1 00:00:00,000 --> 00:00:02,714 Hooray. So, we understand something about how 2 00:00:02,714 --> 00:00:05,614 gravity affects light, how gravity affects time. 3 00:00:05,614 --> 00:00:08,513 Because we're dealing in a relativistic theory. 4 00:00:08,513 --> 00:00:13,017 It affects distances as well. And Einstein thinks when he figures this 5 00:00:13,017 --> 00:00:16,472 out, that's he's well on his way to resolving the theory. 6 00:00:16,472 --> 00:00:19,926 It turns out that the mathematics involved is difficult. 7 00:00:19,926 --> 00:00:24,738 luckily for him, unlike Newton after a few years of worrying about it, he 8 00:00:24,738 --> 00:00:29,545 discovers that mathematicians had already pretty much solved the problem he was 9 00:00:29,545 --> 00:00:32,227 trying to solve, albeit in a different context. 10 00:00:32,227 --> 00:00:35,958 And making the connection in conversations with Riemann, he very 11 00:00:35,958 --> 00:00:39,689 quickly is led to what becomes the Theory of General Relativity. 12 00:00:39,689 --> 00:00:43,887 And it, he comes up with this understanding that gravity is in a very 13 00:00:43,887 --> 00:00:46,157 technical sense, a description of geometry. 14 00:00:46,157 --> 00:00:50,116 And as I said, the mathematics was hairy. And we're not going to do it here. 15 00:00:50,116 --> 00:00:54,179 So, this is the hard part of the class where I get to do a lot of vigorous hand 16 00:00:54,179 --> 00:00:57,367 flapping and tell you the answers. But I'll try to motivate it. 17 00:00:57,367 --> 00:00:59,578 So, what does, what do we know about gravity? 18 00:00:59,578 --> 00:01:02,920 What are the characteristics of a relativistic theory of gravity? 19 00:01:02,920 --> 00:01:05,543 Well, in small regions of space-time, if you observe, 20 00:01:05,543 --> 00:01:08,628 there are a collection of inertial freely falling observers. 21 00:01:08,628 --> 00:01:11,560 And if they measure, then in their local region, the don't, 22 00:01:11,560 --> 00:01:16,093 that they claim that there is no gravity. These are the freely falling observers. 23 00:01:16,093 --> 00:01:19,210 Of course, there are many of them. They, they can have 24 00:01:19,210 --> 00:01:22,610 all different velocities, and their different experiences locally. 25 00:01:22,610 --> 00:01:26,743 Because they have rel, constant relative velocities relative to each other are 26 00:01:26,743 --> 00:01:29,987 related by the Lorentz transformations that we know about. 27 00:01:29,987 --> 00:01:32,603 And then, the reason that gravity is for real, 28 00:01:32,603 --> 00:01:34,852 what's really going on is the tidal effect. 29 00:01:34,852 --> 00:01:39,090 The fact that the acceleration of gravity at different positions in space and at 30 00:01:39,090 --> 00:01:42,020 different times, is not the same. This means that what 31 00:01:42,020 --> 00:01:46,235 but as an inertial observer here is not the same as what an inertial observer 32 00:01:46,235 --> 00:01:48,343 there. So, if two people are freely falling 33 00:01:48,343 --> 00:01:52,157 towards Earth say, a near to each other, they jump off the same spaceship and 34 00:01:52,157 --> 00:01:55,921 they're freely falling towards Earth. They will notice that they are slowly 35 00:01:55,921 --> 00:01:59,785 accelerating towards each other and each will blame the other for being that 36 00:01:59,785 --> 00:02:02,793 inertia, I'm in free fall. You can't be because you're accelerating 37 00:02:02,793 --> 00:02:04,802 towards me. Of course, what's really going on is 38 00:02:04,802 --> 00:02:07,710 they're both accelerating towards the center of the Earth. 39 00:02:07,710 --> 00:02:11,014 and it's the tidal effect that is the real gravity. 40 00:02:11,014 --> 00:02:15,652 on the other hand, if I give an object a velocity, any initial velocity at any 41 00:02:15,652 --> 00:02:20,522 given point and then just let it go then gravitation ensures that it knows where 42 00:02:20,522 --> 00:02:23,652 it's going to go. At that point, gravity is the only force 43 00:02:23,652 --> 00:02:26,319 acting. So, given any initial velocity, there is 44 00:02:26,319 --> 00:02:31,073 a unique inertial world line that starts with that initial velocity, starts moving 45 00:02:31,073 --> 00:02:33,566 at that slop relative to the vertical axis. 46 00:02:33,566 --> 00:02:35,943 And then, from there on let gravity do it. 47 00:02:35,943 --> 00:02:40,246 Gravity will tell this thing where to go. Why am I telling you all these stories? 