# Invariance of coincidence

I am reading the book "Space and Time in Special Relativity" by David Mermin. In chapter 13, at page 128 in my print, he says the following (screenshot): I'm referring specifically to the sentence "Note that this is the principle of the invariance of coincidence again". I had to reread it to see that he meant spacetime coincidence, not just time coincidence. Mathematically, that makes sense; a single point with a single spacetime coördinate will always have just 1 coördinate, no matter what coördinate system is used.

But why is this a principle at all? How can two events occupy the same spacetime coördinate?

I noticed someone asked a similar question here: Coincidence of spacetime events & Lorentz invariance

And one user answered that in this case, coincidence means identical. However, it seems to be used as an argument to say that the principle has no meaning in reality.

Are there any examples of differrent events that occur in the same spacetime coördinate?

• This is the usual problem with defining the word "event". Sometimes it means "a thing that actually happens" and sometimes it means "a point in spacetime". And a lot of books just freely switch back and forth. Dec 27 '18 at 13:21
• That seems strange, I see no need to refer to a "point" as an "event" - Nothing adds to the clarity of its meaning. Why would it be preferable to use the word "event" for a point? edit - I just read your link. This explains a lot. Seems like the mathematical physicists did some terminology mumbo-jumbo. Dec 27 '18 at 15:30
• I'm sorry, this is completely blowing my mind. I just thought a bit more about what happens by calling it "coincidence" and "events" and a "principle of invariance" - this gives a mathematical generalization for a thing that does not exist at all! How did this get into any physics book? Dec 27 '18 at 15:39

Examples of “different events that occur at the same spacetime coordinate”:

When [and where] “I (my worldline) sent a light signal”, “my wristwatch read 2 seconds”.

When [and where] “I (my worldline) received the light signal [for example, its echo]”, “my wristwatch read 8 seconds”.

Such examples are used in radar measurements.

Note: “different events that occur at the same spacetime coordinate” means
different physical situations that mark the same “mathematical point” in the Spacetime manifold.

• I think I understand what you're saying, but just to clarify; do you mean that the spacetime coördinate from where a lightsignal departs can be the same as the coördinate where a different light signal arrives? That would make a lot of sense already, actually. Dec 27 '18 at 14:14
• Yes. In fact, given an event E, its past light cone can be thought of as the directions of “different light signals arriving at E” and its future light cone as the directions of “different light signals emitted at E”. Dec 27 '18 at 14:21
• Would it be correct to assume that only 2 events can ever really happen at the same coördinate in spacetime (one from the past, one going out to the future)? Dec 27 '18 at 14:23
• There may be numerous “physical situations” that mark the same “mathematical point” in the Spacetime manifold. In the earlier example, my wristwatch can read 8 seconds when I receive a light signal and immediately re-emit. It may be that I met another observer’s worldline at this event and that observer’s wristwatch read 7 seconds. Etc... Dec 27 '18 at 15:05
• How does one meet another person's worldline? That would mean that we have to occupy the same spacetime - Is this not forbidden by Pauli's exclusion principle? Dec 27 '18 at 15:27