1 00:00:00,080 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,550 Your support will help MIT OpenCourseWare continue 4 00:00:06,550 --> 00:00:10,160 to offer high quality educational resources for free. 5 00:00:10,160 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,327 at ocw.mit.edu. 8 00:00:21,837 --> 00:00:22,420 PROFESSOR: OK. 9 00:00:22,420 --> 00:00:23,769 In that case let's get going. 10 00:00:23,769 --> 00:00:25,310 In today's lecture, we're going to be 11 00:00:25,310 --> 00:00:27,320 sort of splitting the lecture of things, 12 00:00:27,320 --> 00:00:29,910 if the timing goes as I plan. 13 00:00:29,910 --> 00:00:31,880 We're going to start by finishing 14 00:00:31,880 --> 00:00:34,320 talking about the geodesic equation. 15 00:00:34,320 --> 00:00:36,070 And then if all goes well, we will 16 00:00:36,070 --> 00:00:39,090 start talking about the energy of radiation-- 17 00:00:39,090 --> 00:00:42,420 completely changing topics altogether. 18 00:00:42,420 --> 00:00:45,890 I want to begin, as usual, by reviewing quickly 19 00:00:45,890 --> 00:00:49,960 what we talked about last time, just to remind us where we are. 20 00:00:49,960 --> 00:00:53,630 Last time, we at first, at the beginning of lecture, 21 00:00:53,630 --> 00:00:56,910 talked about how to add time into the Robertson-Walker 22 00:00:56,910 --> 00:00:57,770 Metric. 23 00:00:57,770 --> 00:01:02,990 And this is the formula that we claimed was the correct one. 24 00:01:02,990 --> 00:01:06,850 For a spacetime metric, ds squared, 25 00:01:06,850 --> 00:01:09,930 the meaning is closely analogous to the meaning 26 00:01:09,930 --> 00:01:12,120 that it would have in special relativity. 27 00:01:12,120 --> 00:01:15,320 The main difference being that in special relativity 28 00:01:15,320 --> 00:01:17,550 we always talk about what is observed 29 00:01:17,550 --> 00:01:22,460 by inertial frames of reference and inertial observers. 30 00:01:22,460 --> 00:01:26,130 In general relativity, the concept of an inertial observer 31 00:01:26,130 --> 00:01:28,480 is not so clear cut, but we can talk 32 00:01:28,480 --> 00:01:31,630 about observers for whom there is no forces acting on them 33 00:01:31,630 --> 00:01:34,580 other than possibly gravitational forces. 34 00:01:34,580 --> 00:01:36,690 And whether or not there are gravitational forces 35 00:01:36,690 --> 00:01:39,630 is always, itself, a framed dependent question. 36 00:01:39,630 --> 00:01:42,380 So it does not have a definite answer. 37 00:01:42,380 --> 00:01:45,040 So observers for which there is no forces acting on them 38 00:01:45,040 --> 00:01:47,500 other than gravitational forces are 39 00:01:47,500 --> 00:01:49,600 called free-falling observers. 40 00:01:49,600 --> 00:01:51,740 And they play the role of inertial observers 41 00:01:51,740 --> 00:01:56,890 that the inertial observers play in special relativity. 42 00:01:56,890 --> 00:01:58,980 So if ds squared is positive, it's 43 00:01:58,980 --> 00:02:01,740 the square of the spatial separation measured 44 00:02:01,740 --> 00:02:04,800 by a local free-falling observer, for whom the two 45 00:02:04,800 --> 00:02:07,350 events happen at the same time. 46 00:02:07,350 --> 00:02:08,850 Last time, I think, I did not really 47 00:02:08,850 --> 00:02:11,480 mention or emphasize the word local. 48 00:02:11,480 --> 00:02:15,140 But the point is that in general relativity 49 00:02:15,140 --> 00:02:17,235 we expect in any small region one 50 00:02:17,235 --> 00:02:20,190 can construct an accelerating coordinate system in which 51 00:02:20,190 --> 00:02:22,250 the effects of gravity are canceled out, 52 00:02:22,250 --> 00:02:24,690 as the equivalence principle tells us we can do. 53 00:02:24,690 --> 00:02:27,540 And then you essentially see the effects of special relativity. 54 00:02:27,540 --> 00:02:29,490 But it's only a small region, in principle, 55 00:02:29,490 --> 00:02:31,860 on an infinitesimal region. 56 00:02:31,860 --> 00:02:34,460 So these measurements that correspond 57 00:02:34,460 --> 00:02:36,360 to special relativity measurements 58 00:02:36,360 --> 00:02:39,510 are always made locally by an observer who 59 00:02:39,510 --> 00:02:42,210 is, in principle, arbitrarily close to the events being 60 00:02:42,210 --> 00:02:44,360 measured. 61 00:02:44,360 --> 00:02:47,080 If ds squared is negative, then it's equal to minus c 62 00:02:47,080 --> 00:02:50,160 squared times the square of the time separation 63 00:02:50,160 --> 00:02:53,330 that would be measured by a local free-falling observer 64 00:02:53,330 --> 00:02:56,440 for whom the two events happen at the same location. 65 00:02:56,440 --> 00:02:58,990 I should point out that a special case of this 66 00:02:58,990 --> 00:03:01,190 is an observer looking at his own wristwatch. 67 00:03:01,190 --> 00:03:04,040 His own wristwatch is always at the same location 68 00:03:04,040 --> 00:03:08,630 relevant to him, so it's a special case of this statement. 69 00:03:08,630 --> 00:03:12,050 So it says that ds squared is equal to minus c squared times 70 00:03:12,050 --> 00:03:16,430 the time that a free-falling observer would 71 00:03:16,430 --> 00:03:18,752 read on his own wrist watch. 72 00:03:18,752 --> 00:03:21,290 And If ds squared is 0, it means that the two events 73 00:03:21,290 --> 00:03:25,220 can be joined by a light pulse going from one to the other. 74 00:03:25,220 --> 00:03:27,470 Having said this, we can go back to this formula 75 00:03:27,470 --> 00:03:31,140 and understand why the formula is what it is. 76 00:03:31,140 --> 00:03:33,830 The spatial part is what it is because 77 00:03:33,830 --> 00:03:37,230 any homogeneous and isotropic spatial metric can 78 00:03:37,230 --> 00:03:39,010 be written in this form. 79 00:03:39,010 --> 00:03:41,950 And we are assuming that the universe we're describing 80 00:03:41,950 --> 00:03:44,620 is homogeneous and isotropic. 81 00:03:44,620 --> 00:03:49,260 The dc squared piece is really dictated by item two here. 82 00:03:49,260 --> 00:03:51,940 We want the t that we write in this metric 83 00:03:51,940 --> 00:03:55,510 to be the cosmic time variable that we've been speaking about. 84 00:03:55,510 --> 00:03:57,970 And that means that it is the time variable measured 85 00:03:57,970 --> 00:04:00,050 on the watches of observers who are 86 00:04:00,050 --> 00:04:02,420 at rest in this coordinate system. 87 00:04:02,420 --> 00:04:04,620 And that means that it has to be simply 88 00:04:04,620 --> 00:04:05,990 minus c squared dt squared. 89 00:04:05,990 --> 00:04:08,800 Or else dt would not have the right relationship 90 00:04:08,800 --> 00:04:11,760 to a ds squared to be consistent with what 91 00:04:11,760 --> 00:04:14,460 the s squared is supposed to be. 92 00:04:14,460 --> 00:04:16,230 And then we also talked about why 93 00:04:16,230 --> 00:04:24,510 there are no dt dr terms, or dt d theta, or dt d phi. 94 00:04:24,510 --> 00:04:28,350 We said that any such term would violate isotropy. 95 00:04:28,350 --> 00:04:31,160 If you had a dt dr term, for example, 96 00:04:31,160 --> 00:04:33,090 it would make the positive dr direction 97 00:04:33,090 --> 00:04:35,330 different from the negative dr direction. 98 00:04:35,330 --> 00:04:38,250 And that can't be something that happens 99 00:04:38,250 --> 00:04:39,530 in an isotropic universe. 100 00:04:42,290 --> 00:04:45,870 That then is our metric for cosmology, 101 00:04:45,870 --> 00:04:48,654 the Roberrtson-Walker Metric. 102 00:04:48,654 --> 00:04:50,945 And another important thing is what is it good for, now 103 00:04:50,945 --> 00:04:53,070 that we decided that's the right metric? 104 00:04:53,070 --> 00:04:55,150 What use is to us? 105 00:04:55,150 --> 00:04:57,210 And what we haven't done yet, but it's actually 106 00:04:57,210 --> 00:05:00,120 on the homework, we need the full spacetime metric 107 00:05:00,120 --> 00:05:02,180 to be able to find geodesics, to be 108 00:05:02,180 --> 00:05:04,530 able to learn the paths of particles 109 00:05:04,530 --> 00:05:08,570 moving through this model universe. 110 00:05:08,570 --> 00:05:12,200 So we will be making important use of this Roberrtson-Walker 111 00:05:12,200 --> 00:05:15,725 Metric with its spacetime contributions. 112 00:05:18,011 --> 00:05:18,510 OK. 113 00:05:18,510 --> 00:05:19,530 Any questions about that? 114 00:05:19,530 --> 00:05:21,405 Now I'm ready to change gears to some extent. 115 00:05:21,405 --> 00:05:23,280 Yes, Ani? 116 00:05:23,280 --> 00:05:26,140 AUDIANCE: So in general, the spatial part of the metric, 117 00:05:26,140 --> 00:05:27,850 we can get from the geometry? 118 00:05:27,850 --> 00:05:29,790 And in general, can you just add a minus 119 00:05:29,790 --> 00:05:32,464 c squared dt square for the temporal part? 120 00:05:32,464 --> 00:05:33,880 PROFESSOR: It's not quite general. 121 00:05:33,880 --> 00:05:37,020 Remember we used an argument based on isotropy here. 122 00:05:39,602 --> 00:05:42,900 So I think it's safe to say that any metric you'll 123 00:05:42,900 --> 00:05:46,104 find in this class is likely to have 124 00:05:46,104 --> 00:05:48,520 the time entering, and nothing more complicated than minus 125 00:05:48,520 --> 00:05:50,110 c squared dt squared. 126 00:05:50,110 --> 00:05:52,820 But it's not a general statement about general relativity. 127 00:05:57,042 --> 00:05:57,875 Any other questions? 128 00:06:02,089 --> 00:06:02,589 Yes. 129 00:06:02,589 --> 00:06:11,970 AUDIANCE: [INAUDIBLE], 130 00:06:11,970 --> 00:06:14,520 PROFESSOR: OK, the question is, what would be a circumstance 131 00:06:14,520 --> 00:06:17,470 where we would have to deal with something more complicated? 132 00:06:17,470 --> 00:06:19,000 The answer would be, I think, all 133 00:06:19,000 --> 00:06:26,220 you need is to add to this model universe perturbations that 134 00:06:26,220 --> 00:06:27,910 break the uniformity. 135 00:06:27,910 --> 00:06:30,080 If we tried to describe the real universe instead 136 00:06:30,080 --> 00:06:33,350 of this ideal universe, where our ideal universe is perfectly 137 00:06:33,350 --> 00:06:35,580 isotropic and homogeneous, if it said 138 00:06:35,580 --> 00:06:38,910 we wanted to describe the lumps and bumps of the real universe, 139 00:06:38,910 --> 00:06:40,710 then it would become more complicated. 140 00:06:40,710 --> 00:06:42,930 And we would probably need a dt, dr term. 141 00:06:51,110 --> 00:06:55,090 OK, next we went on to talking about the geodesic equation. 142 00:06:55,090 --> 00:06:59,290 According to General Relativity, the trajectories of particles 143 00:06:59,290 --> 00:07:02,220 that have no forces acting on them other than gravity, 144 00:07:02,220 --> 00:07:05,870 these free-falling observers, are geodesics 145 00:07:05,870 --> 00:07:08,250 in the spacetime.. 146 00:07:08,250 --> 00:07:11,260 So that means we want to learn how to calculate geodesics, 147 00:07:11,260 --> 00:07:14,280 which means paths whose length is stationary 148 00:07:14,280 --> 00:07:17,160 under small variations. 149 00:07:17,160 --> 00:07:19,370 So we considered first just simple geodesics 150 00:07:19,370 --> 00:07:22,380 in the spatial metric, because that's easier to think about. 151 00:07:22,380 --> 00:07:24,780 What is the shortest distance between two points 152 00:07:24,780 --> 00:07:28,500 in a space that's described by some arbitrary metric? 153 00:07:28,500 --> 00:07:30,780 So first we talked about how we describe the metric. 154 00:07:30,780 --> 00:07:34,940 And we introduced two features in this first formula here. 155 00:07:34,940 --> 00:07:37,520 One is that instead of calling the coordinates XYZ 156 00:07:37,520 --> 00:07:38,750 or something like that. 157 00:07:38,750 --> 00:07:42,190 We called them x1, x2, x3, so that we could talk about them 158 00:07:42,190 --> 00:07:45,890 all together in one formula without writing separate pieces 159 00:07:45,890 --> 00:07:48,640 for the different coordinates. 