1 00:00:00,000 --> 00:00:01,497 [SQUEAKING] 2 00:00:01,497 --> 00:00:03,493 [RUSTLING] 3 00:00:03,493 --> 00:00:05,988 [CLICKING] 4 00:00:10,490 --> 00:00:13,410 SCOTT HUGHES: So in this final lecture, 5 00:00:13,410 --> 00:00:16,910 I want to think a bit sort of with an eye 6 00:00:16,910 --> 00:00:20,960 towards thinking about how one might actually 7 00:00:20,960 --> 00:00:24,300 make measurements that prove the nature of the black hole 8 00:00:24,300 --> 00:00:26,740 spacetime that was discussed the previous lecture. 9 00:00:26,740 --> 00:00:31,166 I'm going to discuss motion in a black hole spacetime. 10 00:00:41,880 --> 00:00:45,620 We touched on this a little bit in the previous lecture, 11 00:00:45,620 --> 00:00:50,870 where we discussed the motion of radial light rays, OK? 12 00:00:50,870 --> 00:00:56,840 We, in fact, used radial light rays as a critical tool 13 00:00:56,840 --> 00:00:59,690 for describing the properties of the spacetime. 14 00:00:59,690 --> 00:01:01,610 We use that to help us understand the location 15 00:01:01,610 --> 00:01:02,819 of events horizons. 16 00:01:02,819 --> 00:01:04,819 But I want to think a little bit more generally. 17 00:01:04,819 --> 00:01:06,194 What might it look like if I have 18 00:01:06,194 --> 00:01:08,475 material orbiting in the vicinity of one 19 00:01:08,475 --> 00:01:09,350 of these black holes? 20 00:01:09,350 --> 00:01:10,308 What if it's not light? 21 00:01:10,308 --> 00:01:12,420 What if it's made out of matter? 22 00:01:12,420 --> 00:01:14,330 And so what this is going to boil down to 23 00:01:14,330 --> 00:01:18,500 is understanding the behavior of geodesics in a black hole 24 00:01:18,500 --> 00:01:21,300 spacetime. 25 00:01:21,300 --> 00:01:28,330 And the naive approach to doing this is not wrong, but naive. 26 00:01:28,330 --> 00:01:32,820 What you do is you would just take the spacetime-- 27 00:01:32,820 --> 00:01:35,670 take your Schwarzschild or take your Kerr spacetime-- 28 00:01:35,670 --> 00:01:41,280 and turn a very large crank, grind out all of the connection 29 00:01:41,280 --> 00:01:55,170 coefficients, evaluate the geodesic equation, 30 00:01:55,170 --> 00:01:57,360 integrate it up. 31 00:01:57,360 --> 00:02:03,080 Solve for the geodesics. 32 00:02:03,080 --> 00:02:05,390 Boom, you got yourself your motion, OK? 33 00:02:05,390 --> 00:02:07,170 And that is absolutely correct. 34 00:02:07,170 --> 00:02:08,960 You can do that using your Kerr space time 35 00:02:08,960 --> 00:02:11,810 or your Schwarzschild spacetime. 36 00:02:11,810 --> 00:02:16,030 In fact, if you do that for-- 37 00:02:16,030 --> 00:02:18,477 you can get the connection coefficients describing this. 38 00:02:18,477 --> 00:02:20,060 Those are relatively easy to work out. 39 00:02:20,060 --> 00:02:23,510 I think they are listed in Carroll, in equation 5.53, 40 00:02:23,510 --> 00:02:24,992 according to my notes. 41 00:02:24,992 --> 00:02:26,450 That may just be for Schwarzschild. 42 00:02:26,450 --> 00:02:31,460 But at any rate, they're all listed there, and have a blast. 43 00:02:31,460 --> 00:02:36,470 This approach is not wrong, but-- 44 00:02:36,470 --> 00:02:38,152 my notes say, but it is not useful. 45 00:02:38,152 --> 00:02:39,110 That's not really true. 46 00:02:39,110 --> 00:02:42,860 I'll just say that there is a more useful approach to this. 47 00:02:42,860 --> 00:02:56,930 A more fruitful approach is to exploit the fact that these are 48 00:02:56,930 --> 00:02:58,690 highly symmetric spacetimes. 49 00:03:04,110 --> 00:03:10,030 Exploit the symmetries and the Killing 50 00:03:10,030 --> 00:03:22,430 vectors, and see how they can be used 51 00:03:22,430 --> 00:03:25,337 to reduce the number of degrees of freedom 52 00:03:25,337 --> 00:03:26,420 that you need to describe. 53 00:03:42,850 --> 00:03:44,830 In this lecture, I'm going to go through this 54 00:03:44,830 --> 00:03:47,980 in quite a bit of detail for Schwarzschild. 55 00:03:47,980 --> 00:03:51,490 The concepts that I'm going to apply work 56 00:03:51,490 --> 00:03:55,375 for Kerr and for Kerr-Newman as well, OK? 57 00:03:55,375 --> 00:03:57,500 Schwarzschild is just a little easier to work with. 58 00:03:57,500 --> 00:04:00,802 It's something that I can fit into a single lecture. 59 00:04:00,802 --> 00:04:03,260 In particular, one of the nice things about Schwarzschild-- 60 00:04:03,260 --> 00:04:05,485 so let's go ahead and write down that spacetime. 61 00:04:20,109 --> 00:04:22,530 So one of the nice things about Schwarzschild 62 00:04:22,530 --> 00:04:24,660 is it is spherically symmetric. 63 00:04:31,390 --> 00:04:37,560 This means I can always rotate coordinates such that-- 64 00:04:37,560 --> 00:04:39,060 well, when me think about it-- let's 65 00:04:39,060 --> 00:04:40,490 just back up for a second. 66 00:04:40,490 --> 00:04:44,540 Imagine that I have some kind of a body orbiting a Schwarzschild 67 00:04:44,540 --> 00:04:46,280 black hole. 68 00:04:46,280 --> 00:04:49,770 Spherical symmetry tells me that there 69 00:04:49,770 --> 00:04:53,820 must be some notion of a conserved angular momentum such 70 00:04:53,820 --> 00:04:57,348 that that orbit always lies within a given plane. 71 00:04:57,348 --> 00:04:58,140 Put it another way. 72 00:04:58,140 --> 00:05:02,650 Because it is fairly symmetric, there cannot exist a torque. 73 00:05:06,260 --> 00:05:12,560 The black hole cannot exert a torque that changes 74 00:05:12,560 --> 00:05:23,200 the orientation of the orbital plane. 75 00:05:29,220 --> 00:05:31,360 In particular, because it is fairly symmetric, 76 00:05:31,360 --> 00:05:34,720 there is no unique notion of an equator to this object. 77 00:05:34,720 --> 00:05:39,730 And so you might as well define any orbits to live in the theta 78 00:05:39,730 --> 00:05:41,230 equals pi over 2 plane. 79 00:05:41,230 --> 00:05:56,420 You can always rotate your coordinates 80 00:05:56,420 --> 00:06:06,880 to put any orbit in the theta equals pi over 2 plane. 81 00:06:06,880 --> 00:06:09,850 It's actually a pretty simple exercise. 82 00:06:09,850 --> 00:06:12,340 I won't do this, but if you start 83 00:06:12,340 --> 00:06:15,910 with an orbit that is in the theta equals pi over 2 plane, 84 00:06:15,910 --> 00:06:21,730 and it is moving such that its initial velocity would keep it 85 00:06:21,730 --> 00:06:23,710 in a theta equals pi over 2 plane, 86 00:06:23,710 --> 00:06:27,880 it's a very simple exercise using the geodesic equation 87 00:06:27,880 --> 00:06:31,540 to show that it will always be in the theta 88 00:06:31,540 --> 00:06:33,280 equals pi over 2 plane. 89 00:06:33,280 --> 00:06:35,260 So spherical symmetry says, you know what, 90 00:06:35,260 --> 00:06:38,410 let's just forget about the theta degree of freedom. 91 00:06:38,410 --> 00:06:40,840 I can always define my coordinates in such a way 92 00:06:40,840 --> 00:06:44,740 that it lives in the theta equals pi over 2 plane. 93 00:06:44,740 --> 00:06:48,400 Boom, I have reduced my motion from, in general, 94 00:06:48,400 --> 00:06:52,067 being three spatial dimensions to two spatial dimensions. 95 00:06:52,067 --> 00:06:53,650 That's another one of the reasons why, 96 00:06:53,650 --> 00:06:55,630 for pedagogical purposes, it's nice to start 97 00:06:55,630 --> 00:06:57,060 with Schwarzschild. 98 00:06:57,060 --> 00:07:00,180 For Kerr, this is not the case, OK? 99 00:07:00,180 --> 00:07:02,240 Kerr is a little bit more complicated. 100 00:07:02,240 --> 00:07:06,610 You have to treat the theta motion separately. 101 00:07:06,610 --> 00:07:08,860 It's not strictly symmetric, and so you can't do that. 102 00:07:10,992 --> 00:07:12,700 I have actually spent a tremendous amount 103 00:07:12,700 --> 00:07:14,940 of my career studying the orbits of objects 104 00:07:14,940 --> 00:07:15,940 around Kerr black holes. 105 00:07:15,940 --> 00:07:19,570 And I do have to say that the additional complications that 106 00:07:19,570 --> 00:07:21,730 arise from this lack of sphericity, 107 00:07:21,730 --> 00:07:23,020 they're really beautiful, OK? 108 00:07:23,020 --> 00:07:26,940 There's an amazing amount of fun stuff you can do with it. 109 00:07:26,940 --> 00:07:30,630 You know, there's a reason why I just 110 00:07:30,630 --> 00:07:32,380 keep coming back to this research problem, 111 00:07:32,380 --> 00:07:36,820 and part of it is it's just bloody fun. 112 00:07:36,820 --> 00:07:39,460 But if you're teaching this stuff for the first time, 113 00:07:39,460 --> 00:07:41,470 it's not where you want to begin. 114 00:07:41,470 --> 00:07:43,510 All right, so Schwarzschild allows 115 00:07:43,510 --> 00:07:46,780 us to reduce it from a three-dimensional problem 116 00:07:46,780 --> 00:07:48,710 to a two-dimensional problem. 117 00:07:48,710 --> 00:07:53,230 And Schwarzschild also has two Killing vectors. 