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Split from https://www.physicsforums.com/threads/question-about-relativity-of-simultaneity.847761/
Alonso has in fact raised a relevant issue. Given any clock synchronization in the embankment frame, there is a unique pair of points at the embankment with coordinates x=a and x=b, and a unique time t=T, such that at t=T in the embankment frame, these points are next to the back end and the front end of the train, respectively, and x=0 in the embankment frame is next to the midpoint of the train.
In this case, it is assumed that a=-b, that is, that the midpoint between the two points at t is next to the midpoint of the train. But in fact, this cannot be proved from the two postulates alone. We need also use that the Lorentz transformation is linear to prove this (more precisely, affine, if the origins (0,0) are not mapped to each other), and this linearity cannot be proved from the postulates alone. I think that sweet springs with "homogeneity" and FactChecker with "uniformity" mean the same thing.
To derive the linearity from more fundamental or intuitively obvious principles is a very tricky issue. See https://www.physicsforums.com/threa...formations-are-the-only-ones-possible.651640/
In this case, it is assumed that a=-b, that is, that the midpoint between the two points at t is next to the midpoint of the train. But in fact, this cannot be proved from the two postulates alone. We need also use that the Lorentz transformation is linear to prove this (more precisely, affine, if the origins (0,0) are not mapped to each other), and this linearity cannot be proved from the postulates alone. I think that sweet springs with "homogeneity" and FactChecker with "uniformity" mean the same thing.
To derive the linearity from more fundamental or intuitively obvious principles is a very tricky issue. See https://www.physicsforums.com/threa...formations-are-the-only-ones-possible.651640/