48 00:02:40,246 --> 00:02:43,752 These are all things we know. Because they all have an analogy in these 49 00:02:43,752 --> 00:02:48,246 theories of geometry that were being developed in the 19th and early 20th 50 00:02:48,246 --> 00:02:50,074 century. And what do we know about? 51 00:02:50,074 --> 00:02:52,790 What did, what did these people teach us about geometry? 52 00:02:52,790 --> 00:02:55,309 You're talking about the geometry of curved spaces. 53 00:02:55,309 --> 00:02:57,680 And we'll give an example and try to make it 54 00:02:57,680 --> 00:03:00,421 more clear. But, small regions of occurred flay, 55 00:03:00,421 --> 00:03:04,913 space always appear like flat space and can be described at the normal x, y, z. 56 00:03:04,913 --> 00:03:09,346 Whatever number of coordinates your dimension is we all live on a curved 57 00:03:09,346 --> 00:03:11,855 space. For example, the surface of the Earth. 58 00:03:11,855 --> 00:03:15,749 But we all use coordinates like street and avenue number 59 00:03:15,749 --> 00:03:20,197 in the context of a city, and we can easily describe Manhattan ignoring the 60 00:03:20,197 --> 00:03:24,705 curvature of the Earth and navigate around Manhattan to very good precision. 61 00:03:24,705 --> 00:03:29,866 Because the curvature of the Earth takes affects us only over large distances. 62 00:03:29,866 --> 00:03:34,255 A small chunk of the Earth, of the spherical Earth, is effectively flat and 63 00:03:34,255 --> 00:03:37,458 we can use maps. But, we can't use a map, as we all know 64 00:03:37,458 --> 00:03:41,832 who've, when you look at, at trying to draw projections. A map of the entire 65 00:03:41,832 --> 00:03:45,657 Earth needs to be either a globe or have large distortions on it. 66 00:03:45,657 --> 00:03:50,012 Now again, just as with gravity, when you pick, sit at any point on a curved 67 00:03:50,012 --> 00:03:54,660 surface, you can draw the, these little straight lines and pretending that these 68 00:03:54,660 --> 00:03:58,923 things are flat, pick any direction. You can draw the beginnings of a straight 69 00:03:58,923 --> 00:04:01,442 line in any direction in the little flat region, 70 00:04:01,442 --> 00:04:05,640 and those lines are related by rotations. You rotate a line through one point in 71 00:04:05,640 --> 00:04:08,159 one direction. You rotate it, you get a line in a 72 00:04:08,159 --> 00:04:11,307 different direction. on the other hand, the lines that are 73 00:04:11,307 --> 00:04:14,008 straight. In these local patches, in one patch, in 74 00:04:14,008 --> 00:04:18,466 one little piece of space are not and we'll see this, the same as the straight, 75 00:04:18,466 --> 00:04:23,324 do not look straight as viewed from a different position in a particular not 76 00:04:23,324 --> 00:04:27,382 related to the straight lines through this point by, the first point by 77 00:04:27,382 --> 00:04:30,125 rotation. And third, given a point in an original 78 00:04:30,125 --> 00:04:34,009 direction, there is a unique geodesic, the analog in a curved space of a 79 00:04:34,009 --> 00:04:36,716 straight line. And we'll see an example of how to do 80 00:04:36,716 --> 00:04:38,850 that. But in a, any curved manifold at any 81 00:04:38,850 --> 00:04:41,037 point, there is sort of this analog of the 82 00:04:41,037 --> 00:04:44,524 straightest thing there is. The closest there is to a straight line. 83 00:04:44,524 --> 00:04:47,543 So, we'll define geometric object. It's called the geodesic. 84 00:04:47,543 --> 00:04:52,047 And at any point if you pick a direction, there is a unique geodesic that starts at 85 00:04:52,047 --> 00:04:56,117 that point moving in that direction. Just as there was a unique inertial world 86 00:04:56,117 --> 00:04:59,813 line starting at any point with any velocity, you're beginning to make the 87 00:04:59,813 --> 00:05:03,809 analog that the shape of a space isn't coded exactly in the fact that what are 88 00:05:03,809 --> 00:05:07,405 straight or, or if it's curved. That what are straight lines at one point 89 00:05:07,405 --> 00:05:11,100 are not straight lines at another. Or it's encoded in the distance between 90 00:05:11,100 --> 00:05:15,346 points as some of course, the distances between points depends on how you label 91 00:05:15,346 --> 00:05:17,393 the points. But, some coordinate invariant 92 00:05:17,393 --> 00:05:20,840 information there and in geometric terms, this is called curvature. 