 Quote by strangerep
But how do you know that a priori if you're starting from the relativity principle alone, and trying to derive the relativity group(s)? You can't appeal to geometric intuitions from Minkowski spacetime since the latter is really only an afterthought -- a homogeneous space constructed from a given relativity group.
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I think I explained how I know that, but feel free to ask about the details if I need to clarify something. Note that I didn't use the Minkowski metric. I just used what I know about linear transformations. There isn't a whole lot of things that a linear transformation can do to a simultaneity hyperplane. It can preserve it, or it can tilt it. If a transformation tilts a simultaneity hyperplane, that always favors a direction in space: the direction of the tilt.
A linear transformation can also stretch or rotate a simultaneity hyperplane, but the preservation of simultaneity depends only on whether the hyperplane gets tilted or not.
Quote by strangerep
(BTW, your term "zero-velocity transformation" is a bit misleading. I think "velocity-preserving transformation" is clearer.)
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I define the velocity of a transformation ##\Lambda## as the vector with components ##(\Lambda^{-1})_{i0}/(\Lambda^{-1})_{00}##. I think this terminology is appropriate. The velocity of the transformation is the velocity of the second observer in the coordinates of the first.
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