1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,251 at ocw.mit.edu. 8 00:00:21,770 --> 00:00:23,380 PROFESSOR: OK, let us start. 9 00:00:23,380 --> 00:00:29,200 So let me first remind you what we did in last lecture. 10 00:00:29,200 --> 00:00:43,910 We proved-- we showed that the Weinberg-Witten theorem for 11 00:00:43,910 --> 00:01:01,445 beats existence of massless spin-2 particles. 12 00:01:08,760 --> 00:01:12,770 Massless spin-2 particles are hallmark of gravity, 13 00:01:12,770 --> 00:01:14,160 so that's why we look for them. 14 00:01:17,180 --> 00:01:26,310 In the same-- so emphasize in the same space time, 15 00:01:26,310 --> 00:01:29,560 say a QFT lives. 16 00:01:34,160 --> 00:01:35,850 So this theorem essentially cites this. 17 00:01:38,810 --> 00:01:42,470 You will never have emergent gravity start 18 00:01:42,470 --> 00:01:46,355 from a [INAUDIBLE] quantum field theory, which is 19 00:01:46,355 --> 00:01:47,750 a well-defined stress tensor. 20 00:01:50,870 --> 00:01:55,300 So as we already mentioned, it's a new pole 21 00:01:55,300 --> 00:02:11,520 of this theorem is that actually emergent gravity can actually 22 00:02:11,520 --> 00:02:16,330 live in a different space time. 23 00:02:28,040 --> 00:02:32,170 So as in a holographic duality. 24 00:02:44,140 --> 00:02:48,630 In fact, in holographic duality the gravity 25 00:02:48,630 --> 00:02:53,860 lives in one dimension higher, but we are not 26 00:02:53,860 --> 00:02:56,240 ready to go there yet. 27 00:02:56,240 --> 00:02:57,900 So in order to describe this thing 28 00:02:57,900 --> 00:03:02,930 we still need to do some preparations. 29 00:03:02,930 --> 00:03:10,870 So the preparations-- let me just outline the preparations 30 00:03:10,870 --> 00:03:11,660 we need to do. 31 00:03:17,292 --> 00:03:19,750 So the first thing we will do is black hole thermodynamics. 32 00:03:32,600 --> 00:03:36,290 So these will give hint for something 33 00:03:36,290 --> 00:03:49,620 called the holographic principle, which is actually 34 00:03:49,620 --> 00:03:53,610 more general than the holographic duality discovered 35 00:03:53,610 --> 00:03:54,110 so far. 36 00:03:57,550 --> 00:04:03,010 And the second thing is, we will also quickly 37 00:04:03,010 --> 00:04:05,500 go over the large N gauge theory-- 38 00:04:05,500 --> 00:04:08,745 the properties of the large N gauge theories. 39 00:04:16,050 --> 00:04:27,075 So this give hints-- something called a gauge string duality. 40 00:04:31,000 --> 00:04:34,670 So the behavior of the large N gauge theory 41 00:04:34,670 --> 00:04:37,840 give you a strong hint that actually there's 42 00:04:37,840 --> 00:04:42,950 a string theory description for ordinary gauge theories. 43 00:04:42,950 --> 00:04:48,550 OK, and then when you combine these two things together, 44 00:04:48,550 --> 00:04:51,890 then you get what we have currently, 45 00:04:51,890 --> 00:04:55,290 the holographic duality. 46 00:04:55,290 --> 00:05:02,120 And then we will also talk a little bit of sting theory. 47 00:05:02,120 --> 00:05:03,320 A tiny bit. 48 00:05:03,320 --> 00:05:08,040 Not a lot, so don't be scared. 49 00:05:08,040 --> 00:05:13,100 So we also will talk a little bit about string theory. 50 00:05:13,100 --> 00:05:16,410 So in principle, actually, I can talk about holographic duality 51 00:05:16,410 --> 00:05:20,130 right now, but going over those aspect 52 00:05:20,130 --> 00:05:23,450 can help you to build some intuitions, 53 00:05:23,450 --> 00:05:29,030 and also to have a broader perspective than just 54 00:05:29,030 --> 00:05:33,370 presenting the duality directly. 55 00:05:33,370 --> 00:05:37,090 So before I start today's lecture, 56 00:05:37,090 --> 00:05:40,530 do you have any questions regarding our last lecture 57 00:05:40,530 --> 00:05:46,800 and regarding this-- yeah, general remarks here? 58 00:05:46,800 --> 00:05:47,370 Yes? 59 00:05:47,370 --> 00:05:48,828 AUDIENCE: So moving forward, are we 60 00:05:48,828 --> 00:05:51,000 going to define emergent gravity as this existence 61 00:05:51,000 --> 00:05:53,470 of a massless spin-2 particle? 62 00:05:53,470 --> 00:05:55,037 PROFESSOR: No, that we will not do. 63 00:05:55,037 --> 00:05:57,120 AUDIENCE: So how are we defining emergent gravity? 64 00:05:59,717 --> 00:06:01,300 PROFESSOR: You construct Newton's Law? 65 00:06:01,300 --> 00:06:04,860 Yeah, you will see it. 66 00:06:04,860 --> 00:06:06,890 You will see it when we have it. 67 00:06:06,890 --> 00:06:12,470 Yeah, essentially you see-- essentially it's 68 00:06:12,470 --> 00:06:13,580 handed over to you. 69 00:06:13,580 --> 00:06:16,015 You don't have to go through that step. 70 00:06:16,015 --> 00:06:19,650 You don't have to go through that step. 71 00:06:19,650 --> 00:06:20,150 Yeah. 72 00:06:24,402 --> 00:06:25,235 Any other questions? 73 00:06:29,510 --> 00:06:33,540 OK, good, let's start with black hole thermodynamics. 74 00:06:59,460 --> 00:07:01,640 So let me start by doing a little bit 75 00:07:01,640 --> 00:07:08,580 of dimensional analysis, just to remind you important scales 76 00:07:08,580 --> 00:07:09,285 for the gravity. 77 00:07:20,350 --> 00:07:23,360 So the first thing is what we called the Planck scale. 78 00:07:30,470 --> 00:07:43,890 So in nature we have fundamental constant H-bar, 79 00:07:43,890 --> 00:07:49,030 Newton constant, and the C, speed of light. 80 00:07:49,030 --> 00:07:57,050 And immediately after Planck himself introduced this H-bar, 81 00:07:57,050 --> 00:08:00,740 he realized you can actually combine the three of them 82 00:08:00,740 --> 00:08:09,605 to come up with a map scale, which is just hc divided by GN. 83 00:08:09,605 --> 00:08:12,830 And this, if you plug in the expressed numerical value, 84 00:08:12,830 --> 00:08:20,160 it's about 1.2 to the 10, to the 19 gev, divided by c squared. 85 00:08:20,160 --> 00:08:24,080 You can also write it in terms of the gram, which is about 2.2 86 00:08:24,080 --> 00:08:26,940 10 to the minus 5 gram. 87 00:08:29,630 --> 00:08:32,440 And then you can also have-- we can also 88 00:08:32,440 --> 00:08:41,390 construct the length scale, which is hGN divide c cubed, 89 00:08:41,390 --> 00:08:46,110 which is about 1.6 to 10 the minus 33 centimeter. 90 00:08:51,590 --> 00:08:57,170 And tp equals lp divided by c, OK? 91 00:09:00,400 --> 00:09:06,590 So this was discovered-- Planck introduced them 92 00:09:06,590 --> 00:09:12,270 in 1899, so most of you may know the story. 93 00:09:12,270 --> 00:09:15,170 And then, just immediately after he introduced H-bar-- so 94 00:09:15,170 --> 00:09:18,990 he introduced H-bar in 1899. 95 00:09:18,990 --> 00:09:25,110 In the same year he realized you can write down those numbers. 96 00:09:25,110 --> 00:09:28,930 And of course, those numbers meant nothing to him 97 00:09:28,930 --> 00:09:34,070 because at that time he didn't know special relativity, 98 00:09:34,070 --> 00:09:37,410 he didn't know quantum mechanics-- essentially, 99 00:09:37,410 --> 00:09:39,960 he didn't know anything. 100 00:09:39,960 --> 00:09:49,460 But he famously said that this unit-- so he 101 00:09:49,460 --> 00:09:53,300 claimed those should be basic units of physics, 102 00:09:53,300 --> 00:09:56,980 and he also said these are the units that 103 00:09:56,980 --> 00:10:01,740 would retain their significance for all times and all cultures. 104 00:10:05,320 --> 00:10:10,125 He even said they even will apply to aliens. 105 00:10:15,220 --> 00:10:21,310 But only after 50 years-- so after 1950s people 106 00:10:21,310 --> 00:10:22,260 get some sense. 107 00:10:22,260 --> 00:10:25,530 So many years after special relativity, 108 00:10:25,530 --> 00:10:27,790 many years after general relativity, 109 00:10:27,790 --> 00:10:30,460 and also quantum mechanics, et cetera, people 110 00:10:30,460 --> 00:10:35,700 started grasping the meaning of those scales. 111 00:10:35,700 --> 00:10:38,630 And so let me briefly review them. 112 00:10:42,450 --> 00:10:46,750 So you can-- OK, the feeling of the meaning of those scales 113 00:10:46,750 --> 00:10:50,740 by looking at the strength of gravity. 