1 00:00:00,012 --> 00:00:05,007 So having established that if we're going to study the entire universe, we're going 2 00:00:05,007 --> 00:00:08,714 to assume that we are studying a homogeneous isotropic place. 3 00:00:08,714 --> 00:00:13,404 We are going to try to understand what that tells us about the nature of general 4 00:00:13,404 --> 00:00:16,731 relativity in a space that is isotropic and homogeneous. 5 00:00:16,731 --> 00:00:21,482 And so the first thing we need to ask is, what does this tell us about space-time? 6 00:00:21,482 --> 00:00:26,360 So what kind of a space time does describes a universe in which space is 7 00:00:26,360 --> 00:00:31,505 homogeneous and isotropic at all times. And it turns out that in the context of 8 00:00:31,505 --> 00:00:35,257 general relativity. Isotropic and homogeneous means that 9 00:00:35,257 --> 00:00:40,719 there is a particular set of coordinates that we can find in which the curvature 10 00:00:40,719 --> 00:00:46,072 is constant at every point, and this need not mean by the way, that just because a 11 00:00:46,072 --> 00:00:48,817 space is the same everywhere, that it is flat. 12 00:00:48,817 --> 00:00:53,387 In fact, I have drawn here, or someone has drawn here on the right, figures of 13 00:00:53,387 --> 00:00:58,067 three two-dimensional spaces which have the property that they look the same as 14 00:00:58,067 --> 00:01:01,852 observed from every point. These are three surfaces of constant 15 00:01:01,852 --> 00:01:04,727 curvature. The one on the bottom is of course the 16 00:01:04,727 --> 00:01:08,291 Euclidean Plane. It's curvature is exactly zero, and, this 17 00:01:08,291 --> 00:01:12,820 is a familiar space, and it's one of the properties of the Euclidean plane, that 18 00:01:12,820 --> 00:01:16,939 it's invariant under translations. It looks the same from every point. 19 00:01:16,939 --> 00:01:20,832 It's invariant under rotations, has no preferred direction. 20 00:01:20,832 --> 00:01:25,369 Over here at the top, we have a 2 sphere, a 2-dimensional surface given by the 21 00:01:25,369 --> 00:01:29,069 outside of a ball. This is again, a universe if this were 22 00:01:29,069 --> 00:01:32,550 our 2-dimensional space, in which every point is the same as every 23 00:01:32,550 --> 00:01:34,839 other point. In this case of the sphere, this is 24 00:01:34,839 --> 00:01:37,639 obvious because of its invariance under rotations in 3 dimensions. 25 00:01:37,639 --> 00:01:41,455 You can just take this ball and twist it around, making any point the North Pole. 26 00:01:41,455 --> 00:01:44,182 The North Pole is not distinguished by anything. 27 00:01:44,182 --> 00:01:48,411 Unless you're on earth, in which case it's distinguished by the rotation. 28 00:01:48,411 --> 00:01:51,719 But, the ball in itself has no distinguished North Pole. 29 00:01:51,719 --> 00:01:56,304 And, this is a surface of constant positive curvature and then over here is 30 00:01:56,304 --> 00:02:02,356 this, saddle-like shape which has the property that its curvature is constant 31 00:02:02,356 --> 00:02:08,185 and negative. It is far less obvious, why it is that all points on this shape are 32 00:02:08,185 --> 00:02:12,953 the same as every, as all other points, or why there is not a preferred 33 00:02:12,953 --> 00:02:19,250 direction, but there is indeed a, notion of rotation in this, hyperbolic plane 34 00:02:19,250 --> 00:02:24,996 that, preserves, all of the geometry. And so these in fact in 2 dimensions are 35 00:02:24,996 --> 00:02:28,590 the only 3 examples of spaces of constant curvature. 36 00:02:28,590 --> 00:02:33,398 Any space of constant curvature is one of these 3 and in 4 dimensions, in 3 37 00:02:33,398 --> 00:02:38,206 dimensions sorry. We live in a 3 dimensional space, so our space at any 38 00:02:38,206 --> 00:02:43,927 given time, it turns out is analogous, is one of three types of spaces, analogous 39 00:02:43,927 --> 00:02:47,395 to these three. It can be a space of constant positive 40 00:02:47,395 --> 00:02:52,306 curvature, which is a three dimensional extension of the surface of a ball. 41 00:02:52,306 --> 00:02:56,402 In fact, it's called the three sphere, if you could imagine. 42 00:02:56,402 --> 00:03:01,040 A 3-dimsion- a 4-dimensional ball. This would be the 3-dimensional outside 43 00:03:01,040 --> 00:03:05,358 of a 4-dimensional ball and like the 2-dimensional outside of a 3-D ball it 44 00:03:05,358 --> 00:03:10,213 has positive curvature that's constant and all points in all directions are the 45 00:03:10,213 --> 00:03:13,802 same as all others. There's a negative sort of hyperbolic 46 00:03:13,802 --> 00:03:17,775 version of the same thing and then there's plain old flat Newtonian, 47 00:03:17,775 --> 00:03:21,893 Euclidian 3-dimensional space. And space at any given time is one of 48 00:03:21,893 --> 00:03:26,300 these, how do we distinguish these? Well here remember objects are drawn as 49 00:03:26,300 --> 00:03:29,878 surfaces embedded. You're meant to imagine that ball sitting 50 00:03:29,878 --> 00:03:34,032 inside flat three-dimensional space because that's where we're used to 51 00:03:34,032 --> 00:03:37,223 thinking. Remember that I am talking about the ball 52 00:03:37,223 --> 00:03:40,138 in and of itself. You cannot, there is no inside to this 53 00:03:40,138 --> 00:03:43,199 ball or outside. There's no, anymore than there is below 54 00:03:43,199 --> 00:03:45,950 or above the plane. Our, universe is 3 dimensional. 