References on how dimensionality relates to inverse square laws

Question

This wikipedia article claims that the existence of inverse square laws (like Newton's Law of Gravitation) has a connection with the fact that our universe has three large scale spatial dimensions.

Can someone point me to some book/paper where this topic is treated in a detailed and formal manner?

score 4 · Accepted Answer · answered Feb 21 '18 at 01:38

I think what you're looking for is the divergence theorem: https://en.wikipedia.org/wiki/Divergence_theorem

It tells you that the flux of a vector field through a surface is equal to the integral over the divergence inside the Volume.

Since in 3 Dimensions Objects have surfaces scaling with $r^2$ and mass, charge etc. can be seen as positive divergence, or sources, of the corresponding force fields, we get inverse square laws. in Dimension d however surfaces scale with $r^{d-1}$ and we get $\frac{1}{r^{d-1}}$ laws.

Bob Bee · Answer 2 · 2018-02-21T01:49:47.683

2

It's pretty simple. In 3 dimensions when you transmit a wave uniformly in all directions the energy at any distance as it propagates is the same. Since the area of a sphere at distance r is $4\pi r^2$ the energy per unit area at any distance is inversely proportional to $r^2$. If the spatial dimensions were n it'd be inversely proportional to $r^{n-1}$

edited Feb 21 '18 at 01:49

answered Feb 21 '18 at 01:36

Bob Bee

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References on how dimensionality relates to inverse square laws

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