1 00:00:00,090 --> 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 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 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 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:21,210 --> 00:00:24,860 HONG LIU: So last time, we introduced this threshold 9 00:00:24,860 --> 00:00:28,210 metric for the black hole. 10 00:00:28,210 --> 00:00:36,310 And then this rs gives the horizon size of the black hole. 11 00:00:36,310 --> 00:00:41,390 And then, there are two very important geometric quantities 12 00:00:41,390 --> 00:00:42,260 for the black hole. 13 00:00:42,260 --> 00:00:47,190 One is the so-called area of the horizon, 14 00:00:47,190 --> 00:00:53,070 which is the area of this s2 at the horizon. 15 00:00:53,070 --> 00:00:54,980 And that the other is what is called 16 00:00:54,980 --> 00:00:59,400 the surface gravity, which is essentially 17 00:00:59,400 --> 00:01:02,800 given by 1/2 of the derivative of this function, f, 18 00:01:02,800 --> 00:01:04,930 evaluated at the horizon. 19 00:01:04,930 --> 00:01:09,430 And if you value it on this, then this gives you 1 over 2 20 00:01:09,430 --> 00:01:11,460 rs. 21 00:01:11,460 --> 00:01:12,180 OK. 22 00:01:12,180 --> 00:01:14,730 So those quantities, we will see them very often. 23 00:01:22,530 --> 00:01:25,960 So now we talk a little bit more about the causal structure 24 00:01:25,960 --> 00:01:26,850 of the black hole. 25 00:01:32,520 --> 00:01:34,610 So before I start, do you have any questions? 26 00:02:01,991 --> 00:02:02,490 OK. 27 00:02:05,530 --> 00:02:09,389 So to understand the structure, the space-time structure 28 00:02:09,389 --> 00:02:12,050 of a black hole, a little bit better, 29 00:02:12,050 --> 00:02:16,320 let us look at the geometry view of the horizon. 30 00:02:16,320 --> 00:02:21,130 Because horizon is clearly one of the most important part 31 00:02:21,130 --> 00:02:23,324 of the black hole geometry. 32 00:02:23,324 --> 00:02:25,240 So let's consider the region near the horizon. 33 00:02:39,140 --> 00:02:45,640 So this is for r greater than but close to the Schwarzschild 34 00:02:45,640 --> 00:02:46,140 radius. 35 00:02:51,990 --> 00:03:01,360 So, for this purpose, you can expand this function, f, 36 00:03:01,360 --> 00:03:14,450 so near the horizon you can say rs plus f prime, et cetera. 37 00:03:14,450 --> 00:03:20,080 And this sum is equal to 0, so we essentially, 38 00:03:20,080 --> 00:03:22,950 when you expanded this function f, 39 00:03:22,950 --> 00:03:25,154 so that's essentially what you have. 40 00:03:25,154 --> 00:03:26,118 OK. 41 00:03:26,118 --> 00:03:29,120 Then here, then they have all the terms, 42 00:03:29,120 --> 00:03:33,470 but they can be neglected if we are close to the horizon. 43 00:03:33,470 --> 00:03:38,860 So let's put this insight into here. 44 00:03:38,860 --> 00:03:42,990 But to write this metric in a slightly more transparent form, 45 00:03:42,990 --> 00:03:45,740 it's convenient to consider the proper distance. 46 00:03:48,294 --> 00:03:49,710 Also consider the proper distance. 47 00:03:57,670 --> 00:04:15,390 Say suppose you are a stationary observer sitting at some, 48 00:04:15,390 --> 00:04:18,430 say, some value of r, then you're asking 49 00:04:18,430 --> 00:04:21,890 what is the proper distance from the horizon? 50 00:04:21,890 --> 00:04:29,010 And that can be just read from, by integrating this distance. 51 00:04:29,010 --> 00:04:34,420 So essentially, d rho, so rho, the proper distance, 52 00:04:34,420 --> 00:04:38,850 should given just by dr divided by square root of f. 53 00:04:38,850 --> 00:04:41,090 OK? 54 00:04:41,090 --> 00:04:42,610 You can read it from here. 55 00:04:42,610 --> 00:04:44,110 AUDIENCE: Can you-- can you give us 56 00:04:44,110 --> 00:04:47,640 a reminder what's proper distance? 57 00:04:47,640 --> 00:04:54,880 HONG LIU: The proper distance is the distance 58 00:04:54,880 --> 00:04:56,160 of a local observer. 59 00:04:56,160 --> 00:04:59,510 Just from local observer the [INAUDIBLE] 60 00:04:59,510 --> 00:05:02,375 looks like a flat space-time, and then that's 61 00:05:02,375 --> 00:05:04,000 the distance measured by that observer. 62 00:05:06,830 --> 00:05:07,330 Yeah. 63 00:05:07,330 --> 00:05:09,690 Similarly, which is the local proper time, et cetera. 64 00:05:12,800 --> 00:05:21,960 So then, by using this, then we find here the horizon, 65 00:05:21,960 --> 00:05:34,053 then we can write this as, say, dr plus higher order 66 00:05:34,053 --> 00:05:39,940 corrections, so now if you integrate this rho, 67 00:05:39,940 --> 00:05:50,740 and the setting rho equal to 0 to be the horizon, 68 00:05:50,740 --> 00:05:54,020 then we find rho equal to-- yeah, 69 00:05:54,020 --> 00:05:59,310 I just integrate that equation, which can be easily integrated, 70 00:05:59,310 --> 00:06:01,110 under the integration constant is chosen, 71 00:06:01,110 --> 00:06:06,970 so rather rho is equal to 0 add to the horizon, 72 00:06:06,970 --> 00:06:09,160 then you find the proper distance from the horizon 73 00:06:09,160 --> 00:06:16,570 is proportional to the square root r minus rs of this. 74 00:06:16,570 --> 00:06:17,070 Yeah. 75 00:06:20,440 --> 00:06:27,910 So now we could rewrite to the f itself, in terms of this rho, 76 00:06:27,910 --> 00:06:29,390 OK? 77 00:06:29,390 --> 00:06:33,440 So if we rewrite the f in terms of rho itself, 78 00:06:33,440 --> 00:06:36,540 so f is just f prime, r minus rs. 79 00:06:36,540 --> 00:06:42,580 And a square root r minus rs equal to rho times this factor. 80 00:06:42,580 --> 00:06:50,620 So we said this be algebra, you can see 81 00:06:50,620 --> 00:06:52,290 plus higher order corrections. 82 00:06:52,290 --> 00:06:52,790 OK? 83 00:06:56,690 --> 00:06:58,530 And this oh, so now you recognize 84 00:06:58,530 --> 00:07:00,545 this is essentially just the surface gravity. 85 00:07:03,580 --> 00:07:07,950 So near the horizon, this function 86 00:07:07,950 --> 00:07:11,610 f can be written as kappa square, rho square, 87 00:07:11,610 --> 00:07:13,525 then plus higher order corrections. 88 00:07:18,250 --> 00:07:21,110 So now we can rewrite to the horizon. 89 00:07:21,110 --> 00:07:24,070 Now we can rewrite this metric in terms of this rho. 90 00:07:28,410 --> 00:07:30,960 The ds squared. 91 00:07:30,960 --> 00:07:34,750 Now equal to [INAUDIBLE] Kappa square, 92 00:07:34,750 --> 00:07:39,310 rho square, dt square, plus the rho square. 93 00:07:39,310 --> 00:07:40,990 So you're thinking about definition, 94 00:07:40,990 --> 00:07:43,100 this term just becomes 0 square. 95 00:07:43,100 --> 00:07:46,600 And f I replace by this kappa square rho square. 96 00:07:49,974 --> 00:07:57,660 And to the leading order, I just replace this r square by rs, 97 00:07:57,660 --> 00:07:59,160 because we are close to the horizon. 98 00:08:02,120 --> 00:08:06,237 So now I can slightly rewrite this. 99 00:08:06,237 --> 00:08:07,320 So I introduce a variable. 100 00:08:09,830 --> 00:08:17,350 So let me introduce a variable, eta, equal to kappa times t. 101 00:08:17,350 --> 00:08:21,390 So essentially just t divided by 2 rs. 102 00:08:24,110 --> 00:08:26,410 So with that range of eta, then this 103 00:08:26,410 --> 00:08:30,710 becomes minus rho square to eta square, 104 00:08:30,710 --> 00:08:37,274 plus d rho square, then plus rs square d omega 2 square. 105 00:08:42,580 --> 00:08:45,590 So this is just a s2. 106 00:08:45,590 --> 00:08:52,870 This is a two-dimensional sphere with a radius 107 00:08:52,870 --> 00:08:57,320 given by the horizon sides. 108 00:08:57,320 --> 00:09:00,615 And then these, some of you may already recognize. 109 00:09:05,580 --> 00:09:08,960 This is, in fact, just a Minkowski spacetime. 110 00:09:08,960 --> 00:09:13,430 This is a 1 plus 1 dimensional Minkowski spacetime. 111 00:09:13,430 --> 00:09:14,560 OK? 112 00:09:14,560 --> 00:09:16,800 So this is actually a 1 plus 1 dimension 113 00:09:16,800 --> 00:09:21,750 Minkowski's time retain in the special coordinate. 114 00:09:21,750 --> 00:09:24,790 So, in the so-called Rindler form. 115 00:09:28,758 --> 00:09:30,960 The so-called Rindler form. 116 00:09:30,960 --> 00:09:33,190 OK? 117 00:09:33,190 --> 00:09:42,310 So we see then either the horizon, the spacetime geometry 118 00:09:42,310 --> 00:09:45,660 have the structure of a 1 plus 1 dimensional 119 00:09:45,660 --> 00:09:51,330 Minkowski spacetime, times a two-dimensional sphere. 120 00:09:51,330 --> 00:09:58,750 So now let me elaborate this a little bit, 121 00:09:58,750 --> 00:10:01,676 on why this is Minkowski's space. 122 00:10:01,676 --> 00:10:02,175 OK? 123 00:10:07,440 --> 00:10:09,270 So do you have any questions so far? 124 00:10:12,240 --> 00:10:12,740 OK. 125 00:10:12,740 --> 00:10:13,535 Let's just start. 126 00:10:16,980 --> 00:10:19,000 Let's consider 1 plus 1 dimensional Minkowski 127 00:10:19,000 --> 00:10:19,500 spacetime. 128 00:10:24,960 --> 00:10:29,170 OK, now you're going to see 1 plus 1 to here, done. 129 00:10:29,170 --> 00:10:32,450 I will write it in the following form. 130 00:10:32,450 --> 00:10:35,620 I call my T-- So to distinguish with the t 131 00:10:35,620 --> 00:10:40,980 and there-- so let me call it capital T, and capital X. OK? 132 00:10:40,980 --> 00:10:46,000 So this is our familiar 1 plus 1 dimension Minkowski's time. 133 00:10:46,000 --> 00:10:51,130 So now I will introduce a new variable. 134 00:10:51,130 --> 00:10:52,440 So make a coordinate change. 135 00:11:01,390 --> 00:11:03,650 OK? 136 00:11:03,650 --> 00:11:05,010 So I just plug this into there. 137 00:11:07,492 --> 00:11:08,450 So let me call this m2. 138 00:11:11,795 --> 00:11:13,920 So you just plug in them there, and you immediately 139 00:11:13,920 --> 00:11:19,680 see that ds square m2 is just precisely we see there. 140 00:11:24,070 --> 00:11:26,770 OK? 141 00:11:26,770 --> 00:11:32,210 So this metric, so this part of the metric, 142 00:11:32,210 --> 00:11:33,340 you see, in fact, flat. 143 00:11:37,350 --> 00:11:42,260 But this coordinate, rho eta actually 144 00:11:42,260 --> 00:11:46,460 does not cover the full Minkowski space, 145 00:11:46,460 --> 00:11:53,330 because, just by definition, this coordinate satisfy x 146 00:11:53,330 --> 00:11:56,284 square minus t square equal to rho square. 