118 00:07:53,230 --> 00:07:57,940 The time derivative of every metric component is equal to 0. 119 00:07:57,940 --> 00:08:01,015 That means p downstairs t is constant. 120 00:08:09,510 --> 00:08:11,910 So there is a timelike Killing vector. 121 00:08:11,910 --> 00:08:15,810 So p downstairs t is constant every level along the orbit. 122 00:08:15,810 --> 00:08:18,780 It means there exists a timelike Killing vector. 123 00:08:18,780 --> 00:08:22,780 And so what we do is we associate this 124 00:08:22,780 --> 00:08:25,690 with the energy of the orbit. 125 00:08:25,690 --> 00:08:28,780 We call this up to a minus sign. 126 00:08:28,780 --> 00:08:30,970 And we choose that minus sign because if we imagine 127 00:08:30,970 --> 00:08:32,740 orbits have very, very large radius, 128 00:08:32,740 --> 00:08:35,804 the spacetime is nearly flat, and we 129 00:08:35,804 --> 00:08:37,929 want to sort of clear out the minus sign associated 130 00:08:37,929 --> 00:08:39,789 with lowering our index here. 131 00:08:39,789 --> 00:08:42,159 We're going to call that constant negative 132 00:08:42,159 --> 00:08:43,590 of the energy of the orbit. 133 00:08:51,880 --> 00:08:58,680 The spacetime is also independent of the angle phi. 134 00:08:58,680 --> 00:09:07,560 And so p sub phi is a constant. 135 00:09:07,560 --> 00:09:10,650 This means that the spacetime has an axial Killing 136 00:09:10,650 --> 00:09:13,975 vector, something associated with motions around a symmetry 137 00:09:13,975 --> 00:09:14,475 axis. 138 00:09:16,990 --> 00:09:18,520 So I'm going to call this L-- well, 139 00:09:18,520 --> 00:09:20,850 actually, I will tend to call it L sub z. 140 00:09:20,850 --> 00:09:23,470 You can kind of think of this as, after I 141 00:09:23,470 --> 00:09:26,410 have put everything in the theta equals pi over 2 plane, 142 00:09:26,410 --> 00:09:32,080 this is like an angular momentum on the-- 143 00:09:32,080 --> 00:09:34,920 angular momentum parallel to the axis normal to that plane, 144 00:09:34,920 --> 00:09:37,420 and I call that the z-axis. 145 00:09:37,420 --> 00:09:40,180 It's worth noting that all three of these things 146 00:09:40,180 --> 00:09:43,450 are also true for Reissner-Nordstrom black holes. 147 00:09:43,450 --> 00:09:48,620 This one is not true for Kerr, but these two are, OK? 148 00:09:48,620 --> 00:09:54,130 So although the details change, many of the concepts 149 00:09:54,130 --> 00:09:56,920 will carry over when you look at different, more complicated 150 00:09:56,920 --> 00:09:58,620 classes of black holes. 151 00:09:58,620 --> 00:10:00,370 Let's now think about the forward momentum 152 00:10:00,370 --> 00:10:02,740 of a body moving in a Schwarzschild spacetime. 153 00:10:16,470 --> 00:10:22,820 So let's say it's got a rest mass m, 154 00:10:22,820 --> 00:10:25,890 then it will have three components. 155 00:10:25,890 --> 00:10:27,770 Remember, I have put this thing in a plane 156 00:10:27,770 --> 00:10:29,090 where there is no theta motion. 157 00:10:32,130 --> 00:10:35,530 It will have three theta motions describing its motion 158 00:10:35,530 --> 00:10:37,030 with respect to the time coordinate, 159 00:10:37,030 --> 00:10:42,070 the radial coordinate, and the actual coordinate phi, OK? 160 00:10:42,070 --> 00:10:43,690 So before I do anything with this, 161 00:10:43,690 --> 00:10:47,410 let's take advantage of the fact that we have these quantities 162 00:10:47,410 --> 00:10:49,790 that are constants. 163 00:10:49,790 --> 00:10:52,870 So using the fact that p downstairs mu 164 00:10:52,870 --> 00:10:55,240 is what I get when I hit this guy-- 165 00:10:55,240 --> 00:10:59,210 that be a mu, pardon me. 166 00:10:59,210 --> 00:11:03,680 This is what I get when I hit this guy with the metric. 167 00:11:03,680 --> 00:11:08,245 I can write out a p sub t and p sub phi. 168 00:11:08,245 --> 00:11:10,010 So let's do that on another board. 169 00:11:10,010 --> 00:11:12,050 Well, let's do a p sub t first. 170 00:11:12,050 --> 00:11:20,932 So p sub t is going to be gtt, p sub-- 171 00:11:20,932 --> 00:11:21,432 whoops. 172 00:11:28,730 --> 00:11:32,430 OK, this is minus. 173 00:11:32,430 --> 00:11:34,350 There's an m from that-- 174 00:11:34,350 --> 00:11:35,860 minus sign from my metric. 175 00:11:45,560 --> 00:11:52,352 And this whole thing I define as the negative energy, e. 176 00:12:00,240 --> 00:12:03,210 OK, so that combination-- 177 00:12:03,210 --> 00:12:05,500 we'll do something at the rest mass in just a moment. 178 00:12:05,500 --> 00:12:07,930 But you should basically look at this 179 00:12:07,930 --> 00:12:12,370 and saying that dt d tau, which tells me 180 00:12:12,370 --> 00:12:16,060 about how the body moves with respect to the Schwarzschild 181 00:12:16,060 --> 00:12:18,000 time coordinate-- 182 00:12:18,000 --> 00:12:21,240 that complement of the four velocity times 1 minus 183 00:12:21,240 --> 00:12:25,390 2gm over r is a constant. 184 00:12:25,390 --> 00:12:27,730 Let's look at p downstairs phi. 185 00:12:30,472 --> 00:12:36,180 So this is m r squared sine squared theta. 186 00:12:36,180 --> 00:12:44,870 Ah, we've chosen our orbital plane, 1 times d phi d tau. 187 00:12:44,870 --> 00:12:52,560 This is equal to [INAUDIBLE] momentum L sub z. 188 00:12:52,560 --> 00:12:55,260 In my notes, I might flip around a little bit between L sub z 189 00:12:55,260 --> 00:12:57,910 and L. 190 00:12:57,910 --> 00:13:02,900 So let's massage these a little bit more. 191 00:13:02,900 --> 00:13:05,790 So I can take these two expressions 192 00:13:05,790 --> 00:13:09,980 and I can use them to write dt d tau 193 00:13:09,980 --> 00:13:17,990 and d phi d tau in terms of these conserved quantities. 194 00:13:17,990 --> 00:13:20,870 So dt d tau is equal to-- 195 00:13:20,870 --> 00:13:33,725 I'm going to call it e hat divided by 1 minus 2gm over r. 196 00:13:33,725 --> 00:13:40,930 d phi d tau is going to be Lz hat over our squared. 197 00:13:40,930 --> 00:13:44,110 And so these quantities with a hat 198 00:13:44,110 --> 00:13:52,920 are just the conserved values normalized to the rest mass, 199 00:13:52,920 --> 00:13:53,868 OK? 200 00:13:53,868 --> 00:13:55,660 They all are proportional to the rest mass, 201 00:13:55,660 --> 00:13:58,830 but it's actually if you want to know what the-- 202 00:13:58,830 --> 00:14:01,330 this is essentially telling me about how clocks on the orbit 203 00:14:01,330 --> 00:14:04,750 tick relative to clocks that are infinitely far away. 204 00:14:04,750 --> 00:14:08,840 And that can't depend on the mass of the orbiting object. 205 00:14:08,840 --> 00:14:11,080 This is telling me about how this small body is 206 00:14:11,080 --> 00:14:14,458 moving according to the clock of the orbiting observer. 207 00:14:14,458 --> 00:14:16,750 And again, that can't depend on the mass of the object. 208 00:14:19,400 --> 00:14:21,260 OK, so that's kind of cool. 209 00:14:21,260 --> 00:14:24,430 So I've now managed to relate three-- 210 00:14:24,430 --> 00:14:27,130 excuse me, two of the three components 211 00:14:27,130 --> 00:14:32,020 of the forward momentum to functions of r and quantities 212 00:14:32,020 --> 00:14:34,330 that are known to be constant. 213 00:14:34,330 --> 00:14:36,160 That's good. 214 00:14:36,160 --> 00:14:38,170 We have one more overall constraint. 215 00:14:40,760 --> 00:14:53,960 We know that if I take p dot p, I get negative of the rest mass 216 00:14:53,960 --> 00:14:54,460 squared. 217 00:14:58,150 --> 00:15:01,360 So this, when I write it out-- 218 00:15:32,190 --> 00:15:37,850 OK, notice every single term is proportional to m squared. 219 00:15:37,850 --> 00:15:40,180 So I can divide that out. 220 00:15:40,180 --> 00:15:43,930 I can insert dt d tau. 221 00:15:43,930 --> 00:15:47,560 I can replace for this e divided by this guy. 222 00:15:47,560 --> 00:15:52,420 I can replace d phi d tau by my Lz hat divided by r squared. 223 00:15:59,080 --> 00:16:09,810 Doing so, manipulating a little bit, I get-- 224 00:16:28,842 --> 00:16:31,710 we get something like this. 225 00:16:31,710 --> 00:16:35,190 And let me rearrange this a tiny bit. 226 00:17:13,099 --> 00:17:16,800 All right, I have managed to reduce this 227 00:17:16,800 --> 00:17:20,310 to a one-dimensional problem, OK? 228 00:17:20,310 --> 00:17:25,380 So going from that line over to there, basically all I did 229 00:17:25,380 --> 00:17:31,560 was insert the relationship between dt d tau and E, d phi 230 00:17:31,560 --> 00:17:35,550 d tau and L, cleared out some overall factors of things 231 00:17:35,550 --> 00:17:39,610 like 1 minus 2GM/r, manipulate, manipulate, manipulate, 232 00:17:39,610 --> 00:17:42,180 and what you finally get is this lovely equation here 233 00:17:42,180 --> 00:17:47,460 that tells you how the radial velocity, the radial velocity 234 00:17:47,460 --> 00:17:50,640 with respect to proper time, how it 235 00:17:50,640 --> 00:17:57,420 depends as a function of r given the energy and the angular 236 00:17:57,420 --> 00:17:58,840 momentum. 