1 00:00:00,000 --> 00:00:01,976 [SQUEAKING] 2 00:00:01,976 --> 00:00:03,952 [RUSTLING] 3 00:00:03,952 --> 00:00:04,940 [CLICKING] 4 00:00:10,975 --> 00:00:12,850 SCOTT HUGHES: So let me just do a quick recap 5 00:00:12,850 --> 00:00:15,560 of what we did last time. 6 00:00:15,560 --> 00:00:17,480 So today, we're going to move into things that 7 00:00:17,480 --> 00:00:18,730 are a little bit more physics. 8 00:00:18,730 --> 00:00:20,950 Last time we were really doing some things that 9 00:00:20,950 --> 00:00:23,890 allows us to establish some of the critical mathematical 10 00:00:23,890 --> 00:00:28,150 concepts we need to study the tensors that 11 00:00:28,150 --> 00:00:31,750 are going to be used for physics on a curved manifold. 12 00:00:31,750 --> 00:00:34,210 So one of the things that we saw is 13 00:00:34,210 --> 00:00:36,670 that if I wanted to formulate differential equations 14 00:00:36,670 --> 00:00:40,510 on a curved manifold, if I just defined my derivative the most 15 00:00:40,510 --> 00:00:43,240 naive way you might think of, you 16 00:00:43,240 --> 00:00:45,940 end up with objects that are not tensorial. 17 00:00:45,940 --> 00:00:48,608 And so mathematically you might say, well, that's fine. 18 00:00:48,608 --> 00:00:49,900 It's just not a tensor anymore. 19 00:00:49,900 --> 00:00:52,780 But we really want tensors for our physics 20 00:00:52,780 --> 00:00:55,390 because we want to be working with quantities 21 00:00:55,390 --> 00:00:58,100 that have frame-independent geometric meaning to them. 22 00:00:58,100 --> 00:01:00,880 So that notion of a derivative-- if I just do it the naive way-- 23 00:01:00,880 --> 00:01:02,555 isn't the best. 24 00:01:02,555 --> 00:01:04,180 And so I argued that what we need to do 25 00:01:04,180 --> 00:01:06,940 is to find some kind of a transport operation 26 00:01:06,940 --> 00:01:12,250 in which there is a linear mapping between things 27 00:01:12,250 --> 00:01:13,660 like my vector field or my tensor 28 00:01:13,660 --> 00:01:17,950 field and the displacement, which allows me to cancel out 29 00:01:17,950 --> 00:01:20,380 the bits of the partial derivative transformation 30 00:01:20,380 --> 00:01:23,130 laws that are non tensorial. 31 00:01:23,130 --> 00:01:24,690 There's a lot of freedom to do that. 32 00:01:24,690 --> 00:01:27,480 One of the ways I suggest we do that is by demanding that when 33 00:01:27,480 --> 00:01:30,450 I do this, that derivative-- when applied to the metric-- 34 00:01:30,450 --> 00:01:31,390 give me 0. 35 00:01:31,390 --> 00:01:33,480 And if we do that, we see right away 36 00:01:33,480 --> 00:01:35,473 that the transport law that emerges 37 00:01:35,473 --> 00:01:37,140 gives me the covariant derivative as one 38 00:01:37,140 --> 00:01:37,930 of my examples. 39 00:01:37,930 --> 00:01:39,305 Now this shouldn't be a surprise. 40 00:01:39,305 --> 00:01:40,980 We introduced the covariant derivative 41 00:01:40,980 --> 00:01:44,340 by thinking about flat spacetime operations, 42 00:01:44,340 --> 00:01:47,100 but with all my basis objects being functionals. 43 00:01:47,100 --> 00:01:52,088 And this in some way is sort of a continuation of that notion. 44 00:01:52,088 --> 00:01:54,630 The other thing which I talked about is telling you we're not 45 00:01:54,630 --> 00:01:57,180 going to use a tremendous amount here except to motivate one 46 00:01:57,180 --> 00:02:01,080 very important result. And that is if I define transport 47 00:02:01,080 --> 00:02:03,750 by basically imagining that I slide my vectors in order 48 00:02:03,750 --> 00:02:07,020 to make the comparison-- along some specified vector field-- 49 00:02:07,020 --> 00:02:08,880 I get what's known as the Lie derivative. 50 00:02:08,880 --> 00:02:14,610 And so this is an example of the Lie derivative of a vector. 51 00:02:14,610 --> 00:02:17,790 And you get this form that looks like a commutator 52 00:02:17,790 --> 00:02:19,740 between the vector field you're sliding along 53 00:02:19,740 --> 00:02:21,930 and the vector field you are differentiating. 54 00:02:21,930 --> 00:02:23,997 Similar forms-- which are not really 55 00:02:23,997 --> 00:02:25,830 the form of a commutator-- but similar forms 56 00:02:25,830 --> 00:02:29,520 can be written down for general tensors. 57 00:02:29,520 --> 00:02:33,480 The key thing that you should be aware of 58 00:02:33,480 --> 00:02:36,720 is that it's got a similar form to the covariant derivative, 59 00:02:36,720 --> 00:02:38,100 in that you have one term-- let's 60 00:02:38,100 --> 00:02:39,520 focus on the top line for the second-- 61 00:02:39,520 --> 00:02:40,895 you have one term that looks just 62 00:02:40,895 --> 00:02:43,740 like the ordinary vector contracted 63 00:02:43,740 --> 00:02:46,830 onto a partial derivative of your field. 64 00:02:46,830 --> 00:02:49,980 And then you have terms which correct every free index 65 00:02:49,980 --> 00:02:50,730 of your field-- 66 00:02:50,730 --> 00:02:53,700 one free index if it's a vector, one free index 67 00:02:53,700 --> 00:02:58,140 if it's a one form, and corrections for an end index 68 00:02:58,140 --> 00:02:59,130 tensor-- 69 00:02:59,130 --> 00:03:02,250 with the sign doing something opposite to the sign 70 00:03:02,250 --> 00:03:04,860 that appears in the covariant derivative. 71 00:03:04,860 --> 00:03:08,228 What's interesting about this is that so defined, 72 00:03:08,228 --> 00:03:10,020 the Lie derivative is only written in terms 73 00:03:10,020 --> 00:03:11,790 of partial derivatives. 74 00:03:11,790 --> 00:03:14,663 But if you just imagine-- you promote 75 00:03:14,663 --> 00:03:16,830 those partial derivatives to covariant derivatives-- 76 00:03:16,830 --> 00:03:19,110 you find the exact same result holds 77 00:03:19,110 --> 00:03:21,762 because all of your Christoffel symbols-- or connections 78 00:03:21,762 --> 00:03:23,970 as we like to think of them when we're using parallel 79 00:03:23,970 --> 00:03:27,737 transport-- all the connective objects cancel each other out. 80 00:03:27,737 --> 00:03:29,320 And this is nice because this tells me 81 00:03:29,320 --> 00:03:32,170 that even though this object, strictly speaking, 82 00:03:32,170 --> 00:03:35,230 only involves partial derivatives, what emerges out 83 00:03:35,230 --> 00:03:36,780 of it is in fact tensorial. 84 00:03:36,780 --> 00:03:40,110 And it's an object that I can use for a lot of things 85 00:03:40,110 --> 00:03:41,110 I want to do in physics. 86 00:03:41,110 --> 00:03:43,402 In particular where we're going to use it the most-- 87 00:03:43,402 --> 00:03:45,360 and I said you're going to do this on the PSET, 88 00:03:45,360 --> 00:03:47,030 but I was wrong-- you're going to do something related to one 89 00:03:47,030 --> 00:03:47,613 of the PSETs-- 90 00:03:47,613 --> 00:03:50,002 but I'm going to actually-- if all goes well-- 91 00:03:50,002 --> 00:03:52,210 derive an important result involving these symmetries 92 00:03:52,210 --> 00:03:54,040 in today's lecture. 93 00:03:54,040 --> 00:03:55,540 We can use this to understand things 94 00:03:55,540 --> 00:03:58,255 that are related to conserved quantities in your space time. 95 00:03:58,255 --> 00:03:59,880 And where this comes from is that there 96 00:03:59,880 --> 00:04:02,920 is a definition of an object we call the Killing vector, which 97 00:04:02,920 --> 00:04:07,480 is an object where if your metric is Lie transported 98 00:04:07,480 --> 00:04:11,620 along some field C, we call C a Killing vector. 99 00:04:11,620 --> 00:04:16,519 And from the fact that the covariant derivative the metric 100 00:04:16,519 --> 00:04:21,769 is 0, you can turn the equation governing 101 00:04:21,769 --> 00:04:24,593 the Lie derivative along C into what I wrote down just there. 102 00:04:24,593 --> 00:04:25,760 And I should write its name. 103 00:04:29,810 --> 00:04:31,920 There's the result known as Killing's Equation. 104 00:04:43,270 --> 00:04:46,960 If a vector has a Killing vector-- 105 00:04:46,960 --> 00:04:48,910 if a metric has a Killing vector-- 106 00:04:48,910 --> 00:04:50,830 then you know that your metric is 107 00:04:50,830 --> 00:04:53,080 independent of some kind of a parameter that 108 00:04:53,080 --> 00:04:54,790 characterizes that spacetime. 109 00:04:54,790 --> 00:04:56,290 The converse also holds. 110 00:04:56,290 --> 00:04:58,270 If your metric is independent of something-- 111 00:04:58,270 --> 00:04:59,980 like say the time coordinate-- you 112 00:04:59,980 --> 00:05:02,620 know that there is a Killing vector corresponding 113 00:05:02,620 --> 00:05:04,230 to that independent thing. 114 00:05:04,230 --> 00:05:06,682 And so you'll often see this described as a differing 115 00:05:06,682 --> 00:05:08,140 amorphism of the spacetime-- if you 116 00:05:08,140 --> 00:05:10,307 want to dig into some of the more advanced textbooks 117 00:05:10,307 --> 00:05:11,170 on the subject. 118 00:05:11,170 --> 00:05:14,110 We'll come back to that in a few more details hopefully shortly 119 00:05:14,110 --> 00:05:15,890 before the end of today's lecture. 120 00:05:15,890 --> 00:05:17,587 So where I concluded last time, was we 121 00:05:17,587 --> 00:05:19,420 started talking about these quantities known 122 00:05:19,420 --> 00:05:21,370 as tensor densities, which are given 123 00:05:21,370 --> 00:05:24,760 the less-than-helpful definition-- quantities 124 00:05:24,760 --> 00:05:27,730 that are like tensors, but not quite. 125 00:05:27,730 --> 00:05:30,280 The example I gave of this-- where we were starting-- 126 00:05:30,280 --> 00:05:33,160 was the Levi-Civita symbol. 127 00:05:33,160 --> 00:05:37,180 So let me just write down again what resulted from that. 128 00:05:37,180 --> 00:05:40,950 So if I have Levi-Civita-- and the tilde here is going 129 00:05:40,950 --> 00:05:44,680 to reflect the fact that this is not really a tensor-- 130 00:05:47,580 --> 00:05:54,070 this guy in some prime coordinates 131 00:05:54,070 --> 00:06:01,020 is related to this guy in the unprime coordinates 132 00:06:01,020 --> 00:06:02,340 via the following-- 133 00:06:10,452 --> 00:06:12,160 let's get the primes in the right place-- 134 00:06:16,440 --> 00:06:19,310 the following mess of quantities. 135 00:06:21,870 --> 00:06:23,620 So I'm not going to go through this again. 136 00:06:23,620 --> 00:06:25,630 This is basically a theorem from linear algebra 137 00:06:25,630 --> 00:06:29,020 that relates the determinant of a matrix-- 138 00:06:29,020 --> 00:06:31,660 not metric, but matrix-- 139 00:06:31,660 --> 00:06:34,360 to what you get when you contract a bunch of matrices 140 00:06:34,360 --> 00:06:37,092 onto the Levi-Civita symbol. 141 00:06:37,092 --> 00:06:39,550 And so the key thing to note is that if this were not here, 142 00:06:39,550 --> 00:06:41,850 this would look just like a tensor transformation. 143 00:06:41,850 --> 00:06:42,890 But that is there. 144 00:06:42,890 --> 00:06:44,650 So it's not. 145 00:06:44,650 --> 00:06:48,010 And so we call this a tensor density of weight 1. 146 00:06:56,290 --> 00:06:58,810 So the other one-- which I hinted at the end of the last 147 00:06:58,810 --> 00:07:02,530 lecture, but did not have time to get into-- 148 00:07:02,530 --> 00:07:05,060 is suppose we look at the metric. 