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Time dilation is an elementary consequence of SR, usually derived as part of an introduction to the subject. The material you reference here is advanced particle physics, which has SR (and especially four-vectors) as a key building block.msumm21 said:My question boils down to seeing how time dilation manifests in spontaneous transitions between energy levels (i.e. why looking at spontaneous transitions in a moving frame would appear to slow). Some good surrounding info in above posts, the link below might give the answer I'm looking for around page 9, but I've not yet had time to understand it yet.
https://web2.ph.utexas.edu/~vadim/Classes/2022f/FGR.pdf
I can't reconcile your elementary question with this advanced material.