0:00:07.978,0:00:24.421 Let's review a little bit. And then I want to move on to generalizations of what we've talked about so far. 0:00:24.441,0:00:29.864 I think we worked out the equations of an expanding universe. 0:00:29.864,0:00:33.096 They were Newton's equations ... let's talk about something else first. 0:00:33.096,0:00:35.445 Does Newton's equations really get it right? 0:00:35.445,0:00:39.694 Yeah. Newton's equations does get it right , for the most part. 0:00:39.694,0:00:42.018 Let me explain why. 0:00:42.018,0:00:45.708 Einstein's equations have to do with curved spacetime. 0:00:45.708,0:00:53.996 Now, the universe that we're ultimately going to study, has curved spacetime, alright? 0:00:53.996,0:00:58.683 And, in fact, some versions of it even have curved space. 0:00:58.683,0:01:12.010 That simply means that space itself, forget spacetime, just space itself, if you measure triangles on it, if you do various kinds of geometric exercises on it, you'll discover, perhaps, that space is curved. 0:01:12.010,0:01:18.883 At the moment it looks pretty flat, but it's possible that it will turn out on the average to be curved. 0:01:18.883,0:01:26.608 And, if it is curved, well, maybe it looks like a three-dimensional version, let's say, of a sphere. 0:01:26.608,0:01:44.109 Well we're going to study later, not tonight, maybe partly tonight, that's a portion of a sphere, we're over here, we look out, we can only see so much, we can't even really see that the sphere is curved. 0:01:44.109,0:01:52.052 But at a large enough difference we may be able to see that the sphere is curved 0:01:52.052,0:01:57.593 On the other hand supposing we just decide to look at very neighboring galaxies. 0:01:57.593,0:02:01.810 Now very neighboring galaxies can mean a billion light-years from us now. 0:02:01.810,0:02:10.463 Very neighboring galaxies much smaller than what we think the radius of curvature of this universe is. 0:02:10.463,0:02:17.850 Well then it looks flat, and if it looks flat it should mean 0:02:17.850,0:02:21.750 that at least for that portion, if we're not interested in the whole thing 0:02:21.750,0:02:28.254 but we're just interested in the local nearby behaviour, we should not have to worry about the fact that it's curved. 0:02:28.254,0:02:34.900 If that's correct, then it means that the way these galaxies move relative to each other 0:02:34.900,0:02:40.432 and how they move apart from each other, at least in the small here can be studied using Newton's equations. 0:02:40.432,0:02:42.548 That's what we've been doing. 0:02:42.548,0:02:46.664 We've been looking at the universe in the small 0:02:46.664,0:02:57.141 and studying how a small little fraction of it is expanding, or not expanding - whatever it's doing, 0:02:57.141,0:03:01.454 and it's perfectly legitimate and in fact entirely consistent with Einstein 0:03:01.454,0:03:05.493 with relativity, except for one thing 0:03:05.493,0:03:14.160 Except for one thing - we would run into trouble if the galaxies or whatever is present 0:03:14.160,0:03:19.638 galaxies, particles, whatever is present, if they were really moving past each other, 0:03:19.638,0:03:22.768 with a significant fraction of the speed of light. 0:03:22.768,0:03:31.259 One of the assumptions is that the neighboring things are moving relatively slowly with respect to each other. 0:03:31.259,0:03:36.915 Something very far away would be moving with a large velocity relative to you, 0:03:36.915,0:03:43.293 but as long as the things nearby are moving with non-relativistic velocities, 0:03:43.293,0:03:45.801 you can study - relative to you - 0:03:45.801,0:03:48.597 you could take a small patch of it, 0:03:48.597,0:03:51.608 now small could mean 10 billion light years, okay. 0:03:51.608,0:03:58.408 But you can take a small part of it and study it without using any relativity, really. 0:03:58.408,0:04:06.183 If we discover, that there are particles moving with close to the speed of light, past each other 0:04:06.183,0:04:09.157 then of course we would have to modify the equations. 