147 00:11:56,284 --> 00:11:57,450 Because this is [INAUDIBLE]. 148 00:12:00,560 --> 00:12:03,015 So this, by definition, satisfy x square minus t square 149 00:12:03,015 --> 00:12:04,390 actually greater than equal to 0. 150 00:12:07,160 --> 00:12:15,610 And also, just by definition, because we will take rho 151 00:12:15,610 --> 00:12:20,710 to be positive, and by definition, the x is positive. 152 00:12:20,710 --> 00:12:21,960 OK, we will take rho positive. 153 00:12:27,700 --> 00:12:30,050 Greater/equal to 0. 154 00:12:30,050 --> 00:12:32,310 So let's take rho greater/equal to 0. 155 00:12:35,160 --> 00:12:38,090 And which is what we are having here, 156 00:12:38,090 --> 00:12:42,080 because this rho is related to the proper distance 157 00:12:42,080 --> 00:12:43,660 from the horizon is, by definition, 158 00:12:43,660 --> 00:12:46,610 a non-elective quantity. 159 00:12:46,610 --> 00:12:52,430 So here we also want to restrict to rho greater/equal to 0. 160 00:12:52,430 --> 00:12:58,960 And so it's easy to draw what part of Minkowski 161 00:12:58,960 --> 00:13:02,120 spacetime this is. 162 00:13:02,120 --> 00:13:08,900 Let's plot it-- T, X-- so we have the light cone. 163 00:13:17,700 --> 00:13:23,570 So this occupies the region which x-- so x can be positive, 164 00:13:23,570 --> 00:13:28,220 so have to be on the right side. 165 00:13:28,220 --> 00:13:31,610 But also, we have x, the absolute value 166 00:13:31,610 --> 00:13:34,670 of x must greater than absolute value of t, 167 00:13:34,670 --> 00:13:37,340 because this thing is greater than 0. 168 00:13:37,340 --> 00:13:42,080 So that means it must be cover this region. 169 00:13:47,100 --> 00:13:52,030 And this is, let's say, this is [INAUDIBLE] x equal to t. 170 00:13:52,030 --> 00:13:55,400 And this is T minus equal to x. 171 00:13:55,400 --> 00:13:58,630 So these are the two light cone, and so they can only 172 00:13:58,630 --> 00:13:59,620 occupied that region. 173 00:14:06,690 --> 00:14:13,420 You can also show-- let me erase those things here, 174 00:14:13,420 --> 00:14:14,980 so it's only in this quadrant. 175 00:14:14,980 --> 00:14:18,600 So now let me erase these, so that I can draw further lines. 176 00:14:18,600 --> 00:14:20,090 Not to make things too messy. 177 00:14:27,490 --> 00:14:31,030 So you can also just invert this relation. 178 00:14:31,030 --> 00:14:33,380 So if we invert this relation, then 179 00:14:33,380 --> 00:14:36,670 you see that the rho square equal to x 180 00:14:36,670 --> 00:14:43,190 square minus t square under the tangent times 181 00:14:43,190 --> 00:14:45,890 eta equal to t divided by x. 182 00:14:49,150 --> 00:14:53,600 So that means that the constant rho should correspond 183 00:14:53,600 --> 00:14:55,817 into constant x square minus t square, 184 00:14:55,817 --> 00:14:56,900 and that's just hyperbola. 185 00:14:59,570 --> 00:15:02,610 So this is a constant rho. 186 00:15:02,610 --> 00:15:05,470 Say this is rho equal to constant. 187 00:15:08,130 --> 00:15:10,240 OK? 188 00:15:10,240 --> 00:15:14,040 And from here, the constant eta will corresponding 189 00:15:14,040 --> 00:15:16,742 to constant slope between t and x, 190 00:15:16,742 --> 00:15:18,860 so the constant eta will be corresponding 191 00:15:18,860 --> 00:15:21,270 to the straight line, like this. 192 00:15:24,960 --> 00:15:27,210 OK? 193 00:15:27,210 --> 00:15:33,460 So, essentially, we see that this metric 194 00:15:33,460 --> 00:15:36,810 covers this part of the Minkowski 1 plus 1 195 00:15:36,810 --> 00:15:38,960 dimensional them Minkowski's time, 196 00:15:38,960 --> 00:15:45,470 but 48 the Minkowski's time using coordinate like this. 197 00:15:50,470 --> 00:15:53,100 OK? 198 00:15:53,100 --> 00:16:05,340 So now, on this light cone, so there 199 00:16:05,340 --> 00:16:07,970 are two special lines here. 200 00:16:07,970 --> 00:16:11,650 One is x equal to t, which is essentially 201 00:16:11,650 --> 00:16:16,605 the boundary of this region, so here is when rho equal to 0. 202 00:16:22,960 --> 00:16:30,110 So now you look at is formula, where rho equal to 0, 203 00:16:30,110 --> 00:16:35,610 when rho close to 0-- oh, maybe we should do it in the limit. 204 00:16:35,610 --> 00:16:39,940 So when rho approach 0, in order for x and t finite-- 205 00:16:39,940 --> 00:16:42,300 so on this [INAUDIBLE], of course, x and t are finite. 206 00:16:42,300 --> 00:16:45,250 Then the eta has to go to infinity. 207 00:16:45,250 --> 00:16:49,040 So that means the light cone [INAUDIBLE] rho equal to 0, 208 00:16:49,040 --> 00:16:52,080 and the eta goes to plus infinity. 209 00:16:52,080 --> 00:16:53,720 OK? 210 00:16:53,720 --> 00:16:56,815 And the [INAUDIBLE] rho exponential eta finite. 211 00:17:03,610 --> 00:17:14,130 And similarity, t equal to minus x, this line, 212 00:17:14,130 --> 00:17:18,160 then goes 1 into rho goes to 0, but now 213 00:17:18,160 --> 00:17:21,605 eta goes to minus infinity. 214 00:17:21,605 --> 00:17:22,980 And when eta goes minus infinity, 215 00:17:22,980 --> 00:17:29,470 then the minus sign here will dominate the t and the rho 216 00:17:29,470 --> 00:17:32,250 exponential minus eta finite. 217 00:17:35,130 --> 00:17:35,630 OK? 218 00:17:41,940 --> 00:17:50,365 And so there's a special point here 219 00:17:50,365 --> 00:17:54,370 that goes 1 into x equal to t equal to 0. 220 00:17:54,370 --> 00:17:58,770 So this goes 1 into rho equal to zero, 221 00:17:58,770 --> 00:18:07,590 and any finite eta, say, will correspond into that point. 222 00:18:07,590 --> 00:18:08,090 OK? 223 00:18:23,410 --> 00:18:32,680 So we see that this is precisely what describes the black hole 224 00:18:32,680 --> 00:18:33,770 horizon. 225 00:18:33,770 --> 00:18:35,089 OK? 226 00:18:35,089 --> 00:18:36,880 And if back here you've got rho equal to 0, 227 00:18:36,880 --> 00:18:41,950 precisely lie on the black hole horizon, 228 00:18:41,950 --> 00:18:51,360 so from here, we conclude, so the black hole 229 00:18:51,360 --> 00:19:18,161 horizon is corresponding to x equal to plus minus t. 230 00:19:18,161 --> 00:19:18,660 OK? 231 00:19:23,160 --> 00:19:27,930 Here you also see, so this rho and the eta 232 00:19:27,930 --> 00:19:31,250 is essentially simple transformation of what 233 00:19:31,250 --> 00:19:33,896 we have here, r and the t. 234 00:19:33,896 --> 00:19:54,660 So we also see, just from this discussion-- 235 00:19:54,660 --> 00:20:00,270 so we see that the black hole horizon just 236 00:20:00,270 --> 00:20:03,440 is mapped to the light cone in the 1 plus 1 237 00:20:03,440 --> 00:20:05,990 dimensional Minkowski's time, and that the region 238 00:20:05,990 --> 00:20:07,780 outside the black hole just mapped 239 00:20:07,780 --> 00:20:15,470 to that quadrant of the Minkowski spacetime. 240 00:20:15,470 --> 00:20:21,080 So we conclude that the little black hole geometry, 241 00:20:21,080 --> 00:20:24,800 near horizon black hole geometry. 242 00:20:27,580 --> 00:20:29,105 So this is called Rindler space. 243 00:20:37,210 --> 00:20:41,550 So we see that the near horizon black hole geometry 244 00:20:41,550 --> 00:20:47,070 is just Rindler times s2. 245 00:20:47,070 --> 00:20:49,700 OK? 246 00:20:49,700 --> 00:20:56,690 In particular, it's precisely have this causal structure. 247 00:20:56,690 --> 00:21:01,610 You have a light cone which comes 1 into the horizon. 248 00:21:01,610 --> 00:21:04,420 So that also shows that the horizon is a long surface, 249 00:21:04,420 --> 00:21:08,330 because the horizon comes 1 into the light cone. 250 00:21:08,330 --> 00:21:14,910 And so we can just directly translate that picture from rho 251 00:21:14,910 --> 00:21:17,550 into t and r. 252 00:21:17,550 --> 00:21:18,320 OK? 253 00:21:18,320 --> 00:21:19,940 So this is just r. 254 00:21:19,940 --> 00:21:23,180 You go to rs. 255 00:21:23,180 --> 00:21:27,630 So eta goes to-- So here, is that the rho 256 00:21:27,630 --> 00:21:32,406 equal to 0, of the eta goes to plus infinity. 257 00:21:32,406 --> 00:21:40,390 So here is the rho goes to 0, and eta goes to minus infinity. 258 00:21:40,390 --> 00:21:43,620 So that means, for the black hole, 259 00:21:43,620 --> 00:21:46,160 this will cause 1 into i equal to i s, 260 00:21:46,160 --> 00:21:49,000 and the t goes to positive infinity. 261 00:21:49,000 --> 00:21:53,230 And t and eta just differ by this constant factor, and so 262 00:21:53,230 --> 00:21:55,240 essentially the same thing. 263 00:21:55,240 --> 00:22:00,522 And here, rs t equal to minus infinity. 264 00:22:04,590 --> 00:22:06,930 And then similarly, near the horizon, 265 00:22:06,930 --> 00:22:11,250 the constant r surface will be like this, 266 00:22:11,250 --> 00:22:20,477 just like a hyperbola in the-- and then the constant t surface 267 00:22:20,477 --> 00:22:21,310 will also like this. 268 00:22:24,640 --> 00:22:25,140 OK? 269 00:22:28,240 --> 00:22:34,620 And then, of course, the full spacetime would be times s2, 270 00:22:34,620 --> 00:22:38,820 so you should imagine each point on this plot, which 271 00:22:38,820 --> 00:22:42,480 only plot r on the t plane, so each point there 272 00:22:42,480 --> 00:22:47,320 this s2 with this radius over i rs. 273 00:22:47,320 --> 00:22:47,820 OK? 274 00:22:51,870 --> 00:22:53,120 Any questions on this? 275 00:22:53,120 --> 00:22:53,620 Yes? 276 00:22:53,620 --> 00:22:55,328 AUDIENCE: What's the meaning of the arrow 277 00:22:55,328 --> 00:22:58,474 on r equals constant hyperbola? 278 00:22:58,474 --> 00:22:59,140 HONG LIU: Sorry? 279 00:22:59,140 --> 00:23:01,970 AUDIENCE: What's the meaning of the arrow on the r equals 280 00:23:01,970 --> 00:23:03,930 constant hyperbola? 