93 00:05:20,840 --> 00:05:24,456 This is a lot of fancy words about geometry, but I haven't really told you 94 00:05:24,456 --> 00:05:26,948 anything. Let's do an example that might help us 95 00:05:26,948 --> 00:05:30,955 understand what we are talking about and the example is, let's think about Earth's 96 00:05:30,955 --> 00:05:33,570 surface. It's a two dimensional surface we live on 97 00:05:33,570 --> 00:05:37,858 a surface and it's a curved space. The surface of a sphere is not flat, try 98 00:05:37,858 --> 00:05:41,914 to wrap a piece of paper around a ball or make a, a, a planer, a map two 99 00:05:41,914 --> 00:05:46,433 dimensional map of the Earth on a flat piece of paper and you'll realize that 100 00:05:46,433 --> 00:05:49,852 the Earth is curved. So, get an approximation it's spherical, 101 00:05:49,852 --> 00:05:54,198 spherical is special and it makes some of the observations sort of trivial. 102 00:05:54,198 --> 00:05:57,153 On a sphere, every point is the same as every other. 103 00:05:57,153 --> 00:06:01,556 And so, if we think of the Earth as a sphere, then might as well start at the 104 00:06:01,556 --> 00:06:04,865 pole because we want a point. Every point is like any other. 105 00:06:04,865 --> 00:06:07,628 If you start off in any direction, you're going south. 106 00:06:07,628 --> 00:06:11,695 And now, we understand on Earth exactly what it means to go in a straight line. 107 00:06:11,695 --> 00:06:14,035 Sit at the pole, pick some direction and keep going 108 00:06:14,035 --> 00:06:16,013 straight. What are you going to do? Well, you'll 109 00:06:16,013 --> 00:06:19,110 start out getting south along some meridian, and following that meridian 110 00:06:19,110 --> 00:06:22,207 you'd be going in a straight line, turning east or west of that meridian 111 00:06:22,207 --> 00:06:25,346 would not be going in a straight line. We all have this understanding and 112 00:06:25,346 --> 00:06:28,701 eventually, of course, you'll come to the other, cross the other pole, keep going 113 00:06:28,701 --> 00:06:31,109 around the Earth and come back to the place you started. 114 00:06:31,109 --> 00:06:33,991 what you will have completed is what is called the Great Circle. 115 00:06:33,991 --> 00:06:35,970 These are those geodesics that I talked about. 116 00:06:35,970 --> 00:06:39,324 And you notice that starting at the pole, going off in any direction, there are 117 00:06:39,324 --> 00:06:42,593 meridians, there are these Great Circles. Nothing special about the pole, as I 118 00:06:42,593 --> 00:06:44,442 said, every point is like every other point. 119 00:06:44,442 --> 00:06:48,941 So, indeed, on Earth, there are precisely straight lines, great 120 00:06:48,941 --> 00:06:52,238 circles, geodesics. Unique one starting at any point, heading 121 00:06:52,238 --> 00:06:55,279 off in any direction. Now initially, as you mentioned, a bunch 122 00:06:55,279 --> 00:06:59,282 of penguins walking off from the north, exact North Pole in all directions, they 123 00:06:59,282 --> 00:07:03,388 won't notice they're on a curved planet. They will be walking on a straight sheet 124 00:07:03,388 --> 00:07:05,973 of ice. And everything will look completely flat. 125 00:07:05,973 --> 00:07:09,875 And, in particular, if they measure the distances between them, they will match 126 00:07:09,875 --> 00:07:13,626 what you'd expect from the angle, the small angle approximation if they're 127 00:07:13,626 --> 00:07:17,224 moving in small angles to each other and the distance they've traveled. 128 00:07:17,224 --> 00:07:20,924 As they get farther and farther away, they will find that each successive 129 00:07:20,924 --> 00:07:24,987 penguin is farther and farther apart. on the other hand, as they get really 130 00:07:24,987 --> 00:07:28,815 far, they'll notice that the distances are not increasing sufficiently. 131 00:07:28,815 --> 00:07:33,078 this is where the curvature of Earth comes in and perhaps a demonstration of 132 00:07:33,078 --> 00:07:35,800 this would be helpful. So, 133 00:07:35,800 --> 00:07:38,690 here's our demonstration. It's our old friend 134 00:07:38,690 --> 00:07:42,759 right, right assention declination demonstrator from UNL. 135 00:07:42,759 --> 00:07:45,405 But, we are going to re-purpose it for what 136 00:07:45,405 --> 00:07:49,807 are our own devious goals here. And so what we're going to do is we're going to 137 00:07:49,807 --> 00:07:53,221 imagine this collection of ants or penguins that we talked about. 