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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).

When we say 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.

Let me state a theorem: If A and B coincide in S, then they will the coincide in S' (hence in every and any IRF i.e. two events being coincident is NOT a relative concept)

Q1 - Why is this theorem? Is there a deeper assumption and understanding regarding space-time behind this concept? (I'm not looking for an answer basis Lorentz Transformation - but a more physical / maybe more basic argument). Or is this just an assumption of Special Relativity?

Q2 - If 2 balls A and B collide - they will collide in every IRF. How can I derive this basis above theorem? i.e. how can I "precisely" express collision of 2 balls as two events which coincide?

(I'm asking the above question to better understand space-time, events etc at a little conceptual level and I'm having difficulty in understanding them, Thanks for your help)

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It's a lot simpler than you think. Suppose an event has coordinates $x$ in some reference frame, where $x$ contains both space and time coordinates within it. To get the coordinates $x'$ of the same event in some other reference frame, you apply some function, $$x' \equiv f(x).$$ This works in nonrelativistic physics, special relativity, and even general relativity. In special relativity the function is called a Lorentz transformation. The key (essentially only) assumption here is that the location of an event in spacetime is completely specified by its coordinates.

If two events $A$ and $B$ coincide, their coordinates are the same, $x_A = x_B$. You want a proof of the "theorem" that in any other reference frame, $x_A' = x_B'$. Now hold on to your seat, because this profound result can be beautifully proven in airtight, perfectly rigorous formal mathematics as: $$x_A' = f(x_A) = f(x_B) = x_B'.$$ That's it.

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  • $\begingroup$ Thanks. So the theorem proof is quite good for me (I guess the only "basic" assumption is (x') depends only on x i.e. is a f(x) and nothing outside of x. Do you have an ans to my Q2 also please - maybe I'm struggling to understand the concept of event? $\endgroup$
    – aman_cc
    Dec 15, 2019 at 6:43
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    $\begingroup$ @aman_cc The path of each ball is a set of events (at this position at this time, at that position at that time...). If the balls collide, that means there exists an event that is in both sets. Then, by the logic of my answer, this holds irrespective of reference frame. $\endgroup$
    – knzhou
    Dec 15, 2019 at 6:51
  • $\begingroup$ yes i guessed that :) I think im struggling with the concept of event and how to precisely define it - for e.g. I appreciate that path of a ball is a collection of events - I'm unable to fully understand how exactly will I describe one particular event of this set. i guess i need to play with the subject a bit more to develop that understanding, thanks for your response $\endgroup$
    – aman_cc
    Dec 15, 2019 at 6:59

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