1 00:00:00,012 --> 00:00:04,557 So, we now have a description of how in the context of gravity as geometry, we 2 00:00:04,557 --> 00:00:09,352 can describe universe in which space at any time is isotropic and homogeneous. 3 00:00:09,352 --> 00:00:13,367 it's a space of constant curvature and all that changes is a scale. 4 00:00:13,367 --> 00:00:17,002 And we have a natural set of coordinates to describe this. 5 00:00:17,002 --> 00:00:21,382 These are coordinates and which are, what we call co-moving. 6 00:00:21,382 --> 00:00:26,632 You pick where something will be at a given time, say the present, if it is 7 00:00:26,632 --> 00:00:32,237 freely moving and take a collection of freely falling observers that start out 8 00:00:32,237 --> 00:00:36,322 addressed at some point and use their positions today. 9 00:00:36,322 --> 00:00:40,170 If they're freely falling througout the history of the universe, to denote the 10 00:00:40,170 --> 00:00:42,693 name of a point. And that observer, is as far as I'm 11 00:00:42,693 --> 00:00:46,432 concerned, always at the same position. So these are observers are at rest. 12 00:00:46,432 --> 00:00:49,357 And then relative to them you can have peculiar velocities. 13 00:00:49,357 --> 00:00:53,467 It is in these coordinates that the space looks, nice and isotropic and homogenous. 14 00:00:53,467 --> 00:00:57,890 It would be nice if we had such observers, we do! By and large galaxies, 15 00:00:57,890 --> 00:01:01,989 we claim, are such observers. They are at rest, they only, their 16 00:01:01,989 --> 00:01:06,218 average motion is the Hubble motion so galaxies are precisely such an object. 17 00:01:06,218 --> 00:01:10,020 And now, having described the kinematics, we now turn to the dynamics. 18 00:01:10,020 --> 00:01:15,112 In other words, we have to understand the solutions to Einstein's equations and to 19 00:01:15,112 --> 00:01:17,920 we've studied sort of a left hand side, the geometry. 20 00:01:17,920 --> 00:01:20,222 Now we need to study what matter looks like. 21 00:01:20,222 --> 00:01:23,942 You would not expect an isotropic homogeneous universe to contain, for 22 00:01:23,942 --> 00:01:28,243 example, stars, because certainly if you have a star then the place where the star 23 00:01:28,243 --> 00:01:31,999 is, is not the place where the star isn't, so matter in an isotropic 24 00:01:31,999 --> 00:01:35,441 homogeneous universe has to be itself isotropic and homogeneous. 25 00:01:35,441 --> 00:01:38,748 Now one way to do that is to have nothing at all in the universe. 26 00:01:38,748 --> 00:01:43,173 That does not describe our universe, and so what we mean is, imagine, averaging 27 00:01:43,173 --> 00:01:48,437 out over large distances, ignoring little fluctuations, like galaxies and clusters, 28 00:01:48,437 --> 00:01:52,332 on average, we should be able to describe, at large distances, the 29 00:01:52,332 --> 00:01:56,063 distribution of matter in the universe, as completely uniform. 30 00:01:56,063 --> 00:02:00,890 So that means there is a constant density of energy, because mass is energy, so 31 00:02:00,890 --> 00:02:05,362 we're talking relativistically. There's a constant energy density, rho, 32 00:02:05,362 --> 00:02:10,117 or mass, if you want, density that is the same everywhere. Otherwise the universe 33 00:02:10,117 --> 00:02:13,562 wouldn't be homogeneous. And if there is a gas with a constant 34 00:02:13,562 --> 00:02:18,047 density, then we know enough about thermodynamics to assume that there will 35 00:02:18,047 --> 00:02:21,612 be a constant pressure. And these are constants everywhere in 36 00:02:21,612 --> 00:02:24,300 space, but of course they can depend on time. 37 00:02:24,300 --> 00:02:28,556 So all of the crazy right-hand side of Einstein's equation about energy and 38 00:02:28,556 --> 00:02:32,938 momentum is contained in two functions of only time. There's no variation with 39 00:02:32,938 --> 00:02:35,683 position, the equations are going to simplify. 40 00:02:35,683 --> 00:02:39,637 in further, and furthermore, we have density and we have pressure. Density and 41 00:02:39,637 --> 00:02:44,626 pressure are related by the properties of whatever the stuff is that you fill you 42 00:02:44,626 --> 00:02:48,125 space with. for example, if you have an ideal gas, 43 00:02:48,125 --> 00:02:52,401 then we know that pressure and density are related to temperature. 