138 00:07:53,221 --> 00:07:57,319 They start at the North Pole, and as they move off in all directions, initially 139 00:07:57,319 --> 00:07:59,682 they don't notice the curvature of the Earth. 140 00:07:59,682 --> 00:08:03,832 And what that means is that, one walks along the green meridian, one walks along 141 00:08:03,832 --> 00:08:06,932 the red meridian and they will notice that the distance 142 00:08:06,932 --> 00:08:11,134 between them is given by the distance they have traveled. and then, determined 143 00:08:11,134 --> 00:08:15,284 by the small angle for them any angle between them. But as they get farther and 144 00:08:15,284 --> 00:08:21,416 further apart, you notice that these lines that, that one of the penguins, as 145 00:08:21,416 --> 00:08:25,417 they get farther and farther down their lines, you'll notice that their lines are 146 00:08:25,417 --> 00:08:28,254 not getting separated by a sufficient distance. 147 00:08:28,254 --> 00:08:33,256 They're, in fact, failing to get to be far enough to satisfy the small angle 148 00:08:33,256 --> 00:08:36,334 formula. In fact, as they go farther, the distance 149 00:08:36,334 --> 00:08:41,207 stops increasing all together when they hit the pole and then it starts the 150 00:08:41,207 --> 00:08:44,350 equator, and then it starts decreasing eventually. 151 00:08:44,350 --> 00:08:49,169 These two lines that went off at different angles do something they would 152 00:08:49,169 --> 00:08:53,662 never do in a flat planer world, they collide again at the South Pole. 153 00:08:53,662 --> 00:08:58,677 So, the fact that these two geodesics are not going far away enough from each 154 00:08:58,677 --> 00:09:02,975 other, not going as far from each other as they would in the plane. 155 00:09:02,975 --> 00:09:07,925 And then, eventually, they meet as an indication of what we call the positive 156 00:09:07,925 --> 00:09:13,655 curvature of the surface of the Earth. Less we get too caught up in all curved 157 00:09:13,655 --> 00:09:16,313 surfaces are like the surfaces of the earth. 158 00:09:16,313 --> 00:09:20,963 we have here this nice demonstration of the surface, again two dimensional. 159 00:09:20,963 --> 00:09:25,667 We can only really imagine curved two dimensional spaces because we think of 160 00:09:25,667 --> 00:09:30,443 them as sitting inside our three dimensional space. But there are curved 161 00:09:30,443 --> 00:09:34,821 three and four and six dimensional spaces, at least mathematically. 162 00:09:34,821 --> 00:09:40,195 And so you can this surface, this so-called one-sheeted hyperboloid is a 163 00:09:40,195 --> 00:09:44,706 surface with negative curvature. And I'll make it a little more 164 00:09:44,706 --> 00:09:51,510 transparent so we can see and the geodesic that I'm this program very 165 00:09:51,510 --> 00:09:56,656 nicely draws geodesics through a chosen point and I can vary both the location of 166 00:09:56,656 --> 00:10:00,547 the point in the angle. What I want you to see is what happens 167 00:10:00,547 --> 00:10:03,559 when I make a small change in the initial angle. 168 00:10:03,559 --> 00:10:08,328 See, I'm rotating the initial angle at which this geodesic comes out, ever so 169 00:10:08,328 --> 00:10:11,278 slightly. And what I'm seeing is that geodesics 170 00:10:11,278 --> 00:10:16,361 that start very nearby, here are notice that the change far off is much larger 171 00:10:16,361 --> 00:10:21,193 than the change I am making nearby. So, geodesics that start very near each 172 00:10:21,193 --> 00:10:26,481 other will eventually grow very far. The distance between geodesics grows more 173 00:10:26,481 --> 00:10:32,485 than you would expect in on on the plane. These are getting farther apart more than 174 00:10:32,485 --> 00:10:37,738 the small angle formula specifies a negative curvature leads to diverging 175 00:10:37,738 --> 00:10:43,263 geodesics where as a positive curvature leads to converging geodesics as they do 176 00:10:43,263 --> 00:10:47,698 on the surface of the earth. So, we have here this analogy between 177 00:10:47,698 --> 00:10:52,746 geometry for which we have some intuition, and for which mathematicians 178 00:10:52,746 --> 00:10:54,520 have the formalism. And 179 00:10:54,520 --> 00:11:00,245 the theory of relativistic gravity the role of inertial world lines is played by 180 00:11:00,245 --> 00:11:03,403 geodesics. Logan's transformations correspond to 181 00:11:03,403 --> 00:11:07,184 rotations, rotations preserve the distance between two points. 