An unexpected consequence of time dilation concerns objects that carry a clock with them. If you launch a bomb fitted with a timer that ticks its own proper time, the time the bomb measures during the journey is less than the time you measure, standing still watching it leave and arrive.
The idea is simple but must be handled with care. The timer does not “know” how much distance the bomb covers in the lab: it only counts its own proper time , the invariant along its world line. If we want the timer to go off exactly when the bomb reaches the target, we must set it not to the flight time measured on the ground, but to the proper time , which is always shorter. A naive “Galilean” setting, (ground distance divided by speed), would use lab time and would make the timer go off too late: the bomb would have already passed the target. The relativistic correction is precisely the factor .
A proper-time timer is thus, in a sense, a clock that “lives less” the faster it travels: the same logic as the twin who ages less along the moving trajectory.
Deep dive — The muon: nature's bomb
A muon is an elementary particle produced in the upper layers of the atmosphere (at km altitude) by the interaction of cosmic rays. It “decays” — explodes — after of proper time. If time dilation did not exist, travelling at a speed close to it could cover at most m before decaying: far less than the 15 km separating it from the ground. Yet most muons do reach the ground! For us standing still, the muon is like a relativistic bomb with its own timer: its travel time measured from the ground, , is long enough to let it cross the entire atmosphere. The detection of muons at ground level is one of the most direct experimental confirmations of time dilation.
Links
Topics: Special relativity Concepts: Time dilation · Spacetime invariant Skills: Changing reference frame Objects: Light clock
Related exercises: Mr Rossi goes to the theatre · Problem — The reference frame of minimum time · The rabbit race