1 00:00:00,500 --> 00:00:02,450 [SQUEAKING] 2 00:00:02,450 --> 00:00:04,410 [RUSTLING] 3 00:00:04,410 --> 00:00:06,370 [CLICKING] 4 00:00:11,080 --> 00:00:12,080 SCOTT HUGHES: All right. 5 00:00:12,080 --> 00:00:14,420 Good morning, 8.962. 6 00:00:14,420 --> 00:00:16,340 This is a very weird experience. 7 00:00:16,340 --> 00:00:20,210 I am standing in here talking to an empty classroom. 8 00:00:20,210 --> 00:00:22,670 I have some experience talking to myself, 9 00:00:22,670 --> 00:00:26,180 because like many of us, I am probably a little weirder 10 00:00:26,180 --> 00:00:28,130 than the average. 11 00:00:28,130 --> 00:00:29,600 But that does not change the fact 12 00:00:29,600 --> 00:00:31,470 that this is awkward and a little strange, 13 00:00:31,470 --> 00:00:34,860 and we already miss having you around here. 14 00:00:34,860 --> 00:00:40,820 So I hope we all get through MIT's current weirdness 15 00:00:40,820 --> 00:00:44,600 in a healthy and quick fashion so we can get back 16 00:00:44,600 --> 00:00:48,590 to doing this work we love with the people we 17 00:00:48,590 --> 00:00:49,970 love to have here. 18 00:00:49,970 --> 00:00:53,360 All right, so all that being said, 19 00:00:53,360 --> 00:00:57,020 it's time for us to get back to the business of 8.962, which 20 00:00:57,020 --> 00:00:58,820 is learning about general relativity. 21 00:00:58,820 --> 00:01:01,400 And today, the lecture that I am recording 22 00:01:01,400 --> 00:01:03,920 is one in which we will take all the tools that 23 00:01:03,920 --> 00:01:05,840 have been developed and we will turn this 24 00:01:05,840 --> 00:01:07,610 into a theory of gravity. 25 00:01:11,070 --> 00:01:14,460 Let me go over a quick recap of some of the things 26 00:01:14,460 --> 00:01:19,630 that we talked about in our previous lecture, 27 00:01:19,630 --> 00:01:22,215 and I want to emphasize this because this quantity 28 00:01:22,215 --> 00:01:23,840 that we derived about two lectures ago, 29 00:01:23,840 --> 00:01:26,330 the Riemann curvature tensor, is going 30 00:01:26,330 --> 00:01:29,540 to play an extremely important role in things 31 00:01:29,540 --> 00:01:31,290 that we do moving forward. 32 00:01:31,290 --> 00:01:34,170 So I'll just quickly remind you that in our previous lecture, 33 00:01:34,170 --> 00:01:37,910 we counted up the symmetries that this tensor has. 34 00:01:37,910 --> 00:01:39,740 And so the four most-- 35 00:01:39,740 --> 00:01:43,880 the four that are important for understanding its properties, 36 00:01:43,880 --> 00:01:45,900 its four main symmetries are first of all, 37 00:01:45,900 --> 00:01:50,970 if you exchange indices 3 and 4, it comes in with a minus sign, 38 00:01:50,970 --> 00:01:55,220 so it's anti-symmetric under exchange of indices 3 and 4. 39 00:01:55,220 --> 00:01:56,900 If you lower that first and next so 40 00:01:56,900 --> 00:01:59,330 that they are all in the downstairs position 41 00:01:59,330 --> 00:02:01,820 and you exchange indices 1 and 2, 42 00:02:01,820 --> 00:02:04,772 you likewise pick up a minus sign. 43 00:02:04,772 --> 00:02:06,980 Again, keeping everything in the downstairs position, 44 00:02:06,980 --> 00:02:10,070 if you just wholesale swap indices 1 and 2 45 00:02:10,070 --> 00:02:13,232 for indices 3 and 4 like so, that's symmetric, 46 00:02:13,232 --> 00:02:15,440 and so you get the whole thing back with a plus sign. 47 00:02:20,970 --> 00:02:26,482 And finally, one that's a little bit non-obvious but can be seen 48 00:02:26,482 --> 00:02:28,940 if you're sort of pigheaded enough to sort of stare at this 49 00:02:28,940 --> 00:02:32,990 thing and kind of pound on the algebra a little bit, 50 00:02:32,990 --> 00:02:36,572 if you take the Riemann curvature tensor-- 51 00:02:36,572 --> 00:02:38,530 and this can be at the end of-- the first index 52 00:02:38,530 --> 00:02:40,430 can be either upstairs or downstairs, 53 00:02:40,430 --> 00:02:44,450 but if you cyclically permute indices 2, 3, and 4 54 00:02:44,450 --> 00:02:47,480 and add them up, they sum to 0, OK? 55 00:02:47,480 --> 00:02:50,300 So that tells you that it's a constraint on this thing when 56 00:02:50,300 --> 00:02:52,258 I look at the behavior, this thing with respect 57 00:02:52,258 --> 00:02:54,530 to indices 2, 3, and 4. 58 00:02:54,530 --> 00:02:57,530 We introduced a variant of the Riemann curvature tensor 59 00:02:57,530 --> 00:02:59,360 called the Ricci curvature. 60 00:02:59,360 --> 00:03:02,720 So the way I do this is if I take the trace on the Riemann 61 00:03:02,720 --> 00:03:06,628 curvature tensor on indices 1 and 3, which 62 00:03:06,628 --> 00:03:08,420 is equivalent to taking it with the indices 63 00:03:08,420 --> 00:03:10,795 on the downstairs position and hitting it with the metric 64 00:03:10,795 --> 00:03:16,100 like so, I get this quantity, which I forgot to write down, 65 00:03:16,100 --> 00:03:17,552 is symmetric. 66 00:03:20,150 --> 00:03:22,820 One of the major important physical applications 67 00:03:22,820 --> 00:03:24,350 of the Riemann curvature tensor is 68 00:03:24,350 --> 00:03:27,020 that it allows us to describe the way in which two 69 00:03:27,020 --> 00:03:30,740 neighboring geodesics-- if I have two geodesics that 70 00:03:30,740 --> 00:03:34,310 are separated by a four vector c, 71 00:03:34,310 --> 00:03:38,000 and I look at how that separation evolves as they move 72 00:03:38,000 --> 00:03:41,810 forward along their geodesic paths, this differential 73 00:03:41,810 --> 00:03:43,460 equation describes how it behaves. 74 00:03:43,460 --> 00:03:45,290 And the key thing is that what we see 75 00:03:45,290 --> 00:03:49,670 is that the rate of separation is proportional to the Riemann 76 00:03:49,670 --> 00:03:50,570 curvature. 77 00:03:50,570 --> 00:03:52,430 It ends up playing the role-- when 78 00:03:52,430 --> 00:03:56,360 we think about-- what this tells us is this ends up, remember, 79 00:03:56,360 --> 00:03:58,610 geodesics describe free fall. 80 00:03:58,610 --> 00:04:02,270 And so what this is telling me is a way in which two nearby 81 00:04:02,270 --> 00:04:05,960 but somewhat separated-- separated by a distance c-- 82 00:04:05,960 --> 00:04:08,450 nearby but slightly separated geodesics-- both 83 00:04:08,450 --> 00:04:11,630 are in free fall, but their free fall trajectories 84 00:04:11,630 --> 00:04:13,490 are diverging from one other and perhaps 85 00:04:13,490 --> 00:04:14,900 being focused towards one another 86 00:04:14,900 --> 00:04:18,110 depending upon how R is actually behaving, 87 00:04:18,110 --> 00:04:19,810 this is the behavior of tides. 88 00:04:19,810 --> 00:04:21,860 Free fall is gravity, and this is 89 00:04:21,860 --> 00:04:24,050 saying that the free fall trajectory's 90 00:04:24,050 --> 00:04:27,920 change of separation is governed by the Riemann tensor, 91 00:04:27,920 --> 00:04:31,220 and that's telling me about the action of gravitational tides. 92 00:04:31,220 --> 00:04:35,390 The last thing that we did in Tuesday's-- excuse me-- 93 00:04:35,390 --> 00:04:38,690 in Thursday's lecture was I went through and I developed this 94 00:04:38,690 --> 00:04:41,180 proof of what is known as the Bianchi identity, 95 00:04:41,180 --> 00:04:45,260 which is an identity on the-- 96 00:04:45,260 --> 00:04:47,270 it's an identity on the covariant derivative 97 00:04:47,270 --> 00:04:48,630 of the Riemann tensor. 98 00:04:48,630 --> 00:04:50,120 And so notice what's going on here. 99 00:04:50,120 --> 00:04:53,360 I'm leaving indices 3 and 4 on the Riemann tensor-- 100 00:04:53,360 --> 00:04:56,160 oh shoot. 101 00:04:56,160 --> 00:04:58,620 I'm actually changing my notation halfway through, 102 00:04:58,620 --> 00:05:01,020 let me fix that. 103 00:05:01,020 --> 00:05:02,052 My apologies. 104 00:05:06,210 --> 00:05:07,640 OK. 105 00:05:07,640 --> 00:05:08,660 Apologies for that. 106 00:05:08,660 --> 00:05:13,690 Are you leave indices 3 and 4 unchanged, and what you do 107 00:05:13,690 --> 00:05:16,538 is you cyclically permute the index against-- 108 00:05:16,538 --> 00:05:18,580 in the direction which you're taking a derivative 109 00:05:18,580 --> 00:05:19,940 with indices 1 and 2. 110 00:05:19,940 --> 00:05:24,190 So my first term, it goes alpha, beta, gamma; then beta, gamma, 111 00:05:24,190 --> 00:05:25,962 alpha; gamma, alpha, beta. 112 00:05:25,962 --> 00:05:28,420 OK, notice the way they are cyclically permuting like that. 113 00:05:28,420 --> 00:05:32,680 Sum them up and you get 0. 114 00:05:32,680 --> 00:05:34,120 So let's take it from here. 115 00:05:34,120 --> 00:05:37,540 We're going to start with this Bianchi identity. 116 00:05:37,540 --> 00:05:42,610 What I want to do now is contract the Bianchi identity 117 00:05:42,610 --> 00:05:43,490 in the following way. 118 00:05:48,760 --> 00:05:51,090 So let's take this form that I've written out here-- 119 00:05:51,090 --> 00:05:53,160 and let me just make sure I've left it now in a form that 120 00:05:53,160 --> 00:05:54,180 comports with my notes. 121 00:05:54,180 --> 00:05:56,050 I did, good. 122 00:05:56,050 --> 00:05:58,980 So what I'm going to do is multiply the entire thing. 123 00:06:05,770 --> 00:06:07,620 Using the metric, I'm going to contract it 124 00:06:07,620 --> 00:06:10,190 on indices beta and mu. 125 00:06:13,330 --> 00:06:17,320 So remember, the metric commutes with the covariant derivative. 126 00:06:17,320 --> 00:06:20,140 So unless the derivative itself is with respect 127 00:06:20,140 --> 00:06:22,000 to either the beta or the mu index, 128 00:06:22,000 --> 00:06:24,640 that g just sort of waltzes right in top of there. 