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 --> 00:00:17,255 at ocw.mit.edu. 8 00:00:21,770 --> 00:00:28,320 PROFESSOR: So last time we talked about how to describe 9 00:00:28,320 --> 00:00:34,180 say for the boundary CFT is at a finite temperature what would 10 00:00:34,180 --> 00:00:35,475 be its gravity description? 11 00:00:39,010 --> 00:00:45,090 So what we described was a thermal theory. 12 00:00:45,090 --> 00:00:57,510 Say a Thermal CFT on the flat space, OK? 13 00:01:02,070 --> 00:01:14,180 We show that this is related to a black hole with a planar 14 00:01:14,180 --> 00:01:14,680 horizon. 15 00:01:22,300 --> 00:01:25,360 The key is that the horizon should have the same topology 16 00:01:25,360 --> 00:01:27,115 as the boundary space. 17 00:01:30,495 --> 00:01:37,870 So in terms of the picture this is your boundary, 18 00:01:37,870 --> 00:01:48,060 so this is z equal to 0, and then the horizon somewhere say, 19 00:01:48,060 --> 00:01:51,550 at some value of z0. 20 00:01:51,550 --> 00:01:54,570 So horizon has some value of z0 and then the horizon 21 00:01:54,570 --> 00:02:00,585 have the same topology as the boundary on space. 22 00:02:03,730 --> 00:02:07,800 And then you can do some of the [? lamicks, ?] et cetera. 23 00:02:07,800 --> 00:02:08,960 Any questions about that? 24 00:02:13,520 --> 00:02:16,290 Good. 25 00:02:16,290 --> 00:02:19,520 So previously we also briefly discussed 26 00:02:19,520 --> 00:02:22,342 that the Ads also allow you to put the field 27 00:02:22,342 --> 00:02:25,050 theory on the sphere. 28 00:02:25,050 --> 00:02:27,550 So if you put the field theory on the sphere 29 00:02:27,550 --> 00:02:29,810 then that will be due to the global Ads. 30 00:02:32,950 --> 00:02:34,980 Now you can ask the same question. 31 00:02:34,980 --> 00:02:38,190 What happens if we put the boundary theory on the sphere 32 00:02:38,190 --> 00:02:39,940 and then put there the finite temperature. 33 00:02:43,810 --> 00:02:57,237 So let us put a CFT on the sphere 34 00:02:57,237 --> 00:02:58,320 at the finite temperature. 35 00:03:07,100 --> 00:03:13,320 So the story here is much richer for a very simple reason. 36 00:03:13,320 --> 00:03:16,690 For a very simple reason. 37 00:03:16,690 --> 00:03:32,290 So if you have a CFT just on the Rd minus 1 so this Rd minus 1, 38 00:03:32,290 --> 00:03:34,200 this euclidean space, euclidean flat space 39 00:03:34,200 --> 00:03:41,300 does not have a scale and the CFT does not have a scale. 40 00:03:41,300 --> 00:03:44,890 So if you put it at a finite temperature 41 00:03:44,890 --> 00:03:47,790 then the temperature is the only scale, 42 00:03:47,790 --> 00:03:50,700 and essentially provide a unit over the scale. 43 00:03:54,550 --> 00:04:01,200 So the temperature is the only scale, 44 00:04:01,200 --> 00:04:11,375 so essentially providing an energy unit. 45 00:04:18,820 --> 00:04:23,270 So this means that for CFT on the finite temperature, CFT, 46 00:04:23,270 --> 00:04:42,000 on the flat space then physics at all temperatures 47 00:04:42,000 --> 00:04:42,590 are the same. 48 00:04:50,480 --> 00:04:59,890 So they are just related by a scaling of units. 49 00:05:08,040 --> 00:05:14,930 So in some sense you can see it seems that the physics is very 50 00:05:14,930 --> 00:05:19,710 simple whether you have like 0.0001 degree 51 00:05:19,710 --> 00:05:22,380 or you have 10,000 degrees, doesn't matter. 52 00:05:22,380 --> 00:05:25,252 And it's only the relative scale that matters. 53 00:05:25,252 --> 00:05:27,210 So the physics at all temperatures is the same. 54 00:05:27,210 --> 00:05:28,620 There's no difference between the low temperature 55 00:05:28,620 --> 00:05:31,300 and the high temperature, and the temperature just 56 00:05:31,300 --> 00:05:32,490 provide the units. 57 00:05:35,860 --> 00:05:39,360 But if you put the CFT on the sphere 58 00:05:39,360 --> 00:05:42,630 now the story's different. 59 00:05:42,630 --> 00:05:51,210 So for CFT on the sphere because the sphere has a side itself. 60 00:05:51,210 --> 00:05:56,160 OK, so let's take the sphere have both sides r. 61 00:05:56,160 --> 00:06:00,470 Again when you put the CFT on the sphere 62 00:06:00,470 --> 00:06:04,050 the sides of the sphere does not matter because again 63 00:06:04,050 --> 00:06:06,350 the sides of the sphere are essentially 64 00:06:06,350 --> 00:06:09,970 provide by the units because the theory's getting [? warrant. ?] 65 00:06:09,970 --> 00:06:12,700 So let's just put all the sides to be r. 66 00:06:16,520 --> 00:06:18,620 So let's take sides to be r. 67 00:06:18,620 --> 00:06:20,902 And the r can be chosen to be the same 68 00:06:20,902 --> 00:06:22,610 as the curvature radius in the [? back ?] 69 00:06:22,610 --> 00:06:25,480 just to make formula simple, and you 70 00:06:25,480 --> 00:06:28,400 can choose it to be any radius. 71 00:06:28,400 --> 00:06:32,470 And now since the sphere already have [? all ?] sides 72 00:06:32,470 --> 00:06:34,770 and now you put it at a finite temperature. 73 00:06:34,770 --> 00:06:37,660 Now you have the dimensionless number. 74 00:06:37,660 --> 00:06:45,490 Then at the finite temperature in the physics 75 00:06:45,490 --> 00:06:59,650 is controlled by a dimensionless number, r times t. 76 00:07:03,310 --> 00:07:07,820 So now the temperature makes a difference. 77 00:07:07,820 --> 00:07:09,530 Now different temperature that r is 78 00:07:09,530 --> 00:07:13,420 is different because now you have a relative scale 79 00:07:13,420 --> 00:07:17,190 and if rt is small then essentially you're 80 00:07:17,190 --> 00:07:19,555 at the low temperature, and then when rt is big 81 00:07:19,555 --> 00:07:20,930 then you have a high temperature. 82 00:07:20,930 --> 00:07:22,790 Now there's a dimensionless number 83 00:07:22,790 --> 00:07:25,200 to characterize whether you're in the low temperature 84 00:07:25,200 --> 00:07:27,210 or in the high temperature. 85 00:07:27,210 --> 00:07:30,460 So the physics essentially become much richer because 86 00:07:30,460 --> 00:07:35,627 of the and then you have a you have a whole parameter 87 00:07:35,627 --> 00:07:36,710 the physics can depend on. 88 00:07:40,450 --> 00:07:49,390 And so indeed the story become rather intricate 89 00:07:49,390 --> 00:07:51,160 whether we can see the finite temperature 90 00:07:51,160 --> 00:07:53,690 theory on the sphere so now let me just 91 00:07:53,690 --> 00:07:55,150 mention some important features. 92 00:08:04,970 --> 00:08:12,300 So I will walk you not too slowly but also not too quickly 93 00:08:12,300 --> 00:08:13,920 about the physics on the sphere. 94 00:08:16,970 --> 00:08:21,650 We cannot afford to do it very slowly at this time. 95 00:08:21,650 --> 00:08:23,250 Yeah, but if I do it too quickly, 96 00:08:23,250 --> 00:08:25,590 of course you won't learn anything. 97 00:08:25,590 --> 00:08:31,320 OK, so let me just point out some important features. 98 00:08:31,320 --> 00:08:33,919 So first, when we talked about the story 99 00:08:33,919 --> 00:08:36,860 last week-- when we talked about the finite temperature 100 00:08:36,860 --> 00:08:41,120 on the flat space-- we said, oh, probably there 101 00:08:41,120 --> 00:08:46,810 are two possible descriptions for finite temperature 102 00:08:46,810 --> 00:08:49,220 theory from the [? back ?] point of view. 103 00:08:49,220 --> 00:08:52,520 So one is that you can have a thermal gas. 104 00:08:57,191 --> 00:08:58,440 You have a thermal gas in Ads. 105 00:09:01,710 --> 00:09:04,100 But then we argue that for a theory 106 00:09:04,100 --> 00:09:06,970 on the sphere that's not allowed because you 107 00:09:06,970 --> 00:09:11,210 encompass singularity because the metric becomes singular. 108 00:09:11,210 --> 00:09:15,740 And the one big difference for the theory on the sphere 109 00:09:15,740 --> 00:09:19,170 is that actually now this is allowed. 110 00:09:19,170 --> 00:09:22,873 So the thermal gas in Ads is now allowed. 111 00:09:27,220 --> 00:09:30,230 So now let's go through the argument. 112 00:09:30,230 --> 00:09:32,955 So now first let me just write down the global Ads metric. 113 00:09:39,600 --> 00:09:42,500 Of course you can write in many different coordinates. 114 00:09:42,500 --> 00:09:43,960 So the most convenient coordinates 115 00:09:43,960 --> 00:09:49,220 to write global Ads for our purpose is the following. 116 00:09:54,530 --> 00:10:14,840 So this is just a pure Ads and r goes to 0 to infinite. 117 00:10:14,840 --> 00:10:17,630 So this is like somewhat analog of a, say, 118 00:10:17,630 --> 00:10:21,370 sprinkle coordinate in flat space. 119 00:10:21,370 --> 00:10:23,180 But the difference is of course now 120 00:10:23,180 --> 00:10:27,010 you have nontrivial factors which tell 121 00:10:27,010 --> 00:10:30,090 you're in the curve space time. 122 00:10:36,690 --> 00:10:40,430 So now to go to the thermal Ads we 123 00:10:40,430 --> 00:10:45,890 do the trick you normally do for thermal field theory. 124 00:10:45,890 --> 00:10:51,470 Is that you go to euclidean space 125 00:10:51,470 --> 00:10:55,110 and then you put the euclidean time 126 00:10:55,110 --> 00:10:58,290 to be periodic in the inverse temperature. 127 00:11:01,365 --> 00:11:05,940 Actually, let me write down here the analogous 128 00:11:05,940 --> 00:11:08,955 metric on the sphere. 129 00:11:15,770 --> 00:11:17,720 So this is a situation we can see that 130 00:11:17,720 --> 00:11:23,940 before for the theory on the flat space, on the plane, 131 00:11:23,940 --> 00:11:27,870 and then that would be the metric to be on the sphere. 132 00:11:27,870 --> 00:11:31,230 So we've talked about last time if you 133 00:11:31,230 --> 00:11:36,720 make tau to be periodic then as z 134 00:11:36,720 --> 00:11:41,400 goes to infinity then this circle shrink to zero sides. 135 00:11:41,400 --> 00:11:45,040 Because the prefactor goes to 0. 136 00:11:45,040 --> 00:11:48,680 So the tau circle will have a sides beta 137 00:11:48,680 --> 00:11:52,640 but the proper densities controlled by the prefactor. 138 00:11:52,640 --> 00:11:57,370 And because when z goes to infinity then the proper lens 139 00:11:57,370 --> 00:11:59,740 of the circle goes to 0. 140 00:11:59,740 --> 00:12:02,060 So whenever you have a circle we shrink to 0 141 00:12:02,060 --> 00:12:04,380 then you have a singular behavior, et cetera, 142 00:12:04,380 --> 00:12:07,730 and then the metric's singular. 143 00:12:07,730 --> 00:12:10,670 So you can have [? where ?] it is a singular behavior 144 00:12:10,670 --> 00:12:15,240 and so that's why this is not the amount. 145 00:12:15,240 --> 00:12:19,860 But now in this case, when you do such energy configuration 146 00:12:19,860 --> 00:12:21,950 this becomes well defined. 