160 00:07:48,640 --> 00:07:51,570 So i and j represent 1, 2, and 3, or just 1, 161 00:07:51,570 --> 00:07:56,760 and 2, which is the labeling of the spatial coordinates. 162 00:07:56,760 --> 00:07:59,830 And the other important piece of notation 163 00:07:59,830 --> 00:08:01,820 that is introduced in that formula 164 00:08:01,820 --> 00:08:04,930 is the Einstein summation convention. 165 00:08:04,930 --> 00:08:08,680 Whenever there's an index, like i and j here, 166 00:08:08,680 --> 00:08:11,340 which are repeated with one index lower 167 00:08:11,340 --> 00:08:13,590 and one index upper, they're automatically 168 00:08:13,590 --> 00:08:17,670 summed over all of the values that the coordinates take, 169 00:08:17,670 --> 00:08:20,565 without writing summation sign. 170 00:08:20,565 --> 00:08:22,230 It saves a lot of writing. 171 00:08:22,230 --> 00:08:25,930 And it turns out that one always sums under those circumstances, 172 00:08:25,930 --> 00:08:30,740 so there's no need to write the sums with the summation sign. 173 00:08:30,740 --> 00:08:34,349 Next we want to ask ourselves, how 174 00:08:34,349 --> 00:08:36,470 are we going to describe the path? 175 00:08:36,470 --> 00:08:37,970 Before we can find the minimum path, 176 00:08:37,970 --> 00:08:41,250 we need at least a language to talk about paths. 177 00:08:41,250 --> 00:08:43,600 And we could describe a path going from some point A 178 00:08:43,600 --> 00:08:48,960 to some point B, by giving a function x supra i of lambda. 179 00:08:48,960 --> 00:08:50,890 Well, lambda is an arbitrary parameter, 180 00:08:50,890 --> 00:08:53,700 that parametrizes the path. 181 00:08:53,700 --> 00:08:55,910 x supra i are a set of coordinates. 182 00:08:55,910 --> 00:08:58,540 i runs over the values of all the coordinates 183 00:08:58,540 --> 00:09:01,430 of whatever system you're dealing with. 184 00:09:01,430 --> 00:09:05,290 And you construct such a function where xi of 0 185 00:09:05,290 --> 00:09:09,330 is the starting point, which are the coordinates of the point A. 186 00:09:09,330 --> 00:09:14,930 And xi of some value lambda f, where f just stands for final, 187 00:09:14,930 --> 00:09:16,406 will be the end of the path. 188 00:09:16,406 --> 00:09:17,780 And it's supposed to end at point 189 00:09:17,780 --> 00:09:20,520 B. So the final coordinates of the path 190 00:09:20,520 --> 00:09:28,170 should be x supra i sub b, the coordinates of the point B. 191 00:09:28,170 --> 00:09:31,310 Then we want to use this description of the path 192 00:09:31,310 --> 00:09:33,850 to figure out what the length is of a segment of the path. 193 00:09:33,850 --> 00:09:36,820 And then the full length will be the sum of the segments. 194 00:09:36,820 --> 00:09:40,060 So for each segment, we just apply the metric 195 00:09:40,060 --> 00:09:42,890 to the change in coordinates. 196 00:09:42,890 --> 00:09:45,680 The change in coordinates, as lambda is varied, 197 00:09:45,680 --> 00:09:49,690 is just the derivative of xi with respect to lambda 198 00:09:49,690 --> 00:09:52,740 times the change in lambda. 199 00:09:52,740 --> 00:09:57,810 And putting that in for both dxi and dxj 200 00:09:57,810 --> 00:10:00,870 one gets this formula, relating ds 201 00:10:00,870 --> 00:10:02,800 squared, the square of the length 202 00:10:02,800 --> 00:10:06,370 of an infinitesimal segment to d lambda squared, 203 00:10:06,370 --> 00:10:09,338 the square of the parameter that describes that length. 204 00:10:12,090 --> 00:10:15,090 Then the full length is gotten by, first of all, taking 205 00:10:15,090 --> 00:10:16,720 the square root of this equation to get 206 00:10:16,720 --> 00:10:19,580 the infinitesimal length, ds. 207 00:10:19,580 --> 00:10:22,840 And then taking the integral of that over the path 208 00:10:22,840 --> 00:10:24,400 from beginning to end. 209 00:10:24,400 --> 00:10:27,090 And that, then, gives us the full length of the path, 210 00:10:27,090 --> 00:10:29,760 thinking of it as the sum of the length of each 211 00:10:29,760 --> 00:10:32,220 of infinitesimal segment. 212 00:10:32,220 --> 00:10:33,080 OK? 213 00:10:33,080 --> 00:10:36,180 Fair enough? 214 00:10:36,180 --> 00:10:38,390 Now that we have this formula for the length, now 215 00:10:38,390 --> 00:10:40,250 we have the next challenge, which 216 00:10:40,250 --> 00:10:43,750 is to figure out how to calculate the path which 217 00:10:43,750 --> 00:10:45,950 minimizes that length. 218 00:10:45,950 --> 00:10:47,660 And I didn't use the word last time, 219 00:10:47,660 --> 00:10:50,310 but that what is called the calculus of variations. 220 00:10:50,310 --> 00:10:53,680 And I looked up a little bit of the history in the Wikipedia. 221 00:10:53,680 --> 00:10:57,190 The calculus of variations dates back to 1696, 222 00:10:57,190 --> 00:10:59,720 when Johann Bernoulli invented it, 223 00:10:59,720 --> 00:11:03,840 applied it to the brachistochrone problem, 224 00:11:03,840 --> 00:11:08,120 which is the problem of finding a path for which a frictionless 225 00:11:08,120 --> 00:11:11,290 object will slide and get to its destination in the least 226 00:11:11,290 --> 00:11:12,760 possible time. 227 00:11:12,760 --> 00:11:14,460 And it turns out to be a cycloid, 228 00:11:14,460 --> 00:11:17,590 just like the cycloid that describes our closed universes, 229 00:11:17,590 --> 00:11:21,160 closed matter dominating the universe. 230 00:11:21,160 --> 00:11:25,390 And the problem was also solved by-- Johann Bernoulli 231 00:11:25,390 --> 00:11:28,060 then announced this problem to the world 232 00:11:28,060 --> 00:11:31,200 and challenged other mathematicians to solve it. 233 00:11:31,200 --> 00:11:37,400 There's a famous story that Newton noticed this question 234 00:11:37,400 --> 00:11:39,395 in his mail when he got home at 4:00 AM, 235 00:11:39,395 --> 00:11:41,770 or something like that, from the mint-- he was apparently 236 00:11:41,770 --> 00:11:45,899 a hardworking guy-- but nonetheless when 237 00:11:45,899 --> 00:11:47,690 he seen this problem he couldn't go to bed. 238 00:11:47,690 --> 00:11:51,780 He went ahead and solved it by morning, 239 00:11:51,780 --> 00:11:55,340 which is a good MIT student kind of thing to do. 240 00:11:58,710 --> 00:12:03,400 So the technique is to consider a small variation from whatever 241 00:12:03,400 --> 00:12:06,460 path you're hoping to be the minimum. 242 00:12:06,460 --> 00:12:08,670 And we're going to calculate the first order 243 00:12:08,670 --> 00:12:11,340 change in the length of the path, 244 00:12:11,340 --> 00:12:14,170 starting from our original path, x of lambda, 245 00:12:14,170 --> 00:12:18,230 to some new path, x tilde of lambda. 246 00:12:18,230 --> 00:12:23,200 And we parametrize the new path by writing it 247 00:12:23,200 --> 00:12:26,360 as the old path, plus a correction. 248 00:12:26,360 --> 00:12:28,280 And I've introduced a factor, alpha, 249 00:12:28,280 --> 00:12:30,530 multiplying the correction, because it makes it easier 250 00:12:30,530 --> 00:12:32,910 to talk about derivatives. 251 00:12:32,910 --> 00:12:36,950 And wi of lambda is just some arbitrary deviation 252 00:12:36,950 --> 00:12:39,577 from the original path. 253 00:12:39,577 --> 00:12:41,910 But we want to always go through the same starting point 254 00:12:41,910 --> 00:12:43,701 to the same endpoint, because there's never 255 00:12:43,701 --> 00:12:46,520 going to be a minimum if we're allowed to move the endpoints. 256 00:12:46,520 --> 00:12:48,010 So the endpoints are fixed. 257 00:12:48,010 --> 00:12:50,230 And that means that this path deviation, 258 00:12:50,230 --> 00:12:54,640 w super i in my notation, has to vanish at the two endpoints. 259 00:12:54,640 --> 00:13:00,750 So we impose these two equations on the variation wi. 260 00:13:00,750 --> 00:13:04,930 Then what do I do is take the derivative of the path 261 00:13:04,930 --> 00:13:09,730 length of the varied path, x tilde with respect to alpha, 262 00:13:09,730 --> 00:13:13,090 and if we had a minimum length to start with, 263 00:13:13,090 --> 00:13:15,560 the derivative should always vanish. 264 00:13:15,560 --> 00:13:17,250 That is, the minimum should always 265 00:13:17,250 --> 00:13:20,110 occur when alpha equals 0, if the original path 266 00:13:20,110 --> 00:13:23,240 of the true path, the true minimum path. 267 00:13:23,240 --> 00:13:26,150 And if alpha equals 0 is the minimum, 268 00:13:26,150 --> 00:13:29,270 the derivative should always vanish at alpha equals 0. 269 00:13:29,270 --> 00:13:30,370 And vice versa. 270 00:13:30,370 --> 00:13:34,270 If we know that this happens for every variation wi, 271 00:13:34,270 --> 00:13:36,940 then we know that our path is at least an extremum, 272 00:13:36,940 --> 00:13:40,590 and, presumably, a minimum. 273 00:13:40,590 --> 00:13:43,800 And the path itself is just written by the same formulas 274 00:13:43,800 --> 00:13:49,070 we had before, except for x tilde instead of x itself. 275 00:13:49,070 --> 00:13:52,800 And I've introduced an axillary quantity, a of lambda alpha, 276 00:13:52,800 --> 00:13:55,404 which is just what appears inside the square root. 277 00:13:55,404 --> 00:13:57,070 That just saves some writing, because it 278 00:13:57,070 --> 00:13:58,540 has to be written a number of times 279 00:13:58,540 --> 00:13:59,998 in the course of the manipulations. 280 00:14:02,970 --> 00:14:06,100 So our goal now is to carry out this derivative. 281 00:14:06,100 --> 00:14:08,430 And the derivative acts only on the integrand, 282 00:14:08,430 --> 00:14:09,920 because the limits of integration 283 00:14:09,920 --> 00:14:11,980 do not depend on alpha. 284 00:14:11,980 --> 00:14:15,410 So just carry the derivative into the integrand 285 00:14:15,410 --> 00:14:17,350 and differentiate this square root 286 00:14:17,350 --> 00:14:20,340 of a of lambda, which is, itself, a product of factors 287 00:14:20,340 --> 00:14:22,360 that we have to use-- product rule and chain 288 00:14:22,360 --> 00:14:25,030 rule and various manipulations. 289 00:14:25,030 --> 00:14:28,040 And after we carry out those manipulations, 290 00:14:28,040 --> 00:14:31,680 we end up with this expression in a straightforward 291 00:14:31,680 --> 00:14:36,980 way involving a few steps, which I won't show again. 292 00:14:36,980 --> 00:14:39,910 And the complication is that what we want to do 293 00:14:39,910 --> 00:14:43,760 is to figure out for what paths that expression 294 00:14:43,760 --> 00:14:45,760 will vanish for all wi. 295 00:14:45,760 --> 00:14:48,580 We want it to vanish for all possible variations 296 00:14:48,580 --> 00:14:50,000 of the path. 297 00:14:50,000 --> 00:14:55,190 And what's complicated is that wi appears here 298 00:14:55,190 --> 00:14:58,490 as a multiplicative factor in the first term, 299 00:14:58,490 --> 00:15:03,010 but as a differentiated factor in the second term. 300 00:15:03,010 --> 00:15:05,860 And that makes it very hard to know, initially, 301 00:15:05,860 --> 00:15:08,400 when those two terms might cancel each other to give you 302 00:15:08,400 --> 00:15:11,220 0, which is what we're looking for. 303 00:15:11,220 --> 00:15:13,020 But the brilliant trick that, I guess, 304 00:15:13,020 --> 00:15:16,780 Newton invented, along with Bernoulli and others, 305 00:15:16,780 --> 00:15:18,807 is to integrate by parts. 306 00:15:18,807 --> 00:15:21,390 Integration by parts, I'm sure, was not a well-known procedure 307 00:15:21,390 --> 00:15:23,480 at that time. 308 00:15:23,480 --> 00:15:25,910 But if we integrate the second term by parts, 309 00:15:25,910 --> 00:15:28,760 we could remove the derivative acting on w, 310 00:15:28,760 --> 00:15:31,420 and arrange for w to be a multiplicative factor 311 00:15:31,420 --> 00:15:33,540 in both terms. 