WEBVTT 00:00:00.080 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.820 under a Creative Commons license. 00:00:03.820 --> 00:00:06.550 Your support will help MIT OpenCourseWare continue 00:00:06.550 --> 00:00:10.160 to offer high quality educational resources for free. 00:00:10.160 --> 00:00:12.700 To make a donation or to view additional materials 00:00:12.700 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.327 at ocw.mit.edu. 00:00:21.837 --> 00:00:22.420 PROFESSOR: OK. 00:00:22.420 --> 00:00:23.769 In that case let's get going. 00:00:23.769 --> 00:00:25.310 In today's lecture, we're going to be 00:00:25.310 --> 00:00:27.320 sort of splitting the lecture of things, 00:00:27.320 --> 00:00:29.910 if the timing goes as I plan. 00:00:29.910 --> 00:00:31.880 We're going to start by finishing 00:00:31.880 --> 00:00:34.320 talking about the geodesic equation. 00:00:34.320 --> 00:00:36.070 And then if all goes well, we will 00:00:36.070 --> 00:00:39.090 start talking about the energy of radiation-- 00:00:39.090 --> 00:00:42.420 completely changing topics altogether. 00:00:42.420 --> 00:00:45.890 I want to begin, as usual, by reviewing quickly 00:00:45.890 --> 00:00:49.960 what we talked about last time, just to remind us where we are. 00:00:49.960 --> 00:00:53.630 Last time, we at first, at the beginning of lecture, 00:00:53.630 --> 00:00:56.910 talked about how to add time into the Robertson-Walker 00:00:56.910 --> 00:00:57.770 Metric. 00:00:57.770 --> 00:01:02.990 And this is the formula that we claimed was the correct one. 00:01:02.990 --> 00:01:06.850 For a spacetime metric, ds squared, 00:01:06.850 --> 00:01:09.930 the meaning is closely analogous to the meaning 00:01:09.930 --> 00:01:12.120 that it would have in special relativity. 00:01:12.120 --> 00:01:15.320 The main difference being that in special relativity 00:01:15.320 --> 00:01:17.550 we always talk about what is observed 00:01:17.550 --> 00:01:22.460 by inertial frames of reference and inertial observers. 00:01:22.460 --> 00:01:26.130 In general relativity, the concept of an inertial observer 00:01:26.130 --> 00:01:28.480 is not so clear cut, but we can talk 00:01:28.480 --> 00:01:31.630 about observers for whom there is no forces acting on them 00:01:31.630 --> 00:01:34.580 other than possibly gravitational forces. 00:01:34.580 --> 00:01:36.690 And whether or not there are gravitational forces 00:01:36.690 --> 00:01:39.630 is always, itself, a framed dependent question. 00:01:39.630 --> 00:01:42.380 So it does not have a definite answer. 00:01:42.380 --> 00:01:45.040 So observers for which there is no forces acting on them 00:01:45.040 --> 00:01:47.500 other than gravitational forces are 00:01:47.500 --> 00:01:49.600 called free-falling observers. 00:01:49.600 --> 00:01:51.740 And they play the role of inertial observers 00:01:51.740 --> 00:01:56.890 that the inertial observers play in special relativity. 00:01:56.890 --> 00:01:58.980 So if ds squared is positive, it's 00:01:58.980 --> 00:02:01.740 the square of the spatial separation measured 00:02:01.740 --> 00:02:04.800 by a local free-falling observer, for whom the two 00:02:04.800 --> 00:02:07.350 events happen at the same time. 00:02:07.350 --> 00:02:08.850 Last time, I think, I did not really 00:02:08.850 --> 00:02:11.480 mention or emphasize the word local. 00:02:11.480 --> 00:02:15.140 But the point is that in general relativity 00:02:15.140 --> 00:02:17.235 we expect in any small region one 00:02:17.235 --> 00:02:20.190 can construct an accelerating coordinate system in which 00:02:20.190 --> 00:02:22.250 the effects of gravity are canceled out, 00:02:22.250 --> 00:02:24.690 as the equivalence principle tells us we can do. 00:02:24.690 --> 00:02:27.540 And then you essentially see the effects of special relativity. 00:02:27.540 --> 00:02:29.490 But it's only a small region, in principle, 00:02:29.490 --> 00:02:31.860 on an infinitesimal region. 00:02:31.860 --> 00:02:34.460 So these measurements that correspond 00:02:34.460 --> 00:02:36.360 to special relativity measurements 00:02:36.360 --> 00:02:39.510 are always made locally by an observer who 00:02:39.510 --> 00:02:42.210 is, in principle, arbitrarily close to the events being 00:02:42.210 --> 00:02:44.360 measured. 00:02:44.360 --> 00:02:47.080 If ds squared is negative, then it's equal to minus c 00:02:47.080 --> 00:02:50.160 squared times the square of the time separation 00:02:50.160 --> 00:02:53.330 that would be measured by a local free-falling observer 00:02:53.330 --> 00:02:56.440 for whom the two events happen at the same location. 00:02:56.440 --> 00:02:58.990 I should point out that a special case of this 00:02:58.990 --> 00:03:01.190 is an observer looking at his own wristwatch. 00:03:01.190 --> 00:03:04.040 His own wristwatch is always at the same location 00:03:04.040 --> 00:03:08.630 relevant to him, so it's a special case of this statement. 00:03:08.630 --> 00:03:12.050 So it says that ds squared is equal to minus c squared times 00:03:12.050 --> 00:03:16.430 the time that a free-falling observer would 00:03:16.430 --> 00:03:18.752 read on his own wrist watch. 00:03:18.752 --> 00:03:21.290 And If ds squared is 0, it means that the two events 00:03:21.290 --> 00:03:25.220 can be joined by a light pulse going from one to the other. 00:03:25.220 --> 00:03:27.470 Having said this, we can go back to this formula 00:03:27.470 --> 00:03:31.140 and understand why the formula is what it is. 00:03:31.140 --> 00:03:33.830 The spatial part is what it is because 00:03:33.830 --> 00:03:37.230 any homogeneous and isotropic spatial metric can 00:03:37.230 --> 00:03:39.010 be written in this form. 00:03:39.010 --> 00:03:41.950 And we are assuming that the universe we're describing 00:03:41.950 --> 00:03:44.620 is homogeneous and isotropic. 00:03:44.620 --> 00:03:49.260 The dc squared piece is really dictated by item two here. 00:03:49.260 --> 00:03:51.940 We want the t that we write in this metric 00:03:51.940 --> 00:03:55.510 to be the cosmic time variable that we've been speaking about. 00:03:55.510 --> 00:03:57.970 And that means that it is the time variable measured 00:03:57.970 --> 00:04:00.050 on the watches of observers who are 00:04:00.050 --> 00:04:02.420 at rest in this coordinate system. 00:04:02.420 --> 00:04:04.620 And that means that it has to be simply 00:04:04.620 --> 00:04:05.990 minus c squared dt squared. 00:04:05.990 --> 00:04:08.800 Or else dt would not have the right relationship 00:04:08.800 --> 00:04:11.760 to a ds squared to be consistent with what 00:04:11.760 --> 00:04:14.460 the s squared is supposed to be. 00:04:14.460 --> 00:04:16.230 And then we also talked about why 00:04:16.230 --> 00:04:24.510 there are no dt dr terms, or dt d theta, or dt d phi. 00:04:24.510 --> 00:04:28.350 We said that any such term would violate isotropy. 00:04:28.350 --> 00:04:31.160 If you had a dt dr term, for example, 00:04:31.160 --> 00:04:33.090 it would make the positive dr direction 00:04:33.090 --> 00:04:35.330 different from the negative dr direction. 00:04:35.330 --> 00:04:38.250 And that can't be something that happens 00:04:38.250 --> 00:04:39.530 in an isotropic universe. 00:04:42.290 --> 00:04:45.870 That then is our metric for cosmology, 00:04:45.870 --> 00:04:48.654 the Roberrtson-Walker Metric. 00:04:48.654 --> 00:04:50.945 And another important thing is what is it good for, now 00:04:50.945 --> 00:04:53.070 that we decided that's the right metric? 00:04:53.070 --> 00:04:55.150 What use is to us? 00:04:55.150 --> 00:04:57.210 And what we haven't done yet, but it's actually 00:04:57.210 --> 00:05:00.120 on the homework, we need the full spacetime metric 00:05:00.120 --> 00:05:02.180 to be able to find geodesics, to be 00:05:02.180 --> 00:05:04.530 able to learn the paths of particles 00:05:04.530 --> 00:05:08.570 moving through this model universe. 00:05:08.570 --> 00:05:12.200 So we will be making important use of this Roberrtson-Walker 00:05:12.200 --> 00:05:15.725 Metric with its spacetime contributions. 00:05:18.011 --> 00:05:18.510 OK. 00:05:18.510 --> 00:05:19.530 Any questions about that? 00:05:19.530 --> 00:05:21.405 Now I'm ready to change gears to some extent. 00:05:21.405 --> 00:05:23.280 Yes, Ani? 00:05:23.280 --> 00:05:26.140 AUDIANCE: So in general, the spatial part of the metric, 00:05:26.140 --> 00:05:27.850 we can get from the geometry? 00:05:27.850 --> 00:05:29.790 And in general, can you just add a minus 00:05:29.790 --> 00:05:32.464 c squared dt square for the temporal part? 00:05:32.464 --> 00:05:33.880 PROFESSOR: It's not quite general. 00:05:33.880 --> 00:05:37.020 Remember we used an argument based on isotropy here. 00:05:39.602 --> 00:05:42.900 So I think it's safe to say that any metric you'll 00:05:42.900 --> 00:05:46.104 find in this class is likely to have 00:05:46.104 --> 00:05:48.520 the time entering, and nothing more complicated than minus 00:05:48.520 --> 00:05:50.110 c squared dt squared. 00:05:50.110 --> 00:05:52.820 But it's not a general statement about general relativity. 00:05:57.042 --> 00:05:57.875 Any other questions? 00:06:02.089 --> 00:06:02.589 Yes. 00:06:02.589 --> 00:06:11.970 AUDIANCE: [INAUDIBLE], 00:06:11.970 --> 00:06:14.520 PROFESSOR: OK, the question is, what would be a circumstance 00:06:14.520 --> 00:06:17.470 where we would have to deal with something more complicated? 00:06:17.470 --> 00:06:19.000 The answer would be, I think, all 00:06:19.000 --> 00:06:26.220 you need is to add to this model universe perturbations that 00:06:26.220 --> 00:06:27.910 break the uniformity. 00:06:27.910 --> 00:06:30.080 If we tried to describe the real universe instead 00:06:30.080 --> 00:06:33.350 of this ideal universe, where our ideal universe is perfectly 00:06:33.350 --> 00:06:35.580 isotropic and homogeneous, if it said 00:06:35.580 --> 00:06:38.910 we wanted to describe the lumps and bumps of the real universe, 00:06:38.910 --> 00:06:40.710 then it would become more complicated. 00:06:40.710 --> 00:06:42.930 And we would probably need a dt, dr term. 00:06:51.110 --> 00:06:55.090 OK, next we went on to talking about the geodesic equation. 00:06:55.090 --> 00:06:59.290 According to General Relativity, the trajectories of particles 00:06:59.290 --> 00:07:02.220 that have no forces acting on them other than gravity, 00:07:02.220 --> 00:07:05.870 these free-falling observers, are geodesics 00:07:05.870 --> 00:07:08.250 in the spacetime.. 00:07:08.250 --> 00:07:11.260 So that means we want to learn how to calculate geodesics, 00:07:11.260 --> 00:07:14.280 which means paths whose length is stationary 00:07:14.280 --> 00:07:17.160 under small variations. 00:07:17.160 --> 00:07:19.370 So we considered first just simple geodesics 00:07:19.370 --> 00:07:22.380 in the spatial metric, because that's easier to think about. 00:07:22.380 --> 00:07:24.780 What is the shortest distance between two points 00:07:24.780 --> 00:07:28.500 in a space that's described by some arbitrary metric? 00:07:28.500 --> 00:07:30.780 So first we talked about how we describe the metric. 