### Abstract

A 3+1 formulation of complex Einstein's equation is first obtained on a real 4-manifold M, topologically σ×R, where σ is an arbitrary 3-manifold. The resulting constraint and evolution equations are then simplified by using variables that capture the (anti-) self dual part of the 4-dimensional Weyl curvature. As a result, to obtain a vacuum self-dual solution, one has just to solve one constraint and one "evolution" equation on a field of triads on σ: {Mathematical expression} where Div denotes divergence with respect to a fixed, non-dynamical volume element. If the triad is real, the resulting self-dual metric is real and positive definite. This characterization of self-dual solutions in terms of triads appears to be particularly well suited for analysing the issues of exact integrability of the (anti-) self-dual Einstein system. Finally, although the use of a 3+1 decomposition seems artificial from a strict mathematical viewpoint, as David C. Robinson has recently shown, the resulting triad description is closely related to the hyperkähler geometry that (anti-) self-dual vacuum solutions naturally admit.

Original language | English (US) |
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Pages (from-to) | 631-648 |

Number of pages | 18 |

Journal | Communications in Mathematical Physics |

Volume | 115 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1988 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*115*(4), 631-648. https://doi.org/10.1007/BF01224131