1 00:00:00,000 --> 00:00:01,464 [SQUEAKING] 2 00:00:01,464 --> 00:00:02,928 [RUSTLING] 3 00:00:02,928 --> 00:00:05,368 [CLICKING] 4 00:00:09,760 --> 00:00:11,410 SCOTT HUGHES: Last Thursday, we began 5 00:00:11,410 --> 00:00:13,420 the work of moving from special relativity 6 00:00:13,420 --> 00:00:16,540 to general relativity, and we spent 7 00:00:16,540 --> 00:00:19,390 a lot of time unpacking two formulations of the principle 8 00:00:19,390 --> 00:00:20,050 of equivalence. 9 00:00:20,050 --> 00:00:21,930 So one, which goes under the name 10 00:00:21,930 --> 00:00:24,760 "weak equivalence principle"-- 11 00:00:24,760 --> 00:00:26,380 a simpler way of saying that is that, 12 00:00:26,380 --> 00:00:30,160 at least over a sufficiently small region, 13 00:00:30,160 --> 00:00:32,890 if there is nothing but gravity acting, 14 00:00:32,890 --> 00:00:37,600 I cannot distinguish between freefall under the influence 15 00:00:37,600 --> 00:00:40,090 of gravity or a uniform acceleration. 16 00:00:40,090 --> 00:00:42,250 The two things are equivalent to one another. 17 00:00:42,250 --> 00:00:45,040 Basically, this is a reflection of the fact 18 00:00:45,040 --> 00:00:48,190 that the gravitational charge and the inertial mass 19 00:00:48,190 --> 00:00:49,360 are the same thing. 20 00:00:49,360 --> 00:00:51,550 That is the main thing that really underlies 21 00:00:51,550 --> 00:00:54,160 the weak equivalence principle. 22 00:00:54,160 --> 00:00:56,740 When I gave my tenure talk a number of years ago here, 23 00:00:56,740 --> 00:01:00,760 I pointed out that there was this wonderful program called 24 00:01:00,760 --> 00:01:03,220 the Apollo program that was put together to test this. 25 00:01:03,220 --> 00:01:04,989 And the way it was done was that they put astronauts 26 00:01:04,989 --> 00:01:06,406 on the moon, and you actually show 27 00:01:06,406 --> 00:01:08,920 that, if you drop a hammer and a feather on the moon, 28 00:01:08,920 --> 00:01:10,900 they fall at the same rate. 29 00:01:10,900 --> 00:01:12,668 Of course, the Apollo program probably 30 00:01:12,668 --> 00:01:13,960 did a few other things as well. 31 00:01:13,960 --> 00:01:16,180 But I'm a general relativity theorist, 32 00:01:16,180 --> 00:01:18,430 so for me, that was the outcome of the Apollo program, 33 00:01:18,430 --> 00:01:21,340 was test of the equivalence principle. 34 00:01:21,340 --> 00:01:23,290 We also have a different variation 35 00:01:23,290 --> 00:01:26,740 of this we called the Einstein equivalence principle, which 36 00:01:26,740 --> 00:01:28,420 leads us to a calculation that we went 37 00:01:28,420 --> 00:01:34,930 through last time, which states that we can find 38 00:01:34,930 --> 00:01:36,430 a representation over a sufficiently 39 00:01:36,430 --> 00:01:39,190 small region of spacetime such that the laws of physics 40 00:01:39,190 --> 00:01:42,080 are reduced to those of special relativity. 41 00:01:42,080 --> 00:01:44,120 And we did a calculation to examine 42 00:01:44,120 --> 00:01:52,780 this, where we showed, given an arbitrary spacetime metric, 43 00:01:52,780 --> 00:01:58,420 I can find a coordinate system such that this can be written 44 00:01:58,420 --> 00:02:03,090 in the form metric of flat spacetime 45 00:02:03,090 --> 00:02:09,330 plus terms that are of order so we have coordinate distance 46 00:02:09,330 --> 00:02:10,960 squared. 47 00:02:10,960 --> 00:02:14,370 Let's put it this way-- so this, in the vicinity of a point pl-- 48 00:02:14,370 --> 00:02:16,820 I'll make that clear in just a moment. 49 00:02:16,820 --> 00:02:19,830 It ends up looking like of order of coordinate displacement 50 00:02:19,830 --> 00:02:23,280 squared with corrections that scale as 1 51 00:02:23,280 --> 00:02:25,860 over second derivative of the metric. 52 00:02:25,860 --> 00:02:28,313 That sets the scale for what these end up looking like, 53 00:02:28,313 --> 00:02:29,730 or maybe it's actually times that. 54 00:02:29,730 --> 00:02:29,870 Sorry. 55 00:02:29,870 --> 00:02:30,750 It's times that. 56 00:02:33,910 --> 00:02:35,010 Why did I divide? 57 00:02:35,010 --> 00:02:35,625 I don't know. 58 00:02:35,625 --> 00:02:37,500 Oh, I know why, because I wanted to point out 59 00:02:37,500 --> 00:02:40,900 that that is what the scale of 1 over this thing-- 60 00:02:40,900 --> 00:02:42,270 for God's sake, Scott! 61 00:02:42,270 --> 00:02:43,980 Stop putting your square roots in there! 62 00:02:43,980 --> 00:02:46,060 So it looks like that. 63 00:02:46,060 --> 00:02:48,060 And so this is what I was saying. 64 00:02:48,060 --> 00:02:53,900 You have a curvature scale that is 65 00:02:53,900 --> 00:02:59,887 on the order of square root of 1 over the second derivative 66 00:02:59,887 --> 00:03:00,470 of the metric. 67 00:03:00,470 --> 00:03:02,428 Apologies for botching that as I was writing it 68 00:03:02,428 --> 00:03:03,407 up there quickly. 69 00:03:03,407 --> 00:03:05,240 OK, so we did a calculation that shows that. 70 00:03:05,240 --> 00:03:06,990 And indeed, what we did is we went through 71 00:03:06,990 --> 00:03:09,720 and we showed that a general coordinate transformation has 72 00:03:09,720 --> 00:03:11,760 more than enough degrees of freedom 73 00:03:11,760 --> 00:03:15,210 to make the metric flat to get the flat spacetime 74 00:03:15,210 --> 00:03:16,890 metric at a particular point. 75 00:03:16,890 --> 00:03:19,410 And in fact, there are six degrees of freedom left over, 76 00:03:19,410 --> 00:03:23,895 corresponding to six rotations and six boosts that 77 00:03:23,895 --> 00:03:26,490 are allowable at that particular point or event. 78 00:03:26,490 --> 00:03:29,610 Bear in mind we're working in spacetime. 79 00:03:29,610 --> 00:03:32,640 We have exactly enough degrees of freedom to cancel out 80 00:03:32,640 --> 00:03:36,360 the first order term, but we cannot cancel out the second 81 00:03:36,360 --> 00:03:37,320 order term. 82 00:03:37,320 --> 00:03:39,030 And in fact, we find there are 20 degrees 83 00:03:39,030 --> 00:03:40,380 of freedom left over. 84 00:03:40,380 --> 00:03:43,770 And in a future lecture, we will derive a geometric object 85 00:03:43,770 --> 00:03:46,260 that characterizes the curvature that 86 00:03:46,260 --> 00:03:47,970 has indeed 20 degrees of freedom in it 87 00:03:47,970 --> 00:03:51,657 or 20 independent components that come out of it. 88 00:03:51,657 --> 00:03:53,990 So this is the foundation of where we're moving forward. 89 00:03:53,990 --> 00:03:57,390 And so what this basically tells us 90 00:03:57,390 --> 00:04:00,360 is that, in a spacetime like this, 91 00:04:00,360 --> 00:04:02,130 we have what we call curvature. 92 00:04:02,130 --> 00:04:04,800 Trajectories that start out parallel to one another 93 00:04:04,800 --> 00:04:07,800 are not going to remain parallel as they move forward. 94 00:04:07,800 --> 00:04:10,230 And where we concluded last time, 95 00:04:10,230 --> 00:04:12,030 we were dealing with the problem that, 96 00:04:12,030 --> 00:04:14,665 if I want to take derivatives-- and we're going 97 00:04:14,665 --> 00:04:15,790 to start with vector field. 98 00:04:15,790 --> 00:04:17,399 If I want to differentiate a vector 99 00:04:17,399 --> 00:04:21,839 field on a curved manifold, it doesn't work. 100 00:04:21,839 --> 00:04:24,570 If I do the naive thing of just taking a partial derivative, 101 00:04:24,570 --> 00:04:26,070 it does not work. 102 00:04:26,070 --> 00:04:27,770 So I'll remind you where we left off. 103 00:04:39,760 --> 00:04:47,855 So we found partial derivatives of vectors. 104 00:04:47,855 --> 00:04:49,180 Let me put that in there. 105 00:04:49,180 --> 00:04:50,900 Partial derivatives of vectors, and it 106 00:04:50,900 --> 00:04:51,980 won't take much work to show it's 107 00:04:51,980 --> 00:04:53,490 true for one forms or any tensor, 108 00:04:53,490 --> 00:04:59,810 so let's say, partial derivatives of tensors 109 00:04:59,810 --> 00:05:01,300 do not yield tensors. 110 00:05:06,160 --> 00:05:07,800 We're familiar with this to some extent 111 00:05:07,800 --> 00:05:09,258 because we already encountered this 112 00:05:09,258 --> 00:05:11,920 when we began thinking about the behavior of, even 113 00:05:11,920 --> 00:05:14,280 in flat spacetime, flat spacetime and curvilinear 114 00:05:14,280 --> 00:05:14,780 coordinates. 115 00:05:14,780 --> 00:05:16,947 So that's the fact that the basis objects themselves 116 00:05:16,947 --> 00:05:19,170 have some functional dependence associated with them. 117 00:05:19,170 --> 00:05:21,900 We're interpreting it a little bit differently now. 118 00:05:21,900 --> 00:05:23,483 And so what we're doing is we're going 119 00:05:23,483 --> 00:05:25,192 to say that what's going on here is that, 120 00:05:25,192 --> 00:05:26,640 on my curved manifold-- and I want 121 00:05:26,640 --> 00:05:28,015 you to visualize something that's 122 00:05:28,015 --> 00:05:32,410 like a bumpy surface or, if you like, maybe a sphere. 123 00:05:32,410 --> 00:05:34,950 Think about if you are a two-dimensional being confined 124 00:05:34,950 --> 00:05:36,450 to the surface of a sphere. 