Remember that the caesium atom has been chosen because it's natural resonating frequency is equal to a second as defined by the movements of the earth around the sun,Only if you believe that 9,192,631,770 = 1, and I think you may be alone in that belief.
Those who know a bit more would say that cesium is chosen because 9 GHz microwaves are easy to generate and manipulate, and the cesium ground state hyperfine transition is very stable and easy to identify within the cesium absorption spectrum. The magic number was chosen as the best estimate of the length of an astronomical ephemeris second in terms of the frequency of a cesium clock, and from that moment on, the clock defined the second and all astronomical measurements became secondary measures.
Not a good idea to talk about the "frequency of the cesium atom" lest deBroglie fans think you mean the deBroglie frequency of the atom, which depends on its kinetic energy. The hyperfine splitting energy doesn't.
As the energy (and hence frequency: E=hf) of the cesium clock transition is fixed, the phenomenon of one clock running fast or slow compared with another is due to the relativistic gravitational and/or motion dilation of time between observers.
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