Some Consequences of the Lorentz Equations — Question 10

Question 10

Density enhancement and lifetime dilation cancel in a local decay rate

At corresponding events, a uniform unstable cloud has proper number density $n_0$ and proper mean lifetime $\tau_0$. It moves uniformly with Lorentz factor $\gamma$. For a small laboratory volume and short interval, ignore depletion and gradients.

1. Combine length contraction and time dilation to find decays per laboratory volume per laboratory time.

2. Check the expected count in a small spacetime cell using the unit Jacobian of a Lorentz boost.

Question 10 - Solution

Density enhancement and lifetime dilation cancel in a local decay rate

(a) Method and calculation

$$n=\gamma n_0,\qquad r_{\mathrm{per\ particle}}=1/(\gamma\tau_0).$$$$\boxed{\mathcal R=n\,r_{\mathrm{per\ particle}}=n_0/\tau_0}.$$

(b) Interpretation and further calculation

$$dN=\mathcal R\,dV\,dt,\qquad dV\,dt=dV\prime\,dt\prime.$$$$\boxed{dN=(n_0/\tau_0)\,dV\prime\,dt\prime}.$$

Physical check

The statement is local at corresponding events. It does not claim that densities on different distant simultaneity slices have identical histories.