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## Quantitative explanation for weakness of gravity

The hierarchy problem of weakness of gravity versus gauge forces can be solved in the S4 picture extremely simply.  The core issue is that while in four space dimensions the Yang-Mills functional

$\int_{S^4} trace F\wedge *F dx$

is conformally invariant, the sensitivity of the gravitational counterpart

$\int_{S^4} S dx$

where $S$ is scalar curvature to rescaling has a $R^2$ dependence.  In particular weakness of gravity relative to gauge forces is described by the scale of the universe.  Since in the S4 picture the scale is tied to $1/h$ where $h$ is Planck’s constant, we have a precise quantitative ‘relative weakness’ figure for gravity.  If we consider gravity to be defined on the entire $S^4(1/h)$ then it will be $h^{-2}$; if instead we consider gravity to be defined only on hypersurfaces then it will be $1/h$.

The above explanation for weakness of gravity is extremely simple.  I would suggest that this explanation has not been absorbed due to a model of the universe that is not compact.  It was Weyl’s focus on the conformal invariance of the Maxwell’s equations that led ultimately to non-Abelian gauge theories and the successful Yang-Mills gauge theory.  This success underpins the idea of absorbing gravity within the same framework.  Once the evidence is examined for compactness of the universe is taken into account: the CBR uniform lower bound of 2.7 Kelvin for temperature can be used to show that the universe assumed to be stationary and noncompact leads to contradiction.  A compact four (space) dimensional stationary universe will have conformally invariant Yang-Mills functional but conformally non-invariant total scalar curvature (gravity functional).

The mathematical explanation requires deeper physical interpretation but what it points out is that although gravity and gauge forces both share an inverse-square distance force (in three dimensions) they are radically different phenomena with regards to the metricity of space.  Indeed, Weyl’s attempted unification in 1918 and later focused on the conformal invariance intuitively at the expense of gravity.  In the S4 picture, which would contend that quantization of energy is explained by the 4-sphere geometry, the relative weakness of gravity is trivial to explain unlike the Kaluza-Klein models where the extra space dimension is compactified and it is unclear how to explain weakness of gravity.  Indeed unified field theory attempts had not focused on the metricity difference between gravity and the gauge forces.