Newton Galilean spacetime as fiber bundle

In summary, "Newton Galilean spacetime as fiber bundle" explores the mathematical framework of representing Newtonian mechanics within the context of fiber bundles. It examines how the spatial and temporal dimensions can be modeled as a fiber bundle structure, emphasizing the relationship between points in space and time. This approach allows for a more coherent understanding of the underlying geometric properties of Newtonian physics, facilitating the integration of classical mechanics with modern geometric concepts. The study highlights the implications of this representation for concepts such as simultaneity and the uniformity of space, contributing to a deeper understanding of Newtonian spacetime.
  • #1
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
  • 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
Penrose claims then that the Galilean structure of fiber bundle can be assigned through an affine connection on it. Can you help me clarify this point ? Thanks.
 
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  • #4
haushofer said:
A reference without any talk about fibre bundles is my PhD thesis
Looking at your paper it seems to me that the general Galilean group transformation (2.1) is not actually an affine transformation.
 
  • #5
cianfa72 said:
Looking at your paper it seems to me that the general Galilean group transformation (2.1) is not actually an affine transformation.
Why not? You're referring specifically to the boosts?
 
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  • #6
haushofer said:
Why not? You're referring specifically to the boosts?
Sorry, you are right. All kind of Galilean transformations are actually affine ones.
 
  • #7
robphy said:
II. GALILEAN STRUCTURES 4 Galilean spacetimes. (p.7)
Interesting paper. They starts from a Leibnizian spacetime and endow it with an Galilean connection ##\nabla##. A Galilean connection ##\nabla## is defined as an affine connection such that its parallel transport maps Galilean basis onto Galilean basis.

To me is not clear if either we can just pick one of them or if there is actually a canonical one associated to the Leibnizian spacetime.
 
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  • #8
Afaik the connection is only defined up to an exact 2-form. In the gauging procedure this 2-form is part of the field strength of the gauge field belonging to the central extension.
 
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  • #9
haushofer said:
Afaik the connection is only defined up to an exact 2-form.
An affine connection is a covariant derivative operator that when applied for instance to a vector field produces a (1,1) tensor. A 2-form, instead, take one vector field and return a covector (1-form).

What does it mean your quoted sentence ?
 
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  • #10
cianfa72 said:
An affine connection is a covariant derivative operator that when applied for instance to a vector field produces a (1,1) tensor. A 2-form, instead, take one vector field and return a covector (1-form).

What does it mean your quoted sentence ?
In Newton-Cartan theory one derives a connection from two metric compatibility conditions (one spatial and one temporal metric). These imply no unique connection (as metric compatibility does in GR without torsion), but determine the connection up to a two form ##dK##.
 
  • #11
haushofer said:
These imply no unique connection (as metric compatibility does in GR without torsion), but determine the connection up to a two form ##dK##.
Sorry, can you help me in understanding the claim "up to a two form ##dK##" ? Thanks.
 
  • #12
cianfa72 said:
Sorry, can you help me in understanding the claim "up to a two form ##dK##" ? Thanks.
This might help...

...from poking around...

If you take a peek at text page 62 in
https://pure.rug.nl/ws/portalfiles/portal/34926446/Complete_thesis.pdf
it says
1702333705453.png


You'd probably have to start before this passage to see how to get here.

In the references,
1702333828173.png


which appears to be accessible at
https://www.actaphys.uj.edu.pl/index_n.php?I=R&V=21&N=10#755

Hope this gives you a starting point.

Update...
chasing down another reference:
Dautcourt references Kunzle,
which appears to be accessible at
http://www.numdam.org/item/AIHPA_1972__17_4_337_0/

1702334289605.png
 
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  • #14
cianfa72 said:
Sorry, can you help me in understanding the claim "up to a two form ##dK##" ? Thanks.
I made a typo (been a while :P ); the notation is that the 2-form is written as ##K##, and that it turns out to be exact: ##K=dm##. So the connection depends not only on the spatial and temporal metrics and their inverses, but also on an extra vector field ##m = m_{\mu} dx^{\mu}##. In the gauging procedure in my thesis, this vector field is the gauge field belonging to the central extension (hence the name "m").
 
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  • #15
haushofer said:
So the connection depends not only on the spatial and temporal metrics and their inverses, but also on an extra vector field ##m = m_{\mu} dx^{\mu}##.
Ok, as connection you mean the Christoffel symbols in a basis. And ##m## should be actually a covector field (i.e. a 1-form). ##K=dm## therefore it is the exterior derivative of the 1-form ##m##.

Btw I don't understand the notation in 4.23: $$\mathbf K_{ {\lambda} ({\mu} \tau_{\nu})}$$
 
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  • #16
cianfa72 said:
Ok, as connection you mean the Christoffel symbols in a basis. And ##m## should be actually a covector field (i.e. a 1-form). ##K=dm## therefore it is the exterior derivative of the 1-form ##m##.

Btw I don't understand the notation in 4.23: $$\mathbf K_{ {\lambda} ({\mu} \tau_{\nu})}$$
The tau there is not an index, but the covector ##\tau_{\mu}## that makes up the covariant temporal metric: ##\tau_{\mu\nu}=\tau_{\mu}\tau_{\nu}## (maybe it's confusing to give both the metric and this covector the same letter tau, but in coordinate dependent notation the difference should be clear). You can do that because the 'metric' has rank one, i.e. there is one time direction.

