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cianfa72
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- TL;DR Summary
- About the Fiber bundle structure of Newton/Galilean spacetime
Hi, Penrose in his book "The Road to Reality" claims that Newton/Galilean spacetime has actually a structure of fiber bundle. The base is one-dimensional Euclidean space (time) and each fiber is a copy of ##\mathbb E^3##. The projection on the base space is the "universal time mapping" that assign at each event its universal time.
He claims that such fiber bundle has an affine structure (i.e. the vector space ##\mathbb R^4## acts on it to define affine transformations).
Then I looked at Arnold's Mathematical Methods of Classic Mechanics chapter 1. He defines the galilean group as the group of affine transformations preserving the galilean structure. As I understand it, that means that the set of all affine transformations on fiber bundle is actually restricted to those affine transformations such that
He claims that such fiber bundle has an affine structure (i.e. the vector space ##\mathbb R^4## acts on it to define affine transformations).
Then I looked at Arnold's Mathematical Methods of Classic Mechanics chapter 1. He defines the galilean group as the group of affine transformations preserving the galilean structure. As I understand it, that means that the set of all affine transformations on fiber bundle is actually restricted to those affine transformations such that
- send a pair of simultaneous events to events with the same universal time ##t## preserving their euclidean distance in the ##\mathbb E^3## copy/fiber (i.e. uniform motion, translation and rotation inside the same ##\mathbb E^3## copy/fiber)
- send simultaneous events with universal time ##t## in simultaneous events with the same different universal time (say ##t+t_0##) adding a constant vector displacement from ##\mathbb R^4##
- composition of the above transformations