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Gauge theory requires physical observables to be gauge-independent. However,ever since Lamb noticed the problem of gauge selection in calculating atomicspontaneous radiation spectrum, the problem of gauge dependence was encounteredin many fields of physics research. Therefore, it is important to test theself-consistency of gauge symmetry for various physical systems. In this paper,we calculate the transient spontaneous radiation spectrum of a relativistichydrogen atom in the non-uniform electrostatic field under the atomicself-reference frame. The physical system studied in this paper is aframe-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 whilethe charge is moving relativistically. The obtained peak frequency can differby about $413$ $\mathrm{KHz}$ or larger for the commonly used Coulomb, Lorentz,and multipolar gauges. This observation can be significant not only forstudying how the gauge field interacts with the quantum system in theory, butalso for practical experimental applications, such as the timing accuracy ofatomic clocks in the external electromagnetic field.

Gauge dependence of spontaneous radiation spectrum of relativistic atomic beam under non-uniform electrostatic field