129 00:06:24,640 --> 00:06:27,090 So when I do this, it's going to beta-- 130 00:06:27,090 --> 00:06:30,280 g beta mu nu upstairs is going to walk right through this, 131 00:06:30,280 --> 00:06:32,800 it's going to raise the beta index, 132 00:06:32,800 --> 00:06:36,670 and what I wind up with here is this first term 133 00:06:36,670 --> 00:06:43,170 becomes the covariant derivative of the Ricci tensor, OK? 134 00:06:43,170 --> 00:06:46,710 I have contracted on indices beta and mu. 135 00:06:46,710 --> 00:06:48,570 When it hits this one, is just going 136 00:06:48,570 --> 00:06:55,910 to raise the index on that covariant derivative. 137 00:06:55,910 --> 00:06:57,910 So I've got a term here now that looks something 138 00:06:57,910 --> 00:07:00,940 like the divergence of the Riemann tensor, divergence 139 00:07:00,940 --> 00:07:02,110 with respect to index 3. 140 00:07:04,870 --> 00:07:08,640 When I hit this term, it walks through 141 00:07:08,640 --> 00:07:10,810 the covariant derivative again, and you 142 00:07:10,810 --> 00:07:16,220 see what I'm doing is a trace on indexes 2 and 3. 143 00:07:16,220 --> 00:07:18,700 Now I can take advantage of the anti-symmetry 144 00:07:18,700 --> 00:07:20,290 here-- let's reverse this. 145 00:07:20,290 --> 00:07:21,790 And so it's like doing a-- 146 00:07:21,790 --> 00:07:23,920 throwing in a minus sign and then doing 147 00:07:23,920 --> 00:07:26,270 a trace on indices 1 and 2-- 148 00:07:26,270 --> 00:07:29,080 excuse me, doing a trace on-- 149 00:07:29,080 --> 00:07:31,370 rewind, back up for just a second. 150 00:07:31,370 --> 00:07:33,970 I'm going to take advantage that anti-symmetry, I'll 151 00:07:33,970 --> 00:07:37,630 exchange indices 1 and 2, and then I 152 00:07:37,630 --> 00:07:40,600 am doing a trace on indices 1 and 3, which is going 153 00:07:40,600 --> 00:07:42,250 to give me the Ricci tensor. 154 00:07:42,250 --> 00:07:44,170 But because I have used that-- 155 00:07:44,170 --> 00:07:47,600 or that anti-symmetry, I will do so with a minus sign. 156 00:07:47,600 --> 00:07:52,900 So what I get here is this. 157 00:07:52,900 --> 00:07:56,380 I'm going to probably bobble more than once in this lecture, 158 00:07:56,380 --> 00:08:00,130 because again, doing this in an empty room is a little weird. 159 00:08:06,800 --> 00:08:08,130 All right. 160 00:08:08,130 --> 00:08:12,450 So I really want to get a relationship that simplifies 161 00:08:12,450 --> 00:08:13,525 the Riemann tensor, OK? 162 00:08:13,525 --> 00:08:17,430 A Riemann tensor's got four indices on it. 163 00:08:17,430 --> 00:08:20,610 I'm not scared, but I don't like it, OK? 164 00:08:20,610 --> 00:08:23,100 So we're going to do one more contraction operation 165 00:08:23,100 --> 00:08:24,150 to try to simplify this. 166 00:08:29,320 --> 00:08:39,039 Let's now contract once more using the metric 167 00:08:39,039 --> 00:08:43,240 on indices g and nu, OK? 168 00:08:43,240 --> 00:08:46,060 So when I do it with the first one, it walks right through-- 169 00:08:50,893 --> 00:08:52,560 right through that covariant derivative, 170 00:08:52,560 --> 00:08:55,285 and I get the trace of the Riemann tensor, the Ricci-- 171 00:08:55,285 --> 00:08:57,285 excuse me-- I get the trace of the Ricci tensor, 172 00:08:57,285 --> 00:09:00,290 the Ricci scalar. 173 00:09:00,290 --> 00:09:03,260 When I do it on the second term, OK? 174 00:09:03,260 --> 00:09:07,190 I am now tracing on indices 1 and 4. 175 00:09:07,190 --> 00:09:09,230 I will invoke anti-symmetry to change that 176 00:09:09,230 --> 00:09:21,280 into a trace on indices 1 in 3, and I get the Ricci tenser 177 00:09:21,280 --> 00:09:22,030 with a minus sign. 178 00:09:24,620 --> 00:09:29,080 And then the next one, I just trace on this 179 00:09:29,080 --> 00:09:35,810 and I wind up with something that looks like this. 180 00:09:35,810 --> 00:09:36,310 OK. 181 00:09:36,310 --> 00:09:41,830 Now these two terms are both divergences on the-- 182 00:09:41,830 --> 00:09:44,520 they are both divergences on the second index. 183 00:09:44,520 --> 00:09:46,570 The second index is a dummy index, 184 00:09:46,570 --> 00:09:48,760 so I can put these two together. 185 00:09:48,760 --> 00:09:50,220 So this is equivalent to-- 186 00:10:00,640 --> 00:10:03,510 or dividing out a factor of minus 2, 187 00:10:03,510 --> 00:10:06,070 the way this is more often written. 188 00:10:08,722 --> 00:10:11,358 You can also factor out that derivative. 189 00:10:11,358 --> 00:10:12,400 Let's write it like this. 190 00:10:20,850 --> 00:10:23,570 OK? 191 00:10:23,570 --> 00:10:27,450 So what I'm doing here is I divided by a minus 2 192 00:10:27,450 --> 00:10:30,240 so that I can put this guy in front and I get a minus 1/2 193 00:10:30,240 --> 00:10:32,545 in front of my Ricci term. 194 00:10:32,545 --> 00:10:34,920 And because I want to factor out my covariant derivative, 195 00:10:34,920 --> 00:10:36,628 I need to throw in a factor of the metric 196 00:10:36,628 --> 00:10:38,760 there so that the indices line up right. 197 00:10:38,760 --> 00:10:41,622 So what we do at this point is we do what-- whenever you reach 198 00:10:41,622 --> 00:10:43,080 a certain point in your calculation 199 00:10:43,080 --> 00:10:46,110 where you've got something good, you do what every mathematician 200 00:10:46,110 --> 00:10:49,260 or physicist would do, you give this guy a name. 201 00:10:49,260 --> 00:10:58,310 So switching my indices a tiny bit, 202 00:10:58,310 --> 00:11:06,070 we define g mu nu to be the Ricci tensor minus 1/2 203 00:11:06,070 --> 00:11:09,730 metric Ricci scalar, and this is an entity 204 00:11:09,730 --> 00:11:12,190 known as the Einstein tensor. 205 00:11:22,560 --> 00:11:25,840 This is a course on Einstein's gravity, 206 00:11:25,840 --> 00:11:27,590 so the name alone should tell you, 207 00:11:27,590 --> 00:11:30,510 this guy is going to matter. 208 00:11:30,510 --> 00:11:32,760 One quick side note. 209 00:11:32,760 --> 00:11:37,658 So suppose I take the trace of the Einstein tensor. 210 00:11:37,658 --> 00:11:39,450 When we took the trace of the Ricci tensor, 211 00:11:39,450 --> 00:11:41,867 I didn't write it down, but if I take a trace of this guy, 212 00:11:41,867 --> 00:11:45,450 I just get the Ricci scalar R, which I used over here. 213 00:11:45,450 --> 00:11:46,580 So when I do this here-- 214 00:11:49,461 --> 00:11:49,961 oops. 215 00:11:58,190 --> 00:12:03,390 Suppose I just want to call this the Einstein scalar g. 216 00:12:03,390 --> 00:12:06,650 Well, applying that to its definition, 217 00:12:06,650 --> 00:12:12,845 this is going to be equal to the Ricci scalar minus 1/2 trace 218 00:12:12,845 --> 00:12:16,760 of the metric times R. 219 00:12:16,760 --> 00:12:20,870 And it's a general rule in any theory of spacetime 220 00:12:20,870 --> 00:12:23,450 that the trace of the metric is equal to the number 221 00:12:23,450 --> 00:12:27,340 of dimensions in your spacetime, OK? 222 00:12:27,340 --> 00:12:29,720 You can easily work it out in special relativity, 223 00:12:29,720 --> 00:12:34,350 you're basically just raising one index, and as we'll see, 224 00:12:34,350 --> 00:12:35,600 it holds completely generally. 225 00:12:35,600 --> 00:12:37,700 In fact, it follows directly from the fact 226 00:12:37,700 --> 00:12:41,270 that the upstairs metric is the matrix inverse 227 00:12:41,270 --> 00:12:43,400 of the downstairs metric. 228 00:12:43,400 --> 00:12:48,690 So this is equal to 4, so this whole thing 229 00:12:48,690 --> 00:12:52,980 is just the negative of the Ricci scalar. 230 00:12:52,980 --> 00:12:56,190 What this means is that the Einstein 231 00:12:56,190 --> 00:13:13,083 tensor is the trace-reversed Ricci tensor, OK? 232 00:13:13,083 --> 00:13:14,500 I just want to plant that for now. 233 00:13:14,500 --> 00:13:16,542 This is a fact that we're going to take advantage 234 00:13:16,542 --> 00:13:19,990 of a little bit later, but for now, it's 235 00:13:19,990 --> 00:13:23,980 just a mathematical fact that I want to point out, 236 00:13:23,980 --> 00:13:24,950 I want to set aside. 237 00:13:24,950 --> 00:13:27,070 We'll come back to it when it matters. 238 00:13:34,100 --> 00:13:34,980 OK. 239 00:13:34,980 --> 00:13:37,560 We now have everything that we need 240 00:13:37,560 --> 00:13:40,100 to take all of the framework that we 241 00:13:40,100 --> 00:13:42,870 have been developing all term and turn it 242 00:13:42,870 --> 00:13:45,272 into a theory of gravity. 243 00:13:45,272 --> 00:13:46,230 I just had a nightmare. 244 00:13:46,230 --> 00:13:47,900 Am I being recorded?-- yes, OK. 245 00:13:47,900 --> 00:13:50,025 Sorry, just suddenly thought I might have forgotten 246 00:13:50,025 --> 00:13:51,620 to turn my microphone on! 247 00:13:51,620 --> 00:13:54,940 So let's turn this into a theory of gravity. 248 00:14:16,670 --> 00:14:21,380 Ingredient 1 is something that we have discussed quite a bit 249 00:14:21,380 --> 00:14:22,340 before. 250 00:14:22,340 --> 00:14:24,950 I want to restate it and I want to sort of remind us. 251 00:14:24,950 --> 00:14:26,750 Several times over the past couple of 252 00:14:26,750 --> 00:14:30,140 lectures I have implicitly used this rule already, 253 00:14:30,140 --> 00:14:32,920 but I want to make it a little bit more explicit now. 254 00:14:32,920 --> 00:14:36,011 We're going to use the principle of equivalence. 255 00:14:42,935 --> 00:14:44,310 In particular, we're going to use 256 00:14:44,310 --> 00:14:50,736 what is known as the minimal coupling principle. 257 00:14:56,633 --> 00:14:58,550 So here's the way-- what this basically means. 