125 00:05:36,450 --> 00:05:38,490 There's only two directions at any given point. 126 00:05:38,490 --> 00:05:42,270 You can go up, or you can go on the left-right axis 127 00:05:42,270 --> 00:05:44,540 or the north-south axis, right? 128 00:05:44,540 --> 00:05:48,070 And you would define unit vectors pointing along there. 129 00:05:48,070 --> 00:05:51,330 But as you move around that sphere, those of us who 130 00:05:51,330 --> 00:05:54,570 have a three-dimensional life, and can step back and see this, 131 00:05:54,570 --> 00:05:58,650 we see that these basis objects point in different directions 132 00:05:58,650 --> 00:06:00,730 at different locations on the sphere. 133 00:06:00,730 --> 00:06:02,730 The two-dimensional beings aren't aware of that. 134 00:06:02,730 --> 00:06:05,022 They just know that they're on a surface that's curved. 135 00:06:05,022 --> 00:06:07,140 And so they would say that the tangent 136 00:06:07,140 --> 00:06:09,270 space is different for all these objects. 137 00:06:09,270 --> 00:06:11,700 They've imagined that every one of these basis objects 138 00:06:11,700 --> 00:06:14,700 lives in a plane that is tangent to the sphere at any given 139 00:06:14,700 --> 00:06:17,330 point, and that plane is different at every point 140 00:06:17,330 --> 00:06:18,550 along the sphere. 141 00:06:18,550 --> 00:06:21,750 So we interpret this by saying that all of our basis objects 142 00:06:21,750 --> 00:06:24,060 live in this tangent space, and the tangent space 143 00:06:24,060 --> 00:06:26,385 is different at every point on the surface. 144 00:06:29,250 --> 00:06:34,490 So let me just write out one equation here. 145 00:06:34,490 --> 00:06:39,050 So when we looked at the transformation 146 00:06:39,050 --> 00:06:40,940 of a partial derivative of a vector, 147 00:06:40,940 --> 00:06:44,360 if we looked at just the components-- 148 00:06:51,160 --> 00:06:52,660 so this calculation is in the notes, 149 00:06:52,660 --> 00:06:58,370 so I would just write down the result. What we found 150 00:06:58,370 --> 00:07:02,060 was that, if I'm taking, say, the beta derivative 151 00:07:02,060 --> 00:07:24,830 of component A alpha, what I found when 152 00:07:24,830 --> 00:07:27,350 I want to go into a coordinate transformation 153 00:07:27,350 --> 00:07:30,590 is that there's an extra term that 154 00:07:30,590 --> 00:07:32,420 ruins the tensoriality of this. 155 00:07:36,570 --> 00:07:39,030 So this goes over to something that looks like-- 156 00:08:09,490 --> 00:08:12,910 So the first term is what we'd expect if this were a tensor 157 00:08:12,910 --> 00:08:13,840 relationship. 158 00:08:13,840 --> 00:08:17,560 That's exactly the matrix of the Jacobian between two 159 00:08:17,560 --> 00:08:19,060 different coordinate representations 160 00:08:19,060 --> 00:08:22,360 that we expect to describe how components 161 00:08:22,360 --> 00:08:25,180 change if they indeed obey a tensorial relationship. 162 00:08:25,180 --> 00:08:27,370 This extra thing here I wrote on the second line-- 163 00:08:27,370 --> 00:08:30,430 that is spoiling it for us. 164 00:08:30,430 --> 00:08:32,840 So I began to give you the physical notion of what we 165 00:08:32,840 --> 00:08:34,470 were going to do to fix this. 166 00:08:34,470 --> 00:08:36,650 So let me just reset that up again. 167 00:08:36,650 --> 00:08:40,809 So let's imagine I have a particular curve that 168 00:08:40,809 --> 00:08:44,470 goes along my manifold. 169 00:08:44,470 --> 00:08:47,860 I have a point P here on the curve, 170 00:08:47,860 --> 00:08:50,920 and I have a point Q over here. 171 00:08:50,920 --> 00:08:55,000 And let's say that P is at event x alpha. 172 00:08:55,000 --> 00:08:59,950 Q is at x alpha plus dx alpha. 173 00:09:06,870 --> 00:09:12,580 Here's my vector A at the event Q. 174 00:09:12,580 --> 00:09:19,480 And here's my vector A at the event P. 175 00:09:19,480 --> 00:09:22,375 So what we discussed last time is that, to get this thing, 176 00:09:22,375 --> 00:09:24,250 I'm just doing the normal partial derivative. 177 00:09:24,250 --> 00:09:28,030 I'm basically imagining that these are close to one another, 178 00:09:28,030 --> 00:09:34,318 that I can just subtract A at Q from A at P, divide by dx 179 00:09:34,318 --> 00:09:35,110 and take the limit. 180 00:09:35,110 --> 00:09:37,630 That's the definition of a derivative. 181 00:09:37,630 --> 00:09:40,360 And what this is telling us is, mathematically, yeah, 182 00:09:40,360 --> 00:09:42,580 it's a derivative, but it's not a derivative 183 00:09:42,580 --> 00:09:44,830 that yields a tensor quantity. 184 00:09:44,830 --> 00:09:47,080 And so we are beginning to discuss the fact that, 185 00:09:47,080 --> 00:09:49,930 to compare things that have different tangent spaces, that 186 00:09:49,930 --> 00:09:52,450 live in different points in my curve manifold, 187 00:09:52,450 --> 00:09:55,080 I need a notion of transport to take one from the other 188 00:09:55,080 --> 00:09:56,720 in order to compare them. 189 00:09:56,720 --> 00:09:58,540 So there are two notions of transport 190 00:09:58,540 --> 00:10:00,620 that we're going to talk about here. 191 00:10:00,620 --> 00:10:04,920 The first one is called parallel transport. 192 00:10:04,920 --> 00:10:06,960 So transport notion one-- 193 00:10:20,400 --> 00:10:23,160 we call this parallel transport. 194 00:10:23,160 --> 00:10:26,820 I'm going to actually focus a little bit on the math first 195 00:10:26,820 --> 00:10:30,840 and then come back to what is parallel about this afterwards. 196 00:10:36,610 --> 00:10:40,140 So what I essentially need to do is say, what I want to do 197 00:10:40,140 --> 00:10:42,690 is find some kind of a way of imagining 198 00:10:42,690 --> 00:10:46,800 that I take the vector at P and transport it over 199 00:10:46,800 --> 00:10:52,080 to the point Q, and I will compare the transported object 200 00:10:52,080 --> 00:11:21,240 rather than the object originally at point P. 201 00:11:21,240 --> 00:11:22,995 Abstractly, what I'm essentially going 202 00:11:22,995 --> 00:11:24,620 to do is I'm going to do what we always do in this. 203 00:11:24,620 --> 00:11:26,220 I'm going to imagine that there is 204 00:11:26,220 --> 00:11:28,380 some kind of an operation that is 205 00:11:28,380 --> 00:11:30,870 linear in the separation between the two of them 206 00:11:30,870 --> 00:11:35,020 that allows me to define this transport. 207 00:11:35,020 --> 00:11:36,529 So let's do the following. 208 00:11:43,570 --> 00:11:51,690 So I'm going to assume that we can define an object, which 209 00:11:51,690 --> 00:12:05,202 I will call Pi, capital Pi, alpha, beta, mu, which 210 00:12:05,202 --> 00:12:06,410 is going to do the following. 211 00:12:06,410 --> 00:12:12,207 So what I'm going to do is say that A alpha transported-- 212 00:12:12,207 --> 00:12:14,540 and to make it even clearer, how it's being transported. 213 00:12:14,540 --> 00:12:21,480 Let's say it's being transported from P to Q. 214 00:12:21,480 --> 00:12:25,470 This is given by alpha at P, and what 215 00:12:25,470 --> 00:12:30,250 I'm going to do is say that, whatever this object is, 216 00:12:30,250 --> 00:12:34,830 it is linear in both the coordinate separation 217 00:12:34,830 --> 00:12:37,410 of those two events and the vector field. 218 00:12:45,293 --> 00:12:47,460 So far, I've said nothing about physics, by the way. 219 00:12:47,460 --> 00:12:49,543 I'm just laying out some mathematical definitions. 220 00:12:49,543 --> 00:12:51,840 I'm going to bring it all together in a few moments. 221 00:12:51,840 --> 00:12:53,460 I'm then going to say, OK, I know 222 00:12:53,460 --> 00:12:56,400 that I had trouble with my standard partial derivative. 223 00:12:56,400 --> 00:13:01,260 Let's define a derivative operator in the following way. 224 00:13:12,270 --> 00:13:29,980 Define a derivative by comparing the transported vector 225 00:13:29,980 --> 00:13:37,860 to the field at Q. So what I'm going 226 00:13:37,860 --> 00:13:40,590 to do for the moment is just denote this notion 227 00:13:40,590 --> 00:13:43,980 of a new kind of derivative with a capital D. Right now, 228 00:13:43,980 --> 00:13:46,260 it's just another symbol that we would 229 00:13:46,260 --> 00:13:48,990 pronounce with a "duh" sound so that it sounds like derivative. 230 00:13:48,990 --> 00:13:50,907 So don't read too much into that for a moment. 231 00:13:54,460 --> 00:14:01,350 So I'm going to define this as A at Q minus A 232 00:14:01,350 --> 00:14:13,340 transported from P to Q and then divide by the separation. 233 00:14:13,340 --> 00:14:16,660 Take the limit-- usual thing. 234 00:14:16,660 --> 00:14:23,710 And when you do this, you're going 235 00:14:23,710 --> 00:14:26,440 to get something that looks like the partial derivative 236 00:14:26,440 --> 00:14:29,010 plus an additional term on here. 237 00:14:34,730 --> 00:14:37,370 It looks like this. 238 00:14:37,370 --> 00:14:39,440 So I've said nothing about what properties 239 00:14:39,440 --> 00:14:41,240 I'm going to demand of this thing. 