1 00:00:00,000 --> 00:00:04,920 [SQUEAKING] [RUSTLING] [CLICKING] 2 00:00:11,320 --> 00:00:14,710 SCOTT HUGHES: So this is unfortunately a rather sad 3 00:00:14,710 --> 00:00:17,350 lecture, [LAUGHS] since this will be the last one that 4 00:00:17,350 --> 00:00:19,650 I'm giving to a live audience here, 5 00:00:19,650 --> 00:00:21,400 although any of you who are grad students, 6 00:00:21,400 --> 00:00:24,670 if you want to come and keep me company while I'm recording 7 00:00:24,670 --> 00:00:28,200 my next couple of lectures-- 8 00:00:28,200 --> 00:00:29,200 I'm actually not joking. 9 00:00:29,200 --> 00:00:30,992 It's really weird talking to an empty room, 10 00:00:30,992 --> 00:00:33,640 and so it might be kind of nice to at least have a face 11 00:00:33,640 --> 00:00:34,810 to react to. 12 00:00:34,810 --> 00:00:38,920 So I will be recording these, and making them available. 13 00:00:38,920 --> 00:00:41,920 And per the announcements that I sent out today, 14 00:00:41,920 --> 00:00:44,710 there are some additional assignments 15 00:00:44,710 --> 00:00:45,785 that I've begun posting. 16 00:00:45,785 --> 00:00:47,410 I've begun putting things like that up. 17 00:00:47,410 --> 00:00:50,333 But for the time being, ignore due dates. 18 00:00:50,333 --> 00:00:52,000 In fact, I would like to say, especially 19 00:00:52,000 --> 00:00:54,880 if you are getting ready to leave town, just 20 00:00:54,880 --> 00:00:56,990 ignore the assignments. 21 00:00:56,990 --> 00:01:00,550 There will be updates posted fairly soon. 22 00:01:00,550 --> 00:01:03,430 You know, [CHUCKLES] sometimes-- there's 23 00:01:03,430 --> 00:01:05,820 a saying that Martin Schmidt, our provost, 24 00:01:05,820 --> 00:01:10,060 said one time in respect to a particular initiative 25 00:01:10,060 --> 00:01:12,460 as it was unwinding, and I feel like it applies here: 26 00:01:12,460 --> 00:01:15,055 we're basically sewing together the parachute after we've 27 00:01:15,055 --> 00:01:16,180 jumped out of the airplane. 28 00:01:16,180 --> 00:01:19,570 [CHUCKLES] And so that's kind of where we are right now, 29 00:01:19,570 --> 00:01:22,930 and hopefully it'll be assembled before we hit the ground. 30 00:01:22,930 --> 00:01:25,220 But we're all kind of figuring out the way this goes. 31 00:01:25,220 --> 00:01:27,652 So there will be updates with respect to assignments 32 00:01:27,652 --> 00:01:29,110 and things like that, and you know, 33 00:01:29,110 --> 00:01:30,870 we're just trying to figure out-- 34 00:01:30,870 --> 00:01:32,320 I'll just be blunt-- how the hell 35 00:01:32,320 --> 00:01:34,906 we're going to actually pull this off. 36 00:01:34,906 --> 00:01:39,210 [LAUGHS] It's not curved enough. 37 00:01:39,210 --> 00:01:41,790 All right, so let me get back to where we were. 38 00:01:41,790 --> 00:01:42,630 I want to begin-- 39 00:01:42,630 --> 00:01:44,520 so what we worked with yesterday was 40 00:01:44,520 --> 00:01:47,220 we derived this mathematical object called the Riemann 41 00:01:47,220 --> 00:01:48,480 curvature tensor. 42 00:01:48,480 --> 00:01:52,080 And the way we discussed it was by thinking about some vector 43 00:01:52,080 --> 00:01:55,290 that I parallel transported around a closed loop. 44 00:01:55,290 --> 00:01:58,410 I chose a parallelogram because it allowed me to sort of nicely 45 00:01:58,410 --> 00:02:00,450 formulate the mathematics. 46 00:02:00,450 --> 00:02:02,353 So if I imagine a parallelogram who's 47 00:02:02,353 --> 00:02:04,020 got sides that are sort of close enough, 48 00:02:04,020 --> 00:02:06,600 that parallel is a well-defined term, 49 00:02:06,600 --> 00:02:10,169 and the sides are in two different directions, 50 00:02:10,169 --> 00:02:12,270 one's of length delta x in the lambda direction, 51 00:02:12,270 --> 00:02:14,910 one is a delta x in the sigma direction, 52 00:02:14,910 --> 00:02:17,130 when I take this vector all the way around the loop, 53 00:02:17,130 --> 00:02:18,940 I find the vector has changed. 54 00:02:18,940 --> 00:02:21,320 It's pointing in a different direction, 55 00:02:21,320 --> 00:02:25,130 and this describes the way in which it has sort of moved 56 00:02:25,130 --> 00:02:27,230 as I go around that loop. 57 00:02:27,230 --> 00:02:30,070 That object is-- even though it's created-- 58 00:02:30,070 --> 00:02:33,118 it's assembled from the Christoffel symbols, which 59 00:02:33,118 --> 00:02:34,910 are not tensor, but it's done in such a way 60 00:02:34,910 --> 00:02:37,970 that their nontensorialness cancels out, 61 00:02:37,970 --> 00:02:41,270 and this four-index object R is a tensor. 62 00:02:41,270 --> 00:02:42,710 As you can see, it involves terms 63 00:02:42,710 --> 00:02:45,740 like derivatives of the Christoffels and Christoffel 64 00:02:45,740 --> 00:02:46,550 times Christoffel. 65 00:02:46,550 --> 00:02:50,150 So that's where nonlinearity is going to enter. 66 00:02:50,150 --> 00:02:52,155 I want to sort of fairly quickly-- 67 00:02:52,155 --> 00:02:54,530 because I do want to get into some other stuff that we do 68 00:02:54,530 --> 00:02:55,310 with this-- 69 00:02:55,310 --> 00:02:58,493 talk a bit about the symmetry of this thing. 70 00:02:58,493 --> 00:03:00,410 So I discussed last time how you look at this, 71 00:03:00,410 --> 00:03:02,120 and naively, you think you've got-- 72 00:03:02,120 --> 00:03:04,328 in four dimensions, four dimensions, four [INAUDIBLE] 73 00:03:04,328 --> 00:03:04,828 states. 74 00:03:04,828 --> 00:03:06,500 It looks like you have 256 components. 75 00:03:10,130 --> 00:03:12,793 But in fact, there are quite a few symmetries 76 00:03:12,793 --> 00:03:14,960 associated with this which reduces it significantly. 77 00:03:14,960 --> 00:03:16,640 I kind of gave you the answer last time, 78 00:03:16,640 --> 00:03:18,920 but let me talk about where that comes from. 79 00:03:18,920 --> 00:03:22,340 I'm going to describe a way to see these symmetries, which 80 00:03:22,340 --> 00:03:23,840 there's a couple ways you can do it, 81 00:03:23,840 --> 00:03:25,010 and I'm going to pick the one that's 82 00:03:25,010 --> 00:03:26,052 essentially the simplest. 83 00:03:26,052 --> 00:03:27,710 It's perhaps not as-- 84 00:03:27,710 --> 00:03:31,670 there's a few ways to do it that a purist might prefer, 85 00:03:31,670 --> 00:03:33,170 but in the interest of time, I think 86 00:03:33,170 --> 00:03:34,520 this is sort of the best one. 87 00:03:34,520 --> 00:03:39,320 So the first thing is that just by inspecting this answer, 88 00:03:39,320 --> 00:03:45,950 you should see that this object is antisymmetric if you 89 00:03:45,950 --> 00:03:49,150 exchange the final two indices. 90 00:03:49,150 --> 00:03:52,880 OK, just look at that formula, switch the indices 91 00:03:52,880 --> 00:03:55,440 sigma and lambda, and you get a minus sign. 92 00:03:55,440 --> 00:03:57,440 Physically, that actually hopefully makes sense, 93 00:03:57,440 --> 00:03:59,120 because what that is basically telling you-- remember the way 94 00:03:59,120 --> 00:04:01,460 we did this, this operation of parallel transporting 95 00:04:01,460 --> 00:04:02,420 around a parallelogram. 96 00:04:02,420 --> 00:04:05,660 We sort of went along, say, the lambda direction, then 97 00:04:05,660 --> 00:04:08,015 the sigma, then the lambda, than the sigma. 98 00:04:08,015 --> 00:04:09,735 If I exchange those indices, it's 99 00:04:09,735 --> 00:04:12,110 kind of like I actually go in the opposite direction, OK? 100 00:04:12,110 --> 00:04:13,580 I go in the other direction first. 101 00:04:13,580 --> 00:04:15,650 I go on the sigma first, then the lambda. 102 00:04:15,650 --> 00:04:19,700 And so this corresponds to doing this operation, which 103 00:04:19,700 --> 00:04:21,510 I'll remind you is called a holonomy. 104 00:04:21,510 --> 00:04:23,978 It corresponds to reversing direction. 105 00:04:39,914 --> 00:04:40,650 OK? 106 00:04:40,650 --> 00:04:42,990 So that's a fairly easy one to say. 107 00:04:42,990 --> 00:04:46,950 The others are a little bit tougher, and like I said, 108 00:04:46,950 --> 00:04:49,920 what I'm going to do is follow kind of a shortcut to do this. 109 00:04:49,920 --> 00:04:53,550 One can do it using sort of the full glory of the Riemann 110 00:04:53,550 --> 00:04:56,490 tensor here, but life gets a little bit easier for us 111 00:04:56,490 --> 00:04:57,885 if we do the following. 112 00:05:03,010 --> 00:05:07,005 So what you want to do first is lower the first index. 113 00:05:11,100 --> 00:05:14,400 So what I'm going to do is look at R, 114 00:05:14,400 --> 00:05:19,740 alpha in the downstairs, mu lambda sigma. 115 00:05:19,740 --> 00:05:27,520 This is what I get when I contract, like so. 116 00:05:27,520 --> 00:05:29,020 And then what I'm going to do is I'm 117 00:05:29,020 --> 00:05:32,330 going to do all of my analysis that follows in a local Lorentz 118 00:05:32,330 --> 00:05:32,830 frame. 119 00:05:39,500 --> 00:05:41,720 When I go into the local Lorentz frame, 120 00:05:41,720 --> 00:05:44,840 the metric at a particular point is 121 00:05:44,840 --> 00:05:46,250 the metric of flat spacetime. 122 00:05:46,250 --> 00:05:47,570 It's the A to mu nu. 123 00:05:56,830 --> 00:06:02,333 My Christoffel symbols all vanish, 124 00:06:02,333 --> 00:06:04,000 but you have to be a little bit careful, 125 00:06:04,000 --> 00:06:06,490 because the derivatives of the Christoffel do not vanish, 126 00:06:06,490 --> 00:06:06,990 right? 127 00:06:06,990 --> 00:06:08,532 So one of the key points is that when 128 00:06:08,532 --> 00:06:10,680 you go into those local Lorentz frame, 129 00:06:10,680 --> 00:06:13,840 there is a little bit of curvature associated with it. 130 00:06:13,840 --> 00:06:15,060 So the derivatives do not-- 131 00:06:24,810 --> 00:06:26,780 OK, so the reason I'm doing this-- like I said, 132 00:06:26,780 --> 00:06:29,570 you actually could do the little counting 133 00:06:29,570 --> 00:06:32,600 exercise I'm about to go through with this whole thing. 