Time dilation for radioactive particles

I understand the reasoning behind the idea that a beam of radioactive particles in motion will decay more slowly than one at rest due to time dilation. It's also easy to imagine how this is tested in an accelerator by comparing the decay rates from two beams of - say - muons, which are moving at different speeds.

It gets puzzling for me, however, when I try to imagine two observers moving at different speeds both measuring decay rates from the same beam of muons. Obviously this isn't possible, since in practice the two observers would never detect any of the same decays - each could only detect some of them, so neither observer gets an accurate measure of the decay rate. It seems odd though that the number of decays detected by a single observer is dependent on the velocity of that observer relative to the beam.

I imagine that the explanation falls into one of the following categories:

1. It's explained by relativity, and the fact that if I am moving along with the beam, it'll appear that the individual decays I detect have less energy than if the beam is moving very quickly, and this accounts somehow for the fact that I am detecting more decays. If so, how?

2. The explanation is actually just standard quantum weirdness. If so, can it be recast directly in terms of some other more familiar complementarity phenomenon?

• Are you comfortable with the idea of each observer capturing half of the muons for study? If you are comfortable measuring the anisotropy (if any) of the radioactive beams, and then doing the relativity experiment accordingly, it's just a quick easy statistics problem to relate the two. – Cort Ammon Oct 3 '16 at 16:15
• So with each observer detecting decays through 180 degrees around the beam, (and accounting for any anisotropy if necessary) each one could just double their estimate of the decay rate to get their "measured value" for the full beam. I'm essentially a layperson with a math background and not much technical experience in quantum mechanics - do you think you could describe the the physical principle behind that should relate the two values, or the statistics calculation that comes from it? Either or both, posted as an answer, would be accepted. – xanderflood Oct 3 '16 at 16:42
• What makes you think that you can't make and accurate measurement of the decay rate with only a sample? Real detectors do that all the time. They have to since you can't have either the whole potential incident space sensitive nor get 100 percent detection efficiency. Enter terms for acceptance and efficiency (which you can often measure in situ). – dmckee Oct 3 '16 at 17:24
• @dmckee - nothing but ignorance, and Cort's comment corrected that. However, that ignorance isn't (as far as I can tell) really relevant to my question: what's the physical phenomenon or statistical principle that explains how the motion of the detector influences the number of decays it detects while avoiding any inconsistencies? For instance, the rest mass of the beam changes depending on how much it decays, but the rest mass ought to be well-defined independent of our frame of reference or how we detect the emissions. What gives, or what am I misunderstanding? – xanderflood Oct 3 '16 at 21:26
• The (rest or invariant) mass of the individual particles remains the same, but if you want the "mass of the beam" (not a quantity usually considered) to stay constant you have to include the contributions from the decay products of any particles that decay. – dmckee Oct 3 '16 at 21:33