:22.400 --> 00:25:23.942 Maybe I want to know how much time it 00:25:23.942 --> 00:25:30.510 takes for the light pulse to go from A to B and back to A. 00:25:30.510 --> 00:25:34.540 So let's turn this into an equation for the interval dt. 00:25:37.590 --> 00:25:42.540 So dt, I can do a little bit of this trick called division. 00:25:48.830 --> 00:25:54.580 Here is-- question is, did I do it correctly? 00:25:54.580 --> 00:25:55.080 Yes. 00:25:55.080 --> 00:25:56.705 So I'm doing the old physicist's trick. 00:25:56.705 --> 00:25:59.000 I'm just imagining that an interval of time-- 00:25:59.000 --> 00:26:00.650 an interval of affine parameter d 00:26:00.650 --> 00:26:03.050 lambda, the amount of dt that accumulates 00:26:03.050 --> 00:26:05.000 as it moves to an interval dx. 00:26:05.000 --> 00:26:06.330 They are related by so. 00:26:08.840 --> 00:26:11.480 And, remember, I am assuming that the perturbation 00:26:11.480 --> 00:26:13.280 from space-time, from flat space-time, 00:26:13.280 --> 00:26:16.340 is small enough that I can always linearize. 00:26:16.340 --> 00:26:21.790 So this can be-- 00:26:21.790 --> 00:26:24.493 I can take that square root using the binomial expansion. 00:26:27.880 --> 00:26:38.690 So suppose I want to integrate up the time it takes 00:26:38.690 --> 00:27:15.880 for the thing to go from A to B, and then back from B to A. 00:27:15.880 --> 00:27:19.500 Well, this is going to be given by integrating this guy's 00:27:19.500 --> 00:27:27.270 motion from 0 to epsilon, like so. 00:27:31.860 --> 00:27:33.530 And then integrating it back again. 00:27:38.420 --> 00:27:39.360 My tt is in here. 00:27:44.475 --> 00:27:46.550 It's going to switch direction when I do that, 00:27:46.550 --> 00:27:48.258 so I'm going to flip the sign on that dx. 00:28:18.850 --> 00:28:20.350 I'm leaving it written in this form, 00:28:20.350 --> 00:28:21.570 so you have to be a little bit careful. 00:28:21.570 --> 00:28:22.740 You guys will play with this a little bit more 00:28:22.740 --> 00:28:23.880 on a problem set. 00:28:23.880 --> 00:28:27.420 Bear in mind that as I move through x, the time argument 00:28:27.420 --> 00:28:29.160 of these guys is changing. 00:28:29.160 --> 00:28:31.210 And so that's where there's a little bit of work 00:28:31.210 --> 00:28:33.340 that you'll explore in a problem set. 00:28:33.340 --> 00:28:36.900 But I'm writing it in this form so that we get some-- 00:28:36.900 --> 00:28:39.240 naively, you might think, ah, I'll just flip the sign 00:28:39.240 --> 00:28:40.410 and I'll combine these. 00:28:40.410 --> 00:28:42.743 In a certain limit, that is indeed a good approximation. 00:28:42.743 --> 00:28:45.960 You got to be a little bit more careful in general though. 00:28:45.960 --> 00:28:49.290 This is the time it would take for light 00:28:49.290 --> 00:28:53.910 to go from A to B and back in the absence 00:28:53.910 --> 00:28:57.780 of this gravitational wave, this hxxtt. 00:28:57.780 --> 00:28:59.970 Notice there is an inference on the time 00:28:59.970 --> 00:29:03.120 of arrival of these pulses that depends on hxxtt. 00:29:12.495 --> 00:29:19.660 Time of arrival of the pulses depends 00:29:19.660 --> 00:29:26.650 on this particular piece of the space-time metric. 00:29:26.650 --> 00:29:29.440 One could imagine, if you had a precise clock, 00:29:29.440 --> 00:29:32.950 and you sat there and you timed the arrival of these things-- 00:29:32.950 --> 00:29:35.110 let's say you sent a pulse of light out-- bloop-- 00:29:35.110 --> 00:29:38.740 and you kept track of how long it took with a very precise 00:29:38.740 --> 00:29:41.578 clock, you could look for-- in the absence of a gravitational 00:29:41.578 --> 00:29:43.870 wave, that would be, if you imagine that your pulse was 00:29:43.870 --> 00:29:45.550 sent out perfectly periodically-- 00:29:45.550 --> 00:29:51.030 let's say every millisecond or something like that, 00:29:51.030 --> 00:29:52.890 you sent out a little pulse-- 00:29:52.890 --> 00:29:57.780 if it arrives back and your arrival pulses were just as 00:29:57.780 --> 00:30:00.420 spaced by 1 millisecond, you would say, great, 00:30:00.420 --> 00:30:02.730 I am sitting in empty space-time. 00:30:02.730 --> 00:30:09.980 But if you found that their arrivals varied, 00:30:09.980 --> 00:30:13.142 you would think, hmm, there's something else going on here. 00:30:13.142 --> 00:30:14.850 Gravitational waves is one of things that 00:30:14.850 --> 00:30:16.210 could lead to that variation. 00:30:19.870 --> 00:30:22.760 So this, at least, is kind of a proof-- 00:30:22.760 --> 00:30:24.330 this calculation I just did-- 00:30:24.330 --> 00:30:27.215 you guys are actually, on one of the upcoming problems sets-- 00:30:27.215 --> 00:30:28.215 I believe it's problem-- 00:30:28.215 --> 00:30:31.050 I can't remember if it's 7 or 8-- 00:30:31.050 --> 00:30:33.060 you do a little analysis where you 00:30:33.060 --> 00:30:36.990 will use this to understand how a detector like LIGO 00:30:36.990 --> 00:30:40.800 actually functions to measure out 00:30:40.800 --> 00:30:44.370 the effects due to a gravitational wave on the arms 00:30:44.370 --> 00:30:46.260 of a detector like this. 00:30:46.260 --> 00:30:52.110 Let me just do one more way of understanding 00:30:52.110 --> 00:30:56.610 how it is the gravitational wave leaves an imprint on these two 00:30:56.610 --> 00:30:58.145 separated particles. 00:31:03.490 --> 00:31:07.110 So when you look at that first naive calculation I did, 00:31:07.110 --> 00:31:10.650 the fact that I got no effect, I sort of 00:31:10.650 --> 00:31:13.620 joked that we did a dumb calculation. 00:31:13.620 --> 00:31:15.243 To be fair, it wasn't that dumb. 00:31:15.243 --> 00:31:17.910 What we're-- the first thing you usually think of doing when you 00:31:17.910 --> 00:31:20.410 want to understand the motion of a body in space-time, 00:31:20.410 --> 00:31:22.500 is you look at geodesics. 00:31:22.500 --> 00:31:27.390 Geodesics just tell me that this guy's acceleration with respect 00:31:27.390 --> 00:31:29.397 to the free fall frame is 0. 00:31:29.397 --> 00:31:31.230 And that's exactly what we ended up getting. 