147 00:12:21,950 --> 00:12:28,630 Because of the proper sides of the tau circle so now you 148 00:12:28,630 --> 00:12:37,790 local proper lens local proper sides over the tau circle, 149 00:12:37,790 --> 00:12:44,140 it's just given by this prefactor 1 plus 150 00:12:44,140 --> 00:12:49,200 r squared divided by 1 squared by r squared beta. 151 00:12:54,120 --> 00:12:56,640 And since a greater equal here so 152 00:12:56,640 --> 00:12:58,080 this is greater equal to beta. 153 00:13:02,230 --> 00:13:04,390 So actually this is bounded from below. 154 00:13:08,020 --> 00:13:10,780 So this is a circle [? label ?] will become too small. 155 00:13:10,780 --> 00:13:12,720 Essentially the minimal side of the circle 156 00:13:12,720 --> 00:13:15,502 is the same as the beta itself. 157 00:13:18,100 --> 00:13:21,860 So this euclidean metric is perfectly well defined. 158 00:13:21,860 --> 00:13:32,410 OK, so euclidean metric well defined. 159 00:13:45,400 --> 00:14:02,240 So you can trust here that the euclidean metric is singular 160 00:14:02,240 --> 00:14:03,600 as z goes to infinity. 161 00:14:10,050 --> 00:14:12,037 If you put the 0 in the flat space then 162 00:14:12,037 --> 00:14:13,870 that's not allowed but in this case allowed. 163 00:14:19,320 --> 00:14:21,590 So for the serial in the sphere-- 164 00:14:21,590 --> 00:14:24,356 but still there's an analog of a black hole solution. 165 00:14:24,356 --> 00:14:25,730 You can also find the black hole. 166 00:14:36,250 --> 00:14:39,580 Again you just write down the standards, 167 00:14:39,580 --> 00:14:43,720 sprinkle's metric answers, for black hole metric 168 00:14:43,720 --> 00:14:46,060 and then you solve Einstein equation, 169 00:14:46,060 --> 00:14:50,970 then you find-- so let me just write down the answer. 170 00:14:50,970 --> 00:14:55,525 So you find that this space also allows a black hole solution. 171 00:14:55,525 --> 00:14:56,870 Can be written as the following. 172 00:15:09,180 --> 00:15:12,695 So now f just slight generalization of this factor. 173 00:15:23,790 --> 00:15:26,670 So the mu is primarily related to black hole mass. 174 00:15:34,430 --> 00:15:44,170 And the horizon is that i equal to r0 and r 175 00:15:44,170 --> 00:15:46,330 satisfy that this factor goes to 0. 176 00:15:59,940 --> 00:16:02,460 So you can also find out what is temperature associated 177 00:16:02,460 --> 00:16:03,550 with this black hole. 178 00:16:06,920 --> 00:16:08,580 So here that beta can be anything. 179 00:16:12,609 --> 00:16:14,025 There's no constraint on the beta. 180 00:16:22,430 --> 00:16:25,462 So here using the standard technique to find what 181 00:16:25,462 --> 00:16:27,420 is the Hawking temperature for this black hole. 182 00:16:27,420 --> 00:16:29,480 Let me just write down the answer. 183 00:16:29,480 --> 00:16:33,480 So this is just 4 pi divided by f prime. 184 00:16:33,480 --> 00:16:38,590 You've already add to the horizon standard formula 185 00:16:38,590 --> 00:16:42,830 and you just find the 0 of this guy, 186 00:16:42,830 --> 00:16:47,297 and then you just take the derivative 187 00:16:47,297 --> 00:16:49,380 so you find the beta can be written in terms of r0 188 00:16:49,380 --> 00:16:49,879 as follows. 189 00:16:55,330 --> 00:16:56,920 So this is just a simple algebra. 190 00:17:02,430 --> 00:17:05,359 So you find that if you take the derivative, 191 00:17:05,359 --> 00:17:07,520 we express the derivative is r0, then 192 00:17:07,520 --> 00:17:10,130 you find the Hawking temperature of the following form. 193 00:17:16,960 --> 00:17:18,565 I'm not doing this calculation here. 194 00:17:18,565 --> 00:17:19,940 You can use it to check yourself. 195 00:17:22,700 --> 00:17:26,620 Good, any questions so far? 196 00:17:26,620 --> 00:17:28,245 So now it seems like we have a problem. 197 00:17:30,840 --> 00:17:32,570 Because we have a black hole we also 198 00:17:32,570 --> 00:17:38,640 have a standard thermal Ads. 199 00:17:38,640 --> 00:17:43,140 Then which one is the right description 200 00:17:43,140 --> 00:17:46,435 for the field theory at the finite temperature? 201 00:17:51,540 --> 00:17:55,540 Seems like we have a choice. 202 00:17:55,540 --> 00:18:01,730 But now if you look at this function 203 00:18:01,730 --> 00:18:04,360 the story's even more tricky than that. 204 00:18:04,360 --> 00:18:06,900 So this product of this function this is a rough behavior 205 00:18:06,900 --> 00:18:08,040 of this function. 206 00:18:08,040 --> 00:18:11,180 So this is the inverse temperature 207 00:18:11,180 --> 00:18:14,090 and then this is the sides of the horizon. 208 00:18:14,090 --> 00:18:16,850 And the side of the horizon essentially 209 00:18:16,850 --> 00:18:18,830 determines your entropy. 210 00:18:18,830 --> 00:18:21,010 So it's really a physical thing here 211 00:18:21,010 --> 00:18:24,520 and because of the area over the horizon it's the entropy. 212 00:18:24,520 --> 00:18:27,960 The area of the horizon is a entropy. 213 00:18:27,960 --> 00:18:32,740 So this r0 it can be considered as something standard 214 00:18:32,740 --> 00:18:33,240 for entropy. 215 00:18:35,970 --> 00:18:38,110 So now just to get the intrusion, 216 00:18:38,110 --> 00:18:41,450 this is a complicated function, it's 217 00:18:41,450 --> 00:18:43,590 because intrusion and how this beta depend 218 00:18:43,590 --> 00:18:45,600 on this r0, the horizon side. 219 00:18:54,740 --> 00:18:58,460 Suppose horizon side is very small, r0 goes to 0, 220 00:18:58,460 --> 00:18:59,720 then downstairs is finite. 221 00:18:59,720 --> 00:19:03,470 Upstairs goes to 0 so you just go to 0. 222 00:19:08,880 --> 00:19:13,580 And the r0 is very large so the downstairs 223 00:19:13,580 --> 00:19:17,090 proportionate to r0 square but upstairs only 224 00:19:17,090 --> 00:19:22,350 proportionate to r0 so when r0 is very large 225 00:19:22,350 --> 00:19:26,720 we go to 0 as the r0. 226 00:19:26,720 --> 00:19:29,150 And this is a small function so there 227 00:19:29,150 --> 00:19:30,470 must be a maximum somewhere. 228 00:19:32,980 --> 00:19:35,640 And because you can easily find the maximum just 229 00:19:35,640 --> 00:19:38,080 to take the derivative of that function. 230 00:19:38,080 --> 00:19:42,185 Anyway, so that's how the beta will depend on r. 231 00:19:45,510 --> 00:19:50,710 But now this plot is highly peculiar. 232 00:19:50,710 --> 00:19:51,617 Why? 233 00:19:51,617 --> 00:19:52,450 Can you tell me why? 234 00:20:02,950 --> 00:20:08,030 There's a maximum on theta, which means? 235 00:20:08,030 --> 00:20:09,040 That's right. 236 00:20:09,040 --> 00:20:12,110 So there's a beta max. 237 00:20:12,110 --> 00:20:26,009 So that means this is a beta max which tells you there's 238 00:20:26,009 --> 00:20:26,925 a minimum temperature. 239 00:20:29,744 --> 00:20:31,410 That means that black hole solution only 240 00:20:31,410 --> 00:20:36,220 exist above a certain minimal temperature. 241 00:20:36,220 --> 00:20:38,730 And you can easily find out what is that minimal temperature 242 00:20:38,730 --> 00:20:41,420 just by taking the derivative of this function 243 00:20:41,420 --> 00:20:42,690 and it require to be 0. 244 00:20:49,870 --> 00:20:52,210 Is there any other thing peculiar regarding this plot? 245 00:20:54,970 --> 00:20:56,590 AUDIENCE: When r0-- 246 00:20:56,590 --> 00:20:57,215 PROFESSOR: Hmm? 247 00:20:57,215 --> 00:21:00,630 AUDIENCE: When r0, [INAUDIBLE] black hole. 248 00:21:00,630 --> 00:21:02,697 PROFESSOR: Well, r0 is 0-- the black hole becomes 249 00:21:02,697 --> 00:21:03,530 smaller and smaller. 250 00:21:03,530 --> 00:21:05,229 It means there's no black hole. 251 00:21:05,229 --> 00:21:06,978 AUDIENCE: But the temperature is infinity. 252 00:21:06,978 --> 00:21:07,434 PROFESSOR: Hmm? 253 00:21:07,434 --> 00:21:09,433 AUDIENCE: But the temperature there is infinity. 254 00:21:11,682 --> 00:21:12,550 That's weird. 255 00:21:12,550 --> 00:21:14,337 PROFESSOR: No, that's not weird. 256 00:21:14,337 --> 00:21:16,420 No, that's the standard, the flat space black hole 257 00:21:16,420 --> 00:21:19,032 behaves that way. 258 00:21:19,032 --> 00:21:21,240 Just look at a Schwarzschild black hole in flat space 259 00:21:21,240 --> 00:21:23,670 when the horizon side becomes smaller and smaller 260 00:21:23,670 --> 00:21:27,500 the temperature become higher and higher. 261 00:21:27,500 --> 00:21:31,105 That's a standard feature of flat space. 262 00:21:31,105 --> 00:21:32,886 AUDIENCE: It must be the other way. 263 00:21:32,886 --> 00:21:34,510 If you have a huge black hole, then why 264 00:21:34,510 --> 00:21:37,312 is it that the beta goes down? 265 00:21:37,312 --> 00:21:38,615 That seems bizarre. 266 00:21:38,615 --> 00:21:40,240 Maybe it can't also be standard, right? 267 00:21:45,608 --> 00:21:50,500 [LAUGHTER] 268 00:21:50,500 --> 00:21:55,690 PROFESSOR: Both things you said are correct and are important 269 00:21:55,690 --> 00:21:58,150 but there's something more elementary 270 00:21:58,150 --> 00:21:59,250 you're not pointing out. 271 00:21:59,250 --> 00:22:02,200 You're pointing out the higher order of important things. 272 00:22:02,200 --> 00:22:04,625 But there's one lower order important thing that which 273 00:22:04,625 --> 00:22:05,815 is not yet pointed out. 274 00:22:07,710 --> 00:22:09,085 AUDIENCE: There are two solutions 275 00:22:09,085 --> 00:22:10,950 of the same temperature. 276 00:22:10,950 --> 00:22:12,690 PROFESSOR: Exactly. 277 00:22:12,690 --> 00:22:16,145 In terms of the temperature this is a [INAUDIBLE] 278 00:22:16,145 --> 00:22:17,770 So if you look at the given temperature 279 00:22:17,770 --> 00:22:19,061 they're actually two solutions. 280 00:22:24,990 --> 00:22:31,255 So for any given temperature there are two solutions. 281 00:22:35,810 --> 00:22:40,640 Say T greater than T min so this is 282 00:22:40,640 --> 00:22:42,800 for the beta smaller than beta max 283 00:22:42,800 --> 00:22:45,305 means that T greater than T min There are two solutions. 284 00:22:51,590 --> 00:22:55,130 And then we have to label them so let's 285 00:22:55,130 --> 00:23:00,010 very imaginatively call this one the big black hole and then 286 00:23:00,010 --> 00:23:02,410 this one the small black hole. 287 00:23:02,410 --> 00:23:04,450 Because this one have a larger size and this one 288 00:23:04,450 --> 00:23:07,390 has a smaller size. 289 00:23:07,390 --> 00:23:11,620 So this corresponding to the r0 of those two. 290 00:23:14,039 --> 00:23:15,580 So you have two black hole solutions. 291 00:23:15,580 --> 00:23:17,580 One have a larger radius than the other one. 292 00:23:20,130 --> 00:23:25,766 So we have big black hole and a small black hole. 293 00:23:25,766 --> 00:23:28,370 And now this is an important thing 294 00:23:28,370 --> 00:23:35,490 you mentioned that for the small black hole 295 00:23:35,490 --> 00:23:44,690 the temperature when you decrease the sides 296 00:23:44,690 --> 00:23:45,775 the beta decrease. 