237 00:17:58,840 --> 00:18:03,990 I have written it in this form because the problem 238 00:18:03,990 --> 00:18:10,800 is very strikingly reminiscent to the Newtonian problem 239 00:18:10,800 --> 00:18:15,990 of understanding the motion of a particle in a 1/r potential, 240 00:18:15,990 --> 00:18:19,950 which we often describe as having an effective potential 241 00:18:19,950 --> 00:18:22,230 that has a gravitational term-- 242 00:18:22,230 --> 00:18:23,850 Newton's gravity, or if you're doing 243 00:18:23,850 --> 00:18:26,280 things like quantum mechanics, the Coulomb potential, 244 00:18:26,280 --> 00:18:29,010 and a Coulomb barrier associated with the angular 245 00:18:29,010 --> 00:18:32,340 momentum of an orbit. 246 00:18:32,340 --> 00:18:35,430 So one way to approach what we've got now, 247 00:18:35,430 --> 00:18:40,750 one thing that we could do, is essentially just 248 00:18:40,750 --> 00:18:48,410 pick your energy and your Lz-- 249 00:18:48,410 --> 00:18:50,870 pick your energy and your angular momentum-- 250 00:18:50,870 --> 00:18:52,265 pick an initial position-- 251 00:19:00,935 --> 00:19:05,940 let's imagine you synchronize the clocks at t equals 0-- 252 00:19:05,940 --> 00:19:07,876 and then just integrate. 253 00:19:17,810 --> 00:19:20,780 You've got your dr d tau given here. 254 00:19:20,780 --> 00:19:27,500 Don't forget, you also have d phi d tau and dt d tau related 255 00:19:27,500 --> 00:19:29,880 to the energy and your angular momentum, like so. 256 00:19:29,880 --> 00:19:30,380 Boom. 257 00:19:30,380 --> 00:19:31,190 It's a closed system. 258 00:19:31,190 --> 00:19:33,170 You can always do just sort of a little numerical integration 259 00:19:33,170 --> 00:19:33,670 of this. 260 00:19:37,780 --> 00:19:39,340 In a certain sense, this completely 261 00:19:39,340 --> 00:19:40,760 specifies the problem. 262 00:19:40,760 --> 00:19:43,660 But there's so much more we can do. 263 00:19:43,660 --> 00:19:46,870 In particular, what we see is that all 264 00:19:46,870 --> 00:19:50,530 of the interesting behavior associated with this orbit 265 00:19:50,530 --> 00:19:54,354 is bound up in this function V effective. 266 00:20:09,670 --> 00:20:13,000 So something that's really useful for us to do 267 00:20:13,000 --> 00:20:17,440 is to take a look at what this V effective looks like. 268 00:20:17,440 --> 00:20:19,750 So suppose you are given a particular value 269 00:20:19,750 --> 00:20:26,410 for E hat and Lz hat, and you plot V effective versus r. 270 00:20:35,200 --> 00:20:39,310 Well, what you typically find is that it's 271 00:20:39,310 --> 00:20:45,710 got a behavior that looks kind of like this, where this value 272 00:20:45,710 --> 00:20:57,190 right here is V effective equal to 1. 273 00:20:57,190 --> 00:21:01,020 Notice as r goes to infinity, you get 1 times 1. 274 00:21:01,020 --> 00:21:02,610 So this asymptotes to 1. 275 00:21:05,940 --> 00:21:11,040 Notice E hat squared has the same dimensions as V effective. 276 00:21:11,040 --> 00:21:13,670 In fact, in the unit choices I've used here, 277 00:21:13,670 --> 00:21:16,050 they are both dimensionless. 278 00:21:16,050 --> 00:21:19,200 So what we can do is plot-- 279 00:21:19,200 --> 00:21:21,920 let's imagine that we have-- 280 00:21:25,737 --> 00:21:26,320 you know what? 281 00:21:26,320 --> 00:21:28,390 I'm going to want to sketch this on a different board. 282 00:21:28,390 --> 00:21:29,223 Let me go over here. 283 00:21:32,930 --> 00:21:36,990 So I'm going to want to look at a couple of different values 284 00:21:36,990 --> 00:21:38,325 of V effective-- 285 00:21:41,328 --> 00:21:43,370 excuse me, a couple of different values of E hat. 286 00:22:12,457 --> 00:22:12,980 OK. 287 00:22:12,980 --> 00:22:14,020 Here's an example. 288 00:22:14,020 --> 00:22:16,230 This guy is asymptoting at 1. 289 00:22:16,230 --> 00:22:20,660 So since E hat, as I said, has the same units as V 290 00:22:20,660 --> 00:22:34,770 effective, let's plot them together. 291 00:22:41,860 --> 00:22:49,420 So example one-- imagine if E hat lies right here. 292 00:22:53,030 --> 00:22:53,530 OK. 293 00:22:53,530 --> 00:22:57,340 So let's call this E hat 1. 294 00:23:01,070 --> 00:23:06,160 So this is some value that is greater than 1. 295 00:23:06,160 --> 00:23:07,970 What is the point of doing this? 296 00:23:07,970 --> 00:23:11,492 Well, notice-- I'm going to flip back 297 00:23:11,492 --> 00:23:13,450 and forth between these two middle boards here. 298 00:23:21,620 --> 00:23:23,540 Let's look at the equation that governs 299 00:23:23,540 --> 00:23:25,400 the radial motion of this body. 300 00:23:35,630 --> 00:23:39,360 dr d tau has to be a real number. 301 00:23:49,270 --> 00:23:51,190 This has to be a real number. 302 00:23:51,190 --> 00:23:55,200 So we have to have E hat squared greater than 303 00:23:55,200 --> 00:24:01,400 or equal to V effective in order for dr d tau 304 00:24:01,400 --> 00:24:02,650 to have a meaningful solution. 305 00:24:18,960 --> 00:24:21,090 So let's look at my example here, E hat 1. 306 00:24:27,410 --> 00:24:32,860 E hat 1 is greater than my effective potential 307 00:24:32,860 --> 00:24:38,580 everywhere at all radii until I get down to here. 308 00:24:43,062 --> 00:24:45,430 Let's call that r1. 309 00:24:45,430 --> 00:24:53,890 So in this case, dr d tau, if I think about this thing-- 310 00:24:53,890 --> 00:24:56,380 so note that defines dr d tau squared. 311 00:24:56,380 --> 00:24:59,250 Let's suppose we take the negative square root. 312 00:24:59,250 --> 00:25:03,005 dr d tau is positive and inward-- 313 00:25:03,005 --> 00:25:05,057 or, well, it's negative-- 314 00:25:05,057 --> 00:25:07,390 negative and real, negative and real, negative and real, 315 00:25:07,390 --> 00:25:09,430 negative and real, negative and real-- boom. 316 00:25:09,430 --> 00:25:10,080 It's 0. 317 00:25:12,720 --> 00:25:14,310 Can it go into here? 318 00:25:14,310 --> 00:25:15,360 No. 319 00:25:15,360 --> 00:25:19,290 It cannot go into there, because there E hat squared is less-- 320 00:25:22,580 --> 00:25:23,080 sorry. 321 00:25:23,080 --> 00:25:24,430 That should have been squared. 322 00:25:24,430 --> 00:25:26,380 Inside here, E hat squared is less 323 00:25:26,380 --> 00:25:29,440 than the effective potential. dr d tau is imaginary. 324 00:25:29,440 --> 00:25:31,300 That doesn't make any sense. 325 00:25:31,300 --> 00:25:35,890 The only option is for this guy to change sign and trundle 326 00:25:35,890 --> 00:25:37,760 right back out. 327 00:25:37,760 --> 00:25:41,470 So E hat greater than 1 corresponds 328 00:25:41,470 --> 00:25:56,270 to a body that comes in from large radius, 329 00:25:56,270 --> 00:26:08,010 turns around at particular radius where E hat is 330 00:26:08,010 --> 00:26:11,020 the square root of the effective potential, 331 00:26:11,020 --> 00:26:18,380 and then goes back out to infinity or back out to-- let's 332 00:26:18,380 --> 00:26:20,500 not say infinity-- goes back out to large radius. 333 00:26:25,813 --> 00:26:27,270 OK? 334 00:26:27,270 --> 00:26:31,890 In Newtonian gravity, we would call this a hyperbolic orbit. 335 00:26:31,890 --> 00:26:33,670 This corresponds to-- so remember, 336 00:26:33,670 --> 00:26:35,070 when I did this I have not-- 337 00:26:35,070 --> 00:26:37,860 I'm not actually-- I'm only computing the radial motion. 338 00:26:37,860 --> 00:26:39,880 I'm not looking at the phi motion. 339 00:26:39,880 --> 00:26:43,140 So this actually, when you look at both the radial 340 00:26:43,140 --> 00:26:45,450 motion and the phi motion, what you see 341 00:26:45,450 --> 00:26:48,628 is that this is a body that comes in and then sort of whips 342 00:26:48,628 --> 00:26:50,670 around that small radius and goes right back out. 343 00:26:57,260 --> 00:26:59,330 Let's look at another example. 344 00:27:04,870 --> 00:27:07,130 Let's call this E2-- 345 00:27:11,650 --> 00:27:12,970 some value that is less than 1. 346 00:27:16,750 --> 00:27:17,250 OK. 347 00:27:17,250 --> 00:27:22,690 Well, for E hat 2, it's a potential. 348 00:27:28,840 --> 00:27:33,810 The potential is underneath E hat 2 squared only 349 00:27:33,810 --> 00:27:41,710 between these two radii, which I will call r sub p and r sub a. 350 00:27:45,680 --> 00:27:49,610 What we expect in this case is motion of this body essentially 351 00:27:49,610 --> 00:27:53,000 going back and forth and turning around 352 00:27:53,000 --> 00:27:58,130 at periastron and apoastron. 353 00:27:58,130 --> 00:28:04,670 This is a relativistic generalization 354 00:28:04,670 --> 00:28:06,170 of an elliptical orbit. 