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 --> 00:00:10,150 to offer high quality educational resources for free. 5 00:00:10,150 --> 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 00:00:21,227 --> 00:00:21,810 PROFESSOR: OK. 9 00:00:21,810 --> 00:00:23,160 Good morning, everybody. 10 00:00:23,160 --> 00:00:26,870 Welcome to lecture 12 of A286. 11 00:00:26,870 --> 00:00:29,360 I can't think of any announcements for today, 12 00:00:29,360 --> 00:00:31,140 but let me begin by asking if there 13 00:00:31,140 --> 00:00:35,160 are any questions either about logistics or about physics. 14 00:00:39,810 --> 00:00:40,310 OK. 15 00:00:40,310 --> 00:00:42,030 In that case, let's get started. 16 00:00:42,030 --> 00:00:46,810 I want to begin by having a rapid run through of the things 17 00:00:46,810 --> 00:00:49,690 we talked about last time just to firm everything 18 00:00:49,690 --> 00:00:52,880 up in our minds and get us ready to go on. 19 00:00:52,880 --> 00:00:57,570 So, last time we were talking about non-euclidean geometry 20 00:00:57,570 --> 00:00:59,510 in a serious way. 21 00:00:59,510 --> 00:01:02,920 We began by considering the surface of a sphere, just 22 00:01:02,920 --> 00:01:04,810 a two-dimensional sphere embedded 23 00:01:04,810 --> 00:01:08,040 in a three-dimensional space, described 24 00:01:08,040 --> 00:01:11,430 by the simple equation x squared plus y squared plus z squared 25 00:01:11,430 --> 00:01:14,090 equals R squared. 26 00:01:14,090 --> 00:01:17,630 We said that if we wanted to talk about the surface itself, 27 00:01:17,630 --> 00:01:20,300 we'd want to have coordinates for the surface 28 00:01:20,300 --> 00:01:24,930 and not just speak of things in terms of x, y, and z. 29 00:01:24,930 --> 00:01:28,380 So we introduced the standard polar coordinates-- 30 00:01:28,380 --> 00:01:31,560 theta and phi, which are related to x, y, 31 00:01:31,560 --> 00:01:36,160 and z by these fairly well-known equations. 32 00:01:36,160 --> 00:01:38,020 Then we wanted to know the metric 33 00:01:38,020 --> 00:01:40,280 in terms of our new variables theta and phi, 34 00:01:40,280 --> 00:01:44,020 which is the main goal-- to figure out the metric. 35 00:01:44,020 --> 00:01:49,010 So we first considered varying the two variables one 36 00:01:49,010 --> 00:01:50,530 at a time. 37 00:01:50,530 --> 00:01:53,900 By varying theta, we see that the point described 38 00:01:53,900 --> 00:01:58,950 by theta phi would sweep out a circle whose radius is R, 39 00:01:58,950 --> 00:02:02,600 and the angle subtended is d theta. 40 00:02:02,600 --> 00:02:06,310 So the arc length is just R times d theta. 41 00:02:06,310 --> 00:02:09,289 So for varying theta, the arc length 42 00:02:09,289 --> 00:02:12,370 is given by that simple equation. 43 00:02:12,370 --> 00:02:15,100 Similarly, we went on to ask ourselves 44 00:02:15,100 --> 00:02:17,560 what happens when we vary phi. 45 00:02:17,560 --> 00:02:20,970 As we vary phi, the point described by theta 46 00:02:20,970 --> 00:02:24,130 comma phi again sweeps out a circle, 47 00:02:24,130 --> 00:02:27,350 but this time it's a circle in the horizontal plane whose 48 00:02:27,350 --> 00:02:31,090 radius is not R but whose radius has this projection 49 00:02:31,090 --> 00:02:34,610 factor that's R times sine theta. 50 00:02:34,610 --> 00:02:37,770 So the angle is again-- excuse me, 51 00:02:37,770 --> 00:02:42,030 the arc length is again d phi, the angles, times the radius, 52 00:02:42,030 --> 00:02:44,760 but the radius is R sine theta. 53 00:02:44,760 --> 00:02:51,050 So ds, the total arc length, is R times sine theta d phi. 54 00:02:51,050 --> 00:02:52,789 Then, to put them together, we notice 55 00:02:52,789 --> 00:02:55,330 that these two variations are orthogonal to each other, which 56 00:02:55,330 --> 00:02:57,882 you could see pretty directly from the diagram. 57 00:02:57,882 --> 00:02:59,590 So if we do both of them at the same time 58 00:02:59,590 --> 00:03:03,320 and ask what's the total length of the displacement, 59 00:03:03,320 --> 00:03:06,290 it's just a simple application of the Pythagorean theorem. 60 00:03:06,290 --> 00:03:09,600 And we get the sum of the squares. 61 00:03:09,600 --> 00:03:12,500 So, varying theta gives us Rd theta. 62 00:03:12,500 --> 00:03:16,054 Varying phi gives us R sine theta d phi. 63 00:03:16,054 --> 00:03:17,470 And putting them together, we just 64 00:03:17,470 --> 00:03:20,380 get ds squared is the sum of the squares of those, which 65 00:03:20,380 --> 00:03:24,400 is R squared times d theta squared plus sine squared 66 00:03:24,400 --> 00:03:26,310 theta d phi squared. 67 00:03:26,310 --> 00:03:28,770 And that's the standard metric for the service 68 00:03:28,770 --> 00:03:31,720 of a sphere in polar coordinates. 69 00:03:31,720 --> 00:03:32,900 So that was a warm up. 70 00:03:32,900 --> 00:03:35,520 What we really want to do is to elevate that problem one more 71 00:03:35,520 --> 00:03:38,370 dimension, and then we have a model for the universe. 72 00:03:38,370 --> 00:03:40,830 We can use the same method to construct 73 00:03:40,830 --> 00:03:44,870 a three-dimensional space, which is a three-dimensional surface 74 00:03:44,870 --> 00:03:47,920 of a sphere embedded in four euclidean dimensions, 75 00:03:47,920 --> 00:03:50,590 and that becomes a perfectly viable homogeneous, 76 00:03:50,590 --> 00:03:55,690 isotropic, non-euclidean metric that can describe a universe 77 00:03:55,690 --> 00:03:57,310 and, in particular, describes the type 78 00:03:57,310 --> 00:04:00,720 of universe called a closed universe. 79 00:04:00,720 --> 00:04:04,760 So to do that, we introduce one more axis, w. 80 00:04:04,760 --> 00:04:07,930 And we consider the sphere described by x squared plus y 81 00:04:07,930 --> 00:04:11,862 squared plus z squared plus w squared equals R squared. 82 00:04:11,862 --> 00:04:13,320 So it's a three-dimensional surface 83 00:04:13,320 --> 00:04:15,960 of a sphere in four dimensions. 84 00:04:15,960 --> 00:04:18,130 We then need to introduce one more variable 85 00:04:18,130 --> 00:04:20,120 to describe points on the surface, 86 00:04:20,120 --> 00:04:23,540 and we introduce this in the form of a new angle. 87 00:04:23,540 --> 00:04:26,960 The new angle I chose to call psi. 88 00:04:26,960 --> 00:04:31,520 And we measure that angle from the new axis, from the w-axis. 89 00:04:31,520 --> 00:04:33,680 So the new angle psi is simply the angle 90 00:04:33,680 --> 00:04:38,060 from the w-axis, which means that the projection 91 00:04:38,060 --> 00:04:43,600 of our vector from the origin to the point in the w direction 92 00:04:43,600 --> 00:04:49,050 is just the R cosine psi and the projection into the x, y, z 93 00:04:49,050 --> 00:04:53,180 subspace is R times sine psi. 94 00:04:53,180 --> 00:04:57,260 And the four equations that describe x, y, z, and w 95 00:04:57,260 --> 00:04:59,590 are shown there. 96 00:04:59,590 --> 00:05:03,360 And all we did is we set the w-coordinate equal to R times 97 00:05:03,360 --> 00:05:06,510 cosine psi, which is just the statement 98 00:05:06,510 --> 00:05:08,840 that psi measures the angle from the w-axis. 99 00:05:08,840 --> 00:05:10,680 Nothing more. 100 00:05:10,680 --> 00:05:16,370 And then we multiplied x, y, and z by a factor of sine psi, 101 00:05:16,370 --> 00:05:19,330 so that now we still have maintained the condition that x 102 00:05:19,330 --> 00:05:22,030 squared plus y squared plus z squared plus w squared 103 00:05:22,030 --> 00:05:25,180 is equal to R squared, which you can prove directly 104 00:05:25,180 --> 00:05:28,750 by manipulating this using the famous identity sine 105 00:05:28,750 --> 00:05:30,960 squared plus cosine squared equals 1. 106 00:05:30,960 --> 00:05:34,240 Nothing more profound than that. 107 00:05:34,240 --> 00:05:40,249 So, we're now ready to go ahead and find the new metric, 108 00:05:40,249 --> 00:05:42,790 and this time it'll really be something nontrivial, something 109 00:05:42,790 --> 00:05:47,510 you didn't already know from high school. 110 00:05:47,510 --> 00:05:51,420 The new displacement is to vary psi. 111 00:05:51,420 --> 00:05:53,885 If we vary psi, it's really the same story 112 00:05:53,885 --> 00:05:56,460 as we've seen before except in a different plane. 113 00:05:56,460 --> 00:05:59,330 ds is just equal to R times d psi. 114 00:05:59,330 --> 00:06:01,500 I guess, as we vary psi, the point 115 00:06:01,500 --> 00:06:04,450 described by these coordinates makes a full circle of radius 116 00:06:04,450 --> 00:06:06,610 R. 117 00:06:06,610 --> 00:06:10,550 OK, now what we want to do is put all this together. 118 00:06:10,550 --> 00:06:14,760 If we vary psi, we know that ds is equal to R times deep psi. 119 00:06:14,760 --> 00:06:17,340 If we just vary theta or phi, it's 120 00:06:17,340 --> 00:06:18,700 the same thing we had before. 121 00:06:18,700 --> 00:06:20,039 We don't need to rethink it. 122 00:06:20,039 --> 00:06:21,580 All we need to do is remember there's 123 00:06:21,580 --> 00:06:24,880 an extra factor of sine psi in front of all [INAUDIBLE] 124 00:06:24,880 --> 00:06:28,060 in the XYZ subspace. 125 00:06:28,060 --> 00:06:31,080 So if you vary theta or phi, ds squared 126 00:06:31,080 --> 00:06:34,590 is just equal to what we had before for the metric 127 00:06:34,590 --> 00:06:40,340 multiplied by the extra factor of sine squared of psi. 128 00:06:40,340 --> 00:06:44,620 Then to put them together, if we assume for the moment 129 00:06:44,620 --> 00:06:47,690 that they are orthogonal to each other, 130 00:06:47,690 --> 00:06:49,692 then we just add the sum of the squares. 131 00:06:49,692 --> 00:06:50,900 And that is the right answer. 132 00:06:50,900 --> 00:06:53,970 But I'll justify it in a minute. 133 00:06:53,970 --> 00:06:56,990 But jumping ahead and making the assumption 134 00:06:56,990 --> 00:07:00,440 that these separate displacements are always 135 00:07:00,440 --> 00:07:03,130 orthogonal to each other, ds squared 136 00:07:03,130 --> 00:07:04,860 is then just the sum of the squares, 137 00:07:04,860 --> 00:07:10,360 and we get this matrix to describe our closed universe 138 00:07:10,360 --> 00:07:15,790 in terms of the variables psi, theta, and phi. 139 00:07:15,790 --> 00:07:18,090 To prove this orthogonality, which 140 00:07:18,090 --> 00:07:22,770 is crucial for believing that result, 141 00:07:22,770 --> 00:07:24,730 I gave an argument last time, and I'll 142 00:07:24,730 --> 00:07:27,740 outline again on the slides here. 143 00:07:27,740 --> 00:07:30,830 We can consider the two displacement vectors 144 00:07:30,830 --> 00:07:33,970 that we're trying to show to be orthogonal. 145 00:07:33,970 --> 00:07:37,910 dR sub psi is a four-dimensional vector, 146 00:07:37,910 --> 00:07:40,200 which represents the displacement of the point being 147 00:07:40,200 --> 00:07:43,920 described by these coordinates when psi is changed 148 00:07:43,920 --> 00:07:47,600 to psi plus deep psi, infinitesimal change in the psi 149 00:07:47,600 --> 00:07:49,490 coordinate. 150 00:07:49,490 --> 00:07:53,650 Similarly, I'm going to let dR sub theta be 151 00:07:53,650 --> 00:07:56,535 the displacement vector that the point described 152 00:07:56,535 --> 00:08:00,030 by these coordinates undergoes when theta is varied 153 00:08:00,030 --> 00:08:02,397 by an infinitesimal amount, d theta. 154 00:08:02,397 --> 00:08:04,355 And what we're trying to show is that these two 155 00:08:04,355 --> 00:08:05,563 are orthogonal to each other. 156 00:08:05,563 --> 00:08:08,960 So if we do them both, the magnitude of the change 157 00:08:08,960 --> 00:08:11,310 is just the sum of squares, the square root 158 00:08:11,310 --> 00:08:13,050 of the sum of squares. 159 00:08:13,050 --> 00:08:17,220 So, first looking at dR sub theta, 160 00:08:17,220 --> 00:08:21,240 we notice that dR sub theta has no w-component. 161 00:08:21,240 --> 00:08:23,470 And to make that clear, we should go back a couple 162 00:08:23,470 --> 00:08:26,900 slides and look at how w is defined. 163 00:08:26,900 --> 00:08:29,370 w is defined as R times cosine psi. 164 00:08:29,370 --> 00:08:31,550 So if we vary theta, w doesn't change. 165 00:08:31,550 --> 00:08:34,740 It doesn't depend on theta. 