240 00:14:41,240 --> 00:14:43,130 And in fact, there are many ways that one 241 00:14:43,130 --> 00:14:47,300 could define a transport of an object like this. 242 00:14:47,300 --> 00:14:52,280 In general, when you do this, this thing I'm calling Pi here 243 00:14:52,280 --> 00:14:54,111 is known as the connection. 244 00:14:58,440 --> 00:15:10,080 It is the object that connects point P to point Q. 245 00:15:10,080 --> 00:15:13,327 So let's make a couple demands on its properties. 246 00:15:13,327 --> 00:15:15,410 So now I'll start to put a bit of physics in this. 247 00:15:28,990 --> 00:15:40,010 So demand one is I'm going to require that, when I change 248 00:15:40,010 --> 00:15:51,500 my coordinate representation, when I evaluate this 249 00:15:51,500 --> 00:15:56,160 in my new coordinate system, that I get something 250 00:15:56,160 --> 00:15:57,190 that looks like this. 251 00:16:34,470 --> 00:16:39,530 If I do this, I'm going to find that, when 252 00:16:39,530 --> 00:16:43,910 I change coordinates and apply it to the entire derivative 253 00:16:43,910 --> 00:16:47,750 I've defined over here, that little extra bit of schmutz 254 00:16:47,750 --> 00:16:51,170 that's on the second line there is exactly what you need 255 00:16:51,170 --> 00:16:55,940 to cancel out this annoying bugger so that you have 256 00:16:55,940 --> 00:16:58,280 a nice tensor relationship left. 257 00:16:58,280 --> 00:17:01,070 So I'm going to demand that a key part of whatever this guy 258 00:17:01,070 --> 00:17:05,510 turns out to be is something that cancels out the irritating 259 00:17:05,510 --> 00:17:07,866 garbage that came along with the partial derivative. 260 00:17:45,490 --> 00:17:48,490 Combine that with what I'm about to say in just a moment-- 261 00:17:48,490 --> 00:17:51,430 that pins things down significantly. 262 00:17:51,430 --> 00:17:54,040 I'm going to make one more demand, 263 00:17:54,040 --> 00:17:56,205 and this demand is going to then be connected 264 00:17:56,205 --> 00:17:58,330 to the physical picture that I'm going to introduce 265 00:17:58,330 --> 00:17:59,920 in about five minutes. 266 00:17:59,920 --> 00:18:04,270 My final demand is I'm going to require that 267 00:18:04,270 --> 00:18:07,040 whatever this derivative is-- 268 00:18:07,040 --> 00:18:11,960 when I apply it to the metric, I get 0. 269 00:18:11,960 --> 00:18:23,630 If I do that, then it turns out that the connection is exactly 270 00:18:23,630 --> 00:18:28,370 the Christoffel symbol that we worked out earlier. 271 00:18:28,370 --> 00:18:50,920 If I put in this demand, the connection is the Christoffel, 272 00:18:50,920 --> 00:18:53,260 and this derivative is nothing more 273 00:18:53,260 --> 00:18:55,000 than the covariant derivative. 274 00:18:55,000 --> 00:18:56,050 Der-iv-a-tive. 275 00:19:00,920 --> 00:19:02,600 This is just the covariant derivative 276 00:19:02,600 --> 00:19:03,517 we worked out earlier. 277 00:19:11,560 --> 00:19:13,020 So much for mathematics. 278 00:19:13,020 --> 00:19:15,192 The key thing which I want to emphasize 279 00:19:15,192 --> 00:19:16,650 at this point in the conversation-- 280 00:19:16,650 --> 00:19:18,870 I hope we can see the logic behind the first demand. 281 00:19:18,870 --> 00:19:21,328 That's just something which I'm going to introduce in order 282 00:19:21,328 --> 00:19:23,580 to clean this guy up. 283 00:19:23,580 --> 00:19:25,020 This is my choice. 284 00:19:25,020 --> 00:19:26,520 Not just my choice-- it's the choice 285 00:19:26,520 --> 00:19:29,160 of a lot of people who have helped to develop this subject. 286 00:19:29,160 --> 00:19:33,240 But it's a good one, because, when I do this, 287 00:19:33,240 --> 00:19:38,730 there is a particularly good physical interpretation to what 288 00:19:38,730 --> 00:19:40,850 this notion of transport means. 289 00:19:40,850 --> 00:19:42,570 And bear with me for a second while I 290 00:19:42,570 --> 00:19:43,830 gather my notes together. 291 00:19:49,020 --> 00:19:56,280 So let's do the top one first. 292 00:20:38,990 --> 00:20:42,140 OK, so let's make that curve again. 293 00:20:45,210 --> 00:20:49,260 And actually, let's just go ahead and redraw my vector 294 00:20:49,260 --> 00:20:49,980 field. 295 00:20:49,980 --> 00:20:51,772 I do want to have a different copy of this. 296 00:20:59,210 --> 00:21:02,360 Let me introduce one other object. 297 00:21:02,360 --> 00:21:04,022 So I'm going to give this curve a name. 298 00:21:04,022 --> 00:21:05,480 I'm going to call this curve gamma. 299 00:21:11,062 --> 00:21:12,520 And this is a notion that I'm going 300 00:21:12,520 --> 00:21:14,820 to make much more precise in a future lecture, 301 00:21:14,820 --> 00:21:18,710 but imagine that there is some kind of a tape measure 302 00:21:18,710 --> 00:21:25,340 that reads out along the curve gamma that is uniformly 303 00:21:25,340 --> 00:21:27,800 ticked in a way that we will make precise 304 00:21:27,800 --> 00:21:28,740 in a future lecture. 305 00:21:36,940 --> 00:21:38,950 I will call the parameter that is uniform 306 00:21:38,950 --> 00:21:40,900 that denotes these uniform tick marks-- 307 00:21:45,990 --> 00:21:47,577 I will call that lambda. 308 00:21:47,577 --> 00:21:49,410 If you want to read a little bit about this, 309 00:21:49,410 --> 00:21:52,140 this is what's known as an affine parameter. 310 00:21:52,140 --> 00:21:54,870 We will make this a little bit more precise very soon. 311 00:21:59,590 --> 00:22:02,800 With this in mind, you should be able to convince yourself 312 00:22:02,800 --> 00:22:06,760 that this field U defines the tangent vector to this curve. 313 00:22:21,170 --> 00:22:26,140 So let's say that my original point here, that this 314 00:22:26,140 --> 00:22:29,830 is at lambda equals 2-- 315 00:22:29,830 --> 00:22:33,790 and let's say this is at lambda equals 7. 316 00:22:33,790 --> 00:22:36,610 And what I want to do is transport the vector 317 00:22:36,610 --> 00:22:39,510 from 2 to 7. 318 00:22:39,510 --> 00:22:41,570 Well, a way that I can do this is by saying, 319 00:22:41,570 --> 00:22:44,500 OK, I now know that the derivative 320 00:22:44,500 --> 00:22:47,890 that goes over here-- this is just the covariant derivative. 321 00:22:47,890 --> 00:22:50,590 So I forgot to write this down but from now on, 322 00:22:50,590 --> 00:22:52,000 this derivative I wrote down-- 323 00:22:52,000 --> 00:22:54,430 I'm going to go back to the gradient symbol we used 324 00:22:54,430 --> 00:22:55,923 for the covariant derivative. 325 00:22:59,240 --> 00:23:04,590 What I can do is take the covariant derivative 326 00:23:04,590 --> 00:23:10,560 of my vector field, contract it with this tangent vector. 327 00:23:10,560 --> 00:23:11,405 And what this does-- 328 00:23:17,050 --> 00:23:20,050 I'm going to define this as capital D alpha D lambda. 329 00:23:20,050 --> 00:23:24,370 This is a covariant derivative with respect to the parameter 330 00:23:24,370 --> 00:23:27,860 lambda as I move along this constrained trajectory. 331 00:23:27,860 --> 00:23:42,240 What this does is this tells me how A changes as it 332 00:23:42,240 --> 00:23:47,250 is transported along the curve. 333 00:24:19,590 --> 00:24:20,820 I'm now going to argue-- 334 00:24:20,820 --> 00:24:23,220 what parallel transport comes down to 335 00:24:23,220 --> 00:24:42,840 is when you require that, as you slide this thing along here, 336 00:24:42,840 --> 00:24:45,810 the covariant total derivative of this thing as you 337 00:24:45,810 --> 00:24:49,280 move along the curve, that it's equal to 0. 338 00:24:49,280 --> 00:24:51,340 OK, now let me motivate where that's coming from. 339 00:24:51,340 --> 00:24:52,110 Why is that? 340 00:25:00,970 --> 00:25:04,930 So let's put all of our definitions back in. 341 00:25:04,930 --> 00:25:07,865 Let's apply the definition of covariant derivative 342 00:25:07,865 --> 00:25:09,490 that we discussed in a previous lecture 343 00:25:09,490 --> 00:25:10,990 and we've all come to know and love. 344 00:25:28,610 --> 00:25:30,610 Now, this is the bit where we begin to introduce 345 00:25:30,610 --> 00:25:32,210 a little bit of physics. 346 00:25:32,210 --> 00:25:35,320 Let's imagine that points P and Q 347 00:25:35,320 --> 00:25:37,420 are sufficiently close to each other that they 348 00:25:37,420 --> 00:25:40,210 fit within a single, freely falling frame. 349 00:25:40,210 --> 00:25:42,340 They can go into the same local Lorentz frame. 350 00:26:05,820 --> 00:26:08,730 So remember, when I go into my local Lorentz frame, 351 00:26:08,730 --> 00:26:11,790 the metric becomes the metric of flat spacetime. 352 00:26:11,790 --> 00:26:15,230 G goes over to eta. 353 00:26:15,230 --> 00:26:17,730 There is a second order correction to that. 354 00:26:17,730 --> 00:26:19,660 So the second derivatives-- 355 00:26:19,660 --> 00:26:22,340 I cannot find a [INAUDIBLE] that gets rid of them. 356 00:26:22,340 --> 00:26:25,070 So there'll be a little bit of that there, which tells me 357 00:26:25,070 --> 00:26:28,200 how large this Lorentz frame can be. 358 00:26:28,200 --> 00:26:31,390 But I can get rid of all the first derivatives. 