WEBVTT 00:00:00.000 --> 00:00:01.497 [SQUEAKING] 00:00:01.497 --> 00:00:03.493 [RUSTLING] 00:00:03.493 --> 00:00:05.988 [CLICKING] 00:00:10.490 --> 00:00:13.410 SCOTT HUGHES: So in this final lecture, 00:00:13.410 --> 00:00:16.910 I want to think a bit sort of with an eye 00:00:16.910 --> 00:00:20.960 towards thinking about how one might actually 00:00:20.960 --> 00:00:24.300 make measurements that prove the nature of the black hole 00:00:24.300 --> 00:00:26.740 spacetime that was discussed the previous lecture. 00:00:26.740 --> 00:00:31.166 I'm going to discuss motion in a black hole spacetime. 00:00:41.880 --> 00:00:45.620 We touched on this a little bit in the previous lecture, 00:00:45.620 --> 00:00:50.870 where we discussed the motion of radial light rays, OK? 00:00:50.870 --> 00:00:56.840 We, in fact, used radial light rays as a critical tool 00:00:56.840 --> 00:00:59.690 for describing the properties of the spacetime. 00:00:59.690 --> 00:01:01.610 We use that to help us understand the location 00:01:01.610 --> 00:01:02.819 of events horizons. 00:01:02.819 --> 00:01:04.819 But I want to think a little bit more generally. 00:01:04.819 --> 00:01:06.194 What might it look like if I have 00:01:06.194 --> 00:01:08.475 material orbiting in the vicinity of one 00:01:08.475 --> 00:01:09.350 of these black holes? 00:01:09.350 --> 00:01:10.308 What if it's not light? 00:01:10.308 --> 00:01:12.420 What if it's made out of matter? 00:01:12.420 --> 00:01:14.330 And so what this is going to boil down to 00:01:14.330 --> 00:01:18.500 is understanding the behavior of geodesics in a black hole 00:01:18.500 --> 00:01:21.300 spacetime. 00:01:21.300 --> 00:01:28.330 And the naive approach to doing this is not wrong, but naive. 00:01:28.330 --> 00:01:32.820 What you do is you would just take the spacetime-- 00:01:32.820 --> 00:01:35.670 take your Schwarzschild or take your Kerr spacetime-- 00:01:35.670 --> 00:01:41.280 and turn a very large crank, grind out all of the connection 00:01:41.280 --> 00:01:55.170 coefficients, evaluate the geodesic equation, 00:01:55.170 --> 00:01:57.360 integrate it up. 00:01:57.360 --> 00:02:03.080 Solve for the geodesics. 00:02:03.080 --> 00:02:05.390 Boom, you got yourself your motion, OK? 00:02:05.390 --> 00:02:07.170 And that is absolutely correct. 00:02:07.170 --> 00:02:08.960 You can do that using your Kerr space time 00:02:08.960 --> 00:02:11.810 or your Schwarzschild spacetime. 00:02:11.810 --> 00:02:16.030 In fact, if you do that for-- 00:02:16.030 --> 00:02:18.477 you can get the connection coefficients describing this. 00:02:18.477 --> 00:02:20.060 Those are relatively easy to work out. 00:02:20.060 --> 00:02:23.510 I think they are listed in Carroll, in equation 5.53, 00:02:23.510 --> 00:02:24.992 according to my notes. 00:02:24.992 --> 00:02:26.450 That may just be for Schwarzschild. 00:02:26.450 --> 00:02:31.460 But at any rate, they're all listed there, and have a blast. 00:02:31.460 --> 00:02:36.470 This approach is not wrong, but-- 00:02:36.470 --> 00:02:38.152 my notes say, but it is not useful. 00:02:38.152 --> 00:02:39.110 That's not really true. 00:02:39.110 --> 00:02:42.860 I'll just say that there is a more useful approach to this. 00:02:42.860 --> 00:02:56.930 A more fruitful approach is to exploit the fact that these are 00:02:56.930 --> 00:02:58.690 highly symmetric spacetimes. 00:03:04.110 --> 00:03:10.030 Exploit the symmetries and the Killing 00:03:10.030 --> 00:03:22.430 vectors, and see how they can be used 00:03:22.430 --> 00:03:25.337 to reduce the number of degrees of freedom 00:03:25.337 --> 00:03:26.420 that you need to describe. 00:03:42.850 --> 00:03:44.830 In this lecture, I'm going to go through this 00:03:44.830 --> 00:03:47.980 in quite a bit of detail for Schwarzschild. 00:03:47.980 --> 00:03:51.490 The concepts that I'm going to apply work 00:03:51.490 --> 00:03:55.375 for Kerr and for Kerr-Newman as well, OK? 00:03:55.375 --> 00:03:57.500 Schwarzschild is just a little easier to work with. 00:03:57.500 --> 00:04:00.802 It's something that I can fit into a single lecture. 00:04:00.802 --> 00:04:03.260 In particular, one of the nice things about Schwarzschild-- 00:04:03.260 --> 00:04:05.485 so let's go ahead and write down that spacetime. 00:04:20.109 --> 00:04:22.530 So one of the nice things about Schwarzschild 00:04:22.530 --> 00:04:24.660 is it is spherically symmetric. 00:04:31.390 --> 00:04:37.560 This means I can always rotate coordinates such that-- 00:04:37.560 --> 00:04:39.060 well, when me think about it-- let's 00:04:39.060 --> 00:04:40.490 just back up for a second. 00:04:40.490 --> 00:04:44.540 Imagine that I have some kind of a body orbiting a Schwarzschild 00:04:44.540 --> 00:04:46.280 black hole. 00:04:46.280 --> 00:04:49.770 Spherical symmetry tells me that there 00:04:49.770 --> 00:04:53.820 must be some notion of a conserved angular momentum such 00:04:53.820 --> 00:04:57.348 that that orbit always lies within a given plane. 00:04:57.348 --> 00:04:58.140 Put it another way. 00:04:58.140 --> 00:05:02.650 Because it is fairly symmetric, there cannot exist a torque. 00:05:06.260 --> 00:05:12.560 The black hole cannot exert a torque that changes 00:05:12.560 --> 00:05:23.200 the orientation of the orbital plane. 00:05:29.220 --> 00:05:31.360 In particular, because it is fairly symmetric, 00:05:31.360 --> 00:05:34.720 there is no unique notion of an equator to this object. 00:05:34.720 --> 00:05:39.730 And so you might as well define any orbits to live in the theta 00:05:39.730 --> 00:05:41.230 equals pi over 2 plane. 00:05:41.230 --> 00:05:56.420 You can always rotate your coordinates 00:05:56.420 --> 00:06:06.880 to put any orbit in the theta equals pi over 2 plane. 00:06:06.880 --> 00:06:09.850 It's actually a pretty simple exercise. 00:06:09.850 --> 00:06:12.340 I won't do this, but if you start 00:06:12.340 --> 00:06:15.910 with an orbit that is in the theta equals pi over 2 plane, 00:06:15.910 --> 00:06:21.730 and it is moving such that its initial velocity would keep it 00:06:21.730 --> 00:06:23.710 in a theta equals pi over 2 plane, 00:06:23.710 --> 00:06:27.880 it's a very simple exercise using the geodesic equation 00:06:27.880 --> 00:06:31.540 to show that it will always be in the theta 00:06:31.540 --> 00:06:33.280 equals pi over 2 plane. 00:06:33.280 --> 00:06:35.260 So spherical symmetry says, you know what, 00:06:35.260 --> 00:06:38.410 let's just forget about the theta degree of freedom. 00:06:38.410 --> 00:06:40.840 I can always define my coordinates in such a way 00:06:40.840 --> 00:06:44.740 that it lives in the theta equals pi over 2 plane. 00:06:44.740 --> 00:06:48.400 Boom, I have reduced my motion from, in general, 00:06:48.400 --> 00:06:52.067 being three spatial dimensions to two spatial dimensions. 00:06:52.067 --> 00:06:53.650 That's another one of the reasons why, 00:06:53.650 --> 00:06:55.630 for pedagogical purposes, it's nice to start 00:06:55.630 --> 00:06:57.060 with Schwarzschild. 00:06:57.060 --> 00:07:00.180 For Kerr, this is not the case, OK? 00:07:00.180 --> 00:07:02.240 Kerr is a little bit more complicated. 00:07:02.240 --> 00:07:06.610 You have to treat the theta motion separately. 00:07:06.610 --> 00:07:08.860 It's not strictly symmetric, and so you can't do that. 00:07:10.992 --> 00:07:12.700 I have actually spent a tremendous amount 00:07:12.700 --> 00:07:14.940 of my career studying the orbits of objects 00:07:14.940 --> 00:07:15.940 around Kerr black holes. 00:07:15.940 --> 00:07:19.570 And I do have to say that the additional complications that 00:07:19.570 --> 00:07:21.730 arise from this lack of sphericity, 00:07:21.730 --> 00:07:23.020 they're really beautiful, OK? 00:07:23.020 --> 00:07:26.940 There's an amazing amount of fun stuff you can do with it. 00:07:26.940 --> 00:07:30.630 You know, there's a reason why I just 00:07:30.630 --> 00:07:32.380 keep coming back to this research problem, 00:07:32.380 --> 00:07:36.820 and part of it is it's just bloody fun. 00:07:36.820 --> 00:07:39.460 But if you're teaching this stuff for the first time, 00:07:39.460 --> 00:07:41.470 it's not where you want to begin. 00:07:41.470 --> 00:07:43.510 All right, so Schwarzschild allows 00:07:43.510 --> 00:07:46.780 us to reduce it from a three-dimensional problem 00:07:46.780 --> 00:07:48.710 to a two-dimensional problem. 