The Relativity of Length — Question 8

Question 8

A collinear photograph samples a retarded endpoint separation

A longitudinal rod recedes along +x, entirely to the right of a stationary camera at the origin. Its simultaneous laboratory length is $L=L_0/\gamma$. Define the apparent axial separation using the two endpoint emission positions whose light reaches the camera at the same time T. Use $\gamma=(1-v^2/c^2)^{-1/2}$ and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Derive the apparent separation $L_{\mathrm{app}}$ in terms of L and $\beta=v/c$.

2. Evaluate it for $L_0=120\,\mathrm m$ and $v=0.60c$.

Question 8 - Solution

A collinear photograph samples a retarded endpoint separation

(a) Method and calculation

$$T=t_F+x_F/c=t_R+x_R/c\quad\Rightarrow\quad t_F-t_R=-L_{\mathrm{app}}/c.$$$$L_{\mathrm{app}}=L+v(t_F-t_R)=L-\beta L_{\mathrm{app}}.$$

(b) Interpretation and further calculation

$$\boxed{L_{\mathrm{app}}=L/(1+\beta)}.$$$$L=96.0\,\mathrm m,\qquad\boxed{L_{\mathrm{app}}=60.0\,\mathrm m}.$$

Physical check

This retarded axial-coordinate separation is not an angular image size and is not the simultaneous Lorentz-contracted length.