312 00:15:33,540 --> 00:15:37,570 And a crucial thing that makes the whole thing useful 313 00:15:37,570 --> 00:15:40,080 is that when you do integrate by parts, 314 00:15:40,080 --> 00:15:43,650 you discover that you don't get any endpoint contributions, 315 00:15:43,650 --> 00:15:45,510 because the endpoint contributions would 316 00:15:45,510 --> 00:15:48,250 be proportional to wi at the endpoints. 317 00:15:48,250 --> 00:15:50,290 And remember, wi has to vanish at the endpoints, 318 00:15:50,290 --> 00:15:52,780 because that's the condition that we're not 319 00:15:52,780 --> 00:15:55,350 changing the points A and B. We're always 320 00:15:55,350 --> 00:15:58,250 talking about paths that have the same starting point 321 00:15:58,250 --> 00:16:01,430 and the same ending point. 322 00:16:01,430 --> 00:16:09,460 So integrating by parts, we get this expression, 323 00:16:09,460 --> 00:16:12,940 where now wi multiplies everything, as just 324 00:16:12,940 --> 00:16:14,385 simply a multiplicative factor. 325 00:16:14,385 --> 00:16:15,760 To write it in this form, you had 326 00:16:15,760 --> 00:16:17,690 to do a little bit of juggling of indices. 327 00:16:17,690 --> 00:16:20,240 The other important trick in these manipulations 328 00:16:20,240 --> 00:16:23,530 is to juggle indices, which I'll not show you explicitly. 329 00:16:23,530 --> 00:16:25,850 But the thing to remember is that these indices that 330 00:16:25,850 --> 00:16:28,190 are being summed over can be called anything 331 00:16:28,190 --> 00:16:31,010 and it's still the same sum. 332 00:16:31,010 --> 00:16:33,660 So when you want to get terms to cancel each other, 333 00:16:33,660 --> 00:16:35,820 you may have to change the names of indices 334 00:16:35,820 --> 00:16:38,145 to get them to just cancel identically. 335 00:16:38,145 --> 00:16:41,370 But that's straightforward. 336 00:16:41,370 --> 00:16:43,822 So we get this expression. 337 00:16:43,822 --> 00:16:45,530 And now we want this expression to vanish 338 00:16:45,530 --> 00:16:48,550 for every possible wi of lambda. 339 00:16:48,550 --> 00:16:50,140 And we argued that the only way it 340 00:16:50,140 --> 00:16:53,290 could vanish for every possible wi of lambda 341 00:16:53,290 --> 00:16:57,180 is if the expression in curly brackets, itself, vanishes. 342 00:16:57,180 --> 00:16:59,867 Yeah, if we only know the values for some particular wi 343 00:16:59,867 --> 00:17:01,450 of lambda, then there are lots of ways 344 00:17:01,450 --> 00:17:02,824 it could vanish, because it could 345 00:17:02,824 --> 00:17:05,609 be positive in some places and negative in others. 346 00:17:05,609 --> 00:17:08,069 But the only way it could vanish for all wi 347 00:17:08,069 --> 00:17:10,990 is for the quantity in curly brackets to vanish. 348 00:17:10,990 --> 00:17:12,940 So that gives us our final, or at least, 349 00:17:12,940 --> 00:17:15,445 almost final expression of the geodesic equation. 350 00:17:15,445 --> 00:17:21,010 And that's where we left off last time, with that equation. 351 00:17:21,010 --> 00:17:23,650 So note that this is just an equation that 352 00:17:23,650 --> 00:17:27,269 would either be obeyed or not obeyed by the function x super 353 00:17:27,269 --> 00:17:28,780 i of lambda. 354 00:17:28,780 --> 00:17:30,330 It's just a differential equation 355 00:17:30,330 --> 00:17:33,270 involving x super i of lambda and the metric, which 356 00:17:33,270 --> 00:17:35,890 we assume is given. 357 00:17:35,890 --> 00:17:36,390 OK. 358 00:17:36,390 --> 00:17:37,973 So are there any questions about that? 359 00:17:41,600 --> 00:17:43,670 Everybody happy? 360 00:17:43,670 --> 00:17:44,542 Great. 361 00:17:44,542 --> 00:17:46,375 OK, now we'll continue on on the blackboard. 362 00:17:57,722 --> 00:17:59,180 OK, the first thing I want to do is 363 00:17:59,180 --> 00:18:03,020 to simplify the equation a bit. 364 00:18:03,020 --> 00:18:05,120 This equation is fairly complicated, 365 00:18:05,120 --> 00:18:08,467 because of those square roots of A's in the denominators. 366 00:18:08,467 --> 00:18:10,550 The square root of A is a pretty complicated thing 367 00:18:10,550 --> 00:18:12,332 to start with, and the square root of A 368 00:18:12,332 --> 00:18:14,040 here is even differentiated, because it's 369 00:18:14,040 --> 00:18:16,460 got the lambda making an incredible mess, 370 00:18:16,460 --> 00:18:18,610 if you understand all that. 371 00:18:18,610 --> 00:18:20,430 So it would be nice to simplify that. 372 00:18:20,430 --> 00:18:23,440 And we do have one trick which we can still do, 373 00:18:23,440 --> 00:18:25,420 which we haven't done yet. 374 00:18:25,420 --> 00:18:29,800 We originally constructed our path, xi of lambda, 375 00:18:29,800 --> 00:18:33,300 as a function of some arbitrary parameter, lambda. 376 00:18:33,300 --> 00:18:37,630 Lambda just measures arbitrary points along the path. 377 00:18:37,630 --> 00:18:39,460 There are many, many ways to do that, 378 00:18:39,460 --> 00:18:42,169 an infinite number of ways that you can do that. 379 00:18:42,169 --> 00:18:43,960 And this formula will work for all of them, 380 00:18:43,960 --> 00:18:45,250 it's completely general. 381 00:18:45,250 --> 00:18:47,120 The formula, when we derived it, we 382 00:18:47,120 --> 00:18:52,170 didn't make any assumptions about how lambda was chosen. 383 00:18:52,170 --> 00:18:56,290 But we can simplify the formula by making a particular choice 384 00:18:56,290 --> 00:18:57,390 for lambda. 385 00:18:57,390 --> 00:18:59,410 And the choice that simplifies things 386 00:18:59,410 --> 00:19:02,780 is to choose lambda to be the arc length itself. 387 00:19:02,780 --> 00:19:05,160 Lambda should be the distance along the path. 388 00:19:05,160 --> 00:19:07,460 And then we're trying to express xi 389 00:19:07,460 --> 00:19:11,700 as a function of how far you've already gone. 390 00:19:11,700 --> 00:19:22,290 And that has the effect, if we go back to what Ai was, 391 00:19:22,290 --> 00:19:31,150 A of lambda really is just the path length per lambda. 392 00:19:31,150 --> 00:19:36,370 So if lambda is the path length itself, A is just equal to 1. 393 00:19:40,528 --> 00:19:43,080 I'm trying to get a formula that shows that more clearly. 394 00:19:43,080 --> 00:19:43,680 Here. 395 00:19:43,680 --> 00:19:45,880 If we remember that this quantity is A, 396 00:19:45,880 --> 00:19:48,670 this tells us that ds squared is equal to A times d 397 00:19:48,670 --> 00:19:49,780 lambda squared. 398 00:19:49,780 --> 00:19:52,600 So if ds is the same as d lambda, 399 00:19:52,600 --> 00:19:55,530 as you've chosen your parameter to be the path length, 400 00:19:55,530 --> 00:19:57,840 this formula makes it clear that that's 401 00:19:57,840 --> 00:20:01,510 equivalent to A equal to 1. 402 00:20:01,510 --> 00:20:04,780 So going back to the formula, if A is 1, 403 00:20:04,780 --> 00:20:08,121 we would just drop it from both sides of the equation. 404 00:20:08,121 --> 00:20:10,620 And all that really matters, I should point out here, maybe, 405 00:20:10,620 --> 00:20:13,480 because we'll be using it later, is that A is a constant. 406 00:20:13,480 --> 00:20:16,700 As long as A is a constant, it will not be differentiated, 407 00:20:16,700 --> 00:20:19,350 and then it will cancel on the left side and the right side. 408 00:20:19,350 --> 00:20:20,780 So we don't necessarily care that it is 1, 409 00:20:20,780 --> 00:20:22,279 but we do care that it's a constant. 410 00:20:22,279 --> 00:20:24,590 And then it just disappears from the formula. 411 00:20:24,590 --> 00:20:27,350 And then we get the simpler formula. 412 00:20:27,350 --> 00:20:33,040 And now we'll continue on the blackboard. 413 00:20:33,040 --> 00:20:51,900 The simpler formula is just dds of gij dxj ds 414 00:20:51,900 --> 00:21:01,530 is equal to 1/2 times the derivative of gjk, with respect 415 00:21:01,530 --> 00:21:15,020 to xi, times dxj ds dxk ds, where 416 00:21:15,020 --> 00:21:16,870 s is equal to the path length. 417 00:21:22,840 --> 00:21:24,750 So I've replaced lambda by s, because we 418 00:21:24,750 --> 00:21:26,095 set lambda equal to s. 419 00:21:26,095 --> 00:21:29,290 And s has a more specific meaning than lambda did. 420 00:21:29,290 --> 00:21:31,740 Lambda was a completely arbitrary parametrization 421 00:21:31,740 --> 00:21:32,320 of the path. 422 00:21:35,940 --> 00:21:40,680 So this one deserves a big box, because it really 423 00:21:40,680 --> 00:21:43,072 is the final formula for geodesics. 424 00:21:43,072 --> 00:21:45,030 Once we write it in terms of different letters, 425 00:21:45,030 --> 00:21:49,260 we will later, but this actually is the formula. 426 00:21:49,260 --> 00:21:51,770 Now I should mention just largely 427 00:21:51,770 --> 00:21:55,360 for the sake of your knowing what's going on, 428 00:21:55,360 --> 00:21:58,580 if you ever look at some other general relativity books, 429 00:21:58,580 --> 00:22:01,630 this is not the formula that the geodesic equation is usually 430 00:22:01,630 --> 00:22:03,070 written in. 431 00:22:03,070 --> 00:22:04,360 Frankly, it is the best form. 432 00:22:04,360 --> 00:22:06,130 If you want to find the geodesic, 433 00:22:06,130 --> 00:22:09,560 usually this form of writing the equation is the easiest. 434 00:22:09,560 --> 00:22:12,030 But most general relativity books 435 00:22:12,030 --> 00:22:14,160 prefer instead to just give a formula 436 00:22:14,160 --> 00:22:16,560 for the second derivative, here. 437 00:22:16,560 --> 00:22:18,820 Which involves just expanding this term, 438 00:22:18,820 --> 00:22:21,390 and then when we shuffle things, to try 439 00:22:21,390 --> 00:22:24,410 to simplify the expressions. 440 00:22:24,410 --> 00:22:40,110 So one can write, to start, d ds of gij dxj ds. 441 00:22:40,110 --> 00:22:41,760 We're just going to expand it. 442 00:22:41,760 --> 00:22:43,680 Now we're going to be making use of all the rules of calculus 443 00:22:43,680 --> 00:22:44,350 that we've learn. 444 00:22:44,350 --> 00:22:46,183 Every rule you've ever learned will probably 445 00:22:46,183 --> 00:22:47,980 get used in this calculation. 446 00:22:47,980 --> 00:22:51,140 So it will be using product rule, 447 00:22:51,140 --> 00:22:53,760 of course, because we have a product of two things here. 448 00:22:53,760 --> 00:22:55,850 But we also have the complication 449 00:22:55,850 --> 00:22:59,530 that gij is not explicitly a function of s. 450 00:22:59,530 --> 00:23:03,270 But gij is a function of position. 451 00:23:03,270 --> 00:23:07,040 And the position that one is that for any given value of s 452 00:23:07,040 --> 00:23:10,240 depends on s, because we're moving along the path, 453 00:23:10,240 --> 00:23:12,350 x super i of s. 454 00:23:12,350 --> 00:23:19,970 So the gij here, is evaluated at x super i of s. 455 00:23:22,510 --> 00:23:24,240 I should give this a new letter. 456 00:23:24,240 --> 00:23:27,730 x super k of s. 457 00:23:27,730 --> 00:23:32,750 So it depends on s, through the argument of its argument. 458 00:23:32,750 --> 00:23:36,280 So that's a chain rule situation. 459 00:23:36,280 --> 00:23:41,560 And what we get here is, from just differentiating 460 00:23:41,560 --> 00:23:43,820 the second factor, that's easy. 461 00:23:43,820 --> 00:23:54,320 We get gij DC d squared xj ds squared. 462 00:23:54,320 --> 00:23:58,380 And then, from the derivative of the derivative chain rule 463 00:23:58,380 --> 00:24:11,600 piece, we get the partial of gij, with respect to xk times 464 00:24:11,600 --> 00:24:19,030 the dxj ds times dxk ds. 465 00:24:26,100 --> 00:24:29,840 And then to continue, this piece gets brought over 466 00:24:29,840 --> 00:24:32,500 to the other side, because we're trying to get an equation just 467 00:24:32,500 --> 00:24:34,145 for the second derivative of the path. 468 00:24:54,420 --> 00:25:04,690 So then we get g sub ij d squared x super j ds squared is 469 00:25:04,690 --> 00:25:14,470 equal to 1/2 di-- I'll define that in a second-- g sub jk 470 00:25:14,470 --> 00:25:29,120 minus 2 dk gij dxi ds dxj ds. 471 00:25:34,910 --> 00:25:38,970 where this partial derivative with the subscript 472 00:25:38,970 --> 00:25:43,270 is just an abbreviation for the derivative with respect 473 00:25:43,270 --> 00:25:44,910 to the coordinate with that index. 