The Relativity of Time — Question 8

Question 8

A clock-accuracy requirement gives an exact speed bound

Over laboratory duration T, a uniformly moving clock must lose no more than fraction $\epsilon$ of T: $(T-\tau)/T\le\epsilon$, with $0<\epsilon<1$. Use $d\tau=dt\sqrt{1-v^2/c^2}$ for an ideal clock, with $c=3.00\times10^8\,\mathrm{m/s}$ when needed.

1. Derive the exact allowed speed interval.

2. Evaluate the bound for $\epsilon=10^{-4}$ and compare with the leading small-error approximation.

Question 8 - Solution

A clock-accuracy requirement gives an exact speed bound

(a) Method and calculation

$$1-\sqrt{1-\beta^2}\le\epsilon\quad\Rightarrow\quad\beta^2\le2\epsilon-\epsilon^2.$$$$\boxed{|v|/c\le\sqrt{2\epsilon-\epsilon^2}}.$$

(b) Interpretation and further calculation

$$\epsilon=10^{-4}:\quad\boxed{|v|/c\le0.0141418}.$$$$\epsilon\ll1:\quad\beta_{\max}\simeq\sqrt{2\epsilon}=0.0141421.$$

Physical check

The exact bound uses loss relative to laboratory duration, not loss relative to the moving clock duration.