359 00:26:31,390 --> 00:26:34,160 And if you get rid of all the first derivatives, 360 00:26:34,160 --> 00:26:37,740 you zero out the Christoffel symbols in that frame. 361 00:26:37,740 --> 00:26:40,550 So in that frame, there are no Christoffel symbols. 362 00:26:50,110 --> 00:27:16,060 So what this means is that, in this frame, 363 00:27:16,060 --> 00:27:21,645 this just becomes the idea that a simple total derivative 364 00:27:21,645 --> 00:27:24,270 of this object-- you don't need to include all the garbage that 365 00:27:24,270 --> 00:27:27,150 comes along with the covariant derivative-- 366 00:27:27,150 --> 00:27:28,230 this is equal to 0. 367 00:27:28,230 --> 00:27:37,810 And that's equivalent to saying that, as I take this vector 368 00:27:37,810 --> 00:27:42,250 and transport it along, I hold all of its components 369 00:27:42,250 --> 00:27:46,150 constant as I slide it from one step to the other. 370 00:28:09,410 --> 00:28:14,757 So I start out with my A at point P here. 371 00:28:14,757 --> 00:28:16,340 And then I slide it over a little bit, 372 00:28:16,340 --> 00:28:19,790 holding all the components constant just like this-- 373 00:28:19,790 --> 00:28:23,750 slide over again, slide it over again. 374 00:28:23,750 --> 00:28:25,788 Da, da, da, da da. 375 00:28:25,788 --> 00:28:27,216 Da, da, da, da, da. 376 00:28:31,498 --> 00:28:33,040 Till finally, I get it over to there. 377 00:28:33,040 --> 00:28:34,240 What this is doing is that-- 378 00:28:37,360 --> 00:28:42,730 any two vectors in the middle of this transport process-- 379 00:28:42,730 --> 00:28:45,100 I am holding them as parallel as it 380 00:28:45,100 --> 00:28:48,850 is possible to hold them, given that, to be blunt, 381 00:28:48,850 --> 00:28:51,130 you can't even really define a notion 382 00:28:51,130 --> 00:28:54,250 of two objects being parallel on a curved surface 383 00:28:54,250 --> 00:28:57,140 if there's a macroscopic separation between them. 384 00:28:57,140 --> 00:28:59,640 But if you think about just a little region that's 385 00:28:59,640 --> 00:29:02,740 sufficiently small, that it's flat 386 00:29:02,740 --> 00:29:05,620 up to quadratic corrections, then a notion 387 00:29:05,620 --> 00:29:08,710 of these things being parallel to each other makes sense. 388 00:29:08,710 --> 00:29:13,060 And this idea, that I'm going to demand that, upon transport, 389 00:29:13,060 --> 00:29:15,370 the derivative of the metric equals 0, 390 00:29:15,370 --> 00:29:18,820 thereby yielding my connection being the Christoffel 391 00:29:18,820 --> 00:29:21,700 and this derivative being the covariant derivative-- 392 00:29:21,700 --> 00:29:24,040 it tells us that this notion of transport 393 00:29:24,040 --> 00:29:26,080 is one in which objects are just kept 394 00:29:26,080 --> 00:29:29,230 as parallel as possible as they slide along here. 395 00:29:29,230 --> 00:29:33,020 And that is why this is called "parallel transport." 396 00:29:33,020 --> 00:29:35,530 It's as parallel as it's possible to be, 397 00:29:35,530 --> 00:29:36,806 given the curvature. 398 00:29:59,140 --> 00:30:00,690 So as I switch gears, I just want 399 00:30:00,690 --> 00:30:05,730 to emphasize I went through the mathematics of that 400 00:30:05,730 --> 00:30:09,000 with a fair amount of care because it 401 00:30:09,000 --> 00:30:14,190 is important to keep this stuff as rigorous as possible here. 402 00:30:14,190 --> 00:30:15,930 This notion of parallel transport 403 00:30:15,930 --> 00:30:19,680 gets used a lot when we start talking about things moving 404 00:30:19,680 --> 00:30:24,090 around in a curved spacetime. 405 00:30:24,090 --> 00:30:26,820 In particular, if you think about an object 406 00:30:26,820 --> 00:30:29,280 that is freely falling, and it is experiencing 407 00:30:29,280 --> 00:30:32,100 no forces other than gravity, which 408 00:30:32,100 --> 00:30:35,160 we are going to no longer regard as a force before too long-- 409 00:30:35,160 --> 00:30:38,170 if you just go back to Newtonian intuition, what does it do? 410 00:30:38,170 --> 00:30:41,190 Well, you give it an initial velocity or initial momentum, 411 00:30:41,190 --> 00:30:42,060 and it maintains it. 412 00:30:42,060 --> 00:30:44,880 It just continues going in a straight line. 413 00:30:44,880 --> 00:30:47,280 In spacetime, going in a straight line 414 00:30:47,280 --> 00:30:51,120 basically means, at every step, I move 415 00:30:51,120 --> 00:30:55,620 and I take the tangent to my world line, my four-velocity, 416 00:30:55,620 --> 00:30:58,060 and I move it parallel to itself. 417 00:30:58,060 --> 00:30:59,730 So this notion of parallel transport 418 00:30:59,730 --> 00:31:01,350 is going to be the key thing that we 419 00:31:01,350 --> 00:31:03,990 use to actually define the kinetics of bodies 420 00:31:03,990 --> 00:31:06,470 in curved spacetime. 421 00:31:06,470 --> 00:31:09,380 There's a tremendous amount of work 422 00:31:09,380 --> 00:31:13,040 being done by all sorts of things these days 423 00:31:13,040 --> 00:31:16,190 that's based on studies of orbits in curved spacetime, 424 00:31:16,190 --> 00:31:19,500 and they all come back to this notion of parallel transport. 425 00:31:19,500 --> 00:31:20,000 All right. 426 00:31:20,000 --> 00:31:20,620 Bear with me a second. 427 00:31:20,620 --> 00:31:22,160 I just want to take a sip of water. 428 00:31:22,160 --> 00:31:28,270 And then I'm going to talk about the other notion of transport, 429 00:31:28,270 --> 00:31:30,173 which we are going to discuss-- 430 00:31:30,173 --> 00:31:32,590 I always say, briefly, and then I spend three pages on it, 431 00:31:32,590 --> 00:31:34,970 so we're going to discuss. 432 00:31:34,970 --> 00:31:36,090 All right. 433 00:31:36,090 --> 00:31:39,970 So parallel transport is extremely important, 434 00:31:39,970 --> 00:31:41,620 and there's a huge amount of physics 435 00:31:41,620 --> 00:31:45,330 that is tied up in this, but one thing 436 00:31:45,330 --> 00:31:49,330 which I really want to emphasize is that it is not unique. 437 00:31:49,330 --> 00:31:51,025 And there is one other one which we 438 00:31:51,025 --> 00:31:54,490 are going to really use to define one particularly 439 00:31:54,490 --> 00:31:57,970 important notion, instead of quantities, for our class. 440 00:32:10,210 --> 00:32:14,020 So suppose I've got my curve gamma, 441 00:32:14,020 --> 00:32:16,300 and I'm going to, again, take advantage of the fact 442 00:32:16,300 --> 00:32:23,680 that I can define a set of tick marks along it 443 00:32:23,680 --> 00:32:28,190 and make that vector U be the tangent to this curve. 444 00:32:28,190 --> 00:32:32,770 And I'm going to, again, have my favorite points, 445 00:32:32,770 --> 00:32:43,520 x alpha at point P plus dx alpha at point Q. 446 00:32:43,520 --> 00:32:46,760 There's another notion of transport that is-- basically 447 00:32:46,760 --> 00:32:50,030 what you do is you cheat, and you 448 00:32:50,030 --> 00:32:56,840 imagine that moving from point x alpha to x alpha plus dx alpha 449 00:32:56,840 --> 00:32:59,910 is a kind of coordinate transformation. 450 00:32:59,910 --> 00:33:01,790 So let's do the following. 451 00:33:01,790 --> 00:33:09,772 Let's say that x alpha plus dx alpha-- 452 00:33:09,772 --> 00:33:11,480 we're going to take advantage of the fact 453 00:33:11,480 --> 00:33:16,040 that, since we have this tangent notion built into the symbols 454 00:33:16,040 --> 00:33:21,380 we've defined, we'll just say that it's 455 00:33:21,380 --> 00:33:25,580 going to be the tangent times the interval of lambda. 456 00:33:25,580 --> 00:33:29,180 And what I'm going to do is define 457 00:33:29,180 --> 00:33:33,750 this as a new coordinate system, x prime. 458 00:33:33,750 --> 00:33:38,750 So that's the alpha component of coordinate system x prime. 459 00:33:38,750 --> 00:33:41,540 It's a little bit weird because your x prime has 460 00:33:41,540 --> 00:33:43,250 a differential built into it. 461 00:33:43,250 --> 00:33:44,000 Just bear with me. 462 00:33:46,700 --> 00:33:52,390 So what we're going to do is regard the shift, 463 00:33:52,390 --> 00:33:58,150 or the transport, if you prefer, from P to Q 464 00:33:58,150 --> 00:33:59,727 as a coordinate transformation. 465 00:34:17,780 --> 00:34:20,000 It's the best eraser, so I'll just keep using it. 466 00:34:26,889 --> 00:34:36,040 So what I mean by that is I'm going to regard x alpha, 467 00:34:36,040 --> 00:34:38,830 and I'm going to use a slightly different symbol. 468 00:34:38,830 --> 00:34:40,364 I will define what the L is. 469 00:34:40,364 --> 00:34:42,739 So this is transported, but I'm going to put an L in here 470 00:34:42,739 --> 00:34:44,739 for reasons that I will define in just a moment. 