1 00:00:00,000 --> 00:00:11,760 [SQUEAKING] [RUSTLING] [CLICKING] 2 00:00:11,760 --> 00:00:13,750 SCOTT HUGHES: In the previous lecture, 3 00:00:13,750 --> 00:00:17,070 I began with the Einstein field equations 4 00:00:17,070 --> 00:00:18,690 in the most general form. 5 00:00:18,690 --> 00:00:21,130 And then I specialized to linearized gravity, 6 00:00:21,130 --> 00:00:24,172 to taking the spacetime metric, assuming 7 00:00:24,172 --> 00:00:26,130 that I can find a coordinate system in which it 8 00:00:26,130 --> 00:00:29,460 is close to the eta alpha beta form 9 00:00:29,460 --> 00:00:35,588 that we use for flat space time, plus a small perturbation, 10 00:00:35,588 --> 00:00:36,630 something that's defined. 11 00:00:36,630 --> 00:00:40,020 So I set whenever I have that h multiplied by itself 12 00:00:40,020 --> 00:00:41,580 I can neglect it. 13 00:00:41,580 --> 00:00:45,120 Doing so I was able to recast the Einstein field 14 00:00:45,120 --> 00:00:48,090 equation as a simple linear equation, which is the wave 15 00:00:48,090 --> 00:00:51,240 operator acting on a variant to that perturbation, 16 00:00:51,240 --> 00:00:53,810 the trace reverse version of that perturbation. 17 00:00:53,810 --> 00:00:57,570 It is minus 16 pi g times the stress energy tensor. 18 00:00:57,570 --> 00:01:00,210 So in doing this recasting, I have-- 19 00:01:00,210 --> 00:01:04,680 this is essentially just a trick of reorganizing the variables. 20 00:01:04,680 --> 00:01:07,680 I have taken my h alpha beta, which 21 00:01:07,680 --> 00:01:10,623 describes the way in which spacetime is shifted away 22 00:01:10,623 --> 00:01:13,290 from that of flat spacetime, the perturbation of flat spacetime, 23 00:01:13,290 --> 00:01:14,957 and I've just rearranged that to give it 24 00:01:14,957 --> 00:01:17,370 the trace with the opposite sign. 25 00:01:17,370 --> 00:01:20,730 And I've chosen a gauge such that the divergence 26 00:01:20,730 --> 00:01:25,440 of that trace reverse metric perturbation is zero. 27 00:01:25,440 --> 00:01:28,890 If we take this equation, which is completely 28 00:01:28,890 --> 00:01:31,447 general for linearized spacetime, 29 00:01:31,447 --> 00:01:34,030 the only thing which I had to do is choose a particular gauge. 30 00:01:34,030 --> 00:01:35,322 And there's no physics in that. 31 00:01:35,322 --> 00:01:38,100 That's essentially a coordinate choice. 32 00:01:40,730 --> 00:01:50,240 If I then imagine that my source is static, 33 00:01:50,240 --> 00:01:51,965 I will have static fields. 34 00:01:56,300 --> 00:01:58,100 And so my wave operator goes over 35 00:01:58,100 --> 00:02:00,380 to a simple Laplace operator. 36 00:02:00,380 --> 00:02:05,330 And that it's not too hard to show that the solution that 37 00:02:05,330 --> 00:02:28,280 emerges from this looks like this, where phi n is 38 00:02:28,280 --> 00:02:36,990 Newtonian gravitational potential, which 39 00:02:36,990 --> 00:02:47,470 arises from a distribution of mass like so. 40 00:02:47,470 --> 00:02:50,650 So this is the leading solution. 41 00:02:50,650 --> 00:02:53,840 And it's one that is very powerful and very useful. 42 00:02:53,840 --> 00:02:57,430 And it's actually using tremendous number 43 00:02:57,430 --> 00:03:01,790 of physics and astronomical applications. 44 00:03:04,600 --> 00:03:07,750 This restriction to a static source, though, is-- 45 00:03:07,750 --> 00:03:09,340 well, it's restrictive. 46 00:03:09,340 --> 00:03:11,840 What if I do not want to simply consider a static source? 47 00:03:11,840 --> 00:03:13,450 What if I am interested in sources 48 00:03:13,450 --> 00:03:17,440 of the gravitational interaction that are themselves dynamical? 49 00:03:17,440 --> 00:03:22,150 And so I emphasized last time that this wave equation, my box 50 00:03:22,150 --> 00:03:26,050 h bar alpha beta equals minus 16 pi GT alpha beta, 51 00:03:26,050 --> 00:03:29,080 this is something that I hope you have already 52 00:03:29,080 --> 00:03:32,020 seen in the context of electrodynamics, 53 00:03:32,020 --> 00:03:51,830 in particular whenever we have a wave equation, who's not even 54 00:03:51,830 --> 00:03:54,740 a wave equation, but any differential equation 55 00:03:54,740 --> 00:04:15,690 that is of the form linear differential operator on field 56 00:04:15,690 --> 00:04:17,955 is equal to a source. 57 00:04:21,820 --> 00:04:25,650 OK, so I'm separating out time and spatial behavior here. 58 00:04:25,650 --> 00:04:27,900 Whenever I have a situation in which I'm 59 00:04:27,900 --> 00:04:32,540 interested in studying a field in which some differential 60 00:04:32,540 --> 00:04:36,450 linear operator acting on that field is equal to a source, 61 00:04:36,450 --> 00:04:41,281 we can solve this using the technique of Green's functions. 62 00:04:55,040 --> 00:04:58,160 I am not going to be able to go through the technique 63 00:04:58,160 --> 00:05:00,140 of Green's functions in detail. 64 00:05:00,140 --> 00:05:02,750 I will quickly give you a synopsis of how it works 65 00:05:02,750 --> 00:05:05,480 and then I will look at the particular Green's function 66 00:05:05,480 --> 00:05:07,890 that applies to the wave operator. 67 00:05:07,890 --> 00:05:10,740 Students who would like to read more about this, 68 00:05:10,740 --> 00:05:13,550 I would point you to the textbook by Arfken. 69 00:05:21,118 --> 00:05:24,210 I believe it is called Mathematical Methods 70 00:05:24,210 --> 00:05:27,360 for Physicists. 71 00:05:27,360 --> 00:05:35,370 And in the third edition, Green's functions 72 00:05:35,370 --> 00:05:42,780 are described in Section 16.5 to 16.6. 73 00:05:42,780 --> 00:05:46,150 So in a nutshell, let me just give you a very brief synopsis 74 00:05:46,150 --> 00:05:47,510 of how this technique works. 75 00:05:54,110 --> 00:05:56,120 Suppose we take our source, which 76 00:05:56,120 --> 00:06:05,565 may be some complicated, very ornate entity. 77 00:06:05,565 --> 00:06:08,910 And what we're going to do is replace that source 78 00:06:08,910 --> 00:06:09,990 with a delta function. 79 00:06:22,840 --> 00:06:29,360 So imagine that I take s of t and x 80 00:06:29,360 --> 00:06:38,000 over 2a delta at some time t prime 81 00:06:38,000 --> 00:06:44,155 and a three-dimensional delta at some location x prime. 82 00:06:44,155 --> 00:06:45,530 What I'm basically saying here is 83 00:06:45,530 --> 00:06:48,240 I'm imagining I might have some source that's extended in time 84 00:06:48,240 --> 00:06:50,330 and extended in space, and I'm just 85 00:06:50,330 --> 00:06:55,790 going to look at the little blip at one particular event 86 00:06:55,790 --> 00:07:03,230 and see if I replace my source with this one single blip, 87 00:07:03,230 --> 00:07:05,990 what field emerges from that? 88 00:07:05,990 --> 00:07:09,110 What we're going to do is assert that a solution exists. 89 00:07:17,670 --> 00:07:19,980 And we will call the field that arises 90 00:07:19,980 --> 00:07:22,660 when my source is replaced with a delta function, 91 00:07:22,660 --> 00:07:39,740 we'll call it capital G. OK? 92 00:07:39,740 --> 00:07:47,400 This notation means that this is the amount of field at t, x, 93 00:07:47,400 --> 00:07:51,730 arising from my source at t prime x prime. 94 00:07:55,370 --> 00:08:13,610 So my differential equation for this solution G 95 00:08:13,610 --> 00:08:14,470 now looks like so. 96 00:08:21,550 --> 00:08:25,937 So whatever my linear operator is when it acts on this G, 97 00:08:25,937 --> 00:08:28,270 it gives me the source, which is now the delta function. 98 00:08:31,260 --> 00:08:33,669 Let me just introduce a little bit of notation here. 99 00:08:33,669 --> 00:08:41,010 So t and x, I call this the field point. 100 00:08:41,010 --> 00:08:43,590 This is the point at which the field G is being measured. 101 00:08:51,210 --> 00:08:55,520 t prime, x prime is my source point. 102 00:08:58,330 --> 00:09:03,267 This is the point at which the source is non-zero. 103 00:09:03,267 --> 00:09:04,850 In this particular case, it's actually 104 00:09:04,850 --> 00:09:07,790 a little delta function spike. 