1 00:00:00,500 --> 00:00:02,874 [SQUEAKING] 2 00:00:02,874 --> 00:00:04,311 [RUSTLING] 3 00:00:04,311 --> 00:00:10,285 [CLICKING] 4 00:00:10,285 --> 00:00:12,160 SCOTT HUGHES: The key textbook for this class 5 00:00:12,160 --> 00:00:16,372 is Sean Carroll's textbook on general relativity. 6 00:00:19,150 --> 00:00:22,270 It's now almost 20 years old. 7 00:00:22,270 --> 00:00:23,830 I would say-- I think it's listed 8 00:00:23,830 --> 00:00:25,400 on the website as required. 9 00:00:25,400 --> 00:00:28,950 I would actually call it sort of semi-required. 10 00:00:28,950 --> 00:00:34,120 It is where I will tend to post most of the readings 11 00:00:34,120 --> 00:00:35,980 to the course. 12 00:00:35,980 --> 00:00:41,740 It's a really good, complete textbook for a one semester 13 00:00:41,740 --> 00:00:44,193 course, which is what we have, and I will not 14 00:00:44,193 --> 00:00:45,610 be going through the entire thing. 15 00:00:45,610 --> 00:00:47,940 You can't in a one semester course. 16 00:00:47,940 --> 00:00:50,680 And I will, from time to time, there will be a few topics that 17 00:00:50,680 --> 00:00:54,195 I cannot go as in into as deeply as I would like, 18 00:00:54,195 --> 00:00:56,320 and if we had two semesters maybe I'd be able to do 19 00:00:56,320 --> 00:00:56,890 a little bit more. 20 00:00:56,890 --> 00:00:58,390 And so for those who are interested, 21 00:00:58,390 --> 00:01:00,730 I will suggest readings in this. 22 00:01:00,730 --> 00:01:04,209 My personal favorite supplement to this is a textbook 23 00:01:04,209 --> 00:01:08,358 by Bernard Schutz , A First Course in General Relativity. 24 00:01:12,849 --> 00:01:15,895 OK, so these are all things where-- so the MIT bookstore, 25 00:01:15,895 --> 00:01:17,270 I'm not sure how much they carry. 26 00:01:17,270 --> 00:01:18,650 They're all available through Amazon, 27 00:01:18,650 --> 00:01:20,692 and you can definitely find these kind of things. 28 00:01:20,692 --> 00:01:23,570 If you get Schutz's textbook, definitely 29 00:01:23,570 --> 00:01:24,950 get the second edition. 30 00:01:29,730 --> 00:01:31,770 The first edition contains errors. 31 00:01:31,770 --> 00:01:34,860 There is actually a very important geometric object 32 00:01:34,860 --> 00:01:37,320 that we are going to introduce in a couple of weeks 33 00:01:37,320 --> 00:01:40,092 that his textbook has a really clever derivation of. 34 00:01:40,092 --> 00:01:41,800 I remember seeing that and thinking, wow, 35 00:01:41,800 --> 00:01:42,690 that's really clever. 36 00:01:42,690 --> 00:01:45,232 The reason he was able do it so simply is because it's wrong. 37 00:01:45,232 --> 00:01:46,080 AUDIENCE: [LAUGH] 38 00:01:46,080 --> 00:01:49,050 SCOTT HUGHES: It's corrected in the second edition. 39 00:01:49,050 --> 00:01:54,020 Another one is Gravitation by Misner, Thorne and Wheeler. 40 00:01:54,020 --> 00:01:59,200 This is sort of a bit of personal history for me. 41 00:01:59,200 --> 00:02:01,920 I was Thorne's graduate student, and I used this textbook 42 00:02:01,920 --> 00:02:05,130 when I learned this subject originally. 43 00:02:05,130 --> 00:02:07,560 I, frankly, do not recommend this textbook 44 00:02:07,560 --> 00:02:11,670 to somebody who is learning the subject for the first time. 45 00:02:11,670 --> 00:02:14,040 It's a good place for a reference for certain things. 46 00:02:14,040 --> 00:02:15,230 It's available in the reading room. 47 00:02:15,230 --> 00:02:17,190 Know where you can get a copy and pick it up. 48 00:02:17,190 --> 00:02:18,565 So first of all, picking it up is 49 00:02:18,565 --> 00:02:20,653 kind of a-- it's good exercise. 50 00:02:20,653 --> 00:02:21,570 It's actually like a-- 51 00:02:21,570 --> 00:02:23,790 you know, it's a huge book. 52 00:02:23,790 --> 00:02:27,030 It gravitates. 53 00:02:27,030 --> 00:02:30,420 So it's a great reference and has 54 00:02:30,420 --> 00:02:33,000 a couple of good sections in it for new students. 55 00:02:44,438 --> 00:02:46,480 But I will indicate from time to time, especially 56 00:02:46,480 --> 00:02:48,940 some of the stuff we do in the second half of the class, 57 00:02:48,940 --> 00:02:52,330 my lectures are kind of inspired by Misner, Thorne and Wheeler-- 58 00:02:52,330 --> 00:02:54,640 known as MTW. 59 00:02:54,640 --> 00:02:57,310 And for those of you who have sort of a more mathematically 60 00:02:57,310 --> 00:03:01,960 minded approach to things, General Relativity by Wald. 61 00:03:01,960 --> 00:03:04,210 In the course syllabus I call this the uber book. 62 00:03:04,210 --> 00:03:06,430 This really is sort of the self-contained book 63 00:03:06,430 --> 00:03:10,720 that really pins down the subject very, very well. 64 00:03:10,720 --> 00:03:17,090 It's quite terse and formal, but also very, very clear. 65 00:03:17,090 --> 00:03:19,232 If you are a mathematically minded thinker, 66 00:03:19,232 --> 00:03:21,190 you will find this to be a really good textbook 67 00:03:21,190 --> 00:03:22,600 to refer to. 68 00:03:22,600 --> 00:03:26,200 There is one particular derivation 69 00:03:26,200 --> 00:03:28,837 that I'm going to do in about a month and a half 70 00:03:28,837 --> 00:03:30,670 that I essentially take from Wald's textbook 71 00:03:30,670 --> 00:03:33,190 because it's just beautiful. 72 00:03:33,190 --> 00:03:34,023 Other things in it-- 73 00:03:34,023 --> 00:03:36,440 to me, it's a little too terse from my own personal taste, 74 00:03:36,440 --> 00:03:38,260 but others may find it to be pretty good. 75 00:03:46,920 --> 00:03:49,530 Two other quick things. 76 00:03:49,530 --> 00:03:55,200 So there are 11 problem sets and your grade 77 00:03:55,200 --> 00:03:58,560 is determined entirely on these problems sets. 78 00:03:58,560 --> 00:04:01,230 The syllabus gives the schedule for when they will be posted 79 00:04:01,230 --> 00:04:03,570 and when they are to be handed in. 80 00:04:03,570 --> 00:04:07,560 11 does not divide evenly into 100, so what we do is we 81 00:04:07,560 --> 00:04:14,970 have 10 that are worth 9% of your grade and the 11th 82 00:04:14,970 --> 00:04:17,940 is worth 10% of your grade. 83 00:04:17,940 --> 00:04:20,735 Is that 11th one really 1% longer? 84 00:04:20,735 --> 00:04:22,110 I don't know. 85 00:04:22,110 --> 00:04:25,020 Again, I'm taking the viewpoint that you are graduate students, 86 00:04:25,020 --> 00:04:28,238 or at least you're playing one for the next hour and a half, 87 00:04:28,238 --> 00:04:30,780 and if you're going to sweat the difference of one percentage 88 00:04:30,780 --> 00:04:32,160 point on that thing, come on. 89 00:04:32,160 --> 00:04:33,660 This is about learning the material, 90 00:04:33,660 --> 00:04:37,080 don't worry about those little details that much. 91 00:04:37,080 --> 00:04:39,013 That is where all of your assessment 92 00:04:39,013 --> 00:04:39,930 is going to come from. 93 00:04:42,657 --> 00:04:45,240 The very first time I lectured this course I had a final exam, 94 00:04:45,240 --> 00:04:47,610 and it just turned to be a complete waste of my time 95 00:04:47,610 --> 00:04:49,560 and the students' time. 96 00:04:49,560 --> 00:04:51,830 You really cannot write-- 97 00:04:51,830 --> 00:04:53,400 you either write problems that are 98 00:04:53,400 --> 00:04:55,980 so easy that you can do them in your sleep, 99 00:04:55,980 --> 00:04:57,840 or they're so difficult you can't do them 100 00:04:57,840 --> 00:04:59,290 in the time period of an exam. 101 00:04:59,290 --> 00:05:02,040 So we're just going to stick with problem sets 102 00:05:02,040 --> 00:05:04,000 and that's fine. 103 00:05:04,000 --> 00:05:06,720 Let's move into, then, the way the course 104 00:05:06,720 --> 00:05:08,610 is going to be structured. 105 00:05:08,610 --> 00:05:11,467 So my presentation of this material-- 106 00:05:15,850 --> 00:05:31,270 the first half of the course basically up to spring break 107 00:05:31,270 --> 00:05:34,030 is essentially the mathematical foundations 108 00:05:34,030 --> 00:05:35,322 of general relativity. 109 00:05:45,940 --> 00:05:48,343 There are several choices that need to be 110 00:05:48,343 --> 00:05:49,510 made when you're doing this. 111 00:05:49,510 --> 00:05:50,885 This often, in some universities, 112 00:05:50,885 --> 00:05:52,930 if they have multiple semester sequences, what 113 00:05:52,930 --> 00:05:54,472 I'm going to cover in this first half 114 00:05:54,472 --> 00:05:57,258 of our semester in some places goes for a full semester. 115 00:05:57,258 --> 00:05:59,800 And what this means is there's a couple of things that I just 116 00:05:59,800 --> 00:06:01,800 cannot cover in quite as much depth. 117 00:06:01,800 --> 00:06:03,800 Those will be things where, for those of you who 118 00:06:03,800 --> 00:06:07,300 are interested in it, have that kind of a mathematical thinking 119 00:06:07,300 --> 00:06:09,490 of things, happy to push you to additional readings. 120 00:06:09,490 --> 00:06:11,800 We can dive in and look at it a bit more depth. 121 00:06:11,800 --> 00:06:14,350 My goal is to give you just enough stuff 122 00:06:14,350 --> 00:06:17,523 that we can do the most important applications 123 00:06:17,523 --> 00:06:18,190 of this subject. 124 00:06:18,190 --> 00:06:19,750 And I'm an astrophysicist, and to be blunt, 125 00:06:19,750 --> 00:06:21,460 most of my really interesting applications 126 00:06:21,460 --> 00:06:24,002 tend to be things that have to do with things like cosmology, 127 00:06:24,002 --> 00:06:26,300 black holes, dense stars, and things like that. 128 00:06:26,300 --> 00:06:28,630 And so I want to get enough formalism together 129 00:06:28,630 --> 00:06:31,510 that we can get to that part of things. 