Special Relativity Fiber Bundle: R^3 & Lorentz Metric

[Thytanium]

TL;DR

I want to know if in special relativity, space-time as a fiber bundle, the base space is time and the fibers are R3 space.

In the space-time of special relativity considered as fiber bundle, could it be stated that the base space is time and the fibers are space $R^3$ related to each other by the Lorentz metric as a connection and in this case would there be parallelism, and in this case: how would this fiber space be represented geometrically?

 

[robphy]

From https://www.physicsforums.com/threads/galilean-spacetime-as-a-fiber-bundle.986579/post-6328519 ,
consult this paper by Trautman.:

http://trautman.fuw.edu.pl/publications/Papers-in-pdf/25.pdf
Fibre bundles associated with space-time, Rep. Math. Phys. (Torun) 1, 29–34 (1970)
from the site http://trautman.fuw.edu.pl/publications/scientific-articles.html .

(Hmm… not the reference I wanted… to be updated:
Better:

http://trautman.fuw.edu.pl/publications/Papers-in-pdf/30_Andrzej_Trautman.pdf
see p. 190
)Galilean and Newton-Cartan Spacetime have that kind of structure,
but not Minkowski and general-relativistic Spacetimes.

 

[Thytanium]

From https://www.physicsforums.com/threads/galilean-spacetime-as-a-fiber-bundle.986579/post-6328519 ,
consult this paper by Trautman.:

http://trautman.fuw.edu.pl/publications/Papers-in-pdf/25.pdf
Fibre bundles associated with space-time, Rep. Math. Phys. (Torun) 1, 29–34 (1970)
from the site http://trautman.fuw.edu.pl/publications/scientific-articles.html .

(Hmm… not the reference I wanted… to be updated:
Better:

http://trautman.fuw.edu.pl/publications/Papers-in-pdf/30_Andrzej_Trautman.pdf
see p. 190
)Galilean and Newton-Cartan Spacetime have that kind of structure,
but not Minkowski and general-relativistic Spacetimes.

Thank you Robphy. Thank you for answering my question and clarifying my doubts. And thanks for the links.

 

[Thytanium]

Hello friends. Excuse me. I have been reading the links that Robphy sent me but I don't understand some concepts. Please, if you can recommend me some books on basic concepts of manifolds and fiber bundles for beginners that I can download for free from the web, I would appreciate it. And thanks you and sorry.

 

[fresh_42]

Hello friends. Excuse me. I have been reading the links that Robphy sent me but I don't understand some concepts. Please, if you can recommend me some books on basic concepts of manifolds and fiber bundles for beginners that I can download for free from the web, I would appreciate it. And thanks you and sorry.

You can start right here on PF:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
is a five-part article that can give you an overview and has also a list of references (mostly books available at Amazon).

You will see, that it is not so easy to jump right in. You normally start with a course on ordinary Riemann manifolds. You can find lecture notes via the search keys:
Riemann Manifolds + pdf
Differential Geometry + pdf
Vector Bundles + pdf
possibly extended by + introduction.

 

[Thytanium]

You can start right here on PF:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
is a five-part article that can give you an overview and has also a list of references (mostly books available at Amazon).

You will see, that it is not so easy to jump right in. You normally start with a course on ordinary Riemann manifolds. You can find lecture notes via the search keys:
Riemann Manifolds + pdf
Differential Geometry + pdf
Vector Bundles + pdf
possibly extended by + introduction.

Thanks fresh_42. Grateful friend.

 

[fresh_42]

Thanks fresh_42. Grateful friend.

If you are looking for lecture notes, try to find those written by physics professors. Their treatment should be better suited in this case than the purely mathematical approach, e.g.
http://people.uncw.edu/lugo/COURSES/DiffGeom/DG1.pdf
looks good.

 

[martinbn]

Summary: I want to know if in special relativity, space-time as a fiber bundle, the base space is time and the fibers are R3 space.

In the space-time of special relativity considered as fiber bundle, could it be stated that the base space is time and the fibers are space $R^3$

Yes, but since it is a trivial bundle it may not be that useful. Also there isn't any preferred way to do that. That's the whole point of relativity.

related to each other by the Lorentz metric as a connection and in this case would there be parallelism, and in this case: how would this fiber space be represented geometrically?

This is hard to understand. The metric is not a connection. It would be better if you tried to rephrase it.

 

[Thytanium]

If you are looking for lecture notes, try to find those written by physics professors. Their treatment should be better suited in this case than the purely mathematical approach, e.g.
http://people.uncw.edu/lugo/COURSES/DiffGeom/DG1.pdf
looks good.

Hello friends fresh_42 and martinbn very grateful for your information. Wonderful differential geometry book you sent me. Thanks. Since yesterday I have had internet problems and i still have them. As for my question, I imagined the quadrivectors $x^i$(x,y,z,ct) of the Minkowski space and its parallel transportation and thought the connection for that transport was the relation $ds^2 = dx^2 + dy^2 + dz^2 - (ct)^2$. That is to say, make parallel transport of those vectors. That is what I thought. Thanks friends. Very thankful.

 

[vanhees71]

A connection is something different. In a Riemannian or pseudo-Riemannian space it's given by the Christoffel symbols, which are derivatives of the components of the metric (pseudo-metric). It defines covariant derivatives of tensor components and the parallel transport of vectors.

 

[Thytanium]

A connection is something different. In a Riemannian or pseudo-Riemannian space it's given by the Christoffel symbols, which are derivatives of the components of the metric (pseudo-metric). It defines covariant derivatives of tensor components and the parallel transport of vectors.

Thanks you vanhees71. You have clarified my doubts. Very grateful to you friend.