A (Sophomoric) Idea on Dimensionality

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As an introduction to it, imagine yourself looking down at a 2 dimensional universe. One day, while watching the inhabitants flit about from point (x1,y1) to some point (x2,y2), you decide to play with them. You pull out an old ping pong ball you were playing with the other day and drop it straight through the 2D universe.

Imagine the shock that the inhabitants would have! A strange sight it would be! First, they would first see a point and then a small circle that grew to the size of the diameter of the ping pong ball and then it would get smaller again, disappearing from sight.

While watching it drop through the 2D universe, you would see that they are merely seeing infinitely thin "slices" of it moving through their plane. A point, when it is just barely touching their universe. A large circle, when half way through, and finally, a point again as it fell away.

Now imagine yourself walking to another fine physics lecture, and lo and behold! You see a ball appear from thin air, get bigger and then shrink back down to nothing! Odd, indeed!

Stop for a moment, and assume that time is the 4th dimension. A 4 dimensional object is now nothing more than a 3 dimensional object changing over time! What you have seen suddenly makes sense. You saw nothing more than a hypersphere acting through the 3 dimensional universe that we perceive!

And here comes the idea I had tonight. The folks in the 2 dimensional universe would "see something" by how far it was in the x and y coordinates, and would never be able to perceive anything moving along the z coordinates. Quantum mechanics states that every dimension must have a discrete, minimum value. Theoretically, there is a smallest distance that we can measure called the Planck length.

I'm not fully trained yet so maybe I'm completely off base here but wouldn't the "thinnest slice" of a ball falling through a 2 dimensional universe, or a 4 dimensional object acting through a 3 dimensional universe have to be the same amount of distance as this Planck length ? This means that from the viewpoint of a 3D observer, the two dimensional universe could not truly be two dimensional, but instead has a very, very small 3rd dimension.

Now we've assumed that the 4th dimension is time, but that doesn't seem to agree with what we just said about there being a minimal Planck length to the universe, because the units don't match up! We have the 4th dimension being measured in time, but the Planck length is a distance! Meters just don't equal seconds!

But wait! If you are familiar with Einstein's relativity, you know that physicists don't always measure time in seconds. In relativity, it is quite common to state that the 4th dimension has the units of ct, where c is the speed of light and t is time in seconds. The final units of c times t is measured in meters! Thus, the Planck length still has the correct units. This leads directly to the conclusion that time is quantized, just like momentum and position.

That time is quantized is already known from the Heisenberg uncertainty principle; specfically, that the precision of a known energy times the precision of the known interval of time must be greater than or equal to one half Planck's constant over 2 pi.

There is one issue with that idea. We are making an assumption about the 4th dimension that the 2D inhabitants could also make (somewhat incorrectly). They could assume that time is the "3rd dimension" but according to us, the 3D observers, the 3rd dimension is purely positional. Perhaps the 4th dimension is in reality just some positional dimension (as string theory says, up through the 10 or more dimensions) but we can't comprehend a 4D positional space while thinking of time being a "0th" (zeroth) dimension that runs its course in all of the spaces. Therefore, the "3D object changing over time" may be useful for our thinking but we may be incorrect. Our understanding is severly limited by our limited ability to test for more dimensions. There might be many dimensions that we simply can't observe.

So how could a n dimensional observer try and see if there were truly was another greater than n dimension (or dimensions)? Well, one suggestion proposed is to look and see if any energy "leaks" into the other dimension(s). Work is already under way to do so. In fact, there was a small Scientific American article about a research team doing just that last year. Need to find that..

Now that I've written this, this sounds a hell of a lot like what those damn superstring theorists keep writing about. "Dimensions curled up so small that we don't realize they are there" didn't really click until just now. As I hypothesized, this was probably first thought up by people smarter than I a long time ago. Hunh..

last change 2004-04-18 18:45:20

April 2004
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A bland discussion of dimensionality.

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© 2004, Jeff Hodges