WEBVTT 00:00:00.500 --> 00:00:02.874 [SQUEAKING] 00:00:02.874 --> 00:00:04.311 [RUSTLING] 00:00:04.311 --> 00:00:10.285 [CLICKING] 00:00:10.285 --> 00:00:12.160 SCOTT HUGHES: The key textbook for this class 00:00:12.160 --> 00:00:16.372 is Sean Carroll's textbook on general relativity. 00:00:19.150 --> 00:00:22.270 It's now almost 20 years old. 00:00:22.270 --> 00:00:23.830 I would say-- I think it's listed 00:00:23.830 --> 00:00:25.400 on the website as required. 00:00:25.400 --> 00:00:28.950 I would actually call it sort of semi-required. 00:00:28.950 --> 00:00:34.120 It is where I will tend to post most of the readings 00:00:34.120 --> 00:00:35.980 to the course. 00:00:35.980 --> 00:00:41.740 It's a really good, complete textbook for a one semester 00:00:41.740 --> 00:00:44.193 course, which is what we have, and I will not 00:00:44.193 --> 00:00:45.610 be going through the entire thing. 00:00:45.610 --> 00:00:47.940 You can't in a one semester course. 00:00:47.940 --> 00:00:50.680 And I will, from time to time, there will be a few topics that 00:00:50.680 --> 00:00:54.195 I cannot go as in into as deeply as I would like, 00:00:54.195 --> 00:00:56.320 and if we had two semesters maybe I'd be able to do 00:00:56.320 --> 00:00:56.890 a little bit more. 00:00:56.890 --> 00:00:58.390 And so for those who are interested, 00:00:58.390 --> 00:01:00.730 I will suggest readings in this. 00:01:00.730 --> 00:01:04.209 My personal favorite supplement to this is a textbook 00:01:04.209 --> 00:01:08.358 by Bernard Schutz , A First Course in General Relativity. 00:01:12.849 --> 00:01:15.895 OK, so these are all things where-- so the MIT bookstore, 00:01:15.895 --> 00:01:17.270 I'm not sure how much they carry. 00:01:17.270 --> 00:01:18.650 They're all available through Amazon, 00:01:18.650 --> 00:01:20.692 and you can definitely find these kind of things. 00:01:20.692 --> 00:01:23.570 If you get Schutz's textbook, definitely 00:01:23.570 --> 00:01:24.950 get the second edition. 00:01:29.730 --> 00:01:31.770 The first edition contains errors. 00:01:31.770 --> 00:01:34.860 There is actually a very important geometric object 00:01:34.860 --> 00:01:37.320 that we are going to introduce in a couple of weeks 00:01:37.320 --> 00:01:40.092 that his textbook has a really clever derivation of. 00:01:40.092 --> 00:01:41.800 I remember seeing that and thinking, wow, 00:01:41.800 --> 00:01:42.690 that's really clever. 00:01:42.690 --> 00:01:45.232 The reason he was able do it so simply is because it's wrong. 00:01:45.232 --> 00:01:46.080 AUDIENCE: [LAUGH] 00:01:46.080 --> 00:01:49.050 SCOTT HUGHES: It's corrected in the second edition. 00:01:49.050 --> 00:01:54.020 Another one is Gravitation by Misner, Thorne and Wheeler. 00:01:54.020 --> 00:01:59.200 This is sort of a bit of personal history for me. 00:01:59.200 --> 00:02:01.920 I was Thorne's graduate student, and I used this textbook 00:02:01.920 --> 00:02:05.130 when I learned this subject originally. 00:02:05.130 --> 00:02:07.560 I, frankly, do not recommend this textbook 00:02:07.560 --> 00:02:11.670 to somebody who is learning the subject for the first time. 00:02:11.670 --> 00:02:14.040 It's a good place for a reference for certain things. 00:02:14.040 --> 00:02:15.230 It's available in the reading room. 00:02:15.230 --> 00:02:17.190 Know where you can get a copy and pick it up. 00:02:17.190 --> 00:02:18.565 So first of all, picking it up is 00:02:18.565 --> 00:02:20.653 kind of a-- it's good exercise. 00:02:20.653 --> 00:02:21.570 It's actually like a-- 00:02:21.570 --> 00:02:23.790 you know, it's a huge book. 00:02:23.790 --> 00:02:27.030 It gravitates. 00:02:27.030 --> 00:02:30.420 So it's a great reference and has 00:02:30.420 --> 00:02:33.000 a couple of good sections in it for new students. 00:02:44.438 --> 00:02:46.480 But I will indicate from time to time, especially 00:02:46.480 --> 00:02:48.940 some of the stuff we do in the second half of the class, 00:02:48.940 --> 00:02:52.330 my lectures are kind of inspired by Misner, Thorne and Wheeler-- 00:02:52.330 --> 00:02:54.640 known as MTW. 00:02:54.640 --> 00:02:57.310 And for those of you who have sort of a more mathematically 00:02:57.310 --> 00:03:01.960 minded approach to things, General Relativity by Wald. 00:03:01.960 --> 00:03:04.210 In the course syllabus I call this the uber book. 00:03:04.210 --> 00:03:06.430 This really is sort of the self-contained book 00:03:06.430 --> 00:03:10.720 that really pins down the subject very, very well. 00:03:10.720 --> 00:03:17.090 It's quite terse and formal, but also very, very clear. 00:03:17.090 --> 00:03:19.232 If you are a mathematically minded thinker, 00:03:19.232 --> 00:03:21.190 you will find this to be a really good textbook 00:03:21.190 --> 00:03:22.600 to refer to. 00:03:22.600 --> 00:03:26.200 There is one particular derivation 00:03:26.200 --> 00:03:28.837 that I'm going to do in about a month and a half 00:03:28.837 --> 00:03:30.670 that I essentially take from Wald's textbook 00:03:30.670 --> 00:03:33.190 because it's just beautiful. 00:03:33.190 --> 00:03:34.023 Other things in it-- 00:03:34.023 --> 00:03:36.440 to me, it's a little too terse from my own personal taste, 00:03:36.440 --> 00:03:38.260 but others may find it to be pretty good. 00:03:46.920 --> 00:03:49.530 Two other quick things. 00:03:49.530 --> 00:03:55.200 So there are 11 problem sets and your grade 00:03:55.200 --> 00:03:58.560 is determined entirely on these problems sets. 00:03:58.560 --> 00:04:01.230 The syllabus gives the schedule for when they will be posted 00:04:01.230 --> 00:04:03.570 and when they are to be handed in. 00:04:03.570 --> 00:04:07.560 11 does not divide evenly into 100, so what we do is we 00:04:07.560 --> 00:04:14.970 have 10 that are worth 9% of your grade and the 11th 00:04:14.970 --> 00:04:17.940 is worth 10% of your grade. 00:04:17.940 --> 00:04:20.735 Is that 11th one really 1% longer? 00:04:20.735 --> 00:04:22.110 I don't know. 00:04:22.110 --> 00:04:25.020 Again, I'm taking the viewpoint that you are graduate students, 00:04:25.020 --> 00:04:28.238 or at least you're playing one for the next hour and a half, 00:04:28.238 --> 00:04:30.780 and if you're going to sweat the difference of one percentage 00:04:30.780 --> 00:04:32.160 point on that thing, come on. 00:04:32.160 --> 00:04:33.660 This is about learning the material, 00:04:33.660 --> 00:04:37.080 don't worry about those little details that much. 00:04:37.080 --> 00:04:39.013 That is where all of your assessment 00:04:39.013 --> 00:04:39.930 is going to come from. 00:04:42.657 --> 00:04:45.240 The very first time I lectured this course I had a final exam, 00:04:45.240 --> 00:04:47.610 and it just turned to be a complete waste of my time 00:04:47.610 --> 00:04:49.560 and the students' time. 00:04:49.560 --> 00:04:51.830 You really cannot write-- 00:04:51.830 --> 00:04:53.400 you either write problems that are 00:04:53.400 --> 00:04:55.980 so easy that you can do them in your sleep, 00:04:55.980 --> 00:04:57.840 or they're so difficult you can't do them 00:04:57.840 --> 00:04:59.290 in the time period of an exam. 00:04:59.290 --> 00:05:02.040 So we're just going to stick with problem sets 00:05:02.040 --> 00:05:04.000 and that's fine. 00:05:04.000 --> 00:05:06.720 Let's move into, then, the way the course 00:05:06.720 --> 00:05:08.610 is going to be structured. 00:05:08.610 --> 00:05:11.467 So my presentation of this material-- 00:05:15.850 --> 00:05:31.270 the first half of the course basically up to spring break 00:05:31.270 --> 00:05:34.030 is essentially the mathematical foundations 00:05:34.030 --> 00:05:35.322 of general relativity. 00:05:45.940 --> 00:05:48.343 There are several choices that need to be 00:05:48.343 --> 00:05:49.510 made when you're doing this. 00:05:49.510 --> 00:05:50.885 This often, in some universities, 00:05:50.885 --> 00:05:52.930 if they have multiple semester sequences, what 00:05:52.930 --> 00:05:54.472 I'm going to cover in this first half 00:05:54.472 --> 00:05:57.258 of our semester in some places goes for a full semester. 00:05:57.258 --> 00:05:59.800 And what this means is there's a couple of things that I just 00:05:59.800 --> 00:06:01.800 cannot cover in quite as much depth. 00:06:01.800 --> 00:06:03.800 Those will be things where, for those of you who 00:06:03.800 --> 00:06:07.300 are interested in it, have that kind of a mathematical thinking 00:06:07.300 --> 00:06:09.490 of things, happy to push you to additional readings. 00:06:09.490 --> 00:06:11.800 We can dive in and look at it a bit more depth. 00:06:11.800 --> 00:06:14.350 My goal is to give you just enough stuff 00:06:14.350 --> 00:06:17.523 that we can do the most important applications 00:06:17.523 --> 00:06:18.190 of this subject. 00:06:18.190 --> 00:06:19.750 And I'm an astrophysicist, and to be blunt, 00:06:19.750 --> 00:06:21.460 most of my really interesting applications 00:06:21.460 --> 00:06:24.002 tend to be things that have to do with things like cosmology, 00:06:24.002 --> 00:06:26.300 black holes, dense stars, and things like that. 00:06:26.300 --> 00:06:28.630 And so I want to get enough formalism together 00:06:28.630 --> 00:06:31.510 that we can get to that part of things. 