1 00:00:00,000 --> 00:00:11,400 [SQUEAKING] [RUSTLING] [CLICKING] 2 00:00:11,400 --> 00:00:12,280 SCOTT HUGHES: OK. 3 00:00:12,280 --> 00:00:14,353 Let's move on to the next lecture. 4 00:00:14,353 --> 00:00:16,770 Might have gotten a little bit of me getting prepped here. 5 00:00:16,770 --> 00:00:18,360 I really have to give a lot of credit 6 00:00:18,360 --> 00:00:22,560 to the people in the Office of Digital Learning for making 7 00:00:22,560 --> 00:00:24,870 this entire thing possible. 8 00:00:24,870 --> 00:00:28,590 So I'm really grateful particularly 9 00:00:28,590 --> 00:00:34,270 to Elaine Mello, who is at home running the show remotely right 10 00:00:34,270 --> 00:00:34,770 now. 11 00:00:34,770 --> 00:00:40,160 This is-- a lot of people at MIT are doing a tremendous amount 12 00:00:40,160 --> 00:00:43,940 to keep this place functioning, and everyone really 13 00:00:43,940 --> 00:00:47,060 deserves a huge pat on the back. 14 00:00:47,060 --> 00:00:47,570 All right. 15 00:00:47,570 --> 00:00:50,480 So in the lecture that I just completed a few moments ago, 16 00:00:50,480 --> 00:00:53,480 we described-- 17 00:00:53,480 --> 00:00:57,980 I described how to characterize the gauge-invariant radiative 18 00:00:57,980 --> 00:01:01,355 degrees of freedom, which we call gravitational radiation. 19 00:01:01,355 --> 00:01:04,400 With gravitational radiation, we found at the end-- 20 00:01:04,400 --> 00:01:06,830 there's a formula that I will write down again a little 21 00:01:06,830 --> 00:01:08,960 bit later in this lecture, but we essentially 22 00:01:08,960 --> 00:01:14,300 found that it looks like certain projections of a term that 23 00:01:14,300 --> 00:01:18,255 involves two time derivatives of the quadrupole moment 24 00:01:18,255 --> 00:01:18,755 of a source. 25 00:01:21,340 --> 00:01:26,500 This has in recent years led to a-- 26 00:01:26,500 --> 00:01:28,600 well, we've now been able to directly measure 27 00:01:28,600 --> 00:01:30,040 this form of radiation. 28 00:01:30,040 --> 00:01:35,528 I'm wearing my LIGO hat in honor of the facility that first 29 00:01:35,528 --> 00:01:36,820 directly measured these things. 30 00:01:36,820 --> 00:01:39,160 And we have inaugurated an entirely new field 31 00:01:39,160 --> 00:01:42,442 of observational astronomy based on being 32 00:01:42,442 --> 00:01:43,650 able to measure those things. 33 00:01:46,190 --> 00:01:48,160 It's pretty exciting. 34 00:01:48,160 --> 00:01:51,560 I hope, though, that a few people 35 00:01:51,560 --> 00:01:56,030 noticed how restricted the analysis that I did is. 36 00:01:56,030 --> 00:01:59,240 I defined gravitational waves only 37 00:01:59,240 --> 00:02:06,540 in the context of linearized theory on a flat background. 38 00:02:06,540 --> 00:02:09,335 We haven't talked much about other solutions 39 00:02:09,335 --> 00:02:13,880 to the Einstein field equations yet, but you know it's coming. 40 00:02:13,880 --> 00:02:18,440 And I hope you can appreciate that, in fact, our universe is 41 00:02:18,440 --> 00:02:21,620 not described by a flat background. 42 00:02:21,620 --> 00:02:25,250 Many of the sources that we study, 43 00:02:25,250 --> 00:02:28,045 it is only weakly curved away from a flat background. 44 00:02:28,045 --> 00:02:29,420 So for instance, our solar system 45 00:02:29,420 --> 00:02:31,837 is accurately described as something that is weakly curved 46 00:02:31,837 --> 00:02:33,910 away from a flat background. 47 00:02:33,910 --> 00:02:36,760 And in that context, much of what I described 48 00:02:36,760 --> 00:02:38,470 carries over very cleanly. 49 00:02:38,470 --> 00:02:40,510 And so you essentially have-- 50 00:02:40,510 --> 00:02:42,190 in that case, you need to-- 51 00:02:42,190 --> 00:02:44,440 if you want to analyze things like gravitational waves 52 00:02:44,440 --> 00:02:47,980 propagating in our solar system or through a galaxy which 53 00:02:47,980 --> 00:02:52,870 has a spacetime metric, it's also accurately described 54 00:02:52,870 --> 00:02:56,680 as flat spacetime plus a weak deviation away from it. 55 00:02:56,680 --> 00:02:58,183 You can take advantage of the fact 56 00:02:58,183 --> 00:02:59,600 that your gravitational wave tends 57 00:02:59,600 --> 00:03:02,260 to be something that's rapidly varying 58 00:03:02,260 --> 00:03:04,240 and your solar system and your galaxy 59 00:03:04,240 --> 00:03:06,550 is something that has very slow time variation, 60 00:03:06,550 --> 00:03:07,888 or it's even static. 61 00:03:07,888 --> 00:03:09,430 So you can take advantage of the fact 62 00:03:09,430 --> 00:03:10,805 that there's these different time 63 00:03:10,805 --> 00:03:13,690 derivatives to separate terms. 64 00:03:13,690 --> 00:03:15,310 That actually leads me naturally to, 65 00:03:15,310 --> 00:03:17,260 what do we do if I want to describe 66 00:03:17,260 --> 00:03:21,040 the very realistic scenario of gravitational waves propagating 67 00:03:21,040 --> 00:03:22,990 not on a flat background, but on some kind 68 00:03:22,990 --> 00:03:24,700 of a curved background? 69 00:03:24,700 --> 00:03:27,610 Perhaps they propagate near a black hole 70 00:03:27,610 --> 00:03:30,690 where spacetime is very different from the spacetime 71 00:03:30,690 --> 00:03:31,927 of special relativity. 72 00:03:31,927 --> 00:03:33,760 Perhaps we need to describe them propagating 73 00:03:33,760 --> 00:03:36,880 across a large sector of our universe. 74 00:03:36,880 --> 00:03:38,530 We haven't talked about cosmology yet, 75 00:03:38,530 --> 00:03:42,430 but it's the subject of the next lecture I intend to record. 76 00:03:42,430 --> 00:03:47,200 And the spacetime that describes our universe on large scales 77 00:03:47,200 --> 00:03:50,570 does not look like the metric of special relativity. 78 00:03:50,570 --> 00:04:00,680 So how do I describe gravitational radiation, GWs, 79 00:04:00,680 --> 00:04:07,750 Gravitational Waves, on a non-flat background? 80 00:04:15,150 --> 00:04:19,110 Schematically, you can imagine that in this case, 81 00:04:19,110 --> 00:04:22,440 you're working with a spacetime that can 82 00:04:22,440 --> 00:04:27,143 be broken into a background. 83 00:04:27,143 --> 00:04:29,310 I'll put a little hat on it to say that this denotes 84 00:04:29,310 --> 00:04:30,972 my background spacetime. 85 00:04:33,570 --> 00:04:35,850 And then some kind of a wave field propagating on it. 86 00:04:38,920 --> 00:04:41,890 This background is no longer going 87 00:04:41,890 --> 00:04:43,330 to be that of special relativity. 88 00:04:43,330 --> 00:04:45,520 This could be something that varies 89 00:04:45,520 --> 00:04:46,660 with both space and time. 90 00:04:58,970 --> 00:05:07,020 This will be something that in general is small in the sense 91 00:05:07,020 --> 00:05:10,320 that in most of the cases, certainly all the cases 92 00:05:10,320 --> 00:05:12,220 we're going to study in this class, 93 00:05:12,220 --> 00:05:14,220 any term that involves an h times an h 94 00:05:14,220 --> 00:05:17,910 will be second order in a very small number, 95 00:05:17,910 --> 00:05:20,123 and so we can discard it. 96 00:05:20,123 --> 00:05:22,290 But it's still going to be challenging to figure out 97 00:05:22,290 --> 00:05:27,240 how I can even define what a wave means in this case. 98 00:05:30,300 --> 00:05:31,930 So to wrap your heads around that, 99 00:05:31,930 --> 00:05:34,690 think about what a local measurement might do, OK? 100 00:05:34,690 --> 00:05:38,150 So suppose I am in some region of spacetime. 101 00:05:38,150 --> 00:05:40,422 So first of all, we had a parable 102 00:05:40,422 --> 00:05:42,130 that we looked at in our previous lecture 103 00:05:42,130 --> 00:05:45,490 where we looked at geodesics in a spacetime that was flat 104 00:05:45,490 --> 00:05:48,100 plus a gravitational wave. 105 00:05:48,100 --> 00:05:50,610 And of course, geodesics are geodesics, 106 00:05:50,610 --> 00:05:53,020 and so we just found that they are unaccelerated relative 107 00:05:53,020 --> 00:05:54,040 to freefall frames. 108 00:05:54,040 --> 00:05:57,040 That sort of is-- you know, more or less 109 00:05:57,040 --> 00:06:00,850 defines what a geodesic is, so it was not too surprising. 110 00:06:00,850 --> 00:06:03,250 But we can then look at, for example, two relatively 111 00:06:03,250 --> 00:06:04,960 nearby geodesics, and we can look 112 00:06:04,960 --> 00:06:07,300 at how their geodesic separation changes. 113 00:06:07,300 --> 00:06:08,860 We can look at the behavior of light 114 00:06:08,860 --> 00:06:11,050 as it bounces back and forth. 115 00:06:11,050 --> 00:06:12,860 Something that's important to bear in mind 116 00:06:12,860 --> 00:06:14,560 is that when I make a measurement like 117 00:06:14,560 --> 00:06:20,760 that, I measure this or some quantity that 118 00:06:20,760 --> 00:06:22,830 is derived from this, the curvature tensor 119 00:06:22,830 --> 00:06:26,490 or an integrated light propagation time. 120 00:06:36,710 --> 00:06:40,430 We may know that the spacetime metric 121 00:06:40,430 --> 00:06:44,398 has two vary conceptually different contributions to it. 122 00:06:44,398 --> 00:06:46,190 It's got a background, and it's got a wave. 123 00:06:52,480 --> 00:06:56,300 How do we-- conceptually, we might see how to separate them, 124 00:06:56,300 --> 00:07:03,790 but how do we define some kind of a toolkit that 125 00:07:03,790 --> 00:07:06,610 allows us to separate? 126 00:07:06,610 --> 00:07:11,560 How do we define the separation between background and wave? 127 00:07:17,518 --> 00:07:18,310 Put it another way. 128 00:07:18,310 --> 00:07:21,263 Suppose what you are sensitive to is spacetime curvature. 129 00:07:21,263 --> 00:07:22,930 Suppose you are sensitive to things that 130 00:07:22,930 --> 00:07:25,090 depend on the Riemann tensor. 131 00:07:25,090 --> 00:07:28,120 How do you know from your measurement that you are 132 00:07:28,120 --> 00:07:31,240 measuring the Riemann tensor associated to the gravitational 133 00:07:31,240 --> 00:07:34,020 wave and not a Riemann tensor associated with a curvature-- 134 00:07:34,020 --> 00:07:34,560 excuse me-- 135 00:07:34,560 --> 00:07:37,840 associated with your background? 136 00:07:37,840 --> 00:07:43,070 That is an extremely important issue. 137 00:07:43,070 --> 00:07:46,357 If all I can measure is this, but I want to get this, 138 00:07:46,357 --> 00:07:48,940 what is the trick that I use to distinguish between these two? 139 00:07:56,420 --> 00:07:58,008 I call this a trick in my notes, but I 140 00:07:58,008 --> 00:07:59,300 don't like that term, actually. 141 00:07:59,300 --> 00:08:00,380 This is not a trick. 142 00:08:00,380 --> 00:08:03,560 This is a fundamentally important point 143 00:08:03,560 --> 00:08:07,250 when you are trying to measure some kind of a perturbation 144 00:08:07,250 --> 00:08:08,450 to a-- 145 00:08:08,450 --> 00:08:10,918 not just to spacetime, but to any kind of a field 146 00:08:10,918 --> 00:08:12,710 that you are studying, any kind of quantity 147 00:08:12,710 --> 00:08:15,440 you're studying which is itself spatially and temporally 148 00:08:15,440 --> 00:08:18,110 varying. 