258 00:14:58,550 --> 00:15:02,260 We're going to try to take laws of physics that are 259 00:15:02,260 --> 00:15:05,080 well-understood from laboratory experiments, 260 00:15:05,080 --> 00:15:08,740 from special relativity, everything that we have known 261 00:15:08,740 --> 00:15:10,690 and loved and tested for-- 262 00:15:10,690 --> 00:15:12,430 since we started studying physics, 263 00:15:12,430 --> 00:15:15,520 and we're going to try to see how that can be carried over 264 00:15:15,520 --> 00:15:18,730 to working in the curved spacetime that will describe 265 00:15:18,730 --> 00:15:23,350 gravity with as little additional sort of coupling 266 00:15:23,350 --> 00:15:27,200 to spacetime entities as is possible. 267 00:15:27,200 --> 00:15:29,290 So here's what we're going to do. 268 00:15:29,290 --> 00:15:40,670 Take a law of physics that is valid in inertial coordinates 269 00:15:40,670 --> 00:15:54,410 and flat spacetime, or equivalently, the local Lorentz 270 00:15:54,410 --> 00:16:01,910 frame, which corresponds to the local region 271 00:16:01,910 --> 00:16:34,910 of a freely-falling frame or a freely-falling observer, 272 00:16:34,910 --> 00:16:39,470 take that law of physics that is good in that form 273 00:16:39,470 --> 00:16:47,480 and rewrite it in a coordinate-invariant tensorial 274 00:16:47,480 --> 00:16:47,980 form. 275 00:17:03,690 --> 00:17:07,460 This is one the reasons why throughout this term, 276 00:17:07,460 --> 00:17:14,569 we have been brutally didactic about insisting on getting all 277 00:17:14,569 --> 00:17:18,950 of our laws of physics expressed using tensors, quantities which 278 00:17:18,950 --> 00:17:21,200 have exactly the transformation laws 279 00:17:21,200 --> 00:17:24,619 that we demand in order for them to be true tensors that 280 00:17:24,619 --> 00:17:28,407 live in the curve manifold that we use to describe spacetime. 281 00:17:28,407 --> 00:17:29,990 The last time I actually did something 282 00:17:29,990 --> 00:17:31,615 like this was when I derived-- 283 00:17:31,615 --> 00:17:33,740 I've erased it now, but when I derived the equation 284 00:17:33,740 --> 00:17:35,180 of geodesic deviation, OK? 285 00:17:35,180 --> 00:17:37,612 I first did it using very, very simple language, 286 00:17:37,612 --> 00:17:39,320 and then I sort of looked at it and said, 287 00:17:39,320 --> 00:17:42,350 well this is fine according to that local Lorentz frame, 288 00:17:42,350 --> 00:17:44,630 according to that freely-falling observer. 289 00:17:44,630 --> 00:17:47,000 But this is not tensorial, it's actually 290 00:17:47,000 --> 00:17:49,682 as written only good in that frame. 291 00:17:49,682 --> 00:17:51,890 And so what we did was we took another couple minutes 292 00:17:51,890 --> 00:17:54,710 and said, let's see how I can change this acceleration 293 00:17:54,710 --> 00:17:59,900 operator that describes my equation of geodesic deviation, 294 00:17:59,900 --> 00:18:02,570 put in the extra structure necessary so 295 00:18:02,570 --> 00:18:06,080 that the acceleration operator is tensorial, 296 00:18:06,080 --> 00:18:10,040 and when we did that, we saw that the result was actually 297 00:18:10,040 --> 00:18:14,640 exactly what the Riemann tensor looks like in the local Lorentz 298 00:18:14,640 --> 00:18:15,140 frame. 299 00:18:15,140 --> 00:18:17,380 We said, if it holds a local Lorentz frame, 300 00:18:17,380 --> 00:18:19,760 I'm going to assert it holds in all other frames. 301 00:18:19,760 --> 00:18:23,780 And that indeed is the final step in this procedure. 302 00:18:23,780 --> 00:18:34,954 We assert that the resulting law holds in curved spacetime. 303 00:18:42,370 --> 00:18:44,240 OK? 304 00:18:44,240 --> 00:18:47,990 So this is the procedure by which general relativity 305 00:18:47,990 --> 00:18:50,840 takes the laws of physics, good and flat spacetime, 306 00:18:50,840 --> 00:18:55,340 and rejiggers them so that they work in curved spacetime. 307 00:18:55,340 --> 00:18:59,150 Ultimately, this is physics, and so ultimately 308 00:18:59,150 --> 00:19:01,880 the test for these things are experiments. 309 00:19:01,880 --> 00:19:03,890 And I will simply say at this point 310 00:19:03,890 --> 00:19:06,920 that this procedure has passed all experimental tests 311 00:19:06,920 --> 00:19:09,440 that we have thrown at it so far, 312 00:19:09,440 --> 00:19:14,022 and so we're happy with it. 313 00:19:14,022 --> 00:19:15,480 So let me just describe one example 314 00:19:15,480 --> 00:19:18,063 of where we did this-- actually, I'm going to do two examples. 315 00:19:18,063 --> 00:19:26,020 So if I consider the force-free motion of an object 316 00:19:26,020 --> 00:19:28,170 in the freely-falling frame-- 317 00:19:28,170 --> 00:19:32,850 so recall, in the freely-falling frame, 318 00:19:32,850 --> 00:19:36,730 everything is being acted upon by gravity in an equal way. 319 00:19:51,990 --> 00:19:56,070 If I am in the local Lorentz frame, 320 00:19:56,070 --> 00:20:00,360 I can simply say that my object feels-- 321 00:20:00,360 --> 00:20:03,450 my freely-falling object, freely-flying observer-- 322 00:20:03,450 --> 00:20:05,760 feels no acceleration. 323 00:20:05,760 --> 00:20:09,540 That is a perfectly rigorous expression of the idea 324 00:20:09,540 --> 00:20:13,200 that this observer or object is undergoing force-free motion 325 00:20:13,200 --> 00:20:14,730 in this frame. 326 00:20:14,730 --> 00:20:17,131 This is not tensorial, though. 327 00:20:17,131 --> 00:20:25,310 And so we look at this and say, well, 328 00:20:25,310 --> 00:20:28,490 if I want to make this tensorial, what I'm going to do 329 00:20:28,490 --> 00:20:33,985 is note that the tensor operator that describes-- now 330 00:20:33,985 --> 00:20:36,050 let me keep my indices consistent here. 331 00:20:36,050 --> 00:20:43,010 My tensor operator that describes this equation 332 00:20:43,010 --> 00:20:53,720 is given by taking the covariant derivative of the four velocity 333 00:20:53,720 --> 00:21:00,050 and contracting it with the four velocity itself. 334 00:21:00,050 --> 00:21:02,880 These say the exact same thing in the local Lorentz frame. 335 00:21:02,880 --> 00:21:05,320 This one is tensorial, though, that one is not. 336 00:21:05,320 --> 00:21:07,210 And so we then say, OK, well this 337 00:21:07,210 --> 00:21:09,130 is the version that is tensorial, 338 00:21:09,130 --> 00:21:12,120 I'm going to assert that it holds in general. 339 00:21:12,120 --> 00:21:21,610 Another example in flat spacetime, local conservation 340 00:21:21,610 --> 00:21:29,630 of energy and momentum was expressed by the idea 341 00:21:29,630 --> 00:21:39,460 that my stress energy tensor had no divergence 342 00:21:39,460 --> 00:21:42,300 in the local Lorentz frame. 343 00:21:42,300 --> 00:21:44,730 Well, if I want to make this tensorial, 344 00:21:44,730 --> 00:21:47,060 all I do is I promote that partial derivative I used 345 00:21:47,060 --> 00:21:51,973 to define the divergence to a covariant derivative. 346 00:21:55,028 --> 00:21:57,070 This is how we are going to define conservation-- 347 00:21:57,070 --> 00:21:59,770 we're going to define local conservation of energy 348 00:21:59,770 --> 00:22:02,680 and momentum in a general spacetime theory. 349 00:22:24,380 --> 00:22:26,690 So that's step 1. 350 00:22:26,690 --> 00:22:28,550 We need-- or sorry, ingredient 1. 351 00:22:28,550 --> 00:22:34,700 Ingredient 2 is-- well, let's just step back for a second. 352 00:22:34,700 --> 00:22:39,520 We have done a lot of work to describe the behavior 353 00:22:39,520 --> 00:22:40,818 of curved spacetimes, OK? 354 00:22:40,818 --> 00:22:42,610 Spacetimes that are not just the spacetimes 355 00:22:42,610 --> 00:22:44,830 of special relativity, spacetimes 356 00:22:44,830 --> 00:22:48,910 when my basis objects are functional, where the Riemann 357 00:22:48,910 --> 00:22:53,110 curvature tensor is non-zero. 358 00:22:53,110 --> 00:22:55,750 We've done a lot to do that, but I haven't said anything 359 00:22:55,750 --> 00:23:00,020 about where that curved spacetime actually comes from. 360 00:23:00,020 --> 00:23:07,780 So the next thing which I need is a field equation 361 00:23:07,780 --> 00:23:25,150 which connects my spacetime to sources of matter and energy. 362 00:23:34,300 --> 00:23:35,368 That's a tall order. 363 00:23:35,368 --> 00:23:36,910 The way we're going to do this, we're 364 00:23:36,910 --> 00:23:39,050 actually going to do it two different ways. 365 00:23:39,050 --> 00:23:41,710 So in this current lecture, I'm going 366 00:23:41,710 --> 00:23:44,290 to do it using a method that parallels 367 00:23:44,290 --> 00:23:47,115 how Einstein originally did it when he derived-- 368 00:23:47,115 --> 00:23:48,490 what resulted out of this is what 369 00:23:48,490 --> 00:23:50,490 we call the field equation of general relativity 370 00:23:50,490 --> 00:23:54,460 or the Einstein field equation, and in this first presentation 371 00:23:54,460 --> 00:23:58,150 of this material, I'm going to do it the way Einstein did it. 372 00:23:58,150 --> 00:24:07,000 So what we are going to do is we will require that whatever 373 00:24:07,000 --> 00:24:17,670 emerges from this procedure, it must recover Newtonian gravity 374 00:24:17,670 --> 00:24:19,070 in an appropriate limit. 375 00:24:29,390 --> 00:24:33,740 This is a philosophical point about physics. 376 00:24:33,740 --> 00:24:35,540 When you come up with a new theory, 377 00:24:35,540 --> 00:24:40,880 you may conceptually overturn what came before. 