130 00:06:31,510 --> 00:06:34,450 And so the goal of this is that by the week right 131 00:06:34,450 --> 00:06:44,380 before spring break what we will do is "derive"-- 132 00:06:44,380 --> 00:06:48,020 I put that in quotes, and you'll see why a little bit later-- 133 00:06:48,020 --> 00:06:51,920 the Einstein field equations that govern 134 00:06:51,920 --> 00:06:53,950 gravity in general relativity. 135 00:06:59,482 --> 00:07:00,940 So there are several things that we 136 00:07:00,940 --> 00:07:04,360 could do that are not strictly necessary to get there, 137 00:07:04,360 --> 00:07:06,140 and just because of time limitations 138 00:07:06,140 --> 00:07:09,290 I'm going to choose to elide a few of these topics. 139 00:07:09,290 --> 00:07:13,030 The second half will then be applications. 140 00:07:27,100 --> 00:07:29,800 We will use everything we derived in the first half 141 00:07:29,800 --> 00:07:33,070 to see how general relativity gives us a relativistic theory 142 00:07:33,070 --> 00:07:34,237 of gravity. 143 00:07:34,237 --> 00:07:35,320 We will begin applying it. 144 00:07:35,320 --> 00:07:37,870 We'll see how Newton's law is encoded in these field 145 00:07:37,870 --> 00:07:38,423 equations. 146 00:07:38,423 --> 00:07:40,090 We'll see how we go beyond Newton's law, 147 00:07:40,090 --> 00:07:42,550 get some of the classic tests of general relativity, 148 00:07:42,550 --> 00:07:44,050 and then start looking at solving it 149 00:07:44,050 --> 00:07:45,460 for more interesting systems. 150 00:07:45,460 --> 00:07:47,310 Looking at the evolution of the universe as a whole, 151 00:07:47,310 --> 00:07:48,610 looking at the behavior of black holes, 152 00:07:48,610 --> 00:07:50,485 looking for gravitational waves, constructing 153 00:07:50,485 --> 00:07:53,450 the spacetime of neutron stars, things like that. 154 00:07:53,450 --> 00:07:55,480 So it's a fun semester. 155 00:07:55,480 --> 00:07:58,277 It sort of works well to fit these two things in like this. 156 00:07:58,277 --> 00:08:00,610 And for those of you who are interested in taking things 157 00:08:00,610 --> 00:08:03,790 more deeply, there's a lot of room to grow after this. 158 00:08:03,790 --> 00:08:06,880 And it does look like-- so people who particularly would 159 00:08:06,880 --> 00:08:09,855 like to go a little bit more detail on some of the math, 160 00:08:09,855 --> 00:08:11,230 if you've looked at the syllabus, 161 00:08:11,230 --> 00:08:13,360 I have one of my absolute favorite quotes 162 00:08:13,360 --> 00:08:15,730 from a course evaluation is put on there. 163 00:08:15,730 --> 00:08:19,120 Where a student in 2007 or so wrote, 164 00:08:19,120 --> 00:08:21,670 "The course was fine as it was, but Professor Hughes 165 00:08:21,670 --> 00:08:23,530 as an astrophysicist tended to focus 166 00:08:23,530 --> 00:08:27,932 on really mundane topics like cosmology and black holes." 167 00:08:27,932 --> 00:08:29,890 If you think those are mundane topics, what can 168 00:08:29,890 --> 00:08:31,613 I say, guilty as charged. 169 00:08:31,613 --> 00:08:34,030 But for those you who do want to take a different approach 170 00:08:34,030 --> 00:08:35,572 these sorts of things, we'll probably 171 00:08:35,572 --> 00:08:38,169 alternate lecturing this course in the future between someone 172 00:08:38,169 --> 00:08:40,780 from the CTP who works more in quantum gravity 173 00:08:40,780 --> 00:08:42,820 and things related to that. 174 00:08:42,820 --> 00:08:48,600 That'll be Netta Engelhardt in spring of 2021. 175 00:08:48,600 --> 00:08:50,280 We're now ready to start talking about, 176 00:08:50,280 --> 00:08:52,160 after doing all this sort of prep, 177 00:08:52,160 --> 00:08:54,767 we can actually talk about some of the foundations 178 00:08:54,767 --> 00:08:55,350 of the theory. 179 00:08:55,350 --> 00:08:57,225 So before I dive in, are there any questions? 180 00:09:00,180 --> 00:09:00,680 All right. 181 00:09:04,760 --> 00:09:07,820 What we're going to begin doing for the first couple of weeks-- 182 00:09:07,820 --> 00:09:09,320 well, not really first couple weeks. 183 00:09:09,320 --> 00:09:11,870 The first couple of lectures, is we're 184 00:09:11,870 --> 00:09:16,850 going to begin by discussing special relativity, 185 00:09:16,850 --> 00:09:26,410 but we're going to do special relativity using 186 00:09:26,410 --> 00:09:33,580 mathematical language that emphasizes the geometric nature 187 00:09:33,580 --> 00:09:36,409 of this form of relativity. 188 00:09:55,970 --> 00:10:00,000 What this does is it allows us to introduce basically 189 00:10:00,000 --> 00:10:05,270 the formalism, the notation, all the different tools that 190 00:10:05,270 --> 00:10:09,450 are important for when things get more complicated. 191 00:10:09,450 --> 00:10:11,870 When we apply a lot of these tools to special relativity, 192 00:10:11,870 --> 00:10:13,537 like we will be doing in the first three 193 00:10:13,537 --> 00:10:15,350 or so lectures of this course, it's 194 00:10:15,350 --> 00:10:17,780 kind of like swatting a mosquito with a sledgehammer. 195 00:10:17,780 --> 00:10:21,290 You really don't need that much mathematical structure 196 00:10:21,290 --> 00:10:23,330 to discuss special relativity. 197 00:10:23,330 --> 00:10:25,760 But you're going to be grateful for that sledgehammer 198 00:10:25,760 --> 00:10:28,580 when we start talking about strong field orbits of rotating 199 00:10:28,580 --> 00:10:29,480 black holes, right. 200 00:10:29,480 --> 00:10:34,760 And so the whole idea of this is to develop the framework 201 00:10:34,760 --> 00:10:39,530 in terms of a physical system where 202 00:10:39,530 --> 00:10:41,630 it's simple to understand what is going on. 203 00:10:41,630 --> 00:10:43,130 These are sort of a way to introduce 204 00:10:43,130 --> 00:10:45,050 the mathematical tools in a place where 205 00:10:45,050 --> 00:10:48,260 the physics is straightforward and then kind of carry forward 206 00:10:48,260 --> 00:10:49,350 from there. 207 00:10:49,350 --> 00:10:51,200 I will caution that as a consequence 208 00:10:51,200 --> 00:10:54,530 of this many students find these first three lectures to be 209 00:10:54,530 --> 00:10:55,650 a little on the dull side. 210 00:10:55,650 --> 00:10:58,072 So, sorry. 211 00:10:58,072 --> 00:11:00,530 It's just some stuff that we kind of have to get out there, 212 00:11:00,530 --> 00:11:03,140 and then as we generalize to more interesting 213 00:11:03,140 --> 00:11:05,900 mathematical objects, more interesting physical settings, 214 00:11:05,900 --> 00:11:07,770 it gets more interesting. 215 00:11:07,770 --> 00:11:09,510 All right, so let's dive in. 216 00:11:09,510 --> 00:11:13,370 So the setting for everything that we will be doing 217 00:11:13,370 --> 00:11:17,270 is a geometric concept known as spacetime. 218 00:11:23,160 --> 00:11:25,730 So we give you a precise mathematical definition 219 00:11:25,730 --> 00:11:28,580 of spacetime. 220 00:11:28,580 --> 00:11:37,580 A spacetime is a manifold of events 221 00:11:37,580 --> 00:11:45,245 that is endowed with a metric. 222 00:11:48,370 --> 00:11:50,920 So, a wonderful mathematical definition, and I've 223 00:11:50,920 --> 00:11:53,800 written it in a way that requires me to carefully define 224 00:11:53,800 --> 00:11:56,430 three additional terms. 225 00:11:56,430 --> 00:11:58,840 The concepts that I've underlined here, 226 00:11:58,840 --> 00:12:00,400 I've not defined them yet precisely 227 00:12:00,400 --> 00:12:02,510 exactly what I mean by them. 228 00:12:02,510 --> 00:12:06,370 So let's go over to the sideboard 229 00:12:06,370 --> 00:12:09,190 and talk about what exactly these are. 230 00:12:09,190 --> 00:12:14,628 So a manifold-- if you are a mathematician, 231 00:12:14,628 --> 00:12:16,420 you might twitch a little bit about the way 232 00:12:16,420 --> 00:12:20,500 that I am going to define this, and I 233 00:12:20,500 --> 00:12:23,110 will point you to a reading that does it a little bit more 234 00:12:23,110 --> 00:12:23,980 precisely. 235 00:12:23,980 --> 00:12:26,960 But for the purposes of our class, 236 00:12:26,960 --> 00:12:29,260 a manifold is essentially just a set 237 00:12:29,260 --> 00:12:44,810 or a collection of points with well understood 238 00:12:44,810 --> 00:12:46,340 connectedness properties. 239 00:12:57,320 --> 00:12:59,480 What I mean by that, is I'm going 240 00:12:59,480 --> 00:13:03,798 to talk about manifolds of space and time. 241 00:13:03,798 --> 00:13:05,840 I haven't defined an event yet, but I'm about to, 242 00:13:05,840 --> 00:13:07,610 but I'm going to say that there's a bunch of events that 243 00:13:07,610 --> 00:13:09,830 happen at this place and at this time and a bunch of events that 244 00:13:09,830 --> 00:13:11,810 happen at this place and at this time. 245 00:13:11,810 --> 00:13:14,270 And the manifold, the spacetime, gives me 246 00:13:14,270 --> 00:13:17,150 some notion of how I connect the events over here 247 00:13:17,150 --> 00:13:18,770 to the events over here. 248 00:13:18,770 --> 00:13:21,410 A manifold is a tautological concept. 249 00:13:21,410 --> 00:13:26,750 It's all about how one connects one region to another one. 250 00:13:26,750 --> 00:13:28,880 So if you're working on a manifold that 251 00:13:28,880 --> 00:13:30,530 lives on the surface of a donut, you 252 00:13:30,530 --> 00:13:32,180 have a particular topology associated with that. 253 00:13:32,180 --> 00:13:33,763 If it lives on the surface of a sphere 254 00:13:33,763 --> 00:13:36,060 you have a different topology associated with it. 255 00:13:36,060 --> 00:13:38,690 If you would like to see more careful and more 256 00:13:38,690 --> 00:13:41,880 rigorous discussion of this, this 257 00:13:41,880 --> 00:13:43,880 is one of the places where Carroll is very good. 258 00:13:51,370 --> 00:13:53,380 So go into Carroll, at least in the edition 259 00:13:53,380 --> 00:13:57,170 that I have it's on pages 54 to 62. 