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

February 22, 2016 by zulfahmed

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 ) sphere. So then I ask myself, is the electromagnetic tensor really the curvature of a 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 . Then I look at the Gauss and Codazzi equations for the hypersurface

Now these are analogous to the vector potential of electromagnetism. Recall that in the latex curvature picture one looks at vector potentials and considers . In submanifold geometry the second fundamental form terms give us 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 follows from the Bianchi identity.https://wordpress.com/post/zulfahmed.wordpress.com/8477

Let . Then check that

by using the product rule and the Codazzi equation by collecting terms by the non-differentiated factor, i.e.

then for example the term with is .

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

with . I am not sure if this formula has experimental support but it seems strange because it is a function of 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.

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