 World
path of a rod at rest in \( \bar{\cal O} \)
Length in \( \bar{\cal O} \) is \( \Delta s_{AC}^2
\)
and in \( {\cal O} \) is \( \Delta s_{AB}^2 \)
In \( \bar{\cal O} \), the coordinates of C are
$$
\bar{t}_C=0, \quad \bar{x}_C=\ell
$$
while in \( {\cal O} \), the coordinates of C are
$$
x_C = \ell/\sqrt{1v^2} \quad t_C= \ell v
/ \sqrt{1v^2}
$$
but the interval is the same in both systems
so
$$
x_C^2  t_C^2 = \ell^2
$$
and
$$
t_C = v x_C
$$
Also,
$$
\frac{x_Cx_B}{t_Ct_B}=v
$$

Schutz
 We want \( x_B \) when \( t_B =0 \)
$$
x_B = x_C  v t_C = \ell \sqrt{1v^2}
$$
 This is length contraction