182 00:11:07,184 --> 00:11:11,512 We know that making a rotation does not change the lengths of things. 183 00:11:11,512 --> 00:11:14,819 Lorentz transformation has also preserved something. 184 00:11:14,819 --> 00:11:18,847 They preserve the value of this invariant interval that we defined. 185 00:11:18,847 --> 00:11:23,896 the minus sign here, as opposed to the plus sign here, has an important role. 186 00:11:23,896 --> 00:11:27,864 It means that the geometry, technically we're going to describe is 187 00:11:27,864 --> 00:11:34,540 pseudo-maronian rather than maronian, and it makes many differences. Gravity is not 188 00:11:34,540 --> 00:11:40,723 the same as the description of a curve of, of [UNKNOWN] spaces. 189 00:11:40,723 --> 00:11:46,230 But, gravitational forces are therefore the changes in 190 00:11:46,230 --> 00:11:51,139 that the, the, the, the fact that the title forces that make what seems to be 191 00:11:51,139 --> 00:11:54,870 inertial at one point, not look inertial at another point, 192 00:11:54,870 --> 00:11:59,485 play the role of the curvature. So, the idea is the presence of a massive 193 00:11:59,485 --> 00:12:05,191 object deforms time and it's vicinity and the motion of inertial freely falling 194 00:12:05,191 --> 00:12:09,807 objects is a long geodesic, these straightest of lines that you can draw. 195 00:12:09,807 --> 00:12:15,031 Remember, inertial world lines were those straight lines and the flat case absent 196 00:12:15,031 --> 00:12:19,196 gravity and now they're going to be as straight as they can be in this curved 197 00:12:19,196 --> 00:12:21,973 space-time. And the picture that is often brought up 198 00:12:21,973 --> 00:12:24,429 as a visualization is this rubber sheet image. 199 00:12:24,429 --> 00:12:28,754 You put a bowling ball in the middle of a rubber sheet, it bends the rubber sheet 200 00:12:28,754 --> 00:12:32,866 down and then rolling a ping pong ball will cause the ping pong ball to move 201 00:12:32,866 --> 00:12:35,536 around it. This is misleading on a whole number of 202 00:12:35,536 --> 00:12:37,961 levels. But, I guess the general intuition in 203 00:12:37,961 --> 00:12:42,340 there in some sense the presence of a massive object in particular the main 204 00:12:42,340 --> 00:12:46,000 ingredient here that is completely absent from the real case is 205 00:12:46,000 --> 00:12:49,513 gravitational field of the Earth pulling the bowling ball down. 206 00:12:49,513 --> 00:12:53,784 This, it's preferred down direction is completely non-existent in the actual 207 00:12:53,784 --> 00:12:57,165 case, but I will give you a better visualization in a minute. 208 00:12:57,165 --> 00:13:00,208 a few caveats to taking this picture too seriously. 209 00:13:00,208 --> 00:13:04,716 First of all, the shape of that deformed surface, the rubber sheet is not related 210 00:13:04,716 --> 00:13:08,210 to curved space-time. I'll describe the curved space-time in a 211 00:13:08,210 --> 00:13:10,746 minute. In general relativity, it is not space 212 00:13:10,746 --> 00:13:14,634 that deforms, but space-time. This 4-dimensional object, or if you want 213 00:13:14,634 --> 00:13:17,677 the, in the 2-dimensional case that we've been drawing, 214 00:13:17,677 --> 00:13:21,058 the whole 2-dimensional sheet is curved, not just the x-axis. 215 00:13:21,058 --> 00:13:23,820 It is not space that is curving, it is space-time. 216 00:13:23,820 --> 00:13:29,551 In some circumstances particular cases called static space-time where nothing 217 00:13:29,551 --> 00:13:34,384 changes, essentially in a static space-time, you can define something 218 00:13:34,384 --> 00:13:37,991 called space. And think of space at any given time as 219 00:13:37,991 --> 00:13:41,802 we could think of the entire universe at any given time. 220 00:13:41,802 --> 00:13:45,410 Those horizontal lines in our space-time diagrams. 