00:06:31.510 --> 00:06:34.450 And so the goal of this is that by the week right 00:06:34.450 --> 00:06:44.380 before spring break what we will do is "derive"-- 00:06:44.380 --> 00:06:48.020 I put that in quotes, and you'll see why a little bit later-- 00:06:48.020 --> 00:06:51.920 the Einstein field equations that govern 00:06:51.920 --> 00:06:53.950 gravity in general relativity. 00:06:59.482 --> 00:07:00.940 So there are several things that we 00:07:00.940 --> 00:07:04.360 could do that are not strictly necessary to get there, 00:07:04.360 --> 00:07:06.140 and just because of time limitations 00:07:06.140 --> 00:07:09.290 I'm going to choose to elide a few of these topics. 00:07:09.290 --> 00:07:13.030 The second half will then be applications. 00:07:27.100 --> 00:07:29.800 We will use everything we derived in the first half 00:07:29.800 --> 00:07:33.070 to see how general relativity gives us a relativistic theory 00:07:33.070 --> 00:07:34.237 of gravity. 00:07:34.237 --> 00:07:35.320 We will begin applying it. 00:07:35.320 --> 00:07:37.870 We'll see how Newton's law is encoded in these field 00:07:37.870 --> 00:07:38.423 equations. 00:07:38.423 --> 00:07:40.090 We'll see how we go beyond Newton's law, 00:07:40.090 --> 00:07:42.550 get some of the classic tests of general relativity, 00:07:42.550 --> 00:07:44.050 and then start looking at solving it 00:07:44.050 --> 00:07:45.460 for more interesting systems. 00:07:45.460 --> 00:07:47.310 Looking at the evolution of the universe as a whole, 00:07:47.310 --> 00:07:48.610 looking at the behavior of black holes, 00:07:48.610 --> 00:07:50.485 looking for gravitational waves, constructing 00:07:50.485 --> 00:07:53.450 the spacetime of neutron stars, things like that. 00:07:53.450 --> 00:07:55.480 So it's a fun semester. 00:07:55.480 --> 00:07:58.277 It sort of works well to fit these two things in like this. 00:07:58.277 --> 00:08:00.610 And for those of you who are interested in taking things 00:08:00.610 --> 00:08:03.790 more deeply, there's a lot of room to grow after this. 00:08:03.790 --> 00:08:06.880 And it does look like-- so people who particularly would 00:08:06.880 --> 00:08:09.855 like to go a little bit more detail on some of the math, 00:08:09.855 --> 00:08:11.230 if you've looked at the syllabus, 00:08:11.230 --> 00:08:13.360 I have one of my absolute favorite quotes 00:08:13.360 --> 00:08:15.730 from a course evaluation is put on there. 00:08:15.730 --> 00:08:19.120 Where a student in 2007 or so wrote, 00:08:19.120 --> 00:08:21.670 "The course was fine as it was, but Professor Hughes 00:08:21.670 --> 00:08:23.530 as an astrophysicist tended to focus 00:08:23.530 --> 00:08:27.932 on really mundane topics like cosmology and black holes." 00:08:27.932 --> 00:08:29.890 If you think those are mundane topics, what can 00:08:29.890 --> 00:08:31.613 I say, guilty as charged. 00:08:31.613 --> 00:08:34.030 But for those you who do want to take a different approach 00:08:34.030 --> 00:08:35.572 these sorts of things, we'll probably 00:08:35.572 --> 00:08:38.169 alternate lecturing this course in the future between someone 00:08:38.169 --> 00:08:40.780 from the CTP who works more in quantum gravity 00:08:40.780 --> 00:08:42.820 and things related to that. 00:08:42.820 --> 00:08:48.600 That'll be Netta Engelhardt in spring of 2021. 00:08:48.600 --> 00:08:50.280 We're now ready to start talking about, 00:08:50.280 --> 00:08:52.160 after doing all this sort of prep, 00:08:52.160 --> 00:08:54.767 we can actually talk about some of the foundations 00:08:54.767 --> 00:08:55.350 of the theory. 00:08:55.350 --> 00:08:57.225 So before I dive in, are there any questions? 00:09:00.180 --> 00:09:00.680 All right. 00:09:04.760 --> 00:09:07.820 What we're going to begin doing for the first couple of weeks-- 00:09:07.820 --> 00:09:09.320 well, not really first couple weeks. 00:09:09.320 --> 00:09:11.870 The first couple of lectures, is we're 00:09:11.870 --> 00:09:16.850 going to begin by discussing special relativity, 00:09:16.850 --> 00:09:26.410 but we're going to do special relativity using 00:09:26.410 --> 00:09:33.580 mathematical language that emphasizes the geometric nature 00:09:33.580 --> 00:09:36.409 of this form of relativity. 00:09:55.970 --> 00:10:00.000 What this does is it allows us to introduce basically 00:10:00.000 --> 00:10:05.270 the formalism, the notation, all the different tools that 00:10:05.270 --> 00:10:09.450 are important for when things get more complicated. 00:10:09.450 --> 00:10:11.870 When we apply a lot of these tools to special relativity, 00:10:11.870 --> 00:10:13.537 like we will be doing in the first three 00:10:13.537 --> 00:10:15.350 or so lectures of this course, it's 00:10:15.350 --> 00:10:17.780 kind of like swatting a mosquito with a sledgehammer. 00:10:17.780 --> 00:10:21.290 You really don't need that much mathematical structure 00:10:21.290 --> 00:10:23.330 to discuss special relativity. 00:10:23.330 --> 00:10:25.760 But you're going to be grateful for that sledgehammer 00:10:25.760 --> 00:10:28.580 when we start talking about strong field orbits of rotating 00:10:28.580 --> 00:10:29.480 black holes, right. 00:10:29.480 --> 00:10:34.760 And so the whole idea of this is to develop the framework 00:10:34.760 --> 00:10:39.530 in terms of a physical system where 00:10:39.530 --> 00:10:41.630 it's simple to understand what is going on. 00:10:41.630 --> 00:10:43.130 These are sort of a way to introduce 00:10:43.130 --> 00:10:45.050 the mathematical tools in a place where 00:10:45.050 --> 00:10:48.260 the physics is straightforward and then kind of carry forward 00:10:48.260 --> 00:10:49.350 from there. 00:10:49.350 --> 00:10:51.200 I will caution that as a consequence 00:10:51.200 --> 00:10:54.530 of this many students find these first three lectures to be 00:10:54.530 --> 00:10:55.650 a little on the dull side. 00:10:55.650 --> 00:10:58.072 So, sorry. 00:10:58.072 --> 00:11:00.530 It's just some stuff that we kind of have to get out there, 00:11:00.530 --> 00:11:03.140 and then as we generalize to more interesting 00:11:03.140 --> 00:11:05.900 mathematical objects, more interesting physical settings, 00:11:05.900 --> 00:11:07.770 it gets more interesting. 00:11:07.770 --> 00:11:09.510 All right, so let's dive in. 00:11:09.510 --> 00:11:13.370 So the setting for everything that we will be doing 00:11:13.370 --> 00:11:17.270 is a geometric concept known as spacetime. 00:11:23.160 --> 00:11:25.730 So we give you a precise mathematical definition 00:11:25.730 --> 00:11:28.580 of spacetime. 00:11:28.580 --> 00:11:37.580 A spacetime is a manifold of events 00:11:37.580 --> 00:11:45.245 that is endowed with a metric. 00:11:48.370 --> 00:11:50.920 So, a wonderful mathematical definition, and I've 00:11:50.920 --> 00:11:53.800 written it in a way that requires me to carefully define 00:11:53.800 --> 00:11:56.430 three additional terms. 00:11:56.430 --> 00:11:58.840 The concepts that I've underlined here, 00:11:58.840 --> 00:12:00.400 I've not defined them yet precisely 00:12:00.400 --> 00:12:02.510 exactly what I mean by them. 00:12:02.510 --> 00:12:06.370 So let's go over to the sideboard 00:12:06.370 --> 00:12:09.190 and talk about what exactly these are. 00:12:09.190 --> 00:12:14.628 So a manifold-- if you are a mathematician, 00:12:14.628 --> 00:12:16.420 you might twitch a little bit about the way 00:12:16.420 --> 00:12:20.500 that I am going to define this, and I 00:12:20.500 --> 00:12:23.110 will point you to a reading that does it a little bit more 00:12:23.110 --> 00:12:23.980 precisely. 00:12:23.980 --> 00:12:26.960 But for the purposes of our class, 00:12:26.960 --> 00:12:29.260 a manifold is essentially just a set 00:12:29.260 --> 00:12:44.810 or a collection of points with well understood 00:12:44.810 --> 00:12:46.340 connectedness properties. 00:12:57.320 --> 00:12:59.480 What I mean by that, is I'm going 00:12:59.480 --> 00:13:03.798 to talk about manifolds of space and time. 00:13:03.798 --> 00:13:05.840 I haven't defined an event yet, but I'm about to, 00:13:05.840 --> 00:13:07.610 but I'm going to say that there's a bunch of events that 00:13:07.610 --> 00:13:09.830 happen at this place and at this time and a bunch of events that 00:13:09.830 --> 00:13:11.810 happen at this place and at this time. 00:13:11.810 --> 00:13:14.270 And the manifold, the spacetime, gives me 00:13:14.270 --> 00:13:17.150 some notion of how I connect the events over here 00:13:17.150 --> 00:13:18.770 to the events over here. 00:13:18.770 --> 00:13:21.410 A manifold is a tautological concept. 00:13:21.410 --> 00:13:26.750 It's all about how one connects one region to another one. 00:13:26.750 --> 00:13:28.880 So if you're working on a manifold that 00:13:28.880 --> 00:13:30.530 lives on the surface of a donut, you 00:13:30.530 --> 00:13:32.180 have a particular topology associated with that. 00:13:32.180 --> 00:13:33.763 If it lives on the surface of a sphere 00:13:33.763 --> 00:13:36.060 you have a different topology associated with it. 00:13:36.060 --> 00:13:38.690 If you would like to see more careful and more 00:13:38.690 --> 00:13:41.880 rigorous discussion of this, this 00:13:41.880 --> 00:13:43.880 is one of the places where Carroll is very good. 00:13:51.370 --> 00:13:53.380 So go into Carroll, at least in the edition 00:13:53.380 --> 00:13:57.170 that I have it's on pages 54 to 62. 