1 00:00:00,000 --> 00:00:11,340 [SQUEAKING] [RUSTLING] [CLICKING] 2 00:00:11,340 --> 00:00:14,290 SCOTT HUGHES: All right, so in today's recorded lecture, 3 00:00:14,290 --> 00:00:17,260 I would like to pick up where we started-- 4 00:00:17,260 --> 00:00:17,760 excuse me. 5 00:00:17,760 --> 00:00:20,400 I'd like to pick up where we stopped last time. 6 00:00:20,400 --> 00:00:24,660 So I discussed the Einstein field equations 7 00:00:24,660 --> 00:00:26,010 in the previous two lectures. 8 00:00:26,010 --> 00:00:29,520 I derived them first from the method 9 00:00:29,520 --> 00:00:32,910 that was used by Einstein in his original work on the subject. 10 00:00:32,910 --> 00:00:36,960 And then I laid out the way of coming to the Einstein field 11 00:00:36,960 --> 00:00:38,940 equations using an action principle, 12 00:00:38,940 --> 00:00:42,360 using what we call the Einstein Hilbert action. 13 00:00:42,360 --> 00:00:50,920 Both of them lead us to this remarkably simple equation, 14 00:00:50,920 --> 00:00:56,730 if you think about it in terms simply of the curvature tensor. 15 00:00:56,730 --> 00:01:01,860 This is saying that a particular version of the curvature. 16 00:01:01,860 --> 00:01:03,470 You start with the Riemann tensor. 17 00:01:03,470 --> 00:01:05,600 You trace over two indices. 18 00:01:05,600 --> 00:01:09,620 You reverse the trace such that this whole thing 19 00:01:09,620 --> 00:01:10,760 has zero divergence. 20 00:01:10,760 --> 00:01:14,630 And you simply equate that to the stress energy tensor 21 00:01:14,630 --> 00:01:18,380 with a coupling factor with a complex constant 22 00:01:18,380 --> 00:01:20,120 proportionality that ensures that this 23 00:01:20,120 --> 00:01:22,430 recovers the Newtonian limit. 24 00:01:22,430 --> 00:01:24,950 The Einstein Hilbert exercise demonstrated 25 00:01:24,950 --> 00:01:29,270 that this is in a very quantifiable way, the simplest 26 00:01:29,270 --> 00:01:32,420 possible way of developing a theory of gravity 27 00:01:32,420 --> 00:01:34,528 in this framework. 28 00:01:34,528 --> 00:01:36,070 The remainder of this course is going 29 00:01:36,070 --> 00:01:38,780 to be dedicated to solving this equation, 30 00:01:38,780 --> 00:01:41,000 and exploring the properties of the solutions that 31 00:01:41,000 --> 00:01:42,578 arise from this. 32 00:01:42,578 --> 00:01:44,120 And so let me continue the discussion 33 00:01:44,120 --> 00:01:48,000 I began at the end of the previous lecture. 34 00:01:48,000 --> 00:01:50,120 We are going to find it very useful 35 00:01:50,120 --> 00:02:08,740 to regard this as a set of differential equations 36 00:02:08,740 --> 00:02:17,090 for the spacetime metric given a source. 37 00:02:17,090 --> 00:02:21,080 That, after all, is how we typically 38 00:02:21,080 --> 00:02:24,080 think of solving for fields given a particular source. 39 00:02:24,080 --> 00:02:26,080 And just pardon me while I make sure this is on. 40 00:02:26,080 --> 00:02:26,600 It is. 41 00:02:29,330 --> 00:02:31,190 I give you a distribution of mass. 42 00:02:31,190 --> 00:02:33,350 You compute the Newtonian gravitational potential 43 00:02:33,350 --> 00:02:33,850 for that. 44 00:02:33,850 --> 00:02:36,650 I give you a distribution of currents and fields. 45 00:02:36,650 --> 00:02:38,720 You calculate the electric and magnetic fields 46 00:02:38,720 --> 00:02:40,100 that arise from that. 47 00:02:40,100 --> 00:02:43,220 So I give you some distribution of mass and energy. 48 00:02:43,220 --> 00:02:45,820 You compute the spacetime that arises from that. 49 00:02:45,820 --> 00:02:47,690 But let's stop before we dig into this, 50 00:02:47,690 --> 00:02:52,940 and look at what this actually means 51 00:02:52,940 --> 00:02:55,460 given the mathematical equations that we have. 52 00:02:55,460 --> 00:03:01,310 So G alpha beta is the Einstein tensor. 53 00:03:01,310 --> 00:03:05,060 I construct it by taking several derivatives of the metric. 54 00:03:05,060 --> 00:03:06,770 I first make my Christoffel symbols. 55 00:03:06,770 --> 00:03:08,900 I combine those Christoffel symbols and derivatives 56 00:03:08,900 --> 00:03:11,600 of the Christoffel symbols to make the Riemann tensor. 57 00:03:11,600 --> 00:03:13,550 I hit it with another power of the metric 58 00:03:13,550 --> 00:03:16,157 in order to trace and get the Ricci tensor. 59 00:03:16,157 --> 00:03:18,740 I combine it with the trace of the Ricci tensor and the metric 60 00:03:18,740 --> 00:03:20,270 to get the Einstein. 61 00:03:20,270 --> 00:03:24,800 Schematically, I can think of G alpha beta 62 00:03:24,800 --> 00:03:29,420 as some very complicated linear-- 63 00:03:29,420 --> 00:03:32,600 excuse me, some very complicated nonlinear differential 64 00:03:32,600 --> 00:03:36,640 operator acting on the metric. 65 00:03:55,910 --> 00:03:58,760 So thinking about this is just a differential equation 66 00:03:58,760 --> 00:03:59,750 for the metric. 67 00:03:59,750 --> 00:04:03,360 The left hand side of this equation is a bit of a mess. 68 00:04:03,360 --> 00:04:05,680 Unfortunately, the right hand side can be a mess too. 69 00:04:05,680 --> 00:04:07,680 Let's think about this for a particular example. 70 00:04:07,680 --> 00:04:11,430 Suppose I choose as my force-- 71 00:04:11,430 --> 00:04:12,780 my source-- a perfect fluid. 72 00:04:21,000 --> 00:04:27,653 Well, then my right hand side is going 73 00:04:27,653 --> 00:04:30,070 to be something that involves the density and the pressure 74 00:04:30,070 --> 00:04:41,060 of that fluid, the fluid velocity, and then the metric. 75 00:04:41,060 --> 00:04:43,460 OK, so if I'm thinking about this as a differential 76 00:04:43,460 --> 00:04:45,860 equation for the metric, the metric 77 00:04:45,860 --> 00:04:49,080 is appearing under this differential operator 78 00:04:49,080 --> 00:04:51,030 on the left hand side, and explicitly 79 00:04:51,030 --> 00:04:52,980 in the source on the right hand side. 80 00:04:52,980 --> 00:04:55,710 Oh, and by the way, don't forget my fluid needs 81 00:04:55,710 --> 00:04:56,490 to be normalized. 82 00:04:56,490 --> 00:04:59,040 My fluid flow velocity needs to be normalized. 83 00:04:59,040 --> 00:05:06,560 So I have a further constraint, that the complement to the four 84 00:05:06,560 --> 00:05:10,370 velocity are related to each other by the spacetime metric. 85 00:05:10,370 --> 00:05:14,390 So if I am going to regard this as just a differential equation 86 00:05:14,390 --> 00:05:19,040 for the space time metric. 87 00:05:19,040 --> 00:05:22,840 In general, we're in for a world of pain. 88 00:05:22,840 --> 00:05:27,720 So as I described at the end of the previous lecture, 89 00:05:27,720 --> 00:05:31,370 we are going to examine how to solve this equation. 90 00:05:31,370 --> 00:05:33,650 In what's left of 8.962, we're going 91 00:05:33,650 --> 00:05:36,850 to look at three routes to solving this thing. 92 00:05:36,850 --> 00:05:42,300 The one that we will begin to talk about today 93 00:05:42,300 --> 00:05:48,280 is we solve this for what I will define a little bit more 94 00:05:48,280 --> 00:05:50,125 precisely in a moment as weak gravity. 95 00:05:55,670 --> 00:05:57,670 And what this is going to mean is that I only 96 00:05:57,670 --> 00:06:05,320 consider space times that are in a way that can be quantified 97 00:06:05,320 --> 00:06:11,480 close to flat space time. 98 00:06:11,480 --> 00:06:15,350 Method two will be to consider symmetric solutions. 99 00:06:23,460 --> 00:06:27,210 Part of the reason the general framework is so complicated 100 00:06:27,210 --> 00:06:30,330 is that there are in general, 10 of these 101 00:06:30,330 --> 00:06:32,850 coupled non-linear differential equations. 102 00:06:32,850 --> 00:06:34,530 When we introduce symmetries, or we 103 00:06:34,530 --> 00:06:37,950 consider things like static solutions 104 00:06:37,950 --> 00:06:41,730 or stationary solutions that don't have any time dynamics. 105 00:06:41,730 --> 00:06:43,742 That at least reduces the number of equations 106 00:06:43,742 --> 00:06:44,700 we need to worry about. 107 00:06:44,700 --> 00:06:49,590 They may still be coupled non-linear and complicated, 108 00:06:49,590 --> 00:06:51,428 but hopefully, maybe we can reduce 109 00:06:51,428 --> 00:06:52,970 those from 10 of these things we need 110 00:06:52,970 --> 00:06:56,280 to worry about to just a small number them, one, or two, 111 00:06:56,280 --> 00:06:58,350 or three. 