Einstein's Equations and The Embedding of 3-dimensional CR Manifolds

• Published in: Indiana University mathematics journal 2008-01, Vol.57 (7), p.3131-3176
• Main Authors: Hill, C. Denson, Lewandowski, Jerzy, Nurowski, Paweł
• Format: Article
• Language: English
• Subjects: Algebra; Einstein equations; Exact sciences and technology; General mathematics; General, history and biography; Geodesy; Manifolds and cell complexes; Mathematical analysis; Mathematical congruence; Mathematical manifolds; Mathematical theorems; Mathematics; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Several complex variables and analytic spaces; Spacetime; Tensors; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Vector fields
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2008.57.3473

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.