00:14:01.360 --> 00:14:03.640 He introduces a bit of additional mathematical 00:14:03.640 --> 00:14:06.190 machinery, discusses things with a little more rigor 00:14:06.190 --> 00:14:07.510 than I'm doing here-- 00:14:07.510 --> 00:14:09.880 this isn't rigorous at all, so significantly more rigor 00:14:09.880 --> 00:14:11.320 than I'm doing here. 00:14:11.320 --> 00:14:13.900 Those of you are into that, that should be something 00:14:13.900 --> 00:14:15.590 that you enjoy. 00:14:15.590 --> 00:14:16.570 So an event. 00:14:19.910 --> 00:14:26.045 This is when and where something happens. 00:14:32.580 --> 00:14:34.140 Could be anything. 00:14:34.140 --> 00:14:36.540 From our point of view, the event essentially 00:14:36.540 --> 00:14:38.910 is going to be the fundamental notion 00:14:38.910 --> 00:14:41.030 of a coordinate in space time. 00:14:43.590 --> 00:14:45.780 We will actually, in many cases-- 00:14:45.780 --> 00:14:47.940 actually, that's bad word choice, I should say. 00:14:47.940 --> 00:14:50.160 Coordinates are actually sort of labels 00:14:50.160 --> 00:14:52.540 that we attach to events. 00:14:52.540 --> 00:14:54.180 We are going to be free to adjust 00:14:54.180 --> 00:14:58.200 those labels, but the underlying geometric idea that the event 00:14:58.200 --> 00:15:01.660 is here, that's independent of the coordinates that we choose. 00:15:01.660 --> 00:15:04.909 So we will label these things with coordinates. 00:15:15.390 --> 00:15:28.160 But the event itself exists independent of these labels. 00:15:44.580 --> 00:15:46.620 Just to give an example, there might 00:15:46.620 --> 00:15:49.890 be one event which is I punch myself in the head. 00:15:49.890 --> 00:15:52.980 And so I'm very egotistic, so I will say 00:15:52.980 --> 00:15:55.905 this event happened at time 0-- 00:15:55.905 --> 00:15:57.300 x, y, and z equals 0. 00:15:57.300 --> 00:15:59.490 Because I define this corner of my skull 00:15:59.490 --> 00:16:00.990 as the origin the coordinate system, 00:16:00.990 --> 00:16:03.032 and I always think whatever's happening right now 00:16:03.032 --> 00:16:04.440 is the origin of time. 00:16:04.440 --> 00:16:07.140 Those of you out in the room are also egocentric 00:16:07.140 --> 00:16:09.840 and you would say whatever, I'm going to call that-- 00:16:09.840 --> 00:16:12.810 let's say you are at y of 3 meters 00:16:12.810 --> 00:16:15.960 and I'm going to put the floor as the origin of my z-axis, 00:16:15.960 --> 00:16:18.700 so z of 1.7 meters or whatever. 00:16:18.700 --> 00:16:20.700 You will come with your own independent labeling 00:16:20.700 --> 00:16:22.560 of these things. 00:16:22.560 --> 00:16:24.060 So you're all familiar with the idea 00:16:24.060 --> 00:16:26.453 that we can just change coordinate systems. 00:16:26.453 --> 00:16:28.120 I'm going to harp on this a bit, though, 00:16:28.120 --> 00:16:30.662 because there's going to be a really important distinction we 00:16:30.662 --> 00:16:32.880 make between geometrical objects that 00:16:32.880 --> 00:16:34.830 live in this manifold of spacetime 00:16:34.830 --> 00:16:38.490 and how we represent them using labels that might be attached 00:16:38.490 --> 00:16:39.773 to coordinate systems. 00:16:39.773 --> 00:16:41.190 And I'm going to come back to this 00:16:41.190 --> 00:16:43.770 when we start talking about some additional geometric objects 00:16:43.770 --> 00:16:46.420 in just a couple of minutes. 00:16:46.420 --> 00:16:49.410 So the last object that I have introduced into here 00:16:49.410 --> 00:16:51.330 is one that we will begin talking 00:16:51.330 --> 00:16:53.730 about in a lot more detail in the next lecture, 00:16:53.730 --> 00:16:56.700 but let me put it into here right away. 00:16:56.700 --> 00:16:58.440 And so that is the metric. 00:16:58.440 --> 00:17:01.980 Metric comes from a root meaning to measure, 00:17:01.980 --> 00:17:04.050 and what this is is it's something 00:17:04.050 --> 00:17:18.589 it gives me a notion of distance between events in the manifold. 00:17:25.190 --> 00:17:29.060 For physics to work, this has to exist, right? 00:17:29.060 --> 00:17:31.670 But it's worth knowing that the idea of a manifold 00:17:31.670 --> 00:17:34.250 is in some way more primitive than this. 00:17:34.250 --> 00:17:36.110 You can have a manifold without any notion 00:17:36.110 --> 00:17:37.320 of a metric attached to it. 00:17:37.320 --> 00:17:38.408 And if that's the case-- 00:17:38.408 --> 00:17:39.950 people like to joke that if you don't 00:17:39.950 --> 00:17:42.170 know what the difference is between a metric 00:17:42.170 --> 00:17:43.980 with a manifold and without a man-- 00:17:43.980 --> 00:17:47.690 excuse me, a manifold with a metric and without a metric, 00:17:47.690 --> 00:17:49.940 feel free to drink coffee out of a donut. 00:17:49.940 --> 00:17:52.610 Because topologically, those are the exact same thing, 00:17:52.610 --> 00:17:55.100 but their geometry, which is encoded in the metric, which 00:17:55.100 --> 00:17:57.710 tells me how the different points on that manifold 00:17:57.710 --> 00:18:00.040 are arranged, are rather different. 00:18:02.810 --> 00:18:04.790 What this basically does is it's going 00:18:04.790 --> 00:18:09.702 to give me a mathematical object that enforces or really conveys 00:18:09.702 --> 00:18:11.660 the idea that different events in this manifold 00:18:11.660 --> 00:18:13.490 have a particular distance between them. 00:18:13.490 --> 00:18:28.330 So without this, a manifold has no notion 00:18:28.330 --> 00:18:32.000 of distance encoded in it. 00:18:32.000 --> 00:18:33.820 So the two things together really make 00:18:33.820 --> 00:18:36.460 this concept come to life. 00:18:39.430 --> 00:18:40.930 You can see a lot more, like I said. 00:18:43.385 --> 00:18:45.760 You can get more information about many of these concepts 00:18:45.760 --> 00:18:49.300 from the readings in Carroll, Wald's textbook 00:18:49.300 --> 00:18:53.650 also goes into quite a lot of detail about this stuff. 00:18:53.650 --> 00:18:55.280 So this is the venue. 00:18:55.280 --> 00:18:58.300 This is the setting in which we are going to talk about things. 00:18:58.300 --> 00:19:01.020 And just cutting forward roughly 2 00:19:01.020 --> 00:19:03.760 and 1/2 months' worth of lectures, 00:19:03.760 --> 00:19:08.680 what we are going to find is that part of Einstein's genius 00:19:08.680 --> 00:19:11.890 is that it turns out that this notion of the metric 00:19:11.890 --> 00:19:13.502 ends up encoding gravity. 00:19:13.502 --> 00:19:14.960 So that's kind of where we're going 00:19:14.960 --> 00:19:16.127 to end up going with things. 00:19:16.127 --> 00:19:18.580 The idea that the mathematical structure 00:19:18.580 --> 00:19:21.202 that tells me how far apart two events are 00:19:21.202 --> 00:19:23.410 is intimately connected to the properties of gravity. 00:19:23.410 --> 00:19:27.140 It's pretty cool, and it is something that-- 00:19:27.140 --> 00:19:29.500 physics is an experimental science. 00:19:29.500 --> 00:19:32.430 All of our measurements are consistent with it. 00:19:32.430 --> 00:19:34.000 So that's cool. 00:19:38.670 --> 00:19:42.630 So everything I have said so far, nothing but math. 00:19:42.630 --> 00:19:44.490 Nothing but definitions. 00:19:44.490 --> 00:19:48.490 So let's start working with a particular form of physics. 00:19:48.490 --> 00:19:52.066 So we are going to begin, as I said, with special relativity. 00:19:57.660 --> 00:20:00.720 This is the simplest theory of spacetime 00:20:00.720 --> 00:20:04.500 that is compatible with physics as we know it. 00:20:04.500 --> 00:20:13.610 Not fully compatible, but does a pretty good job. 00:20:13.610 --> 00:20:17.090 And we'll see that it turns out to correspond 00:20:17.090 --> 00:20:25.220 to general relativity when there is no gravity. 00:20:37.680 --> 00:20:39.540 So to set this up we need to have 00:20:39.540 --> 00:20:43.700 some kind of a way of labeling our events. 00:20:43.700 --> 00:20:48.660 And so I'm going to introduce kind of a conceptual-- 00:20:48.660 --> 00:20:50.370 you almost think of it as scaffolding, 00:20:50.370 --> 00:20:53.640 which we're going to use to build a lot of our concepts 00:20:53.640 --> 00:20:54.990 around. 00:20:54.990 --> 00:20:56.490 And in this one-- 00:20:56.490 --> 00:20:59.035 I mean that in a kind of an abstract sense. 00:20:59.035 --> 00:21:00.660 There's going to be sort of the edifice 00:21:00.660 --> 00:21:02.660 that we use to help us build the building that's 00:21:02.660 --> 00:21:04.800 going to be the mathematics of general relativity. 00:21:04.800 --> 00:21:06.883 But this one really is kind of like a scaffolding. 00:21:06.883 --> 00:21:09.030 Because what I want to introduce here 00:21:09.030 --> 00:21:17.670 is a notion of what is called an inertial reference frame. 