00:07:30.780 --> 00:07:34.940 And we introduced two features in this first formula here. 00:07:34.940 --> 00:07:37.520 One is that instead of calling the coordinates XYZ 00:07:37.520 --> 00:07:38.750 or something like that. 00:07:38.750 --> 00:07:42.190 We called them x1, x2, x3, so that we could talk about them 00:07:42.190 --> 00:07:45.890 all together in one formula without writing separate pieces 00:07:45.890 --> 00:07:48.640 for the different coordinates. 00:07:48.640 --> 00:07:51.570 So i and j represent 1, 2, and 3, or just 1, 00:07:51.570 --> 00:07:56.760 and 2, which is the labeling of the spatial coordinates. 00:07:56.760 --> 00:07:59.830 And the other important piece of notation 00:07:59.830 --> 00:08:01.820 that is introduced in that formula 00:08:01.820 --> 00:08:04.930 is the Einstein summation convention. 00:08:04.930 --> 00:08:08.680 Whenever there's an index, like i and j here, 00:08:08.680 --> 00:08:11.340 which are repeated with one index lower 00:08:11.340 --> 00:08:13.590 and one index upper, they're automatically 00:08:13.590 --> 00:08:17.670 summed over all of the values that the coordinates take, 00:08:17.670 --> 00:08:20.565 without writing summation sign. 00:08:20.565 --> 00:08:22.230 It saves a lot of writing. 00:08:22.230 --> 00:08:25.930 And it turns out that one always sums under those circumstances, 00:08:25.930 --> 00:08:30.740 so there's no need to write the sums with the summation sign. 00:08:30.740 --> 00:08:34.349 Next we want to ask ourselves, how 00:08:34.349 --> 00:08:36.470 are we going to describe the path? 00:08:36.470 --> 00:08:37.970 Before we can find the minimum path, 00:08:37.970 --> 00:08:41.250 we need at least a language to talk about paths. 00:08:41.250 --> 00:08:43.600 And we could describe a path going from some point A 00:08:43.600 --> 00:08:48.960 to some point B, by giving a function x supra i of lambda. 00:08:48.960 --> 00:08:50.890 Well, lambda is an arbitrary parameter, 00:08:50.890 --> 00:08:53.700 that parametrizes the path. 00:08:53.700 --> 00:08:55.910 x supra i are a set of coordinates. 00:08:55.910 --> 00:08:58.540 i runs over the values of all the coordinates 00:08:58.540 --> 00:09:01.430 of whatever system you're dealing with. 00:09:01.430 --> 00:09:05.290 And you construct such a function where xi of 0 00:09:05.290 --> 00:09:09.330 is the starting point, which are the coordinates of the point A. 00:09:09.330 --> 00:09:14.930 And xi of some value lambda f, where f just stands for final, 00:09:14.930 --> 00:09:16.406 will be the end of the path. 00:09:16.406 --> 00:09:17.780 And it's supposed to end at point 00:09:17.780 --> 00:09:20.520 B. So the final coordinates of the path 00:09:20.520 --> 00:09:28.170 should be x supra i sub b, the coordinates of the point B. 00:09:28.170 --> 00:09:31.310 Then we want to use this description of the path 00:09:31.310 --> 00:09:33.850 to figure out what the length is of a segment of the path. 00:09:33.850 --> 00:09:36.820 And then the full length will be the sum of the segments. 00:09:36.820 --> 00:09:40.060 So for each segment, we just apply the metric 00:09:40.060 --> 00:09:42.890 to the change in coordinates. 00:09:42.890 --> 00:09:45.680 The change in coordinates, as lambda is varied, 00:09:45.680 --> 00:09:49.690 is just the derivative of xi with respect to lambda 00:09:49.690 --> 00:09:52.740 times the change in lambda. 00:09:52.740 --> 00:09:57.810 And putting that in for both dxi and dxj 00:09:57.810 --> 00:10:00.870 one gets this formula, relating ds 00:10:00.870 --> 00:10:02.800 squared, the square of the length 00:10:02.800 --> 00:10:06.370 of an infinitesimal segment to d lambda squared, 00:10:06.370 --> 00:10:09.338 the square of the parameter that describes that length. 00:10:12.090 --> 00:10:15.090 Then the full length is gotten by, first of all, taking 00:10:15.090 --> 00:10:16.720 the square root of this equation to get 00:10:16.720 --> 00:10:19.580 the infinitesimal length, ds. 00:10:19.580 --> 00:10:22.840 And then taking the integral of that over the path 00:10:22.840 --> 00:10:24.400 from beginning to end. 00:10:24.400 --> 00:10:27.090 And that, then, gives us the full length of the path, 00:10:27.090 --> 00:10:29.760 thinking of it as the sum of the length of each 00:10:29.760 --> 00:10:32.220 of infinitesimal segment. 00:10:32.220 --> 00:10:33.080 OK? 00:10:33.080 --> 00:10:36.180 Fair enough? 00:10:36.180 --> 00:10:38.390 Now that we have this formula for the length, now 00:10:38.390 --> 00:10:40.250 we have the next challenge, which 00:10:40.250 --> 00:10:43.750 is to figure out how to calculate the path which 00:10:43.750 --> 00:10:45.950 minimizes that length. 00:10:45.950 --> 00:10:47.660 And I didn't use the word last time, 00:10:47.660 --> 00:10:50.310 but that what is called the calculus of variations. 00:10:50.310 --> 00:10:53.680 And I looked up a little bit of the history in the Wikipedia. 00:10:53.680 --> 00:10:57.190 The calculus of variations dates back to 1696, 00:10:57.190 --> 00:10:59.720 when Johann Bernoulli invented it, 00:10:59.720 --> 00:11:03.840 applied it to the brachistochrone problem, 00:11:03.840 --> 00:11:08.120 which is the problem of finding a path for which a frictionless 00:11:08.120 --> 00:11:11.290 object will slide and get to its destination in the least 00:11:11.290 --> 00:11:12.760 possible time. 00:11:12.760 --> 00:11:14.460 And it turns out to be a cycloid, 00:11:14.460 --> 00:11:17.590 just like the cycloid that describes our closed universes, 00:11:17.590 --> 00:11:21.160 closed matter dominating the universe. 00:11:21.160 --> 00:11:25.390 And the problem was also solved by-- Johann Bernoulli 00:11:25.390 --> 00:11:28.060 then announced this problem to the world 00:11:28.060 --> 00:11:31.200 and challenged other mathematicians to solve it. 00:11:31.200 --> 00:11:37.400 There's a famous story that Newton noticed this question 00:11:37.400 --> 00:11:39.395 in his mail when he got home at 4:00 AM, 00:11:39.395 --> 00:11:41.770 or something like that, from the mint-- he was apparently 00:11:41.770 --> 00:11:45.899 a hardworking guy-- but nonetheless when 00:11:45.899 --> 00:11:47.690 he seen this problem he couldn't go to bed. 00:11:47.690 --> 00:11:51.780 He went ahead and solved it by morning, 00:11:51.780 --> 00:11:55.340 which is a good MIT student kind of thing to do. 00:11:58.710 --> 00:12:03.400 So the technique is to consider a small variation from whatever 00:12:03.400 --> 00:12:06.460 path you're hoping to be the minimum. 00:12:06.460 --> 00:12:08.670 And we're going to calculate the first order 00:12:08.670 --> 00:12:11.340 change in the length of the path, 00:12:11.340 --> 00:12:14.170 starting from our original path, x of lambda, 00:12:14.170 --> 00:12:18.230 to some new path, x tilde of lambda. 00:12:18.230 --> 00:12:23.200 And we parametrize the new path by writing it 00:12:23.200 --> 00:12:26.360 as the old path, plus a correction. 00:12:26.360 --> 00:12:28.280 And I've introduced a factor, alpha, 00:12:28.280 --> 00:12:30.530 multiplying the correction, because it makes it easier 00:12:30.530 --> 00:12:32.910 to talk about derivatives. 00:12:32.910 --> 00:12:36.950 And wi of lambda is just some arbitrary deviation 00:12:36.950 --> 00:12:39.577 from the original path. 00:12:39.577 --> 00:12:41.910 But we want to always go through the same starting point 00:12:41.910 --> 00:12:43.701 to the same endpoint, because there's never 00:12:43.701 --> 00:12:46.520 going to be a minimum if we're allowed to move the endpoints. 00:12:46.520 --> 00:12:48.010 So the endpoints are fixed. 00:12:48.010 --> 00:12:50.230 And that means that this path deviation, 00:12:50.230 --> 00:12:54.640 w super i in my notation, has to vanish at the two endpoints. 00:12:54.640 --> 00:13:00.750 So we impose these two equations on the variation wi. 00:13:00.750 --> 00:13:04.930 Then what do I do is take the derivative of the path 00:13:04.930 --> 00:13:09.730 length of the varied path, x tilde with respect to alpha, 00:13:09.730 --> 00:13:13.090 and if we had a minimum length to start with, 00:13:13.090 --> 00:13:15.560 the derivative should always vanish. 00:13:15.560 --> 00:13:17.250 That is, the minimum should always 00:13:17.250 --> 00:13:20.110 occur when alpha equals 0, if the original path 00:13:20.110 --> 00:13:23.240 of the true path, the true minimum path. 00:13:23.240 --> 00:13:26.150 And if alpha equals 0 is the minimum, 00:13:26.150 --> 00:13:29.270 the derivative should always vanish at alpha equals 0. 00:13:29.270 --> 00:13:30.370 And vice versa. 00:13:30.370 --> 00:13:34.270 If we know that this happens for every variation wi, 00:13:34.270 --> 00:13:36.940 then we know that our path is at least an extremum, 00:13:36.940 --> 00:13:40.590 and, presumably, a minimum. 00:13:40.590 --> 00:13:43.800 And the path itself is just written by the same formulas 00:13:43.800 --> 00:13:49.070 we had before, except for x tilde instead of x itself. 00:13:49.070 --> 00:13:52.800 And I've introduced an axillary quantity, a of lambda alpha, 00:13:52.800 --> 00:13:55.404 which is just what appears inside the square root. 00:13:55.404 --> 00:13:57.070 That just saves some writing, because it 00:13:57.070 --> 00:13:58.540 has to be written a number of times 00:13:58.540 --> 00:13:59.998 in the course of the manipulations. 00:14:02.970 --> 00:14:06.100 So our goal now is to carry out this derivative. 00:14:06.100 --> 00:14:08.430 And the derivative acts only on the integrand, 00:14:08.430 --> 00:14:09.920 because the limits of integration 00:14:09.920 --> 00:14:11.980 do not depend on alpha. 00:14:11.980 --> 00:14:15.410 So just carry the derivative into the integrand 00:14:15.410 --> 00:14:17.350 and differentiate this square root 00:14:17.350 --> 00:14:20.340 of a of lambda, which is, itself, a product of factors 00:14:20.340 --> 00:14:22.360 that we have to use-- product rule and chain 00:14:22.360 --> 00:14:25.030 rule and various manipulations. 00:14:25.030 --> 00:14:28.040 And after we carry out those manipulations, 00:14:28.040 --> 00:14:31.680 we end up with this expression in a straightforward 00:14:31.680 --> 00:14:36.980 way involving a few steps, which I won't show again. 00:14:36.980 --> 00:14:39.910 And the complication is that what we want to do 00:14:39.910 --> 00:14:43.760 is to figure out for what paths that expression 00:14:43.760 --> 00:14:45.760 will vanish for all wi. 00:14:45.760 --> 00:14:48.580 We want it to vanish for all possible variations 00:14:48.580 --> 00:14:50.000 of the path. 00:14:50.000 --> 00:14:55.190 And what's complicated is that wi appears here 00:14:55.190 --> 00:14:58.490 as a multiplicative factor in the first term, 00:14:58.490 --> 00:15:03.010 but as a differentiated factor in the second term. 00:15:03.010 --> 00:15:05.860 And that makes it very hard to know, initially, 00:15:05.860 --> 00:15:08.400 when those two terms might cancel each other to give you 00:15:08.400 --> 00:15:11.220 0, which is what we're looking for. 00:15:11.220 --> 00:15:13.020 But the brilliant trick that, I guess, 00:15:13.020 --> 00:15:16.780 Newton invented, along with Bernoulli and others, 00:15:16.780 --> 00:15:18.807 is to integrate by parts. 