134 00:06:32,600 --> 00:06:33,828 It's just messy, OK? 135 00:06:33,828 --> 00:06:35,870 And so this is sort of a quick way to see the way 136 00:06:35,870 --> 00:06:36,578 it all comes out. 137 00:06:36,578 --> 00:06:39,625 Because this is a tensor relationship, 138 00:06:39,625 --> 00:06:41,000 you are guaranteed that something 139 00:06:41,000 --> 00:06:43,430 you conclude in a convenient reference frame 140 00:06:43,430 --> 00:06:45,050 will hold in all of them, OK? 141 00:06:45,050 --> 00:06:47,055 So it's a nice way to do this. 142 00:06:47,055 --> 00:06:48,680 Those of you who are sort of, you know, 143 00:06:48,680 --> 00:06:52,238 particularly purist, knock yourselves out 144 00:06:52,238 --> 00:06:54,530 trying to figure out how to do this sort of in general. 145 00:06:54,530 --> 00:06:58,290 This is adequate for our purposes here. 146 00:06:58,290 --> 00:07:01,800 So when I go and I do this, here's 147 00:07:01,800 --> 00:07:06,300 what my all downstairs Riemann tensor turns into. 148 00:07:08,980 --> 00:07:11,800 I'm going to lose the Christoffel squared 149 00:07:11,800 --> 00:07:15,820 terms because Christoffel vanishes, 150 00:07:15,820 --> 00:07:19,930 and the only thing that is left are 151 00:07:19,930 --> 00:07:24,628 the terms that involve the derivative of the Christoffel, 152 00:07:24,628 --> 00:07:26,910 OK? 153 00:07:26,910 --> 00:07:28,840 Let's do something a little bit further. 154 00:07:28,840 --> 00:07:34,680 So let's now plug in the definition of the Christoffel, 155 00:07:34,680 --> 00:07:38,580 and we'll use the metric before we actually go into the frame, 156 00:07:38,580 --> 00:07:40,897 so that we can sort of get the form that we 157 00:07:40,897 --> 00:07:41,730 would get with this. 158 00:08:04,760 --> 00:08:08,700 I should have made myself be a little bit more careful here. 159 00:08:08,700 --> 00:08:12,560 So I'm going to put LLF on this, to make it clear that I'm doing 160 00:08:12,560 --> 00:08:14,890 this in a local Lorentz frame. 161 00:08:18,080 --> 00:08:23,540 So I do this in a local Lorentz frame, 162 00:08:23,540 --> 00:08:26,450 and what this turns into-- so there's a lot of algebra 163 00:08:26,450 --> 00:08:27,710 that I'm not writing out. 164 00:08:37,580 --> 00:08:49,490 This is an alpha, g sigma mu minus d sigma d mu g alpha 165 00:08:49,490 --> 00:08:59,370 lambda plus d sigma d alpha g lambda mu, OK? 166 00:08:59,370 --> 00:09:02,445 So notice, it is only the second derivative of the metric 167 00:09:02,445 --> 00:09:03,570 that is appearing here, OK? 168 00:09:03,570 --> 00:09:05,958 If I go into the local Lorentz frame, that's 169 00:09:05,958 --> 00:09:07,500 the one degree of freedom we were not 170 00:09:07,500 --> 00:09:09,522 able to transform away. 171 00:09:09,522 --> 00:09:11,730 Anything involving the metric in the first derivative 172 00:09:11,730 --> 00:09:13,105 dies in this particular frame. 173 00:09:13,105 --> 00:09:14,730 But this is good enough for us to begin 174 00:09:14,730 --> 00:09:17,820 to count up symmetries, and to see what 175 00:09:17,820 --> 00:09:19,420 things are going to look like. 176 00:09:19,420 --> 00:09:24,328 And so I don't have any great guidance for how to do this. 177 00:09:24,328 --> 00:09:26,370 This is one of those things where, literally, you 178 00:09:26,370 --> 00:09:28,203 just sort of stare at it for a little while, 179 00:09:28,203 --> 00:09:32,010 and you see what happens when you play around with exchanging 180 00:09:32,010 --> 00:09:35,012 various pairs of indices. 181 00:09:35,012 --> 00:09:36,220 So stare at this for a while. 182 00:09:39,260 --> 00:09:41,080 OK, so you can see right away-- 183 00:09:41,080 --> 00:09:43,680 just for counting purposes, let me write down 184 00:09:43,680 --> 00:09:47,010 the one that I argued was kind of obvious even 185 00:09:47,010 --> 00:09:48,360 in the full form. 186 00:09:48,360 --> 00:09:51,000 It's the one you get when you exchange 187 00:09:51,000 --> 00:09:52,530 the last pairs of indices. 188 00:09:52,530 --> 00:09:54,120 We'll call that symmetry one. 189 00:09:57,380 --> 00:10:00,380 Oh, bugger, thank you. 190 00:10:00,380 --> 00:10:05,030 [CHUCKLES] 191 00:10:05,030 --> 00:10:05,550 Yeah. 192 00:10:05,550 --> 00:10:07,100 [CHUCKLES] That's a good way to-- that's 193 00:10:07,100 --> 00:10:09,183 a very complicated way of writing zero, otherwise. 194 00:10:09,183 --> 00:10:10,670 [LAUGHING] 195 00:10:10,670 --> 00:10:13,905 All right, if you exchange the first ones, 196 00:10:13,905 --> 00:10:15,530 you also-- if you look at this formula, 197 00:10:15,530 --> 00:10:17,380 you can see when you exchange alpha and mu, you 198 00:10:17,380 --> 00:10:18,140 get a minus sign. 199 00:10:29,940 --> 00:10:33,190 So one that might be-- 200 00:10:33,190 --> 00:10:35,690 this one takes a little bit of creativity to sort of see it, 201 00:10:35,690 --> 00:10:36,640 where it comes out. 202 00:10:42,250 --> 00:10:45,900 Suppose what I do is I exchange the first two indices 203 00:10:45,900 --> 00:10:48,190 for the second two indices. 204 00:10:48,190 --> 00:10:50,140 Well, it turns out, if you do that, you just 205 00:10:50,140 --> 00:10:51,140 get the expression back. 206 00:10:55,116 --> 00:10:56,110 OK? 207 00:10:56,110 --> 00:10:59,842 So if I just swap alpha mu, make them my last two, 208 00:10:59,842 --> 00:11:01,800 make lambda sigma my first two, [CLICKS TONGUE] 209 00:11:01,800 --> 00:11:03,450 I get Riemann right back. 210 00:11:03,450 --> 00:11:04,590 So that's another symmetry. 211 00:11:10,680 --> 00:11:15,980 And then finally, someday, when I've 212 00:11:15,980 --> 00:11:19,393 got a little bit more bandwidth in my class to do this, 213 00:11:19,393 --> 00:11:20,810 I'd really like to sort of justify 214 00:11:20,810 --> 00:11:23,268 where you can derive this one a little bit more rigorously. 215 00:11:23,268 --> 00:11:25,100 But for now, it's sufficient just 216 00:11:25,100 --> 00:11:27,210 to sort of say, look at that formula, 217 00:11:27,210 --> 00:11:29,720 and you'll see that it's true. 218 00:11:29,720 --> 00:11:34,490 If you take-- and then you cyclically 219 00:11:34,490 --> 00:11:54,180 permute the second, third, and fourth indices, 220 00:11:54,180 --> 00:11:59,090 you get zero, OK? 221 00:11:59,090 --> 00:12:02,680 There is a variant of this particular symmetry, which 222 00:12:02,680 --> 00:12:09,260 is that if you take this and you antisymmetrize on these things, 223 00:12:09,260 --> 00:12:10,340 you get zero, OK? 224 00:12:10,340 --> 00:12:11,960 And I've got some words in the notes 225 00:12:11,960 --> 00:12:13,900 that, in the interest of time, I'm 226 00:12:13,900 --> 00:12:15,993 not going to go through them, but they are posted, 227 00:12:15,993 --> 00:12:17,660 demonstrating why those two are actually 228 00:12:17,660 --> 00:12:18,980 equivalent to one another. 229 00:12:18,980 --> 00:12:22,688 What it boils down to is, expand out this antisymmetrization-- 230 00:12:22,688 --> 00:12:24,980 and I'm going to do that again for a three-index object 231 00:12:24,980 --> 00:12:26,840 a little bit later in this lecture-- 232 00:12:26,840 --> 00:12:30,800 and then occasionally invoke one of these other ones, 233 00:12:30,800 --> 00:12:34,220 and you'll see that these two, four and four prime, are 234 00:12:34,220 --> 00:12:37,110 saying the same thing. 235 00:12:37,110 --> 00:12:41,360 So go through all of these different symmetries, 236 00:12:41,360 --> 00:12:45,860 and you'll find n to the 4 independent components 237 00:12:45,860 --> 00:12:53,042 goes over to n squared times n squared minus 1 over 12, OK? 238 00:12:53,042 --> 00:12:54,250 Not terribly hard to do that. 239 00:12:54,250 --> 00:12:57,650 It's a little exercise in combinatorics. 240 00:12:57,650 --> 00:13:01,490 And when you do that for n equals 20, as I mentioned, 241 00:13:01,490 --> 00:13:04,031 you end up with-- 242 00:13:04,031 --> 00:13:05,505 excuse me, not n equals 20. 243 00:13:05,505 --> 00:13:12,980 When you do this with n equals 4, you end up with 20. 244 00:13:16,190 --> 00:13:18,680 And just for completeness, let me actually write out 245 00:13:18,680 --> 00:13:27,430 that if you actually-- if you go into the local Lorentz frame, 246 00:13:27,430 --> 00:13:36,130 your spacetime metric is in fact minus 1, 1, 1, 247 00:13:36,130 --> 00:13:38,060 1 on the diagonal. 248 00:13:38,060 --> 00:13:41,680 But you, of course, have these quadratic corrections, 249 00:13:41,680 --> 00:13:44,503 and one can in fact write them out explicitly. 250 00:13:49,820 --> 00:13:52,950 So that is what the time piece ends up looking like. 251 00:13:56,440 --> 00:13:59,700 There is an off-diagonal piece, which 252 00:13:59,700 --> 00:14:02,370 only enters at quadratic order. 253 00:14:07,463 --> 00:14:10,780 It looks like this, and move this down a little bit lower. 254 00:14:28,640 --> 00:14:30,750 That's what your space-based piece will look like. 255 00:14:30,750 --> 00:14:34,010 So this explicitly constructs the coordinate system 256 00:14:34,010 --> 00:14:38,990 used in a freely falling frame, including these second order 257 00:14:38,990 --> 00:14:39,920 corrections. 258 00:14:39,920 --> 00:14:45,077 So this particular form is known as Riemann normal coordinates. 259 00:14:50,810 --> 00:14:52,993 So if you are-- 260 00:14:52,993 --> 00:14:55,160 this is discussed in a little bit more detail in one 261 00:14:55,160 --> 00:15:00,620 of the optional readings in the textbook 262 00:15:00,620 --> 00:15:04,520 by Eric Poisson, A Relativist's Toolkit. 263 00:15:04,520 --> 00:15:05,875 Very nice discussion. 264 00:15:14,800 --> 00:15:29,750 All right, so now that we have the curvature tensor in hand, 265 00:15:29,750 --> 00:15:32,900 we really have essentially every tool 266 00:15:32,900 --> 00:15:35,713 that matters for 8.962, OK? 267 00:15:35,713 --> 00:15:37,130 There's a little bit more analysis 268 00:15:37,130 --> 00:15:44,210 we need to do with this guy, but we now have all the pieces. 