00:07:48.710 --> 00:07:53.230 And Schwarzschild also has two Killing vectors. 00:07:53.230 --> 00:07:57.940 The time derivative of every metric component is equal to 0. 00:07:57.940 --> 00:08:01.015 That means p downstairs t is constant. 00:08:09.510 --> 00:08:11.910 So there is a timelike Killing vector. 00:08:11.910 --> 00:08:15.810 So p downstairs t is constant every level along the orbit. 00:08:15.810 --> 00:08:18.780 It means there exists a timelike Killing vector. 00:08:18.780 --> 00:08:22.780 And so what we do is we associate this 00:08:22.780 --> 00:08:25.690 with the energy of the orbit. 00:08:25.690 --> 00:08:28.780 We call this up to a minus sign. 00:08:28.780 --> 00:08:30.970 And we choose that minus sign because if we imagine 00:08:30.970 --> 00:08:32.740 orbits have very, very large radius, 00:08:32.740 --> 00:08:35.804 the spacetime is nearly flat, and we 00:08:35.804 --> 00:08:37.929 want to sort of clear out the minus sign associated 00:08:37.929 --> 00:08:39.789 with lowering our index here. 00:08:39.789 --> 00:08:42.159 We're going to call that constant negative 00:08:42.159 --> 00:08:43.590 of the energy of the orbit. 00:08:51.880 --> 00:08:58.680 The spacetime is also independent of the angle phi. 00:08:58.680 --> 00:09:07.560 And so p sub phi is a constant. 00:09:07.560 --> 00:09:10.650 This means that the spacetime has an axial Killing 00:09:10.650 --> 00:09:13.975 vector, something associated with motions around a symmetry 00:09:13.975 --> 00:09:14.475 axis. 00:09:16.990 --> 00:09:18.520 So I'm going to call this L-- well, 00:09:18.520 --> 00:09:20.850 actually, I will tend to call it L sub z. 00:09:20.850 --> 00:09:23.470 You can kind of think of this as, after I 00:09:23.470 --> 00:09:26.410 have put everything in the theta equals pi over 2 plane, 00:09:26.410 --> 00:09:32.080 this is like an angular momentum on the-- 00:09:32.080 --> 00:09:34.920 angular momentum parallel to the axis normal to that plane, 00:09:34.920 --> 00:09:37.420 and I call that the z-axis. 00:09:37.420 --> 00:09:40.180 It's worth noting that all three of these things 00:09:40.180 --> 00:09:43.450 are also true for Reissner-Nordstrom black holes. 00:09:43.450 --> 00:09:48.620 This one is not true for Kerr, but these two are, OK? 00:09:48.620 --> 00:09:54.130 So although the details change, many of the concepts 00:09:54.130 --> 00:09:56.920 will carry over when you look at different, more complicated 00:09:56.920 --> 00:09:58.620 classes of black holes. 00:09:58.620 --> 00:10:00.370 Let's now think about the forward momentum 00:10:00.370 --> 00:10:02.740 of a body moving in a Schwarzschild spacetime. 00:10:16.470 --> 00:10:22.820 So let's say it's got a rest mass m, 00:10:22.820 --> 00:10:25.890 then it will have three components. 00:10:25.890 --> 00:10:27.770 Remember, I have put this thing in a plane 00:10:27.770 --> 00:10:29.090 where there is no theta motion. 00:10:32.130 --> 00:10:35.530 It will have three theta motions describing its motion 00:10:35.530 --> 00:10:37.030 with respect to the time coordinate, 00:10:37.030 --> 00:10:42.070 the radial coordinate, and the actual coordinate phi, OK? 00:10:42.070 --> 00:10:43.690 So before I do anything with this, 00:10:43.690 --> 00:10:47.410 let's take advantage of the fact that we have these quantities 00:10:47.410 --> 00:10:49.790 that are constants. 00:10:49.790 --> 00:10:52.870 So using the fact that p downstairs mu 00:10:52.870 --> 00:10:55.240 is what I get when I hit this guy-- 00:10:55.240 --> 00:10:59.210 that be a mu, pardon me. 00:10:59.210 --> 00:11:03.680 This is what I get when I hit this guy with the metric. 00:11:03.680 --> 00:11:08.245 I can write out a p sub t and p sub phi. 00:11:08.245 --> 00:11:10.010 So let's do that on another board. 00:11:10.010 --> 00:11:12.050 Well, let's do a p sub t first. 00:11:12.050 --> 00:11:20.932 So p sub t is going to be gtt, p sub-- 00:11:20.932 --> 00:11:21.432 whoops. 00:11:28.730 --> 00:11:32.430 OK, this is minus. 00:11:32.430 --> 00:11:34.350 There's an m from that-- 00:11:34.350 --> 00:11:35.860 minus sign from my metric. 00:11:45.560 --> 00:11:52.352 And this whole thing I define as the negative energy, e. 00:12:00.240 --> 00:12:03.210 OK, so that combination-- 00:12:03.210 --> 00:12:05.500 we'll do something at the rest mass in just a moment. 00:12:05.500 --> 00:12:07.930 But you should basically look at this 00:12:07.930 --> 00:12:12.370 and saying that dt d tau, which tells me 00:12:12.370 --> 00:12:16.060 about how the body moves with respect to the Schwarzschild 00:12:16.060 --> 00:12:18.000 time coordinate-- 00:12:18.000 --> 00:12:21.240 that complement of the four velocity times 1 minus 00:12:21.240 --> 00:12:25.390 2gm over r is a constant. 00:12:25.390 --> 00:12:27.730 Let's look at p downstairs phi. 00:12:30.472 --> 00:12:36.180 So this is m r squared sine squared theta. 00:12:36.180 --> 00:12:44.870 Ah, we've chosen our orbital plane, 1 times d phi d tau. 00:12:44.870 --> 00:12:52.560 This is equal to [INAUDIBLE] momentum L sub z. 00:12:52.560 --> 00:12:55.260 In my notes, I might flip around a little bit between L sub z 00:12:55.260 --> 00:12:57.910 and L. 00:12:57.910 --> 00:13:02.900 So let's massage these a little bit more. 00:13:02.900 --> 00:13:05.790 So I can take these two expressions 00:13:05.790 --> 00:13:09.980 and I can use them to write dt d tau 00:13:09.980 --> 00:13:17.990 and d phi d tau in terms of these conserved quantities. 00:13:17.990 --> 00:13:20.870 So dt d tau is equal to-- 00:13:20.870 --> 00:13:33.725 I'm going to call it e hat divided by 1 minus 2gm over r. 00:13:33.725 --> 00:13:40.930 d phi d tau is going to be Lz hat over our squared. 00:13:40.930 --> 00:13:44.110 And so these quantities with a hat 00:13:44.110 --> 00:13:52.920 are just the conserved values normalized to the rest mass, 00:13:52.920 --> 00:13:53.868 OK? 00:13:53.868 --> 00:13:55.660 They all are proportional to the rest mass, 00:13:55.660 --> 00:13:58.830 but it's actually if you want to know what the-- 00:13:58.830 --> 00:14:01.330 this is essentially telling me about how clocks on the orbit 00:14:01.330 --> 00:14:04.750 tick relative to clocks that are infinitely far away. 00:14:04.750 --> 00:14:08.840 And that can't depend on the mass of the orbiting object. 00:14:08.840 --> 00:14:11.080 This is telling me about how this small body is 00:14:11.080 --> 00:14:14.458 moving according to the clock of the orbiting observer. 00:14:14.458 --> 00:14:16.750 And again, that can't depend on the mass of the object. 00:14:19.400 --> 00:14:21.260 OK, so that's kind of cool. 00:14:21.260 --> 00:14:24.430 So I've now managed to relate three-- 00:14:24.430 --> 00:14:27.130 excuse me, two of the three components 00:14:27.130 --> 00:14:32.020 of the forward momentum to functions of r and quantities 00:14:32.020 --> 00:14:34.330 that are known to be constant. 00:14:34.330 --> 00:14:36.160 That's good. 00:14:36.160 --> 00:14:38.170 We have one more overall constraint. 00:14:40.760 --> 00:14:53.960 We know that if I take p dot p, I get negative of the rest mass 00:14:53.960 --> 00:14:54.460 squared. 00:14:58.150 --> 00:15:01.360 So this, when I write it out-- 00:15:32.190 --> 00:15:37.850 OK, notice every single term is proportional to m squared. 00:15:37.850 --> 00:15:40.180 So I can divide that out. 00:15:40.180 --> 00:15:43.930 I can insert dt d tau. 00:15:43.930 --> 00:15:47.560 I can replace for this e divided by this guy. 00:15:47.560 --> 00:15:52.420 I can replace d phi d tau by my Lz hat divided by r squared. 00:15:59.080 --> 00:16:09.810 Doing so, manipulating a little bit, I get-- 00:16:28.842 --> 00:16:31.710 we get something like this. 00:16:31.710 --> 00:16:35.190 And let me rearrange this a tiny bit. 00:17:13.099 --> 00:17:16.800 All right, I have managed to reduce this 00:17:16.800 --> 00:17:20.310 to a one-dimensional problem, OK? 00:17:20.310 --> 00:17:25.380 So going from that line over to there, basically all I did 00:17:25.380 --> 00:17:31.560 was insert the relationship between dt d tau and E, d phi 00:17:31.560 --> 00:17:35.550 d tau and L, cleared out some overall factors of things 00:17:35.550 --> 00:17:39.610 like 1 minus 2GM/r, manipulate, manipulate, manipulate, 00:17:39.610 --> 00:17:42.180 and what you finally get is this lovely equation here 00:17:42.180 --> 00:17:47.460 that tells you how the radial velocity, the radial velocity 00:17:47.460 --> 00:17:50.640 with respect to proper time, how it 00:17:50.640 --> 00:17:57.420 depends as a function of r given the energy and the angular 00:17:57.420 --> 00:17:58.840 momentum. 