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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Smoothness of the twistor space of a lorentzian manifold, or “convexity wrt null geodesics”

+ 1 like - 0 dislike
57 views

Null lines in Minkowski space form a 5-dimensional manifold, represented as a (real) quadric $\mathbf{PN}\subset\mathbb{C}\mathbf{P}^3$. This is a well-known fact, on which R. Penrose’s twistor programme is based. It is also known that the space of null lines can be constructed locally (albeit without Cauchy–Riemann structure in the curved case) and that the resulting space $\mathfrak{N}$ of (non-parametrized) null geodesics possesses a natural contact structure, co-oriented one if original manifold is time-oriented.

As of global constructions, a Cauchy hypersurface $M$ permits to describe $\mathfrak{N} = ST^*M$; see e.g. arxiv:0810.5091. This requires global hyperbolicity, a strong condition on the original manifold, that isn’t anywhere near necessity for $\mathfrak{N}$ to be a manifold.

On the other hand, for a lorentzian manifold $X$ (or d+1 pseudo-Riemann for arbitrary dimension) let’s define $\mathfrak{S}_x$ (called the sky of $x$) as the projectivization of all null vectors in $T_x X$, diffeomorphic (and conformly equivalent) to the sphere $S^{d-1}$. Let $\mathfrak{S}X$ be the bundle of skies for all $x\in X$, with (d−1)-dimensional fibres. Its total space (2 d-dimensional) has a natural foliation with one-dimensional leaves, namely null geodesics. Now we build $\mathfrak{N}$ as the space of leaves (in other words, the quotient space of $\mathfrak{S}X$ by equivalence relation to lie on the same null geodesic) and for a strongly causal lorentzian manifold it should result in a Hausdorff space. But it isn’t necessary a manifold, as the following 1+1-dimensional example shows:

 \\
\\\\
\\\\
\\\\
\\\\
\\\\\\\\\
\\\\\\\\\\
 \\\\\\\\

Intuitively, the property looked for smoothness of $\mathfrak{N}$ is convexity of $X$ wrt null geodesics, but Ī̲’m unsure how to say it exactly. Convexity of subsets in aforementioned sense is well-known in lorentzian geometry, but Ī̲ failed to find in papers anything like “convex”, “light-convex” or “causally convex” referring to entire manifold. Can anybody suggest a strict formulation that is weaker than global hyperbolicity?

This post imported from StackExchange MathOverflow at 2015-12-22 18:43 (UTC), posted by SE-user Incnis Mrsi
asked Dec 17, 2015 in Theoretical Physics by Incnis Mrsi (-15 points) [ no revision ]
retagged Dec 22, 2015

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$\hbar$ysicsOv$\varnothing$rflow
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
...