00:15:18.807 --> 00:15:21.390 Integration by parts, I'm sure, was not a well-known procedure 00:15:21.390 --> 00:15:23.480 at that time. 00:15:23.480 --> 00:15:25.910 But if we integrate the second term by parts, 00:15:25.910 --> 00:15:28.760 we could remove the derivative acting on w, 00:15:28.760 --> 00:15:31.420 and arrange for w to be a multiplicative factor 00:15:31.420 --> 00:15:33.540 in both terms. 00:15:33.540 --> 00:15:37.570 And a crucial thing that makes the whole thing useful 00:15:37.570 --> 00:15:40.080 is that when you do integrate by parts, 00:15:40.080 --> 00:15:43.650 you discover that you don't get any endpoint contributions, 00:15:43.650 --> 00:15:45.510 because the endpoint contributions would 00:15:45.510 --> 00:15:48.250 be proportional to wi at the endpoints. 00:15:48.250 --> 00:15:50.290 And remember, wi has to vanish at the endpoints, 00:15:50.290 --> 00:15:52.780 because that's the condition that we're not 00:15:52.780 --> 00:15:55.350 changing the points A and B. We're always 00:15:55.350 --> 00:15:58.250 talking about paths that have the same starting point 00:15:58.250 --> 00:16:01.430 and the same ending point. 00:16:01.430 --> 00:16:09.460 So integrating by parts, we get this expression, 00:16:09.460 --> 00:16:12.940 where now wi multiplies everything, as just 00:16:12.940 --> 00:16:14.385 simply a multiplicative factor. 00:16:14.385 --> 00:16:15.760 To write it in this form, you had 00:16:15.760 --> 00:16:17.690 to do a little bit of juggling of indices. 00:16:17.690 --> 00:16:20.240 The other important trick in these manipulations 00:16:20.240 --> 00:16:23.530 is to juggle indices, which I'll not show you explicitly. 00:16:23.530 --> 00:16:25.850 But the thing to remember is that these indices that 00:16:25.850 --> 00:16:28.190 are being summed over can be called anything 00:16:28.190 --> 00:16:31.010 and it's still the same sum. 00:16:31.010 --> 00:16:33.660 So when you want to get terms to cancel each other, 00:16:33.660 --> 00:16:35.820 you may have to change the names of indices 00:16:35.820 --> 00:16:38.145 to get them to just cancel identically. 00:16:38.145 --> 00:16:41.370 But that's straightforward. 00:16:41.370 --> 00:16:43.822 So we get this expression. 00:16:43.822 --> 00:16:45.530 And now we want this expression to vanish 00:16:45.530 --> 00:16:48.550 for every possible wi of lambda. 00:16:48.550 --> 00:16:50.140 And we argued that the only way it 00:16:50.140 --> 00:16:53.290 could vanish for every possible wi of lambda 00:16:53.290 --> 00:16:57.180 is if the expression in curly brackets, itself, vanishes. 00:16:57.180 --> 00:16:59.867 Yeah, if we only know the values for some particular wi 00:16:59.867 --> 00:17:01.450 of lambda, then there are lots of ways 00:17:01.450 --> 00:17:02.824 it could vanish, because it could 00:17:02.824 --> 00:17:05.609 be positive in some places and negative in others. 00:17:05.609 --> 00:17:08.069 But the only way it could vanish for all wi 00:17:08.069 --> 00:17:10.990 is for the quantity in curly brackets to vanish. 00:17:10.990 --> 00:17:12.940 So that gives us our final, or at least, 00:17:12.940 --> 00:17:15.445 almost final expression of the geodesic equation. 00:17:15.445 --> 00:17:21.010 And that's where we left off last time, with that equation. 00:17:21.010 --> 00:17:23.650 So note that this is just an equation that 00:17:23.650 --> 00:17:27.269 would either be obeyed or not obeyed by the function x super 00:17:27.269 --> 00:17:28.780 i of lambda. 00:17:28.780 --> 00:17:30.330 It's just a differential equation 00:17:30.330 --> 00:17:33.270 involving x super i of lambda and the metric, which 00:17:33.270 --> 00:17:35.890 we assume is given. 00:17:35.890 --> 00:17:36.390 OK. 00:17:36.390 --> 00:17:37.973 So are there any questions about that? 00:17:41.600 --> 00:17:43.670 Everybody happy? 00:17:43.670 --> 00:17:44.542 Great. 00:17:44.542 --> 00:17:46.375 OK, now we'll continue on on the blackboard. 00:17:57.722 --> 00:17:59.180 OK, the first thing I want to do is 00:17:59.180 --> 00:18:03.020 to simplify the equation a bit. 00:18:03.020 --> 00:18:05.120 This equation is fairly complicated, 00:18:05.120 --> 00:18:08.467 because of those square roots of A's in the denominators. 00:18:08.467 --> 00:18:10.550 The square root of A is a pretty complicated thing 00:18:10.550 --> 00:18:12.332 to start with, and the square root of A 00:18:12.332 --> 00:18:14.040 here is even differentiated, because it's 00:18:14.040 --> 00:18:16.460 got the lambda making an incredible mess, 00:18:16.460 --> 00:18:18.610 if you understand all that. 00:18:18.610 --> 00:18:20.430 So it would be nice to simplify that. 00:18:20.430 --> 00:18:23.440 And we do have one trick which we can still do, 00:18:23.440 --> 00:18:25.420 which we haven't done yet. 00:18:25.420 --> 00:18:29.800 We originally constructed our path, xi of lambda, 00:18:29.800 --> 00:18:33.300 as a function of some arbitrary parameter, lambda. 00:18:33.300 --> 00:18:37.630 Lambda just measures arbitrary points along the path. 00:18:37.630 --> 00:18:39.460 There are many, many ways to do that, 00:18:39.460 --> 00:18:42.169 an infinite number of ways that you can do that. 00:18:42.169 --> 00:18:43.960 And this formula will work for all of them, 00:18:43.960 --> 00:18:45.250 it's completely general. 00:18:45.250 --> 00:18:47.120 The formula, when we derived it, we 00:18:47.120 --> 00:18:52.170 didn't make any assumptions about how lambda was chosen. 00:18:52.170 --> 00:18:56.290 But we can simplify the formula by making a particular choice 00:18:56.290 --> 00:18:57.390 for lambda. 00:18:57.390 --> 00:18:59.410 And the choice that simplifies things 00:18:59.410 --> 00:19:02.780 is to choose lambda to be the arc length itself. 00:19:02.780 --> 00:19:05.160 Lambda should be the distance along the path. 00:19:05.160 --> 00:19:07.460 And then we're trying to express xi 00:19:07.460 --> 00:19:11.700 as a function of how far you've already gone. 00:19:11.700 --> 00:19:22.290 And that has the effect, if we go back to what Ai was, 00:19:22.290 --> 00:19:31.150 A of lambda really is just the path length per lambda. 00:19:31.150 --> 00:19:36.370 So if lambda is the path length itself, A is just equal to 1. 00:19:40.528 --> 00:19:43.080 I'm trying to get a formula that shows that more clearly. 00:19:43.080 --> 00:19:43.680 Here. 00:19:43.680 --> 00:19:45.880 If we remember that this quantity is A, 00:19:45.880 --> 00:19:48.670 this tells us that ds squared is equal to A times d 00:19:48.670 --> 00:19:49.780 lambda squared. 00:19:49.780 --> 00:19:52.600 So if ds is the same as d lambda, 00:19:52.600 --> 00:19:55.530 as you've chosen your parameter to be the path length, 00:19:55.530 --> 00:19:57.840 this formula makes it clear that that's 00:19:57.840 --> 00:20:01.510 equivalent to A equal to 1. 00:20:01.510 --> 00:20:04.780 So going back to the formula, if A is 1, 00:20:04.780 --> 00:20:08.121 we would just drop it from both sides of the equation. 00:20:08.121 --> 00:20:10.620 And all that really matters, I should point out here, maybe, 00:20:10.620 --> 00:20:13.480 because we'll be using it later, is that A is a constant. 00:20:13.480 --> 00:20:16.700 As long as A is a constant, it will not be differentiated, 00:20:16.700 --> 00:20:19.350 and then it will cancel on the left side and the right side. 00:20:19.350 --> 00:20:20.780 So we don't necessarily care that it is 1, 00:20:20.780 --> 00:20:22.279 but we do care that it's a constant. 00:20:22.279 --> 00:20:24.590 And then it just disappears from the formula. 00:20:24.590 --> 00:20:27.350 And then we get the simpler formula. 00:20:27.350 --> 00:20:33.040 And now we'll continue on the blackboard. 00:20:33.040 --> 00:20:51.900 The simpler formula is just dds of gij dxj ds 00:20:51.900 --> 00:21:01.530 is equal to 1/2 times the derivative of gjk, with respect 00:21:01.530 --> 00:21:15.020 to xi, times dxj ds dxk ds, where 00:21:15.020 --> 00:21:16.870 s is equal to the path length. 00:21:22.840 --> 00:21:24.750 So I've replaced lambda by s, because we 00:21:24.750 --> 00:21:26.095 set lambda equal to s. 00:21:26.095 --> 00:21:29.290 And s has a more specific meaning than lambda did. 00:21:29.290 --> 00:21:31.740 Lambda was a completely arbitrary parametrization 00:21:31.740 --> 00:21:32.320 of the path. 00:21:35.940 --> 00:21:40.680 So this one deserves a big box, because it really 00:21:40.680 --> 00:21:43.072 is the final formula for geodesics. 00:21:43.072 --> 00:21:45.030 Once we write it in terms of different letters, 00:21:45.030 --> 00:21:49.260 we will later, but this actually is the formula. 00:21:49.260 --> 00:21:51.770 Now I should mention just largely 00:21:51.770 --> 00:21:55.360 for the sake of your knowing what's going on, 00:21:55.360 --> 00:21:58.580 if you ever look at some other general relativity books, 00:21:58.580 --> 00:22:01.630 this is not the formula that the geodesic equation is usually 00:22:01.630 --> 00:22:03.070 written in. 00:22:03.070 --> 00:22:04.360 Frankly, it is the best form. 00:22:04.360 --> 00:22:06.130 If you want to find the geodesic, 00:22:06.130 --> 00:22:09.560 usually this form of writing the equation is the easiest. 00:22:09.560 --> 00:22:12.030 But most general relativity books 00:22:12.030 --> 00:22:14.160 prefer instead to just give a formula 00:22:14.160 --> 00:22:16.560 for the second derivative, here. 00:22:16.560 --> 00:22:18.820 Which involves just expanding this term, 00:22:18.820 --> 00:22:21.390 and then when we shuffle things, to try 00:22:21.390 --> 00:22:24.410 to simplify the expressions. 00:22:24.410 --> 00:22:40.110 So one can write, to start, d ds of gij dxj ds. 00:22:40.110 --> 00:22:41.760 We're just going to expand it. 00:22:41.760 --> 00:22:43.680 Now we're going to be making use of all the rules of calculus 00:22:43.680 --> 00:22:44.350 that we've learn. 00:22:44.350 --> 00:22:46.183 Every rule you've ever learned will probably 00:22:46.183 --> 00:22:47.980 get used in this calculation. 00:22:47.980 --> 00:22:51.140 So it will be using product rule, 00:22:51.140 --> 00:22:53.760 of course, because we have a product of two things here. 00:22:53.760 --> 00:22:55.850 But we also have the complication 00:22:55.850 --> 00:22:59.530 that gij is not explicitly a function of s. 00:22:59.530 --> 00:23:03.270 But gij is a function of position. 00:23:03.270 --> 00:23:07.040 And the position that one is that for any given value of s 00:23:07.040 --> 00:23:10.240 depends on s, because we're moving along the path, 00:23:10.240 --> 00:23:12.350 x super i of s. 00:23:12.350 --> 00:23:19.970 So the gij here, is evaluated at x super i of s. 00:23:22.510 --> 00:23:24.240 I should give this a new letter. 00:23:24.240 --> 00:23:27.730 x super k of s. 00:23:27.730 --> 00:23:32.750 So it depends on s, through the argument of its argument. 00:23:32.750 --> 00:23:36.280 So that's a chain rule situation. 00:23:36.280 --> 00:23:41.560 And what we get here is, from just differentiating 00:23:41.560 --> 00:23:43.820 the second factor, that's easy. 00:23:43.820 --> 00:23:54.320 We get gij DC d squared xj ds squared. 