44 00:02:52,401 --> 00:02:57,587 in particular, we will deal with, two extreme cases of equations of state, 45 00:02:57,587 --> 00:03:02,314 relations between pressure and density. We have one case, which in cosmology 46 00:03:02,314 --> 00:03:07,045 lingo is called dust, and so we will be cosmologists and call it dust. 47 00:03:07,045 --> 00:03:12,258 What is, dust has nothing to do with the kind of dust that causes extinction. 48 00:03:12,258 --> 00:03:16,762 the particles, if you will, of this dust are galaxy clusters. 49 00:03:16,762 --> 00:03:22,204 dust is what describes a collection of uniform distribution of slow massive 50 00:03:22,204 --> 00:03:27,632 particles moving freely and interacting only through gravitationally. 51 00:03:27,632 --> 00:03:32,682 Each with each other or more importantly with a whole collection of uniform 52 00:03:32,682 --> 00:03:38,257 distribution because they're moving very slowly we're going to imagine that this 53 00:03:38,257 --> 00:03:42,057 is a 0 temperature object. The pressure is essentially 0. 54 00:03:42,057 --> 00:03:47,507 so this is the canonical example of dust of course is galaxies, slowly moving 55 00:03:47,507 --> 00:03:50,614 objects A collection of dust is isotropic in a 56 00:03:50,614 --> 00:03:54,238 frame that moves with it. In other words, in a frame in which you 57 00:03:54,238 --> 00:03:58,097 measure the velocity zero. This is not something fancy, every pilot 58 00:03:58,097 --> 00:04:02,775 knows that your air speed is constant, whether you're going upwind or downwind. 59 00:04:02,775 --> 00:04:07,193 In other words, the air, the moving air, is isotropic in a frame that's moving 60 00:04:07,193 --> 00:04:09,959 with the air. It's only when you're measuring ground 61 00:04:09,959 --> 00:04:13,741 speed that you realize that you might not be making head, any headway trying to go 62 00:04:13,741 --> 00:04:16,001 upwind. Your air speed of your craft is the same. 63 00:04:16,001 --> 00:04:18,358 The airplane flies the same way upwind as downwind. 64 00:04:18,358 --> 00:04:21,870 It's only relative to the ground that things change, so in a frame, where the 65 00:04:21,870 --> 00:04:25,457 velocity of the fluid is zero, at any given time, at any given place. So at any 66 00:04:25,457 --> 00:04:28,992 event, flew this dust has a preferred rest 67 00:04:28,992 --> 00:04:31,434 frame. That's the rest frame in which the dust 68 00:04:31,434 --> 00:04:35,610 at that event appears to be addressed. And of course since we're going to make 69 00:04:35,610 --> 00:04:39,214 this dust out of galaxies that are comoving in the Robertson Walker 70 00:04:39,214 --> 00:04:43,492 coordinates then what we're going to say is that the preferred Robertson Walker 71 00:04:43,492 --> 00:04:47,196 frame at any point is the prefer, is the preferred, is the rest frame of the dust. 72 00:04:47,196 --> 00:04:49,940 This is be, that way, they will both be isotropic. 73 00:04:49,940 --> 00:04:53,442 Both space and matter will be isotropic in the same frame. 74 00:04:53,442 --> 00:04:58,317 And then the other extreme case is very relativistic particles. 75 00:04:58,317 --> 00:05:03,442 In other words the extreme case is particles moving at the speed of light 76 00:05:03,442 --> 00:05:08,017 and we'll call such particles radiation. This can include photons. 77 00:05:08,017 --> 00:05:12,592 It can include gravitons. It can include essentially neutrinos. 78 00:05:12,592 --> 00:05:16,476 Which even if they're not exactly mass-less are very light and move, under 79 00:05:16,476 --> 00:05:20,392 most circumstances, at very near the speed of light or highly relativistic 80 00:05:20,392 --> 00:05:24,105 speeds, and you can make a calculation for an ideal gas of relativistic 81 00:05:24,105 --> 00:05:26,806 particles. Some of you have followed through in the 82 00:05:26,806 --> 00:05:30,772 context of electron degeneracy learning this and it turns out that for a 83 00:05:30,772 --> 00:05:34,770 relativistic gas, since the speed with which the particles are moving is the 84 00:05:34,770 --> 00:05:38,419 speed of light. density determines the pressure, and even 85 00:05:38,419 --> 