105 00:09:07,790 --> 00:09:14,540 The reason why we do this is pretty much by definition. 106 00:09:17,816 --> 00:09:20,650 Pardon me one second. 107 00:09:20,650 --> 00:09:23,200 Yes, pretty much by definition. 108 00:09:23,200 --> 00:09:26,860 If I take the source and integrate it 109 00:09:26,860 --> 00:09:29,006 against the delta function-- 110 00:09:35,960 --> 00:09:41,060 integrate over all time, integrate over all space-- 111 00:09:59,640 --> 00:10:00,960 I must get the source back. 112 00:10:00,960 --> 00:10:02,040 This is a tautology. 113 00:10:02,040 --> 00:10:03,600 All I'm doing here is really saying 114 00:10:03,600 --> 00:10:08,660 this is the properties of delta functions. 115 00:10:08,660 --> 00:10:13,810 However, what's sort of remarkable is 116 00:10:13,810 --> 00:10:17,140 if instead of integrating the source 117 00:10:17,140 --> 00:10:18,987 against the delta functions-- 118 00:10:26,360 --> 00:10:27,860 if instead of integrating the source 119 00:10:27,860 --> 00:10:30,940 against the delta functions-- 120 00:10:30,940 --> 00:10:56,390 I integrate the source against the Green's function, 121 00:10:56,390 --> 00:11:03,320 if I do this, then what I must get out of it 122 00:11:03,320 --> 00:11:06,170 is the solution f. 123 00:11:06,170 --> 00:11:09,680 This is actually very easy to prove. 124 00:11:09,680 --> 00:11:12,650 Simply take this equation-- 125 00:11:12,650 --> 00:11:15,080 let's call this equation A-- 126 00:11:15,080 --> 00:11:21,210 if you take a equation A and operate on it 127 00:11:21,210 --> 00:11:32,340 with your differential operator D, 128 00:11:32,340 --> 00:11:38,420 so if I hit equation A with D-- remember D, 129 00:11:38,420 --> 00:11:43,200 it's a differential operator that acts on only the unprimed 130 00:11:43,200 --> 00:11:44,320 coordinates. 131 00:11:57,550 --> 00:11:58,935 You know what? 132 00:11:58,935 --> 00:12:00,310 Let's just go and write this out. 133 00:12:05,330 --> 00:12:34,770 So D on this integral, when I hit D on this, 134 00:12:34,770 --> 00:12:37,350 it goes straight through the integral. 135 00:12:37,350 --> 00:12:39,720 This only acts on the unprimed coordinates. 136 00:12:39,720 --> 00:12:41,010 So it ignores the integrals. 137 00:12:41,010 --> 00:12:42,090 It ignores the s. 138 00:13:02,840 --> 00:13:06,610 The only thing that operator hits when I act on the integral 139 00:13:06,610 --> 00:13:09,450 is the Green's function itself. 140 00:13:09,450 --> 00:13:12,550 But I have asserted that the Green's function is such 141 00:13:12,550 --> 00:13:15,070 that when D acts on it, I get the delta functions out. 142 00:13:36,960 --> 00:13:38,803 And by the definition of delta functions, 143 00:13:38,803 --> 00:13:40,220 this just gives me my source back. 144 00:13:43,340 --> 00:13:50,180 So what this means is that if we can find the Green's function 145 00:13:50,180 --> 00:13:55,670 for our differential operator, all we need to do 146 00:13:55,670 --> 00:13:58,640 is integrate it against our source. 147 00:13:58,640 --> 00:14:03,940 And the result must be the solution to our field equation. 148 00:14:03,940 --> 00:14:05,490 It's a beautiful technique. 149 00:14:05,490 --> 00:14:07,560 So if you haven't seen this before, 150 00:14:07,560 --> 00:14:13,070 I highly recommend that you turn to that reading in Arfken. 151 00:14:13,070 --> 00:14:15,290 I'm going to take advantage of this. 152 00:14:15,290 --> 00:14:17,480 I'm going to use this technique now. 153 00:14:17,480 --> 00:14:19,730 In particular, I'm going to take advantage of the fact 154 00:14:19,730 --> 00:14:25,520 that if my differential operator is in fact the flat spacetime 155 00:14:25,520 --> 00:14:29,960 wave operator, there is a very well-known solution 156 00:14:29,960 --> 00:14:31,317 for the Green's functions. 157 00:14:31,317 --> 00:14:33,650 It's called the radiative Green's function in this case. 158 00:15:05,327 --> 00:15:07,410 I'm just going to write down what the solution is. 159 00:15:40,208 --> 00:15:42,250 So I'm going to talk about the properties of this 160 00:15:42,250 --> 00:15:43,240 in just a moment. 161 00:15:43,240 --> 00:15:45,040 So I'll just comment that this is something 162 00:15:45,040 --> 00:15:49,900 that you should be able to find developed in any good quality, 163 00:15:49,900 --> 00:15:53,400 advanced electrodynamics textbook. 164 00:15:53,400 --> 00:15:56,620 In my notes here I suggest looking at this place where 165 00:15:56,620 --> 00:15:58,990 I originally went through this carefully 166 00:15:58,990 --> 00:16:02,540 was in the second edition of Jackson. 167 00:16:02,540 --> 00:16:05,440 It's presumably also present in the latest edition. 168 00:16:05,440 --> 00:16:07,410 But it might be in a slightly different place. 169 00:16:07,410 --> 00:16:08,868 In the second edition, you can find 170 00:16:08,868 --> 00:16:11,350 this discussed in Section 6.6. 171 00:16:11,350 --> 00:16:14,695 When I have taught MIT's junior level electrodynamics course, 172 00:16:14,695 --> 00:16:15,880 I actually go through this. 173 00:16:15,880 --> 00:16:19,955 And by essentially introducing a Fourier transform, 174 00:16:19,955 --> 00:16:23,320 a Fourier decomposition, turning the wave operator 175 00:16:23,320 --> 00:16:27,010 into an algebraic operator, just rearrange a bunch of terms, 176 00:16:27,010 --> 00:16:28,510 then invert a Fourier transform. 177 00:16:28,510 --> 00:16:31,600 It's not too hard to show that this is the result that 178 00:16:31,600 --> 00:16:32,350 comes out of that. 179 00:16:38,470 --> 00:16:43,050 Notice the argument of that delta function. 180 00:16:43,050 --> 00:16:45,040 And please bear in mind, we are working 181 00:16:45,040 --> 00:16:48,790 in units in which we have set the speed of light equal to 1. 182 00:17:07,920 --> 00:17:12,154 So the delta function, it has support only at-- 183 00:17:15,349 --> 00:17:18,829 well, what's going on here is this 184 00:17:18,829 --> 00:17:22,819 is taking into account the amount of time 185 00:17:22,819 --> 00:17:27,020 it takes for information to travel from the source point 186 00:17:27,020 --> 00:17:28,430 to the field point. 187 00:17:56,920 --> 00:17:59,300 OK, if I can illustrate this with a little cartoon, 188 00:17:59,300 --> 00:18:04,760 imagine I have some dynamical source of mass and energy 189 00:18:04,760 --> 00:18:06,080 that's down here wiggling away. 190 00:18:12,200 --> 00:18:16,970 Here is my one example of a source point 191 00:18:16,970 --> 00:18:19,940 that is going to end up inside the integral 192 00:18:19,940 --> 00:18:25,740 when I integrate up the Green's function against my source. 193 00:18:25,740 --> 00:18:28,080 Here I am. 194 00:18:28,080 --> 00:18:32,250 I'm an observer making my measurement out here 195 00:18:32,250 --> 00:18:33,210 at position x. 196 00:18:38,550 --> 00:18:42,360 Whatever is happening at x prime takes time 197 00:18:42,360 --> 00:18:46,860 to be communicated to me making my measurements out here 198 00:18:46,860 --> 00:18:47,850 at distance x. 199 00:18:52,460 --> 00:18:59,660 The time lag is the distance that has to be traveled, 200 00:18:59,660 --> 00:19:01,043 x minus x prime. 201 00:19:01,043 --> 00:19:02,460 And, again, I'll remind you, we're 202 00:19:02,460 --> 00:19:04,860 working in units with a speed of light is equal to 1. 203 00:19:04,860 --> 00:19:08,960 We would divide by c if we were working in normal SI units. 204 00:19:08,960 --> 00:19:09,460 OK? 205 00:19:09,460 --> 00:19:12,510 So this factor in the Green's function is building in-- 206 00:19:12,510 --> 00:19:14,550 I'm going to do the integral in just a moment-- 207 00:19:14,550 --> 00:19:18,330 it is building in the fact that we 208 00:19:18,330 --> 00:19:21,360 need to account for the amount of time 209 00:19:21,360 --> 00:19:24,510 it takes for information about the source's dynamics 210 00:19:24,510 --> 00:19:29,490 to be radiatively communicated to observers far away 211 00:19:29,490 --> 00:19:30,450 from that source. 212 00:19:39,320 --> 00:19:42,490 All right, so we are now going to apply this 213 00:19:42,490 --> 00:19:43,470 to the linearized-- 214 00:19:46,760 --> 00:19:51,250 the Einstein field equation for linearized gravity. 215 00:20:00,340 --> 00:20:05,890 So here is our field equation. 216 00:20:10,650 --> 00:20:13,850 Sadly, the letter capital G is doing multiple duties 217 00:20:13,850 --> 00:20:14,520 in this lecture. 218 00:20:26,310 --> 00:20:39,280 So when we solve this, what we need to do 219 00:20:39,280 --> 00:20:45,670 is integrate t alpha beta regarded as a function 220 00:20:45,670 --> 00:20:48,100 of prime-- 221 00:20:48,100 --> 00:20:54,980 time t prime, position x prime against the radiative Green's 222 00:20:54,980 --> 00:20:55,480 function. 223 00:21:03,260 --> 00:21:05,975 Plug in the radiative Green's function, which we use. 224 00:21:08,313 --> 00:21:10,730 This is the Green's function that corresponds to that wave 225 00:21:10,730 --> 00:21:11,540 operator. 226 00:21:11,540 --> 00:21:15,170 We have a minus 1 over 4 pi hitting our minus 16 227 00:21:15,170 --> 00:21:16,210 pi out front there. 228 00:21:22,360 --> 00:21:26,650 We're going to do the integral over the time variable. 229 00:21:26,650 --> 00:21:29,860 And we're going to leave our result expressed 230 00:21:29,860 --> 00:21:33,250 as an integral over space. 231 00:22:06,690 --> 00:22:12,310 This is an exact solution of the linearized field equations 232 00:22:12,310 --> 00:22:14,260 of general relativity. 233 00:22:14,260 --> 00:22:17,270 This is telling me that what happens-- 234 00:22:17,270 --> 00:22:22,600 the field that is measured, the trace reverse perturbation that 235 00:22:22,600 --> 00:22:24,640 is measured at time t and position 236 00:22:24,640 --> 00:22:29,380 x prime is given by integrating over my source, all 237 00:22:29,380 --> 00:22:33,730 the dynamics that happened earlier than that time t. 238 00:22:33,730 --> 00:22:37,240 What I need to do is take into account that over this source I 239 00:22:37,240 --> 00:22:40,030 have to fold in the amount of time 240 00:22:40,030 --> 00:22:42,220 it takes for information from source 241 00:22:42,220 --> 00:22:46,170 point x prime to go out to x. 242 00:22:46,170 --> 00:22:47,650 I integrate over the source. 243 00:22:47,650 --> 00:22:49,360 At every point in the integrand, I 244 00:22:49,360 --> 00:22:53,500 divide by the distance between the field point and the source 245 00:22:53,500 --> 00:22:54,360 point. 246 00:22:54,360 --> 00:22:56,560 Do that integral, multiply by 4G. 247 00:22:56,560 --> 00:22:57,460 There's my solution. 248 00:23:02,420 --> 00:23:03,420 This is wonderful. 249 00:23:03,420 --> 00:23:06,480 It's actually an exact solution. 250 00:23:06,480 --> 00:23:13,180 Like most exact solutions, it's not always 251 00:23:13,180 --> 00:23:16,380 the most useful thing in the world. 252 00:23:16,380 --> 00:23:19,000 There's one problem which we need to diagnose, 253 00:23:19,000 --> 00:23:25,820 which is that in the way that this has been formulated, 254 00:23:25,820 --> 00:23:33,800 it looks like every component of this trace reverse metric looks 255 00:23:33,800 --> 00:23:34,836 radiative. 256 00:24:00,660 --> 00:24:03,120 What do I mean by it looks radiative? 257 00:24:03,120 --> 00:24:05,640 Well, it came from a wave equation. 258 00:24:05,640 --> 00:24:10,110 They all depend on information that traveled out 259 00:24:10,110 --> 00:24:12,420 at the speed of light with a time lag 260 00:24:12,420 --> 00:24:14,430 appropriate to the speed with which radiation 261 00:24:14,430 --> 00:24:15,848 moves in flat spacetime. 262 00:24:15,848 --> 00:24:17,640 You know, you look at that, and you sort of 263 00:24:17,640 --> 00:24:21,090 say, wow, the entire metric has this sort of character 264 00:24:21,090 --> 00:24:25,020 that we associate with a propagating wave. 265 00:24:25,020 --> 00:24:29,280 All of spacetime is radiative. 266 00:24:29,280 --> 00:24:31,740 We need to be careful about that, because here 267 00:24:31,740 --> 00:24:33,900 is where we need to revisit and think 268 00:24:33,900 --> 00:24:38,993 carefully about what happens when we choose our gauge. 269 00:24:38,993 --> 00:24:41,160 What I'm going to show you is a little demonstration 270 00:24:41,160 --> 00:24:48,570 that in this case, the gauge we chose-- 271 00:24:51,830 --> 00:24:56,540 and let me emphasize that was a wonderful gauge for putting 272 00:24:56,540 --> 00:25:00,290 the horribly messy field equation into a form where 273 00:25:00,290 --> 00:25:02,330 we could actually solve it-- 274 00:25:02,330 --> 00:25:10,850 but the gauge we chose is unfortunately 275 00:25:10,850 --> 00:25:20,451 masking some of the physical character of the spacetime. 276 00:25:28,150 --> 00:25:34,750 So this Lorenz gauge was great for producing 277 00:25:34,750 --> 00:25:36,640 a set of field equations that we could 278 00:25:36,640 --> 00:25:38,590 bring to bear powerful mathematical techniques 279 00:25:38,590 --> 00:25:41,560 and get a closed form actually exact 280 00:25:41,560 --> 00:25:44,800 within the context of linearized theory, an exact solution. 281 00:25:44,800 --> 00:25:47,830 But it might be misleading us as to what 282 00:25:47,830 --> 00:25:49,470 the resulting solution means. 283 00:25:53,940 --> 00:25:57,032 Let me illustrate this with an example from electrodynamics. 284 00:26:03,440 --> 00:26:11,310 Suppose I gave you a vector potential that was of the form 285 00:26:11,310 --> 00:26:21,370 at a time-like component that was q over r minus q omega sine 286 00:26:21,370 --> 00:26:28,850 q dot r minus omega t capital R over r. 287 00:26:32,410 --> 00:26:37,630 Little r is square root of x squared 288 00:26:37,630 --> 00:26:40,210 plus y squared plus z squared. 289 00:26:40,210 --> 00:26:44,590 And capital R is some parameter with the dimensions of length. 290 00:26:47,330 --> 00:26:51,550 And my three spatial components of this thing, 291 00:26:51,550 --> 00:26:59,040 I'm going to write them as kqi sine k 292 00:26:59,040 --> 00:27:07,705 dot r minus omega t capital R over r minus qxi. 293 00:27:18,300 --> 00:27:23,480 OK, this is the potential that has a form that kind of looks 294 00:27:23,480 --> 00:27:30,040 like if I were to sort of give this to you on an oral exam 295 00:27:30,040 --> 00:27:33,830 and say without doing any calculation, what would 296 00:27:33,830 --> 00:27:35,798 you guess this is? 297 00:27:35,798 --> 00:27:37,340 And that would actually be, as you'll 298 00:27:37,340 --> 00:27:39,350 see in a moment, that'd be a very unfair question of me 299 00:27:39,350 --> 00:27:39,800 to ask. 300 00:27:39,800 --> 00:27:41,633 And I look at this and I say, you know what? 301 00:27:41,633 --> 00:27:43,880 That looks kind of like a Coulomb field. 302 00:27:43,880 --> 00:27:46,970 And then there is something radiative 303 00:27:46,970 --> 00:27:49,068 that it's embedded in, kind of a funny amplitude 304 00:27:49,068 --> 00:27:50,360 associated with that radiation. 305 00:27:50,360 --> 00:27:55,100 But maybe this is sort of like a dipole radiator 306 00:27:55,100 --> 00:27:59,370 with a net monopole moment going on it. 307 00:27:59,370 --> 00:28:02,510 You know, we've got things that involve fields that 308 00:28:02,510 --> 00:28:04,850 oscillate in space and time. 309 00:28:04,850 --> 00:28:06,240 They fall off as 1 over r's. 310 00:28:06,240 --> 00:28:08,600 A component falls off as 1 over r cubed. 311 00:28:08,600 --> 00:28:11,480 Yeah, that would be my guess-- a point charged with radiation. 312 00:28:22,110 --> 00:28:24,950 So your oral examiner, the next thing they say 313 00:28:24,950 --> 00:28:28,580 is, great, work out the electromagnetic field tensor 314 00:28:28,580 --> 00:28:29,630 corresponding to this. 315 00:28:40,570 --> 00:28:43,023 So you do this by taking some derivatives. 316 00:28:47,190 --> 00:29:07,420 And what results is this. 317 00:29:07,420 --> 00:29:11,440 This is the electromagnetic field tensor corresponding 318 00:29:11,440 --> 00:29:25,350 to a Coulomb electric field with no magnetic field. 