00:23:54.320 --> 00:23:58.380 And then, from the derivative of the derivative chain rule 00:23:58.380 --> 00:24:11.600 piece, we get the partial of gij, with respect to xk times 00:24:11.600 --> 00:24:19.030 the dxj ds times dxk ds. 00:24:26.100 --> 00:24:29.840 And then to continue, this piece gets brought over 00:24:29.840 --> 00:24:32.500 to the other side, because we're trying to get an equation just 00:24:32.500 --> 00:24:34.145 for the second derivative of the path. 00:24:54.420 --> 00:25:04.690 So then we get g sub ij d squared x super j ds squared is 00:25:04.690 --> 00:25:14.470 equal to 1/2 di-- I'll define that in a second-- g sub jk 00:25:14.470 --> 00:25:29.120 minus 2 dk gij dxi ds dxj ds. 00:25:34.910 --> 00:25:38.970 where this partial derivative with the subscript 00:25:38.970 --> 00:25:43.270 is just an abbreviation for the derivative with respect 00:25:43.270 --> 00:25:44.910 to the coordinate with that index. 00:25:47.910 --> 00:25:49.213 So that's just an abbreviation. 00:25:54.690 --> 00:25:59.890 Now you could think of this as a matrix times a vector 00:25:59.890 --> 00:26:01.820 is equal to an expression. 00:26:01.820 --> 00:26:05.230 What we like to do is just get an expression for this vector. 00:26:05.230 --> 00:26:08.210 So if we think of it as a matrix times a vector, 00:26:08.210 --> 00:26:10.080 all we have to do is invert the matrix 00:26:10.080 --> 00:26:12.671 to be able to get an expression for the vector itself. 00:26:12.671 --> 00:26:13.170 Yes! 00:26:13.170 --> 00:26:14.878 AUDIANCE: Should that closing parenthesis 00:26:14.878 --> 00:26:18.840 be more [INAUDIBLE]? 00:26:18.840 --> 00:26:20.799 PROFESSOR: Oh, Yeah, I think you're right, 00:26:20.799 --> 00:26:21.715 it doesn't look right. 00:26:26.770 --> 00:26:27.750 Yeah. 00:26:27.750 --> 00:26:37.960 Thank you This has to multiply everything. 00:26:41.820 --> 00:26:44.600 Oops! 00:26:44.600 --> 00:26:47.295 OK, OK. 00:26:47.295 --> 00:26:48.920 Given enough chances I'll get it right. 00:26:53.205 --> 00:26:58.710 OK, now everybody happy this time? 00:26:58.710 --> 00:27:03.100 Thank you very much for getting it straight. 00:27:03.100 --> 00:27:06.280 OK, So as I was saying, we want to isolate 00:27:06.280 --> 00:27:07.450 this second derivative. 00:27:07.450 --> 00:27:09.033 We're hoping to get just an expression 00:27:09.033 --> 00:27:10.460 for the second derivative. 00:27:10.460 --> 00:27:14.482 And this can be interpreted as a matrix times a vector 00:27:14.482 --> 00:27:15.190 equals something. 00:27:15.190 --> 00:27:17.090 We want to just invert that matrix. 00:27:17.090 --> 00:27:18.392 Yes? 00:27:18.392 --> 00:27:21.679 AUDIANCE: Isn't the ds and [? the idx ?] [INAUDIBLE]? 00:27:21.679 --> 00:27:23.720 PROFESSOR: Oh, do I have that wrong too, perhaps? 00:27:32.360 --> 00:27:35.630 I think we want j and k there, that don't we? 00:27:35.630 --> 00:27:42.370 OK, attempt number four, or did I lose count as well? 00:27:42.370 --> 00:27:45.770 j and k are the indices and the i 00:27:45.770 --> 00:27:48.520 matches the free i on the left. 00:27:48.520 --> 00:27:51.350 And all the other indices are sound. 00:27:51.350 --> 00:27:55.470 I think, probably, I finally achieved the right formula. 00:27:59.460 --> 00:28:00.460 Thanks for all the help. 00:28:04.220 --> 00:28:08.223 So inverting a matrix, the principal 00:28:08.223 --> 00:28:10.890 is a straightforward mathematical operation. 00:28:10.890 --> 00:28:14.690 In general relativity, we give a name to the inverse metric, 00:28:14.690 --> 00:28:17.890 and it's the same letter g with indices, 00:28:17.890 --> 00:28:20.140 with superscripts instead of subscripts. 00:28:20.140 --> 00:28:23.410 And that's defined to be the matrix inverse. 00:28:23.410 --> 00:28:38.440 So g super ij is defined to be the matrix inverse of g sub ij. 00:28:41.020 --> 00:28:44.790 And to put that into an equation, 00:28:44.790 --> 00:28:48.870 we could say that if we take g with upper indices-- 00:28:48.870 --> 00:28:52.720 and I'll write those upper indices as i and l-- 00:28:52.720 --> 00:29:00.100 and multiply it by a g with lower indices l and j, 00:29:00.100 --> 00:29:04.680 when you sum over adjacent indices in this index 00:29:04.680 --> 00:29:07.060 notation, that's exactly what corresponds 00:29:07.060 --> 00:29:10.660 to the definition of matrix multiplication. 00:29:10.660 --> 00:29:13.980 So this is just the matrix g with upper indices times 00:29:13.980 --> 00:29:18.260 the matrix g with lower indices, and it's the i j'th element 00:29:18.260 --> 00:29:19.820 of that matrix. 00:29:19.820 --> 00:29:22.290 And we're saying it should be the identity matrix, 00:29:22.290 --> 00:29:26.620 and that means that the i j'th element should be 0 if it's off 00:29:26.620 --> 00:29:31.180 diagonal, and 1 if it's diagonal, if i equals j. 00:29:31.180 --> 00:29:32.650 And that's exactly the definition 00:29:32.650 --> 00:29:34.680 of a chronic or a delta. 00:29:34.680 --> 00:29:39.060 So this is equal to delta ij. 00:29:39.060 --> 00:29:43.670 We remember that delta ij is 0 if i is not equal to j, 1 00:29:43.670 --> 00:29:44.860 if i is equal to j. 00:29:44.860 --> 00:29:46.280 That's the definition. 00:29:46.280 --> 00:29:48.120 And it corresponds to that identity matrix 00:29:48.120 --> 00:29:48.925 in matrix language. 00:29:52.100 --> 00:29:57.680 So this is the relationship that actually defines g super il. 00:29:57.680 --> 00:30:01.790 And it is just the statement that g with upper indices 00:30:01.790 --> 00:30:04.130 is the matrix inverse of g with lower indices. 00:30:06.940 --> 00:30:11.560 Using this, we can bring this g to the other side 00:30:11.560 --> 00:30:14.940 essentially by multiplying by g inverse. 00:30:14.940 --> 00:30:19.660 And I will save a little time by not writing that out 00:30:19.660 --> 00:30:25.700 in gory detail, but rather I'll just write the result. 00:30:25.700 --> 00:30:28.110 And the result is written in terms of a new symbol that 00:30:28.110 --> 00:30:29.770 gets defined, which is an absolutely 00:30:29.770 --> 00:30:32.910 standard symbol in General Relativity. 00:30:32.910 --> 00:30:38.820 The formula is d squared x i, ds squared 00:30:38.820 --> 00:30:42.590 is equal to-- we know it's going to be equal to stuff 00:30:42.590 --> 00:30:46.860 times the product of two derivatives. 00:30:46.860 --> 00:30:49.330 And the stuff that appears is just 00:30:49.330 --> 00:30:54.050 given a name, capital gamma, which has an upper index 00:30:54.050 --> 00:30:57.670 i, which matches the left hand side of the equation. 00:30:57.670 --> 00:31:01.230 And two lower indices, which I'm calling j and k, which 00:31:01.230 --> 00:31:04.810 will get summed with the derivatives that follow, 00:31:04.810 --> 00:31:10.755 d x j ds, dx k, ds. 00:31:16.100 --> 00:31:20.080 And this quantity, gamma super i sub jk 00:31:20.080 --> 00:31:21.720 are just the terms that would appear 00:31:21.720 --> 00:31:24.130 when we do these manipulations. 00:31:24.130 --> 00:31:28.070 And I'll write what they are. 00:31:28.070 --> 00:31:39.155 Gamma super i sub jk is equal to 1/2 g super il 00:31:39.155 --> 00:31:42.900 times the derivative with respect 00:31:42.900 --> 00:31:52.290 to j of g sub lk, plus the derivative 00:31:52.290 --> 00:31:58.740 with respect to k of g sub lj. 00:31:58.740 --> 00:32:05.530 And then minus the derivative with respect to l of g sub jk. 00:32:14.840 --> 00:32:20.240 And this quantity has several different names. 00:32:20.240 --> 00:32:22.430 Everybody agrees how to define it up to the sign. 00:32:22.430 --> 00:32:23.929 There are different sign conventions 00:32:23.929 --> 00:32:26.620 that are used in different books. 00:32:26.620 --> 00:32:28.680 And there are also different names for it. 00:32:28.680 --> 00:32:31.070 It's often called the affine connection. 00:32:34.694 --> 00:32:36.860 If you look, for example in Steve Weinberg's General 00:32:36.860 --> 00:32:39.990 Relativity book, he calls it the affine connection. 00:32:39.990 --> 00:32:43.090 It's also very often called the Christofel connection, 00:32:43.090 --> 00:32:44.300 or the Christofel symbol. 00:33:00.610 --> 00:33:03.660 And frankly those are the only names for it 00:33:03.660 --> 00:33:07.340 that I've seen, personally. 00:33:07.340 --> 00:33:09.400 But there's a book about [INAUDIBLE] 00:33:09.400 --> 00:33:12.170 by Sean Carroll which is a very good book. 00:33:12.170 --> 00:33:14.900 And he claims that it's sometimes also called 00:33:14.900 --> 00:33:17.550 the Riemann connection And it's also sometimes called 00:33:17.550 --> 00:33:19.930 the Levi-Civita connection. 00:33:19.930 --> 00:33:23.250 So it's got lots of names, which I guess means lots of people's 00:33:23.250 --> 00:33:24.540 independently invented it. 00:33:27.840 --> 00:33:30.050 But in any case, that's the answer. 00:33:30.050 --> 00:33:33.380 And it's just a way of rewriting the formula we have up there. 00:33:33.380 --> 00:33:34.970 And for solving problems, the formula, 00:33:34.970 --> 00:33:36.540 the way we wrote up there, is almost always 00:33:36.540 --> 00:33:37.560 the best way to do it. 00:33:37.560 --> 00:33:40.160 So this is really just window dressing, largely 00:33:40.160 --> 00:33:42.556 for the purpose of making contact with other books 00:33:42.556 --> 00:33:43.680 that you might come across. 00:33:47.360 --> 00:33:48.860 OK, so that finishes the derivation 00:33:48.860 --> 00:33:50.330 of the geodesic equation. 00:33:50.330 --> 00:33:52.760 Now I'd like to give an example of its use. 00:33:52.760 --> 00:33:54.399 But before I do that, let me just 00:33:54.399 --> 00:33:56.940 pause to ask if there are any questions about the derivation? 00:34:02.350 --> 00:34:03.490 OK. 00:34:03.490 --> 00:34:05.460 So on your homework, you will, in fact, 00:34:05.460 --> 00:34:10.900 be applying this formalism to the Robertson-Walker metric. 00:34:10.900 --> 00:34:15.210 And you'll learn how moving particles slow down 00:34:15.210 --> 00:34:19.650 as they move through an expanding universe, 00:34:19.650 --> 00:34:23.520 completely in an analogy to the way photons, which we've 00:34:23.520 --> 00:34:26.000 already learned, lose energy as they 00:34:26.000 --> 00:34:28.489 travel through an expanding universe. 00:34:28.489 --> 00:34:31.770 So particles with mass also lose energy 00:34:31.770 --> 00:34:33.799 in a well-defined way, which you'll 00:34:33.799 --> 00:34:36.780 be calculating on the homework. 00:34:36.780 --> 00:34:39.879 For example, though, I'll do something different. 00:34:39.879 --> 00:34:44.330 A fun metric to talk about is the Schwarzschild metric, 00:34:44.330 --> 00:34:47.730 which describes, among other things, black holes. 00:34:47.730 --> 00:34:49.310 It in principle, describes anything 00:34:49.310 --> 00:34:52.250 which is spherically symmetric and has a gravitational field. 00:34:52.250 --> 00:34:54.760 But black holes are the most interesting example, 00:34:54.760 --> 00:34:58.930 because it's where the most surprises lie. 00:34:58.930 --> 00:35:10.140 So the Schwarzschild metric has the form 00:35:10.140 --> 00:35:18.170 ds squared is equal to minus c squared d tau squared, which 00:35:18.170 --> 00:35:24.810 is equal to-- this is just a definition, it defines d tau-- 00:35:24.810 --> 00:35:27.560 but in terms of the coordinates, it's 00:35:27.560 --> 00:35:32.100 minus 1 minus 2 G, Newton's constant, M, 00:35:32.100 --> 00:35:35.035 the mass of the object we're discussing-- 00:35:35.035 --> 00:35:38.910 the mass of the black hole, if it is a black hole-- divided 00:35:38.910 --> 00:35:43.270 by r times c squared, r is the radial coordinate, 00:35:43.270 --> 00:35:57.520 times c squared dt squared, plus 1 minus 2 GM over rc 00:35:57.520 --> 00:36:07.970 squared times dr squared plus r squared times d theta 00:36:07.970 --> 00:36:14.705 squared plus sine squared theta d phi squared. 