00:05:42,688 dimensional analysis will tell you that pressure is the density times the square 86 00:05:42,688 --> 00:05:46,726 of the speed of light, and it turns out, the, the coefficient is 1/3, so the 87 00:05:46,726 --> 00:05:50,863 pressure of a relativistic gas is, related to its density by the square of 88 00:05:50,863 --> 00:05:54,569 the speed light over 3. There's no, in, temperature dependence, 89 00:05:54,569 --> 00:05:58,374 because a relativistic gas, by definition, everything is moving at the 90 00:05:58,374 --> 00:06:00,302 speed of light. And again, 91 00:06:00,302 --> 00:06:04,512 you cannot go to the rest frame of relativistic gas in which the speed of 92 00:06:04,512 --> 00:06:09,047 the particles is zero, because they're always moving at the speed of light in 93 00:06:09,047 --> 00:06:12,817 all different directions. But there will be a frame, a specific 94 00:06:12,817 --> 00:06:15,992 frame, in which the radiation field looks isotropic. 95 00:06:15,992 --> 00:06:20,553 This is the frame in which, if you look to the right and look to the left, you 96 00:06:20,553 --> 00:06:23,917 see exactly the same spectrum. And of course, if you are in that frame 97 00:06:23,917 --> 00:06:27,722 and someone is at the same place you are in the same radiation field but they are 98 00:06:27,722 --> 00:06:30,285 moving, they will observe a blueshift, Dop-, blue Doppler shift in one 99 00:06:30,285 --> 00:06:34,142 direction, a red Doppler shift in the other direction, and then say aha, the 100 00:06:34,142 --> 00:06:37,275 spectrum is not isotropic. Only one observer in any event sees an 101 00:06:37,275 --> 00:06:40,442 isotropic spectrum. That is the instantaneous rest frame. 102 00:06:40,442 --> 00:06:45,377 So we have our description of matter and we're going to assert that the matter is 103 00:06:45,377 --> 00:06:50,507 at rest in the Friedmann coordinates, so that space and matter are isotropic and 104 00:06:50,507 --> 00:06:55,507 homogeneous in one system of coordinates. And with that in mind, we write down our 105 00:06:55,507 --> 00:07:00,875 friend the Einstein's equations, this time written correctly. These, the, 106 00:07:00,875 --> 00:07:06,469 latex error that ruined things, so right here the Einstein equation correctly, and 107 00:07:06,469 --> 00:07:10,543 we insert what we know. On the right hand side we insert that 108 00:07:10,543 --> 00:07:16,448 the, geometry, remember these R's represent curvature, and the geometry is 109 00:07:16,448 --> 00:07:22,629 the geometry of a space of constant curvature with curvature R zero which 110 00:07:22,629 --> 00:07:27,633 scales with when the distance is increased, like 1 over a squared. 111 00:07:27,633 --> 00:07:32,795 And then this coefficient, k, is simply given by 0 or plus or minus 1. 112 00:07:32,795 --> 00:07:38,496 0 means, we're in the flat space case. 1 means we're in the positive curvature 113 00:07:38,496 --> 00:07:41,036 case. -1, negative curvature case, 114 00:07:41,036 --> 00:07:46,112 and in all cases I can give the same Parametrization in the curved cases our 0 115 00:07:46,112 --> 00:07:50,420 represents the curvature at T equals 0. At the presence at T equals T0, when A is 116 00:07:50,420 --> 00:07:52,765 1. At present in the case that K equals 0, 117 00:07:52,765 --> 00:07:56,678 you can put whatever you want for R equals 0, cause it's multiplying 0. 118 00:07:56,678 --> 00:08:01,332 That's the simplification on the left hand, on the right, left hand side. 119 00:08:01,332 --> 00:08:05,437 Whereas on the right hand side, we insert what we know about matter. 120 00:08:05,437 --> 00:08:10,222 And what we know about matter again is that it's the same everywhere and it's 121 00:08:10,222 --> 00:08:15,947 determined by this one function rho, and then the pressure is which appears inside 122 00:08:15,947 --> 00:08:21,252 the energy momentum tensor is a, some given function of rho, depending on what 123 00:08:21,252 --> 00:08:24,022 you put in. Okay, so you take all this data, 124 00:08:24,022 --> 00:08:28,576 you plug it into Einstein's equations, you do the mathematics. This is one of 125 00:08:28,576 --> 00:08:32,209 the cases in this class where I say, you do the calculation. 