221 00:13:45,410 --> 00:13:49,542 Those make sense in, in the sense that all of these horizontal lines are 222 00:13:49,542 --> 00:13:52,470 essentially the same in the static space-time case, 223 00:13:52,470 --> 00:13:55,684 but not always. most of the space-time we'll need to 224 00:13:55,684 --> 00:14:00,047 consider are going to be static. inertial orbits are, when you look at the 225 00:14:00,047 --> 00:14:04,237 space, the inertial orbits, the ellipses with the sun at one focus, are not 226 00:14:04,237 --> 00:14:07,164 geodesics of the curved space that the sun creates, 227 00:14:07,164 --> 00:14:11,125 they're geodesics in space-time. It's like helical motion that I drew. 228 00:14:11,125 --> 00:14:15,512 That is the straightest line in the curved space, that is a geodesic in the 229 00:14:15,512 --> 00:14:17,939 curved space time, about a massive object. 230 00:14:17,939 --> 00:14:22,022 So remember, we drew that helix. That helix is a representation of the 231 00:14:22,022 --> 00:14:25,100 geodesic motion in the presence of a massive object. 232 00:14:25,100 --> 00:14:28,592 That's the straightest line you can draw. And, in general 233 00:14:28,592 --> 00:14:33,090 a set of coordinates that allows you to describe all of space-time does not 234 00:14:33,090 --> 00:14:35,813 exist. And we'll see some times when that is a 235 00:14:35,813 --> 00:14:39,186 problem, and we'll actually have to pay attention to this. 236 00:14:39,186 --> 00:14:44,394 A better visualization of, in fact, the gravitational effect the curved, curved 237 00:14:44,394 --> 00:14:49,788 space around space-time. The curved space around a stationary gravitating object 238 00:14:49,788 --> 00:14:54,892 like a star is this so called Flams Parabloid and, this actually is related 239 00:14:54,892 --> 00:14:58,360 to the curvature of space. In the following sense 240 00:14:58,360 --> 00:15:03,463 the space around a star is spherically symmectric, so you can, break it up into 241 00:15:03,463 --> 00:15:06,604 balls. And you can assign each ball a size based 242 00:15:06,604 --> 00:15:09,924 on its area. And what we do here is to each, instead 243 00:15:09,924 --> 00:15:14,473 of this, we draw circles and the circumference of a circle is two pi R 244 00:15:14,473 --> 00:15:20,191 where the surface of that ball is four pi R squared. And if the space is flat then 245 00:15:20,191 --> 00:15:25,520 the distances between the balls are the differences in their Rs that you get from 246 00:15:25,520 --> 00:15:28,894 their areas. But in this curved space, the distances 247 00:15:28,894 --> 00:15:33,333 between Jason falls are not given by the differences in their Rs. 248 00:15:33,333 --> 00:15:37,703 In fact, there is variation in the distance between them. 249 00:15:37,703 --> 00:15:43,030 And as you get closer and closer to the gravitation object, in this case the 250 00:15:43,030 --> 00:15:45,966 star, then the distance between adjacent 251 00:15:45,966 --> 00:15:49,175 spheres, or in this picture circles, gets large. 252 00:15:49,175 --> 00:15:52,431 And the way we, telative to the change in their areas. 253 00:15:52,431 --> 00:15:57,128 And the way we do this here is we pull down the bottom of this sheet so that 254 00:15:57,128 --> 00:16:01,767 these two circles, if this were flat, these two circles which have radii that 255 00:16:01,767 --> 00:16:06,406 are circumferences that are very close to each other would actually be right next 256 00:16:06,406 --> 00:16:11,162 to each other and were extending it so the distance between adjacent circles is 257 00:16:11,162 --> 00:16:14,156 different. And this actually does represent, in a 258 00:16:14,156 --> 00:16:17,680 somewhat reliable way, the space around a gravitating object. 259 00:16:17,680 --> 00:16:21,114 So, I've been very hand wavy about how gravity is curvature. 