reference frames - Special Relativity - When events coincide?

Consider 1-D space. Let S and S' be two inertial reference frames. Let A and B be two events.

Co-ordinates of A and B under S are A = (xA,tA) and B = (xB,tB).

Let me define what I mean by when I say two events coincide - it simply means they have same space-time co-ordinates.

i.e. if (xA = xB) and (tA = tB), then w.r.t S, events A and B coincide.

I have two Qs:

Q1. Is my above definition of events that coincide correct?

Q2. If A and B coincide in S, then will the coincide in S' (i.e. two events being coincident is NOT a relative concept)

Thanks

Answer

Now first of all let me clarify what an event is. An event is something which happens at a particular point in disce and time, so an event in the reference frame S can be described by $(\mathbf{r},t)$.

Answer to the first question is yes. If two events coincide, then they happen at the same place at the same time with respect to a reference frame. i.e. the coincidence of two events, A and B, in reference frame S is defined as $$\mathbf{r_A}=\mathbf{r_B}$$ And $$t_A=t_B$$

For the second question, if two events are coincident in one reference frame, they should be coincident in all inertial reference frames. That's because the Lorentz transformation for both the events give the same change.

There is no proper physical/intuitive proof for this. Sometimes it's just good to stick to abstract mathematical principles, rather than trying to use our intuitions.