474 00:25:47,910 --> 00:25:49,213 So that's just an abbreviation. 475 00:25:54,690 --> 00:25:59,890 Now you could think of this as a matrix times a vector 476 00:25:59,890 --> 00:26:01,820 is equal to an expression. 477 00:26:01,820 --> 00:26:05,230 What we like to do is just get an expression for this vector. 478 00:26:05,230 --> 00:26:08,210 So if we think of it as a matrix times a vector, 479 00:26:08,210 --> 00:26:10,080 all we have to do is invert the matrix 480 00:26:10,080 --> 00:26:12,671 to be able to get an expression for the vector itself. 481 00:26:12,671 --> 00:26:13,170 Yes! 482 00:26:13,170 --> 00:26:14,878 AUDIANCE: Should that closing parenthesis 483 00:26:14,878 --> 00:26:18,840 be more [INAUDIBLE]? 484 00:26:18,840 --> 00:26:20,799 PROFESSOR: Oh, Yeah, I think you're right, 485 00:26:20,799 --> 00:26:21,715 it doesn't look right. 486 00:26:26,770 --> 00:26:27,750 Yeah. 487 00:26:27,750 --> 00:26:37,960 Thank you This has to multiply everything. 488 00:26:41,820 --> 00:26:44,600 Oops! 489 00:26:44,600 --> 00:26:47,295 OK, OK. 490 00:26:47,295 --> 00:26:48,920 Given enough chances I'll get it right. 491 00:26:53,205 --> 00:26:58,710 OK, now everybody happy this time? 492 00:26:58,710 --> 00:27:03,100 Thank you very much for getting it straight. 493 00:27:03,100 --> 00:27:06,280 OK, So as I was saying, we want to isolate 494 00:27:06,280 --> 00:27:07,450 this second derivative. 495 00:27:07,450 --> 00:27:09,033 We're hoping to get just an expression 496 00:27:09,033 --> 00:27:10,460 for the second derivative. 497 00:27:10,460 --> 00:27:14,482 And this can be interpreted as a matrix times a vector 498 00:27:14,482 --> 00:27:15,190 equals something. 499 00:27:15,190 --> 00:27:17,090 We want to just invert that matrix. 500 00:27:17,090 --> 00:27:18,392 Yes? 501 00:27:18,392 --> 00:27:21,679 AUDIANCE: Isn't the ds and [? the idx ?] [INAUDIBLE]? 502 00:27:21,679 --> 00:27:23,720 PROFESSOR: Oh, do I have that wrong too, perhaps? 503 00:27:32,360 --> 00:27:35,630 I think we want j and k there, that don't we? 504 00:27:35,630 --> 00:27:42,370 OK, attempt number four, or did I lose count as well? 505 00:27:42,370 --> 00:27:45,770 j and k are the indices and the i 506 00:27:45,770 --> 00:27:48,520 matches the free i on the left. 507 00:27:48,520 --> 00:27:51,350 And all the other indices are sound. 508 00:27:51,350 --> 00:27:55,470 I think, probably, I finally achieved the right formula. 509 00:27:59,460 --> 00:28:00,460 Thanks for all the help. 510 00:28:04,220 --> 00:28:08,223 So inverting a matrix, the principal 511 00:28:08,223 --> 00:28:10,890 is a straightforward mathematical operation. 512 00:28:10,890 --> 00:28:14,690 In general relativity, we give a name to the inverse metric, 513 00:28:14,690 --> 00:28:17,890 and it's the same letter g with indices, 514 00:28:17,890 --> 00:28:20,140 with superscripts instead of subscripts. 515 00:28:20,140 --> 00:28:23,410 And that's defined to be the matrix inverse. 516 00:28:23,410 --> 00:28:38,440 So g super ij is defined to be the matrix inverse of g sub ij. 517 00:28:41,020 --> 00:28:44,790 And to put that into an equation, 518 00:28:44,790 --> 00:28:48,870 we could say that if we take g with upper indices-- 519 00:28:48,870 --> 00:28:52,720 and I'll write those upper indices as i and l-- 520 00:28:52,720 --> 00:29:00,100 and multiply it by a g with lower indices l and j, 521 00:29:00,100 --> 00:29:04,680 when you sum over adjacent indices in this index 522 00:29:04,680 --> 00:29:07,060 notation, that's exactly what corresponds 523 00:29:07,060 --> 00:29:10,660 to the definition of matrix multiplication. 524 00:29:10,660 --> 00:29:13,980 So this is just the matrix g with upper indices times 525 00:29:13,980 --> 00:29:18,260 the matrix g with lower indices, and it's the i j'th element 526 00:29:18,260 --> 00:29:19,820 of that matrix. 527 00:29:19,820 --> 00:29:22,290 And we're saying it should be the identity matrix, 528 00:29:22,290 --> 00:29:26,620 and that means that the i j'th element should be 0 if it's off 529 00:29:26,620 --> 00:29:31,180 diagonal, and 1 if it's diagonal, if i equals j. 530 00:29:31,180 --> 00:29:32,650 And that's exactly the definition 531 00:29:32,650 --> 00:29:34,680 of a chronic or a delta. 532 00:29:34,680 --> 00:29:39,060 So this is equal to delta ij. 533 00:29:39,060 --> 00:29:43,670 We remember that delta ij is 0 if i is not equal to j, 1 534 00:29:43,670 --> 00:29:44,860 if i is equal to j. 535 00:29:44,860 --> 00:29:46,280 That's the definition. 536 00:29:46,280 --> 00:29:48,120 And it corresponds to that identity matrix 537 00:29:48,120 --> 00:29:48,925 in matrix language. 538 00:29:52,100 --> 00:29:57,680 So this is the relationship that actually defines g super il. 539 00:29:57,680 --> 00:30:01,790 And it is just the statement that g with upper indices 540 00:30:01,790 --> 00:30:04,130 is the matrix inverse of g with lower indices. 541 00:30:06,940 --> 00:30:11,560 Using this, we can bring this g to the other side 542 00:30:11,560 --> 00:30:14,940 essentially by multiplying by g inverse. 543 00:30:14,940 --> 00:30:19,660 And I will save a little time by not writing that out 544 00:30:19,660 --> 00:30:25,700 in gory detail, but rather I'll just write the result. 545 00:30:25,700 --> 00:30:28,110 And the result is written in terms of a new symbol that 546 00:30:28,110 --> 00:30:29,770 gets defined, which is an absolutely 547 00:30:29,770 --> 00:30:32,910 standard symbol in General Relativity. 548 00:30:32,910 --> 00:30:38,820 The formula is d squared x i, ds squared 549 00:30:38,820 --> 00:30:42,590 is equal to-- we know it's going to be equal to stuff 550 00:30:42,590 --> 00:30:46,860 times the product of two derivatives. 551 00:30:46,860 --> 00:30:49,330 And the stuff that appears is just 552 00:30:49,330 --> 00:30:54,050 given a name, capital gamma, which has an upper index 553 00:30:54,050 --> 00:30:57,670 i, which matches the left hand side of the equation. 554 00:30:57,670 --> 00:31:01,230 And two lower indices, which I'm calling j and k, which 555 00:31:01,230 --> 00:31:04,810 will get summed with the derivatives that follow, 556 00:31:04,810 --> 00:31:10,755 d x j ds, dx k, ds. 557 00:31:16,100 --> 00:31:20,080 And this quantity, gamma super i sub jk 558 00:31:20,080 --> 00:31:21,720 are just the terms that would appear 559 00:31:21,720 --> 00:31:24,130 when we do these manipulations. 560 00:31:24,130 --> 00:31:28,070 And I'll write what they are. 561 00:31:28,070 --> 00:31:39,155 Gamma super i sub jk is equal to 1/2 g super il 562 00:31:39,155 --> 00:31:42,900 times the derivative with respect 563 00:31:42,900 --> 00:31:52,290 to j of g sub lk, plus the derivative 564 00:31:52,290 --> 00:31:58,740 with respect to k of g sub lj. 565 00:31:58,740 --> 00:32:05,530 And then minus the derivative with respect to l of g sub jk. 566 00:32:14,840 --> 00:32:20,240 And this quantity has several different names. 567 00:32:20,240 --> 00:32:22,430 Everybody agrees how to define it up to the sign. 568 00:32:22,430 --> 00:32:23,929 There are different sign conventions 569 00:32:23,929 --> 00:32:26,620 that are used in different books. 570 00:32:26,620 --> 00:32:28,680 And there are also different names for it. 571 00:32:28,680 --> 00:32:31,070 It's often called the affine connection. 572 00:32:34,694 --> 00:32:36,860 If you look, for example in Steve Weinberg's General 573 00:32:36,860 --> 00:32:39,990 Relativity book, he calls it the affine connection. 574 00:32:39,990 --> 00:32:43,090 It's also very often called the Christofel connection, 575 00:32:43,090 --> 00:32:44,300 or the Christofel symbol. 576 00:33:00,610 --> 00:33:03,660 And frankly those are the only names for it 577 00:33:03,660 --> 00:33:07,340 that I've seen, personally. 578 00:33:07,340 --> 00:33:09,400 But there's a book about [INAUDIBLE] 579 00:33:09,400 --> 00:33:12,170 by Sean Carroll which is a very good book. 580 00:33:12,170 --> 00:33:14,900 And he claims that it's sometimes also called 581 00:33:14,900 --> 00:33:17,550 the Riemann connection And it's also sometimes called 582 00:33:17,550 --> 00:33:19,930 the Levi-Civita connection. 583 00:33:19,930 --> 00:33:23,250 So it's got lots of names, which I guess means lots of people's 584 00:33:23,250 --> 00:33:24,540 independently invented it. 585 00:33:27,840 --> 00:33:30,050 But in any case, that's the answer. 586 00:33:30,050 --> 00:33:33,380 And it's just a way of rewriting the formula we have up there. 587 00:33:33,380 --> 00:33:34,970 And for solving problems, the formula, 588 00:33:34,970 --> 00:33:36,540 the way we wrote up there, is almost always 589 00:33:36,540 --> 00:33:37,560 the best way to do it. 590 00:33:37,560 --> 00:33:40,160 So this is really just window dressing, largely 591 00:33:40,160 --> 00:33:42,556 for the purpose of making contact with other books 592 00:33:42,556 --> 00:33:43,680 that you might come across. 593 00:33:47,360 --> 00:33:48,860 OK, so that finishes the derivation 594 00:33:48,860 --> 00:33:50,330 of the geodesic equation. 595 00:33:50,330 --> 00:33:52,760 Now I'd like to give an example of its use. 596 00:33:52,760 --> 00:33:54,399 But before I do that, let me just 597 00:33:54,399 --> 00:33:56,940 pause to ask if there are any questions about the derivation? 598 00:34:02,350 --> 00:34:03,490 OK. 599 00:34:03,490 --> 00:34:05,460 So on your homework, you will, in fact, 600 00:34:05,460 --> 00:34:10,900 be applying this formalism to the Robertson-Walker metric. 601 00:34:10,900 --> 00:34:15,210 And you'll learn how moving particles slow down 602 00:34:15,210 --> 00:34:19,650 as they move through an expanding universe, 603 00:34:19,650 --> 00:34:23,520 completely in an analogy to the way photons, which we've 604 00:34:23,520 --> 00:34:26,000 already learned, lose energy as they 605 00:34:26,000 --> 00:34:28,489 travel through an expanding universe. 606 00:34:28,489 --> 00:34:31,770 So particles with mass also lose energy 607 00:34:31,770 --> 00:34:33,799 in a well-defined way, which you'll 608 00:34:33,799 --> 00:34:36,780 be calculating on the homework. 609 00:34:36,780 --> 00:34:39,879 For example, though, I'll do something different. 610 00:34:39,879 --> 00:34:44,330 A fun metric to talk about is the Schwarzschild metric, 611 00:34:44,330 --> 00:34:47,730 which describes, among other things, black holes. 612 00:34:47,730 --> 00:34:49,310 It in principle, describes anything 613 00:34:49,310 --> 00:34:52,250 which is spherically symmetric and has a gravitational field. 614 00:34:52,250 --> 00:34:54,760 But black holes are the most interesting example, 615 00:34:54,760 --> 00:34:58,930 because it's where the most surprises lie. 616 00:34:58,930 --> 00:35:10,140 So the Schwarzschild metric has the form 617 00:35:10,140 --> 00:35:18,170 ds squared is equal to minus c squared d tau squared, which 618 00:35:18,170 --> 00:35:24,810 is equal to-- this is just a definition, it defines d tau-- 619 00:35:24,810 --> 00:35:27,560 but in terms of the coordinates, it's 620 00:35:27,560 --> 00:35:32,100 minus 1 minus 2 G, Newton's constant, M, 621 00:35:32,100 --> 00:35:35,035 the mass of the object we're discussing-- 622 00:35:35,035 --> 00:35:38,910 the mass of the black hole, if it is a black hole-- divided 623 00:35:38,910 --> 00:35:43,270 by r times c squared, r is the radial coordinate, 624 00:35:43,270 --> 00:35:57,520 times c squared dt squared, plus 1 minus 2 GM over rc 625 00:35:57,520 --> 00:36:07,970 squared times dr squared plus r squared times d theta 626 00:36:07,970 --> 00:36:14,705 squared plus sine squared theta d phi squared. 