Einstein's equations and the embedding of 3-dimensional CR manifolds

TL;DR

This paper explores the link between 3D CR manifold embeddability and Einstein's equations, showing how Einstein's equations reduce to CR-invariant equations and lead to the construction of CR functions and a circle bundle structure.

Contribution

It establishes a connection between Einstein's equations and CR geometry, providing new methods to analyze spacetime metrics via CR invariants and circle bundles.

Findings

Reduced Einstein equations to CR-invariant equations.

Constructed two independent CR functions from Einstein equations.

Showed the spacetime metric induces a circle bundle over the CR manifold.

Abstract

We prove several theorems concerning the connection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dimensional CR manifold defined by the fields. Using the reduced Einstein equations we construct two independent CR functions for the corresponding CR manifold. We also point out that the Einstein equations, imposed on spacetimes associated with a 3-dimensional CR manifold, imply that the spacetime metric, after an appropriate rescaling, becomes well defined on a circle bundle over the CR manifold. The circle bundle itself emerges as a consequence of Einstein's equations.