269 00:15:44,210 --> 00:15:47,840 OK, there's no major new mathematical object 270 00:15:47,840 --> 00:15:50,210 or no object described in geometry 271 00:15:50,210 --> 00:15:51,922 that I need to build in order for us 272 00:15:51,922 --> 00:15:53,630 to make a relativistic theory of gravity. 273 00:15:55,845 --> 00:15:58,220 I do want to talk about the curvature tensor a little bit 274 00:15:58,220 --> 00:16:01,430 more, because there are a couple of properties about it 275 00:16:01,430 --> 00:16:03,905 that are very important for us. 276 00:16:03,905 --> 00:16:05,780 And I think we'll have just enough time today 277 00:16:05,780 --> 00:16:06,738 to sort of get to them. 278 00:16:06,738 --> 00:16:10,220 That will allow us to set up, and in the first lecture 279 00:16:10,220 --> 00:16:11,900 that I will record in an empty room, 280 00:16:11,900 --> 00:16:15,990 actually derive a relativistic gravity equation. 281 00:16:15,990 --> 00:16:18,568 So let me talk about a few variants 282 00:16:18,568 --> 00:16:19,610 of this curvature tensor. 283 00:16:25,570 --> 00:16:35,140 So first, suppose you take the trace on some pair of indices. 284 00:16:35,140 --> 00:16:38,280 So what I mean by the trace is, essentially, 285 00:16:38,280 --> 00:16:41,040 imagine all the indices are in the downstairs position, 286 00:16:41,040 --> 00:16:43,110 and I contract it with the metric, 287 00:16:43,110 --> 00:16:44,852 so that I'm summing over them. 288 00:16:44,852 --> 00:16:46,560 Well, first of all, you'll note that when 289 00:16:46,560 --> 00:16:50,130 you do this, if you were to take the trace on indices 1 and 2, 290 00:16:50,130 --> 00:16:51,660 they are antisymmetric. 291 00:16:51,660 --> 00:16:55,710 The metric is symmetric, so contracting metric with indices 292 00:16:55,710 --> 00:16:57,720 1 and 2 is going to give me 0. 293 00:16:57,720 --> 00:17:02,580 If I can track symmetric with indices 3 and 4, I get 0, OK? 294 00:17:02,580 --> 00:17:05,099 So the only trace that makes sense 295 00:17:05,099 --> 00:17:09,359 is do it on either indices 1 and 3 or on indices 2 and 4. 296 00:17:09,359 --> 00:17:10,920 Let's do it on indices 1 and 3. 297 00:17:21,300 --> 00:17:28,740 So what I'm going to do is evaluate R, alpha mu alpha nu-- 298 00:17:28,740 --> 00:17:35,922 which if you like, you can write this as R alpha beta of R. 299 00:17:35,922 --> 00:17:37,610 Here, I'll turn it like this. 300 00:17:37,610 --> 00:17:40,500 Beta mu alpha nu-- 301 00:17:40,500 --> 00:17:48,410 we're going to define this as R with two indices, OK? 302 00:17:48,410 --> 00:17:50,042 Just little r and mu, nu. 303 00:17:50,042 --> 00:17:52,250 This actually shows up enough that it's given a name. 304 00:17:52,250 --> 00:17:55,900 This is called the Ricci curvature tensor. 305 00:18:13,370 --> 00:18:16,580 If you were to like, I said, if you do your trace on indices 1 306 00:18:16,580 --> 00:18:18,630 and 2, you get 0. 307 00:18:18,630 --> 00:18:24,120 If you do it on 3 and 4, you get 0. 308 00:18:24,120 --> 00:18:26,810 If you do it on 2 and 4, it's exactly the same 309 00:18:26,810 --> 00:18:29,810 as doing it on 1 and 3, because of all these other symmetries. 310 00:18:29,810 --> 00:18:31,970 So in fact, this ends up, when you count up 311 00:18:31,970 --> 00:18:33,680 all your different symmetries, this 312 00:18:33,680 --> 00:18:36,108 is the only trace that is meaningful, OK? 313 00:18:36,108 --> 00:18:38,150 There are a few other pairs where you get a minus 314 00:18:38,150 --> 00:18:40,860 sign, but still the same thing. 315 00:18:40,860 --> 00:18:44,040 So this is the only trace that ends up being meaningful. 316 00:18:44,040 --> 00:18:47,120 This is actually, in fact, it's not 317 00:18:47,120 --> 00:18:48,925 too hard to show that this is symmetric. 318 00:18:48,925 --> 00:18:50,300 So even though, you know, Riemann 319 00:18:50,300 --> 00:18:52,910 had all these crazy antisymmetries and symmetries, 320 00:18:52,910 --> 00:18:54,440 this one is simpler, OK? 321 00:19:07,070 --> 00:19:10,280 One way to do this is just to take the exact expression 322 00:19:10,280 --> 00:19:14,390 that we wrote down for the Riemann tensor 323 00:19:14,390 --> 00:19:16,730 in terms of derivatives of Christoffel and Christoffel 324 00:19:16,730 --> 00:19:20,930 squared, and just write it out in this traced over form. 325 00:19:20,930 --> 00:19:29,570 So when you do this, you get this. 326 00:19:29,570 --> 00:19:33,050 This guy is symmetric on the bottom two indices. 327 00:19:40,680 --> 00:19:43,210 Now, this isn't obviously symmetric, 328 00:19:43,210 --> 00:19:45,340 but recall, a lecture or two ago, I 329 00:19:45,340 --> 00:19:47,650 worked through a couple of identities that involved 330 00:19:47,650 --> 00:19:49,465 the determinant of the metric. 331 00:19:49,465 --> 00:19:52,420 It turns out, if you go back and you look up these identities, 332 00:19:52,420 --> 00:19:59,550 you can rewrite this term as d nu of d mu of the log of j-- 333 00:19:59,550 --> 00:20:02,230 excuse me, log of square root of j. 334 00:20:02,230 --> 00:20:04,298 Symmetric when you exchange mu and nu. 335 00:20:07,440 --> 00:20:21,710 And then you get two terms that look like this. 336 00:20:21,710 --> 00:20:23,600 OK, this one, pretty obviously symmetric 337 00:20:23,600 --> 00:20:25,268 on exchange of mu and nu. 338 00:20:25,268 --> 00:20:27,560 When you do this one, you might think to yourself, hmm, 339 00:20:27,560 --> 00:20:29,143 that one doesn't quite look symmetric. 340 00:20:29,143 --> 00:20:32,760 But remember, alpha and beta are both dummy indices. 341 00:20:32,760 --> 00:20:35,120 And so in fact, when you exchange mu and nu, 342 00:20:35,120 --> 00:20:37,948 this last term just turns back into itself. 343 00:20:37,948 --> 00:20:39,740 So that's kind of interesting, because R mu 344 00:20:39,740 --> 00:20:42,020 nu-- the reason I went through that little exercise-- 345 00:20:42,020 --> 00:20:47,360 this is a symmetric tensor, and we're 346 00:20:47,360 --> 00:20:49,620 going to work in four dimensions of spacetime here. 347 00:20:49,620 --> 00:20:56,180 So symmetric 4 by 4 has 10 independent components. 348 00:21:01,710 --> 00:21:02,880 Riemann had 20. 349 00:21:02,880 --> 00:21:06,510 Somehow, this Ricci has-- 350 00:21:06,510 --> 00:21:09,570 in some sense, you can of it as having 10 of-- 351 00:21:09,570 --> 00:21:11,880 the 20 components associated with Riemann 352 00:21:11,880 --> 00:21:15,230 are encoded in this guy. 353 00:21:15,230 --> 00:21:16,340 Where are the other 10? 354 00:21:16,340 --> 00:21:18,548 I'm going to talk about that in just a moment or two. 355 00:21:35,520 --> 00:21:37,290 Before I do that, I will just note 356 00:21:37,290 --> 00:21:39,060 that we are going to want to also know 357 00:21:39,060 --> 00:21:43,440 about the trace of the Ricci tensor. 358 00:21:43,440 --> 00:21:56,090 So if you compute this, this is often just abbreviated R 359 00:21:56,090 --> 00:21:59,000 with no indices whatsoever. 360 00:21:59,000 --> 00:22:01,660 This is called the Ricci scalar or the curvature scalar. 361 00:22:16,510 --> 00:22:19,930 So there's another variant of curvature which we're not 362 00:22:19,930 --> 00:22:22,490 going to use very much, but I wanted to just talk about very, 363 00:22:22,490 --> 00:22:25,400 very briefly. 364 00:22:25,400 --> 00:22:29,173 The derivation of this is highly non-obvious, 365 00:22:29,173 --> 00:22:30,673 so let me just write it out, and I'm 366 00:22:30,673 --> 00:22:32,840 going to talk about it briefly. 367 00:22:32,840 --> 00:22:37,880 So suppose I define a four-index tensor, 368 00:22:37,880 --> 00:22:45,240 C alpha mu lambda sigma, to be the Riemann tensor-- 369 00:22:45,240 --> 00:22:50,780 minus-- so working in n dimensions-- 370 00:22:50,780 --> 00:22:56,492 2 over n minus 2, g alpha-- 371 00:22:56,492 --> 00:22:59,620 I'm going to antisymmetrize here. 372 00:22:59,620 --> 00:23:01,910 So antisymmetrizing on lambda and sigma-- 373 00:23:33,340 --> 00:23:42,460 OK, so in the famous words of Rabi, who ordered that? 374 00:23:42,460 --> 00:23:47,050 So the way that this has been constructed, if you go through 375 00:23:47,050 --> 00:23:48,940 and you carefully look at the way this thing 376 00:23:48,940 --> 00:23:52,000 behaves under exchange of any pairs of indices, 377 00:23:52,000 --> 00:23:54,790 it has exactly the same symmetries as Riemann. 378 00:24:06,510 --> 00:24:11,160 But if you take the trace of this, turns out to be zero. 379 00:24:20,512 --> 00:24:22,470 In fact, when you go through, when you count up 380 00:24:22,470 --> 00:24:24,330 how many independent components it has, 381 00:24:24,330 --> 00:24:29,291 because it has no trace, it has 10 independent components. 382 00:24:48,440 --> 00:24:50,500 So this tensor has a name. 383 00:24:50,500 --> 00:24:55,710 It was first formulated by the mathematician Hermann Weyl. 384 00:24:55,710 --> 00:24:57,220 And so although it's a German name 385 00:24:57,220 --> 00:24:59,860 and it's not spelled that way, it is appropriately the "vile" 386 00:24:59,860 --> 00:25:00,360 tensor. 387 00:25:00,360 --> 00:25:02,710 It is a pretty vile thing to look at, 388 00:25:02,710 --> 00:25:04,190 but it plays an important role. 389 00:25:14,570 --> 00:25:16,270 So in keeping with the idea that it's 390 00:25:16,270 --> 00:25:18,670 got 10 independent components, Ricci 391 00:25:18,670 --> 00:25:22,720 has 10 independent components, Riemann has 20. 392 00:25:22,720 --> 00:25:28,390 Heuristically, you can sort of think and put big quotes 393 00:25:28,390 --> 00:25:29,560 around all these objects. 394 00:25:32,330 --> 00:25:33,720 This doesn't mean approximately. 395 00:25:33,720 --> 00:25:34,960 Let's just put it this way. 396 00:25:34,960 --> 00:25:45,020 This is, got all the same information as Ricci plus Weyl, 397 00:25:45,020 --> 00:25:45,610 OK? 398 00:25:45,610 --> 00:25:48,692 One can, in fact-- by bringing in appropriate powers 399 00:25:48,692 --> 00:25:50,150 in the metric and things like that, 400 00:25:50,150 --> 00:25:52,370 one can actually write down a real equation relating these. 