00:31:31.230 --> 00:31:33.990 It's basically just saying, ah, even in a gravitational wave, 00:31:33.990 --> 00:31:37.080 my two bodies are moving on geodesics. 00:31:37.080 --> 00:31:40.590 If I have two separate bodies, like I 00:31:40.590 --> 00:31:44.580 have in this calculation where I just calculated the variation 00:31:44.580 --> 00:31:47.610 in the time of arrival of light pulses, 00:31:47.610 --> 00:31:49.500 one of the tools we've learned is 00:31:49.500 --> 00:31:54.490 that I can have two geodesics, two free fall trajectories that 00:31:54.490 --> 00:31:55.720 deviate from one another. 00:31:58.250 --> 00:32:05.580 So let's look at the geodesic deviation of the free fall 00:32:05.580 --> 00:32:07.230 world lines of these two bodies. 00:32:21.960 --> 00:32:23.100 So I'm going to-- 00:32:23.100 --> 00:32:25.482 as I did when I did my geodesic calculation, 00:32:25.482 --> 00:32:26.940 I'm going to take them to initially 00:32:26.940 --> 00:32:29.940 be close enough that they have the same 4-velocity. 00:32:43.210 --> 00:32:47.280 And if you want to be careful about counting orders here, 00:32:47.280 --> 00:32:49.922 you can imagine, when the gravitation wave comes along, 00:32:49.922 --> 00:32:52.380 that one or both of them might pick up a correction of this 00:32:52.380 --> 00:32:54.030 that's of order h. 00:32:54.030 --> 00:32:56.280 Depends on what kind of coordinate system you pick up. 00:32:56.280 --> 00:32:58.450 But the key thing is that they start out like this. 00:32:58.450 --> 00:33:00.120 The gravitational waves may come along 00:33:00.120 --> 00:33:01.975 and it will move them around a little bit. 00:33:01.975 --> 00:33:03.600 But the amount it's going to move them, 00:33:03.600 --> 00:33:05.240 the amount of velocity that they're going to pick up, 00:33:05.240 --> 00:33:06.657 and the displacement they're going 00:33:06.657 --> 00:33:09.177 to get from their initial position if they're at rest, 00:33:09.177 --> 00:33:11.010 it can only be of order of the h that you're 00:33:11.010 --> 00:33:12.930 trying to measure there. 00:33:12.930 --> 00:33:16.530 So I'll remind you, the equation of deviation, 00:33:16.530 --> 00:33:28.150 equation of geodesic deviation, it 00:33:28.150 --> 00:33:32.725 defines a sort of proper acceleration acting on-- 00:33:35.930 --> 00:33:38.710 whoops, that's a typo. 00:33:38.710 --> 00:33:44.510 It is the proper acceleration on a vector xi 00:33:44.510 --> 00:33:48.290 that defines, in a precise geometric way, 00:33:48.290 --> 00:33:50.030 the separation of two geodesics. 00:34:02.430 --> 00:34:04.500 And what it looks like is the Riemann curvature 00:34:04.500 --> 00:34:10.080 of your space-time contracted on the 4-velocity of your bodies, 00:34:10.080 --> 00:34:15.239 and that separation vector xi. 00:34:15.239 --> 00:34:22.340 So, linearizing everything, bearing in mind 00:34:22.340 --> 00:34:31.330 that R will itself be of order h, 00:34:31.330 --> 00:34:34.870 and that any difference between the proper time 00:34:34.870 --> 00:34:38.500 along these things and the coordinate time 00:34:38.500 --> 00:34:41.860 of a particular clock is also going to be of order h, 00:34:41.860 --> 00:34:44.680 they are small enough that this equation becomes-- 00:35:01.370 --> 00:35:03.110 Something that looks like this. 00:35:03.110 --> 00:35:05.950 Notice I'm only looking at the spatial components 00:35:05.950 --> 00:35:08.680 of the separation vector xi. 00:35:08.680 --> 00:35:10.777 And that's because, since I'm working, 00:35:10.777 --> 00:35:12.610 since this is what my 4-velocity looks like, 00:35:12.610 --> 00:35:15.710 I'm going to pick out the 00 components of these things. 00:35:15.710 --> 00:35:18.010 And, don't forget, Riemann is antisymmetric 00:35:18.010 --> 00:35:23.320 when I exchange either indices 1 and 2 or indices 3 and 4. 00:35:23.320 --> 00:35:26.830 And because of that, the time-like component of this guy 00:35:26.830 --> 00:35:28.870 is going to be an uninteresting thing. 00:35:28.870 --> 00:35:31.510 I will just get that is equal to the acceleration 00:35:31.510 --> 00:35:33.190 of the time-like component is 0. 00:35:33.190 --> 00:35:35.470 So it doesn't really matter. 00:35:35.470 --> 00:35:45.810 So this is telling me that the geodesic separation 00:35:45.810 --> 00:35:47.880 of these two world lines is-- 00:35:47.880 --> 00:35:50.670 it's a second-order equation in time, 00:35:50.670 --> 00:35:53.310 and it's proportional to the separation itself. 00:35:53.310 --> 00:35:56.514 We need to work out all of these components. 00:36:04.060 --> 00:36:05.810 That is a fairly straightforward exercise, 00:36:05.810 --> 00:36:07.560 and I will just report to you the results. 00:36:20.490 --> 00:36:21.660 A brief aside. 00:36:21.660 --> 00:36:24.210 If you want to sort of verify the results I'm about to write 00:36:24.210 --> 00:36:26.730 out here, you could go into-- 00:36:26.730 --> 00:36:29.220 you could look up the definition of the Riemann tensor. 00:36:29.220 --> 00:36:32.670 Bear in mind that you are doing things at linear orders. 00:36:32.670 --> 00:36:35.410 You can throw away the terms of, like, connection times 00:36:35.410 --> 00:36:35.910 connection. 00:36:35.910 --> 00:36:38.160 It's just derivatives of the connection that are going 00:36:38.160 --> 00:36:39.930 to matter at linear order. 00:36:39.930 --> 00:36:43.800 But, even so, it's a bit of a grotty calculation. 00:36:43.800 --> 00:36:48.510 I am-- as I begin to reorganize the problem sets associated 00:36:48.510 --> 00:36:50.730 with this class under the new system 00:36:50.730 --> 00:36:53.250 that MIT is currently operating on, 00:36:53.250 --> 00:36:57.360 I will soon be releasing a few Mathematica notebooks that 00:36:57.360 --> 00:37:00.780 are very useful tools for doing tedious calculations, 00:37:00.780 --> 00:37:06.180 like computation of Riemann curvature tensor components. 00:37:06.180 --> 00:37:08.550 So I will put a few of those things up there. 