[工程科技]Spincoefficientsforfour-dimensionalneutralmetrics_andnullgeometry.pdfVIP

Spin Coefficients for Four-Dimensional Neutral Metrics, and Null Geometry Abstract Notation for spin coefficients for metrics of neutral signature in four dimensions is introduced. The utility and interpretation of spin coefficients is explored through themes in null geometry familiar from (complex) general relativity. Four-dimensional Walker geometry is exploited to provide examples and the generalization 8 of the real neutral version of Pleba˜nski’s [22] second heavenly equation to certain Walker geometries given 0 0 in [16] is extended further. 2 b e Peter R Law. 1 Mack Place, Monroe, NY 10950, USA. prldb@member.ams.org F 3 1 2000 MSC: 53B30, 53C50 ] Key Words and Phrases: neutral geometry, Walker geometry, four dimensions, spinors. G D . h t a m [ 1 v 1 6 7 1 . 2 0 8 0 : v i X r a 1 1. Introduction Anticipating the value of spinors in the study of four-dimensional neutral geometry, I presented in [15] the algebraic classification of the Weyl curvature spinors for neutral metrics in four dimensions, complementing the Petrov-like classification of the Weyl curvature tensor already given in [14] (see also [5], §§39 - 40, and [21], §18). A spinor approach has proven very natural in describing four-dimensional Walker geometry, see [16]. Much of the two-component spinor formalism expounded in [19] carries over with only obvious modifications to the case of neutral signature. In fact, the spinor formalism appropriate for neutral signature is best regarded as a real version of the spinor formalism for complex space-times and in large part can be formally obtained by complexifying the Lorentzian spinor theory and then regarding all objects as real valued. See [15], however, for a brief expos