166 00:08:34,740 --> 00:08:38,159 So if dR sub theta has no w-component, 167 00:08:38,159 --> 00:08:41,460 it means that when we take the dot product of dR psi 168 00:08:41,460 --> 00:08:43,260 with the dR theta, we want to show 169 00:08:43,260 --> 00:08:45,630 that this is 0 to show that they're orthogonal. 170 00:08:45,630 --> 00:08:48,540 The w-components won't enter, because one of the two 171 00:08:48,540 --> 00:08:50,930 w-components is 0, and the dot product 172 00:08:50,930 --> 00:08:53,881 is a sum of the product of the x-components 173 00:08:53,881 --> 00:08:55,380 plus the product of the y-components 174 00:08:55,380 --> 00:08:57,160 plus the product of the z-components 175 00:08:57,160 --> 00:08:59,360 plus the product of the w-components. 176 00:08:59,360 --> 00:09:01,120 So w-compontents only enter as a product 177 00:09:01,120 --> 00:09:02,610 of the two w-components. 178 00:09:02,610 --> 00:09:06,270 So as long as one of them is 0, there's no contribution there. 179 00:09:06,270 --> 00:09:08,710 So the four-dimensional dot product 180 00:09:08,710 --> 00:09:11,510 reduces to a three-dimensional dot product. 181 00:09:11,510 --> 00:09:14,420 And here I'm introducing a peculiar notation 182 00:09:14,420 --> 00:09:18,060 when I put a subscript-- a superscript, rather-- 3 183 00:09:18,060 --> 00:09:18,729 in a vector. 184 00:09:18,729 --> 00:09:20,520 I just mean take the first three components 185 00:09:20,520 --> 00:09:24,622 and ignore the fourth and think of it as a 3 vector. 186 00:09:24,622 --> 00:09:26,830 So the dot product that we're trying to calculate now 187 00:09:26,830 --> 00:09:29,470 is just the dot product of two 3 vectors-- 188 00:09:29,470 --> 00:09:31,650 the one that we get when we vary psi and the one 189 00:09:31,650 --> 00:09:35,160 that we get when we vary theta. 190 00:09:35,160 --> 00:09:37,790 Next thing to notice is that we can 191 00:09:37,790 --> 00:09:40,690 look at the properties of these two vectors. 192 00:09:40,690 --> 00:09:46,560 And dR psi, the vector we get when we vary psi, I claim 193 00:09:46,560 --> 00:09:49,480 is in the radial direction in this three-dimensional 194 00:09:49,480 --> 00:09:50,620 subspace. 195 00:09:50,620 --> 00:09:52,410 And we can see that by looking again 196 00:09:52,410 --> 00:09:55,630 at these formulas that relate the angles 197 00:09:55,630 --> 00:09:57,670 to the Cartesian coordinates. 198 00:09:57,670 --> 00:10:02,880 When we vary psi, sine psi changes, 199 00:10:02,880 --> 00:10:08,110 but sine psi multiplies x, y, and z all by the same amount. 200 00:10:08,110 --> 00:10:09,870 So sine psi changes. 201 00:10:09,870 --> 00:10:12,775 It changes x, y, and z proportionally. 202 00:10:12,775 --> 00:10:14,650 And if you change x, y, and z proportionally, 203 00:10:14,650 --> 00:10:16,750 it means you're moving in the radial direction 204 00:10:16,750 --> 00:10:18,425 in this three-dimensional subspace. 205 00:10:21,110 --> 00:10:26,560 On the other hand, dR theta is what we get when we vary theta. 206 00:10:26,560 --> 00:10:28,830 And from the beginning, theta was 207 00:10:28,830 --> 00:10:32,450 defined in a way that parametrized the sphere. 208 00:10:32,450 --> 00:10:34,745 So varying theta only moves you along the sphere. 209 00:10:34,745 --> 00:10:38,330 It does not change your distance from the origin. 210 00:10:38,330 --> 00:10:43,200 So varying theta is purely tangential. 211 00:10:43,200 --> 00:10:45,470 So we have a dot product between a radial vector 212 00:10:45,470 --> 00:10:47,360 and a tangential vector, and those are always 213 00:10:47,360 --> 00:10:49,400 orthogonal to each other. 214 00:10:49,400 --> 00:10:53,650 So we get a dot product of 0, as claimed. 215 00:10:53,650 --> 00:10:56,410 So the two original four vectors are 216 00:10:56,410 --> 00:10:59,552 orthogonal to each other, which is what we're trying to prove. 217 00:10:59,552 --> 00:11:01,940 OK, everybody happy with that? 218 00:11:01,940 --> 00:11:02,890 It is a crucial step. 219 00:11:02,890 --> 00:11:04,460 You haven't really gotten the results 220 00:11:04,460 --> 00:11:08,230 unless you know these vectors are orthogonal. 221 00:11:08,230 --> 00:11:11,090 OK, almost done now. 222 00:11:11,090 --> 00:11:14,380 We then later in lecture talked about the implications 223 00:11:14,380 --> 00:11:16,520 of general relativity, and here we 224 00:11:16,520 --> 00:11:19,209 didn't prove what we were claiming. 225 00:11:19,209 --> 00:11:21,000 We just admitted that there are some things 226 00:11:21,000 --> 00:11:23,920 in general relativity that we're just going to have to assume, 227 00:11:23,920 --> 00:11:26,860 and this really is almost the only one. 228 00:11:26,860 --> 00:11:32,990 General relativity tells us how matter causes space to curve. 229 00:11:32,990 --> 00:11:35,720 And it does that in the form of what are called 230 00:11:35,720 --> 00:11:37,730 the Einstein field equations. 231 00:11:37,730 --> 00:11:40,280 And we're not going to learn the Einstein field equations. 232 00:11:40,280 --> 00:11:43,660 That's the subject of a general relativity course. 233 00:11:43,660 --> 00:11:46,520 So we're just going to have to assume 234 00:11:46,520 --> 00:11:50,250 what general relativity tells us about how space curves, 235 00:11:50,250 --> 00:11:53,320 and in particular in this instance, what it tells 236 00:11:53,320 --> 00:11:57,950 us is that the radius of curvature R-- this R that we've 237 00:11:57,950 --> 00:12:00,320 introduced into our metric-- is the radius of curvature 238 00:12:00,320 --> 00:12:07,140 of the space, is related to the matter and motion by R squared 239 00:12:07,140 --> 00:12:10,320 being equal to a squared of t divided by k. 240 00:12:10,320 --> 00:12:13,880 And we did argue last time that, that k in the denominator 241 00:12:13,880 --> 00:12:17,150 really is necessary just to make the units turn out right. 242 00:12:17,150 --> 00:12:19,360 So we really know by dimensional analysis 243 00:12:19,360 --> 00:12:23,200 that this formula has to hold up to some factor. 244 00:12:23,200 --> 00:12:26,870 The fact that the factor is 1 is a fact 245 00:12:26,870 --> 00:12:29,000 about general relativity, which we're not 246 00:12:29,000 --> 00:12:31,520 showing at this point. 247 00:12:31,520 --> 00:12:34,770 When one puts this back into the metric 248 00:12:34,770 --> 00:12:38,620 to express the metric in terms of a of t, 249 00:12:38,620 --> 00:12:42,250 we find finally that the metric can 250 00:12:42,250 --> 00:12:44,950 be written as shown in the box here, 251 00:12:44,950 --> 00:12:47,370 and this is the last equation, where 252 00:12:47,370 --> 00:12:49,460 I've made a substitution of variables. 253 00:12:49,460 --> 00:12:54,430 I replaced the angle psi by a radial coordinate little r, 254 00:12:54,430 --> 00:12:56,940 which is defined to be sine of psi 255 00:12:56,940 --> 00:12:59,640 divided by the square root of k. 256 00:12:59,640 --> 00:13:01,560 And this form of the metric is what's 257 00:13:01,560 --> 00:13:03,660 called the Robertson-Walker metric. 258 00:13:03,660 --> 00:13:05,760 And it's a famous form of the metric. 259 00:13:05,760 --> 00:13:08,840 This is what people normally use. 260 00:13:08,840 --> 00:13:12,700 So that finishes everything we said last time, I think. 261 00:13:12,700 --> 00:13:13,325 Any questions? 262 00:13:16,230 --> 00:13:17,860 Yes. 263 00:13:17,860 --> 00:13:20,935 AUDIENCE: What is the motivation for saying that the-- where 264 00:13:20,935 --> 00:13:23,480 you can describe space as a three-dimensional sphere 265 00:13:23,480 --> 00:13:24,790 in a four space. 266 00:13:24,790 --> 00:13:27,663 Is it because it's only real geometry that, 267 00:13:27,663 --> 00:13:29,532 where there's isotropian [INAUDIBLE]? 268 00:13:29,532 --> 00:13:30,740 PROFESSOR: Yes, that's right. 269 00:13:30,740 --> 00:13:34,330 I was going to be saying that shortly, but yes. 270 00:13:34,330 --> 00:13:37,270 This metric and its open universe counterpart 271 00:13:37,270 --> 00:13:41,240 and flat space together make up the most general 272 00:13:41,240 --> 00:13:45,262 possible metric, which is homogeneous and isotropic. 273 00:13:45,262 --> 00:13:46,720 At this stage, I'm really not going 274 00:13:46,720 --> 00:13:48,110 to claim that anyway, but at this stage, 275 00:13:48,110 --> 00:13:49,890 what we do know is that this metric is 276 00:13:49,890 --> 00:13:51,667 homogeneous and isotropic. 277 00:13:51,667 --> 00:13:53,500 And certainly what we're trying to construct 278 00:13:53,500 --> 00:13:55,670 is metrics, which are homogeneous and isotropic. 279 00:13:55,670 --> 00:13:59,330 But this also is actually the only possibility 280 00:13:59,330 --> 00:14:02,040 within that small class. 281 00:14:02,040 --> 00:14:03,065 Any other questions? 282 00:14:08,890 --> 00:14:09,514 OK. 283 00:14:09,514 --> 00:14:11,555 In that case, we will continue on the blackboard. 284 00:14:24,420 --> 00:14:26,380 So what we have derived so far is 285 00:14:26,380 --> 00:14:28,405 the metric for a closed universe. 286 00:14:31,460 --> 00:14:33,487 Maybe I'll start by getting on the blackboard 287 00:14:33,487 --> 00:14:35,070 the same formula that's up there, just 288 00:14:35,070 --> 00:14:36,480 so I can see it better. 289 00:14:36,480 --> 00:14:39,910 Even though you can probably see equally well either way. 290 00:14:39,910 --> 00:14:46,610 A closed universe is described by ds squared 291 00:14:46,610 --> 00:14:58,870 is equal to a squared of t times dr squared over 1 minus kr 292 00:14:58,870 --> 00:15:14,000 squared plus r squared d theta squared plus sine squared theta 293 00:15:14,000 --> 00:15:14,650 d phi squared. 294 00:15:21,360 --> 00:15:28,070 And to relate this variable r to our previous definitions, 295 00:15:28,070 --> 00:15:34,150 little r is equal to the sine of psi 296 00:15:34,150 --> 00:15:35,910 divided by the square root of k. 297 00:15:40,850 --> 00:15:41,860 OK. 298 00:15:41,860 --> 00:15:42,900 Question back there? 299 00:15:42,900 --> 00:15:48,804 AUDIENCE: Psi is-- sorry this is just [INAUDIBLE] think. 300 00:15:48,804 --> 00:15:51,267 Psi is which angle? 301 00:15:51,267 --> 00:15:53,350 PROFESSOR: OK, the question is psi is which angle. 302 00:15:53,350 --> 00:15:55,150 Psi is the angle we introduced when 303 00:15:55,150 --> 00:15:59,210 we went from two-dimension sphere embedded in three 304 00:15:59,210 --> 00:16:01,010 dimensions to one dimension higher. 305 00:16:01,010 --> 00:16:02,950 Psi is the angle from the new axis. 306 00:16:02,950 --> 00:16:04,540 The angle from the w axis. 307 00:16:04,540 --> 00:16:05,081 AUDIENCE: OK. 308 00:16:09,230 --> 00:16:09,880 PROFESSOR: OK. 309 00:16:09,880 --> 00:16:13,200 So, we've covered a lot of ground here. 310 00:16:13,200 --> 00:16:15,440 We have our first non-euclidean metric 311 00:16:15,440 --> 00:16:17,670 that's visibly important. 312 00:16:17,670 --> 00:16:22,200 But we know from our work on the Newtonian model of the universe 313 00:16:22,200 --> 00:16:25,280 that this little k doesn't have to be positive. 314 00:16:25,280 --> 00:16:29,210 It can be positive, negative, or 0. 315 00:16:29,210 --> 00:16:32,810 If k is 0, the metric is actually 316 00:16:32,810 --> 00:16:35,140 just the metric of a flat space. 317 00:16:35,140 --> 00:16:36,830 But when k is negative, it's a case 318 00:16:36,830 --> 00:16:38,920 that we haven't talked about yet. 319 00:16:38,920 --> 00:16:42,750 So we want to know, what will we write for a metric 320 00:16:42,750 --> 00:16:44,760 if k were negative? 321 00:16:44,760 --> 00:16:49,820 And the answer to that turns out to be perfectly simple. 322 00:16:49,820 --> 00:16:53,440 This formula has a k in it. 323 00:16:53,440 --> 00:16:55,500 Lots of times in our experience-- 324 00:16:55,500 --> 00:16:59,280 I'm sure we all know-- when we write an equation for one 325 00:16:59,280 --> 00:17:01,710 sine of a variable, we find that the same equation 326 00:17:01,710 --> 00:17:04,510 works even if the variable has the other sine. 327 00:17:04,510 --> 00:17:07,960 So if you're buying and selling stocks, if the stocks go up 328 00:17:07,960 --> 00:17:11,880 and the stocks go down, you can use the same equations. 329 00:17:11,880 --> 00:17:14,620 The price today is the price yesterday plus the increment, 330 00:17:14,620 --> 00:17:16,619 and the increment could be positive or negative. 331 00:17:16,619 --> 00:17:19,690 But that equation-- price today equals price yesterday 332 00:17:19,690 --> 00:17:21,930 plus income-- it still works. 