281 00:23:03,930 --> 00:23:04,920 HONG LIU: Ah. 282 00:23:04,920 --> 00:23:07,990 That's just come from here. 283 00:23:07,990 --> 00:23:12,030 So from here, once we map to these coordinates, 284 00:23:12,030 --> 00:23:14,040 we see that the rho constant because 1 285 00:23:14,040 --> 00:23:16,460 to hyperbola like that. 286 00:23:16,460 --> 00:23:19,550 And the rho is essentially just related to r 287 00:23:19,550 --> 00:23:21,240 by a simple transformation. 288 00:23:21,240 --> 00:23:25,080 So constant the rho surface is also the constant the r 289 00:23:25,080 --> 00:23:26,550 surface, because they are related 290 00:23:26,550 --> 00:23:28,450 to a similar transformation. 291 00:23:28,450 --> 00:23:31,850 So the constant r surface must also be a hyperbola like that. 292 00:23:31,850 --> 00:23:34,880 AUDIENCE: But why is there an arrow pointing to that? 293 00:23:34,880 --> 00:23:37,890 HONG LIU: Oh, the arrow just to show that the time goes up. 294 00:23:37,890 --> 00:23:42,510 Yeah, Here you can also show arrow, 295 00:23:42,510 --> 00:23:45,890 which is the direction which the eta increases. 296 00:23:45,890 --> 00:23:48,640 Yeah, so this is the direction which time goes up. 297 00:23:48,640 --> 00:23:51,600 And the time goes up, and, yeah. 298 00:23:54,934 --> 00:23:55,600 Other questions? 299 00:24:00,970 --> 00:24:02,274 OK. 300 00:24:02,274 --> 00:24:03,815 Now let me make some further remarks. 301 00:24:13,150 --> 00:24:18,390 So some of them just repeat what I just said, just so 302 00:24:18,390 --> 00:24:19,540 that we have it here. 303 00:24:25,590 --> 00:24:39,655 So observer at r equal to constant, but r 304 00:24:39,655 --> 00:24:48,760 say slightly outside the horizon for black hole, black hole 305 00:24:48,760 --> 00:25:07,020 geometry corresponds to an observer with rho 306 00:25:07,020 --> 00:25:23,060 equal to constant in Rindler patch of m2, 307 00:25:23,060 --> 00:25:24,860 OK, of two-dimensional Minkowski spacetime. 308 00:25:28,190 --> 00:25:32,785 And then this is observer which follows 309 00:25:32,785 --> 00:25:34,590 a hyperbolic trajectory. 310 00:25:34,590 --> 00:25:35,220 OK? 311 00:25:35,220 --> 00:25:37,980 So it goes like that. 312 00:25:37,980 --> 00:25:40,291 So this is not the standard observer. 313 00:25:40,291 --> 00:25:41,540 This is a hyperbolic observer. 314 00:25:50,700 --> 00:25:51,620 Yeah? 315 00:25:51,620 --> 00:25:56,812 So such observer which has a constant r, 316 00:25:56,812 --> 00:26:00,980 they're not the standard initial Minkowski observer. 317 00:26:00,980 --> 00:26:04,740 They travel in hyperbola. 318 00:26:04,740 --> 00:26:09,010 So now you can check easily, just 319 00:26:09,010 --> 00:26:11,590 by working with this flat space, because we know 320 00:26:11,590 --> 00:26:13,620 everything about flat space. 321 00:26:13,620 --> 00:26:15,310 You can easily work out. 322 00:26:15,310 --> 00:26:16,810 So you can easily check yourself. 323 00:26:16,810 --> 00:26:18,270 It's an elementary calculation. 324 00:26:18,270 --> 00:26:19,353 Take you a couple minutes. 325 00:26:22,090 --> 00:26:45,730 Such observer has a constant proper acceleration, 326 00:26:45,730 --> 00:26:47,670 which is given by a equal to 1 over rho. 327 00:26:50,860 --> 00:26:51,360 OK? 328 00:26:55,830 --> 00:27:00,760 So you can also translate into black hole language, when 329 00:27:00,760 --> 00:27:05,240 we converted this rho into rs. 330 00:27:05,240 --> 00:27:12,930 So this becomes 1/2 rs. 331 00:27:21,330 --> 00:27:21,830 Yeah. 332 00:27:21,830 --> 00:27:27,760 Then they make us just including the whole thing. 333 00:27:32,370 --> 00:27:35,370 So you can see from this formula, 334 00:27:35,370 --> 00:27:38,600 so this is a local proper acceleration of observer, 335 00:27:38,600 --> 00:27:41,410 as we discussed yesterday. 336 00:27:41,410 --> 00:27:48,450 If you want to be-- because a black hole attracts things, 337 00:27:48,450 --> 00:27:50,530 so if you wanted to stay at a fixed 338 00:27:50,530 --> 00:27:55,110 radius outside the black hole, you need to accelerate. 339 00:27:55,110 --> 00:27:57,830 You need to have some acceleration. 340 00:27:57,830 --> 00:28:00,310 And this is just that the acceleration 341 00:28:00,310 --> 00:28:01,466 close to the black hole. 342 00:28:01,466 --> 00:28:03,340 And, in particular, you see this accelerating 343 00:28:03,340 --> 00:28:07,490 goes to infinity when you approach the horizon. 344 00:28:07,490 --> 00:28:08,760 OK? 345 00:28:08,760 --> 00:28:10,570 When you approach the horizon. 346 00:28:10,570 --> 00:28:17,875 So just a side remark-- let me put here, side remark. 347 00:28:22,280 --> 00:28:27,310 So this is related once the last remark, the last thing we said 348 00:28:27,310 --> 00:28:31,550 yesterday, is that the surface gravity is 349 00:28:31,550 --> 00:28:35,270 the acceleration of an observer at the horizon, 350 00:28:35,270 --> 00:28:37,910 near the horizon viewed from infinity. 351 00:28:37,910 --> 00:28:39,540 OK? 352 00:28:39,540 --> 00:28:42,650 So the infinity. 353 00:28:42,650 --> 00:28:46,800 So that formula, so that ar, so that formula 354 00:28:46,800 --> 00:28:49,750 is the local acceleration. 355 00:28:49,750 --> 00:28:51,250 And you can easily convince yourself 356 00:28:51,250 --> 00:28:53,020 the acceleration at infinity is related 357 00:28:53,020 --> 00:28:59,930 to the local acceleration by a [INAUDIBLE] factor precisely 358 00:28:59,930 --> 00:29:02,790 given by this. 359 00:29:02,790 --> 00:29:05,630 And if you just take that formula, 360 00:29:05,630 --> 00:29:09,150 and multiply it by this, then you find precisely 361 00:29:09,150 --> 00:29:12,950 1/2 f prime rs, which is kappa. 362 00:29:12,950 --> 00:29:14,176 OK? 363 00:29:14,176 --> 00:29:14,800 Which is kappa. 364 00:29:19,640 --> 00:29:21,720 So this also give you a derivation 365 00:29:21,720 --> 00:29:25,300 of this surface gravity, this formula for the surface 366 00:29:25,300 --> 00:29:26,013 gravity. 367 00:29:26,013 --> 00:29:27,180 AUDIENCE: I have a question. 368 00:29:27,180 --> 00:29:27,971 Why is [INAUDIBLE]? 369 00:29:31,986 --> 00:29:34,360 HONG LIU: Yeah, that's where I ask you to check yourself. 370 00:29:37,272 --> 00:29:40,070 Yeah, I said it's a couple minutes calculation. 371 00:29:40,070 --> 00:29:41,289 It's flat space. 372 00:29:41,289 --> 00:29:42,580 So you should be able to do it. 373 00:29:45,360 --> 00:29:47,991 Any other question? 374 00:29:47,991 --> 00:29:48,490 Yes? 375 00:29:48,490 --> 00:29:51,940 AUDIENCE: What's the definition of proper acceleration? 376 00:29:51,940 --> 00:29:54,452 HONG LIU: Yeah, it's acceleration in the local rest 377 00:29:54,452 --> 00:29:55,410 frame of that observer. 378 00:30:00,610 --> 00:30:04,160 Yeah, anything you call the proper 379 00:30:04,160 --> 00:30:08,324 is something which is defined in the local rest frame 380 00:30:08,324 --> 00:30:09,490 of that particular observer. 381 00:30:11,930 --> 00:30:12,430 Yeah. 382 00:30:12,430 --> 00:30:14,596 AUDIENCE: What's the expression for the [INAUDIBLE]? 383 00:30:18,215 --> 00:30:18,840 HONG LIU: Yeah. 384 00:30:18,840 --> 00:30:23,700 It's, say the a mu, say it's a vector, 385 00:30:23,700 --> 00:30:29,770 say it can be defined a mu equal to u nu plus nu u nu. 386 00:30:29,770 --> 00:30:40,147 And so, this u nu is the local trajectory of that observer. 387 00:30:40,147 --> 00:30:41,230 So this is the definition. 388 00:30:41,230 --> 00:30:42,480 You can see it in the TR book. 389 00:30:47,410 --> 00:30:49,610 So that's the local-- so that's a velocity 390 00:30:49,610 --> 00:30:52,460 vector of the observer. 391 00:30:52,460 --> 00:30:54,011 U mu. 392 00:30:54,011 --> 00:30:54,510 OK. 393 00:30:54,510 --> 00:30:57,790 So this is the first thing. 394 00:30:57,790 --> 00:31:08,170 The second thing-- Yeah, so if you check in this Waltz 395 00:31:08,170 --> 00:31:10,150 book which I wrote last time. 396 00:31:10,150 --> 00:31:14,220 I put the reference in the Waltz book, and he can find it. 397 00:31:14,220 --> 00:31:16,345 Or he can find it in any GR book, they explain it. 398 00:31:19,620 --> 00:31:42,890 So now, a free-fall observer near the black hole horizon, 399 00:31:42,890 --> 00:31:45,520 they just gave you, just goes one 400 00:31:45,520 --> 00:31:56,180 into inertial observer in Minkowski spacetime, OK, 401 00:31:56,180 --> 00:31:58,650 in M cube. 402 00:31:58,650 --> 00:32:00,950 So this is easy because they map to each other. 403 00:32:00,950 --> 00:32:03,900 So free-fall observer is observer in the black hole 404 00:32:03,900 --> 00:32:06,970 spacetime, which just follow from, 405 00:32:06,970 --> 00:32:09,990 say some time like geodesics. 406 00:32:09,990 --> 00:32:12,490 And we know that in the time like geodesics in the Minkowski 407 00:32:12,490 --> 00:32:14,690 space is just straight line. 408 00:32:14,690 --> 00:32:15,880 Just straight line. 409 00:32:15,880 --> 00:32:21,707 So for object fall into to the black hole, 410 00:32:21,707 --> 00:32:23,040 you just follow a straight line. 411 00:32:25,900 --> 00:32:29,156 So this provide the very simple description of an [INAUDIBLE] 412 00:32:29,156 --> 00:32:30,280 observer near a black hole. 413 00:32:34,950 --> 00:32:36,255 So this is the second remark. 414 00:32:39,730 --> 00:32:50,860 So the third remark is that this Rindler coordinates rho 415 00:32:50,860 --> 00:32:55,655 eta, as we see from here, become singular. 416 00:33:03,510 --> 00:33:04,680 And the rho equal to 0. 417 00:33:09,440 --> 00:33:13,200 Yeah, you see to describe those rho equal to 0 418 00:33:13,200 --> 00:33:14,650 to describe the horizon, you have 419 00:33:14,650 --> 00:33:18,040 to take a [INAUDIBLE] limit. 420 00:33:18,040 --> 00:33:19,590 There's no longer one-to-one relation 421 00:33:19,590 --> 00:33:24,820 between the capital X and T, and the rho and the eta. 422 00:33:24,820 --> 00:33:30,971 So, for this just explains that the Schwarzschild coordinates-- 423 00:33:30,971 --> 00:33:32,970 so the same thing happens with the Schwarzschild 424 00:33:32,970 --> 00:33:34,290 coordinate at the horizon. 425 00:33:37,680 --> 00:33:39,740 In the same way the Schwarzschild 426 00:33:39,740 --> 00:33:42,670 coordinates become singular at the horizon. 