Thank you Robphy. Thank you for answering my question and clarifying my doubts. And thanks for the links.robphy said: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.
You can start right here on PF:Thytanium said: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.
Thanks fresh_42. Grateful friend.fresh_42 said: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.
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.Thytanium said:Thanks fresh_42. Grateful friend.
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.Thytanium said: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##
This is hard to understand. The metric is not a connection. It would be better if you tried to rephrase it.Thytanium said: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?
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.fresh_42 said: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.
Thanks you vanhees71. You have clarified my doubts. Very grateful to you friend.vanhees71 said: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.
The Special Relativity Fiber Bundle is a mathematical concept used to describe the relationship between space and time in the theory of special relativity. It combines the three-dimensional Euclidean space (R^3) with the Lorentz metric, which is used to measure distances in four-dimensional spacetime.
The Lorentz Metric is a non-Euclidean metric that takes into account the effects of special relativity, such as time dilation and length contraction. Unlike the Euclidean Metric, which measures distances in a straight line, the Lorentz Metric measures distances in four dimensions, taking into account the curvature of spacetime.
The use of a Fiber Bundle in Special Relativity allows for a more comprehensive understanding of the relationship between space and time. It allows us to describe how objects move and interact in four-dimensional spacetime, taking into account the effects of special relativity.
The Special Relativity Fiber Bundle is used in various fields, such as physics, astrophysics, and engineering, to make accurate predictions and calculations involving high speeds and strong gravitational fields. It is also used in the development of technologies such as GPS systems and particle accelerators.
While the Special Relativity Fiber Bundle is a powerful mathematical tool, it does have its limitations. It does not take into account the effects of general relativity, which describes the relationship between gravity and spacetime. It also does not apply to objects moving at speeds close to the speed of light, where the effects of special relativity become significant.