333 00:17:21,930 --> 00:17:24,030 And same thing here. 334 00:17:24,030 --> 00:17:27,430 If k it happens to be negative, there's 335 00:17:27,430 --> 00:17:30,766 nothing wrong with this formula. 336 00:17:30,766 --> 00:17:35,070 It, in fact, describes an open universe just as well 337 00:17:35,070 --> 00:17:38,300 as it describes a closed universe. 338 00:17:38,300 --> 00:17:42,090 Now notice, however, that things are a little bit tricky. 339 00:17:42,090 --> 00:17:45,800 If you look at the equation that I wrote immediately below here, 340 00:17:45,800 --> 00:17:47,760 if k is negative, we have the square root 341 00:17:47,760 --> 00:17:51,480 of a negative number here, so the denominator would 342 00:17:51,480 --> 00:17:55,250 be imaginary, and what would that say about r and psi? 343 00:17:55,250 --> 00:17:57,710 It would obviously confuse us. 344 00:17:57,710 --> 00:18:02,020 So you really have to write the metric and the correct form 345 00:18:02,020 --> 00:18:08,370 before you can just change the sign of k. 346 00:18:08,370 --> 00:18:11,090 If we had written the metric in terms of psi 347 00:18:11,090 --> 00:18:14,650 and not made the substitution, we could just as well 348 00:18:14,650 --> 00:18:17,760 have written the metric for a closed universe 349 00:18:17,760 --> 00:18:25,370 as a squared of t divided by k times d psi squared 350 00:18:25,370 --> 00:18:34,080 plus sine squared psi d theta squared plus sine squared 351 00:18:34,080 --> 00:18:39,240 theta d phi squared. 352 00:18:39,240 --> 00:18:41,640 This is an alternative metric for the closed universe. 353 00:18:41,640 --> 00:18:43,160 It's, in fact, where we started. 354 00:18:43,160 --> 00:18:48,170 We then made a substitution, replacing sine psi by little r. 355 00:18:48,170 --> 00:18:50,830 If we had the metric in this form and we said, 356 00:18:50,830 --> 00:18:54,540 well, let k be negative instead of positive, 357 00:18:54,540 --> 00:18:58,200 then notice that a squared is certainly positive, 358 00:18:58,200 --> 00:19:01,150 so we'd have a negative number out here times things which 359 00:19:01,150 --> 00:19:02,970 are also manifestly positive. 360 00:19:02,970 --> 00:19:04,640 We would have a negative definite metric 361 00:19:04,640 --> 00:19:07,340 instead of a positive definite metric. 362 00:19:07,340 --> 00:19:11,420 So we could not change the sign of little k in this formula 363 00:19:11,420 --> 00:19:14,190 and get what we want. 364 00:19:14,190 --> 00:19:16,160 So you have to careful. 365 00:19:16,160 --> 00:19:17,670 It doesn't always work. 366 00:19:17,670 --> 00:19:22,110 But it does work when you write the metric in this form. 367 00:19:22,110 --> 00:19:24,370 Now since it doesn't always work, 368 00:19:24,370 --> 00:19:27,620 and since we haven't really made any sound arguments yet, 369 00:19:27,620 --> 00:19:30,440 I'd like to spend a little time describing-- 370 00:19:30,440 --> 00:19:33,100 I'm not going to do the calculation because it's 371 00:19:33,100 --> 00:19:34,950 too messy, but I'd like spend little time 372 00:19:34,950 --> 00:19:40,700 describing how you would show that this metric works 373 00:19:40,700 --> 00:19:43,960 for an open universe. 374 00:19:43,960 --> 00:19:47,730 So first thing to recognize is-- what 375 00:19:47,730 --> 00:19:51,230 do we actually mean when we say it "works"? 376 00:19:51,230 --> 00:19:54,310 Can somebody tell me what I probably mean when I say that? 377 00:19:58,140 --> 00:19:58,640 Yes. 378 00:19:58,640 --> 00:20:00,890 AUDIENCE: It doesn't have any glaring contradictions? 379 00:20:00,890 --> 00:20:03,240 PROFESSOR: Doesn't have any glaring contradictions. 380 00:20:03,240 --> 00:20:07,547 Yeah, that's good, but we can be more specific, especially 381 00:20:07,547 --> 00:20:09,880 since I'm going to try to describe how we would actually 382 00:20:09,880 --> 00:20:12,130 show it, and it's a little hard to show that something doesn't 383 00:20:12,130 --> 00:20:13,296 have glaring contradictions. 384 00:20:16,980 --> 00:20:19,890 What do we actually care about in constructing these metrics? 385 00:20:22,701 --> 00:20:23,200 Yes? 386 00:20:23,200 --> 00:20:24,991 AUDIENCE: Does the goal that they hold well 387 00:20:24,991 --> 00:20:28,619 in limits, i.e., the flat universes? 388 00:20:28,619 --> 00:20:31,160 PROFESSOR: That the physics will hold well in certain limits, 389 00:20:31,160 --> 00:20:33,530 like they should approach the flat universe and limit. 390 00:20:33,530 --> 00:20:35,960 We certainly do want that to happen, 391 00:20:35,960 --> 00:20:37,430 but there is something else that we 392 00:20:37,430 --> 00:20:39,430 want that doesn't involve taking limits, 393 00:20:39,430 --> 00:20:41,055 because you have different things which 394 00:20:41,055 --> 00:20:43,526 all approach the same limit, of course. 395 00:20:43,526 --> 00:20:46,150 Making sure an answer approaches the right limits is a good way 396 00:20:46,150 --> 00:20:48,960 to test the answer, because most wrong answers will not 397 00:20:48,960 --> 00:20:50,110 have the right limits. 398 00:20:50,110 --> 00:20:52,340 But merely knowing you have the right limits 399 00:20:52,340 --> 00:20:54,750 does not prove that you have the right answer. 400 00:20:54,750 --> 00:20:55,250 Yes? 401 00:20:55,250 --> 00:20:57,749 AUDIENCE: It could also reflect an isotropic and homogeneous 402 00:20:57,749 --> 00:20:58,570 non-euclidean? 403 00:20:58,570 --> 00:20:59,370 PROFESSOR: Exactly. 404 00:20:59,370 --> 00:20:59,890 Exactly. 405 00:20:59,890 --> 00:21:02,270 What we're looking for is a homogeneous and isotropic 406 00:21:02,270 --> 00:21:04,467 non-euclidean space, because that's 407 00:21:04,467 --> 00:21:05,800 what we know about our universe. 408 00:21:05,800 --> 00:21:07,670 It's homogeneous and isotropic, and we're 409 00:21:07,670 --> 00:21:11,410 trying to build a mathematical model of those facts. 410 00:21:11,410 --> 00:21:14,940 So we want homogeneity and isotropy. 411 00:21:14,940 --> 00:21:19,280 If we limit isotropy to isotropy about the origin, which 412 00:21:19,280 --> 00:21:21,910 is enough if we're going to later prove homogeneity, which 413 00:21:21,910 --> 00:21:24,420 will prove that all points are equivalent, 414 00:21:24,420 --> 00:21:27,590 isotropy about the origin is obvious here, 415 00:21:27,590 --> 00:21:30,810 because the angular part is just exactly what we 416 00:21:30,810 --> 00:21:34,890 had for a sphere in three euclidean dimensions. 417 00:21:34,890 --> 00:21:39,790 So it behaves on angles exactly like a euclidean problem, 418 00:21:39,790 --> 00:21:42,580 so we know that it's isotropic. 419 00:21:42,580 --> 00:21:44,500 If you point out that algebraically 420 00:21:44,500 --> 00:21:47,570 as you look at it, it's not obvious that it's isotropic. 421 00:21:47,570 --> 00:21:50,510 We just know where that expression came from. 422 00:21:50,510 --> 00:21:53,100 It came from the sphere. 423 00:21:53,100 --> 00:22:00,500 In terms of theta and phi, it's not manifestly isotropic. 424 00:22:00,500 --> 00:22:02,840 And that's because in choosing theta and phi, 425 00:22:02,840 --> 00:22:05,620 we chose a special point, the North Pole, 426 00:22:05,620 --> 00:22:08,340 to measure our angle theta from. 427 00:22:08,340 --> 00:22:13,550 And the choice of that special point for our coordinate system 428 00:22:13,550 --> 00:22:16,860 broke the isotropy. 429 00:22:16,860 --> 00:22:19,890 But we know that deep down it is isotropic. 430 00:22:19,890 --> 00:22:24,110 And that idea, that you can have such isotropy without having 431 00:22:24,110 --> 00:22:28,740 manifest isotropy is also crucial to how 432 00:22:28,740 --> 00:22:31,950 homogeneity plays out in this metric. 433 00:22:31,950 --> 00:22:34,650 I claim, and we know really, that this metric 434 00:22:34,650 --> 00:22:35,421 is homogeneous. 435 00:22:35,421 --> 00:22:36,920 At least we know where it came from. 436 00:22:36,920 --> 00:22:39,120 It came, again, from the surface of a sphere 437 00:22:39,120 --> 00:22:42,560 one dimension higher than just the angular part. 438 00:22:42,560 --> 00:22:45,830 And then spherical picture--it's obviously homogeneous. 439 00:22:45,830 --> 00:22:48,970 But nonetheless, in building our coordinate system, 440 00:22:48,970 --> 00:22:50,690 we had to break the homogeneity. 441 00:22:50,690 --> 00:22:54,020 We chose a special point-- again, 442 00:22:54,020 --> 00:22:55,770 what we might call the North Pole-- 443 00:22:55,770 --> 00:22:59,330 in this case, the point where w has its maximum value 444 00:22:59,330 --> 00:23:00,860 and made that point special. 445 00:23:00,860 --> 00:23:03,270 The point where w had its maximum value in the x, y, z, w 446 00:23:03,270 --> 00:23:06,970 space is the point which is now the origin 447 00:23:06,970 --> 00:23:09,940 of this coordinate system. 448 00:23:09,940 --> 00:23:15,620 So, if we wanted to prove that this metric really 449 00:23:15,620 --> 00:23:19,360 is homogeneous, we would like to prove it 450 00:23:19,360 --> 00:23:21,720 for the k equals minus 1 case. 451 00:23:21,720 --> 00:23:24,320 But let's first, imaging, what would we 452 00:23:24,320 --> 00:23:26,850 do if we wanted to prove that it was homogeneous 453 00:23:26,850 --> 00:23:30,520 for the k equals plus 1 case or k positive case? 454 00:23:30,520 --> 00:23:32,480 The case that we really think we do understand. 455 00:23:32,480 --> 00:23:34,590 The closed universe. 456 00:23:34,590 --> 00:23:37,370 For the closed universe, this does not look homogeneous. 457 00:23:37,370 --> 00:23:38,910 It looks like the origin is special. 458 00:23:38,910 --> 00:23:41,140 r equals 0 is special. 459 00:23:41,140 --> 00:23:43,560 But we know that it came from the sphere, 460 00:23:43,560 --> 00:23:50,270 and if somebody asked us to prove that, that metric was 461 00:23:50,270 --> 00:23:53,960 homogeneous-- more particularly, somebody 462 00:23:53,960 --> 00:23:57,000 might, for example, challenge us to construct 463 00:23:57,000 --> 00:24:01,760 a coordinate transformation, which would preserve the metric 464 00:24:01,760 --> 00:24:10,330 and map some arbitrary point r 0 theta 0 phi 0 to the origin. 465 00:24:10,330 --> 00:24:12,959 We might undertake that challenge. 466 00:24:12,959 --> 00:24:13,750 It's a lot of work. 467 00:24:13,750 --> 00:24:14,630 We're not going to actually do it. 468 00:24:14,630 --> 00:24:16,406 And I promise I'll never ask you to do it. 469 00:24:16,406 --> 00:24:18,780 But I want to talk a little bit about how we would do it, 470 00:24:18,780 --> 00:24:22,267 because we do have a method, which we know will work. 471 00:24:22,267 --> 00:24:24,100 And knowing that we have a method that works 472 00:24:24,100 --> 00:24:26,570 is all we really need to know. 473 00:24:26,570 --> 00:24:55,360 So suppose we wanted a coordinate transformation 474 00:24:55,360 --> 00:25:08,800 that preserves the form of the metric 475 00:25:08,800 --> 00:25:19,972 and maps some arbitrary point, and I'll 476 00:25:19,972 --> 00:25:24,310 give the coordinates to this arbitrary point a name. 477 00:25:24,310 --> 00:25:29,820 I'll call it r 0, theta 0, and phi 0. 478 00:25:29,820 --> 00:25:31,480 These are coordinates of a point. 479 00:25:31,480 --> 00:25:35,110 So we're going to map this arbitrary point to the origin. 480 00:25:39,800 --> 00:25:42,494 Notice that this is a concrete statement about homogeneity. 481 00:25:42,494 --> 00:25:44,410 If we can map an arbitrary point to the origin 482 00:25:44,410 --> 00:25:45,830 while preserving the metric, we're 483 00:25:45,830 --> 00:25:47,530 really proving that an arbitrary point 484 00:25:47,530 --> 00:25:49,100 is equivalent to the origin. 485 00:25:49,100 --> 00:25:51,350 And if an arbitrary point is equivalent to the origin, 486 00:25:51,350 --> 00:25:53,312 then all points are equivalent to the origin 487 00:25:53,312 --> 00:25:54,520 and equivalent to each other. 488 00:25:54,520 --> 00:25:55,820 We're done. 489 00:25:55,820 --> 00:25:58,410 That proves homogeneity. 490 00:25:58,410 --> 00:26:01,250 So, suppose we wanted to do this. 491 00:26:01,250 --> 00:26:03,280 How would we do it? 492 00:26:03,280 --> 00:26:06,710 The point is that knowing how we got this metric from the sphere 493 00:26:06,710 --> 00:26:10,150 allows us to go back to sphere and rotate the sphere 494 00:26:10,150 --> 00:26:15,970 and rotate back and derive the coordinate transformation 495 00:26:15,970 --> 00:26:17,140 that we want. 496 00:26:17,140 --> 00:26:20,770 And I'll just describe that in slightly more detail. 