1 00:00:01,200 --> 00:00:04,946 Theory of relativity. In theory of space, time, and gravity 2 00:00:04,946 --> 00:00:08,563 provides the theoretical framework for modern cosmology. 3 00:00:08,563 --> 00:00:13,214 Here, I will assume for the purpose of this class that you actually know 4 00:00:13,214 --> 00:00:16,832 something about this and here is just a quick refresher. 5 00:00:16,832 --> 00:00:21,935 The special theory of relatively, which came first, postulates that all inertial 6 00:00:21,935 --> 00:00:26,004 and accelerated frames of references must be equivalent. 7 00:00:26,004 --> 00:00:29,105 And that has a number of important consequences. 8 00:00:29,105 --> 00:00:32,206 It does unify space and time in a novel fashion. 9 00:00:32,206 --> 00:00:35,029 It. postulates the speed of light, is the 10 00:00:35,029 --> 00:00:40,032 maximum speed that can be had in a space time or rather, light moves at the 11 00:00:40,032 --> 00:00:44,072 maximum possible speed. It predicts Lorentz contraction and time 12 00:00:44,072 --> 00:00:47,215 dilation, those are two important concepts and if 13 00:00:47,215 --> 00:00:51,127 you don't know what they are, you really need to look them up. 14 00:00:51,127 --> 00:00:55,040 And, of course, it predicts the equivalence of mass and energy, 15 00:00:55,040 --> 00:00:59,160 the famous EMC^2 = mc^2 formula, as well as a number of other effects. 16 00:00:59,160 --> 00:01:04,125 Here is the front page of Einstein's original special relativity paper from 17 00:01:04,125 --> 00:01:06,702 1905. The special theory of relativity is 18 00:01:06,702 --> 00:01:11,038 limited in its application, because it talks only about unaccelerated 19 00:01:11,038 --> 00:01:14,558 frames of references. What happens with accelerated ones? 20 00:01:14,558 --> 00:01:17,763 Well, this is where general theory of relativity is. 21 00:01:17,763 --> 00:01:22,854 Einstein finished that in 1915 and that was an even more important change in our 22 00:01:22,854 --> 00:01:26,374 view of spacetime. Here, it's assumed that all frames of 23 00:01:26,374 --> 00:01:32,069 references are equivalent, and through equivalence principle, the gravitational 24 00:01:32,069 --> 00:01:36,936 mass and inertia mass are the same. A one sentence summary of what general 25 00:01:36,936 --> 00:01:41,691 relativity says is that the presence of mass or energy, which, remember are 26 00:01:41,691 --> 00:01:44,840 equivalent, determine the local geometry of space. 27 00:01:44,840 --> 00:01:49,339 And local geometry of space determines where would the mass and energy move. 28 00:01:49,339 --> 00:01:53,773 The two have to be computed in a consistent fashion and that is what 29 00:01:53,773 --> 00:01:58,233 general relativity is all about. So general relativity introduced 30 00:01:58,233 --> 00:02:01,838 curvature of space, which also then immediately had 31 00:02:01,838 --> 00:02:06,220 predictions of light bending in gravitational field and so on. 32 00:02:06,220 --> 00:02:11,310 Here is the front page of Einstein's general theory of relativity paper. 33 00:02:11,310 --> 00:02:16,292 However, even before this was published, he already had some of these ideas, 34 00:02:16,292 --> 00:02:22,352 including gravitational bending of light. The equivalence principle postulates that 35 00:02:22,352 --> 00:02:25,180 gravitational and inertial mass are the same. 36 00:02:25,180 --> 00:02:29,276 In addition to that, there is Mach's principle which postulates that 37 00:02:29,276 --> 00:02:33,189 gravitational interaction of all masses is what creates inertia. 38 00:02:33,189 --> 00:02:37,530 In an empty universe, there will be no inertia, there will be no mass, which is 39 00:02:37,530 --> 00:02:41,002 sort of obvious. But Einstein argued that gravity is an 40 00:02:41,002 --> 00:02:45,737 inertial force and there is a famous experiment with elevators and rockets. 41 00:02:45,737 --> 00:02:49,903 There are two important consequences of the equivalence principle. 42 00:02:49,903 --> 00:02:54,701 First, that light should be blue shifted coming in to gravitational potential 43 00:02:54,701 --> 00:03:00,383 wells and redshift coming out of the gravitational potential wells and the 44 00:03:00,383 --> 00:03:03,540 light path would be curved in, space in general. 45 00:03:03,540 --> 00:03:08,507 The presence of mass will introduce local curvature and light moves to what 46 00:03:08,507 --> 00:03:12,705 [INAUDIBLE] is the shortest line. Einstein was already aware of the 47 00:03:12,705 --> 00:03:18,093 possibility gravitational bending of light. And in 1911, he wrote to George 48 00:03:18,093 --> 00:03:22,361 Ellery Hale, famous astronomer, asking him if this can be observed. 49 00:03:22,361 --> 00:03:25,650 Well, apparently, it wasn't possible at the time. 50 00:03:25,650 --> 00:03:28,686 However, it was actually possible to do it during 51 00:03:28,686 --> 00:03:32,794 solar eclipses and there is a famous experiment where Eddington and 52 00:03:32,794 --> 00:03:37,626 collaborators measured positions of stars around solar discs during the maximum 53 00:03:37,626 --> 00:03:41,604 eclipse and then compared them to sunny somewhere, else, else in the sky. 54 00:03:41,604 --> 00:03:46,710 This completely confirmed Einsteins prediction of bending of light by about 55 00:03:46,710 --> 00:03:51,481 almost twoarc arc seconds near the edge of the solar disc and this is why 56 00:03:51,481 --> 00:03:56,319 Einstein really became famous. So let's talk a little bit about geometry 57 00:03:56,319 --> 00:03:59,382 of spacetime. In general, space can be curved. 