00:36:20.120 --> 00:36:24.280 Now here, theta and phi are the usual polar angles. 00:36:24.280 --> 00:36:26.940 We're using a polar coordinate system. 00:36:26.940 --> 00:36:33.880 So as usual, theta lies between 0 and pi. 00:36:33.880 --> 00:36:36.300 0, what we might call the North Pole, and pi 00:36:36.300 --> 00:36:39.760 what we might call the South Pole. 00:36:39.760 --> 00:36:44.792 And phi is what is often called an azimuthal angle, 00:36:44.792 --> 00:36:46.840 it goes around. 00:36:46.840 --> 00:36:48.740 And the way one describes coordinates 00:36:48.740 --> 00:36:50.360 on the surface of the Earth, phi would 00:36:50.360 --> 00:36:54.860 be the longitude variable. 00:36:54.860 --> 00:36:59.460 So 0 is less than or equal to phi is less than 00:36:59.460 --> 00:37:04.690 or equal to 2 pi where phi equals 00:37:04.690 --> 00:37:08.380 2 pi is identified with phi equals 0. 00:37:08.380 --> 00:37:12.150 And you can go around and come back to where you started. 00:37:17.650 --> 00:37:20.670 Now notice that if we set capital M, the mass of this 00:37:20.670 --> 00:37:24.540 object equal to 0, the metric becomes the trivial metric 00:37:24.540 --> 00:37:29.050 of Special Relativity written in spherical polar coordinates. 00:37:29.050 --> 00:37:32.080 So all complications go away if there's no mass. 00:37:32.080 --> 00:37:34.930 The object disappears. 00:37:34.930 --> 00:37:36.800 But as long as the mass is non-zero 00:37:36.800 --> 00:37:40.300 there are factors that multiply the dr squared term 00:37:40.300 --> 00:37:42.080 and the c squared dt squared term. 00:37:45.330 --> 00:37:49.314 Notice that the factors that do that multiplying-- now one 00:37:49.314 --> 00:37:50.480 of these should be inverted. 00:37:54.630 --> 00:37:58.420 Important inverse, it's a minus 1 power for that factor. 00:38:01.240 --> 00:38:06.029 Notice that r can be small enough so 00:38:06.029 --> 00:38:07.320 that these factors will vanish. 00:38:10.110 --> 00:38:12.940 And the place where that happens is called the Schwarzschild 00:38:12.940 --> 00:38:17.620 radius after the same person who invented the metric. 00:38:17.620 --> 00:38:26.400 So r sub Schwarzschild is equal to 2 GM divided by c squared. 00:38:26.400 --> 00:38:31.460 When little r is equal to that, this quantity in parentheses 00:38:31.460 --> 00:38:34.090 vanishes, which means we get infinity here, 00:38:34.090 --> 00:38:38.760 because it's inverted, and we get a 0 there. 00:38:38.760 --> 00:38:44.860 Now when a term in the metric is either 0 or infinite, 00:38:44.860 --> 00:38:48.300 one calls that a singularity. 00:38:48.300 --> 00:38:51.820 In this case, it's a removable singularity. 00:38:51.820 --> 00:38:53.850 That is, the Schwarzschild singularity 00:38:53.850 --> 00:38:56.590 is only there because Schwarzschild 00:38:56.590 --> 00:38:59.402 chose to use these particular coordinates. 00:38:59.402 --> 00:39:01.110 These are simpler than other coordinates. 00:39:01.110 --> 00:39:03.530 He wasn't foolish to use them. 00:39:03.530 --> 00:39:05.250 But the appearance of that singularity 00:39:05.250 --> 00:39:08.600 is really caused solely by the choice of coordinates. 00:39:08.600 --> 00:39:15.070 There really is no singularity at the Schwarzschild horizon. 00:39:15.070 --> 00:39:17.520 And that was shown some years later 00:39:17.520 --> 00:39:21.090 by other people constructing other coordinate systems. 00:39:21.090 --> 00:39:23.760 The coordinate system is best known today 00:39:23.760 --> 00:39:27.370 that avoids the Schwarzschild singularity is 00:39:27.370 --> 00:39:29.840 a coordinate system called the Kruskal coordinate system. 00:39:56.404 --> 00:39:58.570 But we will not be looking at the Kruskal coordinate 00:39:58.570 --> 00:40:00.820 system in this class. 00:40:00.820 --> 00:40:03.260 Leave that for the GR class that you'll take some time. 00:40:08.080 --> 00:40:14.320 OK, now the masses sum parameter, 00:40:14.320 --> 00:40:15.820 notice that the metric is completely 00:40:15.820 --> 00:40:17.400 determined by the mass. 00:40:17.400 --> 00:40:21.450 And that's the same situation as we found in Newtonian gravity. 00:40:21.450 --> 00:40:25.460 The metric outside of the spherically symmetric object, 00:40:25.460 --> 00:40:28.220 by the gravitational field in Newtonian Physics outside 00:40:28.220 --> 00:40:30.295 of a spherical symmetric object, depends only 00:40:30.295 --> 00:40:31.890 on the total mass, which does not 00:40:31.890 --> 00:40:33.770 depend, at all, on how it's distributed 00:40:33.770 --> 00:40:35.630 as long as it's spherically symmetric. 00:40:35.630 --> 00:40:36.970 And the same thing here. 00:40:36.970 --> 00:40:40.080 As long as an object is spherically symmetric, 00:40:40.080 --> 00:40:42.870 the gravitational field outside of the object 00:40:42.870 --> 00:40:46.820 will always look like that formula. 00:40:46.820 --> 00:40:50.660 Now there are still two cases-- outside 00:40:50.660 --> 00:40:53.440 of the object could be larger than or smaller 00:40:53.440 --> 00:40:57.060 than this Schwarzschild radius. 00:40:57.060 --> 00:41:00.340 So for an object like the sun, the Schwarzschild radius, 00:41:00.340 --> 00:41:02.940 we could calculate it-- and it's calculated 00:41:02.940 --> 00:41:05.390 in the notes-- it's about two or three kilometers. 00:41:08.300 --> 00:41:10.710 Hold on and I'll tell you more accurately. 00:41:10.710 --> 00:41:13.380 It's 2.95 kilometers, the Schwarzschild radius 00:41:13.380 --> 00:41:15.790 of the sun. 00:41:15.790 --> 00:41:18.090 But the sun, of course, is much bigger than that. 00:41:18.090 --> 00:41:21.020 And that means that the sun doesn't have a Schwarzschild 00:41:21.020 --> 00:41:22.670 horizon. 00:41:22.670 --> 00:41:28.350 That is, at 2.95 kilometers from the center of the sun 00:41:28.350 --> 00:41:30.030 there's still sun. 00:41:30.030 --> 00:41:31.110 It's not outside the sun. 00:41:31.110 --> 00:41:33.443 This metric only holds outside the spherically symmetric 00:41:33.443 --> 00:41:34.990 object. 00:41:34.990 --> 00:41:36.610 So it does not hold inside the sun. 00:41:36.610 --> 00:41:39.830 The place where this has the apparent singularity 00:41:39.830 --> 00:41:41.852 the metric is not valid at all. 00:41:41.852 --> 00:41:43.310 So there is nothing that even comes 00:41:43.310 --> 00:41:46.150 close to anything worth talking about, 00:41:46.150 --> 00:41:47.800 as far as the Schwarzschild singularity 00:41:47.800 --> 00:41:49.770 for an object like the sun. 00:41:49.770 --> 00:41:53.820 But if the sun were compressed to a size smaller than 2.95 00:41:53.820 --> 00:42:00.040 kilometers with the same mass, then these factors 00:42:00.040 --> 00:42:03.950 would be relevant at the places where they vanish. 00:42:03.950 --> 00:42:09.290 And whatever consequences they have, we would be dealing with. 00:42:09.290 --> 00:42:12.930 Even though r equals r Schwarzschild is not a singular 00:42:12.930 --> 00:42:15.670 point, it is still a special point. 00:42:15.670 --> 00:42:19.655 What you can show-- we won't-- but what we can show is that 00:42:19.655 --> 00:42:21.840 that is the horizon. 00:42:21.840 --> 00:42:24.600 Meaning that if an object falls inside this Schwarzschild 00:42:24.600 --> 00:42:30.191 radius, there is no trajectory that will ever get it out. 00:42:30.191 --> 00:42:30.690 Yes? 00:42:30.690 --> 00:42:34.866 AUDIANCE: Say a star is just incredibly dense at its core. 00:42:34.866 --> 00:42:36.850 Is it possible to have suppression 00:42:36.850 --> 00:42:41.082 of some fractional life of a star that's from that mass 00:42:41.082 --> 00:42:42.059 that it's contained? 00:42:42.059 --> 00:42:43.808 Or like a fusion reaction that is going on 00:42:43.808 --> 00:42:46.220 with the net radius? 00:42:46.220 --> 00:42:49.572 PROFESSOR: OK, could there be a horizon inside of a star? 00:42:49.572 --> 00:42:51.280 I think is what you're asking, basically. 00:42:51.280 --> 00:42:52.870 AUDIANCE: One that actually affects the-- 00:42:52.870 --> 00:42:53.885 PROFESSOR: One that really is a horizon. 00:42:53.885 --> 00:42:54.926 AUDIANCE: That's outside. 00:42:54.926 --> 00:42:57.050 PROFESSOR: Right. 00:42:57.050 --> 00:42:58.790 If this were the sun you were describing, 00:42:58.790 --> 00:43:00.623 this formula would just not be valid inside. 00:43:00.623 --> 00:43:02.010 There would be no horizon inside. 00:43:02.010 --> 00:43:05.070 But you're asking a real valid question. 00:43:05.070 --> 00:43:09.930 If a star had, for some reason, a very dense spot 00:43:09.930 --> 00:43:12.350 in the middle, could it actually form 00:43:12.350 --> 00:43:14.870 a horizon inside the material? 00:43:14.870 --> 00:43:16.412 And the answer is, yes, it could. 00:43:16.412 --> 00:43:17.370 It would not be stable. 00:43:17.370 --> 00:43:27.960 The material would ultimately fall in, but it could happen. 00:43:27.960 --> 00:43:29.003 Yes? 00:43:29.003 --> 00:43:31.461 AUDIANCE: So like our galaxy has a super massive black hole 00:43:31.461 --> 00:43:32.180 in the center. 00:43:32.180 --> 00:43:32.790 PROFESSOR: That's right. 00:43:32.790 --> 00:43:34.748 Our galaxy does have a super massive black hole 00:43:34.748 --> 00:43:35.760 in the center. 00:43:35.760 --> 00:43:35.896 AUDIANCE: Yeah. 00:43:35.896 --> 00:43:37.604 So you can consider that as like a larger 00:43:37.604 --> 00:43:40.570 mass that has black hole, area? 00:43:40.570 --> 00:43:42.180 PROFESSOR: Right! 00:43:42.180 --> 00:43:43.860 Right! 00:43:43.860 --> 00:43:44.650 That's right. 00:43:44.650 --> 00:43:47.114 The comment is that if we go from a star 00:43:47.114 --> 00:43:48.530 to something bigger than a star we 00:43:48.530 --> 00:43:52.030 have perfectly good example in our own galaxy, 00:43:52.030 --> 00:43:53.780 where there is a black hole in the center, 00:43:53.780 --> 00:43:56.090 but there is still mass that continues outside of that. 00:43:56.090 --> 00:43:58.300 And the black hole is accreting, more matter 00:43:58.300 --> 00:44:00.940 does keep falling in, it's not really stable. 00:44:00.940 --> 00:44:05.510 But it certainly does exist, and can exist. 00:44:09.082 --> 00:44:09.915 Any other questions? 00:44:17.840 --> 00:44:21.135 Well, our goal now is to calculate a geodesic. 00:44:28.050 --> 00:44:32.380 And I will just calculate one geodesic. 00:44:32.380 --> 00:44:36.480 I will calculate what happens if an object starts at some fixed 00:44:36.480 --> 00:44:41.270 radius at rest and is released and falls into this black hole. 00:44:47.370 --> 00:44:50.890 I first want to just rewrite the geodesic equation in terms 00:44:50.890 --> 00:44:54.720 of variables that are more appropriate for this case. 00:44:54.720 --> 00:44:57.970 When I wrote that, I had a mind just calculating the geodesics 00:44:57.970 --> 00:45:00.010 in space, looking for the shortest 00:45:00.010 --> 00:45:02.731 path between two points. 