149 00:08:18,110 --> 00:08:25,670 What you need to do is take advantage of a separation 150 00:08:25,670 --> 00:08:28,023 of length and timescales. 151 00:08:46,270 --> 00:08:48,040 When I'm talking about general relativity 152 00:08:48,040 --> 00:08:50,560 and I'm talking about gravitational waves, 153 00:08:50,560 --> 00:08:55,250 I am going to use the fact that a gravitational wave is 154 00:08:55,250 --> 00:08:56,441 oscillatory. 155 00:09:01,260 --> 00:09:16,610 And as such, it varies on a much smaller length and timescales-- 156 00:09:16,610 --> 00:09:21,020 it varies on much smaller length and timescales 157 00:09:21,020 --> 00:09:22,290 than the background does. 158 00:09:34,170 --> 00:09:35,190 OK? 159 00:09:35,190 --> 00:09:37,620 It's useful to introduce an analogy here. 160 00:09:37,620 --> 00:09:42,450 Think about a water wave propagating on the ocean. 161 00:09:49,650 --> 00:09:51,370 In almost all cases when you look 162 00:09:51,370 --> 00:09:53,200 at a water wave propagating on the ocean, 163 00:09:53,200 --> 00:09:56,770 it's obvious what is the wave and what 164 00:09:56,770 --> 00:09:59,147 is the curvature in the ocean. 165 00:09:59,147 --> 00:10:00,730 It's actually associated with the fact 166 00:10:00,730 --> 00:10:03,580 that the Earth is round. 167 00:10:03,580 --> 00:10:07,120 So it is clear from the separation of both length 168 00:10:07,120 --> 00:10:08,635 scales and timescales-- 169 00:10:08,635 --> 00:10:19,650 not clean-- clear how to separate wave 170 00:10:19,650 --> 00:10:30,810 from the curvature of the Earth bending 171 00:10:30,810 --> 00:10:37,710 due to geological structures and things like that. 172 00:10:43,880 --> 00:10:46,450 A component of the curvature of the ocean 173 00:10:46,450 --> 00:10:50,322 that has a curvature scale of, say, 6,000 kilometers-- 174 00:10:50,322 --> 00:10:52,030 well, that's just the fact that the Earth 175 00:10:52,030 --> 00:10:54,050 is a sphere with a radius of 6,000 kilometers. 176 00:10:54,050 --> 00:10:54,550 OK? 177 00:10:54,550 --> 00:10:55,750 So you can kind of see that. 178 00:10:55,750 --> 00:10:57,550 But if you see something that is varying 179 00:10:57,550 --> 00:10:59,133 with a period of about a second that's 180 00:10:59,133 --> 00:11:01,120 got a wavelength of a meter or two, 181 00:11:01,120 --> 00:11:03,412 that has nothing to do with the curvature of the Earth. 182 00:11:03,412 --> 00:11:05,200 That's a wave. 183 00:11:05,200 --> 00:11:08,980 We want to introduce a similar concept here and apply it 184 00:11:08,980 --> 00:11:12,100 to our general relativistic calculation. 185 00:11:12,100 --> 00:11:24,190 So let's introduce two sets of length and timescales-- 186 00:11:24,190 --> 00:11:26,800 let's call them scales-- 187 00:11:26,800 --> 00:11:28,180 we will use in our analysis. 188 00:11:38,240 --> 00:11:44,470 So capital L and capital T, these 189 00:11:44,470 --> 00:11:46,780 are my long length in timescales. 190 00:11:54,360 --> 00:11:59,725 These describe the variation of my background. 191 00:12:06,860 --> 00:12:14,260 Lambda and tau are my short scales, 192 00:12:14,260 --> 00:12:24,980 and they describe the wavelength in the period 193 00:12:24,980 --> 00:12:27,090 of gravitational waves. 194 00:12:27,090 --> 00:12:29,990 So to put this in the context of the kind of measurements 195 00:12:29,990 --> 00:12:35,550 that LIGO makes, the LIGO interferometers 196 00:12:35,550 --> 00:12:40,200 make measurements of behavior of light 197 00:12:40,200 --> 00:12:43,080 moving in the spacetime near the Earth. 198 00:12:43,080 --> 00:12:44,760 Now, there is a component of that 199 00:12:44,760 --> 00:12:48,480 that varies on a scale that has to do 200 00:12:48,480 --> 00:12:52,500 with the curvature of spacetime near the surface of the Earth 201 00:12:52,500 --> 00:12:55,470 due to the Earth's mass distribution. 202 00:12:55,470 --> 00:12:58,060 That occurs on-- so first of all, 203 00:12:58,060 --> 00:13:00,870 that is practically static. 204 00:13:00,870 --> 00:13:04,320 It does vary somewhat because of the motion 205 00:13:04,320 --> 00:13:07,860 of the moon around the Earth and because 206 00:13:07,860 --> 00:13:12,890 of the behavior of the fluid in the Earth's core. 207 00:13:12,890 --> 00:13:16,740 There are small variations there which, interestingly enough, 208 00:13:16,740 --> 00:13:18,630 can be measured. 209 00:13:18,630 --> 00:13:20,940 But those tend to occur on timescales 210 00:13:20,940 --> 00:13:24,150 on the order of hours at the shortest. 211 00:13:24,150 --> 00:13:26,250 For the moon's orbit, the moon goes around 212 00:13:26,250 --> 00:13:29,070 in a time period of about a month. 213 00:13:29,070 --> 00:13:32,180 The Earth's rotating. 214 00:13:32,180 --> 00:13:33,780 It's turning on its axis. 215 00:13:33,780 --> 00:13:35,520 And so at a particular point, you're 216 00:13:35,520 --> 00:13:40,320 sort of casing the gravitational field of the moon as the Earth 217 00:13:40,320 --> 00:13:44,370 rotates under it, and so your time thing here 218 00:13:44,370 --> 00:13:47,940 is on the order of hours to about a day. 219 00:13:47,940 --> 00:13:50,040 The length scale associated with this-- 220 00:13:50,040 --> 00:13:52,320 you can calculate things like the Riemann curvature 221 00:13:52,320 --> 00:13:55,560 or tensor components near the surface of the Earth. 222 00:13:55,560 --> 00:13:58,390 That has the units of 1 over length squared. 223 00:13:58,390 --> 00:14:00,630 So take the square root and inverse it, 224 00:14:00,630 --> 00:14:03,000 and you're going to find that this varies on a length 225 00:14:03,000 --> 00:14:05,978 scale that is thousands or-- 226 00:14:05,978 --> 00:14:08,520 I should probably work this out before I try quoting numbers, 227 00:14:08,520 --> 00:14:12,150 but it's many, many, many, many, many kilometers, 228 00:14:12,150 --> 00:14:16,050 tens of thousands or millions of kilometers. 229 00:14:16,050 --> 00:14:17,760 Tens of thousands, probably. 230 00:14:17,760 --> 00:14:20,310 My gravitational wave, by contrast, 231 00:14:20,310 --> 00:14:23,220 it's got wavelengths that are on the order of-- they 232 00:14:23,220 --> 00:14:25,890 are also on the order of thousands of kilometers, 233 00:14:25,890 --> 00:14:28,260 but there is a time variation on them 234 00:14:28,260 --> 00:14:31,680 that, for the LIGO detectors, is on the order of seconds 235 00:14:31,680 --> 00:14:33,657 at most, really tenths of a second 236 00:14:33,657 --> 00:14:35,490 at the current sensitivity of the detectors, 237 00:14:35,490 --> 00:14:38,550 up to milliseconds. 238 00:14:38,550 --> 00:14:40,750 Way faster variation. 239 00:14:40,750 --> 00:14:43,890 So by looking for pieces of what I can measure 240 00:14:43,890 --> 00:14:46,482 that vary on these length scales and these time scales, 241 00:14:46,482 --> 00:14:47,940 I can separate it from effects that 242 00:14:47,940 --> 00:14:53,490 are varying on these long time scales and long length scales. 243 00:14:53,490 --> 00:14:57,237 So as I move forward in this lecture, 244 00:14:57,237 --> 00:14:59,070 let me emphasize that some of the things I'm 245 00:14:59,070 --> 00:15:01,148 going to introduce here are-- 246 00:15:01,148 --> 00:15:03,190 first of all, they're a little bit more advanced. 247 00:15:03,190 --> 00:15:06,930 You are certainly not responsible for knowing 248 00:15:06,930 --> 00:15:09,400 the gory details of how some of these tools I'm 249 00:15:09,400 --> 00:15:12,380 going to introduce are used. 250 00:15:12,380 --> 00:15:12,880 All right. 251 00:15:12,880 --> 00:15:14,590 So the point is, once I've introduced 252 00:15:14,590 --> 00:15:17,680 these long scales and these short scales, 253 00:15:17,680 --> 00:15:23,050 what we can then do is we can always remove the wave, 254 00:15:23,050 --> 00:15:30,410 remove the oscillation by introducing an averaging 255 00:15:30,410 --> 00:15:31,438 procedure. 256 00:15:37,300 --> 00:15:42,870 So if you imagine that you average over a timescale that 257 00:15:42,870 --> 00:15:46,470 is several times the wave-- excuse me-- 258 00:15:46,470 --> 00:15:54,530 several times the period of the wave and over a length scale 259 00:15:54,530 --> 00:15:58,940 that is several times the wavelength of the wave. 260 00:15:58,940 --> 00:16:02,120 These are taken to be shorter than the long scales, 261 00:16:02,120 --> 00:16:05,790 but longer than the short scales. 262 00:16:05,790 --> 00:16:07,260 And so when you do that-- 263 00:16:10,380 --> 00:16:13,590 let me put it this way. 264 00:16:13,590 --> 00:16:17,670 If I average-- so these angle brackets 265 00:16:17,670 --> 00:16:20,450 are what I'm going to define as my averaging procedure. 266 00:16:20,450 --> 00:16:22,950 I will make this a little bit more quantitative in a moment. 267 00:16:36,210 --> 00:16:39,840 On length scale l and timescale t, 268 00:16:39,840 --> 00:16:42,000 the background basically doesn't change. 269 00:16:50,350 --> 00:16:53,630 So when I average it, I just get the background back. 270 00:16:53,630 --> 00:16:57,480 On the other hand, my wave oscillates a couple of times. 271 00:16:57,480 --> 00:17:01,410 So if I average something that oscillates 272 00:17:01,410 --> 00:17:03,810 over a couple of cycles, I get 0. 273 00:17:06,690 --> 00:17:08,250 So given what I can actually measure, 274 00:17:08,250 --> 00:17:10,980 which is the spacetime or things related to the spacetime, 275 00:17:10,980 --> 00:17:14,354 this averaging procedure lets me pick out the background. 276 00:17:41,620 --> 00:17:43,760 The radiation is then what I get when 277 00:17:43,760 --> 00:17:46,790 I subtract that average bit from the field. 278 00:18:02,240 --> 00:18:05,663 So these are all things that, as a theorist thinking about how 279 00:18:05,663 --> 00:18:07,580 someone is going to go in and actually measure 280 00:18:07,580 --> 00:18:09,122 properties of the gravitational wave, 281 00:18:09,122 --> 00:18:12,980 these are tools I can use to separate one spatially 282 00:18:12,980 --> 00:18:15,580 and temporally varying quantity from another. 283 00:18:15,580 --> 00:18:17,983 Now, here I'm going to describe something that's 284 00:18:17,983 --> 00:18:20,150 a little bit more advanced than we need to get into, 285 00:18:20,150 --> 00:18:21,660 but it's important to discuss. 286 00:18:21,660 --> 00:18:24,105 How do you actually average a tensor like this? 287 00:18:24,105 --> 00:18:25,230 That's a little bit tricky. 288 00:18:28,880 --> 00:18:31,310 So the averaging was first made rigorous. 289 00:18:31,310 --> 00:18:38,280 And there have been various other mathematical formulations 290 00:18:38,280 --> 00:18:40,030 of this that are presented over the years, 291 00:18:40,030 --> 00:18:42,613 but I like this particular one because it's conceptually quite 292 00:18:42,613 --> 00:18:43,880 simple. 293 00:18:43,880 --> 00:18:51,230 It was first made rigorous by Dieter Brill and James Hartle-- 294 00:18:51,230 --> 00:18:54,610 Hartle is the author of a nice elementary textbook 295 00:18:54,610 --> 00:18:57,680 on general relativity-- 296 00:18:57,680 --> 00:18:59,330 in 1964. 