355 00:28:06,170 --> 00:28:07,670 In general relativity, they turn out 356 00:28:07,670 --> 00:28:10,310 generally not to be ellipses. 357 00:28:10,310 --> 00:28:11,870 So we call this an eccentric orbit. 358 00:28:35,790 --> 00:28:38,970 In the weak field limit, if you imagine r being very, very 359 00:28:38,970 --> 00:28:44,190 large, it's not hard to show that the motion is nearly 360 00:28:44,190 --> 00:28:48,120 an ellipse, but it's an ellipse whose long axis is slowly 361 00:28:48,120 --> 00:28:49,770 precessing. 362 00:28:49,770 --> 00:28:52,980 This actually leads to the famous perihelion precession 363 00:28:52,980 --> 00:28:57,160 of Mercury that Einstein first calculated. 364 00:28:57,160 --> 00:29:00,270 And this is an exercise that I am 365 00:29:00,270 --> 00:29:04,620 asking you to do on one of the final P-sets of this course. 366 00:29:04,620 --> 00:29:06,540 Using what I have set up here, it's really not 367 00:29:06,540 --> 00:29:08,960 that difficult to do. 368 00:29:08,960 --> 00:29:10,350 Let me go to another board. 369 00:29:16,060 --> 00:29:26,530 And note that one could imagine an energy such that dr d tau 370 00:29:26,530 --> 00:29:27,715 is exactly 0. 371 00:29:40,940 --> 00:29:43,640 So if you choose your energy so that you 372 00:29:43,640 --> 00:29:48,020 sit right here at the minimum of the potential-- 373 00:29:48,020 --> 00:29:49,450 I will label this as point s-- 374 00:29:52,140 --> 00:29:54,520 the energy that corresponds to exactly that 375 00:29:54,520 --> 00:29:57,280 point is what would be a-- 376 00:29:57,280 --> 00:30:01,990 that is, there is a single point at which dr d tau equals 0. 377 00:30:01,990 --> 00:30:03,520 Anywhere away from that, dr d tau 378 00:30:03,520 --> 00:30:06,010 would be imaginary, so that's not going to work. 379 00:30:06,010 --> 00:30:12,640 But right at that point, dr d tau equals 0, 380 00:30:12,640 --> 00:30:13,930 and you get a circular orbit. 381 00:30:16,988 --> 00:30:19,280 Notice there's a second point at which that can happen. 382 00:30:19,280 --> 00:30:20,450 Let's call this point u. 383 00:30:28,630 --> 00:30:32,090 Perhaps you can guess why I called these points s and u. 384 00:30:32,090 --> 00:30:34,935 If you imagine that you add a tiny amount of energy 385 00:30:34,935 --> 00:30:36,310 right here, well, what it will do 386 00:30:36,310 --> 00:30:39,850 is it will execute small oscillations around the point 387 00:30:39,850 --> 00:30:40,385 s. 388 00:30:40,385 --> 00:30:42,010 It will sort of move it up to something 389 00:30:42,010 --> 00:30:44,830 that's similar to what I drew up there as E2, 390 00:30:44,830 --> 00:30:47,570 but with a very small eccentricity. 391 00:30:47,570 --> 00:30:52,960 So if I slightly disturb a circular orbit down here at s, 392 00:30:52,960 --> 00:30:55,030 I essentially just oscillate in the vicinity 393 00:30:55,030 --> 00:30:57,190 of that circular orbit. 394 00:30:57,190 --> 00:30:58,930 S stands for stable. 395 00:31:01,730 --> 00:31:05,440 If I have an orbit up here at u and I very slightly perturb 396 00:31:05,440 --> 00:31:12,070 it, well, it'll either go in and eventually reach r equals 0, 397 00:31:12,070 --> 00:31:15,102 or it'll go out, and then it's completely unbound, 398 00:31:15,102 --> 00:31:17,560 and it will just keep trundling all the way out essentially 399 00:31:17,560 --> 00:31:19,750 forever. 400 00:31:19,750 --> 00:31:21,730 This guy is unstable. 401 00:31:26,230 --> 00:31:31,700 Stable orbits are particularly interesting and important. 402 00:31:31,700 --> 00:31:35,440 So let's look at these orbits with a little bit more care. 403 00:31:51,230 --> 00:31:53,570 So the very definition of a circular orbit 404 00:31:53,570 --> 00:31:56,060 is that dr d tau equals 0. 405 00:31:56,060 --> 00:31:57,615 Its radius does not change. 406 00:32:05,230 --> 00:32:09,160 If dr d tau equals 0, then it must 407 00:32:09,160 --> 00:32:17,067 have E hat equal the square root of the effective. 408 00:32:19,750 --> 00:32:23,350 Both of these orbits happen to live 409 00:32:23,350 --> 00:32:27,820 at either a minimum or a maximum of this potential curve. 410 00:32:38,710 --> 00:32:46,410 So I'm going to require that the partial derivative 411 00:32:46,410 --> 00:32:48,600 of that potential with respect r be equal to 0. 412 00:32:53,070 --> 00:32:54,910 So let's take a look at this condition. 413 00:33:01,160 --> 00:33:03,590 My effective potential is given up here. 414 00:33:03,590 --> 00:33:07,640 Do a little bit of algebra with this, set this guy equal to 0. 415 00:33:07,640 --> 00:33:18,090 What you'll find after your algebraic smoke clears 416 00:33:18,090 --> 00:33:22,020 is that you get this condition on the angular momentum. 417 00:33:40,090 --> 00:33:42,840 Notice as r gets really large-- 418 00:33:42,840 --> 00:33:43,340 oh, shoot. 419 00:33:47,240 --> 00:33:48,320 Try it again. 420 00:33:48,320 --> 00:33:51,560 Notice as r gets really large that this 421 00:33:51,560 --> 00:33:58,560 asymptotes to plus or minus the square root of GM r. 422 00:33:58,560 --> 00:34:02,280 That is indeed exactly what you get for the angular momentum 423 00:34:02,280 --> 00:34:05,600 of a circular Newtonian orbit. 424 00:34:05,600 --> 00:34:06,450 OK. 425 00:34:06,450 --> 00:34:08,040 So it's a nice sanity check. 426 00:34:13,510 --> 00:34:15,820 It appears to be somewhat pathological 427 00:34:15,820 --> 00:34:18,969 as r approaches 3GM, though. 428 00:34:18,969 --> 00:34:19,719 Hold that thought. 429 00:34:41,920 --> 00:34:43,110 OK. 430 00:34:43,110 --> 00:34:48,719 So now let's take that value of L, 431 00:34:48,719 --> 00:34:53,820 plug it back into the potential, and set 432 00:34:53,820 --> 00:35:00,720 E equal to the square root of that. 433 00:35:00,720 --> 00:35:02,340 A little bit of algebra ensues. 434 00:35:02,340 --> 00:35:21,010 And what you find is that this equals 1 minus 2GM/r 435 00:35:21,010 --> 00:35:25,240 over, again, that factor under a square root of 1 minus 3GM/r. 436 00:35:25,240 --> 00:35:28,090 Again, we sort of see something a little bit 437 00:35:28,090 --> 00:35:33,060 pathological happening as r goes to 3GM. 438 00:35:33,060 --> 00:35:35,690 Let me make two comments about this. 439 00:35:35,690 --> 00:35:39,610 So first of all, notice that this energy is smaller than 1. 440 00:35:46,120 --> 00:35:55,310 I can intuitively-- you can sort of imagine-- remember, 441 00:35:55,310 --> 00:35:57,440 E hat is the energy per unit rest mass. 442 00:36:02,440 --> 00:36:04,180 You can think of this as something 443 00:36:04,180 --> 00:36:08,200 like total energy over M-- 444 00:36:08,200 --> 00:36:11,170 so the energy associated with the orbiting body. 445 00:36:11,170 --> 00:36:18,310 It's got a rest energy, a kinetic energy, 446 00:36:18,310 --> 00:36:25,050 and a potential energy divided by m. 447 00:36:25,050 --> 00:36:29,070 For an orbit to be bound, the potential energy, 448 00:36:29,070 --> 00:36:31,260 which is negative, must have larger magnitude 449 00:36:31,260 --> 00:36:33,150 than the kinetic energy. 450 00:36:33,150 --> 00:36:37,080 So for a bound orbit, E kinetic plus E potential 451 00:36:37,080 --> 00:36:39,660 will be a negative quantity. 452 00:36:39,660 --> 00:36:42,150 So the numerator is going to be something 453 00:36:42,150 --> 00:36:44,250 that, when normalized to m, is less than 1. 454 00:36:44,250 --> 00:36:45,730 So this is exactly what we expect 455 00:36:45,730 --> 00:36:48,180 to describe a bound orbit. 456 00:36:48,180 --> 00:36:50,880 Notice, also-- so if you take this formula 457 00:36:50,880 --> 00:36:59,160 and look at it in the large r limit, it goes to 1 minus GM 458 00:36:59,160 --> 00:37:00,930 over 2r. 459 00:37:00,930 --> 00:37:03,600 This is in fact exactly what you get 460 00:37:03,600 --> 00:37:08,570 when you look at the energy per unit mass throwing in-- 461 00:37:08,570 --> 00:37:09,690 sort of by hand-- 462 00:37:09,690 --> 00:37:11,220 a rest mass. 463 00:37:11,220 --> 00:37:14,160 The minus GM/2r exactly corresponds 464 00:37:14,160 --> 00:37:19,257 to kinetic plus potential for a Newtonian circular orbit. 465 00:37:19,257 --> 00:37:21,090 So lots of stuff is hanging together nicely. 466 00:37:26,270 --> 00:37:30,110 So we've just learned that we can characterize the energy 467 00:37:30,110 --> 00:37:35,270 and angular momentum of circular orbits around my black hole. 468 00:37:35,270 --> 00:37:37,700 Let's look at a couple other things associated with this. 469 00:37:37,700 --> 00:37:41,420 So these plots where I look at the radial motion, 470 00:37:41,420 --> 00:37:46,410 this effective potential, as I mentioned a few moments ago, 471 00:37:46,410 --> 00:37:47,930 there's additional sort of degrees 472 00:37:47,930 --> 00:37:50,750 of freedom in the motion that are being suppressed here. 473 00:37:50,750 --> 00:37:54,050 So this thing is also moving in that plane. 