471 00:34:48,210 --> 00:35:02,830 This, from P to Q, is what I get if I regard the change 472 00:35:02,830 --> 00:35:05,620 from point P to point Q as a simple coordinate 473 00:35:05,620 --> 00:35:09,030 transformation and do my usual rule for changing 474 00:35:09,030 --> 00:35:11,670 coordinate representation. 475 00:35:11,670 --> 00:35:15,950 So expand what the definition of x prime is there, 476 00:35:15,950 --> 00:35:19,770 and what you'll see is that you get a term that's just 477 00:35:19,770 --> 00:35:22,080 basically dx alpha dx beta. 478 00:35:25,570 --> 00:35:27,000 Then you're going to get something 479 00:35:27,000 --> 00:35:29,790 that looks like the partial derivative 480 00:35:29,790 --> 00:35:30,840 of that tangent vector. 481 00:35:36,990 --> 00:35:41,030 And remember, this is being acted on. 482 00:35:41,030 --> 00:35:47,980 I should've said this is this thing evaluated at P. 483 00:35:47,980 --> 00:35:48,480 Great. 484 00:35:48,480 --> 00:35:49,530 So we fill this out. 485 00:36:07,960 --> 00:36:08,460 OK. 486 00:36:14,920 --> 00:36:19,760 So that's what I get when I use this notion to transport 487 00:36:19,760 --> 00:36:22,100 the field from P to Q. 488 00:36:22,100 --> 00:36:24,110 Let's think about it in another way. 489 00:36:24,110 --> 00:36:28,080 Now, these fields are all just functions. 490 00:36:28,080 --> 00:36:33,260 So I can also express the field at Q in terms of the field 491 00:36:33,260 --> 00:36:37,078 at P using a Taylor expansion. 492 00:36:39,710 --> 00:36:42,950 I'm assuming that these are close enough that everything 493 00:36:42,950 --> 00:37:37,840 is accurate to first order in small quantities, 494 00:37:37,840 --> 00:37:40,000 so nothing controversial about this. 495 00:37:40,000 --> 00:37:47,945 I'm assuming dx is small enough that I can do this. 496 00:38:01,990 --> 00:38:05,260 But now I'm going to get rid of my dx beta 497 00:38:05,260 --> 00:38:27,290 using the tangent field U. 498 00:38:27,290 --> 00:38:32,810 Now, before I move on, I just want to emphasize-- 499 00:38:32,810 --> 00:38:35,750 these two boards here, over the way the left-- 500 00:38:35,750 --> 00:38:38,610 we're talking about two rather different quantities. 501 00:38:38,610 --> 00:38:40,250 The one I just moved to the top-- 502 00:38:40,250 --> 00:38:49,670 that actually is the field-- if you were some kind of a gadget 503 00:38:49,670 --> 00:38:51,620 that managed your field A-- 504 00:38:51,620 --> 00:38:53,540 that would tell you what the value is 505 00:38:53,540 --> 00:38:58,520 that you measure at point Q. This would tell 506 00:38:58,520 --> 00:39:00,710 you-- what do we get if you picked P 507 00:39:00,710 --> 00:39:05,180 up and, via this transport mechanism, moved it over to Q? 508 00:39:05,180 --> 00:39:09,560 They are two potentially different things. 509 00:39:09,560 --> 00:39:13,088 So this motivates another kind of derivative. 510 00:39:35,040 --> 00:39:41,400 So suppose I look at A-- 511 00:39:41,400 --> 00:39:44,160 value it at a Q-- 512 00:39:44,160 --> 00:39:49,530 minus A transported-- whoops, that's supposed to be 513 00:39:49,530 --> 00:39:50,730 transported from P to Q-- 514 00:39:59,490 --> 00:40:02,210 defined by D lambda. 515 00:40:02,210 --> 00:40:04,440 I will expand this out in just a moment. 516 00:40:04,440 --> 00:40:06,970 Now I will, at last, give this a name. 517 00:40:06,970 --> 00:40:14,750 This is written with a script L. This 518 00:40:14,750 --> 00:40:24,580 is known as the Lie derivative of the vector A 519 00:40:24,580 --> 00:40:30,120 along U. Anyone heard of the Lie derivative before? 520 00:40:30,120 --> 00:40:31,110 Yeah. 521 00:40:31,110 --> 00:40:34,375 So at least in the context where we're going to be using it, 522 00:40:34,375 --> 00:40:36,750 this is a good way to understand what's going on with it. 523 00:40:36,750 --> 00:40:39,420 We'll see how it is used, at least in 8.962 524 00:40:39,420 --> 00:40:41,730 in just a few moments. 525 00:40:41,730 --> 00:40:44,670 Filling in the details-- so plug in these definitions, 526 00:40:44,670 --> 00:40:47,040 subtract, take limits, blah, blah, blah. 527 00:40:47,040 --> 00:40:54,540 What you find is that this turns out 528 00:40:54,540 --> 00:41:02,530 to be U contracted on the partial derivative of A 529 00:41:02,530 --> 00:41:13,320 minus A contracted on the partial derivative of U. 530 00:41:13,320 --> 00:41:15,530 Exercise for the reader-- 531 00:41:15,530 --> 00:41:18,540 it is actually really easy to show 532 00:41:18,540 --> 00:41:22,400 that you can promote these partial derivatives 533 00:41:22,400 --> 00:41:23,977 to covariant derivatives. 534 00:41:40,050 --> 00:41:42,755 And what this means is that, when you evaluate the Lie 535 00:41:42,755 --> 00:41:44,630 derivative-- so notice, nowhere in here 536 00:41:44,630 --> 00:41:46,838 did I introduce anything with a covariant derivative. 537 00:41:46,838 --> 00:41:48,890 There was no connection, nothing going on there. 538 00:41:48,890 --> 00:41:50,432 If you just go ahead and work it out, 539 00:41:50,432 --> 00:41:53,600 basically, when you expand this guy out, 540 00:41:53,600 --> 00:41:56,312 you'll find you have connection coefficients or Christoffel 541 00:41:56,312 --> 00:41:57,770 symbols that are equal and opposite 542 00:41:57,770 --> 00:41:59,270 and so they cancel each other. 543 00:41:59,270 --> 00:42:01,930 So you can just go from partials to covariants. 544 00:42:01,930 --> 00:42:04,070 Give me just a second, Trey. 545 00:42:04,070 --> 00:42:05,990 And this is telling us that the Lie derivative 546 00:42:05,990 --> 00:42:08,800 is perfectly tensorial. 547 00:42:08,800 --> 00:42:11,330 So the Lie derivative of the vector field 548 00:42:11,330 --> 00:42:13,610 is also a tensor quantity. 549 00:42:13,610 --> 00:42:16,681 You were asking a question, Trey. 550 00:42:16,681 --> 00:42:20,078 AUDIENCE: In the second term, did you miss the D lambda? 551 00:42:24,297 --> 00:42:25,130 SCOTT HUGHES: I did. 552 00:42:25,130 --> 00:42:25,630 Yes, I did. 553 00:42:25,630 --> 00:42:26,300 Thank you. 554 00:42:26,300 --> 00:42:28,970 There should be a D lambda right here. 555 00:42:28,970 --> 00:42:29,850 Thank you. 556 00:42:29,850 --> 00:42:30,540 Yes. 557 00:42:30,540 --> 00:42:31,040 Yeah. 558 00:42:31,040 --> 00:42:34,010 If you don't have that, then you get what 559 00:42:34,010 --> 00:42:35,670 is technically called "crap." 560 00:42:35,670 --> 00:42:38,360 So thank you for pointing that out. 561 00:42:38,360 --> 00:42:48,230 For reasons that I hope you have probably seen before, 562 00:42:48,230 --> 00:42:49,850 you always compute the Lie derivative 563 00:42:49,850 --> 00:42:53,630 of some kind of an object along a vector field. 564 00:42:57,698 --> 00:42:59,990 So when you're computing the Lie derivative of a vector 565 00:42:59,990 --> 00:43:06,690 field along a vector, sometimes this 566 00:43:06,690 --> 00:43:09,762 is written using a commutator. 567 00:43:09,762 --> 00:43:11,970 I just throw that out there because you may encounter 568 00:43:11,970 --> 00:43:13,220 this in some of your readings. 569 00:43:22,550 --> 00:43:25,820 It looks like this. 570 00:43:25,820 --> 00:43:28,245 So let me just do a few more things 571 00:43:28,245 --> 00:43:30,620 that are essentially fleshing out the definition of this. 572 00:43:30,620 --> 00:43:33,120 So I'm not going to go through and apply this definition 573 00:43:33,120 --> 00:43:35,840 very carefully to higher order objects. 574 00:43:35,840 --> 00:43:37,550 What I will just say is that, if I repeat 575 00:43:37,550 --> 00:43:43,687 this exercise and, instead of having a vector field 576 00:43:43,687 --> 00:43:45,770 that I'm transporting from point to point, suppose 577 00:43:45,770 --> 00:43:46,937 I do it for a scalar field-- 578 00:44:02,360 --> 00:44:04,750 well, what you actually get-- pardon me for a second-- 579 00:44:04,750 --> 00:44:11,468 is this on the partial, but the partial derivative of a scalar 580 00:44:11,468 --> 00:44:13,260 is the covariant derivative because there's 581 00:44:13,260 --> 00:44:14,700 no Christoffel that couples in. 582 00:44:21,840 --> 00:44:29,320 If you do this for a one-form, where it's a 1 indexed 583 00:44:29,320 --> 00:44:48,910 object in the downstairs position, 584 00:44:48,910 --> 00:44:51,030 you get something that looks like this. 585 00:44:51,030 --> 00:44:54,980 And again, when you expand out your covariant derivatives, 586 00:44:54,980 --> 00:44:58,530 you find that your Christoffel symbols cancel each other out. 587 00:44:58,530 --> 00:45:00,440 And so, if you like, you can just 588 00:45:00,440 --> 00:45:05,540 go ahead and replace these with partials. 589 00:45:05,540 --> 00:45:08,381 And likewise, let me just write one more out for completeness. 590 00:45:11,030 --> 00:45:12,080 Apply this to a tensor. 591 00:45:27,040 --> 00:45:30,280 So it's a very similar kind of structure 592 00:45:30,280 --> 00:45:36,190 to what you saw when we did the covariant derivative in which 593 00:45:36,190 --> 00:45:40,900 every index essentially gets corrected by a factor that 594 00:45:40,900 --> 00:45:43,270 looks like the covariant derivative of the field 595 00:45:43,270 --> 00:45:44,888 that you are differentiating along. 