1 00:00:00,500 --> 00:00:01,446 [SQUEAKING] 2 00:00:01,446 --> 00:00:04,338 [RUSTLING] 3 00:00:04,338 --> 00:00:06,266 [CLICKING] 4 00:00:11,053 --> 00:00:12,720 SCOTT HUGHES: All right, good afternoon. 5 00:00:12,720 --> 00:00:18,540 So today's lecture is a particularly important one. 6 00:00:18,540 --> 00:00:22,890 We are going to introduce the main physical principle that 7 00:00:22,890 --> 00:00:25,680 underlies general relativity. 8 00:00:25,680 --> 00:00:28,320 We'll be spending a bunch of time connecting that principle 9 00:00:28,320 --> 00:00:31,050 to the mathematics as the rest of the term unwinds. 10 00:00:31,050 --> 00:00:33,210 But today is where we're going to really lay out 11 00:00:33,210 --> 00:00:35,400 where the physics is in what is known 12 00:00:35,400 --> 00:00:37,230 as the principle of equivalence. 13 00:00:37,230 --> 00:00:39,360 So before I get into that, just a quick recap. 14 00:00:39,360 --> 00:00:42,700 So in terms of technical stuff we did last time, 15 00:00:42,700 --> 00:00:44,700 the most important thing we did was 16 00:00:44,700 --> 00:00:46,830 to introduce these mathematical objects called 17 00:00:46,830 --> 00:00:48,750 Christoffel symbols. 18 00:00:48,750 --> 00:00:52,590 Christoffel symbols are those capital gammas. 19 00:00:52,590 --> 00:00:54,660 We began by thinking of them as just what 20 00:00:54,660 --> 00:00:56,940 you get when you look at the derivative of your basis 21 00:00:56,940 --> 00:00:58,920 objects. 22 00:00:58,920 --> 00:01:00,578 Pardon me a second. 23 00:01:00,578 --> 00:01:02,370 There was a lot of chalk on the chalkboard, 24 00:01:02,370 --> 00:01:06,330 and I think I inhaled half of it while I was cleaning it. 25 00:01:06,330 --> 00:01:08,160 So take derivatives of your basis vectors. 26 00:01:08,160 --> 00:01:11,520 And there, you get something that's linearly related 27 00:01:11,520 --> 00:01:12,690 to your basis objects. 28 00:01:12,690 --> 00:01:17,190 And the gamma is the set of mathematical quantities 29 00:01:17,190 --> 00:01:18,900 that sets that linear relationship 30 00:01:18,900 --> 00:01:22,860 between the derivatives and the objects themselves. 31 00:01:22,860 --> 00:01:25,320 One can build under the demand-- 32 00:01:25,320 --> 00:01:27,150 so recall how we did this. 33 00:01:27,150 --> 00:01:29,340 We have a physical argument that tells us 34 00:01:29,340 --> 00:01:32,880 that a quantity we call the covariant derivative, which 35 00:01:32,880 --> 00:01:36,300 is a way of using these Christoffel symbols in order 36 00:01:36,300 --> 00:01:38,560 to get tensorial derivatives. 37 00:01:38,560 --> 00:01:42,630 So if I look at just the partial derivative of a tensor, 38 00:01:42,630 --> 00:01:45,900 if I'm working in a general curvilinear coordinate system, 39 00:01:45,900 --> 00:01:47,730 the set of partial derivatives do not 40 00:01:47,730 --> 00:01:50,200 constitute components of a tensor. 41 00:01:50,200 --> 00:01:50,700 OK? 42 00:01:50,700 --> 00:01:55,020 I posted some notes online, demonstrating explicitly 43 00:01:55,020 --> 00:01:56,400 why that is. 44 00:01:56,400 --> 00:01:57,960 But in a nutshell, the main reason 45 00:01:57,960 --> 00:02:00,730 is that, when you try to work this out 46 00:02:00,730 --> 00:02:02,733 or when you go and you work this out, 47 00:02:02,733 --> 00:02:04,650 you will find that there are additional terms. 48 00:02:04,650 --> 00:02:07,710 It was like the derivative of the transformation matrix that 49 00:02:07,710 --> 00:02:10,110 come in and spoil the tensoriality 50 00:02:10,110 --> 00:02:11,568 of this relationship. 51 00:02:11,568 --> 00:02:13,110 And what the Christoffel symbol does, 52 00:02:13,110 --> 00:02:15,200 as you guys are going to prove on P set three, 53 00:02:15,200 --> 00:02:17,580 it introduces terms that are exactly 54 00:02:17,580 --> 00:02:19,470 the same but with the opposite sign 55 00:02:19,470 --> 00:02:21,160 and cancel those things out. 56 00:02:21,160 --> 00:02:23,700 So that the covariant derivative is 57 00:02:23,700 --> 00:02:25,740 what I get when I correct each index 58 00:02:25,740 --> 00:02:28,110 with an appropriate factor of the Christoffel symbol. 59 00:02:28,110 --> 00:02:31,540 It is something that is a component of a tensor. 60 00:02:31,540 --> 00:02:33,750 In other words, it obeys the transformation laws 61 00:02:33,750 --> 00:02:36,480 that tensors must have. 62 00:02:36,480 --> 00:02:38,790 We have a physical argument that essentially 63 00:02:38,790 --> 00:02:42,330 is something that is tensorial, in other words, both 64 00:02:42,330 --> 00:02:45,960 sides of the equations are good tensor quantities in one 65 00:02:45,960 --> 00:02:48,210 representation, have to be tensorial 66 00:02:48,210 --> 00:02:50,010 in any other representation. 67 00:02:50,010 --> 00:02:52,440 And from that, we deduced that the covariant derivative 68 00:02:52,440 --> 00:02:55,080 of the metric must always be 0. 69 00:02:55,080 --> 00:02:57,160 And by taking appropriate covariant derivatives 70 00:02:57,160 --> 00:03:00,030 of the metric, sort of doing a bit of gymnastics with indices 71 00:03:00,030 --> 00:03:02,320 and sort of wiggling things around a little bit, 72 00:03:02,320 --> 00:03:04,350 we found that the Christoffel symbol can itself 73 00:03:04,350 --> 00:03:08,130 be built out of partial derivatives of the metric. 74 00:03:08,130 --> 00:03:10,710 And the result looks like this. 75 00:03:10,710 --> 00:03:14,070 So that's a little bit more of an in-depth recap 76 00:03:14,070 --> 00:03:15,917 than I typically do. 77 00:03:15,917 --> 00:03:18,000 But I really want to hammer that home because this 78 00:03:18,000 --> 00:03:19,690 is an important point. 79 00:03:19,690 --> 00:03:20,967 We then began switching gears. 80 00:03:20,967 --> 00:03:22,300 And we're going to do something. 81 00:03:22,300 --> 00:03:27,030 So I introduced, at the end of the previous lecture, a fairly 82 00:03:27,030 --> 00:03:29,970 silly argument, which, nonetheless, 83 00:03:29,970 --> 00:03:33,550 has an important physical concept to it. 84 00:03:33,550 --> 00:03:36,630 So where we were going with was the fact that, 85 00:03:36,630 --> 00:03:57,660 when we have gravity, we cannot cover all of spacetime with 86 00:03:57,660 --> 00:04:02,950 inertial frames, or I should say-- 87 00:04:02,950 --> 00:04:04,520 I should watch my wording here-- 88 00:04:04,520 --> 00:04:07,620 with an inertial frame. 89 00:04:07,620 --> 00:04:08,120 OK? 90 00:04:08,120 --> 00:04:09,200 We're, in fact, going to see that you 91 00:04:09,200 --> 00:04:10,820 can have a sequence of them. 92 00:04:10,820 --> 00:04:13,050 And we're going to find ways of linking them up. 93 00:04:13,050 --> 00:04:15,260 But for now, if you think of special relativity 94 00:04:15,260 --> 00:04:17,180 as the theory of physics in which we 95 00:04:17,180 --> 00:04:20,360 can have Lorentz coordinates that cover the entire universe, 96 00:04:20,360 --> 00:04:22,140 that's out the window. 97 00:04:22,140 --> 00:04:24,650 OK, and so I'm going to skip my silly derivation. 98 00:04:24,650 --> 00:04:27,650 Part one of it is essentially just a motivation 99 00:04:27,650 --> 00:04:29,765 that a gravitational redshift exists. 100 00:04:38,170 --> 00:04:39,910 So I gave a silly little demonstration 101 00:04:39,910 --> 00:04:42,550 where you imagined dropping a rock off of a tower. 102 00:04:42,550 --> 00:04:44,800 You had some kind of a magical device that converts it 103 00:04:44,800 --> 00:04:46,390 into a photon that shoots it back up 104 00:04:46,390 --> 00:04:48,460 and then converts it back into a rock. 105 00:04:48,460 --> 00:04:50,360 And you showed that, as it climbed out 106 00:04:50,360 --> 00:04:52,390 of the gravitational potential, it 107 00:04:52,390 --> 00:04:54,295 had to lose some of its energy. 108 00:04:54,295 --> 00:04:56,170 Or else, you would get more energy at the top 109 00:04:56,170 --> 00:04:58,087 than you put in when you initially dropped it. 110 00:04:58,087 --> 00:05:00,580 It's a way of preventing you from making perpetual motion 111 00:05:00,580 --> 00:05:02,298 machines. 112 00:05:02,298 --> 00:05:04,090 That is, without a doubt, a flaky argument. 113 00:05:04,090 --> 00:05:05,840 It can be made a little bit more rigorous. 114 00:05:05,840 --> 00:05:08,380 But I didn't want to. 115 00:05:08,380 --> 00:05:11,860 And I will justify the reason I can do this is that this is, 116 00:05:11,860 --> 00:05:17,430 in fact, a highly tested experimental fact. 117 00:05:23,520 --> 00:05:25,020 Every single one of you has probably 118 00:05:25,020 --> 00:05:27,450 used this if you've ever used GPS 119 00:05:27,450 --> 00:05:29,070 to find your way to some location 120 00:05:29,070 --> 00:05:31,390 where you're supposed to meet a friend. 121 00:05:31,390 --> 00:05:34,800 And what this basically tells me is that, 122 00:05:34,800 --> 00:05:40,040 if I imagine a tower of height h, 123 00:05:40,040 --> 00:05:45,290 if I shoot light with frequency at the bottom, omega 124 00:05:45,290 --> 00:05:49,720 b, when it reaches the top, it'll 125 00:05:49,720 --> 00:05:52,020 have a different frequency, omega t, 126 00:05:52,020 --> 00:05:55,270 where these are related by omega t 127 00:05:55,270 --> 00:06:01,300 equals, and I'm going to put my c squareds back in there, just 128 00:06:01,300 --> 00:06:03,410 to keep it all complete. 129 00:06:03,410 --> 00:06:03,910 OK? 130 00:06:03,910 --> 00:06:09,610 So this is, I emphasize, a very precisely 131 00:06:09,610 --> 00:06:12,280 calibrated experimental result. There 132 00:06:12,280 --> 00:06:15,050 are many ways of doing it, some silly, some serious. 133 00:06:15,050 --> 00:06:16,300 The silly one was sort of fun. 134 00:06:16,300 --> 00:06:18,600 But it does actually capture the experimental fact. 135 00:06:18,600 --> 00:06:19,100 OK? 136 00:06:19,100 --> 00:06:22,450 The key experimental fact is that it's necessary for light 137 00:06:22,450 --> 00:06:25,690 to lose energy as it climbs out of the gravitational field, 138 00:06:25,690 --> 00:06:29,170 in essence, because it's possible, thanks to quantum 139 00:06:29,170 --> 00:06:32,230 field theory, to convert photons into particles and particles 140 00:06:32,230 --> 00:06:33,040 into photons. 