260 00:14:01,360 --> 00:14:03,640 He introduces a bit of additional mathematical 261 00:14:03,640 --> 00:14:06,190 machinery, discusses things with a little more rigor 262 00:14:06,190 --> 00:14:07,510 than I'm doing here-- 263 00:14:07,510 --> 00:14:09,880 this isn't rigorous at all, so significantly more rigor 264 00:14:09,880 --> 00:14:11,320 than I'm doing here. 265 00:14:11,320 --> 00:14:13,900 Those of you are into that, that should be something 266 00:14:13,900 --> 00:14:15,590 that you enjoy. 267 00:14:15,590 --> 00:14:16,570 So an event. 268 00:14:19,910 --> 00:14:26,045 This is when and where something happens. 269 00:14:32,580 --> 00:14:34,140 Could be anything. 270 00:14:34,140 --> 00:14:36,540 From our point of view, the event essentially 271 00:14:36,540 --> 00:14:38,910 is going to be the fundamental notion 272 00:14:38,910 --> 00:14:41,030 of a coordinate in space time. 273 00:14:43,590 --> 00:14:45,780 We will actually, in many cases-- 274 00:14:45,780 --> 00:14:47,940 actually, that's bad word choice, I should say. 275 00:14:47,940 --> 00:14:50,160 Coordinates are actually sort of labels 276 00:14:50,160 --> 00:14:52,540 that we attach to events. 277 00:14:52,540 --> 00:14:54,180 We are going to be free to adjust 278 00:14:54,180 --> 00:14:58,200 those labels, but the underlying geometric idea that the event 279 00:14:58,200 --> 00:15:01,660 is here, that's independent of the coordinates that we choose. 280 00:15:01,660 --> 00:15:04,909 So we will label these things with coordinates. 281 00:15:15,390 --> 00:15:28,160 But the event itself exists independent of these labels. 282 00:15:44,580 --> 00:15:46,620 Just to give an example, there might 283 00:15:46,620 --> 00:15:49,890 be one event which is I punch myself in the head. 284 00:15:49,890 --> 00:15:52,980 And so I'm very egotistic, so I will say 285 00:15:52,980 --> 00:15:55,905 this event happened at time 0-- 286 00:15:55,905 --> 00:15:57,300 x, y, and z equals 0. 287 00:15:57,300 --> 00:15:59,490 Because I define this corner of my skull 288 00:15:59,490 --> 00:16:00,990 as the origin the coordinate system, 289 00:16:00,990 --> 00:16:03,032 and I always think whatever's happening right now 290 00:16:03,032 --> 00:16:04,440 is the origin of time. 291 00:16:04,440 --> 00:16:07,140 Those of you out in the room are also egocentric 292 00:16:07,140 --> 00:16:09,840 and you would say whatever, I'm going to call that-- 293 00:16:09,840 --> 00:16:12,810 let's say you are at y of 3 meters 294 00:16:12,810 --> 00:16:15,960 and I'm going to put the floor as the origin of my z-axis, 295 00:16:15,960 --> 00:16:18,700 so z of 1.7 meters or whatever. 296 00:16:18,700 --> 00:16:20,700 You will come with your own independent labeling 297 00:16:20,700 --> 00:16:22,560 of these things. 298 00:16:22,560 --> 00:16:24,060 So you're all familiar with the idea 299 00:16:24,060 --> 00:16:26,453 that we can just change coordinate systems. 300 00:16:26,453 --> 00:16:28,120 I'm going to harp on this a bit, though, 301 00:16:28,120 --> 00:16:30,662 because there's going to be a really important distinction we 302 00:16:30,662 --> 00:16:32,880 make between geometrical objects that 303 00:16:32,880 --> 00:16:34,830 live in this manifold of spacetime 304 00:16:34,830 --> 00:16:38,490 and how we represent them using labels that might be attached 305 00:16:38,490 --> 00:16:39,773 to coordinate systems. 306 00:16:39,773 --> 00:16:41,190 And I'm going to come back to this 307 00:16:41,190 --> 00:16:43,770 when we start talking about some additional geometric objects 308 00:16:43,770 --> 00:16:46,420 in just a couple of minutes. 309 00:16:46,420 --> 00:16:49,410 So the last object that I have introduced into here 310 00:16:49,410 --> 00:16:51,330 is one that we will begin talking 311 00:16:51,330 --> 00:16:53,730 about in a lot more detail in the next lecture, 312 00:16:53,730 --> 00:16:56,700 but let me put it into here right away. 313 00:16:56,700 --> 00:16:58,440 And so that is the metric. 314 00:16:58,440 --> 00:17:01,980 Metric comes from a root meaning to measure, 315 00:17:01,980 --> 00:17:04,050 and what this is is it's something 316 00:17:04,050 --> 00:17:18,589 it gives me a notion of distance between events in the manifold. 317 00:17:25,190 --> 00:17:29,060 For physics to work, this has to exist, right? 318 00:17:29,060 --> 00:17:31,670 But it's worth knowing that the idea of a manifold 319 00:17:31,670 --> 00:17:34,250 is in some way more primitive than this. 320 00:17:34,250 --> 00:17:36,110 You can have a manifold without any notion 321 00:17:36,110 --> 00:17:37,320 of a metric attached to it. 322 00:17:37,320 --> 00:17:38,408 And if that's the case-- 323 00:17:38,408 --> 00:17:39,950 people like to joke that if you don't 324 00:17:39,950 --> 00:17:42,170 know what the difference is between a metric 325 00:17:42,170 --> 00:17:43,980 with a manifold and without a man-- 326 00:17:43,980 --> 00:17:47,690 excuse me, a manifold with a metric and without a metric, 327 00:17:47,690 --> 00:17:49,940 feel free to drink coffee out of a donut. 328 00:17:49,940 --> 00:17:52,610 Because topologically, those are the exact same thing, 329 00:17:52,610 --> 00:17:55,100 but their geometry, which is encoded in the metric, which 330 00:17:55,100 --> 00:17:57,710 tells me how the different points on that manifold 331 00:17:57,710 --> 00:18:00,040 are arranged, are rather different. 332 00:18:02,810 --> 00:18:04,790 What this basically does is it's going 333 00:18:04,790 --> 00:18:09,702 to give me a mathematical object that enforces or really conveys 334 00:18:09,702 --> 00:18:11,660 the idea that different events in this manifold 335 00:18:11,660 --> 00:18:13,490 have a particular distance between them. 336 00:18:13,490 --> 00:18:28,330 So without this, a manifold has no notion 337 00:18:28,330 --> 00:18:32,000 of distance encoded in it. 338 00:18:32,000 --> 00:18:33,820 So the two things together really make 339 00:18:33,820 --> 00:18:36,460 this concept come to life. 340 00:18:39,430 --> 00:18:40,930 You can see a lot more, like I said. 341 00:18:43,385 --> 00:18:45,760 You can get more information about many of these concepts 342 00:18:45,760 --> 00:18:49,300 from the readings in Carroll, Wald's textbook 343 00:18:49,300 --> 00:18:53,650 also goes into quite a lot of detail about this stuff. 344 00:18:53,650 --> 00:18:55,280 So this is the venue. 345 00:18:55,280 --> 00:18:58,300 This is the setting in which we are going to talk about things. 346 00:18:58,300 --> 00:19:01,020 And just cutting forward roughly 2 347 00:19:01,020 --> 00:19:03,760 and 1/2 months' worth of lectures, 348 00:19:03,760 --> 00:19:08,680 what we are going to find is that part of Einstein's genius 349 00:19:08,680 --> 00:19:11,890 is that it turns out that this notion of the metric 350 00:19:11,890 --> 00:19:13,502 ends up encoding gravity. 351 00:19:13,502 --> 00:19:14,960 So that's kind of where we're going 352 00:19:14,960 --> 00:19:16,127 to end up going with things. 353 00:19:16,127 --> 00:19:18,580 The idea that the mathematical structure 354 00:19:18,580 --> 00:19:21,202 that tells me how far apart two events are 355 00:19:21,202 --> 00:19:23,410 is intimately connected to the properties of gravity. 356 00:19:23,410 --> 00:19:27,140 It's pretty cool, and it is something that-- 357 00:19:27,140 --> 00:19:29,500 physics is an experimental science. 358 00:19:29,500 --> 00:19:32,430 All of our measurements are consistent with it. 359 00:19:32,430 --> 00:19:34,000 So that's cool. 360 00:19:38,670 --> 00:19:42,630 So everything I have said so far, nothing but math. 361 00:19:42,630 --> 00:19:44,490 Nothing but definitions. 362 00:19:44,490 --> 00:19:48,490 So let's start working with a particular form of physics. 363 00:19:48,490 --> 00:19:52,066 So we are going to begin, as I said, with special relativity. 364 00:19:57,660 --> 00:20:00,720 This is the simplest theory of spacetime 365 00:20:00,720 --> 00:20:04,500 that is compatible with physics as we know it. 366 00:20:04,500 --> 00:20:13,610 Not fully compatible, but does a pretty good job. 367 00:20:13,610 --> 00:20:17,090 And we'll see that it turns out to correspond 368 00:20:17,090 --> 00:20:25,220 to general relativity when there is no gravity. 369 00:20:37,680 --> 00:20:39,540 So to set this up we need to have 370 00:20:39,540 --> 00:20:43,700 some kind of a way of labeling our events. 371 00:20:43,700 --> 00:20:48,660 And so I'm going to introduce kind of a conceptual-- 372 00:20:48,660 --> 00:20:50,370 you almost think of it as scaffolding, 373 00:20:50,370 --> 00:20:53,640 which we're going to use to build a lot of our concepts 374 00:20:53,640 --> 00:20:54,990 around. 375 00:20:54,990 --> 00:20:56,490 And in this one-- 376 00:20:56,490 --> 00:20:59,035 I mean that in a kind of an abstract sense. 377 00:20:59,035 --> 00:21:00,660 There's going to be sort of the edifice 378 00:21:00,660 --> 00:21:02,660 that we use to help us build the building that's 379 00:21:02,660 --> 00:21:04,800 going to be the mathematics of general relativity. 380 00:21:04,800 --> 00:21:06,883 But this one really is kind of like a scaffolding. 381 00:21:06,883 --> 00:21:09,030 Because what I want to introduce here 382 00:21:09,030 --> 00:21:17,670 is a notion of what is called an inertial reference frame. 383 00:21:25,960 --> 00:21:28,440 So I'm going to sketch this quickly, 384 00:21:28,440 --> 00:21:31,440 and I'm going to post to the course website 385 00:21:31,440 --> 00:21:38,040 a chapter from an early draft textbook 386 00:21:38,040 --> 00:21:39,920 by Roger Blandford and Kip Thorne. 