401 00:25:52,370 --> 00:25:53,080 It's a bit of a mess. 402 00:25:53,080 --> 00:25:54,070 It's not that interesting. 403 00:25:54,070 --> 00:25:55,695 The key thing I want you to be aware of 404 00:25:55,695 --> 00:25:59,470 is that all of the curvature content of Riemann 405 00:25:59,470 --> 00:26:02,137 is sort of split into Ricci and Weyl. 406 00:26:02,137 --> 00:26:04,720 Looking ahead a little bit-- and because I'm going to be doing 407 00:26:04,720 --> 00:26:07,960 this in an empty room, I just want to make this point now-- 408 00:26:07,960 --> 00:26:14,590 we are soon going to see that Ricci is very closely 409 00:26:14,590 --> 00:26:23,015 related to sources of gravity. 410 00:26:27,470 --> 00:26:30,650 So we're going to-- when we formulate a theory of gravity, 411 00:26:30,650 --> 00:26:32,893 working with things, we're going to find 412 00:26:32,893 --> 00:26:34,310 that there's a tensor that is just 413 00:26:34,310 --> 00:26:37,010 a slight modification of the Ricci tensor, that 414 00:26:37,010 --> 00:26:38,510 is equal to the stress energy tensor 415 00:26:38,510 --> 00:26:39,920 that we use as our source. 416 00:26:39,920 --> 00:26:42,830 Stress energy tensor is a symmetric 4 by 4 object. 417 00:26:42,830 --> 00:26:46,550 Those 10 degrees of freedom in the stress energy tensor 418 00:26:46,550 --> 00:26:48,530 essentially determine the 10 curvature degrees 419 00:26:48,530 --> 00:26:50,002 of freedom encoded in Ricci. 420 00:26:50,002 --> 00:26:51,710 But what that tells you is that if you're 421 00:26:51,710 --> 00:26:56,630 in a region of vacuum, where there is no stress energy, 422 00:26:56,630 --> 00:26:59,160 there's no Ricci, OK? 423 00:26:59,160 --> 00:27:00,750 But there is curvature. 424 00:27:00,750 --> 00:27:01,770 We measure tides. 425 00:27:01,770 --> 00:27:03,810 We see gravitational effects. 426 00:27:03,810 --> 00:27:07,260 Weyl ends up being the quantity that describes behavior 427 00:27:07,260 --> 00:27:11,130 of gravity in vacuum regions. 428 00:27:11,130 --> 00:27:13,860 Ricci ends up very closely related to how it describes it 429 00:27:13,860 --> 00:27:15,277 in regions with matter. 430 00:27:15,277 --> 00:27:17,235 One reason I mentioned this-- so there's only-- 431 00:27:17,235 --> 00:27:19,140 [CHUCKLES] only one LIGO student here now. 432 00:27:19,140 --> 00:27:20,193 Hi, Sylvia. 433 00:27:20,193 --> 00:27:23,010 [CHUCKLES] But in fact, when one is describing 434 00:27:23,010 --> 00:27:25,830 gravitational radiation, the behavior of the Weyl tensor 435 00:27:25,830 --> 00:27:28,320 ends up being very important for characterizing 436 00:27:28,320 --> 00:27:30,390 the degrees of freedom associated with radiation 437 00:27:30,390 --> 00:27:31,960 and general relativity. 438 00:27:31,960 --> 00:27:33,173 So that's a little bit ahead. 439 00:27:33,173 --> 00:27:34,590 There's a few other things you can 440 00:27:34,590 --> 00:27:36,548 do with it which are related to what are called 441 00:27:36,548 --> 00:27:38,140 conformal transformations. 442 00:27:38,140 --> 00:27:40,042 I have a few notes on them, but they're not 443 00:27:40,042 --> 00:27:41,250 that important for our class. 444 00:27:41,250 --> 00:27:42,875 It's discussed a little bit in Carroll, 445 00:27:42,875 --> 00:27:46,333 so I'll leave that as a reading if you are interested, 446 00:27:46,333 --> 00:27:48,000 but we don't need to go through it here. 447 00:27:50,753 --> 00:27:52,170 So there are two other things that 448 00:27:52,170 --> 00:27:54,420 are much more important for us to discuss with respect 449 00:27:54,420 --> 00:27:57,870 to Riemann first, and I'm going to focus 450 00:27:57,870 --> 00:28:01,830 the time that we have on them. 451 00:28:01,830 --> 00:28:04,880 So one of the really important aspects 452 00:28:04,880 --> 00:28:07,100 of curvature that I've emphasized a few times now 453 00:28:07,100 --> 00:28:12,320 is the idea that initially parallel geodesics 454 00:28:12,320 --> 00:28:14,690 become no longer parallel when they're moving 455 00:28:14,690 --> 00:28:16,790 on a manifold that is curved. 456 00:28:16,790 --> 00:28:18,740 We're going to use the Riemann curvature 457 00:28:18,740 --> 00:28:22,640 tensor to quantify what the breakdown of parallelism 458 00:28:22,640 --> 00:28:23,450 actually means. 459 00:28:37,130 --> 00:28:39,800 There's a couple of different discussions of this 460 00:28:39,800 --> 00:28:42,450 that you'll see in various places. 461 00:28:42,450 --> 00:28:46,460 Carroll's discussion is very brief. 462 00:28:46,460 --> 00:28:47,210 It's rigorous. 463 00:28:47,210 --> 00:28:47,930 It's very brief. 464 00:28:47,930 --> 00:28:51,823 I'm doing something that's a little bit-- 465 00:28:51,823 --> 00:28:53,240 to my mind, it's a little bit more 466 00:28:53,240 --> 00:28:54,740 physically motivated OK so I'm going 467 00:28:54,740 --> 00:28:56,782 to do it in a slightly different way from the way 468 00:28:56,782 --> 00:28:57,870 it's done in Carroll. 469 00:28:57,870 --> 00:29:02,930 It's a little close to the way it's done in Schutz's textbook. 470 00:29:02,930 --> 00:29:05,930 So what we want to do is imagine we have initially 471 00:29:05,930 --> 00:29:15,830 parallel geodesics, and what we want to do 472 00:29:15,830 --> 00:29:22,700 is characterize how they become nonparallel. 473 00:29:30,775 --> 00:29:33,960 Well, we'll put it this way, how they deviate 474 00:29:33,960 --> 00:29:37,658 as one moves along their world lines-- 475 00:29:37,658 --> 00:29:38,700 around these world lines. 476 00:29:41,380 --> 00:29:43,740 OK, so here's what I want to do. 477 00:29:43,740 --> 00:29:46,270 That's the idea of the calculation that I want to do. 478 00:29:46,270 --> 00:29:48,570 So what I want to start out with is-- 479 00:29:48,570 --> 00:29:53,580 let's consider two nearby geodesics. 480 00:30:01,140 --> 00:30:03,140 And what I'm going to mean by nearby 481 00:30:03,140 --> 00:30:06,080 is that they are close enough that they're essentially 482 00:30:06,080 --> 00:30:08,540 in the same local Lorentz frame, OK? 483 00:30:08,540 --> 00:30:10,260 So they're going to have the same metric. 484 00:30:10,260 --> 00:30:12,010 I'm going to be able to choose coordinates 485 00:30:12,010 --> 00:30:14,480 such that the Christoffel symbol is zero for both of them. 486 00:30:14,480 --> 00:30:16,420 I will not be able to get rid of the second derivative, 487 00:30:16,420 --> 00:30:17,210 though, OK? 488 00:30:17,210 --> 00:30:21,010 So that will be where a bit of a difference begins to enter. 489 00:30:21,010 --> 00:30:25,310 Two nearby geodesics-- and I'm going 490 00:30:25,310 --> 00:30:28,010 to use just lambda as the affine parameter along them. 491 00:30:37,610 --> 00:30:38,980 So here's my first one. 492 00:30:43,660 --> 00:30:45,910 I'm going to make a few definitions on the next board. 493 00:30:45,910 --> 00:30:53,920 I'm going to call this gamma sub v. And here's my next one. 494 00:30:53,920 --> 00:30:56,800 I'm going to call it gamma sub u. 495 00:31:01,740 --> 00:31:02,840 Define two points here. 496 00:31:05,500 --> 00:31:08,200 OK, I'm going to make a couple of definitions 497 00:31:08,200 --> 00:31:08,950 on the next board. 498 00:31:23,310 --> 00:31:25,560 OK, so definitions-- 499 00:31:25,560 --> 00:31:31,690 I'm going to call u the tangent vector to the curve gamma sub 500 00:31:31,690 --> 00:31:32,190 u. 501 00:31:37,760 --> 00:31:38,260 OK? 502 00:31:38,260 --> 00:31:43,720 So this is equal to dx d lambda on that curve. 503 00:31:43,720 --> 00:31:52,505 v is the tangent vector to gamma sub 504 00:31:52,505 --> 00:32:05,660 v. The point A is at lambda 0 on curve gamma sub u. 505 00:32:08,810 --> 00:32:21,160 A prime is at lambda sub 0 on gamma v. 506 00:32:21,160 --> 00:32:24,910 So they're both parameterized by a parameter lambda, 507 00:32:24,910 --> 00:32:26,310 and I'm going to set the-- 508 00:32:26,310 --> 00:32:29,250 they're basically both synchronizing their clocks 509 00:32:29,250 --> 00:32:30,160 at the same time. 510 00:32:30,160 --> 00:32:32,320 Like, at those points, they're going 511 00:32:32,320 --> 00:32:34,990 to find their starting points as A and A prime. 512 00:32:34,990 --> 00:32:36,700 What I'm going to do now is I'm going 513 00:32:36,700 --> 00:32:39,640 to define what's called a geodesic displacement 514 00:32:39,640 --> 00:32:43,780 factor that points from lambda on the u curve 515 00:32:43,780 --> 00:32:47,405 to lambda on the v curve. 516 00:32:47,405 --> 00:32:50,930 And I'm going to use my favorite Greek letter, xi. 517 00:32:54,570 --> 00:33:00,730 So this points from lambda on gamma u 518 00:33:00,730 --> 00:33:04,050 to lambda that is basically to the same value. 519 00:33:07,350 --> 00:33:09,450 Let me make this a little bit more precise. 520 00:33:09,450 --> 00:33:19,410 Points from the event at lambda on gamma u 521 00:33:19,410 --> 00:33:23,840 to the event at lambda on gamma v-- 522 00:33:23,840 --> 00:33:24,340 OK? 523 00:33:24,340 --> 00:33:27,540 Apologies for being somewhat didactic there, 524 00:33:27,540 --> 00:33:29,880 but we need to define things carefully. 525 00:33:29,880 --> 00:33:34,810 So on this-- so here's my initial-- 526 00:33:38,400 --> 00:33:43,440 so that's what xi looks like at parameter lambda 0. 527 00:33:43,440 --> 00:33:47,340 And what we want to do is examine 528 00:33:47,340 --> 00:33:50,340 how xi evolves as one moves along these geodesics. 529 00:33:50,340 --> 00:33:53,365 That's going to be our goal. 530 00:33:53,365 --> 00:33:55,740 So let's make things a little bit more quantitative here. 531 00:34:12,340 --> 00:34:20,650 Xi is going to be equal to x gamma v at lambda 532 00:34:20,650 --> 00:34:26,290 minus x on gamma u at lambda. 533 00:34:26,290 --> 00:34:29,817 Finally, I'm going to assume that the curves begin parallel 534 00:34:29,817 --> 00:34:30,400 to each other. 535 00:34:38,790 --> 00:34:54,409 That's a statement that u of lambda 0 equals v of lambda 0. 536 00:34:54,409 --> 00:34:57,995 And it also tells me that I can use-- 537 00:35:07,280 --> 00:35:11,190 that this must equal 0 at the initial point. 