  We extend the "gauge choice" problem Lamb noticed to include a time-dependent
relativistic non-perturbative Coulomb field, which can be produced by a cluster
of relativistic charged particles. If adiabatic conditions are carefully
maintained, such a field must be included along side the nuclear Coulomb
potential when defining the atomic state. We reveal that when taking the
external field approximation, the gauge choice for this time-dependent
relativistic non-perturbative Coulomb field cannot be overcome by previous
method, and leads to considerable gauge-dependence of the transient spontaneous
radiation spectrum. We calculate explicitly with a simple one-dimensional
charged harmonic oscillator that such a gauge-dependence can be of a measurable
magnitude of 10 MHz or larger for the commonly used Coulomb, Lorentz, and
multipolar gauges. Contrary to the popular view, we explain that this gauge
dependence is not really a disaster, but actually an advantage here: The
relativistic bound-state problem is so complicated that a fully quantum-field
method is still lacking, thus the external field approximation cannot be
derived and hence not guaranteed. However, by fitting to the experimental data,
one may always define an effective external field, which may likely be
parameterized with the gauge potential in a particular gauge. This effective
external field would not only be of phenomenological use, but also shed light
on the physical significance of the gauge field.