297 00:18:59,330 --> 00:19:01,235 And the reference is given in the notes. 298 00:19:01,235 --> 00:19:04,100 And what you basically do is to define 299 00:19:04,100 --> 00:19:08,900 the average of a quantity like the spacetime metric, 300 00:19:08,900 --> 00:19:11,450 you integrate it over some proper volume 301 00:19:11,450 --> 00:19:18,980 in spacetime, proper 4 volume with a particular weighting 302 00:19:18,980 --> 00:19:21,600 function. 303 00:19:21,600 --> 00:19:25,700 So this is something that is defined such 304 00:19:25,700 --> 00:19:35,070 that if you integrate it over that same region, you get 1. 305 00:19:35,070 --> 00:19:37,020 And it's taken to be something that is-- 306 00:19:37,020 --> 00:19:39,720 you can sort of think of as a Gaussian in all four spacetime 307 00:19:39,720 --> 00:19:45,750 directions that is peaked around-- 308 00:19:45,750 --> 00:19:49,140 it's peaked at a particular location, 309 00:19:49,140 --> 00:19:57,916 and it has a width of l in all directions, l and t. 310 00:20:01,810 --> 00:20:03,760 We spent a lot of time talking about what 311 00:20:03,760 --> 00:20:05,860 makes a tensor a tensor. 312 00:20:05,860 --> 00:20:08,020 And one of the things that we discussed 313 00:20:08,020 --> 00:20:13,000 was the fact that when I am working in curved space time 314 00:20:13,000 --> 00:20:17,010 and my basis objects are functions, 315 00:20:17,010 --> 00:20:20,700 I can't really add a tensor here to a tensor 316 00:20:20,700 --> 00:20:23,940 there because they live in different tangent spaces. 317 00:20:23,940 --> 00:20:27,680 That's a poorly defined operation. 318 00:20:27,680 --> 00:20:30,700 An integral is nothing more than a sum on steroids. 319 00:20:30,700 --> 00:20:35,350 So how is it that I'm allowed to do this? 320 00:20:35,350 --> 00:20:38,570 Well, the idea here is this is not exact. 321 00:20:38,570 --> 00:20:39,780 OK? 322 00:20:39,780 --> 00:20:41,820 When one does this, you will find-- 323 00:20:41,820 --> 00:20:53,740 what comes out-- your average tensor modulo corrections 324 00:20:53,740 --> 00:21:08,430 or errors that are of order short length scale 325 00:21:08,430 --> 00:21:11,212 over long time length scale. 326 00:21:11,212 --> 00:21:13,170 This is the best you're going to be able to do. 327 00:21:13,170 --> 00:21:14,850 You're always going to be a little bit 328 00:21:14,850 --> 00:21:17,190 off when you define an averaging procedure like this 329 00:21:17,190 --> 00:21:19,248 because by definition, an average can't 330 00:21:19,248 --> 00:21:20,040 capture everything. 331 00:21:20,040 --> 00:21:21,207 That's why you're averaging. 332 00:21:21,207 --> 00:21:23,650 You're trying to actually throw away some piece of it. 333 00:21:23,650 --> 00:21:28,140 So this allows us to define our tensor calculus 334 00:21:28,140 --> 00:21:34,080 in this averaged way in a very useful framework provided 335 00:21:34,080 --> 00:21:34,650 with-- 336 00:21:34,650 --> 00:21:36,870 we just need to accept the fact that there's always 337 00:21:36,870 --> 00:21:37,620 going to be-- 338 00:21:37,620 --> 00:21:40,800 it really only makes sense in this regime where 339 00:21:40,800 --> 00:21:43,210 we can cleanly separate our length scales and our time 340 00:21:43,210 --> 00:21:44,700 scales. 341 00:21:44,700 --> 00:21:47,962 I emphasize this because sometimes one does an analysis, 342 00:21:47,962 --> 00:21:49,920 and one finds things are really weird going on. 343 00:21:49,920 --> 00:21:51,503 It's often useful to sort of step back 344 00:21:51,503 --> 00:21:52,550 and say, wait a minute. 345 00:21:52,550 --> 00:21:54,300 I'm trying to describe gravitational waves 346 00:21:54,300 --> 00:21:55,930 in a particular regime. 347 00:21:55,930 --> 00:21:58,110 Are these actually gravitational waves? 348 00:21:58,110 --> 00:21:59,652 And you sort of look at this, and you 349 00:21:59,652 --> 00:22:02,190 realize you've put yourself into a regime where 350 00:22:02,190 --> 00:22:06,540 your perturbation is varying on the same timescales and length 351 00:22:06,540 --> 00:22:08,250 scales as your background. 352 00:22:08,250 --> 00:22:11,670 And in fact, your amplitude is no longer really small. 353 00:22:11,670 --> 00:22:12,900 You sort of go, oh, crap. 354 00:22:12,900 --> 00:22:14,910 What I've done here is I've actually pushed this 355 00:22:14,910 --> 00:22:18,900 beyond the regime in which the radiative approximation is 356 00:22:18,900 --> 00:22:20,050 valid. 357 00:22:20,050 --> 00:22:20,980 So bear this in mind. 358 00:22:20,980 --> 00:22:22,620 You're always working-- you're always 359 00:22:22,620 --> 00:22:26,880 going to be working in this approximative form. 360 00:22:26,880 --> 00:22:28,170 Suppose we have now done this. 361 00:22:28,170 --> 00:22:30,570 We have separated length scales. 362 00:22:30,570 --> 00:22:40,240 So we now have cleanly and conceptually 363 00:22:40,240 --> 00:22:43,600 separated into background and perturbation. 364 00:22:53,690 --> 00:22:55,490 Now what I would like to do is just 365 00:22:55,490 --> 00:23:01,988 run through my exercise of computing all the quantities 366 00:23:01,988 --> 00:23:04,030 that I need to do to describe radiation-- really, 367 00:23:04,030 --> 00:23:07,400 all of my quantities that I need to do to develop and then solve 368 00:23:07,400 --> 00:23:09,990 the Einstein field equations. 369 00:23:09,990 --> 00:23:12,470 And I'm going to do so linearizing in h 370 00:23:12,470 --> 00:23:15,860 about this curved background rather than 371 00:23:15,860 --> 00:23:19,430 about the flat background. 372 00:23:19,430 --> 00:23:26,720 So let's develop all of our spacetime curvature 373 00:23:26,720 --> 00:23:37,270 tools, all the various quantities, 374 00:23:37,270 --> 00:23:45,130 linearizing in h around g hat. 375 00:23:48,620 --> 00:23:50,400 So I'm still doing linearized theory, 376 00:23:50,400 --> 00:23:54,270 but I am not linearizing around a flat background anymore. 377 00:23:54,270 --> 00:23:55,950 So to give you an idea, I'm going 378 00:23:55,950 --> 00:24:00,637 to do one term associated with this and a little bit of-- 379 00:24:00,637 --> 00:24:02,970 one term associated with this and a little bit of detail 380 00:24:02,970 --> 00:24:05,880 so you can just see what the complications end up looking 381 00:24:05,880 --> 00:24:06,660 like. 382 00:24:06,660 --> 00:24:10,057 So first thing you might want to do is compute your connection. 383 00:24:15,910 --> 00:24:18,452 Go back to your definition. 384 00:24:18,452 --> 00:24:20,910 That should be in your notes from a couple of lectures ago. 385 00:24:37,240 --> 00:24:38,400 So you insert this. 386 00:24:38,400 --> 00:24:41,870 Here is the definition of the connection. 387 00:24:41,870 --> 00:24:48,620 Let's now insert this split. 388 00:24:48,620 --> 00:24:54,500 So the inverse metric is going to have 389 00:24:54,500 --> 00:24:56,630 a form that looks very reminiscent to how 390 00:24:56,630 --> 00:24:59,630 we did the inverse metric in linearizing 391 00:24:59,630 --> 00:25:03,350 around a flat background, the exact same logic. 392 00:25:03,350 --> 00:25:06,590 I am raising indices using the background, 393 00:25:06,590 --> 00:25:09,170 so h alpha beta in the upstairs position 394 00:25:09,170 --> 00:25:13,410 is what I get when I raise using g hat. 395 00:25:13,410 --> 00:25:14,680 And you get a minus sign. 396 00:25:14,680 --> 00:25:17,860 Again, this is sort of like the tensor equivalent 397 00:25:17,860 --> 00:25:20,230 of binomial expansion of 1 over 1 plus epsilon. 398 00:25:27,770 --> 00:25:28,860 Pardon me just one moment. 399 00:25:42,460 --> 00:25:43,713 So I'm going to get two terms. 400 00:25:43,713 --> 00:25:45,630 So here's the first thing where I'm raising it 401 00:25:45,630 --> 00:25:47,477 to the inverse metric. 402 00:25:47,477 --> 00:25:49,060 This is going to split into two terms. 403 00:25:49,060 --> 00:25:51,268 Let me just write them all out for completeness here. 404 00:26:08,740 --> 00:26:11,380 So let's pause for just a second here. 405 00:26:11,380 --> 00:26:13,710 So when I multiply all of this out, 406 00:26:13,710 --> 00:26:16,560 I'm going to get one term that involves background 407 00:26:16,560 --> 00:26:20,080 inverse metric hitting all the derivatives of my background 408 00:26:20,080 --> 00:26:20,580 metric. 409 00:26:27,050 --> 00:26:28,720 I'm going to call it-- 410 00:26:28,720 --> 00:26:32,730 I'm going to call that Christoffel with a hat. 411 00:26:32,730 --> 00:26:33,230 OK? 412 00:26:33,230 --> 00:26:35,230 That is nothing more than the Christoffel symbol 413 00:26:35,230 --> 00:26:38,585 that you would have gotten if you were just working 414 00:26:38,585 --> 00:26:39,710 with the background metric. 415 00:26:43,170 --> 00:26:46,990 You're going to get another term that 416 00:26:46,990 --> 00:27:05,540 involves h alpha beta acting on the Christoffel symbol 417 00:27:05,540 --> 00:27:08,000 with all the indices in the downstairs position 418 00:27:08,000 --> 00:27:11,750 and constructed from the background. 419 00:27:11,750 --> 00:27:15,290 Then you're going to get another term that 420 00:27:15,290 --> 00:27:23,580 involves your background inverse hitting 421 00:27:23,580 --> 00:27:28,560 various derivatives of your wave or of your probation 422 00:27:28,560 --> 00:27:29,759 around the background. 423 00:27:36,470 --> 00:27:40,582 At this point-- oh, and there'll be a term of order h 424 00:27:40,582 --> 00:27:42,040 squared which I'm going to discard. 425 00:27:45,465 --> 00:27:47,340 At this point, you just have to kind of stare 426 00:27:47,340 --> 00:27:48,460 at this for a little bit. 427 00:27:48,460 --> 00:27:54,030 And when you do so, you find out something of a miracle occurs. 428 00:27:54,030 --> 00:28:06,620 You can write this as background plus a perturbation 429 00:28:06,620 --> 00:28:07,870 to the connection. 430 00:28:07,870 --> 00:28:10,700 I shift to the connection. 431 00:28:10,700 --> 00:28:13,910 And this shift-- when you stare at and you organize all 432 00:28:13,910 --> 00:28:28,230 the terms that appear here, this turns out-- whoops-- 433 00:28:28,230 --> 00:28:36,950 this turns out to look like something 434 00:28:36,950 --> 00:28:40,210 that is very reminiscent of the formula for the Christoffel 435 00:28:40,210 --> 00:28:45,140 symbols, but only acting on the h's and using 436 00:28:45,140 --> 00:28:49,023 covariant derivatives rather than partial derivatives. 437 00:28:53,347 --> 00:28:55,680 I don't have a good way to prove this other than to say, 438 00:28:55,680 --> 00:28:56,670 you know what? 439 00:28:56,670 --> 00:29:01,700 Expand out all of these covariant derivatives 440 00:29:01,700 --> 00:29:04,750 and then compare these terms. 441 00:29:04,750 --> 00:29:08,390 And this ends up being what you get when the smoke clears. 442 00:29:08,390 --> 00:29:10,670 I go through this one in some detail 443 00:29:10,670 --> 00:29:15,830 because simplifications of this sort happen at every level 444 00:29:15,830 --> 00:29:16,630 as we move on. 445 00:29:16,630 --> 00:29:19,610 What we're basically going to do is sort of brute force expand 446 00:29:19,610 --> 00:29:20,540 things. 