Circular LineLand — revision 39366

This is one page of the [[SolSeed Book of Days]] == December 27th : Curved Space == === Quote of the Day === "Mass tells space-time how to curve, and space-time tells mass how to move." --John Wheeler "There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the curvature of space. " -- John Wheeler (1957) Quoted in New Scientist, 26 Sep 1974. "What is the shape of space? Is it flat, or is it bent? Is it nicely laid out, or is it warped and shrunken? Is it finite, or is it infinite? Which of the following does space resemble more: (a) a sheet of paper, (b) an endless desert, (c) a soap bubble, (d) a doughnut, (e) an Escher drawing, (f) an ice cream cone, (g) the branches of a tree, or (h) a human body? " -- Rudy von Bitter Rucker, The Fourth Dimension: And How to Get There (1985), 91. === Thoughts for the Day === A couple of days ago we meditated on 4 dimensional geometry and a few days before that we imagined time as the curved surface of a sphere. But what does curved space mean? If we live in a curved universe, then this question is one of the Big Metaphysical questions. There are various images of grids with the lines all curved in different ways that are supposed to explain what curved space is. But do they really help? Can you imagine interacting with curved space by looking at these wobbly grids? If you can't, then we hope that today's mediation will help you. === Meditation for the Day === Imagine you are the line segment living in Lineland that we talked about a few days ago. You can travel forward and backward but you cannot even conceive of any other directions. Now imagine that the line you live in is curved. Everything in your universe can only go forwards and backwards along the line and is physically incapable of interacting in any other direction. Imagine you have eyes on each end of your body that can emit light. You emit a burst of light from your forward eye. The light, being made of Lineland matter-energy, heads off in the forward direction along the line. But even if the line is curved to the left, the light will follow the line. [[File:LineSegmentCreatureInLineLand.png]] You could imagine this as the light driving along a single lane freeway with concrete sides and which curves to the left. The light tries to go straight forward and rubs against the concrete sides and is redirected slightly to the left. But this would be wrong. The light doesn't bump against any kind of wall. If Lineland is curved then the line the light is traveling along is curved and the light will curve without interacting with anything to the side; remember nothing in Lineland can interact with anything to the left or right; left and right are not in their physics. Now let's imagine another line segment is hiding somewhere off around the curve of Lineland. Your burst of light hits that other segment in the face and some of it bounces back. The light comes back along the same path and hits you in the eye. Of course you weren't so foolish as to emit a burst that was so bright it would blind you when it came back. By the colours that were removed from the light before it returned you conclude that your neighbour has a green eye. Can you also tell how far away your neighbour is? If you could measure the time it took for your burst of light to return to you, then, yes, provided light travels at a constant speed in Lineland just as it does here. Can you also tell which direction they are from you? Yes and no. The light came from forward so they must be in front of you. Can you tell that they are to the left? No. You can't even conceive of the direction left. When you look ahead you do not see something like that freeway curving to the left. Just as in our universe you see by light but in your universe all light comes from directly ahead or behind. All you see is light that bounces off or is emitted by your neighbour on each side. The light can't miss you because it can't move to the left or right out of the curved line that is Lineland. So, no matter how far to the left or right the curvature of Lineland takes them, light from your neighbours always comes from directly ahead or behind you. To you they are directly ahead and behind you, there is no way that you can know that they are also to the left or right of you. Seeing as they can't move aside to let anyone else get past, you can never have a different set of neighbours. Lineland is a boring place to live. Now imagine instead that you are alone in Lineland and that Lineland is so curved that it is bent into a circle. You go forward. As far as you can tell you are going in a straight line but because the linear 'space' of Lineland is curved into a circle you eventually come back to the same place. Of course, Lineland being such a boring place, there is no way that you can tell you have arrived back at the same place because all places are the same in Lineland. So you emit a burst of light from your forward eye. It goes all the way around Lineland and hits you in your rear eye. Some of it reflects off your rear eye and goes all the way around Lineland and hits you in your forward eye. Again from the missing frequencies you can tell the colour of your rear eye. Essentially, you can see your own eye staring back at you. From your forward eye, you can see your rear eye and