• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Twistors in Curved Spacetime

+ 15 like - 0 dislike

I am looking for good and recent references to constructing twistor space for curved spacetime. This could be a general spacetime, or specific ones (say maximally symmetric spaces different from Minkowski). This could be in he context of the twistor correspondence, or the twistor transform of field equations, either subject generalized to curved spacetime.

The references I am familiar with are the standard ones from about 30-40 years ago, where most constructions involve flat spacetime. Some generalizations are mentioned, but my impression is that the community had not settled at the time on a single approach. Many things happened since, and one of the things I am hoping to get is some understanding of the landscape of current approaches to the subject.

This post has been migrated from (A51.SE)
asked Nov 2, 2011 in Theoretical Physics by Moshe (2,405 points) [ no revision ]
retagged Apr 19, 2014 by dimension10

3 Answers

+ 8 like - 0 dislike

I'm not sure if this is exactly what you are looking for or perhaps you already know what I am about to say.

There is a geometric notion of a twistor spinor (or conformal Killing spinor): one which is in the kernel of the Penrose operator (see below). Then one defines the twistor space as the projectivisation of the space of twistor spinors. Doing this for Minkowski spacetime recovers the usual twistor space.

Let $(M,g)$ be a riemannian spin manifold. (When I say riemannian I include also the case of a metric with indefinite signature.) Let $S$ denote the complex spinor bundle. The spin connection defines a map $$ \nabla: \Gamma(S) \to \Omega^1(S) $$ from spinor fields to one-forms with values in $S$. Now $\Omega^1(S) = \Gamma(T^*M \otimes S)$ and Clifford action of one-forms on spinors gives a map $$ \Omega^1(S) \to \Gamma(S) $$ The composition of the previous two maps is the Dirac operator. The Penrose operator is in some sense the complement of the Dirac operator $D$. The kernel of the Clifford map $T^*M \otimes S \to S$ defines a subbundle $W$, say, of $T^*M \otimes S$. Composing the covariant derivative with the projection $\Omega^1(S) \to \Gamma(W)$ defines the Penrose operator $P: \Gamma(S) \to \Gamma(W)$: explicitly, $$ P_X \psi = \nabla_X \psi + \frac1n X \cdot D\psi $$ for all vector fields $X$ and spinor fields $\psi$, and where $n = \dim M$. (My Clifford algebra conventions are $X^2 = - |X|^2$.) Notice that the "gamma trace" of the Penrose operator vanishes.

There is a sizeable literature on twistor spinors mostly in riemannian and lorentzian signatures. This is the work of Helga Baum and collaborators in Berlin. A search for "twistor spinors" in MathSciNet should give you many links.

One important property of the twistor spinor equation is that it is conformally invariant, whence the twistor spinors of conformally related riemannian spin manifolds correspond in a simple way. Since you mention maximally symmetric lorentzian manifolds, this observation might be of use because such spaces are conformally flat, hence you can write down the twistor spinors simply by rescaling the twistor spinors in Minkowski spacetime. In riemannian signature (hence for round spheres and hyperbolic spaces) this is described in the 1990 Humboldt University Seminarberichte Twistor and Killing spinors on riemannian manifolds by Baum, Friedrich, Grunewald and Kath, later published by Teubner.

This post has been migrated from (A51.SE)
answered Nov 2, 2011 by José Figueroa-O'Farrill (2,315 points) [ no revision ]
Thanks, this is useful. I am mainly looking for an entry point to the literature, or a good survey. My actual confusion is not well-formed enough to make a good question at this point. Sorry for being a bit vague.

This post has been migrated from (A51.SE)
Helga Baum has written a couple of surveys about twistor spinors in lorentzian geometry, but I do not think that they address the twistor transform. The twistorial people I know work mostly in riemannian signature.

This post has been migrated from (A51.SE)
+ 1 like - 0 dislike

I am not too comfortable with this subject, but anyway maybe the list may be useful

  1. R. Penrose, W. Rindler, Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (1988). (certainly known, I suppose)
  2. R. S. Ward, R.O. Wells, Twistor geometry and field theory (CUP, 1990) (Chapter 9)
  3. S. A. Huggett, K. P. Tod, An introduction to twistor theory (CUP, 1994) (Chapter 13 ?)
  4. M. Dunajski, Solitons, instantons, and twistors (OUP, 2010) (Chapter 10.5 ?)
This post has been migrated from (A51.SE)
answered Nov 3, 2011 by Alex V (300 points) [ no revision ]
I am not familiar with the last one, but own first three. For twistors in curved spacetime, most extensive and up to dated discussion is in Ward and Wells among them.

This post has been migrated from (A51.SE)
+ 1 like - 0 dislike

This may not be exactly what you are looking for, and I am certainly not an expert in this. But, I happened to be interested in current state of art in Penrose's non linear graviton program and did a quick (~ 30 min.) literature search last year.

My impression is that there has not been large activities nor a break through. Also, as we know, twistor community is a small group and they probably don't feel a need of review articles for wider readership. (My impression was further reinforced by chatting with Penrose few weeks later, even though he seems very excited about development in twistor string theory.)

Having said that, here are my finding:

  1. A useful overview, not so current, is Penrose's article written in 99.

The central program of twistor theory in Chaos, Solitons & Fractals Vol 10, No. 2-3. pp 581-611, 1999

or you can get it in Andrew Hodges's twistor diagram page.

Penrose devoted last few sections on basic ideas and difficulties of defining twistors in curved spacetime.

  1. A collection of article edited by Lionel Mason.

Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces probably contains much more technical developments upto 2000.

Let me know if you find any useful stuff on this.

This post has been migrated from (A51.SE)
answered Nov 3, 2011 by Demian Cho (285 points) [ no revision ]

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:
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