387 00:22:05,300 --> 00:22:07,660 So when I talk about the inertial reference frame, 388 00:22:07,660 --> 00:22:10,360 I want you to sort of visualize in your head 389 00:22:10,360 --> 00:22:11,620 a lattice of clocks-- 390 00:22:18,730 --> 00:22:20,200 clocks and measuring rods-- 391 00:22:26,110 --> 00:22:36,150 that allows us to label, in other words, to assign 392 00:22:36,150 --> 00:22:48,492 coordinates any event that happens in spacetime. 393 00:22:52,717 --> 00:22:54,300 So just sort of, in your head, imagine 394 00:22:54,300 --> 00:22:58,710 that there's this grid of little clocks and measuring rods, 395 00:22:58,710 --> 00:23:03,420 and a mosquito lands on your head. 396 00:23:03,420 --> 00:23:06,510 It's right near a particular rod and a particular clock. 397 00:23:06,510 --> 00:23:07,890 It bites you, that's an event. 398 00:23:07,890 --> 00:23:09,750 You slap it, that's another event. 399 00:23:09,750 --> 00:23:11,220 And the measuring rods and events 400 00:23:11,220 --> 00:23:13,110 are what allows you to sort of keep 401 00:23:13,110 --> 00:23:15,240 track of the ordering of those events 402 00:23:15,240 --> 00:23:17,520 and where they happen in this four dimensional 403 00:23:17,520 --> 00:23:21,240 manifold of spacetime. 404 00:23:21,240 --> 00:23:22,740 So I'm going to require this lattice 405 00:23:22,740 --> 00:23:24,746 to have a certain set of properties. 406 00:23:29,510 --> 00:23:39,910 First, I'm going to say that this lattice moves freely 407 00:23:39,910 --> 00:23:41,220 through spacetime. 408 00:23:45,920 --> 00:23:47,900 What do I mean by moving freely? 409 00:23:47,900 --> 00:23:51,410 I mean no forces act on it, it does not rotate, 410 00:23:51,410 --> 00:23:52,790 it is inertial. 411 00:23:52,790 --> 00:23:57,040 Every clock and every measuring rod has no inertia, 412 00:23:57,040 --> 00:23:58,850 no force is acting on it at all. 413 00:24:19,633 --> 00:24:21,800 You look at that, you might think to yourself, well, 414 00:24:21,800 --> 00:24:23,810 why don't you make it at rest? 415 00:24:23,810 --> 00:24:24,630 Well, I did. 416 00:24:24,630 --> 00:24:27,485 It's at rest in respect to someone, 417 00:24:27,485 --> 00:24:29,860 but we might have a different observer who's coming along 418 00:24:29,860 --> 00:24:33,610 who has no forces acting on her, and she's moving relative to me 419 00:24:33,610 --> 00:24:35,630 at three quarters the speed of light. 420 00:24:35,630 --> 00:24:37,930 It's not at rest with respects to that observer. 421 00:24:37,930 --> 00:24:39,472 That's actually kind of the key here. 422 00:24:42,040 --> 00:24:45,280 So this inertial reference frame is at rest with respect 423 00:24:45,280 --> 00:24:52,764 to someone who feels no forces but not to all observers. 424 00:25:06,130 --> 00:25:11,810 I'm going to require that my measuring rods are 425 00:25:11,810 --> 00:25:13,040 orthogonal to each other. 426 00:25:20,170 --> 00:25:28,000 So they define an orthogonal coordinate system, 427 00:25:28,000 --> 00:25:31,323 and I am also going to require that the little markings 428 00:25:31,323 --> 00:25:32,990 on them that tell me where things happen 429 00:25:32,990 --> 00:25:34,880 are uniformly ticked. 430 00:25:34,880 --> 00:25:37,580 In other words, I'm going to just make sure 431 00:25:37,580 --> 00:25:39,093 that the spacing between tick marks 432 00:25:39,093 --> 00:25:41,510 here is exactly the same as the spacing between tick marks 433 00:25:41,510 --> 00:25:42,440 here. 434 00:25:42,440 --> 00:25:46,730 You may sort of think well, that's a result or an idea 435 00:25:46,730 --> 00:25:50,270 worthy of the journal Duh, but it's important to specify this. 436 00:25:50,270 --> 00:25:52,520 You want to make sure that the standard you are using 437 00:25:52,520 --> 00:25:56,000 to define length is the same in this region of spacetime 438 00:25:56,000 --> 00:25:58,463 as it is over in this region of spacetime. 439 00:25:58,463 --> 00:26:00,380 When we start getting into general relativity, 440 00:26:00,380 --> 00:26:02,750 we start to see there can be concerns about this coming 441 00:26:02,750 --> 00:26:05,090 about, so it's worth spelling it out and making 442 00:26:05,090 --> 00:26:06,718 it clear at the beginning. 443 00:26:23,320 --> 00:26:27,280 I'm also going to require that my clocks tick uniformly. 444 00:26:31,978 --> 00:26:33,520 We're going to make this lattice that 445 00:26:33,520 --> 00:26:35,590 fills all of spacetime using the best 446 00:26:35,590 --> 00:26:37,450 thing that Swiss engineers can make for us, 447 00:26:37,450 --> 00:26:41,380 we want to make sure that one second, an interval of one 448 00:26:41,380 --> 00:26:43,660 second, is the same here in this classroom 449 00:26:43,660 --> 00:26:45,940 as it is somewhere off in the Andromeda Galaxy. 450 00:26:45,940 --> 00:26:48,730 We want to make sure that there is no evolution to the time 451 00:26:48,730 --> 00:26:51,280 standard when we do this. 452 00:26:54,690 --> 00:26:57,870 Finally, I'm going to synchronize all these clocks 453 00:26:57,870 --> 00:26:59,820 with each other in the following way. 454 00:27:09,380 --> 00:27:13,040 This is going to use the Einstein synchronization 455 00:27:13,040 --> 00:27:13,996 procedure. 456 00:27:23,040 --> 00:27:26,430 This is the first place where a little bit of physics 457 00:27:26,430 --> 00:27:29,560 is actually beginning to finally enter our discussion. 458 00:27:29,560 --> 00:27:30,463 I'll comment that-- 459 00:27:30,463 --> 00:27:32,380 I'm going to go through what this procedure is 460 00:27:32,380 --> 00:27:34,470 in just a second. 461 00:27:34,470 --> 00:27:36,210 But an Easter egg here. 462 00:27:36,210 --> 00:27:38,963 Whenever you see a name that has Einstein in it, 463 00:27:38,963 --> 00:27:40,380 your ears should perk up a little. 464 00:27:40,380 --> 00:27:42,630 Because it probably means this is something important. 465 00:27:42,630 --> 00:27:45,270 This is, after all, a course in relativity, 466 00:27:45,270 --> 00:27:48,300 and that tends to be the things that end up mattering. 467 00:27:48,300 --> 00:27:52,540 Even when they end up being really easy to-- 468 00:27:52,540 --> 00:27:57,840 things we look at now and kind of see as fairly obvious, 469 00:27:57,840 --> 00:28:01,350 it's important, OK, and we often attach Einstein's name to this. 470 00:28:01,350 --> 00:28:04,680 So this Einstein synchronization procedure, this 471 00:28:04,680 --> 00:28:07,502 takes advantage of the fact-- 472 00:28:07,502 --> 00:28:09,960 and by the way, we're not going to teach special relativity 473 00:28:09,960 --> 00:28:11,543 in this course, 8.962. 474 00:28:11,543 --> 00:28:13,710 I assume you have already studied special relativity 475 00:28:13,710 --> 00:28:17,460 and you're all experts in this, and so I can freely borrow 476 00:28:17,460 --> 00:28:19,680 from its important results. 477 00:28:19,680 --> 00:28:29,440 This procedure takes advantage of the fact 478 00:28:29,440 --> 00:28:42,380 that the speed of light is the same to all observers, 479 00:28:42,380 --> 00:28:44,152 no matter what inertial reference 480 00:28:44,152 --> 00:28:45,110 frame they might be in. 481 00:28:52,270 --> 00:28:56,980 So the speed of light is a key invariant. 482 00:28:56,980 --> 00:28:59,410 It connects-- because it's a speed, 483 00:28:59,410 --> 00:29:02,050 it connects space and time, and because it 484 00:29:02,050 --> 00:29:07,360 is the same to all observers, it defines a particular standard 485 00:29:07,360 --> 00:29:10,360 for relating space and time that is 486 00:29:10,360 --> 00:29:13,450 going to have important invariant meaning associated 487 00:29:13,450 --> 00:29:14,822 with it. 488 00:29:14,822 --> 00:29:16,780 Just to remind you what this means-- let's see, 489 00:29:16,780 --> 00:29:18,633 do I have a laser pointer with me? 490 00:29:18,633 --> 00:29:20,050 I might, but don't worry about it. 491 00:29:20,050 --> 00:29:21,700 I have a pretend laser pointer. 492 00:29:21,700 --> 00:29:23,650 My chalk's a laser pointer. 493 00:29:23,650 --> 00:29:25,150 I point my laser pointer at the wall 494 00:29:25,150 --> 00:29:28,120 and you all see it dashing across the room at 300,000 495 00:29:28,120 --> 00:29:29,950 kilometers per second. 496 00:29:29,950 --> 00:29:33,640 I then start jogging at half the speed of light, as is my wont, 497 00:29:33,640 --> 00:29:36,233 and continuing to point that, you guys measure the light 498 00:29:36,233 --> 00:29:37,150 going across the room. 499 00:29:37,150 --> 00:29:40,330 You still measure 300,000 kilometers per second. 500 00:29:40,330 --> 00:29:41,990 I, on the other hand, measure the light 501 00:29:41,990 --> 00:29:43,490 coming out of my laser pointer and I 502 00:29:43,490 --> 00:29:46,630 get 300,000 kilometers per second. 503 00:29:46,630 --> 00:29:49,600 So just because my laser pointer is 504 00:29:49,600 --> 00:29:51,913 moving at half the speed of light according to you 505 00:29:51,913 --> 00:29:53,830 doesn't mean the light that's coming out of it 506 00:29:53,830 --> 00:29:55,485 is boosted to a higher speed. 507 00:29:55,485 --> 00:29:56,860 If you studied special relativity 508 00:29:56,860 --> 00:29:59,260 you'll know that its energy is boosted to a higher energy 509 00:29:59,260 --> 00:30:03,610 level, but the speed of light is always just C. 510 00:30:03,610 --> 00:30:05,440 So we're going to take advantage of that 511 00:30:05,440 --> 00:30:10,270 to come up with a way of synchronizing our clocks. 