58 00:03:59,382 --> 00:04:03,696 It's hard to draw four dimensional space time, so we use, use these 59 00:04:03,696 --> 00:04:07,531 two-dimensional analogies embedded in three--dimensional space. 60 00:04:07,531 --> 00:04:13,214 Positive curvature is like curvature of a sphere and that for response physical 61 00:04:13,214 --> 00:04:17,471 closed universe. the curvature will depend on the amount 62 00:04:17,471 --> 00:04:22,544 of mass and energy is present. If, it's exact right critical amount it 63 00:04:22,544 --> 00:04:27,984 will be flat geometry of familiar space. If it's less than that, it'll be 64 00:04:27,984 --> 00:04:32,690 hyperbolic geometry which is also open in existence to infinity. 65 00:04:32,690 --> 00:04:39,071 So how do we quantity geometry of space? The most general approach is to define a 66 00:04:39,071 --> 00:04:45,531 metric, which is a formula that gives the distance between any two points and it's 67 00:04:45,531 --> 00:04:51,698 usually expressed in form of curvature like here, that is the sum of products of 68 00:04:51,698 --> 00:04:57,931 increments in each of the two coordinates multiplied together, sum over all of them 69 00:04:57,931 --> 00:05:03,446 and some coefficients multiply them. So, these coefficients are the metric 70 00:05:03,446 --> 00:05:09,165 answer and its indices in four-dimensional spacetime run from zero 71 00:05:09,165 --> 00:05:15,133 through three, zero is the, the time, one, two, and three are usual spatial XYZ 72 00:05:15,133 --> 00:05:19,442 dimensions. This is a matrix 4x4 or tensor, if you're 73 00:05:19,442 --> 00:05:23,670 familiar with that and for, for plain familiar geometry. 74 00:05:23,670 --> 00:05:30,218 The coefficients are one on diagonal, its the sum of squares in increments in three 75 00:05:30,218 --> 00:05:34,920 spacial coordinates and all of the diagonal limits is zero. 76 00:05:34,920 --> 00:05:38,890 So this is the simple measure of distance in 3D space, 77 00:05:38,890 --> 00:05:44,404 but in general, it can be different and values of these coefficients can be 78 00:05:44,404 --> 00:05:47,492 derived from general theory of relativity. 79 00:05:47,492 --> 00:05:53,006 So, if we know what the metric is, then the geometry completely specified by it. 80 00:05:53,006 --> 00:05:58,980 In the special relativlistic space, which is still flat in Euclidean. 81 00:05:58,980 --> 00:06:06,103 We have to introduce the time element, and, for reasons that we don't need to go 82 00:06:06,103 --> 00:06:12,425 into, it is expressed as square of the time increment minus the spatial 83 00:06:12,425 --> 00:06:13,582 increment of, what that [INAUDIBLE] to be coordinates. 84 00:06:14,651 --> 00:06:21,418 But the very general case, in general activity for homogeneous and isotropic 85 00:06:21,418 --> 00:06:28,186 universe, is the Robertson-Walker metric. Remember that we introduced homogeneity 86 00:06:28,186 --> 00:06:32,283 and isotropy as means of simplifying general relativity. 87 00:06:32,283 --> 00:06:37,259 Homogeneous universe means the same everywhere, and isotropic in all 88 00:06:37,259 --> 00:06:42,014 directions, and therefore, only the radial co, coordinate would matter. 89 00:06:42,014 --> 00:06:48,307 So now, Minkowksi metric from above still has the same old time element and then 90 00:06:48,307 --> 00:06:54,228 has a spatial element which is multiplied by a function r(t), which is called a 91 00:06:54,228 --> 00:06:58,173 scale factor. And the rest of it is expression that is 92 00:06:58,173 --> 00:07:04,090 universally true for different, different values of spatial curvature. 93 00:07:04,090 --> 00:07:08,546 So, Robertson-Walker Metric is expressed in polar coordinates, 94 00:07:08,546 --> 00:07:12,418 this is useful because only radial direction matters. 95 00:07:12,418 --> 00:07:18,116 And, it can be also written in this form where the function S, depending on the 96 00:07:18,116 --> 00:07:22,937 different curvature, is sign, hyperbolic sign for the radius itself. 97 00:07:22,937 --> 00:07:29,445 Remember that, that whole spatial increment can be changing in time, in 98 00:07:29,445 --> 00:07:33,630 principle and this is what the scale factor is all about. 99 00:07:33,630 --> 00:07:37,593 Sometimes it's denoted as R of the [INAUDIBLE] a(t), 100 00:07:37,593 --> 00:07:41,924 I'll be using both of those. You just have to pay attention. 101 00:07:41,924 --> 00:07:47,576 Basically, that's the ratio of any given distance at some time to what it was 102 00:07:47,576 --> 00:07:52,404 relative to the zero point reference, like if you start with an expanding 103 00:07:52,404 --> 00:07:56,513 universe with distance between two galaxies of a gigaparsec, then sometime 104 00:07:56,513 --> 00:08:01,241 later, it will be more than a gigaparsec. So this is just a thumbnail sketch of 105 00:08:01,241 --> 00:08:05,631 what general relativity is all about. Next time, we will talk a little bit more 106 00:08:05,631 --> 00:08:07,320 how it's applied in cosmology.