149 00:07:05,060 --> 00:07:08,427 Now, the metric-- no ifs, ands, or buts about it-- 150 00:07:08,427 --> 00:07:09,010 it's a tensor. 151 00:07:15,081 --> 00:07:17,870 And it's actually the first tensor 152 00:07:17,870 --> 00:07:22,890 we've started talking about back in our toddler 153 00:07:22,890 --> 00:07:26,470 years of studying flat spacetime, which by the way, 154 00:07:26,470 --> 00:07:29,367 was about three weeks ago. 155 00:07:29,367 --> 00:07:30,450 Obviously that's a tensor. 156 00:07:30,450 --> 00:07:32,100 It's a simple tensor relationship. 157 00:07:32,100 --> 00:07:34,230 Let's take the determinant of both sides of this. 158 00:07:47,293 --> 00:07:49,710 You might look at this and go, why do you want to do that? 159 00:07:49,710 --> 00:07:52,730 Well when I do this, I'm going to call 160 00:07:52,730 --> 00:07:55,710 the determinant of the metric in the primed representation. 161 00:07:55,710 --> 00:07:58,700 Let's call that G prime. 162 00:07:58,700 --> 00:08:00,050 I get 2 powers-- 163 00:08:03,530 --> 00:08:07,200 2 powers of this Jacobian matrix is determinant. 164 00:08:11,810 --> 00:08:16,190 And I get the determinant in my original representation. 165 00:08:16,190 --> 00:08:18,080 Now I want to write this in a way that's 166 00:08:18,080 --> 00:08:19,747 similar to the way I wrote it over here. 167 00:08:19,747 --> 00:08:21,500 Notice I have all my primed objects 168 00:08:21,500 --> 00:08:22,880 over here on the left-hand side. 169 00:08:22,880 --> 00:08:26,900 And my factor of this determinant relates-- 170 00:08:26,900 --> 00:08:29,360 it's got primed indices in the upstairs position, unprimed 171 00:08:29,360 --> 00:08:30,830 in the downstairs. 172 00:08:30,830 --> 00:08:34,850 But the determinant of 1 over a metric is just 1 173 00:08:34,850 --> 00:08:36,289 over the determinant of-- 174 00:08:36,289 --> 00:08:39,799 the determinant of the inverse of a matrix is just 1 175 00:08:39,799 --> 00:08:42,080 over the determinant of that matrix. 176 00:08:42,080 --> 00:08:52,790 And so I can really simply just say this looks like so. 177 00:08:52,790 --> 00:09:00,440 So the determinant of the metric is a tensor density 178 00:09:00,440 --> 00:09:01,790 of weight minus 2. 179 00:09:23,550 --> 00:09:28,170 What this basically tells us is I now have two of these things. 180 00:09:28,170 --> 00:09:30,120 I've been arguing basically this entire course 181 00:09:30,120 --> 00:09:32,670 that we want to use tensors because of the fact 182 00:09:32,670 --> 00:09:37,590 that they give me a covariant way of encoding 183 00:09:37,590 --> 00:09:40,110 geometric concepts. 184 00:09:40,110 --> 00:09:43,350 I've got these two things that are not quite tensors. 185 00:09:43,350 --> 00:09:46,560 I can put them together and get a tensor out of this. 186 00:09:46,560 --> 00:09:52,440 So what this tells me now is I can convert any tensor density 187 00:09:52,440 --> 00:09:53,625 into a proper tensor. 188 00:09:56,540 --> 00:09:58,820 So suppose I have a tensor density of weight 189 00:09:58,820 --> 00:10:11,160 W. I can convert this into a proper tensor 190 00:10:11,160 --> 00:10:21,430 by multiplying by a power of that G. So multiply it 191 00:10:21,430 --> 00:10:26,940 by G to the W over 2. 192 00:10:26,940 --> 00:10:31,140 One slight subtlety here, when we work in spacetime-- 193 00:10:31,140 --> 00:10:35,620 let's just stop for a second and think about special relativity. 194 00:10:35,620 --> 00:10:41,020 In special relativity in an inertial reference frame, 195 00:10:41,020 --> 00:10:44,410 my metric is minus 1 1 1 1 on the diagonal. 196 00:10:44,410 --> 00:10:46,880 So its determinant is negative 1. 197 00:10:46,880 --> 00:10:51,610 And when I take negative 1 to some power that involves 198 00:10:51,610 --> 00:10:54,190 a square root, I get sad. 199 00:10:54,190 --> 00:10:56,480 We all know how to work with complex numbers. 200 00:10:56,480 --> 00:10:58,330 You might think that's all OK. 201 00:10:58,330 --> 00:10:59,260 It's not in this case. 202 00:10:59,260 --> 00:11:03,370 But the way I can fix that is that equation's still true 203 00:11:03,370 --> 00:11:06,353 if I multiply both sides by minus 1. 204 00:11:06,353 --> 00:11:08,020 I want this to be a positive number when 205 00:11:08,020 --> 00:11:08,978 I take the square root. 206 00:11:08,978 --> 00:11:13,240 So I'm allowed just to take the absolute value. 207 00:11:13,240 --> 00:11:16,240 So we take the absolute value to clear out the fact 208 00:11:16,240 --> 00:11:19,820 that in spacetime, we tend to have an indeterminate metric, 209 00:11:19,820 --> 00:11:22,210 where the sign depends on the interval. 210 00:11:33,753 --> 00:11:35,670 So remember the only reason we're doing this-- 211 00:11:35,670 --> 00:11:36,520 this is just a-- 212 00:11:36,520 --> 00:11:38,640 I don't want to say it's a trick. 213 00:11:38,640 --> 00:11:42,320 But it's not that far off from a trick. 214 00:11:42,320 --> 00:11:44,810 I'm just combining two tensor densities in order 215 00:11:44,810 --> 00:11:46,250 to get a tensor out of it. 216 00:11:46,250 --> 00:11:49,940 And minus a tensor density is still a tensor density. 217 00:11:49,940 --> 00:11:50,990 So I'm OK to do that. 218 00:11:50,990 --> 00:11:52,907 And I'm just doing this so that my square root 219 00:11:52,907 --> 00:11:55,310 doesn't go haywire on me. 220 00:11:55,310 --> 00:11:56,630 So a particular example-- 221 00:11:56,630 --> 00:11:59,960 in fact the one that in my career has come up the most-- 222 00:12:02,850 --> 00:12:08,550 is making a proper volume element 223 00:12:08,550 --> 00:12:16,677 converting my Levi-Civita symbol into a tensor that 224 00:12:16,677 --> 00:12:17,760 gives me a volume element. 225 00:12:22,570 --> 00:12:25,930 So my Levi-Civita symbol has a tensor density of weight 1. 226 00:12:25,930 --> 00:12:28,630 If I want to make that into a proper tensor, 227 00:12:28,630 --> 00:12:33,010 I multiply by the square root of the determinant of the metric. 228 00:12:33,010 --> 00:12:36,040 So now I will no longer have that tilde 229 00:12:36,040 --> 00:12:38,260 on there, which was meant to be a signpost that this 230 00:12:38,260 --> 00:12:39,760 as a quantity is a little bit goofy. 231 00:12:46,990 --> 00:12:49,570 You wind up with something like this. 232 00:12:49,570 --> 00:12:51,670 When you go-- and by the way, sometimes 233 00:12:51,670 --> 00:12:52,510 when you're working with this, you 234 00:12:52,510 --> 00:12:54,370 need to have this thing with indices in the upstairs 235 00:12:54,370 --> 00:12:55,150 position. 236 00:12:55,150 --> 00:12:56,260 You have to be a little bit careful. 237 00:12:56,260 --> 00:12:57,718 But I'll just give you one example. 238 00:12:57,718 --> 00:13:02,440 If you raise all four of the indices, what 239 00:13:02,440 --> 00:13:04,025 you find when everything goes through, 240 00:13:04,025 --> 00:13:06,400 this one is not that hard to see because you're basically 241 00:13:06,400 --> 00:13:08,317 playing with a similar relationship to the one 242 00:13:08,317 --> 00:13:09,850 that I wrote down over here-- 243 00:13:09,850 --> 00:13:13,008 just a short homework exercise to demonstrate this. 244 00:13:13,008 --> 00:13:14,800 And then you end up with the tensor density 245 00:13:14,800 --> 00:13:16,260 of the opposite sign. 246 00:13:16,260 --> 00:13:18,760 Weight minus 1, you wind up with a 1 over square root there. 247 00:13:23,430 --> 00:13:25,020 So like I said one of the reasons 248 00:13:25,020 --> 00:13:30,030 why this is an important example is that we use it to form 249 00:13:30,030 --> 00:13:32,088 covariant volume operators. 250 00:13:55,520 --> 00:13:59,440 So in four-dimensional space-- 251 00:13:59,440 --> 00:14:06,860 so imagine here's my basis direction for spatial direction 252 00:14:06,860 --> 00:14:10,610 1, spatial direction 2, spatial direction 3-- 253 00:14:10,610 --> 00:14:12,860 you guys can figure out how to write spatial direction 254 00:14:12,860 --> 00:14:17,060 0 on your own time-- 255 00:14:17,060 --> 00:14:20,790 I would define a covariant 4 volume-- 256 00:14:20,790 --> 00:14:22,430 4 volume element from this. 257 00:14:34,755 --> 00:14:36,250 It'll look like this. 258 00:14:36,250 --> 00:14:40,080 And if this is an orthogonal basis, 259 00:14:40,080 --> 00:14:52,930 this simply turns into something like-- 260 00:14:55,745 --> 00:14:57,650 these are meant to be superscripts 261 00:14:57,650 --> 00:15:00,265 because these are coordinates. 262 00:15:00,265 --> 00:15:02,480 So it just turns into something like this 263 00:15:02,480 --> 00:15:04,210 if I'm working in an orthogonal basis. 264 00:15:09,210 --> 00:15:12,470 And again for intuition, I suggest go down 265 00:15:12,470 --> 00:15:16,685 to 3-dimensional spherical coordinates. 266 00:15:16,685 --> 00:15:17,810 And I wrote this last time. 267 00:15:17,810 --> 00:15:19,490 But let me just quickly write it up. 268 00:15:19,490 --> 00:15:23,750 I mean everything I did here, I tend to-- since this is 269 00:15:23,750 --> 00:15:25,670 a course on spacetime-- 270 00:15:25,670 --> 00:15:28,250 by default I write down all my formulas 271 00:15:28,250 --> 00:15:32,415 for three space dimensions, one time dimension. 272 00:15:32,415 --> 00:15:35,180 But it's perfectly good in 3 spatial dimensions, 273 00:15:35,180 --> 00:15:39,703 2 spatial dimensions, 17 spatial dimensions-- whatever crazy 274 00:15:39,703 --> 00:15:41,870 spacetime your physics want you to put yourself in-- 275 00:15:41,870 --> 00:15:43,620 or space your physics wants to put you in. 276 00:15:48,460 --> 00:15:49,620 So I'll just remind you-- 277 00:16:03,610 --> 00:16:05,360 that when you do this, you've got yourself 278 00:16:05,360 --> 00:16:12,650 a metric across the diagonal of 1 r 279 00:16:12,650 --> 00:16:17,690 squared r squared sine squared theta. 280 00:16:17,690 --> 00:16:19,560 And just to be consistent-- 281 00:16:19,560 --> 00:16:24,572 I usually use Latin letters for only spatial things. 282 00:16:24,572 --> 00:16:25,280 So let's do that. 283 00:16:35,640 --> 00:16:39,685 This would be how I would then write my volume element. 284 00:16:39,685 --> 00:16:40,560 Did I miss something? 285 00:16:40,560 --> 00:16:41,185 AUDIENCE: Yeah. 286 00:16:41,185 --> 00:16:42,270 [INAUDIBLE] 287 00:16:42,270 --> 00:16:43,340 SCOTT HUGHES: Absolutely. 288 00:16:43,340 --> 00:16:45,720 Yeah. 289 00:16:45,720 --> 00:16:46,767 Thank you. 290 00:16:46,767 --> 00:16:47,600 I'm writing quickly. 291 00:16:47,600 --> 00:16:48,100 Yeah? 292 00:16:48,100 --> 00:16:51,827 AUDIENCE: Is there [INAUDIBLE]? 293 00:16:51,827 --> 00:16:52,910 SCOTT HUGHES: This is dx-- 294 00:16:52,910 --> 00:16:53,940 oh, bugger. 295 00:16:53,940 --> 00:16:55,912 Yep. 296 00:16:55,912 --> 00:16:57,370 I'm trying to get to something new. 297 00:16:57,370 --> 00:16:59,037 And I'm afraid I'm rushing a little bit. 298 00:16:59,037 --> 00:17:01,053 So thank you for catching this. 299 00:17:01,053 --> 00:17:03,220 And so with this, take the determiner of this thing. 300 00:17:03,220 --> 00:17:09,130 And sure enough you get r squared sine theta d r d theta 301 00:17:09,130 --> 00:17:11,170 d phi. 302 00:17:11,170 --> 00:17:12,640 So this is the main thing that we 303 00:17:12,640 --> 00:17:15,460 are going to use this result for-- this thing with tensor 304 00:17:15,460 --> 00:17:16,480 densities. 305 00:17:16,480 --> 00:17:18,760 I want to go on a brief aside, which 306 00:17:18,760 --> 00:17:21,790 is relevant to the problem that I delayed on 307 00:17:21,790 --> 00:17:22,890 this week's problem 7. 308 00:17:26,640 --> 00:17:29,370 So there are three parts of problem 7 309 00:17:29,370 --> 00:17:41,940 that I moved from PSET 3 to PSET 4 because they rely on a result 310 00:17:41,940 --> 00:17:43,230 that I want to talk about now. 311 00:17:46,010 --> 00:17:49,000 So the main thing that we use the determinant of the metric 312 00:17:49,000 --> 00:17:51,880 for in a formal way is this-- that it's a tensor 313 00:17:51,880 --> 00:17:53,950 density of weight minus 2. 314 00:17:53,950 --> 00:17:55,750 And so it's a really useful quantity 315 00:17:55,750 --> 00:17:59,023 for converting tensor densities into proper tensors. 316 00:17:59,023 --> 00:18:00,940 And really the most common application of this 317 00:18:00,940 --> 00:18:02,842 tends to be to volume elements. 318 00:18:02,842 --> 00:18:04,300 But it turns out that it's actually 319 00:18:04,300 --> 00:18:07,930 really useful for what a former professor of mine 320 00:18:07,930 --> 00:18:10,450 used to like to call party tricks. 321 00:18:10,450 --> 00:18:15,760 There's some really-- it offers a really nice shortcut 322 00:18:15,760 --> 00:18:18,610 to computing certain Christoffel symbols. 323 00:18:18,610 --> 00:18:21,190 So in honor of Saul Teukolsky let's call this a party trick. 324 00:18:26,480 --> 00:18:36,660 So we're using the determinant of the metric 325 00:18:36,660 --> 00:18:38,470 to compute certain Christoffels. 326 00:18:47,590 --> 00:18:49,340 So this is going to rely on the following. 327 00:18:49,340 --> 00:18:55,100 So suppose I calculate the Christoffel symbol, 328 00:18:55,100 --> 00:18:58,243 but I'm going to sum on the raised index. 329 00:18:58,243 --> 00:19:00,410 And bearing in mind it's symmetric in the lower one, 330 00:19:00,410 --> 00:19:02,630 I'm going to do a contraction of the raised index with one 331 00:19:02,630 --> 00:19:03,505 of the lower indices. 332 00:19:06,260 --> 00:19:10,630 So let's just throw in a couple of definitions. 333 00:19:10,630 --> 00:19:17,280 This is equivalent to the following. 334 00:19:17,280 --> 00:19:21,150 And so throwing in the definition of the Christoffel 335 00:19:21,150 --> 00:19:23,417 with all the indices in the downstairs position-- 336 00:19:43,820 --> 00:19:45,470 this formula, by the way, is something 337 00:19:45,470 --> 00:19:50,240 that I've been writing down now for about 27 years. 338 00:19:50,240 --> 00:19:51,830 And I have to look it up every time. 339 00:19:51,830 --> 00:19:54,447 Usually by the end of a semester of teaching 8.962, 340 00:19:54,447 --> 00:19:55,280 I have it memorized. 341 00:19:55,280 --> 00:19:56,660 But it decays by then. 342 00:19:56,660 --> 00:19:59,135 So if you're wondering how to go from here to here-- 343 00:19:59,135 --> 00:20:00,260 this is the kind of thing-- 344 00:20:00,260 --> 00:20:02,340 just look it up. 345 00:20:02,340 --> 00:20:04,320 So let's pause for a second. 346 00:20:04,320 --> 00:20:08,090 Remember that the metric is-- 347 00:20:08,090 --> 00:20:10,400 it's itself symmetric. 348 00:20:10,400 --> 00:20:12,640 So in keeping with that, I'm going 349 00:20:12,640 --> 00:20:19,860 to flip the indices on this last term, which-- hang on a second. 350 00:20:19,860 --> 00:20:21,420 That was stupid. 351 00:20:21,420 --> 00:20:21,920 Wait. 352 00:20:24,960 --> 00:20:25,460 Pardon me. 353 00:20:32,055 --> 00:20:33,930 This is the term I want to switch indices on. 354 00:20:33,930 --> 00:20:36,693 My apologies. 355 00:20:36,693 --> 00:20:38,110 So the reason I did that is I want 356 00:20:38,110 --> 00:20:39,610 to have both of these guys ending 357 00:20:39,610 --> 00:20:43,312 with the alpha because notice this and this-- 358 00:20:43,312 --> 00:20:44,020 they're the same. 359 00:20:44,020 --> 00:20:48,490 But I have interchanged the beta and the mu. 360 00:20:48,490 --> 00:20:49,850 So these two terms-- 361 00:20:49,850 --> 00:20:51,970 the first term and the third term-- 362 00:20:51,970 --> 00:20:56,970 are anti symmetric upon exchange of beta and mu. 363 00:20:56,970 --> 00:20:58,690 They are contracted with the metric, 364 00:20:58,690 --> 00:21:02,380 which is symmetric upon exchange of beta and mu. 365 00:21:02,380 --> 00:21:02,880 Question? 366 00:21:02,880 --> 00:21:05,080 AUDIENCE: Does the metric have to be symmetric? 367 00:21:05,080 --> 00:21:07,482 SCOTT HUGHES: The metric has to be symmetric. 368 00:21:07,482 --> 00:21:16,330 [LAUGHS] I don't want to get into that right now, but yes 369 00:21:16,330 --> 00:21:17,620 [LAUGHS]. 370 00:21:17,620 --> 00:21:19,780 So these guys are anti symmetric. 371 00:21:19,780 --> 00:21:21,080 This guy is symmetric. 372 00:21:21,080 --> 00:21:23,050 And remember the rule. 373 00:21:23,050 --> 00:21:25,720 Whenever you contract some kind of a symmetric object 374 00:21:25,720 --> 00:21:28,900 with an anti symmetric m you get 0. 375 00:21:28,900 --> 00:21:32,050 So that means this term and this term die. 376 00:21:47,751 --> 00:21:51,590 And what we are left with is gamma mu 377 00:21:51,590 --> 00:22:02,520 mu alpha is 1/2 g u beta, and the alpha derivative of g u 378 00:22:02,520 --> 00:22:04,960 beta. 379 00:22:04,960 --> 00:22:07,240 There is the way it is contracting 380 00:22:07,240 --> 00:22:09,460 the indices in the 2 metric with the other one. 381 00:22:09,460 --> 00:22:11,918 Well here's a theorem that I'm going to prove in a second-- 382 00:22:11,918 --> 00:22:16,100 or at least motivate-- that it's going to rely on a result 383 00:22:16,100 --> 00:22:17,680 that I will pull out of thin air, 384 00:22:17,680 --> 00:22:22,320 but can be found in most linear algebra textbooks. 385 00:22:22,320 --> 00:22:26,820 It's not too hard to show that this can be further written 386 00:22:26,820 --> 00:22:34,390 as 1 over square root of the determinant times the partial 387 00:22:34,390 --> 00:22:44,400 derivative of the square root of the determinant, 388 00:22:44,400 --> 00:22:47,090 which is sometimes-- depending on your applications-- 389 00:22:47,090 --> 00:22:50,700 this can be written very nicely as the derivative 390 00:22:50,700 --> 00:22:54,030 of the logarithm of the absolute value of the-- 391 00:22:54,030 --> 00:22:56,489 the square root of the absolute value of the determinant. 392 00:23:01,650 --> 00:23:05,500 So before I go on and actually demonstrate this, 393 00:23:05,500 --> 00:23:08,770 you can see why this is actually a pretty-- 394 00:23:08,770 --> 00:23:10,150 so this actually comes up. 395 00:23:10,150 --> 00:23:11,650 I'm going to show a few applications 396 00:23:11,650 --> 00:23:13,880 as to why this particular combination of Christoffel 397 00:23:13,880 --> 00:23:16,480 symbols shows up more often than you might guess. 398 00:23:16,480 --> 00:23:19,190 It's really important for certain important calculations. 399 00:23:19,190 --> 00:23:20,920 And this is telling me that I can get it 400 00:23:20,920 --> 00:23:23,380 by just taking one partial derivative of a scalar 401 00:23:23,380 --> 00:23:24,900 function. 402 00:23:24,900 --> 00:23:27,916 And if you know your metric, that's easy. 403 00:23:27,916 --> 00:23:31,000 So this becomes really easy thing to calculate. 404 00:23:31,000 --> 00:23:31,750 So let's prove it. 405 00:23:41,170 --> 00:23:45,107 So the proof of this relies on a few results 406 00:23:45,107 --> 00:23:45,940 from linear algebra. 407 00:24:03,370 --> 00:24:05,650 So let's not think about tensors for a second. 408 00:24:05,650 --> 00:24:07,150 And let's just think about matrices. 409 00:24:07,150 --> 00:24:09,850 So imagine I've got some matrix m. 410 00:24:12,910 --> 00:24:15,410 I'm going to be agnostic about the dimensions of this thing. 411 00:24:18,860 --> 00:24:21,040 And suppose I look at the following variation 412 00:24:21,040 --> 00:24:21,665 of this matrix. 413 00:24:31,240 --> 00:24:34,340 So suppose I imagine doing a little variation. 414 00:24:34,340 --> 00:24:37,210 So suppose every element of m is a function. 415 00:24:37,210 --> 00:24:43,390 And I look at a little variation of the log of the determinant 416 00:24:43,390 --> 00:24:45,750 of that matrix. 417 00:24:45,750 --> 00:24:58,280 Well this can be written as log is basically 418 00:24:58,280 --> 00:25:06,250 a definition of this. 419 00:25:08,850 --> 00:25:13,260 Now, if I exploit properties of logarithms, 420 00:25:13,260 --> 00:25:20,440 this can be written as the log of the determinant-- 421 00:25:20,440 --> 00:25:23,240 m plus delta m-- 422 00:25:23,240 --> 00:25:26,010 divided by the determinant of m. 423 00:25:33,060 --> 00:25:34,920 Now I'm going to use the fact that 1 424 00:25:34,920 --> 00:25:38,317 over the determinant of m is the determinant 425 00:25:38,317 --> 00:25:39,150 of the inverse of m. 426 00:26:00,850 --> 00:26:04,700 So taking advantage of that, I can further write this guy 427 00:26:04,700 --> 00:26:20,000 as something like this. 428 00:26:20,000 --> 00:26:23,495 Now I'm going to invoke an identity, which 429 00:26:23,495 --> 00:26:26,120 I believe you can find proven in many linear algebra textbooks. 430 00:26:26,120 --> 00:26:27,770 It just occurred to me as I'm thinking about this, 431 00:26:27,770 --> 00:26:30,620 I don't know if I've ever seen it explicitly proven myself. 432 00:26:30,620 --> 00:26:32,870 But it's something that's very easy to demonstrate 433 00:26:32,870 --> 00:26:35,200 with just a quick calculation. 434 00:26:35,200 --> 00:26:36,463 You can just do-- 435 00:26:36,463 --> 00:26:37,130 I'm a physicist. 436 00:26:37,130 --> 00:26:38,280 So for me I'll use Mathematica. 437 00:26:38,280 --> 00:26:40,060 I'll look at six or seven examples and go, 438 00:26:40,060 --> 00:26:41,130 it seems right. 439 00:26:41,130 --> 00:26:42,890 And so I've definitely done that. 440 00:26:42,890 --> 00:26:44,557 But I believe this is something that you 441 00:26:44,557 --> 00:26:46,100 can find proven explicitly-- 442 00:26:46,100 --> 00:26:47,930 like I said-- in most books. 443 00:26:47,930 --> 00:26:49,920 So remember these are all matrices. 444 00:26:49,920 --> 00:26:51,360 So this isn't the number 1. 445 00:26:51,360 --> 00:26:53,360 We want to think of this as the identity matrix. 446 00:26:59,900 --> 00:27:02,090 Oh and I'm also going to regard this variation 447 00:27:02,090 --> 00:27:04,580 as a small quantity. 448 00:27:04,580 --> 00:27:15,400 So if I regard epsilon as a small matrix-- 449 00:27:15,400 --> 00:27:17,290 this can be made formal by defining something 450 00:27:17,290 --> 00:27:19,290 like condition number associated with the matrix 451 00:27:19,290 --> 00:27:20,080 or something like that. 452 00:27:20,080 --> 00:27:21,480 But generally what I want to mean 453 00:27:21,480 --> 00:27:23,220 by that is if I take this epsilon 454 00:27:23,220 --> 00:27:26,640 and I add it to 1, all of this-- 455 00:27:26,640 --> 00:27:29,010 so my identity is 1 on the diagonal-- 456 00:27:29,010 --> 00:27:31,380 0s everywhere else-- all the things 457 00:27:31,380 --> 00:27:33,480 that are put into the sum of 1 plus epsilon 458 00:27:33,480 --> 00:27:36,390 are much, much smaller than that 1 that's on the diagonal. 