00:17:58.840 --> 00:18:03.990 I have written it in this form because the problem 00:18:03.990 --> 00:18:10.800 is very strikingly reminiscent to the Newtonian problem 00:18:10.800 --> 00:18:15.990 of understanding the motion of a particle in a 1/r potential, 00:18:15.990 --> 00:18:19.950 which we often describe as having an effective potential 00:18:19.950 --> 00:18:22.230 that has a gravitational term-- 00:18:22.230 --> 00:18:23.850 Newton's gravity, or if you're doing 00:18:23.850 --> 00:18:26.280 things like quantum mechanics, the Coulomb potential, 00:18:26.280 --> 00:18:29.010 and a Coulomb barrier associated with the angular 00:18:29.010 --> 00:18:32.340 momentum of an orbit. 00:18:32.340 --> 00:18:35.430 So one way to approach what we've got now, 00:18:35.430 --> 00:18:40.750 one thing that we could do, is essentially just 00:18:40.750 --> 00:18:48.410 pick your energy and your Lz-- 00:18:48.410 --> 00:18:50.870 pick your energy and your angular momentum-- 00:18:50.870 --> 00:18:52.265 pick an initial position-- 00:19:00.935 --> 00:19:05.940 let's imagine you synchronize the clocks at t equals 0-- 00:19:05.940 --> 00:19:07.876 and then just integrate. 00:19:17.810 --> 00:19:20.780 You've got your dr d tau given here. 00:19:20.780 --> 00:19:27.500 Don't forget, you also have d phi d tau and dt d tau related 00:19:27.500 --> 00:19:29.880 to the energy and your angular momentum, like so. 00:19:29.880 --> 00:19:30.380 Boom. 00:19:30.380 --> 00:19:31.190 It's a closed system. 00:19:31.190 --> 00:19:33.170 You can always do just sort of a little numerical integration 00:19:33.170 --> 00:19:33.670 of this. 00:19:37.780 --> 00:19:39.340 In a certain sense, this completely 00:19:39.340 --> 00:19:40.760 specifies the problem. 00:19:40.760 --> 00:19:43.660 But there's so much more we can do. 00:19:43.660 --> 00:19:46.870 In particular, what we see is that all 00:19:46.870 --> 00:19:50.530 of the interesting behavior associated with this orbit 00:19:50.530 --> 00:19:54.354 is bound up in this function V effective. 00:20:09.670 --> 00:20:13.000 So something that's really useful for us to do 00:20:13.000 --> 00:20:17.440 is to take a look at what this V effective looks like. 00:20:17.440 --> 00:20:19.750 So suppose you are given a particular value 00:20:19.750 --> 00:20:26.410 for E hat and Lz hat, and you plot V effective versus r. 00:20:35.200 --> 00:20:39.310 Well, what you typically find is that it's 00:20:39.310 --> 00:20:45.710 got a behavior that looks kind of like this, where this value 00:20:45.710 --> 00:20:57.190 right here is V effective equal to 1. 00:20:57.190 --> 00:21:01.020 Notice as r goes to infinity, you get 1 times 1. 00:21:01.020 --> 00:21:02.610 So this asymptotes to 1. 00:21:05.940 --> 00:21:11.040 Notice E hat squared has the same dimensions as V effective. 00:21:11.040 --> 00:21:13.670 In fact, in the unit choices I've used here, 00:21:13.670 --> 00:21:16.050 they are both dimensionless. 00:21:16.050 --> 00:21:19.200 So what we can do is plot-- 00:21:19.200 --> 00:21:21.920 let's imagine that we have-- 00:21:25.737 --> 00:21:26.320 you know what? 00:21:26.320 --> 00:21:28.390 I'm going to want to sketch this on a different board. 00:21:28.390 --> 00:21:29.223 Let me go over here. 00:21:32.930 --> 00:21:36.990 So I'm going to want to look at a couple of different values 00:21:36.990 --> 00:21:38.325 of V effective-- 00:21:41.328 --> 00:21:43.370 excuse me, a couple of different values of E hat. 00:22:12.457 --> 00:22:12.980 OK. 00:22:12.980 --> 00:22:14.020 Here's an example. 00:22:14.020 --> 00:22:16.230 This guy is asymptoting at 1. 00:22:16.230 --> 00:22:20.660 So since E hat, as I said, has the same units as V 00:22:20.660 --> 00:22:34.770 effective, let's plot them together. 00:22:41.860 --> 00:22:49.420 So example one-- imagine if E hat lies right here. 00:22:53.030 --> 00:22:53.530 OK. 00:22:53.530 --> 00:22:57.340 So let's call this E hat 1. 00:23:01.070 --> 00:23:06.160 So this is some value that is greater than 1. 00:23:06.160 --> 00:23:07.970 What is the point of doing this? 00:23:07.970 --> 00:23:11.492 Well, notice-- I'm going to flip back 00:23:11.492 --> 00:23:13.450 and forth between these two middle boards here. 00:23:21.620 --> 00:23:23.540 Let's look at the equation that governs 00:23:23.540 --> 00:23:25.400 the radial motion of this body. 00:23:35.630 --> 00:23:39.360 dr d tau has to be a real number. 00:23:49.270 --> 00:23:51.190 This has to be a real number. 00:23:51.190 --> 00:23:55.200 So we have to have E hat squared greater than 00:23:55.200 --> 00:24:01.400 or equal to V effective in order for dr d tau 00:24:01.400 --> 00:24:02.650 to have a meaningful solution. 00:24:18.960 --> 00:24:21.090 So let's look at my example here, E hat 1. 00:24:27.410 --> 00:24:32.860 E hat 1 is greater than my effective potential 00:24:32.860 --> 00:24:38.580 everywhere at all radii until I get down to here. 00:24:43.062 --> 00:24:45.430 Let's call that r1. 00:24:45.430 --> 00:24:53.890 So in this case, dr d tau, if I think about this thing-- 00:24:53.890 --> 00:24:56.380 so note that defines dr d tau squared. 00:24:56.380 --> 00:24:59.250 Let's suppose we take the negative square root. 00:24:59.250 --> 00:25:03.005 dr d tau is positive and inward-- 00:25:03.005 --> 00:25:05.057 or, well, it's negative-- 00:25:05.057 --> 00:25:07.390 negative and real, negative and real, negative and real, 00:25:07.390 --> 00:25:09.430 negative and real, negative and real-- boom. 00:25:09.430 --> 00:25:10.080 It's 0. 00:25:12.720 --> 00:25:14.310 Can it go into here? 00:25:14.310 --> 00:25:15.360 No. 00:25:15.360 --> 00:25:19.290 It cannot go into there, because there E hat squared is less-- 00:25:22.580 --> 00:25:23.080 sorry. 00:25:23.080 --> 00:25:24.430 That should have been squared. 00:25:24.430 --> 00:25:26.380 Inside here, E hat squared is less 00:25:26.380 --> 00:25:29.440 than the effective potential. dr d tau is imaginary. 00:25:29.440 --> 00:25:31.300 That doesn't make any sense. 00:25:31.300 --> 00:25:35.890 The only option is for this guy to change sign and trundle 00:25:35.890 --> 00:25:37.760 right back out. 00:25:37.760 --> 00:25:41.470 So E hat greater than 1 corresponds 00:25:41.470 --> 00:25:56.270 to a body that comes in from large radius, 00:25:56.270 --> 00:26:08.010 turns around at particular radius where E hat is 00:26:08.010 --> 00:26:11.020 the square root of the effective potential, 00:26:11.020 --> 00:26:18.380 and then goes back out to infinity or back out to-- let's 00:26:18.380 --> 00:26:20.500 not say infinity-- goes back out to large radius. 00:26:25.813 --> 00:26:27.270 OK? 00:26:27.270 --> 00:26:31.890 In Newtonian gravity, we would call this a hyperbolic orbit. 00:26:31.890 --> 00:26:33.670 This corresponds to-- so remember, 00:26:33.670 --> 00:26:35.070 when I did this I have not-- 00:26:35.070 --> 00:26:37.860 I'm not actually-- I'm only computing the radial motion. 00:26:37.860 --> 00:26:39.880 I'm not looking at the phi motion. 00:26:39.880 --> 00:26:43.140 So this actually, when you look at both the radial 00:26:43.140 --> 00:26:45.450 motion and the phi motion, what you see 00:26:45.450 --> 00:26:48.628 is that this is a body that comes in and then sort of whips 00:26:48.628 --> 00:26:50.670 around that small radius and goes right back out. 00:26:57.260 --> 00:26:59.330 Let's look at another example. 00:27:04.870 --> 00:27:07.130 Let's call this E2-- 00:27:11.650 --> 00:27:12.970 some value that is less than 1. 00:27:16.750 --> 00:27:17.250 OK. 00:27:17.250 --> 00:27:22.690 Well, for E hat 2, it's a potential. 00:27:28.840 --> 00:27:33.810 The potential is underneath E hat 2 squared only 00:27:33.810 --> 00:27:41.710 between these two radii, which I will call r sub p and r sub a. 00:27:45.680 --> 00:27:49.610 What we expect in this case is motion of this body essentially 00:27:49.610 --> 00:27:53.000 going back and forth and turning around 00:27:53.000 --> 00:27:58.130 at periastron and apoastron. 00:27:58.130 --> 00:28:04.670 This is a relativistic generalization 00:28:04.670 --> 00:28:06.170 of an elliptical orbit. 