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## Conformal invariance of the Yang-Mills functional and weakness of gravity

This is an attempt to understand the relative weakness of gravity with respect to gauge forces (we pretend that electromagnetism is approximately a $U(1)$-gauge theory even though we claim that this is not true in an S4 universe).  So we want to understand two functionals.  The first is the Einstein-Hilbert action
$S_{EH}(g) = \int_M S(g) d\mu_g$

We could consider the functional over hypersurfaces of $S^4(1)$ and then consider scaling by $R= 1/h$.  The scaling will increase the functional by a factor of $h^{-4}$.  Next consider the Yang-Mills action

$S_{YM}(A) = \int_M trace F_A \wedge *F_A d\mu_g$

The integrand is conformally invariant because we are dealing with 2-forms in a 4-manifold.

We would like to explain the relative weakness of gravity in terms of the scaling $R=1/h$ as a consequence of the conformal invariance of the matter portion and the volume normalization of the gravitational portion.

In order to understand the geometric reason for the relative weakness of gravity relative to gauge forces, it is useful to consider what happens when we consider the Yang-Mills functional versus the integral of scalar curvature when we scale from $S^4(1)$ to $S^4(1/h)$.  The Yang-Mills functional is conformally invariant in four space dimensions so it remains the same.  The integral of the scalar curvature

$\int_{S^4(1/h)} S d\mu_{S^4(1/h)} = \int_{S^4(1)} h^2 S h^{-4} d\mu_{S^4(1)}$

So we have a factor of $h^{-2}$ change in the total scalar curvature.  This can explain why gravity should be weaker than the gauge forces in an $S^4(1/h)$ universe simply.  In particular, in producing a resolution of this hierarchy problem, the S4 picture is different from the Kaluza-Klein thoeries which have not addressed this problem.