474 00:37:54,050 --> 00:37:56,600 It's whirling around with respect to phi. 475 00:37:56,600 --> 00:38:00,290 We've lost that information in the way we've drawn this here. 476 00:38:00,290 --> 00:38:07,120 Let's define omega to be the angular velocity of this orbit 477 00:38:07,120 --> 00:38:09,260 as seen by a distant observer. 478 00:38:22,450 --> 00:38:23,300 OK. 479 00:38:23,300 --> 00:38:26,900 Why am I doing it as seen by a distant observer? 480 00:38:26,900 --> 00:38:29,620 Well, when things orbit, there tend 481 00:38:29,620 --> 00:38:34,580 to be periodicities that imprint themselves on observables. 482 00:38:34,580 --> 00:38:36,673 It could be the period associated 483 00:38:36,673 --> 00:38:38,840 with the gravitational wave that arises out of this. 484 00:38:38,840 --> 00:38:41,900 It could be the period associated 485 00:38:41,900 --> 00:38:46,040 with peaks and a light curve if this is a star orbiting 486 00:38:46,040 --> 00:38:47,900 around a black hole. 487 00:38:47,900 --> 00:38:52,400 It could be oscillations in the X-ray flux 488 00:38:52,400 --> 00:38:54,410 if this is some kind of a lump in an accretion 489 00:38:54,410 --> 00:38:57,210 disk of material orbiting a black hole. 490 00:38:57,210 --> 00:38:59,720 So if this is the angular velocity seen by distant 491 00:38:59,720 --> 00:39:04,010 observers-- remember, the time that distant observers use to-- 492 00:39:04,010 --> 00:39:07,100 the time in which the distant observers' clocks run 493 00:39:07,100 --> 00:39:10,280 is the Schwarzschild time t. 494 00:39:10,280 --> 00:39:19,070 So this will be d phi dt, which I can write as d phi d tau-- 495 00:39:19,070 --> 00:39:23,690 this is the angular velocity according to the orbit itself-- 496 00:39:23,690 --> 00:39:29,120 normalized to dt d tau. 497 00:39:29,120 --> 00:39:32,810 Now, these are both quantities that are simply 498 00:39:32,810 --> 00:39:36,470 related to constants of motion. 499 00:39:36,470 --> 00:39:41,040 What I've got in the numerator here is L hat over r squared. 500 00:39:41,040 --> 00:39:42,650 And what I've got in the denominator 501 00:39:42,650 --> 00:39:47,980 here is E hat over 1 minus-- 502 00:39:47,980 --> 00:39:49,698 I dropped my t. 503 00:39:49,698 --> 00:39:50,990 It would happen at some point-- 504 00:39:50,990 --> 00:39:52,355 1 minus 2GM/r. 505 00:39:58,040 --> 00:40:01,460 So let's go ahead and take our solution here. 506 00:40:04,070 --> 00:40:09,920 My E hat is 1 minus 2GM/r divided by square root of blah, 507 00:40:09,920 --> 00:40:10,640 blah, blah. 508 00:40:10,640 --> 00:40:13,550 The 1 minus 2GM/r cancels. 509 00:40:13,550 --> 00:40:20,630 My L hat is square root GM r divided by, again, 510 00:40:20,630 --> 00:40:23,510 that square root 1 minus 3GM/r. 511 00:40:23,510 --> 00:40:28,250 Notice, the square root 1 minus 3GM/r factors all cancel. 512 00:40:28,250 --> 00:40:38,160 So this becomes plus or minus 1 over r squared square root GM 513 00:40:38,160 --> 00:40:38,660 r. 514 00:40:44,920 --> 00:40:45,730 Looks like this. 515 00:40:45,730 --> 00:40:49,180 Plus and minus basically just correspond to 516 00:40:49,180 --> 00:40:51,880 whether this motion sort of is going 517 00:40:51,880 --> 00:40:53,770 in the same sense as your phi coordinate 518 00:40:53,770 --> 00:40:55,203 or in the opposite sense. 519 00:40:55,203 --> 00:40:56,620 There's really no physics in that. 520 00:40:56,620 --> 00:41:00,850 It just comes along for the ride that both behave the same. 521 00:41:00,850 --> 00:41:02,530 If you guys get interested in this, 522 00:41:02,530 --> 00:41:05,830 and you do a similar calculation around a Kerr black hole, 523 00:41:05,830 --> 00:41:08,200 you'll find that your prograde solution gives you 524 00:41:08,200 --> 00:41:10,930 a different frequency than your retrograde solution because 525 00:41:10,930 --> 00:41:13,270 of the fact that the dragging of inertial frames 526 00:41:13,270 --> 00:41:15,730 due to the spin of the black hole kind of breaks 527 00:41:15,730 --> 00:41:17,880 that symmetry. 528 00:41:17,880 --> 00:41:20,960 Something which is interesting and-- 529 00:41:20,960 --> 00:41:23,460 well, I'll make a comment about this in just a second, which 530 00:41:23,460 --> 00:41:24,490 is kind of interesting. 531 00:41:24,490 --> 00:41:26,865 And here's what I'll say-- sometimes people think this is 532 00:41:26,865 --> 00:41:28,620 more profound than it should be-- 533 00:41:28,620 --> 00:41:34,410 is this is, in fact, exactly the same frequency law 534 00:41:34,410 --> 00:41:37,050 that you get using Newtonian gravity. 535 00:41:37,050 --> 00:41:45,570 This is actually exactly the same as Kepler's law. 536 00:41:52,490 --> 00:41:54,150 That seems really, really cool. 537 00:41:54,150 --> 00:41:54,650 And it is. 538 00:41:54,650 --> 00:41:55,817 It's actually really useful. 539 00:41:55,817 --> 00:41:57,830 It makes it very easy to remember this. 540 00:41:57,830 --> 00:42:01,310 But don't read too much into it. 541 00:42:01,310 --> 00:42:03,470 More than anything, it is a statement 542 00:42:03,470 --> 00:42:07,220 about a particular quality of this radial coordinate. 543 00:42:07,220 --> 00:42:12,170 So remember, in Newtonian gravity r tells me 544 00:42:12,170 --> 00:42:14,780 the distance between-- if I have an orbit at r1 545 00:42:14,780 --> 00:42:19,590 and an orbit at r2, then I know that the distance between them 546 00:42:19,590 --> 00:42:22,257 is r2 minus r1. 547 00:42:22,257 --> 00:42:24,840 In the Schwarzschild spacetime, the distance between these two 548 00:42:24,840 --> 00:42:27,990 orbits is not r2 minus r1. 549 00:42:27,990 --> 00:42:34,170 However, r2 labels a sphere of surface area 4 pi r2 squared. 550 00:42:34,170 --> 00:42:38,730 And r1 labels a sphere of surface area 4 pi r1 squared. 551 00:42:38,730 --> 00:42:42,510 It's easy to also show that the circumference of the orbit 552 00:42:42,510 --> 00:42:45,330 at r2 is 2 pi r2, the circumference 553 00:42:45,330 --> 00:42:48,630 of the orbit at r1 is 2 pi r1. 554 00:42:48,630 --> 00:42:50,820 That, more than anything, is why we end up 555 00:42:50,820 --> 00:42:52,650 reproducing Kepler's law here, is 556 00:42:52,650 --> 00:42:56,880 that this areal coordinate is nicely 557 00:42:56,880 --> 00:42:59,296 amenable to this interpretation. 558 00:43:08,720 --> 00:43:13,110 So is this orbit-- 559 00:43:13,110 --> 00:43:15,980 so I described over here an orbit that is unstable 560 00:43:15,980 --> 00:43:19,100 and an orbit that is stable. 561 00:43:19,100 --> 00:43:20,870 I have described how to compute-- 562 00:43:20,870 --> 00:43:23,780 if I wanted to find a circular orbit at a given 563 00:43:23,780 --> 00:43:28,680 radius, those formulas that I derived over there 564 00:43:28,680 --> 00:43:31,950 on the right-most blackboards, they tell me 565 00:43:31,950 --> 00:43:34,320 what the energy and the angular momentum 566 00:43:34,320 --> 00:43:37,650 need to be as a function of r. 567 00:43:37,650 --> 00:43:40,560 Is that orbit stable? 568 00:43:40,560 --> 00:43:47,370 Well, if it is, I can check that by computing 569 00:43:47,370 --> 00:43:51,558 the second derivative of my effective potential. 570 00:43:58,110 --> 00:44:09,090 So my orbits are stable if the second derivative with respect 571 00:44:09,090 --> 00:44:20,690 to r is greater than 0, unstable if this turns out 572 00:44:20,690 --> 00:44:23,420 to be negative. 573 00:44:23,420 --> 00:44:26,550 Let's look at the crossover point from one to the other. 574 00:44:26,550 --> 00:44:29,600 What if there is a radius where, in fact, 575 00:44:29,600 --> 00:44:33,410 the stable and the unstable orbits coincide? 576 00:44:33,410 --> 00:44:35,630 In fact, what one finds, if you look 577 00:44:35,630 --> 00:44:38,270 at the effective potential-- you imagine just sort of playing 578 00:44:38,270 --> 00:44:39,680 with L sub z. 579 00:44:39,680 --> 00:44:41,690 So let's say we take that L sub z, 580 00:44:41,690 --> 00:44:45,440 and we just explore it for lots of different radii 581 00:44:45,440 --> 00:44:46,880 of the orbits. 582 00:44:46,880 --> 00:44:48,950 You find that as the orbits radius gets 583 00:44:48,950 --> 00:44:52,762 smaller and smaller, your stable orbit tends to go up, 584 00:44:52,762 --> 00:44:54,470 and this minimum sort of becomes flatter, 585 00:44:54,470 --> 00:44:56,470 and your unstable orbit just kind of comes down. 586 00:44:56,470 --> 00:44:58,970 They sort of approach one another. 587 00:44:58,970 --> 00:45:02,090 There is a point just when they coincide-- 588 00:45:04,777 --> 00:45:06,360 this should have an "effective" on it. 589 00:45:06,360 --> 00:45:07,442 My apologies. 590 00:45:14,190 --> 00:45:16,830 Right when they coincide, this defines 591 00:45:16,830 --> 00:45:19,190 what we call the marginally stable orbit. 