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## KANT-WEBER-RIEMANN-MIE-HILBERT-EINSTEIN ACTION

The long chain that led to general relativity clarified that the Ricci curvature should be the gravitational Lagrangian (see weyl-purely-infinitesimal-geometry)  We know now that the universe is a compact sphere of fixed radius, so the gravitational action can be specialized to hypersurfaces of $S^4(1/h)$.

The Einstein-Hilbert functional whose Euler-Lagrange equations produce the gravitational field equations in this case take the form

$L(g) = \int_M Scal(g) \sqrt{-g} dx = vol(M) Scal(S^4(1/h)) + 2 \int_M (9H^2 - S) \sqrt{-g} dx$

where $H$ is the mean curvature and $S$ is the squared norm of the second fundamental form.  The formula for scalar curvature of a hypersurface of a sphere is well-known to geometric analysts (see 2.4 of ScalarCurvatureWithConstantMeanCurvature2009)

Regardless of the correct field equations, here we have a unification of electromagnetism and gravity so long as the second fundamental form terms describe the electromagnetic potential.

Formally, one could follow the established derivation of the field equations as in Klainerman’s notes from 2009 (KlainermanGR2009)

Doing so would take $\dot{g}_{\mu\nu}$ to be normal to the hypersurface (the physical universe) but the novelty in my approach is that matter fields can be treated quantitatively via the second fundamental form and its derivatives.  This gives us a view of all matter as curvature of physical space in an ambient four-sphere universe.