459 00:27:36,390 --> 00:27:38,260 That will be sufficient. 460 00:27:38,260 --> 00:27:46,230 So if epsilon is a small matrix, then the determinant 461 00:27:46,230 --> 00:27:55,540 of 1 plus epsilon is approximately equal to 1 462 00:27:55,540 --> 00:27:59,130 plus the trace of epsilon. 463 00:28:05,840 --> 00:28:08,000 What that approximately refers to is-- of course 464 00:28:08,000 --> 00:28:09,180 you can take that further. 465 00:28:09,180 --> 00:28:10,370 And you'll get additional corrections 466 00:28:10,370 --> 00:28:12,860 that involve epsilon times epsilon, epsilon times epsilon, 467 00:28:12,860 --> 00:28:14,570 times epsilon. 468 00:28:14,570 --> 00:28:17,150 I believe when you do that, the coefficient is 469 00:28:17,150 --> 00:28:18,020 no longer universal. 470 00:28:18,020 --> 00:28:20,990 But it depends upon the dimensions of the matrix. 471 00:28:20,990 --> 00:28:23,000 But leading order it's independent of dimensions 472 00:28:23,000 --> 00:28:23,600 of the matrix. 473 00:28:23,600 --> 00:28:24,500 And that's something that you can 474 00:28:24,500 --> 00:28:26,460 you can play with a little bit yourself. 475 00:28:26,460 --> 00:28:28,960 Like I said, this is sufficient for what we want to do here. 476 00:28:35,070 --> 00:28:42,460 So I'm going to think of my small matrix 477 00:28:42,460 --> 00:28:50,350 as the matrix of inverse m times a variation of m. 478 00:28:50,350 --> 00:28:51,940 This is our epsilon. 479 00:28:55,010 --> 00:28:58,490 So we're going to apply it to the line that I have up here. 480 00:28:58,490 --> 00:29:04,180 And this tells me that my delta on the log of the derivative 481 00:29:04,180 --> 00:29:17,690 of m is the log of 1 plus the trace of m to the minus 1 482 00:29:17,690 --> 00:29:20,380 on the matrix m. 483 00:29:20,380 --> 00:29:23,130 Log of 1 plus a small number is that small number. 484 00:29:33,000 --> 00:29:34,440 Now the application. 485 00:29:34,440 --> 00:29:36,900 So this is-- like I said, this the theorem that you 486 00:29:36,900 --> 00:29:41,730 can find in books that I don't know about but truly exist. 487 00:29:41,730 --> 00:29:45,690 This is something I've seen documented in a lot of places. 488 00:29:45,690 --> 00:29:50,115 Let's treat our m as the metric of spacetime. 489 00:29:55,700 --> 00:29:58,300 So my m will be g alpha beta. 490 00:29:58,300 --> 00:30:01,010 My m inverse will be g in the upstairs position. 491 00:30:06,030 --> 00:30:09,070 And I will write this something like so. 492 00:30:09,070 --> 00:30:12,600 And I'm going to apply this by looking 493 00:30:12,600 --> 00:30:14,070 at variations in my metric. 494 00:30:50,510 --> 00:30:54,070 So delta log-- 495 00:30:54,070 --> 00:30:56,330 I'm going to throw my absolute values in here. 496 00:30:56,330 --> 00:31:00,320 That's perfectly allowed to go ahead and put that into there. 497 00:31:00,320 --> 00:31:03,350 Applying this to what I've got, this 498 00:31:03,350 --> 00:31:15,338 is going to be the trace of g mu beta times the variation of g 499 00:31:15,338 --> 00:31:16,935 beta gamma. 500 00:31:16,935 --> 00:31:19,310 And I forgot to say, how do I take the trace of a matrix? 501 00:31:33,482 --> 00:31:35,190 So the trace that we're going to use-- we 502 00:31:35,190 --> 00:31:37,710 want it to be something that has geometric meaning 503 00:31:37,710 --> 00:31:40,240 and has a tensorial meaning to it. 504 00:31:40,240 --> 00:31:48,200 So we're going to call the trace of this thing g alpha 505 00:31:48,200 --> 00:31:50,360 beta epsilon alpha beta. 506 00:31:53,217 --> 00:31:54,800 If you think about what this is doing, 507 00:31:54,800 --> 00:31:57,380 you're essentially going to take your-- 508 00:31:57,380 --> 00:31:59,915 let's say I apply this to the metric itself. 509 00:31:59,915 --> 00:32:01,790 I put one index in the upstairs position, one 510 00:32:01,790 --> 00:32:05,120 the downstairs position, and then I am summing along 511 00:32:05,120 --> 00:32:08,390 the diagonal when I do this. 512 00:32:08,390 --> 00:32:15,990 You will sometimes see this written as something like that. 513 00:32:21,300 --> 00:32:25,790 So in this case, when I'm taking the trace of this guy here, 514 00:32:25,790 --> 00:32:28,810 that is going to force me to-- 515 00:32:28,810 --> 00:32:31,810 let's see. 516 00:32:31,810 --> 00:32:33,560 So this gives me a quantity where 517 00:32:33,560 --> 00:32:35,910 I'm summing over my betas. 518 00:32:35,910 --> 00:32:38,630 And then I'm just going to sum over the diagonal indices. 519 00:32:38,630 --> 00:32:42,780 I'm forcing my two remaining indices to be the same. 520 00:32:42,780 --> 00:32:46,880 So putting this together, this tells me-- 521 00:32:59,160 --> 00:33:02,630 so now what I'm going to do is say, 522 00:33:02,630 --> 00:33:06,070 I basically have part of a partial derivative here. 523 00:33:06,070 --> 00:33:18,050 All I need to do is now divide by a variation in my coordinate 524 00:33:18,050 --> 00:33:19,010 and take the limit. 525 00:33:39,340 --> 00:33:43,633 So it comes out of this as the partial derivative-- 526 00:33:53,310 --> 00:33:54,670 looks like this. 527 00:33:54,670 --> 00:33:56,940 Now let's trace it back to our Christoffel symbol. 528 00:34:02,060 --> 00:34:05,270 My Christoffel symbol-- the thing 529 00:34:05,270 --> 00:34:07,100 which I'm trying to compute-- is one half 530 00:34:07,100 --> 00:34:08,110 of this right-hand side. 531 00:34:15,630 --> 00:34:17,922 So it's one half of the left-hand side. 532 00:34:17,922 --> 00:34:20,464 And I can take that one half, march it through my derivative, 533 00:34:20,464 --> 00:34:23,300 and use the fact that 1/2 the log of x 534 00:34:23,300 --> 00:34:24,830 is the log of the square root of x. 535 00:34:40,909 --> 00:34:43,260 Check. 536 00:34:43,260 --> 00:34:46,860 So like I said, this is what an old mentor of mine 537 00:34:46,860 --> 00:34:48,570 used to like to call a party trick. 538 00:34:48,570 --> 00:34:52,440 It is a really useful party trick for certain calculations. 539 00:34:52,440 --> 00:34:54,949 So I want to make sure you saw where that comes from. 540 00:34:54,949 --> 00:34:57,360 This is something you will now use on the problem 541 00:34:57,360 --> 00:35:00,087 that I just moved from PSET 3 to PSET 4. 542 00:35:00,087 --> 00:35:02,170 It's useful for you to know where this comes from. 543 00:35:02,170 --> 00:35:03,630 You're certainly not going to need to go through this 544 00:35:03,630 --> 00:35:04,130 yourself. 545 00:35:04,130 --> 00:35:06,720 But this is a good type of calculation 546 00:35:06,720 --> 00:35:10,440 to be comfortable with. 547 00:35:10,440 --> 00:35:12,960 Those of you who are more rigorous in your math than me, 548 00:35:12,960 --> 00:35:15,660 you might want to run off and verify 549 00:35:15,660 --> 00:35:17,910 a couple of these identities that I used. 550 00:35:17,910 --> 00:35:24,600 But this is very nice for physics level rigor-- 551 00:35:24,600 --> 00:35:26,730 at least astrophysicists level rigor. 552 00:35:26,730 --> 00:35:29,070 So let me talk about one of the places where this 553 00:35:29,070 --> 00:35:32,550 shows up and is quite useful. 554 00:35:39,310 --> 00:35:42,000 So a place where I've seen this show up the most 555 00:35:42,000 --> 00:35:44,190 is when you're looking at the spacetime divergence 556 00:35:44,190 --> 00:35:44,970 of a vector field. 557 00:35:55,320 --> 00:35:58,550 So when you're calculating the covariant derivative 558 00:35:58,550 --> 00:36:00,980 of alpha contracting on the indices-- 559 00:36:00,980 --> 00:36:01,940 let's just throw in-- 560 00:36:05,140 --> 00:36:07,030 expand out the full definition of things-- 561 00:36:13,920 --> 00:36:16,080 all you've gotta do is correct one index. 562 00:36:16,080 --> 00:36:21,090 And voila, this is exactly the things that change-- 563 00:36:21,090 --> 00:36:23,190 where is it-- change my alpha to a mu-- that's 564 00:36:23,190 --> 00:36:26,610 exactly what I've got before. 565 00:36:26,610 --> 00:36:32,680 And so-- hang on just one moment. 566 00:36:32,680 --> 00:36:34,440 I know what I'm doing. 567 00:36:34,440 --> 00:36:36,750 These are all dummy indices. 568 00:36:36,750 --> 00:36:39,610 So in order to keep things from getting crossed, 569 00:36:39,610 --> 00:36:41,300 I'm going to relabel these over here. 570 00:36:47,660 --> 00:36:51,340 So I can take advantage of this identity and write this. 571 00:37:08,150 --> 00:37:17,260 So stare at this for a second. 572 00:37:17,260 --> 00:37:21,200 And you'll see that the whole thing can be rewritten 573 00:37:21,200 --> 00:37:22,560 in a very simple form. 574 00:37:47,560 --> 00:37:49,660 Ta-da. 575 00:37:49,660 --> 00:37:52,360 You haven't done as much work with covariant in your lives 576 00:37:52,360 --> 00:37:53,350 as I have. 577 00:37:53,350 --> 00:37:56,205 So let me just emphasize that ordinarily when 578 00:37:56,205 --> 00:37:57,580 you see an expression like you've 579 00:37:57,580 --> 00:37:59,950 got up there on the top line, you look at that, 580 00:37:59,950 --> 00:38:02,388 and you kind of go [GROANS] because you look at that, 581 00:38:02,388 --> 00:38:04,180 and the first thing that comes to your mind 582 00:38:04,180 --> 00:38:07,360 is you've got to work out every one of those Christoffel 583 00:38:07,360 --> 00:38:10,470 symbols and sum it up to get those things. 584 00:38:10,470 --> 00:38:13,870 And in a general spacetime, there will be 40 of them. 585 00:38:13,870 --> 00:38:16,960 And before Odin gave us Mathematica, 586 00:38:16,960 --> 00:38:18,310 that was a fair amount of labor. 587 00:38:18,310 --> 00:38:20,870 Even with Mathematica it's not necessarily trivial 588 00:38:20,870 --> 00:38:23,520 because it's really easy to screw things up. 589 00:38:23,520 --> 00:38:26,350 With this you calculate the determinant of the metric, 590 00:38:26,350 --> 00:38:28,990 you take its square root, you multiply your guy, 591 00:38:28,990 --> 00:38:31,330 and you take a partial derivative, and you divide. 592 00:38:31,330 --> 00:38:33,580 That is something that most of us 593 00:38:33,580 --> 00:38:35,410 learned how to do quite a long time ago. 594 00:38:35,410 --> 00:38:38,892 It cleans the hell out of this up. 595 00:38:38,892 --> 00:38:41,350 So the fact that this gives us something that only involves 596 00:38:41,350 --> 00:38:43,794 partial derivatives is awesome. 597 00:38:55,660 --> 00:38:59,590 This also-- it turns out-- so when you have things like this, 598 00:38:59,590 --> 00:39:03,730 it gives us a nice way to express Gauss's theorem 599 00:39:03,730 --> 00:39:04,740 in a curved manifold. 600 00:39:08,830 --> 00:39:11,225 So Gauss's theorem-- if I just look at the integrals 601 00:39:11,225 --> 00:39:13,600 that-- or rather the integral for Gauss's theorem-- let's 602 00:39:13,600 --> 00:39:15,070 put it that way. 603 00:39:15,070 --> 00:39:17,560 Let's say a Gauss's-type integral. 