447 00:29:20,540 --> 00:29:23,690 We'll find a term that looks like just a pure background 448 00:29:23,690 --> 00:29:26,420 piece, and then there'll be a whole bunch 449 00:29:26,420 --> 00:29:30,880 of little bits of crap that are linear in the field h. 450 00:29:30,880 --> 00:29:32,960 And you can gather them together and often 451 00:29:32,960 --> 00:29:35,370 write them in a form that's prettier, kind of like this, 452 00:29:35,370 --> 00:29:38,720 where it's just sort of a shift to the connection involving 453 00:29:38,720 --> 00:29:40,550 covariant derivatives. 454 00:29:40,550 --> 00:29:44,060 There's no great algorithm for actually working all these guys 455 00:29:44,060 --> 00:29:45,230 out. 456 00:29:45,230 --> 00:29:47,360 Basically, you just have a bit of labor 457 00:29:47,360 --> 00:29:49,590 to do to put them together. 458 00:29:49,590 --> 00:30:01,760 So let me write out a couple of other examples of what 459 00:30:01,760 --> 00:30:02,810 follows from this. 460 00:30:24,490 --> 00:30:24,990 Let's see. 461 00:30:24,990 --> 00:30:27,090 There's a point I want to get to in just a moment. 462 00:30:27,090 --> 00:30:28,840 I'll go through that in a bit more detail. 463 00:30:32,910 --> 00:30:34,260 OK. 464 00:30:34,260 --> 00:30:39,840 So when I take my Christoffel and I work out my Riemann 465 00:30:39,840 --> 00:30:47,610 tensor, I find a piece that looks 466 00:30:47,610 --> 00:30:53,487 just like Riemann computed solely from the background. 467 00:30:56,230 --> 00:31:00,090 And then I get a correction to it. 468 00:31:00,090 --> 00:31:35,515 And to linear order in h, this looks like when you work out-- 469 00:31:39,530 --> 00:31:40,030 OK. 470 00:31:40,030 --> 00:31:42,440 Hang on just a second. 471 00:31:42,440 --> 00:31:44,440 So at this point, we can now go ahead 472 00:31:44,440 --> 00:31:47,990 and start making things like the Ricci tensor and the Einstein 473 00:31:47,990 --> 00:31:48,490 tensor. 474 00:31:51,050 --> 00:31:53,300 I got slightly ahead of myself here 475 00:31:53,300 --> 00:31:56,160 because there's an important point to make at this point. 476 00:31:56,160 --> 00:32:04,730 So now, as we move forward, if we're 477 00:32:04,730 --> 00:32:07,130 assuming Einstein's general relativity, 478 00:32:07,130 --> 00:32:13,670 our curvature tensors end up coupling to the Einstein field 479 00:32:13,670 --> 00:32:23,160 equations, a couple things like the Ricci tensor 480 00:32:23,160 --> 00:32:27,690 and the Einstein tensor to the stress energy tensor. 481 00:32:37,150 --> 00:32:41,560 Accordingly, I am going to make the following simplification. 482 00:32:49,350 --> 00:32:56,040 Let's set the stress energy tensor equal to 0. 483 00:32:56,040 --> 00:33:01,650 So the analysis which I'm going to do in the remaining 484 00:33:01,650 --> 00:33:05,610 time in this lecture, it only describes vacuum regions 485 00:33:05,610 --> 00:33:06,750 of spacetime. 486 00:33:06,750 --> 00:33:10,080 One doesn't need to do this, and I strongly, 487 00:33:10,080 --> 00:33:11,880 strongly emphasize this. 488 00:33:11,880 --> 00:33:15,210 Doing so makes this lecture about one third as long 489 00:33:15,210 --> 00:33:17,640 as it would be if I did not assume this was equal to 0. 490 00:33:17,640 --> 00:33:20,520 So this is solely being introduced to simplify things. 491 00:33:20,520 --> 00:33:23,370 And what this means is that I am essentially describing 492 00:33:23,370 --> 00:33:27,111 gravitational waves as I-- 493 00:33:27,111 --> 00:33:30,620 I'm describing gravitational waves far from their source. 494 00:33:30,620 --> 00:33:32,370 Unfortunately, it does mean it complicates 495 00:33:32,370 --> 00:33:35,780 a little bit how they work in some cosmological spacetimes. 496 00:33:35,780 --> 00:33:37,590 But it is still sufficient for me 497 00:33:37,590 --> 00:33:40,650 to introduce some of the key concepts that I want to do. 498 00:33:43,770 --> 00:33:46,740 By assuming that t mu nu is equal to 0, 499 00:33:46,740 --> 00:33:49,590 what I'm going to do is I'm essentially, then, 500 00:33:49,590 --> 00:33:56,910 assuming that the background is equal to 0. 501 00:33:56,910 --> 00:34:00,450 The background spacetime arises from an Einstein tensor 502 00:34:00,450 --> 00:34:02,640 that is equal to 0. 503 00:34:02,640 --> 00:34:06,360 And it's not hard to show that this corresponds to Ricci being 504 00:34:06,360 --> 00:34:08,250 equal to 0, and of course, then, the Ricci 505 00:34:08,250 --> 00:34:10,130 scalar being equal to 0. 506 00:34:15,969 --> 00:34:20,230 So as I move forward here, my background Riemann tensor 507 00:34:20,230 --> 00:34:23,290 is nonzero, but my background Ricci tensor 508 00:34:23,290 --> 00:34:24,300 will be equal to 0. 509 00:34:24,300 --> 00:34:28,960 And I strongly emphasize that this is just a simplification 510 00:34:28,960 --> 00:34:32,080 that I am introducing in order to make today's analysis 511 00:34:32,080 --> 00:34:33,651 a little bit more tractable. 512 00:34:50,400 --> 00:34:56,550 So my background Ricci tensor is equal to 0. 513 00:34:56,550 --> 00:35:01,460 Let's compute the perturbation in my Ricci tensor. 514 00:35:11,990 --> 00:35:19,960 This is what I get when I trace over indices 1 and 3. 515 00:35:19,960 --> 00:35:23,500 I will skip over much of the algebra that is in this. 516 00:35:23,500 --> 00:35:30,010 But what one finds going through this sea of various definitions 517 00:35:30,010 --> 00:35:33,160 is that what results is a delta r mu nu. 518 00:35:40,720 --> 00:35:42,600 I will define this symbol in just a moment. 519 00:36:31,270 --> 00:36:32,810 OK. 520 00:36:32,810 --> 00:36:34,270 So a couple of definitions. 521 00:36:34,270 --> 00:36:36,270 This box operator that I'm introducing here, 522 00:36:36,270 --> 00:36:39,065 this is a covariance wave operator. 523 00:36:46,670 --> 00:36:57,380 And the trace is what I get when I contract my perturbation 524 00:36:57,380 --> 00:37:00,180 h with the background metric. 525 00:37:00,180 --> 00:37:00,760 OK? 526 00:37:00,760 --> 00:37:02,320 So I just want to quickly emphasize 527 00:37:02,320 --> 00:37:05,620 that there is nothing very profound about what 528 00:37:05,620 --> 00:37:07,090 has gone into this. 529 00:37:07,090 --> 00:37:11,050 This was all done by essentially just throwing together 530 00:37:11,050 --> 00:37:14,243 all the various definitions and linearizing 531 00:37:14,243 --> 00:37:15,910 in all the quantities that I care about. 532 00:37:28,470 --> 00:37:33,165 So what I would like to do now is take-- 533 00:37:38,590 --> 00:37:44,890 I'm going to take my Ricci tensor and my Ricci scalar. 534 00:37:44,890 --> 00:37:53,050 I will assemble them to make my Einstein tensor. 535 00:37:55,990 --> 00:37:59,350 And by requiring that that be equal to 0, 536 00:37:59,350 --> 00:38:02,320 I will get an equation that describes radiation 537 00:38:02,320 --> 00:38:06,130 as it's propagating in this curved background. 538 00:38:06,130 --> 00:38:09,490 But before I do this, I want to just pause for a second 539 00:38:09,490 --> 00:38:15,430 here and note that I'm going to get a mess when I do this. 540 00:38:15,430 --> 00:38:18,550 You can already sort of see I've got some interesting structures 541 00:38:18,550 --> 00:38:19,430 here. 542 00:38:19,430 --> 00:38:20,830 So here's a trace. 543 00:38:20,830 --> 00:38:22,390 Here's a trace. 544 00:38:22,390 --> 00:38:24,370 Here is a term that looks like a divergence. 545 00:38:24,370 --> 00:38:26,830 Here is a term that looks like a divergence. 546 00:38:26,830 --> 00:38:29,810 When we were doing linearize theory on a flat background 547 00:38:29,810 --> 00:38:33,340 spacetime, we got rid of those degrees of freedom. 548 00:38:33,340 --> 00:38:38,560 We sort of deduced that these were just kind of annoying 549 00:38:38,560 --> 00:38:40,040 for doing our analysis. 550 00:38:40,040 --> 00:38:44,080 And by changing gauge, we were able to rewrite the equations 551 00:38:44,080 --> 00:38:46,270 in a way that allowed us to get rid of them. 552 00:38:49,280 --> 00:38:51,470 Before I assemble this, I want to explore 553 00:38:51,470 --> 00:38:53,630 what changing gauge means now that I'm 554 00:38:53,630 --> 00:38:57,320 working on a curved background. 555 00:38:57,320 --> 00:39:03,440 So brief aside: let's generalize our notion 556 00:39:03,440 --> 00:39:05,037 of a gauge transformation. 557 00:39:11,500 --> 00:39:13,330 So what I'm going to do is just as when 558 00:39:13,330 --> 00:39:16,060 I was linearizing around a flat background spacetime, 559 00:39:16,060 --> 00:39:20,230 I began by introducing an infinitesimal coordinate 560 00:39:20,230 --> 00:39:21,286 displacement. 561 00:39:25,390 --> 00:39:29,755 And when I did this, I then had a matrix-- 562 00:39:39,370 --> 00:39:47,800 a matrix that affected this coordinate transformation, 563 00:39:47,800 --> 00:39:49,565 which looks like so. 564 00:39:52,510 --> 00:39:55,300 Just as when I-- whoops, typo. 565 00:39:55,300 --> 00:40:01,490 Just as when I was doing this around a flat background, 566 00:40:01,490 --> 00:40:06,580 I'm going to assume that these elements tend to be small. 567 00:40:15,330 --> 00:40:17,880 So let's apply this generalized gauge transformation 568 00:40:17,880 --> 00:40:18,660 to my metric. 569 00:40:26,760 --> 00:40:29,280 Something worth highlighting at this point-- 570 00:40:29,280 --> 00:40:31,470 my metric depends on space and time. 571 00:40:31,470 --> 00:40:32,323 It's not flat. 572 00:40:32,323 --> 00:40:33,990 So I have to build in the fact that it's 573 00:40:33,990 --> 00:40:38,250 a function of these coordinates. 574 00:40:38,250 --> 00:40:54,200 So this-- pardon me just one second. 575 00:41:20,520 --> 00:41:23,400 So now I need to find the inverse metric-- excuse me-- 576 00:41:23,400 --> 00:41:25,500 the inverse coordinate transformation 577 00:41:25,500 --> 00:41:27,955 that will essentially give me the same thing, 578 00:41:27,955 --> 00:41:29,580 flipping my indices around a little bit 579 00:41:29,580 --> 00:41:32,020 and introducing a minus sign. 580 00:41:32,020 --> 00:41:34,520 And when I expand this out-- 581 00:41:34,520 --> 00:41:37,220 again, bearing in mind I have to be careful about the coordinate 582 00:41:37,220 --> 00:41:56,160 that this is being evaluated at, OK?-- 583 00:41:56,160 --> 00:41:59,790 So I'm going to get one term here that involves my Kronecker 584 00:41:59,790 --> 00:42:01,260 deltas hitting this guy. 585 00:42:01,260 --> 00:42:02,910 Again, be careful with that argument. 586 00:42:36,950 --> 00:42:38,000 OK? 587 00:42:38,000 --> 00:42:40,000 So I'm going to get another term that involves-- 588 00:42:40,000 --> 00:42:41,770 I'm going to get two terms that involve 589 00:42:41,770 --> 00:42:45,250 my derivative of my infinitesimal displacement 590 00:42:45,250 --> 00:42:47,290 hitting the background. 591 00:42:47,290 --> 00:42:50,590 I'm going to get terms that involve the Kronecker hitting 592 00:42:50,590 --> 00:42:52,830 my metric perturbation. 