141 00:06:33,040 --> 00:06:36,070 This is a way of conserving energy. 142 00:06:36,070 --> 00:06:38,920 OK, so this was just part one of this argument that we cannot 143 00:06:38,920 --> 00:06:42,140 cover all of spacetime with an inertial frame. 144 00:06:45,542 --> 00:06:47,000 So here's part two of the argument. 145 00:06:50,750 --> 00:06:53,945 Suppose there was, in fact, a very large region of space-- 146 00:06:53,945 --> 00:06:55,820 now let's say it's near the Earth's surface-- 147 00:06:55,820 --> 00:06:58,310 that could be covered by a single Lorentz frame. 148 00:07:23,260 --> 00:07:23,780 OK? 149 00:07:23,780 --> 00:07:24,950 So I want you to imagine that you're 150 00:07:24,950 --> 00:07:26,460 sending your beam of light up. 151 00:07:26,460 --> 00:07:28,335 And we're going to take advantage of the fact 152 00:07:28,335 --> 00:07:29,870 that light is wave like. 153 00:07:29,870 --> 00:07:31,430 So what I want you to do is think 154 00:07:31,430 --> 00:07:36,110 of the spacetime trajectory of successive crests of this wave 155 00:07:36,110 --> 00:07:38,470 as it climbs out of the gravitational potential. 156 00:07:58,163 --> 00:08:00,330 So I"m going to make my spacetime diagram the way we 157 00:08:00,330 --> 00:08:04,670 like to do it with time running up. 158 00:08:04,670 --> 00:08:08,400 And so this direction represents height. 159 00:08:08,400 --> 00:08:10,770 So down here is the bottom of the tower. 160 00:08:10,770 --> 00:08:12,592 So let's just make the drawing clearer. 161 00:08:12,592 --> 00:08:14,550 Let's let this tick be the bottom of the tower. 162 00:08:19,210 --> 00:08:20,710 This tick over here will be the top. 163 00:08:26,030 --> 00:08:27,860 Now if we were just in special relativity, 164 00:08:27,860 --> 00:08:30,318 there was no gravity, we know light would just move on a 45 165 00:08:30,318 --> 00:08:33,179 degree angle on this thing. 166 00:08:33,179 --> 00:08:36,270 We don't know yet what gravity is going to do to this light. 167 00:08:36,270 --> 00:08:36,770 OK? 168 00:08:36,770 --> 00:08:38,900 But you can imagine that whatever it's going to do, 169 00:08:38,900 --> 00:08:42,260 it's going to bend it away in some way from the trajectory 170 00:08:42,260 --> 00:08:44,510 that special relativity would predict. 171 00:08:44,510 --> 00:08:49,930 So let's just imagine that crest one of this light, 172 00:08:49,930 --> 00:08:52,220 it follows some trajectory in spacetime 173 00:08:52,220 --> 00:08:53,510 that maybe looks like this. 174 00:08:57,740 --> 00:09:02,420 So here is the world line of crest one. 175 00:09:09,510 --> 00:09:12,720 So what about the world line of credit two? 176 00:09:19,480 --> 00:09:21,730 So let's think about what the world line of credit two 177 00:09:21,730 --> 00:09:25,690 must be under this assumption that we can, in fact, cover 178 00:09:25,690 --> 00:09:28,123 all of spacetime with a single Lorentz frame. 179 00:09:28,123 --> 00:09:28,810 OK? 180 00:09:28,810 --> 00:09:49,390 If that is the case, so if that is the case, 181 00:09:49,390 --> 00:09:52,550 no position and no moment is special. 182 00:09:52,550 --> 00:09:53,050 OK? 183 00:09:53,050 --> 00:09:55,150 Everything is actually translation invariant 184 00:09:55,150 --> 00:09:56,350 in both time and space. 185 00:10:25,490 --> 00:10:27,300 So the second crest is going to be 186 00:10:27,300 --> 00:10:30,720 emitted one wave period later. 187 00:10:30,720 --> 00:10:33,690 But there's absolutely nothing special about spacetime one 188 00:10:33,690 --> 00:10:34,700 wave period later. 189 00:10:34,700 --> 00:10:37,270 And as it moves along, there can't 190 00:10:37,270 --> 00:10:38,700 be anything special about this. 191 00:10:38,700 --> 00:10:42,060 If it is to be a global Lorentz frame, whatever 192 00:10:42,060 --> 00:10:45,630 the trajectory is that the crest two follows through spacetime, 193 00:10:45,630 --> 00:10:48,900 it's got to be congruent with the trajectory of crest one. 194 00:10:48,900 --> 00:10:51,210 I've got to be able to just simply slide them 195 00:10:51,210 --> 00:10:52,487 on top of each other. 196 00:10:52,487 --> 00:10:53,710 OK? 197 00:10:53,710 --> 00:10:55,665 That is what this assumption demands. 198 00:11:11,280 --> 00:11:19,580 So this is going to look something like this. 199 00:11:19,580 --> 00:11:20,290 OK? 200 00:11:20,290 --> 00:11:21,790 Artwork a little bit off. 201 00:11:21,790 --> 00:11:23,500 The key thing which I want to emphasize, 202 00:11:23,500 --> 00:11:29,780 though, is that if this guy is congruent, 203 00:11:29,780 --> 00:11:32,850 the spacing between crest one and crest two at the bottom, 204 00:11:32,850 --> 00:11:37,220 let's call it delta tb, there'll be some spacing between the two 205 00:11:37,220 --> 00:11:37,820 at the top. 206 00:11:37,820 --> 00:11:38,780 We'll call it delta tt. 207 00:11:42,093 --> 00:11:43,010 They must be the same. 208 00:12:05,660 --> 00:12:06,670 But that's [INAUDIBLE]. 209 00:12:06,670 --> 00:12:08,750 That is just a period of the wave. 210 00:12:08,750 --> 00:12:10,460 And that's just up to a factor of 2 pi. 211 00:12:10,460 --> 00:12:12,574 That's 1 over the frequency. 212 00:12:20,640 --> 00:12:28,550 So this implies delta tb is not equal to delta t at the top. 213 00:12:28,550 --> 00:12:31,220 So a frequency at the top is smaller. 214 00:12:31,220 --> 00:12:35,000 So the period will actually be a little bit larger. 215 00:12:35,000 --> 00:12:38,030 So when you think about this argument carefully, 216 00:12:38,030 --> 00:12:40,790 the weak point is this assumption 217 00:12:40,790 --> 00:12:43,270 that I can actually cover the region with a large Lorentz 218 00:12:43,270 --> 00:12:43,770 frame. 219 00:12:43,770 --> 00:12:48,340 Remember what this is telling me is that, essentially, there 220 00:12:48,340 --> 00:12:50,840 is no special direction, right? 221 00:12:50,840 --> 00:12:53,030 I can do translation, variance in time and space, 222 00:12:53,030 --> 00:12:54,710 and they're all the same. 223 00:12:54,710 --> 00:12:57,330 But that's actually completely contradicted by common sense. 224 00:12:57,330 --> 00:13:00,980 If there were no special direction, why did it go down? 225 00:13:00,980 --> 00:13:03,550 There's obviously something special about down, right? 226 00:13:11,670 --> 00:13:16,740 So we conclude we cannot have a global Lorentz frame when 227 00:13:16,740 --> 00:13:17,520 we've got gravity. 228 00:13:29,420 --> 00:13:33,120 And in case you want to read a little bit more about it, 229 00:13:33,120 --> 00:13:35,780 I'll just give a highlight that this argument was originally 230 00:13:35,780 --> 00:13:45,045 developed by a gentleman named Alfred Shield. 231 00:13:55,870 --> 00:14:04,280 So that's unfortunate, OK, from sort 232 00:14:04,280 --> 00:14:07,520 of a philosophical perspective because the fact 233 00:14:07,520 --> 00:14:13,070 that we don't have a global Lorentz frame makes you think, 234 00:14:13,070 --> 00:14:13,670 oh, crap. 235 00:14:13,670 --> 00:14:15,680 What are we going to do with that framework we 236 00:14:15,680 --> 00:14:17,320 spent all this time developing? 237 00:14:17,320 --> 00:14:18,630 OK? 238 00:14:18,630 --> 00:14:22,160 The existence of a Lorentz frame and many of the mathematics 239 00:14:22,160 --> 00:14:25,430 we've been developing for the past few lectures, 240 00:14:25,430 --> 00:14:28,640 they all center around things you can do in Lorentz frames. 241 00:14:28,640 --> 00:14:33,540 So it's kind of like, OK, so do we just throw all that out? 242 00:14:33,540 --> 00:14:39,450 Well, here is where Einstein's key physical insight 243 00:14:39,450 --> 00:14:43,290 came in and gave us the physical tools 244 00:14:43,290 --> 00:14:46,260 that then needed to be coupled to some mathematics in order 245 00:14:46,260 --> 00:14:48,730 to turn it into something that can be worked with 246 00:14:48,730 --> 00:14:50,610 but, nonetheless, really save the day. 247 00:15:00,540 --> 00:15:03,660 So the key insight has there's a few steps in it. 248 00:15:03,660 --> 00:15:07,588 So first thing, I was very careful to use the word global 249 00:15:07,588 --> 00:15:09,630 when I ruled out the existence of Lorentz frames. 250 00:15:20,420 --> 00:15:23,890 But I'm allowed to have local Lorentz frames. 251 00:15:23,890 --> 00:15:24,390 Let's see. 252 00:15:24,390 --> 00:15:25,650 What does that actually mean? 253 00:15:35,820 --> 00:15:40,220 So I'll remind that a Lorentz frame is a tool that we use 254 00:15:40,220 --> 00:15:42,270 to describe inertial observers. 255 00:15:42,270 --> 00:15:46,008 In fact, we often call them inertial frames 256 00:15:46,008 --> 00:15:48,050 because they're sort of the constant coordinates. 257 00:15:48,050 --> 00:15:50,480 They represent the coordinates of an inertial observer 258 00:15:50,480 --> 00:15:53,965 who happens to be at rest in that Lorentz coordinate system. 259 00:15:53,965 --> 00:15:54,840 So let's switch over. 260 00:15:54,840 --> 00:15:56,750 Let's start calling them inertial frames for just 261 00:15:56,750 --> 00:15:57,290 a moment. 262 00:16:04,130 --> 00:16:17,150 So an inertial frame means that there is nothing accelerating, 263 00:16:17,150 --> 00:16:24,010 so no accelerations on observers or objects at rest 264 00:16:24,010 --> 00:16:24,670 in that frame. 265 00:16:31,750 --> 00:16:33,350 In other words, no forces are acting. 266 00:16:44,790 --> 00:16:49,770 Einstein's insight was to recognize 267 00:16:49,770 --> 00:16:56,330 that the next best thing is a freely falling frame. 268 00:17:15,674 --> 00:17:16,670 OK? 269 00:17:16,670 --> 00:17:23,395 If you are in a freely falling frame, to you, it sure as hell 270 00:17:23,395 --> 00:17:25,020 seems like there's a force acting on it 271 00:17:25,020 --> 00:17:27,880 and there's some accelerations acting upon it. 272 00:17:27,880 --> 00:17:30,750 But let's imagine you're in this freely falling frame. 