604 00:39:23,330 --> 00:39:25,520 So go back to when you're talking 605 00:39:25,520 --> 00:39:26,750 about conservation laws. 606 00:39:30,210 --> 00:39:38,400 If I imagine I'm integrating the divergence of some vector field 607 00:39:38,400 --> 00:39:41,130 over a four-dimensional volume, look at that, 608 00:39:41,130 --> 00:39:44,820 I get a nice cancellation. 609 00:39:44,820 --> 00:39:57,180 So this turns into an integral of that nice, clean derivative 610 00:39:57,180 --> 00:40:00,780 over my four coordinates. 611 00:40:00,780 --> 00:40:03,570 And then you can take advantage of the actual content 612 00:40:03,570 --> 00:40:05,977 of Gauss's Theorem to turn that into an integral 613 00:40:05,977 --> 00:40:08,310 over the three-dimensional surface that bounds that four 614 00:40:08,310 --> 00:40:08,810 volume. 615 00:40:13,330 --> 00:40:14,320 It's a good point-- 616 00:40:14,320 --> 00:40:15,825 so you're emboldened by this. 617 00:40:15,825 --> 00:40:16,950 You say, yay, look at that. 618 00:40:16,950 --> 00:40:19,230 We can do all this awesome stuff with this identity. 619 00:40:19,230 --> 00:40:20,875 It gives me a great way to express some 620 00:40:20,875 --> 00:40:23,400 of these conservation laws. 621 00:40:23,400 --> 00:40:24,767 You might think to yourself-- 622 00:40:24,767 --> 00:40:26,850 and I realized as I was looking over these notes-- 623 00:40:26,850 --> 00:40:30,390 I'm about to I think give away a part of one of the problems 624 00:40:30,390 --> 00:40:32,510 on the PSET-- but c'est la vie. 625 00:40:32,510 --> 00:40:35,510 It's an important point. 626 00:40:35,510 --> 00:40:38,432 Can we do something similar for tensors? 627 00:40:38,432 --> 00:40:40,640 So this is great that you have this form for vectors. 628 00:40:43,220 --> 00:40:46,287 The divergence of a vector is a mathematical notion that 629 00:40:46,287 --> 00:40:47,495 comes up in various contexts. 630 00:40:47,495 --> 00:40:49,320 So this is important. 631 00:40:49,320 --> 00:40:51,410 But we've already talked about the fact 632 00:40:51,410 --> 00:40:53,150 that things like energy and momentum 633 00:40:53,150 --> 00:40:55,670 are described by a stress energy tensor. 634 00:40:55,670 --> 00:40:57,080 So can we do this for tensors? 635 00:41:05,190 --> 00:41:09,000 Well the answer turns out to be no, 636 00:41:09,000 --> 00:41:10,245 except in a handful of cases. 637 00:41:13,070 --> 00:41:17,260 And I have a comment about those handful of cases. 638 00:41:17,260 --> 00:41:19,510 So suppose I take this-- 639 00:41:19,510 --> 00:41:21,040 and I'm taking the divergence on say 640 00:41:21,040 --> 00:41:22,360 the first index of this guy-- 641 00:41:27,150 --> 00:41:31,540 so there's the bit involves my partial derivative-- 642 00:41:31,540 --> 00:41:36,450 I'm going to have a bit that involves 643 00:41:36,450 --> 00:41:38,640 correcting the first index. 644 00:41:51,600 --> 00:41:58,160 So the first correction is it's of a form that 645 00:41:58,160 --> 00:41:59,720 does in fact involve this guy we just 646 00:41:59,720 --> 00:42:00,963 worked out this identity for. 647 00:42:00,963 --> 00:42:02,630 And in principle we could take advantage 648 00:42:02,630 --> 00:42:05,480 of that to massage this and use this identity. 649 00:42:05,480 --> 00:42:08,508 But the second one there's nothing to do with that. 650 00:42:08,508 --> 00:42:10,050 This you just have to go and work out 651 00:42:10,050 --> 00:42:12,840 all of your 40 different Christoffel symbols 652 00:42:12,840 --> 00:42:14,310 and sit down and slog through it. 653 00:42:14,310 --> 00:42:15,935 This spoils your ability to do anything 654 00:42:15,935 --> 00:42:21,270 with it, with one exception. 655 00:42:21,270 --> 00:42:25,102 What if a is an anti-symmetric tensor? 656 00:42:25,102 --> 00:42:26,630 If a is an anti-symmetric tensor, 657 00:42:26,630 --> 00:42:29,870 you've got symmetry, anti symmetry, and it dies. 658 00:42:29,870 --> 00:42:32,420 So that is one example of where you can actually apply it. 659 00:42:32,420 --> 00:42:35,840 And I had you guys play with that a little bit on the PSET. 660 00:42:35,840 --> 00:42:38,780 It's worth noting though that the main reason why one often 661 00:42:38,780 --> 00:42:41,300 finds this to be a useful thing to do 662 00:42:41,300 --> 00:42:44,810 is that when you take the divergence of something 663 00:42:44,810 --> 00:42:47,570 like a vector, you get a scalar out. 664 00:42:47,570 --> 00:42:50,570 You get a quantity that is-- 665 00:42:50,570 --> 00:42:52,490 really its transformation properties 666 00:42:52,490 --> 00:42:55,400 between different inertial frames or freely-falling frames 667 00:42:55,400 --> 00:42:55,940 is simple. 668 00:43:06,230 --> 00:43:09,290 So even when you can do this and take advantage of this thing, 669 00:43:09,290 --> 00:43:13,028 working with the divergence of a tensor-- 670 00:43:13,028 --> 00:43:15,320 exploiting a trick like this turns out to generally not 671 00:43:15,320 --> 00:43:16,240 be all that useful. 672 00:43:16,240 --> 00:43:20,000 And I'll use the example of the stress energy tensor. 673 00:43:20,000 --> 00:43:24,390 So conservation of stress energy in special relativity-- 674 00:43:24,390 --> 00:43:27,802 it was the partial derivative-- 675 00:43:27,802 --> 00:43:29,260 the divergence of the stress energy 676 00:43:29,260 --> 00:43:31,093 tensor expressed with the partial derivative 677 00:43:31,093 --> 00:43:32,200 was equal to 0. 678 00:43:32,200 --> 00:43:34,555 We're going to take this over to covariant derivative 679 00:43:34,555 --> 00:43:36,430 of the stress energy tensor being equal to 0. 680 00:43:36,430 --> 00:43:37,810 That's what the equivalence principle tells us 681 00:43:37,810 --> 00:43:38,435 that we can do. 682 00:43:40,960 --> 00:43:43,600 Now when I take the divergence of something like the stress 683 00:43:43,600 --> 00:43:45,820 energy tensor, I get a 4 vector. 684 00:43:48,580 --> 00:43:51,130 Every 4 vector always has implicitly 685 00:43:51,130 --> 00:43:54,138 a set of basis objects attached to it. 686 00:43:54,138 --> 00:43:55,930 When I've got basis objects attached to it, 687 00:43:55,930 --> 00:43:59,008 those are defined with respect to the tangent space 688 00:43:59,008 --> 00:44:00,550 at a particular point in the manifold 689 00:44:00,550 --> 00:44:02,260 where you are currently working. 690 00:44:02,260 --> 00:44:05,680 And so if I want to try to do something like an integral like 691 00:44:05,680 --> 00:44:09,190 this-- where I add up the four vector I get by taking 692 00:44:09,190 --> 00:44:12,220 the divergence of stress energy and integrate it over 693 00:44:12,220 --> 00:44:13,330 a volume-- 694 00:44:13,330 --> 00:44:16,180 I'm going to get nonsense because what's going on 695 00:44:16,180 --> 00:44:18,880 is I'm combining vector fields that are defined 696 00:44:18,880 --> 00:44:22,060 in different tangent spaces that can't be properly compared 697 00:44:22,060 --> 00:44:23,710 to one another. 698 00:44:23,710 --> 00:44:25,400 In order to do that kind of comparison, 699 00:44:25,400 --> 00:44:28,520 you have to introduce a transport law. 700 00:44:28,520 --> 00:44:30,160 And when you start doing transports 701 00:44:30,160 --> 00:44:34,060 over macroscopic regions, you run into trouble. 702 00:44:34,060 --> 00:44:36,370 They turn out to be path dependent. 703 00:44:36,370 --> 00:44:38,430 And this is where we run into ambiguities 704 00:44:38,430 --> 00:44:40,180 that have to do with the curvature content 705 00:44:40,180 --> 00:44:41,560 of your manifold. 706 00:44:41,560 --> 00:44:44,440 We'll discuss where that comes into our calculations 707 00:44:44,440 --> 00:44:45,610 a little bit later. 708 00:44:45,610 --> 00:44:49,270 But what it basically boils down to is 709 00:44:49,270 --> 00:44:51,850 if I use a stress energy tensor as an example, 710 00:44:51,850 --> 00:44:56,050 this equation tells me about local conservation 711 00:44:56,050 --> 00:44:59,520 of energy and momentum. 712 00:44:59,520 --> 00:45:03,720 In general relativity I cannot take the local conservation 713 00:45:03,720 --> 00:45:06,120 of energy and momentum and promote it to a global 714 00:45:06,120 --> 00:45:08,130 conservation of energy and momentum. 715 00:45:08,130 --> 00:45:10,510 It's ambiguous. 716 00:45:10,510 --> 00:45:15,910 We'll deal with that and the conceptual difficulties 717 00:45:15,910 --> 00:45:18,100 that that presents a little bit later in the course. 718 00:45:18,100 --> 00:45:21,850 But it's a good see the plant at this point. 719 00:45:21,850 --> 00:45:23,350 So let's switch gears. 720 00:45:23,350 --> 00:45:25,570 We have a new set of mathematical tools. 721 00:45:30,340 --> 00:45:35,860 I want to take a detour away from thinking about some more 722 00:45:35,860 --> 00:45:39,070 abstract mathematical notions and start thinking about how 723 00:45:39,070 --> 00:45:41,300 we actually do some physics. 724 00:45:41,300 --> 00:45:44,200 So what I want to do is talk today about how 725 00:45:44,200 --> 00:45:47,980 do we formulate the kinematics of a body 726 00:45:47,980 --> 00:45:49,698 moving in curved spacetime? 727 00:46:08,427 --> 00:46:10,010 So I've already hinted at this in some 728 00:46:10,010 --> 00:46:11,000 of my previous lectures. 729 00:46:11,000 --> 00:46:12,375 And what I want to do now is just 730 00:46:12,375 --> 00:46:14,990 basically fill in some of the gaps. 731 00:46:14,990 --> 00:46:18,470 The way that we do this really just 732 00:46:18,470 --> 00:46:21,940 builds on Einstein's insight about what 733 00:46:21,940 --> 00:46:24,500 the weak equivalence principle means. 734 00:46:24,500 --> 00:46:27,320 So go into a freely falling frame. 735 00:46:34,910 --> 00:46:36,870 Go in that freely-falling frame. 736 00:46:36,870 --> 00:46:42,160 Put things into locally Lorentz coordinates. 737 00:46:42,160 --> 00:46:44,538 In other words perform that little calculation 738 00:46:44,538 --> 00:46:46,330 that make spacetime look like the spacetime 739 00:46:46,330 --> 00:46:49,219 of special relativity of the curvature corrections. 740 00:46:59,210 --> 00:47:01,490 And to start with, let's consider what we always 741 00:47:01,490 --> 00:47:04,440 do in physics, is we'll look at the simplest body first. 742 00:47:04,440 --> 00:47:06,440 We're going to look at what we call a test body. 743 00:47:15,410 --> 00:47:22,150 So this is the body that has no charge, no spatial extent, 744 00:47:22,150 --> 00:47:29,920 it's of zero dimensional point, no spin-- 745 00:47:29,920 --> 00:47:33,580 nothing interesting, except a mass. 746 00:47:42,350 --> 00:47:44,360 So if you want to think about this-- 747 00:47:44,360 --> 00:47:46,430 I use a way that I find to think about this 748 00:47:46,430 --> 00:47:51,200 is all these various aspects to it, you're adding additional-- 749 00:47:51,200 --> 00:47:54,800 either charges to it or additional multipolar structure 750 00:47:54,800 --> 00:47:56,090 to this body. 751 00:47:56,090 --> 00:47:59,090 I'm thinking of this-- this is sort of like a pure monopole. 752 00:47:59,090 --> 00:48:00,920 It's nothing but mass concentrated 753 00:48:00,920 --> 00:48:03,680 in a single zero size point. 754 00:48:03,680 --> 00:48:05,299 Obviously it's an idealization. 