1 00:00:00,000 --> 00:00:01,944 [SQUEAKING] 2 00:00:01,944 --> 00:00:03,888 [RUSTLING] 3 00:00:03,888 --> 00:00:06,804 [CLICKING] 4 00:00:09,982 --> 00:00:10,690 SCOTT HUGHES: OK. 5 00:00:10,690 --> 00:00:12,160 So let's just dive in. 6 00:00:12,160 --> 00:00:15,580 And I'll just quickly recap where we were last time. 7 00:00:15,580 --> 00:00:18,160 So what I did last time was, we are now 8 00:00:18,160 --> 00:00:20,290 beginning the adventure, so to speak, 9 00:00:20,290 --> 00:00:24,730 of looking at how one examines-- 10 00:00:24,730 --> 00:00:29,140 how one finds and then studies the properties of solutions 11 00:00:29,140 --> 00:00:30,550 to the Einstein field equations. 12 00:00:30,550 --> 00:00:32,710 In other words, given a particular source, 13 00:00:32,710 --> 00:00:35,423 what is the space-time that emerges? 14 00:00:35,423 --> 00:00:37,840 What is the space-time that is consistent with that source 15 00:00:37,840 --> 00:00:41,562 of gravity in Einstein's relativistic theory here? 16 00:00:41,562 --> 00:00:43,270 So we begin-- we're going to look at this 17 00:00:43,270 --> 00:00:44,650 in a couple of different ways. 18 00:00:44,650 --> 00:00:47,110 And the first one is where we're going 19 00:00:47,110 --> 00:00:49,490 to-- what we do-- linearize the Einstein equation. 20 00:00:49,490 --> 00:00:51,198 So we're going to imagine that space-time 21 00:00:51,198 --> 00:00:53,770 is a small deviation from the flat space-time that 22 00:00:53,770 --> 00:00:55,930 contains no gravity. 23 00:00:55,930 --> 00:00:59,590 And we are going to expand the Einstein field equations, 24 00:00:59,590 --> 00:01:04,720 discarding all terms that are non-linear in the perturbation 25 00:01:04,720 --> 00:01:06,710 away from flat space-time. 26 00:01:06,710 --> 00:01:11,410 So last time we did this, we developed our field equations, 27 00:01:11,410 --> 00:01:13,480 we recast them in a form that's particularly 28 00:01:13,480 --> 00:01:16,840 convenient for making solutions, and we 29 00:01:16,840 --> 00:01:19,450 noted that the coordinate degrees of freedom 30 00:01:19,450 --> 00:01:21,610 can be thought of as a kind of gauge freedom 31 00:01:21,610 --> 00:01:25,690 that, in some ways, it's very useful for us, 32 00:01:25,690 --> 00:01:28,780 because it allows us to write the equations in a form that 33 00:01:28,780 --> 00:01:30,820 is particularly amenable to solving, 34 00:01:30,820 --> 00:01:32,500 but can also be a little bit dangerous, 35 00:01:32,500 --> 00:01:34,420 because you can sometimes wind up 36 00:01:34,420 --> 00:01:37,630 with a solution whose physical character is masked 37 00:01:37,630 --> 00:01:40,720 by the nature of that gauge. 38 00:01:40,720 --> 00:01:42,580 I went through a somewhat advanced topic 39 00:01:42,580 --> 00:01:44,770 for the purposes of 8.962, and I do not 40 00:01:44,770 --> 00:01:47,860 expect students to follow this in detail. 41 00:01:47,860 --> 00:01:50,140 But it's good to at least survey this and understand 42 00:01:50,140 --> 00:01:51,307 what was going on with this. 43 00:01:51,307 --> 00:01:56,740 What I did was I decomposed the perturbation to flat space-time 44 00:01:56,740 --> 00:01:58,660 into irreducible degrees of freedom. 45 00:01:58,660 --> 00:02:00,850 I sort of chose a time coordinate. 46 00:02:00,850 --> 00:02:04,270 I then imagined the other three coordinates in our space, 47 00:02:04,270 --> 00:02:07,740 and I looked at how, having chosen time, 48 00:02:07,740 --> 00:02:10,259 the different degrees of freedom decompose 49 00:02:10,259 --> 00:02:14,440 in two scalars with respect to coordinate rotations, 50 00:02:14,440 --> 00:02:16,540 vectors with respect to coordinate rotations, 51 00:02:16,540 --> 00:02:18,010 and two-index tensors with respect 52 00:02:18,010 --> 00:02:20,370 to coordinate rotations. 53 00:02:20,370 --> 00:02:23,320 I then took those, particularly the vectors and the tensors, 54 00:02:23,320 --> 00:02:26,230 and I further broke them up into degrees of freedom that 55 00:02:26,230 --> 00:02:34,660 are either divergence-free vector fields or the gradients 56 00:02:34,660 --> 00:02:37,390 of scalar fields. 57 00:02:37,390 --> 00:02:39,580 And in doing this, what we found was 58 00:02:39,580 --> 00:02:43,050 that the 10 degrees of freedom that are specified-- excuse me, 59 00:02:43,050 --> 00:02:46,700 the 10 degrees of freedom that are present in the perturbation 60 00:02:46,700 --> 00:02:48,730 the space time, can be broken down 61 00:02:48,730 --> 00:02:54,130 into six functions that are gauge-invariant plus four gauge 62 00:02:54,130 --> 00:02:56,050 degrees of freedom. 63 00:02:56,050 --> 00:02:59,890 Applying a similar decomposition to the source, 64 00:02:59,890 --> 00:03:03,610 we found that those six gauge invariant degrees of freedom 65 00:03:03,610 --> 00:03:06,900 are governed by four Poisson type equations. 66 00:03:06,900 --> 00:03:09,870 And I will refer you to the previous lecture 67 00:03:09,870 --> 00:03:11,620 for exact definitions of these things, 68 00:03:11,620 --> 00:03:14,620 but there is a scalar field phi that 69 00:03:14,620 --> 00:03:22,385 is related to the mean density of energy in your space-time 70 00:03:22,385 --> 00:03:24,010 and pressures in your space-time-- sort 71 00:03:24,010 --> 00:03:26,410 of isotropic stresses. 72 00:03:26,410 --> 00:03:29,500 This DTS is an additional degree of freedom 73 00:03:29,500 --> 00:03:32,500 that is defined in that previous lecture. 74 00:03:32,500 --> 00:03:35,050 You find that a Poisson equation for a scalar 75 00:03:35,050 --> 00:03:40,330 field I called theta is directly related to the energy density. 76 00:03:40,330 --> 00:03:46,450 There are two degrees of freedom associated 77 00:03:46,450 --> 00:03:49,960 with sort of off diagonal terms in the metric-- the time-space 78 00:03:49,960 --> 00:03:51,130 terms in the metric. 79 00:03:51,130 --> 00:03:53,980 And they are related to the momentum flow. 80 00:03:53,980 --> 00:03:56,020 These are only two degrees of freedom, 81 00:03:56,020 --> 00:04:00,910 because the psi is a divergence-free field. 82 00:04:00,910 --> 00:04:03,430 And finally, what we found is if you 83 00:04:03,430 --> 00:04:06,940 take the spatial piece of the space-time metric, 84 00:04:06,940 --> 00:04:09,460 you project out the bits that are 85 00:04:09,460 --> 00:04:12,080 transverse-- in other words, it's divergence-free. 86 00:04:12,080 --> 00:04:14,860 We will come to a more physical meaning of this a little bit 87 00:04:14,860 --> 00:04:16,329 later in today's lecture. 88 00:04:16,329 --> 00:04:19,089 And you require that this be traceless-- 89 00:04:19,089 --> 00:04:24,340 then those components of the spatial space-time perturbation 90 00:04:24,340 --> 00:04:27,340 are already gauge invariant, and they're 91 00:04:27,340 --> 00:04:31,050 related to sort of anisotropic stresses in our source. 92 00:04:31,050 --> 00:04:33,100 They are governed by a wave equation. 93 00:04:33,100 --> 00:04:36,940 And as such, these represent a radiative degree of freedom. 94 00:04:36,940 --> 00:04:38,710 Because this is [coughs] (excuse me!)-- 95 00:04:38,710 --> 00:04:42,640 because this is trace-free and divergence-right, 96 00:04:42,640 --> 00:04:45,940 naively you would think that this has six numbers in it. 97 00:04:45,940 --> 00:04:49,120 A symmetric 3 by 3 matrix is how you would represent it. 98 00:04:49,120 --> 00:04:52,210 But because it is trace-free and because it is divergenceless, 99 00:04:52,210 --> 00:04:53,590 there are four constraints. 100 00:04:53,590 --> 00:04:57,400 And so those six independent numbers are reduced to two. 101 00:04:57,400 --> 00:04:59,110 And as we'll see in this lecture, 102 00:04:59,110 --> 00:05:01,990 those two degrees of freedom correspond to the polarizations 103 00:05:01,990 --> 00:05:04,210 of gravitational waves. 104 00:05:04,210 --> 00:05:07,780 Today's lecture is all about understanding these guys. 105 00:05:07,780 --> 00:05:10,120 Given our kind of funny schedule that we're on today, 106 00:05:10,120 --> 00:05:12,370 I got up early and have not had time to take a shower, 107 00:05:12,370 --> 00:05:16,510 so I am wearing my LIGO hat. 108 00:05:16,510 --> 00:05:18,580 And what we're going to be doing today 109 00:05:18,580 --> 00:05:22,630 is the foundations of how one calculates this 110 00:05:22,630 --> 00:05:25,510 and why one builds an observatory like LIGO 111 00:05:25,510 --> 00:05:28,090 to measure these things. 112 00:05:28,090 --> 00:05:28,700 All right. 113 00:05:28,700 --> 00:05:29,950 So let me just write that out. 114 00:05:29,950 --> 00:05:32,195 So our goal for today-- 115 00:05:32,195 --> 00:05:33,820 I should say our goal for this lecture, 116 00:05:33,820 --> 00:05:35,610 because this is the first of three lectures 117 00:05:35,610 --> 00:05:36,693 I'm going to record today. 118 00:05:39,190 --> 00:05:49,310 Our goal is to understand hij TT in terms of quantities 119 00:05:49,310 --> 00:05:59,680 that we can observe and also to understand 120 00:05:59,680 --> 00:06:04,120 how to compute this field-- 121 00:06:04,120 --> 00:06:15,015 this hij TT given us source. 122 00:06:19,590 --> 00:06:20,960 So let's begin by just assuming. 123 00:06:20,960 --> 00:06:23,720 Let's begin by focusing on how we can understand 124 00:06:23,720 --> 00:06:27,342 what hij TT actually means-- how we can go and measure this. 125 00:06:27,342 --> 00:06:29,300 So what I'm going to do is just we'll start out 126 00:06:29,300 --> 00:06:33,290 by let's just hand ourselves an hij TT 127 00:06:33,290 --> 00:06:35,117 and study its properties. 128 00:06:37,980 --> 00:06:49,290 So what I'm going to do is begin with a space-time perturbation 129 00:06:49,290 --> 00:06:53,580 that has everything except hij TT equal to 0. 130 00:06:53,580 --> 00:07:01,860 So this is something that has phi equals 0, theta equals 0, 131 00:07:01,860 --> 00:07:04,710 psi equal to 0. 