Non-Relativistic Twistor Theory and Newton–Cartan Geometry

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Dunajski, M 
Gundry, J 

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 OO(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.

hep-th, hep-th, gr-qc, math.DG
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Communications in Mathematical Physics
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Springer Science and Business Media LLC
Science and Technology Facilities Council (ST/L000636/1)
We are grateful to Christian Duval, George Sparling and Paul Tod for helpful discussions. This work started when MD was visiting the Institute for Fundamental Sciences (IMP) in Tehran in April 2010. MD is grateful to IMP for the extended hospitality when volcanic eruption in Iceland halted air travel in Europe. The work of JG has been supported by an STFC studentship.