Let $S$ and $S^{\prime}$ be inertial frames in 2-dimensional spacetime with coordinate systems $(t, x)$ and $\left(t^{\prime}, x^{\prime}\right)$ respectively. Suppose that $S^{\prime}$ moves with positive velocity $v$ relative to $S$ and the spacetime origins of $S$ and $S^{\prime}$ coincide. Write down the Lorentz transformation relating the coordinates of any event relative to the two frames.

Show that events which occur simultaneously in $S$ are not generally seen to be simultaneous when viewed in $S^{\prime}$.

In $S$ two light sources $A$ and $B$ are at rest and placed a distance $d$ apart. They simultaneously each emit a photon in the positive $x$ direction. Show that in $S^{\prime}$ the photons are separated by a constant distance $d \sqrt{\frac{c+v}{c-v}}$.