Planck scale in extra dimensions

[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 :)

Homework Statement

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?

Thanks for your time!
Nathan

 

[Jhero]

Dear Nathan,

First of all, welcome to the forum and thank you for your interesting question! The relationship between the Planck length and the gravitational constant is indeed a tricky one, especially in higher dimensions. Let's take a closer look at the equations you mentioned.

The first equation, M_{4+n}=\sqrt{\frac{\hbar c^5}{G_{4+n}}}, is the standard relationship between the mass M of a particle and the gravitational constant G in 4+n dimensions. This equation is derived from the general expression for the Schwarzschild radius in n extra dimensions, which is given by r_{S}^{n}=\frac{2GM}{c^{2}}\left(\frac{c}{\hbar}\right)^{n-2}. By setting this expression equal to the size of the extra dimensions, which is typically taken to be on the order of the Planck length, we can solve for the mass M in terms of G_{4+n} and the Planck length. This is where your first equation comes from.

Now, let's move on to the second equation, M_{4+n}^{2+n}\approx G_{4+n}^{-1}. This equation is a bit more complicated, but it essentially comes from the fact that in higher dimensions, the gravitational force is not just proportional to the inverse square of the distance, but rather to the inverse (n+2)th power. This means that the contribution of a particle to the gravitational potential is proportional to its mass raised to the power of (n+2). When we combine this with the standard expression for the gravitational potential, which is given by V=-\frac{GM}{r^{n+1}}, we can solve for the mass in terms of the gravitational constant and the Planck length, giving us the second equation.

I hope this helps to clarify the origin of the extra factor of M_{4+n}^n in the second equation. Keep up the good work and don't hesitate to ask for further clarification if needed.