Worked length contraction of a rod of rest length 5 at speed 3c/5

In an inertial chart $x$ let the chart $x^{\mathrm{\prime }}$ be related to $x$ by a boost of speed $v=3c\mathrm{/}5$ along the first spatial axis, $\gamma =5\mathrm{/}4$, so that $x^{\mathrm{\prime }0}=\frac{5}{4}(x^0-\frac{3}{5}{x}^1)$ and $x^{\mathrm{\prime }1}=\frac{5}{4}(x^1-\frac{3}{5}{x}^0)$ (Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation, The passive Lorentz boost with parameter beta). Let $L_0=5$ and let the rod at rest in $x^{\mathrm{\prime }}$ have endpoints on the worldlines $x^{\mathrm{\prime }1}=0$ and $x^{\mathrm{\prime }1}=5$, with vanishing transverse coordinates. Solving the transition for $x^1$ as a function of $x^0$ gives $x^1=\frac{3}{5}{x}^0$ on the first worldline and $x^1=\frac{3}{5}{x}^0+4$ on the second, so at every equal $x$-time the endpoints are separated by $4=L_0\mathrm{/}\gamma$ in the frame $x$. Explicit witness: the events $p:=x^{-1}(5,3,0,0)$ and $r:=x^{-1}(5,7,0,0)$ are simultaneous in $x$ (both have $x^0=5$) and separated by $4$; their $x^{\mathrm{\prime }}$-coordinates are $x^{\mathrm{\prime }}(p)=(4,0,0,0)$ and $x^{\mathrm{\prime }}(r)=(1,5,0,0)$, so they are not simultaneous in $x^{\mathrm{\prime }}$ but their $x^{\mathrm{\prime }}$-separation is $5=L_0$. Comparing events at equal $x^{\mathrm{\prime }}$-time instead: the events of the two worldlines with $x^{\mathrm{\prime }0}=4$ are $x^{-1}(5,3,0,0)$ and $x^{-1}(35\mathrm{/}4,37\mathrm{/}4,0,0)$, the second of which has $x^{\mathrm{\prime }1}=5$; at equal $x^{\mathrm{\prime }}$-time $4$ the endpoints are separated by $5$, the rest length. Thus the frame $x$ assigns length $4$ to a rod of rest length $5$ at rest in $x^{\mathrm{\prime }}$, exactly the contraction $L_0\mathrm{/}\gamma$ of clause (ii) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation, and the comparison uses the simultaneity slice of the measuring frame in the sense of Simultaneity slices and rest distance relative to a future unit timelike vector.

• Which events are being compared. The two witness events of step 1.2 are simultaneous in $x$ and not in $x^{\mathrm{\prime }}$, while the two events of step 1.3 are simultaneous in $x^{\mathrm{\prime }}$ and not in $x$; the contracted and the rest lengths therefore refer to different pairs of events on the same two worldlines, which is exactly the content of clause (ii) of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation.

• No optical claim. The length $4$ assigned by $x$ is the distance of the rod's endpoints on a simultaneity slice of $x$; it is not an appearance or a photograph, and no light-propagation or measurement model is involved.