55 00:03:45,950 --> 00:03:49,770 There, it's not sitting, as far as I know, inside any higher dimensional 56 00:03:49,770 --> 00:03:51,174 universe. Though, maybe, 57 00:03:51,174 --> 00:03:54,741 on an optional talk, we'll have that discussion about maybe it is. 58 00:03:54,741 --> 00:03:57,168 at the moment, as far as we know it is not, 59 00:03:57,168 --> 00:04:01,706 so all there is is the 3 dimensions. So what intrinsic property of these 60 00:04:01,706 --> 00:04:06,891 spaces can tell us about the curvature? remember, that it's the properties of the 61 00:04:06,891 --> 00:04:12,194 geodesics, the weird straight lines on curved spaces, and the easiest sort of 62 00:04:12,194 --> 00:04:17,708 most beautiful distinction in the, that the geodesics tell us between 0, 63 00:04:17,708 --> 00:04:22,514 negative, and positive curvature is that while geodesics in the plane are our 64 00:04:22,514 --> 00:04:27,166 favorite Euclidean straight lines and they satisfy the nice axioms of euclidean 65 00:04:27,166 --> 00:04:31,406 geometry. For example, the sum of the angles of a triangle is 180 degrees. 66 00:04:31,406 --> 00:04:36,258 on this saddle point, and it's not completely clear in this picture, but on 67 00:04:36,258 --> 00:04:40,130 the saddle points, geodesic. I'll, I'm going to exaggerate the effect. 68 00:04:41,252 --> 00:04:45,912 Geodesics, in fact, diverge, which tells you that the sum of the angles of a 69 00:04:45,912 --> 00:04:50,522 triangle drawn on a space of negative curvature is going to be less than 180 70 00:04:50,522 --> 00:04:53,072 degrees. On the other hand, on the sphere, 71 00:04:53,072 --> 00:04:56,962 currently geodesics converge. Think about two lines of longitude 72 00:04:56,962 --> 00:05:00,057 leaving the pole. They meet again at the other pole. 73 00:05:00,057 --> 00:05:05,159 And so, the sum of the angles, of a triangle drawn on a sphere will in fact 74 00:05:05,159 --> 00:05:09,955 be larger than 180 degrees. But note that a very small triangle drawn 75 00:05:09,955 --> 00:05:14,985 on the sphere, or a very small triangle drawn on the hyperbola, will appear 76 00:05:14,985 --> 00:05:20,402 approximately to have, will have a sum of the angles, which is very close to 180 77 00:05:20,402 --> 00:05:26,291 because a small segment of anything looks like flat space. the order of magnitude 78 00:05:26,291 --> 00:05:31,498 that tells you how big a triangle needs to be before you measure distinguished 79 00:05:31,498 --> 00:05:36,044 before you measure a distinction between flat space and one of these curved 80 00:05:36,044 --> 00:05:39,694 spaces, is the magnitude of the curvature denoted here by R0. 81 00:05:39,694 --> 00:05:43,302 There is no curvature here. If R0 is big, then the curvature is 82 00:05:43,302 --> 00:05:46,106 large. If R0 is big here, this is a very curved 83 00:05:46,106 --> 00:05:51,688 sphere, which means roughly that the ball it is wrapping is rather small, so curve, 84 00:05:51,688 --> 00:05:54,835 it's more curved. And so all of the positive curvature, 85 00:05:54,835 --> 00:06:00,197 positive constant curvature spaces that can possibly exist are not exactly all 86 00:06:00,197 --> 00:06:03,440 the same. They are all a 3-dimensional version of 87 00:06:03,440 --> 00:06:07,172 the sphere, but of course they can have different sizes. 88 00:06:07,172 --> 00:06:11,927 And so space that if, you decide that space is positively curved, then space at 89 00:06:11,927 --> 00:06:16,741 any given time is given by from the 3-dimensional analog of the sphere, and 90 00:06:16,741 --> 00:06:21,454 then the size of the sphere can change. And as the size of the sphere changes its 91 00:06:21,454 --> 00:06:25,246 curvature can change. A bigger sphere, as I said before, has a 92 00:06:25,246 --> 00:06:28,712 smaller curvature. You need a bigger triangle before you 93 00:06:28,712 --> 00:06:33,477 measure deviation from 180 degrees. Likewise, you can make this hyperbola, 94 00:06:33,477 --> 00:06:37,002 bigger or smaller. And a bigger hyperbola had, had a smaller 95 00:06:37,002 --> 00:06:40,207 curvature. And if you look at the dimensions, it's 96 00:06:40,207 --> 00:06:44,244 the area of the triangle that determines the deviation of the angles. 97 00:06:44,244 --> 00:06:48,272 So if you make this feel twice as big. The curvature becomes 1/4. 98 00:06:48,272 --> 00:06:53,905 the curvature it was previously, so we have these 3 types of universes. 99 00:06:53,905 --> 00:06:58,557 Positive curvature. Positive curvature universes are closed 100 00:06:58,557 --> 00:07:04,685 just like this 2 sphere in the sense that there is no way you can get 2 observers 101 00:07:04,685 --> 00:07:08,454 cannot get as far as they wish from each other. 102 00:07:08,454 --> 00:07:12,194 If you try to go too far. You'd come back around the world, and 103 00:07:12,194 --> 00:07:15,660 come back to where you were. this is not the case for the hyperbola. 104 00:07:15,660 --> 00:07:18,119 In the hyperbola, it is clearly open and infinite. 105 00:07:18,119 --> 00:07:21,585 there is no way to close that. and the geodesics are diverging. 106 00:07:21,585 --> 00:07:24,847 And then we could possibly live in a flat, three dimensional space. 107 00:07:24,847 --> 00:07:27,022 If we live in a flat three dimensional space. 