Two Lorentz Boosts Causing Rotation - Measurable w/ Gyroscope?

[Ookke]

It seems that combining two Lorentz boosts cause rotation:

https://en.wikipedia.org/wiki/Lorentz_transformation#Composition_of_two_boosts

Do you think this rotation is something that could be measured by gyroscope? Or is it like space rotates around the accelerating observer, and the observer has no means to measure this rotation locally?

If I get this right, we could accelerate two observers differently so that they finally have the same velocity vector (and common rest frame), and their final orientation is not the same, although the initial orientation is. So the accelerations have somehow created a difference in orientation, but can we determine, which one of the observers has performed this rotation, or how this should be interpreted? Thanks.

 

[WannabeNewton]

Do you think this rotation is something that could be measured by gyroscope?

Yes: https://en.wikipedia.org/wiki/Thomas_precession

So the accelerations have somehow created a difference in orientation, but can we determine, which one of the observers has performed this rotation, or how this should be interpreted?

Yes: just have each carry a gyroscope and see which one precesses relative to the distant stars, as well as by what angle, when they reconvene.

 

[Ookke]

Ok, nice. But if no single boost causes any rotation, why would two boosts in a row do that? It's like space remembers that we already had boost in one direction, and when we have another, this mysterious Thomas rotation takes place.

Maybe I'm thinking this rotation too literally. Assuming that we have sensitive enought test equipment, should we detect us turning in a rocket that first uses its rear engine, then its side engine? At which point we should expect to detect this turning, if at all?