WEBVTT 00:00:00.500 --> 00:00:01.932 [SQUEAKING] 00:00:01.932 --> 00:00:03.381 [RUSTLING] 00:00:03.381 --> 00:00:04.830 [CLICKING] 00:00:11.130 --> 00:00:13.460 SCOTT HUGHES: I'm going to record two lectures today. 00:00:13.460 --> 00:00:19.970 These are the final two lectures I will be recording for 8.962. 00:00:19.970 --> 00:00:24.170 I actually have notes on an additional two lectures. 00:00:24.170 --> 00:00:26.600 I have notes for an additional two lectures. 00:00:26.600 --> 00:00:29.540 But those additional two lectures 00:00:29.540 --> 00:00:32.180 are somewhat advanced material. 00:00:32.180 --> 00:00:36.200 It's sort of fun to go over them in the last week of the course, 00:00:36.200 --> 00:00:37.130 for certain students. 00:00:37.130 --> 00:00:38.720 It can be a great introduction to some 00:00:38.720 --> 00:00:41.180 of the most important topics in modern research. 00:00:41.180 --> 00:00:42.788 But I am not going to come into campus 00:00:42.788 --> 00:00:45.080 and present those lectures, OK, given everything that's 00:00:45.080 --> 00:00:46.760 going on right now. 00:00:46.760 --> 00:00:49.250 To be blunt, they are kind of bonus material. 00:00:49.250 --> 00:00:51.480 And this isn't the time. 00:00:51.480 --> 00:00:54.290 This is not the semester for us to go through our bonuses. 00:00:54.290 --> 00:00:57.570 I will make those notes available. 00:00:57.570 --> 00:01:00.620 I will be happy to discuss them in a saner moment 00:01:00.620 --> 00:01:03.150 with any student who is interested in them. 00:01:03.150 --> 00:01:07.100 But this semester, let's just focus on the core material. 00:01:07.100 --> 00:01:12.150 So that brings us to the topic of what we are studying today. 00:01:12.150 --> 00:01:14.490 Pardon me while I correct my handwriting. 00:01:14.490 --> 00:01:17.640 So if I can do a recap of what we discussed last time, 00:01:17.640 --> 00:01:23.220 we took a look at a spacetime that has this Schwarzschild 00:01:23.220 --> 00:01:25.420 solution written in the Schwarzschild 00:01:25.420 --> 00:01:28.080 coordinates everywhere. 00:01:28.080 --> 00:01:30.000 In our previous lecture, we looked 00:01:30.000 --> 00:01:34.830 at a spacetime that described a fluid body, a spherically 00:01:34.830 --> 00:01:36.900 symmetric fluid body. 00:01:36.900 --> 00:01:38.230 And it had a surface. 00:01:38.230 --> 00:01:40.990 This was the solution that we used for its exterior. 00:01:40.990 --> 00:01:45.840 This ends up describing a solution that 00:01:45.840 --> 00:01:47.580 has zero stress energy tensor. 00:01:47.580 --> 00:01:50.320 So it describes a vacuum situation. 00:01:50.320 --> 00:01:50.820 OK? 00:01:50.820 --> 00:01:56.370 So if I imagine a spacetime that looks like this everywhere, 00:01:56.370 --> 00:02:00.000 well, what I end up finding is that this is a vacuum solution. 00:02:00.000 --> 00:02:02.880 It has T-mu-nu equals 0 everywhere. 00:02:02.880 --> 00:02:05.820 However, it also has a mass m. 00:02:05.820 --> 00:02:07.710 So this is the vacuum solution with mass, 00:02:07.710 --> 00:02:11.970 which is, well, that's weird. 00:02:11.970 --> 00:02:14.880 We examined some of its curvature properties, 00:02:14.880 --> 00:02:18.650 and we found that at r equals 0, there is a tidal singularity, 00:02:18.650 --> 00:02:20.220 OK, in invariant quantity that we 00:02:20.220 --> 00:02:23.910 constructed from the tidal tensors, blows up at r 00:02:23.910 --> 00:02:26.040 equals 0. 00:02:26.040 --> 00:02:28.690 R equals 2GM looks a little bit funny. 00:02:28.690 --> 00:02:31.920 And it turns out tides are well behaved there. 00:02:31.920 --> 00:02:34.687 OK, there's nothing pathological in the spacetime there. 00:02:34.687 --> 00:02:36.270 But there is a coordinate singularity. 00:02:36.270 --> 00:02:38.700 Our coordinate t is behaving oddly. 00:02:38.700 --> 00:02:40.290 And where we left things last time 00:02:40.290 --> 00:02:41.850 is I did a little diagnosis of this 00:02:41.850 --> 00:02:47.070 by imagining a body that falls into spacetime 00:02:47.070 --> 00:02:49.663 from some starting radius R0. 00:02:49.663 --> 00:02:51.330 And I looked at the motion of this thing 00:02:51.330 --> 00:02:53.310 as a function of time. 00:02:53.310 --> 00:02:56.700 What we find is that if we look at the motion of this thing 00:02:56.700 --> 00:03:00.090 as a function of the proper time of that in-falling body, 00:03:00.090 --> 00:03:04.500 it crosses 2GM in finite proper time. 00:03:04.500 --> 00:03:07.530 And shortly afterwards, again, in finite proper time, 00:03:07.530 --> 00:03:10.590 it reaches the r equals 0 tidal singularity. 00:03:10.590 --> 00:03:14.565 If I look at that same motion as a function 00:03:14.565 --> 00:03:17.300 of the coordinate time t, it never 00:03:17.300 --> 00:03:22.620 even reaches, never even reaches r equals 2GM. 00:03:22.620 --> 00:03:24.180 We found a solution, what you can 00:03:24.180 --> 00:03:26.010 see in notes that I've put online 00:03:26.010 --> 00:03:29.263 and that are presented in the previous lecture. 00:03:29.263 --> 00:03:30.930 But we found a solution in which it just 00:03:30.930 --> 00:03:34.350 asymptotically approaches r equals 2GM, 00:03:34.350 --> 00:03:39.210 only reaching that radius in the t goes to infinity limit. 00:03:39.210 --> 00:03:41.720 These are two starkly different pictures 00:03:41.720 --> 00:03:43.790 of the kinematics of this in-falling body. 00:03:43.790 --> 00:03:46.120 So the question is, what is going on here? 00:03:46.120 --> 00:03:48.260 And as food for thought, I reminded us 00:03:48.260 --> 00:03:51.710 that these coordinates, if we sort of look at the way 00:03:51.710 --> 00:03:54.860 that spacetime behaves for a very, very, very large M-- 00:03:54.860 --> 00:03:56.960 Excuse me, very, very, very large R, 00:03:56.960 --> 00:04:01.220 not M, but for a very large R, R much, much greater than 2GM, 00:04:01.220 --> 00:04:03.830 this turns into the line element of flat spacetime. 00:04:03.830 --> 00:04:06.530 Flat spacetime is what we use in special relativity. 00:04:06.530 --> 00:04:08.240 And our time coordinate there, it 00:04:08.240 --> 00:04:12.110 is designed using this Einstein synchronization 00:04:12.110 --> 00:04:16.279 procedure, which means that the properties of light 00:04:16.279 --> 00:04:18.350 as it propagates through spacetime 00:04:18.350 --> 00:04:21.740 are built into the coordinate system. 00:04:21.740 --> 00:04:23.218 So that suggests what we might want 00:04:23.218 --> 00:04:25.760 to start doing in order to try to get some insight as to what 00:04:25.760 --> 00:04:30.260 is going on with this weird spacetime is 00:04:30.260 --> 00:04:33.770 to think about light as it propagates into spacetime. 00:04:33.770 --> 00:04:37.134 Let's think about what happens to radiation as it propagates. 00:04:39.860 --> 00:04:53.480 So let's imagine as the body falls that it emits a radio 00:04:53.480 --> 00:05:16.630 pulse with a frequency as measured 00:05:16.630 --> 00:05:18.070 according to the in falling. 00:05:18.070 --> 00:05:19.445 Let's say there's an observer who 00:05:19.445 --> 00:05:21.820 is falling in who's got a little radio transmitter that's 00:05:21.820 --> 00:05:24.010 beaming this message out. 00:05:24.010 --> 00:05:27.820 And according to that observer, this thing is emitted, 00:05:27.820 --> 00:05:30.880 the pulse has a frequency omega-- 00:05:33.520 --> 00:05:42.440 very far away, we can describe the momentum 00:05:42.440 --> 00:05:51.680 of this radio pulse like so. 00:05:51.680 --> 00:05:58.440 Imagine the thing is propagating out radially and very, 00:05:58.440 --> 00:06:02.510 very far away where the spacetime is approximately 00:06:02.510 --> 00:06:03.680 flat. 00:06:03.680 --> 00:06:06.920 It's simply the four-momentum that 00:06:06.920 --> 00:06:11.180 describes a null geodesic moving in the radial direction, OK? 00:06:11.180 --> 00:06:14.000 Minus sign here because with the index in the upstairs position 00:06:14.000 --> 00:06:16.190 and this asymptotically flat region, 00:06:16.190 --> 00:06:18.030 that would be the energy. 00:06:18.030 --> 00:06:20.480 So a couple of facts to bear in mind. 00:06:27.770 --> 00:06:38.820 The energy measured by an observer with four-velocity 00:06:38.820 --> 00:06:48.090 U is given by, let's call it E sub U, the energy measured 00:06:48.090 --> 00:06:57.420 by observer U. This is minus P-dot-U. 00:06:57.420 --> 00:07:02.372 We developed this in special relativity. 00:07:02.372 --> 00:07:04.830 But remember the way that we use the equivalence principle. 00:07:04.830 --> 00:07:06.955 We're gonna use the Einstein equivalence principle. 00:07:06.955 --> 00:07:10.470 And any law that holds in a freely falling frame, if I 00:07:10.470 --> 00:07:12.570 can write it in a tensorial way that 00:07:12.570 --> 00:07:15.300 works in that freely falling frame, it works in any frame. 