108 00:07:27,022 --> 00:07:33,628 Then, it looks at least infinite and open and so, the meaning of this question mark 109 00:07:33,628 --> 00:07:37,800 will become clear. So, space at any given time in a universe 110 00:07:37,800 --> 00:07:42,988 that is homogeneous and isotropic Is either positively curved with constant 111 00:07:42,988 --> 00:07:46,976 curvature negatively curved with constant curvature or flat. 112 00:07:46,976 --> 00:07:52,229 And our job will be to determine which of these three situations obtains in our 113 00:07:52,229 --> 00:07:56,864 universe and if it's not flat then to determine the actual curvature r 0. 114 00:07:56,864 --> 00:08:01,691 Now the space time geometry of a space time that at any given time is given by 115 00:08:01,691 --> 00:08:06,229 one of these kinds of spaces was described by a whole collection of 116 00:08:06,229 --> 00:08:10,630 people, Friedman, Lemaitre, Robertson, Walker, and goes under the name 117 00:08:10,630 --> 00:08:14,922 Robertson-Walker geometry. Robertson-Walker geometry is the geometry 118 00:08:14,922 --> 00:08:19,694 of space time. So now we are talking about 4-dimensional space-time, in which 119 00:08:19,694 --> 00:08:23,055 space is homogeneous. So at any given time space is say, a 120 00:08:23,055 --> 00:08:27,657 3-sphere, but the size of the sphere, 3-sphere, can vary with time, and so 121 00:08:27,657 --> 00:08:31,767 there is a scale factor that relates. Pick the size of one sphere to be one, 122 00:08:31,767 --> 00:08:36,092 and then bigger spheres will have a bigger than one, smaller times one, space 123 00:08:36,092 --> 00:08:40,652 is smaller, will have a smaller than one. And, because all we're doing in scaling 124 00:08:40,652 --> 00:08:44,967 all of space, the distances between observers that occupy in some sense a 125 00:08:44,967 --> 00:08:49,047 fixed position on the sphere. Pick a point on the sphere and then stay 126 00:08:49,047 --> 00:08:51,832 at that same point when you enlarge the sphere. 127 00:08:51,832 --> 00:08:56,782 the distances between observers sitting at fixed points on this fixed space, so 128 00:08:56,782 --> 00:09:01,707 label points of space by observers and then distances between them will all 129 00:09:01,707 --> 00:09:07,912 scale together when I make space bigger. And by convention we choose the scale 130 00:09:07,912 --> 00:09:13,062 factor A, normalized so that A at T 0 which we call the present, is 1. 131 00:09:13,062 --> 00:09:18,917 In other words, D 0 are the distances at T=0 and then if A of at some future time 132 00:09:18,917 --> 00:09:24,382 is 2, that means that by that time, all distances have doubled and so on. 133 00:09:24,382 --> 00:09:28,576 Now in this Robertson-Walker geometry, you can solve the equations of 134 00:09:28,576 --> 00:09:33,150 electromagnetism for example and you find that light moves and this is not an 135 00:09:33,150 --> 00:09:37,906 obvious fact but it's true, despite the fact that space is growing, light moves 136 00:09:37,906 --> 00:09:42,332 along geodesics. In other words, the motion of a light beam is along these 137 00:09:42,332 --> 00:09:46,053 straightest straight lines that you can draw in space. 138 00:09:46,053 --> 00:09:49,871 In, if space is flat, like moves simply along straight lines, 139 00:09:49,871 --> 00:09:53,621 if space is a sphere, like moves along great circles, and so on. 140 00:09:53,621 --> 00:09:57,074 And it moves of course, this is a physical obvious fact, 141 00:09:57,074 --> 00:10:00,732 with velocity c as measured by any observer anywhere. 142 00:10:00,732 --> 00:10:04,862 So any observer measuring the motion of a light being in their vicinity will 143 00:10:04,862 --> 00:10:09,027 measure of course the speed of light because that's the universal property. 144 00:10:09,027 --> 00:10:11,422 We're talking about a relativistic theory. 145 00:10:11,422 --> 00:10:15,192 And, put these two together, you find these spaces exhibit the kind of 146 00:10:15,192 --> 00:10:19,367 cosmological red-shift that we already discovered underlies the Hubble 147 00:10:19,367 --> 00:10:22,077 discovery. In other words, light that is emitted 148 00:10:22,077 --> 00:10:27,477 from some position in space, at some time t, and is emitted as observed by an 149 00:10:27,477 --> 00:10:33,947 observer at that event with wavelength lambda emission, will then be observed at 150 00:10:33,947 --> 00:10:39,687 certainly some later time, obviously because emission always precedes 151 00:10:39,687 --> 00:10:45,925 absorption or observation, and the ratio between the observed wavelength and the 152 00:10:45,925 --> 00:10:48,724 emitted wave length is the ratio of the scale factors. 153 00:10:48,724 --> 00:10:54,520 In other words, it is the property of the equations of electromagnetism, and we'll 154 00:10:54,520 --> 00:10:57,724 see an example of this much later of how this works out. 155 00:10:57,724 --> 00:11:02,464 That as a light beam travels through space time, if space time is, if space 156 00:11:02,464 --> 00:11:07,443 itself is being scaled by some factor, than the wavelengths of light scale by 157 00:11:07,443 --> 00:11:10,952 exactly the same factor. And this gives rise to the cosmological, 158 00:11:12,003 --> 00:11:15,589 redshift. If a is increasing, then, the wavelength 159 00:11:15,589 --> 00:11:21,071 at which you observe something at a later time, is larger than the wavelength that 160 00:11:21,071 --> 00:11:25,538 which it was emitted. This gives you the cosmological redshift. 