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I read that if two events happen at the same time to some referential, they necessarily do not happen at the same time to a referential with relative velocity to the first.

I disagreed with that thinking about this situation: a woman in the middle of a platform moving away from a man on earth (as the image shows, following the perpendicular bisector of AB) . Two bombs explode on the extremities of the platform. The man sees the flashes at the same time, so the explosions were similtaneous to him, undoubtedly. To me, the explosions were also simultaneous to the woman. I am trying to equationate, but people are telling me that sentence I read, and I don't know why what I am thinking is not right.

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  • $\begingroup$ perhaps you could try to read the example of the train, and ask specifically what part you do not understand en.wikipedia.org/wiki/Relativity_of_simultaneity $\endgroup$
    – user65081
    Commented Nov 8, 2018 at 14:53
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    $\begingroup$ How does the distance to the marks have anything to do with when they were produced? Also: The marks moving in her frame, so that kinds confuses using the marks. $\endgroup$
    – JEB
    Commented Nov 8, 2018 at 15:02
  • $\begingroup$ The marks are to say that, as the image shows, if the distances are the same to him and the speed of light is constant, the flashes happen at the same time to him, undoubtedly. $\endgroup$ Commented Nov 8, 2018 at 15:09
  • $\begingroup$ My thinking is different from the Wikipedia's example of the train because on that example, the train is moving horizonally and the explosions occur on different x positions. On my example, the moving is vertical and the explosions occur on the same y position. $\endgroup$ Commented Nov 8, 2018 at 15:12
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    $\begingroup$ The claim, "If two events happen at the same time to some referential, they necessarily do not happen at the same time to a referential with relative velocity to the first," is wrong. It should say, "If two events happen at the same time to some referential, they do not necessarily happen at the same time to a referential with relative velocity to the first." The distinction is subtle but important, because disagreement on simultaneity depends on the boost direction. In the case given, the boost is normal to the platform, so both observes will say the two events happened simultaneously. $\endgroup$
    – JM1
    Commented Nov 8, 2018 at 15:32

2 Answers 2

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You're absolutely right. If the woman observes the bombs going off simultaneously, so, then, will the man.

Your mistake is the statement that "if two events happen at the same time to some referential, they necessarily do not happen at the same time to a referential with relative velocity to the first." This statement is true in one (spatial) dimension, but fails to be true in higher dimensions. The statement should read "if two events happen at the same time to some referential, they do not necessarily happen at the same time to a referential with relative velocity to the first."

The precise reason for why this example fails is that the velocity of relative motion is perpendicular to the displacement between the two bombs. If you offset the velocity of the woman's platform to be anywhere off that perpendicular, the man will not see the explosions as simultaneous.

A great way to see this is to "collapse" the problem into a one-dimensional one, by projecting the entire poblem onto the woman's axis of motion. In this new problem, the bombs are at the same point in space and explode at the same time. That is, in the 1D version, they are the same event. However, if the woman's velocity is off the perpendicular, the bombs are projected onto different points, and the logic of 1D simultaneity applies once more.

I hope this helps! (I'd draw diagrams, but I'm typing this on my phone).

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For the specific configuration that you've done, the events will be simultaneous in both frames. This is because you've chosen a relative velocity which is orthogonal to the separation between the two events.

However, the relativity of simultaneity is very much a thing, and if the relative velocity between the two frames had a nonzero horizontal component along the separation between the two events, then the two observers would disagree on the simultaneity of the two events.

The specific formulation you've used, though,

if two events happen at the same time to some referential, they necessarily do not happen at the same time to a referential with relative velocity to the first.

is incorrect. It's perfectly possible for events to be simultaneous in separate frames of reference, so long as their separation vector is orthogonal to the relative velocity. If you want to see this in a more concrete form, the tool to use is the vector formulation of the Lorentz transformations,

\begin{aligned}t'&=\gamma \left(t-{\frac {v\mathbf {n} \cdot \mathbf {r} }{c^{2}}}\right)\,,\\\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} -\gamma tv\mathbf {n} \,.\end{aligned}

where it's clear that an event happening at the spacetime origin and an event happening at position $\mathbf r$ at time $t=0$ will be simultaneous in the primed frame (i.e. $t'=0$) if and only if $\mathbf r$ is orthogonal to the relative velocity vector $v\mathbf n$.

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