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## GAUSS AND CODAZZI EQUATIONS FOR A FOUR-SPHERE AND MY CONFUSION REGARDING THE STRESS-ENERGY OF EM IN GRT

I have a fairly simple picture of how our actual universe ought to work.  I know that the universe has four macroscopic space dimensions and it is embedded in a large radius (radius $1/h$) sphere.  So then I ask myself, is the electromagnetic tensor really the curvature of a $U(1)$ connection?  There’s a physical circle normal to every point in the physical universe in my view.  A nice theorem tells me that the second fundamental form is globally diagonalizable for a hypersurface of $S^4$.  Then I look at the Gauss and Codazzi equations for the hypersurface
$R_{kjih} = c( g_{kh}g_{ji} - g_{ki}g_{jh}) + H_{kh} H_{ji} - H_{ki} H_{jh}$

$\nabla_k H_{ji} = \nabla_j H_{ki}$

Now these $H_{ki}$ are analogous to the vector potential of electromagnetism.  Recall that in the latex $U(1)$ curvature picture one looks at vector potentials $\Phi_a$ and considers $F = [\nabla_a + ie \Phi_a, \nabla_b + i e \Phi_b]$.  In submanifold geometry the second fundamental form terms give us $A_i = \sum H_{ij} e_j$ as a modification to the Levi-Civita connection on the hypersurface and formally are identical in form to the electromagnetic potential.  From this point of view, a natural candidate for the electromagnetic tensor should be from here but this is standard:

There is thus a natural 2-form that occurs and we want to know whether this could be a good candidate for the electromagnetic 2-form.

The easy part of Maxwell’s equation $\nabla_{[a} G_{b c]} = 0$ follows from the Bianchi identity.https://wordpress.com/post/zulfahmed.wordpress.com/8477

Let $\Omega_{abcd} = H_{ad}H_{bc} - H_{ac} H_{bd}$.  Then check that

$\Omega_{ab[cd;e]} =0$ by using the product rule and the Codazzi equation by collecting terms by the non-differentiated factor, i.e.

$\Omega_{ab[cd;e]} = \nabla_e H_{ad} H_{bc} + \nabla_c H_{ae}H_{bd} + \nabla_d H_{ac} H_{be} - \nabla_e H_{ac} H_{bd} - \nabla_c H_{ad} H_{be} - \nabla_d H_{ae} H_{bc}$

then for example the term with $H_{bc}$ is $H_{bc} ( \nabla_e H_{ad} - \nabla_d H_{ae})$.

This is still sloppy.  However, the comparison is to the gravitational field equations with the electromagnetic stress-energy tensor.  This reads

$R_{ab} - 1/2 R g_{ab} = C T^{ab}$

with $T^{ab} = (1/\mu_0) [ F^{a k} F^b_k - 1/4 g^{ab} F_{jk}F^{jk}]$.  I am not sure if this formula has experimental support but it seems strange because it is a function of $F$ rather than it’s potential.  I wonder if the relationship between gravity and electromagnetism is simpler if one considers the electromagnetic potential directly as a real physical object which would make geometric sense.