00:37:08.550 --> 00:37:11.100 If you would like to validate some of the things 00:37:11.100 --> 00:37:14.670 that I am computing, that is where you will 00:37:14.670 --> 00:37:18.300 find a good way to test this. 00:37:18.300 --> 00:37:28.030 So, anyway, the non-zero Riemann components. 00:37:34.730 --> 00:37:40.378 So at linear order, you have R x0 x0. 00:37:40.378 --> 00:37:42.920 Bear in mind that, linear order, you are raising and lowering 00:37:42.920 --> 00:37:46.700 indices with the metric of flat space-time. 00:37:46.700 --> 00:37:49.930 So that's the same thing as this. 00:37:49.930 --> 00:37:53.750 And this turns out to be minus 1/2 two time 00:37:53.750 --> 00:37:56.570 derivatives of hxx. 00:38:01.370 --> 00:38:04.972 Y0 y0-- that's your yo-yo component-- 00:38:14.130 --> 00:38:15.790 pretty much looks exactly the same, 00:38:15.790 --> 00:38:18.960 but it's two time derivatives of the yy piece. 00:38:18.960 --> 00:38:22.080 Remember though-- I've erased it a while ago-- 00:38:22.080 --> 00:38:25.590 there are some constraints that we found due to the fact 00:38:25.590 --> 00:38:29.370 that my original h mu nu has to be-- its spatial piece has 00:38:29.370 --> 00:38:31.440 to be traceless. 00:38:31.440 --> 00:38:35.160 In particular, we found hyy was minus hxx. 00:38:41.370 --> 00:38:43.717 So I can remove that additional function, 00:38:43.717 --> 00:38:45.300 and just leave things in terms of hxx. 00:38:53.890 --> 00:39:04.240 And our final non-zero component looks like this. 00:39:04.240 --> 00:39:08.290 All others are either 0 or related by a symmetry. 00:39:22.570 --> 00:39:26.290 So notice, although I've written down three different 00:39:26.290 --> 00:39:28.510 complements here, there's really only two, 00:39:28.510 --> 00:39:31.190 because the symmetry that's-- 00:39:31.190 --> 00:39:33.190 it's not a symmetry of Riemann, but the symmetry 00:39:33.190 --> 00:39:36.610 associated with the trace-free nature of hijtt 00:39:36.610 --> 00:39:38.780 made these two equal to each other. 00:39:38.780 --> 00:39:41.523 So this and this are really the only independent degrees 00:39:41.523 --> 00:39:42.190 of freedom here. 00:40:11.900 --> 00:40:12.820 All right. 00:40:12.820 --> 00:40:15.490 Let's plug this into the equation of geodesic deviation, 00:40:15.490 --> 00:40:19.745 this guy, and see what we get. 00:40:19.745 --> 00:40:23.680 And what we get is an equation governing 00:40:23.680 --> 00:40:26.260 the acceleration of the x component of deviation. 00:41:27.240 --> 00:41:27.740 Here it is. 00:41:30.923 --> 00:41:32.340 Let's make a couple of assumptions 00:41:32.340 --> 00:41:34.600 and try to understand what this is telling us. 00:41:34.600 --> 00:41:36.390 So what I'm going to do is-- let's 00:41:36.390 --> 00:41:41.370 assume this is-- the first assumption is necessitated 00:41:41.370 --> 00:41:48.103 by the idea that I can do linearized gravity. 00:41:48.103 --> 00:41:50.520 So I'm going to assume that these guys are all very small. 00:41:54.900 --> 00:42:00.180 And I will imagine that my displacement 00:42:00.180 --> 00:42:06.420 vectors have some piece which-- 00:42:06.420 --> 00:42:08.470 basically, it's time constant. 00:42:08.470 --> 00:42:09.315 It does not change. 00:42:13.910 --> 00:42:24.560 And that there's a bit that will be of order h that describes 00:42:24.560 --> 00:42:27.530 how the separation of my two bodies 00:42:27.530 --> 00:42:31.803 responds to this incoming field. 00:42:31.803 --> 00:42:34.220 So what I'm going to do, when you go and you plug this in, 00:42:34.220 --> 00:42:46.490 assume that this guy is constant in time, and this one varies. 00:43:00.840 --> 00:43:05.450 What I'm going to do is imagine I don't just have two bodies. 00:43:05.450 --> 00:43:07.070 Imagine I have a ring of bodies. 00:43:22.630 --> 00:43:27.550 So suppose I have got my coordinate axes here. 00:43:27.550 --> 00:43:31.480 Horizontal is x, vertical on the board is y. 00:43:31.480 --> 00:43:39.155 And so imagine I have a ring of bodies, like so. 00:43:39.155 --> 00:43:40.780 And let's consider the following limit. 00:43:40.780 --> 00:43:46.720 Let's imagine that my hxx is some function that 00:43:46.720 --> 00:43:57.440 is sinusoidally varying in time, and my hxy is 0. 00:43:57.440 --> 00:43:58.750 I just want to call that the-- 00:43:58.750 --> 00:44:00.500 as we saw, there's really only two degrees 00:44:00.500 --> 00:44:02.450 of freedom in this field. 00:44:02.450 --> 00:44:04.070 And in the way that we formulated it, 00:44:04.070 --> 00:44:07.737 it's going to encapsulated by these two functions, xx and xy. 00:44:07.737 --> 00:44:08.820 So let's just look at one. 00:44:08.820 --> 00:44:12.320 Let's just look at the influence of the hxx. 00:44:12.320 --> 00:44:16.040 So when you do this, you essentially 00:44:16.040 --> 00:44:19.940 have the second derivative of xi x 00:44:19.940 --> 00:44:24.290 looks like two derivatives of your x field, your xx field 00:44:24.290 --> 00:44:26.210 times xi x. 00:44:26.210 --> 00:44:29.270 And two derivatives, your sort of acceleration 00:44:29.270 --> 00:44:33.100 in the y direction, is minus those two derivatives 00:44:33.100 --> 00:44:35.030 of your field times xi y. 00:44:37.682 --> 00:44:39.140 So what this is going to do, if you 00:44:39.140 --> 00:44:42.420 think about the way this is going to act on these things, 00:44:42.420 --> 00:44:47.360 you're going to get a sinusoidal acceleration associated 00:44:47.360 --> 00:44:52.550 with the displacement of this little sea of freely 00:44:52.550 --> 00:44:56.330 falling bodies as the gravitational wave comes along. 00:44:56.330 --> 00:44:58.370 And it's going to do so in such a way 00:44:58.370 --> 00:45:01.220 that the displacement in the x direction 00:45:01.220 --> 00:45:03.830 does the opposite of what the displacement in the y direction 00:45:03.830 --> 00:45:04.740 is doing. 00:45:04.740 --> 00:45:15.060 So as time passes, you'll find that-- so, initially, 00:45:15.060 --> 00:45:19.100 let's say it stretches along x, but then squeezes along y. 00:45:22.890 --> 00:45:30.070 Then it's a sinusoid, so, a quarter of a cycle later, 00:45:30.070 --> 00:45:33.070 it's back to being a circle. 