161 00:11:25,538 --> 00:11:30,354 Note, this is not a doppler shift. this is the doppler, the, the, the, this 162 00:11:30,354 --> 00:11:34,908 is the, redshift is observed, by observers and emitters and absorbers that 163 00:11:34,908 --> 00:11:39,313 are at rest, if there are perculiar velocities, there will be a doppler shift 164 00:11:39,313 --> 00:11:43,931 on top of this, but these are observers that are as far as I can discuss, address 165 00:11:43,931 --> 00:11:47,944 this. They said at fixed points in this space, and it is the space, if you will, 166 00:11:47,944 --> 00:11:51,161 that is growing. and so it does not require any motion, 167 00:11:51,161 --> 00:11:54,023 and I don't want to think about it as a doppler shift. 168 00:11:54,023 --> 00:11:57,762 And in particular, it is never a blueshift unless spaces shrink. 169 00:11:57,762 --> 00:12:01,867 it is indeed a gravitational effect, but it is a little bit different from the 170 00:12:01,867 --> 00:12:05,106 gravitational redshift we saw in the Pound-Rebka experiment. 171 00:12:05,106 --> 00:12:09,072 Because in the Pound-Rebka experiment, again we had a redshift when light was 172 00:12:09,072 --> 00:12:12,450 propagating up, but a blueshift when light was propagating down. 173 00:12:12,450 --> 00:12:16,009 In this case in some sense, in an expanding universe, all light is 174 00:12:16,009 --> 00:12:18,727 propagating up. There is only a redshift, in a 175 00:12:18,727 --> 00:12:22,201 contracting universe. All light is propagating down, there is 176 00:12:22,201 --> 00:12:25,068 only a blueshift. But if we're both in the same say 177 00:12:25,068 --> 00:12:28,406 expanding universe. and A sees B's light redshifted, then B 178 00:12:28,406 --> 00:12:32,960 will see A's light redshifted, because in both case it takes positive time for the 179 00:12:32,960 --> 00:12:34,760 light to get from A to B or to get from A to B, or to get from B to A. 180 00:12:36,210 --> 00:12:42,949 So does our observation of the Hubble expansion fit into this framework? Well 181 00:12:42,949 --> 00:12:48,830 we have our scale factor, which describes the scale of the universe at time T. 182 00:12:48,830 --> 00:12:54,199 It's a function of T, and for times near to the time T0, which by convention, 183 00:12:54,199 --> 00:12:57,797 remember, is the present, I can approximate this by linear function. 184 00:12:57,797 --> 00:13:01,725 I can approximate anything by linear function and I will call, so it's 1 when 185 00:13:01,725 --> 00:13:04,595 t is equal to t0. This is the first order deviation. We're 186 00:13:04,595 --> 00:13:07,085 going to skate very close to calculus this week. 187 00:13:07,085 --> 00:13:11,144 I'll try to be as clear as I can when we get close to there and I'll designate the 188 00:13:11,144 --> 00:13:15,998 coefficient by the letter H naught, and again this is expected to be valid in 189 00:13:15,998 --> 00:13:20,529 some kind of Newton approximation when this coefficient is small. 190 00:13:20,529 --> 00:13:23,211 I'll justify calling it H naught in a moment. 191 00:13:23,211 --> 00:13:28,450 In this case, by our cosmological redshift understanding, that, if light is 192 00:13:28,450 --> 00:13:33,461 emitted at some time before t 0 and observed, by convention, when I observe 193 00:13:33,461 --> 00:13:38,762 something, I'll be observing it at the present time t0 unless I indicate 194 00:13:38,762 --> 00:13:44,010 otherwise. And so the wavelength at which light is observed divided by the 195 00:13:44,010 --> 00:13:49,545 wavelength at which it is emitted is the ratio of the scale factors. Here should 196 00:13:49,545 --> 00:13:52,323 be A of t0, but a of t0 by convention is 1. 197 00:13:52,323 --> 00:13:59,417 And, what is this? This is one over the scale factor at t emission. So it's 1 198 00:13:59,417 --> 00:14:06,912 over 1 plus h0 times t emission minus t0. But, because this number is small, I can 199 00:14:06,912 --> 00:14:13,992 use Newton's approximation. This is, 1 plus h0 times this inverse. 200 00:14:13,992 --> 00:14:24,282 This is, with Newton's approximation, the same as 1-H0(Tem-T0) and 1-H0(Tem-T0) is 201 00:14:24,282 --> 00:14:32,027 the same as 1+H0(T0-Tem). So this is the expression I'm going to 202 00:14:32,027 --> 00:14:38,387 use it's valid. When H0 times t minus t0 is a small 203 00:14:38,387 --> 00:14:50,147 number, and so to this order I can set 1 plus z equal to that, in other words, z 204 00:14:50,147 --> 00:14:56,005 equals H0 times t0 minus t emission This is remember, a positive number t emission 205 00:14:56,005 --> 00:14:58,264 proceeds t0. So, this is the red shift. 206 00:14:58,264 --> 00:15:02,682 On the other hand, to the same order it'll turn out that the distance light 207 00:15:02,682 --> 00:15:06,169 traveled is the speed of light times the time it is traveling. 208 00:15:06,169 --> 00:15:10,016 This will not always be true because, the universe is changing. 209 00:15:10,016 --> 00:15:14,710 We'll see how that gets corrected. And so, comparing this expression to that 210 00:15:14,710 --> 00:15:20,195 expression, I find that indeed I obtained the Hubble Relation as a low order Newton 211 00:15:20,195 --> 00:15:23,899 type approximation. That small distances and times, small 212 00:15:23,899 --> 00:15:29,190 relative to what? To the amount, to the distance light could have traveled since 213 00:15:29,190 --> 00:15:33,311 the Big Bang, times small relative to the time since the Big Bang. 214 00:15:33,311 --> 00:15:37,722 And that those distances, and therefore, redshifts, small compared 215 00:15:37,722 --> 00:15:41,657 to one, and for under those circumstances, I find the Hubble, 216 00:15:41,657 --> 00:15:44,377 condition. And so, the Hubble constant is the 217 00:15:44,377 --> 00:15:47,397 coefficient of this expansion of the scale factor. 