538 00:35:11,190 --> 00:35:12,960 Not going to equal 0 everywhere. 539 00:35:12,960 --> 00:35:13,710 In fact, it won't. 540 00:35:17,230 --> 00:35:19,230 But I want to use this as a boundary condition 541 00:35:19,230 --> 00:35:20,688 in the calculation I'm about to do. 542 00:35:31,840 --> 00:35:32,340 OK. 543 00:35:40,470 --> 00:35:44,000 So what we're going to do is essentially say, you know, 544 00:35:44,000 --> 00:35:46,670 as we move along these two curves, these are geodesics. 545 00:35:46,670 --> 00:35:48,800 We're going to use the geodesic equation to slide 546 00:35:48,800 --> 00:35:50,030 along these two curves. 547 00:35:50,030 --> 00:35:52,520 We know what the equation is that governs 548 00:35:52,520 --> 00:35:54,230 u as it moves along gamma u. 549 00:35:54,230 --> 00:35:56,030 We know what the equation is that governs 550 00:35:56,030 --> 00:35:58,110 v it moves along gamma v. We're just 551 00:35:58,110 --> 00:35:59,860 going to take the difference between them, 552 00:35:59,860 --> 00:36:03,800 and we're going to kind of use that to develop an acceleration 553 00:36:03,800 --> 00:36:08,757 equation that governs that displacement xi. 554 00:36:08,757 --> 00:36:11,090 Here's the bit where I differ a little bit from Carroll. 555 00:36:11,090 --> 00:36:13,940 So Carroll, like I said, Carroll has a very brief discussion, 556 00:36:13,940 --> 00:36:15,300 which is-- 557 00:36:15,300 --> 00:36:16,873 it's absolutely right. 558 00:36:16,873 --> 00:36:19,040 But I want to give a little bit of physical insight, 559 00:36:19,040 --> 00:36:23,435 and I think you can do that by choosing to work 560 00:36:23,435 --> 00:36:24,560 in the local Lorentz frame. 561 00:36:48,510 --> 00:36:51,170 So we're going to work in the local Lorentz frame, 562 00:36:51,170 --> 00:36:52,920 and I'm going to center this local Lorentz 563 00:36:52,920 --> 00:37:05,170 frame on the event A. The reason why I'm doing that is this then 564 00:37:05,170 --> 00:37:12,990 allows me to say, g mu nu at the event is A mu nu. 565 00:37:23,140 --> 00:37:24,290 Pardon me a second. 566 00:37:24,290 --> 00:37:31,130 My Christoffel symbols at event A are all going to be zero. 567 00:37:36,480 --> 00:37:43,702 g mu nu at A prime is also going to be A mu nu. 568 00:37:43,702 --> 00:37:45,160 We have to be a little bit careful. 569 00:37:45,160 --> 00:37:49,950 Our Christoffel symbols do not vanish at point A prime 570 00:37:49,950 --> 00:37:51,700 because there's a little bit of curvature. 571 00:37:51,700 --> 00:38:06,010 They are in fact related to the fact that I'm sort of slightly 572 00:38:06,010 --> 00:38:08,708 far away, and I'm picking up that second order correction, 573 00:38:08,708 --> 00:38:11,150 OK? 574 00:38:11,150 --> 00:38:16,082 This is sort of where all the important bits of the analysis 575 00:38:16,082 --> 00:38:17,040 are going to come from. 576 00:38:17,040 --> 00:38:19,590 This is the fact that they're close to each other. 577 00:38:19,590 --> 00:38:21,260 I can set up a local Lorentz frame, 578 00:38:21,260 --> 00:38:23,660 but it's got that little bit of sort of schmutz 579 00:38:23,660 --> 00:38:25,280 at second order that's coming in, 580 00:38:25,280 --> 00:38:27,610 and kind of pushing me away from a simple local Lorentz 581 00:38:27,610 --> 00:38:28,800 frame there-- 582 00:38:28,800 --> 00:38:31,380 the simple mathematical form right there. 583 00:38:31,380 --> 00:38:34,430 OK, let's put that up. 584 00:38:37,010 --> 00:38:37,510 OK. 585 00:38:51,600 --> 00:38:55,320 So let's look at the equations that govern 586 00:38:55,320 --> 00:38:56,820 motion along these two curves. 587 00:39:02,730 --> 00:39:09,160 So the geodesic equation along curve gamma u-- 588 00:39:09,160 --> 00:39:17,360 and we'll just look at it at A. It's a second order-- 589 00:39:17,360 --> 00:39:20,630 I'm going to write it in terms of the coordinates. 590 00:39:20,630 --> 00:39:22,790 It looks like this. 591 00:39:22,790 --> 00:39:27,620 We're evaluating this at A. At A, my Christoffel symbols are 592 00:39:27,620 --> 00:39:30,740 all zero, so this is just zero. 593 00:39:38,830 --> 00:39:46,760 Let's look at it along the other curve, at A prime. 594 00:40:16,888 --> 00:40:17,388 OK? 595 00:40:28,280 --> 00:40:32,765 So in just a second, I am going to-- 596 00:40:37,210 --> 00:40:40,270 wait a second. 597 00:40:40,270 --> 00:40:45,340 I'm going to substitute in that derivative of Christoffel 598 00:40:45,340 --> 00:40:46,630 that's going to go in there. 599 00:40:46,630 --> 00:40:48,400 Before I do that-- so notice, this also 600 00:40:48,400 --> 00:40:50,800 depends on the velocity as I move along at that point, 601 00:40:50,800 --> 00:40:51,300 right? 602 00:40:51,300 --> 00:40:52,697 Typical geodesic-- 603 00:40:52,697 --> 00:40:54,280 So I'm going to substitute in the fact 604 00:40:54,280 --> 00:40:59,740 that both the x mu and the x nu velocity here, 605 00:40:59,740 --> 00:41:04,516 this is defined as v mu. 606 00:41:04,516 --> 00:41:06,910 But remember, these guys are defined 607 00:41:06,910 --> 00:41:08,350 as being initially parallel. 608 00:41:14,655 --> 00:41:16,120 OK? 609 00:41:16,120 --> 00:41:18,805 So what this tells me is-- 610 00:41:25,238 --> 00:41:26,420 OK, that's an alpha. 611 00:41:43,032 --> 00:41:43,990 So this is interesting. 612 00:41:43,990 --> 00:41:46,780 What I'm seeing is I end up with an equation 613 00:41:46,780 --> 00:41:50,200 where the derivative of the Christoffel symbol 614 00:41:50,200 --> 00:41:56,170 is coupling to sort of my motion along that world 615 00:41:56,170 --> 00:41:58,810 line and the displacement. 616 00:41:58,810 --> 00:42:01,480 Now, let me just remind you, what I really wanted to do 617 00:42:01,480 --> 00:42:05,350 was get an equation that governs how this guy changes, OK? 618 00:42:05,350 --> 00:42:08,560 But this guy is just defined as the difference 619 00:42:08,560 --> 00:42:13,600 between the position along curve v minus the position 620 00:42:13,600 --> 00:42:16,758 along curve u. 621 00:42:16,758 --> 00:42:18,800 So if I take the difference of these two things-- 622 00:42:27,980 --> 00:42:32,230 pardon me, forgot to label which curve this one is on. 623 00:42:32,230 --> 00:42:40,570 So I'm doing this at A prime, doing this one at A. 624 00:42:40,570 --> 00:42:48,560 This is an equation governing the acceleration 625 00:42:48,560 --> 00:42:49,960 of this displacement. 626 00:43:00,460 --> 00:43:01,960 OK? 627 00:43:01,960 --> 00:43:03,390 So this is interesting, OK? 628 00:43:03,390 --> 00:43:06,260 So what you're sort of seeing is that the geodesic displacement 629 00:43:06,260 --> 00:43:10,310 looks something like two derivatives 630 00:43:10,310 --> 00:43:13,820 of the metric coupling in the forward velocity 631 00:43:13,820 --> 00:43:16,370 and the displacement itself. 632 00:43:16,370 --> 00:43:22,630 Now, as written, this equation is fine if all you are doing 633 00:43:22,630 --> 00:43:24,830 is living life in the local Lorentz frame, OK? 634 00:43:24,830 --> 00:43:26,830 We want to do a little better than that, though. 635 00:43:26,830 --> 00:43:30,520 So I'm going to do a little bit more massaging of this, 636 00:43:30,520 --> 00:43:33,610 but I kind of want to emphasize that this already brings out 637 00:43:33,610 --> 00:43:35,228 the key physical point, OK? 638 00:43:35,228 --> 00:43:37,520 I can't yet-- you know, you're sort of looking at this, 639 00:43:37,520 --> 00:43:39,062 and you're thinking to yourself, that 640 00:43:39,062 --> 00:43:41,620 looks like a piece of Riemann, but it ain't Riemann, right? 641 00:43:41,620 --> 00:43:43,370 It's a derivative of a Christoffel symbol, 642 00:43:43,370 --> 00:43:44,370 and that's not a tensor. 643 00:43:44,370 --> 00:43:46,453 So this isn't quite the kind of thing we want yet. 644 00:43:46,453 --> 00:43:48,050 We need to do a little bit more work. 645 00:43:52,430 --> 00:43:55,035 And so what I would sort of say is, 646 00:43:55,035 --> 00:43:56,410 everything I did up to here, this 647 00:43:56,410 --> 00:43:58,032 is like the key important physics. 648 00:43:58,032 --> 00:43:59,740 Now I'm going to do a little bit of stuff 649 00:43:59,740 --> 00:44:03,140 to sort of put the suit and tie that a tensor is supposed 650 00:44:03,140 --> 00:44:03,640 to wear. 651 00:44:03,640 --> 00:44:05,348 I'm going to dress it up a little bit, so 652 00:44:05,348 --> 00:44:07,462 that it's wearing the clothes that all quantities 653 00:44:07,462 --> 00:44:08,920 in this class are supposed to wear. 654 00:44:19,070 --> 00:44:21,580 So we want to make this tensorial. 655 00:44:30,860 --> 00:44:34,175 And so for guidance of that, these time derivatives 656 00:44:34,175 --> 00:44:36,050 or derivatives with respect to the parameter, 657 00:44:36,050 --> 00:44:38,510 it's not really nicely formulated, OK? 658 00:44:38,510 --> 00:44:41,300 The thing which we should note is, 659 00:44:41,300 --> 00:44:45,710 d by d lambda, that's what I get when 660 00:44:45,710 --> 00:44:48,890 I contract the forward velocity with a partial derivative. 661 00:44:48,890 --> 00:44:52,400 What we should do is replace this with something 662 00:44:52,400 --> 00:45:03,580 like what I'm going to call capital D by d lambda, which 663 00:45:03,580 --> 00:45:06,920 is what I'm going to get when I do derivatives using-- 664 00:45:06,920 --> 00:45:08,630 when I use forward velocity contracted 665 00:45:08,630 --> 00:45:12,270 on a covariant derivative, OK? 666 00:45:12,270 --> 00:45:12,770 So-- 667 00:45:21,250 --> 00:45:22,590 Suppose I just-- 668 00:45:22,590 --> 00:45:24,510 I'm agnostic about what xi actually is. 669 00:45:24,510 --> 00:45:26,400 I just know it's a vector field, and I 670 00:45:26,400 --> 00:45:27,660 want to compute this thing. 