378 00:24:40,880 --> 00:24:43,140 You may have an entirely new way of thinking about it. 379 00:24:43,140 --> 00:24:46,130 You may go from saying that there is a potential that 380 00:24:46,130 --> 00:24:48,660 is sourced by mass that fills all of space 381 00:24:48,660 --> 00:24:51,650 and that objects react to to saying something like, 382 00:24:51,650 --> 00:24:55,280 we now decide that the manifold of events 383 00:24:55,280 --> 00:24:57,470 has a curvature that is determined 384 00:24:57,470 --> 00:25:00,010 by the distribution of matter and energy in spacetime. 385 00:25:00,010 --> 00:25:02,690 It's very different philosophical and ultimately 386 00:25:02,690 --> 00:25:04,790 mathematical ways of formulating this, 387 00:25:04,790 --> 00:25:07,460 but they have to give consistent predictions, 388 00:25:07,460 --> 00:25:10,070 because at the end of the day, Newtonian gravity 389 00:25:10,070 --> 00:25:12,200 works pretty damn well, OK? 390 00:25:12,200 --> 00:25:14,570 We can't just throw that away. 391 00:25:14,570 --> 00:25:18,440 So what we're going to do is demand 392 00:25:18,440 --> 00:25:23,630 that in an appropriate limit, both the field equation 393 00:25:23,630 --> 00:25:26,580 for Newtonian gravity-- 394 00:25:26,580 --> 00:25:31,430 so this is the Laplace operator now, 395 00:25:31,430 --> 00:25:37,190 which I'm going to write in a semi-coordinate-invariant form, 396 00:25:37,190 --> 00:25:39,760 as the chronic or delta contracted-- basically 397 00:25:39,760 --> 00:25:43,310 it's the trace on a matrix of partial derivatives acting 398 00:25:43,310 --> 00:25:45,170 on a potential. 399 00:25:45,170 --> 00:25:51,710 This equals 4 pi rho, and I call this semi-coordinate invariant 400 00:25:51,710 --> 00:25:54,230 because part of what goes into this 401 00:25:54,230 --> 00:25:58,100 is this Newtonian limit only works if everything 402 00:25:58,100 --> 00:26:02,300 is sufficiently slowly varying in time, 403 00:26:02,300 --> 00:26:04,520 that things having to do with time derivatives 404 00:26:04,520 --> 00:26:06,290 can be neglected, OK? 405 00:26:06,290 --> 00:26:07,490 It's never really been-- 406 00:26:07,490 --> 00:26:10,250 prior to some of the more modern experiments that we've 407 00:26:10,250 --> 00:26:12,830 had to do, time-varying sources of gravity 408 00:26:12,830 --> 00:26:14,600 are very hard to work with. 409 00:26:14,600 --> 00:26:19,570 And so Newton was never really tested in that way. 410 00:26:19,570 --> 00:26:23,890 Nonetheless, whatever emerges from Einstein 411 00:26:23,890 --> 00:26:25,670 had best agree with this. 412 00:26:25,670 --> 00:26:39,420 And we are also going to require that the equation of motion 413 00:26:39,420 --> 00:26:43,572 in this framework agree with Newtonian gravity. 414 00:26:43,572 --> 00:26:45,780 We actually went through this-- this was a little bit 415 00:26:45,780 --> 00:26:49,950 of a preview of this lecture, we did this in our-- 416 00:26:49,950 --> 00:26:52,320 we concluded our discussion of geodesics. 417 00:26:52,320 --> 00:26:54,900 Let me just recap the result that came out of this. 418 00:27:03,145 --> 00:27:06,370 So our equation of motion was that-- 419 00:27:11,850 --> 00:27:18,420 you can write it as the acceleration of an observer 420 00:27:18,420 --> 00:27:27,277 is related to the gradient of the potential. 421 00:27:30,860 --> 00:27:32,660 All right. 422 00:27:32,660 --> 00:27:35,812 So let's follow in the footsteps of Einstein and do this. 423 00:27:35,812 --> 00:27:38,270 So what we're going to do-- let's do the equation of motion 424 00:27:38,270 --> 00:27:38,600 first. 425 00:27:38,600 --> 00:27:40,320 I've already gone through this briefly, 426 00:27:40,320 --> 00:27:41,820 but I want to go over it again and I 427 00:27:41,820 --> 00:27:44,460 want to update the notation slightly. 428 00:27:44,460 --> 00:27:46,790 So let's do the equation of motion 429 00:27:46,790 --> 00:27:50,150 by beginning with the geodesic equation. 430 00:27:58,590 --> 00:28:10,220 We will start with the acceleration 431 00:28:10,220 --> 00:28:21,220 coupled to the four velocity by the Christoffel symbols. 432 00:28:24,760 --> 00:28:27,040 All tests of Newtonian gravity, especially 433 00:28:27,040 --> 00:28:28,983 those that Einstein had available at the time 434 00:28:28,983 --> 00:28:31,150 that he was formulating this, were slow motion ones. 435 00:28:31,150 --> 00:28:33,790 We were considering objects moving 436 00:28:33,790 --> 00:28:36,100 at best in our solar system. 437 00:28:36,100 --> 00:28:39,250 And so things there on a human scale certainly move quickly, 438 00:28:39,250 --> 00:28:42,140 but they're slow compared to the speed of light. 439 00:28:42,140 --> 00:28:49,260 And so let's impose the slow motion limit, which tells us 440 00:28:49,260 --> 00:29:03,370 that the 0th component of the four velocity 441 00:29:03,370 --> 00:29:06,080 is much larger than the spatial components of the four 442 00:29:06,080 --> 00:29:07,377 velocity, OK? 443 00:29:07,377 --> 00:29:09,460 Remember working in units where the speed of light 444 00:29:09,460 --> 00:29:10,600 is equal to 1. 445 00:29:10,600 --> 00:29:13,455 And so if this is being measured in human units, kilometers 446 00:29:13,455 --> 00:29:14,830 per second, and things like that, 447 00:29:14,830 --> 00:29:18,710 this is on the order of the speed of light. 448 00:29:18,710 --> 00:29:22,280 So when we throw this in, we see that we expand this out, 449 00:29:22,280 --> 00:29:24,980 that the contributions from the dt 450 00:29:24,980 --> 00:29:29,280 d tau terms here are going to be vastly larger than any others. 451 00:29:29,280 --> 00:29:33,260 And so we can simplify our equation 452 00:29:33,260 --> 00:29:54,930 to a form that looks like this. 453 00:30:02,590 --> 00:30:03,090 OK? 454 00:30:03,090 --> 00:30:09,300 In the spirit of being uber complete, 455 00:30:09,300 --> 00:30:11,270 let's write out that Christoffel symbol. 456 00:30:28,600 --> 00:30:31,795 So dig back into your previous lectures' notes, 457 00:30:31,795 --> 00:30:34,420 remind yourself what the formula for the Christoffel symbol is. 458 00:30:54,980 --> 00:30:57,140 OK. 459 00:30:57,140 --> 00:31:00,330 Notice, two of the terms here are time derivatives. 460 00:31:00,330 --> 00:31:03,220 The Newtonian limit-- all the tests that were available 461 00:31:03,220 --> 00:31:05,680 when Einstein was formulating this, the limit that we care 462 00:31:05,680 --> 00:31:07,797 about here, the gravitational field, 463 00:31:07,797 --> 00:31:09,880 the gravitational potentials that he was studying, 464 00:31:09,880 --> 00:31:13,360 what the Newtonian limit emerges from, they are static. 465 00:31:13,360 --> 00:31:19,160 So we're going to do is neglect time derivatives to recover 466 00:31:19,160 --> 00:31:19,660 this limit. 467 00:31:23,890 --> 00:31:42,710 And when we do this, what we find is that the component-- 468 00:31:42,710 --> 00:31:44,110 the Christoffel component that we 469 00:31:44,110 --> 00:31:47,170 care about looks like one derivative of the 0 0 470 00:31:47,170 --> 00:31:50,710 piece of the spacetime metric. 471 00:31:54,290 --> 00:31:56,950 It's not too hard to convince yourself 472 00:31:56,950 --> 00:32:03,050 that this, in fact, reduces-- 473 00:32:06,650 --> 00:32:08,502 oops, pardon me. 474 00:32:08,502 --> 00:32:09,210 I skipped a step. 475 00:32:09,210 --> 00:32:10,320 Pardon me just one moment. 476 00:32:10,320 --> 00:32:13,590 Just one moment, my apologies. 477 00:32:13,590 --> 00:32:17,610 I'm going to write the spacetime metric-- 478 00:32:17,610 --> 00:32:20,730 I'm going to work in a coordinate system such 479 00:32:20,730 --> 00:32:26,840 that spacetime looks like the flat space 480 00:32:26,840 --> 00:32:30,880 time of special relativity plus a little bit else, OK? 481 00:32:30,880 --> 00:32:33,430 This is consistent with the idea that every system 482 00:32:33,430 --> 00:32:36,370 we have studied in Newtonian gravity 483 00:32:36,370 --> 00:32:38,590 is one where the predictions of special relativity 484 00:32:38,590 --> 00:32:40,600 actually work really, really well, OK? 485 00:32:40,600 --> 00:32:42,340 Gravity is new, it's special, it's 486 00:32:42,340 --> 00:32:45,290 why we have a whole other course describing it. 487 00:32:45,290 --> 00:32:47,860 But clearly it can't be too far different 488 00:32:47,860 --> 00:32:49,530 from special relativity or we wouldn't 489 00:32:49,530 --> 00:32:51,447 have been able to formulate special relativity 490 00:32:51,447 --> 00:32:52,250 in the first place. 491 00:32:52,250 --> 00:32:55,610 So my apologies, I sort of jumped ahead here for a second. 492 00:32:55,610 --> 00:32:57,220 We're going to treat the g mu nu that 493 00:32:57,220 --> 00:33:01,330 goes into this as the metric of flat spacetime plus something 494 00:33:01,330 --> 00:33:03,850 else where I'm going to imagine that all 495 00:33:03,850 --> 00:33:06,520 the different components of this-- 496 00:33:06,520 --> 00:33:09,760 so a typical component of this h mu nu 497 00:33:09,760 --> 00:33:12,190 has an absolute value that is much smaller than 1. 498 00:33:16,000 --> 00:33:24,770 It's not too hard to prove that when you invert this, what you 499 00:33:24,770 --> 00:33:29,880 wind up with is a form that looks like so where this h mu 500 00:33:29,880 --> 00:33:38,530 nu with the indices in the upstairs position 501 00:33:38,530 --> 00:33:42,040 is given by raising h's indices using 502 00:33:42,040 --> 00:33:43,313 the metric of flat spacetime. 