Spin coefficients for four-dimensional neutral metrics, and null geometry

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 of the real neutral version of Plebański’s second heavenly equation to certain Walker geometries given in [P.R. Law, Y. Matsushita, A Spinor approach to Walker geometry, Comm. Math. Phys. 282 (2008) 577–623] is extended further.