A complete correspondence between the Newman-Penrose and 1+1+2 formalisms

We establish a correspondence between the Newman-Penrose and 1+1+2 semitetrad covariant formalisms by expressing all Newman-Penrose spin coefficients, Ricci scalars, and Weyl scalars in terms of the scalar, vector, and tensor variables of the 1+1+2 decomposition. This provides a direct dictionary between two widely used approaches to general relativity and gives a geometrical interpretation of Newman-Penrose quantities in terms of covariantly defined 1+1+2 variables. As a simple demonstration, we use this mapping to derive inequalities on the Newman-Penrose scalars and the cosmological constant, which constrains the existence of future outer trapping horizons in spacetimes exhibiting local rotational symmetry.