Twistor Bundles, Einstein Equations and Real Structures

Abstract

We consider sphere bundles P and P' of totally null planes of maximal dimension and opposite self-duality over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on PP'. The formulation is suitable for both complex- and real-valued metrics. It unifies results for all three possible real signatures. In the purely Riemannian positive definite case it implies the existence of a natural almost hermitian structure on PP' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost hermitian structure on PP'.

Paper accepted in Classical and Quantum Gravity, Special issue in honour of Professor Andrzej Trautman

Cited by in corpus (10)

• Twistor space for rolling bodies
• From Spinor Geometry to Complex General Relativity
• Twisting non-shearing congruences of null geodesics, almost CR structures, and Einstein metrics in even dimensions
• Complex Geometry of Nature and General Relativity
• Almost Robinson geometries
• Toric hyperkahler manifolds with quaternionic Kahler bases and supergravity solutions
• On a certain formulation of the Einstein equations
• Elliptic fibrations associated with the Einstein spacetimes
• Self-dual two-forms and divergence-free vector fields
• Extension of bundles of null directions