00:45:33.070 --> 00:45:40.700 And, a quarter of a cycle later, it stretches along y 00:45:40.700 --> 00:45:41.760 and squeezes along x. 00:45:52.080 --> 00:45:56.760 Let's consider the limit now where hxx is equal to 0, 00:45:56.760 --> 00:45:59.770 and hxy is some sinusoid. 00:46:24.100 --> 00:46:25.630 So I start out with my circle. 00:46:28.990 --> 00:46:30.428 Look at your equations over here. 00:46:30.428 --> 00:46:31.970 Think about what they're going to do. 00:46:31.970 --> 00:46:36.460 Now we're going to find that the change in x and the change in y 00:46:36.460 --> 00:46:38.420 is correlated. 00:46:38.420 --> 00:46:45.540 So this guy stretches it out into an ellipse 00:46:45.540 --> 00:46:48.670 along a 45-degree line. 00:46:48.670 --> 00:46:55.170 A quarter of a cycle later, it's back to being a circle. 00:46:55.170 --> 00:46:59.360 And a quarter of a cycle later, it does the same thing 00:46:59.360 --> 00:47:00.480 but with opposite sign. 00:47:05.090 --> 00:47:08.840 These two fields, my hxx and my hxy, 00:47:08.840 --> 00:47:11.830 are what we call the fundamental polarizations 00:47:11.830 --> 00:47:12.890 of a gravitational wave. 00:47:19.130 --> 00:47:21.980 Polarization states are only defined with respect 00:47:21.980 --> 00:47:26.610 to a particular set of basis vectors. 00:47:26.610 --> 00:47:29.840 So if I make my basis vectors be this x and y, 00:47:29.840 --> 00:47:34.420 we call this one, for reasons that I hope are obvious, 00:47:34.420 --> 00:47:35.920 the plus polarization. 00:47:40.890 --> 00:47:45.300 The gravitational wave acts as a tidal stretch and squeeze. 00:47:45.300 --> 00:47:46.620 Notice what's going on here. 00:47:46.620 --> 00:47:49.830 I stretch along this axis, squeeze along this one. 00:47:49.830 --> 00:47:53.070 A quarter of a cycle later, I squeeze along this one, 00:47:53.070 --> 00:47:54.180 stretch along that one. 00:47:58.470 --> 00:48:05.890 So I get a tidal stretch and squeeze, 00:48:05.890 --> 00:48:08.410 where the main directions along which the stretch 00:48:08.410 --> 00:48:11.080 and squeeze are happening is plus shaped 00:48:11.080 --> 00:48:12.310 with respect to these axes. 00:48:24.090 --> 00:48:28.937 This one is known as the cross polarization. 00:48:33.230 --> 00:48:41.310 You also have a tidal stretch and squeeze, 00:48:41.310 --> 00:48:43.200 but the lines of force in this case 00:48:43.200 --> 00:48:46.870 are along axes that have an X shape. 00:49:18.430 --> 00:49:22.300 So at the level of 8.962, this is 00:49:22.300 --> 00:49:26.710 all we're going to say about the way that these things act on-- 00:49:26.710 --> 00:49:29.380 how gravitational waves act and what their observables are. 00:49:29.380 --> 00:49:32.980 And I deliberately chose to wear my LIGO today, 00:49:32.980 --> 00:49:38.110 because what LIGO does is look for these tidal stretches 00:49:38.110 --> 00:49:39.400 and squeezes. 00:49:39.400 --> 00:49:42.130 And the way it does so-- so, obviously, it's 00:49:42.130 --> 00:49:44.650 technologically not really feasible 00:49:44.650 --> 00:49:46.000 to set up a ring like this. 00:49:46.000 --> 00:49:49.533 But what we can do is sample the ring here and here. 00:49:49.533 --> 00:49:50.950 Actually, it's a little bit better 00:49:50.950 --> 00:49:52.990 to do on the plus polarization plot. 00:49:57.880 --> 00:50:00.790 So you make an experimental setup that 00:50:00.790 --> 00:50:02.825 samples the ring here and here. 00:50:02.825 --> 00:50:04.450 And if a gravitational wave comes along 00:50:04.450 --> 00:50:07.090 and it's lined up right, you see, ah, look at that. 00:50:07.090 --> 00:50:09.130 I'm getting that stretch and squeeze. 00:50:09.130 --> 00:50:12.820 The way you actually measure it out is essentially a timing 00:50:12.820 --> 00:50:16.060 experiment, exactly like this little calculation I did here. 00:50:16.060 --> 00:50:19.930 You bounce light between mirrors, 00:50:19.930 --> 00:50:25.390 and you very carefully time the round-trip travel time 00:50:25.390 --> 00:50:28.730 from one mirror to the other. 00:50:28.730 --> 00:50:32.590 And what's particularly cool is the experimental aspect 00:50:32.590 --> 00:50:34.040 of this. 00:50:34.040 --> 00:50:37.900 So how does one measure the time so precisely? 00:50:37.900 --> 00:50:41.448 Well, you use a laser beam as your clock. 00:50:41.448 --> 00:50:43.990 And what's beautiful about doing that is the way that you can 00:50:43.990 --> 00:50:45.940 actually sort of do the metrology at the level 00:50:45.940 --> 00:50:50.470 necessary to get the precision to measure your hxx's in here 00:50:50.470 --> 00:50:53.300 is by interference. 00:50:53.300 --> 00:50:55.510 So you can treat the incoming laser beam 00:50:55.510 --> 00:50:57.760 as defining your time standard, and then 00:50:57.760 --> 00:51:01.060 by interfering the light that has traveled down the arm 00:51:01.060 --> 00:51:05.090 and bounced back with that laser beam, 00:51:05.090 --> 00:51:08.620 basically the laws of nature allow 00:51:08.620 --> 00:51:10.840 you to automatically check and see 00:51:10.840 --> 00:51:15.010 whether there is a shift in the time of arrival associated 00:51:15.010 --> 00:51:18.310 with the action of the gravitational wave. 00:51:18.310 --> 00:51:21.860 So I imagine there will be some questions about this. 00:51:21.860 --> 00:51:24.987 There is a homework exercise that allows you to sort of test 00:51:24.987 --> 00:51:27.070 drive some of these concepts and allows you to see 00:51:27.070 --> 00:51:29.990 how it is that LIGO works. 00:51:29.990 --> 00:51:31.750 It really just sort of touches the-- it 00:51:31.750 --> 00:51:33.310 just scratches the surface of what 00:51:33.310 --> 00:51:36.430 an amazing experimental feat it is. 00:51:36.430 --> 00:51:40.928 But it should give you a good idea as to what goes into that. 00:51:40.928 --> 00:51:42.720 Now, there is an important part to all this 00:51:42.720 --> 00:51:45.210 that we've not talked about yet. 00:51:45.210 --> 00:51:48.660 I handed you hij. 00:51:48.660 --> 00:51:49.920 Really, I handed you h mu nu. 00:52:01.270 --> 00:52:03.580 The big thing that we care about next 00:52:03.580 --> 00:52:07.840 is, given a particular source, how do I 00:52:07.840 --> 00:52:10.060 compute this radiation field? 