114 00:10:50,740 --> 00:10:54,981 So let me first start this example for electromagnetism, 115 00:10:54,981 --> 00:10:57,340 which you have a potential, which 116 00:10:57,340 --> 00:11:00,940 is e squared divided by r. 117 00:11:00,940 --> 00:11:04,150 So if you have two charged particle of charge e, 118 00:11:04,150 --> 00:11:07,360 then the potential between them is e squared divided by r. 119 00:11:11,350 --> 00:11:14,424 And then for particle of mass m then 120 00:11:14,424 --> 00:11:15,840 you also have Compton wavelengths. 121 00:11:20,560 --> 00:11:23,194 So if you have a particle with mass m, 122 00:11:23,194 --> 00:11:24,610 you also have Compton wavelengths. 123 00:11:24,610 --> 00:11:26,120 It's H-bar divided my mc. 124 00:11:28,820 --> 00:11:37,630 So you can get rough measure of the strength 125 00:11:37,630 --> 00:11:40,850 of electromagnetism-- the actual strength 126 00:11:40,850 --> 00:11:45,940 of the electromagnetism by conceiving 127 00:11:45,940 --> 00:11:49,980 the following [INAUDIBLE] dimension number, which 128 00:11:49,980 --> 00:11:56,780 I call lambda e, which is the potential evaluated 129 00:11:56,780 --> 00:11:59,600 at the minimal-- in quantum mechanics, 130 00:11:59,600 --> 00:12:03,570 essentially this is the minimal distance you can make sense of. 131 00:12:03,570 --> 00:12:05,570 Because once you go distance smaller than these, 132 00:12:05,570 --> 00:12:08,770 you can no longer-- and then the quantum uncertainty 133 00:12:08,770 --> 00:12:12,500 will create the energy uncertainty bigger than m, 134 00:12:12,500 --> 00:12:17,680 and you can no longer talk about single particle 135 00:12:17,680 --> 00:12:19,280 in a sensible way. 136 00:12:19,280 --> 00:12:21,260 So the essentially the minimal length 137 00:12:21,260 --> 00:12:24,160 scale you can talk about single particles. 138 00:12:24,160 --> 00:12:29,160 So this is essentially some energy scale. 139 00:12:29,160 --> 00:12:31,880 So you can compare these to the, say, the static mass 140 00:12:31,880 --> 00:12:33,360 of the particle. 141 00:12:33,360 --> 00:12:36,990 So this give you a measure of the strength 142 00:12:36,990 --> 00:12:39,670 of electromagnetism. 143 00:12:39,670 --> 00:12:42,470 Of course, if you plug this in you 144 00:12:42,470 --> 00:12:46,000 just get e squared divided by H-bar c. 145 00:12:46,000 --> 00:12:49,150 And of course, we know this is the fine structure constant, 146 00:12:49,150 --> 00:12:54,680 which is indeed the coupling. 147 00:12:54,680 --> 00:13:01,670 Say you would do-- indeed, it's a coupling in QED, OK? 148 00:13:01,670 --> 00:13:04,777 So you can do the same thing for gravity. 149 00:13:04,777 --> 00:13:06,360 You can do the same thing for gravity. 150 00:13:15,120 --> 00:13:26,905 So for gravity we have essentially g, and say 151 00:13:26,905 --> 00:13:28,550 if you take two particle of masses 152 00:13:28,550 --> 00:13:34,720 m then the potential between them is gm, m squared, 153 00:13:34,720 --> 00:13:37,120 divided by r. 154 00:13:37,120 --> 00:13:40,750 And then again, you can define effective strength 155 00:13:40,750 --> 00:13:42,130 for the gravity. 156 00:13:42,130 --> 00:13:46,090 I evaluate this potential at Compton wavelengths 157 00:13:46,090 --> 00:13:51,890 divided by the static energy of the particle. 158 00:13:51,890 --> 00:13:54,570 And then you can just plug this in. 159 00:13:54,570 --> 00:14:01,350 So this is GN m squared, divided by H-bar, divided by mc, 160 00:14:01,350 --> 00:14:06,430 then divided by mc squared. 161 00:14:06,430 --> 00:14:16,970 And then you find this is just equal to GN m squared, divided 162 00:14:16,970 --> 00:14:20,020 by H-bar c. 163 00:14:20,020 --> 00:14:27,490 OK, so now if you compare with this equation-- 164 00:14:27,490 --> 00:14:32,280 OK, so this is just given by m squared divided by mp squared. 165 00:14:32,280 --> 00:14:40,590 OK, so for gravity this effective strength 166 00:14:40,590 --> 00:14:44,460 is just given by the mass of the particle. 167 00:14:44,460 --> 00:14:45,880 So because for gravity the mass is 168 00:14:45,880 --> 00:14:47,730 like some kind of effective charge, 169 00:14:47,730 --> 00:14:49,810 and then divided by Planck scale-- 170 00:14:49,810 --> 00:14:52,050 this Planck mass-- squared. 171 00:14:52,050 --> 00:15:06,190 OK, and you can also write it as lp squared divided 172 00:15:06,190 --> 00:15:12,970 by rc squared, or just as this Planck length 173 00:15:12,970 --> 00:15:16,740 divided by [INAUDIBLE] wavelengths of this particle. 174 00:15:16,740 --> 00:15:23,350 OK, so for most elementary particles-- 175 00:15:23,350 --> 00:15:25,380 so for typical elementary particles 176 00:15:25,380 --> 00:15:30,570 we know the m is always much smaller than mp. 177 00:15:30,570 --> 00:15:37,270 And then the lambda g would be typically much smaller than 1. 178 00:15:37,270 --> 00:15:44,500 So for example, say, if you can see the electron, 179 00:15:44,500 --> 00:15:48,540 the electron mass would be 5 to the 10 180 00:15:48,540 --> 00:15:53,870 to the minus 4 gev divided by c squared. 181 00:15:53,870 --> 00:15:57,690 Of course this is much smaller than this Planck mass. 182 00:15:57,690 --> 00:16:00,860 And then you find that this ratio-- so you 183 00:16:00,860 --> 00:16:02,140 can work out this ratio. 184 00:16:02,140 --> 00:16:06,110 So let's compare it to the corresponding strength 185 00:16:06,110 --> 00:16:08,070 for electromagnetism. 186 00:16:08,070 --> 00:16:14,730 So this is about 10 to the minus 43 if you work this out. 187 00:16:14,730 --> 00:16:16,720 So this tells you the gravity's really weak. 188 00:16:21,140 --> 00:16:27,690 So for ordinary elementary particles-- 189 00:16:27,690 --> 00:16:31,420 so the gravity is really weak, and so we 190 00:16:31,420 --> 00:16:38,070 can forget all about gravity until you reach the Planck 191 00:16:38,070 --> 00:16:41,620 mass, or your Compton wavelengths 192 00:16:41,620 --> 00:16:44,110 reach the Planck length. 193 00:16:44,110 --> 00:16:49,815 And then the fact of the quantum gravity will be important. 194 00:16:52,480 --> 00:16:57,540 So from this exercise we know that the mp is the energy 195 00:16:57,540 --> 00:17:06,380 scale, that the effective gravity strength become 196 00:17:06,380 --> 00:17:16,740 of order one-- become of order one-- that is, quantum gravity 197 00:17:16,740 --> 00:17:25,606 fact becomes significant. 198 00:17:34,070 --> 00:17:45,474 And similarly-- so lp is the corresponding length scale 199 00:17:45,474 --> 00:17:46,640 associated with such energy. 200 00:17:54,290 --> 00:17:59,360 So this give you a heuristic feeling-- 201 00:17:59,360 --> 00:18:02,730 give you a heuristic indication that the meaning 202 00:18:02,730 --> 00:18:05,020 of those Planck scales. 203 00:18:05,020 --> 00:18:11,570 OK, so there's another important scale associated with gravity. 204 00:18:11,570 --> 00:18:12,800 So any questions on this? 205 00:18:16,390 --> 00:18:20,380 Good, there's another important scale associated with gravity. 206 00:18:20,380 --> 00:18:34,650 It's called Schwarzschild radius-- Schwarzschild radius. 207 00:18:34,650 --> 00:18:37,320 So just from dimensional analysis, 208 00:18:37,320 --> 00:18:41,070 the Schwarzschild radius can be argued as follows. 209 00:18:41,070 --> 00:18:43,360 So can see that-- again, we can just even 210 00:18:43,360 --> 00:18:47,000 see from Newtonian gravity. 211 00:18:47,000 --> 00:18:54,490 So I would say let's consider the object of mass m. 212 00:18:59,450 --> 00:19:09,170 Then we ask is the distance-- at what distance, 213 00:19:09,170 --> 00:19:26,420 maybe I should-- at what distance from it 214 00:19:26,420 --> 00:19:28,110 the classical gravity becomes strong. 215 00:19:45,230 --> 00:19:49,240 So for this purpose, let's consider, 216 00:19:49,240 --> 00:19:53,680 say, a probe mass-- say, m prime. 217 00:19:53,680 --> 00:19:54,865 So, can see the probe mass. 218 00:20:00,290 --> 00:20:04,910 So I define a scale which I call rs, 219 00:20:04,910 --> 00:20:09,760 as I require the potential energy between m and m prime. 