297 00:23:45,775 --> 00:23:47,525 That then means the temperature increases. 298 00:23:52,490 --> 00:23:55,660 So this as I said it's like an entropy. 299 00:23:55,660 --> 00:23:58,260 So if you lower the entropy somehow 300 00:23:58,260 --> 00:24:01,020 your temperature increases. 301 00:24:01,020 --> 00:24:02,520 So when you increase the temperature 302 00:24:02,520 --> 00:24:04,853 you actually lower the entropy for the small black hole. 303 00:24:04,853 --> 00:24:06,722 A small black hole you go like this. 304 00:24:06,722 --> 00:24:07,460 Go like this. 305 00:24:07,460 --> 00:24:11,960 When you reduce the sides, you reduce the beta, 306 00:24:11,960 --> 00:24:13,890 means you increase the temperature. 307 00:24:13,890 --> 00:24:16,865 But this is actually the standard Schwarzschild black 308 00:24:16,865 --> 00:24:18,700 hole behavior in flat space. 309 00:24:18,700 --> 00:24:23,430 And this just tells you have [? a lack ?] of specific heat. 310 00:24:26,940 --> 00:24:28,954 So this is a situation we understand. 311 00:24:28,954 --> 00:24:30,370 This is a situation we can come up 312 00:24:30,370 --> 00:24:33,060 with a flat space, a Schwarzchild black hole. 313 00:24:33,060 --> 00:24:39,110 And it make sense why it's happening here because here 314 00:24:39,110 --> 00:24:43,800 roughly is order r. 315 00:24:43,800 --> 00:24:46,190 Because the r is the scale here so 316 00:24:46,190 --> 00:24:50,130 the maximum location where the maximum is controlled by the r. 317 00:24:50,130 --> 00:24:55,330 So this region corresponding to sides of the black hole 318 00:24:55,330 --> 00:24:59,126 to be much, much smaller than Ads curvature radius. 319 00:24:59,126 --> 00:25:00,500 So when something is much smaller 320 00:25:00,500 --> 00:25:04,348 than the curvature radius what you do see? 321 00:25:04,348 --> 00:25:05,260 Hmm? 322 00:25:05,260 --> 00:25:06,614 AUDIENCE: [INAUDIBLE] 323 00:25:06,614 --> 00:25:07,780 PROFESSOR: Yeah, flat space. 324 00:25:07,780 --> 00:25:11,510 Yeah, it just like even though we live in the curved universe 325 00:25:11,510 --> 00:25:14,190 but we actually see flat space because of course our scale 326 00:25:14,190 --> 00:25:17,720 is much more than curvature radius of universe. 327 00:25:17,720 --> 00:25:20,600 And a similar key thing here even though the black hole 328 00:25:20,600 --> 00:25:24,240 have a size much smaller than the curvature radius of Ads 329 00:25:24,240 --> 00:25:26,562 they essentially behave like flat space black hole. 330 00:25:26,562 --> 00:25:28,770 And so that's exactly what you see for the flat space 331 00:25:28,770 --> 00:25:30,600 black hole. 332 00:25:30,600 --> 00:25:32,300 And indeed, in a more interesting 333 00:25:32,300 --> 00:25:39,180 is this big black hole, the [? interesting ?] 334 00:25:39,180 --> 00:25:41,140 is a big black hole. 335 00:25:41,140 --> 00:25:45,670 And the big black hole when r0 increases the temperature 336 00:25:45,670 --> 00:25:47,780 increase. 337 00:25:47,780 --> 00:25:49,680 So this is the right way. 338 00:25:49,680 --> 00:25:58,920 So this is indeed the standard, just positive specific heat. 339 00:26:03,560 --> 00:26:06,050 You can check it has positive specific heat. 340 00:26:06,050 --> 00:26:10,020 So this is indeed what you'd expect from a thermal system. 341 00:26:10,020 --> 00:26:13,140 And so this is indeed what you expect from a thermal system. 342 00:26:17,090 --> 00:26:21,100 So now we are encountering an even weirder situation. 343 00:26:23,860 --> 00:26:25,160 We not only have so more ideas. 344 00:26:25,160 --> 00:26:26,120 We're not only have black hole. 345 00:26:26,120 --> 00:26:27,495 We actually have two black holes. 346 00:26:31,187 --> 00:26:33,270 So what's actually described with the field theory 347 00:26:33,270 --> 00:26:35,880 at finite temperature? 348 00:26:35,880 --> 00:26:38,940 So what does this really mean? 349 00:26:38,940 --> 00:26:40,780 Number two is a black hole solution. 350 00:26:40,780 --> 00:26:44,680 Now let's look at the number three. 351 00:26:44,680 --> 00:26:47,254 So now let's look at what are the indications 352 00:26:47,254 --> 00:26:47,920 those solutions. 353 00:26:52,140 --> 00:26:55,260 So first thing you can just guess. 354 00:26:55,260 --> 00:26:58,200 So what do you think what these three solutions should mean? 355 00:27:01,580 --> 00:27:03,710 If you believe this, duality has to be true. 356 00:27:06,250 --> 00:27:07,700 Yes, so that the game we play here 357 00:27:07,700 --> 00:27:12,810 is that you believe that the duality is always true. 358 00:27:12,810 --> 00:27:15,310 And then whenever you see some phenomenon, the gravity side, 359 00:27:15,310 --> 00:27:16,700 you say this, I try to integrate. 360 00:27:16,700 --> 00:27:19,000 I try to understand on the field theory side. 361 00:27:19,000 --> 00:27:22,690 And it makes sense and then we just carry out the guess 362 00:27:22,690 --> 00:27:24,105 and check it, et cetera. 363 00:27:26,940 --> 00:27:30,430 So what would the three solution mean? 364 00:27:30,430 --> 00:27:37,430 Three gravity solutions, or in principle for example, 365 00:27:37,430 --> 00:27:40,835 for a temperature at here we can have three solutions 366 00:27:40,835 --> 00:27:42,085 all have the same temperature. 367 00:27:45,640 --> 00:27:48,390 What would this mean? 368 00:27:48,390 --> 00:27:56,120 So the simplest guess is that it tells you 369 00:27:56,120 --> 00:28:05,470 that for CFT and the ST minus y at this temperature 370 00:28:05,470 --> 00:28:06,755 there's three possible phases. 371 00:28:14,365 --> 00:28:15,615 There's three possible phases. 372 00:28:18,990 --> 00:28:22,800 It's just like water, ice, or vapor. 373 00:28:28,550 --> 00:28:29,800 There's three possible phases. 374 00:28:38,125 --> 00:28:39,500 Now, the interpretation is clear. 375 00:28:41,888 --> 00:28:42,929 Then we should just find. 376 00:28:49,290 --> 00:28:51,640 So to decide what is the right solution then 377 00:28:51,640 --> 00:28:52,750 we should find what? 378 00:28:55,900 --> 00:28:58,140 That's right, we should find the phase, 379 00:28:58,140 --> 00:29:03,709 find the one with the lowest free energy. 380 00:29:19,200 --> 00:29:21,330 And then to decide which is the right one then 381 00:29:21,330 --> 00:29:24,760 we just need to find the one with the lowest free energy. 382 00:29:27,740 --> 00:29:37,325 So now recall that the partition function the field theory 383 00:29:37,325 --> 00:29:40,167 side should always be identified with the partition 384 00:29:40,167 --> 00:29:41,375 function on the gravity side. 385 00:29:45,160 --> 00:29:49,670 And then this side the partition function by definition 386 00:29:49,670 --> 00:29:51,510 is minus beta, the free energy. 387 00:30:05,140 --> 00:30:08,449 [? Seems ?] like going more and more. 388 00:30:08,449 --> 00:30:08,990 that's weird. 389 00:30:13,300 --> 00:30:16,860 But on the gravity side we write this 390 00:30:16,860 --> 00:30:19,460 by saddle-point approximation. 391 00:30:19,460 --> 00:30:21,340 We integrate over all possible field 392 00:30:21,340 --> 00:30:26,980 on the gravity side weighted by the euclidean action, 393 00:30:26,980 --> 00:30:32,250 and the leading order in the saddle-point approximation 394 00:30:32,250 --> 00:30:33,560 you just evaluate it. 395 00:30:46,350 --> 00:30:49,245 You just evaluate it on the classical solutions. 396 00:30:55,160 --> 00:30:58,190 So now you can just equate them. 397 00:30:58,190 --> 00:31:00,590 You just need the free energy must 398 00:31:00,590 --> 00:31:09,590 be equal to se to the to the [? Ic. ?] 399 00:31:09,590 --> 00:31:12,830 So this is just the same principle we discussed before 400 00:31:12,830 --> 00:31:14,000 on how we just apply it. 401 00:31:18,510 --> 00:31:21,860 So now if we believe the duality and we 402 00:31:21,860 --> 00:31:24,470 believe with some of the [? lamicks ?] then 403 00:31:24,470 --> 00:31:44,132 we should just find the solution with now just SE. 404 00:31:47,010 --> 00:31:49,600 So we should calculate the euclidean action for all three 405 00:31:49,600 --> 00:31:53,150 solutions and then we find the corresponding free energy 406 00:31:53,150 --> 00:31:57,810 for each of them, and then the one with the largest, SE, 407 00:31:57,810 --> 00:32:00,590 will correspond to the one with lowest free energy. 408 00:32:00,590 --> 00:32:02,380 Then that will be the stable solution. 409 00:32:02,380 --> 00:32:08,550 And the other phases presumably will be unstable or not stable. 410 00:32:08,550 --> 00:32:09,910 just like in the standard story. 411 00:32:14,620 --> 00:32:19,940 So this conclusion was drawn from some of the [? lamrick ?] 412 00:32:19,940 --> 00:32:21,930 itself. 413 00:32:21,930 --> 00:32:24,370 But what will be nice we can actually draw this conclusion 414 00:32:24,370 --> 00:32:27,760 without using thermal dynamics. 415 00:32:27,760 --> 00:32:31,530 So we say, so this [INAUDIBLE] was 416 00:32:31,530 --> 00:32:34,620 drawn to identify this with the free energy 417 00:32:34,620 --> 00:32:38,090 and then we say according to the sum of dynamics, 418 00:32:38,090 --> 00:32:41,187 and then we should choose the one with the lowest energy. 419 00:32:41,187 --> 00:32:42,770 But will be better if you can actually 420 00:32:42,770 --> 00:32:47,240 derive this thing from the gravity side, 421 00:32:47,240 --> 00:32:50,950 and you can and for a very simple reason. 422 00:32:50,950 --> 00:33:03,150 So this also follows from the standard rules 423 00:33:03,150 --> 00:33:17,180 of saddle-point approximation as follows. 424 00:33:25,600 --> 00:33:27,620 Oh, I forgot to write on this board. 425 00:33:40,730 --> 00:33:43,200 So when you evaluate this [? possible ?] 426 00:33:43,200 --> 00:33:47,830 so when you evaluate your integral 427 00:33:47,830 --> 00:33:51,320 you're in the saddle-point approximation, 428 00:33:51,320 --> 00:33:55,080 so normally you are instructed to add the contribution 429 00:33:55,080 --> 00:33:57,286 of all saddle points. 430 00:33:57,286 --> 00:33:59,410 So the fact that you have three different solutions 431 00:33:59,410 --> 00:34:03,270 it tells you have three separate saddle points. 432 00:34:03,270 --> 00:34:06,935 And then just from the standard rule 433 00:34:06,935 --> 00:34:08,310 of the saddle-point approximation 434 00:34:08,310 --> 00:34:11,370 you just need to add all of them together. 435 00:34:11,370 --> 00:34:17,630 So you add the expression SE, you evaluate the thermal Ads. 436 00:34:17,630 --> 00:34:20,170 And the expression, SE you evaluate it 437 00:34:20,170 --> 00:34:26,495 as a big black hole, and then you add the SE 438 00:34:26,495 --> 00:34:31,396 to the small black hole, et cetera. 439 00:34:31,396 --> 00:34:32,770 OK, you add all of them together. 