00:52:18.492 --> 00:52:21.900 So bearing in mind that hijtt, this 00:52:21.900 --> 00:52:25.380 is the gauge-invariant radiative degree 00:52:25.380 --> 00:52:31.810 of freedom in my space-time, how do I compute this guy? 00:52:37.080 --> 00:52:40.620 And what I'm going to do is borrow sort of one 00:52:40.620 --> 00:52:43.500 of the most important lessons of that gauge-invariant formalism. 00:53:02.010 --> 00:53:07.230 So we need to be cautious of the fact that, in some gauges-- 00:53:07.230 --> 00:53:08.790 for example the Lorentz gauge, which 00:53:08.790 --> 00:53:11.571 is so convenient for solving the Einstein equation-- 00:53:22.650 --> 00:53:30.100 components of the metric appear to be radiative whether they 00:53:30.100 --> 00:53:30.960 are or they aren't. 00:53:42.040 --> 00:53:44.440 This is a consequence of our gauge. 00:53:50.320 --> 00:54:05.470 But we also know only hijtt is radiation in all gauges. 00:54:09.950 --> 00:54:11.950 So that was what the previous lecture was about. 00:54:11.950 --> 00:54:13.960 As I've emphasized, it was a slightly advanced one. 00:54:13.960 --> 00:54:15.970 I don't expect everyone to follow all details. 00:54:15.970 --> 00:54:19.690 But if you can sort of grok these two main concepts, 00:54:19.690 --> 00:54:23.120 you're ready to now exploit this in the following way. 00:54:28.760 --> 00:54:32.925 What you do is, step a, compute-- 00:54:36.380 --> 00:54:43.160 find some metric perturbation in any convenient gauge 00:54:43.160 --> 00:54:44.390 by any convenient method. 00:55:05.580 --> 00:55:10.200 Once you have that, extract hijtt. 00:55:15.280 --> 00:55:15.780 You're done. 00:55:33.857 --> 00:55:35.440 So that's what we're going to do right 00:55:35.440 --> 00:55:40.960 now in the last 20 or so minutes of this recorded lecture. 00:56:07.720 --> 00:56:10.750 So the way we're going to do it is let's 00:56:10.750 --> 00:56:13.420 take the linearized Einstein equation in Lorentz gauge. 00:56:31.000 --> 00:56:32.650 As we showed in-- 00:56:32.650 --> 00:56:36.750 I believe it was two lectures ago, 00:56:36.750 --> 00:56:38.470 the Einstein field equations in this case 00:56:38.470 --> 00:56:40.840 reduce to a flat space-time wave operator 00:56:40.840 --> 00:56:45.520 on the trace-reversed metric perturbation, 00:56:45.520 --> 00:56:47.650 being up to a factor of minus 16 pi 00:56:47.650 --> 00:56:52.000 G of the stress-energy tensor. 00:56:52.000 --> 00:56:55.540 And this has an exact solution. 00:57:21.488 --> 00:57:25.200 So I'll remind you of this notation. 00:57:25.200 --> 00:57:34.050 So this is the field at time t and spatial location x. 00:57:40.994 --> 00:57:43.060 x and t-- well, let's write it like this. 00:57:43.060 --> 00:57:47.385 t and x are my field point. 00:57:47.385 --> 00:57:48.760 That is where I measure my field. 00:57:54.320 --> 00:57:59.300 x prime is my source point. 00:57:59.300 --> 00:58:02.090 That is where I am looking at my particular contribution 00:58:02.090 --> 00:58:04.250 to the integral. 00:58:04.250 --> 00:58:06.710 As we're doing this calculation, there was a t prime, 00:58:06.710 --> 00:58:09.440 but the radiative Green's function turned that t prime 00:58:09.440 --> 00:58:10.730 into this retarded time. 00:58:10.730 --> 00:58:13.790 So what we are seeing is that the field at t 00:58:13.790 --> 00:58:16.970 depends upon what's going on at the source at t 00:58:16.970 --> 00:58:20.210 minus the interval of time it takes for information 00:58:20.210 --> 00:58:22.670 to propagate from x prime to x. 00:58:26.860 --> 00:58:29.860 So what I'm going to do is, first, I'm 00:58:29.860 --> 00:58:33.640 going to introduce a couple of approximations 00:58:33.640 --> 00:58:36.130 that reflect the reality of the conditions on which we 00:58:36.130 --> 00:58:39.160 typically evaluate this integral. 00:58:39.160 --> 00:58:42.820 And then we're going to talk about how to solve this thing, 00:58:42.820 --> 00:58:47.140 and, given that we have now solved it, how to project out 00:58:47.140 --> 00:58:50.320 the gauge-invariant degrees of freedom associated 00:58:50.320 --> 00:58:51.530 with the radiation. 00:59:05.740 --> 00:59:06.667 Whoops, pardon me. 00:59:06.667 --> 00:59:07.417 Let's put this up. 00:59:16.860 --> 00:59:19.470 So first is I introduce this approximation. 00:59:19.470 --> 00:59:23.280 It's worth bearing in mind that when you are calculating this, 00:59:23.280 --> 00:59:25.458 x minus x prime, which is kind of like the distance 00:59:25.458 --> 00:59:27.000 from a particular point in the source 00:59:27.000 --> 00:59:29.010 to where you're making your measurement, 00:59:29.010 --> 00:59:32.480 it is generally a lot larger than the size of the source. 00:59:46.165 --> 00:59:48.040 So the kind of situation we're thinking about 00:59:48.040 --> 00:59:51.530 is, here we are, making our measurements at some point x 00:59:51.530 --> 00:59:52.030 here. 00:59:55.320 --> 01:00:00.220 Here's my source, and here is a point x 01:00:00.220 --> 01:00:03.100 prime inside the source. 01:00:03.100 --> 01:00:04.750 So the thing to bear in mind is that, 01:00:04.750 --> 01:00:07.410 for the kind of calculations where that-- 01:00:07.410 --> 01:00:11.492 when one is actually doing this, x minus x prime-- 01:00:11.492 --> 01:00:13.450 well, if I think about the kind of measurements 01:00:13.450 --> 01:00:15.550 that LIGO is doing, for example, that 01:00:15.550 --> 01:00:17.200 is typically hundreds of millions 01:00:17.200 --> 01:00:20.380 to billions of light years. 01:00:20.380 --> 01:00:23.560 The size of the source itself is of order tens 01:00:23.560 --> 01:00:25.875 to hundreds of kilometers. 01:00:25.875 --> 01:00:28.480 Bit of a separation of length scales there, you would say. 01:00:28.480 --> 01:00:33.320 So what I'm going to do is approximate the solution 01:00:33.320 --> 01:00:38.373 of that integral, and, for pedantic sake, 01:00:38.373 --> 01:00:40.040 I'm going to use an approximately equal. 