273 00:17:30,750 --> 00:17:34,320 And you have a bunch of small objects 274 00:17:34,320 --> 00:17:35,748 that you release near you. 275 00:17:35,748 --> 00:17:37,831 Let's see if I can find some bits of broken chalk. 276 00:17:42,545 --> 00:17:44,420 So when you're in this really falling frame-- 277 00:17:48,172 --> 00:17:49,880 this was a lot easier when I was younger. 278 00:17:49,880 --> 00:17:52,460 All right, when you're in this really falling frame, OK, 279 00:17:52,460 --> 00:17:55,500 and these objects are all falling, 280 00:17:55,500 --> 00:18:02,360 there is no acceleration of these objects relative to you. 281 00:18:02,360 --> 00:18:05,810 The reason for this is that the gravitational force 282 00:18:05,810 --> 00:18:08,630 is proportional to the mass. 283 00:18:08,630 --> 00:18:18,830 And so the fact that we have f equals ma. 284 00:18:18,830 --> 00:18:26,670 And fg proportional to mass means 285 00:18:26,670 --> 00:18:33,665 that all objects in that freely falling frame-- 286 00:18:33,665 --> 00:18:35,790 this is a term that I'm going to start using a lot. 287 00:18:40,550 --> 00:18:44,538 So I'm just going to abbreviate it FFF. 288 00:18:44,538 --> 00:18:46,580 If I get really lazy, I might call it an F cubed. 289 00:18:46,580 --> 00:18:50,480 Anyway, l objects that are in that freely falling frame, 290 00:18:50,480 --> 00:18:52,850 they're experiencing the same acceleration. 291 00:18:52,850 --> 00:18:59,950 And so they experience 0 relative acceleration 292 00:18:59,950 --> 00:19:02,757 at least in the absence of other forces, OK? 293 00:19:02,757 --> 00:19:04,090 Perhaps one of them is charging. 294 00:19:04,090 --> 00:19:05,800 There's an electric force. 295 00:19:05,800 --> 00:19:08,770 Well, then it sort of suggests that the interesting force 296 00:19:08,770 --> 00:19:14,290 is the extra force provided to that relative to what 297 00:19:14,290 --> 00:19:17,170 we call the gravitational force that is driving this freefall. 298 00:19:44,590 --> 00:19:51,233 So I urge you to start getting comfortable with this idea 299 00:19:51,233 --> 00:19:53,900 because we're going to find this to be-- so one thing, which I'm 300 00:19:53,900 --> 00:19:56,570 going to do in a couple of minutes, 301 00:19:56,570 --> 00:20:02,300 is actually a little calculation that shows me 302 00:20:02,300 --> 00:20:11,010 I can, in fact, always find a Lorentz 303 00:20:11,010 --> 00:20:14,426 frame in the vicinity of any point in spacetime. 304 00:20:14,426 --> 00:20:15,100 OK? 305 00:20:15,100 --> 00:20:17,970 And so we're going to actually regard that Lorentz frame 306 00:20:17,970 --> 00:20:21,120 as being the preferred coordinate system of a freely 307 00:20:21,120 --> 00:20:23,490 falling observer, one who is not accelerated 308 00:20:23,490 --> 00:20:26,460 relative to freefall at that point. 309 00:20:26,460 --> 00:20:27,930 We are not in freefall right now. 310 00:20:27,930 --> 00:20:30,090 The damn floor is pushing us out of the way. 311 00:20:30,090 --> 00:20:33,540 But in this way of doing things, we 312 00:20:33,540 --> 00:20:36,730 would actually regard us as being the complicated people, 313 00:20:36,730 --> 00:20:37,380 OK? 314 00:20:37,380 --> 00:20:39,900 Someone who is merrily plummeting to their death, 315 00:20:39,900 --> 00:20:41,650 they're actually the simple observers, 316 00:20:41,650 --> 00:20:44,340 who are doing what they should be doing, in some sense. 317 00:20:47,730 --> 00:20:51,670 So the key thing, a way of saying this, 318 00:20:51,670 --> 00:20:53,700 so coming back to this and just one more point 319 00:20:53,700 --> 00:20:57,840 about that, because there are no relative 320 00:20:57,840 --> 00:21:06,830 accelerations, within this frame, objects 321 00:21:06,830 --> 00:21:10,040 maintain their relative velocities. 322 00:21:20,670 --> 00:21:22,567 That's basically the definition of something 323 00:21:22,567 --> 00:21:23,400 that is an inertial. 324 00:21:23,400 --> 00:21:26,400 If I have a frame where everything is-- 325 00:21:26,400 --> 00:21:28,697 two objects are moving with respect to each other at 1 326 00:21:28,697 --> 00:21:30,780 meter per second, they're always moving at 1 meter 327 00:21:30,780 --> 00:21:33,090 per second, that's inertial. 328 00:21:33,090 --> 00:21:35,700 And so this sort of demands that we do all of our experiments 329 00:21:35,700 --> 00:21:38,385 in plummeting elevators or in space, right? 330 00:21:38,385 --> 00:21:39,760 OK, obviously, you can't do that. 331 00:21:39,760 --> 00:21:41,218 So there's some complications we're 332 00:21:41,218 --> 00:21:43,733 going to have to learn the math to describe. 333 00:21:43,733 --> 00:21:45,900 But this is the way we're going to from now on think 334 00:21:45,900 --> 00:21:48,810 about things is that the freely falling frame is 335 00:21:48,810 --> 00:21:52,230 the most natural generalization of an inertial frame 336 00:21:52,230 --> 00:21:55,980 that we describe using Lorentz coordinates. 337 00:21:55,980 --> 00:21:59,550 Now before I discuss a few details about this, 338 00:21:59,550 --> 00:22:01,410 an important thing that is worth noting 339 00:22:01,410 --> 00:22:05,880 is that a very important aspect of any realistic source 340 00:22:05,880 --> 00:22:09,794 of gravity is that it is not uniform. 341 00:22:09,794 --> 00:22:10,650 OK? 342 00:22:10,650 --> 00:22:14,490 Gravity here is a little bit stronger 343 00:22:14,490 --> 00:22:16,710 than gravity on the ceiling, OK? 344 00:22:16,710 --> 00:22:21,560 And it's a lot stronger than gravity in geostationary orbit. 345 00:22:21,560 --> 00:22:24,390 We call this variation in gravity tides. 346 00:22:29,370 --> 00:22:38,910 So tides break down the notion of uniform freely falling 347 00:22:38,910 --> 00:22:39,410 frames. 348 00:22:49,570 --> 00:22:52,780 What this is telling us is that this idea, 349 00:22:52,780 --> 00:22:55,300 if I'm going to create this inertial frame that 350 00:22:55,300 --> 00:22:57,860 is essentially a freely falling frame, 351 00:22:57,860 --> 00:23:00,400 it will have to be a finite size. 352 00:23:00,400 --> 00:23:03,010 Hence, it's local and not global. 353 00:23:03,010 --> 00:23:05,740 OK, physically, this is telling me 354 00:23:05,740 --> 00:23:12,840 that, if I have a really tall freely falling elevator, 355 00:23:12,840 --> 00:23:22,140 and I'm here, I have one friend here and one friend down there, 356 00:23:22,140 --> 00:23:25,137 even if we start absolutely at rest, let's say we're at rest. 357 00:23:25,137 --> 00:23:27,470 We measure reference back to the walls of this elevator. 358 00:23:27,470 --> 00:23:29,550 It's a very stiff elevator. 359 00:23:29,550 --> 00:23:31,650 We will see the three of us diverge away 360 00:23:31,650 --> 00:23:33,360 from each other with time, OK? 361 00:23:33,360 --> 00:23:35,820 Because this person, let's say earth is down here, 362 00:23:35,820 --> 00:23:37,920 this person is feeling the-- 363 00:23:37,920 --> 00:23:41,460 and using Newtonian intuition, gravity is a little stronger. 364 00:23:41,460 --> 00:23:43,200 Gravity is medium. 365 00:23:43,200 --> 00:23:44,520 Gravity is weak. 366 00:23:44,520 --> 00:23:47,130 And so as viewed in that freely falling frame, 367 00:23:47,130 --> 00:23:49,983 let's say we center it on me here in the middle, 368 00:23:49,983 --> 00:23:52,650 I will see the person in the top go up, the person at the bottom 369 00:23:52,650 --> 00:23:53,150 go down. 370 00:23:55,580 --> 00:24:01,330 Another way of saying this, so let 371 00:24:01,330 --> 00:24:05,800 me just say that, so a very tall elevator, 372 00:24:05,800 --> 00:24:13,420 we'll see a separation of freefall 373 00:24:13,420 --> 00:24:14,830 since gravity is not uniform. 374 00:24:28,720 --> 00:24:32,770 But there's another way of saying this. 375 00:24:32,770 --> 00:24:35,160 And what this is telling me is imagine 376 00:24:35,160 --> 00:24:40,710 I made a spacetime diagram that shows trajectories of the three 377 00:24:40,710 --> 00:24:45,550 of us, so the three of us that are plummeting 378 00:24:45,550 --> 00:24:46,690 in this elevator, OK? 379 00:24:57,800 --> 00:24:59,690 What we find, if we make our trajectories 380 00:24:59,690 --> 00:25:01,773 as we fall-- in fact, let's just go ahead and make 381 00:25:01,773 --> 00:25:03,950 a little sketch of this. 382 00:25:03,950 --> 00:25:07,550 OK, so let's say here is the middle. 383 00:25:10,330 --> 00:25:11,488 Here is the top. 384 00:25:11,488 --> 00:25:12,280 Here is the bottom. 385 00:25:12,280 --> 00:25:16,750 I'm going to draw this in coordinates that are fixed 386 00:25:16,750 --> 00:25:19,060 on the person in the middle. 387 00:25:19,060 --> 00:25:22,660 So the person the middle, in their freely falling frame, 388 00:25:22,660 --> 00:25:25,870 they think they're standing still. 389 00:25:25,870 --> 00:25:29,330 They see the person at the top moving up and away, 390 00:25:29,330 --> 00:25:33,308 the person at the bottom moving up and away. 391 00:25:33,308 --> 00:25:36,900 It's a very similar story to the little parable I sketched 392 00:25:36,900 --> 00:25:38,430 of that light pole moving up. 393 00:25:38,430 --> 00:25:41,098 These are not congruent trajectories. 394 00:25:51,140 --> 00:25:54,440 Another way to say this is that, in any moment, if I think 395 00:25:54,440 --> 00:25:58,280 of these as world lines, I can, at any moment 396 00:25:58,280 --> 00:26:01,400 along the world line, I can draw its tangent vector. 397 00:26:05,960 --> 00:26:11,715 The tangents do not remain parallel. 398 00:26:16,420 --> 00:26:21,397 Now I'm going to take you back to probably, 399 00:26:21,397 --> 00:26:23,230 I don't know when you guys all learned this, 400 00:26:23,230 --> 00:26:25,355 but when you first started learning about geometry. 401 00:26:25,355 --> 00:26:26,800 For me, it was in middle school. 402 00:26:26,800 --> 00:26:29,070 And when you first started learning about geometry, 403 00:26:29,070 --> 00:26:31,150 you were usually doing geometry on the plane. 404 00:26:31,150 --> 00:26:34,690 And they gave you various axioms about the way things behaved. 