592 00:45:28,230 --> 00:45:30,000 I may have put this one on a problem set. 593 00:45:30,000 --> 00:45:32,083 But it might be one of the ones I decided to drop. 594 00:45:32,083 --> 00:45:34,200 So I'm just going to go ahead and do the analysis. 595 00:45:34,200 --> 00:45:39,180 When you compute this, bearing in mind that your angular 596 00:45:39,180 --> 00:45:41,070 momentum is a constant-- 597 00:45:53,990 --> 00:45:58,250 so take this, substitute in now your solution for L sub z, 598 00:45:58,250 --> 00:45:59,750 which I've written down over there-- 599 00:46:18,880 --> 00:46:22,240 what you find is that the marginally stable orbit-- 600 00:46:24,860 --> 00:46:26,830 let's call it r sub ms-- 601 00:46:26,830 --> 00:46:29,540 it is located at a radius of 6GM. 602 00:46:39,860 --> 00:46:44,330 This is a profoundly new behavior 603 00:46:44,330 --> 00:46:49,640 that doesn't even come close to existing 604 00:46:49,640 --> 00:46:52,530 in Newtonian spacetime-- 605 00:46:52,530 --> 00:46:54,830 spacetime-- doesn't come close to existing in Newtonian 606 00:46:54,830 --> 00:46:57,469 gravity, excuse me. 607 00:46:57,469 --> 00:47:02,210 [SIGHS] I'm getting tired. 608 00:47:07,860 --> 00:47:09,570 The message I want you to understand 609 00:47:09,570 --> 00:47:17,420 is that, what this tells us is that no stable circular orbits 610 00:47:17,420 --> 00:47:30,155 exist inside r equals 6GM. 611 00:47:34,070 --> 00:47:37,700 So this is very, very different behavior. 612 00:47:37,700 --> 00:47:42,560 If I have-- let's just say I have a very compact body 613 00:47:42,560 --> 00:47:45,140 but Newtonian gravity rules. 614 00:47:45,140 --> 00:47:46,693 I can make circular orbits around it, 615 00:47:46,693 --> 00:47:48,860 basically go all the way down until they essentially 616 00:47:48,860 --> 00:47:50,485 touch the surface of that body. 617 00:47:50,485 --> 00:47:51,860 And you might think based on this 618 00:47:51,860 --> 00:47:53,902 that you would want to make orbits that basically 619 00:47:53,902 --> 00:47:56,510 go all the way down, that sort of kiss the edge of r 620 00:47:56,510 --> 00:47:58,520 equals 2GM. 621 00:47:58,520 --> 00:48:02,300 Well, this is telling you you can't do that. 622 00:48:02,300 --> 00:48:03,830 When you start trying to make orbits 623 00:48:03,830 --> 00:48:05,997 that go inside-- at least, circular orbits-- that go 624 00:48:05,997 --> 00:48:10,190 inside 6GM, they're not stable. 625 00:48:10,190 --> 00:48:14,000 If someone sneezes on them, they either 626 00:48:14,000 --> 00:48:16,340 are sort of blown out to infinity, 627 00:48:16,340 --> 00:48:19,100 or they fall into the event horizon. 628 00:48:19,100 --> 00:48:21,800 And in fact, one of the consequences of this 629 00:48:21,800 --> 00:48:24,500 is that, in astrophysical systems, 630 00:48:24,500 --> 00:48:27,710 we generically expect there to be kind of-- if you imagine 631 00:48:27,710 --> 00:48:30,560 material falling into a black hole, 632 00:48:30,560 --> 00:48:33,020 imagine that there's like a star or something that's just 633 00:48:33,020 --> 00:48:36,560 dumping gas into orbit around a black hole, well, 634 00:48:36,560 --> 00:48:39,765 it will tend to form a disk that orbits around this thing. 635 00:48:39,765 --> 00:48:41,390 And the elements of the disk are always 636 00:48:41,390 --> 00:48:43,370 rubbing against each other and radiating. 637 00:48:43,370 --> 00:48:44,870 That makes them get hot. 638 00:48:44,870 --> 00:48:48,030 They lose energy because of this radiation. 639 00:48:48,030 --> 00:48:49,940 And so they will very slowly sort of fall in. 640 00:48:49,940 --> 00:48:51,195 But they make this kind of-- 641 00:48:51,195 --> 00:48:52,820 it's thought that in most cases they'll 642 00:48:52,820 --> 00:48:54,620 make this kind of thick disk that 643 00:48:54,620 --> 00:48:58,275 fills much of the spacetime surrounding the black hole. 644 00:48:58,275 --> 00:48:59,900 But there will be a hole in the center. 645 00:48:59,900 --> 00:49:02,425 Not just because the thing is a black hole. 646 00:49:02,425 --> 00:49:03,800 I don't mean that kind of a hole. 647 00:49:03,800 --> 00:49:06,258 There'll actually be something surrounding the black hole's 648 00:49:06,258 --> 00:49:09,380 event horizon, because there are no stable circular orbits. 649 00:49:09,380 --> 00:49:12,770 Once the material comes in and hits this particular radius, 650 00:49:12,770 --> 00:49:17,120 6GM in the Schwarzschild case, it very rapidly falls in, 651 00:49:17,120 --> 00:49:19,910 reduces the density of that material tremendously. 652 00:49:19,910 --> 00:49:21,950 And you get this much thinner region of the disk 653 00:49:21,950 --> 00:49:24,920 where essentially things fall in practically instantly. 654 00:49:24,920 --> 00:49:30,840 It should be noted that this 6GM is, of course, only 655 00:49:30,840 --> 00:49:32,040 for Schwarzschild. 656 00:49:32,040 --> 00:49:34,200 If you talk about Kerr, you actually 657 00:49:34,200 --> 00:49:35,970 have two different radii corresponding 658 00:49:35,970 --> 00:49:38,460 to material that goes around parallel to the black hole's 659 00:49:38,460 --> 00:49:41,100 spin and material that goes anti-parallel 660 00:49:41,100 --> 00:49:42,930 to the black hole's spin. 661 00:49:42,930 --> 00:49:44,400 And it complicates things somewhat. 662 00:49:44,400 --> 00:49:46,080 There's two different radii there. 663 00:49:46,080 --> 00:49:48,590 The one that goes parallel tends to get a little bit closer. 664 00:49:48,590 --> 00:49:50,007 The one that's anti-parallel tends 665 00:49:50,007 --> 00:49:52,230 to go out a little bit farther. 666 00:49:52,230 --> 00:49:54,210 But the general prediction that there's 667 00:49:54,210 --> 00:49:57,700 an innermost orbit beyond which stable orbits do not exist, 668 00:49:57,700 --> 00:50:03,850 that's robust and holds across the domain of black holes. 669 00:50:03,850 --> 00:50:07,740 So let me conclude by talking about one 670 00:50:07,740 --> 00:50:09,450 final category of orbits-- 671 00:50:13,300 --> 00:50:14,310 photon orbits. 672 00:50:20,100 --> 00:50:22,770 So let's recall that when we were talking 673 00:50:22,770 --> 00:50:40,040 about null geodesics, we parametrized them in such a way 674 00:50:40,040 --> 00:50:41,880 that d-- 675 00:50:41,880 --> 00:50:43,460 sort of the tangent to the world line 676 00:50:43,460 --> 00:50:55,530 has an affine parameter attached to it, such that we can write p 677 00:50:55,530 --> 00:51:00,750 equals dx d lambda. 678 00:51:00,750 --> 00:51:01,845 These guys are null. 679 00:51:05,130 --> 00:51:07,690 So p dot p equals 0. 680 00:51:12,020 --> 00:51:13,850 For the case of orbiting bodies, that 681 00:51:13,850 --> 00:51:15,440 was equal to minus mass squared. 682 00:51:15,440 --> 00:51:17,927 Mass is 0 here. 683 00:51:17,927 --> 00:51:20,010 So we're going to follow a very similar procedure. 684 00:51:20,010 --> 00:51:22,910 The spacetime is still time independent, 685 00:51:22,910 --> 00:51:25,820 so there is still a notion of a conserved energy. 686 00:51:25,820 --> 00:51:28,850 It is still actually symmetric, so there is still a notion 687 00:51:28,850 --> 00:51:32,030 of axial angular momentum. 688 00:51:32,030 --> 00:51:35,870 But because p dot p is 0 now, rather than minus m squared, 689 00:51:35,870 --> 00:51:39,920 when we go through the exercise of-- that sort of parallels 690 00:51:39,920 --> 00:51:41,990 what we did for our massive particle, 691 00:51:41,990 --> 00:51:44,360 we're going to derive a different potential. 692 00:51:44,360 --> 00:51:52,940 So let's go ahead and evaluate this guy again. 693 00:51:57,030 --> 00:52:06,130 And I get 0 equals minus 1 minus 2GM/r dt d lambda 694 00:52:06,130 --> 00:52:14,610 squared plus 1 minus 2GM/r dr d lambda squared. 695 00:52:14,610 --> 00:52:16,110 I'm still going to require the thing 696 00:52:16,110 --> 00:52:18,300 to be in the theta equals pi/2 plane 697 00:52:18,300 --> 00:52:20,620 so that we know theta term. 698 00:52:20,620 --> 00:52:22,620 And I have set theta to pi/2. 699 00:52:22,620 --> 00:52:24,810 So my sine squared theta is just one. 700 00:52:29,740 --> 00:52:30,580 So I get this. 701 00:52:33,550 --> 00:52:35,500 I'll remind you that I can relate-- 702 00:52:35,500 --> 00:52:40,290 so the relationship between the time-light component 703 00:52:40,290 --> 00:52:43,960 of momentum and energy, the axial component of momentum, 704 00:52:43,960 --> 00:52:46,720 and the angular momentum, it's exactly the same as before. 705 00:52:46,720 --> 00:52:48,430 There's no rest mass appearing. 