627 00:36:20,120 --> 00:36:24,280 Now here, theta and phi are the usual polar angles. 628 00:36:24,280 --> 00:36:26,940 We're using a polar coordinate system. 629 00:36:26,940 --> 00:36:33,880 So as usual, theta lies between 0 and pi. 630 00:36:33,880 --> 00:36:36,300 0, what we might call the North Pole, and pi 631 00:36:36,300 --> 00:36:39,760 what we might call the South Pole. 632 00:36:39,760 --> 00:36:44,792 And phi is what is often called an azimuthal angle, 633 00:36:44,792 --> 00:36:46,840 it goes around. 634 00:36:46,840 --> 00:36:48,740 And the way one describes coordinates 635 00:36:48,740 --> 00:36:50,360 on the surface of the Earth, phi would 636 00:36:50,360 --> 00:36:54,860 be the longitude variable. 637 00:36:54,860 --> 00:36:59,460 So 0 is less than or equal to phi is less than 638 00:36:59,460 --> 00:37:04,690 or equal to 2 pi where phi equals 639 00:37:04,690 --> 00:37:08,380 2 pi is identified with phi equals 0. 640 00:37:08,380 --> 00:37:12,150 And you can go around and come back to where you started. 641 00:37:17,650 --> 00:37:20,670 Now notice that if we set capital M, the mass of this 642 00:37:20,670 --> 00:37:24,540 object equal to 0, the metric becomes the trivial metric 643 00:37:24,540 --> 00:37:29,050 of Special Relativity written in spherical polar coordinates. 644 00:37:29,050 --> 00:37:32,080 So all complications go away if there's no mass. 645 00:37:32,080 --> 00:37:34,930 The object disappears. 646 00:37:34,930 --> 00:37:36,800 But as long as the mass is non-zero 647 00:37:36,800 --> 00:37:40,300 there are factors that multiply the dr squared term 648 00:37:40,300 --> 00:37:42,080 and the c squared dt squared term. 649 00:37:45,330 --> 00:37:49,314 Notice that the factors that do that multiplying-- now one 650 00:37:49,314 --> 00:37:50,480 of these should be inverted. 651 00:37:54,630 --> 00:37:58,420 Important inverse, it's a minus 1 power for that factor. 652 00:38:01,240 --> 00:38:06,029 Notice that r can be small enough so 653 00:38:06,029 --> 00:38:07,320 that these factors will vanish. 654 00:38:10,110 --> 00:38:12,940 And the place where that happens is called the Schwarzschild 655 00:38:12,940 --> 00:38:17,620 radius after the same person who invented the metric. 656 00:38:17,620 --> 00:38:26,400 So r sub Schwarzschild is equal to 2 GM divided by c squared. 657 00:38:26,400 --> 00:38:31,460 When little r is equal to that, this quantity in parentheses 658 00:38:31,460 --> 00:38:34,090 vanishes, which means we get infinity here, 659 00:38:34,090 --> 00:38:38,760 because it's inverted, and we get a 0 there. 660 00:38:38,760 --> 00:38:44,860 Now when a term in the metric is either 0 or infinite, 661 00:38:44,860 --> 00:38:48,300 one calls that a singularity. 662 00:38:48,300 --> 00:38:51,820 In this case, it's a removable singularity. 663 00:38:51,820 --> 00:38:53,850 That is, the Schwarzschild singularity 664 00:38:53,850 --> 00:38:56,590 is only there because Schwarzschild 665 00:38:56,590 --> 00:38:59,402 chose to use these particular coordinates. 666 00:38:59,402 --> 00:39:01,110 These are simpler than other coordinates. 667 00:39:01,110 --> 00:39:03,530 He wasn't foolish to use them. 668 00:39:03,530 --> 00:39:05,250 But the appearance of that singularity 669 00:39:05,250 --> 00:39:08,600 is really caused solely by the choice of coordinates. 670 00:39:08,600 --> 00:39:15,070 There really is no singularity at the Schwarzschild horizon. 671 00:39:15,070 --> 00:39:17,520 And that was shown some years later 672 00:39:17,520 --> 00:39:21,090 by other people constructing other coordinate systems. 673 00:39:21,090 --> 00:39:23,760 The coordinate system is best known today 674 00:39:23,760 --> 00:39:27,370 that avoids the Schwarzschild singularity is 675 00:39:27,370 --> 00:39:29,840 a coordinate system called the Kruskal coordinate system. 676 00:39:56,404 --> 00:39:58,570 But we will not be looking at the Kruskal coordinate 677 00:39:58,570 --> 00:40:00,820 system in this class. 678 00:40:00,820 --> 00:40:03,260 Leave that for the GR class that you'll take some time. 679 00:40:08,080 --> 00:40:14,320 OK, now the masses sum parameter, 680 00:40:14,320 --> 00:40:15,820 notice that the metric is completely 681 00:40:15,820 --> 00:40:17,400 determined by the mass. 682 00:40:17,400 --> 00:40:21,450 And that's the same situation as we found in Newtonian gravity. 683 00:40:21,450 --> 00:40:25,460 The metric outside of the spherically symmetric object, 684 00:40:25,460 --> 00:40:28,220 by the gravitational field in Newtonian Physics outside 685 00:40:28,220 --> 00:40:30,295 of a spherical symmetric object, depends only 686 00:40:30,295 --> 00:40:31,890 on the total mass, which does not 687 00:40:31,890 --> 00:40:33,770 depend, at all, on how it's distributed 688 00:40:33,770 --> 00:40:35,630 as long as it's spherically symmetric. 689 00:40:35,630 --> 00:40:36,970 And the same thing here. 690 00:40:36,970 --> 00:40:40,080 As long as an object is spherically symmetric, 691 00:40:40,080 --> 00:40:42,870 the gravitational field outside of the object 692 00:40:42,870 --> 00:40:46,820 will always look like that formula. 693 00:40:46,820 --> 00:40:50,660 Now there are still two cases-- outside 694 00:40:50,660 --> 00:40:53,440 of the object could be larger than or smaller 695 00:40:53,440 --> 00:40:57,060 than this Schwarzschild radius. 696 00:40:57,060 --> 00:41:00,340 So for an object like the sun, the Schwarzschild radius, 697 00:41:00,340 --> 00:41:02,940 we could calculate it-- and it's calculated 698 00:41:02,940 --> 00:41:05,390 in the notes-- it's about two or three kilometers. 699 00:41:08,300 --> 00:41:10,710 Hold on and I'll tell you more accurately. 700 00:41:10,710 --> 00:41:13,380 It's 2.95 kilometers, the Schwarzschild radius 701 00:41:13,380 --> 00:41:15,790 of the sun. 702 00:41:15,790 --> 00:41:18,090 But the sun, of course, is much bigger than that. 703 00:41:18,090 --> 00:41:21,020 And that means that the sun doesn't have a Schwarzschild 704 00:41:21,020 --> 00:41:22,670 horizon. 705 00:41:22,670 --> 00:41:28,350 That is, at 2.95 kilometers from the center of the sun 706 00:41:28,350 --> 00:41:30,030 there's still sun. 707 00:41:30,030 --> 00:41:31,110 It's not outside the sun. 708 00:41:31,110 --> 00:41:33,443 This metric only holds outside the spherically symmetric 709 00:41:33,443 --> 00:41:34,990 object. 710 00:41:34,990 --> 00:41:36,610 So it does not hold inside the sun. 711 00:41:36,610 --> 00:41:39,830 The place where this has the apparent singularity 712 00:41:39,830 --> 00:41:41,852 the metric is not valid at all. 713 00:41:41,852 --> 00:41:43,310 So there is nothing that even comes 714 00:41:43,310 --> 00:41:46,150 close to anything worth talking about, 715 00:41:46,150 --> 00:41:47,800 as far as the Schwarzschild singularity 716 00:41:47,800 --> 00:41:49,770 for an object like the sun. 717 00:41:49,770 --> 00:41:53,820 But if the sun were compressed to a size smaller than 2.95 718 00:41:53,820 --> 00:42:00,040 kilometers with the same mass, then these factors 719 00:42:00,040 --> 00:42:03,950 would be relevant at the places where they vanish. 720 00:42:03,950 --> 00:42:09,290 And whatever consequences they have, we would be dealing with. 721 00:42:09,290 --> 00:42:12,930 Even though r equals r Schwarzschild is not a singular 722 00:42:12,930 --> 00:42:15,670 point, it is still a special point. 723 00:42:15,670 --> 00:42:19,655 What you can show-- we won't-- but what we can show is that 724 00:42:19,655 --> 00:42:21,840 that is the horizon. 725 00:42:21,840 --> 00:42:24,600 Meaning that if an object falls inside this Schwarzschild 726 00:42:24,600 --> 00:42:30,191 radius, there is no trajectory that will ever get it out. 727 00:42:30,191 --> 00:42:30,690 Yes? 728 00:42:30,690 --> 00:42:34,866 AUDIANCE: Say a star is just incredibly dense at its core. 729 00:42:34,866 --> 00:42:36,850 Is it possible to have suppression 730 00:42:36,850 --> 00:42:41,082 of some fractional life of a star that's from that mass 731 00:42:41,082 --> 00:42:42,059 that it's contained? 732 00:42:42,059 --> 00:42:43,808 Or like a fusion reaction that is going on 733 00:42:43,808 --> 00:42:46,220 with the net radius? 734 00:42:46,220 --> 00:42:49,572 PROFESSOR: OK, could there be a horizon inside of a star? 735 00:42:49,572 --> 00:42:51,280 I think is what you're asking, basically. 736 00:42:51,280 --> 00:42:52,870 AUDIANCE: One that actually affects the-- 737 00:42:52,870 --> 00:42:53,885 PROFESSOR: One that really is a horizon. 738 00:42:53,885 --> 00:42:54,926 AUDIANCE: That's outside. 739 00:42:54,926 --> 00:42:57,050 PROFESSOR: Right. 740 00:42:57,050 --> 00:42:58,790 If this were the sun you were describing, 741 00:42:58,790 --> 00:43:00,623 this formula would just not be valid inside. 742 00:43:00,623 --> 00:43:02,010 There would be no horizon inside. 743 00:43:02,010 --> 00:43:05,070 But you're asking a real valid question. 744 00:43:05,070 --> 00:43:09,930 If a star had, for some reason, a very dense spot 745 00:43:09,930 --> 00:43:12,350 in the middle, could it actually form 746 00:43:12,350 --> 00:43:14,870 a horizon inside the material? 747 00:43:14,870 --> 00:43:16,412 And the answer is, yes, it could. 748 00:43:16,412 --> 00:43:17,370 It would not be stable. 749 00:43:17,370 --> 00:43:27,960 The material would ultimately fall in, but it could happen. 750 00:43:27,960 --> 00:43:29,003 Yes? 751 00:43:29,003 --> 00:43:31,461 AUDIANCE: So like our galaxy has a super massive black hole 752 00:43:31,461 --> 00:43:32,180 in the center. 753 00:43:32,180 --> 00:43:32,790 PROFESSOR: That's right. 754 00:43:32,790 --> 00:43:34,748 Our galaxy does have a super massive black hole 755 00:43:34,748 --> 00:43:35,760 in the center. 756 00:43:35,760 --> 00:43:35,896 AUDIANCE: Yeah. 757 00:43:35,896 --> 00:43:37,604 So you can consider that as like a larger 758 00:43:37,604 --> 00:43:40,570 mass that has black hole, area? 759 00:43:40,570 --> 00:43:42,180 PROFESSOR: Right! 760 00:43:42,180 --> 00:43:43,860 Right! 761 00:43:43,860 --> 00:43:44,650 That's right. 762 00:43:44,650 --> 00:43:47,114 The comment is that if we go from a star 763 00:43:47,114 --> 00:43:48,530 to something bigger than a star we 764 00:43:48,530 --> 00:43:52,030 have perfectly good example in our own galaxy, 765 00:43:52,030 --> 00:43:53,780 where there is a black hole in the center, 766 00:43:53,780 --> 00:43:56,090 but there is still mass that continues outside of that. 767 00:43:56,090 --> 00:43:58,300 And the black hole is accreting, more matter 768 00:43:58,300 --> 00:44:00,940 does keep falling in, it's not really stable. 769 00:44:00,940 --> 00:44:05,510 But it certainly does exist, and can exist. 770 00:44:09,082 --> 00:44:09,915 Any other questions? 771 00:44:17,840 --> 00:44:21,135 Well, our goal now is to calculate a geodesic. 772 00:44:28,050 --> 00:44:32,380 And I will just calculate one geodesic. 773 00:44:32,380 --> 00:44:36,480 I will calculate what happens if an object starts at some fixed 774 00:44:36,480 --> 00:44:41,270 radius at rest and is released and falls into this black hole. 775 00:44:47,370 --> 00:44:50,890 I first want to just rewrite the geodesic equation in terms 776 00:44:50,890 --> 00:44:54,720 of variables that are more appropriate for this case. 777 00:44:54,720 --> 00:44:57,970 When I wrote that, I had a mind just calculating the geodesics 778 00:44:57,970 --> 00:45:00,010 in space, looking for the shortest 779 00:45:00,010 --> 00:45:02,731 path between two points. 780 00:45:02,731 --> 00:45:04,480 The geodesic that we're talking about when 781 00:45:04,480 --> 00:45:07,500 we're talking about an object in general relativity 782 00:45:07,500 --> 00:45:09,870 moving along the geodesic is a geodesic 783 00:45:09,870 --> 00:45:12,570 that's a time-like geodesic. 