220 00:20:09,760 --> 00:20:14,900 at such a scale rs, then this become of order, 221 00:20:14,900 --> 00:20:21,550 say again, the static energy of this probe political, 222 00:20:21,550 --> 00:20:24,390 or of this probe mass. 223 00:20:24,390 --> 00:20:26,830 So if you cancel things out then you 224 00:20:26,830 --> 00:20:35,280 find rs is of order GNm divided by c squared. 225 00:20:35,280 --> 00:20:46,780 OK, so this give you-- I'll give you a scale. 226 00:20:46,780 --> 00:20:51,350 You can also ask-- so this is from Newtonian gravity. 227 00:20:51,350 --> 00:20:55,730 Of course, when your gravity becomes strong 228 00:20:55,730 --> 00:20:58,090 you should replace the Newtonian gravity 229 00:20:58,090 --> 00:21:01,530 by Einstein, the relativity. 230 00:21:01,530 --> 00:21:05,960 And then when you go to relativity-- 231 00:21:05,960 --> 00:21:10,040 when you go to relativity, general relativity, 232 00:21:10,040 --> 00:21:14,170 then you find then there's a Schwarzschild radius. 233 00:21:14,170 --> 00:21:16,070 So there's a Schwarzschild radius 234 00:21:16,070 --> 00:21:22,130 which given by 2 gm divided by c squared, which corresponding 235 00:21:22,130 --> 00:21:23,710 to the sides of a black hole. 236 00:21:30,340 --> 00:21:35,040 OK, corresponding to the sides of a black hole. 237 00:21:35,040 --> 00:21:37,570 So this is a classical scale-- purely classical scale. 238 00:21:37,570 --> 00:21:39,390 Just the scale which the classical gravity 239 00:21:39,390 --> 00:21:42,270 becomes strong. 240 00:21:42,270 --> 00:21:53,350 In particular, rs can be considered 241 00:21:53,350 --> 00:22:20,040 as the minimal length scale one can probe an object of mass m, 242 00:22:20,040 --> 00:22:22,870 OK? 243 00:22:22,870 --> 00:22:28,140 So classically, black hole absorb everything. 244 00:22:28,140 --> 00:22:31,330 So once you fall into black hole you can never come back. 245 00:22:31,330 --> 00:22:34,910 And so the minimal distance you can 246 00:22:34,910 --> 00:22:39,820 approach an object of mass m, it is given by the Schwarzschild 247 00:22:39,820 --> 00:22:40,654 radius. 248 00:22:40,654 --> 00:22:42,570 So when you go inside the Schwarzschild radius 249 00:22:42,570 --> 00:22:44,186 we just fall into the black hole, 250 00:22:44,186 --> 00:22:45,810 and you can never send information out. 251 00:22:48,430 --> 00:22:51,967 And the one interesting thing compare-- 252 00:22:51,967 --> 00:22:54,300 yeah, one interesting thing regarding this Schwarzschild 253 00:22:54,300 --> 00:22:55,720 radius. 254 00:22:55,720 --> 00:22:59,840 You said Schwarzschild radius increase with mass. 255 00:22:59,840 --> 00:23:04,240 OK, if you increase the mass the Schwarzschild radius increases. 256 00:23:04,240 --> 00:23:06,670 So in principal it can be very large 257 00:23:06,670 --> 00:23:09,029 when you can see the very large object-- when you can 258 00:23:09,029 --> 00:23:10,195 see the very massive object. 259 00:23:15,660 --> 00:23:17,585 So now-- so let us summarize. 260 00:23:21,650 --> 00:23:24,780 Let us summarize. 261 00:23:24,780 --> 00:23:37,040 So for object of mass m there are two important scales. 262 00:23:42,270 --> 00:23:43,555 Two important scales. 263 00:23:51,980 --> 00:24:01,876 So one is just the standard Compton wavelengths, 264 00:24:01,876 --> 00:24:03,625 and the other is the Schwarzschild radius. 265 00:24:11,980 --> 00:24:15,440 So one is quantum and the other is classical. 266 00:24:15,440 --> 00:24:17,310 The other is classical. 267 00:24:17,310 --> 00:24:20,750 So let's take the ratio between them. 268 00:24:20,750 --> 00:24:22,484 Let's take the ratio between them. 269 00:24:25,970 --> 00:24:31,710 So if you take the ratio between them-- so let's 270 00:24:31,710 --> 00:24:37,212 forget about these two, just to consider of order 271 00:24:37,212 --> 00:24:39,100 up to order 1. 272 00:24:39,100 --> 00:24:44,470 So c squared, divided by H-bar, divided by mc. 273 00:24:44,470 --> 00:24:51,445 And then this again give you gm m squared divided by H-bar c. 274 00:24:54,670 --> 00:24:59,223 And then this is again just m squared divided by mp squared. 275 00:25:04,170 --> 00:25:06,200 This again is m squared divided mp squared, 276 00:25:06,200 --> 00:25:09,280 and p is the Planck mass. 277 00:25:09,280 --> 00:25:13,590 So let's consider the different scenarios. 278 00:25:13,590 --> 00:25:15,120 So the first, let's consider-- just 279 00:25:15,120 --> 00:25:20,050 suppose the mass of the object is much, much greater than mp. 280 00:25:20,050 --> 00:25:24,935 OK, so in this case then the Schwarzschild radius 281 00:25:24,935 --> 00:25:26,810 are much larger than the Compton wavelengths. 282 00:25:30,220 --> 00:25:34,920 Much larger than the Compton wavelengths, And essentially, 283 00:25:34,920 --> 00:25:40,160 all of physics is controlled by the classical gravity, 284 00:25:40,160 --> 00:25:45,190 because you can no longer probe-- yeah, 285 00:25:45,190 --> 00:25:47,040 because a Compton wavelengths is much, 286 00:25:47,040 --> 00:25:51,106 much inside the Schwarzschild radius, which you cannot probe. 287 00:25:51,106 --> 00:25:53,105 So the physics is essentially classical gravity. 288 00:26:01,350 --> 00:26:06,134 And the quantum effect is not important, 289 00:26:06,134 --> 00:26:08,300 so you don't have to worry about the quantum effect. 290 00:26:16,450 --> 00:26:17,690 I will put quote here. 291 00:26:20,290 --> 00:26:25,032 Not quote, I will to put some-- yeah, some quote here. 292 00:26:25,032 --> 00:26:26,240 You will see what this means. 293 00:26:32,130 --> 00:26:35,410 And the second possibility is for mass much, 294 00:26:35,410 --> 00:26:36,435 much smaller than p. 295 00:26:39,510 --> 00:26:43,980 So in this case, then the Compton wavelengths 296 00:26:43,980 --> 00:26:49,400 will be much, much greater than the Schwarzschild radius. 297 00:26:49,400 --> 00:26:52,980 OK, so this is a quantum object, so the quantum size 298 00:26:52,980 --> 00:26:56,280 is much larger than the Schwarzschild radius. 299 00:26:56,280 --> 00:27:00,540 But we also know, precisely in this region, 300 00:27:00,540 --> 00:27:02,149 this lambda g is also very small. 301 00:27:02,149 --> 00:27:03,690 The effective strength of the gravity 302 00:27:03,690 --> 00:27:04,740 is also very, very small. 303 00:27:07,520 --> 00:27:10,570 So we also have found before that the lambda g is also 304 00:27:10,570 --> 00:27:13,380 very, very small. 305 00:27:13,380 --> 00:27:18,400 So in this case, as what we said here-- 306 00:27:18,400 --> 00:27:30,414 the gravity is very weak and not important. 307 00:27:30,414 --> 00:27:33,600 It's much, much weaker than other interactions, 308 00:27:33,600 --> 00:27:35,180 so you can essentially ignore them. 309 00:27:38,030 --> 00:27:41,540 Then we're only left with the single scale 310 00:27:41,540 --> 00:27:45,100 which mass is of all the mp. 311 00:27:45,100 --> 00:27:48,680 And then, as we said before, the quantum gravity is important. 312 00:27:48,680 --> 00:27:51,350 So let me just say, quantum gravity important. 313 00:28:02,862 --> 00:28:08,630 If this were the full story, then life would be very boring. 314 00:28:08,630 --> 00:28:10,130 Even though it would be very simple. 315 00:28:12,930 --> 00:28:18,470 Because the only scale you need to worry about quantum gravity 316 00:28:18,470 --> 00:28:19,700 is essentially natural zero. 317 00:28:19,700 --> 00:28:21,730 It's only one scale. 318 00:28:21,730 --> 00:28:24,380 And it will take us maybe hundred 319 00:28:24,380 --> 00:28:27,180 if not thousand years to reach that scale 320 00:28:27,180 --> 00:28:30,850 by whatever accelerator or other probes. 321 00:28:30,850 --> 00:28:33,430 So there's really no urgency to think about the quantum 322 00:28:33,430 --> 00:28:35,060 gravity. 323 00:28:35,060 --> 00:28:39,420 Because right now we are at this kind of scale. 324 00:28:39,420 --> 00:28:41,277 Right now we are this kind of scale. 325 00:28:41,277 --> 00:28:43,110 It's very, very far from this kind of scale. 326 00:28:46,300 --> 00:28:52,990 But the remarkable thing about black hole-- so this part 327 00:28:52,990 --> 00:28:57,270 of the physics is essentially controlled 328 00:28:57,270 --> 00:28:59,760 by Schwarzschild radius, because the Schwarzschild 329 00:28:59,760 --> 00:29:05,830 radius is the minimal classical radius you can achieve. 