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## Purely numerical exploration of relative weakness of gravity in S4(1/h)

Although in the S4 picture, gravity is naturally explained by minimization of an Einstein-Hilbert action over hypersurfaces rather than over all possible metrics with a fixed base space, we assume that the formerly will produce a gravitational field equations approximately equal to the classical one, i.e. ( see here ):

$R_{ij} - \frac{1}{2}S g_{ij} = \frac{8 \pi G}{c^4} T_{ij} + \Lambda g_{ij}$

These are Lagrange equations  (in the usual functional rather than the S4 functional) for the term

$- \frac{c^3}{16 \pi G} \int ( R + 2\Lambda)$ (1)

where the integral is taken over a fixed base with varying metrics.  In the S4 physics side, the right cosmological constant should be $\Lambda = h^2$ because this is the term that appears naturally when calculating the Ricci curvature of a hypersurface using the Ricci curvature formula for hypersurfaces with induced metric from $S^4(1/h)$.

For exploration, let us consider $x$ such that \$latex $\frac{c^3}{16 \pi G}\Lamda x = 1$, that is, $x = \frac{ 16 \pi G}{c^3 h^2}$.  Now numerically, $c^3 h^2 = O(10^{-6})$ and $G = 6.6\times 10^{-11} N\cdot (m/kg)^2$ but now we have to insert the value in terms of electron volts, which gives $G' \sim 6.6 \times 1.6 \times 10^{-11+19 = 8}$.  Using this $G'$, we have $x \sim 10^{14}$.  Now this figure of $x$ is not $vol(M)$ but rather $O(1/h)$.

We do not draw any conclusions from the above but note rather, that there are several ways of finding natural relations relative strengths of gravity and electromagnetism in an $S^4(1/h)$ universe.  Either we can normalize the scalar curvature and obtain a relative difference in strength like $O(h^3)$ or we can consider scaling by $O(1/h)$ thought of as the ‘length of fibers’ for $U(1)$ electromagnetism even though we are not actually dealing with a $U(1)$ fiber bundle but rather the space $S^4(1/h)$.