132 00:07:04,710 --> 00:07:07,410 I am going to pick hij TT. 133 00:07:11,736 --> 00:07:14,885 If we have a form such that it only depends-- 134 00:07:17,530 --> 00:07:19,180 it's a function that depends only 135 00:07:19,180 --> 00:07:21,670 on the combination T minus z. 136 00:07:21,670 --> 00:07:25,520 So this represents something propagating-- 137 00:07:25,520 --> 00:07:27,340 as we see, it actually describes radiation. 138 00:07:37,860 --> 00:07:41,440 in the z-direction. 139 00:07:41,440 --> 00:07:52,626 So this assumed form can be represented 140 00:07:52,626 --> 00:07:54,115 by the following matrix. 141 00:08:21,120 --> 00:08:23,810 So if you take a look at this, I'm 142 00:08:23,810 --> 00:08:26,547 going to justify a few details of this later. 143 00:08:26,547 --> 00:08:27,630 There's a few constraints. 144 00:08:27,630 --> 00:08:29,910 So it looks like I have four numbers here. 145 00:08:29,910 --> 00:08:37,440 Because my space-time metric is always symmetric under-- 146 00:08:37,440 --> 00:08:39,150 we'll just say symmetry-- 147 00:08:39,150 --> 00:08:51,000 under exchange of indices, h xy is the same thing as h yx. 148 00:08:51,000 --> 00:08:53,790 So these two things are the same. 149 00:08:53,790 --> 00:09:00,240 And from the requirement that this construction be 150 00:09:00,240 --> 00:09:01,530 trace-free-- 151 00:09:01,530 --> 00:09:04,500 so from the way I've written this here, 152 00:09:04,500 --> 00:09:13,640 that tells me that h xx plus hyy h yy must equal 0. 153 00:09:19,350 --> 00:09:22,110 So these two quantities on the diagonal are not independent. 154 00:09:22,110 --> 00:09:24,250 They must be of opposite sign. 155 00:09:24,250 --> 00:09:26,400 So there are my two degrees of freedom. 156 00:09:26,400 --> 00:09:29,250 Why have I set the z part equal to 0? 157 00:09:29,250 --> 00:09:32,880 We'll look at that in a little bit more detail a little later 158 00:09:32,880 --> 00:09:33,580 in the lecture. 159 00:09:33,580 --> 00:09:35,205 But essentially, it comes from the fact 160 00:09:35,205 --> 00:09:38,130 that I'm making it propagate in the z-direction. 161 00:09:38,130 --> 00:09:41,100 What we're going to see is that if-- 162 00:09:41,100 --> 00:09:44,310 whatever direction the wave is propagating in, 163 00:09:44,310 --> 00:09:46,530 my field has to be orthogonal to it. 164 00:09:46,530 --> 00:09:50,350 And we'll prove that a little bit more carefully later. 165 00:09:50,350 --> 00:09:53,740 So, for now let's just take this as handed to us. 166 00:09:53,740 --> 00:09:56,620 We can see this is at least consistent with the properties 167 00:09:56,620 --> 00:09:58,450 I've written down here, from the fact 168 00:09:58,450 --> 00:10:01,360 that it depends on T minus z, it's 169 00:10:01,360 --> 00:10:04,350 not hard to show that this thing will satisfy the wave equation. 170 00:10:10,810 --> 00:10:15,690 So we have a good source for exploration in front of us-- 171 00:10:15,690 --> 00:10:18,340 excuse me-- a good space-time for us 172 00:10:18,340 --> 00:10:20,540 to explore the properties of. 173 00:10:20,540 --> 00:10:22,940 So let's begin by saying great. 174 00:10:22,940 --> 00:10:26,290 One of the most important things we've learned how to do 175 00:10:26,290 --> 00:10:30,010 is, given a space-time, I can compute 176 00:10:30,010 --> 00:10:34,420 how a freely falling body moves in that space-time. 177 00:10:34,420 --> 00:10:40,690 So let's consider what happens if I have freely falling bodies 178 00:10:40,690 --> 00:11:01,780 in a space-time, in a space-time such that this-- 179 00:11:01,780 --> 00:11:05,467 in a space-time that is this-- 180 00:11:05,467 --> 00:11:07,050 so I am not saying this very fluently. 181 00:11:12,480 --> 00:11:13,730 So this is what we want to do. 182 00:11:13,730 --> 00:11:16,272 We're going to consider freely falling bodies in a space-time 183 00:11:16,272 --> 00:11:17,905 that involves this h mu nu. 184 00:11:17,905 --> 00:11:19,280 And the way I'm going to do this, 185 00:11:19,280 --> 00:11:20,923 my first pass is to sort of say, OK, 186 00:11:20,923 --> 00:11:23,090 I know all about the motion of freely falling bodies 187 00:11:23,090 --> 00:11:24,080 in space-time. 188 00:11:24,080 --> 00:11:25,280 They are geodesics. 189 00:11:36,270 --> 00:11:38,020 Geodesics describe such bodies. 190 00:11:38,020 --> 00:11:48,360 So let's go ahead and look at the geodesic equation 191 00:11:48,360 --> 00:11:49,900 for this setup. 192 00:11:49,900 --> 00:11:51,408 Now I wanted to-- 193 00:11:51,408 --> 00:11:53,200 let's give ourselves a couple of conditions 194 00:11:53,200 --> 00:11:54,620 to try and understand this. 195 00:11:54,620 --> 00:11:57,580 Let's imagine that these are-- we're initially 196 00:11:57,580 --> 00:12:02,071 in a space-time that is just flat space-time. 197 00:12:07,150 --> 00:12:12,039 So we're going to imagine that h mu nu is 0 initially. 198 00:12:16,930 --> 00:12:19,290 And perhaps it then-- 199 00:12:19,290 --> 00:12:21,520 we're sort of imagining a space-time in which there's 200 00:12:21,520 --> 00:12:24,440 just you falling freely in empty space. 201 00:12:24,440 --> 00:12:26,320 Absolutely nothing going on here. 202 00:12:26,320 --> 00:12:29,670 And this h mu nu field passes over you. 203 00:12:29,670 --> 00:12:33,247 A wave packet passes over your freely falling particles. 204 00:12:38,980 --> 00:12:43,590 So h mu nu, not equal 0, comes along, 205 00:12:43,590 --> 00:12:45,554 and passes over your particles. 206 00:12:52,480 --> 00:12:56,270 I'm going to imagine that these bodies are initially at rest. 207 00:12:56,270 --> 00:13:00,070 So when I'm in this regime where h mu nu is equal to 0, 208 00:13:00,070 --> 00:13:02,290 these guys are just sitting stationary with respect 209 00:13:02,290 --> 00:13:04,515 to whoever is making these measurements. 210 00:13:15,273 --> 00:13:16,440 So we're going to say great. 211 00:13:16,440 --> 00:13:18,660 I've got these bodies sitting at rest in what 212 00:13:18,660 --> 00:13:20,610 is initially empty space-time. 213 00:13:20,610 --> 00:13:22,540 A wave passes along. 214 00:13:22,540 --> 00:13:23,870 What happens to these bodies? 215 00:13:30,630 --> 00:13:33,920 So if these guys are at rest, then I know that, 216 00:13:33,920 --> 00:13:36,860 at initial times-- so let's just say that this comes along, 217 00:13:36,860 --> 00:13:41,360 and it becomes h mu nu not equal to 0 at some time-- 218 00:13:41,360 --> 00:13:43,040 what is defined as the origin of time. 219 00:13:46,460 --> 00:13:51,100 So t equals 0 is when the non-zero wave comes along. 220 00:13:57,840 --> 00:14:02,340 So my space-time is just eta mu nu for all earlier times. 221 00:14:02,340 --> 00:14:04,920 My particles are sitting here at rest at all earlier times, 222 00:14:04,920 --> 00:14:07,290 then h mu nu comes along. 223 00:14:07,290 --> 00:14:10,320 So what we want to do, then, is look 224 00:14:10,320 --> 00:14:16,410 at how does this guy behave at t equals 0. 225 00:14:26,050 --> 00:14:31,540 Now, initially, the only component that will get picked 226 00:14:31,540 --> 00:14:34,661 out of this is the 00 component. 227 00:14:38,990 --> 00:14:43,205 And, working to linear order, I can write this. 228 00:14:49,878 --> 00:14:51,670 Just wanted to write one more step in here. 229 00:14:51,670 --> 00:14:58,160 Let's say that this, I'm going to write as eta alpha beta beta 230 00:14:58,160 --> 00:14:58,748 00. 231 00:14:58,748 --> 00:15:00,290 Go back to your earlier lecture, when 232 00:15:00,290 --> 00:15:02,000 we defined the Christoffel symbol to look up 233 00:15:02,000 --> 00:15:03,125 the definition of this guy. 234 00:15:15,860 --> 00:15:18,100 So these are all the only components 235 00:15:18,100 --> 00:15:22,150 of the space-time that enter into this Christoffel symbol. 236 00:15:22,150 --> 00:15:24,490 Every single one-- let's go back and take a look 237 00:15:24,490 --> 00:15:26,750 at the space-time that we're looking down here. 238 00:15:26,750 --> 00:15:32,500 Notice this involves beta 0, 0 beta, and 00. 239 00:15:32,500 --> 00:15:36,130 That is the space-time parts, all of which are 0. 240 00:15:36,130 --> 00:15:38,530 The other space-time parts, all of which are 0. 241 00:15:38,530 --> 00:15:42,140 And the time-time part, which is 0. 242 00:15:42,140 --> 00:15:46,100 So the initial acceleration is 0. 243 00:15:46,100 --> 00:15:53,550 That means du alpha d tau is equal to 0. 244 00:15:59,620 --> 00:16:02,320 Which means that my particle begins on a geodesic, 245 00:16:02,320 --> 00:16:04,510 and it stays on the geodesic. 246 00:16:04,510 --> 00:16:07,210 In these coordinates, it does not move at all. 247 00:16:26,880 --> 00:16:30,030 This seems to suggest that gravitational waves have 248 00:16:30,030 --> 00:16:31,290 no effect. 249 00:16:31,290 --> 00:16:32,850 I have a small body here. 250 00:16:32,850 --> 00:16:35,780 Gravitational waves come along. 251 00:16:35,780 --> 00:16:36,943 It's unaccelerated. 252 00:16:43,898 --> 00:16:44,690 So what's this for? 253 00:16:47,840 --> 00:16:51,620 Well, if you think about this for a second, what you should 254 00:16:51,620 --> 00:16:54,308 convince yourself is that the calculation I just did, 255 00:16:54,308 --> 00:16:55,850 it's probably the first one you would 256 00:16:55,850 --> 00:16:57,225 think of doing when you're trying 257 00:16:57,225 --> 00:16:59,360 to ascertain how does a body move 258 00:16:59,360 --> 00:17:01,040 in a particular space-time. 259 00:17:01,040 --> 00:17:04,890 But, in some ways, it was kind of a dumb calculation to do. 260 00:17:04,890 --> 00:17:12,750 So let's bear in mind, what this is doing 261 00:17:12,750 --> 00:17:17,339 is tell me the geodesic equation describes motion 262 00:17:17,339 --> 00:17:20,910 with respect to some given coordinate system. 