00:07:15.300 --> 00:07:17.820 So this tensorial statement holds true, 00:07:17.820 --> 00:07:20.850 even though we are now working in the spacetime that 00:07:20.850 --> 00:07:23.640 is distinctly different from special relativity. 00:07:23.640 --> 00:07:40.850 We also know that in a time independent spacetime, 00:07:40.850 --> 00:07:46.470 the downstairs T component of four momentum is constant. 00:07:53.480 --> 00:08:01.130 So what this basically means is if I think about this radio 00:08:01.130 --> 00:08:03.290 pulse with its light propagating out 00:08:03.290 --> 00:08:08.270 through the spacetime, the value P0 associated with this thing's 00:08:08.270 --> 00:08:10.220 four momentum, it's the same everywhere 00:08:10.220 --> 00:08:11.868 along its trajectory. 00:08:18.570 --> 00:08:25.620 So let's consider a static observer sitting 00:08:25.620 --> 00:08:27.099 in the Schwarzschild spacetime. 00:08:41.280 --> 00:08:50.130 OK, I'm going to require G-alpha-beta U-alpha 00:08:50.130 --> 00:08:52.864 U-beta to be equal to minus 1. 00:08:52.864 --> 00:08:58.730 And I will require that this have some timelike piece, 00:08:58.730 --> 00:09:00.580 that this observer is static. 00:09:00.580 --> 00:09:03.450 So their spatial components of their four-velocity 00:09:03.450 --> 00:09:04.880 are equal to 0. 00:09:04.880 --> 00:09:07.890 It looks like, by the way-- just pause one moment here. 00:09:07.890 --> 00:09:10.110 It looks like the projector is on because of-- 00:09:17.127 --> 00:09:18.710 I should probably just leave it as is. 00:09:18.710 --> 00:09:20.585 All right, I'm not going to worry about that. 00:09:20.585 --> 00:09:21.970 Just leave that as it is. 00:09:21.970 --> 00:09:23.690 Sorry, let's go back to what I'm talking about here. 00:09:23.690 --> 00:09:25.482 So I have a static observer, so an observer 00:09:25.482 --> 00:09:28.330 who is not moving in space. 00:09:28.330 --> 00:09:29.830 They're only moving through time. 00:09:29.830 --> 00:09:32.823 And I need to normalize their four-velocity. 00:09:32.823 --> 00:09:34.990 Bear in mind, this is not a freely-falling observer. 00:09:34.990 --> 00:09:38.185 OK, this is an observer who must be accelerated, in some sense. 00:09:38.185 --> 00:09:41.140 And there must be some kind of a mechanism that 00:09:41.140 --> 00:09:44.410 is allowing this observer to hover at the fixed location 00:09:44.410 --> 00:09:47.140 in spacetime where they are at. 00:09:47.140 --> 00:09:50.380 So when I put these two constraints together 00:09:50.380 --> 00:09:55.670 and I tie it into that spacetime, 00:09:55.670 --> 00:10:02.420 what I find is that the timelike component is 1 square root 00:10:02.420 --> 00:10:04.550 of 1 minus 2GM over r. 00:10:09.730 --> 00:10:17.680 So let's compare the energy that is emitted at radius 00:10:17.680 --> 00:10:26.163 r to the energy that is absorbed at some radius big R. So 00:10:26.163 --> 00:10:27.080 I'm going to imagine-- 00:10:42.146 --> 00:10:44.260 And so the reason I formulate it in this way, what 00:10:44.260 --> 00:10:47.560 I want to do now is ask myself, well, 00:10:47.560 --> 00:10:52.690 what is the energy that an observer at little r 00:10:52.690 --> 00:10:53.500 would measure here? 00:11:10.200 --> 00:11:12.190 What is the energy that is observed 00:11:12.190 --> 00:11:18.650 by the person at radius capital R? 00:11:18.650 --> 00:11:21.500 And now, we know, if I imagine that these are both 00:11:21.500 --> 00:11:37.710 being measured by static observers, 00:11:37.710 --> 00:11:41.010 because P sub T, P sub 0 or P sub T, 00:11:41.010 --> 00:11:59.320 because it is a constant as the light propagates out, 00:11:59.320 --> 00:12:02.600 I can take the ratio of these two. 00:12:02.600 --> 00:12:11.240 And the energy observed at capital R versus the energy 00:12:11.240 --> 00:12:12.350 emitted at little r-- 00:12:17.292 --> 00:12:18.000 Hang on a second. 00:12:27.950 --> 00:12:29.570 As I was looking over my notes, there 00:12:29.570 --> 00:12:35.310 was a result that made no sense because I had a typo. 00:12:39.310 --> 00:12:40.810 So for those of you following along, 00:12:40.810 --> 00:12:43.660 the four-velocity component should have been a 1 00:12:43.660 --> 00:12:45.857 over the square root of that quantity I had earlier. 00:12:45.857 --> 00:12:47.440 Suddenly, what I'm about to write down 00:12:47.440 --> 00:12:48.400 makes a lot more sense. 00:12:52.222 --> 00:12:52.930 So this becomes-- 00:13:05.210 --> 00:13:07.130 Let's imagine that the observer is 00:13:07.130 --> 00:13:10.730 so far away that they are effectively infinitely 00:13:10.730 --> 00:13:11.760 far away. 00:13:11.760 --> 00:13:13.910 So the question I'm asking is, if I 00:13:13.910 --> 00:13:17.780 imagine that the light pulse is emitted 00:13:17.780 --> 00:13:24.020 at some radius little r, what would a very distant observer 00:13:24.020 --> 00:13:26.080 measure the energy of that pulse to be? 00:13:26.080 --> 00:13:34.390 OK, so my light pulse or radio pulse 00:13:34.390 --> 00:13:39.570 is emitted at some finite radius. 00:13:39.570 --> 00:13:42.540 And we'll call e infinity, the value that is measured 00:13:42.540 --> 00:13:43.860 by a very distant observer. 00:13:47.830 --> 00:13:54.380 This becomes square root 1 minus 2GM over r. 00:13:54.380 --> 00:13:57.050 Notice that the energy, no matter 00:13:57.050 --> 00:14:00.290 how energetic the light is when you emit it, 00:14:00.290 --> 00:14:03.350 I described it as being a radio pulse, 00:14:03.350 --> 00:14:05.270 but it could be a laser pointer. 00:14:05.270 --> 00:14:06.410 It could be ultraviolet. 00:14:06.410 --> 00:14:07.835 It could be a gamma ray. 00:14:07.835 --> 00:14:09.950 You know, you could just hawk whatever 00:14:09.950 --> 00:14:12.665 massively-powerful source of photons you want 00:14:12.665 --> 00:14:15.180 and you're pointing it out in the radial direction 00:14:15.180 --> 00:14:18.530 and hoping that your distant friend can measure it. 00:14:18.530 --> 00:14:22.400 No matter what energy you emit as you are falling in, 00:14:22.400 --> 00:14:26.390 as you approach r equals 2GM, the amount 00:14:26.390 --> 00:14:31.190 of energy in that beam that reaches a distant observer 00:14:31.190 --> 00:14:32.000 goes to 0. 00:14:49.660 --> 00:15:04.970 No matter how energetic my pulse of light, 00:15:04.970 --> 00:15:16.670 no energy reaches distant observers 00:15:16.670 --> 00:15:28.310 as the emitter approaches R equals 2GM. 00:15:28.310 --> 00:15:31.460 In a similar way, imagine that as you are falling in 00:15:31.460 --> 00:15:33.710 and you've got this little beacon that you are sending 00:15:33.710 --> 00:15:37.910 messages out to your distant friends, 00:15:37.910 --> 00:15:41.450 imagine you send them out with pulses that are separated 00:15:41.450 --> 00:16:04.180 by delta T. So the person in the falling frame 00:16:04.180 --> 00:16:06.610 turns his beacon on for a moment every delta 00:16:06.610 --> 00:16:08.410 T. Let's say delta T is every second. 00:16:21.200 --> 00:16:24.770 The interval between pulses, as measured far away, 00:16:24.770 --> 00:16:27.050 one can show using a similar kind of calculation. 00:16:52.000 --> 00:16:57.220 You pick up a factor of 1 over square root 1 minus 2GM over r. 00:16:57.220 --> 00:17:03.800 This goes to infinity as r goes 2GM. 00:17:09.790 --> 00:17:13.099 So remember, the time coordinate was originally 00:17:13.099 --> 00:17:17.060 defined by imagining that I can synchronize all of my clocks 00:17:17.060 --> 00:17:19.750 using light pulses that are bouncing around. 00:17:19.750 --> 00:17:23.540 But what we can kind of see here is that light pulses that 00:17:23.540 --> 00:17:27.740 are emitted in the vicinity of r equals 2GM, 00:17:27.740 --> 00:17:30.650 they're kind of going to hell, OK? 00:17:30.650 --> 00:17:33.120 So if I imagine, let's just say for the sake of argument, 00:17:33.120 --> 00:17:35.480 I'm using a green laser pointer as the thing 00:17:35.480 --> 00:17:39.110 that I use for my Einstein synchronization procedure. 00:17:39.110 --> 00:17:43.760 And suppose that I communicate what time is on the clock 00:17:43.760 --> 00:17:46.280 by a modulation of the signal. 00:17:46.280 --> 00:17:49.077 So suppose that I modulate it by putting 00:17:49.077 --> 00:17:51.410 little spaces on it that tend to be about a second long. 00:17:51.410 --> 00:17:53.327 Or let's just say it's like a microsecond long 00:17:53.327 --> 00:17:55.850 so I can really pack some information into that. 00:17:55.850 --> 00:18:02.920 Well, the laser pointers that are emitting from r equals 2GM, 00:18:02.920 --> 00:18:05.350 their signal infinitely redshifts away. 00:18:05.350 --> 00:18:06.630 They lose all their energy. 00:18:06.630 --> 00:18:07.330 So they're not green. 00:18:07.330 --> 00:18:08.710 If they're really close to it, maybe they'll 00:18:08.710 --> 00:18:10.335 be red when they get to the next thing. 00:18:10.335 --> 00:18:13.660 Instead of being a pulse every microsecond, 00:18:13.660 --> 00:18:15.550 it'll be every two microseconds. 