512 00:30:10,270 --> 00:30:12,620 The way it works is this. 513 00:30:12,620 --> 00:30:15,310 So let's look at a two dimensional slice of my lattice 514 00:30:15,310 --> 00:30:16,810 here. 515 00:30:16,810 --> 00:30:20,600 Time, and let's make this be the x-axis. 516 00:30:20,600 --> 00:30:28,390 And so I will have, let's say, this is where clock one exists 517 00:30:28,390 --> 00:30:31,855 and this is where clock two exists. 518 00:30:34,635 --> 00:30:36,010 Let's go into the reference frame 519 00:30:36,010 --> 00:30:38,530 that is at rest with respect to this lattice. 520 00:30:38,530 --> 00:30:41,950 We want to synchronize clock one with clock two. 521 00:30:41,950 --> 00:30:45,580 So as time marches on, these guys stand still. 522 00:30:45,580 --> 00:30:47,710 So here's the path in spacetime-- 523 00:30:47,710 --> 00:30:50,920 the world line, traced out by clock one. 524 00:30:50,920 --> 00:30:54,280 Here's the clock path in spacetime, the world line, 525 00:30:54,280 --> 00:30:57,480 traced out by clock two. 526 00:30:57,480 --> 00:31:03,280 So let's say-- let's call this event-- 527 00:31:03,280 --> 00:31:09,100 let's say that this event happens at a time t1e. 528 00:31:09,100 --> 00:31:13,840 t1 is when clock one emits a pulse of light. 529 00:31:30,040 --> 00:31:33,730 So this light will just follow a little trajectory a bit 530 00:31:33,730 --> 00:31:35,020 through spacetime. 531 00:31:35,020 --> 00:31:44,110 This goes out and strikes clock two at which point 532 00:31:44,110 --> 00:31:48,070 it is bounced back to clock one. 533 00:31:48,070 --> 00:31:50,150 Let's ignore this point for just a second. 534 00:31:50,150 --> 00:31:52,990 So let's just say for the moment that this is then 535 00:31:52,990 --> 00:32:08,010 reflected back and then it is received back at clock one 536 00:32:08,010 --> 00:32:10,680 at a time t1r. 537 00:32:19,570 --> 00:32:21,760 Let's make this a little bit neater. 538 00:32:21,760 --> 00:32:23,340 t1e, t1r. 539 00:32:26,360 --> 00:32:30,300 Clock one receives the reflected pulse. 540 00:32:36,630 --> 00:32:40,510 So the moment at which it bounces, 541 00:32:40,510 --> 00:32:48,450 we'll call that t2b, that is the moment at which the light 542 00:32:48,450 --> 00:32:50,890 bounces off of clock two. 543 00:32:50,890 --> 00:32:52,440 And the way we synchronize our clocks 544 00:32:52,440 --> 00:32:53,970 is just by requiring that this be 545 00:32:53,970 --> 00:33:03,090 equal to the average of the emission and the reception 546 00:33:03,090 --> 00:33:03,810 time. 547 00:33:03,810 --> 00:33:05,940 Totally trivial idea, right? 548 00:33:05,940 --> 00:33:09,510 All I'm saying is I'm going to just require that 549 00:33:09,510 --> 00:33:11,310 in order for clock one and clock two 550 00:33:11,310 --> 00:33:13,180 to be synchronized to one another, 551 00:33:13,180 --> 00:33:17,460 let's make sure that when I bounce light between any two 552 00:33:17,460 --> 00:33:20,430 pairs of clocks they are set such that when 553 00:33:20,430 --> 00:33:22,560 the light bounces it's the midway point 554 00:33:22,560 --> 00:33:25,020 between the total light-- 555 00:33:25,020 --> 00:33:27,000 the halfway of the total light travel 556 00:33:27,000 --> 00:33:29,850 time that the light moves along. 557 00:33:29,850 --> 00:33:32,190 Very, very simple concept. 558 00:33:32,190 --> 00:33:33,810 So there's nothing particularly deep 559 00:33:33,810 --> 00:33:36,543 here, but notice Einstein's name is attached to this. 560 00:33:36,543 --> 00:33:37,710 And I don't say that to be-- 561 00:33:37,710 --> 00:33:39,960 I'm not being sarcastic. 562 00:33:39,960 --> 00:33:41,460 It really points to the fact that we 563 00:33:41,460 --> 00:33:45,000 are using one of the most fundamental results 564 00:33:45,000 --> 00:33:46,800 of special relativity in designing 565 00:33:46,800 --> 00:33:50,910 how these clocks work in this inertial reference frame. 566 00:33:50,910 --> 00:33:55,680 Believe it or not, this thing, this really simple concept, it 567 00:33:55,680 --> 00:33:58,440 comes back to sort of bite us on the butt a little bit later 568 00:33:58,440 --> 00:33:59,550 in this course. 569 00:33:59,550 --> 00:34:02,280 Because when we throw gravity into the mix, 570 00:34:02,280 --> 00:34:04,080 we're going to learn that gravity 571 00:34:04,080 --> 00:34:07,560 impacts the way light travels through spacetime. 572 00:34:07,560 --> 00:34:10,500 We're going to get to some objects where gravity is 573 00:34:10,500 --> 00:34:13,150 so strong that light cannot escape them. 574 00:34:13,150 --> 00:34:16,830 And we're going to find that our perhaps most naive 575 00:34:16,830 --> 00:34:20,940 ways of labeling time in the spacetime of such objects 576 00:34:20,940 --> 00:34:23,489 kind of goes completely haywire. 577 00:34:23,489 --> 00:34:25,949 Fundamentally, the reason why time is going haywire 578 00:34:25,949 --> 00:34:27,989 when you have really strong gravity 579 00:34:27,989 --> 00:34:30,510 is because we use light as our tool 580 00:34:30,510 --> 00:34:32,300 for synchronizing all of our clocks, 581 00:34:32,300 --> 00:34:34,889 and if the way that light moves in your space time 582 00:34:34,889 --> 00:34:37,110 is affected, the way you're going to label time 583 00:34:37,110 --> 00:34:38,590 is going to be affected. 584 00:34:38,590 --> 00:34:41,370 So this is a very simple concept. 585 00:34:41,370 --> 00:34:43,620 Again, I sort of emphasize these first couple lectures 586 00:34:43,620 --> 00:34:45,929 you're swatting a mosquito with a sledgehammer, 587 00:34:45,929 --> 00:34:47,760 but we're setting up this edifice 588 00:34:47,760 --> 00:34:49,260 because this will come back and it's 589 00:34:49,260 --> 00:34:50,760 important to bear this in mind when 590 00:34:50,760 --> 00:34:52,949 things get a little more interesting later 591 00:34:52,949 --> 00:34:55,860 in the course. 592 00:34:55,860 --> 00:34:59,250 So let's start setting up some geometrical objects here. 593 00:35:06,808 --> 00:35:08,600 Pardon me, let me do one other thing really 594 00:35:08,600 --> 00:35:11,340 quickly before I start setting up some geometrical objects. 595 00:35:11,340 --> 00:35:15,290 So when I sketched this thing called a spacetime diagram 596 00:35:15,290 --> 00:35:18,950 here, I should have talked a little bit about the units 597 00:35:18,950 --> 00:35:21,200 that I'm going to use to describe 598 00:35:21,200 --> 00:35:23,660 the ticking of my clocks and the spacing of tick marks 599 00:35:23,660 --> 00:35:24,560 on my measuring rods. 600 00:35:28,360 --> 00:35:31,990 What we will generally do in this course 601 00:35:31,990 --> 00:35:47,270 is choose the basic unit of length 602 00:35:47,270 --> 00:36:02,050 to be the distance light travels in your basic unit of time. 603 00:36:07,310 --> 00:36:10,580 While you parse that sentence, what that's basically saying is 604 00:36:10,580 --> 00:36:14,450 suppose I set my clock so that they tick every second. 605 00:36:14,450 --> 00:36:17,090 Well, if my clocks tick once per second, 606 00:36:17,090 --> 00:36:20,060 then my basic unit of length will be the light second. 607 00:36:39,290 --> 00:36:41,380 Do you want to put this into more familiar units? 608 00:36:41,380 --> 00:36:45,160 That's about 300,000 kilometers. 609 00:36:45,160 --> 00:36:46,933 One of my personal favorites-- 610 00:36:50,560 --> 00:36:59,640 if the time unit is 1 nanosecond, 611 00:36:59,640 --> 00:37:05,760 the length unit is, of course, one I will call it LNS-- 612 00:37:05,760 --> 00:37:06,930 Light Nanosecond. 613 00:37:09,610 --> 00:37:11,440 Students who are in 8.033 with me are not 614 00:37:11,440 --> 00:37:13,273 allowed to answer this question, does anyone 615 00:37:13,273 --> 00:37:17,330 know what one light nanosecond is? 616 00:37:17,330 --> 00:37:20,060 Actually, this is a little bit ridiculous. 617 00:37:20,060 --> 00:37:24,080 But to within far greater than a percent accuracy 618 00:37:24,080 --> 00:37:25,940 it is one foot. 619 00:37:25,940 --> 00:37:29,000 The English unit that comes on these asinine rulers 620 00:37:29,000 --> 00:37:30,950 that those of us educated the United States 621 00:37:30,950 --> 00:37:34,040 learned in all of our European friends sneer at us about. 622 00:37:34,040 --> 00:37:36,660 The speed of light is to incredible precision 623 00:37:36,660 --> 00:37:38,537 1 foot per nanosecond. 624 00:37:44,508 --> 00:37:46,300 To be fair, let's make that a wiggly equal. 625 00:37:51,100 --> 00:37:53,170 So what this means is that in the units that I'm 626 00:37:53,170 --> 00:37:59,590 going to be working with, if I then 627 00:37:59,590 --> 00:38:03,685 want to express the speed of light in these units-- 628 00:38:10,090 --> 00:38:22,360 So C is one light time unit per time unit, which 629 00:38:22,360 --> 00:38:24,640 we are just going to call 1. 630 00:38:24,640 --> 00:38:27,460 So we will generally set the speed of light equal to 1. 631 00:38:27,460 --> 00:38:29,410 Just bear in mind what this essentially means 632 00:38:29,410 --> 00:38:31,618 is that you can think of, if you want to then convert 633 00:38:31,618 --> 00:38:35,530 to your favorite meters per second, furlongs per fortnight, 634 00:38:35,530 --> 00:38:38,380 whatever it is that you're most comfortable with, 635 00:38:38,380 --> 00:38:41,670 C is effectively a conversion factor then. 636 00:38:41,670 --> 00:38:43,870 And so what this means is that when we do this 637 00:38:43,870 --> 00:38:47,792 all velocities that we measure are going to be dimensionless. 638 00:38:47,792 --> 00:38:49,750 Really what we're doing is we're measuring them 639 00:38:49,750 --> 00:38:51,208 as fractions of the speed of light. 640 00:39:00,540 --> 00:39:03,600 Now, with this system of units defined, 641 00:39:03,600 --> 00:39:06,490 let's talk about a geometric object. 