A new characterization of half-flat solutions to Einstein's equation
type: Item
instance of: scholarly article
title: A new characterization of half-flat solutions to Einstein's equation
zbMATH Open document ID: 0642.53079
DOI: 10.1007/BF01224131
author: Abhay Ashtekar
author: Ted Jacobson
author: Lee Smolin
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: A four-dimensional metric is said to be half-flat if its Riemann tensor is proportional to its dual. This implies that the metric is also Ricci flat and thus solves the vacuum Einstein equations. If the metric signature is Lorentzian, this also implies that the metric is flat and thus physically trivial, but this is not necessarily so for Euclidean metrics or those with complex components. This paper studies half-flat complex valued metrics in terms of a \(3+1\) decomposition of the manifold and a triad of vectors on the three-manifolds. The condition of half- flatness is shown to be equivalent to pair of equations, constraint plus ``evolution'', for each vector, although here ``evolution'' naturally does not have the same meaning as in the hyperbolic case for real Lorentzian metrics. Examples of special classes of solutions are given.
Mathematics Subject Classification ID: 53B50
Mathematics Subject Classification ID: 83C20
Mathematics Subject Classification ID: 83C05
zbMATH DE Number: 4046617
zbMATH Keywords: Ricci flat
zbMATH Keywords: Einstein equations
zbMATH Keywords: half-flat complex valued metrics
zbMATH Keywords: triad of vectors
zbMATH Keywords: classes of solutions
reviewed by: Carl H. Brans
MaRDI profile type: Publication
cites work: Asymptotically flat ℋ spaces
cites work: Nonlinear gravitons and curved twistor theory
cites work: Q3324593
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cites work: Hyperkähler metrics and supersymmetry
full work available at URL: https://doi.org/10.1007/bf01224131
OpenAlex ID: W1983857848
description: scientific article; zbMATH DE number 4046617