263 00:17:37,950 --> 00:17:40,110 Satisfying this equation is just saying, 264 00:17:40,110 --> 00:17:43,020 yup, you're following a geodesic. 265 00:17:43,020 --> 00:17:44,400 And guess what? 266 00:17:44,400 --> 00:17:48,330 These are-- this radiation is a form of gravity. 267 00:17:48,330 --> 00:17:51,990 Even if the body responds to it, it is still 268 00:17:51,990 --> 00:17:53,700 going to follow a geodesic. 269 00:17:53,700 --> 00:17:55,988 It's going to be in free fall. 270 00:17:55,988 --> 00:17:57,780 The gravitational field may be a little bit 271 00:17:57,780 --> 00:17:59,238 different from the ones that you're 272 00:17:59,238 --> 00:18:02,220 more used to from your previous experiences with gravity, 273 00:18:02,220 --> 00:18:06,090 but it is still just following the free fall trajectory 274 00:18:06,090 --> 00:18:07,470 that gravity demands. 275 00:18:15,230 --> 00:18:22,920 So free fall in these coordinates 276 00:18:22,920 --> 00:18:27,099 means the body is remaining fixed in these coordinates. 277 00:18:37,860 --> 00:18:40,620 This is one of the most important lessons 278 00:18:40,620 --> 00:18:41,550 in general relativity. 279 00:18:41,550 --> 00:18:44,340 When you ask a statement that depends 280 00:18:44,340 --> 00:18:47,100 on the coordinate system you've written down, 281 00:18:47,100 --> 00:18:48,870 you have to understand your coordinates. 282 00:18:48,870 --> 00:18:50,040 And I would submit we don't really 283 00:18:50,040 --> 00:18:52,680 understand the coordinates that we wrote down here with this. 284 00:18:52,680 --> 00:19:02,970 These coordinates essentially follows the body. 285 00:19:02,970 --> 00:19:08,470 Even if the body is moving, they essentially 286 00:19:08,470 --> 00:19:10,750 wiggle right along with the body itself. 287 00:19:26,570 --> 00:19:28,420 So if you want to understand what 288 00:19:28,420 --> 00:19:32,480 the impact of gravitational waves is on things, 289 00:19:32,480 --> 00:19:35,680 you have to try to formulate what you're 290 00:19:35,680 --> 00:19:38,860 able to measure using coordinate-independent language 291 00:19:38,860 --> 00:19:41,440 as much as possible. 292 00:19:41,440 --> 00:19:43,840 At the very least, you need to understand 293 00:19:43,840 --> 00:19:47,185 your coordinate system a lot better than I did in setting up 294 00:19:47,185 --> 00:19:48,310 this calculation over here. 295 00:19:59,400 --> 00:20:03,500 So let's reframe the way we do this analysis. 296 00:20:03,500 --> 00:20:05,750 Let's imagine-- so one of the things 297 00:20:05,750 --> 00:20:09,380 that we learned in thinking about how 298 00:20:09,380 --> 00:20:13,310 gravity works in general relativity is I can always, 299 00:20:13,310 --> 00:20:17,210 if I'm just focusing on a single body, a single point mass, 300 00:20:17,210 --> 00:20:20,060 I can always go into freely falling coordinates. 301 00:20:20,060 --> 00:20:22,910 And space-time is essentially the space-time 302 00:20:22,910 --> 00:20:24,810 of special relativity. 303 00:20:24,810 --> 00:20:27,530 I can find a representation so that everything looks just 304 00:20:27,530 --> 00:20:29,330 like special relativity near there. 305 00:20:29,330 --> 00:20:35,490 Gravity is basically completely thrown away, 306 00:20:35,490 --> 00:20:37,830 at least as far as the mathematical representation 307 00:20:37,830 --> 00:20:39,600 is concerned. 308 00:20:39,600 --> 00:20:40,700 I'm in free fall. 309 00:20:40,700 --> 00:20:41,700 I don't really see much. 310 00:20:41,700 --> 00:20:45,840 If I wanted to see the impact of these waves, what I need to do 311 00:20:45,840 --> 00:20:50,370 is think about how things behave over some finite region. 312 00:20:50,370 --> 00:20:56,850 I wanted to put together a setup such that I have two bodies, 313 00:20:56,850 --> 00:21:00,450 and the relative effect of things between those two bodies 314 00:21:00,450 --> 00:21:03,190 will allow me to see the effect of space-time curvature 315 00:21:03,190 --> 00:21:03,690 in action. 316 00:21:06,810 --> 00:21:09,660 So let's now-- with this in mind, 317 00:21:09,660 --> 00:21:21,624 let's consider two nearby bodies that each follow geodesics. 318 00:21:29,210 --> 00:21:34,560 So body A is right here. 319 00:21:34,560 --> 00:21:36,570 And I'm going to define body A as living 320 00:21:36,570 --> 00:21:38,462 at the origin of my coordinates. 321 00:21:42,320 --> 00:21:47,230 And we've got a body B over here. 322 00:21:47,230 --> 00:21:54,950 And we're going to say that this guy is displaced 323 00:21:54,950 --> 00:22:04,640 from the origin by a small amount, some epsilon, in what I 324 00:22:04,640 --> 00:22:07,250 will call the x direction. 325 00:22:07,250 --> 00:22:10,100 And what I want to do is say, OK, I've got these two guys. 326 00:22:10,100 --> 00:22:11,870 They are moving through a space-time. 327 00:22:11,870 --> 00:22:15,112 Let's make this be the space-time that we had just 328 00:22:15,112 --> 00:22:16,070 discussed a moment ago. 329 00:22:19,220 --> 00:22:24,950 So they are each moving on geodesics of this space time. 330 00:22:24,950 --> 00:22:28,250 We know that if I set up my coordinate system in the way 331 00:22:28,250 --> 00:22:30,740 that I've done it here, I am not going 332 00:22:30,740 --> 00:22:34,050 to see any acceleration on each of these bodies. 333 00:22:34,050 --> 00:22:40,070 Let's now try to focus on some kind of a quantity that 334 00:22:40,070 --> 00:22:43,210 tells me about the separation between these two. 335 00:22:43,210 --> 00:22:44,733 I'm going to do this in two ways. 336 00:22:44,733 --> 00:22:46,400 So the first thing which I'm going to do 337 00:22:46,400 --> 00:22:53,840 is say, let's imagine that I am bouncing a light pulse back 338 00:22:53,840 --> 00:23:42,730 and forth between A and B. What I'm going to do 339 00:23:42,730 --> 00:23:48,820 is write the 4-momentum associated with that light. 340 00:23:48,820 --> 00:23:53,340 So recall I can't really define a 4-velocity for light 341 00:23:53,340 --> 00:23:55,440 because proper time is not defined. 342 00:23:55,440 --> 00:24:03,270 But I can introduce some kind of an affine parameter 343 00:24:03,270 --> 00:24:07,050 such that d by d lambda tells me about motion on the world line 344 00:24:07,050 --> 00:24:08,582 that this light follows. 345 00:24:08,582 --> 00:24:11,040 And this is just bouncing back and forth between these two. 346 00:24:11,040 --> 00:24:14,430 So it moves in the x direction. 347 00:24:14,430 --> 00:24:23,950 Because this is light, p dot T equals 0. 348 00:24:45,860 --> 00:24:47,540 So these components, the 4-momentum 349 00:24:47,540 --> 00:24:50,360 associated with the light, the dt d lambda and the dx d 350 00:24:50,360 --> 00:24:54,440 lambda, they are related to each other like so. 351 00:24:54,440 --> 00:24:55,930 And look. 352 00:24:55,930 --> 00:24:58,190 The space-time metric, the gauge-invariant piece 353 00:24:58,190 --> 00:25:03,360 of the space-time, hxxtt, it is appearing in here. 354 00:25:03,360 --> 00:25:06,470 So let's say, OK, what I am interested now 355 00:25:06,470 --> 00:25:09,050 is computing what-- 356 00:25:09,050 --> 00:25:12,680 suppose I'm sitting at A, and I want 357 00:25:12,680 --> 00:25:17,000 to compute the time, according to A's clock, 358 00:25:17,000 --> 00:25:21,108 that it takes for a pulse of light to travel from A to B. 359 00:25:21,108 --> 00:25:22,400 Maybe I'll make it bounce back. 360 00:25:22,400 --> 00:25:23,942 Maybe I want to know how much time it 361 00:25:23,942 --> 00:25:30,510 takes for the light pulse to go from A to B and back to A. 362 00:25:30,510 --> 00:25:34,540 So let's turn this into an equation for the interval dt. 363 00:25:37,590 --> 00:25:42,540 So dt, I can do a little bit of this trick called division. 364 00:25:48,830 --> 00:25:54,580 Here is-- question is, did I do it correctly? 365 00:25:54,580 --> 00:25:55,080 Yes. 366 00:25:55,080 --> 00:25:56,705 So I'm doing the old physicist's trick. 367 00:25:56,705 --> 00:25:59,000 I'm just imagining that an interval of time-- 368 00:25:59,000 --> 00:26:00,650 an interval of affine parameter d 369 00:26:00,650 --> 00:26:03,050 lambda, the amount of dt that accumulates 370 00:26:03,050 --> 00:26:05,000 as it moves to an interval dx. 371 00:26:05,000 --> 00:26:06,330 They are related by so. 372 00:26:08,840 --> 00:26:11,480 And, remember, I am assuming that the perturbation 373 00:26:11,480 --> 00:26:13,280 from space-time, from flat space-time, 374 00:26:13,280 --> 00:26:16,340 is small enough that I can always linearize. 375 00:26:16,340 --> 00:26:21,790 So this can be-- 376 00:26:21,790 --> 00:26:24,493 I can take that square root using the binomial expansion. 377 00:26:27,880 --> 00:26:38,690 So suppose I want to integrate up the time it takes 378 00:26:38,690 --> 00:27:15,880 for the thing to go from A to B, and then back from B to A. 379 00:27:15,880 --> 00:27:19,500 Well, this is going to be given by integrating this guy's 380 00:27:19,500 --> 00:27:27,270 motion from 0 to epsilon, like so. 381 00:27:31,860 --> 00:27:33,530 And then integrating it back again. 382 00:27:38,420 --> 00:27:39,360 My tt is in here. 383 00:27:44,475 --> 00:27:46,550 It's going to switch direction when I do that, 384 00:27:46,550 --> 00:27:48,258 so I'm going to flip the sign on that dx. 385 00:28:18,850 --> 00:28:20,350 I'm leaving it written in this form, 386 00:28:20,350 --> 00:28:21,570 so you have to be a little bit careful. 387 00:28:21,570 --> 00:28:22,740 You guys will play with this a little bit more 388 00:28:22,740 --> 00:28:23,880 on a problem set. 389 00:28:23,880 --> 00:28:27,420 Bear in mind that as I move through x, the time argument 390 00:28:27,420 --> 00:28:29,160 of these guys is changing. 