497 00:26:20,770 --> 00:26:30,930 What we would first do is-- I claim 498 00:26:30,930 --> 00:26:32,900 we can do it in three steps, each of which 499 00:26:32,900 --> 00:26:36,810 we know how to do although they're messy. 500 00:26:36,810 --> 00:26:38,260 I guess I'll start over here. 501 00:26:46,060 --> 00:26:54,510 So step one is to find the x, y, z, w-coordinates 502 00:26:54,510 --> 00:26:56,280 that go with this point. 503 00:26:56,280 --> 00:26:58,430 Because once we have the x, y, z, w-coordinates, 504 00:26:58,430 --> 00:26:59,990 we're in our four-dimensional space 505 00:26:59,990 --> 00:27:01,920 where we know how to do rotations. 506 00:27:01,920 --> 00:27:08,560 So the first thing we do is we just find 507 00:27:08,560 --> 00:27:12,310 the corresponding x, y, z, w-coordinates where x0 is just 508 00:27:12,310 --> 00:27:19,065 the x-coordinate that goes with r0, theta0, and phi0. 509 00:27:19,065 --> 00:27:26,360 And y0 is the y-coordinate that goes with r0, theta0, and phi0. 510 00:27:26,360 --> 00:27:31,984 And z0 is the z-coordinate, and w0 is the w-coordinate. 511 00:27:31,984 --> 00:27:33,650 And these are just points we had before. 512 00:27:33,650 --> 00:27:35,191 I'm writing them symbolically, but we 513 00:27:35,191 --> 00:27:37,740 know how to express the Cartesian coordinates 514 00:27:37,740 --> 00:27:39,961 in terms of the angles. 515 00:27:39,961 --> 00:27:40,460 Yes? 516 00:27:40,460 --> 00:27:44,150 AUDIENCE: Do we have a psi0 as well? 517 00:27:44,150 --> 00:27:46,820 PROFESSOR: r0 replaced psi0. 518 00:27:46,820 --> 00:27:50,784 r0 is the sine of psi divided by root k. 519 00:27:50,784 --> 00:27:52,200 So we only need three coordinates. 520 00:27:52,200 --> 00:27:54,720 We could have different choices of what we call them. 521 00:27:54,720 --> 00:27:56,580 We could've used psi here. 522 00:27:56,580 --> 00:28:00,470 The reason I'm using little r is that I want to, when I'm done, 523 00:28:00,470 --> 00:28:03,410 describe what we would do if k were a negative. 524 00:28:03,410 --> 00:28:05,200 And if k were a negative, we already 525 00:28:05,200 --> 00:28:08,730 said that psi does not actually work. 526 00:28:08,730 --> 00:28:10,605 We have to use different coordinates in order 527 00:28:10,605 --> 00:28:15,750 to smoothly write an open universe coordinate system. 528 00:28:18,820 --> 00:28:19,320 Yes? 529 00:28:19,320 --> 00:28:20,986 AUDIENCE: How is that not like cheating? 530 00:28:20,986 --> 00:28:24,442 Because, I mean, you did define r with the root k, 531 00:28:24,442 --> 00:28:26,150 and now we're just kind of ignoring them. 532 00:28:26,150 --> 00:28:28,025 PROFESSOR: Well that's exactly-- the question 533 00:28:28,025 --> 00:28:29,680 is why is this not cheating? 534 00:28:29,680 --> 00:28:31,190 And the reason it's not cheating is 535 00:28:31,190 --> 00:28:34,290 because of what I'm about to show you. 536 00:28:34,290 --> 00:28:38,300 What I'm saying really is that just setting k negative 537 00:28:38,300 --> 00:28:41,290 is something you might expect to probably work. 538 00:28:41,290 --> 00:28:43,790 I think you have good grounds to expect it to probably work. 539 00:28:43,790 --> 00:28:44,690 Now what we're talking about is how 540 00:28:44,690 --> 00:28:45,981 to actually show that it works. 541 00:28:48,720 --> 00:28:49,220 Yes? 542 00:28:49,220 --> 00:28:51,132 AUDIENCE: Professor, what's the necessity 543 00:28:51,132 --> 00:28:55,912 of defining w as opposed to, I don't know, t? 544 00:28:55,912 --> 00:28:59,367 The traditional, like, t-- 545 00:28:59,367 --> 00:28:59,950 PROFESSOR: OK. 546 00:28:59,950 --> 00:29:02,840 The question is why did I call the fourth variable w and not 547 00:29:02,840 --> 00:29:03,740 t. 548 00:29:03,740 --> 00:29:06,250 The answer is that the variable we're talking about here 549 00:29:06,250 --> 00:29:08,630 is not time. 550 00:29:08,630 --> 00:29:11,650 It's another spatial coordinate. 551 00:29:11,650 --> 00:29:13,910 So, for that reason I think it's better 552 00:29:13,910 --> 00:29:15,700 to call it w than to call it t. 553 00:29:15,700 --> 00:29:18,740 Of course, needless to say, the name of a variable 554 00:29:18,740 --> 00:29:21,060 doesn't have any actual significance. 555 00:29:21,060 --> 00:29:24,040 So I certainly could have called it t equally well, 556 00:29:24,040 --> 00:29:27,399 but I think that would have caused some confusion by people 557 00:29:27,399 --> 00:29:28,940 thinking it was time, which it's not. 558 00:29:31,452 --> 00:29:32,285 Any other questions? 559 00:29:37,640 --> 00:29:38,160 OK. 560 00:29:38,160 --> 00:29:40,550 So, what I'm outlining is the steps 561 00:29:40,550 --> 00:29:43,430 that you would use to prove that this is homogeneous. 562 00:29:43,430 --> 00:29:45,367 I'm doing it for the closed case. 563 00:29:45,367 --> 00:29:47,700 Now the point is that if we can do it for the open case, 564 00:29:47,700 --> 00:29:49,330 we'll prove that the open metric is 565 00:29:49,330 --> 00:29:51,560 what we want it to be-- homogeneous and isotropic. 566 00:29:51,560 --> 00:29:53,220 And that's the only criteria that we 567 00:29:53,220 --> 00:29:56,580 have for goodness of a metric. 568 00:29:56,580 --> 00:29:59,020 So, continuing with the closed case 569 00:29:59,020 --> 00:30:02,520 in mind, the first step of doing this mapping, 570 00:30:02,520 --> 00:30:07,140 to map some arbitrary point to the origin, 571 00:30:07,140 --> 00:30:11,245 is to first find its x, y, z, w-coordinates, 572 00:30:11,245 --> 00:30:14,110 its Cartesian coordinates. 573 00:30:14,110 --> 00:30:17,220 Once we have the Cartesian coordinates, 574 00:30:17,220 --> 00:30:20,020 we know that in the four-dimensional space 575 00:30:20,020 --> 00:30:23,262 we can perform ordinary euclidean rotations. 576 00:30:23,262 --> 00:30:24,970 Rotations in four dimensions [INAUDIBLE], 577 00:30:24,970 --> 00:30:26,040 not three dimensions. 578 00:30:26,040 --> 00:30:27,650 And even rotations in three dimensions 579 00:30:27,650 --> 00:30:30,180 are not all that simple, but nonetheless in principle, 580 00:30:30,180 --> 00:30:33,290 we know how to do rotations in four dimensions. 581 00:30:33,290 --> 00:30:35,140 And we know what we're trying to do 582 00:30:35,140 --> 00:30:37,970 in the four-dimensional picture. 583 00:30:37,970 --> 00:30:39,540 We're trying to rotate this point 584 00:30:39,540 --> 00:30:42,470 to the origin of our coordinate system. 585 00:30:42,470 --> 00:30:44,630 And the origin of our coordinate system-- the way 586 00:30:44,630 --> 00:30:49,350 we've done this mapping-- is to make w equals capital R, 587 00:30:49,350 --> 00:30:52,270 w equal to its maximum value, the center 588 00:30:52,270 --> 00:30:53,720 of our coordinate system. 589 00:30:53,720 --> 00:30:55,970 That's where psi was equal to 0, and now 590 00:30:55,970 --> 00:30:59,480 where our new variable little r is equal to 0. 591 00:30:59,480 --> 00:31:04,110 So we'd like to map this point, whatever it is, to the point 592 00:31:04,110 --> 00:31:07,380 where w has a maximum value, and the other coordinates all 593 00:31:07,380 --> 00:31:08,740 vanish. 594 00:31:08,740 --> 00:31:09,490 So we can do that. 595 00:31:09,490 --> 00:31:11,320 We can find a rotation that does that. 596 00:31:11,320 --> 00:31:13,580 It's not even unique, because you can always 597 00:31:13,580 --> 00:31:16,730 rotate about the final axis. 598 00:31:16,730 --> 00:31:19,100 But in any case, we imagine that we can do that. 599 00:31:19,100 --> 00:31:26,345 And that's step two-- is to find the right rotation. 600 00:31:30,580 --> 00:31:36,140 And a general rotation is a linear transformation, 601 00:31:36,140 --> 00:31:42,440 so you can write it as x prime, y prime, z prime, w prime, 602 00:31:42,440 --> 00:31:45,600 as a four vector, is equal to some 4 603 00:31:45,600 --> 00:32:06,290 by 4 rotation matrix times the original four coordinates. 604 00:32:06,290 --> 00:32:07,710 So an equation of this form would 605 00:32:07,710 --> 00:32:10,260 describe a four-dimensional rotation. 606 00:32:10,260 --> 00:32:16,850 And we in particular want the four-dimensional rotation which 607 00:32:16,850 --> 00:32:28,240 maps-- maybe I'll write it as a similar matrix equation-- what 608 00:32:28,240 --> 00:32:35,440 we want is that the matrix, when it operates on x0, y0, z0, 609 00:32:35,440 --> 00:32:38,947 and w0, which remember are just the four coordinates 610 00:32:38,947 --> 00:32:40,530 that correspond to our original point, 611 00:32:40,530 --> 00:32:49,200 r0, theta0, phi0, we want this to map into 0, 0, 0, 612 00:32:49,200 --> 00:32:53,910 r, which is the four-dimensional description of the origin 613 00:32:53,910 --> 00:32:55,410 of our new coordinate system. 614 00:33:04,531 --> 00:33:06,530 And then finally, step three is the obvious one. 615 00:33:06,530 --> 00:33:09,200 Once we've found the transformation that 616 00:33:09,200 --> 00:33:12,550 maps to the origin, now we just go back 617 00:33:12,550 --> 00:33:16,220 to our original angular coordinates. 618 00:33:16,220 --> 00:33:29,430 So, now we set r prime equal to the radius function 619 00:33:29,430 --> 00:33:35,340 of x prime, y prime, z prime, and w prime. 620 00:33:35,340 --> 00:33:37,080 And this just means the r-coordinate-- 621 00:33:37,080 --> 00:33:40,750 that corresponds to those four euclidean 622 00:33:40,750 --> 00:33:45,520 coordinates and similarly for the other variables. 623 00:33:45,520 --> 00:33:48,570 Theta prime is the theta function 624 00:33:48,570 --> 00:33:52,490 of x prime, y prime, z prime, and w prime. 625 00:33:52,490 --> 00:33:57,930 And phi prime is equal to the phi function. 626 00:33:57,930 --> 00:33:59,940 All these are functions that we know. 627 00:33:59,940 --> 00:34:01,898 I just don't want to write them out explicitly, 628 00:34:01,898 --> 00:34:03,620 because that's a lot of work. 629 00:34:03,620 --> 00:34:10,980 So it's phi of x prime, y prime, z prime, w prime. 630 00:34:10,980 --> 00:34:13,510 So now with the three steps, we have our mapping. 631 00:34:13,510 --> 00:34:18,000 We could start with an arbitrary point, perform the rotation, 632 00:34:18,000 --> 00:34:21,324 and then calculate the angular variables again. 633 00:34:24,489 --> 00:34:26,860 Ok, do people understand what I'm talking about here? 634 00:34:26,860 --> 00:34:27,711 Oh good. 635 00:34:27,711 --> 00:34:28,210 OK. 636 00:34:28,210 --> 00:34:30,777 And the good things is I promise I won't make you do it. 637 00:34:30,777 --> 00:34:32,610 And I've never done it either, to be honest. 638 00:34:32,610 --> 00:34:34,639 But it's obvious that we can do it. 639 00:34:34,639 --> 00:34:39,070 And if we can do this, this would prove and especially 640 00:34:39,070 --> 00:34:40,908 demonstrate homogeneity. 641 00:34:40,908 --> 00:34:42,949 It would be a mapping that would map an arbitrary 642 00:34:42,949 --> 00:34:46,230 point to the origin, proving that, that arbitrary point was 643 00:34:46,230 --> 00:34:50,429 equivalent to the origin as far as the metric is concerned. 644 00:34:50,429 --> 00:34:52,921 And now my claim, I mean, I'm not really 645 00:34:52,921 --> 00:34:54,420 going to prove this either, but it's 646 00:34:54,420 --> 00:34:56,830 a claim that could be verified by going through things. 647 00:34:56,830 --> 00:34:58,829 And it also seems highly plausible-- 648 00:34:58,829 --> 00:35:00,620 that if you looked at each of these steps-- 649 00:35:00,620 --> 00:35:01,994 these are all just algebra steps, 650 00:35:01,994 --> 00:35:04,490 these are all just algebraic equations-- 651 00:35:04,490 --> 00:35:06,570 that they would work just as well for negative k 652 00:35:06,570 --> 00:35:09,440 as they will for positive k. 653 00:35:09,440 --> 00:35:12,825 And by doing these same series of manipulations 654 00:35:12,825 --> 00:35:15,910 for negative k, you would prove that 655 00:35:15,910 --> 00:35:17,750 the Robertson-Walker metric for negative k 656 00:35:17,750 --> 00:35:19,440 is homogeneous, which is our goal. 657 00:35:19,440 --> 00:35:20,731 We already know it's isotropic. 658 00:35:20,731 --> 00:35:22,980 If we can prove it's homogeneous, we're home free. 659 00:35:22,980 --> 00:35:25,120 It's [INAUDIBLE]. 