Time dilation..with particles
One way of checking out the predictions of the time dilation formula is to find a clock which is moving very fast: then the differences between τ and τ0 should be apparent. Here Nature is kind: it supplies us with an abundance of very-fast-moving clocks...we just have to recognise them!
The point is that anything which has a ‘natural lifespan’ can be thought of as constituting a clock. Thus the particles which populate the microworld and ‘decay’ after some typical lifetime (into some other species) can be thought of as carrying a clock which tells them that its time to ‘go’. (If you think that description might apply to you too, you are right: and so does time dilation!) You probably know that the decay of particles is ‘statistical’ in the sense that one may speak only of a ‘typical’ lifetime (or half-life, if you prefer).
If one has a sample containing a given number of particles (at time t=0) the number still there (alive!) at time t falls off exponentially with t...as the particles ‘die’. If the lifetime is big the fall off with t is slow; if the lifetime is small the fall off with t is fast.
Experiments show that the decay rate is different according to whether the particles are at rest or moving rapidly (in each case ‘relative’ to the experimentalist).
The moving particles have a longer lifetime; their clocks run slow; they decay more slowly; they ‘live longer’.