319 00:29:25,350 --> 00:29:30,180 This potential is nothing more than a Coulomb point 320 00:29:30,180 --> 00:29:37,740 charge in a really dumb gauge. 321 00:29:37,740 --> 00:29:40,590 The way I actually generated this was I started with the q 322 00:29:40,590 --> 00:29:43,145 over r potential of a Coulomb point charge. 323 00:29:43,145 --> 00:29:46,080 And I just applied some crazy gauge generator to it 324 00:29:46,080 --> 00:29:48,140 to see what happened. 325 00:29:48,140 --> 00:29:52,430 So the parable of this story is whenever 326 00:29:52,430 --> 00:29:54,500 you are working with quantities that 327 00:29:54,500 --> 00:29:59,800 are subject to gauge transformations, 328 00:29:59,800 --> 00:30:03,620 you have to be careful about drawing conclusions about what 329 00:30:03,620 --> 00:30:05,990 those quantities mean in terms of the physics you 330 00:30:05,990 --> 00:30:07,120 want to get out of it. 331 00:30:07,120 --> 00:30:11,090 Gauge can obscure the physics if we are not careful. 332 00:30:11,090 --> 00:30:13,460 And in particular-- there's almost like a conservation 333 00:30:13,460 --> 00:30:15,140 of pain principle here-- 334 00:30:15,140 --> 00:30:17,990 it is often the case that the best gauge 335 00:30:17,990 --> 00:30:22,190 for formulating tractable equations for solving 336 00:30:22,190 --> 00:30:25,910 your problem turn out to be gauges that give you 337 00:30:25,910 --> 00:30:28,790 just kind of crappy results if you want to interpret what's 338 00:30:28,790 --> 00:30:31,314 going on physically. 339 00:30:31,314 --> 00:30:33,723 In the remainder of this lecture, 340 00:30:33,723 --> 00:30:35,390 I am going to introduce something that's 341 00:30:35,390 --> 00:30:38,270 a slightly advanced topic. 342 00:30:38,270 --> 00:30:42,713 And I pretty shamelessly stole this from a paper that I wrote, 343 00:30:42,713 --> 00:30:44,630 something that I did with a colleague of mine, 344 00:30:44,630 --> 00:30:45,380 Eanna Flanagan. 345 00:30:45,380 --> 00:30:48,290 And it's based on an idea that was developed 346 00:30:48,290 --> 00:30:53,395 in a different context by my MIT colleague Ed Bertschinger. 347 00:30:53,395 --> 00:30:54,770 So here is what we're going to do 348 00:30:54,770 --> 00:31:00,430 in the remainder of this lecture. 349 00:31:00,430 --> 00:31:11,510 What we're going to do is recast the metric and the source 350 00:31:11,510 --> 00:31:24,380 in a form, which allows us to categorize 351 00:31:24,380 --> 00:31:27,860 the radiative and non-radiative degrees of freedom 352 00:31:27,860 --> 00:31:29,103 of the spacetime. 353 00:31:49,073 --> 00:31:51,240 This will, in fact-- we're going to do this in a way 354 00:31:51,240 --> 00:31:56,850 where the results are, in fact, completely generic, 355 00:31:56,850 --> 00:31:59,670 provided we stick to linearized gravity, 356 00:31:59,670 --> 00:32:05,700 gravity linearized around flat spacetime, 357 00:32:05,700 --> 00:32:08,400 which will be sufficient for our discussion right now. 358 00:32:13,310 --> 00:32:16,920 I will give you the punch line upfront. 359 00:32:21,190 --> 00:32:28,500 What we will discover is that spacetime 360 00:32:28,500 --> 00:32:41,990 has four physical degrees of freedom that are non-radiative. 361 00:32:49,848 --> 00:32:51,390 And the hallmark of this is that they 362 00:32:51,390 --> 00:32:54,580 are going to be governed by differential equations that 363 00:32:54,580 --> 00:32:56,320 look like the Poisson equation. 364 00:32:56,320 --> 00:32:59,600 They will look like the Laplace operator on some potential 365 00:32:59,600 --> 00:33:00,475 is equal to a source. 366 00:33:05,041 --> 00:33:06,432 Let me rewrite that. 367 00:33:06,432 --> 00:33:07,640 This handwriting is terrible. 368 00:33:31,210 --> 00:33:37,450 Spacetime has two more degrees of freedom that are radiative. 369 00:33:47,070 --> 00:33:49,182 These are governed by wave equations. 370 00:34:07,910 --> 00:34:11,870 The metric tensor of spacetime has 10 components, 371 00:34:11,870 --> 00:34:14,469 10 unique components. 372 00:34:14,469 --> 00:34:16,810 It's represented by about 4 by 4 symmetric matrix. 373 00:34:16,810 --> 00:34:17,310 OK? 374 00:34:17,310 --> 00:34:19,520 So it has 10 components in it. 375 00:34:19,520 --> 00:34:21,145 4 plus 2 does not equal 10. 376 00:34:21,145 --> 00:34:22,520 You might look at this and think, 377 00:34:22,520 --> 00:34:25,159 what about the other four components of this thing? 378 00:34:25,159 --> 00:34:27,320 I've got 4 degrees of freedom. 379 00:34:27,320 --> 00:34:30,767 That must correspond in some sense to 4 of those components. 380 00:34:30,767 --> 00:34:32,350 I've got another 2 degrees of freedom. 381 00:34:32,350 --> 00:34:34,350 That must correspond to two of those components. 382 00:34:34,350 --> 00:34:35,949 What's going on the other four? 383 00:34:35,949 --> 00:34:38,270 Well, the other 4 degrees of freedom 384 00:34:38,270 --> 00:34:41,429 are essentially eaten up by our gauge freedom. 385 00:34:41,429 --> 00:34:44,540 OK, we are free to specify our coordinates 386 00:34:44,540 --> 00:34:46,520 via an infinitesimal coordinate shift. 387 00:34:46,520 --> 00:34:50,719 And so we have 4 degrees of freedom under our control. 388 00:34:50,719 --> 00:34:53,750 General relativity insists on giving physics 389 00:34:53,750 --> 00:34:56,120 to the other six. 390 00:34:56,120 --> 00:34:58,700 These four non-radiative degrees of freedom, the two 391 00:34:58,700 --> 00:35:01,910 radiative ones, and the four gauge degrees of freedom 392 00:35:01,910 --> 00:35:04,550 completely describe the spacetime metric. 393 00:35:04,550 --> 00:35:07,550 And just to use an analogy, these non-radiative 394 00:35:07,550 --> 00:35:09,380 of degrees of freedom are kind of 395 00:35:09,380 --> 00:35:12,770 like the non-radiative electric and magnetic fields 396 00:35:12,770 --> 00:35:14,740 that you get in electrodynamics. 397 00:35:14,740 --> 00:35:16,310 Your two radiative degrees of freedom 398 00:35:16,310 --> 00:35:20,210 are going to turn out to two polarization states associated 399 00:35:20,210 --> 00:35:24,140 with waves in the gravitational field in the same way 400 00:35:24,140 --> 00:35:27,350 that in electrodynamics, the electric and magnetic field 401 00:35:27,350 --> 00:35:32,670 also have two polarizations associated with them. 402 00:35:32,670 --> 00:35:38,090 So this is all worked out in great detail in a paper 403 00:35:38,090 --> 00:35:40,400 that Eanna Flanagan of Cornell University and I 404 00:35:40,400 --> 00:35:44,670 wrote about 16 years ago. 405 00:35:44,670 --> 00:35:50,120 And so I will make available through the 8.962 website 406 00:35:50,120 --> 00:35:53,210 a copy of that paper. 407 00:35:53,210 --> 00:35:55,700 And it was Flanagan who sort of got this started. 408 00:35:55,700 --> 00:35:58,700 And he and I then figured out all the details together. 409 00:35:58,700 --> 00:36:02,030 Ed Bertschinger is sort of the grandfather of this idea 410 00:36:02,030 --> 00:36:04,490 since he did something very similar 411 00:36:04,490 --> 00:36:10,580 in an analysis that characterized perturbations 412 00:36:10,580 --> 00:36:13,250 to cosmological spacetimes. 413 00:36:13,250 --> 00:36:14,910 All right, so here's how we proceed. 414 00:36:18,180 --> 00:36:22,500 Let's consider h mu nu as a tensor 415 00:36:22,500 --> 00:36:24,469 field on a flat background. 416 00:36:36,950 --> 00:36:40,460 In the previous lecture that I recorded, 417 00:36:40,460 --> 00:36:45,560 I described how when I am working in linearized theory, 418 00:36:45,560 --> 00:36:47,507 it's a useful fiction to think of h mu nu. 419 00:36:47,507 --> 00:36:49,340 Even though we know it's actually telling us 420 00:36:49,340 --> 00:36:50,630 about the curvature of spacetime, 421 00:36:50,630 --> 00:36:52,670 it's a useful fiction to think of it as a tensor 422 00:36:52,670 --> 00:36:54,200 field in a flat background. 