1 00:00:00,880 --> 00:00:05,903 Let us continue this brief introduction into relativistic cosmology. 2 00:00:05,903 --> 00:00:10,335 An important notion is the so-called cosmological principle. 3 00:00:10,335 --> 00:00:13,881 It states that universe is same, at a given time, 4 00:00:13,881 --> 00:00:17,500 it's same in all directions and at all locations. 5 00:00:17,500 --> 00:00:21,046 This is what's described as homogeinity and isotropy. 6 00:00:21,046 --> 00:00:26,956 In a sense, cosmological principle is the generalization of Copernican principle, 7 00:00:26,956 --> 00:00:32,158 but on a global, universal scale. Now, there is a perfect cosmological 8 00:00:32,158 --> 00:00:36,573 principle proposed by Hoyle and collaborators which says that universe 9 00:00:36,573 --> 00:00:41,361 should also be same at all times and that was the basis of this steady state 10 00:00:41,361 --> 00:00:45,403 cosmology, which unfortunately did not survive experimental tests. 11 00:00:45,403 --> 00:00:48,325 So, what do we mean by isotropy and homogeneity? 12 00:00:48,325 --> 00:00:53,362 The picture on the left is homogeneous. You can be anywhere in the infinite plain 13 00:00:53,362 --> 00:00:56,533 covered with the stripes and it will look the same, 14 00:00:56,533 --> 00:01:00,140 but it's not isotropic. The stripes have a given direction. 15 00:01:00,140 --> 00:01:04,179 So it's not the same along the stripes or orthogonal to it. 16 00:01:04,179 --> 00:01:08,835 The picture on the right is isotropic from the point in the middle, 17 00:01:08,835 --> 00:01:13,080 same in all radial direction, but it's clearly not homogeneous. 18 00:01:14,220 --> 00:01:19,808 And, here, we have an example, it would be homogeneous and isotropic texture that 19 00:01:19,808 --> 00:01:25,839 is same everywhere in all directions in all places and the one on the right which 20 00:01:25,839 --> 00:01:30,857 is neither homogeneous nor isotropic. Why is this important? 21 00:01:30,857 --> 00:01:35,619 Well, is it really true? Well, it is true on scales larger than 22 00:01:35,619 --> 00:01:40,850 about 100 megaparsecs and there are a couple of experimental 23 00:01:40,850 --> 00:01:46,060 pieces of evidence for that. The map on the right shows positions of 24 00:01:46,060 --> 00:01:50,887 radio sources in the sky. Those tend to be very far away and so 25 00:01:50,887 --> 00:01:55,560 there are a fair sampler or of [INAUDIBLE] universe's volume. 26 00:01:55,560 --> 00:01:59,621 As you can see, it is, it does look pretty homogeneous. 27 00:01:59,621 --> 00:02:04,908 The ellipse on the lower left is indicative of what cosmic microwave 28 00:02:04,908 --> 00:02:10,807 background sky looks like, very uniform. There are fluctuations in it, but they 29 00:02:10,807 --> 00:02:15,875 are parts in a million, so that tells us that indeed the universe 30 00:02:15,875 --> 00:02:19,810 of very large scales is homogeneous and isotropic. 31 00:02:19,810 --> 00:02:25,029 Not so on scales smaller than about hundred megaparsecs where we see all the 32 00:02:25,029 --> 00:02:28,554 large scale structure, filaments, and voids, and so on. 33 00:02:28,554 --> 00:02:33,842 Turns out that doesn't really matter so much as far as cosmological tests are 34 00:02:33,842 --> 00:02:39,197 concerned and cosmology really operates on scales of gigaparsecs and larger. 35 00:02:39,197 --> 00:02:43,535 So, it's true that locally, universe is not homogeneous, isotropic, 36 00:02:43,535 --> 00:02:48,890 you are standing on a planet after all, but on large enough scales it is a good 37 00:02:48,890 --> 00:02:52,263 approximation. Let's go back to general activity. 