You symmetrize in that expression over mu and nu.
 
  • #17
haushofer said:
The tau there is not an index, but the covector ##\tau_{\mu}## that makes up the covariant temporal metric: ##\tau_{\mu\nu}=\tau_{\mu}\tau_{\nu}## (maybe it's confusing to give both the metric and this covector the same letter tau, but in coordinate dependent notation the difference should be clear). You can do that because the 'metric' has rank one, i.e. there is one time direction.

You symmetrize in that expression over mu and nu.
Ok, so ##\tau_{\mu\nu}=\tau_{\mu}\tau_{\nu}## is actually the tensor product ##\tau \otimes \tau## (of course this metric tensor has rank 1). Then symmetrize it as ##\tau_{(\mu \nu)}##.

Now which is the operation that follows involving the 2-form ##K## ?
 
  • #18
cianfa72 said:
Ok, so ##\tau_{\mu\nu}=\tau_{\mu}\tau_{\nu}## is actually the tensor product ##\tau \otimes \tau## (of course this metric tensor has rank 1). Then symmetrize it as ##\tau_{(\mu \nu)}##.

Now which is the operation that follows involving the 2-form ##K## ?
It's all explained in the thesis: metric compatibility applied to ##h^{\mu\nu}## and ##\tau_{\mu\nu}##. So you write down ##\nabla_{\rho} h^{\mu\nu} = \nabla_{\rho}\tau_{\mu\nu}=0## and try to solve for the connection. That won't work without gauge-fixing, because these metric compatibility conditions cannot be solved for uniquely (as in GR); they are invariant under the shift of connection described in eqn. 4.23.

By the way, you don't need to explicitly symmetrize the tensor product; the tensor product of two (co)vectors is symmetric per definition. If you mean the symmetrization in the formula 4.23, never mind.
 
  • #19
As far as I can tell in eqn. 4.23 the latter term on RHS is the tensor product ##K \otimes \tau## that has got 3 lower indices ##\lambda##, ##\mu## and ##\nu##. Then the two indices ##\mu## and ##\nu## are symmetrized.

Is the above correct ?
 
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  • #20
cianfa72 said:
As far as I can tell in eqn. 4.23 the latter term on RHS is the tensor product ##K \otimes \tau## that has got 3 lower indices ##\lambda##, ##\mu## and ##\nu##. Then the two indices ##\mu## and ##\nu## are symmetrized.

Is the above correct ?
Yes. You need the symmetrization, because the connection is defined to be torsionless. So the metric compatibility conditions will only be invariant under the shift if you symmetrize in mu and nu.
 
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  • #21
From this lecture notes Newton-Cartan the general Galilean connection has a non-zero torsion. However in case of Galilean absolute time (i.e. ##d\tau =0##) then the Galilean connection becomes torsion-free.
 
  • #22
I like those notes; the first two references refer to my PhD-research :P

In the Poincare gauge theory the torsion is given by the field strength of spacetime translations. This field strengt/curvature is put to zero in order to solve the spin connection for the tetrads and remove the local translations, as explained here,

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

The torsion of the Christoffel symbol is related to the curvatures via the vielbein postulate.

Newton-Cartan theory can similarly be derived as the gauge theory of the Bargmann algebra. Then one has field strengths for spatial (P) and temporal (H) transformations separately. The vielbein postulate relates this torsion to the field strengths ##R(H)## and ##R(P)##. Again you have to put these two curvatures to zero in order to solve the spin connection(s) for the tetrads and remove the local translations.

For how to introduce torsion in this framework, see e.g.

https://arxiv.org/abs/1409.5555

Afaik, you need an extra term in the curvature ##R(H)##, which is provided in the gauge theory of the Schrodinger algebra. In eqn.2.4 of that paper you see that ##[D,H]=-2H## which introduces an extra term in ##R(H)##, leading to the ##R(H)## in eqn. 2.13.
 
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  • #23
Reading this thread, reminds me of an interesting conversation between Yuval Ne'eman and Ivor Robinson - "General Relativity [in the mid-1950s] had become mostly a playground for mathematical sophistication, but with very little new physical content."
 
  • #24
dx said:
Reading this thread, reminds me of an interesting conversation between Yuval Ne'eman and Ivor Robinson - "General Relativity [in the mid-1950s] had become mostly a playground for mathematical sophistication, but with very little new physical content."
Indeed, the 50s was followed by the
"Golden Age of Relativity" https://en.wikipedia.org/wiki/History_of_general_relativity#Golden_age

A Golden Age of General Relativity? Some remarks on the history of general relativity
Hubert F. M. Goenner
https://arxiv.org/abs/1607.03319

Robinson became active in 1960.
Here is a paper "Spherical Gravitational Waves" (1960)
http://trautman.fuw.edu.pl/publications/Papers-in-pdf/13.pdf
in collaboration with A. Trautman,
who wrote
"Comparison of Newtonian and Relativistic Theories of Space-Time" (1966)
http://trautman.fuw.edu.pl/publications/Papers-in-pdf/22.pdf
and
"Lectures in Relativity" (Brandeis)
http://trautman.fuw.edu.pl/publications/Papers-in-pdf/21.pdf
(which was how I learned about Newton-Cartan spacetime).