0:04:09.157,0:04:14.930 But there are particles moving fast by comparison with the speed of light past us. 0:04:14.930,0:04:17.919 What are they? 0:04:17.919,0:04:21.868 [audience] Neutrinos are [inaudible] 0:04:21.868,0:04:24.022 Well, neutrinos for one. But, photons. 0:04:24.022,0:04:27.570 Now, I don't mean photons from the Sun, 0:04:27.570,0:04:29.184 I mean photons that would be there even if there was no Sun, 0:04:29.184,0:04:32.716 the Universe is filled in the same way that it's filled with galaxies, 0:04:32.716,0:04:34.553 it's also filled with radiation. 0:04:34.553,0:04:36.743 Homogeneous radiation. 0:04:36.743,0:04:39.989 And that homogeneous radiation does move with the speed of light. 0:04:39.989,0:04:47.860 That means that we have to modify our equations somehow to account for these very very fastly moving, 0:04:47.860,0:04:50.655 rapidly moving photons. 0:04:50.655,0:04:56.428 We're going to do that tonight, but I want to... 0:04:56.428,0:04:58.126 ... uh ... 0:04:58.126,0:05:03.168 I want to just review what we did last time, quickly. 0:05:03.168,0:05:06.116 We first of all said: 0:05:06.116,0:05:12.050 suppose that space is homogeneous and filled with galaxies - I'm not going to try to draw all the galaxies, 0:05:12.050,0:05:15.129 they form a gas, if you like. 0:05:15.129,0:05:18.120 They kind of fill the blackboard, 0:05:18.120,0:05:22.450 with a certain number of particles per cubic metre. 0:05:22.450,0:05:24.763 In other words, a density. 0:05:24.763,0:05:26.586 A density called rho. 0:05:35.365,0:05:36.827 And that was the content of the Universe in kilograms per cubic metre if you like. 0:05:38.268,0:05:41.804 You could use some other units, but whatever units you like. 0:05:45.377,0:05:48.154 Physical units, kilograms per cubic metre, and we called it rho. 0:05:51.996,0:05:55.224 We laid down a grid on this Universe, 0:05:55.507,0:05:59.223 and laying down the grid, there was clear ambiguity 0:06:00.670,0:06:02.249 imagine that we laid down the grid at some specific time - like today. 0:06:03.296,0:06:08.013 We laid down the grid, and you could ask, 0:06:08.935,0:06:10.834 what is the spacing between the grid - a coordinate system 0:06:13.139,0:06:19.376 let's call it coordinates X 0:06:19.722,0:06:23.181 and the distance between X equals something, and X equals something plus one 0:06:23.406,0:06:27.646 in other words, one grid - uh - one grid separation here 0:06:27.784,0:06:28.921 one lot of separation - there's a certain distance associated with it 0:06:29.262,0:06:31.618 how big is that distance? 0:06:33.102,0:06:35.787 Well, we called it 'a', but how big is 'a'? 0:06:36.197,0:06:39.259 That depends on the grid that we laid down. 0:06:39.495,0:06:41.666 If we laid down a very coarse grid, it would be one thing. 0:06:41.956,0:06:47.610 If we laid down a fine grid, it would be another thing. 0:06:47.969,0:06:52.541 And so it would better be that our equations - at least at the moment - 0:06:52.873,0:06:55.315 do not prefer any specific value of 'a'. 0:06:55.547,0:07:00.623 We could lay down a different grid. 0:07:00.851,0:07:07.735 A different grid could be twice as dense, 0:07:08.025,0:07:12.204 so here's a black - forms one grid 0:07:12.574,0:07:16.095 and the black and green together form another grid. 0:07:16.263,0:07:19.699 If we looked at the more dense grid, 0:07:19.869,0:07:24.471 we would also invent an 'a', let's call it 'a prime'. 0:07:24.638,0:07:28.720 'a prime' is the distance between neighboring points on the dense grid, 0:07:28.879,0:07:30.880 that would be one half 'a'. 0:07:31.026,0:07:35.051 So if you ask me, what is the value of 'a'? 0:07:35.188,0:07:38.681 I'm gonna say I can't tell you until I know precisely what grid is laid down. 0:07:39.433,0:07:49.334 And so 'a' itself does not have a physical meaning, 9:59:59.000,9:59:59.000 At least at this stage. [br]Later on we'll discuss more what a means.