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One big difference between GR & EM is that the physical fields
in GR are the 2nd derivatives of the potential fields, rather
than the first derivatives, as they are in EM. In EM the
A-potential components don't have a physical meaning themselves,
rather the E & B fields, which are the derivatives of the
A-potential components, are the physical force fields acting on
the charged sources and whose values are constrained by the
requirements of Maxwell's equations.