405 00:26:34,690 --> 00:26:37,780 One of them is now known as Euclid's parallelism axiom. 406 00:26:48,730 --> 00:26:52,780 And it states that if I have two lines that start parallel, 407 00:26:52,780 --> 00:26:54,100 they remain parallel. 408 00:27:11,470 --> 00:27:13,700 If you dig into the history of mathematics, 409 00:27:13,700 --> 00:27:19,400 this axiom bugged the crap out of people for many, many years, 410 00:27:19,400 --> 00:27:20,300 OK? 411 00:27:20,300 --> 00:27:24,110 Because when you look at pictures on a piece of paper 412 00:27:24,110 --> 00:27:27,900 or on a chalkboard or something like that, it seems right, OK? 413 00:27:27,900 --> 00:27:30,770 But it really can't be justified quite as rigorously 414 00:27:30,770 --> 00:27:33,380 as many other axioms that Euclid wrote down. 415 00:27:33,380 --> 00:27:35,950 And it turns out, there's an underlying assumption. 416 00:27:35,950 --> 00:27:39,200 The assumption is that you are drawing your lines 417 00:27:39,200 --> 00:27:40,850 on a flat manifold. 418 00:27:57,130 --> 00:28:00,710 There's a ready counterexample. 419 00:28:00,710 --> 00:28:03,830 Go to the Earth's equator. 420 00:28:03,830 --> 00:28:06,440 You start, say, somewhere in Brazil. 421 00:28:06,440 --> 00:28:08,420 Your friend starts somewhere in Africa. 422 00:28:08,420 --> 00:28:12,440 Both of you stand on the equator and go due north. 423 00:28:12,440 --> 00:28:15,500 You are moving on parallel trajectories. 424 00:28:15,500 --> 00:28:18,760 You will intersect at the North Pole. 425 00:28:18,760 --> 00:28:19,260 OK? 426 00:28:19,260 --> 00:28:19,890 Well, guess what? 427 00:28:19,890 --> 00:28:21,210 You ain't moving on a flat manifold. 428 00:28:21,210 --> 00:28:22,918 You're moving on the surface of a sphere. 429 00:28:22,918 --> 00:28:26,670 OK, curvature causes initially parallel trajectories 430 00:28:26,670 --> 00:28:29,940 to become non-parallel. 431 00:28:29,940 --> 00:28:33,660 What this is telling us is that the manifold I'm going to want 432 00:28:33,660 --> 00:28:39,300 to use to describe events when I have gravity cannot be flat. 433 00:28:39,300 --> 00:28:41,940 I'm going to have to introduce curvature into it. 434 00:28:41,940 --> 00:28:44,670 Take us a little while to unpack precisely mathematically what 435 00:28:44,670 --> 00:28:46,210 that means. 436 00:28:46,210 --> 00:28:50,790 But this is sort of a sign that things got more complicated. 437 00:28:50,790 --> 00:28:53,670 And tides are the way in which that complication 438 00:28:53,670 --> 00:28:56,930 is being introduced. 439 00:28:56,930 --> 00:29:03,000 So switch pages here. 440 00:29:17,820 --> 00:29:22,340 So what we have basically danced around and put together here 441 00:29:22,340 --> 00:29:25,520 is one formulation of what is known 442 00:29:25,520 --> 00:29:29,765 as the principle of equivalence. 443 00:29:56,990 --> 00:30:00,620 The principle of equivalence, or one formulation of it, 444 00:30:00,620 --> 00:30:23,420 tells me that over sufficiently small regions, 445 00:30:23,420 --> 00:30:42,490 the motion of freely falling particles due to gravity cannot 446 00:30:42,490 --> 00:30:50,099 be distinguished from uniform acceleration. 447 00:30:55,843 --> 00:30:57,260 The physical manifestation of this 448 00:30:57,260 --> 00:30:58,635 is that if you're an observer who 449 00:30:58,635 --> 00:31:01,050 is in that freely falling frame moving due to gravity, 450 00:31:01,050 --> 00:31:03,720 you are also experiencing that uniform acceleration. 451 00:31:03,720 --> 00:31:06,650 So you see no net acceleration. 452 00:31:06,650 --> 00:31:08,090 This particular formulation of it 453 00:31:08,090 --> 00:31:10,616 is known as the weak equivalence principle. 454 00:31:16,928 --> 00:31:18,970 I'm going to give you a couple variations on this 455 00:31:18,970 --> 00:31:21,880 in just a second. 456 00:31:21,880 --> 00:31:27,040 One thing that's important about it is, 457 00:31:27,040 --> 00:31:32,750 essentially, it's a statement that, if you 458 00:31:32,750 --> 00:31:38,100 think about Newton's laws, the idea that f equals ma 459 00:31:38,100 --> 00:31:40,670 and the force of gravity is proportional to m, 460 00:31:40,670 --> 00:31:43,460 it's a fairly precise statement that you 461 00:31:43,460 --> 00:31:46,190 think of that m as the gravitational charge. 462 00:31:46,190 --> 00:31:48,410 It's saying that the gravitational charge, 463 00:31:48,410 --> 00:31:50,900 the gravitational mass and the inertial mass 464 00:31:50,900 --> 00:31:52,610 are the same thing. 465 00:31:52,610 --> 00:31:54,923 That's actually a testable statement. 466 00:31:54,923 --> 00:31:56,840 What you can do is look at materials that have 467 00:31:56,840 --> 00:31:58,000 very different compositions. 468 00:31:58,000 --> 00:31:59,417 In particular, what you want to do 469 00:31:59,417 --> 00:32:01,130 is look at things that maybe have 470 00:32:01,130 --> 00:32:03,920 different ratios of neutrons to protons 471 00:32:03,920 --> 00:32:07,730 or have highly bound nuclei with lots of gluons in them 472 00:32:07,730 --> 00:32:09,260 or various kinds of fields that you 473 00:32:09,260 --> 00:32:11,810 can imagine perhaps couple to gravity differently 474 00:32:11,810 --> 00:32:14,040 than the quarks do. 475 00:32:14,040 --> 00:32:16,760 And so there are what are called free fall experiments to test 476 00:32:16,760 --> 00:32:24,640 this, which really basically just boils down 477 00:32:24,640 --> 00:32:27,290 to dropping lots of different elements of the periodic table 478 00:32:27,290 --> 00:32:30,040 and making sure they all fall at the same rate. 479 00:32:30,040 --> 00:32:34,910 And the answer is that they fall at exactly the same rate 480 00:32:34,910 --> 00:32:36,670 within experimental precision. 481 00:32:36,670 --> 00:32:42,850 And the last time I looked it up, well, what they do 482 00:32:42,850 --> 00:32:50,020 is they demonstrate the WEP, the Weak Equivalence Principle, 483 00:32:50,020 --> 00:32:53,980 is valid to about, I believe, it's about a part in 10 484 00:32:53,980 --> 00:32:54,700 to the minus 13. 485 00:32:54,700 --> 00:32:56,075 Might be a little bit better now. 486 00:32:58,640 --> 00:33:03,600 So bear with me just one second here. 487 00:33:09,530 --> 00:33:10,850 OK, so good. 488 00:33:10,850 --> 00:33:13,360 We've got this notion here. 489 00:33:13,360 --> 00:33:15,340 And you might now think, OK, great. 490 00:33:15,340 --> 00:33:17,780 I have a way of generalizing what 491 00:33:17,780 --> 00:33:19,890 the notion of an inertial frame is. 492 00:33:19,890 --> 00:33:22,570 Is this enough for me to move forward? 493 00:33:22,570 --> 00:33:34,350 Can we now do all of physics by just applying 494 00:33:34,350 --> 00:33:42,900 the laws of special relativity in freely falling frames 495 00:33:42,900 --> 00:33:45,660 or, putting it this way, our new notion of an inertial frame? 496 00:33:54,560 --> 00:33:56,270 For two reasons that are closely related, 497 00:33:56,270 --> 00:33:57,710 this doesn't quite work. 498 00:33:57,710 --> 00:34:04,377 And it comes down to the fact that because of tides, we 499 00:34:04,377 --> 00:34:04,960 can't do that. 500 00:34:28,620 --> 00:34:31,250 What this basically means is that the transformation that 501 00:34:31,250 --> 00:34:35,192 puts you into a freely falling frame here is not the same 502 00:34:35,192 --> 00:34:36,650 as the transformation that puts you 503 00:34:36,650 --> 00:34:39,500 into a freely falling frame 10,000 miles over the Earth's 504 00:34:39,500 --> 00:34:40,909 surface, OK? 505 00:34:40,909 --> 00:34:42,739 There are different transformations 506 00:34:42,739 --> 00:34:45,320 at different locations in spacetime. 507 00:34:45,320 --> 00:34:47,840 This comes down the idea that it's not a global Lorentz 508 00:34:47,840 --> 00:34:48,380 frame, OK? 509 00:34:48,380 --> 00:34:51,230 We had a sequence of local Lorentz frames 510 00:34:51,230 --> 00:34:52,850 that we have to link up. 511 00:35:16,808 --> 00:35:19,100 So I'm going to do a calculation in just a moment where 512 00:35:19,100 --> 00:35:24,020 we explicitly show that we can always go into a frame that 513 00:35:24,020 --> 00:35:29,120 is locally Lorentz, but that there's a term that I'm going 514 00:35:29,120 --> 00:35:30,680 to very strongly argue is essentially 515 00:35:30,680 --> 00:35:33,590 the curvature associated with the spacetime metric that 516 00:35:33,590 --> 00:35:36,230 sets the size of what that region actually is. 517 00:35:40,320 --> 00:35:43,940 So what we are going to do, what is 518 00:35:43,940 --> 00:35:49,570 going to guide our physics as we move forward is, 519 00:35:49,570 --> 00:35:53,080 essentially, we're going to take advantage of a reformulation-- 520 00:35:56,665 --> 00:36:05,047 there we go-- of the equivalence principle, which 521 00:36:05,047 --> 00:36:06,255 I'm going to word as follows. 522 00:36:11,250 --> 00:36:25,160 In sufficiently small regions of spacetime, 523 00:36:25,160 --> 00:36:37,470 we can find a representation or a coordinate system such that-- 524 00:36:37,470 --> 00:36:39,220 I don't want to cram this into the margin. 525 00:36:39,220 --> 00:36:40,220 So I'll switch boards-- 526 00:36:58,800 --> 00:37:13,930 the laws of physics reduce to those of special relativity. 527 00:37:13,930 --> 00:37:19,390 This is known as the Einstein equivalence principle. 528 00:37:19,390 --> 00:37:22,923 At least, that's the name that Carroll uses in his textbook. 529 00:37:22,923 --> 00:37:24,340 A couple books use different ones. 530 00:37:24,340 --> 00:37:26,215 But this is a very nice one because it really 531 00:37:26,215 --> 00:37:30,700 is the principle by which Einstein guided us to rewriting 532 00:37:30,700 --> 00:37:33,230 the laws of physics. 533 00:37:33,230 --> 00:37:35,830 So we've got a weak equivalence principle, an Einstein 534 00:37:35,830 --> 00:37:37,030 equivalence principle. 535 00:37:37,030 --> 00:37:39,010 Just as an aside, there is something called 536 00:37:39,010 --> 00:37:40,886 a strong equivalence principle. 