For any single nucleus (of an isotope that undergoes
spontaneous decay), while a large population will decay at
a characteristic rate related to half life and population, a
single nucleus may decay instantly, or survive for an
infinite time before decay. But, given any
two of these
nuclei (and we might take tritium as a simple one): if they
are in the same quantum state to start with, they are
absolutely identical - they are in some ways quantum
clones.

How can a nucleus “know” when to decay? I understand
there are incomprehensible behaviours of the strong and
weak nuclear forces, and the probability functions of
quantum mechanics at play.

I propose an experiment in which nuclei (with a known
probability/frequency of spontaneous decay) are
accelerated to a significant proportion of c, to see whether
their decay frequency is affected by time dilation.

The mechanics of the experiment would be really quite
simple.

My design would be a simple linear accelerator
with sensors distributed along its length. A
population of atoms of known half-life are ionised
and accelerated to a significant proportion of c.
Depending on their velocity, a shift should be
observed (for a static observer or sensor) in their
rate of decay. So, sensor/observer is “static”.

You're going to have a population of nucleii with a range of velocities, even with the tightest engineering controls. That will affect your data.

The nucleii and the observer need to be in the same reference frame, and the sample needs to be cooled to near 0K to minimize translational velocity within the population.

//How can a nucleus “know” when to decay? // It doesn't.
The chances of its decaying in the next second are (for any
given isotope) the same whether it's just been formed or its
been sitting around for many half-lives.

Also, unstable particles (including unstable nuclei) have
longer half-lives when accelerated to relativistic velocities.
So, if your particles have a half-life of a microsecond at rest
(relative to the observer), and you send them round an
accelerator for a microsecond, a lot more than half will
remain and come out the far end. It's been done, and in
fact is a common factor in many accelerator experiments.

Muons produced 20km up in the upper atmosphere by
cosmic radiation impact have a half life of 2.2 microseconds
and yet more are observed at the Earth's surface because of
time dilation effects. See link...

// have longer half-lives when accelerated to relativistic velocities. //

No, they don't.

Their half life ** measured by a "stationary" observer ** is longer, but that's because time passes more slowly for the particles at their high relativistic velicity.

It's an example if the well-known "twin paradox".

As you correctly point out, the statistical probability of decay is a constant. If the observer is in the same reference frame as the sample, then the observed decay rate should be the same independent if their translational velocity. But if the observer and sample are in different reference frames, then relativistic time distortion has to be factored in.

[Frankx]'s original question is valid but only if the sample and observer travel together. It can't be resolved experimentally by s "static" observer.

OK, I'm moving this to Halfbakery Archive, as the effect has been experimentally measured as noted.

Rossi–Hall experiment and others [link]

How does a nucleus "know" when to decay: it seems to be down to the probability function for quantum tunneling. It's a random event, but with a defined probability (over a certain period of time)

In terms of patisseriological topology, [@] looks more like a Danish Pastry or a Cinnamon Roll than a bun. Although, in the grand Linnaean taxonomy of baked goods, are these sub-classes of 'bun'?

//probability function for quantum tunneling// So if God doesn't play craps, an energy valley of the space-time fabric. Another variable not isolated by current technology or theory.