112 00:06:58,350 --> 00:07:00,390 Makes it at least a little easier. 113 00:07:00,390 --> 00:07:05,340 In truth, symmetric solutions-- 114 00:07:05,340 --> 00:07:13,310 if you can then add techniques for perturbing around them, 115 00:07:13,310 --> 00:07:15,280 these turn out to be tremendously powerful. 116 00:07:20,910 --> 00:07:24,210 My own research career is-- 117 00:07:24,210 --> 00:07:26,370 uses this technique a tremendous amount. 118 00:07:28,960 --> 00:07:33,713 Finally, just basically say, you know what? 119 00:07:33,713 --> 00:07:35,380 Let's just dive in and solve this puppy. 120 00:07:41,150 --> 00:07:46,900 Just do a numerical solution of the whole monster, 121 00:07:46,900 --> 00:07:48,482 no simplifications. 122 00:07:55,100 --> 00:07:59,030 Eminent members of our field have dedicated entire careers 123 00:07:59,030 --> 00:08:00,920 to item three. 124 00:08:00,920 --> 00:08:03,260 I will give you an introduction to the main concepts 125 00:08:03,260 --> 00:08:05,660 in the last lecture. 126 00:08:05,660 --> 00:08:07,240 But it's not something we're going 127 00:08:07,240 --> 00:08:11,210 to be able to explore in much detail in this class. 128 00:08:11,210 --> 00:08:13,403 We're going to do one in a fair amount of detail. 129 00:08:13,403 --> 00:08:15,320 We will look at a couple of the most important 130 00:08:15,320 --> 00:08:16,820 symmetric solutions, so that you can 131 00:08:16,820 --> 00:08:21,315 see how these techniques work. 132 00:08:21,315 --> 00:08:22,690 And then in my last two lectures, 133 00:08:22,690 --> 00:08:24,020 I'm going to describe a little bit about what 134 00:08:24,020 --> 00:08:26,252 happens when you perturb some of the most interesting 135 00:08:26,252 --> 00:08:27,085 symmetric solutions. 136 00:08:27,085 --> 00:08:33,970 And we'll talk about numerical solutions for the general case. 137 00:08:33,970 --> 00:08:35,850 All right, so let's begin. 138 00:08:35,850 --> 00:08:37,370 We'll begin with choice one. 139 00:08:37,370 --> 00:08:44,900 Look at weak gravity, also known as 140 00:08:44,900 --> 00:08:47,156 linearized general relativity. 141 00:08:54,460 --> 00:08:57,550 So linearized general relativity is a situation 142 00:08:57,550 --> 00:09:00,100 in which we are only going to consider space times that 143 00:09:00,100 --> 00:09:01,360 are nearly flat. 144 00:09:15,650 --> 00:09:28,250 If I am in this situation, then I can choose coordinates, 145 00:09:28,250 --> 00:09:37,890 such that my space time metric is the metric of flat space 146 00:09:37,890 --> 00:09:49,120 time plus a tensor H-alpha beta, all of whose components-- 147 00:09:52,338 --> 00:09:54,380 so this notation that I'm sort of inventing here, 148 00:09:54,380 --> 00:09:57,320 double bars around H-alpha beta. 149 00:09:57,320 --> 00:10:07,830 This means the magnitude of H-alpha beta's components. 150 00:10:13,790 --> 00:10:18,770 These all must be much, much less than one. 151 00:10:18,770 --> 00:10:22,190 When you are in such a coordinate system 152 00:10:22,190 --> 00:10:28,070 you are in what we call nearly Lorenz coordinates. 153 00:10:39,040 --> 00:10:41,110 Such a coordinate system is as close 154 00:10:41,110 --> 00:10:43,510 to a globally inertial coordinate system 155 00:10:43,510 --> 00:10:45,610 as is possible to make. 156 00:10:45,610 --> 00:10:47,660 There are other coordinate choices we could make. 157 00:10:47,660 --> 00:10:50,050 So for instance, you're working in a system like that. 158 00:10:50,050 --> 00:10:52,000 This basically boils down to coordinates 159 00:10:52,000 --> 00:10:56,320 that are Cartesian like on their spatial slices. 160 00:10:56,320 --> 00:10:58,240 You could work in other ones. 161 00:10:58,240 --> 00:10:59,740 These are particularly convenient. 162 00:10:59,740 --> 00:11:02,680 Because for instance, if I work in a coordinate system whose 163 00:11:02,680 --> 00:11:05,145 spatial sector is spherical like, 164 00:11:05,145 --> 00:11:07,270 well, then there's going to be some components that 165 00:11:07,270 --> 00:11:12,940 grow very large as I go to large radius away from some-- 166 00:11:12,940 --> 00:11:15,280 the source of my gravitation. 167 00:11:15,280 --> 00:11:18,730 And this just makes my analysis quite convenient. 168 00:11:18,730 --> 00:11:22,180 In particular, where I'm going to take advantage of this. 169 00:11:28,030 --> 00:11:30,400 Whenever I come across a term that 170 00:11:30,400 --> 00:11:38,420 involves the perturbations squared or to a higher power, 171 00:11:38,420 --> 00:11:40,040 I'm just going approximate it as zero. 172 00:11:40,040 --> 00:11:43,850 I will always neglect terms beyond linear, 173 00:11:43,850 --> 00:11:53,510 hence the term linearized GR, in my analysis. 174 00:12:06,980 --> 00:12:08,630 Now, there are a couple of properties, 175 00:12:08,630 --> 00:12:14,030 before I get into how to develop weak gravity, linearized GR. 176 00:12:14,030 --> 00:12:17,390 I want to discuss a little bit some of the properties 177 00:12:17,390 --> 00:12:22,100 of spacetimes of this form. 178 00:12:22,100 --> 00:12:25,710 What are the particularly important properties of these? 179 00:12:25,710 --> 00:12:28,269 So let's consider coordinate transformations. 180 00:12:50,730 --> 00:12:54,870 My space time metric is a tensor, like any other. 181 00:12:54,870 --> 00:13:00,300 And so the usual rules pertain here, that I can change my 182 00:13:00,300 --> 00:13:09,360 coordinates using some matrix that 183 00:13:09,360 --> 00:13:11,670 relates my original coordinate system, which 184 00:13:11,670 --> 00:13:17,770 I've denoted without bars, to some new coordinate system that 185 00:13:17,770 --> 00:13:18,280 is barred. 186 00:13:23,780 --> 00:13:29,400 Now, recall that when we worked in flat space time, 187 00:13:29,400 --> 00:13:32,190 there was one category of coordinate transformations 188 00:13:32,190 --> 00:13:33,540 that was special. 189 00:13:33,540 --> 00:13:35,834 Those were the Lorentz transformations. 190 00:14:10,920 --> 00:14:15,110 So we are not working in flat space time. 191 00:14:15,110 --> 00:14:18,640 So on the face of it, we don't expect Lorentz transformations 192 00:14:18,640 --> 00:14:20,140 to play a particularly special role, 193 00:14:20,140 --> 00:14:23,260 except perhaps in the domain of a freely falling frame. 194 00:14:27,130 --> 00:14:28,210 But you know what? 195 00:14:28,210 --> 00:14:31,310 This is a nearly flat space time, so just for giggles, 196 00:14:31,310 --> 00:14:34,210 let's see what happens if you apply the Lorentz 197 00:14:34,210 --> 00:14:38,716 transformation to your nearly flat spacetime. 198 00:14:53,620 --> 00:15:05,510 So if I look at G mu bar nu bar as being 199 00:15:05,510 --> 00:15:20,740 a Lorentz transformation applied to my nearly flat metric. 200 00:15:20,740 --> 00:15:39,493 Well, what I get [INAUDIBLE] side is this. 201 00:15:39,493 --> 00:15:41,910 Now, one of the reasons why the Lorentz transformation was 202 00:15:41,910 --> 00:15:44,820 special in flat spacetime is that it 203 00:15:44,820 --> 00:15:49,740 leaves the metric of flat spacetime unchanged. 204 00:15:49,740 --> 00:15:57,480 And so this just maps to eta mu nu. 205 00:16:01,780 --> 00:16:05,850 I'm going to define what comes out here as h mu nu. 206 00:16:11,600 --> 00:16:14,030 I'm doing this in a fairly, I hope, obvious way. 207 00:16:23,190 --> 00:16:24,860 This is interesting. 208 00:16:24,860 --> 00:16:31,040 What this has told me is that when I apply the Lorentz 209 00:16:31,040 --> 00:16:35,600 transformation to my nearly flat space time, 210 00:16:35,600 --> 00:16:39,330 the background is unchanged. 211 00:16:39,330 --> 00:16:41,750 And the perturbation to the background 212 00:16:41,750 --> 00:16:45,260 transforms just like any tensor field 213 00:16:45,260 --> 00:16:47,900 would transform in flat space. 214 00:16:53,970 --> 00:16:57,270 Now, it should be emphasized, we are not working in flat space. 215 00:17:00,530 --> 00:17:03,620 We can compute curvature tensors. 216 00:17:03,620 --> 00:17:05,869 If we parallel transport-- 217 00:17:05,869 --> 00:17:09,410 If we consider two geodesics moving through the space time, 218 00:17:09,410 --> 00:17:12,150 we will see that parallel transport along those two 219 00:17:12,150 --> 00:17:14,150 geodesics, if they start out initially parallel, 220 00:17:14,150 --> 00:17:15,900 they do not remain parallel. 