00:21:25.960 --> 00:21:28.440 So I'm going to sketch this quickly, 00:21:28.440 --> 00:21:31.440 and I'm going to post to the course website 00:21:31.440 --> 00:21:38.040 a chapter from an early draft textbook 00:21:38.040 --> 00:21:39.920 by Roger Blandford and Kip Thorne. 00:22:05.300 --> 00:22:07.660 So when I talk about the inertial reference frame, 00:22:07.660 --> 00:22:10.360 I want you to sort of visualize in your head 00:22:10.360 --> 00:22:11.620 a lattice of clocks-- 00:22:18.730 --> 00:22:20.200 clocks and measuring rods-- 00:22:26.110 --> 00:22:36.150 that allows us to label, in other words, to assign 00:22:36.150 --> 00:22:48.492 coordinates any event that happens in spacetime. 00:22:52.717 --> 00:22:54.300 So just sort of, in your head, imagine 00:22:54.300 --> 00:22:58.710 that there's this grid of little clocks and measuring rods, 00:22:58.710 --> 00:23:03.420 and a mosquito lands on your head. 00:23:03.420 --> 00:23:06.510 It's right near a particular rod and a particular clock. 00:23:06.510 --> 00:23:07.890 It bites you, that's an event. 00:23:07.890 --> 00:23:09.750 You slap it, that's another event. 00:23:09.750 --> 00:23:11.220 And the measuring rods and events 00:23:11.220 --> 00:23:13.110 are what allows you to sort of keep 00:23:13.110 --> 00:23:15.240 track of the ordering of those events 00:23:15.240 --> 00:23:17.520 and where they happen in this four dimensional 00:23:17.520 --> 00:23:21.240 manifold of spacetime. 00:23:21.240 --> 00:23:22.740 So I'm going to require this lattice 00:23:22.740 --> 00:23:24.746 to have a certain set of properties. 00:23:29.510 --> 00:23:39.910 First, I'm going to say that this lattice moves freely 00:23:39.910 --> 00:23:41.220 through spacetime. 00:23:45.920 --> 00:23:47.900 What do I mean by moving freely? 00:23:47.900 --> 00:23:51.410 I mean no forces act on it, it does not rotate, 00:23:51.410 --> 00:23:52.790 it is inertial. 00:23:52.790 --> 00:23:57.040 Every clock and every measuring rod has no inertia, 00:23:57.040 --> 00:23:58.850 no force is acting on it at all. 00:24:19.633 --> 00:24:21.800 You look at that, you might think to yourself, well, 00:24:21.800 --> 00:24:23.810 why don't you make it at rest? 00:24:23.810 --> 00:24:24.630 Well, I did. 00:24:24.630 --> 00:24:27.485 It's at rest in respect to someone, 00:24:27.485 --> 00:24:29.860 but we might have a different observer who's coming along 00:24:29.860 --> 00:24:33.610 who has no forces acting on her, and she's moving relative to me 00:24:33.610 --> 00:24:35.630 at three quarters the speed of light. 00:24:35.630 --> 00:24:37.930 It's not at rest with respects to that observer. 00:24:37.930 --> 00:24:39.472 That's actually kind of the key here. 00:24:42.040 --> 00:24:45.280 So this inertial reference frame is at rest with respect 00:24:45.280 --> 00:24:52.764 to someone who feels no forces but not to all observers. 00:25:06.130 --> 00:25:11.810 I'm going to require that my measuring rods are 00:25:11.810 --> 00:25:13.040 orthogonal to each other. 00:25:20.170 --> 00:25:28.000 So they define an orthogonal coordinate system, 00:25:28.000 --> 00:25:31.323 and I am also going to require that the little markings 00:25:31.323 --> 00:25:32.990 on them that tell me where things happen 00:25:32.990 --> 00:25:34.880 are uniformly ticked. 00:25:34.880 --> 00:25:37.580 In other words, I'm going to just make sure 00:25:37.580 --> 00:25:39.093 that the spacing between tick marks 00:25:39.093 --> 00:25:41.510 here is exactly the same as the spacing between tick marks 00:25:41.510 --> 00:25:42.440 here. 00:25:42.440 --> 00:25:46.730 You may sort of think well, that's a result or an idea 00:25:46.730 --> 00:25:50.270 worthy of the journal Duh, but it's important to specify this. 00:25:50.270 --> 00:25:52.520 You want to make sure that the standard you are using 00:25:52.520 --> 00:25:56.000 to define length is the same in this region of spacetime 00:25:56.000 --> 00:25:58.463 as it is over in this region of spacetime. 00:25:58.463 --> 00:26:00.380 When we start getting into general relativity, 00:26:00.380 --> 00:26:02.750 we start to see there can be concerns about this coming 00:26:02.750 --> 00:26:05.090 about, so it's worth spelling it out and making 00:26:05.090 --> 00:26:06.718 it clear at the beginning. 00:26:23.320 --> 00:26:27.280 I'm also going to require that my clocks tick uniformly. 00:26:31.978 --> 00:26:33.520 We're going to make this lattice that 00:26:33.520 --> 00:26:35.590 fills all of spacetime using the best 00:26:35.590 --> 00:26:37.450 thing that Swiss engineers can make for us, 00:26:37.450 --> 00:26:41.380 we want to make sure that one second, an interval of one 00:26:41.380 --> 00:26:43.660 second, is the same here in this classroom 00:26:43.660 --> 00:26:45.940 as it is somewhere off in the Andromeda Galaxy. 00:26:45.940 --> 00:26:48.730 We want to make sure that there is no evolution to the time 00:26:48.730 --> 00:26:51.280 standard when we do this. 00:26:54.690 --> 00:26:57.870 Finally, I'm going to synchronize all these clocks 00:26:57.870 --> 00:26:59.820 with each other in the following way. 00:27:09.380 --> 00:27:13.040 This is going to use the Einstein synchronization 00:27:13.040 --> 00:27:13.996 procedure. 00:27:23.040 --> 00:27:26.430 This is the first place where a little bit of physics 00:27:26.430 --> 00:27:29.560 is actually beginning to finally enter our discussion. 00:27:29.560 --> 00:27:30.463 I'll comment that-- 00:27:30.463 --> 00:27:32.380 I'm going to go through what this procedure is 00:27:32.380 --> 00:27:34.470 in just a second. 00:27:34.470 --> 00:27:36.210 But an Easter egg here. 00:27:36.210 --> 00:27:38.963 Whenever you see a name that has Einstein in it, 00:27:38.963 --> 00:27:40.380 your ears should perk up a little. 00:27:40.380 --> 00:27:42.630 Because it probably means this is something important. 00:27:42.630 --> 00:27:45.270 This is, after all, a course in relativity, 00:27:45.270 --> 00:27:48.300 and that tends to be the things that end up mattering. 00:27:48.300 --> 00:27:52.540 Even when they end up being really easy to-- 00:27:52.540 --> 00:27:57.840 things we look at now and kind of see as fairly obvious, 00:27:57.840 --> 00:28:01.350 it's important, OK, and we often attach Einstein's name to this. 00:28:01.350 --> 00:28:04.680 So this Einstein synchronization procedure, this 00:28:04.680 --> 00:28:07.502 takes advantage of the fact-- 00:28:07.502 --> 00:28:09.960 and by the way, we're not going to teach special relativity 00:28:09.960 --> 00:28:11.543 in this course, 8.962. 00:28:11.543 --> 00:28:13.710 I assume you have already studied special relativity 00:28:13.710 --> 00:28:17.460 and you're all experts in this, and so I can freely borrow 00:28:17.460 --> 00:28:19.680 from its important results. 00:28:19.680 --> 00:28:29.440 This procedure takes advantage of the fact 00:28:29.440 --> 00:28:42.380 that the speed of light is the same to all observers, 00:28:42.380 --> 00:28:44.152 no matter what inertial reference 00:28:44.152 --> 00:28:45.110 frame they might be in. 00:28:52.270 --> 00:28:56.980 So the speed of light is a key invariant. 00:28:56.980 --> 00:28:59.410 It connects-- because it's a speed, 00:28:59.410 --> 00:29:02.050 it connects space and time, and because it 00:29:02.050 --> 00:29:07.360 is the same to all observers, it defines a particular standard 00:29:07.360 --> 00:29:10.360 for relating space and time that is 00:29:10.360 --> 00:29:13.450 going to have important invariant meaning associated 00:29:13.450 --> 00:29:14.822 with it. 00:29:14.822 --> 00:29:16.780 Just to remind you what this means-- let's see, 00:29:16.780 --> 00:29:18.633 do I have a laser pointer with me? 00:29:18.633 --> 00:29:20.050 I might, but don't worry about it. 00:29:20.050 --> 00:29:21.700 I have a pretend laser pointer. 00:29:21.700 --> 00:29:23.650 My chalk's a laser pointer. 00:29:23.650 --> 00:29:25.150 I point my laser pointer at the wall 00:29:25.150 --> 00:29:28.120 and you all see it dashing across the room at 300,000 00:29:28.120 --> 00:29:29.950 kilometers per second. 00:29:29.950 --> 00:29:33.640 I then start jogging at half the speed of light, as is my wont, 00:29:33.640 --> 00:29:36.233 and continuing to point that, you guys measure the light 00:29:36.233 --> 00:29:37.150 going across the room. 00:29:37.150 --> 00:29:40.330 You still measure 300,000 kilometers per second. 00:29:40.330 --> 00:29:41.990 I, on the other hand, measure the light 00:29:41.990 --> 00:29:43.490 coming out of my laser pointer and I 00:29:43.490 --> 00:29:46.630 get 300,000 kilometers per second. 00:29:46.630 --> 00:29:49.600 So just because my laser pointer is 00:29:49.600 --> 00:29:51.913 moving at half the speed of light according to you 00:29:51.913 --> 00:29:53.830 doesn't mean the light that's coming out of it 00:29:53.830 --> 00:29:55.485 is boosted to a higher speed. 00:29:55.485 --> 00:29:56.860 If you studied special relativity 00:29:56.860 --> 00:29:59.260 you'll know that its energy is boosted to a higher energy 00:29:59.260 --> 00:30:03.610 level, but the speed of light is always just C. 00:30:03.610 --> 00:30:05.440 So we're going to take advantage of that 00:30:05.440 --> 00:30:10.270 to come up with a way of synchronizing our clocks. 