440 00:34:36,760 --> 00:34:40,980 And now each action is in exponential. 441 00:34:40,980 --> 00:34:43,230 And as I said, they are weighted. 442 00:34:43,230 --> 00:34:45,350 All this action are weighted by n square. 443 00:34:45,350 --> 00:34:48,139 Because the action is always proponent to law 444 00:34:48,139 --> 00:34:50,750 of [? Newton ?] and the law of newton is [? proponent ?] to n 445 00:34:50,750 --> 00:34:51,553 square. 446 00:34:51,553 --> 00:34:53,469 So there's a big parameter in the exponential. 447 00:34:57,900 --> 00:35:10,660 So the saddle is not just SE dominates in the large N limit. 448 00:35:21,320 --> 00:35:23,460 You just add three exponentials, which one 449 00:35:23,460 --> 00:35:26,640 is the biggest will dominate. 450 00:35:26,640 --> 00:35:30,170 Because each one of them is proponent to n square and n 451 00:35:30,170 --> 00:35:32,700 goes to infinity. 452 00:35:32,700 --> 00:35:39,240 And n goes infinite means it's greater than the other guy 453 00:35:39,240 --> 00:35:41,880 by a tiny bit. 454 00:35:41,880 --> 00:35:44,640 in the prefactor of n squared they will become predominant. 455 00:35:47,560 --> 00:35:51,010 So we see that the sum of the [? lammicks. ?] Sum 456 00:35:51,010 --> 00:35:54,390 of the [? lammicks tells ?] us we should define the smallest 457 00:35:54,390 --> 00:35:58,730 free energy and then that statement can be essentially 458 00:35:58,730 --> 00:36:02,770 derived if I don't know that statement. 459 00:36:02,770 --> 00:36:06,140 I can actually derive that rule, say finding the smallest 460 00:36:06,140 --> 00:36:09,080 free energy, just by using the standard saddle-point 461 00:36:09,080 --> 00:36:10,740 approximation from the gravity side. 462 00:36:13,830 --> 00:36:16,350 So this is a nice consistency check. 463 00:36:16,350 --> 00:36:19,750 This a nice consistent check that the duality 464 00:36:19,750 --> 00:36:23,610 which we believe to be true should be true. 465 00:36:23,610 --> 00:36:24,640 Yes? 466 00:36:24,640 --> 00:36:25,910 AUDIENCE: So the one thing I don't totally understand 467 00:36:25,910 --> 00:36:27,868 is that, so say that you give me this Euclidean 468 00:36:27,868 --> 00:36:33,090 integral over metrics on the gravity side. 469 00:36:33,090 --> 00:36:36,265 So the idea that there's really only three contributions-- 470 00:36:36,265 --> 00:36:39,060 because we found three solutions with a particular temperature-- 471 00:36:39,060 --> 00:36:41,451 how do we know that there aren't any other solutions? 472 00:36:41,451 --> 00:36:43,700 Like, how do we know there aren't any other terms that 473 00:36:43,700 --> 00:36:44,970 were not accounted for? 474 00:36:44,970 --> 00:36:48,150 PROFESSOR: Oh yeah, there are three-- 475 00:36:48,150 --> 00:36:52,930 so this is a rule of the saddle-point approximation, 476 00:36:52,930 --> 00:36:55,670 you find the saddle-point. 477 00:36:55,670 --> 00:37:00,690 So you find the solution of the equation of motion 478 00:37:00,690 --> 00:37:03,860 with the right boundary conditions and these 479 00:37:03,860 --> 00:37:05,810 are the only ones. 480 00:37:05,810 --> 00:37:08,244 Yeah, and then of course, each of them 481 00:37:08,244 --> 00:37:09,285 include the fluctuations. 482 00:37:25,062 --> 00:37:27,270 This is just the standard saddle-point approximation. 483 00:37:30,040 --> 00:37:31,160 Any other questions? 484 00:37:31,160 --> 00:37:36,980 So now the task just boils down to-- yeah, 485 00:37:36,980 --> 00:37:39,750 let me just emphasize here-- yeah, 486 00:37:39,750 --> 00:37:43,750 so now, the task just boils down to find 487 00:37:43,750 --> 00:37:46,710 what is the lowest free energy. 488 00:37:46,710 --> 00:37:47,210 OK? 489 00:37:53,230 --> 00:37:55,510 So now it's 3-- no, it's 4. 490 00:37:58,400 --> 00:38:00,930 So now, let's try to decide which one is more stable. 491 00:38:00,930 --> 00:38:10,670 So first, let me just emphasize SE is proportional to the 1 492 00:38:10,670 --> 00:38:13,220 Newton-- 1/G Newton. 493 00:38:13,220 --> 00:38:16,550 And then it's always proportional to N squared. 494 00:38:16,550 --> 00:38:17,120 OK? 495 00:38:17,120 --> 00:38:19,700 Because 1/G Newton is proportional to N squared 496 00:38:19,700 --> 00:38:22,820 if you translate from the [INAUDIBLE] language. 497 00:38:22,820 --> 00:38:27,020 And now, if you vary the free energy for them-- 498 00:38:27,020 --> 00:38:31,720 for this-- here let me call this-- thermal [? gas and ?] 499 00:38:31,720 --> 00:38:36,430 AdS, I always just call it TAdS, OK, Thermal AdS. 500 00:38:36,430 --> 00:38:42,200 I think, yeah, just-- the shorthand is just thermal AdS. 501 00:38:42,200 --> 00:38:52,080 So for the thermal AdS, this is actually 0. 502 00:38:52,080 --> 00:38:53,015 This is actually 0. 503 00:38:56,320 --> 00:39:00,290 So thermal AdS is the free energy-- I should say this way. 504 00:39:00,290 --> 00:39:04,860 SE is actually the 0 times N squared. 505 00:39:04,860 --> 00:39:06,650 And then you can have fluctuations then 506 00:39:06,650 --> 00:39:09,620 that can contribute to the order and to the power 0 507 00:39:09,620 --> 00:39:12,030 [? contribution ?]. 508 00:39:12,030 --> 00:39:15,915 So the 0 in this-- 0 is simple. 509 00:39:15,915 --> 00:39:22,860 It is because, when we go from the [? pure ?] AdS 510 00:39:22,860 --> 00:39:25,890 into the thermal AdS, essentially 511 00:39:25,890 --> 00:39:28,920 we just changing the global structure. 512 00:39:28,920 --> 00:39:29,910 OK? 513 00:39:29,910 --> 00:39:34,840 We just make the Euclidean circle so for the pure AdS, 514 00:39:34,840 --> 00:39:39,600 when you go to Euclidean space, the tau have infinite size. 515 00:39:39,600 --> 00:39:41,490 And then when you go to the thermal AdS, 516 00:39:41,490 --> 00:39:44,330 we just make the tau to be a compact circle. 517 00:39:44,330 --> 00:39:48,200 So you have not changed the-- the solution is identical. 518 00:39:48,200 --> 00:39:51,170 The solution is identical go to the pure AdS. 519 00:39:51,170 --> 00:39:59,620 Just the global structure is different in terms 520 00:39:59,620 --> 00:40:04,660 of the periodicity of the Euclidean time circle. 521 00:40:04,660 --> 00:40:07,590 So now, if you vary the classical action-- 522 00:40:07,590 --> 00:40:10,260 so if you're reference point is that the pure AdS, 523 00:40:10,260 --> 00:40:13,160 the classical action is 0. 524 00:40:13,160 --> 00:40:15,910 And then, for the thermal AdS will be 0. 525 00:40:15,910 --> 00:40:17,032 OK? 526 00:40:17,032 --> 00:40:17,740 Will be the same. 527 00:40:17,740 --> 00:40:19,800 It would be the same-- the classical action will 528 00:40:19,800 --> 00:40:24,020 be the same as pure AdS, OK? 529 00:40:24,020 --> 00:40:29,400 So [INAUDIBLE] order N squared, but then you 530 00:40:29,400 --> 00:40:31,025 can have fluctuations which contribute. 531 00:40:36,800 --> 00:40:41,170 And the fluctuations is to the-- yeah, 532 00:40:41,170 --> 00:40:42,920 so the fluctuations can be interpreted 533 00:40:42,920 --> 00:40:49,230 as the, say, the graviton symbol [? and ?] division 534 00:40:49,230 --> 00:40:52,410 from the symbol, graviton gas. 535 00:40:52,410 --> 00:40:53,620 OK? 536 00:40:53,620 --> 00:41:01,460 And so, say, in the themal AdS sites 537 00:41:01,460 --> 00:41:03,790 essentially, have a thermal gas graviton. 538 00:41:03,790 --> 00:41:07,350 And there, free energy is independent of N 539 00:41:07,350 --> 00:41:09,690 because they are the fluctuations. 540 00:41:09,690 --> 00:41:11,310 And just as usual, whenever you can 541 00:41:11,310 --> 00:41:13,130 see that the fluctuations there, they're 542 00:41:13,130 --> 00:41:20,100 independent of a coupling constant. 543 00:41:20,100 --> 00:41:21,110 Is this clear? 544 00:41:21,110 --> 00:41:23,450 AUDIENCE: Why is the first term 0? 545 00:41:23,450 --> 00:41:24,296 PROFESSOR: Hm? 546 00:41:24,296 --> 00:41:25,990 AUDIENCE: Why is the first term 0? 547 00:41:25,990 --> 00:41:27,781 PROFESSOR: The first term is 0 because this 548 00:41:27,781 --> 00:41:30,995 has the same classical solution as the pure AdS. 549 00:41:30,995 --> 00:41:35,162 AUDIENCE: The pure AdS, then, has a nonzero [INAUDIBLE]. 550 00:41:37,670 --> 00:41:39,812 PROFESSOR: Yeah. 551 00:41:39,812 --> 00:41:41,390 Yeah, that's a very good question. 552 00:41:41,390 --> 00:41:42,500 AUDIENCE: [INAUDIBLE] 553 00:41:42,500 --> 00:41:43,620 PROFESSOR: Yeah. 554 00:41:43,620 --> 00:41:44,969 I will-- yeah. 555 00:41:44,969 --> 00:41:45,760 Wait a few minutes. 556 00:41:45,760 --> 00:41:47,766 I will talk about this more precisely. 557 00:41:47,766 --> 00:41:48,640 It's a good question. 558 00:41:48,640 --> 00:41:49,130 AUDIENCE: Oh, OK. 559 00:41:49,130 --> 00:41:49,796 PROFESSOR: Yeah. 560 00:41:52,800 --> 00:41:55,030 Let me just tell you the answer first, 561 00:41:55,030 --> 00:41:58,210 and then I will say a few things. 562 00:42:01,450 --> 00:42:02,760 Any other questions? 563 00:42:02,760 --> 00:42:05,480 OK. 564 00:42:05,480 --> 00:42:09,700 And then if you calculated the Euclidean action 565 00:42:09,700 --> 00:42:14,490 for the big black hole and the small black hole, 566 00:42:14,490 --> 00:42:17,330 you find that they are always greater than that. 567 00:42:17,330 --> 00:42:20,800 It's always larger than action for the small black hole. 568 00:42:24,770 --> 00:42:29,640 And this is some number, nonzero number, times order N squared. 569 00:42:33,140 --> 00:42:34,540 So this is always the case. 570 00:42:39,170 --> 00:42:42,390 We also [INAUDIBLE] way for you to calculate in a few minutes. 571 00:42:42,390 --> 00:42:46,360 But I will not do this explicit calculation here 572 00:42:46,360 --> 00:42:49,130 because it takes some time. 573 00:42:49,130 --> 00:42:51,030 So there is a simple way, physical way, 574 00:42:51,030 --> 00:42:54,380 to understand this-- to understand why this is true 575 00:42:54,380 --> 00:42:56,370 because this is a free energy. 576 00:42:56,370 --> 00:43:00,470 So this tells that the free energy of the big black hole 577 00:43:00,470 --> 00:43:02,990 is always much more-- is always smaller than the free energy 578 00:43:02,990 --> 00:43:04,030 of the small black hole. 579 00:43:07,670 --> 00:43:09,680 This is not a precise way to understand it, 580 00:43:09,680 --> 00:43:13,280 but it is a heuristic way-- is that if you look at these two 581 00:43:13,280 --> 00:43:16,450 solutions-- so these two solutions have 582 00:43:16,450 --> 00:43:19,360 the same temperature, but the big black hole 583 00:43:19,360 --> 00:43:21,300 have a larger size. 584 00:43:21,300 --> 00:43:25,070 When you have a larger size and the entropy 585 00:43:25,070 --> 00:43:27,710 is proportional to the area of the horizon-- 586 00:43:27,710 --> 00:43:31,165 and the entropy is much higher, so the big black hole 587 00:43:31,165 --> 00:43:35,110 has much higher entropy than the small black hole. 588 00:43:35,110 --> 00:43:38,960 And so you choose-- you would expect that this should 589 00:43:38,960 --> 00:43:40,650 have a smaller free energy. 