671 00:45:27,660 --> 00:45:30,230 Well, we learned how to do that a couple lectures ago. 672 00:45:34,960 --> 00:45:36,320 I'm going to work this guy out. 673 00:45:44,280 --> 00:45:55,070 And-- [EXCLAIMS] I couple in a term that looks like this. 674 00:45:55,070 --> 00:45:56,570 You might be tempted at this point 675 00:45:56,570 --> 00:45:58,400 to go, oh, I'm in a local Lorentz frame. 676 00:45:58,400 --> 00:46:00,230 I can get rid of that Christoffel. 677 00:46:00,230 --> 00:46:02,180 Don't do that quite yet, because we're 678 00:46:02,180 --> 00:46:04,190 going to want to take one more derivative. 679 00:46:04,190 --> 00:46:06,725 Christoffel does vanish, but its derivative does not. 680 00:46:06,725 --> 00:46:08,600 Wait till you've done all of your derivatives 681 00:46:08,600 --> 00:46:11,587 before you insert that relationship. 682 00:46:30,100 --> 00:46:31,600 Just getting another piece of chalk. 683 00:46:40,150 --> 00:46:43,420 So my covariant derivative along the trajectory 684 00:46:43,420 --> 00:46:51,375 with my parameter is the usual total derivative plus-- 685 00:46:57,831 --> 00:46:59,630 that, OK? 686 00:46:59,630 --> 00:47:02,079 Now, I'm going to want to take one more derivative. 687 00:47:29,524 --> 00:47:31,060 OK? 688 00:47:31,060 --> 00:47:34,990 So there's a lot of junk in here. 689 00:47:34,990 --> 00:47:36,100 I'm going to get one term. 690 00:47:40,830 --> 00:47:43,110 It just looks like this. 691 00:47:43,110 --> 00:47:45,600 But now I'm going to get a whole bunch of other terms 692 00:47:45,600 --> 00:47:46,738 that involve-- 693 00:47:46,738 --> 00:47:48,780 so this is what I get when I expand this guy out, 694 00:47:48,780 --> 00:47:49,590 and I just have-- 695 00:47:51,957 --> 00:47:53,290 well, I should hang on a second. 696 00:47:53,290 --> 00:47:54,748 So first, I'm going to get one that 697 00:47:54,748 --> 00:47:59,340 involves essentially the first term, u on the partial, 698 00:47:59,340 --> 00:48:01,000 hitting both of these terms. 699 00:48:01,000 --> 00:48:02,700 OK, so when I do that-- 700 00:48:06,210 --> 00:48:07,610 pardon me just one moment. 701 00:48:13,430 --> 00:48:14,290 OK, never mind. 702 00:48:19,232 --> 00:48:20,690 Let me just write it down, and I'll 703 00:48:20,690 --> 00:48:21,898 describe where it comes from. 704 00:48:26,450 --> 00:48:32,360 OK, so this first term, this is basically the connection term 705 00:48:32,360 --> 00:48:33,950 coupling to this, OK? 706 00:48:33,950 --> 00:48:36,077 Associated with this covariant derivative. 707 00:48:36,077 --> 00:48:37,910 Now I'm going to have a whole bunch of terms 708 00:48:37,910 --> 00:48:43,570 that involve this guy operating on these terms over here. 709 00:48:43,570 --> 00:48:46,652 And my apologies that the board is not clearing as well as I 710 00:48:46,652 --> 00:48:47,360 would like today. 711 00:48:55,980 --> 00:48:56,740 And you know what? 712 00:48:56,740 --> 00:48:59,670 I'm going to just write it out like this, 713 00:48:59,670 --> 00:49:01,590 and move to a different board in a second. 714 00:49:09,430 --> 00:49:09,930 OK? 715 00:49:16,023 --> 00:49:18,440 All right, so I'm going to put this over on the other side 716 00:49:18,440 --> 00:49:20,390 here. 717 00:49:20,390 --> 00:49:22,610 What I have there-- so the first term, like I said, 718 00:49:22,610 --> 00:49:25,395 I'm basically just operating that thing 719 00:49:25,395 --> 00:49:26,520 on the two different terms. 720 00:49:26,520 --> 00:49:29,450 So it's a little easy when I first hit it on the d xi d 721 00:49:29,450 --> 00:49:31,617 lambda, and it's going to be messy when I expand out 722 00:49:31,617 --> 00:49:33,575 all the derivatives that operate on here, which 723 00:49:33,575 --> 00:49:35,480 is why I'm writing it with some care, 724 00:49:35,480 --> 00:49:38,390 because we're about to make a lot of mess on the board. 725 00:49:49,288 --> 00:49:49,788 OK. 726 00:50:19,250 --> 00:50:25,842 OK, so first, I'm going to do what happens when this guy hits 727 00:50:25,842 --> 00:50:26,800 the Christoffel symbol. 728 00:50:46,290 --> 00:50:48,030 Then I'm going to get a term that 729 00:50:48,030 --> 00:50:56,680 involves this combination of guys 730 00:50:56,680 --> 00:50:58,668 acting on the four velocity. 731 00:51:13,330 --> 00:51:18,440 And then I get these guys acting on my displacement. 732 00:51:18,440 --> 00:51:21,220 OK, [BLOWS AIR] now I'm ready to simplify. 733 00:51:21,220 --> 00:51:24,520 OK, so let me just emphasize, everything I have here, 734 00:51:24,520 --> 00:51:27,850 all I did was expand out those derivatives 735 00:51:27,850 --> 00:51:30,430 and write them in a form where I want to basically call out 736 00:51:30,430 --> 00:51:33,550 different terms and see how they behave. 737 00:51:33,550 --> 00:51:35,590 Two things to bear in mind, I am going 738 00:51:35,590 --> 00:51:38,590 to do this in the local Lorenz frame 739 00:51:38,590 --> 00:51:52,080 and in the vicinity of the point A. 740 00:51:52,080 --> 00:51:55,970 That's going to allow me to get rid of a lot of terms. 741 00:51:58,968 --> 00:52:00,760 I'll use the-- here's an open box of chalk. 742 00:52:06,550 --> 00:52:10,780 OK, so earlier, I didn't want to get rid of my Christoffel 743 00:52:10,780 --> 00:52:13,360 symbols because I was going to take another derivative. 744 00:52:13,360 --> 00:52:14,780 I'm done with that. 745 00:52:14,780 --> 00:52:17,890 So anything that's just a Christoffel on its own, 746 00:52:17,890 --> 00:52:19,510 I go into local Lorentz form-- 747 00:52:19,510 --> 00:52:22,060 Christoffel, you die! 748 00:52:22,060 --> 00:52:24,348 Die! 749 00:52:24,348 --> 00:52:25,640 Can't really do much with this. 750 00:52:25,640 --> 00:52:27,150 Let's set this aside. 751 00:52:27,150 --> 00:52:29,730 Let's look at this guy. 752 00:52:29,730 --> 00:52:34,400 What do we know about the forward vector u? 753 00:52:34,400 --> 00:52:35,550 It's a geodesic. 754 00:52:35,550 --> 00:52:37,200 It obeys the geodesic equation. 755 00:52:37,200 --> 00:52:40,950 The geodesic equation is, this thing in parentheses equals 0. 756 00:52:40,950 --> 00:52:42,570 You die! 757 00:52:42,570 --> 00:52:46,260 Finally, I have this condition here at the beginning. 758 00:52:46,260 --> 00:52:54,050 That is essentially the velocity at point A 759 00:52:54,050 --> 00:52:58,070 minus the velocity at point B. Our initial condition 760 00:52:58,070 --> 00:53:02,420 is that these things start out parallel to each other. 761 00:53:02,420 --> 00:53:05,495 These die because these are initially parallel. 762 00:53:09,040 --> 00:53:13,900 So initially parallel geodesic-- 763 00:53:13,900 --> 00:53:15,910 local Lorentz frame-- 764 00:53:15,910 --> 00:53:18,160 So the only derivative I'm going to need to expand out 765 00:53:18,160 --> 00:53:21,820 is this guy, and because I'm working in a local Lorentz 766 00:53:21,820 --> 00:53:24,490 frame, that's actually not that bad. 767 00:53:37,960 --> 00:53:42,030 So what I finally get after all the smoke clears 768 00:53:42,030 --> 00:53:45,590 is that my covariant acceleration 769 00:53:45,590 --> 00:53:48,450 of this displacement vector is related 770 00:53:48,450 --> 00:53:55,040 to my noncovariant acceleration, with a term 771 00:53:55,040 --> 00:53:57,980 that basically ends up just being-- 772 00:53:57,980 --> 00:54:05,540 oops-- a derivative of the Christoffel symbol. 773 00:54:05,540 --> 00:54:08,940 It couples in two powers of the four 774 00:54:08,940 --> 00:54:15,210 velocity and the displacement. 775 00:54:15,210 --> 00:54:16,980 But we actually already have-- 776 00:54:16,980 --> 00:54:22,830 we worked out earlier what this guy was, OK? 777 00:54:22,830 --> 00:54:27,960 So my noncovariant acceleration was a different derivative 778 00:54:27,960 --> 00:54:29,920 of the Christoffel symbol. 779 00:54:29,920 --> 00:54:31,740 So if I go and I plug this in-- 780 00:54:55,560 --> 00:54:56,880 My apologies for pausing here. 781 00:54:56,880 --> 00:54:58,960 There are a lot of indices on this page. 782 00:55:03,880 --> 00:55:06,670 Look at this, I've got a derivative of Christoffel 783 00:55:06,670 --> 00:55:09,550 minus a derivative of a Christoffel. 784 00:55:09,550 --> 00:55:11,800 That is exactly what the Riemann curvature 785 00:55:11,800 --> 00:55:14,957 tensor looks like in the local Lorentz frame. 786 00:55:14,957 --> 00:55:17,540 Now, if you actually go back and you look up your definitions, 787 00:55:17,540 --> 00:55:19,390 you'll see it's not quite right. 788 00:55:19,390 --> 00:55:23,350 There's a couple of indices that are sort of a little bit off, 789 00:55:23,350 --> 00:55:25,840 but they turn out to all be dummy indices. 790 00:55:25,840 --> 00:55:27,730 So what you should do at this point 791 00:55:27,730 --> 00:55:31,900 is just relabel a couple of your dummy indices. 792 00:55:40,000 --> 00:55:57,300 So if on the second term, you take beta to mu, mu to gamma, 793 00:55:57,300 --> 00:56:23,190 and nu to beta, what you finally wind up with is-- 794 00:56:32,470 --> 00:56:34,470 We'll write it first in the local Lorentz frame. 795 00:56:50,740 --> 00:56:54,080 This is nothing but the Riemann tensor in the local Lorentz 796 00:56:54,080 --> 00:56:54,580 frame. 797 00:56:54,580 --> 00:56:57,130 And from the principle that a tensor equation that 798 00:56:57,130 --> 00:57:00,670 holds in one frame must hold in all, 799 00:57:00,670 --> 00:57:15,560 we can deduce from this that this 800 00:57:15,560 --> 00:57:18,320 is the way the displacement behaves. 801 00:57:18,320 --> 00:57:21,190 So I start out with two geodesics 802 00:57:21,190 --> 00:57:23,830 that are perfectly parallel to one another. 803 00:57:23,830 --> 00:57:25,940 If they are in a spacetime that is curved, 804 00:57:25,940 --> 00:57:28,970 the Riemann curvature tensor tells how those initially 805 00:57:28,970 --> 00:57:34,370 parallel trajectories evolve as I move along those geodesics. 806 00:57:34,370 --> 00:57:36,450 So I want to make two remarks. 807 00:57:36,450 --> 00:57:41,240 One is just sort of a way of thinking about the calculation 808 00:57:41,240 --> 00:57:42,080 I just did. 809 00:57:42,080 --> 00:57:43,580 It may not-- you might be slightly 810 00:57:43,580 --> 00:57:44,700 dissatisfied with the fact that I 811 00:57:44,700 --> 00:57:46,220 decided to do the whole calculation 812 00:57:46,220 --> 00:57:48,110 in a whole special frame. 