00:28:06.170 --> 00:28:07.670 In general relativity, they turn out 00:28:07.670 --> 00:28:10.310 generally not to be ellipses. 00:28:10.310 --> 00:28:11.870 So we call this an eccentric orbit. 00:28:35.790 --> 00:28:38.970 In the weak field limit, if you imagine r being very, very 00:28:38.970 --> 00:28:44.190 large, it's not hard to show that the motion is nearly 00:28:44.190 --> 00:28:48.120 an ellipse, but it's an ellipse whose long axis is slowly 00:28:48.120 --> 00:28:49.770 precessing. 00:28:49.770 --> 00:28:52.980 This actually leads to the famous perihelion precession 00:28:52.980 --> 00:28:57.160 of Mercury that Einstein first calculated. 00:28:57.160 --> 00:29:00.270 And this is an exercise that I am 00:29:00.270 --> 00:29:04.620 asking you to do on one of the final P-sets of this course. 00:29:04.620 --> 00:29:06.540 Using what I have set up here, it's really not 00:29:06.540 --> 00:29:08.960 that difficult to do. 00:29:08.960 --> 00:29:10.350 Let me go to another board. 00:29:16.060 --> 00:29:26.530 And note that one could imagine an energy such that dr d tau 00:29:26.530 --> 00:29:27.715 is exactly 0. 00:29:40.940 --> 00:29:43.640 So if you choose your energy so that you 00:29:43.640 --> 00:29:48.020 sit right here at the minimum of the potential-- 00:29:48.020 --> 00:29:49.450 I will label this as point s-- 00:29:52.140 --> 00:29:54.520 the energy that corresponds to exactly that 00:29:54.520 --> 00:29:57.280 point is what would be a-- 00:29:57.280 --> 00:30:01.990 that is, there is a single point at which dr d tau equals 0. 00:30:01.990 --> 00:30:03.520 Anywhere away from that, dr d tau 00:30:03.520 --> 00:30:06.010 would be imaginary, so that's not going to work. 00:30:06.010 --> 00:30:12.640 But right at that point, dr d tau equals 0, 00:30:12.640 --> 00:30:13.930 and you get a circular orbit. 00:30:16.988 --> 00:30:19.280 Notice there's a second point at which that can happen. 00:30:19.280 --> 00:30:20.450 Let's call this point u. 00:30:28.630 --> 00:30:32.090 Perhaps you can guess why I called these points s and u. 00:30:32.090 --> 00:30:34.935 If you imagine that you add a tiny amount of energy 00:30:34.935 --> 00:30:36.310 right here, well, what it will do 00:30:36.310 --> 00:30:39.850 is it will execute small oscillations around the point 00:30:39.850 --> 00:30:40.385 s. 00:30:40.385 --> 00:30:42.010 It will sort of move it up to something 00:30:42.010 --> 00:30:44.830 that's similar to what I drew up there as E2, 00:30:44.830 --> 00:30:47.570 but with a very small eccentricity. 00:30:47.570 --> 00:30:52.960 So if I slightly disturb a circular orbit down here at s, 00:30:52.960 --> 00:30:55.030 I essentially just oscillate in the vicinity 00:30:55.030 --> 00:30:57.190 of that circular orbit. 00:30:57.190 --> 00:30:58.930 S stands for stable. 00:31:01.730 --> 00:31:05.440 If I have an orbit up here at u and I very slightly perturb 00:31:05.440 --> 00:31:12.070 it, well, it'll either go in and eventually reach r equals 0, 00:31:12.070 --> 00:31:15.102 or it'll go out, and then it's completely unbound, 00:31:15.102 --> 00:31:17.560 and it will just keep trundling all the way out essentially 00:31:17.560 --> 00:31:19.750 forever. 00:31:19.750 --> 00:31:21.730 This guy is unstable. 00:31:26.230 --> 00:31:31.700 Stable orbits are particularly interesting and important. 00:31:31.700 --> 00:31:35.440 So let's look at these orbits with a little bit more care. 00:31:51.230 --> 00:31:53.570 So the very definition of a circular orbit 00:31:53.570 --> 00:31:56.060 is that dr d tau equals 0. 00:31:56.060 --> 00:31:57.615 Its radius does not change. 00:32:05.230 --> 00:32:09.160 If dr d tau equals 0, then it must 00:32:09.160 --> 00:32:17.067 have E hat equal the square root of the effective. 00:32:19.750 --> 00:32:23.350 Both of these orbits happen to live 00:32:23.350 --> 00:32:27.820 at either a minimum or a maximum of this potential curve. 00:32:38.710 --> 00:32:46.410 So I'm going to require that the partial derivative 00:32:46.410 --> 00:32:48.600 of that potential with respect r be equal to 0. 00:32:53.070 --> 00:32:54.910 So let's take a look at this condition. 00:33:01.160 --> 00:33:03.590 My effective potential is given up here. 00:33:03.590 --> 00:33:07.640 Do a little bit of algebra with this, set this guy equal to 0. 00:33:07.640 --> 00:33:18.090 What you'll find after your algebraic smoke clears 00:33:18.090 --> 00:33:22.020 is that you get this condition on the angular momentum. 00:33:40.090 --> 00:33:42.840 Notice as r gets really large-- 00:33:42.840 --> 00:33:43.340 oh, shoot. 00:33:47.240 --> 00:33:48.320 Try it again. 00:33:48.320 --> 00:33:51.560 Notice as r gets really large that this 00:33:51.560 --> 00:33:58.560 asymptotes to plus or minus the square root of GM r. 00:33:58.560 --> 00:34:02.280 That is indeed exactly what you get for the angular momentum 00:34:02.280 --> 00:34:05.600 of a circular Newtonian orbit. 00:34:05.600 --> 00:34:06.450 OK. 00:34:06.450 --> 00:34:08.040 So it's a nice sanity check. 00:34:13.510 --> 00:34:15.820 It appears to be somewhat pathological 00:34:15.820 --> 00:34:18.969 as r approaches 3GM, though. 00:34:18.969 --> 00:34:19.719 Hold that thought. 00:34:41.920 --> 00:34:43.110 OK. 00:34:43.110 --> 00:34:48.719 So now let's take that value of L, 00:34:48.719 --> 00:34:53.820 plug it back into the potential, and set 00:34:53.820 --> 00:35:00.720 E equal to the square root of that. 00:35:00.720 --> 00:35:02.340 A little bit of algebra ensues. 00:35:02.340 --> 00:35:21.010 And what you find is that this equals 1 minus 2GM/r 00:35:21.010 --> 00:35:25.240 over, again, that factor under a square root of 1 minus 3GM/r. 00:35:25.240 --> 00:35:28.090 Again, we sort of see something a little bit 00:35:28.090 --> 00:35:33.060 pathological happening as r goes to 3GM. 00:35:33.060 --> 00:35:35.690 Let me make two comments about this. 00:35:35.690 --> 00:35:39.610 So first of all, notice that this energy is smaller than 1. 00:35:46.120 --> 00:35:55.310 I can intuitively-- you can sort of imagine-- remember, 00:35:55.310 --> 00:35:57.440 E hat is the energy per unit rest mass. 00:36:02.440 --> 00:36:04.180 You can think of this as something 00:36:04.180 --> 00:36:08.200 like total energy over M-- 00:36:08.200 --> 00:36:11.170 so the energy associated with the orbiting body. 00:36:11.170 --> 00:36:18.310 It's got a rest energy, a kinetic energy, 00:36:18.310 --> 00:36:25.050 and a potential energy divided by m. 00:36:25.050 --> 00:36:29.070 For an orbit to be bound, the potential energy, 00:36:29.070 --> 00:36:31.260 which is negative, must have larger magnitude 00:36:31.260 --> 00:36:33.150 than the kinetic energy. 00:36:33.150 --> 00:36:37.080 So for a bound orbit, E kinetic plus E potential 00:36:37.080 --> 00:36:39.660 will be a negative quantity. 00:36:39.660 --> 00:36:42.150 So the numerator is going to be something 00:36:42.150 --> 00:36:44.250 that, when normalized to m, is less than 1. 00:36:44.250 --> 00:36:45.730 So this is exactly what we expect 00:36:45.730 --> 00:36:48.180 to describe a bound orbit. 00:36:48.180 --> 00:36:50.880 Notice, also-- so if you take this formula 00:36:50.880 --> 00:36:59.160 and look at it in the large r limit, it goes to 1 minus GM 00:36:59.160 --> 00:37:00.930 over 2r. 00:37:00.930 --> 00:37:03.600 This is in fact exactly what you get 00:37:03.600 --> 00:37:08.570 when you look at the energy per unit mass throwing in-- 00:37:08.570 --> 00:37:09.690 sort of by hand-- 00:37:09.690 --> 00:37:11.220 a rest mass. 00:37:11.220 --> 00:37:14.160 The minus GM/2r exactly corresponds 00:37:14.160 --> 00:37:19.257 to kinetic plus potential for a Newtonian circular orbit. 00:37:19.257 --> 00:37:21.090 So lots of stuff is hanging together nicely. 00:37:26.270 --> 00:37:30.110 So we've just learned that we can characterize the energy 00:37:30.110 --> 00:37:35.270 and angular momentum of circular orbits around my black hole. 00:37:35.270 --> 00:37:37.700 Let's look at a couple other things associated with this. 00:37:37.700 --> 00:37:41.420 So these plots where I look at the radial motion, 00:37:41.420 --> 00:37:46.410 this effective potential, as I mentioned a few moments ago, 00:37:46.410 --> 00:37:47.930 there's additional sort of degrees 00:37:47.930 --> 00:37:50.750 of freedom in the motion that are being suppressed here. 00:37:50.750 --> 00:37:54.050 So this thing is also moving in that plane. 