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## Maxwell’s equations with differential forms

It is quite well-known that Maxwell’s equations can be expressed via differential forms and there are many presentations for such.  So flux densities for electric and magnetic fields $D, B$ are 2-forms; the current density $J$ is a 2-form.  The electric and magnetic field intensities $E, H$ are 1-forms and the Maxwell’s equations are:
$dE = - \partial_t B$

$dH = \partial_t D + J$

$dD = 0$

$dB = 0$

These equations have perfect sense on $S^4(1/h)$.  The last two equations say that the flux densities are closed forms which implies, because the sphere has no cohomology in dimension 2 that in fact both are exact.  This note is simply to observe that electromagnetism on a 4-sphere can be understood extremely simply on the entire 4-sphere.  I will come back to the issue of how it is described via differential forms for a principal $U(1)$-bundle in terms of a connection and then use the description of connections via a 1-forms on the total space of the bundle and then note that for a hypersurface of a sphere, the analogue of the circle-bundle occurs from the geodesic normal circles and the connection 1-form has the analogue of a 1-form on the 4-sphere.  This latter issue should have testable implications in the real universe because it would imply that actual electromagnetism in our universe does NOT have full $U(1)$ gauge invariance.

The standard presentation for the principal bundle approach is to use the electric potential $\phi$ and the magnetic potential $(A_1, A_2, A_3)$ to form, on the trivial circle bundle over space-time the differential 1-form
$A = d\theta + \phi dt + A_1 dx^1 + A_2 dx^2 + A_3 dx^3$ so that the Maxwell’s equations become $d*\Omega=0$ where $\Omega = dA$.  There is no difficulty in producing such a 1-form for $S^4(1/h) \times \mathbf{R}$ where $\theta$ is a coordinate along the normal direction.  However, since the four-spere is not the total space of a principal circle bundle, one cannot use the first coordinate globally.