00:18:15.550 --> 00:18:19.900 And the closer we get to it, the more redshifted 00:18:19.900 --> 00:18:22.850 it becomes and the more extended the interval between pulses 00:18:22.850 --> 00:18:23.350 becomes. 00:18:26.710 --> 00:18:29.890 I'm going to post to the 8.962 website 00:18:29.890 --> 00:18:31.570 a set of notes that sort of cleans up 00:18:31.570 --> 00:18:33.260 this calculation a little bit, just goes through this 00:18:33.260 --> 00:18:34.560 in a little bit more detail. 00:18:34.560 --> 00:18:38.260 It was something I wrote in the Spring 2019 00:18:38.260 --> 00:18:39.970 semester in response to a good question 00:18:39.970 --> 00:18:41.910 that a student had asked me about this. 00:18:41.910 --> 00:18:44.260 And I think they do a nice job of just going over 00:18:44.260 --> 00:18:45.130 this calculation. 00:18:45.130 --> 00:18:46.547 And they give a couple of examples 00:18:46.547 --> 00:18:47.770 of the way this is behaving. 00:18:47.770 --> 00:18:49.690 The key thing which I want to emphasize 00:18:49.690 --> 00:18:57.895 is that as r goes to 2GM, what we see is that-- 00:18:57.895 --> 00:18:59.270 well, let's just put it this way. 00:18:59.270 --> 00:19:01.030 In fact, let's remove the word "as." 00:19:01.030 --> 00:19:11.090 The surface r equals 2GM corresponds 00:19:11.090 --> 00:19:12.320 to infinite redshift. 00:19:22.170 --> 00:19:24.470 OK, we've talked already a little bit about, 00:19:24.470 --> 00:19:28.730 if I have light climbing out of a gravitational field, 00:19:28.730 --> 00:19:32.000 the light gets a little bit redder as it climbs out. 00:19:32.000 --> 00:19:33.560 Well, at this particular surface, 00:19:33.560 --> 00:19:37.830 the sphere of radius 2GM in this spacetime, 00:19:37.830 --> 00:19:40.953 if you can get down to there, that redshift 00:19:40.953 --> 00:19:41.870 then becomes infinite. 00:19:41.870 --> 00:19:46.560 All of the energy is drained out of it as it climbs out. 00:19:46.560 --> 00:19:48.800 So what this basically tells us is 00:19:48.800 --> 00:19:52.190 that this surface breaks the Einstein synchronization 00:19:52.190 --> 00:19:56.120 procedure and it renders that time coordinate bad, 00:19:56.120 --> 00:19:59.600 at least if we are concerned about understanding things 00:19:59.600 --> 00:20:01.020 right at this surface. 00:20:16.530 --> 00:20:20.460 OK, as long as we're concerned with the exterior 00:20:20.460 --> 00:20:23.430 of the surface, that's not a problem, OK? 00:20:23.430 --> 00:20:25.680 Everything works fine as long as we move away 00:20:25.680 --> 00:20:27.120 from this coordinate singularity, 00:20:27.120 --> 00:20:29.220 sort of in the same way that many 00:20:29.220 --> 00:20:31.800 of the pathologies associated with the spherical 00:20:31.800 --> 00:20:33.870 coordinate system here, the north and south pole, 00:20:33.870 --> 00:20:36.162 they are fine as long as you're not trying to do things 00:20:36.162 --> 00:20:42.850 like measure the longitude angle corresponding to the North 00:20:42.850 --> 00:20:43.350 Pole. 00:20:43.350 --> 00:20:45.083 So this is not even well-defined. 00:20:45.083 --> 00:20:46.500 In the same way, this is basically 00:20:46.500 --> 00:20:49.380 telling us that the coordinate T in which we wrote down 00:20:49.380 --> 00:20:55.180 this spacetime isn't really well-defined at r equals 2GM. 00:20:55.180 --> 00:21:01.930 So what we need to do, if we want to try to get some-- hey, 00:21:01.930 --> 00:21:03.340 I'm over here, camera!-- 00:21:03.340 --> 00:21:05.650 if we want to try to get a little bit of insight 00:21:05.650 --> 00:21:09.794 as to what is going on here, we need a better time coordinate. 00:21:21.430 --> 00:21:23.070 So I'm going to talk about a couple. 00:21:23.070 --> 00:21:25.710 And the way that we're going to formulate 00:21:25.710 --> 00:21:30.210 these is all the pathologies are revealed 00:21:30.210 --> 00:21:32.280 when we look at the behavior of light propagating 00:21:32.280 --> 00:21:33.047 in this spacetime. 00:21:33.047 --> 00:21:34.380 So let's play around with light. 00:21:34.380 --> 00:21:37.140 Let's look at null geodesics in this spacetime. 00:21:37.140 --> 00:21:40.480 So to begin with, let's stick, for just the next couple 00:21:40.480 --> 00:21:43.292 of moments, with the original Schwarzschild coordinates. 00:21:54.880 --> 00:21:57.510 OK, so I'm going to look at null geodesics. 00:21:57.510 --> 00:22:01.510 So I'm going to set 0, and I'm gonna ask myself, 00:22:01.510 --> 00:22:04.440 how does it move through an interval at dt 00:22:04.440 --> 00:22:05.910 and an interval of dr? 00:22:20.540 --> 00:22:29.570 OK, so we can solve this to find how, 00:22:29.570 --> 00:22:32.090 if an object is moving on a radial null geodesic, 00:22:32.090 --> 00:22:35.000 how do dt and dr behave? 00:22:35.000 --> 00:22:35.983 How does dt dr behave? 00:22:35.983 --> 00:22:36.650 Put it that way. 00:23:01.150 --> 00:23:04.570 OK, so that's what my dt dr looks like. 00:23:04.570 --> 00:23:07.330 Plus, the plus sign corresponds to a solution 00:23:07.330 --> 00:23:09.700 that is moving the outward in all direction. 00:23:09.700 --> 00:23:13.780 Minus sign is an inward directed null geodesic. 00:23:13.780 --> 00:23:17.290 These define what we consider to be the opening angle. 00:23:17.290 --> 00:23:23.240 So dt dr defines the opening angle of a light cone. 00:23:36.330 --> 00:23:45.620 So if we go to very, very, very large r, 00:23:45.620 --> 00:23:47.990 OK, we get dt dr equals 1. 00:23:52.750 --> 00:23:56.470 And this corresponds in units in which c equals 1 00:23:56.470 --> 00:24:01.570 to light moving on a 45 degree angle in a spacetime diagram. 00:24:01.570 --> 00:24:04.675 This is familiar behavior from special relativity. 00:24:19.030 --> 00:24:28.505 But as r goes to 2GM, we see dt dr going to 0. 00:24:28.505 --> 00:24:29.880 Let's make a sketch and see what, 00:24:29.880 --> 00:24:34.020 sort of, dt dr, what the tangent to a null geodesic 00:24:34.020 --> 00:24:38.580 looks like in the t-r plane as a function of radius. 00:25:20.420 --> 00:25:22.490 So this little dash here means I'm 00:25:22.490 --> 00:25:26.000 sort of imagining that I'm going to stretch my r axis so 00:25:26.000 --> 00:25:29.030 that, out here, you're in the asymptotically 00:25:29.030 --> 00:25:32.332 flat region where things look like special relativity. 00:25:40.060 --> 00:25:43.290 So here we are out in this asymptotically flat region. 00:25:43.290 --> 00:25:45.540 My outward-going light ray goes off 00:25:45.540 --> 00:25:48.040 at 45 degree angle in the r-t plane. 00:25:48.040 --> 00:25:52.110 Inward one goes at a 45 degree angle pointing inside. 00:25:52.110 --> 00:25:54.320 Down here in the stronger field, it's 00:25:54.320 --> 00:25:57.370 going to be a little bit steeper than this. 00:25:57.370 --> 00:26:03.360 And so the opening angle of my light cone is closing up. 00:26:10.140 --> 00:26:14.550 Here, we'll have closed up a lot more. 00:26:19.600 --> 00:26:21.570 Here, it's closed up practically all the way. 00:26:21.570 --> 00:26:25.420 Now, as I approach r equals 2GM, both the inward 00:26:25.420 --> 00:26:27.040 and the outward direction in these 00:26:27.040 --> 00:26:32.160 coordinates go parallel to 2GM. 00:26:32.160 --> 00:26:35.580 So you can see the collapse of the light cone 00:26:35.580 --> 00:26:47.720 in these coordinates as you approach 00:26:47.720 --> 00:26:49.820 this coordinate singularity. 00:26:49.820 --> 00:26:52.680 So we need a healthier coordinate system. 00:26:52.680 --> 00:26:57.980 One thing that we can do is we can move the pathology out 00:26:57.980 --> 00:27:01.070 of our time coordinate and into our radial coordinate 00:27:01.070 --> 00:27:02.717 with the following definition. 00:27:09.880 --> 00:27:16.480 Suppose you choose a radial coordinate r-star such 00:27:16.480 --> 00:27:19.960 that dt equals plus or minus dr-star everywhere. 00:27:19.960 --> 00:27:20.560 OK? 00:27:20.560 --> 00:27:27.310 So if I replace my horizontal axis with-- 00:27:27.310 --> 00:27:30.670 pardon me-- if I replace my horizontal axis 00:27:30.670 --> 00:27:35.400 with the r-star, this will be 45 degree angles everywhere. 00:27:35.400 --> 00:27:36.190 OK? 00:27:36.190 --> 00:27:51.500 But to make this work, what you find is r-star 00:27:51.500 --> 00:27:54.360 must look like this. 00:27:54.360 --> 00:27:57.620 What you basically see is that this coordinate system 00:27:57.620 --> 00:28:00.650 takes r equals 2GM and it moves it 00:28:00.650 --> 00:28:03.878 to r-star equals minus infinity. 00:28:15.850 --> 00:28:18.780 So the way that this coordinate representation, 00:28:18.780 --> 00:28:22.880 that this different radial coordinate, 00:28:22.880 --> 00:28:26.190 the way that this makes the light cones always 00:28:26.190 --> 00:28:27.373 have 45 degree opening-- 00:28:27.373 --> 00:28:29.790 really 90 degrees opening-- it makes the light rays always 00:28:29.790 --> 00:28:34.470 go off at 45 degree angles is by essentially constantly 00:28:34.470 --> 00:28:37.380 stretching the radial axis so that this guy just 00:28:37.380 --> 00:28:39.780 gets stretched out so that it's opening at 90 degrees. 