Question

Show, by Minkowski diagrams, that (1) time-like events can occur at the same position, but not at the same time, (2) space-like events can occur at the same time, but not at the same position, (3) space-like events can have any order/sequence.

Answer 1

I'll do the whole analysis in c=1 units.

(1) Consider the following Minkowski diagram. Forget about the blue lines and the t'-x' axis. Focus on the events A and B. Convince yourself that events A and B are time-like and they occur at different times and positions in x-t frame. The dashed line is the 45 degree line corresponding to speed of light, which is 1.

Now consider a boost in the +ve x direction. In x-t frame the x'-t' line move towards each other (they actually are orthogonal in their own frame) and are represented by blue lines. Note that these are the lines that an observer in x-t frame will draw. So the observer just has to find the right line that passes through the A and B events, which will become the t' axis. Clearly, events A and B have the same x' coordinate (x'=0), which is exactly what we wanted. For A and B to occur at the same time we will need an x'-axis to pass through them, but this can't happen because we can't move faster than light!

(2)Something similar is to happen here as well. This time we just need to find an x'-axis which passes through A and B and it is possible for space-like events only. In x'-t' frame both A and B occur at t'=0, which is what we wanted. Again they can't occur at same positions because it is impossible to cross the c=1 barrier.

(3) In part (2) if we go a little faster, then A and B would reverse order. For simplicity, consider the event B at origin O. The red lines correspond to x"-t" frame which is even faster than x'-t' frame. The following can be concluded for space-like events OA

$\begin{align*} t_A>0\\ t'_A=0\\ t''_A<0 \end{align*}$