01:00:43.030 --> 01:00:47.490 What I'm going to do is replace x minus x prime with r. 01:00:47.490 --> 01:00:50.640 Let's just say that this distance here 01:00:50.640 --> 01:00:54.780 is effectively the same for all points in the source. 01:00:54.780 --> 01:00:56.280 I can pull that out of the integral. 01:01:11.480 --> 01:01:14.120 If you like, you can refine this. 01:01:14.120 --> 01:01:16.155 There's a more careful expansion. 01:01:16.155 --> 01:01:17.780 And if you've done any electrodynamics, 01:01:17.780 --> 01:01:19.155 you should be familiar with this. 01:01:25.270 --> 01:01:38.820 And it uses the fact that 1 over x minus x prime 01:01:38.820 --> 01:01:43.350 can be expanded in Legendre polynomials, 01:01:43.350 --> 01:01:45.403 in a sum of Legendre polynomials. 01:01:58.230 --> 01:02:00.270 So if you do that, you can introduce refinements 01:02:00.270 --> 01:02:02.250 to what I'm about to derive right here. 01:02:02.250 --> 01:02:05.657 What this does, then, is it defines a multipolar expansion. 01:02:22.240 --> 01:02:23.370 I am not going to do that. 01:02:23.370 --> 01:02:27.520 I'm just going to look at the very first term. 01:02:27.520 --> 01:02:28.950 And you can sort of think of that 01:02:28.950 --> 01:02:31.560 as saying that I am just doing the leading, the most 01:02:31.560 --> 01:02:34.530 important multipole. 01:02:34.530 --> 01:02:38.190 Additional multipoles beyond that are certainly important, 01:02:38.190 --> 01:02:41.580 and people who work in the field of gravitational radiation, 01:02:41.580 --> 01:02:44.670 we certainly do not neglect them. 01:02:44.670 --> 01:02:48.420 In fact, much of my current research 01:02:48.420 --> 01:02:54.360 is based on sort of high-order calculations associated 01:02:54.360 --> 01:02:57.560 with looking at behavior of some of those multipoles. 01:02:57.560 --> 01:03:02.170 This will be fine for initial pedagogical purposes. 01:03:02.170 --> 01:03:06.470 Next, we're going to use the fact that only the spatial-- 01:03:06.470 --> 01:03:06.970 whoops. 01:03:06.970 --> 01:03:08.512 That's not supposed to be an h tilde. 01:03:08.512 --> 01:03:10.162 My apologies. 01:03:10.162 --> 01:03:11.870 That's supposed to be h bar, because this 01:03:11.870 --> 01:03:13.018 is the trace-reverse thing. 01:03:20.680 --> 01:03:23.110 Weird times, folks. 01:03:23.110 --> 01:03:23.910 Where was I at? 01:03:23.910 --> 01:03:24.410 Yes. 01:03:24.410 --> 01:03:28.390 So I am only going to care about the spatial pieces of this. 01:03:28.390 --> 01:03:30.790 You might protest, hey, isn't it the spatial part 01:03:30.790 --> 01:03:33.070 of the metric, not the trace-reverse metric, 01:03:33.070 --> 01:03:33.903 that matters? 01:03:33.903 --> 01:03:34.945 And you would be correct. 01:03:38.260 --> 01:03:40.860 If you do this carefully, to be perfectly blunt, 01:03:40.860 --> 01:03:42.610 you have to sort of finish the calculation 01:03:42.610 --> 01:03:45.642 before you can kind of justify the step I'm about to make. 01:03:45.642 --> 01:03:48.100 And we're not going to have time to do that very carefully, 01:03:48.100 --> 01:03:50.590 but I'll make a comment about it. 01:03:50.590 --> 01:03:53.170 For now, it suffices to say we only care 01:03:53.170 --> 01:03:54.670 about the spatial components. 01:04:07.410 --> 01:04:11.170 So I'm going to take my alpha beta over to an ij. 01:04:25.760 --> 01:04:27.590 So I now have something that involves 01:04:27.590 --> 01:04:29.330 the integral of the spatial pieces 01:04:29.330 --> 01:04:32.870 of the stress-energy tensor over my source. 01:04:32.870 --> 01:04:35.100 And now I'm going to take advantage of an Easter egg 01:04:35.100 --> 01:04:39.330 that was hidden in an early problem set. 01:04:39.330 --> 01:04:41.760 Back on problem set 2, I asked you 01:04:41.760 --> 01:04:44.230 to prove a result that I called the tensor virial theorem. 01:04:53.370 --> 01:04:55.900 One of the reasons why I assigned that was that I knew I 01:04:55.900 --> 01:04:58.353 was going to need it in this lecture. 01:04:58.353 --> 01:04:59.770 And what the tensor virial theorem 01:04:59.770 --> 01:05:09.680 told us is that if I integrate that, 01:05:09.680 --> 01:05:16.867 this ends up being equivalent to 1/2 two time derivatives. 01:05:29.310 --> 01:05:33.500 It's equal to the integral of two time derivatives-- 01:05:33.500 --> 01:05:34.000 excuse me. 01:05:34.000 --> 01:05:36.730 It's two time derivatives of the integral of the 00 piece 01:05:36.730 --> 01:05:38.280 of my stress-energy tensor. 01:05:38.280 --> 01:05:43.223 So it's the second moment of that, xi prime, xj prime. 01:05:43.223 --> 01:05:45.890 This is great, because what I've got in the left-hand side here, 01:05:45.890 --> 01:05:47.554 that's exactly what my integral is. 01:05:51.260 --> 01:05:52.260 Actually, you know what? 01:05:52.260 --> 01:05:53.550 This is going to give me a result that's 01:05:53.550 --> 01:05:54.518 sufficiently clean. 01:05:54.518 --> 01:05:55.310 I want a new board. 01:06:02.040 --> 01:06:23.000 So when that is done, what I finally see 01:06:23.000 --> 01:06:29.300 is that my hij, trace reversed, is 01:06:29.300 --> 01:06:34.550 2 G over r two time derivatives of a tensor 01:06:34.550 --> 01:06:37.550 that I will call Iij. 01:06:44.220 --> 01:06:52.680 This is what I get when I integrate T 00, 01:06:52.680 --> 01:06:56.250 two moments of T 00. 01:06:56.250 --> 01:07:02.000 This is called the quadrupole moment of the source. 01:07:09.730 --> 01:07:12.220 The asterisk is because, as you guys will 01:07:12.220 --> 01:07:14.050 see on one of the problem sets, there's 01:07:14.050 --> 01:07:16.413 a slight tweak that goes in the exact definition 01:07:16.413 --> 01:07:17.830 of the quadrupole moment which has 01:07:17.830 --> 01:07:24.250 to do with the behavior of the trace of Iij. 01:07:24.250 --> 01:07:26.710 I will just leave this like so for now, 01:07:26.710 --> 01:07:30.880 and leave you guys to explore that on a problem set. 01:07:30.880 --> 01:07:32.110 So we're almost done. 01:07:32.110 --> 01:07:41.590 What I now need to do is pull out 01:07:41.590 --> 01:07:43.470 the transverse and traceless piece of this. 01:08:00.133 --> 01:08:02.050 Let's consider the transverse condition first. 