590 00:43:40,650 --> 00:43:43,550 But, of course, you have to do a precise calculation 591 00:43:43,550 --> 00:43:47,130 to really check it because their energy is not the same. 592 00:43:47,130 --> 00:43:50,000 Their energy is also not the same. 593 00:43:50,000 --> 00:43:52,280 Yeah, but this is a heuristic way to understand. 594 00:43:52,280 --> 00:43:54,092 Anyway, so you find that the free energy-- 595 00:43:54,092 --> 00:43:56,300 so you find the Euclidean action, or the [INAUDIBLE], 596 00:43:56,300 --> 00:43:58,410 is always much-- is always greater 597 00:43:58,410 --> 00:44:00,510 than the small black hole. 598 00:44:00,510 --> 00:44:02,670 And that is good. 599 00:44:02,670 --> 00:44:04,200 That is good because that tells you 600 00:44:04,200 --> 00:44:07,210 the small back hole is never dominate 601 00:44:07,210 --> 00:44:10,330 because you always find something bigger than that. 602 00:44:10,330 --> 00:44:13,400 And the small black hole have a [? lack of ?] specific heat. 603 00:44:13,400 --> 00:44:15,740 That's not something you want because our Field 604 00:44:15,740 --> 00:44:18,570 Theory definitely have quality of specific heat. 605 00:44:18,570 --> 00:44:21,550 Our Field Theory definitely has quality of specific heat. 606 00:44:21,550 --> 00:44:23,990 So that tells you that this is never-- 607 00:44:23,990 --> 00:44:25,910 can be low-- that would be a dominate phase. 608 00:44:36,340 --> 00:44:39,735 So now, if you calculated a number-- 609 00:44:39,735 --> 00:44:41,610 so now if you look at the number explicitly-- 610 00:44:53,560 --> 00:45:00,680 So what you'd find actually-- so you also have-- let me just 611 00:45:00,680 --> 00:45:04,030 first tell you the results, OK? 612 00:45:04,030 --> 00:45:09,736 You find there exists a temperature, Tc, 613 00:45:09,736 --> 00:45:12,785 which is greater than the minimal temperature. 614 00:45:17,120 --> 00:45:19,890 Then, you find that the-- when you're smaller than the minimal 615 00:45:19,890 --> 00:45:20,431 temperature-- 616 00:45:35,790 --> 00:45:36,830 Yeah. 617 00:45:36,830 --> 00:45:39,115 Right. 618 00:45:39,115 --> 00:45:39,615 OK. 619 00:45:43,910 --> 00:45:49,270 So when your T is smaller than the minimal temperature-- 620 00:45:49,270 --> 00:45:49,770 oh, no. 621 00:45:49,770 --> 00:45:50,930 No. 622 00:45:50,930 --> 00:45:53,464 T is smaller than Tc. 623 00:45:53,464 --> 00:45:55,630 So you find that there exists a Tc, which is greater 624 00:45:55,630 --> 00:45:58,840 than the minimal temperature. 625 00:45:58,840 --> 00:46:03,440 So let me just say this. 626 00:46:03,440 --> 00:46:08,090 So when T is smaller than T minimum, of course, 627 00:46:08,090 --> 00:46:12,887 there's only thermal AdS exists. 628 00:46:12,887 --> 00:46:14,720 And, of course, that would be the only phase 629 00:46:14,720 --> 00:46:18,850 which can describe the Field Theory result. 630 00:46:18,850 --> 00:46:22,430 But, now, there exists another temperature [? then, ?] 631 00:46:22,430 --> 00:46:26,080 within T, smaller than-- greater than T minimum 632 00:46:26,080 --> 00:46:27,235 and smaller than Tc. 633 00:46:30,400 --> 00:46:37,700 Then, you find that the SE big black hole is smaller than 0. 634 00:46:40,930 --> 00:46:42,560 And the thermal AdS, essentially, 635 00:46:42,560 --> 00:46:45,310 by definition to be 0. 636 00:46:45,310 --> 00:46:46,850 So that means that the thermal AdS 637 00:46:46,850 --> 00:46:50,170 will have lower free energy. 638 00:46:50,170 --> 00:46:52,220 So in this range, the thermal AdS will dominate. 639 00:47:02,020 --> 00:47:09,550 And when T is greater than Tc, then you 640 00:47:09,550 --> 00:47:16,947 find that the big black hole now has a positive action 641 00:47:16,947 --> 00:47:18,030 and [INAUDIBLE] dominates. 642 00:47:28,760 --> 00:47:36,902 So if I draw this-- and then there's a Tc 643 00:47:36,902 --> 00:47:43,920 and there's a T minimum-- so this is the temperature axis. 644 00:47:43,920 --> 00:47:47,590 So here, of course, you only have a thermal AdS. 645 00:47:50,770 --> 00:47:53,700 But within this range, the thermal AdS 646 00:47:53,700 --> 00:48:04,020 dominates over big black hole and the small black hole. 647 00:48:07,550 --> 00:48:10,873 And when you're above, then the big black hole dominates. 648 00:48:15,510 --> 00:48:19,950 So you actually have a transition at Tc. 649 00:48:19,950 --> 00:48:22,185 So below here, it's just thermal AdS. 650 00:48:22,185 --> 00:48:34,910 And then you have a transition from thermal AdS 651 00:48:34,910 --> 00:48:38,340 to the big black hole at Tc. 652 00:49:00,420 --> 00:49:01,720 So a quick side remark. 653 00:49:01,720 --> 00:49:04,795 How we-- how would you find this result? 654 00:49:13,380 --> 00:49:18,144 So this would be the result if you calculated the-- so 655 00:49:18,144 --> 00:49:19,810 you just calculate the Euclidean action. 656 00:49:19,810 --> 00:49:20,540 You compare them. 657 00:49:20,540 --> 00:49:21,665 That's what you would find. 658 00:49:32,390 --> 00:49:34,080 So any questions about that? 659 00:49:34,080 --> 00:49:34,580 Yes? 660 00:49:34,580 --> 00:49:36,746 AUDIENCE: Do you ever get the small black hole phase 661 00:49:36,746 --> 00:49:40,080 at all somehow artificially? 662 00:49:40,080 --> 00:49:43,185 PROFESSOR: Yeah, you can always get artificially. 663 00:49:43,185 --> 00:49:44,490 Yeah, artificially, always. 664 00:49:44,490 --> 00:49:45,330 Yeah. 665 00:49:45,330 --> 00:49:48,840 And let me not go into that. 666 00:49:48,840 --> 00:49:53,690 So now, let me just say a few remarks, say, in finding SE. 667 00:50:00,240 --> 00:50:06,237 So when you find the Euclidean action, 668 00:50:06,237 --> 00:50:07,570 you find it is always divergent. 669 00:50:10,330 --> 00:50:13,954 It's the same thing as what we encountered before. 670 00:50:13,954 --> 00:50:15,370 If you calculate the whole action, 671 00:50:15,370 --> 00:50:17,494 you have to integrate the-- over the-- essentially, 672 00:50:17,494 --> 00:50:19,880 the volume of the AdS. 673 00:50:19,880 --> 00:50:21,850 But the volume of AdS goes to infinity 674 00:50:21,850 --> 00:50:23,950 at-- near the boundary, essentially, 675 00:50:23,950 --> 00:50:25,932 you always get infinity. 676 00:50:25,932 --> 00:50:27,640 So in order to get a [? finite ?] answer, 677 00:50:27,640 --> 00:50:28,681 you need renormalization. 678 00:50:38,900 --> 00:50:41,540 So when you do renormalization, essentially, you 679 00:50:41,540 --> 00:50:45,200 put the [? cut ?] off and then you subtract covariant, 680 00:50:45,200 --> 00:50:46,200 the local counter terms. 681 00:50:58,162 --> 00:50:58,870 You put a cutoff. 682 00:51:07,550 --> 00:51:10,566 And then you subtract covariant counter terms at the cutoff. 683 00:51:10,566 --> 00:51:11,940 And then you get a finite answer. 684 00:51:11,940 --> 00:51:14,880 Then, you take the cutoff to the boundary. 685 00:51:14,880 --> 00:51:19,470 So we will not go through this procedure, 686 00:51:19,470 --> 00:51:22,840 but it's something you can do. 687 00:51:22,840 --> 00:51:24,870 But there's an alternative shortcut 688 00:51:24,870 --> 00:51:28,090 to not go through these. 689 00:51:28,090 --> 00:51:34,540 Or you can just subtract-- just calculate the Euclidean action 690 00:51:34,540 --> 00:51:40,550 for the cutoff, then subtract the value of pure AdS. 691 00:51:46,460 --> 00:51:47,920 So if you calculate the pure AdS, 692 00:51:47,920 --> 00:51:50,780 you will find the answer is also divergent. 693 00:51:50,780 --> 00:51:53,080 You will find the answer is also divergent. 694 00:51:53,080 --> 00:51:56,010 And you just subtract it. 695 00:51:56,010 --> 00:51:57,430 And then when you do the subtract, 696 00:51:57,430 --> 00:52:01,980 you will find the difference is [? finite. ?] 697 00:52:01,980 --> 00:52:04,870 And then you find the difference is finite. 698 00:52:04,870 --> 00:52:08,180 And this is a slightly simple way to do than that, 699 00:52:08,180 --> 00:52:10,860 but we also not do this because, still, you 700 00:52:10,860 --> 00:52:14,780 have to calculate Euclidean action [? and such. ?] 701 00:52:14,780 --> 00:52:27,060 So the synchron-- shortcut is to assume thermal dynamics. 702 00:52:27,060 --> 00:52:29,390 That's what we did last time. 703 00:52:29,390 --> 00:52:32,480 Because you can find the entropy density very easy. 704 00:52:32,480 --> 00:52:36,400 So entropy density-- did I erase my-- 705 00:52:36,400 --> 00:52:38,340 so the entropy density is just, essentially, 706 00:52:38,340 --> 00:52:41,024 the area of the horizon. 707 00:52:41,024 --> 00:52:43,690 So the entropy-- right now, it's actually the total entropy, not 708 00:52:43,690 --> 00:52:46,064 the entropy density because now we are [? almost here. ?] 709 00:52:46,064 --> 00:52:49,960 So the entropy would be-- and size of the sphere 710 00:52:49,960 --> 00:52:55,250 will be r0 to the power d minus 1 times omega d minus 1. 711 00:52:55,250 --> 00:53:00,540 Omega is the volume of a unit sphere. 712 00:53:00,540 --> 00:53:03,569 And then divided by 4 pi 4GN. 713 00:53:03,569 --> 00:53:05,110 So this is the entropy of the system. 714 00:53:07,930 --> 00:53:12,040 And you can express this in terms of the temperature 715 00:53:12,040 --> 00:53:13,991 because r0 is a function of temperature. 716 00:53:17,290 --> 00:53:18,640 r0 is a function of temperature. 717 00:53:18,640 --> 00:53:20,250 If you invert this, you can imagine r0 718 00:53:20,250 --> 00:53:22,710 is a function of temperature. 719 00:53:22,710 --> 00:53:24,460 So you have-- essentially you have entropy 720 00:53:24,460 --> 00:53:27,720 as a function of temperature. 721 00:53:27,720 --> 00:53:29,600 And-- Hm? 722 00:53:29,600 --> 00:53:32,420 AUDIENCE: d minus 1, what's that? 723 00:53:32,420 --> 00:53:33,992 PROFESSOR: [? Formula ?] d minus 1 724 00:53:33,992 --> 00:53:35,530 is the volume of the unit sphere. 725 00:53:49,490 --> 00:53:50,660 So you know the entropy. 726 00:53:50,660 --> 00:53:56,100 You can just use the formula S equal to minus F minus T 727 00:53:56,100 --> 00:54:03,850 divided by T to-- so you can also 728 00:54:03,850 --> 00:54:09,220 write this as minus F r0, partial r0, partial T, 729 00:54:09,220 --> 00:54:11,250 et cetera. 730 00:54:11,250 --> 00:54:14,140 And then you can integrate this equation 731 00:54:14,140 --> 00:54:17,930 to find the-- to find F as a function of r0 732 00:54:17,930 --> 00:54:23,270 because we know how the r0 depends on T from here. 733 00:54:23,270 --> 00:54:26,070 So you just need to integrate this equation. 