How To Measure The Dimension Of Space
A simple way to think about the gravitational field of an object is to imagine a fixed number of "lines of force" that radiate from the object evenly into space. The density of the lines at any given point in space represents the strength of the gravitational field at that point.
Now, with this simple model in mind, we write down Newton's famous formula for the gravitational force f between two masses m and q at a distance r apart..
G is just a constant which goes by the impressive name "universal gravitational constant", but it's not really interesting. Its value depends on which unit system we're using to measure f, m and r and we can pick a unit system so that G=1, then..
We can make q a unit mass, q=1, and imagine we're using q to probe the gravitational field generated by m, so..
This is the essence on Newton's theory of gravitation. But what is this equation really saying? If we think about our lines of force model in a 1D space and then in a 2D space we realize that this equation is actually..
where n=number of dimensions of the space containing the mass m.
We can solve this equation for n..
The expression on the right is an experimentally measurable quantity. But it's the dimension of the space!
Can we push our luck further? Hey, why not..
In physics we know that the log function shows up in discussions of information entropy. But we've expressed dimension in terms of the log function. So perhaps dimension is derived from the information content of a space. Of course, we need to define "information content" in a strict mathematical way, but once that is done we might have a concept more fundamental than dimension. Now that would be a breakthrough!
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Content written and posted by Ken Abbott email@example.com