01:08:31.210 --> 01:08:32.740 So, to do the transverse condition, 01:08:32.740 --> 01:08:34.615 I'm going to take advantage of the fact that, 01:08:34.615 --> 01:08:50.620 far from the source, box of h bar ij is equal to 0, 01:08:50.620 --> 01:08:52.870 because I'm in a region where the source is equal to 0 01:08:52.870 --> 01:08:54.617 when I'm far away from it. 01:08:54.617 --> 01:08:56.200 This suggests that what we ought to do 01:08:56.200 --> 01:08:59.229 is expand our solution plane waves. 01:09:13.960 --> 01:09:19.240 So what I can do is write h bar. 01:09:19.240 --> 01:09:21.880 And I'm going to choose my indices carefully here, 01:09:21.880 --> 01:09:23.830 call it jl. 01:09:23.830 --> 01:09:28.630 It's some amplitude that does not depend on space or time, 01:09:28.630 --> 01:09:37.510 e to the i omega t minus a wave vector dotted with x. 01:09:37.510 --> 01:09:41.950 So this amplitude may depend on omega, may depend on k. 01:09:41.950 --> 01:09:44.340 It does not depend on t or on x. 01:09:44.340 --> 01:09:47.529 And the reason I changed my indices from i and j 01:09:47.529 --> 01:09:50.100 to j and l, and I did not use the index k, 01:09:50.100 --> 01:09:53.297 is just to avoid confusion between square root of minus 1 01:09:53.297 --> 01:09:54.130 and the wave vector. 01:09:58.200 --> 01:10:01.052 To be absolutely careful, this is a single mode. 01:10:01.052 --> 01:10:03.510 If you like, maybe you should think about this as something 01:10:03.510 --> 01:10:04.950 that I have-- 01:10:04.950 --> 01:10:07.933 I'm doing a sum or an integral over omega and k 01:10:07.933 --> 01:10:09.600 in order to reconstruct the whole thing. 01:10:09.600 --> 01:10:14.340 This is just a single Fourier mode in the wave field. 01:10:14.340 --> 01:10:20.360 So if I want this to be transverse, 01:10:20.360 --> 01:10:25.890 that condition is that the divergence of this guy 01:10:25.890 --> 01:10:28.460 be equal to 0. 01:10:28.460 --> 01:10:34.890 But when I take the divergence of this, this becomes-- 01:10:34.890 --> 01:10:38.540 I pull down a factor of minus i, and I pull down 01:10:38.540 --> 01:10:41.870 the j-th component of my wave vector k. 01:10:54.930 --> 01:10:55.740 And I get my wave-- 01:10:55.740 --> 01:10:56.550 my Fourier term. 01:11:04.470 --> 01:11:06.540 That can be thought of as-- 01:11:06.540 --> 01:11:08.580 so we can clear out the factor of minus i. 01:11:38.330 --> 01:11:55.198 The transverse condition boils down to saying that kj-- 01:11:55.198 --> 01:11:55.698 whoops. 01:12:03.372 --> 01:12:04.080 No, that's right. 01:12:04.080 --> 01:12:05.800 That's right. 01:12:05.800 --> 01:12:10.450 kj contracted on the wave field is equal to 0. 01:12:10.450 --> 01:12:13.420 In other words, the wave field is 01:12:13.420 --> 01:12:17.230 orthogonal to the wave vector. 01:12:32.020 --> 01:12:33.790 If you have studied electrodynamics, 01:12:33.790 --> 01:12:36.800 this should be familiar to you. 01:12:36.800 --> 01:12:38.050 This is the same-- 01:12:38.050 --> 01:12:40.090 it's exactly the same as the idea 01:12:40.090 --> 01:12:42.130 that when I have an electromagnetic wave, 01:12:42.130 --> 01:12:46.060 my E field and my B field are orthogonal to the direction 01:12:46.060 --> 01:12:49.210 of propagation. 01:12:49.210 --> 01:13:01.298 Let's define a unit vector n sub j to be kj but just normalized. 01:13:07.280 --> 01:13:11.630 So my wave is propagating in the direction of the unit vector n. 01:13:14.655 --> 01:13:16.530 Go back to another Easter egg that was hidden 01:13:16.530 --> 01:13:17.760 on an earlier problem set. 01:13:17.760 --> 01:13:19.470 You guys studied projection tensors. 01:13:28.710 --> 01:13:41.250 If I introduce the projection tensor Pij, 01:13:41.250 --> 01:13:45.840 Pij will project any spatial vector. 01:13:45.840 --> 01:13:53.190 It will project out the components normal, orthogonal 01:13:53.190 --> 01:13:54.660 to the direction of propagation n. 01:14:23.470 --> 01:14:28.870 You can in fact think of Pij as defining the metric 01:14:28.870 --> 01:14:32.350 for the subspace that is orthogonal to n. 01:14:32.350 --> 01:14:38.090 So imagine that your wave is propagating in the n direction. 01:14:38.090 --> 01:14:40.090 There's a plane being carried along with it. 01:14:40.090 --> 01:14:43.690 The plane is orthogonal to n, and Pij 01:14:43.690 --> 01:14:47.470 is the metric you use to define spatial distances 01:14:47.470 --> 01:14:48.200 in that plane. 01:14:51.950 --> 01:14:57.980 Using Pij, we can now make my h trace-- excuse me. 01:14:57.980 --> 01:14:59.210 I can make it transverse. 01:15:33.520 --> 01:15:34.960 All I need to do-- 01:15:34.960 --> 01:15:38.380 I'll put one T on this to denote Transverse. 01:15:44.860 --> 01:15:46.690 Whoops. 01:15:46.690 --> 01:15:57.940 All I need to do is project each of its indices 01:15:57.940 --> 01:15:59.770 within that tensor. 01:15:59.770 --> 01:16:01.210 So we're almost there. 01:16:01.210 --> 01:16:04.435 It's transverse, but not yet trace free. 01:16:11.860 --> 01:16:15.370 So if I take the trace on this guy, 01:16:15.370 --> 01:16:18.580 taking the trace on this guy amounts to-- 01:16:18.580 --> 01:16:30.050 I am going to require that this be equal to 0. 01:16:33.610 --> 01:16:37.810 Now, suppose I have some tensor that is not trace free, 01:16:37.810 --> 01:16:39.670 and I want to remove its trace. 01:16:53.750 --> 01:16:55.500 I'm going to make this a two-index tensor. 01:17:02.130 --> 01:17:05.497 So I'm not going to prove the following result. 01:17:05.497 --> 01:17:06.955 You can easily verify for yourself. 01:17:10.230 --> 01:17:29.850 The formula for doing so is that aij trace free is aij minus 1 01:17:29.850 --> 01:17:41.410 over N akl gkl gij. 01:17:41.410 --> 01:17:51.590 G is the metric that describes the manifold 01:17:51.590 --> 01:17:52.730 in which this tensor lives. 01:18:07.340 --> 01:18:09.571 N is its dimension. 01:18:22.840 --> 01:18:26.390 Now, for us, the tensor field whose trace I want to remove 01:18:26.390 --> 01:18:26.890 is hijT. 01:18:52.860 --> 01:19:00.150 So hij bar T is the field whose trace I wish to remove. 