427 00:33:42,670 --> 00:33:44,860 OK? 428 00:33:44,860 --> 00:33:47,690 At the same way. 429 00:33:47,690 --> 00:33:53,080 But we do understand in Minkowski spacetime, 430 00:33:53,080 --> 00:34:00,450 we know that the reason this-- we do 431 00:34:00,450 --> 00:34:02,260 know the actual Minkowski spacetime have 432 00:34:02,260 --> 00:34:05,940 four patches, rather than just a single patch, rather than just 433 00:34:05,940 --> 00:34:08,330 the single quadrant. 434 00:34:08,330 --> 00:34:12,719 And so, just by changing the [INAUDIBLE] 435 00:34:12,719 --> 00:34:16,159 to a different coordinate, this capital X and capital T, 436 00:34:16,159 --> 00:34:20,530 we can actually describe the full Minkowski spacetime, not 437 00:34:20,530 --> 00:34:21,969 just the single quadrant. 438 00:34:21,969 --> 00:34:23,230 OK? 439 00:34:23,230 --> 00:34:28,440 So let me call this region one. 440 00:34:28,440 --> 00:34:31,770 So now let me call this, so this Minkowski spacetime 441 00:34:31,770 --> 00:34:38,810 separated by light cone into four regions-- I, II, III, IV. 442 00:34:38,810 --> 00:34:39,310 OK. 443 00:34:39,310 --> 00:34:41,230 I hope this is clear. 444 00:34:41,230 --> 00:34:43,659 There are too many things on the blackboard. 445 00:34:43,659 --> 00:34:48,590 But by this quadrant is I, this top quadrant is II, 446 00:34:48,590 --> 00:34:51,790 this right quadrant is III, and the bottom is IV. 447 00:34:51,790 --> 00:34:52,290 OK? 448 00:34:57,720 --> 00:34:59,600 So even though this Rindler coordinate 449 00:34:59,600 --> 00:35:10,030 become singular rho equal to 0, by using Minkowski coordinates, 450 00:35:10,030 --> 00:35:16,030 standard Minkowski coordinates, x and t, 451 00:35:16,030 --> 00:35:27,770 we can actually extend this region I to the full Minkowski 452 00:35:27,770 --> 00:35:28,441 spacetime. 453 00:35:28,441 --> 00:35:28,940 OK? 454 00:35:31,900 --> 00:35:34,560 To the full m2. 455 00:35:34,560 --> 00:35:36,030 To the full Minkowski spacetime. 456 00:35:42,140 --> 00:35:45,950 So now he's becoming mechanical. 457 00:35:45,950 --> 00:35:51,530 So now, you must be, because this is essentially 458 00:35:51,530 --> 00:35:55,704 what describes the black hole near the horizon. 459 00:35:55,704 --> 00:35:57,120 Then in the black hole you must be 460 00:35:57,120 --> 00:35:59,280 able to find the set of coordinates 461 00:35:59,280 --> 00:36:01,740 to actually extend beyond the horizon. 462 00:36:01,740 --> 00:36:03,350 OK? 463 00:36:03,350 --> 00:36:28,650 So, similarly, by changing to suitable coordinates, 464 00:36:28,650 --> 00:36:37,140 one can-- so this is normally called the Kruskal coordinate-- 465 00:36:37,140 --> 00:36:41,360 we will not need to use it, so I will not write it down 466 00:36:41,360 --> 00:36:43,530 explicitly. 467 00:36:43,530 --> 00:36:48,860 It's just a generalization of this-- going to this capital 468 00:36:48,860 --> 00:36:54,440 X and T. So in the black hole geometry, 469 00:36:54,440 --> 00:36:59,050 by going to these Kruskal coordinates, 470 00:36:59,050 --> 00:37:00,460 the counterpart of this Minkowski 471 00:37:00,460 --> 00:37:19,010 coordinates, one can also extend the Schwarzschild geometry 472 00:37:19,010 --> 00:37:22,240 to four regions. 473 00:37:22,240 --> 00:37:22,740 OK? 474 00:37:26,660 --> 00:37:29,360 Just like this region in Minkowski 475 00:37:29,360 --> 00:37:32,550 can be extended to four regions, and this region 476 00:37:32,550 --> 00:37:37,070 outside the black hole horizon can be extend to four regions. 477 00:37:37,070 --> 00:37:37,940 OK? 478 00:37:37,940 --> 00:37:41,510 And we can, in fact, immediately just modeled on here, 479 00:37:41,510 --> 00:37:44,020 going from here to the full Minkowski spacetime, 480 00:37:44,020 --> 00:37:51,710 we can just immediately extend this, 481 00:37:51,710 --> 00:37:56,390 so we can immediately extend the black hole geometry 482 00:37:56,390 --> 00:37:57,170 into four regions. 483 00:38:00,060 --> 00:38:07,030 The only difference-- of course, this picture 484 00:38:07,030 --> 00:38:11,240 is only precise when you are close to the horizon. 485 00:38:11,240 --> 00:38:18,480 So once are you far away from the horizon, 486 00:38:18,480 --> 00:38:21,480 so those corrections will become important. 487 00:38:21,480 --> 00:38:22,190 OK? 488 00:38:22,190 --> 00:38:25,750 But those don't change the basic causal structure 489 00:38:25,750 --> 00:38:28,890 of the black hole, which are essentially determined 490 00:38:28,890 --> 00:38:30,670 by the horizon structure. 491 00:38:30,670 --> 00:38:35,422 So essentially, just add to the-- 492 00:38:35,422 --> 00:38:36,990 in terms of the causal structure, 493 00:38:36,990 --> 00:38:41,300 this picture just extends to the full black hole spacetime. 494 00:38:41,300 --> 00:38:42,270 OK? 495 00:38:42,270 --> 00:38:45,100 And in particular, say if go this way, go to r, 496 00:38:45,100 --> 00:38:49,970 go to infinity, and there's a slight subtlety, 497 00:38:49,970 --> 00:38:52,340 slight difference, from this picture, 498 00:38:52,340 --> 00:38:55,080 it's because black hole eventually becomes 499 00:38:55,080 --> 00:38:56,495 singular when r equal to 0. 500 00:38:59,450 --> 00:39:03,380 So black hole have a singularity at i equal to 0. 501 00:39:03,380 --> 00:39:06,090 And that happens inside the horizon, 502 00:39:06,090 --> 00:39:08,300 so this happens in this region. 503 00:39:08,300 --> 00:39:10,310 So eventually you will reach i equal to 0. 504 00:39:10,310 --> 00:39:13,360 You will reach some singularity, so we will draw it 505 00:39:13,360 --> 00:39:15,440 by some wavy lines. 506 00:39:15,440 --> 00:39:17,715 And also there's a singularity in the past. 507 00:39:20,395 --> 00:39:22,130 AUDIENCE: Question. 508 00:39:22,130 --> 00:39:26,244 Is the Gordon transformation the same as from Rindler to flat? 509 00:39:26,244 --> 00:39:27,410 HONG LIU: It's very similar. 510 00:39:31,980 --> 00:39:38,310 Of course, it's not identical, because that Rindler is only 511 00:39:38,310 --> 00:39:41,100 valid close to the horizon, and we're 512 00:39:41,100 --> 00:39:46,130 away from the horizon things become different. 513 00:39:46,130 --> 00:39:51,980 Yeah, but qualitatively, they're very similar. 514 00:39:51,980 --> 00:39:55,680 The precise mathematical form, of course, are not the same. 515 00:39:55,680 --> 00:39:57,699 AUDIENCE: And on the horizon, they are the same. 516 00:39:57,699 --> 00:39:59,240 HONG LIU: Yeah, essentially the same. 517 00:40:01,800 --> 00:40:03,104 Any other questions. 518 00:40:03,104 --> 00:40:04,270 AUDIENCE: I have a question. 519 00:40:04,270 --> 00:40:05,000 HONG LIU: Yes. 520 00:40:05,000 --> 00:40:06,416 AUDIENCE: Does it make sense, just 521 00:40:06,416 --> 00:40:08,288 from a mathematical point of view, 522 00:40:08,288 --> 00:40:12,690 to have a piece of Minkowski spacetime on its own? 523 00:40:12,690 --> 00:40:15,880 You just, the manifold is only that quadrant? 524 00:40:15,880 --> 00:40:17,311 HONG LIU: Yeah, sure. 525 00:40:17,311 --> 00:40:17,810 Sure. 526 00:40:17,810 --> 00:40:19,660 AUDIENCE: Could a similar thing happen 527 00:40:19,660 --> 00:40:23,090 in the case of a black hole, that the inside is simply 528 00:40:23,090 --> 00:40:24,850 the manifold does not extend into it? 529 00:40:24,850 --> 00:40:25,521 HONG LIU: Sure. 530 00:40:25,521 --> 00:40:26,020 Yeah. 531 00:40:26,020 --> 00:40:27,350 It can happen. 532 00:40:27,350 --> 00:40:28,230 Yeah. 533 00:40:28,230 --> 00:40:35,650 But, one thing, so here we are making a basic assumption, 534 00:40:35,650 --> 00:40:40,270 is that you can fall into a black hole. 535 00:40:40,270 --> 00:40:44,010 And from this picture, we know that falling 536 00:40:44,010 --> 00:40:46,570 into a black hole, from Minkowski point of view, 537 00:40:46,570 --> 00:40:47,820 means something very trivial. 538 00:40:47,820 --> 00:40:49,980 Just cross that light cone. 539 00:40:49,980 --> 00:40:55,420 And so this region must exist. 540 00:40:55,420 --> 00:40:59,660 So let me call this I, II, III, IV. 541 00:40:59,660 --> 00:41:01,290 So this region must exist. 542 00:41:01,290 --> 00:41:02,740 OK? 543 00:41:02,740 --> 00:41:05,490 Yeah, so let me make some further remark. 544 00:41:05,490 --> 00:41:11,140 In this three, then it may become clear to you. 545 00:41:11,140 --> 00:41:18,210 Say these are the remarks for this diagram. 546 00:41:18,210 --> 00:41:24,890 He said when you fall into the black hole, 547 00:41:24,890 --> 00:41:25,960 you enter region II. 548 00:41:31,770 --> 00:41:35,390 Because time goes up, you enter region II. 549 00:41:35,390 --> 00:41:39,970 And clearly, because this is a light cone, 550 00:41:39,970 --> 00:41:43,920 so if you draw a light cone inside the horizon, 551 00:41:43,920 --> 00:41:45,770 no matter what point you draw the light cone 552 00:41:45,770 --> 00:41:47,186 inside the horizon, there's no way 553 00:41:47,186 --> 00:41:50,310 you can cross outside across this surface. 554 00:41:50,310 --> 00:41:52,530 OK? 555 00:41:52,530 --> 00:42:08,660 So clearly, no information or observer in region 556 00:42:08,660 --> 00:42:22,240 II, can reach region I. OK? 557 00:42:25,450 --> 00:42:27,280 So we call this a future horizon. 558 00:42:33,130 --> 00:42:37,065 We call this a future horizon, because once you go beyond it, 559 00:42:37,065 --> 00:42:39,880 you cannot influence here anymore. 560 00:42:39,880 --> 00:42:41,560 So we call this a future horizon. 561 00:42:41,560 --> 00:42:42,060 OK? 562 00:42:47,600 --> 00:42:51,650 So another comment is that regions III and IV, 563 00:42:51,650 --> 00:43:00,560 just like in Minkowski case, can be obtained. 564 00:43:06,220 --> 00:43:13,435 from I and II by time reversal. 565 00:43:20,541 --> 00:43:21,040 OK? 566 00:43:25,350 --> 00:43:27,210 Yeah, actually, not precisely. 567 00:43:27,210 --> 00:43:28,290 Only region IV. 568 00:43:33,760 --> 00:43:37,090 Only region II. 569 00:43:37,090 --> 00:43:38,630 Let me see. 570 00:43:38,630 --> 00:43:53,716 Region IV can be obtained from II by time reversal. 