784 00:45:12,570 --> 00:45:14,510 That is, any increment along the geodesic 785 00:45:14,510 --> 00:45:17,300 is a time-like interval, or following a particle. 786 00:45:17,300 --> 00:45:22,030 Particles travel on time-like trajectories in relativity. 787 00:45:22,030 --> 00:45:26,700 So the usual notation for time is something like tau 788 00:45:26,700 --> 00:45:30,260 rather than s, which is why I wrote it this way. 789 00:45:30,260 --> 00:45:33,140 ds squared is just defined to be minus c squared d tau squared. 790 00:45:33,140 --> 00:45:36,680 So d tau squared has no more or less information 791 00:45:36,680 --> 00:45:40,050 than ds squared, but it has the opposite sign and a difference 792 00:45:40,050 --> 00:45:42,850 by a factor of c squared, as well. 793 00:45:42,850 --> 00:45:48,230 And another change in notation which is a rather universal 794 00:45:48,230 --> 00:45:53,200 convention is that, when we talk about space alone we use Latin 795 00:45:53,200 --> 00:45:55,810 indices, ijk.. 796 00:45:55,810 --> 00:45:58,260 When we talk about spacetime, where one of the indices 797 00:45:58,260 --> 00:46:01,500 might be 0 referring to the time direction, 798 00:46:01,500 --> 00:46:05,840 then we usually use Greek indices, mu, nu, lambda. 799 00:46:05,840 --> 00:46:11,280 So I'm going to rewrite the geodesic equation using tau 800 00:46:11,280 --> 00:46:13,790 as my parameter instead of s, since we're 801 00:46:13,790 --> 00:46:16,430 talking about proper time along the trajectory instead 802 00:46:16,430 --> 00:46:17,890 of distances. 803 00:46:17,890 --> 00:46:21,250 And using Greek letters instead of Latin letters, because we're 804 00:46:21,250 --> 00:46:24,131 talking about spacetime rather than just space. 805 00:46:24,131 --> 00:46:25,630 So otherwise what I'm going to write 806 00:46:25,630 --> 00:46:26,713 is just identical to that. 807 00:46:26,713 --> 00:46:30,020 So really is nothing more than a change in notation. 808 00:46:30,020 --> 00:46:43,090 d d tau of g mu nu, dx super nu d tau. 809 00:46:43,090 --> 00:46:49,520 And it is equal to 1/2 times the partial of g lambda 810 00:46:49,520 --> 00:47:04,135 sigma with respect to x nu dx lambda d tau dx sigma d tau. 811 00:47:08,221 --> 00:47:10,220 Now you might want to go through the calculation 812 00:47:10,220 --> 00:47:12,540 and make sure of the fact that now we're 813 00:47:12,540 --> 00:47:14,960 dealing with a metric which is not positive, 814 00:47:14,960 --> 00:47:17,210 definite, doesn't make any difference. 815 00:47:17,210 --> 00:47:18,444 But it doesn't. 816 00:47:18,444 --> 00:47:19,860 It does mean that now we certainly 817 00:47:19,860 --> 00:47:24,040 have possibilities of getting maxima and stationary points as 818 00:47:24,040 --> 00:47:28,260 well as minima, because of the variety of signs 819 00:47:28,260 --> 00:47:30,220 that appear in the metric. 820 00:47:30,220 --> 00:47:32,590 But otherwise, the calculations of the geodesic equation 821 00:47:32,590 --> 00:47:35,040 goes through exactly as we calculated it. 822 00:47:35,040 --> 00:47:38,240 And the only thing I'm doing here, 823 00:47:38,240 --> 00:47:40,320 relative to what we have there, is just 824 00:47:40,320 --> 00:47:43,125 changing the notation a bit to conform to the notaion that 825 00:47:43,125 --> 00:47:45,680 is usually used for talking about spacetime trajectories. 826 00:48:04,060 --> 00:48:07,300 Since we're talking about radio trajectories, 827 00:48:07,300 --> 00:48:09,640 we're just going to release a particle at rest 828 00:48:09,640 --> 00:48:11,690 and then it will fall straight towards the center 829 00:48:11,690 --> 00:48:15,860 of our spherical object, we know by symmetry 830 00:48:15,860 --> 00:48:17,580 that it's not going to be deflected 831 00:48:17,580 --> 00:48:19,750 in the positive theta or the negative theta, 832 00:48:19,750 --> 00:48:22,720 or the positive phi or negative phi directions, 833 00:48:22,720 --> 00:48:24,735 because that would violate isotropy. 834 00:48:24,735 --> 00:48:26,455 It would violate the rotational symmetry 835 00:48:26,455 --> 00:48:28,260 that we know as part of this metric. 836 00:48:28,260 --> 00:48:31,800 This Is just the metric of the surface of the sphere. 837 00:48:31,800 --> 00:48:34,220 So theta and phi will just stay whatever values they have 838 00:48:34,220 --> 00:48:35,920 when you drop this object. 839 00:48:35,920 --> 00:48:38,600 So we will not even talk about theta and phi. 840 00:48:38,600 --> 00:48:41,680 We will only talk about r and t, how 841 00:48:41,680 --> 00:48:44,880 particle falls in as a function of time. 842 00:48:44,880 --> 00:48:49,770 And then it turns out to be useful to just first write down 843 00:48:49,770 --> 00:48:51,600 what the metric itself tells us. 844 00:48:51,600 --> 00:48:53,015 And we'll divide by d tau. 845 00:48:53,015 --> 00:48:55,280 So we could talk about derivatives with respect 846 00:48:55,280 --> 00:48:57,130 to tau. 847 00:48:57,130 --> 00:48:59,200 So changing an overall sign, since everything's 848 00:48:59,200 --> 00:49:00,700 going to be negative and we'd rather 849 00:49:00,700 --> 00:49:03,780 have everything be positive, we can just 850 00:49:03,780 --> 00:49:08,750 rewrite the metric equation as saying, 851 00:49:08,750 --> 00:49:16,750 that c squared is equal to 1 minus 2 GM over rc 852 00:49:16,750 --> 00:49:22,240 squared, times c squared times dt 853 00:49:22,240 --> 00:49:37,580 d tau squared minus 1 minus 2 GM over rc squared inverse times 854 00:49:37,580 --> 00:49:41,180 dr d tau squared. 855 00:49:45,070 --> 00:49:47,290 So this is nothing more than rewriting this equation 856 00:49:47,290 --> 00:49:49,915 saying d theta is equal to 0 and d phi will be 0. 857 00:49:52,490 --> 00:49:54,100 Written this way, though, it tells us 858 00:49:54,100 --> 00:49:59,285 that we can find dt d tau, for example, if we know dr d tau. 859 00:49:59,285 --> 00:50:01,720 And we also know where we are, you know, little r. 860 00:50:01,720 --> 00:50:03,950 And we'll be using that, shortly. 861 00:50:06,590 --> 00:50:08,060 To continue a little further, we're 862 00:50:08,060 --> 00:50:10,185 going to introduce some abbreviations just so we're 863 00:50:10,185 --> 00:50:12,100 don't have to write so much. 864 00:50:12,100 --> 00:50:16,760 I'm going to define little h of r 865 00:50:16,760 --> 00:50:22,680 as just one minus r Schwarzschild over r. 866 00:50:26,500 --> 00:50:33,580 And this is also 1 minus 2 GM over rc squared. 867 00:50:33,580 --> 00:50:36,030 That's a factor that keeps recurring 868 00:50:36,030 --> 00:50:38,281 in our expression for the metric. 869 00:50:38,281 --> 00:50:38,780 Yes? 870 00:50:38,780 --> 00:50:40,321 AUDIANCE: The second to last equation 871 00:50:40,321 --> 00:50:43,837 is supposed to be a c squared in between the two parenthesis? 872 00:50:43,837 --> 00:50:44,670 PROFESSOR: Probably. 873 00:50:47,780 --> 00:50:48,600 Yes, thank you. 874 00:50:52,610 --> 00:50:55,359 G squared, right? 875 00:50:55,359 --> 00:50:55,900 Thanks a lot. 876 00:51:09,660 --> 00:51:11,510 In terms of h of r, we can rewrite 877 00:51:11,510 --> 00:51:15,370 that equation slightly more simply. 878 00:51:15,370 --> 00:51:17,720 I'm going to bring things to the other side 879 00:51:17,720 --> 00:51:20,970 and write it as c squared times dt 880 00:51:20,970 --> 00:51:32,680 d tau squared is equal to c squared h inverse of r plus h 881 00:51:32,680 --> 00:51:41,720 to the minus 2 of r times dr d tau squared. 882 00:51:45,656 --> 00:51:48,710 This is just a rewriting of the above equation, 883 00:51:48,710 --> 00:51:52,057 making use of the new notation that we've introduced. 884 00:51:52,057 --> 00:51:53,640 And this is the form we will be using. 885 00:51:53,640 --> 00:51:55,056 It explicitly tells us how to find 886 00:51:55,056 --> 00:51:57,455 dt d tau in terms of other things. 887 00:51:57,455 --> 00:52:00,450 So dt d tau is not independent. 888 00:52:04,750 --> 00:52:07,870 Since we know dt d tau in terms of dr d tau. 889 00:52:07,870 --> 00:52:11,400 If We get an expression for dr d tau we're sort of finished. 890 00:52:11,400 --> 00:52:14,420 We could find everything we want to know about t 891 00:52:14,420 --> 00:52:17,250 from the equation we just wrote. 892 00:52:17,250 --> 00:52:20,440 So it turns out that all we need to do 893 00:52:20,440 --> 00:52:23,340 to calculate this radial trajectory 894 00:52:23,340 --> 00:52:26,710 is to look at the component of the metric 895 00:52:26,710 --> 00:52:32,140 where that free index, mu, mu is the index that's not summed, 896 00:52:32,140 --> 00:52:34,880 we're going to set mu equal to r. 897 00:52:34,880 --> 00:52:37,740 Remember mu is a number that corresponds to a coordinate. 898 00:52:37,740 --> 00:52:39,573 And we're going to set mu equal to the value 899 00:52:39,573 --> 00:52:41,170 that corresponds to the r coordinate. 900 00:52:41,170 --> 00:52:46,040 And that will be sufficient to get us our answer. 901 00:52:46,040 --> 00:52:56,300 When we do that, the equation becomes d d tau of g sub r. 902 00:52:56,300 --> 00:53:00,540 Now the second index, nu in the original expression, 903 00:53:00,540 --> 00:53:04,965 is summed from 0 to 3 for the gr case, where 904 00:53:04,965 --> 00:53:07,180 we have four coordinates, one time and three 905 00:53:07,180 --> 00:53:11,030 spatial coordinates, but we only need 906 00:53:11,030 --> 00:53:16,210 to write the terms where gr nu variable is non-zero. 907 00:53:16,210 --> 00:53:19,410 And the metric itself is diagonal. 908 00:53:19,410 --> 00:53:22,700 So if one index is a little r, the other index 909 00:53:22,700 --> 00:53:25,330 has to also be r, or else it vanishes. 910 00:53:25,330 --> 00:53:27,960 So the only value of nu that contributes to the sum 911 00:53:27,960 --> 00:53:30,890 is when nu is also equal to the r coordinate. 912 00:53:30,890 --> 00:53:37,900 So we get g sub rr d xr-- which, in fact I'll 913 00:53:37,900 --> 00:53:43,540 write it as just dr. x super r is just the r coordinate, which 914 00:53:43,540 --> 00:53:54,974 we also just call r times d tau is equal to 1/2 dr. 915 00:53:54,974 --> 00:53:57,270 And now, on the right-hand side, we're 916 00:53:57,270 --> 00:54:00,780 summing over lambda and sigma. 917 00:54:00,780 --> 00:54:04,940 And lambda and sigma have to have the property that g sub 918 00:54:04,940 --> 00:54:07,866 lambda sigma depends on r, or else the first factor 919 00:54:07,866 --> 00:54:08,365 will vanish. 920 00:54:11,030 --> 00:54:14,466 And furthermore, g sub lambda sigma 921 00:54:14,466 --> 00:54:16,590 has be non-zero, for the values of lambda and sigma 922 00:54:16,590 --> 00:54:19,190 that you want, which means that lambda and sigma for this case 923 00:54:19,190 --> 00:54:20,898 has to be equal to each other, because we 924 00:54:20,898 --> 00:54:23,980 have no off-diagonal terms to our metric. 925 00:54:23,980 --> 00:54:29,410 So the only contributions we get are from g sub rr and g sub tt. 926 00:54:29,410 --> 00:54:37,280 So you get the derivative with respect to r of g sub rr times 927 00:54:37,280 --> 00:54:42,740 dr d tau squared. 928 00:54:42,740 --> 00:54:46,980 This become squared, because lambda is equal to sigma. 929 00:54:46,980 --> 00:55:04,070 And then plus 1/2 drg sub tt times dt d tau squared. 930 00:55:15,860 --> 00:55:20,970 And note that buried in here is, if we expand this, 931 00:55:20,970 --> 00:55:22,810 the second derivative of r with respect 932 00:55:22,810 --> 00:55:26,330 to time-- respect to tau. 933 00:55:26,330 --> 00:55:30,760 So we can extract that and solve for it. 934 00:55:30,760 --> 00:55:35,870 And things like dt d tau will appear in our answer, 935 00:55:35,870 --> 00:55:38,856 initially, because it's here already. 