813 00:57:48,110 --> 00:57:52,940 If that's-- you would prefer to see something a little bit more 814 00:57:52,940 --> 00:57:56,480 rigorous, feel free to explore some of the other textbooks 815 00:57:56,480 --> 00:57:58,460 that we have for this class. 816 00:57:58,460 --> 00:58:01,215 They do treat this a little bit more rigorously than this. 817 00:58:01,215 --> 00:58:02,840 One reason why I wanted to do this is I 818 00:58:02,840 --> 00:58:04,880 wanted to really emphasize this idea 819 00:58:04,880 --> 00:58:06,680 that where this effect comes from is 820 00:58:06,680 --> 00:58:10,040 that if I think about the freely falling frame describing two 821 00:58:10,040 --> 00:58:12,645 nearby geodesics, it is that derivative, 822 00:58:12,645 --> 00:58:15,020 it's that secondary derivative of the metric, that really 823 00:58:15,020 --> 00:58:18,320 plays a fundamental role in driving these two things apart. 824 00:58:18,320 --> 00:58:22,010 What we get out of it is a completely frame-independent 825 00:58:22,010 --> 00:58:24,890 equation, and this ends up really 826 00:58:24,890 --> 00:58:27,620 being the key equation describing the behavior 827 00:58:27,620 --> 00:58:30,590 of tides in general relativity. 828 00:58:30,590 --> 00:58:33,943 We're going to-- when you go into a freely falling frame, 829 00:58:33,943 --> 00:58:35,360 we've basically at that point said 830 00:58:35,360 --> 00:58:38,960 there's no longer such a thing as gravitational acceleration. 831 00:58:38,960 --> 00:58:40,960 And if there's no gravitational acceleration 832 00:58:40,960 --> 00:58:43,310 that's coming, well, what the hell does gravity do? 833 00:58:43,310 --> 00:58:46,580 General relativity, tides become the key thing 834 00:58:46,580 --> 00:58:50,330 that really defines what the action of gravity is. 835 00:58:50,330 --> 00:58:52,880 One of the ones I am personally very interested in applying 836 00:58:52,880 --> 00:58:54,710 this to is that this equation gives you 837 00:58:54,710 --> 00:58:56,870 a very rigorous and frame-independent way 838 00:58:56,870 --> 00:59:00,530 to describe how a gravitational wave detector responds 839 00:59:00,530 --> 00:59:02,747 to the impact of a gravitational wave. 840 00:59:02,747 --> 00:59:04,580 If you imagine you have two test masses that 841 00:59:04,580 --> 00:59:09,590 are moving through spacetime, they just follow geodesics, OK? 842 00:59:09,590 --> 00:59:10,730 They feel gravity. 843 00:59:10,730 --> 00:59:12,170 They are free falling. 844 00:59:12,170 --> 00:59:13,415 They don't do anything. 845 00:59:13,415 --> 00:59:15,457 If you look at them, if you're falling with them, 846 00:59:15,457 --> 00:59:18,770 you don't see any effect whatsoever. 847 00:59:18,770 --> 00:59:21,590 But if they're sufficiently far away from one another 848 00:59:21,590 --> 00:59:24,860 that their displacement kind of takes the curvature 849 00:59:24,860 --> 00:59:28,550 to that freely falling frame, then this equation 850 00:59:28,550 --> 00:59:30,890 governs how the separation between the two of them 851 00:59:30,890 --> 00:59:32,687 actually evolves with time. 852 00:59:32,687 --> 00:59:34,520 And so for instance, if you're a student who 853 00:59:34,520 --> 00:59:36,853 works in LIGO, where some of the many important analyses 854 00:59:36,853 --> 00:59:40,220 describing how gravitational wave detectors respond 855 00:59:40,220 --> 00:59:41,820 to an [INAUDIBLE] radiation field, 856 00:59:41,820 --> 00:59:44,120 they are based on this equation. 857 00:59:44,120 --> 00:59:45,950 If you're interested in understanding 858 00:59:45,950 --> 00:59:49,340 how an extended object is tidally 859 00:59:49,340 --> 00:59:52,040 distorted due to the fact that it's sort of big enough 860 00:59:52,040 --> 00:59:54,410 that it doesn't all sort of live at a single point, 861 00:59:54,410 --> 00:59:57,830 but its shape spills across the local Lorentz frame, 862 00:59:57,830 --> 00:59:59,390 this equation governs-- it allows 863 00:59:59,390 --> 01:00:01,015 you to set up some of the stresses that 864 01:00:01,015 --> 01:00:04,700 act upon that body and change it from a simple point mass. 865 01:00:04,700 --> 01:00:07,250 So I'm waxing slightly rhapsodic here, 866 01:00:07,250 --> 01:00:10,250 because this is one of the more important physical equations 867 01:00:10,250 --> 01:00:13,390 that come out of the class at this point. 868 01:00:13,390 --> 01:00:21,940 All right, in our last 12 or 13 minutes, 869 01:00:21,940 --> 01:00:26,020 I would like to do one other little identity which 870 01:00:26,020 --> 01:00:40,510 is extremely important, but rather messy. 871 01:00:40,510 --> 01:00:43,427 In fact, I think I'm going to just-- if I have the time-- 872 01:00:43,427 --> 01:00:44,260 yeah, you know what? 873 01:00:44,260 --> 01:00:45,302 I think I will have time. 874 01:00:45,302 --> 01:00:51,050 I'm going to set up the first page of the next lecture. 875 01:00:51,050 --> 01:00:54,310 So let's move away from-- oh, by the way, 876 01:00:54,310 --> 01:00:56,990 I forgot to give the name of this. 877 01:00:56,990 --> 01:01:01,200 This is the equation of geodesic deviation. 878 01:01:20,270 --> 01:01:25,810 OK, so recall when we first talked about the Riemann 879 01:01:25,810 --> 01:01:26,310 tensor. 880 01:01:35,940 --> 01:01:37,612 There's a very mathematical action 881 01:01:37,612 --> 01:01:38,820 that the Riemann tensor does. 882 01:01:38,820 --> 01:01:40,278 You can think of it as what happens 883 01:01:40,278 --> 01:01:43,260 when you have a commutator of covariant derivatives acting 884 01:01:43,260 --> 01:01:43,830 on a vector. 885 01:01:58,630 --> 01:02:02,790 So if I evaluate the commentator, sigma lambda-- 886 01:02:02,790 --> 01:02:09,270 excuse me-- covariant of lambda covariant with sigma 887 01:02:09,270 --> 01:02:18,460 acting on v alpha, OK, it ends up looking like this. 888 01:02:18,460 --> 01:02:21,900 And I argue that this definition is kind of the differential 889 01:02:21,900 --> 01:02:24,870 version of that little holonomy operation by which I derived 890 01:02:24,870 --> 01:02:26,550 Riemann in the first place. 891 01:02:26,550 --> 01:02:29,370 A more generalized form of this is, 892 01:02:29,370 --> 01:02:38,420 suppose I act on some object with two indices, one 893 01:02:38,420 --> 01:02:40,100 in the upstairs, one in the downstairs. 894 01:02:50,446 --> 01:02:50,946 Oops. 895 01:03:02,310 --> 01:03:03,750 OK? 896 01:03:03,750 --> 01:03:06,720 So these are just some reminders of a couple definitions. 897 01:03:10,560 --> 01:03:11,700 I want to use this. 898 01:03:11,700 --> 01:03:15,465 I'm going to apply these things to two relationships 899 01:03:15,465 --> 01:03:17,990 that I'm going to write down, and I'm 900 01:03:17,990 --> 01:03:20,840 going to do something kind of crazy, 901 01:03:20,840 --> 01:03:24,230 then I'm going to do something a little bit crazier, 902 01:03:24,230 --> 01:03:27,950 and something kind of cool is going to emerge from this. 903 01:03:27,950 --> 01:03:30,650 And this is something that deserves a clean board. 904 01:03:50,050 --> 01:03:55,570 So I'm going to write down relationship A. That's 905 01:03:55,570 --> 01:03:58,160 what I get when I do the commutator 906 01:03:58,160 --> 01:04:06,000 of the alpha beta derivatives on the gamma derivative 907 01:04:06,000 --> 01:04:08,850 of some one form, OK? 908 01:04:08,850 --> 01:04:12,480 Now, if I apply those little definitions I worked out, 909 01:04:12,480 --> 01:04:17,940 this is Riemann mu gamma alpha beta-- 910 01:04:32,975 --> 01:04:33,970 OK? 911 01:04:33,970 --> 01:04:36,760 So this is relationship A. 912 01:04:36,760 --> 01:04:42,750 Relationship B, I'm going to take 913 01:04:42,750 --> 01:04:54,540 the covariant derivative along alpha, of the beta gamma 914 01:04:54,540 --> 01:04:59,550 commutator on p delta, OK? 915 01:04:59,550 --> 01:05:00,800 So notice the difference here. 916 01:05:00,800 --> 01:05:02,630 One, I'm doing the commutator acting 917 01:05:02,630 --> 01:05:04,613 on this particular derivative. 918 01:05:04,613 --> 01:05:06,030 Next one, I'm doing the derivative 919 01:05:06,030 --> 01:05:07,310 of commutator acting on this. 920 01:05:10,720 --> 01:05:14,480 And so I'm going to skip a line or two that are in my notes, 921 01:05:14,480 --> 01:05:19,460 but this ends up turning into minus p mu-- 922 01:05:25,870 --> 01:05:36,145 beta alpha delta minus R mu and delta gamma. 923 01:05:39,260 --> 01:05:40,150 OK? 924 01:05:40,150 --> 01:05:42,880 So the way I went to get this line, so the two lines 925 01:05:42,880 --> 01:05:47,190 that I skipped over, one is I just straightforwardly applied 926 01:05:47,190 --> 01:05:49,690 the commutator derivative, and then took a derivative. 927 01:05:49,690 --> 01:05:51,640 And then I took advantage of the fact 928 01:05:51,640 --> 01:06:00,370 that I can commute the metric with covariant derivatives, 929 01:06:00,370 --> 01:06:03,110 and sort of raising the next, move things-- 930 01:06:03,110 --> 01:06:05,573 move an index up on one side and down on the other, 931 01:06:05,573 --> 01:06:07,740 and take advantage of some symmetries of the Riemann 932 01:06:07,740 --> 01:06:11,420 tensor to slough my indices around. 933 01:06:11,420 --> 01:06:13,575 OK, so this board's kind of a disaster, 934 01:06:13,575 --> 01:06:15,120 so I'm going to go to a cleaner board 935 01:06:15,120 --> 01:06:17,360 here for what I want to do next. 936 01:06:17,360 --> 01:06:19,110 I'm just going to clean this so you're not 937 01:06:19,110 --> 01:06:20,193 distracted by its content. 938 01:06:23,720 --> 01:06:28,060 So two equations, what the hell do they 939 01:06:28,060 --> 01:06:30,440 have to do with each other? 940 01:06:30,440 --> 01:06:43,130 Here is where I'm going to do something which you might-- 941 01:06:43,130 --> 01:06:44,790 let's do the other board. 