734 00:54:26,070 --> 00:54:28,990 So this is a simple exercise because now 735 00:54:28,990 --> 00:54:33,540 you just need to do an integral of some function. 736 00:54:33,540 --> 00:54:39,670 And then you find omega d minus 1 divided by 16 737 00:54:39,670 --> 00:54:47,570 pi GN, r0 d minus 2 minus r0 d divided by R squared. 738 00:54:50,321 --> 00:54:51,945 You get actually a rather simple answer 739 00:54:51,945 --> 00:54:53,070 if you do that integration. 740 00:54:56,716 --> 00:54:58,590 So we have chosen the integration constant so 741 00:54:58,590 --> 00:55:00,350 that when r0 equals to 0, this is 742 00:55:00,350 --> 00:55:03,455 0 because r0 equal to 0, if the black hole has 743 00:55:03,455 --> 00:55:06,450 0 size, essentially, [? there's ?] no black hole. 744 00:55:06,450 --> 00:55:08,740 And then we have chosen the free energy 745 00:55:08,740 --> 00:55:13,780 to be 0 for the [? no ?] black hole. 746 00:55:13,780 --> 00:55:15,570 So this free energy, again, should 747 00:55:15,570 --> 00:55:18,010 be interpreted as the difference with the pure AdS. 748 00:55:21,850 --> 00:55:28,130 So for the black hole-- this is the black hole free energy 749 00:55:28,130 --> 00:55:30,570 because, by definition, this way, the pure AdS, 750 00:55:30,570 --> 00:55:31,380 you just have 0. 751 00:55:36,300 --> 00:55:38,150 Any questions about this? 752 00:55:38,150 --> 00:55:42,970 So now you see here, clearly, this expression. 753 00:55:42,970 --> 00:55:44,460 So now this is a free energy now. 754 00:55:44,460 --> 00:55:47,200 This is not the Euclidean action. 755 00:55:47,200 --> 00:55:52,280 So they differ by the amount [? assigned ?] 756 00:55:52,280 --> 00:55:53,340 in my convention. 757 00:55:56,080 --> 00:56:01,010 So now you see that there exists r0, critical r0. 758 00:56:09,960 --> 00:56:14,240 F black hole is greater than 0 Free energy is greater 759 00:56:14,240 --> 00:56:26,990 than 0 for r0 is smaller than R. 760 00:56:26,990 --> 00:56:31,520 So you can write it as formula d minus 1-- so let me just write. 761 00:56:31,520 --> 00:56:38,000 Yeah, you can put the omega-- the r0 to that [? out ?] 762 00:56:38,000 --> 00:56:39,420 for the overall factor r0. 763 00:56:39,420 --> 00:56:43,440 Yeah, put d minus 2 out, and this just 764 00:56:43,440 --> 00:56:47,150 becomes 1 minus r0 squared. 765 00:56:50,970 --> 00:56:53,200 So you find that the free energy is greater than 0. 766 00:56:53,200 --> 00:56:54,780 But r0 is greater than 1. 767 00:56:54,780 --> 00:57:01,170 That's greater than the-- so this 768 00:57:01,170 --> 00:57:05,470 is greater than the value of the thermal AdS. 769 00:57:05,470 --> 00:57:09,220 And the black hole is smaller than 0. 770 00:57:09,220 --> 00:57:16,200 The free energy [? smaller ?] than 0 become great than R. 771 00:57:16,200 --> 00:57:19,478 So the r0 is somewhere here. 772 00:57:19,478 --> 00:57:20,960 So r0 is somewhere here. 773 00:57:20,960 --> 00:57:24,881 It has to exist above the-- it's somewhere here. 774 00:57:24,881 --> 00:57:28,120 This is r0. 775 00:57:28,120 --> 00:57:33,490 This is this r0 critical [? as you go ?] to R. 776 00:57:33,490 --> 00:57:38,890 This is r0 critical [? as you go ?] to R, 777 00:57:38,890 --> 00:57:41,500 somewhere here. 778 00:57:41,500 --> 00:57:42,920 So it's always a big black hole. 779 00:57:50,020 --> 00:57:53,600 And you can find out what is this critical temperature. 780 00:57:53,600 --> 00:57:58,120 So the critical temperature is just the beta evaluate at r0 781 00:57:58,120 --> 00:58:02,670 equal to R. And then you can find out 782 00:58:02,670 --> 00:58:06,460 that this is 2 pi R divided by d minus 1. 783 00:58:06,460 --> 00:58:09,675 Just using this formula you can find out it's that. 784 00:58:17,940 --> 00:58:22,785 So now I emphasize that this is a phase transition. 785 00:58:25,330 --> 00:58:27,110 And this is not only a phase transition, 786 00:58:27,110 --> 00:58:30,770 this is actually a first order phase transition 787 00:58:30,770 --> 00:58:34,070 for the following reason. 788 00:58:34,070 --> 00:58:38,510 So in our unit, below here, the free energy 789 00:58:38,510 --> 00:58:41,065 is 0 times order N squared. 790 00:58:41,065 --> 00:58:42,740 It's 0 times N squared. 791 00:58:47,100 --> 00:58:54,100 But about here-- well, actually, above Tc-- at Tc, 792 00:58:54,100 --> 00:58:57,350 the free energy is exactly equal to 0. 793 00:58:57,350 --> 00:59:02,814 So [? r0c ?] is equal to R at Tc. 794 00:59:02,814 --> 00:59:03,730 There's a free energy. 795 00:59:03,730 --> 00:59:04,780 It's exactly 0. 796 00:59:04,780 --> 00:59:10,060 So at Tc, the-- a big black hole and the thermal AdS 797 00:59:10,060 --> 00:59:11,420 have the same free energy. 798 00:59:11,420 --> 00:59:14,320 So they can exist together. 799 00:59:14,320 --> 00:59:20,420 And above here, then the free energy of the big black hole 800 00:59:20,420 --> 00:59:24,210 is some negative number times order-- times N squared. 801 00:59:28,630 --> 00:59:32,280 So your free energy has a huge change. 802 00:59:32,280 --> 00:59:36,010 So essentially, the first derivative of free energy, 803 00:59:36,010 --> 00:59:40,390 is discontinuous across the phase transition. 804 00:59:45,509 --> 00:59:47,633 So this is actually a first order phase transition. 805 00:59:54,255 --> 00:59:56,510 So this is-- yeah, let me just emphasize that. 806 01:00:08,628 --> 01:00:19,490 So this is a first order phase transition, 807 01:00:19,490 --> 01:00:28,180 with the free energy equal to order N to the power 0 808 01:00:28,180 --> 01:00:32,235 when T is smaller than Tc and some nonzero number times N 809 01:00:32,235 --> 01:00:32,735 squared. 810 01:00:35,265 --> 01:00:36,420 Well, T greater than Tc. 811 01:00:51,507 --> 01:00:52,090 Any questions? 812 01:01:03,475 --> 01:01:07,930 AUDIENCE: When r0 is equal to R, then F is equal to 0. 813 01:01:07,930 --> 01:01:08,920 PROFESSOR: Yeah. 814 01:01:08,920 --> 01:01:12,275 AUDIENCE: Then, why is the-- oh. 815 01:01:12,275 --> 01:01:15,620 PROFESSOR: No, the derivative of F should be discontinuous. 816 01:01:15,620 --> 01:01:18,140 No, F is continuous. 817 01:01:18,140 --> 01:01:22,280 No, this is a definition of the-- for the phase transition, 818 01:01:22,280 --> 01:01:24,550 the free energy is always continuous. 819 01:01:24,550 --> 01:01:27,910 And then the free-- so the first derivative here will 820 01:01:27,910 --> 01:01:29,920 be proportional to N squared. 821 01:01:29,920 --> 01:01:31,570 And you can check-- Yeah? 822 01:01:31,570 --> 01:01:34,080 AUDIENCE: So the point is-- OK, so just see if I understand. 823 01:01:34,080 --> 01:01:36,000 So the point is that it's discontinuous 824 01:01:36,000 --> 01:01:38,010 in the limit of large N or something? 825 01:01:38,010 --> 01:01:38,580 I mean-- 826 01:01:38,580 --> 01:01:39,330 PROFESSOR: Yeah, that's right. 827 01:01:39,330 --> 01:01:39,790 That's right. 828 01:01:39,790 --> 01:01:40,290 Exactly. 829 01:01:44,120 --> 01:01:45,250 So let me emphasize. 830 01:01:45,250 --> 01:01:47,580 So this is N goes to infinity limit. 831 01:02:09,957 --> 01:02:11,165 Is there any other questions? 832 01:02:13,690 --> 01:02:17,320 So we see there's a lot of [? rich ?] story when 833 01:02:17,320 --> 01:02:20,210 you go to the sphere. 834 01:02:20,210 --> 01:02:22,920 And, actually, there's a phase transition. 835 01:02:22,920 --> 01:02:25,750 And the phase transition roughly [? adds to-- ?] 836 01:02:25,750 --> 01:02:29,860 the black hole size is the AdS radius. 837 01:02:29,860 --> 01:02:33,260 Exactly the black hole size is AdS radius. 838 01:02:33,260 --> 01:02:38,070 And this is, more or less, what you expected because that's 839 01:02:38,070 --> 01:02:41,140 where the physics become nontrivial because when 840 01:02:41,140 --> 01:02:44,360 the horizon is much, much smaller than the black hole 841 01:02:44,360 --> 01:02:47,400 [? size-- ?] than the curvature radius, as I said, 842 01:02:47,400 --> 01:02:52,361 just should reduce to the flat space black hole and et cetera. 843 01:02:52,361 --> 01:02:54,610 Yeah, because the curvature radius is only scale here. 844 01:03:00,220 --> 01:03:02,830 So now, let me make another remark. 845 01:03:06,510 --> 01:03:07,680 Now, I erase this. 846 01:03:07,680 --> 01:03:08,360 I think it's OK. 847 01:03:12,510 --> 01:03:14,570 Yeah So you should actually check yourself. 848 01:03:22,500 --> 01:03:27,840 So below Tc, the F prime is always 0. 849 01:03:27,840 --> 01:03:29,690 When you take the derivative, it's 850 01:03:29,690 --> 01:03:34,010 always 0 at order N squared. 851 01:03:34,010 --> 01:03:41,250 Let me, again, write this answer, 0 times N squared. 852 01:03:41,250 --> 01:03:49,992 So you should check that the first derivative of F 853 01:03:49,992 --> 01:03:50,968 is discontinuous. 854 01:03:59,264 --> 01:04:01,860 AUDIENCE: So F itself is not discontinuous? 855 01:04:01,860 --> 01:04:06,450 PROFESSOR: No, F is not because the F is 0 precisely 856 01:04:06,450 --> 01:04:08,490 at T equal to Tc. 857 01:04:08,490 --> 01:04:10,790 So F is equal to 0. 858 01:04:10,790 --> 01:04:13,490 And the-- Yeah. 859 01:04:16,380 --> 01:04:20,440 So the reason the first derivative is discontinuous 860 01:04:20,440 --> 01:04:25,380 is very clear because below Tc, this is 0. 861 01:04:25,380 --> 01:04:30,460 So the first derivative is 0 times N squared. 862 01:04:30,460 --> 01:04:33,990 And above Tc, the derivative is the derivative 863 01:04:33,990 --> 01:04:41,160 of the free energy of the black hole at I equal to r0. 864 01:04:41,160 --> 01:04:44,040 And, clearly, the derivative is nonzero. 865 01:04:44,040 --> 01:04:47,250 At I equal to r0, the derivative is [? nonzero. ?] OK? 866 01:04:47,250 --> 01:04:50,060 And so you see, they're discontinuous. 867 01:05:02,050 --> 01:05:09,845 So since physics only depends on the [? dimensionless ?] number, 868 01:05:09,845 --> 01:05:29,050 R times T-- So large R, small T-- large R at fixed T 869 01:05:29,050 --> 01:05:35,194 essentially is the same as the, say, large T fixed R. 870 01:05:35,194 --> 01:05:36,860 It can only depend on the ratio of them. 871 01:05:36,860 --> 01:05:37,360 OK? 872 01:05:39,884 --> 01:05:41,300 It depends on the product of them. 873 01:05:44,000 --> 01:05:48,340 So you can either think of-- you can either 874 01:05:48,340 --> 01:05:50,410 think you fix the temperature, you 875 01:05:50,410 --> 01:05:53,500 increase the size of the sphere. 876 01:05:53,500 --> 01:05:56,000 Or you fix the size of the sphere, 877 01:05:56,000 --> 01:05:57,770 you increase the temperature. 878 01:05:57,770 --> 01:06:02,070 [? In ?] fact, the same is to increase this guy. 879 01:06:02,070 --> 01:06:05,785 And this is essentially the limit going to the flat space. 