01:19:00.150 --> 01:19:06.830 This lives in the two-dimensional manifold 01:19:06.830 --> 01:19:20.380 whose metric is the projection tensor Pij. 01:19:20.380 --> 01:19:22.910 And we are working in a space where raising and lowering 01:19:22.910 --> 01:19:24.910 indices, there's really no important distinction 01:19:24.910 --> 01:19:26.050 between the two of them, because we're 01:19:26.050 --> 01:19:27.258 working in linearized theory. 01:19:29.660 --> 01:19:47.110 So I remove the trace by taking this guy and subtracting off-- 01:19:47.110 --> 01:19:48.430 pardon me just one moment. 01:19:51.364 --> 01:19:52.342 Yes, sorry. 01:20:02.140 --> 01:20:03.270 Subtracting off like so. 01:20:03.270 --> 01:20:03.770 Pardon me. 01:20:03.770 --> 01:20:04.840 It should have a T on it. 01:20:12.680 --> 01:20:15.270 Plugging in my various definitions, 01:20:15.270 --> 01:20:24.620 this turns into hlk Pli Pkj minus 1/2. 01:20:42.062 --> 01:20:44.270 So you might want to go and check some algebra there, 01:20:44.270 --> 01:20:45.000 but this is all-- 01:20:45.000 --> 01:20:47.760 I'm very confident in this. 01:20:47.760 --> 01:20:50.940 There's an overall factor of h bar lk. 01:20:50.940 --> 01:20:52.440 And the last thing which I will note 01:20:52.440 --> 01:20:56.080 is that, once I have removed the trace, 01:20:56.080 --> 01:21:00.350 this bar becomes meaningless. 01:21:00.350 --> 01:21:03.715 It's trace free, so the trace reversing of it has no action. 01:21:03.715 --> 01:21:05.840 Now, you should be a little bit careful about that. 01:21:05.840 --> 01:21:07.640 There's a few different notions of trace 01:21:07.640 --> 01:21:09.620 that went into this definition. 01:21:09.620 --> 01:21:12.650 Without proof, and just stating that by being a little bit more 01:21:12.650 --> 01:21:14.090 careful of some of the definitions 01:21:14.090 --> 01:21:17.990 we put down, the difference between-- 01:21:17.990 --> 01:21:19.640 when you count this up, you'll find 01:21:19.640 --> 01:21:21.450 that you may be making a small error, 01:21:21.450 --> 01:21:24.830 but it is on the order of terms that are 01:21:24.830 --> 01:21:26.776 quadratic in the perturbation. 01:21:34.400 --> 01:21:37.520 Putting all these pieces together, 01:21:37.520 --> 01:21:41.030 we finally obtain our transverse and traceless 01:21:41.030 --> 01:21:42.395 metric perturbation. 01:21:55.273 --> 01:21:56.190 Which looks like this. 01:22:27.700 --> 01:22:31.370 This is a result known as the quadrupole formula 01:22:31.370 --> 01:22:32.450 for gravitational waves. 01:22:41.500 --> 01:22:43.630 This is-- so given-- 01:22:43.630 --> 01:22:47.050 just take a stock of the final result. 01:22:47.050 --> 01:22:51.490 Suppose you have some dynamic gravitating source. 01:22:51.490 --> 01:22:56.870 This is telling me, what I do is I compute the quadrupole moment 01:22:56.870 --> 01:22:59.420 associated with that source. 01:22:59.420 --> 01:23:02.840 I take two time derivatives. 01:23:02.840 --> 01:23:05.420 I hit it with this combination of projection tensors, which 01:23:05.420 --> 01:23:07.567 has to do-- that tells me things about the geometry 01:23:07.567 --> 01:23:08.150 of the source. 01:23:08.150 --> 01:23:10.372 That's saying that my source, the waves that 01:23:10.372 --> 01:23:12.830 come from the source propagate along a particular direction 01:23:12.830 --> 01:23:13.790 to me. 01:23:13.790 --> 01:23:16.520 This picks out the components that 01:23:16.520 --> 01:23:18.960 are transverse with respect to that direction 01:23:18.960 --> 01:23:22.370 and are traceless, so that I don't have any gauge degrees 01:23:22.370 --> 01:23:24.500 of freedom in what results. 01:23:24.500 --> 01:23:28.100 Multiply by twice Newton's constant, divide by distance. 01:23:28.100 --> 01:23:31.430 And this is a wave field that our friends 01:23:31.430 --> 01:23:36.500 over in Building NW22 can now measure pretty much every day. 01:23:36.500 --> 01:23:38.750 You're going to do a few homework exercises associated 01:23:38.750 --> 01:23:39.250 with. 01:23:39.250 --> 01:23:46.320 One thing which is worth commenting on before I move on 01:23:46.320 --> 01:23:51.810 is, when you keep factors of c in this, 01:23:51.810 --> 01:23:55.887 that G goes over to G over c to the-- 01:23:55.887 --> 01:23:56.470 wait a minute. 01:23:56.470 --> 01:23:58.342 Let me double-check that. 01:23:58.342 --> 01:24:00.850 Is it G over c to the fourth or G over c squared? 01:24:00.850 --> 01:24:03.210 Anyhow, it's G divided by c to some power. 01:24:06.140 --> 01:24:08.390 Let's just say that it's missing factors of c. 01:24:12.510 --> 01:24:14.180 I may comment on that in some notes 01:24:14.180 --> 01:24:16.980 that I will add to the board. 01:24:16.980 --> 01:24:18.980 And so that kind of tells you that this is going 01:24:18.980 --> 01:24:23.930 to be a really small quantity. 01:24:23.930 --> 01:24:26.570 Makes it very hard for these things to be measured. 01:24:26.570 --> 01:24:29.530 Nonetheless, it turns out even though they have-- 01:24:29.530 --> 01:24:32.960 if you think about the impact they have on the detector, 01:24:32.960 --> 01:24:35.900 even though it's small, they carry a tremendous amount 01:24:35.900 --> 01:24:37.760 of energy. 01:24:37.760 --> 01:24:39.590 I'm going to take a break before I begin 01:24:39.590 --> 01:24:41.600 recording my next lecture. 01:24:41.600 --> 01:24:43.400 But the next lecture is actually going 01:24:43.400 --> 01:24:45.770 to lead us to an understanding of a different version 01:24:45.770 --> 01:24:47.180 of the quadrupole formula. 01:24:47.180 --> 01:24:50.000 It's based on this, but it describes the energy content 01:24:50.000 --> 01:24:52.070 of gravitational waves. 01:24:52.070 --> 01:24:56.232 Both of these formulas go by the name the quadrupole formula. 01:24:56.232 --> 01:24:59.720 It's a little bit confusing sometimes if both are used. 01:24:59.720 --> 01:25:01.850 And you're going to use both of them 01:25:01.850 --> 01:25:03.990 on upcoming homework assignments. 01:25:03.990 --> 01:25:04.490 All right. 01:25:04.490 --> 01:25:07.270 So I will end this recording here.