Relative simultaneity in Special Relativity: was it ever used in real world examples? I'm working on researches in the area of Special and General Relativity (SR/GR) focused on time dilation and I have a question to pose.
Lorentz transformations (LT) introduce a time transformation such that  two events occurring at same time in one frame not being at same time in another frame (relative simultaneity).
A "special/singular" case for LT is when the clock in the frame at rest is also fixed. In this case the dependency on the position for transforming a delta time is dropped, and the conversion is only based on gamma (i.e. delta t' = gamma * delta t). This is the setup is fully equivalent to other relativity theories, based on absolute synchronization methods, and  more specifically based on an absolute simultaneity. So this aspect is not a SR peculiarity anymore.
Looking at the SR literature for Earth and space time drift analysis real world experiments, all the models used are always based on the above "singular" case, i.e. an absolute-like frame is defined with a clock is at rest there. This allows to correlate delta times in different frames through coordinate vs. proper time comparisons, under certain assumptions for clock synchronization. But again it's not a specific feature of SR with respect to other theories! The same can be obtained with absolute frame & time models, with in some case much more easy and clear steps.
Does anyone know an example where the relative simultaneity is used for modeling a real experiment, i.e. where also the term -v*x/c^2 plays a role in the tranformation?
 A: How about this :

We investigate the speed and lifetime of cosmic-ray muons.  The speed of cosmic-ray muons wasdetermined by measuring time-of-flight between parallel scintillator paddles for various separations

...

Relativistic kinematics was found to give a muchbetter fit, than Newtonian kinematics, between our experimental results and existing data on theenergies and momenta of cosmic-ray muons.

Actually all the analyses of the enormous amount of data in particle physics depend crucially on using relativistic kinematics.
A: I think you're asking for experiments in which there are two different physical reference frames, i.e., two arrays of Einstein-synchronized clocks in relative motion. The muon experiment doesn't count because there are no sets of comoving muons with Einstein-synchronized decay times.
You're right, it's hard to think of such experiments. Normally there's one convenient reference frame (the lab frame) and constructing another is prohibitively difficult.
It doesn't matter because relativity isn't about reference frames, even though they were a big part of Einstein's original paper and they've dominated introductions to it ever since. It's really about spacetime geometry. In Euclidean geometry, you can do some constructions without any coordinates at all. In many other cases it's easier to set up a Cartesian coordinate system. But it's rarely helpful to use two different Cartesian coordinate systems when solving the same problem. You don't need to, since one coordinate system gives you coordinates for everything, and that's enough to do the algebra.
The crucial thing in special relativity is the distance formula $\sqrt{Δt^2-Δx^2-Δy^2-Δz^2}$, which plays the same role as the Pythagorean formula. Lorentz transformations are important only because they preserve that formula. If you use rapidity (spacetime angle) instead of velocity (spacetime slope) in the Lorentz transformation, it looks like a transformation between Cartesian coordinate systems with a common origin. If you rewrite the Cartesian formula in terms of slope, it looks like a Lorentz transformation.
If we did an experiment with a second set of Einstein-synchronized clocks, it could still be analyzed from the perspective of a single reference frame (probably the lab frame). The results would be consistent with that analysis. And the results wouldn't prove anything, because you could still argue that the lab frame is the only true frame and special relativity only appears to be true because of distortions of the moving clocks, etc., just as everyone believed before 1905. The universe doesn't know what a clock is or what Einstein synchronization is, so it would just follow the same rules as always.