221 00:17:15,900 --> 00:17:18,630 So this is not flat space time. 222 00:17:18,630 --> 00:17:22,040 But in many ways, it's close enough 223 00:17:22,040 --> 00:17:25,579 that we can borrow many of the mathematical tools that 224 00:17:25,579 --> 00:17:28,220 were used in flat space time. 225 00:17:28,220 --> 00:17:31,910 In particular, we can introduce the following. 226 00:17:31,910 --> 00:17:39,990 Think of it as a useful fiction, which 227 00:17:39,990 --> 00:17:48,200 is that in this framework, we can regard 228 00:17:48,200 --> 00:17:51,260 the perturbation that pushes us away from flat 229 00:17:51,260 --> 00:18:04,090 spacetime as just an ordinary tensor field 230 00:18:04,090 --> 00:18:06,700 living in the manifold of special relativity, 231 00:18:06,700 --> 00:18:09,160 living in the eta alpha beta metric. 232 00:18:12,380 --> 00:18:16,990 It's worth bearing in mind in a fundamental sense, it's not. 233 00:18:16,990 --> 00:18:18,910 Space time is curved. 234 00:18:18,910 --> 00:18:22,240 h alpha beta is telling me about that curvature. 235 00:18:22,240 --> 00:18:24,700 But the mathematics works in such a way 236 00:18:24,700 --> 00:18:26,380 that you can borrow a lot of tools 237 00:18:26,380 --> 00:18:27,880 that we used in special relativity. 238 00:18:27,880 --> 00:18:30,490 And just imagine h as a tensor field 239 00:18:30,490 --> 00:18:33,760 living in that special relativity manifold. 240 00:18:39,530 --> 00:18:43,430 OK, so that's useful fact one that we 241 00:18:43,430 --> 00:18:51,470 want to bear in mind as we work in this nearly flat space time. 242 00:18:51,470 --> 00:18:57,600 Useful fact two is we want to think 243 00:18:57,600 --> 00:19:00,980 about what happens when we raise and lower indices. 244 00:19:15,260 --> 00:19:17,630 So suppose now that I'm going to regard 245 00:19:17,630 --> 00:19:19,940 h alpha beta as an ordinary tensor 246 00:19:19,940 --> 00:19:22,040 field living in this thing. 247 00:19:22,040 --> 00:19:24,470 I might want to know what it looks 248 00:19:24,470 --> 00:19:27,145 like with its indices raised. 249 00:19:27,145 --> 00:19:28,520 So I'm going to do what I usually 250 00:19:28,520 --> 00:19:33,478 do when I want to raise the indices on a tensor. 251 00:19:33,478 --> 00:19:34,520 I hit it with the metric. 252 00:19:38,190 --> 00:19:42,480 We're going to talk about what my upstairs metric looks 253 00:19:42,480 --> 00:19:43,590 like in just a moment. 254 00:19:43,590 --> 00:19:45,360 But clearly, it's going to be something 255 00:19:45,360 --> 00:19:49,440 that looks like the metric of flat space time plus terms 256 00:19:49,440 --> 00:19:51,600 that are on the order of h. 257 00:19:51,600 --> 00:19:57,420 Because I always drop terms of order h squared and higher, 258 00:19:57,420 --> 00:20:03,210 I can immediately say that this must simply 259 00:20:03,210 --> 00:20:10,410 be the metric of flat space time with the indices 260 00:20:10,410 --> 00:20:13,260 in the upstairs position acting on h. 261 00:20:13,260 --> 00:20:20,270 In other words, at least when I am 262 00:20:20,270 --> 00:20:24,830 acting on tensors that are built from the space time 263 00:20:24,830 --> 00:20:40,440 metric itself, I'm going to always want 264 00:20:40,440 --> 00:20:56,550 to raise and lower them using my flat space time, eta alpha 265 00:20:56,550 --> 00:20:57,870 beta. 266 00:20:57,870 --> 00:21:01,780 Bearing this in mind, let's carefully 267 00:21:01,780 --> 00:21:04,750 think about what the metric inverse actually looks like. 268 00:21:11,050 --> 00:21:15,550 I actually used this in one of my calculations 269 00:21:15,550 --> 00:21:16,820 in the previous lecture. 270 00:21:16,820 --> 00:21:19,935 And I said, I'm going to justify this in the next lecture. 271 00:21:19,935 --> 00:21:20,560 So here we are. 272 00:21:20,560 --> 00:21:22,850 Now we're going to justify it. 273 00:21:22,850 --> 00:21:25,346 So let's use this definition. 274 00:21:32,800 --> 00:21:35,700 So the upstairs metric is defined such 275 00:21:35,700 --> 00:21:38,160 that when it contracts with the downstairs metric, 276 00:21:38,160 --> 00:21:39,810 I get the identity back. 277 00:21:39,810 --> 00:21:46,590 This is the definition of the metric inverse. 278 00:21:46,590 --> 00:21:49,320 Working in linear theory, I know that this thing 279 00:21:49,320 --> 00:21:53,910 is going to be something like eta alpha beta 280 00:21:53,910 --> 00:21:57,450 plus a term of order h. 281 00:21:57,450 --> 00:21:59,700 I don't know what that term is yet, so let me just 282 00:21:59,700 --> 00:22:00,990 give it a new name. 283 00:22:00,990 --> 00:22:06,406 I'm going to call it m alpha beta. 284 00:22:06,406 --> 00:22:06,906 Whoops. 285 00:22:20,288 --> 00:22:21,830 Hopefully, the math will soon show me 286 00:22:21,830 --> 00:22:24,110 what this m actually is. 287 00:22:24,110 --> 00:22:27,335 It will be of order h, but as of yet, unknown. 288 00:22:56,500 --> 00:22:57,432 OK, so you know what? 289 00:22:57,432 --> 00:22:58,640 Let's rewrite that over here. 290 00:23:11,990 --> 00:23:14,320 So let's now multiply this guy out. 291 00:23:14,320 --> 00:23:18,340 Eta alpha beta hitting eta beta gamma. 292 00:23:18,340 --> 00:23:21,860 That gives me delta alpha gamma. 293 00:23:21,860 --> 00:23:23,770 m hitting eta. 294 00:23:23,770 --> 00:23:25,660 Now, remember what I just said over here. 295 00:23:28,360 --> 00:23:30,160 When I'm working with spacetime tensors, 296 00:23:30,160 --> 00:23:34,240 I always raise and lower indices using eta alpha beta. 297 00:23:34,240 --> 00:23:38,500 So when m alpha beta hits eta beta gamma, 298 00:23:38,500 --> 00:23:43,600 this is going to give me m alpha gamma. 299 00:23:43,600 --> 00:23:47,860 This eta hits that h. 300 00:23:47,860 --> 00:23:53,520 I get h, alpha upstairs, gamma downstairs. 301 00:23:53,520 --> 00:23:57,120 And then this guy is of order h, that guy is of order h. 302 00:23:57,120 --> 00:24:00,940 So additional terms of order h squared. 303 00:24:04,290 --> 00:24:06,860 So these guys cancel. 304 00:24:06,860 --> 00:24:11,690 And what I am left with is m alpha gamma 305 00:24:11,690 --> 00:24:16,940 equals negative h alpha gamma. 306 00:24:16,940 --> 00:24:20,400 I can raise my two indices, raise the gammas on both sides 307 00:24:20,400 --> 00:24:20,900 here. 308 00:24:20,900 --> 00:24:26,060 And we deduce from this that my inverse metric 309 00:24:26,060 --> 00:24:28,220 is eta alpha beta in the upstairs 310 00:24:28,220 --> 00:24:33,290 position, minus h alpha beta. 311 00:24:33,290 --> 00:24:34,940 At least two linear order in h. 312 00:24:45,560 --> 00:24:47,490 I'll just comment that what this is-- 313 00:24:47,490 --> 00:24:50,730 essentially, the matrix or tensor equivalent 314 00:24:50,730 --> 00:24:53,130 of a binomial expansion. 315 00:24:53,130 --> 00:24:57,450 1 over 1 plus epsilon is approximately 1 minus epsilon. 316 00:24:57,450 --> 00:24:59,390 That's all this is. 317 00:24:59,390 --> 00:25:01,380 But this is important to do. 318 00:25:01,380 --> 00:25:05,940 In fact, I will just sort of remark somewhat anecdotally 319 00:25:05,940 --> 00:25:08,640 that when I work with graduate students on projects, 320 00:25:08,640 --> 00:25:10,710 where we have to do things in linearized theory, 321 00:25:10,710 --> 00:25:13,410 getting a sign wrong here is one of the most common mistakes 322 00:25:13,410 --> 00:25:16,100 that people make. 323 00:25:16,100 --> 00:25:18,390 All right, one final detail. 324 00:25:45,440 --> 00:25:49,170 So this detail, as I just labeled it, 325 00:25:49,170 --> 00:25:51,100 is a particularly important one. 326 00:25:51,100 --> 00:25:55,815 We talked about general coordinate transformations. 327 00:25:55,815 --> 00:25:57,790 And I immediately went in to discuss a Lorentz 328 00:25:57,790 --> 00:25:59,500 transformation. 329 00:25:59,500 --> 00:26:03,040 There's a different category of coordinate transformation 330 00:26:03,040 --> 00:26:10,270 that plays a very important role in understanding 331 00:26:10,270 --> 00:26:13,893 the physics of systems that we analyze in linearized theory. 332 00:26:13,893 --> 00:26:15,310 So let's consider a different kind 333 00:26:15,310 --> 00:26:16,900 of coordinate transformation. 