Newton Galilean spacetime as fiber bundle

[cianfa72]

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.

 

[robphy]

possibly useful:

II. GALILEAN STRUCTURES 4 Galilean spacetimes. (p.7)
in
https://arxiv.org/abs/gr-qc/0211030
Leibnizian, Galilean and Newtonian structures of spacetime
Antonio N. Bernal, Miguel Sánchez
J.Math.Phys. 44 (2003) 1129-1149 https://doi.org/10.1063/1.1541120

 

[haushofer]

A reference without any talk about fibre bundles is my PhD thesis,

https://research.rug.nl/en/publications/newton-cartan-gravity-revisited

It discusses also the metric structure of "Newtonian" spacetimes, because the connection on it is related to metric compatibility.

 

[cianfa72]

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.

 

[haushofer]

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?

 

[cianfa72]

Why not? You're referring specifically to the boosts?

Sorry, you are right. All kind of Galilean transformations are actually affine ones.

 

[cianfa72]

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.

 

[haushofer]

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.

 

[cianfa72]

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 ?

 

[haushofer]

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 $dK$.

 

[cianfa72]

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.

 

[robphy]

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

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

In the references,

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/

 

[haushofer]

Exactly, that eqn. 4.23 is key.

 

[haushofer]

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").

 

[cianfa72]

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: $$\mathbf K_{ {\lambda} ({\mu} \tau_{\nu})}$$

 

[haushofer]

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.

 

[cianfa72]

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 $K$ ?

 

[haushofer]

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.

 

[cianfa72]

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 ?

 

[haushofer]

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.

 

[cianfa72]

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.

 

[haushofer]

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.

 

[dx]

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."

 

[robphy]

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

 

[haushofer]

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.

 

[dx]

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.

 

[vanhees71]

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).

 

[cianfa72]

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 $E\times \tau(V)$.

 

[robphy]

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 ?

Yes
from
https://www.physicsforums.com/threads/what-exactly-are-invariants.1045392/post-6799064,

• https://www.nbc.com/saturday-night-live/video/shimmer-floor-wax/2721424
• https://snltranscripts.jt.org/75/75ishimmer.phtml

In short: it's both a fiber bundle and an affine space.

 

[PeterDonis]

In short: it's both a fiber bundle and an affine space.

It's a floor wax and a dessert topping!

https://snltranscripts.jt.org/75/75ishimmer.phtml