571 00:43:57,668 --> 00:44:00,420 Yeah, by time reversal. 572 00:44:00,420 --> 00:44:16,860 So for the real back hole, for region III and IV 573 00:44:16,860 --> 00:44:31,150 do not exist for real black holes formed from collapse. 574 00:44:31,150 --> 00:44:38,601 Formed from collapse. 575 00:44:38,601 --> 00:44:39,100 OK? 576 00:44:41,990 --> 00:44:44,530 So in some sense, regions III and IV 577 00:44:44,530 --> 00:44:48,100 are artifact of we are working with a time reversal 578 00:44:48,100 --> 00:44:49,410 in [INAUDIBLE] geometry. 579 00:44:58,740 --> 00:45:05,010 So also, again, if you draw the light cone in the region I, 580 00:45:05,010 --> 00:45:08,160 there's no way you can send signals to region IV. 581 00:45:08,160 --> 00:45:11,060 OK, again, because of the light cone, it's going that, 582 00:45:11,060 --> 00:45:14,160 so there's no way you can cross into the-- so the light radius 583 00:45:14,160 --> 00:45:16,100 no way can reach region IV. 584 00:45:20,190 --> 00:45:40,990 So observer in region I cannot influence-- oh, 585 00:45:40,990 --> 00:45:43,260 this is already finished. 586 00:45:43,260 --> 00:45:44,510 Influence IV. 587 00:45:44,510 --> 00:45:45,805 Maybe I should go back to here. 588 00:46:02,630 --> 00:46:09,370 Cannot influence region events in region IV. 589 00:46:13,500 --> 00:46:16,940 So that's why this one is called the past horizon. 590 00:46:23,620 --> 00:46:24,120 OK? 591 00:46:24,120 --> 00:46:27,390 So this is called the past horizon. 592 00:46:27,390 --> 00:46:35,860 So the final remark r the diagram, is that r equal to 0. 593 00:46:35,860 --> 00:46:37,740 Of course should be. 594 00:46:37,740 --> 00:46:40,650 So if you go into the r equal to 0 595 00:46:40,650 --> 00:46:43,720 cause light inside to the horizon. 596 00:46:43,720 --> 00:46:47,570 And so r equal to 0 is the black hole singularity. 597 00:46:54,080 --> 00:47:00,870 So notice that outside the horizon, 598 00:47:00,870 --> 00:47:05,450 the constant r surface goes like this, OK. 599 00:47:05,450 --> 00:47:08,440 But when you go inside the horizon, as we remark before, 600 00:47:08,440 --> 00:47:15,000 the f will change sine, then the rho between t and r 601 00:47:15,000 --> 00:47:17,330 will switched. 602 00:47:17,330 --> 00:47:19,670 So r actually become time. 603 00:47:19,670 --> 00:47:25,010 So that's why r equal to 0 like a constant time surface, OK? 604 00:47:25,010 --> 00:47:25,970 And so is like this. 605 00:47:25,970 --> 00:47:28,040 And then this just a time reversal of that. 606 00:47:32,657 --> 00:47:34,490 So this is called the spacetime singularity. 607 00:47:37,150 --> 00:47:39,145 So this is a spacetime singularity, 608 00:47:39,145 --> 00:47:40,820 because r become time. 609 00:47:49,650 --> 00:47:50,480 Yes? 610 00:47:50,480 --> 00:47:54,840 AUDIENCE: So this is a time reversal in variant geometry, 611 00:47:54,840 --> 00:47:59,735 but so is it possible for there to be time reversal, a time 612 00:47:59,735 --> 00:48:04,910 reversal in processes that are not time reversal? 613 00:48:04,910 --> 00:48:08,040 That you cannot time reverse in a time reversal in variant 614 00:48:08,040 --> 00:48:08,880 geometry? 615 00:48:08,880 --> 00:48:17,410 HONG LIU: Of course, in the time reversal in variant geometry, 616 00:48:17,410 --> 00:48:22,750 there certainly exists processes which it does not respect. 617 00:48:22,750 --> 00:48:24,720 Time reversal in variants. 618 00:48:24,720 --> 00:48:26,040 That can happen. 619 00:48:26,040 --> 00:48:29,550 But for each such process, there's a mirror process. 620 00:48:29,550 --> 00:48:33,500 AUDIENCE: So in the case of crossing from I to III, 621 00:48:33,500 --> 00:48:36,620 I guess you can cross from I to II, but not from II to I, 622 00:48:36,620 --> 00:48:39,680 but you can cross from IV to I? 623 00:48:39,680 --> 00:48:41,310 HONG LIU: Yeah, you can cross IV to I. 624 00:48:41,310 --> 00:48:43,060 AUDIENCE: So that's like a mirror process? 625 00:48:43,060 --> 00:48:44,320 HONG LIU: Yeah, That's right. 626 00:48:47,670 --> 00:48:50,640 AUDIENCE: [INAUDIBLE] what does it really 627 00:48:50,640 --> 00:48:56,080 mean about the inside the horizon the space and time 628 00:48:56,080 --> 00:48:57,565 [INAUDIBLE]? 629 00:48:57,565 --> 00:49:00,257 I mean, mathematically it's [INAUDIBLE]? 630 00:49:06,327 --> 00:49:08,910 HONG LIU: Physically, if you are a local observer you will not 631 00:49:08,910 --> 00:49:11,090 feel anything. 632 00:49:11,090 --> 00:49:14,150 From a local observer point of view, 633 00:49:14,150 --> 00:49:17,570 when you cross the horizon-- So this 634 00:49:17,570 --> 00:49:22,300 is just related to our choice of coordinates. 635 00:49:22,300 --> 00:49:28,170 Just imagine from here, we'll cross from here. 636 00:49:28,170 --> 00:49:30,740 Of course, nothing really happens. 637 00:49:30,740 --> 00:49:32,690 Yeah. 638 00:49:32,690 --> 00:49:33,450 Yeah. 639 00:49:33,450 --> 00:49:39,350 Here, the reason-- yeah, just the t, 640 00:49:39,350 --> 00:49:42,099 we just related to the rho of t and r switch. 641 00:49:42,099 --> 00:49:43,390 T and r, they just coordinated. 642 00:49:43,390 --> 00:49:49,910 They're not-- Yeah, it's just how you describe the system. 643 00:49:49,910 --> 00:49:52,110 You can prefer not to use t and r. 644 00:49:52,110 --> 00:49:56,115 Then cross the horizon, nothing really happens. 645 00:49:59,340 --> 00:50:01,210 AUDIENCE: [INAUDIBLE] euclidean formula, 646 00:50:01,210 --> 00:50:04,469 you cannot cross the horizon. 647 00:50:04,469 --> 00:50:06,760 HONG LIU: Yeah, and we will talk about euclidean later. 648 00:50:09,247 --> 00:50:10,080 Any other questions? 649 00:50:13,230 --> 00:50:13,730 Yes? 650 00:50:13,730 --> 00:50:15,500 AUDIENCE: Why do you specify that it's 651 00:50:15,500 --> 00:50:17,340 real black holes from collapse? 652 00:50:17,340 --> 00:50:19,530 HONG LIU: Maybe I should just equal black holes, not 653 00:50:19,530 --> 00:50:20,740 imaginary black holes. 654 00:50:23,340 --> 00:50:26,770 Yeah, just say black holes. 655 00:50:26,770 --> 00:50:29,583 Yeah, real life black holes. 656 00:50:29,583 --> 00:50:30,470 [LAUGHTER] 657 00:50:30,470 --> 00:50:35,530 Yeah, then I was getting neglected [INAUDIBLE]. 658 00:50:35,530 --> 00:50:37,250 Any other questions? 659 00:50:37,250 --> 00:50:39,015 AUDIENCE: [INAUDIBLE] space lights. 660 00:50:39,015 --> 00:50:40,431 HONG LIU: Sir, you have something. 661 00:50:40,431 --> 00:50:42,808 AUDIENCE: And don't they take infinite time to form? 662 00:50:42,808 --> 00:50:45,069 Don't they take infinite time to form? 663 00:50:45,069 --> 00:50:46,110 HONG LIU: The black hole? 664 00:50:49,010 --> 00:50:55,960 No, they take-- gravitational collapse always take finite, 665 00:50:55,960 --> 00:50:59,860 always take finite proper time. 666 00:50:59,860 --> 00:51:02,860 Say, from the observer in infinity, 667 00:51:02,860 --> 00:51:05,980 and you will have the phenomenon because 668 00:51:05,980 --> 00:51:07,690 of the infinite redshift. 669 00:51:07,690 --> 00:51:11,580 And then, seems like they will never pass through the horizon. 670 00:51:11,580 --> 00:51:13,840 But eventually they will. 671 00:51:13,840 --> 00:51:17,191 Eventually you cannot see them, because the light ray will 672 00:51:17,191 --> 00:51:18,690 become dimmer and dimmer, et cetera. 673 00:51:18,690 --> 00:51:20,280 Yeah. 674 00:51:20,280 --> 00:51:23,750 Anyway, so this is a standard question 675 00:51:23,750 --> 00:51:26,352 in GR of what happens before the black hole collapse. 676 00:51:26,352 --> 00:51:27,310 Yeah, they do collapse. 677 00:51:30,490 --> 00:51:33,170 Does this answer your questions? 678 00:51:33,170 --> 00:51:33,670 No? 679 00:51:33,670 --> 00:51:34,170 OK. 680 00:51:34,170 --> 00:51:35,870 [LAUGHTER] 681 00:51:35,870 --> 00:51:36,580 Yeah. 682 00:51:36,580 --> 00:51:38,570 I cannot do the whole GR course here. 683 00:51:38,570 --> 00:51:40,055 Yeah. 684 00:51:40,055 --> 00:51:42,530 AUDIENCE: I just wanted to ask-- so 685 00:51:42,530 --> 00:51:47,480 how long does it take for the observer's frame? 686 00:51:47,480 --> 00:51:51,405 The time it takes from the horizon to hits 687 00:51:51,405 --> 00:51:53,469 the r equals to 0 singularity. 688 00:51:53,469 --> 00:51:55,260 HONG LIU: Well, then you just calculate it. 689 00:51:55,260 --> 00:51:56,550 AUDIENCE: Is it finite? 690 00:51:56,550 --> 00:51:57,758 HONG LIU: Yeah, it is finite. 691 00:51:57,758 --> 00:51:59,315 AUDIENCE: Then how does it feel? 692 00:51:59,315 --> 00:51:59,815 I mean-- 693 00:51:59,815 --> 00:52:00,801 [LAUGHTER] 694 00:52:00,801 --> 00:52:04,816 [INAUDIBLE] suddenly stop when you hit the singularity? 695 00:52:04,816 --> 00:52:05,440 HONG LIU: Yeah. 696 00:52:05,440 --> 00:52:07,815 Before you hit the singularity, you'll already be killed. 697 00:52:07,815 --> 00:52:09,830 [LAUGHTER] 698 00:52:09,830 --> 00:52:12,940 So the curvature will become very big. 699 00:52:12,940 --> 00:52:15,040 The curvature will be very big. 700 00:52:15,040 --> 00:52:16,640 And, yeah. 701 00:52:16,640 --> 00:52:20,870 AUDIENCE: [INAUDIBLE] quantum fluctuation spacetime. 702 00:52:20,870 --> 00:52:22,240 HONG LIU: Yeah. 703 00:52:22,240 --> 00:52:25,840 So what we believe is that this singularity-- so the curvature 704 00:52:25,840 --> 00:52:28,500 will become infinite here. 705 00:52:28,500 --> 00:52:30,454 So before you reach here, you're already, say, 706 00:52:30,454 --> 00:52:32,120 be killed by the tidal force, et cetera. 707 00:52:36,810 --> 00:52:40,530 But we do believe that the curvature, 708 00:52:40,530 --> 00:52:45,120 this curvature singularity, must be just a mathematical artifact 709 00:52:45,120 --> 00:52:47,340 of using the classical GR. 710 00:52:47,340 --> 00:52:50,080 Say, if you are able to use quantum gravity, 711 00:52:50,080 --> 00:52:54,930 then there should not be a singularity. 712 00:52:54,930 --> 00:52:57,130 But there are even debate whether they're even 713 00:52:57,130 --> 00:52:59,160 reaching behind the horizon. 714 00:52:59,160 --> 00:53:01,140 Maybe you can never cross the horizon. 