00:37:54.050 --> 00:37:56.600 It's whirling around with respect to phi. 00:37:56.600 --> 00:38:00.290 We've lost that information in the way we've drawn this here. 00:38:00.290 --> 00:38:07.120 Let's define omega to be the angular velocity of this orbit 00:38:07.120 --> 00:38:09.260 as seen by a distant observer. 00:38:22.450 --> 00:38:23.300 OK. 00:38:23.300 --> 00:38:26.900 Why am I doing it as seen by a distant observer? 00:38:26.900 --> 00:38:29.620 Well, when things orbit, there tend 00:38:29.620 --> 00:38:34.580 to be periodicities that imprint themselves on observables. 00:38:34.580 --> 00:38:36.673 It could be the period associated 00:38:36.673 --> 00:38:38.840 with the gravitational wave that arises out of this. 00:38:38.840 --> 00:38:41.900 It could be the period associated 00:38:41.900 --> 00:38:46.040 with peaks and a light curve if this is a star orbiting 00:38:46.040 --> 00:38:47.900 around a black hole. 00:38:47.900 --> 00:38:52.400 It could be oscillations in the X-ray flux 00:38:52.400 --> 00:38:54.410 if this is some kind of a lump in an accretion 00:38:54.410 --> 00:38:57.210 disk of material orbiting a black hole. 00:38:57.210 --> 00:38:59.720 So if this is the angular velocity seen by distant 00:38:59.720 --> 00:39:04.010 observers-- remember, the time that distant observers use to-- 00:39:04.010 --> 00:39:07.100 the time in which the distant observers' clocks run 00:39:07.100 --> 00:39:10.280 is the Schwarzschild time t. 00:39:10.280 --> 00:39:19.070 So this will be d phi dt, which I can write as d phi d tau-- 00:39:19.070 --> 00:39:23.690 this is the angular velocity according to the orbit itself-- 00:39:23.690 --> 00:39:29.120 normalized to dt d tau. 00:39:29.120 --> 00:39:32.810 Now, these are both quantities that are simply 00:39:32.810 --> 00:39:36.470 related to constants of motion. 00:39:36.470 --> 00:39:41.040 What I've got in the numerator here is L hat over r squared. 00:39:41.040 --> 00:39:42.650 And what I've got in the denominator 00:39:42.650 --> 00:39:47.980 here is E hat over 1 minus-- 00:39:47.980 --> 00:39:49.698 I dropped my t. 00:39:49.698 --> 00:39:50.990 It would happen at some point-- 00:39:50.990 --> 00:39:52.355 1 minus 2GM/r. 00:39:58.040 --> 00:40:01.460 So let's go ahead and take our solution here. 00:40:04.070 --> 00:40:09.920 My E hat is 1 minus 2GM/r divided by square root of blah, 00:40:09.920 --> 00:40:10.640 blah, blah. 00:40:10.640 --> 00:40:13.550 The 1 minus 2GM/r cancels. 00:40:13.550 --> 00:40:20.630 My L hat is square root GM r divided by, again, 00:40:20.630 --> 00:40:23.510 that square root 1 minus 3GM/r. 00:40:23.510 --> 00:40:28.250 Notice, the square root 1 minus 3GM/r factors all cancel. 00:40:28.250 --> 00:40:38.160 So this becomes plus or minus 1 over r squared square root GM 00:40:38.160 --> 00:40:38.660 r. 00:40:44.920 --> 00:40:45.730 Looks like this. 00:40:45.730 --> 00:40:49.180 Plus and minus basically just correspond to 00:40:49.180 --> 00:40:51.880 whether this motion sort of is going 00:40:51.880 --> 00:40:53.770 in the same sense as your phi coordinate 00:40:53.770 --> 00:40:55.203 or in the opposite sense. 00:40:55.203 --> 00:40:56.620 There's really no physics in that. 00:40:56.620 --> 00:41:00.850 It just comes along for the ride that both behave the same. 00:41:00.850 --> 00:41:02.530 If you guys get interested in this, 00:41:02.530 --> 00:41:05.830 and you do a similar calculation around a Kerr black hole, 00:41:05.830 --> 00:41:08.200 you'll find that your prograde solution gives you 00:41:08.200 --> 00:41:10.930 a different frequency than your retrograde solution because 00:41:10.930 --> 00:41:13.270 of the fact that the dragging of inertial frames 00:41:13.270 --> 00:41:15.730 due to the spin of the black hole kind of breaks 00:41:15.730 --> 00:41:17.880 that symmetry. 00:41:17.880 --> 00:41:20.960 Something which is interesting and-- 00:41:20.960 --> 00:41:23.460 well, I'll make a comment about this in just a second, which 00:41:23.460 --> 00:41:24.490 is kind of interesting. 00:41:24.490 --> 00:41:26.865 And here's what I'll say-- sometimes people think this is 00:41:26.865 --> 00:41:28.620 more profound than it should be-- 00:41:28.620 --> 00:41:34.410 is this is, in fact, exactly the same frequency law 00:41:34.410 --> 00:41:37.050 that you get using Newtonian gravity. 00:41:37.050 --> 00:41:45.570 This is actually exactly the same as Kepler's law. 00:41:52.490 --> 00:41:54.150 That seems really, really cool. 00:41:54.150 --> 00:41:54.650 And it is. 00:41:54.650 --> 00:41:55.817 It's actually really useful. 00:41:55.817 --> 00:41:57.830 It makes it very easy to remember this. 00:41:57.830 --> 00:42:01.310 But don't read too much into it. 00:42:01.310 --> 00:42:03.470 More than anything, it is a statement 00:42:03.470 --> 00:42:07.220 about a particular quality of this radial coordinate. 00:42:07.220 --> 00:42:12.170 So remember, in Newtonian gravity r tells me 00:42:12.170 --> 00:42:14.780 the distance between-- if I have an orbit at r1 00:42:14.780 --> 00:42:19.590 and an orbit at r2, then I know that the distance between them 00:42:19.590 --> 00:42:22.257 is r2 minus r1. 00:42:22.257 --> 00:42:24.840 In the Schwarzschild spacetime, the distance between these two 00:42:24.840 --> 00:42:27.990 orbits is not r2 minus r1. 00:42:27.990 --> 00:42:34.170 However, r2 labels a sphere of surface area 4 pi r2 squared. 00:42:34.170 --> 00:42:38.730 And r1 labels a sphere of surface area 4 pi r1 squared. 00:42:38.730 --> 00:42:42.510 It's easy to also show that the circumference of the orbit 00:42:42.510 --> 00:42:45.330 at r2 is 2 pi r2, the circumference 00:42:45.330 --> 00:42:48.630 of the orbit at r1 is 2 pi r1. 00:42:48.630 --> 00:42:50.820 That, more than anything, is why we end up 00:42:50.820 --> 00:42:52.650 reproducing Kepler's law here, is 00:42:52.650 --> 00:42:56.880 that this areal coordinate is nicely 00:42:56.880 --> 00:42:59.296 amenable to this interpretation. 00:43:08.720 --> 00:43:13.110 So is this orbit-- 00:43:13.110 --> 00:43:15.980 so I described over here an orbit that is unstable 00:43:15.980 --> 00:43:19.100 and an orbit that is stable. 00:43:19.100 --> 00:43:20.870 I have described how to compute-- 00:43:20.870 --> 00:43:23.780 if I wanted to find a circular orbit at a given 00:43:23.780 --> 00:43:28.680 radius, those formulas that I derived over there 00:43:28.680 --> 00:43:31.950 on the right-most blackboards, they tell me 00:43:31.950 --> 00:43:34.320 what the energy and the angular momentum 00:43:34.320 --> 00:43:37.650 need to be as a function of r. 00:43:37.650 --> 00:43:40.560 Is that orbit stable? 00:43:40.560 --> 00:43:47.370 Well, if it is, I can check that by computing 00:43:47.370 --> 00:43:51.558 the second derivative of my effective potential. 00:43:58.110 --> 00:44:09.090 So my orbits are stable if the second derivative with respect 00:44:09.090 --> 00:44:20.690 to r is greater than 0, unstable if this turns out 00:44:20.690 --> 00:44:23.420 to be negative. 00:44:23.420 --> 00:44:26.550 Let's look at the crossover point from one to the other. 00:44:26.550 --> 00:44:29.600 What if there is a radius where, in fact, 00:44:29.600 --> 00:44:33.410 the stable and the unstable orbits coincide? 00:44:33.410 --> 00:44:35.630 In fact, what one finds, if you look 00:44:35.630 --> 00:44:38.270 at the effective potential-- you imagine just sort of playing 00:44:38.270 --> 00:44:39.680 with L sub z. 00:44:39.680 --> 00:44:41.690 So let's say we take that L sub z, 00:44:41.690 --> 00:44:45.440 and we just explore it for lots of different radii 00:44:45.440 --> 00:44:46.880 of the orbits. 00:44:46.880 --> 00:44:48.950 You find that as the orbits radius gets 00:44:48.950 --> 00:44:52.762 smaller and smaller, your stable orbit tends to go up, 00:44:52.762 --> 00:44:54.470 and this minimum sort of becomes flatter, 00:44:54.470 --> 00:44:56.470 and your unstable orbit just kind of comes down. 00:44:56.470 --> 00:44:58.970 They sort of approach one another. 00:44:58.970 --> 00:45:02.090 There is a point just when they coincide-- 00:45:04.777 --> 00:45:06.360 this should have an "effective" on it. 00:45:06.360 --> 00:45:07.442 My apologies. 00:45:14.190 --> 00:45:16.830 Right when they coincide, this defines 00:45:16.830 --> 00:45:19.190 what we call the marginally stable orbit. 