391 00:28:29,160 --> 00:28:31,210 And so that's where there's a little bit of work 392 00:28:31,210 --> 00:28:33,340 that you'll explore in a problem set. 393 00:28:33,340 --> 00:28:36,900 But I'm writing it in this form so that we get some-- 394 00:28:36,900 --> 00:28:39,240 naively, you might think, ah, I'll just flip the sign 395 00:28:39,240 --> 00:28:40,410 and I'll combine these. 396 00:28:40,410 --> 00:28:42,743 In a certain limit, that is indeed a good approximation. 397 00:28:42,743 --> 00:28:45,960 You got to be a little bit more careful in general though. 398 00:28:45,960 --> 00:28:49,290 This is the time it would take for light 399 00:28:49,290 --> 00:28:53,910 to go from A to B and back in the absence 400 00:28:53,910 --> 00:28:57,780 of this gravitational wave, this hxxtt. 401 00:28:57,780 --> 00:28:59,970 Notice there is an inference on the time 402 00:28:59,970 --> 00:29:03,120 of arrival of these pulses that depends on hxxtt. 403 00:29:12,495 --> 00:29:19,660 Time of arrival of the pulses depends 404 00:29:19,660 --> 00:29:26,650 on this particular piece of the space-time metric. 405 00:29:26,650 --> 00:29:29,440 One could imagine, if you had a precise clock, 406 00:29:29,440 --> 00:29:32,950 and you sat there and you timed the arrival of these things-- 407 00:29:32,950 --> 00:29:35,110 let's say you sent a pulse of light out-- bloop-- 408 00:29:35,110 --> 00:29:38,740 and you kept track of how long it took with a very precise 409 00:29:38,740 --> 00:29:41,578 clock, you could look for-- in the absence of a gravitational 410 00:29:41,578 --> 00:29:43,870 wave, that would be, if you imagine that your pulse was 411 00:29:43,870 --> 00:29:45,550 sent out perfectly periodically-- 412 00:29:45,550 --> 00:29:51,030 let's say every millisecond or something like that, 413 00:29:51,030 --> 00:29:52,890 you sent out a little pulse-- 414 00:29:52,890 --> 00:29:57,780 if it arrives back and your arrival pulses were just as 415 00:29:57,780 --> 00:30:00,420 spaced by 1 millisecond, you would say, great, 416 00:30:00,420 --> 00:30:02,730 I am sitting in empty space-time. 417 00:30:02,730 --> 00:30:09,980 But if you found that their arrivals varied, 418 00:30:09,980 --> 00:30:13,142 you would think, hmm, there's something else going on here. 419 00:30:13,142 --> 00:30:14,850 Gravitational waves is one of things that 420 00:30:14,850 --> 00:30:16,210 could lead to that variation. 421 00:30:19,870 --> 00:30:22,760 So this, at least, is kind of a proof-- 422 00:30:22,760 --> 00:30:24,330 this calculation I just did-- 423 00:30:24,330 --> 00:30:27,215 you guys are actually, on one of the upcoming problems sets-- 424 00:30:27,215 --> 00:30:28,215 I believe it's problem-- 425 00:30:28,215 --> 00:30:31,050 I can't remember if it's 7 or 8-- 426 00:30:31,050 --> 00:30:33,060 you do a little analysis where you 427 00:30:33,060 --> 00:30:36,990 will use this to understand how a detector like LIGO 428 00:30:36,990 --> 00:30:40,800 actually functions to measure out 429 00:30:40,800 --> 00:30:44,370 the effects due to a gravitational wave on the arms 430 00:30:44,370 --> 00:30:46,260 of a detector like this. 431 00:30:46,260 --> 00:30:52,110 Let me just do one more way of understanding 432 00:30:52,110 --> 00:30:56,610 how it is the gravitational wave leaves an imprint on these two 433 00:30:56,610 --> 00:30:58,145 separated particles. 434 00:31:03,490 --> 00:31:07,110 So when you look at that first naive calculation I did, 435 00:31:07,110 --> 00:31:10,650 the fact that I got no effect, I sort of 436 00:31:10,650 --> 00:31:13,620 joked that we did a dumb calculation. 437 00:31:13,620 --> 00:31:15,243 To be fair, it wasn't that dumb. 438 00:31:15,243 --> 00:31:17,910 What we're-- the first thing you usually think of doing when you 439 00:31:17,910 --> 00:31:20,410 want to understand the motion of a body in space-time, 440 00:31:20,410 --> 00:31:22,500 is you look at geodesics. 441 00:31:22,500 --> 00:31:27,390 Geodesics just tell me that this guy's acceleration with respect 442 00:31:27,390 --> 00:31:29,397 to the free fall frame is 0. 443 00:31:29,397 --> 00:31:31,230 And that's exactly what we ended up getting. 444 00:31:31,230 --> 00:31:33,990 It's basically just saying, ah, even in a gravitational wave, 445 00:31:33,990 --> 00:31:37,080 my two bodies are moving on geodesics. 446 00:31:37,080 --> 00:31:40,590 If I have two separate bodies, like I 447 00:31:40,590 --> 00:31:44,580 have in this calculation where I just calculated the variation 448 00:31:44,580 --> 00:31:47,610 in the time of arrival of light pulses, 449 00:31:47,610 --> 00:31:49,500 one of the tools we've learned is 450 00:31:49,500 --> 00:31:54,490 that I can have two geodesics, two free fall trajectories that 451 00:31:54,490 --> 00:31:55,720 deviate from one another. 452 00:31:58,250 --> 00:32:05,580 So let's look at the geodesic deviation of the free fall 453 00:32:05,580 --> 00:32:07,230 world lines of these two bodies. 454 00:32:21,960 --> 00:32:23,100 So I'm going to-- 455 00:32:23,100 --> 00:32:25,482 as I did when I did my geodesic calculation, 456 00:32:25,482 --> 00:32:26,940 I'm going to take them to initially 457 00:32:26,940 --> 00:32:29,940 be close enough that they have the same 4-velocity. 458 00:32:43,210 --> 00:32:47,280 And if you want to be careful about counting orders here, 459 00:32:47,280 --> 00:32:49,922 you can imagine, when the gravitation wave comes along, 460 00:32:49,922 --> 00:32:52,380 that one or both of them might pick up a correction of this 461 00:32:52,380 --> 00:32:54,030 that's of order h. 462 00:32:54,030 --> 00:32:56,280 Depends on what kind of coordinate system you pick up. 463 00:32:56,280 --> 00:32:58,450 But the key thing is that they start out like this. 464 00:32:58,450 --> 00:33:00,120 The gravitational waves may come along 465 00:33:00,120 --> 00:33:01,975 and it will move them around a little bit. 466 00:33:01,975 --> 00:33:03,600 But the amount it's going to move them, 467 00:33:03,600 --> 00:33:05,240 the amount of velocity that they're going to pick up, 468 00:33:05,240 --> 00:33:06,657 and the displacement they're going 469 00:33:06,657 --> 00:33:09,177 to get from their initial position if they're at rest, 470 00:33:09,177 --> 00:33:11,010 it can only be of order of the h that you're 471 00:33:11,010 --> 00:33:12,930 trying to measure there. 472 00:33:12,930 --> 00:33:16,530 So I'll remind you, the equation of deviation, 473 00:33:16,530 --> 00:33:28,150 equation of geodesic deviation, it 474 00:33:28,150 --> 00:33:32,725 defines a sort of proper acceleration acting on-- 475 00:33:35,930 --> 00:33:38,710 whoops, that's a typo. 476 00:33:38,710 --> 00:33:44,510 It is the proper acceleration on a vector xi 477 00:33:44,510 --> 00:33:48,290 that defines, in a precise geometric way, 478 00:33:48,290 --> 00:33:50,030 the separation of two geodesics. 479 00:34:02,430 --> 00:34:04,500 And what it looks like is the Riemann curvature 480 00:34:04,500 --> 00:34:10,080 of your space-time contracted on the 4-velocity of your bodies, 481 00:34:10,080 --> 00:34:15,239 and that separation vector xi. 482 00:34:15,239 --> 00:34:22,340 So, linearizing everything, bearing in mind 483 00:34:22,340 --> 00:34:31,330 that R will itself be of order h, 484 00:34:31,330 --> 00:34:34,870 and that any difference between the proper time 485 00:34:34,870 --> 00:34:38,500 along these things and the coordinate time 486 00:34:38,500 --> 00:34:41,860 of a particular clock is also going to be of order h, 487 00:34:41,860 --> 00:34:44,680 they are small enough that this equation becomes-- 488 00:35:01,370 --> 00:35:03,110 Something that looks like this. 489 00:35:03,110 --> 00:35:05,950 Notice I'm only looking at the spatial components 490 00:35:05,950 --> 00:35:08,680 of the separation vector xi. 491 00:35:08,680 --> 00:35:10,777 And that's because, since I'm working, 492 00:35:10,777 --> 00:35:12,610 since this is what my 4-velocity looks like, 493 00:35:12,610 --> 00:35:15,710 I'm going to pick out the 00 components of these things. 494 00:35:15,710 --> 00:35:18,010 And, don't forget, Riemann is antisymmetric 495 00:35:18,010 --> 00:35:23,320 when I exchange either indices 1 and 2 or indices 3 and 4. 496 00:35:23,320 --> 00:35:26,830 And because of that, the time-like component of this guy 497 00:35:26,830 --> 00:35:28,870 is going to be an uninteresting thing. 498 00:35:28,870 --> 00:35:31,510 I will just get that is equal to the acceleration 499 00:35:31,510 --> 00:35:33,190 of the time-like component is 0. 500 00:35:33,190 --> 00:35:35,470 So it doesn't really matter. 501 00:35:35,470 --> 00:35:45,810 So this is telling me that the geodesic separation 502 00:35:45,810 --> 00:35:47,880 of these two world lines is-- 503 00:35:47,880 --> 00:35:50,670 it's a second-order equation in time, 504 00:35:50,670 --> 00:35:53,310 and it's proportional to the separation itself. 505 00:35:53,310 --> 00:35:56,514 We need to work out all of these components. 506 00:36:04,060 --> 00:36:05,810 That is a fairly straightforward exercise, 507 00:36:05,810 --> 00:36:07,560 and I will just report to you the results. 508 00:36:20,490 --> 00:36:21,660 A brief aside. 509 00:36:21,660 --> 00:36:24,210 If you want to sort of verify the results I'm about to write 510 00:36:24,210 --> 00:36:26,730 out here, you could go into-- 511 00:36:26,730 --> 00:36:29,220 you could look up the definition of the Riemann tensor. 512 00:36:29,220 --> 00:36:32,670 Bear in mind that you are doing things at linear orders. 513 00:36:32,670 --> 00:36:35,410 You can throw away the terms of, like, connection times 514 00:36:35,410 --> 00:36:35,910 connection. 515 00:36:35,910 --> 00:36:38,160 It's just derivatives of the connection that are going 516 00:36:38,160 --> 00:36:39,930 to matter at linear order. 517 00:36:39,930 --> 00:36:43,800 But, even so, it's a bit of a grotty calculation. 