How can a dimension be hidden?
Extra dimensions pop up everywhere in physics. From string theory to attempts at unifying general relativity with quantum theory. But how can it be that we can’t see them?
Before we answer this question, let’s answer a different one: What does it mean to ‘live’ in a dimension? Imagine a being with tentacles that lives in only one dimension – when it waves its tentacles, they go back and forth on a straight line. They have nowhere else to go.
Compare that to an ant walking on a table that lives in two dimensions – length and breadth, but not height. When the ant waves its legs, they can move back and forth AND sideways to the left or right.
Rolled up
Now we answer the question we started with: extra dimensions can be hidden if they are ‘rolled up’. Consider again the ant that can see only two dimensions, but this time it is walking on a narrow strip of paper. The strip has a very small breadth but infinite length, so if we roll up the strip along its breadth, the ant can still move infinitely far back and forth along the length of the tube (remember that for the ant there is no third dimension, so it won’t notice what happens to the strip).
Things are very different when the ant moves to the left or right. When it moves sideways it will walk in circles around the tube so that it appears in the same place every time it completes a circle. What will we see when we look at the tube with the ant on it from a great distance?
Going South
If the diameter of the tube is very small compared to the distance between us and the tube, it will appear as if the ant only moves back and forth along the length of the tube, because the diameter of the tube is too small to see (just as a clothesline in the distance seems like a one-dimensional line while it’s actually a three-dimensional object).
But what do we see when the ant moves sideways? Each time it goes around the tube, it will disappear and reappear from behind the tube as seen from our perspective*. It would seem to pop into and out of existence, so the dimension isn’t hidden at all, because we can see its effect! Unless…
The diameter of the tube is very small indeed, so that the ant appears and disappears at such an immensely high rate that the change is invisible to us (just like some living room lights that flicker at such a high rate that we see it as a continuous source of light).
Why does it Matter?
We care about this because hidden dimensions can ‘hide’ kinetic energy – a hidden dimension could explain why we measure too little kinetic energy when we analyse collisions between elementary particles in a particle-accelerator like CERN.
Say we see some particle A bumping into particle B and we assume the conservation of momentum. If A and B have the same mass, while A has the same speed and opposite direction before and after the collision, then the conservation of momentum allows us to predict that particle B will also have kept its speed and changed direction.
But it could also be that the particles have started to rotate in some invisible rolled up dimension as a result of the collision. If the extra-dimensional rotation is too fast to measure (like the ant circling the tube), the conservation of momentum is of little use, and we can say nothing about the velocities of the particles after the collision.
The usefulness of the principle of momentum conservation suggests that very little (if any) momentum/kinetic energy ‘leaks away’ to hidden dimensions, but future experiments with increasing resolution might show otherwise…
Read this post to learn how hidden dimensions can make quantum entanglement understandable.
*) The argument is the same whether we consider the tube as embedded in some higher dimensional space, or whether we think of ourselves as very far away on the tube.
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