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December 28th : 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.
Thoughts for the Day
Yesterday we contemplated how lucky we are not to live in a one dimensional universe because one dimensional universes are boring. Today we will apply what we learned from the circular finite one dimensional universe that we imagined yesterday to our eleven dimensional universe.
Yes I said eleven not three. Our universe has eleven dimensions but string theorist say that eight of them are rolled up tight so we don't notice them. How can we fail to notice dimensions? They are directions after all. We can understand how a FlatLand creature like the Square can fail to notice the up-down dimension. But how can we as eleven dimensional creatures (which is what string theorists say we are) fail to notice 8 of our own dimensions? This is a Big Question worth contemplating.
Contemplation for the Day
Remember the circular finite LineLand that we imagined yesterday. 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 contemplate the nature of rolled up dimensions.