593 00:42:52,830 --> 00:42:58,330 I'm going to discard terms that are quadratically small. 594 00:42:58,330 --> 00:43:01,310 Now, here is where we need to be a little bit careful. 595 00:43:04,830 --> 00:43:08,220 Notice I have shifted this guy here. 596 00:43:08,220 --> 00:43:11,747 I'm going to expand this with a Taylor expansion. 597 00:43:45,543 --> 00:43:48,430 Move this a little higher so that it's not blocking my view. 598 00:44:01,845 --> 00:44:02,345 OK. 599 00:44:02,345 --> 00:44:03,370 That looks pretty gross. 600 00:44:06,035 --> 00:44:07,160 We can clean it up, though. 601 00:44:25,020 --> 00:44:28,440 If I use the fact that-- 602 00:44:28,440 --> 00:44:33,980 for example, looking at one of these terms, 603 00:44:33,980 --> 00:44:42,200 this can be written as the covariant derivative 604 00:44:42,200 --> 00:44:49,926 of my infinitesimal displacement minus a connection coefficient. 605 00:44:56,250 --> 00:44:58,620 You can gather together a whole bunch of terms. 606 00:45:06,150 --> 00:45:07,300 And what you find-- 607 00:45:36,920 --> 00:45:39,470 something that looks remarkably similar to the form 608 00:45:39,470 --> 00:45:41,000 that we got. 609 00:45:41,000 --> 00:45:43,520 By the way, I may have forgotten to define that back here. 610 00:45:43,520 --> 00:45:47,210 My sincere apologies to those of you following along at home! 611 00:45:47,210 --> 00:45:49,490 My covariant derivatives with a hat 612 00:45:49,490 --> 00:45:51,260 mean this is a covariant derivative being 613 00:45:51,260 --> 00:45:54,540 taken with respect to the background spacetime. 614 00:45:54,540 --> 00:45:57,125 So just do your normal recipe for evaluating 615 00:45:57,125 --> 00:45:59,300 a covariant derivative, but compute 616 00:45:59,300 --> 00:46:03,223 all of your Christoffels using the background spacetime g hat. 617 00:46:03,223 --> 00:46:04,640 Carrying back over to here, what I 618 00:46:04,640 --> 00:46:07,070 find is that doing this infinitesimal coordinate 619 00:46:07,070 --> 00:46:10,370 transformation, it looks just like, why did you 620 00:46:10,370 --> 00:46:11,750 do all these steps properly? 621 00:46:11,750 --> 00:46:13,880 And this little bit of doing the Taylor expansion 622 00:46:13,880 --> 00:46:14,990 in the metric-- 623 00:46:14,990 --> 00:46:18,440 I speak from experience-- it's easy to overlook that bit. 624 00:46:18,440 --> 00:46:21,050 This looks just like what we did when 625 00:46:21,050 --> 00:46:24,530 we applied an infinitesimal coordinate transformation 626 00:46:24,530 --> 00:46:29,150 to a perturbation around flat spacetime. 627 00:46:29,150 --> 00:46:33,680 And it tells us that there is a kind of gauge that 628 00:46:33,680 --> 00:46:36,200 can be applied to my gravitational waves 629 00:46:36,200 --> 00:46:42,290 around a curved background provided I promote 630 00:46:42,290 --> 00:46:49,120 my derivatives of the infinitesimal coordinate 631 00:46:49,120 --> 00:46:53,460 generator from partials to covariance. 632 00:46:53,460 --> 00:47:00,350 So this defines-- if I introduce a generalized coordinate 633 00:47:00,350 --> 00:47:04,320 transformation, this generalizes the notion of a gauge 634 00:47:04,320 --> 00:47:07,680 transformation when I'm examining things like radiation 635 00:47:07,680 --> 00:47:09,840 on a curved background. 636 00:47:09,840 --> 00:47:12,947 This is my generalized gauge transformation. 637 00:47:21,900 --> 00:47:40,680 It will also prove useful for us to have a notion of a trace 638 00:47:40,680 --> 00:47:42,912 reversed perturbation. 639 00:47:53,740 --> 00:47:59,750 So I'm going to introduce an h bar mu nu. 640 00:47:59,750 --> 00:48:01,980 This will be my original h mu nu, the perturbation 641 00:48:01,980 --> 00:48:03,676 around my curved background. 642 00:48:07,010 --> 00:48:10,970 And I subtract off 1/2 of the trace like so. 643 00:48:10,970 --> 00:48:14,420 If you take the trace of this in the way that I defined earlier, 644 00:48:14,420 --> 00:48:17,840 this will give you minus little h back. 645 00:48:29,230 --> 00:48:32,110 So again, at this point, I'm going 646 00:48:32,110 --> 00:48:35,560 to skip over a couple of lines in my lecture notes 647 00:48:35,560 --> 00:48:38,560 because they're straightforward but a little bit tedious. 648 00:48:38,560 --> 00:48:45,370 What we're going to do is take my Ricci tensor, 649 00:48:45,370 --> 00:48:48,340 my Ricci scalar, my background spacetime. 650 00:48:48,340 --> 00:48:52,520 I'm going to assemble my Einstein tensor associated 651 00:48:52,520 --> 00:48:54,800 with this metric perturbation, but I'm 652 00:48:54,800 --> 00:48:57,716 going to write it in terms of the trace reverse [INAUDIBLE].. 653 00:49:01,070 --> 00:49:18,500 So let's write out delta g mu nu in terms of this guy. 654 00:49:24,570 --> 00:49:26,990 So assembling all the pieces, doing the algebra. 655 00:50:21,740 --> 00:50:22,240 OK. 656 00:50:25,002 --> 00:50:26,960 So pardon while I just write all that junk out. 657 00:50:31,090 --> 00:50:32,950 So this, again, I hope, reminds us 658 00:50:32,950 --> 00:50:35,110 of a step when we were doing perturbations 659 00:50:35,110 --> 00:50:36,675 around flat spacetime. 660 00:50:50,048 --> 00:50:51,590 We have all these terms that are sort 661 00:50:51,590 --> 00:50:54,800 of annoying divergences of that trace reverse metric 662 00:50:54,800 --> 00:50:56,390 perturbation. 663 00:50:56,390 --> 00:51:00,110 And so a line or two of algebra will justify this. 664 00:51:00,110 --> 00:51:03,740 I can change gauge. 665 00:51:03,740 --> 00:51:08,450 I can find a gauge in which my perturbation has 666 00:51:08,450 --> 00:51:17,150 no divergence if I choose these generators such 667 00:51:17,150 --> 00:51:23,240 that they satisfy a wave equation in which the source is 668 00:51:23,240 --> 00:51:28,730 the divergence of my old metric perturbation. 669 00:51:28,730 --> 00:51:31,460 Let's suppose we've done this. 670 00:51:31,460 --> 00:51:34,370 By the way, we will call this generalized Lorenz gauge. 671 00:51:45,900 --> 00:51:53,440 When you do this, what you find is that 672 00:51:53,440 --> 00:51:56,350 your metric perturbation-- 673 00:51:56,350 --> 00:52:21,760 oh, shoot-- is governed by the following Einstein tensor. 674 00:52:21,760 --> 00:52:24,700 And if you imagine that this guy is propagating in vacuum, 675 00:52:24,700 --> 00:52:28,690 what this tells you is that the wave satisfies 676 00:52:28,690 --> 00:52:31,368 something that's similar to the flat spacetime wave equation. 677 00:52:31,368 --> 00:52:33,160 Now, the operator is a little bit different 678 00:52:33,160 --> 00:52:39,590 because you have assembled it out of covariant derivatives 679 00:52:39,590 --> 00:52:41,270 and with a correction, which shows 680 00:52:41,270 --> 00:52:43,870 that your wave is actually now coupling 681 00:52:43,870 --> 00:52:46,149 to the curvature of spacetime. 682 00:52:51,140 --> 00:52:53,320 So I have a few more notes about this. 683 00:52:53,320 --> 00:52:55,690 It's not really important that we go through them 684 00:52:55,690 --> 00:52:56,998 in great detail here. 685 00:52:56,998 --> 00:52:58,540 Those of you who are interested, they 686 00:52:58,540 --> 00:53:02,020 will certainly be made available through the website. 687 00:53:02,020 --> 00:53:05,620 The key thing which I want to emphasize is essentially some 688 00:53:05,620 --> 00:53:07,620 of the key bits of-- an important piece 689 00:53:07,620 --> 00:53:12,810 of the technique, which is that what we are doing is using-- 690 00:53:12,810 --> 00:53:16,330 we are linearizing around this flat background. 691 00:53:16,330 --> 00:53:19,390 We have generalized our gauge transformation 692 00:53:19,390 --> 00:53:22,450 to allow us to simplify the mathematical structure 693 00:53:22,450 --> 00:53:25,300 of this equation. 694 00:53:25,300 --> 00:53:28,570 And then what results is an interesting correction 695 00:53:28,570 --> 00:53:30,520 to the usual wave equation that one sees. 696 00:53:37,060 --> 00:53:41,330 So I will also emphasize this is a somewhat more advanced topic. 697 00:53:41,330 --> 00:53:42,940 So one of the reasons why I'm eliding 698 00:53:42,940 --> 00:53:46,720 over some of these details is they are somewhat tedious. 699 00:53:46,720 --> 00:53:49,930 You don't need to know them in detail. 700 00:53:49,930 --> 00:53:51,430 It's good to be familiar with them 701 00:53:51,430 --> 00:53:53,320 and to be able to follow along here. 702 00:53:53,320 --> 00:53:55,630 I am going to post a couple of papers 703 00:53:55,630 --> 00:53:58,990 to the 8.962 website that lay out 704 00:53:58,990 --> 00:54:01,307 some of the foundations of this stuff. 705 00:54:01,307 --> 00:54:02,890 The key thing which I want to use this 706 00:54:02,890 --> 00:54:09,090 for is an important aspect of gravitational waves 707 00:54:09,090 --> 00:54:12,390 is that they carry energy. 708 00:54:12,390 --> 00:54:15,840 Electromagnetic waves carry energy. 709 00:54:15,840 --> 00:54:19,277 It shouldn't be a surprise if I have some kind of a source-- 710 00:54:19,277 --> 00:54:20,360 we're in a room right now. 711 00:54:20,360 --> 00:54:22,170 There's light shining down on me. 712 00:54:22,170 --> 00:54:26,790 The energy that is going on me, it heats me slightly. 713 00:54:26,790 --> 00:54:29,765 It is being drained from some power source somewhere else. 714 00:54:29,765 --> 00:54:31,140 We're all familiar with the story 715 00:54:31,140 --> 00:54:34,850 of electromagnetic radiation. 716 00:54:34,850 --> 00:54:38,380 For gravitational waves, ascertaining 717 00:54:38,380 --> 00:54:42,410 what the energy content of the wave actually is 718 00:54:42,410 --> 00:54:44,230 is a little bit more subtle. 719 00:54:44,230 --> 00:54:52,630 And indeed, understanding that there is an unambiguous energy 720 00:54:52,630 --> 00:54:55,630 carried by these waves is something 721 00:54:55,630 --> 00:54:59,140 that occupied a lot of the foundations 722 00:54:59,140 --> 00:55:04,180 of gravitational wave theory for a couple of decades. 723 00:55:04,180 --> 00:55:06,040 In part, this is driven by the fact 724 00:55:06,040 --> 00:55:08,260 that these things are so hard to measure 725 00:55:08,260 --> 00:55:09,525 that it was difficult to-- 726 00:55:09,525 --> 00:55:11,650 this is one of those things where if you could just 727 00:55:11,650 --> 00:55:14,685 go out and measure it, even if your theory was a little bit 728 00:55:14,685 --> 00:55:16,560 uncertain, you would say, well, goddammit it, 729 00:55:16,560 --> 00:55:18,480 it left an imprint on my detector here. 730 00:55:18,480 --> 00:55:20,178 It must have carried energy. 731 00:55:20,178 --> 00:55:21,970 And so you would know where you were going. 732 00:55:21,970 --> 00:55:23,553 But because there were no measurements 733 00:55:23,553 --> 00:55:26,140 of these things for years and years and years, 734 00:55:26,140 --> 00:55:30,400 no one quite knew which direction to step in. 735 00:55:30,400 --> 00:55:41,200 Part of what makes this complicated 736 00:55:41,200 --> 00:55:45,970 is the fact that when we are in general relativity, 737 00:55:45,970 --> 00:55:48,790 no matter what your spacetime is, 738 00:55:48,790 --> 00:55:52,180 you can always go into a local Lorenz frame. 