936 00:55:38,856 --> 00:55:43,220 But we could replace dt d tau by this top equation 937 00:55:43,220 --> 00:55:46,090 and eliminate it from our results. 938 00:55:46,090 --> 00:55:50,400 And I'm going to skip the algebra, which 939 00:55:50,400 --> 00:55:52,600 is straightforward, although tedious. 940 00:55:52,600 --> 00:55:55,950 I urge you to go through it in the notes. 941 00:55:55,950 --> 00:55:58,580 But the end result ends up being remarkably simple, 942 00:55:58,580 --> 00:56:02,530 after a number of cancellations that look like surprises. 943 00:56:02,530 --> 00:56:05,730 And what you find in the end-- and it's just 944 00:56:05,730 --> 00:56:09,200 the simplification of this formula, nothing more-- you 945 00:56:09,200 --> 00:56:15,490 find that d squared r d tau squared is 946 00:56:15,490 --> 00:56:22,810 just equal to minus Newton's constant times the mass 947 00:56:22,810 --> 00:56:23,910 divided by r squared. 948 00:56:31,950 --> 00:56:34,100 Now this is rather shocking, and even looks exactly 949 00:56:34,100 --> 00:56:35,285 like Newtonian mechanics. 950 00:56:38,129 --> 00:56:40,420 However, even though it looks like Newtonian mechanics, 951 00:56:40,420 --> 00:56:43,670 it's not really the same as Newtonian mechanics, 952 00:56:43,670 --> 00:56:47,330 because the variables don't mean quite the same thing. 953 00:56:47,330 --> 00:56:49,850 First of all, even r does not really 954 00:56:49,850 --> 00:56:55,390 mean radius in the same sense as radius is defined by Newton. 955 00:56:55,390 --> 00:56:57,030 In Newtonian mechanics, radius is 956 00:56:57,030 --> 00:56:58,860 the distance from the origin. 957 00:56:58,860 --> 00:57:01,560 If we wanted to know the distance from the origin, 958 00:57:01,560 --> 00:57:04,690 we would have to integrate this metric. 959 00:57:04,690 --> 00:57:07,214 And in fact, there isn't even an actual origin here, 960 00:57:07,214 --> 00:57:09,380 because you would have to go through the singularity 961 00:57:09,380 --> 00:57:10,254 before you get there. 962 00:57:10,254 --> 00:57:12,122 And you really can't. 963 00:57:12,122 --> 00:57:16,581 That integral is not really even defined. 964 00:57:16,581 --> 00:57:18,830 Although, of course, if we had something like the sun, 965 00:57:18,830 --> 00:57:20,871 where the metric was different from this small r, 966 00:57:20,871 --> 00:57:22,850 then we could integrate from r equals 0, 967 00:57:22,850 --> 00:57:24,910 and that would define the true radius, 968 00:57:24,910 --> 00:57:26,450 distance from the center. 969 00:57:26,450 --> 00:57:27,480 But it would not be r. 970 00:57:27,480 --> 00:57:31,992 It would be what you got by integrating with the metric. 971 00:57:31,992 --> 00:57:33,450 So r has a different interpretation 972 00:57:33,450 --> 00:57:37,100 than it does for Newtonian physics. 973 00:57:37,100 --> 00:57:40,310 I might add, it still has a simple interpretation. 974 00:57:40,310 --> 00:57:46,260 If you look at this metric, the tangential part, the angular 975 00:57:46,260 --> 00:57:50,970 part, is exactly what you have for Euclidean geometry. 976 00:57:50,970 --> 00:57:54,050 It's just r squared times the same combination of d theta 977 00:57:54,050 --> 00:57:57,820 and d phi as appears on the surface of a sphere. 978 00:57:57,820 --> 00:58:01,870 So little r is sometimes called the circumferential radius, 979 00:58:01,870 --> 00:58:06,790 because it really does give you the circumference of circles 980 00:58:06,790 --> 00:58:09,160 at that radial coordinate. 981 00:58:09,160 --> 00:58:11,650 If we went around in a circle at a fixed r, 982 00:58:11,650 --> 00:58:14,450 the circle would involve varying phi, for example, 983 00:58:14,450 --> 00:58:16,440 over a range of 2 pi, we really would 984 00:58:16,440 --> 00:58:21,460 see a total circumference of 2 pi little r. 985 00:58:21,460 --> 00:58:22,990 So r is related to circumferences 986 00:58:22,990 --> 00:58:26,420 in exactly the way as it is in Euclidean geometry. 987 00:58:26,420 --> 00:58:27,920 But it's not related to the distance 988 00:58:27,920 --> 00:58:31,330 from the origin in the same way as it is in Euclidean geometry. 989 00:58:34,350 --> 00:58:38,700 In addition, tau, here, is not the universal time 990 00:58:38,700 --> 00:58:41,910 that Newton imagined. 991 00:58:41,910 --> 00:58:48,780 But rather, tau is measured along the geodesic. 992 00:58:51,690 --> 00:58:53,190 It is just ds squared, but remember, 993 00:58:53,190 --> 00:58:56,490 ds squared is being measured along the geodesic, which 994 00:58:56,490 --> 00:58:58,450 means that it is, in fact, the proper time 995 00:58:58,450 --> 00:59:01,550 as it would be measured by the person falling 996 00:59:01,550 --> 00:59:05,310 with the object towards the black hole. 997 00:59:05,310 --> 00:59:09,514 So tau is proper time as measured by the falling object. 998 00:59:09,514 --> 00:59:10,930 And that follows from what we know 999 00:59:10,930 --> 00:59:12,800 about the meaning of the metric itself. 1000 00:59:17,660 --> 00:59:21,767 OK, that said we would now like to just study 1001 00:59:21,767 --> 00:59:22,975 this equation more carefully. 1002 00:59:50,417 --> 00:59:51,875 And since the equation itself still 1003 00:59:51,875 --> 00:59:55,150 has the same form as what you get from Newton, if you 1004 00:59:55,150 --> 00:59:57,700 remember what you would have done if this was 801, 1005 00:59:57,700 --> 01:00:00,480 you can, in fact, do exactly the same thing here. 1006 01:00:00,480 --> 01:00:03,230 And what you probably would have done, if this was 801, 1007 01:00:03,230 --> 01:00:08,270 would be to recognize that this equation can be integrated. 1008 01:00:08,270 --> 01:00:17,200 We can write the equation as d d tau of 1/2 1009 01:00:17,200 --> 01:00:29,720 dr d tau squared minus GM/r equals 0. 1010 01:00:29,720 --> 01:00:31,180 I you carry out these derivatives 1011 01:00:31,180 --> 01:00:32,796 you would get that equation. 1012 01:00:32,796 --> 01:00:34,170 And this is just the conservation 1013 01:00:34,170 --> 01:00:38,780 of energy version of the force equation. 1014 01:00:55,410 --> 01:00:59,710 And that tells us that this quantity is a constant. 1015 01:00:59,710 --> 01:01:14,731 If we drop the object from some initial position, r sub 0, 1016 01:01:14,731 --> 01:01:18,040 and we drop it with no initial velocity, 1017 01:01:18,040 --> 01:01:21,800 we just let go of it at r sub 0, that tells us 1018 01:01:21,800 --> 01:01:24,620 what this quantity is when we drop it. 1019 01:01:24,620 --> 01:01:27,490 It's minus GM over r sub 0. 1020 01:01:27,490 --> 01:01:30,120 This piece vanishes if there is no initial velocity. 1021 01:01:30,120 --> 01:01:32,036 And that means it will always have that value. 1022 01:01:35,390 --> 01:01:41,750 And knowing that, we can write dr d tau 1023 01:01:41,750 --> 01:01:46,000 is equal to-- just solving for that-- 1024 01:01:46,000 --> 01:02:03,980 minus the square root of 2GM times r0 minus r over r r0 1025 01:02:03,980 --> 01:02:08,100 I've collected two terms and put them over a common denominator 1026 01:02:08,100 --> 01:02:09,480 and added them. 1027 01:02:09,480 --> 01:02:12,440 So this is not quite as obvious as it might be. 1028 01:02:12,440 --> 01:02:15,004 But this is just the statement that that quantity 1029 01:02:15,004 --> 01:02:16,920 has the same value as it did when you started. 1030 01:02:23,700 --> 01:02:27,550 Now this can be further integrated. 1031 01:02:27,550 --> 01:02:35,840 We can write it as dr over-- bringing 1032 01:02:35,840 --> 01:02:39,420 all this to the other side-- is equal to d tau. 1033 01:02:39,420 --> 01:02:40,950 And then integrate both sides. 1034 01:02:45,480 --> 01:02:47,850 Notice when I bring this to the other side 1035 01:02:47,850 --> 01:02:50,150 and bring the d tau to the right., everything 1036 01:02:50,150 --> 01:02:52,190 on the left-hand side now only depends on r. 1037 01:02:52,190 --> 01:02:56,080 So this is just an explicit integral over r that we can do. 1038 01:03:16,160 --> 01:03:20,120 And I will just tell you that when the integral is done 1039 01:03:20,120 --> 01:03:24,550 we get a formula for tau as a function of r. 1040 01:03:32,529 --> 01:03:35,570 And it's equal to the square root 1041 01:03:35,570 --> 01:03:46,920 of r sub 0 over 2GM times r0 times 1042 01:03:46,920 --> 01:03:55,900 the inverse tangent of the square root of r0 1043 01:03:55,900 --> 01:04:08,070 minus r over r plus the square root of r times r0 minus r. 1044 01:04:14,500 --> 01:04:18,780 So when r equals r0, this gives us 0, and that's what we want. 1045 01:04:18,780 --> 01:04:24,470 When we start we're at r0, or time 0, or proper time 0. 1046 01:04:24,470 --> 01:04:29,280 And then as r gets smaller, as it falls in, time grows. 1047 01:04:29,280 --> 01:04:31,920 And this gives us the time as a function of r. 1048 01:04:31,920 --> 01:04:33,920 We might prefer to have r as a function of time, 1049 01:04:33,920 --> 01:04:37,440 but that formula can't really be inverted analytically. 1050 01:04:37,440 --> 01:04:38,710 So that's the best we can do. 1051 01:04:43,680 --> 01:04:46,000 Now one thing that you notice from this 1052 01:04:46,000 --> 01:04:49,100 is that nothing special happens as r 1053 01:04:49,100 --> 01:04:51,907 decreases all the way to 0. 1054 01:04:51,907 --> 01:04:53,490 Even when you plug in r equals 0 here, 1055 01:04:53,490 --> 01:04:55,270 you just get some finite number. 1056 01:04:55,270 --> 01:04:58,270 So in a finite amount of time, the observer 1057 01:04:58,270 --> 01:05:00,790 would find himself falling through the Schwarzchild 1058 01:05:00,790 --> 01:05:04,830 horizon and all the way to r equals 0. 1059 01:05:04,830 --> 01:05:07,880 I didn't mention it but r equals 0 is a true singularity. 1060 01:05:07,880 --> 01:05:11,340 Our metric is also singular when r equals 0. 1061 01:05:11,340 --> 01:05:14,820 These quantities all become infinite. 1062 01:05:14,820 --> 01:05:19,610 And physically what would happen is 1063 01:05:19,610 --> 01:05:24,530 that, as the object falling in approaches r equals 0, 1064 01:05:24,530 --> 01:05:26,732 the tidal forces, that is the difference 1065 01:05:26,732 --> 01:05:29,190 in the gravitational force on one part of the object verses 1066 01:05:29,190 --> 01:05:32,810 another, will get stronger and stronger. 1067 01:05:32,810 --> 01:05:35,310 And objects will just be ripped apart. 1068 01:05:35,310 --> 01:05:40,050 And the ripping apart occurs as being spaghetti-ized, that is, 1069 01:05:40,050 --> 01:05:42,054 the force on the front gets to be very strong 1070 01:05:42,054 --> 01:05:43,470 compared to the force on the back. 1071 01:05:43,470 --> 01:05:46,011 So I'll just get stretched out along the direction of motion. 1072 01:05:53,500 --> 01:05:56,920 Now the curious thing is what this 1073 01:05:56,920 --> 01:05:59,240 looks like if we think of it not as a function 1074 01:05:59,240 --> 01:06:01,140 of the proper time measured by the wrist 1075 01:06:01,140 --> 01:06:05,950 watch of the object falling in but rather, 1076 01:06:05,950 --> 01:06:10,070 we could try to describe it in terms of our external time 1077 01:06:10,070 --> 01:06:11,010 variable. 1078 01:06:11,010 --> 01:06:15,800 The variable t that appears in the Schwarzchild metric. 1079 01:06:15,800 --> 01:06:18,470 And to do that, to make the conversion, 1080 01:06:18,470 --> 01:06:23,630 we want to calculate what the dr dt is, instead of dr d tau. 1081 01:06:23,630 --> 01:06:26,450 Like maybe an analogous formula, in terms of t. 1082 01:06:29,690 --> 01:06:33,540 And to get that, we use simply chain rule here. 1083 01:06:33,540 --> 01:06:38,080 dr dt is equal to dr d tau-- which we've already 1084 01:06:38,080 --> 01:06:45,540 calculated-- times d tau dt. 1085 01:06:45,540 --> 01:06:49,870 And d tau dt is 1 over dt d tau. 1086 01:06:49,870 --> 01:06:52,330 If you just have two variables that depend on each other. 