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I know that thought experiment about trains when a flash of light in the middle reaches the both end simultaneously for a passenger but different times for the bystander.

So were there (non-thought) experiments that directly checked this?

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    $\begingroup$ If you have time dialation, then you're going to have to lose simultaneity. Time dilation is very very well tested. $\endgroup$ Commented May 31, 2013 at 19:21

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I think the answer to this depends a lot on your definition of "directly." Relativity of simultaneity is built into the Lorentz transformation, and Lorentz invariance is one of the most precisely tested physical theories in all of history. Essentially you're asking for an experiment that verifies one element of the matrix involved in the Lorentz transformation, but every element of the matrix is present in all cases. I would consider the Sagnac effect to be a fairly direct test, and the Sagnac effect was one of the effects observed in the Hafele-Keating experiment, as well as many other, earlier tests of relativity. Every time you fly on a commercial jet, you're benefiting from a ring laser gyro, which works based on the Sagnac effect.

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The most direct experiment confirming this is the Kennedy-Thorndike experiment. It used two different length arms so modifying the Michelson-Morley experiment that had shown that no local "aether" explained observations calling for an explanation that Einstein provided. P.S. If you want seek more technical explanation of the methods, it is readily found on the Web using those names.

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[...] that thought experiment about trains when a flash of light in the middle reaches the both end[s] simultaneously for a passenger but different times for the bystander.

That's not really a great/correct description of (a variant of) the well-known thought experiment, closely related to Einstein's (final and enduring) definition of how to measure "simultaneity".

Instead: if some (third) passenger in the middle between two given passengers on a train (e.g. the two ends of the train) stated a signal (flash of light) then the corresponding indications of the two given passengers/ends receiving this signal are simultaneous; and if some railroad tie in the middle between two given bystanders stated a signal (flash of light) then the corresponding indications of the two given bystanders receiving this signal are simultaneous as well.

It is important to understand that the railroad tie is not identified as middle between the two given passengers, and that the mentioned third passenger is not identified as middle between the two given bystanders; not even "at the moment they pass each other". The railroad tie always moves relative to the ("moving") train with all its passengers; and all passengers (in particular the third one, in the middle between the given two passengers) always move relative to the bystanders.

So were there (non-thought) experiments that directly checked this?

There are no meaningful experiments without underlying thought experiments. There cannot be any measurement values obtained from observational data without first deciding on (and then sticking to) some particular method of measurement. To paraphrase Niels Bohr: We must be able to tell our friends what we have done (or at least intended to do), so they'll also listen to what we have found.

And of course there is no experimental check of definitions (such as Einstein's definition mentioned above), or of terminology, or of "rules to play by"; there are ("only") experimental checks of hypotheses expressed in terms of definitions and terminology and "rules" (which one has decided to use, and which ought to be stuck to), such as:
whether bystander Alice took a sigh simultaneously to bystander Bob picking his nose, or not; or
whether the (third) passenger's name in the middle between the two given passengers Reese and Dewey was named "Malcolm", or not; or
whether all passengers moved equally (at the same velocity) relative to the bystanders, or not; etc.

In short: asking whether experiments "checked this" is absurd without thought-experimental definition of what "this" is supposed to be.

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  • $\begingroup$ None of this seems to address the question. $\endgroup$
    – user4552
    Commented May 31, 2013 at 19:16
  • $\begingroup$ My submitted answer (or rather reply) addresses the statement and related question of the OP: it is ill-posed. (I'll amend my answer to express this still more explicitly.) $\endgroup$
    – user12262
    Commented May 31, 2013 at 19:20
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It is logically wrong to state that the experiments mentioned in the other answers prove the relativity of simultaneity, since there are alternative theories to special relativity in agreement with all experiments to date in which absolute simultaneity is preserved. In other words no experiment has allowed to observe the relativity of simultaneity.

See the Mansouri and Sexl paper "A Test Theory of Special Relativity", I'm quoting the abstract:

An ether theory is constructed that maintains absolute simultaneity and is kinematically equivalent to special relativity

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    $\begingroup$ After some more thought I'm coming back to this. What the Mansouri/Sexl paper shows, among other things, is that experiments cannot discriminate between special relativity (which implies relativity of simultaneity) and a particular ether theory (which implies absolute simultaneity). Experiments have not invalidated relativity of simultaneity, just as they have not invalidated absolute simultaneity, either can be preserved depending on the assumptions we start from. Relativity of simultaneity cannot be directly observed for the very reason that there is no instantaneous travel of [...] $\endgroup$
    – user44558
    Commented Feb 7, 2015 at 15:41
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    $\begingroup$ [...] information that could allow us to determine unambiguously when the flash of light reaches both end of the train. Upon reaching the ends of the train both flashes are reflected towards our eyes (whether we are inside or outside) and only then do we determine when the flashes reached the ends of the train by applying assumptions. With relativity's assumptions, light travels at c relative to the passenger and to the bystander, so the passenger infers that light reaches both ends at the same time while the bystander infers that light reached the back of the train first, leading to [...] $\endgroup$
    – user44558
    Commented Feb 7, 2015 at 15:42
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    $\begingroup$ [...] relativity of simultaneity. With the assumption that the bystander is in a preferred frame in which light travels at c in all directions and in which moving objects in that frame are length contracted and time dilated, light travels at c-v towards the front of the train and c+v towards the back of the train, so both passenger and bystander infer that the flash has reached the back of the train first, preserving absolute simultaneity. This latter assumption is consistent with all tests of special relativity. Relativity of simultaneity is the result of a particular convention of [...] $\endgroup$
    – user44558
    Commented Feb 7, 2015 at 15:43
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    $\begingroup$ [...] clock synchronization, another convention (as described in the Mansouri/Sexl paper) leads to absolute simultaneity, and I'm sure you will agree that conventions do not change the way the universe behaves, they only change our interpretations. $\endgroup$
    – user44558
    Commented Feb 7, 2015 at 15:43
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    $\begingroup$ So relativity of simultaneity has not been directly observed, "directly observed" in the sense that the passenger sees the flash of light reach the back of the train at the same time as the front while the bystander sees it reach the back first, because neither passenger nor bystander knows when the flash of light reaches the ends of the train, and because the flash reaching the back of the train first for both passenger and bystander can remain consistent with all experimental tests of special relativity. $\endgroup$
    – user44558
    Commented Feb 7, 2015 at 16:06

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