Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 7 / 1

The official notes give only “Part 1: as in textbooks” (PDF p. 13, handwritten p. 8). The requested derivation is supplied here.

Put a light source at the rod midpoint in its rest frame $S$ and emit two pulses simultaneously toward its ends, each a distance $\ell$ away. In $S$, the reflection events are $(t,x)=(\ell /c,\pm \ell )$. Invariance of light speed, the Einstein's principle of relativity and linearity give a Lorentz transformation $x^{\prime }=A(x-vt)$, $t^{\prime }=A(t-vx/c^2)$. Applying the inverse transformation with speed $-v$ gives $A^2(1-v^2/c^2)=1$, hence $A=\gamma$.

The transformed reflection events haveThese events are not simultaneous in $S^{\prime }$. Both rod ends move at $-v$ there, so extrapolating each endpoint to $t^{\prime }=0$ gives $x_{\pm }^{\prime }(0)=x_{\pm }^{\prime }+v{t}_{\pm }^{\prime }=\pm \ell /\gamma$. Measuring simultaneous endpoints therefore yields length contraction: