Planck scale in extra dimensions

1. Jan 2, 2009

toipot

Hi everybody, it's my first post on here but as i seem to have hit a brick wall with my work i'm hoping somebody might be able to help me :)

1. The problem statement, all variables and given/known data
I'm looking at how the planck scale is reduced in higher dimensions (ADD theories) and i've managed to reproduce an expression relating the 4-d gravitational constant $$G_{4}$$ and a fundamental gravitational constant $$G_{4+n}$$ by invoking gauss's law for gravity in extra dimensions around a line/plane of mass (due to the compactification of the extra dimensions). Here n represents the number of extra dimensions. With this i've had no problems and the answer I get seems to match the answers found in the literature.

The issue i'm having is with relating this change of gravitational constant to a change in planck length. I've just been using the normal relation for converting G into $$M_{pl}$$ i.e $$M_{4+n}=\sqrt{\frac{\hbar c^5}{G_{4+n}}}$$ but all the papers on the subject use the relation $$M_{4+n}^{2+n}\approx G_{4+n}^{-1}$$ with this relation I get all the right answers but I can't for the life of me figure out where it comes from. Can anyone let me know where the extra factor of $$M_{4+n}^n$$ arises?