Past exam of the symmetry and relativity course of the University of Oxford / 2015 / 4 / b

Let $A=(c{t}_A,f(t_A))$ on the rear rocket and $B=(c{t}_B,h(t_B))$ on the front rocket. The instantaneous inertial rest frame of the rear rocket at $A$ moves at $v=f^{\prime }(t_A)$ relative to S. Transform the time difference of these two events:Setting $\Delta t^{\prime }=0$ givesThe velocity belongs to the rear event $A$. It is not to be evaluated at laboratory time $t_B$, nor does the simultaneity condition mean $t_A=t_B$ in S. At nonzero velocity the front event on the same instantaneous-rest slice is at a different laboratory time.

Use the same event pair already made simultaneous in the rear instantaneous rest frame. Its proper spatial separation is the Minkowski interval length, soThe inequality selects the front rocket rather than the equally long backward extension. Equivalently the Lorentz-transformed spatial difference obeys $\gamma [h(t_B)-f(t_A)-v(t_B-t_A)]=L$. This is not the condition $h(t)-f(t)=L$ at equal laboratory times, which would describe the stretching arrangement in a.

Write $\Delta x=h(t_B)-f(t_A)$. Substituting $\Delta t=v\Delta x/c^2$ into ii gives $L^2=\Delta x^2(1-v^2/c^2)$. The positive-separation branch therefore hasThe Born-rigid front trajectory from a rear worldline follows by setting $t_A=t$:Every $v$ and $\gamma$ in this formula is evaluated at the rear event with laboratory time $t$, not at the shifted argument of $h$.

For completeness, a smooth future-directed front trajectory requires this time mapping to be locally invertible. If the signed laboratory acceleration is $a_S=dv/dt$ and the rear proper acceleration is ${\alpha }_F={\gamma }^3{a}_S$, thenWhen $1+L{\alpha }_F/c^2>0$, their ratio gives $dh/d{t}_B=v$, so the two simultaneous points indeed share their instantaneous velocity. If that factor reaches zero, an acceleration-horizon obstruction appears; an arbitrary rear trajectory cannot be extended to an arbitrary rigid length while retaining both rockets future-directed. The positive-acceleration example below satisfies the condition.