Relativity of Simultaneity

Question

Relativity of Simultaneity seems to be about OBSERVING two events simultaneously (please correct me if I am wrong).

However, as long as the two events are separated by a distance (any distance) then two observers (in the same frame) cannot agree that they happen simultaneously unless they are equidistant from the two events within the frame.

Consider the example of A and B at rest with respect to another observer C. A and B are stationary in space but separated by 1 light second of distance.

An event occurs simultaneously at A and B, the time should be the same since they are at rest with respect to each other. However, A will OBSERVE event at A and exactly after 1 second, the event at B. B will OBSERVE the event at B and exactly after 1 second, the event at A.

C may observe A first, B first or A and B simultaneously (which is fine).

My questions are:

1. Are A and B in the same inertial frame? And if so, if simultaneity is about OBSERVING an event, then how do we account for the different observation times within the same frame? An event can never be simultaneous for A and B unless they are at the same location.
2. If A and B are not in the same frame, then they are not in inertial frames as there is no motion with respect to each other.
3. If simultaneity is not about OBSERVING two events, then how do we separate the visual aspect (time taken for light to travel) from the temporal aspects (time at which the two events actually occurred.

Answer 1

This is actually a rather subtle point never fully appreciated by students the first time they learn relativity.

There is a difference between someone at A receiving the photons from event X, and event X actually occurring. In physics classes (as distinct from astronomy), we are almost always talking about the occurrence of the event, not its observation. The idea is you set up your canonical grid of rulers and clocks throughout space. Events happen. Later on, at your own convenience, you go and visit the clocks (or have them send you data) and assemble the whole picture. This God's-eye perspective is what we usually have, where we witness "simultaneous" events simultaneously only because we are post-processing the data.

To emphasize: events are single points in spacetime, and they do not change place or time. Event X may have occurred at B's location - fine - but then don't confuse X with Y = (the observation of X by A), which is another event entirely, occurring at a different place (A rather than B) at a different time (1 light-second later than X, in the frame of A/B).

So:

1. Yes they are in the same frame. Both will agree, in post-processing, that the simultaneous event happened at the same time. A will see the flash at A first, and ditto for B, but there won't be any confusion after the fact when they get together and discuss what their grids of clocks said.

2. See 1.

3. Carefully. If you ever take your physics "rocketships and grids of rulers" training and go into astronomy, you'll learn to appreciate the fact that we are looking out along our past light cone, rather than "horizontally" in a spacetime diagram. We don't have an infinite grid of clocks spread throughout the universe to query, but rather just our telescopes right here. You already know that the further away an event, the longer the elapsed time between it and its observation. But the key is to internalize that, and get used to adding and subtracting corrections to account for it.

Answer 2

My favorite quote related to the problem:

"Now it may seem that we have been paying too much deference to the astronomers: 'After all, they did not discover time. Time is something of which we are immediately conscious.' I venture to differ and to suggest that (subject to certain reservations) time as now understood was discovered by an astronomer - Römer. By our sense of vision it appears to us that we are present at events far distant from us, so that they seem to occur in instants of which we are immediately conscious. Römer's discovery of the finite velocity of light has forced us to abandon that view; we still like to think of world-wide instants, but the location of distant events among them is a matter of hypothetical calculation, not of perception. Since Römer, time has become a mathematical construction devised to give the least disturbance to the old illusion that the instants in our consciousness are world-wide."

Arthur S. Eddington, The Relativity of Time (Nature 106:802, 1921)