00:30:10.270 --> 00:30:12.620 The way it works is this. 00:30:12.620 --> 00:30:15.310 So let's look at a two dimensional slice of my lattice 00:30:15.310 --> 00:30:16.810 here. 00:30:16.810 --> 00:30:20.600 Time, and let's make this be the x-axis. 00:30:20.600 --> 00:30:28.390 And so I will have, let's say, this is where clock one exists 00:30:28.390 --> 00:30:31.855 and this is where clock two exists. 00:30:34.635 --> 00:30:36.010 Let's go into the reference frame 00:30:36.010 --> 00:30:38.530 that is at rest with respect to this lattice. 00:30:38.530 --> 00:30:41.950 We want to synchronize clock one with clock two. 00:30:41.950 --> 00:30:45.580 So as time marches on, these guys stand still. 00:30:45.580 --> 00:30:47.710 So here's the path in spacetime-- 00:30:47.710 --> 00:30:50.920 the world line, traced out by clock one. 00:30:50.920 --> 00:30:54.280 Here's the clock path in spacetime, the world line, 00:30:54.280 --> 00:30:57.480 traced out by clock two. 00:30:57.480 --> 00:31:03.280 So let's say-- let's call this event-- 00:31:03.280 --> 00:31:09.100 let's say that this event happens at a time t1e. 00:31:09.100 --> 00:31:13.840 t1 is when clock one emits a pulse of light. 00:31:30.040 --> 00:31:33.730 So this light will just follow a little trajectory a bit 00:31:33.730 --> 00:31:35.020 through spacetime. 00:31:35.020 --> 00:31:44.110 This goes out and strikes clock two at which point 00:31:44.110 --> 00:31:48.070 it is bounced back to clock one. 00:31:48.070 --> 00:31:50.150 Let's ignore this point for just a second. 00:31:50.150 --> 00:31:52.990 So let's just say for the moment that this is then 00:31:52.990 --> 00:32:08.010 reflected back and then it is received back at clock one 00:32:08.010 --> 00:32:10.680 at a time t1r. 00:32:19.570 --> 00:32:21.760 Let's make this a little bit neater. 00:32:21.760 --> 00:32:23.340 t1e, t1r. 00:32:26.360 --> 00:32:30.300 Clock one receives the reflected pulse. 00:32:36.630 --> 00:32:40.510 So the moment at which it bounces, 00:32:40.510 --> 00:32:48.450 we'll call that t2b, that is the moment at which the light 00:32:48.450 --> 00:32:50.890 bounces off of clock two. 00:32:50.890 --> 00:32:52.440 And the way we synchronize our clocks 00:32:52.440 --> 00:32:53.970 is just by requiring that this be 00:32:53.970 --> 00:33:03.090 equal to the average of the emission and the reception 00:33:03.090 --> 00:33:03.810 time. 00:33:03.810 --> 00:33:05.940 Totally trivial idea, right? 00:33:05.940 --> 00:33:09.510 All I'm saying is I'm going to just require that 00:33:09.510 --> 00:33:11.310 in order for clock one and clock two 00:33:11.310 --> 00:33:13.180 to be synchronized to one another, 00:33:13.180 --> 00:33:17.460 let's make sure that when I bounce light between any two 00:33:17.460 --> 00:33:20.430 pairs of clocks they are set such that when 00:33:20.430 --> 00:33:22.560 the light bounces it's the midway point 00:33:22.560 --> 00:33:25.020 between the total light-- 00:33:25.020 --> 00:33:27.000 the halfway of the total light travel 00:33:27.000 --> 00:33:29.850 time that the light moves along. 00:33:29.850 --> 00:33:32.190 Very, very simple concept. 00:33:32.190 --> 00:33:33.810 So there's nothing particularly deep 00:33:33.810 --> 00:33:36.543 here, but notice Einstein's name is attached to this. 00:33:36.543 --> 00:33:37.710 And I don't say that to be-- 00:33:37.710 --> 00:33:39.960 I'm not being sarcastic. 00:33:39.960 --> 00:33:41.460 It really points to the fact that we 00:33:41.460 --> 00:33:45.000 are using one of the most fundamental results 00:33:45.000 --> 00:33:46.800 of special relativity in designing 00:33:46.800 --> 00:33:50.910 how these clocks work in this inertial reference frame. 00:33:50.910 --> 00:33:55.680 Believe it or not, this thing, this really simple concept, it 00:33:55.680 --> 00:33:58.440 comes back to sort of bite us on the butt a little bit later 00:33:58.440 --> 00:33:59.550 in this course. 00:33:59.550 --> 00:34:02.280 Because when we throw gravity into the mix, 00:34:02.280 --> 00:34:04.080 we're going to learn that gravity 00:34:04.080 --> 00:34:07.560 impacts the way light travels through spacetime. 00:34:07.560 --> 00:34:10.500 We're going to get to some objects where gravity is 00:34:10.500 --> 00:34:13.150 so strong that light cannot escape them. 00:34:13.150 --> 00:34:16.830 And we're going to find that our perhaps most naive 00:34:16.830 --> 00:34:20.940 ways of labeling time in the spacetime of such objects 00:34:20.940 --> 00:34:23.489 kind of goes completely haywire. 00:34:23.489 --> 00:34:25.949 Fundamentally, the reason why time is going haywire 00:34:25.949 --> 00:34:27.989 when you have really strong gravity 00:34:27.989 --> 00:34:30.510 is because we use light as our tool 00:34:30.510 --> 00:34:32.300 for synchronizing all of our clocks, 00:34:32.300 --> 00:34:34.889 and if the way that light moves in your space time 00:34:34.889 --> 00:34:37.110 is affected, the way you're going to label time 00:34:37.110 --> 00:34:38.590 is going to be affected. 00:34:38.590 --> 00:34:41.370 So this is a very simple concept. 00:34:41.370 --> 00:34:43.620 Again, I sort of emphasize these first couple lectures 00:34:43.620 --> 00:34:45.929 you're swatting a mosquito with a sledgehammer, 00:34:45.929 --> 00:34:47.760 but we're setting up this edifice 00:34:47.760 --> 00:34:49.260 because this will come back and it's 00:34:49.260 --> 00:34:50.760 important to bear this in mind when 00:34:50.760 --> 00:34:52.949 things get a little more interesting later 00:34:52.949 --> 00:34:55.860 in the course. 00:34:55.860 --> 00:34:59.250 So let's start setting up some geometrical objects here. 00:35:06.808 --> 00:35:08.600 Pardon me, let me do one other thing really 00:35:08.600 --> 00:35:11.340 quickly before I start setting up some geometrical objects. 00:35:11.340 --> 00:35:15.290 So when I sketched this thing called a spacetime diagram 00:35:15.290 --> 00:35:18.950 here, I should have talked a little bit about the units 00:35:18.950 --> 00:35:21.200 that I'm going to use to describe 00:35:21.200 --> 00:35:23.660 the ticking of my clocks and the spacing of tick marks 00:35:23.660 --> 00:35:24.560 on my measuring rods. 00:35:28.360 --> 00:35:31.990 What we will generally do in this course 00:35:31.990 --> 00:35:47.270 is choose the basic unit of length 00:35:47.270 --> 00:36:02.050 to be the distance light travels in your basic unit of time. 00:36:07.310 --> 00:36:10.580 While you parse that sentence, what that's basically saying is 00:36:10.580 --> 00:36:14.450 suppose I set my clock so that they tick every second. 00:36:14.450 --> 00:36:17.090 Well, if my clocks tick once per second, 00:36:17.090 --> 00:36:20.060 then my basic unit of length will be the light second. 00:36:39.290 --> 00:36:41.380 Do you want to put this into more familiar units? 00:36:41.380 --> 00:36:45.160 That's about 300,000 kilometers. 00:36:45.160 --> 00:36:46.933 One of my personal favorites-- 00:36:50.560 --> 00:36:59.640 if the time unit is 1 nanosecond, 00:36:59.640 --> 00:37:05.760 the length unit is, of course, one I will call it LNS-- 00:37:05.760 --> 00:37:06.930 Light Nanosecond. 00:37:09.610 --> 00:37:11.440 Students who are in 8.033 with me are not 00:37:11.440 --> 00:37:13.273 allowed to answer this question, does anyone 00:37:13.273 --> 00:37:17.330 know what one light nanosecond is? 00:37:17.330 --> 00:37:20.060 Actually, this is a little bit ridiculous. 00:37:20.060 --> 00:37:24.080 But to within far greater than a percent accuracy 00:37:24.080 --> 00:37:25.940 it is one foot. 00:37:25.940 --> 00:37:29.000 The English unit that comes on these asinine rulers 00:37:29.000 --> 00:37:30.950 that those of us educated the United States 00:37:30.950 --> 00:37:34.040 learned in all of our European friends sneer at us about. 00:37:34.040 --> 00:37:36.660 The speed of light is to incredible precision 00:37:36.660 --> 00:37:38.537 1 foot per nanosecond. 00:37:44.508 --> 00:37:46.300 To be fair, let's make that a wiggly equal. 00:37:51.100 --> 00:37:53.170 So what this means is that in the units that I'm 00:37:53.170 --> 00:37:59.590 going to be working with, if I then 00:37:59.590 --> 00:38:03.685 want to express the speed of light in these units-- 00:38:10.090 --> 00:38:22.360 So C is one light time unit per time unit, which 00:38:22.360 --> 00:38:24.640 we are just going to call 1. 00:38:24.640 --> 00:38:27.460 So we will generally set the speed of light equal to 1. 00:38:27.460 --> 00:38:29.410 Just bear in mind what this essentially means 00:38:29.410 --> 00:38:31.618 is that you can think of, if you want to then convert 00:38:31.618 --> 00:38:35.530 to your favorite meters per second, furlongs per fortnight, 00:38:35.530 --> 00:38:38.380 whatever it is that you're most comfortable with, 00:38:38.380 --> 00:38:41.670 C is effectively a conversion factor then. 00:38:41.670 --> 00:38:43.870 And so what this means is that when we do this 00:38:43.870 --> 00:38:47.792 all velocities that we measure are going to be dimensionless. 00:38:47.792 --> 00:38:49.750 Really what we're doing is we're measuring them 00:38:49.750 --> 00:38:51.208 as fractions of the speed of light. 