Planck distance and less dimensional gravity

[exponent137]

If we imagine 2+1 dimensional gravity, I think that this influences on Planck distance and time.
Because attraction between two bodies can be described:

$$F = m_1 m_2 G/r$$

F= force, m are masses, G is changed gravitational constant and r is distance between two bodies.
So G does not have the same dimension as a common our 4D G. So calculation of Planck's distance is different.

How it is with this?

 

[mfb]

You would have to modify other forces, too - or you get a really strange universe.
While this is no problem for Coulomb's law, the other features of the Standard Model are a bit tricky, as they are made for our 3+1 dimensions.

Anyway, you could get a different mass in some way.
This is usually done with a higher-dimensional gravity. If these additional dimensions are compact, you can introduce a new scale for them and calculate a modified Planck mass. This is usually done to lie within reach of the LHC ;).

 

[exponent137]

In 3+1 dimensions, dimensionless masses of particles are calculated as:
$$\mu_e^2=m_e^2 G/(\hbar c)$$
In 2+1 this is not possible with the above mentioned 2+1 dimensional G. So, what is changed, $\hbar$, m, or the above formula, or what else?