423 00:36:54,200 --> 00:36:57,080 And we're going to take advantage of that. 424 00:36:57,080 --> 00:37:03,427 We are going to choose time and space coordinates. 425 00:37:13,380 --> 00:37:16,800 Once I have chosen time and space coordinates, 426 00:37:16,800 --> 00:37:23,160 I then think about how this 4 by 4 tensor, how 427 00:37:23,160 --> 00:37:25,050 its different components break up 428 00:37:25,050 --> 00:37:28,690 and how they behave with respect to rotations in the spatial 429 00:37:28,690 --> 00:37:29,190 coordinates. 430 00:37:29,190 --> 00:37:34,610 I can imagine going in and tweak my spatial coordinate system. 431 00:37:34,610 --> 00:37:51,510 And what I want to do is examine how the components break 432 00:37:51,510 --> 00:38:05,560 up into subgroups with respect to spatial only 433 00:38:05,560 --> 00:38:07,123 coordinate transformations. 434 00:38:18,730 --> 00:38:24,790 What we'll see is h mu nu is going to break up into-- you're 435 00:38:24,790 --> 00:38:26,560 going to get one piece. 436 00:38:26,560 --> 00:38:28,670 It's your time-time piece. 437 00:38:28,670 --> 00:38:29,380 OK? 438 00:38:29,380 --> 00:38:32,170 If I have chosen my time and my space coordinates 439 00:38:32,170 --> 00:38:36,160 and I imagine messing around with my spatial coordinate 440 00:38:36,160 --> 00:38:39,010 system, htt is unchanged. 441 00:38:39,010 --> 00:38:39,848 And so this-- 442 00:38:39,848 --> 00:38:41,390 I'm actually going to give it a name. 443 00:38:41,390 --> 00:38:44,610 I'm going to call it minus 2 phi-- 444 00:38:44,610 --> 00:38:46,210 this behaves as a scalar. 445 00:38:49,020 --> 00:38:55,650 My spacetime piece-- so components of this, 446 00:38:55,650 --> 00:38:56,820 of the form hti-- 447 00:38:59,873 --> 00:39:02,290 I'm going to characterize this in more detail a little bit 448 00:39:02,290 --> 00:39:03,340 later. 449 00:39:03,340 --> 00:39:05,290 But these three numbers are going 450 00:39:05,290 --> 00:39:08,260 to behave with respect to spatial only coordinate 451 00:39:08,260 --> 00:39:10,060 transformations like a vector. 452 00:39:16,620 --> 00:39:20,880 Likewise hij is going to behave like a 3 by 3 tensor. 453 00:39:33,030 --> 00:39:36,060 So let's break this down a little bit further. 454 00:39:36,060 --> 00:39:41,130 Whenever I am dealing with a function that 455 00:39:41,130 --> 00:39:44,880 is vector in nature-- 456 00:39:44,880 --> 00:39:46,950 and bear mind it's a field. 457 00:39:46,950 --> 00:39:48,450 So I'm dealing with a vector field-- 458 00:40:00,850 --> 00:40:07,150 I'm going to write this as a divergence-free function 459 00:40:07,150 --> 00:40:09,340 plus the gradient of some scalar. 460 00:40:12,990 --> 00:40:17,470 So what I'm going to do is say hti 461 00:40:17,470 --> 00:40:23,590 is equal to beta i plus the gradient of some gamma. 462 00:40:28,005 --> 00:40:30,380 And I'm going to require that the divergence of that beta 463 00:40:30,380 --> 00:40:32,060 be equal to 0. 464 00:40:32,060 --> 00:40:36,800 Notice, since all of my indices are spatial indices 465 00:40:36,800 --> 00:40:39,878 and I am working in nearly Lorenz coordinates-- 466 00:40:44,155 --> 00:40:46,280 the placement of the indices, whether it's upstairs 467 00:40:46,280 --> 00:40:48,095 or downstairs, is immaterial-- 468 00:40:57,500 --> 00:41:00,200 I will tend to write them all in the downstairs position. 469 00:41:00,200 --> 00:41:02,752 And so repeated indices-- 470 00:41:02,752 --> 00:41:04,460 we're going to sort of abuse the Einstein 471 00:41:04,460 --> 00:41:07,002 notation a little bit-- repeated indices will be summed over. 472 00:41:21,120 --> 00:41:24,510 Let's extend this logic to the tensor piece, to hij. 473 00:41:29,822 --> 00:41:31,280 This takes a little bit of thought. 474 00:41:31,280 --> 00:41:33,440 So let me just write out the answer 475 00:41:33,440 --> 00:41:36,650 and then describe the character of every piece that 476 00:41:36,650 --> 00:41:37,370 goes into this. 477 00:41:52,620 --> 00:41:59,630 So hij can be written as hij tt-- 478 00:41:59,630 --> 00:42:02,660 I will define that in a moment-- 479 00:42:02,660 --> 00:42:16,760 plus 1/3 hij plus d, sort of a symmetrized gradient 480 00:42:16,760 --> 00:42:27,650 of a vector epsilon, plus delta i delta j minus 1/3 delta 481 00:42:27,650 --> 00:42:32,420 ij plus operator on some function lambda. 482 00:42:35,420 --> 00:42:37,340 So let me go through and describe 483 00:42:37,340 --> 00:42:39,470 what each of these things mean. 484 00:42:39,470 --> 00:42:41,870 And while I'm at it, let me count up 485 00:42:41,870 --> 00:42:44,390 the number of degrees of freedom associated with them. 486 00:42:44,390 --> 00:42:47,540 Let's back up for just a second and do that over here. 487 00:42:47,540 --> 00:42:48,800 Gamma is a scalar. 488 00:42:52,500 --> 00:42:55,280 So there's 1 degree of freedom associated with gamma. 489 00:42:58,570 --> 00:42:59,910 Let me move this over to here. 490 00:42:59,910 --> 00:43:01,340 So that's 1 degree of freedom. 491 00:43:05,130 --> 00:43:13,252 Beta is a vector, but it's divergenceless. 492 00:43:20,490 --> 00:43:22,850 Because it's a vector, I have three components. 493 00:43:25,470 --> 00:43:28,110 But being divergence-free, those three components 494 00:43:28,110 --> 00:43:30,690 obey a constraint. 495 00:43:30,690 --> 00:43:34,314 So that's 3 minus 1. 496 00:43:34,314 --> 00:43:38,150 So this scalar and this divergence-free vector 497 00:43:38,150 --> 00:43:40,100 give me the 3 degrees of freedom I 498 00:43:40,100 --> 00:43:43,540 need to specify the 3 independent components of hti. 499 00:43:46,790 --> 00:43:50,270 Over here, as I look through all of these different functions, 500 00:43:50,270 --> 00:43:51,410 h is a scalar. 501 00:43:58,520 --> 00:44:00,920 I have defined it in such a way-- 502 00:44:06,250 --> 00:44:07,250 hold on just one second. 503 00:44:07,250 --> 00:44:09,500 I'm going to come back to that point in just a moment. 504 00:44:12,000 --> 00:44:17,850 Epsilon j is a vector. 505 00:44:17,850 --> 00:44:33,720 It is defined such that its divergence is 0. 506 00:44:33,720 --> 00:44:35,970 OK? 507 00:44:35,970 --> 00:44:40,217 So what is going on here is this contribution to hij, 508 00:44:40,217 --> 00:44:42,050 it gives me a piece of this thing that looks 509 00:44:42,050 --> 00:44:43,217 like the gradient of vector. 510 00:44:47,560 --> 00:44:52,210 This has 3 degrees of freedom minus 1 constraint. 511 00:44:52,210 --> 00:44:54,040 This is a scalar. 512 00:44:54,040 --> 00:44:57,550 It only has 1 degree of freedom associated with it. 513 00:45:01,110 --> 00:45:03,810 Lambda is another scalar. 514 00:45:09,560 --> 00:45:15,260 Notice whereas h feeds directly into hij its derivatives 515 00:45:15,260 --> 00:45:16,640 of lambda that feed into this. 516 00:45:23,740 --> 00:45:25,230 So this has one degree of freedom. 517 00:45:28,710 --> 00:45:30,360 But this is defined in such a way 518 00:45:30,360 --> 00:45:37,240 that if I take the trace of hij, this operator gives me 0. 519 00:45:37,240 --> 00:45:37,740 Right? 520 00:45:37,740 --> 00:45:55,320 So the trace-- what the trace will do 521 00:45:55,320 --> 00:46:01,120 is sum over elements when i and j are equal. 522 00:46:01,120 --> 00:46:05,850 So this will give me essentially the Laplace operator. 523 00:46:05,850 --> 00:46:09,490 But the trace of delta ij is 3. 524 00:46:09,490 --> 00:46:12,310 So I get Laplace operator minus Laplace operator, 525 00:46:12,310 --> 00:46:16,230 I get the 0 operator acting on lambda. 526 00:46:16,230 --> 00:46:19,970 So this gives me 1 degree of freedom. 527 00:46:25,320 --> 00:46:29,100 This gave me 1 degree of freedom. 528 00:46:29,100 --> 00:46:32,880 hij tt-- and you know what, I'm going 529 00:46:32,880 --> 00:46:34,652 to go to a separate board for this one. 530 00:47:04,710 --> 00:47:07,180 This is a tensor. 531 00:47:07,180 --> 00:47:13,650 tt stands for transverse and traceless. 