537 00:37:49,125 --> 00:37:51,000 We're not going to talk about this very much, 538 00:37:51,000 --> 00:37:53,625 but I'm going to give it to you because it's just kind of cool. 539 00:37:56,950 --> 00:38:07,420 It tells me that gravity falls in a gravitational field 540 00:38:07,420 --> 00:38:09,490 in a way that is indistinguishable from mass. 541 00:38:22,000 --> 00:38:24,730 That's a little weird. 542 00:38:24,730 --> 00:38:27,940 I'm not going to explain it very much right now. 543 00:38:27,940 --> 00:38:29,770 A way to think about it will become clearer 544 00:38:29,770 --> 00:38:31,870 in some future problem sets. 545 00:38:31,870 --> 00:38:33,910 Basically what it's telling us is that when 546 00:38:33,910 --> 00:38:35,410 you make very strong-- 547 00:38:35,410 --> 00:38:38,440 when you make any kind of a bound object, 548 00:38:38,440 --> 00:38:40,930 some of the mass of that object, in the sense 549 00:38:40,930 --> 00:38:43,750 of the mass that is measured by orbits, 550 00:38:43,750 --> 00:38:46,690 can be thought of as gravitational energy, OK? 551 00:38:46,690 --> 00:38:49,000 Energy and mass are equivalent to one another. 552 00:38:49,000 --> 00:38:51,580 So that gravitational energy should respond to gravity. 553 00:38:51,580 --> 00:38:54,777 And in fact, it falls just like any other mass, OK? 554 00:38:54,777 --> 00:38:55,610 This is another one. 555 00:38:55,610 --> 00:38:56,620 This is actually very-- 556 00:38:56,620 --> 00:39:03,220 well, prior to September 2015, this 557 00:39:03,220 --> 00:39:05,890 was very difficult to test. 558 00:39:05,890 --> 00:39:08,230 There were some very precise measurements 559 00:39:08,230 --> 00:39:11,140 of the moon's orbit that were done to test this. 560 00:39:11,140 --> 00:39:13,390 And there were observations of binary pulsar systems 561 00:39:13,390 --> 00:39:16,440 that were done to test this. 562 00:39:16,440 --> 00:39:19,690 Now every time LIGO measures a pair of binary black holes, 563 00:39:19,690 --> 00:39:22,170 which are nothing but gravitational energy, 564 00:39:22,170 --> 00:39:24,490 and our theoretical models match the wave forms, 565 00:39:24,490 --> 00:39:27,290 sort of like, boom, equivalence principle. 566 00:39:27,290 --> 00:39:27,810 Drop mic. 567 00:39:27,810 --> 00:39:28,710 Leave room. 568 00:39:28,710 --> 00:39:38,620 All right, so let's get precise. 569 00:39:43,520 --> 00:39:50,620 What I want to do is show that we can always 570 00:39:50,620 --> 00:39:53,160 find a local Lorentz frame. 571 00:40:06,830 --> 00:40:09,660 And what I mean by that is that I 572 00:40:09,660 --> 00:40:12,990 want to be able to show that I can always 573 00:40:12,990 --> 00:40:17,370 do a change of my coordinates, a change of my representation 574 00:40:17,370 --> 00:40:23,220 such that, at some point, I can convert the metric of spacetime 575 00:40:23,220 --> 00:40:26,220 into the metric we used in special relativity, 576 00:40:26,220 --> 00:40:28,530 at least over some finite region. 577 00:40:58,100 --> 00:41:00,382 So let me define a couple of quantities 578 00:41:00,382 --> 00:41:02,590 and then let me formulate the way this calculation is 579 00:41:02,590 --> 00:41:03,190 going to work. 580 00:41:06,290 --> 00:41:14,560 Let us let coordinates with unbarred Greek indices 581 00:41:14,560 --> 00:41:16,470 be the coordinate system that we start in. 582 00:41:24,460 --> 00:41:29,260 And let's say that, in this representation, 583 00:41:29,260 --> 00:41:32,720 the metric is G alpha beta. 584 00:41:35,710 --> 00:41:38,840 Let's demand that there exists a set of 585 00:41:38,840 --> 00:41:40,880 coordinates that I will represent 586 00:41:40,880 --> 00:41:41,960 with bars on the indices. 587 00:41:45,340 --> 00:41:46,990 When we transform to these coordinates, 588 00:41:46,990 --> 00:42:10,060 I want spacetime to be Lorentz at least in the vicinity 589 00:42:10,060 --> 00:42:11,590 of some point or some event. 590 00:42:19,318 --> 00:42:20,886 We'll call that event p. 591 00:42:47,690 --> 00:42:50,200 So we will assume that there is some mapping 592 00:42:50,200 --> 00:42:53,140 between these coordinates, so there's 593 00:42:53,140 --> 00:42:55,040 no nasty singularities there. 594 00:42:58,830 --> 00:43:01,650 And it's nothing that we can't deal with. 595 00:43:07,000 --> 00:43:09,300 So we can do it in either direction, 596 00:43:09,300 --> 00:43:14,820 but let's say x alpha is x alpha of p written 597 00:43:14,820 --> 00:43:16,590 as some function of these guys. 598 00:43:16,590 --> 00:43:19,215 And so there's a transformation matrix between the two of them. 599 00:43:26,930 --> 00:43:31,890 It takes the usual form, OK? 600 00:43:31,890 --> 00:43:42,050 So mathematically, the way I'm going to show this is I 601 00:43:42,050 --> 00:43:56,400 want to show that we can find a coordinate system such 602 00:43:56,400 --> 00:44:02,390 that, if I compute the spacetime metric 603 00:44:02,390 --> 00:44:07,190 and the barred representation, this is always 604 00:44:07,190 --> 00:44:12,065 going to be given by doing this coordinate transformation. 605 00:44:16,030 --> 00:44:20,060 Then I get the metric of flat spacetime 606 00:44:20,060 --> 00:44:22,445 over as large a region as possible. 607 00:44:33,664 --> 00:44:37,685 OK, we know we're not going to be able to do it everywhere. 608 00:44:37,685 --> 00:44:39,060 What I want to show is that I can 609 00:44:39,060 --> 00:44:42,540 make that happen at the point p and that the functional 610 00:44:42,540 --> 00:44:44,780 behavior of this thing is sufficiently flat 611 00:44:44,780 --> 00:44:49,237 that it remains at this over some region. 612 00:44:49,237 --> 00:44:51,570 So let me just sketch what the logic of this calculation 613 00:44:51,570 --> 00:44:52,195 is going to be. 614 00:45:16,630 --> 00:45:21,320 So what we're going to do is expand. 615 00:45:21,320 --> 00:45:27,430 So remember g is a function. 616 00:45:27,430 --> 00:45:29,650 All of these l's are functions. 617 00:45:29,650 --> 00:45:31,150 So what I'm going to do is I'm going 618 00:45:31,150 --> 00:45:33,820 to think of this whole thing as itself 619 00:45:33,820 --> 00:45:35,430 being some kind of a function. 620 00:45:35,430 --> 00:45:37,150 I'm going to expand in the Taylor series. 621 00:45:56,673 --> 00:45:58,340 There's something really cool that we're 622 00:45:58,340 --> 00:45:59,080 going to have to do with this. 623 00:45:59,080 --> 00:46:00,280 I mean, if I do this in general, you're 624 00:46:00,280 --> 00:46:02,160 going to get something pretty vomitous It's 625 00:46:02,160 --> 00:46:03,200 a giant, giant mess. 626 00:46:03,200 --> 00:46:03,700 OK? 627 00:46:03,700 --> 00:46:06,360 You might worry, what the hell are we going to do with this? 628 00:46:06,360 --> 00:46:08,568 OK, but there's something really cool that we can do. 629 00:46:08,568 --> 00:46:11,650 So we are given the metric g. 630 00:46:11,650 --> 00:46:14,440 But we are free to pick our coordinate transformation to be 631 00:46:14,440 --> 00:46:17,000 whatever we want it to be. 632 00:46:17,000 --> 00:46:23,620 So what we are going to do is compare 633 00:46:23,620 --> 00:46:31,030 the degrees of freedom offered by 634 00:46:31,030 --> 00:46:32,950 the coordinate transformation. 635 00:46:32,950 --> 00:46:34,840 Which again, I emphasize, we are free to make 636 00:46:34,840 --> 00:46:36,132 that whatever we need it to be. 637 00:46:44,382 --> 00:46:46,840 So the coordinate transformation and its derivative, right? 638 00:46:46,840 --> 00:46:48,940 We're free to play around with this thing. 639 00:46:48,940 --> 00:46:49,915 It's under our control. 640 00:47:00,230 --> 00:47:03,530 To the constraints that are imposed 641 00:47:03,530 --> 00:47:05,378 by the metric and its derivatives, which 642 00:47:05,378 --> 00:47:05,920 we are given. 643 00:47:23,470 --> 00:47:27,150 So what we're essentially going to do-- 644 00:47:27,150 --> 00:47:29,730 so in a moment or two, I'm going to write out 645 00:47:29,730 --> 00:47:34,140 very schematically what this coordinate transformation is 646 00:47:34,140 --> 00:47:37,170 going to look like when I do the Taylor expansion. 647 00:47:37,170 --> 00:47:39,420 And what I want to do is, essentially, 648 00:47:39,420 --> 00:47:42,690 just count up how many degrees of freedom 649 00:47:42,690 --> 00:47:45,030 the coordinate system offers me, count up 650 00:47:45,030 --> 00:47:46,860 the number of constraints that need 651 00:47:46,860 --> 00:47:49,620 to be matched in order to effect this transformation, 652 00:47:49,620 --> 00:47:52,220 see whether I've got enough. 653 00:47:52,220 --> 00:47:54,162 Assess what we learn from that, OK? 654 00:48:02,610 --> 00:48:05,070 We just set up with a few more things 655 00:48:05,070 --> 00:48:06,270 to help define the logic. 656 00:48:10,000 --> 00:48:17,670 So I'm going to write G alpha beta as G alpha beta 657 00:48:17,670 --> 00:48:22,360 at the point p plus x gamma. 658 00:48:22,360 --> 00:48:23,110 And you know what? 659 00:48:23,110 --> 00:48:26,950 Let's do the expansion in the bar coordinate. 660 00:48:26,950 --> 00:48:34,200 Minus x gamma bar at point p times the derivative 661 00:48:34,200 --> 00:48:34,850 at point p. 662 00:48:39,502 --> 00:48:40,960 And then I'm going to get something 663 00:48:40,960 --> 00:48:49,180 that involves a complicated form of x 664 00:48:49,180 --> 00:48:59,150 squared and second derivatives. 665 00:48:59,150 --> 00:49:00,160 OK, this can keep going. 666 00:49:00,160 --> 00:49:02,500 But this is going to be enough for our purposes. 667 00:49:02,500 --> 00:49:09,650 I am likewise going to expand my coordinate transformation 668 00:49:09,650 --> 00:49:10,150 matrix. 669 00:49:28,940 --> 00:49:31,204 Make sure I got all my bars in place. 670 00:49:31,204 --> 00:49:31,704 OK. 671 00:49:47,615 --> 00:49:48,115 OK. 672 00:49:54,262 --> 00:49:55,720 So let me go over to another board, 673 00:49:55,720 --> 00:49:57,740 just write down a few more important things. 674 00:49:57,740 --> 00:49:58,990 And then we'll start counting. 