260 00:16:21,114 --> 00:16:24,956 I can write you the equation. So, here is what Einstein tells us. 261 00:16:24,956 --> 00:16:29,555 Here is Einstein's Theory of General Relativity and the discovery that gravity 262 00:16:29,555 --> 00:16:33,979 is Geometry is only the first step. the next step is writing the equation 263 00:16:33,979 --> 00:16:38,163 which tells you that the way in which mass energy generates a source of 264 00:16:38,163 --> 00:16:42,560 curvature, its a source of space-time curvature in the same way that electric 265 00:16:42,560 --> 00:16:47,016 charge is a source of electric field. And when you change the distribution of 266 00:16:47,016 --> 00:16:51,472 energy and momentum, it is the curvature of space-time that is this field the 267 00:16:51,472 --> 00:16:55,985 ripples in space time are the waves in this field that carry this information 268 00:16:55,985 --> 00:16:59,019 off at the speed of light. And this is the equation, 269 00:16:59,019 --> 00:17:03,224 this is Maxwell's equation for gravity. It looks simpler but only because all 270 00:17:03,224 --> 00:17:06,829 these symbols are con, confusing. We will not do anything with this 271 00:17:06,829 --> 00:17:10,598 equation except write it down, but I just want to write it down for you, 272 00:17:10,598 --> 00:17:14,577 this is Einstein's equation. these are looking objects are of the 273 00:17:14,577 --> 00:17:18,138 Riemann Tensor. They express the curvature of space at a 274 00:17:18,138 --> 00:17:22,864 particular point in space and time. The curvature of space-time, G, we know 275 00:17:22,864 --> 00:17:28,173 that is Newton's constant, C is the speed of light. And this T object is the energy 276 00:17:28,173 --> 00:17:30,245 moment of stress tensor. This is 277 00:17:30,245 --> 00:17:35,351 our characterization of the flow and amount of energy and momentum at a given 278 00:17:35,351 --> 00:17:38,272 point in space and time or at a given event. 279 00:17:38,272 --> 00:17:43,250 So, at a given event, you have this acting like the electric charge and this 280 00:17:43,250 --> 00:17:48,560 tells this is a differential equation that tells the gravitational field how to 281 00:17:48,560 --> 00:17:51,680 respond. You can solve this equation. One of the 282 00:17:51,680 --> 00:17:56,724 consequences of this equation is that inertial motion, the motion of objects 283 00:17:56,724 --> 00:18:01,851 under the influence of nothing but gravity is along geodesics of this curved 284 00:18:01,851 --> 00:18:05,425 space-time that you obtained by solving the equation. 285 00:18:05,425 --> 00:18:10,212 So, just as Maxwell's equa, equations imply charge conservation, these 286 00:18:10,212 --> 00:18:15,000 equations imply energy momentum conservation, but also the fact that 287 00:18:15,000 --> 00:18:20,870 objects upon which only gravity acts move along geodesics of the space-time that 288 00:18:20,870 --> 00:18:24,729 has been created. there is of course, as always, there has 289 00:18:24,729 --> 00:18:29,147 to be a limit in which you reproduce Newton's theory, because Newton's theory 290 00:18:29,147 --> 00:18:33,680 has worked so well for us in describing, say, the motion of Earth around the sun, 291 00:18:33,680 --> 00:18:36,238 and binary stars, and so on. What is the limit? 292 00:18:36,238 --> 00:18:40,495 The limit is that if the curvatures are small, if you don't get too close to 293 00:18:40,495 --> 00:18:44,416 objects that are too massive, the gravitational fields are not extreme. 294 00:18:44,416 --> 00:18:47,610 And if those speeds at which objects are moving are slow, 295 00:18:47,610 --> 00:18:52,592 then you reproduce Newton's formula. You can express, for example, the motion 296 00:18:52,592 --> 00:18:57,309 of a planet about a stationary star in terms of, sort of, some effective 297 00:18:57,309 --> 00:19:01,826 potential energy function which will appear somewhat familiar. 298 00:19:01,826 --> 00:19:05,945 It's a function of R. It starts off with our favorite familiar 299 00:19:05,945 --> 00:19:09,864 term, -GmM/R, this is just the Newtonian potential 300 00:19:09,864 --> 00:19:12,190 energy. It's got this weird looking. 301 00:19:12,190 --> 00:19:16,110 Think others the angular momentum, we'll see in a minute what that is. 302 00:19:16,110 --> 00:19:20,697 And then, the first correction, there's a whole series of corrections. 303 00:19:20,697 --> 00:19:27,645 But the leading correction in Newtonian physics is this L^2/c^2r^3 correction. 304 00:19:27,645 --> 00:19:32,771 This becomes a little more sensible if you make the somewhat unjustified 305 00:19:32,771 --> 00:19:35,672 statement that the angular momentum is mvr. 306 00:19:35,672 --> 00:19:38,640 This is true for motion in a circular orbit. 307 00:19:38,640 --> 00:19:43,497 Plugging L = mvr into this equation simplifies things a great deal. 308 00:19:43,497 --> 00:19:49,787 For example, L^2/2mr^2 becomes. Let's write it out so we know what we are 309 00:19:49,787 --> 00:19:53,816 doing so you don't accuse me of cheating. L^2 m^2v^2r^2/2mr^2. 