00:28:39.780 --> 00:28:41.260 This guy is stretched a little bit less. 00:28:41.260 --> 00:28:42.927 This guy is stretched a little bit less. 00:28:42.927 --> 00:28:45.280 Basically, not stretched when you're really far away. 00:28:45.280 --> 00:28:48.590 But as you approach r equals 2GM, 00:28:48.590 --> 00:28:52.050 you're infinitely stretching at these coordinates. 00:28:52.050 --> 00:29:11.270 These are known as tortoise coordinates, basically, 00:29:11.270 --> 00:29:13.100 because you start walking, taking ever 00:29:13.100 --> 00:29:15.650 slower and slower steps. 00:29:15.650 --> 00:29:18.140 Even steps, if you imagine you're stepping evenly 00:29:18.140 --> 00:29:21.620 in r-star, you're taking ever smaller and smaller steps 00:29:21.620 --> 00:29:26.150 in r, as you approach the infinite redshift surface. 00:29:29.120 --> 00:29:42.620 So with that tortoise coordinate defined, 00:29:42.620 --> 00:29:45.110 you use that as an intermediary to define 00:29:45.110 --> 00:29:48.530 a couple of new coordinates for your spacetime that 00:29:48.530 --> 00:29:49.970 are adapted to radiation. 00:29:55.100 --> 00:30:03.980 So we're going to define v to be t plus r-star. 00:30:03.980 --> 00:30:06.320 And the importance of this is that this 00:30:06.320 --> 00:30:09.290 is a coordinate that's not hard to convince yourself 00:30:09.290 --> 00:30:13.780 this is constant on an in-going radial null ray. 00:30:26.240 --> 00:30:31.520 I'm going to define a coordinate U to be t minus r-star. 00:30:31.520 --> 00:30:44.884 And this is constant on outgoing radial null rays. 00:30:44.884 --> 00:30:45.384 OK? 00:30:48.472 --> 00:30:49.930 So this basically means that if I'm 00:30:49.930 --> 00:30:55.330 working in this coordinate system, r-star, 00:30:55.330 --> 00:30:57.970 if I want to know the behavior of this guy, 00:30:57.970 --> 00:30:59.660 well, an outgoing radial null ray, 00:30:59.660 --> 00:31:01.660 you just might say, ah, that's the null ray that 00:31:01.660 --> 00:31:04.160 has U equals 17. 00:31:04.160 --> 00:31:04.660 OK? 00:31:04.660 --> 00:31:07.450 And that will then pick out, basically, 00:31:07.450 --> 00:31:09.280 a whole sequence of events along which 00:31:09.280 --> 00:31:12.008 that null ray has propagated. 00:31:15.150 --> 00:31:17.430 And, you know, as you can see, it vastly 00:31:17.430 --> 00:31:19.410 simplifies how we describe it. 00:31:19.410 --> 00:31:24.030 So once you've defined these two coordinates, 00:31:24.030 --> 00:31:26.040 you can rewrite the Schwarzschild spacetime 00:31:26.040 --> 00:31:27.090 in terms of them. 00:31:27.090 --> 00:31:29.983 It is generally best to choose-- 00:31:29.983 --> 00:31:32.400 So we're going to take this another step in just a moment, 00:31:32.400 --> 00:31:33.390 but we'll start-- 00:31:33.390 --> 00:31:38.770 you choose either V or U, and you replace the Schwarzschild 00:31:38.770 --> 00:31:39.270 time. 00:31:52.300 --> 00:31:58.730 So let's use V to replace time. 00:32:04.600 --> 00:32:07.800 So when you go and you look at what your new coordinate system 00:32:07.800 --> 00:32:08.850 looks like-- 00:32:08.850 --> 00:32:11.760 and remember, the way you do this is the usual thing, 00:32:11.760 --> 00:32:14.820 you're going to make your matrix of partial derivatives 00:32:14.820 --> 00:32:16.380 between your old coordinate system. 00:32:16.380 --> 00:32:23.400 So your old coordinates are t, r, theta, and phi. 00:32:23.400 --> 00:32:28.662 And then they go over to v, r, theta, and phi. 00:32:28.662 --> 00:32:29.162 OK? 00:32:33.145 --> 00:32:35.520 So you make your matrix of partial derivatives describing 00:32:35.520 --> 00:32:36.120 this. 00:32:36.120 --> 00:32:39.060 And here's what you find when you 00:32:39.060 --> 00:32:45.843 change to the metric in the new representation. 00:32:55.520 --> 00:32:58.115 Notice, there's no dr squared term at all. 00:32:58.115 --> 00:32:59.330 OK? 00:32:59.330 --> 00:33:01.430 We do still see something that's, you know, 00:33:01.430 --> 00:33:04.280 at least a coordinate singularity at r equals 2GM. 00:33:04.280 --> 00:33:07.430 We haven't quite gotten rid of it entirely here. 00:33:07.430 --> 00:33:12.650 But we've definitely mollified the impact 00:33:12.650 --> 00:33:14.480 of this coordinate singularity. 00:33:14.480 --> 00:33:17.270 So in this coordinate system, you can then 00:33:17.270 --> 00:33:22.640 set ds squared equal to 0, and you find two solutions 00:33:22.640 --> 00:33:24.290 describing radial null curves. 00:33:34.800 --> 00:33:38.600 So dv dr equals 0 for in-going. 00:33:41.460 --> 00:33:44.710 And, you know, by definition, these things 00:33:44.710 --> 00:33:47.080 are constant on an in-going radial null ray. 00:33:47.080 --> 00:33:49.540 And so as you move along it, V remains constant. 00:34:02.920 --> 00:34:05.690 OK, and you get something a little bit more complicated 00:34:05.690 --> 00:34:06.580 for the outgoing one. 00:34:15.710 --> 00:34:19.796 Let's redraw this using my new coordinates, OK? 00:34:31.260 --> 00:34:36.996 So I'm going to leave my horizontal axis as r. 00:34:36.996 --> 00:34:46.810 So I'm going to make my time axis be v. So out here, 00:34:46.810 --> 00:34:49.110 here's my in-going null ray. 00:34:53.639 --> 00:34:55.409 And here's my, eh-- let's make that 00:34:55.409 --> 00:34:58.170 a little closer to 45 degrees. 00:34:58.170 --> 00:35:00.720 Here's my outgoing null ray. 00:35:00.720 --> 00:35:02.040 OK? 00:35:02.040 --> 00:35:06.920 As I move in to smaller and smaller values of r, 00:35:06.920 --> 00:35:11.370 notice in the limit, as r goes to 2GM, 00:35:11.370 --> 00:35:13.790 the slope becomes infinite. 00:35:13.790 --> 00:35:16.410 OK, so this thing gets steeper. 00:35:16.410 --> 00:35:25.050 This one keeps pointing in, so this guy gets steeper, steeper. 00:35:25.050 --> 00:35:33.668 Right here, it lies pointing exactly straight up. 00:35:33.668 --> 00:35:35.710 What's kind of cool is that in these coordinates, 00:35:35.710 --> 00:35:39.750 I can actually look at what it looks like inside this thing. 00:35:39.750 --> 00:35:42.490 And so inside, this guy tips over and gets a negative slope 00:35:42.490 --> 00:35:43.390 and looks like this. 00:35:46.770 --> 00:35:49.160 Now, bear in mind, these two things, 00:35:49.160 --> 00:35:51.530 they denote the null rays. 00:35:51.530 --> 00:35:59.750 All radial timelike trajectories, 00:35:59.750 --> 00:36:01.450 and indeed all timelike trajectories, 00:36:01.450 --> 00:36:08.530 not just the radial ones, all timelike trajectories 00:36:08.530 --> 00:36:16.690 must follow a world line that is bounded by these two sides. 00:36:29.410 --> 00:36:34.050 OK, so if I'm out here, everything in here 00:36:34.050 --> 00:36:38.670 describes trajectories that a timelike observer can follow. 00:36:38.670 --> 00:36:40.920 Everything in here describes a trajectory 00:36:40.920 --> 00:36:44.790 a timelike observer can follow, everything in here, 00:36:44.790 --> 00:36:46.230 everything in here. 00:36:48.990 --> 00:36:55.200 Notice, when I am at this one here, right at r equals 2GM, 00:36:55.200 --> 00:37:00.660 every allowed timelike trajectory points towards r 00:37:00.660 --> 00:37:03.770 equals 0. 00:37:03.770 --> 00:37:07.820 At best, I can imagine an observer 00:37:07.820 --> 00:37:11.030 who's very close to the speed of light who sort of skims 00:37:11.030 --> 00:37:13.052 along inside of this thing. 00:37:13.052 --> 00:37:14.510 But they are timelike, so they will 00:37:14.510 --> 00:37:17.460 have a little bit of a slope that points them inward. 00:37:50.560 --> 00:37:51.420 OK. 00:37:51.420 --> 00:37:53.700 As you move inside r equals 2GM, it's even more so. 00:37:53.700 --> 00:37:55.695 OK, you can't even sort of go parallel to the r 00:37:55.695 --> 00:37:56.580 equals 2GM line. 00:37:56.580 --> 00:37:58.230 They all point towards this thing. 00:38:01.290 --> 00:38:04.320 Once you get to r equals 2GM, all trajectories, 00:38:04.320 --> 00:38:07.740 as they move to the future, must move to smaller radius. 00:38:33.360 --> 00:38:36.690 What this tells us, in particular, 00:38:36.690 --> 00:38:43.050 is that once you have reached r equals 2GM, 00:38:43.050 --> 00:38:44.920 you are never coming back. 00:39:10.830 --> 00:39:13.140 All allowed trajectories, everything 00:39:13.140 --> 00:39:15.480 that is permissible by the laws of physics, 00:39:15.480 --> 00:39:17.550 moves along a trajectory that points towards r 00:39:17.550 --> 00:39:20.940 equals 0 once you hit that r equals 2GM line. 00:39:20.940 --> 00:39:25.020 Because of this, this surface r, this surface 00:39:25.020 --> 00:39:27.510 of infinite redshift is given the name-- 00:39:33.888 --> 00:39:35.680 Let me come back to the point I was making. 00:39:35.680 --> 00:39:39.420 So the surface r equal 2GM, nothing that crosses 00:39:39.420 --> 00:39:41.550 it is ever going to come back. 00:39:41.550 --> 00:39:47.530 This surface of infinite redshift, 00:39:47.530 --> 00:39:49.700 we call an event horizon. 00:40:00.080 --> 00:40:02.290 OK? 00:40:02.290 --> 00:40:05.725 No events that are on the other side of r 00:40:05.725 --> 00:40:10.030 equals 2GM can have any causal influence on events 00:40:10.030 --> 00:40:10.700 on the outside. 00:40:42.000 --> 00:40:44.910 If you have a spacetime with an event horizon like this 00:40:44.910 --> 00:40:47.020 and this one that we are talking about right here, 00:40:47.020 --> 00:40:49.170 I'm going to talk at the end of my final lecture 00:40:49.170 --> 00:40:51.435 that I record about-- 00:40:51.435 --> 00:40:53.310 Maybe, actually, I'm gonna do it in this one. 00:40:53.310 --> 00:40:54.030 Yeah, I am. 00:40:54.030 --> 00:40:55.770 So at the end of this lecture that I'm recording, 00:40:55.770 --> 00:40:57.812 I'm going to go over a couple of other spacetimes 00:40:57.812 --> 00:40:59.590 that have this structure. 00:40:59.590 --> 00:41:02.790 Such spacetimes are called-- 00:41:09.970 --> 00:41:32.080 wait for it-- such spacetimes are called black holes. 00:41:32.080 --> 00:41:36.040 They are black because light cannot get out of them. 00:41:36.040 --> 00:41:39.010 And they are holes because you just jump into 'em, 00:41:39.010 --> 00:41:42.180 and you ain't never coming out. 00:41:42.180 --> 00:41:43.720 So let me pause for just one moment. 00:41:43.720 --> 00:41:46.720 I want to send a quick note to the ODL person who 00:41:46.720 --> 00:41:49.916 is helping me out here. 00:41:49.916 --> 00:41:52.030 I just want to let her know that this lecture may 00:41:52.030 --> 00:42:00.033 run a tiny bit long, since I spent a moment 00:42:00.033 --> 00:42:02.200 chatting with a police officer who checked in on me. 00:42:30.968 --> 00:42:34.920 OK, if you're watching, Elaine, hi! 00:42:34.920 --> 00:42:36.860 So I just sent you a quick note, letting 00:42:36.860 --> 00:42:41.150 you know that I'm likely to run a little bit long. 00:42:41.150 --> 00:42:43.530 All right, let's get back to black holes. 00:42:43.530 --> 00:42:49.010 So to sort of call out some of the structure 00:42:49.010 --> 00:42:53.090 of this spacetime, I want to spend just a few minutes 00:42:53.090 --> 00:42:58.160 talking about one final coordinate transformation that 00:42:58.160 --> 00:43:02.880 is very useful, but looks really weird. 00:43:02.880 --> 00:43:06.170 So just bear with me as I go through this-- 00:43:06.170 --> 00:43:09.560 a very useful, but unquestionably somewhat 00:43:09.560 --> 00:43:11.613 obtuse coordinate transformation. 00:43:20.500 --> 00:43:23.288 What I'm going to do is I'm going 00:43:23.288 --> 00:43:24.580 to define a coordinate v-prime. 00:43:32.490 --> 00:43:35.260 This is given by taking the exponential 00:43:35.260 --> 00:43:45.010 of the in-going coordinate time V, normalized for GM. 00:43:45.010 --> 00:43:49.570 U prime will be the exponent of U, 00:43:49.570 --> 00:43:53.020 a time that works well for the outgoing coordinate system 00:43:53.020 --> 00:43:55.090 divided by 4GM. 00:43:55.090 --> 00:44:00.420 I'm then going to define capital T 00:44:00.420 --> 00:44:08.720 to be 1/2 V-prime plus U-prime, capital 00:44:08.720 --> 00:44:13.400 R to be 1/2 V-prime minus U-prime. 00:44:23.500 --> 00:44:27.770 It's then simple to show, where simple 00:44:27.770 --> 00:44:30.440 is professor speak for "a student can probably 00:44:30.440 --> 00:44:31.895 do it in an hour or so." 00:44:31.895 --> 00:44:34.670 It's sort of tedious, but straightforward, 00:44:34.670 --> 00:44:36.620 just hooking together lots of definitions 00:44:36.620 --> 00:44:41.660 and slogging through a couple of identities. 00:44:50.160 --> 00:44:56.400 It's simple to show that capital T relates 00:44:56.400 --> 00:45:03.010 to Schwarzschild time T and Schwarzschild radius r like so. 00:45:03.010 --> 00:45:03.885 There's two branches. 00:45:42.790 --> 00:45:47.580 OK, so this is how one relates capital T and capital 00:45:47.580 --> 00:45:54.460 R to Schwarzschild t and Schwarzschild r in the region 00:45:54.460 --> 00:45:57.490 r greater than or equal to 2GM. 00:46:03.150 --> 00:46:48.990 You find a somewhat different solution 00:46:48.990 --> 00:46:51.768 in the region r less than 2GM. 00:46:51.768 --> 00:46:53.310 So if you're looking at this and kind 00:46:53.310 --> 00:46:56.250 of going, "what the hell are you talking about here," 00:46:56.250 --> 00:46:56.810 that's fine. 00:47:03.446 --> 00:47:06.496 Let me just write down two more relationships. 00:47:11.460 --> 00:47:13.560 And then I'll describe what this is good for. 00:47:26.110 --> 00:47:28.660 So a particularly clean inversion 00:47:28.660 --> 00:47:48.460 between TR and the original Schwarzschild tr, 00:47:48.460 --> 00:47:56.720 both the r greater than 2GM and r less than 2GM branches 00:47:56.720 --> 00:47:58.055 can be subsumed into this. 00:48:07.570 --> 00:48:16.360 And you find T over R looks like the hyperbolic tangent 00:48:16.360 --> 00:48:25.920 T over 4GM when you're in the exterior 00:48:25.920 --> 00:48:34.060 and the hyperbolic cotangent in the interior. 00:48:34.060 --> 00:48:38.780 So these rather bizarre-looking coordinates, 00:48:38.780 --> 00:48:52.806 these are known as Kruskal-Szekeres coordinates. 00:48:59.663 --> 00:49:01.080 I'll just leave that down like so. 00:49:04.740 --> 00:49:06.987 So when one goes into this, I'm not going to deny it, 00:49:06.987 --> 00:49:08.820 this is a bizarre looking coordinate system. 00:49:08.820 --> 00:49:11.490 OK, but it's got several features 00:49:11.490 --> 00:49:15.960 that make it very useful for understanding 00:49:15.960 --> 00:49:18.260 what is going on physically in this spacetime. 00:49:44.110 --> 00:49:46.990 So first, if you rewrite your metric 00:49:46.990 --> 00:49:53.050 in terms of capital T and capital R, what you get 00:49:53.050 --> 00:49:55.680 is a form that has no singularities. 00:49:55.680 --> 00:49:59.620 It's well-behaved everywhere, except at r equals 0. 00:50:10.100 --> 00:50:12.530 So you do get a singularity there, 00:50:12.530 --> 00:50:14.810 things blow up as r goes to 0. 00:50:22.840 --> 00:50:24.903 There's no other coordinate pathologies. 00:50:27.748 --> 00:50:29.540 And then you get sort of an angular sector. 00:50:29.540 --> 00:50:32.480 It's actually cleanest in terms of the Schwarzschild radius r, 00:50:32.480 --> 00:50:34.330 so we'll leave it in terms of that. 00:50:34.330 --> 00:50:36.400 One thing which is nice is notice 00:50:36.400 --> 00:50:51.050 that radial null geodesics, they simply 00:50:51.050 --> 00:50:59.085 obey dt equals plus or minus dr everywhere, OK? 00:50:59.085 --> 00:51:01.460 The only place where you run into a little bit of problem 00:51:01.460 --> 00:51:05.360 is as r goes to 0. 00:51:05.360 --> 00:51:07.910 And that's special, OK? 00:51:07.910 --> 00:51:10.910 So I got that just by setting ds squared equal to 0. 00:51:10.910 --> 00:51:13.280 It's radial, so my D-omega goes to 0. 00:51:13.280 --> 00:51:19.080 And then just dt equals plus or minus dr. 00:51:19.080 --> 00:51:20.840 So that's really nice, OK? 00:51:20.840 --> 00:51:22.960 I'm gonna make a sketch in just a moment. 00:51:22.960 --> 00:51:26.940 And the fact that I know light always moves along 00:51:26.940 --> 00:51:29.220 45 degree lines in this coordinate system 00:51:29.220 --> 00:51:32.310 is going to help me to understand the causal structure 00:51:32.310 --> 00:51:33.784 of this spacetime. 00:51:40.210 --> 00:51:43.670 The causal structure is what I mean by which events 00:51:43.670 --> 00:51:45.170 can influence other events. 00:51:51.840 --> 00:51:54.450 What can exert a causal influence on what? 00:52:03.410 --> 00:52:06.890 So as I move on, I'm going to make a sketch in just a moment, 00:52:06.890 --> 00:52:10.010 I want to highlight a couple of behaviors 00:52:10.010 --> 00:52:12.920 that we see that are really sort of called out 00:52:12.920 --> 00:52:16.805 in this mapping between the two coordinate systems. 00:52:20.480 --> 00:52:29.360 So notice that a surface of constant Schwarzschild 00:52:29.360 --> 00:52:37.780 radius, constant r forms a hyperbola 00:52:37.780 --> 00:52:40.311 in the Kruskal-Szekeres coordinates. 00:53:03.410 --> 00:53:09.020 Notice that surfaces of constant time, 00:53:09.020 --> 00:53:11.330 they form lines in that they form 00:53:11.330 --> 00:53:14.870 lines of slope t over r equal to some constant. 00:53:32.020 --> 00:53:39.370 So they are lines with t over r equaling, on the exterior, 00:53:39.370 --> 00:53:41.510 let's just focus on the exterior, 00:53:41.510 --> 00:53:45.530 they have a slope that's given by the hyperbolic tangent of t 00:53:45.530 --> 00:53:46.030 over 4GM. 00:53:53.720 --> 00:53:56.450 On the interior-- replaced with cotangent, 00:53:56.450 --> 00:53:57.997 hyperbolic cotangent. 