880 01:06:09,310 --> 01:06:13,400 It's going to the R to the T. If you take R go to infinity, 881 01:06:13,400 --> 01:06:16,755 you go to a [? theory ?] on R d minus 1. 882 01:06:19,350 --> 01:06:22,370 So we see that the theory on R d minus 1 883 01:06:22,370 --> 01:06:29,430 is mapped to the high temperature limits 884 01:06:29,430 --> 01:06:30,610 of the theory on the sphere. 885 01:06:39,199 --> 01:06:40,990 So essentially, just corresponding to the 1 886 01:06:40,990 --> 01:06:45,750 point for the [? theory ?] on the sphere 887 01:06:45,750 --> 01:06:49,490 and this infinite temperature of the [? theory ?] on the sphere. 888 01:06:49,490 --> 01:06:51,010 So that's why, as we said before, 889 01:06:51,010 --> 01:06:53,650 because there's no scale here, the [INAUDIBLE] temperature 890 01:06:53,650 --> 01:06:56,840 from here is essentially the same in the flat space. 891 01:06:59,782 --> 01:07:04,750 And in particular, this is described by a big black hole. 892 01:07:07,450 --> 01:07:09,250 And this is exactly what we see before 893 01:07:09,250 --> 01:07:12,965 because when the black hole become very big, 894 01:07:12,965 --> 01:07:14,970 then you can approximate it to a plane 895 01:07:14,970 --> 01:07:18,010 because, locally, it's like a plane [? anymore. ?] 896 01:07:18,010 --> 01:07:20,310 And the spherical horizon is just like a plane. 897 01:07:20,310 --> 01:07:22,830 And then you go to the black plane-- 898 01:07:22,830 --> 01:07:27,450 you go to the black hole with the flat horizon. 899 01:07:35,270 --> 01:07:41,080 AUDIENCE: Why are the phases only dependent on R times T? 900 01:07:41,080 --> 01:07:46,280 PROFESSOR: Yeah, because this a CFT and this 901 01:07:46,280 --> 01:07:48,580 is only dimensions number. 902 01:07:48,580 --> 01:07:52,180 It can only depend on them through this dimensions number. 903 01:07:52,180 --> 01:07:53,300 There is no other scale. 904 01:07:58,000 --> 01:08:00,470 Good? 905 01:08:00,470 --> 01:08:02,150 Other questions? 906 01:08:02,150 --> 01:08:03,080 Yes? 907 01:08:03,080 --> 01:08:04,580 AUDIENCE: So on the side of the CFT, 908 01:08:04,580 --> 01:08:06,566 what are the different phases? 909 01:08:06,566 --> 01:08:07,157 I didn't- 910 01:08:07,157 --> 01:08:07,865 PROFESSOR: Sorry? 911 01:08:07,865 --> 01:08:12,380 AUDIENCE: What are the different phases on the CFT side? 912 01:08:12,380 --> 01:08:15,861 PROFESSOR: Yeah, that's what we are going to explain. 913 01:08:20,550 --> 01:08:22,859 So now, we have to describe the black hole story. 914 01:08:22,859 --> 01:08:25,399 Or now, we have described the gravity story. 915 01:08:25,399 --> 01:08:28,120 Just by looking at the gravity size, 916 01:08:28,120 --> 01:08:30,750 we saw there are three possible phases. 917 01:08:30,750 --> 01:08:34,470 At low temperature, you get this thermal AdS phase. 918 01:08:34,470 --> 01:08:38,330 At high temperature, you get big black hole phase, 919 01:08:38,330 --> 01:08:40,850 and then there is a phase transition between them. 920 01:08:40,850 --> 01:08:43,670 And then, also, you have unstable small black hole 921 01:08:43,670 --> 01:08:49,944 phase, which never appears as a stable phase 922 01:08:49,944 --> 01:08:50,735 at any temperature. 923 01:08:54,010 --> 01:08:56,420 So, now, let's try to see whether we can understand this 924 01:08:56,420 --> 01:08:57,545 from the Field Theory side. 925 01:08:57,545 --> 01:08:59,660 What does this mean from the Field Theory side? 926 01:08:59,660 --> 01:09:05,474 Why somehow-- if I, for example, put [INAUDIBLE] on a sphere, 927 01:09:05,474 --> 01:09:07,859 do I expect such kind of phase transition? 928 01:09:07,859 --> 01:09:10,226 Does this make sense? 929 01:09:10,226 --> 01:09:11,600 Oh, by the way, I should say this 930 01:09:11,600 --> 01:09:16,840 is the-- so this is called a Hawking-Page transition. 931 01:09:16,840 --> 01:09:18,760 So this is called a Hawking-Page transition. 932 01:09:28,870 --> 01:09:30,390 So remarkably, this was discovered 933 01:09:30,390 --> 01:09:35,850 in 1980-- I think '81 or '82, almost 20 years 934 01:09:35,850 --> 01:09:37,518 before this AdS/CFT conjecture. 935 01:09:39,746 --> 01:09:41,370 But they already figured out that there 936 01:09:41,370 --> 01:09:43,515 is a phase transition. 937 01:09:43,515 --> 01:09:45,640 They were-- so they were looking at the black holes 938 01:09:45,640 --> 01:09:48,069 [? and ?] AdS, and they said, oh, there's two black holes. 939 01:09:48,069 --> 01:09:50,040 But there's also some AdS. 940 01:09:50,040 --> 01:09:51,645 And then, somehow, there's some kind 941 01:09:51,645 --> 01:09:53,100 of phase transition between them. 942 01:09:53,100 --> 01:09:55,317 But because they don't know the Field Theory, 943 01:09:55,317 --> 01:09:56,900 they don't know this should correspond 944 01:09:56,900 --> 01:09:59,335 into some kind of Field Theory system. 945 01:09:59,335 --> 01:10:01,460 But they figured out this gravity story essentially 946 01:10:01,460 --> 01:10:04,457 in 1981. 947 01:10:04,457 --> 01:10:06,540 Now, let's explain what should be the Field Theory 948 01:10:06,540 --> 01:10:13,927 interpretation of this-- The Field Theory 949 01:10:13,927 --> 01:10:14,760 explanation of this. 950 01:10:38,190 --> 01:10:40,065 Physical reasons for Hawking-Page transition. 951 01:10:55,600 --> 01:11:00,600 So now, I will consider a toy example. 952 01:11:00,600 --> 01:11:02,580 So I won't consider-- so [INAUDIBLE] 953 01:11:02,580 --> 01:11:05,680 on the sphere, that's a little bit too complicated. 954 01:11:05,680 --> 01:11:08,950 But I'm going to consider toy example. 955 01:11:08,950 --> 01:11:12,070 And you will see, from this toy example, 956 01:11:12,070 --> 01:11:15,655 that the physics behind this is very simple and actually, also, 957 01:11:15,655 --> 01:11:16,155 [INAUDIBLE]. 958 01:11:19,390 --> 01:11:25,899 So let's consider you have N squared harmonic oscillator-- 959 01:11:25,899 --> 01:11:26,940 free harmonic oscillator. 960 01:11:30,250 --> 01:11:31,720 So let's just imagine you can take 961 01:11:31,720 --> 01:11:33,540 them to be different frequencies-- [? let's ?] just 962 01:11:33,540 --> 01:11:35,725 even for simplicity-- or for the same frequency. 963 01:11:45,230 --> 01:11:48,040 Say, omega-- I have some omega, which I just take to be 1. 964 01:11:48,040 --> 01:11:48,876 Doesn't matter. 965 01:11:52,260 --> 01:11:54,520 So just consider N squared free harmonic oscillator. 966 01:11:57,710 --> 01:12:02,240 So I claim when N-- when N goes infinite limit, 967 01:12:02,240 --> 01:12:04,530 this system have the same phase transition-- 968 01:12:04,530 --> 01:12:07,000 have exactly the same phase transition described there. 969 01:12:09,610 --> 01:12:12,490 So, now, I will explain. 970 01:12:12,490 --> 01:12:14,450 So first, let's look just at-- this 971 01:12:14,450 --> 01:12:16,700 is a system we know how to solve exactly, so let's 972 01:12:16,700 --> 01:12:18,570 look at the spectrum. 973 01:12:18,570 --> 01:12:22,140 So this is total spectrum, the total energy spectrum of this N 974 01:12:22,140 --> 01:12:23,880 squared harmonic oscillators. 975 01:12:23,880 --> 01:12:25,800 So let's call the 0, the ground state. 976 01:12:30,150 --> 01:12:32,870 So let me normalize the energy so that the ground 977 01:12:32,870 --> 01:12:35,990 state have energy 0. 978 01:12:35,990 --> 01:12:39,580 And then you can excite one harmonic oscillator, 979 01:12:39,580 --> 01:12:41,200 and then you have state of 1. 980 01:12:48,411 --> 01:12:48,910 Yeah. 981 01:12:48,910 --> 01:12:53,980 Actually-- Yeah, have state of 1. 982 01:12:53,980 --> 01:12:56,660 And you can have two harmonic , oscillator et cetera. 983 01:13:01,204 --> 01:13:03,370 And then you'll have many harmonic oscillators, say, 984 01:13:03,370 --> 01:13:07,400 of all-- then, you're going to have almost every harmonic 985 01:13:07,400 --> 01:13:08,710 oscillator excited. 986 01:13:08,710 --> 01:13:11,580 And then you have energy of order N squared, et cetera. 987 01:13:21,410 --> 01:13:26,835 So now you have to imagine a slightly nontrivial condition. 988 01:13:26,835 --> 01:13:29,210 Now, you have to imagine a slightly nontrivial condition, 989 01:13:29,210 --> 01:13:30,876 which this is the only thing [? reach ?] 990 01:13:30,876 --> 01:13:32,190 beyond the harmonic oscillator. 991 01:13:42,350 --> 01:13:42,850 Yeah. 992 01:13:42,850 --> 01:13:49,390 Actually, I just realized this. 993 01:13:53,570 --> 01:13:55,635 I slightly oversimplified the story a little bit. 994 01:14:03,590 --> 01:14:04,090 Yeah. 995 01:14:06,840 --> 01:14:08,340 Yeah, let me say this way. 996 01:14:08,340 --> 01:14:10,070 Yeah. 997 01:14:10,070 --> 01:14:10,570 Yeah. 998 01:14:10,570 --> 01:14:14,290 Let's consider 2N harmonic oscillators. 999 01:14:14,290 --> 01:14:18,950 So let me arrange it into two matrix, A and B. 1000 01:14:18,950 --> 01:14:20,820 And they are all free. it doesn't matter. 1001 01:14:20,820 --> 01:14:23,319 For this purpose, it doesn't matter how many harmonic-- just 1002 01:14:23,319 --> 01:14:27,080 imagine I have arranged them into two matrices, 1003 01:14:27,080 --> 01:14:30,630 so I have two N squared harmonic oscillators. 1004 01:14:30,630 --> 01:14:33,010 And then I can excite them. 1005 01:14:33,010 --> 01:14:35,830 But, now, I have a condition. 1006 01:14:35,830 --> 01:14:39,620 So I have a condition, which is analog to the condition 1007 01:14:39,620 --> 01:14:42,010 of the gauge invariance. 1008 01:14:42,010 --> 01:14:45,750 Instead, I want all the states to be 1009 01:14:45,750 --> 01:14:51,950 the trace singlet created by A and B. 1010 01:14:51,950 --> 01:14:54,080 You have A, B, et cetera. 1011 01:14:54,080 --> 01:14:57,580 So A have N squared creation and annihilation operator, 1012 01:14:57,580 --> 01:15:01,080 and B have N squared annihilation-- creation 1013 01:15:01,080 --> 01:15:03,660 and annihilation operator. 1014 01:15:03,660 --> 01:15:05,930 And then you can act them-- then, 1015 01:15:05,930 --> 01:15:09,590 you can [? accurately ?] form the spectrum that it is. 1016 01:15:09,590 --> 01:15:12,600 But there's a-- but also I need to add a gauge invariance 1017 01:15:12,600 --> 01:15:13,100 condition. 1018 01:15:18,860 --> 01:15:24,860 It's that the state [? acted ?] by A and B 1019 01:15:24,860 --> 01:15:27,400 have to be SU(N) singlet. 1020 01:15:33,080 --> 01:15:34,960 And SU(N) is the conjugate. 1021 01:15:34,960 --> 01:15:37,780 Just imagine A and b transformed under some SU(N) 1022 01:15:37,780 --> 01:15:40,820 [? join representation. ?] And they 1023 01:15:40,820 --> 01:15:44,520 have to be the SU(N) singlet. 1024 01:15:44,520 --> 01:15:49,350 So equivalent statement say that the trace A and B-- the order 1025 01:15:49,350 --> 01:15:53,210 state has to be created by operators 1026 01:15:53,210 --> 01:15:57,170 inside the trace, single trace or multiple traces, et cetera. 