1087 01:06:52,330 --> 01:06:56,490 The derivatives are just the inverse of each other. 1088 01:06:56,490 --> 01:07:00,220 So this could be written as dr d tau-- which 1089 01:07:00,220 --> 01:07:04,430 we've calculated-- divided by dt d tau. 1090 01:07:07,410 --> 01:07:10,900 And dt d tau we've really already calculated as well, 1091 01:07:10,900 --> 01:07:13,740 because it's just given by this formula here. 1092 01:07:26,530 --> 01:07:29,470 So we could write out what that is and figure out 1093 01:07:29,470 --> 01:07:33,550 how it's going to behave as the object approaches 1094 01:07:33,550 --> 01:07:34,645 the Schwarzchild radius. 1095 01:07:54,430 --> 01:08:00,080 So it becomes dr dt is equal to, I'll 1096 01:08:00,080 --> 01:08:07,520 just write the numerator as dr d tau given by that expression. 1097 01:08:07,520 --> 01:08:10,180 But what's behaving in a more peculiar way 1098 01:08:10,180 --> 01:08:17,890 is the denominator, which is h inverse of r plus c 1099 01:08:17,890 --> 01:08:29,820 to the minus 2, h to the minus 2 of r times dr d tau squared. 1100 01:08:35,149 --> 01:08:39,160 So now we want to look at this function h inverse of r. 1101 01:08:39,160 --> 01:08:41,145 And this just means 1/h of r. 1102 01:08:41,145 --> 01:08:44,460 It doesn't mean functional inverse. 1103 01:08:44,460 --> 01:08:51,773 That is just equal to r over r minus r Schwarzchild. 1104 01:08:51,773 --> 01:08:53,439 And we're going to be interested in what 1105 01:08:53,439 --> 01:08:56,255 happens when r gets to be very near r Schwarzchild, 1106 01:08:56,255 --> 01:08:58,950 because that's where the interesting things happen, 1107 01:08:58,950 --> 01:09:02,148 as you're approaching the Schwarzchild horizon. 1108 01:09:02,148 --> 01:09:04,189 And that means that the behavior of the numerator 1109 01:09:04,189 --> 01:09:04,890 won't be important. 1110 01:09:04,890 --> 01:09:06,264 The denominator will be going up, 1111 01:09:06,264 --> 01:09:08,399 and that's what will control everything. 1112 01:09:08,399 --> 01:09:11,899 So we can approximate this as just 1113 01:09:11,899 --> 01:09:16,510 r Schwarzchild over r minus r Schwarzchild. 1114 01:09:16,510 --> 01:09:18,540 And this is for r near r Schwarzchild. 1115 01:09:22,653 --> 01:09:24,444 We've replaced the numerator by a constant. 1116 01:09:28,520 --> 01:09:30,069 And then if we look at this formula, 1117 01:09:30,069 --> 01:09:34,319 this is going to blow up as we approach the horizon. 1118 01:09:34,319 --> 01:09:35,970 This is the square of that quantity. 1119 01:09:35,970 --> 01:09:40,040 It will blow up faster than the first power of that quantity. 1120 01:09:40,040 --> 01:09:41,939 And therefore, this will dominate, 1121 01:09:41,939 --> 01:09:44,950 the denominator of the expression. 1122 01:09:44,950 --> 01:09:47,130 We can ignore this. 1123 01:09:47,130 --> 01:09:50,840 When this dominates, the dr d tau pieces cancel. 1124 01:09:50,840 --> 01:09:51,520 So that's nice. 1125 01:09:51,520 --> 01:09:54,770 We don't even need to think about what the dr d tau is. 1126 01:09:54,770 --> 01:10:06,890 And what we get near the horizon is simply 1127 01:10:06,890 --> 01:10:13,564 a factor of c times r minus r Schwarzchild over r 1128 01:10:13,564 --> 01:10:14,105 Schwarzchild. 1129 01:10:18,290 --> 01:10:20,770 It's basically just h. 1130 01:10:20,770 --> 01:10:22,810 This becomes upstairs with a plus sign. 1131 01:10:22,810 --> 01:10:25,470 And the square root turns it into h instead of h squared. 1132 01:10:29,350 --> 01:10:30,700 So this is the inverse of that. 1133 01:10:37,220 --> 01:10:40,290 OK, now if we try to play the same game here 1134 01:10:40,290 --> 01:10:44,780 as we did here, to determine what our time variable behaves 1135 01:10:44,780 --> 01:10:48,180 as a function of r, instead of the proper time variable 1136 01:10:48,180 --> 01:10:59,340 tau, what we find is that t of r-- this is for r near r 1137 01:10:59,340 --> 01:11:07,830 Schwarzchild-- is about equal to minus 1138 01:11:07,830 --> 01:11:18,895 r sub s over c times the integral up to r 1139 01:11:18,895 --> 01:11:25,665 of dr prime over r prime minus rs. 1140 01:11:37,080 --> 01:11:37,915 This is dr dt. 1141 01:11:42,126 --> 01:11:43,990 Yeah, this was dr dt from the beginning. 1142 01:11:43,990 --> 01:11:46,200 I forgot to write the r somehow. 1143 01:11:46,200 --> 01:11:47,659 AUDIANCE: Doesn't that [INAUDIBLE]? 1144 01:11:47,659 --> 01:11:50,283 PROFESSOR: Yeah, I didn't write the lower limit of integration. 1145 01:11:50,283 --> 01:11:51,800 I was about to comment on that. 1146 01:11:51,800 --> 01:11:54,700 The integrand that we're writing is only a good approximation 1147 01:11:54,700 --> 01:11:57,910 whenever we're near r. 1148 01:11:57,910 --> 01:12:01,030 So whatever happens near the lower limit of integration, 1149 01:12:01,030 --> 01:12:02,830 we just haven't done accurately. 1150 01:12:02,830 --> 01:12:06,010 So I'm going to just not write a lower limit of integration 1151 01:12:06,010 --> 01:12:08,020 here, meaning that we're interested only 1152 01:12:08,020 --> 01:12:12,586 in what happens as the upper limit of integration r 1153 01:12:12,586 --> 01:12:13,960 becomes very near r Schwarzchild. 1154 01:12:13,960 --> 01:12:15,626 And everything will be dominated by what 1155 01:12:15,626 --> 01:12:18,067 happens near the upper limit of integration. 1156 01:12:18,067 --> 01:12:19,817 AUDIANCE: So would you just integrate over 1157 01:12:19,817 --> 01:12:21,380 on [INAUDIBLE] for that? 1158 01:12:21,380 --> 01:12:22,963 PROFESSOR: That's right, that's right. 1159 01:12:22,963 --> 01:12:26,640 We just integrated over a small region near, r Schwarzchild. 1160 01:12:53,670 --> 01:12:56,050 Nu r, which is also about equal to r Schwarzchild. 1161 01:12:59,530 --> 01:13:02,280 And the point is, that this diverges logarithmically 1162 01:13:02,280 --> 01:13:05,680 as r approaches r Schwarzchild. 1163 01:13:05,680 --> 01:13:14,920 So it behaves approximately as minus r Schwarzchild 1164 01:13:14,920 --> 01:13:20,640 over c times the logarithm of r minus r Schwarzchild. 1165 01:13:25,960 --> 01:13:28,220 So as r approaches r Schwarzchild, 1166 01:13:28,220 --> 01:13:30,820 this quantity that's the argument of the logarithm 1167 01:13:30,820 --> 01:13:32,470 gets closer and closer to 0. 1168 01:13:32,470 --> 01:13:36,294 It gets smaller and smaller approaching 0. 1169 01:13:36,294 --> 01:13:37,960 But the logarithm of a very small number 1170 01:13:37,960 --> 01:13:40,672 is a negative number, a large negative number. 1171 01:13:40,672 --> 01:13:42,130 And then there's a minus sign here. 1172 01:13:42,130 --> 01:13:44,680 You get a large positive number and it diverges. 1173 01:13:44,680 --> 01:13:48,020 As r approaches r Schwarzchild the time variable 1174 01:13:48,020 --> 01:13:50,520 approaches infinity. 1175 01:13:50,520 --> 01:13:53,590 And that means that at no finite time 1176 01:13:53,590 --> 01:13:57,665 does the object ever reach the Schwarzchild horizon. 1177 01:13:57,665 --> 01:14:00,510 But as seen from the outside, it takes an infinite amount 1178 01:14:00,510 --> 01:14:04,100 of time for the object to reach the Schwarzchild horizon. 1179 01:14:04,100 --> 01:14:07,600 As time gets larger and larger, the object gets and closer 1180 01:14:07,600 --> 01:14:11,200 to the Schwarzchild horizon, asymptotically approaching it 1181 01:14:11,200 --> 01:14:12,809 but never reaching it. 1182 01:14:12,809 --> 01:14:14,350 So this, of course, is very peculiar, 1183 01:14:14,350 --> 01:14:17,620 because from the point of view of the person 1184 01:14:17,620 --> 01:14:19,500 falling into the black hole, all this 1185 01:14:19,500 --> 01:14:21,984 just happens in a finite amount of time and is over with. 1186 01:14:21,984 --> 01:14:23,400 From the outside, it looks like it 1187 01:14:23,400 --> 01:14:25,480 takes an infinite amount of time. 1188 01:14:25,480 --> 01:14:28,920 And weird things like this can happen because of the fact that 1189 01:14:28,920 --> 01:14:33,250 in general relativity time is a locally measured variable. 1190 01:14:33,250 --> 01:14:35,490 You measure your time, I measure my time. 1191 01:14:35,490 --> 01:14:36,810 They don't have to agree. 1192 01:14:36,810 --> 01:14:40,510 And in this case, they can disagree by an infinite amount, 1193 01:14:40,510 --> 01:14:44,460 which is rather bizarre, but that's what happens. 1194 01:14:44,460 --> 01:14:47,610 So according to classical general relativity, 1195 01:14:47,610 --> 01:14:50,200 when an object falls into a black hole, 1196 01:14:50,200 --> 01:14:53,020 from the point of view of the object nothing 1197 01:14:53,020 --> 01:14:56,900 special would happen as that object crossed the Schwarzchild 1198 01:14:56,900 --> 01:14:59,020 horizon. 1199 01:14:59,020 --> 01:15:01,020 Everybody believed that that was really the case 1200 01:15:01,020 --> 01:15:02,570 until maybe a couple years ago. 1201 01:15:02,570 --> 01:15:04,696 Now it's controversial, actually. 1202 01:15:04,696 --> 01:15:06,070 At the classical level, everybody 1203 01:15:06,070 --> 01:15:07,195 believes that's still true. 1204 01:15:07,195 --> 01:15:08,880 I mean, classical general relativity 1205 01:15:08,880 --> 01:15:10,470 says that an object can fall through the Schwarzchild 1206 01:15:10,470 --> 01:15:12,180 horizon and then nothing happens. 1207 01:15:12,180 --> 01:15:14,820 It's not really a singularity. 1208 01:15:14,820 --> 01:15:18,140 But the issue is that when one incorporates, or attempts 1209 01:15:18,140 --> 01:15:20,700 to incorporate, the effects of quantum theory, which nobody 1210 01:15:20,700 --> 01:15:23,660 really knows how to do in a totally reliable way, 1211 01:15:23,660 --> 01:15:27,330 then there are indications that there's 1212 01:15:27,330 --> 01:15:30,600 something dramatic happening at the Schwarzchild horizon. 1213 01:15:30,600 --> 01:15:33,100 The phrase that's often used for what people think 1214 01:15:33,100 --> 01:15:37,360 might be happening at the horizon is the word firewall. 1215 01:15:37,360 --> 01:15:40,240 So whether or not there is a firewall at the horizon, 1216 01:15:40,240 --> 01:15:42,932 is not settled at this point. 1217 01:15:42,932 --> 01:15:44,890 Certainly, though, classical general relativity 1218 01:15:44,890 --> 01:15:46,750 does not predict the firewall. 1219 01:15:46,750 --> 01:15:49,410 If it exists, all the arguments that say it might exist 1220 01:15:49,410 --> 01:15:51,450 are based on the quantum physics of black holes, 1221 01:15:51,450 --> 01:15:54,655 and black hole evaporation, and things like that. 1222 01:15:54,655 --> 01:15:56,030 As you know quantum mechanically, 1223 01:15:56,030 --> 01:15:57,740 the black holes are not stable, either, 1224 01:15:57,740 --> 01:16:01,210 if they evaporate-- as was derived 1225 01:16:01,210 --> 01:16:05,479 by Stephen Hawking in, I think, 1974. 1226 01:16:05,479 --> 01:16:07,020 But that's strictly a quantum effect. 1227 01:16:07,020 --> 01:16:11,005 It would go to 0 as h bar goes to 0, and, at the moment, 1228 01:16:11,005 --> 01:16:13,580 we're only talking about classical general relativity. 1229 01:16:13,580 --> 01:16:16,220 So the black hole that we're describing is perfectly stable. 1230 01:16:16,220 --> 01:16:18,480 And nothing happens if you fall through the horizon. 1231 01:16:18,480 --> 01:16:20,120 Except from the outside, it looks 1232 01:16:20,120 --> 01:16:22,203 like it would take an infinite amount of time just 1233 01:16:22,203 --> 01:16:24,087 to reach the horizon. 1234 01:16:24,087 --> 01:16:24,920 So we'll stop there. 1235 01:16:24,920 --> 01:16:26,420 I guess I'm not going to get to talk 1236 01:16:26,420 --> 01:16:32,360 about the energy associated with radiation. 1237 01:16:32,360 --> 01:16:35,380 But we'll get to that on Thursday. 1238 01:16:35,380 --> 01:16:37,900 So see you folks on Thursday.