00:39:00.540 --> 00:39:03.600 Now, with this system of units defined, 00:39:03.600 --> 00:39:06.490 let's talk about a geometric object. 00:39:06.490 --> 00:39:18.640 So let's imagine that O is an observer 00:39:18.640 --> 00:39:25.190 in the inertial reference frame that I 00:39:25.190 --> 00:39:26.390 defined a few moments ago. 00:39:31.070 --> 00:39:33.810 This is a mouthful to say, it's even more of a mouthful 00:39:33.810 --> 00:39:36.710 to write, so I'm going to typically abbreviate this IRF. 00:39:50.450 --> 00:39:53.360 So o observes two events, which I 00:39:53.360 --> 00:40:14.710 will label P and Q. I'll just go to a clean board for this. 00:40:33.730 --> 00:40:37.363 So let's say here in spacetime-- so imagine 00:40:37.363 --> 00:40:39.280 I've got coordinate axes that have been drawn, 00:40:39.280 --> 00:40:40.720 it's going to be three dimensional, 00:40:40.720 --> 00:40:43.053 for simplicity I'm not going to actually write them out. 00:40:43.053 --> 00:40:50.650 Let's say I've got event P here and event Q over here. 00:40:50.650 --> 00:40:53.792 Now if we were just doing Euclidean geometry in three 00:40:53.792 --> 00:40:55.750 space, you guys have all known that once you've 00:40:55.750 --> 00:40:57.490 got two events written down on a plane 00:40:57.490 --> 00:40:58.795 or in a three dimensional space, something 00:40:58.795 --> 00:41:01.212 like that, you can define the displacement vector from one 00:41:01.212 --> 00:41:01.858 to the other. 00:41:01.858 --> 00:41:03.650 We're going to the same thing in spacetime. 00:41:07.490 --> 00:41:10.100 So let's call delta x-- 00:41:12.960 --> 00:41:15.770 and I'm going to make a comment on notation in just a moment-- 00:41:15.770 --> 00:41:34.300 this is the displacement in spacetime from P to Q. 00:41:34.300 --> 00:41:36.400 We're going to define the components 00:41:36.400 --> 00:41:45.220 of this displacement vector as seen by O. So when 00:41:45.220 --> 00:41:51.190 I write equals with a dot on it that means the geometric object 00:41:51.190 --> 00:41:53.380 that I've written on the left hand side 00:41:53.380 --> 00:41:57.760 is given according to the specified 00:41:57.760 --> 00:42:01.090 observer by the following set of complements, which 00:42:01.090 --> 00:42:03.490 I'm about to write down. 00:42:03.490 --> 00:42:18.910 So this looks like so. 00:42:22.260 --> 00:42:24.935 So, couple things that I want to emphasize that I'm introducing 00:42:24.935 --> 00:42:26.310 here, bits of notation that we're 00:42:26.310 --> 00:42:28.860 going to use over and over again in this term. 00:42:28.860 --> 00:42:32.340 Notice I am using an over arrow to denote 00:42:32.340 --> 00:42:34.770 a vector in spacetime. 00:42:34.770 --> 00:42:36.600 Different texts, different professors 00:42:36.600 --> 00:42:38.700 use slightly different notations for this. 00:42:38.700 --> 00:42:42.150 Those of you who took 8.033 at MIT with Sal Vitale he 00:42:42.150 --> 00:42:45.930 preferred to write a little under tilde when he wrote that. 00:42:45.930 --> 00:42:48.810 For us working in four dimensional space time, 00:42:48.810 --> 00:42:51.250 it is going to be of paramount importance to us, 00:42:51.250 --> 00:42:53.820 and so we're going to use this over arrow which you probably 00:42:53.820 --> 00:42:56.250 have all seen for ordinary three dimensional vectors. 00:42:56.250 --> 00:42:59.700 For us it's going to represent a four dimensional vector. 00:42:59.700 --> 00:43:04.290 Now in truth we're not going to use it all that much 00:43:04.290 --> 00:43:07.190 after the first couple of weeks of the class, 00:43:07.190 --> 00:43:08.970 our first couple lectures even. 00:43:08.970 --> 00:43:11.220 Occasionally we'll bust it out, but we 00:43:11.220 --> 00:43:25.180 will tend to use a more compact notation in which we 00:43:25.180 --> 00:43:33.040 say delta x, that displacement factor has the components 00:43:33.040 --> 00:43:42.610 delta x mu, where mu lies is in either t, x, y, and z, 00:43:42.610 --> 00:43:48.195 or 0, 1, 2, 3. 00:43:48.195 --> 00:43:49.570 When we set up a problem, we need 00:43:49.570 --> 00:43:52.380 to make a mapping to what the numerical correspondences 00:43:52.380 --> 00:43:52.880 between-- 00:43:52.880 --> 00:43:57.760 I need to tell you that mu equals 0 corresponds to time, 00:43:57.760 --> 00:43:59.913 and mu equals 1 corresponds to x. 00:43:59.913 --> 00:44:01.580 We'll switch to other coordinate systems 00:44:01.580 --> 00:44:04.180 and I'll have to be careful to say, almost always, 00:44:04.180 --> 00:44:06.460 mu equals 0 will be time. 00:44:06.460 --> 00:44:08.877 But what the other three correspond to, 00:44:08.877 --> 00:44:10.460 that depends on the coordinate system. 00:44:10.460 --> 00:44:12.627 It might be a radius, it might be a different angle, 00:44:12.627 --> 00:44:15.130 some things like that. 00:44:15.130 --> 00:44:17.920 Again, just sort of being a little overly cautious 00:44:17.920 --> 00:44:19.960 and careful defining these. 00:44:19.960 --> 00:44:27.210 I will note, though, that generally Greek indices 00:44:27.210 --> 00:44:38.390 in most textbooks, they tend to be used 00:44:38.390 --> 00:44:40.220 to label spacetime indices. 00:44:45.155 --> 00:44:46.530 And then there are times when you 00:44:46.530 --> 00:44:48.510 might want to just sort of imagine 00:44:48.510 --> 00:44:51.120 you've chosen a particular moment in time, 00:44:51.120 --> 00:44:55.030 and you want to look at what space looks like at that time. 00:44:55.030 --> 00:45:10.750 And so you might then go down to Latin indices 00:45:10.750 --> 00:45:14.450 to pick out spatial components at some moment in time. 00:45:32.373 --> 00:45:34.290 We'll see that come up from time to time, just 00:45:34.290 --> 00:45:36.248 want you to be aware there is this distinction. 00:45:36.248 --> 00:45:37.840 And as you read other textbooks, there 00:45:37.840 --> 00:45:40.300 are a few others that are used. 00:45:40.300 --> 00:45:41.785 Always just check, usually in some 00:45:41.785 --> 00:45:43.660 of the introductory chapters of the textbook, 00:45:43.660 --> 00:45:45.587 they will define these things very carefully. 00:45:45.587 --> 00:45:47.920 Wald is an example of someone who actually does anything 00:45:47.920 --> 00:45:48.878 a little bit different. 00:45:48.878 --> 00:45:53.140 He tends to use lower case Latin letters 00:45:53.140 --> 00:45:57.670 from the top of the alphabet to denote spacetime indices, 00:45:57.670 --> 00:46:03.460 and those from i, j, k, he uses them to denote Latin indices. 00:46:03.460 --> 00:46:04.880 If you're old enough to get this, 00:46:04.880 --> 00:46:08.110 this is often called by some of us who grew up in the Dark Ages 00:46:08.110 --> 00:46:10.240 it is sometimes called the Fortran convention. 00:46:10.240 --> 00:46:13.210 If you've ever programmed in Fortran, you know why that is. 00:46:13.210 --> 00:46:15.250 If you didn't, please don't bother learning it, 00:46:15.250 --> 00:46:18.820 it's really not worth the brain cells it would take. 00:46:18.820 --> 00:46:21.970 So we've got this geometric object 00:46:21.970 --> 00:46:25.698 that is viewed by observer o. 00:46:25.698 --> 00:46:28.240 Let's now think about what this looks like from the viewpoint 00:46:28.240 --> 00:46:29.860 of a different observer. 00:46:29.860 --> 00:46:32.730 A different inertial observer. 00:46:55.940 --> 00:47:00.800 Let's say somebody comes dashing through the room here, 00:47:00.800 --> 00:47:05.990 and observer o sees them running across the room at something 00:47:05.990 --> 00:47:09.050 like 87% of the speed of light. 00:47:09.050 --> 00:47:11.090 You know, since, as I have assumed, 00:47:11.090 --> 00:47:12.890 you are all experts in special relativity, 00:47:12.890 --> 00:47:16.640 that they will measure intervals of time and intervals of space 00:47:16.640 --> 00:47:18.650 differently than observer o does. 00:47:24.690 --> 00:47:33.570 So here's event P. Here's event Q. Here is delta x. 00:47:33.570 --> 00:47:39.180 This is all as measured by observer o bar. 00:47:42.380 --> 00:47:45.110 Something which I really want to strongly emphasize 00:47:45.110 --> 00:47:51.800 at this point is that this P, this Q, and this delta x, 00:47:51.800 --> 00:47:54.250 notice I haven't put bars on any of them. 00:47:54.250 --> 00:47:56.210 I haven't put primes or anything like that. 00:47:56.210 --> 00:48:01.310 It is the exact same P and Q and delta x as this over here. 00:48:01.310 --> 00:48:11.790 That is because P, Q, and delta x are geometric objects whose 00:48:11.790 --> 00:48:15.330 meaning transcends the particular inertial reference 00:48:15.330 --> 00:48:20.320 frame used to define the coordinates at which P exists, 00:48:20.320 --> 00:48:22.860 at which Q exists, and that then defined the delta x. 00:48:29.000 --> 00:48:31.670 These geometric objects exist independent 00:48:31.670 --> 00:48:33.186 of the representation. 00:48:41.620 --> 00:48:46.040 If I can use an intuitive example-- 00:48:46.040 --> 00:48:48.370 if I take and I hold-- let's be careful the pose 00:48:48.370 --> 00:48:51.190 I do with this-- let's say I stick my arm out. 00:48:51.190 --> 00:48:56.493 I say that my arm is pointing to the left, right? 