00:45:02.731 --> 00:45:04.480 The geodesic that we're talking about when 00:45:04.480 --> 00:45:07.500 we're talking about an object in general relativity 00:45:07.500 --> 00:45:09.870 moving along the geodesic is a geodesic 00:45:09.870 --> 00:45:12.570 that's a time-like geodesic. 00:45:12.570 --> 00:45:14.510 That is, any increment along the geodesic 00:45:14.510 --> 00:45:17.300 is a time-like interval, or following a particle. 00:45:17.300 --> 00:45:22.030 Particles travel on time-like trajectories in relativity. 00:45:22.030 --> 00:45:26.700 So the usual notation for time is something like tau 00:45:26.700 --> 00:45:30.260 rather than s, which is why I wrote it this way. 00:45:30.260 --> 00:45:33.140 ds squared is just defined to be minus c squared d tau squared. 00:45:33.140 --> 00:45:36.680 So d tau squared has no more or less information 00:45:36.680 --> 00:45:40.050 than ds squared, but it has the opposite sign and a difference 00:45:40.050 --> 00:45:42.850 by a factor of c squared, as well. 00:45:42.850 --> 00:45:48.230 And another change in notation which is a rather universal 00:45:48.230 --> 00:45:53.200 convention is that, when we talk about space alone we use Latin 00:45:53.200 --> 00:45:55.810 indices, ijk.. 00:45:55.810 --> 00:45:58.260 When we talk about spacetime, where one of the indices 00:45:58.260 --> 00:46:01.500 might be 0 referring to the time direction, 00:46:01.500 --> 00:46:05.840 then we usually use Greek indices, mu, nu, lambda. 00:46:05.840 --> 00:46:11.280 So I'm going to rewrite the geodesic equation using tau 00:46:11.280 --> 00:46:13.790 as my parameter instead of s, since we're 00:46:13.790 --> 00:46:16.430 talking about proper time along the trajectory instead 00:46:16.430 --> 00:46:17.890 of distances. 00:46:17.890 --> 00:46:21.250 And using Greek letters instead of Latin letters, because we're 00:46:21.250 --> 00:46:24.131 talking about spacetime rather than just space. 00:46:24.131 --> 00:46:25.630 So otherwise what I'm going to write 00:46:25.630 --> 00:46:26.713 is just identical to that. 00:46:26.713 --> 00:46:30.020 So really is nothing more than a change in notation. 00:46:30.020 --> 00:46:43.090 d d tau of g mu nu, dx super nu d tau. 00:46:43.090 --> 00:46:49.520 And it is equal to 1/2 times the partial of g lambda 00:46:49.520 --> 00:47:04.135 sigma with respect to x nu dx lambda d tau dx sigma d tau. 00:47:08.221 --> 00:47:10.220 Now you might want to go through the calculation 00:47:10.220 --> 00:47:12.540 and make sure of the fact that now we're 00:47:12.540 --> 00:47:14.960 dealing with a metric which is not positive, 00:47:14.960 --> 00:47:17.210 definite, doesn't make any difference. 00:47:17.210 --> 00:47:18.444 But it doesn't. 00:47:18.444 --> 00:47:19.860 It does mean that now we certainly 00:47:19.860 --> 00:47:24.040 have possibilities of getting maxima and stationary points as 00:47:24.040 --> 00:47:28.260 well as minima, because of the variety of signs 00:47:28.260 --> 00:47:30.220 that appear in the metric. 00:47:30.220 --> 00:47:32.590 But otherwise, the calculations of the geodesic equation 00:47:32.590 --> 00:47:35.040 goes through exactly as we calculated it. 00:47:35.040 --> 00:47:38.240 And the only thing I'm doing here, 00:47:38.240 --> 00:47:40.320 relative to what we have there, is just 00:47:40.320 --> 00:47:43.125 changing the notation a bit to conform to the notaion that 00:47:43.125 --> 00:47:45.680 is usually used for talking about spacetime trajectories. 00:48:04.060 --> 00:48:07.300 Since we're talking about radio trajectories, 00:48:07.300 --> 00:48:09.640 we're just going to release a particle at rest 00:48:09.640 --> 00:48:11.690 and then it will fall straight towards the center 00:48:11.690 --> 00:48:15.860 of our spherical object, we know by symmetry 00:48:15.860 --> 00:48:17.580 that it's not going to be deflected 00:48:17.580 --> 00:48:19.750 in the positive theta or the negative theta, 00:48:19.750 --> 00:48:22.720 or the positive phi or negative phi directions, 00:48:22.720 --> 00:48:24.735 because that would violate isotropy. 00:48:24.735 --> 00:48:26.455 It would violate the rotational symmetry 00:48:26.455 --> 00:48:28.260 that we know as part of this metric. 00:48:28.260 --> 00:48:31.800 This Is just the metric of the surface of the sphere. 00:48:31.800 --> 00:48:34.220 So theta and phi will just stay whatever values they have 00:48:34.220 --> 00:48:35.920 when you drop this object. 00:48:35.920 --> 00:48:38.600 So we will not even talk about theta and phi. 00:48:38.600 --> 00:48:41.680 We will only talk about r and t, how 00:48:41.680 --> 00:48:44.880 particle falls in as a function of time. 00:48:44.880 --> 00:48:49.770 And then it turns out to be useful to just first write down 00:48:49.770 --> 00:48:51.600 what the metric itself tells us. 00:48:51.600 --> 00:48:53.015 And we'll divide by d tau. 00:48:53.015 --> 00:48:55.280 So we could talk about derivatives with respect 00:48:55.280 --> 00:48:57.130 to tau. 00:48:57.130 --> 00:48:59.200 So changing an overall sign, since everything's 00:48:59.200 --> 00:49:00.700 going to be negative and we'd rather 00:49:00.700 --> 00:49:03.780 have everything be positive, we can just 00:49:03.780 --> 00:49:08.750 rewrite the metric equation as saying, 00:49:08.750 --> 00:49:16.750 that c squared is equal to 1 minus 2 GM over rc 00:49:16.750 --> 00:49:22.240 squared, times c squared times dt 00:49:22.240 --> 00:49:37.580 d tau squared minus 1 minus 2 GM over rc squared inverse times 00:49:37.580 --> 00:49:41.180 dr d tau squared. 00:49:45.070 --> 00:49:47.290 So this is nothing more than rewriting this equation 00:49:47.290 --> 00:49:49.915 saying d theta is equal to 0 and d phi will be 0. 00:49:52.490 --> 00:49:54.100 Written this way, though, it tells us 00:49:54.100 --> 00:49:59.285 that we can find dt d tau, for example, if we know dr d tau. 00:49:59.285 --> 00:50:01.720 And we also know where we are, you know, little r. 00:50:01.720 --> 00:50:03.950 And we'll be using that, shortly. 00:50:06.590 --> 00:50:08.060 To continue a little further, we're 00:50:08.060 --> 00:50:10.185 going to introduce some abbreviations just so we're 00:50:10.185 --> 00:50:12.100 don't have to write so much. 00:50:12.100 --> 00:50:16.760 I'm going to define little h of r 00:50:16.760 --> 00:50:22.680 as just one minus r Schwarzschild over r. 00:50:26.500 --> 00:50:33.580 And this is also 1 minus 2 GM over rc squared. 00:50:33.580 --> 00:50:36.030 That's a factor that keeps recurring 00:50:36.030 --> 00:50:38.281 in our expression for the metric. 00:50:38.281 --> 00:50:38.780 Yes? 00:50:38.780 --> 00:50:40.321 AUDIANCE: The second to last equation 00:50:40.321 --> 00:50:43.837 is supposed to be a c squared in between the two parenthesis? 00:50:43.837 --> 00:50:44.670 PROFESSOR: Probably. 00:50:47.780 --> 00:50:48.600 Yes, thank you. 00:50:52.610 --> 00:50:55.359 G squared, right? 00:50:55.359 --> 00:50:55.900 Thanks a lot. 00:51:09.660 --> 00:51:11.510 In terms of h of r, we can rewrite 00:51:11.510 --> 00:51:15.370 that equation slightly more simply. 00:51:15.370 --> 00:51:17.720 I'm going to bring things to the other side 00:51:17.720 --> 00:51:20.970 and write it as c squared times dt 00:51:20.970 --> 00:51:32.680 d tau squared is equal to c squared h inverse of r plus h 00:51:32.680 --> 00:51:41.720 to the minus 2 of r times dr d tau squared. 00:51:45.656 --> 00:51:48.710 This is just a rewriting of the above equation, 00:51:48.710 --> 00:51:52.057 making use of the new notation that we've introduced. 00:51:52.057 --> 00:51:53.640 And this is the form we will be using. 00:51:53.640 --> 00:51:55.056 It explicitly tells us how to find 00:51:55.056 --> 00:51:57.455 dt d tau in terms of other things. 00:51:57.455 --> 00:52:00.450 So dt d tau is not independent. 00:52:04.750 --> 00:52:07.870 Since we know dt d tau in terms of dr d tau. 00:52:07.870 --> 00:52:11.400 If We get an expression for dr d tau we're sort of finished. 00:52:11.400 --> 00:52:14.420 We could find everything we want to know about t 00:52:14.420 --> 00:52:17.250 from the equation we just wrote. 00:52:17.250 --> 00:52:20.440 So it turns out that all we need to do 00:52:20.440 --> 00:52:23.340 to calculate this radial trajectory 00:52:23.340 --> 00:52:26.710 is to look at the component of the metric 00:52:26.710 --> 00:52:32.140 where that free index, mu, mu is the index that's not summed, 00:52:32.140 --> 00:52:34.880 we're going to set mu equal to r. 00:52:34.880 --> 00:52:37.740 Remember mu is a number that corresponds to a coordinate. 00:52:37.740 --> 00:52:39.573 And we're going to set mu equal to the value 00:52:39.573 --> 00:52:41.170 that corresponds to the r coordinate. 00:52:41.170 --> 00:52:46.040 And that will be sufficient to get us our answer. 00:52:46.040 --> 00:52:56.300 When we do that, the equation becomes d d tau of g sub r. 00:52:56.300 --> 00:53:00.540 Now the second index, nu in the original expression, 00:53:00.540 --> 00:53:04.965 is summed from 0 to 3 for the gr case, where 00:53:04.965 --> 00:53:07.180 we have four coordinates, one time and three 00:53:07.180 --> 00:53:11.030 spatial coordinates, but we only need 00:53:11.030 --> 00:53:16.210 to write the terms where gr nu variable is non-zero. 00:53:16.210 --> 00:53:19.410 And the metric itself is diagonal. 00:53:19.410 --> 00:53:22.700 So if one index is a little r, the other index 00:53:22.700 --> 00:53:25.330 has to also be r, or else it vanishes. 00:53:25.330 --> 00:53:27.960 So the only value of nu that contributes to the sum 00:53:27.960 --> 00:53:30.890 is when nu is also equal to the r coordinate. 00:53:30.890 --> 00:53:37.900 So we get g sub rr d xr-- which, in fact I'll 00:53:37.900 --> 00:53:43.540 write it as just dr. x super r is just the r coordinate, which 00:53:43.540 --> 00:53:54.974 we also just call r times d tau is equal to 1/2 dr. 00:53:54.974 --> 00:53:57.270 And now, on the right-hand side, we're 00:53:57.270 --> 00:54:00.780 summing over lambda and sigma. 00:54:00.780 --> 00:54:04.940 And lambda and sigma have to have the property that g sub 00:54:04.940 --> 00:54:07.866 lambda sigma depends on r, or else the first factor 00:54:07.866 --> 00:54:08.365 will vanish. 00:54:11.030 --> 00:54:14.466 And furthermore, g sub lambda sigma 00:54:14.466 --> 00:54:16.590 has be non-zero, for the values of lambda and sigma 00:54:16.590 --> 00:54:19.190 that you want, which means that lambda and sigma for this case 00:54:19.190 --> 00:54:20.898 has to be equal to each other, because we 00:54:20.898 --> 00:54:23.980 have no off-diagonal terms to our metric. 00:54:23.980 --> 00:54:29.410 So the only contributions we get are from g sub rr and g sub tt. 00:54:29.410 --> 00:54:37.280 So you get the derivative with respect to r of g sub rr times 00:54:37.280 --> 00:54:42.740 dr d tau squared. 00:54:42.740 --> 00:54:46.980 This become squared, because lambda is equal to sigma. 00:54:46.980 --> 00:55:04.070 And then plus 1/2 drg sub tt times dt d tau squared. 00:55:15.860 --> 00:55:20.970 And note that buried in here is, if we expand this, 00:55:20.970 --> 00:55:22.810 the second derivative of r with respect 00:55:22.810 --> 00:55:26.330 to time-- respect to tau. 00:55:26.330 --> 00:55:30.760 So we can extract that and solve for it. 00:55:30.760 --> 00:55:35.870 And things like dt d tau will appear in our answer, 00:55:35.870 --> 00:55:38.856 initially, because it's here already. 