518 00:36:43,800 --> 00:36:48,510 I am-- as I begin to reorganize the problem sets associated 519 00:36:48,510 --> 00:36:50,730 with this class under the new system 520 00:36:50,730 --> 00:36:53,250 that MIT is currently operating on, 521 00:36:53,250 --> 00:36:57,360 I will soon be releasing a few Mathematica notebooks that 522 00:36:57,360 --> 00:37:00,780 are very useful tools for doing tedious calculations, 523 00:37:00,780 --> 00:37:06,180 like computation of Riemann curvature tensor components. 524 00:37:06,180 --> 00:37:08,550 So I will put a few of those things up there. 525 00:37:08,550 --> 00:37:11,100 If you would like to validate some of the things 526 00:37:11,100 --> 00:37:14,670 that I am computing, that is where you will 527 00:37:14,670 --> 00:37:18,300 find a good way to test this. 528 00:37:18,300 --> 00:37:28,030 So, anyway, the non-zero Riemann components. 529 00:37:34,730 --> 00:37:40,378 So at linear order, you have R x0 x0. 530 00:37:40,378 --> 00:37:42,920 Bear in mind that, linear order, you are raising and lowering 531 00:37:42,920 --> 00:37:46,700 indices with the metric of flat space-time. 532 00:37:46,700 --> 00:37:49,930 So that's the same thing as this. 533 00:37:49,930 --> 00:37:53,750 And this turns out to be minus 1/2 two time 534 00:37:53,750 --> 00:37:56,570 derivatives of hxx. 535 00:38:01,370 --> 00:38:04,972 Y0 y0-- that's your yo-yo component-- 536 00:38:14,130 --> 00:38:15,790 pretty much looks exactly the same, 537 00:38:15,790 --> 00:38:18,960 but it's two time derivatives of the yy piece. 538 00:38:18,960 --> 00:38:22,080 Remember though-- I've erased it a while ago-- 539 00:38:22,080 --> 00:38:25,590 there are some constraints that we found due to the fact 540 00:38:25,590 --> 00:38:29,370 that my original h mu nu has to be-- its spatial piece has 541 00:38:29,370 --> 00:38:31,440 to be traceless. 542 00:38:31,440 --> 00:38:35,160 In particular, we found hyy was minus hxx. 543 00:38:41,370 --> 00:38:43,717 So I can remove that additional function, 544 00:38:43,717 --> 00:38:45,300 and just leave things in terms of hxx. 545 00:38:53,890 --> 00:39:04,240 And our final non-zero component looks like this. 546 00:39:04,240 --> 00:39:08,290 All others are either 0 or related by a symmetry. 547 00:39:22,570 --> 00:39:26,290 So notice, although I've written down three different 548 00:39:26,290 --> 00:39:28,510 complements here, there's really only two, 549 00:39:28,510 --> 00:39:31,190 because the symmetry that's-- 550 00:39:31,190 --> 00:39:33,190 it's not a symmetry of Riemann, but the symmetry 551 00:39:33,190 --> 00:39:36,610 associated with the trace-free nature of hijtt 552 00:39:36,610 --> 00:39:38,780 made these two equal to each other. 553 00:39:38,780 --> 00:39:41,523 So this and this are really the only independent degrees 554 00:39:41,523 --> 00:39:42,190 of freedom here. 555 00:40:11,900 --> 00:40:12,820 All right. 556 00:40:12,820 --> 00:40:15,490 Let's plug this into the equation of geodesic deviation, 557 00:40:15,490 --> 00:40:19,745 this guy, and see what we get. 558 00:40:19,745 --> 00:40:23,680 And what we get is an equation governing 559 00:40:23,680 --> 00:40:26,260 the acceleration of the x component of deviation. 560 00:41:27,240 --> 00:41:27,740 Here it is. 561 00:41:30,923 --> 00:41:32,340 Let's make a couple of assumptions 562 00:41:32,340 --> 00:41:34,600 and try to understand what this is telling us. 563 00:41:34,600 --> 00:41:36,390 So what I'm going to do is-- let's 564 00:41:36,390 --> 00:41:41,370 assume this is-- the first assumption is necessitated 565 00:41:41,370 --> 00:41:48,103 by the idea that I can do linearized gravity. 566 00:41:48,103 --> 00:41:50,520 So I'm going to assume that these guys are all very small. 567 00:41:54,900 --> 00:42:00,180 And I will imagine that my displacement 568 00:42:00,180 --> 00:42:06,420 vectors have some piece which-- 569 00:42:06,420 --> 00:42:08,470 basically, it's time constant. 570 00:42:08,470 --> 00:42:09,315 It does not change. 571 00:42:13,910 --> 00:42:24,560 And that there's a bit that will be of order h that describes 572 00:42:24,560 --> 00:42:27,530 how the separation of my two bodies 573 00:42:27,530 --> 00:42:31,803 responds to this incoming field. 574 00:42:31,803 --> 00:42:34,220 So what I'm going to do, when you go and you plug this in, 575 00:42:34,220 --> 00:42:46,490 assume that this guy is constant in time, and this one varies. 576 00:43:00,840 --> 00:43:05,450 What I'm going to do is imagine I don't just have two bodies. 577 00:43:05,450 --> 00:43:07,070 Imagine I have a ring of bodies. 578 00:43:22,630 --> 00:43:27,550 So suppose I have got my coordinate axes here. 579 00:43:27,550 --> 00:43:31,480 Horizontal is x, vertical on the board is y. 580 00:43:31,480 --> 00:43:39,155 And so imagine I have a ring of bodies, like so. 581 00:43:39,155 --> 00:43:40,780 And let's consider the following limit. 582 00:43:40,780 --> 00:43:46,720 Let's imagine that my hxx is some function that 583 00:43:46,720 --> 00:43:57,440 is sinusoidally varying in time, and my hxy is 0. 584 00:43:57,440 --> 00:43:58,750 I just want to call that the-- 585 00:43:58,750 --> 00:44:00,500 as we saw, there's really only two degrees 586 00:44:00,500 --> 00:44:02,450 of freedom in this field. 587 00:44:02,450 --> 00:44:04,070 And in the way that we formulated it, 588 00:44:04,070 --> 00:44:07,737 it's going to encapsulated by these two functions, xx and xy. 589 00:44:07,737 --> 00:44:08,820 So let's just look at one. 590 00:44:08,820 --> 00:44:12,320 Let's just look at the influence of the hxx. 591 00:44:12,320 --> 00:44:16,040 So when you do this, you essentially 592 00:44:16,040 --> 00:44:19,940 have the second derivative of xi x 593 00:44:19,940 --> 00:44:24,290 looks like two derivatives of your x field, your xx field 594 00:44:24,290 --> 00:44:26,210 times xi x. 595 00:44:26,210 --> 00:44:29,270 And two derivatives, your sort of acceleration 596 00:44:29,270 --> 00:44:33,100 in the y direction, is minus those two derivatives 597 00:44:33,100 --> 00:44:35,030 of your field times xi y. 598 00:44:37,682 --> 00:44:39,140 So what this is going to do, if you 599 00:44:39,140 --> 00:44:42,420 think about the way this is going to act on these things, 600 00:44:42,420 --> 00:44:47,360 you're going to get a sinusoidal acceleration associated 601 00:44:47,360 --> 00:44:52,550 with the displacement of this little sea of freely 602 00:44:52,550 --> 00:44:56,330 falling bodies as the gravitational wave comes along. 603 00:44:56,330 --> 00:44:58,370 And it's going to do so in such a way 604 00:44:58,370 --> 00:45:01,220 that the displacement in the x direction 605 00:45:01,220 --> 00:45:03,830 does the opposite of what the displacement in the y direction 606 00:45:03,830 --> 00:45:04,740 is doing. 607 00:45:04,740 --> 00:45:15,060 So as time passes, you'll find that-- so, initially, 608 00:45:15,060 --> 00:45:19,100 let's say it stretches along x, but then squeezes along y. 609 00:45:22,890 --> 00:45:30,070 Then it's a sinusoid, so, a quarter of a cycle later, 610 00:45:30,070 --> 00:45:33,070 it's back to being a circle. 611 00:45:33,070 --> 00:45:40,700 And, a quarter of a cycle later, it stretches along y 612 00:45:40,700 --> 00:45:41,760 and squeezes along x. 613 00:45:52,080 --> 00:45:56,760 Let's consider the limit now where hxx is equal to 0, 614 00:45:56,760 --> 00:45:59,770 and hxy is some sinusoid. 615 00:46:24,100 --> 00:46:25,630 So I start out with my circle. 616 00:46:28,990 --> 00:46:30,428 Look at your equations over here. 617 00:46:30,428 --> 00:46:31,970 Think about what they're going to do. 618 00:46:31,970 --> 00:46:36,460 Now we're going to find that the change in x and the change in y 619 00:46:36,460 --> 00:46:38,420 is correlated. 620 00:46:38,420 --> 00:46:45,540 So this guy stretches it out into an ellipse 621 00:46:45,540 --> 00:46:48,670 along a 45-degree line. 622 00:46:48,670 --> 00:46:55,170 A quarter of a cycle later, it's back to being a circle. 623 00:46:55,170 --> 00:46:59,360 And a quarter of a cycle later, it does the same thing 624 00:46:59,360 --> 00:47:00,480 but with opposite sign. 625 00:47:05,090 --> 00:47:08,840 These two fields, my hxx and my hxy, 626 00:47:08,840 --> 00:47:11,830 are what we call the fundamental polarizations 627 00:47:11,830 --> 00:47:12,890 of a gravitational wave. 628 00:47:19,130 --> 00:47:21,980 Polarization states are only defined with respect 629 00:47:21,980 --> 00:47:26,610 to a particular set of basis vectors. 630 00:47:26,610 --> 00:47:29,840 So if I make my basis vectors be this x and y, 631 00:47:29,840 --> 00:47:34,420 we call this one, for reasons that I hope are obvious, 632 00:47:34,420 --> 00:47:35,920 the plus polarization. 633 00:47:40,890 --> 00:47:45,300 The gravitational wave acts as a tidal stretch and squeeze. 634 00:47:45,300 --> 00:47:46,620 Notice what's going on here. 635 00:47:46,620 --> 00:47:49,830 I stretch along this axis, squeeze along this one. 636 00:47:49,830 --> 00:47:53,070 A quarter of a cycle later, I squeeze along this one, 637 00:47:53,070 --> 00:47:54,180 stretch along that one. 638 00:47:58,470 --> 00:48:05,890 So I get a tidal stretch and squeeze, 639 00:48:05,890 --> 00:48:08,410 where the main directions along which the stretch 640 00:48:08,410 --> 00:48:11,080 and squeeze are happening is plus shaped 641 00:48:11,080 --> 00:48:12,310 with respect to these axes. 642 00:48:24,090 --> 00:48:28,937 This one is known as the cross polarization. 643 00:48:33,230 --> 00:48:41,310 You also have a tidal stretch and squeeze, 644 00:48:41,310 --> 00:48:43,200 but the lines of force in this case 645 00:48:43,200 --> 00:48:46,870 are along axes that have an X shape. 646 00:49:18,430 --> 00:49:22,300 So at the level of 8.962, this is 647 00:49:22,300 --> 00:49:26,710 all we're going to say about the way that these things act on-- 648 00:49:26,710 --> 00:49:29,380 how gravitational waves act and what their observables are. 649 00:49:29,380 --> 00:49:32,980 And I deliberately chose to wear my LIGO today, 650 00:49:32,980 --> 00:49:38,110 because what LIGO does is look for these tidal stretches 651 00:49:38,110 --> 00:49:39,400 and squeezes. 