715 00:53:01,140 --> 00:53:03,200 Maybe you already get killed at the horizon. 716 00:53:03,200 --> 00:53:04,390 [LAUGHTER] 717 00:53:04,390 --> 00:53:06,260 That's a lot of possibilities. 718 00:53:06,260 --> 00:53:07,660 That's a lot of possibilities. 719 00:53:07,660 --> 00:53:08,160 Yeah. 720 00:53:08,160 --> 00:53:10,220 So we don't really know. 721 00:53:10,220 --> 00:53:13,740 So here, we are using the assumption 722 00:53:13,740 --> 00:53:16,760 that the things, so there's nothing wrong, 723 00:53:16,760 --> 00:53:19,160 at least from the classical metric, 724 00:53:19,160 --> 00:53:22,154 nothing wrong on the horizon, then you can cross it. 725 00:53:22,154 --> 00:53:24,570 But there have been debate whether you can really cross it 726 00:53:24,570 --> 00:53:25,070 or not. 727 00:53:25,070 --> 00:53:27,040 AUDIENCE: But you'll get killed anyway. 728 00:53:27,040 --> 00:53:27,590 [LAUGHTER] 729 00:53:29,430 --> 00:53:31,132 HONG LIU: But nobody knows. 730 00:53:31,132 --> 00:53:34,770 Yeah, even if you get killed, no other people will know. 731 00:53:34,770 --> 00:53:36,626 [LAUGHTER] 732 00:53:39,525 --> 00:53:42,525 AUDIENCE: Are some people working on experiments? 733 00:53:42,525 --> 00:53:45,090 [INAUDIBLE] we have an academia response. 734 00:53:45,090 --> 00:53:49,720 So we add a prohibition to a system to see how it responses. 735 00:53:49,720 --> 00:53:55,310 For a black hole, if [INAUDIBLE] what kind of experiments? 736 00:53:55,310 --> 00:53:55,960 HONG LIU: Yeah. 737 00:53:55,960 --> 00:53:57,910 I think in the second half of the class, 738 00:53:57,910 --> 00:54:00,950 we will see such examples. 739 00:54:00,950 --> 00:54:01,450 We will. 740 00:54:01,450 --> 00:54:03,360 Yeah. 741 00:54:03,360 --> 00:54:07,000 Some process, some scattering process reached the black hole. 742 00:54:07,000 --> 00:54:10,730 That can be considered as some kind of linear response. 743 00:54:10,730 --> 00:54:12,604 Yeah. 744 00:54:12,604 --> 00:54:13,270 Other questions? 745 00:54:18,920 --> 00:54:20,320 OK, good. 746 00:54:20,320 --> 00:54:25,410 So I'm running a little bit short of time. 747 00:54:25,410 --> 00:54:28,870 Maybe I should-- So does everybody 748 00:54:28,870 --> 00:54:31,029 know what's a Penrose diagram? 749 00:54:31,029 --> 00:54:31,570 AUDIENCE: No. 750 00:54:34,420 --> 00:54:35,530 HONG LIU: So not? 751 00:54:35,530 --> 00:54:36,480 Half? 752 00:54:36,480 --> 00:54:38,560 Maybe half? 753 00:54:38,560 --> 00:54:40,750 Raise your hand, people who know Penrose diagram? 754 00:54:43,510 --> 00:54:44,190 OK. 755 00:54:44,190 --> 00:54:46,141 Yeah, let me maybe skip it now-- 756 00:54:46,141 --> 00:54:46,640 [LAUGHTER] 757 00:54:48,700 --> 00:54:51,460 Because we will not immediately use it. 758 00:54:51,460 --> 00:54:53,900 So let me just skip it, just because I want 759 00:54:53,900 --> 00:54:56,270 to reach the next topic today. 760 00:54:58,810 --> 00:55:00,719 It's a little bit more interesting 761 00:55:00,719 --> 00:55:01,760 than the Penrose diagram. 762 00:55:04,840 --> 00:55:05,340 OK. 763 00:55:05,340 --> 00:55:09,095 So these are all essentially the classical physics 764 00:55:09,095 --> 00:55:12,132 of a black hole. 765 00:55:12,132 --> 00:55:13,840 So now we can go to a little bit quantum. 766 00:55:30,200 --> 00:55:35,900 So Hawking, in 1974, made the great discovery 767 00:55:35,900 --> 00:55:38,230 that actually, you don't need to treat 768 00:55:38,230 --> 00:55:40,800 the black hole as quantum. 769 00:55:40,800 --> 00:55:44,530 If you just treat the matter field, treated the matter, 770 00:55:44,530 --> 00:55:46,390 whatever matter outside the black hole-- so 771 00:55:46,390 --> 00:55:51,130 photon, electron, neutrino-- treated those things as quantum 772 00:55:51,130 --> 00:55:57,200 in the back hole geometry, then actually, then black hole 773 00:55:57,200 --> 00:56:01,030 does not appear as completely black. 774 00:56:01,030 --> 00:56:03,590 Actually, black hole radiates like a black body. 775 00:56:03,590 --> 00:56:06,550 So actually, black hole have a well-defined temperature. 776 00:56:06,550 --> 00:56:08,440 OK? 777 00:56:08,440 --> 00:56:14,840 So now, we will try to derive this black hole temperature. 778 00:56:14,840 --> 00:56:18,190 There are many ways to derive this. 779 00:56:18,190 --> 00:56:24,110 The most authentic away is essentially 780 00:56:24,110 --> 00:56:26,440 Hawking's original derivation. 781 00:56:26,440 --> 00:56:31,380 To really show that if you have a gravitational collapse, 782 00:56:31,380 --> 00:56:37,660 and then long after the black hole has formed, 783 00:56:37,660 --> 00:56:40,360 actually there is a still [INAUDIBLE] radiation 784 00:56:40,360 --> 00:56:44,420 coming out of the black hole, from the perspective 785 00:56:44,420 --> 00:56:46,430 of the observer at infinity. 786 00:56:46,430 --> 00:56:52,010 And so you can deduce that actually black hole 787 00:56:52,010 --> 00:56:56,960 is like a black body with a finite temperature. 788 00:56:56,960 --> 00:56:59,920 And you can show that the radiation is precisely 789 00:56:59,920 --> 00:57:03,980 [INAUDIBLE] in the semi-classical approximation. 790 00:57:03,980 --> 00:57:06,480 Say treating the geometry as classical, 791 00:57:06,480 --> 00:57:08,810 but treat the quantum field as quantum. 792 00:57:08,810 --> 00:57:11,572 And then you can show that the spectrum 793 00:57:11,572 --> 00:57:12,530 is exactly [INAUDIBLE]. 794 00:57:16,430 --> 00:57:18,440 But we will not go through such a derivation, 795 00:57:18,440 --> 00:57:20,630 because it takes a little bit of time. 796 00:57:20,630 --> 00:57:28,690 And also, we will not lead that particular technical tour 797 00:57:28,690 --> 00:57:29,570 de force. 798 00:57:29,570 --> 00:57:34,380 Yeah, it's a technical tour de force, but we will not need it. 799 00:57:34,380 --> 00:57:36,990 So I will give you a different derivation. 800 00:57:36,990 --> 00:57:41,040 And this derivation is very simple, 801 00:57:41,040 --> 00:57:43,610 but it's a little bit sleek. 802 00:57:43,610 --> 00:57:45,550 It hides a lot of things. 803 00:57:45,550 --> 00:57:49,590 And so I will just try to give you a derivation first, 804 00:57:49,590 --> 00:57:54,280 and then we will talk about the funny things which 805 00:57:54,280 --> 00:57:57,220 are hidden by this derivation. 806 00:57:57,220 --> 00:57:58,370 OK? 807 00:57:58,370 --> 00:58:02,060 So first, let me remind you, that in the quantum field 808 00:58:02,060 --> 00:58:21,820 theory, if we want to describe a system at finite temperature, 809 00:58:21,820 --> 00:58:28,450 we go to euclidean, one way to do it is 810 00:58:28,450 --> 00:58:29,658 to go to euclidean signature. 811 00:58:33,870 --> 00:58:37,270 We take t equal to minus i tau. 812 00:58:37,270 --> 00:58:47,850 And then you periodically identify tau to have a radius, 813 00:58:47,850 --> 00:58:52,300 to have a period identify tau to have lens h bar times beta, 814 00:58:52,300 --> 00:58:54,650 and the beta is 1 over temperature. 815 00:58:54,650 --> 00:58:56,010 OK? 816 00:58:56,010 --> 00:58:58,895 So there's a key thing, this h bar here. 817 00:59:01,630 --> 00:59:04,590 And we will always take the Boltzmann constant to be 1. 818 00:59:10,000 --> 00:59:14,870 So conversely, if you have a euclidean theory which 819 00:59:14,870 --> 00:59:20,360 have a periodic time, then when you analytic continue back 820 00:59:20,360 --> 00:59:23,570 to the Lorentzian signature, then that system 821 00:59:23,570 --> 00:59:25,320 should be at a finite temperature. 822 00:59:25,320 --> 00:59:26,460 OK? 823 00:59:26,460 --> 00:59:29,810 Should be at a final temperature. 824 00:59:29,810 --> 00:59:32,460 So now let's try to analytic continue. 825 00:59:32,460 --> 00:59:34,830 So now the logic is the following. 826 00:59:34,830 --> 00:59:37,070 So now the logic is that, let's now 827 00:59:37,070 --> 00:59:40,040 try to analytic continue the black hole spacetime 828 00:59:40,040 --> 00:59:42,520 to euclidean signature. 829 00:59:42,520 --> 00:59:47,840 And then I will show that the euclidean time of a black hole 830 00:59:47,840 --> 00:59:49,865 is forced to be periodic. 831 00:59:53,550 --> 00:59:55,770 With some specific periods. 832 00:59:55,770 --> 00:59:58,130 And so this tells you, by consistency, 833 00:59:58,130 --> 01:00:00,070 the black hole must have a finite temperature. 834 01:00:00,070 --> 01:00:00,570 OK? 835 01:00:00,570 --> 01:00:03,330 So that's the logic. 836 01:00:03,330 --> 01:00:04,710 So now let's do a [INAUDIBLE]. 837 01:00:13,590 --> 01:00:16,590 And let me keep this here. 838 01:00:16,590 --> 01:00:24,110 So, for black hole, so let's take t equals to minus i tau. 839 01:00:24,110 --> 01:00:24,915 OK? 840 01:00:24,915 --> 01:00:26,540 Then the euclidean black hole geometry. 841 01:00:40,790 --> 01:00:44,450 So now, again, I go to the horizon. 842 01:00:44,450 --> 01:00:47,000 I go to the horizon, and I just take this thing. 843 01:00:47,000 --> 01:00:50,390 So this is a Lorentzian metric near the horizon. 844 01:00:50,390 --> 01:00:52,440 So now I now go to the horizon. 845 01:01:00,670 --> 01:01:01,780 Let's go to the horizon. 846 01:01:01,780 --> 01:01:05,240 Then we find that this euclidean metric, then just you 847 01:01:05,240 --> 01:01:07,900 take this t equal to minus i tau, 848 01:01:07,900 --> 01:01:11,400 then you find rho square kappa square tau 849 01:01:11,400 --> 01:01:20,450 square plus t rho square plus r square d omega 2 square. 850 01:01:20,450 --> 01:01:23,050 OK? 851 01:01:23,050 --> 01:01:28,325 So again, in analog with here, I introduce a euclidean version 852 01:01:28,325 --> 01:01:32,730 of this eta, so I introduce a euclidean version of this eta, 853 01:01:32,730 --> 01:01:35,640 which is [INAUDIBLE] theta. 854 01:01:35,640 --> 01:01:51,200 So I introduce, let me define, introduce, theta 855 01:01:51,200 --> 01:01:53,850 equal to kappa tau. 856 01:01:53,850 --> 01:01:55,920 OK? 857 01:01:55,920 --> 01:02:05,870 So you do that-- so now I can erase here-- then you 858 01:02:05,870 --> 01:02:11,030 find this becomes rho square d theta 859 01:02:11,030 --> 01:02:17,448 square plus d rho square plus r s square d omega 2 square. 