00:48:56.493 --> 00:48:57.910 You guys will look at this and say 00:48:57.910 --> 00:48:59.360 your arm is pointing to the right, 00:48:59.360 --> 00:49:01.985 because you're using a slightly different system of coordinates 00:49:01.985 --> 00:49:04.600 to orient yourself in this room. 00:49:04.600 --> 00:49:05.890 We're both right. 00:49:05.890 --> 00:49:09.100 We have represented this geometric object, my arm, 00:49:09.100 --> 00:49:11.070 in different ways. 00:49:11.070 --> 00:49:12.888 But me calling this pointing to the left 00:49:12.888 --> 00:49:15.180 and you calling it pointing to the right doesn't change 00:49:15.180 --> 00:49:17.970 the basic nature of my arm. 00:49:17.970 --> 00:49:20.790 It doesn't mean that my blood cells 00:49:20.790 --> 00:49:23.410 changed because of something like this happening. 00:49:23.410 --> 00:49:25.620 This has an independent existence. 00:49:25.620 --> 00:49:27.620 In the same way, this delta x, it 00:49:27.620 --> 00:49:30.210 is the displacement, these two events. 00:49:30.210 --> 00:49:32.280 This might be mosquito lands on my head, 00:49:32.280 --> 00:49:34.410 this might be me smacking it with my hand, 00:49:34.410 --> 00:49:36.870 or flying near my head and me smocking with my hand. 00:49:36.870 --> 00:49:39.480 Those are events that exist independent of how 00:49:39.480 --> 00:49:42.700 we choose to represent them. 00:49:42.700 --> 00:49:45.810 So the key thing is we preserve that notion 00:49:45.810 --> 00:49:48.240 of the geometric object's independent existence. 00:49:48.240 --> 00:49:51.870 What does change is the representation 00:49:51.870 --> 00:49:53.100 that the two observers use. 00:50:02.970 --> 00:50:06.485 So I'm going to jump to this Greek index notation. 00:50:17.820 --> 00:50:22.450 And so what I'm going to say is that according to observer o 00:50:22.450 --> 00:50:30.780 bar, they are going to represent this object by a collection 00:50:30.780 --> 00:50:34.080 of components that are not the same as the complaints that 00:50:34.080 --> 00:50:36.092 are used by observer o. 00:50:36.092 --> 00:50:37.550 And to keep my notation consistent, 00:50:37.550 --> 00:50:44.280 let's play a little o underneath this arrow. 00:50:44.280 --> 00:50:47.040 So this is just sort of shorthand for delta 00:50:47.040 --> 00:50:50.970 x is represented according to o by those components. 00:50:50.970 --> 00:50:55.750 Delta x is represented according to o bar by these components. 00:50:55.750 --> 00:50:57.300 And again, since I'm assuming you all 00:50:57.300 --> 00:50:59.610 are experts in special relativity, 00:50:59.610 --> 00:51:04.830 we already know how to relate the barred components 00:51:04.830 --> 00:51:06.060 to the unbarred components. 00:51:06.060 --> 00:51:08.539 They're related by a Lorentz transformation. 00:51:35.500 --> 00:51:40.480 So what we would say is delta x is zero bar component, or t 00:51:40.480 --> 00:51:42.050 bar, if you prefer. 00:51:42.050 --> 00:51:43.600 It's given by gamma-- 00:51:43.600 --> 00:51:47.260 I will define gamma in just a moment, you can probably guess. 00:51:49.890 --> 00:51:51.670 So I'm imagining an observer that 00:51:51.670 --> 00:51:56.035 is just moving along the x-axis or the coordinate one axis. 00:52:35.330 --> 00:52:38.640 So what went on in that transformation 00:52:38.640 --> 00:52:51.390 I just wrote down is O bar moves with v along spatial axis 00:52:51.390 --> 00:53:04.340 1 with speed v as seen by O. And of course gamma 00:53:04.340 --> 00:53:10.200 is 1 divided by square root of 1 minus v squared. 00:53:10.200 --> 00:53:11.460 Remember, speed of light is 1. 00:53:15.410 --> 00:53:18.540 We don't want to be writing this crap out every time we have 00:53:18.540 --> 00:53:20.820 to transform different representations, 00:53:20.820 --> 00:53:23.990 so we're going to introduce more compact notation for this. 00:53:30.750 --> 00:53:34.960 So we're going to say delta x mu bar-- 00:53:38.264 --> 00:53:40.270 this is what I get when I sum over 00:53:40.270 --> 00:53:45.760 index nu from 0 to 3 of lambda. 00:53:45.760 --> 00:53:51.840 Mu bar nu, delta x nu. 00:53:51.840 --> 00:53:54.447 So that's defining a matrix multiplication, , 00:53:54.447 --> 00:53:56.530 and you can read out the components of this lambda 00:53:56.530 --> 00:53:58.072 matrix from what I've got over there. 00:54:03.530 --> 00:54:21.850 And even better yet, delta x mu bar 00:54:21.850 --> 00:54:28.090 is lambda mu bar nu delta x nu. 00:54:28.090 --> 00:54:30.630 So in this last line, if you haven't seen this before, 00:54:30.630 --> 00:54:33.703 I am using Einstein's summation convention. 00:54:45.790 --> 00:54:49.300 If I have an index that appears in one geometric 00:54:49.300 --> 00:54:51.260 object in the downstairs position 00:54:51.260 --> 00:54:53.050 and an adjacent geometric objects 00:54:53.050 --> 00:54:57.370 in the upstairs position and it's 00:54:57.370 --> 00:55:00.520 repeated-- so repeated indices in the upstairs and downstairs 00:55:00.520 --> 00:55:03.400 position are assumed to be summed over their full range, 00:55:03.400 --> 00:55:04.110 from 0 to 3. 00:55:10.278 --> 00:55:12.320 We're going to talk about this a little bit more, 00:55:12.320 --> 00:55:15.230 what's going on with this, after I've built up a little bit more 00:55:15.230 --> 00:55:17.410 of the mathematical structure. 00:55:17.410 --> 00:55:19.310 In particular, what is the distinction 00:55:19.310 --> 00:55:22.286 between through the upstairs and downstairs positions. 00:55:37.913 --> 00:55:39.330 Some of you might be saying, well, 00:55:39.330 --> 00:55:42.390 isn't one way of writing it what we call a covariant component 00:55:42.390 --> 00:55:44.820 and wasn't one a contravariant component? 00:55:44.820 --> 00:55:47.730 If you know those terms, mazel tov. 00:55:47.730 --> 00:55:49.973 They're not actually really helpful, 00:55:49.973 --> 00:55:51.390 and so I kind of deliberately like 00:55:51.390 --> 00:55:53.520 to use this more primitive wording of calling it 00:55:53.520 --> 00:55:55.380 just upstairs and downstairs. 00:55:55.380 --> 00:55:57.630 Because what we're going to find, the goal of physics, 00:55:57.630 --> 00:56:00.240 is to understand the universe in a way that 00:56:00.240 --> 00:56:03.390 allows us to connect this understanding to measurements. 00:56:03.390 --> 00:56:05.940 And measurements don't care about contravariant 00:56:05.940 --> 00:56:08.460 versus covariant, and all these things are essentially 00:56:08.460 --> 00:56:12.000 just ways of representing objects with our mathematics 00:56:12.000 --> 00:56:14.940 that is sort of a go between from some of our physical ideas 00:56:14.940 --> 00:56:16.920 to what can eventually be measured. 00:56:16.920 --> 00:56:20.093 So covariant, contravariant, eh, whatever. 00:56:20.093 --> 00:56:21.510 At the end of the day, we're going 00:56:21.510 --> 00:56:23.700 to see as we put these sort of things together 00:56:23.700 --> 00:56:26.940 it's how these terms connect to one another that matters. 00:56:26.940 --> 00:56:30.285 The name is not that important. 00:56:30.285 --> 00:56:32.160 I do want to make one little point about this 00:56:32.160 --> 00:56:33.970 as I move forward. 00:56:33.970 --> 00:56:39.400 It is sort of worth noting that if I 00:56:39.400 --> 00:56:44.830 think of how I relate the displacement components 00:56:44.830 --> 00:56:48.040 according to my barred observer relative to those as measured 00:56:48.040 --> 00:56:54.460 by my unbarred observer, I can think of this Lorentz 00:56:54.460 --> 00:56:59.890 transformation matrix as what I get when I differentiate 00:56:59.890 --> 00:57:03.040 one representation's coordinates with respect 00:57:03.040 --> 00:57:06.700 to the other representation's coordinates. 00:57:06.700 --> 00:57:09.160 Kind of trivial in this case, and when 00:57:09.160 --> 00:57:10.870 we are doing special relativity there 00:57:10.870 --> 00:57:13.120 is a particular form of the Lorentz transformation 00:57:13.120 --> 00:57:15.220 that we tend to use, but I just want 00:57:15.220 --> 00:57:18.280 to highlight this because this relationship between two 00:57:18.280 --> 00:57:21.068 different representations of a reference frame 00:57:21.068 --> 00:57:23.110 is going to come up over and over and over again. 00:57:23.110 --> 00:57:25.120 This is a more general form, this idea 00:57:25.120 --> 00:57:27.500 that you are essentially taking the derivative. 00:57:27.500 --> 00:57:31.000 You're looking at how one representation varies according 00:57:31.000 --> 00:57:32.230 to the other representation. 00:57:32.230 --> 00:57:34.870 So using that to think about how to move 00:57:34.870 --> 00:57:37.910 between one inertial reference frame to another 00:57:37.910 --> 00:57:40.610 is going to be very important to us. 00:57:40.610 --> 00:57:43.175 This is a more general form. 00:57:58.950 --> 00:58:01.320 I want to make one further point, 00:58:01.320 --> 00:58:06.330 and then I will introduce a more careful definition of what 00:58:06.330 --> 00:58:08.910 is meant by a vector and I think that will 00:58:08.910 --> 00:58:11.140 be a good place for us to stop. 00:58:11.140 --> 00:58:13.130 So when I look at this particular form-- 00:58:13.130 --> 00:58:16.260 so let's look at the last thing I wrote there, 00:58:16.260 --> 00:58:19.196 where I use the Einstein summation convention. 