334 00:26:16,900 --> 00:26:27,790 Let's consider a coordinate shift, 335 00:26:27,790 --> 00:26:36,130 which I'm going to define by x alpha prime. 336 00:26:36,130 --> 00:26:37,930 Really, they should be x prime alpha, 337 00:26:37,930 --> 00:26:41,080 but you'll see why I need to name it this way right now. 338 00:26:41,080 --> 00:26:47,330 So this is my original coordinate, x alpha, 339 00:26:47,330 --> 00:26:52,490 plus some little offset that's a function of the coordinates x 340 00:26:52,490 --> 00:26:53,240 beta. 341 00:26:53,240 --> 00:26:57,740 So think of this as suppose I have a coordinate grid that 342 00:26:57,740 --> 00:26:58,550 looks like this. 343 00:27:04,550 --> 00:27:07,340 And I have some function that says, 344 00:27:07,340 --> 00:27:09,830 I want to consider a different system of coordinates that 345 00:27:09,830 --> 00:27:14,150 maybe pushes me a little bit along away from each of these 346 00:27:14,150 --> 00:27:16,848 coordinate lines in a way that varies 347 00:27:16,848 --> 00:27:18,140 as a function of position here. 348 00:27:21,960 --> 00:27:27,230 This notation is kind of an abuse of the indices. 349 00:27:27,230 --> 00:27:29,820 I am really not trying to define a coordinate invariant 350 00:27:29,820 --> 00:27:30,720 relationship here. 351 00:27:30,720 --> 00:27:33,360 I am just trying to connect two quantities, 352 00:27:33,360 --> 00:27:35,670 and I'm trying to connect quantities in two 353 00:27:35,670 --> 00:27:37,080 specific coordinate systems. 354 00:27:37,080 --> 00:27:41,220 And as we'll see, even though this is a bit ugly. 355 00:27:41,220 --> 00:27:44,005 It works well for what we want to do. 356 00:27:44,005 --> 00:27:45,630 So my coordinate transformation matrix. 357 00:28:03,060 --> 00:28:06,760 OK, so I just take the matrix of-- 358 00:28:06,760 --> 00:28:08,410 I developed the Jacobian-- my matrix 359 00:28:08,410 --> 00:28:11,410 of partial derivatives-- of the new coordinate 360 00:28:11,410 --> 00:28:14,350 with respect to the old coordinate in the usual way. 361 00:28:14,350 --> 00:28:18,530 This is going to be my first term. 362 00:28:18,530 --> 00:28:20,578 It's just a Kronecker delta. 363 00:28:20,578 --> 00:28:22,120 And then I'm going to get a term that 364 00:28:22,120 --> 00:28:28,450 looks like matrix of derivatives of the function that 365 00:28:28,450 --> 00:28:31,692 defines my infinitesimal-- defines my shift. 366 00:28:31,692 --> 00:28:33,400 I just gave away what I was about to say. 367 00:28:33,400 --> 00:28:36,190 I'm going to require [in?] my work in these nearly Lorentz 368 00:28:36,190 --> 00:28:39,190 coordinates, all of these entries need to be small. 369 00:28:44,000 --> 00:28:46,750 These will all be much, much less than 1. 370 00:28:46,750 --> 00:28:48,520 And so we call this an infinitesimal 371 00:28:48,520 --> 00:28:50,635 coordinate transformation. 372 00:29:08,980 --> 00:29:11,980 We are going to need to use the inverse of this guy. 373 00:29:24,570 --> 00:29:27,000 And using the definition of the inverse of this, 374 00:29:27,000 --> 00:29:30,580 saying essentially that when I take this, and I contract-- 375 00:29:30,580 --> 00:29:31,680 Let me put it this way. 376 00:29:31,680 --> 00:29:33,450 I'll just write it out. 377 00:29:33,450 --> 00:29:40,210 So if I compute this. 378 00:29:40,210 --> 00:29:43,860 I get my Kronecker delta back. 379 00:29:43,860 --> 00:29:47,250 Taking advantage of the smallness 380 00:29:47,250 --> 00:29:50,860 of the transformation. 381 00:29:50,860 --> 00:30:07,560 It's not terribly hard to demonstrate that 382 00:30:07,560 --> 00:30:09,570 what comes out of it is this. 383 00:30:18,810 --> 00:30:21,720 The minus sign is the key thing which I want to emphasize here. 384 00:30:21,720 --> 00:30:24,420 That minus sign is very similar to the minus sign 385 00:30:24,420 --> 00:30:25,218 that I have here. 386 00:30:25,218 --> 00:30:26,760 What we're doing is, again, just kind 387 00:30:26,760 --> 00:30:30,870 of the matrix equivalent of expanding 1 over 1 388 00:30:30,870 --> 00:30:32,490 plus epsilon for small epsilon. 389 00:30:35,880 --> 00:30:38,040 The reason that I am doing this is 390 00:30:38,040 --> 00:30:40,920 that I would now like to look at how 391 00:30:40,920 --> 00:30:45,458 the metric changes under this coordinate transformation. 392 00:30:55,720 --> 00:31:00,910 So what I'm going to do is define g mu nu 393 00:31:00,910 --> 00:31:02,230 in the new coordinate system. 394 00:31:11,650 --> 00:31:14,200 Usual operation. 395 00:31:14,200 --> 00:31:18,910 Let's now insert the many different definitions 396 00:31:18,910 --> 00:31:21,910 that we have introduced here. 397 00:31:21,910 --> 00:31:27,280 Notice that what I am using for my transformation matrix 398 00:31:27,280 --> 00:31:29,500 there is the inverse that I just wrote down. 399 00:31:45,290 --> 00:31:48,630 So let's fill that in. 400 00:31:48,630 --> 00:31:51,990 So I'm going to get a term that involves 401 00:31:51,990 --> 00:31:59,420 a Kronecker minus a matrix of partial derivatives. 402 00:31:59,420 --> 00:32:07,070 My other one gives me a nether Kronecker matrix 403 00:32:07,070 --> 00:32:09,230 of partial derivatives. 404 00:32:09,230 --> 00:32:13,400 And then finally, don't forget we 405 00:32:13,400 --> 00:32:17,150 are working in this nearly flat space time metric. 406 00:32:21,010 --> 00:32:23,260 And so I insert in my last term, eta alpha 407 00:32:23,260 --> 00:32:26,300 beta plus h alpha beta. 408 00:32:26,300 --> 00:32:30,620 So now, let's go and expand all of these terms out. 409 00:32:30,620 --> 00:32:32,440 My Kronecker, first, I get a term 410 00:32:32,440 --> 00:32:40,510 where both of the Kroneckers hit the metric of flat space time. 411 00:32:40,510 --> 00:32:42,810 So what I get is eta mu nu. 412 00:32:42,810 --> 00:32:45,240 Then I get a term which both the Kroneckers hit, 413 00:32:45,240 --> 00:32:48,400 the perturbation h alpha beta. 414 00:32:48,400 --> 00:32:50,740 Gives me h mu nu. 415 00:32:50,740 --> 00:32:52,720 Then, I'm going to get terms that 416 00:32:52,720 --> 00:32:59,200 involve these matrices of partial derivatives hitting 417 00:32:59,200 --> 00:33:00,450 the metric of flat space time. 418 00:33:02,980 --> 00:33:07,390 And what that's going to do is in keeping with our principle 419 00:33:07,390 --> 00:33:10,360 that when we're dealing with spacetime quantities, we raise 420 00:33:10,360 --> 00:33:14,200 and lower indices with eta. 421 00:33:14,200 --> 00:33:17,415 This is going to now give me-- 422 00:33:17,415 --> 00:33:18,540 pardon me just one moment-- 423 00:33:26,450 --> 00:33:28,863 a term that looks like partial-- 424 00:33:28,863 --> 00:33:30,530 everything in the downstairs position, d 425 00:33:30,530 --> 00:33:35,090 mu xi nu minus d nu xi mu. 426 00:33:35,090 --> 00:33:39,920 And then all the other terms are on the order 427 00:33:39,920 --> 00:33:47,000 of h times derivatives of the generators 428 00:33:47,000 --> 00:33:50,190 of my coordinate transmission. 429 00:33:50,190 --> 00:33:51,510 Small times small. 430 00:33:51,510 --> 00:33:52,870 These are infinitesimal squared. 431 00:33:52,870 --> 00:33:56,300 We are going to neglect them. 432 00:33:56,300 --> 00:33:58,970 Suppose that I insist that I have gone from one nearly 433 00:33:58,970 --> 00:34:01,127 flat spacetime to another. 434 00:34:01,127 --> 00:34:02,210 Bear in mind this picture. 435 00:34:02,210 --> 00:34:04,252 I'm just changing my representation a little bit. 436 00:34:04,252 --> 00:34:05,810 I've not changed the physics. 437 00:34:05,810 --> 00:34:13,639 So if I write this as eta mu prime nu prime, 438 00:34:13,639 --> 00:34:16,130 plus h mu prime nu prime. 439 00:34:16,130 --> 00:34:17,330 Well, I've got etas on both. 440 00:34:23,960 --> 00:34:25,480 The thing which is interesting is 441 00:34:25,480 --> 00:34:30,370 that I have generated a shift to my perturbation to the metric. 442 00:34:50,761 --> 00:34:52,219 Let's drop the primes for a second. 443 00:34:52,219 --> 00:34:54,420 And I'll just say that my nu, h mu 444 00:34:54,420 --> 00:35:09,980 nu is the old h mu nu minus the symmetrized combination 445 00:35:09,980 --> 00:35:12,560 of derivatives of-- 446 00:35:12,560 --> 00:35:16,430 the symmetrized combination of derivatives of infinitesimal 447 00:35:16,430 --> 00:35:17,813 coordinate transformation. 