sci.physics.research

The Helmholtz decomposition of a three-vector field is described in a
Wiki article at:

/wiki/Helmholtz_decomposition

and a generalization to any arbitrary manifold via the Hodge
decomposition is discussed in:

/wiki/De_Rham_cohomology#Hodge_decomposition.

While the above specify mathematical properties of vector fields
generally, I am interested in their (possible) application to physics.
Thus:

Has anyone ever specified these decompositions in a generally-covariant
fashion, using vectors and tensors, for the two
relativistically-interdependent three-vector fields E and B of
electricity and magnetism, in a four-dimensional spacetime with a
Minkowski metric tensor signature?

Also, whereas the Helmoltz decomposition contains two additive terms,
the Hodge decomposition contains a third, additive, harmonic term,
labeled gamma in the Wiki article. Is it known whether this third term
would come into play in four-dimensional spacetime with E and B fields,
whether this term (or some other term) might contain time derivatives of
the Newtonian potentials (since Helmholtz already contains space
derivatives which from both the known theory of EM potentials and from
Maxwell's equations have time derivative counterparts when specified in
spacetime), and whether we would find the E field "crossing over" into
some terms of the decomposition of B, or vice versa?

Thanks,

Jay.
____________________________
Jay R. Yablon
Email: jyablon@
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