00:45:28.230 --> 00:45:30.000 I may have put this one on a problem set. 00:45:30.000 --> 00:45:32.083 But it might be one of the ones I decided to drop. 00:45:32.083 --> 00:45:34.200 So I'm just going to go ahead and do the analysis. 00:45:34.200 --> 00:45:39.180 When you compute this, bearing in mind that your angular 00:45:39.180 --> 00:45:41.070 momentum is a constant-- 00:45:53.990 --> 00:45:58.250 so take this, substitute in now your solution for L sub z, 00:45:58.250 --> 00:45:59.750 which I've written down over there-- 00:46:18.880 --> 00:46:22.240 what you find is that the marginally stable orbit-- 00:46:24.860 --> 00:46:26.830 let's call it r sub ms-- 00:46:26.830 --> 00:46:29.540 it is located at a radius of 6GM. 00:46:39.860 --> 00:46:44.330 This is a profoundly new behavior 00:46:44.330 --> 00:46:49.640 that doesn't even come close to existing 00:46:49.640 --> 00:46:52.530 in Newtonian spacetime-- 00:46:52.530 --> 00:46:54.830 spacetime-- doesn't come close to existing in Newtonian 00:46:54.830 --> 00:46:57.469 gravity, excuse me. 00:46:57.469 --> 00:47:02.210 [SIGHS] I'm getting tired. 00:47:07.860 --> 00:47:09.570 The message I want you to understand 00:47:09.570 --> 00:47:17.420 is that, what this tells us is that no stable circular orbits 00:47:17.420 --> 00:47:30.155 exist inside r equals 6GM. 00:47:34.070 --> 00:47:37.700 So this is very, very different behavior. 00:47:37.700 --> 00:47:42.560 If I have-- let's just say I have a very compact body 00:47:42.560 --> 00:47:45.140 but Newtonian gravity rules. 00:47:45.140 --> 00:47:46.693 I can make circular orbits around it, 00:47:46.693 --> 00:47:48.860 basically go all the way down until they essentially 00:47:48.860 --> 00:47:50.485 touch the surface of that body. 00:47:50.485 --> 00:47:51.860 And you might think based on this 00:47:51.860 --> 00:47:53.902 that you would want to make orbits that basically 00:47:53.902 --> 00:47:56.510 go all the way down, that sort of kiss the edge of r 00:47:56.510 --> 00:47:58.520 equals 2GM. 00:47:58.520 --> 00:48:02.300 Well, this is telling you you can't do that. 00:48:02.300 --> 00:48:03.830 When you start trying to make orbits 00:48:03.830 --> 00:48:05.997 that go inside-- at least, circular orbits-- that go 00:48:05.997 --> 00:48:10.190 inside 6GM, they're not stable. 00:48:10.190 --> 00:48:14.000 If someone sneezes on them, they either 00:48:14.000 --> 00:48:16.340 are sort of blown out to infinity, 00:48:16.340 --> 00:48:19.100 or they fall into the event horizon. 00:48:19.100 --> 00:48:21.800 And in fact, one of the consequences of this 00:48:21.800 --> 00:48:24.500 is that, in astrophysical systems, 00:48:24.500 --> 00:48:27.710 we generically expect there to be kind of-- if you imagine 00:48:27.710 --> 00:48:30.560 material falling into a black hole, 00:48:30.560 --> 00:48:33.020 imagine that there's like a star or something that's just 00:48:33.020 --> 00:48:36.560 dumping gas into orbit around a black hole, well, 00:48:36.560 --> 00:48:39.765 it will tend to form a disk that orbits around this thing. 00:48:39.765 --> 00:48:41.390 And the elements of the disk are always 00:48:41.390 --> 00:48:43.370 rubbing against each other and radiating. 00:48:43.370 --> 00:48:44.870 That makes them get hot. 00:48:44.870 --> 00:48:48.030 They lose energy because of this radiation. 00:48:48.030 --> 00:48:49.940 And so they will very slowly sort of fall in. 00:48:49.940 --> 00:48:51.195 But they make this kind of-- 00:48:51.195 --> 00:48:52.820 it's thought that in most cases they'll 00:48:52.820 --> 00:48:54.620 make this kind of thick disk that 00:48:54.620 --> 00:48:58.275 fills much of the spacetime surrounding the black hole. 00:48:58.275 --> 00:48:59.900 But there will be a hole in the center. 00:48:59.900 --> 00:49:02.425 Not just because the thing is a black hole. 00:49:02.425 --> 00:49:03.800 I don't mean that kind of a hole. 00:49:03.800 --> 00:49:06.258 There'll actually be something surrounding the black hole's 00:49:06.258 --> 00:49:09.380 event horizon, because there are no stable circular orbits. 00:49:09.380 --> 00:49:12.770 Once the material comes in and hits this particular radius, 00:49:12.770 --> 00:49:17.120 6GM in the Schwarzschild case, it very rapidly falls in, 00:49:17.120 --> 00:49:19.910 reduces the density of that material tremendously. 00:49:19.910 --> 00:49:21.950 And you get this much thinner region of the disk 00:49:21.950 --> 00:49:24.920 where essentially things fall in practically instantly. 00:49:24.920 --> 00:49:30.840 It should be noted that this 6GM is, of course, only 00:49:30.840 --> 00:49:32.040 for Schwarzschild. 00:49:32.040 --> 00:49:34.200 If you talk about Kerr, you actually 00:49:34.200 --> 00:49:35.970 have two different radii corresponding 00:49:35.970 --> 00:49:38.460 to material that goes around parallel to the black hole's 00:49:38.460 --> 00:49:41.100 spin and material that goes anti-parallel 00:49:41.100 --> 00:49:42.930 to the black hole's spin. 00:49:42.930 --> 00:49:44.400 And it complicates things somewhat. 00:49:44.400 --> 00:49:46.080 There's two different radii there. 00:49:46.080 --> 00:49:48.590 The one that goes parallel tends to get a little bit closer. 00:49:48.590 --> 00:49:50.007 The one that's anti-parallel tends 00:49:50.007 --> 00:49:52.230 to go out a little bit farther. 00:49:52.230 --> 00:49:54.210 But the general prediction that there's 00:49:54.210 --> 00:49:57.700 an innermost orbit beyond which stable orbits do not exist, 00:49:57.700 --> 00:50:03.850 that's robust and holds across the domain of black holes. 00:50:03.850 --> 00:50:07.740 So let me conclude by talking about one 00:50:07.740 --> 00:50:09.450 final category of orbits-- 00:50:13.300 --> 00:50:14.310 photon orbits. 00:50:20.100 --> 00:50:22.770 So let's recall that when we were talking 00:50:22.770 --> 00:50:40.040 about null geodesics, we parametrized them in such a way 00:50:40.040 --> 00:50:41.880 that d-- 00:50:41.880 --> 00:50:43.460 sort of the tangent to the world line 00:50:43.460 --> 00:50:55.530 has an affine parameter attached to it, such that we can write p 00:50:55.530 --> 00:51:00.750 equals dx d lambda. 00:51:00.750 --> 00:51:01.845 These guys are null. 00:51:05.130 --> 00:51:07.690 So p dot p equals 0. 00:51:12.020 --> 00:51:13.850 For the case of orbiting bodies, that 00:51:13.850 --> 00:51:15.440 was equal to minus mass squared. 00:51:15.440 --> 00:51:17.927 Mass is 0 here. 00:51:17.927 --> 00:51:20.010 So we're going to follow a very similar procedure. 00:51:20.010 --> 00:51:22.910 The spacetime is still time independent, 00:51:22.910 --> 00:51:25.820 so there is still a notion of a conserved energy. 00:51:25.820 --> 00:51:28.850 It is still actually symmetric, so there is still a notion 00:51:28.850 --> 00:51:32.030 of axial angular momentum. 00:51:32.030 --> 00:51:35.870 But because p dot p is 0 now, rather than minus m squared, 00:51:35.870 --> 00:51:39.920 when we go through the exercise of-- that sort of parallels 00:51:39.920 --> 00:51:41.990 what we did for our massive particle, 00:51:41.990 --> 00:51:44.360 we're going to derive a different potential. 00:51:44.360 --> 00:51:52.940 So let's go ahead and evaluate this guy again. 00:51:57.030 --> 00:52:06.130 And I get 0 equals minus 1 minus 2GM/r dt d lambda 00:52:06.130 --> 00:52:14.610 squared plus 1 minus 2GM/r dr d lambda squared. 00:52:14.610 --> 00:52:16.110 I'm still going to require the thing 00:52:16.110 --> 00:52:18.300 to be in the theta equals pi/2 plane 00:52:18.300 --> 00:52:20.620 so that we know theta term. 00:52:20.620 --> 00:52:22.620 And I have set theta to pi/2. 00:52:22.620 --> 00:52:24.810 So my sine squared theta is just one. 00:52:29.740 --> 00:52:30.580 So I get this. 00:52:33.550 --> 00:52:35.500 I'll remind you that I can relate-- 00:52:35.500 --> 00:52:40.290 so the relationship between the time-light component 00:52:40.290 --> 00:52:43.960 of momentum and energy, the axial component of momentum, 00:52:43.960 --> 00:52:46.720 and the angular momentum, it's exactly the same as before. 00:52:46.720 --> 00:52:48.430 There's no rest mass appearing. 