1 00:00:05,220 --> 00:00:10,121 Let us now address the question of the expanding universe and what does it 2 00:00:10,121 --> 00:00:13,454 really mean. This is usually something that baffles 3 00:00:13,454 --> 00:00:17,441 many people when they first hear about expanding universe. 4 00:00:17,441 --> 00:00:22,277 What is it expanding into and how? The important notion here is that there 5 00:00:22,277 --> 00:00:26,721 really are two different kind of coordinates in general relativity 6 00:00:26,721 --> 00:00:29,597 cosmology. There are, there are the comoving 7 00:00:29,597 --> 00:00:32,800 coordinates which expand as the universe expands. 8 00:00:32,800 --> 00:00:37,810 Examples of those would be unbound systems like any two distant galaxies 9 00:00:37,810 --> 00:00:42,821 that are not gravitationally bound together or wavelengths of massless 10 00:00:42,821 --> 00:00:46,755 quanta like photons. But they're expanding relative to what? 11 00:00:46,755 --> 00:00:51,044 There has to be a fixed set of coordinates and those are called proper 12 00:00:51,044 --> 00:00:54,004 coordinates, those was what you usually think of as 13 00:00:54,004 --> 00:00:57,810 coordinates, which are fixed and space expands relative to them. 14 00:00:57,810 --> 00:01:02,869 If it wasn't the case, we couldn't tell there was an expansion, because 15 00:01:02,869 --> 00:01:07,151 everything would be expanding. Well, things that do not expand with 16 00:01:07,151 --> 00:01:12,470 expanding space or atoms or molecules or solid bodies like planets or in fact any 17 00:01:12,470 --> 00:01:17,076 gravitationally bound systems like the solar system, stars and galaxies. 18 00:01:17,076 --> 00:01:21,811 So, galaxies don't puff up with the expanding universe, they stay the same 19 00:01:21,811 --> 00:01:23,498 size, but they do go apart. 20 00:01:23,498 --> 00:01:26,093 Okay, so what's the universe expanding into? 21 00:01:26,093 --> 00:01:30,310 Well its, in expanding into itself. There is nothing outside to it, 22 00:01:30,310 --> 00:01:35,828 except for purely hypothetical multiverse, but same would apply in each 23 00:01:35,828 --> 00:01:42,124 one of its constituent bubble universes. Consider a positive curvature universe, 24 00:01:42,124 --> 00:01:47,253 that's like the surface of a sphere except in one extra dimension. 25 00:01:47,253 --> 00:01:51,606 It has finite volume and finite surface, but it can grow. 26 00:01:51,606 --> 00:01:56,060 So there is no edge and it can expand. If you will, 27 00:01:56,060 --> 00:02:00,540 you know, the fourth dimension into which it's expanding is time. 28 00:02:00,540 --> 00:02:06,015 Now, flat or negative curvature universe is simply infinite in all directions and 29 00:02:06,015 --> 00:02:09,713 you can think of a grid stretching out everywhere. 30 00:02:09,713 --> 00:02:14,833 So, in, in either case, there is no edge and there is no center, because of 31 00:02:14,833 --> 00:02:19,242 homogeneity and isotropy. So the universe is expanding into itself 32 00:02:19,242 --> 00:02:23,980 in the sense and there is no, nothing outside of the universe. 33 00:02:23,980 --> 00:02:29,061 Oh, well how does it expand. The function describes that is called the 34 00:02:29,061 --> 00:02:34,290 scale factor which we mentioned earlier in [INAUDIBLE] function inside 35 00:02:34,290 --> 00:02:37,972 Robertson-Walker matrix, usually called R(t) or a(t). 36 00:02:37,972 --> 00:02:43,496 You take any two distance points and measure how their distance changes in 37 00:02:43,496 --> 00:02:46,810 time, that's essentially what scale factor is. 38 00:02:46,810 --> 00:02:52,554 And because universe is homogeneous and isotropic, it same exact same function 39 00:02:52,554 --> 00:02:55,515 will apply to all points in space and time. 40 00:02:55,515 --> 00:03:00,813 So if we knew what that function was then we can tell exactly how the universe is 41 00:03:00,813 --> 00:03:04,580 going to evolve. Being able to do this is exceptionally 42 00:03:04,580 --> 00:03:09,253 what cosmological models are. And we do this by solving the Friedman 43 00:03:09,253 --> 00:03:14,276 equation, which is what Einstein equations are reduced to after assuming 44 00:03:14,276 --> 00:03:19,668 homogeneity and isotropy. But first, let's think about cosmological 45 00:03:19,668 --> 00:03:22,683 redshift. This is the usual balloon analogy, 46 00:03:22,683 --> 00:03:26,560 inflate the balloon and points on it go further apart. 47 00:03:26,560 --> 00:03:31,515 Well, if you glued little buttons on the surface of the balloon, buttons will not 48 00:03:31,515 --> 00:03:35,294 be expanding as balloon expands, those would be like galaxies. 49 00:03:35,294 --> 00:03:39,816 But if you drew wiggle lines on, on the balloon, those will stretch as the 50 00:03:39,816 --> 00:03:42,480 balloon expands. Those will be like photons. 51 00:03:42,480 --> 00:03:48,258 So we talk about the redshift since the wavelength tends to increase, moves to 52 00:03:48,258 --> 00:03:51,841 the red. And familiar notion is the Doppler shift 53 00:03:51,841 --> 00:03:57,693 which in nonrelativistic case is just the ratio of speed by which something is 54 00:03:57,693 --> 00:04:03,909 moving to the speed of light or speed of sound in case of sound Doppler shift. 