675 00:50:09,510 --> 00:50:13,150 So first key thing, which I want to note 676 00:50:13,150 --> 00:50:17,400 before I dig into the calculation, 677 00:50:17,400 --> 00:50:25,640 is the metric at this point, its derivative, 678 00:50:25,640 --> 00:50:29,590 its second derivative. 679 00:50:29,590 --> 00:50:31,790 These have been handed to us. 680 00:50:31,790 --> 00:50:32,290 OK? 681 00:50:32,290 --> 00:50:34,240 So we have no freedom to play with these. 682 00:50:34,240 --> 00:50:35,835 These are going to give us constraints 683 00:50:35,835 --> 00:50:36,835 that we need to satisfy. 684 00:50:51,320 --> 00:51:01,610 The coordinate transformation in its various derivatives, 685 00:51:01,610 --> 00:51:03,310 we are free to specify these. 686 00:51:08,166 --> 00:51:11,700 These are our degrees of freedom. 687 00:51:20,670 --> 00:51:23,630 OK, so let me just restate the calculation 688 00:51:23,630 --> 00:51:24,630 that we are going to do. 689 00:51:24,630 --> 00:51:27,960 Basically, we wanted to do this. 690 00:51:27,960 --> 00:51:30,173 Schematically, here's what this calculation 691 00:51:30,173 --> 00:51:31,090 is going to look like. 692 00:51:33,610 --> 00:51:35,115 And if you look at this right now, 693 00:51:35,115 --> 00:51:36,990 you might think this is going to be horrible. 694 00:51:36,990 --> 00:51:40,020 But we don't need to specify every step of this thing 695 00:51:40,020 --> 00:51:41,250 in absolutely gory detail. 696 00:51:46,040 --> 00:51:57,490 What we want to do is make this, which 697 00:51:57,490 --> 00:52:15,810 I can write as this guy at point p, this guy at point p, 698 00:52:15,810 --> 00:52:27,150 this guy at point p plus a term that is going to be linear 699 00:52:27,150 --> 00:52:30,600 in the distance from the spacetime displacement away 700 00:52:30,600 --> 00:52:31,380 from point p. 701 00:52:36,920 --> 00:52:39,420 If you want to go and multiply this out, knock yourself out. 702 00:52:39,420 --> 00:52:41,880 But it suffices to know that you are 703 00:52:41,880 --> 00:52:43,350 going to get some new things that 704 00:52:43,350 --> 00:52:46,180 involve first derivatives of all these quantities. 705 00:52:46,180 --> 00:52:56,080 So this will involve these two things. 706 00:53:10,310 --> 00:53:14,677 And this will involve second derivatives 707 00:53:14,677 --> 00:53:15,510 of all these things. 708 00:53:20,272 --> 00:53:21,730 Now what we're going to do is we're 709 00:53:21,730 --> 00:53:24,490 going to try to make what we would get multiplying all 710 00:53:24,490 --> 00:53:28,295 this out look as much like minus 1, 1, 1, 711 00:53:28,295 --> 00:53:30,560 1 of the diagonal as possible. 712 00:53:30,560 --> 00:53:34,143 And like I said, it really just involves accounting argument. 713 00:53:34,143 --> 00:53:36,310 It's actually one of my favorite little calculations 714 00:53:36,310 --> 00:53:39,793 we do in this class because it's my favorite word for this 715 00:53:39,793 --> 00:53:40,960 is it really looks vomitous. 716 00:53:40,960 --> 00:53:43,040 But it's quite elegant. 717 00:53:45,700 --> 00:53:49,895 And I have an eight-year-old kid. 718 00:53:49,895 --> 00:53:51,520 And I'm constantly trying to teach her, 719 00:53:51,520 --> 00:53:52,860 math is just all about counting. 720 00:53:52,860 --> 00:53:54,110 And this is a perfect example. 721 00:53:54,110 --> 00:53:56,770 This is really just about counting. 722 00:53:56,770 --> 00:53:58,680 So let's look at this thing at 0th order. 723 00:54:02,930 --> 00:54:07,760 So what I'm trying to do is guarantee that I've got some l, 724 00:54:07,760 --> 00:54:09,380 which I can choose, which will make 725 00:54:09,380 --> 00:54:15,260 the components of the g at that point be minus 1, 1, 1, 1. 726 00:54:15,260 --> 00:54:22,550 So my constraints, the conditions I must satisfy 727 00:54:22,550 --> 00:54:28,760 involve things that hit these 4 by 4 symmetric tensor 728 00:54:28,760 --> 00:54:30,240 components. 729 00:54:30,240 --> 00:54:32,150 So this guy is something that is symmetric. 730 00:54:36,280 --> 00:54:38,830 Nature has just handed me a symmetric tensor 731 00:54:38,830 --> 00:54:41,230 with a symmetric 4 by 4 object. 732 00:54:41,230 --> 00:54:44,440 Those are 10 constraints that my transformation must satisfy. 733 00:54:55,590 --> 00:55:03,270 What I'm going to use to do this is this matrix at the point p. 734 00:55:03,270 --> 00:55:09,870 And I'll remind you, this matrix is just 735 00:55:09,870 --> 00:55:14,280 following set of partial derivatives at that point p. 736 00:55:14,280 --> 00:55:16,290 This is also 4 by 4. 737 00:55:16,290 --> 00:55:20,460 It is not symmetric. 738 00:55:20,460 --> 00:55:29,670 So this actually gives me 16 degrees of freedom 739 00:55:29,670 --> 00:55:31,691 to satisfy these constraints. 740 00:55:39,868 --> 00:55:40,840 Ta da! 741 00:55:40,840 --> 00:55:44,980 We can easily make this thing look Lorentzian at the point p. 742 00:55:44,980 --> 00:55:47,290 In fact, we can do so and have six degrees 743 00:55:47,290 --> 00:55:49,180 of freedom left over. 744 00:55:49,180 --> 00:55:52,840 Any idea what those six degrees of freedom are? 745 00:55:52,840 --> 00:55:55,406 Remember you're going into a Lorentz frame. 746 00:55:55,406 --> 00:55:57,180 Yeah? 747 00:55:57,180 --> 00:55:58,830 Three rotations, three boosts. 748 00:55:58,830 --> 00:55:59,460 Exactly right. 749 00:56:13,640 --> 00:56:14,820 Boom. 750 00:56:14,820 --> 00:56:18,940 So not only does it work mathematically, 751 00:56:18,940 --> 00:56:21,030 but there's a little bit of leftover stuff 752 00:56:21,030 --> 00:56:22,890 which hopefully hits our intuition 753 00:56:22,890 --> 00:56:25,512 as to how things should behave in special relativity. 754 00:56:28,956 --> 00:56:31,400 OK, that's not really good enough though, right? 755 00:56:31,400 --> 00:56:34,140 So I just showed that I can do it at this point. 756 00:56:34,140 --> 00:56:38,132 And if I move a centimeter away from that point, suppose, 757 00:56:38,132 --> 00:56:39,590 there's some really steep gradient, 758 00:56:39,590 --> 00:56:41,040 and then it goes completely to hell. 759 00:56:41,040 --> 00:56:41,540 OK? 760 00:56:41,540 --> 00:56:42,538 Then we're in trouble. 761 00:56:42,538 --> 00:56:43,580 So we have to keep going. 762 00:56:43,580 --> 00:56:45,740 We have to look at the additional terms 763 00:56:45,740 --> 00:56:46,984 in this expansion. 764 00:56:52,420 --> 00:56:55,630 So when we do things at 0th order, we have now-- 765 00:56:55,630 --> 00:56:57,400 so g has been handed to us. 766 00:56:57,400 --> 00:57:01,900 We have now specified behavior of l at that point. 767 00:57:01,900 --> 00:57:05,710 So l has been completely soaked up. 768 00:57:05,710 --> 00:57:09,340 I don't have any more freedom to mess around with it. 769 00:57:09,340 --> 00:57:17,990 When I go to the next order, OK, so the quantity 770 00:57:17,990 --> 00:57:23,330 that is setting my constraints is the first derivative 771 00:57:23,330 --> 00:57:25,350 of this. 772 00:57:25,350 --> 00:57:30,806 So these are four derivatives of my 10 metric functions. 773 00:57:38,100 --> 00:57:44,920 4 by 4 symmetric by 4 components. 774 00:57:47,770 --> 00:57:51,165 So I have got 40 constraints I must satisfy. 775 00:57:59,010 --> 00:58:01,290 OK, well, the tool that I have available, 776 00:58:01,290 --> 00:58:04,800 I am free to specify the derivatives at this point. 777 00:58:07,560 --> 00:58:10,233 Again, remember this is itself a partial derivative. 778 00:58:20,860 --> 00:58:33,540 So I've got 4 components alpha, 4 derivatives with gamma bar, 4 779 00:58:33,540 --> 00:58:35,640 derivatives with mu bar. 780 00:58:35,640 --> 00:58:39,450 Partial derivatives, it should not matter what the order is. 781 00:58:39,450 --> 00:58:42,510 So this needs to be symmetric on mu and gamma. 782 00:58:52,430 --> 00:58:55,720 So I, in fact, have 40 degrees of freedom at 1st order. 783 00:59:03,400 --> 00:59:04,820 Perfect match. 784 00:59:04,820 --> 00:59:05,320 OK? 785 00:59:05,320 --> 00:59:06,970 So I can make my coordinate system. 786 00:59:06,970 --> 00:59:11,140 Not only can I make it equal to the Lorentz form at-- 787 00:59:11,140 --> 00:59:14,110 I can make the spacetime metric Lorentz at the point p, 788 00:59:14,110 --> 00:59:16,180 I can also make it flat at that point p. 789 00:59:19,820 --> 00:59:21,950 All right, now we're feeling cocky. 790 00:59:21,950 --> 00:59:24,860 So let's move on. 791 00:59:24,860 --> 00:59:26,130 How far can I go? 792 00:59:29,950 --> 00:59:31,060 I lost the page I need. 793 00:59:50,657 --> 00:59:52,740 OK, so now you have a flavor for what we're doing. 794 00:59:52,740 --> 00:59:53,240 OK? 795 00:59:53,240 --> 00:59:55,250 I want to count up the number of degrees of-- 796 00:59:55,250 --> 00:59:57,080 sorry-- the number of constraints 797 00:59:57,080 --> 00:59:59,550 that are in the metric object, the new metric 798 00:59:59,550 --> 01:00:02,073 object I work with at this order and compare it 799 01:00:02,073 --> 01:00:04,490 to the degrees of freedom in the coordinate transformation 800 01:00:04,490 --> 01:00:05,073 at this order. 801 01:00:09,383 --> 01:00:11,300 So when I go to 2nd order, my new metric thing 802 01:00:11,300 --> 01:00:13,430 that I'm going to be messing around with 803 01:00:13,430 --> 01:00:16,050 is the second derivative. 804 01:00:16,050 --> 01:00:22,840 So I take two derivatives of g alpha beta at the point p. 805 01:00:22,840 --> 01:00:25,770 So this again needs to be symmetric in the derivative. 806 01:00:25,770 --> 01:00:38,820 So I have a symmetric 4 by 4 on these two guys. 807 01:00:38,820 --> 01:00:49,030 And the metric is itself 4 by 4, one alpha and beta. 808 01:00:49,030 --> 01:00:51,125 So there is 100 conditions, 100 constraints 809 01:00:51,125 --> 01:00:52,000 that we must satisfy. 810 01:00:55,750 --> 01:01:07,190 So let's look at the second derivative of this. 811 01:01:07,190 --> 01:01:09,190 OK, so I'm going to move this down a little bit. 812 01:01:21,440 --> 01:01:24,550 This is now going to look like the third derivative 813 01:01:24,550 --> 01:01:25,800 when I do this transformation. 