596 00:45:44,888 --> 00:45:46,430 The signs are a little bit different. 597 00:45:46,430 --> 00:45:51,290 So it's a similar tune, but it's in a different key. 598 00:45:57,500 --> 00:46:04,490 OK, so that's great. 599 00:46:04,490 --> 00:46:08,117 And if you get your jollies just understanding 600 00:46:08,117 --> 00:46:09,950 different mathematical transport operations, 601 00:46:09,950 --> 00:46:13,368 maybe this is already fun enough. 602 00:46:13,368 --> 00:46:15,410 But we're in a physics class, and so the question 603 00:46:15,410 --> 00:46:18,020 that should be to your mind is, is there 604 00:46:18,020 --> 00:46:20,910 a point to all this analysis? 605 00:46:20,910 --> 00:46:27,887 So in fact, the most important application of the Lie 606 00:46:27,887 --> 00:46:30,340 derivative for our purposes-- 607 00:46:33,040 --> 00:46:36,130 in probably the last lecture or two, 608 00:46:36,130 --> 00:46:42,610 I will describe some stuff related to modern research 609 00:46:42,610 --> 00:46:43,810 that uses it quite heavily. 610 00:46:43,810 --> 00:46:47,980 But to begin with in our class, the most important application 611 00:46:47,980 --> 00:47:01,610 will be when we consider cases where, 612 00:47:01,610 --> 00:47:05,880 when I compute the Lie derivative of some tensor 613 00:47:05,880 --> 00:47:09,200 along a vector U and I get 0. 614 00:47:24,932 --> 00:47:26,890 I'm just going to leave it schematic like that. 615 00:47:26,890 --> 00:47:30,400 So L U of the tensor is equal to 0. 616 00:47:30,400 --> 00:47:37,104 If this is the case, we say that the tensor is Lie transported. 617 00:47:52,600 --> 00:47:54,350 This is, incidentally, just a brief aside. 618 00:47:54,350 --> 00:47:56,900 It shows up a lot in fluid dynamics. 619 00:47:56,900 --> 00:48:01,670 In that case, U often defines the flow lines 620 00:48:01,670 --> 00:48:03,110 associated with the velocity field 621 00:48:03,110 --> 00:48:04,735 of some kind of a fluid that is flowing 622 00:48:04,735 --> 00:48:06,808 through your physical situation. 623 00:48:06,808 --> 00:48:08,600 And you would be interested in the behavior 624 00:48:08,600 --> 00:48:13,530 of all sorts of quantities that are embedded in that fluid. 625 00:48:13,530 --> 00:48:15,350 And as we're going to see, when you 626 00:48:15,350 --> 00:48:19,010 find that those quantities are Lie transported in this way, 627 00:48:19,010 --> 00:48:21,693 there is a powerful physical outcome 628 00:48:21,693 --> 00:48:23,360 associated with that, which we are going 629 00:48:23,360 --> 00:48:25,410 to derive in just a moment. 630 00:48:25,410 --> 00:48:34,819 So suppose I, in fact, have a tensor that is Lie transported. 631 00:49:05,290 --> 00:49:08,170 So suppose I have some tensor that is Lie transported. 632 00:49:08,170 --> 00:49:16,830 If that's the case, what I can do 633 00:49:16,830 --> 00:49:19,740 is define a particular coordinate system 634 00:49:19,740 --> 00:49:22,765 centered on the curve for which U is the tangent. 635 00:49:42,840 --> 00:49:47,250 So what I'm going to do is I'm going to define this curve such 636 00:49:47,250 --> 00:49:55,350 that x0 is equal to lambda, that parameter that defines 637 00:49:55,350 --> 00:49:58,530 my length along the curve in a way that, I will admit I've 638 00:49:58,530 --> 00:50:00,540 not made very precise yet but will soon. 639 00:50:07,130 --> 00:50:13,370 And then I'm going to require that my other three coordinates 640 00:50:13,370 --> 00:50:15,080 are all constant on that curve. 641 00:50:20,820 --> 00:50:47,780 So if I do that, then my tangent vector is simply delta x0. 642 00:50:47,780 --> 00:50:50,980 In other words, it's only got one non-trivial component, 643 00:50:50,980 --> 00:50:52,780 and its value of that component is 1. 644 00:50:58,210 --> 00:50:59,650 And this is the constant. 645 00:50:59,650 --> 00:51:03,580 So the derivatives of the tangent field 646 00:51:03,580 --> 00:51:06,520 are all equal to 0. 647 00:51:06,520 --> 00:51:26,360 And when you trace this through all of our various definitions, 648 00:51:26,360 --> 00:51:28,790 what you find is that it boils down to just 649 00:51:28,790 --> 00:51:31,970 looking at how the tensor field varies 650 00:51:31,970 --> 00:51:36,350 with respect to that parameter along the curve itself. 651 00:51:36,350 --> 00:51:39,375 If it's Lie transported, then this is equal to 0. 652 00:51:42,600 --> 00:51:47,870 And so this means that, whatever x0 represents, 653 00:51:47,870 --> 00:51:53,430 it's going to be a constant along that curve with respect 654 00:51:53,430 --> 00:51:54,776 to this tensor field. 655 00:51:59,060 --> 00:52:01,003 Oh, excuse me. 656 00:52:01,003 --> 00:52:01,670 Screwed that up. 657 00:52:05,470 --> 00:52:13,980 The tensor does not vary with this parameter along the curve. 658 00:52:27,070 --> 00:52:30,100 This was a lot, so let's just step back and think 659 00:52:30,100 --> 00:52:32,380 about what this is saying. 660 00:52:32,380 --> 00:52:37,255 One of the most important things that we do in physics when 661 00:52:37,255 --> 00:52:39,040 we're trying to analyze systems is 662 00:52:39,040 --> 00:52:40,960 we try to identify quantities that 663 00:52:40,960 --> 00:52:44,080 are constants of the motion. 664 00:52:44,080 --> 00:52:47,340 This is really tricky in a curved spacetime 665 00:52:47,340 --> 00:52:52,440 because much of our intuition gets 666 00:52:52,440 --> 00:52:56,010 garbled by all of the facts that different points have 667 00:52:56,010 --> 00:52:58,530 different tangent spaces. 668 00:52:58,530 --> 00:53:00,750 You worry about whether something being true, 669 00:53:00,750 --> 00:53:03,240 and is it just a function of the coordinate system 670 00:53:03,240 --> 00:53:04,560 that I wrote this out in? 671 00:53:04,560 --> 00:53:07,980 What the hell is going on here? 672 00:53:07,980 --> 00:53:12,390 The Lie derivative is giving us a covariant, frame-independent 673 00:53:12,390 --> 00:53:18,250 way of identifying things that are constants in our spacetime. 674 00:53:18,250 --> 00:53:22,950 So we're going to wrap up this discussion. 675 00:53:22,950 --> 00:53:25,890 Let's suppose that the tensor that I'm looking at here 676 00:53:25,890 --> 00:53:27,180 is called the metric. 677 00:53:51,470 --> 00:54:06,750 Suppose there exists a vector C such that the metric is Lie 678 00:54:06,750 --> 00:54:08,960 transported along this thing. 679 00:54:27,310 --> 00:54:28,420 What does this tell us? 680 00:54:28,420 --> 00:54:43,570 So first, it means there exists some coordinate such 681 00:54:43,570 --> 00:54:52,270 that the metric does not vary. 682 00:54:52,270 --> 00:54:55,090 The metric is constant with respect to that coordinate. 683 00:54:55,090 --> 00:54:58,120 Essentially, if you go through what I sketched a moment ago, 684 00:54:58,120 --> 00:54:59,920 this is telling us that the existence 685 00:54:59,920 --> 00:55:02,470 of this kind of a vector, which I'm going to give a name to 686 00:55:02,470 --> 00:55:05,240 in just a moment-- 687 00:55:05,240 --> 00:55:07,000 the existence of this thing demands 688 00:55:07,000 --> 00:55:10,274 that my metric is constant with respect to some coordinate. 689 00:55:14,360 --> 00:55:16,650 I am not going to prove the following statement. 690 00:55:16,650 --> 00:55:19,920 I will just state it, because, in some ways, 691 00:55:19,920 --> 00:55:23,025 the converse of that statement is even more powerful. 692 00:55:38,690 --> 00:55:54,640 If there is a coordinate, such that dgd, 693 00:55:54,640 --> 00:56:01,318 whatever that coordinate is, is equal to 0, 694 00:56:01,318 --> 00:56:02,985 then a vector field of this type exists. 695 00:56:14,860 --> 00:56:18,423 So the second thing I want to do is expand the Lie derivative. 696 00:56:25,530 --> 00:56:32,810 So if I require that my metric be transported along the vector 697 00:56:32,810 --> 00:56:44,047 C, well, insert my definition of the Lie derivative. 698 00:57:02,570 --> 00:57:05,120 Now, what is the main defining characteristic 699 00:57:05,120 --> 00:57:06,980 of the covariant derivative? 700 00:57:06,980 --> 00:57:10,142 How did I get my connection in the first place? 701 00:57:10,142 --> 00:57:11,850 In other words, what is this going to be? 702 00:57:14,750 --> 00:57:17,300 OK, students who took undergraduate classes with me, 703 00:57:17,300 --> 00:57:19,300 I'll remind you of one of the key bits of wisdom 704 00:57:19,300 --> 00:57:20,175 I always tell people. 705 00:57:20,175 --> 00:57:24,040 If the professor asks you a question, 90% of the time, 706 00:57:24,040 --> 00:57:27,220 if you just shout out, "0," you are likely to be right. 707 00:57:27,220 --> 00:57:28,973 [LAUGHING] 708 00:57:28,973 --> 00:57:30,640 Usually, there's some kind of a symmetry 709 00:57:30,640 --> 00:57:33,190 that we want you to understand, which allows you to go, 710 00:57:33,190 --> 00:57:34,448 oh, it's equal to 0. 