755 00:48:05,299 --> 00:48:06,716 But you've got to start somewhere. 756 00:48:14,810 --> 00:48:18,920 So since it's got no charge, no spatial extent, 757 00:48:18,920 --> 00:48:22,803 it's got nothing but mass, nothing's 758 00:48:22,803 --> 00:48:23,720 going to couple to it. 759 00:48:23,720 --> 00:48:26,685 It's not going to basically do anything but freefall. 760 00:48:37,040 --> 00:48:49,906 In this frame the body moves on a purely inertial trajectory. 761 00:49:04,262 --> 00:49:06,470 And what does a purely inertial trajectory look like? 762 00:49:06,470 --> 00:49:11,360 Well you take whatever your initial conditions are. 763 00:49:15,330 --> 00:49:17,490 And you move in a straight line with respect 764 00:49:17,490 --> 00:49:20,560 to time as measured on your own clock. 765 00:49:20,560 --> 00:49:23,670 Simplest, stupidest possible motion that you can. 766 00:49:37,320 --> 00:49:39,890 So we would obviously call that a straight line with respect 767 00:49:39,890 --> 00:49:41,390 to the parameterization that's being 768 00:49:41,390 --> 00:49:43,740 used in this representation. 769 00:49:43,740 --> 00:49:46,640 So what does that mean in a more general sense 770 00:49:46,640 --> 00:49:47,730 of the representation? 771 00:49:47,730 --> 00:49:50,300 So if we think about this a little bit more geometrically, 772 00:49:50,300 --> 00:49:53,780 when a body is moving in a straight line, that basically 773 00:49:53,780 --> 00:49:58,250 means that whatever the tangent vector to its world line is, 774 00:49:58,250 --> 00:50:02,540 it's essentially moving such that the tangent vector at time 775 00:50:02,540 --> 00:50:08,550 T1 is parallel to the tangent vector at T1 plus delta T1, 776 00:50:08,550 --> 00:50:10,160 provided that's actually small enough 777 00:50:10,160 --> 00:50:13,010 that they're sort of within the same local Lorentz frame. 778 00:50:16,950 --> 00:50:20,120 So a more geometric way of thinking about this motion 779 00:50:20,120 --> 00:50:22,110 is that it's parallel transporting its tangent 780 00:50:22,110 --> 00:50:22,610 vector. 781 00:51:13,858 --> 00:51:15,650 Let's make this a little bit more rigorous. 782 00:51:15,650 --> 00:51:17,150 So let's imagine this body's moving 783 00:51:17,150 --> 00:51:19,983 on a particular trajectory through spacetime. 784 00:51:25,780 --> 00:51:28,840 So it's a trajectory parameterized. 785 00:51:28,840 --> 00:51:33,250 I will define its parameterization a little bit 786 00:51:33,250 --> 00:51:34,440 more carefully very soon. 787 00:51:38,660 --> 00:51:42,090 So for now, just think of lambda as some kind of a quantity. 788 00:51:42,090 --> 00:51:44,420 It's a scale that just accumulates uniformly as it 789 00:51:44,420 --> 00:51:48,000 moves along the world line. 790 00:51:48,000 --> 00:51:50,620 So I'm going to say the small body has 791 00:51:50,620 --> 00:51:55,900 a path through spacetime, given by u x of lambda. 792 00:51:55,900 --> 00:52:08,190 Its tangent is given by this. 793 00:52:08,190 --> 00:52:10,590 And if it is parallel transporting its own tangent 794 00:52:10,590 --> 00:52:24,753 vector, that is-- 795 00:52:24,753 --> 00:52:26,170 I'll remind you that the condition 796 00:52:26,170 --> 00:52:31,950 for parallel transport was that you 797 00:52:31,950 --> 00:52:34,080 take the covariant derivative your field. 798 00:52:34,080 --> 00:52:38,670 And as you are moving along, you contract it 799 00:52:38,670 --> 00:52:41,370 along the tangent vector of the trajectory you're moving on. 800 00:52:41,370 --> 00:52:43,260 And you get 0. 801 00:52:43,260 --> 00:52:48,600 So in my notes, there's a couple of equivalent ways 802 00:52:48,600 --> 00:52:49,280 of writing this. 803 00:52:49,280 --> 00:52:51,000 So you will sometimes see this written 804 00:52:51,000 --> 00:52:55,320 as the gradient along u of u. 805 00:52:59,090 --> 00:53:06,970 And you'll sometimes see this written as capital u 806 00:53:06,970 --> 00:53:08,108 u lambda equals 0. 807 00:53:08,108 --> 00:53:08,900 So these are just-- 808 00:53:08,900 --> 00:53:09,790 I just throw that out because these 809 00:53:09,790 --> 00:53:11,915 are different forms that are common in the notation 810 00:53:11,915 --> 00:53:12,842 that you will see. 811 00:53:12,842 --> 00:53:14,050 So let's expand this guy out. 812 00:53:39,018 --> 00:53:40,060 It's something like this. 813 00:53:40,060 --> 00:53:43,870 So what we're going to do-- so remember this is dx d lambda. 814 00:53:43,870 --> 00:53:46,438 This is d by dx. 815 00:53:46,438 --> 00:53:48,730 That's a total derivative with respect to the parameter 816 00:53:48,730 --> 00:53:49,230 lambda. 817 00:53:51,712 --> 00:53:52,420 So this becomes-- 818 00:54:05,500 --> 00:54:07,590 I'm going to write it in two forms. 819 00:54:07,590 --> 00:54:11,070 This is often written expanding out the u into a second order 820 00:54:11,070 --> 00:54:11,570 form. 821 00:54:18,890 --> 00:54:21,230 This is obvious but sufficiently important. 822 00:54:21,230 --> 00:54:26,900 It's worth calling it out. 823 00:54:26,900 --> 00:54:30,410 And this has earned itself a box. 824 00:54:39,570 --> 00:54:42,930 This result is known as the geodesic equation. 825 00:55:03,020 --> 00:55:05,040 The trajectories which solve these equations 826 00:55:05,040 --> 00:55:06,520 are known as geodesics. 827 00:55:22,083 --> 00:55:25,580 One of the reasons why I highlight this is it's-- 828 00:55:29,790 --> 00:55:32,610 I'm trying to keep a straight face with the comment I 829 00:55:32,610 --> 00:55:33,600 want to make. 830 00:55:33,600 --> 00:55:37,350 A tremendous amount of research in general relativity 831 00:55:37,350 --> 00:55:41,670 is based around doing solutions of this equation 832 00:55:41,670 --> 00:55:45,060 for various spacetimes that go in to make the Christoffel 833 00:55:45,060 --> 00:55:47,370 symbols. 834 00:55:47,370 --> 00:55:54,240 My career-- [LAUGHS] it's probably not false to say that 835 00:55:54,240 --> 00:56:00,720 about 65% of my papers have this equation at its centerpiece 836 00:56:00,720 --> 00:56:03,540 at some point with the thing that goes into making 837 00:56:03,540 --> 00:56:04,860 my gammas-- 838 00:56:04,860 --> 00:56:08,190 things related to black hole spacetimes. 839 00:56:08,190 --> 00:56:12,450 This is really important because this gives me the motion 840 00:56:12,450 --> 00:56:14,260 of a freely-falling frame. 841 00:56:14,260 --> 00:56:16,380 What does a freely-falling frame describe? 842 00:56:16,380 --> 00:56:18,610 Somebody who's moving under gravity. 843 00:56:18,610 --> 00:56:21,420 So when you're doing things like describing orbits, for example, 844 00:56:21,420 --> 00:56:23,700 this is your tool. 845 00:56:23,700 --> 00:56:25,350 A tremendous number of applications 846 00:56:25,350 --> 00:56:27,840 where if what you care about is the motion 847 00:56:27,840 --> 00:56:30,450 of a body due to relativistic gravity, 848 00:56:30,450 --> 00:56:32,580 this gives you a leading solution. 849 00:56:32,580 --> 00:56:35,190 Now bear in mind when I did this, 850 00:56:35,190 --> 00:56:36,900 this is the motion of a test body. 851 00:56:36,900 --> 00:56:38,880 This is an object with no charge, 852 00:56:38,880 --> 00:56:41,730 no spatial extent, no spin-- 853 00:56:41,730 --> 00:56:45,570 that describes no object. 854 00:56:45,570 --> 00:56:49,080 So it should be borne in mind that this is 855 00:56:49,080 --> 00:56:52,110 the leading solution to things. 856 00:56:52,110 --> 00:56:54,170 Suppose the body is charged. 857 00:56:54,170 --> 00:56:56,130 And there is an electromagnetic field 858 00:56:56,130 --> 00:56:58,410 that this body is interacting with. 859 00:56:58,410 --> 00:57:01,170 Then what you do is you are no longer going 860 00:57:01,170 --> 00:57:03,900 to be parallel transporting this tangent factor. 861 00:57:03,900 --> 00:57:05,733 It will be pushed away-- we like to say-- 862 00:57:05,733 --> 00:57:06,900 from the parallel transport. 863 00:57:06,900 --> 00:57:09,180 And you'll replace the 0 on the right hand side 864 00:57:09,180 --> 00:57:11,550 here with a properly-constructed force 865 00:57:11,550 --> 00:57:14,730 that describes the interactions of those charges 866 00:57:14,730 --> 00:57:17,280 with the fields. 867 00:57:17,280 --> 00:57:19,770 Suppose the body has some size. 868 00:57:19,770 --> 00:57:22,650 Well then what ends up happening is that the body actually 869 00:57:22,650 --> 00:57:25,020 doesn't just couple to a single-- remember what's 870 00:57:25,020 --> 00:57:29,190 going on here is that in the freely falling frame, 871 00:57:29,190 --> 00:57:32,490 I'm imagining that spacetime is flat at some point. 872 00:57:32,490 --> 00:57:35,183 And in a decent enough vicinity of that point, 873 00:57:35,183 --> 00:57:36,600 the first order corrections are 0. 874 00:57:36,600 --> 00:57:38,610 But there might be second order corrections. 875 00:57:38,610 --> 00:57:41,040 Well imagine a body is so big that it fills 876 00:57:41,040 --> 00:57:42,800 that freely-falling frame. 877 00:57:42,800 --> 00:57:46,368 And it actually tastes those second order corrections. 878 00:57:46,368 --> 00:57:47,910 Then what's going to happen is you're 879 00:57:47,910 --> 00:57:50,320 going to get additional terms on this equation, which 880 00:57:50,320 --> 00:57:53,200 have to do with the coupling of the spatial extent of that body 881 00:57:53,200 --> 00:57:55,780 to the curvature of the spacetime. 882 00:57:55,780 --> 00:58:00,580 That is where-- so for people who study astrophysical systems 883 00:58:00,580 --> 00:58:03,970 involving binaries, when you have spinning bodies, 884 00:58:03,970 --> 00:58:06,400 that ends up actually-- you cannot describe a body 885 00:58:06,400 --> 00:58:08,950 that's spinning without it having some spatial extent. 886 00:58:08,950 --> 00:58:11,830 And you find terms here that involve coupling of those 887 00:58:11,830 --> 00:58:14,720 spins to the curvature of the spacetime. 888 00:58:14,720 --> 00:58:18,910 So this is the leading piece of the motion of a body moving 889 00:58:18,910 --> 00:58:20,680 in the current spacetime. 890 00:58:20,680 --> 00:58:22,460 And it's enough to do a tremendous amount. 891 00:58:22,460 --> 00:58:27,470 Basically because gravity is just so bloody strong that all 892 00:58:27,470 --> 00:58:28,792 of these various things-- 893 00:58:28,792 --> 00:58:30,250 it's the weakest fundamental force. 894 00:58:30,250 --> 00:58:33,340 But it adds up because it's only got one sine. 895 00:58:33,340 --> 00:58:36,190 And when you're dealing with some of these things, 896 00:58:36,190 --> 00:58:38,770 it really ends up being the coupling to the monopole-- 897 00:58:38,770 --> 00:58:40,540 the most important thing. 898 00:58:40,540 --> 00:58:43,210 So all these other terms that come in and correct this 899 00:58:43,210 --> 00:58:45,430 are small enough that we can add them in. 900 00:58:45,430 --> 00:58:48,610 And that, to be blunt, is modern research. 901 00:58:48,610 --> 00:58:51,370 So let me make a couple of comments about this. 902 00:58:54,750 --> 00:58:56,145 A more general form-- 903 00:58:59,300 --> 00:59:02,930 this will help to clarify what the meaning of that lambda 904 00:59:02,930 --> 00:59:03,650 actually is. 905 00:59:14,940 --> 00:59:21,700 Suppose that as my vector is transported along itself-- 906 00:59:21,700 --> 00:59:24,940 so one way is recall how we derive parallel transport. 