448 00:35:22,650 --> 00:35:25,260 Does this remind us of anything? 449 00:35:25,260 --> 00:35:33,870 This is starkly reminiscent of the way 450 00:35:33,870 --> 00:35:38,110 in which when we work with electromagnetic fields, 451 00:35:38,110 --> 00:35:48,480 I can take a potential, and shift it 452 00:35:48,480 --> 00:35:56,410 by the gradient of some scalar to generate a new potential. 453 00:35:56,410 --> 00:35:59,140 In so doing, what we find is that this 454 00:35:59,140 --> 00:36:03,370 leaves the fields unchanged. 455 00:36:03,370 --> 00:36:07,480 If you compute your Faraday tensor associated with this, 456 00:36:07,480 --> 00:36:09,320 it is unchanged. 457 00:36:12,250 --> 00:36:13,870 Similarly, we're going to write out 458 00:36:13,870 --> 00:36:16,240 the details of this in just a moment. 459 00:36:16,240 --> 00:36:31,840 When I generate the Riemann curvature from this, 460 00:36:31,840 --> 00:36:35,230 we find that although the metric has been tweaked a little bit 461 00:36:35,230 --> 00:36:37,210 by this coordinate transformation, 462 00:36:37,210 --> 00:36:39,370 Riemann is left unchanged. 463 00:36:55,970 --> 00:37:00,990 In acknowledgment of this, we call an infinitesimal 464 00:37:00,990 --> 00:37:04,020 coordinate transformation of this kind a gauge 465 00:37:04,020 --> 00:37:05,235 transformation. 466 00:37:40,440 --> 00:37:42,210 What the gauge transformation does 467 00:37:42,210 --> 00:37:44,910 is it allows us to change the metric, 468 00:37:44,910 --> 00:37:47,742 or change the way that we are representing our metric. 469 00:37:47,742 --> 00:37:49,200 And it's going to turn out to leave 470 00:37:49,200 --> 00:37:52,710 curvature tensors unchanged, in the same way that 471 00:37:52,710 --> 00:37:55,380 changing the potential and electrodynamics 472 00:37:55,380 --> 00:37:58,728 with a gauge transformation leaves our fields unchanged. 473 00:37:58,728 --> 00:38:01,020 And we're going to exploit this in exactly the same way 474 00:38:01,020 --> 00:38:02,700 that we exploit this in electrodynamics. 475 00:38:02,700 --> 00:38:05,490 We use this in electrodynamics in order 476 00:38:05,490 --> 00:38:08,790 to recast the equations governing our potentials 477 00:38:08,790 --> 00:38:11,640 into a form that is maximally convenient for whatever 478 00:38:11,640 --> 00:38:13,098 calculation we are doing right now. 479 00:38:13,098 --> 00:38:15,473 We're going to find-- and then we're going to derive this 480 00:38:15,473 --> 00:38:17,070 probably in about 20 minutes-- 481 00:38:17,070 --> 00:38:21,430 that the equations that govern h mu nu. 482 00:38:21,430 --> 00:38:23,210 If we leave things as general as possible, 483 00:38:23,210 --> 00:38:24,210 they're a bit of a mess. 484 00:38:24,210 --> 00:38:27,420 But by choosing the right gauge, we 485 00:38:27,420 --> 00:38:29,700 can simplify them, and wind up with a set 486 00:38:29,700 --> 00:38:32,640 of equations that are-- 487 00:38:32,640 --> 00:38:35,490 they cover all physical situations that matter, 488 00:38:35,490 --> 00:38:39,480 and that allow us to just cast things 489 00:38:39,480 --> 00:38:44,040 into a form that is much better for us to work with. 490 00:38:44,040 --> 00:38:45,420 All right. 491 00:38:45,420 --> 00:38:50,130 So we have now developed all of the sort of linguistics 492 00:38:50,130 --> 00:38:56,640 of linearized geometry that I want to use. 493 00:38:56,640 --> 00:38:58,560 Let's now go from linearized geometry 494 00:38:58,560 --> 00:39:01,410 to linearized gravity by running this through, and making 495 00:39:01,410 --> 00:39:02,740 some physics. 496 00:39:02,740 --> 00:39:05,730 What I want to do is look at the field 497 00:39:05,730 --> 00:39:08,033 equations in this framework. 498 00:39:11,350 --> 00:39:14,260 I am not going to run through every step of the next couple 499 00:39:14,260 --> 00:39:17,260 calculations. 500 00:39:17,260 --> 00:39:21,070 Doing so is a good illustration of the kind of calculation 501 00:39:21,070 --> 00:39:23,680 that a physicist likes to call straightforward but tedious. 502 00:39:23,680 --> 00:39:26,170 So I'm going to just write down what the results turned out 503 00:39:26,170 --> 00:39:27,590 to be. 504 00:39:27,590 --> 00:39:33,220 So let's run the metric through the machinery 505 00:39:33,220 --> 00:39:43,570 that we need to make all of our curvature tensors. 506 00:39:55,030 --> 00:39:58,350 OK, I'll remind you when we do this, we are linearizing. 507 00:39:58,350 --> 00:40:01,470 So anytime we see a term that looks like h squared, it dies. 508 00:40:01,470 --> 00:40:05,020 So we're only keeping things to linear order in h. 509 00:40:05,020 --> 00:40:14,270 So the first thing we find is the Riemann tensor turns 510 00:40:14,270 --> 00:40:22,280 into the following combination of partial derivatives 511 00:40:22,280 --> 00:40:23,840 of the metric perturbation h. 512 00:40:43,220 --> 00:40:47,300 In my notes, I have written out what 513 00:40:47,300 --> 00:40:54,230 happens when you switch from some original tensor h 514 00:40:54,230 --> 00:40:58,340 to a modified one using this gauge transformation. 515 00:40:58,340 --> 00:41:01,710 And what I show is that-- 516 00:41:01,710 --> 00:41:02,690 just a quick aside-- 517 00:41:08,520 --> 00:41:13,560 the gauge transformation generates 518 00:41:13,560 --> 00:41:21,772 a delta Riemann that looks like it's 519 00:41:21,772 --> 00:41:22,980 a whole bunch of-- let's see. 520 00:41:22,980 --> 00:41:23,813 Let's count them up. 521 00:41:23,813 --> 00:41:25,470 1, 2, 3, 4, 5, 6, 7, 8. 522 00:41:25,470 --> 00:41:26,785 You have eight terms. 523 00:41:26,785 --> 00:41:29,160 Of course there's eight, because there's four terms here, 524 00:41:29,160 --> 00:41:30,632 and you get two more for each one. 525 00:41:30,632 --> 00:41:33,090 So you're going to wind up with eight additional terms that 526 00:41:33,090 --> 00:41:35,610 involve three partial derivatives of the gauge 527 00:41:35,610 --> 00:41:36,547 generator. 528 00:41:48,740 --> 00:41:52,460 So they're of the form d cubed on xi. 529 00:41:52,460 --> 00:41:53,630 And it's not hard to show. 530 00:41:53,630 --> 00:41:54,920 You just sort of look at them. 531 00:41:54,920 --> 00:41:56,225 They cancel in pairs. 532 00:42:05,870 --> 00:42:08,970 And so delta Riemann is zero. 533 00:42:08,970 --> 00:42:13,352 The Riemann tensor is invariant to the gauge transformation. 534 00:42:17,420 --> 00:42:20,090 All right, we want to take this Riemann 535 00:42:20,090 --> 00:42:21,890 and use it to build the Einstein tensor. 536 00:42:21,890 --> 00:42:26,840 Our goal here is to make the field equation 537 00:42:26,840 --> 00:42:28,406 in linearized coordinates. 538 00:42:42,020 --> 00:42:47,540 So let's start by making the Ricci tensor. 539 00:42:47,540 --> 00:42:49,010 So we're going to raise and lower 540 00:42:49,010 --> 00:42:57,010 indices in linearized theory with the flat spacetime. 541 00:42:57,010 --> 00:43:07,690 So when we make this guy, what we get is this. 542 00:43:26,840 --> 00:43:29,960 I've introduced a couple of definitions here. 543 00:43:29,960 --> 00:43:31,940 One of them, you've seen before. 544 00:43:31,940 --> 00:43:36,980 The box operator is just a flat spacetime wave operator. 545 00:43:36,980 --> 00:43:42,900 And h with no indices is what I get 546 00:43:42,900 --> 00:43:50,885 when I trace over h using the flat background spacetime. 547 00:43:53,773 --> 00:43:54,690 And let's do one more. 548 00:44:01,520 --> 00:44:08,160 Evaluating r, I get one further contraction. 549 00:44:08,160 --> 00:44:17,140 And this turns out to be d alpha, d mu, h alpha mu 550 00:44:17,140 --> 00:44:21,760 minus box of h. 551 00:44:21,760 --> 00:44:24,670 So we now have all the pieces we need 552 00:44:24,670 --> 00:44:27,258 to make the Einstein tensor. 553 00:44:27,258 --> 00:44:28,800 So I'm going to write out the result. 554 00:44:28,800 --> 00:44:31,383 And then we're going to stop and just look at it for a second. 555 00:44:48,440 --> 00:44:56,360 Einstein is Ricci minus 1/2 metric Ricci scalar. 556 00:44:56,360 --> 00:44:58,790 Keeping things to leading order in h. 557 00:45:01,694 --> 00:45:05,060 This becomes flat spacetime metric going into there. 