- #36
Saw
Gold Member
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Thanks, for me you are answering the question and would only continue the thread if robphy has more to say.vanhees71 said:The null vector never is an eigenvector of any matrix. Because triviallyfor any matrix .
As I said, for me these pseudo-4D extension is just a convenient way to realize all Galilei transformations (including Galilei boosts) by-matrix multiplications, but that's all there is to it. It doesn't add any additional structure to the description of Galilean spacetime than is already there when interpreting it as a fiber bundle, which is the natural choice for the Galilei-Newton spacetime.
Concerning the Galilei boost, let's to for 1D motion. Then the "vector" is
and the boost is described by
This "Jordan matrix" has only one eigenvectorwith eigenvalue , which simply tells you that at time all the points on the axis are unchanged by the transformation, which is, however trivial anyway.
Just two comments:
- The convenient way to realize the matrix multiplication... does it have to do with "homogeneous coordinates", in the vein of what you can do with a translation? (See my post #12.)
- You may want to put the missing negative sign by the v term in the transformation matrix.