How does superstring theory explain the inverse square gravity law, given that it requires 9 spatial dimension?

In superstring theory, the spacetime dimension is either 10, one of them is time, the rest are spatial dimensions.

But based on geometrical argument, we can say that $F\propto r^{1-D}$, where $D$ is the space dimension. So that means if spatial dimensions are more than $10$ according to string theory, then gravity will decay as $F\propto r^{-9}$ or more.

I've read the description of how superstring handles it here:

String and Superstring theory (really, M-theory, see http://en.wikipedia.org/wiki/M-theory, and http://en.wikipedia.org/wiki/String_theory#Extra_dimensions), require a minimum of 9 spatial dimensions. If those, or some of those are big dimensions, then the weakness of gravity might be explained, and then if we then look at smaller and smaller distances, gravity is (relatively) stronger. It is also one reason there are attempts to measure the strength of gravity at smaller and smaller distances -- to see if it does not go as $1/r^2$. So far, as I said above, it's only been down to about 1 millimeter, and nothing strange has been found.

String Theory mostly has it (because the strings that cause gravity were thought to be able to extend in all dimensions, whereas normal forces like nuclear and electromagnetism strings are constrained to move in our 3D brane) that gravity propagates in the 10 spatial dimensions. String Theory also assumed that the other dimensions are small, microscopic and we can't see them. Then you need to calculate how much it dilutes gravity. But some String Theory developments assume 1 or more large extra dimensions, and then it dilutes (and gets relatively stronger in the much smaller domain).

My question is how can this ever be possible that the gravity varies as $F\propto r^{-8}$ at small distance but $F\propto r^{-2}$ at large distance? There must be at some point where these two meets, and at those points, gravitational forces values are not unique ( and not continuous), how can this be? How does string theory ( or any theories ) explain this?

Answer

Consider a manifold with 3 macroscopic spatial dimensions and 6 extra spatial dimensions which are curled up on a lengthscale $l$. Let's try to apply Gauss' law for a closed hypersurface of spatial size $r$ around a point mass, where $r<

Then the interior of the Gaussian hypersurface looks like 10-dimensional Euclidean space, so the 'area' of the hypersurface is proportional to $r^{9}$.

By symmetry, the field is isotropic (the same in all directions). Sure, there are macrosopic space directions and curled-up directions, but the curling-up scale $l$ is much larger than our hypersurface, so this distinction shouldn't matter. Now Gauss' law tells us the total flux does not depend on r, so we conclude that the field strength is proportional $r^{-9}$.

Note that we've made two approximations. Did you spot them?

1. The area of the hypersurface is proportional to $r^{9}$.



2. The symmetry/isotropy argument, which asserts that there's no difference between a displacement $r$ along the macroscopic direction and a displacement $r$ into the curled-up direction

These approximations are good for $r << l$. But as $r$ increases, they get increasingly inaccurate. Both approximations break down when $r \sim l$. Thus our result $F \propto r^{-9}$ is only an approximation valid at $r << l$. By a similar argument, the relationship $F \propto r^{-2}$ is only an approximation valid for $r >> l$.

The crux of your question is what happens when $r \sim l$. Well, for these distances, neither of the two power laws would be accurate. We would see a gradual transition between the two.