558 00:45:08,170 --> 00:45:10,800 So when you put all these ingredients together, 559 00:45:10,800 --> 00:45:14,550 there's an overall prefactor of 1/2. 560 00:45:14,550 --> 00:45:18,143 And then there are 1, 2, 3, 4, 5, 6 terms. 561 00:45:18,143 --> 00:45:19,060 Let me write them out. 562 00:45:51,980 --> 00:45:52,480 OK. 563 00:45:57,010 --> 00:46:00,020 So recall at the beginning of the lecture, 564 00:46:00,020 --> 00:46:04,030 I pointed out that when one regards G alpha beta 565 00:46:04,030 --> 00:46:06,980 as just a differential operator on the spacetime metric, 566 00:46:06,980 --> 00:46:09,130 it's kind of a mess. 567 00:46:09,130 --> 00:46:12,370 Bearing in mind that what I have here is a simplified 568 00:46:12,370 --> 00:46:14,120 version of that, I have discarded 569 00:46:14,120 --> 00:46:18,410 all of the terms that are higher order in h than linear. 570 00:46:18,410 --> 00:46:21,310 This is already pretty much a bloody mess as it is. 571 00:46:21,310 --> 00:46:23,200 So you can sort of see my point there. 572 00:46:23,200 --> 00:46:25,570 If this were done in its full generality, 573 00:46:25,570 --> 00:46:27,145 it would be kind of a disaster. 574 00:46:30,630 --> 00:46:36,930 Now, in linearized theory, there is a bit of sleight of hand 575 00:46:36,930 --> 00:46:39,880 that lets us clean this up a little bit. 576 00:46:39,880 --> 00:46:42,810 Let me emphasize that the next few lines of calculation I'm 577 00:46:42,810 --> 00:46:45,450 going to write down, there's nothing profound. 578 00:46:45,450 --> 00:46:46,920 All I'm going to do is show a way 579 00:46:46,920 --> 00:46:49,620 of reorganizing the terms, which simplifies 580 00:46:49,620 --> 00:46:51,130 this in an important way. 581 00:46:55,200 --> 00:46:58,670 So what we're going to do is define the following tensor. 582 00:46:58,670 --> 00:47:10,390 h bar is h minus 1/2 eta alpha beta h. 583 00:47:10,390 --> 00:47:13,450 So this is a good point to go, well, who ordered that? 584 00:47:13,450 --> 00:47:16,120 Let's take the trace of this. 585 00:47:16,120 --> 00:47:19,420 Let's define h bar with no indices 586 00:47:19,420 --> 00:47:22,840 is what I get when I trace on this. 587 00:47:22,840 --> 00:47:25,793 That's going to be the trace of h. 588 00:47:25,793 --> 00:47:27,460 This would be the trace of h alpha beta, 589 00:47:27,460 --> 00:47:35,200 so I just get h back, minus 1/2 h times the trace 590 00:47:35,200 --> 00:47:36,620 of eta alpha beta. 591 00:47:36,620 --> 00:47:38,260 And the trace of eta alpha beta. 592 00:47:38,260 --> 00:47:41,510 This is what I get when I raise one index, 593 00:47:41,510 --> 00:47:43,180 and sum over the diagonal. 594 00:47:43,180 --> 00:47:43,990 That is 4. 595 00:47:47,870 --> 00:47:51,620 So the trace of h bar is negative the trace of h. 596 00:47:57,150 --> 00:48:01,110 We call h bar alpha beta the trace reversed 597 00:48:01,110 --> 00:48:02,610 metric perturbation. 598 00:48:02,610 --> 00:48:05,070 It's got exactly the same information 599 00:48:05,070 --> 00:48:06,970 as my original metric perturbation, 600 00:48:06,970 --> 00:48:10,800 but I've just redefined a couple terms in order 601 00:48:10,800 --> 00:48:16,230 to give it a trace that has the opposite sign 602 00:48:16,230 --> 00:48:19,170 of the original perturbation h. 603 00:48:19,170 --> 00:48:22,020 The reason why this is useful is recall 604 00:48:22,020 --> 00:48:26,550 the Einstein tensor is itself the trace reversed Ricci 605 00:48:26,550 --> 00:48:28,200 tensor. 606 00:48:28,200 --> 00:48:30,540 What we're going to see is that if we-- 607 00:48:30,540 --> 00:48:33,870 in acknowledgment that it's sort of a trace reverse thing, 608 00:48:33,870 --> 00:48:36,900 if I plug in a trace reverse metric perturbation, 609 00:48:36,900 --> 00:48:39,520 a couple of terms are going to get cleaned up. 610 00:48:39,520 --> 00:48:40,600 So here's how we do this. 611 00:48:40,600 --> 00:48:46,560 So let's now insert h bar. 612 00:48:46,560 --> 00:48:48,820 This guy is going to be equal to-- 613 00:48:48,820 --> 00:48:51,190 oops, pardon me. 614 00:48:51,190 --> 00:48:53,970 Insert h. 615 00:48:53,970 --> 00:49:03,350 This guy is h bar plus 1/2 eta alpha beta h. 616 00:49:03,350 --> 00:49:05,360 So just move that to the other side. 617 00:49:05,360 --> 00:49:07,160 All I'm doing is taking the definition, 618 00:49:07,160 --> 00:49:10,070 and I am moving part of it to the other side, 619 00:49:10,070 --> 00:49:13,210 so that I can substitute in for h. 620 00:49:13,210 --> 00:49:14,830 When you plug this into here, you'll 621 00:49:14,830 --> 00:49:16,780 see that there are certain cancellations. 622 00:49:16,780 --> 00:49:21,130 In particular, every term that involves the trace of h, 623 00:49:21,130 --> 00:49:23,920 h without any indices, is canceled out. 624 00:49:38,870 --> 00:49:42,710 And so what you find doing this algebra is that your Einstein 625 00:49:42,710 --> 00:49:47,595 tensor turns into-- 626 00:49:58,607 --> 00:49:59,440 that can't be right. 627 00:50:21,490 --> 00:50:22,580 OK. 628 00:50:22,580 --> 00:50:25,700 So now, my Einstein tensor has no trace of h in it. 629 00:50:25,700 --> 00:50:28,760 Every h that appears on its right hand side 630 00:50:28,760 --> 00:50:30,890 is the tensor with both of the indices. 631 00:50:30,890 --> 00:50:35,020 But now, it's the trace reverse version of that. 632 00:50:35,020 --> 00:50:40,000 This is still a bit of a mess. 633 00:50:40,000 --> 00:50:44,810 Now, we're going to do something that's got a little bit more-- 634 00:50:44,810 --> 00:50:46,230 it's not just sleight of hand. 635 00:50:46,230 --> 00:50:48,340 This is something that's got a little bit more 636 00:50:48,340 --> 00:50:50,650 of sort of the meaning of some of these manipulations 637 00:50:50,650 --> 00:50:51,567 that we've worked out. 638 00:50:51,567 --> 00:50:55,540 It's going to play a role in helping us to understand this. 639 00:50:55,540 --> 00:51:02,250 Notice this term involves delta mu on h mu. 640 00:51:02,250 --> 00:51:07,380 Excuse me, partial mu on h mu, partial mu on h mu. 641 00:51:07,380 --> 00:51:11,540 Partial mu and partial nu on h mu. 642 00:51:11,540 --> 00:51:16,940 This is the only term that does not look like a divergence. 643 00:51:28,100 --> 00:51:29,920 Three of the terms in my Einstein tensor 644 00:51:29,920 --> 00:51:39,660 look like divergences of the trace reverse metric. 645 00:51:44,650 --> 00:51:47,845 Wouldn't it be nice if we could eliminate them somehow? 646 00:51:55,070 --> 00:51:57,140 Well, if you studied gauge transformations 647 00:51:57,140 --> 00:51:59,120 and electrodynamics, you'll note that there's 648 00:51:59,120 --> 00:52:01,470 something similar that is done. 649 00:52:01,470 --> 00:52:05,690 You can choose a gauge, such the divergence of the vector 650 00:52:05,690 --> 00:52:06,680 for potential vanishes. 651 00:52:12,170 --> 00:52:21,640 Can we set the divergence of this guy equal to zero? 652 00:52:21,640 --> 00:52:25,660 So if you look at this, mu is a dummy index. 653 00:52:25,660 --> 00:52:29,230 This is four conditions that we are trying to set. 654 00:52:29,230 --> 00:52:31,425 This has to happen for mu-- 655 00:52:31,425 --> 00:52:32,800 well, we're going to sum over mu. 656 00:52:32,800 --> 00:52:33,300 Pardon me. 657 00:52:33,300 --> 00:52:35,260 It's going to happen for nu equal time, 658 00:52:35,260 --> 00:52:38,710 and for my three spaces. 659 00:52:38,710 --> 00:52:40,503 These are four conditions. 660 00:52:48,240 --> 00:53:03,050 My gauge generators, my xi nu are four free functions. 661 00:53:12,420 --> 00:53:15,120 That suggests that the gauge generators give me 662 00:53:15,120 --> 00:53:19,410 enough freedom that I can adjust my gauge such 663 00:53:19,410 --> 00:53:22,950 that if I start out with some original, 664 00:53:22,950 --> 00:53:27,300 I have an h old that is not divergence free. 665 00:53:27,300 --> 00:53:29,430 Perhaps I can make an h new that is. 666 00:53:32,308 --> 00:53:33,100 Well, let's try it. 667 00:53:55,910 --> 00:53:58,250 So remember, I just erased it. 668 00:53:58,250 --> 00:54:00,470 But in fact, I'll just write it down right now. 669 00:54:00,470 --> 00:54:03,440 The shift to the metric perturbation arising 670 00:54:03,440 --> 00:54:05,990 from the gauge transformation, it's on h. 671 00:54:05,990 --> 00:54:07,520 We need to look at how it affects 672 00:54:07,520 --> 00:54:08,660 the trace reverse stage. 