652 00:49:39,400 --> 00:49:42,130 And the way it does so-- so, obviously, it's 653 00:49:42,130 --> 00:49:44,650 technologically not really feasible 654 00:49:44,650 --> 00:49:46,000 to set up a ring like this. 655 00:49:46,000 --> 00:49:49,533 But what we can do is sample the ring here and here. 656 00:49:49,533 --> 00:49:50,950 Actually, it's a little bit better 657 00:49:50,950 --> 00:49:52,990 to do on the plus polarization plot. 658 00:49:57,880 --> 00:50:00,790 So you make an experimental setup that 659 00:50:00,790 --> 00:50:02,825 samples the ring here and here. 660 00:50:02,825 --> 00:50:04,450 And if a gravitational wave comes along 661 00:50:04,450 --> 00:50:07,090 and it's lined up right, you see, ah, look at that. 662 00:50:07,090 --> 00:50:09,130 I'm getting that stretch and squeeze. 663 00:50:09,130 --> 00:50:12,820 The way you actually measure it out is essentially a timing 664 00:50:12,820 --> 00:50:16,060 experiment, exactly like this little calculation I did here. 665 00:50:16,060 --> 00:50:19,930 You bounce light between mirrors, 666 00:50:19,930 --> 00:50:25,390 and you very carefully time the round-trip travel time 667 00:50:25,390 --> 00:50:28,730 from one mirror to the other. 668 00:50:28,730 --> 00:50:32,590 And what's particularly cool is the experimental aspect 669 00:50:32,590 --> 00:50:34,040 of this. 670 00:50:34,040 --> 00:50:37,900 So how does one measure the time so precisely? 671 00:50:37,900 --> 00:50:41,448 Well, you use a laser beam as your clock. 672 00:50:41,448 --> 00:50:43,990 And what's beautiful about doing that is the way that you can 673 00:50:43,990 --> 00:50:45,940 actually sort of do the metrology at the level 674 00:50:45,940 --> 00:50:50,470 necessary to get the precision to measure your hxx's in here 675 00:50:50,470 --> 00:50:53,300 is by interference. 676 00:50:53,300 --> 00:50:55,510 So you can treat the incoming laser beam 677 00:50:55,510 --> 00:50:57,760 as defining your time standard, and then 678 00:50:57,760 --> 00:51:01,060 by interfering the light that has traveled down the arm 679 00:51:01,060 --> 00:51:05,090 and bounced back with that laser beam, 680 00:51:05,090 --> 00:51:08,620 basically the laws of nature allow 681 00:51:08,620 --> 00:51:10,840 you to automatically check and see 682 00:51:10,840 --> 00:51:15,010 whether there is a shift in the time of arrival associated 683 00:51:15,010 --> 00:51:18,310 with the action of the gravitational wave. 684 00:51:18,310 --> 00:51:21,860 So I imagine there will be some questions about this. 685 00:51:21,860 --> 00:51:24,987 There is a homework exercise that allows you to sort of test 686 00:51:24,987 --> 00:51:27,070 drive some of these concepts and allows you to see 687 00:51:27,070 --> 00:51:29,990 how it is that LIGO works. 688 00:51:29,990 --> 00:51:31,750 It really just sort of touches the-- it 689 00:51:31,750 --> 00:51:33,310 just scratches the surface of what 690 00:51:33,310 --> 00:51:36,430 an amazing experimental feat it is. 691 00:51:36,430 --> 00:51:40,928 But it should give you a good idea as to what goes into that. 692 00:51:40,928 --> 00:51:42,720 Now, there is an important part to all this 693 00:51:42,720 --> 00:51:45,210 that we've not talked about yet. 694 00:51:45,210 --> 00:51:48,660 I handed you hij. 695 00:51:48,660 --> 00:51:49,920 Really, I handed you h mu nu. 696 00:52:01,270 --> 00:52:03,580 The big thing that we care about next 697 00:52:03,580 --> 00:52:07,840 is, given a particular source, how do I 698 00:52:07,840 --> 00:52:10,060 compute this radiation field? 699 00:52:18,492 --> 00:52:21,900 So bearing in mind that hijtt, this 700 00:52:21,900 --> 00:52:25,380 is the gauge-invariant radiative degree 701 00:52:25,380 --> 00:52:31,810 of freedom in my space-time, how do I compute this guy? 702 00:52:37,080 --> 00:52:40,620 And what I'm going to do is borrow sort of one 703 00:52:40,620 --> 00:52:43,500 of the most important lessons of that gauge-invariant formalism. 704 00:53:02,010 --> 00:53:07,230 So we need to be cautious of the fact that, in some gauges-- 705 00:53:07,230 --> 00:53:08,790 for example the Lorentz gauge, which 706 00:53:08,790 --> 00:53:11,571 is so convenient for solving the Einstein equation-- 707 00:53:22,650 --> 00:53:30,100 components of the metric appear to be radiative whether they 708 00:53:30,100 --> 00:53:30,960 are or they aren't. 709 00:53:42,040 --> 00:53:44,440 This is a consequence of our gauge. 710 00:53:50,320 --> 00:54:05,470 But we also know only hijtt is radiation in all gauges. 711 00:54:09,950 --> 00:54:11,950 So that was what the previous lecture was about. 712 00:54:11,950 --> 00:54:13,960 As I've emphasized, it was a slightly advanced one. 713 00:54:13,960 --> 00:54:15,970 I don't expect everyone to follow all details. 714 00:54:15,970 --> 00:54:19,690 But if you can sort of grok these two main concepts, 715 00:54:19,690 --> 00:54:23,120 you're ready to now exploit this in the following way. 716 00:54:28,760 --> 00:54:32,925 What you do is, step a, compute-- 717 00:54:36,380 --> 00:54:43,160 find some metric perturbation in any convenient gauge 718 00:54:43,160 --> 00:54:44,390 by any convenient method. 719 00:55:05,580 --> 00:55:10,200 Once you have that, extract hijtt. 720 00:55:15,280 --> 00:55:15,780 You're done. 721 00:55:33,857 --> 00:55:35,440 So that's what we're going to do right 722 00:55:35,440 --> 00:55:40,960 now in the last 20 or so minutes of this recorded lecture. 723 00:56:07,720 --> 00:56:10,750 So the way we're going to do it is let's 724 00:56:10,750 --> 00:56:13,420 take the linearized Einstein equation in Lorentz gauge. 725 00:56:31,000 --> 00:56:32,650 As we showed in-- 726 00:56:32,650 --> 00:56:36,750 I believe it was two lectures ago, 727 00:56:36,750 --> 00:56:38,470 the Einstein field equations in this case 728 00:56:38,470 --> 00:56:40,840 reduce to a flat space-time wave operator 729 00:56:40,840 --> 00:56:45,520 on the trace-reversed metric perturbation, 730 00:56:45,520 --> 00:56:47,650 being up to a factor of minus 16 pi 731 00:56:47,650 --> 00:56:52,000 G of the stress-energy tensor. 732 00:56:52,000 --> 00:56:55,540 And this has an exact solution. 733 00:57:21,488 --> 00:57:25,200 So I'll remind you of this notation. 734 00:57:25,200 --> 00:57:34,050 So this is the field at time t and spatial location x. 735 00:57:40,994 --> 00:57:43,060 x and t-- well, let's write it like this. 736 00:57:43,060 --> 00:57:47,385 t and x are my field point. 737 00:57:47,385 --> 00:57:48,760 That is where I measure my field. 738 00:57:54,320 --> 00:57:59,300 x prime is my source point. 739 00:57:59,300 --> 00:58:02,090 That is where I am looking at my particular contribution 740 00:58:02,090 --> 00:58:04,250 to the integral. 741 00:58:04,250 --> 00:58:06,710 As we're doing this calculation, there was a t prime, 742 00:58:06,710 --> 00:58:09,440 but the radiative Green's function turned that t prime 743 00:58:09,440 --> 00:58:10,730 into this retarded time. 744 00:58:10,730 --> 00:58:13,790 So what we are seeing is that the field at t 745 00:58:13,790 --> 00:58:16,970 depends upon what's going on at the source at t 746 00:58:16,970 --> 00:58:20,210 minus the interval of time it takes for information 747 00:58:20,210 --> 00:58:22,670 to propagate from x prime to x. 748 00:58:26,860 --> 00:58:29,860 So what I'm going to do is, first, I'm 749 00:58:29,860 --> 00:58:33,640 going to introduce a couple of approximations 750 00:58:33,640 --> 00:58:36,130 that reflect the reality of the conditions on which we 751 00:58:36,130 --> 00:58:39,160 typically evaluate this integral. 752 00:58:39,160 --> 00:58:42,820 And then we're going to talk about how to solve this thing, 753 00:58:42,820 --> 00:58:47,140 and, given that we have now solved it, how to project out 754 00:58:47,140 --> 00:58:50,320 the gauge-invariant degrees of freedom associated 755 00:58:50,320 --> 00:58:51,530 with the radiation. 756 00:59:05,740 --> 00:59:06,667 Whoops, pardon me. 757 00:59:06,667 --> 00:59:07,417 Let's put this up. 758 00:59:16,860 --> 00:59:19,470 So first is I introduce this approximation. 759 00:59:19,470 --> 00:59:23,280 It's worth bearing in mind that when you are calculating this, 760 00:59:23,280 --> 00:59:25,458 x minus x prime, which is kind of like the distance 761 00:59:25,458 --> 00:59:27,000 from a particular point in the source 762 00:59:27,000 --> 00:59:29,010 to where you're making your measurement, 763 00:59:29,010 --> 00:59:32,480 it is generally a lot larger than the size of the source. 764 00:59:46,165 --> 00:59:48,040 So the kind of situation we're thinking about 765 00:59:48,040 --> 00:59:51,530 is, here we are, making our measurements at some point x 766 00:59:51,530 --> 00:59:52,030 here. 767 00:59:55,320 --> 01:00:00,220 Here's my source, and here is a point x 768 01:00:00,220 --> 01:00:03,100 prime inside the source. 769 01:00:03,100 --> 01:00:04,750 So the thing to bear in mind is that, 770 01:00:04,750 --> 01:00:07,410 for the kind of calculations where that-- 771 01:00:07,410 --> 01:00:11,492 when one is actually doing this, x minus x prime-- 772 01:00:11,492 --> 01:00:13,450 well, if I think about the kind of measurements 773 01:00:13,450 --> 01:00:15,550 that LIGO is doing, for example, that 774 01:00:15,550 --> 01:00:17,200 is typically hundreds of millions 775 01:00:17,200 --> 01:00:20,380 to billions of light years. 776 01:00:20,380 --> 01:00:23,560 The size of the source itself is of order tens 777 01:00:23,560 --> 01:00:25,875 to hundreds of kilometers. 778 01:00:25,875 --> 01:00:28,480 Bit of a separation of length scales there, you would say. 779 01:00:28,480 --> 01:00:33,320 So what I'm going to do is approximate the solution 780 01:00:33,320 --> 01:00:38,373 of that integral, and, for pedantic sake, 781 01:00:38,373 --> 01:00:40,040 I'm going to use an approximately equal. 