739 00:56:01,090 --> 00:56:06,190 When you do that, spacetime is flat at a point. 740 00:56:21,900 --> 00:56:24,900 Furthermore, all the derivatives associated with it-- 741 00:56:24,900 --> 00:56:29,290 the first derivatives associated with it are 0. 742 00:56:29,290 --> 00:56:31,160 So it's sort of-- 743 00:56:31,160 --> 00:56:35,830 how the hell can I have a field h, a gravitational wave 744 00:56:35,830 --> 00:56:41,260 h that carries energy if I am free to change coordinates 745 00:56:41,260 --> 00:56:44,630 and make it equal to 0? 746 00:56:44,630 --> 00:56:47,550 Well, there's a couple of comments I'll make about that. 747 00:56:47,550 --> 00:56:50,215 So in sort of the same way-- 748 00:56:50,215 --> 00:56:51,590 well, let's back up for a moment. 749 00:56:51,590 --> 00:56:57,470 We made an analogy earlier to the electromagnetic potential 750 00:56:57,470 --> 00:57:00,150 when I was describing gauge transformations. 751 00:57:00,150 --> 00:57:02,270 So the metric is a quantity that is 752 00:57:02,270 --> 00:57:05,020 subject to gauge transformations, 753 00:57:05,020 --> 00:57:07,130 but the Riemann curvature tensor-- 754 00:57:07,130 --> 00:57:09,550 when I linearize around a flat background, 755 00:57:09,550 --> 00:57:11,770 I found that the Riemann curvature 756 00:57:11,770 --> 00:57:14,463 tensor was invariant to those transformations. 757 00:57:14,463 --> 00:57:15,880 It's a little bit more complicated 758 00:57:15,880 --> 00:57:19,330 when I linearize around a curved background, 759 00:57:19,330 --> 00:57:22,570 but it still remains the case that I can never 760 00:57:22,570 --> 00:57:24,460 make my curvature go away. 761 00:57:24,460 --> 00:57:27,850 It's reminiscent of the fact that I can choose a gauge that 762 00:57:27,850 --> 00:57:30,460 lets my electrostatic-- 763 00:57:30,460 --> 00:57:35,485 my electromagnetic potentials do any number of crazy-ass things. 764 00:57:35,485 --> 00:57:38,110 I must be getting tired because I'm swearing a little bit more. 765 00:57:38,110 --> 00:57:39,970 So I can let my electromagnetic potential 766 00:57:39,970 --> 00:57:43,540 do any sorts of crazy things, but my fields 767 00:57:43,540 --> 00:57:46,300 are the quantities that ultimately carry energy. 768 00:57:46,300 --> 00:57:48,560 They carry energy and momentum. 769 00:57:48,560 --> 00:57:50,320 So in the same way, the fact that I 770 00:57:50,320 --> 00:57:53,470 can get rid of the metric by going into a local Lorenz 771 00:57:53,470 --> 00:57:55,450 frame, that shouldn't bother us too much. 772 00:57:55,450 --> 00:57:57,850 That's kind of like going into a frame where-- 773 00:57:57,850 --> 00:58:02,080 or it's going into a gauge where I just make my potentials go 774 00:58:02,080 --> 00:58:03,460 away or make my potentials become 775 00:58:03,460 --> 00:58:06,250 static or something like that. 776 00:58:06,250 --> 00:58:09,310 This is basically telling me that the energy 777 00:58:09,310 --> 00:58:12,582 must be something that is bound up in the curvature. 778 00:58:19,820 --> 00:58:22,430 Furthermore, the fact that I can always make things 779 00:58:22,430 --> 00:58:24,890 look flat at a particular point-- 780 00:58:24,890 --> 00:58:37,120 I must use non-locality to actually pull 781 00:58:37,120 --> 00:58:41,995 out and understand the energy content. 782 00:58:47,980 --> 00:58:50,890 This sort of means that in a very fundamental sense, 783 00:58:50,890 --> 00:58:55,410 I am never going to be able to define, in a completely 784 00:58:55,410 --> 00:58:58,480 gauge-invariant way, the notion of energy 785 00:58:58,480 --> 00:59:02,760 in a gravitational wave at an event in spacetime. 786 00:59:02,760 --> 00:59:07,390 I may actually come up with gauges in which there 787 00:59:07,390 --> 00:59:10,648 is some notion of an energy like quantity 788 00:59:10,648 --> 00:59:12,190 that's defined at a particular point. 789 00:59:12,190 --> 00:59:13,870 It will not be gauge-invariant, though. 790 00:59:17,400 --> 00:59:19,950 I am going to need to-- if I want to really rigorously 791 00:59:19,950 --> 00:59:22,260 define what the energy content gravitation wave is, 792 00:59:22,260 --> 00:59:24,052 it's going to have to be based on something 793 00:59:24,052 --> 00:59:27,090 where I'm averaging over a region that is large compared 794 00:59:27,090 --> 00:59:29,580 to a wavelength, but small compared 795 00:59:29,580 --> 00:59:32,578 to the scales associated with my background. 796 00:59:42,630 --> 00:59:45,960 So let me sketch for you how we can understand this. 797 00:59:45,960 --> 00:59:47,820 And this will conclude with the derivation 798 00:59:47,820 --> 00:59:52,710 of a second quantity that is often 799 00:59:52,710 --> 00:59:54,000 called the quadrupole formula. 800 01:00:00,010 --> 01:00:03,220 Again turning to my electromagnetic analogy, 801 01:00:03,220 --> 01:00:06,100 the energy content of-- 802 01:00:06,100 --> 01:00:08,343 well, [INAUDIBLE] the content of the energy momentum 803 01:00:08,343 --> 01:00:09,760 carried by an electromagnetic wave 804 01:00:09,760 --> 01:00:12,070 is described by a pointing vector. 805 01:00:12,070 --> 01:00:14,410 And the pointing vector looks like the e field 806 01:00:14,410 --> 01:00:15,840 times the b field. 807 01:00:15,840 --> 01:00:18,670 It is quadratic in the field. 808 01:00:18,670 --> 01:00:20,800 In a similar way, we are going to expect 809 01:00:20,800 --> 01:00:23,950 the gravitational wave energy content 810 01:00:23,950 --> 01:00:27,040 to be something that is going to be quadratic in a field. 811 01:00:27,040 --> 01:00:29,550 So to do this properly, we're going 812 01:00:29,550 --> 01:00:33,700 to need to think about how to go in the second order 813 01:00:33,700 --> 01:00:52,670 in our theory, so second order in perturbations 814 01:00:52,670 --> 01:00:54,035 around the background. 815 01:01:01,470 --> 01:01:05,500 So what I'm going to do is imagine 816 01:01:05,500 --> 01:01:13,270 that my spacetime looks like some background 817 01:01:13,270 --> 01:01:16,690 plus some epsilon times h alpha beta. 818 01:01:16,690 --> 01:01:18,490 This h alpha beta is what we just spent 819 01:01:18,490 --> 01:01:20,340 the past hour or so computing. 820 01:01:25,140 --> 01:01:29,920 And imagine that there is some additional term j alpha beta. 821 01:01:29,920 --> 01:01:33,100 This epsilon that I've introduced here 822 01:01:33,100 --> 01:01:35,256 is just an order counting parameter. 823 01:01:40,980 --> 01:01:43,360 Its value is actually 1. 824 01:01:43,360 --> 01:01:44,070 OK? 825 01:01:44,070 --> 01:01:46,260 But what it does is it allows us to keep 826 01:01:46,260 --> 01:01:48,960 track of the order in perturbation theory 827 01:01:48,960 --> 01:01:49,680 to which I go. 828 01:01:49,680 --> 01:01:51,660 So if the gravitational waves are typically 829 01:01:51,660 --> 01:01:55,570 on the order of, say, 10 to the minus 22, 830 01:01:55,570 --> 01:01:56,890 here's my background curvature. 831 01:01:56,890 --> 01:01:59,590 These terms are all in the order of 1 or so. 832 01:01:59,590 --> 01:02:02,740 These are all in the order of 10 to the minus 22 or so. 833 01:02:02,740 --> 01:02:05,470 These are all in the order of 10 to the minus 44 or so. 834 01:02:08,290 --> 01:02:12,190 What I want to do is run this through-- 835 01:02:12,190 --> 01:02:15,220 and let's just focus on vacuum spacetime for now. 836 01:02:15,220 --> 01:02:18,130 Let's run this through the vacuum Einstein equation. 837 01:02:21,100 --> 01:02:24,110 So let's expand g alpha beta. 838 01:02:24,110 --> 01:02:26,920 Actually, let's not have too much crosstalk between indices. 839 01:02:26,920 --> 01:02:27,880 I'll call this g mu nu. 840 01:02:36,890 --> 01:02:39,170 And I'm going to require this to be equal to vacuum, 841 01:02:39,170 --> 01:02:41,240 so I'll set that equal to 0. 842 01:02:41,240 --> 01:02:44,230 What I want to do is expand this guy. 843 01:02:54,040 --> 01:03:05,220 That's a tremendous amount of work, 844 01:03:05,220 --> 01:03:06,720 but I'm going to sketch for you what 845 01:03:06,720 --> 01:03:08,095 the highlights of this look like. 846 01:03:22,180 --> 01:03:27,077 So when I expand my Einstein tensor, 847 01:03:27,077 --> 01:03:28,160 I'm going to get one term. 848 01:03:31,670 --> 01:03:36,380 That is basically just saying that my background spacetime 849 01:03:36,380 --> 01:03:37,340 is a vacuum solution. 850 01:03:41,250 --> 01:03:45,290 I'm going to get a term that looks like-- 851 01:03:45,290 --> 01:03:51,090 I'll call it g1 alpha beta. 852 01:03:51,090 --> 01:03:57,270 And this is going to depend on my first-order perturbation 853 01:03:57,270 --> 01:03:58,816 in the background. 854 01:04:02,290 --> 01:04:06,580 g1 is basically the delta g that I worked out 855 01:04:06,580 --> 01:04:10,340 in the first hour or so of this lecture. 856 01:04:10,340 --> 01:04:21,340 I will get another term that looks like the same g1, 857 01:04:21,340 --> 01:04:26,000 but in which my second-order term is coming along. 858 01:04:33,210 --> 01:04:34,850 But I'm going to get an additional term 859 01:04:34,850 --> 01:04:39,440 that results from non-linear coupling of h to h. 860 01:04:44,130 --> 01:04:45,270 I'll call that g2. 861 01:05:00,080 --> 01:05:03,000 So I emphasize again, this is just my ordinary Einstein 862 01:05:03,000 --> 01:05:03,500 equation. 863 01:05:03,500 --> 01:05:09,560 This is basically telling me my background satisfies the-- 864 01:05:09,560 --> 01:05:12,470 my background is a vacuum solution. 865 01:05:12,470 --> 01:05:15,822 This is the linear wave operator on a curved background 866 01:05:15,822 --> 01:05:17,780 that we just worked out in the first hour or so 867 01:05:17,780 --> 01:05:19,190 of this lecture. 868 01:05:19,190 --> 01:05:21,860 This is that same linear operator, 869 01:05:21,860 --> 01:05:25,340 but now applied to the second-order perturbation. 870 01:05:25,340 --> 01:05:27,950 And this is something new. 871 01:05:27,950 --> 01:05:30,320 It is very messy. 872 01:05:30,320 --> 01:05:33,410 It involves lots of terms that involve 873 01:05:33,410 --> 01:05:37,820 h hitting covariant derivatives of h, covariant derivatives 874 01:05:37,820 --> 01:05:41,730 of h hitting each other. 875 01:05:41,730 --> 01:05:44,120 I will post a paper by Richard Isaacson that steps 876 01:05:44,120 --> 01:05:46,160 through this in some detail. 877 01:05:46,160 --> 01:05:55,990 What we're going to now do is we require the Einstein equation 878 01:05:55,990 --> 01:06:02,900 to hold order by order. 879 01:06:02,900 --> 01:06:05,260 In other words, at every order in epsilon, 880 01:06:05,260 --> 01:06:08,030 this equation must work. 881 01:06:08,030 --> 01:06:11,260 So at order epsilon to the 0 or at order 1-- 882 01:06:23,535 --> 01:06:25,310 this is what I just said-- 883 01:06:25,310 --> 01:06:26,930 the background is a vacuum solution. 884 01:06:32,890 --> 01:06:35,640 Groovy. 