126 00:08:32,209 --> 00:08:36,813 And the equations that you are, you find are two equations for 1, whatever 127 00:08:36,813 --> 00:08:40,279 variables here. The variables are the time dependance of 128 00:08:40,279 --> 00:08:44,038 the scale constant. There's no space dependance of anything. 129 00:08:44,038 --> 00:08:48,742 So we're trying to determine the time dependance of the scale factor The time 130 00:08:48,742 --> 00:08:53,787 dependence of the energy density, and that's it, we have 2 functions, and we're 131 00:08:53,787 --> 00:08:56,697 trying to find the time dependence of these 2. 132 00:08:56,697 --> 00:09:01,462 And so we indeed find 2 equations, they are physics equations, they are 133 00:09:01,462 --> 00:09:06,672 differential equations, so watch out, I said we would skate near calculus, so 134 00:09:06,672 --> 00:09:12,942 there's this little dot here, this dot means the rate of change of a, with 135 00:09:12,942 --> 00:09:19,497 respect to time and we already said, if a were a position this would be it's 136 00:09:19,497 --> 00:09:25,592 velocity and we already noted that the rate of change of a divided by either 137 00:09:25,592 --> 00:09:30,678 relative change of a is what we call the Hubble constant, only we no longer expect 138 00:09:30,678 --> 00:09:34,365 it to be constant. So in general this will be a function of 139 00:09:34,365 --> 00:09:39,071 time, and this is given by Einsteins equation tells us that this is given by 140 00:09:39,071 --> 00:09:42,672 the right hand side 8 pi G. The c squares end up cancelling, 141 00:09:42,672 --> 00:09:47,087 times rho of t. It's the same everywhere in space, but it varies with time. 142 00:09:47,087 --> 00:09:51,477 And then, this is actually not a part of the right hand side of it, obviously. 143 00:09:51,477 --> 00:09:56,062 This is a curvature piece, it's a another piece of the left hand side, and we've 144 00:09:56,062 --> 00:10:00,207 put it on the right hand side to get an equation for the Hubble Constant. 145 00:10:00,207 --> 00:10:05,986 And so the curvature of space and the density of energy both together determine 146 00:10:05,986 --> 00:10:10,794 the rate at which the scale constant, the scale factor changes. 147 00:10:10,794 --> 00:10:14,894 That's one equation. We need another equation, and so again 148 00:10:14,894 --> 00:10:18,522 this tells us that the curvature of space if you want. 149 00:10:18,522 --> 00:10:23,225 Another way to think about it, is if you take the density and the velocity of 150 00:10:23,225 --> 00:10:26,442 everything. So that gives you the Hubble Constant, 151 00:10:26,442 --> 00:10:29,195 that will determine the curvature of space. 152 00:10:29,195 --> 00:10:33,873 Space curves in response to the density of energy and momentum. And so, the 153 00:10:33,873 --> 00:10:38,457 density of energy rho and the rate at which everything is moving, which is 154 00:10:38,457 --> 00:10:42,111 determined by a dot. put both of those in, and you can 155 00:10:42,111 --> 00:10:45,787 determine the curvature of space. But of course, like all equations, if you 156 00:10:45,787 --> 00:10:47,884 know one thing, you can determine another. 157 00:10:47,884 --> 00:10:51,123 Everything ends up determined. What's the second equation? Second 158 00:10:51,123 --> 00:10:54,021 equation looks even scarier. I have here an a with two dots. 159 00:10:54,021 --> 00:10:56,607 if A were a position, this would be the velocity, 160 00:10:56,607 --> 00:10:59,946 this would be the acceleration. It's the rate of change of the rate of 161 00:10:59,946 --> 00:11:03,615 change of a. And it's related to the Hubble Constant 162 00:11:03,615 --> 00:11:08,418 by this expression, where this is the rate of change of the Hubble Constant, 163 00:11:08,418 --> 00:11:12,443 and this its square. And this is given in term of the matter 164 00:11:12,443 --> 00:11:17,148 distribution by a combination of the matter, the energy density and the 165 00:11:17,148 --> 00:11:21,840 pressure. notice that for a relativistic gas, both 166 00:11:21,840 --> 00:11:28,487 of these are in fact equal because for a relativistic gas, p is equal to rho c 167 00:11:28,487 --> 00:11:34,370 squared over 3, whereas for the case of dust, of slow-moving matter, p was 168 00:11:34,370 --> 00:11:40,239 actually 0 so this is just rho. And now This is really a lot of calculus. 