330 00:29:05,830 --> 00:29:09,700 Yeah, you can probe the system, and the quantum physics 331 00:29:09,700 --> 00:29:11,940 is relevant. 332 00:29:11,940 --> 00:29:19,852 But the remarkable thing is that it turns out 333 00:29:19,852 --> 00:29:21,310 that this statement is not correct. 334 00:29:25,195 --> 00:29:26,445 This statement is not correct. 335 00:29:29,060 --> 00:29:32,250 Actually, quantum effect is important. 336 00:29:32,250 --> 00:29:49,400 So the remarkable thing is that the black hole 337 00:29:49,400 --> 00:30:06,280 can have quantum effect manifest at the microscopic level. 338 00:30:12,810 --> 00:30:24,520 Say, at length scale of order Schwarzschild radius, OK? 339 00:30:24,520 --> 00:30:28,040 So that's why it makes the black hole so interesting, 340 00:30:28,040 --> 00:30:34,725 and also makes why black hole is such a rich source of insight 341 00:30:34,725 --> 00:30:38,900 and information if you want to know about quantum gravity. 342 00:30:38,900 --> 00:30:41,410 And as we will see, actually, we also 343 00:30:41,410 --> 00:30:42,880 contain a rich source information 344 00:30:42,880 --> 00:30:46,640 about ordinary many body systems, due to this duality. 345 00:30:53,640 --> 00:30:56,350 Any questions regarding this? 346 00:30:56,350 --> 00:30:57,190 Yes? 347 00:30:57,190 --> 00:31:01,370 AUDIENCE: So when I talk about the m much larger than mp, 348 00:31:01,370 --> 00:31:05,550 so the m is not only an elementary particle, 349 00:31:05,550 --> 00:31:07,510 it can be just and object? 350 00:31:07,510 --> 00:31:10,760 PROFESSOR: Yeah, it can be a bound state. 351 00:31:10,760 --> 00:31:14,550 So it can be-- here, we always talking about quantum object. 352 00:31:17,450 --> 00:31:19,190 But it can have mass very large. 353 00:31:19,190 --> 00:31:21,830 AUDIENCE: That's still a quantum object? 354 00:31:21,830 --> 00:31:22,690 PROFESSOR: Yeah. 355 00:31:22,690 --> 00:31:27,780 What you will not see from a traditional way, 356 00:31:27,780 --> 00:31:30,800 for such a large mass object, you 357 00:31:30,800 --> 00:31:33,260 will not see its quantum uncertainty, 358 00:31:33,260 --> 00:31:35,970 because quantum uncertainty is tiny. 359 00:31:35,970 --> 00:31:37,780 The Compton wavelengths is very tiny, 360 00:31:37,780 --> 00:31:39,980 and so the fluctuations are very small. 361 00:31:39,980 --> 00:31:42,460 And so you have to probe very, very small in scale 362 00:31:42,460 --> 00:31:45,430 to see its quantum-- from traditional point of view, 363 00:31:45,430 --> 00:31:47,720 we have to probe very, very small in scale 364 00:31:47,720 --> 00:31:50,350 to see its quantum fluctuations. 365 00:31:50,350 --> 00:31:52,090 And that scale is much, much smaller 366 00:31:52,090 --> 00:31:53,631 than the Schwarzschild radius. 367 00:31:53,631 --> 00:31:54,172 AUDIENCE: OK. 368 00:31:57,205 --> 00:31:58,455 PROFESSOR: Any other question? 369 00:32:01,600 --> 00:32:06,800 Good, so let me-- before we talk more specific 370 00:32:06,800 --> 00:32:13,215 about black holes, let me just make one final remark. 371 00:32:15,800 --> 00:32:22,520 It's that in a sense, this Planck length-- 372 00:32:22,520 --> 00:32:25,490 this length scale defined by Planck 373 00:32:25,490 --> 00:32:30,340 can be considered as a minimal localization length. 374 00:32:37,890 --> 00:32:40,460 OK, for the following reason. 375 00:32:40,460 --> 00:32:45,900 So let's firstly imagine in non gravitational physics-- 376 00:32:45,900 --> 00:32:57,350 so in non gravitational physics if you 377 00:32:57,350 --> 00:33:01,120 want to probe some short distance scales then 378 00:33:01,120 --> 00:33:03,760 it's easy if you are rich enough. 379 00:33:03,760 --> 00:33:07,600 Then you just accelerate the particles 380 00:33:07,600 --> 00:33:10,230 to very high energies. 381 00:33:10,230 --> 00:33:17,810 Say e plus, e minus, with p and minus p. 382 00:33:20,740 --> 00:33:25,790 Then that can probe the distance scale, 383 00:33:25,790 --> 00:33:27,765 then this can probe length scales. 384 00:33:31,030 --> 00:33:35,576 Say of order h divided by p. 385 00:33:38,180 --> 00:33:45,820 OK, so if you make the particle energy high enough 386 00:33:45,820 --> 00:33:52,900 you can, in principle, probe as short as any scale as you want. 387 00:33:52,900 --> 00:33:56,090 Anything, as far as you can make this p as large as you want. 388 00:33:59,420 --> 00:34:09,460 So in principle, you can take l all the way to zero. 389 00:34:09,460 --> 00:34:11,580 So the scale comes all the way to 0. 390 00:34:11,580 --> 00:34:12,949 If you take p, go to infinity. 391 00:34:17,659 --> 00:34:19,416 But in gravity, this is not so. 392 00:34:23,820 --> 00:34:42,920 So with gravity the square of the distance [INAUDIBLE] 393 00:34:42,920 --> 00:34:58,280 so when your energy is much, much greater than ep-- 394 00:34:58,280 --> 00:35:06,800 say, the Planck mass-- then, as we 395 00:35:06,800 --> 00:35:12,340 discussed from there-- so this is a central mass energy, OK? 396 00:35:12,340 --> 00:35:15,680 Say if your central mass image becomes 397 00:35:15,680 --> 00:35:18,460 much, much larger than the Planck scale-- Planck 398 00:35:18,460 --> 00:35:31,260 mass-- then rs controlled by the image-- yeah, 399 00:35:31,260 --> 00:35:32,590 and [INAUDIBLE] p. 400 00:35:35,740 --> 00:35:38,002 Yeah, let me just forget about c. 401 00:35:38,002 --> 00:35:43,840 Let me just say-- OK? 402 00:35:43,840 --> 00:35:46,120 Then the Schwarzschild radius from now 403 00:35:46,120 --> 00:35:55,960 on-- so you go to y, OK? 404 00:35:55,960 --> 00:36:07,060 So the Schwarzschild radius will take over 405 00:36:07,060 --> 00:36:13,690 as the minimal length scale. 406 00:36:17,370 --> 00:36:21,330 OK, so what's going to happen is that if you collided these two 407 00:36:21,330 --> 00:36:25,510 particles at very high energies, then at a certain point, 408 00:36:25,510 --> 00:36:28,790 even before these two particles meet together, 409 00:36:28,790 --> 00:36:33,320 they already form a black holes over the Schwarzschild sites, 410 00:36:33,320 --> 00:36:34,460 OK? 411 00:36:34,460 --> 00:36:39,390 If this energy is high enough, then we will form black hole, 412 00:36:39,390 --> 00:36:43,060 and then you can no longer probe inside the Schwarzschild 413 00:36:43,060 --> 00:36:44,730 radius of that black hole. 414 00:36:44,730 --> 00:36:49,970 So that defines a new scale which you can probe, OK? 415 00:36:53,010 --> 00:36:55,480 So the funny thing about this Schwarzschild radius 416 00:36:55,480 --> 00:36:59,970 is that it's proportional to energy, rather 417 00:36:59,970 --> 00:37:03,200 inverse proportion to energy as the standard Compton 418 00:37:03,200 --> 00:37:05,910 wavelengths, OK? 419 00:37:05,910 --> 00:37:11,280 So after a certain point, when you go beyond the Planck mass, 420 00:37:11,280 --> 00:37:14,210 when you further increase in the energy, 421 00:37:14,210 --> 00:37:17,025 then you're actually probing the larger distance scales rather 422 00:37:17,025 --> 00:37:19,700 than smaller distance scales, due to the funny thing 423 00:37:19,700 --> 00:37:23,800 about the gravity and the funny thing about black hole. 424 00:37:23,800 --> 00:37:34,890 OK, so actually, this scale increases with p-- increase 425 00:37:34,890 --> 00:37:36,510 with you center of mass energy. 426 00:37:36,510 --> 00:37:49,280 OK, and the high energies those two longer length scales. 427 00:37:55,980 --> 00:38:00,610 OK, this essentially defines the Planck length 428 00:38:00,610 --> 00:38:02,420 as a minimal scale one can probe. 429 00:38:20,168 --> 00:38:22,000 OK, so when your center of mass energy 430 00:38:22,000 --> 00:38:26,180 is smaller than the Planck scale-- than the Planck mass-- 431 00:38:26,180 --> 00:38:28,860 then your Compton wavelength of course is larger than lp. 432 00:38:32,630 --> 00:38:35,990 But when this is greater than mp, 433 00:38:35,990 --> 00:38:39,340 then as we discussed here-- then the Schwarzschild radius, 434 00:38:39,340 --> 00:38:44,200 of course, [INAUDIBLE] object will be greater than the Planck 435 00:38:44,200 --> 00:38:49,330 side, and will break through the common wavelengths 436 00:38:49,330 --> 00:38:54,220 and will be greater than the Planck sides. 