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gravity is one of the four forces of the universe. but at the quantum level gravity is very weak in comparison to the other forces. so does gravity leak out and go to higher dimensions? and if so is the reverse true? can other forces leak out to come here?

The four canonical forces are gravity, electromagnetism, the strong nuclear force (binds quarks into protons and neutrons, and binds protons and neutrons into atomic nuclei), and the weak nuclear force, which governs radioactive decay.

I wouldn't say gravity is weak as much as I'd say that protons and electrons don't have a lot of mass. There's no way to directly compare gravity and electromagnetism without resorting to a charged, massive particle, and thus the imbalance is in the particle used rather than in the forces themselves. It's like comparing the performance of a red car running on octane with a blue car running on fumes. The red car is probably going to do better, but that has nothing to do with the color.
If you compare the electromagnetic pull of planets on each other compared to their gravity, you'd find that gravity has a much larger effect than electromagnetism, because in this case the masses of the planets are much greater than their electrical charges.
In terms of the "standard" units of physics, gravity and electromagnetism are equally strong.

Besides which, if gravity were escaping along other dimensions than the 3+1 we normally observe, then intuitively (naively?) we'd get something other than an inverse square law; we'd get an inverse (n-1)-th power law, where n is the number of spatial dimensions we have. I'm not entirely sure how Randall's brane theory gets around this.

You're completely right, but then there is nothing unkosher in making such comparisons through stable elementary particles. One usually says that a force is strong (weak) in that the corresponding coupling constant is large (small), which definitely makes a lot of difference when one employs perturbative methods. So, for EM, you have a Planck charge Qp = sqrt(4πℏcε₀), and the corresponding coupling constant as α = (e/Qp)². Doing the same thing with the Planck mass Mp and an elementary mass (say electron, for consistency), you have a gravitational coupling that's absurdly tiny by comparison.

Then with the understood dependence on elementary particles, these are all equivalent questions:
-- "Why is gravity so much weaker than other forces?"
-- "Why is the gravitational coupling so much smaller than that of other forces?"
-- "Why is the Planck mass so freakin' big compared to elementary particles?"


Yes, but it's modified over large distances when the extra dimensions is compact. Intuitively, a mass smeared over a small compact dimension won't look too different from a point-mass at large distances, so we can pretend to spread it throughout the volume of the extra dimensions, which reproduce inverse-squared behavior with an altered gravitational constant (divided by the volume of the extra dimensions).

Gauss's law is correct in all dimensions; writing it in terms of a potential gives the standard Poisson's equation ∇²V = 4πGρ. In Cartesian coordinates (w,x,y,z), let R=x²+y²+z² and r²=w²+R², the potential of a point-mass m should be
[1] V = -Gm/(πr²)
Compactifying the w-direction into a circle of radius a, the identification w~(w+2πa) wraps around the potential, repeating it over w+2πan, producing a summation over all integers n:
[2] V = -(Gm/π) Sum[ 1/(R²+(w+2πan)²),
Now, if I let w = 0, then the summation is (R/2a) coth(R/2a), so for very large R, this is essentially.
[4] V₀ = -G'm/R,
where G' = G/(2πa). Nonzero w might also has a closed form, but I'm not sure how to get it. It's probably something hellish involving digamma functions.

Edit: I don't know what the hell I was thinking here... that's it; I'm going to sleep instead.

I can understand that it's approximately the same with small compactified extra dimensions, but Randall's inter-brane dimensions are not small.

There's an exponential drop in metric along the inter-brane dimension? Wow. That's...really weird.