  Gauge theory requires physical observables to be gauge-independent. However,
ever since Lamb noticed the problem of gauge selection in calculating atomic
spontaneous radiation spectrum, the problem of gauge dependence was encountered
in many fields of physics research. Therefore, it is important to test the
self-consistency of gauge symmetry for various physical systems. In this paper,
we calculate the transient spontaneous radiation spectrum of a relativistic
hydrogen atom in the non-uniform electrostatic field under the atomic
self-reference frame. The physical system studied in this paper is a
frame-transformed version of our recent work
[\href{https://link.springer.com/paper/10.1140/epjd/s10053-022-00407-5}{Euro.
J. Phys. D \textbf{76}, 84(2022)}] where the radiating object is static while
the charge is moving relativistically. The obtained peak frequency can differ
by about $413$ $\mathrm{KHz}$ or larger for the commonly used Coulomb, Lorentz,
and multipolar gauges. This observation can be significant not only for
studying how the gauge field interacts with the quantum system in theory, but
also for practical experimental applications, such as the timing accuracy of
atomic clocks in the external electromagnetic field.