860 01:02:26,730 --> 01:02:32,910 So now, this we recognize without any explanation. 861 01:02:32,910 --> 01:02:35,230 This is just euclidean flat space. 862 01:02:35,230 --> 01:02:41,420 Euclidean flat two-dimensional space in polar coordinates. 863 01:02:41,420 --> 01:02:43,930 OK? 864 01:02:43,930 --> 01:02:50,670 But with one difference, is that, in the standard euclidean 865 01:02:50,670 --> 01:02:54,960 coordinate, the theta is periodic in 2 pi, 866 01:02:54,960 --> 01:03:02,630 but here, for the black hole case, this tau, in principal, 867 01:03:02,630 --> 01:03:03,730 is uncompact. 868 01:03:03,730 --> 01:03:06,010 OK? 869 01:03:06,010 --> 01:03:20,470 But this metric, as a singularity, 870 01:03:20,470 --> 01:03:33,585 has a conical singularity at the rho 871 01:03:33,585 --> 01:03:42,290 equal to 0 for any theta not equal to 2 pi. 872 01:03:42,290 --> 01:03:44,030 OK? 873 01:03:44,030 --> 01:03:47,400 For any theta not equal to 2 pi, there's a conical singularity. 874 01:03:47,400 --> 01:03:48,780 So this is easy to understand. 875 01:03:48,780 --> 01:03:50,738 If you have things which are now equal to 2 pi, 876 01:03:50,738 --> 01:03:53,485 you can fold it, then become a cone, and then tip of the cone 877 01:03:53,485 --> 01:03:53,985 is singular. 878 01:04:00,155 --> 01:04:04,100 For any theta whose periods not equal to 2 pi. 879 01:04:04,100 --> 01:04:05,600 I think you understand the sentence. 880 01:04:13,480 --> 01:04:17,110 A singular-- yeah, let me write proper English. 881 01:04:19,890 --> 01:04:24,810 It has a conical singularity unless theta 882 01:04:24,810 --> 01:04:26,970 is periodic in 2 pi. 883 01:04:30,100 --> 01:04:31,125 This is proper English. 884 01:04:36,340 --> 01:04:42,050 So, now the little horizon metric will become like this. 885 01:04:42,050 --> 01:04:43,890 And then we see that this metric is actually 886 01:04:43,890 --> 01:04:47,720 singular at rho equal to 0. 887 01:04:47,720 --> 01:04:51,440 If theta is not periodic in 2 pi. 888 01:04:51,440 --> 01:04:53,840 But as we said, in the Lorentzian signature, 889 01:04:53,840 --> 01:04:57,490 the horizon is a completely smooth place. 890 01:04:57,490 --> 01:05:00,140 Nothing should happen there. 891 01:05:00,140 --> 01:05:02,050 In particular, from this Rindler picture, 892 01:05:02,050 --> 01:05:04,690 you can just pass through the horizon. 893 01:05:04,690 --> 01:05:05,790 This just flat space. 894 01:05:05,790 --> 01:05:07,990 It's nothing really happening there. 895 01:05:12,190 --> 01:05:25,520 So since to the horizon non-singular in Lorentzian 896 01:05:25,520 --> 01:05:47,130 picture, in Lorentz signature, it 897 01:05:47,130 --> 01:05:59,840 should not be singular in euclidean. 898 01:06:03,250 --> 01:06:05,580 OK? 899 01:06:05,580 --> 01:06:12,720 So we conclude that this tau that must be periodic, 900 01:06:12,720 --> 01:06:15,830 so that this redefine, this theta, 901 01:06:15,830 --> 01:06:17,480 should have period in 2 pi. 902 01:06:17,480 --> 01:06:18,160 OK? 903 01:06:18,160 --> 01:06:27,940 So that means tau must be periodic, with a period given 904 01:06:27,940 --> 01:06:31,820 by 2 pi divided by kappa. 905 01:06:31,820 --> 01:06:32,320 OK? 906 01:06:38,620 --> 01:06:41,090 So that theta has period in 2 pi. 907 01:06:41,090 --> 01:06:41,885 Yes? 908 01:06:41,885 --> 01:06:43,454 AUDIENCE: Isn't that metric slightly 909 01:06:43,454 --> 01:06:45,370 different from the original one, because here, 910 01:06:45,370 --> 01:06:49,240 rho is forced to be greater than 0? 911 01:06:49,240 --> 01:06:53,740 HONG LIU: No, rho equal to 0 is the horizon, right? 912 01:06:53,740 --> 01:06:57,256 AUDIENCE: But negative rhos are not covered in this subject. 913 01:06:57,256 --> 01:06:59,285 But negative rhos are excluded. 914 01:06:59,285 --> 01:06:59,910 HONG LIU: Yeah. 915 01:06:59,910 --> 01:07:01,940 In the euclidean spacetime, negative rho 916 01:07:01,940 --> 01:07:04,345 is also not allowed. 917 01:07:04,345 --> 01:07:07,250 In euclidean flat space. 918 01:07:07,250 --> 01:07:08,980 It's the same thing here. 919 01:07:08,980 --> 01:07:11,735 Rho is a radial coordinate. 920 01:07:11,735 --> 01:07:14,854 AUDIENCE: In Minkowskian time, we had the region I and II, 921 01:07:14,854 --> 01:07:15,770 like inside and out. 922 01:07:15,770 --> 01:07:16,436 HONG LIU: Right. 923 01:07:16,436 --> 01:07:18,470 But once you write in this coordinate, 924 01:07:18,470 --> 01:07:21,530 once your write in this form, euclidean form, 925 01:07:21,530 --> 01:07:25,120 and then this is just really a flat space. 926 01:07:25,120 --> 01:07:28,830 And rho is really a radial coordinate. 927 01:07:28,830 --> 01:07:31,355 You cannot extend rho to negative value. 928 01:07:35,850 --> 01:07:42,710 So recall that this thing should be identified as h beta. 929 01:07:42,710 --> 01:07:44,000 OK? 930 01:07:44,000 --> 01:07:46,690 So now, if you consider, say, a quantum field theory 931 01:07:46,690 --> 01:07:52,160 in this black hole spacetime, then that quantum field theory 932 01:07:52,160 --> 01:07:57,890 must have a periodic imaginary time. 933 01:07:57,890 --> 01:08:00,760 And then this should be identified with this h bar 934 01:08:00,760 --> 01:08:02,610 beta. 935 01:08:02,610 --> 01:08:13,790 So this tells us-- let me just write-- so 936 01:08:13,790 --> 01:08:18,550 let me say recall t is the proper time. 937 01:08:18,550 --> 01:08:26,319 So this tau is the euclidean version of this t, 938 01:08:26,319 --> 01:08:28,124 but each observer a different r, they 939 01:08:28,124 --> 01:08:32,850 have a different proper time, and t is only the proper time 940 01:08:32,850 --> 01:08:34,240 for the observer at infinity. 941 01:08:34,240 --> 01:08:36,870 OK? 942 01:08:36,870 --> 01:08:49,116 [INAUDIBLE] the proper time for observers 943 01:08:49,116 --> 01:08:50,074 at r equal to infinity. 944 01:08:53,180 --> 01:09:13,410 So we deduce from here that observers 945 01:09:13,410 --> 01:09:27,386 at r equal to infinity must feel a finite temperature given by. 946 01:09:31,040 --> 01:09:34,060 So, by identify this thing to be h bar beta and beta 947 01:09:34,060 --> 01:09:39,410 is 1 over t, so that means t is equal to 1 over beta 948 01:09:39,410 --> 01:09:42,950 is equal to h bar kappa divided by 2 pi. 949 01:09:46,109 --> 01:09:46,660 OK? 950 01:09:46,660 --> 01:09:49,434 So this kappa is 1 over 2 s. 951 01:09:49,434 --> 01:09:56,893 So this is 2 pi r s, and this is h bar 8 pi GNm. 952 01:10:04,260 --> 01:10:07,290 So we conclude if you want to put a quantum field 953 01:10:07,290 --> 01:10:10,640 theory in this black hole spacetime, 954 01:10:10,640 --> 01:10:13,110 then this quantum field theory is automatically 955 01:10:13,110 --> 01:10:17,850 at the finite temperature, with this temperature. 956 01:10:17,850 --> 01:10:20,570 OK? 957 01:10:20,570 --> 01:10:23,110 With this temperature. 958 01:10:23,110 --> 01:10:26,150 So to emphasize, this is a temperature 959 01:10:26,150 --> 01:10:30,920 to the observer at r equal to infinity. 960 01:10:30,920 --> 01:10:39,732 So you can easily work out the local temperature for observer 961 01:10:39,732 --> 01:10:41,780 at some r. 962 01:10:44,530 --> 01:10:47,860 Then you just work out the redshift factor, 963 01:10:47,860 --> 01:10:51,750 since the t local, so as we discussed yesterday, 964 01:10:51,750 --> 01:10:57,420 is one half r d t of the infinity. 965 01:10:57,420 --> 01:11:01,450 So that tells you, and the temperature should be 966 01:11:01,450 --> 01:11:04,370 conjugated with the time, [INAUDIBLE] time, 967 01:11:04,370 --> 01:11:09,780 so that means that the t local r should be, 968 01:11:09,780 --> 01:11:16,600 that the t at infinity minus one half r. ? 969 01:11:16,600 --> 01:11:20,180 OK Then t for infinity is this one. 970 01:11:20,180 --> 01:11:22,000 So, yeah, just call it t. 971 01:11:22,000 --> 01:11:31,840 So this is just h bar kappa divided by 2 pi f minus 1/2 r. 972 01:11:31,840 --> 01:11:36,290 And in particular, this will goes to infinity, 973 01:11:36,290 --> 01:11:38,750 as r goes to r s. 974 01:11:38,750 --> 01:11:42,240 Because this f goes to 0 when you approach horizon, 975 01:11:42,240 --> 01:11:45,460 because it lacked in power, so this goes to infinity. 976 01:11:45,460 --> 01:11:50,880 So the local temperature at the horizon is very, very hot. 977 01:11:50,880 --> 01:11:53,260 So at infinity, you have this temperature. 978 01:11:53,260 --> 01:11:56,080 But when you get closer to, closer to horizon, 979 01:11:56,080 --> 01:11:59,494 for any station observer, you feel hotter and hotter, 980 01:11:59,494 --> 01:12:01,660 and then become infinite temperature of the horizon. 981 01:12:01,660 --> 01:12:02,160 OK? 982 01:12:05,050 --> 01:12:07,160 So black hole is actually a very hot place, 983 01:12:07,160 --> 01:12:10,710 if you want to get close to it. 984 01:12:10,710 --> 01:12:11,490 Yes? 985 01:12:11,490 --> 01:12:14,725 AUDIENCE: But didn't-- But this metric is still the one near 986 01:12:14,725 --> 01:12:15,710 the horizon. 987 01:12:15,710 --> 01:12:16,524 HONG LIU: Yeah. 988 01:12:16,524 --> 01:12:18,690 AUDIENCE: So how can we make sure that that's indeed 989 01:12:18,690 --> 01:12:19,872 the temperature at infinity? 990 01:12:19,872 --> 01:12:21,830 I mean, it seems that there's other corrections 991 01:12:21,830 --> 01:12:23,700 that we have to do. 992 01:12:23,700 --> 01:12:24,650 [INAUDIBLE] 993 01:12:24,650 --> 01:12:26,316 HONG LIU: Yeah, this is a good question, 994 01:12:26,316 --> 01:12:30,370 The only thing matters, the only thing matters, 995 01:12:30,370 --> 01:12:35,030 is what time you use to periodically identify. 996 01:12:35,030 --> 01:12:39,074 So the time we identify is the time which is used at infinity. 997 01:12:39,074 --> 01:12:39,740 AUDIENCE: I see. 998 01:12:39,740 --> 01:12:44,320 HONG LIU: Yeah, because we analytic continue this time, 999 01:12:44,320 --> 01:12:47,390 and it's this tau get identified. 