814 01:01:25,800 --> 01:01:27,467 So it's going to be the third derivative 815 01:01:27,467 --> 01:01:43,420 of x alpha with respect to mu bar, delta bar, and gamma bar. 816 01:01:43,420 --> 01:01:48,850 So I've got 4 degrees of freedom for my alpha. 817 01:01:52,690 --> 01:01:56,680 And this must be perfectly symmetric 818 01:01:56,680 --> 01:02:00,190 under the any interchange of the indices mu bar, 819 01:02:00,190 --> 01:02:02,022 delta bar, and gamma bar. 820 01:02:02,022 --> 01:02:03,710 OK? 821 01:02:03,710 --> 01:02:07,400 This is a little exercise in combinatorics. 822 01:02:07,400 --> 01:02:12,170 So the number of equivalent ways of arranging this 823 01:02:12,170 --> 01:02:17,570 turns out to be n times m plus 1 times n 824 01:02:17,570 --> 01:02:22,860 plus 2 over 3 factorial. 825 01:02:22,860 --> 01:02:24,770 And I'm in 4 dimensions for n equals 4. 826 01:02:27,620 --> 01:02:30,020 Work that out, and you will find that you 827 01:02:30,020 --> 01:02:34,870 have 80 degrees of freedom. 828 01:02:42,210 --> 01:02:45,050 So what we can do is we can always 829 01:02:45,050 --> 01:02:47,780 find a coordinate transformation that makes it flat. 830 01:02:47,780 --> 01:02:50,570 It makes it Lorentzian at point p. 831 01:02:50,570 --> 01:02:52,290 It is flat in that region. 832 01:02:52,290 --> 01:02:56,470 In other words, there is no first derivative there. 833 01:02:56,470 --> 01:02:57,020 Sorry. 834 01:02:57,020 --> 01:03:00,020 Flat's not really the right word. 835 01:03:00,020 --> 01:03:03,170 There is no slope right at that point. 836 01:03:03,170 --> 01:03:06,430 But we cannot transfer away the second derivative. 837 01:03:06,430 --> 01:03:13,100 And so what this tells me is the coordinate freedom that we have 838 01:03:13,100 --> 01:03:17,260 means that we can always put our metric into the following form. 839 01:03:56,198 --> 01:03:57,990 I'm just going to write this schematically. 840 01:03:57,990 --> 01:04:02,580 Second derivative to the metric and sort 841 01:04:02,580 --> 01:04:06,701 of the quadratic separation in spacetime coordinate. 842 01:04:29,810 --> 01:04:35,110 So a couple things about this are pretty interesting. 843 01:04:35,110 --> 01:04:36,870 So this is basically telling me that we 844 01:04:36,870 --> 01:04:40,590 can make it Lorentz only up to terms that look, essentially, 845 01:04:40,590 --> 01:04:42,570 like the second derivative of the metric. 846 01:04:42,570 --> 01:04:44,640 Well, the second derivative of any function 847 01:04:44,640 --> 01:04:46,690 tells you about the curvature of that function. 848 01:04:46,690 --> 01:04:49,232 So this word curvature is going to be coming up over and over 849 01:04:49,232 --> 01:04:50,560 and over again. 850 01:04:50,560 --> 01:04:53,010 We are, in fact, because this is general relativity, 851 01:04:53,010 --> 01:04:53,940 we can't do everything simply. 852 01:04:53,940 --> 01:04:56,340 We're going to actually find that there is a rigorous way 853 01:04:56,340 --> 01:04:59,550 to define a notion of curvature that we're going to play with, 854 01:04:59,550 --> 01:05:02,370 which indeed looks like two derivatives 855 01:05:02,370 --> 01:05:04,350 of the spacetime metric. 856 01:05:04,350 --> 01:05:07,440 And we are going to find it has, I mean, it's actually 857 01:05:07,440 --> 01:05:10,690 going to end up looking like a 4 index tensor. 858 01:05:10,690 --> 01:05:11,190 OK? 859 01:05:11,190 --> 01:05:14,850 So it's going to be an object that's got 4 indices on it. 860 01:05:14,850 --> 01:05:19,540 Each of those indices goes over the 4 spacetime coordinates. 861 01:05:19,540 --> 01:05:23,100 And so naively, it looks like it's got 4 to the 4th power 862 01:05:23,100 --> 01:05:24,840 independent components. 863 01:05:24,840 --> 01:05:26,862 256. 864 01:05:26,862 --> 01:05:29,070 It has certain symmetries we're going to see, though. 865 01:05:29,070 --> 01:05:31,028 And when you take into account those symmetries 866 01:05:31,028 --> 01:05:32,730 and you count up how many of those 867 01:05:32,730 --> 01:05:35,130 components are actually independent, 868 01:05:35,130 --> 01:05:37,870 any guesses what the number is going to turn out to be? 869 01:05:37,870 --> 01:05:39,310 20. 870 01:05:39,310 --> 01:05:42,825 Yes, it exactly compensates for the number that cannot be 871 01:05:42,825 --> 01:05:44,700 zeroed out by this coordinate transformation. 872 01:05:44,700 --> 01:05:46,690 It's called the Riemann curvature tensor, 873 01:05:46,690 --> 01:05:48,490 and we will get to that fairly-- 874 01:05:48,490 --> 01:05:50,810 actually, really, just in a couple lectures. 875 01:05:50,810 --> 01:05:54,520 The other thing which this is useful for 876 01:05:54,520 --> 01:06:01,180 is in all of my discussion of the equivalence principle 877 01:06:01,180 --> 01:06:04,690 up to now, there's been a weasel word 878 01:06:04,690 --> 01:06:06,940 that I have inserted into much of the physics. 879 01:06:06,940 --> 01:06:10,250 I always said over sufficiently small regions. 880 01:06:10,250 --> 01:06:10,750 OK? 881 01:06:10,750 --> 01:06:13,630 I say that trajectories begin to deviate from one another 882 01:06:13,630 --> 01:06:16,290 when they get sufficiently far away from one another. 883 01:06:16,290 --> 01:06:16,790 OK? 884 01:06:16,790 --> 01:06:18,385 And you should be going, what the hell 885 01:06:18,385 --> 01:06:20,343 does sufficiently small, sufficiently far away, 886 01:06:20,343 --> 01:06:22,070 what do all these things mean? 887 01:06:22,070 --> 01:06:22,600 OK? 888 01:06:22,600 --> 01:06:24,880 And the issue is you need to have a scale. 889 01:06:24,880 --> 01:06:30,057 Well, this scale is going to be set by the second derivative 890 01:06:30,057 --> 01:06:30,640 of the metric. 891 01:06:38,730 --> 01:06:47,080 So the size of the region over which spacetime 892 01:06:47,080 --> 01:06:49,600 is inertial in this coordinate system, 893 01:06:49,600 --> 01:07:03,890 in this representation is approximately so 894 01:07:03,890 --> 01:07:08,060 imagine it's 1 over-- we can think of d2g 895 01:07:08,060 --> 01:07:11,375 as being 1 over a length squared. 896 01:07:11,375 --> 01:07:12,800 So 1 over the square root of that 897 01:07:12,800 --> 01:07:15,037 gives me a rough idea of how long 898 01:07:15,037 --> 01:07:17,120 the curvature scale associated with your spacetime 899 01:07:17,120 --> 01:07:17,730 actually is. 900 01:07:17,730 --> 01:07:20,900 And it tells you how big your inertial region actually is. 901 01:07:25,160 --> 01:07:27,530 So we're going to make this notion of curvature regress 902 01:07:27,530 --> 01:07:30,410 very, very soon. 903 01:07:30,410 --> 01:07:32,600 As a prelude to this, we now need 904 01:07:32,600 --> 01:07:35,990 to start thinking about how we do mathematics and physics 905 01:07:35,990 --> 01:07:37,100 on a curved manifold. 906 01:07:40,290 --> 01:07:46,125 So I'm going to start to set up some of the issues 907 01:07:46,125 --> 01:07:47,000 that we need to face. 908 01:08:18,807 --> 01:08:20,640 So we're going to need to define what I mean 909 01:08:20,640 --> 01:08:22,439 by a manifold that is curved. 910 01:08:22,439 --> 01:08:24,660 So a curved manifold is going to simply 911 01:08:24,660 --> 01:08:30,479 be one in which initially parallel trajectories do not 912 01:08:30,479 --> 01:08:32,069 remain parallel. 913 01:09:00,830 --> 01:09:04,819 So an example is the surface of a sphere, as I illustrated. 914 01:09:04,819 --> 01:09:07,189 You start somewhere on the equator in Brazil. 915 01:09:07,189 --> 01:09:10,170 Your friend starts somewhere on the equator in Africa. 916 01:09:10,170 --> 01:09:12,290 The two of you start walking north. 917 01:09:12,290 --> 01:09:14,120 You are exactly parallel when you take 918 01:09:14,120 --> 01:09:15,652 that first step of the equator. 919 01:09:15,652 --> 01:09:17,569 But your trajectories cross at the North Pole. 920 01:09:25,600 --> 01:09:27,370 Interestingly, an example that looks 921 01:09:27,370 --> 01:09:32,890 curved but is not, the surface of a cylinder. 922 01:09:40,858 --> 01:09:41,359 OK? 923 01:09:41,359 --> 01:09:43,580 If I take two lines, again, I need 924 01:09:43,580 --> 01:09:46,262 to imagine that this region continues all the way up here. 925 01:09:46,262 --> 01:09:48,470 I make two lines that are parallel to each other here 926 01:09:48,470 --> 01:09:50,840 and I have them extend around this thing, 927 01:09:50,840 --> 01:09:53,899 they would remain parallel the entire way out, OK? 928 01:09:53,899 --> 01:09:56,150 Another way of stating that is that you can always 929 01:09:56,150 --> 01:10:00,500 take a cylinder and, with an appropriate cut, 930 01:10:00,500 --> 01:10:03,710 but without tearing it, you can flatten it out and make it 931 01:10:03,710 --> 01:10:07,372 into a simple sheet, a perfectly flat sheet. 932 01:10:07,372 --> 01:10:09,830 You cannot do that with a sphere without tearing it in some 933 01:10:09,830 --> 01:10:10,330 places. 934 01:10:14,040 --> 01:10:20,510 So what is going to begin to complicate things 935 01:10:20,510 --> 01:10:24,890 is that we want to work with vectors and tensors that 936 01:10:24,890 --> 01:10:26,825 live in this curved manifold. 937 01:10:39,170 --> 01:10:43,210 We haven't really thought too carefully yet about the space. 938 01:10:43,210 --> 01:10:45,230 And let's just focus on vectors for now. 939 01:10:45,230 --> 01:10:47,630 We haven't thought too carefully about the space in which 940 01:10:47,630 --> 01:10:49,640 the vectors actually live. 941 01:10:49,640 --> 01:10:51,140 So implicit to everything we have 942 01:10:51,140 --> 01:11:02,530 talked about up until now is that we often regard vectors 943 01:11:02,530 --> 01:11:07,750 as objects that themselves reside in a tangent space. 944 01:11:11,580 --> 01:11:13,500 OK, if I'm working in a manifold that is flat, 945 01:11:13,500 --> 01:11:15,667 say it's the surface of this board, OK, and just two 946 01:11:15,667 --> 01:11:19,050 dimensions, every point on this board 947 01:11:19,050 --> 01:11:20,580 has the same tangent, all right? 