218 00:15:47,397 --> 00:15:51,077 That's the Hubble H0, because it's the Hubble constant today. 219 00:15:51,077 --> 00:15:55,472 The Hubble constant need not, in fact, be a constant. At any other time, 220 00:15:55,472 --> 00:16:01,638 I will define the Hubble constant, erasing all my scribbles, by if you have 221 00:16:01,638 --> 00:16:05,336 a time t*. And for times t not to far from t*, 222 00:16:05,336 --> 00:16:11,956 the Hubble, the skill factor is given by the skill constant at t* times 1 plus 223 00:16:11,956 --> 00:16:17,589 some number times t minus t* , then I would designate this number as the Hubble 224 00:16:17,589 --> 00:16:21,122 constant at time t*. So now imagine, if you will, that this 225 00:16:21,122 --> 00:16:25,577 expression is in fact correct. It's not an approximation, it's an exact equation, 226 00:16:25,577 --> 00:16:28,142 as we might have mistakenly thought last week. 227 00:16:28,142 --> 00:16:32,237 Then what does this tell me about the Hubble Constant? Well you'd think the 228 00:16:32,237 --> 00:16:35,812 Hubble Constant's not changing. Look, this equation always holds. 229 00:16:35,812 --> 00:16:40,636 Yes, but remember the Hubble Constant is the coefficient of the fractional change 230 00:16:40,636 --> 00:16:43,674 in a. So take some later time, and notice that 231 00:16:43,674 --> 00:16:49,527 in a given time interval after that, for every second past, t*, the scale factor 232 00:16:49,527 --> 00:16:54,466 increases by the same amount. But if t* is later than the present, it's 233 00:16:54,466 --> 00:16:59,974 later than t0, then, a, by that time will have grown. And so, the Hubble constant, 234 00:16:59,974 --> 00:17:03,217 which is, the, rate of relative growth of the scale 235 00:17:03,217 --> 00:17:07,044 factor will in fact have decreased. So if you want to understand constant 236 00:17:07,044 --> 00:17:11,181 expansion, to mean this equation where the scale factor is a linear function of 237 00:17:11,181 --> 00:17:14,160 t, then that in fact means a decreasing Hubble constant. 238 00:17:14,160 --> 00:17:17,873 In particular, we have no reason to assume the Hubble constant does not 239 00:17:17,873 --> 00:17:21,935 depend on time and understanding it's time dependance will be something we'll 240 00:17:21,935 --> 00:17:26,547 spend some work on, to find out what it does in our real 241 00:17:26,547 --> 00:17:28,697 work. So, in our real universe. 242 00:17:28,697 --> 00:17:33,647 So this is how Hubble fits into Robertson-Walker as a first approximation 243 00:17:33,647 --> 00:17:38,822 to the actual cosmological redshift. we said that one of the issues in a 244 00:17:38,822 --> 00:17:43,882 universe where distances are changing is what do you mean by distance? 245 00:17:43,882 --> 00:17:50,011 So we will have 3, at least, different ways to define the distance between 2 246 00:17:50,011 --> 00:17:53,849 points. One is, remember we have these co-moving 247 00:17:53,849 --> 00:18:00,237 points and so we label each point on, say the sphere by some name, x and then the 248 00:18:00,237 --> 00:18:04,036 distances are, the distances between the x's, and then 249 00:18:04,036 --> 00:18:09,103 at previous times we have the same sphere with the same points except we considered 250 00:18:09,103 --> 00:18:13,787 all sizes smaller, and in the future we'll have the same sphere with the same 251 00:18:13,787 --> 00:18:18,403 points, but all sizes are bigger. And so the distance alone, the canonical 252 00:18:18,403 --> 00:18:21,856 sphere, is D0. This is the coordinate distance, we call 253 00:18:21,856 --> 00:18:23,622 it. you might think of that, 254 00:18:23,622 --> 00:18:27,179 as the distance right now, because right now the scale factor is 1. 255 00:18:27,179 --> 00:18:31,039 So if you want to define something that measures the actual distance between 2 256 00:18:31,039 --> 00:18:34,816 objects right now, and in particular, will reduce for objects that are near 257 00:18:34,816 --> 00:18:38,427 enough for me to make a measurement. For my right hand and my left hand, if 258 00:18:38,427 --> 00:18:42,362 you ask what's the distance between my right hand and my left hand, if you apply 259 00:18:42,362 --> 00:18:46,212 this crazy Friedman's, Robertson-Walker coordinate system, then this will be 260 00:18:46,212 --> 00:18:50,264 measured by the distance in coordinates between the coordinates of this finger 261 00:18:50,264 --> 00:18:54,248 and the coordinates of that finger, evaluated at the scale factor of 1, which 262 00:18:54,248 --> 00:18:56,153 is, corresponds to right now. Okay. 263 00:18:56,153 --> 00:19:00,112 But it's very rare that we want distances between objects that are very near. 264 00:19:00,112 --> 00:19:04,345 And, in general, the coordinate distance between those 2 things is not going to be 265 00:19:04,345 --> 00:19:08,136 a physical observable, because you cannot have time to lay a ruler, a large 266 00:19:08,136 --> 00:19:12,507 fraction of the size of the universe, before the universe changes on you. 267 00:19:12,507 --> 00:19:17,659 So what can you actually measure? Well, one thing we measure that is associated 268 00:19:17,659 --> 00:19:21,030 to distance, is we measure the small angle formula. 