# What happens in a simultaneous event with a change of observer from another observer?

Okay, so I'm considering Einstein's train thing (at least... I hope I am... and if not this is my question anyway). We have an observer on the train (A), an observer off the train (B), and two trees (E1 and E2). The E1 and E2 are, according to B, struck simultaneously, but according to A E2 is hit first, then E1. Now, in addition to being able to observe E1 and E2, A and B can observe each other as well. So suppose after B detects E2, B sneezes (I know, this doesn't work for actual things going at light speed since the distances involved are too short). My question, then, is when does A record B sneezing with respect to E1 and E2? Is it simultaneous with E1 and E2 since, for B, the event was in between, and there is no 'in between' from A's perspective?

You are confusing - it's quite a common confusion - E1 and E2 with the observers' detection of E1 and E2.

A will perceive that B detected E1 and E2 simultaneously (it's OK to use the word as these two detections happened at the same place) and then sneezed. But A knows that E1 and E2 happened some time back and will do the sums to calculate how long the light signals took to get to B and - because one tree was travelling towards them and the other tree travelling away - will get two different numbers, in contrast to B who is midway between two stationary trees and says the time lapses must have been the same.

Observers agree - even in relativity - about what they actually observe. They must. They differ when they argue back from their primary observations to reconstruct remote events.

Anyway: the sneeze happened after B detected E1 and E2, and is therefore after E1 and after E2 in all reference frames.

I like to make a time ordered table of events for the reference frames:

B:

1. $$E_1$$ and $$E_2$$ occur.
2. A detects $$E_2$$.
3. B detects $$E_1$$ and $$E_2$$.
4. B sneezes.
5. A detects $$E_1$$.

A:

1. $$E_2$$ occurs.
2. $$E_1$$ occurs.
3. A detects $$E_2$$.
4. B detects $$E_1$$ and $$E_2$$.
5. B sneezes.
6. A detects $$E_1$$.

When considering observers in this kind of experiment, remember they are allowed to have a lattice of synchronized clocks and rulers with which they can reconstruct conditions at various moments (aka: now), at all locations (aka: everywhere). In other words, they can split spacetime into their own versions of space and time, after the fact, with complete knowledge.

From this, one can then reconstruct what a single observer "sees" based on signals propagating at $$c$$.

This is part of the power of the Gedankenexperiment.

Meanwhile, when considering real experiments, we tend to reverse the order: what does an observer at a single location detect, and how do we reconstruct reality from that?

Failure to distinguish the two cases can lead to confusion.