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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?

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  • $\begingroup$ It's not "observation error". Events that are simultaneous in at least one frame don't have to be simultaneous for other observers, but the deeper significance is that if events are simultaneous for some observer, then the events cannot be casually related (one cannot cause the other to happen). $\endgroup$ – Xeren Narcy May 28 '15 at 6:57
  • $\begingroup$ I think you're right about this question being useful. I can see why the answer at physics.stackexchange.com/questions/181269/… doesn't answer you question. I just recently rewrote my old answer to this question entirely now that I believe I have a better sense of what's a good answer than I had then explaining why we say simultaneity is relative and not just explaining how you're deriving different conclusions because you're assuming simultaneity is absolute. I think that as somebody who had that confusion, if you read it, you'll see why that $\endgroup$ – Timothy Oct 25 '19 at 20:54
  • $\begingroup$ answer doesn't explain it properly. I also just deleted my answer to that question because I now see it as not being a useful answer. $\endgroup$ – Timothy Oct 25 '19 at 20:54
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It boils down to how you define simultaneous. You can define simultaneous as "events that are guaranteed, by the nature of an experimental setup (further assumed to be infallible), to occur simultaneously in the frame of reference in which that experiment is conducted", then it will indeed be the case that the lamps will light up "simultaneously" in all reference frames. But you haven't meaningfully contradicted relatively because that's not the definition of simultaneity in relativity. In relativity, an observer calculates, at the time at which light from an event reaches the observer, how long ago the event occurred by dividing the distance to its source by the speed of light. Two events are considered simultaneous if they get the same time calculation by this method. Every observer is assumed to have a chronometer and exact knowledge of the distance to the source of the event.

http://upload.wikimedia.org/wikipedia/commons/9/96/Einstein_train_relativity_of_simultaneity.png

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  • $\begingroup$ If this is so, then in Einstein's experiment, when the observer on the train had moved closer to A than to B, he should have calculated his distance from A at the instant when he received light from A and his distance from B when he received light from B, and allowing for the time taken for the light to travel to him from A and B, he would have for sure determined that the events really were simultaneous in the train's frame of reference (or in all frames of reference moving relative to each other). But he did not do that, so he saw the events as non-simultaneous. That just proves my point. $\endgroup$ – Khushro Shahookar May 28 '15 at 10:11
  • $\begingroup$ If you do the math on it, you will see that all observers moving at the same speed as the train will agree that the lamps lit at the same time, while all observers having non-zero velocity relative to the train will not. $\endgroup$ – Atsby May 28 '15 at 10:18
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I will use the example of a spaceship going at half the speed of light because it's not subject to the centrifugal force of going around Earth or the gravitational time dilation of Earth. In this case, general relativity simplifies to special relativity. Suppose you're in the centre of a cylindrical space ship whose inside is all a single empty room, going at half the speed of light in outer space and you can't see outside the spaceship. Suppose you also have a gun that shoots bullets in opposite directions at 0.5c when it's stationary. Now take that gun into the spaceship and shoot one bullet the same direction as you're going and the other bullet in the opposite direction. According to special relativity, the bullet shot in the opposite direction will be stationary and the one shot in the same direction will travel at 0.8c. Now you and the spaceship are subject to length contraction by a factor of $\sqrt{\frac{4}{3}}$ and to time dilation by a factor of $\sqrt{\frac{4}{3}}$. Also although the bullet you shot in the same direction reaches the frond end of the spaceship later than the bullet you shot in the opposite direction, the light generated by the event of the bullet hitting the back end also takes longer to get back to you than the light generated by the event of the bullet hitting the front end. The math shows that you observe it exactly the same way as you would if the spaceship weren't moving at all.

How about if you try creating a stronger gun that shoots bullets in opposite directions at 48c when you're stationary. Surely, both events of a bullet hitting an end of the ship will occur later than the event of shooting them but by only a tiny bit. If that is what happens, from the information about the timing of your observations and the fact that the both of the events of bullet hitting an end of the spaceship must occur later than the event of firing them, you can then determine that you are indeed moving. It turns out that according to special relativity, no matter can ever reach the speed of light in the first place. That's because as an object gets closer to the speed of light, its time dilation and length contraction factors become larger preventing an object from ever being accelerated all the way to the speed of light.

We define simultaneity to be relative because without access to information about the velocity of the spaceship, you have no way to determine which of two events you observed occurred later when their difference in time is less than their difference in position divided by c.

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