503 00:33:43,313 --> 00:33:45,730 We're going to talk about this in a little bit more detail 504 00:33:45,730 --> 00:33:47,740 in a future lecture, it doesn't really 505 00:33:47,740 --> 00:33:49,020 matter too much right now. 506 00:33:49,020 --> 00:33:51,460 I just want to point out that the inverse g, which 507 00:33:51,460 --> 00:33:54,220 we need to use here, also has this form that 508 00:33:54,220 --> 00:34:00,300 looks like flat spacetime metric and this h coupling into it. 509 00:34:00,300 --> 00:34:04,020 Now the reason I'm going through all this 510 00:34:04,020 --> 00:34:06,420 is that in order to work out this Christoffel symbol, 511 00:34:06,420 --> 00:34:08,489 I need to take a derivative. 512 00:34:08,489 --> 00:34:12,179 The derivative of eta is 0, OK? 513 00:34:12,179 --> 00:34:14,880 So the only thing that gets differentiated is h. 514 00:34:33,510 --> 00:34:44,570 So when you work out this Christoffel symbol, 515 00:34:44,570 --> 00:34:46,139 what you get is this. 516 00:34:46,139 --> 00:34:50,449 If you're being-- keeping score, there are corrections 517 00:34:50,449 --> 00:34:53,719 of order h squared, and pardon me, 518 00:34:53,719 --> 00:35:00,140 I should have actually noted, there 519 00:35:00,140 --> 00:35:01,820 are corrections of order h squared 520 00:35:01,820 --> 00:35:04,003 that go into this inverse. 521 00:35:04,003 --> 00:35:05,420 Let me move this over so I can fit 522 00:35:05,420 --> 00:35:06,587 that in a little bit better. 523 00:35:14,860 --> 00:35:19,050 But in keeping with the idea that-- 524 00:35:19,050 --> 00:35:22,810 in keeping with the idea that for the Newtonian limit, 525 00:35:22,810 --> 00:35:24,533 the h squared could-- 526 00:35:24,533 --> 00:35:26,950 h is small, we're going to treat the h squared corrections 527 00:35:26,950 --> 00:35:28,570 as negligible and we will drop them. 528 00:35:33,660 --> 00:35:34,230 OK. 529 00:35:34,230 --> 00:35:37,740 So let's look at what motion in this limit turns into, then. 530 00:35:37,740 --> 00:35:41,757 We now have enough pieces to compute all the bits 531 00:35:41,757 --> 00:35:42,840 of the equation of motion. 532 00:35:48,160 --> 00:35:51,510 So in keeping with the idea that I am going to neglect all time 533 00:35:51,510 --> 00:35:55,970 derivatives, this tells me that the gamma 0, 534 00:35:55,970 --> 00:35:58,710 00 term is equal to 0. 535 00:36:02,030 --> 00:36:12,330 And from this, we find that there is a simple equation 536 00:36:12,330 --> 00:36:13,170 describing time. 537 00:36:18,280 --> 00:36:32,810 In our equation describing space, OK? 538 00:36:32,810 --> 00:36:38,050 So what I've done there is just taken this geodesic equation, 539 00:36:38,050 --> 00:36:42,320 plugged in that result for the Christoffel symbol, 540 00:36:42,320 --> 00:36:43,660 and expanded this guy out. 541 00:37:05,580 --> 00:37:10,270 So what results, I can divide both sides now 542 00:37:10,270 --> 00:37:12,835 by two powers of dt d tau. 543 00:37:26,420 --> 00:37:26,920 All right. 544 00:37:29,960 --> 00:37:35,230 If you take a look at what we've got here, 545 00:37:35,230 --> 00:37:37,380 this prediction of the-- no even a prediction, this 546 00:37:37,380 --> 00:37:39,690 result from the geodesic equation 547 00:37:39,690 --> 00:37:43,860 is identical to our Newtonian equation of motion 548 00:37:43,860 --> 00:37:46,593 provided we make the following identification. 549 00:37:58,670 --> 00:38:04,040 h00 must be minus 2 phi where phi 550 00:38:04,040 --> 00:38:06,800 was Newtonian gravitational potential. 551 00:38:06,800 --> 00:38:24,070 Or equivalently, g00 is the negative of 1 plus 2 phi. 552 00:38:24,070 --> 00:38:24,570 All right. 553 00:38:24,570 --> 00:38:26,670 So that's step 1. 554 00:38:26,670 --> 00:38:33,140 We have made for ourselves a correspondence 555 00:38:33,140 --> 00:38:40,295 between what the metric should be and the equation of motion. 556 00:38:43,510 --> 00:38:46,408 We still have to do the field equation, 557 00:38:46,408 --> 00:38:47,450 so let's talk about that. 558 00:39:11,270 --> 00:39:13,520 So very helpfully I've already got the Newtonian field 559 00:39:13,520 --> 00:39:14,728 equation right above me here. 560 00:39:14,728 --> 00:39:20,975 Let me rewrite it because I'm going 561 00:39:20,975 --> 00:39:23,533 to want to tweak my notation a tiny bit. 562 00:39:23,533 --> 00:39:25,700 I don't want to think about what this is telling me. 563 00:39:32,130 --> 00:39:37,683 So eta ij is the same thing as delta ij, 564 00:39:37,683 --> 00:39:39,100 I just want to put it in this form 565 00:39:39,100 --> 00:39:42,890 so that it looks like a piece of a spacetime tensor. 566 00:39:42,890 --> 00:39:46,285 This is manifestly not a tensorial equation. 567 00:39:48,880 --> 00:39:53,050 I have a bunch of derivatives on my potential being 568 00:39:53,050 --> 00:39:53,930 set equal to-- 569 00:39:53,930 --> 00:39:57,960 OK, there's a couple constants, but this, OK? 570 00:39:57,960 --> 00:40:00,605 When we learned about quantities like this 571 00:40:00,605 --> 00:40:02,230 in undergraduate physics, usually we're 572 00:40:02,230 --> 00:40:03,440 told that this is a-- 573 00:40:03,440 --> 00:40:05,140 excuse me, this is a scalar. 574 00:40:05,140 --> 00:40:08,650 But we now know, rho is not a scalar, 575 00:40:08,650 --> 00:40:15,550 it is the mass density, which up to a factor of c 576 00:40:15,550 --> 00:40:21,040 squared, is the same thing as the energy density. 577 00:40:21,040 --> 00:40:23,110 And when we examined how this behaves 578 00:40:23,110 --> 00:40:26,110 as we change between inertial reference frames, 579 00:40:26,110 --> 00:40:31,130 we found this transforms like a particular component 580 00:40:31,130 --> 00:40:31,630 of a tensor. 581 00:40:49,800 --> 00:40:53,310 And as I sort of emphasize, not that long ago we 582 00:40:53,310 --> 00:40:57,450 have been in something of a didactic fury insisting 583 00:40:57,450 --> 00:41:00,030 that everything be formulated in terms of tensors. 584 00:41:00,030 --> 00:41:02,700 Pulling out a particular component of a tensor 585 00:41:02,700 --> 00:41:04,455 is bad math and bad physics. 586 00:41:17,280 --> 00:41:20,077 So we want to promote this to something tensorial. 587 00:41:36,980 --> 00:41:40,100 So s on the right-hand side, we've 588 00:41:40,100 --> 00:41:45,380 got one component of the stress energy tensor. 589 00:41:45,380 --> 00:41:48,830 We would like whatever is going to be 590 00:41:48,830 --> 00:41:52,980 on the right-hand side of this equation 591 00:41:52,980 --> 00:41:54,620 to be the stress energy tensor, OK? 592 00:41:54,620 --> 00:41:56,037 We can sort of imagine that what's 593 00:41:56,037 --> 00:42:00,320 going on in Newton's gravity is that there is one particular-- 594 00:42:00,320 --> 00:42:02,780 maybe there's one component of this equation 595 00:42:02,780 --> 00:42:05,652 that in all the analyses that were done 596 00:42:05,652 --> 00:42:07,610 that led to our formation of Newtonian gravity, 597 00:42:07,610 --> 00:42:09,735 there may be one component that was dominant, which 598 00:42:09,735 --> 00:42:12,620 is how it was that Newton and everyone since then 599 00:42:12,620 --> 00:42:14,795 was able to sort of pick out a particular component 600 00:42:14,795 --> 00:42:18,290 of this equation as being important. 601 00:42:18,290 --> 00:42:25,680 Over here on the left-hand side, we saw earlier that 602 00:42:25,680 --> 00:42:28,190 the equation of motion we're going to look for corresponds 603 00:42:28,190 --> 00:42:33,180 to the Newtonian limit if the metric plays the same role-- 604 00:42:33,180 --> 00:42:36,540 up to factors of 2 and offsets by 1 and things like that-- 605 00:42:36,540 --> 00:42:41,460 the metric must play the same role as the Newtonian 606 00:42:41,460 --> 00:42:43,567 gravitational potential. 607 00:42:49,420 --> 00:42:51,880 So if I look at-- 608 00:42:51,880 --> 00:42:56,370 if I look at the Newtonian field equation, 609 00:42:56,370 --> 00:43:01,720 I see two derivatives acting on the potential. 610 00:43:01,720 --> 00:43:09,820 So I want my metric to stand in for the potential, 611 00:43:09,820 --> 00:43:21,930 we expect there to be two derivatives of metric 612 00:43:21,930 --> 00:43:23,473 entering this relationship. 613 00:43:27,180 --> 00:43:29,820 So now two derivatives of the metric 614 00:43:29,820 --> 00:43:32,480 is going to give me something that smells like a curvature. 615 00:44:16,270 --> 00:44:31,190 So we want to put a curvature tensor on the left-hand side 616 00:44:31,190 --> 00:44:34,080 of this equation. 617 00:44:34,080 --> 00:44:35,630 We have several to choose from, OK? 618 00:44:35,630 --> 00:44:38,243 It clearly can't be the Riemann tensor. 619 00:44:38,243 --> 00:44:40,160 There's too many indices, it just doesn't fit. 620 00:44:40,160 --> 00:44:42,590 It could be the Ricci curvature, OK? 621 00:44:42,590 --> 00:44:45,350 The Ricci curvature has two indices. 622 00:44:45,350 --> 00:44:47,880 That has two indices, that's a candidate. 623 00:44:47,880 --> 00:44:49,880 But it's worth stopping and reminding ourselves, 624 00:44:49,880 --> 00:44:52,157 wait a minute, this guy has some properties 625 00:44:52,157 --> 00:44:53,240 that I already know about. 626 00:45:03,690 --> 00:45:05,940 t mu nu tells me about the properties 627 00:45:05,940 --> 00:45:12,630 of energy and momentum in my spacetime, 628 00:45:12,630 --> 00:45:15,900 and as such, conservation-- local conservation 629 00:45:15,900 --> 00:45:19,530 of energy and momentum requires that it be divergence-free. 630 00:45:19,530 --> 00:45:22,020 So whatever this curvature tensor is here 631 00:45:22,020 --> 00:45:27,180 on the left-hand side, we need it 632 00:45:27,180 --> 00:45:36,520 to be a divergence-free 2-index mathematical object. 