Retarded Potential - Potentials in The Coulomb Gauge

Potentials in The Coulomb Gauge

In the coulomb gauge, Maxwell's equations are

although the solutions contrast the above, since A is a retarded potential yet φ changes instantly, given by:

This is presents an advantage and a disadvantage of the coulomb gauge - φ is easily calculable from the charge distribution ρ but A is not so easily calculable from the current distribution j. However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:

Thus the potentials in the Coulomb gauge are unique and depend on the instantaneous values of fields in the whole space. This may seem as if it was possible to use this for superluminal communication; varying the fields on Mars influences immediately the potentials on Earth. However, the potentials are not measurable quantities according to present theory; they are just useful mathematical functions. Hence currently there is no possibility of using this for superluminal communication.