1027 01:15:59,790 --> 01:16:00,695 So is this clear? 1028 01:16:06,400 --> 01:16:07,370 AUDIENCE: So one thing. 1029 01:16:07,370 --> 01:16:09,385 So A and B are operators? 1030 01:16:09,385 --> 01:16:09,965 Or they are-- 1031 01:16:09,965 --> 01:16:12,090 PROFESSOR: They're operators, harmonic oscillators. 1032 01:16:12,090 --> 01:16:12,570 AUDIENCE: Oh, I see. 1033 01:16:12,570 --> 01:16:13,236 PROFESSOR: Yeah. 1034 01:16:13,236 --> 01:16:17,540 Imagine you have two matrices under-- and two 1035 01:16:17,540 --> 01:16:20,730 harmonic oscillators that form two matrices. 1036 01:16:20,730 --> 01:16:26,140 And each of them have N squared harmonic oscillator. 1037 01:16:26,140 --> 01:16:28,350 And if I just say they are free, then you 1038 01:16:28,350 --> 01:16:31,009 don't even have to think about the matrix structure. 1039 01:16:31,009 --> 01:16:33,550 AUDIENCE: So the only thing, I'm confused about the matrices. 1040 01:16:33,550 --> 01:16:37,295 So are the matrices acting on some [INAUDIBLE] So in other 1041 01:16:37,295 --> 01:16:38,304 words-- 1042 01:16:38,304 --> 01:16:38,970 PROFESSOR: Yeah. 1043 01:16:38,970 --> 01:16:40,910 Let me be a little bit more explicit. 1044 01:16:43,630 --> 01:16:45,990 [INAUDIBLE] Think about the foreign system. 1045 01:16:49,570 --> 01:16:56,530 The Lagrangian 1/2 Tr A dot squared, Tr 1/2, 1046 01:16:56,530 --> 01:17:07,160 Tr B dot squared minus 1/2 Tr A square 1/2 Tr B squared. 1047 01:17:07,160 --> 01:17:10,150 So this is a free harmonic oscillator. 1048 01:17:10,150 --> 01:17:12,460 So these are two N squared free harmonic oscillators 1049 01:17:12,460 --> 01:17:13,460 with frequency 1. 1050 01:17:17,660 --> 01:17:19,687 So if I don't impose this singlet condition, 1051 01:17:19,687 --> 01:17:21,645 this is just a purely free harmonic oscillator. 1052 01:17:26,650 --> 01:17:28,500 But now, I [? require ?] all the state-- 1053 01:17:28,500 --> 01:17:34,960 and so SU(N) transformation-- [? yeah, you can act them. ?] 1054 01:17:34,960 --> 01:17:38,890 But now, I impose the condition that they will have-- 1055 01:17:38,890 --> 01:17:42,030 they should be SU(N) singlets. 1056 01:17:42,030 --> 01:17:44,540 So they can [? create ?] it by A and B, et cetera. 1057 01:17:44,540 --> 01:17:47,598 But they should be SU(N) singlet. 1058 01:17:52,320 --> 01:17:54,120 Good? 1059 01:17:54,120 --> 01:17:58,040 So now one thing you can convince 1060 01:17:58,040 --> 01:18:06,170 yourself is that because of the trace condition-- 1061 01:18:06,170 --> 01:18:09,290 so if you think about the state of energy 1 or 2, 1062 01:18:09,290 --> 01:18:15,520 et cetera-- so as far as the energy is order N to the power 1063 01:18:15,520 --> 01:18:21,970 0, so as far as the energy does not scale with N, 1064 01:18:21,970 --> 01:18:25,460 you can check yourself the density of state-- the way 1065 01:18:25,460 --> 01:18:28,320 you can-- the degeneracy of those states 1066 01:18:28,320 --> 01:18:34,050 will always be of order N to the power 0. 1067 01:18:34,050 --> 01:18:37,062 So this is a fact. 1068 01:18:37,062 --> 01:18:38,145 You can convince yourself. 1069 01:18:56,270 --> 01:19:01,800 And when you go to the energy of order N squared, 1070 01:19:01,800 --> 01:19:04,220 then you can check that the density of state 1071 01:19:04,220 --> 01:19:06,400 become exponential of order N squared. 1072 01:19:19,030 --> 01:19:22,704 So the reason for this is very simple. 1073 01:19:22,704 --> 01:19:24,870 So the reason for this is very simple heuristically. 1074 01:19:30,150 --> 01:19:33,330 So the reason for this is very simple 1075 01:19:33,330 --> 01:19:35,710 because if you have a state of order 1, which does not 1076 01:19:35,710 --> 01:19:41,750 scale with N, means there's only order 1 states-- order 1077 01:19:41,750 --> 01:19:43,390 1 oscillators are excited. 1078 01:19:47,802 --> 01:19:49,510 And then, of course, the density of state 1079 01:19:49,510 --> 01:19:53,640 will be independent of N. And if you do have energy of order 1080 01:19:53,640 --> 01:19:56,670 N squared because each of-- each oscillator 1081 01:19:56,670 --> 01:20:00,520 has a frequency of 1, and then you have energy N squared. 1082 01:20:00,520 --> 01:20:04,100 And then that means order N squared oscillator is excited. 1083 01:20:04,100 --> 01:20:08,400 And then you have order Ns-- if you have order N squared 1084 01:20:08,400 --> 01:20:11,910 oscillator excited, then how many ways you can choose them 1085 01:20:11,910 --> 01:20:14,691 to construct the [? state of ?] energy order N square and that 1086 01:20:14,691 --> 01:20:16,232 [? exponential ?] of order N squared. 1087 01:20:23,600 --> 01:20:28,190 So imagine each of the states, you 1088 01:20:28,190 --> 01:20:33,290 can have N possible oscillator to act, say, for each of them 1089 01:20:33,290 --> 01:20:35,200 for twice. 1090 01:20:35,200 --> 01:20:37,330 Then, you have 2 to the order N squared 1091 01:20:37,330 --> 01:20:40,170 probabilities of doing that. 1092 01:20:40,170 --> 01:20:42,260 So that's where this [? exponential ?] order 1093 01:20:42,260 --> 01:20:43,640 N squared come out. 1094 01:20:43,640 --> 01:20:44,390 So this is a fact. 1095 01:20:44,390 --> 01:20:46,040 You should try to convince yourself 1096 01:20:46,040 --> 01:20:49,980 if it's not obvious to you now. 1097 01:20:49,980 --> 01:20:53,840 So the reason I add this singlet condition is for here. 1098 01:20:53,840 --> 01:20:56,150 If I don't add a singlet condition, 1099 01:20:56,150 --> 01:20:59,220 then there's some kind of independence 1100 01:20:59,220 --> 01:21:01,370 here, which I don't want. 1101 01:21:01,370 --> 01:21:03,460 There can be [? log ?] independence. 1102 01:21:03,460 --> 01:21:05,040 Yeah, just-- yeah. 1103 01:21:05,040 --> 01:21:08,940 So this makes [? your ?] story a little bit [? convenient. ?] 1104 01:21:08,940 --> 01:21:15,640 Oh, so now-- yeah. 1105 01:21:15,640 --> 01:21:19,440 We're running out of time unfortunately. 1106 01:21:19,440 --> 01:21:25,000 So now let's consider the free energy, 1107 01:21:25,000 --> 01:21:26,960 which, roughly, you can consider as integrates 1108 01:21:26,960 --> 01:21:32,500 of all possible states with this weight factor and times 1109 01:21:32,500 --> 01:21:33,380 the density of state. 1110 01:21:37,472 --> 01:21:38,930 [INAUDIBLE] because the free energy 1111 01:21:38,930 --> 01:21:45,080 sum of all states, minus BH, and the sum 1112 01:21:45,080 --> 01:21:48,040 translated into the integral with a density of states. 1113 01:21:48,040 --> 01:21:51,950 And then that translates into that. 1114 01:21:51,950 --> 01:21:59,300 So naively-- so we always consider-- 1115 01:21:59,300 --> 01:22:04,087 so our beta is always-- does not scale with N. That's order 1. 1116 01:22:04,087 --> 01:22:04,920 It's always order 1. 1117 01:22:07,550 --> 01:22:20,880 So if you look at here naively, only state 1118 01:22:20,880 --> 01:22:23,880 of energy of order that's not scale N 1119 01:22:23,880 --> 01:22:34,550 will contribute significantly because of this suppression. 1120 01:22:38,930 --> 01:22:41,220 Because of the thermal suppression, say, 1121 01:22:41,220 --> 01:22:43,110 if you have energy of state of order 1122 01:22:43,110 --> 01:22:46,785 N squared, then this is a huge suppression. 1123 01:22:49,310 --> 01:22:50,720 Then, they should not contribute. 1124 01:22:50,720 --> 01:22:53,270 Then, they should not make a visible expression-- 1125 01:22:53,270 --> 01:22:58,840 a contribution to your free energy. 1126 01:22:58,840 --> 01:23:04,565 Expect, except when those states have a huge density of states. 1127 01:23:07,600 --> 01:23:14,440 So except when those states have a huge density of states. 1128 01:23:14,440 --> 01:23:19,690 So, here, we see those states have actually 1129 01:23:19,690 --> 01:23:21,232 order N squared density of states. 1130 01:23:25,170 --> 01:23:32,480 So that means, in here, in that integral, you have beta times 1131 01:23:32,480 --> 01:23:36,910 order N squared-- something of order N squared and then times 1132 01:23:36,910 --> 01:23:41,710 something positive of order N squared. 1133 01:23:41,710 --> 01:23:44,330 When you can see the contribution of the state 1134 01:23:44,330 --> 01:23:46,740 of energy of order N squared. 1135 01:23:46,740 --> 01:23:51,390 So when this factor dominate over this factor, 1136 01:23:51,390 --> 01:23:54,580 then the entropy will overwhelm the thermal suppression. 1137 01:23:54,580 --> 01:23:56,200 And then, such state will dominate. 1138 01:23:59,690 --> 01:24:02,770 And that precisely happens when beta becomes small enough. 1139 01:24:02,770 --> 01:24:04,855 Because when beta becomes small enough, 1140 01:24:04,855 --> 01:24:06,480 when you got the high temperature, then 1141 01:24:06,480 --> 01:24:08,200 beta decrease. 1142 01:24:08,200 --> 01:24:10,570 So eventually, this will overwhelm this one 1143 01:24:10,570 --> 01:24:12,301 because the beta will decrease. 1144 01:24:12,301 --> 01:24:14,300 And that's precisely the Hawking-Page transition 1145 01:24:14,300 --> 01:24:15,960 we'll see. 1146 01:24:15,960 --> 01:24:23,290 So at low beta-- at big beta or small temperature, 1147 01:24:23,290 --> 01:24:24,870 thermal-- this one dominates. 1148 01:24:28,870 --> 01:24:34,970 And then, the [? V ?] would be of order N to the power 0 1149 01:24:34,970 --> 01:24:37,947 because only state of order-- energy order zero 1150 01:24:37,947 --> 01:24:38,613 will contribute. 1151 01:24:41,490 --> 01:24:53,290 But the beta-- when beta is sufficiently small, then 1152 01:24:53,290 --> 01:24:58,995 the [? log ?] D(E) minus BE can become greater than 0. 1153 01:25:02,420 --> 01:25:05,115 And then the order N squared states will dominate.| 1154 01:25:10,510 --> 01:25:12,265 And then you find the-- then, you find 1155 01:25:12,265 --> 01:25:14,890 the [? partial function ?] will be [? exponential ?] of order N 1156 01:25:14,890 --> 01:25:15,390 squared. 1157 01:25:18,480 --> 01:25:23,710 And that's precisely what we see in the Hawking-Page transition. 1158 01:25:23,710 --> 01:25:26,010 So I emphasize that physics here have 1159 01:25:26,010 --> 01:25:30,150 nothing to do with the details of the system, 1160 01:25:30,150 --> 01:25:34,610 only related to the large N and to the density of states. 1161 01:25:34,610 --> 01:25:38,030 So if you have interactions, if you have more complicated 1162 01:25:38,030 --> 01:25:39,920 system, it doesn't matter. 1163 01:25:39,920 --> 01:25:40,670 It doesn't matter. 1164 01:25:40,670 --> 01:25:45,340 As far as your [INAUDIBLE], this phenomenon will happen. 1165 01:25:45,340 --> 01:25:48,080 This phenomenon will happen. 1166 01:25:48,080 --> 01:25:52,070 And so this is the essence of the Hawking-Page transition. 1167 01:25:52,070 --> 01:25:53,748 We will stop here.