269 00:19:21,030 --> 00:19:25,581 We love our small angle formula. So, imagine that somewhere out in the 270 00:19:25,581 --> 00:19:30,207 universe is a galaxy say, and the galaxy has size D, and the galaxy is at 271 00:19:30,207 --> 00:19:35,081 coordinate distance, D0 for me. So imagine that we're in flat 272 00:19:35,081 --> 00:19:41,687 space, in a flat 2-dimensional universe, so this is the universe at present, and 273 00:19:41,687 --> 00:19:45,735 here is the galaxy and the galaxy is at a distance D0. 274 00:19:45,735 --> 00:19:51,853 OK, now suppose that I want to make a small angle approximation and measure the 275 00:19:51,853 --> 00:19:56,310 apparent angular size of the galaxy. Everything works fine. 276 00:19:56,310 --> 00:20:01,202 Remember, light moves in straight lines in this coordinate plane. 277 00:20:01,202 --> 00:20:06,258 that was our assertion and that is the facts about Robertson-Walker. 278 00:20:06,258 --> 00:20:12,145 But however, when I look at this galaxy, remember, I am looking at this galaxy as 279 00:20:12,145 --> 00:20:17,529 it was a long time ago. And so in fact, the distance that I am 280 00:20:17,529 --> 00:20:24,043 looking at, if you will, is not the size of the galaxy itself, but in fact, the 281 00:20:24,043 --> 00:20:28,683 size of the galaxy as it looks in today's coordinates. 282 00:20:28,683 --> 00:20:35,412 So in other words, that galaxy is sitting at a particular point in the plane. 283 00:20:35,412 --> 00:20:37,894 This is a measurement made in today's plane. 284 00:20:37,894 --> 00:20:42,370 however, the two sides of the galaxy in today's plane the galaxy of course 285 00:20:42,370 --> 00:20:45,290 doesn't expand because it's gravitationally bound. 286 00:20:45,290 --> 00:20:48,826 We'll talk about that. But were the, did a, had I put two points 287 00:20:48,826 --> 00:20:53,180 on the 2 sides of the galaxy, they would have been much bigger today, than they 288 00:20:53,180 --> 00:20:55,967 were then. Therefore the distance between the two 289 00:20:55,967 --> 00:21:00,252 sides of the galaxy in 2 day's coordinates is the size of the galaxy 290 00:21:00,252 --> 00:21:06,032 times the ratio of today's scale factor. But the scale factor at the time of 291 00:21:06,032 --> 00:21:09,694 emission, and if I measure our redshift z for the 292 00:21:09,694 --> 00:21:13,406 light, then the ratio of the scale factors is 1 plus z. 293 00:21:13,406 --> 00:21:19,583 In other words, everything looks, bigger in an expanding universe, because it 294 00:21:19,583 --> 00:21:25,537 looks as big as it would have been, would have looked, had its 2 edges expanded 295 00:21:25,537 --> 00:21:30,581 with the universe. that gives me this expression for the, 296 00:21:30,581 --> 00:21:35,112 small angle formula. I define an angular size distance by, 297 00:21:35,112 --> 00:21:40,557 angular size distance is the distance that you'd put in a small angle formula, 298 00:21:40,557 --> 00:21:45,219 if you wanted it to make sense. If D over D were alpha over 1 radian, 299 00:21:45,219 --> 00:21:49,703 over 206265 arc seconds. And then, from this expression, we see 300 00:21:49,703 --> 00:21:55,073 that the angular size distance is related to coordinate distance or physical 301 00:21:55,073 --> 00:22:00,436 distance today by the expression, take that, divide 1+z, where z is the redshift 302 00:22:00,436 --> 00:22:04,831 at which you observe the light from the object whose angular size you are 303 00:22:04,831 --> 00:22:07,479 measuring. Of course for nearby objects, z is 0. 304 00:22:07,479 --> 00:22:11,798 This is the same old small angle formula. You don't have to change your 305 00:22:11,798 --> 00:22:16,606 understanding of the dimensions of the moon but for distant objects, this is 306 00:22:16,606 --> 00:22:21,516 what the expansion of the universe tells you. You measure an angle, that is the 307 00:22:21,516 --> 00:22:26,564 angle that you would have seen were the two sides of the objects freely moving 308 00:22:26,564 --> 00:22:31,104 and freely falling and expanding. So this is one notion of distance. 309 00:22:31,104 --> 00:22:36,058 There's another equation in which we use distance a lot. One was the small angle 310 00:22:36,058 --> 00:22:39,922 formula. The other of course is the brightness equation. 311 00:22:39,922 --> 00:22:44,223 B is L over 4 pi D squared. is it, which D should we plug in, if we 312 00:22:44,223 --> 00:22:50,692 have an object of no luminosity, and we measure it's brightness, to what D does 313 00:22:50,692 --> 00:22:55,902 that core, th equation correspond. In other words, for what definition of D 314 00:22:55,902 --> 00:23:01,832 does this equation hold? Is it D0, is it Da? It turns out, it's none of the above. 