At two points of the reference frame K two events occurred separated by a time interval `Deltat`. Demonstrate that if these events obey the cause-and-effect relationship in the frame K (e.g. a shot fired and a bullet hitting a target), they obey that relationship in any other inertial reference frame `K^'`.

Explanation

We can take the coordinates of the two events to be
`A:(0,0,0,0) B: (Deltat, a, 0, 0)`
For B to be the effect and A to be cause we must have `Deltat gt (|a|)/(c)`.
In the moving frame the coordinates of A and B become
`A: (0,0,0,0), B: [gamma(Deltat-(aV)/(c^2)), gamma(a-VDeltat), 0, 0]` where `gamma=(1)/(sqrt(1-((V^2)/(c^2))))`
Since
`(Deltat^')^2-(a^('^2))/(c^2)=gamma^2[(Deltat-(aV)/(c^2))^2-1/c^2(a-VDeltat)^2)=(Deltat)^2-a^2/c^2 gt 0`
we must have `Deltat^' gt (|a^'|)/(c)`