00:54:03.700 --> 00:54:06.200 The last thing which I'd like to note before I make a sketch 00:54:06.200 --> 00:54:10.520 here is let's look at the special surface 00:54:10.520 --> 00:54:12.770 of infinite redshift, this event horizon. 00:54:16.790 --> 00:54:35.150 So if I plug in r equals 2GM, plug this in over here, 00:54:35.150 --> 00:54:38.810 I get t squared minus r squared equals 0. 00:54:38.810 --> 00:54:41.300 This is the asymptotic limit to those hyperbolae. 00:54:41.300 --> 00:54:50.260 They just become lines t equals plus or minus r. 00:54:50.260 --> 00:54:53.980 So the event horizon in this coordinate representation 00:54:53.980 --> 00:54:57.790 is just going to be a pair of lines crossing 00:54:57.790 --> 00:55:00.470 in the origin of these coordinate systems, 00:55:00.470 --> 00:55:02.530 OK, a pair of 45 degree lines crossing 00:55:02.530 --> 00:55:05.260 into this coordinate system. 00:55:05.260 --> 00:55:10.690 Notice, also, that t equals plus or minus r, 00:55:10.690 --> 00:55:25.260 this corresponds to Schwarzschild t going to 00:55:25.260 --> 00:55:28.720 plus or minus infinity. 00:55:28.720 --> 00:55:34.260 So this, indeed, is a weird, singular limit 00:55:34.260 --> 00:55:37.560 of the Schwarzschild time coordinate. 00:55:37.560 --> 00:55:42.030 So you can find much prettier versions of this figure 00:55:42.030 --> 00:55:46.800 than the one I'm about to attempt to sketch. 00:55:46.800 --> 00:55:49.300 Let's see what I can do with this. 00:55:49.300 --> 00:55:56.630 So horizontal will be the Kruskal-Szekeres coordinate 00:55:56.630 --> 00:56:00.620 r, vertical will be the coordinate t. 00:56:16.190 --> 00:56:28.070 Here is the event horizon, r equals t or little r 00:56:28.070 --> 00:56:30.960 equals 2GM. 00:56:30.960 --> 00:56:44.870 Some different surface of r equal to some constant value 00:56:44.870 --> 00:56:50.500 greater than 2GM will live on a hyperbola like so. 00:56:57.810 --> 00:57:02.630 Some value of r equals constant, but less than 2GM, 00:57:02.630 --> 00:57:04.500 lies on a hyperbola like so. 00:57:14.760 --> 00:57:17.550 In particular, there is one special hyperbole 00:57:17.550 --> 00:57:19.380 corresponding to r equals 0. 00:57:22.390 --> 00:57:30.530 And this is where my artistry is going to truly fail me. 00:57:30.530 --> 00:57:33.442 This is an infinite tidal singularity. 00:57:36.610 --> 00:57:38.110 Now, the thing which is particularly 00:57:38.110 --> 00:57:42.010 useful about this particular coordinate system 00:57:42.010 --> 00:57:47.140 is, remember, light always moves in the capital R, 00:57:47.140 --> 00:57:49.660 capital T coordinates. 00:57:49.660 --> 00:57:56.650 It always moves on lines that go dt equals plus or minus dr. So 00:57:56.650 --> 00:58:01.960 what you can see is that imagine I start here 00:58:01.960 --> 00:58:05.332 and I send out a little light pulse, 00:58:05.332 --> 00:58:10.790 OK, a radially outgoing light pulse, it will always 00:58:10.790 --> 00:58:17.090 go away and go to larger and larger values of r, just sort 00:58:17.090 --> 00:58:20.840 of constantly moves along this particular trajectory. 00:58:20.840 --> 00:58:23.960 Let me write out again what I'm doing here. 00:58:23.960 --> 00:58:36.670 So a radial outgoing light ray, it will follow dt equals dr. 00:58:36.670 --> 00:58:40.840 But notice that this line goes parallel to the event horizon. 00:58:40.840 --> 00:58:43.720 If I am on the inside of this guy 00:58:43.720 --> 00:58:48.310 and I try to make a light pulse that goes outside, 00:58:48.310 --> 00:58:51.430 points in the radial direction, all it does 00:58:51.430 --> 00:58:53.470 is, in these coordinates, it moves parallel 00:58:53.470 --> 00:58:54.430 to the event horizon. 00:58:54.430 --> 00:58:56.780 It can never cross it. 00:58:56.780 --> 00:58:59.560 And in fact, because this is a hyperbola, 00:58:59.560 --> 00:59:01.480 one finds that even though you have 00:59:01.480 --> 00:59:05.530 tried to make this guy as outgoing as outgoing can be, 00:59:05.530 --> 00:59:13.920 it will eventually intersect the r equals 0 tidal singularity. 00:59:13.920 --> 00:59:17.260 Since I cannot have any event, here, 00:59:17.260 --> 00:59:20.570 that communicates with any event on the other side of this, 00:59:20.570 --> 00:59:47.170 this region, everything at r less than 2GM, 00:59:47.170 --> 00:59:52.060 these will be causally disconnected-- 00:59:52.060 --> 00:59:54.460 not "casually," pardon me. 01:00:04.520 --> 01:00:13.010 They are causally disconnected from the world 01:00:13.010 --> 01:00:14.880 that lies outside of r is 2GM. 01:00:18.740 --> 01:00:21.340 So in my notes and in Carroll's textbook, 01:00:21.340 --> 01:00:23.080 there is another coordinate system 01:00:23.080 --> 01:00:25.920 that you can do which essentially takes 01:00:25.920 --> 01:00:27.660 points that are infinitely far away 01:00:27.660 --> 01:00:32.160 and brings them into a finite coordinate location. 01:00:32.160 --> 01:00:35.802 And that final thing, it puts it in what 01:00:35.802 --> 01:00:37.635 are called Penrose coordinates and it allows 01:00:37.635 --> 01:00:40.560 you make what's called the Penrose diagram, which 01:00:40.560 --> 01:00:46.200 displays, in a very simple way, how different events are either 01:00:46.200 --> 01:00:50.610 causally connected or causally disconnected from the others. 01:00:50.610 --> 01:00:54.150 Fairly advanced stuff, not important, but many of you 01:00:54.150 --> 01:00:55.530 may find it interesting. 01:00:55.530 --> 01:00:58.110 Happy to talk further, once we all 01:00:58.110 --> 01:01:00.825 have the bandwidth to have those kinds of conversations. 01:01:06.880 --> 01:01:08.218 So let me summarize. 01:01:27.250 --> 01:01:31.440 So the summary is that this spacetime, so earlier we 01:01:31.440 --> 01:01:34.740 were looking at the spacetime of a spherically symmetric fluid 01:01:34.740 --> 01:01:36.390 object. 01:01:36.390 --> 01:01:39.480 We found a particularly clean form 01:01:39.480 --> 01:01:42.480 for the exterior of that object, where it was a vacuum solution. 01:02:01.900 --> 01:02:06.490 If we imagine a spacetime that has this everywhere, 01:02:06.490 --> 01:02:16.220 then we get this solution that we call a black hole. 01:02:16.220 --> 01:02:18.520 So I emphasize this is vacuum everywhere, 01:02:18.520 --> 01:02:23.890 but sort of the analysis kind of goes to hell at r equals 0. 01:02:23.890 --> 01:02:26.890 So there are some singular field equations there trying 01:02:26.890 --> 01:02:28.600 to describe the stress energy. 01:02:28.600 --> 01:02:31.810 Its behavior as you approach there, 01:02:31.810 --> 01:02:36.460 let's just say we're not quite sure what's going on. 01:02:36.460 --> 01:02:40.810 In this coordinate system, we see weird things 01:02:40.810 --> 01:02:49.730 happening as we approach r equals 2GM. 01:02:49.730 --> 01:02:51.500 This is simply a coordinate singularity. 01:02:51.500 --> 01:02:55.850 There's really nothing going bad with the physics here. 01:02:55.850 --> 01:02:59.060 But our attempts to use a time coordinate 01:02:59.060 --> 01:03:02.510 that's based on, essentially, the way light 01:03:02.510 --> 01:03:05.270 moves in empty space, it's failing in this region. 01:03:05.270 --> 01:03:08.600 And all of this lecture is about uncovering this and seeing 01:03:08.600 --> 01:03:13.700 that, in fact, this is a surface of infinite redshift beyond 01:03:13.700 --> 01:03:17.000 which things cannot communicate. 01:03:17.000 --> 01:03:24.290 So this is one of the big discoveries that came out 01:03:24.290 --> 01:03:27.200 of general relativity, OK, this creature we 01:03:27.200 --> 01:03:29.860 call the black hole. 01:03:29.860 --> 01:03:33.140 It is not the only solution of the Einstein field equations 01:03:33.140 --> 01:03:35.600 that we call a black hole. 01:03:35.600 --> 01:03:38.690 Let me talk briefly about two others. 01:03:54.010 --> 01:03:58.960 So another one has a spacetime that looks like this. 01:04:28.410 --> 01:04:30.030 So where did this come from? 01:04:30.030 --> 01:04:32.240 Well, suppose I bung this through the Einstein field 01:04:32.240 --> 01:04:37.520 equations, what I find is that it comes from a non-zero stress 01:04:37.520 --> 01:04:38.300 energy tensor. 01:04:38.300 --> 01:04:44.660 In fact, it's a stress energy tensor that looks like this. 01:04:44.660 --> 01:04:45.644 Pardon me. 01:05:14.220 --> 01:05:18.990 This is a stress energy tensor of a Coulomb electric field 01:05:18.990 --> 01:05:20.130 with total charge q. 01:05:28.720 --> 01:05:31.480 This represents a charged black hole. 01:06:09.540 --> 01:06:11.790 It turns out if you analyze this thing carefully, 01:06:11.790 --> 01:06:13.980 you find it has an event horizon. 01:06:19.570 --> 01:06:30.140 It turns out to be located at G quantity that 01:06:30.140 --> 01:06:34.110 involves the square root of m squared minus q squared. 