907 00:59:24,940 --> 00:59:28,810 We imagine going into a freely-falling frame 908 00:59:28,810 --> 00:59:30,580 and a Lorentz representation. 909 00:59:30,580 --> 00:59:32,782 And we said, in that frame, I'm going 910 00:59:32,782 --> 00:59:34,240 to imagine moving this thing along, 911 00:59:34,240 --> 00:59:37,240 holding all the components constants-- that 912 00:59:37,240 --> 00:59:39,430 defined parallel transport. 913 00:59:39,430 --> 00:59:43,900 Imagine that I don't keep the components constant, 914 00:59:43,900 --> 00:59:46,270 but I hold them all in a constant ratio with respect 915 00:59:46,270 --> 00:59:49,090 to each other, but I allow the overall magnitude 916 00:59:49,090 --> 00:59:50,380 to expand or contract. 917 00:59:56,050 --> 01:00:03,420 So suppose we allow the vector's normalization 918 01:00:03,420 --> 01:00:05,220 to change as it slides along. 919 01:00:18,740 --> 01:00:20,330 Well the way I would mathematically 920 01:00:20,330 --> 01:00:24,860 formulate this is I'm going to use a notation that 921 01:00:24,860 --> 01:00:25,540 looks like this. 922 01:00:25,540 --> 01:00:27,920 So recall this capital D-- it's a shorthand 923 01:00:27,920 --> 01:00:38,630 for this combination of the tangent 924 01:00:38,630 --> 01:00:41,997 and the covariant derivative. 925 01:00:41,997 --> 01:00:43,580 I'm going to call the parameterization 926 01:00:43,580 --> 01:00:45,650 I use when I set up like this lambda star, 927 01:00:45,650 --> 01:00:47,775 for reasons that I hope will be clear in just about 928 01:00:47,775 --> 01:00:49,100 two minutes. 929 01:00:49,100 --> 01:00:52,070 So what I'm basically saying is that as I move along, 930 01:00:52,070 --> 01:00:54,098 I don't keep the components constant. 931 01:00:54,098 --> 01:00:55,640 But I keep them proportional to where 932 01:00:55,640 --> 01:00:57,238 they were on the previous step. 933 01:00:57,238 --> 01:00:59,780 But I allow their magnitude to change by some function, which 934 01:00:59,780 --> 01:01:00,530 I'll call a kappa. 935 01:01:08,573 --> 01:01:10,490 So you might look at that and think, you know, 936 01:01:10,490 --> 01:01:13,220 that's a more general kind of transport law. 937 01:01:13,220 --> 01:01:16,970 It seems to describe physically a very similar situation here. 938 01:01:16,970 --> 01:01:20,000 It's kind of annoying that this normalization is changing. 939 01:01:20,000 --> 01:01:22,940 Is there anything going on with this? 940 01:01:22,940 --> 01:01:29,930 Well what you guys are going to do as a homework exercise, 941 01:01:29,930 --> 01:01:32,525 you're going to prove that if this 942 01:01:32,525 --> 01:01:34,310 is the situation you're in, you've 943 01:01:34,310 --> 01:01:36,140 chosen a dumb parameterization. 944 01:01:36,140 --> 01:01:38,780 And you can actually convert this 945 01:01:38,780 --> 01:01:41,060 to the normal geodesic parameterization 946 01:01:41,060 --> 01:01:43,520 by just relabeling your lambda. 947 01:01:51,310 --> 01:01:59,910 So we can always reparameterize this, such 948 01:01:59,910 --> 01:02:03,480 that the right-hand side is 0. 949 01:02:03,480 --> 01:02:05,460 And right-hand side being 0 corresponds 950 01:02:05,460 --> 01:02:08,010 to the transport vector remaining constant 951 01:02:08,010 --> 01:02:09,670 as it moves along. 952 01:02:09,670 --> 01:02:11,070 So I'll just quickly sketch-- 953 01:02:11,070 --> 01:02:18,050 so imagine there exists some different parameterization, 954 01:02:18,050 --> 01:02:19,790 which I will call lambda. 955 01:02:29,030 --> 01:02:32,480 So imagine something that gives me my normal parallel transport 956 01:02:32,480 --> 01:02:35,180 exists. 957 01:02:35,180 --> 01:02:40,970 And I have a different one that involves the star parameter. 958 01:02:40,970 --> 01:02:43,700 You can actually show that these two things describe exactly 959 01:02:43,700 --> 01:02:50,630 the same motion, but with lambda and the dumb parameterization, 960 01:02:50,630 --> 01:02:56,090 lambda star, related to each other by a particular integral. 961 01:03:04,680 --> 01:03:07,820 So what this shows us is we can always-- 962 01:03:07,820 --> 01:03:10,400 as long as I'm talking about motion where I'm in this 963 01:03:10,400 --> 01:03:12,260 regime-- where there's no forces acting-- 964 01:03:12,260 --> 01:03:14,510 it's not an extended body-- it's just a test body-- 965 01:03:14,510 --> 01:03:16,557 I can always put it into a regime 966 01:03:16,557 --> 01:03:18,890 where it'll [INAUDIBLE] geodesic and the right-hand side 967 01:03:18,890 --> 01:03:20,030 is equal to 0. 968 01:03:20,030 --> 01:03:21,860 If I'm finding that's not the case, 969 01:03:21,860 --> 01:03:25,820 I need to adjust my parameterization. 970 01:03:25,820 --> 01:03:28,550 When you are, in fact, in a prioritization 971 01:03:28,550 --> 01:03:33,650 such as the right-hand side is 0, 972 01:03:33,650 --> 01:03:36,822 you are using what is called an affine parameterization. 973 01:03:36,822 --> 01:03:38,530 That's a name that's worth knowing about. 974 01:03:57,550 --> 01:04:01,910 So your intuition is that the affine parameterization-- 975 01:04:01,910 --> 01:04:03,410 I described this in words last time. 976 01:04:03,410 --> 01:04:06,035 And this just helps to make it a little bit more mathematically 977 01:04:06,035 --> 01:04:07,440 precise what those words mean. 978 01:04:17,210 --> 01:04:25,340 Affine parameters correspond to the tick marks 979 01:04:25,340 --> 01:04:38,065 on the world line, being uniformly spaced 980 01:04:38,065 --> 01:04:39,190 in the local Lorentz frame. 981 01:04:45,550 --> 01:04:47,670 If you are working with time-like trajectories-- 982 01:04:47,670 --> 01:04:51,420 which if you're a physicist, you will be much of the time-- 983 01:04:51,420 --> 01:04:54,030 a really good choice of the affine parameter 984 01:04:54,030 --> 01:04:58,020 is the proper time of a body moving through the spacetime. 985 01:04:58,020 --> 01:05:01,050 That is something that is uniformly spaced, 986 01:05:01,050 --> 01:05:02,010 assuming that's-- 987 01:05:02,010 --> 01:05:03,030 you don't have to assume anything. 988 01:05:03,030 --> 01:05:04,405 Just by definition it's the thing 989 01:05:04,405 --> 01:05:07,080 that uniformly measures the time as experienced 990 01:05:07,080 --> 01:05:07,995 by that observer. 991 01:05:51,130 --> 01:05:54,130 So this is-- you guys are going to do on PSET 4-- 992 01:05:54,130 --> 01:05:56,140 this is the exercise you need to do to convert 993 01:05:56,140 --> 01:05:58,360 a nonaffine parameterized geodesic 994 01:05:58,360 --> 01:06:01,270 to an affine parameterized one. 995 01:06:01,270 --> 01:06:03,100 That kind of parameterization is not 996 01:06:03,100 --> 01:06:09,960 too hard to show that if we adjust the parameterization 997 01:06:09,960 --> 01:06:10,920 in a linear fashion-- 998 01:06:20,580 --> 01:06:25,160 so in other words, let's say I go from lambda to some lambda 999 01:06:25,160 --> 01:06:29,220 prime, which is equal to a lambda plus b, 1000 01:06:29,220 --> 01:06:31,700 where and b are both constants-- 1001 01:06:37,450 --> 01:06:40,195 we get a new affine parameterization. 1002 01:06:43,450 --> 01:06:45,510 But that's the only class of reparamterizations 1003 01:06:45,510 --> 01:06:46,593 that allows me to do that. 1004 01:06:49,288 --> 01:06:50,580 And hopefully that makes sense. 1005 01:06:50,580 --> 01:06:53,100 If you imagine that you're using proper time 1006 01:06:53,100 --> 01:06:55,440 as your reparameterization, this is basically 1007 01:06:55,440 --> 01:06:57,690 saying that you just chose a different origin for when 1008 01:06:57,690 --> 01:06:58,648 you started your clock. 1009 01:06:58,648 --> 01:07:00,690 And this means you changed the units in which you 1010 01:07:00,690 --> 01:07:01,560 are measuring time. 1011 01:07:01,560 --> 01:07:02,060 That's all. 1012 01:07:08,210 --> 01:07:11,780 So I'm going to skip a bunch of the details. 1013 01:07:11,780 --> 01:07:13,940 But I'm going to scan and put up the notes 1014 01:07:13,940 --> 01:07:16,580 corresponding to one other route to getting 1015 01:07:16,580 --> 01:07:20,227 to the geodesic equation, which I think it's definitely worth 1016 01:07:20,227 --> 01:07:20,810 knowing about. 1017 01:07:28,880 --> 01:07:36,328 It connects very nicely to other work in classical mechanics. 1018 01:07:36,328 --> 01:07:38,870 So it's a bit of a shame we're going to need to skip over it. 1019 01:07:38,870 --> 01:07:41,030 But we're a little bit behind pace. 1020 01:07:41,030 --> 01:07:42,590 And this is straightforward enough 1021 01:07:42,590 --> 01:07:45,950 that I feel OK posting the notes that you can read it. 1022 01:07:45,950 --> 01:07:50,910 So there is a second path to geodesics. 1023 01:07:56,040 --> 01:07:59,380 So recall the way that we argued how to get the geodesic 1024 01:07:59,380 --> 01:08:02,205 equation, which we said we're going to go 1025 01:08:02,205 --> 01:08:04,330 into-- it's actually in the board right above where 1026 01:08:04,330 --> 01:08:05,410 I'm writing right now-- 1027 01:08:05,410 --> 01:08:07,120 go into the freely-falling frame. 1028 01:08:07,120 --> 01:08:09,070 I have a body that isn't coupling to anything 1029 01:08:09,070 --> 01:08:10,270 but gravity. 1030 01:08:10,270 --> 01:08:11,870 Therefore in the freely-falling frame, 1031 01:08:11,870 --> 01:08:13,270 it just maintains its momentum. 1032 01:08:13,270 --> 01:08:15,190 It's going to go in a straight line. 1033 01:08:15,190 --> 01:08:17,590 Straight means parallel transporting tangent vector-- 1034 01:08:17,590 --> 01:08:20,830 math, math, math-- and that's how we get all that. 1035 01:08:20,830 --> 01:08:22,569 So what this boiled down to is I was 1036 01:08:22,569 --> 01:08:26,109 trying to make rigorous in a geometric sense what straight 1037 01:08:26,109 --> 01:08:28,620 meant. 1038 01:08:28,620 --> 01:08:30,222 There's another notion of straight 1039 01:08:30,222 --> 01:08:31,930 that one can imagine applying when you're 1040 01:08:31,930 --> 01:08:33,013 working in a curved space. 1041 01:08:35,850 --> 01:08:37,740 So your intuition for-- 1042 01:08:37,740 --> 01:08:40,529 if you're talking about how do I make a straight line 1043 01:08:40,529 --> 01:08:42,960 between two points on a globe-- 1044 01:08:42,960 --> 01:08:45,689 your intuition is you say, oh, well the straightest line 1045 01:08:45,689 --> 01:08:48,330 that I can make is the path that is shortest. 1046 01:09:19,290 --> 01:09:21,460 We're going to formulate-- and I'll 1047 01:09:21,460 --> 01:09:23,165 leave the details and the calculation 1048 01:09:23,165 --> 01:09:24,790 to the notes-- we're going to formulate 1049 01:09:24,790 --> 01:09:26,920 how one can apply a similar thing 1050 01:09:26,920 --> 01:09:28,600 to the notion of geodesics. 1051 01:09:28,600 --> 01:09:36,760 So imagine I've got an event p here and event q up here. 1052 01:09:36,760 --> 01:09:43,750 And I ask myself, what is the accumulated proper time 1053 01:09:43,750 --> 01:09:48,939 experienced by all possible paths that take me 1054 01:09:48,939 --> 01:09:50,368 from event p to event q? 1055 01:09:50,368 --> 01:09:51,910 I'm going to need to restrict myself. 