Galilean spacetime as a fiber bundle

In summary, the book discusses the notion of spacetime affine structure and how it captures Newton's first law. It discusses the idea of identifying free motions (inertial motions) in this structure by a linear equation.
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cianfa72
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TL;DR 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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cianfa72 said:
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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1. What is Galilean spacetime as a fiber bundle?

Galilean spacetime as a fiber bundle is a mathematical concept that combines the principles of Galilean spacetime (a 4-dimensional space-time continuum) and fiber bundles (a mathematical tool used to describe the structure of space). It is used to model the motion of objects in a non-relativistic framework.

2. How is Galilean spacetime represented as a fiber bundle?

Galilean spacetime is represented as a fiber bundle by considering the 4-dimensional space-time continuum as the base space and the set of all possible velocities as the fiber space. This means that for every point in the base space, there is a corresponding set of velocities in the fiber space.

3. What are the advantages of using Galilean spacetime as a fiber bundle?

One advantage of using Galilean spacetime as a fiber bundle is that it allows for a more intuitive understanding of the motion of objects in space. It also provides a mathematical framework for analyzing non-relativistic systems and can be extended to include more complex physical phenomena.

4. How is Galilean spacetime as a fiber bundle different from other spacetime models?

Galilean spacetime as a fiber bundle differs from other spacetime models, such as Minkowski spacetime, in that it does not incorporate the principles of special relativity. This means that it is limited to describing non-relativistic systems and cannot account for phenomena such as time dilation or length contraction.

5. Are there any practical applications of Galilean spacetime as a fiber bundle?

Yes, Galilean spacetime as a fiber bundle has practical applications in fields such as classical mechanics, fluid dynamics, and electromagnetism. It can also be used to model the motion of objects in engineering and physics experiments that do not involve high speeds or strong gravitational fields.

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