673 00:54:12,400 --> 00:54:16,320 So if I start with my new perturbation 674 00:54:16,320 --> 00:54:21,670 is related to my old perturbation as follows. 675 00:54:27,270 --> 00:54:36,760 It's not too hard to show that your trace reversed 676 00:54:36,760 --> 00:54:38,276 metric perturbation. 677 00:54:45,092 --> 00:54:46,050 Pardon, pardon, pardon. 678 00:54:49,200 --> 00:54:57,170 My trace reverse perturbation transforms 679 00:54:57,170 --> 00:54:59,235 in almost the exact same way. 680 00:54:59,235 --> 00:55:00,110 I get one extra term. 681 00:55:08,940 --> 00:55:11,660 So now, what I want to do is look 682 00:55:11,660 --> 00:55:15,710 at how the divergence of this transforms. 683 00:55:29,770 --> 00:55:32,980 So I'm going to get one term here, d mu of this. 684 00:55:32,980 --> 00:55:39,917 It gives me a wave operator acting on my gauge generator. 685 00:55:42,780 --> 00:55:53,022 And then I get another term here that looks like-- 686 00:55:53,022 --> 00:55:54,605 remember, partial derivatives commute. 687 00:55:54,605 --> 00:56:12,210 So you can think of this as d nu of the divergence of xi, eta mu 688 00:56:12,210 --> 00:56:20,090 nu acting on this changes this into d 689 00:56:20,090 --> 00:56:22,670 nu on the convergence of xi. 690 00:56:22,670 --> 00:56:26,220 And I messed up the sign, my apologies. 691 00:56:26,220 --> 00:56:29,330 That plus sign should have come down here. 692 00:56:29,330 --> 00:56:32,000 These are equal but opposite. 693 00:56:32,000 --> 00:56:32,570 They cancel. 694 00:56:35,850 --> 00:57:03,250 So let me just highlight the result. 695 00:57:03,250 --> 00:57:10,760 So what this tells me is if I choose my gauge generators just 696 00:57:10,760 --> 00:57:18,570 right, I can adjust my trace reverse metric, 697 00:57:18,570 --> 00:57:20,430 so that it is divergence free. 698 00:57:51,910 --> 00:57:58,970 If I do that, then the first three terms in my Einstein 699 00:57:58,970 --> 00:58:00,410 tensor here vanish. 700 00:58:11,320 --> 00:58:15,420 And if I do that, then here is my Einstein tensor. 701 00:58:23,790 --> 00:58:30,640 So just as in e and m, all that you need to do is say, 702 00:58:30,640 --> 00:58:32,650 I'm going to change my gauge such 703 00:58:32,650 --> 00:58:35,045 that the following condition holds. 704 00:58:38,530 --> 00:58:41,440 The condition that describes going 705 00:58:41,440 --> 00:58:43,900 into this gauge such that the divergence 706 00:58:43,900 --> 00:58:48,070 of your trace reverse perturbation vanishes. 707 00:58:48,070 --> 00:58:50,170 This is a simple wave equation. 708 00:58:50,170 --> 00:58:53,185 So solutions to this are guaranteed to exist. 709 00:59:16,783 --> 00:59:18,200 If you sit down and you ask, can I 710 00:59:18,200 --> 00:59:20,600 come up with some kind of a pathological spacetime, 711 00:59:20,600 --> 00:59:22,020 or a pathological-- 712 00:59:22,020 --> 00:59:22,870 no. 713 00:59:22,870 --> 00:59:24,530 Imagine I'm in some original spacetime 714 00:59:24,530 --> 00:59:27,992 sufficiently pathological that doesn't allow me to do this. 715 00:59:27,992 --> 00:59:29,450 If you do that, you're going end up 716 00:59:29,450 --> 00:59:32,450 violating the conditions that define weak spacetime. 717 00:59:32,450 --> 00:59:34,970 You can't do that in linearized gravity anyway. 718 00:59:34,970 --> 00:59:38,510 So in practice, we can always choose the gauge 719 00:59:38,510 --> 00:59:41,600 that puts in linearized gravity my Einstein 720 00:59:41,600 --> 00:59:44,490 tensor in this form. 721 00:59:44,490 --> 00:59:48,740 This form is exactly analogous to the Lorentz gauge 722 00:59:48,740 --> 00:59:51,580 condition that is used in electrodynamics. 723 00:59:51,580 --> 00:59:58,330 And so we call this Lorentz gauge in linearized gravity. 724 01:00:08,220 --> 01:00:18,940 Once we've done that, here's what my Einstein field 725 01:00:18,940 --> 01:00:19,890 equations turn into. 726 01:01:03,600 --> 01:01:10,325 In the next lecture, we will solve this exactly. 727 01:01:13,020 --> 01:01:15,920 See, what you want to emphasize is 728 01:01:15,920 --> 01:01:21,600 this is one of those situations where the answer is 729 01:01:21,600 --> 01:01:26,310 so easy and simple for us all to work out, 730 01:01:26,310 --> 01:01:29,710 we don't actually really need to even do that much calculation. 731 01:01:29,710 --> 01:01:38,190 I'll remind you that in electrodynamics, 732 01:01:38,190 --> 01:01:47,160 if you work in Lorentz gauge of electrodynamics, the wave 733 01:01:47,160 --> 01:01:59,020 equation that governs the electromagnetic potential 734 01:01:59,020 --> 01:02:00,828 turns out to be-- 735 01:02:00,828 --> 01:02:02,620 could be factors of c and things like that, 736 01:02:02,620 --> 01:02:04,810 depending on which units you're working in. 737 01:02:04,810 --> 01:02:07,720 But we find an equation that has exactly 738 01:02:07,720 --> 01:02:10,150 the same mathematical structure. 739 01:02:10,150 --> 01:02:11,900 Possibly there's a plus sign. 740 01:02:11,900 --> 01:02:14,530 I should have looked that up. 741 01:02:14,530 --> 01:02:18,620 Wave operator on my vector potential is a source. 742 01:02:18,620 --> 01:02:20,658 And this is very easily solved using 743 01:02:20,658 --> 01:02:22,450 what's called a radiative Green's function. 744 01:02:38,380 --> 01:02:40,670 I will discuss this in the next lecture. 745 01:02:40,670 --> 01:02:44,690 You can look up the details in any advanced electrodynamics 746 01:02:44,690 --> 01:02:45,190 textbook. 747 01:02:45,190 --> 01:02:48,640 Jackson has very nice discussion of this. 748 01:02:48,640 --> 01:02:50,230 I have an extra index. 749 01:02:50,230 --> 01:02:51,790 I have a different coefficient. 750 01:02:51,790 --> 01:02:56,240 But the mathematical structure is identical. 751 01:02:56,240 --> 01:02:58,700 So as far as linearized theory is concerned, 752 01:02:58,700 --> 01:03:01,270 we're basically done. 753 01:03:01,270 --> 01:03:03,580 So I'm going to talk about the exact solution of this 754 01:03:03,580 --> 01:03:05,740 in the next lecture. 755 01:03:05,740 --> 01:03:09,170 To wrap up today's lecture, to wrap up this current lecture. 756 01:03:09,170 --> 01:03:13,420 Let me look at the solution of this in a particular limit. 757 01:03:35,130 --> 01:03:41,560 So I'm going to take my source to be 758 01:03:41,560 --> 01:03:50,430 a static, non-relativistic, perfect fluid. 759 01:03:58,450 --> 01:04:02,800 The fact that it is static means that all of my time derivatives 760 01:04:02,800 --> 01:04:03,400 will be zero. 761 01:04:08,245 --> 01:04:09,620 And if that's true for my source, 762 01:04:09,620 --> 01:04:12,320 it has to be true for the field that arises from it. 763 01:04:12,320 --> 01:04:23,300 Non-relativistic tells me that the fluid density greatly 764 01:04:23,300 --> 01:04:26,160 exceeds its pressure. 765 01:04:26,160 --> 01:04:30,000 And as a consequence, I can write my stress energy 766 01:04:30,000 --> 01:04:38,990 tensor as approximately density four velocity four velocity. 767 01:04:38,990 --> 01:04:41,690 And when you go and you look at the magnitude of these things, 768 01:04:41,690 --> 01:04:42,830 I sort of looked at this a little bit 769 01:04:42,830 --> 01:04:43,830 in the previous lecture. 770 01:04:46,530 --> 01:04:50,828 T00 is approximately rho. 771 01:04:56,020 --> 01:04:59,560 All others will be negligible. 772 01:04:59,560 --> 01:05:01,780 Probably there's a small correction to this, 773 01:05:01,780 --> 01:05:04,090 but we can neglect that on a first pass. 774 01:05:28,110 --> 01:05:35,970 So my field equation is dominated by the zero zero 775 01:05:35,970 --> 01:05:36,570 component. 776 01:05:43,460 --> 01:05:46,640 That's going to be the most important piece of this. 777 01:05:46,640 --> 01:05:49,310 Since this is static, I can immediately say-- 778 01:05:53,150 --> 01:05:57,559 I can change that wave operator into the plus operator. 779 01:06:02,610 --> 01:06:07,020 And we now notice this is exactly the equation 780 01:06:07,020 --> 01:06:10,610 governing the Newtonian gravitational potential, 781 01:06:10,610 --> 01:06:11,880 modulo a factor of four. 782 01:06:25,020 --> 01:06:27,250 Pardon me, factor of minus four. 783 01:06:27,250 --> 01:06:32,360 And so we see from this that h bar zero zero 784 01:06:32,360 --> 01:06:35,700 is just negative 4 times the Newtonian gravitational 785 01:06:35,700 --> 01:06:36,659 potential. 