660 00:35:25,120 --> 00:35:25,620 Yes? 661 00:35:25,620 --> 00:35:30,540 PROFESSOR: Does the necessity of placing the origin at r, 662 00:35:30,540 --> 00:35:35,944 like big R, on w is that just because that thing is 663 00:35:35,944 --> 00:35:36,444 expanding? 664 00:35:36,444 --> 00:35:40,380 Or-- I'm kind of having trouble understanding 665 00:35:40,380 --> 00:35:43,824 why it wouldn't be just straight 0s. 666 00:35:43,824 --> 00:35:46,780 Like, why there's that [INAUDIBLE] value [INAUDIBLE]? 667 00:35:46,780 --> 00:35:50,510 PROFESSOR: OK, the question is why does the origin look 668 00:35:50,510 --> 00:35:53,410 like this as opposed to just being all 0s. 669 00:35:53,410 --> 00:35:55,602 The answer is that all 0s is not even in our space. 670 00:35:55,602 --> 00:35:57,560 Because, remember the space we're interested in 671 00:35:57,560 --> 00:35:59,072 is the surface of the sphere. 672 00:35:59,072 --> 00:36:00,530 And the surface of the sphere obeys 673 00:36:00,530 --> 00:36:03,370 x squared plus y squared plus z squared plus w squared 674 00:36:03,370 --> 00:36:05,730 equals r squared. 675 00:36:05,730 --> 00:36:08,830 So if all the coordinates were 0, 676 00:36:08,830 --> 00:36:10,970 it's not part of our space at all. 677 00:36:10,970 --> 00:36:13,090 So we're using this four-dimensional space, 678 00:36:13,090 --> 00:36:15,469 the embedding space, to make things simple. 679 00:36:15,469 --> 00:36:17,010 But in the end, we're only interested 680 00:36:17,010 --> 00:36:18,750 in the three-dimensional surface. 681 00:36:18,750 --> 00:36:20,614 So the origin of our coordinate system 682 00:36:20,614 --> 00:36:22,030 for that three-dimensional surface 683 00:36:22,030 --> 00:36:24,360 had better be in the three-dimensional surface. 684 00:36:24,360 --> 00:36:26,360 So of course, other choices we could have made-- 685 00:36:26,360 --> 00:36:28,568 we could have put it anywhere we want on the surface. 686 00:36:28,568 --> 00:36:30,860 Choosing to put it where w has its maximum value 687 00:36:30,860 --> 00:36:33,910 is just an arbitrary convention. 688 00:36:33,910 --> 00:36:34,410 Yes? 689 00:36:34,410 --> 00:36:36,860 AUDIENCE: So you were saying that most metrics are not 690 00:36:36,860 --> 00:36:38,149 homogeneous? 691 00:36:38,149 --> 00:36:38,940 PROFESSOR: Oh yeah. 692 00:36:38,940 --> 00:36:39,440 Sure. 693 00:36:39,440 --> 00:36:41,370 Most metrics are not homogeneous. 694 00:36:41,370 --> 00:36:44,640 Most objects are not around. 695 00:36:44,640 --> 00:36:46,140 AUDIENCE: When we're doing this math 696 00:36:46,140 --> 00:36:47,726 to show that the Robertson-Walker was 697 00:36:47,726 --> 00:36:50,570 homogeneous, it didn't seem that we 698 00:36:50,570 --> 00:36:53,990 use the exact form of the Robertson-Walker metric 699 00:36:53,990 --> 00:36:54,700 at all in it. 700 00:36:54,700 --> 00:36:56,797 We just said, it is possible to do these-- 701 00:36:56,797 --> 00:36:57,630 PROFESSOR: Well, no. 702 00:36:57,630 --> 00:37:00,690 We did use the form when we made this rotation. 703 00:37:00,690 --> 00:37:03,400 We used the form that, in the euclidean formulation, 704 00:37:03,400 --> 00:37:06,040 we knew that it was rotationally invariant. 705 00:37:06,040 --> 00:37:08,890 And the rotational invariance in the euclidean formulation 706 00:37:08,890 --> 00:37:12,970 is homogeneity and guarantees homogeneity, 707 00:37:12,970 --> 00:37:14,180 but it's a special property. 708 00:37:14,180 --> 00:37:17,510 If it was ellipsoidal shaped instead of spherical, 709 00:37:17,510 --> 00:37:20,830 when you rotated it, it would not be invariant. 710 00:37:20,830 --> 00:37:23,450 If it had any bumps or lumps when you rotated it, 711 00:37:23,450 --> 00:37:26,485 it would not be invariant. 712 00:37:26,485 --> 00:37:26,985 Yes, Aviv. 713 00:37:26,985 --> 00:37:29,615 AUDIENCE: I feel like there should be a fourth step where 714 00:37:29,615 --> 00:37:33,920 you show that the metric doesn't change forms [INAUDIBLE]. 715 00:37:33,920 --> 00:37:35,160 PROFESSOR: Yeah. 716 00:37:35,160 --> 00:37:37,730 We do need to know the metric does not change form, 717 00:37:37,730 --> 00:37:39,980 but I think we do have a guaranteed-- 718 00:37:39,980 --> 00:37:41,564 maybe we should've said some words. 719 00:37:41,564 --> 00:37:43,730 I don't think it's really a fourth step in the sense 720 00:37:43,730 --> 00:37:46,270 that I don't think it requires anymore algebra. 721 00:37:46,270 --> 00:37:50,260 But the point is that we know that the metric is invariant, 722 00:37:50,260 --> 00:37:52,780 that the four-dimensional metric is 723 00:37:52,780 --> 00:37:55,430 invariant under this rotation, which was really 724 00:37:55,430 --> 00:37:57,510 the only non-trivial step. 725 00:37:57,510 --> 00:38:00,420 And otherwise, besides the rotation, all we did 726 00:38:00,420 --> 00:38:07,550 is we went from the r theta phi variables 727 00:38:07,550 --> 00:38:09,150 to the utility euclidean variables, 728 00:38:09,150 --> 00:38:11,233 and then we went back from the euclidean variables 729 00:38:11,233 --> 00:38:13,070 to the r theta phi variables. 730 00:38:13,070 --> 00:38:16,470 But we already know how to go from euclidean variables 731 00:38:16,470 --> 00:38:19,710 to r theta phi variables, and it results in that metric. 732 00:38:22,216 --> 00:38:24,090 And it will still result in that metric when, 733 00:38:24,090 --> 00:38:25,890 if a prime is on all of the coordinates. 734 00:38:25,890 --> 00:38:28,645 AUDIENCE: So you don't have to say, like, 735 00:38:28,645 --> 00:38:31,591 suppose we have a [INAUDIBLE] byproduct. 736 00:38:31,591 --> 00:38:35,354 And do the mapping, figure out what 737 00:38:35,354 --> 00:38:37,728 the ds squared is in terms of r theta phi 738 00:38:37,728 --> 00:38:39,447 to show that it's the same? 739 00:38:39,447 --> 00:38:42,671 Do we have to do that [INAUDIBLE]? 740 00:38:42,671 --> 00:38:44,170 PROFESSOR: OK, the question is do we 741 00:38:44,170 --> 00:38:46,710 have to explicitly show that ds squared is 742 00:38:46,710 --> 00:38:48,210 the same for the new variables as it 743 00:38:48,210 --> 00:38:49,910 was for the old variables. 744 00:38:49,910 --> 00:38:52,480 We certainly want to be convinced that that's true, 745 00:38:52,480 --> 00:38:55,592 and we certainly want to have an argument which convinces us. 746 00:38:55,592 --> 00:38:57,050 But I would claim that if you think 747 00:38:57,050 --> 00:38:59,330 about what underlies these steps, 748 00:38:59,330 --> 00:39:02,740 I only gave a schematic description of them. 749 00:39:02,740 --> 00:39:05,810 If you think about what underlies these steps, 750 00:39:05,810 --> 00:39:09,990 I think it's implicit that the form of the metric 751 00:39:09,990 --> 00:39:11,810 is what we had. 752 00:39:11,810 --> 00:39:15,360 The form of the metric that we had was completely dictated 753 00:39:15,360 --> 00:39:17,540 by the transformation, which expressed 754 00:39:17,540 --> 00:39:22,240 r theta and phi in terms of x, y, z, and w. 755 00:39:22,240 --> 00:39:24,160 And as long as you know the metric in x, y, z, 756 00:39:24,160 --> 00:39:27,870 and w, and that's the euclidean metric both before 757 00:39:27,870 --> 00:39:31,790 and after our rotation, then when you 758 00:39:31,790 --> 00:39:33,568 use the same equations to go from x, y, z, 759 00:39:33,568 --> 00:39:38,270 w to r, theta, and phi, you'll always get the same metric. 760 00:39:38,270 --> 00:39:41,340 So I think we are guaranteed by this process 761 00:39:41,340 --> 00:39:44,170 to get a metric for our new variables, r 762 00:39:44,170 --> 00:39:45,750 prime, theta prime, and phi prime, 763 00:39:45,750 --> 00:39:49,950 which has exactly the same form as the original metric. 764 00:39:49,950 --> 00:39:51,666 Because it's the same calculation again. 765 00:39:51,666 --> 00:39:53,790 The only difference is that this time the variables 766 00:39:53,790 --> 00:39:57,060 all have primes on them. 767 00:39:57,060 --> 00:39:59,560 And the crucial step, the step that was nontrivial, 768 00:39:59,560 --> 00:40:02,580 is the fact that this rotation did not change the metric. 769 00:40:02,580 --> 00:40:04,520 That's where the homogeneity was built in, 770 00:40:04,520 --> 00:40:06,220 that we started with a sphere that we 771 00:40:06,220 --> 00:40:08,800 knew was rotationally invariant. 772 00:40:08,800 --> 00:40:14,060 And this whole calculation just extracts that homogeneity 773 00:40:14,060 --> 00:40:15,635 that we built in from the beginning. 774 00:40:20,070 --> 00:40:20,570 Yes? 775 00:40:20,570 --> 00:40:24,800 AUDIENCE: For k negative, it's r squared over k, 776 00:40:24,800 --> 00:40:26,540 so is r negative? 777 00:40:26,540 --> 00:40:27,350 PROFESSOR: No. 778 00:40:27,350 --> 00:40:32,690 r is still positive when expressed 779 00:40:32,690 --> 00:40:35,800 in terms of x, y, z, and w. 780 00:40:35,800 --> 00:40:38,630 Let me think if I can show that. 781 00:40:42,650 --> 00:40:45,030 I'll show that next time. 782 00:40:45,030 --> 00:40:49,865 It's a little involved, but it will be positive. 783 00:40:49,865 --> 00:40:54,620 I might add that if we look at this formula 784 00:40:54,620 --> 00:40:58,070 and ask what's going on, what's going on-- 785 00:40:58,070 --> 00:41:01,280 we don't necessarily need to know this-- but what's going on 786 00:41:01,280 --> 00:41:04,916 is that in going from the closed case to the open case, 787 00:41:04,916 --> 00:41:07,540 k goes from positive to negative and therefore square root of k 788 00:41:07,540 --> 00:41:09,670 becomes imaginary. 789 00:41:09,670 --> 00:41:12,780 But psi also becomes imaginary. 790 00:41:12,780 --> 00:41:15,920 To describe the relationship between the closed metric 791 00:41:15,920 --> 00:41:18,740 and the open metric, if you're using psi, 792 00:41:18,740 --> 00:41:20,750 you have to say that for the closed metric, 793 00:41:20,750 --> 00:41:23,600 you'll use real values of psi, and for the open metric, 794 00:41:23,600 --> 00:41:26,710 you'll use imaginary values of psi. 795 00:41:26,710 --> 00:41:30,920 And that makes r real and makes this formula work. 796 00:41:30,920 --> 00:41:33,330 And you could also then see how this formula works. 797 00:41:33,330 --> 00:41:35,970 If psi is assigned imaginary values, 798 00:41:35,970 --> 00:41:37,820 then deep psi squared is negative, 799 00:41:37,820 --> 00:41:40,070 so this negative sign cancels that negative sign, 800 00:41:40,070 --> 00:41:43,680 and you again get a positive definite metric. 801 00:41:43,680 --> 00:41:44,180 Yes? 802 00:41:44,180 --> 00:41:47,474 AUDIENCE: So how can we choose the imaginary [INAUDIBLE] 803 00:41:47,474 --> 00:41:48,140 of sine and psi. 804 00:41:48,140 --> 00:41:52,380 Is that just to reflect or is that a choice that we're 805 00:41:52,380 --> 00:41:55,337 making for our model? 806 00:41:55,337 --> 00:41:55,920 PROFESSOR: OK. 807 00:41:55,920 --> 00:41:56,130 Yeah. 808 00:41:56,130 --> 00:41:57,860 The question is why do we choose to use 809 00:41:57,860 --> 00:41:59,970 the imaginary value of psi. 810 00:41:59,970 --> 00:42:06,000 And the answer is perhaps that I failed 811 00:42:06,000 --> 00:42:09,380 to state all the conditions we're interested in when I said 812 00:42:09,380 --> 00:42:13,000 what properties we want this metric to have. 813 00:42:13,000 --> 00:42:15,040 We want the metric that we're seeking 814 00:42:15,040 --> 00:42:18,330 to be homogeneous and isotropic, as we said. 815 00:42:18,330 --> 00:42:20,777 What we didn't say, but what I kind of 816 00:42:20,777 --> 00:42:22,610 had in the back of my mind as an assumption, 817 00:42:22,610 --> 00:42:26,150 is that the metric should also be positive definite. 818 00:42:26,150 --> 00:42:28,200 You can construct other metrics which 819 00:42:28,200 --> 00:42:30,860 are homogeneous and isotropic but not positive definite. 820 00:42:30,860 --> 00:42:32,495 In fact, you would if you let psi 821 00:42:32,495 --> 00:42:34,507 be real and let k be negative. 822 00:42:34,507 --> 00:42:36,840 You'd have a negative definite metric, which would still 823 00:42:36,840 --> 00:42:39,670 be homogeneous and isotropic. 824 00:42:39,670 --> 00:42:41,840 So to enforce all three properties, 825 00:42:41,840 --> 00:42:46,380 you have to use some imagination. 