Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

180 submissions , 140 unreviewed
4,479 questions , 1,782 unanswered
5,147 answers , 21,898 comments
1,470 users with positive rep
704 active unimported users
More ...

  How to select which differentiable manifold to use to model spacetime?

+ 1 like - 0 dislike
53 views

I've asked this question on SE as well, but haven't gotten a satisfactory answer. Would be really grateful for your help:

I'm studying differential geometry basics for general relativity. I know that spacetime is modeled as a 4-dimensional smooth manifold. Smooth manifold means that we consider a restriction of the maximal atlas such that all charts in it are compatible. A smooth manifold is specified once we choose an equivalence class of compatible atlases. Among all coordinate charts among all of those atlases (in the equivalence class), the differentiability notion is well-defined - a curve that's differentiable in one coordinate system of one atlas will be differentiable in any other coordinate system of any other atlas in the equivalence class.

This much is clear.

I'm watching a lecture on the same topic and the lecturer discusses about an issue here (the link starts at the relevant timestamp and it's just a 1.5 min watch till 24:00). For the 4-D case, there exist uncountably many smooth atlases up to diffeomorphism.

At 23:25 timestamp, he mentions that the differentiability of curves depends on the atlas we're using. So a curve may be differentiable w.r.t. one atlas and not to another (not in the same equivalence class but another smooth atlas nonetheless).
So this amounts to saying that depending on the choice of smooth atlas equivalence class (which amounts to choosing what differentiable manifold to use), a curve may be differentiable in one choice and not in another.

This makes it seem that the differentiability notion is ill-defined. The lecturer talks about the same issue briefly here as well, but doesn't get around to discussing the resolution. How is this issue resolved?
 

asked Jul 19 in Theoretical Physics by Shirish (5 points) [ no revision ]
recategorized Jul 21 by Shirish

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverfl$\varnothing$w
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...