Non-relativistic twistor theory and Newton--Cartan geometry

TL;DR

This paper develops a non-relativistic twistor theory linking Newton--Cartan structures to complex three-manifolds, exploring their stability, reconstruction, and incorporation of Coriolis force, and connecting to non-relativistic limits of gravitational instantons.

Contribution

It introduces a novel non-relativistic twistor framework for Newton--Cartan geometry, including stability analysis and methods for reconstructing connections and forces.

Findings

Newton--Cartan structures correspond to complex three-manifolds with specific rational curves.

Newton--Cartan space-times are unstable under general Kodaira deformations.

Connections can be reconstructed using Merkulov's map and holomorphic line bundles.

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.