On Trautman's page ( http://trautman.fuw.edu.pl/publications/scientific-articles.html ),
this is the last article mentioned with Robinson as co-author
"The conformal geometry of complex quadrics and the fractional-linear form of Mobius transformations" (1993)
http://trautman.fuw.edu.pl/publications/Papers-in-pdf/81.pdf

---
update:

Trautman's book "Spacetime and Gravitation"
introduces the Galilean Spacetime in Ch 3 (pdf page 30)
http://trautman.fuw.edu.pl/publications/Books/Spacetime_and_gravitation_Kopczynski_Trautman.pdf

"Fiber Bundles associated with Space-time" has little bit in
Section 6 "Connections" , 6.1 "Affine Spaces" (p.51)
http://trautman.fuw.edu.pl/publications/Papers-in-pdf/25.pdf
 
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  • #25
dx said:
Reading this thread, reminds me of an interesting conversation between Yuval Ne'eman and Ivor Robinson - "General Relativity [in the mid-1950s] had become mostly a playground for mathematical sophistication, but with very little new physical content."
I'm not sure what you mean here, but nowadays people use newton-cartan geometry for holographic reasons. To me it is/was a playground to understand GR better (general covariance, diffeom.invariance) (and, e.g., supersymmetry). The main advantage to this gauge approach is that it allows for the easy construction of other theories of gravity.
 
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  • #26
Regarding what haushofer said, if you use the spin connection and veilbein as the gravitational field, then general relativity is the gauge theory corresponding to the inhomogeneous Lorentz group.
 
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  • #27
For that, see

T. W. B. Kibble, Lorentz Invariance and the Gravitational
Field, Jour. Math. Phys. 2, 212 (1960),
https://doi.org/10.1063/1.1703702

Another famous work in this direction is

R. Utiyama, Invariant theoretical interpretation of
interaction, Phys. Rev. 101, 1597 (1956),
https://doi.org/10.1103/PhysRev.101.1597

A good textbook treatment is found in

P. Ramond, Field Theory: A Modern Primer,
Addison-Wesley, Redwood City, Calif., 2 edn. (1989).
 
  • #28
Sorry to resume this thread. From the references in posts above my understanding is that Galilean spacetime is both a fiber bundle and an affine space.

Did I get that correctly ?

Edit: from Trautman source the principal fiber bundle ##P(E)## over the Galilean affine space ##E## is naturally identified (i.e. there is a natural isomorphism) with the product bundle ##E \times P(V)##.

I'd say that the tangent bundle ##\tau(E)## itself over ##E## is also naturally identified with the product bundle ##E\times \tau(V)##.
 
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  • #29

FAQ: Newton Galilean spacetime as fiber bundle

What is Newton-Galilean spacetime in the context of a fiber bundle?

Newton-Galilean spacetime, when viewed as a fiber bundle, is a mathematical framework that combines the concepts of classical mechanics with the geometric structure of fiber bundles. In this context, spacetime is modeled as a 4-dimensional manifold with a 3-dimensional spatial fiber attached to each point in a 1-dimensional time base. This allows for a clear separation between space and time, which is a hallmark of Galilean relativity.

How does the fiber bundle structure help in understanding Newtonian mechanics?

The fiber bundle structure provides a rigorous and geometric way to describe Newtonian mechanics. By treating space and time as separate entities, it simplifies the analysis of physical phenomena in a way that aligns with our intuitive understanding of classical mechanics. This structure also facilitates the description of physical fields and particles moving through spacetime, allowing for a more organized and systematic approach to solving mechanical problems.

What are the key components of the Newton-Galilean fiber bundle?

The key components of the Newton-Galilean fiber bundle include the base manifold, which represents time, and the fiber, which represents the 3-dimensional spatial coordinates. The total space of the bundle is the Cartesian product of these two components. Additionally, there is a projection map that links points in the total space to points in the base manifold, ensuring that each point in time is associated with a corresponding spatial configuration.

How does the concept of a connection on a fiber bundle apply to Newton-Galilean spacetime?

In the context of Newton-Galilean spacetime, a connection on the fiber bundle can be interpreted as a way to define how spatial coordinates change as one moves through time. This connection provides a rule for parallel transport, which is essential for defining the motion of particles and fields in a consistent manner. It effectively encodes the laws of motion and can be used to derive equations of motion for particles in this spacetime framework.

Can Newton-Galilean spacetime as a fiber bundle be extended to include relativistic effects?

While Newton-Galilean spacetime as a fiber bundle is inherently non-relativistic, the concept of a fiber bundle can indeed be extended to include relativistic effects. In the relativistic case, one would use a different base manifold and fiber structure, such as the Lorentzian spacetime used in general relativity. This involves a more complex connection that respects the principles of special and general relativity, allowing for the incorporation of relativistic phenomena into the geometric framework.

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