437 00:38:54,220 --> 00:38:57,730 And this give you, essentially, the minimal radius to probe. 438 00:39:04,880 --> 00:39:11,500 Alternatively, we can also just reach the same argument. 439 00:39:11,500 --> 00:39:14,370 Simply, I can just write down a couple equations. 440 00:39:14,370 --> 00:39:15,995 So let's consider you have uncertainty. 441 00:39:19,540 --> 00:39:27,710 So suppose you have a position, data x, and then 442 00:39:27,710 --> 00:39:30,670 the answer in the energy or momentum associated 443 00:39:30,670 --> 00:39:33,400 with the data x is data p. 444 00:39:37,570 --> 00:39:42,020 But on the other hand, the distance you can probe 445 00:39:42,020 --> 00:39:47,110 must be greater than the Schwarzschild radius associated 446 00:39:47,110 --> 00:39:49,580 with lp, data p. 447 00:39:55,050 --> 00:39:58,030 OK, data p. 448 00:39:58,030 --> 00:40:01,640 So if you combine these two equations together-- so 449 00:40:01,640 --> 00:40:09,310 this is greater than GN H-bar divided by lp. 450 00:40:09,310 --> 00:40:12,110 So now I have suppressed the c. 451 00:40:12,110 --> 00:40:13,650 So you can see from this equation-- 452 00:40:13,650 --> 00:40:16,990 you can see that data x must be greater 453 00:40:16,990 --> 00:40:23,820 than H-bar GN, which is lp. 454 00:40:23,820 --> 00:40:26,870 OK, which is lp here. 455 00:40:31,790 --> 00:40:35,220 This is the same argument as this one, 456 00:40:35,220 --> 00:40:37,100 but this is a little bit formal. 457 00:40:37,100 --> 00:40:40,340 So the essence is that once you're energy is big enough, 458 00:40:40,340 --> 00:40:41,930 then you will create the black hole, 459 00:40:41,930 --> 00:40:44,420 and then your physics will completely change. 460 00:40:44,420 --> 00:40:50,907 AUDIENCE: [INAUDIBLE] black hole evaporates [INAUDIBLE] 461 00:40:50,907 --> 00:40:54,900 they do not contain that information. 462 00:40:54,900 --> 00:41:01,870 PROFESSOR: Right, yeah so we will go into that. 463 00:41:01,870 --> 00:41:03,690 When the black hole evaporate, we still 464 00:41:03,690 --> 00:41:05,530 don't probe the short-- it's still harder 465 00:41:05,530 --> 00:41:08,990 to probe the short distance scale. 466 00:41:08,990 --> 00:41:10,550 Yeah, we will talk about that later. 467 00:41:13,990 --> 00:41:16,700 So yeah, here just a heuristic argument 468 00:41:16,700 --> 00:41:21,452 to tell you that because of this, actually 469 00:41:21,452 --> 00:41:22,660 the physics are very special. 470 00:41:25,250 --> 00:41:27,240 The physics of the gravity is very special. 471 00:41:30,848 --> 00:41:31,975 Any another questions? 472 00:41:37,290 --> 00:41:39,618 AUDIENCE: As a probe for quantum gravity, 473 00:41:39,618 --> 00:41:42,280 I was thinking-- what if there were some phenomenon 474 00:41:42,280 --> 00:41:46,680 in which gravity is weak, and it's a macroscopic scale, 475 00:41:46,680 --> 00:41:51,110 but there's some kind of coherence happening 476 00:41:51,110 --> 00:41:54,902 that will-- maybe on the scale of galaxies or something 477 00:41:54,902 --> 00:41:57,860 like that-- that will allow quantum effects to manifest. 478 00:41:57,860 --> 00:42:01,804 Like an analogy of what happens in a laser or something 479 00:42:01,804 --> 00:42:04,270 like that. 480 00:42:04,270 --> 00:42:07,640 PROFESSOR: Yeah, that is black hole. 481 00:42:07,640 --> 00:42:11,850 The black hole is the way which gravity 482 00:42:11,850 --> 00:42:15,690 can manifest at the quantum effect that matches in scales. 483 00:42:15,690 --> 00:42:19,640 And we don't know any other ways for gravity 484 00:42:19,640 --> 00:42:23,445 to manifest such quantum physics at large distance scales. 485 00:42:26,740 --> 00:42:32,470 OK, so let me conclude this I generally discussion. 486 00:42:32,470 --> 00:42:38,510 Again, by reminding you various regimes of gravity. 487 00:42:38,510 --> 00:42:41,915 Various regimes of the gravity or quantum gravity. 488 00:43:01,720 --> 00:43:05,130 So in this discussion, you should 489 00:43:05,130 --> 00:43:07,230 always-- in the discussion I'm going 490 00:43:07,230 --> 00:43:11,400 to do in the next minutes, you should always think, 491 00:43:11,400 --> 00:43:14,890 when I take the limit I always keep my energy fixed. 492 00:43:14,890 --> 00:43:18,630 I keep the energy scale I'm interested in fixed. 493 00:43:18,630 --> 00:43:21,350 I keep that fixed. 494 00:43:21,350 --> 00:43:29,250 So the classical gravity regime is the regime 495 00:43:29,250 --> 00:43:31,990 in which you take H-bar equals zero-- 496 00:43:31,990 --> 00:43:40,470 take Newton constant finite-- and the regime of a particle 497 00:43:40,470 --> 00:43:43,010 physics, we would normally be, sometimes, 498 00:43:43,010 --> 00:43:45,670 say, QFT in fixed space time. 499 00:43:52,180 --> 00:43:54,990 So this is a quantum field theory 500 00:43:54,990 --> 00:43:56,530 in the fixed space time, including 501 00:43:56,530 --> 00:44:05,980 curved-- including curved. 502 00:44:05,980 --> 00:44:10,580 So this is a regime in which H-bar is finite, 503 00:44:10,580 --> 00:44:12,650 while you take the Newton constant go to 0. 504 00:44:17,610 --> 00:44:22,850 And then there's, of course, the quantum gravity regime, 505 00:44:22,850 --> 00:44:25,960 which is the GN and the H-bar both finite. 506 00:44:29,930 --> 00:44:31,980 And then there's a very interesting regime, 507 00:44:31,980 --> 00:44:35,750 which is actually the regime most of us work with. 508 00:44:38,640 --> 00:44:47,917 So there's also something called the semiclassical regime 509 00:44:47,917 --> 00:44:48,750 for quantum gravity. 510 00:44:51,300 --> 00:44:59,790 So this is the regime you keep H-bar finite, 511 00:44:59,790 --> 00:45:10,480 and you expand this system in Newton constant, OK? 512 00:45:10,480 --> 00:45:13,290 So around-- so you expanded whatever 513 00:45:13,290 --> 00:45:17,260 quantity Newton constant around-- 514 00:45:17,260 --> 00:45:19,960 say G Newton equal to zero. 515 00:45:19,960 --> 00:45:24,830 So G Newton equal to 0 is the classical regime. 516 00:45:24,830 --> 00:45:29,520 It's the regime the gravity's not important, 517 00:45:29,520 --> 00:45:32,000 but including H-bar. 518 00:45:32,000 --> 00:45:36,590 And now you can take into account the quantum gravity 519 00:45:36,590 --> 00:45:40,550 fact, semiclassically, by expanding around the GN. 520 00:45:40,550 --> 00:45:43,600 OK, so this is normally what we call semiclassical regime 521 00:45:43,600 --> 00:45:44,180 of gravity. 522 00:45:52,380 --> 00:45:53,580 Yes? 523 00:45:53,580 --> 00:45:57,402 AUDIENCE: Do we know that G goes to 0 [INAUDIBLE] 524 00:45:57,402 --> 00:45:59,390 and it's not some other part? 525 00:45:59,390 --> 00:46:03,300 PROFESSOR: Yeah, this is a very good question. 526 00:46:03,300 --> 00:46:10,090 So this is indeed the question. 527 00:46:10,090 --> 00:46:13,160 Indeed, most of the particles regarding black hole 528 00:46:13,160 --> 00:46:14,760 is in this regime. 529 00:46:14,760 --> 00:46:17,940 So our current understanding of quantum physics or black hole 530 00:46:17,940 --> 00:46:26,160 is in the semiclassical regime, and you treat any matter field 531 00:46:26,160 --> 00:46:29,030 H-bar finite, but the gravity's weak. 532 00:46:29,030 --> 00:46:31,650 And so there are various indication 533 00:46:31,650 --> 00:46:34,070 that this limit is actually not smooth, 534 00:46:34,070 --> 00:46:36,085 but the only for very subtle questions. 535 00:46:38,777 --> 00:46:40,610 For simple questions, for typical questions, 536 00:46:40,610 --> 00:46:42,150 actually this is a limit. 537 00:46:42,150 --> 00:46:45,500 This limit is smooth, but there can be very subtle questions 538 00:46:45,500 --> 00:46:47,400 which this limit is not smooth. 539 00:46:47,400 --> 00:46:49,950 And one such question is this so-called the black hole 540 00:46:49,950 --> 00:46:52,700 information loss, and the subtle limit 541 00:46:52,700 --> 00:46:58,012 of taking this-- yeah, you [INAUDIBLE] taking this limit. 