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

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.

• Thanks. Is there a more fundamental reason / physical agrument for Ans 2 than Lorentz transformation Dec 14, 2019 at 15:26
• The more fundamental reason is that you can't change the properties of a thing by changing its name. Dec 14, 2019 at 15:58
• @WillO thanks, but I really did not understand your comment? Any more help / pointer will be quite useful Dec 15, 2019 at 4:21
• @aman_cc An event is a point in spacetime. A pair of coordinates is a name for that point. Changing coordinate systems means changing the names of all the points, including the one you're interested in. If I change your name from aman_cc to aman_dd, you will still be the same person. If I change the name of a point from (x=3,t=2) to (x=9/4, t=1/4), it's still the same point. Dec 15, 2019 at 4:52
• A point is an event. No event is ever in motion. The whole concept of "in motion" does not apply to events. A point is something like your presence at a concert. I can describe that event as "Latitutde 50 degrees, longitude 30 degrees, 6PM Eastern Standard Time" or as "300 yards due north of Times Square, 7.3 hours after sunrise" or just "aman_cc at the concert". Those are different ways of naming the same event. The event is what it is. Dec 15, 2019 at 6:07

Let me define what I mean by when I say two events coincide - it simply means they have same space-time co-ordinates. [...] Q1. Is my above definition of events that coincide correct? [...]

No: In the given context of the theory of relativity, where events are considered distinguishable elements (points) of spacetime, the phrase "two events that are coincident" is an oxymoron.

Instead: Either there are exactly two distinct events to be considered, or exactly only one event (in which several distinct material points may have taken part together, a.k.a. in coincidence; and in which each participant may have gathered several distinct perceptions together, a.k.a. in coincidence).

This understanding that one-and-the-same event should not be referred to as "two distinct events", nor vice versa, is basic, constitutive, indenpendent of any coordinate assignments; while it remains of course possible to label one-and-the-same event with various different coordinate labels.

If two events happen at the same time at the same place then you can call them two coincident events but they are really one and the same event. They will also be coincident in any other reference frame.

In your comment to the accepted answer you suppose that spacetime consists of a set of points independent of any reference frame, and ask whether it is possible, therefore, to talk of frames moving in relation to those points. The problem is that you have no way of knowing how frames are moving in an absolute sense. All you can say is that in Frame A the point has one set of coordinates and in Frame B it has another set, and that Frame A and Frame B are moving with respect to each other.