642 00:39:06,490 --> 00:39:18,640 So let's imagine that O is an observer 643 00:39:18,640 --> 00:39:25,190 in the inertial reference frame that I 644 00:39:25,190 --> 00:39:26,390 defined a few moments ago. 645 00:39:31,070 --> 00:39:33,810 This is a mouthful to say, it's even more of a mouthful 646 00:39:33,810 --> 00:39:36,710 to write, so I'm going to typically abbreviate this IRF. 647 00:39:50,450 --> 00:39:53,360 So o observes two events, which I 648 00:39:53,360 --> 00:40:14,710 will label P and Q. I'll just go to a clean board for this. 649 00:40:33,730 --> 00:40:37,363 So let's say here in spacetime-- so imagine 650 00:40:37,363 --> 00:40:39,280 I've got coordinate axes that have been drawn, 651 00:40:39,280 --> 00:40:40,720 it's going to be three dimensional, 652 00:40:40,720 --> 00:40:43,053 for simplicity I'm not going to actually write them out. 653 00:40:43,053 --> 00:40:50,650 Let's say I've got event P here and event Q over here. 654 00:40:50,650 --> 00:40:53,792 Now if we were just doing Euclidean geometry in three 655 00:40:53,792 --> 00:40:55,750 space, you guys have all known that once you've 656 00:40:55,750 --> 00:40:57,490 got two events written down on a plane 657 00:40:57,490 --> 00:40:58,795 or in a three dimensional space, something 658 00:40:58,795 --> 00:41:01,212 like that, you can define the displacement vector from one 659 00:41:01,212 --> 00:41:01,858 to the other. 660 00:41:01,858 --> 00:41:03,650 We're going to the same thing in spacetime. 661 00:41:07,490 --> 00:41:10,100 So let's call delta x-- 662 00:41:12,960 --> 00:41:15,770 and I'm going to make a comment on notation in just a moment-- 663 00:41:15,770 --> 00:41:34,300 this is the displacement in spacetime from P to Q. 664 00:41:34,300 --> 00:41:36,400 We're going to define the components 665 00:41:36,400 --> 00:41:45,220 of this displacement vector as seen by O. So when 666 00:41:45,220 --> 00:41:51,190 I write equals with a dot on it that means the geometric object 667 00:41:51,190 --> 00:41:53,380 that I've written on the left hand side 668 00:41:53,380 --> 00:41:57,760 is given according to the specified 669 00:41:57,760 --> 00:42:01,090 observer by the following set of complements, which 670 00:42:01,090 --> 00:42:03,490 I'm about to write down. 671 00:42:03,490 --> 00:42:18,910 So this looks like so. 672 00:42:22,260 --> 00:42:24,935 So, couple things that I want to emphasize that I'm introducing 673 00:42:24,935 --> 00:42:26,310 here, bits of notation that we're 674 00:42:26,310 --> 00:42:28,860 going to use over and over again in this term. 675 00:42:28,860 --> 00:42:32,340 Notice I am using an over arrow to denote 676 00:42:32,340 --> 00:42:34,770 a vector in spacetime. 677 00:42:34,770 --> 00:42:36,600 Different texts, different professors 678 00:42:36,600 --> 00:42:38,700 use slightly different notations for this. 679 00:42:38,700 --> 00:42:42,150 Those of you who took 8.033 at MIT with Sal Vitale he 680 00:42:42,150 --> 00:42:45,930 preferred to write a little under tilde when he wrote that. 681 00:42:45,930 --> 00:42:48,810 For us working in four dimensional space time, 682 00:42:48,810 --> 00:42:51,250 it is going to be of paramount importance to us, 683 00:42:51,250 --> 00:42:53,820 and so we're going to use this over arrow which you probably 684 00:42:53,820 --> 00:42:56,250 have all seen for ordinary three dimensional vectors. 685 00:42:56,250 --> 00:42:59,700 For us it's going to represent a four dimensional vector. 686 00:42:59,700 --> 00:43:04,290 Now in truth we're not going to use it all that much 687 00:43:04,290 --> 00:43:07,190 after the first couple of weeks of the class, 688 00:43:07,190 --> 00:43:08,970 our first couple lectures even. 689 00:43:08,970 --> 00:43:11,220 Occasionally we'll bust it out, but we 690 00:43:11,220 --> 00:43:25,180 will tend to use a more compact notation in which we 691 00:43:25,180 --> 00:43:33,040 say delta x, that displacement factor has the components 692 00:43:33,040 --> 00:43:42,610 delta x mu, where mu lies is in either t, x, y, and z, 693 00:43:42,610 --> 00:43:48,195 or 0, 1, 2, 3. 694 00:43:48,195 --> 00:43:49,570 When we set up a problem, we need 695 00:43:49,570 --> 00:43:52,380 to make a mapping to what the numerical correspondences 696 00:43:52,380 --> 00:43:52,880 between-- 697 00:43:52,880 --> 00:43:57,760 I need to tell you that mu equals 0 corresponds to time, 698 00:43:57,760 --> 00:43:59,913 and mu equals 1 corresponds to x. 699 00:43:59,913 --> 00:44:01,580 We'll switch to other coordinate systems 700 00:44:01,580 --> 00:44:04,180 and I'll have to be careful to say, almost always, 701 00:44:04,180 --> 00:44:06,460 mu equals 0 will be time. 702 00:44:06,460 --> 00:44:08,877 But what the other three correspond to, 703 00:44:08,877 --> 00:44:10,460 that depends on the coordinate system. 704 00:44:10,460 --> 00:44:12,627 It might be a radius, it might be a different angle, 705 00:44:12,627 --> 00:44:15,130 some things like that. 706 00:44:15,130 --> 00:44:17,920 Again, just sort of being a little overly cautious 707 00:44:17,920 --> 00:44:19,960 and careful defining these. 708 00:44:19,960 --> 00:44:27,210 I will note, though, that generally Greek indices 709 00:44:27,210 --> 00:44:38,390 in most textbooks, they tend to be used 710 00:44:38,390 --> 00:44:40,220 to label spacetime indices. 711 00:44:45,155 --> 00:44:46,530 And then there are times when you 712 00:44:46,530 --> 00:44:48,510 might want to just sort of imagine 713 00:44:48,510 --> 00:44:51,120 you've chosen a particular moment in time, 714 00:44:51,120 --> 00:44:55,030 and you want to look at what space looks like at that time. 715 00:44:55,030 --> 00:45:10,750 And so you might then go down to Latin indices 716 00:45:10,750 --> 00:45:14,450 to pick out spatial components at some moment in time. 717 00:45:32,373 --> 00:45:34,290 We'll see that come up from time to time, just 718 00:45:34,290 --> 00:45:36,248 want you to be aware there is this distinction. 719 00:45:36,248 --> 00:45:37,840 And as you read other textbooks, there 720 00:45:37,840 --> 00:45:40,300 are a few others that are used. 721 00:45:40,300 --> 00:45:41,785 Always just check, usually in some 722 00:45:41,785 --> 00:45:43,660 of the introductory chapters of the textbook, 723 00:45:43,660 --> 00:45:45,587 they will define these things very carefully. 724 00:45:45,587 --> 00:45:47,920 Wald is an example of someone who actually does anything 725 00:45:47,920 --> 00:45:48,878 a little bit different. 726 00:45:48,878 --> 00:45:53,140 He tends to use lower case Latin letters 727 00:45:53,140 --> 00:45:57,670 from the top of the alphabet to denote spacetime indices, 728 00:45:57,670 --> 00:46:03,460 and those from i, j, k, he uses them to denote Latin indices. 729 00:46:03,460 --> 00:46:04,880 If you're old enough to get this, 730 00:46:04,880 --> 00:46:08,110 this is often called by some of us who grew up in the Dark Ages 731 00:46:08,110 --> 00:46:10,240 it is sometimes called the Fortran convention. 732 00:46:10,240 --> 00:46:13,210 If you've ever programmed in Fortran, you know why that is. 733 00:46:13,210 --> 00:46:15,250 If you didn't, please don't bother learning it, 734 00:46:15,250 --> 00:46:18,820 it's really not worth the brain cells it would take. 735 00:46:18,820 --> 00:46:21,970 So we've got this geometric object 736 00:46:21,970 --> 00:46:25,698 that is viewed by observer o. 737 00:46:25,698 --> 00:46:28,240 Let's now think about what this looks like from the viewpoint 738 00:46:28,240 --> 00:46:29,860 of a different observer. 739 00:46:29,860 --> 00:46:32,730 A different inertial observer. 740 00:46:55,940 --> 00:47:00,800 Let's say somebody comes dashing through the room here, 741 00:47:00,800 --> 00:47:05,990 and observer o sees them running across the room at something 742 00:47:05,990 --> 00:47:09,050 like 87% of the speed of light. 743 00:47:09,050 --> 00:47:11,090 You know, since, as I have assumed, 744 00:47:11,090 --> 00:47:12,890 you are all experts in special relativity, 745 00:47:12,890 --> 00:47:16,640 that they will measure intervals of time and intervals of space 746 00:47:16,640 --> 00:47:18,650 differently than observer o does. 747 00:47:24,690 --> 00:47:33,570 So here's event P. Here's event Q. Here is delta x. 748 00:47:33,570 --> 00:47:39,180 This is all as measured by observer o bar. 749 00:47:42,380 --> 00:47:45,110 Something which I really want to strongly emphasize 750 00:47:45,110 --> 00:47:51,800 at this point is that this P, this Q, and this delta x, 751 00:47:51,800 --> 00:47:54,250 notice I haven't put bars on any of them. 752 00:47:54,250 --> 00:47:56,210 I haven't put primes or anything like that. 753 00:47:56,210 --> 00:48:01,310 It is the exact same P and Q and delta x as this over here. 754 00:48:01,310 --> 00:48:11,790 That is because P, Q, and delta x are geometric objects whose 755 00:48:11,790 --> 00:48:15,330 meaning transcends the particular inertial reference 756 00:48:15,330 --> 00:48:20,320 frame used to define the coordinates at which P exists, 757 00:48:20,320 --> 00:48:22,860 at which Q exists, and that then defined the delta x. 758 00:48:29,000 --> 00:48:31,670 These geometric objects exist independent 759 00:48:31,670 --> 00:48:33,186 of the representation. 760 00:48:41,620 --> 00:48:46,040 If I can use an intuitive example-- 761 00:48:46,040 --> 00:48:48,370 if I take and I hold-- let's be careful the pose 762 00:48:48,370 --> 00:48:51,190 I do with this-- let's say I stick my arm out. 763 00:48:51,190 --> 00:48:56,493 I say that my arm is pointing to the left, right? 