00:52:48.430 --> 00:52:51.160 And so now I find-- 00:52:51.160 --> 00:52:54.792 so my energy, I don't put a hat on it. 00:52:54.792 --> 00:52:57.250 It's not energy bringing it mass, because there is no mass. 00:53:06.850 --> 00:53:08.620 That looks like so. 00:53:08.620 --> 00:53:16.650 And my L looks like so. 00:53:31.290 --> 00:53:37.020 Using them, I can now derive an equation 00:53:37.020 --> 00:53:41.120 governing the radial motion of my light ray. 00:53:50.080 --> 00:53:56.650 And sparing you the line or two of algebra, it looks like this. 00:54:13.377 --> 00:54:14.460 Pardon me just one moment. 00:54:18.780 --> 00:54:19.280 OK. 00:54:22.670 --> 00:54:23.730 OK. 00:54:23.730 --> 00:54:27.070 So kind of similar to what we had before, if you look at it 00:54:27.070 --> 00:54:29.760 you'll see the term involving the angular momentum 00:54:29.760 --> 00:54:33.050 is a little bit different. 00:54:33.050 --> 00:54:35.480 As you stare at this for a moment, 00:54:35.480 --> 00:54:37.220 something should be disturbing you. 00:54:40.460 --> 00:54:42.800 Notice that the equation of motion 00:54:42.800 --> 00:54:47.640 appears to depend on the energy. 00:54:47.640 --> 00:54:49.170 OK. 00:54:49.170 --> 00:54:51.660 That should really bug you. 00:54:51.660 --> 00:54:57.060 Why should gamma rays and infrared radiation 00:54:57.060 --> 00:55:00.540 follow different trajectories? 00:55:00.540 --> 00:55:03.390 As long as I am truly in a geometric optics limit, 00:55:03.390 --> 00:55:05.130 I shouldn't. 00:55:05.130 --> 00:55:08.320 Now, it is true that if you consider very long wavelength 00:55:08.320 --> 00:55:10.320 radiation, you might need to worry about things. 00:55:10.320 --> 00:55:13.200 You might need to solve the wave equation in the spacetime. 00:55:13.200 --> 00:55:18.060 But as long as the wave nature of this radiation is-- 00:55:18.060 --> 00:55:20.790 if the wavelength is small enough that it's negligible 00:55:20.790 --> 00:55:22.530 compared to 2GM-- 00:55:22.530 --> 00:55:25.140 I shouldn't care what the energy is. 00:55:25.140 --> 00:55:27.900 Energy should not be influencing this. 00:55:27.900 --> 00:55:31.020 So what's going on is I need to reparametrize 00:55:31.020 --> 00:55:32.170 this a little bit. 00:55:32.170 --> 00:55:37.710 What I'm going to do to wash away my dependence on-- 00:55:37.710 --> 00:55:40.710 wash away my apparent dependence on the energy here, 00:55:40.710 --> 00:55:43.507 is I'm going to redefine my affine parameter. 00:55:51.800 --> 00:55:55.270 So let's take lambda-- 00:55:55.270 --> 00:55:57.300 so L divided by lambda. 00:55:57.300 --> 00:56:07.215 And I am going to define b to be L divided by E. 00:56:07.215 --> 00:56:11.448 This is the quantity that I will call the impact parameter. 00:56:15.678 --> 00:56:17.220 And I'll describe why I am calling it 00:56:17.220 --> 00:56:21.100 that in just a few minutes. 00:56:21.100 --> 00:56:24.120 So I'm going to take this entire equation, divide both sides 00:56:24.120 --> 00:56:25.005 by L squared. 00:56:45.790 --> 00:56:49.060 And I get something that looks like this. 00:57:04.190 --> 00:57:08.450 What I'm going to do is say that the impact parameter, b, 00:57:08.450 --> 00:57:10.262 is what I can control. 00:57:10.262 --> 00:57:11.720 It's the parameter that-- all I can 00:57:11.720 --> 00:57:15.830 control that defines the photon that I am studying here. 00:57:15.830 --> 00:57:20.150 And everything after this, this is my photon potential. 00:57:23.080 --> 00:57:34.250 Notice that the photon potential has no free parameters in it. 00:57:58.370 --> 00:58:04.166 If I plot this guy as a function of r, 00:58:04.166 --> 00:58:07.760 it kind it just looks like this. 00:58:07.760 --> 00:58:09.710 Two aspects of it are worth calling out. 00:58:13.710 --> 00:58:21.340 This peak occurs at r equals 3GM. 00:58:26.820 --> 00:58:31.667 Remember the way in which our energy-- oh, still have it 00:58:31.667 --> 00:58:34.000 on the board here-- things like the energy per unit mass 00:58:34.000 --> 00:58:36.420 and the angular momentum per unit mass all 00:58:36.420 --> 00:58:40.040 blew up when r equal 3GM. 00:58:40.040 --> 00:58:44.400 3GM is showing up now when I look 00:58:44.400 --> 00:58:49.260 at the motion of radiation, look at massless-- 00:58:49.260 --> 00:58:54.000 radiation corresponding to a massless particle, so to speak. 00:58:54.000 --> 00:58:58.530 If I look at the energy per unit mass, and the mass goes to 0, 00:58:58.530 --> 00:59:00.250 I get infinity. 00:59:00.250 --> 00:59:02.400 So the fact that I was actually seeing 00:59:02.400 --> 00:59:05.460 sort of things blowing up as r goes to 3GM 00:59:05.460 --> 00:59:08.940 was kind of like the equations hinting to me in advance 00:59:08.940 --> 00:59:13.620 that there was a hidden solution corresponding to radiation that 00:59:13.620 --> 00:59:15.600 could be regarded as a particular limit 00:59:15.600 --> 00:59:17.100 that they were sort of struggling 00:59:17.100 --> 00:59:19.440 to communicate to me. 00:59:19.440 --> 00:59:20.940 The other thing which I want to note 00:59:20.940 --> 00:59:24.480 is that the potential, the height of the potential right 00:59:24.480 --> 00:59:42.190 here, it has a peak value of 1 over 27 G squared M squared. 00:59:42.190 --> 00:59:45.640 Hold that thought for just a moment. 00:59:45.640 --> 00:59:49.000 Actually, let me write it in a slightly different way. 00:59:49.000 --> 00:59:56.640 This equals 1 over 3 root 3 GM squared. 01:00:09.650 --> 01:00:14.410 So now, to wrap this up I need to tell you what is really 01:00:14.410 --> 01:00:16.998 meant by this impact parameter. 01:00:36.530 --> 01:00:42.710 So go back to freshman mechanics. 01:00:42.710 --> 01:00:46.220 And impact parameter there is-- like, let's say 01:00:46.220 --> 01:00:58.080 I have a problem where I am looking at an object that 01:00:58.080 --> 01:01:03.220 is on an infall trajectory. 01:01:06.860 --> 01:01:11.580 There is a momentum p that is describing this. 01:01:14.520 --> 01:01:17.040 And it's moving in such a way-- it's not moving 01:01:17.040 --> 01:01:19.140 on a real trajectory, right? 01:01:19.140 --> 01:01:21.900 It's actually moving in a sense that 01:01:21.900 --> 01:01:28.170 is a little bit off of true to the radial direction. 01:01:28.170 --> 01:01:34.560 This guy actually has an angular momentum that is given by-- 01:01:34.560 --> 01:01:38.360 let's say that this is equal to px x hat. 01:01:38.360 --> 01:01:41.430 Let's say this is b y hat. 01:01:41.430 --> 01:01:52.488 This guy's got an angular momentum of b cross p. 01:01:52.488 --> 01:01:54.780 That's the impact parameter associated with this thing. 01:01:54.780 --> 01:02:02.670 It sort of tells me about the offset of this momentum 01:02:02.670 --> 01:02:05.780 from being directly radial towards the center 01:02:05.780 --> 01:02:06.405 of this object. 01:02:12.760 --> 01:02:17.680 Well, I'm going to use a similar notion 01:02:17.680 --> 01:02:21.280 to give me a geometric sense of what this impact parameter here 01:02:21.280 --> 01:02:22.270 means. 01:02:22.270 --> 01:02:25.810 Suppose here is my black hole. 01:02:25.810 --> 01:02:29.860 It's a little circle of radius 2GM. 01:02:29.860 --> 01:02:34.353 And I'm sitting way the heck out here, safely far away 01:02:34.353 --> 01:02:35.020 from this thing. 01:02:41.280 --> 01:02:43.890 And what I'm going to do is shoot light-- 01:02:43.890 --> 01:02:45.065 not quite radial. 01:02:45.065 --> 01:02:46.440 What I'm going to do is I'm going 01:02:46.440 --> 01:02:50.990 to have sort of an array of laser beams 01:02:50.990 --> 01:02:53.630 that kind of come up here-- 01:02:57.200 --> 01:02:59.300 an array of laser beams. 01:02:59.300 --> 01:03:03.650 And I'm going to shoot them towards this black hole. 01:03:03.650 --> 01:03:06.540 And what I'm going to do is I'm going to offset the laser beams 01:03:06.540 --> 01:03:07.860 by a distance b. 01:03:10.272 --> 01:03:12.480 And I'm going to fire it straight towards this thing. 01:03:16.012 --> 01:03:17.970 Go through sort of a careful definition of what 01:03:17.970 --> 01:03:19.560 angular momentum means, and you'll 01:03:19.560 --> 01:03:23.700 see that that definition of impact parameter 01:03:23.700 --> 01:03:27.750 gives you a very nice sense of the energy associated 01:03:27.750 --> 01:03:31.050 with the trajectory of this beam and a notion of angular 01:03:31.050 --> 01:03:34.470 momentum associated with this. 01:03:34.470 --> 01:03:37.230 So let's flip back and forth between a couple 01:03:37.230 --> 01:03:39.960 of different boards here. 01:03:39.960 --> 01:03:42.383 There are three cases that are interesting. 