532 00:47:13,650 --> 00:47:17,310 This is defined so that if I take its trace-- 533 00:47:17,310 --> 00:47:19,250 oops, I said this would all be downstairs-- 534 00:47:19,250 --> 00:47:26,260 if I evaluate this, I get 0. 535 00:47:26,260 --> 00:47:26,760 OK? 536 00:47:26,760 --> 00:47:30,660 So hij is a tensor but with no trace. 537 00:47:30,660 --> 00:47:37,320 And it's defined so that if I take it's divergence, I get 0. 538 00:47:37,320 --> 00:47:43,320 So this guy has 6 independent components. 539 00:47:43,320 --> 00:47:45,242 This is 1 constraint. 540 00:47:48,480 --> 00:47:50,220 This is 3 constraints, right? 541 00:47:50,220 --> 00:47:52,400 Because it holds for every value of j. 542 00:47:56,950 --> 00:48:02,740 So this ends up giving me 2 free functions. 543 00:48:02,740 --> 00:48:06,304 So as we sum all these up-- 544 00:48:06,304 --> 00:48:08,190 let's see, did I miss anyone here? 545 00:48:14,730 --> 00:48:15,540 Right. 546 00:48:15,540 --> 00:48:30,330 So I have totally characterized h mu nu 547 00:48:30,330 --> 00:48:40,464 by a set of scalar, vector, and tensor functions. 548 00:48:45,310 --> 00:48:49,390 I have a scalar phi, which I have over here 549 00:48:49,390 --> 00:48:51,850 with my time-time piece. 550 00:48:51,850 --> 00:48:55,580 I have a scalar gamma. 551 00:48:55,580 --> 00:48:58,930 I have a vector, beta i. 552 00:48:58,930 --> 00:49:08,220 I have a scalar h, a scalar lambda, a vector epsilon i, 553 00:49:08,220 --> 00:49:13,910 and finally, a tensor hij tt. 554 00:49:13,910 --> 00:49:14,410 OK? 555 00:49:14,410 --> 00:49:16,553 So what I've done is-- 556 00:49:16,553 --> 00:49:18,640 let me just step back for a second. 557 00:49:18,640 --> 00:49:20,320 What I've been doing in this exercise 558 00:49:20,320 --> 00:49:23,140 is trying to figure out with respect 559 00:49:23,140 --> 00:49:26,410 to its behavior under rotations and then 560 00:49:26,410 --> 00:49:29,010 just because it'll prove to be convenient in a moment 561 00:49:29,010 --> 00:49:34,330 its behavior when I sort of look at casting quantities 562 00:49:34,330 --> 00:49:37,210 in this irreducible form that involves 563 00:49:37,210 --> 00:49:40,655 divergence-free vectors and gradients of scalars. 564 00:49:40,655 --> 00:49:42,280 This is sort of the equivalent of doing 565 00:49:42,280 --> 00:49:44,950 that for a tensor function. 566 00:49:44,950 --> 00:49:48,820 The 10 independent components in h mu nu 567 00:49:48,820 --> 00:49:50,560 have now been encapsulated by-- 568 00:49:50,560 --> 00:50:01,470 I've got one function here 1 here, 2 here, 1 here, 1 here 2 569 00:50:01,470 --> 00:50:04,200 here, 2 here. 570 00:50:04,200 --> 00:50:09,720 1, 2, 4, 5, 6, 8 and 10. 571 00:50:09,720 --> 00:50:13,530 So all I've done is rewrite those 10 independent components 572 00:50:13,530 --> 00:50:17,340 of h mu nu into a form, that as we're going to see, 573 00:50:17,340 --> 00:50:21,570 is particularly convenient for allowing us to understand what 574 00:50:21,570 --> 00:50:24,312 the gauge invariant degrees of freedom in spacetime 575 00:50:24,312 --> 00:50:25,770 are, at least in linearized theory. 576 00:50:33,880 --> 00:50:38,920 All right, so, so far, all I've really done 577 00:50:38,920 --> 00:50:48,040 is rewrite my metric using a set of auxiliary variables 578 00:50:48,040 --> 00:50:51,610 that might look a little bit crazy. 579 00:50:51,610 --> 00:50:59,650 What we would like to do is examine 580 00:50:59,650 --> 00:51:01,960 what my linearized Einstein field 581 00:51:01,960 --> 00:51:06,250 equation looks like in terms of all of these new fields. 582 00:51:18,920 --> 00:51:31,050 So in terms of the phi gamma beta h lambda epsilon hij tt, 583 00:51:31,050 --> 00:51:33,480 I would like to run this through the mechanism, 584 00:51:33,480 --> 00:51:35,700 make my Einstein field equation, but express it 585 00:51:35,700 --> 00:51:37,110 in terms of these things. 586 00:51:37,110 --> 00:51:39,475 We'll see why that is in just a moment. 587 00:51:39,475 --> 00:51:41,850 But before I do this, we have to be a little bit careful. 588 00:51:41,850 --> 00:51:44,250 We have 10 functions here. 589 00:51:44,250 --> 00:51:46,630 We have 4 gauge degrees of freedom. 590 00:51:46,630 --> 00:51:48,130 And the whole point of this exercise 591 00:51:48,130 --> 00:51:51,450 is to try to understand how to deal with the fact 592 00:51:51,450 --> 00:51:54,720 that a gauge that is convenient for doing my calculation 593 00:51:54,720 --> 00:51:57,690 may leave me with a result that is 594 00:51:57,690 --> 00:51:59,580 confusing in terms of the physics 595 00:51:59,580 --> 00:52:01,780 I'm trying to understand. 596 00:52:01,780 --> 00:52:07,960 So what we're going to do is say that I 597 00:52:07,960 --> 00:52:21,880 can take my generator of a gauge transformation, xi alpha. 598 00:52:21,880 --> 00:52:24,820 I can write this since I have chosen time 599 00:52:24,820 --> 00:52:27,520 and space coordinates. 600 00:52:27,520 --> 00:52:30,310 I can break it into a time-like piece and a spatial piece. 601 00:52:33,440 --> 00:52:36,260 I'm going to say that the timeline piece is some scalar 602 00:52:36,260 --> 00:52:42,910 field A. And spatial piece looks like a vector field B 603 00:52:42,910 --> 00:52:48,820 sub i plus the gradient of some scalar c. 604 00:52:48,820 --> 00:52:55,140 And I'm going to require Di Bi to be equal to 0. 605 00:53:11,607 --> 00:53:22,520 The reason why I'm doing this is these functions that I came up 606 00:53:22,520 --> 00:53:26,930 with, my phis and gammas and betas and epsilons, whatever, 607 00:53:26,930 --> 00:53:33,460 they are going to change if I introduce a gauge 608 00:53:33,460 --> 00:53:34,701 transformation. 609 00:53:42,070 --> 00:53:44,760 So suppose I change gauge as we learned how 610 00:53:44,760 --> 00:53:46,010 to do in the previous lecture. 611 00:53:57,980 --> 00:54:00,440 When you do this, what you discover 612 00:54:00,440 --> 00:54:08,450 is the function phi changes to the original phi, 613 00:54:08,450 --> 00:54:11,820 so something that looks like the time derivative of that scalar 614 00:54:11,820 --> 00:54:23,390 A. The beta picks up a term that looks like the time 615 00:54:23,390 --> 00:54:33,620 derivative of the vector field B. Gamma 616 00:54:33,620 --> 00:54:38,570 goes to gamma minus A minus time derivative of c. 617 00:54:41,360 --> 00:54:45,920 H goes to H minus 2 nabla square root of c. 618 00:55:02,910 --> 00:55:05,430 So all of these functions that we sort of cleverly 619 00:55:05,430 --> 00:55:10,080 introduced trying to write these different pieces of the metric 620 00:55:10,080 --> 00:55:14,400 of spacetime in an irreducible form, when we change gauge, 621 00:55:14,400 --> 00:55:18,790 they change in this way. 622 00:55:18,790 --> 00:55:20,210 I left one out. 623 00:55:20,210 --> 00:55:32,760 It turns out when you change gauge, hij tt goes to hij tt. 624 00:55:32,760 --> 00:55:37,366 This piece is actually gauge invariant. 625 00:55:49,200 --> 00:55:51,200 That's really interesting, because that tells me 626 00:55:51,200 --> 00:55:56,600 that once I have worked out my field 627 00:55:56,600 --> 00:56:00,860 equations, the piece of it that describes 628 00:56:00,860 --> 00:56:03,470 this transverse and traceless piece 629 00:56:03,470 --> 00:56:08,720 of the metric perturbation, it has 630 00:56:08,720 --> 00:56:11,300 meaning in any representation that I write down. 631 00:56:11,300 --> 00:56:15,260 So the piece that looks radiative in one 632 00:56:15,260 --> 00:56:17,120 representation, in fact, is radiative 633 00:56:17,120 --> 00:56:18,703 in all representations. 634 00:56:18,703 --> 00:56:20,120 We still have to figure out what's 635 00:56:20,120 --> 00:56:21,370 going on with all this stuff. 636 00:56:21,370 --> 00:56:21,550 OK? 637 00:56:21,550 --> 00:56:23,210 So we have a little bit more work to do. 638 00:56:23,210 --> 00:56:25,752 But we've just learned something very interesting about this. 639 00:56:33,450 --> 00:56:41,780 So at this point, there's really no simple way 640 00:56:41,780 --> 00:56:43,610 to describe what you do next. 641 00:56:43,610 --> 00:56:45,950 Honest to god, what you essentially do 642 00:56:45,950 --> 00:56:50,540 is you just stare at these things for a while. 