38 00:02:52,263 --> 00:02:57,608 Remember, the most important notion is that presence of mass and energy, the 39 00:02:57,608 --> 00:03:02,313 term is geometry and geometry determines where mass and energy go. 40 00:03:02,313 --> 00:03:07,658 And the two have to be consistent, so getting a consistent solution to that 41 00:03:07,658 --> 00:03:12,948 is essentially Einstein's equations. We will not derive them in here in any 42 00:03:12,948 --> 00:03:16,418 shape or form. Just need to show you a little bit of 43 00:03:16,418 --> 00:03:20,422 what they look like. So we start first with Poisson equation 44 00:03:20,422 --> 00:03:25,895 which describes gravitational potential phi as a function of density rho and this 45 00:03:25,895 --> 00:03:30,633 is true in Newtonian physics as well. Now, general relativity says that 46 00:03:30,633 --> 00:03:35,771 potential, gravitational potential can be also made equivalent to geometrical 47 00:03:35,771 --> 00:03:41,177 description through a metric transfer. And, there is the plane density, there is 48 00:03:41,177 --> 00:03:47,753 so-called matter energy density tensor. And that is a matrix four by four and we 49 00:03:47,753 --> 00:03:53,803 need not go into its details. So a short connotation, so using these 50 00:03:53,803 --> 00:04:01,208 two indices, mu and nu, which go from zero to three, this represents 16 partial 51 00:04:01,208 --> 00:04:08,280 differential equations of this form and those are the Einstein equations. 52 00:04:08,280 --> 00:04:13,309 So basically, thing to remember. On the one side, there is a term that is 53 00:04:13,309 --> 00:04:18,125 all about spacetime geometry. And on the other side, there is a term 54 00:04:18,125 --> 00:04:21,738 that is all about mass and energy and for the move. 55 00:04:21,738 --> 00:04:25,775 So this is what Einstein equations are really all about. 56 00:04:25,775 --> 00:04:30,436 So there are 16 of them, but we made those important assumptions 57 00:04:30,436 --> 00:04:35,127 that universe is homogeneous and isotropic and that means that only one 58 00:04:35,127 --> 00:04:37,864 coordinate would matter, radial coordinate. 59 00:04:37,864 --> 00:04:43,076 So that makes things much simpler, instead of 16 equations we only have one. 60 00:04:43,076 --> 00:04:48,028 And that is what's called the Friedmann equation which is the basic staple of 61 00:04:48,028 --> 00:04:52,640 cosmology. Just one more thing. 62 00:04:52,640 --> 00:04:58,717 Gravity as we know it is an attractive force and this is certainly true in 63 00:04:58,717 --> 00:05:03,294 [INAUDIBLE] as well. But Einstein's equations allow for 64 00:05:03,294 --> 00:05:06,895 introduction of an additional term in potential. 65 00:05:06,895 --> 00:05:12,072 And you can think of it, as say integration constant or a new face of 66 00:05:12,072 --> 00:05:16,649 gravity just like electricity and magnetism are two faces of 67 00:05:16,649 --> 00:05:21,141 electromagnetic interaction, but something that's only apparent of 68 00:05:21,141 --> 00:05:25,377 cosmological scales. Essentially, this corresponds to constant 69 00:05:25,377 --> 00:05:30,160 energy density term and that is what's called a cosmological constant. 70 00:05:30,160 --> 00:05:35,075 Now, if it is seen as, say integration constant, we have no idea what it is. 71 00:05:35,075 --> 00:05:39,597 The theory doesn't give us its value and we don't even know its sign. 72 00:05:39,597 --> 00:05:42,612 It can be attractive force or repulsive force. 73 00:05:42,612 --> 00:05:47,723 The result, it is a repulsive force and it is actually very important, but we'll 74 00:05:47,723 --> 00:05:51,918 cover that later in the class. So, those are the basic notions of 75 00:05:51,918 --> 00:05:55,064 relativistic cosmology. Next time we'll talk more about the 76 00:05:55,064 --> 00:05:57,620 expanding universe and what that means.