706 00:52:48,430 --> 00:52:51,160 And so now I find-- 707 00:52:51,160 --> 00:52:54,792 so my energy, I don't put a hat on it. 708 00:52:54,792 --> 00:52:57,250 It's not energy bringing it mass, because there is no mass. 709 00:53:06,850 --> 00:53:08,620 That looks like so. 710 00:53:08,620 --> 00:53:16,650 And my L looks like so. 711 00:53:31,290 --> 00:53:37,020 Using them, I can now derive an equation 712 00:53:37,020 --> 00:53:41,120 governing the radial motion of my light ray. 713 00:53:50,080 --> 00:53:56,650 And sparing you the line or two of algebra, it looks like this. 714 00:54:13,377 --> 00:54:14,460 Pardon me just one moment. 715 00:54:18,780 --> 00:54:19,280 OK. 716 00:54:22,670 --> 00:54:23,730 OK. 717 00:54:23,730 --> 00:54:27,070 So kind of similar to what we had before, if you look at it 718 00:54:27,070 --> 00:54:29,760 you'll see the term involving the angular momentum 719 00:54:29,760 --> 00:54:33,050 is a little bit different. 720 00:54:33,050 --> 00:54:35,480 As you stare at this for a moment, 721 00:54:35,480 --> 00:54:37,220 something should be disturbing you. 722 00:54:40,460 --> 00:54:42,800 Notice that the equation of motion 723 00:54:42,800 --> 00:54:47,640 appears to depend on the energy. 724 00:54:47,640 --> 00:54:49,170 OK. 725 00:54:49,170 --> 00:54:51,660 That should really bug you. 726 00:54:51,660 --> 00:54:57,060 Why should gamma rays and infrared radiation 727 00:54:57,060 --> 00:55:00,540 follow different trajectories? 728 00:55:00,540 --> 00:55:03,390 As long as I am truly in a geometric optics limit, 729 00:55:03,390 --> 00:55:05,130 I shouldn't. 730 00:55:05,130 --> 00:55:08,320 Now, it is true that if you consider very long wavelength 731 00:55:08,320 --> 00:55:10,320 radiation, you might need to worry about things. 732 00:55:10,320 --> 00:55:13,200 You might need to solve the wave equation in the spacetime. 733 00:55:13,200 --> 00:55:18,060 But as long as the wave nature of this radiation is-- 734 00:55:18,060 --> 00:55:20,790 if the wavelength is small enough that it's negligible 735 00:55:20,790 --> 00:55:22,530 compared to 2GM-- 736 00:55:22,530 --> 00:55:25,140 I shouldn't care what the energy is. 737 00:55:25,140 --> 00:55:27,900 Energy should not be influencing this. 738 00:55:27,900 --> 00:55:31,020 So what's going on is I need to reparametrize 739 00:55:31,020 --> 00:55:32,170 this a little bit. 740 00:55:32,170 --> 00:55:37,710 What I'm going to do to wash away my dependence on-- 741 00:55:37,710 --> 00:55:40,710 wash away my apparent dependence on the energy here, 742 00:55:40,710 --> 00:55:43,507 is I'm going to redefine my affine parameter. 743 00:55:51,800 --> 00:55:55,270 So let's take lambda-- 744 00:55:55,270 --> 00:55:57,300 so L divided by lambda. 745 00:55:57,300 --> 00:56:07,215 And I am going to define b to be L divided by E. 746 00:56:07,215 --> 00:56:11,448 This is the quantity that I will call the impact parameter. 747 00:56:15,678 --> 00:56:17,220 And I'll describe why I am calling it 748 00:56:17,220 --> 00:56:21,100 that in just a few minutes. 749 00:56:21,100 --> 00:56:24,120 So I'm going to take this entire equation, divide both sides 750 00:56:24,120 --> 00:56:25,005 by L squared. 751 00:56:45,790 --> 00:56:49,060 And I get something that looks like this. 752 00:57:04,190 --> 00:57:08,450 What I'm going to do is say that the impact parameter, b, 753 00:57:08,450 --> 00:57:10,262 is what I can control. 754 00:57:10,262 --> 00:57:11,720 It's the parameter that-- all I can 755 00:57:11,720 --> 00:57:15,830 control that defines the photon that I am studying here. 756 00:57:15,830 --> 00:57:20,150 And everything after this, this is my photon potential. 757 00:57:23,080 --> 00:57:34,250 Notice that the photon potential has no free parameters in it. 758 00:57:58,370 --> 00:58:04,166 If I plot this guy as a function of r, 759 00:58:04,166 --> 00:58:07,760 it kind it just looks like this. 760 00:58:07,760 --> 00:58:09,710 Two aspects of it are worth calling out. 761 00:58:13,710 --> 00:58:21,340 This peak occurs at r equals 3GM. 762 00:58:26,820 --> 00:58:31,667 Remember the way in which our energy-- oh, still have it 763 00:58:31,667 --> 00:58:34,000 on the board here-- things like the energy per unit mass 764 00:58:34,000 --> 00:58:36,420 and the angular momentum per unit mass all 765 00:58:36,420 --> 00:58:40,040 blew up when r equal 3GM. 766 00:58:40,040 --> 00:58:44,400 3GM is showing up now when I look 767 00:58:44,400 --> 00:58:49,260 at the motion of radiation, look at massless-- 768 00:58:49,260 --> 00:58:54,000 radiation corresponding to a massless particle, so to speak. 769 00:58:54,000 --> 00:58:58,530 If I look at the energy per unit mass, and the mass goes to 0, 770 00:58:58,530 --> 00:59:00,250 I get infinity. 771 00:59:00,250 --> 00:59:02,400 So the fact that I was actually seeing 772 00:59:02,400 --> 00:59:05,460 sort of things blowing up as r goes to 3GM 773 00:59:05,460 --> 00:59:08,940 was kind of like the equations hinting to me in advance 774 00:59:08,940 --> 00:59:13,620 that there was a hidden solution corresponding to radiation that 775 00:59:13,620 --> 00:59:15,600 could be regarded as a particular limit 776 00:59:15,600 --> 00:59:17,100 that they were sort of struggling 777 00:59:17,100 --> 00:59:19,440 to communicate to me. 778 00:59:19,440 --> 00:59:20,940 The other thing which I want to note 779 00:59:20,940 --> 00:59:24,480 is that the potential, the height of the potential right 780 00:59:24,480 --> 00:59:42,190 here, it has a peak value of 1 over 27 G squared M squared. 781 00:59:42,190 --> 00:59:45,640 Hold that thought for just a moment. 782 00:59:45,640 --> 00:59:49,000 Actually, let me write it in a slightly different way. 783 00:59:49,000 --> 00:59:56,640 This equals 1 over 3 root 3 GM squared. 784 01:00:09,650 --> 01:00:14,410 So now, to wrap this up I need to tell you what is really 785 01:00:14,410 --> 01:00:16,998 meant by this impact parameter. 786 01:00:36,530 --> 01:00:42,710 So go back to freshman mechanics. 787 01:00:42,710 --> 01:00:46,220 And impact parameter there is-- like, let's say 788 01:00:46,220 --> 01:00:58,080 I have a problem where I am looking at an object that 789 01:00:58,080 --> 01:01:03,220 is on an infall trajectory. 790 01:01:06,860 --> 01:01:11,580 There is a momentum p that is describing this. 791 01:01:14,520 --> 01:01:17,040 And it's moving in such a way-- it's not moving 792 01:01:17,040 --> 01:01:19,140 on a real trajectory, right? 793 01:01:19,140 --> 01:01:21,900 It's actually moving in a sense that 794 01:01:21,900 --> 01:01:28,170 is a little bit off of true to the radial direction. 795 01:01:28,170 --> 01:01:34,560 This guy actually has an angular momentum that is given by-- 796 01:01:34,560 --> 01:01:38,360 let's say that this is equal to px x hat. 797 01:01:38,360 --> 01:01:41,430 Let's say this is b y hat. 798 01:01:41,430 --> 01:01:52,488 This guy's got an angular momentum of b cross p. 799 01:01:52,488 --> 01:01:54,780 That's the impact parameter associated with this thing. 800 01:01:54,780 --> 01:02:02,670 It sort of tells me about the offset of this momentum 801 01:02:02,670 --> 01:02:05,780 from being directly radial towards the center 802 01:02:05,780 --> 01:02:06,405 of this object. 803 01:02:12,760 --> 01:02:17,680 Well, I'm going to use a similar notion 804 01:02:17,680 --> 01:02:21,280 to give me a geometric sense of what this impact parameter here 805 01:02:21,280 --> 01:02:22,270 means. 806 01:02:22,270 --> 01:02:25,810 Suppose here is my black hole. 807 01:02:25,810 --> 01:02:29,860 It's a little circle of radius 2GM. 808 01:02:29,860 --> 01:02:34,353 And I'm sitting way the heck out here, safely far away 809 01:02:34,353 --> 01:02:35,020 from this thing. 810 01:02:41,280 --> 01:02:43,890 And what I'm going to do is shoot light-- 811 01:02:43,890 --> 01:02:45,065 not quite radial. 812 01:02:45,065 --> 01:02:46,440 What I'm going to do is I'm going 813 01:02:46,440 --> 01:02:50,990 to have sort of an array of laser beams 814 01:02:50,990 --> 01:02:53,630 that kind of come up here-- 815 01:02:57,200 --> 01:02:59,300 an array of laser beams. 816 01:02:59,300 --> 01:03:03,650 And I'm going to shoot them towards this black hole. 817 01:03:03,650 --> 01:03:06,540 And what I'm going to do is I'm going to offset the laser beams 818 01:03:06,540 --> 01:03:07,860 by a distance b. 819 01:03:10,272 --> 01:03:12,480 And I'm going to fire it straight towards this thing. 820 01:03:16,012 --> 01:03:17,970 Go through sort of a careful definition of what 821 01:03:17,970 --> 01:03:19,560 angular momentum means, and you'll 822 01:03:19,560 --> 01:03:23,700 see that that definition of impact parameter 823 01:03:23,700 --> 01:03:27,750 gives you a very nice sense of the energy associated 824 01:03:27,750 --> 01:03:31,050 with the trajectory of this beam and a notion of angular 825 01:03:31,050 --> 01:03:34,470 momentum associated with this. 826 01:03:34,470 --> 01:03:37,230 So let's flip back and forth between a couple 827 01:03:37,230 --> 01:03:39,960 of different boards here. 828 01:03:39,960 --> 01:03:42,383 There are three cases that are interesting. 