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I'm trying to clarify the definition of simultaneity and/or relative simultaneity - for now, only with respect to two different reference frames that are not moving relative to each other.

There should be no confusion, when we say two events are simultaneous such as lightening striking two trees simultaneously according to an observer/machine midway between them that can measure such events to the required accuracy.

But if there is another observer closer to one tree than the other such a machine/observer would detect the nearer strike sooner but do we still say the strikes are simultaneous to that observer because that observer/machine knows that he is closer to one tree than the other and would have to compensate? This really is a question about definition: If we have to take into account the speed of light and distance to the observer for simultaneity when NO ONE is moving, if moving, yes, it all changes.

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The lighting strikes are simultaneous in your scenario. The observer may not observe the flash hitting their eyeballs/detectors at the same time, but when they work out conceptually when the lightning flashes had to have occurred in order to cause the observed results, they come out to the same result: the lighting strikes occurred at the same moment in time.

In fact, to be extremely formal, these observers are actually in the same frame, but they are measuring with different coordinate systems. The vectors describing space, time, and events are the same vectors for both observers. They may assign them different numeric values. One may assign a vector for the tree's position as <0, 10, 0> and the other might assign it as <5, 23, -9>, but the vectors describing the physics of the problem are the same.

Were one of the observers to be in relative motion to the other, some of the vectors would actually be different, not just measured differently. This is where we could arive at the result that the two lightnning strikes were simultanious in one frame, and not simultanious in another.

This, by the way, is a tremendously frustrating subject, even when we're not accounting for relativity. Very few people properly handle the difference between frames/vectors and coordinate-systems/coordinate-vectors. More than one prototype vehicle has been flung unceremoniously to the ground by mistakes like these, So don't be too bothered if it takes a while to get it.

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    $\begingroup$ Thanks but now I'm curious what are you referring to with "More than one prototype vehicle has been flung unceremoniously to the ground by mistakes like these" $\endgroup$ Oct 5, 2020 at 1:58
  • $\begingroup$ @ParityViolator well, there was the Mars Climate Orbiter that was lost due to one component reporting impulse in pound-force seconds and another component expecting it in newton seconds, leading it to place itself in too low of an orbit, where it likely hit the atmosphere and was lost, either " destroyed in the atmosphere or re-entered heliocentric space after leaving Mars’ atmosphere" $\endgroup$ Oct 5, 2020 at 7:18
  • $\begingroup$ There might be something else that is being referred to here, but that's one of the more high-profile ones that I've heard of. Definitely a lesson in "never assume" especially for coordinate systems and unit/measurement systems. $\endgroup$ Oct 5, 2020 at 7:19
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It's simultaneous for both of them. The one closer to the tree will just say that he's closer to the tree and that's why he saw light from one earlier. Observer's don't say events are simultaneous if they see light from two events reaching them at the same time. They say it's simultaneous if they take the distance and speed of light into account to measure at exactly what time light was emitted from the point in question and if the calculated times match.

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