A relativistic timer

Problem

A bomb equipped with a proper-time timer is launched at $v=5c/6$ towards a target a distance $L=30$~ls away. The timer must go off exactly when the bomb reaches the target. To what value of proper time $\Delta\tau$ should it be set? Would a “Galilean” setting $\Delta\tau=L/v$ give the same result?

Solution

Natural units $c=1$, so $v=\tfrac{5}{6}$ and $L=30$~ls.

Laboratory time of the journey. In the laboratory’s SR (where $L$ and the velocity are measured) the bomb takes: $t=\frac{L}{v}=\frac{30}{5/6}=30\cdot\frac{6}{5}=36\ \text{s}$

Proper time of the bomb. The timer counts proper time, related to the laboratory time by the Lorentz factor $\gamma$: $\gamma=\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-\tfrac{25}{36}}}=\frac{1}{\sqrt{11/36}}=\frac{6}{\sqrt{11}}\approx 1{,}809$ $\Delta\tau=\frac{t}{\gamma}=\frac{36}{1{,}809}\approx 19{,}9\ \text{s}$ Equivalently, with the invariant: $\Delta\tau=\sqrt{t^2-L^2}=\sqrt{36^2-30^2}=\sqrt{1296-900}=\sqrt{396}\approx 19{,}9$~s. $\ev{\Delta\tau\approx 19{,}9\ \text{s}}$

“Galilean” setting. The value $L/v=36$~s is the laboratory time $t$, not the proper time: setting the timer to $36$~s would make it go off too late (by the time $36$~s have passed on the bomb’s clock, the bomb has already overshot the target by a good margin). Time dilation makes the proper time $\Delta\tau\approx 19{,}9$~s markedly smaller than the $36$ Galilean seconds. The two results do not coincide.

Links

Topics: Special relativity Concepts: Time dilation · Spacetime invariant Skills: Symbolic setup Methods: Natural units