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## An extremely simple approach to the hierarchy problem in physics

The problem is to explain the relative weakness of gravity compared to electromanetism.  The difference in order of magnitude is quite large, around $10^{-40}$. From my view, it is not accidental that this approximately $1/vol(M)$ where where $M$ is a hypersurface in $S^4(1/h)$ and represents the model of a physical universe in the S4 picture.

Now consider the construction of gravity: the field equations result generally from the critical points of the Einstein-Hilbert action

$S^0_{EH}(g) = \int_{M} S(g) d\mu_g$

where $S(g)$ is the scalar curvature of metric $g$ for the hypersurface $M$ and $d\mu_g$ is its riemannian volume element.  instead of this action, we consider the normalize Einstein-Hilbert action

$S_{EH}(g) = \int_{M} \frac{S(g)}{vol(M)} d\mu_g$

So we normalize the scalar curvature but do no scaling to the matter terms:

$S_{EH}(g, A) = \int_{M} \frac{S(g)}{vol(M)} + L(A) d\mu_g$

It is clear that purely numerically this normalization would produce $\frac{1}{vol(M)} \sim h^3 \sim 10^{-45}$ which gives a quantitity that is off from measured by around 5 orders of magnitude at most.  This is my initial proposal for ‘solving’ or providing an explanation for the hierarchy problem.  The explanation is geometrically meaningful, in the sense that the normalized scalar curvature is geometrically more natural as a Lagrangian density than the un-normalized version.   Of course, this solution is strongly tied to the $S^4(1/h)$ geometry for value of $1/vol(M)$ but more generally of the compactness of the universe, allowing us to find critical points of $S_{EH}(g)$ among hypersurfaces of $S^4(1/h)$.