I think this is a very important question because if simultaneity is absolute, then it shakes the foundation of relativity. But it was asked here a month ago, and no one answered it. Neither does anyone say that yes, they believe simultaneity is absolute, nor does any expert give any convincing proof that it is relative. I know I am not supposed to repeat questions, but I have no other forum to turn to, so I am presenting this as a challenge to all the experts on this forum to try and find an answer. If there is no answer here, can anyone suggest where and how I can get an answer?
(a) Generation of simultaneous events: Consider the situation of an embankment and train, moving at a uniform relativistic velocity. Consider two locations on the embankment, A and B, with mid-point X. Place a battery and single switch at X, and connect it by pairs of wires to lamps at A and B. If the pairs of wires XA and XB, and the lamps at A and B are of identical construction and are activated by a single switch, then we can conclude that both lamps will produce a flash of light simultaneously when the switch is pressed momentarily. That means the events are generated simultaneously (or “are” simultaneous), regardless of how any observer observes them.
The observer on the train will see the events as being generated simultaneously because he will see shorter wires XA and XB, and his clock will be running slower than the clock on the embankment, but still he will see both pairs of wires and both lamps of identical construction, so he will have to conclude that both flashes of light A and B must have been generated simultaneously. There would be no logical reason for one to be generated earlier than or later than the other, as far as the observer in the other frame can observe. So the observers of all inertial frames of reference moving at relativistic speeds must conclude that the events were generated simultaneously and were absolutely simultaneous. If we were to mount the above apparatus on a train, then the observer on the embankment would reach the same conclusion. So we can conclude that the events are generated simultaneously, or are fundamentally simultaneous for all frames of reference regardless of how an observer observes them. Now if a particular observer observes them as non-simultaneous, then that can only be seen as an observation error. If I am measuring everything using a half-metre rod labelled as a 1-metre rod, I do not conclude that everything in the world has suddenly expanded to twice the size. I conclude that everything is still the same size, and it is my observation error that I see everything as twice the size when I measure it with my half-metre rod labelled as a 1-metre rod.
(b) Observation of simultaneous events in the same frame of reference: First consider just one frame of reference and use Einstein’s method of observation (Einstein’s relativity book, chapter Relativity of Simultaneity). The events are observed as simultaneous if an observer at the mid-point X observers the flashes of light at the same time. Of course, observers at other locations closer to A or B in the same frame of reference as the lamps will observe the events as non-simultaneous, because light from the events will take a different amount of time to reach them. That does not make the events non-simultaneous, it is just an observation error due to the locations of different observers. Also, relativity theory does not say that simultaneity is relative for different observers in the same frame of reference as the events. But only observers at equal distance from A and B are able to make the correct observation of simultaneous events, without any observation error.
(c) Observation of simultaneous events in different frames of reference: From the previous sentence we note that it is important that the observer be at the mid-point X at the instant when he makes the observation, ie at the instant when light from both A and B reaches him. It does not matter if he is not at X but somewhere else at any other time, just as long as he is at X at the time the flashes of light reach X. Now consider the lamps at A and B on the embankment and an observer on the train. The mistake that Einstein made was that his observer on the train was at X at the moment when the flashes of light were produced, and subsequently he moved closer to A and further away from B when the flashes of light reached him, so of course he observed the flashes of light as non-simultaneous – observation error. If we station several observers throughout the length of the train, we find that one of the observers on the train will be at X (i.e. at equal distance from A and B) at the instant when the flashes of light from both A and B reach him, and that observer will see both events as simultaneous, even though he is on the moving train, because light from both A and B will reach him at X simultaneously. That observer will be the one able to make the correct observation that the events A and B in the embankment’s frame of reference are also simultaneous in the train’s frame of reference, or that the events are simultaneous in all frames of reference moving at uniform relativistic speeds.
In the same frame of reference as the events, there is one observer who sees the events as simultaneous and another observer who sees them as non-simultaneous although this can only be an observation error. The events themselves cannot be both simultaneous and non-simultaneous in the same frame of reference. In another frame of reference moving relative to the frame of the events, one of the observers who happens to be at the correct location observes the events as simultaneous, while other observers have the observation error. Then how can anyone say that simultaneity is relative?