[beefield]

I have never quite understood why there can't be stable stallites in dimensions above 3.

I mean, I know the argument that gravity inverse square law becomes inverse cube law in 4d, but what I do not understand is that what/why enforces that. Why in a hypothetical 4d world there just can not be a gravity-like force that is inverse square? Would that cause some kind of contradiction?

[alexey-salmin]

If you make a uniform flash of light in 3d space it spreads around you in a shape of a sphere of increasing radius. Energy gets distributed evenly across the sphere's surface which is growing with time proportional to a square of the radius. So energy density (think intensity) decreases as inverse square. In 4d space the sphere's surface grows with the cube of the radius.

This sort of intuition. Applies to electromagnetic waves, sound and gravity all alike.

[beefield]

But there are forces even in our 3d world that do not follow inverse square law (strong/weak nuclear forces). That kind of proves that all forces do not need to follow this intuition?

[qnleigh]

In particle physics, fundamental forces are generated by the exchange of virtual particles (I have been reading a quantum field theory textbook for the last few months to try to understand precisely what this means, among other questions, but this is accepted fact in the field). The Coulomb force comes from the exchange of photons. So this 1/r^2 argument for intensity leads to the Coulomb force falling off like 1/r^2.

This argument doesn't obviously apply to gravity (though presumably it would for a quantum theory of gravity), but the equations for gravity (general relativity) give the same result.

At a higher level, it turns out that when you try to combine quantum mechanics with special relativity, the resulting theories are highly constrained. It's not like classical mechanics, where you can just say 'suppose there's a 1/r^12 force.' You get mathematical inconsistencies if you stay too far. Weird stuff

[alexey-salmin]

All long-distance forces seem to follow it though. I think it's related to the energy conservation law.

It proves nothing of course. When we speak of these N+T universes, we try to imagine a system that follow the same "fundamental laws" but with different N and T. What exactly is fundamental is up to debate. You can even imagine a system that has different math, but it will be very hard to reason about it.

[qnleigh]

Those forces are also mediated by particles (called gluons, W, and Z bosons). But these particles are massive, charged and interacting, which caused them to behave quite differently and only act on short length scales.