00:55:38.856 --> 00:55:43.220 But we could replace dt d tau by this top equation 00:55:43.220 --> 00:55:46.090 and eliminate it from our results. 00:55:46.090 --> 00:55:50.400 And I'm going to skip the algebra, which 00:55:50.400 --> 00:55:52.600 is straightforward, although tedious. 00:55:52.600 --> 00:55:55.950 I urge you to go through it in the notes. 00:55:55.950 --> 00:55:58.580 But the end result ends up being remarkably simple, 00:55:58.580 --> 00:56:02.530 after a number of cancellations that look like surprises. 00:56:02.530 --> 00:56:05.730 And what you find in the end-- and it's just 00:56:05.730 --> 00:56:09.200 the simplification of this formula, nothing more-- you 00:56:09.200 --> 00:56:15.490 find that d squared r d tau squared is 00:56:15.490 --> 00:56:22.810 just equal to minus Newton's constant times the mass 00:56:22.810 --> 00:56:23.910 divided by r squared. 00:56:31.950 --> 00:56:34.100 Now this is rather shocking, and even looks exactly 00:56:34.100 --> 00:56:35.285 like Newtonian mechanics. 00:56:38.129 --> 00:56:40.420 However, even though it looks like Newtonian mechanics, 00:56:40.420 --> 00:56:43.670 it's not really the same as Newtonian mechanics, 00:56:43.670 --> 00:56:47.330 because the variables don't mean quite the same thing. 00:56:47.330 --> 00:56:49.850 First of all, even r does not really 00:56:49.850 --> 00:56:55.390 mean radius in the same sense as radius is defined by Newton. 00:56:55.390 --> 00:56:57.030 In Newtonian mechanics, radius is 00:56:57.030 --> 00:56:58.860 the distance from the origin. 00:56:58.860 --> 00:57:01.560 If we wanted to know the distance from the origin, 00:57:01.560 --> 00:57:04.690 we would have to integrate this metric. 00:57:04.690 --> 00:57:07.214 And in fact, there isn't even an actual origin here, 00:57:07.214 --> 00:57:09.380 because you would have to go through the singularity 00:57:09.380 --> 00:57:10.254 before you get there. 00:57:10.254 --> 00:57:12.122 And you really can't. 00:57:12.122 --> 00:57:16.581 That integral is not really even defined. 00:57:16.581 --> 00:57:18.830 Although, of course, if we had something like the sun, 00:57:18.830 --> 00:57:20.871 where the metric was different from this small r, 00:57:20.871 --> 00:57:22.850 then we could integrate from r equals 0, 00:57:22.850 --> 00:57:24.910 and that would define the true radius, 00:57:24.910 --> 00:57:26.450 distance from the center. 00:57:26.450 --> 00:57:27.480 But it would not be r. 00:57:27.480 --> 00:57:31.992 It would be what you got by integrating with the metric. 00:57:31.992 --> 00:57:33.450 So r has a different interpretation 00:57:33.450 --> 00:57:37.100 than it does for Newtonian physics. 00:57:37.100 --> 00:57:40.310 I might add, it still has a simple interpretation. 00:57:40.310 --> 00:57:46.260 If you look at this metric, the tangential part, the angular 00:57:46.260 --> 00:57:50.970 part, is exactly what you have for Euclidean geometry. 00:57:50.970 --> 00:57:54.050 It's just r squared times the same combination of d theta 00:57:54.050 --> 00:57:57.820 and d phi as appears on the surface of a sphere. 00:57:57.820 --> 00:58:01.870 So little r is sometimes called the circumferential radius, 00:58:01.870 --> 00:58:06.790 because it really does give you the circumference of circles 00:58:06.790 --> 00:58:09.160 at that radial coordinate. 00:58:09.160 --> 00:58:11.650 If we went around in a circle at a fixed r, 00:58:11.650 --> 00:58:14.450 the circle would involve varying phi, for example, 00:58:14.450 --> 00:58:16.440 over a range of 2 pi, we really would 00:58:16.440 --> 00:58:21.460 see a total circumference of 2 pi little r. 00:58:21.460 --> 00:58:22.990 So r is related to circumferences 00:58:22.990 --> 00:58:26.420 in exactly the way as it is in Euclidean geometry. 00:58:26.420 --> 00:58:27.920 But it's not related to the distance 00:58:27.920 --> 00:58:31.330 from the origin in the same way as it is in Euclidean geometry. 00:58:34.350 --> 00:58:38.700 In addition, tau, here, is not the universal time 00:58:38.700 --> 00:58:41.910 that Newton imagined. 00:58:41.910 --> 00:58:48.780 But rather, tau is measured along the geodesic. 00:58:51.690 --> 00:58:53.190 It is just ds squared, but remember, 00:58:53.190 --> 00:58:56.490 ds squared is being measured along the geodesic, which 00:58:56.490 --> 00:58:58.450 means that it is, in fact, the proper time 00:58:58.450 --> 00:59:01.550 as it would be measured by the person falling 00:59:01.550 --> 00:59:05.310 with the object towards the black hole. 00:59:05.310 --> 00:59:09.514 So tau is proper time as measured by the falling object. 00:59:09.514 --> 00:59:10.930 And that follows from what we know 00:59:10.930 --> 00:59:12.800 about the meaning of the metric itself. 00:59:17.660 --> 00:59:21.767 OK, that said we would now like to just study 00:59:21.767 --> 00:59:22.975 this equation more carefully. 00:59:50.417 --> 00:59:51.875 And since the equation itself still 00:59:51.875 --> 00:59:55.150 has the same form as what you get from Newton, if you 00:59:55.150 --> 00:59:57.700 remember what you would have done if this was 801, 00:59:57.700 --> 01:00:00.480 you can, in fact, do exactly the same thing here. 01:00:00.480 --> 01:00:03.230 And what you probably would have done, if this was 801, 01:00:03.230 --> 01:00:08.270 would be to recognize that this equation can be integrated. 01:00:08.270 --> 01:00:17.200 We can write the equation as d d tau of 1/2 01:00:17.200 --> 01:00:29.720 dr d tau squared minus GM/r equals 0. 01:00:29.720 --> 01:00:31.180 I you carry out these derivatives 01:00:31.180 --> 01:00:32.796 you would get that equation. 01:00:32.796 --> 01:00:34.170 And this is just the conservation 01:00:34.170 --> 01:00:38.780 of energy version of the force equation. 01:00:55.410 --> 01:00:59.710 And that tells us that this quantity is a constant. 01:00:59.710 --> 01:01:14.731 If we drop the object from some initial position, r sub 0, 01:01:14.731 --> 01:01:18.040 and we drop it with no initial velocity, 01:01:18.040 --> 01:01:21.800 we just let go of it at r sub 0, that tells us 01:01:21.800 --> 01:01:24.620 what this quantity is when we drop it. 01:01:24.620 --> 01:01:27.490 It's minus GM over r sub 0. 01:01:27.490 --> 01:01:30.120 This piece vanishes if there is no initial velocity. 01:01:30.120 --> 01:01:32.036 And that means it will always have that value. 01:01:35.390 --> 01:01:41.750 And knowing that, we can write dr d tau 01:01:41.750 --> 01:01:46.000 is equal to-- just solving for that-- 01:01:46.000 --> 01:02:03.980 minus the square root of 2GM times r0 minus r over r r0 01:02:03.980 --> 01:02:08.100 I've collected two terms and put them over a common denominator 01:02:08.100 --> 01:02:09.480 and added them. 01:02:09.480 --> 01:02:12.440 So this is not quite as obvious as it might be. 01:02:12.440 --> 01:02:15.004 But this is just the statement that that quantity 01:02:15.004 --> 01:02:16.920 has the same value as it did when you started. 01:02:23.700 --> 01:02:27.550 Now this can be further integrated. 01:02:27.550 --> 01:02:35.840 We can write it as dr over-- bringing 01:02:35.840 --> 01:02:39.420 all this to the other side-- is equal to d tau. 01:02:39.420 --> 01:02:40.950 And then integrate both sides. 01:02:45.480 --> 01:02:47.850 Notice when I bring this to the other side 01:02:47.850 --> 01:02:50.150 and bring the d tau to the right., everything 01:02:50.150 --> 01:02:52.190 on the left-hand side now only depends on r. 01:02:52.190 --> 01:02:56.080 So this is just an explicit integral over r that we can do. 01:03:16.160 --> 01:03:20.120 And I will just tell you that when the integral is done 01:03:20.120 --> 01:03:24.550 we get a formula for tau as a function of r. 01:03:32.529 --> 01:03:35.570 And it's equal to the square root 01:03:35.570 --> 01:03:46.920 of r sub 0 over 2GM times r0 times 01:03:46.920 --> 01:03:55.900 the inverse tangent of the square root of r0 01:03:55.900 --> 01:04:08.070 minus r over r plus the square root of r times r0 minus r. 01:04:14.500 --> 01:04:18.780 So when r equals r0, this gives us 0, and that's what we want. 01:04:18.780 --> 01:04:24.470 When we start we're at r0, or time 0, or proper time 0. 01:04:24.470 --> 01:04:29.280 And then as r gets smaller, as it falls in, time grows. 01:04:29.280 --> 01:04:31.920 And this gives us the time as a function of r. 01:04:31.920 --> 01:04:33.920 We might prefer to have r as a function of time, 01:04:33.920 --> 01:04:37.440 but that formula can't really be inverted analytically. 01:04:37.440 --> 01:04:38.710 So that's the best we can do. 01:04:43.680 --> 01:04:46.000 Now one thing that you notice from this 01:04:46.000 --> 01:04:49.100 is that nothing special happens as r 01:04:49.100 --> 01:04:51.907 decreases all the way to 0. 01:04:51.907 --> 01:04:53.490 Even when you plug in r equals 0 here, 01:04:53.490 --> 01:04:55.270 you just get some finite number. 01:04:55.270 --> 01:04:58.270 So in a finite amount of time, the observer 01:04:58.270 --> 01:05:00.790 would find himself falling through the Schwarzchild 01:05:00.790 --> 01:05:04.830 horizon and all the way to r equals 0. 01:05:04.830 --> 01:05:07.880 I didn't mention it but r equals 0 is a true singularity. 01:05:07.880 --> 01:05:11.340 Our metric is also singular when r equals 0. 01:05:11.340 --> 01:05:14.820 These quantities all become infinite. 01:05:14.820 --> 01:05:19.610 And physically what would happen is 01:05:19.610 --> 01:05:24.530 that, as the object falling in approaches r equals 0, 01:05:24.530 --> 01:05:26.732 the tidal forces, that is the difference 01:05:26.732 --> 01:05:29.190 in the gravitational force on one part of the object verses 01:05:29.190 --> 01:05:32.810 another, will get stronger and stronger. 01:05:32.810 --> 01:05:35.310 And objects will just be ripped apart. 01:05:35.310 --> 01:05:40.050 And the ripping apart occurs as being spaghetti-ized, that is, 01:05:40.050 --> 01:05:42.054 the force on the front gets to be very strong 01:05:42.054 --> 01:05:43.470 compared to the force on the back. 01:05:43.470 --> 01:05:46.011 So I'll just get stretched out along the direction of motion. 01:05:53.500 --> 01:05:56.920 Now the curious thing is what this 01:05:56.920 --> 01:05:59.240 looks like if we think of it not as a function 01:05:59.240 --> 01:06:01.140 of the proper time measured by the wrist 01:06:01.140 --> 01:06:05.950 watch of the object falling in but rather, 01:06:05.950 --> 01:06:10.070 we could try to describe it in terms of our external time 01:06:10.070 --> 01:06:11.010 variable. 01:06:11.010 --> 01:06:15.800 The variable t that appears in the Schwarzchild metric. 01:06:15.800 --> 01:06:18.470 And to do that, to make the conversion, 01:06:18.470 --> 01:06:23.630 we want to calculate what the dr dt is, instead of dr d tau. 01:06:23.630 --> 01:06:26.450 Like maybe an analogous formula, in terms of t. 01:06:29.690 --> 01:06:33.540 And to get that, we use simply chain rule here. 01:06:33.540 --> 01:06:38.080 dr dt is equal to dr d tau-- which we've already 01:06:38.080 --> 01:06:45.540 calculated-- times d tau dt. 01:06:45.540 --> 01:06:49.870 And d tau dt is 1 over dt d tau. 01:06:49.870 --> 01:06:52.330 If you just have two variables that depend on each other. 