May migrate to principal-bundle-connection in the future?
There are many possible connections
There are connections that cannot be local flat. For any local region, the connection field cannot be eliminated by changing the gauge.
Similar to flat-metric-iff-curvature-0
There exist local bundle coordinates or phases, where the connection is zero <==> the curvature is zero where , in coordinates
When not exist flat connection coordinate, choose minimal curvature, based on metric-volume-form
Example
case. is valued, and it is commutative . In this case , in coordinates
Transforming from local flat-connection coordinates to general coordinates gives a PDE
PDE solvable condition
From the solution of the PDE , integration gives and phase transformation
[electromagnetic-field]
In
There are many possible connections; select by minimizing curvature.
Note that means the definition of this action requires spacetime metric
eq
In coordinates
In spacetime decomposition coordinates
Of course, this decomposition method is not invariant
Question How to make obviously imply the form of the electromagnetic field equations? [Maxwell-equation]
Also
Where
Note that and exterior derivative are related to Hodge star. can also be related to Hodge star. It may be related to the behavior of Hodge star in the spatial coordinates of spacetime decomposition.
Note that Hodge star requires a metric
Using the special specification [Lorentz-gauge] , the equation becomes