885 01:06:35,640 --> 01:06:51,650 At order epsilon, this is my wave equation 886 01:06:51,650 --> 01:06:52,998 on my curved background. 887 01:07:00,980 --> 01:07:07,250 At order epsilon squared, now something new happens. 888 01:07:37,470 --> 01:07:39,330 Now, that is interesting. 889 01:07:39,330 --> 01:07:42,630 What we are seeing is that the terms which 890 01:07:42,630 --> 01:07:47,340 involve quadratic things with the linear perturbation, things 891 01:07:47,340 --> 01:07:51,150 that are quadratic in h are acting 892 01:07:51,150 --> 01:07:54,730 as the source to the wave equation that governs j. 893 01:08:35,149 --> 01:08:38,279 Let's try to make some headway on this. 894 01:08:38,279 --> 01:08:43,529 Let's go back to our separation of length in timescales. 895 01:08:43,529 --> 01:08:54,120 So let's define delta j mu nu to be 896 01:08:54,120 --> 01:09:00,090 j mu nu minus what I get when I average 897 01:09:00,090 --> 01:09:06,399 j mu nu on an intermediate length scale. 898 01:09:06,399 --> 01:09:10,210 So lambda-- I'll remind you that lambda and tau are sort of 899 01:09:10,210 --> 01:09:11,800 coming along for the ride here. 900 01:09:11,800 --> 01:09:14,200 This is my gw short scale. 901 01:09:17,410 --> 01:09:20,649 l is my background long scale. 902 01:09:25,200 --> 01:09:26,720 And so what I'm going to do is say, 903 01:09:26,720 --> 01:09:28,479 I'm going to take my j mu nu-- 904 01:09:28,479 --> 01:09:31,220 I don't really know too much about it quite yet. 905 01:09:31,220 --> 01:09:37,819 And I do know that by definition, my h only 906 01:09:37,819 --> 01:09:39,740 varies on the short scale. 907 01:09:39,740 --> 01:09:40,353 OK? 908 01:09:40,353 --> 01:09:41,270 So I'm going to take-- 909 01:09:41,270 --> 01:09:44,970 I'm going to do this averaging procedure here. 910 01:09:44,970 --> 01:09:46,290 Pardon me. 911 01:09:46,290 --> 01:09:48,670 That's an error. 912 01:09:48,670 --> 01:09:51,420 And so my l is my intermediate averaging scale. 913 01:09:58,590 --> 01:10:01,560 So I'm going to average things on the intermediate scale l 914 01:10:01,560 --> 01:10:02,490 here. 915 01:10:02,490 --> 01:10:07,410 And so this will be something that only varies 916 01:10:07,410 --> 01:10:10,780 on the short scale lambda. 917 01:10:10,780 --> 01:10:15,970 This, I will take out everything that varies in the short scale. 918 01:10:15,970 --> 01:10:17,910 This only varies on the long scale. 919 01:10:29,830 --> 01:10:30,880 So this is interesting. 920 01:10:30,880 --> 01:10:31,380 OK? 921 01:10:31,380 --> 01:10:34,570 By now doing this, let's regroup my metric. 922 01:10:45,680 --> 01:10:46,680 So this guy-- 923 01:10:56,127 --> 01:10:56,960 I'm going to first-- 924 01:11:02,258 --> 01:11:02,800 sorry, folks. 925 01:11:02,800 --> 01:11:03,758 Getting a little tired. 926 01:11:09,750 --> 01:11:12,530 Here are all the terms that vary on the long scale. 927 01:11:30,380 --> 01:11:32,850 And here are all the terms that vary on my short scale. 928 01:11:47,120 --> 01:11:49,970 With this idea than I am now-- so it's kind of interesting. 929 01:11:49,970 --> 01:11:52,940 What we see is that there is a piece 930 01:11:52,940 --> 01:11:56,990 of the second-order perturbation that has kind of become 931 01:11:56,990 --> 01:11:59,260 a correction to the background. 932 01:11:59,260 --> 01:12:03,950 So when I take my perturbation theory to the next order, 933 01:12:03,950 --> 01:12:05,090 there's no reason-- 934 01:12:05,090 --> 01:12:08,450 so at linear order, we define the linear order perturbation 935 01:12:08,450 --> 01:12:11,690 as being only the bit that varies on the short scale. 936 01:12:11,690 --> 01:12:14,205 But there is no reason why it has 937 01:12:14,205 --> 01:12:16,580 to remain the case that it only varies on the short scale 938 01:12:16,580 --> 01:12:18,110 when I go to the next order. 939 01:12:18,110 --> 01:12:22,640 And any bit that does, in fact, vary on the long scale, 940 01:12:22,640 --> 01:12:25,160 bearing in mind that the way we do our measurements-- 941 01:12:25,160 --> 01:12:27,740 we separate things and lengthen time scales-- 942 01:12:27,740 --> 01:12:31,830 it's going to look to us like a correction to the background. 943 01:12:31,830 --> 01:12:36,888 So with that in mind, let's revisit the second-order term 944 01:12:36,888 --> 01:12:37,930 in the Einstein equation. 945 01:12:55,190 --> 01:13:02,910 So here is my second-order Einstein equation. 946 01:13:05,577 --> 01:13:07,160 You should just assume that these also 947 01:13:07,160 --> 01:13:08,090 depend on the background. 948 01:13:08,090 --> 01:13:09,941 I'm not going to write that out explicitly. 949 01:13:22,730 --> 01:13:24,620 What I would like to do now is-- 950 01:13:24,620 --> 01:13:28,500 let's apply this averaging to this equation. 951 01:13:28,500 --> 01:13:34,520 So the average of this guy equals the average of this guy. 952 01:13:34,520 --> 01:13:37,028 Does this get us anywhere? 953 01:13:37,028 --> 01:13:38,570 Well, it definitely gets us somewhere 954 01:13:38,570 --> 01:13:42,320 with this, because this is a linear operator. 955 01:13:42,320 --> 01:13:43,270 OK? 956 01:13:43,270 --> 01:13:46,555 So when I go and-- 957 01:13:51,500 --> 01:13:53,770 I should be a little bit careful there. 958 01:13:53,770 --> 01:13:58,480 A useful trick which I forgot to state-- my apologies-- 959 01:13:58,480 --> 01:14:03,550 and this trick is worked out in the paper by Isaacson 960 01:14:03,550 --> 01:14:04,970 that I'm going to post-- 961 01:14:04,970 --> 01:14:10,470 is that whenever you take the average of second derivatives 962 01:14:10,470 --> 01:14:16,330 of various kinds of fields, it is 963 01:14:16,330 --> 01:14:21,630 equal to second derivatives of the average of that field 964 01:14:21,630 --> 01:14:28,620 plus corrections that scale as the square of the short length 965 01:14:28,620 --> 01:14:31,270 scale divided by the long length scale. 966 01:14:31,270 --> 01:14:35,130 So what that tells me is that at least up to these corrections, 967 01:14:35,130 --> 01:14:38,520 which the separation of length scales by default 968 01:14:38,520 --> 01:14:42,210 is assuming is a small number, I can take my averaging operator 969 01:14:42,210 --> 01:14:42,930 inside here. 970 01:15:13,495 --> 01:15:15,370 I can't do that trick on the right-hand side. 971 01:15:15,370 --> 01:15:17,370 I've not written out what this equation actually 972 01:15:17,370 --> 01:15:18,660 looks like here. 973 01:15:18,660 --> 01:15:21,302 It's a nonlinear operator, so it's going to be a lot messier. 974 01:15:21,302 --> 01:15:23,010 But this is something that's linear in j. 975 01:15:23,010 --> 01:15:27,270 And so as long as I bear in mind that I incur a small error 976 01:15:27,270 --> 01:15:28,980 by taking various-- 977 01:15:28,980 --> 01:15:34,830 by taking my averaging operation inside the differential 978 01:15:34,830 --> 01:15:40,890 operator, this is telling me that the wave equation-- 979 01:15:40,890 --> 01:15:41,390 sorry. 980 01:15:41,390 --> 01:15:43,760 My Einstein equation applied to this operator 981 01:15:43,760 --> 01:15:47,772 here is set by the quantity that I have there 982 01:15:47,772 --> 01:15:48,730 on the right-hand side. 983 01:15:56,220 --> 01:16:01,635 Now, remember, my background is itself a vacuum solution. 984 01:16:29,000 --> 01:16:33,550 Since my background is a vacuum solution, 985 01:16:33,550 --> 01:16:39,550 I can rewrite this whole thing as the Einstein tensor 986 01:16:39,550 --> 01:16:54,930 on my background metric plus my average quadratic correction. 987 01:17:05,323 --> 01:17:06,990 This is going to be equal to the average 988 01:17:06,990 --> 01:17:10,360 of that second-order piece. 989 01:17:10,360 --> 01:17:14,980 This is just the Einstein field equation with a strange source. 990 01:17:17,502 --> 01:17:19,919 If you look at this-- let's make the following definition. 991 01:17:28,730 --> 01:17:30,260 I'm going to define a stress energy 992 01:17:30,260 --> 01:17:36,750 tensor for gravitational waves as minus 1 993 01:17:36,750 --> 01:17:42,005 over 8 pi g times the average of this operator. 994 01:17:47,460 --> 01:17:51,600 What this is now telling me is that as my gravitational wave 995 01:17:51,600 --> 01:17:55,080 propagates through spacetime, it carries, 996 01:17:55,080 --> 01:17:58,050 it generates a stress energy that 997 01:17:58,050 --> 01:18:01,470 changes the background in an amount that 998 01:18:01,470 --> 01:18:04,330 is quadratic in the wave's amplitude. 999 01:18:26,380 --> 01:18:31,450 Now, to get the details of what that operator actually is, 1000 01:18:31,450 --> 01:18:33,910 I have a formula or two in my notes 1001 01:18:33,910 --> 01:18:36,100 that will be scanned and made available. 1002 01:18:36,100 --> 01:18:38,260 I want to cut to the punchline of this. 1003 01:18:38,260 --> 01:18:42,790 The derivation of this is given in a paper by Richard Isaacson 1004 01:18:42,790 --> 01:18:44,920 that I will be posting to the course website. 1005 01:18:49,330 --> 01:18:51,664 So expand out this operator. 1006 01:18:56,410 --> 01:18:59,350 Place in this the transverse traceless field. 1007 01:19:15,353 --> 01:19:16,145 And what you find-- 1008 01:19:19,030 --> 01:19:21,280 fill it with a transverse traceless field. 1009 01:19:21,280 --> 01:19:23,440 That sort of simplifies a few things, 1010 01:19:23,440 --> 01:19:52,430 and you get this remarkably simple result. 1011 01:19:52,430 --> 01:19:56,190 The angle brackets, I'll remind you, 1012 01:19:56,190 --> 01:19:58,990 they tell you that this quantity is only 1013 01:19:58,990 --> 01:20:03,393 defined under the aegis of a particular averaging operation. 1014 01:20:10,160 --> 01:20:14,570 That aside, what we now have is a quantity 1015 01:20:14,570 --> 01:20:18,080 that tells us about how gravitational waves carry 1016 01:20:18,080 --> 01:20:20,965 energy and momentum away from their sources. 1017 01:20:20,965 --> 01:20:23,090 This is known as the Isaacson stress energy tensor. 1018 01:20:32,490 --> 01:20:34,410 Let me comment, especially for students 1019 01:20:34,410 --> 01:20:37,890 who work on things related to gravitational waves-- 1020 01:20:37,890 --> 01:20:40,470 you will encounter many different notions 1021 01:20:40,470 --> 01:20:43,938 of how to compute the energy, things like energy and angular 1022 01:20:43,938 --> 01:20:45,355 momentum and things like that that 1023 01:20:45,355 --> 01:20:48,120 are carried from a radiating source of gravitational waves. 1024 01:20:48,120 --> 01:20:50,910 There is wonderful discussion of this point in a textbook 1025 01:20:50,910 --> 01:20:52,900 by Poisson and Will. 1026 01:20:52,900 --> 01:20:57,510 It's just entitled Gravity, I believe. 1027 01:20:57,510 --> 01:21:03,540 The Isaacson stress tensor has the virtue that it is-- 1028 01:21:03,540 --> 01:21:06,000 it makes it very clear that the waves are-- 1029 01:21:06,000 --> 01:21:07,440 so it's a tensor. 1030 01:21:07,440 --> 01:21:10,470 It makes it very clear that it arises out 1031 01:21:10,470 --> 01:21:15,030 of quadratic derivatives of the wave field. 