A new characterization of half-flat solutions to Einstein's equation
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.
scientific article; zbMATH DE number 4046617