643 00:56:50,540 --> 00:56:55,520 And you start to notice that there are certain combinations 644 00:56:55,520 --> 00:57:00,980 of these different functions that, like hij tt, 645 00:57:00,980 --> 00:57:03,880 certain combinations of them are gauge invariant. 646 00:57:47,880 --> 00:57:57,370 So if I define capital Phi to be little phi plus the time 647 00:57:57,370 --> 00:58:07,550 derivative of my gamma minus 1/2 2 derivatives of lambda, 648 00:58:07,550 --> 00:58:17,270 if I define theta as 1/2 h minus Laplace operator on lambda. 649 00:58:17,270 --> 00:58:23,760 I define the vector field, psi i to be 650 00:58:23,760 --> 00:58:29,040 beta i minus 1/2 epsilon i-- 651 00:58:29,040 --> 00:58:35,620 note, this vector field is divergence-free-- 652 00:58:35,620 --> 00:58:38,340 and, of course, hij tt. 653 00:58:45,390 --> 00:58:49,200 Every one of these combinations is unchanged under a gauge 654 00:58:49,200 --> 00:58:50,417 transformation. 655 00:59:32,180 --> 00:59:34,100 But let's note something. 656 00:59:34,100 --> 00:59:35,600 Phi is a scalar field. 657 00:59:39,980 --> 00:59:41,450 Theta is a scalar field. 658 00:59:45,020 --> 00:59:48,590 So I have 1 plus 1 functions associated with them. 659 00:59:52,610 --> 00:59:54,460 This is a divergence-free vector. 660 01:00:03,150 --> 01:00:08,500 So it has 3 minus 1 degrees of freedom associated with it. 661 01:00:11,690 --> 01:00:17,840 And hij, this is traceless and divergence free. 662 01:00:17,840 --> 01:00:22,320 And as I already counted up, this guy 663 01:00:22,320 --> 01:00:31,540 has 6 minus 1 minus 3 degrees of freedom in it, also 2. 664 01:00:31,540 --> 01:00:34,060 So when I characterize the gauge invariant 665 01:00:34,060 --> 01:00:36,070 degrees of freedom in the spacetime, 666 01:00:36,070 --> 01:00:38,040 I've only got 6 functions left. 667 01:00:46,450 --> 01:00:48,650 But this is good, right? 668 01:00:48,650 --> 01:00:52,610 My original 10 degrees of freedom in the spacetime metric 669 01:00:52,610 --> 01:00:55,670 are characterized by these sort of 6 fundamental degrees 670 01:00:55,670 --> 01:00:57,620 of freedom in the gravitational field 671 01:00:57,620 --> 01:01:02,000 plus 4 gauge degrees of freedom that are bound up 672 01:01:02,000 --> 01:01:04,735 in my gauge generators. 673 01:01:12,510 --> 01:01:14,730 So what I would like to do is see 674 01:01:14,730 --> 01:01:17,280 if I can write the Einstein field 675 01:01:17,280 --> 01:01:20,868 equations in terms of these gauge invariant 676 01:01:20,868 --> 01:01:21,660 degrees of freedom. 677 01:01:39,850 --> 01:01:42,600 And you know what, I'm going to write it in its full form. 678 01:01:42,600 --> 01:01:44,350 We're, of course, going to linearize this. 679 01:01:48,177 --> 01:01:50,010 But what I'm going to do is take this thing, 680 01:01:50,010 --> 01:01:57,510 write it in linearized gravity, using my gauge invariant 681 01:01:57,510 --> 01:01:58,140 variables. 682 01:02:08,740 --> 01:02:11,140 Before I do this, it's really helpful 683 01:02:11,140 --> 01:02:14,260 to first decompose the stress energy 684 01:02:14,260 --> 01:02:19,435 tensor in a manner similar to how we decomposed our metric. 685 01:02:29,780 --> 01:02:34,080 So what I'm going to do is I've chosen time and space 686 01:02:34,080 --> 01:02:36,353 directions. 687 01:02:36,353 --> 01:02:38,020 So I'm going to call the time-time piece 688 01:02:38,020 --> 01:02:39,530 rho in energy density. 689 01:02:43,170 --> 01:02:45,530 The timespace piece, we know that this tells me 690 01:02:45,530 --> 01:02:47,090 something about the flow of energy 691 01:02:47,090 --> 01:02:48,790 or the density of momentum. 692 01:02:48,790 --> 01:02:57,440 And I'm going to write this as a divergence-free vector 693 01:02:57,440 --> 01:03:00,150 plus a gradient of some scalar. 694 01:03:00,150 --> 01:03:05,840 And I am going to describe the space-space piece 695 01:03:05,840 --> 01:03:12,700 as an isotropic pressure, an anisotropic term-- 696 01:03:12,700 --> 01:03:14,450 I'm going to give you a constraint on this 697 01:03:14,450 --> 01:03:15,200 in just a moment-- 698 01:03:20,370 --> 01:03:29,070 some kind of a gradient of a vector field, 699 01:03:29,070 --> 01:03:32,700 and then a second order trace free operator 700 01:03:32,700 --> 01:03:34,500 acting on a scalar. 701 01:03:41,700 --> 01:03:47,760 So, yeah, I'm going to want to introduce 702 01:03:47,760 --> 01:03:49,110 a couple of constraints here. 703 01:03:53,440 --> 01:03:56,624 I'm going to require-- 704 01:03:56,624 --> 01:03:58,950 I've already written out that the ISI is 0-- 705 01:03:58,950 --> 01:04:05,320 I'm also going to require that the divergence of sigma 706 01:04:05,320 --> 01:04:13,970 be equal to 0, the divergence of the sigma ij be equal to 0. 707 01:04:13,970 --> 01:04:18,420 And I'm going to require that the trace of this 708 01:04:18,420 --> 01:04:20,490 trace be equal to zero. 709 01:04:20,490 --> 01:04:25,890 Let me strongly emphasize that really all I'm doing 710 01:04:25,890 --> 01:04:27,750 is rearranging terms. 711 01:04:27,750 --> 01:04:30,390 I'm just trying to rewrite the components of my stress energy 712 01:04:30,390 --> 01:04:34,620 tensor using this decomposition under rotations 713 01:04:34,620 --> 01:04:36,420 and then looking at fields that can 714 01:04:36,420 --> 01:04:39,300 be written as divergence-free vectors plus gradients 715 01:04:39,300 --> 01:04:40,610 of scalars. 716 01:04:40,610 --> 01:04:43,420 And I have a typo. 717 01:04:43,420 --> 01:04:47,220 I'm just trying to do this in a way that parallels 718 01:04:47,220 --> 01:04:49,022 what I did for the metric. 719 01:04:49,022 --> 01:04:49,522 OK? 720 01:05:16,050 --> 01:05:19,020 Before I do this, don't forget that I 721 01:05:19,020 --> 01:05:26,510 am required to make sure that my stress energy tensor satisfies 722 01:05:26,510 --> 01:05:31,070 a law of local conservation of energy and momentum. 723 01:05:31,070 --> 01:05:40,360 When we do this, what we find is that these various things 724 01:05:40,360 --> 01:05:44,180 that I introduced here, some of them are related to each other. 725 01:05:44,180 --> 01:05:50,530 So in particular, what you find is the Laplace operator on s 726 01:05:50,530 --> 01:05:56,290 is equal to the time derivative of the density, the energy 727 01:05:56,290 --> 01:05:57,920 density. 728 01:05:57,920 --> 01:06:05,450 Laplace operator on the scalar sigma 729 01:06:05,450 --> 01:06:12,950 is related to the pressure and the time derivative of this s. 730 01:06:20,900 --> 01:06:25,360 Finally, Laplace operator on that sigma i 731 01:06:25,360 --> 01:06:27,680 is related to also the time derivative 732 01:06:27,680 --> 01:06:33,020 of this derivative of si. 733 01:06:33,020 --> 01:06:34,803 What this tells us is that-- 734 01:06:34,803 --> 01:06:37,220 so I really want to make sure people don't get too hung up 735 01:06:37,220 --> 01:06:37,720 on this. 736 01:06:37,720 --> 01:06:41,350 This is really just sort of the a convenient way 737 01:06:41,350 --> 01:06:43,565 of reorganizing the information in those terms. 738 01:06:43,565 --> 01:06:45,190 But if you do want to think about this, 739 01:06:45,190 --> 01:06:54,940 this is telling us that only rho, p, si, and sigma ij 740 01:06:54,940 --> 01:06:56,530 are freely specifiable. 741 01:07:02,770 --> 01:07:06,450 If you know these, there's a total of 6 functions here-- 742 01:07:06,450 --> 01:07:11,220 1 scalar, 1 scalar, 3 vectors minus 1 constraint, 743 01:07:11,220 --> 01:07:13,750 6 tensors minus 4 constraints, because this is 744 01:07:13,750 --> 01:07:16,060 divergenceless and trace-free. 745 01:07:16,060 --> 01:07:18,730 These are the only ones that are freely specifiable. 746 01:07:18,730 --> 01:07:25,580 They determine the other 4 fields. 