00:58:23.010 --> 00:58:25.800 This is just a chance for me to introduce a little bit more 00:58:25.800 --> 00:58:26.985 terminology in notation. 00:58:32.410 --> 00:58:35.580 So when I wrote down the relationship delta 00:58:35.580 --> 00:58:45.670 x mu bar with lambda mu bar nu, delta x nu, 00:58:45.670 --> 00:58:49.220 how I labeled the index that I was summing over 00:58:49.220 --> 00:58:50.729 is kind of irrelevant. 00:59:06.420 --> 00:59:14.460 This is exactly the same as lambda mu bar alpha delta 00:59:14.460 --> 00:59:15.960 x alpha. 00:59:15.960 --> 00:59:22.400 I can switch to something else if you 00:59:22.400 --> 00:59:25.830 have enough fonts available. 00:59:25.830 --> 00:59:28.950 You can use smiley faces as your index, whatever. 00:59:28.950 --> 00:59:32.280 The key bit is that as long as you are summing over it, 00:59:32.280 --> 00:59:34.292 it's kind of irrelevant how you label it. 00:59:34.292 --> 00:59:36.750 I think that looks silly, so I'm going to go back to alpha. 00:59:49.520 --> 00:59:54.020 When I have an index like that that is being summed over, 00:59:54.020 --> 00:59:56.840 it's going to sort of disappear at the end of my analysis. 00:59:56.840 --> 01:00:00.510 Its only role is to serve as a place holder. 01:00:00.510 --> 01:00:02.510 It allows me to keep things lined up properly 01:00:02.510 --> 01:00:05.490 so that I can do a particular mathematical operation. 01:00:05.490 --> 01:00:20.140 So in this equation, nu or alpha is called a dummy index. 01:00:25.778 --> 01:00:27.320 Now, when I'm doing it with something 01:00:27.320 --> 01:00:30.200 like this, where I'm just relating one set of one 01:00:30.200 --> 01:00:33.020 indexed objects to another set of one indexed objects, 01:00:33.020 --> 01:00:35.067 it's kind of trivial to move these things around. 01:00:35.067 --> 01:00:37.400 We're going to make some much more complicated equations 01:00:37.400 --> 01:00:40.270 later in this class, and we'll find in those cases 01:00:40.270 --> 01:00:42.020 that sometimes it's actually really useful 01:00:42.020 --> 01:00:44.120 to have the freedom to relabel our dummy indices. 01:00:44.120 --> 01:00:46.070 It allows us to sort of pick out patterns 01:00:46.070 --> 01:00:49.730 that might exist among things and see 01:00:49.730 --> 01:00:53.280 how to simplify a relationship in a really useful way. 01:00:53.280 --> 01:00:56.060 On the other hand, I do not have the freedom 01:00:56.060 --> 01:00:59.060 to change that mu bar that appears there, right. 01:00:59.060 --> 01:01:01.340 I have to have that being the same 01:01:01.340 --> 01:01:03.030 on both sides of the equation. 01:01:03.030 --> 01:01:07.730 Because I'm free to mix around things where it's just 01:01:07.730 --> 01:01:11.020 going to be summed over and not play a role, 01:01:11.020 --> 01:01:14.600 but in this my mu bar-- 01:01:14.600 --> 01:01:16.250 pardon me for a second-- 01:01:16.250 --> 01:01:20.440 mu bar is not a dummy index, and so I do not have that freedom. 01:01:24.500 --> 01:01:31.660 We sometimes call this the free index. 01:01:35.650 --> 01:01:38.150 As I write that down, it seems like a bit of a strange name, 01:01:38.150 --> 01:01:39.608 actually, and it's really not free. 01:01:39.608 --> 01:01:41.755 You're actually constrained in what it can be. 01:01:41.755 --> 01:01:43.880 What can I say, probably there's some history there 01:01:43.880 --> 01:01:44.880 that I don't know about. 01:01:58.490 --> 01:02:01.290 So let's do our last concept for the day. 01:02:08.070 --> 01:02:11.370 So let's carefully define a spacetime vector. 01:02:23.520 --> 01:02:27.720 So a spacetime vector is going to be any quartet of numbers, 01:02:27.720 --> 01:02:55.390 those numbers we will call components, which transforms 01:02:55.390 --> 01:03:00.242 between inertial reference frames 01:03:00.242 --> 01:03:01.450 like the displacement vector. 01:03:14.950 --> 01:03:19.490 So if I represent some spacetime vector A 01:03:19.490 --> 01:03:25.740 as some collection of numbers that as observed by o 01:03:25.740 --> 01:03:29.580 has components A0, A1, A2, A3. 01:03:37.540 --> 01:03:42.430 If a second inertial observer relates their components 01:03:42.430 --> 01:03:43.300 to these by-- 01:04:04.880 --> 01:04:07.798 if that describes the components for the observer o bar, 01:04:07.798 --> 01:04:08.590 then it's a vector. 01:04:23.400 --> 01:04:26.740 If you have taken any mathematics, 01:04:26.740 --> 01:04:28.950 they carefully defined vector spaces, 01:04:28.950 --> 01:04:30.520 this should be familiar to you. 01:04:30.520 --> 01:04:32.460 It's a very similar operation to what 01:04:32.460 --> 01:04:38.700 is done in a lot of other kinds of analysis. 01:04:38.700 --> 01:04:43.020 The key to making this notion of a vector a sensible one 01:04:43.020 --> 01:04:46.830 is this transformation law. 01:04:46.830 --> 01:04:52.950 So if I had a quartet of numbers, which is say, 01:04:52.950 --> 01:04:57.090 the number of batteries in my pocket, the number 01:04:57.090 --> 01:05:01.440 of times my dog sneezed this morning, how many toes I 01:05:01.440 --> 01:05:02.827 have on my left foot, and-- 01:05:02.827 --> 01:05:04.410 I'm sick of counting so let's just say 01:05:04.410 --> 01:05:07.020 0 for the fourth component-- 01:05:07.020 --> 01:05:10.020 that does not transform between reference frames 01:05:10.020 --> 01:05:11.680 by a Lorentz transformation. 01:05:11.680 --> 01:05:14.250 It's just a collection of random numbers. 01:05:14.250 --> 01:05:16.530 So not any old quartet of numbers 01:05:16.530 --> 01:05:18.600 will constitute the components of a vector. 01:05:18.600 --> 01:05:21.810 It has to be things that have a physical meaning that you 01:05:21.810 --> 01:05:24.852 connect to what you measure by a Lorentz transformation. 01:05:28.740 --> 01:05:32.850 For it to be a good vector, it also has to-- that A 01:05:32.850 --> 01:05:36.030 has to obey the various linearity laws that 01:05:36.030 --> 01:05:37.570 define a vector space. 01:05:48.090 --> 01:06:02.160 So if I had two vectors and I add them together, 01:06:02.160 --> 01:06:04.620 then their sum is a vector. 01:06:13.180 --> 01:06:16.875 If I have a vector and I multiply it by some scalar-- 01:06:27.572 --> 01:06:29.530 by the way, you have to be a little bit careful 01:06:29.530 --> 01:06:33.730 when I say scalar here, because you might think 01:06:33.730 --> 01:06:36.700 to yourself something like, you know, 01:06:36.700 --> 01:06:39.570 the mass of my shoe, that's a scalar. 01:06:39.570 --> 01:06:42.060 But you be careful when you're talking about things 01:06:42.060 --> 01:06:44.250 in relativity, whether the scale you're dealing with 01:06:44.250 --> 01:06:47.622 is actually a quantity that is Lorentz in variant. 01:06:47.622 --> 01:06:49.830 With quantity like mass, if you're talking about rest 01:06:49.830 --> 01:06:51.630 mass-- we'll get into the distinction among these things 01:06:51.630 --> 01:06:52.000 a bit later-- 01:06:52.000 --> 01:06:52.830 OK, you're good. 01:06:55.380 --> 01:06:56.880 You just have be careful to pick out 01:06:56.880 --> 01:06:59.430 something that actually is the same according 01:06:59.430 --> 01:07:03.790 to all observers. 01:07:03.790 --> 01:07:09.850 So when I say scalar, this means same to all observers. 01:07:15.040 --> 01:07:17.100 If this is the case, then I can define 01:07:17.100 --> 01:07:24.180 D to be that scalar times a vector and it is also a vector. 01:07:24.180 --> 01:07:25.130 Question? 01:07:25.130 --> 01:07:26.010 Oh, stretching, OK. 01:07:33.650 --> 01:07:35.720 I think, actually, I'm going to stop there. 01:07:35.720 --> 01:07:39.590 There is one topic that I just don't 01:07:39.590 --> 01:07:42.925 feel like I can get enough of it for it to be useful right now, 01:07:42.925 --> 01:07:43.550 so I think yes. 01:07:43.550 --> 01:07:45.620 I'm going to stop there for today. 01:07:45.620 --> 01:07:48.430 When we pick it up on Thursday we're 01:07:48.430 --> 01:07:51.050 going to wrap up this discussion of vectors-- 01:07:51.050 --> 01:07:52.550 again, I want to kind of emphasize, 01:07:52.550 --> 01:07:55.230 I can already see in some places you're getting the 1,000 yard 01:07:55.230 --> 01:07:55.730 stare. 01:07:55.730 --> 01:07:59.358 There's no question we're being excessively careful 01:07:59.358 --> 01:08:00.650 with some of these definitions. 01:08:00.650 --> 01:08:02.233 They are very straightforward, there's 01:08:02.233 --> 01:08:03.650 nothing challenging here. 01:08:03.650 --> 01:08:07.460 But there will be a payoff when we do get to times where 01:08:07.460 --> 01:08:09.470 the analyses, the geometries we're looking at, 01:08:09.470 --> 01:08:12.440 get pretty messed up. 01:08:12.440 --> 01:08:15.410 Having this formal foundation very carefully laid 01:08:15.410 --> 01:08:17.520 will help us significantly. 01:08:17.520 --> 01:08:19.950 So all right, I'm going to stop there for today, 01:08:19.950 --> 01:08:23.590 and we will pick it up on Thursday.