The Relativity of Time — Question 4

Question 4

A ratio of survival fractions eliminates source normalization

Two species have proper mean lifetimes $\tau_A=1.0\,\mu\mathrm s$ and $\tau_B=2.0\,\mu\mathrm s$. They share one constant beam speed. Across a known laboratory flight time $T=4.0\,\mu\mathrm s$, the ratio of normalized survival fractions is $P_A/P_B=e^{-1}$.

1. Infer the common Lorentz factor from the ratio.

2. Find the speed and each survival fraction.

Question 4 - Solution

A ratio of survival fractions eliminates source normalization

(a) Method and calculation

$$\ln(P_A/P_B)=-\frac{T}{\gamma}\left(\frac1{\tau_A}-\frac1{\tau_B}\right).$$$$-1=-2/\gamma\quad\Rightarrow\quad\boxed{\gamma=2}.$$

(b) Interpretation and further calculation

$$\boxed{|v|=\frac{\sqrt3}{2}c\simeq0.866025c}.$$$$\boxed{P_A=e^{-2}},\qquad\boxed{P_B=e^{-1}}.$$

Physical check

These are normalized survival fractions, not raw detected counts; species-dependent efficiencies must already be corrected.