826 00:42:46,380 --> 00:42:49,160 And the easiest way to do it is to write the metric 827 00:42:49,160 --> 00:42:53,320 in the magical form where you can just let k go to minus k, 828 00:42:53,320 --> 00:42:56,900 and that's the Robertson-Walker form here. 829 00:42:56,900 --> 00:43:00,569 And if you write it this way, when you let k go to minus k, 830 00:43:00,569 --> 00:43:02,110 it becomes negative definite, and you 831 00:43:02,110 --> 00:43:03,194 have to scratch your head. 832 00:43:03,194 --> 00:43:05,110 And if you really clever, you might say, well, 833 00:43:05,110 --> 00:43:06,580 if I assign negative-- excuse me, 834 00:43:06,580 --> 00:43:08,410 if I assign imaginary values to psi, 835 00:43:08,410 --> 00:43:11,250 it'll become positive definite again. 836 00:43:11,250 --> 00:43:15,300 That works, but it's less straightforward. 837 00:43:15,300 --> 00:43:15,800 Yes? 838 00:43:15,800 --> 00:43:17,800 AUDIENCE: Do we want it to be positive definite 839 00:43:17,800 --> 00:43:18,800 so that it's visible? 840 00:43:18,800 --> 00:43:19,300 PROFESSOR: Exactly. 841 00:43:19,300 --> 00:43:21,424 We want it to be positive definite so it's visible. 842 00:43:21,424 --> 00:43:24,250 AUDIENCE: Is there any way we can have a negative definite 843 00:43:24,250 --> 00:43:27,066 for a multidimensional space that's reflects 844 00:43:27,066 --> 00:43:30,180 a 3D positive definite space? 845 00:43:30,180 --> 00:43:32,430 PROFESSOR: Is there anyway in higher dimensional space 846 00:43:32,430 --> 00:43:36,980 or something that we can have it maybe have mixed signs 847 00:43:36,980 --> 00:43:38,004 or be negative. 848 00:43:38,004 --> 00:43:39,420 Well in fact, we will see shortly, 849 00:43:39,420 --> 00:43:41,530 because we're going to add in time, 850 00:43:41,530 --> 00:43:43,330 time will occur with the opposite sign, 851 00:43:43,330 --> 00:43:45,710 and it will not be positive definite anymore. 852 00:43:45,710 --> 00:43:48,060 But that will nonetheless correspond to real physics. 853 00:43:48,060 --> 00:43:48,779 AUDIENCE: Right. 854 00:43:48,779 --> 00:43:53,179 So why do we have to force that ds squared into spaces? 855 00:43:53,179 --> 00:43:55,470 PROFESSOR: Because now we are talking only about space, 856 00:43:55,470 --> 00:43:59,320 and certainly for our universe, space is positive. 857 00:43:59,320 --> 00:44:03,600 Now, I might add since you brought this up, 858 00:44:03,600 --> 00:44:08,350 that in relativity, there's no clear distinction between space 859 00:44:08,350 --> 00:44:11,190 and time, so you might wonder why should I 860 00:44:11,190 --> 00:44:14,420 be saying that space is positive and time is negative. 861 00:44:14,420 --> 00:44:17,050 And perhaps I'm oversimplifying a bit when 862 00:44:17,050 --> 00:44:20,200 I say that space is positive and time is negative. 863 00:44:20,200 --> 00:44:25,404 But what is a requirement for general relativity 864 00:44:25,404 --> 00:44:27,820 to match our universe, and therefore a requirement that we 865 00:44:27,820 --> 00:44:30,760 impose on the general relativity theory, 866 00:44:30,760 --> 00:44:34,710 is that the metric have three positive eigenvalues and one 867 00:44:34,710 --> 00:44:35,892 negative eigenvalue. 868 00:44:35,892 --> 00:44:37,600 And that's how it's described, and that's 869 00:44:37,600 --> 00:44:39,830 called the signature of the metric-- the number 870 00:44:39,830 --> 00:44:43,190 of positive eigenvalues and the number of negative eigenvalues. 871 00:44:43,190 --> 00:44:46,690 And reality clearly has-- well, I shouldn't say clearly. 872 00:44:46,690 --> 00:44:48,670 String theory is more dimensions-- 873 00:44:48,670 --> 00:44:52,760 but to describe our macroscopic world, clearly, 874 00:44:52,760 --> 00:44:57,070 we have quantities that we intuitively 875 00:44:57,070 --> 00:44:59,020 identify as three space dimensions in one time 876 00:44:59,020 --> 00:44:59,744 dimension. 877 00:44:59,744 --> 00:45:01,160 And the metric that describes that 878 00:45:01,160 --> 00:45:03,915 is a metric whose signature is three positive eigenvalues 879 00:45:03,915 --> 00:45:05,687 and one negative eigenvalue. 880 00:45:05,687 --> 00:45:07,770 Although, some people reverse the sign conventions 881 00:45:07,770 --> 00:45:10,980 and say it's three negative and 1 positive, 882 00:45:10,980 --> 00:45:12,339 which works just as well. 883 00:45:12,339 --> 00:45:14,130 You can define the metric with either sign. 884 00:45:14,130 --> 00:45:15,694 But this is the one that we're using, 885 00:45:15,694 --> 00:45:17,735 the one that corresponds to space being positive. 886 00:45:20,201 --> 00:45:21,200 OK, any other questions? 887 00:45:24,740 --> 00:45:25,240 OK. 888 00:45:25,240 --> 00:45:28,686 In that case, let's move onward. 889 00:45:28,686 --> 00:45:30,338 What did I want to talk about next? 890 00:45:35,330 --> 00:45:36,482 OK. 891 00:45:36,482 --> 00:45:37,940 I wanted to write on the blackboard 892 00:45:37,940 --> 00:45:42,080 a statement which came about earlier due to questioning 893 00:45:42,080 --> 00:45:44,260 but hasn't been written on the blackboard yet, which 894 00:45:44,260 --> 00:45:48,330 is that there's a theorem which says 895 00:45:48,330 --> 00:45:52,350 that the most general possible three-dimensional metric, which 896 00:45:52,350 --> 00:45:56,475 is homogeneous and isotropic, is this space. 897 00:46:01,740 --> 00:46:18,220 So any three-dimensional, homogeneous, and isotropic 898 00:46:18,220 --> 00:46:31,480 space can be described by the Robertson-Walker metric. 899 00:46:46,100 --> 00:46:50,375 We are not going to prove this, but it is a theorem. 900 00:46:50,375 --> 00:46:52,750 If you want to see a proof, there's a proof, for example, 901 00:46:52,750 --> 00:46:58,330 in Steve Weinberg's gravitation and cosmology textbook. 902 00:46:58,330 --> 00:47:01,600 And in a lot of other books, I'm sure. 903 00:47:01,600 --> 00:47:03,370 But we'll take it for granted. 904 00:47:03,370 --> 00:47:06,050 These are certainly the only homogeneous and isotopic spaces 905 00:47:06,050 --> 00:47:07,670 that we know how to construct. 906 00:47:07,670 --> 00:47:10,100 And in fact, it's the only ones that exist. 907 00:47:10,100 --> 00:47:13,490 Now, I emphasize that if the space is 908 00:47:13,490 --> 00:47:15,780 homogeneous and isotropic, obviously 909 00:47:15,780 --> 00:47:17,840 the metric does not have to look exactly 910 00:47:17,840 --> 00:47:21,880 like that, because you can choose different coordinates. 911 00:47:21,880 --> 00:47:24,330 We could take these coordinates and make some arbitrary 912 00:47:24,330 --> 00:47:26,830 transformation and make the metric look incredibly ugly. 913 00:47:26,830 --> 00:47:29,440 It would still be a homogeneous and isotropic space. 914 00:47:29,440 --> 00:47:32,270 But the claim is that any homogeneous and isotropic space 915 00:47:32,270 --> 00:47:34,530 can be written in metric that looks exactly 916 00:47:34,530 --> 00:47:36,470 like this with a proper choice of coordinates. 917 00:47:45,800 --> 00:47:46,430 OK. 918 00:47:46,430 --> 00:47:49,695 Next thing I want to discuss is the size of these universes. 919 00:47:52,770 --> 00:47:55,740 Size in the generalized sense of, actually 920 00:47:55,740 --> 00:47:57,850 the question of whether it's positive-- 921 00:47:57,850 --> 00:48:00,630 whether it's infinite or finite. 922 00:48:00,630 --> 00:48:04,030 Notice that our closed universe, one 923 00:48:04,030 --> 00:48:08,340 can see from the embedding in four euclidean dimensions, 924 00:48:08,340 --> 00:48:09,390 is finite. 925 00:48:09,390 --> 00:48:12,390 The surface of a sphere is finite. 926 00:48:12,390 --> 00:48:17,207 And one can also see it from the form of the metric, 927 00:48:17,207 --> 00:48:19,540 although one has to think a little bit about how exactly 928 00:48:19,540 --> 00:48:20,910 it works. 929 00:48:20,910 --> 00:48:25,370 If we start at the origin and let r get bigger, 930 00:48:25,370 --> 00:48:27,190 clearly something funny's going to happen 931 00:48:27,190 --> 00:48:30,290 when r is equal to 1 over the square root of k. 932 00:48:30,290 --> 00:48:32,320 That is, when kr squared is equal to 1, 933 00:48:32,320 --> 00:48:34,720 this metric will become singular. 934 00:48:34,720 --> 00:48:38,910 If one goes back to the angular description in terms of psi, 935 00:48:38,910 --> 00:48:41,610 it's clearer what's going on there. 936 00:48:41,610 --> 00:48:49,484 When kr squared is 1, that's exactly where sine psi is one. 937 00:48:49,484 --> 00:48:50,900 And that just means you've reached 938 00:48:50,900 --> 00:48:53,140 the equator of your sphere. 939 00:48:57,902 --> 00:48:58,485 Quick picture. 940 00:49:06,300 --> 00:49:09,530 Size measured from the w-axis. 941 00:49:09,530 --> 00:49:13,680 The equator corresponds to sine psi equals 1. 942 00:49:13,680 --> 00:49:18,910 And then if you continue, psi gets bigger up to pi, 943 00:49:18,910 --> 00:49:21,110 but r starts getting smaller again. 944 00:49:21,110 --> 00:49:22,790 So r is double valued in the sense 945 00:49:22,790 --> 00:49:26,240 that there are two latitudes at which r has the same value-- 946 00:49:26,240 --> 00:49:28,000 one of the northern hemisphere and one 947 00:49:28,000 --> 00:49:29,124 in the southern hemisphere. 948 00:49:31,590 --> 00:49:35,030 But in any case, r starts from 0, goes to some maximum value, 949 00:49:35,030 --> 00:49:36,680 and then goes back to 0. 950 00:49:36,680 --> 00:49:38,982 Everything is finite. 951 00:49:38,982 --> 00:49:41,065 And one could integrate and find the total volume, 952 00:49:41,065 --> 00:49:42,050 and it's finite. 953 00:49:42,050 --> 00:49:46,080 And I think it was or will be a homework problem where 954 00:49:46,080 --> 00:49:49,220 you do exactly that integral at the volume of a closed 955 00:49:49,220 --> 00:49:51,840 universe. 956 00:49:51,840 --> 00:49:56,720 On the other hand, if k-- let's first set it equal to 0. 957 00:49:56,720 --> 00:50:01,180 If k is 0, we have here just the euclidean metric and polar 958 00:50:01,180 --> 00:50:04,640 coordinates describing flat space. 959 00:50:04,640 --> 00:50:09,390 And there's no limit on r. r can become as large as you want. 960 00:50:09,390 --> 00:50:14,440 You can hypothesize that space somehow ends, 961 00:50:14,440 --> 00:50:19,520 but we don't believe-- we don't know of any end to space. 962 00:50:19,520 --> 00:50:23,130 And there are-- in any case, a precise way you could describe 963 00:50:23,130 --> 00:50:25,680 what it would mean for space to end in general relativity 964 00:50:25,680 --> 00:50:29,000 and the usual postulates of general relativity is that , 965 00:50:29,000 --> 00:50:32,410 that doesn't happen-- that space doesn't just have an arbitrary 966 00:50:32,410 --> 00:50:33,600 end. 967 00:50:33,600 --> 00:50:37,650 So the flat case is an infinite space when k is equal to 0. 968 00:50:37,650 --> 00:50:40,260 It's just an infinite euclidean space. 969 00:50:40,260 --> 00:50:44,810 Similarly, if k is negative, then nothing funny 970 00:50:44,810 --> 00:50:47,770 happens as r increases. 971 00:50:47,770 --> 00:50:51,120 So there's no reason not to let r increase to infinity. 972 00:50:51,120 --> 00:50:53,530 Anything else would just be putting in an arbitrary 973 00:50:53,530 --> 00:50:56,010 wall into space without any motivation 974 00:50:56,010 --> 00:50:58,980 for believing that such a wall is there. 975 00:50:58,980 --> 00:51:03,320 And one has to remember that r is not physical distance. 976 00:51:03,320 --> 00:51:05,040 So the fact that r can go to infinity 977 00:51:05,040 --> 00:51:09,064 doesn't necessarily make the space infinite, 978 00:51:09,064 --> 00:51:10,730 but you can calculate physical distance. 979 00:51:25,140 --> 00:51:27,660 We can calculate the physical distance 980 00:51:27,660 --> 00:51:31,139 from the origin to the radius r, and we get that 981 00:51:31,139 --> 00:51:32,430 just by integrating the metric. 982 00:51:32,430 --> 00:51:34,610 The metric tells us what the actual physical length 983 00:51:34,610 --> 00:51:36,990 is for an infinitesimal segment. 984 00:51:36,990 --> 00:51:40,140 That's what the metric meant in the first place. 985 00:51:40,140 --> 00:51:45,840 So the integral that we'd be doing of an a of t out front, 986 00:51:45,840 --> 00:51:51,670 and then we'll be integrating dr prime over the square root 987 00:51:51,670 --> 00:52:00,290 of 1 minus kr prime squared from 0 up to r. 988 00:52:00,290 --> 00:52:02,562 Remember k is negative. 