  The concept of gauge invariance in classical electrodynamics assumes tacitly
that Maxwell's equations have unique solutions. By calculating the
electromagnetic field of a moving particle both in Lorenz and in Coulomb gauge
and directly from the field equations we obtain, however, contradicting
solutions. We conclude that the tacit assumption of uniqueness is not
justified. The reason for this failure is traced back to the inhomogeneous wave
equations which connect the propagating fields and their sources at the same
time.