310 00:19:57,579 --> 00:20:00,785 And now, I do all my handy dandy cancellations. 311 00:20:00,785 --> 00:20:05,385 The r^2 is cancel, one of the m is cancel, and this is a half mv^2. 312 00:20:06,640 --> 00:20:11,940 So, this term in the case of circular orbits is a fancy way of saying Kinetic 313 00:20:11,940 --> 00:20:15,353 energy, this term is Newtonian potential energy. 314 00:20:15,353 --> 00:20:22,252 This term, when you plug everything in, turns just into v^2/c^2 times this term. 315 00:20:22,252 --> 00:20:27,670 And so this is the leading correction to Newtonian physics, and of course, there 316 00:20:27,670 --> 00:20:32,075 are higher order corrections. But when v is small compared to c and the 317 00:20:32,075 --> 00:20:36,778 curvatures are not too extreme, we see that we reproduce precisely Newtonian 318 00:20:36,778 --> 00:20:39,576 physics. So, good. We expect Einstein's theory, as 319 00:20:39,576 --> 00:20:44,636 we said, to reduce to Newtonian theory in suitable limits, and this is the suitable 320 00:20:44,636 --> 00:20:46,720 limit in which that works. And then, 321 00:20:46,720 --> 00:20:50,468 the last thing I want to point out is that there's a complication in 322 00:20:50,468 --> 00:20:54,271 gravitational theory that makes the mathematics of solving Einstein's 323 00:20:54,271 --> 00:20:58,671 equation and understanding the structure of its solutions vibrant, and lively, and 324 00:20:58,671 --> 00:21:02,040 interesting field of mathematical research as well as physics. 325 00:21:02,040 --> 00:21:06,424 Which is a complication that is not present in in Maxwellian electrodynamics. 326 00:21:06,424 --> 00:21:10,865 And the idea is this. Take the mass of say the Earth and the Sun, The Earth-Sun 327 00:21:10,865 --> 00:21:13,597 system. The mass of the Earth-Sun system I claim 328 00:21:13,597 --> 00:21:17,754 is a little bit less than the mass of the Earth plus the mass of the sun. 329 00:21:17,754 --> 00:21:21,796 And the reason is that Earth is bound to the sun, there's some negative 330 00:21:21,796 --> 00:21:24,928 gravitational potential binding energy between the two. 331 00:21:24,928 --> 00:21:29,425 Remember, if I took the Earth and the Sun very far away, as they fell, they would 332 00:21:29,425 --> 00:21:32,232 release energy. Having lost energy, that reduces their 333 00:21:32,232 --> 00:21:34,493 mass because mass and energy are equivalent. 334 00:21:34,493 --> 00:21:38,398 So, the mass of the earth and the sun together is a little bit less than the 335 00:21:38,398 --> 00:21:42,765 mass of the earth and the sun separately just as the mass of a helium atom nucleus 336 00:21:42,765 --> 00:21:45,540 is less than the mass of two neutrons and two protons. 337 00:21:45,540 --> 00:21:50,237 But, that means that the gravitational field that the Earth-Sun system together 338 00:21:50,237 --> 00:21:54,518 creates is different from the gravitational field that would be created 339 00:21:54,518 --> 00:21:58,281 by the earth and the sun. There are some contribution due to the 340 00:21:58,281 --> 00:22:02,187 gravitational binding energy. In other words, since gravitational 341 00:22:02,187 --> 00:22:05,880 fields carry energy, gravitational fields gravitate. 342 00:22:05,880 --> 00:22:08,429 Which is, almost as if you had the photon, 343 00:22:08,429 --> 00:22:13,217 the particle that carries light, being a charged particle so creating its own 344 00:22:13,217 --> 00:22:16,450 electric field. This would be like an electric field 345 00:22:16,450 --> 00:22:20,492 creating an electric field. This makes Einstein's equation, unlike 346 00:22:20,492 --> 00:22:22,357 Maxwell's equation, non-linear. 347 00:22:22,357 --> 00:22:25,466 A superposition principle for waves does not hold. 348 00:22:25,466 --> 00:22:30,378 These equations are very complicated. And as I said, studying their solutions 349 00:22:30,378 --> 00:22:34,436 is an ongoing enterprise. We won't get to do a lot more of the meat 350 00:22:34,436 --> 00:22:37,856 of general relativity. As I said, the mathematics there is more 351 00:22:37,856 --> 00:22:41,386 than we can afford to do. But, we'll see some of the consequences 352 00:22:41,386 --> 00:22:45,524 and hopefully that will be worth the effort to try to understand something.