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Suppose $\phi, A_i$ are the components of the electromagnetic potential.  Given that we know that the universe has four space dimensions, the most intuitive electromagnetic 1-form to consider is, using orthonormal coordinates adapted to the physical hypersurface $M$, i.e. $(x_1, x_2, x_3, x_4)$ with $x_4 = 0$ on $M$,
$A = \phi dx_4 + A_1 dx_1 + A_2 dx_2 + A_3 dx_3$
This is different from the convenient 1-form used for Maxwell’s equation for principal bundles where $\phi$ is the coefficient for time.  The formulation above is extremely appealing geometrically and is natural for a four-dimensional universe.  One can read off the electromagnetic field components from $dA$ although Maxwell’s equation will introduce time separately.
Macroscopically, there are just two fundamental forces: gravity and electromagnetism.  The question that must be answered in a parsimonious way is how the two evolution equations relate to the evolution of $M_t$.  A very simplistic answer that is appealing is that $M_t$ evolves according to the flow of the electromagnetic vector field (the dual of the one-form above) and the gravitational equation is tabulating the changes in the Ricci curvature.  This would be the simplest sort of picture.