989 00:52:02,562 --> 00:52:04,770 So this is a positive quantity under the square root. 990 00:52:04,770 --> 00:52:08,760 It's not going to cause any problems by vanishing on us. 991 00:52:08,760 --> 00:52:11,900 And this is an integral, which is in fact, doable. 992 00:52:11,900 --> 00:52:16,260 And it's just equal to, still the a 993 00:52:16,260 --> 00:52:19,780 of t in front, which I think I left out in the notes, 994 00:52:19,780 --> 00:52:25,490 times an inverse hyperbolic cinch of the square root 995 00:52:25,490 --> 00:52:34,992 of minus kr over square root of minus k. 996 00:52:34,992 --> 00:52:35,950 Remember k is negative. 997 00:52:35,950 --> 00:52:39,210 See these are all square roots of positive numbers. 998 00:52:39,210 --> 00:52:41,390 And that cinch function, the inverse cinch 999 00:52:41,390 --> 00:52:43,260 can get to be as large as one wants 1000 00:52:43,260 --> 00:52:46,230 by letting r be as large as one wants. 1001 00:52:46,230 --> 00:52:47,697 It grows without bound. 1002 00:52:47,697 --> 00:52:50,030 And that means the physical distance grows without bound 1003 00:52:50,030 --> 00:52:52,960 as r grows to infinity. 1004 00:52:52,960 --> 00:52:54,750 And it grows faster than linear, I think. 1005 00:53:00,392 --> 00:53:01,100 I take that back. 1006 00:53:01,100 --> 00:53:01,690 I'm not sure. 1007 00:53:07,080 --> 00:53:07,580 Yes? 1008 00:53:07,580 --> 00:53:09,080 AUDIENCE: I guess I'm still confused 1009 00:53:09,080 --> 00:53:12,400 because r is sine of psi over square root of k, 1010 00:53:12,400 --> 00:53:15,774 so sine psi is bounded by 1 and negative 1, 1011 00:53:15,774 --> 00:53:19,640 so if we're letting r go to infinity, how-- 1012 00:53:19,640 --> 00:53:23,330 PROFESSOR: That formula only works for the closed case. 1013 00:53:23,330 --> 00:53:26,400 r equals sine psi over root k. 1014 00:53:26,400 --> 00:53:31,870 We can apply it to the open case if we let psi become imaginary. 1015 00:53:31,870 --> 00:53:35,070 But then the bounds that you said no longer apply. 1016 00:53:37,790 --> 00:53:41,210 The sine of an imaginary variable 1017 00:53:41,210 --> 00:53:43,400 is, in fact, the cinch of a real variable. 1018 00:53:58,730 --> 00:54:00,100 OK? 1019 00:54:00,100 --> 00:54:03,160 Next thing I want to point out is 1020 00:54:03,160 --> 00:54:11,150 that the Gauss-Bolyai-Lobachevski 1021 00:54:11,150 --> 00:54:15,180 geometry that you did a homework problem about or it 1022 00:54:15,180 --> 00:54:19,100 was an extra credit problem, so some of you did not. 1023 00:54:19,100 --> 00:54:21,335 But we talked about the Gauss-Bolyai-Lobachevski 1024 00:54:21,335 --> 00:54:21,835 geometry. 1025 00:54:29,200 --> 00:54:37,050 That really is just an open Robertson-Walker, RW 1026 00:54:37,050 --> 00:54:44,725 Robertson-Walker metric, but in two space dimensions. 1027 00:54:53,220 --> 00:54:56,640 But it's completely analogous to the Robertson-Walker metric 1028 00:54:56,640 --> 00:55:00,390 in three space dimensions that we're talking about here. 1029 00:55:00,390 --> 00:55:03,030 So you might recall that Felix Klein construction 1030 00:55:03,030 --> 00:55:04,880 looked very complicated. 1031 00:55:04,880 --> 00:55:07,210 That's because of the coordinates that he used. 1032 00:55:07,210 --> 00:55:09,990 Those coordinates might be simple from some point of view, 1033 00:55:09,990 --> 00:55:12,730 but from the point of view of illustrating homogeneity, 1034 00:55:12,730 --> 00:55:15,170 they're very complicated coordinates. 1035 00:55:15,170 --> 00:55:18,630 And anyway, to physicists, the Robertson-Walker open 1036 00:55:18,630 --> 00:55:21,660 coordinate system is familiar, and the Felix Klein coordinate 1037 00:55:21,660 --> 00:55:22,470 system is not. 1038 00:55:28,550 --> 00:55:29,170 OK. 1039 00:55:29,170 --> 00:55:30,890 If there are no further questions 1040 00:55:30,890 --> 00:55:34,830 about these spatial metrics, the next thing I want to talk about 1041 00:55:34,830 --> 00:55:36,780 is adding time to the picture. 1042 00:55:36,780 --> 00:55:37,570 Because in the end, we're going to be 1043 00:55:37,570 --> 00:55:38,980 interested in a spacetime metric, 1044 00:55:38,980 --> 00:55:40,720 because that's what general relativity is 1045 00:55:40,720 --> 00:55:42,535 all about-- spacetime metrics. 1046 00:56:26,970 --> 00:56:27,470 OK. 1047 00:56:27,470 --> 00:56:30,720 Everything is going to hinge on an important fact 1048 00:56:30,720 --> 00:56:34,880 from special relativity, which we are going to assume but not 1049 00:56:34,880 --> 00:56:38,700 prove, because most you have had courses 1050 00:56:38,700 --> 00:56:40,490 about special relativity elsewhere. 1051 00:56:40,490 --> 00:56:43,800 And for those you who have not, you can either, 1052 00:56:43,800 --> 00:56:46,510 if you wish, read an appendix to lecture 1053 00:56:46,510 --> 00:56:49,540 notes five, in which the fact that I'm about to show you 1054 00:56:49,540 --> 00:56:52,130 is derived, or you could just assume it. 1055 00:56:52,130 --> 00:56:55,100 Whichever you prefer, depending on how much time you have. 1056 00:56:55,100 --> 00:56:58,470 This is not a course about special relativity. 1057 00:56:58,470 --> 00:57:02,250 You're not required to learn how to derive 1058 00:57:02,250 --> 00:57:03,950 the fact that I'm about to write. 1059 00:57:03,950 --> 00:57:06,350 And what I'm about to write starts 1060 00:57:06,350 --> 00:57:11,030 with a definition given any two events. 1061 00:57:11,030 --> 00:57:15,750 An event is a point in spacetime. 1062 00:57:15,750 --> 00:57:18,400 Snapping my finger-- this clearly speaking event. 1063 00:57:18,400 --> 00:57:21,550 It happens at a certain place in a certain time. 1064 00:57:21,550 --> 00:57:27,410 And every real events occupies some small volume of spacetime. 1065 00:57:27,410 --> 00:57:29,870 An ideal event is a point in spacetime. 1066 00:57:29,870 --> 00:57:31,910 And we'll be talking about ideal events. 1067 00:57:31,910 --> 00:57:36,190 Which, our model, as I said, has points in spacetime. 1068 00:57:36,190 --> 00:57:38,230 So given any two events, one can talk 1069 00:57:38,230 --> 00:57:40,800 about a separation between those events. 1070 00:57:40,800 --> 00:57:43,910 And they will be separated in both space and time, 1071 00:57:43,910 --> 00:57:46,330 although either one of those could be 0. 1072 00:57:46,330 --> 00:57:49,500 But they're not both 0, or it's the same event. 1073 00:57:49,500 --> 00:57:55,690 And it's possible to define an interesting quantity, which 1074 00:57:55,690 --> 00:57:59,220 is the difference between the x-coordinates of the two 1075 00:57:59,220 --> 00:58:00,970 events. 1076 00:58:00,970 --> 00:58:03,500 This will be the separation between two events, which I'll 1077 00:58:03,500 --> 00:58:08,100 call a and b, and xa and xb are the x-coordinates 1078 00:58:08,100 --> 00:58:09,055 of those two events. 1079 00:58:11,630 --> 00:58:15,460 And probably you all have enough imagination 1080 00:58:15,460 --> 00:58:20,016 to guess that ya and yb are the y-coordinates of those two 1081 00:58:20,016 --> 00:58:20,515 events. 1082 00:58:24,301 --> 00:58:29,540 And za and zb are the z-coordinates of those events. 1083 00:58:29,540 --> 00:58:33,530 And now here's a surprising one, if you haven't already seen it. 1084 00:58:33,530 --> 00:58:36,030 We're going to have minus c squared 1085 00:58:36,030 --> 00:58:43,570 times ta minus tb squared. 1086 00:58:43,570 --> 00:58:45,335 Now, this is all in special relativity. 1087 00:58:45,335 --> 00:58:47,070 I maybe should clarify. 1088 00:58:47,070 --> 00:58:51,670 We haven't gotten to general relativity or cosmology yet. 1089 00:58:51,670 --> 00:58:53,140 But we need to understand something 1090 00:58:53,140 --> 00:58:54,595 from special relativity first. 1091 00:58:57,330 --> 00:58:59,830 So in special relativity, it's natural to define 1092 00:58:59,830 --> 00:59:02,850 that interval between two events. 1093 00:59:02,850 --> 00:59:04,910 And the magical property, which is 1094 00:59:04,910 --> 00:59:07,510 why we define this integral in the first place, 1095 00:59:07,510 --> 00:59:12,710 is that if we had two different inertial observers, 1096 00:59:12,710 --> 00:59:16,370 we could calculate how the coordinates 1097 00:59:16,370 --> 00:59:18,050 as seen by one observer are related 1098 00:59:18,050 --> 00:59:20,170 to the coordinates as seen by the other observer. 1099 00:59:20,170 --> 00:59:23,700 And that's called the Lorentz transformation. 1100 00:59:23,700 --> 00:59:26,950 And one finds that this particular quantity 1101 00:59:26,950 --> 00:59:31,710 will have exactly the same value to both observers always. 1102 00:59:31,710 --> 00:59:36,350 The two observers will, in general, find different values 1103 00:59:36,350 --> 00:59:39,350 for every one of the four quantities here. 1104 00:59:39,350 --> 00:59:43,020 But when the four quantities are added up with a minus sign 1105 00:59:43,020 --> 00:59:48,750 in front of the time term, the calculations 1106 00:59:48,750 --> 00:59:53,330 would show that you get the same value for inertial observers. 1107 00:59:53,330 --> 00:59:55,640 So this quantity is called Lorentz invariant, 1108 00:59:55,640 --> 00:59:58,400 meaning it's invariant under Lorentz transformations. 1109 00:59:58,400 --> 01:00:00,000 And it makes it a very important thing 1110 01:00:00,000 --> 01:00:03,440 to talk about because in the end, 1111 01:00:03,440 --> 01:00:06,350 physical things have to be essentially Lorentz 1112 01:00:06,350 --> 01:00:09,550 invariant because the laws of physics are Lorentz invariant. 1113 01:00:09,550 --> 01:00:12,710 The laws of physics are the same in all Lorentz frames, 1114 01:00:12,710 --> 01:00:15,190 so ultimately they have to involve quantities, which 1115 01:00:15,190 --> 01:00:17,632 have some simple relationship from one Lorentz 1116 01:00:17,632 --> 01:00:18,740 frame to another. 1117 01:00:22,900 --> 01:00:32,540 OK now, we'd still like to have a clearer notion, I think, 1118 01:00:32,540 --> 01:00:35,317 of what this quantity means. 1119 01:00:35,317 --> 01:00:37,150 It's defined by that equation and principle, 1120 01:00:37,150 --> 01:00:39,610 but it would be nice if we had some understanding of what 1121 01:00:39,610 --> 01:00:40,340 it means. 1122 01:00:40,340 --> 01:00:42,890 And I think the easiest way to describe 1123 01:00:42,890 --> 01:00:47,860 what it means is to look at special frames, 1124 01:00:47,860 --> 01:00:51,250 even though the important feature of this quantity 1125 01:00:51,250 --> 01:00:55,939 is that it has the same numerical value in all frames. 1126 01:00:55,939 --> 01:00:57,980 So the numerical value is the same in all frames, 1127 01:00:57,980 --> 01:01:00,730 but some frames make it easier to interpret it. 1128 01:01:00,730 --> 01:01:02,740 That's what I'm claiming. 1129 01:01:02,740 --> 01:01:07,730 So, what that frame is depends on the value of s squared, 1130 01:01:07,730 --> 01:01:10,670 or at least the sine of it. 1131 01:01:10,670 --> 01:01:16,910 For sab squared greater than 0, which 1132 01:01:16,910 --> 01:01:18,840 means it's dominated by the spatial terms 1133 01:01:18,840 --> 01:01:22,710 there, because those are the positive ones. 1134 01:01:22,710 --> 01:01:29,470 And therefore, the separation is called spacelike. 1135 01:01:36,650 --> 01:01:40,500 Some books put a hyphen between space and like and some don't. 1136 01:01:40,500 --> 01:01:41,045 I don't. 1137 01:01:44,930 --> 01:01:50,270 And there's a theorem that says that if the separation 1138 01:01:50,270 --> 01:01:54,040 between two events is spacelike, there always 1139 01:01:54,040 --> 01:01:56,964 exists a Lorentz frame, an inertial frame, in which 1140 01:01:56,964 --> 01:01:58,505 the two events happen simultaneously. 1141 01:02:10,140 --> 01:02:12,700 Backwards E is a There Exists symbol. 1142 01:02:12,700 --> 01:02:26,940 Then there exists an inertial frame 1143 01:02:26,940 --> 01:02:30,385 in which a and b are simultaneous. 1144 01:02:41,000 --> 01:02:43,120 In that frame, we could look at what 1145 01:02:43,120 --> 01:02:46,000 that formula tells us sab squared is. 1146 01:02:46,000 --> 01:02:48,920 Since they're simultaneous, ta equals tb, 1147 01:02:48,920 --> 01:02:52,020 and therefore the last term does not contribute. 