169 00:11:40,239 --> 00:11:45,013 So let's try to understand what it is that we're trying to say. 170 00:11:45,013 --> 00:11:50,217 So imagine that I have some time t* and I expand a times t, and express it for 171 00:11:50,217 --> 00:11:56,443 times near t*, and the first term in the expansion was the Hubble expansion term 172 00:11:56,443 --> 00:12:01,212 that we talked about before. The next term in the order will be a 173 00:12:01,212 --> 00:12:06,937 quadratic term proportional to t minus t* squared, and of course in general there, 174 00:12:06,937 --> 00:12:09,787 you can expand and get higher order terms. 175 00:12:09,787 --> 00:12:15,037 And I will, for dimensional reasons, factor out the factor of H, H squared and 176 00:12:15,037 --> 00:12:20,122 then the coefficient is called q over 2 and notice I put a minus sign here. 177 00:12:20,122 --> 00:12:26,368 Because I understand that gravity will decelerate things and indeed, if I write 178 00:12:26,368 --> 00:12:31,067 the expression in this way. So a of t for t near t star is a of t 179 00:12:31,067 --> 00:12:36,822 star times 1, which is the value when t is equal to t star plus the 1st order 180 00:12:36,822 --> 00:12:43,027 term which, who's coefficient I called H, and then the 2nd order term whose 181 00:12:43,027 --> 00:12:45,567 coefficient I call minus H squared, q over 2. 182 00:12:45,567 --> 00:12:52,922 q is called the deceleration parameter for good reason, because what it 183 00:12:52,922 --> 00:13:00,332 indicates is that if you take since you're subtracting a positive quantity, 184 00:13:00,332 --> 00:13:05,831 it means that if you start as a function of T and you plot a. 185 00:13:05,831 --> 00:13:11,290 It starts at t0, at the value a0. The line determined by the Hubble 186 00:13:11,290 --> 00:13:17,668 expression would be this constant increase, and look what the deceleration 187 00:13:17,668 --> 00:13:23,661 parameter does is it slows it down. It decreases below the straight line, 188 00:13:23,661 --> 00:13:29,156 and in terms of this deceleration parameter, you can rewrite this crazy a 189 00:13:29,156 --> 00:13:34,548 double dot thing as just minus H squared q. It's just twice this coefficient. 190 00:13:34,548 --> 00:13:39,450 And notice that the sign here is well justified, because pressure being 191 00:13:39,450 --> 00:13:44,895 positive, density being positive, this sign being negative, gravity, [SOUND] 192 00:13:44,895 --> 00:13:48,836 decelerates expansion. This is not a surprise, but this 193 00:13:48,836 --> 00:13:54,000 non-surprising result is derivable with great work by following through 194 00:13:54,000 --> 00:13:58,344 Einstein's equations. So Einstein's equations not surprisingly 195 00:13:58,344 --> 00:14:04,469 tell us that both density and pressure which is, the, remember, the pressure 196 00:14:04,469 --> 00:14:09,142 results from the velocity The momentum that the particles are carrying. 197 00:14:09,142 --> 00:14:13,752 This momentum contributes to gravitational attraction because momentum 198 00:14:13,752 --> 00:14:16,872 gravitates just in the same way that energy does. 199 00:14:16,872 --> 00:14:22,027 So remember, notice that the pressure of the relativistic gas actually increases 200 00:14:22,027 --> 00:14:26,119 the deceleration, contributes to deceleration, as opposed to one would 201 00:14:26,119 --> 00:14:28,838 think of pressure leads to, makes thing try to expand. 202 00:14:28,838 --> 00:14:32,851 Well pressure makes things expand, if you put something in a container, but in this 203 00:14:32,851 --> 00:14:36,633 case we're talking about a universe where the pressure is the same everywhere. 204 00:14:36,633 --> 00:14:40,261 There's no pressure gradient between inside and outside for the pressure to 205 00:14:40,261 --> 00:14:43,107 put us, push on. There's no container for the universe, no 206 00:14:43,107 --> 00:14:45,462 edge of the universe for the pressure to push on. 207 00:14:45,462 --> 00:14:48,942 Every bit of the universe experiences the same pressure on both sides. 208 00:14:48,942 --> 00:14:53,809 What is going on is that pressure contributes to the gravitational 209 00:14:53,809 --> 00:14:59,546 attraction, because remember, momentum as well as energy is a source for gravity, 210 00:14:59,546 --> 00:15:03,165 couldn't be otherwise consistent with relativity. 