711 00:57:34,448 --> 00:57:36,490 By the way, whenever I point that out to a class, 712 00:57:36,490 --> 00:57:39,040 I then work really hard to make a non-zero answer 713 00:57:39,040 --> 00:57:40,310 for the next time I ask it. 714 00:57:40,310 --> 00:57:45,640 So the covariant derivative of g is 0, so this term dies. 715 00:57:45,640 --> 00:57:48,850 Because the covariant derivative of g is 0, 716 00:57:48,850 --> 00:57:52,940 I can always commute the metric with covariant derivatives. 717 00:57:52,940 --> 00:57:55,540 So I can take this, move it inside the derivative. 718 00:57:55,540 --> 00:57:58,129 I can take this, move it inside the derivative. 719 00:58:19,180 --> 00:58:23,590 So what this means is this Lie derivative equation, 720 00:58:23,590 --> 00:58:35,090 after all the smoke clears, can be written like this. 721 00:58:35,090 --> 00:58:38,210 Or, if I recall, there's this notation 722 00:58:38,210 --> 00:58:42,633 for symmetry of indices, which I introduced 723 00:58:42,633 --> 00:58:43,550 in a previous lecture. 724 00:58:43,550 --> 00:58:47,660 The symmetric covariant derivative of this C 725 00:58:47,660 --> 00:58:48,920 is equal to 0. 726 00:58:48,920 --> 00:58:58,350 This equation is known as Killing's equation, 727 00:58:58,350 --> 00:59:07,620 and C is a Killing vector. 728 00:59:07,620 --> 00:59:10,710 Now, this was a fair amount of formalism. 729 00:59:10,710 --> 00:59:14,520 I was really laying out a lot of the details to get this right. 730 00:59:14,520 --> 00:59:20,807 So to give you some context as to why this matters, 731 00:59:20,807 --> 00:59:22,890 there's a bit more that needs to come out of this, 732 00:59:22,890 --> 00:59:26,460 but we're going to get to it very soon. 733 00:59:26,460 --> 00:59:28,560 Suppose I have a body that is freely 734 00:59:28,560 --> 00:59:30,900 falling through some spacetime. 735 00:59:30,900 --> 00:59:31,650 And you know what? 736 00:59:31,650 --> 00:59:32,858 I'm going to leave this here. 737 00:59:32,858 --> 00:59:35,427 So this is a slightly advanced tangent, 738 00:59:35,427 --> 00:59:36,510 so I'll start a new board. 739 01:00:00,370 --> 01:00:03,130 So if I have some body that is freely falling, 740 01:00:03,130 --> 01:00:05,840 what we are going to show in, probably, 741 01:00:05,840 --> 01:00:10,720 Thursday's lecture is that the equation of motion that governs 742 01:00:10,720 --> 01:00:12,830 it is-- you can argue this on physical grounds, 743 01:00:12,830 --> 01:00:14,510 and that's all I will do for now-- 744 01:00:14,510 --> 01:00:18,710 it's a trajectory that parallel transports its own tangent 745 01:00:18,710 --> 01:00:19,580 factor. 746 01:00:19,580 --> 01:00:22,010 For intuition, go into the freely falling frame 747 01:00:22,010 --> 01:00:24,500 where it's just the trajectory from special relativity. 748 01:00:24,500 --> 01:00:26,900 It's a straight line in that frame, 749 01:00:26,900 --> 01:00:29,580 and parallel transporting its own tangent vector 750 01:00:29,580 --> 01:00:34,430 basically means it just moves on whatever course it is going. 751 01:00:34,430 --> 01:00:49,010 So this is a trajectory for which 752 01:00:49,010 --> 01:00:52,610 I demand that the four-velocity governing it parallel 753 01:00:52,610 --> 01:00:55,790 transports along itself. 754 01:00:55,790 --> 01:00:59,150 Now, suppose you are moving in a spacetime that 755 01:00:59,150 --> 01:01:00,530 has a Killing vector. 756 01:01:15,470 --> 01:01:17,240 So this will be Thursday's lecture. 757 01:01:22,080 --> 01:01:24,600 Suppose the spacetime has a Killing vector. 758 01:01:24,600 --> 01:01:27,990 Well, so there will be some goofy C that you know exists, 759 01:01:27,990 --> 01:01:31,860 and you know C obeys this equation. 760 01:01:31,860 --> 01:01:33,900 By combining these things, you can 761 01:01:33,900 --> 01:01:41,220 show that there is some quantity, C, 762 01:01:41,220 --> 01:01:43,830 which is given by taking the inner product 763 01:01:43,830 --> 01:01:47,400 of the four-velocity of this freely falling thing 764 01:01:47,400 --> 01:01:49,260 and the Killing vector. 765 01:01:49,260 --> 01:01:51,960 And you can prove that this is a constant of the motion. 766 01:02:06,018 --> 01:02:08,560 So let's think about where this goes with some of the physics 767 01:02:08,560 --> 01:02:11,500 that you presumably all know and love already. 768 01:02:11,500 --> 01:02:15,100 Suppose you look at a spacetime. 769 01:02:15,100 --> 01:02:18,465 So you climb a really high mountain. 770 01:02:18,465 --> 01:02:20,590 You discover that there's a spacetime metric carved 771 01:02:20,590 --> 01:02:22,660 into the stone into the top of it. 772 01:02:22,660 --> 01:02:24,760 You think, OK, this probably matters. 773 01:02:24,760 --> 01:02:28,870 You look at it and you notice it depends on, say, time, radius, 774 01:02:28,870 --> 01:02:29,710 and two angles. 775 01:02:34,150 --> 01:02:39,250 Suppose you have a metric that is time-independent. 776 01:02:46,160 --> 01:02:49,040 Hey, if it's time-independent, then I 777 01:02:49,040 --> 01:02:53,450 know that the derivative of that thing with respect to time 778 01:02:53,450 --> 01:02:54,500 is 0. 779 01:02:54,500 --> 01:02:57,320 There must exist a Killing factor 780 01:02:57,320 --> 01:03:00,020 that is related to the fact that there is no time 781 01:03:00,020 --> 01:03:01,150 dependence in this metric. 782 01:03:22,800 --> 01:03:25,675 So you go and you calculate it. 783 01:03:25,675 --> 01:03:27,800 So this thing, that C is a constant of the motion-- 784 01:03:35,838 --> 01:03:36,880 I believe that's P set 4. 785 01:03:41,560 --> 01:03:42,340 It's not hard. 786 01:03:42,340 --> 01:03:44,170 You combine that equation that we're going to derive, 787 01:03:44,170 --> 01:03:46,000 called the geodesic equation, with Killing's equation. 788 01:03:46,000 --> 01:03:46,600 Math happens. 789 01:03:46,600 --> 01:03:47,408 You got it. 790 01:03:47,408 --> 01:03:49,575 So suppose you've got a metric that's time-dependent 791 01:03:49,575 --> 01:03:51,210 and you know you've got this thing. 792 01:03:51,210 --> 01:03:51,918 So you know what? 793 01:03:51,918 --> 01:03:53,860 Let's work it out and look at it. 794 01:03:53,860 --> 01:03:56,080 It becomes clear, after studying this 795 01:03:56,080 --> 01:04:10,340 for a little bit, that the C for this Killing vector is energy. 796 01:04:10,340 --> 01:04:12,620 In the same way that, if you have a time-independent 797 01:04:12,620 --> 01:04:15,710 Lagrangian, your system has a conserved energy, 798 01:04:15,710 --> 01:04:18,110 if you have a time-independent metric, 799 01:04:18,110 --> 01:04:19,777 there is a Killing vector, which-- 800 01:04:19,777 --> 01:04:21,110 the language we like to use is-- 801 01:04:21,110 --> 01:04:23,690 we say the motion of that spacetime 802 01:04:23,690 --> 01:04:26,660 emits a conserved energy. 803 01:04:26,660 --> 01:04:30,890 Suppose you find that the metric is independent of some angle. 804 01:04:30,890 --> 01:04:33,710 We'll call it phi. 805 01:04:33,710 --> 01:04:35,310 Three guesses what's going to happen. 806 01:04:35,310 --> 01:04:37,580 In this case-- just one guess, actually. 807 01:04:37,580 --> 01:04:38,105 Conserved-- 808 01:04:38,105 --> 01:04:39,230 AUDIENCE: Angular momentum. 809 01:04:39,230 --> 01:04:41,490 SCOTT HUGHES: Angular momentum pops out in that case. 810 01:04:41,490 --> 01:04:45,290 So this ends up being the way in which we, essentially, 811 01:04:45,290 --> 01:04:47,750 make very rigorous and geometric the idea 812 01:04:47,750 --> 01:04:53,572 that conservation laws are put into general relativity, OK? 813 01:04:53,572 --> 01:04:55,530 So I realize there's a lot of abstraction here. 814 01:04:55,530 --> 01:04:57,238 So I want to go on a bit of an aside just 815 01:04:57,238 --> 01:04:58,970 to tie down where we are going with this 816 01:04:58,970 --> 01:05:01,760 and why this actually matters. 817 01:05:01,760 --> 01:05:04,575 OK. 818 01:05:04,575 --> 01:05:05,110 Let's see. 819 01:05:05,110 --> 01:05:07,230 So we got about 10 minutes left. 820 01:05:07,230 --> 01:05:12,000 So what we're going to do at the very end of today-- 821 01:05:12,000 --> 01:05:14,050 and we'll pick this up beginning of next time. 822 01:05:14,050 --> 01:05:17,870 So for the people who walked in a few minutes late, 823 01:05:17,870 --> 01:05:21,165 the stuff that I'm actually about to start talking about we 824 01:05:21,165 --> 01:05:23,540 need to get through before you can do one of the problems 825 01:05:23,540 --> 01:05:24,370 on the P set. 826 01:05:24,370 --> 01:05:26,120 So I'm probably going to take that problem 827 01:05:26,120 --> 01:05:28,016 and move it on to P set 4, but I'm 828 01:05:28,016 --> 01:05:30,560 going to start talking about it right now. 829 01:05:30,560 --> 01:05:39,980 So we've really focused a lot, so far, on tensors. 830 01:05:42,650 --> 01:05:48,530 We're going to now start talking about a related quantity called 831 01:05:48,530 --> 01:05:49,850 tensor densities. 