829 01:03:53,010 --> 01:03:56,538 Suppose b is small. 830 01:03:56,538 --> 01:04:04,560 In particular, suppose I have b less than 3 root 3 GM. 831 01:04:04,560 --> 01:04:08,185 So I start over here. 832 01:04:08,185 --> 01:04:09,060 Let's take the limit. 833 01:04:09,060 --> 01:04:10,200 What if b equals 0? 834 01:04:10,200 --> 01:04:13,560 Well, if b equals 0, this guy just goes, zoom, 835 01:04:13,560 --> 01:04:16,100 straight into the black hole. 836 01:04:16,100 --> 01:04:21,570 As long as it is anything less than 3 root 3 GM, 837 01:04:21,570 --> 01:04:24,740 what will happen is, when I shoot this thing in, 838 01:04:24,740 --> 01:04:26,810 it goes in, and it bends, perhaps, 839 01:04:26,810 --> 01:04:29,360 but it ends up going into the black hole. 840 01:04:29,360 --> 01:04:36,758 Let's look at this in the context 841 01:04:36,758 --> 01:04:48,510 of the equation of motion, the potential, and the impact 842 01:04:48,510 --> 01:04:50,370 parameter. 843 01:04:50,370 --> 01:04:58,310 If b is less than 1 over 3 root 3 GM, then 1 over b squared 844 01:04:58,310 --> 01:05:00,530 will be higher than this peak. 845 01:05:00,530 --> 01:05:02,290 And this thing is just going to be, 846 01:05:02,290 --> 01:05:03,832 whoo [fast motion sound effect], it's 847 01:05:03,832 --> 01:05:05,740 going to go right over the top of the peak. 848 01:05:05,740 --> 01:05:07,145 And it shoots into small radius. 849 01:05:17,340 --> 01:05:17,840 OK. 850 01:05:17,840 --> 01:05:19,990 Remember, L and E are constants of the motion. 851 01:05:19,990 --> 01:05:23,680 So that b is a parameter that defines this light 852 01:05:23,680 --> 01:05:25,300 for the this entire trajectory. 853 01:05:25,300 --> 01:05:28,820 1 over b squared is less than V phot everywhere-- 854 01:05:28,820 --> 01:05:33,535 excuse me, greater than V phot everywhere. 855 01:05:33,535 --> 01:05:35,580 A rather crucial typo. 856 01:05:35,580 --> 01:05:37,560 And as such, it shoots the light ray, 857 01:05:37,560 --> 01:05:50,020 and it goes right over the peak into the black hole, 858 01:05:50,020 --> 01:05:51,730 eventually winds up at r equals 0. 859 01:05:54,780 --> 01:06:06,880 If b is greater than 3 root 3 GM, then 1 over b squared 860 01:06:06,880 --> 01:06:11,920 is less than V phot at the peak. 861 01:06:15,980 --> 01:06:19,970 This corresponds, in this drawing, 862 01:06:19,970 --> 01:06:23,600 to a beam of light that follows a trajectory that kind of comes 863 01:06:23,600 --> 01:06:27,310 in here at this point. 864 01:06:27,310 --> 01:06:31,790 dr d tau, if we were to continue to go into smaller radii, 865 01:06:31,790 --> 01:06:33,530 it would become imaginary. 866 01:06:33,530 --> 01:06:34,760 That's not allowed. 867 01:06:34,760 --> 01:06:37,430 So it switches direction and trundles right back 868 01:06:37,430 --> 01:06:38,621 out to infinity. 869 01:06:43,810 --> 01:06:50,840 On this diagram, that corresponds to a light ray 870 01:06:50,840 --> 01:06:51,860 that's perhaps out here. 871 01:06:51,860 --> 01:06:54,890 This guy comes in, and he gets bent by the gravity 872 01:06:54,890 --> 01:06:57,348 a little bit, and then just, shoo [light ray sound effect], 873 01:06:57,348 --> 01:06:58,910 shoots back off to infinity. 874 01:06:58,910 --> 01:07:14,770 The critical point, b equals 3 GM, 875 01:07:14,770 --> 01:07:18,090 is right where the impact parameter hits the peak. 876 01:07:22,070 --> 01:07:40,060 And so what happens in this plot is this guy comes in, 877 01:07:40,060 --> 01:07:43,350 hits the peak, and just sits there forever. 878 01:07:48,745 --> 01:07:50,620 You get a little bit more of a physical sense 879 01:07:50,620 --> 01:07:53,230 as to what's going on by thinking about it here. 880 01:07:57,610 --> 01:08:00,990 This guy comes in, [INAUDIBLE] this, 881 01:08:00,990 --> 01:08:06,670 and then just whirls around, and lies on what 882 01:08:06,670 --> 01:08:09,070 we call the photon orbit. 883 01:08:09,070 --> 01:08:11,320 What's the radius of that photon orbit? 884 01:08:11,320 --> 01:08:12,865 r equals 3 GM. 885 01:08:25,979 --> 01:08:30,270 Now, astrophysically, a more interesting situation 886 01:08:30,270 --> 01:08:33,420 is, imagine you had some source of light that 887 01:08:33,420 --> 01:08:37,359 dumps a lot of photons in the vicinity of a black hole. 888 01:08:37,359 --> 01:08:40,380 Some of those photons are going to tend to-- 889 01:08:40,380 --> 01:08:42,460 some of them are going to go into the black hole, 890 01:08:42,460 --> 01:08:44,043 some are going to scatter a little bit 891 01:08:44,043 --> 01:08:45,453 and shoot off to infinity. 892 01:08:45,453 --> 01:08:46,870 But if you imagine that there will 893 01:08:46,870 --> 01:08:49,210 be some population of them that get stuck 894 01:08:49,210 --> 01:08:53,470 right on the r equals 3GM orbit-- 895 01:08:53,470 --> 01:08:56,000 now, that is an unstable orbit. 896 01:08:56,000 --> 01:08:58,390 And in fact, if you look at it, what you find is that 897 01:08:58,390 --> 01:09:04,479 your typical photon is likely to whirl around a bunch of times, 898 01:09:04,479 --> 01:09:07,420 and then, you know, if it's not precisely at 3GM 899 01:09:07,420 --> 01:09:14,859 but it's 3.00000000000000001GM, it will whirl around maybe 10 900 01:09:14,859 --> 01:09:19,107 or 15 times, and then it'll go off to infinity. 901 01:09:19,107 --> 01:09:21,399 So what we actually expect is if we have an object that 902 01:09:21,399 --> 01:09:24,550 is illuminated like this, then we 903 01:09:24,550 --> 01:09:27,069 will actually see these things come out here, 904 01:09:27,069 --> 01:09:28,960 and we will see-- 905 01:09:28,960 --> 01:09:32,779 so bear in mind, this is circularly-- this is symmetric. 906 01:09:32,779 --> 01:09:38,729 So take this and rotate around the symmetry axis. 907 01:09:38,729 --> 01:09:47,850 We expect to see a ring whose radius is twice 908 01:09:47,850 --> 01:09:50,229 the critical impact parameter. 909 01:09:50,229 --> 01:09:54,229 So it would have a diameter of 6 root 3 GM. 910 01:09:54,229 --> 01:09:58,680 It would be essentially a ring or a circle of radius 3 root 3 911 01:09:58,680 --> 01:09:59,880 GM. 912 01:09:59,880 --> 01:10:04,560 This is, in fact, what the Event Horizon Telescope measured. 913 01:10:04,560 --> 01:10:07,020 So last year when I was lecturing this class, 914 01:10:07,020 --> 01:10:09,390 this was announced almost-- 915 01:10:09,390 --> 01:10:10,950 I mean, they timed it well. 916 01:10:10,950 --> 01:10:13,920 They basically timed it to about two or three lectures 917 01:10:13,920 --> 01:10:18,598 before I discussed this aspect of black holes. 918 01:10:18,598 --> 01:10:20,640 So that was-- thank you, Event Horizon Telescope. 919 01:10:20,640 --> 01:10:23,620 That was very nice of them. 920 01:10:23,620 --> 01:10:28,350 And of course, we don't expect-- so Schwarzschild black holes 921 01:10:28,350 --> 01:10:31,410 are probably a mathematical curiosity. 922 01:10:31,410 --> 01:10:33,600 Objects in the real universe all rotate. 923 01:10:33,600 --> 01:10:36,330 We expect Kerr to be the generic solution. 924 01:10:36,330 --> 01:10:37,830 And so there's a fair amount of work 925 01:10:37,830 --> 01:10:40,460 that goes into how you correct this to do-- 926 01:10:40,460 --> 01:10:42,030 so doing this for Schwarzschild is, 927 01:10:42,030 --> 01:10:44,613 because of spherical symmetry, it's beautiful and it's simple. 928 01:10:44,613 --> 01:10:46,440 Kerr is a little bit more complicated. 929 01:10:46,440 --> 01:10:48,960 But, you know, it's a problem that can be solved. 930 01:10:48,960 --> 01:10:51,570 And a lot of very smart people spend a lot of time 931 01:10:51,570 --> 01:10:53,340 doing this to sort of map out what 932 01:10:53,340 --> 01:10:56,640 the shadow of the black hole looks like, 933 01:10:56,640 --> 01:11:00,030 what this ring would look like in the case of light 934 01:11:00,030 --> 01:11:04,230 coming off of a spinning black hole. 935 01:11:04,230 --> 01:11:05,360 So that's it. 936 01:11:05,360 --> 01:11:09,290 This is all that I'm going to present for 8.962 937 01:11:09,290 --> 01:11:11,690 in the spring of 2020 semester. 938 01:11:11,690 --> 01:11:15,320 So to everyone, as you are scattered 939 01:11:15,320 --> 01:11:20,300 around the world attempting to sort of stay connected 940 01:11:20,300 --> 01:11:24,080 to physics and your friends and your classwork, 941 01:11:24,080 --> 01:11:25,250 I wish you good health. 942 01:11:25,250 --> 01:11:28,550 And I hope to see you again at a time when the world is 943 01:11:28,550 --> 01:11:29,930 a little less crazy. 944 01:11:29,930 --> 01:11:34,480 In the meantime, enjoy our little beautiful black holes.