1000 01:12:47,390 --> 01:12:50,290 And this tau is the tau used by the observer at infinity. 1001 01:12:50,290 --> 01:12:50,960 AUDIENCE: I see. 1002 01:12:50,960 --> 01:12:51,585 HONG LIU: Yeah. 1003 01:12:51,585 --> 01:12:55,250 So observer in other positions, they 1004 01:12:55,250 --> 01:12:58,840 use different tau obtained by analytic continuation 1005 01:12:58,840 --> 01:12:59,940 of this guy. 1006 01:12:59,940 --> 01:13:03,844 So that's why, then, you will see a different temperature. 1007 01:13:03,844 --> 01:13:05,810 AUDIENCE: Well, no. 1008 01:13:05,810 --> 01:13:08,830 I'm asking, this is not the actual-- 1009 01:13:08,830 --> 01:13:10,316 we're using a [INAUDIBLE] metric, 1010 01:13:10,316 --> 01:13:12,664 or [INAUDIBLE] metric, right? 1011 01:13:12,664 --> 01:13:13,580 HONG LIU: We're using? 1012 01:13:13,580 --> 01:13:16,350 AUDIENCE: We're not using the exact metric, are we? 1013 01:13:16,350 --> 01:13:18,360 HONG LIU: Oh, we are using the exact metric. 1014 01:13:18,360 --> 01:13:23,400 I mean, this doesn't-- so I know what you're asking. 1015 01:13:27,770 --> 01:13:31,120 This tau is the exact tau for the-- t. 1016 01:13:31,120 --> 01:13:32,386 We have not changed the t. 1017 01:13:32,386 --> 01:13:33,094 AUDIENCE: Oh, OK. 1018 01:13:33,094 --> 01:13:34,135 I see what you're saying. 1019 01:13:34,135 --> 01:13:34,800 Yes. 1020 01:13:34,800 --> 01:13:35,820 HONG LIU: We just look at the behavioral 1021 01:13:35,820 --> 01:13:37,590 of the metric near the horizon. 1022 01:13:37,590 --> 01:13:41,830 But t is still the t used in the regional metric, 1023 01:13:41,830 --> 01:13:44,370 and which is the t used by the observer at infinity. 1024 01:13:47,454 --> 01:13:48,120 Other questions? 1025 01:13:54,610 --> 01:13:55,110 OK. 1026 01:13:55,110 --> 01:13:57,410 Good. 1027 01:13:57,410 --> 01:14:00,509 So you may say, r, OK, fine, the black hole 1028 01:14:00,509 --> 01:14:01,800 will have a finite temperature. 1029 01:14:05,610 --> 01:14:10,600 But now, if we use this relation with a Rindler spacetime, 1030 01:14:10,600 --> 01:14:16,620 then we immediately conclude, in Minkowski spacetime, 1031 01:14:16,620 --> 01:14:19,720 some observer must also see a finite temperature. 1032 01:14:19,720 --> 01:14:20,470 OK? 1033 01:14:20,470 --> 01:14:42,200 So let's just-- So let's do, for the Rindler space-- OK? 1034 01:14:42,200 --> 01:14:51,265 So let's write is as so this is a Rindler space Lorentzian 1035 01:14:51,265 --> 01:14:51,765 metric. 1036 01:14:54,550 --> 01:15:00,320 Now we take, again, go to the euclidean signature. 1037 01:15:00,320 --> 01:15:04,270 Let me call eta equal to [INAUDIBLE] i theta. 1038 01:15:04,270 --> 01:15:09,437 And then I get the euclidean metric, which just-- yeah, 1039 01:15:09,437 --> 01:15:11,520 of course it's just exactly the same thing that we 1040 01:15:11,520 --> 01:15:14,580 did for the black hole, because they are the same thing. 1041 01:15:14,580 --> 01:15:16,450 Near the horizon is same thing. 1042 01:15:16,450 --> 01:15:21,900 So that means the theta must be identified with 2 pi. 1043 01:15:24,280 --> 01:15:24,780 OK? 1044 01:15:29,110 --> 01:15:38,280 But now, you say, observer who use this time, eta, must 1045 01:15:38,280 --> 01:15:40,050 feel at the finite temperature. 1046 01:15:40,050 --> 01:15:40,870 OK? 1047 01:15:40,870 --> 01:15:44,170 So the time, which time you use, is crucial. 1048 01:15:44,170 --> 01:15:47,710 And this eta is not the standard Minkowski spacetime. 1049 01:15:47,710 --> 01:15:52,640 And this is the time which-- yeah, 1050 01:15:52,640 --> 01:15:58,390 I forgot-- which is the time. 1051 01:15:58,390 --> 01:16:07,810 So for the Rindler space, is the time which goes like this. 1052 01:16:07,810 --> 01:16:10,210 Yeah, it's the time which goes like this. 1053 01:16:13,130 --> 01:16:16,270 This is not the standard Minkowski time. 1054 01:16:16,270 --> 01:16:20,520 So for observer who are using that time, 1055 01:16:20,520 --> 01:16:24,710 then they should feel to be at a finite temperature. 1056 01:16:24,710 --> 01:16:28,160 Of course, the observer who used that time 1057 01:16:28,160 --> 01:16:31,135 is precisely the observer who are at the constant rho. 1058 01:16:31,135 --> 01:16:33,980 So if you have a constant rho, then d rho equal to 0, 1059 01:16:33,980 --> 01:16:35,280 then that's the time you use. 1060 01:16:35,280 --> 01:16:36,780 OK? 1061 01:16:36,780 --> 01:16:44,040 So, but with a slight correction-- so for observer. 1062 01:16:50,420 --> 01:16:53,250 So this is normally called a Rindler observer 1063 01:16:53,250 --> 01:17:00,030 at constant rho, at rho equal to constant, 1064 01:17:00,030 --> 01:17:08,690 then it's local proper time as given by this guy, then t, 1065 01:17:08,690 --> 01:17:14,680 d t local is equal to just rho d eta. 1066 01:17:14,680 --> 01:17:16,980 OK? 1067 01:17:16,980 --> 01:17:20,040 Then that means the corresponding euclidean 1068 01:17:20,040 --> 01:17:29,080 time is related to this theta, also by this factor of rho, 1069 01:17:29,080 --> 01:17:32,110 is rho d theta. 1070 01:17:32,110 --> 01:17:36,770 And that means that this tau local will 1071 01:17:36,770 --> 01:17:42,950 be periodic in 2 pi times rho. 1072 01:17:42,950 --> 01:17:45,460 OK? 1073 01:17:45,460 --> 01:17:47,980 And again, you identify this with h bar beta. 1074 01:17:51,470 --> 01:18:02,030 So that's tell you for Rindler observer, at location rho, 1075 01:18:02,030 --> 01:18:06,660 you will see a temperature which is h bar divided by 2 pi rho. 1076 01:18:09,810 --> 01:18:13,050 And, as we said before, 1 over rho-- 1077 01:18:13,050 --> 01:18:16,310 I just erased it-- is essentially acceleration. 1078 01:18:16,310 --> 01:18:21,850 So this is just h a divided by 2 pi, and a is the acceleration. 1079 01:18:21,850 --> 01:18:25,000 So we call a equals to 1 over rho. 1080 01:18:25,000 --> 01:18:28,530 So the acceleration is equal to one over rho. 1081 01:18:28,530 --> 01:18:36,620 So this tells you, once you write it in this form, 1082 01:18:36,620 --> 01:18:42,620 once you write in this form, you can forget about Rindler space, 1083 01:18:42,620 --> 01:18:46,430 because the only physics is now using the acceleration. 1084 01:18:46,430 --> 01:18:47,270 OK? 1085 01:18:47,270 --> 01:18:52,090 Now you can say any Minkowski observer 1086 01:18:52,090 --> 01:18:55,080 with a non-zero acceleration, will 1087 01:18:55,080 --> 01:18:57,760 feel to be at the finite temperature. 1088 01:18:57,760 --> 01:18:59,050 OK? 1089 01:18:59,050 --> 01:19:01,680 Will feel at the finite temperature. 1090 01:19:01,680 --> 01:19:04,610 And the finite temperature is proportional to 1091 01:19:04,610 --> 01:19:08,140 its acceleration and the h bar. 1092 01:19:08,140 --> 01:19:08,845 Yes? 1093 01:19:08,845 --> 01:19:12,225 AUDIENCE: And so an observer free-falling into a black hole 1094 01:19:12,225 --> 01:19:13,620 will not-- 1095 01:19:13,620 --> 01:19:15,370 HONG LIU: Yeah, will not feel temperature. 1096 01:19:15,370 --> 01:19:15,970 AUDIENCE: So 0 temperature? 1097 01:19:15,970 --> 01:19:17,970 So even like when you're approaching the horizon 1098 01:19:17,970 --> 01:19:19,880 and you're close to the horizon, usually you 1099 01:19:19,880 --> 01:19:20,910 would have an infinite temperature, 1100 01:19:20,910 --> 01:19:22,300 but here you're going to have a 0 temperature? 1101 01:19:22,300 --> 01:19:22,600 HONG LIU: Yeah. 1102 01:19:22,600 --> 01:19:23,141 That's right. 1103 01:19:23,141 --> 01:19:26,410 Yeah, so here, and besides, this must be observer 1104 01:19:26,410 --> 01:19:28,680 is some constant value of r. 1105 01:19:28,680 --> 01:19:30,160 Yeah, must be a stationary observer 1106 01:19:30,160 --> 01:19:31,306 outside the black hole. 1107 01:19:35,109 --> 01:19:37,275 Yeah, so let me just write down this final sentence. 1108 01:19:45,060 --> 01:20:07,600 You said, in Minkowski spacetime an accelerated observer will 1109 01:20:07,600 --> 01:20:25,022 feel the temperature proportional to acceleration. 1110 01:20:34,420 --> 01:20:36,830 So this was first discovered by Unruh. 1111 01:20:36,830 --> 01:20:40,640 So this is called Unruh temperature. 1112 01:20:40,640 --> 01:20:45,030 So this was discovered by Unruh, I think also around 1974. 1113 01:20:45,030 --> 01:20:47,680 Maybe 1976. 1114 01:20:47,680 --> 01:20:50,650 Around the time of Hawking's discovery of Hawking radiation. 1115 01:20:54,360 --> 01:20:58,411 So Unruh, actually, was very close to derive 1116 01:20:58,411 --> 01:20:59,660 the Hawking radiation himself. 1117 01:21:02,260 --> 01:21:03,745 Yeah, there were several people who 1118 01:21:03,745 --> 01:21:06,650 were very close to derive the Hawking radiation, 1119 01:21:06,650 --> 01:21:10,390 but only Hawking, in the end, did it. 1120 01:21:10,390 --> 01:21:13,580 But then, Unruh realized, actually 1121 01:21:13,580 --> 01:21:17,060 Hawking radiation is actually the same as this phenomena 1122 01:21:17,060 --> 01:21:18,590 in flat space. 1123 01:21:18,590 --> 01:21:23,790 And so, now this is called Unruh temperature. 1124 01:21:23,790 --> 01:21:24,290 OK. 1125 01:21:24,290 --> 01:21:27,050 So as I said earlier, that this derivation 1126 01:21:27,050 --> 01:21:31,770 is very simple and powerful, and we will use it again and again 1127 01:21:31,770 --> 01:21:34,570 in the future. 1128 01:21:34,570 --> 01:21:39,850 But it's quite sleek. 1129 01:21:39,850 --> 01:21:42,420 It does not tell you where this radiation, 1130 01:21:42,420 --> 01:21:44,560 where this temperature comes from. 1131 01:21:44,560 --> 01:21:46,190 And if you have a black body radiation, 1132 01:21:46,190 --> 01:21:48,680 where does the radiation come from, et cetera. 1133 01:21:48,680 --> 01:21:53,960 And so next time, we will try to explain, physically, 1134 01:21:53,960 --> 01:21:57,520 where does this temperature come from? 1135 01:21:57,520 --> 01:21:58,020 Yeah. 1136 01:22:00,620 --> 01:22:01,520 OK. 1137 01:22:01,520 --> 01:22:03,670 Let's stop here.