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## Relative weakness of gravity in terms of volume of the universe

In page 108 of Carroll’s CarrollLectureNotesGeneralRelativity, we find that if we consider the Maxwell’s equations in the form

$\nabla_\mu ( 1 + \alpha S) F^{\mu\nu} = 4 \pi J^\nu$

and $\alpha \sim l_P^2$

where

$l_p = (\frac{G\hbar}{c^3})^{1/2} = 1.6 \times 10^{-33} cm$ (2)

Now note that $vol(S^4(1/h)) = vol(S^4(1)) \times (h^{-4}) = B (\frac{1}{h^2})^2$.  Thus dividing (2) by this term produces a term comparable to O(1):

$l_p \sqrt( B ) \frac{1}{h^2} \sim \frac{G}{c^4h^3} \sim O(1) cm$

Thus it is sensible to try to explain the weakness of gravity in an $S^4(1/h)$ universe in terms of a volume scaling either of the entire S4 universe or the physical hypersurface.  The issue of why there might be a normalization is yet unclear to me but geometrically the normalization of scalar curvature in a compact manifold is a reasonable step.