942 01:06:44,790 --> 01:06:47,940 I'm going through something that you might legitimately 943 01:06:47,940 --> 01:06:49,320 think is crazy. 944 01:06:49,320 --> 01:06:52,680 What I want to do is take equation A-- 945 01:06:56,450 --> 01:06:58,800 actually, I'm going to look at both of these equations. 946 01:06:58,800 --> 01:07:05,350 And I am going to antisymmetrize on the indices alpha, beta, 947 01:07:05,350 --> 01:07:05,850 and gamma. 948 01:07:24,622 --> 01:07:25,970 OK? 949 01:07:25,970 --> 01:07:27,220 So just bear with me a second. 950 01:07:30,050 --> 01:07:32,447 So the way I'm going to do this-- 951 01:07:32,447 --> 01:07:33,530 here's what I'm doing now. 952 01:07:36,140 --> 01:07:37,593 Here's my commutator. 953 01:07:49,930 --> 01:07:51,730 So if I look at equation A-- 954 01:07:56,340 --> 01:07:56,840 OK? 955 01:07:56,840 --> 01:07:58,520 So I need to expand this guy out. 956 01:08:02,260 --> 01:08:05,681 Let me just write out a step of this analysis. 957 01:08:10,980 --> 01:08:20,581 The way you do this is you add up every even permutation 958 01:08:20,581 --> 01:08:21,289 of those indices. 959 01:08:39,370 --> 01:08:40,360 That's a beta. 960 01:08:46,340 --> 01:08:51,016 And then you subtract off every odd permutation 961 01:08:51,016 --> 01:08:51,724 of these indices. 962 01:09:21,300 --> 01:09:23,790 Oh, and don't forget-- 963 01:09:23,790 --> 01:09:26,180 exactly on that one form, OK? 964 01:09:29,680 --> 01:09:32,210 So now we can-- 965 01:09:32,210 --> 01:09:39,170 if you expand these guys out, and gather terms exactly 966 01:09:39,170 --> 01:09:45,130 correctly, it's not hard to show that this turns out 967 01:09:45,130 --> 01:09:58,620 to be what you get if you just slide those commutators over 968 01:09:58,620 --> 01:09:59,190 by one. 969 01:10:41,435 --> 01:10:43,423 OK. 970 01:10:43,423 --> 01:10:47,290 You know, if you're really feeling motivated, just try it, 971 01:10:47,290 --> 01:10:50,020 OK? 972 01:10:50,020 --> 01:10:55,360 What this is is, the right hand side-- 973 01:10:55,360 --> 01:10:58,870 sorry, this is the left hand side-- 974 01:10:58,870 --> 01:11:02,140 we'll put this way. 975 01:11:02,140 --> 01:11:09,870 So when I apply this to the left hand side of equation A, 976 01:11:09,870 --> 01:11:13,440 what emerges is the antisymmetrized left hand 977 01:11:13,440 --> 01:11:38,160 side of equation B. 978 01:11:38,160 --> 01:11:43,590 So when I antisymmetrize on the indices alpha, beta, gamma, 979 01:11:43,590 --> 01:11:45,210 these two relationships become-- 980 01:11:45,210 --> 01:11:46,440 they say the same thing. 981 01:12:03,430 --> 01:12:08,170 So that means the right hand side must be the same as well. 982 01:12:08,170 --> 01:12:10,270 So let's apply this and see what happens 983 01:12:10,270 --> 01:12:14,440 if I require the antisymmetrized right hand side of A 984 01:12:14,440 --> 01:12:30,600 to equal the antisymmetrized right hand side of B. 985 01:12:30,600 --> 01:12:34,650 OK, so first, if I remind myself how I order these equations-- 986 01:12:34,650 --> 01:12:35,150 OK, yeah. 987 01:12:35,150 --> 01:13:32,580 So let's do-- here's A, and here is B. 988 01:13:32,580 --> 01:13:36,830 OK, so a few things to notice. 989 01:13:36,830 --> 01:13:38,900 This term, we actually showed when 990 01:13:38,900 --> 01:13:42,410 we looked at the different symmetries of the Riemann 991 01:13:42,410 --> 01:13:43,900 tensor. 992 01:13:43,900 --> 01:13:46,400 This is-- I didn't actually show this, but it's in the notes 993 01:13:46,400 --> 01:13:47,670 and I stated it. 994 01:13:47,670 --> 01:13:49,610 This is one of those symmetries. 995 01:13:49,610 --> 01:13:51,560 If I take this thing, and I just add up 996 01:13:51,560 --> 01:13:56,150 what I get when I permute the three final indices, 997 01:13:56,150 --> 01:13:57,622 I get zero. 998 01:13:57,622 --> 01:13:59,330 And that's equivalent to antisymmetrizing 999 01:13:59,330 --> 01:14:00,690 on these things. 1000 01:14:00,690 --> 01:14:02,810 So this is zero by Riemann symmetry. 1001 01:14:11,190 --> 01:14:16,890 This term here and this term here are exactly the same, OK? 1002 01:14:16,890 --> 01:14:19,055 Because the only indices that are 1003 01:14:19,055 --> 01:14:21,430 in a slightly different order are alpha, beta, and gamma. 1004 01:14:21,430 --> 01:14:22,870 This one goes alpha, beta, gamma. 1005 01:14:22,870 --> 01:14:25,245 Where I wrote it here, it just became beta, gamma, alpha. 1006 01:14:25,245 --> 01:14:27,420 That's a cyclic permutation I've antisymmetrized. 1007 01:14:27,420 --> 01:14:29,323 They are exactly the same. 1008 01:14:29,323 --> 01:14:31,490 So these are on the same side-- or on opposite sides 1009 01:14:31,490 --> 01:14:32,610 of the equation. 1010 01:14:32,610 --> 01:14:35,350 So they cancel each other out. 1011 01:14:35,350 --> 01:14:40,370 And so what arises from all this is p mu-- 1012 01:14:44,700 --> 01:14:46,678 pardon me, I missed the derivative. 1013 01:15:04,990 --> 01:15:09,880 I've set no properties on p, so the only way 1014 01:15:09,880 --> 01:15:22,090 this can always hold is if in fact, this is equal to zero. 1015 01:15:24,640 --> 01:15:29,562 This is a result known as the Bianchi identity. 1016 01:15:37,756 --> 01:15:40,250 Let me just write another form of it, 1017 01:15:40,250 --> 01:15:42,920 and then I'm fairly pressed for time, 1018 01:15:42,920 --> 01:15:44,630 so I think I'm just going to write down 1019 01:15:44,630 --> 01:15:47,060 the result of the next thing, and I 1020 01:15:47,060 --> 01:15:51,010 will put the details of this into 1021 01:15:51,010 --> 01:15:52,490 the first prerecorded lecture. 1022 01:16:01,310 --> 01:16:04,400 If you expand out that antisymmetry 1023 01:16:04,400 --> 01:16:11,330 and take advantage of the Riemann symmetry-- 1024 01:16:11,330 --> 01:16:23,630 the various Riemann symmetry relationships, 1025 01:16:23,630 --> 01:16:24,820 this is an equivalent form. 1026 01:16:30,170 --> 01:16:33,760 So these two things both are an important geometric 1027 01:16:33,760 --> 01:16:37,870 relationship the curvature tensor must go by. 1028 01:16:37,870 --> 01:16:44,270 Now, in some notes that I am going to-- well, 1029 01:16:44,270 --> 01:16:49,030 they're actually already scanned and on the web. 1030 01:16:49,030 --> 01:16:53,980 What I do is take the Bianchi identity and contract 1031 01:16:53,980 --> 01:17:02,610 on some of the indices to convert my Riemann tensor-- 1032 01:17:02,610 --> 01:17:05,040 my Riemann curvature, into a Ricci tensor-- 1033 01:17:05,040 --> 01:17:06,215 Ricci curvature. 1034 01:17:17,860 --> 01:17:20,965 So in particular, if I use this second form of it, 1035 01:17:20,965 --> 01:17:22,590 it's just covariant derivatives, right? 1036 01:17:22,590 --> 01:17:24,780 And the metric commutes with covariant derivatives, 1037 01:17:24,780 --> 01:17:27,750 so I can just sort of walk it through. 1038 01:17:27,750 --> 01:17:32,940 What you find when you do this is that you can write 1039 01:17:32,940 --> 01:17:40,020 this relationship in the following form, 1040 01:17:40,020 --> 01:17:42,920 and the derivation of this will be scanned and posted, 1041 01:17:42,920 --> 01:17:46,040 and I will step through it in the first recorded lecture. 1042 01:17:48,660 --> 01:17:51,390 The divergence of a particular combination 1043 01:17:51,390 --> 01:17:56,040 of the Ricci curvature and the Ricci scalar is equal to zero. 1044 01:17:56,040 --> 01:18:00,060 This is a sufficiently interesting tensor 1045 01:18:00,060 --> 01:18:01,920 that it is now given a name. 1046 01:18:05,350 --> 01:18:12,550 We write this G, and we call this the Einstein curvature 1047 01:18:12,550 --> 01:18:13,050 tensor. 1048 01:18:26,180 --> 01:18:28,520 I sort of mentioned that one of our governing principles 1049 01:18:28,520 --> 01:18:31,670 here is, we're going to change the source of gravitation 1050 01:18:31,670 --> 01:18:34,850 from just matter density to the stress energy tensor. 1051 01:18:34,850 --> 01:18:38,720 That is a two-index divergence-free tensor. 1052 01:18:38,720 --> 01:18:40,393 We want our gravitational-- 1053 01:18:40,393 --> 01:18:41,810 the left hand side of the equation 1054 01:18:41,810 --> 01:18:43,280 to look like two derivatives in the metric, which 1055 01:18:43,280 --> 01:18:43,905 is a curvature. 1056 01:18:43,905 --> 01:18:47,760 So we need a divergence-free two-index curvature. 1057 01:18:47,760 --> 01:18:49,227 Ta-da! 1058 01:18:49,227 --> 01:18:50,810 It's got Einstein's name on it, so you 1059 01:18:50,810 --> 01:18:52,520 know it's got to be good. 1060 01:18:52,520 --> 01:18:57,030 All right, so that is where, unfortunately, the COVID virus 1061 01:18:57,030 --> 01:18:59,390 is requiring us to stop for the semester, 1062 01:18:59,390 --> 01:19:01,680 at least as far as in-person goes. 1063 01:19:01,680 --> 01:19:07,100 So look for notes to be posted, videos to be posted, 1064 01:19:07,100 --> 01:19:11,300 and I, of course, will be in contact, 1065 01:19:11,300 --> 01:19:12,860 behind a sick wall or something. 1066 01:19:12,860 --> 01:19:14,960 But anyway, [CHUCKLES] certainly, 1067 01:19:14,960 --> 01:19:16,730 I am not going out of contact. 1068 01:19:16,730 --> 01:19:24,170 And so those of you who are scattering to parts unknown-- 1069 01:19:24,170 --> 01:19:26,570 known to you, that is, unknown to me-- 1070 01:19:26,570 --> 01:19:29,630 good luck with your travels and getting yourselves settled. 1071 01:19:29,630 --> 01:19:31,793 I really feel-- everyone feels terrible 1072 01:19:31,793 --> 01:19:33,710 that you're going through this crap right now, 1073 01:19:33,710 --> 01:19:38,330 but hopefully, this will flatten the curve, as they say, 1074 01:19:38,330 --> 01:19:40,610 and it will be the right thing to do. 1075 01:19:40,610 --> 01:19:42,020 But stay in touch, OK? 1076 01:19:42,020 --> 01:19:45,590 This campus is going to be weird and sad without the students 1077 01:19:45,590 --> 01:19:49,090 here, and so we want to hear from you.