211 00:15:03,165 --> 00:15:10,455 So these are our equations, and they are going to, in the end, determine the 212 00:15:10,455 --> 00:15:17,667 evolution both of the density and of the scale constant. 213 00:15:17,667 --> 00:15:25,039 However in general, we will have differing sources of density. 214 00:15:25,039 --> 00:15:30,054 Remember, we can have a universe that contains both dust, slow moving things, 215 00:15:30,054 --> 00:15:35,236 and, say, photons, relativistic things. And one can imagine a universe in which 216 00:15:35,236 --> 00:15:40,332 these don't interact, and then that means that the energy in dust is conserved. 217 00:15:40,332 --> 00:15:44,725 Because no dust is created, the total mass, remember the energy of the dust is 218 00:15:44,725 --> 00:15:48,907 just is mass, it's addressed, and so the total mass of the dust is conserved. 219 00:15:48,907 --> 00:15:52,133 And separately the total energy in photons is conserved. 220 00:15:52,133 --> 00:15:56,340 And if this is the situation when they do not interact, then I know the time 221 00:15:56,340 --> 00:15:59,654 evolution of rho of t if I know the time evolution of a of t. 222 00:15:59,654 --> 00:16:02,982 and this is because, if you take the given volume, 223 00:16:02,982 --> 00:16:07,937 it has some dust in it, and then as this volume expands, it has the same amount of 224 00:16:07,937 --> 00:16:12,542 dust in it. But if the distances have grown by a factor of 2, the volume of 225 00:16:12,542 --> 00:16:17,737 this region has grown by a factor of 3. So when distance by a factor of 8, and so 226 00:16:17,737 --> 00:16:22,682 the density, mass conservation says density decreases like the third power of 227 00:16:22,682 --> 00:16:27,097 the scale factor, when space doubles in length and all 228 00:16:27,097 --> 00:16:30,342 lengths double, the density decreases by a factor of 8. 229 00:16:30,342 --> 00:16:34,420 How does this affect relativistic particles? Well for relativistic 230 00:16:34,420 --> 00:16:39,151 particles, think again about photons, but of course the theory is universal. 231 00:16:39,151 --> 00:16:43,833 In a, the case of a relativistic particle moving at the speed of light, the same 232 00:16:43,833 --> 00:16:48,353 property of Thames, if I take the volume, containing a certain number of them, 233 00:16:48,353 --> 00:16:52,848 their number is conserved when I, that volume increases, it carries the same 234 00:16:52,848 --> 00:16:56,842 amount, number of particles in twice the, eight times the volume. 235 00:16:56,842 --> 00:17:02,158 However, they're all redshifted, so the energy carried by each of them is also 236 00:17:02,158 --> 00:17:06,745 decreased by a factor of 8. And so the behavior of the density of 237 00:17:06,745 --> 00:17:11,556 nonrelativist-, the density of nonrelativistic matter is that as a 238 00:17:11,556 --> 00:17:16,937 function of time, if there is no interaction and no creation of matter or 239 00:17:16,937 --> 00:17:21,696 loss of matter, like say the loss of stars, if you wish, convert mass dust 240 00:17:21,696 --> 00:17:26,101 energy into radiation energy, but stars are a small perturbation in the 241 00:17:26,101 --> 00:17:29,758 large scale of things. Most of the matter of the universe is 242 00:17:29,758 --> 00:17:34,181 inter cluster of gas, which is not participating in stellar processes, and 243 00:17:34,181 --> 00:17:37,142 even when there are, when dust isn't a star, 244 00:17:37,142 --> 00:17:41,663 only a few percent of the mass of a star over 10 billion years will be converted 245 00:17:41,663 --> 00:17:46,031 to energy. But, so this is a small violation of this conservation of energy. 246 00:17:46,031 --> 00:17:50,669 Ignoring little effects like stars, then the energy density of dust scales with 247 00:17:50,669 --> 00:17:55,051 the scale factor, like the negative 3rd power, whereas the energy density of 248 00:17:55,051 --> 00:17:58,738 radiation scales like the negative 4th power. 249 00:17:58,738 --> 00:18:04,730 Remember, three of those are because of the volume, and then one extra factor 250 00:18:04,730 --> 00:18:09,530 because each photon's energy is decreased by the redshift. 