542 00:46:58,012 --> 00:46:58,845 Any other questions? 543 00:47:04,630 --> 00:47:07,770 And this is a regime, actually, we will work with most. 544 00:47:07,770 --> 00:47:10,850 OK, this is a regime we will work with most. 545 00:47:10,850 --> 00:47:14,570 So we will always-- in particular in nature. 546 00:47:14,570 --> 00:47:16,570 So right there I'm keeping the H-bar explicitly, 547 00:47:16,570 --> 00:47:19,300 but later I will also set the H-bar equal to 1. 548 00:47:19,300 --> 00:47:22,450 And so H-bar will equal to-- so we are always typically working 549 00:47:22,450 --> 00:47:23,640 with this regime. 550 00:47:23,640 --> 00:47:25,300 So H-bar equal to one. 551 00:47:25,300 --> 00:47:33,140 And then you will have-- yeah, then 552 00:47:33,140 --> 00:47:35,514 you take into account the fact of GN 553 00:47:35,514 --> 00:47:36,680 in the perturbations series. 554 00:47:42,490 --> 00:47:46,720 Good, so now let's move to the black hole. 555 00:47:46,720 --> 00:47:50,790 OK, with this-- yes? 556 00:47:50,790 --> 00:47:54,610 AUDIENCE: [INAUDIBLE] Also working 557 00:47:54,610 --> 00:47:57,870 in the other sermiclassical regime. 558 00:47:57,870 --> 00:48:02,490 I mean finite GN, but expand H-bar. 559 00:48:02,490 --> 00:48:11,380 PROFESSOR: Yeah, this is not so much. 560 00:48:11,380 --> 00:48:17,340 It's because it's easy-- in some sense it's easy. 561 00:48:20,310 --> 00:48:24,700 So in a sense, we are doing a little bit both. 562 00:48:24,700 --> 00:48:28,320 Yeah, so later-- right now I don't want to go into that. 563 00:48:28,320 --> 00:48:31,860 You will see the effective coupling constant 564 00:48:31,860 --> 00:48:37,670 and show the quantum gravity fact is in fact H-bar times GN. 565 00:48:37,670 --> 00:48:39,590 It's H-bar times GN. 566 00:48:39,590 --> 00:48:42,070 And so the quantum gravity fact will be important 567 00:48:42,070 --> 00:48:44,300 when the H-bar times-- when you do perturbation 568 00:48:44,300 --> 00:48:46,680 series H-bar times GN. 569 00:48:46,680 --> 00:48:49,260 So when I say you are doing same thing GN, 570 00:48:49,260 --> 00:48:51,630 essentially it's because I'm fixing H-bar. 571 00:48:51,630 --> 00:48:54,300 So you're actually doing perturbation series H-bar GN. 572 00:48:54,300 --> 00:48:55,500 So you have to do both. 573 00:48:58,182 --> 00:48:59,015 Any other questions? 574 00:49:10,711 --> 00:49:11,210 Good. 575 00:49:14,690 --> 00:49:21,410 Right, so now let's talk about the black hole. 576 00:49:24,430 --> 00:49:26,150 Let's talk about classical geometry. 577 00:49:31,440 --> 00:49:39,550 OK, so here I will assume you already 578 00:49:39,550 --> 00:49:43,250 have some background in GR-- in general relativity-- 579 00:49:43,250 --> 00:49:46,000 and for example, you have seen since Schwarzschild 580 00:49:46,000 --> 00:49:48,970 metric, et cetera. 581 00:49:48,970 --> 00:49:51,110 And if you have not seen a Schwarzschild metric, 582 00:49:51,110 --> 00:49:51,785 it's also OK. 583 00:49:54,630 --> 00:50:01,000 I think you should be able to follow what I'm going to say. 584 00:50:01,000 --> 00:50:03,616 So for simplicity, right now let me consider 585 00:50:03,616 --> 00:50:04,740 zero cosmological constant. 586 00:50:09,700 --> 00:50:13,750 OK, zero cosmological constant. 587 00:50:17,610 --> 00:50:20,150 OK, zero cosmological constant corresponding to-- we 588 00:50:20,150 --> 00:50:25,600 consider symptotically Minkowski space time. 589 00:50:25,600 --> 00:50:30,920 And so in the space time of zero cosmological constant entered 590 00:50:30,920 --> 00:50:45,120 in the space time metric due to an object of mass m. 591 00:50:49,736 --> 00:51:04,545 It can be written as-- so this is the famous Schwarzschild 592 00:51:04,545 --> 00:51:05,045 metric. 593 00:51:33,350 --> 00:51:38,980 And so of course here we are assuming 594 00:51:38,980 --> 00:51:50,000 this is a spherical symmetric, and a neutral, et cetera. 595 00:51:50,000 --> 00:51:52,630 So this object does not carry any charge. 596 00:51:55,480 --> 00:52:04,610 And this m and this f-- it's given by 1 minus 2m, 597 00:52:04,610 --> 00:52:10,810 mass divided by r, or it's equal to rs divided by r. 598 00:52:10,810 --> 00:52:13,910 OK, so this rs is the Schwarzschild radius. 599 00:52:14,620 --> 00:52:17,150 So now c is always equal to 1. 600 00:52:24,280 --> 00:52:27,980 So if this object-- so we consider this object 601 00:52:27,980 --> 00:52:29,320 is very close metric center. 602 00:52:29,320 --> 00:52:35,280 If this object have finite sides, then of course 603 00:52:35,280 --> 00:52:37,935 this metric only is varied outside this object. 604 00:52:40,830 --> 00:52:45,140 But if the sides of this object is 605 00:52:45,140 --> 00:52:49,350 smaller than the Schwarzschild radius, 606 00:52:49,350 --> 00:52:50,480 then this is a black hole. 607 00:52:50,480 --> 00:53:04,510 OK, and the black hold is distinguished by event horizon 608 00:53:04,510 --> 00:53:07,980 and r equal to rs. 609 00:53:07,980 --> 00:53:10,220 OK, and the r equal to rs. 610 00:53:10,220 --> 00:53:15,610 So at r equal to rs, you see that this f becomes 0. 611 00:53:15,610 --> 00:53:18,220 OK, so f equals become 0. 612 00:53:18,220 --> 00:53:26,800 So essentially at here gtt-- so the metric for the tt component 613 00:53:26,800 --> 00:53:35,720 is becomes 0, and the grr, the metric before the r component 614 00:53:35,720 --> 00:53:36,445 become infinite. 615 00:53:41,760 --> 00:53:49,300 And another thing is that when r becomes smaller than rs, 616 00:53:49,300 --> 00:53:50,845 the f switch sign. 617 00:53:57,540 --> 00:54:04,270 So f become active, and then in this case then the r become 618 00:54:04,270 --> 00:54:07,850 time coordinates and the t becomes spatial coordinates, 619 00:54:07,850 --> 00:54:12,206 when you go inside to the r equal to rs. 620 00:54:12,206 --> 00:54:14,450 OK, this is just a feature of this metric. 621 00:54:17,490 --> 00:54:24,720 So now let me say some simple fact about this metric. 622 00:54:34,420 --> 00:54:38,660 So most of them I expect you know-- 623 00:54:38,660 --> 00:54:40,120 I expect you know of them. 624 00:54:45,710 --> 00:54:52,290 But just to remind you some of those facts 625 00:54:52,290 --> 00:54:54,670 will be important later. 626 00:54:54,670 --> 00:54:59,800 OK, so these are mostly reminders. 627 00:55:03,490 --> 00:55:10,992 So first, this metric-- if you look at this metric itself, 628 00:55:10,992 --> 00:55:12,200 it's time reversal invariant. 629 00:55:20,860 --> 00:55:26,350 OK, because if you take t go to minus t, 630 00:55:26,350 --> 00:55:31,070 of course the invariant on the t goes to minus t. 631 00:55:31,070 --> 00:55:35,350 OK, so this-- actually, because of this, 632 00:55:35,350 --> 00:55:39,320 this cannot describe a real black hole. 633 00:55:39,320 --> 00:55:45,010 So the real life black hole arise from the gravitational 634 00:55:45,010 --> 00:55:48,650 collapse, and the gravitational collapse cannot be a time 635 00:55:48,650 --> 00:55:50,270 symmetric process, OK? 636 00:55:50,270 --> 00:55:53,200 So this cannot describe-- so this does not describe a real 637 00:55:53,200 --> 00:55:57,830 black hole, but it's a good approximation to the real life 638 00:55:57,830 --> 00:56:01,019 black hole after this object have stabilized. 639 00:56:01,019 --> 00:56:03,060 So after the gravitational collapse has finished. 640 00:56:05,620 --> 00:56:10,710 So this is a mathematical-- in some sense, 641 00:56:10,710 --> 00:56:18,859 it is a mathematical idealization 642 00:56:18,859 --> 00:56:24,230 of real life black hole. 643 00:56:27,510 --> 00:56:30,712 OK, so this is first remark. 644 00:56:35,980 --> 00:56:42,770 So the second remark is that despite this grr goes infinity, 645 00:56:42,770 --> 00:56:44,290 this metric component goes infinity. 646 00:56:46,840 --> 00:56:54,650 So the space time is non-singular at the horizon. 647 00:57:02,130 --> 00:57:07,110 OK, so you can check it by computing, 648 00:57:07,110 --> 00:57:11,240 say, curvature invariants of this metric. 