55 00:04:03,909 --> 00:04:09,614 In special relativity, there is little bit of relativistic correction, shown 56 00:04:09,614 --> 00:04:10,560 here. So, 57 00:04:10,560 --> 00:04:15,183 we do think of cosmological redshift often as being a Doppler shift. 58 00:04:15,183 --> 00:04:20,083 Galaxies are moving away from us therefore there is a Doppler shift in 59 00:04:20,083 --> 00:04:25,467 wavelength stretching, but really more correct explanation or more correct way 60 00:04:25,467 --> 00:04:30,160 of thinking of it is that it corresponds to the stretching of space. 61 00:04:31,960 --> 00:04:36,917 Well, if any distance stretches this according to the scale factor, r(t), then 62 00:04:36,917 --> 00:04:41,810 the wavelengths will do the same. And our definition of redshift is change 63 00:04:41,810 --> 00:04:44,543 in wavelength divided by wavelength itself. 64 00:04:44,543 --> 00:04:48,420 And so, if we do either these two, we get that the scale factor. 65 00:04:48,420 --> 00:04:56,470 Ratio between scale factored now, and some different time, is one plus ratchet. 66 00:04:56,470 --> 00:05:01,870 So, we essentially use one + red shift as the measure of cosmic expansion. 67 00:05:01,870 --> 00:05:08,021 This turns out to be exactly equivalent to the Doppler shift expan, explanation, 68 00:05:08,021 --> 00:05:12,147 but this is probably a better way of thinking about it. 69 00:05:12,147 --> 00:05:17,923 Consider an observer at a radius of zero and some radial light rays. 70 00:05:17,923 --> 00:05:23,474 since they move at [INAUDIBLE] six, that means that some of the spatial 71 00:05:23,474 --> 00:05:27,880 displacement is equal to the temporal placement. 72 00:05:27,880 --> 00:05:32,720 And the following equation must apply according to Robertson-Walker metric. 73 00:05:32,720 --> 00:05:39,466 So if we emit our array of light at time t1, and its observed at t of zero, then 74 00:05:39,466 --> 00:05:46,380 integral over those two paths has to match, and according to this formula. 75 00:05:46,380 --> 00:05:52,291 So if a co-moving source is some distance R and its coordinates is fixed, well 76 00:05:52,291 --> 00:05:57,907 then, the relative displacements of, or relative increments in time, scaled by 77 00:05:57,907 --> 00:06:02,584 the scale factor must be the same. So therefore, there's going to be a 78 00:06:02,584 --> 00:06:06,914 redshift in expanding universe. You can divide it to and it's exact same 79 00:06:06,914 --> 00:06:11,861 result as we've gotten before. Well, lets consider Hubble's law in the 80 00:06:11,861 --> 00:06:17,173 light of this explanation. For nearby source we can write this as a 81 00:06:17,173 --> 00:06:23,118 following expression, R dot is the time derivative of scale factor of this time. 82 00:06:23,118 --> 00:06:28,350 So, there are actually is essentially time derivative of the scale factor 83 00:06:28,350 --> 00:06:31,680 divided by the log of scale factor itself. 84 00:06:31,680 --> 00:06:37,943 In other words, it is dimensionless fractional change in scale factor and 85 00:06:37,943 --> 00:06:42,829 that is what Hubble constant is. So we can derive Hubble's law directly 86 00:06:42,829 --> 00:06:47,405 from Robertson-Walker metric and Hubble constasnt is not really a constant, 87 00:06:47,405 --> 00:06:51,696 it changes in time. But at any given time, this is a constant 88 00:06:51,696 --> 00:06:55,497 obviously and so, it's an instantaneous expansion rate. 89 00:06:55,497 --> 00:07:00,134 Todays values usually do not get h not, little zero subscript. 90 00:07:00,134 --> 00:07:04,543 Well, finally here is something interesting to think about. 91 00:07:04,543 --> 00:07:08,040 The holy cow of physics is a ration of energy, 92 00:07:08,040 --> 00:07:13,852 but turns out energy is not conserved in an expanding or contracting universe. 93 00:07:13,852 --> 00:07:19,287 The couple simple examples you, those protons that stretch with redshift, but 94 00:07:19,287 --> 00:07:24,421 where does their energy go? A proton may be emitted, able to ionize a 95 00:07:24,421 --> 00:07:28,291 hydrogen atom say, but when it's received it's energy's much 96 00:07:28,291 --> 00:07:31,700 less because wavelength is less. Where did that energy go? 97 00:07:31,700 --> 00:07:34,992 Or, the other way around. Take two galaxies, there not 98 00:07:34,992 --> 00:07:40,120 gravitationally bound but they still have some potential energy relative to each 99 00:07:40,120 --> 00:07:43,582 other. Expansion of universe carries them apart, 100 00:07:43,582 --> 00:07:48,800 the potential energy drops so the system has gained energy, from where. 101 00:07:48,800 --> 00:07:54,796 So, the upshot is that energy's actually not conserved, in an expanding universe. 102 00:07:54,796 --> 00:07:59,958 It's something to think about. So next time we'll talk about Friedmann 103 00:07:59,958 --> 00:08:05,500 Equation which is the actual way in which we compute cosmological models.