01:06:34.110 --> 01:06:37.820 Now, if q-- don't even ask me what the units are that this is 01:06:37.820 --> 01:06:40.490 being measured in, they're pretty goofy units-- 01:06:40.490 --> 01:06:43.580 but if, in these units, the magnitude of q 01:06:43.580 --> 01:06:47.000 is greater than m, there is no horizon. 01:06:47.000 --> 01:06:52.340 There is still, however, an infinite tidal singularity at r 01:06:52.340 --> 01:06:53.780 equals 0. 01:06:53.780 --> 01:06:56.540 So such a solution would give us what is 01:06:56.540 --> 01:06:58.530 known as a naked singularity. 01:07:01.290 --> 01:07:04.530 I have a few comments and I have a couple notes on these. 01:07:04.530 --> 01:07:06.710 But they're not as important as other things 01:07:06.710 --> 01:07:07.870 I'd like to talk about. 01:07:07.870 --> 01:07:10.570 You might wonder, where does that r horizon actually 01:07:10.570 --> 01:07:11.458 come from? 01:07:11.458 --> 01:07:13.750 OK, that is what you get when you find the route, where 01:07:13.750 --> 01:07:17.793 you look for the place where the metric function vanishes. 01:07:17.793 --> 01:07:19.960 OK, you notice there's only one metric function that 01:07:19.960 --> 01:07:22.960 appears in there, 1 minus 2GM over r plus q 01:07:22.960 --> 01:07:24.800 squared over r squared. 01:07:24.800 --> 01:07:27.490 So in general, if you have what's 01:07:27.490 --> 01:07:31.330 known as a stationary spacetime, you can find 01:07:31.330 --> 01:07:36.440 coordinates such that surfaces of constant r 01:07:36.440 --> 01:07:40.110 are spacelike surfaces. 01:07:40.110 --> 01:07:43.590 If you look for the place where a surface of constant r 01:07:43.590 --> 01:07:49.290 makes a transition from being a spacelike surface to being 01:07:49.290 --> 01:07:55.980 a null surface, that tells you that you 01:07:55.980 --> 01:08:00.470 have located an event horizon. 01:08:32.580 --> 01:08:34.640 OK, this is discussed in a little bit more detail 01:08:34.640 --> 01:08:35.359 in my notes. 01:08:35.359 --> 01:08:36.950 There's also some very nice discussion 01:08:36.950 --> 01:08:40.040 in Carroll's textbook. 01:08:40.040 --> 01:08:41.840 What it boils down to is that if you 01:08:41.840 --> 01:08:45.680 can find a radial coordinate that allows you to do this, 01:08:45.680 --> 01:08:48.649 then the condition G upstairs r upstairs r 01:08:48.649 --> 01:08:52.514 equals 0 defines your event horizon, OK? 01:08:52.514 --> 01:08:55.010 It just so happens that it's also 01:08:55.010 --> 01:08:57.800 equal to G downstairs t downstairs t equals 0, 01:08:57.800 --> 01:08:59.899 in this case and in the Schwarzschild case. 01:08:59.899 --> 01:09:02.840 But it's not like that for all black holes 01:09:02.840 --> 01:09:04.310 that you can write down. 01:09:04.310 --> 01:09:07.760 In particular, let me write down the final black hole 01:09:07.760 --> 01:09:11.479 spacetime I want to discuss in this lecture. 01:09:15.564 --> 01:09:17.439 This is gonna take a minute, so bear with me. 01:10:47.090 --> 01:10:53.500 OK, so in this spacetime, the symbol delta I've written here, 01:10:53.500 --> 01:11:00.430 this is r-squared minus 2GM r plus a squared. 01:11:00.430 --> 01:11:07.093 Rho squared is r squared plus a squared cosine 01:11:07.093 --> 01:11:08.260 of the square root of theta. 01:11:11.040 --> 01:11:14.940 This thing turns out, if you compute 01:11:14.940 --> 01:11:17.850 the inverse metric components, you 01:11:17.850 --> 01:11:23.760 find that g upstairs r upstairs r is proportional to delta. 01:11:23.760 --> 01:11:31.230 And so there is a horizon where delta equals 0. 01:11:46.660 --> 01:11:47.160 Sorry. 01:11:59.547 --> 01:12:02.130 That turns out to be located at a radius that looks like this. 01:12:05.760 --> 01:12:13.330 This represents the spacetime of a spinning black hole. 01:12:13.330 --> 01:12:18.870 It was discovered by Roy Kerr, a mathematician from New Zealand. 01:12:21.570 --> 01:12:25.100 I think this was actually a big part of his PhD work. 01:12:35.570 --> 01:12:41.590 So it is known as a Kerr black hole. 01:12:41.590 --> 01:12:49.780 The parameter a is related to the angular momentum, the spin 01:12:49.780 --> 01:12:53.720 angular momentum of the black hole in the units 01:12:53.720 --> 01:12:56.090 that we measure these, normalized to the mass. 01:12:58.790 --> 01:13:09.310 So notice that in order for this to actually have a horizon, 01:13:09.310 --> 01:13:13.900 you need that a to be less than or equal to GM. 01:13:13.900 --> 01:13:16.990 If it saturates that bound, then you 01:13:16.990 --> 01:13:19.750 get what's known as a maximal black hole. 01:13:19.750 --> 01:13:21.550 So it has a couple of noteworthy features. 01:13:31.180 --> 01:13:35.570 First, it is not spherically symmetric. 01:13:44.070 --> 01:13:46.530 If it were spherically symmetric, 01:13:46.530 --> 01:13:50.400 we could write the d theta squared d phi squared piece, 01:13:50.400 --> 01:13:53.130 we could find some radial coordinate such 01:13:53.130 --> 01:14:06.190 that there was some simple radius 01:14:06.190 --> 01:14:13.840 such that G theta theta was simply sine 01:14:13.840 --> 01:14:16.420 squared G theta theta. 01:14:16.420 --> 01:14:19.870 This is the condition that defines spherical symmetry. 01:14:19.870 --> 01:14:25.280 And there is no angle-independent radial 01:14:25.280 --> 01:14:28.040 coordinate that allows you to do that. 01:14:28.040 --> 01:14:37.730 Notice also that there is a connection 01:14:37.730 --> 01:14:40.260 in this coordinate system between t and phi. 01:14:46.230 --> 01:14:58.070 Gt phi is equal to minus 2 GM a r sine squared theta over rho 01:14:58.070 --> 01:14:59.230 squared. 01:14:59.230 --> 01:15:00.770 Why minus 2 and not minus 4? 01:15:00.770 --> 01:15:06.510 Well, remember what I have there is Gt phi dt d phi plus G phi 01:15:06.510 --> 01:15:10.150 t d phi dt. 01:15:10.150 --> 01:15:15.040 One can show that this term, it reflects the kind 01:15:15.040 --> 01:15:19.180 of physics in which the spin of the black hole 01:15:19.180 --> 01:15:22.900 introduces a spinning, almost magnetic-like element 01:15:22.900 --> 01:15:24.760 to gravitation. 01:15:24.760 --> 01:15:26.810 If you guys do the homework assignment 01:15:26.810 --> 01:15:33.040 I have assigned in which you compute the linearized effect 01:15:33.040 --> 01:15:36.350 on a spacetime of a spinning body, 01:15:36.350 --> 01:15:37.790 you'll get a flavor of this, OK? 01:15:37.790 --> 01:15:41.060 This ends up giving you, that calculation gives you 01:15:41.060 --> 01:15:44.360 a similar term in the spacetime which further analysis 01:15:44.360 --> 01:15:46.370 shows leads to bodies. 01:15:46.370 --> 01:15:49.850 Essentially, what you find is that if you have an orbit that 01:15:49.850 --> 01:15:53.240 goes in the same sense as the body's 01:15:53.240 --> 01:15:56.380 spin versus an orbit that goes in the opposite sense 01:15:56.380 --> 01:15:58.430 of the body's spin, there's a splitting 01:15:58.430 --> 01:16:00.390 in the orbit's properties due to that. 01:16:00.390 --> 01:16:00.890 OK? 01:16:00.890 --> 01:16:02.720 So it breaks the symmetry between what 01:16:02.720 --> 01:16:07.400 we call a prograde orbit and a retrograde orbit. 01:16:07.400 --> 01:16:11.660 This is one of the most important solutions 01:16:11.660 --> 01:16:15.260 that we know of in general relativity because of a result 01:16:15.260 --> 01:16:16.520 that I'm going to discuss now. 01:16:20.970 --> 01:16:23.540 Oh, first of all, I should mention that, in fact, you 01:16:23.540 --> 01:16:28.520 can combine charge with spin. 01:16:28.520 --> 01:16:32.390 I'm not going to write down the result because it's just 01:16:32.390 --> 01:16:33.060 kind of messy. 01:16:33.060 --> 01:16:35.060 But it does exist in closed form. 01:16:43.623 --> 01:16:45.040 If you're interested in this, read 01:16:45.040 --> 01:16:51.680 about what is called the Kerr-Newman solution. 01:16:51.680 --> 01:16:55.030 So if you're keeping score, we have this spherically symmetric 01:16:55.030 --> 01:16:58.030 black hole, which only has a mass, 01:16:58.030 --> 01:17:02.030 you have the charged black hole whose name I forgot to list. 01:17:02.030 --> 01:17:02.530 Ah! 01:17:02.530 --> 01:17:03.730 Sorry about that. 01:17:03.730 --> 01:17:21.843 This guy is known as the Reissner-Nordstrom black hole. 01:17:21.843 --> 01:17:23.510 One of those O's, I believe, is supposed 01:17:23.510 --> 01:17:24.677 to have a stroke through it. 01:17:24.677 --> 01:17:27.580 Those of you who speak Scandinavian languages 01:17:27.580 --> 01:17:31.310 can probably spell it and pronounce it better than I can. 01:17:31.310 --> 01:17:32.860 So we have Schwarzschild, which is 01:17:32.860 --> 01:17:37.300 only mass, Reissner-Nordstrom, which is mass in charge, 01:17:37.300 --> 01:17:41.310 Kerr, which is mass and spin, and Kerr-Newman, which 01:17:41.310 --> 01:17:44.110 is mass, spin, and charge. 01:17:44.110 --> 01:17:47.010 You might start thinking, all right, well, 01:17:47.010 --> 01:17:48.400 does this keep going? 