Large Extra Dims and a derived Planck Mass

[robousy]

Hey folks,

I just had a thought I wanted to share with you guys.

Just to review first the idea of Large extra dims before I make the comment:

The higher dimensional Einstein-Hilbert action is given by

$$S_{bulk}=-\frac{1}{2}\int d^{4+n}x\sqrt{g^{4+n}}\tilde{M}^{n+2}\tilde{R}$$
where
$$\tilde{M}$$
is the n-dimensional Planck mass and
$$\tilde{R}$$
is the $$4+n$$
dimensional Ricci scalar. Integrate out the extra dims (which we assume to be toroidal):
$$S_{bulk} = -\frac{1}{2}\tilde{M}^{n+2}\int d^{4}x\int d\Omega_nr^n\sqrt{g^{(4)}}R^{(4)}$$
and simplify:
$$= -\frac{1}{2}\tilde{M}^{n+2}(2\pi r)^n\int d^{4}x\sqrt{g^{(4)}}R^{(4)}$$.
We can see from this equation that what we perceive as the Planck scale is, in fact a quantity that is derived from a more fundamental quantum gravity scale and the volume of the extra dimensions:
$$M_{Pl}^2=(2\pi r)^n\tilde{M}^{n+2}$$.
Ok, you can find that derivation easily on arXiv.

So, here is the comment:

If the Planck mass is a derived quantity whose origin is ultimately higher dimensional then does that not also imply that the speed of light, and Plancks constant are also derived quantities because;

$$M_{Pl}=\sqrt{\frac{\hbar c}{G}}$$

I think this is somewhat interesting because it would imply that relativity and quantum mechanics would have a different behaviour in the bulk.

What do you guys think?

 

[starkind]

I'm not sure there is any behavior in the bulk. Or perhaps there is every behavior, which amounts to the same thing. There is no differentiable behavior in either case, is there?

 

[BenTheMan]

Hey rich---

Maybe you should go and put the hbar's and c's back into the Einstein Hilbert action you wrote down. Then I think what you'd find is that the constant sitting out front IS Newton's constant, plus factors of hbar and c.

Or, in other words, G changes along with M, and h and c go along for the ride.

I think :)

 

[robousy]

Hey Ben,

Whats up?

Hmmmm, but the definition of the Planck mass is

$$M_{Pl}=\sqrt{\frac{\hbar c}{G}}$$

eg

$$\frac{M_{Pl}^2}{(2\pi r)^{2n}}={\frac{\tilde{\hbar} \tilde{c}}{\tilde{G}}}$$

so the way I'm seeing it they are all potentially derived from the higher dimensional quantity. I'll have a bit more of a think about what you said when I get back to the office tomorrow!

Rich

 

[BenTheMan]

Hey Ben,

Whats up?

Hmmmm, but the definition of the Planck mass is

$$M_{Pl}=\sqrt{\frac{\hbar c}{G}}$$

Sure, this is the four dimensional Planck Mass.

$$\frac{M_{Pl}^2}{(2\pi r)^{2n}}={\frac{\tilde{\hbar} \tilde{c}}{\tilde{G}}}$$

so the way I'm seeing it they are all potentially derived from the higher dimensional quantity. I'll have a bit more of a think about what you said when I get back to the office tomorrow!

Rich

Sure---but you have to use the appropriate Newton's constant, too. The Newton's constant now contains factors of $$2 \pi R$$ that go away in the dimensional reduction from N dimensions to 4 dimensions. In fact, what you wrote is (I think) the exact relationship between Newton's Constant in N dimensions and the N-dimensional Planck constant.

 

[robousy]

Aaah, ok I see what you are saying. Thanks for your insights! :)

 

[granpa]

Aaah, ok I see what you are saying. Thanks for your insights! :)

could you explain it to the rest of us? is there something to this or not?

 

[robousy]

Hey Grandpa,

Yes, sure there is something to this. Its a demonstration of why gravity is so much weaker than all the other four forces, or rather its an possible explanation. Sadly its not my idea. :)

As for the speed of light and Plancks constant being derived quantities, I think that's not the case.