782 01:00:43,030 --> 01:00:47,490 What I'm going to do is replace x minus x prime with r. 783 01:00:47,490 --> 01:00:50,640 Let's just say that this distance here 784 01:00:50,640 --> 01:00:54,780 is effectively the same for all points in the source. 785 01:00:54,780 --> 01:00:56,280 I can pull that out of the integral. 786 01:01:11,480 --> 01:01:14,120 If you like, you can refine this. 787 01:01:14,120 --> 01:01:16,155 There's a more careful expansion. 788 01:01:16,155 --> 01:01:17,780 And if you've done any electrodynamics, 789 01:01:17,780 --> 01:01:19,155 you should be familiar with this. 790 01:01:25,270 --> 01:01:38,820 And it uses the fact that 1 over x minus x prime 791 01:01:38,820 --> 01:01:43,350 can be expanded in Legendre polynomials, 792 01:01:43,350 --> 01:01:45,403 in a sum of Legendre polynomials. 793 01:01:58,230 --> 01:02:00,270 So if you do that, you can introduce refinements 794 01:02:00,270 --> 01:02:02,250 to what I'm about to derive right here. 795 01:02:02,250 --> 01:02:05,657 What this does, then, is it defines a multipolar expansion. 796 01:02:22,240 --> 01:02:23,370 I am not going to do that. 797 01:02:23,370 --> 01:02:27,520 I'm just going to look at the very first term. 798 01:02:27,520 --> 01:02:28,950 And you can sort of think of that 799 01:02:28,950 --> 01:02:31,560 as saying that I am just doing the leading, the most 800 01:02:31,560 --> 01:02:34,530 important multipole. 801 01:02:34,530 --> 01:02:38,190 Additional multipoles beyond that are certainly important, 802 01:02:38,190 --> 01:02:41,580 and people who work in the field of gravitational radiation, 803 01:02:41,580 --> 01:02:44,670 we certainly do not neglect them. 804 01:02:44,670 --> 01:02:48,420 In fact, much of my current research 805 01:02:48,420 --> 01:02:54,360 is based on sort of high-order calculations associated 806 01:02:54,360 --> 01:02:57,560 with looking at behavior of some of those multipoles. 807 01:02:57,560 --> 01:03:02,170 This will be fine for initial pedagogical purposes. 808 01:03:02,170 --> 01:03:06,470 Next, we're going to use the fact that only the spatial-- 809 01:03:06,470 --> 01:03:06,970 whoops. 810 01:03:06,970 --> 01:03:08,512 That's not supposed to be an h tilde. 811 01:03:08,512 --> 01:03:10,162 My apologies. 812 01:03:10,162 --> 01:03:11,870 That's supposed to be h bar, because this 813 01:03:11,870 --> 01:03:13,018 is the trace-reverse thing. 814 01:03:20,680 --> 01:03:23,110 Weird times, folks. 815 01:03:23,110 --> 01:03:23,910 Where was I at? 816 01:03:23,910 --> 01:03:24,410 Yes. 817 01:03:24,410 --> 01:03:28,390 So I am only going to care about the spatial pieces of this. 818 01:03:28,390 --> 01:03:30,790 You might protest, hey, isn't it the spatial part 819 01:03:30,790 --> 01:03:33,070 of the metric, not the trace-reverse metric, 820 01:03:33,070 --> 01:03:33,903 that matters? 821 01:03:33,903 --> 01:03:34,945 And you would be correct. 822 01:03:38,260 --> 01:03:40,860 If you do this carefully, to be perfectly blunt, 823 01:03:40,860 --> 01:03:42,610 you have to sort of finish the calculation 824 01:03:42,610 --> 01:03:45,642 before you can kind of justify the step I'm about to make. 825 01:03:45,642 --> 01:03:48,100 And we're not going to have time to do that very carefully, 826 01:03:48,100 --> 01:03:50,590 but I'll make a comment about it. 827 01:03:50,590 --> 01:03:53,170 For now, it suffices to say we only care 828 01:03:53,170 --> 01:03:54,670 about the spatial components. 829 01:04:07,410 --> 01:04:11,170 So I'm going to take my alpha beta over to an ij. 830 01:04:25,760 --> 01:04:27,590 So I now have something that involves 831 01:04:27,590 --> 01:04:29,330 the integral of the spatial pieces 832 01:04:29,330 --> 01:04:32,870 of the stress-energy tensor over my source. 833 01:04:32,870 --> 01:04:35,100 And now I'm going to take advantage of an Easter egg 834 01:04:35,100 --> 01:04:39,330 that was hidden in an early problem set. 835 01:04:39,330 --> 01:04:41,760 Back on problem set 2, I asked you 836 01:04:41,760 --> 01:04:44,230 to prove a result that I called the tensor virial theorem. 837 01:04:53,370 --> 01:04:55,900 One of the reasons why I assigned that was that I knew I 838 01:04:55,900 --> 01:04:58,353 was going to need it in this lecture. 839 01:04:58,353 --> 01:04:59,770 And what the tensor virial theorem 840 01:04:59,770 --> 01:05:09,680 told us is that if I integrate that, 841 01:05:09,680 --> 01:05:16,867 this ends up being equivalent to 1/2 two time derivatives. 842 01:05:29,310 --> 01:05:33,500 It's equal to the integral of two time derivatives-- 843 01:05:33,500 --> 01:05:34,000 excuse me. 844 01:05:34,000 --> 01:05:36,730 It's two time derivatives of the integral of the 00 piece 845 01:05:36,730 --> 01:05:38,280 of my stress-energy tensor. 846 01:05:38,280 --> 01:05:43,223 So it's the second moment of that, xi prime, xj prime. 847 01:05:43,223 --> 01:05:45,890 This is great, because what I've got in the left-hand side here, 848 01:05:45,890 --> 01:05:47,554 that's exactly what my integral is. 849 01:05:51,260 --> 01:05:52,260 Actually, you know what? 850 01:05:52,260 --> 01:05:53,550 This is going to give me a result that's 851 01:05:53,550 --> 01:05:54,518 sufficiently clean. 852 01:05:54,518 --> 01:05:55,310 I want a new board. 853 01:06:02,040 --> 01:06:23,000 So when that is done, what I finally see 854 01:06:23,000 --> 01:06:29,300 is that my hij, trace reversed, is 855 01:06:29,300 --> 01:06:34,550 2 G over r two time derivatives of a tensor 856 01:06:34,550 --> 01:06:37,550 that I will call Iij. 857 01:06:44,220 --> 01:06:52,680 This is what I get when I integrate T 00, 858 01:06:52,680 --> 01:06:56,250 two moments of T 00. 859 01:06:56,250 --> 01:07:02,000 This is called the quadrupole moment of the source. 860 01:07:09,730 --> 01:07:12,220 The asterisk is because, as you guys will 861 01:07:12,220 --> 01:07:14,050 see on one of the problem sets, there's 862 01:07:14,050 --> 01:07:16,413 a slight tweak that goes in the exact definition 863 01:07:16,413 --> 01:07:17,830 of the quadrupole moment which has 864 01:07:17,830 --> 01:07:24,250 to do with the behavior of the trace of Iij. 865 01:07:24,250 --> 01:07:26,710 I will just leave this like so for now, 866 01:07:26,710 --> 01:07:30,880 and leave you guys to explore that on a problem set. 867 01:07:30,880 --> 01:07:32,110 So we're almost done. 868 01:07:32,110 --> 01:07:41,590 What I now need to do is pull out 869 01:07:41,590 --> 01:07:43,470 the transverse and traceless piece of this. 870 01:08:00,133 --> 01:08:02,050 Let's consider the transverse condition first. 871 01:08:31,210 --> 01:08:32,740 So, to do the transverse condition, 872 01:08:32,740 --> 01:08:34,615 I'm going to take advantage of the fact that, 873 01:08:34,615 --> 01:08:50,620 far from the source, box of h bar ij is equal to 0, 874 01:08:50,620 --> 01:08:52,870 because I'm in a region where the source is equal to 0 875 01:08:52,870 --> 01:08:54,617 when I'm far away from it. 876 01:08:54,617 --> 01:08:56,200 This suggests that what we ought to do 877 01:08:56,200 --> 01:08:59,229 is expand our solution plane waves. 878 01:09:13,960 --> 01:09:19,240 So what I can do is write h bar. 879 01:09:19,240 --> 01:09:21,880 And I'm going to choose my indices carefully here, 880 01:09:21,880 --> 01:09:23,830 call it jl. 881 01:09:23,830 --> 01:09:28,630 It's some amplitude that does not depend on space or time, 882 01:09:28,630 --> 01:09:37,510 e to the i omega t minus a wave vector dotted with x. 883 01:09:37,510 --> 01:09:41,950 So this amplitude may depend on omega, may depend on k. 884 01:09:41,950 --> 01:09:44,340 It does not depend on t or on x. 885 01:09:44,340 --> 01:09:47,529 And the reason I changed my indices from i and j 886 01:09:47,529 --> 01:09:50,100 to j and l, and I did not use the index k, 887 01:09:50,100 --> 01:09:53,297 is just to avoid confusion between square root of minus 1 888 01:09:53,297 --> 01:09:54,130 and the wave vector. 889 01:09:58,200 --> 01:10:01,052 To be absolutely careful, this is a single mode. 890 01:10:01,052 --> 01:10:03,510 If you like, maybe you should think about this as something 891 01:10:03,510 --> 01:10:04,950 that I have-- 892 01:10:04,950 --> 01:10:07,933 I'm doing a sum or an integral over omega and k 893 01:10:07,933 --> 01:10:09,600 in order to reconstruct the whole thing. 894 01:10:09,600 --> 01:10:14,340 This is just a single Fourier mode in the wave field. 895 01:10:14,340 --> 01:10:20,360 So if I want this to be transverse, 896 01:10:20,360 --> 01:10:25,890 that condition is that the divergence of this guy 897 01:10:25,890 --> 01:10:28,460 be equal to 0. 898 01:10:28,460 --> 01:10:34,890 But when I take the divergence of this, this becomes-- 899 01:10:34,890 --> 01:10:38,540 I pull down a factor of minus i, and I pull down 900 01:10:38,540 --> 01:10:41,870 the j-th component of my wave vector k. 901 01:10:54,930 --> 01:10:55,740 And I get my wave-- 902 01:10:55,740 --> 01:10:56,550 my Fourier term. 903 01:11:04,470 --> 01:11:06,540 That can be thought of as-- 904 01:11:06,540 --> 01:11:08,580 so we can clear out the factor of minus i. 905 01:11:38,330 --> 01:11:55,198 The transverse condition boils down to saying that kj-- 906 01:11:55,198 --> 01:11:55,698 whoops. 907 01:12:03,372 --> 01:12:04,080 No, that's right. 908 01:12:04,080 --> 01:12:05,800 That's right. 909 01:12:05,800 --> 01:12:10,450 kj contracted on the wave field is equal to 0. 910 01:12:10,450 --> 01:12:13,420 In other words, the wave field is 911 01:12:13,420 --> 01:12:17,230 orthogonal to the wave vector. 