The Relativity of Length — Question 10

Question 10

Contraction increases number density without creating particles

A uniform cloud has rest number density $n_0$ and rest volume $V_0$. It moves at $v=0.60c$ along x. Count particles on simultaneous slices; a boundary fixed in the laboratory has transverse area A. Use $\gamma=(1-v^2/c^2)^{-1/2}$ and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the laboratory number density.

2. Find the crossing rate through the boundary and verify particle number is unchanged for a comoving cloud volume.

Question 10 - Solution

Contraction increases number density without creating particles

(a) Method and calculation

$$N=n_0V_0,\qquad V=V_0/\gamma.$$$$\boxed{n=N/V=\gamma n_0=1.25n_0}.$$

(b) Interpretation and further calculation

$$\boxed{\dot N_{\mathrm{cross}}=nvA=0.750n_0cA}.$$$$nV=(\gamma n_0)(V_0/\gamma)=\boxed{n_0V_0}.$$

Physical check

Number is invariant for the chosen collection of particles. Density changes because the simultaneous volume used in the count changes.