1148 01:02:52,020 --> 01:02:57,190 So in that frame, s squared is just xa minus xb squared 1149 01:02:57,190 --> 01:03:01,329 plus ya minus yb squared plus za minus zb squared. 1150 01:03:01,329 --> 01:03:02,370 And we know what that is. 1151 01:03:02,370 --> 01:03:05,320 That's just the euclidean length, the euclidean distance 1152 01:03:05,320 --> 01:03:08,330 between the two points. 1153 01:03:08,330 --> 01:03:20,130 So, in this frame, sab squared is 1154 01:03:20,130 --> 01:03:35,799 just equal to the distance between events squared. 1155 01:03:35,799 --> 01:03:37,590 Or you take the square root because they're 1156 01:03:37,590 --> 01:03:38,400 positive numbers. 1157 01:03:38,400 --> 01:03:41,270 You could say sab is equal to the distance between the two 1158 01:03:41,270 --> 01:03:42,400 events. 1159 01:03:42,400 --> 01:03:46,490 So when sab squared is positive, sab 1160 01:03:46,490 --> 01:03:48,740 is just the distance between the two events 1161 01:03:48,740 --> 01:03:50,531 in the frame in which they're simultaneous. 1162 01:04:05,160 --> 01:04:11,050 If sab squared is not positive, it could be negative or 0. 1163 01:04:11,050 --> 01:04:13,535 Let me go to the negative case first. 1164 01:04:18,840 --> 01:04:23,800 For sab squared less than 0, for it to be less than 0, 1165 01:04:23,800 --> 01:04:26,280 it means that this expression is dominated by the time term 1166 01:04:26,280 --> 01:04:28,450 because that's the negative term. 1167 01:04:28,450 --> 01:04:30,700 And therefore the separation is called timelike. 1168 01:04:46,880 --> 01:04:49,350 And again, there's a theorem. 1169 01:04:49,350 --> 01:04:52,240 The theorem says that if the separation between two events 1170 01:04:52,240 --> 01:04:54,275 is timelike, there exists a frame in which it 1171 01:04:54,275 --> 01:04:55,525 happened in the same location. 1172 01:05:31,254 --> 01:05:32,712 If they're at the same location, we 1173 01:05:32,712 --> 01:05:35,380 could again look at that equation that 1174 01:05:35,380 --> 01:05:38,490 defines sab squared and ask what form 1175 01:05:38,490 --> 01:05:40,660 does it take in this special frame 1176 01:05:40,660 --> 01:05:42,840 where the two events are the same location. 1177 01:05:42,840 --> 01:05:44,590 That means the first three terms are all 0 1178 01:05:44,590 --> 01:05:46,620 because-- same location. 1179 01:05:46,620 --> 01:05:49,950 It means in that frame, sab squared is negative, 1180 01:05:49,950 --> 01:05:52,190 and it's just minus c squared times the time 1181 01:05:52,190 --> 01:05:53,120 separation squared. 1182 01:06:00,900 --> 01:06:15,330 So in that frame, sab squared is equal to minus 1183 01:06:15,330 --> 01:06:22,860 c squared tau ab squared, where tau ab 1184 01:06:22,860 --> 01:06:24,470 is just equal to the time separation. 1185 01:06:35,750 --> 01:06:39,030 So when sab squared is negative, its meaning 1186 01:06:39,030 --> 01:06:42,370 is as minus c squared times the square of the time separation 1187 01:06:42,370 --> 01:06:44,120 in the frame where the two events happened 1188 01:06:44,120 --> 01:06:46,770 at the same place. 1189 01:06:46,770 --> 01:06:48,880 Now, this notion of the two events happening 1190 01:06:48,880 --> 01:06:54,620 at the same place has a particularly simple intuition 1191 01:06:54,620 --> 01:06:57,340 if the two events that we're talking about happen 1192 01:06:57,340 --> 01:07:02,780 on the same object, like two flashes of the same strobe 1193 01:07:02,780 --> 01:07:05,360 if that strobe is moving at a constant velocity. 1194 01:07:05,360 --> 01:07:06,960 Otherwise, all bets are off. 1195 01:07:06,960 --> 01:07:09,150 But if that strobe is moving at a constant velocity 1196 01:07:09,150 --> 01:07:13,350 so that the frame of the strobe is an inertial frame, 1197 01:07:13,350 --> 01:07:15,530 then the frame of the strobe is, in fact, 1198 01:07:15,530 --> 01:07:17,190 the frame in which the two events 1199 01:07:17,190 --> 01:07:18,590 happened at the same place. 1200 01:07:18,590 --> 01:07:23,100 They both happened at the bulb of the strobe light, which 1201 01:07:23,100 --> 01:07:27,030 in the frame of the strobe is just one point. 1202 01:07:27,030 --> 01:07:32,740 So this time interval, which the sab squared measures, 1203 01:07:32,740 --> 01:07:35,820 is simply the time interval as measured by the object itself, 1204 01:07:35,820 --> 01:07:38,750 is measured by an observer following the strobe. 1205 01:07:38,750 --> 01:07:42,780 And if we place the strobe by a person with a wristwatch, 1206 01:07:42,780 --> 01:07:45,560 this notion of time, which is called proper time, 1207 01:07:45,560 --> 01:07:48,490 is simply the time measured by the person's wristwatch. 1208 01:07:48,490 --> 01:07:51,007 A clock that follows the object so that anything that 1209 01:07:51,007 --> 01:07:53,215 happened to that object happens at the same location. 1210 01:08:03,790 --> 01:08:19,069 So if events happen to the same object, 1211 01:08:19,069 --> 01:08:32,675 ta ab is just the time interval measured by that object. 1212 01:08:52,109 --> 01:08:58,760 And as you give these things names for the spacelike case, 1213 01:08:58,760 --> 01:09:17,410 sab is often called the proper distance between the events, 1214 01:09:17,410 --> 01:09:25,379 and ta ab is the proper time interval between the events. 1215 01:09:39,960 --> 01:09:40,700 OK. 1216 01:09:40,700 --> 01:09:45,069 One more case to do, which is, if it's not 1217 01:09:45,069 --> 01:09:48,859 positive or negative, there's only one remaining 1218 01:09:48,859 --> 01:09:51,359 choice, which is it's got to be 0. 1219 01:09:51,359 --> 01:09:55,340 If sab squared is 0, then again looking back 1220 01:09:55,340 --> 01:09:58,260 at the original definition, it means that the spatial piece 1221 01:09:58,260 --> 01:10:02,580 is equal to minus c squared times the time piece 1222 01:10:02,580 --> 01:10:04,750 so that they all cancel. 1223 01:10:04,750 --> 01:10:07,030 If you think about it, that's precisely the statement 1224 01:10:07,030 --> 01:10:09,210 that these two events are located 1225 01:10:09,210 --> 01:10:12,270 in just the right situation so the light beam that leaves one 1226 01:10:12,270 --> 01:10:14,340 will just arrive at the other. 1227 01:10:14,340 --> 01:10:18,510 Because it says that some of the first three terms, 1228 01:10:18,510 --> 01:10:19,910 which is the distance squared, is 1229 01:10:19,910 --> 01:10:22,145 equal to c squared times the time interval squared. 1230 01:10:22,145 --> 01:10:23,550 And that just says that something 1231 01:10:23,550 --> 01:10:24,800 travels at the speed of light. 1232 01:10:24,800 --> 01:10:28,370 It could travel that distance in that time and go from point a 1233 01:10:28,370 --> 01:10:31,670 to point b or vice versa. 1234 01:10:31,670 --> 01:10:33,010 Only one or the other, not both. 1235 01:10:33,010 --> 01:10:35,800 But it's always one or the other. 1236 01:10:35,800 --> 01:10:41,920 So, for that reason, the interval is called lightlike. 1237 01:10:53,600 --> 01:11:06,010 And it means that a light pulse can travel from a to b, 1238 01:11:06,010 --> 01:11:07,974 or I could have interchanged a or b. 1239 01:11:07,974 --> 01:11:08,890 Everything is squared. 1240 01:11:08,890 --> 01:11:10,265 It doesn't matter which is which. 1241 01:11:12,800 --> 01:11:14,560 Now, there's a peculiar thing here. 1242 01:11:14,560 --> 01:11:18,320 You would think that if a light pulse can travel from a to b, 1243 01:11:18,320 --> 01:11:20,870 there would still be some relevant measure 1244 01:11:20,870 --> 01:11:24,430 to how far apart a and b are. 1245 01:11:24,430 --> 01:11:28,130 However, what we're basically seeing here 1246 01:11:28,130 --> 01:11:31,250 is that if a and b are lightlike separated, 1247 01:11:31,250 --> 01:11:34,050 in any given reference frame you could talk about what the time 1248 01:11:34,050 --> 01:11:36,450 interval is and that will be equal to the space interval 1249 01:11:36,450 --> 01:11:38,920 up to a factor of c. 1250 01:11:38,920 --> 01:11:40,520 But if we imagine looking at this 1251 01:11:40,520 --> 01:11:43,420 at different frames, different inertial frames, 1252 01:11:43,420 --> 01:11:45,220 these two points can get arbitrarily 1253 01:11:45,220 --> 01:11:47,130 close together or arbitrarily far apart, 1254 01:11:47,130 --> 01:11:49,320 depending on what frame we look at them in. 1255 01:11:49,320 --> 01:11:51,990 There is no Lorentz invariant measure 1256 01:11:51,990 --> 01:11:54,230 of how far apart they look. 1257 01:11:54,230 --> 01:11:56,707 The Lorentz invariant-- the only Lorentz invariant measure 1258 01:11:56,707 --> 01:11:58,290 simply tells us that they're lightlike 1259 01:11:58,290 --> 01:11:59,900 related to each other. 1260 01:11:59,900 --> 01:12:01,800 And this leads to some very peculiar issues 1261 01:12:01,800 --> 01:12:05,450 when you try to prove rigorous theorems about relativity. 1262 01:12:05,450 --> 01:12:08,110 You can't really say whether two lightlike points, two 1263 01:12:08,110 --> 01:12:10,500 lightlike separated points are close or far. 1264 01:12:10,500 --> 01:12:14,180 Because there's no real meaning for them to be close or far. 1265 01:12:22,181 --> 01:12:22,680 OK. 1266 01:12:22,680 --> 01:12:25,180 Let me just say one more fact about special relativity, 1267 01:12:25,180 --> 01:12:28,850 and then we'll quit for today and come back on Thursday 1268 01:12:28,850 --> 01:12:31,320 and then talk about how to extend this 1269 01:12:31,320 --> 01:12:32,492 into general relativity. 1270 01:12:32,492 --> 01:12:33,450 OK, there's a question. 1271 01:12:33,450 --> 01:12:33,820 Yes? 1272 01:12:33,820 --> 01:12:34,986 AUDIENCE: Just really quick. 1273 01:12:34,986 --> 01:12:37,271 For the Lorentz invariant to equal to 0, 1274 01:12:37,271 --> 01:12:41,215 does that mean that the objects should 1275 01:12:41,215 --> 01:12:44,666 be moving at the speed of light relative to each other? 1276 01:12:44,666 --> 01:12:46,373 Is it like that? 1277 01:12:46,373 --> 01:12:46,956 PROFESSOR: OK. 1278 01:12:46,956 --> 01:12:49,880 The question is, if the separation is lightlike, 1279 01:12:49,880 --> 01:12:50,750 s squared is 0. 1280 01:12:50,750 --> 01:12:52,309 Does that mean that these two objects 1281 01:12:52,309 --> 01:12:54,600 are moving at the speed of light relative to each other 1282 01:12:54,600 --> 01:12:56,270 or something like that? 1283 01:12:56,270 --> 01:12:57,440 No, it does not. 1284 01:12:57,440 --> 01:12:59,380 It only talks about their positions. 1285 01:12:59,380 --> 01:13:03,530 It doesn't say anything about the motion of these objects. 1286 01:13:03,530 --> 01:13:06,340 It's only a statement about their x and t-coordinates 1287 01:13:06,340 --> 01:13:07,140 at some instant. 1288 01:13:09,850 --> 01:13:11,930 OK, let me still write one more equation 1289 01:13:11,930 --> 01:13:14,867 on the blackboard to kind of finish the special relativity 1290 01:13:14,867 --> 01:13:15,825 part of the discussion. 1291 01:13:20,250 --> 01:13:23,190 In the end, we are interested in the metric. 1292 01:13:23,190 --> 01:13:26,260 And what makes a metric a little bit 1293 01:13:26,260 --> 01:13:28,650 different from a distance function 1294 01:13:28,650 --> 01:13:31,630 is that metrics refer to infinitesimal distances. 1295 01:13:31,630 --> 01:13:34,660 So we're going to want to know the infinitesimal form of that. 1296 01:13:34,660 --> 01:13:36,640 And it's obvious, so it's nothing 1297 01:13:36,640 --> 01:13:37,809 to make a big deal about. 1298 01:13:37,809 --> 01:13:39,850 But I think it's worth writing on the blackboard. 1299 01:13:39,850 --> 01:13:42,080 The infinitesimal form of that equation 1300 01:13:42,080 --> 01:13:47,590 is that ds squared is equal to dx squared plus dy squared 1301 01:13:47,590 --> 01:13:54,110 plus dz squared minus c squared dt squared where dxdy, dz 1302 01:13:54,110 --> 01:13:56,600 and dt are the infinitesimal coordinate 1303 01:13:56,600 --> 01:14:00,420 differences between two events. 1304 01:14:00,420 --> 01:14:05,150 And it's in that form that we'll be beginning from and taking 1305 01:14:05,150 --> 01:14:08,510 off into the world of general relativity 1306 01:14:08,510 --> 01:14:13,200 and the metric of general relativistic spacetimes. 1307 01:14:13,200 --> 01:14:15,610 So we'll continue with this on Thursday.