633 00:45:41,120 --> 00:45:43,990 At the beginning of today's lecture, 634 00:45:43,990 --> 00:45:47,370 I showed how by contracting on the Bianchi identity, 635 00:45:47,370 --> 00:45:50,910 you can, in fact, deduce that there exists exactly such 636 00:45:50,910 --> 00:45:53,160 a mathematical object. 637 00:45:53,160 --> 00:46:01,590 So let us suppose that our equation that 638 00:46:01,590 --> 00:46:04,290 relates the properties of the spacetime 639 00:46:04,290 --> 00:46:07,260 to the sources of energy and momentum of my spacetime 640 00:46:07,260 --> 00:46:10,350 is essentially that that Einstein tensor, g mu nu, 641 00:46:10,350 --> 00:46:13,360 be equal to the stress energy tensor. 642 00:46:13,360 --> 00:46:16,450 Now in fact, they don't have the same dimensions as each other, 643 00:46:16,450 --> 00:46:28,370 so let's throw in a kappa, some kind of a constant to make sure 644 00:46:28,370 --> 00:46:31,250 that we get the right units, the right dimensions, 645 00:46:31,250 --> 00:46:33,110 and that we recover the Newtonian limit. 646 00:46:40,580 --> 00:46:44,240 The way we're going to deduce how well this works is 647 00:46:44,240 --> 00:46:46,670 see whether an equation of this form 648 00:46:46,670 --> 00:46:51,800 gives me something that looks like the Newtonian limit 649 00:46:51,800 --> 00:46:54,980 when I go to what I'm going to call the weak gravity limit, 650 00:46:54,980 --> 00:46:58,070 and I'm going to then use, assuming it does work-- 651 00:47:00,620 --> 00:47:02,830 not to give away the plot, it does-- 652 00:47:02,830 --> 00:47:06,110 we'll use that to figure out what this constant kappa must 653 00:47:06,110 --> 00:47:06,610 be. 654 00:47:35,340 --> 00:47:40,840 So if we do, in fact, have a field equation of the form g 655 00:47:40,840 --> 00:47:44,040 mu nu is some constant t mu nu, it's 656 00:47:44,040 --> 00:47:47,580 not too hard to figure out that an equivalent form of this 657 00:47:47,580 --> 00:47:58,230 is to say that the Ricci tensor is k times t mu nu minus 1/2 g 658 00:47:58,230 --> 00:48:04,560 mu nu t where this t is just the trace of the stress energy 659 00:48:04,560 --> 00:48:05,550 tensor. 660 00:48:05,550 --> 00:48:08,252 Remember, I spent a few moments after we derived the Einstein 661 00:48:08,252 --> 00:48:09,960 tensor pointing out that it's essentially 662 00:48:09,960 --> 00:48:13,030 the same thing as Ricci but with the trace reversed. 663 00:48:13,030 --> 00:48:17,010 This is just a trace-reversed equivalent to that equation. 664 00:48:17,010 --> 00:48:18,780 This step that I'm introducing here, 665 00:48:18,780 --> 00:48:20,453 basically it just makes the algebra 666 00:48:20,453 --> 00:48:21,870 for the next calculation I'm going 667 00:48:21,870 --> 00:48:25,200 to do a little bit easier, OK? 668 00:48:25,200 --> 00:48:29,190 So I just want to emphasize that this and that are exactly 669 00:48:29,190 --> 00:48:31,460 the same content. 670 00:48:31,460 --> 00:48:33,980 All right. 671 00:48:33,980 --> 00:48:39,000 So to make some headway, we need to choose a form for a stress 672 00:48:39,000 --> 00:48:40,200 energy tensor. 673 00:48:40,200 --> 00:48:43,140 Our goal is to recover the Newtonian limit, 674 00:48:43,140 --> 00:48:45,630 and so what we want to do is make the stress energy tensor 675 00:48:45,630 --> 00:48:49,830 of a body that corresponds to the sort of sources of gravity 676 00:48:49,830 --> 00:48:54,690 that are used in studies of Newtonian gravity. 677 00:48:54,690 --> 00:48:59,160 So let's do something very simple for us. 678 00:48:59,160 --> 00:49:02,620 Let's pick a static-- 679 00:49:02,620 --> 00:49:06,190 in other words, no time variation, 680 00:49:06,190 --> 00:49:18,110 a static perfect fluid as our source of gravity. 681 00:49:22,270 --> 00:49:24,580 So I'm going to choose for my t mu nu-- 682 00:49:27,190 --> 00:49:38,500 dial yourself back to lectures where we talk about this, OK? 683 00:49:38,500 --> 00:49:42,270 So this is the perfect fluid stress energy tensor. 684 00:49:42,270 --> 00:49:44,610 We're working in the Newtonian limit, 685 00:49:44,610 --> 00:49:47,460 and we are working in units where the speed of light 686 00:49:47,460 --> 00:49:48,690 is equal to 1. 687 00:49:48,690 --> 00:49:50,758 If you put speed of light back into these things, 688 00:49:50,758 --> 00:49:52,800 you explicitly include it, this is actually a rho 689 00:49:52,800 --> 00:49:54,705 c squared that appears here. 690 00:49:54,705 --> 00:49:56,970 And so what this tells me is that if I'm studying sort 691 00:49:56,970 --> 00:50:01,810 of Newtonian limit problems, rho is much, much, 692 00:50:01,810 --> 00:50:05,310 much greater than P in the limit that we care about. 693 00:50:20,030 --> 00:50:22,595 Furthermore, I am treating this fluid as being static. 694 00:50:27,840 --> 00:50:32,790 So that means that my four velocity only 695 00:50:32,790 --> 00:50:35,550 has one component, OK? 696 00:50:35,550 --> 00:50:38,220 The fluid is not flowing. 697 00:50:38,220 --> 00:50:40,740 You might be tempted to say, oh, OK, I can just put a 1 698 00:50:40,740 --> 00:50:41,760 in for this. 699 00:50:41,760 --> 00:50:45,513 Not so fast, OK? 700 00:50:45,513 --> 00:50:47,430 Let's be a little bit more careful about that. 701 00:51:07,280 --> 00:51:10,190 One of the key governing properties of a four velocity 702 00:51:10,190 --> 00:51:12,238 is that it is properly normalized. 703 00:51:15,660 --> 00:51:21,038 So this equals g mu nu u mu u nu is minus 1. 704 00:51:21,038 --> 00:51:24,800 We know that the only components of this that matter, 705 00:51:24,800 --> 00:51:28,370 so to speak, are the mu and nu equal 0. 706 00:51:28,370 --> 00:51:35,160 So this becomes g00 mu 0 squared equals minus 1. 707 00:51:35,160 --> 00:51:43,720 But g00 is-- well, let's write it this way-- 708 00:51:43,720 --> 00:51:49,390 Negative 1 plus h00. 709 00:51:53,070 --> 00:51:59,640 Go through this algebra, and what it tells you is u0 710 00:51:59,640 --> 00:52:04,170 equals 1 plus 1/2 h00. 711 00:52:08,000 --> 00:52:11,760 Again, I'm doing my algebra at leading order in h here. 712 00:52:16,020 --> 00:52:18,060 We raise and lower indices. 713 00:52:18,060 --> 00:52:23,850 So in my calculation, I'm going to want to know 714 00:52:23,850 --> 00:52:26,790 the downstairs version of this. 715 00:52:26,790 --> 00:52:32,520 And if I, again, treat this thing-- 716 00:52:32,520 --> 00:52:39,620 treat this thing consistently, OK. 717 00:52:39,620 --> 00:52:41,840 What I'll find is I just pick up a minus sign there. 718 00:52:41,840 --> 00:52:42,340 OK. 719 00:52:59,335 --> 00:52:59,835 OK. 720 00:53:03,177 --> 00:53:04,760 Let's now put all the pieces together. 721 00:53:07,880 --> 00:53:12,590 The only component of my stress energy tensor 722 00:53:12,590 --> 00:53:15,410 that's going to now really matter 723 00:53:15,410 --> 00:53:21,050 is rho u0 u0, which, putting all these ingredients back 724 00:53:21,050 --> 00:53:27,030 together, is rho 1 plus h00. 725 00:53:30,200 --> 00:53:43,660 The trace of this guy, putting all these pieces together, 726 00:53:43,660 --> 00:53:46,910 is just equal to negative rho. 727 00:53:46,910 --> 00:53:49,268 Since I only have one component that's 728 00:53:49,268 --> 00:53:50,810 going to end up mattering, let's just 729 00:53:50,810 --> 00:53:53,750 focus on one component of my proposed field equation. 730 00:54:05,110 --> 00:54:05,610 OK? 731 00:54:05,610 --> 00:54:07,277 So this is the guy that I want to solve. 732 00:54:14,290 --> 00:54:16,800 I'll let you digest that and set up the calculation. 733 00:54:34,930 --> 00:54:46,370 We've got T00 minus 1/2 T00 T. This is going to be, 734 00:54:46,370 --> 00:54:49,910 plugging in these bits that I worked out on the other board, 735 00:54:49,910 --> 00:54:50,540 here's my T00. 736 00:54:58,182 --> 00:54:59,990 Just make sure I did that right earlier-- 737 00:54:59,990 --> 00:55:00,490 I did. 738 00:55:21,500 --> 00:55:22,000 OK. 739 00:55:22,000 --> 00:55:27,840 So this is my right-hand side of my field equation. 740 00:55:27,840 --> 00:55:32,120 It will actually be sufficient for our purposes 741 00:55:32,120 --> 00:55:34,918 to neglect this term, OK? 742 00:55:34,918 --> 00:55:36,210 We'll see why in just a moment. 743 00:55:58,796 --> 00:56:03,840 So plugging that in, I need to work out the 00 component 744 00:56:03,840 --> 00:56:05,340 of my Ricci. 745 00:56:09,750 --> 00:56:13,160 So I go back to its foundational definition. 746 00:56:13,160 --> 00:56:17,450 This is what I get when I take the trace on indices 1 747 00:56:17,450 --> 00:56:18,770 and 3 of the Riemann tensor. 748 00:56:23,550 --> 00:56:26,010 I can simplify that to just doing the trace over 749 00:56:26,010 --> 00:56:29,550 the spatial indices, because the term I'm leaving out is the one 750 00:56:29,550 --> 00:56:33,200 that is of the form 00 here, which by the anti-symmetry, 751 00:56:33,200 --> 00:56:35,100 on exchange of those indices, must vanish. 752 00:56:38,500 --> 00:56:56,870 Plugging in my definition, what I find 753 00:56:56,870 --> 00:56:59,550 is it is going to look like this here. 754 00:56:59,550 --> 00:57:02,050 So I'm just going to neglect the order of gamma squared term 755 00:57:02,050 --> 00:57:03,925 because I'm working in a limit where I assume 756 00:57:03,925 --> 00:57:06,047 that all these h's are small. 757 00:57:06,047 --> 00:57:08,630 This is going to vanish because of my assumption of everything 758 00:57:08,630 --> 00:57:13,350 being static in this limit. 759 00:57:13,350 --> 00:57:19,140 So this, I then go and plug in my definitions. 