01:17:48.400 --> 01:17:54.520 Do I have a solution for a black hole that's got, you know, 01:17:54.520 --> 01:17:58.080 northern hemisphere bigger than the southern hemisphere? 01:17:58.080 --> 01:18:00.100 You know, every time you think about adding 01:18:00.100 --> 01:18:02.230 a bit of extra sort of schmutz to this thing, 01:18:02.230 --> 01:18:05.200 do I need another solution? 01:18:05.200 --> 01:18:07.990 Well, let me describe a remarkable theorem. 01:18:16.770 --> 01:18:40.080 The only stationary spacetimes in 3 plus 1 dimensions with 01:18:40.080 --> 01:18:47.980 event horizons are the Kerr-Newman black holes-- 01:19:02.480 --> 01:19:19.520 completely parameterized by mass, spin, and charge. 01:19:23.270 --> 01:19:26.720 If you take the Kerr-Newman solution, you set charge to 0, 01:19:26.720 --> 01:19:27.700 you get Kerr. 01:19:27.700 --> 01:19:30.230 If you take Kerr and you set spin to 0, 01:19:30.230 --> 01:19:32.010 you get Schwarzschild. 01:19:32.010 --> 01:19:33.600 So the Kerr-Newman solution gives me 01:19:33.600 --> 01:19:36.860 something that includes these other ones as sort of a subset. 01:19:36.860 --> 01:19:39.540 And what this theorem states is that the only-- so 01:19:39.540 --> 01:19:42.240 stationary means time-independent. 01:19:59.070 --> 01:20:01.650 So in other words, the only spacetimes 01:20:01.650 --> 01:20:05.760 that are not dynamical, but that have event horizons, 01:20:05.760 --> 01:20:09.050 at least with three space and one time dimension, 01:20:09.050 --> 01:20:10.650 are the Kerr-Newman black holes. 01:20:10.650 --> 01:20:14.340 Once you know these, you have characterized all black holes 01:20:14.340 --> 01:20:15.850 you can care about. 01:20:15.850 --> 01:20:20.610 And in fact, in any astrophysical context, 01:20:20.610 --> 01:20:23.070 any macroscopic object with charge 01:20:23.070 --> 01:20:26.580 is rapidly neutralized by ambient plasma that just sort 01:20:26.580 --> 01:20:28.520 of fills all of space. 01:20:28.520 --> 01:20:33.840 And so this Kerr solution, in fact, 01:20:33.840 --> 01:20:38.610 gives an exact mathematical description to every black hole 01:20:38.610 --> 01:20:41.740 that we observe in the universe. 01:20:41.740 --> 01:20:44.840 That is an amazing statement. 01:20:44.840 --> 01:20:45.340 OK? 01:20:45.340 --> 01:20:48.430 Of course, as a physicist, you want to test this. 01:20:48.430 --> 01:20:50.710 And this, in fact, is much of what 01:20:50.710 --> 01:20:53.220 my research and research of many of my colleagues is about. 01:20:53.220 --> 01:20:56.470 Can we actually formulate tests of this Kerr hypothesis? 01:20:56.470 --> 01:20:58.210 And many of us have spent our careers 01:20:58.210 --> 01:21:00.010 coming up with such things. 01:21:00.010 --> 01:21:03.280 Suffice it to say, in the roughly negative 5 minutes 01:21:03.280 --> 01:21:06.850 I have left in this lecture, that the Kerr metric has 01:21:06.850 --> 01:21:09.310 survived every test that we have thrown at it. 01:21:12.660 --> 01:21:16.710 So this metric, like I said, was essentially 01:21:16.710 --> 01:21:21.000 derived by the mathematician Roy Kerr as his PhD thesis. 01:21:21.000 --> 01:21:26.880 And it has really earned him a place in physicist Valhalla. 01:21:31.245 --> 01:21:32.620 Let me just conclude this lecture 01:21:32.620 --> 01:21:33.910 by making one comment here. 01:21:39.105 --> 01:21:40.480 An important word in this theorem 01:21:40.480 --> 01:21:45.280 is the statement that the only stationary spacetimes 01:21:45.280 --> 01:21:47.320 are the Kerr-Newman ones, stationary spacetimes 01:21:47.320 --> 01:21:49.810 with event horizons, so the Kerr-Newman black holes. 01:21:49.810 --> 01:21:56.380 What this means is that when a black hole forms, 01:21:56.380 --> 01:22:02.920 it may be dynamical, it may not yet be stationary. 01:22:02.920 --> 01:22:06.610 And so the way that this theorem, which 01:22:06.610 --> 01:22:25.150 is known as the No-Hair theorem, the way that it is enforced 01:22:25.150 --> 01:22:31.240 is that, imagine I have some kind of an object that 01:22:31.240 --> 01:22:34.090 due to physics that we don't have time to go into here, 01:22:34.090 --> 01:22:38.260 imagine that this thing, its physics changes in such a way 01:22:38.260 --> 01:22:41.350 that its fluid can no longer support 01:22:41.350 --> 01:22:45.510 its own mass against gravity, and it 01:22:45.510 --> 01:22:47.680 collapses to a black hole. 01:22:47.680 --> 01:22:48.180 OK? 01:22:48.180 --> 01:22:51.375 Initially, this could be a huge, complicated mess. 01:22:57.960 --> 01:23:07.510 So we have a mass, charge, spin, magnetic fields, who knows? 01:23:15.790 --> 01:23:22.120 The No-Hair theorem guarantees that after some period of time, 01:23:22.120 --> 01:23:25.280 it will be totally characterized by three numbers-- the mass, 01:23:25.280 --> 01:23:28.470 the spin parameter a and the charge q. 01:23:28.470 --> 01:23:38.500 What goes on is that during the collapse process, 01:23:38.500 --> 01:23:40.208 radiation is generated. 01:23:43.560 --> 01:23:46.650 What this radiation does is it carries away 01:23:46.650 --> 01:23:49.890 gravitational waves, carries away electromagnetic waves. 01:23:49.890 --> 01:23:53.100 Some of this is actually absorbed by this black hole. 01:23:53.100 --> 01:23:58.080 And it does so in such a way that it precisely cancels out 01:23:58.080 --> 01:24:00.570 everything in the spacetime that does not 01:24:00.570 --> 01:24:03.210 fit the Kerr-Newman form. 01:24:03.210 --> 01:24:10.360 You wind up-- so this is one of these things where we really 01:24:10.360 --> 01:24:14.830 can only probe this either with observations that sort of look 01:24:14.830 --> 01:24:17.770 at things like black holes and compact bodies 01:24:17.770 --> 01:24:20.500 colliding and forming black holes 01:24:20.500 --> 01:24:22.540 and looking at what the end state looks like. 01:24:22.540 --> 01:24:24.640 Or we can do this with numerical experiments 01:24:24.640 --> 01:24:27.310 where we simulate very complicated collapsing 01:24:27.310 --> 01:24:29.440 or colliding objects on a supercomputer 01:24:29.440 --> 01:24:31.850 and look at what the end result is. 01:24:31.850 --> 01:24:35.470 And what we always find is that the complex radiation that 01:24:35.470 --> 01:24:38.860 is generated in the collapse and the collision process 01:24:38.860 --> 01:24:42.550 always shaves away every bit of structure, 01:24:42.550 --> 01:24:45.070 except for exactly what is left to leave it 01:24:45.070 --> 01:24:47.087 in the Kerr-Newman form at the end. 01:24:47.087 --> 01:24:48.670 Really, when we do these calculations, 01:24:48.670 --> 01:24:50.462 we generally wind up with a Kerr black hole 01:24:50.462 --> 01:24:52.840 because we tend to study astrophysical problems that 01:24:52.840 --> 01:24:54.820 are electrically neutral. 01:24:54.820 --> 01:24:57.760 So this is a result that is sometimes 01:24:57.760 --> 01:25:05.000 called Price's theorem, based on sort 01:25:05.000 --> 01:25:08.510 of foundational calculations that were done 01:25:08.510 --> 01:25:10.340 by my friend Richard Price. 01:25:10.340 --> 01:25:15.770 He did much of this right around the time I was born in the days 01:25:15.770 --> 01:25:19.700 of being a PhD student and looking at the behavior 01:25:19.700 --> 01:25:22.880 of highly distorted black holes and seeing how the No-Hair 01:25:22.880 --> 01:25:24.230 theorem-- 01:25:24.230 --> 01:25:25.940 You can imagine making a spacetime that 01:25:25.940 --> 01:25:27.320 contains what should be a black hole, 01:25:27.320 --> 01:25:28.445 but you somehow distort it. 01:25:28.445 --> 01:25:31.430 What you find is it becomes dynamical 01:25:31.430 --> 01:25:34.682 and it vibrates in such a way as to get rid of that distortion. 01:25:34.682 --> 01:25:36.140 And you leave behind something that 01:25:36.140 --> 01:25:40.600 is precisely Kerr or Kerr-Newman if you have charge. 01:25:40.600 --> 01:25:42.770 Price's theorem is a semi-facetious statement 01:25:42.770 --> 01:25:48.290 that tells me everything in the spacetime that can be radiated, 01:25:48.290 --> 01:25:50.030 is radiated. 01:25:50.030 --> 01:25:59.280 In other words, any bit of structure, any bit of structure 01:25:59.280 --> 01:26:01.500 in the spacetime that does not comport 01:26:01.500 --> 01:26:05.280 with the Kerr-Newman solution radiates away 01:26:05.280 --> 01:26:06.730 and only Kerr-Newman is left. 01:26:09.980 --> 01:26:12.340 So that concludes this lecture. 01:26:12.340 --> 01:26:17.050 My final lecture, which I will record in about 15 minutes, 01:26:17.050 --> 01:26:20.050 is one in which we are going to look 01:26:20.050 --> 01:26:22.480 at one of the ways in which we test this spacetime, which 01:26:22.480 --> 01:26:25.900 is by studying the behavior of orbits 01:26:25.900 --> 01:26:27.830 going around a black hole. 01:26:27.830 --> 01:26:30.210 So I will stop here.