251 00:18:09,530 --> 00:18:13,693 Dense slide. Now, Einstein had understood all of these 252 00:18:13,693 --> 00:18:17,520 equations. it's the usual case that the, what you 253 00:18:17,520 --> 00:18:19,825 get out an equation is what you put into it. 254 00:18:19,825 --> 00:18:22,735 You put in a lot of symmetry, the equation simplifies. 255 00:18:22,735 --> 00:18:26,813 This complicated set of coupled partial differential equations becomes this 256 00:18:26,813 --> 00:18:30,775 relatively simple set of ordinary differential equations for functions of 257 00:18:30,775 --> 00:18:33,837 one variable, time. And Einstein was trying to write the 258 00:18:33,837 --> 00:18:38,214 solution that describes our universe, and what he knew about our universe was that 259 00:18:38,214 --> 00:18:41,066 it's static. So what he wanted was a universe in which 260 00:18:41,066 --> 00:18:44,846 the skill factor didn't change, in which nothing essentially changed. 261 00:18:44,846 --> 00:18:48,619 He had no reason to imagine that distances were changing. 262 00:18:48,619 --> 00:18:54,292 He did not know at the time of Hubble's result, which by the way had previously 263 00:18:54,292 --> 00:18:59,530 been, it turns out, measured by [FOREIGN] and understood in a deep way. 264 00:18:59,530 --> 00:19:04,732 And so he tried to set this to 0, and it's possible to set the rate of change 265 00:19:04,732 --> 00:19:08,976 of A to 0 by making balancing the energy density against curvatures. So you need 266 00:19:08,976 --> 00:19:13,675 to have a curved space, if you have any matter in it, but if you have positive 267 00:19:13,675 --> 00:19:17,869 curvature, and positive energy density, you can balance them to get zero 268 00:19:17,869 --> 00:19:20,891 velocity. Glorious, however, you cannot make the 269 00:19:20,891 --> 00:19:26,002 acceleration zero, so the zero velocity because of course both of these terms are 270 00:19:26,002 --> 00:19:30,503 all negative And so the acceleration is always negative. So the 0 velocity 271 00:19:30,503 --> 00:19:34,904 solution you found out is sort of like the 0 velocity of my little rubber ball 272 00:19:34,904 --> 00:19:37,999 at the top of its flight. It stops for a minute but it's 273 00:19:37,999 --> 00:19:42,181 acceleration is still negative, it won't stay at 0 velocity, it will fall. 274 00:19:42,181 --> 00:19:46,008 So, indeed if you have 0a., then the negative acceleration means that 275 00:19:46,008 --> 00:19:50,422 you have universe that is not expanding but is about to start contracting. 276 00:19:50,422 --> 00:19:54,930 So Einstein was very saddened by his inability to find a description of what 277 00:19:54,930 --> 00:19:58,580 he thought to be the universe, namely a static universe. 278 00:19:58,580 --> 00:20:02,981 But Einstein was nothing if not creative. He looked at his equation, and he 279 00:20:02,981 --> 00:20:08,374 realized that without ruining any of the axioms or the logic that led him, to the 280 00:20:08,374 --> 00:20:13,143 equations of general relativity, he could make a modification that would have 281 00:20:13,143 --> 00:20:18,024 absolutely no impact on the connection, the agreement with Newtonian physics, 282 00:20:18,024 --> 00:20:21,248 but would fix this, would allow a static universe. 283 00:20:21,248 --> 00:20:26,204 And the correction, in terms of Einstein's equation, involved adding one 284 00:20:26,204 --> 00:20:31,112 more term to the left hand side. So, modifying the way that gravity, the, 285 00:20:31,112 --> 00:20:35,687 metric function G is determined by the distribution of energy momentum tensor. 286 00:20:35,687 --> 00:20:40,202 He called this the cosmological term, we today call it the cosmological constant. 287 00:20:40,202 --> 00:20:44,562 the properties of lambda is that for this to be a consistent modification of 288 00:20:44,562 --> 00:20:47,042 Einstein's equation lambda is just a number. 289 00:20:47,042 --> 00:20:51,692 A constant, it has the dimensions if you work through of inverse length squared. 290 00:20:51,692 --> 00:20:55,127 So lamba is measured in 1 over meters squared, 291 00:20:55,127 --> 00:20:59,447 and this is the modified version of Einstein's equation. 