315 00:23:01,832 --> 00:23:07,157 Its something that we call luminosity distance and lets compute the luminosity 316 00:23:07,157 --> 00:23:11,607 distance. So what really happens is that well, 4 pi 317 00:23:11,607 --> 00:23:15,632 is to radians 4 pi. but if you have an object that's at a 318 00:23:15,632 --> 00:23:19,052 distance, that's at a distance D0 in coordinates, 319 00:23:19,052 --> 00:23:24,864 then of course you find D0 squared here. However, remember this is the energy that 320 00:23:24,864 --> 00:23:30,327 the objects emits per second. Now all of those photons that the object 321 00:23:30,327 --> 00:23:36,155 is emitting are getting red shifted. That means every photon that gets emitted 322 00:23:36,155 --> 00:23:41,162 I am observing with an energy that is less than it was emitted with. 323 00:23:41,162 --> 00:23:46,496 And so that means that the factor is the ratio of the scale factors, so that means 324 00:23:46,496 --> 00:23:51,459 that brightness is less than you would have expected at that distance, by a 325 00:23:51,459 --> 00:23:53,978 factor of 1 plus z because of the redshift. 326 00:23:53,978 --> 00:23:58,694 Further more if the object is emitting say, one photon a second and I'm 327 00:23:58,694 --> 00:24:02,472 observing those photons that coming in redshifted. 328 00:24:02,472 --> 00:24:08,006 They're also separated by more than a second, because the redshift means that 329 00:24:08,006 --> 00:24:12,967 all times are dilated in the past. So, there is a, the photons are coming 330 00:24:12,967 --> 00:24:16,220 less frequently, and each carries less energy. 331 00:24:16,220 --> 00:24:19,062 That gives me a total of 2 factors of 1+z. 332 00:24:19,062 --> 00:24:23,338 And again, nothing you can do in a slightly more rigorous way by a 333 00:24:23,338 --> 00:24:28,374 relativistic calculation but this is exactly the logic that applies so that 334 00:24:28,374 --> 00:24:33,954 the luminosity distance is given by the distance that you put in the denominator 335 00:24:33,954 --> 00:24:39,214 that relates brightness to luminosity. It has two factors of 1+Z in it, and so I 336 00:24:39,214 --> 00:24:44,533 have two different definitions of distance that agree. 337 00:24:44,533 --> 00:24:51,888 Of course they're both d0 at small distance but the luminosity distance is 338 00:24:51,888 --> 00:24:55,334 d0 times 1 plus z, because its square is d0 times 1 plus z squared. 339 00:24:55,334 --> 00:25:00,512 And the angular the size distance is z0 over 1 plus z. 340 00:25:00,512 --> 00:25:05,322 this means objects appear dimmer because of redshift than they would, and this 341 00:25:05,322 --> 00:25:09,994 means that objects appear bigger because of redshift than they otherwise would. 342 00:25:09,994 --> 00:25:14,199 And so, the bottom line is since we can't really measure this, but you can 343 00:25:14,199 --> 00:25:18,543 sometimes measure both this and this. They are related by 1+Z^2, a relative 344 00:25:18,543 --> 00:25:21,099 factor, and this sometimes you can measure. 345 00:25:21,099 --> 00:25:24,856 You can use that to verify that we actually observe, live, in a 346 00:25:24,856 --> 00:25:28,848 Robertson-Walker universe. Furthermore, there's another property 347 00:25:28,848 --> 00:25:31,912 that is okay. z, tells you what happens to spectral 348 00:25:31,912 --> 00:25:35,666 lines emitted by a star. a spectral line, emitted from a star at a 349 00:25:35,666 --> 00:25:39,801 particular distance, will be redshifted by the redshift appropriate to that 350 00:25:39,801 --> 00:25:42,591 distance given the distance redshift relation. 351 00:25:42,591 --> 00:25:46,473 What happens to the continuum blocked by the radiation of a star? Does a star, of 352 00:25:46,473 --> 00:25:50,097 course it will all be reddened, so the light that the star emits in blue might 353 00:25:50,097 --> 00:25:52,816 appear as yellow. The light that appears as yellow might 354 00:25:52,816 --> 00:25:55,311 appear as red. What kind of a weird spectrum is it? 355 00:25:55,311 --> 00:25:58,997 This is beautiful. It turns out, that the Black body, the Planck Black body 356 00:25:58,997 --> 00:26:02,739 spectrum has the wonderful property, that if you take every wavelength, and 357 00:26:02,739 --> 00:26:06,422 multiply it by a factor, so stretch out all the waves by some factor a, it's 358 00:26:06,422 --> 00:26:09,375 still a Black body spectrum. But it's a Black body spectrum 359 00:26:09,375 --> 00:26:12,908 corresponding of course to a different temperature, and you wouldn't be 360 00:26:12,908 --> 00:26:18,731 surprised because of Veen's law, that lambda max is 0.0029 meters divided 361 00:26:18,731 --> 00:26:24,334 by the temperature in Kelvin. That if all wavelengths get stretched, 362 00:26:24,334 --> 00:26:31,516 then the temperature, all wavelengths get larger. The temperature decreases by the 363 00:26:31,516 --> 00:26:36,284 same factor as the factor by which wavelengths increase. 364 00:26:36,284 --> 00:26:41,648 And indeed a redshifted Black body spectrum is a Black body spectrum with a 365 00:26:41,648 --> 00:26:46,501 temperature that is decreased by a factor, by the same factor by which 366 00:26:46,501 --> 00:26:49,842 wavelengths are stretched. This will play a role. 367 00:26:49,842 --> 00:26:54,310 So, a star viewed at large redshift, looks like a star but a redder star. 368 00:26:54,310 --> 00:26:58,415 And you have to be careful when you're doing color magnitude diagrams of 369 00:26:58,415 --> 00:27:03,011 clusters, if they're far enough that the cosmological red shift is not trivial, 370 00:27:03,011 --> 00:27:07,355 you will find that this is a meaningful correction and you'll have to take it 371 00:27:07,355 --> 00:27:10,528 into account. Now, one more thing about the kinematics 372 00:27:10,528 --> 00:27:13,232 of relativity. We talked about the red shift. 