912 01:12:32,020 --> 01:12:33,790 If you have studied electrodynamics, 913 01:12:33,790 --> 01:12:36,800 this should be familiar to you. 914 01:12:36,800 --> 01:12:38,050 This is the same-- 915 01:12:38,050 --> 01:12:40,090 it's exactly the same as the idea 916 01:12:40,090 --> 01:12:42,130 that when I have an electromagnetic wave, 917 01:12:42,130 --> 01:12:46,060 my E field and my B field are orthogonal to the direction 918 01:12:46,060 --> 01:12:49,210 of propagation. 919 01:12:49,210 --> 01:13:01,298 Let's define a unit vector n sub j to be kj but just normalized. 920 01:13:07,280 --> 01:13:11,630 So my wave is propagating in the direction of the unit vector n. 921 01:13:14,655 --> 01:13:16,530 Go back to another Easter egg that was hidden 922 01:13:16,530 --> 01:13:17,760 on an earlier problem set. 923 01:13:17,760 --> 01:13:19,470 You guys studied projection tensors. 924 01:13:28,710 --> 01:13:41,250 If I introduce the projection tensor Pij, 925 01:13:41,250 --> 01:13:45,840 Pij will project any spatial vector. 926 01:13:45,840 --> 01:13:53,190 It will project out the components normal, orthogonal 927 01:13:53,190 --> 01:13:54,660 to the direction of propagation n. 928 01:14:23,470 --> 01:14:28,870 You can in fact think of Pij as defining the metric 929 01:14:28,870 --> 01:14:32,350 for the subspace that is orthogonal to n. 930 01:14:32,350 --> 01:14:38,090 So imagine that your wave is propagating in the n direction. 931 01:14:38,090 --> 01:14:40,090 There's a plane being carried along with it. 932 01:14:40,090 --> 01:14:43,690 The plane is orthogonal to n, and Pij 933 01:14:43,690 --> 01:14:47,470 is the metric you use to define spatial distances 934 01:14:47,470 --> 01:14:48,200 in that plane. 935 01:14:51,950 --> 01:14:57,980 Using Pij, we can now make my h trace-- excuse me. 936 01:14:57,980 --> 01:14:59,210 I can make it transverse. 937 01:15:33,520 --> 01:15:34,960 All I need to do-- 938 01:15:34,960 --> 01:15:38,380 I'll put one T on this to denote Transverse. 939 01:15:44,860 --> 01:15:46,690 Whoops. 940 01:15:46,690 --> 01:15:57,940 All I need to do is project each of its indices 941 01:15:57,940 --> 01:15:59,770 within that tensor. 942 01:15:59,770 --> 01:16:01,210 So we're almost there. 943 01:16:01,210 --> 01:16:04,435 It's transverse, but not yet trace free. 944 01:16:11,860 --> 01:16:15,370 So if I take the trace on this guy, 945 01:16:15,370 --> 01:16:18,580 taking the trace on this guy amounts to-- 946 01:16:18,580 --> 01:16:30,050 I am going to require that this be equal to 0. 947 01:16:33,610 --> 01:16:37,810 Now, suppose I have some tensor that is not trace free, 948 01:16:37,810 --> 01:16:39,670 and I want to remove its trace. 949 01:16:53,750 --> 01:16:55,500 I'm going to make this a two-index tensor. 950 01:17:02,130 --> 01:17:05,497 So I'm not going to prove the following result. 951 01:17:05,497 --> 01:17:06,955 You can easily verify for yourself. 952 01:17:10,230 --> 01:17:29,850 The formula for doing so is that aij trace free is aij minus 1 953 01:17:29,850 --> 01:17:41,410 over N akl gkl gij. 954 01:17:41,410 --> 01:17:51,590 G is the metric that describes the manifold 955 01:17:51,590 --> 01:17:52,730 in which this tensor lives. 956 01:18:07,340 --> 01:18:09,571 N is its dimension. 957 01:18:22,840 --> 01:18:26,390 Now, for us, the tensor field whose trace I want to remove 958 01:18:26,390 --> 01:18:26,890 is hijT. 959 01:18:52,860 --> 01:19:00,150 So hij bar T is the field whose trace I wish to remove. 960 01:19:00,150 --> 01:19:06,830 This lives in the two-dimensional manifold 961 01:19:06,830 --> 01:19:20,380 whose metric is the projection tensor Pij. 962 01:19:20,380 --> 01:19:22,910 And we are working in a space where raising and lowering 963 01:19:22,910 --> 01:19:24,910 indices, there's really no important distinction 964 01:19:24,910 --> 01:19:26,050 between the two of them, because we're 965 01:19:26,050 --> 01:19:27,258 working in linearized theory. 966 01:19:29,660 --> 01:19:47,110 So I remove the trace by taking this guy and subtracting off-- 967 01:19:47,110 --> 01:19:48,430 pardon me just one moment. 968 01:19:51,364 --> 01:19:52,342 Yes, sorry. 969 01:20:02,140 --> 01:20:03,270 Subtracting off like so. 970 01:20:03,270 --> 01:20:03,770 Pardon me. 971 01:20:03,770 --> 01:20:04,840 It should have a T on it. 972 01:20:12,680 --> 01:20:15,270 Plugging in my various definitions, 973 01:20:15,270 --> 01:20:24,620 this turns into hlk Pli Pkj minus 1/2. 974 01:20:42,062 --> 01:20:44,270 So you might want to go and check some algebra there, 975 01:20:44,270 --> 01:20:45,000 but this is all-- 976 01:20:45,000 --> 01:20:47,760 I'm very confident in this. 977 01:20:47,760 --> 01:20:50,940 There's an overall factor of h bar lk. 978 01:20:50,940 --> 01:20:52,440 And the last thing which I will note 979 01:20:52,440 --> 01:20:56,080 is that, once I have removed the trace, 980 01:20:56,080 --> 01:21:00,350 this bar becomes meaningless. 981 01:21:00,350 --> 01:21:03,715 It's trace free, so the trace reversing of it has no action. 982 01:21:03,715 --> 01:21:05,840 Now, you should be a little bit careful about that. 983 01:21:05,840 --> 01:21:07,640 There's a few different notions of trace 984 01:21:07,640 --> 01:21:09,620 that went into this definition. 985 01:21:09,620 --> 01:21:12,650 Without proof, and just stating that by being a little bit more 986 01:21:12,650 --> 01:21:14,090 careful of some of the definitions 987 01:21:14,090 --> 01:21:17,990 we put down, the difference between-- 988 01:21:17,990 --> 01:21:19,640 when you count this up, you'll find 989 01:21:19,640 --> 01:21:21,450 that you may be making a small error, 990 01:21:21,450 --> 01:21:24,830 but it is on the order of terms that are 991 01:21:24,830 --> 01:21:26,776 quadratic in the perturbation. 992 01:21:34,400 --> 01:21:37,520 Putting all these pieces together, 993 01:21:37,520 --> 01:21:41,030 we finally obtain our transverse and traceless 994 01:21:41,030 --> 01:21:42,395 metric perturbation. 995 01:21:55,273 --> 01:21:56,190 Which looks like this. 996 01:22:27,700 --> 01:22:31,370 This is a result known as the quadrupole formula 997 01:22:31,370 --> 01:22:32,450 for gravitational waves. 998 01:22:41,500 --> 01:22:43,630 This is-- so given-- 999 01:22:43,630 --> 01:22:47,050 just take a stock of the final result. 1000 01:22:47,050 --> 01:22:51,490 Suppose you have some dynamic gravitating source. 1001 01:22:51,490 --> 01:22:56,870 This is telling me, what I do is I compute the quadrupole moment 1002 01:22:56,870 --> 01:22:59,420 associated with that source. 1003 01:22:59,420 --> 01:23:02,840 I take two time derivatives. 1004 01:23:02,840 --> 01:23:05,420 I hit it with this combination of projection tensors, which 1005 01:23:05,420 --> 01:23:07,567 has to do-- that tells me things about the geometry 1006 01:23:07,567 --> 01:23:08,150 of the source. 1007 01:23:08,150 --> 01:23:10,372 That's saying that my source, the waves that 1008 01:23:10,372 --> 01:23:12,830 come from the source propagate along a particular direction 1009 01:23:12,830 --> 01:23:13,790 to me. 1010 01:23:13,790 --> 01:23:16,520 This picks out the components that 1011 01:23:16,520 --> 01:23:18,960 are transverse with respect to that direction 1012 01:23:18,960 --> 01:23:22,370 and are traceless, so that I don't have any gauge degrees 1013 01:23:22,370 --> 01:23:24,500 of freedom in what results. 1014 01:23:24,500 --> 01:23:28,100 Multiply by twice Newton's constant, divide by distance. 1015 01:23:28,100 --> 01:23:31,430 And this is a wave field that our friends 1016 01:23:31,430 --> 01:23:36,500 over in Building NW22 can now measure pretty much every day. 1017 01:23:36,500 --> 01:23:38,750 You're going to do a few homework exercises associated 1018 01:23:38,750 --> 01:23:39,250 with. 1019 01:23:39,250 --> 01:23:46,320 One thing which is worth commenting on before I move on 1020 01:23:46,320 --> 01:23:51,810 is, when you keep factors of c in this, 1021 01:23:51,810 --> 01:23:55,887 that G goes over to G over c to the-- 1022 01:23:55,887 --> 01:23:56,470 wait a minute. 1023 01:23:56,470 --> 01:23:58,342 Let me double-check that. 1024 01:23:58,342 --> 01:24:00,850 Is it G over c to the fourth or G over c squared? 1025 01:24:00,850 --> 01:24:03,210 Anyhow, it's G divided by c to some power. 1026 01:24:06,140 --> 01:24:08,390 Let's just say that it's missing factors of c. 1027 01:24:12,510 --> 01:24:14,180 I may comment on that in some notes 1028 01:24:14,180 --> 01:24:16,980 that I will add to the board. 1029 01:24:16,980 --> 01:24:18,980 And so that kind of tells you that this is going 1030 01:24:18,980 --> 01:24:23,930 to be a really small quantity. 1031 01:24:23,930 --> 01:24:26,570 Makes it very hard for these things to be measured. 1032 01:24:26,570 --> 01:24:29,530 Nonetheless, it turns out even though they have-- 1033 01:24:29,530 --> 01:24:32,960 if you think about the impact they have on the detector, 1034 01:24:32,960 --> 01:24:35,900 even though it's small, they carry a tremendous amount 1035 01:24:35,900 --> 01:24:37,760 of energy. 1036 01:24:37,760 --> 01:24:39,590 I'm going to take a break before I begin 1037 01:24:39,590 --> 01:24:41,600 recording my next lecture. 1038 01:24:41,600 --> 01:24:43,400 But the next lecture is actually going 1039 01:24:43,400 --> 01:24:45,770 to lead us to an understanding of a different version 1040 01:24:45,770 --> 01:24:47,180 of the quadrupole formula. 1041 01:24:47,180 --> 01:24:50,000 It's based on this, but it describes the energy content 1042 01:24:50,000 --> 01:24:52,070 of gravitational waves. 1043 01:24:52,070 --> 01:24:56,232 Both of these formulas go by the name the quadrupole formula. 1044 01:24:56,232 --> 01:24:59,720 It's a little bit confusing sometimes if both are used. 1045 01:24:59,720 --> 01:25:01,850 And you're going to use both of them 1046 01:25:01,850 --> 01:25:03,990 on upcoming homework assignments. 1047 01:25:03,990 --> 01:25:04,490 All right. 1048 01:25:04,490 --> 01:25:07,270 So I will end this recording here.