Circular LineLand

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December 27th : Curved Space

Quote of the Day

"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 contemplated 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 contemplation will help you.

Contemplation for the Day

Let us imagine

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.

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.

Contemplate how lucky we are to have three space dimensions which we can explore.

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