1032 01:21:15,030 --> 01:21:19,380 And I really enjoy going through this derivation 1033 01:21:19,380 --> 01:21:24,630 because you can see the way in which a term that is-- 1034 01:21:24,630 --> 01:21:26,160 at least at the schematic level, you 1035 01:21:26,160 --> 01:21:29,850 can see the way in which the second-order contribution 1036 01:21:29,850 --> 01:21:35,070 to the Einstein tensor leads to a source that modifies 1037 01:21:35,070 --> 01:21:36,210 your background spacetime. 1038 01:21:36,210 --> 01:21:37,668 That's exactly what a stress energy 1039 01:21:37,668 --> 01:21:39,030 tensor is supposed to do. 1040 01:21:39,030 --> 01:21:40,488 For certain practical applications, 1041 01:21:40,488 --> 01:21:42,572 it turns out to be a little bit hard to work with. 1042 01:21:42,572 --> 01:21:44,880 And so you'll see this discussed in some detail in some 1043 01:21:44,880 --> 01:21:46,172 of the more advanced textbooks. 1044 01:21:46,172 --> 01:21:49,080 I'm happy to discuss this with students who are interested. 1045 01:21:49,080 --> 01:21:53,280 For our purposes, this is not of concern. 1046 01:21:53,280 --> 01:21:56,460 Let me just conclude this lecture 1047 01:21:56,460 --> 01:22:00,780 by evaluating this for the solution that we worked out. 1048 01:22:00,780 --> 01:22:05,880 So let us imagine that we go into a nearly flat region where 1049 01:22:05,880 --> 01:22:08,010 those covariant derivatives become simple. 1050 01:22:17,660 --> 01:22:22,180 And let's just compute one piece of the stress energy tensor. 1051 01:22:22,180 --> 01:22:25,320 So let's look at the energy flux in a nearly flat region. 1052 01:22:40,870 --> 01:22:43,710 So what I'm going to compute is the 0, 1053 01:22:43,710 --> 01:22:54,980 0 piece of this which defines how 1054 01:22:54,980 --> 01:22:56,560 power flows through spacetime. 1055 01:23:01,960 --> 01:23:06,280 So doing this in this region basically 1056 01:23:06,280 --> 01:23:17,090 is going to look like the time derivative of my wave 1057 01:23:17,090 --> 01:23:19,850 field contracted onto itself. 1058 01:23:19,850 --> 01:23:21,710 Let's get the total amount of energy 1059 01:23:21,710 --> 01:23:26,000 that is flowing through a large sphere 1060 01:23:26,000 --> 01:23:31,640 by evaluating t0 0 integrated over a large sphere 1061 01:23:31,640 --> 01:23:33,440 of radius r. 1062 01:23:33,440 --> 01:23:38,775 And for my hij, I will plug in that quadrupole amplitude 1063 01:23:38,775 --> 01:23:40,400 that I derived in the previous lecture. 1064 01:23:49,110 --> 01:23:50,970 Over dots denote d by dt. 1065 01:24:06,880 --> 01:24:10,540 So if you're interested in doing this integral yourself, 1066 01:24:10,540 --> 01:24:14,374 it's actually not that hard, and that's a pretty-- 1067 01:24:14,374 --> 01:24:15,803 it's a salubrious exercise. 1068 01:24:15,803 --> 01:24:16,720 Let's put it that way. 1069 01:24:20,905 --> 01:24:22,530 So I'll remind you that this projection 1070 01:24:22,530 --> 01:24:25,230 tensor is built from-- 1071 01:24:28,850 --> 01:24:30,350 built from the vectors that describe 1072 01:24:30,350 --> 01:24:32,570 the direction of propagation. 1073 01:24:32,570 --> 01:24:37,370 And a few useful tools to know about-- 1074 01:24:37,370 --> 01:24:57,200 so should you choose to work this guy out, 1075 01:24:57,200 --> 01:24:59,300 it's worth knowing that if I integrate 1076 01:24:59,300 --> 01:25:03,740 ni nj over the sphere, this can be written in terms 1077 01:25:03,740 --> 01:25:06,420 of just sines and cosines. 1078 01:25:06,420 --> 01:25:08,390 These are basically direction sines-- 1079 01:25:08,390 --> 01:25:11,420 or direction cosines. 1080 01:25:11,420 --> 01:25:14,390 I get a simple result that depends only 1081 01:25:14,390 --> 01:25:17,990 on the value of the indices i and j. 1082 01:25:28,210 --> 01:25:31,450 If I integrate this over this sphere, I get 0. 1083 01:25:31,450 --> 01:25:35,530 And any odd power of the ends multiplied together 1084 01:25:35,530 --> 01:25:39,503 when integrated over the sphere will give me 0. 1085 01:25:39,503 --> 01:25:41,920 And one more is necessary if you want to do this integral. 1086 01:25:44,800 --> 01:25:57,050 If you integrate four of these buggers multiplied together, 1087 01:25:57,050 --> 01:26:01,460 you get this object totally symmetrized 1088 01:26:01,460 --> 01:26:05,111 on the indices i, j, k, and l. 1089 01:26:14,370 --> 01:26:16,750 So throw it all together. 1090 01:26:16,750 --> 01:26:18,090 Take the derivatives. 1091 01:26:18,090 --> 01:26:20,140 Note that the only thing that depends on time 1092 01:26:20,140 --> 01:26:22,360 is that quadrupole moment. 1093 01:26:22,360 --> 01:26:29,498 And what you wind up with is this remarkably simple result. 1094 01:26:29,498 --> 01:26:29,998 Whoops. 1095 01:26:51,670 --> 01:26:53,520 This is a result that also is known 1096 01:26:53,520 --> 01:26:54,740 as the quadrupole formula. 1097 01:27:01,800 --> 01:27:04,800 It's worth comparing this to the dipole formula 1098 01:27:04,800 --> 01:27:06,480 you get in electrodynamics. 1099 01:27:06,480 --> 01:27:09,150 So the dipole formula that describes 1100 01:27:09,150 --> 01:27:11,760 the leading electromagnetic-- the leading power 1101 01:27:11,760 --> 01:27:14,550 in electromagnetic radiation that comes off of a source 1102 01:27:14,550 --> 01:27:16,890 looks like what you get when you multiply 1103 01:27:16,890 --> 01:27:22,620 two derivatives of the dipole moment with each other. 1104 01:27:22,620 --> 01:27:26,970 The leading radiation in gravity is quadripolar. 1105 01:27:26,970 --> 01:27:29,418 And so rather than being second derivatives, 1106 01:27:29,418 --> 01:27:31,710 you have to take one more derivative out of this thing. 1107 01:27:31,710 --> 01:27:36,210 There are three time derivatives of a quadrupole moment leading 1108 01:27:36,210 --> 01:27:38,220 to this formula. 1109 01:27:38,220 --> 01:27:40,830 So this result was actually originally derived 1110 01:27:40,830 --> 01:27:42,030 by Albert Einstein. 1111 01:27:42,030 --> 01:27:44,258 It's one of the first calculations that he did. 1112 01:27:44,258 --> 01:27:45,300 He actually did it twice. 1113 01:27:45,300 --> 01:27:48,720 He first derived something kind of like this in, I believe, 1114 01:27:48,720 --> 01:27:50,860 1916. 1115 01:27:50,860 --> 01:27:52,510 And it was totally wrong. 1116 01:27:52,510 --> 01:27:54,780 He just basically made a huge mistake. 1117 01:27:54,780 --> 01:27:58,380 But then he did it more or less correctly in 1918. 1118 01:27:58,380 --> 01:27:59,387 So he got this right. 1119 01:27:59,387 --> 01:28:00,220 And it was a small-- 1120 01:28:00,220 --> 01:28:00,870 a small error. 1121 01:28:00,870 --> 01:28:02,400 He was off by a factor of 2, which 1122 01:28:02,400 --> 01:28:04,440 is a simple algebra mistake. 1123 01:28:04,440 --> 01:28:07,290 It's actually really enjoyable to count up 1124 01:28:07,290 --> 01:28:09,180 the number of minor errors that Einstein 1125 01:28:09,180 --> 01:28:10,630 made in some of these works. 1126 01:28:10,630 --> 01:28:13,240 It just makes you feel a little bit better about yourself. 1127 01:28:13,240 --> 01:28:17,370 So anyway, Einstein got this essentially right, 1128 01:28:17,370 --> 01:28:19,680 and then it was hugely controversial 1129 01:28:19,680 --> 01:28:22,740 for quite a few decades after that. 1130 01:28:22,740 --> 01:28:25,650 Essentially, going back to these conceptual issues with which I 1131 01:28:25,650 --> 01:28:28,920 began this lecture, people just grappled with the idea of, 1132 01:28:28,920 --> 01:28:31,183 what does it mean for gravitational radiation 1133 01:28:31,183 --> 01:28:33,600 to carry energy in the first place given that I can always 1134 01:28:33,600 --> 01:28:35,370 go into a freely falling frame? 1135 01:28:35,370 --> 01:28:39,120 It took a little while for those concepts to really solidify. 1136 01:28:39,120 --> 01:28:42,390 Once they did, though, we were off to the races. 1137 01:28:42,390 --> 01:28:45,690 And it's worth noting that if I take this formula 1138 01:28:45,690 --> 01:28:47,640 and I apply it to a binary system-- 1139 01:28:47,640 --> 01:28:50,132 I imagine I have two stars that are orbiting each other. 1140 01:28:50,132 --> 01:28:52,590 I can compute the quadrupole moment and the time derivative 1141 01:28:52,590 --> 01:28:53,970 of the quadrupole moment. 1142 01:28:53,970 --> 01:28:58,510 I can compute the rate at which power is leaving these systems. 1143 01:28:58,510 --> 01:29:02,700 Now, a somewhat more complicated variant of this calculation-- 1144 01:29:02,700 --> 01:29:03,965 well, let me back up. 1145 01:29:03,965 --> 01:29:05,340 So you go ahead, and you do that. 1146 01:29:05,340 --> 01:29:08,100 You compute this for two stars that are orbiting each other. 1147 01:29:08,100 --> 01:29:10,290 You will do this on a future problem set. 1148 01:29:10,290 --> 01:29:12,870 What you find is that gravitational radiation carries 1149 01:29:12,870 --> 01:29:15,150 energy and angular momentum away from the system, 1150 01:29:15,150 --> 01:29:17,580 and it causes them to fall towards one another, which 1151 01:29:17,580 --> 01:29:19,770 causes their orbital frequency to evolve 1152 01:29:19,770 --> 01:29:22,380 in a very predictable fashion. 1153 01:29:22,380 --> 01:29:26,190 That law that arises to very good approximation 1154 01:29:26,190 --> 01:29:28,500 directly from this formula-- there are corrections 1155 01:29:28,500 --> 01:29:30,292 that people have worked out over the years, 1156 01:29:30,292 --> 01:29:33,600 but what you get just doing this quadrupole formula applied 1157 01:29:33,600 --> 01:29:38,010 to two stars orbiting each other describes this chirp signal 1158 01:29:38,010 --> 01:29:41,530 that I have on my hat to very good accuracy. 1159 01:29:41,530 --> 01:29:45,360 And this is pretty much exactly what the LIDO detectors 1160 01:29:45,360 --> 01:29:46,500 measure these days. 1161 01:29:46,500 --> 01:29:49,350 They measure the evolution of systems 1162 01:29:49,350 --> 01:29:52,740 that are evolving under the aegis of this formula. 1163 01:29:52,740 --> 01:29:53,280 All right. 1164 01:29:53,280 --> 01:29:55,260 So that is all we are going to say 1165 01:29:55,260 --> 01:30:00,300 about gravitational radiation in 8.962 this term. 1166 01:30:00,300 --> 01:30:02,040 As we figure out the way things are 1167 01:30:02,040 --> 01:30:04,470 going to proceed moving forward, I 1168 01:30:04,470 --> 01:30:06,720 will make myself available to answer 1169 01:30:06,720 --> 01:30:08,200 some questions about this. 1170 01:30:08,200 --> 01:30:10,140 I do want to emphasize as I conclude 1171 01:30:10,140 --> 01:30:13,270 this lecture that many of these topics are fairly advanced. 1172 01:30:13,270 --> 01:30:15,700 I don't expect you to be familiar with them. 1173 01:30:15,700 --> 01:30:17,790 But I hope you understood the punchline of this. 1174 01:30:17,790 --> 01:30:19,560 And I expect you to be able to apply 1175 01:30:19,560 --> 01:30:21,900 some of these formulas that arise at the end. 1176 01:30:21,900 --> 01:30:25,400 And I will conclude this lecture here.