649 00:57:11,240 --> 00:57:14,304 You find the number of the-- all the curvature invariant 650 00:57:14,304 --> 00:57:15,220 that are well-defined. 651 00:57:18,040 --> 00:57:22,570 So this horizon is just the coordinate-- 652 00:57:22,570 --> 00:57:24,600 you can show that this horizon is just 653 00:57:24,600 --> 00:57:25,700 a coordinate singularity. 654 00:57:32,620 --> 00:57:36,360 Which we will see-- actually, we will see it 655 00:57:36,360 --> 00:57:42,310 in maybe next lecture, or maybe at the end of today's lecture, 656 00:57:42,310 --> 00:57:45,430 that just this coordinate, r and the t, 657 00:57:45,430 --> 00:57:48,717 coordinate become singular. 658 00:57:48,717 --> 00:57:50,300 The coordinate itself become singular. 659 00:58:08,820 --> 00:58:12,660 So this t, we normally call it Schwarzschild time. 660 00:58:12,660 --> 00:58:14,640 So let me just introduce a name. 661 00:58:14,640 --> 00:58:17,325 So this t, we call it the Schwarzschild time. 662 00:58:30,620 --> 00:58:33,050 So Schwarzschild found this solution 663 00:58:33,050 --> 00:58:40,860 while fighting first World War, really in the battlefield. 664 00:58:40,860 --> 00:58:44,760 And a couple months after he finished this metric, 665 00:58:44,760 --> 00:58:46,150 he died from some disease. 666 00:58:53,490 --> 00:58:54,850 Right, this is that. 667 00:58:58,740 --> 00:59:05,000 So another thing, you can easily check yourself 668 00:59:05,000 --> 00:59:12,706 with r equal to rs-- the horizon is a null surface-- is 669 00:59:12,706 --> 00:59:13,580 a null hyper surface. 670 00:59:22,090 --> 00:59:24,160 It's a null hyper surface. 671 00:59:24,160 --> 00:59:32,670 The null hyper surface just say this surface contains 672 00:59:32,670 --> 00:59:34,690 geodesics which are null. 673 00:59:39,600 --> 00:59:44,690 And this is a-- the third remark is an extremely important one, 674 00:59:44,690 --> 00:59:50,680 which we will use many times in the future. 675 00:59:50,680 --> 01:00:03,750 We said the horizon is a surface of infinite red shift compared 676 01:00:03,750 --> 01:00:13,290 to the-- infinite red shift from the perspective of observed 677 01:00:13,290 --> 01:00:15,930 infinity, OK? 678 01:00:15,930 --> 01:00:19,500 So let me save time, now, to add to this qualifying remark 679 01:00:19,500 --> 01:00:23,440 from the perspective of observed infinity. 680 01:00:23,440 --> 01:00:28,100 So now let me just illustrate this point a little bit more 681 01:00:28,100 --> 01:00:30,000 explicitly. 682 01:00:30,000 --> 01:00:39,550 So let's consider observer-- consider an observer-- so 683 01:00:39,550 --> 01:00:46,550 let me call oh-- at some hyper surface r, which 684 01:00:46,550 --> 01:00:51,430 is close to the horizon. 685 01:00:51,430 --> 01:00:57,060 Yeah, say as someplace which is close to the horizon. 686 01:01:01,920 --> 01:01:05,030 Slightly outside the horizon. 687 01:01:05,030 --> 01:01:08,620 And let us consider another observer, 688 01:01:08,620 --> 01:01:16,890 which I call o infinity, which at i go to infinity. 689 01:01:16,890 --> 01:01:21,630 OK, very far away from the black hole. 690 01:01:21,630 --> 01:01:26,300 So let's first look at this observer. 691 01:01:26,300 --> 01:01:34,490 So the i equals infinity, then your metric-- so at i 692 01:01:34,490 --> 01:01:37,120 equals infinity, this f just becomes one, 693 01:01:37,120 --> 01:01:41,140 because when your r goes to infinity, this r just become 1. 694 01:01:41,140 --> 01:01:44,469 And then this become the standard Minkowski space 695 01:01:44,469 --> 01:01:46,260 time, written in the spherical coordinates. 696 01:01:49,250 --> 01:01:59,130 So we just have the standard Minkowski time written 697 01:01:59,130 --> 01:02:01,340 in spherical coordinates. 698 01:02:01,340 --> 01:02:03,133 And then from here you can immediately 699 01:02:03,133 --> 01:02:08,050 see-- so this t is what we call Schwarzschild time. 700 01:02:08,050 --> 01:02:16,100 t is the proper time for this observed infinity. 701 01:02:16,100 --> 01:02:18,580 Say, for o infinity. 702 01:02:18,580 --> 01:02:30,210 OK, so now let's look at someplace i equal to rh. 703 01:02:30,210 --> 01:02:37,330 So at i equal to rh, then the metric 704 01:02:37,330 --> 01:02:44,690 is given by minus f rh dt squared with the rest. 705 01:02:44,690 --> 01:02:53,090 OK, and to define the proper time for the observer at-- we 706 01:02:53,090 --> 01:02:55,875 can just directly write it as minus d tau squared. 707 01:02:55,875 --> 01:02:59,020 So that's the proper time observed by observer 708 01:02:59,020 --> 01:03:02,360 at this hyper surface, OK? 709 01:03:02,360 --> 01:03:06,370 So then we concludes-- so let me call it tau h. 710 01:03:06,370 --> 01:03:12,818 So we conclude that the problem time for oh 711 01:03:12,818 --> 01:03:18,610 is given by f 1/2 rh times dt. 712 01:03:18,610 --> 01:03:22,050 OK, so it relates to the proper time at infinity 713 01:03:22,050 --> 01:03:22,884 by this factor. 714 01:03:28,460 --> 01:03:31,670 So if I write it more explicitly-- so this 715 01:03:31,670 --> 01:03:37,670 is 1 minus rh, divided by rs 1/2 times dt. 716 01:03:46,090 --> 01:04:12,860 So we see that as rh-- suppose this observer-- 717 01:04:12,860 --> 01:04:19,520 the location of this observer approach the horizon, say, 718 01:04:19,520 --> 01:04:26,850 if rh approach to rs, then this d tau h divided by dt 719 01:04:26,850 --> 01:04:27,760 will go to 0. 720 01:04:35,730 --> 01:04:44,560 So that means compared to the time 721 01:04:44,560 --> 01:05:02,740 at infinity the time at r equal to rs becomes infinite now. 722 01:05:02,740 --> 01:05:06,200 So-- or approximate, let me say, becomes infinite now. 723 01:05:18,860 --> 01:05:24,730 So that means any finite interval-- 724 01:05:24,730 --> 01:05:30,550 any finite proper time interval for observer at oh-- 725 01:05:30,550 --> 01:05:35,560 for this oh-- when you view [INAUDIBLE] infinity 726 01:05:35,560 --> 01:05:37,640 become infinitely null, OK? 727 01:05:37,640 --> 01:05:39,890 Become very long time scale. 728 01:05:39,890 --> 01:05:41,860 OK, become very long time scale. 729 01:05:41,860 --> 01:05:49,710 So you can also invert this relation. 730 01:05:49,710 --> 01:05:54,170 OK, you can also invert the recent relation. 731 01:05:54,170 --> 01:06:02,980 Say, suppose some event of energy 732 01:06:02,980 --> 01:06:15,282 with energy eh-- with proper energy 733 01:06:15,282 --> 01:06:23,790 eh-- say, for this observer at oh, for this observer oh. 734 01:06:23,790 --> 01:06:31,750 Then because of the time relation between them, 735 01:06:31,750 --> 01:06:34,800 because of the time relation between them, 736 01:06:34,800 --> 01:06:45,170 then from the perspective of this observer at infinity 737 01:06:45,170 --> 01:06:50,470 the energy is given just but you invert the [INAUDIBLE] 738 01:06:50,470 --> 01:06:54,600 between the time, because of the energy and time conjugate. 739 01:06:54,600 --> 01:07:00,040 So the e infinity becomes eh times this f 1/2 rh. 740 01:07:05,100 --> 01:07:10,100 So for this just again says, in a slightly different way, 741 01:07:10,100 --> 01:07:19,780 that for any finite eh local proper energy-- so this 742 01:07:19,780 --> 01:07:26,000 is a local proper energy for the observer at oh-- 743 01:07:26,000 --> 01:07:30,160 this e infinity-- the same event viewed 744 01:07:30,160 --> 01:07:35,130 form the perspective of the observer at infinity goes to 0. 745 01:07:35,130 --> 01:07:39,500 S rh goes to rs. 746 01:07:39,500 --> 01:07:52,960 So that e is infinity red shifted-- 747 01:07:52,960 --> 01:07:54,430 become infinite red shift. 748 01:07:54,430 --> 01:08:00,930 So any process with local proper energy 749 01:08:00,930 --> 01:08:02,940 viewed from infinity corresponding 750 01:08:02,940 --> 01:08:06,520 to very, very low energy process. 751 01:08:06,520 --> 01:08:09,130 So this actually will play a very important role. 752 01:08:09,130 --> 01:08:11,630 This feature will play a very important role 753 01:08:11,630 --> 01:08:14,303 when later we talk about holographic duality 754 01:08:14,303 --> 01:08:18,382 and it's implication, say, for the field series, et cetera. 