Past exam of the symmetry and relativity course of the University of Oxford / 2017 / 2 / b

Let $\Delta t=t_d-t_g$ and $\Delta \mathbf{x}={\mathbf{x}}_d-{\mathbf{x}}_g$. A spacelike separation obeysUnder a boost with velocity $\mathbf{V}$, the time separation is $\Delta t^{\prime }=\Gamma (\Delta t-\mathbf{V}\cdot \Delta \mathbf{x}/c^2)$. ChooseThen $\Delta t^{\prime }=0$, while $|\mathbf{V}|/c=c|\Delta t|/|\Delta \mathbf{x}|<1$ follows from the spacelike inequality. Therefore an inertial simultaneity frame exists. This choice has no transverse boost component and is not claimed to be the unique such frame. If the events are already simultaneous, $\mathbf{V}=0$ suffices.