01:06:52.330 --> 01:06:56.490 The derivatives are just the inverse of each other. 01:06:56.490 --> 01:07:00.220 So this could be written as dr d tau-- which 01:07:00.220 --> 01:07:04.430 we've calculated-- divided by dt d tau. 01:07:07.410 --> 01:07:10.900 And dt d tau we've really already calculated as well, 01:07:10.900 --> 01:07:13.740 because it's just given by this formula here. 01:07:26.530 --> 01:07:29.470 So we could write out what that is and figure out 01:07:29.470 --> 01:07:33.550 how it's going to behave as the object approaches 01:07:33.550 --> 01:07:34.645 the Schwarzchild radius. 01:07:54.430 --> 01:08:00.080 So it becomes dr dt is equal to, I'll 01:08:00.080 --> 01:08:07.520 just write the numerator as dr d tau given by that expression. 01:08:07.520 --> 01:08:10.180 But what's behaving in a more peculiar way 01:08:10.180 --> 01:08:17.890 is the denominator, which is h inverse of r plus c 01:08:17.890 --> 01:08:29.820 to the minus 2, h to the minus 2 of r times dr d tau squared. 01:08:35.149 --> 01:08:39.160 So now we want to look at this function h inverse of r. 01:08:39.160 --> 01:08:41.145 And this just means 1/h of r. 01:08:41.145 --> 01:08:44.460 It doesn't mean functional inverse. 01:08:44.460 --> 01:08:51.773 That is just equal to r over r minus r Schwarzchild. 01:08:51.773 --> 01:08:53.439 And we're going to be interested in what 01:08:53.439 --> 01:08:56.255 happens when r gets to be very near r Schwarzchild, 01:08:56.255 --> 01:08:58.950 because that's where the interesting things happen, 01:08:58.950 --> 01:09:02.148 as you're approaching the Schwarzchild horizon. 01:09:02.148 --> 01:09:04.189 And that means that the behavior of the numerator 01:09:04.189 --> 01:09:04.890 won't be important. 01:09:04.890 --> 01:09:06.264 The denominator will be going up, 01:09:06.264 --> 01:09:08.399 and that's what will control everything. 01:09:08.399 --> 01:09:11.899 So we can approximate this as just 01:09:11.899 --> 01:09:16.510 r Schwarzchild over r minus r Schwarzchild. 01:09:16.510 --> 01:09:18.540 And this is for r near r Schwarzchild. 01:09:22.653 --> 01:09:24.444 We've replaced the numerator by a constant. 01:09:28.520 --> 01:09:30.069 And then if we look at this formula, 01:09:30.069 --> 01:09:34.319 this is going to blow up as we approach the horizon. 01:09:34.319 --> 01:09:35.970 This is the square of that quantity. 01:09:35.970 --> 01:09:40.040 It will blow up faster than the first power of that quantity. 01:09:40.040 --> 01:09:41.939 And therefore, this will dominate, 01:09:41.939 --> 01:09:44.950 the denominator of the expression. 01:09:44.950 --> 01:09:47.130 We can ignore this. 01:09:47.130 --> 01:09:50.840 When this dominates, the dr d tau pieces cancel. 01:09:50.840 --> 01:09:51.520 So that's nice. 01:09:51.520 --> 01:09:54.770 We don't even need to think about what the dr d tau is. 01:09:54.770 --> 01:10:06.890 And what we get near the horizon is simply 01:10:06.890 --> 01:10:13.564 a factor of c times r minus r Schwarzchild over r 01:10:13.564 --> 01:10:14.105 Schwarzchild. 01:10:18.290 --> 01:10:20.770 It's basically just h. 01:10:20.770 --> 01:10:22.810 This becomes upstairs with a plus sign. 01:10:22.810 --> 01:10:25.470 And the square root turns it into h instead of h squared. 01:10:29.350 --> 01:10:30.700 So this is the inverse of that. 01:10:37.220 --> 01:10:40.290 OK, now if we try to play the same game here 01:10:40.290 --> 01:10:44.780 as we did here, to determine what our time variable behaves 01:10:44.780 --> 01:10:48.180 as a function of r, instead of the proper time variable 01:10:48.180 --> 01:10:59.340 tau, what we find is that t of r-- this is for r near r 01:10:59.340 --> 01:11:07.830 Schwarzchild-- is about equal to minus 01:11:07.830 --> 01:11:18.895 r sub s over c times the integral up to r 01:11:18.895 --> 01:11:25.665 of dr prime over r prime minus rs. 01:11:37.080 --> 01:11:37.915 This is dr dt. 01:11:42.126 --> 01:11:43.990 Yeah, this was dr dt from the beginning. 01:11:43.990 --> 01:11:46.200 I forgot to write the r somehow. 01:11:46.200 --> 01:11:47.659 AUDIANCE: Doesn't that [INAUDIBLE]? 01:11:47.659 --> 01:11:50.283 PROFESSOR: Yeah, I didn't write the lower limit of integration. 01:11:50.283 --> 01:11:51.800 I was about to comment on that. 01:11:51.800 --> 01:11:54.700 The integrand that we're writing is only a good approximation 01:11:54.700 --> 01:11:57.910 whenever we're near r. 01:11:57.910 --> 01:12:01.030 So whatever happens near the lower limit of integration, 01:12:01.030 --> 01:12:02.830 we just haven't done accurately. 01:12:02.830 --> 01:12:06.010 So I'm going to just not write a lower limit of integration 01:12:06.010 --> 01:12:08.020 here, meaning that we're interested only 01:12:08.020 --> 01:12:12.586 in what happens as the upper limit of integration r 01:12:12.586 --> 01:12:13.960 becomes very near r Schwarzchild. 01:12:13.960 --> 01:12:15.626 And everything will be dominated by what 01:12:15.626 --> 01:12:18.067 happens near the upper limit of integration. 01:12:18.067 --> 01:12:19.817 AUDIANCE: So would you just integrate over 01:12:19.817 --> 01:12:21.380 on [INAUDIBLE] for that? 01:12:21.380 --> 01:12:22.963 PROFESSOR: That's right, that's right. 01:12:22.963 --> 01:12:26.640 We just integrated over a small region near, r Schwarzchild. 01:12:53.670 --> 01:12:56.050 Nu r, which is also about equal to r Schwarzchild. 01:12:59.530 --> 01:13:02.280 And the point is, that this diverges logarithmically 01:13:02.280 --> 01:13:05.680 as r approaches r Schwarzchild. 01:13:05.680 --> 01:13:14.920 So it behaves approximately as minus r Schwarzchild 01:13:14.920 --> 01:13:20.640 over c times the logarithm of r minus r Schwarzchild. 01:13:25.960 --> 01:13:28.220 So as r approaches r Schwarzchild, 01:13:28.220 --> 01:13:30.820 this quantity that's the argument of the logarithm 01:13:30.820 --> 01:13:32.470 gets closer and closer to 0. 01:13:32.470 --> 01:13:36.294 It gets smaller and smaller approaching 0. 01:13:36.294 --> 01:13:37.960 But the logarithm of a very small number 01:13:37.960 --> 01:13:40.672 is a negative number, a large negative number. 01:13:40.672 --> 01:13:42.130 And then there's a minus sign here. 01:13:42.130 --> 01:13:44.680 You get a large positive number and it diverges. 01:13:44.680 --> 01:13:48.020 As r approaches r Schwarzchild the time variable 01:13:48.020 --> 01:13:50.520 approaches infinity. 01:13:50.520 --> 01:13:53.590 And that means that at no finite time 01:13:53.590 --> 01:13:57.665 does the object ever reach the Schwarzchild horizon. 01:13:57.665 --> 01:14:00.510 But as seen from the outside, it takes an infinite amount 01:14:00.510 --> 01:14:04.100 of time for the object to reach the Schwarzchild horizon. 01:14:04.100 --> 01:14:07.600 As time gets larger and larger, the object gets and closer 01:14:07.600 --> 01:14:11.200 to the Schwarzchild horizon, asymptotically approaching it 01:14:11.200 --> 01:14:12.809 but never reaching it. 01:14:12.809 --> 01:14:14.350 So this, of course, is very peculiar, 01:14:14.350 --> 01:14:17.620 because from the point of view of the person 01:14:17.620 --> 01:14:19.500 falling into the black hole, all this 01:14:19.500 --> 01:14:21.984 just happens in a finite amount of time and is over with. 01:14:21.984 --> 01:14:23.400 From the outside, it looks like it 01:14:23.400 --> 01:14:25.480 takes an infinite amount of time. 01:14:25.480 --> 01:14:28.920 And weird things like this can happen because of the fact that 01:14:28.920 --> 01:14:33.250 in general relativity time is a locally measured variable. 01:14:33.250 --> 01:14:35.490 You measure your time, I measure my time. 01:14:35.490 --> 01:14:36.810 They don't have to agree. 01:14:36.810 --> 01:14:40.510 And in this case, they can disagree by an infinite amount, 01:14:40.510 --> 01:14:44.460 which is rather bizarre, but that's what happens. 01:14:44.460 --> 01:14:47.610 So according to classical general relativity, 01:14:47.610 --> 01:14:50.200 when an object falls into a black hole, 01:14:50.200 --> 01:14:53.020 from the point of view of the object nothing 01:14:53.020 --> 01:14:56.900 special would happen as that object crossed the Schwarzchild 01:14:56.900 --> 01:14:59.020 horizon. 01:14:59.020 --> 01:15:01.020 Everybody believed that that was really the case 01:15:01.020 --> 01:15:02.570 until maybe a couple years ago. 01:15:02.570 --> 01:15:04.696 Now it's controversial, actually. 01:15:04.696 --> 01:15:06.070 At the classical level, everybody 01:15:06.070 --> 01:15:07.195 believes that's still true. 01:15:07.195 --> 01:15:08.880 I mean, classical general relativity 01:15:08.880 --> 01:15:10.470 says that an object can fall through the Schwarzchild 01:15:10.470 --> 01:15:12.180 horizon and then nothing happens. 01:15:12.180 --> 01:15:14.820 It's not really a singularity. 01:15:14.820 --> 01:15:18.140 But the issue is that when one incorporates, or attempts 01:15:18.140 --> 01:15:20.700 to incorporate, the effects of quantum theory, which nobody 01:15:20.700 --> 01:15:23.660 really knows how to do in a totally reliable way, 01:15:23.660 --> 01:15:27.330 then there are indications that there's 01:15:27.330 --> 01:15:30.600 something dramatic happening at the Schwarzchild horizon. 01:15:30.600 --> 01:15:33.100 The phrase that's often used for what people think 01:15:33.100 --> 01:15:37.360 might be happening at the horizon is the word firewall. 01:15:37.360 --> 01:15:40.240 So whether or not there is a firewall at the horizon, 01:15:40.240 --> 01:15:42.932 is not settled at this point. 01:15:42.932 --> 01:15:44.890 Certainly, though, classical general relativity 01:15:44.890 --> 01:15:46.750 does not predict the firewall. 01:15:46.750 --> 01:15:49.410 If it exists, all the arguments that say it might exist 01:15:49.410 --> 01:15:51.450 are based on the quantum physics of black holes, 01:15:51.450 --> 01:15:54.655 and black hole evaporation, and things like that. 01:15:54.655 --> 01:15:56.030 As you know quantum mechanically, 01:15:56.030 --> 01:15:57.740 the black holes are not stable, either, 01:15:57.740 --> 01:16:01.210 if they evaporate-- as was derived 01:16:01.210 --> 01:16:05.479 by Stephen Hawking in, I think, 1974. 01:16:05.479 --> 01:16:07.020 But that's strictly a quantum effect. 01:16:07.020 --> 01:16:11.005 It would go to 0 as h bar goes to 0, and, at the moment, 01:16:11.005 --> 01:16:13.580 we're only talking about classical general relativity. 01:16:13.580 --> 01:16:16.220 So the black hole that we're describing is perfectly stable. 01:16:16.220 --> 01:16:18.480 And nothing happens if you fall through the horizon. 01:16:18.480 --> 01:16:20.120 Except from the outside, it looks 01:16:20.120 --> 01:16:22.203 like it would take an infinite amount of time just 01:16:22.203 --> 01:16:24.087 to reach the horizon. 01:16:24.087 --> 01:16:24.920 So we'll stop there. 01:16:24.920 --> 01:16:26.420 I guess I'm not going to get to talk 01:16:26.420 --> 01:16:32.360 about the energy associated with radiation. 01:16:32.360 --> 01:16:35.380 But we'll get to that on Thursday. 01:16:35.380 --> 01:16:37.900 So see you folks on Thursday.