760 00:57:38,730 --> 00:57:39,593 OK. 761 00:57:39,593 --> 00:57:41,510 Again, I'm going to lose these two derivatives 762 00:57:41,510 --> 00:57:43,218 by the assumption of things being static. 763 00:57:47,270 --> 00:57:53,337 And pardon me just a second-- 764 00:57:53,337 --> 00:57:55,920 yeah, so I'm going to lose these two because of the assumption 765 00:57:55,920 --> 00:57:57,420 of things being static. 766 00:57:57,420 --> 00:57:59,250 The only derivative-- the only term 767 00:57:59,250 --> 00:58:01,860 that's going to matter, the derivative here is h. 768 00:58:01,860 --> 00:58:07,240 And so when I hit it with the inverse metric, 769 00:58:07,240 --> 00:58:22,540 this becomes simply the derivative of the h00 piece, 770 00:58:22,540 --> 00:58:23,040 OK? 771 00:58:25,570 --> 00:58:31,600 I can go from g straight to eta because the correction to this 772 00:58:31,600 --> 00:58:34,070 is of order h squared, which as I've repeatedly emphasized, 773 00:58:34,070 --> 00:58:35,028 we're going to neglect. 774 00:58:43,780 --> 00:58:47,113 All right, we're almost there. 775 00:58:47,113 --> 00:58:49,030 Let me put this board up, I want to keep this. 776 00:59:12,150 --> 00:59:13,090 OK, where was I? 777 00:59:13,090 --> 00:59:14,980 So I've got it down to here, let me just 778 00:59:14,980 --> 00:59:18,550 simplify this one step more. 779 00:59:18,550 --> 00:59:23,380 Eta is-- if mu is not spatial, then this is just 0. 780 00:59:23,380 --> 00:59:27,880 So I can neatly change my mu derivative into a j. 781 00:59:27,880 --> 00:59:31,010 I can just focus on the spatial piece of it. 782 00:59:31,010 --> 00:59:40,480 So this tells me R00 is minus 1/2 Kronecker delta delta i 783 00:59:40,480 --> 00:59:44,800 delta j acting on h00. 784 00:59:44,800 --> 00:59:47,436 This operator is nothing more than-- 785 00:59:47,436 --> 00:59:50,005 it's a Laplace operator. 786 00:59:50,005 --> 00:59:56,560 So this is minus 1/2, our old-fashioned, happily, 787 00:59:56,560 --> 00:59:58,220 well-known from undergrad studies 788 00:59:58,220 --> 01:00:01,220 Laplace operator on h00. 789 01:00:01,220 --> 01:00:04,270 So putting all this together, my field 790 01:00:04,270 --> 01:00:13,020 equation, which I wrote in this form, 791 01:00:13,020 --> 01:00:21,390 reduces down to del squared h00 equals minus kappa rho. 792 01:00:24,050 --> 01:00:41,650 The Newtonian limit that we did for the equation of motion, 793 01:00:41,650 --> 01:00:45,810 the fact that we showed that geodesics correspond to this, 794 01:00:45,810 --> 01:00:50,640 that already led me to deduce that h00 795 01:00:50,640 --> 01:00:54,360 was equal to minus 2 phi. 796 01:00:54,360 --> 01:01:07,170 My Newtonian field equation requires 797 01:01:07,170 --> 01:01:20,050 me to have the Laplace operator acting on the potential phi, 798 01:01:20,050 --> 01:01:24,770 giving me 4 pi g rho. 799 01:01:24,770 --> 01:01:29,520 Put all these pieces together, and what 800 01:01:29,520 --> 01:01:34,980 we see is this proposed field equation 801 01:01:34,980 --> 01:01:40,365 works perfectly provided we choose for that constant. 802 01:01:57,260 --> 01:02:01,560 Kappa equals 8 pi j. 803 01:02:01,560 --> 01:02:10,880 And so we finally get g mu nu equals 8 pi g t mu nu. 804 01:02:14,740 --> 01:02:19,290 This is known as the Einstein field equation. 805 01:02:26,170 --> 01:02:28,900 So before I do a few more things with it, 806 01:02:28,900 --> 01:02:30,670 let us pause and just sort of take 807 01:02:30,670 --> 01:02:34,880 stock of what went into this calculation. 808 01:02:34,880 --> 01:02:37,520 We have a ton of mathematical tools 809 01:02:37,520 --> 01:02:39,020 that we have developed that allow us 810 01:02:39,020 --> 01:02:43,370 to just to describe the behavior of curved manifolds 811 01:02:43,370 --> 01:02:47,510 and the motion of bodies in a moving curved manifolds. 812 01:02:47,510 --> 01:02:50,870 We didn't yet have a tool telling us 813 01:02:50,870 --> 01:02:55,790 how the spacetime metric can be specified, OK? 814 01:02:55,790 --> 01:02:58,640 We didn't have the equivalent of the Newtonian fuel equation 815 01:02:58,640 --> 01:03:01,760 that told me how gravity arises from a source. 816 01:03:01,760 --> 01:03:05,870 So what we did was we looked at the geodesic equation, 817 01:03:05,870 --> 01:03:08,540 we went into a limit where things deviated 818 01:03:08,540 --> 01:03:10,610 just a little bit from flat spacetime, 819 01:03:10,610 --> 01:03:13,210 and we required objects be moving non-relativistically 820 01:03:13,210 --> 01:03:17,480 so that their spatial four velocity components were all 821 01:03:17,480 --> 01:03:18,410 small. 822 01:03:18,410 --> 01:03:23,660 That told us that we were able to reproduce the Newtonian 823 01:03:23,660 --> 01:03:28,310 equation of motion if h00, the little deviation of spacetime 824 01:03:28,310 --> 01:03:31,460 from flat spacetime in the 00 piece, 825 01:03:31,460 --> 01:03:35,330 was equal to negative 2 times the Newtonian potential. 826 01:03:35,330 --> 01:03:40,190 We then said, well, the Newtonian field equation 827 01:03:40,190 --> 01:03:42,590 is sort of sick from a relativistic perspective 828 01:03:42,590 --> 01:03:45,440 because it is working with a particular component 829 01:03:45,440 --> 01:03:47,720 of a tensor rather than with a tensor. 830 01:03:47,720 --> 01:03:49,640 So let's just ask ourselves, how can we 831 01:03:49,640 --> 01:03:53,970 promote this to a properly constructed tensorial equation? 832 01:03:53,970 --> 01:03:56,820 So we insisted the right-hand side be t mu nu. 833 01:03:56,820 --> 01:03:59,480 And then we looked for something that 834 01:03:59,480 --> 01:04:02,690 looks like two derivatives of the potential, 835 01:04:02,690 --> 01:04:06,320 or, more properly, two derivatives of the metric 836 01:04:06,320 --> 01:04:08,600 which is going to give me a curvature tensor, 837 01:04:08,600 --> 01:04:11,150 and say, OK, I want a two-index curvature tensor 838 01:04:11,150 --> 01:04:13,070 on the left-hand side. 839 01:04:13,070 --> 01:04:15,650 Since stress energy tensor is divergence-free, 840 01:04:15,650 --> 01:04:18,350 I am forced to choose a character tensor that 841 01:04:18,350 --> 01:04:21,560 is divergence-free, and that's what leads me to this object, 842 01:04:21,560 --> 01:04:24,180 and there's the Einstein curvature. 843 01:04:24,180 --> 01:04:26,880 And then insisting that that procedure reproduce 844 01:04:26,880 --> 01:04:31,550 the Newtonian limit when things sort of deviate very slightly 845 01:04:31,550 --> 01:04:34,147 from flat spacetime, that insisted 846 01:04:34,147 --> 01:04:36,230 the constant proportionality between the two sides 847 01:04:36,230 --> 01:04:39,290 be 8 pi j. 848 01:04:39,290 --> 01:04:43,640 This, in a nutshell, is how Einstein derived this equation 849 01:04:43,640 --> 01:04:53,900 originally when it was published in 1915. 850 01:04:53,900 --> 01:04:56,000 When I first went through this exercise 851 01:04:56,000 --> 01:04:59,870 and really appreciated this, I was struck 852 01:04:59,870 --> 01:05:01,580 by what a clever guy he was. 853 01:05:01,580 --> 01:05:05,270 And it is worth noting that the mathematics for doing this 854 01:05:05,270 --> 01:05:07,330 was very foreign to Einstein at that time. 855 01:05:07,330 --> 01:05:09,620 There's a reason there's a 10-year gap 856 01:05:09,620 --> 01:05:11,810 between his papers on special relativity 857 01:05:11,810 --> 01:05:14,570 and his presentation of the field equations 858 01:05:14,570 --> 01:05:16,070 of general relativity. 859 01:05:16,070 --> 01:05:21,680 Special relativity was 1905, field equation was 1915. 860 01:05:21,680 --> 01:05:24,860 He was spending most of those intervening 10 years learning 861 01:05:24,860 --> 01:05:30,530 all the math that we have been studying for the past six 862 01:05:30,530 --> 01:05:31,880 or seven weeks, OK? 863 01:05:31,880 --> 01:05:36,033 So we kind of have the luxury of knowing what path to take. 864 01:05:36,033 --> 01:05:38,450 And so we were able to sort of pick out the most important 865 01:05:38,450 --> 01:05:40,490 bits so that we could sort of-- 866 01:05:40,490 --> 01:05:42,057 we knew where we wanted to go. 867 01:05:42,057 --> 01:05:43,890 He had to learn all this stuff from scratch, 868 01:05:43,890 --> 01:05:46,532 and he worked with quite a few mathematicians 869 01:05:46,532 --> 01:05:47,615 to learn all these pieces. 870 01:05:50,420 --> 01:05:52,530 Having said that, though, it did strike me 871 01:05:52,530 --> 01:05:56,822 this is a somewhat ad hoc kind of a derivation. 872 01:05:56,822 --> 01:05:59,030 When you look at this, you might sort of think, well, 873 01:05:59,030 --> 01:06:00,740 could we not-- might there not be 874 01:06:00,740 --> 01:06:03,020 other things I could put on either the left-hand side 875 01:06:03,020 --> 01:06:06,500 or the right-hand side that would still 876 01:06:06,500 --> 01:06:08,390 respect the Newtonian limit? 877 01:06:08,390 --> 01:06:22,830 And indeed, we can add any divergence-free tensor onto-- 878 01:06:22,830 --> 01:06:25,080 depending how you count it-- either the left-hand side 879 01:06:25,080 --> 01:06:27,372 or the right-hand side-- let's say the left-hand side-- 880 01:06:32,130 --> 01:06:36,340 and we would still have a good field equation. 881 01:06:44,620 --> 01:06:46,960 Einstein himself was the first one to note this. 882 01:06:46,960 --> 01:06:50,840 Here's an example of such a divergence-free tensor. 883 01:06:50,840 --> 01:06:52,510 The metric itself, OK? 884 01:06:52,510 --> 01:06:56,380 The metric is compatible with the covariant derivative. 885 01:06:56,380 --> 01:06:59,380 Any covariant derivative of the metric is 0. 