Galilean spacetime as a fiber bundle

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Summary:

Galilean spacetime has been defined as fiber bundle (over absolute time projection).
How to single out physically inertial paths through spacetime
Hi,

reading the book "The Road to Reality" by Roger Penrose I was a bit confused about the notion of Galilean spacetime as fiber bundle (section 17.2).

As explained there, each fiber over absolute time ##t## is a copy of ##\mathbf E^3## (an instance of it over each ##t##), there exist no identification between fibers nevertheless the whole bundle (the spacetime) is actually one "thing".

Now, from a physical point of view, I believe the direction of 'inertial motions' can be singled out by zero reading of accelerometers (inertial paths in spacetime are actually those having zero reading of accelerometers following them).

I'm puzzled about how identify them in each fiber (copy of ##\mathbf E^3##) without reference to a given inertial reference frame IRF
 

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I'm puzzled about how identify them in each fiber (copy of ##\mathbf E^3##) without reference to a given inertial reference frame IRF
Does really make sense defining them in a frame invariant way ?
 
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robphy
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You might be interested in my comment
Thanks for the references.

As far as I can understand, spacetime affine structure basically 'embodies' Newton first law. As explained there 'rectilinear coordinate systems' capture the spacetime affine structure and the particular 'Inertial' class exhibits a time coordinate (coordinate value for ##e_4## base vector) between events numerically the same as the difference in 'absolute time' ##t##.

If we restrict ourselves to just 'rectilinear coordinate systems' I believe that free motions (inertial motions) are described anyhow by a linear equation (in the chosen associated affine base -- repère), don't you ?
 
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