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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

[tex] S_{bulk}=-\frac{1}{2}\int d^{4+n}x\sqrt{g^{4+n}}\tilde{M}^{n+2}\tilde{R}[/tex]

where

[tex]\tilde{M}[/tex]

is the n-dimensional Planck mass and

[tex]\tilde{R}[/tex]

is the [tex]4+n[/tex]

dimensional Ricci scalar. Integrate out the extra dims (which we assume to be toroidal):

[tex]S_{bulk} = -\frac{1}{2}\tilde{M}^{n+2}\int d^{4}x\int d\Omega_nr^n\sqrt{g^{(4)}}R^{(4)}[/tex]

and simplify:

[tex] = -\frac{1}{2}\tilde{M}^{n+2}(2\pi r)^n\int d^{4}x\sqrt{g^{(4)}}R^{(4)} [/tex].

We can see from this equation that what we perceive as the Planck scale is, in fact a quantity that isderivedfrom a more fundamental quantum gravity scale and the volume of the extra dimensions:

[tex] M_{Pl}^2=(2\pi r)^n\tilde{M}^{n+2} [/tex].

Ok, you can find that derivation easily on arXiv.

So, here is the comment:

If the planck mass is aderivedquantity whose origin is ultimately higher dimensional then does that not also imply that the speed of light, and plancks constant are alsoderivedquantities because;

[tex] M_{Pl}=\sqrt{\frac{\hbar c}{G}}[/tex]

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?

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# Large Extra Dims and a derived Planck Mass

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