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## JACOBI COVECTORS TO DERIVE GRAVITATIONAL FIELD EQUATIONS FOR S4 UNIVERSE

Normal variation of a hypersurface $M_0$ of $S^4(1/h)$ with $\kappa= h^2$ can be described by a function $f:M_0\rightarrow \mathbf{R}$ by applying the exponential map along the normal unit vector $N$, i.e. $x\rightarrow \exp_x( N f(x))$.  We are interested in considering deformations of $M_0$ to $M_t = \exp_x( N t f(x))$.  Let $w_1,w_2,w_3$ be tangential coframes and it is known that their images under this deformations are Jacobi vector fields:

$w_i \rightarrow T^i(f,t) w_i(t)$ where $w_i(t)$ are parallel translates of $w_i$ along the normal geodesic where

$T^i(f,t) = c^i \frac{1}{\sqrt{\kappa}}\sin(\frac{tf}{\sqrt{\kappa}}) + d^i \cos(\frac{tf(x)}{\kappa})$

Now the gravitational lagrangian for hypersurfaces of $S^4(1/h)$ is the integral of the scalar curvature which is known explicitly in terms of the second fundamental form

$L(g) = vol(M)\cdot Scal(S^4(1/h)) + \int_M 9 H^2 - S dv_g$

where $H$ is the mean curvature and $S$ is the squared norm of the second fundamental form.  In analogy with the derivation of the Einstein field equations, we consider the normal variations by arbitrary functions $f(x)$ and then consider the critical points of the function $\frac{d}{dt} L(g(t))$ under the variations.  This I will claim will produce the correct graviational field equations because we already know that the universe is a compact scaled four-sphere.

The implications of this picture are obviously extensive — for this points to four dimensional matter that is completely not accounted for by the established three dimensional theories of quantum mechanics and general relativity.  Quantum mechanics is likely a linear approximation of the correct physics.