C3 - intervals

the line with a unit gradient in a $ct-x$ plot is the light line

for an object moving at $c/2$, the interval $d{s}^2>0$, so, it is a time-like interval, and can have causal connection with an event at the origin origin

for an object that falls beyond the light line, $d{s}^2<0$, so it is a space-like interval, and cannot have causal connection

if it lies along $ct=0$, it occurs at the same time as an event at the origin

in a frame of reference $S^{\prime }$ travelling at $c/2$ in the rest frame

since the spacetime interval is invariant under LTs, all inertial observers must agree on whether an interval is time-like, space-like, or light-like

they may disagree about the individual time and distance separations

consequently, there exist a reference frame where two events:

• occur at the same position if they are time-like
• occur at the same time if they are space-like
• neither if light-like