786 01:06:44,810 --> 01:06:47,150 At this order in the calculation, 787 01:06:47,150 --> 01:06:51,620 all other contributions to the trace reverse metric are zero. 788 01:07:04,300 --> 01:07:07,510 OK, so let's go from the trace reverse metric back 789 01:07:07,510 --> 01:07:08,110 to the metric. 790 01:07:11,200 --> 01:07:19,050 We use the fact that h mu nu is the trace reversed h mu nu. 791 01:07:19,050 --> 01:07:22,493 If we trace reverse it, we'll get the original metric back. 792 01:07:22,493 --> 01:07:23,910 Basically, we trace reverse twice. 793 01:07:31,120 --> 01:07:32,440 So the trace of this guy. 794 01:07:46,900 --> 01:07:50,000 OK, and so putting all these ingredients together, 795 01:07:50,000 --> 01:07:54,700 what we see are that the only non-zero contributions here 796 01:07:54,700 --> 01:07:55,753 are 8 0 0. 797 01:07:55,753 --> 01:07:56,920 Let's do this one carefully. 798 01:07:56,920 --> 01:08:13,360 This is minus 4 times Newtonian potential minus 1/2 times 8 0 0 799 01:08:13,360 --> 01:08:15,070 and 4 times Newtonian potential. 800 01:08:15,070 --> 01:08:17,060 I have a c of minus sign here. 801 01:08:17,060 --> 01:08:22,630 This turns into minus 2 phi n. 802 01:08:22,630 --> 01:08:26,479 And h1 1 equals h2 2. 803 01:08:26,479 --> 01:08:29,060 h equals h3 3. 804 01:08:29,060 --> 01:08:37,290 This is going to be 0 minus 1/2 times 1 times 4 phi n. 805 01:08:47,870 --> 01:08:49,130 We put all these together. 806 01:09:19,250 --> 01:09:31,755 And what we get is-- 807 01:09:54,197 --> 01:09:55,030 And I'll remind you. 808 01:09:55,030 --> 01:09:58,990 This is a metric that I quoted in a previous lecture 809 01:09:58,990 --> 01:10:01,530 that I said we would prove in an upcoming one. 810 01:10:01,530 --> 01:10:02,860 Well, here it is. 811 01:10:02,860 --> 01:10:07,078 This is the Newtonian limit of general relativity. 812 01:10:12,230 --> 01:10:17,360 And it's worth remarking that this thing-- 813 01:10:17,360 --> 01:10:20,030 we are now in the very first lecture 814 01:10:20,030 --> 01:10:24,250 after having derived the Einstein field equations. 815 01:10:24,250 --> 01:10:28,940 20 years ago, almost all laboratory tests, laboratory 816 01:10:28,940 --> 01:10:31,700 and astronomical observational tests of general relativity 817 01:10:31,700 --> 01:10:34,880 essentially came from this based on. 818 01:10:34,880 --> 01:10:36,770 This ends up being the foundation 819 01:10:36,770 --> 01:10:38,390 of gravitational lensing. 820 01:10:38,390 --> 01:10:41,780 This is used to look at post-Newtonian corrections 821 01:10:41,780 --> 01:10:44,090 in our solar system. 822 01:10:44,090 --> 01:10:47,270 To a good approximation, it describes a tremendous number 823 01:10:47,270 --> 01:10:51,770 of binary systems that we see in our galaxy 824 01:10:51,770 --> 01:10:53,510 and in a few other galaxies. 825 01:10:53,510 --> 01:10:57,650 You really need to look for a much more extreme systems 826 01:10:57,650 --> 01:11:02,120 before the way in which the analysis changes 827 01:11:02,120 --> 01:11:07,430 due to going beyond linear order starts to become important. 828 01:11:07,430 --> 01:11:10,850 There is an upcoming homework exercise. 829 01:11:10,850 --> 01:11:17,398 And for students taking this course in spring of 2020, 830 01:11:17,398 --> 01:11:18,815 it remains to be determined how we 831 01:11:18,815 --> 01:11:20,750 are going to do problem sets at this point. 832 01:11:20,750 --> 01:11:23,330 I will be making a decision on that in coming days. 833 01:11:23,330 --> 01:11:26,870 But I want to tell you about an exercise on P 834 01:11:26,870 --> 01:11:33,000 set number 7, in which you do a variation of this calculation. 835 01:11:33,000 --> 01:11:39,200 So instead of just having a static body 836 01:11:39,200 --> 01:11:43,930 with a body who has massive density rho, 837 01:11:43,930 --> 01:11:45,360 consider a rotating body. 838 01:11:54,253 --> 01:11:55,670 And the thing which is interesting 839 01:11:55,670 --> 01:11:59,240 here is that in general relativity, 840 01:11:59,240 --> 01:12:04,580 all forms, all fluxes of energy and momentum 841 01:12:04,580 --> 01:12:07,280 contribute to gravity through the stress energy tensor. 842 01:12:07,280 --> 01:12:10,730 So if I have a body that is rotating about an axis, 843 01:12:10,730 --> 01:12:11,870 there's a mass flow. 844 01:12:11,870 --> 01:12:14,420 There are mass currents that arise. 845 01:12:14,420 --> 01:12:17,030 And what you find if you do this calculation correctly 846 01:12:17,030 --> 01:12:19,280 is that there is a correction to the spacetime that 847 01:12:19,280 --> 01:12:25,680 enters into here, which reflects the fact that a rotating 848 01:12:25,680 --> 01:12:27,990 body generates a unique contribution 849 01:12:27,990 --> 01:12:32,460 to the gravity that is manifested in this space time. 850 01:12:32,460 --> 01:12:34,860 Now, when one looks at the behavior 851 01:12:34,860 --> 01:12:36,510 of a body in a spacetime like the one 852 01:12:36,510 --> 01:12:38,550 I've written down right here. 853 01:12:38,550 --> 01:12:40,645 It's very reminiscent of-- 854 01:12:40,645 --> 01:12:42,270 well, it's just like Newtonian gravity. 855 01:12:42,270 --> 01:12:43,830 It's the Newtonian limit. 856 01:12:43,830 --> 01:12:46,740 And Newtonian gravity looks a lot like the Coulomb 857 01:12:46,740 --> 01:12:48,420 electric attraction. 858 01:12:48,420 --> 01:12:51,820 So this is often called a gravito electric field. 859 01:12:51,820 --> 01:12:53,445 People use that term, particularly when 860 01:12:53,445 --> 01:12:57,000 they're talking about linearized general relativity. 861 01:12:57,000 --> 01:13:00,180 If I have a rotating body, I now have mass currents 862 01:13:00,180 --> 01:13:02,280 flowing in this thing. 863 01:13:02,280 --> 01:13:04,530 And the correction to the spacetime 864 01:13:04,530 --> 01:13:07,900 that arises from this, it's qualitatively quite different 865 01:13:07,900 --> 01:13:08,400 from us. 866 01:13:08,400 --> 01:13:11,850 It doesn't have that simple gravito electric Coulombic type 867 01:13:11,850 --> 01:13:12,960 of form. 868 01:13:12,960 --> 01:13:16,480 It, in fact, looks a lot more like a magnetic field. 869 01:13:16,480 --> 01:13:18,570 And in fact, when you ask, how does 870 01:13:18,570 --> 01:13:22,500 this new term that is generated affect the motion of bodies? 871 01:13:22,500 --> 01:13:25,170 You find something that looks a lot like the magnetic Lorentz 872 01:13:25,170 --> 01:13:28,900 force law describing its motion. 873 01:13:28,900 --> 01:13:33,210 So this is a very, very powerful tool. 874 01:13:33,210 --> 01:13:34,620 But it's already not enough. 875 01:13:34,620 --> 01:13:37,770 So we can go a lot further than this. 876 01:13:37,770 --> 01:13:41,700 We have done so far, the simplest possible thing 877 01:13:41,700 --> 01:13:46,270 that we can do with this toolkit that we have derived so far. 878 01:13:46,270 --> 01:13:48,270 In the next lecture that I will record, 879 01:13:48,270 --> 01:13:51,570 we're going to return to my linearized Einstein field 880 01:13:51,570 --> 01:13:53,400 equations. 881 01:13:53,400 --> 01:13:57,660 And I am going to explore general solutions of this. 882 01:13:57,660 --> 01:14:03,540 This is going to lead us into a discussion of how 883 01:14:03,540 --> 01:14:07,060 things behave when my gravitational source is 884 01:14:07,060 --> 01:14:07,560 dynamic. 885 01:14:10,120 --> 01:14:13,320 I do not want to lose the time derivatives that are 886 01:14:13,320 --> 01:14:15,150 present in that wave operator. 887 01:14:15,150 --> 01:14:17,400 And so this is going to lead us quite naturally, then, 888 01:14:17,400 --> 01:14:20,550 to a discussion of gravitational radiation. 889 01:14:20,550 --> 01:14:22,480 And so after the next lecture, we'll 890 01:14:22,480 --> 01:14:25,350 spend a lecture or two discussing the nature 891 01:14:25,350 --> 01:14:26,910 of gravitational radiation. 892 01:14:26,910 --> 01:14:29,340 And there will be an upcoming homework assignment 893 01:14:29,340 --> 01:14:32,340 or two in which you explore the properties 894 01:14:32,340 --> 01:14:34,790 of gravitational radiation.