from your rear eye you can see your forward eye. From your point of view Lineland is a straight line but for some reason if something goes far enough it arrives back at the same point. To you the universe is a straight line of a finite distance (the distance you have to travel to get back the same point; the circumference of the circle that you cannot imagine) where the two ends of the universe just happen to be the same place. Now add two more dimensions. Because we are used to living on the surface of our world it will be easier if we now think of the Lineland finite circular dimension as the vertical dimension and add two flat infinite dimensions to that: forward/backward and left/right. Imagine that there is a forest growing on the plane defined by these two flat infinite dimensions but that the ground out of which the forest is growing is transparent and ethereal so that you can see the roots of the trees extending downward as easily as you can see the branches extending up. If you look to the left or the right or forward or backward then you can see forest extending into the distance until the trees fill up the entire view just as if you were standing in a normal forest. If you look up you will see the branches and leaves and above that, not sky, but roots because that is the circular dimension in this imaginary universe. The roots you see hanging down from above the branches are the roots you also see if you look down through the ground. But if you look down through the roots you will also branches and leaves; the same branches and leaves that you see when you look up. Somewhere in the open space above the branches of the trees is the same place that is in the open space below the roots. It will be as if the forest is stuck between two horizontal mirrors but with one major exception, all the forests appear right side up. If there were two mirrors then the reflected forests would appear upside down. And as if there were mirrors, if the roots and branches have enough open space between them you will see an infinite number of forests, one above the other extending up and down. But each forest will really just be the same forest seen from a different angle through the circular dimension. Each forest is an image made in your eye by light that has travelled around the circular dimension (at some slight angle) a different number of times. Now imagine that the finite distance around the circle shrinks. As it does so the distance you have to travel up or down to reach the same place also shrinks. The roots get closer to the branches until they touch and the trees begin to push on themselves roots against branches. The height of the trees no longer fits in the circle. The trees begin to topple and fall sideways because the infinite flat dimensions give them room to extend but the finite circular dimension does not. Now even lying sideways, the 'upper' branches come in contact with the lower branches. As the circular dimension shrinks the branches and roots of the trees press themselves together from each side. It all becomes uncomfortable for the trees. So let's imagine the trees away. This shrunken universe no longer has room for trees. Now you are standing alone on this transparent ground. Above you, you can see your own feet and as the circle shrinks they are falling toward your face. You look down and get a momentary feeling of falling as you see your feet approaching your downturned head below you. Quickly, before you stomp your own head, you duck and then lie down. But soon like the trees, as you lie on your back, your chest presses into your back from below and your back presses into your chest from above. You are about to crush yourself. This has become some kind of nightmare and that isn't proper for a meditation. So again let us imagine you out of this universe again. There is no longer room for people in this universe. Now imagine that the circle shrinks drastically further. Air molecules now press against themselves from above and below. Each molecule turns so that all of its atoms lie in the flat infinite plane. Let's stop the shrinking of the circular dimension here. At this point there is no room for any atom to get above or below any other atom. Different atoms can only react with each other from the left or the right or from in front or behind never from below or above. This universe is still three dimensional but it acts exactly like two-dimensional Flatland! This is similar to what physicists mean when they say that our universe has eleven dimensions but that all but three of the space dimensions are rolled up very small. The other dimensions are circular like the circular Lineland dimension and the circumferences of their circles are even smaller than atoms. In string theory the tiniest theoretical particles are not spherical but have different lengths in different dimensions. This means that they can only be aligned in one direction in each of the eight rolled up dimensions. Because the shape of space itself controls how particles can interact, string theorists just have to find a set of shapes and sizes for the eight rolled up dimensions that predicts how particles actually interact in this universe and they may find a useful model of this universe. Now go back to the forest and gently meditate on the nature of rolled up dimensions.