Abstract
We develop a non–relativistic twistor theory, in which Newton–Cartan structures of Newtonian gravity correspond to complex three–manifolds with a four–parameter family of rational curves with normal bundle \({\mathcal {O} \oplus \mathcal {O}(2)}\). We show that the Newton–Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton–Cartan connections can nevertheless be reconstructed from Merkulov’s generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non–trivial on twistor lines. The resulting geometries agree with non–relativistic limits of anti-self-dual gravitational instantons.
Article PDF
Similar content being viewed by others
References
Andringa, R., Bergshoeff, E.A., Rosseel, J., Sezgin, E.: Newton-Cartan Supergravity; Classical Quant, Grav. 30, 205005 (2013). arXiv:1305.6737
Cartan E.: Sur les varietes a connexion affine et la theorie de la relativite. Ann EC. Norm. Sup. 40, 325–412 (1923)
Dautcourt G.: On the Newtonian limit of general relativity. Acta. Phys. Pol. B 21, 755 (1989)
Dunajski M.: Solitons, instantons & twistors, Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2009)
Dunajski M.: The Twisted photon associated to hyper-hermitian four manifolds. J. Geom. Phys. 30, 266–281 (1999)
Dunajski M., Mason L.J.: Twistor theory of hyper-Kähler metrics with hidden symmetries. J. Math. Phys. 44, 3430–3454 (2003)
Dunajski M., Hoegner M.: SU(2) solutions to self-duality equations in eight dimensions. J. Geom. Phys. 62, 1747–1759 (2012)
Duval C., Burdet G., Kunzle H.P., Perrin M.: Bargmann structures and Newton-Cartan theory. Phys. Rev. D 31, 18411853 (1985)
Duval C., Horvathy P.A.: Non-relativistic conformal symmetries and Newton-Cartan structures. J. Phys. A 42, 465206 (2009)
Eastwood M.G., Penrose R., Wells R.O.: Cohomology and massless fields. Commun. Math. Phys. 78, 305–351 (1980)
Ehlers, J.: Examples of Newtonian limits of relativistic spacetimes. Class. Quantum Grav. 14 (1997)
Fedoruk S., Kosinski P., Lukierski J., Maslanka P.: Nonrelativistic counterparts of twistors and the realizations of Galilean conformal algebra. Phys. Lett. B 669, 129134 (2011)
Gibbons G.W., Hawking S.W.: Gravitational multi-instantons. Phys. Lett. B 78, 430 (1978)
Maldonado R., Ward R.S.: Geometry of periodic monopoles. Phys. Rev. D 88, 125013 (2013)
Hartnoll S.: Lectures on holographic methods for condensed matter physics. Class. Quant. Grav. 26, 224002 (2009)
Hitchin N.J.: Complex manifolds and Einstein’s equations, Twistor geometry and nonlinear systems (Primorsko, 1980), Lecture Notes in Math. vol. 970, pp. 73–99. Springer, Berlin, New York (1982)
Hitchin, N.J.: Higgs bundles and diffeomorphism groups (2015). arXiv:1501.04989
Kodaira K.: On stability of compact submanifolds of complex manifolds. Am. J. Math. 85, 79–94 (1963)
Künzle H.P.: Covariant Newtonian limit of Lorentz space-times. Gen. Rel. Grav. 7, 445 (1976)
Merkulov, S.A.: Relative deformation theory and differential geometry. In: Twistor theory. Marcel Dekker, New York (1995)
Merkulov S.A.: Geometry of Kodaira moduli spaces, Proc. Am. Math. Soc. 124, 1499–1506 (1996)
Merkulov S.A., Schwachhofer L.: Classification of irreducible holonomies of torsion-free affine connections. Ann. Math. 150, 77–149 (1999)
Penrose R.: Twistor algebra. J. Math. Phys. 8, 345–366 (1967)
Penrose R.: Solutions of the zero-rest-mass equations. J. Math. Phys. 10, 38 (1969)
Penrose R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31–52 (1976)
Penrose, R., Rindler, W.: Spinors and space-time. Two-spinor calculus and relativistic fields. In: Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1987, 1988)
Plebański J.F.: Some solutions of complex Einstein equations. J. Math. Phys. 16, 2395–2402 (1975)
Sommers P.: Space spinors. J. Math. Phys. 21, 2567 (1980)
Son, D.T.: Newton-Cartan Geometry and the Quantum Hall effect (2013). arXiv:1306.0638
Sparling, G.A.J.: Dynamically broken symmetry and global Yang-Mills in Minkowski space Twistor Newsletter (1977)
Tod K.P.: The singularities of H-space. Math. Proc. Camb. Philos. Soc. 92, 331 (1982)
Trautman A.: Sur la theorie newtonienne de la gravitation. Comptes Rendus Acad. Sci. Paris. 247, 617 (1963)
Ward R.S.: On self-dual gauge fields. Phys. Lett. 61, 81–82 (1977)
Ward R.S.: Self-dual space-times with cosmological constant Comm. Math. Phys. 78, 1–17 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. T. Chruściel
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Dunajski, M., Gundry, J. Non-Relativistic Twistor Theory and Newton–Cartan Geometry. Commun. Math. Phys. 342, 1043–1074 (2016). https://doi.org/10.1007/s00220-015-2557-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2557-8