01:03:53.010 --> 01:03:56.538 Suppose b is small. 01:03:56.538 --> 01:04:04.560 In particular, suppose I have b less than 3 root 3 GM. 01:04:04.560 --> 01:04:08.185 So I start over here. 01:04:08.185 --> 01:04:09.060 Let's take the limit. 01:04:09.060 --> 01:04:10.200 What if b equals 0? 01:04:10.200 --> 01:04:13.560 Well, if b equals 0, this guy just goes, zoom, 01:04:13.560 --> 01:04:16.100 straight into the black hole. 01:04:16.100 --> 01:04:21.570 As long as it is anything less than 3 root 3 GM, 01:04:21.570 --> 01:04:24.740 what will happen is, when I shoot this thing in, 01:04:24.740 --> 01:04:26.810 it goes in, and it bends, perhaps, 01:04:26.810 --> 01:04:29.360 but it ends up going into the black hole. 01:04:29.360 --> 01:04:36.758 Let's look at this in the context 01:04:36.758 --> 01:04:48.510 of the equation of motion, the potential, and the impact 01:04:48.510 --> 01:04:50.370 parameter. 01:04:50.370 --> 01:04:58.310 If b is less than 1 over 3 root 3 GM, then 1 over b squared 01:04:58.310 --> 01:05:00.530 will be higher than this peak. 01:05:00.530 --> 01:05:02.290 And this thing is just going to be, 01:05:02.290 --> 01:05:03.832 whoo [fast motion sound effect], it's 01:05:03.832 --> 01:05:05.740 going to go right over the top of the peak. 01:05:05.740 --> 01:05:07.145 And it shoots into small radius. 01:05:17.340 --> 01:05:17.840 OK. 01:05:17.840 --> 01:05:19.990 Remember, L and E are constants of the motion. 01:05:19.990 --> 01:05:23.680 So that b is a parameter that defines this light 01:05:23.680 --> 01:05:25.300 for the this entire trajectory. 01:05:25.300 --> 01:05:28.820 1 over b squared is less than V phot everywhere-- 01:05:28.820 --> 01:05:33.535 excuse me, greater than V phot everywhere. 01:05:33.535 --> 01:05:35.580 A rather crucial typo. 01:05:35.580 --> 01:05:37.560 And as such, it shoots the light ray, 01:05:37.560 --> 01:05:50.020 and it goes right over the peak into the black hole, 01:05:50.020 --> 01:05:51.730 eventually winds up at r equals 0. 01:05:54.780 --> 01:06:06.880 If b is greater than 3 root 3 GM, then 1 over b squared 01:06:06.880 --> 01:06:11.920 is less than V phot at the peak. 01:06:15.980 --> 01:06:19.970 This corresponds, in this drawing, 01:06:19.970 --> 01:06:23.600 to a beam of light that follows a trajectory that kind of comes 01:06:23.600 --> 01:06:27.310 in here at this point. 01:06:27.310 --> 01:06:31.790 dr d tau, if we were to continue to go into smaller radii, 01:06:31.790 --> 01:06:33.530 it would become imaginary. 01:06:33.530 --> 01:06:34.760 That's not allowed. 01:06:34.760 --> 01:06:37.430 So it switches direction and trundles right back 01:06:37.430 --> 01:06:38.621 out to infinity. 01:06:43.810 --> 01:06:50.840 On this diagram, that corresponds to a light ray 01:06:50.840 --> 01:06:51.860 that's perhaps out here. 01:06:51.860 --> 01:06:54.890 This guy comes in, and he gets bent by the gravity 01:06:54.890 --> 01:06:57.348 a little bit, and then just, shoo [light ray sound effect], 01:06:57.348 --> 01:06:58.910 shoots back off to infinity. 01:06:58.910 --> 01:07:14.770 The critical point, b equals 3 GM, 01:07:14.770 --> 01:07:18.090 is right where the impact parameter hits the peak. 01:07:22.070 --> 01:07:40.060 And so what happens in this plot is this guy comes in, 01:07:40.060 --> 01:07:43.350 hits the peak, and just sits there forever. 01:07:48.745 --> 01:07:50.620 You get a little bit more of a physical sense 01:07:50.620 --> 01:07:53.230 as to what's going on by thinking about it here. 01:07:57.610 --> 01:08:00.990 This guy comes in, [INAUDIBLE] this, 01:08:00.990 --> 01:08:06.670 and then just whirls around, and lies on what 01:08:06.670 --> 01:08:09.070 we call the photon orbit. 01:08:09.070 --> 01:08:11.320 What's the radius of that photon orbit? 01:08:11.320 --> 01:08:12.865 r equals 3 GM. 01:08:25.979 --> 01:08:30.270 Now, astrophysically, a more interesting situation 01:08:30.270 --> 01:08:33.420 is, imagine you had some source of light that 01:08:33.420 --> 01:08:37.359 dumps a lot of photons in the vicinity of a black hole. 01:08:37.359 --> 01:08:40.380 Some of those photons are going to tend to-- 01:08:40.380 --> 01:08:42.460 some of them are going to go into the black hole, 01:08:42.460 --> 01:08:44.043 some are going to scatter a little bit 01:08:44.043 --> 01:08:45.453 and shoot off to infinity. 01:08:45.453 --> 01:08:46.870 But if you imagine that there will 01:08:46.870 --> 01:08:49.210 be some population of them that get stuck 01:08:49.210 --> 01:08:53.470 right on the r equals 3GM orbit-- 01:08:53.470 --> 01:08:56.000 now, that is an unstable orbit. 01:08:56.000 --> 01:08:58.390 And in fact, if you look at it, what you find is that 01:08:58.390 --> 01:09:04.479 your typical photon is likely to whirl around a bunch of times, 01:09:04.479 --> 01:09:07.420 and then, you know, if it's not precisely at 3GM 01:09:07.420 --> 01:09:14.859 but it's 3.00000000000000001GM, it will whirl around maybe 10 01:09:14.859 --> 01:09:19.107 or 15 times, and then it'll go off to infinity. 01:09:19.107 --> 01:09:21.399 So what we actually expect is if we have an object that 01:09:21.399 --> 01:09:24.550 is illuminated like this, then we 01:09:24.550 --> 01:09:27.069 will actually see these things come out here, 01:09:27.069 --> 01:09:28.960 and we will see-- 01:09:28.960 --> 01:09:32.779 so bear in mind, this is circularly-- this is symmetric. 01:09:32.779 --> 01:09:38.729 So take this and rotate around the symmetry axis. 01:09:38.729 --> 01:09:47.850 We expect to see a ring whose radius is twice 01:09:47.850 --> 01:09:50.229 the critical impact parameter. 01:09:50.229 --> 01:09:54.229 So it would have a diameter of 6 root 3 GM. 01:09:54.229 --> 01:09:58.680 It would be essentially a ring or a circle of radius 3 root 3 01:09:58.680 --> 01:09:59.880 GM. 01:09:59.880 --> 01:10:04.560 This is, in fact, what the Event Horizon Telescope measured. 01:10:04.560 --> 01:10:07.020 So last year when I was lecturing this class, 01:10:07.020 --> 01:10:09.390 this was announced almost-- 01:10:09.390 --> 01:10:10.950 I mean, they timed it well. 01:10:10.950 --> 01:10:13.920 They basically timed it to about two or three lectures 01:10:13.920 --> 01:10:18.598 before I discussed this aspect of black holes. 01:10:18.598 --> 01:10:20.640 So that was-- thank you, Event Horizon Telescope. 01:10:20.640 --> 01:10:23.620 That was very nice of them. 01:10:23.620 --> 01:10:28.350 And of course, we don't expect-- so Schwarzschild black holes 01:10:28.350 --> 01:10:31.410 are probably a mathematical curiosity. 01:10:31.410 --> 01:10:33.600 Objects in the real universe all rotate. 01:10:33.600 --> 01:10:36.330 We expect Kerr to be the generic solution. 01:10:36.330 --> 01:10:37.830 And so there's a fair amount of work 01:10:37.830 --> 01:10:40.460 that goes into how you correct this to do-- 01:10:40.460 --> 01:10:42.030 so doing this for Schwarzschild is, 01:10:42.030 --> 01:10:44.613 because of spherical symmetry, it's beautiful and it's simple. 01:10:44.613 --> 01:10:46.440 Kerr is a little bit more complicated. 01:10:46.440 --> 01:10:48.960 But, you know, it's a problem that can be solved. 01:10:48.960 --> 01:10:51.570 And a lot of very smart people spend a lot of time 01:10:51.570 --> 01:10:53.340 doing this to sort of map out what 01:10:53.340 --> 01:10:56.640 the shadow of the black hole looks like, 01:10:56.640 --> 01:11:00.030 what this ring would look like in the case of light 01:11:00.030 --> 01:11:04.230 coming off of a spinning black hole. 01:11:04.230 --> 01:11:05.360 So that's it. 01:11:05.360 --> 01:11:09.290 This is all that I'm going to present for 8.962 01:11:09.290 --> 01:11:11.690 in the spring of 2020 semester. 01:11:11.690 --> 01:11:15.320 So to everyone, as you are scattered 01:11:15.320 --> 01:11:20.300 around the world attempting to sort of stay connected 01:11:20.300 --> 01:11:24.080 to physics and your friends and your classwork, 01:11:24.080 --> 01:11:25.250 I wish you good health. 01:11:25.250 --> 01:11:28.550 And I hope to see you again at a time when the world is 01:11:28.550 --> 01:11:29.930 a little less crazy. 01:11:29.930 --> 01:11:34.480 In the meantime, enjoy our little beautiful black holes.