764 00:48:56,493 --> 00:48:57,910 You guys will look at this and say 765 00:48:57,910 --> 00:48:59,360 your arm is pointing to the right, 766 00:48:59,360 --> 00:49:01,985 because you're using a slightly different system of coordinates 767 00:49:01,985 --> 00:49:04,600 to orient yourself in this room. 768 00:49:04,600 --> 00:49:05,890 We're both right. 769 00:49:05,890 --> 00:49:09,100 We have represented this geometric object, my arm, 770 00:49:09,100 --> 00:49:11,070 in different ways. 771 00:49:11,070 --> 00:49:12,888 But me calling this pointing to the left 772 00:49:12,888 --> 00:49:15,180 and you calling it pointing to the right doesn't change 773 00:49:15,180 --> 00:49:17,970 the basic nature of my arm. 774 00:49:17,970 --> 00:49:20,790 It doesn't mean that my blood cells 775 00:49:20,790 --> 00:49:23,410 changed because of something like this happening. 776 00:49:23,410 --> 00:49:25,620 This has an independent existence. 777 00:49:25,620 --> 00:49:27,620 In the same way, this delta x, it 778 00:49:27,620 --> 00:49:30,210 is the displacement, these two events. 779 00:49:30,210 --> 00:49:32,280 This might be mosquito lands on my head, 780 00:49:32,280 --> 00:49:34,410 this might be me smacking it with my hand, 781 00:49:34,410 --> 00:49:36,870 or flying near my head and me smocking with my hand. 782 00:49:36,870 --> 00:49:39,480 Those are events that exist independent of how 783 00:49:39,480 --> 00:49:42,700 we choose to represent them. 784 00:49:42,700 --> 00:49:45,810 So the key thing is we preserve that notion 785 00:49:45,810 --> 00:49:48,240 of the geometric object's independent existence. 786 00:49:48,240 --> 00:49:51,870 What does change is the representation 787 00:49:51,870 --> 00:49:53,100 that the two observers use. 788 00:50:02,970 --> 00:50:06,485 So I'm going to jump to this Greek index notation. 789 00:50:17,820 --> 00:50:22,450 And so what I'm going to say is that according to observer o 790 00:50:22,450 --> 00:50:30,780 bar, they are going to represent this object by a collection 791 00:50:30,780 --> 00:50:34,080 of components that are not the same as the complaints that 792 00:50:34,080 --> 00:50:36,092 are used by observer o. 793 00:50:36,092 --> 00:50:37,550 And to keep my notation consistent, 794 00:50:37,550 --> 00:50:44,280 let's play a little o underneath this arrow. 795 00:50:44,280 --> 00:50:47,040 So this is just sort of shorthand for delta 796 00:50:47,040 --> 00:50:50,970 x is represented according to o by those components. 797 00:50:50,970 --> 00:50:55,750 Delta x is represented according to o bar by these components. 798 00:50:55,750 --> 00:50:57,300 And again, since I'm assuming you all 799 00:50:57,300 --> 00:50:59,610 are experts in special relativity, 800 00:50:59,610 --> 00:51:04,830 we already know how to relate the barred components 801 00:51:04,830 --> 00:51:06,060 to the unbarred components. 802 00:51:06,060 --> 00:51:08,539 They're related by a Lorentz transformation. 803 00:51:35,500 --> 00:51:40,480 So what we would say is delta x is zero bar component, or t 804 00:51:40,480 --> 00:51:42,050 bar, if you prefer. 805 00:51:42,050 --> 00:51:43,600 It's given by gamma-- 806 00:51:43,600 --> 00:51:47,260 I will define gamma in just a moment, you can probably guess. 807 00:51:49,890 --> 00:51:51,670 So I'm imagining an observer that 808 00:51:51,670 --> 00:51:56,035 is just moving along the x-axis or the coordinate one axis. 809 00:52:35,330 --> 00:52:38,640 So what went on in that transformation 810 00:52:38,640 --> 00:52:51,390 I just wrote down is O bar moves with v along spatial axis 811 00:52:51,390 --> 00:53:04,340 1 with speed v as seen by O. And of course gamma 812 00:53:04,340 --> 00:53:10,200 is 1 divided by square root of 1 minus v squared. 813 00:53:10,200 --> 00:53:11,460 Remember, speed of light is 1. 814 00:53:15,410 --> 00:53:18,540 We don't want to be writing this crap out every time we have 815 00:53:18,540 --> 00:53:20,820 to transform different representations, 816 00:53:20,820 --> 00:53:23,990 so we're going to introduce more compact notation for this. 817 00:53:30,750 --> 00:53:34,960 So we're going to say delta x mu bar-- 818 00:53:38,264 --> 00:53:40,270 this is what I get when I sum over 819 00:53:40,270 --> 00:53:45,760 index nu from 0 to 3 of lambda. 820 00:53:45,760 --> 00:53:51,840 Mu bar nu, delta x nu. 821 00:53:51,840 --> 00:53:54,447 So that's defining a matrix multiplication, , 822 00:53:54,447 --> 00:53:56,530 and you can read out the components of this lambda 823 00:53:56,530 --> 00:53:58,072 matrix from what I've got over there. 824 00:54:03,530 --> 00:54:21,850 And even better yet, delta x mu bar 825 00:54:21,850 --> 00:54:28,090 is lambda mu bar nu delta x nu. 826 00:54:28,090 --> 00:54:30,630 So in this last line, if you haven't seen this before, 827 00:54:30,630 --> 00:54:33,703 I am using Einstein's summation convention. 828 00:54:45,790 --> 00:54:49,300 If I have an index that appears in one geometric 829 00:54:49,300 --> 00:54:51,260 object in the downstairs position 830 00:54:51,260 --> 00:54:53,050 and an adjacent geometric objects 831 00:54:53,050 --> 00:54:57,370 in the upstairs position and it's 832 00:54:57,370 --> 00:55:00,520 repeated-- so repeated indices in the upstairs and downstairs 833 00:55:00,520 --> 00:55:03,400 position are assumed to be summed over their full range, 834 00:55:03,400 --> 00:55:04,110 from 0 to 3. 835 00:55:10,278 --> 00:55:12,320 We're going to talk about this a little bit more, 836 00:55:12,320 --> 00:55:15,230 what's going on with this, after I've built up a little bit more 837 00:55:15,230 --> 00:55:17,410 of the mathematical structure. 838 00:55:17,410 --> 00:55:19,310 In particular, what is the distinction 839 00:55:19,310 --> 00:55:22,286 between through the upstairs and downstairs positions. 840 00:55:37,913 --> 00:55:39,330 Some of you might be saying, well, 841 00:55:39,330 --> 00:55:42,390 isn't one way of writing it what we call a covariant component 842 00:55:42,390 --> 00:55:44,820 and wasn't one a contravariant component? 843 00:55:44,820 --> 00:55:47,730 If you know those terms, mazel tov. 844 00:55:47,730 --> 00:55:49,973 They're not actually really helpful, 845 00:55:49,973 --> 00:55:51,390 and so I kind of deliberately like 846 00:55:51,390 --> 00:55:53,520 to use this more primitive wording of calling it 847 00:55:53,520 --> 00:55:55,380 just upstairs and downstairs. 848 00:55:55,380 --> 00:55:57,630 Because what we're going to find, the goal of physics, 849 00:55:57,630 --> 00:56:00,240 is to understand the universe in a way that 850 00:56:00,240 --> 00:56:03,390 allows us to connect this understanding to measurements. 851 00:56:03,390 --> 00:56:05,940 And measurements don't care about contravariant 852 00:56:05,940 --> 00:56:08,460 versus covariant, and all these things are essentially 853 00:56:08,460 --> 00:56:12,000 just ways of representing objects with our mathematics 854 00:56:12,000 --> 00:56:14,940 that is sort of a go between from some of our physical ideas 855 00:56:14,940 --> 00:56:16,920 to what can eventually be measured. 856 00:56:16,920 --> 00:56:20,093 So covariant, contravariant, eh, whatever. 857 00:56:20,093 --> 00:56:21,510 At the end of the day, we're going 858 00:56:21,510 --> 00:56:23,700 to see as we put these sort of things together 859 00:56:23,700 --> 00:56:26,940 it's how these terms connect to one another that matters. 860 00:56:26,940 --> 00:56:30,285 The name is not that important. 861 00:56:30,285 --> 00:56:32,160 I do want to make one little point about this 862 00:56:32,160 --> 00:56:33,970 as I move forward. 863 00:56:33,970 --> 00:56:39,400 It is sort of worth noting that if I 864 00:56:39,400 --> 00:56:44,830 think of how I relate the displacement components 865 00:56:44,830 --> 00:56:48,040 according to my barred observer relative to those as measured 866 00:56:48,040 --> 00:56:54,460 by my unbarred observer, I can think of this Lorentz 867 00:56:54,460 --> 00:56:59,890 transformation matrix as what I get when I differentiate 868 00:56:59,890 --> 00:57:03,040 one representation's coordinates with respect 869 00:57:03,040 --> 00:57:06,700 to the other representation's coordinates. 870 00:57:06,700 --> 00:57:09,160 Kind of trivial in this case, and when 871 00:57:09,160 --> 00:57:10,870 we are doing special relativity there 872 00:57:10,870 --> 00:57:13,120 is a particular form of the Lorentz transformation 873 00:57:13,120 --> 00:57:15,220 that we tend to use, but I just want 874 00:57:15,220 --> 00:57:18,280 to highlight this because this relationship between two 875 00:57:18,280 --> 00:57:21,068 different representations of a reference frame 876 00:57:21,068 --> 00:57:23,110 is going to come up over and over and over again. 877 00:57:23,110 --> 00:57:25,120 This is a more general form, this idea 878 00:57:25,120 --> 00:57:27,500 that you are essentially taking the derivative. 879 00:57:27,500 --> 00:57:31,000 You're looking at how one representation varies according 880 00:57:31,000 --> 00:57:32,230 to the other representation. 881 00:57:32,230 --> 00:57:34,870 So using that to think about how to move 882 00:57:34,870 --> 00:57:37,910 between one inertial reference frame to another 883 00:57:37,910 --> 00:57:40,610 is going to be very important to us. 884 00:57:40,610 --> 00:57:43,175 This is a more general form. 885 00:57:58,950 --> 00:58:01,320 I want to make one further point, 886 00:58:01,320 --> 00:58:06,330 and then I will introduce a more careful definition of what 887 00:58:06,330 --> 00:58:08,910 is meant by a vector and I think that will 888 00:58:08,910 --> 00:58:11,140 be a good place for us to stop. 889 00:58:11,140 --> 00:58:13,130 So when I look at this particular form-- 890 00:58:13,130 --> 00:58:16,260 so let's look at the last thing I wrote there, 891 00:58:16,260 --> 00:58:19,196 where I use the Einstein summation convention. 