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There are many ways of writing the laws of electromagnetism. However, the
most obviously Lorentz invariant way is to write them in terms of the
vector and scalar potentials (see Sect. 4.6). When written in this fashion, Maxwell's
equations reduce to
where is the scalar potential, and the vector potential.
Note that the differential operator appearing in these equations is
the Lorentz invariant d'Alembertian, defined in Eq. (1409).
Thus, the above pair
of equations can be rewritten in the form
Maxwell's equations can be written in Lorentz invariant form provided that
the entity

(1441) 
transforms as a contravariant 4vector. This entity is known as the
potential 4vector. It follows from Eqs. (1433), (1437), and (1438)
that

(1442) 
Thus, the field equations which govern classical electromagnetism can all
be summed up in a single 4vector equation.
Next: Gauge invariance
Up: Relativity and electromagnetism
Previous: The current density 4vector
Richard Fitzpatrick
20060202