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MAXWELL’S EQUATIONS IN AN S4 HYPERSURFACE?

In general doing any explicit computation for hypersurfaces of a four-sphere is intractable.  In 1968 Jim Simons proved that the Laplacian of the squared length of the second fundamental form has a formula for minimal hypersurfaces, i.e. hypersurfaces with zero mean curvature; this was generalized for manifolds with constant mean curvature by Nomizu-Smyth in 1969.  This generalizes to, with $A$ the second fundamental form and $H$ the mean curvature this lemma of G. Huisken:

(i) $|\nabla A|^2 \ge \frac{3}{n+2} |\nabla H|^2$

(ii) $|\nabla A|^2 - \frac{1}{4} | \nabla H | ^2 \ge \frac{2(n-1)}{3n} | \nabla H|^2$

(iii) $\Delta |A|^2 = 2 \langle h_{ij}, \nabla_i\nabla_j H \rangle + 2 |\nabla A|^2 + 2 Z + 2 n K (|A|^2 -\frac{1}{n}H^2)$ with $Z=H \trace(A^3) - |A|^4$

Suppose $w_1,w_2,w_3,w_4,w_5$ is a coframe adapted to $M\times\mathbf{R}\rightarrow S^4(1/h)\times\mathbf{R}$ with $w_5 = dt$ and $w_4$ being the dual of the unit normal vector field.  The Maxwell’s equation is

$dF = 0$

$d*F = 0$

This is just a simple picture.  The idea is that electromagnerism, considered a $U(1)$ gauge theory is considered with a Lagrangian that is invariant under a gauge transformation via a connection on a principal $U(1)$-bundle where the action of $U(1)$ along fibers preserves the Lagrangian.  I believe that in our actual universe, there is no principal $U(1)$-bundle but rather the actual universe is a scaled 4-sphere where the substitute for true $U(1)$ action is translation in the normal direction.  Parallel translation of the frame $(w_1,w_2,w_3)$ along the normal geodesic along $w_4$ is the substitute for the circle action.  The standard geometric interpretation of $F$ is that it is the curvature of a connection of a $U(1)$-bundle; sections of this $U(1)$ bundle represent charged fields.

A simple idea is to consider the normal covector $\omega_4$ as the electromagnetic potential.  So $F = d\omega_4$.  Trivially $dF = 0$.  Now $F = d\omega_4 = h_{ij} \omega_j \wedge \omega_i$.  In a simple situation where the coframe diagonalizes the symmetric second fundamental form $h_{ij}$ globally we just get $F=0$ and the Maxwell’s equations are trivial.  In general, we have the nontrivial condition $d*F = 0$.  Smooth diagonalization of $h_{ij}$ is made possible by results of SmoothDiagonalization

If the physical universe is a compact hypersurface $M$ of $S^4(1/h)$ then how could electromagnetism work?  In the established treatment of electromagnetism, e.g. here), $F=dA$ where $A$ is a $U(1)$-connection so the electromagnetic field $F$ is invariant under $A \rightarrow A + d\Lambda$. Now if the physical universe is $M$ so there is a normal circle of length $2\pi/h$ at each point in the physical universe, then a simple first question is how to describe the analogue of the sections of the $U(1)$-bundle which now are deformations of $M$ in $S^4$.  So suppose $f:M\rightarrow [-2\pi/h, +2\pi/h]$ is a smooth map.

The ‘topological’ part of Maxwell’s equations, $dF=0$ is clearly fine so long as the deformation is a diffeomorphism since the exterior derivative commutes with pullbacks:

$d f^*(F) = f^*(dF) = 0$

So the problematic part is to understand $d *F=0$ where the asterisk is the Hodge operator.

It is useful to know the image of tangent vectors $X\in T_x M$ by a deformation of $M$ to $M_f$.  The map is $x \rightarrow \phi(x) = \exp( f(x) N)$.  For a tangent vector $X \in T_xM$ one finds the Jacobi field with initial condition $J(0) = 0, D_t J(0) = X$ and the image in $T_{\phi(x)} S^4(1/h)$ of $X$ is our interest.  These Jacobi fields are known explicitly for constant sectional curvatures (JacobiFields): in terms of a tangent frame in $T_{\phi(x)}M$ their components will be

$c^i (1/\sqrt{\kappa}) \sin(\sqrt{\kappa} f(x)) + d^i \cos(\sqrt{\kappa} f(x))$