948 01:11:20,580 --> 01:11:25,075 So if I draw a vector here and I draw another vector over here, 949 01:11:25,075 --> 01:11:26,700 it's really easy for me to compare them 950 01:11:26,700 --> 01:11:28,890 because they actually live in the same space that 951 01:11:28,890 --> 01:11:31,410 is tangent to this board. 952 01:11:31,410 --> 01:11:34,410 If I'm on the surface of a sphere, 953 01:11:34,410 --> 01:11:37,110 points that are tangent to the sphere at the North Pole 954 01:11:37,110 --> 01:11:38,730 are very different from points that 955 01:11:38,730 --> 01:11:41,800 are tangent to the sphere on the equator. 956 01:11:41,800 --> 01:11:44,783 So it becomes difficult for me to actually compare fields 957 01:11:44,783 --> 01:11:46,950 when they are defined on a curved surface like this. 958 01:12:23,670 --> 01:12:26,180 So this makes it a little bit-- and so I'm just 959 01:12:26,180 --> 01:12:29,630 going to set up this problem, and I will sketch the issue. 960 01:12:29,630 --> 01:12:32,855 And then we will resolve it in our lecture on Tuesday. 961 01:12:32,855 --> 01:12:34,730 This makes it a little bit complicated for me 962 01:12:34,730 --> 01:12:37,370 to take derivatives of things like vectors 963 01:12:37,370 --> 01:12:40,700 when I'm working in a curved manifold. 964 01:12:40,700 --> 01:12:42,740 So let's consider the following situation. 965 01:12:47,250 --> 01:12:49,760 So I'm going to define some curve, which I would just 966 01:12:49,760 --> 01:12:52,970 call gamma. 967 01:12:52,970 --> 01:12:55,320 It lives in a curved space of some sort. 968 01:12:59,920 --> 01:13:01,990 I'm going to draw it here on the board. 969 01:13:01,990 --> 01:13:06,200 But imagine that this is on the surface of a sphere. 970 01:13:06,200 --> 01:13:07,490 OK? 971 01:13:07,490 --> 01:13:10,686 So here's the curve gamma. 972 01:13:10,686 --> 01:13:19,690 And let's say this point p here has coordinates x alpha. 973 01:13:19,690 --> 01:13:24,060 And this point q here has coordinates 974 01:13:24,060 --> 01:13:27,787 x alpha plus dx alpha. 975 01:13:27,787 --> 01:13:29,370 Suppose there's some vector field that 976 01:13:29,370 --> 01:13:31,630 fills all this manifold. 977 01:13:31,630 --> 01:13:32,830 OK? 978 01:13:32,830 --> 01:13:37,910 And so let's say the vector a looks 979 01:13:37,910 --> 01:13:41,858 like this here at the point p. 980 01:13:41,858 --> 01:13:48,140 And it looks like this here at the point q. 981 01:13:48,140 --> 01:13:50,480 How do I take the derivative of the vector field 982 01:13:50,480 --> 01:13:52,100 as I go from point p to point q? 983 01:14:11,350 --> 01:14:19,320 Well, your first guess should basically just 984 01:14:19,320 --> 01:14:22,170 be do what you always do when you first 985 01:14:22,170 --> 01:14:23,656 learn how to take a derivative. 986 01:15:10,280 --> 01:15:11,360 OK? 987 01:15:11,360 --> 01:15:14,380 So that would be a notion of a derivative. 988 01:15:14,380 --> 01:15:16,740 There's nothing mathematically wrong with that. 989 01:15:16,740 --> 01:15:17,240 OK? 990 01:15:17,240 --> 01:15:18,907 But we're going to find that it gives us 991 01:15:18,907 --> 01:15:22,690 problems for the same reason that we ran into problems when 992 01:15:22,690 --> 01:15:24,900 we began working with vector spaces 993 01:15:24,900 --> 01:15:26,270 and curvilinear coordinates. 994 01:15:26,270 --> 01:15:27,520 OK? 995 01:15:27,520 --> 01:15:31,660 If I want this to be a tensorial object in the same way 996 01:15:31,660 --> 01:15:33,918 that we have been defining tensors all along here, 997 01:15:33,918 --> 01:15:35,210 I'm going to run into problems. 998 01:15:35,210 --> 01:15:36,880 So let me just actually demonstrate what 999 01:15:36,880 --> 01:15:40,268 happens if I try to do this. 1000 01:15:40,268 --> 01:15:42,310 So the key bit, the way we want to think about it 1001 01:15:42,310 --> 01:15:45,790 is that the points p and q don't have the same tangent 1002 01:15:45,790 --> 01:15:56,640 space, which is a fancy way of saying 1003 01:15:56,640 --> 01:15:59,310 that, as I move from point p to point q, 1004 01:15:59,310 --> 01:16:01,260 the basis vectors are moving. 1005 01:16:01,260 --> 01:16:03,379 They're starting to point in different directions. 1006 01:16:17,870 --> 01:16:19,733 So if this were to be-- 1007 01:16:19,733 --> 01:16:21,400 like I said, mathematically, if you just 1008 01:16:21,400 --> 01:16:23,050 want to get that derivative, that 1009 01:16:23,050 --> 01:16:26,810 is a quantity which has a mathematical meaning, OK? 1010 01:16:26,810 --> 01:16:29,140 But it's not the component of a tensor, which 1011 01:16:29,140 --> 01:16:31,990 we have called out as having a particularly important meaning 1012 01:16:31,990 --> 01:16:34,840 in this geometric construction of physics that we are doing. 1013 01:16:34,840 --> 01:16:37,905 And so if this were to be tensorial, 1014 01:16:37,905 --> 01:16:40,280 then I should be able to switch to new coordinates, which 1015 01:16:40,280 --> 01:16:57,000 I'll designate with primes, such that the following was true. 1016 01:17:04,390 --> 01:17:08,050 The reason why this doesn't actually work 1017 01:17:08,050 --> 01:17:10,000 is I'm going to demand that alpha-- 1018 01:17:10,000 --> 01:17:13,330 excuse me-- that a is, in fact, actually already tensorial. 1019 01:17:13,330 --> 01:17:13,830 OK? 1020 01:17:13,830 --> 01:17:15,330 It's the compound of a vector, which 1021 01:17:15,330 --> 01:17:16,830 is a particular kind of tensor. 1022 01:17:23,400 --> 01:17:30,540 So I'm going to demand that the following be true. 1023 01:17:33,470 --> 01:17:40,760 And I'm going to demand that my derivatives, 1024 01:17:40,760 --> 01:17:41,970 they are just derivatives. 1025 01:17:41,970 --> 01:17:44,340 They do the usual rule Jacobian rule when 1026 01:17:44,340 --> 01:17:47,310 I switch coordinate systems. 1027 01:17:47,310 --> 01:17:49,302 And so skipping a line of algebra, 1028 01:17:49,302 --> 01:17:50,760 which you can get in my notes, this 1029 01:17:50,760 --> 01:17:53,220 is actually very similar to the notes I've already posted. 1030 01:17:53,220 --> 01:17:54,720 When you work this out, you're going 1031 01:17:54,720 --> 01:18:15,280 to find you get one term that's correct, 1032 01:18:15,280 --> 01:18:18,790 but you get another term that involves 1033 01:18:18,790 --> 01:18:21,190 a derivative of your coordinate transformation matrix. 1034 01:18:26,220 --> 01:18:29,432 And this is an actual term, spoils the tensorialness 1035 01:18:29,432 --> 01:18:30,140 of this quantity. 1036 01:18:34,100 --> 01:18:40,510 The way we are going to cure this is we 1037 01:18:40,510 --> 01:18:46,350 are going to demand that if we want our derivatives to be 1038 01:18:46,350 --> 01:18:51,420 derivatives that comport with a notion of taking tensor objects 1039 01:18:51,420 --> 01:18:55,110 and getting other tensor objects out of them, before we take 1040 01:18:55,110 --> 01:18:58,530 the derivative, we have to have some way of transporting 1041 01:18:58,530 --> 01:19:01,020 the objects to the same location in the manifold, 1042 01:19:01,020 --> 01:19:02,850 where I can then compare them. 1043 01:19:34,190 --> 01:19:39,400 So here's one example of a notion of transport 1044 01:19:39,400 --> 01:19:40,180 that we could use. 1045 01:19:45,920 --> 01:19:48,920 So here's a at point q. 1046 01:19:48,920 --> 01:19:51,490 There's a at point p. 1047 01:19:51,490 --> 01:19:55,637 Notion one that we'll talk about is known as parallel transport. 1048 01:20:03,750 --> 01:20:06,550 What that's essentially going to mean is I'm going to say, 1049 01:20:06,550 --> 01:20:09,465 well, let's take one of these guys, either q or p. 1050 01:20:09,465 --> 01:20:11,490 The way it's drawn in my notes, I've used q. 1051 01:20:11,490 --> 01:20:12,810 But it could be either. 1052 01:20:12,810 --> 01:20:15,090 And let's imagine I slide it parallel 1053 01:20:15,090 --> 01:20:42,120 to itself until this gives me a alpha from q transported to p. 1054 01:20:42,120 --> 01:20:45,140 And then I define a derivative with that transported notion 1055 01:20:45,140 --> 01:20:46,370 of the vector. 1056 01:20:46,370 --> 01:20:49,970 Now notice I've called this notion one. 1057 01:20:49,970 --> 01:20:51,890 You can take from that this idea of how 1058 01:20:51,890 --> 01:20:54,210 I transport the vector from one place to the other 1059 01:20:54,210 --> 01:20:55,430 to do this comparison. 1060 01:20:55,430 --> 01:20:57,800 I cannot uniquely define it. 1061 01:20:57,800 --> 01:21:00,087 There are, in fact, multiple ways you can do this. 1062 01:21:00,087 --> 01:21:01,670 We're going to talk about two that are 1063 01:21:01,670 --> 01:21:04,720 useful at the level of 8.962. 1064 01:21:04,720 --> 01:21:06,470 In principle, I imagine you could probably 1065 01:21:06,470 --> 01:21:08,760 come up with a whole butt load of these things. 1066 01:21:08,760 --> 01:21:11,150 These are two that the physics picks out 1067 01:21:11,150 --> 01:21:12,740 as being particularly useful for us 1068 01:21:12,740 --> 01:21:14,930 for the analysis that we are going to do. 1069 01:21:14,930 --> 01:21:15,530 OK? 1070 01:21:15,530 --> 01:21:16,910 So we'll pick it up there on Tuesday. 1071 01:21:16,910 --> 01:21:18,368 We're going to start by coming back 1072 01:21:18,368 --> 01:21:20,510 to this notion of I want to differentiate a vector 1073 01:21:20,510 --> 01:21:22,540 field in a curved manifold. 1074 01:21:22,540 --> 01:21:24,950 And let me just state before we conclude, 1075 01:21:24,950 --> 01:21:29,030 now that I've transported a from q to p, 1076 01:21:29,030 --> 01:21:31,220 they share the same tangent space. 1077 01:21:31,220 --> 01:21:33,650 Since they share the same tangent space, 1078 01:21:33,650 --> 01:21:35,030 I can compare them more easily. 1079 01:21:35,030 --> 01:21:37,520 That allows me to make a sensible derivative that 1080 01:21:37,520 --> 01:21:40,280 respects all the notions of what a tensor should be. 1081 01:21:40,280 --> 01:21:42,940 We'll pick it up from there on Tuesday.