747 01:07:29,210 --> 01:07:30,290 OK? 748 01:07:30,290 --> 01:07:33,980 So density, pressure, kind of an energy flow, 749 01:07:33,980 --> 01:07:35,405 and anisotropic stresses. 750 01:07:40,930 --> 01:07:43,210 OK, redemption is at hand. 751 01:07:56,610 --> 01:08:01,790 Take all of the framework that we developed 752 01:08:01,790 --> 01:08:04,190 and that we discussed in the previous lecture. 753 01:08:04,190 --> 01:08:06,920 And let's grind out the components 754 01:08:06,920 --> 01:08:07,920 of the Einstein tensors. 755 01:08:14,490 --> 01:08:17,702 Doing so to linear order in h, which 756 01:08:17,702 --> 01:08:20,160 is equivalent to saying linear order in all of these fields 757 01:08:20,160 --> 01:08:23,279 that we've introduced, what you find 758 01:08:23,279 --> 01:08:31,560 is Gtt minus Laplace operator times this field theta. 759 01:08:35,490 --> 01:08:44,109 Gti is minus 1/2 Laplace operator on this vector field 760 01:08:44,109 --> 01:08:52,448 psi minus the time derivative, the gradient of the time 761 01:08:52,448 --> 01:08:53,490 derivative of your theta. 762 01:08:58,450 --> 01:09:06,729 And Gij, it's like a wave operator 763 01:09:06,729 --> 01:09:08,140 on the tt piece of your metric. 764 01:09:41,529 --> 01:09:42,490 OK, that's a lot. 765 01:09:46,327 --> 01:09:47,660 Let's equate them to the source. 766 01:10:04,060 --> 01:10:06,990 And we'll take advantage of the fact that when you do this, 767 01:10:06,990 --> 01:10:09,470 you're always going to associate like with like, OK? 768 01:10:09,470 --> 01:10:11,570 Terms that are divergence-free, you're 769 01:10:11,570 --> 01:10:13,862 going to equate to a source that is divergence-free, 770 01:10:13,862 --> 01:10:14,570 things like that. 771 01:11:21,790 --> 01:11:27,550 This completely characterizes the Einstein field equations, 772 01:11:27,550 --> 01:11:29,680 solutions to the Einstein field equations 773 01:11:29,680 --> 01:11:32,540 in linearized gravity. 774 01:11:32,540 --> 01:11:36,460 Now, I want to make a couple of remarks about this. 775 01:11:36,460 --> 01:11:39,190 Part of what was the motivation for this entire lecture 776 01:11:39,190 --> 01:11:44,690 was this observation that when we solve the field equations 777 01:11:44,690 --> 01:11:50,820 using the radiative Green's function, by construction, 778 01:11:50,820 --> 01:11:54,120 the entire spacetime solution had this radiative character 779 01:11:54,120 --> 01:11:55,560 associated with it. 780 01:11:55,560 --> 01:11:58,290 Everything sort of fell off as 1 over r 781 01:11:58,290 --> 01:12:01,770 and had a time dependence that reflected a time 782 01:12:01,770 --> 01:12:04,260 delay, the time it takes for radiation to travel 783 01:12:04,260 --> 01:12:06,960 from the source to the point at which the field is being 784 01:12:06,960 --> 01:12:08,160 measured. 785 01:12:08,160 --> 01:12:14,220 But we saw via this electrodynamic example 786 01:12:14,220 --> 01:12:16,560 that we put up just for intuition's sake, 787 01:12:16,560 --> 01:12:18,690 that it's entirely possible to have 788 01:12:18,690 --> 01:12:23,520 a totally non-radiative field that looks radiative basically 789 01:12:23,520 --> 01:12:28,730 because we chose a gauge that masked its physical character. 790 01:12:28,730 --> 01:12:30,050 This was a lengthy and-- 791 01:12:30,050 --> 01:12:32,420 I'll be blunt-- a somewhat advanced calculation. 792 01:12:32,420 --> 01:12:34,520 I do not expect everyone to be able to follow 793 01:12:34,520 --> 01:12:35,520 this in great detail. 794 01:12:35,520 --> 01:12:39,110 But I want you to understand how we got to this final end result 795 01:12:39,110 --> 01:12:40,160 here. 796 01:12:40,160 --> 01:12:45,640 By decomposing the spacetime, the perturbation 797 01:12:45,640 --> 01:12:47,710 to spacetime in linearized theory, 798 01:12:47,710 --> 01:12:52,690 into sort of as irreducible as possible a set of functions, 799 01:12:52,690 --> 01:12:56,410 we found that there are exactly 6 degrees 800 01:12:56,410 --> 01:12:59,450 of freedom in that spacetime that are completely 801 01:12:59,450 --> 01:13:01,420 gauge invariant. 802 01:13:01,420 --> 01:13:06,080 These functions, they will have this form. 803 01:13:06,080 --> 01:13:07,940 They will obey these equations. 804 01:13:07,940 --> 01:13:09,610 In principle, you can solve these things 805 01:13:09,610 --> 01:13:14,080 and understand how these fields behave no matter 806 01:13:14,080 --> 01:13:16,430 what gauge you are working in. 807 01:13:16,430 --> 01:13:19,600 And what we see is 2 degrees of freedom-- 808 01:13:19,600 --> 01:13:22,510 remember, hij tt is a 3 by 3 tensor, 809 01:13:22,510 --> 01:13:24,430 but its traceless and its divergenceless. 810 01:13:24,430 --> 01:13:27,340 So there's actually only 2 degrees of freedom here-- 811 01:13:27,340 --> 01:13:30,910 it is indeed a wave equation. 812 01:13:30,910 --> 01:13:33,010 It is the only piece of the metric 813 01:13:33,010 --> 01:13:36,460 that is a wave equation in every gauge 814 01:13:36,460 --> 01:13:38,350 and in every representation. 815 01:13:38,350 --> 01:13:41,800 All other gravitational degrees of freedom obey something 816 01:13:41,800 --> 01:13:45,660 more like a Poisson equation. 817 01:13:45,660 --> 01:13:47,030 These are non-radiative. 818 01:13:47,030 --> 01:13:48,337 This is radiative. 819 01:13:51,100 --> 01:13:53,600 The way that we are going to use this-- the next thing which 820 01:13:53,600 --> 01:13:56,450 I want to talk about is gravitational radiation. 821 01:13:56,450 --> 01:13:58,010 And what this calculation told you 822 01:13:58,010 --> 01:14:01,740 is that if you are interested in gravitational radiation, 823 01:14:01,740 --> 01:14:06,080 this is the only piece of the spacetime and the source 824 01:14:06,080 --> 01:14:09,980 that you need to be concerned about. 825 01:14:09,980 --> 01:14:12,020 If you solve that equation, it will give you 826 01:14:12,020 --> 01:14:14,780 the gauge invariant radiative degrees of freedom 827 01:14:14,780 --> 01:14:19,710 in a spacetime no matter what representation you use. 828 01:14:19,710 --> 01:14:24,390 So we are going to-- for the next couple of lectures, 829 01:14:24,390 --> 01:14:27,740 we are going to focus on this. 830 01:14:27,740 --> 01:14:32,390 We're going to show how we can sort of solve the Einstein 831 01:14:32,390 --> 01:14:36,050 field equations in a particularly convenient gauge 832 01:14:36,050 --> 01:14:39,350 and then say, all right, I know I've now 833 01:14:39,350 --> 01:14:42,260 got these 10 functions. 834 01:14:42,260 --> 01:14:44,540 Four of them are just purely things 835 01:14:44,540 --> 01:14:46,240 I can get rid of by choosing my gauge. 836 01:14:46,240 --> 01:14:50,120 There's only six that are sort of truly physical. 837 01:14:50,120 --> 01:14:54,200 Of those six, four of them don't constitute true radiative 838 01:14:54,200 --> 01:14:55,040 degrees of freedom. 839 01:14:55,040 --> 01:14:57,230 There's only 2 degrees of freedom in my solution 840 01:14:57,230 --> 01:14:58,992 that described radiation. 841 01:14:58,992 --> 01:15:00,950 We're going to talk about how to pull them out, 842 01:15:00,950 --> 01:15:05,630 how to characterize them, and the important physical content 843 01:15:05,630 --> 01:15:08,150 that this gravitational radiation carries. 844 01:15:08,150 --> 01:15:11,780 So in a nutshell, you should take this lecture 845 01:15:11,780 --> 01:15:21,080 as demonstrating that in a very deep way spacetime 846 01:15:21,080 --> 01:15:23,790 always has some kind of a radiative component associated 847 01:15:23,790 --> 01:15:24,290 with it. 848 01:15:24,290 --> 01:15:25,790 But it's really only encapsulated 849 01:15:25,790 --> 01:15:29,090 in 2 degrees of freedom in the spacetime. 850 01:15:29,090 --> 01:15:30,230 It may look otherwise. 851 01:15:30,230 --> 01:15:32,630 But just be careful that because your gauge 852 01:15:32,630 --> 01:15:35,786 has confused you essentially. 853 01:15:35,786 --> 01:15:39,670 And with that, I will stop this lecture here.