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

PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a "Wick rotation"?

+ 5 like - 0 dislike
3510 views

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like \begin{equation*} \hat{g} = \pi^*g\oplus g_V \end{equation*} where $\pi: T^{(*)}B \to B$ is the projection and $g_V$ is the metric on the vertical directions of the fibration. In certain cases (i.e. when $B$ is affine), we can compactify by taking the quotient of $T^{(*)}B$ by a (dual) lattice $\Gamma^{(*)}$ to obtain the non-singular torus fibrations $T^{(*)}B/\Gamma^{(*)} \to B$.

In physics, there is a (I don't know how well-defined) notion of "time-compactification" by passing from a Riemannian manifold $(M,\hat{g}_+)$ to a pseudo-Riemannian manifold $(M,\hat{g}_-)$ via a "Wick rotation" (https://en.wikipedia.org/wiki/Wick_rotation). In flat coordinates, if \begin{equation*} \hat{g}_+ = \sum_i dx_i^2 + \sum_j dy_j^2 \ , \end{equation*} then a Wick rotation is equivalent to the procedure of substituting $y_j \to \sqrt{-1} y_j$ since \begin{equation*} \hat{g}_- = \sum_i dx_i^2 - \sum_j dy_j^2 \ . \end{equation*}

I would like to know whether this can be related to the procedure of taking the alternative metric \begin{equation} \hat{g}' = \pi^*g\oplus (-1)g_V \end{equation} on $T^{(*)}B$ and whether this can somehow be seen as equivalent to compactification by taking the quoitient with respect to $\Gamma^{(*)}$.

This post imported from StackExchange MathOverflow at 2015-08-04 14:41 (UTC), posted by SE-user harry
asked Aug 3, 2015 in Mathematics by harry (25 points) [ no revision ]
retagged Aug 4, 2015
A couple of notes. Wick rotation does not a priori have anything to do with compactification, though sometimes the two come together. You can find a mathematically precise notion of Wick rotation in this answer. I'm not sure that what you are doing qualifies, as you are not doing an analytic continuation anywhere. Though, perhaps there is a way to reinterpret your construction in terms of complexification and analytic continuation.

This post imported from StackExchange MathOverflow at 2015-08-04 14:41 (UTC), posted by SE-user Igor Khavkine
@Igor, thanks for the link. Perhaps you could explain how your view of the Wick rotation presented there is ("sometimes") related to compactification. How does a noncompact submanifold get smoothly deformed to a compact submanifold of the complexification?

This post imported from StackExchange MathOverflow at 2015-08-04 14:41 (UTC), posted by SE-user harry
Simple. The complexification $M_\mathbb{C}$ is not unique. For example, the flat metric on $\mathbb{R}$ embeds into both $\mathbb{C}$ (with a non-compact Wick rotation) and $\mathbb{C}/i\mathbb{Z}$ (with a compact Wick rotation).

This post imported from StackExchange MathOverflow at 2015-08-04 14:41 (UTC), posted by SE-user Igor Khavkine

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).
Please complete the anti-spam verification




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

Your rights
...