832 01:05:56,080 --> 01:05:58,523 There's really only two that matter for our purposes, 833 01:05:58,523 --> 01:06:00,190 but I want to go through them carefully. 834 01:06:00,190 --> 01:06:01,780 So I will set up with one, and then we'll 835 01:06:01,780 --> 01:06:03,460 conclude the other one at the beginning 836 01:06:03,460 --> 01:06:04,900 of Thursday's lecture. 837 01:06:04,900 --> 01:06:07,950 So let me define this first. 838 01:06:07,950 --> 01:06:09,700 I'm going to give a definition that I like 839 01:06:09,700 --> 01:06:11,260 but that's actually kind of stupid. 840 01:06:11,260 --> 01:06:24,250 So these are quantities that transform almost like tensors-- 841 01:06:28,890 --> 01:06:31,060 a little bit lame, but, as you'll see in a moment, 842 01:06:31,060 --> 01:06:32,100 it's kind of accurate. 843 01:06:32,100 --> 01:06:35,520 What you'll find is that the transformation law 844 01:06:35,520 --> 01:06:53,003 is off by a factor that is the determinant 845 01:06:53,003 --> 01:06:54,670 of the coordinate transformation matrix. 846 01:07:08,233 --> 01:07:09,150 Take it to some power. 847 01:07:13,493 --> 01:07:15,660 So there's is an infinite number of tensor densities 848 01:07:15,660 --> 01:07:17,130 that one could define. 849 01:07:17,130 --> 01:07:19,800 Two are important for this class. 850 01:07:42,642 --> 01:07:44,350 So the two that are most important for us 851 01:07:44,350 --> 01:07:54,780 are the Levi-Civita symbol and the determinant of the metric. 852 01:08:00,870 --> 01:08:05,580 So we use Levi-Civita already to talk about volumes. 853 01:08:05,580 --> 01:08:08,012 And it was a tensor when we were working 854 01:08:08,012 --> 01:08:10,470 in rectilinear coordinates, where the underlying coordinate 855 01:08:10,470 --> 01:08:14,100 system was essentially Cartesian plus time. 856 01:08:14,100 --> 01:08:15,484 It's not in general, OK? 857 01:08:15,484 --> 01:08:16,859 And we'll go through why that is. 858 01:08:16,859 --> 01:08:19,710 That'll probably be the last thing we can fit in today. 859 01:08:19,710 --> 01:08:21,450 So let me remind you-- 860 01:08:24,677 --> 01:08:29,490 Levi-Civita-- I'm going to write it with a tilde on it 861 01:08:29,490 --> 01:08:34,680 to emphasize that it is not tensorial. 862 01:08:37,229 --> 01:08:40,550 So this is equal to plus 1 if the indices 863 01:08:40,550 --> 01:08:45,529 are 0, 1, 2, 3 and even permutations 864 01:08:45,529 --> 01:08:53,960 of that equals minus 1 for odd permutations of that. 865 01:08:56,710 --> 01:08:59,090 And it's 0 for any index repeated. 866 01:09:11,160 --> 01:09:14,380 Now, this symbol has a really nice property when 867 01:09:14,380 --> 01:09:16,090 you apply it to any matrix. 868 01:09:16,090 --> 01:09:17,979 In fact, this is a theorem. 869 01:09:35,260 --> 01:09:36,979 So I'm working in four-dimensional space. 870 01:09:36,979 --> 01:09:44,920 So let's say I've got a 4-by-4 matrix, which I will call m. 871 01:09:44,920 --> 01:09:48,580 Write a new [INAUDIBLE] notation, m alpha mu. 872 01:09:48,580 --> 01:10:11,500 If I evaluate Levi-Civita, contract it on these guys, 873 01:10:11,500 --> 01:10:18,050 I get Levi-Civita back, multiply it 874 01:10:18,050 --> 01:10:21,360 by the determinant of the matrix m. 875 01:10:31,620 --> 01:10:34,050 Now, suppose what I choose for my matrix 876 01:10:34,050 --> 01:10:36,480 m is my coordinate transformation matrix. 877 01:11:12,780 --> 01:11:14,930 So I'm just going to write down this result, 878 01:11:14,930 --> 01:11:20,480 and I'll leave it since we're running a little short on time. 879 01:11:20,480 --> 01:11:23,908 You can just double check that I've moved things 880 01:11:23,908 --> 01:11:25,700 from one side of the equation to the other, 881 01:11:25,700 --> 01:11:27,825 and you can just double-check I did that correctly. 882 01:11:31,010 --> 01:11:34,430 What that tells me is that Levi-Civita 883 01:11:34,430 --> 01:11:39,450 and a new set of prime coordinates 884 01:11:39,450 --> 01:11:43,070 is equal to this guy in the old, unprimed 885 01:11:43,070 --> 01:11:53,590 coordinates with all my usual factors of transformation 886 01:11:53,590 --> 01:12:11,990 matrices and then an extra bit that 887 01:12:11,990 --> 01:12:14,780 is the determinant of the coordinate transformation 888 01:12:14,780 --> 01:12:16,340 matrix. 889 01:12:16,340 --> 01:12:18,050 If it were just the top line, this 890 01:12:18,050 --> 01:12:20,100 is exactly what you would need for Levi-Civita 891 01:12:20,100 --> 01:12:23,205 to be a tensor in the way that we have defined tensors. 892 01:12:23,205 --> 01:12:23,705 It's not. 893 01:12:44,390 --> 01:12:48,370 So the extra factor pushes away from a tensor relationship. 894 01:12:48,370 --> 01:12:51,100 And so what we would say is, because this is off 895 01:12:51,100 --> 01:13:02,610 by a factor of what's sometimes called the Jacobian, 896 01:13:02,610 --> 01:13:07,290 we call this a tensor density of weight 1. 897 01:13:16,590 --> 01:13:19,320 So in order to do this properly-- 898 01:13:19,320 --> 01:13:21,740 I don't want to rush-- 899 01:13:21,740 --> 01:13:24,390 at the beginning of the next lecture, 900 01:13:24,390 --> 01:13:27,890 we're going to look at how the determinant of the metric 901 01:13:27,890 --> 01:13:29,167 behaves. 902 01:13:29,167 --> 01:13:31,250 And what we'll see is that, although the metric is 903 01:13:31,250 --> 01:13:34,100 a tensor, its determinant is a tensor 904 01:13:34,100 --> 01:13:36,950 density of weight negative 2. 905 01:13:36,950 --> 01:13:39,920 And so what that tells us is that I can actually 906 01:13:39,920 --> 01:13:43,370 put together a combination of the Levi-Civita 907 01:13:43,370 --> 01:13:46,730 and the determinant of the metric in such a way 908 01:13:46,730 --> 01:13:49,328 that their product is tensorial. 909 01:13:49,328 --> 01:13:51,245 And that turns out to be real useful because I 910 01:13:51,245 --> 01:13:54,050 can use this to define, in a curved spacetime, 911 01:13:54,050 --> 01:13:57,170 covariant volume elements, OK? 912 01:13:57,170 --> 01:13:58,923 With this as written, my volume elements-- 913 01:13:58,923 --> 01:14:00,590 if I just use this like I did when we're 914 01:14:00,590 --> 01:14:03,020 taught about special relativity, my volume elements won't 915 01:14:03,020 --> 01:14:06,800 be elements of a tensor, and a lot of the framework 916 01:14:06,800 --> 01:14:09,170 that we've developed goes to hell. 917 01:14:09,170 --> 01:14:11,510 So an extra factor of the determining of the metric 918 01:14:11,510 --> 01:14:13,220 will allow us to correct this. 919 01:14:13,220 --> 01:14:16,770 And this seems kind of abstract. 920 01:14:16,770 --> 01:14:19,490 So let me just, as a really brief aside, before we conclude 921 01:14:19,490 --> 01:14:22,200 today's class-- 922 01:14:22,200 --> 01:14:26,980 suppose I'm just in Euclidean three-space 923 01:14:26,980 --> 01:14:30,070 and I'm working in spherical coordinates. 924 01:14:38,490 --> 01:14:41,390 So here's my line element. 925 01:14:41,390 --> 01:14:51,240 My metric is the diagonal of 1r squared 926 01:14:51,240 --> 01:14:55,080 r squared sine squared theta. 927 01:14:55,080 --> 01:14:59,284 The determinant of the metric, which I will write g-- 928 01:15:02,450 --> 01:15:06,560 it's r to the fourth sine squared theta. 929 01:15:06,560 --> 01:15:08,060 What we're going to learn when we do 930 01:15:08,060 --> 01:15:14,380 this is that the metric is a tensor density of weight 2. 931 01:15:14,380 --> 01:15:16,710 And so to correct it to get something of weight 1, 932 01:15:16,710 --> 01:15:17,670 we take a square root. 933 01:15:23,153 --> 01:15:24,820 If you're working in circle coordinates, 934 01:15:24,820 --> 01:15:26,720 does that look familiar? 935 01:15:26,720 --> 01:15:28,990 This is, in fact, exactly what allows 936 01:15:28,990 --> 01:15:31,853 us to convert differentials of our coordinates. 937 01:15:31,853 --> 01:15:33,770 Remember, we're working in a coordinate basis. 938 01:15:33,770 --> 01:15:36,220 And so we think of our little element 939 01:15:36,220 --> 01:15:38,200 of just the coordinates. 940 01:15:38,200 --> 01:15:41,290 It's just dr, d theta, d phi. 941 01:15:41,290 --> 01:15:42,880 This ends up being the quantity that 942 01:15:42,880 --> 01:15:46,390 allows us to convert the little triple of our coordinates 943 01:15:46,390 --> 01:15:48,730 into something that has the proper dimensions 944 01:15:48,730 --> 01:15:51,880 and form to actually be a real volume element. 945 01:15:51,880 --> 01:15:54,078 And so dr, de theta, d phi-- 946 01:15:54,078 --> 01:15:55,120 that ain't enough volume. 947 01:15:55,120 --> 01:15:58,720 But r squared sine theta, dr, d theta, d phi-- 948 01:15:58,720 --> 01:16:00,273 that's a volume element, OK? 949 01:16:00,273 --> 01:16:02,440 So basically, that's all that we're doing right now, 950 01:16:02,440 --> 01:16:04,510 is we're making that precise and careful. 951 01:16:04,510 --> 01:16:06,135 And that's where I will pick things up. 952 01:16:06,135 --> 01:16:08,190 We'll finish that up on Thursday.