1056 01:09:51,910 --> 01:09:54,285 I want it to be something that an observer can physically 1057 01:09:54,285 --> 01:09:56,730 ride-- so all the time-like trajectories that 1058 01:09:56,730 --> 01:09:59,050 connect event p to event q. 1059 01:09:59,050 --> 01:10:01,660 So I've got one a path that goes like this, 1060 01:10:01,660 --> 01:10:04,735 got a path that goes like this, path that goes like this, path 1061 01:10:04,735 --> 01:10:06,610 goes like this, path goes like-- some of them 1062 01:10:06,610 --> 01:10:08,270 might have just become somewhat space like, 1063 01:10:08,270 --> 01:10:09,460 so I should rule them out. 1064 01:10:09,460 --> 01:10:10,380 But you get the idea. 1065 01:10:10,380 --> 01:10:12,880 Imagine I take all the possible time-like paths 1066 01:10:12,880 --> 01:10:14,230 that connect p and q. 1067 01:10:18,010 --> 01:10:22,600 Some of those paths will involve strong accelerations. 1068 01:10:22,600 --> 01:10:25,690 So they will not be the freefall path. 1069 01:10:25,690 --> 01:10:28,260 Among them there will be one that corresponds exactly 1070 01:10:28,260 --> 01:10:28,760 to freefall. 1071 01:10:43,290 --> 01:10:45,540 So if I were talking about-- imagine I was trying to-- 1072 01:10:45,540 --> 01:10:48,560 and this is something that Muslim astronomers worked out 1073 01:10:48,560 --> 01:10:50,810 long, long ago-- they wanted to know the shortest path 1074 01:10:50,810 --> 01:10:53,660 from some point on earth towards Mecca. 1075 01:10:53,660 --> 01:10:56,680 And so you need to find what the shortest distance was 1076 01:10:56,680 --> 01:10:57,680 for something like that. 1077 01:10:57,680 --> 01:11:00,290 And when you're doing this on the surface of a sphere, 1078 01:11:00,290 --> 01:11:02,137 that's complicated. 1079 01:11:02,137 --> 01:11:03,720 And that's where the qibla arose from, 1080 01:11:03,720 --> 01:11:06,210 was working out the mathematics to know how to do this. 1081 01:11:06,210 --> 01:11:07,730 This is a similar kind of concept. 1082 01:11:07,730 --> 01:11:09,860 I'm trying to define-- 1083 01:11:09,860 --> 01:11:13,740 in this case, it's going to turn out it's not the shortest path, 1084 01:11:13,740 --> 01:11:17,600 but it's the path on which an observer ages the most because 1085 01:11:17,600 --> 01:11:19,400 as soon as you accelerate someone-- 1086 01:11:19,400 --> 01:11:20,000 it's not hard. 1087 01:11:20,000 --> 01:11:21,542 Go back to some of those problem sets 1088 01:11:21,542 --> 01:11:23,840 you guys did where you look at accelerated observers. 1089 01:11:23,840 --> 01:11:26,720 Acceleration tends to decrease the amount of aging you have 1090 01:11:26,720 --> 01:11:29,370 as you move through some interval of spacetime. 1091 01:11:29,370 --> 01:11:33,920 So the path that has no acceleration on it, 1092 01:11:33,920 --> 01:11:36,480 this is going to be the one on which an observer is maximally 1093 01:11:36,480 --> 01:11:36,980 aged. 1094 01:11:47,065 --> 01:11:48,440 Why maximum instead of a minimum? 1095 01:11:48,440 --> 01:11:50,810 Well it comes down to the bloody minus sign 1096 01:11:50,810 --> 01:11:53,780 that enters into the timepiece of an interval 1097 01:11:53,780 --> 01:11:55,052 that we have in relativity. 1098 01:11:55,052 --> 01:11:56,510 And that's all I'll say about that, 1099 01:11:56,510 --> 01:11:58,970 is just boils down to that. 1100 01:11:58,970 --> 01:12:03,350 So what we want to do is say, well along all 1101 01:12:03,350 --> 01:12:06,380 of these trajectories, the amount 1102 01:12:06,380 --> 01:12:09,648 of proper time that's accumulated-- so let's 1103 01:12:09,648 --> 01:12:11,690 just say that every one of these is parameterized 1104 01:12:11,690 --> 01:12:15,440 by some lambda that describes the motion along these things. 1105 01:12:34,480 --> 01:12:35,890 This is the amount of proper time 1106 01:12:35,890 --> 01:12:39,400 that someone accumulates as they move from point p-- 1107 01:12:39,400 --> 01:12:42,688 which is at-- let's say this is defined as lambda equals 0-- 1108 01:12:42,688 --> 01:12:44,980 and it's indeterminate what that top lambda is actually 1109 01:12:44,980 --> 01:12:45,480 going to be. 1110 01:12:45,480 --> 01:12:47,860 It's whatever it takes when you get up to lambda of q. 1111 01:12:53,820 --> 01:12:58,020 So what the notes I'm going to post do, 1112 01:12:58,020 --> 01:13:01,530 is they define an action principle that 1113 01:13:01,530 --> 01:13:05,445 can be applied to understand what the trajectory is 1114 01:13:05,445 --> 01:13:06,570 that allows you to do this. 1115 01:13:23,790 --> 01:13:25,760 So I'll just hit the highlights. 1116 01:13:25,760 --> 01:13:37,060 So in notes to be posted, I show that this delta t-- 1117 01:13:37,060 --> 01:13:38,150 this delta tau rather-- 1118 01:13:43,410 --> 01:13:45,580 this can be used to define an action. 1119 01:14:11,600 --> 01:14:12,410 It looks like this. 1120 01:14:17,370 --> 01:14:19,820 And then if you vary the action-- 1121 01:14:26,970 --> 01:14:29,760 or rather you do a variation of your trajectory-- 1122 01:14:29,760 --> 01:14:33,630 where you require that the action remain stationary under 1123 01:14:33,630 --> 01:14:42,610 that variation-- in other words I require delta i equals 0 as x 1124 01:14:42,610 --> 01:14:43,420 goes over to such-- 1125 01:14:57,820 --> 01:15:08,840 so-- what you wind up with-- 1126 01:15:20,870 --> 01:15:25,715 is delta i equals-- 1127 01:16:10,040 --> 01:16:14,340 Notice what I've got in here. 1128 01:16:14,340 --> 01:16:15,790 This is just a Christoffel symbol. 1129 01:16:21,290 --> 01:16:25,715 So when I do this variation, what I find-- 1130 01:16:25,715 --> 01:16:28,950 and by the way going from essentially that board 1131 01:16:28,950 --> 01:16:33,080 to that board, it's about 2/3 a page of algebra. 1132 01:16:33,080 --> 01:16:34,850 Going down to this one, there's a bunch 1133 01:16:34,850 --> 01:16:36,683 of straightforward but fairly tedious stuff. 1134 01:16:36,683 --> 01:16:38,985 It's one reasons why I'm skipping over the details. 1135 01:16:38,985 --> 01:16:40,610 We've got enough G mu nus on the board. 1136 01:17:00,720 --> 01:17:08,250 So the key point is I am going to require that my action be 1137 01:17:08,250 --> 01:17:11,850 stationary, independent of the nature of the variation 1138 01:17:11,850 --> 01:17:13,150 that I make. 1139 01:17:13,150 --> 01:17:16,110 For that to be true, the quantity in braces 1140 01:17:16,110 --> 01:17:20,057 here must be equal to 0. 1141 01:17:20,057 --> 01:17:21,640 Let me just write that down over here. 1142 01:17:21,640 --> 01:17:23,682 This is a good place to conclude today's lecture. 1143 01:17:42,630 --> 01:17:45,390 So we require this to be 0 for any variation. 1144 01:17:53,040 --> 01:18:00,020 Yet the bracketed term being equal to 0, pull that 1145 01:18:00,020 --> 01:18:07,390 out, and clear out that factor of the metric with an inverse, 1146 01:18:07,390 --> 01:18:09,860 you've got your geodesic equation back. 1147 01:18:09,860 --> 01:18:11,590 So we just quickly wrap this up. 1148 01:18:11,590 --> 01:18:15,080 So it's worth looking over these notes. 1149 01:18:15,080 --> 01:18:17,080 It's not worth going through them in gory detail 1150 01:18:17,080 --> 01:18:18,705 on the board, which is why I'm skipping 1151 01:18:18,705 --> 01:18:20,020 a few pages of these things. 1152 01:18:20,020 --> 01:18:23,500 But what this demonstrates is that geodesics-- 1153 01:18:28,070 --> 01:18:29,780 our original definition is that they 1154 01:18:29,780 --> 01:18:33,380 carry the notion of a straight line in a straightforward way 1155 01:18:33,380 --> 01:18:39,470 from where they are obvious in a locally Lorentz frame 1156 01:18:39,470 --> 01:18:41,240 to a more covariant formulation of that-- 1157 01:18:48,170 --> 01:18:55,170 so a generalized straight line to a curved spacetime. 1158 01:18:55,170 --> 01:19:08,500 And they give the trajectory of extremal aging in other words 1159 01:19:08,500 --> 01:19:12,850 a trajectory along which between two points in spacetime, 1160 01:19:12,850 --> 01:19:15,520 an observer moving from one to the other will 1161 01:19:15,520 --> 01:19:17,060 accumulate the most proper time. 1162 01:19:20,350 --> 01:19:26,380 So I'm going to stop here. 1163 01:19:26,380 --> 01:19:28,748 There's a bit more, which I would like to do, 1164 01:19:28,748 --> 01:19:30,040 but I just don't have the time. 1165 01:19:30,040 --> 01:19:33,940 But I'll tell you the key things that I want to say next. 1166 01:19:33,940 --> 01:19:37,060 Everything that I've done here so far 1167 01:19:37,060 --> 01:19:41,367 is I've really fixated on time-like trajectories. 1168 01:19:41,367 --> 01:19:43,450 I've imagined there's a body with some finite rest 1169 01:19:43,450 --> 01:19:47,500 mass where I can make a sensible notion of proper time. 1170 01:19:47,500 --> 01:19:49,000 We are also going to want to talk 1171 01:19:49,000 --> 01:19:50,750 about the behavior of light. 1172 01:19:50,750 --> 01:19:53,590 Light moves on null trajectories. 1173 01:19:53,590 --> 01:19:58,660 I cannot sensibly define proper time on long such a trajectory. 1174 01:19:58,660 --> 01:19:59,500 They are massless. 1175 01:19:59,500 --> 01:20:01,667 There's all sorts of properties associated with them 1176 01:20:01,667 --> 01:20:03,250 that just make this analysis. 1177 01:20:03,250 --> 01:20:05,260 The way I've done it so far, I'll 1178 01:20:05,260 --> 01:20:08,188 need to tweak things a little bit in order for it to work. 1179 01:20:08,188 --> 01:20:09,230 We will do that tweaking. 1180 01:20:09,230 --> 01:20:11,290 It's actually quite straightforward 1181 01:20:11,290 --> 01:20:14,110 and allows us to also bring in a bit more intuition 1182 01:20:14,110 --> 01:20:16,210 about what affine parameters mean when we do that. 1183 01:20:16,210 --> 01:20:17,680 So that'll be the one thing we do. 1184 01:20:17,680 --> 01:20:19,780 The other-- it's unfortunate I wasn't able to get to it 1185 01:20:19,780 --> 01:20:21,620 today-- but it's a straightforward saying, 1186 01:20:21,620 --> 01:20:24,100 which I think I may include in the notes that I post-- 1187 01:20:24,100 --> 01:20:28,420 is including what happens, if your spacetime-- 1188 01:20:28,420 --> 01:20:31,570 so if the metric you use to generate these Christoffels has 1189 01:20:31,570 --> 01:20:33,860 a Killing factor associated with it, 1190 01:20:33,860 --> 01:20:35,770 you can combine Killing's equation 1191 01:20:35,770 --> 01:20:37,360 with the geodesic equation to prove 1192 01:20:37,360 --> 01:20:40,570 the existence of conserved quantities associated 1193 01:20:40,570 --> 01:20:41,343 with that motion. 1194 01:20:41,343 --> 01:20:42,760 And that's where we start to begin 1195 01:20:42,760 --> 01:20:46,150 to see that if I have a spacetime that 1196 01:20:46,150 --> 01:20:48,850 is independent of time, there's a notion of conserved energy 1197 01:20:48,850 --> 01:20:50,020 associated with it. 1198 01:20:50,020 --> 01:20:52,410 So we will do that on Tuesday.