755 01:08:21,240 --> 01:08:21,819 Yes? 756 01:08:21,819 --> 01:08:24,309 AUDIENCE: So this is just a pedantic comment, 757 01:08:24,309 --> 01:08:27,795 but I think you need a minus sign in front of your f, 758 01:08:27,795 --> 01:08:31,779 just to make sure proper time isn't imaginary. 759 01:08:31,779 --> 01:08:35,418 PROFESSOR: Sorry, which minus sign? 760 01:08:35,418 --> 01:08:38,260 AUDIENCE: In the d tau stage. 761 01:08:38,260 --> 01:08:41,160 PROFESSOR: You mean here? 762 01:08:41,160 --> 01:08:42,154 You mean here or here? 763 01:08:42,154 --> 01:08:43,029 AUDIENCE: Below that. 764 01:08:43,029 --> 01:08:48,580 PROFESSOR: Below that, yeah-- oh, sorry, sorry. 765 01:08:48,580 --> 01:08:52,120 Thank you, I wrote it wrong. 766 01:08:52,120 --> 01:08:55,460 It should be rs rh. 767 01:08:55,460 --> 01:08:58,013 Yeah, because rs is above. 768 01:08:58,013 --> 01:09:04,640 Sorry, yeah-- so rs is above, so varied rh, so rh is downstairs. 769 01:09:04,640 --> 01:09:10,300 Thank you, so rh is-- so I always consider rh 770 01:09:10,300 --> 01:09:14,910 as [INAUDIBLE] equal to rs, OK? 771 01:09:18,547 --> 01:09:19,380 Any other questions? 772 01:09:25,859 --> 01:09:28,520 OK, and so some other fact. 773 01:09:28,520 --> 01:09:32,170 And again, I will just list them. 774 01:09:32,170 --> 01:09:33,859 I will just them. 775 01:09:33,859 --> 01:09:37,750 If you're not familiar with them, 776 01:09:37,750 --> 01:09:41,490 it should be very easy for you to go through them, 777 01:09:41,490 --> 01:09:46,149 to re-derive them, with a little bit knowledge in gr. 778 01:09:51,189 --> 01:10:03,780 So number four is that it takes a free fall-- free fall means 779 01:10:03,780 --> 01:10:12,410 we just follow geodesics-- free fall of a traveler 780 01:10:12,410 --> 01:10:23,160 a finite proper time to reach the horizon-- 781 01:10:23,160 --> 01:10:39,614 say, from the infinity-- but infinite Schwarzschild time. 782 01:10:48,560 --> 01:10:50,440 So also you can easily check that it actually 783 01:10:50,440 --> 01:10:53,770 takes infinite Schwarzschild time for object 784 01:10:53,770 --> 01:10:55,960 to fall through the horizon. 785 01:10:55,960 --> 01:10:58,830 But from the free fall observer itself, 786 01:10:58,830 --> 01:11:02,440 it's actually just finite proper time. 787 01:11:02,440 --> 01:11:06,760 And so from the perspective of the observer at infinity, 788 01:11:06,760 --> 01:11:11,330 it looks like this object never fall into the black hole. 789 01:11:11,330 --> 01:11:14,470 It's just frozen at the horizon. 790 01:11:14,470 --> 01:11:27,740 So another remark is that once inside the horizon-- that 791 01:11:27,740 --> 01:11:33,250 means when r becomes smaller than rs-- 792 01:11:33,250 --> 01:12:03,230 the traveler can not send signals to outside, 793 01:12:03,230 --> 01:12:05,528 nor can he escape. 794 01:12:09,350 --> 01:12:11,970 So that's why this is called even horizon. 795 01:12:11,970 --> 01:12:16,290 So we will see this slightly later. 796 01:12:16,290 --> 01:12:18,155 In next lecture we will see this explicitly. 797 01:12:21,890 --> 01:12:28,530 So finally, there are two important geometric quantities 798 01:12:28,530 --> 01:12:31,197 associated with the horizon. 799 01:12:31,197 --> 01:12:50,159 Two important geometric-- OK? 800 01:12:52,680 --> 01:13:02,130 So the first one is the area of the spatial section. 801 01:13:02,130 --> 01:13:08,410 So suppose-- so let's consider we are at i equal to rs, 802 01:13:08,410 --> 01:13:12,120 and then here you just said r equal to rs, 803 01:13:12,120 --> 01:13:15,357 and then this is a two-dimensional sphere, OK? 804 01:13:15,357 --> 01:13:17,190 So this two-dimensional sphere corresponding 805 01:13:17,190 --> 01:13:19,490 to a spatial section of the horizon. 806 01:13:22,890 --> 01:13:33,499 So first, A is the area of a spatial section. 807 01:13:40,360 --> 01:13:46,600 So you can just say let's look at the area of this part 808 01:13:46,600 --> 01:13:50,210 with the r equal to the horizon radius. 809 01:13:50,210 --> 01:13:55,820 So this just give you a equal to ah ah, equal to 4 810 01:13:55,820 --> 01:13:59,030 pi rs squared. 811 01:13:59,030 --> 01:14:12,370 And this rs is 2GN, so this become 16 pi GN squared. 812 01:14:12,370 --> 01:14:16,520 OK, so this is one of the key quantities of a black hole 813 01:14:16,520 --> 01:14:17,850 horizon. 814 01:14:17,850 --> 01:14:22,919 It's what we call the horizon area, OK? 815 01:14:22,919 --> 01:14:24,210 What we call this horizon area. 816 01:14:28,550 --> 01:14:31,750 And the B is called the surface gravity. 817 01:14:37,270 --> 01:14:39,760 B is called service gravity. 818 01:14:39,760 --> 01:14:43,570 So the surface gravity is defined 819 01:14:43,570 --> 01:14:59,986 by the acceleration of a stationary observer 820 01:14:59,986 --> 01:15:12,870 at the horizon as measured at infinity. 821 01:15:18,280 --> 01:15:22,750 OK, at infinity means at spatial infinity. 822 01:15:22,750 --> 01:15:27,080 So you can-- if you are not familiar with this concept 823 01:15:27,080 --> 01:15:31,920 of surface gravity, you can find it in standard textbooks. 824 01:15:31,920 --> 01:15:48,130 For example, the Wald say page 158, and also section-- OK. 825 01:15:48,130 --> 01:15:52,170 Just try to check it there. 826 01:15:52,170 --> 01:15:54,670 So, say, suppose you have a black hole. 827 01:15:59,490 --> 01:16:02,490 Say this is the Schwarzschild radius. 828 01:16:02,490 --> 01:16:05,710 Of course, things want to fall into the black hole. 829 01:16:05,710 --> 01:16:09,730 So if you want to remain at a fixed 830 01:16:09,730 --> 01:16:12,980 location outside the black hole, then you really 831 01:16:12,980 --> 01:16:14,520 have to accelerate. 832 01:16:14,520 --> 01:16:18,230 You have to fire some engine to keep yourself to stay there. 833 01:16:18,230 --> 01:16:22,250 And you can calculate what is acceleration you need 834 01:16:22,250 --> 01:16:26,120 to be able to stay here, OK? 835 01:16:26,120 --> 01:16:28,330 And once you are closer and closer to the horizon, 836 01:16:28,330 --> 01:16:30,705 that acceleration becomes bigger and bigger-- eventually, 837 01:16:30,705 --> 01:16:34,620 becomes infinity when you approach the horizon. 838 01:16:34,620 --> 01:16:41,220 But because of this red shift effect, 839 01:16:41,220 --> 01:16:48,210 when this acceleration is viewed from the units for observer 840 01:16:48,210 --> 01:16:51,990 at affinity, then you have infinity divided by infinity, 841 01:16:51,990 --> 01:16:54,650 then turns out to be finite. 842 01:16:54,650 --> 01:16:57,330 And this is called a surface gravity. 843 01:16:57,330 --> 01:16:59,810 It's normally called a kappa. 844 01:16:59,810 --> 01:17:01,410 Normally called a kappa. 845 01:17:01,410 --> 01:17:04,090 And this is one of basic quantities-- 846 01:17:04,090 --> 01:17:06,270 basic geometric quantities of the horizon. 847 01:17:06,270 --> 01:17:08,410 So I will not derive it here. 848 01:17:08,410 --> 01:17:10,600 I don't have time. 849 01:17:10,600 --> 01:17:14,240 And if you want to see this, Wald's book. 850 01:17:14,240 --> 01:17:16,600 So you can calculate that this is just 851 01:17:16,600 --> 01:17:22,210 given by 1/2 the derivative of this function 852 01:17:22,210 --> 01:17:28,300 f evaluated at the horizon location. 853 01:17:28,300 --> 01:17:33,920 OK, so f is equal to 0 at the horizon, but f prime is not. 854 01:17:33,920 --> 01:17:36,920 OK, so you can easily calculate. 855 01:17:36,920 --> 01:17:45,730 So this is 1 over 2rs from here, and is equal to 1 over 4GN. 856 01:17:49,482 --> 01:17:53,070 OK, so this is another very important quantity 857 01:17:53,070 --> 01:17:55,440 for the black hole. 858 01:17:55,440 --> 01:18:00,690 Actually, I think right now is a good place to stop. 859 01:18:04,510 --> 01:18:07,480 OK, so let's stop here for today, 860 01:18:07,480 --> 01:18:10,510 and the next time we will describe-- then 861 01:18:10,510 --> 01:18:12,805 from here we will discuss the causal structure 862 01:18:12,805 --> 01:18:16,320 of the black hole, and then you will see explicitly 863 01:18:16,320 --> 01:18:20,480 some of this statement, if you have not seen them before.