373 00:27:13,232 --> 00:27:18,002 That tells you that our clocks run slow. That was the 2nd factor of 1 + z, the 374 00:27:18,002 --> 00:27:22,667 coordinate factor velocity of light. I said light moves relative to every 375 00:27:22,667 --> 00:27:26,587 observer with speed c. This does not mean it moves with speed c 376 00:27:26,587 --> 00:27:29,452 as observed in our coordinates which means. 377 00:27:29,452 --> 00:27:34,569 light beams moving in the dim distant past move with velocity C relative to the 378 00:27:34,569 --> 00:27:38,200 observers back then. Those observers have slow clocks and 379 00:27:38,200 --> 00:27:41,304 small distances. It turns out that in terms of the 380 00:27:41,304 --> 00:27:46,026 positions today of the observers that observe the light a long time ago, the 381 00:27:46,026 --> 00:27:50,955 coordinate velocity of light is actually not the speed of light but it does move 382 00:27:50,955 --> 00:27:53,479 along geodesics, along straight lines. 383 00:27:53,479 --> 00:27:57,882 And then, what about things that are not light? Well, do they move along straight 384 00:27:57,882 --> 00:28:01,738 lines? We observe that they do and indeed, even along long distances it 385 00:28:01,738 --> 00:28:04,836 turns out they do. Massive objects, just like as the case 386 00:28:04,836 --> 00:28:09,192 for light, they move along geodesics of space. So they move in straight lines or 387 00:28:09,192 --> 00:28:11,872 whatever the straight lines are in your curved space. 388 00:28:11,872 --> 00:28:15,591 And it turns out that if the objects are freely falling, in other words, if 389 00:28:15,591 --> 00:28:19,613 they're only moving under the influence of the gravity, not something orbiting a 390 00:28:19,613 --> 00:28:21,969 star. I mean they only move under the influence 391 00:28:21,969 --> 00:28:26,200 of the generalized gravity of this homogeneous space, the curvature of this 392 00:28:26,200 --> 00:28:30,343 homogeneous space, not the local fluctuations that are a little galaxy 393 00:28:30,343 --> 00:28:34,473 clusters, or averaging over all that. And object that moves only under the 394 00:28:34,473 --> 00:28:39,044 generalized gravity of all of space, moves through the peculiar velocity, that 395 00:28:39,044 --> 00:28:43,111 decreases, as space expands. In other words, the peculiar velocity of 396 00:28:43,111 --> 00:28:47,487 a galaxy we observe, means that in the past, it was moving twice as fast when 397 00:28:47,487 --> 00:28:50,512 the distances between galaxies were half what they are now. 398 00:28:50,512 --> 00:28:54,397 So, if we observe galaxies to be moving at all, it means in the past they were 399 00:28:54,397 --> 00:28:57,912 moving quite a bit faster. Remember, most of the peculiar velocities 400 00:28:57,912 --> 00:29:00,937 we observe are actually not freely falling velocities. 401 00:29:00,937 --> 00:29:04,612 They have to do with the orbital motion of a galaxy within its cluster or 402 00:29:04,612 --> 00:29:08,017 something, but if you look at the relative motions of distant large 403 00:29:08,017 --> 00:29:11,822 clusters that have nothing to do with gravitational orbital motion, 404 00:29:11,822 --> 00:29:15,876 then they will have moved faster in the past, then they are moving now. 405 00:29:15,876 --> 00:29:18,520 And this is, the analog of studying kinetics. 406 00:29:18,520 --> 00:29:21,517 We have described Robertson-Walker, geometry. 407 00:29:21,517 --> 00:29:26,403 The geometry of, of expanding or changing in sale, size, homogenous isotropic 408 00:29:26,403 --> 00:29:28,996 space. And now, what we need to do is figure 409 00:29:28,996 --> 00:29:32,571 out, what do Einstein's equations tell us? if 410 00:29:32,571 --> 00:29:38,257 you have some distribution of energy and matter what does that do to the time of 411 00:29:38,257 --> 00:29:44,070 evolution of space? So the curvature of space, the density of matter, and the 412 00:29:44,070 --> 00:29:49,362 evolution of the scale factor, are all related by Einstein's equation and that's 413 00:29:49,362 --> 00:29:54,577 the next thing we're going to, turn to. one, another thing to note is that all of 414 00:29:54,577 --> 00:29:59,127 these solutions to Einstein's equation have the property that they have a Big 415 00:29:59,127 --> 00:30:03,523 Bang, in the sense that at some time the scale factor becomes 0 when you solve 416 00:30:03,523 --> 00:30:07,421 Einstein's equation. Or if you wanted, some time all the distances become 0, but 417 00:30:07,421 --> 00:30:10,081 that could be a question of how you measure distances. 418 00:30:10,081 --> 00:30:14,060 Remember, distance is an ambiguous thing. What's the unambiguous thing? At some 419 00:30:14,060 --> 00:30:16,592 time there's a singularity all curvatures diverge. 420 00:30:16,592 --> 00:30:21,011 they diverge either very far enough into the past, that's what we call a big bang, 421 00:30:21,011 --> 00:30:24,982 in some you can construct solutions that don't have a singularity in the past, 422 00:30:24,982 --> 00:30:27,077 but then they have the singularity in the future. 423 00:30:27,077 --> 00:30:30,017 If you want to be creative you can construct, solutions that have 424 00:30:30,017 --> 00:30:33,292 singularities in the past and the future, but there are not solutions without 425 00:30:33,292 --> 00:30:36,572 singularities, so we're going to have to address this issue of the Big Bang at 426 00:30:36,572 --> 00:30:36,572 some point.