292 00:20:59,447 --> 00:21:05,166 When, of course you can alternatively think of taking this same blue thing, 293 00:21:05,166 --> 00:21:11,274 flipping its sign and putting it on the right-hand side of the equation, and 294 00:21:11,274 --> 00:21:16,670 thinking of good old Einstein's equation in the presence of a different kind of 295 00:21:16,670 --> 00:21:20,309 energy momentum distribution. So what kind of crazy energy momentum 296 00:21:20,309 --> 00:21:24,363 distribution is lambda? Well it's a constant, so it does not depend on space, 297 00:21:24,363 --> 00:21:28,168 so it satisfies our isotropic and homogeneous relations. So I should be 298 00:21:28,168 --> 00:21:30,796 able to describe it by a constant energy density. 299 00:21:30,796 --> 00:21:33,462 Indeed, lambda corresponds to a constant energy. 300 00:21:33,462 --> 00:21:38,351 Related to the value of lambda by this, this was what follows from placing lambda 301 00:21:38,351 --> 00:21:42,696 in the position where rho sits here, and then the cosmological constant 302 00:21:42,696 --> 00:21:46,706 corresponds to a pressure which is negative rho times c squared. 303 00:21:46,706 --> 00:21:51,076 So if you have a positive, and the cosmological constant can have either 304 00:21:51,076 --> 00:21:55,806 sign. If it's positive, then it has a positive corresponds to positive energy 305 00:21:55,806 --> 00:22:00,679 density and a negative pressure, whereas a negative cosmological constant 306 00:22:00,679 --> 00:22:05,953 the roles are reversed and then if you now re-do the calculation that led us to 307 00:22:05,953 --> 00:22:11,318 Friedman's equations in the presence of this cosmological term Einstein finds 308 00:22:11,318 --> 00:22:15,531 that the first equation for H squared is modified and the 2nd equation for minus 309 00:22:15,531 --> 00:22:18,564 QH squared. Remember, here we're determining Q, the 310 00:22:18,564 --> 00:22:23,035 acceleration, the deceleration parameter is also modified, 311 00:22:23,035 --> 00:22:26,678 but it is essentially by exactly the same term. 312 00:22:26,678 --> 00:22:32,957 This has to do with the fact that pressure and density are essentially the 313 00:22:32,957 --> 00:22:37,160 same thing. And so, you plug this thing into that. 314 00:22:37,160 --> 00:22:43,949 And what do you discover, what you find is that now I can find a solution where 315 00:22:43,949 --> 00:22:50,069 the left hand side, the rate of change of the scale factor, and its acceleration, 316 00:22:50,069 --> 00:22:51,715 are both 0. How do I do this? 317 00:22:51,715 --> 00:22:55,736 Well, I'm going to pick a universal which p is equal to 0. 318 00:22:55,736 --> 00:23:01,843 So my matter is going to be dust. I'm going to of course need positive 319 00:23:01,843 --> 00:23:08,062 curvature so that the energy density is balanced against curvature. And then, 320 00:23:08,062 --> 00:23:13,287 it's balanced by, curvature, and I'm going to also have a cosmological 321 00:23:13,287 --> 00:23:18,512 constant, which will allow the negative term here, this is going to be zero, to 322 00:23:18,512 --> 00:23:22,887 be balanced by this term. And it turns out that if you set our 0 323 00:23:22,887 --> 00:23:26,837 equal to lambda. Remember both of these have dimensions of 324 00:23:26,837 --> 00:23:29,802 1 over length squared, so that even makes sense, 325 00:23:29,802 --> 00:23:34,096 equal to this concoction, proportional to the energy density, both of these 326 00:23:34,096 --> 00:23:37,093 equation vanish. This is called the static Einstein 327 00:23:37,093 --> 00:23:41,135 universe, and Einstein thought, well, since this is the homogeneous, isotropic, 328 00:23:41,135 --> 00:23:44,170 static universe, this must describe our universe. 329 00:23:44,170 --> 00:23:48,225 He then was informed about Hubble's measurements and realized that our 330 00:23:48,225 --> 00:23:52,405 universe perhaps is not static. And this cosmological term was 331 00:23:52,405 --> 00:23:56,353 unnecessary. Einstein then regretted introducing this 332 00:23:56,353 --> 00:24:01,667 uglification of what was a beautifully elegant equation before. He called it his 333 00:24:01,667 --> 00:24:05,423 greatest blunder. As we shall see, it might not have been a 334 00:24:05,423 --> 00:24:09,263 blunder, Einstein, even when he blundered, had foresight.