886 01:06:59,380 --> 01:07:12,640 And so I can just put the metric over here, 887 01:07:12,640 --> 01:07:14,263 that's perfectly fine. 888 01:07:14,263 --> 01:07:15,930 Now the dimensions are a little bit off, 889 01:07:15,930 --> 01:07:20,340 so we have to insert a constant of proportionality 890 01:07:20,340 --> 01:07:22,110 to make everything come out right. 891 01:07:22,110 --> 01:07:27,330 This lambda is known as the cosmological constant. 892 01:07:39,350 --> 01:07:41,080 Now what's kind of interesting is 893 01:07:41,080 --> 01:07:43,000 that one can write down the Einstein field 894 01:07:43,000 --> 01:07:50,140 equations in this way, but you could just as easily 895 01:07:50,140 --> 01:07:52,780 take that lambda g mu nu and move it 896 01:07:52,780 --> 01:07:56,050 onto the right-hand side and think of this additional term 897 01:07:56,050 --> 01:07:59,490 as a particularly special source of stress energy. 898 01:08:11,750 --> 01:08:14,100 Let's do that. 899 01:08:14,100 --> 01:08:21,760 So let's define t mu nu lambda equal 900 01:08:21,760 --> 01:08:29,109 negative lambda over 8 pi g times g mu nu. 901 01:08:29,109 --> 01:08:31,689 If we do that, we then just have g 902 01:08:31,689 --> 01:08:35,260 mu nu equals 8 pi gt mu nu with a particular contribution 903 01:08:35,260 --> 01:08:39,939 to our t mu nu being this cosmological constant term. 904 01:08:39,939 --> 01:08:44,050 When we do this, what you see is that t mu 905 01:08:44,050 --> 01:09:00,439 nu is nothing more than a perfect fluid with rho 906 01:09:00,439 --> 01:09:05,609 equals 8 pi g in the freely-falling frame, pressure 907 01:09:05,609 --> 01:09:09,620 of negative lambda over 8 pi j. 908 01:09:14,620 --> 01:09:16,300 Such a stress energy tensor actually 909 01:09:16,300 --> 01:09:19,330 arises in quantum field theories. 910 01:09:19,330 --> 01:09:21,776 This represents a form of zero-point energy 911 01:09:21,776 --> 01:09:22,359 in the vacuum. 912 01:09:34,149 --> 01:09:35,830 You basically need to look for something 913 01:09:35,830 --> 01:09:39,399 that is a stress energy tensor that is isotopic and invariant 914 01:09:39,399 --> 01:09:41,990 to Lorentz transformations and the local Lorentz frame, 915 01:09:41,990 --> 01:09:44,500 and that uniquely picks out a stress energy tensor that 916 01:09:44,500 --> 01:09:47,279 is proportional to the metric in the freely-falling frame. 917 01:09:47,279 --> 01:09:49,029 So this is an argument that was originally 918 01:09:49,029 --> 01:09:50,890 noted by Yakov Zeldovich. 919 01:10:02,600 --> 01:10:03,100 Whoops. 920 01:10:13,250 --> 01:10:15,170 And much of this stuff was considered 921 01:10:15,170 --> 01:10:18,470 to be kind of a curiosity for years 922 01:10:18,470 --> 01:10:21,600 until cosmological observations-- 923 01:10:21,600 --> 01:10:22,850 we haven't done cosmology yet. 924 01:10:22,850 --> 01:10:24,900 We will do this in a couple of weeks-- 925 01:10:24,900 --> 01:10:27,740 a couple of lectures, I should say. 926 01:10:27,740 --> 01:10:30,950 And it turns out that the large-scale structure 927 01:10:30,950 --> 01:10:33,800 of our universe seems to support the existence potentially 928 01:10:33,800 --> 01:10:36,060 of there being a cosmological constant. 929 01:10:36,060 --> 01:10:43,190 So the behavior of all these things is a lot more relevant, 930 01:10:43,190 --> 01:10:45,450 it's been a lot more relevant over the past, 931 01:10:45,450 --> 01:10:48,290 say, 15 or 20 years than it was when I originally 932 01:10:48,290 --> 01:10:51,920 learned the subject in 1993. 933 01:10:51,920 --> 01:10:54,830 So I want to just conclude this lecture 934 01:10:54,830 --> 01:11:01,220 with a couple of remarks about things that are commonly 935 01:11:01,220 --> 01:11:07,130 set equal to 1 when we are doing calculations of this point. 936 01:11:07,130 --> 01:11:21,950 So one often sets G equal to 1 as well as c equal to 1. 937 01:11:21,950 --> 01:11:24,140 Carroll's textbook does not-- 938 01:11:24,140 --> 01:11:26,300 several other modern textbooks do not-- 939 01:11:26,300 --> 01:11:29,480 I personally like for pedagogical purposes leaving 940 01:11:29,480 --> 01:11:33,320 the G in there, because it is very useful for calling 941 01:11:33,320 --> 01:11:36,420 out-- helping to understand the way in which different terms 942 01:11:36,420 --> 01:11:38,123 sort of couple in. 943 01:11:38,123 --> 01:11:40,540 It can-- if nothing else, it serves as a very useful order 944 01:11:40,540 --> 01:11:42,207 counting parameter, something that we'll 945 01:11:42,207 --> 01:11:45,610 see in some of the future calculations that we do. 946 01:11:45,610 --> 01:11:47,570 But there's a reason why one often 947 01:11:47,570 --> 01:11:50,660 works with G equal to 1 in many relativity analyses. 948 01:11:50,660 --> 01:11:55,040 Fundamentally, this is because gravity is a very weak force. 949 01:11:55,040 --> 01:11:58,160 G is the most poorly known of all 950 01:11:58,160 --> 01:12:00,038 of the fundamental constants of nature. 951 01:12:00,038 --> 01:12:01,080 I think it's only known-- 952 01:12:01,080 --> 01:12:02,310 I forget the number right now, but it's 953 01:12:02,310 --> 01:12:04,040 known to about five or six digits. 954 01:12:04,040 --> 01:12:06,890 Contrast this with things like the intrinsic magnetic moment 955 01:12:06,890 --> 01:12:11,270 of the electron, which is known to something like 13 digits. 956 01:12:11,270 --> 01:12:15,230 What this sort of means is that because G is so poorly known-- 957 01:12:19,820 --> 01:12:21,820 well, let me just write that out in words first. 958 01:12:21,820 --> 01:12:23,440 So G is itself poorly known. 959 01:12:30,860 --> 01:12:33,600 And so when we measure the properties 960 01:12:33,600 --> 01:12:48,800 of various large objects using gravity, 961 01:12:48,800 --> 01:12:56,990 we typically find that something like G times an object's mass 962 01:12:56,990 --> 01:13:06,400 is measured much better than M alone. 963 01:13:06,400 --> 01:13:09,400 Basically, the observable that one is probing 964 01:13:09,400 --> 01:13:13,270 is G times M. To get M out of that, you take G times M 965 01:13:13,270 --> 01:13:15,700 and you divide by the value of G that you 966 01:13:15,700 --> 01:13:18,340 have determined independently. 967 01:13:18,340 --> 01:13:20,800 If you only know this guy to five or six digits, 968 01:13:20,800 --> 01:13:23,290 you're only going to know this guy to five or six digits. 969 01:13:23,290 --> 01:13:25,660 Whereas, for instance, for our sun, 970 01:13:25,660 --> 01:13:28,130 GM is known to about nine digits, maybe even 10 digits 971 01:13:28,130 --> 01:13:28,630 now. 972 01:13:33,870 --> 01:13:45,630 When you set both G and c to 1, what you find 973 01:13:45,630 --> 01:13:53,640 is that mass, time, and length all come out 974 01:13:53,640 --> 01:13:55,120 having the same dimension. 975 01:14:18,170 --> 01:14:21,200 And what that means is that certain factors of G and c 976 01:14:21,200 --> 01:14:23,275 can be combined to become convergent factors. 977 01:14:26,120 --> 01:14:31,540 So in particular, the combination over c squared, 978 01:14:31,540 --> 01:14:34,375 it converts a normal mass-- let's say an SI mass-- 979 01:14:37,330 --> 01:14:38,050 into a length. 980 01:14:41,030 --> 01:14:50,500 A very useful one is G times the mass of the sun over c squared 981 01:14:50,500 --> 01:14:55,340 is 1.47 kilometers. 982 01:14:55,340 --> 01:14:57,590 GM over c cubed, OK? 983 01:14:57,590 --> 01:15:01,440 You're going to divide by another factor of velocity. 984 01:15:01,440 --> 01:15:08,280 This takes mass to time. 985 01:15:08,280 --> 01:15:12,630 So a similar one, G mass sun over c cubed, 986 01:15:12,630 --> 01:15:17,825 this is 4.92 times 10 to minus 6 seconds. 987 01:15:26,410 --> 01:15:28,550 One more before I conclude this lecture. 988 01:15:28,550 --> 01:15:52,820 (I can't erase this equation, it's too beautiful!) 989 01:15:52,820 --> 01:15:59,690 If I do g over c to the fourth, this converts energy to length. 990 01:16:06,330 --> 01:16:08,330 I'm not going to give the numeric value of this, 991 01:16:08,330 --> 01:16:10,122 but I'm going to make a comment about this. 992 01:16:10,122 --> 01:16:15,360 So bear in mind that your typical component of T mu nu 993 01:16:15,360 --> 01:16:20,600 has the dimensional form energy per unit volume-- 994 01:16:24,690 --> 01:16:26,550 i.e., energy over length cubed. 995 01:16:29,870 --> 01:16:35,830 So if I take G over c to the fourth times T mu nu, 996 01:16:35,830 --> 01:16:42,130 that is going to give me a length over a length cubed-- 997 01:16:47,540 --> 01:16:50,270 in other words, 1 over length squared, 998 01:16:50,270 --> 01:16:52,504 which is exactly what you get for curvature. 999 01:17:00,570 --> 01:17:02,820 So when one writes the Einstein field equations, 1000 01:17:02,820 --> 01:17:08,587 if you leave your G's and your c's in there, 1001 01:17:08,587 --> 01:17:10,920 the correct coupling factor between your Einstein tensor 1002 01:17:10,920 --> 01:17:12,628 and your stress energy tensor is actually 1003 01:17:12,628 --> 01:17:15,320 8 pi G over c to the fourth. 1004 01:17:15,320 --> 01:17:18,270 And I just want to leave you with the observation 1005 01:17:18,270 --> 01:17:21,180 that G is a pretty small constant, 1006 01:17:21,180 --> 01:17:24,390 c to the fourth is a rather large constant, 1007 01:17:24,390 --> 01:17:28,260 and so we are getting a tiny amount of curvature 1008 01:17:28,260 --> 01:17:30,840 from a tremendous amount of stress energy. 1009 01:17:30,840 --> 01:17:34,190 Spacetime is hard to bend.