892 00:58:23,010 --> 00:58:25,800 This is just a chance for me to introduce a little bit more 893 00:58:25,800 --> 00:58:26,985 terminology in notation. 894 00:58:32,410 --> 00:58:35,580 So when I wrote down the relationship delta 895 00:58:35,580 --> 00:58:45,670 x mu bar with lambda mu bar nu, delta x nu, 896 00:58:45,670 --> 00:58:49,220 how I labeled the index that I was summing over 897 00:58:49,220 --> 00:58:50,729 is kind of irrelevant. 898 00:59:06,420 --> 00:59:14,460 This is exactly the same as lambda mu bar alpha delta 899 00:59:14,460 --> 00:59:15,960 x alpha. 900 00:59:15,960 --> 00:59:22,400 I can switch to something else if you 901 00:59:22,400 --> 00:59:25,830 have enough fonts available. 902 00:59:25,830 --> 00:59:28,950 You can use smiley faces as your index, whatever. 903 00:59:28,950 --> 00:59:32,280 The key bit is that as long as you are summing over it, 904 00:59:32,280 --> 00:59:34,292 it's kind of irrelevant how you label it. 905 00:59:34,292 --> 00:59:36,750 I think that looks silly, so I'm going to go back to alpha. 906 00:59:49,520 --> 00:59:54,020 When I have an index like that that is being summed over, 907 00:59:54,020 --> 00:59:56,840 it's going to sort of disappear at the end of my analysis. 908 00:59:56,840 --> 01:00:00,510 Its only role is to serve as a place holder. 909 01:00:00,510 --> 01:00:02,510 It allows me to keep things lined up properly 910 01:00:02,510 --> 01:00:05,490 so that I can do a particular mathematical operation. 911 01:00:05,490 --> 01:00:20,140 So in this equation, nu or alpha is called a dummy index. 912 01:00:25,778 --> 01:00:27,320 Now, when I'm doing it with something 913 01:00:27,320 --> 01:00:30,200 like this, where I'm just relating one set of one 914 01:00:30,200 --> 01:00:33,020 indexed objects to another set of one indexed objects, 915 01:00:33,020 --> 01:00:35,067 it's kind of trivial to move these things around. 916 01:00:35,067 --> 01:00:37,400 We're going to make some much more complicated equations 917 01:00:37,400 --> 01:00:40,270 later in this class, and we'll find in those cases 918 01:00:40,270 --> 01:00:42,020 that sometimes it's actually really useful 919 01:00:42,020 --> 01:00:44,120 to have the freedom to relabel our dummy indices. 920 01:00:44,120 --> 01:00:46,070 It allows us to sort of pick out patterns 921 01:00:46,070 --> 01:00:49,730 that might exist among things and see 922 01:00:49,730 --> 01:00:53,280 how to simplify a relationship in a really useful way. 923 01:00:53,280 --> 01:00:56,060 On the other hand, I do not have the freedom 924 01:00:56,060 --> 01:00:59,060 to change that mu bar that appears there, right. 925 01:00:59,060 --> 01:01:01,340 I have to have that being the same 926 01:01:01,340 --> 01:01:03,030 on both sides of the equation. 927 01:01:03,030 --> 01:01:07,730 Because I'm free to mix around things where it's just 928 01:01:07,730 --> 01:01:11,020 going to be summed over and not play a role, 929 01:01:11,020 --> 01:01:14,600 but in this my mu bar-- 930 01:01:14,600 --> 01:01:16,250 pardon me for a second-- 931 01:01:16,250 --> 01:01:20,440 mu bar is not a dummy index, and so I do not have that freedom. 932 01:01:24,500 --> 01:01:31,660 We sometimes call this the free index. 933 01:01:35,650 --> 01:01:38,150 As I write that down, it seems like a bit of a strange name, 934 01:01:38,150 --> 01:01:39,608 actually, and it's really not free. 935 01:01:39,608 --> 01:01:41,755 You're actually constrained in what it can be. 936 01:01:41,755 --> 01:01:43,880 What can I say, probably there's some history there 937 01:01:43,880 --> 01:01:44,880 that I don't know about. 938 01:01:58,490 --> 01:02:01,290 So let's do our last concept for the day. 939 01:02:08,070 --> 01:02:11,370 So let's carefully define a spacetime vector. 940 01:02:23,520 --> 01:02:27,720 So a spacetime vector is going to be any quartet of numbers, 941 01:02:27,720 --> 01:02:55,390 those numbers we will call components, which transforms 942 01:02:55,390 --> 01:03:00,242 between inertial reference frames 943 01:03:00,242 --> 01:03:01,450 like the displacement vector. 944 01:03:14,950 --> 01:03:19,490 So if I represent some spacetime vector A 945 01:03:19,490 --> 01:03:25,740 as some collection of numbers that as observed by o 946 01:03:25,740 --> 01:03:29,580 has components A0, A1, A2, A3. 947 01:03:37,540 --> 01:03:42,430 If a second inertial observer relates their components 948 01:03:42,430 --> 01:03:43,300 to these by-- 949 01:04:04,880 --> 01:04:07,798 if that describes the components for the observer o bar, 950 01:04:07,798 --> 01:04:08,590 then it's a vector. 951 01:04:23,400 --> 01:04:26,740 If you have taken any mathematics, 952 01:04:26,740 --> 01:04:28,950 they carefully defined vector spaces, 953 01:04:28,950 --> 01:04:30,520 this should be familiar to you. 954 01:04:30,520 --> 01:04:32,460 It's a very similar operation to what 955 01:04:32,460 --> 01:04:38,700 is done in a lot of other kinds of analysis. 956 01:04:38,700 --> 01:04:43,020 The key to making this notion of a vector a sensible one 957 01:04:43,020 --> 01:04:46,830 is this transformation law. 958 01:04:46,830 --> 01:04:52,950 So if I had a quartet of numbers, which is say, 959 01:04:52,950 --> 01:04:57,090 the number of batteries in my pocket, the number 960 01:04:57,090 --> 01:05:01,440 of times my dog sneezed this morning, how many toes I 961 01:05:01,440 --> 01:05:02,827 have on my left foot, and-- 962 01:05:02,827 --> 01:05:04,410 I'm sick of counting so let's just say 963 01:05:04,410 --> 01:05:07,020 0 for the fourth component-- 964 01:05:07,020 --> 01:05:10,020 that does not transform between reference frames 965 01:05:10,020 --> 01:05:11,680 by a Lorentz transformation. 966 01:05:11,680 --> 01:05:14,250 It's just a collection of random numbers. 967 01:05:14,250 --> 01:05:16,530 So not any old quartet of numbers 968 01:05:16,530 --> 01:05:18,600 will constitute the components of a vector. 969 01:05:18,600 --> 01:05:21,810 It has to be things that have a physical meaning that you 970 01:05:21,810 --> 01:05:24,852 connect to what you measure by a Lorentz transformation. 971 01:05:28,740 --> 01:05:32,850 For it to be a good vector, it also has to-- that A 972 01:05:32,850 --> 01:05:36,030 has to obey the various linearity laws that 973 01:05:36,030 --> 01:05:37,570 define a vector space. 974 01:05:48,090 --> 01:06:02,160 So if I had two vectors and I add them together, 975 01:06:02,160 --> 01:06:04,620 then their sum is a vector. 976 01:06:13,180 --> 01:06:16,875 If I have a vector and I multiply it by some scalar-- 977 01:06:27,572 --> 01:06:29,530 by the way, you have to be a little bit careful 978 01:06:29,530 --> 01:06:33,730 when I say scalar here, because you might think 979 01:06:33,730 --> 01:06:36,700 to yourself something like, you know, 980 01:06:36,700 --> 01:06:39,570 the mass of my shoe, that's a scalar. 981 01:06:39,570 --> 01:06:42,060 But you be careful when you're talking about things 982 01:06:42,060 --> 01:06:44,250 in relativity, whether the scale you're dealing with 983 01:06:44,250 --> 01:06:47,622 is actually a quantity that is Lorentz in variant. 984 01:06:47,622 --> 01:06:49,830 With quantity like mass, if you're talking about rest 985 01:06:49,830 --> 01:06:51,630 mass-- we'll get into the distinction among these things 986 01:06:51,630 --> 01:06:52,000 a bit later-- 987 01:06:52,000 --> 01:06:52,830 OK, you're good. 988 01:06:55,380 --> 01:06:56,880 You just have be careful to pick out 989 01:06:56,880 --> 01:06:59,430 something that actually is the same according 990 01:06:59,430 --> 01:07:03,790 to all observers. 991 01:07:03,790 --> 01:07:09,850 So when I say scalar, this means same to all observers. 992 01:07:15,040 --> 01:07:17,100 If this is the case, then I can define 993 01:07:17,100 --> 01:07:24,180 D to be that scalar times a vector and it is also a vector. 994 01:07:24,180 --> 01:07:25,130 Question? 995 01:07:25,130 --> 01:07:26,010 Oh, stretching, OK. 996 01:07:33,650 --> 01:07:35,720 I think, actually, I'm going to stop there. 997 01:07:35,720 --> 01:07:39,590 There is one topic that I just don't 998 01:07:39,590 --> 01:07:42,925 feel like I can get enough of it for it to be useful right now, 999 01:07:42,925 --> 01:07:43,550 so I think yes. 1000 01:07:43,550 --> 01:07:45,620 I'm going to stop there for today. 1001 01:07:45,620 --> 01:07:48,430 When we pick it up on Thursday we're 1002 01:07:48,430 --> 01:07:51,050 going to wrap up this discussion of vectors-- 1003 01:07:51,050 --> 01:07:52,550 again, I want to kind of emphasize, 1004 01:07:52,550 --> 01:07:55,230 I can already see in some places you're getting the 1,000 yard 1005 01:07:55,230 --> 01:07:55,730 stare. 1006 01:07:55,730 --> 01:07:59,358 There's no question we're being excessively careful 1007 01:07:59,358 --> 01:08:00,650 with some of these definitions. 1008 01:08:00,650 --> 01:08:02,233 They are very straightforward, there's 1009 01:08:02,233 --> 01:08:03,650 nothing challenging here. 1010 01:08:03,650 --> 01:08:07,460 But there will be a payoff when we do get to times where 1011 01:08:07,460 --> 01:08:09,470 the analyses, the geometries we're looking at, 1012 01:08:09,470 --> 01:08:12,440 get pretty messed up. 1013 01:08:12,440 --> 01:08:15,410 Having this formal foundation very carefully laid 1014 01:08:15,410 --> 01:08:17,520 will help us significantly. 1015 01:08:17,520 --> 01:08:19,950 So all right, I'm going to stop there for today, 1016 01:08:19,950 --> 01:08:23,590 and we will pick it up on Thursday.