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

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 e...

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Bibliographic Details
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
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Summary: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.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2008.57.3473